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Essays on social security

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Content ESSAYS ON SOCIAL SECURITY Copyright 2005 by Kaiji Chen A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ECONOMICS) May 2005 Kaiji Chen R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. UMI Number: 3180315 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3180315 Copyright 2005 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. A ck n o w led g em en ts I am greatly indebted to my dissertation advisors, Caroline Betts, Ay§e imrohoroglu, Selahattin imrohoroglu and Vincenzo Quadrini for numerous suggestions and continuous encouragement. Their continuous support and expert guidance are invaluable. I also wish to thank Yong Kim and Michael Magill for helpful discussion throughout my writing of the dissertation. I also would like to express my gratitude to Research Department of Federal Reserve Bank of Minneapolis. I benefited enormously from faculty and students there during my visit. I owe a great deal to Timothy Kehoe, Patrick Kehoe, Ellen McGrattan and Fabrizio Perri for valuable suggestions and encouragement. Many friends contributed to make this thesis possible. At various stages, their encouragement has been essential and I am sincerely grateful to all of them. In particular, I would thanks to Zheng Song, a coauthor and a friend. Finally, my warmest thanks to my parents and my wife, for always supporting my decision in these long years. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. iii T ab les o f C o n ten ts A cknowledgem ents ii List of Tables v List of Figures vi Abstract vii 1 Introduction 1 2 The W elfare Implications of Social Security for Homeowners 6 2.1 The Environment 16 2.1.1 Demographics 16 2.1.2 Production Technologies 17 2.1.3 Preferences and Endowments 19 2.1.4 Social Security System 20 2.1.5 Market Arrangements 20 2.1.6 Timing and Information 22 2.1.7 Equilibrium 23 2.2 Calibration and Computation 27 2.2.1 Demographics 28 2.2.2 Technology 28 2.2.3 Endowments 29 2.2.4 Preference 30 2.2.5 Social Security 31 2.2.6 Market Arrangements 31 2.2.7 One-Asset Economy 33 2.2.8 Solution Methods 34 2.3 Results 36 2.3.1 Policy Reform 37 2.3.2 Welfare Decomposition 47 2.4 Sensitivity Analysis 57 2.4.1 Intratemporal Elasticity of Substitution 57 2.4.2 Initial Housing Endowment 59 2.4.3 Intertemporal elasticity of Substitution 60 2.5 Conclusion 62 2.6 Appendix 65 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 2.6.1 Calibration of the Benchmark Economy 65 2.6.2 Computation and Algorithm 67 2.6.3 Welfare Effects of Privatizing Social Security in a Deterministic Complete-Market Economy 69 2.6.4 Roles of Housing for the Risk Sharing Benefits of PAYG Systems 74 3 Sustaining Social Security: A Unique M arkov Perfect Equilibrium w ith M ajority V oting 80 3.1 The Model Economy 89 3.2 Political Equilibrium 92 3.2.1 Dictatorship 95 3.2.2 Majority Voting 100 3.3 Comparative Statics 102 3.3.1 The Myopic Effect and Social Contract Effects on the Size of Social Security 105 3.3.2 The Social Contract Effect on the Growth of Social Security 109 3.4 Empirical Evidence 110 3.5 Conclusion 117 3.6 Appendix 119 3.6.1 Proof of Lemma 2 119 3.6.2 Proof of Lemma 3 122 3.6.3 Data Source 122 Bibliography 124 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. V L ist o f T ab les 2.1 Parameter Values for the Markov Chain 30 2.2 Benchmark Parameter Definition and Values for Two-Asset Economy 32 2.3 Parameter Values for the One-Asset Economy 34 2.4 The Benchmark Economy with Housing (Initial Steady State) 36 2.5 Aggregate Statistics for Alternative Economies and Social Security Systems 39 2.6 The Overall Welfare Gain of Privatizing Social Security 43 2.7 The Roles of Market Frictions in the Benchmark Economy 46 2.8 The Role of Financial Markets Frictions in the Economies with Perfect Rental Market 47 2.9 Results for Welfare Decomposition 50 2.10 Results for Welfare Decomposition for the economies with Perfect Rental Market 51 2.11 Robustness to Elasticity of Substitution Between c and h 58 2.12 Robustness to Initial Housing Endowment 59 2.13 Welfare Gains of Eliminating Social Security (sigma=1.5) 62 3.1 The Effects of w(u) and n on the Size of Social Security System 108 3.2 Social Security Program in the OECD (Averages) 111 3.3 Social Contract Effect on the Growth of Social Security 113 3.4 Social Contract Effect on the Level of Pension Size 116 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. L ist o f F igu res 2.1 Age Profile of Beginning-of-Period Assets 41 2.2 Proportion of Households with Positive Non-housing Assets (Benchmark Economy with Owner-occupied Housing) 42 2.3 General Equilibrium Effects 55 2.4 Imperfect Annuity Effects 75 2.5 Age Profile of End-of-Period Accidental Bequest (One-Asset Economy) 77 2.6 Age Profile of End-of-Period Accidental Bequest (Economy with Owner- occupied Housing) 78 3.1 The Existence Condition of Social Security System under DMP 99 3.2 Effects of wu on bo and bi 105 3.3 Effects of n on bo and bi 106 3.4 Cross-Country Relationship between Income Inequality and the Average Growth Rate of Social Security (All Sample Countries) 112 3.5 Cross-country Relationship between Income Inequality and the Average Growth Rate of Social Security (Small Open Economies) 114 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. A b stra ct This thesis comprises two essays on social security. Both essays share a common theme: the desirability of social security system. In the first essay, I explore this question from the perspective of a benevolent government; the second explains the sustainability from the perspective of a rational median voter under majority voting. The Welfare Implications o f Social Security for Homeowners explores how an explicit incorporation of owner-occupied housing into the life-cycle framework affects the long-run welfare implications of privatizing social security in a model calibrated to the U.S. economy. It is motivated by the fact that for most households in the U.S., the largest proportion of net worth is owner-occupied housing, which differs from financial assets by being tied to consumption homeowners’ of housing services and by being relatively illiquid. The literature on social security, nonetheless, has so far assumed perfect substitutability between housing and financial assets. The finding is that the welfare gain of privatizing social security is almost twice as much in the economy with housing as in a standard life-cycle economy. The key reason for this difference is that as mandatory savings for future retirement, the social security system is a worse substitute for household savings when a sizable fraction of household assets is held for immediate consumption of housing durable services. Sustaining Social Security: A Unique Markov Perfect Equilibrium with Majority Voting develops a positive theory of social security in a majority voting R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. framework with differentiable Markov perfect equilibrium. I show that even under temporal separation of costs and benefits, there exists a unique equilibrium that self-interested median voters have incentive to sustain social security if expectation on future policy choice depends only on payoff relevant variables. Correspondingly, our model has a number of distinctive implications. First, social security tax rates are increasing over time until converging to a steady state. Secondly, the growth rate of social security benefit per beneficiary is negatively correlated with income inequality. Our empirical evidence shows that these predictions are broadly consistent with the data from the OECD countries. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 1 Chapter 1 Introduction This thesis comprises two essays on social security. Both essays have a com mon theme: is social security desirable? Specifically, is social security desirable for the aggregate economy? If not, which generations will benefit from imposing or elim inating a social security system? And how does this welfare result reconcile with the political choice of social security? To address the above questions, I take two different approaches in these two essays, respectively. The first, called welfare approach, studies the welfare effects of social security from the perspective of a benevolent government, who explores the trade-off between the efficiency and the distortion it creates. The second, called positive approach, treats social security as rational choices of individual generations. Chapter 2 explores how an explicit incorporation of owner-occupied housing into the life-cycle framework affects the long-run welfare implications of eliminating social security. This question is motivated by the fact that for most households in the R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 2 U.S., the largest proportion of net worth is owner-occupied housing, which differs from financial assets by being tied to consumption homeowners’ of housing services and by being relatively illiquid. Moreover, homeowners’ borrowing capacity in financial markets closely interacts with their housing asset holdings. Despite the quantitative importance of owner-occupied housing, the literature on social security has so far as sumed perfect substitutability between housing and financial assets. To measure the roles of owner-occupied housing, I compare the welfare effects of this social security reform between an economy with owner-occupied housing and a standard life-cycle economy in which households save only in one risk-free financial asset. In the economy with owner-occupied housing, I incorporate three types of mar ket frictions related to housing in order to construct the linkage between consumption of housing services and home ownership: missing rental market; down payment re quirement and housing transaction costs. I then calibrate both economies to the US data. My major finding is that the welfare gain of eliminating social security is al most twice as much in the economy with housing as in a standard life-cycle economy. The key reason for this difference is that by incorporating housing market frictions, housing and financial assets serve different purposes for homeowners. That is, home owners hold housing assets for immediate consumption of durable housing services, while they hold financial assets to finance retirement consumption late in life or con sumption in future unforeseen contingencies. Hence, when housing market frictions cause households to hold a large fraction of household assets in owner-occupied hous R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 3 ing, the social security system — as mandatory saving for future retirement — is a much worse substitute for household savings than in the standard life-cycle economy, in which housing and financial assets are perfect substitutable. Correspondingly, in the presence of housing market frictions the welfare gain from eliminating social security is substantially higher. Chapter 3 then ask why social security, despite its long run welfare decreas ing effects, can be sustained in the real economy. Specifically, I develops a positive theory of social security in a majority voting framework with differentiable Markov perfect equilibrium. This approach is interesting to me because one of the key issue in the political sustainability of a Pay-As-You-Go social security system is that as inter- generational transfer program from the working to retirees, social security is featured by temporal separation between costs and benefits. That is, in a standard majority voting framework, for a self-interested median voter to impose a social security tax upon himself, he must expect a social contract where future social security benefits axe somehow linked to the present level of taxes. The literature, however, has so far resorted to non-payoff-relevant expectation to construct the linkage between current so cial security contributions and future benefits. As a result, it suffers from the problem of multiple equilibria. To this end, I construct a three-period overlapping generations model. I show that even under temporal separation of costs and benefits, in the space of linear func tional there exists a unique equilibrium that self-interested median voters have incentive to sustain social security if expectation on future policy choice depends only on payoff R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 4 relevant variables. The key for the existence of PAYG in this setup is the expectation that future social security benefits are positively correlated with current social security payments. This (self-fulfilled) expectation is achieved through fundamental linkages between policy choice and the payoff relevant variable, which in my model is the hu man capital stock. Specifically, the current payroll tax rate has a negative impact on the next period median voter’ s human capital stock due to its distortionary effects on human capital investment and a lower human capital stock held by next period’ s median voter induce him to choose a higher tax rate due to the positive correlation between marginal cost of taxation and human capital stock. Correspondingly, my model has a number of distinctive implications. First, social security tax rates are increasing over time until converging to a steady state. Secondly, the growth rate of social security programme is negatively correlated with income inequality. Third, cross-sectionally, the impact of income inequality on the equilibrium social contrast influences the impact of income inequality on the size of social security systems, making the relationship non-monotonic. My empirical evidence shows that these predictions are broadly consistent with the data from the OECD countries. Both approaches contribute to the policy debate of social security reform. The first approach explores whether social security should be reformed on the long-run wel fare ground. The numerical results in the first essay indicates that in the long run, both rich and poor households will benefit from eliminating a public pension system. How ever, as a steady state analysis, this essay abstracts from the issue of implementability R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 5 of specific reform schemes along transition. The second approach helps to understand how to design a specific reform scheme so as to make it sustainable as a political choice. My results show that not only the retirees but also the middle-age taxpayers tend to favor the status quo. Therefore, according to this study, one of the keys for a re form scheme to be sustainable is to break the (expected) fundamental linkage between current contribution and future benefits by using other policy instruments, say public debts, in a time consistent way. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 6 Chapter 2 The Welfare Im plications of Social Security for Homeowners Many studies on social security indicate that the welfare effects of alternative social security arrangements depend largely on whether the substitution of a particular system for private savings contributes to a more desirable life-cycle consumption pat tern. 1 This literature typically assumes that housing and financial assets are perfect substitutable and thus households save only in the form of financial assets. As a result, it has so far ignored the fact that owner-occupied housing, which differs from financial assets by being tied up to homeowners’ consumption of housing services and by being relatively illiquid, constitutes the largest share in net worth for most households in the U.S. The proportion of families owning a principal residence, for example, reached *A typical finding in this literature is that imposing an unfunded social security system crowds out a large quantity of household savings and aggregate capital. See, among others, Auerbach and Kotlikoff (1987), Hubbard and Judd (1987), Imrohoroglu, Imrohoroglu and Joines (1995) and Storesletten, Telmer and Yaron (1999). R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 7 67.7% in 2001, and home equity accounted for 32.3% of total household net worth in 2000.2 Moreover, homeowners’ borrowing capacity in financial markets closely interacts with their housing asset holdings. Home-secured debt, for instance, accounted for 75% of the total liabilities of all families, and the proportion of families with home-secured debt amounted to 44.6% in 2001.3 These measures of the quantitative importance of owner-occupied housing in most households’ life-cycle saving decisions suggest that incorporating owner-occupied housing into the life-cycle model may potentially has a large impact on the welfare implications of alternative social security systems. This chapter explores the long-run welfare consequences for homeowners of eliminating an unfunded pay-as-you-go (PAYG hereafter) social security system. The welfare implications of this social security reform are compared between an economy with owner-occupied housing and a standard life-cycle economy in which households save only in one risk-free financial asset. Both economies are calibrated to the US data. To construct the linkage between consumption of housing services and hous ing ownership and to explore the roles of this linkage for the welfare effects of social security, I incorporate three types of market frictions related to housing into an oth erwise standard life-cycle model. First, I introduce rental market friction by assuming that the rental market is missing. Implicit in the model is that due to the moral haz ard problems with renting, households prefer purchasing housing to renting for their consumption of housing services. To the opposite, in an economy with perfect rental market, an assumption implicit in the standard life cycle model, consumption of hous 2See Aizcorbe, Kennickell and Moore (2003) and US Census Bureau (2003). 3Home-secured debt includes first and second mortgage, home equity loan and lines of credit secured by primary residence. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 8 ing services is not necessarily linked housing ownership. Hence, as the first attempt to examine the sensitivity of the welfare effects of social security to this assumption, I assume the opposite for the rental market structure so that the only channel for households to consume housing services is housing ownership.4 The second market friction is financial market friction, or limited borrowing, which interacts with housing purchase. This market friction is well supported by the down payment requirements observed in the U.S. housing market. Theoretically, this friction originates in the assumption that the borrower lacks the commitment to repay debt.5 To prevent the borrower from defaulting in the future, the optimal debt contract involves endogenous borrowing capacity, which, in my model, depends on the quantity of housing collateral. An implication of this optimal contract is that the degree of this market friction depends on the alternative social security arrangements, which affects the life cycle profile of housing asset accumulation. The third market friction is housing transaction costs.6 This friction affects households’ incentive for precautionary saving against idiosyncratic income risks.7 In the presence of transaction costs, housing is illiquid relative to financial assets. The illiquidity of housing indicates that variations of housing assets, and therefore, con 4 Several studies explore the roles of government subsidies to owner-occupied housing in house holds’ tenure decision, including Gervais (2002) on the roles of nontaxibility of imputed rents and tax deductibility of mortgage interest payment and Jeske and Krueger (2004) on the role of implicit guarantee for Government Sponsored Enterprises. 5The role of limited commitment has been much discussed in recent literature. See, among others, Kehoe and Levine (1993), Kocherlakota (1996) and Krueger (2001) for efficient risk sharing, Kehoe and Perri (2002) for international business cycle, Alvarez and Jermann (2001) for asset pricing puzzles and Azariadis and Lambertini (2003) for life-cycle consumption and saving. 6The friction behind the presence of transaction cost is asymetric information, or so called “lemon problem”. th r o u g h o u t this paper, precautionary savings against idiosyncratic income risks (or mortality risks) refer to the increase of non-housing assets over the corresponding stock that would prevail in a model with no such risks. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 9 sumption are costly. This increases the demand for precautionary saving when idiosyn cratic income risks exist. As the transaction costs for housing are quite large in the U.S. economy, I consider this market friction a potential channel for eliminating social security to affect aggregate savings and factor prices. To formalize the above idea, I set up a quantitative general equilibrium life cycle model with heterogeneous agents that are subject to both idiosyncratic labor income risks and uncertain lifetimes. There are two production technologies in this economy: the first uses non-housing capital and labor as inputs to produce nondurable goods, which can be either consumed, invested as non-housing assets or invested as residential structures;8 the second combines residential structures and land to produce housing. The presence of land in housing production endogenizes housing prices in my model. My framework builds upon Fernandez-Villaverde and Krueger (2001), who de velop a model of durable consumption with collateral borrowing. However, my model is distinct in several aspects. First, Fernandez-Villaverde and Krueger (2001) focus on the life-cycle patterns of consumption and saving, and abstract from social security sys tems. I incorporate the social security system because my primary interest is the role of owner-occupied housing for the effects of eliminating social security. To my knowledge, I are the first to explore social security in a model with owner-occupied housing or housing market frictions. Second, in their paper housing is not explicitly modeled. In stead, their definition of consumer durables covers a broader range, including vehicles, 8In our economy, the measurement of ‘non-housing capital’ includes consumer durables other than housing, and correspondingly, the measurement of ‘nondurable goods’ includes service flows from con sumer durables other than housing. See the Appendix 7.1 for details of measurement. