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Essays in business cycle economics Galizia, Dana 2015

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Essays in Business Cycle EconomicsbyDana GaliziaB.A., The University of Western Ontario, 2005M.A., McGill University, 2007A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Economics)The University of British Columbia(Vancouver)June 2015c© Dana Galizia, 2015AbstractThis thesis contains three distinct chapters that contribute to our understanding of the causes andconsequences of business cycles. Modern business-cycle models generally feature several differentrandom shock processes that drive business cycles. Being able to reliably evaluate the individualimportance of any one these shocks depends importantly on having accurate estimates of the variancesof the shocks. In the first chapter, it is shown that when a model is a poor approximation to the data,typical variance estimates are biased upward. A simple procedure to identify and partially correct forthese effects is proposed. Applying this procedure to a recent paper from the literature reduces theestimated variances by as much as a third of their respective naive estimates.The second chapter explores a view of recessions (typically associated with Friedrich Hayek)whereby, after a period of rapid accumulation of houses, consumer durables and business capital, theeconomy goes through a period of needed liquidation that results in a decline in economic activity.An alternative (typically associated with Keynes) that is often contrasted with this liquidation view isthat recessions are times of deficient demand. These two views have opposite implications for fiscalpolicy: in the first, fiscal policy simply prolongs the needed adjustment, while in the second fiscalpolicy can prop up demand. This chapter argues that the two views may be more closely linked thanpreviously recognized, in that liquidations can produce periods where the economy is characterizedby deficient demand.The final chapter presents a model in which business-cycle booms and busts are inherently related,whereby a boom causes a subsequent bust, which in turn leads to another boom, and so on. In par-ticular, it is shown how a purely deterministic model can produce fluctuations that persist indefinitely.These cycles exactly repeat themselves, while in the data business cycles are somewhat irregular. Itis shown that by adding a small amount of random variation to the model, it is capable of replicatingbusiness cycle features in the data well, including their irregularity.iiPrefaceAll materials in this dissertation are original, unpublished work. Chapter 2 is exclusively my indepen-dent work. Chapter 3 is joint work with Paul Beaudry and Franck Portier. Initial concept formationwas by Paul Beaudry and Franck Portier. I participated in all stages of its realization. Paul Beaudryand I did most of the analysis of the model together. The manuscript was composed primarily byPaul Beaudry, with edits contributed by me and Franck Portier. I developed the formal proofs of thetheoretical propositions, in most cases after extensive conversations with Paul Beaudry. I created allfigures except those contained in Appendix B, which were created by Franck Portier.The materials in Chapter 4 are primarily though not exclusively my independent work. The basicstructure of the model used throughout the chapter, as well as portions of the text of Section 4.5.1 areused with permission from Beaudry et al. (2014), of which I am a co-author, and which forms Chapter3 in this thesis. The remainder of Chapter 4 is my independent work, including the composition of themanuscript, the theoretical results of Section 4.5.3, and all quantitative results presented throughoutthe paper.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Business cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Chapter one: Misspecification and the causes of business cycles . . . . . . . . . . . 11.3 Chapter two: Reconciling Hayek’s and Keynes’ views of recessions . . . . . . . . . 21.4 Chapter three: Can a limit-cycle model explain business cycle fluctuations? . . . . . 22 Misspecification and the Causes of Business Cycles . . . . . . . . . . . . . . . . . . . . 42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Basic framework and methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.1 Example 1: Missing shocks . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Formal framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.1 Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.2 Integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.3 Orthogonal decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Further analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.1 Econometric model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18iv2.4.2 Data process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.3 Theoretical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.4 Practical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.1 Example 2: Univariate AR(1) model . . . . . . . . . . . . . . . . . . . . . . 232.5.2 Example 3: Medium-scale new Keynesian model . . . . . . . . . . . . . . . 242.6 Application: Investment shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Reconciling Hayek’s and Keynes’ Views of Recessions . . . . . . . . . . . . . . . . . . 383.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Static model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.2 Deriving the value function V (a) . . . . . . . . . . . . . . . . . . . . . . . 453.2.3 Equilibrium in the first sub-period . . . . . . . . . . . . . . . . . . . . . . . 463.2.4 Is there deficient demand in the unemployment regime? . . . . . . . . . . . 533.2.5 Effects of changes in X on welfare . . . . . . . . . . . . . . . . . . . . . . 533.2.6 Allowing for offers of unemployment insurance . . . . . . . . . . . . . . . 543.2.7 Introducing government spending . . . . . . . . . . . . . . . . . . . . . . . 553.3 Further discussions and relaxing of assumptions . . . . . . . . . . . . . . . . . . . 573.3.1 Relaxing functional-form assumptions . . . . . . . . . . . . . . . . . . . . 573.3.2 A version with productive capital . . . . . . . . . . . . . . . . . . . . . . . 603.3.3 Multiple equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.3.4 The role of beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.4.1 Global dynamics for a simple case . . . . . . . . . . . . . . . . . . . . . . . 673.4.2 Local dynamics in the general case . . . . . . . . . . . . . . . . . . . . . . 693.5 Policy trade-offs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764 Can a Limit-Cycle Model Explain Business Cycle Fluctuations? . . . . . . . . . . . . 784.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.2 Literature: Deterministic fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . 814.3 Data: Hours worked . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.4 Conditions for a limit cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.5 The unemployment-risk model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.5.1 Static version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.5.2 Baseline dynamic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92v4.5.3 Limit cycles in the dynamic model . . . . . . . . . . . . . . . . . . . . . . . 934.6 Quantitative exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.6.1 Functional forms, calibration and estimation . . . . . . . . . . . . . . . . . . 984.6.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.6.3 Additional results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.6.4 Multiple equilibria and indeterminacy . . . . . . . . . . . . . . . . . . . . . 1084.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113A Appendix for “Misspecification and the Causes of Business Cycles” . . . . . . . . . . 119A.1 Model details for Example 3 in Section 2.5 (Smets and Wouters (2007)) . . . . . . . 119A.2 Proofs of theoretical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121A.3 Data for JPT model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124A.4 BCF variance decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125B Appendix for “Reconciling Hayek’s and Keynes’ Views of Recessions” . . . . . . . . . 127B.1 Proofs of propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127B.2 Introducing Nash bargaining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146B.3 Noise shock extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150C Appendix for “Can a Limit-Cycle Model Explain Business Cycle Fluctuations?” . . . 151C.1 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151C.1.1 Proof of Proposition 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151C.1.2 Proof of Proposition 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151C.1.3 Proof of Proposition 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153C.1.4 Proof of Proposition 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154C.2 Solution and estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155C.2.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155C.2.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157C.3 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157C.4 Solving the model forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158viList of TablesTable 2.1 Variances of true and smoothed shocks . . . . . . . . . . . . . . . . . . . . . . . 26Table 2.2 Corrected variances of smoothed shocks . . . . . . . . . . . . . . . . . . . . . . 26Table 2.3 Sample correlations for smoothed shocks (case B) . . . . . . . . . . . . . . . . . 27Table 2.4 Corrected variances (case C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Table 2.5 Naive variance decomposition for hours . . . . . . . . . . . . . . . . . . . . . . 30Table 2.6 Corrected estimates (fraction of naive estimate) . . . . . . . . . . . . . . . . . . 36Table 2.7 Corrected BCF variance decomposition for hours . . . . . . . . . . . . . . . . . 37Table 4.1 Parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100viiList of FiguresFigure 2.1 Autocovariance function (Cov (yt, xt−k)) . . . . . . . . . . . . . . . . . . . . . 28Figure 2.2 Correlation between %∆Ct and %∆It+k . . . . . . . . . . . . . . . . . . . . . 31Figure 2.3 Maximum scaled MSRE for smoothed shocks . . . . . . . . . . . . . . . . . . . 32Figure 2.4 Correlation coefficients between smoothed shocks . . . . . . . . . . . . . . . . 33Figure 2.5 Proportion of correlations outside 95% CI . . . . . . . . . . . . . . . . . . . . . 34Figure 3.1 Labor wedge as function of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Figure 3.2 Consumption as function of X . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 3.3 Equilibrium determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 3.4 Cost of funds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 3.5 Equilibrium determination (multiple equilibria) . . . . . . . . . . . . . . . . . . 63Figure 3.6 Consumption as function of X (multiple equilibria) . . . . . . . . . . . . . . . . 64Figure 3.7 Xt+1 as a function of Xt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure 3.8 Response of economy to a noise shock . . . . . . . . . . . . . . . . . . . . . . . 72Figure 4.1 Hours worked data (1960-2012) . . . . . . . . . . . . . . . . . . . . . . . . . . 84Figure 4.2 Conditions for a limit cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Figure 4.3 Static equilibrium determination . . . . . . . . . . . . . . . . . . . . . . . . . . 92Figure 4.4 Deterministic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Figure 4.5 Stochastic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Figure 4.6 Autocovariance: Hours worked (L) and output (y) . . . . . . . . . . . . . . . . 103Figure 4.7 Spectrum: Hours worked (alternative filters) . . . . . . . . . . . . . . . . . . . . 104Figure 4.8 Hours worked in Smets-Wouters (2007) . . . . . . . . . . . . . . . . . . . . . . 105Figure 4.9 Spectrum: Output (data and stochastic model) . . . . . . . . . . . . . . . . . . . 107Figure B.1 The Model with Nash bargaining, consumption as function of X . . . . . . . . . 148Figure B.2 The Model with Nash bargaining, equilibrium determination . . . . . . . . . . . 149Figure B.3 The Model with Nash bargaining, equilibrium determination (multiple equilibria) 149viiiAcknowledgmentsThis thesis could not have been completed without the support graciously given to me by a numberof people. I would like to express my deepest gratitude to my advisor Paul Beaudry for his unendingsupply of patience, guidance and encouragement, and for his generous financial support. I would alsolike to thank my other thesis committee members, Henry Siu and Yaniv Yedid-Levi, for their supportand helpful advice. Thank you also to Franck Portier, Vadim Marmer, Francesco Trebbi, MatildeBombardini and Amartya Lahiri for helpful discussions.Thank you to my parents, who have never wavered in their support for me, and whose contributionsto my arriving at this point have been both innumerable and indispensable. Finally, words cannotexpress my unending gratitude to my wife, Jocelyne, who has been by my side every step of the way.I will be forever thankful for the love she has given me and the sacrifices she has made so that I couldreach this point. If I have accomplished anything, it’s because she made it possible.ixFor my loving wife, Jocelyne.xChapter 1Introduction1.1 Business cyclesSince the early part of the twentieth century, a substantial part of economic research has been devotedto the study of business cycles (i.e., persistent but non-permanent fluctuations in economic aggregatessuch as output or unemployment). Despite the considerable amount of attention this topic has received,there are many questions that remain unanswered, and many more for which the typical answers areoffered only tentatively.This thesis consists of three distinct chapters, each of which seeks to contribute to answeringthe most fundamental questions considered by business-cycle economists: What ultimately causesbusiness cycles, and how do those causes end up manifesting themselves as the patterns in the data thatare typically thought of as characterizing the business cycle? The final two chapters can be viewed asexploring answers to these questions directly, while the first chapter contributes to our understandingof how to answer these questions.1.2 Chapter one: Misspecification and the causes of business cyclesModern macroeconomics models generally feature several different random shock processes that ul-timately drive business cycle fluctuations. A standard tool in the quantitative macroeconomics tool-box for evaluating the individual importance of these shocks is a variance decomposition, by whichthe variance of any given variable is decomposed into individual portions attributable to each of theshocks in the model. The reliability of this tool depends importantly on having accurate estimates ofthe variances of the shocks.In the first chapter of this thesis, I develop a novel framework and use it to show that when a modelis misspecified—roughly speaking, when it is inherently incapable of matching the data to which it isapplied—the shock variances as they are typically estimated will be biased upward. Next, using thesame framework, I propose a simple procedure to identify and partially correct for the effects of model1misspecification on these variance estimates. As an example of its usefulness, I apply this procedureto a recent paper and find that it reduces the estimated variances of the shocks in the model by as muchas a third of their respective naive estimates.1.3 Chapter two: Reconciling Hayek’s and Keynes’ views of recessionsRecessions often happen after periods of rapid accumulation of houses, consumer durables and busi-ness capital. This observation has led some economists, most notably Friedrich Hayek, to concludethat recessions mainly reflect periods of needed liquidation resulting from past over-investment. Ac-cording to the main proponents of this view, government spending should not be used to mitigatesuch a liquidation process, as doing so would simply result in a needed adjustment being postponed.In contrast, ever since the work of Keynes, many economists have viewed recessions as periods ofdeficient demand that should be countered by activist fiscal policy.In the second chapter of this thesis, we reexamine the liquidation perspective of recessions ina setup where prices are flexible but where not all trades are coordinated by centralized markets.We show why and how liquidations can produce periods where the economy functions particularlyinefficiently, with many socially desirable trades between individuals remaining unexploited when theeconomy inherits too many capital goods. In this sense, our model illustrates how liquidations cancause recessions characterized by deficient aggregate demand and accordingly suggests that Keynes’and Hayek’s views of recessions may be much more closely linked than previously recognized. Inour framework, interventions aimed at stimulating aggregate demand face the trade-off emphasizedby Hayek whereby current stimulus mainly postpones the adjustment process and therefore prolongsthe recessions. However, when examining this trade-off, we find that some stimulative policies maynevertheless remain desirable even if they postpone a recovery.1.4 Chapter three: Can a limit-cycle model explain business cyclefluctuations?In conventional models of the business cycle, all fluctuations are ultimately caused by the arrival ofrandom shocks. As a result, individual booms and busts are largely unrelated phenomena. In thefinal chapter of this thesis, I explore an alternative to this viewpoint, which is that booms and bustsare inherently related phenomena. According to this view, fluctuations are at least in part driven bydeterministic cyclical forces, rather than random shocks.The paper shows (1) how a purely deterministic general-equilibrium model can give rise to cyclicalfluctuations that continue indefinitely, and (2) that this model can replicate business cycle features ofthe data once it includes a small amount of random variation. The deterministic cycle arises througha simple micro-founded mechanism in a rational-expectations environment, and does not rely on theexistence of multiple equilibria or dynamic indeterminacy. Since these cycles would indefinitely repeatthemselves in the absence of shocks, a simple productivity shock is introduced in order to create2irregularities. The model is estimated to match US hours data, and is shown to be able to match itclosely. The productivity shock in the model is of a reasonable persistence and relatively small size,accounting for only around a fifth of the standard deviation of hours in the model. This highlights thatmodels capable of generating deterministic fluctuations do not require the addition of large, persistentshocks in order to match patterns in the data, which is a common criticism of conventional models.3Chapter 2Misspecification and the Causes ofBusiness Cycles2.1 IntroductionOne of the central goals of macroeconomic research is to understand the root causes of business cycles.In recent years, much of the research devoted to this goal has been conducted in three steps. First, builda dynamic stochastic general-equilibrium (DSGE) model within which business-cycle fluctuations aregenerated by the random arrival of exogenous shocks1 that are then propagated through endogenousmechanisms. Second, estimate the parameters of this DSGE model using real-world data and someestimation algorithm (such as maximum likelihood or, increasingly, Bayesian methods). Third, withinthe estimated model, evaluate the quantitative importance of each of the shocks in generating business-cycle fluctuations.2If the true data-generating process (DGP) is contained in the set of DGPs spanned by the modelconstructed at step one above, then under some basic regularity conditions we can expect the parame-ters estimated at step two to converge to the true parameters, and in turn we may interpret the resultsof the third step as reflecting the actual importance of each type of shock in generating real-worldeconomic fluctuations. However, when the true DGP is not contained in the set of DGPs spannedby the model—i.e., when the model is misspecified—this reasoning no longer holds. Since even themost optimistic economist would agree that all models are misspecified, this raises some importantquestions. Namely, what effect does misspecification have on shock variance estimates in a model?Does this effect depend on the severity of the misspecification? If misspecification is detected, can1 Throughout this paper, what I refer to as shocks are the innovations to an exogenous process, rather than the processitself. For example, if TFP evolves according to an AR(1) process zt = ρzt−1 + εt, with εt a sequence of i.i.d. randomvariables, then I refer to εt as the (exogenous) TFP shock.2 In the class of models considered in this paper, conditional on all other parameters there will be a one-to-one mappingbetween the variance of each shock and its respective quantitative importance. As such, I will use the terms “quantitativeimportance” and “shock variance” interchangeably.4some correction be made to the variance estimates? To my knowledge, no framework currently existsto help answer these questions. This paper attempts to fill that gap in the literature.The impact of misspecification on understanding the root causes of business cycles is highly rel-evant. Over the last decade or so, DSGE models have become ever richer, with a typical model nowpossessing a variety features that (it is argued) allow it to fit the data reasonably well. Deriving somedegree of confidence from this good fit, it has become common to ask the model which types ofeconomic shocks appear to be important for driving real-world business-cycle fluctuations.3 Specif-ically, given a DSGE model that has been linearized around a non-stochastic steady state, and thatcontains structural shocks that are independent of one another both cross-sectionally and over time, itis a straightforward exercise to obtain the proportion of the variance of any endogenous variable at-tributable to an individual shock, i.e., to perform a variance decomposition. On the basis of a variancedecomposition, one may then conclude, for example, that a particular shock is an important drivingforce for business cycle fluctuations. The usefulness of this variance decomposition tool in under-standing the causes of real-world business cycles clearly depends fundamentally on the accuracy ofthe shock variance estimates. As I clarify in this paper, in the presence of misspecification these vari-ance estimates will be systematically biased upward. While this poses a challenge to the conclusionstypically drawn from a variance decomposition, as we shall see the news is not all bad. First, I showthat as the severity of the misspecification becomes smaller (in a particular sense), this bias shrinksto zero. This reinforces the view that, even though a model may be misspecified, it may still yielduseful conclusions. Second, I propose a simple method of adjusting the variance estimates that willreduce the degree of bias, at least in part. In combination, these two properties also imply that largeradjustments will typically be associated with more severe misspecification, so that this methodologycan be used as an additional tool for measuring the severity of misspecification in a model. Sincethese adjustments will generally be different for each shock in the model, unlike other measures thisone is unique in providing information to the modeller about misspecification at the level of individualshocks. As illustrated in an example below, this information can be useful in diagnosing the source ofmisspecification and, in turn, suggesting potential modifications to the model that may help to alleviateit.The framework I propose revolves around the sequence of “smoothed shocks”, i.e., the sequenceof realized shock values that, when fed into the model, allow it to exactly reproduce the observed data.4As shown below, if the DSGE model (for a given set of parameter values) were the true DGP, thenunder certain regularity assumptions the sequence of smoothed shocks will asymptotically recoverthe true values of the shocks. Since the true shocks are (by construction) independent of one another3 See, for example, Smets and Wouters (2007), Nolan and Thoenissen (2009), Christiano et al. (2010b), Justiniano et al.(2010, 2011), Schmitt-Grohe´ and Uribe (2011, 2012), Barsky and Sims (2012), Blanchard et al. (2013), Christiano et al.(2014), etc.4 In practice there will typically be more than one sequence capable of reproducing the data, in which case the smoothedshocks are taken to be the sequence that is most likely to have occurred (i.e., the one for which the joint probability densityfunction is maximized).5both intertemporally and cross-sectionally, this implies that if the model is the true DGP then thesmoothed shocks should also be asymptotically independent of one another. This observation suggestsa relatively simple diagnostic tool for detecting the presence of misspecification: if the smoothedshocks are correlated (beyond what can be reasonably explained by sampling error), then the modelmust be misspecified.5The basic intuition for this result is quite simple. Any model—including the (unknown) trueDGP—can be thought of as a collection of mechanisms, where each mechanism corresponds to thecombination of a shock and its impulse response function. A model is misspecified if it is missing oneor more mechanisms that are present in the true DGP. If this is the case, the smoothing algorithm—thatis, the algorithm that finds a sequence of shocks that exactly reproduces the data—must systematicallymix together two or more of the mechanisms that are in the model in order to replicate the mechanismsthat are missing. It is precisely this “mixing” process that leads to correlations in the smoothed shocks.I next show that, when the model is misspecified, conventional shock variance estimates willtend to overstate the importance of one or more of the shocks. The intuition for this bias is againquite simple: since a mechanism in the model is being used to explain not only the fluctuations inthe data actually caused by that mechanism, but also fluctuations caused by certain mechanisms thatare not contained in the model, the smoothing algorithm attributes too much variation to the modelmechanisms, and thus tends to overestimate the variance of the model shocks.After highlighting this potential source of bias, I propose a simple procedure to (at least partially)correct for it. First, I orthogonally decompose a smoothed shock into two components: its true value inproducing variation in the data, and an additional component related to misspecification. Next, I showthat the true value of the shock should be unpredictable using values of other smoothed shocks. Thetrue shock is therefore a component of the OLS residual obtained after regressing the correspondingsmoothed shock on other smoothed shocks. The variance of this residual therefore represents a lessbiased estimate of the true variance of the shock.To illustrate a practical application, I apply the above methodology to a recent paper by Justinianoet al. (2010), and show that the source of bias discussed above may indeed have significant implica-tions. Justiniano et al. (2010) construct a medium-scale New Keynesian model featuring a variety ofdifferent shocks. The variance decomposition obtained from their estimated model indicates that themajority (roughly 60 percent) of the variance of hours worked at business-cycle frequencies can beattributed to the investment shock (a shock to the relative productivity of new investment vis-a`-vis theexisting capital stock). This shock is also found to account for the majority of business-cycle variationin output and investment. As I show, however, six of the seven smoothed shocks in their model (in-cluding the investment shock) are significantly correlated, suggesting that the model is misspecifiedand that the naive variance estimates may be overstating the importance of these shocks. Applying5 Note that, under the precise definition of “misspecification” adopted in this paper (see Section 2.3.1), the converse willnot necessarily be true: it will be possible for the smoothed shocks to be uncorrelated, but for the model to nonetheless bemisspecified.6the proposed correction procedure, I estimate that the true variance of the investment shock is smallerthan the naive estimate by at least one-third.I contribute here to two different bodies of literature. First, there is a substantial macroeconometricbody of literature that explores issues of model fit and estimation/inference in potentially misspecifiedDSGE models (e.g., Gourieroux et al. (1993), Watson (1993), Canova (1994), Diebold et al. (1998),Schorfheide (2000), Hall and Inoue (2003), Dridi et al. (2007), Hnatkovska et al. (2012)). In addition,I also contribute to the vast literature that attempts to identify the sources of business cycle fluctua-tions. More specifically, I contribute to the recent literature that has considered investment shocks asa potential source of fluctuations, including the aformentioned Smets and Wouters (2003, 2007) andJustiniano et al. (2010), as well as Greenwood et al. (1988), Greenwood et al. (1997, 2000), Schmitt-Grohe´ and Uribe (2011, 2012), Justiniano et al. (2011), and Jermann and Quadrini (2012).The remainder of the paper proceeds as follows. Section 2.2 introduces in a relatively informalway the basic framework and methodology of the paper, and presents a simple example to illustrateit. Section 2.3 then contains a formal presentation of the framework and methodology, while Section2.4 provides some additional analysis. Section 2.5 then presents two additional examples in whichthe true DGP is known, before Section 2.6 applies the framework and methodology in a real-worldexample using the model of Justiniano et al. (2010). Finally, Section 2.7 concludes.2.2 Basic framework and methodologyIn this section I introduce the basic framework and methodology of the paper in a relatively informalway, with an eye towards conveying as clearly as possible the basic intuition. The ideas of this sectionare then discussed in formal detail in Section 2.3 below.Suppose one has in hand an infinite sequence of past and future observations Y∞ ≡ {Yt}∞t=−∞,where Yt is some jointly normally distributed mean-zero m-variate data process satisfying basic sta-tionarity and ergodicity assumptions. A model for {Yt} is defined here as an expression for Yt as alinear function of a (potentially infinite) history of r-dimensional i.i.d. normal random vectors, whereany two different elements of that random vector are orthogonal to one another. That is, letting Y˜tdenote the model counterpart of Yt, a model is completely summarized by an MA(∞) representationY˜t =∞∑j=0Ψ˜jεt−j (2.1)where{Ψ˜j}is an absolutely-summable sequence of m × r matrices and εt is an r-vector of i.i.d.normal random variables with diagonal covariance matrix, the l-th diagonal element of which is givenby σ˜2l .6 Assuming that the true DGP can also be written in this way, there are two particular models6 Note that the assumption of a diagonal covariance matrix here—i.e., of uncorrelated shocks—is without any loss ofgenerality, as it is always possible to re-cast a model with correlated shocks in this form. For example, suppose we have amodel in which shocks i and j are correlated with one another, with all other shocks uncorrelated. Then it is always possible7of interest: the econometric model (EM), and the true model (TM). Make the following importantassumption about the EM.7Assumption 2.1. (Invertibility) There exists some absolutely summable sequence {ψj} of matricessuch that, given an infinite sequence of past and future values of Y˜t generated by (2.1), we may recoverthe sequence {εt} from8εt =∞∑j=−∞ψj Y˜t−j (2.2)Conditions under which Assumption 2.1 holds are fairly general and will be discussed in detailin Section 2.4.1. For now, we will simply take it as given. Under this assumption, the sequenceof smoothed shocks, denoted εˆt, are obtained by applying the linear filter (2.2) to the data series Yt(instead of its model counterpart Y˜t). That is,εˆt ≡∞∑j=−∞ψjYt−j (2.3)It is straightforward to verify that if one substitutes the sequence εˆt obtained from (2.3) into equation(2.1) for εt one recovers the data series Yt.9 Thus, the sequence εˆt is indeed the sequence of smoothedshocks defined in Section 2.1. If the EM is correctly specified (i.e., if the EM and TM are one and thesame), then one can interpret the sequence of smoothed shocks as capturing the true values of theseshocks in the real world. In general, however, the TM will contain a number of shocks that are absentin the EM, in which case the sequence of smoothed model shocks, εˆt, would in general be differentfrom their “true” values, which I denote by ε∗t .At this point it is worth clarifying what I mean by the “true value” of a shock. This notion will bemade precise in Section 2.3, but for now it will be sufficient to illustrate the concept with an example.In particular, consider the distinction between a real-world TFP shock, and a smoothed TFP shockinferred from some real-world data set10 using a misspecified model. The former is the true valueof the shock, and the one whose variance we would ultimately like to know, but it is not directlyobservable. On the other hand, the latter is observable, but it will in general differ from the true value,and thus its variance will be different as well.to re-cast the model as one with uncorrelated shocks by writing εi,t = εˇi,t + εˇij,t and εj,t = εˇj,t + ρεˇij,t, where the εˇ’sare i.i.d. normal shocks that are all uncorrelated with one another, and ρ and the σˇ’s are chosen so that σˇ2i + σˇ2ij = σ˜2i ,σˇ2j + ρ2σˇ2ij = σ˜2j , and ρσˇ2ij = Cov (εi,t, εj,t). Replacing εi,t and εj,t in the model with εˇi,t, εˇj,t and εˇij,t, the originalmodel has been re-cast as one featuring only orthogonal shocks.7Note that we do not make any such assumption about the TM, which need only in general satisfy (2.1).8 Note that I define invertibility as the ability to recover the structural shocks from some combination of current, pastand future values of Y˜t. This definition is more general than that used in the structural VAR literature (see, for example,Ferna´ndez-Villaverde et al. (2007), Sims (2012)), according to which the MA(∞) representation in (2.1) would be deemedinvertible only if εt can be recovered from current and past values of Y˜t alone.9 See Proposition 2.1 in Section 2.3 for a formal statement and proof of this fact.10 Assume for the sake of argument that this data set does not include the TFP process itself, so that the value of the TFPshocks must be inferred from the behavior of other endogenous variables in the system.8Let σ∗2l ≡ V ar(ε∗l,t)denote the true variance of the l-th model shock. This is the quantity theeconometrician would like to estimate. Further, let σˆ2l ≡ V ar (εˆl,t) be the variance of the corre-sponding smoothed shock. It can be verified that if the model is correctly specified then σˆ2l is themaximum-likelihood estimator of σ∗2l . Since the model is implicitly assumed to be correctly specifiedwhen estimating the parameters via maximum-likelihood, σˆ2l is thus precisely the variance estimatorthat is typically used in the literature and that forms a key input into the variance decomposition.When the model is misspecified, however, we will in general have σˆ2l 6= σ∗2l . We are thus inter-ested in the relationship between σˆ2l and σ∗2l . Let νt ≡ εˆt − ε∗t denote the (unobserved) vector of“shock recovery errors”, and make the following assumption.Assumption 2.2. (Orthogonality) E[ε∗t ν ′t−k]= 0 for all k ∈ N.The framework presented in detail in Section 2.3.1 will directly imply that Assumption 2.2 holds.Nonetheless, as with Assumption 2.1 we will for now simply take Assumption 2.2 as given. Under thisassumption we can orthogonally decompose εˆt into two components: one capturing the true value ofthe shock (ε∗t ), and another entirely reflecting the fact that the EM is misspecified (νt). The immediateconsequence of this orthogonal decomposition is thatσˆ2l = σ∗2l + V ar (νl,t) ≥ σ∗2l (2.4)Thus, when the model is misspecified the naive variance estimate σˆ2l will overstate the true varianceof the shock; that is, misspecification causes the variance estimates to be biased upwards. This resultis the first key contribution of this paper.Next, note that the true values of the shocks are, by definition, independent of one another bothcross-sectionally and intertemporally. That is, we have Cov(ε∗l,t, ε∗i,s)= 0 whenever (l, t) 6= (i, s).It thus follows immediately thatCov (εˆl,t, εˆi,s) = Cov (νl,t, νi,s) (2.5)Thus, if two smoothed shocks exhibit non-zero covariance, this can only be due to misspecification.This is the second key contribution of this paper. Finally, an implication of this result is that anycomponent of εˆl,t that can be predicted using εˆi,s must be attributable to misspecification rather thanto ε∗l,t. This suggests a simple procedure to correct (at least in part) the variance estimates: Regressεˆl,t on date-t values of shocks i 6= l and on leads and lags of all shocks (including leads and lagsof εˆl,t). The error term from this regression is, by construction, the component of εˆl,t that cannot bepredicted using the other shocks. The variance of this error term, which I denote σ¯2l , therefore containsall of the variation in εˆl,t due to ε∗l,t, plus an additional non-negative component. We will thereforehave σˆ2l ≥ σ¯2l ≥ σ∗2l , so that σ¯2l is a less-biased estimate of σ∗2l than the naive maximum-likelihoodestimate. This result is the third key contribution of this paper.92.2.1 Example 1: Missing shocksConsider a simple economy populated by a representative household with preferences given byE0∞∑t=0βt (Ct − ηtLt)Here, Ct is consumption, Lt is labor supplied, β is the constant discount factor, and ηt, which capturesthe relative disutility of labor, follows the exogenous stochastic processlog ηt = ε∗η,twhere ε∗η,t is i.i.d. N(0, σ∗η). The household produces output Xt using technology Xt = AtL1−αt ,where productivity At follows exogenous stochastic processlogAt = ε∗A,twith ε∗A,t i.i.d. N (0, σ∗A). In addition to consuming the output it produces, the household also receivesa stochastic endowment ε∗w,t which is i.i.d. N (0, σ∗w). Thus, the household budget constraint is givenbyCt = AtL1−αt + ε∗w,tTaking first-order conditions and log-linearizing the model around its non-stochastic steady stateyields the following equations for consumption and laborct = −1− αα ε∗η,t + C−1ε∗w,t +1αε∗A,t (2.6)lt = −1αε∗η,t +1αε∗A,t (2.7)where lower-case variables indicate log-deviations from steady state and C is the steady-state value ofCt.Suppose that the value of the parameter α was known to the econometrician, who wishes to es-timate via maximum likelihood the variances of the stochastic processes using data on ct and lt.Suppose, however, that the econometrician wrongly assumes that σ∗A = 0, i.e., that productivity isconstant. This econometrician would then estimate σ ≡ (ση, σw)′ using the misspecified modelc˜t = −1− αα εη,t + C−1εw,t (2.8)l˜t = −1αεη,t (2.9)10The smoothed shocks εˆt ≡ (εˆη,t, εˆw,t)′ can be obtained asεˆt =(−αltC [ct − (1− α) lt])and the maximum-likelihood variance estimates are(σˆ2ησˆ2w)=(α2σ∗2lC2φ∗)whereφ∗ ≡ V ar [ct − (1− α) lt] = σ∗2c − 2 (1− α)σ∗cl + (1− α)2 σ∗2land σ∗2c ≡ V ar (ct), σ∗2l ≡ V ar (lt) and σ∗cl ≡ Cov (ct, lt) are the relevant data moments. Pluggingin the (known) true values for these data moments, we obtain(σˆ2ησˆ2w)=(σ∗2η + σ∗2Aσ∗2w + C2σ∗2A)Thus, the maximum-likelihood estimates overstate the true variances by an amount that is increasingin the degree of misspecification, as captured by a non-zero value of σ∗2A .Next, to obtain the corrected estimates, note thatCov (εˆη,t, εˆw,t) = −Cσ∗2A (2.10)Thus, we obtain the regression equationsεˆη,t = −Cσ∗2Aσ∗2w + C2σ∗2Aεˆw,t + ξη,tεˆw,t = −Cσ∗2Aσ∗2η + σ∗2Aεˆη,t + ξw,twhere the ξ’s are standard OLS residuals that are orthogonal to the regressors. Some algebra yieldsσ∗2η ≤ σ¯2η ≡ V ar (ξη,t) = σ∗2η +σ∗2wσ∗2w + C2σ∗2Aσ∗2A ≤ σˆ2ησ∗2w ≤ σ¯2w ≡ V ar (ξw,t) = σ∗2w +σ∗2ησ∗2η + σ∗2AC2σ∗2A ≤ σˆ2wThat is, the corrected shock variance estimates lie between the true variances and the maximum-likelihood variance estimates.To understand these results, consider what happens when this economy is hit with a positive tech-11nology shock. There are two channels through which this shock affects the economy: the rise inproductivity directly increases output and consumption, while the increase in the marginal productof labor leads the household to both work and consume more. As seen in equations (2.6)-(2.7), thecombined effect of these two channels is that consumption and labor increase in the same proportion.In the misspecified model (2.8)-(2.9), however, there does not exist a mechanism that can produceequiproportional changes in consumption and labor. A negative labor-disutility shock has a similarlabor-leisure substitution effect as the technology shock, but it lacks the direct increase in output, andthus causes consumption to increase by relatively less than labor. Only by combining a negative labor-disutility shock with a positive endowment shock (which increases consumption but has no effect onlabor) can the misspecified model get consumption and labor to increase in the same proportion. This“mixing” of shocks is what produces the negative covariance in (2.10). While maximum-likelihoodestimation explicitly ignores the information conveyed by this mixing of shocks, the methodologyproposed here uses it to correct, in part, the resulting overestimate of the shock variances.2.3 Formal frameworkHaving introduced the framework and methodology in a relatively informal way in the previous sec-tion, this section presents a more rigorous foundation for several of the results of Section 2.2. Inparticular, I make precise the definition of the “true values” of the shocks in the EM, ε∗t , which forma crucial part of the reasoning in Section 2.2, and show how the framework proposed here directlyimplies Assumption 2.2.While formally stating the results of this section requires first setting up a fair amount of mathe-matical machinery, the basic intuition is fairly simple. Crucially, in order to make the concept of the“true value” of an EM shock well-defined, we must first answer the following questions: Under whatcircumstances and in what sense can we consider a shock in the EM and a shock in the TM to befundamentally the same, even if the EM is misspecified? Under what circumstances should we viewtwo such shocks as being fundamentally different? The framework presented in this section answersthese as follows: Two such shocks are fundamentally the same if they are associated with impulseresponse functions (IRFs) that are identical up to a scaling and time-shift factor. Otherwise, they aredifferent.With these answers in hand, it then becomes straightforward to formally define the concept of the“true value” of a shock in the EM: If there is a shock in the TM that is fundamentally the same, thenits value is the true value. If there is no such shock, then its true value is zero.11 Assumption 2.2 willthen follow directly from the fact that shocks that are fundamentally different from one another are11 There is a sense in which this definition is quite demanding, in that a shock in the EM must meet a high standard—its exact associated IRF must exist in the TM—in order for its “true” variance to be non-zero. Thus, there is a sort ofdiscontinuity, in that if the IRF exists in the TM then the true variance is strictly positive, but if it differs by even anarbitrarily small amount from some IRF in the TM then the true variance is zero. However, as shall become clear in Section2.5 below, in practice this discontinuity will not be important, in that the observed degree of misspecification—and thereforeany adjustments made to the variance estimates—will approach zero as the IRFs in the EM approach those in the TM.12associated with independent white noise processes.2.3.1 Set-upThis basic set-up of the framework proceeds as follows. First, the set of all stationary IRFs is parti-tioned into groups that are equivalent up to a particular scaling and time-shift factor, and to each cellof this partition we associate an independent univariate Gaussian white noise process. I then create ameasurable space (G,Γ) from the partition by choosing exactly one IRF from each cell of the partitionto form a set G, and then defining an appropriate σ-algebra Γ on that set.Step 1: Partition the IRFs and associate a white noise to each cellLet `1,m denote the space of absolutely summable sequences in Rm equipped with the product topol-ogy. We will interpret a typical element Ψ of `1,m as an IRF, and writeΨ = (Ψj)∞j=0Next, for Ψ ∈ `1,m, define J (Ψ) = min {j : Ψj 6= 0} as the first non-zero element of Ψ.12 Definean equivalence relation ∼ on `1,m by(Ψ ∼ Ψ¯)⇔(∃a 6= 0 such that ΨJ(Ψ)+j = aΨ¯J(Ψ¯)+j for all j ≥ 0)In words, if Ψ ∼ Ψ¯, then Ψ¯ is a scaled, time-shifted version of Ψ, in which case we will say thatΨ and Ψ¯ are indistinguishable. For convenience, when it exists we will denote the value of a in theabove definition by a(Ψ, Ψ¯).13 For any Ψ ∈ `1,m, we define [Ψ] ≡{Ψ¯ ∈ `1,m : Ψ ∼ Ψ¯}as theequivalence class of Ψ, and letQ denote the set of all equivalence classes in `1,m, which is a partitionof that set. Thus, each element Q ∈ Q is a set of IRFs that are equivalent to one another accordingto ∼. To each such Q ∈ Q, we then associate an independent univariate white noise process ηt (Q),where ηt (Q) ∼ N (0, 1) and E [ηt (Q) ηt−k (Q′)] = 0 for all k ∈ Z when Q 6= Q′.Step 2: Create a measurable space (G,Γ) from the partitionLet G ⊂ `1,m be some collection of IRFs formed by choosing exactly one element from each Q ∈Q. Thus, any collection of IRFs in G will be distinguishable, and for any Ψ ∈ `1,m there exists aunique element in G indistinguishable from it. Denote this element of G by G(Ψ). Next, the integralrepresentation introduced in the subsequent section requires defining a measure on the subsets of G.To do so requires first formally defining a σ-algebra on G. For any γ ⊂ G, letQγ ≡ {[Ψ] : Ψ ∈ γ} ⊂ Q12 If no such element exists, i.e., if Ψj = 0 for all j, we set J (Ψ) = 0.13 Note that the order of the arguments matters here, and in particular, a(Ψ, Ψ¯) = 1/a(Ψ¯,Ψ).13Thus, Qγ is the set of equivalence classes spanned by the elements of γ. We will say that γ is anopen subset of G if and only if there exists some open set H ⊂ `1,m formed by choosing exactlyone element from each Q ∈ Qγ . Letting Γ denote the Borel σ-algebra generated by these open sets,(G,Γ) is a measurable space.2.3.2 Integral representationIn this section, I show that, for any G as constructed above, any model of the form given in (2.1) canbe equivalently recast asY˜t =∫G∞∑j=0Ψjηt−j ([Ψ]) dσ (2.11)for some measure