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Essays in uncertainty and aggregate economic activity

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Essays in uncertainty and aggregate economic activity

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Content ESSAYS IN UNCERTAINTY AND
AGGREGATE ECONOMIC ACTIVITY
by
Diego Vilán
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ECONOMICS)
August 2015
Copyright 2015 Diego Vilán
We had pierced the veneer of outside things. We had suffered, starved and
triumphed, groveled down yet grasped at glory, grown bigger in the bigness of
the whole. We had seen God in His splendors, heard the text that Nature renders.
We had reached the naked soul of man.
Ernest Shackleton (1919)
ii
To my family,
and to those who believed.
iii
Acknowledgments
This dissertation is the culmination of a long journey during which I have accrued
debts to many. Foremost among these is to my advisor Vincenzo Quadrini for his
guidance,patienceandconstantsupportduringmyyearsatUSC.Iamalsoextremely
grateful to the other members of my committee Juan Rubio-Ramírez, Guillaume Van-
denbroucke,SeloImrohoroglu,andMichaelMichauxfortheirextensivefeedbackand
insightful suggestions as well as to Caroline Betts and Elias Albagli, who served in
the qualifying committee. I would also like to thank participants of USC’s Macroeco-
nomic seminar for all their helpful comments and recommendations along the way.
IamalsogreatlyindebtedtoPedroSilosforcountlessdiscussionsandsuggestions
that contributed not only to the quality of this manuscript, but also to my growth as
an economist and as a researcher. Moreover, this dissertation has also been consid-
erably improved by constant interaction with my friends and colleagues, Federico
Mandelman, German Cubas and Enrique Martinez-Garcia.
In addition, I want to acknowledge Young Miller and Morgan Ponder who helped
menavigatethevariousadministrativehurdlesthroughoutmyyearsatUSC.I’malso
gratefulforthedifferentfellowshipsfromUSC’sDornsifeCollegeofLetters,Artsand
Science, the Department of Economics and the Graduate School.
iv
Iwouldalsoliketothankmyfriends,JimenaZuniga,HimaChintalapatiandAriel
Ivanier who, although being in different parts of the world, managed to stay close.
Last, but certainly not least, I extend my heartfelt thanks to my family, for their
constant love and encouragement. My parents Ernesto and Alicia, who made impor-
tant sacrifices in order for me to be here. My sisters, Andrea and Sofia, for their
support and their patience. And to my grandmother, who taught me that there is no
reward without sacrifice.
v
Table of Contents
Epigraph ii
Dedication iii
Acknowledgments iv
List of Tables viii
List of Figures ix
Abstract x
Chapter 1: Demand Shocks and Endogenous Uncertainty 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Empirical Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Time series facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Firm-level facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Entrepreneurs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.2 Production and Demand Uncertainty . . . . . . . . . . . . . . . . 16
1.3.3 Endogenous Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3.4 Risk and Return trade-off . . . . . . . . . . . . . . . . . . . . . . . 24
1.3.5 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.6 Theoretical Propositions . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4 Quantitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.4.1 ShopperTrak data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.5.1 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.5.2 Model Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.5.3 Effect Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.5.4 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.5.4.1 The effects of tau . . . . . . . . . . . . . . . . . . . . . . . 42
1.5.4.2 The effects of capacity utilization . . . . . . . . . . . . . 43
1.5.5 Extension: Persistent Demand Shocks . . . . . . . . . . . . . . . . 46
1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
vi
Chapter 2: Uncertainty and aggregate macroeconomic fluctuations in Small
Open Economies 51
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.2 Characterizing the stochastic volatility process . . . . . . . . . . . . . . . 55
2.2.1 The Law of motion for terms of trade . . . . . . . . . . . . . . . . 56
2.2.2 Estimation Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.2.3 Posterior Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.3 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.3.1 The Household Problem . . . . . . . . . . . . . . . . . . . . . . . . 60
2.3.2 Firms and Technology . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.3.2.1 Production of Consumption goods . . . . . . . . . . . . 63
2.3.2.2 Production of Tradable goods . . . . . . . . . . . . . . . 63
2.3.2.3 ProductionofDomestic(exportableandimportable)inter-
mediate inputs . . . . . . . . . . . . . . . . . . . . . . . . 64
2.3.2.4 Production of Non-Tradable goods . . . . . . . . . . . . 66
2.3.3 Exogenous forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.3.4 Solution concept and Equilibrium definition . . . . . . . . . . . . 68
2.3.5 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.4 Results: The real effects of risk shocks . . . . . . . . . . . . . . . . . . . . 71
2.4.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.5 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Bibliography 78
Chapter A: Appendix to Chapter 1 85
A.1 Omitted Theoretical Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . 85
A.2 Aggregate Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
A.3 Alternatives Measures of Uncertainty . . . . . . . . . . . . . . . . . . . . 91
A.4 Evidence of Corporate Lending . . . . . . . . . . . . . . . . . . . . . . . . 92
A.5 About ShopperTrak data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Chapter B: Appendix to Chapter 2 95
B.1 Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
B.1.1 For the consumption goods sector: . . . . . . . . . . . . . . . . . . 96
B.1.2 For the tradable goods sector: . . . . . . . . . . . . . . . . . . . . . 97
B.1.3 For the nontradable goods sector: . . . . . . . . . . . . . . . . . . 97
B.1.4 For the exportable and importable goods sectors: . . . . . . . . . 97
B.1.5 Market clearing conditions: . . . . . . . . . . . . . . . . . . . . . . 98
vii
List of Tables
1.1 U.S. Compustat Moments 1980-2012 . . . . . . . . . . . . . . . . . . . . . 10
1.2 Calibration Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.3 ShopperTrak Data Moments . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.4 Targeted Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.5 Non-targeted Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.6 Model’s sensitivity to t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.1 Business Cycle Statistics (1959-2009) . . . . . . . . . . . . . . . . . . . . . 52
2.2 Terms of Trade and Business Cycle . . . . . . . . . . . . . . . . . . . . . . 53
2.3 Prior Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.4 Posterior Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5 Calibration Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.6 Targeted Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.7 Non-Targeted Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.8 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.1 ShopperTrak Clients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
viii
List of Figures
1.1 Uncertainty indicators over the Business cycle. . . . . . . . . . . . . . . . 7
1.2 Uncertainty and Economic Activity. . . . . . . . . . . . . . . . . . . . . . 8
1.3 Uncertainty over the business cycle . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Model’s Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 Distribution of sales per worker . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6 Distribution of customers per worker . . . . . . . . . . . . . . . . . . . . 22
1.7 Censoring point: q 2
t
> q 1
t
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.8 Endogenous Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.9 Distribution of customers per firm . . . . . . . . . . . . . . . . . . . . . . 25
1.10 Profit distribution simulation . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.11 Impulse responses for a 1% shock to q t
. . . . . . . . . . . . . . . . . . . 38
1.12 Effect decomposition for a 1% shock to sales . . . . . . . . . . . . . . . . 40
1.13 Effect decomposition for a 1% shock to sales . . . . . . . . . . . . . . . . 41
1.14 Sensitivity to capacity utilization rates . . . . . . . . . . . . . . . . . . . . 44
1.15 Consumer Traffic and Business Cycle . . . . . . . . . . . . . . . . . . . . 46
2.1 Chile’s Terms of Trade volatility and GDP . . . . . . . . . . . . . . . . . . 59
2.2 Responses to an innovation in the volatility of p
X
t
. . . . . . . . . . . . . 73
2.3 Responses to the level and the volatility innovation of p
X
t
. . . . . . . . . 74
A.1 Disagreement amongst professional forecasters. . . . . . . . . . . . . . . 91
A.2 Net Financial Assets in the nonfinancial business sector as a percentage
of total nonfinancial assets. . . . . . . . . . . . . . . . . . . . . . . . . . . 92
A.3 ShopperTrak’s technology . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
ix
Abstract
This dissertation is a collection of essays with the unifying objective being to better
understand the origins of fluctuations in macroeconomic uncertainty as well as its
implications on aggregate economic activity. In turn, I investigate both the genesis
as well as the effects of sharp, unexpected changes in uncertainty (i.e.: uncertainty
shocks) and quantify their impact on economic outcomes.
In the first chapter, I document the negative relationship between uncertainty and
aggregate economic activity in the U.S.. This empirically establishes a pattern, yet
offers no indication as to the direction of the causality. I then explore the hypoth-
esis that fluctuations in uncertainty are not exogenous realizations of a stochastic
process, but rather, endogenous outcomes triggered when fundamental shocks per-
turbaneconomy. Indoingsothestudyfocusesonthesymbioticrelationshipbetween
uncertaintyandeconomicactivitytoexplainhowfirst-momentdisturbancescanabate
orexacerbateaneconomy’slevelofdispersion,butalsotohighlighthowtime-varying
uncertainty can affect mean equilibrium outcomes.
The study proposes a quantitative theory that is consistent with the time-varying
cross sectional properties of U.S. macroeconomic aggregates. I find that in an econ-
omy subject to capacity constraints, level shocks may fuel or abate the degree of
uncertainty faced by economic agents. This, in turn, changes the agent’s ability to
predict future outcomes and affects their decisions today. As such, fluctuations in
x
uncertainty might reinforce the original level shock that initially triggered them.
The second chapter is based on earlier work with Hernan Seoane from Univer-
sidad Carlos III and focuses on the effects of uncertainty fluctuations in emerging
economies. In particular, it examines how fluctuations in the volatility of prices at
which these countries transact internationally contribute to the size of their cyclical
fluctuations. Understanding the effects of uncertainty fluctuations on domestic vari-
ables such as output, investment, consumption and hours worked might help shed
light into the design of optimal policies aimed at smoothing the negative effects usu-
ally associated with increased future risk.
The study begins by documenting the evidence of time-varying volatility in the
terms of trade and international interest rates faced by emerging economies. An esti-
mated stochastic volatility process for interest rates and terms of trade are then fed
into a multi-sector model of a small open economy featuring non-traded goods. The
model is solved using a third-order approximation of the policy function, and cali-
brated to match several moments of the corresponding macroeconomic aggregates.
Results show that increases the volatility of these exogenous prices may trigger losses
inoutput, consumptionandinvestment. Theseresultsdependlargelyonwhetherthe
shocksaffecttheabilitytoborrowinternationallyandwhethersectoralreallocationof
resources is feasible.
xi
Chapter 1
Demand Shocks and Endogenous
Uncertainty
1.1 Introduction
Uncertainty fluctuations are large and strongly countercyclical. In the U.S., uncer-
tainty has been systematically documented as having sizable adverse effects on
economic activity and inflation. In terms of aggregate output, for example, Baker
and Bloom (2011) establish that sudden changes in uncertainty may account for GDP
declines in the vicinity of two percent. Gilchrist et al. (2014) report that uncertainty
shocks can explain about one third of the total variation in industrial output and
payroll employment; while Bachmann et al. (2013) find them responsible for manu-
facturing losses in excess of one percent. Moreover, Bloom (2009) and Bloom et al.
(2012) argue that increased uncertainty makes it optimal for firms to wait, leading to
significant declines in hiring, investment and output; and Fernández-Villaverde et al.
(2013) establish that time-varying risk shocks may also have negative consequences
for price stability.
While it has been well established that uncertainty and aggregate economic
activity are negatively related, it is less evident why or how this occurs. To date most
research efforts have been devoted to documenting, quantifying and understanding
the effects of fluctuations in uncertainty on business conditions. In doing so, studies
have often assumed the existence of sharp exogenous changes in the volatility of
shocks which, mediated by physical (Bloom (2009)), financial (Gilchrist et al. (2014))
1
or nominal (Basu and Bundick (2012)) frictions, negatively impact mean economic
outcomes. By focusing on the effects of fluctuations in uncertainty, however, almost
no attention has been paid to the understanding their probable sources.
Motivated by the above, this study seeks to provide evidence as to the potential
origins of fluctuations in uncertainty. In doing so, it delivers a quantitative theory
that is consistent with the time-varying cross sectional properties of U.S. macroe-
conomic aggregates. The paper will focus on the symbiotic relationship between
uncertainty and economic activity to explain how first-moment disturbances can
abate or exacerbate dispersion, but also to highlight how time-varying uncertainty
can affect mean equilibrium outcomes. I argue that while swings in uncertainty
appear to be endogenously related to aggregate economic activity, fluctuations
in idiosyncratic risk will ultimately affect macroeconomic dynamics. Intuitively
recessions are times of heightened uncertainty, yet greater uncertainty may also
exacerbate a recession. In particular, the study will focus on the widely held notion
that consumer demand uncertainty experienced by firms could be at the heart of
business cycle fluctuations. The analysis is conducted through the lens of a an
incomplete markets, heterogeneous-agents framework which is able to successfully
reproduce the right business cycle co-movements.
The paper has two main goals. The first objective is to further our understanding
of the relationship between uncertainty and mean aggregate activity. In doing so
I focus on the synchronicity between uncertainty and economic outcomes, and
propose an innovative channel through which the former may relate to the business
environment. In particular, firms in the model face uncertainty about the number of