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 10 furniture and etc. I focus exclusively on owner-occupied housing, due to its importance in homeowners’ life-cycle saving decisions and the importance of home-secured debt in homeowners’ total liabilities. Third, they assume a one-to-one transformation between nondurables and consumer durables. I explicitly incorporate land in housing produc tion. As I mention above, this creates an interaction between housing assets and the housing price, which gives rise to a more accurate aggregate response of the housing stock to eliminating social security. Moreover, introducing land makes my character ization of housing production close to that observed in the real economy. Finally, in their paper there are no transaction costs involved in the trade of durables. I choose to incorporate housing transaction costs in my model, and these costs turn out to be a significant factor affecting the saving response to eliminating social security. I calibrate both economies, the economy with owner-occupied housing (or with housing market frictions) and the standard life-cycle economy (or with perfect rental market), to the U.S. data. I then examine the stationary equilibria that are solved for by numerical methods. A detailed exploration of the transition path of eliminating social security in my settings, especially, how to deal with social security liabilities along the transition path, is desirable. However, due to the computational burden it involves, I leave this interesting issue for future research. My major finding is that the magnitude of the long-run welfare gain of elim inating social security is almost twice as much in the economy with housing market frictions as in the economy with perfect rental market, despite the fact that in the pres ence of housing market frictions, eliminating social security creates a smaller increase R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 11 in aggregate wealth. The key reason for this difference is that by incorporating housing market frictions, housing and financial assets serve different purposes for homeowners. That is, homeowners hold housing assets for immediate consumption of durable hous ing services, while they hold financial assets to finance retirement consumption late in life or consumption in future unforeseen contingencies. Hence, when housing market frictions cause households to hold a large fraction of household assets in owner-occupied housing, the social security system — as mandatory saving for future retirement — is a much worse substitute for household savings than in the standard life-cycle economy, in which housing and financial assets are perfect substitutable. Correspondingly, in the presence of housing market frictions the welfare gain from eliminating social security is substantially higher. Specifically, my counterfactual experiments indicates that the main channels for introducing housing market frictions to raise the welfare gains of eliminating social security are the following. First, in a simple deterministic partial equilibrium environ ment, when households are borrowing constrained, the social security system creates a distortion of intertemporal consumption smoothing over the life cycle by postponing consumption until late in life (intertemporal consumption smoothing effects). This is because the taxes paid by young generations reduce their disposable income. The pres ence of down payment requirement in an economy with missing rental market makes this distortion more severe, because in this setting the payroll tax finance of social se curity benefits crowds out a large quantity of housing assets that is linked to immediate consumption of housing services. As a result, in an economy with housing eliminating R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 12 payroll tax finance provides a larger welfare gain in terms of intertemporal consumption smoothing than in a standard life-cycle economy. Second, in the general equilibrium, eliminating social security results in a higher aggregate savings and a lower interest rate. The decrease in the interest rate generates a welfare cost arising from its negative income effect (general equilibrium effects). Nonetheless, due to the missing of rental market housing is no longer an inter esting bearing assets for homeowners.9 Hence, when a sizable proportion of their total assets are held for immediate consumption purposes, homeowners shall accumulate a smaller quantity of interest bearing assets than their counterparts in the economy with perfect rental market. This wealth composition permits a smaller welfare cost caused by the negative income effect of a fall in the interest rate than in the economy with per fect rental market. In other words, a fall in the price of present consumption (interest rate) is more beneficial to homeowners than their counterparts in the standard economy when a sizable proportion of homeowners’ assets is held for immediate consumption. Consequently, all homeowners derive a higher welfare benefit than their counterparts in the standard economy from a change in equilibrium factor returns, ceteris paribus, once social security is privatized.10 This study is related to several strands of literature. First, it adds to the extensive discussion on the welfare implications of alternative social security schemes. 9In an economy with imperfect rental market, the market return for tenant-occupied housing re sponds less than one-to-one to a change in market return for financial assets. 10Our results also demonstrate that when social security is privatized, the increase in the present value of lifetime wage earnings is higher in the economy with housing than in the standard life-cycle economy, because the presence of housing transaction costs leads to a more drastic drop in the interest rate in the economy with housing. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 13 Contributors to this literature include, among others, Feldstein (1985), Auerbach and Kotlikoff (1987), Hubbard and Judd (1987), Imrohoroglu, Imrohoroglu and Joines (1995), Huggett and Ventura (1999), Storesletten, Telmer and Yaron (1999), Fuster, Imrohoroglu, Imrohoroglu (2003) and Pries (2004).1 1 All of these papers examine economies with only financial assets and abstract from owner-occupied housing. Incor porating owner-occupied housing, nonetheless, provides a more precise picture of most households’ asset accumulation over the life cycle, which is the key factor underlying the welfare effects of alternative social security policies. Furthermore, it allows us to ex amine the effects of social security reform on the housing market and the housing credit market, arguably the largest asset market and credit market, respectively, currently in the U.S. In terms of the methodology of welfare decomposition, the closest paper to this study is Storesletten, Telmer and Yaron (1999).1 2 In a standard life-cycle economy, they have found that the magnitude of risk sharing benefits of the PAYG system against both idiosyncratic income risks and mortality risks is as large as 63% of the net welfare gain of eliminating social security, though these risk sharing benefits are outweighed by the welfare gain arising from the general equilibrium effects of eliminating social security. By introducing the three housing market frictions, my results significantly strengthen “ Several other noteworthy papers are Bohn (1998) and Krueger and Kubler (2003), who study the intergenerational risk sharing role of social security system in the presence of aggregate shocks; Huang, Imrohoroglu, and Sargent (1997), Kotlikoff, Smetter, and Walliser (1998), and Conesa and Krueger (1999), who study the transitional dynamics associated with the social security reform; and Diamond and Geanakoplos (2003), who explore the general equilibrium impact of social security portfolio diver sification into private securities. 12In Storesletten et al. (1999), the definition of general equilibrium effects includes welfare effects arising from both a change in equilibrium factor prices and the direct impact of eliminating payroll taxes on the lifetime wealth. Moreover, their welfare decomposition does not involve the welfare gain from intertemporal consumption smoothing. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 14 their finding on the quantitative importance of the general equilibrium effects. The risk sharing benefits of the PAYG system, moreover, are much less important in my economy than in their paper. Specifically, with housing linked to homeowner’s consumption as a result of missing rental market, the welfare cost by losing the partial annuity role of the PAYG system is only half of that in the standard life-cycle economy. A novel feature of my economy, furthermore, is the crowding-out effect of the current social security system on the private risk sharing (via collateral borrowing) against idiosyncratic income risks. As a consequence, in my economy the net benefit arising from the corresponding public risk sharing role of the PAYG system is merely two-thirds of that in the standard life-cycle economy. This study also contributes to the emerging literature on the role of (owner- occupied) housing for a variety of macroeconomic issues. For example, Di'az and Luengo-Prado (2002) and Gruber and Martin (2003) explore the role of housing for precautionary savings and wealth distribution. Different characteristics of housing, in addition, have been recently exploited to explain facts associated with asset prices. 13 Among the business cycle studies, Davis and Heathcote (2003), by explicitly model ing the production of housing, successfully explain both the volatility of residential investment and the co-movement of residential investment with other macro variables observed in the data. Nonetheless, surprisingly few studies have so far been conducted to explore the welfare implications of housing for various fiscal policies. 14 Hence my 13See, among others, Piazzesi, Shneider, and Tuzel (2002) for the role of composition risk, Chetty and Szeidl (2003) for the role of consumption commitment, Nakajima (2003) for the role of the illiquidity of housing, and Lustig and Van Nieuwerburgh (2004) for the role of housing collateral. 14An exception is Gervais (2002), which studies the impact of the preferential tax treatment of housing capital on aggregate capital and individual welfares. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 15 study, which emphasizes the importance of housing for the welfare implications of social security policies, provides a useful complement to the above literature. Another line of the literature that is closely related to this chapter is the study of life-cycle portfolio choice with housing (or consumer durables). Important references include Cocco (2000), Platania and Schlagenhauf (2000), Fernandez-Villaverde and Krueger (2001), Flavin and Yamashita (2002), Ortalo-Magne and Rady (2003), and Yao and Zhang (2003). All these studies have provided strong support for the importance of housing in household life-cycle consumption and saving (portfolio) decisions. However, a common feature of the above models is the absence of any interaction between housing prices and aggregate housing stocks: either housing prices follow an exogenous process, or the total supply of housing assets is fixed. By contrast, I explicitly model the interaction between the price and the quantity of housing assets, which allows us to measure more precisely how the aggregate housing stock responds to eliminating social security. My analysis indicates that in the absence of this interaction, the welfare gain of eliminating social security will be potentially biased up. The chapter is organized as follows. In Section 2.2, I describe the environ ment of the economy with housing and define the stationary equilibrium. Section 2.3 provides a discussion of the calibration procedure and the computation method, and constructs a standard life-cycle economy as the reference point. In Section 2.4, I com pare the welfare implications of eliminating social security between my economy and the standard economy and analyze the roles of housing for the difference of the welfare gains between the two economies. Section 2.5 provides sensitivity analysis and Section 6 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 16 concludes. The Appendix contains details of calibration, an analytical characterization of the welfare effects of eliminating social security in a deterministic complete-market overlapping generations model with housing, and the computational details. 2.1 T h e E nvironm ent I consider a discrete time dynamic general equilibrium life-cycle economy with both idiosyncratic income and lifetime uncertainty. Compared to a standard life-cycle economy, the model has three novel features: first, housing serves as an asset that provides durable services; second, for each household the borrowing limit of non-housing assets depends on the quantity of her housing assets; third, a transaction cost is involved in the sale of housing assets. I call this economy the benchmark economy. 2.1.1 Dem ographics Assume that the demographic structure is stationary. In each period the economy is inhabited by a continuum of ex ante identical individuals, with constant population growth rate n. Each individual can live for a maximum of J period, working only for the first jr — 1 period; the retirement age jr is exogenous. For each j = 1 ,..., J — 1, denote ipj £ (0,1) as the probability of surviving onto age j + 1 conditional on living at age j. Clearly, tp0 — 1 and ipj = 0. The probability of surviving through age s is then TLj=1ipj. Denote {tij}J_1 as the fraction of individuals of age j in the whole / - 1 . . V 1 population. Clearly, the fraction of newborns is /r1 = I 1 + (1 -I- n) n^=1'0i I \ 3= 1 J and the fraction of individuals for age j = 2,..., J — 1 can be computed recursively by R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 17 Mj+i = ( 1 + n ) - 1 ipjfij. Private annuity markets axe assumed to be missing. In addition, accidental bequests of both non-housing and housing assets are 1 0 0 % taxed away and consumed by the government. 2.1.2 Production Technologies The economy has two production technologies. The first uses non-housing capital and labor input to produce nondurable goods. Aggregate output of nondurable goods, denoted as Y, is produced according to the production function Y = AF(K, T V ). K is the quantity of aggregate non-housing capital; T V is aggregate efficient labor input. A is the level of total factor productivity (TFP), which grows at a constant rate g. I assume that F is strictly increasing in both arguments and strictly concave. Further more, F satisfies the Inada conditions and is homogeneous of degree one. Without loss of generality, I assume that in this economy there is a representative firm that hires labor and non-housing capital from households to produce nondurable goods each pe riod. The output can be either consumed, saved as non-housing assets by households, or purchased as residential structures by real estate developers on a one-to-one basis. I normalize the price of nondurable goods to one. Denote Xr and Xa as aggregate investment in residential structures and aggregate investment in non-housing capital, respectively. I assume that non-housing capital and residential structures depreciate at a rate Sk and 8r, respectively. The law of motion for non-housing capital is K' = (l - < 5 f e ) K + Xa R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 18 where the superscript prime throughout this chapter is referred to as end-of-period variables. Assume a representative real estate developer produces new housing each period with the second technology. Under this technology, the construction of new housing each period, denoted as Xh, is decreasing return to scale with respect to the investment in residential structures, X r. In other words, both investment in residential structures and a fixed factor enter into the production of new housing. For calibration purpose, I interpret this fixed factor as a constant acreage of new land provided by a non-profit co-op. After the production of housing takes place, the non-profit co-op distributes the income from land evenly across the currently alive households as lump sum transfers. In each period, the real estate developer solves the following static maximization problem max PhXh - X r - piL Xr,Xh subject to X h < G (Xr,L) (2.1) where ph and pi are the housing price and land price in terms of the nondurable good, respectively. L is the quantity of land, which is normalized to one. Note that the housing asset is modeled as a continuous variable instead of a discrete variable. This assumption stems from the demand in my general equilibrium setting to track the R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 19 aggregate quantity of housing assets, which closely interacts with the housing price and non-housing asset accumulation. Assume the production technology for housing, G, is non-decreasing linearly homogeneous in Xr and L. Correspondingly, I have pi = d2G PhXh — Xr. In addition, < 0. Note that land acts like an adjustment cost for residential investment in my economy. The aggregate stock of housing assets, denoted as H, depreciates at a rate Sh each period. 2.1.3 Preferences and Endowm ents Agents enter into the economy with neither non-housing nor housing assets and axe endowed with one unit of time in each period. 15 Individuals of different ages differ in their labor productivity. Denote {ej}j=1 as the deterministic age profile of average labor productivity. In addition, workers of the same age face idiosyncratic shocks to their labor productivity. The stochastic process of labor productivity are assumed to be identical across individual workers and follows a finite-state Markov process t t (p1 i p) with the state space p S E = {pit-., pN}. Assume 7 r has a unique stationary distribution, denoted by II. Households derive utility from both nondurable goods and housing service flows. For simplicity, I assume one unit of housing stock generates one unit of housing service flow. Furthermore, assume that the rental market is missing so that quantity of housing services consumed by each household is equivalent to his quantity of housing stock at the beginning of each period. Leisure is not valued and in each period labor 15In Section 5.2, we check the robustness of our welfare results to alternative specifications of initial housing endowments. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 20 supply is inelastic. The lifetime utility function can then be written as £ (cj-.M j (2-2) where /3 is the utility discount factor, c and h are nondurable and housing consumption, respectively. The period utility function u is assumed to be strictly increasing in both arguments, strictly concave and obeys the Inada conditions. 2.1.4 Social Security System The social security system in the initial steady state is an unfunded PAYG system. In each period, the government levies a payroll tax on current workers and distributes the tax revenue uniformly across the current retirees. In the interest of computational tractability, I assume that the level of social security benefits received by a retiree is independent of the history of her social security contributions. 16 My specification of the unfunded social security system implies an upper bound for the degree of intra-generational redistribution inherent in it. 2.1.5 Market Arrangem ents Two types of asset markets exist in this economy: the housing asset market and non-housing asset market. Agents can hold housing assets h G [0,h], where h is a sufficiently large number so that it never binds. Moreover, they can save or borrow 16A more realistic assumption is that the level of social security benefits is a concave function of the accumulated contribution of over the working ages; see Storesletten et al. (1999). However, under this assumption the state variable will increase by one dimension, which will tremendously raise the computational costs. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 21 in one risk-free non-housing asset a € [a, + 0 0 ], where a negative a can be regarded as the borrowing limit. A saver can choose to either lend to other homeowners or rent her non-housing assets to the representative firm. 17 For simplicity, there is no difference between interest rates for borrowing and for lending and renegotiation of debt involves no cost. Clearly, under this assumption a household’s allocation between non-housing assets and debt is indeterminate given a. I therefore interpret a as net worth excluding housing assets. In addition, the homeowners can only borrow up to the value of their end-of period housing stock minus a down payment. The borrowing constraint can be written as a' > - ( I - 7 )phti where 7 is the down payment ratio. Finally, a non-convex transaction cost y (h, h!) is incurred each time agents change their holding of housing stock. The non-convex adjustment cost function ensures that the adjustment of housing asset is lumpy and infrequent. In short, in this model I introduce three types of market frictions associated with housing. First, the rental market is missing so that households’ consumption of housing services is linked to their housing assets holdings. Second, non-housing asset market involves limited borrowing, which capacity interacts with housing asset accu mulation. Finally, housing transaction is subject to adjustment costs. ^Alternatively, we can assume that in this economy there is a central bank that pools individual households’ deposit to finance loans to homeowners or use it as non-housing capital to rent to the producers of nondurable goods. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 22 In the description of the housing market above, I abstract from several aspects of housing that have been attracted attention in literature. For example, there is no rental market for housing. 18 Neither do I incorporate tax deductibility of mortgage payments and income tax associated with non-housing assets. On the one hand, the presence of rental choices makes it possible to separate purchase of consumption services from purchase of housing. On the other hand, rent premium, various government sub sidies for owner-occupied housing, and the opportunity for collateral borrowing makes owning a house more attractive than renting a house. Incorporating these features shall significantly increase the computational cost, however. Therefore, I leave this extension for future research. 2.1.6 Tim ing and Inform ation At each period, the events proceed as follows. At the beginning of each period, the idiosyncratic component of labor productivity for each agent is realized. Then agents supply non-housing capital and labor to the representative firm. After nondurable goods are produced, the representative real estate developer purchases res idential structures and land to produce housing. Next agents receive factor payments and government transfers and decide how much to consume, save in non-housing assets (or borrow), or purchase housing. In each period, the amount of housing service flow consumed by a household is predetermined by her beginning-of-period housing stock. Right after the housing service flow is consumed, the housing stock depreciates at rate 18According to Fixed Asset Tables of Bureau of Economic Analysis (BEA), in 2001 owner-occupied housing accounted for 76.17% of the stock of residential fixed assets. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 23 5h. Finally, uncertainty about early death is revealed. All accidental bequests are taxed away and consumed by the government the next period after nondurable good productivity becomes common knowledge once realized. 2.1.7 Equilibrium I restrict my analysis to stationary equilibria. Denote s = {a,h,r],j} as the individual state. Let $ (s) denote the measure of individuals with state s. The household’s problem can be written recursively as where r and w are the interest rate and wage rate, respectively, b is social security benefit per retiree, Tr is the lump-sum transfer by the government. I(j) is an indicator function such that production takes place. All information is publicly observed. The idiosyncratic labor V (a, h,rj,j) = u (c, h) + fiipj ^ 2 7 T (r/ 11 }) v (a h ', rj',j + l) (2.3) s . t . c + Phh’ + a' w (1 - t) ejr] + (1 + r)a +ph ( l - 5h) h - y(h, h!) + 1 (j ) b + Tr a > - ( 1 - 7 )phh! c,h' > 0 0 i f 1 < j < jr — 1 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 24 I now define the stationary equilibrium. Let J = {1,... J} and let S = R x R + x E x J. Let B (R) and B (R+) be the cr-algebra of R and R +, respectively, and P(E) and V{J) be the power set of E and J, respectively. Let S =B (R) x B (R+) x 'P(E) x'P(J) and let M be the set of finite measures over the measurable space (S,<S). Definition 1 Given a replacement rate d, a stationary equilibrium consists of a value function V for the households, a set of individual policy functions c, a', h', production plan {Y, K , N} for the representative firm, production plan (Xh, X r, L) for the real estate developer, a set of prices {r, w, Ph, Pi} and a finite measure < t > £ M, such that 1. Given (r, w}, V solves the individual’ s problem (3), with c, a', h' as the associated policy functions. 