σ on (G,Γ). I will refer to this representation for Y˜t as its integral representationon G.To see how this representation can be constructed, fixG and a model (2.1). Write Ψ˜j =(Ψ˜(1)j , . . . , Ψ˜(r)j),and define Ψ˜(l) ≡(Ψ˜(l)j)∞j=0∈ `1,m for l = 1, . . . , r. Thus, G˜ ≡{Ψ˜(1), . . . , Ψ˜(r)}is the set ofIRFs contained in the model. For simplicity, I will restrict attention to the case where J (Ψ) = 0 forall Ψ ∈ G ∪ G˜.14 It is nonetheless straightforward to extend the argument to the general case.Next, let σ be the (unique) measure on (G,Γ) satisfying (1) σ ({Ψ}) = a(Ψ, G(Ψ)) σ˜l if Ψ =G(Ψ˜(l)), and (2) for any γ ∈ Γ with γ ∩G(G˜)= ∅,15 we have σ (γ) = 0. In words, σ is a measurethat assigns weight a(Ψ, G(Ψ)) σ˜l to the IRF in G that is indistinguishable from the l-th IRF in G˜,and zero elsewhere. We then have∫G∞∑j=0Ψjηt−j ([Ψ]) dσ =r∑l=1σ({G(Ψ˜(l))}) ∞∑j=0G(Ψ˜(l))jηt−j([Ψ˜(l)])=∞∑j=0r∑l=1a(Ψ˜(l), G(Ψ˜(l)))G(Ψ˜(l))jσ˜lηt−j([Ψ˜(l)])=∞∑j=0r∑l=1Ψ˜(l)j σ˜lηt−j([Ψ˜(l)])Settingεl,t ≡ σ˜lηt([Ψ˜(l)])this is clearly equivalent to the representation in (2.1).14 We also assume throughout that Ψ˜(l) ∼ Ψ˜(i) implies l = i, and that Ψ˜(l) 6= 0 for any l. If either of these assumptionsare violated, the model itself contains a fundamental indeterminacy that can be fixed by, for example, eliminating Ψ˜(l) fromthe model in either case.15 With slight abuse of notation, G(G˜)here denotes the unique set of elements of G that are indistinguishable from theelements of G˜.142.3.3 Orthogonal decompositionsIn this section, using the integral form introduced above, I show that for any given EM we may obtaina decomposition of the TM into a component of Yt that is explained by IRFs in the EM, and anorthogonal component that is not. Further, I establish that, in combination with Assumption 2.1, thisresult implies Assumption 2.2.In particular, fix a model, and chooseG such that Ψ˜(l) ∈ G for all l (and thusG(Ψ˜(l))= Ψ˜(l)).16Let σ∗ denote the measure associated with the integral representation of the TM on G, so that the TMcan be writtenYt =∫G∞∑j=0Ψjηt−j ([Ψ]) dσ∗Letting G˜c be the complement of G˜ in G, we can writeYt =∫G˜∞∑j=0Ψjηt−j ([Ψ]) dσ∗ +∫G˜c∞∑j=0Ψjηt−j ([Ψ]) dσ∗=∞∑j=0Ψ˜jε∗t−j +∫G˜c∞∑j=0Ψjηt−j ([Ψ]) dσ∗where Ψ˜j ≡(Ψ˜(1)j , . . . , Ψ˜(r)j), ε∗l,t ≡ σ∗l ηt([Ψ˜(l)]), σ∗l ≡ σ∗(Ψ˜(l)), and ε∗t ≡(ε∗1,t, . . . , ε∗r,t)′.More compactly, we can write this asYt = Y˜ ∗t + Zt (2.12)Y˜ ∗t may be interpreted as the variation in Yt that is generated by the IRFs contained in the EM (or theirequivalents according to ∼), and Zt as the variation in Yt generated by IRFs for which no equivalentexists in the EM. Note that the EM is correctly specified if and only if V ar (Zt) = 0; otherwise, it ismisspecified. Note also that, irrespective of whether the model is the true DGP, E[ε∗tZ ′t−k]= 0 forall k, and therefore in turn E[Y˜ ∗t Z ′t−k]= 0, so that (2.12) is an orthogonal decomposition of Yt.Next, substituting (2.12) into (2.3) we may obtainεˆt =∞∑j=−∞ψj Y˜ ∗t−j +∞∑j=−∞ψjZt−j (2.13)By Assumption 2.1, the first term on the right-hand side of this expression is the true value of the EMshocks ε∗t , and thus the second term is the recovery error νt defined in Section 2.2. Further, these twoterms are clearly orthogonal to one another at all leads and lags, so that E[ε∗t ν ′t−k]= 0 for all k.That is, in combination with Assumption 2.1, the framework developed above necessarily implies that16 This choice of G is for simplicity of exposition only. With the appropriate modifications, the desired result will holdfor any G.15Assumption 2.2 holds. We thus verify the upward-bias result from Section 2.2,σˆ2l = σ∗2l + V ar (νl,t) ≥ σ∗2lFurthermore, it is straightforward to see that when the model is misspecified (i.e., when V ar (Zt) 6=0), we will have V ar (νl,t) > 0 for at least one l, and therefore σˆ2l > σ∗2l ; that is, in the presence ofmisspecification the naive variance estimate will necessarily overstate the true variance of at least oneshock. It can be easily verified that the other results of Section 2.2 also hold.2.3.4 DiscussionIt is worth clarifying in what contexts the framework proposed in this paper is appropriate to use.One of the key properties of the framework is that shocks are identified uniquely up to a scaling andtime-shift factor for their IRFs. Conversely, any two shocks that are the same up to a scaling andtime-shift factor (i.e., that are indistinguishable according to∼) are not identified uniquely. Implicitly,the smoothing algorithm first determines which equivalence class (among those spanned by the EM)an imputed shock has come from using only information about the “shape” of the IRF from its firstnon-zero impact onwards. Once the equivalence class has been identified, the algorithm then choosesthe appropriate date and value of the shock as determined by the specific IRF from that equivalenceclass that appears in the EM. A corolloray to this procedure is that if one were to replace an IRF inthe EM with a different one from the same equivalence class, one would in general obtain a differentvariance estimate for that shock. As such, this framework is not appropriate for addressing the questionof which of two shocks from the same equivalence class—but with potentially different economicinterpretations—are more relevant, since the algorithm views these two shocks as fundamentally thesame. However, as can be easily verified, the estimate of the variance induced in the observablevariables by these two different shocks would nonetheless be identical. To illustrate with a simpleexample, suppose the TM for output is given simply by Xt = At, where At = ρAt−1 + εt, t isan i.i.d. exogenous process with variance σ2, and ρ is known. Using this TM as the EM and givensufficient data on Xt, one would be able to recover the true sequence of shocks εt, would obtain avariance estimate for the exogenous process equal to σ2, and would find that the variance induced inXt by this shock is equal to σ2X ≡ σ2/(1− ρ2). Now suppose one instead uses as the EM the samemodel, except that now it is assumed that Xt = φAt, where φ 6= 0 is some (known) constant. Inthis case, one would recover the sequence of shocks εt/φ and would obtain a variance estimate forthe TFP shock of σ2/φ2, but would nonetheless find that the variance induced in Xt by this shockremains equal to σ2X . As this example makes clear, then, one should view the framework in this papernot as seeking to find the variance of a shock, but as seeking to find the total variance in the observablevariables induced by that shock (or by its equivalents according to ∼).To anticipate one potential objection to the framework proposed in this paper, it should be notedthat, if we were to find a process for the smoothed shocks that exhibits correlation, it may be the case16that the true shocks are in fact correlated. In this case, there may be a desire to view independenceof the shocks as a modelling approximation made for convenience, rather than a property to be takenseriously (as it is in this framework). As such, one might argue, we should not be too concerned ifwe find evidence of its violation. I find this argument unconvincing for several reasons. First, if theshocks truly are correlated, there must be some underlying economic reason for this. Yet correlationamong smoothed shocks is rarely acknowledged in the literature,17 let alone justified using some eco-nomic argument. Second, though it would increase the number of estimated parameters, it would bea relatively straightforward exercise to allow for correlation in the EM,18 but again, such an exerciseappears to be rarely done in the literature.19 Third, if we do allow for such correlation in the model,rational expectations dictates that agents in the model know this. When there are intertemporal corre-lations between the shocks, this has the capacity to significantly alter agents’ behavior, since currentshock realizations would carry information about future values of the shocks. There is little reason tobelieve a priori that estimating such a model would yield even qualitatively similar results to thoseobtained in the case where independence is assumed. Finally, as demonstrated in Section 2.5 below, tothe extent that independence is a reasonable—though not necessarily perfect—approximation to thetrue structure of the exogenous processes, the bias-correction procedure proposed here would have aminimal effect on variance estimates. Thus, if resulting corrections are in fact large, this should betaken at the very least as clear evidence that independence is not a reasonable approximation.Next, it may be helpful to view the framework presented in Sections 2.3.1-2.3.3 as a DSGE ana-logue to a typical regression setting. In the latter, the regression error term is, conceptually, the residualcomponent of the dependent variable after as much of its variation as possible is explained by the re-gressors. In general, no economic meaning is assigned to this residual component. Rather, it is takenas arising from the econometrician’s ignorance about the full process underlying the dependent vari-able. In a DSGE setting, the shocks are often taken to be a similar residual component. Unlike in aregression setting, however, in a DSGE model the shocks are given an economic interpretation andare thus a part of the econometrician’s explanation, not residuals standing in for her ignorance. If webelieve that factors beyond those captured by the model are of some relevance, this would motivate theexplicit consideration of some orthogonal error term analogous to the regression residual. One mayview the framework above as motivating such a residual (see, e.g., Zt in equation (2.12)).To close this section, it is worth briefly mentioning here that there are, conceptually, two distincttypes of misspecification that arise in the above framework. In the first type, the IRFs in the EM arealso found in the TM, but there are IRFs in the TM that are not present in the EM. This is the case17 Ingram et al. (1994) are a notable exception. They extend the canonical neoclassical growth model of King et al.(1988), which features only a shock to productivity, by adding two additional shocks. For several different calibrations ofthe model, they recover the implied series for the shocks using U.S. data on output, consumption and labor, and find that theshocks exhibit substantial correlation, both cross-sectionally and intertemporally.18 In the above framework, this would in practice consist of adding one or more IRFs to the EM that are linear combina-tions of existing IRFs. In other words, this would effectively increase the number of shocks in the model.19 There are some infrequent exceptions to this. For example, in Smets and Wouters (2007), innovations to TFP are alsoallowed to impact the exogenous spending shock.17in Example 1. In the second type of misspecification, there may be shocks in the EM and TM withthe same economic interpretation but for which the associated IRF in the EM differs (however mildly)from the corresponding IRF in the TM. The framework introduced here does not distinguish betweenthese two types of misspecification, despite the fact that there may be some desire to be more forgivingtoward the second type. Nonetheless, as will be illustrated in the examples of Section 2.5 below, if theEM is a reasonable approximation to the TM, we will continue to find an important role for the EMshocks even if their precise IRFs differ somewhat from those in the EM.2.4 Further analysisIn this section, I provide basic conditions on the EM and TM that are sufficient for the key results ofSection 2.2 to hold, then re-state those results formally.2.4.1 Econometric modelConsider the class of linearized DSGE models in state-space form asXt = AXt−1 +Bεt (2.14)Here, Xt is an n-vector of (possibly unobservable) model variables, εt is an m-vector of i.i.d. normalrandom variables with diagonal covariance matrix, the l-th diagonal element of which is given by σ2l ,and A and B are (n× n)- and (n×m)-matrix-valued functions of P , respectively, where P ∈ P isthe vector of “deep” model parameters (excluding σ ≡ (σ1, . . . , σm)′). The eigenvalues of A are allstrictly less than one in modulus, so that {Xt} is a covariance-stationary process.For what follows, I assume that the parameter vector P is given. There is a substantial literaturethat considers methods for obtaining parameterizations in this class of models.20 I wish to sidestepthis issue altogether and take as a starting point a particular parameterization (or a set of parameter-izations, as the case may be) that the econometrician has identified as in some sense best for his orher purposes. The theoretical and methodological considerations that follow should thus be viewedas tools to analyze the implications of a particular value of P in the model, rather than of the modelitself. Henceforth, for the sake of brevity, I shall generally suppress dependence of the analysis onthe parameterization P , with the tacit understanding that all discussion and results are related to thatspecific parameterization only.The m-vector of observable data Yt is presumed by the econometrician to be related to the modelvariables byY˜t = FXt (2.15)where F is an m × n matrix and, as in Section 2.2, Y˜t is the model counterpart of Yt. F and B are20 See, for example, Gourieroux et al. (1993), An and Schorfheide (2007), Canova (2007), Ferna´ndez-Villaverde (2010).18assumed to be such that the matrix FB is invertible.212.4.2 Data processSuppose we have a series of observations drawn from anm-variate process {Yt}. I make the followingassumption about {Yt}.Assumption 2.3. {Yt} is a mean-zero jointly normally distributed covariance-stationary process withV ar (Yi,t) > 0 for all i = 1, . . . ,m.Note that we have not assumed that the TM for Yt is given by the EM (2.14)-(2.15). Rather, asargued in Section 2.3.3, we may writeYt = Y˜ ∗t + Zt (2.16)whereY˜ ∗t = FX∗t (2.17)X∗t = AX∗t−1 +Bε∗t (2.18)Here, ε∗t and Zt have the same interpretation as in Section 2.3.3, and in particular E[ε∗tZ ′t−k]=E[Y˜ ∗t Z ′t−k]= 0 for all k ∈ Z.2.4.3 Theoretical resultsThis section formally presents the theoretical results underpinning the methodology discussed in Sec-tion 2.2. Let C ≡[In −B (FB)−1 F]A. I make the following assumption regarding the eigenvaluesof C.Assumption 2.4. None of the eigenvalues of C lie on the complex unit circle.Assumption 2.4 is related to the condition under which, if the model is the true DGP, then Yt hasa “structural VAR” representation (i.e., a VAR representation with innovations given by FBεt). Astructural VAR representation is inherently a backward-looking process: Yt is a function only of lagsof itself and of the current structural shocks. For this representation to exist, intuitively, an infinitehistory of past values of Yt must be sufficient to recover the previous period’s latent state vector,Xt−1, so that the only “new” information contained in Yt is the vector of current structural shocks. Itcan be shown that, for this to be true, all of the eigenvalues of C must lie strictly inside the complex21 Note that I have implicitly assumed that the model contains the same number of shocks as observable variables; i.e.,m = r in the notation of Section 2.2. If we had r > m, we would be unable to invert the EM’s MA(∞) representation,so that Assumption 2.1 would not hold. On the other hand, if we had r < m, some subset of the elements of Y˜t would belinearly dependent; i.e., the implied autocovariance function would be singular. Since the actual data is unlikely to possessthis feature, the model would be inconsistent with the data in that there would in general be no sequence of shocks thatwould allow the EM to exactly reproduce the data.19unit circle.22 In contrast, while we similarly require that the latent state vector be recoverable (sothat we may in turn recover the structural shocks), we have no need to construct a backward-lookingstructural VAR representation. As such, there is no need for the state vector to be recoverable onthe basis of past values of Yt only; it will be sufficient that some combination of past, present andfuture values of Yt reveal Xt−1, a property that is guaranteed by Assumption 2.4.23 This intuition isestablished formally in the following proposition.Proposition 2.1. Suppose Assumptions 2.3-2.4 hold. Then there exists a sequence of absolutelysummable m×m matrices {ψj}∞j=−∞ such thatε∗t =∞∑j=−∞ψjLj (Yt − Zt) (2.19)where L is the lag operator and ψj is a function of A, B, F and j only. Further,∞∑j=−∞ψjLj(∞∑i=0FAiBLi)= Im =(∞∑i=0FAiBLi)∞∑j=−∞ψjLj (2.20)Proof. All proofs are given in Appendix A.2.Proposition 2.1 establishes conditions under which, given infinite sequences of observations Y∞ ≡{Yt}∞t=−∞ and Z∞ ≡ {Zt}∞t=−∞, the true value of the structural shock, ε∗t , may be recovered. Thenext proposition verifies that Assumptions 2.3 and 2.4 together imply Assumption 2.2.Proposition 2.2. Suppose Assumptions 2.3-2.4 hold. Thenεˆt = ε∗t + νt (2.21)where E[ε∗t ν ′t−k]= 0 for all k ∈ N.Proposition 2.2 orthogonally decomposes a smoothed EM shock into its true value and a residual.This decomposition result has several useful implications, summed up in the following corollary.Corollary 2.1. Suppose Assumptions 2.3-2.4 hold. Then the following are true:(a) σˆ2l ≡ E[εˆ2l,t]≥ σ∗2l for all l.22 See, for example, Ferna´ndez-Villaverde et al. (2007).23 The relaxation of the structural VAR assumption (|eig (C)| < 1) to Assumption 2.4 (|eig (C)| 6= 1) is an economicallyrelevant one. As pointed out by, for example, Leeper et al. (2011) and Sims (2012), the requirement that all eigenvalues ofC lie strictly inside the complex unit circle often excludes models where agents’ information sets are strictly greater thanthe econometrician’s, such as in models with “news” and “noise” shocks (see, for example, Beaudry and Portier (2004),Schmitt-Grohe´ and Uribe (2012), Jaimovich and Rebelo (2009), Blanchard et al. (2013), Christiano et al. (2010a)). Incontrast, Assumption 2.4 will not generally exclude this important class of models.20(b) If E [εˆl,tεˆi,s] 6= 0 and (l, t) 6= (i, s), then the model is misspecified.(c) If the model is misspecified, then there exists an l such that σˆ2l > σ∗2l .(d) For (l, t) 6= (i, s), E[ε∗l,tεˆi,s]= 0 irrespective of whether the model is misspecified.Part (a) of Corollary 2.1 points out that the variance of smoothed shock l is an upper bound forthe true variance, which is itself unobservable. Parts (b) and (c) highlight that if the smoothed shocksexhibit correlation with one another, then the model is certainly misspecified, and thus the variance ofat least one of the shocks will be overstated.24Part (d) forms the basis for the correction procedure that is the principal methodological contribu-tion of this paper. In particular, for any q ∈ N, we may writeεˆl,t =∑i 6=lΘi,0εˆi,t +q∑j=1(Θj εˆt−j + Θ−j εˆt+j) + ξl,t (2.22)where E [ξl,tεˆi,t+j ] = 0 for any (l, t) 6= (i, t+ j) with |j| ≤ q. Here, the Θ’s have the interpretationof population regression coefficients, and ξl,t as the corresponding OLS residual. ξl,t captures thecomponent of εˆl,t that cannot be predicted using q past and future values of the recovered shocks(including the current values of shocks i 6= l). The following proposition establishes how a moreaccurate upper bound for σ∗2l may be obtained from the regression equation (2.22).Proposition 2.3. Suppose Assumptions 2.3-2.4 hold, and let ξl,t be as in equation (2.22). Thenσ∗2l ≤ σ¯2l ≡ E[ξ2l,t]≤ σˆ2l (2.23)Proposition 2.3 establishes that σ¯2l is an upper bound for the unknown value of σ∗2l , and furtherthat this upper bound is a (weakly) better estimator of σ∗2l than σˆ2l .2.4.4 Practical considerationsAssumption 2.4 and the availability of infinite past and future observations on Yt play an importantrole in the theoretical results of Section 2.4.3. In particular, in the case where the EM (2.14)-(2.15)is correctly specified, these properties together guarantee that the shocks may be exactly recovered asa linear combination of the observations on Yt. This is crucial for the orthogonal decomposition ofProposition 2.2, and, since it relies on this result, also for the variance correction suggested at the endof Section 2.4.3.In practice, however, Assumption 2.4 may fail to hold in important cases of interest (the applicationof Section 2.6 explores such an instance), and data sets are usually limited to at most a few hundred24 Note that the converse does not necessarily hold, i.e., if E [εˆl,tεˆi,s] = 0 for all (l, t) 6= (i, s), this does not necessarilyimply that the model is correctly specified, so that the naive estimates σˆ2l may still overstate the importance of one or moreshocks.21periods of observations. In either of these cases, the shocks may be recovered in expectation only,with a generally non-zero and non-diagonal error covariance matrix even when the model is correctlyspecified. Nonetheless, given the linear-Gaussian structure of the EM and TM, the expected valuesof the shocks conditional on the model and observed data (i.e., the smoothed shocks) will be linearcombinations of the sequence of observations on Yt. Thus, as in (2.3), we may express a smoothedshock as a linear filter applied to the time series of data.25 As such, using a similar argument as in theproof of Proposition 2.2, we may generalize the decomposition result asεˆt = εˆ∗t + νtwhere E[εˆ∗t ν ′t−k]= 0, εˆ∗t is defined to be the value of the smoothed shock that would obtain ifthe model were correctly specified, and νt is the linear filter applied only to the sequence of Zt’s(analogous to the second term on the right-hand side of equation 2.13). In general, the errors δ∗t ≡εˆ∗t − ε∗t that would be made in recovering the shocks if the model were correctly specified will becorrelated across different shocks. As a result, the actual smoothed shocks, εˆt, will in general exhibitnon-zero correlation with one another even if the model is correctly specified. Whether the robustvariance estimates proposed in Section 2.3 will continue to be an improvement on the naive onesunder these circumstances depends on the quality of the approximation εˆ∗t ≈ ε∗t . While no analyticalresult is available to check the accuracy of this approximation, given that the researcher has in handthe fully specified model, it is nonetheless straightforward to check it numerically via simulation, asillustrated in the application of Section 2.6. Similarly, critical values for any relevant statistics mayalso be obtained via simulation.2.5 Further examplesIn this section, I present two additional fully-specified examples to illustrate the above methodology.In the first example, which is simple enough to be solved analytically, I highlight a type of misspec-ification that is conceptually distinct from the type present in the example of Section 2.2.1. In thatearlier example, the EM correctly represents the dynamic impacts of the shocks it contains (i.e., theIRFs in the EM also exist in the TM), but is missing an additional source of exogenous variation. Incontrast, in the first example presented in this section the economic interpretations of a shock in theEM and another in the TM are the same, but their IRFs are not, and so by the framework developed inSection 2.3 they are nonetheless considered fundamentally different.Next, in the second example of this section, the TM is taken to be a variant of a standard medium-scale New Keynesian model (that of Smets and Wouters (2007)) that features a variety of real andnominal frictions and seven exogenous shock processes. The EM, on the other hand, will in generalbe missing several frictions and only contain three of the seven shock processes.26 This example most25 Note that, when the data set is finite, the filter for εˆt will generally depend on t.26 Note that both types of misspecification highlighted above will be present in this example.22closely captures quantitative macroeconomic modelling in practice, whereby economists use highlysimplified models to fit data generated by substantially more complicated ones.2.5.1 Example 2: Univariate AR(1) modelSuppose we have a sequence of data on TFP growth, γt, for which the TM is an AR(1) processγt = ργt−1 + ε∗1,t, ε∗1,t ∼ i.i.d.N (0, σ∗1)where 0 < |ρ| < 1. The EM, meanwhile, is given byγ˜t = δγ˜t−1 + ε˜2,t, ε˜2,t ∼ i.i.d.N (0, σ˜2)with |δ| < 1, where δ is some number taken as given by the econometrician and that may potentiallybe different from ρ, in which case the EM would be misspecified. This EM can be easily inverted toobtain ε˜2,t = γ˜t − δγ˜t−1. Substituting the data process γt into the inverted EM for γ˜t, we may obtainthe process for the smoothed shocks,εˆ2,t = (ρ− δ) γt−1 + ε∗1,tand thus the naive estimate of the variance of the TFP shock is given byσˆ22 =1− 2ρδ + δ21− ρ2 σ∗21When δ 6= ρ, the IRF in the EM does not exist in the TM, and thus by the framework of Section2.3 we have σ∗22 = 0, which is clearly less than the naive estimate σˆ22 . The conclusion that σ∗22 = 0 israther stark here, since it will hold for all δ 6= ρ even as δ → ρ, while if δ = ρ we have σ∗22 = σ∗21 .However, this discontinuous behavior is of little practical importance. In particular, misspecificationis identified in this framework as non-zero autocovariance in the sequence of smoothed shocks. Fork 6= 0 we may obtainCov (εˆ2,t, εˆ2,t−k) =(ρ− δ) (1− ρδ)1− ρ2 ρ|k|−1σ∗21Clearly, Cov (εˆ2,t, εˆ2,t−k) → 0 as δ → ρ. Thus, even though the EM continues to be misspecifiedas δ becomes arbitrarily close to ρ (i.e., as the IRF in the EM becomes ever closer to the IRF in theTM), the degree of observed misspecification (and thus any adjustment made using the bias-correctionprocedure) will nonetheless shrink to zero.Next, to apply the bias-correction procedure, we wish to find the component of εˆt that cannot bepredicted using past and future values. In general, one can continue to improve the quality of theadjusted estimate by including a greater number of leads and lags in the regression (i.e., by making qarbitrarily large in equation (2.22)). For simplicity, I focus here on the simple case where q = 1. In23this case, the estimated regression equation is given byεˆ2,t = β (εˆ2,t−1 + εˆ2,t+1) + ξtwhereβ ≡ (ρ− δ) (1− ρδ)(1− ρ2)[1− δ (ρ− δ)](1− 2ρδ + δ2)2 − ρ2 (ρ− δ)2 (1− ρδ)2and ξt is the OLS residual. The bias-corrected estimate of the shock variance is then given by σ¯22 ≡V ar (ξt). Some simple algebra shows thatσ¯22 = σˆ22 − 2βCov (εˆ2,t, εˆ2,t−1)It can be verified that βCov (εˆ2,t, εˆ2,t−1) ≥ 0 and therefore 0 = σ∗22 < σ¯22 ≤ σˆ22 . Thus, the correctedestimate lies between the true value and the naive estimate σˆ2. Furthermore, as expected given thediscussion above, we have βCov (εˆ2,t, εˆ2,t−1) → 0 as δ → ρ, so that the size of the adjustmentapproaches zero as the IRF in the EM approaches the IRF in the TM. That is, notwithstanding thefact that the true variance is zero for all δ 6= ρ, as the degree of misspecification approaches zero thebias-corrected estimate of the variance will approach the desired value of σ∗21 .2.5.2 Example 3: Medium-scale new Keynesian modelIn this example, I consider three different combinations of TM and EM. In all cases, the models willbe variants of the widely-cited medium-scale New Keynesian model of Smets and Wouters (2007),which in its baseline form features a number of real and nominal frictions, as well as seven differentexogenous shock process: exogenous spending, neutral TFP, investment-specific technology (IST),risk premium, price mark-up, wage mark-up, and monetary policy shocks.27 In particular, I considerthe following combinations:• Case A: The EM and TM both feature sticky prices and wages and only the first three shocksabove (exogenous spending, neutral TFP, and IST).• Case B: The EM and TM both feature sticky prices and wages. The EM contains only the firstthree shocks while the TM contains the full complement of seven shocks.• Case C: The EM and TM both contain only the first three shocks. The EM features flexibleprices and wages, while the TM features sticky prices and wages.Note that, while the EM in Case A is correctly specified, Cases B and C correspond to two dis-tinct types of misspecification. As in Example 1, the EM in Case B is missing important sources ofexogenous variation due to the absence of the risk premium, mark-up and monetary policy shocks.27 A summary of the equations characterizing the DGP and model is contained in Appendix A.1. For further details, seeSmets and Wouters (2007).24Meanwhile, as in Example 2, in Case C even though the shocks in the EM have the same economicinterpretation as shocks in the TM, their IRFs will be different since the EM does not feature stickyprices or wages. As such, these shocks are fundamentally different, and thus the true variance of eachEM shock is zero. As we shall see, however, the “less misspecified” is the EM—that is, the moreflexible are prices and wages in the TM—the smaller the degree of observed misspecification will be,with no observed misspecification in the limit as prices and wages become perfectly flexible. In turn,as in Example 2, the size of any bias adjustment will also shrink to zero.I assume the econometrician estimates the variances of the EM shocks using data on three realvariables: consumption, investment, and hours worked. For reasons discussed in Section 2.4.1, I as-sume that, aside possibly from the true degree of price and wage flexibility, the econometrician knowsall of the parameter values of the TM except for the variances of the shock process innovations, whichare then estimated using a large data set on consumption, investment and hours worked, themselvesgenerated from the TM.28,29 The first column of data in Table 2.1 reports, for each of the three shockprocesses in the EM, the variance of the shock with that same economic interpretation in the TM.30The remaining three columns report naive variance estimates for the three different cases.31For Case A, the EM is correctly specified, and as a result the naive variance estimates reportedin the table are the same as the true values reported in the first column.32 For Case B, however, wehave the first type of misspecification discussed above: the IRFs in the EM are the same as theirTM counterparts, but the TM also includes other sources of exogenous variation. As we might haveexpected from (2.4), the shock variances are significantly overestimated in this case, ranging from over13 times the true value for the investment shock, to well over 800 times for the exogenous spendingand TFP shocks.Next, I apply the bias-correction methodology to obtain revised estimates of the variances forCases A and B. Specifically, I regress each of the three smoothed shocks on contemporaneous valuesof the other two smoothed shocks and on four leads and lags of all three shocks. I then compute thevariance of the resulting residual. The results are presented in Table 2.2. For comparison purposes,the first column of data again reports the variance of the shock with that economic interpretation inthe TM. Looking at the second column, the corrected estimates for Case A show no change relative to28 Except where otherwise noted, the model parameters used for generating the data are taken as the mode of the posteriordistribution reported in Smets and Wouters (2007).29 In practice, I simulate 101,000 periods of data, discarding the first 1,000 to minimize the impact of the choice of theinitial state vector—set equal to zero—on the results. I use this large number of data periods to avoid the complications(discussed in Section 2.4.4) that arise when samples are of a more realistic size.30 Note that these variances are the same for each of the three cases.31 Naive estimates are obtained by first running the Kalman smoothing algorithm (with shock variance parameters setequal to values from TM), then computing the sample variances of the resulting smoothed shocks. While these estimateswill in general differ from estimates obtained via maximum likelihood, the two are asymptotically equivalent. Given thevery large data set, it can be verified that the quantitative difference between the two approaches is negligible.32 Technically, there are small discrepancies between the two sets of estimates due to (1) sampling error; and (2) the factthat, in a finite sample, the shocks cannot be exactly recovered. However, because of the large sample size employed here,such discrepancies are quantitatively unimportant.25Table 2.1: Variances of true and smoothed shocksTM 3 shocks,sticky7 shocks,sticky3 shocks,stickyEM 3 shocks,sticky3 shocks,sticky3 shock,flexCase True A B CShock:Spendingt 0.27 0.27 221.52 2.18TFPt 0.21 0.21 179.27 0.61ISTt 0.20 0.20 2.73 0.21Notes: “True” column shows variance of shock in TM. Remaining columns show naive variance estimatesfor relevant TM and EM, obtained by first running Kalman smoothing algorithm (with shock variancesset equal to values from TM), then computing sample variances of resulting smoothed shocks.the naive case from Table 2.1. Intuitively, since in this case the model is the TM, the true sequence ofshocks can be recovered (nearly) exactly. Since the true shocks are, by construction, uncorrelated withone another, so too are the smoothed shocks. Thus, none of the variation in a smoothed shock can beexplained by variation in the other shocks, and we obtain an R2 of essentially zero in the regression.Table 2.2: Corrected variances of smoothed shocksDGP 3 shocks,sticky7 shocks,sticky3 shocks,stickyModel 3 shocks,sticky3 shocks,sticky3 shocks,flexCase True A B CShock:Spendingt 0.27 0.27 0.55 0.01TFPt 0.21 0.21 0.44 0.00ISTt 0.20 0.20 0.23 0.01Notes: “True” column shows variance of shock in TM. Remaining columns show bias-corrected esti-mates. For a given smoothed shock, set of regressors is contemporaneous values of other smoothedshocks plus four leads and lags of all smoothed shocks.For Case B, on the other hand, the corrected estimates in Table 2.2 show a substantial improve-ment over the naive estimates from Table 2.1. For example, whereas the naive estimate of the neutralTFP variance was 875 times the true value, the corrected estimate is only a little more than twicethe true value. The relative discrepancies are even smaller for the other two shocks. A clue for whythe corrected estimates show such a large improvement in Case B can be seen from Table 2.3, whichpresents contemporaneous cross-correlations between the smoothed shocks (first three rows), as wellas the first-order autocorrelation for each shock (final row). Clearly, the smoothed shocks exhibit avery high degree of contemporaneous correlation with one another, especially the exogenous spend-26ing and neutral TFP shocks, which are nearly collinear. The spending and TFP shocks also exhibitsignificantly negative first-order autocorrelation. Because there is such a high degree of correlationbetween the smoothed shocks, they are highly predictable using values of the other smoothed shocks,which in turn results in the regression residual being small.Table 2.3: Sample correlations for smoothed shocks (case B)Spendingt TFPt ISTtSpendingt - 0.997 -0.800TFPt 0.997 - -0.829ISTt -0.800 -0.829 -Lagged variable -0.505 -0.499 -0.043Notes: Table presents correlations among smoothed shocks for Case B. Upper three rows of data showunconditional correlation matrix. Final row shows first-order autocorrelation for that smoothed shock.This high degree of correlation in the smoothed shocks stems from a number of features of theTM, one example of which I focus on here for illustrative purposes. Consider the risk premium shock.In the TM this shock accounts for over 80 percent of the variance of the one-step-ahead forecast errorin consumption, while in the EM it is entirely absent. Thus, the EM is clearly missing an importantfactor in generating high-frequency variation in consumption. This is confirmed in Figure 2.1, whichshows autocovariance functions (ACFs) for the three observable variables using the TM. Each panelof the figure corresponds to Cov (yt, xt−k) for two variables yt and xt. Within each panel there arefour lines plotted, each corresponding to the ACF that arises in the TM when all but one shock isshut down. The four plots correspond to the four “real” shocks (the exogenous spending shock, thetwo technology shocks, and the risk premium shock). The first five lags are shown, and to make thecomparison as clean as possible, the variances of the shocks are normalized so that the unconditionalvariance of consumption is always equal to one.Looking at the top-left panel of the Figure, we see that the autocovariance generated in the con-sumption process by the risk premium shock (the solid line) shows a relatively steep decline. Incontrast, the spending, TFP and IST shocks (the dashed, dotted, and dash-dot lines, respectively) allgenerate highly persistent consumption dynamics. In order for the EM to be able to reproduce thehigh-frequency consumption variation that is a feature of the TM, it would therefore need to be thecase that certain combinations of shocks tend to occur together or in a particular sequence (or both).Since the smoothing algorithm is precisely the process of finding a sequence of shocks that allow theEM to exactly reproduce the data, it should then come as no surprise that we observe correlationsbetween them.Next, consider Case C. In this case we have the second type of misspecification discussed above:the shocks in the TM and the EM have the same economic interpretation, but because the EM does notfeature sticky prices and wages, the IRFs in the model are different from those in the TM. As such,27Figure 2.1: Autocovariance function (Cov (yt, xt−k))0 1 2 3 4 500.−kkc t0 1 2 3 4 50123it−kk0 1 2 3 4 5−0.500.5lt−kk0 1 2 3 4 500.511.522.5ki t0 1 2 3 4 5010203040k0 1 2 3 4 5−101234k0 1 2 3 4 5−0.500.5kl t0 1 2 3 4 5−101234k0 1 2 3 4 500.  Exogenous spending TFP Investment Risk premiumNotes: Figure shows autocovariance functions (ACFs) for observable variables in TM generated by each of fourlisted shocks individually. ACFs are normalized so that unconditional variance of ct is equal to one.the true variances of the EM shocks are actually zero. As seen in Table 2.1, however, using naiveestimation procedures we obtain non-zero estimates.33Turning to the corrected estimates in Table 2.2, we find that they are quite close to their true valueof zero. As in Example 2, however, there is some sense that this result is undesirable, since there areshocks in the TM with the same economic interpretation and that do actually account for a positiveamount of variation. But as also illustrated in Example 2, the extremeness of the adjustment in thiscase is due to the very high degree of misspecification in the EM. In particular, there is a significantamount of price and wage stickiness in the TM: each quarter, only fractions 0.34 and 0.26 of price-and wage-setters, respectively, are allowed to re-set their prices/wages. In the EM, however, thesefractions are incorrectly assumed to be equal to one. To see how the corrected estimates change as the33 Note that, while it turns out the estimates for Case C are greater than the values reported in the first column, this neednot always be true.28degree of stickiness in the TM and EM converge, Table 2.4 shows corrected variance estimates as thefraction of re-optimizers in the TM approaches the EM case of one (so that the EM is closer to beingcorrectly specified). The first column of data in the table again reports the true variance estimates,while the second reproduces the adjusted levels from Table 2.2. The remaining columns report theresults for three different values of the fraction of re-optimizers—0.95, 0.99 and 0.999—which showthat the estimates steadily improve with the quality of the model. Thus, despite the stark fact that thetrue variances of the EM shocks remain zero even as the EM and TM converge, the observed degree ofmisspecification nonetheless converges to zero, and the adjusted shock values converge to the valuesfrom the TM. Put another way, one need only be concerned about this sort of “over-adjustment” if theEM is a poor approximation to the TM, which is precisely the case in which there should be a largeadjustment so as to warn the econometrician that misspecification is likely to be a problem.Table 2.4: Corrected variances (case C)Fraction of re-optimizersTrue Baseline .95 .99 .999Shock:Spendingt 0.27 0.01 0.19 0.33 0.28TFPt 0.21 0.00 0.03 0.11 0.19ISTt 0.20 0.01 0.17 0.20 0.20Notes: “True” column shows variance of shock in TM. “Baseline” column reproduces corrected estimatesunder baseline parameterization of TM from Table 2.2. Remaining columns show corrected estimates forseveral different fractions of re-optimizers in TM.2.6 Application: Investment shocksIn this section, I apply the methodology introduced in this paper to the recent model of Justiniano et al.(2010) (henceforth JPT), itself a variant of the Smets and Wouters (2007) model discussed in Example3 in Section (2.5).34Following the Bayesian estimation procedure used by JPT,35 I re-estimate their model using quar-terly U.S. data on output, consumption, investment, hours, wages, inflation and the nominal interestrate over the period 1954QIV-2004QIV (T = 201).36 Table 2.5 reports resulting variance decompo-sitions for log-hours using the median parameter values from the posterior parameter distribution.3734 While there are a number of differences between JPT and Smets and Wouters (2007), the main source of the divergencein their results is in the data used to estimate the models. Specifically, Smets and Wouters (2007) include consumer durablesin their measure of consumption and exclude the change in inventories from their measure of investment. JPT, on the otherhand, include consumer durables and the change in inventories in their measure of investment, with consumption includingonly non-durables and services. This produces a more volatile investment series and a less volatile consumption series,which largely drives their results.35 See An and Schorfheide (2007) for a review of Bayesian estimation in the context of DSGE models.36 See Appendix A.3 for details about data sources. Bayesian estimation of the DSGE model was done using Dynare (seeAdjemian et al. (2011)).37 The posterior statistics I obtained for the 35 estimated parameters were very close to those reported by JPT and are29Column (1) shows a standard decomposition of the unconditional variance. As noted by JPT, thisdecomposition indicates that the wage mark-up shock (typically interpreted as a labor supply shock)accounts for the majority (52 percent) of the unconditional variance of hours in the model. However,the estimated wage mark-up process is highly persistent (autoregressive parameter of 0.98), suggestingthat the bulk of the variation induced by this shock may be of a long-run nature, and thus potentiallyless important for business cycle variation.Table 2.5: Naive variance decomposition for hours(1) (2)Shock Unconditional BCFMonetary policy 0.03 0.06Neutral technology 0.04 0.11Government spending 0.08 0.02Investment 0.25 0.61Price mark-up 0.06 0.06Wage mark-up 0.52 0.05Patience 0.03 0.08Notes: Table entries show naive variance decompositions for hours, obtained fromJPT’s model using the median of the posterior parameter distribution. Column (1)decomposes the unconditional variance. Column (2) decomposes the BCF variance,defined as the variance associated with periodic fluctuations between 6 and 32 quarters.Columns may not add up to 1 due to rounding.To address this issue, Column (2) of Table 2.5 reports the decomposition of the business-cycle-frequency (BCF) variance38 of log-hours in the model. Echoing the results found by JPT, the resultsindicate that the investment shock, rather than the wage mark-up shock, accounts for the majority(61 percent) of BCF fluctuations in hours. While not shown in the table, the investment shock wasalso found to account for large proportions of the BCF variance of output growth (56 percent) andinvestment growth (88 percent). On the basis of this evidence, one may be tempted to conclude, asJPT do, that “investment shocks are the leading source of business cycles.” (p. 137)As JPT also note, however, the investment shock is found to be a negligible determinant ofBCF fluctuations in consumption growth, accounting for a mere 6 percent of its variance, while theotherwise-irrelevant household patience shock accounts for 61 percent. Figure 2.2 plots the correla-tion between consumption growth and four leads and lags of investment growth in the data (solid line)and in the model (dashed line). The data indicates a positive and significant correlation between con-sumption growth and investment growth within a one-quarter lead/lag. In contrast, in the EM there isa small negative correlation between the two. This EM correlation is small because consumption andavailable upon request.38 BCF variances are obtained by integrating the spectrum of the model over the relevant frequencies (defined here, as inJPT, to be frequencies associated with periods between 6 and 32 quarters). This process effectively removes the varianceassociated with high- and low-frequency fluctuations, leaving only the medium-frequency fluctuations normally associatedwith business cycles. See Appendix A.4 for further details.30investment are driven primarily by two distinct (and orthogonal) shocks: the patience and investmentshocks, respectively. Meanwhile, the correlation is negative primarily because a positive investmentshock increases the return on investment with no change to current productive capacity, causing thehousehold to substitute away from consumption and toward investment.39Figure 2.2: Correlation between %∆Ct and %∆It+k−4 −3 −2 −1 0 1 2 3 4−0.1−0.0500.  