customer they will need to serve each period (idiosyncratic), as well as the amount of
resources these customers may command (aggregate). Being risk averse firm owners
will respond cautiously to changes in macroeconomic conditions, leading to cyclical
2
employment and output fluctuations.
The second goal of this paper is to contribute to the understanding of the
cross-sectional dynamics of business cycles. The availability of highly disaggregated,
longitudinal microeconomic and sectorial data, has recently shed light over the
idiosyncratic responses of economic agents to aggregate shocks. In turn, under-
standing the cross-sectional behavior of individual firms and households becomes
paramount for comprehending aggregate dynamics. In the model endogenous
changes in uncertainty further variations in economic activity allowing it to better
replicate the observed cyclical patterns of higher moments.
Results indicate that time-varying uncertainty has significant effects on the
aggregate economic activity. In the model’s baseline specification, fluctuations in
uncertainty accounted for about one-quarter of the overall variation in employment,
output and consumption at business cycle frequencies. Moreover, uncertainty swings
act as an amplification mechanism reinforcing the original shock to mean level
activity. Overall, a one percent negative shock to credit conditions leads to output
and employment losses of around 0.8 and 0.6 percent respectively.
The paper makes a few additional contributions to the literature. First, it intro-
duces an innovative way of modeling fluctuations in consumer’s demand. Rather
than assuming exogenous changes to a household’s discount factor, the model will
keep track of the distribution of customers visiting a firm. Second, the proposed
framework sheds light on the relationship between uncertainty and risk averse
behavior, in that higher perceived risk might exacerbate the effects of first moment
disturbances hitting the economy. Lastly, the study proposes a parsimonious frame-
work capable of capturing fluctuations in uncertainty which requires no nominal
rigidities and offers a tractable closed form solution.
3
1.1.1 Related Literature
This study is closely related to a fast growing body of literature studying the effects
of time-varying uncertainty on economic activity. It follows Bloom (2009), Basu and
Bundick (2012), and Leduc and Liu (2012) in that fluctuations in second moments
have first order aggregate effects. The overriding idea in this area of research is
that spikes in uncertainty, channeled through some adjustment friction, generate the
observed fluctuations in economic activity. Moreover, the paper also relates to the
scholarly research focusing on uncertainty fluctuations as an endogenous outcome
rather than a cause. In this view, Bachmann and Moscarini (2011) propose a model
in which recessions tend to incentivize firms’ risk taking behavior and hence lead
to higher cross-sectional dispersion. Similarly, Fostel and Geanakoplos (2012) and
D’Erasmo and Boedo (2011) suggest alternative mechanisms capable of generating
countercyclical uncertainty.
The proposed framework also represents a natural extension to Bewley-type
models such as Aiyagari (1994), Huggett (1997) and Krusell and Smith (1998).
These models introduce idiosyncratic risk into an incomplete markets neoclassical
framework, but focus on labor-income risk, rather than demand uncertainty. Fur-
thermore, the paper closely follows Angeletos (2007) and Quadrini (2014), both of
which provide the theoretical underpinnings behind the set-up as well as the chosen
solution method.
The study also relates to the literature seeking to understand the idiosyncratic
effects of aggregate shocks. Higson et al. (2002) and Higson et al. (2004) report that
rapidly growing and rapidly declining firms appear to be less sensitive to negative
macroeconomic disturbances relative to those firms in the middle range of growth.
Thisappearstobeconsistentwiththefactthatthehighermomentsofthedistribution
of firm growth rates have significant cylical patterns. Similarly, Kehrig (2011) finds
4
that the cross-sectional dispersion of firm-level total factor productivity in the U.S.
tends to be greater in recession than in expansions.
In terms of production, some papers assign a productive role to consumer
demand for goods and services. With this in mind, this study follows Bai et al.
(2012) and Petrosky-Nadeau and Wasmer (2011) in that output will not only be a
function of factor inputs (like in any neoclassical framework), but consumer demand
will play a paramount role in determining the level of economic activity. Moreover,
in line with Arellano et al. (2010) the framework also explores the effects of input
pre-commitments in optimal firm behavior.
Finally, the study is also related to the literature highlighting the effects of
financial frictions on the interaction between uncertainty and economic activity.
Gilchrist et al. (2014) argue that increases in firm risk lead to bond premia and the
cost of capital, which in turn, triggers the prolonged decline in investment activity.
It also follows Jermann and Quadrini (2012) in that the financial sector may be the
source of the business cycle and not solely a propagation channel for shocks that hit
other sectors of the economy.
The remainder of this paper is organized as follows: Section 1.2 presents the
empirical motivation and analysis from the Compustat and ShopperTrak data sets.
Section 1.3 explains the model and 1.4 describes its calibration. Finally, Section 1.5
presents the main results while Section 1.6 draws some final conclusions.
5
1.2 Empirical Motivation
1.2.1 Time series facts
The negative association between uncertainty and economic activity finds substantial
empiricalsupportintheU.S.economy. Theabovepatterns,however,arenotexclusive
to it and a plethora of studies have recorded similar realities in countries around
the globe. Bachmann et al. (2013) use German data to provide evidence as to the
detrimental effects of uncertainty in that country. For the UK, Denis and Kannan
(2013) estimate that uncertainty shocks generate industrial production and output
losses, while Bloom et al. (2007) finds evidence that supports the claim that higher
uncertainty reduces domestic firms’ capital expenditures. Similar conclusions have
been reached for developing economies. Arslan et al. (2011) establish that a one
standard deviation increase in aggregate uncertainty generates a 4 percent drop in
Turkey’s GDP growth rate; while Fernández-Villaverde et al. (2011) compute the
negative effects of interest rate volatility for a group of Latin American economies.
Globally, Baker et al. (2012) document the effects of uncertainty in slowing down the
global recovery.
Given its intrinsically unobservable and yet broad nature, uncertainty can be very
hardtomeasure. Itreflectstheambivalenceinthemindsofconsumers,investors,and
policymakers about the likelihood of potential future outcomes. It can also reflect
skepticism about aggregate events such as the growth rate, credit conditions and
exchange rates; or micro phenomena such as industry level legislation or personal
ambiguity. Not surprisingly, a plethora of proxies have been developed over the last
years in an attempt to capture sudden variations in risk. One of these measures is
the Exchange Volatility Index (VIX) which captures the expected thirty days forward
impliedvolatilitybackedoutfromoptionprices. Analternativeproxyforuncertainty
is the corporate bond spread computed as the difference between the Baa 30 year
6
yield and the U.S. Treasury yield at a comparable maturity. Another measure fre-
quently used is the disagreement amongst professional forecasters. Periods or higher
uncertainty usually correlate with greater dispersion in professionals’ opinions. The
intuition is that uncertainty makes it harder for agents to make accurate predictions.
Figure 1.1 plots a selection of commonly used empirical measures of uncertainty over
the business cycle.
Figure 1.1: Uncertainty indicators over the Business cycle.
Independently on which metric is used, virtually every indicator of uncertainty
risesinrecessionsandsubduesduringexpansions. Conversely,measuresofeconomic
activity tend to move in communion with the cycle. Figure 1.2 shows this graphically,
plotting the business cycle evolution of six macroeconomic indicators. Intuitively
as economic activity slows down, jobs are lost, consumption falls and capacity
7
utilization rates plummet. Additionally, as aggregate credit conditions deteriorate,
sales growth slows down and companies’ net-worth suffer. This is the negative
association between uncertainty and economic activity which will be at the core of
this study. In particular, by focusing on consumer demand uncertainty the model
will successfully reproduce the business cycle dynamics in all six macroeconomic
yardsticks mentioned below.
Figure 1.2: Uncertainty and Economic Activity.
Consumption corresponds to the year-over-year changes in Personal Consumption Expenditures (PCE) as recorded by the BEA,
while Employment tracks the year over year changes to the level of total non-farm, quarterly employment. Capacity Utilization
refersthepercentageofindustrialcapacitycurrentlybeingusedbyfirmsdomesticallytoproducefinishedproductsascompiled
by the Board of Governors of the Federal Reserve System. Retail Sales correspond to the yearly change in the level of retail
and food services sales as measured by the U.S. Census Bureau, and Credit Conditions refer to the Federal Reserve Bank of
Chicago’s National Financial Conditions Index (NFCI), where positive values of the index indicate that financial conditions are
tighter than average. Finally, Firm’s Net Worth track the evolution of the non-financial corporate business sector’s net worth as a
percentage of GDP.
8
1.2.2 Firm-level facts
Researchers focusing on the impact of uncertainty on individual firms and house-
holds have found that uncertainty at the firm level is also negatively associated with
growth and economic activity. Kehrig (2011), for example, shows that for US durable
goods manufacturers uncertainty about plant-level TFP rises sharply in recessions
affecting firms’ entry and survival rates. Vavra (2013) establishes that uncertainty
about prices also surges during recessions, making it harder for the Federal Reserve
to conduct monetary policy. Higson et al. (2002) find that risk shocks are negatively
correlated with the cycle, but affect firms in an uneven way. Leahy and Whited
(1996) find a strong negative relationship between uncertainty and investment for US
publicly listed firms.
The primary firm-level data source used in this paper is the US Compustat
database. Compustat North America provides the annual and quarterly Income
Statement, Balance Sheet, Statement of Cash Flows, and supplemental data items
on most publicly held companies in the United States and Canada. Financial data
items are collected from a wide variety of sources including news wire services, news
releases, shareholder reports, direct company contacts, and quarterly and annual
documents filed with the Securities and Exchange Commission. Compustat files
also contain information on aggregates, industry segments, banks, market prices,
dividends, and earnings. Depending upon the data set, coverage may extend as far
back as 1950 through the most recent year-end.
Using Compustat has some advantages versus using census data sets like the
Longitudinal Research Dataset (LRD) or the Annual Survey of Manufacturers (ASM),
because firm-level data are accessible to all researchers in different countries, and the
panel for the US goes as far as the 1950s. Naturally, this data is not not without flaws,
the most commonly recognized being the fact that the firm’s recorded in Compustat
9
account by about one-third of US employment (Davis et al. (2006)).
The data set comprises of 32 years of data (1980-2012), with cross-sections that
have, on average over 3,000 firms per year. From the original Compustat data, I
select firms that report information on gross and net sales, employment and capital
stocks. Following Bloom (2009) I drop firms with missing information as well as
removeoutliers. Tocalculatefirm-levelemploymentgrowthratesIusethesymmetric
adjustment rate definition proposed in Davis et al. (2006):
g
h,t
=
h
i
t
0.5⇤ (h
i
t
+h
i
t 1
)
Firm-level sales growth rates are simple log-differences. To focus on idiosyncratic
changes that do not capture differences in industry-specific responses to aggregate
shocks, I follow Bachmann et al. (2013) in removing firm effects from employment
and sales growth rates. Annual GDP and inflation data come from the Federal
Reserve Economic Data (FRED) database. All moments are robust to different infla-
tion indexes specifications. Table 1.1 summarizes some of the statistical properties of
the US Compustat data set.
Table 1.1: U.S. Compustat Moments 1980-2012
ln(sales) ln(emp)
Cross-sectional dispersion 2.227 2.416
Cross-sectional Skewness -0.187 -0.125
Cross-sectional Kurtosis 2.236 2.484
Dispersion growth rate corr w/ cycle 0.388
⇤⇤
0.293
⇤⇤
Skewness corr w/cycle 0.406
⇤⇤
0.062
⇤⇤
# observations (ave. per year) 3,296 1,585
# observations (total) 114,368 53,875
Source: Compustat and U.S. Bureau of Economic Analysis
10
The table suggests the presence of significant deviations from symmetry and nor-
mality in the data. The skewness of both sales and employment is negative implying
that the mass of the cross-sectional distribution is concentrated towards the right.
This feature is consistent with the fact that most firms surveyed in the Compustat
dataset are primarily well established, publicly traded companies. Moreover, both
variablesexperiencesignificantpositivekurtosis,suggestingthatagreaterproportion
of the variance comes infrequent extreme events.
The data also points towards the existence of a considerable amount of cross-
sectional heterogeneity in the growth rates of U.S. firms which varies with the
aggregate state of the economy. Both the growth rates of the cross-sectional
dispersion of employment and sales appear to be negatively correlated with the
business cycle. Figure 1.3 plots the evolution of the growth rate of the cross-sectional
dispersion of sales and employment over the course of five recessions as defined by
the NBER. Additionally, the degree of skewness appears to be strongly pro-cyclical,
hintingtoanasymmetriceffectoffluctuationsinbusinessconditions. Inotherwords,
during a recession firms not only seem to perform worse on average but a small
group of firms tend to experience larger drops than their average peers.
These observations are in line with what has been documented by other
researchers in the field. Bloom (2009) and Bachmann and Bayer (2014) both report
similar findings even when using alternative datasets. Further, Kehrig (2011) finds
analogous patterns for alternative measures of cross-sectional dispersion. These
stylized facts will all be important empirical regularities to be matched by the
proposed framework.
11
Figure 1.3: Uncertainty over the business cycle
1.3 The Model
The baseline model has two sectors: an entrepreneurial and a household sector.
Entrepreneurs are sole owners of firms and will be responsible for producing
goods in the economy. Households supply labor and will demand consumption
goods. Firms face uncertain demand for their products and hold financial assets to
mitigatetheeffectsofadverseidiosyncraticshocks. Thefullset-upisdescribedbelow.
Time is discrete, indexed by t2{ 0,1,...,• }. There is a continuum of infinitely-
lived households whose preferences are separable in consumption, c
t
, and labor sup-
ply, h
t
, as described by:
U
H
= E
t
• Â t=0
b t

