University of Southern California Dissertations and Theses

Essays on Japanese macroeconomy

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Essays on Japanese macroeconomy

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Content ESSAYS ON JAPANESE MACROECONOMY
by
Kota Nakamura
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ECONOMICS)
May 2024
Copyright 2024 Kota Nakamura

Dedication
To my family and parents
ii

Acknowledgements
Thanks to Professor Robert Dekle for his mentorship and supervision throughout the study.
Thanks to Professor Pablo Kurlat and Professor Selahattin İmrohoroğlu for many helpful comments and
suggestions.
Thanks to Cabinet Office and National Personnel Authority of Government of Japan for giving me an
invaluable opportunity to study during my career as a national public officer.
Thanks to my family–my partner Yoko and my dearest daughter Sara–for supporting me throughout the
study. Thanks to my parents–my father Futoshi, my mother Yoshiko and my elder sister Tomoka.
iii

Table of Contents
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
Chapter 1: Aging, Population Decline, and Fertility in Japan . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Data Sources and Institutional Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Institutional Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Preliminary Empirical Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Results of the Preliminary Empirical Examination . . . . . . . . . . . . . . . . . . . 11
1.4 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.1 Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.1.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.1.2 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.2 Equilibrium Conditions in a Laissez-Faire Economy . . . . . . . . . . . . . . . . . . 16
1.4.3 Solution of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Calibration and Model Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5.1 Calibration of the Agglomeration Elasticity β . . . . . . . . . . . . . . . . . . . . . 18
1.5.2 Calibration of b, z, and γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5.3 Matching the Population Movements . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5.4 Model Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.6 Social Optimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.6.1 Constrained Efficient Representative Agent . . . . . . . . . . . . . . . . . . . . . . 27
1.6.2 Comparison of Laissez Faire and Social Optimum . . . . . . . . . . . . . . . . . . . 29
1.7 Policy Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.7.1 Regional Transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.7.2 Child Allowance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.7.3 Quantitative Analysis of the Full Model . . . . . . . . . . . . . . . . . . . . . . . . 34
1.7.3.1 Incorporating Current Policies in Japan . . . . . . . . . . . . . . . . . . . 34
iv

1.7.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
A.1 An Extended Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
A.2 Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
A.2.1 Laissez Faire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
A.2.2 Constrained Efficient Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
A.3 Alternative Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
A.3.1 Declining Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
A.4 Computation of the Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
A.4.1 Solving Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
A.4.2 Solving Backward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
A.5 The Number of Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
A.6 Generational Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Chapter 2: Large Public Debt under Low Interest Rates: a Welfare Analysis . . . . . . . . . . . . . 52
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.3 Empirics and Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.3.1 Empirics: Non-homothetic Utility from Asset . . . . . . . . . . . . . . . . . . . . . 65
2.3.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.4 Implication of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.4.1 Increase in Public Debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.4.2 Recession and Fiscal Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.4.3 Foreign Investors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.4.4 Reduction of Debt Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.4.5 Economic Growth and g > r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.4.6 Most Preferred Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
B.1 Detailed Overview of Literature on Public Debt and Welfare . . . . . . . . . . . . . . . . . 86
B.2 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
B.3 Alternative Borrowing Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
B.4 Supplement to Instrument Variables in 2.3.1 (Empirics on Non-homothetic Preference) . . 95
B.5 Derivation in 2.4.5 (Economic Growth and g > r ) . . . . . . . . . . . . . . . . . . . . . . . 97
B.6 Optimal Size of the Debt Increase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Chapter 3: Monetary Policy, the Dual Labor Market, and Consumption in Japan . . . . . . . . . . 102
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.2 Monetary Shocks and the Consumption of Regular and Non-regular Workers . . . . . . . . 109
3.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.2.2 Monetary Policy Shocks (Kubota and M. Shintani (2022)) . . . . . . . . . . . . . . . 112
3.2.3 Consumption Response to Monetary Shocks . . . . . . . . . . . . . . . . . . . . . . 113
3.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.3.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.3.2 Labor Market Players . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.3.2.1 Labor Packer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.3.2.2 Labor Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.3.3 Firms in the Goods Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
v

3.3.3.1 Final Goods Firm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.3.3.2 Retail Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.3.3.3 Wholesale Firm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.3.4 Stochastic Processes and Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.4 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.4.1 Frisch Labor Supply Elasticity: χ
R and χ
n
. . . . . . . . . . . . . . . . . . . . . . . 121
3.4.2 Intertemporal Elasticity of Substitution: σ . . . . . . . . . . . . . . . . . . . . . . . 123
3.4.3 Production Substitutability Parameter: µ . . . . . . . . . . . . . . . . . . . . . . . . 125
3.5 Model Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.5.1 Steady State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.5.2 Benchmark Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.5.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.5.4 Welfare Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
C.1 Detail on Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
C.1.1 Data on Labor Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
C.1.2 Wage Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
C.1.3 Labor Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
C.2 Detail of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
C.2.1 Household . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
C.2.1.1 Regular Workers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
C.2.1.2 Non-regular Workers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
C.2.2 Labor Market Players . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
C.2.2.1 Labor Packer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
C.2.3 Firms in Goods Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
C.2.3.1 Final Goods Firm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
C.2.3.2 Retail Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
C.2.3.3 Wholesale Firm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
C.2.4 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
C.2.4.1 Monetary Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
C.2.4.2 Exogenous Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
C.2.5 Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
C.3 Derivation of Regression Equation in 3.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
vi

List of Tables
1.1 47 Prefectures in Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Estimation Results for Gravity Equation of Internal Migration in Japan . . . . . . . . . . . 11
1.3 Estimation Results of Agglomeration Elasticity (β) . . . . . . . . . . . . . . . . . . . . . . 20
1.4 GMM Estimation Results of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5 Comparison of Steady State Allocations in Laissez Faire and Social Optimum . . . . . . . . 29
1.6 Comparison of Steady State Allocations with Different Types of Transfers . . . . . . . . . 31
1.7 Comparison of Steady State Allocations with Varying Transfers Conditional on the
Number of Children per Household (τn) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.8 Comparison of Steady State Allocations with Varying Transfers Conditional on the
Number of Children per Household(τn) in the Full Model . . . . . . . . . . . . . . . . . . . 36
A.9 Comparison of Steady State Allocations with Varying Pension Systems (τa) . . . . . . . . 50
2.1 Descriptive Statistics of JHPS/KHPS Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.2 Marginal Propensity to Consume out of Constructed Permanent Income . . . . . . . . . . 68
2.3 Summary of Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1 Comparison between KHPS/JHPS Data and Japanese Official Statistics . . . . . . . . . . . . 110
3.2 Descriptive Statistics of KHPS/JHPS Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.3 Broader Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.4 Narrower Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.5 List of Model Parameters and Calibrated Values . . . . . . . . . . . . . . . . . . . . . . . . 121
vii

3.6 Estimation of Frisch Labor Supply Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.7 Estimation of Intertemporal Elasticity of Substitution . . . . . . . . . . . . . . . . . . . . . 125
3.8 Estimation of Substitutability of Regular and Non-regular Labor . . . . . . . . . . . . . . . 126
3.9 Steady State of the Quantitative Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.10 Welfare Loss Depending on the Composition of Workers ω . . . . . . . . . . . . . . . . . . 137
viii

List of Figures
1.1 Scatter Plot of Total Fertility Rate and Population Density . . . . . . . . . . . . . . . . . . . 2
1.2 Elderly Population and Net Population Flows by Prefecture . . . . . . . . . . . . . . . . . . 6
1.3 The Hypothetical Elderly Population Share Projected Mechanically . . . . . . . . . . . . . 10
1.4 Scatter Plot of kuni-Level Rice Output (per Capita) and Population . . . . . . . . . . . . . . 19
1.5 Population Trends: By Region and Japan Overall . . . . . . . . . . . . . . . . . . . . . . . . 23
1.6 Transition to Steady State (Log Utility) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.7 Transition to Steady State (CRRA Function with σc = 0.7) . . . . . . . . . . . . . . . . . . 25
1.8 Welfare Change Measured in Terms of Consumption Equivalent Measure . . . . . . . . . . 37
A.9 Transition to Steady State (log utility) in a Two-Region Model . . . . . . . . . . . . . . . . 48
A.10 Transition to Steady State (CRRA Utility with σc = 0.7) in a Two-Region Model . . . . . . 48
A.11 Transtion of Aging Rate with and without a Pension System . . . . . . . . . . . . . . . . . 51
2.1 Net Public Debt to GDP Ratio in Advanced Countries . . . . . . . . . . . . . . . . . . . . . 53
2.2 Short-term Interest Rates (left) and 10 Years Interest Rates (right) in Advanced Countries . 53
2.3 Determination of Asset Supply and Interest Rate in Steady State Equilibrium . . . . . . . . 63
2.4 Calibration: Fit of Euler Equation to Household Data . . . . . . . . . . . . . . . . . . . . . 70
2.5 Comparison of Steady State Welfare with Welfare Including Transition when Public Debt
is Raised . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.6 Comparison of Welfare Including Transition with and without an Increase in Public Debt
when Output Drops (Tax Cut for Both Types) . . . . . . . . . . . . . . . . . . . . . . . . . 75
ix

2.7 Comparison of Welfare Including Transition with and without an Increase in Public Debt
when Output Drops (Tax Cut for Borrowers Only) . . . . . . . . . . . . . . . . . . . . . . . 76
2.8 Comparison of Steady State Welfare with Welfare Including Transition when Public Debt
is Raised (in the Presence of Foreign Investors) . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.9 Comparison of Steady State Welfare with Welfare Including Transition when Debt to GDP
Ratio is Reduced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.10 Comparison of Steady State Welfare with Welfare Including Transition with Economic
Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.11 Comparison of Steady State Welfare with Welfare Including Transition (Most Preferred
Specification) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.12 Comparison of Welfare Including Transition with or without an Increase in Public Debt to
GDP Ratio when Output Drops (Preferred Specification) . . . . . . . . . . . . . . . . . . . 84
B.13 Example Calculation of the Steady State Transition 1 . . . . . . . . . . . . . . . . . . . . . 93
B.14 Example Calculation of the Steady State Transition 2 . . . . . . . . . . . . . . . . . . . . . 94
B.15 Comparison of Steady State Welfare with Welfare Including Transition when Output
Drops (Alternative Borrowing Constraint) . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
B.16 Example Calculation of the Steady State Transition . . . . . . . . . . . . . . . . . . . . . . 96
B.17 Debt Increase Size and Consumption Equivalence . . . . . . . . . . . . . . . . . . . . . . . 101
B.18 Debt Increase Size and Consumption Equivalence when Output Drops . . . . . . . . . . . 101
3.1 Interest Rates and Growth in Real GDP and Consumption (Quarter on Quarter) . . . . . . 103
3.2 Growing Share of Non-regular Employees . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.3 Hourly Wages of Regular and Non-regular Workers . . . . . . . . . . . . . . . . . . . . . . 106
3.4 Wages per Worker: Actual versus Constant 1994 Regular Worker Shares . . . . . . . . . . 107
3.5 Impulse Response to a Monetary Easing Shock . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.6 Impulse Response to a Monetary Easing Shock with Varying Fraction of Non-regular Wokers 131
3.7 Impulse Response to a Monetary Easing Shock with Varying Quality of Non-regular Workers 132
3.8 Impulse Response to a Monetary Easing Shock with Varying Fraction of Non-regular Wokers 133
x

3.9 Impulse Response to a Monetary Easing Shock with Varying Adjustment Cost of Regular
Workers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.10 Impulse Response to a Monetary Easing Shock with Varying Wage Stickiness of Regular
Worker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
xi

Abstract
This thesis consists of three essays on macroeconomic problems faced by Japan.
The first chapter addresses how local heterogeneity in population aging and decline affects the growth
and welfare of a country. This study addresses this question by focusing on Japan, where the aging and
shrinking of the population is characterized by marked heterogeneity across regions. To this end, a dynamic spatial model equipped with endogenous fertility and overlapping generations replicates, through
agglomeration and congestion, this demographic distribution, which is considered to be suboptimal when
compared with an allocation by a constrained planner maximizing the welfare of a representative agent. In
order to internalize the externality of having children on the size of the future generation, several types of
policies are studied. The model calculation incorporating the current policies in Japan suggests that transfers to households based on the number of children has the potential to increase welfare if these transfers
are raised from the current level.
The second chapter discusses the welfare consequence of having a large amount of public debt under
low real rates of return, which has been a recent norm in advanced countries before COVID. We embed the
non-homothetic utility of asset holding in a discrete time two agents perpetual youth model to reconcile
low interest rates with large public debt. Whereas the increase in the level of public debt lowers the
steady state welfare, the rise in public debt temporarily waives the taxation, and this could be a source for
improvement in welfare if the welfare during the transition is taken into account. We find that public debt
indeed has a possibility to raise welfare, especially when there is a drop in output. We quantitatively study
xii

various types of fiscal policies, including the introduction of foreign investors and public debt reduction.
Our most preferred specification suggests that increasing the public debt level itself cannot raise the welfare
compared with staying at the initial steady state as the public debt to GDP ratio reaches about 200%, but
it continues to be a better policy response when there is a drop in output compared with holding the level
of public debt fixed.
The last chapter (joint with Robert Dekle) studies the sluggish response of consumption in Japan during
2013-2019 to the massive monetary expansion by the Bank of Japan. From our observations based on
household panel data that the consumption response of households of non-regular workers are stronger
than those of regular workers, we set up a New Keynesian model featuring the dual labor market in Japan.
There are two types of worker households. One is unionized and monopolistically competitive (regular
workers), and the other participates in a perfectly competitive market (non-regular workers). The model
well replicates the weak rise of aggregate consumption to monetary policy easing. We show that the
widening use of non-regular workers in recent years may have led to the weakening of the consumption
response.
xiii

Chapter 1
Aging, Population Decline, and Fertility in Japan
1.1 Introduction
In many advanced economies, the fertility rate has been well below the replacement rate for decades. For
example, in Japan it fell below 2.0% in 1980 and continued to gradually decline to around 1.4%. As a result,
the working age population (those aged between 15 and 64) has been declining and the share of the elderly
in the population increasing. This, in turn, has led to a rise in the dependency ratio, increasing pressure
on government finances, posing a greater financial burden on the young, and putting government support
for the elderly (such as pensions and health care) at risk.
Another aspect is that population aging and the decline in fertility are characterized by marked heterogeneity across regions. While there is a subset of cities in Japan that continue to register population
growth, other cities and rural regions suffer from a declining population and a high elderly population
share, since many people, especially the young, migrate from areas with a declining population to areas
with a growing population. Moreover, this regional heterogeneity in demographic trends translates into
aggregate population decline through fertility choices. This is illustrated in Figure 1.1, which provides a
I am grateful to professors Robert Dekle, Selahattin İmrohoroğlu, Hiro Ishise, Matthew Kahn, Pablo Kurlat, Monica Morlacco,
and Paulina Oliva for their guidance, comments and suggestions. All errors are mine.
1

scatter plot of the population density and total fertility rate in each of the 47 prefectures of Japan in 2019. A
negative correlation can be observed: the more densely populated or "congested" a prefecture is, the lower
the total fertility rate. While low-density areas tend to have higher fertility rates than high-density areas,
the absolute number of children born in such areas is insufficient to compensate for the low fertility rate
in high-density areas, so that the population overall declines. While the situation in Japan is characterized by certain idiosyncrasies, similar demographic patterns can be observed in other developed countries
experiencing a shrinking population.
.05
.1
.15
.2
.25
Total fertility rate
2 2.5 3 3.5 4
Population density
Figure 1.1: Scatter Plot of Total Fertility Rate and Population Density
While there are studies such as Sato (2007) that provide a theoretical explanation of the mechanisms
underlying this heterogeneity in fertility rates across regions in a country, little is known about the welfare
implications of this heterogeneity and the resulting population decline. This is of interest to policymakers
who wish to change the demographic trajectory. The Japanese government, for example, has introduced
policies to encourage individuals to relocate from Tokyo and its surrounding areas to local areas by providing subsidies, the amount of which rises with the number of children. While this could serve the objective
of the Japanese government to increase the number of young people in local areas to ease congestion in
Tokyo and its surrounding areas and raises the number of young people in regions with high fertility
2

rates, it might also have adverse affects by reducing agglomeration economies and reducing overall welfare. This potential trade-off makes it difficult to determine whether it is desirable for the government to
try to shape the demographic distribution of a country and what kind of policies are effective. This study
seeks to address this issue.
Specifically, this study aims to examine how regional heterogeneity in population aging and shrinking
affects the economic welfare of an entire country. Sato (2007) explains regional fertility differences by
incorporating fertility choices into an economic geography model. In the model, more populated regions
have higher productivity due to Marshallian agglomeration economies and attract more people. At the
same time, however, congestion diseconomies are larger in such regions and reduce the time people can
allocate to child-rearing. The resulting fertility in regions with denser populations is lower than that in
regions with less dense populations. This paper constructs a model based on Sato (2007), presents a welfare
analysis, and postulates a novel source of externalities that the literature on normative analysis in spatial
economics has not addressed to date: fertility choice. Each generation’s fertility choice affects the size of
the next generation and hence economies of scale. However, the role of such scale economies in relation
to geography and fertility is something that the literature has not addressed so far. Since the benefits of
an increased cohort size are shared by everyone, the fertility rate chosen by people in equilibrium in a
laissez-faire economy might be lower than the fertility rate that maximizes economic welfare. Thus, there
is potential for policy intervention to affect demographic heterogeneity across regions to improve welfare.
Japan is one of the best countries to study the welfare implications of policies that affect regional
heterogeneity in demographics. There are a number of reasons. First, excluding Monaco, Japan has the
highest share of elderly people (those aged 65 or older) in the world, with over 28% in 2019, while Germany had the second highest with 21.7%. Second, the Japanese government has made addressing the
over-concentration of resources in Tokyo one of its top policy priorities1
and has introduced subsidies
1
In 2014, a report titled "Local Extinction" by a government committee to discuss the revitalization of regional areas stated
that about half of all municipalities (896 out of 1,799) face the danger of "extinction" by 2040, with the number of women of childbearing age between 20 and 40 expected to decline to half. This report led the government to make the revitalization of regional
3

and incentives to encourage firms and individuals to relocate to regional areas and to promote regional
economies.
The analysis in this study concentrates on broadly two types of policies. The first type is a fiscal
equalization scheme under which resources are transferred from populous, highly productive regions to
less populous regions. The second is transfers (subsidies) to individuals based on the number of children
they have – a type of policy that is seen in practice in many countries. These subsidies are especially
effective even though they distort incentives regarding fertility choices: since fertility choices generate
externalities in the form of affecting the size of future generations, providing subsidies for having children
potentially improves welfare. Calibrating the model developed in this paper to the Japanese economy
and incorporating transfers to "lagging" regions and to households with children, this paper examines the
welfare implications of different sizes of transfers to highlight the effectiveness of regional transfers and
child allowance payments.
The remainder of this study is organized as follows. Section 1.2 provides information on the institutional setting in Japan. An empirical examination of the causes of the heterogeneity in local demographics
is provided in Section 1.3. Section 1.4 presents a model that replicates the regional heterogeneity in demographics. The model is then calibrated and the laissez-faire equilibrium is shown in Section 1.5. After
comparing it with the social optimum equilibrium in Section 1.6 to examine the direction policies should
take, several policies to make the laissez-faire equilibrium closer to the social optimum are studied in
Section 1.7. Finally, Section 1.8 concludes.
prefectures and the correction of the over-concentration of resources in Tokyo one of the top policy priorities and provided the
rationale for the use of subsidies and incentives to encourage individuals and firms to relocate to regional prefectures and boost
their economies.
4

1.2 Data Sources and Institutional Setting
This section provides a description of the main data source used for the analysis, the Report on Internal
Migration in Japan. This is followed by an overview of the administrative structure of Japan and the
classification of prefectures and regions used in this study.
1.2.1 Data Sources
The Report on Internal Migration in Japan provides statistics on prefectural migration and is published by
the Ministry of Internal Affairs and Communication (MIC). The data are compiled at the prefecture level
and are available annually from 2012 to 2019 and every five years from 1955 to 2010. Migration data are
available by sex and ten-year age bracket. For example, the data show that the number of men aged 30 to
39 who moved from Osaka to Tokyo in 2019 is 124,934. Meanwhile, the number of residents by age in each
prefecture is available from the Population Estimates published by MIC.
In addition to the population data, the analysis below employs a number of variables on prefectural
characteristics, such as prefecture-level wages obtained from the Monthly Labour Survey by the Ministry
of Health, Labour and Welfare.
1.2.2 Institutional Setting
Japan consists of 47 prefectures. To facilitate the analysis, these prefectures are grouped into three regions
based on whether they belong to a large agglomerations and on the size of their population. In Japan,
prefectures are conventionally divided into seven regions (such as the Kanto region, which consists of
Tokyo and its six neighboring prefectures); however, this study uses a different grouping. The three major
urban agglomerations, greater Tokyo, greater Osaka, and Aichi (which comprises Nagoya), account for
more than half of the total population. Some of the neighboring prefectures of these areas have relatively
large populations (such as Kanagawa and Kyoto), while others have smaller populations of less than 2
5

(a) Proportion of over 65s (2019) (b) Net inflow (2019)
Figure 1.2: Elderly Population and Net Population Flows by Prefecture
million (Gifu near Aichi and Nara near the Osaka area, for example). In this study, these prefectures, 11 in
total as shown in Table 1, are classified as "Region 1." Next, prefectures that are not in Region 1 but have a
large population (of more than 1.5 million) and a high GDP (above 5 trillion yen) are classified into "Region
2." Some of the prefectures classified into Region 2 are far from Tokyo, such as Hokkaido and Kumamoto,
but have sizeable populations. Finally, the remaining 22 prefectures are classified as "Region 3" and are
prefectures with smaller populations and relatively large elderly population shares (except for Okinawa).
Japan is characterized by considerable heterogeneity across prefectures in terms of the demographic
age structure and rate of population decline. The three major prefectures that form the centers of Region 1
(blue circles in Figure 2) have lower elderly population ratios and register population inflows (or a relatively
small outflow). On the other hand, other prefectures, especially rural ones, suffer from rapid aging and
depopulation, since each year many young people move into the major metropolitan areas. Figure 1.2(a)
shows a map of Japan in which the 47 prefectures are colored based on the share of those aged 65 or over.
The darker the color, the higher is the share of the elderly. The figure shows that prefectures with elderly
populations are those outside the three major agglomerations. While the population share of the elderly
6

Population (mil.) Total Fertility Rate (%) Aging Rate (%) GDP (tril.Yen)
Region 1
Tokyo 13.26 1.24 22.67 110.36
Kanagawa 9.04 1.39 23.86 33.89
Osaka 8.71 1.38 26.15 40.01
Aichi 7.40 1.56 23.79 41.09
Saitama 7.21 1.39 24.82 22.82
Chiba 6.13 1.37 25.86 20.65
Hyogo 5.47 1.47 27.09 21.57
Kyoto 2.56 1.34 27.51 10.37
Gifu 2.02 1.56 28.10 7.57
Mie 1.80 1.55 27.90 8.01
Nara 1.35 1.38 28.70 3.76
Region 2
Hokkaido 5.36 1.31 29.09 19.94
Fukuoka 5.04 1.52 25.90 19.24
Shizuoka 3.67 1.54 27.79 17.53
Ibaraki 2.88 1.48 26.76 13.38
Hiroshima 2.81 1.60 27.53 12.35
Niigata 2.29 1.44 29.86 9.07
Miyagi 2.28 1.36 25.75 9.87
Nagano 2.08 1.58 30.06 8.29
Tochigi 1.96 1.49 25.87 9.02
Gunma 1.96 1.49 27.60 8.94
Fukushima 1.89 1.58 28.68 7.86
Okayama 1.89 1.54 28.66 7.87
Kumamoto 1.78 1.68 28.78 5.87
Kagoshima 1.63 1.70 29.43 5.51
Region 3
Okinawa 1.42 1.96 19.63 4.28
Shiga 1.40 1.60 24.15 6.30
Yamaguchi 1.40 1.60 32.07 5.92
Nagasaki 1.37 1.66 29.60 4.66
Ehime 1.36 1.53 30.62 4.93
Aomori 1.30 1.42 30.14 4.61
Iwate 1.27 1.49 30.38 4.68
Oita 1.16 1.59 30.45 4.43
Ishikawa 1.14 1.54 27.87 4.81
Yamagata 1.12 1.48 30.76 4.06
Miyazaki 1.10 1.70 29.49 3.64
Toyama 1.06 1.51 30.54 4.75
Akita 1.01 1.35 33.84 3.46
Wakayama 0.96 1.53 30.89 3.58
Kagawa 0.96 1.63 29.93 3.85
Saga 0.83 1.64 27.68 2.95
Yamanashi 0.83 1.50 28.41 3.39
Fukui 0.78 1.62 28.63 3.39
Tokushima 0.75 1.53 30.95 3.17
Kochi 0.72 1.50 32.85 2.43
Shimane 0.69 1.78 32.48 2.60
Tottori 0.57 1.64 29.71 1.84
Table 1.1: 47 Prefectures in Japan
7

for Japan as a whole is 28%, there is considerable heterogeneity across prefectures, with the share ranging
from below 25% in Tokyo and Okinawa to over 33% in prefectures such as Akita. These areas with a
high share of elderly people, moreover, are also losing residents at a high rate, as shown in Figure 1.2(b),
which depicts net migration in 2019 calculated as the number of incoming residents minus the number of
outgoing residents. The figure indicates that only a small number of prefectures enjoy net inmigration,
while most prefectures, especially those that already have a large elderly population share, record net
outmigration. The data show that most are young people, especially women, moving to large cities, with
surveys indicating that they are mainly moving to enter university or find a job. For example, of the 80,000
newcomers to Tokyo in 2019, half were women in their twenties.
1.3 Preliminary Empirical Findings
This section provides empirical findings on how the regional heterogeneity in elderly population shares
has come about through internal migration by examining the role demographic structure plays in people’s
migration decisions. Since the main focus of this study is the regional heterogeneity in demographics—i.e.,
elderly population shares and fertility rates—that shape and are shaped by internal migration, the findings
provide the rationale for including the life cycle and interregional migration in the model presented in the
next section.
1.3.1 Specification
This section empirically examines people’s migration decision using a panel dataset of inter-prefectural
migration flows.
8

To examine the importance of how the elderly people share of a region affects prefectural migration,
the following gravity equation of individuals’ migration choice is estimated using a balanced panel:
Modsat = α0 + β1Wdsat + β2Wosat + β3Rdt + β4Rot + γ1Xdsat + γ2Xosat
+θo + δd + φs + ψa + ξt + εodsat.
where o stands for the prefecture of origin, d for the destination prefecture, s for individuals’ sex, a for the
age bin they fall into, and t for time. The dependent variable Modsat is the number of people (characterized
by s and a) who moved from o to d in year t. Wosat denotes the average wage in the prefecture of origin,
while Wdsat is the average wage in the destination prefecture. Likewise, Rot is the proportion of people
aged 65 and over. X is a matrix of covariates and includes the population size of the prefectures of origin
and destination as well as the distance between the two prefectures. The former aims to capture the size
of prefectural economies, while the latter represents moving costs.
The coefficients on Rdt and Rot, β3 and β4, indicate how reluctant or eager people are to choose to
migrate from o to d. A higher proportion of elderly people means that the proportion of younger people
is lower, which limits younger people’s exposure to peers of their own age. Younger people therefore are
more tempted to gather in regions with a lower elderly population ratio and to move away from regions
with a higher ratio. The expected sign is negative for β3 and positive for β4. A potential concern is that the
elderly population ratios in the prefectures of origin and destination may be endogenously determined, so
that the corresponding parameter estimates, β3 and β4, may be biased. That is, while the proportion of
people aged 65 and over may influence migration decisions, the elderly share is also determined by internal
migration, given that it is especially the young that tend to migrate. In order to take this endogeneity
into account, the projected, hypothetical elderly population share is used as an instrumental variable. By
definition, the evolution of the population of a particular prefecture over time is given by the following
equation:
Pj,t+1 = Pj,t +
X
i
ini,j,t+1 −
X
k
outj,k,t+1 − deathj,t+1 + bornj,t+1.
9

where Pj,t is the population of prefecture j in yeart. The population in yeart+1 depends on the population
in the previous period, Pj,t, the number births, birthsj,t+1, the number of deaths, deathsj,t+1, the number
of incoming migrants from other prefectures (across i), ini,j,t+1, and the number of outgoing migrants to
other prefectures (across k), outj,k,t+1. The hypothetical, counterfactual evolution of the population of a
particular prefecture in the absence of internal migration can be calculated as follows:
P˜j,t+1 = P˜j,t − deathj,t+1 + bornj,t+1.
The hypothetical population figures P˜
j,t can be used to compute the elderly population ratios R˜ in the
absence of internal migration. For the calculation here, birth and death rates for Japan overall are used
for deaths and births.
2 This alternative calculation makes it possible to capture the exogenous developments in the elderly population ratio without the influence of migration, which is the source of potential
endogeneity in the regression.
.2
.25
.3
.35
.4
Projected share of elderly people
.2 .25 .3 .35 .4
Actual share of elderly people in 2019
projected from 2012
fitted line on projection from 2012
projected from 2000
fitted line on projection from 2000
projected from 1990
fitted line on projection from 1990
projected from 1980
fitted line on projection from 1980
45 degree line
Figure 1.3: The Hypothetical Elderly Population Share Projected Mechanically
Figure 1.3 shows these hypothetical elderly population ratios for different starting years. The red line
running through the blue dots fits the hypothetical rates starting from 2012 that would have been realized in
2019 in the absence of internal migration. The other dots are computed beginning from various past census
2A similar approach is employed by Maestas, Mullen, and Powell (2023), who use hypothetical demographic variables to
examine the effect of the elderly population ratio on economic growth.
10