2. {r, w} are such that the maximization problem of the representative firm is solved. r = AFk (K, N) — 5k (2.4) w = AFn (K,N) 3 . { p h i P i } a r e s u c h t h a t t h e r e a l e s t a t e d e v e l o p e r ’s p r o b l e m i s s o l v e d . R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 25 Ph = \ G x r { X r , L ) } - 1 (2.5) Pi = P h G i ( X r , L ) 4 ■ Given ph andpi, the real estate developer’ s problem is solved. 5. The social security policies satisfy dwN f < f> (d a x dh x dp x {1 , ...,jr — 1 }) twN = b &(da x dhx dpx { j r , J}) 6. Individual and aggregate behaviors are consistent K ' J a/(s)$ (da x dhx dr) x dj) N = J tji7$ (da x dhx dr) x dj) J c(s)< J> (da x dhx dr) x dj) J h(s)Q (da x dhx dr) x dj) J &(da x dhx dr) x dj) C H Pi = Tr 6. Markets clear R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 26 (a) goods market clears C + X, + X a + J y (h, h'(s)) $ (da x dh x dr] x dj) = Y where X a = J [a'(s) - ( 1 - 5k)a $ (da x dhx dy x dj) (b) housing market clears H' = (1 — 5h)H + X h (c) all factor markets clear 7. The law of motion for $ is stationary T($) = $ where the operator T : M — > M can be explicitly expressed as: a. for all J such that 1 £ J, all A x H x E e B (R) x B (R+) x VfE), and all s — {a, h, r),j} G S R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 27 T($) (A x H x E x J) = J P (s; A x H x E x J) $ (da x dh x drj x dj) where P (s; A x H x E x J) Y V'j7 1 ' (?/ 1 J? ) if j + 1 6 J, € A and h'(s) € 77 rfeE 0 else b. for all A x H x E £ B (R) x 6 (R+) x V(E) T($) (A x 77 x E x 1) (7 7 ) ifO e A and 0 € H r/€E 0 else 2.2 C alibration and C om p u tation In this section, I first calibrate the benchmark model to match the long run average of the U.S. data. I then construct as the reference point an economy that abstracts from housing and calibrate this economy. After that, I briefly discuss my solution methods for the stationary equilibria of both economies. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 28 2.2.1 D em ographics One period in my model corresponds to one year of calendar time. The max imum number of periods that an agent is alive J = 6 6 and households retire at period jr = 46. This maps into an economy in which individuals enter the labor force at age 20 and retire at age 65, with the maximum age 85. The survival probabilities {V ,*}f=i is taken from Faber (1982). Finally, the population growth rate n is set to be 0.01, a number in line with the U.S. long run average. 2.2.2 Technology The production function for both goods is Cobb-Douglas. Specifically, The production function for the nondurable good takes the form Y = AK aN l~a. Non housing capital K is defined as the sum of the private fixed assets, consumer durables, inventory stock and net foreign assets minus the stock of private residential structures. I then calibrate a so that the share of capital income in the output Y matches the U.S. data. This gives a = 0.2732. The productivity growth rate g = 0.015, which is consistent with the long-run average growth rate of U.S. real GNP per capita. Housing production function takes the form Xh = x l ^ L ^ . Following Davis and Heathcote (2003), I set < f > = 0.106.19 I set the annual depreciation rates for non-housing assets and for residential structures to match the investment-capital ratio for each asset. The corresponding val ues of 5r and Sk are 0.0205 and 0.0951, respectively. Finally, the equilibrium definition 19According to Davis and Heathcote (2003), this number is from an unpublished 2000 memo from Dennis Duke to Paul L. Hsen entitled “Summary of the One-family Construction Cost Study” . R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 29 implies that the depreciation rate for the housing asset Sh = 0.0183. The Appendix 7.1 contains a more detailed description of the calibration procedure on these technology parameters. 2.2.3 Endowm ents The deterministic age profiles of labor productivity {ej} ^ = 1 is taken from Hansen (1993). For retirees, tj = 0. I follow Huggett (1996) in parametrizing the idiosyncratic component of labor income process. Huggett (1996) uses the follow AR(1) process for the log of labor income process log Vj+i = P log V j + £j+i (2-6) where s ~ N (0, cr2) and log r/: ~ N (0,ct2 ). The autoregressive coefficient p and variance of innovation < j 2 are set to be 0.96 and 0.045, respectively. The variance of labor income shocks at initial age < r 2 = 0.38. Using the method proposed by Tauchen (1986), I approximate the continuous AR(1) process with a seven-state Markov chain. This results in a value of Gini coefficient for labor income as 0.40, which is broadly consistent with the U.S. data.20 Table 2.1 reports the values of 77 in the seven states, together with the stationary distribution of the Markov chain. 20Quadrini and Rfos-Rull (1997, Table 1) reports a value for Gini index of earnings of .51 for the U.S. household with head aged 35 to 50 years. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 30 Value n Vi 0.1838 0.0637 V ? , 0.2948 0.1283 Vs 0.4728 0.1955 V4 0.7583 0.2250 V n 1.2163 0.1955 V 6 1.9509 0.1283 Vj 3.1290 0.0637 Table 2.1: Parameter Values for the Markov Chain 2.2.4 Preference I parameterize the period utility function with the standard isoelastic specifi cation. 1— a u (c, h) = {6cv + (1 - 0) hv)v - 1 1 — a The coefficient of relative risk aversion a is set to 2, which is standard in macroeconomic literature. I set v — 0 following Fernandez-Villaverde and Krueger (2001) . 21 In the section of sensitivity analysis, I will check the robustness of my results to v = 0.4, a value in line with the estimation results by McGrattan et al. (1997) and Rupert et al. (1995) (under the sample of single male).I then calibrate the utility discount factor /3 and the share of the nondurable consumption in the utility function 0 so that both the ratio y and the ratio is consistent with the U.S. data.22 I choose these two ratios as my targets because quantitatively in an economy with housing market frictions, the welfare impacts of social security depends critically the wealth composition of homeowners. Accordingly, /3 and 6 are set to 0.9613 and 0.8474, respectively. 21Fernhndez-Villaverde and Krueger (2001) list several empirical studies to argue that the hypothesis v = 0 cannot be rejected at 5% level. 22See Appendix 7.1 for a measurement of these two ratios according to NIPA. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 31 2.2.5 Social Security In the initial steady state, I choose the replacement rate d so that the payroll tax rate matches it empirical counterpart. Currently, the OASI (Old-Age and Survivors Insurance) rate is 10.7% . 23 This implies '0 = 48.3%. 2.2.6 M arket A rrangem ents The transaction cost function for selling housing is set as y(h, ti) = Ih> ip ( 1 - Sh) phh, where 1 i f h '^ h 0 i f h' = h Implicit in the above function is that each time a homeowner changes her housing stock, she needs to sell her current housing assets first.24 This selling incurs a loss proportional to the selling price. I choose tp = 0.05, the typical fee charged by the real estate brokers in the U.S. Finally, the down payment ratio 7 is set to 20%, which is the average down payment ratio of primary mortgage loans in the U.S. Table 2.2 summarizes the calibrated parameters. 23Social security payroll tax rate currently in the U.S. is 15.3%. Since we focus on the retirement benefits, we subtract the part of the tax rate due to Medicare and Disability Insurance. 24Alternatively, we could assume that each time h! is not equivalent to ( l — Sh) h , a proportional transaction cost is incurred. Our numerical results shows that this specification increases the welfare gain of eliminating social security. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 32 Symbol Definition Value Demographics J Maximum age 6 6 jr Retirement age 46 A Survival probabilities Faber (1982) n Population growth rate Technology 0 . 0 1 a Capital share in nondurable goods production function 0.2732 9 Net TFP growth rate 0.015 Land share in housing production function 0.106 5r Depreciation rate for residential structures 0.0205 5k Depreciation rate for non-housing capital 0.0951 Sh Depreciation rate for housing stock Endowment 0.0183 Deterministic productivity of agents in age j Hansen (1993) * Variance of innovation to idiosyncratic shock 0.045 Variance of income distribution at initial age 0.38 P Autocorrelation coefficient in stochastic earning process Preference 0.96 P Discount factor in utility function 0.9613 e Share of non-housing consumption in utility function 0.8474 a Coefficient of relative risk aversion Market Arrangement 2 V Transaction cost 0.05 7 Down payment ratio 0 . 2 0 Table 2.2: Benchmark Parameter Definition and Values for Two-Asset Economy R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 33 2.2.7 O ne-A sset Econom y I now construct a standard life-cycle economy as the reference point. In this economy, all households consume a single nondurable good, and save in one risk-free liquid asset. There is a single production sector which hires capital and labor from households to produce the single consumption good. Alternatively, this economy can be regarded as an economy with perfect rental market so that housing and financial assets are perfect substitutable. In this economy, both assets are supplied by households to the (aggregate) production sector for the same market return. Correspondingly, households are indifferent between purchasing housing services from the market and from self holding of owner-occupied housing. Moreover, to compare the roles of financial market friction for the welfare effects of social security between the two economies, I assume that the borrowing market is closed. I call this economy ‘one-asset economy’. I adopt the same parameterization as the benchmark economy in terms of demographic features, labor endowments, and the social security replacement rate at the initial steady state. The capital K in this economy is defined as the sum of the private fixed assets, consumer durables, inventory stock and the net foreign assets. Following the literature, the capital share in the final goods production, a is chosen to 0.36. The period utility function is u (c) = yry. The coefficient of relative risk aversion, cr, is again set to 2 to be consistent with the benchmark economy. The depreciation rate for the capital stock, 5, is set to 0.0651 to match the investment-capital ratio in the U.S. data. According to National Income and Product Accounts (NIPA), the average value of this ratio is 0.0902 between year 1954 and 2000. I calibrate the utility discount R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 34 Symbol Definition Value a Capital share in output production function 0.360 S Depreciation rate for capital 0.0651 P Discount factor in utility function 0.9852 a Lower bound for asset 0 Table 2.3: Parameter Values for the One-Asset Economy factor /3 so that the capital-output ratio matches the wealth-income ratio in the U.S., which is 3.1. This gives (i = 0.9852. Finally, the absence of borrowing opportunities implies a = 0. In Section 2.4 I will show that my results axe robust to the assumption a = — 5, a borrowing limit sufficiently loose so that it never binds for any household. Table 2.3 summarizes the parameter values specific to this economy. 2.2.8 Solution M ethods Since analytical solutions for this problem do not exist, I solve for the station ary equilibria of both economies by numerical methods. For the one-asset economy, I follow the standard approach along the tradition of Auerbach and Kotlikoff (1987) to derive the optimal policies.25 For the economy with housing, because all productions take the Cobb-Douglas forms, a balanced growth path exists in this economy. On the balance growth path, variables {V, K, C, X r,pi} grow at a constant rate (1 + g) (1 + n), Ph grows at a rate ((1 + g) (1 + n))^ and {Xh, H } grows at a rate ((1 + g) (1 + n ))1-^ . Accordingly, housing assets per capita, h, grow at a rate (1 + (1 + n)- ^, the sec ond term of which stems from the facts that with supply fixed in each period, land 25See Rfos-Rull (1998) for details of computing the steady state of Overlapping Generations models. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 35 per capita grows at a rate (1 + n)_1. I detrend all aggregate variables so that the transformed variables are constant at the steady states. The existence of non-convex transaction costs in the benchmark economy in dicates that the value function is not necessarily concave and differentiable. I therefore discretize the asset space and, for each grid point of end-of-period housing assets, use the Golden Section Search method to find the optimal policy for the non-housing as set,26 which may not necessarily lie on the grid points. Then I find the optimal level of end-of-period housing assets by grid search. To speed up the computation, I define q = a + ( 1 — 7 )phh and reformulate the household’s problem in terms of q, so that in computation the lower bound for asset choices becomes exogenous.27 In the tradition of computing general equilibrium overlapping generations models, I solve for the households’ problem by backwards induction. The computation starts with guesses for the interest rate r and housing price ph■ To compute the land price and thus the lump-sum transfers, simply notice that the Cobb-Douglas production function for housing implies that in equilibrium Pi = P h i l-< /> )“7 % (2.7) After solving for the households’ problems, I compute the stationary distribu tion $ by forward recursion. Then, given the stationary distribution, I compute values of various macroeconomic aggregates. Finally, I solve for the interest rate r by equation 26See Chapter 10 of Press et al. (1992) for details of this method. 27This method was first used by Diaz and Luengo-Prado (2003) in an infinite horizon framework. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 36 Variables Ini. St. St. Values U. S. Data Payroll tax rate, r 0.107 0.107 r 6.73% 6.73% K /Y 1.682 1.682 PhH/Y 1.306 1.306 Pi/Y 0.007 - Gini coefficient for wealth 0.71 0.76 Debt to ph,H ratio (for borrowers) 59.4% 56.0% Table 2.4: The Benchmark Economy with Housing (Initial Steady State) (5) and the housing price ph by y4> Ph — z — j (2.8) which corresponds to the definition of Tobin’s q. I iterate on r and ph until convergence is achieved. The Appendix 7.2 summarizes the algorithm and the computational details. 2.3 R esu lts To begin with, Table 2.4 reports some properties of the benchmark economy. Based on my calibration, the values for capital-output ratio K /Y and value of housing- output ratio phH/Y are consistent with the U.S. data. The ratio of lump sum transfers (or land price) to output of nondurable goods is small. My Gini coefficient for wealth, 0.71, is consistent with the U.S. data, which is 0.76 according to Quadrini and Ri'os-Rull (1997, Tablel). The ratio of average debt to housing value among borrowers is close to the median ratio of home secured debt to the value of the principal residence among those with such debt in 2 0 0 1 . 28 28 See Aizcorbe et al. (2002). R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 37 2.3.1 Policy Reform This section explores the long-run welfare effects of eliminating social security, that is, eliminating the PAYG system and allowing households to save through private asset markets for their own retirement. I first report the aggregate statistics and explore the simulated life-cycle profiles of wealth portfolio in the economy with housing. I then compare the welfare effects of eliminating social security between the two economies. Finally, I analyze the role of introducing housing market frictions for the welfare effects of eliminating social security. For both economies, the welfare effects of eliminating social security can be measured by the compensating variations, denoted as CV. For the benchmark economy, the compensating variations measure how much (in percent) the consumption index, defined as ceh1~0, must be increased at each period and each contingency in the PAYG system so that a given type of agent is indifferent between the two systems. The same definition of CV applies to the one-asset economy, except that the percentage compensation in total consumption c is calculated. In the benchmark economy, the welfare gain of eliminating social security for an unborn agent (before the realization of all contingencies), denoted as wq, is where Vf and Vp refer to the value in the privatized and PAYG system, respectively. To better understand the aggregate welfare effects, I classify all agents by the types of shocks to labor productivity they receive at the beginning of the first age. Agents who Zontov/My,!) (2.9) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 38 receives initial productivity shock r]i, i = 1,7, are referred to as type-i agents. For a newborn type-i agent, the welfare gain of eliminating social security, denoted as Wi, is Note that if the given agent experiences welfare loss under the privatized system this number is negative. For the one-asset economy, the calculation of CV also follows equation (8 ) and (9), except that the state space for value function is decreased by one dimension. General Features Table 2.5 summarizes the aggregate statistics of the two economies in the alternative social security systems. I see that, first, eliminating social security leads to several changes specific to the economy with housing. For example, the aggregate demand for housing rises, as reflected by a 34.64% increase in the PhH/Y ratio and a 3.74% rise in the housing price. The fraction of households holding positive non-housing assets, in addition, rises by about 9% when social security is privatized. Second, for both economies eliminating social security boosts household savings in the form of interest bearing assets, as evidenced by a rise in K /Y ratio and a fall in interest rates.29 Third, the increase in aggregate wealth in the one-asset economy (48.6%) is about 5% higher than that in the economy with housing, as measured by K + PhH. 29Note that interest rates at the initial steady state are different between the two economies. This is because in the economy with housing the interest rate stands for the rate of return for non-housing assets only, while in the standard economy it stands for the rate of return for total assets. (2.10) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 39 Benchmark Economy One-Asset Economy Variables Ini. St. St. Fin. St. St. Ini. St. St. Fin. St. St. Replacement rate, d 48.3% 0 % 48.3% 0 % r 6.73% 3.18% 5.1% 2.5% K /Y 1.682 2.152 3.1 4.016 PhH/Y 1.306 1.758 - - K 2.859 4.014 5.082 7.609 Housing price, ph 0.882 0.917 - - PhH 2 . 2 2 0 3.279 - - Prop, of HHs w/ saving 47.1% 56.9% - - Note: r refers to the rate of return for non-housing assets in the benchmark economy, and the rate of return for total assets in the one-asset economy; K/Y (similar for K) denotes non-housing capital output ratio in benchmark economy and aggregate capital output ratio in one-asset economy. Table 2.5: Aggregate Statistics for Alternative Economies and Social Security Systems Life Cycle Profiles of The Benchmark Economy I now explore the life-cycle patterns of wealth portfolio in the benchmark economy. I focus on two questions. First, how does the life-cycle pattern of (owner- occupied) housing assets interact with those of non-housing assets over the life cycle? Second, how does eliminating social security affect these life-cycle patterns? The life cycle patterns of the two assets are derived by averaging over simulation for 1 0 , 0 0 0 households with the initial stochastic earnings distribution following the benchmark calibration. All households start life without either non-housing or housing assets. Figure 2.1 illustrates how the average levels of the two assets evolve over the life cycle. In each steady state, the life-cycle profiles of both assets are hump-shaped. Two points are worth emphasizing, moreover. First, the hump for non-housing assets is much more pronounced than that for housing assets. This difference results from R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 40 the different purpose for homeowners to accumulate these two types of assets due to missing rental market. Households accumulate non-housing assets mainly for financing consumption in retirement, when their income is significantly less than that at work ing periods. As a result, I see that though the average level of non-housing assets is very high around the retirement age, it runs down quickly thereafter. In contrast, the holding of housing assets is largely for consumption of housing services. As I can see, the liquidation of housing assets is slow after retirement age, because housing service flows are still valued by retirees. Second, the impacts of eliminating social security on the life cycle profiles of the two assets exhibit sharp difference. On the one hand, eliminating social security encourages households to accumulate more housing assets throughout the life cycle, especially during the early working periods. On the other hand, this policy experiment leads to more borrowing early in life instead. This indicates that the degree of financial market friction becomes smaller as eliminating social security allows homeowners to accumulate more housing assets. Such a decrease in financial market friction in turns allows homeowners to accumulate more housing, especially early in life. In addition, while the housing assets crowded by social security are largely concentrated in the early part of life, social security crowds out a large quantity of non-housing assets around the retirement age. As housing and financial assets serve different purposes for consumption over the life-cycle, such an asymmetric effect potentially influences the degree of substitutability of social security for these two types of assets. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 41 16 i Non-housing, w/o SS © in o > c (A 3 O 5 C o z o © 3 « > 9 ) C Non-housing, wI SS 3 O X Housing, w/o SS C I < Housing, w/ SS 90 Age Figure 2.1: Age Profile of Beginning-of-Period Assets The above points are also reflected in the life cycle patterns of the proportion of households with positive non-housing assets. As shown by Figure 2.2, less than half of the households have positive non-housing assets early in life. Only when they become middle-aged, most households start to save in the form of non-housing assets. Another interesting point is that during the first half of the working periods eliminating social security significantly lowers the fraction of households with positive non-housing assets. This also indicates that eliminating social security reduces the degree of financial market friction. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 42 1.2 w/o SS C D c '« 3 O X c o Z w/ SS '5 5 0 0 . .c 1 I 0.4 o CL 2 Q . 0.2 - 70 80 90 20 30 40 50 60 Age Figure 2.2: Proportio of Households with Positive Non-housing Assets (Benchmark Economy with Owner-occupied Housing) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 43 CV (%) Benchmark Economy One-Asset Economy Difference Aggregate Welfare Gain wo 17.11 9.12 7.99 Welfare Gain Across Types of Agents Wi 23.08 14.38 8.70 W 2 2 0 . 6 6 12.46 8 . 2 0 w3 17.45 10.32 7.13 W 4 15.90 8.28 7.62 w5 14.41 6.19 8 . 