c
t
g h
1+t t
1+t !
12
where E
0
is the conditional expectation operator, b is the discount factor, g > 0 mea-
sures the relative disutility of labor effort and t > 0 is related to the Frisch elasticity
of labor supply. Household supply labor in a competitive market and allocate their
labor and financial earnings between consumption goods and risk-free assets. Their
budget constraint is:
w
t
h
t
+
b
t+1
R
t
c
t
+b
t
where w
t
h
t
is the period real labor income, R
t
is the gross interest rate and b
t+1
is the
loan contracted in period t and due in period t+1. Balances are settled every period
and there is no default. Households may accumulate intertemporal assets, but face
the following borrowing constraint:
W b
t+1
R
t
8 t
Households will seek to purchase consumption goods before collecting their labor
income. Since the goods are acquired before wages are paid and before the opening
of financial markets for inter-temporal transactions, all purchases are paid with intra-
period credit. This intra-period credit is subject to a limit q t
, which is stochastic and
follows the process:
lnq t
= r lnq t 1
+#
t
: #
t
⇠ N(µ
#
,s 2
#
)
13
This time-varying limit is meant to capture the evolution of aggregate consumer
credit conditions in the economy.
1.3.1 Entrepreneurs
There is a continuum of entrepreneurs indexed by i with lifetime preferences over
consumption streams given by:
U
i
E
= E
0
• Â t=0
b t
lnc
i
t
where E
0
is the expectation operator conditional on the information available at t = 0
and b the discount factor.
Entrepreneurs are individual owners of firms and produce a homogeneous, non-
storable and competitively traded consumption good. Firms have revenue functions
y
i
t
h
i
t
, where the variable h
i
t
is the input of labor and y
i
t
represents output per worker.
A firm’s level of output per unit of labor is an idiosyncratic stochastic variable that
will be defined below. For the moment what matters is that the operation of every
firm is subject to an idiosyncratic shock y
i
t
.
Following Arellano et al. (2010) it is assumed that entrepreneurs choose the
input of labor before observing the actual realization of y
i
t
. Moreover, I assume
that the wage rate cannot be made contingent on the realization of the idiosyncratic
uncertainty. Since labor markets are competitive, this implies that wage rate will
be the same for all firms. Markets are assumed to be incomplete, with only one
asset available for entrepreneurs to self-insure against the idiosyncratic risk: a
non-contingent bond b
i
t
that pays the gross interest rate R
t
. The entrepreneur’s
14
budget constraint is therefore:
y
i
t
h
i
t
+b
i
t
c
i
t
+w
t
h
i
t
+
b
i
t+1
R
t
where h
i
t
is the labor input provided by households to firm i in period t, and y
i
t
is
firm’s i idiosyncratic output per worker. All in all, these assumptions imply that the
firm faces a risk in the choice of labor which cannot be fully insured.
Given the entrepreneur’s preferences over consumption, linear production tech-
nology and distributional assumptions on the idiosyncratic uncertainty, its optimal
policy is characterized by the following proposition:
Proposition 1 Define f t
as the value that satisfies E
y
h
y
i
t
w
t
(y
i
t
w
t
)f t
+1
i
= 0. Then the
entrepreneur’s policy functions will take the form:
h
i
t
= f b
i
t
c
i
t
=(1 b )a
i
t
b
i
t+1
= b R
t
a
i
t
Especially important is that the employment decision will be linear in b
i
t
. The
factor of proportionality f t
depends negatively on the wage w
t
, which is the same for
allfirms,andonthedistributionofy
i
t
,whichisalsothesameforallfirms. Thisallows
15
us to derive the aggregate demand for labor as a linear function of the aggregate
financial wealth of entrepreneurs which is:
H
t
= f t
Z
i
b
i
t
= f t
B
t
The next step is to describe the determination of the idiosyncratic variable y
i
t
which depends on the uncertainty about the demand of goods produced by an
individual firm.
1.3.2 Production and Demand Uncertainty
Every period consumers get randomly distributed among producers. In particular,
assume that each household visits c < 1 producers. Even if each household visits
the same number of producers, the distribution of consumers over producers is not
uniform. This implies that some producers will receive more consumers (per-unit of
labor) than others. As such, the demand uncertainty faced by firms derives from the
randomness in which households get distributed among entrepreneurs.
Denote by n
i
t
the number of consumers per unit of labor received by producer
i in period t. This variable is stochastic with probability density f(n). Since each
household visits c producers, the distribution must satisfy
R
nf(n)dn = c . That
istosay,theaveragenumberofconsumersperworkerreceivedbyeachproducerisc .
Given the choice of h
i
t
, a firm can produce at most ¯ yh
i
t
, where ¯ y> 0 is a constant
and represents a technological constraint. Since all entrepreneurs utilize the same
production technology, ¯ y will be the same for all firms. The quantity ¯ yh
i
t
represents
the firm’s period t production capacity after hiring h
i
t
units of labor. The actual
16
production, however, depends on the quantity of goods that the firm can sell, which
is unknown to the entrepreneur at the time he or she must make the hiring decision.
Each period can be thought of being divided in three subperiods. In the first
subperiod firms choose employment h
i
t
and promise to pay workers the wage w
t
. In
the second subperiod households visit producers shop for consumption goods and
engage in production. In the third subperiod households are allowed to re-trade the
goods acquired from the entrepreneurs in a Walrasian market and all credit/debit
positions, including the promised wages, are settled. Each subperiod is outlined
below.
Subperiod 1: Hiring stage. Entrepreneurs hire labor h
i
t
and set their period
productive capacity ¯ yh
i
t
. The hiring decision takes into account the uncertainty about
the goods that the firm will actually be able to sell in the second subperiod.
Subperiod 2: Decentralized shopping and production. Since a household has a
creditcapacityofq t
andvisitsc firms,thespendingcapacityineachproducerisq t
/c .
Thereforeafirmthatreceives n
i
t
h
i
t
consumerscansellatmost n
i
t
h
i
t
q t
/c unitsofgoods,
that is the number of consumers multiplied by the credit capacity of each consumer.
Assuming that producers have all the bargaining power, the revenue per worker of
firm i is:
y
i
t
=
8
>>>>><
>>>>>:
¯ y if n
i
t
⇣
q t
c ⌘
¯ y
n
i
t
⇣
q t
c ⌘
if n
i
t
⇣
q t
c ⌘
< ¯ y
17
Henceproductionperunitoflaborwillbedeterminedbythenumberofcustomers
that a firm receives, as well as by their purchasing capacity (intra-period credit). Last,
sales for a firm that hires h
i
t
workers is:
Y
i
t
= y
i
t
h
i
t
The assumption that the producers hold all the bargaining power guarantees that,
when the demand is smaller than the production capacity of the firm, the firm does
not sell to customers more goods than their credit capacity. At the same time, the
assumption that households are allowed to re-trade the acquired goods in subperiod
3 (as described below) guarantees that the firm does not charge an interest rate on
the intra-period credit when the demand exceeds the production capacity of the firm.
Notice that charging an interest rate is equivalent to charging a higher price for the
good (units of consumption goods in subperiod 3 per one unit of consumption goods
in subperiod 2).
Subperiod 3: Centralized trading and settlements. Since during the second
subperiod households are randomly matched with producers, the quantity of goods
purchased differs across households. By assuming that at this stage the acquired
goods can be re-traded in a centralized, anonymous market, all households face the
same optimization problem at the end of the period. Specifically, they solve the
recursive problem below:
Let S
t
= {B
t
,q t
} represent the aggregate states of the economy at time t, namely
the extent of credit conditions in the economy and the aggregate level of wealth
1
.
1
I define B
t
as B
t
= B
E
t
+B
H
t
where B
E
t
=
R
b
i
t
dF(i) and B
H
t
= b
t
in equilibrium.
18
Recursively, the household’s optimization problem can be stated as:
V(S,b)= max
c,h,b
0
⇢
c a h
1+t 1+t +b E
q V(S
0
,b
0
)