(1) (2) (3)
destination’s aging rate -0.856∗∗∗ -1.749∗∗∗ -2.523∗∗∗
(0.09) (0.13) (0.40)
origin’s aging rate 0.356∗∗∗ 0.948∗∗∗ 1.339∗∗∗
(0.08) (0.13) (0.40)
destination’s wage 0.380∗∗∗ 0.373∗∗∗ 0.397∗∗∗
(0.02) (0.02) (0.05)
origin’s wage 0.423∗∗∗ 0.428∗∗∗ 0.348∗∗∗
(0.02) (0.02) (0.05)
destination’s population 0.859∗∗∗ 0.857∗∗∗ 0.943∗∗∗
(0.01) (0.01) (0.04)
origin’s population -0.633∗∗∗ -0.631∗∗∗ -0.552∗∗∗
(0.01) (0.01) (0.04)
distance -0.919∗∗∗ -0.919∗∗∗ -1.144∗∗∗
(0.00) (0.00) (0.01)
IV No Yes Yes
additional IV No No Yes
Num obs 187538 187538 23161
R-sqr 0.89 0.73 0.64
∗
p < 0.05,
∗∗ p < 0.01,
∗∗∗ p < 0.001
Table 1.2: Estimation Results for Gravity Equation of Internal Migration in Japan
years (1980, 1990, and 2000) to evolve up to 2019 plotted against the actual elderly population ratio in 2019.
The figure shows that the fitted lines are flatter than the 45-degree line, meaning that prefectures with
high actual elderly population ratios would have had lower elderly population ratios without migration.
Conducting the same exercise using base years further in the past indicates that the predicted elderly
population ratios for prefectures that attracted substantial inmigration are larger than the actual ratios. In
other words, the elderly population ratios of those prefectures are smaller than they would have been in
the absence of migration.
1.3.2 Results of the Preliminary Empirical Examination
Table 1.2 summarizes the results of this estimation. The coefficients on the elderly population ratio variables are as conjectured, indicating that prefectures with a higher elderly ratio experience more outmigration and less inmigration. The coefficient on the population of the destination prefecture, which serves
11

as a proxy for the size of the economy and the attractiveness of the destination, is positive, while the coefficient on the prefecture of origin is negative. These results indicate that a high elderly ratio acts as a
factor encouraging outmigration and a low ratio as a factor encouraging inmigration. Next, the results
in column (1) using the actual elderly ratios and in column (2) using the hypothetical elderly ratios as an
IV are compared. Doing so shows that using the IV results in substantially larger coefficients in absolute
value. This suggests that how gray a prefecture is potentially is an important push factor. Column (3)
uses some more hypothetical elderly population ratios starting from more distant pasts than Column (2)
as additional IVs. Since the estimates in Columns (2) and (3) do not differ very much, this suggests that a
single hypothetical elderly population ratio instrument works to counter the endogeneity problem. The
only unexpected sign is the coefficient on wages in the origin prefecture. Overall, the results show that
the elderly ratio plays a role in regional migration in Japan.
1.4 Model
This section presents the model employed in this study to examine the regional dynamics in fertility rates
and elderly population rates, aggregate demographics, and the impact of policies aiming to affect these
dynamics. The model is based on Sato (2007) and, in addition to multiple life stages and regions, incorporates fertility choices and cohort formation. Specifically, the model proposed here is a three-region,3
three-life-stages overlapping generations model with endogenous fertility. The model will be used for
policy experiments in Section 1.7.
3There are several reasons for the choice of three regions. First, three types of region capture the demographic situation in
Japan well, consisting of (1) the most populated metropolitan areas featuring Tokyo, Osaka and Aichi prefecture, (2) prefectures
far from the major metropolitan areas that, however, each contain one or two cities with a population of more than one million,
such as Miyagi and Fukuoka prefectures, and (3) all other prefectures, which are rural prefectures characterized by a rapidly
shrinking and aging population. Second, while a two-region model would necessarily mean that one region is more populated
and has a lower birthrate and the other is less populated and has a higher birthrate, using a three-region model means that the
second-most populated region could either have a higher or lower birthrate than one or both of the other regions. In fact, whether
the second-most populated region attracts more people or has more children than the other regions is later shown to be dependent
on the type and magnitude of the policies. While the main text discusses only the three-region model, the two-region version of
the model is discussed in the Appendix.
12

1.4.1 Environment
1.4.1.1 Households
Agents’ lives can be divided into three periods: young, middle-aged, and old. Each stage corresponds to
thirty years. The utility function for the generation that are middle-aged at time t and spend their middle
age and old age in Region l is given by the following equation:
Ult = γ
c
1−σc
lt+1
1 − σc
+ (1 − γ)
n
1−σn
lt
1 − σn
,
where clt+1 is consumption in old age, and nlt is the number of children they give birth to when they are
young.4 Note that, for simplicity, consumption in life stages other than old age are omitted. (Alternatively,
it would be possible to allow agents to consume in middle age as well.) Moreover, agents derive utility
from the number of children, not from the utility children enjoy. nlt, the number of children per person,
is not restricted to taking integer values but for simplicity is allowed to take any positive real number.
Further, it is assumed that when they are young, agents do not make choices with regard to consumption and saving, so their only choice focuses on moving across regions, with the choice based on considerations of the expected lifetime utility of living in a particular region. In the model, agents are not allowed
to migrate when they are middle-aged or old. While this abstracts from the migration of the middle-aged
back to their prefecture of origin after they have completed their education in one of the metropolitan
prefectures and of the old after retirement in one of the metropolitan prefectures to spend their retirement
in their prefecture of birth, there are two good reasons for making this simplifying assumption. The first
is that the vast majority of regional migrants migrate before the age of thirty, which corresponds to the
young in the model here. Second, this abstraction from migration in later life substantially simplifies the
optimization problems and facilitates model-solving in the analysis below.
4The model abstracts from the sex of agents and allows all agents to potentially have children.
13

Agents in middle age are endowed with one unit of time. The use of time is given by the following
time constraint:
1 ≥ llt + blNlt + zlnlt.
The middle-aged allocate their time to two activities. The first is work. It is assumed that there is a
representative firm in each region and agents work at the reginal representative firm at the perfectly
competitive wage wlt determined in each region’s labor market, supplying llt hours of work. The second
is child-rearing, based on their fertility choice of how many children to have. Child-rearing is associated
with a time cost, zl
, which is multiplied by the number of children.5 Further, agglomeration diseconomies
are introduced by assuming that a certain amount of middle-aged agents’ time is "eaten up" by congestion
blNlt, which is proportional to the population size of a region. 6 For simply, it is assumed that the middleaged do not consume. That is, they save all of their wage earnings and carry over these savings into old
age:
slt+1 ≤ wltllt.
The old do not work and use their savings for consumption:
clt+1 ≤ (1 + rt+1 − δ)slt+1,
The young can migrate freely. This means that a young person born in Region 1 can choose to spend
the rest of their life not in their region of origin (Region 1) but in another region (Region 2 or 3). Let nlt
be the number of children the middle-aged in Region l (Nlt) give birth to. Thus, the total number of the
5The justification for the assumption that child-rearing is associated with a time cost is that it takes time and effort to find
nursery schools, especially in larger cities in Japan. An alternative would be to incorporate the cost of child-rearing in monetary
terms. Eckstein and Wolpin (1985), for example, assume that child-rearing "costs" some units of goods.
6There are various different ways in which this congestion term could be introduced in the time use constraint. For example,
it could be argued that Figure 1.1 suggests that the relationship between population density and fertility follows an inverted U
shape rather than a straight downward-sloping line. While the functional form of this congestion term is a matter of debate, this
study focuses on the linear case.
14

young at time t is n1tN1t +n2tN2t +n3tN3t
. These young at time t grow up to be middle-aged in the next
period t+ 1, and the only requirement of the model is that the total number of young in period t coincides
with the total number of middle-aged in period t + 1:
X
l=1,2,3
nltNlt =
X
l=1,2,3
Nlt+1.
1.4.1.2 Firms
There is a representative firm in each region, which hires the middle-aged at wage wlt and rents capital at
rental rate rt
. Capital is assumed to be mobile across regions, meaning that the rental rate is equalized.7
The production technology is
Ylt = AltL˜
ltKα
ltL
1−α
lt ,
where productivity consists of two parts: AltL˜
lt, which is the exogenous productivity in region Alt, and
the endogenous knowledge spillover L˜
lt = L
βl
lt . This means that as the number of people residing in
Region l increases, productivity is pushed up. This is a shortcut way of introducing the agglomeration
spillovers discussed in the literature on spatial equilibrium. Since individual agents and firms fail to take
into account the effects they have on these externalities, competitively determined wage and rental rates
do not reflect the term βl
:
wlt = Alt(1 − α)Kα
ltL
−α
lt L˜
lt = Alt(1 − α)Kα
ltL
−α+βl
lt , rt = AltαKα−1
lt L
1−α
lt L˜
lt = AltαKα−1
lt L
1−α+βl
lt
7This assumption of a unified capital market across regions reduces the number of state variables in the model. Since capital
in each period is allocated in equilibrium so that the marginal productivity of capital is equated across regions, there is no need
to keep track of the amount of capital in each region for the most of periods: as long as the total amount of capital is known, its
distribution across regions is uniquely determined and thus there is no need to consider the amount of capital in each region as
state variables. The same applies to the population: as long as the total population is known, the distribution of the population
across regions is determined so that the utility obtained in each region is equalized and there is no need to keep track of the
population in each region as individual state variables for the most of periods.
15

1.4.2 Equilibrium Conditions in a Laissez-Faire Economy
Since consumption of the young and middle-aged is abstracted from, there is no trade-off in intertemporal
consumption. The first order optimality condition describes the trade-off between working in middle-age
and consumption in old age. Since, furthermore, bequests are also abstracted from, individuals’ choice of
how much to work is directly linked to how much they consume when they are old, so that the trade-off
is in fact between having children and consuming:
(1 − γ)
γ
c
σc
lt+1 = n
σn
lt zlwlt(1 + rt+1 − δ).
If the log functional form is used for the utility of consumption and joy of having children, the following
analytical expression for the consumption and the number of children is obtained:
clt+1 = (1 + rt+1 − δ)wt
llt, nlt =
(1 − γ)
zl
(1 − blNlt)
In the second equation regarding the number of children, there is a negative relationship between
congestion blNlt and the number of children nlt, which replicates the negative correlation observed in the
scatter plot of prefectures’ population density and total fertility rate in Figure 1.1. This equation does not
include wages and rental incomes, since here the log utility is used in the specification of preferences: any
increase in income potentially affects the decision regarding the number of children but is cancelled out
by the substitution effect. It would be possible to change this absence of income and substitution effects by
using a different utility function, but then the analytical expression of the optimal fertility choice would
16

no longer be available. 8
. In the following, both a log utility function and a general utility function are
considered.
Since the young are allowed to migrate freely in the model, the equilibrium allocation of the population
across regions needs to be a spatial equilibrium: agents must be indifferent in the choice of region, that is,
U1t = U2t = U3t must hold in equilibrium. Otherwise, agents would want to relocate to a place where
they can derive a higher utility and the population distribution would not be in equilibrium.
There are two other equilibrium conditions. The first is that rental rates are equalized across regions.
The second is that aggregate capital follows the following law of motion:
Kt+1 = St+1 =
X
l=1,2,3
Nltslt+1 =
X
l=1,2,3
Nltwltllt.
1.4.3 Solution of the Model
Both the steady state and the transition of the model are of interest. The steady state is derived by dropping
the time subscripts and solving the system of equations. The stability of the model around this steady
state is confirmed by checking the eigenvalues of the linearized system. To derive the path from the initial
conditions and the steady state, there are two different ways in which the path from the initial conditions
and the steady state could be derived. The first is to employ the log function for utility to simply solve
the model forward to obtain the steady state. The second is to start with the general specification of the
utility function and solve the system backward from the steady state using a procedure akin to the ’reverse
shooting’ in Judd (1998). The equilibrium conditions and solution methods are detailed in the Appendix.
8Many countries experience a decline in the fertility rate as they develop. One explanation proposed by Becker and Lewis
(1973) is the quantity-versus-quality trade-off: introducing a quality dimension in child rearing, they argued that the richer people
become the fewer children they have, and the more they invest in the education of each child. While the simple model here does
not include a quality dimension with regard to the choice of whether to have children and how many, a shortcut way to describe
this ’change in tastes’ would be to introduce a negative income effect on the utility of child-rearing. This paper, however, does
not argue that the root cause of the decline in fertility comes from the quantity-versus-quality trade-off.
17

1.5 Calibration and Model Dynamics
To calibrate the model, the following parameters need to be determined: the coefficient of congestion (bl
),
the cost of child-rearing (zl
), the labor share (αl
); the agglomeration elasticity (βl
), the relative importance
of child-rearing in utility (γ), the depreciation rate (δ), and productivity (Al
). Since the model is rather unconventional and there are not many preceding studies that combine regional choices and fertility choices,
it is desirable to use as many parameters as possible from micro estimation. Specifically, the agglomeration
elasticity βl
is calibrated based on the estimation approach employed by Ciccone and R. Hall (1996), while
bl
, zl
, and γ are calibrated using moment conditions derived from the model and time use survey data. As
for the remaining parameters, α = 0.6 and δ = 0.03 are fixed following conventional estimates, while all
other parameters are determined by matching the model and the data.
1.5.1 Calibration of the Agglomeration Elasticity β
Following Ciccone and R. Hall (1996) and Glaeser and Gottlieb (2009), the agglomeration elasticity is estimated by regressing the log of prefectural per-capita GDP on the log of the prefectural population. This
equation, however, of course does not take into account that population itself is an endogenous variable:
while an increase in the population may raise productivity through agglomeration externalities, higher
productivity also works to attract more people to that region.
The instrumental variable employed in this study is the prefectural population in 1890. The rationale
for using this instrument is the distinct historical situation in Japan around that time. The regional distribution of Japan’s population at the time is well explained by historical patterns of rice production. In
those days, tax payments by farmers mainly took the form of rice. The Tokugawa government (1603-1868)
sent out local public officers to each region (kuni, which were divided slightly differently from the current
administrative units in Japan) and measured the rice production potential of farmlands several times to
collect the rice tax. This meant that regional production capabilities were well known throughout Japan,
18

0 500 1000 1500 2000
pop1834
0 1000 2000 3000
koku
Figure 1.4: Scatter Plot of kuni-Level Rice Output (per Capita) and Population
with regions with a high output, such as the western side of Honshu, attracting farmers and merchants.
For example, Kaga, which more or less corresponds to today’s Ishikawa prefecture, which ranks only 34th
in terms of population among Japan’s 47 prefectures, once ranked 1st because of its high rice output.
Against this background, Figure 1.4 plots on the horizontal axis the ‘koku’ of rice output (1 koku
corresponds to about 280 liters or the amount of rice consumed by one person in one year) and on the
vertical axis the population in 1834. Each dot corresponds to a kuni. The total number of kunis is about 68,
which is larger than the number of current prefectures, which is 47. The figure shows that there is a close
correlation between kuni-level rice output and kuni-level populations. 9 Rice output in the Edo period was
mainly influenced by climate conditions, so that prefectural productivity levels (rice production capability)
in those days are not expected to be directly linked to productivity today. Based on these considerations,
the current prefectural population here is instrumented by the prefectural population in 1873 and per capita
output is regressed on prefectural populations today. The only reason the past population size of a region
affects the present productivity of that region is assumed to be through its linkage to the present population
size, because the major determinant of the past population size, rice production capability, is not likely to
9Another way to construct the instrumental variable would be to directly use koku. However, doing so would require matching the 68 Edo-period kuni with the 47 prefectures today. This would be a rather complicated task, since the administrative
boundaries of the former kuni and current prefectures do not fully coincide.
19

directly affect the productivity of that region today. The data for current prefectural populations used in
the estimation are as of 2019.
(1) (2) (3)
Prefectural income per capita Prefectural income per capita Prefectural income per capita
Population 0.086∗∗∗ 0.192∗
0.174∗
(0.02) (0.08) (0.08)
IV No Yes Yes
Excluding Tokyo No No Yes
Observations 47 47 46
Table 1.3: Estimation Results of Agglomeration Elasticity (β)
Table 1.3 summarizes the estimation result. The use of instrumental variables pushes up the estimate
from 0.086 to 0.192. The difference between the estimates is larger than that between the estimates by
Glaeser and Gottlieb (2009), who obtained estimates of 0.04 using OLS and 0.08 using instrumental variable
estimation. Given that the population of Tokyo is so much larger than that of the other prefectures, the
instrumental variable estimation is repeated dropping Tokyo from the sample. This reduces the estimate
but still yields a higher value than the OLS estimate.10
Glaeser and Gottlieb (2009) level some criticism against the methodology employed by Ciccone and
R. Hall (1996). Specifically, Glaeser and Gottlieb (2009) argue that population levels in the past might influence productivity today, contrary to the argument presented by Ciccone and R. Hall (1996). For instance,
population levels in the past might play a role in productivity today through the accumulation of nontradeable physical capital over time. The estimation in this study potentially is subject to similar criticism.
D. R. Davis and D. E. Weinstein (2002) examine the sizes of cities in Japan spanning a period of 2,000 years,
paying particular attention to their sizes before and after the bombing of cities in World War II (around
1944). They argue that the bombing did not have a lasting impact on the city size distribution, and cities
10In the set up of the model, the time endowment is fixed to be 1. All the parameters are needed to be set in relation to this,
so that the parameters are set into the model with proper adjustments.
20

that had a large population before the war did so again soon after the war. This suggests that cities that
had a larger population in the past might, despite the bombing, have maintained non-tradeable physical
capital and tended to have higher productivity for reasons other than agglomeration externalities.11
1.5.2 Calibration of b, z, and γ
For the calibration of the other parameters, another data source is used, and a relationship in the model
that can be compared with the data will be explored.
Another source of data for the calibration of parameters is the Survey on Time Use and Leisure Activities by the Ministry of Internal Communications. These data are used to calibrate the parameters regarding
congestion, child-rearing time costs, and the relative preference regarding the joy of having children. The
Survey on Time Use provides data on the time spent on commuting and child-rearing, both for the entire
population and for those engaged in these activities. For the analysis here, data from the survey on the
time spent on commuting and child-rearing is used for bNlt and zlnlt. Since the data are cross-section, it
is difficult to estimate different coefficients zl
for different regions. Therefore, the estimation based on this
dataset tries to capture the general effect, not the heterogeneous effect across regions, of the number of
children on child-rearing time. How regional differences in this link are captured is explained in the next
subsection.
Moreover, the relationship between population density and the total fertility rate for each prefecture is
plotted in Figure 1.1, which can be directly compared with the fertility-population relationship in the log
utility function case in Section 1.4.2: nl =
(1−γ)
z
(1−bNl). As discussed in the previous section, a candidate
for the instrumental variable for the current population appearing in the congestion variable blNlt in the
time use constraint and the fertility-population relationship, nl =
(1−γ)
z
(1 − bNl), is the population size
11To test the potential failure of the past population to instrument for the current population, Glaeser and Gottlieb (2009)
suggest finding other types of instrumental variables that do not resort to past economic activity but to exogenous phenomena
such as geological characteristics. As an example, Glaeser and Gottlieb (2009) study the agglomeration economies in the Sun Belt
region in the United States and propose using climate data such as the temperature in January as instrumental variables (Combes,
Mayer, and Thisse (2009)).
21

in the past. These three relationships of congestion time and population, the one of child-rearing time and
congestion, and the relationship between congestion and fertility can be employed as moment conditions,
with the prefectural population in 1873 used as instrument to conduct a generalized method of moments
(GMM) estimation.
(1) (2)
γ 0.934 0.592
(consumption preference) (0.749) (1.137)
b 0.122∗∗∗ 0.112∗
(coeff. on congestion) (0.04) (0.05)
z 0.028 0.142
(coeff. on child-rearing) (0.31) (0.33)
Excluding Tokyo No Yes
Observations 47 46
Table 1.4: GMM Estimation Results of Parameters
The results of the GMM estimation are presented in Table 1.4. They appear to be susceptible to outliers, since the estimates differ considerably depending on whether Tokyo is included (Column (1)) or not
(Column (2)). The estimates in Column (2) are used in the remainder of the study.
1.5.3 Matching the Population Movements
The remaining parameters include the innate productivity of each region Alt, which is the part of productivity that is unconnected to agglomeration spillovers, and possible regional differences in child-rearing
cost zl
, where the homogeneous portion of the cost is estimated in the previous section. These parameters
therefore are calibrated so that the values explain the regional differences in the data. Specifically, the
parameters are chosen by matching the population data and the transition described by the model. This is
done by matching each region’s share in the total population in 2005, the year in which Japan’s population
and the population in each of the regions (calculated using Census data) peaked, to each region’s total
population share in the model at the maximum of the population transition curves observed when the
constant relative risk aversion (CRRA) function with σc < 1 is used, as will be discussed in Section 1.5.4.
22

Figure 1.5: Population Trends: By Region and Japan Overall
The parameterization of the regional differences chosen to match the data shows that productivity
in the most productive region (Region 1) is 10% higher and that in the second-most productive region
(Region 2) 3% higher than in the least productive region (Region 3). Moreover, child-rearing costs are 40%
higher in Region 1 and 20% higher in Region 2 than in Region 3. This parameterization of the differences in
productivity and child-rearing costs is maintained throughout the remainder of the study unless otherwise
stated.
1.5.4 Model Dynamics
Using some initial values for the population and savings for each region, the transition of the model to
the steady state can be computed. The easiest way to do this is to use all the utility functions in log, not
just because the closed form of the fertility choice is available but also because the model can be easily
solved forward. Those born in one period are then “allocated” to the regions in the next period such that
the utility they obtain in each region is equalized. Once the population is determined, the fertility choice
is automatically determined through the closed-form relationship between nlt and Nlt. The amount of
23

0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Transition of Population
All Region
Region1
Region2
Region3
(a) Population
0 5 10 15 20 25 30 35 40 45 50
0.285
0.29
0.295
0.3
0.305
0.31
0.315
0.32
0.325
0.33
0.335
Transition of Aging Rate
All Region
Region1
Region2
Region3
(b) Elderly Population Rate
Figure 1.6: Transition to Steady State (Log Utility)
savings the middle-aged make corresponds to the amount of capital in the next period, which is allocated
across regions so that the rates of return on capital are equalized.
Figure 1.6 presents an example of the transition to the steady state. Panel 1.6(a) shows the developments
in the population in each region and at the aggregate level from the initial levels to the steady state.
Irrespective of the initial distribution of the population, Region 1, the most productive region, ends up
with the largest population. The second most productive region (Region 2) ends up with the secondlargest population, and the least productive region ends up with the smallest population. This population
ranking is preserved throughout the transition to the steady state. Next, Panel 1.6(b) depicts the evolution
of the elderly ratio over time, which is defined as the ratio of the number of the old to the total population
of each region and Japan overall. Since the demographics in the model are simplified and use just three age
groups, the ratios do not differ very much across regions. Moreover, the long-run elderly ratio in the steady
state is one-third by design. Nevertheless, the graph replicates well the observed pattern that Region 3 has
a higher elderly ratio than the other two regions because many of the young leave the region when they
turn middle-aged.
While the transition in Figure 1.6 is quite intuitive, a potential criticism is that the depicted population
trends are not a very accurate description of reality. After all, there is no economy in the world where
24

0 10 20 30 40 50 60 70 80 90 100
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Transition of Population
All Region
Region1
Region2
Region3
(a) Population
0 5 10 15 20 25 30 35 40 45 50
0.2
0.25
0.3
0.35
Transition of Aging Rate
All Region
Region1
Region2
Region3
(b) Elderly Population Ratio
Figure 1.7: Transition to Steady State (CRRA Function with σc = 0.7)
the population reaches a steady state and stops to change over time. Instead, in a number of economies,
including Japan, the population has started shrinking after reaching a peak. The results so far are based on
a log utility function. While this simplifies the model calculation, the use of a log utility function means
that income and substitution effects regarding fertility choices are omitted; that is, how much labor and
capital income the young receive does not affect their choice of the number of children. This absence of
income and substitution effects turns out to be key for determining population dynamics. When the CRRA
parameters in the utility functions σc and σn are set to values other than 1 (log utility), these effects do
play a role and the population dynamics look different.
This can be seen in Figure 1.7. Experimenting with the CRRA parameters shows that when the utility
from consumption is non-homothetic (σc < 1), there is, as seen in Panel 1.7(a), an “overshoot” in the
population, which temporarily increases above the long-run steady state level and then gradually decreases
and converges to the steady state. Why does such an overshoot occur? Due to non-homotheticity in
consumption utility, children are an inferior good: as individuals’ earnings potential increases, they want
to spend less time on child-rearing and increase working hours so that they can consume more. At the
initial stage of the transition where the accumulated capital is small, individuals derive more utility from
child-rearing, and that makes the population grow at a higher pace to eventually overshoot. Capital then
25

gradually accumulates, partly thanks to the higher productivity resulting from the agglomeration of a
greater population, and at the same time the number of children starts to decline. The decline in the
number of children does not instantly reduce the population size, so capital continues to accumulate and
exceeds the long-run level (overshoots). The population size is now at the steady state level, but there is
an over-accumulation of capital, so that capital starts to decline to its long-run level. As a result of these
processes, the overshoot of the population and capital diminishes, and both arrive at their steady state
levels.
This case also generates a "realistic" pattern of the movement of the elderly population ratio. In Figure
1.7(b), Region 3, the least populated region, which experiences a lot of outmigration by the young, has a
higher elderly population ratio than the other regions in the period when the population experiences a
decline (after the peak in population developments shown in Figure 7(a)). This replicates the pattern of
elderly population ratios across regions shown in Figure 1.2 in that Region 3 has a higher elderly population
share than the other regions when the total population starts to decline.
1.6 Social Optimum
While the interest of this study is the potential inefficiency of the laissez-faire equilibrium compared with
the equilibrium that takes the externalities of childbirth into account, it is not straightforward to determine
what the ’optimal’ size of the population is when fertility is endogenized in the model. For example, in an
equilibrium with a larger population there are people who would not have been born in an equilibrium
with a smaller population. Since Pareto optimality is based on the comparison of the welfare of fixed sets
of people, the concept is difficult to apply when the sets of people vary.
The optimal size of the population in an endogenized fertility model is one of the deep questions in
economics, and many criteria for optimality and refinements of these criteria have been proposed so far.
Early studies such as Nerlove, Razin, and Sadka (1986) contrast the laissez-faire equilibrium with one with a
26

Benthamite social welfare function that considers the maximization of the total sum of utility of individuals,
and one with a Millian social welfare function which considers the maximization of the average welfare
of individuals. 12
The criterion employed in this study is that suggested by Eckstein and Wolpin 1985, who propose to
use the welfare of a representative agent to consider the social optimum. In their model with a single
region, the social optimum allocation yields a higher population growth rate, which induces a higher rate
of return. The approach here is to expand this criterion to a multi-region setting and equip regions with
agglomeration economies and diseconomies.
Obtaining the social optimum then requires solving the problem of maximizing the utility of a representative agent who potentially lives in any one of the three regions.
1.6.1 Constrained Efficient Representative Agent
The idea of ’constrained efficiency’ is often used in discussions of the efficiency of an equilibrium where
there are some market failures. While the best solution clearly would be to eliminate the source of market
failure through the creation of a missing market, in a constrained efficiency setting the creation of the
market is not allowed. In this study, therefore, it is assumed that in the constrained efficiency problem
the representative agent is more sophisticated than in laissez-faire in that they take the impact of their
fertility choices on the size of future generations through factor prices, which in turn affect their utility,
into account. This partially internalizes the externalities, and although it does not lead to the first-best
outcome, it does result in a higher welfare than in laissez-faire, where agents fail to take into account that
their fertility choices lead to a smaller population. The equilibrium allocation in the constrained efficiency
12Golosov, L. E. Jones, and Tertilt (2007) propose two efficiency concepts: A-efficiency maximizes the welfare of the initial
generation, while P-efficiency applies Pareto efficiency to everyone, including those born only in some equilibria. However, a
shortcoming of P-efficiency is that the set of P-efficient allocations is large. Another unique perspective in this debate is provided
by De la Croix and Doepke (2021): to take into account the welfare of non-existent individuals, they introduce the idea of a fixed
supply of souls that are reincarnated sooner or later. To evaluate the value of the possibly non-existent agents, they separate the
possible set of populations which could vary from the fixed supply of souls. Their study is unique in evaluating the utility of the
probabilistic life lived by a fixed supply of souls.
27

problems yields the fertility rate for an equilibrium that is closer to the first-best. The different fertility rate
in a constrained efficient equilibrium, at the same time, leads to a different distribution of the population
across regions.
For simplicity, log preferences are used for both consumption and fertility. Since the constrained efficient representative agent is allowed to foresee the pecuniary externalities of their fertility choices, prices
are regarded as a function of fertility choices:
max ¯u
s.t.for each l = 1, 2, 3, γ log (c
′
l
) + (1 − γ) log nl ≥ u, ¯
w(kl
, nl)l(nl) ≥ k
′
l
,
(1 + r(k
′
l
, N′
l
(nl)) − δ)k
′
l ≥ c
′
l
,
P
l=1,2,3 N′
l ≥
P
l=1,2,3 nlNl
.
The optimality condition is
(1 − γ)
1
n
γ
1
c
′
= zw(1 + r
′ − δ)(1 − α + β) + b(1 − α + β)wr′L/L′N
′ − βwr′L/N′
,
which is quite different from the laissez-faire case. The difference between the two conditions is the opportunity cost of rearing children. In laissez-faire, time allocated to child-rearing reduces the time available
for work. Therefore, the only opportunity cost of child-rearing is the labor wage. In contrast, in the constrained efficiency setting, the constrained planner takes the effect of fertility choices on the size of the
next generation, which matters for both congestion and agglomeration, into account. While congestion
raises the opportunity cost of child-rearing, the economic benefits from greater agglomeration through
the addition of children to the next generation offsets the cost to some extent. The second and third terms
28

on the right-hand side of the optimality condition of the constrained optimization problem correspond to
the opportunity cost of congestion and the benefits from agglomeration, respectively.13
1.6.2 Comparison of Laissez Faire and Social Optimum
Table 1.5 compares the allocation of the representative agent in a laissez-faire equilibrium and in the constrained efficient equilibrium discussed in the previous subsection. The comparison shows that the population in the social optimum is larger than that in the laissez-faire equilibrium in all regions. Moreover, the
number of children per household nlt in Regions 2 and 3 is higher in the social optimum than the laissezfaire equilibrium. The reason is that in the constrained efficient equilibrium an agent takes the effect of
their fertility choice on the cohort size into account. A larger cohort pushes up the size of the economy,
which is beneficial for people through the enhanced productivity through agglomeration economies and
consumption. In Region 1, congestion matters more and lowers the number of children, whereas in the
other two regions the agglomeration resulting from the larger size of the cohort in the next period induces
a higher number of children per person.
Laissez-Faire Social Optimum
Number of children: n1 0.84 0.82
Number of children: n2 1.03 1.04
Number of children: n3 1.30 1.34
Population: N1 0.64 0.78
Population: N2 0.45 0.53
Population: N3 0.29 0.35
Capital: k1 3.78 4.92
Capital: k2 1.61 1.90
Capital: k3 0.61 0.69
Table 1.5: Comparison of Steady State Allocations in Laissez Faire and Social Optimum
13It is noted that even if the agglomeration spillover and congestion are muted (β = 0 and b = 0), the first order condition
in the constrained efficient problem is still different from the counterpart in the laissez-faire problem due to the consideration of
the effect of increasing the number of children on the size of the next cohort, which raises the interest rate.
29