2 2 W o 12.74 4.11 8.63 Wj 10.71 2.23 8.48 Table 2.6: The Overall Welfare Gain of Privatizing Social Security O verall W elfare Effects Table 2.6 reports the overall welfare effects of eliminating social security for the economy as a whole, as well as for each type of agents. As its first row shows, by living in the privatized system an unborn agent in the benchmark economy experiences a welfare gain almost twice as much as his counterpart in the one-asset economy. This large difference, moreover, holds for each type of agents. Furthermore, though for both economies the welfare gain is decreasing in the level of initial productivity shock, households with the highest initial productivity in the benchmark economy seem to be much more in favor of eliminating social security than their counterparts in the one-asset economy. To summarize, my results show that incorporating housing into the life-cycle framework substantially increases the overall welfare gains of eliminating social security, despite the fact that in the presence of housing the increase in aggregate wealth is somewhat smaller. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 44 Q uantifying th e R oles o f M arket Frictions Note that the large difference of the welfare gain of eliminating social security mentioned above results from the different degree of market frictions inherent in these two economies. In the economy with owner-occupied housing, three market frictions are introduced: rental market friction, financial market frictions, and housing transaction costs. Moreover, all these three frictions interact with each other. By contrast, in the standard one-asset economy, only financial market friction exist. My question therefore is how the introduction of market frictions related to housing contributes to a higher welfare gain of eliminating social security than the intro duction of financial market friction alone? To measure the quantitative role of market frictions, I set up a baseline economy in which all the above mentioned market frictions are missing. Such an economy corresponds to a one-asset economy with no borrowing constraint. The value for j3 is adjusted accordingly to generate a capital-output ratio of 3.1 in the initial steady state. This results in a = — 5 and (3 = 0.9856.3 0 Finally, borrowing in this economy should be interpreted as choosing a negative amount of net worth. I do the same policy experiment in this economy and denote the compensating variations in this frictionless economy as CVq. I now measure the impact of each market friction on the overall welfare effects of eliminating social security in the economy with owner-occupied housing. For this purpose, I also set up two counterfactual economies in which either housing transaction costs are zero (ip = 0 ) or there is no limited borrowing in the non-housing asset market. 30Under this borrowing limit, the simulation results of 10,000 households show that no households will borrow over an amount of -0.6. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 45 In the latter economy, 7 is set to be a sufficiently large negative number so that no households in simulation would ever hit that borrowing limit. This results in 7 = — 10. To isolate the overall welfare implication of financial market friction, I first compute the compensating variations in the economy without borrowing constraint, denoted as CV),. Then the difference between CVb and the compensating variations in the benchmark economy (denoted as CV2 ) corresponds to the welfare gain caused by the presence of financial market friction. Similarly, the welfare impact of housing transaction costs is measured by the difference of CV between the economy without housing transaction costs, denoted as CVc, and the benchmark economy. Finally, I measure the role of financial market friction in the one-asset economy as the difference of the compensating variations between the frictionless economy and the one-asset economy, denoted as CV\. Table 2.7 summarizes the roles of various market frictions for the welfare ef fects of eliminating social security in an economy with owner-occupied housing. The first column of this table reports the overall welfare effects of eliminating social secu rity in economies with different specifications of market frictions; the second column reports the corresponding difference between the alternative economies and the bench mark economy, thereby indicating the welfare effect of each market friction. The resid ual in the last row is obtained by subtracting the sum of the welfare difference caused by housing transaction costs and limited borrowing from the corresponding difference between the benchmark and the one-asset economy. It can be regarded as a measure ment of the welfare effects of eliminating social security arising from the introduction of missing rental market. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 46 Experiment cvx cv2 - cvx Benchmark Economy (CV2 ) 17.11 0 Frictionless Economy (CVo) 9.05 8.06 No Transaction Costs (CVc) 16.97 0.14 No Limited Borrowing (CV5) 10.64 6.47 Residual - 1.45 Note: CVX stands for CV2, CVi, CVb or CVc for each of the above four economies; the value of Residual is computed as (CV2 — CVi) — (CV2 — cvb ) - (CV2— CVc). Table 2.7: The Roles of Market Frictions in the Benchmark Economy Table 2.7 indicates that in an economy without rental market, the presence of financial market friction is quantitatively most important in affecting the welfare gain of eliminating social security. This is evidenced by the fact that when this market friction is removed, the welfare gain of eliminating social security drops to 10.64, indicating about 80% of the increase in the overall welfare gain of eliminating social security is attributable to the introduction of this friction in an economy without rental market. By contrast, removing transaction costs leads to only a small decrease in the welfare gain of eliminating social security. Finally, the value of the residual indicates that about 18% of the increase in welfare gain of eliminating social security can be accounted for by the missing of the rental market per se. Table 2.8 shows the contribution of financial market friction to the welfare effect of eliminating social security in an economy with perfect rental market. I see that with perfect rental markets, financial market friction only increases the welfare R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 47 Experiments CVX CV\ — CVX One-Asset Economy (CVi) 9.12 0 Frictionless Economy (CVo) 9.05 0.07 Table 2.8: The Role of Financial Markets Frictions in the Economies with Perfect Rental Market gain of eliminating social security by very little. This indicates that degree of rental market friction is the key for financial market friction to affect the welfare effects of eliminating social security. In summary, the much higher welfare gain of eliminating social security in my benchmark economy than in the standard one-asset economy are due to two reasons: First, I introduce two additional market frictions, between which the rental market friction plays the major role. Second with missing rental market, the role of financial market frictions become much larger. 2.3.2 W elfare D ecom position I now decompose the overall welfare effects of eliminating social security into several components. The main purpose of this exercise is to find what are the major mechanisms through which the introduction of missing rental market and transaction cost contribute to the higher welfare gain of eliminating social security and what are the major channels that cause financial market friction to contribute to a higher welfare gain in an economy with missing rental market? R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 48 Method The potential channels for the welfare impact of eliminating social security include the following: 1) General equilibrium effects, that is, welfare effects of elimi nating social security arising from changes in equilibrium factor prices. 2) Imperfect annuity effects. Without private annuity markets, the annuity form of social secu rity benefits provides partial insurance against mortality risks, leading to a reduction in precautionary savings and accidental bequests. Without social security annuities, households in the privatized system bear a welfare loss. 3) Income risk sharing effects. The nonlinear correlation between social security contributions and benefits provides within-cohort redistributions among retirees. As households wish to smooth consump tion over the life cycle, this redistribution further affects consumption at all ages before retirement, thereby providing partial risk sharing against idiosyncratic income uncer tainties. Households in the privatized system, therefore, suffer a loss without this risk sharing opportunity. 4) Intertemporal smoothing effects. When households are credit constrained, as mandatory savings the payroll tax finance of social security benefits forces households to postpone consumption until they retire. As a consequence, the life-cycle consumption and saving behavior is distorted. The elimination of payroll tax ation thus improves households’ capability in achieving a desirable life-cycle pattern of consumption and saving, and raises the individuals’ welfare. 5) Direct wealth effects. When the internal return of social security contributions, i.e. g + n, is not equivalent to the market return on capital, r, eliminating social security will directly change the lifetime wealth, thereby affecting the welfare of households. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 49 To decompose the overall welfare effects, I conduct a number of counterfac- tual experiments similar to those in Storesletten et al. (1999), in which progressively fewer candidates are at work. Specifically, to isolate the general equilibrium effects, I hold prices fixed at their levels in the initial steady state and compute the associated compensating variations, denoted as wp. The difference of the welfare gain between this economy and the benchmark economy, w q — wp, constitutes the magnitude of gen eral equilibrium effects. Along a similar vein, to identify imperfect annuity effects of the PAYG system, I construct an economy with perfect annuity markets and com pute the resulting welfare gain, denoted as wa. The imperfect annuity effects can then computed as wp — wa. Next I shut down the income uncertainty and computed the associated compensating variations, denoted as w( i- The difference wa — w( i measures the income risk sharing effects. Finally, W d measures the welfare gain attributable to either the intertemporal smoothing effects or the direct wealth effects. Decomposition Results I summarizes the results of welfare decomposition for both the benchmark and frictionless economies in Table 2.9. the impact of housing market frictions on each channel is measured by the difference between the two economies of the compensating variations attributable to each channel. The top and the bottom panels of this table report the compensating variations under each alternative experiment, and the welfare gain of eliminating social security attributable to each of the above-mentioned compo nents, respectively. For example, according to the first column of the bottom panel, R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 50 CV (%) Benchmark. Frictionless Difference Welfare Gains Overall Welfare Effects, w q 17.11 9.05 8.06 Partial Equilibrium, wp 6.90 2.87 4.03 Perfect Annuity, wa 7.99 5.10 2.89 No Uncertainty, 9.48 6.99 2.49 Welfare Decomposition General Equilibrium Effects, u;o — wP 10.21 6.18 4.03 Imperfect Annuity Effects, wp — wa -1.09 -2.23 1.14 Income Risk Sharing Effects, wa — wj -1.49 -1.89 0.40 Int. Smoothing +Wealth Effects, w,i 9.48 6.99 2.49 Table 2.9: Results for Welfare Decomposition households in the benchmark economy enjoy a 1 0 .2 1 % of welfare gain due to a change in equilibrium prices, the largest source of the overall welfare gain of eliminating social security. Privatizing social security, moreover, creates a welfare gain of 9.48% due to either improved intertemporal consumption smoothing or an increase in lifetime wealth resulting directly from the elimination of payroll taxes. On the other hand, homeown ers in the privatized system suffer a welfare loss of 1.09% and 1.49%, respectively, when the public insurance roles of the PAYG system against idiosyncratic income risks and mortality risks are missing. As shown by the right column of Table 2.9, the major channels for housing market frictions to affect the welfare gain of eliminating social security are the gen eral equilibrium effects and the channel standing for either intertemporal consumption smoothing effects or direct wealth effects. These two components account for about 81% of the contribution by the introduction of housing market frictions. By contrast, though the presence of housing market frictions significantly lowers the importance of R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 51 CV (%) One-asset Frictionless Difference Welfare Gains Overall Welfare Effects, w q 9.12 9.05 0.07 Partial Equilibrium, wp 2.95 2.87 0.08 Perfect Annuity, wa 5.12 5.10 0 . 0 2 No Uncertainty, W d 7.29 6.99 0.30 Welfare Decomposition General Equilibrium Effects 6.17 6.18 -0 .0 1 Imperfect Annuity Effects -2.17 -2.23 0.06 Income Risk Sharing Effects -2.17 -1.89 -0.28 Int. Smoothing + Wealth Effects 7.29 6.99 0.30 Table 2.10: Results for Welfare Decomposition for the economies with perfect rental market the risk sharing roles of the PAYG system, the corresponding welfare differences are quantitatively small relative to the difference of the overall welfare gain. Table 2.10 indicates the role of financial market frictions in an economy with perfect rental market. I see that the major channel for this market friction to in crease the welfare gain of eliminating social security is the channel standing for either intertemporal consumption smoothing effects or direct wealth effects, though its quan titative role is much smaller than its counterparts in the economy with housing. Also, the introduction of this market friction actually makes the income risk sharing effects of social security larger, contrary to the case in the presence of housing markets frictions. This friction, in addition, has no significant impact on the general equilibrium effects and partial annuity effects. Comparing the right column of Table 2.9 and 2.10, I find that the major channels for the introduction of housing market frictions to create a higher welfare gain R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 52 of eliminating social security than the introduction of financial market friction alone are general equilibrium effects and intertemporal consumption smoothing effects. These two channels accounts for about 78% of the increase in the welfare gain of eliminating social security. Therefore, I now analyze via these two channels how market frictions contribute to a higher welfare gain of eliminating social security.31 General Equilibrium Effects According to my welfare decomposition, the general equilibrium effects of eliminating social security stems from a change in equilibrium factor prices. Appendix 2 . 6 shows that the magnitude of this effect relies mainly on the quantity of household net interest bearing assets over the life cycle, a determinant of interest rate effect, and the sensitivity of the response of the interest rate to eliminating social security, a determinant of both interest rate effect and wage effect. I now explore how the introduction of housing market frictions contributes to a higher welfare gain by affecting these two elements, compared to the introduction of financial market friction alone?. The quantity of household lifetime net interest bearing assets is mainly af fected by the introduction of rental market friction. First, the missing of rental market links consumption of housing services to housing assets. As the life cycle patterns of asset accumulation show, this market friction increases all homeowners’ demand to borrow to finance housing purchase early in life. Moreover, it induces middle-age homeowners to allocate part of their wealth to housing assets, a non-interest-bearing 31The Appendix 7.4 contains an analysis of how housing market frictions lower the net risk sharing benefits of the PAYG system against both idiosyncratic risks and mortality risks. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 53 asset when rental market is missing. To the contrary, in the one-asset economy with financial market friction alone, all households hold their wealth in the form of interest bearing assets. As a result, homeowners, especially those who are young, accumulate a smaller quantity of interest bearing assets than their counterparts in the standard life-cycle economy. This wealth composition permits a smaller welfare cost caused by the negative income effect of a fall in the interest rate when social security is eliminated than in the standard economy. Intuitively, a fall in the price of present consumption (interest rate) is more beneficial to homeowners than their counterparts in the stan dard economy, when a sizable proportion of homeowners’ assets is held for immediate consumption, rather than postponing consumption until late in life. On the other hand, the presence of housing transaction costs leads to a more drastic response of the interest rate to eliminating social security in the benchmark economy (as shown by Table 2.5) . 32 When a transaction cost is linked to the sale of housing, the adjustment of housing assets become costly and housing assets relatively illiquid. Moreover, this adjustment cost introduces a ‘committed expenditure risk’ on the part of nondurable consumption.33 That is, given the non-housing assets avail able today, to avoid liquidating housing asset in case of bad shocks, households tend to adjust nondurable consumption more frequently after they build up housing assets. Therefore, to prevent fluctuations of both types of consumption, the increase in pre 32Obviously, the introduction of missing rental market dampens the response of non-housing assets, and thus, the responses of K / Y ratio and interest rates to privatizing social security. And the intro duction of financial market frictions allowing collateral borrowing gives rise to less increase in interest bearing assets than the introduction of no borrowing constraint. 33The notion of committed expenditure risk was first mentioned by Fratantoni (2001). R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 54 cautionary buffer tends to be higher in the benchmark economy when social security is privatized.34 Note that the interest rate effect is also the key to understand the relative magnitudes of the general equilibrium effects across different types of agents, which are illustrated by Figure 2.3. For each type of agents, the left column in this graph stands for the general equilibrium effects under the one-asset economy and the right column for the corresponding values in the benchmark economy. Two points are noteworthy. First, for both economies the magnitude of general equilibrium effects is decreasing in the level of initial labor productivity. Second, while the general equilibrium effects in the benchmark economy axe strictly positive for all types of agents, under the one-asset economy it causes a welfare loss for the highest two types of agents. The intuition is as follows. Clearly, the lower the initial productivity, the more borrowing or the less saving an agent wishes to take in the early part of life. As a result, when eliminating social security pushes down the interest rate, agents with lower initial productivity tend to derive a higher welfare gain from a decline in the price of present consumption (in the economy with housing), or suffer less from a decline in the marginal return to savings (in the one-asset economy). This explains why the magnitude of general equilibrium effects declines in the level of initial productivity shock in both economies. Moreover, as agents of the highest two types in the one-asset 34Alternatively, Chetty (2003), Chetty and Szeidl (2003) and Postlewaite, Samuelson and Silverman (2004) show that households will be more risk averse in the presence of housing transaction costs. Given this result, it is straightforward that the presence of housing transaction costs increases the need for precautionary savings. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 55 2 0 - r i n n 1 a c s s F L n n O B e n c h m a r k ■ O n e - A s s e t 1 1 1 1 I 1 1 H 1 1 1 1 2 3 4 5 r 1 T y p e o f A g e n t s Figure 2.3: General Equilibrium Effects economy save the most, the negative interest rate effect dominates the positive wage effect when social security is privatized. As a result, they suffer a welfare loss from the general equilibrium effects. Intertemporal Consumption Smoothing Effects If I shut down the idiosyncratic income risks, assuming perfect annuity mar kets and fixed prices, the resulting economywide compensating variations measure the welfare gain associated with either the improved intertemporal consumption smoothing or the increase in the lifetime wealth arising directly from a change in payroll tax rates. As the Table 2.9 and 2.10 shows, the contribution of housing market frictions to this channel (2.49) is much higher than the corresponding contribution by financial market R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 56 friction in an economy with perfect rental market (0.30). Note that the intertemporal temporal consumption smoothing effects does not appear if no households are borrow ing constrained. This indicates that the introduction of housing market frictions leads to a significantly higher intertemporal consumption smoothing effects than the intro duction of financial market friction alone.35. The intuition is as follows. In a simple deterministic partial equilibrium en vironment, when households are borrowing constrained, the PAYG system creates a distortion of intertemporal consumption smoothing over the life cycle by postponing consumption until late in life. This is because the taxes paid by young generations reduce their disposable income. The presence of missing rental market makes this dis tortion more severe. This is because in this setting homeowners hold housing assets for immediate consumption of housing services. As shown by Figure 2 .1 , due to the financial market friction, the payroll tax finance of social security benefits crowds out a large quantity of housing assets, especially in the early part of life. Such a distortion, therefore, will have a more distortionary impact on the intertemporal consumption smoothing than that in an economy with perfect rental markets, where consumption of housing services is not necessarily linked to housing assets holdings. As a result, in an economy with housing market frictions, eliminating payroll tax finance provides a 35The introduction of missing rental markets makes the welfare gain from direct wealth effects smaller. Intuitively, the potential welfare benefit from the direct wealth effects originates from a higher market return for interest bearing assets than the rate of return for social security. In a model without rental market, as a sizable proportion of household savings is used for immediate consumption, the welfare gains arising from a higher return to save in the from of interest bearing assets than in social security shall be lower. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 57 larger welfare gain in terms of intertemporal consumption smoothing than in a standard life-cycle economy. Sum m ary My analysis shows that incorporating housing into the life-cycle framework gives rise to an overall welfare gain of eliminating social security almost twice as much as in the standard life-cycle economy. This difference is mainly attributable to the intro duction of the rental market frictions. This market friction mainly works through the general equilibrium effects and due to its presence, the introduction of financial market frictions significantly raises the magnitude of intertemporal consumption smoothing effects. 