s.t.: wh+
b
0
R
c+b
: W b
0
R
Their optimal policies satisfy the first order conditions:
a h
t t
= w
t
u
c
(c
t
,h
t
) b R
t
E
t
u
c
(c
t+1
,h
t+1
)
where the last condition will satisfied with equality if the inter-temporal borrowing
constraint is binding.
Overall, the model’s timing is as follows: each entrepreneur i enters period t with
risk-freebonds b
i
t
andchoosesthelaborinput h
i
t
knowing q t
butbeforetherealization
of the idiosyncratic matching n
i
t
takes place. Labor markets are competitive and the
real wage w
t
fluctuates to equate demand and supply. Once n
i
t
is known production
takes place, consumers acquire goods on credit, and firms’ profits are realized.
Following households collect their wages and balances are settled. In settling their
liabilities, households may choose to re-trade some of their purchased goods in an
anonymous Walrasian market which opens at the end of every period. Agents who
acquired goods on credit beyond their actual possibilities might seek to sell some of
their purchases to settle claims. Similarly households who were not able to purchase
enough goods from the firms they were matched with, might seek to increase their
consumption via this market. Finally, each agent chooses the next period’s bond
holding b
i
t+1
. Figure 1.4 schematically represents the model’s timing.
19
Figure 1.4: Model’s Timing
1.3.3 Endogenous Uncertainty
Every period entrepreneurs must decide on their optimal level of output and
employment. They must do so aware of the state of aggregate credit conditions in
the economy, but before knowing the actual number of customer that will visit their
store. To make their decision, entrepreneurs will take into account their current level
of assets b
i
t
and form expectations about their future level of sales. Firms will base
these forecasts on the probability distribution of n
i
t
conditional on the realization of
q t
. This conditioning is relevant since the level of aggregate credit will have first and
second moment effects on the distribution of sales per worker as detailed below.
Since the realization of q t
represents an aggregate shock and, given the definition
of sales in the model, positive realizations of this variable will shift the distribu-
tion of sales per unit of labor to the right, while negative ones will do so to the
left. This represents a first moment effect on the distribution of sales, implying a
higher or lower mean, yet a constant level of dispersion. The intuition is simple,
when aggregate credit conditions in the economy are good, agents are able to
demand more goods and firms expect their average period sales to be higher. The
opposite happens during a contraction. The figure below describes the effects of a
20
positiveincreaseinthelevelofaggregatecreditonthedistributionofsalesperworker.
Figure 1.5: Distribution of sales per worker
The realization of q t
, moreover, will also influence the level of idiosyncratic
uncertainty faced by individual producers in the economy. Given that workers are
subject to a technological constraint ¯ y, the aggregate level of credit in the economy
willconditionthemaximumnumberofclientsthateachworker cancarefor. Assume
for example that capacity is set to ¯ y =W units of the consumption good. A worker
in this economy could sell allW units to a sole customer with enough credit limit to
demand the worker’s entire output, orW /n units to n customers with lower credit
limits. Effectively this constraint will produce a censored distribution of clients per
worker as described in Figure 1.6:
A low realization of q t
means that each worker can serve a large number of
clients, yet also implies it will require a substantial number of clients for that
21
Figure 1.6: Distribution of customers per worker
worker to be profitable. This increases the level or risk per worker beard by
the entrepreneur. Conversely, a high realization of q t
suggests that workers will
need to serve fewer customers to become profitable, since each one will demand
a greater number of goods, effectively lowering the risk per worker. Figure 1.7
highlights the change in dispersion for an increase in the aggregate level of credit
from q 1
t
to q 2
t
, where q 2
t
> q 1
t
. The shaded area represents the reallocation of
probability mass into the new censoring point, and hence the overall reduction in the
levelofuncertaintyperunitoflaborthatanyfirmmustsustainwhenhiringaworker.
Overallthesetwofeaturescapturetheeffectsoftheaggregatestateoftheeconomy
in shaping entrepreneurs’ expectations about their future level of sales and in doing
so condition their production and hiring decisions. As credit conditions improve, the
average expected level of sales rises (first moment effect) enticing firms to hire more
workers. At the same time, the variance of the distribution over which agents form
their expectations decreases (second moment effect), furthering the entrepreneurs’
demand for labor. Figure 1.8 represents the combined first and second moment
effects, where the shaded area represents the decrease in dispersion induced by an
22
Figure 1.7: Censoring point: q 2
t
> q 1
t
improvement in aggregate credit conditions.
Figure 1.8: Endogenous Uncertainty
23
In an expansion, higher expected sales coupled with a lower risk per unit of
labor leads entrepreneurs to revise their production plans and increase their hiring.
The opposite happens in a contraction. The final level of production, nonetheless,
will also depend on an entrepreneur’s ability to hedge the total production risk as
described in the next section.
1.3.4 Risk and Return trade-off
Thelevelofaggregatecreditintheeconomywillconditionboth,theexpectedaverage
level of sales and the level of risk per unit of labor. It will also limit the number of
customers that may be cared for per worker. Beyond ¯ n an entrepreneur knows that
those clients will not be served and revenue will be lost. Hiring more employees
allows business owners to server more customers, but also increases the overall level
of risk they must bear. Given that workers collect their wages independently of
the achieved level of sales, the bigger the wage bill the greater the entrepreneur’s
exposure to an adverse realization of n
i
t
. In turn, as more households are hired the
firm’s expected sales rise, but so does the size of a potential loss. This represents
the fundamental trade-off solved by entrepreneurs when confronted with the task of
choosing their optimal level of inputs.
This trade-off has two principal components: the distribution of sales per firm as
well as the size of its wage bill. In terms of sales, the relevant underlying distribution
is that of customers per firm. As more workers are recruited, this distribution
achieves a higher mean and a higher variance. This implies that the firm’s expected
sales will increase, but so will the level of randomness faced by its owner. Figure 1.9
sketches the mentioned changes endured by the distribution of customers for a firm
24
Figure 1.9: Distribution of customers per firm
which increasing its number of employees.
The second element affecting the entrepreneur’s profitability, the wage bill, also
increases as more workers are recruited. Crucially, while the wage bill increases
monotonically with every new employee, the probability of additional customers
does not. Figure 1.10 simulates the profit distribution for three different firm sizes: 2,
4 and 6 employees. As the number of employees grow the resulting distribution has
a higher mean and higher variance, yet more importantly, it begins to increasingly
gain mass on the low outcome events. As such, even when the shape of the customer
distributiontiltsinfavoroftheentrepreneur, theexposuretoahigherwagebilllimits
the realization of potential profits.
Intuitively as entrepreneurs hire more workers, they increase the scale of their
operation. The bigger the size of the firm, the greater its expected sales, but also
the greater the entrepreneur’s potential loss. Conditional on their level of safe assets,
25
Figure 1.10: Profit distribution simulation
entrepreneurs will choose a level of employment consistent with their expected sales
and potential losses. What follows is a characterization of the model’s equilibrium as
well as its steady state dynamics.
26
1.3.5 Equilibrium
Households do not face idiosyncratic risk and maximize lifetime utility by choosing
c
t
,h
t
and b
t+1
for all t = 0,1,2,.... Let S
t
= {B
t
,q t
} represent the aggregate states of
the economy at time t, namely the extent of credit conditions in the economy and the
aggregate level of wealth.
Recursively, the household’s optimization problem can be stated as:
V(S,b)= max
c,h,b
0
⇢
c a h
1+t 1+t +b E
q V(S
0
,b
0
)

s.t.: wh+
b
0
R
c+b
: W b
0
R
Their policies satisfy the first order conditions:
a h
t t
= w
t
u
c
(c
t
,h
t
) b R
t
E
t
u
c
(c
t+1
,h
t+1
)
where the last condition will satisfied with equality if the borrowing constraint is
binding.
Similarly, the recursive problem for firm i at time t could be written as:
V(S,b
i
)= max
h
i
E
n
(
max
b
i
0
"
ln