The comparison in this section highlights that in a laissez-faire economy the population size is smaller
than in the social optimum.14 The next section considers policies to make the allocation in laissez-faire
closer to that in the social optimum.
1.7 Policy Analysis
The study so far has presented a model that makes it possible to consider demographic developments
in a country to show that the population in a laissez-faire equilibrium is smaller than that in the social
optimum. The ultimate goal of this study is to examine how the size and distribution of the population in
a laissez-faire setting can be brought closer to the social optimum through policy interventions. Against
this background, this section considers two possible policies using the model. The first is transfers across
regions which are then distributed equally to each individual, while the second is a subsidy providing
financial support (transfers) to individuals based on the number of children. Finally, a quantitative welfare
assessment of current policies in Japan based on the model presented here is provided.
1.7.1 Regional Transfers
One of the reasons for the transfer of funds across regions is to allow each region to meet the fiscal needs
necessary to provide the minimum level of public services to itsresidents. While it goes without saying that
regions with a larger population collect more taxes and have larger expenditures, there is also considerable
heterogeneity in tax revenue and expenditure on a per capita basis. For example, regions with a larger
share of elderly, i.e., retirees, are likely to have lower labor income tax revenues and higher expenditures on
health and welfare for the elderly. Similarly, businesses tend to locate their headquarters in more populous
regions to benefit from agglomeration economies, resulting in an unequal distribution of corporate tax
14It should be noted that the population values in the laissez-faire equilibrium (e.g., N1 = 0.64) do not have meaning in
themselves: only the proportion across regions is replicated through the calibration discussed in Section 1.5. This paper studies
the difference in population sizes across regions and for different specifications and policies.
30

revenues, part of which accrue to local governments. To addresssuch unequal distribution of tax revenues,
Japan has a fiscal equalization scheme in the form of "Local Allocation Tax Grants," under which a certain
portion of national tax revenue15 (about 18 trillion yen in fiscal 2020) is distributed across regions to make
up for discrepancies between local fiscal expenditure needs and tax revenues. Except for a subset of regions
including the 23 special wards of Tokyo prefecture, which are net contributors, the vast majority of regions
receive such transfers. The purpose of this section is to examine how this policy changes the allocation
and welfare in a laissez-faire economy compared with that without the transfer policy. In this subsection,
it is assumed that Region 1 is a net contributor of funds, while the other two regions are net recipients of
transfers.
To examine the impact of such transfers, they are incorporated into the model in different ways. The
simplest way to do so is to assume that funds are collected in the form of lump-sum taxes and then distributed as lump-sum transfers. Table 1.6a presents the resulting steady state allocations. Compared with
Transfer taken
from Region 1: T
0
5% of
GDP
10% of
GDP
Number of children: n1 0.84 0.81 0.64
Number of children: n2 1.04 1.06 1.25
Number of children: n3 1.30 1.37 1.45
Population: N1 0.64 0.72 1.37
Population: N2 0.45 0.46 0.45
Population: N3 0.29 0.29 0.86
Capital: k1 3.78 5.01 14.31
Capital: k2 1.61 1.45 0.37
Capital: k3 0.61 0.47 0.22
Average utility: u¯ 0.68 0.69 0.62
(a) Lump-sum Transfers
Tax rate on
labor income: τl
0.00 0.10 0.25
Number of children: n1 0.84 0.82 0.63
Number of children: n2 1.04 1.04 1.18
Number of children: n3 1.30 1.34 1.46
Population: N1 0.64 0.61 1.22
Population: N2 0.45 0.42 0.33
Population: N3 0.29 0.27 0.82
Capital: k1 3.78 3.12 7.89
Capital: k2 1.61 1.12 0.29
Capital: k3 0.61 0.37 0.11
Average utility: u¯ 0.68 0.64 0.57
(b) Transfers Funded by Labor Income Taxes (τl)
Table 1.6: Comparison of Steady State Allocations with Different Types of Transfers
the economy without transfers, the transfers result in larger populations. In the equilibria with transfers
the number of children per person is higher in Regions 2 and 3 and lower in Region 1. This is the same
pattern as the social optimum discussed in Section 1.6. Thus, this policy has the potential to make the
15Specifically, 33.1% of income tax and corporate tax revenues, 50% of liquor tax revenues, 19.5% of consumption tax revenues, and the full amount of local corporate tax revenues are earmarked for the Local Allocation Tax Grants.
31

allocation closer to the social optimum. The transfers flowing into Regions 2 and 3 increase consumption
in these regions. Now that Region 1 has fewer resources available due to the transfer burden, welfare in
Region 1 declines while that in the other regions increases. This unequal welfare across regions is equilibrated again if there is an influx of population in Region 1, and that can happen only when the number of
children increases in the other regions and they migrate to Region 1. In this process, the total population
increases and so does the size of the economy. The increased size in turn pushes up production in Region
1. Another point to note is that the relationship between the transfer amount and average utility is not
monotone and eventually turns negative. When transfers reach 10% of gross domestic product (GDP),
they do raise the population size but provide lower utility than when transfers amount to only 5% of GDP.
While the increased population in Region 1 raises the degree of agglomeration, the smaller number of
children in Region 1 fails to raise overall utility. Another interesting finding is that the relative population
sizes in Regions 2 and 3 reverse: when transfers amount to 10% of GDP, production in Regions 2 and 3
falls, so that innate productivity differences begin to stop playing a role in determining the population
ranking.
Next, the effect of regional transfers is examined when they are funded not by a lump sum tax on
Region 1 but by distortionary taxes. A labor income tax of the same rate is imposed on agents in all
regions and the proceeds are distributed back to Regions 2 and 3. Unlike the lump-sum tax, taxation of
labor income discourages labor supply and distorts time allocation decisions. Not only does it push up
the cost of working, it also makes child-rearing more costly, since child-rearing time competes with time
allocated to work.
The results of the policy are shown in Table 1.6b. Unlike lump sum taxation, labor income taxation at
a moderate rate (10%) in all regions fails to push up the population size. Average utility is also reduced. If
the rate is set sufficiently high (25%), transfers funded by labor income taxes push up the population level
in all regions. The fertility rate is lowered in Region 1 and increased in the other regions. Utility, however,
32

does not improve in this case either. The reason for the different outcomes for τl = 0.1 and τl = 0.25
is that in the latter case, working is so costly in most of the regions that people other than those in the
most productive one (Region 1) work less, so that output is lower, just like in the case with a lump sum tax
amount of 10% of GDP on Region 1 .
1.7.2 Child Allowance
Another potential policy option to counteract the aging and shrinking of the population is to provide
subsidies for having children. Providing one-time or long-term incentives is a type of policy used in a
range of countries. In Japan, the government provides a one-time transfer of approximately 500,000 yen
for the birth of each child. In addition, there are regular subsidies for households with children below a
certain age. While in the past such subsidies were limited to 5,000 yen per month and the age limit was
15, the government has recently raised the monthly amount and increased the limit to 18 years old as a
measure to raise the fertility rate.
To examine the impact of the subsidy using the model developed here, it is assumed that it is proportional to the number of children per person nlt. The subsidy is financed through a fixed lump sum tax
imposed on all agents.16 Now the number of children per person has two effects on utility: not only does
it provide people the joy of having a child, people also receive a transfer of an amount proportional to the
number of children.
The results of the policy are shown in Table 1.7. The subsidy amount is expressed as a percentage of
the annual income of individuals in a given region. The policy succeeds in raising the number of children
in Regions 2 and 3, while it reduces the number of children in Region 1. These changes in fertility choices
lead to a larger population in all three regions. All these results are in the same direction as the transfer
policy discussed in the previous subsection. What is notable is that the policy leads to higher utility than
16It could also be assumed that the per capita lump tax differs across regions. For example, it could be made higher in Region
1 and lower in the other regions. This would increase the redistributive effect and would likely increase welfare.
33

Transfer per child: τn 0 5% of annual income 50% of annual income
Number of children: n1 0.84 0.83 0.82
Number of children: n2 1.04 1.04 1.11
Number of children: n3 1.30 1.33 1.40
Population: N1 0.64 0.69 0.99
Population: N2 0.45 0.47 0.49
Population: N3 0.29 0.31 0.31
Capital: k1 3.78 4.06 4.84
Capital: k2 1.61 1.64 1.16
Capital: k3 0.61 0.62 0.45
Average utility: u¯ 0.68 0.69 0.67
Table 1.7: Comparison of Steady State Allocations with Varying Transfers Conditional on the Number of
Children per Household (τn)
without the subsidy. The previous subsection showed that distortionary taxes lead to lower welfare. With
the child allowance, however, the policy focuses on the number of children, which directly affects the
population size, something that individuals do not internalize. The policy helps to partially reduce the
inefficiency resulting from the externalities that fertility choices have on the size of the cohort.
While subsidies corresponding to 5% of recipients’ annual income are realistic, subsidies of 50% of
recipients’ annual income are clearly unrealistic. The reason for considering the latter case is to illustrate
that a subsidy of this type is effective in raising the population size and average utility for a wide range of
amounts. It should be borne in mind, however, that the funding is through a lump sum tax. As mentioned in
Section 1.7.1, labor income taxes lower utility and thus could alter the effectiveness of the child allowance.
1.7.3 Quantitative Analysis of the Full Model
1.7.3.1 Incorporating Current Policies in Japan
While Sections 1.7.1 and 1.7.2 examined the different types of subsidies separately, this subsection incorporates both subsidies simultaneously. Households’ payments and receipts linked to the policies in the
model are set to match those of current policies in Japan as much as possible. In addition to these subsidies, Japan’s pay-as-you-go public pension system is also incorporated in the analysis: people pay into
34

the system when they are middle-aged and receive their pension when they are old. Incorporating public
pensions into the model does not materially alter the results regarding the policies studied so far. However,
since public pension receipts account for a non-negligible portion of individuals’ income, they are included
in the analysis below.17
Starting with households’ payments, it is assumed that households pay a labor income tax rate of
τl = 23%, which corresponds to the rate imposed on the median income level. Taxation on capital is
omitted in the current calculation but could be easily added. Moreover, the labor income tax rate could
be allowed to vary from region to region to take into account that large prefectures in Region 1 contain a
larger number of high-income individuals. For simplicity, however, the calculation here completely omits
progressive tax rates.
Next, public pension contributions of individuals when they are middle-aged amount to about 16,000
yen per month. Since one period corresponds to 30 years in the model, the per person burden on the
middle-aged (the working-age generation in the model) is about 200,000 yen per year, which corresponds
to ca. 2.5% of the average annual income of 8 million yen. It is difficult, however, in the early stage of the
transition of the model economy to let people pay a fixed amount since the income starts at a low level, so
instead it is assumed that contributions amount to a fraction of individuals’ income (i.e., 2.5%).
Turning to households’ receipts, the local allocation tax grant is distributed as regional transfers and
delivered as a lumpsum transfer to households. The amount of transfers in this calculation is based on
transfers in 2015. While Tokyo prefecture does not receive any transfers, all the other prefectures in
Region 1 receive local allocation tax grants. Therefore, for the analysis here, the sum of grant receipts for
each of the three regions is calculated and expressed as a share of regional GDP. The calculations show
that Region 1 receives grants amounting to 0.5% of its GDP, while the shares for Regions 2 and 3 are 2%
and 4%, respectively.
17The numerical analysis including the public pension system is provided in the Appendix.
35

The public pension amount individuals receive when they are old is determined by the government
based on an income substitution ratio, which is about 50% after adjustment.18 The effect of the public
pension scheme on the allocation is discussed in the Appendix.
Transfers to individuals having children, in addition to the one-off payment of about 500,000 yen upon
childbirth, consist of a child allowance that, depending on the municipality, ranges from 10,000 to 15,000
yen per month until graduation from high school. What is more, another 1 million yen per child is added
to the relocation subsidy of 1 million yen for workers relocating from a large city. To reflect these various
transfers, the analysis here introduces a transfer per child of 2% of annual income.
To ensure fiscal balance in all periods, government surpluses or deficits are allocated to all agents
equally.
1.7.3.2 Results
Transfer per child: τn 0 0.020 0.065 0.10
Number of children: n1 0.84 0.82 0.79 0.74
Number of children: n2 1.04 1.06 1.08 1.12
Number of children: n3 1.31 1.36 1.44 1.49
Population: N1 0.71 0.82 0.98 1.17
Population: N2 0.47 0.50 0.51 0.49
Population: N3 0.30 0.32 0.38 0.50
Capital: k1 0.34 0.48 0.72 1.03
Capital: k2 0.14 0.16 0.18 0.15
Capital: k3 0.05 0.06 0.06 0.05
Table 1.8: Comparison of Steady State Allocations with Varying Transfers Conditional on the Number of
Children per Household(τn) in the Full Model
The results are shown in Table 1.8 for different child allowance amounts. The baseline, as set out in
subsection 1.7.3.1, is τn = 0.02 (i.e., child allowance payments equivalent to 2% of people’s annual income).
Various forms of distributional policies transfer resources from the most productive region, Region 1,
18The ratio could be set lower than this in the analysis given that in practice not all elderly people live up to the age of 90.
Since one period of the model corresponds to 30 years, the life of the old ends at 90 years old. Since the model does not incorporate
the probability of death when individuals go on to the next period of their life, it is implicitly assumed that all individuals live
up to the age of 90. Employing the actual current income substitution ratio for Japan in the model means that the total sum of
pension payments individuals receive is larger than in reality, since the average life expectancy is less than 90 years in reality.
36

to the other regions to increase the population compared with the laissez-faire outcome. Without the
child allowance (τn = 0), the size of the population is smaller, since there is no boost to the number of
children in Regions 2 and 3. As τn increases, the total population increases: whereas the total population
(N1 + N2 + N3) is about 1.6 in the result with τn = 2%, it is about 1.9 in that with τn = 6.5%. This is
because the number of children per person in Regions 2 and 3 increases (and fewer children are born in
Region 1).
0 1 2 3 4 5 6 7 8 9 10
transfers proportional to the number of children per household (percent of annual household income)
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
consumption equivalent (percent change from outcome with
n=0.02 )
Figure 1.8: Welfare Change Measured in Terms of Consumption Equivalent Measure
The optimal child allowance transfers in terms of the percentage of annual income are depicted in
Figure 1.8. Setting the average utility when τn = 0.02 as the benchmark, the horizontal axis represents
the level of τn, while the vertical axis represents the associated consumption equivalence measures. For
example, a consumption equivalence of 1% (attained at about τn = 0.03) means that in order to reach
the average utility attained with τn = 0.03, under the policy with τn = 0.02, agents need 1% more
consumption. The negative consumption equivalence at τn = 0 shows that child allowance transfers have
the potential to improve welfare. As can be seen, however, the relationship is not monotone; rather, welfare
peaks at a child allowance level of τn = 0.065. In other words, average utility does not rise limitlessly
but reaches a peak when the transfer per child is set at about 6.5% of households’ annual income. At this
37

level, the consumption equivalence reaches 2%. Beyond this value, an increase in τn does increase the size
of the population but does not lead to a welfare improvement. Instead, the distortions brought about by
child allowance transfers lead to a deterioration in average utility.
The policy implication of these results is that the current size of child benefit transfers may be too
small, and that there is room for increasing the size of transfers to raise the population size and hence
welfare.
1.8 Conclusion
The aging and shrinking of Japan’s population is subject to substantial heterogeneity across regions.
Against this background, the current study showed how the elderly population share functions as an indicator predicting the direction of internal migration. To show that the size and distribution of the population
have implications for the economic growth and welfare of Japan overall, this study presented a model featuring overlapping generations, endogenous fertility and cohort formation, and multiple regions. Using
the utility of a representative agent as a criterion of welfare, the laissez-faire allocation was compared with
the socially optimal allocation in a constrained efficiency setting to show that the latter leads to greater
welfare, since it internalizes the effect of fertility choices on the size of the economy. The analysis showed
that it would be possible to make the laissez-faire allocation closer to the social optimal allocation through
policies that direct resources from more productive to less productive regions, since this would not only
boost consumption in the recipient regions but also increase the number of children born, so that the size
of the economy increases. Given the current set of transfers, taxes, and pensions in Japan, additional child
allowance transfers would be a particularly useful policy tool. The calculations presented here based on
current policies in Japan suggest that raising child allowance transfers would improve welfare by boosting
the number of children.
38

The analysis is based on the calibration of parameters, some of which were obtained through original
estimates based on an instrumental variable estimation. It is hoped that this model will provide a starting
point for future studies focusing on the population distribution within countries and the policies that affect
it.
The current model is kept as simple as possible, so that an array of ingredients can be handled: overlapping generations, endogenous fertility and cohort formation, and multiple regions. One possible way
of extending the current analysis would be to enrich each of these dimensions, for example by considering
more than three regions and/or more than three life stages. Moreover, the assumption that people only
choose once whether to move or stay put is a substantial abstraction, and a possible extension would be
to allow people to move again in old age, for example. While each of these modifications would make
the model far heavier, they would allow a more realistic and detailed analysis of the polices affecting the
demographics of a country.
A.1 An Extended Literature Review
Study on the role of fertility choices on economic growth dates back to Eckstein and Wolpin (1985). By
defining a social optimum as a utility of representative agent, they demonstrate in an overlapping generations model the smaller number of children in a laissez-faire equilibrium than that in a social optimum
because people fail to internalize the contribution of their fertility choice to the population growth, which
lowers the steady state amount of capital through natural interest rate. This paper, through the consideration of multiple regions, distinguishes this contribution as an externality that affects agents in other
regions as well.
The simple model demonstrated in this paper is formulated in Ikazaki (2014) and Muroishi and Yakita
(2021) following Sato (2007). While they study the laissez-faire equilibrium only, this paper departs from
theirs by comparing it with constrained social optimum, and discusses how the heterogeneity in population
39

across the nation affects aggregate growth and welfare. This paper also study multiple formulations of
policies that affect the spatial features of the demographics in Japan.
Many studies have focused on the spatial equilibrium of Japan. Sato (2007) is the first paper that relates
fertility choice to an agglomeration economy and diseconomy. The whole analysis is concentrated on a
steady state and there is no transition analysis. Morita and K. Yamamoto (2018) also begins with a negative
correlation of fertility and congestion, but abstracts from regional migration and introduce monopolistic
competition and free entry of firms. They demonstrate that, in the optimal equilibrium19, there are fewer
entries of firms and larger fertility choices.
Some recent studies combine dynamic migration decisions with overlapping generation models20, and
several of them in fact choose Japan as the setting of the study. Y. Suzuki (2021) studies the dynamic spatial equilibrium of Japan and argues that ongoing aging affects the welfare of local people by delaying the
adjustment of capital when a negative shock hits the local area. K. Suzuki and Y. Doi (2022) studies the
effect of foreign workers in Japan using a static but spatial equilibrium model equipped with geography,
trade, multiple sectors and linkages among them, and worker qualities coming from educational attainment. He argues that the introduction of foreign workers is a better policy than often discussed policy
options such as the facilitation of labor participation of women and elderly people. Giannone et al. (2023)
shares the same question with this paper, and studies the geography of depopulation and aging using a
dynamic spatial equilibrium model of Japan. They highlight the importance of local amenities to shape the
observed concentration of people in big cities and the depopulation and aging in local areas. While both
Y. Suzuki (2021) and Giannone et al. (2023) introduce fertility and take into account the formation of the
next generation based on fertility rates of each age of women and different regions, it is not endogeneized.
This paper features, on the other hand, the different grounds for the choice of the number of children in
different regions and different stages of the population evolution as an important dimension. This is why
19They do not particularly discuss their definition of optimality unlike this paper.
20Komissarova (2022) studies the migration after retirement in the U.S. using an overlapping-generations spatial equilibrium
model.
40

this paper let the fertility choice be relevant to congestion, and possibly to the stage of income level in the
case of non-homothetic utility function.
A normative aspect of spatial development has been one of the important objectives of urban economics. Henderson (1974) is the seminal work, and discusses why a city fails to attain the optimal size,
where a scale economy in production, specialization, and commuting cost are taken into account. This incorporation of both the concentration forces and dispersion forces21 gives rise to the "fundamental trade-off
in urban economics" (Fujita (1989)). Albouy et al. (2019) extends the analysis by endogeneizing the number of cities and discusses that the cities in the most amenable cite tend to be underpopulated because a
decreasing return to scale will emerge both across and within cities. In this paper, the optimal city size is
mainly determined by the scale economy and diseconomy, but they are at the same time affected by the
formation of the next generations through fertility choices as well.
The study of optimal spatial policies is one of the most advancing topics recently. A policy targeting
regions and not people is often called a place-based policy. For example, a government might choose to
subsidize areas experiencing a decline in the population or industry to revitalize those economies. Such
a policy is, however, criticized that the policy tries to invite and lock people into an unproductive area.
Breinlich, Ottaviano, and Temple (2014) cites an often-heard idea that "a policy should protect people rather
than places." In the literature, however, many papers point to the occasions when a place-based policy is
justified. Fajgelbaum and Gaubert (2020) studies an optimal spatial transfer in the presence of knowledge
spillover that is especially strong when a high-skilled worker cooperates with a low-skilled worker. Relocating a high-skilled worker to a region where low-skilled workers populate has the potential to improve
aggregate efficiency. In this way, they justify the use of a place-based policy where the correction of local
externality leads to aggregate efficiency. This paper, on the other hand, posits that a place-based policy
in disadvantaged regions is justifiable because it might help people relocate from big cities to those local
21While congestions and commuting cost are popular ingredients as dispersion forces in spatial models including this study,
Matsuyama and T. Takahashi (1998) is an uncommon work which does not incorporate dispersion forces but successful in describing a situation where the concentration of the activities self-defeat (i.e. concentration in the laissez-faire is suboptimal).
41

cities to have children, which widens the size of the population and its effect on agglomeration is beneficial
for all regions.
A.2 Equilibrium Conditions
A.2.1 Laissez Faire
• Definition of aggregate labor:
L1 = N1l1, (1)
L2 = N2l2, (2)
L3 = N3l3. (3)
• Definition of time allocations:
l1 = 1 − b1N1 − z1n1, (4)
l2 = 1 − b2N2 − z2n2, (5)
l3 = 1 − b3N3 − z3n3. (6)
• Equalization of interest rate:
r = A1αKα−1
1 L
1−α+β1
1
, (7)
r = A2αKα−1
2 L
1−α+β2
2
, (8)
r = A3αKα−1
3 L
1−α+β3
3
. (9)
• Definition of wage rate (firm’s optimality condition):
w1 = A1(1 − α)K
α
1 L
−α+β1
1
, (10)
w2 = A2(1 − α)K
α
2 L
−α+β2
2
, (11)
w3 = A3(1 − α)K
α
3 L
−α+β3
3
. (12)
42

• Budget constraint in old period:
c
′
1 = (1 + r
′ − δ)(1 − τ1)w1l1 + T1, (13)
c
′
2 = (1 + r
′ − δ)(1 − τ2)w2l2 + T2, (14)
c
′
3 = (1 + r
′ − δ)(1 − τ3)w3l3 + T3. (15)
• Equaliazation of utility across regions:
γ
{(1 + r
′ − δ)(1 − τ1)w1l1 + T1}
1−σc
1 − σc
+ (1 − γ)
n
1−σn
1
1 − σn
(16)
− γ
{(1 + r
′ − δ)(1 − τ2)w2l2 + T2}
1−σc
1 − σc
− (1 − γ)
n
1−σn
2
1 − σn
= 0,(17)
γ
{(1 + r
′ − δ)(1 − τ1)w2l2 + T2}
1−σc
1 − σc
+ (1 − γ)
n
1−σn
2
1 − σn
(18)
− γ
{(1 + r
′ − δ)(1 − τ2)w3l3 + T3}
1−σc
1 − σc
− (1 − γ)
n
1−σn
3
1 − σn
= 0.(19)
• Capital demand and supply:
K1 + K2 + K3 − w1L1 − w2L2 − w3L3 = 0. (20)
• Evolution of population:
N
′
1 + N
′
2 + N
′
3 = N1n1 + N2n2 + N3n3. (21)
• Optimal choice of fertility rate and consumption:
1 − γ
γ
{(1 + r
′ − δ)(1 − τ1)w1l1 + T1}
σc = z1(1 + r
′ − δ)(1 − τ1)w1n
σn
1
, (22)
1 − γ
γ
{(1 + r
′ − δ)(1 − τ1)w2l2 + T2}
σc = z2(1 + r
′ − δ)(1 − τ2)w2n
σn
2
, (23)
1 − γ
γ
{(1 + r
′ − δ)(1 − τ1)w3l3 + T3}
σc = z3(1 + r
′ − δ)(1 − τ3)w3n
σn
3
. (24)
A.2.2 Constrained Efficient Problem
Optimal choices of fertility rate and consumption differ from the laissez-faire and are as follows:
43

• Optimal choice of fertility rate and consumption:
1 − γ
γ
{(1 + r
′ − δ)(1 − τ1)w1l1 + T1}
σc =

r
′

(1 − α + β)bN1l1/l′
1 − βl1N1/N′
1

(25)
+ z1(1 + r
′ − δ)(1 − α + β)

(1 − τ1)w1n
σn
1
, (26)
1 − γ
γ
{(1 + r
′ − δ)(1 − τ2)w2l2 + T2}
σc =

r
′

(1 − α + β)bN2l2/l′
2 − βl2N2/N′
2

(27)
+ z2(1 + r
′ − δ)(1 − α + β)

(1 − τ2)w2n
σn
2
, (28)
1 − γ
γ
{(1 + r
′ − δ)(1 − τ3)w3l3 + T3}
σc =

r
′

(1 − α + β)bN3l3/l′
3 − βl3N3/N′
3

(29)
+ z3(1 + r
′ − δ)(1 − α + β)

(1 − τ3)w3n
σn
3
. (30)
A.3 Alternative Specification
A.3.1 Declining Population
One of the restrictive requirements in the analysis in the preceding literature is an assumption of the
’steady state’. After all Japan and other advanced economies in Europe experience a steady decline in the
total population22. There are, in fact, no countries that are in a steady state and keep a constant level of
population.
One way to depart from a constant population level in the steady state is to look at a balanced growth
path: a steady state in terms of per person, with the overall population on the exploding or vanishing
trajectory. However, there is a technical difficulty; while the balanced growth path deals with the (population) level-free allocation, the agglomeration, and congestion featured in the model are governed by the
level of the population. There is some literature that alleviates this problem (Rossi-Hansberg and Wright
(2007), M. A. Davis, Fisher, and Whited (2014)).
22C. I. Jones (2022) features a shrinking population in a semi-endogenous growth model of knowledge spillover. When the
fertility choice of the agents implies a negative population growth, the competitive equilibrium leads to a result named as an
"Empty Planet," where the balanced growth path allocation is on a vanishing path of the population.

Described here is a short-cut way of relating the congestion to the concentration proportion of the
population. While in the baseline case above the congestion is modelled in the use of time as blNlt, it
could be alternatively formulated by:
bl
Nlt
N1t + N2t
.
This is helpful but not without a compromise. This specification says that a city is ’crowded’ as long as
it has more population than the other. This applies to not just the current Tokyo but to the Tokyo at the
beginning of the development, where there is indeed more population concentrated than the other areas
but still not that congested in terms of the size of the population. This suggests that the congestion in this
way might not be what most people have in mind.
A.4 Computation of the Transitions
A.4.1 Solving Forward
To solve the model forward, one needs to give initial conditions for the total number of young people
at t0, Nt
, and the aggregate capital Kt
. Both are allocated to each region so that equilibrium conditions
for time t0 are satisfied, especially the optimality conditions that include variables with time subscript
of one-period future: t0 + 1. Variables in t0 can be determined despite the presence of future variables
clt0+1 and rt0+1. For clt0+1, this is directly replaced using the budget constraint in the old period by
(1 + rt0+1 − δ)wt0
llt0
. Then the remaining future variable is rt0+1. If, however, log specification for the
utility function of consumption is used:
(1 − γ)
γ
(1 + rt0+1 − δ)wt0
llt0 = n
σn
lt zlwlt0
(1 + rt0+1 − δ).
45

the term (1 + rt0+1 − δ) cancels out and drops. Also, in the spatial equilibrium relationship:
γ log c1t+1 + (1 − γ) log n1t = γ log c2t+1 + (1 − γ) log n2t = γ log c3t+1 + (1 − γ) log n3t
,
likewise clt0+1 can be replaced with (1 + rt0+1 − δ)wt0
llt0
, where the term (1 + rt0+1 − δ) drops out from
the equations since the rental rate is equalized and unique. Therefore one can solve the model forward in
this log consumption utility case by distributing Nt and Kt so that equilibrium relationships are satisfied,
and the state variables are updated following Nt+1 =
P
l=1,2,3 nltNlt and Kt+1 =
P
l=1,2,3 Nltwltllt.
A.4.2 Solving Backward
The model can be solved forward only when the specification of utility functions is a log. In other cases
where the CRRA parameter is different from 1 or there are transfers, the model needs to be solved in a
different way. One observation from the constrained social optimal equilibrium is that there are more
future variables (from the viewpoint of the young agent) in the optimality conditions; while in the laissezfaire there is only a subset of future variables like k
′
l
, the constrained efficient problem has a wide range
of future variables k
′
l
, N′
l
, L
′
l
, and w
′
l
since s/he is aware of the effect of the fertility choice on them.
Therefore it would be useful to solve the transition backward; starting from the steady state, given the
future variables, one solves for the current variables (Nls and nls) to determine the series of variables.
The idea of the computation is referred to as ’Reverse Shooting’ in Judd (1998). Judd (1998) demonstrates in a canonical Cass-Koopmans model that the optimal path of the capital from initial values to the
steady state can be shot from the steady state variable (kT +1). Pick a value of kT , and compute kT −1 from
the Euler equation combined with resource constraint (a second order difference equation). Repeating the
procedure, one would eventually get a value of kt close to the initial value but different. Then try a different value of kT −1 to eventually get a sequence of kt that well runs from an initial value to the steady state
value kT +1.
46

Like the Reverse Shooting above, a given trial series of capitals are taken, the model is solved backward,
and the market clearing is verified each period. The problem is far more complicated than that due to the
two sets of state variables Nt and Kt
, and a need to allocate them across regions so that the utility is
equalized.
A.5 The Number of Regions
As discussed in the main text, the number of regions are chosen to be three in order to apply the broad
categories of Japanese prefectures of central, local central, and local areas. Nothing prohibits the model
from choosing the number other than three.
If the number of regions is increased, however, it might affect the computational efficiency or accuracy
because one needs to solve the conditions on multiple regions at a time to allocate the people and capital
in a given period.
In that respect, two region model might have a better potential to be efficiently and accurately solved.
Another point to note, however, is that when there are only two regions the fact that a region has a net
inflow of people is identical to that another region is losing population, while in the three regions setting
whether the second region loses or gains population is dependent on the setting.
Shown below are baseline results with log utility and CRRA utility with σc = 0.7. It should be noted
that the parametrization here just inherits those set in the three regions model.
A.6 Generational Transfer
Generational transfer such as the pension system is ubiquitous in advanced economies. People pay for the
system when they are young and work, and receive benefits when they are old and retired. In Japan, the
public pension system started as a fully funded program in 1960. The system at an initial stage successfully
47

0 10 20 30 40 50 60 70 80 90 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Transition of Population
All Region
Region1
Region2
(a) Population
0 5 10 15 20 25 30 35 40 45 50
0.29
0.295
0.3
0.305
0.31
0.315
0.32
0.325
0.33
0.335
Transition of Aging Rate
All Region
Region1
Region2
(b) Aging Rate
Figure A.9: Transition to Steady State (log utility) in a Two-Region Model
0 10 20 30 40 50 60 70 80 90 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Transition of Capital
All Region
Region1
Region2
(a) Population
0 5 10 15 20 25 30 35 40 45 50
0.22
0.24
0.26
0.28
0.3
0.32
0.34
Transition of Aging Rate
All Region
Region1
Region2
(b) Aging Rate
Figure A.10: Transition to Steady State (CRRA Utility with σc = 0.7) in a Two-Region Model
48

funded the payment thanks to the rich savings of the young generations. As the population aging and
lowering fertility kicked in, however, more people entered eligible ages but the collection of the funding
struggled. In order to sustain the program and honor the payment duty, the Japanese government gradually
shifted the funding scheme from a fully funded one to a pay-as-you-go; the payment from the young
generations is directly used for the benefit of the old generations. Japan’s pension system is still a mixture
of the fully funded one and pay-as-you-go in that a part of the collected money is invested in the capital
market by the fund to finance the benefit of future generations.
The effects of the national pension system on the demographics are the interest of the model. Two
features are crucial to the consequence of the exercise; pay-as-you-go scheme and omission of consumption
in the young period. In the pay as you go pension system, the payment from the middle-aged generation
is directly used for the benefit of the old people, which means that the saving of the economy is simply
reduced. The more pension system transfers, the less capital is accumulated in the economy. Secondly,
since consumption during the middle-aged period is omitted, what matters to the individual for their
consumption is the total amount of the lifetime income; they work in the middle-aged period, and then a
part of it is taken away as a payment to the pension system. All the other income is dedicated to the saving
which earns interest revenue but depreciates a bit, and the benefit is delivered when they are old. The only
possible difference in their lifetime income with and without generational transfers is the difference in
the amount of payment and transfer receipt, and the foregone interest revenue due to the social security
payment. If people also consume in their middle-aged period, the amount of the payment is also important
in the determination of how much to save and consume. Such a dimension would be likely to affect their
decision on where to live because the productive region which pays higher would be more attractive to
live in, but is completely suppressed due to the omission of the middle-aged period consumption.
In this subsection, a consequence of the pension system is studied where the middle-aged are taken
away some amount of money and then they receive as they enter the old period a similar amount. The
49

amount of transfer is set to some percent of annual income earned by region 3, which has the lowest GDP
in the benchmark case due to innate productivity differences and the agglomeration externality spurred
by ex-post concentration of population to those areas23. The lifetime budget constraint of people is as
follows:
c
′ ≤ (1 + r
′ − δ)(wl − payment) + receipt.
The payment amount is unique across regions since typically the pension system is run by a central government. All the money collected from the middle-aged is distributed to the old people. Since the population
evolves from period to period, the amount they pay slightly differs from the amount they receive.
pension payment: τa No pension 10% of
annual income in redion 3
20% of
annual income in redion 3
number of child: n1 0.84 0.85 0.85
number of child: n2 1.04 1.03 1.02
number of child: n3 1.30 1.23 1.23
population: N1 0.64 0.59 0.52
population: N2 0.45 0.42 0.39
population: N3 0.29 0.28 0.26
capital: k1 3.78 2.94 2.10
capital: k2 1.61 1.36 1.06
capital: k3 0.61 0.55 0.46
average utility: u¯ 0.68 0.66 0.62
Table A.9: Comparison of Steady State Allocations with Varying Pension Systems (τa)
Table A.9 shows the consequence of the introduction of the pay-as-you-go pension system. The introduction of pension reduces the capital, and then the population size whereas the amount they pay and
receive almost coincides. This is explained by the loss of capital accumulation. An associated average
utility will decrease as well.
Looking at the transition of the aging rate is useful to understand what is happening in the economy
with generational transfer. Figure A.11 draws two evolutions of the aging rate; one is an economy without
generational transfer, and the other is with the one corresponding to τa = 0.1 in A.9. One observation
23The choice of region 3 is purely a purpose of running the model calculation without errors. If the amount is set according
to the most productive region, there is a possibility that the young people in the poorest region cannot afford the payment.
50