2.4 S en sitiv ity A n alysis In this section, I examine whether my major finding is sensitive to the elasticity of substitution between non-housing consumption and housing consumption, initial housing endowment, and the intertemporal elasticity of substitution. In each sensitivity analysis, I recalibrate the parameters /3 and 9 so that the model generated K /Y and PhH /Y ratios at the initial steady state match the data. 2.4.1 Intratem poral Elasticity of Substitution I first check the sensitivity of my welfare results in the benchmark economy to the elasticity of substitution between c and h in the utility function. Intuitively, as non-housing consumption become more substitutable for housing services, homeowners R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 58 CV (%) v — 0 v = 0.4 Difference Welfare Gains Overall Welfare Effects, w q 17.11 15.74 1.37 Partial Equilibrium, wp 6.90 6.73 0.17 Perfect Annuity, wa 7.99 7.88 0 . 1 1 No Uncertainty, W d 9.48 9.79 -0.31 Welfare Decomposition General Equilibrium Effects 1 0 .2 1 9.01 1 . 2 0 Imperfect Annuity Effects -1.09 -1.15 0.06 Income Risk Sharing Effects -1.49 -1.91 0.42 Int. Smoothing + Wealth Effects 9.48 9.79 -0.31 Table 2.11: Robustness to Elasticity of Substitution Between c and h will depend less on the accumulation of housing assets to satisfy their consumption de mand. As a result, the role of housing market frictions on the welfare costs of imposing social security will become smaller. To check the sensitivity against a higher elasticity of substitution between c and h, I set v — 0.4, a value in line with the estimation results by McGrattan et al. (1997) and Rupert et al. (1995) under the sample of single male. Table 2.11 reports the sensitivity analysis results under different values of v. I find that when v = 0.4, the welfare gain of eliminating social security only decreases by 1.37%. In other words, most of the increase in welfare gain caused by the introduction of housing market frictions still remains even when non-housing consumption becomes more substitutable to housing consumption. The right column of the lower panel shows that the major channel for such a decrease in welfare gain is the general equilibrium effects. Intuitively, as consumption become more substitutable for housing services, households will accumulate less housing assets and more non-housing assets. As a R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 59 ________ CV (%)__________g = 0 £ = 0.1 g = 0.3 Welfare Gains Overall Welfare Effects, w q 17.11 16.05 14.65 Table 2.12: Robustness to Initial Housing Endowment result, the lifetime net interest bearing asset increases. This leads to a larger negative income effect of a fall in the interest rate when social security is eliminated. 2.4.2 Initial H ousing Endowm ent In the benchmark, I assume that households are born with no housing assets. As my period utility function obeys Inada conditions, this assumption may potentially bias up the welfare gain of eliminating social security, especially for agents with low initial labor productivity. This is because a small increase of housing consumption may leads to a large utility gain when consumption level is small. To check the robust of my results to this assumption, I specify the following utility function u (c, h) = {9cv + {1 - 6) {h + e)v) ^ * - 1 1 — a where £ can be regarded as the housing endowment when an agent is born. I try two alternative values of £, 0.1 and 0.3. The latter roughly corresponds to one third of the weighted average wage income of a new-born household or twice the wage income of a newborn agent with the lowest initial productivity. Table 2.12 reports the result. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 60 Table 2.12 shows that when initial housing endowment increase to 0.3, the welfare gain of eliminating social security fall by 2.46%. In other words, about 70% of the increase in welfare caused by the introduction of housing market frictions still remains. Hence, I conclude that my results regarding the roles of housing market frictions on the welfare gain of eliminating social security is robust to the initial housing endowment. 2.4.3 Intertem poral E lasticity o f Substitution I now examine the robustness of my results to the intertemporal elasticity of substitution. Since my utility function takes the CRRA form, the intertemporal elas ticity of substitution is 1/cr. The parameter a can potentially affect the magnitude of the welfare gain through two channels. On the one hand, a lower a implies a higher intertemporal elasticity of substitution for consumption. Hence, households are more willing to postpone their consumption and save more today. As the unfunded social security system can be regarded as a vehicle of mandatory saving, which partially sub stitutes the desired savings, eliminating social security shall lead to a smaller increase in savings than the case with a = 2. Therefore, under a higher intertemporal elasticity of substitution, both the general equilibrium effects and the intertemporal consumption distortion created by the unfunded systems will be lower. As a result, the welfare cost of imposing PAYG systems will be lower at lower a. On the other hand, a lower a means a lower degree of risk aversion to both idiosyncratic income risks and mortality risks. Hence, agents in the privatized system will have less incentive to take advantage of the enhanced opportunities of private risk sharing or self-insurance. Both channels R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 61 leads to a smaller welfare gain of eliminating social security for both economies. The picture is reversed when the parameter a becomes larger. To examine whether the difference of the welfare gain between the two economies remains significant even though its absolute magnitude might decrease under a lower cr, I set a = 1.5.36 I adjust the parameters (5 and 6 in the economy with housing and the parameter (3 in the one-asset economy accordingly to target the same ratios as previous. Table 2.13 reports the welfare results for both economies with a = 1.5. I see that as the magnitude of the overall welfare gain become smaller in both economies, its absolute difference between these two also becomes somewhat smaller than in the case < j = 2. However, now the welfare gain of the economy with housing is more than twice that in the one-asset economy. The welfare decomposition shows that this decrease of the absolute difference in welfare gain stems mainly from the decrease of the gap of the general equilibrium effects between the two economies. In contrast, the differences of the other three components between the two economies are close in values to the case <7 = 2. Therefore, my finding that incorporating housing creates a much higher welfare gain of eliminating social security is robust to alternative specifications of a. 36We also examine the robustness for a = 4. We find that while the magnitude of the absolute welfare gains and of each of the four components move in the opposite direction as in the case for < 7 = 1.5, the difference between these two economies is robust. The results are available from the author upon request. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 62 CV (%) with Housing One-asset Difference Welfare Gains Overall Welfare Effects, w q 12.95 5.26 7.69 Partial Equilibrium, wp 6.34 2.23 4.11 Perfect Annuity, wa 7.63 4.53 3.10 No Uncertainty, wr i 9.17 6.84 2.33 Welfare Decomposition General Equilibrium Effects 6.61 3.03 3.58 Imperfect Annuity Effects -1.29 -2.30 1 .0 1 Income Risk Sharing Effects -1.54 -2.31 0.77 Int. Smoothing -1 -Wealth Effects 9.17 6.84 2.33 Table 2.13: Welfare Gains of Eliminating Social Security (sigma=1.5) 2.5 C on clusion In this chapter, I study the long-run welfare implications of social security for homeowners in a model calibrated to the U.S. economy. To construct the linkage between consumption of housing services and holding of housing assets and to examine the impact of this linkage to social security, I introduce three market frictions: a missing rental market, financial market friction interacting with housing purchase and housing transaction cost. My main finding is that the overall welfare gain of eliminating social security is almost twice as large in an economy with housing market frictions as in a standard life-cycle economy, despite the fact that in the presence of housing, the increase in aggregate wealth created by eliminating social security is smaller. The major reason for this difference is that the incorporation of housing market frictions makes housing and financial assets serve different purposes for homeowners. That is, homeowners hold housing assets to derive immediate consumption from housing durable services, rather R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 63 than to postpone consumption until late in life. Therefore, as mandatory saving for future retirement, the PAYG system is a worse substitute for household savings when a large fraction of household assets is held for this purpose. All previous studies on the welfare effects of alternative social security arrange ments are conducted in models with financial assets alone. The implicit assumption behind these studies is that rental market is perfect so that housing and financial as sets are perfect substitutable. Nevertheless, owner-occupied housing, accounts for the largest share of net worth for most households in the U.S. A contribution of my chap ter is to demonstrate that the presence of owner-occupied housing or housing market frictions has a very large impact on the welfare effects of eliminating social security. My findings thus call for an explicit incorporation of housing market frictions in future research on alternative social security arrangements. It is important to note that my model leaves out several issues that deserve future research. First, in the current framework, it would be interesting to study the transition path of eliminating social security when housing price is endogenous. An explicit exploration of financing the social security liabilities along the transition path, for example, might help us understand why the unfunded social security system is still the status quo, despite the substantial welfare benefits of eliminating it found in this chapter. Especially, which cohorts and wealth quintiles in the current population will be the winners and losers of this reform scheme? Does the majority gain? Second, in my economy bequest motives are absent and all agents were born without housing as sets. The model I present has the potential to distinguish the impact of social security R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 64 reform on two types of individuals: those who receive housing as bequests and those who do not. Third, in my economy all agents were born ex ante homogeneous. How ever, as the importance of housing assets is different for households in different wealth quintiles, it is worthwhile to explore in detail the welfare implication of social security reform on households for different wage groups and wealth quintiles. Fourth, I restrict ourselves to one particular social security reform scheme, i.e. a full privatization of the social security system. An analysis under the current framework of various other reform plans, say, a switch to Personal Security Accounts (PSA) or age-dependent con tribution rates, will enrich my understanding of the role of housing in social security reform and the impact of social security reform on housing markets. Furthermore, the current framework can be extended to study the degree of inefficiency of an unfunded social security system and the optimal design of the social security reform.37 Finally, in my model housing is perfectly divisible and rental markets of housing are missing. It is expected that in the presence of rental markets, which are imperfect due to moral hazard problems, social security reform will affect the households’ welfare via its im pacts on the home ownership ratio. In addition, my life-cycle framework with owner-occupied housing is suitable to address a variety of other macroeconomic and policy issues. Such issues may include the impact of demographics, say, the baby boom and population aging, on housing prices in the low frequency and the welfare implications of other fiscal policies and 37See Conesa and Garriga (2004) and Huggett and Parra (2004) for an exploration of this issue in a standard life-cycle economy. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 65 their reform proposals (e.g. a switch to a more redistributive tax system or a switch of the tax base from capital income to labor income). 2.6 A p p en d ix 2.6.1 Calibration o f the Benchm ark Econom y In this section, I describe the detailed procedure of my calibration for the benchmark economy. I use data from the 2003 revision of National Income and Product Accounts (NIPA) and Fixed Asset Tables (FAT) of Bureau of Economic Analysis (BEA) for the years 1954-2000. The measurement of the macroeconomic aggregates follows Cooley and Prescott (1995) and Diaz and Luengo-Prado (2002) with special attention paid to the following issues. First, I exclude government capital from the definition of the capital stock, as my interest is on the private sector. I define residential structures (denoted as RS) as the stock of the private residential structures, and non-housing capital K as the sum of the private fixed assets, consumer durables, inventory stock and net foreign assets minus the stock of private residential structures. Output Y corresponds to GNP plus service flows from consumer durables minus service flows from housing (denoted as Yh), which is imputed as the rental value of both tenant-occupied and owner-occupied dwelling units. The definitions of R and R correspond to the definition of K and RS. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 66 The capital share ap in private fixed capital (excluding consumer durables and residential structures) is then computed as UCI - Y h + DEP ap ~ GNP - AC I - Yh where UCI = rental income + net interest + corporate profit refers to unambiguous capital income. D EP denote consumption of fixed capital. And AC I = proprietors' income + indirect business taxes. Denote Y^p = ap(GNP — Yh) as the income of non-housing capital (excluding consumer durables) and YS ( i as the service flows from consumer durables, which is computed following Cooley and Prescott (1995). Then the capital share in the output function a is computed as Ykp A Ysd a - G NP + Ysd- Y h This gives a value 0.2732 for a. I compute Sk and Sr according to the laws of motion for K and RS, respec- Jr Jfc lively, on the balanced growth path. The ratios -= -£ and — are 0.0455 and 0.1201, Jib K K RS respectively. Accordingly, Sk = 0.0951 and 5r = 0.0205. Finally, the ratios — and are 1.6819 and 1.0434, respectively, where output Y is measured by G NP + Ysd — Yh- Y This implies an interest rate r = a — — 6k = 6.73%. To compute the depreciation rate for the housing asset 8h, I follow Davis and Heathcote (2003) and assume that residential structures depreciates gradually at rate Sr after it has been combined with land to produce housing, while the dimension of R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 67 land remains unchanged. In equilibrium, the law of motion for housing asset can be written as Ht+1 = X ^ L * + ( ( 1 - <n X rt-!)1 -* L* + ( ( 1 - 5r)2 X rt-r) 1 ” 0 L* + ... = x } - ‘ t,L< t ‘ + ( l - 6r)1-*H t The depreciation rate for housing, Sh, can be defined by 1 _ gh = ( 1 _ jrj W This gives 6h = 0.0183. I now compute the ratio . On the balanced growth path, it is easy to show that the ratio of the stock of residential structures over the value of the housing stock satisfies R S _ ((l+flh)(l + n ) - (1 - 5^)) (1 - 0) PhH (l+gd) ( l + n ) - ( l - 6r) where gh and gd are the net growth rate of housing stock and residential structures per capita on the balanced growth path, which are equal to (1 + g)l~^ ( 1 + n)~< ‘ > — 1 and RS g, respectively. Combined with the empirical value for -p-, this gives a value for the ratio as 1.3058. 2.6.2 C om putation and A lgorithm To economize on computer memory, I define q = a + (1 — 7 )phh as the wealth held in excess of the down payment, following Diaz and Luengo-Prado (2003). Since R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 68 the lower bound of q is zero across all agents, this decreases the computational cost as sociated with the endogeneity of the borrowing constraint on assets a. The individual’s state vectors thus becomes s = {q, h,rj,j} . I can rewrite the household’s problem as v (Q , d, r],j) = max < u (c, h) + /% ^ tt (7/ 1q) v {q\ ti, rf, j + l) c ,q ^ ^ s.t. C + I V h t i + q ' - (1 - I (j))b = I(j)w(l-T)ejT] + (l + r ) q + ci h' > 0 ^ 1 - 5h^ j (1 - Ih,p) - ( 1 - 7 ) ( 1 + r) phh where the resource constraint and the law of motion for the probability measure are reformulated in terms of q. I solve the stationary equilibrium by the following steps: 1. Guess ph and r and solve for pi and Tr. 2. Solve for the individual household’s decision rules by backward recursion. 3. Use forward recursion to compute the distribution < f > , and then compute the aggregate K and X r. 4. Use equation (5) and (8 ) to update Ph and r. 5. Iterate on ph and r until convergence. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 69 2.6.3 W elfare Effects o f Privatizing Social Security in a D eterm inistic C om plete-M arket Econom y In this section, I use an Overlapping Generations model with housing to char acterize the welfare effects of eliminating social security in a deterministic complete- market economy. My purpose is to characterize the mechanisms determining the gen eral equilibrium effects and the direct wealth effects. Note that in a deterministic economy, the risk sharing benefits of the PAYG system are missing. In addition, the assumption of complete markets implies that the distortion of the PAYG system to the intertemporal consumption smoothing is also missing. The economy is inhabited by an infinite sequence of overlapping generations, with constant gross population growth rate N. Each agent was born with no assets and lives for three periods. An individual born at date t supplies one unit of labor inelastically in both date t and date t + 1 in exchange for wage income wt and wl+\, and retires at date t + 2. There are two assets in this economy: non-housing asset and housing asset, which depreciate at rate 5 and Sh, respectively. At the beginning of each period t a representative firm rents non-housing capital (at a rate of return r<) and hires labor from households to produce nondurable goods. The nondurable good can be either consumed, invested into non-housing assets or housing assets on a one-to-one basis. Households value both nondurable consumption and housing service flows. The amount of housing service consumed when agents are middle-aged and old depends on the end-of-period housing stock at the previous age, while I assume that housing service flows consumed when agents are young is a small constant. Finally, suppose R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 70 there exists a PAYG social security system that taxes agents a lump sum amount r at each of their working ages and pays benefits (IV + iV2) r when they retire. The maximization problem of a young household born at time t can be written as V(t) = max U {cuMt) + PU {cx+xMt+i) (A-l) G 2 t+ 1 ,« 3 t+ 2 i f o t + l ,fl3 t+ 2 +I32u (c3f+2, hst+2) s.t. Clt + 02t + l + fi-2 i + l = W t — T C2 t+ i + a-3 t+ 2 + h z t+ 2 = w t+1 + o-2 t+ i (1 + U + 1 — S) + h,2 t+ i ^1 — 6h j — r C3J+2 = a 3t+2 (1 + l't+ 2 — 8 ) + h,3t + 2 ^1 — S h ^ + ( N + N 2) T where cu , an ,and hu denote the level of nondurable consumption, beginning-of-period non-housing asset, and beginning-of-period housing asset for an age i agent at time t, respectively. /3 is utility discount factor. The Euler equations for the above problem are R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 71 Ucu = P ( 1 + n + 1 -~ 3) UC 2 t+ l ^C 2t+1 = p ( i + rt+2 - ~ $) UC 3 t+ 2 U clt = p ( y C2t+l(i - sh) + Uh2t+ 1 U c2 t+ 1 — P (Uc3 t+ 2 ( 1 - sh) "h Uh3 t+ 2 where the subscripts of U refer to the partial derivative of the period utility function with respect to the corresponding variables To derive the welfare impacts of eliminating social security, rewrite the value function in terms of optimal asset choices only (1 + r t + 1 — 6) ci2t+i + w t + 1 + / i 2 t + i ( l — Sh) — r — hst+2 V ( r ) = U(wt - T - a2t+1 - h2t+1 , h i t ) ( A - 2) / + pu — a,3t+ 2, h2 t+i ' +P2U ^(1 + rt+2 — S) a^t+ 2 + h3t+ 2 ( l — + (N + N 2) r , h ,3t+2 j Taking derivative on both side of equation (A-2) with respect to r and using the en velop theorem, I get R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 72 V'(r) = — UC lt — PUC 2 t+ 1 + p2UC 3 t+ 2 (N + N 2) (A-3) , T T d w t , O T T d w t + 1 , , o r r . 5 r t + l , * 2 r r . d r t + 2 C U ~ Q ^ + * ■ P U c 2 t+ ia 2 t + l g r + P t/c3t+2a 3i+2....^ ... By Euler equations, equation (A-3) becomes v '« %,.( i 1 + rt+1_ i + (i + rt+1_ J)(1 + rt+J_ i)) < A ’4) 4-r/ I a2t+i d',r t+i _____________Q 3t+2__________ dr t+2 \ C 1 , 1 + n + i - ^ dr (1 + rt+i - 6) (1 + rt+ 2 - 5) dr ) The three bracket terms on the RHS of equation (A-4) represent three channels of the welfare impact of a change in the payroll tax rate r .38 The first measures the direct impact of a change in r on the lifetime wealth. I call it ‘direct wealth effects’. The second and third terms constitute the welfare effects of eliminating social security due to a change in equilibrium factor returns, or so-called ‘ general equilibrium effects’. In particular, the second measures the impact of a change in the payroll tax on the lifetime wage earnings via its influence on the equilibrium wage rate. I call this effect the wage effect. The third term denotes the welfare effects of a change in r via its impact on the interest rate. I call it the interest rate effect. At the steady states, because all prices are constant, equation (A-4) can be rewritten as 38It is easy to see that the analytical forms of the above components under the one-asset economy are the same as in the economy with housing except that the marginal utility of nondurable goods in Equation (A-4) is replaced by the marginal utility of total consumption. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 73 V'{t) = Ucu 1 + r - 8 ( 1 + f - S )2 1 \ d w 1 + r — 8) dr 1 ___ | N + N 2 (A-5) 1 + r - < 5 + (l + f - S)2 ) dr °2t+i , Q .3 t+ 2 \ dr where r and w denote the steady state interest rate and wage rate, respectively. I now discuss the determinants of the sign and the magnitude of each of the three bracket terms in equation (A-5). The first has been extensively discussed in literature. Its sign depends on whether the economy is dynamic efficient or not. If the economy is dynamic efficient, i.e. n + S < r, where n is net population growth rate, then eliminating social security (or a fall in r) will increase the welfare gain. The magnitude of this argument, in addition, relies on the magnitude of the present value of social security benefits minus the present value of social security contributions, the inverse of so-called ‘net tax burdens’. Because < 0, the sign of the second term is always negative, indicating that the wage effect of eliminating social security is welfare improving. In addition, the larger is | ^ | , or equivalently, the larger is | ^ |, the larger is the welfare gain associated with this effect. To determine the sign of the third term, note that ^ > 0 , because eliminating social security tends to increase savings, thereby decreasing the interest rate. Therefore, the sign of this term depends critically on the sign of ( i+pLa + ' (i+piiijjg) > present value of lifetime net interest bearing assets. The smaller is the present value of lifetime net interest bearing assets, the smaller is the welfare loss (if it is positive) or the larger is the welfare gain (if it is negative) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 74 associated with a fall in the interest rate, ceteris paribus. Moreover, the magnitude of the interest rate effect depends crucially on the magnitude of The stronger is the response of the interest rate to eliminating social security the larger is this effect (in absolute value). 