DataModelTo gain further insight, I proceed with analysis of the smoothed shocks. It can be verified in thecase of JPT’s EM that the matrix C (as defined in Section 2.4.3) contains several eigenvalues equalto one, so that Assumption 2.4 fails to hold.40 To check that the approximation εˆ∗t ≈ ε∗t holds (seeSection 2.4.4) in spite of the finite sample size and the violation of Assumption 2.4, I simulated datafrom the EM,41 then obtained the smoothed shocks from this simulated data. From this sequence ofsmoothed shocks, I then computed the mean squared recovery error (MSRE) for each shock in the EM(i.e., V ar(δ∗l,t) in the notation of Section 2.4.4) at each date t, then scaled the result by the (known)variance of that shock, σˆ2l . Figure 2.3 plots the maximum of this statistic across the seven different39 The failure of many RBC-type models to generate positive comovement between consumption, investment and hoursin response to shocks that affect the expected return to investment without affecting current production technology has beenknown since at least Barro and King (1984); see also Beaudry and Portier (2007).40 Because JPT’s model features a stochastic trend in productivity, the data contains first-differences of several non-stationary variables, while the model state vector contains the corresponding variables in (stationarized) levels. Since avariable cannot be exactly recovered from even an inifinite history of changes in that variable, and since the condition thatC contain no eigenvalues of modulus one is precisely that needed to guarantee exact recovery of the state vector, it shouldbe unsurprising that C contains eigenvalues equal to one for JPT’s model.41 Simulated values throughout this section are based on N simulated data sets, each of T periods in length, whereT = 201 is the length of the actual data set.31shocks for each the first ten periods.42 As the figure shows, the smoothed shocks very quickly attain ahigh degree of accuracy, with the maximum MSRE equal to less than 2 percent of the variance of theshock by t = 4, and less than 1 percent by t = 5. Though not shown in the figure, accuracy continuesto increase, with the maximum MSRE falling below 0.5 percent by t = 17 and remaining belowthis threshold for the remainder of the sample. Thus, beyond the first few periods, the approximationεˆ∗t ≈ ε∗t appears to be sufficient for the principal results of Section 2.4.3 to hold. To ensure that noneof the results are driven by the relatively inaccurate recovery of the early shock values, I drop the firstfour periods of smoothed shocks in all subsequent computation.Figure 2.3: Maximum scaled MSRE for smoothed shocks0 1 2 3 4 5 6 7 8 9 1000. Figure displays maxl V ar(δ∗l,t)/σ∗l , computed by simulating 10,000 data sets.Data point for t = 1 is off the scale in Figure 2.3, with a maximum MSRE equal to 80percent of the variance of the shock.Next, returning to the sequence of smoothed shocks obtained from the actual data, Figure 2.4 plotssample correlations between the smoothed investment shock at date t on the one hand, and each of thesmoothed shocks at a number of leads and lags on the other. Black dots represent sample correlations,while the dashed lines show pointwise simulated 95 percent confidence intervals (based on 100,000draws) for these correlations under the null hypothesis that the EM is correctly specified. Circlesaround dots highlight correlations that fall outside of the confidence intervals.Of the 71 points shown in Figure 2.4, 15 (21 percent) are outside of the 95 percent confidence42 The data point for t = 1 is off the scale in Figure 2.3, with a maximum MSRE equal to 80 percent of the variance ofthe shock.32Figure 2.4: Correlation coefficients between smoothed shocks−6 −4 −2 0 2 4 6−0.200.2Invest.t, Monetaryt+jj−6 −4 −2 0 2 4 6−0.4−, Neut. Tech.t+jj−6 −4 −2 0 2 4 6−0.2−0.100.1Invest.t, Govt.t+jj−6 −4 −2 0 2 4 6−0.2−0.100.1Invest.t, Invest.t+jj−6 −4 −2 0 2 4 6−0.100.1Invest.t, Pricet+jj−6 −4 −2 0 2 4 6−0.100.1Invest.t, Waget+jj−6 −4 −2 0 2 4 6−0.200.2Invest.t, Patiencet+jj  Smoothed Shock Correlations95% Simulated CIOutside 95% CINotes: Dots indicate the correlation coefficient between the smoothed investment shock and the specifiedlead or lag of another smoothed shock. Confidence intervals were computed from 100,000 simulated datasets under the null hypothesis that the EM is correctly specified.bands. If the EM were correctly specified, this would be an extreme result. Figure 2.5 presents asimulated probability mass function for the number of correlations between the investment shock andother shocks within 5 leads or lags that fall outside of the 95 percent confidence bands for the casewhen the EM is correctly specified. The median and mode of this distribution are both 3 and the mean3.5. The maximum number of correlations outside of the confidence bands in 100,000 simulationswas 14, a figure attained by only 2 (0.002 percent) of those simulations. 97 percent of the time, thenumber of such correlations was less than or equal to 7. In this context, 15 such correlations is clearlywell outside what could be considered reasonable if the EM were correctly specified. While I focuson the investment shock in Figure 2.4 because of the important role ascribed to it by JPT, the extreme33number and degree of correlations are not limited exclusively to this shock. For example, the neutraltechnology and patience shocks each had 16 of 71 points outside of the 95 percent confidence bands,while the monetary policy shock had 15.Figure 2.5: Proportion of correlations outside 95% CI0 1 2 3 4 5 6 7 8 9 10 11 12 13 1400. Simulated probability mass function (based on 100, 000 simulations of same numberof periods as actual data set) for number of correlations between investment shock and othershocks (within 5 leads or lags) that fall outside of the simulated 95 percent confidence intervalunder null hypothesis that EM is correctly specified.Note also that not only was the frequency of significant correlations in Figure 2.4 high, but thedegree to which the correlations exceeded the bounds was in some cases also extreme. Most notably,the monetary policy shock at a two-quarter lead, the neutral technology shock contemporaneouslyand at a one-quarter lead and lag, and the household patience shock at a one-quarter lead all exhibitcorrelations with the investment shock that exceeded 0.27 in absolute value, well outside the 95 percentconfidence bounds, which were all less than 0.15 in absolute value.The sample correlation evident in Figure 2.4 between the smoothed investment and patienceshocks is instructive. In particular, it was noted above that, in the estimated EM, investment wasfound to be driven by the former shock while consumption was driven by the latter, despite the factthat the EM was estimated using data that suggests that investment and consumption are correlated.This apparent puzzle is easily resolved if the two smoothed shocks themselves exhibit sample corre-lation, a property that Figure 2.4 clearly establishes. Intuitively, the estimation algorithm is “mixing”the investment and patience shocks together in a systematic way in order to reproduce the autoco-variance patterns in the data. On the other hand, because in the model the shocks are assumed to be34orthogonal, all moments derived from the EM—including those implicit in the computation of vari-ance decompositions—are obtained assuming no such mixing is taking place, leading to the apparentconflict between model and data.Next, as argued in Section 2.3, the correlations between the smoothed shocks are indicative ofmodel misspecification and, as a result, the naive estimates of the variances of the shocks will beoverstated. I turn now to obtaining bias-corrected estimates. In order to do so, a central issue that mustbe addressed is how to obtain estimates of the regression coefficients (i.e., the Θ’s in equation (2.22)).Given the relatively small sample size, using q leads and lags of all the smoothed shocks (plus theother contemporaneous shocks) as regressors very quickly uses up degrees of freedom.43 However,as Figure 2.4 suggests for the case of the investment shock, only a small subset of these potentialregressors will carry significant explanatory power, so that over-fitting is a serious concern. To addressthis issue, I use a simple selection rule, choosing only those regressors that meet the following criteria:(1) are within 5 leads or lags of the dependent variable, and (2) exhibit correlation coefficients with thedependent variable that are outside of a simulated 100 (1− α)% confidence interval. This rule has theadvantage of being both simple to apply and relatively flexible, in that the consequences of differentvalues of α can be explored.For a given value of α and this selection rule, I next compute the corrected estimate of the varianceof shock l, σ¯2l,α, as the unbiased estimator of the variance of the regression residuals from equation(2.22). That is, σ¯2l,α = (Tl,α − kl,α)−1∑ξˆ2l,t, where ξˆl,t is the fitted residual, Tl,α is the numberof observation periods, and kl,α is the number of regression parameters. Table 2.6 presents resultsfor several different values of α. For each value of α, the table reports the corrected estimate, σ¯2l,α,as a fraction of the naive estimate, σˆ2l . The smaller this fraction is, the more of the variance of thesmoothed shock we may attribute to misspecification. For comparison purposes, the bottom row ofthe table reports simulated 90 percent lower bounds for the corresponding statistic obtained assumingthe model is the true DGP.Several things emerge from the results in Table 2.6. First, with the exception of the wage mark-up shock, the corrected variance estimates are well below the simulated lower bounds. Given thetheoretical considerations of Section 2.3, this suggests that misspecification may be an important factorfor this model. Second, for the most conservative case (α = 0.001), the point estimates suggestthat nearly one-third of the variances of the smoothed neutral technology and investment shocks areattributable to misspecification, while over 90 percent of the time this statistic would be zero if themodel were the true DGP. Similarly, over one-fifth of the variance of the smoothed patience shock isattributable to misspecification for this level of α. Third, for α = 0.005, except again for the wagemark-up shock, the point estimates indicate that over one-fifth of the variance of every smoothed shockis attributable to misspecification, and over one-quarter when α = 0.02. Again, these figures are43 When all potential regressors within q leads and lags are used, the number of estimated parameters will be 6 + 14q,while the number of available observations is 197−2q. The number of degrees of freedom is thus 191−16q, which declinesvery rapidly with q.35Table 2.6: Corrected estimates (fraction of naive estimate)αShock 0.001 0.005 0.02Monetary policy 0.86 0.70 0.69Neutral technology 0.67 0.67 0.65Government spending 0.86 0.76 0.75Investment 0.68 0.68 0.67Price mark-up 0.87 0.75 0.69Wage mark-up 1.00 0.96 0.92Patience 0.79 0.77 0.6790% simulated lower bound 1.00 0.94 0.90Notes: Table entries show σ¯2l,α/σˆ2l , where σ¯2l,α is corrected estimated of variance of shock lusing selection rule for α (see text) and σˆ2l is naive estimate. σ¯2l,α is unbiased estimator ofvariance of regression residuals from equation (2.22). That is, σ¯2l,α = (Tl,α−kl,α)−1∑ ξˆ2l,t,where ξˆl,t is fitted residual, Tl,α is number of observation periods, and kl,α is number of re-gression parameters. Bottom row reports simulated 90 percent lower bounds for correspond-ing statistic when EM is correctly specified. Simulations are based on 10,000 draws of thesame size as data set.suggestive of an important degree of misspecification. It should also be noted again that the correctedvariance estimates are upper bounds for the true value. Thus, the actual degree of misspecificationmay indeed be significantly higher than these figures suggest.To check the implications for the hours variance decomposition, Table 2.7 reports the proportion ofthe variance of hours attributable to each of the shocks assuming the shock variances are the correctedestimates computed above.44 Again, 90 percent simulated lower bounds for the case where the EM iscorrectly specified are also reported (in parentheses). Column (1) reproduces the baseline naive BCFvariance decomposition from Table 2.5, while columns (2)-(4) report corrected decompositions of theBCF variance of hours for different values of α. The Table shows substantial declines in the estimatedimportance of the investment shock for the BCF variance of hours, with estimates decreasing by 19-20percentage points relative to the naive case.2.7 ConclusionThis paper makes several contributions to the literature. First, a novel framework is developed that canbe used to analyze the implications of misspecification on estimates of model shock variances. Usingthis framework, I show that if a DSGE model is correctly specified, then under basic stability condi-tions the time-series process for the smoothed shocks should be a vector white noise with diagonalcovariance matrix. Thus, if a realized process for the smoothed shocks does not possess this property,then the model must be misspecified.44 Note that only the numerator of this fraction is different from the baseline case; the denominator remains unchanged.36Table 2.7: Corrected BCF variance decomposition for hours(1) (2) (3) (4)Decomposition: Naive Corrected Corrected Correctedα: 0.001 0.005 0.02ShockMonetary policy 0.06 0.05 0.04 0.04(0.06) (0.06) (0.06)Neutral technology 0.11 0.07 0.07 0.07(0.11) (0.10) (0.09)Government spending 0.02 0.02 0.01 0.01(0.02) (0.02) (0.02)Investment 0.61 0.42 0.42 0.41(0.61) (0.58) (0.55)Price mark-up 0.06 0.05 0.04 0.04(0.06) (0.06) (0.05)Wage mark-up 0.05 0.05 0.05 0.05(0.05) (0.05) (0.05)Patience 0.08 0.07 0.07 0.06(0.09) (0.08) (0.08)Notes: Columns (2)-(4) show corrected BCF variance decompositions for hours. Naive de-composition is reproduced in column (1). Figures in parentheses are 90 percent simulatedlower bounds for case when EM is correctly specified.Next, I showed how, using this framework, one may orthogonally decompose a smoothed shockinto two components: its true value, and an additional component related entirely to misspecification.Since the true values of any two shocks are independent by construction, any non-zero covariancebetween two distinct smoothed shocks must then be entirely attributable to misspecification. Further,if the smoothed shocks do exhibit non-zero covariance, using the sample variance of a smoothed shockas an estimator of the variance of the true shock will lead to estimates that are biased upward. Tocorrect for this potential source of bias, I propose a fairly simple methodology that involves extractingthe component of a given smoothed shock that is unpredictable using other shocks.I apply this framework and methodology to a recent paper by Justiniano et al. (2010), and estimatethat at least one-third of the variance of the investment shock—the leading driver of business cyclefluctuations in their model—can be attributed to misspecification, and as a result, the estimated im-portance of the investment shock in generating business cycle variation in hours declines by around20 percentage points.37Chapter 3Reconciling Hayek’s and Keynes’ Viewsof Recessions3.1 IntroductionThere remains considerable debate regarding the causes and consequences of recessions. Two viewsthat are often presented as opposing, and which created controversy in the recent recession and itsaftermath, are those associated with the ideas of Hayek and Keynes.1 The Hayekian perspective isgenerally associated with viewing recessions as a necessary evil. According to this view, recessionsmainly reflect periods of liquidation resulting from past over-accumulation of capital goods. A situa-tion where the economy needs to liquidate such an excess can quite naturally give rise to a recession,but government spending aimed at stimulating activity, it is argued, is not warranted since it wouldmainly delay the needed adjustment process and thereby postpone the recovery. In contrast, the Key-nesian view suggests that recessions reflect periods of deficient aggregate demand where the economyis not effectively exploiting the gains from trade between individuals. According to this view, policyinterventions aimed at increasing investment and consumption are generally desirable, as they favorthe resumption of mutually beneficial trade between individuals.2In this paper we reexamine the liquidationist perspective of recessions in an environment withdecentralized markets, flexible prices and search frictions. In particular, we examine how the economyadjusts when it inherits from the past an excessive amount of capital goods, which could be in the formof houses, durable goods or productive capital. Our goal is not to focus on why the economy may haveover-accumulated in the past,3 but to ask how it reacts to such an over-accumulation once it is realized.1 In response to the large recession in the US and abroad in 2008-2009, a high-profile debate around these two views wasorganized by Reuters. See http://www.reuters.com/subjects/keynes-hayek. See also Wapshott (2012) for a popular accountof the Hayek-Keynes controversy.2 See Caballero and Hammour (2004) for an alternative view on the inefficiency of liquidations, based on the reductionof cumulative reallocation and inefficient restructuring in recessions.3 There are several reason why an economy may over-accumulate capital. For example, agents may have had overly38As suggested by Hayek, such a situation can readily lead to a recession as less economic activity isgenerally warranted when agents want to deplete past over-accumulation. However, because of theendogenous emergence of unemployment risk in our set-up, the size and duration of the recessionimplied by the need for liquidation is not socially optimal. In effect, the reduced gains from tradeinduced by the need for liquidation creates a multiplier process that leads to an excessive reductionin activity. Although prices are free to adjust, the liquidation creates a period of deficient aggregatedemand where economic activity is too low because people spend too cautiously due to increasedunemployment risk. In this sense, we argue that liquidation and deficient aggregate demand should notbe viewed as alternative theories of recessions but instead should be seen as complements, where pastover-accumulation may be a key driver of periods of deficient aggregate demand. This perspectivealso makes salient the trade-offs faced by policy. In particular, a policy-maker in our environmentfaces an unpleasant trade-off between the prescriptions emphasized by Keynes and Hayek. On theone hand, a policy-maker would want to stimulate economic activity during a liquidation-inducedrecession because precautionary savings is excessively high. On the other hand, the policy-maker alsoneeds to recognize that intervention will likely postpone recovery, since it slows down the neededdepletion of excess capital. The model offers a simple framework where both of these forces arepresent and can be compared.On a more general note, one of the contributions of this paper is to show why an economy canfunction quite efficiently in growth periods when it is far from its steady state, while simultaneouslyfunctioning particularly inefficiently when it is going through a liquidation phase near its steady state.When the economy is far from its steady-state level of capital, demand for capital is very strong andunemployment risk is therefore minimal. In contrast, when there is excessive capital, we show thatreduced labor demand shows up at least in part as increased unemployment even if workers and firmsbargain pair-wise efficiently on wages and hours-worked. The increased unemployment risk thencauses households to increase precautionary savings, which in turn amplifies the initial fall in outputand employment. The result is an over-reaction to the initial impetus induced by a need to liquidatecapital.4 As a presentation device, we show how this process can be represented on a diagram some-what similar to a Keynesian cross, but where the micro-foundation and many comparative statics differsubstantially from the sticky-price interpretation commonly used to discuss multipliers. Moreover, byclarifying why this process does not depend on sticky prices, our analysis suggest that monetary policymay be of limited help in addressing the difficulties associated with a period of liquidation.One potential criticism of a pure liquidationist view of recessions is that, if markets functionedefficiently, such periods should not be socially painful. In particular, if economic agents interact inoptimistic expectations about future expected economic growth that did not materialize, as in Beaudry and Portier (2004),or it could have been the case that credit supply was unduly subsidized either through explicit policy, as argued in Mian andSufi (2010) and Mian et al. (2010), or as a by-product of monetary policy, as studied by Bordo and Landon-Lane (2013).4 It is now common in the macroeconomic literature to summarize the functioning of a model by indicating where andhow it creates distortions or wedges, as exemplified by Chari et al. (2007). Accordingly, one way to view the working ofour model economy is as generating an endogenous labor market wedge driven by unemployment risk, where the size of thewedge reacts to the extent to which inherited capital is above or below the steady state.39perfect markets and realize they have over-accumulated in the past, this should lead them to enjoy atype of holiday paid for by their past excessive work. Looking backwards in such a situation, agentsmay resent the whole episode, but looking forward after a period of over-accumulation, they shouldnonetheless feel content to enjoy the proceeds of the past excessive work, even if it is associated with arecession. In contrast, in our environment we will show that liquidation periods are generally sociallypainful because of the multiplier process induced by precautionary savings and unemployment risk.In effect, we will show that everyone in our model economy can be worse off when they inherit toomany capital goods from the past. This type of effect, whereby abundance creates scarcity, may appearquite counter-intuitive at first pass. To make as clear as possible the mechanism that can cause welfareto be reduced by such abundance, much of our analysis will focus on the case where the inheritedcapital takes the form of a good that directly contributes to utility, such as houses or durable goods.In this situation we will show why inheriting more houses or durables can make everyone worse off.However, as we shall show, this result is a local result that is most likely to be present around aneconomy’s steady state. In contrast, if we were to destroy all capital goods in our model economy,this would always reduce welfare, as the direct effects on utility would out-weight the inefficienciesinduced by unemployment risk. Accordingly, our model has the characteristic that behavior can bequite different when it inherits a large or small amount of capital from the past.The structure of our model builds on the literature related to search models of decentralized trad-ing. In particular, we share with Lucas (1990) and Shi (1998) a model in which households arecomposed of agents that act in different markets without full coordination. Moreover, as in Lagos andWright (2005) and Rocheteau and Wright (2005), we exploit alternating decentralized and centralizedmarkets to allow for a simple characterization of the equilibrium. However, unlike those papers, we donot have money in our setup. The paper also shares key features with the long tradition of macro mod-els emphasizing strategic complementarities, aggregate demand externalities and multipliers, such asDiamond (1982) and Cooper and John (1988), but we do not emphasize multiple equilibrium. Insteadwe focus on situations where the equilibrium remains unique, which allows standard comparativestatics exercises to be conducted without needing to worry about equilibrium-selection issues. Themultiplier process derived in the paper therefore shares similarities with that found in the recent litera-ture with strategic complementarities such as Angeletos and La’O (2013), in the sense that it amplifiesdemand shocks. However, the underlying mechanism in this paper is very different, operating throughunemployment risk rather than through direct demand complementarities as in Angeletos and La’O(2013).Unemployment risk and its effects on consumption decisions is at the core of our model. The em-pirical relevance of precautionary saving related to unemployment risk has been documented by many,starting with Carroll (1992). For example, Carroll and Dunn (1997) have shown that expectations ofunemployment are robustly and negatively correlated with every measure of consumer expenditure(non-durable goods, durable goods and home sales). Carroll et al. (2012) confirm this finding andshow why business cycle fluctuations may be driven to a large extent by changes in unemployment40uncertainty. Alan et al. (2012) use U.K. micro data to show that increases in saving rates in recessionsappear largely driven by uncertainty related to unemployment.5 There are also recent theoretical pa-pers that emphasized how unemployment risk and precautionary savings can amplify shocks and causebusiness cycle fluctuations. These papers are the closest to our work. In particular, our model structureis closely related to that presented in Guerrieri and Lorenzoni (2009). However, their model empha-sizes why the economy may exhibit excessive responses to productivity shocks, while our frameworkoffers a mechanism that amplifies demand-type shocks. Our paper also shares many features withHeathcote and Perri (2012), who develop a model in which unemployment risk and wealth impactconsumption decisions and precautionary savings. Wealth matters in their setup because of financialfrictions that make credit more expensive for wealth-poor agents. They obtain a strong form of de-mand externality that gives rise to multiple equilibria and, accordingly, they emphasize self-fulfillingcycles as the important source of fluctuations.6 Finally, the work by Ravn and Sterk (2012) empha-sizes as we do how unemployment risk and precautionary savings can amplify demand shocks, buttheir mechanism differs substantially from ours since it relies on sticky nominal prices.While the main mechanism in our model has many precursors in the literature, we believe that oursetup illustrates most clearly (i) how unemployment risk gives rise to a multiplier process for demandshocks even in the absence of price stickiness or increasing returns, (ii) how this multiplier processcan be ignited by periods of liquidation, and (iii) how fiscal policy can and cannot be used to counterthe process.The remaining sections of the paper are structured as follows. In Section 3.2, we present a staticmodel where agents inherit from the past different levels of capital goods, and we describe how andwhy high values of inherited capital can lead to poor economic outcomes. The static setup allows fora clear exposition of the nature of the demand externality that arises in our setting with decentralizedtrade. We focus on the case where the inherited capital is in the form of a good which directly increasesutility so as to make clear how more goods can reduce welfare. In Section 3.3, we discuss a set ofextensions, including a discussion of the case where the inherited capital takes the form of a productivegood. In Section 3.4, we extend the model to an infinite-period dynamic setting. We take particularcare in contrasting the behavior of the economy when it is close to and far from its steady state. Finally,in Section 3.5, we discuss the trade-offs faced by a policy-maker when inheriting an excessive amountof capital from the past, while Section 3.6 concludes.5 Using these empirical insights, Challe and Ragot (2013) have recently proposed a tractable quantitative model in whichuninsurable unemployment risk is the source of wealth heterogeneity.6 The existence of aggregate demand externalities and self-fulfilling expectations is also present in the work of Farmer(2010) and in the work of Chamley (2014). In a model with search in both labor and goods markets, Kaplan and Menzio(2013) also obtain multiple equilibria, as employed workers have more income to spend and less time to shop for low prices.As already underlined, and contrarily to those studies, our analysis is restricted to configurations in which the equilibriumis unique.413.2 Static modelIn this section, we present a very stripped-down static model in order to illustrate why an economy mayfunction particularly inefficiently when it inherits a large stock of capital from the past. In particular,we will want to make clear why agents in an economy can be worse off when inherited capital goodsare too high. For the mechanism to be as transparent as possible, we focus mainly on the case wherethe inherited capital produces services which directly enter agents’ utility functions. Accordingly, thistype of capital can be considered as representing houses or other durable consumer goods. In a latersection, we will discuss how the analysis carries over to the case of productive capital.In our model, trades are decentralized, and there are two imperfections which cause unemploy-ment risk to emerge and generate precautionary savings behavior. First, there will be a matchingfriction in the spirit of Diamond-Mortensen-Pissarides, which will create the possibility that a house-hold may not find employment when looking for a job. Second, there will be adverse selection in theinsurance market that will limit the pooling of this risk. Since the adverse selection problem can beanalyzed separately, we will begin the presentation by simply assuming that unemployment insuranceis not available. Later we will introduce the adverse selection problem which rationalizes this miss-ing market, and show that all main results are maintained. The key exogenous variable in the staticmodel will be a stock of consumer durables that households inherit from the past. Our goal is to showwhy and when high values of this stock can cause the economy to function inefficiently and possiblyeven cause a decrease in welfare. We will also explore the role of governement spending in affectingeconomic activity in our setup.3.2.1 SetupConsider an environment populated by a mass L of households indexed by j. In this economy thereare two sub-periods. In the first sub-period, households buy good 1, which we will call clothes, and tryto find employment in the clothing sector. We refer to this good as clothes since in the dynamic versionof the model it will represent a partially durable good. The good produced in the second sub-period,good 2, will be referred to as household services since it will have no durability. As there is no moneyin this economy, when the household buys clothes its bank account is debited, and when (and if) itreceives employment income its bank account is credited. Then, in the second sub-period, householdsbalance their books by repaying any outstanding debts or receiving a payment for any surplus. Thesepayments are made in terms of good 2, which is also the numeraire in this economy.7Preferences for the first sub-period are represented byU(cj)− ν(`j)where c represents consumption of clothes and ` is the labor supplied by households in the production7 We remain agnostic about the precise details of how good 2 is produced for the time being. One possible interpretationis discussed in the following sub-section.42of clothes. The function U(·) is assumed to be increasing in c and strictly concave with limc→∞ U ′ ≤0 and U ′′′ > 0. The dis-utility of work function ν(·) is assumed to be increasing and convex in `, withν(0) = 0. The agents are initially endowed with Xj units of clothes, which they can either consumeor trade. We assume symmetric endowments, so that Xj = X ∀j.8 In the dynamic version of themodel, X will represent the stock of durable goods and will be endogenous.Trade in clothing will be subject to a coordination problem because of frictions in the labor market.At the beginning of the first sub-period, the household splits up responsibilities between two members.The first member, called the buyer, goes to the clothes market to make purchases. The second membersearches for employment opportunities in the labor market. The market for clothes functions in aWalrasian fashion, with both buyers and firms that sell clothes taking prices as given. The marketfor labor in this first sub-period is subject to a matching friction, with sellers of labor searching foremployers and employers searching for labor. The important information assumption is that buyers donot know, when choosing their consumption of clothes, whether the worker member of the householdhas secured a match. This assumption implies that buyers will worry about unemployment risk whenmaking purchases of clothes.There is a large set of potential clothes firms in the economy who can decide to search for work-ers in view of supplying clothes to the market. Each firm can hire one worker and has access to adecreasing-returns-to-scale production function θF (`), where ` is the number of hours worked for thefirm and θ > 0 is a technology shift factor. Production also requires a fixed cost θΦ in terms of theoutput good, so that the net production of a firm hiring ` hours of labor is θ [F (`)− Φ]. For now, wewill normalize θ to 1, and will reintroduce θ in its general form when we want to talk about the effectsof technological change and balanced growth. We will also assume throughout that Ω(`) ≡ F ′(`)`is increasing in `.9 Moreover, we will assume that Φ is sufficiently small such that there exists an`? > 0 satisfying F (`?)−F ′ (`?) `? = Φ. These restrictions on the production technology are alwayssatisfied if, for example, F (`) = `α, with 0 < α < 1.Firms search for workers and, upon finding a worker, they jointly decide on the number of hoursworked and on the wage to be paid. The fixed cost Φ is paid before firms can look for workers. Upona match, the determination of the wage and hours-worked within a firm is done efficiently thougha competitive bargaining process. In effect, upon a match, one can view a Walrasian auctioneer ascalling out a wage w that equilibrates the demand for and supply of labor among the two parties in thematch. Assuming such a process for wage and employment determination has the feature of limitingwithin-pair distortions that could muddle the understanding of the main mechanisms of the model. InAppendix B.2 we show that the main results of the paper are robust to alternative bargaining protocols.Note that we have deliberately chosen a random-matching – rather than a directed-search – framework8In what follows, we will drop the j index except where doing so may cause confusion.9 Because we assume free-entry for clothes firms, the quantity θΩ(`) will equal net output of clothes (after subtractingfirms’ fixed costs) by a single employed worker. The assumption that this quantity is increasing in ` is satisfied, for example,if F is a CES combination of labor and some other input in fixed supply, with an elasticity of substitution between theseinputs of at least 1, which nests the case where F is Cobb-Douglas.43as we want to illustrate how inefficiencies in the labor market can interact with the liquidation processto create periods of deficient aggregate demand. Given the wage, the demand for labor from the firmis described by the marginal productivity conditionpF ′(`) = wwhere p is the relative price of clothes in terms of the non-durable good produced in the second sub-period.10 The supply of labor is chosen optimally by the worker in a manner to be derived shortly.Letting N represent the number of firms who decide to search for workers, the number of matchesis then given by the constant-returns-to-scale matching functionM(N,L), withM(N,L) ≤ min{N,L}.The equilibrium condition for the clothes market is given byL · (c−X) = M(N,L)F (`)−NΦwhere the left-hand side is total purchases of new clothes and the right-hand side is the total availablesupply after subtracting search costs.Firms will enter the market up to the point where expected profits are zero. The zero-profit condi-tion can be written as11MN [pF (`)− w`] =MN [pF (`)− pF′(`)`] = pΦAt the end of the first sub-period, household j’s net asset position aj , expressed in units of good2, is given by w`j − p(cj − X).12 We model the second sub-period so that it is costly to arrive inthat sub-period with debt. For now, we can simply denote the value of entering the second sub-periodwith assets aj by V (aj), where we assume that V (·) is increasing, with V ′(a1) > V ′(a2) whenevera1 < 0 < a2; that is, we are assuming that the marginal value of a unit of assets is greater if one isin debt than if one is in a creditor position. In the following sub-section we specify preferences and amarket structure for the second sub-period that rationalizes this V (·) function.Taking the function V (a) as given, we can specify the household’s consumption decision as wellas his labor-supply decision conditional on a match. The buyer’s problem in household j is given bymaxcjU (cj) + µV (w`j − p (cj −X)) + (1− µ)V (−p (cj −X))where µ is the probability that a worker finds a job and is given by µ ≡ M(N,L)/L. From thisexpression, we can see that the consumption decision is made in the presence of unemployment risk.10 As will become clear, p can be given an interpretation as an interest rate.11 We assume that searching firms pool their ex-post profits and losses so that they make exactly zero profits in equilibrium,regardless of whether they match.12 Note that it is in general possible to have cj − X < 0, i.e., a household may choose to sell off a portion of itsendowment. However, because all households are symmetric in this environment and since aggregate production is subjectto a non-negativity constraint, in equilibrium we will always have cj −X ≥ 0.44The worker’s problem in household j when matched, taking w as given, can be expressed aschoosing a level of hours to supply in the first sub-period so as to solvemax`j−ν(`j) + V (w`j − p (cj −X))3.2.2 Deriving the value function V (a)V (a) represents the value function associated with entering the second sub-period with a net assetposition a. In this subsection, we derive such a value function by specifying primitives in terms ofpreferences, technology and market organization. We choose to model this sub-period in such a waythat if there were no friction in the first sub-period, there would be no trade between agents in thesecond sub-period. For this reason let us call “services” the good produced in the second periodhousehold, with preferences given byU˜(c˜)− ν˜(˜`)where c˜ is consumption of these services, U˜(·) is increasing and strictly concave in c˜, ˜` is the laborused to produced household services, and ν˜(·) is increasing and convex in ˜`.To ensure that a unit of net assets is more valuable when in debt than when in surplus, let us assumethat households in the second sub-period can produce services for their own consumption, using oneunit of labor to produce θ˜ unit of services. However, if a household in the second sub-period has toproduce market services – that is, services that can be sold to others in order to satisfy debt – then toproduce θ˜ units of market services requires them to supply 1 + τ units of labor, τ > 0. To simplifynotation, we can set θ˜ = 1 for now and return to the more general formulation when talking abouteffects of technological change. The continuation value function V (a) can accordingly be defined asV (a) = maxc˜,˜`U˜(c˜)− ν˜(˜`)subject toc˜ = ˜`+ a if a ≥ 0andc˜ = ˜`+ a(1 + τ) if a < 0It is easy to verify that V (a) is increasing in assets and concave. If ν˜(˜`) is strictly convex, thenV (a) will be strictly concave, regardless of the value of τ , with the key property that V ′(a1) > V ′(a2)if a1 < 0 < a2; that is, the marginal value of an increase in assets is greater if one is in debt thanif one is in surplus.13 In the case where ν˜(`) is linear, then V (a) will be piecewise linear and will13 To avoid backward-bending supply curves, we will also assume that ν˜(·) and U˜(·) are such that V ′′′(a) ≥ 0. Thisassumption is sufficient but not necessary for later results. Note that a sufficient condition for V ′′′(a) ≥ 0 is that bothU˜ ′′′(·) ≥ 0 and ν˜′′′(·) < 0.45not be differentiable at zero. Nonetheless, it will maintain the key property that V ′(a1) > V ′(a2) ifa1 < 0 < a2. We will mainly work with this case, and in particular, will assume that ν˜(˜`) = v · ˜`,which implies that V (a) is piecewise linear with a kink at zero.3.2.3 Equilibrium in the first sub-periodGiven the function V (a), a symmetric equilibrium for the first sub-period is represented by five ob-jects: two relative prices (the price of clothing p and the wage rate w), two quantities (consumptionof clothes by each household c and the amount worked in each match `), and a number N of activefirms, such that1. c solves the buyer’s problem taking µ, p, w and ` as given.2. The labor supply ` solves the worker’s problem conditional on a match, taking p, w and c asgiven.3. The demand for labor ` maximizes the firm’s profits given a match, taking p and w as given.4. The goods market clears; that is, L · (c−X) = M(N,L)F (`)−NΦ .5. Firms’ entry decisions ensure zero profits.The equilibrium in the first sub-period can therefore be represented by the following system offive equations:U ′(c) = p{M(N,L)L V′ (w`− p (c−X))+[1− M(N,L)L]V ′ (−p (c−X))} (3.1)ν ′(`) = V ′ (w`− p (c−X))w (3.2)pF ′(`) = w (3.3)M(N,L)F (`) = L(c−X) +NΦ (3.4)M(N,L)[pF (`)− w`] = NpΦ (3.5)In the above system,14 equations (3.1) and (3.2) represent the first-order conditions for the household’schoice of consumption and supply of labor. Equations (3.3) and (3.5) represent a firm’s labor demandcondition and its entry decision. Finally, (3.4) is the goods market clearing condition.14 To ensure that an employed worker’s optimal choice of labor is strictly positive, we assume that limc→0 U ′(c) >lim`→0ν′(`)F ′(`) .46At this level of generality it is difficult to derive many results. Nonetheless, we can combine(3.1), (3.2) and (3.3) to obtain the following important expression regarding a characteristic of theequilibrium,ν ′(`)U ′(c){1 + (1− µ)[ V ′ (−p (c−X))V ′ (w`− p (c−X)) − 1]}= F ′(`) (3.6)From equation (3.6), we see that as long as µ < 1, the marginal rate of substitution between leisureand consumption will not be equal to the marginal productivity of work; that is, the labor market willexhibit a wedge given by(1− µ)[ V ′ (−p (c−X))V ′ (w`− p (c−X)) − 1]In fact, in this environment, the possibility of being unemployed leads to precautionary savings, whichin turn causes the marginal rate of substitution between leisure and consumption to be low relative tothe marginal productivity of labor. As we will see, changes in X will cause this wedge to vary, whichwill cause a feedback effect on economic activity. Obviously, in this environment there would be adesire for agents to share the risk of being unemployed, which could reduce or even eliminate thewedge. As noted earlier, the reason that this type of insurance may be limited is the presence ofadverse selection, an issue to which we will return.Our main goal now is to explore the effects of changes in X on equilibrium outcomes. In partic-ular, we are interested in clarifying why and when an increase in X can actually lead to a reductionin consumption and/or welfare. The reason we are interested in this comparative static is that we areinterested in knowing why periods of liquidations – that is, periods where agents inherit excessivelevels of durable goods from the past – may be socially painful.To clarify the analysis, we will make two simplifying assumptions. First, we will assume that thematching function takes the form M(N,L) = min{N,L}; that is, matches are determined by theshort side of the market. This assumption creates a clear and useful dichotomy, with the economycharacterized as being either in an unemployment regime if L > N or in a full-employment regimeif N > L. We will also assume that V (a) is piece-wise linear, with V ′(a) = v · a if a ≥ 0 andV ′(a) = v · a · (1 + τ) if a < 0, with τ > 0 and v > 0. This form of the V (·) function corresponds tothe case discussed in subsection 3.2.2 where the dis-utility of work in the second sub-period is linear.The important element here is τ . In effect, 1 + τ represents the ratio of the marginal value of an extraunit of assets when one is in debt relative to its value when one is in surplus. A value of τ > 0 canbe justified in many ways, one of which is presented in subsection 3.2.2. Alternatively, τ > 0 couldreflect a financial friction related to the cost of borrowing versus savings.47Under these two functional-form assumptions, the equilibrium conditions can be reduced to thefollowing:U ′(c) = ν′(`)F ′(`)(1 + τ − min{N,L}L τ)(3.7)min{N,L}L =c−XF ′(`)` (3.8)min{N,L}N [F (`)− F′(`)`] = Φ (3.9)w = ν′(`)v (3.10)p = ν′(`)vF ′(`) (3.11)This system of equations now has the feature of being block-recursive. Equations (3.7), (3.8) and(3.9) can be solved for c, ` and N , with equations (3.10) and (3.11) then providing the wage andthe price. From equations (3.7) and (3.8), one can immediately notice the complementarity that canarise between consumption and employment in the case where N < L (the unemployment regime).From (3.7) we see that, if N < L, agents will tend to increase their consumption if they believethere are many firms looking for workers (N expected to be large). Then from equation (3.8) wesee that more firms will be looking to hire workers if they believe that consumption will be high. Sogreater consumption favors greater employment, which in turn reinforces consumption. This feedbackeffect arises as the result of consumption and employment playing the role of strategic complements.Workers demand higher consumption when they believe that many firms are searching to hire, as theyview a high N as reducing their probability of entering the second sub-period in debt. It is importantto notice that this multiplier argument is implicitly taking `, the number of hours worked by agents,as given. But, in the case where the economy is characterized by unemployment, this is precisely theright equilibrium conjecture. In particular, from (3.9) we can see that if the economy is in a state ofunemployment, then ` is simply given by `?, the solution to the equation F (`?)− F ′(`?)`? = Φ, andis therefore locally independent of X or c. Hence, in the presence of unemployment, consumptionand firm hiring will act as strategic complements. As is common in the case of strategic complements,multiple equilibria can arise. This possibility is stated in Proposition 3.1.Proposition 3.1. There exists a τ¯ > 015 such that (a) if τ < τ¯ , then there exists a unique equilibriumfor any value of X; and (b) if τ > τ¯ , then there exists a range of X for which there are multipleequilibria.The proofs of all propositions are presented in Appendix B.1.15 τ¯ = −U ′′(U ′−1(ν′(`?)F ′(`?)))F ′(`?)[F (`?)−Φ]ν′(`?) .48While situations with multiple equilibria may be interesting, in this paper we will mainly focuson the case where the equilibrium is unique, as we believe this is more likely to be the empiricallyrelevant case. Accordingly, Proposition 3.1 tells us that our setup will have a unique equilibrium if themarginal cost of debt is not too large. For the remainder of this section, we will assume that τ < τ¯ .Proposition 3.2 focuses on this case and provides a first step in the characterization of the equilibrium.Proposition 3.2. When τ < τ¯ , there exists an X? such that if X ≤ X? then the equilibrium is char-acterized by full employment, while if X > X? it is characterized by unemployment. Furthermore,there exists an X?? > X? such that if X > X??, then employment is zero and agents simply consumetheir endowment (i.e., c = X).16The content of Proposition 3.2 is very intuitive as it simply states that if agents have a low en-dowment of the consumption good, then there are substantial gains from trade, and that will favor fullemployment. In contrast, if the endowment is very high, this will reduce the demand for the goodsufficiently as to create unemployment. Finally, if X is extremely high, all trade among agents willstop as people are content to simply consume their endowment.Proposition 3.2 can also be used to provide insight regarding the relationship between the laborwedge in this economy and the inherited endowment of X , where the labor wedge is defined as[U ′(c)− ν′(`)F ′(`)]/ ν′(`)F ′(`) . In Figure 3.1, we plot the labor wedge as a function of X . As can be seen, forX < X?,17 the labor wedge is zero, while for X ∈ [X?, X??], the labor wedge rises monotonically,reaching a peak at the point X?? where trade collapses. Then, for X > X??, we enter the no-employment zone and the wedge declines gradually until it reaches zero anew at a point where theno-employment outcome is socially optimal. This figure nicely illustrates that the degree of distortionin this economy varies with X , with low values of X being associated with a more efficient economy,while higher values of X generate a positive and growing wedge as long as trade remains present.From this observation, we can see how a higher inherited capital stock can increase inefficiency.Proposition 3.3 complements Proposition 3.2 by indicating how consumption is determined in eachregime.Proposition 3.3. When the economy exhibits unemployment (X?? > X > X?), the level of consump-tion is given as the unique solution toc = U ′−1( ν ′(`?)F ′(`?)