0 5 10 15 20 25 30 35 40 45 50
0.29
0.295
0.3
0.305
0.31
0.315
0.32
0.325
0.33
0.335
Transition of Aging Rate
with pension
no pension
Figure A.11: Transtion of Aging Rate with and without a Pension System
is that the economy with generational transfer starts at a higher level of aging rate in the process of
transition to the steady state. A higher aging rate means that the increase in the new generations is not
that strong, and so the share of the elderly people is pushed up. The introduction of the generational
transfer pulls down the consumption amount due to the foregone interest revenue in every region by the
same magnitude; the social security cost is set at the same amount throughout the regions, and also the
interest rate is equalized because of perfect capital mobility. This drag-down of the consumption amount
makes people more attracted to live in the most productive region 1, and fewer people choose to live in
regions 2 and 3. This contributes to the reduction in the size of the cohort compared with the economy
in the absence of generational transfer. Another observation is that the transition of the aging rate to the
steady state level (1/3) is slower in the economy with generational transfer. The slow transition of the
aging rate is associated with slow transitions of population and capital. The accumulation of capital is
slower precisely because of the presence of generational transfer, which carries the resources of the young
directly to the old without putting them into the capital market.
To sum up, the generational transfer like the national pay-as-you-go pension system basically reduces
the size of the population and pushes up the national aging rate.
51

Chapter 2
Large Public Debt under Low Interest Rates: a Welfare Analysis
2.1 Introduction
Advanced economies have accumulated increasingly more public debt over the past several decades (Figure
2.1). Some of them record a net public debt1
to GDP ratio of almost 100% or more, with the US and Japan
being 95.1% and 161.5%, respectively in 2022. This is in part made possible by historically low interest
rates in these countries (Figure 2.2). In response to the large economic downturn of the Great Recession
and the COVID-19 pandemic, the short-term rates are lowered to about 0%, and the situation persisted for
several years after the Great Recession.2 Not just short-term rates, which are policy tools of the central
banks, but also the long-term interest rates have been trending downwards over the last 30 years.
I am grateful for Robert Dekle, Pablo Kurlat, and Selahattin İmrohoroğlu for invaluable guidances. I also appreciate useful
comments and suggestions given by Hiro Ishise and participants at macro reading group in USC department of economics. All
remaining errors are mine.
1Definition of the net public debt used by the IMF is "Gross debt minus financial assets corresponding to debt instruments,"
where the gross debt means "All liabilities that require future payment of interest and/or principal by the debtor to the creditor"
(IMF (October 2023), page 32). Basically, the public sector includes all the public enterprises and central banks. Importantly, the
net debt consolidates the government securities issued by the central government and the government securities owned by the
central bank.
2Two years after the onset of the pandemic there came an outstanding level of inflation in many countries. While most central
banks including the Federal Reserve and the European Central Bank, raised policy rates to combat these, the Bank of Japan keeps
the policy rate at almost 0% as of January 2024.
52

Figure 2.1: Net Public Debt to GDP Ratio in Advanced Countries
Figure 2.2: Short-term Interest Rates (left) and 10 Years Interest Rates (right) in Advanced Countries
53

Is the current large amount of public debt a serious problem in terms of welfare? The welfare consequence of large public debt is not evident in an economy characterized by a low interest rate. While large
public debt could entail a higher tax burden for the households, the debt service might not be so large
when public bonds are highly priced and thus the interest rate is almost zero. Blanchard (2019) studies
debt servicing under exogenously low interest rates with g > r, and argues that the welfare cost of public
debt may not be necessarily so large in the low interest rate environment. Moreover, whether debt service
is high or not, it is received by someone in the country as long as the bulk of the outstanding public debt is
held by domestic investors. Whether large public debt matters or not cannot be answered without thinking
about more than one agent.
To answer this question, we set up an economy characterized by a large amount of public debt and
endogenously low equilibrium interest rates, calibrate it to recent Japan, the advanced economy with the
highest public debt to GDP ratio, and study the relationship between the level of public debt, rates of
return, and welfare.
While there are numerous models proposing a situation dubbed as ’secular stagnation’ (Summers
(2014)), A. Mian, Straub, and Sufi (2021) successfully explain the combination of low interest rates, large
(private and public) debt, and output below natural level. Their trick is the non-homothetic utility from
asset holding enjoyed by the top income earners, while the rest of the people borrow from those rich, not
just directly but also indirectly through financial assets. Due to non-homothetic utility from asset holdings, in their model, savers do not increase their spending on consumption one for one upon an increase
in income. In order to fill the gap in demand in the goods market, the equilibrium interest rate needs to
drop and encourage spending by the saver. Their model successfully explains how the United States has
financed an increase in private/public debt under declining real interest rates in the past 50 years.
54

What is the problem of public debt in such an economy? Not only that public debt crowd out the
productive capitals3
, but public debt can also cause resource reallocation from the rich (saver) to the poor
(borrower) through taxation; while taxes fall on both borrower and saver, saver—typically very rich people—receive interest payments. What is more, in the model with non-homothetic utility from asset holdings, savers do not increase their spending on consumption upon an increase in income. Borrowers, on the
other hand, cannot spend as much as they wish since they need to serve for taxes, which finance the interest payment of public debt. In order to fill the gap in demand in the goods market, the equilibrium interest
rate needs to drop, encouraging the spending by the saver. While savers increase spending, borrowers
increase borrowing since the lower interest rate makes it possible, and the result is even more accelerated
resource reallocation from borrowers to savers. Therefore, in the model of A. Mian, Straub, and Sufi (2021),
steady state welfare is decreasing in the level of private and public debt.
Does it mean, however, that we should not raise the level of public debt to GDP ratio? An increase
in public debt temporarily allows the government to reduce the collection of taxes, and this short-run
benefit of raising the public debt to GDP ratio is missing in the former argument. Of course the increased
public debt to GDP ratio could lead to higher future debt services. Under the chronically low interest rate,
however, it should be possible that debt service is not so high, making it possible for the overall welfare
starting from the given public debt to GDP ratio to the increased ratio to be higher than the welfare staying
at given public debt to GDP ratio. This possibility is even more likely when the fiscal policy is designed
so as to redistribute from the saver to the borrower, which is emphasized in A. Mian, Straub, and Sufi
(2021). This paper quantifies both possibilities to discuss the overall welfare consequence of large public
debt under low interest rates.
3While crowding out is a typical source of the welfare cost of public spending, their baseline economy is an endowment
economy and thus there is no crowding out of capital but just an influence on the interest rate. Also, while most of the literature
discussing the welfare cost of public debt takes public bonds as unproductive, there is some work dealing with productive public
bonds, including Chatterjee, Gibson, and Rioja (2017).
55

The contribution of this paper is to deepen the understanding of the welfare consequence of the large
public debt coupled with low interest rates. While A. Mian, Straub, and Sufi (2021) discussed this situation
of ’indebted demand’ in a continuous time economy, we embed thistwo dynasties setting in a discrete time
economy and compare the welfare under different fiscal policies. We calibrate the model to the most relevant economy for this topic, Japan, utilizing not just aggregate macroeconomic moments like the savers’
asset holding relative to GDP but also micro data of Japanese households. Our study gives insights into
how an additional issuance of public debt affects the interest rate and flow of funds in the economy, and
the resulting welfare as a consequence of them.
Understanding the welfare consequence of the large public debt under the low interest rate is important
for the following two reasons. First, this is a relevant topic for many advanced countries. As we have seen
in Figure 2.1, there has been a sequence of large economic downturns over the last two decades, and
most countries coped with them by sizing up the amount of public debt to finance the transfer and public
investment. Among the countries in Figure 2.1, only Germany succeeded in reducing the public debt to
GDP ratio after the Great Recession. Considering the economic turmoils following the expansion of the
pandemic, it should be modest to think that the current level of public debt to GDP ratio will be here to
stay, and it might get even higher than now. Second, what is associated with a large amount of public
debt is one of the major concerns for Japan. As the public debt accumulates, many studies have shown
concerns about the sustainability of the fiscal conditions in Japan. T. Doi, Hoshi, and Okimoto (2011) and
Hoshi and Ito (2014) are among them, and do the simulation on tax reforms Japan must implement as early
as possible. While they express serious concerns about Japan’s risk of ’fiscal default’ if drastic reforms
in taxes and spending are not taken, Japan experiences neither hikes in interest rates nor price levels ten
years after their warning. Nonetheless, it does not mean that piling up a huge amount of public debt is a
free lunch; it is important to deepen our understanding of the potential side effects of a high level of public
debt even in the absence of the pressing risk of default.
56

We study several variants of fiscal policies. First, we compare steady state welfare with welfare when
the public debt level is raised. It is confirmed that increasing public debt might give higher welfare if we
take the transition process into account than staying at the original level of public debt, but this benefit
declines as the debt to GDP ratio increases and eventually vanishes. Second, we consider a drop in output
because deficit financed fiscal expansions are implemented often at the onset of an economic downturn.
We compare the transition welfare with a fixed level of public debt, and the transition welfare with the
increase in public debt. Again we confirm that deficit finance could do better than keeping the debt level
constant if the public debt to GDP ratio is not high enough when the economy is hit by a bad shock.
Third, the same experiment is considered with the introduction of foreign investors to the baseline model.
While foreign investors can be another source of supply of savings other than domestic savers and thus
reduce the increase in the savers’ balance sheet, the debt service paid to them would exit from the domestic
economy and could worsen the welfare of domestic households. We confirm that while the former benefit
widens the range of the public debt to GDP ratio in which deviation from the initial level can be welfareenhancing, the latter effect increases the speed at which such a benefit diminishes as the rise in public
debt to GDP ratio. Fourth, we study the welfare consequence of public debt reduction. In the short run,
an increase in taxation to finance the liquidation of part of public debt is so costly for households. In the
long run, however, a reduced public debt level reverses the indebted demand mechanism and limits the
resource reallocation from borrowers to savers, which is appreciated by the borrowers, the majority in the
economy. While it is difficult to find a case where public debt reduction improves the overall transition
welfare, the cost of the fiscal reconstruction is indeed decreasing as the starting level of the public debt
to GDP ratio is higher. Fifth, while the endowment economy studied so far does not feature economic
growth, the constant growth of endowment is introduced, with a particular interest in g > r. We find that
g > r may not be compatible with the indebted mechanism discussed in the whole paper, but still resulting
interest rate could be kept at a low level and allows welfare analysis of different fiscal policies. Finally, we
57

employ the most preferred specification to derive the policy implication for Japan. We observe that while
an increase in public debt becomes in the end not a desirable policy measure as the rise in the outstanding
public debt to GDP ratio, deficit finance still continues to be a better policy response to the recession than
a fixed level of public debt to GDP ratio for a wide range of outstanding public debt to GDP ratio.
This paper contributes to the long literature discussing the welfare consequences of public debt since
this is the first attempt to cast the problem in an economy characterized by a large public debt and a low
interest rate. In the following, we briefly overview the important findings of the literature. A more detailed
survey can be found in the appendix B.1.
The seminal paper discussing the welfare consequence of public debt is Aiyagari and McGrattan (1998).
They consider an incomplete market model to discuss the optimal level of public debt in steady state equilibrium. While public debt requires distortionary taxation when lump sum tax is unavailable and crowds
out capital accumulation, it could be beneficial since the existence of public debt effectively eases the borrowing constraint, and an intensified interest rate makes it easier to smooth consumption. Calibrating the
model to the United States in the 1990s, they argue that the optimal ratio of public debt to GDP is 60%.
If so, however, does this mean that the United States in reality suboptimally accumulates a higher public
debt to GDP ratio? The subsequent literature starting from Desbonnet and Weitzenblum (2012) rationalizes this departure of public debt from a steady state optimal level. They consider the transition from one
steady state to another with increased public debt, and compute the welfare not just of the steady state but
including the whole steady state transition process. Since an increase in public debt temporarily allows
a reduction in tax collection, deviating from the original level of public debt can be welfare improving.
When considering a 10% increase in public debt to GDP ratio, the economy has a ’profitable deviation’ of
the public debt to GDP ratio until it reaches a remarkable 560%.
The implication of this analysis, however, might not be a helpful guide to considering the situations
faced by current advanced economies, which are characterized by low interest rates and large public debt.
58

When we consider the public debt at the current level in these models, associated interest rates typically
stand at more than 5%, whereas we are enjoying less than 1% in reality. If we are to calibrate the incomplete
market model to obtain interest rates at this much low levels, we would need to set a very high discount
factor, or introduce a very rare and bad state of the economy into income processes (Kocherlakota (2021)),
both of which may not necessarily describe the reality. This facilitates a need to recast the optimal public
debt argument in a model featuring a low interest rate despite a large public debt.
We base the study on Japan, which records the highest public debt to GDP ratio in the advanced
economies of 161.5% in net and 260.1% in gross terms in 2022. Japan also experiences an accelerating
decline in birth rates and population aging, with the ratio of the number of elderly people (over age 65)
to the entire population being as high as 28.4%, and thus need for an increase in social security related
expenditure and a long-run decline in income tax revenues is inevitable. This is why many studies focus on
public finance in Japan (Broda and D. Weinstein (2004) and Hansen and İmrohoroğlu (2016)), and this paper
seeks to add to the strand of literature by offering the welfare evaluation of the coming future. Nakajima
and S. Takahashi (2017) study the optimal level of public debt in Japan by applying Aiyagari and McGrattan
(1998)’s model, and shares a close interest with our study. While they derive an intriguing conclusion that
the optimal steady state level of public debt to GDP ratio is indeed −50% (thus public ’saving’),they do
not consider welfare including transition, and also they allow the endogenous interest rate to reach 1.5%
to 2% in their analysis.
While our model follows A. Mian, Straub, and Sufi (2021), there are several works that are built to
feature low interest rates and discuss the ’secular stagnation.’ Michau (2018) also features a preference for
asset holding, and studies optimal policy. J Caballero and Farhi (2018) feature the shortage of the supply
of safe assets, while Eggertsson, Mehrotra, and Robbins (2019) combine the OLG model with financial
frictions.
59

The rest of the paper is organized as follows. In section 2.2 we set up the model whose key feature
of non-homothetic utility from asset holding follows A. Mian, Straub, and Sufi (2021). In section 2.3 we
show empirical background for embracing non-homothetic utility from asset holding, and calibrate the
model using aggregate and micro data in Japan. In section 2.4 we demonstrate the result of the model, and
associated welfare analysis. Section 2.5 concludes.
2.2 Model
There are two types of dynasties in the economy: the saver, and the borrower. The mass of the total agent
is fixed to 1, and the fraction of saver is µ
s
.
We consider an endowment economy where each dynasty gets ω
i
(ω
s + ω
b = 1, again the total mass
of endowment is 1). These ω
i
s are the only fundamental that differentiates two dynasties, and they are
basically identical in preference. Throughout the analysis, we put the following assumption:
• Assumption: ω
s
µs >
ω
b
µb
,
which means that, when considering each member within the dynasties, the saver receives a larger endowment than the borrower. The period utility function is specified as:
log
c
i
t
µi

+ α
(a
i
t/µi + a¯¯)
1−σ
1 − σ
The key feature of this model is a non-homothetic preference for asset holding. Non-homotheticity
results from subsistence parameter a¯¯, and σ < 1. One interpretation of the utility from asset holding is
a perpetual youth model with bequest utility. In each period some of the saver and borrower die, and
then asset is perfectly inherited within the dynasty to the successor. Holding of assets is useful not only
for themselves if they live long, but also for their successors even if they die. Non-homotheticity in this
utility suggests that agents prefer to hold more assets relative to consumption as the asset accumulates,
60

describing the typical transition from saving as a means to saving as an end4
. We consider the validity of
this non-homotheticity assumption in the next section empirically.
The optimization problem for both is formulated as follows (i = {s, b}):
max
{c
i
t
,ai
{t+1}
}∞t=0
X∞
t=0
β
t

log
c
i
t
µi

+ α
(a
i
t
/µi + ¯a)
1−σ
1 − σ

s.t. ci
t + a
i
t+1 ≤ ω
i
(1 − τ
i
t
) + (1 + rt)a
i
t
,
a
i
t+1 ≥ −a/¯ (1 + rt+1),
a0 given.
Note that we put an ad-hoc borrowing constraint. There are many possible candidates for the borrowing
limit here as long as it is relaxed when the interest rate is dropped. A story for this limit a/¯ (1+rt+1) is that
the lender, which is the saver in this model, can seize the endowment of the borrower up to a¯, and so they
are ready to borrow up to the present value of this amount. In the appendix B.3, alternative borrowing
limit, a/r ¯ t+1, is explored, which is motivated by Aiyagari (1994).
Saver’s Euler equation is
αβ
µs
(a
s
t+1 + ¯a)
−σ + β(1 + rt+1)
1
c
s
t+1
−
1
c
s
t
= 0.
We focus on an equilibrium where borrowers borrow from savers, which is made possible under the assumption stated before (each saver receives more endowment). In this equilibrium borrower borrows up
to the limit:
a
b
t+1 = −
a¯
1 + rt+1
.
4Another interpretation is that households prefer the social status arising from the holding of a large amount of assets.
61

Government budget constraint is:
Gt + (1 + rt)Bt ≤ Bt+1 + τ
s
t ω
s + τ
b
t ω
b
.
Gt
is government outlay, which is considered to either be thrown into the ocean or enter the utility function
in an additively separable way. Bt
is the level of outstanding public debt, which is in the equilibrium owned
entirely by savers. Since we consider an endowment economy, income taxes are equivalent to lump sum
taxes and thus do not cause distortion. In most of the analyses in this paper, we choose and fix a particular
path of public bond {Bt} and outlays {Gt}, while tax is accordingly determined so that government keeps
a balanced budget for each period.
Market clearing of the asset is:
a
s
t + a
b
t = B.
An equilibrium is defined as follows:
• Definition: A competitive equilibrium of this economy given fiscal policy, {Gt
, Bt}, and associated
income taxes, τ
s
t
, τ b
t
, is a set of the price {rt} and quantities {c
s
t
, cb
t
, as
t
, ab
t
} where
– each dynasty maximizes utility
– government keeps balanced budget: Gt + (1 + rt)Bt ≤ Bt+1 + τ
s
t ω
s + τ
b
t ω
b
– asset market clears: a
s
t + a
b
t = B.
Now consider a steady state equilibrium. Saver’s Euler equation becomes:
αβ/µs
(a
s
/µs + ¯a)
−σ
(a
s
r + ω
s
(1 − τ
s
)) + β(1 + r) − 1 = 0.
62

Figure 2.3: Determination of Asset Supply and Interest Rate in Steady State Equilibrium
As in Figure 2.3, the slope of this curve in the a − r plane is dependent on parametrization. It could
be downward sloping, while the saving supply in the canonical model including the neoclassical growth
model is upward sloping. Market clearing turns into:
a
s =
a¯
1 + r
+ B.
These two equations determine the steady state supply of assets a
s
and an associated interest rate r. To
understand the key mechanism of high public debt levels and low interest rate, Figure 2.3 depicts the saving
supply curve (saver’s Euler equation) and saving demand curve (market clearing condition), for the case
with non-homothetic utility for asset holding (left) and homothetic one (right).
With non-homotheticity, the supply of savings is downward sloping in a steady state (left graph in
Figure 2.3). An increase in outstanding public debt level B shifts the debt demand curve to the right,
yielding a lower equilibrium interest rate. This is at odds with canonical economic models where an
increase in public debt increases the required rate of return, or an associated increase in government
outlays crowds out the asset market to raise the interest rate. The right figure depicts the same asset
saving and supply curves in the case of homothetic utility from assets; this is close to typical models
63

featuring utility from durable goods, bond holding, and so on. An increase in outstanding debt level B, in
this case, leads to a higher equilibrium interest rate.
Why does the non-homothetic case result in the opposite result? Intuitively, saving is a luxurious good
in the non-homothetic utility case, while it is not so in homothetic utility case. In non-homothetic case,
households’ desire to increase the savings as they become richer needs to be limited by the increased prices
for savings (which are, in turn, lowered interest rates).
The mechanism is as follows. While taxes to finance the future repayment of the public debt fall on
both borrower and saver, saver (rich people) purchase and own the public bond security and receive the
interest payments. This marks resource flow from borrowers to savers through the debt services. Savers,
on the other hand, do not raise consumption one for one upon an increase in the income receipt through
repayment of the public debt due to non-homotheticity in asset holding utility; marginal utility from asset
holding increases as the holding of asset increases. Borrowers reduce their consumption due to the increase
in tax burden, but savers do not necessarily fill the lack of consumption demand since they prefer saving to
consumption. In order to encourage their consumption, the interest rate needs to drop in the endowment
economy considered here. The story does not end here though, since the reduction in interest rate makes it
possible for borrowers to increase their borrowing in the face of looser borrowing constraints. This yield
further resource flow from borrowers to savers, and reinforces the reduction in interest rate described
above further.
This formulation, however, leaves a serious problem for the model to be used to study welfare. While
utility from asset holding is a key to inducing low interest rates despite a large amount of public debt,
every increase in the debt mechanically pushes up the utility of the saver. Suppose government plans the
following fiscal policy: issue an additional 1 billion dollars of public debt in period t, and give back every
money collected to the purchaser, which is in our formulation the saver. While this transaction is feasible,
the increase in public bonds among the savers’ portfolios inflates the utility from asset holding. It follows
64

that government can attain any level of welfare by manipulating the bond issuance and lump sum transfer,
which leaves the welfare comparison meaningless.
To cope with this, in the welfare analysis in section 2.4 we consider only the borrower’s borrowing
to be the unique source for asset holding utility. For this reason, in the welfare calculation of this paper,
following Michau (2018)
5
, the utility from asset holding becomes as below:6
:
α
(a
i
t
/µi + ¯a − Bi
t
/µi
)
1−σ
1 − σ
where Bi
t
is the holding of the public bonds by the dynasty i. While in the equilibrium we consider Bs
t = Bt
and Bb
t = 0 hold, we assume that dynasties take these terms as exogenous to their decisions.
2.3 Empirics and Calibration
2.3.1 Empirics: Non-homothetic Utility from Asset
The key feature of the model that induces an equilibrium low interest rate despite a high level of public
debt to GDP ratio is a non-homothetic preference for asset holding. In this subsection, we use panel data on
Japanese households to study the validity of the non-homotheticity. We follow Straub (2019), and estimate
the relationship between consumption and the permanent income of households.
Our data source is the Japan Household Panel Survey and Keio Household Panel Survey (JHPS/KHPS)
conducted by the Panel Data Research Center at Keio University in Japan. KHPS and JHPS had been
originally distinct household panel surveys conducted separately since 2004 and 2009, respectively, and
then were combined starting in 2014. The JHPS/KHPS collects income, asset, and consumption data in
addition to demographic information for a stratified random sample of households. One caveat of the
5Michau (2018) introduces to Ramsey model with money (1) the preference for asset holding and (2) downward wage rigidity.
There are two differences in his formulation from ours. First, not just public bonds but also money gets into the utility function.
Second, since his is a representative agent model, the whole amount of public bonds and money is deducted.
6
Since in the equilibrium considered in this paper only the saver purchases public bonds, this subtraction of public bond
holding from asset holding is applied only to the saver.
65

mean sd min max
Age of the Househead 48.89 9.65 30 65
Number of Family Members 3.45 1.39 1 11
Disposable Household Income 523.99 260.06 31 1700
Total Household Expenditure 34.57 18.29 8 174
Household Asset Holding 792.87 1384.21 0 20000
Number of Observations 29537
Number of Households 5730
Note: money amount is in units of ten thousand Yen
Table 2.1: Descriptive Statistics of JHPS/KHPS Dataset
JHPS/KHPS is that consumption is surveyed for only one month (January of the survey year), while income
and assets data are surveyed at an annual frequency. For consumption, we choose the total monthly
expenditures including mortgage payments. To lower the influence of outliers, we drop all observations
below the 5th and above the 95th percentiles in income distribution. Following Straub (2019), we use the
observations of households whose househeads are aged between 30 and 65.
Table 2.1 summarizes the descriptive statistics of the dataset.
Following Straub (2019), we specify the consumption and income to be composed as follows:
log cit = X
′
itβ + φ log wi + ǫit
log yit = X˜
′
itβ + log wi + ηit + ψit + vit.
While cit stands for consumption of household i at time t, we have two concepts of income: current
disposable income yit, and permanent income wi
. As specified in the second equation, current disposable
income is composed of permanent income level, persistent shock ηit, transitory income shock ψit, and
measurement error vit, as well as other factors X˜
it which is observable to the econometricians. The permanent income hypothesis argues that this wit is the main driver of the current consumption cit, with the
coefficient φ = 1. The main idea of this empirical exercise is to consider whether this φ is estimated to
66

be 1 or less than 1, which suggests that an increase in permanent income does not lead to an increase in
consumption one for one.
Following Straub (2019) we use as controls X˜
it in the income equation the household head’s five-year
age bracket dummy, household size dummy, and year dummy, while in Xit we use, on top of these, a
city dummy (living in cities holding more than a million, other cities, or towns and villages) and a region
dummy (8 region blocks) that are expected to capture heterogeneous living costs. Using these controls, we
first regress disposable income log yit on X˜
it to obtain the residuals yˆit.
Since permanent income is a theoretical object, we need to construct a proxy for it. We use a moving
average of residual income:
y¯
T
it ≡
1
T
(T
X−1)/2
τ=−(T −1)/2
yˆi,t+τ ,
with a hope that the procedure of taking average washes away permanent and transitory income shocks.
Using this, the consumption equation specified earlier is tested using the following equation:
log cit = X
′
itβ + φy¯
T
it + ǫit.
Since our dataset spans only from 2008 to 2018, it is preferable to secure as many observations as
possible by setting T not a large value, and thus we set it to T = 5. Straub (2019), however, points out
a possible bias in OLS estimate coming from the concern that the constructed proxy does not clear the
permanent income shock ηit. Since the reaction to persistent income shocks is considered to be weaker
than that to the change in permanent income, this could be a source for a downward bias. Straub (2019)
proposes to overcome this problem by imposing an AR(1) structure on the process of permanent income
shock ηit, and using zit+τ = ˆyit+τ − ρyˆit+τ−1(τ ≥ 2) as an instrument in the form of future socks (more
explanation follows in Appendix). The process of permanent income shocks themselves can be a difficult
67

(1) (2) (3) (4) (5) (6)
no IV ρ = 0.88 ρ = 0.89 ρ = 0.90 ρ = 0.91 ρ = 0.92
log Permanent Income 0.435∗∗∗ 0.466∗∗∗ 0.466∗∗∗ 0.457∗∗∗ 0.426∗∗∗ 0.336∗
(0.01) (0.07) (0.08) (0.10) (0.12) (0.15)
T 5 5 5 5 5 5
Observations 23983 3433 3433 3433 3433 3433
R-squared 0.27 0.22 0.22 0.22 0.22 0.22
∗
p < 0.05,
∗∗ p < 0.01,
∗∗∗ p < 0.001
Table 2.2: Marginal Propensity to Consume out of Constructed Permanent Income
object to estimate, and so following Straub (2019) we try multiple candidates of the values of ρ, with prior
information that it is centered around 0.9. The estimation result is shown in Table 2.2 for baseline OLS
case and IV of varying values of ρ.
Column (1) shows the result of the OLS estimate, and gives an estimate well below 1. To clear the
concern that this might be due to the endogeneity bias, we try instruments using future values of quasi
differences of the income residuals, trying several autocorrelation parameters ρ. While several results give
higher estimates than the OLS estimate, none of the estimates come close to 1. Using household panel in
the U.S. (PSID), Straub (2019) gives the instrumented estimate of around 0.7, which is even higher than
ours. We regard this whole exercise as a motivation to advocate non-homotheticity assumption in the case
of Japan.
2.3.2 Calibration
The calibration of the model is summarized in Table 2.3.
While A. Mian, Straub, and Sufi (2021) sets saver to be top 1% of the income distribution and borrower
to be the rest of the 99% based on the observation in A. R. Mian, Straub, and Sufi (2020) that the top 1%
finances the bottom 99% effectively through financial securities, in this model top 10% is chosen to be
the saver. While a similar exercise is difficult to apply to Japan due to the information restriction7
, the
7Moriguchi and Saez (2008) document the income and wage concentration in Japan. While there was a high income concentration before WWII of top 0.1% holding almost 10% of income share, which was comparable to the U.S, it precipitously dropped
68

Demographics
µs 0.1 fraction of the saver the richest bracket available in WDI
ω
s
0.45
real income share
of the saver WDI database (pre-tax income share)
Preference
β 0.98 discount factor calibration
(standard discount factor times survival rate)
a¯ 15 ad hoc borrowing limit calibration
a¯¯ 20 subsistence level calibration
α
0.02
(/1.8)
relative importance
of utility from asset calibration
σ
0.7
(/0.8)
IES parameter
for utility from asset calibration
Fiscal Policy
B 1.5 Public Bond Initially 1.5 (pre-pandemic debt to GDP)
G 0.15 Government spending Yearly budget to GDP
Table 2.3: Summary of Parametrization
income inequality data prepared by World Inequality Database, which is based on income tax data by the
national tax agency, do document that the top 10% of the income distribution has a share of 45% of pre-tax
national income. If further information on the financial flow by income distribution is available, µ
s
could
be set more precisely.
There are multiple preference parameters regarding the utility of holding asset; utility weight given
to asset holding α, subsistence level in asset holding a¯¯, parameter governing intertemporal elasticity of
substitution σ, and an ad-hoc borrowing limit a¯, which eventually goes to the utility function of borrowers
in the equilibrium we consider. Also, the discount factor β can be distinct from standard literature since the
survival probability of the dynasties should be taken into account. 8 Also, we fix the subsistence parameter
a¯¯ = 5. This a¯¯ is large enough to alleviate the negative entry in the asset holding utility of the borrowers.
to 2%, and long stayed low until recently, while the U.S. saw more than 7% share again in 2001. Therefore, we do not expect as
high concentration of income share as A. Mian, Straub, and Sufi (2021) in our study.
8There is another possible target of calibration of a¯: the household debt to GDP ratio, which is roughly 60% recently. This
a¯ and B give the steady state amount of saving a
s
of 2.1. While this choice for the calibration target also has a background, it
turns out that this total saving is too small when we consider public debt to GDP ratio of more than 300%: the calculation of
stedy state transition does not go well. With the baseline choice of a¯ = 2.1, we successfully cope with public debt to GDP ratio
of up to 500% in section 2.4.
69