y
i
h
i
+b
i
wh
i
b
i
0
R
!
+b E
q V(S
0
,b
i
0
)
#)
where E
q refers to the expectation of q t+1
conditional on q t
and E
n
refers to the
unconditional expectation over all potential realizations of n
i
t
. This difference resides
in that n
i
t
does not exhibit any serial correlation, while q t
does. Given the above, we
27
can define a recursive competitive equilibrium as follows:
Definition 1 A Recursive Competitive Equilibrium consists of the following functions:
(a) A value function V
E
(B,q ,b
i
) and decision rules c
i
(B,q ,b
i
),h
i
(B,q ,b
i
) and b
i
0
(B,q ,b
i
)
for the entrepreneur
(b) A value function V
H
(B,q ,b) and decision rules c(B,q ,b),h(B,q ,b), and b
0
(B,q ,b) for
the household
(c) Price functions w(B, q ) and R(B, q )
(d) A perceived law of motion for the aggregate state S
0
=F (S)=F (B,q )
such that:
(i) Given c) and d), a) solves the entrepreneur’s optimization problem
(ii) Given c) and d), b) solves the household’s optimization problem
(iii) All markets clear:
R
c
i
t
dF(i)+c
H
t
= Y
t
(Goods market)
R
f t
b
i
t
dF(i)= a h
t t
(Labor market)
R
(b
i
t
b
i
t+1
R
t
) dF(i)= b
t
+
b
t+1
R
t
(Financial markets)
(iv) Perceptions about the aggregate states are correct
Implicit in the equilibrium’s definition is the presence of the end of period
Walrasian market described above. In turn, agents may seek to maximize lifetime
consumption, independently on the number of consumption goods they originally
acquired.
Given the equilibrium definition above, the model’s solution is detailed below.
Since the choice of labor h
i
t
is made before the realization of the matching shock n
i
t
,
28
but the saving decision is made after its observation, it will be convenient to define
the entrepreneur’s wealth after production has taken place as:
a
i
t
= b
i
t
+(q t
n
i
t
w
i
t
)h
i
t
1.3.6 Theoretical Propositions
Rewriting per-worker sales y
i
t
in terms of q t
and n
i
t
, and Following Angeletos (2007)
and Quadrini (2014), I state the following propositions.
Proposition 2 Define f t
as the value that satisfies E
n
h
q t
n
i
t
w
t
(q t
n
i
t
w
t
)f t
+1
i
= 0. Then the
entrepreneur’s policy functions will take the form:
h
i
t
= f b
i
t
c
i
t
=(1 b )a
i
t
b
i
t+1
= b R
t
a
i
t
Note that the demand for labor will be linear in the entrepreneur’s wealth b
i
t
.
The factor of proportionality is time-varying, but common to all firms. In turn, the
aggregate demand for labor can be obtained as:
H
t
= f t
Z
i2 N
b
i
t
= f t
B
t
where B
t
denotes the average, per-capita level of wealth. As shown by Quadrini
(2014), the factor of proportionality f t
will depend negatively on the equilibrium
wage rage. In turn, this implies that the aggregate demand of labor will depend
negatively on the wage rate (as in any Walrasian model), but positively on the
29
economy’s level of risk-less assets. For individual producers these assets represent
a firm’s financial net worth. The corresponding theoretical proof can be found in
Appendix A.1.
The above is a unique feature of the model which sheds some light on the
relationship between labor demand and the financial soundness of firms. When
businesses’ net worth suffer (as it does during contractions), the demand for labor
declines inducing a lower equilibrium output and employment. This happens not
as a result of firms lacking the resources to hire employees, or because the value
of their collateral has plummeted and access to financing options are scarce. This
occurspurelyoutofriskconsiderations: withalowernetworthentrepreneurscannot
properly insure against idiosyncratic shocks and seek to reduce their exposure by
limiting their hiring. In other words, given the fact that entrepreneurs are risk
averse and cannot hedge their hiring bets appropriately, they choose to behave
conservatively and revise their production plans downwards. The opposite will
happen in an expansion when a firm’s net worth improves. This a unique and a
crucial feature of the model which will greatly affect the equilibrium dynamics as
described in the next sections.
Another property worth mentioning is that an entrepreneur’s consumption policy
function is linear in wealth. This has two major implications. First, it implies that
entrepreneurs will always consume (and save) a constant proportions of their end
of period assets. As such, during expansions entrepreneurs will not only seek to
consume more but also to increase their stock of savings which will allow them to
increase future production. Second, it makes the problem extremely tractable as
it allows for linear aggregation. Consequently, even when entrepreneurs might be
heterogeneous in asset holdings, in order to understand the aggregate dynamics we
only need to keep track of the average level of wealth B
t
.
30
Proposition 3 In a stationary equilibrium, households will exhaust their credit capacity as
long as b R< 1
Given that entrepreneurs are risk averse and face uninsurable idiosyncratic risks,
they will constantly seek to self-insure. Their desire to smooth consumption would
make them save and hold bonds even if b R = 1. Unfortunately for them, the supply
of these assets is constrained by the borrowing limit of households. Being risk
neutral and solely exposed to an aggregate shock, households need extra incentives
to issue the risk free assets. In turn, in order to induce households to borrow the
equilibrium interest must decline. As long as the interest rate is lower than the
intertemporal discount rate, households will continue to increase their leverage
until their borrowing limit binds setting the steady state interest rate lower than the
intertemporal discount rate.
31
1.4 Quantitative Analysis
I calibrate parameter values of the model economy to match some relevant statistics
fromU.S.data. Therearetwosetsofparameters. Thefirstsetofparametersischosen
externally without using model-generated data while the second set of parameters is
determined jointly by minimizing the distance between the statistics from the model
and the data.
The model period is a year, which corresponds to the data frequency obtained
from Compustat. I set ¯ y to match the U.S. long-run capacity utilization measures
of approximately eighty percent as reported by the Federal Reserve. Following
Reichling and Whalen (2012), I set t = 0.4, implying a labor elasticity of 2.5. This
number is in line with what is used and recommended by the U.S. Congressional
Budget Office. The persistence of the aggregate financial shock is estimated as an
AR(1) process from the survey of senior loan officers available since the second
quarter of 1990. Further, I use customer traffic data to estimate s and set as µ = 0
since the model features constant returns to scale and consequently µ will only have
a scaling effect on the economy. The rest of the parameters (b ,g ,W ) are calibrated to
match the following steady state moments: U.S. long run interest rate of 3%, hours
worked = 1/3 and the ratio of unsecured credit to disposable income as reported by
Herkenhoff (2013). Table 1.2 below summarizes this information.
1.4.1 ShopperTrak data
Paramount to the study’s analysis is an understanding of the distribution of cus-
tomers that firms will care for every period. Proprietary ShopperTrak data was used
to gain such an insight. ShopperTrak is a multinational corporation specialized in
the measurement of consumer traffic flow. The company utilizes electronic traffic
32
Table 1.2: Calibration Values
Parameter Description Value Target/Source
b Discount factor 0.959 Interest rate r = 3%
g Disutility of labor 1.11 Hours worked = 1/3
W Borrowing limit 0.122 Unsecured Credit/ Income = 0.4
t Inv. Frisch elasticity 0.40 CBO estimate (2012)
µ
n
Parameter of matching function 1
2
s 2
n
Consumer Traffic data
s n
Parameter of matching function 0.17 Consumer Traffic data
r Persistence of credit shock 0.884 FRB Senior Loan Officer Survey
s q Stdev of credit shock 0.008 FRB Senior Loan Officer Survey
¯ y Maximum output per worker 0.902 FRB U.S. Capacity utilization rate
counters (ETC)
2
to quantify and monitor customer movements inside as well as
in and out stores. ShopperTrak’s proprietary technology allows their clients to
better understand consumer patterns and manage their resources more effectively.
Currently, thecompanyhassomefiftythousandsETCdevicesinstalledonlyinNorth
America and about seventy thousand world wide.
The company has furnished a dataset containing proprietary consumer traffic
information for almost one thousand stores all of which are located inside the United
States. The data is annual and encompasses a total of four years (2010-2013). The
information is geographically diversified with all fifty U.S. states being represented.
Because of privacy considerations the actual brands included in the sample were
not disclosed, but an anonymous numeric-identifier allows individual stores to be
tracked over time. All in all the dataset forms a balanced panel with a total of 3,840
observations. Table 1.3 lists the key relevant statistics.
As one can see from the table, consumer traffic seems to be pro-cyclical. The data
alsorevealsthatthecross-sectionaldistributionacrosstheU.S.ishighlyasymmetrical
and right skewed, implying that only a handful of stores receive a high volume of
2
See appendix for further details on ShopperTrak and ETCs.
33
Table 1.3: ShopperTrak Data Moments
Statistic Value
Consumer traffic cross-sectional dispersion 1.763
Consumer traffic cross-sectional skewness -0.307
Consumer traffic cross-sectional kurtosis 1.257
Consumer traffic growth rate corr w/ cycle 0.392
⇤⇤
# observations (ave. per year) 985
# observations (total) 3,840
Source: Own calculations based on ShopperTrak data
customers.
1.5 Results
In this section I analyze the quantitative implications of the model. First, I showcase
the model’s ability to successfully match some broad features of the Compustat data.
Second, I describe how a sudden change in aggregate credit conditions may affect
the model’s equilibrium values. Third, I decompose and quantify the contribution of
endogenous uncertainty to the macroeconomic effects of a first moment disturbance
hitting the economy.
1.5.1 General Results
Table 1.4 below reports some fundamental simulation results. The basic strategy
was to calibrate the model utilizing steady state moments and then validating the
framework with non-targeted ones at the business cycle frequency
3
. Overall the
framework does a good job in matching all three targeted steady state moments.
3
Since the framework has a closed form solution, I’m implicitly assuming that the steady state
moments are equal to the model’s ergodic mean; something which in principle is only assured for
linearized models. In turn, I perform a consistency check which can be found in the appendix.
34
Table 1.4: Targeted Moments
Moment Data Model
Steady State interest rate 0.030 0.033
Hours worked 0.333 0.324
Unsecured debt/ Income 0.400 0.397
In addition, the model can successfully replicate several non-targeted moments.
Table 1.5 summarizes some of these results. In the data, both the cross-sectional
dispersion of growth rates of employment and sales are counter cyclical. This empir-
ical regularity has been often documented by other researchers using different data
sets. For example, Bachmann and Bayer (2013) report similar results for Germany
using USTAN data. The model generates the right business cycle co-movement
as an improvement in credit conditions induces firms to raise their sales forecasts
and consequently increase their hiring. As output rises, a greater share of firms
begin producing at their maximum capacity ¯ y, triggering the observed dropped
in cross-sectional dispersion. Furthermore, in the data the correlation with the
business cycle of sales growth dispersion is stronger than that of employment. This
quantitative feature is also correctly matched by the model as sales dispersion tends
to evolve faster than employment.
In addition to the well documented countercyclicality in dispersion, the model is
also able to match higher order moments such as the cross-sectional skewness and
kurtosis of sales and employment growth rates. In the data both sales and employ-
ment are negatively skewed and simulations of the model are able to reproduce these
empirical regularities. In particular, the model’s ergodic distribution of sales has a
negative skewness of -0.272 while that of employment of -0.222. While the model
does slightly overstate the degree of asymmetry in the data, it does quantitatively
match the fact that sales exhibit higher skewness than employment.
35
Table 1.5: Non-targeted Moments
Moment Data Model
Cross-sectional dispersion of employment 2.416 1.675
Cross-sectional dispersion of sales 2.227 1.231
Sales growth rate dispersion correl w/ cycle -0.388 -0.622
Emp growth rate dispersion correl w/ cycle -0.248 -0.586
Employment’s cross-sectional skewness -0.131 -0.222
Sale’s cross-sectional skewness -0.187 -0.272
Also pertinent is the fact that the degree of skewness reported by the model
appears to be pro-cyclical and in line with what is observed in the data. The left
tail of the growth rates distribution tends to expand vigorously during downturns
and less so during expansions, highlighting the fact that a small share of firms seem
to experience a far worse performance than their above average peers. Intuitively
this points out to an asymmetric response in an economy’s downside risk, especially
during recessions. In the model this asymmetric bias seems to be the result of risk
averse and prudent agents, who care more about the deterioration in the outlook
during contractions than the potential gains during an expansion.
In terms of kurtosis, both variables show evidence of heavy tails and peakedness
relative to a Gaussian distribution. The model’s baseline specification successfully
reproduces this positive kurtosis for the cross-sectional distributions of both employ-
ment and sales, although it somewhat understates them both.
1.5.2 Model Dynamics
In addition to the model’s steady state properties, its dynamic features were also
studied. Figure 1.11 plots the response of output, consumption, employment, wages,
interest rates, employment dispersion and capacity utilization to a one percent
positive increase in aggregate credit conditions. Upon impact both output and
36
employment rise. Output does so by almost 0.8 percent, while employment’s reaction
is slightly weaker at 0.6 percent from its steady state value.
Higher production and higher employment puts upward pressure on the real
wage which increases over a quarter of a percent and remains above its steady state
value for about fifteen periods. Similarly, capacity utilization rates rise sharply as
firms update their production plans to meet the expected growth in demand for
consumption goods. The rise in capacity utilization more than doubles that of the
original aggregate shock that propitiated it. In turn, this pushes a greater share of
firms to produce at their maximum per-worker level ¯ y, generating a significant drop
in employment growth dispersion in excess of six percent.
The increase in firms’ profits propitiates a spike in the demand for safe assets as
risk averse entrepreneurs seek to protect themselves from idiosyncratic uncertainty.
The supply of these assets is, nonetheless, constrained by the leverage capacity of the
representative household. A shift in demand coupled with a inelastic supply induce
the price on these assets to rise. Consequently the return on bonds falls as may be
seen in the impulse response function below. In particular, the equilibrium interest
rate falls close to 0.20 percent from its steady state value.
Lastly, therearetheeffectsonconsumption. Improvedaggregatecreditconditions
foster entrepreneur’s profits allowing them to enlarge their demand for consumption
goods. Moreover, household’s also increase their equilibrium consumption allocation
which depends on their labor income net of debt payments. Given that both wages
and hours worked are rising, this increases the household’s revenue. Additionally,
interest rates are falling, so their payment liability is decreased. A combination of
higher incomes and lower interests allows the household to enlarge its consumption
of goods even beyond what the entrepreneur can. Household’s consumption rises
close to 0.8 percent from its original steady state value.
37
Figure 1.11: Impulse responses for a 1% shock to q t
38
1.5.3 Effect Decomposition
Figure 1.11 illustrates how changes in an economy’s credit conditions will have
a direct impact on the equilibrium values of its macroeconomics aggregates. The
dynamic effects discussed above can be thought of having two principal components.
Thefirstoneoftheseeffectsistheresultofthechangeintheaveragelevelofexpected
sales (first moment) while the second one is driven by the fluctuation in the degree of
idiosyncratic uncertainty faced by producers (second moment). This variation in the
level of endogenous uncertainty will be responsible for the difference in magnitude
with traditional models.
To quantify the effects of endogenous uncertainty, Figures 1.12 and 1.13 compare
the framework introduced in Section 1.3 with a version of the same model where
the endogenous uncertainty channel has been shut down. In particular consider the
following framework:
U
i
E
= E
0
• Â t=0
b t
lnc
i
t
y
i
t
h
i
t
+b
i
t
c
i
t
+w
t
h
i
t
+
b
i
t+1
R
t
Where y
i
t
stands for sales per worker, is an iid random variable and represents
an aggregate shock to the economy. The mentioned decomposition is thus done
by comparing the effects of a one percent positive innovation to q t
in the baseline
model, with a one percent increase in y
i
t
in the alternative framework. The latter is
able to capture solely the effects of a level shock, whereas the former one is able
to capture both the first and the second moment effects of a level disturbance. By
comparing impulse responses it is possible to isolate the quantitative contribution
of the endogenous uncertainty channel. Computation shows that On average, about
39
22 percent of the overall effect generated by a change in credit conditions can be
explained by changes in time-varying uncertainty.
Figure 1.12: Effect decomposition for a 1% shock to sales
40
Figure 1.13: Effect decomposition for a 1% shock to sales
41
1.5.4 Sensitivity Analysis
This section explores the sensitivity of the above reported results to variations in
some of the model’s key parameters. Only a few select examples are reported here.
The rest of the analysis can be found in the Appendix 6.
1.5.4.1 The effects of tau
Intrinsically linked to the response of employment supply to variations in credit
conditions, the parameter t plays an important role in the overall dynamics of the
model. Trade-off between fluctuations in the real wage and the labor supply will
condition the response of output in the economy. Table 1.6 below explores the
responses of the model to variations in the parameter governing the labor supply
of households. In each case, the model was recalibrated utilizing the same targets
specified in section 1.4, but each time with a particular value for t . The tabulated
results are the parameter values as well as the response (on impact) of employment
and real wages to a one percent improvement in credit conditions measured as a
percent deviation from their steady state values.
Table 1.6: Model’s sensitivity to t tg b CR Employment Wage
0.2 1.111 0.9596593 0.1217439 0.73% 0.21%
0.4 1.384 0.9596594 0.1217433 0.61% 0.23%
0.5 1.545 0.9596590 0.1217432 0.53% 0.24%
0.7 1.925 0.9596593 0.1217436 0.32% 0.37%
1.0 2.677 0.9596592 0.1217433 0.15% 0.42%
As expected, the response of employment to a positive credit shock becomes
stronger as the Frisch elasticity of labor supply increases. Similarly the lower the
value of t , the weaker the response of the real wage. In other words, as t decreases
42
and the labor supply becomes more elastic, the equilibrium wage becomes less
sensitive to changes in credit conditions.
1.5.4.2 The effects of capacity utilization
The rate at which productive capacity is being used in the economy will be an
important factor governing its cross-sectional dynamics. Interestingly, the lower the
steady state capacity utilization rate, the greater the share of firms that will benefit
from an improvement in credit conditions and yet the worse the mean entrepreneur
might end up being.
This happens because as credit expands and firms seek to increase their produc-
tion, the increase in labor demand puts pressure on the equilibrium wage. For those
firms operating below ¯ y this increase in cost is still profit maximizing. Yet, for those
already operating at their maximum capacity, this increase represents a dent on their
profits. Moreover,thegreatertheshareoffirmsinitiallyoperatingbelowfullcapacity,
the greater the increase in labor demand and hence the greater the rise real wages.
As firm’s operating costs rise, the mean entrepreneur’s profit falls and so does his
equilibrium consumption. Figure 1.14 plots the model’s dynamic response to a one
percent increase in q t
, starting from a steady capacity utilization rate of sixty percent.
On impact, there is a fifty percent stronger response of output than that described
on Figure 1.11. In line with this reaction, equilibrium employment also rises more
thaninthebaselinespecification. Similarly, employmentdispersiondropsascapacity
utilization rates increase in the economy. Overall most variables’ response seems
consistent with the baseline results.
43
Figure 1.14: Sensitivity to capacity utilization rates
In terms of consumption, however, things change substantially. Since labor costs
rise sharply for all firms, the mean entrepreneur’s profit will fall. With less claims
on final production their consumption drops slightly, about 0.2 percent from steady
44
state. With less profits, entrepreneurs reduce their appetite for savings, causing bond
prices to drop and consequently inducing a rise in its yield. Households, on the
other hand, are benefited by the strong increase in wages and hours, although higher
interest rates will act like a dent on their available resources. Consequently their
consumption rises, but less than in the baseline specification.
45
1.5.5 Extension: Persistent Demand Shocks
One of the assumptions present in the model’s baseline specification was that the
idiosyncratic disturbances faced by entrepreneurs presented no serial correlation.
This afforded us a tractable and intuitive closed form solution. However, there
are reasons to believe that demand fluctuations may in fact experience certain
dependence over time.
Figure 1.15: Consumer Traffic and Business Cycle
Source: Own calculations based on ICSC data
To gain a deeper understanding of this assumption I utilize the consumer traffic
diffusion index from the International Consortium of Shopping Centers (ICSC). The
diffusion index is produced monthly by the ICSC from a survey of consumer traffic
reported by shopping center’s executives. Readings over 50 imply a general positive
momentum in the number of customers visiting shopping centers, while readings
below50hintofaslowdown. TheadvantagesofthisdataseriesovertheShopperTrak
data is that it is available for a longer time horizon. The disadvantage, however, is
46
that since it constitutes an aggregated index, individual stores cannot be tracked
across time. Figure 1.15 plots this alternative measure of consumer traffic.
Figure 1.15 suggests that consumer traffic is pro-cyclical and highly persistent.
Even when there is only enough data to capture two U.S. recessions, both the 2001
and 2008 downturns appear clearly visible. Not surprisingly the drop in consumer
traffic related to the 2008 depression appears to be deeper and longer-lasting than
that of 2001. Furthermore, the process appears to be persistent, with increases and
decreases in consumer traffic lasting several months.
With this in mind, this section seeks to extend the baseline specification by includ-
ing correlated idiosyncratic disturbances. In turn I relax the original assumption and
investigate the implications and overall performance changes of the framework pre-
sented in section 1.3 when disturbances are serially correlated. In particular, assume
now that the distribution of customers per worker arriving to a store follows:
lnn
i
t
= r lnn
i
t 1
+y t
: y t
⇠ N(µ
n
,s 2
n
)
where the persistence parameter for the customer traffic process is estimated
utilizing ICSC data.
Let S
t
= {q t
,B
t
} represent the economy’s aggregate states. The household’s opti-
mization problem does not change. However, the entrepreneur’s recursive formula-
tion would now be:
V(S,b
i
,n
i
)= max
h
E
n
(
max
b
i
0
"
ln