2.6.4 R oles of H ousing for the Risk Sharing Benefits o f PAYG Sys tem s Imperfect Annuity Effects To compute the imperfect annuity effects of the PAYG system, I assume a counterfactual economy in which perfect annuity markets exist in the private sector. The household’s budget constraint in this counterfactual economy then becomes cj + ipj(phtij+1 + a 'j+1) = I{j)w ( 1 - r) ejT] + ( 1 + r)aj +ph ( ( l - 8h') hj - y(hj, h'+i)) +(1 -I(j))b + Tr To what extent the introduction of market frictions affects the partial annuity role of social security? As shown by the second row in the bottom panel of Table 2.9 and 2 .1 0 , the introduction of housing market frictions reduces the imperfect annuity effects of social security from 2.23 to 1.09, while the corresponding drop caused by the introduction of financial market frictions alone is 0.06.3 9 This makes the welfare loss of 39 The reason for the presence of borrowing opportunity to enhance the role of imperfect annuity R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 75 H B e n c h m a r k ■ O n e - A s s e t T y p e o f A g e n t s Figure 2.4: Imperfect Annuity Effects shutting down the imperfect annuity effects of social security twice as much in the one- asset economy as in the economy with housing market frictions. Furthermore, Figure 2.4 shows that for each type of agents the magnitude of imperfect annuity effects in the benchmark economy is substantially less than the corresponding value in the one-asset economy. The key factor contributing to the difference of imperfect annuity effects be tween the two economies is the presence of missing rental market. The intuition is as follows. Typically in the one-asset economy, eliminating social security shall force households around the retirement age to hold substantially higher net worth than their effects is that the annuity form of retirement benefits increases the desired consumption of the young, which becomes more easily to finance with borrowing. This feature is emphasized by Hubbard and Judd (1987). R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 76 counterparts in the unfunded system, to prepare for retirement consumption when the annuity benefits are absent. Accordingly, for retirees in the privatized system, as their survival probabilities decline drastically, the amount of accidental bequest is significantly higher than that in the system with social security (Figure 2.5). In the benchmark economy, by contrast, with one component of consumption from housing service, households around the retirement age shall allot part of the saving increase into the increment of housing assets when social security is privatized. For retirees, this in crement in housing assets stems solely from the increasing demand for consumption of housing services due to an increase in lifetime wealth. As a result, the magnitude of this increment tends to be smaller and more persistent relative to that of non-housing assets. This is evidenced by Figure 2.6, which plots the life cycle profiles of the average bequests in these two assets. The presence of missing rental market, therefore, drives down the increase in the accidental bequest for housing assets and for net worth, and thus, lowers the demand for the partial annuity role of the PAYG system. Income Risk Sharing Effects To compute the magnitude of income risk sharing effects, I further shun down the labor income uncertainty, the labor productivity of agents at each age j is the product of ej and the average of its stochastic components at that age. The budget constraint for a representative agent with age j is R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 77 0.6 0.5 w/o SS 1.0.4 £ 3 M O O c 1 0.3 i 0 3 w/o SS 0.2 < 0.1 0 90 70 80 40 50 60 20 30 A g e Figure 2.5: Age Profile of End-of-Period Accidental Bequest (One-Asset Economy) C j + ipjiphh'^ i + a'-+1) = I(j)w (! “ T) ejVj + ( 1 + r)a,j + ph ( ( l - Sh S j hj - y(hj, h’ j+1)) +(1 — I (j))b + Tr where rjj = Ej (r/) denotes the average of the income shocks to labor produc tivity for age j agents. The income risk sharing effects for both economies are reported in the third row of the bottom panel in Table 2.9 and 2.10. As I mentioned above, the introduction R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 78 0.3 Bequest in K, w/o SS 0.25 Bequest in H, w/o SS < 2 I o i c to 3 o - f c o z o ® 3 O ) 3 0.2 Bequest in K, w/ SS 0.05 Bequest in H, w/ SS 0 1 c u 5 I -0.05 C Q O ) I - 0.1 -0.15 - 0.2 Age Figure 2.6: Age Profile of End-of-Period Accidental Bequest (Economy with Owner- occupied Housing) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 79 of housing market frictions leads to an income risk sharing effects of social security .6 8 % lower that their counterparts when financial market friction alone is introduced. Such a difference between the two economies results mainly from the presence of collateral borrowing, which is possible when housing and financial assets are not perfect substitute. In the benchmark economy, the presence of collateral borrowing enables households to share income risk via private contracts, in addition to the public risk sharing offered by the PAYG system. More importantly, this private sharing capa bility is endogenous and depends critically on the quantity of housing assets. Despite its public insurance role, the PAYG system decreases the disposable income of workers, thereby postponing the accumulation of housing assets. This will limit the extent to which homeowners make use of the collateral borrowing to insure against idiosyncratic income shocks. Conversely, eliminating payroll taxes allows households to accumulate more housing assets, thus creating more capacity for private risk sharing through col lateral borrowing. This enhanced ability of private risk sharing, accordingly, provides a partial compensation for the welfare cost due to loss of the public risk sharing role of the PAYG system. In the standard one-asset economy, by contrast, the introduction of financial market friction shuts down private risk sharing. As a consequence, the net benefit of public insurance role of the PAYG system is much higher. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 80 Chapter 3 Sustaining Social Security: A Unique Markov Perfect Equilibrium w ith M ajority Voting One of the key issue in the political sustainability of a Pay-As-You-Go (PAYG hereafter) social security system is that as inter-generational transfer program from the working to retirees, social security is featured by temporal separation between costs and benefits (Sjoblom (1985)). That is, in a standard majority voting framework, for a self- interested median voter to impose a social security tax upon himself, he must expect a social contract where future social security benefits are somehow linked to the present R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 81 level of taxes. 1 To formulate expectation of such a dynamic linkage, the literature has so far relied on two approaches. The first, so called “once-and for all voting”, assumes that the initial median voter expects full commitment by future generations to his choice of tax level for at least his lifetime.2 The second, so called “trigger strategy”, specifies that the expectation of future policy choice is based on a system of rewards and punishments in a infinite dynamic game.3 A common feature of both approaches is that the expected (and therefore actual) choice of future social security tax rate depends on states that axe payoff irrelevant to future median voters. In this chapter, I develop a positive theory of social security in a majority voting framework with differentiable Markov perfect equilibrium. My key result is that even under temporal separation of costs and benefits, in the space of linear functional there exists a unique equilibrium that self-interested median voters can have incen tive to sustain social security. The key for the existence of PAYG in this setup is the expectation that future social security benefits are positively correlated with current social security payments. This (self-fulfilled) expectation is achieved through funda mental linkages between policy choice and the payoff relevant variable, which in my model is the human capital stock.4 Specifically, the current payroll tax rate has a negative impact on the next period median voter’ s human capital stock due to its dis- tortionary effects on human capital investment and a lower human capital stock held by lrTo avoid the problem of temporal seperation of costs and benefits, several studies assume that the welfare of retirees weighs somehow in the preference of the policymaker, resorting, for example, to altruism (Tabellini,1992), probalistic voting (Katuscak, 2002 and Gonzalez-Eires and Niepelt, 2004) or gerontocracy (Mulligan and Sala-i-Martin, 1999). 2See among others, Browning (1975) and Conesa and Krueger (1999). 3See, among others, Cooley and Soares (1999) and Boldrin and Rustichini (1999). 4In a companion paper, Chen and Song (2003), we address the same question using individual physical asset as the fundamental state variable. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 82 next period’s median voter induce him to choose a higher tax rate due to the positive correlation between marginal cost of taxation and human capital stock. To formalize the above idea, I assume a small open economy. There are three generations alive at any time, the young, the middle-aged and the old. Young agents can make human capital investment to increase labor productivity after they observe the realization of the current social security tax. All agents are risk neutral and human capital investment involves a convex utility loss for young agents. At each period of time, the young and the middle-aged work and their labor incomes are taxed to finance the social security benefits of the old in the same period. I introduce intra-cohort het erogeneity by assuming each agent is born with either high or low ability. The wage income in both working ages to be proportional to the ability level and the stock of hu man capital invested at the young age. Benefits are evenly distributed across different types of the old generation. I focus on equilibria in which the middle-aged agents with low ability, called “the middle poor”, are always the decisive voter.5 By assumption of differentiability of Markov strategies, I solve for the closed-form equilibrium voting strategy, which is unique in the space of linear functional. The facts that all agents are risk neutral implies that the only payoff relevant state variables for the middle poor is his own human capital stock. The median voter expects a positive dynamic linkage between current contribution and future benefits, following the reasoning mentioned above. As 5As we will show in Section 3, this fiscal constitution can be achieved by either the assumption of dictatorsip or majority voting given that a reasonable upperbound for social security tax exists. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 83 a result, the current middle poor therefore has an incentive to choose a positive tax which equalizes current marginal cost and future marginal benefits. The usage of differentiable Markov perfect strategy, together with intra-cohort income heterogeneity, then gives rise to two effects of income inequality on the size of social security systems. The first, referred to as myopic effect, is the impact of income inequality on the policy choice of current median voter, given that he myopically ignores the impact of income inequality on the social contract. Fixing the social contract, the returns from intra-generational transfer are obviously negatively correlated with the median voter’s productivity. Therefore, a lower level of income inequality would dis courage current social security payments. On the other hand, income inequality affects the policy choice of current median voter by affecting his expectation of the future social contract. I call this channel, a channel so far ignored in the literature, the social contract effect. It is interesting to see that income inequality and the choice of median voter is negatively correlated via this effect. As income inequality, and therefore, marginal returns to current social security contribution from the channel of intra-generational transfer become smaller, the demand by the median voter for social security as an asset will become less, given a constant marginal returns from inter-generational trans fer. Hence, to restore equilibrium, the marginal returns from inter-generational transfer have to increase. With rational expectation the current median therefore tends to voter for a higher social security tax today to satisfy his increased demand for social security assets. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 84 The uniqueness of the equilibrium in my model generates the following testable theoretical predictions. First, the social security tax rate increases over time until converging to a steady state. The intuition is as follows. Since the chance to vote for social security was unexpected by the initial period middle poor when he was young, his human capital stock was relatively large. As a result, he tends to vote for a relatively small tax level due to the high marginal tax cost associated with a large human capital stock. This tax, however, creates a distortionary effect on the human capital investment by the second period median voter when he was young, making his human capital stock lower than that of the initial period median voter. A lower human capital stock further implies a higher payroll tax rate at the second period by similar argument for the initial median voter. For similar reasoning, the increase of social security tax will continue until it converges to a steady state, as the existence of such a steady state is the prerequisite for the survival of social security along transition. Second, the growth rate of social security programme is negatively correlated with income inequality. This is because the growth rate of the size of social security systems is positively correlated with the marginal response of future social security tax to the current social security tax, a dynamic linkage embedded in the equilibrium social contract. As I mentioned above, income inequality has a negative impact on the sensitivity concerning the reaction of future tax to the current tax at equilibrium. Accordingly, the growth rate of social security is negatively correlated with income inequality. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 85 Third, the steady state social security tax rate is not monotone in income inequality. Due to the opposite direction of the two channels through which income inequality affects the social security tax rate, the sign of the overall impact of income inequality on the level of social security tax depends on which effect eventually domi nates. My numerical results shows that for plausible parameters values, the steady state social security tax rate turns out to be an inverted-U function of income inequality. This indicates that when income inequality is large the social contract effect dominates while the myopic effect starts to dominate when the income inequality drops to some critical level. I then use the data of OECD countries to examine the consistency between my model’s predictions and the facts. The evidence not only provides suggestive sup ports for my model, but also sheds light on the directions for future research, both theoretically and empirically. The first prediction is clearly consistent with the history of social security systems in the OECD. I run regression of the average growth rate of social security benefits on income inequality and dependency ratio to test the second prediction. The results show that the negative impact of income inequality on social security growth can be well supported by the data for small OECD countries. This provides empirical evidence for the existence of the social contract effect. I also find a robust negative relationship between the size of social security systems and income inequality in small OECD countries. The seemingly puzzle, which cannot be explain by the previous models, can be reconciled with the third prediction. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 86 This chapter contributes to the literature on the political sustainability of social security system in several aspects. First, I shows that even in a majority vot ing framework with temporal separation of costs and benefits, social security can be sustained in a differentiable Markov perfect equilibrium. As I mentioned above, in all previous effort to construct the expected linkage between future benefits to current contribution, the expectation of future policy depends on payoff irrelevant variables, variables relevant only because they are expected to be relevant. The assumption of full commitment in Once-and-all-for voting is obviously far from realistic, though it simplifies the analysis of the relevant topics, e.g., the general equilibrium and risk sharing effects of PAYG (Conesa and Krueger, 1999) or the impact of productivity or demographic shock on the identity of the median voter (Tabellini, 2000). Trigger strategy makes a major theoretical advance by allowing repeated voting. However, the dependence of policy choice on voting history, which is payoff irrelevant, leads to indeterminacy of political equilibria. Finally, Azariadis and Galasso (2002) solves a closed-form solution of the political equilibrium associated with strategies of the young depending on aggregate physical capital. However, their usage of the two-period OG model causes nonexistence of payoff relevant state variables in a partial equilibrium for the young generation as the median voter. As a result, their solution also suffers from multiplicity of equilibria. Similar problem occurs in Forni (2005), who uses aggre gate physical capital as states in a general equilibrium framework. In their two-period OG model, the assumption of unit intertemporal elasticity of substitution and 100% depreciation rate of physical capital leads to strict proportional relationship between R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 87 consumption in both periods and current physical capitals. As a result, in their model there is no fundamental impact of aggregate physical capital on the policy choice of the young as median voter: current policy choice depends on current physical capital only because they expect future policy choice depends on future physical capital. By contrast, in my chapter the dependence of expected (and actual) policy choice only on payoff relevant state variables overcomes the indeterminacy of equilibrium solutions.6 Second, my theory provides an explanation for the evolution of the size of social security during the postwar period. Note that in models with Once-and-for-all voting and trigger strategy, the size of social security is constant over time. With the usage of Markov strategy in my model, there exists a unique equilibrium in which social security tax rate increases over time until it converges. This prediction, which is broadly consistently with the experience of most OECD countries in the postwar period, constitutes an additional advantage of the Markov strategy over the previous two approaches. Moreover, this evolution of social security is implied by the positive dynamic linkage of future social security benefits and the current social security con tribution, as required by the political sustainability of social security. Therefore, my model shows that the evolution of social security can be an intrinsic feature of social security in an political equilibrium. This complements to the limited few studies on 6McCallum (1983) finds that in a wide-class of linear rational expectation models, non-uniqueness of solutions occurs because unnecessary or ’extraneous’ components are permitted to influence expected (and therefore actual) values of endogenous variables. Maskin and Tirole (2001) argues that Markov perfect equilibrium, by preventing non-payoff-relevant variables from affecting strategivc behavior, is often very successful in eliminating or reducing a large multiplicity of equilibria in dynamic game. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 88 the dynamics of the size of social security, which resort to interactions between the size of social security and wealth inequality (Song, 2004) or saving rate (Katuscak, 2002) . 7 The methodology used in this chapter is closely related to Hassler, Rodriguez Mora, Storesletten and Zilibotti (2003, henceforth HRSZ), who analyze the evolution of the welfare state with a closed-form Markov perfect equilibrium in an OG model. HRSZ, the identity of the median voter is endogenous. The current median voter could strategically vote for sufficiently low taxation to encourage human capital investment, which is the intertemporal link in HRSZ, and thus ensure the rich to be the median voter in the following period. Since my interest is the political sustainability of social security under temporal separation of costs and benefits, I focus on a specific political equilibrium in which the median voter is always the middle aged poor. Within the context of empirical work, evidences on the relationship between the size of social security systems and income inequality are still far from being con clusive. For example, Tabellini (2000) finds that, by applying cross-country regression (more than 40 countries), the size of social security systems is positively correlated with income inequality.8 However, a significantly negative relationship across the OECD has been repeatedly reported (e.g., Lindert, 1996 and Rodriguez, 1998) and further con firmed in the present chapter. 9 This is in contrast to the conventional wisdom that 7W hile in Song (2004) and Katuscak(2004) social security would survive backward refinement, our model, as well as previous studies under once-and-for-all-voting and trigger strategy does not, that is, social secuity can sustains only if the horizon is infinite. However, we do not view this property as a drawback of the model itself. Rather, the non-backward refinement is a natural result of the temporal seperation of costs and benefits under the assumption that all voters are self-interested. 8Persson and Tabellini (2000, Chapter 6) note that the measure of inequality is bounded to be imperfect for such a large sample of countries. 9The literature also indicate the lack of positve correlation between income inequality and the size of government transfer (see, e.g., Perotti, 1996 and Benabou, 1996). R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 89 social security benefits, as a role of intra-generational redistribution, should increase with income inequality. Encouragingly, my model implies that the puzzling correlation between social security and inequality in the OECD countries can be well explained by the social contract effect ignored in the literature. The chapter is organized as follows. Section 2 describes the economic envi ronment and the voting procedure. I characterize the political equilibrium in Section 3. Section 4 examines the impact of income inequality and dependence ratio on the size of social security and the growth rate of social security, as predicted by out model. In section 5, I checks the consistency of theories with empirical evidences. In section 6 , compares the model’s implications with those under the Ramsey solution. Section 7 concludes. 