17.5 18 18.5 19 19.5
log asset
13
13.5
14
14.5
15
15.5
16
16.5
17
17.5
18
log consumption
Figure 2.4: Calibration: Fit of Euler Equation to Household Data
The rest of the parameters, α, σ and β are jointly determined using micro data. In the previous section
the steady state Euler equation, αβ/µs
(a
s/µs + ¯a)
−σ
c
s + β(1 + r) − 1 = 0 is derived. It is expected that
this relationship also holds to the consumption and asset choice of the household at the top 10% of the
income distribution. We again utilize JHPS/KHPS data of Japanese households to study the relationship
between household consumption and household holding of assets.
Plots in Figure 2.4 corresponds to pair of asset and consumption (log of Yen in 2015) of the top 10%
asset holding households in the dataset. The solid line is a fitted line of the Euler equation using maximum
likelihood estimation. A problem of this approach is that we need a value of steady state interest rate r
to fit the line to the data, but we need values of the parameters to obtain r.Thus we start from a guess
of r to choose the rest of the parameters, α and σ,by estimation, then put the estimates into steady state
calculation to check the equilibrium interest rate, and use this as an updated guess to repeat the procedure.
We aim for the steady state interest rate of r around 0.001, and obtain α = 0.35, β = 0.96 and σ = 0.85.
Finally, the baseline value of B is set to the pre-pandemic level of 1.5 relative to real GDP. In the
experiments, B is changed within the range of 0.5 to 3.1 (public debt to GDP ratio of 310%). G is set to
70

0.15, which is a recent time series average except for special years like the COVID-19 pandemic (2020),
financial crisis (2009), and major earthquakes (2011).
2.4 Implication of the Model
While steady state welfare declines as the level of public debt climbs up, fiscal policy can affect the short run
welfare of households by reducing the tax collection or redistributing between borrowers and savers, and
overall welfare including not just steady state but also transition between steady states could be higher than
the welfare in the absence of debt increase. These possibilities are quantified in this section. The welfare
including transition is obtained by numerically solving the perfect foresight steady state transition of the
model economy, which is detailed in Appendix B.2.
The welfare used in this section is a utilitarian sum expressed as below, with Ws
and Wb being lifetime
welfare of saver and borrower respectively:
W = µ
sWs + (1 − µ
s
)Wb
.
At t = 0 they are in an initial steady state and subject to a sudden change either in real terms (drop in
output) or in prospects (future path of public bond issuance or government outlays are altered by the fiscal
authority) or both. They will reach a new steady state after several periods, and this welfare measure
captures the whole of the transition process.
First, steady state welfares with B1
is compared with welfare when the public debt level is raised from
B1
to B2
, where B1 < B2
. Second, we consider a drop in output of 5%, and compare the transition welfare
with a fixed level of public debt level, and the transition welfare with the increase in public debt. Third, the
same experiment is considered with the introduction of foreign investors to the baseline model. Fourth,
we consider the opposite policy, namely, a reduction in public debt levels. Fifth, while the endowment
71

0.5 1 1.5 2 2.5
initial debt to GDP ratio
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
welfare change in consumption equivalence (in %)
transition welfare with 20% increase in public debt
Figure 2.5: Comparison of Steady State Welfare with Welfare Including Transition when Public Debt is
Raised
economy studied so far does not feature economic growth, the constant growth of endowment is analyzed,
with a particular interest in the current situation: g > r. Finally, we demonstrate the most preferred
specification, and discuss its policy implication.
2.4.1 Increase in Public Debt
While steady state welfare is inversely related to the amount of outstanding public debt, the welfare of
transiting from the initial level of public debt to the higher one could attain higher welfare than the steady
state welfare of staying at the initial level. While the long run welfare definitely goes down due to an
increased level of public debt, in the short run the proceed from public debt can be used to improve welfare.
The objective of this section is to quantitatively study this trade off.
Figure 2.5 depicts the welfare gain or loss of increasing the public debt level compared with keeping
it fixed. The horizontal axis stands for the initial level of public debt to GDP ratio. Our interest is, for
each level (B1), how welfare changes when the government raises the debt to GDP ratio by 20% (to B2),
gradually over several years following the path of public debt as below:
72

Bt = δ
t
· B1 + (1 − δ
t
) · B2.
Throughout the paper, we choose δ = 0.5, which completes the transition of public debt to GDP ratio
almost in 5 periods (corresponding to 5 years).
How the two welfares are different is measured using a consumption equivalence unit: how much
of the consumption increase do we need to give to each dynasty so that s/he is indifferent between the
welfare without change in an increase (steady state welfare with B1) and the welfare with the public debt
to GDP ratio changes from B1 to B2. We compute the consumption equivalence as a blue solid line in the
figure, which is defined as a percentage of the consumption at a steady state with the associated initial debt
level, v in the following expression (basedenotes base of the comparison, and policy denotes the associated
allocations and prices with the policy):
Wbase(B1, B1) = µs
X∞
t=0
β
t
"
log
c
s,base
t
µs
(1 + v
100
)
!
+ α
(a
s,base
t
/µs + ¯a − Bs,base/µs
)
1−σ
1 − σ
#
+µb
X∞
t=0
β
t
"
log
c
b,base
t
µb
(1 + v
100
)
!
+ α
(a
b,base
t
/µb + ¯a − Bb,base/µb
)
1−σ
1 − σ
#
= µs
X∞
t=0
β
t
"
log
c
s,policy
t
µs
!
+ α
(a
s,policy
t
/µs + ¯a − Bs,policy/µs
)
1−σ
1 − σ
#
+µb
X∞
t=0
β
t
"
log
c
b,policy
t
µb
!
+ α
(a
b,policy
t
/µb + ¯a − Bb,policy/µb
)
1−σ
1 − σ
#
= Wpolicy(B1, B2)
A positive value of v means that the policy of changing the public debt to GDP ratio gives higher welfare
than the policy of staying at a given public debt level, while negative consumption equivalence means that
welfare with the change in B1 makes the overall welfare worse off.
The line in Figure 2.5 is decreasing with the public debt to GDP ratio, and takes positive values when
the initial debt level is low, but takes negative values when the initial debt is large (over 200%). This
result suggests that the increase in public debt does have a potency to increase welfare when we take the
transition process into consideration, but does not continue to be so as the starting public debt level rises.
With a low level of public debt, while the increase in it temporarily reduces the tax burden of households,
73

the permanent increase in borrowers’ debt service for private lending and public debt (taxation) is modest,
and thus deviation from the initial steady state is ’profitable’. With a large level of public debt, on the
other hand, debt service increases despite the declining rates of return, and this is not all compensated by
the temporary tax reduction made possible by debt increase, and the resulting welfare including all the
transition is lower than staying at the initial steady state.
Note, however, that the consumption equivalence unit is not that large number; at most 0.03% in
Figure 2.5. Not so large difference in consumption equivalence measure between levels of public debt to
GDP ratios is also confirmed in the leading literature in this field including Aiyagari and McGrattan (1998)
and Röhrs and Winter (2017).
2.4.2 Recession and Fiscal Policy
While the previous subsection considers an increase in public debt only, in reality, deficit finance is implemented especially when the nation is hit by a large negative shock like the Great Recession and the
pandemic. While fiscal policies of this sort seem to have a sound rationale that they could benefit households in the short run amid the shock, a less discussed point is that it could do harm if the long run cost of
having a large public debt exceeds the short run benefit. Figure 2.6 compares two types of fiscal policies.
Suppose at time t = 0 there is a sudden drop in output of −5%, which is comparable to the initial
stage of the Great Recession and the pandemic in Japan. To cope with this shock, the government could
implement a fiscal policy of raising the public debt to GDP ratio to conduct tax reduction, spending increase, and redistribution. While the expansion of government consumption and investment is a typical
policy option, we consider an endowment economy, and thus increase in Gt does not contribute to welfare
improvement. Therefore, here we consider two experiments.
In the first experiment (Figure 2.6), we consider a tax reduction only. We compare the case of a fixed
level of public debt, where government does not change the taxation at all and mechanically impose the
74

0.5 1 1.5 2 2.5
initial debt to GDP ratio
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
welfare change in consumption equivalence (in %)
transition welfare with 20% increase in public debt
Figure 2.6: Comparison of Welfare Including Transition with and without an Increase in Public Debt when
Output Drops (Tax Cut for Both Types)
taxes on dynasties in the same manner as in the absence of recession, with the case of an increased level
of public debt to reduce the taxation temporarily. The green solid line in Figure 2.6 corresponds to the
consumption equivalence measure for the former type of finance to deliver the same utility as the latter
finance. With a lower level of initial public debt to GDP ratio, the resulting welfare is higher with the
help of public debt increase. An increase in public debt temporarily gives households a ’moratorium’ for
tax collection amid the recession, which is appreciated when we consider transition welfare. When the
recession hits an economy with an already high level of public debt, however, the short run benefit is offset
by an intensified long run cost of having a large level of public debt; an increase in the asset in the economy
is associated with a higher level of household debt as well, and both lead to larger debt services imposed
on borrowers. As long as the range of public debt levels in Figure 2.6 is concerned, the fiscal policy of
increasing the public debt levels is always welfare enhancing compared with the case of holding the public
debt level fixed throughout the recession.
In the second experiment (Figure 2.7), we consider the increase in spending and redistribution via
taxation. The government increases the outlays G (which is again considered to be just throwing away
goods into the ocean), and temporarily cuts tax burdens of the less rich households, namely, borrowers.
75

0.5 1 1.5 2 2.5
initial debt to GDP ratio
-2
0
2
4
6
8
10
welfare change in consumption equivalence (in %)
10-3
transition welfare with 20% increase in public debt
Figure 2.7: Comparison of Welfare Including Transition with and without an Increase in Public Debt when
Output Drops (Tax Cut for Borrowers Only)
Government finances this policy either by taxes alone or deficit finance of 20% increase in public debt
level. The green solid line in Figure 2.7 corresponds to the consumption equivalence measure for the
former type of finance to deliver the same utility as the latter finance. Again, the green solid line in Figure
2.7 corresponds to the consumption equivalence measure for the former type of finance to deliver the same
utility as the latter finance. In Figure 2.7, fiscal policy redistributes from savers to borrowers by cutting
the tax burden on the borrowers by half compared with the amount paid in the steady state, which is
gradually going back to the taxation at the steady state level. It is possible to allow more substantial (in
degree and duration) tax cuts for borrowers, and this change moves the crossing point of the green solid
line and dotted line (corresponds to consumption equivalent measure of zero) to the right; namely, deficit
finance could be welfare-improving compared with no increase in public debt on condition that the policy
redistributes more from savers to borrowers. This suggests a policy implication for countries with a high
level of public debt that the deficit finance should be more redistributive for the resulting aggregate welfare
to be improved9
.
9While this redistribution policy is expected to make an increase in public debt to GDP ratio attractive, the range of starting
public debt to GDP ratio at which additional deficit finance is welfare enhancing is not so high (200%)in Figure 2.7 compared
with the first experiment in Figure 2.6 (300%), because of the increased government outlays Gt.
76

2.4.3 Foreign Investors
So far the model economy is a completely closed one, and the only supplier of the saving is the saver. In
reality, however, public bonds are demanded not only by domestic investors but also by foreign investors.
Japan has been long famous as a country with a large home bias for demand in public bonds, but the share
of the foreign investors has been on a rise, and it is 14% in 2021. Foreign investors, however, have a role
other than the supplier of the saving; they receive debt services, which may not be reinvested into the
country and foregone. While public bonds held by domestic investors receive debt services part of which
they do pay in the form of tax obligation, interest payments passed on to foreign investors are resources
lost from the country servicing the debt. This subsection studies how the effects of deficit finance are
sensitive to the introduction of foreign investors10
.
While the presence of foreign investors could have an impact on the determination of rates of return
on the public bonds, we assume that Japan is a ’large’ country, and thus interest rate is determined solely
within the domestic market and foreign investors just demand a fraction of it at the equilibrium price.
Let the fraction of public bonds held by the foreign investors as ζ. Then the market clearing condition is
changed into:
a
s + ζB =
a¯
1 + r
+ B.
Observe that the introduction of foreign investors reduces the size of the public debt held by the savers,
(1−ζ)B. Since the interest rate is determined so that savers are discouraged from increasing their savings,
this introduction of foreign investors has a function of limiting the decrease in the equilibrium interest
10A. Mian, Straub, and Sufi (2021) explored an open economy extension differently from ours in their appendix. They regard
the US lives in a small open economy and imperfect access to world financial market. The foreign investors could borrow from
the U.S. when the domestic interest rate is lower than the world interest rate, and thus play a role as an additional borrower. We,
on the other hand, give foreign investors a role as additional “savers.”
77

0.5 1 1.5 2 2.5
initial debt to GDP ratio
-0.01
0
0.01
0.02
0.03
0.04
0.05
welfare change in consumption equivalence (in %)
transition welfare with 20%
increase in public debt (with foreign investors)
transition welfare with 20%
increase in public debt (no foreign investors)
Figure 2.8: Comparison of Steady State Welfare with Welfare Including Transition when Public Debt is
Raised (in the Presence of Foreign Investors)
rate. The existence of foreign investors, on the other hand, let the debt services paid to them exit from the
economy. The goods market clearing condition is transformed into:
c
s
t + c
b
t + Gt + ζ[B(1 + rt) − Bt+1] = Yt
.
Figure 2.8 depicts the welfare comparison of the same setting as the section 2.4.1 in the presence of
foreign investors who hold a share of public bonds.
While the blue line stands for the same consumption equivalent measure of transition welfare as studied in section 2.4.1, the orange solid line corresponds to the consumption equivalent measure of transition
welfare in the presence of the foreign investors. We observe that the consumption equivalence measure
with foreign investors crosses the dotted zero line at a higherinitial debt to GDP ratio, and also has a steeper
slope than the consumption equivalence measure without foreign investors (Figure 2.5). The participation
of foreign investors limits the drop in equilibrium interest rate by reducing the savers’ portfolio size and,
equivalently, borrowers’ liability. This limited reduction in the equilibrium interest rate leads to a limited
drop in welfare in relation to a rise in the initial debt to GDP ratio, which makes higher the debt level at
78

which consumption equivalent measure takes zero. The limited reduction of interest rate, on the other
hand, makes the debt service payment higher compared with the case without foreign investors. What is
worse, debt services paid to the foreign investors do reduce the income of the two dynasties since foreign
investors here are assumed to only hold a fixed amount of public debt and do not reinvest this. These
intensified debt service payments and resources lost from the domestic economy accelerate the speed at
which the benefit of deficit finance diminishes, namely, the slope of the (orange) line in the graph.
To sum up, in the presence of foreign investors, the benefit from deficit finance is increased, but the
increase in cost is high as the starting level of public debt to GDP is raised. Even if the existing foreign
investors are beneficial in terms of welfare, the presence of them could backfire as the public debt level
rises.
2.4.4 Reduction of Debt Level
Lower debt to GDP ratio is associated with higher steady state welfare in the model economy. Then, is
it useful to reduce the debt to GDP ratio by a temporary increase in the tax burden with a wish to attain
higher welfare? This subsection discusses the welfare comparison of debt reductions.
While the reduced debt to GDP ratio attains higher long run welfare, the increased tax burden to
liquidate the debt is tough for households in the short run. Röhrs and Winter (2017) studies public debt
reduction in an incomplete market economy with production, the same as Aiyagari and McGrattan (1998),
and finds that long run benefit of being closer to an optimal level of public debt disappears once the welfare
during transition is taken into account, and that tax burden should be as backloaded as possible so that
process of debt reduction is slow. Ino and Kobayashi (2020) applies the model to Japan, and assessed
the timing of consumption tax increase to achieve fiscal reconstruction and its impacts on heterogeneous
households. While the key to the results in both studies is the existence of borrowing constrained agents
who suffer most from debt reduction policy, in our model borrowers gain from reduced public debt level
79

1 1.5 2 2.5 3
initial debt to GDP ratio
-0.07
-0.065
-0.06
-0.055
-0.05
-0.045
-0.04
-0.035
-0.03
welfare change in consumption equivalence (in %)
transition welfare with 20% reduction in public debt
Figure 2.9: Comparison of Steady State Welfare with Welfare Including Transition when Debt to GDP Ratio
is Reduced
and an associated increase in interest rate because private debt shrinks and debt service, both private and
public, will be reduced.
Figure 2.9 depicts the transition welfare with debt reduction policy in terms of consumption equivalence measure basing the steady state welfare without debt reduction. The horizontal axis corresponds to
the initial level of public debt to GDP ratio, B1. The blue line is the transition welfare starting from B1
and gradually reducing it by 20%. Since debt reduction policy is more likely to be sought when there is
no negative shock to the economy, we do not consider a reduction in output and reduce the debt to GDP
ratio only. We observe that the blue solid line is always located lower than zero on the vertical axis. This
is a striking result; while debt reduction potentially gives higher welfare in the long run, short run welfare
loss from the increased cost of debt servicing excels. Another observation is the shrinking difference in
welfare between debt reduction policy (blue solid line) and staying with B1 (zero on the vertical axis) as
B1 rises; while debt reduction policy gives lower welfare than staying with original debt to GDP level,
overall welfare gets closer to steady state welfare as the starting point of initial debt rises. This suggests
that debt reduction policy is appreciated more as an economy stands at a higher debt to GDP ratio.
80

While the baseline case does not give welfare improvement from debt reduction policy, it might be
possible to engineer a policy that implements transition welfare exceeding the steady state welfare with
staying at the original debt to GDP ratio.
2.4.5 Economic Growth and g > r
Another feature of recent chronically low interest rates is that they are in most advanced economies lower
than the growth rate. This is often expressed as “g > r”, and a subject of the studies including Barro (2020)
and Reis (2021).
The endowment economy we have studied so far has no economic growth; the amount of total endowment is constant and normalized to be 1. The objective of this subsection is to introduce a simple
exogenous exponential growth into the endowment: Yt = (1 + g)
t
· Y0 = (1 + g)
t
, where we let Y0 = 1
for simplicity. Throughout this exercise, let g = 0.007, an average growth rate of Japan in the past thirty
years. Also, we re-calibrate the economy so that the equilibrium interest rate becomes lower than the
growth rate.
There are some changes in the derivation. To think about a balanced growth path of the model, define
a detrended variable as x˜t =
xt
(1+g)
t
. We leave the detailed setup to the appendix B.3. Euler equation in
steady state is now expressed as:
αβ
µ
(˜a
s + a¯¯)
−σ
{(r − g)˜a
s + ω
s
(1 − τ
s
)} + β(1 + r) − (1 + g) = 0.
Coupled with asset market clearing and government budget constraints, an equilibrium interest rate is
obtained. One striking observation is that the derivative of the equilibrium r with respect to B, under a
reasonable calibration and policy experiment, turns out to be positive. Remember that in the baseline of the
non-growth case we feature a declining interest rate r with the increase in the rising level of public debt
B. Therefore in the case of economic growth, the key feature of ’indebted demand’ does not work. This
81

results from the ever lasting growth of the endowment. While savers’ steady state consumption comes
from interest revenue from asset holding (private and public lending) and endowment, the former turns
into a negative source since the asset holding decision made yesterday does not grow at the rate of g; and
the interest revenue is not large enough to cover this forgone growth since the interest rate is very low.
On the other hand, borrowers’ consumption is now:
c˜
b
t = (g − r)
a¯
1 + r
+ (1 − τ
b
)ω
b
.
In the case of g > r, borrowers’ consumption is larger due to the private lending; the borrowing made
yesterday does not grow at the rate of g, but r. This offers them an additional income source compared
with the case of g < r. This benefit could go up if there is a drop in r, which would be the opposite of the
indebted demand mechanism, and thus it does not operate in this case. Therefore, the interest rate goes
up when additional public debt is issued.
While indebted demand is absent, the economy does characterize a very low interest rate along with
a large public debt. Also, it is possible that aggregate welfare measure W is still decreasing in B. Figure
2.10 shows the consumption equivalence measures of attaining the transition welfare of increasing debt
to GDP ratio by 20% relative to steady state welfare.
We observe that, in the range of public debt to GDP ratio shown in the figure, transition welfare always
attains positive consumption equivalence measures. We interpret this as considerable merit of the positive
growth which leaves ample room for the deficit finance to improve the transition welfare.
2.4.6 Most Preferred Specification
So far we’ve seen several variants of fiscal policy one by one. This subsection shows our most preferred
specification in order to derive policy implications for future fiscal policy in Japan.
82

0.5 1 1.5 2 2.5
initial debt to GDP ratio
0.09
0.1
0.11
0.12
0.13
0.14
0.15
welfare change in consumption equivalence (in %)
transition welfare with 20% increase in public debt
Figure 2.10: Comparison of Steady State Welfare with Welfare Including Transition with Economic Growth
We consider a policy response to a large negative shock akin to the onset of the pandemic, namely,
5% drop of output and gradual recovery, the same scenario as in the section 2.4.2. We include foreign
investors, who inelastically hold a 14% share of issued public bonds. In this simulation, we do not
consider a redistribution policy of applying different tax rates, since the actual policy response in Japan at
the onset of the pandemic mainly took the form of lump sum transfers, not a reduction in the tax rate for
specific income brackets.
Figure 2.11 shows the result of debt increase in the most preferred specification without drop in output.
As in Figure 2.5, an increase in public debt is welfare enhancing if the starting level of public debt to GDP
ratio is below a certain level. The figure shows that such point is about 190% of the public debt to GDP
ratio. This suggests that an increase in public debt at normal times will be eventually out of the policy
options for Japan.
Figure 2.12 shows the result of the policy response to recession in the most preferred specification.
Compared with the section 2.4.2, not just that consumption measures take positive values in the extended
range of 50% to 500% of public debt to GDP ratio, the size of the measure is even larger than in Figure 2.6.
This increased gain comes from the introduction of foreign investors, which limit the drop in interest rate
83

0.5 1 1.5 2 2.5
initial debt to GDP ratio
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
welfare change in consumption equivalence (in %)
transition welfare with 20% increase in public debt
Figure 2.11: Comparison of Steady State Welfare with Welfare Including Transition (Most Preferred Specification)
0.5 1 1.5 2 2.5 3 3.5 4 4.5
initial debt to GDP ratio
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0.28
welfare change in consumption equivalence (in %)
transition welfare with 20% increase in public debt
Figure 2.12: Comparison of Welfare Including Transition with or without an Increase in Public Debt to
GDP Ratio when Output Drops (Preferred Specification)
84

and thus limit the resource reallocation from borrowers to savers. As we’ve discussed in the section 2.4.3,
however, this increased gain could backfire when the debt to GDP ratio at the beginning of the recession
is so high that consumption measures take negative values.
While it is confirmed that this bad scenario of backfire is not the case in the range of up to 500%
of public debt to GDP ratio in Figure 2.12, this simulation does not consider a structural change in the
economy from the current calibration, and is silent about it. For instance, increased inequality like the
United States discussed in A. Mian, Straub, and Sufi (2021) (top 1% and bottom 99%) could strengthen the
resource reallocation from borrowers to savers, and might change the simulation result in Figure 2.12.
2.5 Conclusion
The welfare consequence of large public debt is not evident in an economy characterized by a low interest
rate. While large public debt could entail a higher tax burden for the households, the debt service might
not be so large when public bonds are highly priced and thus interest rate is almost zero, which is the
situation happening in many advanced economies including Japan. This paper studies the welfare cost of
the large public debt under a low interest rate in a less standard model with non-homothetic asset holding
utility. We find that public debt indeed has a possibility to raise welfare, especially when there is a drop in
output. Our most preferred specification suggests that an increase in public debt level itself cannot raise
the welfare compared with the initial steady state when the economy accumulates an already high level
of outstanding public debt (about 200%), but the deficit finance continues to be a better policy response
when there is a drop in output compared with holding the level of public debt fixed.
While the setup of two dynasties substantially simplifies the model and underscores the mechanism
of indebted demand in A. Mian, Straub, and Sufi (2021), how multiple and endogenously changing compositions of households with different income levels could change the result will be one of the immediate
interests. Also, while this paper bears on A. Mian, Straub, and Sufi (2021) to reconcile a large public debt
85

with low interest rates, how the result is sensitive to alternative formulation of secular stagnation including the shortage of safe assets (J Caballero and Farhi (2018)) and aging (Eggertsson, Mehrotra, and Robbins
(2019)), will be interesting to deepen our understanding of the welfare consequence of large public debt
under low interest rates.
B.1 Detailed Overview of Literature on Public Debt and Welfare
We briefly overview the literature studying welfare consequences of the amount of outstanding public debt
in the economy.
In a standard neoclassical growth model, there is not much to be appreciated about public debt. Barro
(1974) gives a striking "Ricardian Equivalence": government finances through lump sum taxation and public debt issuance result in the same allocation since what mattersto the households is the net present value
of the public deficit. If lump sum tax is not available, public debt matters for the allocation, but the focus
is on how to minimize the efficiency loss from distortionary taxation, and typically there is no role discussed for the public bond issuance to improve welfare. Welfare consequences of public debt start to be
meaningful when there is heterogeneity among agents.
Even the minimal heterogeneity, namely, two agents setting gives a rather different implication than
the representative agent model. Mankiw (2000)’s spender-saver model introduces to a neoclassical growth
model two agents; one is engaged in intertemporal optimization, and the other is a hand-to-mouth. This
setup is very close to A. Mian, Straub, and Sufi (2021) and our model. In their model, the public debt
increases the steady state inequality between agents. A higher level of debt means a higher level of taxation
to pay for the interest payments on the debt. The taxes fall on both spenders and savers, but the interest
payments go entirely to the savers, the holder of the outstanding public bonds. Thus, a higher level of
debt raises the steady state income and consumption of the savers, and lowers the steady state income and
86

consumption of the spenders. Public debt influences the distribution of income and consumption, unlike
in a representative agent model.
Starting from Aiyagari and McGrattan (1998), many studies base the analysis on a model of an incomplete market. Agents in the model are heterogeneous in that they receive idiosyncratic shocks to
labor productivity, and the only way to prepare for the shock is to purchase one single bond (incomplete
insurance). Borrowing is constrained, which induces precautionary saving in combination with the incompleteness of the market. This over saving is the potential source where the presence of outstanding public
debt can reduce the amount of private savings. The reduced amount of capital approaches the golden rule
level of capital, which has the potential to improve welfare.
Households’ problem is
max
{ct,lt,at+1}∞
t=0
u (ct
, lt) =

c
µ
t
l
1−µ
t
1−v
1 − v
s.t. ct + at+1 ≤ (1 − τy) wtet (1 − lt) + (1 + rt) at + Tt
ct ≥ 0
at+1 ≥ 0
given (a0, e0)
Households are each period endowed with one unit of time, which they will divide between leisure (lt
) and
labor (1 − lt
). They receive labor wage wtfor their labor per efficiency, and this is subject to idiosyncratic
productivity shocks. They save in at+1and earn the interest revenue rt
, where there is only one single
87

type of asset available (incomplete market). All incomes are subject to a proportional tax rate τy, and they
receive a net transfer Tt
. The government has the budget constraint of:
Gt + (1 + rt) Bt + Tt = Bt+1 + τywtLt
.
Firms rent capital and labor from households at rate rt and wt
, and produce according to Yt = F (Kt
, ztLt),
where zt = (1 + g)
t
is the technological growth rate. The focus of their analysis is a steady state equilibrium. Denote b as the public debt to GDP ratio. After detrending, the budget constraint is:
c˜t + ˜at+1 ≤ (1 − τy) ˜wtet (1 − lt) + (1 + rt) ˜at + T˜
t
In steady state equilibirum, both equations turn into:
G˜ + rb + T = τywL˜
c˜ = (1 − τy) ˜w (1 − l) + ra˜ + T˜
Combining these, and use a new notation of a
∗
t = ˜at − b, the budget constraint and borrowing constraint
is:
c˜ = ˜w (1 − l) + ra∗ − G˜
a
∗
t = ˜at − b ≥ −b
Observe that the public debt to GDP ratio b does not appear in the budget constraint, and appears only
at the borrowing constraint. Compared with an economy with b = 0, this means that in the steady state
88

equilibrium households are faced with looser borrowing constraint. Since the government with positive
public debt to GDP ratio is a net supplier of saving opportunities, the borrowing constraint of the households is looser, which gives households more opportunities to smooth consumption. Also, the introduction
of public bonds pushes up the interest rate, which in the absence of outstanding public debt is lower due
to the precautionary saving to prepare for market incompleteness. This intensified interest rate fosters
the consumption smoothing of households on one hand, but crowds out some of the capital on the other
hand. Compared with the economy in the absence of outstanding public debt (no debt but taxes only),
which is equipped with more capital than the golden rule level, the economy with outstanding public debt
can improve the overall welfare through the channels described above. Aiyagari and McGrattan (1998)
calibrates the model to the U.S. in the 1990s, and numerically finds that the optimal public debt to GDO
ratio is 66%.
The following studies are centered around Aiyagari and McGrattan (1998). While their conclusion
that the steady state optimal quantity of public debt to GDP ratio of 66% surprisingly coincides with the
observed ratio in the U.S. in the 1990s is interesting, does this mean that the government should always
seek this ratio? Obviously, there could be a short run benefit of issuing a public debt and postponing the
liquidation of this fiscal deficit by increased taxation, so that households are exempted from a tax burden
temporarily, especially when there is a shortage of demand due to a downturn resulting from recession
or negative shocks. This deviation from the current level of public debt to GDP ratio can be rationalized,
Desbonnet and Weitzenblum (2012) argues, if we consider welfare not just in steady state equilibrium
but also the whole transition process of moving from one steady state to the other. They consider the
transition from one steady state to another with increased public debt, and compute the welfare not just
of the steady state but including the whole steady state transition process. Since an increase in public
debt temporarily allows a reduction in tax collection, deviating from the original level of public debt can
89

be welfare improving. When considering a 10% increase in public debt to GDP ratio, the economy has a
’profitable deviation’ of the public debt to GDP ratio until it reaches a remarkable 560%.
With the importance of transition welfare in mind, Röhrs and Winter (2017) studies the welfare cost of
public debt reduction in an incomplete market economy. When seen only at the steady state equilibrium,
an equilibrium with a too high level of public debt to GDP ratio gives lower welfare than the one with a
lower ratio since there would be more saving going to the capital stock, and also the tax burden is smaller
since both the outstanding public debt and equilibrium interest rate is lower. Once we take the welfare
during the transition into account, however, the short run cost of public debt reduction is so high that
such a liquidation policy results in lower welfare. Not just the tax burden to pay back the public debt, but
also the liquidation of public debt means fewer saving opportunities in the economy, and lets more and
more poor households to hit the borrowing limit, which would become tighter during the transition. For
the policy to have as small losses as possible, the increase in tax burden and the drop in borrowing limit
should be as postponed as possible. This means that the increase in tax rate should be as slow as possible,
and should not increase in a convex shape but in a concave shape.
Peterman and Sager (2022) considers the incomplete market and optimal quantity of public debt in the
overlapping generations model, instead of the infinitely lived agents model, and finds that the steady state
optimal quantity of public debt is negative, that is, the government should hold government asset. While
increased interest rate in the presence of public debt in Aiyagari and McGrattan (1998) is appreciated by
households who are engaged in consumption smoothing, in the OLG economy households spend most of
their early stage of lives accumulating savings to prepare for their lives after retiring. Compared with the
infinitely lived agent model, agents in the OLG model could enjoy the benefits of a high interest rate only
when they have accumulated a fortune in the middle of their lives. Therefore, unlike an infinitely lived
agent model, households are happy to enjoy lowered interest rates due to the government saving in the
OLG model. Peterman and Sager (2022) argues that this benefit is so large that it could pay to reduce the
90

amount of public debt and suffer the short run pains of increased tax burdens when considering the whole
transition welfare.
While most of the literature studies economies with an incomplete market, we focus more on the
importance of studying low equilibrium interest rates and large outstanding public debt, and (at least
currently) stepping away from an incomplete market model, and drawing more on Mankiw (2000) and A.
Mian, Straub, and Sufi (2021).
B.2 Computation
The steady state of the model is computed by finding the intersection of the asset supply and demand
curve:
αβ/µs
(a
s
/µs + ¯a)
−σ
(a
s
r + ω
s
(1 − τ
s
)) + β(1 + r) − 1 = 0.
a
s =
a¯
1 + r
+ B,
where the tax rate τ
s
is determined so that the government budget constraint G + rB = τ
sω
s + τ
bω
b
.
In the most policy experiments of this study we determine τ
s
and τ
b
so that some constant shares of the
government outlays (the government consumption and the interest payment) are borne for each dynasty.
Steady state transition upon either increase in public debt level or drop in output (or both) is computed
following the procedure below, which we base on the replication code of Guerrieri and Lorenzoni (2017).
1. compute the two steady states with B1
and B2
respectively
2. take a long enough time period T, and guess a sequence of {r
k
t }
T
t=0, where k is the number of trials
91