[1 + τ − c−XF ′(`?)`? τ])When the economy exhibits full employment (X ≤ X?), consumption is the unique solution toc = U ′−1( ν ′(Ω−1(c−X))F ′(Ω−1(c−X)))16 X? = U ′−1(ν′(`?)F ′(`?))− F ′(`?)`? and X?? = U ′−1(ν′(`?)F ′(`?) (1 + τ)).17 We assume here and throughout the remainder of this paper that U ′ (F (`?)− Φ) > ν′(`?)F ′(`?) , so that X? > 0.49Figure 3.1: Labor wedge as function of X .0 0.2 0.4 0.6 0.8⋆ X⋆⋆XWedgeNote: Labor wedge is defined as[U ′(c)− ν′(`)F ′(`)]/ ν′(`)F ′(`) . Example is constructed assuming thefunctional forms U(c) = log(c), ν(`) = ν`1+ω1+ω and F (`) = A`α, with parameters ω = 1,ν = 0.5, α = 0.67, A = 1, Φ = 0.35 and τ = 0.3.Finally, when X ≥ X??, consumption is given by c = X .Given the above propositions, we are now in a position to examine an issue of main interest, whichis how an increase in X affects consumption. In particular, we want to ask whether an increase in X ,which acts as an increase in the supply of goods, can lead to a decrease in the actual consumption ofgoods. Proposition 3.4 addresses this issue.Proposition 3.4. If X?? > X > X?, then c is decreasing in X . If X ≤ X? or X > X??, then c isincreasing in X .The content of Proposition 3.4 is illustrated in Figure 3.2. Proposition 3.4 indicates that, start-ing at X = 0, consumption will continuously increase in X as long as X is compatible with fullemployment. Then, when X is greater than X?, the economy enters the unemployment regime andconsumption starts to decrease as X is increased. Finally, beyond X?? trade collapses and consump-tion becomes equal to X and hence it increases with X . The reason that consumption decreases witha higher supply of X in the unemployment region is precisely because of the multiplier process de-scribed earlier. In this region, an increase in X leads to a fall in expenditures on new consumption,where we define expenditures as e ≡ c − X . The decrease in expenditures reduces the demand forgoods as perceived by firms. Less firms then search for workers, which increases the risk of un-employment. The increase in unemployment risk leads households to cut their expenditures further,which further amplifies the initial effect of an increase in X on expenditures. It is because of this50Figure 3.2: Consumption as function of X .0 0.2 0.4 0.6 0.8 10.80.850.90.9511.⋆ X⋆⋆XcNote: Example is constructed assuming the functional forms U(c) = log(c), ν(`) = ν`1+ω1+ω andF (`) = A`α, with parameters ω = 1, ν = 0.5, α = 0.67, A = 1, Φ = 0.35 and τ = 0.3.type of multiplier process that an increase in the supply of the good can lead to a decrease in its totalconsumption (X + e). Note that such a negative effect does not happen when the economy is at fullemployment, as an increase in X does not cause an increase in precautionary savings, which is thekey mechanism at play causing consumption to fall.The link noted above between household j’s expenditure, which we can denote by ej ≡ cj −Xj ,and its expectation about the expenditures by other agents in the economy, which can denote by e, canbe captured by rewriting the relations determining ej implied by the elements of Proposition 3.3 asej = Z(e)−X (3.12)withZ(e) ≡ U ′−1 (Q(e)) (3.13)andQ (e) ≡ν′(`?)F ′(`?)(1 + τ − τ ee?)if 0 < e < e?ν′(Ω−1(e))F ′(Ω−1(e)) if e ≥ e?(3.14)Here, e? ≡ Ω(`?) is the level of output (net of firms’ search costs) that would be produced if allworkers were employed, with hours per employed worker equal to `?. In equilibrium we have theadditional requirement that ej = e for all j.The equilibrium determination of e is illustrated in Figure 3.3, which somewhat resembles a Key-nesian cross. In the figure, we plot the function ej = Z(e) − X for two values of X: a first value51Figure 3.3: Equilibrium determination0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.900.⋆ee j(e)45-deg.High XLow XNote: Example is constructed assuming the functional forms U(c) = log(c), ν(`) = ν`1+ω1+ω andF (`) = A`α, with parameters ω = 1, ν = 0.5, α = 0.67, A = 1, Φ = 0.35 and τ = 0.3. Valuesof X used were X = 0 for the full-employment equilibrium and X = 0.7 for the unemploymentequilibrium.of X which places the economy in an unemployment regime, and a second value of X which placesthe economy in a full-employment regime. An equilibrium in this figure corresponds to the pointwhere the function ej = Z(e) − X crosses the 45◦ line. Note that changes in X simply move theej = Z(e)−X curve vertically.There are several features to note about Figure 3.3. First, in the case whereX ∈ (X?, X??), so thatthe equilibrium of the economy is in an unemployment regime with positive trade (i.e., 0 < e < e?),the diagram is similar to a Keynesian cross. We can see graphically how an increase in X by oneunit shifts down the Z(e)−X curve and, since the slope of Z(e)−X is positive and less than one, amultiplier process kicks in which causes e to fall by more than one. Because of this multiplier process,total consumption of clothes, which is equal to e+X , decreases, which is the essence of the first part ofProposition 3.4. Second, when X < X?, so that the economy is in a full-employment regime (i.e., theequilibrium is such that e > e?), the diagram is different from the Keynesian cross. The most notabledifference is the negative slope of the function Z(e) −X for values of e > e?. This reflects the factthat unemployment risk is not present in this regime. In fact, when X is sufficiently small so that theeconomy is in the full-employment regime, an increase in X by one unit leads to a decrease in e thatis less than one, compared to a decrease of greater than one as exhibited in the unemployment regime.Here, expenditure by others actually plays the role of a strategic substitute with one’s own expenditure– as opposed to playing the role of a strategic complement as is the case in the unemployment regime– through its effects on real wages and prices. Accordingly, in this region, an increase in X leads to52an increase in total consumption of clothes. Another more subtle difference with the Keynesian crossis in how the intercept of Z(e)−X is determined. The intercept is given by U ′−1( ν′(`?)F ′(`?)(1+ τ))−X .The X term in the intercept can be interpreted as capturing a pure aggregate-demand effect, wherebyhigher values of X reduce aggregate demand. However, the remaining term, U ′−1( ν′(`?)F ′(`?)(1 + τ)),reflects technology and preferences. In particular, we can generalize this term by re-introducing thetechnology parameter θ, in which case the intercept becomes U ′−1( ν′(`?)θF ′(`?)(1 + τ)).18 In this case,we see that an improvement in technology shifts up the intercept, and will lead to an increase inexpenditures. This feature of the Z(e)−X curve illustrates its equilibrium nature, which incorporatesboth demand and supply effects, as opposed to a Keynesian cross that only reflects demand effects.3.2.4 Is there deficient demand in the unemployment regime?In the case where X is large enough for the economy to be in the unemployment regime (X? <X < X??), we have already noted that the marginal rate of substitution between consumption andleisure is greater than the marginal product of labor, with this distortion increasing the larger is X .In this sense, the economy is clearly working inefficiently in the unemployment regime. In this sec-tion, we want to examine whether this regime can also be appropriately characterized as sufferingfrom deficient aggregate demand. In particular, suppose the structure of markets were not changedand X? < X < X??. Now suppose that all households deviated from their equilibrium strategiesby increasing slightly their demand for consumption goods. If in this case the expected utility of thehousehold would be increased, then it appears reasonable to characterize the situation as one of defi-cient demand. Using this definition, Proposition 3.5 indicates that the unemployment regime of ourmodel is in fact characterized by deficient demand.Proposition 3.5. When the economy is in the unemployment regime (X? < X < X??), a coordinatedincrease by households in the purchase of the first sub-period consumption good increases the expectedutility of all households.Proposition 3.5 can alternatively be interpreting as confirming that the consumption choices ofindividual households play the role of strategic complements in the unemployment regime.3.2.5 Effects of changes in X on welfareWe have shown that when X is high enough, then the economy will be in the unemployment regime,where a local increase in X causes consumption to fall. We now want to ask how expected welfare isaffected in these cases, where expected welfare is defined as U(c)+µ [−ν(`) + V (w`− p(c−X))]+(1−µ)V (−p(c−X)). In particular, we want to ask whether welfare can decrease when the economyis endowed with more goods. Proposition 3.6 answers this question in the affirmative. Proposition 3.618 Recall that an increase in θ is associated with a proportional change in the search cost, so that `? remains unchanged.53actually goes a step further and indicates two sufficient conditions for there to exist a range of X inthe unemployment regime where an increase in X leads to a fall in welfare.Proposition 3.6. An increase in X can lead to a fall in expected welfare. In particular, if either (i) τis close enough to τ¯ or (ii) the average cost of work ν(`?)`? is low enough relative to the marginal costof work ν ′(`?), then there is always a range of X ∈ [X?, X??] such that an increase in X leads to adecrease in expected welfare.Proposition 3.6 provides a step toward answering whether more goods can make everyone worseoff. In effect, the proposition indicates that the economy can function in a very perverse fashionwhen households have inherited many goods. We saw from Proposition 3.4 that an increase in Xalways leads to a decrease in consumption when we are in the unemployment regime. In comparison,Proposition 3.6 is weaker as it only indicates the possibility of a fall in welfare in the unemploymentregion when X rises. In response to a rise in X in the unemployment regime, there are three distinctchannels through which expected welfare is affected. First, as discussed above, consumption falls,which tends to directly decrease welfare. Second, this fall in consumption is associated with a fall inthe probability of being employed. It can be verified that the net benefit of being employed is strictlypositive, so that this second effect also tends to decrease welfare. Finally, a rise inX means that a givenquantity of consumption can be obtained with a lower level of expenditure, which increases assets forthe employed and decreases debt for the unemployed, and therefore tends to increase welfare. Whetherthis final effect is outweighed by the first two depends on the factors discussed in Proposition 3.6.As noted in Proposition 3.6, the effects of an increase in X on welfare depends, among otherthings, on the difference between the marginal utility cost of work and the average utility cost of work.This distinction is relevant because an important component of the net benefit of being employed is theutility value of wages earned, net of the value of foregone leisure.19 In the current model, the averageutility cost of work can be arbitrarily small relative to its marginal cost. When the average cost ofwork is low, the net benefit of being employed is large, and therefore a rise in the unemployment ratecaused by a rise in X will have a larger negative effect on welfare (i.e., the second channel discussedabove becomes more important). Hence, in our model, when employment is not perceived as verypainful, and we are in the unemployment regime, then an increase in X leads to decreased welfare.3.2.6 Allowing for offers of unemployment insuranceIn our analysis thus far, we have assumed that agents do not have access to unemployment insurance. Itmay be thought that allowing for the private provision of unemployment insurance would necessarilyeliminate the mechanisms we have highlighted. For this reason, in this subsection we want to brieflyindicate how our analysis can be extended rather trivially to include an adverse selection problem that19 The other component is the net welfare gain that stems from consumption expenditures being made in the positive-assetstate rather than the more costly (in utility terms) negative-asset state.54will justify the absence of unemployment insurance, without changing the main results. In particular,suppose there is a fraction ρ of households that behave as the households we have modeled to date,which we call participant households, and suppose the remaining (1 − ρ) fraction of households,which we can call the non-participant households, are simply not interested in work within the period.These latter households are happy to consume their endowment without wanting to search for work.Now suppose that some private agent wanted to offer unemployment insurance before the matchingprocess, but could not differentiate between the two types of households. In this case, an insurerwill not be able to offer contracts that will only be attractive to the participant households, becauseany unemployment insurance contract with a positive net payment to unemployed individuals will bedesirable to non-participants. Therefore, as indicated in Proposition 3.7, as long as ρ is sufficientlylow, this type of adverse selection problem implies that the only equilibrium outcome is one where noinsurance is offered. Accordingly, in this setup, the mechanisms we have emphasized regarding howchanges in X affect outcomes will directly apply.Proposition 3.7. In the presence of both participant households and non-participant households, ifρ < 11+τ , i.e., if the fraction of participant households is sufficiently low, then no unemployment-insurance contracts are traded in equilibrium.3.2.7 Introducing government spendingWe now turn to examining how changes in government spending can affect economic activity. Todo this, we extend the model by simply adding a government to the first sub-period. The govern-ment undertakes two activities in this sub-period: it buys goods, and it taxes employed individuals.We assume that the government runs a balanced budget so that its expenditure on goods is equal tothe lump-sum tax per employed worker times the number of employed workers. It turns out that theeffects of government spending in this setup depend crucially on what the government does with thegoods. Accordingly, we will consider two types of government purchases: wasteful, and non-wasteful.Wasteful government purchases, denotedGw, are not valued by households,20 while non-wasteful pur-chases, denoted Gn, are assumed to directly affect agents’ utility by entering as a substitute to privateconsumption. Note that Gw and Gn are per-capita government expenditures. If we return to the setof equilibrium conditions given by equations (3.7) to (3.11), the only condition that changes withthe introduction of a government is equation (3.8), the goods-market equilibrium condition. The otherconditions remain the same once the variable c is interpreted as total consumption including consump-tion of non-wasteful government purchases. The goods market equilibrium condition, equation (3.8),20We can as well assume that they are valued by households but that utility is linearly separable in Gw.55therefore has to be rewritten as21min{N,L}L =c−X +GwF ′(`)`since c−X +Gw now represents the total purchases of clothes in the sub-period. If we again allow eto represent these total purchases (e = c−X +Gw), then the determination of e takes a form almostidentical to that described previously by equations (3.12)-(3.14). In fact, the determination of totalexpenditures e is now given by the solution toe = Z(e)−X +Gw (3.15)where Z(e) was defined in equation (3.13).There are two key things to notice about equation (3.15). First, non-wasteful government ex-penditure Gn does not enter into this condition, and therefore does not affect the equilibrium levelof economic activity e; that is, non-wasteful government expenditure crowds out private expenditureone-to-one. Second, in contrast, wasteful government expenditure will tend to stimulate activity in amanner parallel to a decrease in X . To understand why non-wasteful government purchases do notaffect activity, it is helpful consider how people would behave simply if they conjectured the outcome.In this case, since they would conjecture that unemployment risk is not changing, they would want toconsume at the same overall level as before the increase in Gn. But if they consume at the exact sameoverall level, it requires households to decrease their private purchases by exactly the same amount asthe purchases made by the government. Hence, activity will not be increased and agents’ initial con-jecture is rationalized. This is why non-wasteful government purchases do not affect activity in oursetup, even when the economy exhibits unemployment. Note that this logic does not hold in the caseof wasteful government purchases. If government purchases are wasteful, and people conjecture thatunemployment risk is unaffected, their overall consumption will be unchanged, and, with no increasedutility from government purchases, private purchases would also be unchanged. But total purchases– including those made by the government – would necessarily be increased. If the economy were inthe unemployment regime, this additional demand would be met by a rise in the employment rate µ,and hence households’ conjecture that unemployment risk is unchanged would be false. Recognizingthat unemployment risk in fact fell, households would reduce their precautionary savings and increasetheir private purchases, further increasing demand, and leading to a multiplier greater than one. If theeconomy had instead been in the full-employment regime, the additional demand would be met by arise in hours per worker l, which is associated with a rise in the price p and a corresponding fall inprivate purchases, mitigating to some extent the rise in demand caused by the government and leadingto a multiplier less than one. These results are summarized in Proposition 3.8.21 We assume throughout this subsection that τ < τ¯ , and that total government expenditures are sufficiently low so thatthe lump-sum tax on employed workers is not so large as to cause households to prefer to be unemployed.56Proposition 3.8. An increase in non-wasteful government purchases has no effect on economic ac-tivity. An increase in wasteful government purchases leads to an increase in economic activity. Ifthe economy is in the unemployment regime, wasteful government purchases are associated with amultiplier that is greater than one, while if the economy is in the full-employment regime, wastefulgovernment purchases are associated with a multiplier that is less than one.From Proposition 3.8 we see that the multiplier associated with wasteful government purchasesdepends on the state of the economy and the type of purchases. In particular, the multiplier for wastefulgovernment purchases is greater than one when the economy has a high level of X and is thereforein the unemployment regime. In contrast, when the economy has a low level of X and is therefore inthe full-employment regime, the multiplier for wasteful government purchases is less than one. Theinteresting aspect of Proposition 3.8 is that it emphasizes why the effects of government purchasesmay vary drastically, from zero to more than one, depending on the circumstances.While wasteful government purchases increase economic activity, this does not imply that theyincrease welfare. In fact, it can be easily verified that an increase in wasteful government purchasesnecessarily decreases welfare when the economy is in the full-employment regime, as it reduces pri-vate consumption and increases hours worked. On the other hand, when the economy is in the unem-ployment regime (due to a high value of X), the effect on welfare depends on a number of factors,in much the same way that the effect on welfare of a change in X depends on a number of factors.For example, the change in welfare depends on the ratio of the average dis-utility of labor relative tothe marginal dis-utility of labor. As discussed earlier, when this ratio is low, the net benefit to beingemployed is high, and since one of the effects of an increase in wasteful government purchases is toincrease the employment rate, the resulting increase in welfare through this channel is also high. Assuch, welfare is overall more likely to increase when the average dis-utility of work is low.It turns out that sufficient conditions under which an increase in wasteful government purchasesincreases welfare are given by those contained in Proposition 3.6 regarding the welfare effects of achange X . This is stated in Proposition 3.9.Proposition 3.9. If the economy is in the unemployment regime and if X is in the range such thata fall in X would increase welfare, then an increase in wasteful government purchases will increasewelfare.3.3 Further discussions and relaxing of assumptions3.3.1 Relaxing functional-form assumptionsOne of the important simplifying assumptions of our model is the use of a matching function of the“min” form. This specification has the nice feature of creating two distinct employment regimes: one57where there is unemployment and one where there is full employment. However, this stark dichotomy,while useful, is not central to the main results of the model. In fact, as we now discuss, the importantfeature for our purposes is that there be one regime in which expenditures by individual agents playthe role of strategic substitutes, and another in which they play the role of strategic complements. Tosee this, it is helpful to re-examine the equilibrium condition for the determination of expenditure fora general matching function. This is given byU ′(X + ej) = vp(e)[1 + τ − M(N(e), L)L τ](3.16)where M(N,L) is a CRS matching function satisfying M(N,L) ≤ min{N,L}. In (3.16), we havemade explicit the dependence of N and p on e, where this dependence comes from viewing the re-maining four equilibrium conditions as determining N , p w and ` as functions of e.22 Note that theseother equilibrium conditions imply that p(e) andN(e) are always weakly increasing in e. In (3.16) wehave once again made clear that this condition relates the determination of expenditure for agent j, ej ,to the average expenditure of all agents, e. From this equation, we can see that average expenditure canplay either the role of strategic substitute or strategic complement to the expenditure decision of agentj. In particular, through its effect on the price p, e plays the role of a strategic substitute, while throughits effect on firm entry N and, in turn, unemployment, it plays the role of strategic complement. Thesign of the net effect of e on ej therefore depends on whether the price effect or the unemploymenteffect dominates. In the case where M(N,L) = min{N,L}, the equilibrium features the stark di-chotomy whereby ∂p(e)/∂e = 0 and ∂M(N(e), L)/∂e > 0 for e < e?, while ∂p(e)/∂e > 0 and∂M(N(e), L)/∂e = 0 for e > e?. In other words, for low values of e the expenditures of othersplays the role of strategic complement to j’s decision since the price effect is not operative, whilefor high values of e it plays the role of strategic substitute since the risk-of-unemployment channelis non-operative. This reversal in the role of e from acting as a complement to acting as a substi-tute is illustrated in Figure 3.4, where we first plot a cost-of-funds schedule for agents, defined byr = p(e)[1 + τ − min{N(e),L}L τ], where r represents the total cost of funds to agent j when averageexpenditure is e. Our notion of the total cost of funds reflects both the direct cost of borrowing, p(e),and the extra cost associated with the presence of unemployment risk. We superimpose on this figurethe demand for e as a function of the total cost of funds, which is implicitly given by the functionU ′(X + e)/v = r. This latter relationship, which can be interpreted as a type of aggregate demandcurve, is always downward-sloping since U is concave. The important element to note in this figure is22 These remaining four equilibrium conditions can be writtenν′(`) = vwpF ′(`) = wM(N,L)F (`) = L(c−X) +NΦM(N,L)[pF (`)− w`] = NpΦ58Figure 3.4: Cost of fundsecost of funds (r)eU'(X+e)vpethat the cost-of-funds schedule r = p(e)[1 + τ − min{N(e),L}L τ]is first decreasing and then increas-ing in e. Over the range e < e?, the cost of funds to an agent is declining in aggregate e, since N isincreasing while p is staying constant. Therefore, in the range e < e?, a rise in e reduces unemploy-ment and makes borrowing less costly to agents. This is the complementarity zone. In contrast, overthe range e ≥ e?, the effect of e on the cost of funds is positive since the unemployment channel isno longer operative, while the price channel is. This is the strategic substitute zone. In the figure, achange in X moves the demand curve U ′(X + e)/v = r without affecting the cost-of-funds curve. Achange in X therefore has the equilibrium property ∂e/∂X < −1 when e < e? because the cost-of-funds curve is downward-sloping in this region, while ∂e/∂X > −1 in the region e ≥ e? because thecost-of-funds curve is upward-sloping.From the above discussion it should now be clear that our main results do not hinge on the “min”form of the matching function, but instead depend on the existence of two regions: one where thetotal cost of borrowing by agents at low levels of e is decreasing in e because the effect of e onunemployment risk dominates its effect on p, with a second region where the price effect dominatesthe effect running through the unemployment-risk channel. It can be easily verified that a sufficientcondition for this feature is that the elasticity of M(N,L) with respect to N tends towards one whenN becomes sufficiently small, while simultaneously having this elasticity tending to zero when N issufficiently large. This property is clearly captured by the “min” function, but is in fact also capturedby a large class of matching functions, as the following proposition establishes.Proposition 3.10. For any non-trivial matching function M(N,L)23 that is (i) non-decreasing and23By “non-trivial matching function” we mean a function satisfying, for any L > 0, M(N,L) > 0 for some N .59weakly concave in N and (ii) satisfies 0 ≤ M(N,L) ≤ min{N,L}, the elasticity of M with respectto N approaches one as N → 0 and approaches zero as N →∞.While this proposition guarantees under quite general conditions that the cost-of-funds locus willbe negatively sloped at low levels of e and positively sloped at high values of e, it is interesting to askif this non-monotonicity property can be ensured by other means over a region where the matchingfunction has a constant elasticity. In effect, this property can be ensured through assumptions on ν(`)and F (`). In particular, if the elasticities of ν ′(`) and F ′(`) with respect to ` tend toward zero when `is sufficiently low – that is, if ν(`) and F (`) become close to linear when ` is low – this will guaranteea downward-sloping cost-of-funds schedule even if the matching function has a constant elasticity.Furthermore, if the elasticity of either ν ′(`) or F ′(`) with respect to ` tends toward infinity when `is large, this will guarantee that the cost-of-funds schedule will be upward-sloping at high values ofe. While it is an open empirical question whether any of these conditions are met in reality over aneconomically significant range, it appears at least plausible to us that for low values of activity (i)congestion effects in matching associated with increases in N are small, (ii) the returns to labor inproduction exhibit little decreasing returns, and (iii) the dis-utility of work is close to linear. All theseconditions will favor a downward-sloping cost-of-funds curve at low levels of activity, which is whatis needed for the main results of this paper to hold.A second important functional-form assumption we have used to derive our results is that the dis-utility of work in the second sub-period be linear so as to obtain a piecewise linear V (a) function.This restriction is again not necessary to obtain our main results. However, if we depart substantiallyfrom the linearity assumption for second-sub-period dis-utility of labor, income effects can greatlycomplicate our simple characterizations.3.3.2 A version with productive capitalWe have shown how a rise in the supply of the capital goodX , by decreasing demand for employmentand causing households to increase precautionary savings, can perversely lead to a decrease in con-sumption. While thus far we have considered the case whereX enters directly into the utility function,in this section we show that Proposition 3.4 can be extended to the case where X is introduced as aproductive capital good. To explore this in the simplest possible setting, suppose there are now twotypes of firms and that the capital stock X no longer enters directly into the agents’ utility function.The first type of firm remains identical to those in the first version of the model, except that insteadof producing a consumption good they produce an intermediate good, the amount of which is givenbyM. There is also now a continuum of competitive firms who rent the productive capital good Xfrom the households and combine it with goods purchased from the intermediate goods firms in orderto produce the consumption good according to the production function g(X,M). We assume that gis strictly increasing in both arguments and concave, and exhibits constant returns to scale. Given X ,60it can be verified that the equilibrium determination ofM will then be given as the solution togM(X,M)U ′(g(X,M)) = Q(M) (3.17)where Q(·) is defined in equation (3.14).Note the similarity between condition (3.17) and the corresponding equilibrium condition for thedurable-goods version of the model, which can be written U ′(X + e) = Q(e). In fact, if g(X,M) =X +M, so that the elasticity of substitution between capital and the intermediate goodM is infinite,then the two conditions become identical, and thereforeX affects economic activity in the productive-capital version of the model in exactly the same way as it does in the durable-goods model. Thus, arise in X leads to a fall in consumption when the economy is in the unemployment regime. In fact, asstated in Proposition 3.11, this latter result will hold for a more general g as long as g does not featuretoo little substitutability between X andM.24Proposition 3.11. If the equilibrium is in the full-employment regime, then an increase in productivecapital leads to an increase in consumption. If the equilibrium is in the unemployment regime, thenan increase in productive capital leads to a decrease in consumption if and only if the elasticity ofsubstitution between X andM is not too small.The reason for the requirement in Proposition 3.11 that the elasticity of substitution be sufficientlylarge relates to the degree to which an increase in X causes an initial impetus that favors less em-ployment. If the substitutability between X and M is small, so that complementarity is large, theneven though the same level of consumption could be achieved at a lower level of employment, a socialplanner would nonetheless want to increase employment. Since the multiplier process in our modelsimply amplifies – and can never reverse – this initial impetus, strong complementarity would leadto a rise in employment and therefore a rise in consumption, rather than a fall. In contrast, if thiscomplementarity is not too large, then an increase in X generates an initial impetus that favors lessemployment, which is in turn amplified by the multiplier process, so that a decrease in consumptionbecomes more likely.25Let us emphasize that the manner in which we have just introduced productive capital into oursetup is incomplete – and possibly unsatisfying – since we are maintaining a static environment withno investment decision. In particular, it is reasonable to think that the more interesting aspect ofintroducing productive capital into our setup would be its effect on investment demand. To this end,we now consider extending the model to a simple two-period version that features investment. The24 We assume throughout this section that an equilibrium exists and is unique. Conditions under which this is trueare similar to the ones obtained for the durable-goods model, though the presence of non-linearities in g makes explicitlycharacterizing them less straightforward in this case.25 Note that a rise in X also increases output for any given level of employment. To ensure that consumption falls inequilibrium, we require that the substitutability between X andM be large enough so that the drop in employment morethan offsets this effect.61main result from this endeavor is to emphasize that the conditions under which a rise in X leads to afall in consumption are weaker than those required for the same result in the absence of investment.In other words, our results from the previous section extend more easily to a situation where X isinterpreted as physical capital if we simultaneously introduce an investment decision. The reason forthis is that, in the presence of an investment decision, a rise in X is more likely to cause an initialimpetus in favor of less activity.To keep this extension as simple as possible, let us consider a two-period version of our modelwith productive capital (where there remains two sub-periods in each period). In this case, it can beverified that the continuation value for household j for the second period is of the form R(X2) ·X2,j ,where X2,j is capital brought by household j into the second period and X2 is capital brought intothat period by all other households. In order to rule out the possibility of multiple equilibria that couldarise in the presence of strategic complementarity in investment, we assume we are in the case whereR′(X2) < 0. The description of the model is then completed by specifying the capital accumulationequation,X2 = (1− δ)X1 + i (3.18)where i denotes investment in the first period and X1 is the initial capital stock, as well as the newfirst-period resource constraint,c+ i = g(X1,M) (3.19)Given this setup, we need to replace the equilibrium condition from the static model (equation(3.17)) with the constraints (3.18) and (3.19) plus the following two first-order conditions,gM(X1,M)U ′(c) = Q(M) (3.20)U ′(c) = R(X2) (3.21)Equation (3.20) is the household’s optimality condition for its choice of consumption, and is similar toits static counterpart (3.17), while equation (3.21) is the intertemporal optimality condition equatingthe marginal value of consumption with the marginal value of investment.Of immediate interest is whether, in an unemployment-regime equilibrium, a rise in X1 will pro-duce an equilibrium fall in consumption and/or employment in the first period. As Proposition 3.12indicates, the conditions under which our previous results extend are weaker than those required inProposition 3.11 for the static case, in the sense that lower substitution between X andM is possible.Proposition 3.12. In the two-period model with productive capital,26 an increase in capital leads toa decrease in both consumption and investment if and only if the elasticity of substitution between Xand M is not too small. Furthermore, for a given level of equilibrium employment, this minimumelasticity of substitution is lower than that required in Proposition 3.11 in the absence of investment26 We are again assuming that the equilibrium exists, is unique, and is in the unemployment regime.62decisions.The intuition for why consumption and investment fall when the elasticity of substitution is highis similar to in the static case. The addition of the investment decision has the effect of making itmore likely that an increase in X leads to a fall in consumption because the increase in X decreasesinvestment demand, which in turn increases unemployment and precautionary savings.3.3.3 Multiple equilibriaBefore discussing the welfare effects of changes in X , let us briefly discuss how multiple equilibriacan arise in this model when τ > τ¯ . It can be verified that, when τ > τ¯ , the equilibrium determinationof expenditures can still be expressed as the solution to the pair of equations ej = Z(e) − X andej = e. The problem that arises is that this system may no longer have a unique solution. Instead,depending on the value of X , it may have multiple solutions, an example of which is illustrated inFigure 3.5. In the figure, we see that, for this value of X , there are three such solutions.Figure 3.5: Equilibrium determination (multiple equilibria)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.900.⋆ee j(e)45-deg.Note: Example is constructed assuming the functional forms U(c) = log(c), ν(`) = ν`1+ω1+ω andF (`) = A`α, with parameters ω = 1, ν = 0.5, α = 0.67, A = 1, Φ = 0.35, τ = 1.2 andX = 0.3.Figure 3.6 shows how the set of possible equilibrium values of consumption depends on X whenτ > τ¯ . As can be seen, when X is in the right range, there is more than one such equilibrium, withat least one in the unemployment regime and one in the full-employment regime. When this is thecase, the selection of the equilibrium will depend on people’s sentiment. If people are pessimistic,they cut back on consumption, which leads firms to cut back on employment, which can rationalizethe initial pessimism. In contrast, if households are optimistic, they tend to buy more, which justifies63Figure 3.6: Consumption as function of X (multiple equilibria)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8  Full employmentLow unemploymentHigh unemploymentZero employmentNote: Example is constructed assuming the functional forms U(c) = log(c), ν(`) = ν`1+ω1+ω andF (`) = A`α, with parameters ω = 1, ν1 = 0.5, α = 0.67, A = 1, Φ = 0.35 and τ = 1.2.many firms wanting to hire, which reduces unemployment and supports the optimistic beliefs. Thistype of environment featuring multiple equilibria driven by demand externalities is at the core of manypapers. On this front, this paper has little to add. The only novel aspect of the current paper in termsof multiple equilibria is to emphasize how the possibility of multiple equilibria may depend on theeconomy’s holding of capital goods.3.3.4 The role of beliefsThere is another aspect in which the current model differs from a Keyesnian-cross setup, and that iswith respect to the role of beliefs. The current setup should be thought of as part of the family ofcoordination games, and accordingly can potentially be analyzed with the tools and concepts usedin the global games literature. Because of our assumption of homogeneity across households, wehave not been very specific about agents’ beliefs up to now. Nonetheless, it is worth emphasizingthat the type of multiplier process present in the unemployment regime is the equilibrium outcomeof a simultaneous-move game, rather than the outcome of events occurring sequentially over time.As such, prior beliefs of the players in the game are potentially a key driving force in the multiplierprocess. To clarify the potential role of these beliefs in our setup, it is helpful to briefly consider thecase where agents have different holdings of X . For example, suppose that each agent j has an Xjdrawn from a distribution with mean χ. The first-order condition for household j, assuming he thinks64he is in the unemployment regime, can then be stated asU ′(ej +Xj) =ν ′(`?)F ′(`?) (1 + τ − Ej [µ]τ)What is unknown to the household in this setup is the match probability µ, and therefore the expendi-ture decision, ej , depends on household j’s expectation of µ, which we write as Ej [µ]. But µ in turndepends on firms’ entry decisions, which depends on firms’ expectation of aggregate consumption.This latter expectation can be expressed as Ef [∫eidi], where the operator Ef [·] represents expec-tations by firms, and∫eidi is the aggregate level of expenditures. So the first-order condition forhousehold j would be given byU ′(ej +Xj) =ν ′(`?)F ′(`?)(1 + τ − Ej [Ef [∫eidi]/L]Ω(l?) τ)We can now see that agent j’s consumption decision will depend on his expectation of firms’ ex-pectation of the aggregate level of expenditure. This type of setup therefore involves forecasting theforecasts of others. If we assume that U(·) is quadratic and all relevant random variables jointly nor-mally distributed, then this problem can be solved analytically, and will lead agent j to have a decisionrule for consumption which depends on both Xj and χ, the prior about the average level of Xi acrossall other agents. Hence, both actual Xj’s and beliefs regarding the average value of Xj in the econ-omy will be main forces that drive expenditure and employment. For example, if agents believe thatother agents have a high holding of X , this will depress consumption for all agents regardless of theactual holdings of X . Furthermore this effect can potentially be large because of the amplificationmechanisms running through precautionary savings.3.4 DynamicsIn this section we want to explore a dynamic extension of our static durable-goods model wherecurrent consumption contributes to the accumulation of X . In particular, we want to consider the casewhere the accumulation of X obeys the accumulation equationXt+1 = (1− δ)Xt + γet 0 < δ ≤ 1 , 0 < γ ≤ 1− δ (3.22)where the parameter γ represents the fraction of current consumption expenditures, et = ct − Xt,which take the form of durable goods. Since we do not want to allow heterogeneity between individ-uals to expand over time, we will allow individuals to borrow and lend only within a period but notacross periods; in other words, households are allowed to spend more than their income in the first65sub-period of a period, but must repay any resulting debt in the second sub-period.27 The problemfacing a household in the first sub-period of a period is therefore to choose how much clothing tobuy and, conditional on a match, how much labor to supply. We model the second sub-period as insub-section 3.2.2, where households use labor to produce household services either for their own con-sumption or, at a level of productivity that is lower by a factor 1 + τ , for the consumption of others. Ineach second sub-period, then, the household chooses how much to consume of household services andhow much to produce of household services to both satisfy his needs and to pay back any accumulateddebt. In order to keep the model very tractable, we will continue to assume that dis-utility of work inthe second sub-period is linear (i.e., equal to v · ˜`). Under this assumption, all households will choosethe same level of consumption of household services in each second sub-period, while the productionof household services will vary across households depending on whether they entered the sub-periodin debt or in surplus. Since there are no interesting equilibrium interactions in second sub-periods, wecan maintain most of our focus on equilibrium outcomes in the sequence of first sub-periods.Relative to the static case, the only difference in equilibrium relationships (aside from the additionof the accumulation equation (3.22)) is that the first-order condition associated with the households’choice of consumption of clothes is now given by the Euler equationU ′ (Xt + et)−Q (et) = β[(1− δ − γ)U ′ (Xt+1 + et+1)− (1− δ)Q (et+1)](3.23)where Q is as defined in equation (3.14). In this dynamic setting, an equilibrium will be representedas a sequence of the previous equilibrium conditions (3.8) to (3.11) plus the accumulation equation(3.22) and the Euler equation (3.23).There are many complications that arise in the dynamic version of this model, which makes char-acterizing equilibrium behavior difficult. In particular, there can be multiple equilibrium paths andmultiple steady-state solutions. Luckily, the problem can be simplified if we focus on cases where δis small; that is, on cases where the durability of goods is long. In addition to simplifying the analy-sis, focusing on the low-δ case appears reasonable to us, as many consumer durables are long-lived,especially if we include housing in that category. In the case where δ is sufficiently small, as statedin Proposition 3.13, the economy will have only one steady state and that steady state will have theproperty of exhibiting unemployment.Proposition 3.13. If δ is sufficiently small, then the model has a unique steady state and this steadystate is characterized by unemployment.Proposition 3.13 is very useful, as it will allow us to analyze the equilibrium behavior aroundthe steady state without worrying about equilibrium selection. Accordingly, for the remainder of thissection, we will assume that δ is sufficiently small so that Proposition 3.13 applies. However, before27 This lack of borrowing across periods can be rationalized if one assumes that the transaction cost of intermediatingloans across periods is greater than 1 + τ .66examining local properties in some generality, we believe that it is helpful to first illustrate globalequilibrium behavior for a simple case that builds directly on our static analysis. The reason that wewant to illustrate global behavior for at least one example is to emphasize that local behavior in oursetup is likely to differ substantially and meaningfully from global behavior. Moreover, the examplewill allow us to gain some intuition on how the latter local results should best be interpreted.Before discussing the transitional dynamics of the model, we first briefly discuss the conditionsunder which the model would exhibit a balanced growth path. In particular, suppose production in thefirst sub-periods is given by θtF (`t) where θt is a technology index that is assumed to grow at a rategθ. Then it is easy to verify that our economy will admit an equilibrium growth path where both e andX grow at rate gθ if the following three conditions are satisfied (i) the fixed cost of creating jobs growsat rate gθ, (ii) the productivity of labor in the second sub-periods grows at rate gθ, and (iii) the utilityof consumption is represented by the log function. These conditions are not surprising, as they parallelthose needed for a balanced growth path in many common macro models. The important aspect tonote about this balanced-growth property is that the notion of high or low levels of capital should beinterpreted as relative to the balanced growth path. In other words, the key endogenous state variablein the system should be viewed as the ratio of Xt to the growth component of θt.3.4.1 Global dynamics for a simple caseThe difficulty in analyzing the global dynamics for our model is related to the issue of multiple equilib-ria we discussed in the static setting. If the static setting exhibits multiple equilibria then the dynamicsetting will likely exhibit multiple equilibrium paths. To see this, it is useful to recognize that ourproblem of describing equilibrium paths can be reduced to finding the household’s decision rule forconsumption. Since the only state variable in the system is Xt, the household’s decision rule for con-sumption will likely be representable by a relationship (which may be stochastic) of the form c(Xt).Given c(Xt), the equilibrium dynamics of the system are given byXt+1 = (1− δ − γ)Xt + γc(Xt) (3.24)If the relationship c(Xt) is a function, then equilibrium dynamics are deterministic. However, ifwe consider the case with β = 0 – so that households are not forward-looking and thus the dynamicequilibrium is simply a sequence of static equilibria – we already know that the household’s decisionrule c(Xt) may not be a function. For example, if τ > τ¯ , then the household’s decision rule may bea correspondence of the form given in Figure 3.6. Therefore, even for the rather simple case whereβ = 0 and τ > τ¯ we know that the equilibrium dynamics need not be unique, in which case someequilibrium-selection device will be needed to solve the model. In contrast, for the case where β = 0and τ < τ¯ , then we know from Proposition 3.3 that c(Xt) is a function. Hence, in the case whereβ = 0 and τ < τ¯ , we can describe the global dynamics of the system rather easily, and this is what wewill do in this section. In particular, when β = 0 and τ < τ¯ , the stock of durables evolves according67to equation (3.24), with c(Xt) given by the value of c obtained using Proposition 3.3 with Xt in placeof X .Figure 3.7 plots the equilibrium transition function for X for three cases; that is, it plots (1− δ −γ)Xt + γc(Xt) for different possible c(Xt) functions. The figure is drawn so that the steady stateis in the unemployment region, which is consistent with a low value of δ as implied by Proposition3.13. As can be seen from the figure, when Xt is not too great (i.e., less than X?) the economy isFigure 3.7: Xt+1 as a function of Xt0.1 0.2 0.3 0.4 0.5 0.6 0.7 t+145-deg.  −c′(Xt) < 1−δ−γγ1−δ−γγ < −c′(Xt) < 2−δ−γγ−c′(Xt) > 2−δ−γγNote: Figure shows transition functions Xt+1(Xt) for three different decision rules c(Xt) thatall yield the same steady state. Decision rules are identical when the economy features eitherfull employment or zero employment, and differ when the economy features partial unemployment.Legend entries refer to value of c′(Xt) in partial-unemployment regime.in the full-employment regime and Xt+1 > Xt. So if the economy starts with a low value of Xt itwill generally go through a phase of full employment. During this phase, we know from Proposition3.3 that consumption is also increasing. Eventually, Xt will exceed X? and the economy entersthe unemployment regime, at which point the dynamics depend on the derivative of the equilibriumdecision rule, i.e., c′(Xt), where in this regime c(Xt) solvesU ′(c) = ν′(`?)F ′(`?)(1 + τ − τ c−Xte?)If −c′(Xt) < 1−δ−γγ when X? < Xt < X??, then the transition function maintains a positive slope68near the steady state and the economy will converge monotonically to its steady state. However,note that even if X converges monotonically to its steady state in such a case, this will not be thecase for consumption. Again, from Proposition 3.3 we know that consumption is decreasing in Xin the unemployment region. Hence, starting from X = 0, in this case consumption would initiallyincrease, reaching a maximum just as the economy enters the unemployment regime, then declinetowards its eventual steady-state level which is lower than the peak obtained during the transition.If instead −c′(Xt) > 1−δ−γγ , then the transition function for X will exhibit a negative slope in theunemployment regime. In this case, X will no longer converge monotonically to the steady state. Infact, if the slope of this function (which depends on the elasticity of c with respect to X at the steadystate) is negative but greater than -1, the system will converge with oscillations. However, if this slopeis smaller than -1, which can arise for very large negative values of c′(Xt), then the system will notconverge and instead can exhibit rich dynamics, including cycles and chaos. In general, however, evenin the case where c′(Xt) is very negative, the system will not necessarily be explosive, since once itmoves sufficiently far away from the steady state, forces kick in that work to push it back. Such richdynamics, with the possibility of limit cycles, are certainly intriguing, but we will not dwell on themsince it appears unlikely to us that this type of configuration is relevant.There are two main messages to take away from exploring the global dynamics in this special casewith β = 0. First, the behavior of the state variable X can be well-behaved, exhibiting monotonicconvergence throughout. Second, the behavior of consumption (and therefore possibly welfare), cannonetheless exhibit interesting non-monotonic dynamics, with steady-state consumption actually be-ing below the highest level it achieved during the transition. It is worth noting that if the steady statewere to be in the full-employment regime (due, for example, to a higher δ), then from X = 0 both Xtand ct would always converge monotonically to the steady state when β = 0 and τ < τ¯ .The most interesting aspect about the global dynamics in this case is that it allows us to illustratethe following possibility: If the economy is near its steady state, then a small reduction in Xt willincrease consumption and can potentially increase welfare, while a large decrease in Xt will certainlydecrease welfare. In this sense, the model exhibits behavior around the unemployment steady state thatcan differ substantially from behavior far away from the steady state, with the behavior far away fromthe steady state being more akin to that generally associated with classical economics, while behaviorin the unemployment regime being more similar to that suggested by a Keynesian perspective.283.4.2 Local dynamics in the general caseIn this subsection, we explore the local dynamics of the general model when β > 0, still assuming thatδ is sufficiently small so that the steady state is unique and in the unemployment regime. From ouranalysis of the case with β = 0, we know that local dynamics can exhibit convergence or divergence28 It is worth noting that this type of synthesis, which emphasizes differences between being near to the steady state versusfar from the steady state, is substantially different from the new neo-classical synthesis, which emphasizes differences in thelong run and the short run because of sticky prices.69depending on how responsive consumption is toX around the steady state. The one question we couldnot address when β = 0 is whether dynamics could exhibit local indeterminacy. In other words, canforward-looking behavior give rise to an additional potential local source of multiple equilibria in oursetup? Proposition 3.14 indicates that this is not possible; that is, the roots of the system around theunique steady state can not both be smaller than one.29Proposition 3.14. The local dynamics around the steady state can either exhibit monotonic conver-gence in c andX , convergence with oscillations, or divergence. Locally indeterminacy is not possible.Proposition 3.14 is useful as it tells us that the decision rule for consumption around the steadystate is a function.30 Accordingly, we can now examine the sign of the derivative of this function. Thequestion we want to examine is whether the decision rule for consumption around the steady state hasthe property that a larger X leads to a lower level of consumption, as was the case in our static modelwhen in the unemployment regime. In other words, we want to know whether the results regarding theeffect of X on consumption we derived for the static model extend to the steady state of the dynamicsetting with β > 0. Proposition 3.15 indicates that if τ is not too large, then local dynamics willexhibit this property. Note that the condition on τ is a sufficient condition only.Proposition 3.15. If τ is sufficiently small, then in a neighborhood of the unique steady state, con-sumption is decreasing in X , with the dynamics for X converging monotonically to the steady state.From Proposition 3.15 we now know that, as long as τ is not too big, our model has the propertythat when the economy has over-accumulated relative to the steady state (i.e., if X slightly exceeds itssteady-state value), then consumption will be lower than in the steady state throughout the transitionperiod toward the steady state, which we can refer to as a period of liquidation. In this sense, theeconomy is overreacting to its inherited excess of capital goods during this liquidation period, sinceit is reducing its expenditures to such an extent that people are consuming less even though thereare more goods available to them in the economy. While such a response is not socially optimal, itremains unclear whether it is so excessive as to make people worse off in comparison to the steadystate, since they are also working less during the liquidation phase. It turns out that, as in the staticcase, the welfare effect of such a liquidation period depends, among other things, on whether theaverage dis-utility of work is small enough relative to the marginal dis-utility. For example, if theaverage dis-utility of work is sufficiently low relative to its marginal value, then it can be verified thata liquidation period induced by inheriting an excess ofX relative to the steady state will make average29 In this section we only consider local dynamics around a unique unemployment-regime steady state. Nonetheless, it isstraightforward to show that if the unique steady state is in the full-employment regime, then the local dynamics necessarilyexhibit monotonic convergence.30 This is a slight abuse of language since Proposition 3.14 does not rule out the existence of other equilibrium paths awayfrom the steady state.70utility in all periods of the transition lower than the steady state level of utility. This result depends inaddition on the unemployment rate not being too large in the steady state.While we do not have a simple characterization of the global dynamics when β > 0, Propositions3.14 and 3.15 suggest to us that the intuition we gained from the case where β = 0 likely extends tothe more general problem as long as τ is not too large and δ is small. In particular, we take our analysisas suggesting that, starting fromX = 0, the economy will generally go though a phase of full employ-ment, with both X and c increasing over time. The economy then enters into the unemployment rangeonce X is large enough. Then, as long as τ is not too great, X will continue to monotonically in-crease, converging toward its steady state. In contrast to X , upon entering the unemployment regime,consumption starts to decrease as unemployment risk leads to precautionary savings which depressesactivity. Eventually, the economy will reach a steady state where consumption, employment, andpossibly period welfare are below the peak levels reached during the transition.In the above discussion of liquidation, we have taken the level of inherited capital as given andhave only examined how the economy responds over time to a situation where X is initially aboveits steady state. In particular, we have shown that such a liquidation phase can be associated withexcessively low consumption, low welfare and high unemployment, all relative to their steady statevalues. While the focus of the paper is precisely to understand behavior during such a liquidationphase, it nonetheless remains interesting to ask how welfare would behave if we were to view thewhole cycle, both the over-accumulation phase and the liquidation phase together. To briefly examinethis issue, we build on the news-noise literature and consider a case where agents in an economystart at a steady state and then receive information about productivity.31 Agents have to make theirconsumption decision based on the news, and we assume that they subsequently learn that the news isfalse. This leads to an initial high level of consumption during the period where agents are optimistic,followed by a period of low consumption during the liquidation phase after realizing that they hadmistakenly over-accumulated. Details of this extension are presented in Appendix B.3.In Figure 3.8 we report for illustration purposes two impulse responses associated with a simplecalibration of such a noise-driven-boom-followed-by-liquidation model. We plot the dynamics for thestock of durables and the average period utility of households relative to the steady state. From thefigure, we see that during the first period, when agents are acting on optimistic beliefs about produc-tivity, their period welfare increases even if they are working hard to ramp up their stocks of durablegoods. After one period, they realize their error since productivity has not actually improved, andconsequently cut back on their expenditures to start a liquidation process. The welfare of householdsfrom the second period on is lower than in steady state because of the excessively cautious behavior ofhouseholds, which stops the economy from taking advantage of the excessively high inherited capitalstock.It is interesting to contrast this path with that which would happen if unemployment risk wereperfectly insured or if matching frictions were absent. In such a case, the news would still lead to a31 See Beaudry and Portier (2013) for a survey of this literature.71Figure 3.8: Response of economy to a noise shock−1 0 1 2 3 4 5 6 7 8 9 1000.ˆtt−1 0 1 2 3 4 5 6 7 8 9 10−0.1−0.0500. uˆttNote: Impulse is associated with a 10% overly-optimistic belief by shoppers in the first sub-periodof t = 0. Xˆt is the stock of durables and uˆt is average period utility across all households, bothexpressed in deviations from steady state. Example is constructed assuming the functional formsU(c) = log(c), ν(l) = ν1l1+ω1+ω and F (l) = Alα, with parameters β = 0.9, δ = 0.1, γ = 0.1,ω = 1.2, ν1 = ν2 = 0.35, α = 0.67, A = 1.2, Φ = 0.5 and τ = 0.3.boom, and the realization of the error would lead to a recession. However, the dynamics of periodwelfare would be very different. Instead of the boom being associated with high period welfare andthe recession being associated with low period welfare, as in our model with unemployment risk, theopposite would happen. The boom would be associated with low period welfare, as agents wouldbe working harder than normal, while in the recession welfare would be above the steady-state valuesince agents would take a vacation and benefit from past excess work. While evaluating welfare iscertainly difficult, the path for period welfare in our model with unemployment risk appears to us asmore in line with common perceptions about boom-bust cycles than that implied by a situation withno market frictions.3.5 Policy trade-offsIn this last section, we turn to one of our motivating questions and ask whether or not stimulativepolicies should be used when an economy is going through a liquidation phase characterized by high72unemployment. In particular, we consider the case where the economy has inherited from the past alevel of X above its steady-state value and, in the absence of intervention, would experience a periodof liquidation, with consumption below its steady-state level throughout the transition. Obviously,the first-best policies in this environment would be to remove the sources of frictions or to perfectlyinsure agents against unemployment risk. However, for a number of reasons, such-first best policiesmay not be possible. We therefore want to consider the value of a more limited type of policy: onethat seeks only to temporarily boost expenditures. In particular, we are interested in asking whetherwelfare would be increased by stimulating expenditures for one period, knowing that this would implya higherX tomorrow and therefore lower consumption in all subsequent periods until the liquidation iscomplete. This policy question is aimed at capturing the tension between the Keynesian and Hayekianprescriptions in recession. In answering this question, we will be examining the effects of such apolicy without being very explicit about the precise policy tools used to engineer the stimulus, as wethink it could come from several sources. However, it can be verified that the stimulus we considercan be engineered by a one period subsidy to consumption financed by a tax on the employed.Examining how a temporary stimulus to expenditures affects welfare during a liquidation turnsout to be quite involved. For this reason, we break down the question into two parts. First, we askwhether a temporary stimulus would increase welfare if the economy were initially in a steady statecharacterized by unemployment. Second, we ask whether the effect on welfare of such a stimuluswould be greater if the economy were initially in a state of liquidation (i.e., with X0 above its steadystate) than in the case where it is initially at a steady state.When looking at how a temporary boost in expenditures would affect welfare, one may expect itto depend on many factors, including the extent of risk-aversion and the dis-utility of work. However,since the level of expenditures represents a private optimum, the present discounted welfare effectof a temporary boost in expenditures turns out to depend on a quite limited set of factors. In par-ticular, if the economy is initially at a steady state in the unemployment regime, then to a first-orderapproximation the direction of the cumulative welfare effect depends simply on whether the stimulusinduces an increase or decrease in the presented discounted value of the output stream. This is statedin Proposition 3.16.Proposition 3.16. Suppose the economy is in steady state in the unemployment regime. Then, to a first-order approximation, a (feasible) change in the path of expenditures from this steady state equilibriumwill increase the present discounted value of expected welfare if and only if it increases the presenteddiscounted sum of the resulting expenditure path,∑∞i=0 βiet+i.The logic behind Proposition 3.16 derives mainly from the envelope theorem. Since the consump-tion stream is optimally chosen from the individual’s perspective, most of the effects of a change in theconsumption path are only of second order and can therefore be neglected when the change is small.Moreover, in the unemployment region, prices, wages and hours worked are invariant to changes in73expenditures. Hence the only effects needed to be taken into account for welfare purposes are theinduced changes in the match probabilities times the marginal value of changing these match prob-abilities. When the economy is initially in a steady state, the marginal value of changing the matchprobabilities is the same at each point in time. Further, since the match probabilities are proportionalto expenditures, this explains why welfare increases if and only if the perturbed path of expenditureshas a positive presented discounted value. With this result in hand, it becomes rather simple to calcu-late whether, starting from steady state, a one-period increase in expenditures followed by a return toequilibrium decision rules results in an increase in welfare. In particular, recall that the law of motionfor X is given byXt+1 = (1− δ)Xt + γe(Xt) 0 < γ < 1− δwhere the function e(Xt) is the equilibrium policy function for et. Now, beginning from steady state,suppose at t = 0 we stimulate expenditures by  for one period such that the stock at t = 1 is nowgiven byX˜1 = (1− δ)X0 + γ(e+ )As as result of this one-period perturbation, the path for expenditures for all subsequent periods will bechanged even if there is no further policy intervention. The new sequence forX , which we denote X˜t,will be given by X˜t+1 = (1− δ)X˜t + γe(X˜t) for all t ≥ 1. From Proposition 3.16, this perturbationincreases present discounted welfare if and only if > −∞∑t=1βt[e(X˜t)− e](3.25)For  small, we can use the linear approximation of the function e(·) around the steady state to makethis calculation. Note that e′(X) = −(1 − δ − λ1)/γ, where λ1 is the smallest eigenvalue of thedynamic system in modulus.32 Thus, in this case, one may show that condition (3.25) becomes1− β(1− δ)1− βλ1> 0If the system is locally stable, then λ1 < 1, and therefore this condition will always hold. Hence, ifwe are considering a situation where the economy is in an unemployment-regime steady state, andthis steady state is locally stable, then a one-period policy of stimulating household expenditures willincrease welfare. This arises even though most of the effect of the policy is to front-load utility bycreating an initial boom followed by a liquidation bust.33 While we knew that the initial steady statewas sub-optimal, and that a policy that increases expenditures in all periods would likely be desirable,it is interesting to learn that a policy that favors expenditure today over expenditure tomorrow – when32 See the proof of Proposition 3.15.33 Note that this result does not depend on the welfare factors considered earlier in the static model, such as the magnitudesof τ and of the difference between the marginal and average disutility of work.74in the economy is in the unemployment regime – tends to increase welfare.The question we now want to examine is whether the gains in welfare of a temporary stimulus aregreater when the economy is initially in a liquidation phase than in steady state. We believe this is arelevant question since a case for stimulus during a liquidation can best be made if the gains are greaterthan when the economy is in steady state. Otherwise, there is no particular reason to favor stimuli morewhen unemployment is above normal than when it is at a normal level. Somewhat surprisingly to us,as long as U ′′′ is not too big,34 the answer to this question is negative, as stated in Proposition 3.17.Proposition 3.17. Assuming the economy’s steady state is in the unemployment regime and U ′′′ is nottoo big, then, to a second-order approximation around the steady state, a temporary stimulus increasesthe presented discounted value of welfare less when implemented during a liquidation phase then whenimplemented at the steady state.Although a period of liquidation is associated with a higher-than-normal level of unemployment,and the degree of distortion as captured by the labor wedge is higher in such periods when compared tothe steady state, Proposition 3.17 indicates that the gains to a temporary stimulus are not greater duringa liquidation period than in normal (steady-state) times. At first pass, one may be puzzled by thisresult, as one might have expected the gains to be highest when the marginal utility of consumptionis highest. However, when the economy is in a liquidation phase, while the benefits from currentstimulus are high, so are the costs associated with delaying the recovery. In fact, because consumptionlevels are at a private optimum, these two forces essentially cancel each other out. Moreover, when inthe unemployment regime, the direct gain from employing one more individual – that is, the value ofthe additional production, net of the associated dis-utility of work – is the same regardless of whetherunemployment is high or low. Hence, the only remaining difference between the value of stimulus inhigh- versus low-unemployment states relates to the net utility gain from employed workers enteringthe second sub-period in surplus rather than debt. In a lower-unemployment regime, households takeless precaution, so that unemployed workers end up with more debt, which is costly. It is this forcewhich makes postponing an adjustment particularly costly when in a liquidation phase.35With respect to the policy debate between the followers of Hayek and Keynes, we take our resultsare clarifying the scope of the arguments. On the one hand, we have found that a policy that stimulatescurrent consumption at the cost of lower consumption in the future can often be welfare-improvingwhen the economy features unemployment. However, at the same time, we have found that the ratio-nale for such a policy does not increase simply because the level of unemployment is higher. Hence,if one believes that stimulus is not warranted in normal times (because of some currently un-modeledcosts) and that normal times are characterized by excessive unemployment, then stimulus should not34 Note that this condition on U ′′′ is sufficient but not necessary for this result.35 There is an additional force at play here, which relates to the fact that the magnitude of the amplification mechanismwill in general be different when the economy is away from the steady state. However, as long as U ′′′ is not too big, thiseffect can safely be ignored.75be recommended during liquidation periods. While this insight will likely not extinguish the debateon the issue, we believe it can help focus the dialogue.3.6 ConclusionThere are three types of elements that motivated us to write this paper. First, there is the observa-tion that most deep recessions arise after periods of fast accumulation of capital goods, either in theform of houses, consumer durables, or productive capital. This, in our view, gives plausibility to thehypothesis that recessions may often reflect periods of liquidation where the economy is trying todeplete excesses from past over-accumulation.36,37 Second, during these apparent liquidation-drivenrecessions, the process of adjustment seems to be socially painful and excessive, in the sense that thelevel of unemployment does not seem to be consistent with the idea that the economy is simply “takinga vacation” after excessive past work. Instead, the economy seems to be exhibiting some coordina-tion failure that makes the exploitation of gains from trade between individuals more difficult thanin normal times. These two observations capture the tension we believe is often associated with theHayekian and Keynesian views of recessions. Finally, even when monetary authorities try to countersuch recessions by easing policy, this does not seem to be fully effective. This leads us to believethat there are likely mechanisms at play beyond those related to nominal rigidities.38 Hence, our ob-jective in writing this paper was to offer a framework that is consistent with these three observations,and accordingly to provide an environment where the policy trade-offs inherent to the Hayekian andKeynesian views could be discussed.A central contribution of the paper is to provide a simple macro model that explains, using real asopposed to nominal frictions, why an economy may become particularly inefficient when it inheritsan excessive amount of capital goods from the past. The narrative behind the mechanism is quitestraightforward. When the economy inherits a high level of capital, this decreases the desire for tradebetween agents in the economy, leading to less demand. When there are fixed costs associated withemployment, this will generally lead to an increase in unemployment. If the risk of unemploymentcannot be entirely insured away, households will react to the increased unemployment by increasingsaving and thereby further depressing demand. This multiplier process will cause an excess reaction tothe inherited goods and can be large enough to make society worse off even if – in a sense – it is richersince it has inherited a large stock of goods. Within this framework, we have shown that policies aimedat stimulating activity will face an unpleasant trade-off, as the main effect of stimulus will simply be36 Note that this is a fundamentalist view of recessions, in that the main cause of a recession is viewed as an objectivefundamental (in this case, the level of capital relative to technology) rather than a sunspot-driven change in beliefs.37 An alternative interpretation of this observation is that financial imbalances associated with the increase in capitalgoods are the main source of the subsequent recessions.38 We chose to analyze in this paper in an environment without any nominal rigidities so as to clarify the potential roleof real rigidities in understanding behavior in recessions. However, in doing so, we are not claiming that the economy doesnot also exhibit nominal rigidities or that monetary policy is ineffective. We are simply suggesting that explanations basedmainly on nominal rigidities may be missing important forces at play that cannot be easily overcome by monetary policy.76to postpone the adjustment process. Nonetheless, we find that such stimulative policies may remaindesirable even if they postpone recovery, but these gains do not increase simply because the rate ofunemployment is higher. As noted, the mechanisms presented in the paper have many antecedentsin the literature, but we believe that our framework offers a particularly tractable and clear way ofcapturing these ideas and of reconciling diverse views about the functioning of the macro-economy.77Chapter 4Can a Limit-Cycle Model ExplainBusiness Cycle Fluctuations?4.1 IntroductionIn conventional models of the business cycle, all fluctuations are ultimately caused by the arrival ofrandom shocks. One implication of this is that a boom is usually caused by one collection of randomevents, and any subsequent recession is caused by another collection of random events. In this sense,in conventional models individual booms and busts are largely unrelated phenomena.An alternative to this viewpoint is that booms and busts are inherently related, and that, for ex-ample, a protracted boom can sow the seeds for a subsequent recession, which can in turn lead to thenext boom, and so on. According to this view, fluctuations are at least in part driven by deterministiccyclical forces that do not fit neatly into the usual shocks-and-propagation-mechanisms characteriza-tion of conventional models. While an earlier literature made an attempt to formalize the forces thatcan produce these deterministic fluctuations,1 in recent decades this research area has gone dormant.This paper revisits the idea of deterministic fluctuations, with the aim of showing (1) how a purelydeterministic general-equilibrium model featuring strategic complementarity2 near the steady statecan give rise to a stable limit cycle,3 and (2) that this model can replicate business cycle features onceit is augmented to include a small amount of variation from exogenous sources. The limit cycle in themodel arises through a simple micro-founded mechanism in a rational-expectations environment. In1To fix terminology, I will say a dynamic system exhibits “deterministic fluctuations” if, in the absence of stochasticshocks, its state vector neither diverges to infinity nor converges to a single point. See Appendix C.3 for a more formaldefinition.2Two agents’ decisions are strategic complements if a rise in any agent’s choice variable results in a rise in the marginalvalue of the other agent’s choice variable. Their decisions are strategic substitutes if the reverse is true, i.e., if a rise in anyagent’s choice variable leads to a fall in the marginal value of the other agent’s choice variable.3A “limit cycle” is a deterministic fluctuation that exactly repeats itself every k periods. A “stable limit cycle” is a limitcycle that acts as an attractor for the system, so that nearby points converge to it over time. See Appendix C.3 for furtherdetail.78contrast to most conventional models, the model I present does not require shocks in order to generatefluctuations, nor does it rely on the existence of multiple equilibria or dynamic indeterminacy.4 Cyclesemerge endogenously and would indefinitely repeat themselves in the absence of shocks. Shocks areonly introduced into the model to create irregularities in the cycles.The key deterministic mechanism in the model, which is based closely on Beaudry et al. (2014),will center around the accumulation of a stock of capital, interpreted here as a stock of durable goodsand/or housing. When this stock is high following a boom period, agents reduce their demand for newcapital goods with the goal of running down the stock of capital, which leads to a bust. Because of ademand externality in the model, this bust is excessively large, and can lead new purchases of capitalgoods to become sufficiently depressed that the stock of capital overshoots its steady-state level. Oncethis happens, the capital stock is too low and the reverse mechanisms come into play, producing aboom and a subsequent capital stock that is too high, and so on.5The demand externality, meanwhile, arises because of two key imperfections in the model. First,there is a matching friction in the labor market in the spirit of Diamond-Mortensen-Pissarides, whichcreates the possibility that a household may not find employment when looking for a job. Second,households are unable to perfectly insure against this idiosyncratic unemployment risk. The com-bination of these two imperfections causes agents to react to a fall in the unemployment rate byincreasing their demand for new goods, which in turn reduces the unemployment rate further. Ifsufficiently strong, this strategic complementarity—where an initial increase in one agent’s purchasesleads all other agents to increase their own purchases—causes the economy to be very sensitive tosmall changes in the stock of capital goods. In turn, this causes the unique steady state of the model tobe locally unstable, so that even in the absence of shocks the economy will in general fail to convergeto the (unique) steady state. When the economy reaches full employment, however, the above demandexternality is no longer operative. Instead, a rise in one household’s purchases causes the price ofgoods to increase, which tends to favor a decrease in expenditure by other households. As a result ofthis strategic substitutability, when far enough away from the steady state the economy exhibits lo-cally the types of stabilizing forces that are present globally in conventional models. This prevents thesystem from exploding. The combination of this non-explosiveness property with an unstable steadystate yields conditions under which a stable limit cycle may appear.Because the model does not rely exclusively on shocks to drive fluctuations, it is capable of ad-dressing one of the common criticisms of conventional business-cycle models, which is their frequent4Even though they may not require shocks to fundamentals to generate fluctuations, models featuring multiple equilibriaor indeterminacy still require some time-varying equilibrium-selection device, and this device is usually taken to be someexogenous shock that affects agents’ beliefs in a coordinated way (see, e.g., Farmer and Guo (1994), Jaimovich (2007),Kaplan and Menzio (2014)). Thus, these models can still be seen as requiring shocks in order to generate fluctuations.5Because this mechanism operates through a demand channel, to make this channel as clear as possible the stock ofcapital enters directly into the household utility function in the model without affecting the production side of the economy.It is for this reason that I interpret the stock of capital here as including only durables and housing, and not productive capital.Nonetheless, subject to several technical restrictions the important elements of the model also extend to a productive-capitalenvironment (see Beaudry et al. (2014)).79reliance on poorly motivated and/or implausibly large shocks. For example, the widely cited modelof Smets and Wouters (2007) features two “mark-up” shocks that are highly persistent6 and togetheraccount for over half of the 40-quarter forecast-error variance (FEV) of output and hours worked.7Outside of their apparent usefulness in helping the model fit moments of the data, however, littleempirical justification for the size and persistence of these exogenous shocks is offered.Mark-up shocks are just one example drawn from a large number of shocks proposed in the liter-ature that may help a particular model fit the data, but that do not necessarily have any clear empiricaljustification. For example, investment shocks, liquidity shocks, news/noise shocks, risk shocks, fi-nancial shocks, government spending shocks, confidence shocks, intertemporal preference shocks,and ambiguity shocks have all been argued in recent years to be quantitatively important drivers offluctuations, based primarily on their ability to help a particular model fit the data. Without a directcompelling argument for the empirical relevance of these shocks, though, the case can be made that,while the conventional class of models may have had some success in fitting the data, it has perhapsbeen less successful at explaining the data.The second goal of this paper is therefore to show that the model can match fluctuations in US datawith a minimal amount of exogenously-caused variation. With this in mind, I add a single shock—a simple TFP process—to the model, then estimate the model parameters to match as closely aspossible the spectrum of hours worked in US data over the period 1960-2012. I focus on the hoursspectrum here for two reasons. First, as discussed further in section 4.2, one of the main criticismsof earlier deterministic-fluctuations models was that they produced cycles that were far too regular.This regularity shows up clearly as a large spike in the spectrum. The ability for the model to matchthe much flatter spectrum found in the data will thus be an important test of its ability to generaterealistic data. Second, as compared with other data series, hours is arguably less likely to be directlyimpacted by various exogenous shocks. For example, while GDP is directly affected by things likeshocks to total factor productivity, over the business cycle we may expect hours to largely respondonly indirectly to exogenous shocks. To the extent that this is true, variation in hours is more likely tobe caused by the endogenous mechanisms that are the primary focus of this paper. Nonetheless, afterestimating the model, I evaluate its ability to fit several other data series, including GDP.The main quantitative results are as follows. First, I show that the purely deterministic versionof the model (i.e., with the TFP shock shut down) is capable of generating cycles in hours of anempirically reasonable length. For example, at the baseline parameterization, the model generatescycles with a roughly 30-quarter period. This contrasts with the earlier literature on deterministic-fluctuations models, where cycles tended to be either far too short or far too long.Second, these deterministic cycles are highly regular, as evidenced both by a clear repeating pat-6At the reported median posterior parameter estimates, the price mark-up process has persistence 0.90, while the wagemark-up process has persistence 0.97.7Smets and Wouters (2007) are not the only ones to find that mark-up shocks explain a large portion of variation in amodel. For example, Schmitt-Grohe´ and Uribe (2012) report that a wage mark-up shock explains nearly 70% of the varianceof hours growth in their model.80tern in the simulated path of hours as well as by a large spike in its spectrum. Including the TFPprocess, however, easily rectifies this problem, resulting in simulated data that appears realisticallyirregular and a spectrum that closely matches the one estimated from the data.Third, the estimated TFP process is close to that estimated directly from TFP data, with an uncon-ditional variance that is, if anything, slightly smaller. In many conventional business-cycle models,this TFP process on its own would generate fluctuations that are far smaller than those found in thedata. In the model presented here, however, the bulk of the fluctuations come from the deterministicforces, which account for 79% of the standard deviation of hours. Instead, the TFP shock serves pri-marily to accelerate and decelerate the deterministic cycles at random, causing significant fluctuationsin their length while only minimally affecting their amplitude. This result again highlights one ofthe more general insights of the paper: conventional models usually require large persistent shocksin order to produce the types of large persistent fluctuations found in the data. This is the primaryapparent motivation for a large number of the shocks now found throughout the literature. Modelscapable of generating deterministic fluctuations, however, are able to produce large persistent fluctua-tions without any shocks whatsoever. Shocks are only required in order to help match the irregularityof the fluctuations found in the data, and this can be accomplished with shocks that are of a muchmore plausible type and size.The remainder of the paper proceeds as follows. Section 4.2 briefly reviews an older deterministic-fluctuations literature and highlights some of its important failures. Section 4.3 discusses the keyfeatures of the data that the model will attempt to match. Section 4.4 discusses intuitively the basicproperties an economic model must have in order to produce a limit cycle. Section 4.5 introduces themodel and discusses its key properties, and includes several theoretical results establishing conditionsunder which a limit cycle may appear. Section 4.6 presents the main quantitative results of the paper,while section 4.7 concludes.4.2 Literature: Deterministic fluctuationsSince at least as far back as Kaldor (1940), economists have considered models that are capable ofproducing deterministic fluctuations. In the 1970s and 1980s in particular, a large literature emergedthat examined the conditions under which qualitatively and quantitatively reasonable economic fluctu-ations might occur in a purely deterministic setting (see, e.g., Benhabib and Nishimura (1979, 1985),Day (1982, 1983), Grandmont (1985), Boldrin and Montrucchio (1986), Day and Shafer (1987); forsurveys of the literature, see Boldrin and Woodford (1990) and Scheinkman (1990)). By the early1990s, however, this literature seemed to have largely gone dormant.There appear to be several key reasons why interest in deterministic fluctuations may have waned,each of which are addressed in the present paper. First, the earlier literature on deterministic fluctu-ations can be broadly sub-divided into two categories: models with and without fully-microfounded,81forward-looking agents.8 The latter category, which were generally more capable of producing rea-sonable deterministic fluctuations than the former, likely fell out of favor as macro in general movedtoward more microfounded models.Second, in the category of models featuring forward-looking agents, the primary focus was onmodels with a neoclassical, competitive-equilibrium structure.9 Such models were often found torequire relatively extreme parameter values in order to generate deterministic fluctuations. For exam-ple, the Turnpike Theorem of Scheinkman (1976) establishes that, under certain basic conditions metby these models, for a sufficiently high discount factor—i.e., for agents that are “forward-looking”enough—the steady state of the model is globally attractive, so that persistent deterministic fluctua-tions cannot appear.10 While in principle this does not rule out deterministic fluctuations completely,in practice the size of the discount factor needed to generate them was often implausibly low. Forexample, in a survey of deterministic-flucutuations models by Boldrin and Woodford (1990), discountfactors for several of the models they discuss were on the order of 0.3 or less.11 As the present paperillustrates, however, if one departs from the assumptions of a neoclassical, competitive-equilibriumenvironment—for example, if there is a demand externality as in the model presented in section 4.5—then a discount factor arbitrarily close to one can relatively easily support deterministic fluctuations inequilibrium.Third, as suggested above, models producing periodic cycles—that is, cycles which exactly re-peat themselves every k periods—are clearly at odds with the data, where such consistenly regularcycles cannot be found. This can be observed by looking at the spectrum of data generated by such amodel, which will generally feature one or more large spikes at frequencies associated with k-periodcycles. Spectra estimated on actual data generally lack such spikes,12 which suggests less regularityin real-world cycles. To address this issue, papers from the earlier literature largely sought to establishconditions under which such irregular cycles could emerge in a purely deterministic setting (i.e., viachaotic dynamics13). While in a number of cases this was found to be possible, the conditions appear8The first category includes, e.g., Benhabib and Nishimura (1979, 1985) and Boldrin and Montrucchio (1986), while thelatter includes, e.g., Day (1982, 1983).9While there are some exceptions, they are comparatively rare. Perhaps the clearest example is Hammour (1989), chapter1, which is focused on deterministic fluctuations in an environment of increasing returns. Other exceptions include modelsin the search literature that are capable of generating deterministic fluctuations, such as Diamond and Fudenberg (1989),Boldrin et al. (1993), and Coles and Wright (1998). Note however that these search papers were mainly concerned withcharacterizing the set of possible equilibria for a particular model (which for some parameterizations included deterministiccycles), rather than being focused on deterministic cycles directly.10See the discussion in section 4.5.3 for further details.11It is possible in principle to rationalize such low discount factors by choosing a longer period length for the model.However, if households discount the future with a quarterly discount factor of 0.99 or greater—as is frequently the case inthe business-cycle literature—a factor of 0.3 would be associated with a period length of 120+ quarters (30+ years). Sincethe minimum period length of a cycle is two periods, this would generate cycles on the order of 60+ years, well outside ofwhat is normally thought of as the business cycle.12See panel (b) of Figure 4.1 for an example.13Informally, chaotic fluctuations are deterministic fluctuations (see footnote 1) that do not converge to periodic cyclesand for which the paths emanating from two different initial points cannot be made arbitrarily close by choosing those initialpoints sufficiently close together. See, e.g., Glendinning (1994) for a formal definition.82to have been significantly