Precautionary saving and aggregate demand
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Challe, Edouard; Matheron, Julien; Ragot, Xavier; Rubio-Ramírez, Juan Francisco Article Precautionary saving and aggregate demand Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Challe, Edouard; Matheron, Julien; Ragot, Xavier; Rubio-Ramírez, Juan Francisco (2017) : Precautionary saving and aggregate demand, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 8, Iss. 2, pp. 435-478, https://doi.org/10.3982/QE714 This Version is available at: https://hdl.handle.net/10419/195545 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/
Quantitative Economics 8 (2017), 435–478 1759-7331/20170435 Precautionary saving and aggregate demand Edouard Challe Department of Economics, Ecole Polytechnique and CREST (CNRS) Julien Matheron Research Division, Banque de France Xav i e r Ragot Department of Economics, Paris School of Economics Juan F. R ubio-Ramirez Department of Economics, Emory University and Federal Reserve Bank of Atlanta We construct, and then estimate by maximum likelihood, a tractable dynamic stochastic general equilibrium model with incomplete insurance and heterogenous agents. The key feature of our framework is that cross-sectional heterogeneity remains finite dimensional. The solution to the model thus admits a statespace representation that can be used to recover the distribution of the model’s parameters. Household heterogeneity expands the set of observables to crosssectional moments available at the business-cycle frequency (in addition to the Edouard Challe: [email protected] Julien Matheron: [email protected] Xavier Ragot: [email protected] Juan F. Rubio-Ramirez: [email protected] We are grateful to Susanto Basu, Morten Ravn, and Jose-Victor Rios Rull for their feedback on an earlier version of this paper. We also benefited from comments by conference participants at the 2013 (Micro Heterogeneity) and 2014 (Impulse and Propagation) NBER Summer Institutes, the 2013 NBER EFSF Workshop on DGSE Models, the 2014 and 2015 SED Meetings, the 2014 ASSA–AEA Meetings, the 2014 ENSAI Economic Day, the 2014 ES North American Meetings, the INET–Cambridge Conference on Aggregate Demand, the Labor Market and Macroeconomic Policy (September 2014), the 2014 Hydra Workshop on Dynamic Macroeconomics, the 3rd AMSE/BDF Labor Market Conference, the 2015 CREI/IFW/Richmond Fed Conference on New Developments in the Macroeconomics of the Labor Markets, as well by seminar participants at IAE/Barcelona GSE, IMF, EIEF, CEPREMAP, Ecole Polytechnique, Tinbergen Institute, Toulouse School of Economics, San Francisco Fed, USC, UCLA, UC Davis, UC Irvine, University of Warwick, University of Cambridge, and University of Konstanz. Edouard Challe acknowledges financial support from Investissements d’Avenir (ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047) and Chaire FDIR. Xavier Ragot acknowledges financial support from the European Commission FP7-612796 MACFINROBODS. Juan F. Rubio-Ramirez acknowledges financial support from National Science Foundation Grant SES-1227397, Fondation Banque de France pour la Recherche, the Institute for Economic Analysis (IAE), the Programa de Excelencia en Educacion e Investigacion of the Bank of Spain, and the Spanish Ministry of Science and Technology ECO2011-30323-c03-01. The views expressed herein are those of the authors and should not be interpreted as reflecting those of the Federal Reserve Bank of Atlanta, the Federal Reserve System, the Banque de France, the Eurosystem, or OFCE. Copyright ©2017 The Authors. Quantitative Economics. The Econometric Society. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://www.qeconomics.org. DOI: 10.3982/QE714
436 Challe, Matheron, Ragot, and Rubio-Ramirez Quantitative Economics 8 (2017) usual macro and monetary time series). Incomplete insurance gives rise to a precautionary motive for holding wealth that propagates aggregate shocks via (i) a stabilizing aggregate supply effect, working through the supply of capital, and (ii) a destabilizing aggregate demand effect coming from the feedback loop between unemployment risk and precautionary saving. Using the estimated model to measure the contribution of precautionary savings to the propagation of recent recessions, we find strong aggregate demand effects during the Great Recession and, to a lesser extent, during the 1990–1991 recession. In contrast, the supply effect at least offsets the demand effect during the 2001 recession. Keywords. DSGE, incomplete insurance, heterogenous agents, Bayesian estimation. JEL classification. C32, E12, E21, E52. “A