3. starting from consumption policy in the second steady state (obtained by value function iteration),
solve the saver’s problem backward with the interest rate being {r
k
t } using the endogenous grid
points method
4. using the sequence of asset purchase policy of the saver obtained in 3., compute the saver’s asset
holding starting from asset level in the first steady state up to period T
5. check the asset market clearing of the whole period; update the guess of interest rates to be lower
if asset supply (by saver) is too high in a given period, and to be higher otherwise, based on the
following update rule:
r
k+1
t = r
k
t − (a
s∗
t + a
b∗
t − Bt) × δt
. (δt
is some scalar)
6. iterate 3-5 until market clearings are sufficiently satisfied in every period, where the eventual level
of asset supply attains the level in the steady state with B1
(say, k = 1, . . . , 1000)
With torelance for the asset market clearing error being 1e − 6, the calculation basically converges in less
than 3 minutes using a MacBook Air (2020) laptop equipped with a 3.2 GHz Apple M1 chip with 8 cores.
As an example of the calculation, Figure B.13 shows the steady state transition in the section 2.4.2.
Economy is hit by a drop in output that reduces the real GDP level from 1 to 0.95, and gradually reverts
to the original level of 1. It starts with one steady state characterized by the public debt of B1 = 150%,
and deviates from this steady state due to an exogenous drop in output and associated policy response
of deficit finance. The government raises public debt gradually over 10 years to B2 = 170%, where the
economy goes to another steady state.
Since households have perfect foresight, they anticipate the whole path of output and public debt at
time t = 0. For the saver to take on the sudden increase in assets, the interest rate jumps up, but it soon
adjusts to new steady state levels, which are slightly lower than the original one due to the increase in
92

0 10 20 30
t
1.5
1.55
1.6
1.65
1.7
Transition of Public Debt Level
0 10 20 30
t
0.95
0.96
0.97
0.98
0.99
1
Transition of Output
0 10 20 30
t
0
0.02
0.04
0.06
0.08
0.1
Transition of Interest Rate
0 10 20 30
t
2.05
2.1
2.15
2.2
2.25
2.3
Transition of Asset of the Saver
Figure B.13: Example Calculation of the Steady State Transition 1
public debt. While recession itself could lower the asset holding of the saver, a sudden increase in saving
supply excels this drop and eventually the asset holding follows an increasing path.
B.3 Alternative Borrowing Constraints
Figure B.14 shows an example of the calculation of steady state transition upon a drop in output under
fiscal policy financed by deficits, with an alternative borrowing limit of a/r ¯ t+1 instead of a/¯ (1 + rt+1).
This is the borrowing constraint considered in Aiyagari (1994) as one equivalent to limit condition and
nonnegativity of consumption11
.
While steady state transition with the baseline borrowing constraint (Figure B.13) takes less than 20
periods, here it takes more than 30 periods (as many as 150 periods!). We also notice that movements of
the interest rate are so minuscule. This is because with the alternative borrowing constraint −a/r ¯ t only a
small decrease induces a large change in the amount allowed to borrow compared with the baseline budget
constraint of −a/¯ (1 + rt). While the moderate change in interest rate seems reasonable, the transition of
assets of the saver might be too long (150 years). Moreover, this borrowing constraint is not compatible
11Precisely, it is −wlmin/r, where wlmin corresponds to the worst draw of the idiosyncratic income.
93

0 10 20 30
t
1.5
1.55
1.6
1.65
1.7
Transition of Public Debt Level
0 10 20 30
t
0.95
0.96
0.97
0.98
0.99
1
Transition of Output
0 10 20 30
t
1.6
1.65
1.7
1.75
1.8 10-3
Transition of Interest Rate
0 10 20 30
t
3.36
3.38
3.4
3.42
3.44
3.46
Transition of Asset of the Saver
Figure B.14: Example Calculation of the Steady State Transition 2
with g > r, since −a/¯ (rt−g) turns out to be positive. Nonetheless, it is useful to see the transition welfare
with this borrowing constraint.
Figure B.15 shows the comparison of fiscal policy with a fixed level of public debt to GDP ratio and with
an increase in the ratio by 20%, where the latter is measured using the consumption equivalence which
makes dynasties enjoying the former to move to the latter. As seen in the section 2.4.2, (1) consumption
equivalence is decreasing in the level of public debt to GDP ratio, and (2) change in the measure is quite
minuscule.
Another type of borrowing constraint is the one again discussed in Aiyagari (1994), − ¯(a − τ
i
tω
i
)/(1 +
rt). He considers this when introducing government and public spending, and this modification is to
ensure the non-negativity of consumption. It turns out that, in the steady state equilibrium, this adjusted
borrowing constraint induces debt neutrality if the lump sum tax is available. Aiyagari and McGrattan
(1998) also demonstrates that outstanding public debt functions as an additional saving opportunity, which
effectively eases the borrowing constraint in the steady state equilibrium.
94

0.5 1 1.5 2 2.5
initial debt to GDP ratio
0.6985
0.699
0.6995
0.7
0.7005
0.701
0.7015
0.702
0.7025
welfare change in consumption equivalence (in %)
transition welfare with 20% increase in public debt
Figure B.15: Comparison of Steady State Welfare with Welfare Including Transition when Output Drops
(Alternative Borrowing Constraint)
How this borrowing constraint is appreciated in our model? For the first point, there is no gain; savers
in the model are Ricardian equivalent, and the borrowers are effectively hand-to-mouth in the equilibrium
we consider. For the second point, there is no ease of the constraint; savers are always savers, and borrowers always borrow up to the limit. We are thinking about not a steady state where the public bond is
distributed to everyone, but a transition where savers purchases a newly issued public bond and borrowers
borrow an additional private loan. Nonetheless, this borrowing constraint is distinct from the baseline one
in that it will limit the indebted demand mechanism by reducing the borrowing exactly when outstanding
public debt is large. Figure B.16 shows the basic calculation of the transition of steady state under the
borrowing limit of (¯a − τ
bω
b
)/(1 + rt+1).
B.4 Supplement to Instrument Variables in 2.3.1 (Empirics on Non-homothetic
Preference)
In the section 2.3.1, we discuss the following system:
95

Figure B.16: Example Calculation of the Steady State Transition
cˆit = X
′
itβ + φ log wit + ǫit
yˆit = ˆwi + ηit + ψit + vit.
where we use the moving average of the second to impute wˆ, and put it into the first equation. The
concern, however, is that the operation of taking a moving average may not wash away the permanent
ηit and transitory ψit shocks, which in turn bias the regression of the first equation. The direction of
bias is toward zero because the unobservable determinant of consumption εitand shocks ηit and ψit are
considered to be negatively related. Even when there is a positive income shock, this does not change the
permanent income, and they try to reduce the consumption to smooth the consumption12. So our hope is
to kill this correlation of consumption with a permanent component of the shock ηit. Typical instruments
of lagged or future income variables do not work because of the permanent nature of the shock ηit.
12This is named as "consumption smoothing bias" in Straub (2019).
96

Straub (2019) proposes that, while assigning an autoregressive structure on permanent income shocks
ηit = ρηit−1+ε
η
it, the instrument takes a “quasi difference”: zit+τ = ˆyit+τ −ρyˆit+τ−1(τ ≥ 2). Let’s expand
this:
zit+τ = ˆyit+τ − ρyˆit+τ−1
= (1 − ρ) ˆwi + ηit+τ − ρηit+τ−1 + ψit+τ − ρψit+τ−1 + vit+τ − ρvit+τ−1
= (1 − ρ) ˆwi + ε
η
it+τ + ψit+τ − ρψit+τ−1 + vit+τ − ρvit+τ−1.
Thus with τ ≥ 2, zit+τ is uncorrelated with ηit = ρηit−1 + ε
η
it,and other shocks, and so the first stage will
net out the effect of permanent shock.
Two remarks are in order. First, it could be a case that ρ = 1, namely, permanent shock ηit follows
a random walk. This case is also addressed in Straub (2019). Second, autocorrelation coefficient ρin the
process of permanent income shock is also a target of estimation itself. Here we follow Straub (2019)’s
exposition, and try multiple values of ρ and skip the estimation for ρ.
B.5 Derivation in 2.4.5 (Economic Growth and g > r )
The utility function is now:
X∞
t=0
β
t

log
c˜
i
t
µi

+ α
(˜a
i
t
/µi + ¯a)
1−σ
1 − σ

+ const.
Note that log utility of goods consumption allows the trend term to be separate as a constant as long
as β(1 + g) < 1. On the other hand, utility from asset holding does not allow a balanced growth due
to the subsistence part a¯¯. Following A. Mian, Straub, and Sufi (2021)’s treatment in the appendix, we
97

reinterpret the utility from asset holding as asset holding ’in relation to total output Yt
,’ and thus not a
i
tbut
a
i
t/Yt = ˜a
i
t
enters the utility function. The budget constraints of the dynasties are now:
c˜
i
t + (1 + g)˜a
i
t+1 ≤ ω
i
(1 − τ
i
t
) + (1 + rt)˜a
i
t
.
Borrowing constraint13 now is also growing at the same rate as the endowment:
a
i
t+1 ≥ −a¯(1 + g)
t+1/(1 + rt+1),
and this is also detrended as:
a˜
i
t+1 ≥ −a/¯ (1 + rt+1).
Finally, the government budget constraint becomes:
G˜
t + (1 + rt)B˜
t = (1 + g)B˜
t+1 + τ
s
t ω
s + τ
b
t ω
b
.
The Euler equation is:
αβ
µ
( ˜at
s + a¯¯)
−σ
{(r − g) ˜at
s + ω
s
(1 − τ
s
t
)} + β(1 + r) − (1 + g) = 0.
Market clearing for the asset is:
a˜
s
t+1 =
a¯
1 + rt+1
+ Bt
.
Let’s consider a steady state equilibrium. Three equations become:
13There are other possible candidates for borrowing constraint as there are in the case without economic growth, and examples
include a˜
i
t+1 ≥ −a/¯ (1 + rt+1 − g) and a˜
i
t+1 ≥ −a/¯ (rt+1 − g).
98

αβ
µ
(˜a
s + a¯¯)
−σ
{(r − g)˜a
s + ω
s
(1 − τ
s
)} + β(1 + r) − (1 + g) = 0.
G˜ + (1 + r)B˜ = (1 + g)B˜ + τ
sω
s + τ
bω
b
.
a˜
s =
a¯
1 + r
+ B.
Suppose tax obligation is a fixed fraction of the fiscal need: ω
s
τ
s = τ (G˜ + (r − g)B). Eliminating a˜s and
τ
s
, we obtain:
αβ
µ

a¯
1 + r
+ a¯
−σ
(r − g)

a¯
1 + r
+ B

+ τ (G˜ + (r − g)B)

+ β(1 + r) − (1 + g) = 0.
Use the implicit function theorem to derive the slope of r as a function of B,
dr
dB
=
αβ
a¯
1+r + a¯
−σ
(g − r)(1 − τ)
αβ
σ

a¯
1+r + a¯
−σ−1 a¯
(1+r)2
n
(r − g)

a¯
1+r + B

+ ωs (1 − τs)
o
+

a¯
1+r + a¯
−σ

a¯
1+r + B − (r − g)
a¯
(1+r)2 − τB + βµ
.
The two curly brackets in the denominator take positive values in most standard parametrizations. Therefore, the sign of the slope is in fact determined by g−r in the numerator. In the baseline case of no growth,
g − r < 0, and thus the sign is negative: every increase in public bonds contributes to a decline in the
interest rate.
The opposite of this is discussed in section the section 2.4.5. g > r changes the sign of dr/dB to be
positive, and thus the key mechanism discussed in A. Mian, Straub, and Sufi (2021), indebted demand, does
not work.
99

A. Mian, Straub, and Sufi (2021) refer to the case with economic growth inthe appendix. They discuss an
endowment economy (like ours) without government (unlike ours). They demonstrate that in the absence
of government the non-zero growth rate does not affect the allocation, and only pushes up the interest rate
by shifting the saving demand curve and supply curve upward. Clearly their result gives a new interest
rate r
new = g + r
withoutgrowth > g, opposite from the current center of discussion (g > r). Our study in
the section 2.4.5 is distinct from A. Mian, Straub, and Sufi (2021) in that we introduce government and the
effect of public bond on the up and down of interest rate, and study the consequence of g > r.
B.6 Optimal Size of the Debt Increase
In the experiments so far, we have fixed the increase of public debt to be 20% of GDP, and varied the level
of the public debt to GDP ratio at which we consider an experiment. As a distinct experiment, we fix the
level of public debt to GDP ratio to be 150%, and varied instead the size of the debt increase. We measure
the consumption equivalence of each size of the debt increase in reference to the welfare of staying at the
debt to GDP ratio of 150% (no increase in the debt). By this experiment, we study what size of the debt
increase gives the highest transition welfare and derive an implication for the optimal policy response to
a shock.
Figure B.17 shows the result when there is no shock to the economy. Compared with staying at B1 =
150%, blue line shows the consumption equivalence according to the debt increase/decrease size. There
are three observations. First, as the debt is increased, the consumption equivalence rises up to several
sizes, but then turns to decline. Second, the debt increase size of about 40% attains the highest welfare.
Third, debt decrease (negative increase size) necessarily deteriorates the welfare.
Figure B.18 considers the similar exercise to Figure B.17 when output drops by 5%. Compared with
the welfare of staying at B1 = 150%, green line shows the consumption equivalence according to the
debt increase size. Notably, the size of the debt increase that attains the highest welfare is now about 60%.
100

-50% 0% 50% 100% 150%
size of increase in debt level
-0.14%
-0.12%
-0.1%
-0.08%
-0.06%
-0.04%
-0.02%
0%
0.02%
welfare change in consumption equivalence
transition welfare with differential increase in public debt
Figure B.17: Debt Increase Size and Consumption Equivalence
0% 20% 40% 60% 80% 100% 120% 140% 160% 180%
size of increase in debt level
-0.1%
-0.08%
-0.06%
-0.04%
-0.02%
0%
0.02%
welfare change in consumption equivalence
transition welfare with differential increase in public debt
Figure B.18: Debt Increase Size and Consumption Equivalence when Output Drops
This suggests that, as in the section 2.4, debt increase is more appreciated and encouraged when there is a
shock to the economy.
101

Chapter 3
Monetary Policy, the Dual Labor Market, and Consumption in Japan
3.1 Introduction
There is a growing theoretical literature that re-examines how monetary policy stimulates consumption. In
this literature, consumption due to rising labor income accounts for about 50% of the change in consumption from a monetary shock (Auclert (2019), Bilbiie (2020), and Morrison (2024)). If that is the case, then
changing features of the labor market may affect how monetary shocks impact consumption. In this paper,
we explore how the increasing proportion of non-regular workers–those with more informal attachments
to the labor force–has affected the transmission of monetary shocks into consumption in Japan.
One of the prevailing puzzles in the Japanese economy between 2009 and 2019 is the slow growth in
consumption, despite the deep cut in interest rates and the large quantitative easing. This paper relates this
"low consumption response puzzle" to inequality in Japan, specifically to the dual structure of the labor
force.
This is a joint work with Robert Dekle: University of Southern California, dekle@usc.edu. We are grateful to participants
at a workshop at Keio University for helpful comments.
102

We develop a "Two-Agent New Keynesian (TANK)" model with regular and non-regular workers. As
explained below, non-regular workers have a more "casual" attachment to a firm.1 Their salary and benefits
are lower than regular workers. In our TANK model, non-regular workers are paid their marginal product, while regular workers receive negotiated wages that are sticky. We show in our model simulations
that as the proportion of non-regular workers increases, aggregate consumption becomes less sensitive to
monetary shocks.
Loose Monetary Policies and Consumption
Figure 3.1 shows the relationship between the quarter-on-quarter change in real household consumption
Figure 3.1: Interest Rates and Growth in Real GDP and Consumption (Quarter on Quarter)
Notes: real GDP and consumption is seasonally adjusted.
Source: interest rates: BOJ / GDP and consumption: SNA / shadow rates: Krippner’s websitea
a
https://www.ljkmfa.com/visitors/
and short-term (overnight call rate) and long-term (10-year bond rate) interest rates from 2009 to 2019. The
1Non-regular workers are assumed to be different from regular workers in the following respects. First, they have lower
human capital than regular workers. Second, non-regular workers are paid their marginal product, while regular worker wages
are negotiated with the employer through their union. Third, the adjustment (layoffs) of non-regular workers are costless to the
employer, while the adjustment of regular workers entail a cost. Fourth, the wages of regular workers can only adjust at fixed
reset probabilities, while those of non-regular workers can adjust continuously with the shock.
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figure includes the "shadow" interest rates estimated by Krippner (Krippner (2013) and Krippner (2015)).
These "shadow" rates are intended to capture the hypothetical interest rates that would pervade, given the
unconventional monetary policies enacted in Japan near the zero lower bound.
Since 2010, while short-term rates were already at the zero lower bound, long-term rates, as well as the
"shadow" rates declined under the expansionary monetary policies of the Bank of Japan. However, despite
these relaxed monetary conditions, consumption growth has been tepid, always staying below the growth
in real GDP.
There are several papers that discuss the weak response of aggregate consumption to monetary shocks
in Japan. Hausman, Unayama, and Wieland (2021) observes that during the rapid monetary expansion
period from 2012 to 2018, GDP grew at an annual rate of 1.1%, while the growth in consumption was 0.4%
(as shown in Figure 3.1). Hausman, Unayama, and Wieland (2021) carefully examine the heterogeneous
response of household consumption during the rapid monetary growth era using Japanese household data.
They find that what contributes to the high aggregate consumption response in the U.S. such as the larger
response of young people (Wong et al. (2019)) and homeowners (Cloyne, Ferreira, and Surico (2020)) to
monetary shocks, is not observed in Japanese household level data. Our study adds to this explanation of
weak aggregate consumption by focusing on another dimension in workers’ characteristics, namely, the
types of jobs they are engaged in.
Inui, Sudou, and Yamada (2017) construct a New Keynesian model of two types of workers; those
attached to the firm, and those free to move across firms. They then examine the impact of monetary
policies on aggregate consumption. Their argument is that as the proportion of unattached workers rise,
the consumption and income inequality are less affected by monetary shocks, since mobile workers are
better able to move from the rigid price firms to the flexible price firms. In our model of labor market
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dualism, workers engaged in non-regular type of work are stuck in flexible wage jobs and they cannot move
to flexible wage type jobs. Matsui and Yoshimi (2015) also build a model of two distinct labor markets.2
Rise in the Number of Non-regular Workers
Figure 3.2: Growing Share of Non-regular Employees
Notes: non-regular employees are shorter hours workers, mainly consisting of part time workers.
Source:Labor Force Survey
In Figure 3.2, we depict the trends in regular and non-regular workers. In 1984, the proportion of
regular workers was 85% and the proportion of non-regular workers was 15%. By 2017, the proportion
of regular workers declined to about 63%, and the proportion of non-regular workers rose to about 37%.
3
Figure 3.3 depicts the hourly wages of regular and non-regular workers. Before 2004, we do not have
statistics on the wages of non-regular workers, so we used the wages of part-time workers instead. Real
2One is a perfectly competitive market where the wage rate is determined in the market as the marginal productivity of labor,
the other is a unionized labor market where the workers and the firm negotiate the wage rate through Nash bargaining. Our
model adds two changes to the unionized workers. First, we allow unions to monopolistically competitively set the wage rate.
Second, we introduce adjustment costs of the demand for the regular type of workers.
3Non-regular workers include part-time workers, albeit or temporary workers, dispatched workers from temporary labor
agencies, contract workers, and some other minor categories.
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Figure 3.3: Hourly Wages of Regular and Non-regular Workers
Notes: wages are seasonally adjusted.
Source: wages of full timer and part timer: Monthly Labor Survey / wages and composition of regular and non-regular workers:
Basic Survey of Wage Structure
hourly wages of full-time workers have been about 2.2 to 2.3 times higher than the hourly wages of parttime workers. A similar gap persists between the wages of regular and non-regular workers after 2004.
Figure 3.2 and 3.3 show that there has been a dramatic relative increase in the number of non-regular
workers, who are paid less than half of regular workers. If so, then we would expect that the average
worker wage will stagnate in Japan, as lower wage workers start to make up a larger proportion of the
labor force without a rise in the wage of the non-regular workers.
Figure 3.4 shows the average wage if the proportion of regular workers stayed at its 1994 share. The
relative wages between regular and non-regular workers are assumed to be the same as in Figure 3.3.
Figure 3.4 shows that wages per worker would be higher. Thus, the growth in non-regular workers puts
large downward pressure on average wages.4
4Hoshi and Kashyap (2020), while not concerned with the aggregate consumption response to monetary shocks as we do
here, show that the response of aggregate wages to business cycles is impacted by the ratio of regular and non-regular workers.
106

Figure 3.4: Wages per Worker: Actual versus Constant 1994 Regular Worker Shares
Notes: seasonally adjusted hourly wages are multiplied by the hours worked, 12 (number of months), and divided by the number
of workers to prepare a computed yearly wages per worker.
Source: Monthly Labor Survey
In this paper, we show that the rise in the share of non-regular workers has contributed to lowering
the response of aggregate consumption to monetary shocks. The characteristics of Japanese non-regular
workers that differ from regular workers, albeit stylized, are their lower human capital (suggested by their
lower wages), and the flexibility of their wages and the demand for their services. Regular workers are
subject to adjustment costs to their employment and their wages are not as flexible.
Since our focus is on short-run monetary policy responses, we assume that the expansion of nonregular workers over the past three decades is structural and exogenous to the shocks in our model. In
Section 3.2, we show using Japanese household panel data that there is heterogeneity in the consumption
responses between regular and non-regular workers with respect to monetary shocks. In Section 3.3, we
present a New Keynesian model featuring two distinct labor markets, one for regular and the other for
107

non-regular worker households. We assume that these two household types are distinct, and households
cannot switch from one type to another at the business cycle frequency. In Section 3.4, we calibrate the
model using disaggregated Japanese data.
In Section 3.5, we simulate our calibrated New Keynesian model and conduct policy experiments. We
conduct five policy experiments, in addition to our benchmark simulations.In all of our simulations, we
shock our model with a 1% monetary shock according to the calibrated Taylor rule embedded in our model.
In the benchmark simulation, we take our calibrated values, and the current proportion of non-regular
workers, 37%. We then trace out the consumption, labor supplies, and wages of the regular workers,
non-regular workers, and in the aggregate.
In our first policy experiment, we examine the effects of lowering the proportion of non-regular workers to the level in the 1990s, 20%. We find that upon the monetary shock, aggregate wages and aggregate
consumption would have been higher.
In our second experiment, we lower the human capital of non-regularl workers. Compared to the
baseline, where the productivities of the two types of workers are equal, the lower productivity of the
non-regular workers reduces aggregage consumption upon the monetary shock.
In our third experiment, we allow differing compositions of regular workers to non-regular workers
like in the first experiment, but allow the quality of labor of the non-regular type to be lower than that of
the regular type.
In the final two experiments, we 1) raise the costs to firms of adjusting the number of regular workers;
and 2) allow unions to reset the wages of regular workers more frequently. Both of these experiments
correspond to recent changes in Japanese labor market institutions. The higher adjustment costs and the
higher frequency of wage changes both dampen the growth in regular worker employment, and lower the
changes in aggregate wages and in consumption.
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3.2 Monetary Shocks and the Consumption of Regular and Non-regular
Workers
3.2.1 Data
Our data source is the Japan Household Panel Survey (JHPS/KHPS) conducted by the Panel Data Research
Center at Keio University. KHPS and JHPS were originally distinct household panel surveys conducted
separately beginning in 2004 and 2009. The surveys have been combined since 2014. The JHPS/KHPS
collects income, asset and consumption data in addition to demographic information for a stratified random
sample of households; the same households continuously answer the survey questions5
. One caveat of the
JHPS/KHPS is that consumption is surveyed for only one month (January of the survey year), while income
and assets data are surveyed at both monthly (the same timing as consumption) and annual frequencies.
For consumption, we use total monthly expenditures including mortgage payments. To lower the influence
of outliers, we drop all observations below the 10th and above the 90th percentiles in consumption.
As mentioned, the JHPS/KHPS classifies workers based on three steps. First, the respondents were
asked whether they worked or not the year before the survey. For those who worked in the previous
year, the second classification is the type of employment. While some workers are engaged in selfemployed jobs and professional jobs, the vast majority of workers are wage workers. We classify wage
workers into "regular" type of workers (or regular workers) if they (1) are directly employed, (2) have
contracts without a limited term, and (3) work full time. The other wage workers are all classified into
"non-regular" type of workers (or non-regular workers). Non-regular type of workers include parttime jobs, dispatched workers and any other wage workers that violate at least one of the above three
conditions. JHPS/KHPS does these classifications for both primary respondents and their partners if any
5One of the most standard official household surveys in Japan isthe Family Income and Expenditure Survey (FIES) by Ministry
of Internal Affairs and Communication. This survey has coverage of roughly 8,000 households at a monthly frequency, but based
on rotating panels where the same households are surveyed only for up to six months. In this paper, we prefer to control for
household fixed effects, and thus stick to JHPS/KHPS dataset.
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in the household. Since our analysis is based not on individual workers but on households, we classify a
household into a "non-regular" type if neither of the respondents or their partners has a regular type of
job and at least one of them is engaged in a non-regular type of job. However, if either the respondent or a
partner is engaged in a regular type of job, such households are classified as regular types of households.
In our estimation below, we also use another, broader definition of regular and ’non-regular’ workers
that covers every type of worker. This broader category of non-regular workers include the self-employed,
wage workers that work at a family business, or at home without an employee relationship. Professional
workers such as lawyers, accountants, and doctors are included in the regular workers category.
KHPS/JHPS LFS MLS
share of Non-regular workers (%) 38% 37%
- among age 15-24 40% 47%
- among age 25-34 27% 26%
- among age 35-44 31% 29%
- among age 45-54 34% 32%
- among age 55-64 39% 47%
monthly wage of Reg. workers
(1,000 Yen) 34.84 31.06
- among age 15-24 16.38 20.98
- among age 25-34 25.05 26.27
- among age 35-44 33.08 32.81
- among age 45-54 39.10 38.63
- among age 55-64 41.09 35.22
monthly wage of Non-reg. workers
(1,000 Yen) 15.00 21.08
- among age 15-24 14.50 18.37
- among age 25-34 14.85 20.51
- among age 35-44 12.24 21.01
- among age 45-54 12.98 20.61
- among age 55-64 17.46 22.07
Notes: details on LFS (Labor Force Survey) and MLS (Monthly Labor Survey) are in the appendix. The statistics in this table treat
primary respondents and his/her partners in a given household to be separate observations. All the descriptive information is as
of the fiscal year 2017. Percentage share is the fraction among the employed workers (regular workers and non-regular workers).
Table 3.1: Comparison between KHPS/JHPS Data and Japanese Official Statistics
Descriptive Statistics
?? compares some of the descriptive statistics of our dataset built from JHPS/KHPS with the counter part
110

among the official government statistics in Japan in 2017. The statistics in this table treats primary respondents and his/her partner in a given household to be separate observations. Shares of the regular
workers and non-regular workers among employed workers are compared with statistics from more representative datasets such as the Labor Force Survey. We find three broad patterns. First, the composition
of non-regular workers is high for young people ages 15 to 24. This is because workers in this age category
mostly work only part-time. Second, the composition of the non-regular workers is lowest for the group
of age 25 to 34. The main reason is that in Japan, most regular workers are hired as new school graduates.
Third, the composition of non-regular workers rises as workers age. There are several reasons for this.
For example, it is pointed out that women’s labor force participation drops at about the age when they
experience childbirth. After that point, many return to the labor force, but often at lower paid non-regular
jobs. Another reason is that it is harder for those who lose their regular type of jobs in their 40s or 50s to
find a regular type of job.
For wage comparisons, we choose the monthly wage as our measure of income. Therefore we compare
monthly wage receipt data in our dataset with that in the Monthly Labor Survey. The gap in monthly wages
between regular and nonregular workers is wider in the Monthly Wage Survey.
In Table 3.2, we look at several demographic and important variables on household basis. We classify
a household to be ’regular type’ if either of the primary respondent or his/her partner is a regular type of
worker. On the other hand, if neither of the primary respondents nor his/her partner is regular type but
either of them is engaged in non-regular type of job, then such a household is classified as ’non-regular
type.’ For the age of the primary respondent, that of the non-regular workers households is higher than
that of the regular workers households. For the marital status and the family size, however, there is not
much difference between the types of households.
As has been studied in the Figure 3.3, the monthly wage of the primary respondent in regular type of
workers household is almost twice as high as that of the primary respondent in non-regular type of workers
111