y
i
h
i
+b
i
wh
i
b
i
0
R
!
+b E
S
V(S
0
,b
i
0
,n
i
0
)
#)
47
where E
n
refers to the expectation of n
i
t+1
conditional on the current realization of n
i
t
and E
S
represents the equivalent conditional expectation for S
t
. Since the model loses
its closed form solution I solve it by performing a linear approximation around the
agent’s policy function following Covas (2006). Results are described in the table 1.7.
Table 1.7: Results
Moment Data Baseline Extension
Cross-sectional dispersion of employment 2.416 1.675 1.832
Cross-sectional dispersion of sales 2.227 1.231 1.515
Sales growth rate dispersion correl w/ cycle -0.388 -0.622 -0.533
Emp growth rate dispersion correl w/ cycle -0.248 -0.586 -0.397
Employment’s cross-sectional skewness -0.131 -0.222 -0.171
Sale’s cross-sectional skewness -0.187 -0.272 -0.232
Employment’s cross-sectional kurtosis 2.484 1.705 1.811
Sales’s cross-sectional kurtosis 2.236 1.932 2.052
As can be seen from table 1.7, both the sales and employment dispersion growth
rates appear countercyclical in all model specifications. This is in line with the data
and implies that the model’s results are qualitatively robust. Quantitatively, there is
also a slight improvement in the model’s capacity to match the data. The inclusion of
serially correlated customer traffic substantially improves the model’s effectiveness at
at matching the desired moments.
48
1.6 Conclusion
InthisstudyIhaveinvestigatedtheeffectsoffluctuationsinuncertaintyonaggregate
economic activity. In particular, I have done so contemplating the hypothesis that
changesinuncertaintyareendogenoustothecurrentstateoftheeconomy. Thepaper
develops a general equilibrium incomplete markets framework with heterogeneous
firms that account for the asymmetric fluctuations of the U.S. labor market and
output. The fundamental property of the model is that expansions and contractions
in the economy are inititated by shifts in aggregate credit conditions and these, in
turn, may induce changes in uncertainty.
The model generates realistic volatility in aggregate employment and output.
Moreover, I have found that endogenous fluctuations in uncertainty may significant
amplify the real effects of first moment shocks. The uncertainty channel is shown
to be able to propagate approximately thirty percent of a level’s shock initial effect.
The model also predicts that the level of uncertainty varies with the business cycle.
This is in line with what has been documented for the U.S. where every measure of
uncertainty systematically falls in expansions and rises during recessions.
I have also found that aggregate fluctuations will have effects on the cross-
sectional dispersion of output and employment. This highlights the importance
of taking into account the risk tolerance of individual producers which is often
washed away in aggregate figures. Results confirm that the proper understanding of
business cycles requires knowledge of the cross-sectional distributions as well as the
aggregate time-series. There is need for theories that can explain not just the mean
variation of consumption, output, and employment, but also why the distribution of
firm behavior changes considerably over the cycle and how this may (or may not)
matter in determining the amplitude of the cycle and the process of job creation and
49
destruction.
There are several extensions that might be useful to consider. The first one would
be to add capital to the framework. This would allow the model to provide insights
into fluctuations in investment, which is usually a more fundamental contributor to
business cycle dynamics than employment. Moreover, it could also shed light to the
relationship between uncertainty and asset allocation. Under a set-up with capital,
the entrepreneur would now have two instruments in which to save one yielding
a safe but low return, and another one yielding a more risky yet potentially more
rewarding alternative.
Further, there are two potentially interesting extensions regarding the effects of
uncertainty on nominal variables. First, since the model is written in real terms, there
is no explicit role for monetary assets. Adding money to the study’s framework
would allow for the exploration of the effects of fluctuations in uncertainty on
nominal shocks, as well as its effect on the role of monetary policy. Additionally, the
model could provide insights into the effectiveness of monetary policy at different
levels of economic uncertainty throughout the business cycle. These extensions are
left for future research efforts.
50
Chapter 2
Uncertainty and aggregate
macroeconomic fluctuations in Small
Open Economies
2.1 Introduction
It has been widely documented that emerging economies tend to experience far
more volatile business cycles than their more developed counterparts. Uribe and
Schmitt-Grohé (2011) reports that emerging market economies are more than twice
as volatile as developed ones, with private consumption being more volatile than
output in emerging countries but less volatile than output in developed ones. Table
2.1 summarizes a few of these empirical regularities.
Some explanations for this difference in volatility point towards endogenous
reasons such as incipient institutional arrangements and (unintended) pro-cyclical
domestic government policies. More importantly, however, it is believed that emerg-
ing economies are generally exposed to more frequent and more intense shocks than
developed ones. Two characteristics make emerging countries particularly suitable
for this to occur: insufficient domestic capital markets which foster international
borrowing and a natural specialization in the production (and exporting) of a few
commodities such as basic metals, agricultural goods or oil. The first reality exposes
emerging economies to changes in the world interest rate, while the second one to
51
swings in the terms of trade.
Regarding interest rates, Uribe and Schmitt-Grohé (2011) find that business cycles
in emerging markets are correlated with the cost of borrowing that these countries
face in international financial markets. As such, periods of low interest rates are
typically associated with economic expansions and times of high interest rates are
often characterized by depressed levels of economic activity.
With respect to terms of trade, several authors have documented the extent to
which fluctuations in this relative price are responsible for increased volatility in
GDP growth. Mendoza (1995) and Kose (2002) find that terms of trade movements
can account for roughly half of the output volatility in emerging countries. Moreover,
Baxter and Kouparitsas (2000) suggest that terms of trade fluctuations are twice as
large in developing countries as in developed ones.
Table 2.1: Business Cycle Statistics (1959-2009)
Statistic Developed Countries Developing Countries
s (y) 3.7 9.2
s (c) 3.3 8.9
s (G) 6.0 10.9
s (i) 11.7 17.0
s (x) 9.4 21.7
s (m) 9.8 23.9
Source: Uribe and Schmitt-Grohé (2011)
The reasons are twofold. On the one side, economic reliance of developing
countries on commodity exports, whose prices are more volatile than those of man-
ufactured goods, exposes them to fickle changes in external demand. Additionally,
since developing economies generally have a higher degree of openness to foreign
trade, unexpected swings in terms of trade affect a large share of their economy. On
52
the other side, emerging markets tend to have very little, if any, leverage over their
export prices. Broda (2004) documents how world markets dictate the price of the
goodsemergingeconomiesexports,whiledevelopedcountriescanexertasubstantial
influenceontheirexportprices. Inturn,shiftsintermsoftradearelargelyexogenous
and potentially very damaging.
While changes to the mean of these external sources of uncertainty have been well
documented, less attention has been paid to study changes to their volatility. In a
Knightian sense, these spikes in volatility may be regarded as a source of aggregate
risk allowing us to capture events that do not directly relate to the country under
study. In particular, stochastic volatility enables us to pick up the changes in global
risks which have been particularly relevant since 2008 with the economic crisis
spawned in the more developed economies.
The intuition behind this channel is that higher risk triggers precautionary
behaviors, such as postponing consumption or investment decisions, leading to
potential downturns in output even in countries that do not have a direct link with
those countries under stress. In other words, countries with sounds fundamentals
and implementing good policies may still suffer the negative consequences of a more
uncertain global economy.
Table 2.2: Terms of Trade and Business Cycle
Statistic Developed Countries Developing Countries
s (tot) 4.70 10.0
r (tot
t
,tot
t 1
) 0.47 0.40
s (tot)/s (y) 0.45 0.77
Source: Uribe and Schmitt-Grohé (2011)
53
To date most efforts to model swings in volatility have been focused on real,
closed economy set-ups, which seek to mimic US dynamics. Kim and Nelson
(1998), McConnell and Perez-Quiros (2000), Sims and Zha (2006), Stock and Watson
(2003) among others have presented overwhelming evidence that the volatility of
US aggregate time series data has changed over the last forty years in a substantial
way. In particular, Stock and Watson (2003) have labeled the noticeable reduction
in volatility of output that has occurred since the middle of the 1980s the Great
Moderation.
Less attention has been given to the effects of time varying volatility in an open
economy context. Pioneering work in this area correspond to Fernández-Villaverde
et al. (2011), who documented the real effects induced by changes in the volatility of
interest rates at which small open economies may borrow. Mendoza (1997) uses an
endowment economy to document how consumption growth is affected by various
degrees of terms of trade volatility. No work, however, has been done in analyzing
the shocks to the volatility of terms of trade in a structural framework.
This study seeks to complete this gap in the literature and analyze the real
consequences of stochastic volatility in terms of trade using Chile as a case study.
Chile is a small a small open economy with access to international capital markets
and specialized in the production of mining commodities. Its exports of copper
represent over fifty percent of the country’s total exports and about five to ten per-
centoftotalGDP,makingitaparticularlyrelevantexampletotakeintoconsideration.
The rest of the paper proceeds as follows. First, the evolution of the time-varying
volatility associated with Chile’s terms of trade is characterized and presented as
empirical evidence. It is not trivial how to identify the effects of an exogenous
increase in volatility, for macroeconomic risk can change endogenously for various
reasons. However, using the small open economy assumption we can take the terms
54
of trade as being exogenous and estimate the volatility shocks using time series
models.
In the second part of the paper the estimated process is fed into a multi-sector
model of a small open economy with free capital mobility. In particular, the economy
features five goods: a consumption good, tradable and non-tradable goods as well
as two intermediate inputs: one domestically produced while the other is imported.
Capital goods are assumed to be imported and its accumulation is subject to convex
adjustment costs, but are not sector specific. The model is solved with a third-oder
perturbation approach; a non-linear solution method that allows to study the effects
of stochastic volatility (which cannot be directly captured using approximations up
to second order). Parameters are calibrated using Chilean data as reference.
The main finding of the study is that volatility shocks do matter when seeking to
understand domestic fluctuations in output, consumption and investment; as well
as sectoral reallocation of resources. That is to say, second order moments do have
first order effects on aggregate macroeconomic variables. Moreover, volatility shocks
seem to closely relate to their level counterparts in terms of timing and direction, but
differ mostly in terms of their magnitude.
2.2 Characterizing the stochastic volatility process
We assume the terms of trade follow an autoregressive process with stochastic
volatility. The general approach is borrowed from Fernández-Villaverde et al. (2011)
in that this process is first estimated and then fed into a structural model. The data
for export prices for Chile is monthly and ranges from January 1989 to December
2011. Monthly, rather than quarterly data is used because this seems to be more
55
appropriate for capturing the volatility of terms of trade as required by the study.
2.2.1 The Law of motion for terms of trade
ThetermsoftradefacedbydomesticresidentsattimetiswrittenasanAR(1)process:
p
X
t
= r p
p
X
t 1
+e
s e
P
t
e
P
t
(2.1)
The main feature of this process is that the standard deviations are not constant,
as commonly assumed, but also follow an AR(1) process of the form:
s e
P
t
=(1 r s p)¯ s e
P
+r s ps e
P
t 1
+ j X
t X
t
(2.2)
where e
P
t
⇠ N(0,1) and t X
t
⇠ N(0,1). Thus, the process for terms of trade
displays stochastic volatility. The parameters ¯ s e
P
and j X
control the degree of mean
volatility and stochastic volatility in the terms of trade. A high ¯ s e
P
implies a high
mean volatility whereas a high j X
a high degree of stochastic volatility.
Other specifications such as GARCH models have been explored but discarded
since they cannot distinguish between innovations to the terms of trade and to
the volatility: higher volatilities are triggered only by innovations to the price. In
comparison, stochastic volatility has two different types of shocks. In the benchmark
exercise, all innovations are assumed to be independent of each other, although little
changes if this assumption is relaxed.
56
2.2.2 Estimation Strategy
The parameters in the equations above are estimated with a likelihood-based
approach. The likelihood of these processes is difficult to evaluate, because of
the presence of two innovations (to the level and to volatility) that interact in a
non-linear fashion. This problem is address by usage of the particle filter such
as in Fernández-Villaverde et al. (2011). We follow a Bayesian approach to infer-
encebycombiningthelikelihoodfunctionwithasetofpriors,alldetailedintable2.3:
Table 2.3: Prior Distributions
Parameter Distribution Mean St. Dev.
r p
Beta 0.90 0.01
¯ s e
p
Normal -3.09 0.80
r s p Beta 0.90 0.01
j X
Normal+ 0.50 0.50
Overall, the priors are sufficiently loose to accommodate a wide array of small
open economies. Robustness tests were done by increasing the standard deviation of
the priors, but this did not significantly change the model’s predictions.
57
2.2.3 Posterior Estimates
We sample 20,000 draws from the posterior of each estimated process, using a
random walk Metropolis-Hastings algorithm. The draw was run after a search for
initial conditions and an additional 5,000 burn-in draws. The posterior medians are
tabled below:
Table 2.4: Posterior Distributions
Parameter Median
r p
0.8941
¯ s e
p
-2.4150
r s p 0.3028
j X
0.9787
Figure 1 displays the log GDP (red line, right axis) and the posterior median (blue
line, left axis) of the estimated process for s t
. The estimated volatility series does a
remarkablejobatpicking-upperiodsofhighuncertaintyinthecommoditiesmarkets.
Starting 2006 volatility increases after the initial commodity boom. A second episode,
and the most important in size, was during the recent global financial crisis when the
effects of volatility shocks were easily appreciated.
More importantly, however, the series is able to capture the negative relationship
between spikes in terms of trade volatility and GDP. In other words, periods of high
uncertainty regarding international prices seem to constantly match periods of lower
GDP growth. This will be a salient feature that the model will seek to replicate.
58
Figure 2.1: Chile’s Terms of Trade volatility and GDP
2.3 The Model
The model follows a prototypical small open economy model with incomplete asset
markets in the spirit of Mendoza (1991), Neumeyer and Perri (2005), and Uribe
and Yue (2006). Since the economy is small and open, exportable and importable
goods are internationally traded and their prices are taken as exogenous. The
imported good (M) will be used as the numeraire. Households have access to an
internationally traded risk less bond denominated in units of the imported good and
face portfolio-adjustment costs.
59
2.3.1 The Household Problem
A small open economy is inhabited by an infinitely lived representative household
who seeks to maximize lifetime utility:
U = E
0
• Â t=0
b t
u(c
t
µ¯ c
t 1
,h
t
) (2.3)
subject to:
d
t
+w
X
t
h
X
t
+w
N
t
h
N
t
+u
M
t
k
M
t
+u
X
t
k
X
t
+P t
R
t 1
d
t 1
+ p
c
t
c
t
+Y (d
t
)+i
M
t
+i
X
t
(2.4)
lim
j! • E
t
d
t+j+1
P j
s=0
(1+R
t+s
)
 0 (2.5)
where ¯ c
t 1
denotes the cross-sectional average level of consumption in period
t 1 and h
t
the fraction of time devoted to work in period t. The parameter b 2 (0,1)
denotes de discount factor while µ measures the degree of external habit formation
(µ = 0 corresponds to perfect separability). The higher the value of µ the stronger
the degree of habit formation.
Households offer labor services for a wage to the exporting (X) and non-tradable
(N) sectors. They also own the capital stock k
t
which they rent at the sectoral market
rate u
j
t
andreceivedistributedprofitsearnedbyfirmsP t
, whereP t
=Â z2 (c,N,T,X)
P z
t
.
Moreover, u(c µ¯ c,h)=
[c µ¯ c w 1
h
w ]
1 g 1
1 g .
Let d
t
denotethedebtpositionassumedbythehouseholdinperiod t. Agentsmay
borrow or lend freely in international financial markets by buying or selling risk-free
bonds denominated in units of importable goods, which yield a gross interest rate of
60
R
t
.
In order to eliminate the unit root built in the dynamics of the standard formu-
lation of the SOE model, households are assumed to face costs in adjusting their
foreign portfolio asset position. This cost is assumed to be convex and to satisfy
Y (
¯
d)=Y 0
(
¯
d)= 0 for some
¯
d> 0. In particular:Y (d)=
y 2
(d ¯
d)
2
.
Households may also invest in physical capital. The process of capital accumu-
lation displays adjustment costs in the form of gestation lags and convex costs of
installing new capital goods. Investment is sector specific and based on imported
goods. We assume it takes 4 months to build new productive capital. For j = {X,M}
consider:
i
j
t
=
1
4
3
Â n=0
s
j
n,t
(2.6)
Total investment is defined as:
i
t
=
Â j2 (X,M)
i
j
t
=
Â j2 (X,M)
3
Â n=0
s
j
n,t
(2.7)
New investment projects initiated in period t are s
0,t
. This will be the agent’s
choice variable. The evolution of s
n,t
follows:
s
j
n+1,t+1
= s
j
n,t
(2.8)
The capital stock evolves according to:
61
k
j
t+1
=(1 d )k
j
t
+k
j
t
F 0
@ s
j
3,t
k
j
t
1
A (2.9)
where d 2 [0,1],F (d )= d andF 0
(d )= 1. In particular, for f >0:
F (x)= x f 2
(x d )
2
.
Household’schoose{c
t
,h
N
t
,h
X
t
,s
M
0,t
,s
X
0,t
,d
t+1
}
• t=0
tomax(1). Letl t
betheLagrange
multiplieronthebudgetconstraint, l t
q
X
t
and l t
q
N
t
themultipliersonthecapitalstock
and l t
v
X
n,t
and l t
v
N
n,t
on the law of motion of investment projects. The Lagrangian
associated with the household’s optimization problem can be written as:
L = E
0
• Â t=0
b t
(
u(c
t
µ¯ c
t 1
,h
t
)+l t
h
d
t
+w
X
t
h
X
t
+w
N
t
h
N
t
+u
X
t
k
X
t
+u
N
t
k
N
t
R
t 1
d
t 1
Y (d
t
) p
c
t
c
t
1
4
3
Â n=0
s
X
n,t
1
4
3
Â n=0
s
M
n,t
i
+l t
3
Â n=0
v
X
n,t
h
s
X
n,t
s
X
n+1,t+1
i
+l t
3
Â n=0
v
M
n,t
h
s
M
n,t
s
M
n+1,t+1
i
+l t
q
X
t
"
(1 d )k
X
t
+k
X
t
F
s
X
3,t
k
X
t
!
k
X
t+1
#
+l t
q
M
t
"
(1 d )k
M
t
+k
M
t
F
s
M
3,t
k
M
t
!
k
M
t+1
#)
2.3.2 Firms and Technology
The model features the production of five goods: consumption, tradables, non-
tradables, exportables and importables. The production technologies are described
below.
62
2.3.2.1 Production of Consumption goods
Domestic firms employ tradable and non-tradable goods as intermediate inputs to
produce a final consumption good. Technology is given by the following CES func-
tion:
c
t
=[c (c
T
t
)
e
+(1 c )(c
N
t
)
e
]
1/e
(2.10)
Firms in this sector seek to maximize the following profit function subject to the
technological constraint detailed above:
P c
t
= p
c
t
c
t
p
T
t
c
T
t
p
N
t
c
N
t
(2.11)
with p
T
t
and p
N
t
denoting the relative price of tradables and non tradables con-
sumption goods in terms of importable goods.
2.3.2.2 Production of Tradable goods
Production of tradable goods is done via Cobb-Douglass technology which combines
a domestic (X) and a foreign (imported) input (M):
c
T
t
=(c
X
t
)
a (c
M
t
)
1 a (2.12)
where a 2 [0,1]. Firms in the tradable sector hire capital services from perfectly
competitive markets and seek to maximize their profits:
P T
t
= p
T
t
c
T
t
p
X
t
c
X
t
c
M
t
(2.13)
63
2.3.2.3 Production of Domestic (exportable and importable) intermediate inputs
An importable good is either an imported good or a good that is produced domesti-
callybutishighlysubstitutablewithagoodthatisimported. Similarly, anexportable
good is either an exported good or a good that is sold domestically but it highly
substitutable with a good that is exported.
Firmsintheexportablegoodssectorhirelaborandcapitalincompetitivemarkets.
The production process is subject to a working-capital constraint that requires firms
to hold non-interest bearing assets to finance a fraction of the wage bill each period.
Formally:
E
0
• Â t=0
b t
l t
l 0
P X
t
(2.14)
where:
1
P N
t
= p
X
t
F
X
(k
X
t
,h
X
t
)+b
X
t
+k X
t
R
d
t 1
b
X
t 1
w
X
t
h
X
t
u
X
t
k
X
t
k X
t 1
(2.15)
subject to:
y
X
t
= F
X
(k
X
t
,h
X
t
)= A
X
t
⇣
k
X
t
⌘
a X⇣
h
X
t
⌘
1 a X
(2.16)
k X
t
h N
w
X
t
h
X
t
(2.17)
1
Note that given that households own the firms their marginal utility of wealth is used as the
stochastic discount factor.
64
lim
v! • E
t
a
X
t+v
P v
s=0
(1+R
d
t+s
)
 0 (2.18)
R
d
t
=
R
t
1 Y 0
(d)
(2.19)
where h X
0.
Defining the firm’s total net liabilities at the end of period t as a
X
t
= R
d,X
t
b
X
t
k X
t
and assuming the working-capital constraint will always bind (otherwise the firm
wouldbeincurringinunnecessaryfinancialcosts)wecanrewritethefirms’constraint
as:
a
X
t
R
d
t
= a
X
t 1
p
X
t
F
X
(k
X
t
,h
X
t
)+w
N
t
h
X
t
"
1+h
R
d
t
1
R
d
t
!#
+u
X
t
k
X
t
+P X
t
(2.20)
In turn, the firm’s problem can be stated as choosing processes for a
t
, h
t
and k
t
so
as to maximize:
E
0
• Â t=0
b t
l t
l 0
(
a
X
t
R
d
t
a
X
t 1
+ p
X
t
F
X
(k
X
t
,h
X
t
) w
X
t
h
X
t
"
1+h X