rational expectations equilibrium is a likelihood function.” Thomas J. Sargent (in Evans and Honkapohja (2005)). A rational expectations equilibrium is a likelihood function: given preferences and technologies, if the aggregate shocks have a distribution, then there is a likelihood function on the structural parameters that obeys the cross-equation restrictions implied by the model. Following this insight, a growing number of researchers have constructed medium-scale New Keynesian models with enough shocks and wedges to fit the data well, which they have estimated using likelihood-based procedures.1Likelihood-based procedures have two advantages with respect to traditional calibration approaches. First, they deliver estimates of the paths of the shocks that explain the data and, hence, it is possible to construct counterfactuals. Second, they provide a sense of the parameter uncertainty surrounding the estimates. A common feature of this line of work is the assumption of perfect insurance against idiosyncratic income shocks. The reason for this is that imperfect insurance typically generates enormous ex post heterogeneity among agents, which existing solution methods cannot handle without drastically restricting the set of aggregate shocks and endogenous state variables. As a consequence, imperfect-insurance models cannot be estimated by maximum likelihood; rather, they are calibrated by some method of moments using only a subset of the moment conditions.2 In this paper, we formulate a medium-scale, New Keynesian dynamic stochastic general equilibrium model with imperfect insurance, whose solution admits a finitedimensional state-space representation that can be used to compute the likelihood function. The property that the state-space remains finite-dimensional, despite imperfect insurance, follows from two basic assumptions. The first one concerns the extent of risk sharing; namely, we depart from perfect insurance in a minimal way by assuming 1Examples are Boivin and Giannoni (2006), Del Negro, Schorfheide, Smets, and Wouters (2007), Ireland (2004), Lubik and Schorfheide (2004), Canova and Sala (2009), DeJong, Ingram, and Whiteman (2000), Schorfheide (2000), Otrok (2001), Smets and Wouters (2007), Justiniano, Primiceri, and Tambalotti (2010), Aruoba and Schorfheide (2011). 2Examples are Chatterjee, Corbae, Nakajima, and Ríos-Rull (2007), Davila, Hong, Krusell, and Ríos Rull (2012).
Quantitative Economics 8 (2017) Precautionary saving and aggregate demand 437 that households belong to large, representative “families,” within which full risk sharing takes place between employed members only, while the unemployed receive unemployment benefits provided by the government. Our second assumption is that the debt limit faced by households is tighter than the “natural” debt limit, that is, the maximum amount that a household can borrow while being able to repay (and always enjoy positive consumption) in the worst possible income history (see, e.g., Aiyagari (1994)). We show that under these two assumptions—partial risk sharing and tight debt limit— the wealth distribution converges to a distribution with a finite number of mass points, which in turn implies that the aggregate state itself remains finite dimensional.3Incidentally, one attractive feature of our approach is to make it possible to include the timeseries dimension of cross-sectional information into the likelihood estimation of the model, in addition to the usual macro and monetary time series. While household-level data have routinely been used to calibrate imperfect-insurance models (starting with Krusell and Smith (1998)), this information has not yet been used as observable variables when estimating medium-scale New Keynesian models. A crucial step in that direction was made in papers like Iacoviello (2008)orJustiniano, Primiceri, and Tambalotti (2015), who calibrate such models to match some key features of the wealth distribution. We push this approach further by using quarterly household-level data on consumption dispersion at the estimation stage, in addition to lower-frequency household-level data at the calibration stage.4 Our general framework significantly expands the set of macroeconomic questions that can be investigated via structural, likelihood-based estimation, namely, to any issue where imperfect insurance against idiosyncratic shocks is likely to matter. In the present paper we illustrate our approach by focusing on one such issue: the way households’ “precautionary saving” behavior—their rational savings response in the face of imperfect insurance—propagates