Type of statistic Regular type of HH Non-regular type of HH
Age of the Primary Respondent Mean 46.90 56.97
Standard Deviation 10.62 13.49
Proportion of Married
Household Mean 83.18% 66.23%
(Standard Deviation) (0.37) (0.47)
Number of Family Members Mean 3.56 3.06
(Standard Deviation) (1.39) (1.39)
Monthly Wage Earning
of Primary Respondent Mean 325 190
(Standard Deviation) (181) (142)
(Within Variation) (73) (75)
Correlation with
Monetary Policy Shock −0.52% 2.41%
Total Household Expenditure Mean 284 239
(Standard Deviation) (167) (152)
Correlation with
Monetary Policy Shock −1.69% −2.24%
Notes: money amount is in units of ten thousand Yen. Unit of observation is a household, which is classified into ’Regualr type
of HH’ if either of the primary respondent or his/her partner is a regular type of worker, and ’Non-regular type’ if neither of
the primary respondents nor his/her partner is regular type but either of them is engaged in non-regular type of job. ’Within
Variation’ means standard deviation of observations across time.
Table 3.2: Descriptive Statistics of KHPS/JHPS Data
household. While this level difference leads to larger standard deviation of the wage of the regular workers,
when seen on an individual household basis over time, this ’within’ variation of the monthly wage is larger
for the non-regular type of workers.
The total household expenditure is close between regular and non-regular workers household. Their
correlation with the monetary policy shock is, however, larger in magnitude for non-regular workers than
for regular workers.
3.2.2 Monetary Policy Shocks (Kubota and M. Shintani (2022))
As our measure of monetary policy shocks, we use data from the Euro-Yen futures market prepared by
Kubota and M. Shintani (2022). They identify monetary policy shocks from a so called high-frequency
method, which is used in many preceding studies (E. Nakamura and Steinsson (2018), F. Nakamura, Sudo,
Sugisaki, et al. (2021)). The idea is that Euro Yen futures are determined in efficient financial markets.
Kubota and M. Shintani (2022) collect these surprises at a 30-minute window following announcements
112

by the BOJ for the 3-month, 6-month, 9-month, and 12-month Euro-Yen futures market. They use the
following equation regressing stock prices and JGB yields on those ’exogenous’ Euro-yen movements:
∆yt = α + β∆xt + εt
The authors decompose these surprises into 2 factors; the "path factor", which affects the expected path of
future short rates, and the "target factor", which mainly affects the current short-term rates.
We choose to use the path factor in our empirical work below given that consumption decisions are
forward looking. Since the frequency of our dataset is annual, we aggregate these shocks into annual
frequencies.
3.2.3 Consumption Response to Monetary Shocks
The main specification we use in our estimation of the impact of monetary shocks on the consumption of
regular and non-regular households is:
∆ log cht =
X
1
k=0
βk ∗ εt−k + α3Nonht +
X
1
k=0
βN on,kNonh,t−k ∗ εt−k
+ α4F ullht +
X
1
k=0
βF ull,kF ullh,t−k ∗ εt−k + αXht + λh + vht
The dependent variable is the total expenditures of the household. The controls X include the age brackets
of the primary respondent in 3 categories (the young if up to 34 years old, the middle if between 35 and 64
years old, and the elderly otherwise), the number of family member, the number of children, the 8 region
block dummies and city size dummies (living in the 21 largest cities, other cities, and towns and villages),
the city and region dummies are meant to take into account the difference in living costs across regions.
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(1) (2) (3) (4) (5)
Effect on Non-reg. HH -0.760 -0.655 -0.760 -0.655 -0.820
(0.23) (0.23) (0.16) (0.16) (0.25)
[0.111] [0.172] [0.058] [0.105] [0.098]
Effect on Reg. HH -0.242 -0.204 -0.242 -0.204 -0.635
(0.14) (0.14) (0.10) (0.10) (0.16)
[0.573] [0.637] [0.507] [0.579] [0.160]
Difference -0.518 -0.451 -0.518 -0.451 -0.185
(0.09) (0.09) (0.06) (0.06) (0.09)
[0.156] [0.219] [0.083] [0.134] [0.615]
Controls No Yes No Yes Yes
HH Random Effect No No Yes Yes No
HH Fixed Effect No No No No Yes
Num. Obs. 56577 56469 56577 56469 56469
Notes: numbers in parenthesis and brackets are standard errors and p-values, respectively.
Coefficients on contemporaneous and lagged effects are combined using delta method.
Data source: JHPS/KHPS
Table 3.3: Broader Definition
(1) (2) (3) (4) (5)
Effect on Non-reg. HH -0.613 -0.480 -0.613 -0.480 -0.105
(0.15) (0.15) (0.10) (0.11) (0.16)
[0.117] [0.222] [0.055] [0.139] [0.795]
Effect on Reg. HH -0.198 -0.150 -0.198 -0.150 -0.069
(0.00) (0.00) (0.00) (0.00) (0.00)
[0.345] [0.476] [0.261] [0.397] [0.748]
Difference -0.416 -0.330 -0.416 -0.330 -0.036
(0.15) (0.15) (0.10) (0.10) (0.16)
[0.347] [0.456] [0.250] [0.364] [0.936]
Controls No Yes No Yes Yes
HH Random Effect No No Yes Yes No
HH Fixed Effect No No No No Yes
Num. Obs. 52465 52357 52465 52357 52357
Notes: numbers in parenthesis and brackets are standard errors and p-values, respectively.
Coefficients on contemporaneous and lagged effects are combined using delta method.
Data source: JHPS/KHPS
Table 3.4: Narrower Definition
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The results are depicted in Table 3.3 (broader definition of regular and non-regular workers) and in
Table 3.4 (narrower definition of regular and non-regular workers). We find in Table 3 that compared
to the baseline category (the unemployed and the retired), an increase in the magnitude of the monetary
policy shock lowers the consumption of both the non-regular and regular types of households. We find that
in all specifications, non-regular households have a greater decline in household total expenditures than do
regular households. This suggests that expansionary monetary policies do raise household expenditures,
but the most affected were non-regular type of households. The differences between households are less
significantly estimated in Table 3.4 with the narrower definition of regular and non-regular workers.
In the model developed in the next section, we will rationalize this finding of the stronger consumption
response of the non-regular type of households to monetary easing.
3.3 Model
Below we develop a Two Agent New Keynesian (TANK) model with households consisting of regular
workers and non-regular workers respectively. The non-regular workers face a perfectly competitive labor
market. The regular workers, on the other hand, are assumed to belong to unions, which monopolistically
set a wage rate.6 There are a continuum of industries and unions associated with each regular worker.
3.3.1 Households
Non-regular workers take the market wage as given and work for firms, while regular workers, who
each belong to unions, sell their labor not to firms, but to an intermediary labor packer. Flow utility,
discounted by β, for both type of households given by (i = R, I):
U

C
i
t
, Li
t

=
C
i
t
1−σ
1 − σ
− ψ
L
i
t
1+χ
1 + χ
6The formulation of labor unions is based on Sims (2020).
11

The budget constraint facing the regular type of households, written in nominal terms, is:
PtC
i
t + B
i
t ≤ Wi
t L
i
t + Rt−1B
i
t−1 + κBi
t
2
+ DIV i
t
The household can save via a one period bond (Bi
t
) with the gross nominal interest rate Rt
, but selling
and buying them incurs an adjustment cost (κBR
t
2
). Wi
t
is the nominal wage which their monopolistically
competitive unions set. Here the regular type of households are assumed to have a higher fraction of
ownership of the firms in the economy (DIV i
t
), getting the bulk of the dividends.
The problem faced by non-regular workers is the same as that by regular workers. The only difference
is that they are paid at Wn
t
, which is the competitive market wage in the non-regular type of labor market.
While the union to which regular workers belong solves the dynamic problem to choose the reset wage,
non-regular type of workers just take the market-determined Wn
as given.
3.3.2 Labor Market Players
Regular workers belong to labor unions, which are monopolistically competitive and allocate homogeneous labor into differentiated labor for each industry. Labor unions collect labor from regular workers,
and sell them to a labor packer. A labor packer then sells the aggregated (regular type) labor to wholesale firms at a competitive price. Note that while a non-regular worker’s wage is perfectly competitive
and flexible, regular workers’ wage is not only rigid but also includes a markup. Wages of regular workers depart from marginal productivity because of the seniority wage system and un-modeled firm level
benefits such as housing and recreational facilities.
116

3.3.2.1 Labor Packer
There are a continuum of industries indexed by l ∈ [0, 1], and associated labor unions. The labor unions
sell regular type of labor to a labor packer at WR
t
(l). The labor packer combines union labor into a final
labor input available to wholesale firms via a CES technology:
L
R
d,t =
Z 1
0
L
R
t
(l)
ǫw−1
ǫw dl ǫw
ǫw−1
Profit maximization yields a demand curve for each union’s labor and an aggregate wage index for regular
workers:
L
R
t
(l) =
Wt(l)
Wt
−ǫw
Ld,t
WR
t
1−ǫw =
Z 1
0
WR
t
(l)
1−ǫw dl
3.3.2.2 Labor Unions
Unions have market power in one of the industries they belong to and set wages. With probability 1−φw,
a union can update its wage. The problem for a union given the opportunity to update is to pick WR
t
(l)
to maximize the present discounted value of real dividends, where discounting is by the household’s real
SDF, Λ
R
t,t+1, and the probability that the price chosen today will be in effect in the future. The solution to
the maximization problem is the price they set, and is called the reset wage. The maximization problem is
shown in more detail in the Appendix.
117

3.3.3 Firms in the Goods Markets
The only input of production is labor, which has two types. There are three layers of firms. Wholesale
firms collect non-regular labor from non-regular workers directly, and regular workers indirectly through
the labor packer. Retail firms purchase wholesale products, and sell them to a final good firm. Retail firms
transform homogeneous wholesale products into differentiated intermediary products, giving retail firms
market power as in the canonical New Keynesian model. A final goods firm buys these differentiated
products from retail firms, and sells the products to regular and non-regular workers households.
3.3.3.1 Final Goods Firm
The final goods firm combines retail outputs into a final good. Retail output is transformed into final good
via:
Yt =
Z 1
0
Yt(f)
ǫp−1
ǫp df
ǫp
ǫp−1
Profit maximization by the final goods firm yields a demand for each retail output and a price index.
Yt(f) =
Pt(f)
Pt
−ǫp
Yt
P
1−ǫp
t =
Z 1
0
Pt(f)
1−ǫp df
118

3.3.3.2 Retail Firms
Retail firms purchase wholesale output at Pw,t and costlessly repackages the output, and sells the output
to a competitive final goods firm at Pt(f), where retailers are indexed by f ∈ [0, 1]. Retailers can only
adjust their price with probability 1 − φp. Their price-setting problems are:
max
Pt(f)
Et
X∞
j=0
φ
j
pΛ
R
t,t+j
n
Pt(f)
1−ǫp P
ǫp−1
t+j
Yt+j − Pw,t+jPt(f)
−ǫp P
ǫp−1
t+j
Yt+j
o
The solution to this problem is the reset inflation rate, which is typical in New Keynesian model with Calvo
(1983)’s staggered prices, and is in the same spirit as the reset wage rate set by labor unions.
3.3.3.3 Wholesale Firm
A representative wholesale firm hires two types of labor: (1)the regular type of labor sold by a labor packer,
and (2)the non-regular type of labor hired through the competitive labor market. The production function
of wholesale output is:
YW,t = AtLd,t = At
h
ω

A
n
t L
n
d,tµ + (1 − ω)

A
R
t L
R
d,tµ
i 1
µ
,
where At
is stochastic TFP to be specified later. An
t
and AR
t
are meant to capture the quality of labor
input for each type, and µ governs the substitutability of the two types of labor. We also assume that labor
demand for regular type of workers is subject to a quadratic adjustment cost of the form:
θ(L
R
d,t − L
R
d,t−1
)
2
.
1

This term is meant to capture the sluggish adjustment in labor demand in the regular type of workers
resulting from difficulties in firing. The resulting profit maximizing problem of the firm is set up as:
max
Ln
d,t,LR
d,t
πW,t = Pw,tYW,t − Wn
t L
n
t − WR
t L
R
t − θPw,t(L
R
d,t − L
R
d,t−1
)
2
.
3.3.4 Stochastic Processes and Aggregation
We assume that the gross nominal interest rate, Rt
, is set according to a Taylor type rule:
ln Rt = (1 − ρR) ln R + ρR ln Rt−1 + (1 − ρR) θπ (ln Πt − ln Π) + sRεR,t (3.1)
Variables without time subscripts, R and Π, denote non-stochastic steady state values.
The only exogenous variable is productivity At
. We assume that it follows an AR(1) in log with nonstochastic mean normalized to unity:
ln At = ρA ln At−1 + sAεA,t
We assume that there is a mass of ω of non-regular workers, and 1−ω of regular workers in the economy.
We assume zero net supply of bonds.
3.4 Calibration
This section details the calibration of the model described in the previous section. Table 3.5 summarizes
the figures and sources for each.
120

Description Value Origin
β discount factor 0.99 standard
χ Frisch elasticity of labor supply 1.83 our estimate
χ
n Frisch elasticity of labor supply 1.10 our estimate
σ
inverse of inter-temporal elasticity
of substitution (CRRA parameter) 1.12 our estimate
εp price elasticity of demand 10.0 Muto and K. Shintani (2017)
εw price elasticity of labor demand 6.23 Muto and K. Shintani (2017)
φp
probability that retail firms fail
to change the price 0.49 Iwasaki, Muto, and M. Shintani (2021)
φw
probability that labor unions fail
to change the wage 0.81 Iwasaki, Muto, and M. Shintani (2021)
µ
production substitutability of regular
and non-regular type of workers 0.75 our estimate
ω proportion of non-egular type of workers 0.37
Labor Force Survey
(Ministry of Internal Affairs
and Communications)
κ adjustment cost of bond holding 10 Inui, Sudou, and Yamada (2017)
Table 3.5: List of Model Parameters and Calibrated Values
While some parameters (β, κ, εp, εw, φp, and φw) are chosen from previous work, we calibrate χ, χ
I
,
σ and µ using our own estimates, as described below. .
3.4.1 Frisch Labor Supply Elasticity: χ
R and χ
n
We define the Frisch labor supply elasticity as the wage elasticity of hours supplied holding consumption
constant. Thus it captures the substitution effects, but not the income effects from the wage change. It is
typically estimated based on the intra-temporal optimality condition of consumption and labor, which is
derived as below in our model:
ψLχ
t = C
−σ
t Wt
.
Taking logarithms, first differencing, and adding unobserved disturbances εt
:
∆ log Lt = const. +
1
χ
∆ log Wt + α∆ log Ct + εt
.
121

One of the challenges for this estimation is that we need to measure the response of labor to predictable
wage changes, not raw wage changes, which may include unpredictable movements in wages. Thus the
literature starting from MaCurdy (1981) uses instruments to obtain predictable wages, examples of which
are (workers’ own or their parents’) education, age, and where they live. In earlier work for the case of
Japan, Kuroda, I. Yamamoto, et al. (2007) use panel data of young women in the Japanese Panel Surveys
of Consumers to estimate the Frisch parameter to be 0.306 for households with regular workers, and 0.305
for non-regular workers.
We use our JHPS/KHPS data to estimate this parameter. One of the advantages of our dataset is that it
not only includes the record of hours worked and wages but also collects the type of jobs they are engaged
in, which allows us to obtain estimates for the regular type of workers and the non-regular type of workers
separately.
(1) (2) (3) (4)
All workers 0.222∗∗∗ 0.731∗∗∗
(9.79) (6.24)
Regular workers 0.148∗∗∗ 0.331
(5.20) (0.88)
Non-regular workers 0.285∗∗∗ 1.330∗∗
(8.54) (2.59)
control yes yes yes yes
IV no no yes yes
p-val. of Regular - Irregular 0.002 0.241
Number of Observations 31824 31824 31824 31824
Notes: for this analysis we treat working husbands and working wives in a given household as distinct observations, and cluster
standard errors at household levels. The coefficient "All workers" are obtained from a regression of pooling both types of workers
and running log hours changes on log wages changes, whereas "Regular workers" and "Non-regular workers" are obtained from
a regression of running log hours changes on interaction of dummy variables indicating regular type and non-regular type and
log wages changes. For regression (2) and (4) the difference of coefficients are tested with lincom (linear combination) in Stata,
and the p-values are shown.
∗
p < 0.05,
∗∗ p < 0.01,
∗∗∗ p < 0.001
Table 3.6: Estimation of Frisch Labor Supply Elasticity
Table 3.6 summarizes the estimation results using education, age, sex, city size dummy (living in (a)
a city holding over 1 million of population, (b) other cities, and (c) towns or villages), and 8 region block
dummy variables as instrumental variables. Columns (1) and (2) give OLS estimates, while (3) and (4)
122

give IV estimates, which are our preferred specifications. The estimate for non-regular workers is higher
than that for regular workers, and that for all workers lies in between. All estimates are higher in IV
cases, suggesting that IV overcomes the downward bias. Although the estimate for regular workers is not
significant, and thus the test for the difference in the estimates between the effects on regular workers
and those on non-regular workers do not reject the null, we take these results as suggestive that the labor
supply of non-regular workers is more elastic than that of regular workers.
3.4.2 Intertemporal Elasticity of Substitution: σ
The Intertemporal elasticity of substitution (IES) is estimated using the Euler equation:
C
−σ
t = βRtEt

π
−1
t+1C
−σ
t+1
,
which is transformed into log differences (growth rates):
∆ log Ct = const. +
1
−σ
(πt + ∆rt) + εt
.
Traditionally this equation is estimated using aggregate time series data, seminal work of which are R. E.
Hall (1988) and Campbell and Mankiw (1989). The challenge here is the endogeneity between the real
interest rate and consumption7
. We overcome this challenge by using an exogenous policy shift as an
instrument and use micro data.
Cashin and Unayama (2016) earlier measured the IES using the consumption tax hike in Japan. This was
a good natural experiment in three respects. Firstly, consumption taxes in Japan are applied to almost all
7
Early work such as R. E. Hall (1988) and Campbell and Mankiw (1989) used lagged variables as IV, but this is pointed out to
be subject to weak instruments problem by Yogo (2004).
123

expenditures8
at the same rate9
. Thus a rise in the consumption tax rate can be regarded as a proportional
change in the price level. Secondly, the tax hike is announced prior to implementation, so that households
can engage in intertemporal optimization. Thirdly, given that interest rates in Japan have been near zero
and stable for the last 20 years, we can attribute the change in real interest rates almost entirely to price
changes.
While Cashin and Unayama (2016) uses the Family Income and Expenditure Survey10 and utilize the
consumption tax hike in 1997, we apply the JHPS/KHPS data to the consumption tax hike in April 2014.
Since our dataset has household expenditure data for only January, we regard data in January 2014 as the
expenditure before the tax hike, and data in January 2015 as the expenditure after the tax hike. How was
the policy announced and anticipated? The policy change was announced in June 2012 and was agreed
to be implemented among the ruling Democratic Party at the time and the other two parties, the Liberal
Democratic Party and Komei Party. The two parties defeated the Democratic Party in the election at the
end of the year, but maintained the earlier agreement; the consumption tax hike from 5% to 8% was
implemented as scheduled. Thus we can regard the period between January 2013 and January 2014 as the
announcement period, when households shifted their expenditures.
Table 3.7 summarizes the estimation results. Each specification uses time differenced demographic
data on the number of children and the number of family members as controls. Column (2) has lower
estimates on the tax hike dummy (dummy taking 1 in 2015) than column (1), the only difference between
the two columns being the inclusion of the announcement dummy which takes 1 only in 2013 and 2014.
The resulting estimate of the IES parameter is 0.067 in column (2), which is a low estimate comparable to
R. E. Hall (1988). Cashin and Unayama (2016) gives 0.21 as the most preferred estimate. The difference
8
Examples of exemption are raised in Cashin and Unayama (2016).
9When rates were 3% (since April 1989), 5% (since April 1997) and 8% (since April 2014), there was almost no heterogeneity
in applied rates. When rates were raised to 10% (since October 2019), some items including food and newspapers were under
reduced rates (8%).
10This is one of the broadest surveys on household consumption and income. While it holds on average 8,000 households and
their detailed information, each household is surveyed for only 6 months, thus this is a so-called rotating panel.
124

(1) (2)
consumption tax hike -0.00316 -0.00201
(-0.46) (-0.29)
announce 0.0110
(1.63)
IES estimate (converted from estimate) 0.105 0.067
(0.230) (0.231)
Number of Observations 54507 54507
Notes: "consumption tax hike" dummy takes 1 only in year 2015. "announce" dummy takes 1 in year 2013. IES estimate is negative
of coefficients on "consumption tax hike" dummy divided by 0.03 (rate increase).
∗
p < 0.05,
∗∗ p < 0.01,
∗∗∗ p < 0.001
Table 3.7: Estimation of Intertemporal Elasticity of Substitution
between their estimate and ours might be an interesting topic in itself, but we proceed with our estimate
of 0.06711
.
3.4.3 Production Substitutability Parameter: µ
In the model we consider labor to be the only factor of production. There are two types of labor. One type
is the regular type L
R
t
, and the other is the non-regular type L
n
t
, and both are imperfect substitutes. The
production function for intermediate goods (wholesale output) is YW,t. The µ governs the substitutability
between the regular type and the non-regular type of workers, and we estimate this based on the "canonical
model" by Katz and Murphy (1992):
ln
w
R
t
wn
t

= µ ln
AR
t
An
t

+ (µ − 1) ln
L
R
t
L
n
t

+ const. + εt
.
This µ − 1 is estimated as −0.71, thus σˆ =
1
1−µˆ = 1.41 as in Katz and Murphy (1992) and is considered to
be robust in the US regarding the substitutability between high-skilled and low-skilled workers.
We employ the specification of Katz and Murphy, and use wage and labor input data from the Monthly
Labor Survey. The survey records data on workers in establishments including wages, hours, numbers
11While our estimate in Table 3.7 is not statistically significant, Cashin and Unayama (2016) also gives results not statistically
significant.
125

and so on, for the regular type of workers and part-time workers. We use monthly wages exclusive of
bonus payments to make seasonality as small as possible. We choose hours worked in a given month for
variables L
R
t
and L
n
t
. Note that part-time workers are a narrower definition than the non-regular type of
workers, which includes fixed-term workers.
As an IV to instrument for the ratio of labor inputs, we employ a policy change in Japan on the regulation of the Temporary Staffing Services Law. Originally introduced in 1985, the law allowed several
industries to use temporary workers, who are a type of non-regular workers. The 1999 amendment to
the Law substantially expanded the range of industries in which the law is applicable12. Moreover, in the
2004 amendment, temporary workers were allowed to be used in the manufacturing industry, the second
largest industry in terms employees. These exogenous policy changes should have affected the ratio of
regular workers to non-regular workers in most industries.
(1) (2) (3) (4)
average all industry all industry all industry
L
R/Ln
-0.165∗∗∗ 0.595∗∗∗ 0.922∗∗∗ -0.735∗∗
(-3.75) (54.50) (64.48) (-2.92)
AR/An
-0.00000713 -0.00441∗∗∗ -0.00475∗∗∗ -0.00358∗∗∗
(-0.03) (-16.58) (-31.01) (-9.79)
IV no no no yes
Number of Observations 352 4024 4024 4024
Notes: L
R/Ln
is the log of the ratio of hours worked by regular type of workers divided by those by non-regular type of workers.
A
R/An
is treated as a yearly time trend. Each regression includes full set of month dummies to take care of seasonality. Specification "average" means the average across industry is used for regression. "all industry", on the other hand, include all industry
averages into the estimation and form a panel.
∗
p < 0.05,
∗∗ p < 0.01,
∗∗∗ p < 0.001
Table 3.8: Estimation of Substitutability of Regular and Non-regular Labor
The estimation result is shown in Table 3.8. Column (1) is a time series regression of the total industry
average, and gives the estimate of −0.165, which corresponds to σˆ = 6.06. While Katz and Murphy (1992)
use a time series regression, we can increase the size of the sample by using industry level data. However,
12While the law used a positive list before the 1999 amendment, which specified types of professional jobs and industries in
which temporary staffing service is allowed, including interpreters and secretary services, the 1999 amendment used a negative
list, which specified the types of jobs and industries in which temporary staff servicing are not allowed, including manufacturing
and medical industries.
126

Description Value (% of output)
Y output 1.02 100%
C aggregate consumption 0.74 68%
C
R consumption of regular type 0.38 35%
C
n
consumption of non-regular type 0.35 32%
L
R labor supply of regular type 0.62 -
L
n
labor supply of non-regular type 0.51 -
W aggregate wage rate 0.34 -
WR wage rate of regular type 0.40 -
Wn wage rate of non-regular type 0.23 -
π price inflation 0.00 -
div dividened 0.71 65%
G government expenditure 0.26 26%
Table 3.9: Steady State of the Quantitative Model
pooled OLS in column (2) and panel OLS in column (3) give positive estimates, which are difficult to
interpret. Column (4) is the result utilizing IVs, and again gives a negative estimate of −0.735. This
corresponds to σˆ = 1.36, and happens to be very close to Katz and Murphy (1992)’s estimate on the
substitution elasticity between high-skilled and low-skilled workers.
3.5 Model Simulation
We first show the steady state of the model introduced in section 3.3, and the parameters calibrated in
section 3.4. Then we show the impulse responses of the model in our monetary easing exercise. Finally,
we compute and compare the welfare losses from the business cycles from the model, by varying the
composition of the regular and non-regular workers.
3.5.1 Steady State
Table 3.9 summarizes the steady state of the variables in the model. Our model replicates well the wage
difference between the regular type of workers and the non-regular type of workers, which we observe in
Section 3.1. The feature of the model which contributes to this is exactly the duality of the labor market; a
127

competitive non-regular type labor market with flexible wage adjustment, and an unionized regular type
labor market where the wage includes a markup and therefore is higher than the marginal product of the
worker.
Since the regular type of worker exerts monopolistic power on the wage setting in their industry, the
regular worker secures higher wages than non-regular workers. This wage difference is reflected in the
consumption levels for each type of worker.
Government expenditures are set at 26% of GDP, as a result of taxes on labor and capital income. Note
that government expenditures are endogeneized in this model through flat-rate income taxation. This
share in GDP of government expenditures is comparable to the fraction of actual Japanese government
outlays in recent years.
3.5.2 Benchmark Results
Figure 3.5 depicts the impulse responses to expansionary monetary policies. We impose a 1% monetary
shock according to the Taylor rule (Equation (3.1) in our model). First, the consumption response of the
non-regular type of workers is higher than that of the regular type of workers. This agrees with what we
establish in Section 3.2 with the household panel data. Since the majority of the workers in the economy,
however, are the regular type of workers, the resulting aggregate consumption response is mitigated. In
fact, the real wage of regular workers goes down upon the impact of the monetary easing because the wage
is not only slowly responding but also not indexed to inflation. Second, the labor supply of the non-regular
workers fluctuates while that of the regular type of workers moves smoothly. The monetary easing in this
exercise is a temporary one. Also, the regular type of labor input is costly to adjust. Firms adjust the labor
demand for the regular type of workers smoothly and adjust the labor demand for the non-regular type of
workers from time to time; on impact of monetary easing it is raised to meet the increase in demand for
128

final goods, but then it is reduced quickly since the regular type of workers’ labor is adjusted slowly and
continued to be used more.
0 10 20 30
-0.4
-0.2
0
R
0 10 20 30
0
0.5
1
Output
0 10 20 30
0
0.1
0.2
0.3
Government Expenditure
0 10 20 30
0
0.5
1
Inflation
0 10 20 30
0
0.5
1
Aggregate Consumption
0 10 20 30
0
0.5
1
R. Consumption
0 10 20 30
0
0.5
1
N. Consumption
0 10 20 30
0
1
2
R. Labor
0 10 20 30
-0.5
0
0.5
N. Labor
0 10 20 30
0
0.1
0.2
Aggregate Wage
0 10 20 30
-0.5
0
0.5
R. Wage
0 10 20 30
0
0.5
1
N. Wage
Impulse Responses to Monetary Easing Shock (lowering by 1.00%)
Figure 3.5: Impulse Response to a Monetary Easing Shock
Notes: Numbers in the vertical axis are percentage deviations from steady state values.
129

3.5.3 Experiments
In our first experiment, we ask the question; had it not been for an expansion in the use of non-regular
workers, what would be the consumption response to the monetary policy shocks? We study the impulse
responses with differing compositions of regular workers and non-regular workers. This corresponds to
a comparison of consumption responses to the monetary policy shocks in the 1980-1990 period, when the
use of non-regular workers was not so prevalent (ω = 0.20), with the consumption responses in 2017,
when the use of non-regular workers was more prevalent (ω = 0.37). Specifically, we regard the impulse
responses in Figure 3.5 with ω = 0.37 to be the benchmark, and we newly compute alternative impulse
responses with ω = 0.20. In Figure 3.6 the former is depicted in solid lines, and the latter in dotted lines.
We observe that the reaction of aggregate consumption to monetary easing is lower in its peak of 1.5%
when ω = 0.20 than that of 1% when ω = 0.37.
When non-regular worker households are more prevalent (high ω), firms can adjust their labor inputs
more easily by the use of non-regular workers. This makes the wages of non-regular workers respond
more to the monetary easing. The response of the wages of regular workers, however, does not differ
much from the wages in the low ω case. Since the adjustment of the regular workers’ wage is slow due
to the limited chance of wage revision, it cannot move differently at the beginning of the adjustment.
Since this wage response of non-regular workers has more influence on the aggregate wage response in
the case with a large fraction of non-regular workers, the aggregate wage response is higher on impact of
the monetary easing but weaker thereafter. At the same time, the response of the labor supply of regular
workers household is weakened when the fraction of non-regular workers is large because firms can use the
non-regular workers more and reduce the adjustment of the labor of the regular workers, which entails an
adjustment cost. However, given higher wages and the larger composition of the regular workers (3.2 and
3.3), the higher weight of regular worker wages in aggregate wages dominates and dampens the aggregate
wage and consumption response to monetary shocks.
130

0 10 20 30
-0.6
-0.4
-0.2
0
R
0 10 20 30
0
0.5
1
1.5
2
Output
0 10 20 30
-0.2
0
0.2
0.4
0.6
Government Expenditure
0 10 20 30
0
0.5
1
1.5
2
Inflation
0 10 20 30
-0.5
0
0.5
1
1.5
2
Aggregate Consumption
0 10 20 30
-0.5
0
0.5
1
1.5
2
R. Consumption
0 10 20 30
-1
0
1
2
3
N. Consumption
0 10 20 30
0
0.5
1
1.5
2
R. Labor
0 10 20 30
-1
0
1
2
N. Labor
0 10 20 30
-0.4
-0.2
0
0.2
0.4
0.6
Aggregate Wage
0 10 20 30
-1
-0.5
0
0.5
R. Wage
0 10 20 30
-0.5
0
0.5
1
1.5
N. Wage
smaller fraction of non-regular workers ( =0.20) larger fraction of non-regular workers ( =0.38)
Impulse Responses to Monetary Easing Shock (lowering by 1.00%)
Figure 3.6: Impulse Response to a Monetary Easing Shock with Varying Fraction of Non-regular Wokers
Notes: Numbers in the vertical axis are percentage deviations from steady state values.
In our second experiment, we vary the relative productivity of non-regular workers. We take the
impulse responses in Figure 3.5 with An = 1, no difference in the productivity of labor between regular
and non-regular workers to be the benchmark. As our experiment, we take An = 0.5, that non-regular
workers have a lower productivity (or can be regarded as quality or human capital) of labor than regular
workers. In Figure 3.7 the former is depicted in dotted lines, and the latter in solid lines.
We observe that, compared with the baseline (dotted line) where non-regular workers have the same
productivity (AR = An = 1), aggregate consumption reacts less strongly to monetary easing when the
productivity of non-regular workers is lower AR > An = 0.5 (solid line). Because of the CES production
function, a lower An has two effects; it decreases both the share of the non-regular workers and their productivity. This lowers aggregate productivity and the responses of aggregate wages and consumption to
positive monetary shocks. The decreased share of the non-regular labor input will be An µω
An µω+1−ω
, while the
131

associated overall productivity will be A [ An µω + 1]1/µ, which will be lower than the aggregate productivity A if Anµ is lower than the benchmark. This decline in aggregate productivity lowers the marginal
productivities of both regular and non-regular workers, which in turn lowers the wage response to the
monetary easing of both types of workers.
0 10 20 30
-0.6
-0.4
-0.2
0
R
0 10 20 30
-0.5
0
0.5
1
1.5
Output
0 10 20 30
-0.1
0
0.1
0.2
0.3
Government Expenditure
0 10 20 30
0
0.5
1
1.5
2
Inflation
0 10 20 30
0
0.2
0.4
0.6
0.8
1
Aggregate Consumption
0 10 20 30
0
0.2
0.4
0.6
0.8
1
R. Consumption
0 10 20 30
-1
0
1
2
3
N. Consumption
0 10 20 30
-0.5
0
0.5
1
1.5
2
R. Labor
0 10 20 30
-1
-0.5
0
0.5
1
N. Labor
0 10 20 30
-0.1
0
0.1
0.2
0.3
0.4
Aggregate Wage
0 10 20 30
-1
-0.5
0
0.5
R. Wage
0 10 20 30
-0.5
0
0.5
1
1.5
N. Wage
no difference in quality of labor (A n
=1) quality of labor is lower among non-regular workers (A n
=0.75)
Impulse Responses to Monetary Easing Shock (lowering by 1.00%)
Figure 3.7: Impulse Response to a Monetary Easing Shock with Varying Quality of Non-regular Workers
Notes: Numbers in the vertical axis are percentage deviations from steady state values.
In our third experiment, we allow differing compositions of regular workers to non-regular workers
just like in the first experiment, but this time we let the quality of labor of the non-regular type to be lower
(An = 0.5) than that of the regular type (AR = 1). In Figure 3.8 the case with a lower proportion of the
non-regular type of workers is depicted in dotted lines, and the case with a higher proportion is depicted
in solid lines. We observe that the reaction of aggregate consumption to monetary easing continues to be
lower in its peak from about 1.5% when ω = 0.20 to about 1% when ω = 0.37, as in Figure 3.6. We can
see, however, that all the responses in this low quality of labor for non-regular type case have a smaller
132