more restrictive even than those required to generate simple periodic cycles.In constrast, rather than restricting attention to a purely deterministic setting, this paper embeds deter-ministic (but highly regular) cyclical mechanisms into a stochastic environment for which irregularityemerges naturally.Finally, being inherently highly non-linear, economic models that are capable of generating deter-ministic fluctuations are often difficult to work with analytically beyond the very simplest of settings,and quantitative results often require computationally-expensive solution algorithms. Prior to rela-tively recent advances in computing technology, obtaining these quantitative results may have beeninfeasible and, as a result, a number of potentially fruitful areas of research—such as, for example,combining deterministic and stochastic cyclical forces—may have gone unexplored.4.3 Data: Hours workedAs noted above, the focus of this paper is on attempting to explain patterns in hours worked. In thissection, I discuss the key properties of the hours data series used as a focal point for both the qualitativeand quantitative discussion that follows.The hours series I use is the quarterly index of nonfarm business hours worked from the USBureau of Labor Statistics, divided by civilian noninstitutional population obtained from the FREDdatabase. The full sample period is from 1948Q1 to 2014Q1, though I focus on the 1960Q1-2012Q4subsample.14 The series was transformed by taking logs and then running the result through a band-pass (BP) filter15 in order to remove long-run trend components, defined here as cyclical componentswith periods greater than 80 quarters (20 years).16,17 The filter was applied to the full 1948Q1-2014Q1sample, after which observations outside of the 1960Q1-2012Q4 sub-sample were discarded.Panel (a) of Figure 4.1 plots the resulting series, with NBER-dated recessions indicated by shadedareas. Two things should be noted from the figure. First, it confirms that the BP-filtered hours seriesexhibits fluctuations that correspond closely to conventional business cycle definitions, as evidencedby the large downward movements during NBER recessions. Second, with the exception of the 1971-1983 period where fluctuations were somewhat more frequent, over the past half-century a full cycle inhours appears to have taken anywhere from 8 to 11 years (32 to 44 quarters) to complete. This pattern,which suggests some degree of regularity, can be confirmed by looking at the spectrum, which isobtained by first orthogonally decomposing the BP-filtered data series into sinusoidal components ofdifferent period lengths, then computing the variance of each such component. Panel (b) of Figure 4.114As alluded to below, prior to 1960 the business cycle appears to have been more irregular. The analysis considered herewill be concerned with the more regular post-1960 period.15See Baxter and King (1999) and Christiano and Fitzgerald (2003). Code to implement Christiano and Fitzgerald’s(2003) filter in several environments is available at https://www.frbatlanta.org/cqer/researchcq/bpf/.16Understanding the mechanisms underyling these low-frequency fluctuations, such as cultural and demographic factors,are beyond the scope of this paper.17Note that none of the high-frequency fluctuations were removed, i.e., a BP(2,80) filter was used. Note also that thechoice of this particular filter is not crucial. In section 4.6, I verify that the key results are robust to a number of alternativefiltering choices.83Figure 4.1: Hours worked data (1960-2012)(a) Data SeriesQuarter  1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010−0.08−0.06−0.04−0.0200.020.040.06 Hours WorkedNBER Recessions(b) Spectrum4 6 8 12 16 24 32 40 8000.511.52x 10−3Period of component (quarters)Notes: Hours Worked series is the log of BLS nonfarm hours worked divided by population, detrended with a BP filterto remove fluctuations with periods greater than 80 quarters. In panel (a), shaded areas are NBER-dated recessions. Forpanel (b), raw spectrum in obtained as the squared modulus of the discrete Fourier transform of the data series (scaled sothat the integral with respect to angular frequency over the interval [−pi, pi] equals the variance of the series). Spectrumin figure is kernel-smoothed raw spectrum. Kernel is a Hamming window with bandwidth parameter 11.plots the result, with the period of the component on the horizontal axis and the associated variance onthe vertical one. From the figure, we see that the bulk of the variation in hours occurs at periodicitiesgreater than 24 quarters, with a peak at around 40 quarters. It should be noted that cycles of thislength are outside the typical range conventionally associated with business cycles. For example, intheir Handbook article, Stock and Watson (1999) define business cycles as fluctuations between 6 and32 quarters in length. While this range appropriately reflected the length of the business cycle over thebroad historical time frame considered in that paper,18 Figure 4.1 suggests that over the more recenttime frame considered here the business cycle has become longer and, apparently, somewhat moreregular.4.4 Conditions for a limit cycleThe basic conditions under which a limit cycle may appear are most easily understood by way ofexample. The example presented here is for a continuous-time bivariate system, but the basic intuitionextends to discrete-time systems and to systems with an arbitrary number of state variables.1918Stock and Watson (1999) report that of the 30 full cycles (peak to peak) identified by the NBER over the period 1858to 1996, 90% were 32 quarters in length or shorter, with the shortest being six quarters. This observation formed the basisfor their definition of the business cycle.19There is a technical issue here that I will sidestep throughout this paper. Certain types of deterministic fluctuationsshare many of the basic qualitative features of a limit cycle, but never exactly repeat themselves. For example, in a bivariate84Suppose we have a bivariate system whose evolution is characterized by the differential equationsz˙ (t) = g (z (t)) and the boundary condition z (0) = z0, where (suppressing explicit dependence ont when no confusion will arise) z ≡ (x, y)′, g : R2 → R2 is some function, and z˙ indicates thetime-derivative of z. It turns out to be more convenient for our purposes to re-cast this system intopolar coordinates, expressing the state as (ϕ, r)′, where ϕ is the angle between the vector z and thepositive x-axis, and r ≥ 0 is the magnitude of z. Since it will not play an important role in the basicconditions for a limit cycle, assume for simplicity that ϕ evolves asϕ˙ = θ¯for some constant rate of rotation θ¯ 6= 0. Next, suppose we haver˙ = −r (ar + b)for some parameters a and b, with ab 6= 0.20 Note that, regardless of the values of a and b, this systemalways has a unique steady state at r = 0, i.e., at the point x = y = 0. Thus, assuming r > 0, thevector z rotates around the origin (in (x, y)-space) at a constant rate, but with a potentially fluctuatinglength. Depending on the signs of a and b, however, the system will exhibit different qualitativeproperties.Conventional economic models generally feature a globally stable steady state. This correspondshere to the case where r˙ < 0 for all r > 0, so that the length of z is always shrinking. This propertyholds when a, b ≥ 0. Panel (a) of Figure 4.2 shows a phase plot of such a case. The individual arrowsindicate the forces acting on the length r at that point in the state space. Also plotted are paths trackingthe evolution of the system beginning from two different starting points: one close to the steady stateand the other far away from it. We can see in this case that there are forces everywhere that tendto push the system inward towards the steady state. As a result, starting from any point the systemconverges to the steady state.The case of a globally unstable steady state (i.e., r˙ > 0 for all r > 0), meanwhile, corresponds tothe case where a, b ≤ 0, an example of which is plotted in panel (b) of Figure 4.2. Here we see thatforces exist everywhere which push the system outwards, away from the steady state, so that if thesystem begins anywhere but at the steady state it will diverge to infinity.The conditions under which a limit cycle may appear correspond to the remaining cases, i.e.,where ab < 0. When this is true there now exists, in addition to the steady-state value r = 0, a seconddiscrete-time system characterized by rotation around the unit circle by θ radians per period, if θ/pi is irrational then thesystem will never return to the same point twice. I will ignore these uninteresting technicalities and apply the term “limitcycle” loosely, with the understanding that the term may not always strictly apply.20N.B.: The fact that ϕ˙ does not depend on r, and vice versa, will not generally hold for an arbitrary g.85Figure 4.2: Conditions for a limit cycle−1 −0.5 0 0.5 1−1−0.500.51(a) Globally Stable SS−1 −0.5 0 0.5 1−1−0.500.51(b) Globally Unstable SS−1 −0.5 0 0.5 1−1−0.500.51(c) Stable Limit Cycle−1 −0.5 0 0.5 1−1−0.500.51(d) Unstable Limit CycleNotes: Panel (a): a = 0.1, b = 0.2. Panel (b): a = −0.1, b = −0.2. Panel (c): a = 1, b = −0.5. Panel(d): a = −1, b = 0.5. In all cases, θ¯ = 0.6pi.(non-negative) solution to r˙ = 0, given byr = rˆ ≡ − ba > 0When the system begins with r = rˆ, it will rotate in the plane at the constant rate θ¯, but move neithertoward nor away from the steady state. The set of points z such that ‖z‖ = rˆ is the limit cycle of thissystem.Of particular interest is the case where the limit cycle is stable, i.e., where neighboring points will86tend to converge to the limit cycle over time. This will occur if (1) when the system begins inside thelimit cycle, forces tend to push it outwards; and (2) when the system begins outside the limit cycle,forces tend to push it inwards. This corresponds to the case where a > 0 and b < 0, so that when ris small r˙ ≈ −br > 0, while for r large r˙ ≈ −ar2 < 0. An example of this is shown in panel (c) ofFigure 4.2. Here we see that, regardless of where the system begins (as long as it is not exactly at thesteady state) it will converge to the limit cycle.For the sake of completeness, the final case where a < 0 and b > 0 is shown in panel (d) of Figure4.2. This corresponds to an unstable limit cycle: if the system begins on the limit cycle (shown asthe dashed circle in the figure), it will remain there forever, but if it begins off the limit cycle, it willeither converge to the steady state (if it begins inside the limit cycle) or diverge to infinity (if it beginsoutside the limit cycle).The preceding analysis highlights the two key properties needed to obtain a stable limit cyclein a general setting: (1) when the system is close to the steady state, there are forces which tendto push it away from the steady state (i.e., the steady state is unstable); and (2) when the system isfar away from the steady state, there are forces which tend to push it towards the steady state (i.e.,the system is non-explosive). A limit cycle then emerges as the set of points where these outwardand inward forces precisely balance. A natural question to ask, then, is whether there are reasonableconditions under which a dynamic economy may exhibit these two key properties. It turns out thatone possible set of such economic conditions is as follows. First, suppose that in a neighborhood ofthe steady state, individual agents’ actions are strategic complements. As is well known, strategiccomplementarity often leads to situations where small changes in state variables can lead to relativelylarge changes in equilibrium outcomes, which is precisely the state of affairs needed for a system tobe (locally) unstable. Second, suppose that when far enough away from the steady state, individualagents’ actions act as strategic substitutes. When this is the case, small changes in state variablestend to produce small changes in equilibrium outcomes, and as a result the system tends to drift backtowards the steady state. In the following section, an economic model is presented which possessesthese two features and which, as a result, will have the potential to generate limit cycles.4.5 The unemployment-risk modelIn this section I present a simple economic model that is capable of generating limit cycles. In themodel, which is based closely on Beaudry et al. (2014), households begin each period with a stockof durable goods and must decide how many additional goods to purchase in the goods market. Ab-stracting for the moment from forward-looking behavior, there are two key static factors that affectthis decision. First, household demand is decreasing in the size of the current stock of durables: whentheir existing stock of durables is low, households want to purchase more, and vice versa. Second,because of a self-insurance motive, household demand is decreasing in the unemployment rate. Thereare two imperfections in the model that cause this self-insurance behavior to emerge. First, there is87a matching friction in the spirit of Diamond-Mortensen-Pissarides, which creates the possibility thata household may not find employment when looking for a job. Second, households are unable toperfectly insure against this idiosyncratic unemployment risk. The upshot is that an increase in theunemployment rate causes them to reduce their demand for new goods.The combination of these two factors produces the following mechanism by which deterministicfluctuations emerge in the model: if households have an excess stock of durables, they reduce theirdemand for new goods. This fall in demand then increases the unemployment rate, which causeshouseholds to further reduce their demand, further increasing the unemployment rate, and so on, sothat, in equilibrium, output falls by significantly more than the initial fall in demand. This mul-tiplier mechanism—which occurs because of strategic complementarity in households’ purchasingdecisions—drives the excess sensitivity in the dynamic system which is a pre-condition for local insta-bility. Once the economy reaches full or zero employment, however, the self-insurance mechanism isnot operative, and thus the excessive sensitivity that creates instability disappears. In its place, inwardforces—arising because of strategic substitutability, which in turn operates through the price of newgoods—emerge that prevent the economy from exploding. The combination of these locally-outwardand globally-inward forces creates the conditions for a limit cycle to occur.4.5.1 Static versionBefore presenting the full dynamic model in detail, I begin by briefly presenting a simpler version ofthe model that is static in nature, highlighting the key properties that will be important in generatinglimit cycles in a dynamic setting. Further details and in-depth analysis of this static model can befound in Beaudry et al. (2014).Consider an environment populated by a mass one of households. In this economy there aretwo sub-periods. In the first sub-period, households purchase consumption goods and try to findemployment. As there is no money in this economy, when the household buys consumption goodsits bank account is debited, and when (and if) it receives employment income its bank account iscredited. As we shall see, households will in general end the first sub-period with a non-zero bankaccount balance. Thus, in the second sub-period, households resolve their net asset positions byrepaying any outstanding debts or receiving a payment for any surplus. These payments are made interms of a second good, referred to here for simplicity as household services. Household services arealso the numeraire in this economy.Preferences for the first sub-period are represented byU (c)− ν (`)where c represents consumption of clothes and ` ∈[0, ¯`]is the labor supplied by households in theproduction of goods, with ¯` the agent’s total time endowment. U is assumed to be strictly increasingand strictly concave, while the dis-utility of work function ν is assumed to be strictly increasing and88strictly convex, with ν(0) = 0. Households are initially endowed with X units of consumption goods,which they can either consume or trade. In the dynamic version of the model, X will represent astock of durable goods and will be endogenous. Trade in consumption goods is subject to a coordi-nation problem because of frictions in the labor market. At the beginning of the first sub-period, thehousehold splits up responsibilities between two members. The first member, called the buyer, goesto the goods market to make purchases. The second member searches for employment opportunitiesin the labor market. The goods market functions in a Walrasian fashion, with both buyers and firmstaking the price of these goods p (in units of household services) as given. The market for labor inthis first sub-period is subject to a matching friction, with sellers of labor searching for employersand employers searching for labor. The important information assumption is that buyers do not know,when choosing how much to buy, whether the worker member of the household has secured a match.This assumption implies that buyers make purchase decisions in the presence of unemployment risk.There is a large set of potential consumption goods firms in the economy who can decide to searchfor workers in view of supplying goods to the market. Each firm can hire one worker and has accessto a decreasing-returns-to-scale production function F (`), where ` is the number of hours workedfor the firm.21 Production also requires a fixed cost k in terms of the output good, so that the netproduction of a firm hiring ` hours of labor is F (`) − k. Firms search for workers and, upon findinga worker, they jointly decide on the number of hours worked and on the wage to be paid. The fixedcost k is paid before firms can look for workers. Upon a match, the determination of the wage andhours worked within a firm is done efficiently though a competitive bargaining process,22 so that inequilibrium pF ′ (`) = w, where w is the wage, expressed in terms of household services.23The labor market operates as follows. All workers are assumed to search for employment. Lettingn represent the number of firms who decide to search for workers, the number of matches φ is thengiven by the short side of the market, i.e., φ = min {n, 1}. The equilibrium condition for the goodsmarket is then given byc−X = φF (`)− nkwhere the left-hand side is total purchases of consumption goods and the right-hand side is the totalavailable supply after subtracting firms’ fixed costs. Firms enter the market up to the point where21It is also assumed that F is such that both F ′ (`) ` and [F (`)− F ′ (`) `] are strictly increasing functions of `. Thisproperty is exhibited, for example, by the Cobb-Douglas function F (`) = A`α.22By “competitive bargaining”, I mean any bargaining process such that the equilibrium outcome satisfies (1) that workersare paid their marginal product in a match, and (2) that, conditional on being matched, workers supply and firms hire theindividually-optimal number of hours at the equilibrium wage. This can be microfounded by assuming, for example, thatall “matched” firms and workers meet in a secondary labor market, and that this secondary market operates in a Walrasianfashion.23As discussed in Beaudry et al. (2014), the assumption of a competitive bargaining process is for simplicity. The mainmechanisms are robust to alternative bargaining protocols.89expected profits are zero. This condition can be written as24φn[F (`)− wp `]= kAt the end of the first sub-period, a household’s net asset position a, expressed in units of house-hold services, is given by a = w` − p (c−X) if the worker was employed, and a = −p (c−X) ifthe worker was unemployed. Rather than explicitly modelling the second sub-period, for simplicityassume that the continuation value function for the second sub-period, V , is given by25V (a) =va if a ≥ 0(1 + τ) va if a < 0where v, τ > 0 are parameters. This function is piecewise linear and concave, with a kink at a = 0.26Here, the marginal value of assets is given by v when assets are positive and (1 + τ) v when assets arenegative. Since buyers in general face unemployment risk when making their purchase decisions, thewedge between the marginal value of assets when in deficit and that when in surplus generates self-insurance behavior, whereby a fall in the employment rate causes buyers to reduce their purchasesout of increased concern that they will end up in the costly unemployment state. This mechanism iscentral to the strategic complementarity that emerges in the model, which in turn is what will allowthe dynamic version of the model to generate limit-cycle behavior. The strength of this mechanism,meanwhile, is governed by the parameter τ . Given the above value function V , the buyer’s problem isto choose e to maximizeU (X + e) + φ [−ν (`) + v (w`− pe)]− (1− φ) (1 + τ) vpesubject to e ≥ 0, where e ≡ c−X is purchases of new goods. The worker’s problem, meanwhile, isto choose ` to maximize −ν (`) + v (w`− pe).EquilibriumLetting ej denote purchases by household j and e the average level of purchases in the economy, onemay show that household j’s optimal consumption-choice decision is characterized by27U ′ (X + ej) = p (e) v [1 + τ − τφ (e)] (4.1)24As in Beaudry et al. (2014), assume that searching firms pool their ex-post profits and losses so that they each makeexactly zero profits in equilibrium, regardless of whether they are matched with a worker.25See Section 2.2 in Beaudry et al. (2014) for a discussion of how to microfound such a value function.26As noted in Beaudry et al. (2014), what matters here is that the marginal value of assets be smaller in surplus than indeficit. The piecewise linearity property is assumed only for tractability.27See Beaudry et al. (2014).90where p (·) and φ (·) are the price of consumption goods and the employment rate, respectively,expressed as functions of aggregate purchases. The left-hand side of (4.1) is simply householdj’s marginal utility of consumption. The right-hand side, meanwhile, captures buyer j’s expectedmarginal-utility cost of funds. When the economy is at full employment (φ (e) = 1), this is simplyequal to the price p (e) of consumption goods in terms of household services, times the marginal valuev of those services when assets are non-negative. When there is unemployment, however, the buyerfaces some positive probability of ending up in the negative-asset state, which is associated with ahigher marginal value of assets (i.e., (1 + τ) v). As a result, the expected marginal-utility cost offunds is higher and, all else equal, household j would choose a lower level of purchases.An equilibrium for this economy is given by a solution to (4.1) with the additional restriction thatej = e. To understand how the equilibrium is affected by shifts in X , note the following properties ofthe equilibrium functions p (·) and φ (·). First, one may show that φ (e) = min {e/e∗, 1}, where e∗is the output (net of fixed costs) produced per firm when there is a positive level of unemployment.28Second, one may show that p (·) is a continuous function of e, with p′ (e) = 0 for e < e∗, andp′ (e) > 0 for e > e∗.29 The consequences of these two properties for the marginal-utility cost offunds (i.e., the right-hand side of (4.1)) are illustrated by the curve labelled “cost of funds” in panel (a)of Figure 4.3. For e sufficiently small, the curve is downward-sloping: as e rises, output is increasedalong the extensive labor margin, lowering the unemployment rate and making purchases feel lessexpensive to households. Once e reaches the full-employment level e∗, however, additional increasesin output come via the intensive labor margin, which is associated with a rising price and thus anincreased cost of funds.The two regimes—unemployment and full employment—are associated with different equilibriumresponses to a rise in the endowmentX .30 Panel (a) of Figure 4.3 shows the case for the unemploymentregime. The economy is initially in equilibrium at the level e1 of purchases, which occurs at theintersection of the cost of funds curve and the solid marginal-utility function U ′ (X + e). A rise in theendowment by ∆X then shifts this marginal-utility function to the left by ∆X units, as representedby the dashed curve in the figure. We see that the equilibrium level of purchases falls as a result ofthe rise in X , and furthermore that it falls by more than ∆X (so that total consumption c = X + efalls). This amplified response is due to the strategic complementarity that exists in the unemploymentregime: a rise in the endowment causes households to reduce their demand for new goods which, via28When there is unemployment, the “min” matching function and the firm’s zero-profit condition together imply F (`)−F ′ (`) ` = k. Since k is a constant, conditional on there being unemployment this implies that ` = `∗, where `∗ solves thisequation. Output net of fixed costs is then e∗ ≡ F (`∗)− k.29Combining the household’s labor supply condition and the firm’s labor demand condition, one may obtain p =ν′ (`) / [vF ′ (`)]. As pointed out in footnote 28, when e < e∗ we have ` = `∗, so that p = p∗ ≡ ν′ (`∗) / [vF ′ (`∗)].Further, once the economy achieves full employment, a rise in output must come through the intensive margin of labor (i.e.,through a rise in `), which causes p (·) to be increasing in e on e > e∗.30As shown in Beaudry et al. (2014), if τ is sufficiently large there may be more than one equilibrium. While this is aninteresting theoretical possibility, the evidence obtained from the quantitative exercise of section 4.6, though not conclusive,gives no indication that multiple equilibria are of concern. I therefore restrict attention throughout this paper to the casewhere the equilibrium is unique, i.e., where τ is not too large.91Figure 4.3: Static equilibrium determinationecost offundse1Xe2U'(X+e)ΔXecost offundse1Xe2U'(X+e)ΔX(a) Unemployment Regime (b) Full-employment Regimean extensive labor margin adjustment, lowers the employment rate φ, which in turn raises the cost offunds, causing households to reduce purchases further, further lowering the employment rate, etc.In contrast, panel (b) of Figure 4.3 shows the same experiment but beginning from the full-employment regime. In this case, we again see that a rise in X is associated with a fall in equilibriumpurchases, but in this case the fall is by less than ∆X (so that total consumption rises). This dampedresponse occurs as a result of the strategic substitutability that exists when the economy is at full em-ployment: a rise in the endowment causes households to reduce their demand for new goods which,via an intensive labor margin adjustment, lowers hours-per-worker, which lowers the price p, in turnlowering the cost of funds and causing households to increase their purchases.The sensitivity of purchases to changes in X in the unemployment regime because of strate-gic complementarity, and the corresponding insensitivity in the full-employment regime because ofstrategic substitutability, will play a crucial part in generating limit cycles in the dynamic version ofthe model. Note also that the sensitivity of e to X in the unemployment regime is increasing in thesteepness of the slope of the cost of funds schedule in that regime. Since this steepness in turn dependspositively on the parameter τ , we see that τ captures the degree of strategic complementarity in theunemployment regime.4.5.2 Baseline dynamic modelConsider now a dynamic version of the above economy. Time is discrete, and each period is dividedinto two sub-periods, with the economy operating in each such sub-period as in the static case. Theprincipal difference from the static model is that the stock of durable goods brought into a period is92now endogenous, accumulating according toXt+1 = (1− δ) (Xt + γet) (4.2)where Xt is the stock of durables brought into period t and et is quantity of consumption-goodspurchases in period t. For simplicity, I assume that a constant fraction γ ∈ (0, 1] of these purchasesare durable.31 δ ∈ (0, 1] is the depreciation rate.The household’s labor supply decision is entirely static and therefore the same as in the previoussubsection (i.e., `t is chosen each period to maximize −ν (`t) + v (wt`t − ptet)). Buyers, meanwhile,face a dynamic optimization problem, choosing ct and et to maximize the objective function∞∑t=0βt {U (ct) + φt [−ν (`t) + v (wt`t − ptet)]− (1− φt) (1 + τ) vptet} (4.3)subject to ct = Xt + et and the accumulation equation (4.2), and taking `t as given.32I assume that a steady state for this economy exists and is unique. As is the case in the staticmodel, it can be verified that this is true as long as τ is not too large.33 I further assume that this steadystate satisfies ` < ¯`, so that the household’s time constraint is not binding at the steady state.4.5.3 Limit cycles in the dynamic modelThe myopic caseConditions under which limit cycles may appear in this model can be understood most easily in themyopic case where β = 0. In this case, we simply have a repeated sequence of the static economydiscussed in section 4.5.1, with the only linkage between them being the inherited stock of durablegoods. We may characterize the equilibrium evolution of the stock of durables over time asXt+1 = (1− δ) [Xt + γe (Xt)] ≡ g (Xt)where e (Xt) expresses the equilibrium level of purchases at date t as a function of the only statevariable, Xt. This equilibrium is determined entirely as it was in Figure 4.3, with the unemploy-ment regime characterized by strategic complementarity and the full-employment regime by strategicsubstitutability.31In the quantitative exercise below, I will interpret “durables” as including both conventional durable goods as well asresidential investment, which is conceptually similar.32In order to avoid expanding heterogeneity between individuals over time, individuals are assumed to borrow and lendvia their bank account balances only within a period but not across periods. In other words, households are allowed tospend more than their income in the first sub-period of a period, but must repay any resulting debt in the second sub-period.Similar assumptions were used in Lagos and Wright (2005) and Rocheteau and Wright (2005), and more recently in Kaplanand Menzio (2014), in order to avoid having to track the asset positions of all agents in the economy over time.33See Beaudry et al. (2014). See also the comments in footnote 30, which apply equally here.93Recall the two basic conditions discussed in section 4.4 which are required to generate a stablelimit cycle: (1) a locally unstable steady state, and (2) global non-explosiveness. Letting X¯ denote thesteady state level of durables, these two conditions correspond mathematically to (1)∣∣g′(X¯)∣∣ > 1,and (2) |g′ (X)| < 1 for∣∣X − X¯∣∣ sufficiently large, whereg′ (X) = (1− δ)[1 + γe′ (X)]It is straightforward to verify that the second condition necessarily holds here, as follows. Suppose Xis sufficiently small so that the economy is in the full-employment regime. As was shown earlier, a risein X in this regime is associated with a fall in e, but by less than the rise in X , i.e., −1 < e′ (X) < 0.Thus, (1− δ) (1− γ) < g′ (X) < 1− δ, and therefore |g′ (X)| < 1 clearly holds.34 Suppose insteadthat X is very large. In this case it can be verified that the non-negativity constraint e ≥ 0 binds, sothat e′ (X) = 0 and therefore g′ (X) = 1− δ, and thus again |g′ (X)| < 1 holds.Next, suppose the steady state of the system is in the unemployment regime. Then from theanalysis for the static model, we know that e′(X¯)< −1, i.e., a rise in the stock of durables leads toa more than one-for-one fall in purchases. Whether or not e falls sufficiently so that the first conditionfor a stable limit cycle holds will depend on the strength of the complementarity in this regime, i.e.,on τ . For smaller values of τ , i.e., those for whiche′(X¯)> − 2− δγ (1− δ) ≡ κthe complementarity is relatively weak, and thus g′(X¯)> −1. In this case, the steady state is stable,so that a limit cycle will not appear. On the other hand, for larger values of τ (i.e., those for whiche′(X¯)< κ), we will have g′(X¯)< −1, and thus the steady state is unstable. In combination withthe fact that the system is non-explosive (as argued above), we see that in general a stable limit cyclewill emerge in this case.The general caseThe previous subsection showed that, when β = 0, limit cycles can emerge in the unemployment riskmodel. While the myopic case was useful for building intuition, of more general interest is whetherlimit cycles may occur for an arbitrary β. It is not immediately obvious that this should hold, andindeed, as a “Turnpike Theorem” (due to Scheinkman (1976)) below highlights, in a class of modelswidely used in the literature, for β sufficiently close to one limit cycles cannot occur.In particular, consider a general deterministic dynamic economy with date-t state vector zt ∈ Rn.LetW (zt, zt+1) denote the period-t return function when the current state is zt and the subsequent pe-34It is worth emphasizing that strategic substitutability in the full-employment regime is the key property generating thisrelative insensitivity of e to changes in X .94riod’s state is zt+1.35 The following theorem characterizes the solution to the problem of maximizinglifetime utility∑βtW (zt, zt+1), where β is the discount factor.Turnpike Theorem. (Scheinkman (1976)) If W is concave, then there exists a β¯ < 1 such that ifβ¯ ≤ β ≤ 1 then the steady state is unique and globally stable.36The key property that ensures global stability in this theorem is the assumption thatW is concave.Since, all else equal, fluctuations are sub-optimal whenW is concave, when β is sufficiently close toone it is in general optimal to take temporarily costly action in the present in order to avoid permanentfluctuations in the future. This in turn implies global convergence to the steady state, so that limitcycles cannot occur. Concavity of W is a property that holds in a wide variety of economic modelsthat have become standard in the literature, including nearly all quantitative models of the businesscycle. As we shall see, however, in the unemployment-risk model discussed above, concavity of Wmay be violated, in which case global stability may not obtain.As a first step in establishing the potential for limit cycles in the unemployment-risk model, thefollowing proposition verifies that the system satisfies the second condition needed for a stable limitcycle (i.e., non-explosiveness).Proposition 4.1. Given any initial endowment of durables X0, lim supt→∞ |Xt| <∞.Proof. All proofs in Appendix C.1.Proposition 4.1 ensures that in the limit the system either exhibits deterministic fluctuations (suchas a limit cycle) or converges to a fixed point. The following proposition establishes that, in contrastto models for which the Turnpike Theorem applies, local instability is possible in this model for anarbitrarily high discount factor.Proposition 4.2. There exists parameter values and functional forms such that, for some β¯ < 1, ifβ¯ ≤ β < 1 then the (unique) steady state is locally unstable.In combination with Proposition 4.1, Proposition 4.2 confirms that there are parameter values andfunctional forms for which the model will generate deterministic fluctuations even if β is arbitrarilyclose to one. The reasons for the failure of the Turnpike Theorem to hold for this model can beclarified as follows. Suppose the steady state of the model is in the unemployment regime, and letW (Xt, Xt+1) be a period-t return function such that the solution to the problemmax{Xt+1}∞∑t=0βtW (Xt, Xt+1) (4.4)35Note that, in this formulation, W implicitly incorporates any constraints and static-equilibrium outcomes, so thatW (zt, zt+1) is the equilibrium period-t return conditional on the current and next-period state being zt and zt+1, re-spectively.36For a proof and more formal statement of the theorem, see Scheinkman (1976) Theorem 3.95implements the equilibrium of the model in a neighborhood of this steady state.37 If it turns out thatWis concave, then the Turnpike Theorem implies that the model cannot generate limit-cycle dynamics.The following proposition establishes that in factW may not be concave.Proposition 4.3. There exists parameter values and functional forms such that, in the neighborhoodof an unemployment-regime steady state,W is not concave.Intuitively, non-concavity of W can arise as a result of a “bunching” mechanism in the model:because unemployment risk is low, when other agents are purchasing lots of goods it is a good timefor an individual agent to purchase goods. Similarly, when other agents are purchasing few goods, itis a bad time for an individual agent to buy goods. If sufficiently strong, this bunching mechanism—which arises precisely because of the strategic complementarity in the unemployment regime—leadsto a tendency to have periods of high durables accumulation alternating with periods of low durablesaccumulation, i.e., deterministic fluctuations.The final proposition of this section clarifies the importance of the parameter τ in controlling thestrength of this bunching mechanism, and therefore in influencing whether or not the economy will beable to generate limit-cycle dynamics.Proposition 4.4. For τ sufficiently close to zero, the steady state is stable.Proposition 4.4 thus confirms that, if τ is not sufficiently large, the degree of strategic complemen-tarity is too small to produce an unstable steady state.4.6 Quantitative exerciseThis section presents the main quantitative results of the paper. I estimate a version of the dynamicmodel discussed above, with the primary goal of establishing that it is capable of matching the keyquantitative features of the hours data discussed in section 4.3.The baseline dynamic model presented in section 4.5 was constructed with an eye toward ana-lytical tractability. As a result, that model lacks many of the features which are known to be helpfulin quantitatively matching the data, and includes several others which, while not central to the keymechanisms, turn out to be restrictive in a quantitative setting. Since the main purpose of the exercisein this section is quantitative in nature, I make several adjustments to the model designed to help it inthat regard.First, as is well known, dynamic systems with a single state variable have considerable difficultyin producing deterministic fluctuations with the basic qualitative properties that we observe in macroe-conomic aggregates. In particular, deterministic fluctuations in such models tend to be erratic, with thesystem often jumping back and forth from one side of the steady state to the other every few periodsor less. Thus, if the unemployment-risk model is to have any chance of successfully replicating key37An example of such aW is found in the proof of Proposition 4.3.96features of the data, it will require the addition of at least one other state variable. To this end, andfollowing much of the quantitative business cycle literature, I now assume that the household exhibitsinternal habit-formation in consumption,38 so that its period utility for consumption is now given byU (ct − hct−1)Here, h ∈ [0, 1) is a parameter controlling the degree of habit persistence.Second, the relatively simple structure of the baseline model produces a stark dichotomy, wherebyin the unemployment regime all output adjustments occur along the extensive labor margin, while inthe full-employment regime all adjustments occur along the intensive margin. In order to relax thisstark dichotomy, in the quantitative version of the model I allow firms to be heterogeneous in termsof their fixed costs. That is, rather than assuming that all firms have fixed cost k, I assume that then-th firm has fixed cost k (n) ≥ 0, where k (·) is a non-decreasing function. This will allow for thepossibility of there being regions where both extensive and intensive labor margin adjustments mayoccur.39,40Third, as discussed earlier and in contrast to what is observed in the data, purely deterministicmodels of economic fluctuations tend to yield cycles of a constant length. This can be observed eitheras a very regular pattern in a plot of time series data generated from the model, or as one or more largespikes in the spectrum estimated from that data.41 One of the key contributions of this paper is toshow that by introducing a relatively small amount of randomness into a limit-cycle model it becomespossible to produce realistically irregular fluctuations. To this end, I also include in the model anexogenous random TFP process, θ˜t.4238The key desirable property for a second state variable here is that it introduces momentum into the dynamics of X , sothat movements from a high to a low level of X and back are gradual, rather than rapid as they are when X is the only statevariable. Consumption habit exhibits this property by reducing period-to-period fluctuations in household demand, with theadded advantages that it maintains tractability and keeps the model as close as possible to the baseline version discussedearlier. Nonetheless, there are likely a number of other choices (e.g., adjustment costs in investment or employment) thatcould have been made instead and that would have delivered similar qualitative dynamics.39To see this, note that the marginal firm entrant must earn zero expected profit, which in the unemployment regime isequivalent to the condition F (`)−F ′ (`) ` = k(n), where n is the index of the marginal entrant. A rise in the employmentrate is associated with a rise in n, which (weakly) increases the right-hand side of this expression. Since the left-hand sideof this expression is strictly increasing in `, this then implies that a rise in the emploment rate is in general also associatedwith a rise in hours-per-worker, i.e., both extensive and intensive labor margin adjustments occur.40The functional form chosen for this k (·) (discussed below) will nest the baseline case of a constant fixed cost. Sincethe parameters of this function will be estimated, the data will ultimately choose the degree to which k (·) is non-constant.41One may show that the spectrum associated with any limit cycle is infinitely high at a countable number of points (i.e.,a countable sum of Dirac delta functions), and zero everywhere else.42For convenience, in order to retain certain analytical properties that are helpful in a computational setting, I assumethat firms’ fixed costs and households’ second-sub-period value functions also fluctuate with the TFP process. Output, fixedcosts, and the value function are thus given by θ˜tF (·), θ˜tk (·), and θ˜−1t V (·), respectively.974.6.1 Functional forms, calibration and estimationProduction is assumed to be of the Cobb-Douglas formF (`) = A`αUtility over consumption (net of habit) is assumed to be of the formU (C) = aC − b2C2while disutility of labor is taken to be of the formν (`) = ν11 + ω`1+ωThe fixed cost of the n-th firm is assumed to be given byk (n) =0 n ≤ n0n−n0η k¯ n0 < n < n0 + ηk¯ n ≥ n0 + ηwhere n0, η and k¯ are parameters. This function is piecewise linear with three regimes: low-n firmshave fixed cost zero, high-n firms have fixed cost k¯, and over the intermediate range k rises linearlyfrom zero to k¯.43 Finally, I assume the TFP process is given byθt ≡ log(θ˜t)= ρθt−1 + t, t ∼ N(0,( σ100)2)Several of the model parameters were directly calibrated. In particular, I set the labor share ata standard value of α = 2/3. The inverse Frisch elasticity was calibrated at the widely used levelω = 1. I set the depreciation rate and discount factor at standard values of δ = 0.025 and β = 0.99,respectively, and normalize the maximum fixed cost and scale parameter in the second sub-periodvalue function at k¯ = 1 and v = 1, respectively. Finally, the fraction of purchases entering the durablesstock was calibrated at γ = 0.192, which is the average ratio of durables to total consumption in theNational Income and Product Accounts data.44 The remaining parameters were estimated.Solving the model for a particular parameterization was done using the parameterized expecta-tions (PE) approach.45 Given this solution, a large data set (T = 100, 000 periods in length) was43Quadratic utility and the piecewise-linear form for k(·) were assumed for tractability and computational efficiency.None of the key properties of the model rely on these assumptions.44As noted above, I include the conceptually-similar residential investment under the heading of “durables”. The figureof 0.192 can thus be obtained from NIPA data as the average of (Durable goods + Residential investment)/(Consumption +Residential investment) over the sample period 1960Q1-2012Q4.45See, for example, den Haan and Marcet (1990) and Marcet and Marshall (1994). Details can be found in Appendix C.2.98simulated and, after taking logs of the resulting hours series and detrending it with the same BP filteras used for the data, the spectrum of log-hours was estimated. The non-calibrated parameters werethen estimated so as to minimize the average squared difference between the model spectrum and thespectrum estimated from the data (see panel (b) of Figure 4.1). Further details of the solution andestimation procedure are presented in Appendix C.2.Estimated parameter values are reported in Table 4.1. Several things should be noted. First, theTFP process is close to the process that would be estimated directly from productivity data. For exam-ple, using John Fernald’s (2014) measure of business-sector labor productivity growth over the sampleperiod (1960Q1-2012Q4),46 after cumulating, linearly detrending, and fitting an AR(1) process, oneobtains a persistence estimate of 0.974 and an innovation standard deviation of 0.713%, yielding anunconditional productivity standard deviation of 3.16%.47 The corresponding parameters estimatedfor the unemployment-risk model, meanwhile, are ρ = 0.969 and σ = 0.570, respectively, whichyields an unconditional standard deviation of 2.30%. The fact that the model only features a singleshock, and that the variance of that shock in the model is, if anything, smaller than its data counterparthighlights the more general observation that models featuring deterministic fluctuations may not re-quire the presence of large amounts of exogenous variation in order to generate empirically reasonablebusiness cycles.The only other parameter with a clear comparator in the data or literature is habit persistence,which is estimated here to be h = 0.76, well within the range of standard estimates obtained elsewherein the literature. For example, Smets and Wouters (2007) report a 90% confidence interval for habit of(0.64, 0.78), while Justiniano et al. (2010) report a 90% confidence interval of (0.72, 0.84).The remaining parameters in Table 4.1 are composed mainly of uninteresting scale parameters,and parameters for which few if any precedents exist. The parameter τ , which captures the strengthof the household’s desire to reduce spending in response to a rise in unemployment risk, falls into thelatter category. Given its central role in the