R
d
t
1
R
d
t
!#
u
X
t
k
X
t
)
(2.21)
Firms in the importable sector face a static optimization problem:
P M
t
= F
M
(k
M
t
) u
M
t
k
M
t
(2.22)
65
subject to:
y
M
t
= F
M
(k
M
t
,h
M
t
)= A
M
t
⇣
k
M
t
⌘
a M
(2.23)
(2.24)
2.3.2.4 Production of Non-Tradable goods
A non-tradable good is a good that is neither exportable nor importable. Similar to
firms in the exporting sector, businesses in the non-tradable sector hire labor services
in competitive markets and are subject to a working capital constraint:
E
0
• Â t=0
b t
l t
l 0
P N
t
(2.25)
P N
t
= p
N
t
F
j
(k
N
t
,h
N
t
)+b
N
t
+k N
t
R
d
t 1
b
N
t 1
w
N
t
h
N
t
u
N
t
k
N
t
k N
t 1
(2.26)
subject to:
y
N
t
= F
N
(h
N
t
)= A
N
t
⇣
h
N
t
⌘
1 a N
(2.27)
k N
t
h N
w
N
t
h
N
t
(2.28)
lim
v! • E
t
a
N
t+v
P v
s=0
(1+R
d
t+s
)
 0 (2.29)
66
R
d
t
=
R
t
1 Y 0
(d)
(2.30)
where h N
0.
Again, defining the firm’s total net liabilities at the end of period t as a
N
t
=
R
d,N
t
b
N
t
k N
t
we rewrite the firm’s constraint as:
a
N
t
R
d
t
= a
N
t 1
p
N
t
F
N
(h
N
t
)+w
N
t
h
N
t
"
1+h
R
d
t
1
R
d
t
!#
+P N
t
(2.31)
and re-state the problem as choosing processes for a
t
and h
t
so as to maximize:
E
0
• Â t=0
b t
l t
l 0
(
a
N
t
R
d
t
a
N
t 1
+ p
N
t
F
N
(h
N
t
) w
N
t
h
N
t
"
1+h
R
d
t
1
R
d
t
!#)
(2.32)
(2.33)
67
2.3.3 Exogenous forces
There are two main source of uncertainty in this economy: an interest rate shock and
a shock to the terms of trade. The shock to the terms of trade exhibits stochastic
volatility, while the interest rate one doesn’t.
lnp
X
t
=(1 r p
)ln ¯ p
X
+r lnp
X
t 1
+s e
P
t
e
P
t
(2.34)
lns e
P
t
=(1 r s p)ln ¯ s e
P
+r s plns e
P
t 1
+jt
t
(2.35)
lnR
t
=(1 r r
)ln
¯
R+r r
lnR
t 1
+s R
t
e
R
t
(2.36)
lns R
t
=(1 r s r)ln ¯ s e
R
+r s r lns R
t 1
+qg
t
(2.37)
A
X
t
= r X
A
X
t 1
+u t
(2.38)
A
N
t
= r N
A
N
t 1
+V t
(2.39)
A
M
t
= r M
A
M
t 1
+i t
(2.40)
where e
P
t
⇠ N(0,1), e
R
t
⇠ N(0,1), t t
⇠ N(0,1) u t
⇠ N(0,1), V t
⇠ N(0,1), i t
⇠ N(0,1) and g t
⇠ N(0,1).
2.3.4 Solution concept and Equilibrium definition
The model’s solution is approximated using a third-order taylor expansion to the
policy function around the non-stochastic steady state. In particular, letting y
t
col-
lect all the variables in the model (both exogenous and endogenous), measured as
differences from their respective steady state values, and letting z
t
collect the prede-
termined variables and the innovations (also as differences from their steady state),
the approximated solution takes the form:
y
t
= G
0
+G
1
z
t
+G
2
(z
t
⌦ z
t
)+G
3
(z
t
⌦ z
t
⌦ z
t
) (2.41)
Inotherwords, thepolicyfunctionisnodifferentthanthatfromatraditionalRBC
model with the exception that it includes some additional parameters (s e
P
t
) which
68
allow the agents in the economy to keep track of the distribution of the shocks. This
is not trivial since, for example, a higher/lower variance will make the distribution
of outcomes more/less dispersed and the agents will take that into account when
making their optimal decisions. In other words, the parameters describing the
distribution of the shocks now also become state variables of the agent’s problem.
2.3.5 Calibration
The parameters characterizing the evolution of the exogenous variables are calibrated
using the estimation procedure described in the previous section. The calibration for
the remaining parameters is presented in Table 3. The time period is one month.
In terms of the utility parameters we follow Fernandez-Villaverde et al. (2001b) or
Reinhart and Vegh (1995) when setting the value for risk aversion and choose the
Frisch elasticity in line with the literature. The shares of capital in both production
functions are chosen following Medina and Naudon (2011) to represent values in line
with Chile’s data. The depreciation rate d implies an annual rate of near 6 percent,
which is generally used in these types of models. The rest of the parameters are
detailed in Table 3:
69
Table 2.5: Calibration Values
Parameter Description Value
a Share of c
X
in c
T
0.15
a X
Share of k
X
in y
X
0.571
a N
Share of k
N
in y
N
0.66
a M
Share of k
N
in y
M
0.698
b Discount Factor 0.9995
d X
Depreciation rate 0.008
d M
Depreciation rate 0.008
w 1
(w 1)
= Labor supply elasticity 1.455
g Inverse of intertemporal E.o.S 2
c Share of c
T
in c 0.48
e CES parameter -0.5
µ Degree of external habit formation 0.204
r X
Persistence of TFP shock in X 0.9
r N
Persistence of TFP shock in N 0.9
r M
Persistence of TFP shock in M 0.9
y Portfolio adjustment cost 1.4
h X
Fraction of wages to finance in X 0.4
h N
Fraction of wages to finance in N 0.4
f X
Capital adjustment cost 2.2
f M
Capital adjustment cost 2.2
The parameters h , f , µ and y were calibrated to match five moments in the data
via SGMM: (i) the standard deviation of output (s y
), (ii) the standard deviation of
consumption (s c
) , (iii) the standard deviation of investment (s I
), (iv) the correlation
between output and consumption (r y,c
) as well as the correlation between output and
investment (r y,I
).
The data comes from the OECD and IMF databases as well as Chile’s Central
Bank (BCL). The data is HP-filtered and ranges from January 1995 until December
2011 (7 years, 84 data points). Following Fernandez-Villaverde et al. (2011b), these
moments are matched with moments computed from the ergodic distribution of
the model. In particular, the model is simulated starting from the steady state for
5000 periods and the moments are computed using the last 1000 observations. Note
70
that to guarantee a non-explosive simulated path, we apply the pruning procedure
suggested by Den Haan and De Wind (2012).
2.4 Results: The real effects of risk shocks
The model’s calibration allows the targeted moments to be matched with a great deal
of precision as described in table 2.6.
Table 2.6: Targeted Moments
Moment Model Data
s y
22.33 24.18
s c
23.34 22.29
s i
21.12 23.17
r y,c
0.932 0.966
r y,i
0.877 0.818
Moreover, the model is also able to replicate some moments from the data which
were not originally targeted, testifying to the accuracy and robustness of the results.
These moments are described in table 2.7.
Table 2.7: Non-Targeted Moments
Moment Model Data
s tb
16.66 15.83
r tb,y
-0.345 -0.579
r tb
t
,tb
t 1
0.334 0.245
r y
t
,y
t 1
0.798 0.691
r c
t
,c
t 1
0.543 0.627
r i
t
,i
t 1
0.597 0.665
Regarding the model’s dynamics, the effects of the volatility shocks are presented
viaanimpulse-responseanalysis. Giventhenon-linearityofthepolicyfunction,com-
puting these impulse responses is not trivial. They are calculated in the following
71
fashion, following Koop et al. (1996) and Fernández-Villaverde et al. (2011). Let y
t
denote the variables of the model (both exogenous and endogenous), x
t
collects the
predetermined variables (endogenous and exogenous) while v
t
denotes the innova-
tions to the exogenous component x
t
. The impulse response to a unit shock in the i
element of the innovation vector v
t
is:
E(y
t+j
|x
t
,v
i,t
= 1,v
i,t
= 0) E(y
t+j
|x
t
,v
t
= 0) (2.42)
To implement this, the models needs first to be simulated for the desired amount
of periods starting from the ergodic mean of x
t
, giving a value of one to the desired
innovations in the initial period and setting all other innovations in that period and
all innovations in the following periods equal to zero. In addition, another sequence
of endogenous variables needs to be simulated, starting also from the ergodic mean
of x
t
but setting all innovations in all periods equal to zero (this approximates
E(y
t+j
|x
t
,v
t
= 0). The reported impulse responses are the difference between the
former and the later simulated sequences
2
.
The responses to a sudden, unanticipated increase in volatility (a volatility shock)
to terms of trade are described in Figure 2.2. On impact, consumption and invest-
ment drop, the trade balance improves and foreign debt decreases. These immediate
reactions are qualitatively in line with a precautionary savings story generated by the
one-sector model in Fernández-Villaverde et al. (2011).
Important differences, nonetheless, arise in this multi-sector model featuring
non-tradable goods. As mentioned, precautionary motives generate a desire to
reduce consumption. Due to risk aversion, agents when facing a spike in the
volatility will tend to care more about potential losses than positive outcomes. In
turn, when faced with an increase in volatility and due to the homotheticity of
2
Note that given the non-linearity of the solution, E(y
t+j
|x
t
,v
t
= 0) will not equal zero, as it gener-
ally occurs when the steady state is taken to be the initial point.
72
preferences, households contract the consumption demand for all types of goods
in their consumption bundle. For traded goods, this only implies a change in the
trade balance. Non-tradable goods, however, must clear domestically inducing a
drop in their relative price generating in effect a real exchange rate depreciation.
This depreciation, in turn, generates incentives to reassign resources from the
N sector to the production of X goods. This sectoral reallocation of resources
helps to mitigate the drop in GDP, by expanding the output of the exportable
good sector. It also helps to explain why investment in sector X is not negatively
affected. Last, because the debt is denominated in units of foreign goods, the
depreciation makes servicing the debt much more costly, so the indebtedness
level of the economy falls. All together, higher exports and lower foreign financing
leadstoanoverallimprovementofthetradebalanceasseenintheimpulseresponses.
,left
Figure 2.2: Responses to an innovation in the volatility of p
X
t
73
Figure 2.3: Responses to the level and the volatility innovation of p
X
t
Toconclude,giventhatmostoftheliteratureuptodatehasdealtwithinnovations
to the level, comparing these shocks with volatility ones seems relevant. The model
presented in this study is able to capture both effects which provides an uniform
structure for comparison. As may be appreciated from Figure 2.3, level and volatility
shocks are intimately related in both direction and timing.
74
2.4.1 Sensitivity Analysis
To test the model’s ability to match non-targeted moments, we perform he following
sensitivity analysis. We compare the model’s baseline specification with two other
variations including only one source of uncertainty with stochastic volatility. That is
to say we simulate moments utilizing a model with only terms-of-trade shocks and a
model with only i-rate shocks. The results are plotted in table 2.8 below.
Moment Data Model tot only i-rate only
s c
22.29 24.11 18.33 12.43
s i
23.17 26.66 17.55 10.17
r y,c
0.966 0.911 0.799 0.534
r y,i
0.818 0.892 0.657 0.434
s tb
15.83 19.11 11.97 8.74
r tb,y
-0.579 -0.371 -0.417 -0.222
r tb
t
,tb
t 1
0.245 0.142 0.110 0.007
r y
t
,y
t 1
0.691 0.543 0.490 0.201
r c
t
,c
t 1
0.627 0.732 0.525 0.222
r i
t
,i
t 1
0.665 0.554 0.415 0.278
Table 2.8: Sensitivity Analysis
All in all, the baseline specification definitively seems to deliver a closer match to
the data. This was not unexpected since the presence of an additional disturbance
affords the model a greater degree of freedom. Interestingly, nonetheless, is the fact
that when comparing the contribution of both disturbances towards the model’s
fit with the data, the terms of trade seems to do a better job than the interest rate.
Without exception, every moment in the data seemed to be better identified when
the model can at a minimum count with stochastic volatility in shocks to the terms
of trade. This is especially interesting if one considers that the terms of trade operate
mainly through a precautionary savings channel, while the interest rate enters
directly in the agent’s Euler equation.
75
2.5 Final Remarks
Sharp swings in a developing country’s terms of trade can seriously disrupt output
growth. However, shocks to the mean level of the variable are quite different to
shocks to its variance. Since variance shocks operate mainly through a precautionary
motive type of behavior, a good way to understand their dynamics is to identify what
happens after an adverse level shock occurs and identify how agents seek to react or
hedge against that outcome.
In the case of a negative realization of p
X
t
, agents faced a negative wealth shock
that reduced aggregate consumption. A substitution effect will also take place as
non-tradable goods become relative more expensive. To hedge against this potential
scenario (which becomes more likely should p
X
t
become more volatile) agents might
seek to increase production in that sector, and to do so they need to increase the
capital stock in that sector.
To implement this hedging strategy two options are available in principle:
borrowing from the rest of the world and reallocating some resources from the non-
tradable goods sector. However, because this reallocation induces a real depreciation,
borrowing internationally becomes relatively more expensive and hence the trade
balance improves as the overall stock of debt falls (seen Figure 2.2). In turn, the main
mechanism available to agents points towards the reallocation of capital from sector
the non-tradables to the tradables sector.
The richness in the presented framework is able to capture these internal
dynamics, which would otherwise go unnoticed in a standard small open economy
set-up. The effects of uncertainty on risk aversion and precautionary savings are well
understood. However, taking into account the internal reallocation dynamics of an
economy facing fluctuations in uncertainty has been shown to be as important.
76
With this in mind, two potential extensions to this study would be to study
the effects of uncertainty on international borrowing constraints. With tighter
international borrowing conditions, agents in the economy would find it harder
to hedge against swings in uncertainty and would in turn seek to increase their
precautionary savings. This could amplify the detrimental effects of a negative
realization of uncertainty as the negative effects on consumption and investment
would be magnified.
A second potential extension would seek to understand the effects of trade policy
on the effects of terms of trade uncertainty. The first idea is to study whether having
a more diversified exporting base would help ameliorate the effects of terms of
trade shocks. The second, whether tariffs or quotas could be used as a policy tool
against fluctuations in uncertainty. As disturbances hitting the economy become
more volatile, for example, subsidies could be applied to export to encourage the
sectoral reallocation that mitigates the fall in GDP. Similarly, tariffs could be used to
encourage the import substitution process reinforcing the original effects of the real
depreciation of the exchange rate. These questions are all left for future research
efforts.
77
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84
Appendix A
Appendix to Chapter 1
A.1 Omitted Theoretical Proofs
Proposition 1. Individual labor demand is linear in financial wealth (b
i
t
), while con-
sumption and savings are linear in total assets (a
i
t
):
h
i
t
= f t
b
i
t
b
i
t+1
= R
t
b a
i
t
c
i
t
=(1 b )a
i
t
Proof Proposition 1.
The recursive formulation of the entrepreneur’s problem presented in section
1.3.5 can also be written in terms of the information available to the agent at the time
of making a decision. In turn, I define the following two stages or sub-problems:
Stage I:
V
t
(q t
,B
t
,b
i
t
)= max
h
i
t
E
n
t
ˆ
V
t
(q t
,B
t
,a
i
t
)
s.t.: a
i
t
=(q t
n
i
t
w
t
)h
i
t
+b
i
t
Stage II:
ˆ
V
t
(q t
,B
t
,a
i
t
)= max
c
i
t
[lnc
t
+b E
q t+1
V
t+1
(q t+1
,B
t+1
,b
i
t+1
)]
s.t.: a
i
t
c
i
t
+
b
i
t+1
R
t
85
where E
q t+1
stands for the expectation of q t+1
conditional on the realization of q t
.
In stage I the entrepreneur chooses its labor inputs aware of the extent of credit
conditions, yet uncertain about the level of demand that he will receive that period.
In stage II, the entrepreneur observes the realization of n
i
t
and allocates the end of
period wealth between consumption and savings. The stage I first order condition is:
d V
t
d h
i
t
() E
n
t
"
d ˆ
V
t
d a
i
t
d a
i
t
d h
i
t
#
= 0
Theenvelopecondition d ˆ
V
t
/d a
i
t
= 1/c
i
t
isderivedandthenusedintheexpression
above to yield:
d V
t
d h
i
t
() E
n
t
"
q t
n
i
t
w
t
c
i
t
#
= 0
The stage II first order condition is:
d ˆ
V
t
d c
i
t
= 0 ()
1
c
i
t
+b E
n
t
"
E
q t+1
d V
t+1
d b
i
t+1
( R
t
)
#
= 0
Substituting the relevant envelope condition, and denoting E
t
as conditional expecta-
tion given the information set at time t yields the following Euler equation:
1
c
i
t
= b E
t
R
t