3.1 T h e M od el E conom y Consider a small open economy inhabited by an infinite sequence of overlapping- generations. Each generation lives for three periods. An agent of generation t works in the first two periods of her life and is retired in the last. Labour supply in each of the first two periods is inelastic and normalized to unity. Young agents can make human capital investment to increase labour productivity. There exists heterogeneity within each cohort. Agents are associated with either high ability wa or low ability wu, referred to as the rich or poor, respectively. Let hJ t be the human capital investment of a young agent born at time t and associated R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 90 with type j, j = u,s. Human capital and wage incomes in her working ages equal h3 t and u > J h3 t , respectively. 10 The life-time wealth A\ follows Ai = { i - r tW K A 1 - r^ )wlh{ (3.1) where Tt is a proportional wage income tax rate levied on working generations and Pt+2 is the social security benefits to the retirees born at time t. To obtain analytical solution, I assume that agents have a linear-quadratic preference over life-time wealth and costs of human capital investment (e.g. Hassler et aL, 2003): ^ ^ ( ' ■ O ’ < 3 - 2 ) subject to (3.1). Solving (3.2) yields ' l ‘ = (1 _ n + 1T ± 1) to 3 ( 3 ’ 3 ) Human capital investment increases in ability and decreases in tax rates. The propor tion of the poor is a constant A in each cohort. As usual, I assume A > 1/2 such that the poor occupy the majority of population. The average productivity is normalized to unity so that wh = (1 — A wu) / (1 — A ). The weighted average wage incomes for agents born at time t, denoted by w1 , are equal to wt = ™ t+ 1 = \w uh % + (1 - A ) wshs t (3.4) 10We may assume human capital depreciates over time. Then wage incomes for the middle-aged are equal to Sw^hl, where S is the depreciation rate. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 91 Plug (3.3) into (3.4), I obtain the current tax base yf. yt = n l ( _ 1 + n2w\ - * G ^ + n ( i - T , + i z H ) (3-5) where 4 > = O n (wu)2 + (1 — A ) (wh)2^j. I use the fact that = ws/w u implied by (3.3). The future tax base yt + 1 is independent on current human capital stock: Vt+i = n - $ ( r t + ( n + Tt+1 + ^ r t + 2 ^ (3-6) where II = $ (1 + n) (l + ^ ) . The gross population growth rate is a constant n > 1. The budget balance of social security system requires that at each period social security benefits paid to the old generation should be equal to social security contributions collected from the working generations Pt = r tyt (3.7) Substitute (3.6) and (3.5) into (3.1), the indirect utility functions of the middle-aged and old with type j, denoted by and J , can be written as V m ’J ( h 3 t_ 1, T U T t+ l,T t+2) = ( l - T t j w ^ h j ^ (3.8) Tt+ 1 (n - $ (rt + (n + ^) rt+1 + % T f+2)) R v°'i (AJLi,rt, r m ) = r t4 > + n ^ - r t + 1 (3-9) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 92 Note that v°'s = v°'u since social security benefits pt+\ are evenly distributed across retirees. 3.2 P olitical E quilibrium The sequence of social security tax rates {t(}^0, or equivalently the the se quence of social security benefits * s determined through certain repeated po litical decision process at the beginning of each period. More specifically, Tt is chosen by the decisive voter (e.g., the median voter) by maximizing her indirect utility. There exists a cap on tj, denoted by t . In words, a social security tax rate greater than r is prohibited, r can be considered as the upper boundary of politically acceptable tax rates in the legislative process. Following Hassler et al. (2003), I assume that young agents do not vote. This reflects, though a bit excessively, the phenomenon that the older are more influential in the determination of public policies. 11 Voting takes place at the beginning of each period, before the young decides human capital investment. Old households have no interests at stake and thus abstain from voting. 12 For expositional ease, I shall keep the former interpretation in the context. “ For instance, Mulligan and Sala-i-Martin (1999) argue that the old have more influence in the political decision process because they have lower cost of time. Empirically, voting turnout is indeed lower for younger households (e.g. Wolfinger and Rosenstone, 1980). 12See Hassler et al. (2003) for discussions of this assumption. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 93 In Markov perfect equilibrium, r< evolves according to a policy rule T that can be written as a function of the payoff-relevant state variable h“_ i: 13 (3.10) where T : [h,h\ — ► [0, r], h = 0wu (l + (1 — r) and h = 9wu (l + axe the lower bound and upper bound of respectively. Social security benefits at time t + 1 are not predetermined for the decisive voter at time t. But given the policy rule T, she can manipulate r t+i by hf via r t. Plug (3.10) into (3.3), I have with B : [0, t ] — > [0, r]. In the following text, B will be referred to as a social contract (3.11) (3.11) defines the young’s human capital investment strategy H : [0, r] — * ■ [h, h], which solves (3.12) Combining (3.10) and (3.12) yields Tt+ 1 = T o H ( r t ) = B (rt) (3.13) 13The payoff-relevant state variable for a middle rich is her human capital h t - i - But it can be transformed to h t - \ by h ? _ i/h “_x = w s / w u . R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 94 that characterizes the evolution of social security system. Given (3.13), rt solves Tj = arg max vd ec (3-14) T t € [ 0 , T ] subject to Tt+i = B (rt) and Tt+2 = B (B (rt)). vdec is the indirect utility function of the decisive voter who determines r t . (3.8) and (3.9) show that the decisive voter’ s choice of rt only depends on whoever is decisive. Define rt = T (h“ _i) the solution of (3.14). T is a equilibrium policy rule if and only if T = T. For analytical convenience, I assume that T and H are differentiable. The cor responding equilibrium is referred to as the differentiable Markov perfect equilibrium. The definition is given by D efinition 2 A (differentiable) Markov perfect political equilibrium is a pair of differ entiable functions (T,H), where T : [h, ~ h ] — > [0,r] is the policy rule of social security tax rate and H : [0, r] — ► [h, ~ h \ is a private decision rule of human capital investment, that solve the following functional equations: (1) T (h“_i) = arg maxT t € [0 ^ ] vdev, subject to r t+ 1 = T o H (rt) and r ( + 2 = T o H (T o H (r t)). (2) H (n) = 9wu (l - rt + — f ^ 1) • In this chapter throughout, I focus my attention to a particular political equi librium, in which the middle poor turn out to be decisive for rt under majority voting. This sharply contrasts the approach that the old play at least some roles on the political R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 95 decision of social security benefits. 14 My work shows that the social security system can survive this much “tougher” political environment with expectation of future policy choice depending only on payoff relevant variables. For expositional reason, let us first solve a dictatorship equilibrium, in which the political power rests in the hands of the middle poor. The equilibrium under the dictatorship coincides with the equilibrium under majority voting, with some restrictions on parameter values. 3.2.1 D ictatorship I define "dictatorship of the middle poor" (DMP) as the regime where the social security tax rate is chosen at the beginning of each period by the middle poor. It is immediate that the policy rule T (h“) = 0 for all € [h,h\ trivially satisfies the conditions in Definition 1. This is referred to as the trivial equilibrium. I say that an equilibrium is nontrivial if T (/),“) ^ 0 at least under some hf € [h, h]. Given a nontrivial equilibrium social contract B, if exists, the future tax base (3.6) can be written in a recursive fashion yt+1 = Y (rt) = IL-<t>(rt + (n + ^ B (rt) + ^ B (B (rt))) (3.15) Differentiating Y (r<) yields Y' (rt) = — $ ^1 + (n + 1/R) B' (rt) + n (B ' (r t ) ) 2 /llj < 0 for any B' (rt). That is to say, increasing r* reduces yt+i under any equilibrium B. 14According to the United Nations, the four eldest countries in the world were Monaco, Italy, greece and Swden with 71, 78, 78 and 78 percent of their population under age 60, respectively. Therefore, the voter of median age is not a retiree but a taypaying worker, as pointed out by Mulligan and Sala-i-Martin (1999b). R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 96 Maximizing the indirect utility of the middle poor, (3.14) solves ^ = B'MYM + B W M (316) R where LHS of (3.16) captures the marginal costs of social security tax rate while the marginal returns are given by the first term in RHS. I suppress two multipliers asso ciated with the constraints rt > 0 and tj < r, since corner solutions can be ruled out by the differentiability of T ( - ) - 15 The following lemma establishes the monotonicity in the nontrivial equilibrium. Lemma 3 The nontrivial equilibrium under DMP, if exists, has T' (■ ) < 0, H' (•) < 0 and B' (•) > 0. Proof: Suppose B' (rt) < 0 for some 6 (0,r). By Y' (•) < 0, (3.16) cannot hold. Contradiction. H' (•) < 0 and T' (•) < 0 are straightforward according to (3.12) and (3.13). □ The negative sign of H' (•) is due to the distortional effect of payroll tax on human capital investment. The negative sign of T' (•) also stems from the fundamental impact of the policymaker’s human capital stock on his political choice, instead of pure expectation. Intuitively, a low human capital h“ _x is associated with low marginal costs of ti, as captured by LHS of (3.16). This makes the decisive voter raise her preferred tax rate. Combining with H' (•) < 0 ,1 obtain a positive relationship between Tt+\ and l5If the constraints are binding, i.e., Tt = 0 or r, for some h t - \ 6 then rt must be equal to 0 or 1 for all h t - i € [h, h\ by the differentiability of T . T (•) = 0 is a trivial equilibrium. If T (•) = r instead, B ' (•) = 0. Then (3.16) implies rt = 0. Contradiction. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 97 t t‘ That is to say, the more the working generations contribute to the social security system, the more they expect to receive from it in the future. The negative T' (•) plays an essential role in the sustainability and evolution of social security system. B' (•) > 0 actually provides the incentive for the decisive voter to support social security system, since otherwise she would rationally choose zero contribution. Under the quasi-linear preference, the equilibrium can be solved analytically. Lemma 4 Let q = ^2 — 27n2R9 (wu)2 /2<&j /27n3 and define fa, fa, bo, b\, -ko, recursively 7 T l = + _ _ i - (3.17) IT , = - * ( 1 + ( 3 1 8 ) S --------------nt.-» ^ )2(l + fl)------------- (3 . 19) $b1(n + l/R + n/R(l + b1) ) - n 1(l + 2b1)-6 (w u)2 7 T 0 = n - $ ( ^ ( n + ^ J b0 + ^b 0( l+ h ) j (3.20) Rwu * s (3'21) s -h IW r L ( 3 ' 2 2 ) (1) A nontrivial DMP equilibrium exists if < f )0 " f - fah ^ 0 (3.23) fa + fah, < T (3-24) (2) Assume (3.23) and (3.24). There exists a unique linear nontrivial Dy- R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 98 namic Markov Perfect equilibrium. T{hU) = + (3.25) 1 + 6wu(f>i/R B (tt) = b0 + biTt (3.26) (3.27) y (n) = ^ r 0 + 7T1 Tt (3.28) (3) Assume (3.23) and (3.24). In the nontrivial DMP equilibrium, I have b o > 0 and bo + b\ < r. The social security tax rates monotonically converge to the steady state bo/ ( 1 — & i) € [ 0 , r]. Proof: See appendix. The first part of the lemma gives the sustainability of social security system under DMP. Given (3.23), the middle poor chooses non-negative social security tax rate at h. Since < / > i < 0, (3.23) provides a necessary condition of sustaining social security system under DMP. (3.24) ensures the differentiability of T for sufficiently large r. It turns out that (3.23) and (3.24) can be satisfied under a wide variety of parameter values. An example is plotted in Figure 3.1, in which I set R = 1.0430, A = 0.6 and t = 0.5. Condition (3.24) can be easily satisfied unless for implausibly large n (e.g. n > 20). (3.23) holds for all wu and n in the diagram except in the triangle with high wu and low n. An immediate observation is that social security system under DMP can exist in a dynamic efficient economy, as long as there exists sufficient intracohort R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 99 1.5 1.4 1.3 the triangle in which / (23) does not hold / 1.2 1.1 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 3.1: The Existence Condition of Social Security System under DMP heterogeneity, i.e., a sufficiently low wu.u Condition (3.23), thus, highlights the role of intra-generational redistribution on the sustainability of social security system in a dynamic efficient economy. The second part of the lemma gives the uniqueness of linear Markov perfect equilibrium. The uniqueness contrasts the multiplicity of linear equilibrium in Azariadis and Galasso (2002), though their model shares a number of common features with ours (say the median voter framework and linear production technology). Azariadis and Galasso construct a policy rule contingent on the capital held by old households. However, due to the linear technology, old households’ capital is not a payoff-relevant 16A dynamic efficient economy in our model corresponds to R > n. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 100 state variable for the decisive voter, the young households, in their model. Therefore, the sustainability of social security system relies on an indeterministic self-fulfilling expectation of future social security benefits. The policy rule T in my model, instead, is a rigorous Markovian strategy since human capital is payoff-relevant for the middle poor. The third part of the lemma characterizes the evolution of social security system under DMP. Suppose there is no social security taxes before time 0 and DMP arises unanticipatedly at time 0. Then I have K q = h and T (h) < bo/ (1 — b\). This implies that { t( } ^ 0 be an increasing sequence. Put in words, the increasing sizes of social security system can be an intrinsic feature in a dynamic political equilibrium under DMP. Intuitively, without the positive link b\ between the current social security contributions and future benefits, coordination failure among generations rooting in the temporal separation of social security contributions and benefits (Sjoblom, 1985), would destroy the welfare state. Therefore, the equilibrium social contract B can be regarded as an incentive mechanism that props the political support of the middle-aged for social security system. 3.2.2 M ajority V oting Now turn to majority voting, in which the median voter is decisive for Tt■ Let tT'S i tT ’ U an(l Tt be the preferred tax rate of the middle rich, the middle poor and the old, respectively. The equilibrium under DMP survives majority voting if and only if R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 101 the median voter is always the middle poor, i.e., Tm,s < Tm,u < To ^ (3 2 g ) The first inequality is straightforward, since the middle rich get the same social security benefits as the middle poor, but have to pay higher taxes ros/ij_1. So I only need to show the second inequality in (3.29). Given the equilibrium social contract B, the current tax base yt can be written as V, = n W - 1 ,r.) = * + „ ( i _ r, + 1 * (r<)) ) (3.30) Qwu \ R Maximizing the indirect utility of the old, (3.14) solves t° = m in{r, — Yc (h“ _1,rt) /dYc /drt}. In words, the old choose t° to attain the top of the Laffer curve. Substitute the social contract B (3.27) into (3.30), I have . f_ 1 + 1/R -bi/R + h^Jdnw * T‘ = m m ' T' ---------- 2(1 + b1/R)------------1 (331) where bi is defined in (3.17). Since r° increases in h“_i, the minimum t° locates at h“_i = h. So t° must be binding by the upper boundary r if the following inequality holds L ^ 1 - R + (2-R + ( 1 + R) /n) ( 1 - r) . * 5 -----------------T T zf ----------------- (3'32) (3.32) gives a sufficient condition for the middle poor to be the median voter. A smaller f> i makes the tax base yt less elastic with respect to Tt and thus induces the old R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 102 to impose a higher Tt. If (3.32) holds, (3.29) is satisfied and the political equilibrium under repeated majority voting coincides with the one under DMP. Proposition 5 Assume (3.23), (3.24) a n < d (3.32). Then there exists a (differentiable) Markov perfect equilibrium under repeated majority voting, in which the median voter is always the middle poor. The equilibrium outcomes are the same as those under the nontrivial DMP equilibrium, as stated in Lemma 1. Put in other words, I need to add the restriction (3.32) on parameter values to pin down the identity of the median voter under majority voting. Like (3.23) and (3.24), condition (3.32) can be easily satisfied. In fact, (3.32) holds for all wu and n in the diagram.1 7 The intuition for the middle to be the median voter is simple. On the one hand, increasing rj raises the tax burden of the middle, while the tax burden of the old is always zero. On the other hand, the elasticities of current tax base yt and future tax base yt + 1 with respect to Tt are different. Increasing Tt discourages not only h f, but h’ )+1. So yt+i tends to be more elastic than yt- 3.3 C om parative S tatics The uniqueness of linear Markov perfect equilibrium allows us to run com parative statics analyses, which are not fully secure under multiple equilibria. Since social security system works as a redistributive policy between cohorts and within co hort, I are particularly interested in the impacts of income inequality and demographic 17The third part of Lemma 1 implies b\ < 1. So it is sufficient for RHS of (3.32) to be greater than unity. Take r = 0.5 as an example. This requires R — 1 > 2 (n — 1), which seems quite reasonable in a dynamic efficient economy. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 103 structure on the size of social security system. Before preceding, let us first investigate the impact of wu and n on two parameters & o and b\ in the social contract B that characterizes the size of social security system in the dynamics. Lemma 6 Assume (3.23), (3.24) and (3.32). Then db\/ dwu > 0 and db\/dn < 0. Proof: See appendix. At a first look, the negative correlation between b\ and income inequality or population growth rate at equilibrium seems somewhat counter-intuitive. However, I argue that it arises naturally in the politico-economic equilibrium. In the social contract B (3.27), b\ can be considered as marginal returns to Tt via inter-generational transfer. If I raise wu, given b\ fixed, the middle poor would be less willing to pay social security contributions since they receive a low level of intra-generational transfers. To sustain the social security system, marginal returns of current contribution via intergenerational transfer, b\ has to be increased in equilibrium to attract social security contributions. Note that the expectation of an increase in the marginal response of future policy choice Tt+ 1 to Tt by the current median voter is supported by the following fundamental linkage: an increase of wu intensifies the negative responsiveness of the human capital investment by the current young poor to a marginal increase in r t, due to a larger distortionary effect of Tt on h“; as wu increases, the responsiveness of Tt+i on a decrease in also increases because a larger wu implies a larger drop of the marginal cost of Tt+x in response to a drop in h”, as indicated by Equation (17). Similarly, if I reduce n, the middle poor receives a low level of inter-generational transfers. In R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 104 equilibrium, b\ has to be increased to sustain the social security system. Put in words, b\ is higher in the economy with low level of inequality or low population growth rate. Due to the complexity of 60 in (3.19), it is hard to get analytical details on dbo/dx, where x = wu or n. So I resort to numerical experiments. Note that there are two effects of x on b p : dbp _ db0 dx dx + W 7 T < 3-33> 6l dbi dx The first term on RHS of (3.33) is the direct effect of x , given the marginal returns b\. The second term captures the indirect effect through bi. The following results turn out to be robust under all experiments I have done so far. Num erical Result 1: Assume (3.23), (3.24) (3.32). Then (1) dbo/dwu\bi < 0 and dbp/dn\b i > 0 . (2) dbp/dbi > 0 . Fixing 6 1, a high wu implies a low level of intra-cohort transfers, while a low n implies a low level of inter-generational transfers. These induce the middle poor to reduce social security contributions and thus leads to a low constant term bp in the social contract. On the other hand, b p is increasing in b\. Intuitively, a high marginal return on r < + i provides incentive for the tax-payers to raise t *. This leads to a positive dbp/dbi. Since db\/dwu > 0 and db\/dn < 0, the indirect effect is opposite to the direct effect. It turns out that the indirect effect of wu dominates for low wu while the direct effect dominates for high wu. Correspondingly, b p is an inverted-U function of wu. Figure 3.2 gives an example. The direct effect of n, however, always dominates R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 105 Panel D' steady state labour inco Figure 3.2: Effects of wu on b o and b\ the indirect effect. So bo is monotonically increasing in n, as plotted in Figure 3.3. 3.3.1 T he M yopic Effect and Social Contract Effects on th e Size o f Social Security In this subsection I distinguish two channels through which wu and n may affect the size of social security system. They are referred to as the "myopic" effect and social contract effect, respectively. The first order condition (3.16) implies that the optimal tax rate for the middle poor at time t equal R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 106 el A 0.5 teady state tax rate Panel C: steady state output per capital 1 05 0.95 0.9 1 5 2 Panel D: steady state labour income H a l Figure 3.3: Effects of n on b o and b\ R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 107 R w V f l ^ - & 1 7 T O - & 07T 1 , OQ/A T ‘ = 2 ( 3 ' 3 4 ) The impacts of wu and n on t* can be written as drt dri . drt dbx drt db0 dx dx 60,61 db\ dx dbo dx v—— - v — ^ V V ^ the myopic effect the social contract effect I define the impact of x under fixed bo and 6 1 , drt/dx , as the myopic effect in the sense that the median voter myopically ignore the impact of x on future policy outcomes via the endogenous social contract B. The last two terms on RHS of (3.35) captures this indirect channel, which is referred to as the social contract effect. Let us investigate these two effects in order. T he M yopic Effect. Fixed the social contract bo and b\, (3.34) indicates that the size of social security system be negatively correlated with wu but positively correlated with n. Given the stock of human capital h“ _x, the myopic effect implies that social security taxes should be higher in the economy associated with higher income inequality and population growth rate. This coincides with the conventional wisdom on the determination of social security size. T he Social C ontract Effects. The myopic effect is just the one side of the story. The social contract per se is an endogenous equilibrium outcome, as I can see from Lemma 1. The social contract effects are two-fold, via bo and b\. The last term on RHS of (3.35), (drt/dbi) / (dbi/dx), reflects the social contract effect of x via 6 1. drt/dbi > 0 is straightforward by (3.34). Together with Lemma 1, (drt/dbi) / (db\/dx) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 108 wu n the myopic effect - + the social contract effect via b\ + - the social contract effect via b o + /~ + Table 3.1: The Effects of w(u) and n on the Size of Social Security System should be positive and negative for x = wu and n, respectively. That is to say, a higher income inequality or a larger proportion of retirees tend to reduce the size of social security according to the social contract effect via b\. The social contract effect via bo, i.e. (drt/dbo) / (dbo/dx), is more compli cated. Since drt/dbo > 0, the sign of this effect coincides with sgn(dbo/dx). Numerical Result 1 shows that sgn(dbo/dx) is determined by two opposite effect. For x = wu, the social contract effect via bo is positive for small wu and negative for large wu. For x = n, the direct effect always dominates. Consequently, I get a positive (drt/dbo) / (dbo/dn). Table 3.1 summarizes the above analyses. The aggregate impact of wu or n on the social security size is not obvious due to the conflicting effects stated in Table 3.1. In particular, if I look at the steady state social security tax rate r* = bo/ (1 — b\), r* is non-monotonically related to wu, see Figure 3.2 for an example. The non-monotonicity roots in the inverted-U b o with respect to wu. Figure 3.3 plots r* with respect to n. It turns out that the positive myopic effect and social contract effect via bo dominate the negative social contract effect via b\ in the steady state. So r* is monotonically increasing in n. To conclude, I distinguish the social contract effects that have been long ignored in the literature. The R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 109 social contract effects not only have quantitative impacts, but can qualitatively change the relationship between the size of social security system and income inequality. 