model, however, it deserves some comment. If interpretednarrowly as a one-period financial premium on debt vis-a`-vis saving, the estimate of τ = 0.27, or27%, clearly exceeds typical borrowing-lending spreads as reported in the literature. However, thereare several reasons to think this view of τ may be overly restrictive. First, in order to avoid signifi-cantly complicating the model, conditional on the employment rate an individual worker’s probabilityof being employed is assumed to be independent from quarter to quarter. If the actual employmentstate of an individual exhibits persistence, then considering only one-period financial costs may un-derstate households’ desire to reduce spending in response to an increase in unemployment. Second,borrowing-lending spreads that reflect average borrowing rates faced by all households may not ac-curately reflect rates faced by unemployed individuals, which are likely to be higher. Third, manyunemployed individuals may in fact be unable to access financial markets at all, instead being forced46Available at http://www.frbsf.org/economic-research/economists/jfernald/quarterly tfp.xls.47Similar values are obtained when using Fernald’s TFP or utilization-adjusted TFP measures instead of labor productiv-ity.99Table 4.1: Parameter valuesParameter Value DescriptionEstimated Parametersa 12.535 Marginal utility of consumption, interceptb 2.247 Maginal utility of consumption, slopeh 0.761 Habit persistenceν1 13.274 Labor disutility scaling factorτ 0.270 Premium on debtA 3.199 Constant productivity factorn0 0.843 Measure of firms with zero fixed costη 0.091 Measure of firms over which fixed cost is risingρ 0.969 Persistence of TFPσ 0.570 100 × s.d. of innnovation to TFPCalibrated Parametersα 0.667 Labor shareω 1 Inverse Frisch elasticityδ 0.025 Depreciation of durablesβ 0.99 Discount factork¯ 1 Maximum firm fixed costγ 0.192 Fraction of purchases entering durables stockto rely on costly asset liquidations and/or reduced consumption levels in order to meet their obliga-tions, the potential for either of which may cause households to strongly reduce their desired spending.To the extent that any or all of these factors should be subsumed into τ , the value estimated here maynot be unreasonable.4.6.2 Main resultsTo illustrate the deterministic mechanisms, I first report results obtained when shutting down the TFPshock (i.e., setting σ = 0).48 Panel (a) of Figure 4.4 plots a simulated 212-quarter sample49 of log-hours generated from this deterministic model. Two key properties should be noted. First, the modelis clearly capable of generating cycles of a reasonable length, which in this case is approximately 30quarters. As noted in section 4.2, the apparent inability of models of deterministic flutcutations togenerate cycles of quantitatively reasonable lengths appears to have been one of the factors leading to48In particular, I first obtained the PE coefficients from the full stochastic model. The simulation results for the determin-istic model were then generated using these stochastic PE coefficients, but feeding in a constant value θt = 0 for the TFPprocess. In other words, agents in the deterministic model implicitly behave as though they live in the stochastic world. Asa result, any differences between the deterministic and the stochastic results in this section are due exclusively to differencesin the realized sequence of TFP shocks, rather than differences in, say, agents’ beliefs about the underlying data-generatingprocess.49This is equal to the length of the sample period of the data.100the abandonment of this literature. As this exercise demonstrates, however, unreasonable cycle lengthsare by no means an unavoidable property of these models. Second, notwithstanding the reasonablecycle length, it is clear when comparing the simulated data in Figure 4.4 to the actual data in Figure 4.1that the fluctuations in the deterministic unemployment-risk model are far too regular,50 a shortcomingshared by many earlier models of deterministic fluctuations.Figure 4.4: Deterministic model(a) Sample of Simulated Hours Worked0 20 40 60 80 100 120 140 160 180 200−0.04−0.03−0.02−0.0100. Spectrum of Hours Worked4 6 8 12 16 24 32 40 8000.511.522.533.54x 10−3Period of component (quarters)  DataDeterministic ModelNotes: Panel (a) shows 212-quarter simulated sample (same size as data set) of BP-filtered log(hours worked) (φt`t)generated from the deterministic model. Initial simulated series was 252 quarters long, with first and last 20 quartersdiscarded after BP-filtering. Details for computation of model spectrum in panel (b) can be found in Appendix C.2.These properties of the deterministic model—i.e., a highly regular 30-quarter cycle—can also beseen clearly in the frequency domain. Panel (b) of Figure 4.4 plots the spectrum for the deterministicmodel (dashed line), along with the spectrum for the data (solid line) for comparison.51 Consistentwith the pattern in the time domain, the spectrum exhibits a peak at around 30 quarters. Further,the regularity of the cycle is manifested as a large spike in the spectrum. In contrast, the spectrumestimated from the data is much flatter.Re-introducing the TFP shock into the model, we see a markedly different picture in both the timeand frequency domains. Panel (a) of Figure 4.5 plots a 212-quarter sample of log-hours generated fromthe stochastic model. While clear cyclical patterns are evident in the figure, it is immediately obviousthat the inclusion of the TFP shock results in fluctuations that are significantly less regular than thosegenerated in the deterministic model, appearing qualitatively quite similar to the fluctuations found in50Note that the cycles clearly do not exactly repeat themselves. As alluded to in footnote 19, this property is due to thediscrete-time formulation of the model. In a continuous-time version of the model, the cycles would necessarily repeatthemselves, a direct consequence of the Poincare´-Bendixson Theorem (see, e.g., Guckenheimer and Holmes (2002), p. 44).51Note that the model was not re-estimated after shutting down the TFP shock. As such, there may be alternative param-eterizations of the deterministic model that are better able to match the spectrum in the data.101Figure 4.1 for actual data. This is confirmed by the spectrum, which is plotted in panel (b) of Figure4.5 alongside the data spectrum. Also plotted is a pointwise 90% simulated confidence interval fromthe model for data sets of the same length as the data (i.e., 212 quarters).52 The stochastic modelclearly matches the data quite well in this dimension, including possessing a peak near 40 quartersand, as compared to the deterministic model, lacking any large spike. The good fit of the model canalso be seen by looking at the autocovariance function (ACF) of hours, i.e., Cov (Lt, Lt−k), where kis the lag (in quarters). Panel (a) of Figure 4.6 plots the result for the first 40 lags for both the data andmodel, along with pointwise 90% confidence intervals. As the figure shows, the curves lie nearly ontop of one another, indicating that the model matches the data very well in this dimension also.53Figure 4.5: Stochastic model(a) Sample of Simulated Hours Worked0 20 40 60 80 100 120 140 160 180 200−0.06−0.04−0.0200.020.040.06Quarters(b) Spectrum of Hours Worked4 6 8 12 16 24 32 40 8000.511.522.533.5x 10−3Period of component (quarters)  DataStochastic Model90% CINotes: Panel (a) shows 212-quarter simulated sample (same size as data set) of BP-filtered log(hours worked) (φt`t)generated from the stochastic model. Initial simulated series was 252 quarters long, with first and last 20 quartersdiscarded after BP-filtering. Details for computation of model spectrum in panel (b) can be found in Appendix C.2.Dotted lines show a pointwise 90% confidence interval for the spectrum that would be estimated from a model-generateddata set of the same length as the actual data set (i.e., 212 quarters).To verify that the good fit of the spectrum is not driven by the choice of filter, Figure 4.7 plotsthe data and model spectra for hours under four alternative filtering choices.54 Panels (a)-(c) presentresults for three alternative band-pass filters with different upper bounds (100, 60, and 40 quarters,respectively), while panel (d) plots spectra using a Hodrick-Prescott filter with parameter 1600. Asthe figure shows, the model fits the data very well in all cases.52That is, if the model were the true data-generating process, then at each periodicity the spectrum estimated from thedata would lie inside the confidence interval 90% of the time.53Note that the ACF is simply the inverse Fourier transform of the spectrum. Since the spectrum of the model and dataare similar, we would expect the ACF to be similar as well, a property clearly verified in Figure 4.6.54Note that the model spectra were obtained using the baseline model parameters as reported in Table 4.1.102Figure 4.6: Autocovariance: Hours worked (L) and output (y)0 10 20 30 40−50510x 10−4k(a) Cov(Lt,Lt−k)  DataStochastic Model90% CI0 10 20 30 40−50510x 10−4k(b) Cov(Lt,yt−k)0 10 20 30 40−50510x 10−4k(c) Cov(yt,Lt−k)0 10 20 30 40−50510x 10−4k(d) Cov(yt,yt−k)Notes: Figure shows autocovariances of BP(2,80)-filtered hours and output in the data and stochastic model. kis the lag in quarters. Data series for output is the log of nominal GDP, deflated by population and the GDPdeflator. Output in the model is the sum of wage earnings and firm profits, which is equal to total productionnet of fixed costs, i.e., θ˜t[φtF (`t)−∫ nt0 k (x) dx], where nt is the number of firm entrants at date t. Dottedlines show pointwise 90% confidence intervals for the autocovariance functions that would be estimated from amodel-generated data set of the same length as the actual data set (i.e., 212 quarters).Next, it should be emphasized that the exogenous shock process in this model primarily accel-erates and decelerates the endogenous cyclical dynamics, causing significant random fluctuations inthe length of the cycle while only modestly affecting its amplitude. For example, in the deterministicversion of the model the standard deviation of log-hours is 0.026, while in the stochastic model itis 0.033, implying that 79% of the standard deviation of hours is due to deterministic mechanisms.In contrast, if this TFP process were the only shock process operating in the widely-cited model ofSmets and Wouters (2007), for example, it would generate a standard deviation of log-hours of only0.005. This again suggests the more general point that, if one is willing to consider the class of modelscapable of generating deterministic fluctuations, then a very parsimonious set of shocks that are small103Figure 4.7: Spectrum: Hours worked (alternative filters)4 6 8 12 16 24 32 40 8001234 x 10−3Period of component (quarters)(a) BP(2,100)  DataStochastic Model90% CI4 6 8 12 16 24 32 40 8000.511.522.533.5x 10−3Period of component (quarters)(b) BP(2,60)4 6 8 12 16 24 32 40 8000.511.522.5x 10−3Period of component (quarters)(c) BP(2,40)4 6 8 12 16 24 32 40 8000. 10−3Period of component (quarters)(d) HP(1600)Notes: Each panel plots corresponding data (solid) and model (dashed) spectrum using the reported filter insteadof the baseline BP(2,80) filter. Dotted lines show pointwise 90% confidence intervals for the spectrum thatwould be estimated from a model-generated data set of the same length as the actual data set (i.e., 212 quarters).in magnitude can potentially yield qualitatively and quantitatively reasonable fluctuations.As a final exercise in this section, it is worth briefly further comparing the above results to thoseof Smets and Wouters (2007). Their model has received much attention in the literature for its abilityto fit well a number of key macroeconomic data series. Panel (a) of Figure 4.8 shows the spectrum forhours worked as generated by the Smets and Wouters (2007) model at the reported median posteriorparameter values. As suggested by the relatively close fit, their model also matches patterns in thehours data reasonably well, though not quite as well as the unemployment-risk model.55More insight into the drivers of fluctuations in the Smets and Wouters (2007) model can be ob-tained by looking at a spectral variance decomposition; that is, by decomposing the total variance at55This should not be too surprising, as the unemployment-risk model was estimated to match only the hours series, whilethe Smets and Wouters (2007) was estimated to simultaneously match seven different data series (including hours).104Figure 4.8: Hours worked in Smets-Wouters (2007)(a) Data and Smets-Wouters Spectrum4 6 8 12 16 24 32 40 64 8000.511.52x 10−3Period of component (quarters)  DataSmets−Wouters(b) Spectral Variance DecompositionPeriod of component (quarters)  4 6 8 12 16 24 32 40 64 8001234567 x 10−4 Mark−upBond premiumTechnologyMonetary policyGov’t spendingNotes: Data spectrum is as in Figure 4.1. Spectrum for Smets-Wouters (SW) obtained by simulating 10,000 data sets ofthe same size as the actual data series. For each simulation, the data was de-trended and the spectrum estimated usingthe same procedures as for the actual data. A point-wise average was taken across all simulated spectra. Because thehours series used by SW for their estimation differs somewhat from the series used here, for purposes of comparability,in panel (a) the SW spectrum was scaled by a constant so that the total variance is the same as in the data. Panel(b) shows portion of variance at each periodicity attributable to each of the following shock groupings: “Mark-up” –price and wage mark-up shocks; “Bond Premium” – bond premium shock; “Technology” – TFP and investment-specifictechnology shocks; “Monetary policy” – monetary policy shock; “Gov’t spending” – government spending shock.each individual periodicity into the portions that are attributable to each of the shocks in that model.Panel (b) of Figure 4.8 presents such a decomposition. It is clear from the figure that, in the range ofperiodicities reponsible for the bulk of the variance of hours, the two mark-up shocks (price and wage)in the Smets and Wouters (2007) model account for by far the largest portion. In fact, the proportion ofthe total hours variance that is explained by the mark-up shocks rises monotonically with periodicity,explaining around a third of the variance of hours by the 24-quarter periodicity and over half by the36-quarter periodicity.56 In contrast, the unemployment-risk model presented here is equally capableof matching the spectrum in hours, but does so with only a reasonably-sized TFP shock and withoutrelying on poorly motivated mark-up shocks.4.6.3 Additional resultsTo this point, I have focused on the fit of the model with respect to the target series, hours worked.In this subsection, I evaluate how well the model performs in several other dimensions that were notdirectly targeted.56The importance of the mark-up shocks is not exclusive to hours within the Smets and Wouters (2007) model. Forexample, as reported in that paper, at a 40-quarter horizon the mark-up shocks together account for over half of the forecast-error variance (FEV) of output and over 80% of the FEV of inflation.105Panel (a) of Figure 4.9 compares the spectrum of output for the data and the stochastic model.57As shown in the figure, the model spectrum matches the data reasonably well, though it is somewhattoo large (indicating too much output variance in the model), and the average periodicity is somewhattoo low. The second observation should not be too surprising, as the model does not include capitalas a factor of production. Since productive capital tends to exhibit lower-frequency fluctuations thanlabor (the other factor of production), all else equal its omission from the model will cause the averageperiodicity of output to be too small. Panel (d) of Figure 4.6, meanwhile, plots the ACF for output,which confirms the first observation: the variance of output in the model (i.e., the autocovariance atlag k = 0) is slightly larger than in the data. Notwithstanding this, however, the spectrum and ACFfor output in the data lies well within a 90% confidence interval for the model, suggesting a relativelygood overall fit.Next, panel (b) of Figure 4.9 plots the coherence between hours and output for the data and for thestochastic model.58 Coherence is analogous to a regression R2, giving the proportion of the varianceof hours that can be linearly predicted by output at a given periodicity. A coherence of one would thusindicate that hours and output are perfectly correlated at that periodicity, while a coherence of zerowould indicate that hours and output are orthogonal. In the data (solid line in the figure), we see thatat the lowest periodicities hours and output are modestly correlated, with coherence around 0.4-0.5.As the periodicity rises, the coherence initially increases relatively rapidly, reaching a peak of 0.87 ataround 13 quarters. Over this range, as indicated by the dashed line in the figure the model coherencematches the data very well. Beyond the 13-quarter periodicity, however, the data and model begin todiverge somewhat. The data coherence largely flattens out, with a gradual downward slope, reaching0.82 at the 80-quarter periodicity. The model coherence, meanwhile, rises somewhat over this range.As with the spectrum of output, the discrepancy between the data and model coherences at higherperiodicities can be explained by the lack of productive capital in the model.59 Notwithstanding thisdiscrepancy, however, the basic qualitative properties of the relationship between hours and output inthe data—namely, moderate correlation at higher frequencies but significant correlation at medium-to-low frequencies (including the range of frequencies in which the bulk of variation occurs)—arewell-captured by the model.While coherence measures the strength of the relationship between two series at a given periodic-ity, it provides no information about the sign of this relationship or whether one series tends to lead theother. To address how well the model fits in these dimensions, panels (b) and (c) of Figure 4.6 plot the57Data series for output is the log of nominal GDP, deflated by population and the GDP deflator, then de-trended usinga BP(2,80) filter using the same procedure as with hours worked (see section 4.3). Output in the model is the sum of wageearnings and firm profits, which is equal to total production net of fixed costs, i.e., θ˜t[φtF (`t)−∫ nt0 k (x) dx], where ntis the number of firm entrants at date t.58The coherence at a periodicity P is given by |sL,y (P )|2 / [sL (P ) sy (P )], where sL is the spectrum of hours, sy is thespectrum of output, and sL,y is the cross-spectrum.59Including capital would tend to reduce the coherence between output and hours by introducing another factor of pro-duction which is imperfectly correlated with hours. Since fluctuations in capital tend to be much more important at higherperiodicities, the coherence would tend to fall by more at the upper end of the range of periodicities.106Figure 4.9: Spectrum: Output (data and stochastic model)4 6 8 12 16 24 32 40 8000.511.522.533.5x 10−3Period of component (quarters)(a) Output Spectrum  DataStochastic Model90% CI4 6 8 12 16 24 32 40 8000. of component (quarters)(b) Hours−Output Coherence  DataStochastic Model90% CI4 6 8 12 16 24 32 40 8000.511.52 x 10−3Period of component (quarters)(c) Empl., Hrs./Worker Spectrum  Employment Rate (Data)Employment Rate (Stochastic Model)Hours−per−Worker (Data)Hours−per−Worker (Stochastic Model)Notes: Data series for output is the log of nominal GDP, deflated by population and the GDP deflator. Data seriesfor the employment rate is the log of the BLS’s index of nonfarm business employment divided by population.Data series for hours-per-worker is the log of nonfarm business hours divided by nonfarm business employment.All series were de-trended using a BP(2,80) filter using the same procedure as with hours worked. Output inthe model is the sum of wage earnings and firm profits, which is equal to total production net of fixed costs, i.e.,θ˜t[φtF (`t)−∫ nt0 k (x) dx], where nt is the number of firm entrants at date t. Spectrum for data and modelcomputed as with hours. Raw coherence at a periodicity p is given by |sL,y (p)|2 / [sL (p) sy (p)], where sLis the spectrum of hours, sy is the spectrum of output, and sL,y is the cross-spectrum. Coherence was thenkernel-smoothed using a Hamming window with bandwidth parameter 51. In panels (a) and (b), dotted linesshow pointwise 90% confidence intervals for the spectrum and coherence, respectively, that would be estimatedfrom a model-generated data set of the same length as the actual data set (i.e., 212 quarters).cross-covariance function (CCF) for hours and output. Two things should be noted from these plots.First, hours and output are positively correlated in both the model and data. Second, in the modelhours and output are in phase (i.e., the peak of the CCF occurs at a lag of k = 0), while in the data thepeak occurs at the point where output leads hours by one quarter. Nonetheless, the CCF is close to flat107in the data between its peak and k = 0,60 suggesting that any lead of output is weak at best. Further,as suggested by the reported 90% confidence intervals, over all the cross-covariance between outputand hours is well-captured by the model.Finally, while we have established that the model does a good job of matching patterns in totalhours, consider the model’s implications for its two component parts, the employment rate, φt, andhours-per-worker, `t. Panel (c) of Figure 4.9 shows spectra for the data and stochastic model forthese two series.61 From the figure, we see that the spectrum of the employment rate from the modelmatches fairly well the one from the data, and in particular the employment rate exhibits an overalllevel of volatility that is close to the volatility in the data. Thus, this model addresses one of thefrequent criticisms of many models of unemployment in the literature, which is that they generate toolittle employment volatility.62On the other hand, the model does a relatively poor job of matching behavior in hours-per-worker.In particular, while the basic pattern of the model spectrum is close to that in the data, the modelspectrum is in most places too small, especially beyond the lowest periodicities. This suggests thatthe model features too little in the way of movements along the intensive labor margin.63 To under-stand why, recall that when the economy moves into a region where the fixed-cost function k (·) isincreasing, forces come into play which cause output fluctuations to occur on both intensive and ex-tensive labor margins. Recall also that the former are associated with strategic substitutability (throughchanges in the price of goods), while the latter are associated with strategic complementarity (throughchanges in unemployment risk). If a given change in output occurs too much along the intensive mar-gin (as is the case for this parameterization of the model), the associated strategic substitutability tendsto push the economy back towards the steady state quickly, so that any change in hours-per-worker isrelatively small and short-lived.644.6.4 Multiple equilibria and indeterminacyIn the estimation exercise conducted above, I only considered parameter combinations for which (a)there exists a unique steady state, and (b) the probability of having multiple static equilibria (i.e.,multiple equilibria in a period, conditional on the current state and on agents’ beliefs about the fu-ture) was negligible. As mentioned briefly above, these two constraints can be expressed as upper60The peak of the data CCF is only 0.28% greater than it is at k = 0.61Data series for the employment rate is the log of the BLS’s index of nonfarm business employment divided by popula-tion. Data series for hours-per-worker is the log of nonfarm business hours divided by nonfarm business employment. Bothseries were de-trended using a BP(2,80) filter using the same procedure as with hours worked.62See for example Shimer (2005).63As Figure 4.9 shows, extensive-margin fluctuations are an order of magnitude larger than intensive-margin fluctuationsin both the model and the data. As a result, even though the model does not capture well the intensive-margin fluctuations,this has little impact on the fit of total hours, which is driven primarily by extensive-margin fluctuations.64One way to increase the variance of hours-per-worker is thus to have the upward-sloping part of the fixed-cost functionbe less steep. Since hours-per-worker was not a target of the estimation algorithm, however, there is no reason why it shouldhave favored a flatter k (·). Improving the fit of the model in this dimension by including hours-per-worker information aspart of the estimation objective function is a task for future work.108bounds on τ . Intuitively, multiple steady states and multiple equilibria may arise in this model if thestrategic complementarity between agents’ actions is too strong. Since τ governs the strength of thiscomplementarity, ruling out multiple equilibria is equivalent to limiting the size of τ .In particular, defineτ∗ ≡ αk¯b(1− α) vp∗τ¯ ≡ [1− β (1− γ) (1− δ)] [(1− δ) γ + δ] (1− βh) (1− h)[1− β (1− δ)] δ τ∗The following proposition characterizes sufficient (though not necessary) conditions under which thesteady state and static equilibria are unique.Proposition 4.5. The steady state of the unemployment-risk model is unique if τ < τ¯ . The period-tstatic equilibrium is unique if τ < θ˜2t τ∗.Ex ante, it is not clear whether imposing the constraints on τ from Proposition 4.5 is restrictivein practice. The results from the estimation reported above, however, give no indication that theseconstraints are binding. In particular, at the parameter values reported in Table 4.1, we have τ¯ = 2.61and τ∗ = 0.82, both well above the value of τ = 0.27. Clearly, the constraint ensuring a unique steadystate is not binding at the optimal parameter values. The constraint ensuring a static equilibrium,meanwhile, depends on the level of productivity θ˜t, which can in principle be arbitrarily small, andthus the constraint may be violated with strictly positive probability. Nonetheless, given the size ofthe estimated TFP shock, this probability is negligible in practice. For example, in 100,000 simulatedperiods, the smallest value of θ˜2t τ∗ that occurred was 0.69, still more than twice the value of τ .While there are relatively simple analytical conditions that can be obtained to ensure uniquenessof the steady state and of static equilibrium, verifying dynamic determinacy—that is, the presenceof a unique path converging to the limit cycle for a given initial state—is more challenging, sinceno analytical results are available in general. Nonetheless, in numerical simulations I was unableto find any evidence of indeterminacy. In particular, given initial values for the state variables andarbitrary initial values for the jump variables (chosen in practice from some neighborhood of thePE solution), one may simulate a non-stochastic version of the model forward.65 If, for a given initialstate, the system were to converge to the limit cycle for multiple combinations of initial jump variables,this would indicate the presence of indeterminacy. However, performing this experiment many timesbeginning from different initial conditions, in all cases the system eventually exploded, which suggeststo me that indeterminacy is not likely to be an issue here.4.7 ConclusionConventional models of the business cycle usually feature fluctuations driven exclusively by exoge-nous shocks around a unique stable steady state. In these models, booms and subsequent recessions are65See Appendix C.4 for details.109typically unrelated, each being driven by different independent realizations of the underlying shocks.A contrasting view is that booms and busts are inherently related, with a boom sowing the seeds ofa subsequent bust, which then sets the stage for the next boom. This second view received someattention in an older literature, but formal attempts to model underlying mechanisms appear to havebeen largely abandoned due to the perception that implausible assumptions or parameter values werenecessary to generate quantitatively reasonable fluctuations.In this paper, I present a purely deterministic general-equilibrium model featuring strategic com-plementarity near the steady state and show that it can give rise to a stable limit cycle. The limit cyclearises through a simple micro-founded mechanism in a rational-expectations environment. Cyclesemerge endogenously, and thus the model does not require shocks in order to generate fluctuations,nor does it rely on the existence of multiple equilibria or dynamic indeterminacy.Since cycles would indefinitely repeat themselves in the absence of shocks, a TFP shock is in-troduced into the model in order to create irregularities. The model is then estimated to match thespectrum of US hours. In contrast to results suggested in the earlier literature, I find that the model isable to match this spectrum quite closely. The TFP shock in the model is also shown to be of a rea-sonable persistence and relatively small size, accounting for around a fifth of the standard deviationof hours in the model. This result highlights the important insight that models capable of generatingdeterministic fluctuations do not require the addition of large, persistent, poorly-motivated shocks inorder to match the patterns in the data, which is a common criticism of conventional models.110Chapter 5ConclusionIn this thesis, I have contributed to the literature devoted to understanding the causes and consequencesof business cycles. In the first chapter, I sought to answer the questions, What effect does misspecifi-cation have on shock variance estimates in a DSGE model? Does this effect depend on the severity ofthe misspecification? If misspecification is detected, can some correction be made to the variance esti-mates? To answer these questions, I first developed a novel framework that allows a particular DSGEmodel to be compared in a meaningful and well-defined way to the “true” (but unknown) model. Us-ing this framework, I then showed that if a DSGE model is correctly specified, then the smoothedshocks should follow a vector white noise process with diagonal covariance matrix, and that if anobserved process for the smoothed shocks does not possess this property, then the model must bemisspecified. I further showed that if the model is misspecified then the shock variance estimates arebiased upward, before proposing a simple procedure to correct, in part, for this bias. Finally, I appliedthis framework and methodology to a recent paper by Justiniano et al. (2010), and found that at leastone-third of the variance of the investment shock—the leading driver of business cycle fluctuations intheir model—can be attributed to misspecification.In the second chapter, my co-authors and I proceeded from three observations. First, most deeprecessions arise after periods of fast accumulation of capital goods, suggesting that recessions mayoften reflect periods of liquidation, a viewpoint often associated with the economist Friedrich Hayek.Second, recessions appear to be socially painful phenomena, suggesting that there are mechanisms atplay that are causing the economy to function especially inefficiently, a viewpoint typically associatedwith John Maynard Keynes. Finally, even when monetary authorities try to counter such recessionsby easing policy, this does not seem to be fully effective, suggesting that nominal rigidities may notbe the only important source of inefficiency.Our paper has offered a framework that is consistent with these three observations, and that ac-cordingly provides an environment where the policy trade-offs inherent to the Hayekian and Keyne-sian views can be discussed. The narrative underpinning the model is quite straightforward. Whenthe economy inherits a high level of capital, this decreases the desire for trade between agents in the111economy, leading to less demand. When there are fixed costs associated with employment, this willgenerally lead to an increase in unemployment. If the risk of unemployment cannot be entirely insuredaway, households will react to the increased unemployment by increasing saving and thereby furtherdepressing demand. This multiplier process will cause an excess reaction to the inherited goods andcan be large enough to make society worse off even if—in a sense—it is richer since it has inheriteda large stock of goods. Within this framework, we showed that policies aimed at stimulating activitywill face an unpleasant trade-off, as the main effect of stimulus will simply be to postpone the adjust-ment process. Nonetheless, we find that such stimulative policies may remain desirable even if theypostpone recovery, but these gains do not increase simply because the rate of unemployment is higher.In the final chapter, I have investigated the potential role of deterministic mechanisms in generatingbusiness cycle fluctuations. Such mechanisms received some attention in an older literature, but formalattempts to incorporate them into the standard macroeconomic modelling paradigm appear to havebeen largely abandoned due to the perception that implausible assumptions or parameter values werenecessary to generate quantitatively reasonable fluctuations. In this paper, I have shown that thisneed not be the case. In particular, I presented a purely deterministic general-equilibrium modelfeaturing strategic complementarity near the steady state and showed that it can easily give rise tolimit cycles under plausible parameter values. Furthermore, adding a simple and reasonably-sizedsource of exogenous variation (in the form of a TFP shock) to the model, I found that the model isable to match data on US business cycles very well. 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(1993). “Measures of Fit for Calibrated Models.” Journal of Political Economy,101(6):1011–1041.118Appendix AAppendix for “Misspecification and theCauses of Business Cycles”A.1 Model details for Example 3 in Section 2.5 (Smets and Wouters(2007))The equations for the Smets and Wouters (2007) model are reproduced here for convenience. SeeSmets and Wouters (2007) for specifics regarding the model set-up and the derivation of these equa-tions. The resource constraint:yt = cyct + iyit + zyzt + εgtThe consumption Euler equation:ct = c1ct−1 + (1− c1)Etct+1 + c2 (lt − Etlt+1)− c3(rt − Etpit+1 + εbt)The investment Euler equation:it = i1it−1 + (1− i1)Etit+1 + i2qt + εitThe value of the capital stock evolves according to:qt = q1Etqt+1 + (1− q1)Etrkt+1 −(rt − Etpit+1 + εbt)The aggregate production function equation:yt = φp [αkst + (1− α) lt + εat ]The capital utilization equation:kst = kt−1 + zt119The capital utilization first-order condition:zt = z1rktThe capital accumulation equation:kt = k1kt−1 + (1− k1) it + k2εitThe intermediate goods firm’s real mark-up:µpt = α (kst − lt) + εat − wtThe New Keynesian Phillips curve:pit = pi1pit−1 + pi2Etpit+1 − pi3µpt + εptThe firm’s capital-labor ratio:kst − lt = wt − rktThe labor market real mark-up:µwt = wt − σllt −γγ − λct +λγ − λct−1The wage Phillips curve:wt = w1wt−1 + (1− w1) (Etwt+1 + Etpit+1)− w2pit + w3pit−1 − w4µwt + εwtFinally, the Taylor rule:rt = ρrt−1 + (1− ρ) [rpipit + ry (yt − ypt )] + r∆y[yt − ypt − yt−1 + ypt−1]+ εrtAbove, yt is output, ct is consumption, it is investment, zt are capital utilization costs, lt is hoursworked, kt is the end-of-period capital stock, kst is employed capital, qt is the shadow value of capital,µpt is the intermediate goods firms’ real mark-up, rkt is the rental rate on capital, wt is the real wagerate, µwt is the real wage mark-up, rt is the nominal interest rate, pit is inflation from date (t− 1) todate t, and ypt is potential output (as obtained from the RBC version of the model). All variables arelog-linearized around the deterministic trend.120The RBC version of the model is obtained by settingξw = 0ξp = 0and removing the risk premium, two mark-up, and monetary policy shocks. All other features ofthe DGP remain unchanged, including all of the real frictions and the monopolistically competitiveintermediate goods market. The New Keynesian Phillips curve, wage Phillips curve and Taylor ruleare indeterminate and thus dropped from the system, while the intermediate goods and labor marketmark-up equations are replaced, respectively, with0 = α (kst − lt) + εat − wt0 = wt − σllt −γγ − λct +λγ − λct−1The consumption and marginal value of capital Euler equations are also combined to obtainct = c1ct−1 + (1− c1) cet + c2 (lt − let )− c3q1qet − c3 (1− q1) rket + c3qtWe have thus reduced the system by four equations, and correspondingly drop four variables: rt, pit,µpt and µwt .A.2 Proofs of theoretical resultsProof of Proposition 2.1From (2.16)-(2.18), we can writeε∗t = (FB)−1 (Yt − Zt)− (FB)−1 FAX∗t−1 (A.1)Substituting this into equation (2.18) for ε∗t yieldsX∗t = CX∗t−1 +B (FB)−1 (Yt − Zt) (A.2)Next, we may write C = QJQ−1, where J is the Jordan normal form of C and Q is a matrixwhose columns are the corresponding generalized eigenvectors of C. In particular, J is an n × nmatrix with the eigenvalues of C on the main diagonal ordered by increasing modulus, and, if there isa non-zero non-diagonal entry, then it is equal to one, lies immediately above the main diagonal, andsatisfies that the entry immediately to the left of it is equal to the entry immediately below it. Note thatQ and J are functions of A, B and F only. Note also that, since J is an upper-triangular matrix, its121eigenvalues are its diagonal elements, and thus it has the same eigenvalues as C. Under Assumption2.4, J can therefore be partitioned asJ =(J1 00 J2)where J1 is an n1 × n1 matrix having eigenvalues strictly less than one in modulus, and J2 is ann2 × n2 matrix having eigenvalues strictly greater than one in modulus, with n1 + n2 = n. Note thatJ2 is non-singular by construction. Partition Q and Q−1 conformably asQ =(Q1 Q2)Q−1 =(Q1Q2)Letting xt ≡ Q−1X∗t and yt ≡ Q−1B (FB)−1 (Yt − Zt), premultiplying equation (A.2) by Q−1yieldsxt = Jxt−1 + ytor, partitioning xt and yt conformably with J ,(x1,tx2,t)=(J1x1,t−1J2x2,t−1)+(y1,ty2,t)We may thus writex1,t =∞∑j=0J j1Ljy1,tandx2,t =−1∑j=−∞J j2Ljy2,twhere convergence of the sums follows from the fact that the eigenvalues of J1 and J−12 are all lessthan one in modulus. ThusX∗t = Qxt=∞∑j=0Q1J j1Q1B (FB)−1 Lj (Yt − Zt) +−1∑j=−∞Q2J j2Q2B (FB)−1 Lj (Yt − Zt)Backing this expression up one period and substituting into equation (A.1) for Xt−1 yieldsε∗t =∞∑j=−∞ψjLj (Yt − Zt)122whereψj =− (FB)−1 FAQ2J j−12 Q2B (FB)−1 if j < 0(FB)−1[Im − FAQ2J−12 Q2B (FB)−1]if j = 0− (FB)−1 FAQ1J j−11 Q1B (FB)−1 if j > 0Again, since the eigenvalues of J1 and J−12 are all strictly less than one in modulus, the sequence {ψj}is absolutely summable.Finally, note thatX∗t =∞∑i=0AiBLiε∗tand thus (∞∑i=0FAiBLi)ε∗t = Yt − Zt (A.3)Pre-multiplying (A.3) by∑∞j=−∞ ψjLj yields∞∑j=−∞ψjLj(∞∑i=0FAiBLi)ε∗t =∞∑j=−∞ψjLj (Yt − Zt) = ε∗twhich confirms the first equality in (2.20). Substituting (A.3) into (2.19) for (Yt − Zt) yieldsε∗t =∞∑j=−∞ψjLj(∞∑i=0FAiBLi)ε∗twhich confirms the second equality in (2.20).Proof of Proposition 2.2Since the EM can be inverted by Proposition 2.1, Proposition 2.2 follows directly from (2.13) and thesubsequent discussion.Proof of Corollary 2.1Part (a) follows obviously from independence of ε∗t and νt. To see part (b), suppose the EM is correctlyspecified, so that V ar (Zt) = 0. Then εˆt = ε∗t and therefore E [εˆl,tεˆi,s] = E[ε∗l,tε∗i,s]= 0 for all(l, t) 6= (i, s). To see part (c), suppose the EM is misspecified. Then V ar (Zt) 6= 0 and thereforethere exists some l such that V ar (νl,t) 6= 0. From Proposition 2.2, this implies that σˆ2l > σ∗2l . Finally,part (d) follows from independence of ε∗l,t on the one hand and both ε∗i,s and νi,s on the other.123Proof of Proposition 2.3By independence of ε∗l,t and νl,t (Proposition 2.2) and of ε∗l,t and εˆi,s for (l, t) 6= (i, s) (Corollary2.1(d)), we haveE[ξ2l,t]= σ∗2l + Eνl,t −∑i 6=lΘi,0εˆi,t −q∑j=1(Θj εˆt−j + Θ−j εˆt+j)2≥ σ∗2lThe second part of the Proposition (i.e., that E[ξ2l,t]≤ σˆ2l ) follows directly from standard regressionresults.A.3 Data for JPT modelQuarterly U.S. data spanning 1954QIV to 2009QI was constructed for seven observable variables: realper capita GDP growth, real per capita consumption growth, real per capita investment growth, (logof) per capita hours, real wage growth, the inflation rate, and the nominal interest rate.Raw data were obtained from several different sources. NIPA data were obtained from the Bureauof Economic Analysis. Data on the federal funds rate and population (civilian noninstitutional) wereobtained from the St. Louis Fed’s FRED database. An index of hours worked in the non-farm businesssector was obtained from the Bureau of Labor Statistics, as was an index of compensation in the non-farm business sector.Real per capital GDP was constructed as nominal GDP divided by the GDP deflator and popula-tion. Real per capita consumption was constructed as nominal purchases of non-durable goods andservices, divided by the GDP deflator and population. Real per capita investment was constructedas nominal gross private domestic investment plus purchases of durable goods, divided by the GDPdeflator and population. Per capita hours was constructed as the hours index divided by population,and normalized by a constant factor chosen so that the average of the log of the series over the period1954QIV-2004QIV was zero. The real wage was constructed as the compensation index divided bythe GDP deflator. The inflation rate was constructed as growth rate of the GDP deflator. Finally, thenominal interest rate was constructed as one-quarter of the federal funds rate in annual terms.124A.4 BCF variance decompositionLet Φk ≡ E[XtX ′t−k]denote the k-th autocovariance of Xt in the EM of Section 2.4.1. Theautocovariance-generating function (AGF) for {Xt} is given by1GX (z) = (In −Az)−1BΣB′(In −A′z−1)−1where Σ ≡ E [εtε′t]. The spectrum for {Xt} at frequency ω is given bysX (ω) = (2pi)−1GX(e−iω)= (2pi)−1(In −Ae−iω)−1BΣB′(In −A′eiω)−1(A.4)where i ≡√−1. It can be shown that, for all integers kΦk =∫ pi−pisX (ω) eiωkdωSince we are interested in doing a decomposition of the unconditional variances of the endogenousvariables, of particular interest is the case where k = 0. i.e.Φ0 =∫ pi−pisX (ω) dω (A.5)Let sjjX (ω) denote the jj-th element of the spectrum. Equation A.5 implies that the integral of sjjX (ω)over all frequencies ω ∈ [−pi, pi] yields the unconditional variance of Xj,t (the j-th element of Xt).For ω0 ∈ [0, pi], we can interpret∫ ω0−ω0sjjX (ω) dω as the portion of the unconditional variance of Xj,tattributable to fluctuations with frequencies between 0 and ω0. For 0 ≤ ω0 ≤ ω1 ≤ pi, definevj (ω0, ω1) ≡∫ ω1−ω1sjjX (ω) dω −∫ ω0−ω0sjjX (ω) dω (A.6)If we choose ω0 and ω1 so that the interval [ω0, ω1] gives the set of BCFs, then vj (ω0, ω1) can beinterpreted as the portion of the unconditional variance of Xj,t attributable to BCFs, i.e. its BCFvariance.Next, letting v(l)j (ω0, ω1) be the quantity in (A.6) obtained when the variances of all but the l-thshock are set to zero, it is straightforward to show that∑ml=1 v(l)j (ω0, ω1) = vj (ω0, ω1). Thus,v(l)j (ω0, ω1)vj (ω0, ω1)can be interpreted as the contribution of the shock l to the BCF variance of Xj,t.1 See, for example, Hamilton (1994).125In practice, the integral in (A.6) is approximated using 500 bins spanning the relevant frequencies.Note that the period of a cycle with frequency ω is given by 2piω . Following Stock and Watson (1999)and JPT, I define business cycle frequencies to be those with periods between 6 and 32 quarters. Thus,I set ω0 = 2pi32 and ω1 = 2pi6 .126Appendix BAppendix for “Reconciling Hayek’s andKeynes’ Views of Recessions”B.1 Proofs of propositionsProof of Proposition 3.1We first establish that there always exists an equilibrium of this model. Substituting equation (3.8)into equation (3.7) and letting e ≡ c−X yieldsU ′(X + e) = ν′(`)F ′(`)(1 + τ − τ eΩ(`))(B.1)where Ω(`) ≡ F ′(`)` is output net of search costs per employed worker, which is assumed to bestrictly increasing. When N < L (i.e., the full-employment constraint is not binding), equation (3.9)implies that ` = `?, and equation (3.8) implies that e < e?, where e? ≡ Ω(`?). On the other hand,when N > L (i.e., the full-employment constraint binds), equation (3.8) implies that ` = Ω−1(e).Further, since min{N,L} < N and F (`)− F ′(`)` is assumed to be strictly increasing in `, equation(3.9) implies that ` > `?, and thus, by strict increasingness of Ω, we also have e > e?. Substitutingthese results into equation (B.1) yields that e > 0 is an equilibrium of this model if it satisfiesU ′(X + e) = Q(e) (B.2)where the function Q(e), defined in equation (3.14), is the expected marginal utility cost of consump-tion when aggregate expenditures are e = c − X . Note that Q is continuous, strictly decreasing on[0, e?], and strictly increasing on [e?,∞).Lemma B.1. If U ′(X) ≤ Q(0), then there is an equilibrium with e = 0.127Proof. To see this, suppose aggregate conditions are that e = 0. Then the marginal utility of consump-tion when the household simply consumes its endowment is no greater than its expected marginal cost,and thus households respond to aggregate conditions by making no purchases, which in turn validatese = 0.Lemma B.2. If U ′(X) > Q(0), then there is an equilibrium with e > 0.Proof. We have that minQ(e) = ν ′(`?)/F ′(`?) > 0. Since we have assumed limc→∞ U ′(c) ≤ 0, itnecessarily follows that for anyX , there exists an e sufficiently large that U ′(X+e) < minQ(e), andtherefore, by the intermediate value theorem, there must exist a solution e > 0 to equation (B.2).Lemmas B.1 and B.2 together imply that an equilibrium necessarily exists. We turn now to show-ing under what conditions this equilibrium is unique for all values of X . As in equation (3.12),we may represent household j’s optimal expenditure when aggregate expenditure is e as ej(e) =U ′−1(Q(e)) − X , so that equilibrium is a fixed point ej(e) = e. The function ej(e) is continuouseverywhere, and differentiable everywhere except at e = e?, withe′j(e) =Q′(e)U ′′ (U ′−1(Q(e)))Note that e′j(e) is independent of X , strictly increasing on [0, e?] and strictly decreasing on [e?,∞).Lemma B.3. Iflime↑e?e′j(e) < 1 (B.3)then e′j(e) < 1 for all e.Proof. Note first that e′j(e) < 0 for e > e?, so that this condition is obviously satisfied in that case.For e < e?, note thate′′j (e) =Q′′(e)− U ′′′ (X + ej(e))[e′j(e)]2U ′′ (X + ej(e))Since Q′′(e) = 0 on this range and U ′′′ > 0, we have e′′j (e) > 0, and thus e′j(e) < lime↑e? e′j(e),which completes the proof.Lemma B.4. Inequality (B.3) holds if and only ifτ < τ¯ ≡ −U ′′(U ′−1(ν ′(`?)f ′(`?))) f ′(`?) [f(`?)− Φ]ν ′(`?)Proof. We have thatlime↑e?e′j(e) =ν ′(`?)τ−U ′′(U ′−1(ν′(`?)f ′(`?)))f ′(`?) [f(`?)