0 10 20 30
-0.8
-0.6
-0.4
-0.2
0
0.2
R
0 10 20 30
0
0.5
1
1.5
2
Output
0 10 20 30
-0.2
0
0.2
0.4
0.6
Government Expenditure
0 10 20 30
0
0.5
1
1.5
Inflation
0 10 20 30
-0.5
0
0.5
1
1.5
Aggregate Consumption
0 10 20 30
-0.5
0
0.5
1
1.5
2
R. Consumption
0 10 20 30
-0.5
0
0.5
1
1.5
2
N. Consumption
0 10 20 30
0
0.5
1
1.5
2
R. Labor
0 10 20 30
-2
-1
0
1
2
3
N. Labor
0 10 20 30
-0.2
0
0.2
0.4
Aggregate Wage
0 10 20 30
-0.5
0
0.5
R. Wage
0 10 20 30
-0.2
0
0.2
0.4
0.6
0.8
N. Wage
smaller fraction of non-regular workers ( =0.20) larger fraction of non-regular workers ( =0.38)
Impulse Responses to Monetary Easing Shock (lowering by 1.00%)
Figure 3.8: Impulse Response to a Monetary Easing Shock with Varying Fraction of Non-regular Wokers
Notes: Numbers in the vertical axis are percentage deviations from steady state values.
magnitude than that in the same quality of labor case. As explained in the second experiment, a lower
quality of labor effectively lowers the composition of that type of labor as a production input and also
lowers overall productivity even in the baseline case. This lowered overall productivity in the baseline
case is responsible for the smaller magnitude of responses in Figure 3.8 than those in Figure 3.6.
In our fourth experiment, we study impulse responses with varying adjustment costs of labor of the
regular type of workers (θ). We take θ = 2.0 to be the benchmark (as in Figure 3.5). We compare the
benchmark with θ = 5.0,which means that adjusting regular labor has becomes more expensive. We
observe that compared with the baseline (dotted line), the case with higher adjustment costs yields a weaker
response of aggregate consumption to the monetary easing shock. Institutionally, the protection of the
regular type of workers has increased in Japan over time. For example, starting in the early 2000s, the
rights of employers to dismiss regular workers has been enshrined by law, not only by case law. Later
133

0 10 20 30
-0.4
-0.3
-0.2
-0.1
0
R
0 10 20 30
-0.5
0
0.5
1
1.5
Output
0 10 20 30
-0.1
0
0.1
0.2
0.3
Government Expenditure
0 10 20 30
0
0.5
1
1.5
2
Inflation
0 10 20 30
0
0.2
0.4
0.6
0.8
1
Aggregate Consumption
0 10 20 30
-1.5
-1
-0.5
0
0.5
1
R. Consumption
0 10 20 30
-1
0
1
2
3
4
N. Consumption
0 10 20 30
-0.5
0
0.5
1
1.5
2
R. Labor
0 10 20 30
-0.4
-0.2
0
0.2
0.4
0.6
N. Labor
0 10 20 30
-0.1
0
0.1
0.2
0.3
0.4
Aggregate Wage
0 10 20 30
-1.5
-1
-0.5
0
0.5
1
R. Wage
0 10 20 30
-0.5
0
0.5
1
1.5
2
N. Wage
baseline adjustment cost of regular type of workers ( =0.90) higher adjustment cost of regular type of workers ( =1.3)
Impulse Responses to Monetary Easing Shock (lowering by 1.00%)
Figure 3.9: Impulse Response to a Monetary Easing Shock with Varying Adjustment Cost of Regular Workers
Notes: Numbers in the vertical axis are percentage deviations from steady state values.
these rights were written in one of the labor laws so that the regular type of workers’ rights was properly
secured.
In Figure 3.9 the benchmark is depicted in dotted lines, and the result with θ = 5.0 in solid lines. We
observe that compared with the baseline (dotted line), the case with higher adjustment costs (solid line)
yields a weaker response of aggregate consumption to monetary easing shock. While the labor supply and
wage of the non-regular type of workers are essentially unchanged, the rise in the labor of regular workers
is suppressed due to the high adjustment costs imposed on the wholesale firm.
The response in the wages of the regular type of workers is slightly more persistent with higher adjustment costs. The slower adjustment of the regular type of workers’ wage and labor also corresponds
to a longer expansion of consumption of a regular type of workers (after plunging on impact). Thus, their
consumption does not rise so much and this drags down aggregate consumption.
134

In our last experiment, we study the impulse responses with varying wage reset probabilities of the
regular workers φw. We take the impulse responses in Figure 3.5 with φw = 0.81 to be the benchmark.
We compare the benchmark with alternative impulse responses with φw = 0.50, implying that the labor
unions have more opportunities to reset the wage rate. Institutionally, at the start of the Abe administration, the government emphasized wage increases as a key to overcoming deflation, and the government
moved aggressively to push the business commuity to raise wages and to introduce tax At the start of the
Abenomics, the government emphasized wage increases as a key to overcoming deflation, and the government moved aggressively to ask the business community to raise wages and to introduce tax breaks to
encourage this. Notably, the Prime Minister repeatedly called for a 3% wage increase to business community.
In Figure 3.10 the former is depicted in dotted lines, and the latter in solid lines. We observe that the
wage rate of the regular type of workers in fact responds more to a monetary easing shock, which raises
the aggregate wage response as well. The response of aggregate consumption, however, is weakened.
While the wage rate of the regular type of workers is raised more in response to the monetary stimulus,
the wholesale firm reduces the labor demand compared with the benchmark because now the regular type
of workers are more costly to hire. This leads to a reduced response of the labor income of the regular type
of workers despite the upward revision of their wage rate. Hiring of non-regular labor does not respond
sufficiently to raise the total labor rsponse To sum up, while a more flexible adjustment of the regular
worker’s wage seems to lead to expansion of labor income and consumption, it in fact leads to a lower
aggregate consumption response to an expansionary monetary policy shock.
135

0 10 20 30
-0.4
-0.3
-0.2
-0.1
0
R
0 10 20 30
-0.5
0
0.5
1
1.5
Output
0 10 20 30
-0.1
0
0.1
0.2
0.3
Government Expenditure
0 10 20 30
0
0.5
1
1.5
2
Inflation
0 10 20 30
0
0.2
0.4
0.6
0.8
1
Aggregate Consumption
0 10 20 30
-0.5
0
0.5
1
R. Consumption
0 10 20 30
-1
0
1
2
3
N. Consumption
0 10 20 30
-0.5
0
0.5
1
1.5
2
R. Labor
0 10 20 30
-0.5
0
0.5
N. Labor
0 10 20 30
0
0.2
0.4
0.6
0.8
Aggregate Wage
0 10 20 30
-1
-0.5
0
0.5
1
R. Wage
0 10 20 30
-0.5
0
0.5
1
1.5
N. Wage
baseline wage updating probability of regular type of workers (
w
=0.81) higher probability of regular type of workers (
w
=0.5)
Impulse Responses to Monetary Easing Shock (lowering by 1.00%)
Figure 3.10: Impulse Response to a Monetary Easing Shock with Varying Wage Stickiness of Regular
Worker
Notes: Numbers in the vertical axis are percentage deviations from steady state values.
136

Regular HH Non-regular HH
ω 0.20 0.38 0.20 0.38
Φ 0.22 0.14 0.65 0.71
Table 3.10: Welfare Loss Depending on the Composition of Workers ω
3.5.4 Welfare Implications
We finally consider the welfare of each type of worker when they are hit by monetary policy shocks. We
conduct second order Taylor expansions to the utility functions of each type of worker, and compute a
certainty consumption equivalence. Applying second order Taylor expansions to U

C
i
t
, Li
t

and taking
the expectation operator, we compare this with the steady state utility from certainty equivalence Φ
i
(which we define as a ’welfare loss’):
U

(1 − Φ
i
)C
i
, Li

= E(U

C
i
t
, Li
t

) ≃ U

C
i
, Li

−
1
2
h
σCi1−σV ar(Cˆi
t
) + χ
iL
i1+χ
i
V ar(Lˆi
t
)
i
,
where variables with hats Cˆi
t
,Lˆi
t means percentage deviation from the steady state values. This Φ
i measures the welfare loss due to the business cycle in second order; as a household is faced with higher volatility in their quantities, they are less happy compared with non-stochastic stationary welfare U. In Table
3.10, we calculate these welfare losses for each type of household undertwo different parameters; ω = 0.20,
which corresponds to the proportion of the non-regular type of households in the 1990s, and ω = 0.37,
which corresponds to the current proportion. We observe, comparing two ωs, that while the regular type
of households are enjoying lower Φ (0.22 → 0.14) as the proportion of non-regular workers widen, nonregular workers are worse off (0.65 → 0.71) as ω gets higher. This is because firms are better able to
adjust the amount of labor depending on the business cycle since they do not have to rely so much on the
costly-adjustment regular type of workers. All the fluctuations in the adjustment of the labor supply go to
non-regular workers, and this is why non-regular workers are worse off when ω increases.

3.6 Conclusion
The weak response of aggregate consumption to expansionary monetary policy shocks during the recent
period has been a puzzle. Starting from our observation based on Japanese household panel data that the
consumption response of non-regular worker households is stronger than that of regular worker households, we construct a New Keynesian model with dual labor markets. Our model has two types of workers,
one unionized and monopolistically competitive (regular workers), and the other in a perfectly competitive market (non-regular workers). We calibrate our model with Japanese household level panel data. We
find that our model well replicates the weak rise of aggregate consumption to monetary policy easing, and
the widening use of non-regular type of workers is aligned with the weakening of consumption response.
Moreover, since non-regular workers are more affected by business cycles, the increased proportion of
non-regular type of jobs in the economy has led to a reduction in welfare.
While our two agent model replicates an important fact that the consumption response to monetary
policy shocks is larger for non-regular workers than for regular workers, there are several possible extentions. First, we do not consider the extensive margin of labor supply but incorporate only the intensive
margin. The composition of the two types of labor is treated as exogenous. Monetary policy, however,
could affect this composition. This consideration of the extensive margin naturally extends the household’s
problem to the choice of which labor markets to enter. For this decision, human capital accumulation or
the distribution of skills across agents are among the features that need to be added. Moreover, since our
model does not have capital accumulation just like the textbook New Keynesian model, the movement of
income and consumption is tough to decouple. Although our model tries at least to make a level difference
by introducing income taxes, the introduction of saving for some of the agents, as in Kaplan, Moll, and
Violante (2018), would enrich the agents’ response to monetary policy shocks.
138

C.1 Detail on Data
C.1.1 Data on Labor Force
The main data source is Labor Force Survey (LFS) by Ministry of Internal Affairs and Communication
(MIC). This is a monthly survey on status of employment and unemployment for randomly selected households. While data on number of employed workers is available on monthly basis since 1953, what comes
in each employment status (regular workers and non-regular workers) dates back to 1984. Seasonally
adjusted data is, however, available only from 2013. Below is the classification rule listed in the website:13
Type of employment
Employees, excluding Executive of company or corporation are classified into seven categories of "Regular employee", "Part-time worker", "Arbeit (temporary worker)", "Dispatched worker from temporary labour agency",
"Contract employee", "Entrusted employee" and "Other" according to how they are called at their workplaces.
Six categories except "Regular employee" are classified into "Non-regular employee".
Another source is Monthly Labor Survey (MLS) by Ministry of Health, Labor and Welfare (MHLW).
This is a monthly survey for establishments and collects records of number of days and hours worked,
number of workers, and wages paid. Number of workers data come in two broad category of ’regular
workers’ and ’part-time workers’. We cite the definition of each as below from the website:14
Number of Regular Employees : Regular Employees are workers who satisfy one of the following conditions:
(1) persons hired for an indefinite period or for longer than one month
(2) persons hired by the day or for less than one month and who were hired for 18 days or more in each of the
two preceding months
Note : If the board-directors of corporations satisfy above mentioned condition, work regularly and are paid
13Page 3 of https://www.stat.go.jp/english/data/roudou/pdf/definite.pdf
14Page 4 of https://www.mhlw.go.jp/english/database/db-slms/dl/slms-01.pdf
139

a salary based on the same salary rules as normal workers, they are regarded as regular workers. If family
members of the owner of a business satisfy above mentioned condition, work regularly and are paid a salary
based on the same salary rules as normal workers, they are regarded as regular employees.
Definition of Part-time workers
Part-time workers are the persons who satisfy either of the following : a.whose scheduled working hours per
day is shorter than ordinary workers, b.whose scheduled working hours per day is the same as ordinary workers, but whose number of scheduled working days per week is fewer than ordinary workers.
Full-time employees are regular employees who are not a part-time worker.
Careful consideration is needed regarding some of the employment status categories. The notable one
is ’part-time.’ While LFS classifies a worker into ’part time’ just because workers are referred to as "parttime" or "part-san" in the work place, MLS classifies a worker as ’part-time’ if the scheduled work hours
or scheduled number of work days are shorter than normal workers (roughly corresponding to regular
workers).
C.1.2 Wage Data
MLS collects various scopes of wages.
Contractual cash earnings : earnings paid according to method and conditions previously determined by
labor contracts, collective agreements, or wage regulations of establishments.
Scheduled cash earnings : contractual cash earnings other than non-scheduled cash earnings.
Non-scheduled cash earnings are the wages paid for work performed outside scheduled working hours, and on
days off or night work, that is allowances for working outside work hours, night work, early morning work,
and overnight duty.
140

Special cash earnings : amount actually paid to the employee during the survey period for temporary or unforeseen reasons not based on any previous agreement, contract, or rule. Also included in this category are
retroactive payment of wages as a result of a new agreement, and payments such as summer and year end
bonuses which, though terms and amounts are fixed by collective agreements, are calculated over a period
exceeding three months, and such as allowances (e. g., marriage allowance) paid with respect to unforeseen
events.
Total cash earnings : total for contractual cash earnings and special cash earnings?
As explained in the previous section, definition of ’part-time’ workers in MLS may not be appealing
to every researcher. Wage data for more minute subsets of workers can be obtained in Basic Survey of
Wage Structure (BSWS) by MHLW. The survey is conducted annually and survey the result in June. We
should note, however, that the current classification of regular and non-regular workers (each come both
in normal and shorter hours) starts from 2005 only.
C.1.3 Labor Data
MLS reports working hours of several definition.
Scheduled hours worked : actual number of hours worked between starting and ending hours of employment determined by the work regulations of the establishment.
Non-scheduled hours worked : actual number of hours worked (ex. early morning work, overtime work, or
work on a day off ).
Total hours worked : total for scheduled hours worked and non-scheduled hours worked.
141

C.2 Detail of the Model
C.2.1 Household
C.2.1.1 Regular Workers
Flow utility is given by:
U

C
R
t
, LR
t

=
C
R
t
1−σ
1 − σ
− ψ
L
R
t
1+χ
R
1 + χR
Flow utility is discounted by β. The budget constraint facing the household, written in nominal terms, is:
PtC
R
t + B
R
t ≤ WR
t L
R
t + Rt−1B
R
t−1 − κBR
t
2
+ DIVt
(2)
The household can save via a one period bond (BR
t
) with gross nominal interest rate Rt
, but selling and
buying them incurs costs (κBR
t
2
). WR
t
is the nominal wage earned by regular type of workers. Here
regular type of households are assumed to have full ownership of the firms in the economy (DIVt
), but
this is easily relaxed and some of the share can just be passed to non-regular workers.
While regular type of households are homogeneous and supply the same amount of labour, they supply
labour to continuum of monopolistically competitive markets, where wage is set by the unions, which are
not an independent decision maker but belong to households. Unions sell the labour to wholesale firm (to
be defined later), and the industry j is faced with a labor demand function of:
L
j
t =

W
j
t
Wt
!−εw
L
R
d,t.
Households make a decision not just on consumption C
R
t
and hours L
R
t
, but also, as a labor union, on
the wage W
j
t where j ∈ [0, 1]. However, they cannot set the wages every period: they have a probability
φw with which they are stacked with current prices.
14

The Lagrangian of the household looks like as follows:
L =E0
X∞
t=0
β
t
(
C
R
t
1−σ
1 − σ
− ψ
L
R
t
1+χ
R
1 + χR
+
λtwt
µ˜t
"
ht − h
d
t
Z 1
0

w
i
t
wt
−η˜
di#
+λt
"
Ld,t Z 1
0
Wi
t

Wi
t
Wt
−εw
di − PtC
R
t − B
R
t + Rt−1B
R
t−1 − κBR
t
2
+ DIVt
#)
The first order conditions with respect to C
R
t
, LR
t
, and BR
t
are:
C
R
t
−σ = µtPt
ψLR
t
χ
R
= µtMRSt
µt(1 + 2κBR
t
) = βRtEtµt+1
Re-written in real terms, where Πt = Pt/Pt−1, we have:
ψLR
t
χ
R
= C
R
t
−σmrst
(3)
1 + 2κBR
t = RtEtΛ
R
t,t+1Π
−1
t+1 (4)
Λ
R
t,t+1 = β

C
R
t+1
CR
t
!−σ
(5)
where Λ
R
t,t+1 is the real stochastic discount factor.
Now let’s study the determination of w
R
t
(the real wage, w
R
t = WR
t
/Pt
). The first observation is that
while wage is set so as to maximise the discounted flow of profit if possible; otherwise it is just preserved
and the value is exposed to the inflation:
w
i
t =



w˜t
if w
i
t
is set optimally in t
w
i
t−1
/πt otherwise
143

Next, we determine the optimal reset wage w˜t
. The determination of w˜t takes into account not just the
period t but also the further period than t+ 1 in which households are stacked with the period determined
at period t. So we can write wt+s=w˜t/(Πs
k=1πt+k) in this calculation. Below is the Lagrangean set up
before in which terms relevant to the determination of w˜t
is extracted:
L
w = Et
X∞
s=0
(φwβ)
sλt+s


Qs
k=1
1
πt+k

wt+s


−η˜
Ld,t+s
"
w˜
1−εw
t
Ys
k=1

1
πt+k

− w˜
−εw
t
wt+s
µ˜t+s
#
.
the FOC is:
Et
X∞
s=0
(˜αβ)
sλt+s


w˜t
Qs
k=1
µz∗ πt+k−1
πt+k

wt+s


−η˜
h
d
t+s
"
(˜η − 1)
η˜
w˜t
Ys
k=1

µz
∗πt+k−1
πt+k

−
−Uh(t + s)
λt+s
#
= 0
Let’s define the following f
1
t
and f
2
t
:
f
1
t =

η˜ − 1
η˜

w˜tEt
X∞
s=0
(βα˜)
sλt+s

wt+s
w˜t
η˜
h
d
t+s
Ys
k=1

πt+k
µz
∗ πt+k−1
η˜−1
f
2
t = −w˜
−η˜
t Et
X∞
s=0
(βα˜)
sw
η˜
t+sh
d
t+sUh (ct+s − bct+s−1, ht+s)
Ys
k=1

πt+k
µz
∗ πt+k−1
η˜
These can be rewritten in recursive forms:
f
1
t =

η˜ − 1
η˜

w˜tλt

wt
w˜t
η˜
h
d
t + ˜αβEt

πt+1
µz
∗ πt
η˜−1
w˜t+1
w˜t
η˜−1
f
1
t+1,
f
2
t = −Uh (ct − bct−1, ht)

w˜t
wt
−η˜
h
d
t + ˜αβEt

w˜t+1πt+1
µz
∗w˜tπt
η˜
f
2
t+1.
144

The FOCs correspond to the following holds for each period:
f
1
t = f
2
t
C.2.1.2 Non-regular Workers
Basically the problem faced by non-regular workers is the same asthat by regular workers. The flow utility
is
U (C
n
t
, Ln
t
) = C
n
t
1−σ
1 − σ
− ψ
L
n
t
1+χ
n
1 + χn
The budget constraint facing the household, written in nominal terms, is:
PtC
n
t + B
n
t ≤ Wn
t L
n
t + Rt−1B
n
t−1 − κBn
t
2
. (6)
Note that they are paid at Wn
t
, which is the competitive market wage in non-regular labor market. The
optimality conditions are listed below:
ψLn
t
χ
n
= C
n
t
−σWn
t
(7)
1 + 2κBn
t = RtEtΛ
n
t,t+1Π
−1
t+1 (8)
Λ
n
t,t+1 = β

C
n
t+1
Cn
t
−σ
(9)
C.2.2 Labor Market Players
We need labor packers to combine the differentiated labor into a single labor which can be sold to the
wholesale firm.
145

C.2.2.1 Labor Packer
There are a continuum of labor unions indexed by l ∈ [0, 1]. They hire regular type of labor from the
household at MRSt and sell to a labor packer at WR
t
(l). The labor packer combines union labor into a
final labor input available to wholesale firms via a CES technology. In particular:
L
R
d,t =
Z 1
0
L
R
t
(l)
ǫw−1
ǫw dl ǫw
ǫw−1
Profit maximization yields a demand curve for each union’s labor and an aggregate wage index of regular
workers:
L
R
t
(l) =
Wt(l)
Wt
−ǫw
Ld,t
WR
t
1−ǫw =
Z 1
0
WR
t
(l)
1−ǫw dl
C.2.3 Firms in Goods Markets
C.2.3.1 Final Goods Firm
The final goods firm combines retail output into a final output good. Retail output is transformed into final
output via:
Yt =
Z 1
0
Yt(f)
ǫp−1
ǫp df
ǫp
ǫp−1
Profit maximization by the final goods firm yields a demand for each retail output and a price index.
Yt(f) =
Pt(f)
Pt
−ǫp
Yt
P
1−ǫp
t =
Z 1
0
Pt(f)
1−ǫp df
146

C.2.3.2 Retail Firms
The retail firms purchase wholesale output at Pw,t costlessly repackage it, and sell it to a competitive final
goods firm at Pt(f), where retailers are indexed by f ∈ [0, 1]. Their nominal dividend is:
DIVr,t(f) = Pt(f)Yt(f) − Pw,tYt(f)
Using the demand function, this is:
DIVr,t(f) = Pt(f)
1−ǫp P
ǫp
t Yt − Pw,tPt(f)
−ǫp P
ǫp
t Yt
Or, in real terms:
divr,t(f) = Pt(f)
1−ǫp P
ǫp−1
t Yt − Pw,tPt(f)
−ǫp P
ǫp−1
t Yt
Retailers can only adjust their price with probability 1 − φp. This makes their price-setting problem dynamic, where future real dividends are discounted by the household’s stochastic discount factor as well as
the probability that a price chosen in period t remains in effect in the future. The price-setting problem is:
max
Pt(f)
Et
X∞
j=0
φ
j
pΛ
R
t,t+j
n
Pt(f)
1−ǫp P
ǫp−1
t+j
Yt+j − Pw,t+jPt(f)
−ǫp P
ǫp−1
t+j
Yt+j
o
Calculating this, we obtain the following reset inflation rate:
Π
#
t =
ǫp
ǫp − 1
x1,t
x2,t
(10)
x1,t = pw,tYt + φpEtΛt,t+1Π
ǫp
t+1x1,t+1 (11)
x2,t = Yt + φpEtΛt,t+1Π
ǫp−1
t+1 x2,t+1 (12)
147

C.2.3.3 Wholesale Firm
A representative wholesale firm hires two types of labor: (1) regular type of labor sold by a labor packer,
and (2) non-regular type of labor which is directory hired through competitive labor markets at marginal
cost. Production function of wholesale output is:
YW,t = AtLd,t = At
h
ωLn
d,t
µ + (1 − ω)L
R
d,t
µ
i 1
µ
, (13)
where At
is stochastic TFP to be specified later. µ governs the substitutability of two types of labor; 1
1−µ
is the elasticity of substitution. Cost minimization problem is:
min
Ln
d,t,LR
d,t
Wn
t L
n
d,t + WR
t L
R
d,t + θPw,t(L
R
d,t − L
R
d,t−1
)
2
,
s.t. YW,t ≤ At
h
ωLn
d,t
µ + (1 − ω)L
R
d,t
µ
i 1
µ
.
Solving this, we obtain the following condition regarding labor demands:
1 − ω
ω

L
R
d,t
L
n
d,t!µ−1

AR
An
µ
w
n
t = w
R
t + 2w
R
t
θ

L
R
d,t − L
R
d,t−1

. (14)
Wholesale firm produces wholesale output and sells it to a continuum of retail firms at Pw,t. His
dividend is expressed as follows using the redult from cost minimization: Its nominal dividend is:
DIVW,t = Pw,tYW,t − Wn
t L
n
t − WR
t L
R
t − θPw,t(L
R
d,t − L
R
d,t−1
)
2
.
The optimality condition is:
Wt = Pw,t
14

Or, in real terms:
wt = pw,t (15)
C.2.4 Stochastic Processes
C.2.4.1 Monetary Policy
Assuming the gross nominal rate, Rt
, is set according to a Taylor type rule:
ln Rt = (1 − ρR) ln R + ρR ln Rt−1 + (1 − ρR) θπ (ln Πt − ln Π) + sRεR,t (16)
Variables without time subscripts denote non-stochastic steady state values.
C.2.4.2 Exogenous Process
At
is the only exogenous variable. Assume it follows an AR(1) in the log with non-stochastic mean normalized to unity:
ln At = ρA ln At−1 + sAεA,t (17)
C.2.5 Aggregation
The aggregate inflation rate and real wage evolve according to the following expressions, which can be
derived using properties of Calvo pricing:
1 = (1 − φp)

Π
#
t
1−ǫp
+ φpΠ
ǫp−1
(18)
w
R
t
1−ǫw = (1 − φw)

w
R
t
#
1−ǫw
+ φwΠ
ǫw−1
t w
R
t−1
1−ǫw (19)
149

Goods market-clearing requires that wholesale output by sold to unions in the aggregate, or:
YW,t =
Z 1
0
Yt(f)df
Given the demand function for each retailers output, this works out to:
YW,t = Ytv
p
t
(20)
Where v
p
t
is a measure of price dispersion:
v
p
t = (1 − φp)

Π
#
t
−ǫp
+ φpΠ
ǫp
t
v
p
t−1
(21)
Labor supplied by the regular workers must equal labor used by the union:
L
R
t =
Z 1
0
L
R
u,t(l)dl
Reminding that unions’ labor demand is L
R
t
(l) =
Wt(l)
Wt
−ǫw
Ld,t, this reduces to:
Lt = Ld,tv
w
t
, (22)
where v
w
t
is a measure of wage dispersion:
v
w
t = (1 − φw)

w
#
t
wt
!−ǫw
+ φwΠ
ǫw
t

wt
wt−1
ǫw
v
w
t−1
(23)
Also, labor market for non-regular workers needs to clear:
L
n
d,t = L
n
t
. (24)
150

We consider that there are mass of ω of non-regular workers, and 1 − ω of regular workers in the
economy. We assume zero net supply of bond, so bond market clearing is:
ωBn
t + (1 − ω)B
R
t = 0. (25)
Although redundant due to Warlas Law, goods market clearing is expressed as follows:
Yt = ω(C
n
t − κBn
t
2
) + (1 − ω)(C
R
t − κBR
t
2
)
C.3 Derivation of Regression Equation in 3.4.3
FOCs are:
∂YW,t
∂Ln
t
= Y
1−µ
W,t A
µ
t ωAn
t
µL
n
t
µ−1 = w
n
t
∂YW,t
∂LR
t
= Y
1−µ
W,t A
µ
t
(1 − ω)A
R
t
µL
R
t
µ−1 = w
R
t
Taking the ratio of the two equations,
w
R
t
wn
t
=
1 − ω
ω

AR
t
An
t
µ
L
R
t
L
n
t
µ−1
The regression equation is obtained by taking logarithm of the both sides.
151

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Abstract (if available)

Abstract This thesis consists of three essays on macroeconomic problems faced by Japan.

The first chapter addresses how local heterogeneity in population aging and decline affects the growth and welfare of a country. This study addresses this question by focusing on Japan, where the aging and shrinking of the population is characterized by marked heterogeneity across regions. To this end, a dynamic spatial model equipped with endogenous fertility and overlapping generations replicates, through agglomeration and congestion, this demographic distribution, which is considered to be suboptimal when compared with an allocation by a constrained planner maximizing the welfare of a representative agent. In order to internalize the externality of having children on the size of the future generation, several types of policies are studied. The model calculation incorporating the current policies in Japan suggests that transfers to households based on the number of children has the potential to increase welfare if these transfers are raised from the current level.

The second chapter discusses the welfare consequence of having a large amount of public debt under low real rates of return, which has been a recent norm in advanced countries before COVID. We embed the non-homothetic utility of asset holding in a discrete time two agents perpetual youth model to reconcile low interest rates with large public debt. Whereas the increase in the level of public debt lowers the steady state welfare, the rise in public debt temporarily waives the taxation, and this could be a source for improvement in welfare if the welfare during the transition is taken into account. We find that public debt indeed has a possibility to raise welfare, especially when there is a drop in output. We quantitatively study various types of fiscal policies, including the introduction of foreign investors and public debt reduction. Our most preferred specification suggests that increasing the public debt level itself cannot raise the welfare compared with staying at the initial steady state as the public debt to GDP ratio reaches about 200%, but it continues to be a better policy response when there is a drop in output compared with holding the level of public debt fixed.

The last chapter (joint with Robert Dekle) studies the sluggish response of consumption in Japan during 2013-2019 to the massive monetary expansion by the Bank of Japan. From our observations based on household panel data that the consumption response of households of non-regular workers are stronger than those of regular workers, we set up a New Keynesian model featuring the dual labor market in Japan. There are two types of worker households. One is unionized and monopolistically competitive (regular workers), and the other participates in a perfectly competitive market (non-regular workers). The model well replicates the weak rise of aggregate consumption to monetary policy easing. We show that the widening use of non-regular workers in recent years may have led to the weakening of the consumption response.

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Asset Metadata

Creator Nakamura, Kota (author)

Core Title Essays on Japanese macroeconomy

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Economics

Degree Conferral Date 2024-05

Publication Date 03/25/2024

Defense Date 03/08/2024

Publisher Los Angeles, California (original), University of Southern California (original), University of Southern California. Libraries (digital)

Tag dual labor market,fiscal policy,Japanese economy,macroeconomic policy,monetary policy,OAI-PMH Harvest,population aging,population decline,public debt

Format theses (aat)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Dekle, Robert (committee chair), Imrohoroglu, Selahattin (committee member), Kurlat, Pablo (committee member)

Creator Email kota.nakamura.w3b@cao.go.jp,kotas.nkmr@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC113858372

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Document Type Dissertation

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Type texts

Source 20240327-usctheses-batch-1131 (batch), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright.

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA

Repository Email cisadmin@lib.usc.edu