3.3.2 T he Social Contract Effect on th e Growth o f Social Security The growth rate of a social security system gt is defined as gt = ln r4 — hiTt_i (3.36) Solving the first order difference equation rj = bo + i, I have Tt = bo (l — b\) / (1 — hi) + & ir o - If f is sufficiently large such that b\ro is close to zero, the growth rate of Tt can be approximated measured by * ”l n (r^ ) ( 3 ' 3 7 ) It is easy to check that RHS of (3.37) increases in b\ for any positive integer f.1 8 By Lemma 1, I find that gt is positively correlated with wu but negatively correlated with n. This seemingly counter-intuitive relationship is in fact a natural implication of the equilibrium social contract I discussed above. It also implies that the diversified growth patterns of the social security system across countries can be simply explained by the various levels of income inequality and dependency ratio via the social contract effect. This helps us to distinguish the present model from many others. I will see below that the prediction of my model is well consistent with a sample of OECD countries. l8Numerical experiments show that the approximation (3.37) is reliable, i.e., the growth rate of rt is indeed positively correlated with bi. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 110 3.4 E m pirical E vid en ce In this section, I would like to examine whether my model’s predictions de scribed in the previous section are consistent with the empirical evidence. There are three main predictions of the model. (1) The size of the social security system increases over time. (2) The growth rate of social security benefits is negatively correlated with income inequality and positively correlated with the dependency ratio. (3) The sign of the correlation between the size of social security systems and income inequality depends on whether the myopic effect or the social contract effect dominates. The first prediction is in line with the increasing generosity of social security system during the postwar period, which is believed to be a stylized fact of the evo lution of the social security system (see Mulligan and Sala-i-Martin, 1999a, for more details). Prom Table 3.2, I see that both the absolute size and relative size of social security system expand over time in the OECD countries. Particularly, real benefit per social security beneficiary in 1990 amounts to seven thousand US Dollars, more than three times larger than in 1960. Public social security per social security beneficiary increases approximately three folds as well, from $1,546 at 1960 to $ 4,653 at 1980.1 9 Although ageing population tends to raise the absolute size of social security system, say Benefits per GNP in the table, current literatures on the social security system, 19Note that the growth rate of GDP Per Capita in OECD countries is far below the growth rate of pension benefits and hence cannot explain the rapid expansion of social security system. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. I l l 1960 1970 1980 1990 Benefits Per GNP 0.0633 0.0708 0.110 0.119 Public Pensions Per GNP 0.0462 0.0588 0.0924 - Real Benefits Per Capita 0.293 0.530 0.976 1.362 Public Pension Per Capita 0.228 0.433 0.810 - Real Benefit Per Pensioner 1.962 3.186 5.713 7.157 Public Pension Per Pensioner 1.546 2.583 4.653 - Source: Breyer and Craig (1997, table 2). Absolute amounts in 1000 of 1982 US Dollars. Table 3.2: Social Security Program in the OECD (Averages) without consideration of the social contract effect, cannot well explain the increasingly generous payments to each social security beneficiary. I now use data for 20 OECD countries over the period of 1960-1985 to test the second prediction. The sample countries are divided into two groups: G7 and small OECD countries (the definition of small OECD countries follows the OECD Analytical Database).2 0 The measure of income inequality is average Gini coefficient and dependency ratio is defined as the ratio of the population of age 65 above to the population of age 15-64. The data source are in the Appendix. It is clear from Figure 3.4 that Income inequality and average growth rate of social security are negatively correlated across all sample countries. To test the third prediction, I run cross-country regression of the average growth rate of real social security benefit per beneficiary during 1960-1985 on both income inequality and dependency ratio. I notice that the drastic demographic changes in many OECD countries over the past four decades is 20Some small OECD countries, such as Switherland, are dropped from our samples due to missing observations of either pension benefits or inequality. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 112 ♦ Spain ♦ Japan ♦ Finland ♦ Ireland ♦ Austria 4 1 ♦ Italy ♦ Norway ♦ G reece ♦ Belgiui ♦ Sweden ♦ Netherlam 4 ♦ France 3 ♦ Australia ♦ Germany ♦ UK o 2 9 ♦ Portugal A verage Gini (*100) Figure 3.4: Cross-Country Relationship between Income Inequality and the Average Growth Rate of Social Security (All Sample Countries) not consistent with the prerequisite of cross-section regression, i.e., dependency ratio should be rather stable during the relevant period. So I collect the data of dependency ratio for both 1960 and 1985 and run regression on these two years’ dependency ratio separately, to see if there is any significant discrepancy for the estimated coefficients. Table 3.3 summarizes the results. Column (1) and (2) report the group of all sample countries, with dependency ratio of 1960 and 1985 included, respectively. The sign of the estimated coefficients on Gini is negative in both regressions, indicating that the average growth rate is negatively correlated with income inequality. And the estimated coefficients on Gini in both cases are significantly different from zero at 10 percent level. In addition, regression on the dependency ratio in different years only R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 113 All Sample Countries Small Open Economies (1) (2) (3) (4) Gini -0.138* -0.147* -0.325** -0.329** (0.098) (0.102) (0.130) (0.134) dependency ratio (1960) -9.109 (14.09) - 9.131 (16.91) - dependency ratio (1985) - -6.290 (12.97) - 2.446 (14.01) R 2 0.118 0.109 0.412 0.396 Note: Columns (1) and (2) report OLS regressions for all 20 sample countries; column (3) and (4) report OLS regressions for 13 small OECD countries. ***, ** and * refer to significance at 1%, 5% and 10%, respectively. Standard errors are in parenthesis. All specifications include constant intercept. Table 3.3: Social Contract Effect on the Growth of Social Security slightly changes the estimated coefficients on Gini. All these suggest that my model’s prediction on the social contract effect of income inequality on social security growth is well supported by the empirical evidence. However, when it turns to the dependency ratio, I see that the signs of estimated coefficients are negative, which is opposite to the model’s prediction. Moreover, the coefficient in neither case is significantly different from zero. The insignificance might imply that cross section regression is not appropriate to estimate the impact of dependency ratio on social security growth if there are vast changes in demographic structure during the relevant period. Since my model is developed for small countries, where factor prices are ex ogenous, I would like to run regression for the subgroup of small OECD countries to assess the role of endogenous factor prices. Column (3) and (4) report the results. Consistent with the model’s prediction, the sign of the estimated coefficients on Gini and dependency ratio is negative and positive, respectively. Moreover, the significance R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 114 £ 4 n £ 4 1 ■ ? 3 2 O • 5 f 2 S % < 0 ♦ Spain ♦ Ireland ♦ Austria ♦ Norway ♦ G reece ♦ Belgium /eden ♦ Netherland ♦ Denmark ♦ New Zealand ♦ Australia 3 5 3 7 3 9 ♦ Portugal A verage Gini (*100) Figure 3.5: Cross-country Relationship between Income Inequality and the Average Growth Rate of Social Security (Small Open Economies) level of the estimated coefficients on Gini reaches 5% and B? jumps from nearly 10% to around 40%. Figure 3.5 plots social security growth against Gini for small countries. The figure reflects an impressively close relationship between them. This indicates that the social contract effect can well explain the growth of social security system among small OECD countries. Compared with the results of Column (3) and (4) with (1) and (2), one can see that the it might be necessary to include general equilibrium consider ation to better capture the social security growth for those large countries. I leave this interesting work for future research. The third prediction states that the size of social security systems is negatively correlated with the income inequality, if the social contract effect exists and dominates R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 115 the direct effect. This prediction is closely related to the intriguing puzzle of the negative correlation of welfare spending and inequality recently found in the literature (see, e.g., Lindert, 1996 and Rodriguez, 1998). Note that the level of the size of social security systems increases with dependency ratio. Hence, the puzzle could be spurious due to the negative relationship between inequality and dependency ratio. To control the impact of dependency ratio, I run regression of the share of social security expenditure as a percentage of GDP in 1980 on Gini and dependency ratio. The reason I choose the year of 1980 is that the social security share starts to level off in the 1980s after several decades of growth.2 1 To assess the sensitivity of demographic changes, I again collect data of dependency ratio for 1960 and 1980. Table 3.4 summarizes the results. Column (1) and (2) reports the results for all sample countries. Notice that the sign of the estimated coefficients on Gini are not only insignificant but also opposite to each other in the two regressions. This indicates that the puzzle disappears if I control the impact of dependency ratio. Then I repeat the regressions for the subgroup of small OECD countries. The results are reported in Column (3) and (4). Now the sign of the estimated coefficients on Gini becomes negative in both cases. Moreover, it is significantly different from zero at 5% and 1%, respectively. This indicates that the negative relationship between the size of social security systems and inequality remains in the small OECD countries even after controlling the impact of dependency ratio. The negative relationship, which is contradicted to the prediction of previous models, can be well reconciled in my model, 21Later we will check the robustness of estim ation results by using data for 1971. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 116 All Sample Countries Small Open Economies (1) (2) (3) (4) Gini 0.064 -0.052 -0.421** -0.488*** (0.145) (0.154) (0.155) (0.150) dependency ratio (1960) 54.24** 44 97** - (22.18) - (19.42) dependency ratio (1980) 56.16*** (16.54) - 37.15** (15.80) - R2 0.411 0.269 0.651 0.647 Note: The same as Table 3.3. Table 3.4: Social Contract Effect on the Level of Pension Size as stated in the third prediction. It also implies that the social contract effect not only determines social security growth, but also dominates the direct effect on the level of the size of social security systems for small OECD countries. Compared with the results of Column (3) and (4) with (1) and (2), again one might guess that to explore the interaction of factor prices and income inequality, which is absent in my model, will explain the determination of the level of social security benefits in large countries. I leave this interesting work for future research. Finally, my model implies that, to run an unbiased estimation of the sign of demography effect, I have to control for the potential growth trend of the size of social security systems. This is because social security growth is an intrinsic feature of the social contract implied in my model. It is noteworthy that Razin et al. (2002) reports a opposite sign of demography effect from a panel of OECD data. They pointed out that the average dependency ratio fell from 58% to 54% during 1970 and 1991. Then, together with the increasing social security benefits, it would be natural to expect a R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 117 negative relationship between social security and dependency ratio. This result, at a first sight, seems inconsistent with my model since the fifth prediction says that a unanticipated negative shock to the dependency ratio would raise current social security benefits. However my model implies that without controlling of the potential growth trend of the size of social security systems, the panel data approach could be misleading. I summarize the above empirical results as follows. First, the model’ s predic tion on the social contract effect of income inequality on both of the growth and level of the size of social security systems can well be supported by the data for small OECD countries. Second, incorporating the change of demographic structure is necessary, the oretically and empirically, to understand the influence of demography on the evolution of the social security system. Third, the interaction between endogenous factor prices and income inequality seems to be indispensable to explore the evolution of the social security system for large countries. 3.5 C on clusion In this chapter, I develop a positive theory of social security in a majority voting framework with differentiable Markov perfect equilibrium. My model is a three- period overlapping generations model with the middle aged poor agents as median voters. Payroll taxes distorts the human capital investment of the young, which is proportional to wage incomes at both young and middle ages. In this setup, the (self fulfilled) expectation of social contract relating future benefits to current contribution is achieved through two fundamental linkages: a negative reaction function of the next R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 118 period median voter’ s human capital stock to the current payroll tax rate due to its distortionary effects on human capital investment and a negative reaction function of the payroll tax rate chosen by next period’ s median voter to his human capital stock due to the positive correlation between marginal cost of taxation and human capital stock My usage of the differentiable Markov perfect equilibrium and the method of undetermined coefficients enables us to solve for the close-form solution of the Markov strategy in this dynamic game. The major theoretical contributions of this chapter are two folds. First, I show that even with temporal separation of costs and benefits, self-interested median voters can have incentive to sustain social security with expectation of the dynamic linkage between current contribution and future benefits depending only on payoff relevant variables. Second, by expectation based only on fundamentals dynamic linkage, I can achieve the uniqueness of Markov perfect equilibrium in which social security is sustained by self-interested voters. My model, therefore, contrasts to the previous studies on the political sustainability of social security which either resorts to imperfect temporal separability between costs and benefits to sustain social security or builds expectation of future policy choice on non-fundamental linkages. Moreover, the uniqueness of the equilibrium provides several sharp predictions that can be tested by the data. First, social security tax rates are increasing over time until converging to a steady states. Secondly, the growth rate of social security programme is negatively correlated with income inequality. Third, cross-sectionally, the impact of income inequality on the equilibrium social contrast influences makes R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 119 the relationship between income inequality and the size of social security systems non monotonic. This makes my theory a useful framework to explore the dynamics of social security in politico-economic equilibrium. My empirical evidence shows that these predictions are broadly consistent with the data from the OECD countries. Several interesting extensions can be made in the future research, as implied by my empirical evidences. First, I would like to assume continuously changing population growth rates. This can help us to analyze the dynamics of social security system with anticipated demographic changes (say increasing dependency ratio). Second, general equilibrium effects are ignored in the model. This leads to the difficulty in accounting for the evolution of the social security system in a large economy such as the United States. So I plan to introduce endogenous capital accumulation in the model. Tentative results show that both of the growth and level of the size of social security systems are crucially dependent on factor prices. 3.6 A p p en d ix 3.6.1 P roof o f Lemma 2 Due to the linear-quadratic preference, it would be natural to guess that the policy rule T is linear T (h U ) = ^ 0 + « _ i (3.38) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 120 where fa and fa are two undetermined coefficients. Substitute into (3.11) H ^ = (1+1- i r - T <)1+eSMR (3'39> Combine (3.38) and (3.39), I obtain a linear social contract B B(Tt) = b0 + b1Tt (3.40) where b0 = fa + l+o°X0l/it (* + < I > 1 anc* h = - 1+g* ^ x/Rfa. Plug (3.40) into (3.15), the function tax base is Y {t t) = 7T 0 + 7TlT t (3-41) where 7 r0 = II - $ ((n + ^) b0 + j|f>o (1 + 6i)) and 7 T i = - $ (l + (n + ^) 6i + % b\). The first order condition (3.16) yields Mo + b 0m , Rwu L U T ‘ ----- 2 b ^ ~ + 26i7r7 < 3 '42) (3.42) pins down fa and fa in (3.38) fa = - b' Z ~ b Q 7 ri (3-43) 2&17T1 U 26i 7 T i , Rwu , . 4 > r = X T (3.44) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 121 (3.44) implies a four-order polynomial of b\ R Factorizing (??), one root of b\ equals — R, which should be omitted by Lemma 1. The other three roots solve where 0 = R 6 (wu)2 /2 < f> . Rearrange It is straightforward to see that LHS and RHS have an unique cross for b\ > 0, which gives the only positive root of b\ (3.17) satisfying Lemma 1. The other undetermined coefficients can be easily solved. Then I need to check if T ( h ^ ) € (0,r) for all h“_1 € (h, h). This gives the existence conditions (3.23) and (3.24). Finally, T (/i) > 0 requires ( f > 0 + fifa > 0. This implies b o > 0. On the other side, T (h) < r implies bg + bi < t. Together with & i > 0 implied by Lemma 1, I have nb\ + b\ — 0 = 0 (3.46) b\ S (0,1) and ^ € (0,7=). □ R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 122 3.6.2 P roof of Lemma 3 Differentiating (3.46) with respect to x: db\ _ 9 0 / dx dx 3 nb\ + 2b\ where x refers to wu and n, respectively. Since b\ > 0, sgn(db\/dx) = sgn(dQ/dx). It is immediate that dQ/dwu > 0 and dO/dn < 0. 3.6.3 D ata Source The cross-country data on both the average growth rate of real social security benefits per beneficiary and the social security benefit as percentage of GDP are taken from OECD (1988). As I lack the data for the average growth rate of social security benefit per beneficiary for France and Greece, I approximate the average growth rate of payroll tax rates by the average growth rate of the share of social security expen diture in GDP for these two countries. For most countries, the data covers the period 1960-1985.2 2 The data of average Gini coefficients are from the updated data set of Deininger and Squire (1996, Table l)23. The data shows that inequality does not vary that much during the sample period, though there is a trend for increase in inequality. In addition, 22The starting year for Australia is 1961 and the ending year for Portugal and Sweden is 1984. The average growth rates for Belgium and Spain are for periods 1971-1984 and 1974-1985, respectively. 23For Austria, the calculation of the statistics excludes the fraction of population who are self- employed. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 123 the coverage period for the inequality statistics are broadly consistent with the coverage period for the average growth rate of real social security benefits per beneficiary. The data for the dependency ratio in 1960, 1971 and 1980, 1985 are from the author’ calculation based on the demographic data in United Nations (2000). For those countries missing data of the above specific years, I use the data of the following year. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 124 Bibliography [1 ] Aizcorbe, A. M., A. B. Kennickell, and K. B. Moore (2003), “Recent Changes in U.S. Family Finances: Evidence from the 1998 and 2001 Survey of Consumer Finances,” Federal Reserve Bulletin, January, 1-32. [2 ] Alvarez, F. and U. Jermann (2000), “Efficiency, Equilibrium and Asset pricing with Risk of Default,” Econometrica, 68, 775-797. [3 ] Auerbach, A. J. and L. J. Kotlikoff (1987), “ Dynamic Fiscal Policy”, New York: Cambridge University Press. [4 ] Azariadis, C.s and V. Galasso, 2002, Fiscal Constitutions, Journal of Economic Theory, 103, 255-281. [5 ] Azariadis, C. and L. 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Creator Chen, Kaiji (author)

Core Title Essays on social security

School Graduate School

Degree Doctor of Philosophy

Degree Program Economics

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

Tag Economics, General,Economics, Theory,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Betts, Caroline Marie (committee member), Imrohoroglu, Ayse (committee member), Imrohoroglu, Selahattin (committee member), Quadrini, Vincenzo (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-339627

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Rights Chen, Kaiji

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