1
c
i
t+1
!
86
NextIproveProposition1followingaguess-and-verifyapproach. Beginbyguess-
ing the following policy functions:
h
i
t
= f t
b
i
t
(A.1)
b
i
t+1
= R
t
b a
i
t
(A.2)
Replacing (A.2) in the stage II budget constraint
c
i
t
= a
i
t
b
i
t+1
R
t
(A.3)
yields the policy function for consumption:
c
i
t
=(1 b )a
i
t
(A.4)
From the Euler equation (FOC of stage II) we have that
1
c
i
t
= b R
t
E
t

1
c
i
t+1
!
)
1
a
i
t
= b R
t
E
t

1
a
i
t+1
!
(A.5)
Combining the definition of a
i
t
and (A.1) yields
a
i
t+1
=[(q t+1
n
i
t+1
w
t+1
)f t+1
+1]b
i
t+1
(A.6)
which implies that (A.5) can be written as:
1
a
i
t
=

b R
t
b
i
t+1
!
E
t

1
1+(q t+1
n
i
t+1
w
t+1
)f t+1
!
) 1 = E
t

1
1+(q t+1
n
i
t+1
w
t+1
)f t+1
!
(A.7)
87
For the proof to be complete I need to verify that (A.7) satisfies the problem’s
FOCs:
E
t
"
q t
n
i
t
w
t
(q t
n
i
t
w
t
)f t
+1
#
= 0 (A.8)
In turn, from (A.7)
E
t
"
1
1+(q t
n
i
t
w
t
)f t
#
1 = 0 (A.9)
) E
t
"
1 1 (q t
n
i
t
w
t
)f 1+(q t
n
i
t
w
t
)f t
#
= 0
) ( f )E
t
"
q t
n
i
t
w
t
1+(q t
n
i
t
w
t
)f t
#
= 0
) E
t
"
q t
n
i
t
w
t
(q t
n
i
t
w
t
)f t
+1
#
= 0 (A.10)
which satisfies (A.8).
88
A.2 Aggregate Measures
For this economy, aggregate real income will equal the profits of the entrepreneurs
and the labor income of the representative household. In turn:
Y
t
=
Z
(y
i
t
w
t
)h
i
t
dF(i)+w
t
h
t
(A.11)
=
Z
y
i
t
h
i
t
dF(i) Z
w
t
h
i
t
dF(i)+w
t
h
t
=
Z
y
i
t
h
i
t
dF(i)
In terms of real consumption:
c
e,i
t
dF(i)=(y
i
t
w
t
)h
i
t
+b
i
t
b
i
t+1
R
t
)
Z
c
e,i
t
dF(i)=
Z
(y
i
t
w
t
)h
i
t
dF(i)+
Z
b
i
t
dF(i) Z
b
i
t+1
R
t
dF(i)
This implies that the aggregate consumption of entrepreneurs can be written as:
C
E
t
= Y
t
w
t
h
t
+b
e
t
b
e
t+1
R
t
and aggregate consumption of the households as
C
H
t
= w
t
h
t
+
b
H
t+1
R
t
b
H
t
89
Hence total consumption in the economy would be equal to:
C
E
t
+C
H
t
= Y
t
w
t
h
t
+b
e
t
b
e
t+1
R
t
+w
t
h
t
+
b
H
t+1
R
t
b
H
t
= Y
t
+b
e
t
b
e
t+1
R
t
+
b
H
t+1
R
t
b
H
t
= Y
t
which is the total income/production described by expression A.11.
90
A.3 Alternatives Measures of Uncertainty
Figure A.1: Disagreement amongst professional forecasters.
The figure above plots the cross sectional dispersion in private sector forecasts over
the business cycle. The data comes from the Federal Reserve Bank of Philadelphia’s
survey of professional forecasters from 1968Q4 - 2014Q3 for the first four variables
and 1981Q3 - 2014Q3 for the remaining two. Beginning from top left we have the
forecasts for Real GDP, the Price Deflator, Industrial Production, the Unemployment rate,
Real Consumption and Non-residential fixed investment. In times of higher uncertainty
forecasts become less precise and dispersion amongst predictions increases. Not sur-
prisingly, recessions tend to be periods of greatest disagreement amongst forecasters.
91
A.4 Evidence of Corporate Lending
IntheframeworkintroducedinSection1.3resourceswould,inequilibrium,flowfrom
the entrepreneursto thehouseholds sector. At firstthis resultmight seemlike an odd
feature of the model. However, in the U.S., the private corporate sector has been a
net lender since the beginning of the 2000’s as seen in figure A.2. The only exception
to date has been the year 2008 at the height of the Big Recession, when the financial
assets held by most corporations dropped in value.
Figure A.2: Net Financial Assets in the nonfinancial business sector as a percentage
of total nonfinancial assets.
Source: Federal Reserve Flow of Funds Report.
Interestingly the reversal from net borrower to lender has so far only occurred in
theU.S.Corporatesector,andnotintheNon-corporateone. Theevidencereportedon
thefigureaboveshowsthatalargefractionofthebusinesssectorisself-financingand
no longer dependent on outside sources. And even when the aggregate figures may
mask some firm level heterogeneity, they do paint a general picture of the evolution
of the overall trend across time.
92
A.5 About ShopperTrak data
Founded in 1989 and headquartered in Chicago, Illinois; ShopperTrak Corporation
is the world’s largest retail traffic counter. The company provides shopper insights
and analytics solutions to improve retail profitability and effectiveness. ShopperTrak
helps companies identify, understand, and maximize their total shopper conversion
rate (the percentage of shoppers who actually purchase something) and improves
store performance through shopper behavior insights. It also helps retailers with
solutions for store traffic counting, interior analytics, and industry benchmarking;
and provides unique data benchmark tools that help retailers understand their
performance in context of the market. ShopperTrak is the leader in its industry
and the only one to provide an end-to-end service: from device installation to data
analysis. The company serves major brands, retailers, mall owners, and financial
institutions. Table A.1 provides a small selection of its customer base.
Table A.1: ShopperTrak Clients
Apparel Home Improvement Technology Food
GAP Inc. (Old Navy, GAP, BR) Home Depot Apple Inc. Godiva
Crocs Lowe’s
Victorias Secret Crate & Barrel
Payless Shoes
American Eagle Outfitters
J. Crew
Journeys
Thomas Sabo
Source: ShopperTrak’s website and specialized press
ShopperTrak utilizes proprietary technology to analyze and monitor customer
traffic. Itsfifth-generationdevice,calledtheOrbit,issmartenoughtodetectshoppers
who enter side- by-side or in groups, distinguish children from adults and ignore
shopping carts or strollers. Figure A.3 provides a few examples of this technology.
93
All images were taken from local stores in Santa Monica, CA.
Figure A.3: ShopperTrak’s technology
94
Appendix B
Appendix to Chapter 2
B.1 Optimality conditions
Assuming an interior solution, for j = {X,M} the corresponding first order condi-
tions for the household are:
[c
t
] : p
c
t
l t
= U
c
(c
t
µ¯ c
t 1
,h
t
) (B.1)
[h
N
t
] : w
t
l t
= U
h
(c
t
µ¯ c
t 1
,h
t
) (B.2)
[h
X
t
] : w
t
l t
= U
h
(c
t
µ¯ c
t 1
,h
t
) (B.3)
[d
t
] : l t
[1 Y 0
(d
t
)] = b R
t
E
t
l t+1
(B.4)
[s
j
0,t
] : v
j
0,t
=
1
4
(B.5)
95
[s
j
1,t
] : bl
t
v
j
1,t
=
b 4
l t
+l t 1
v
j
0,t 1
(B.6)
[s
j
2,t
] : bl
t
v
j
2,t
=
b 4
l t
+l t 1
v
j
1,t 1
(B.7)
[s
j
3,t
] : b 2
4
l t
q
j
t
F 0
0
@ s
j
3,t
k
t
1
A 3
5
=
b 4
l t
+l t 1
v
j
2,t 1
(B.8)
[k
j
t+1
] : l t
q
j
t
= b E
t
(
l t+1
q
t+1
"
1 d +F
s
3,t+1
k
t+1
!
s
3,t+1
k
t+1
F 0

s
3,t+1
k
t+1
!#
+l t+1
u
j
t+1
)
(B.9)
The first order conditions for the firms are:
B.1.1 For the consumption goods sector:
[c
T
t
] :
c
N
t
c
T
t
=
✓
1 c c ◆ 1
1+e
✓
p
T
t
p
N
t
◆
1
1+e
(B.10)
[c
N
t
] :
c
t
c
T
t
=
✓
1
c ◆ 1
1+e
✓
p
T
t
p
c
t
◆
1
1+e
(B.11)
96
B.1.2 For the tradable goods sector:
[c
X
t
] :
p
X
t
c
X
t
p
T
t
c
T
t
= a (B.12)
[c
M
t
] :
c
M
t
p
T
t
c
T
t
= 1 a (B.13)
These optimality conditions state that the shares of domestic and foreign inputs
in the total production of tradable goods are constant and equal to a and (1 a )
respectively.
B.1.3 For the nontradable goods sector:
The first order conditions associated with these sectors are (B.4), (2.20), the no-Ponzi-
game condition (2.29) and:
[h
N
t
] : p
N
t
F
N
h
(h
N
t
)= w
t
"
1+h
R
d
t
1
R
d
t
!#
(B.14)
B.1.4 For the exportable and importable goods sectors:
[k
X
t
] : p
X
t
F
X
k
(k
X
t
)= u
X
t
(B.15)
[k
M
t
] : F
M
k
(k
M
t
)= u
M
t
(B.16)
97
B.1.5 Market clearing conditions:
k
t
= k
X
t
+k
M
t
(B.17)
h
t
= h
N
t
(B.18)
c
N
t
= y
N
t
(B.19)
y
t
= p
N
t
y
N
t
+ p
T
t
y
T
t
+y
M
t
(B.20)
i
t
= i
M
t
+i
X
t
(B.21)
tby
t
=
p
X
t
y
X
t
p
X
t
c
X
t
c
M
t
i
t
Y (d
t
)
y
t
(B.22)
d
t+1
= R
t
d
t
Y (d
t
) p
N
t
y
N
t
p
X
t
y
X
t
y
M
t
+ p
N
t
c
N
t
+ p
c
t
c
t
+i
t
(B.23)
Note that in an economy like the one described here, where the debt-adjustment
cost is faced by households, the national income accounts would measure private
consumption as c
t
+Y (d
t
) and not simply c
t
.
98

Abstract (if available)

Abstract This dissertation is a collection of essays with the unifying objective being to better understand the origins of fluctuations in macroeconomic uncertainty as well as its implications on aggregate economic activity. In turn, I investigate both the genesis as well as the effects of sharp, unexpected changes in uncertainty (i.e.: uncertainty shocks) and quantify their impact on economic outcomes.

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Asset Metadata

Creator Vilán, Diego (author)

Core Title Essays in uncertainty and aggregate economic activity

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Economics

Publication Date 08/03/2015

Defense Date 04/15/2015

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

Tag endogenous uncertainty,macroeconomic activity,OAI-PMH Harvest,stochastic volatility,terms of trade

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Quadrini, Vincenzo (committee chair), Imrohoroglu, Selahattin (committee member), Michaux, Michael (committee member), Rubio-Ramirez, Juan (committee member), Vandenbroucke, Guillaume (committee member)

Creator Email diegovilan@hotmail.com,vilan@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-624205

Unique identifier UC11303320

Identifier etd-VilanDiego-3787.pdf (filename),usctheses-c3-624205 (legacy record id)

Legacy Identifier etd-VilanDiego-3787.pdf

Dmrecord 624205

Document Type Dissertation

Format application/pdf (imt)

Rights Vilán, Diego; Vilan, Diego

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

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