− Φ]128which is clearly less than one if and only if τ < τ¯ .Lemma B.5. If τ < τ¯ , then there always exists a unique equilibrium regardless of the value of X . Ifτ > τ¯ , then there exists values of X ∈ R such that there are multiple equilibria.Proof. We have already established that there always exists an equilibrium. Note that equilibriumoccurs at the point where the ej = ej(e) locus intersects with the locus characterizing the equilibriumcondition, i.e., ej = e. To see the first part of the lemma, suppose τ < τ¯ so that inequality (B.3) holds.Then since the slope of the equilibrium locus is one, and the slope of the ej = ej(e) locus is strictlyless than one by Lemma B.3, there can be at most one intersection, and therefore the equilibrium isunique.To see the second part of the lemma, suppose that τ > τ¯ and thus (B.3) does not hold. Then bystrict convexity of ej(e) on (0, e?), there exists a value e < e? such that e′j(e) > 1 on (e, e?). DefineX˜(e) ≡ U ′−1(Q(e)) − e, and note that e is an equilibrium when X = X˜(e). We show that thereare at least two equilibria when X = X˜(e) with e ∈ (e, e?). To see this, choose e0 ∈ (e, e?), andnote that, for X = X˜(e0), ej(e0) = e0 and e′j(e) > 1 on (e0, e?). Thus, it must also be the case thatej(e?) > e?. But since ej(e) is continuous everywhere and strictly decreasing on e > e?, this impliesthat there exists some value e > e? such that ej(e) = e, which would represent an equilibrium. Sincee0 < e? is also an equilibrium, there are at least two equilibria.This completes the proof of Proposition 3.1.Proof of Proposition 3.2Lemma B.6. If τ < τ¯ and X is such that e > 0, then de/dX < 0.Proof. Totally differentiating equilibrium condition (B.2) with respect to X yieldsdedX =U ′′(X + e)Q′(e)− U ′′(X + e) (B.4)From Lemma B.4, we see that Q′(e) > U ′′(U ′−1(Q(e))). In equilibrium, U ′−1(Q(e)) = X + e,so that this inequality becomes Q′(e) > U ′′(X + e), and thus the desired conclusion follows byinspection.Given Lemma B.6 and the fact that the economy exhibits unemployment when e < e? and fullemployment when e ≥ e?, it is clear that the economy will exhibit unemployment if and only if X issmaller than the level such that e = e? is the equilibrium; that is, if X ≤ X?, whereX? ≡ U ′−1( ν ′(`?)F ′(`?))− F ′(`?)`?This completes the proof of the first part of the proposition.129Next, from Lemma B.1, we see that there is a zero-employment equilibrium if and only ifU ′(X) ≤ν′(`?)F ′(`?)(1 + τ), which holds when X ≥ X??, whereX?? ≡ U ′−1( ν ′(`?)F ′(`?)(1 + τ))This completes the proof of Proposition 3.2.Proof of Proposition 3.3If X < X??, we know from Proposition 3.2 that e > 0, and therefore e solves equation (B.2).Substituting e = c−X for e yields the desired result in this case. From Proposition 3.2, we also knowthat if X ≥ X?? then e = 0, in which case c = X , which completes the proof.Proof of Proposition 3.4If X > X??, so that the economy features zero employment and therefore c = X , then clearly cis increasing in X . Thus, suppose X < X??, so that e > 0. Totally differentiating the expressionc = X + e with respect to X and using equation (B.4), we obtaindcdX =Q′(e)Q′(e)− U ′′(X + e) (B.5)Since the denominator of this expression is positive (see the proof of Lemma B.6), the sign of dc/dXis given by the sign of Q′(e), which is negative if e < e? (i.e., if X? < X < X??) and positive ife > e? (i.e., if X < X?). This completes the proof.Proof of Proposition 3.5Letting U(e) denote welfare conditional on the coordinated level of e, we may obtain thatU(e) = U(X + e) + µ(e)[L? −ν ′(`?)F ′(`?)e]− [1− µ(e)](1 + τ) ν′(`?)F ′(`?)ewhere µ(e) = e/[F ′(`?)`?] denotes employment conditional on e, and L? ≡ ν ′(`?)`? − ν(`?) ≥ 0.Using the envelope theorem, it is straightforward to see that the only welfare effects of a marginalchange in e from its decentralized equilibrium value are those that occur through the resulting changein employment. Thus,U ′(e) =[L? + τ ν′(`?)F ′(`?)e]µ′(e) > 0and therefore a coordinated rise in e would increase expected utility of all households.130Proof of Proposition 3.6Denote welfare as a function of X byU(X) ≡ U(X + e) + µ [−ν(`) + V (w`− pe)] + (1− µ)V (−pe)If X < X?, so that the economy is in the full-employment regime, or if X > X??, so that theeconomy is in the zero-employment regime, we may show that U ′(X) > 0 always holds. Thus, wefocus on the case where X ∈ (X?, X??). When this is true, some algebra yieldsU(X) = U(X + e) +{`?[ν ′(`?)− ν(`?)`?]+ ν′(`?)F ′(`?)τe}µ− ν′(`?)F ′(`?)(1 + τ)eUsing the envelope theorem, we may differentiate this expression with respect to X to obtainU ′(X) = U ′(X + e) +[L? + ν′(`?)F ′(`?)τe] dµdX (B.6)where L? ≡ ν ′(`?)`? − ν(`?) ≥ 0.Lemma B.7. U ′′(X) > 0 on (X?, X??).Proof. Substituting the equilibrium condition (B.2) into (B.6) and using the fact thatdµdX =1F ′(`?)`?dedXafter some algebra, we obtainU ′(X) = ν′(`?)F ′(`?)[1 + τ + τµ( dedX − 1)]+ L?F ′(`?)`?dedX (B.7)From (B.4), we may also obtain thatdedX =( ν ′(`?)τ−U ′′(X + e) [F ′(`?)]2 `?− 1)−1d2edX2 =U ′′′(X + e)U ′′(X + e)dedX[ dcdX]2> 0and thereforeU ′′(X) = ν′(`?)F ′(`?)τdµdX( dedX − 1)+[ ν ′(`?)F ′(`?)τµ+L?F ′(`?)l?] d2edX2Since de/dX < 0, dµ/dx < 0, and thus the first term is positive, as is the second term, and the proofis complete.131Lemma B.8. Ifτ > τ ≡ ν(`?)ν ′(`?)`?( τ¯1 + τ¯)then there exists a range of X such that U ′(X) < 0.Proof. Since U is convex by Lemma B.7, U ′(X) < 0 for some values ofX if and only if limX↓X? U ′(X) <0. Taking limits of equation (B.7), and using the facts thatlimX↓X?dedX = −τ¯τ¯ − τand limX↓X? µ = 1, we obtain thatlimX↓X?U ′(X) = ν′(`?)F ′(`?)(1− τ τ¯τ¯ − τ)−L?F ′(`?)`?( τ¯τ¯ − τ)Substituting in from the definition of L?, straightforward algebra yields that this expression is lessthan one if and only if τ > τ .Note that, by convexity of ν(`) and the fact that ν(0) = 0, we have ν(`?) ≤ ν ′(`?)`?, and thusτ < τ¯ , so that there always exists values of τ such that τ < τ < τ¯ . From the definition of τ , wealso see that, holding τ and ν ′(`?) constant, if ν(`?)/`? is small, this inequality is more likely to besatisfied.Proof of Proposition 3.7We suppose there is a competitive insurance industry offering a menu of unemployment insurancecontracts. A typical contract is denoted (h, q), where h is the premium, paid in all states, and q is thecoverage, which the purchaser of the contract receives if and only if he is unemployed. Both h andq are expressed in units of good 1. Since insurance is only potentially useful when 0 < µ < 1, wehenceforth assume that this is true. Note also that zero profit of insurers requires that h = (1− µρˆ)q,where ρˆ is the fraction of purchasers of the contract that are participant households. This implies thatnon-participant households will not purchase any such zero-profit contract featuring q < 0.Lemma B.9. In any separating equilibrium, no contracts are purchased by participant households.1Proof. Suppose there is a separating equilibrium, and let (hp, qp) denote the contract purchased byparticipant households, and (hn, qn) that purchased by non-participant households. From the insurer’szero-profit condition, we must have hp = (1 − µ)qp and hn = qn. Since non-participant householdswill always deviate to any contract with hp < qp, this implies that we must have qp < 0 in such anequilibrium.1 Technically, agents are always indifferent between not purchasing a contract and purchasing the contract (0, 0). Forease of terminology, we will assume that the contract (0, 0) does not exist.132Next, for any zero-profit separating contract, the assets of employed participant households aregiven by Ae = w`−p[(1−µ)qp+ e] and of unemployed participant households by Au = p(µqp− e).Note that, since qp < 0 and from the resource constraint wl > pc, we must have Au < 0 < Ae. Also,the derivative of the household’s objective function with respect to qp along the locus of zero-profitcontracts is given by∂U∂qp= pµ(1− µ)[V ′(Au)− V ′(Ae)]> 0wherever such a derivative exists. Since Au < 0 < Ae, this derivative must exist at the candidateequilibrium, and therefore in a neighborhood of that equilibrium the objective function is strictlyincreasing on qp < 0. Thus, given any candidate zero-profit equilibrium contract with qp < 0, thereexists an alternative contract (h′p, q′p) with q′p > qp which satisfies that h′p−(1−µ)q′p is strictly greaterthan but sufficiently close to zero so that participant households would choose it over (hp, qp), whilenon-participant households would not choose it, and therefore insurers could make a positive profitselling it. Thus, (hp, qp) cannot be an equilibrium contract. Since this holds for all qp < 0, it followsthat no separating equilibrium exists in which contracts are purchased by participant households.Next, consider a pooling equilibrium, so that ρˆ = ρ. As argued above, we must have q ≥ 0 inany such equilibrium. Assets of an employed worker when choosing a zero-profit pooling contract(h, q) = ((1− µρ)q, q) are given by Ae = w`− p[(1− µρ)q + e], while Au = p(µρq − e) are thoseof an unemployed worker. Let U(q) denote the value of the household’s objective function whenchoosing such a zero-profit pooling contract.Lemma B.10. If U(q) is strictly decreasing in q whenever Ae > Au, then a pooling equilibrium doesnot exist.Proof. Note first that if Ae ≤ Au, then being unemployed is always strictly preferred to being em-ployed by participant households, so that this cannot represent an equilibrium. Furthermore, as arguedabove, we must have q ≥ 0 in any pooling equilibrium. Thus, suppose Ae > Au and q > 0. We showthat such a q cannot represent an equilibrium. To see this, let (h′, q′) denote an alternative contractwith 0 < q′ < q and h′ = (1 − µρ)q′. Since U is strictly decreasing in q, this contract is strictlypreferred by participant households. Furthermore, since non-participant households would get netpayment µρ(q′ − q) < 0 from deviating to this new contract, only participant households would devi-ate to it, and therefore the expected profit to an insurer offering it would be (1− ρ)µq′ > 0. Thus, thisdeviation is mutually beneficial for participants and insurers, and so q cannot be an equilibrium.Lemma B.11. If ρ < 1/(1 + τ), then there is no equilibrium in which an insurance contract ispurchased by participant households.Proof. Note that U(q) is continuous, withU ′(q) = pµ[(1− µ)ρV ′ (Au)− (1− µρ)V ′ (Ae)]133wherever this derivative exists (i.e., whenever AeAu 6= 0). If AeAu > 0, then V ′(Ae) = V ′(Au), andtherefore U ′(q) = −pµ(1− ρ)V ′(Ae) < 0. Suppose on the other hand that AeAu < 0. If in additionAe > Au, we must have Au < 0 < Ae, and therefore U ′(q) = −pvµ{1 − ρ[1 + τ(1 − µ)]}. Sinceρ < 1/(1 + τ), it follows that U ′(q) < 0. Thus, U(q) is strictly decreasing whenever Ae > Au, andtherefore by Lemma B.10, no pooling equilibrium exists. Since, by Lemma B.9, there does not exista separating equilibrium either, no equilibrium exists.Proof of Proposition 3.8We may re-write the equilibrium condition (3.15) asU ′(X + e−Gw) = Q(e) (B.8)where Q(e) is as defined in equation (3.14).That non-wasteful government purchases have no effect on economic activity can be seen directlyfrom the fact that Gn does not appear in equation (B.8). Totally differentiating equation (B.8) withrespect to Gw, we obtaindedGw= −U′′(X + e−Gw)Q′(e)− U ′′(X + e−Gw)Under the assumption that τ < τ¯ , the denominator of this expression is positive, and thus de/dGw >0. Further, if the economy is in the unemployment regime, thenQ′(e) < 0 and therefore de/dGw > 1,while if the economy is in the unemployment regime, then Q′(e) > 0 and therefore de/dGw < 1,which completes the proof.Proof of Proposition 3.9First, note that a balanced budget requires that employed workers be taxed (Gn + Gw)/µ. Lettingep = e−Gn −Gw denote private expenditures, we may therefore obtain welfare as a function of X ,Gn and Gw asU(X,Gn, Gw) = U(X + ep +Gn)+ ν′(`?)F ′(`?)[(F ′(`?)ν ′(`?) L? + τep)µ−Gn −Gw − (1 + τ)ep](B.9)where as before L? ≡ ν ′(`?)`? − ν(`?). Taking derivatives with respect to Gw and applying theenvelope theorem, we may obtain thatU3(X,Gn, Gw) =ν ′(`?)F ′(`?)[(F ′(`?)ν ′(`?) L? + τep) dµdGw− 1](B.10)134Meanwhile, differentiating (B.9) with respect to X , applying the envelope theorem and using theequilibrium condition U ′(X + ep +Gn) = Q(ep +Gn +Gw), we may obtain thatF ′(`?)ν ′(`?) L? + τep =F ′(`?)ν ′(`?)[U1(X,Gn, Gw)−ν ′(`?)F ′(`?)(1 + τ − τµ)]( dµdX)−1(B.11)We may also obtain from the equilibrium condition that dµ/dGw = −dµ/dX . Substituting this and(B.11) into (B.10), we may obtainU3(X,Gn, Gw) =ν ′(`?)F ′(`?)(1− µ)τ − U1(X,Gn, Gw)Since the first term on the left-hand side is positive, if X is in the range such that U1(X,Gn, Gw) < 0,then we necessarily have U3(X,Gn, Gw) > 0, which completes the proof.Proof of Proposition 3.10Letm(N) ≡M(N,L) and note that the restrictions onM imply, among other things, that (a)m(0) =0, (b) m′(0) ∈ (0, 1], and (c) limN→∞m′(N) = 0. We have thatlimN→0m′ (N)Nm (N) = limN→0m′ (N)[m (N)−m (0)] /Nwhere we have used property (a). The limit of the numerator is clearly just m′(0), while the limit ofthe denominator is, by definition, also equal tom′(0) and thus, since by property (b)m′(0) is non-zeroand bounded, we have thatlimN→0m′ (N)Nm (N) =m′ (0)m′ (0) = 1Next, suppose limN→∞m′(N)N/m(N) > 0. Since 0 < limN→∞m(N) < ∞, this impliesthat limN→∞N/g(N) > 0 where g(N) ≡ 1/m′(N). This in turn implies that g(N) = O(N) asN →∞, or, equivalently, that there exists an N0 > 0 such that, for N ≥ N0,g′(N)g(N) ≤1Nwhere the right-hand side of this inequality is simply the growth rate of N . We may therefore obtain,135for N ≥ N0,g(N) = g(N0) exp{∫ NN0g′(s)g(s) ds}≤ g(N0) exp{∫ NN01sds}= g(N0)NN0and thus m′(N) ≥ m′(N0)N0/N . Butm(N) = m(N0) +∫ NN0m′(s)ds≥ m(N0) +m′(N0)N0∫ NN01sds= m(N0) +m′(N0)N0 [log(N)− log(N0)]The expression on the last line above is clearly unbounded as N → ∞, which would imply thesame for m(N), a clear contradiction of the requirement that M(N,L) ≤ L. Thus, we cannot havelimN→∞m′(N)N/m(N) > 0, i.e., we must have limN→∞m′(N)N/m(N) = 0.Proof of Proposition 3.11The following result will be useful.Lemma B.12. Let EgXM denote the elasticity of substitution between X andM embodied in g. ThenEgXM =gX(X,M)gM(X,M)gXM(X,M)g(X,M)(B.12)Proof. Letting Hk denote homogeneity of degree k, note first that, since g is H1, for a, b ∈ {X,M},ga is H0 and gab is H−1.Next, by definition, we haveEgXM ≡[d log (gX(X,M)/gM(X,M))d log (M/X)]−1136Letting M˜ ≡M/X and using H0 of gX and gM, we may obtainEgXM =gX(1,M˜)gI(1,M˜)M˜[ddM˜(gX(1,M˜)gM(1,M˜))]−1= gX(1,M˜)gM(1,M˜)M˜[gXM(1,M˜)gI(1,M˜)− gX(1,M˜)gMM(1,M˜)]= gX(X,M)gM(X,M)M [gXM(X,M)gM(X,M)− gX(X,M)gMM(X,M)]where the last line follows fromH0 of ga andH−1 of gab. Adding and subtracting gXM(X,M)gX(X,M)Xin the denominator and grouping terms yieldsEgXM =gX(X,M)gM(X,M)gXM(X,M) [gX(X,M)X + gM(X,M)M]− gX(X,M) [gXM(X,M)X + gMM(X,M)M]The first bracketed term in the denominator equals g(X,M) by H1 of g, while the second bracketedterm equals 0 by H0 of gM, and thus equation (B.12) follows.Next, let W (X,M) ≡ U(g(X,M)). Then the equilibrium condition (3.17) can be writtenWM(X,M) = Q(M) (B.13)Note thatWMM(X,M) = [gM(X,M)]2 U ′′(g(X,M)) + gMMU ′(g(X,M)) < 0so that the left-hand side of equation (B.13) is strictly decreasing in M. To ensure the existenceof an equilibrium with M > 0, we assume that WM(X, 0) > Q(0). We further assume thatgMMM(X,M) ≥ 0, which ensures that QMMM > 0, and therefore, similar to in the durable-goods model, there are at most three equilibria: at most two in the unemployment regime, and at mostone in the full-employment regime. Additional conditions under which we can ensure that there existsa unique equilibrium are similar in flavor to in the durable-goods case, though less easily characterizedexplicitly. We henceforth simply assume conditions are such that the equilibrium is unique, and notethat this implies thatWMM(X,M) < Q′(M) (B.14)at the equilibrium value ofM. Define alsoEQM ≡Q′(M)MQ(M)as the elasticity of Q with respect toM.137Lemma B.13. dc/dX < 0 if and only if− EQMEgXM > 1 (B.15)Proof. Totally differentiating the equilibrium condition (B.13) with respect to X yields thatdMdX =WXM(X,M)Q′(M)−WMM(X,M)(B.16)Doing the same with the equilibrium condition c = g(X,M) yieldsdcdX = gX(X,M) + gM(X,M)dMdX= gX(X,M) [Q′(M)−WMM(X,M)] + gM(X,M)WXM(X,M)Q′(M)−WMM(X,M)where the second line has used (B.16). By (B.14), the denominator is positive, so that this expressionis of the same sign as the numerator. Substituting in for WMM and WXM and using the equilibriumcondition (B.13), we may obtain that dc/dX < 0 if and only if[gXM(X,M)gX(X,M)−gMM(X,M)gM(X,M)]M < −EQM (B.17)The term in square brackets, meanwhile, can be written asgXM(X,M) [gX(X,M)X + gM(X,M)M]− gX(X,M) [gXM(X,M)X + gMM(X,M)M]gX(X,M)gM(X,M)MBy H0 of gM, the second term in the numerator equals zero, and thus by H1 of g, we have thatgXM(X,M)gX(X,M)−gMM(X,M)gM(X,M)= gXM(X,M)g(X,M)gX(X,M)gM(X,M)MSubstituting this into (B.17) and using (B.12) yields (B.15).If the economy is in the full-employment regime, EQM > 0 and therefore, since EgXM > 0, condi-tion (B.15) cannot hold. Thus, from Lemma B.13, if the economy is in the full-employment regime,dc/dX > 0. If instead the economy is in the unemployment regime, then EQM < 0, and thereforecondition (B.15) can hold as long as EgXM is sufficiently large, which completes the proof of theproposition.Proof of Proposition 3.12Let y = g(X1,M) denote output of the final good in the first period. Furthermore, let B(X2) ≡U ′−1(R(X2)) + X2 denote the total resources (output plus undepreciated first-period capital) that138would be required for the choiceX2 to satisfy the constraints (3.18) and (3.19) as well as the intertem-poral optimality condition (3.21), and note thatB′(X2) =R′(X2)U ′′(c) + 1 > 1 (B.18)where the inequality follows from the assumption made that R′(X2) < 0. Since total resourcesactually available are (1 − δ)X1 + g(X1,M), we have X2 = B−1((1 − δ)X1 + g(X1,M)), andtherefore from condition (3.20) equilibrium can be characterized by a solution toG(X1,M) = Q(M) (B.19)forM, where G(X,M) ≡ gM(X,M)R(B−1((1− δ)X + g(X,M))). Note thatGM(X1,M) = gMM(X1,M)R(X2) +R′(X2) [gM(X1,M)]2B′(X2)< 0Similar to in the static case, we assume that G(X, 0) > Q(0) so that there is an equilibrium withM > 0, and further, conditions are such that this equilibrium is unique, which implies thatGM(X1,M) < Q′(M) (B.20)at the equilibrium value ofM.Lemma B.14. If dX2/dX1 < 0 then dc/dX1 < 0 and di/dX1 < 0.Proof. Since in equilibrium c+X2 = B(X2), we have thatdcdX1=[B′(X2)− 1] dX2dX1Since B′(X2) > 1, if dX2/dX1 < 0 then dc/dX1 < 0. Further, if X2 falls when X1 rises, from thecapital accumulation equation (3.18) we see that i must also fall.Lemma B.15. dX2/dX1 < 0 if and only if{−EQM +(1− δ)gXM(X,M)gX(X,M) [gX(X,M) + 1− δ]}EgXM > 1 (B.21)Proof. Totally differentiating the equilibrium condition (B.19) with respect to X1 yields thatdMdX1= GX(X1,M)Q′(M)−GM(X1,M)(B.22)139Doing the same with y = g(X,M) yieldsdydX1= gX(X1,M) + gM(X1,M)dMdX1(B.23)while differentiating X2 = B−1((1− δ)X1 + g(X1,M)) yieldsdX2dX1= 1B′(X2)(1− δ + dydX1)= [1− δ + gX(X1,M)] [Q′(M)−GM(X1,M)] + gM(X1,M)GX(X1,M)B′(X2) [Q′(M)−GM(X1,M)]where the second line has used equations (B.22) and (B.23). Since the denominator of this expressionis positive by (B.18) and (B.20), the sign of dX2/dX1 is given by the sign of the numerator. Substi-tuting in for GM and GX and using (B.19), some algebra yields that this expression is negative if andonly if condition (B.21) holds.Lemmas B.14 and B.15 together indicate that dc/dX1 < 0 and di/dX1 < 0 both hold if and onlyif condition (B.21) holds. Further, for a given equilibrium level of M, it is clear that the minimumlevel of EgXM needed to satisfy (B.21) is (weakly) greater than that needed to satisfy (B.15) in thestatic case.Proof of Proposition 3.13It can be verified that the steady-state level of purchases e solvesU ′(δ + γδ e)= ζQ(e) (B.24)whereζ ≡ 1− β(1− δ)1− β(1− δ) + βγ ∈ (0, 1)Lemma B.16. For δ sufficiently small, a steady state exists and is unique.Proof. Similar to in the static case, we may express individual j’s optimal choice of steady-stateexpenditure ej given aggregate steady-state expenditure e asej(e) =δδ + γU′−1 (ζQ(e))As before, we can verify that e′j(e) < 0 for e > e?, while e′j(e) > 0 and e′′j (e) > 0 for e < e?. Thus,an equilibrium necessarily exists and is unique if e′j(e) < 1 for e < e?, which is equivalent to the140condition that lime↑e? e′j(e) < 1. This is in turn equivalent to the condition τ < τ˜ , whereτ˜ ≡ −δ + γδζ U′′(U ′−1(ζ ν′(`?)F ′(`?))) F ′(`?) [F (`?)− Φ]ν ′(`?) (B.25)As δ → 0, τ˜ approaches infinity, and thus it will hold for any τ , which completes the proof.Note for future reference that if e′j(e) < 1 then(δ + γ)U ′′ (X + e) < δζQ′(e) (B.26)Lemma B.17. For δ sufficiently small, there exists a steady state in the unemployment regime.Proof. Since U ′(0) > Q(0) by assumption, we also have U ′(0) > ζQ(0). Thus, ifU ′(δ + γδ e?)< ζQ(e?)then by the intermediate value theorem, equation B.24 holds for at least one value of e < e?. Notethatlimδ→0δ + γδ e? =∞and limδ→0 ζt = (1−β)/(1−β+βγ) > 0. Thus, since limc→∞ U ′(c) ≤ 0 by assumption, it followsthatlimδ→0U ′(δ + γδ e?)≤ 0 < limδ→0ζZ (e?)and thus the desired property holds for δ close enough to zero.Lemmas B.16 and B.17 together prove the proposition.Proof of Proposition 3.14Linearizing the system in et and Xt around the steady state and letting variables with hats denotedeviations from steady state and variables without subscripts denote steady-state quantities, we haveXˆt+1 = (1− δ)Xˆt + γeˆteˆt+1 = −[1− β(1− δ)(1− δ − γ)]U ′′X + e)β [(1− δ)Q′(e)− (1− δ − γ)U ′′(X + e)]Xˆt+ Q′(e)− [1− βγ(1− δ − γ)]U ′′(X + e)β [(1− δ)Q′(e)− (1− δ − γ)U ′′(X + e)] eˆt141orxˆt+1 ≡(Xˆt+1eˆt+1)=(1− δ γaeX aee)(Xˆteˆt)≡ Axˆtwhere aeX and aee are the coefficients on Xˆt and eˆt in the expression for eˆt+1. The eigenvalues of Aare then given byλ1 ≡1− δ + aee −√(1− δ + aee)2 − 4β−12λ2 ≡1− δ + aee +√(1− δ + aee)2 − 4β−12We may obtain thatλ1λ2 = β−1 > 1 (B.27)so that |λi| > 1 for at least one i ∈ {1, 2}. Thus, this system cannot exhibit local indeterminacy (see,e.g., Blanchard and Kahn (1980)), which completes the proof.Proof of Proposition 3.15Note for future reference that (B.27) implies that if the eigenvalues are real then they are of the samesign, with λ2 > λ1.Lemma B.18. The system is saddle-path stable if and only if|1− δ + aee| >1 + ββ (B.28)in which case the eigenvalues are real and of the same sign as 1− δ + aee.Proof. To see the “if” part, suppose (B.28) holds, and note that this implies(1− δ + aee)2 >(1 + ββ)2> 4β−1and therefore the eigenvalues are real. If 1− δ+aee > (1 +β)/β, then this implies that λ2 > λ1 > 0,and therefore the system is stable as long as λ1 < 1, which is equivalent to the condition(1− δ + aee)− 2 <√(1− δ + aee)2 − 4β−1 (B.29)Since 1− δ + aee > (1 + β)/β > 2, both sides of this inequality are positive, and therefore, squaringboth sides and rearranging, it is equivalent to1− δ + aee >1 + ββ (B.30)which holds by hypothesis. A similar argument can be used to establish the claim for the case that142−(1− δ + aee) > (1 + β)/β.To see the “only if” part, suppose the system is stable. If the eigenvalues had non-zero complexpart, then |λ1| = |λ2| > 1, in which case the system would be unstable. Thus, the eigenvalues mustbe real, i.e., (1− δ + aee)2 > 4β−1, which in turn implies that|1− δ + aee| > 2√β−1If 1 − δ + aee > 2√β−1, then, reasoning as before, λ2 > λ1 > 0, and therefore if the system isstable then (B.29) must hold. Since (1 − δ + aee) > 2√β−1 > 2, then again both sides of (B.29)are positive, and thus that inequality is equivalent to (B.30), which in turn implies (B.28). Similararguments establish (B.28) for the case where −(1− δ + aee) > 2√β−1.Lemma B.19. The system is saddle-path stable with positive eigenvalues if and only if(1− δ − γ)U ′′(X + e) < (1− δ)Q′(e) (B.31)Proof. Note that the system is stable with positive eigenvalues if and only if (B.30) holds. We havethat1− δ + aee −1 + ββ =[1− β(1− δ − γ)][δζQ′(e)− (δ + γ)U ′′(X + e)]β[(1− δ)Q′(e)− (1− δ − γ)U ′′(X + e)]Since the numerator is positive by (B.26), inequality (B.30) holds if and only if (B.31) holds.Lemma B.20. Ifτ < τ˜? ≡ −1− δ − γ1− δ U′′(U ′−1(ζ ν′(`?)F ′(`?))) F ′(`?)[F (`?)− Φ]ν ′(`?)then the system is saddle-path stable with positive eigenvalues.Proof. Note that condition (B.31) always holds around a full-employment steady state. If the steadystate is in the unemployment regime, then it can be verified that condition (B.31) holds if and only ife′j(e) <δδ + γ ζ1− δ − γ1− δ ∈ (0, 1)where ej(e) is as defined in Lemma B.16. As before, this condition holds for all e if it holds forlime↑e? e′j(e), which it can be verified is equivalent to the condition τ < τ˜?. Note also that τ˜? < τ˜ ,where τ˜ was defined in equation (B.25), so that this condition is strictly stronger than the one requiredto ensure the existence of a unique steady state.Lemmas B.19 and B.20 together establish that, for τ sufficiently small (e.g., τ < τ˜?), the systemconverges monotonicaly to the steady state. It remains to show that consumption is decreasing in143the stock of durables. Assuming τ is sufficiently small so that the system is saddle-path stable withpositive eigenvalues, it is straightforward to obtain the solutionXˆt = λt1Xˆ0eˆt = ψXˆtcˆt = (1 + ψ) Xˆtwhere ψ ≡ −(1− δ − λ1)/γ. Thus, consumption is decreasing in the stock of durables if and only ifψ < −1.Lemma B.21. If (B.31) holds and the steady state is in the unemployment regime, then ψ < −1.Proof. We may write1− δ − γ − λ1=√[aee + 2γ − (1− δ)]2 + 4β−1[β(1− δ − γ)(aee + γ)− 1]− [aee + 2γ − (1− δ)]2Now, aee + 2γ − (1 − δ) > aee − (1 − δ) > 0, so that 1 − δ − γ − λ1 is positive if and only ifβ(1− δ − γ)(aee + γ) > 1. We haveβ(1− δ − γ)(aee + γ) =[1 + βγ(1− δ)]Q′(e)− U ′′(X + e)(1−δ1−δ−γ)Q′(e)− U ′′(X + e)Note by earlier assumptions that this expression is strictly positive, and that1− δ1− δ − γ − [1 + βγ(1− δ)] = γ1− β(1− δ)(1− δ − γ)1− δ − γ > 0Thus, ifQ′(e) < 0 (i.e., the steady state is in the unemployment regime) then β(1−δ−γ)(aee+γ) > 1,in which case 1− δ − γ − λ1 > 0 and therefore ψ < −1.Proof of Proposition 3.16Without loss of generality, assume the alternative path begins at t = 0, and let e˜t(∆) ≡ e + ∆ · tdenote the alternative feasible path of expenditures, where t is the change in the path of expenditures,and ∆ is a perturbation parameter, which is equal to zero in the steady-state equilibrium and equal toone for the alternative path. Let X˜t(∆) denote the associated path for the stock of durables, and notethat X˜0(∆) = X , i.e., this alternative path does not affect the initial stock of durables. Welfare can144then be written as a function of ∆ asU(∆) =∞∑t=0βt{U(X˜t(∆) + e˜t(∆)) +e˜t(∆)F ′(`?)`? [−ν(`?) + V (w?`? − p?e˜t(∆))]+(1− e˜t(∆)F ′(`?)l?)V (−p?e˜t(∆))}From the envelope theorem, beginning from the steady state path (i.e., ∆ = 0), for a marginal changein ∆ the net effect on welfare through the resulting changes inU and V in each period is zero. Thus, weneed only consider effects that occur through changes in the employment rate term, e˜t(∆)/[F ′(`?)`?].A first-order approximation to U(1) around U(0) is therefore given byU(1) ≈ U(0) + 1F ′(`?)`?[L? + τ ν′(`?)F ′(`?)e] ∞∑t=0βte˜′t(0)Substituting in e˜′t(0) = t, the desired result obtains.Proof of Proposition 3.17Let e˜t() and X˜t() denote alternative paths for expenditure and the stock of durables, with e˜t() ≡e(X˜t()) + t and X˜t+1() = (1 − δ)X˜t() + γe˜t(). Here, e(·) is the equilibrium policy functionfor expenditures, while 0 =  and t = 0 for t ≥ 1. Letting U(X0, ) denote the correspondingwelfare as a function of X0 and , we may write a second-order approximation to this function around(X0, ) = (X, 0) asU(X0, ) ≈ U(X, 0) + UXXˆ0 + U+12[UXXXˆ20 + U2]+ UXXˆ0where variables with hats indicate deviations from steady state and partial derivatives of U are evalu-ated at the point (X0, ) = (X, 0). Clearly, to a second-order approximation, the welfare effect of atemporary stimulus is smaller when the economy is in a liquidation phase if and only if UX < 0.Next, using the envelope condition as in the proof of Proposition 3.16, it is straightforward toobtain thatU(X0, 0) =1F ′(`?)`?∞∑t=0βt[L? + τ ν′(`?)F ′(`?)e(Xt)]e˜′t(0)where Xt = X˜t(0) is the stock of durables that would occur in the absence of stimulus. One may alsoobtain thate˜′t(0) =1 : t = 0γe′(Xt){∏t−1i=1 [1− δ + γe′(Xt−i)]}: t ≥ 1145so thatU(X0, 0) =1F ′(`?)`?{[L? + τ ν′(`?)F ′(`?)e(X0)]+ γ∞∑t=1βt[L? + τ ν′(`?)F ′(`?)e(Xt(X0))]·e′(Xt(X0))(t−1∏i=1[1− δ + γe′(Xt−i(X0))])}whereXt(X0) indicates the equilibrium value ofXt givenX0. Taking the derivative of this expressionwith respect to X0 and evaluating at X0 = X yieldsUX(X, 0) =1F ′(`?)`? τν ′(`?)F ′(`?) ·1− βλ1(1− δ)1− βλ21ψ + Ξe′′(X)where ψ ≡ e′(X) < 0, which was computed above, and Ξ is some strictly positive number. Sinceλ1 < 1, the first term on the right-hand side of this expression is clearly negative. Thus, there is astrictly positive number ξ such that if e′′(X) < ξ we will have UX(X, 0) < 0, which is the desiredresult.Letting χ(Xt) denote the equilibrium value of Xt+1 given Xt, we may re-express the equilibriumequations governing the dynamics of the system (i.e., equations (3.22) and (3.23)) asχ(Xt) = (1− δ)Xt + γe(Xt)andU ′(Xt + e(Xt))−Q(e(Xt)) = β[(1− δ − γ)U ′(χ(Xt) + e(χ(Xt)))− (1− δ)Q(e(χ(Xt)))]Taking derivatives of both sides of these equations twice with respect to Xt, evaluating at Xt = Xand solving for e′′(X), we may obtain that e′′(X) = bU ′′′(X + e), where b is some number that doesnot depend on U ′′′(X + e). Thus, if U ′′′ is sufficiently close to zero, e′′(X) < ξ and the desired resultholds.B.2 Introducing Nash bargainingHere we consider the static model of section 3.2 and replace the “competitive” determination of w and` within a match by Nash bargaining.The gain from a match for a firm is pF (`) − w` while outside option is zero. The gain for thehousehold is−ν(`)+V (w`−p(c−X) while the outside option is V (−p(c−X)). Using the piecewiselinear specification for V , the Nash-Bargaining criterionW is:W =(pF (`)− w`)ψ(− ν(`) + vw`+ vτp(c−X))ψ146MaximizingW w.r.t. ` and w gives the following F.O.C.:ψWpF (`)− w`(pF ′(`)− w)= (1− ψ)W−ν(`) + vw`+ vτp(c−X)(vw − ν ′(`))ψWpF (`)− w` =(1− ψ)W−ν(`) + vw`+ vτp(c−X)vRearranging gives the two equationsvpF ′(`) = ν ′(`)vw` = (1− ψ)vpF (`) + ψν(`)− ψvτp(c−X)Assuming that the matching function is “min”, the equilibrium is given by the five following equations:u′(c) = ν′(`)F ′(`)(1 + τ − min{N,L}L τ)(B.32)w` = (1− ψ)pF (`) + ψv ν(`)− ψτp(c−X) (B.33)vpF ′(`) = ν ′(`) (B.34)min{N,L}F (`) = L(c−X) +NΦ (B.35)min{N,L}(pF (`)− w`)= pNΦ (B.36)Equations (B.33) and (B.34) determine p and w once N , c and ` are determined by the three otherequations. After some manipulations, those three equations (B.32), (B.35) and (B.36) can be written:u′(c) = ν′(`)F ′(`)(1 + τ − min{N,L}L τ)(B.37)min{N,L}L =(c−X)(1− ψ)F (`) + ψF ′(`) ν(`)ν′(`) − ψτ(c−X)(B.38)ψmin{N,L}N = Φ(F (`)− F ′(`) ν(`)ν ′(`) + τ(c−X))−1(B.39)In the unemployment regime, those equations writeu′(c) = ν′(`)F ′(`)(1 + τ − NL τ)(B.40)NL =(c−X)(1− ψ)F (`) + ψF ′(`) ν(`)ν′(`) − ψτ(c−X)(B.41)Φψ = F (`)− F′(`) ν(`)ν ′(`) + τ(c−X) (B.42)Main difference with the model of the main text is that (B.42) does not determine ` independently of147(B.40) and (B.41). But it is still the case that, assuming ` is fixed, (B.40) implies that if N is high, cwill be high and (B.41) implies that if c is high, N will be high. As far as (B.42) implies that ` doesnot vary too much, subsequent results of section 3.2 hold. This can be illustrated with a numericalexample that reproduces Figures 3.2, 3.3 and 3.5.Consider the functional forms ν(`) = ν1`1+ω1+ω , F (`) = θ1A`α, u(c) = ln c, V (a) is ν2/θ2 if a ≥ 0and (1 + τ)ν2/θ2 if a < 0. Common parameters values are ψ = .5, ω = 1.2, ν1 = .5, α = .67,A = 1, Φ = .35, L = 1. Solving for the equilibrium in such a case produce Figures B.1, B.2 and B.3,which are qualitatively similar to Figures 3.2, 3.3 and 3.5.Figure B.1: The Model with Nash bargaining, consumption as function of X .0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 ⋆ X ⋆⋆Note: Example is constructed assuming the functional forms ν(`) = ν`1+ω1+ω , F (`) = A`α, u(c) =ln c, V (a) is av if a ≥ 0 and (1 + τ)av if a < 0. Parameters values are ψ = .5, ω = 1.2,ν = v = .5, α = .67, A = 1, Φ = .35, L = 1 and τ = .05.148Figure B.2: The Model with Nash bargaining, equilibrium determination0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.800. j(e)Low XHigh X45 degNote: Example is constructed assuming the functional forms ν(`) = ν`1+ω1+ω , F (`) = A`α, u(c) =ln c, V (a) is va if a ≥ 0 and (1 + τ)va if a < 0. Parameters values are ψ = .5, ω = 1.2,ν = v = .5, α = .67, A = 1, Φ = .35, L = 1 and τ = .05. Values of X used were X = .3 forthe full-employment equilibrium and X = 0.9 for the unemployment equilibrium.Figure B.3: The Model with Nash bargaining, equilibrium determination (multiple equilibria)0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.800. j(e)Low XHigh X45 degNote: Example is constructed assuming the functional forms ν(`) = ν`1+ω1+ω , F (`) = A`α, u(c) =ln c, V (a) is va if a ≥ 0 and (1 + τ)va if a < 0. Parameters values are ψ = .5, ω = 1.2,ν = v = .5, α = .67, A = 1, Φ = .35, L = 1, τ = .4, X = .6 or X = 1.149B.3 Noise shock extensionFor the extension discussed at the end of Section 3.4.2, we re-introduce the first-sub-period (θ) andsecond-sub-period (θ˜) productivity factors to the model, and assume that θ˜t = θt. We assume that theeconomy is always in the unemployment regime, and that all agents come into the first sub-period ofperiod t with the same belief about the value of θt, but that after the household splits to go to market,the true value is revealed to the workers and firms, while the shoppers retain their initial belief.To abstract from issues relating to uncertainty about the true value of θt, we assume that all agentsare subjectively certain – though possibly incorrect – about the entire stream of productivity values θt,only updating such a belief if they receive some information that contradicts it. One may verify that,in the unemployment regime, shoppers’ prior beliefs are never contradicted until re-uniting with theworkers after making their purchases. We denote agents’ belief about θt at the beginning of date s byθ¯t|sIn the example constructed, we assume that productivity is constant at θt = 1 for all t ∈ Z, butthat at the beginning of t = 0, agents receive information such that θ¯t|0 = θ > 1 for all t ≥ 0, i.e.,that productivity has risen permanently. After the households split, workers and firms learn that infact productivity has not changed, nor will it in the future. Shoppers do not receive this informationuntil after making their purchases, so that for one shopping period they are overly optimistic. In allsubsequent periods s ≥ 1, however, we have θ¯t|s = θt = 1.150Appendix CAppendix for “Can a Limit-Cycle ModelExplain Business Cycle Fluctuations?”C.1 ProofsC.1.1 Proof of Proposition 4.1Recall thatXt+1 = (1− δ) (Xt + et)Since et ≥ 0, if lim supt→∞ |Xt| =∞ then lim supt→∞Xt =∞. Suppose then thatlim supt→∞Xt =∞Since δ ∈ (0, 1], this necessarily implies that lim supt→∞ et = ∞. But et is bounded above by thelevel of output, the maximum feasible level of which occurs when φt = 1 and `t = ¯`, in which casetotal output is given by F(¯`) < ∞. Thus we clearly cannot have lim supt→∞ et = ∞, and thus wecannot have lim supt→∞ |Xt| =∞.C.1.2 Proof of Proposition 4.2The proof proceeds by example, showing that, for the case where γ = 1 and U (c) = ac− b2c2, thereexists parameter values and functional forms such that for β close enough to one the steady state isunstable.With γ = 1 and U (c) = ac − b2c2, we may characterize the evolution of this system by the151conditionsa− b (Xt + et) = vp (et) [1 + τ − τφ (et)]− β (1− δ) vp (et+1) [1 + τ − τφ (et+1)] (C.1)Xt+1 = (1− δ) (Xt + et) (C.2)where p (·) and φ (·) are as in the static model. For a given state Xt and a given anticipated level ofet+1, a sufficient condition to ensure that (C.1) has a unique solution is given byb > vp∗ τe∗ ≡ b0 (C.3)where e∗ is output per firm (net of fixed costs) when the economy is in the unemployment regime andp∗ is the price in the unemployment regime, as described in section 4.5.1 for the static model (seefootnote 29 regarding p∗). I henceforth assume that (C.3) holds.Next, the steady-state level of e is given by the solution e¯ toa− bδ e¯ = [1− β (1− δ)] vp (e¯) [1 + τ − τφ (e¯)]with the steady-state level of X then given byX¯ = 1− δδ e¯Note that a sufficient condition for the steady state to be unique is given byb > δ [1− β (1− δ)] b0which is clearly implied by (C.3).Next, note that, for any e ∈ (0, e∗) (i.e., in the unemployment regime), the level of a that imple-ments e¯ = e is given bybδ e+ [1− β (1− δ)] vp∗(1 + τ − τ ee∗)Note also that e¯ is continuous in β. Thus, choose some e¯1 ∈ (0, e∗), and let a = a1, where a1 is thevalue of a that would implement e¯ = e¯1 when β = 1, i.e.,a1 ≡bδ e¯1 + δvp∗(1 + τ − τ e¯1e∗)Thus, if β = 1 the steady state is in the unemployment regime by construction, and by continuityof e¯ in β the steady state is also necessarily in the unemployment regime for β sufficiently close toone. This implies the existence of a β < 1 such that the steady state is in the unemployment regimewhen β > β. Assume henceforth that β ∈(β, 1)and note that this implies that p′ (e¯) = 0 andφ′ (e¯) = 1/e∗.152Next, linearizing equations (C.2)-(C.3) around this steady state and solving, we may obtain inmatrix form (Xˆt+1eˆt+1)=(1− δ 1− δ− bβ(1−δ)b0 −b−b0β(1−δ)b0)(Xˆteˆt)≡ A(Xˆteˆt)Thus, the steady state is locally stable if and only if at least one of the two eigenvalues of A lies insidethe complex unit circle. These eigenvalues are given byλi =[1− δ − b−b0β(1−δ)b0]±√[1− δ − b−b0β(1−δ)b0]2− 4β−12Note that λ1λ2 = β−1 > 1, so that if the eigenvalues are complex then both must lie outside the unitcircle. Supposeb =[1 + q (1− δ)2]b0 (C.4)for some q > 0, and note that as long as δ < 1, which I henceforth assume, such a value of b satisfies(C.3). One may then show that the eigenvalues are complex as long as(1− δ)2 (β − q)2 < 4βClearly, for β close enough to q this condition necessarily holds, and thus, if q is close enough to one(e.g., if q = 1), then for β arbitrarily close to one the eigenvalues are complex and therefore outsidethe unit circle, in which case the steady state is unstable.C.1.3 Proof of Proposition 4.3LetV (et;Xt) ≡ U (Xt + et)− vp∗[(1 + τ) et −12τe2te∗]where e∗ is output per firm (net of fixed costs) when the economy is in the unemployment regime andp∗ is the price in the unemployment regime, as described in section 4.5.1 for the static model (seefootnote 29 regarding p∗). It can be verified that maximizing∞∑t=0βtV (et;Xt)subject to (4.2) implements the de-centralized equilibrium outcome in the neighborhood of an unemployment-regime steady state. Thus, usingW (Xt, Xt+1) ≡ V( 1γ (1− δ)Xt+1 −1γXt;Xt)153in problem (4.4) satisfies the desired properties. Next, we may obtainW11(X¯, X¯)= (1− γ)2γ2 U′′ (X¯ + e¯)+ 1γ2vp∗τe∗Thus,W11(X¯, X¯)> 0 ifvp∗τe∗ > − (1− γ)2 U ′′(X¯ + e¯)This condition can clearly hold for certain parameter values (e.g., for γ sufficiently close to one), inwhich caseW is not concave.C.1.4 Proof of Proposition 4.4We show that the steady state is locally stable when τ = 0. By continuity of all relevant functions, itthen follows that the steady state is locally stable for τ > 0 sufficiently small.When τ = 0, equilibrium is characterized by the equationsU ′ (Xt + et)− vp (et) = β (1− δ) (1− γ)U ′ (Xt+1 + et+1)− β (1− δ) vp (et+1)Xt+1 = (1− δ) (Xt + γet)Assume the steady state is in the unemployment regime, so that p (et) = p∗ in a neighborhood of thesteady state.1 Linearizing around this steady state, assuming β (1− δ) (1− γ) > 0 we may obtain inmatrix form(Xˆt+1eˆt+1)=1− δ (1− δ) γ[1−β(1−δ)2(1−γ)]β(1−δ)(1−γ)[1−β(1−δ)2(1−γ)γ]β(1−δ)(1−γ)(Xˆteˆt)≡ A(Xˆteˆt)The steady state is locally stable if at least one of the eigenvalues of A lies inside the unit circle. It isstraightforward to show that the smallest eigenvalue of A is given by λ1 = (1− δ) (1− γ), which isclearly less than one in modulus. Thus, the steady state is locally stable. If instead β (1− δ) (1− γ) =0, then eˆt = −Xˆt and thus Xˆt+1 = λ1Xˆt, which is clearly a stable system as well.1It is straightforward to verify that a full-employment-regime steady state must be stable.154C.2 Solution and estimationC.2.1 SolutionTo solve the model for a given parameterization, letting e˜t ≡ et/θ˜t equilibrium in the economy ischaracterized by the following equations:a− b(Xt + θ˜te˜t − hct−1)+ (1− δ) γλt = θ˜−1tν1αA [` (e˜t)]ω+1−α [1 + τ − τφ (e˜t)] + µt (C.5)µt = Et{βh[a− b(Xt+1 + θ˜t+1e˜t+1 − hct)]}(C.6)λt = Et{β[a− b(Xt+1 + θ˜t+1e˜t+1 − hct)+ (1− δ)λt+1 − µt+1]}(C.7)ct = Xt + θ˜te˜t (C.8)Xt+1 = (1− δ)(Xt + γθ˜te˜t)(C.9)Here, φ (e˜) and ` (e˜) are the equlibrium levels of the employment rate and hours-per-worker condi-tional on total purchases e˜, and are given byφ (e˜) ≡12(n0 +√n20 + 4η e˜e˜∗)if 0 < e˜ ≤ e¯e˜e˜∗ if e¯ < e˜ < e∗1 if e˜ ≥ e∗` (e˜) ≡ 2e˜αA(n0+√n20+4ηe˜e˜∗)1αif 0 < e˜ ≤ e¯(e∗αA) 1α if e¯ < e˜ < e∗(e˜αA) 1α if e˜ ≥ e∗where e∗ ≡ α1−α k¯ and e¯ ≡ (n0 + η) e∗. Meanwhile, µt and λt are the Lagrange multipliers on thedefinition of consumption and the durables accumulation equations ((C.8) and (C.9)), respectively.Conditional on the state variables Xt, ct−1 and θt, and on values of the Lagrange multipliers µtand λt, equation (C.5) can be solved for e˜t. To obtain values of µt and λt, I employ the method ofparameterized expectations as follows. Let Yt ≡(Xt − X¯, ct−1 − c¯, θt)′denote the vector of statevariables (expressed as deviations from steady state). The expectations in equations (C.6) and (C.7)are assumed to be functions only of Yt, i.e.,Et{βh[a− b(Xt+1 + θ˜t+1e˜t+1 − hct)]}= gµ (Yt)155Et{β[a− b(Xt+1 + θ˜t+1e˜t+1 − hct)+ (1− δ)λt+1 − µt+1]}= gλ (Yt)I parameterize the functions gj (·) by assuming that they are well-approximated byN -th-degree multi-variate polynomials in the state variables. In particular, let Y (N)t denote the vector whose first elementis 1 and whose remaining elements are obtained by collecting all multivariate polynomial terms in Yt(e.g., Xt, ct−1, θt, X2t , Xtct−1, Xtθt, c2t , ctθt, etc.) up to degree N . I assume thatgj (Yt) = Θ′jY(N)twhere Θj is a vector of coefficients on the polynomial terms. Thus, given Θµ, Θλ and the state Yt, µtand λt are obtained asµt = Θ′jY(N)tλt = Θ′jY(N)tThese values and values for the state variables can be plugged into (C.5) to yield a solution for e˜t,which can then be replaced in (C.8) and (C.9) to obtain values for the subsequent period’s state. Inpractice, I use N = 2.2To obtain Θµ and Θλ, I proceed iteratively as follows. Begin with some initial guesses Θµ,0and Θλ,0,3 and generate a sample of length T = 100, 000 of the exogenous process θt. Next, givenΘµ,i and Θλ,i, assume that gj (Yt) = Θ′j,iY(N)t and simulate the path of the economy for T periods.Given this simulated path, let Y(N) denote the matrix whose t-th row is given by Y (N)′t , and constructT -vectors g˜µ and g˜λ, the t-th elements of which are given respectively byβh[a− b(Xt+1 + θ˜t+1e˜t+1 − hct)]andβ[a− b(Xt+1 + θ˜t+1e˜t+1 − hct)+ (1− δ)λt+1 − µt+1]i.e., the terms inside the conditional-expectation operators in equations (C.6) and (C.7). Then updatethe guesses of Θj viaΘj,i+1 =(Y(N)′Y(N))−1Y(N)′g˜jand iterate until convergence.2I experimented with larger values ofN and found that it resulted in a substantial increase in computational time withoutsignificantly affecting the results.3In practice, I set the first elements of Θµ,0 and Θλ,0 to the steady-state values µ¯ and λ¯, respectively, and the remainingelements to zero. This corresponds to an initial belief that the gj’s are constant and equal to their steady-state levels.156C.2.2 EstimationAs discussed in section 4.6.1, estimation was done by searching for parameters to minimize S2, theaverage squared difference between the model spectrum and the spectrum estimated from the data.To obtain S2 given a solution to the model for a parameterization, T = 100, 000 periods ofdata were simulated. This simulated sample was then subdivided into Nsim = 1, 000 overlappingsubsamples. For each subsample, the log of hours was BP-filtered, after which 20 quarters from eitherend of the subsample were removed, leaving a series of the same length as the actual data sample. Thespectrum was then estimated on each individual subsample in the same way as for the actual data, andthe results then averaged across all subsamples to yield the spectrum for the model.C.3 DefinitionsA deterministic dynamic system characterizing the evolution of a state vector z (t) over time can beexpressed as a function G : T × Z → Z . Here, T is the set of dates at which the system is defined(e.g., T = R in a continuous-time formulation, and T = Z in a discrete-time formulation), whileZ ⊂ Rn is the n-dimensional state space. The function G takes a date t ∈ T and a date-0 statez (0) = z0 as inputs and returns z (t) = G (t, z0).4 We focus here on time-invariant dynamic systems,i.e., systems for which G (t+ ∆t, z0) = G (∆t, G (t, z0)). We have the following definition.Definition C.1. G exhibits deterministic fluctuations if, for some z0, the following hold.(a) lim supt→∞ ‖G (t, z0)‖ <∞.(b) limt→∞G (t, z0) does not exist.In words, the system exhibits deterministic fluctuations if it neither diverges to infinity nor con-verges to a single point. Of particular interest for us will be one type of deterministic fluctuation, thelimit cycle, defined as follows.Definition C.2. A subset L ⊂ Z is a limit cycle of G with prime period k > 0 if the following hold.(a) For any z ∈ L and ∆t ∈ (0, k), we have G (k, z) = z and G (∆t, z) 6= z.(b) For any z, z′ ∈ L there is some ∆t ≥ 0 such that G (∆t, z) = z′.(c) Let Γ (t, z) denote the distance between G (t, z) and the closest point in L.5 Then there existssome z /∈ L such that either limt→∞ Γ (t, z) = 0 or limt→−∞ Γ (t, z) = 0.The first property here says that, if the system starts at a point in L, then it will return to that pointk > 0 periods later (and no sooner). The second property says that as the system evolves beginningfrom any point in L it will visit every other point in L at some subsequent date. Finally, the third4Note that by definition G (0, z0) = z0.5Properties (a) and (b) ensure that L is necessarily a closed set, so that such a closest point always exists.157property says that there is some point not in the limit cycle such that, beginning from that point, thesystem will eventually converge to the limit cycle as it moves either forward or backward throughtime.C.4 Solving the model forwardIn the non-stochastic case, we may re-arrange equation (C.5) to yielda− bXt + bhct−1 + (1− δ) γλt − µt =ν1αA [l (et)]ω+1−α [1 + τ − τφ (et)] + bet ≡ H (et)Thus, given the state variables Xt and ct−1 and current values of µt and λt, we may obtainet = H−1 (a− bXt + bhct−1 + (1− δ) γλt − µt)where the conditions in Proposition 4.5 ensure that H is an invertible function. This value of et thengives ct and Xt+1 via equations (C.8) and (C.9), respectively. From equations (C.6) and (C.7) we canthen solve for µt+1 and λt+1 asµt+1 =a− bXt+1 + bhct −H(1b(a− bXt+1 + bhct − 1βhµt))− γ(1βhµt − 1βλt)1− γλt+1 =a− bXt+1 + bhct −H(1b(a− bXt+1 + bhct − 1βhµt))−(1βhµt − 1βλt)(1− δ) (1− γ)158


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