aggregate shocks. We focus on precautionary savings against unemployment risk, the main source of time-varying idiosyncratic risk at the business-cycle frequency. While there may be other sources of business-cycle driven change in idiosyncratic risk (e.g., wage risk), those would almost certainly magnify the response of the precautionary motive and hence strengthen the effects that we are after. To capture the main channels by which precautionary saving may affect outcomes, our framework combines three basic frictions: (i) nominal rigidities (in prices and wages), (ii) labor-market frictions, and (iii) imperfect insurance against idiosyncratic unemployment risk. All three frictions are known, even in isolation, to capture some important features of the business cycle. Importantly, their interactions give rise to a feedback loop between precautionary savings and aggregate demand: following aggregate shocks 3Challe and Ragot (2014) construct and calibrate a real business-cycle (RBC) model with imperfect insurance that also features a finite-dimensional wealth distribution, using a period utility function that is linear above a threshold (rather than assuming a form of partial cross-household insurance). The framework that they use ignores aggregate demand effects by construction, and is not suitable for structural estimation. 4More specifically, we estimate the model using as an observable a moment of the cross-sectional distribution of consumption (constructed from the U.S. Consumption Expenditure Survey). In a similar spirit, but focusing on firms’ capital structure rather than household heterogeneity, Ajello (2014) uses firm-level data on the capital structure at the estimation stage of a dynamic stochastic general equilibrium (DSGE) model with credit frictions.
438 Challe, Matheron, Ragot, and Rubio-Ramirez Quantitative Economics 8 (2017) that lower demand, job creation is discouraged, unemployment persistently rises, and hence so does idiosyncratic unemployment risk. Imperfectly insured households rationally respond to this rise in idiosyncratic unemployment risk by increasing precautionary wealth, thereby cutting consumption and degrading demand even more. This “aggregate demand” effect of time-varying precautionary savings is, however, usually not the only one at work in economies with imperfect insurance. As is now well understood, time-varying precautionary savings also have an “aggregate supply” effect that tends to reduce, not increase, aggregate volatility. Indeed, in a typical recession, as unemployment risk rises, imperfectly insured households save more (for precautionary purposes) than they would if they were perfectly insured. These additional savings lower the equilibrium interest rate relative to the perfect-insurance benchmark, which tends to limit the contraction in investment and the capital stock. Conversely, the reduction in unemployment risk in a typical boom leads to a fall in precautionary savings that raises the equilibrium interest rate and lowers the demand for capital, relative to the perfect-insurance benchmark. The aggregate supply effect of precautionary savings against unemployment risk thus tends to smooth fluctuations in investment, capital, and, ultimately, output (see, e.g., Krusell and Smith (1998)). Hence, in the presence of both the aggregate demand and supply effects of precautionary savings, determining which effect dominates, and hence whether time-varying precautionary savings ultimately makes the economy more or less responsive to aggregate shocks, becomes an empirical question. Our framework allows us to incorporate both effects (see Krueger, Mitman, and Perri (2015)) and to measure their relative strength from the data. Once the joint posterior distribution of the structural parameters of the model has been recovered, we ask whether the precautionary motive mattered in the propagation of the recent U.S. recessions, including the Great Recession? In these instances, has the aggregate demand effect dominated the aggregate supply effect, making the precautionary motive inherently destabilizing? To answer these questions, we extract the aggregate shocks that affected the U.S. economy during these periods and then feed them into a counterfactual perfect-insurance model; hence, the precautionary motive due to imperfect insurance is not present by construction. For the Great Recession, we find evidence of a powerful feedback loop between idiosyncratic unemployment risk and consumption demand, so that the aggregate demand effect largely dominates the aggregate supply effect (not only does the precautionary motive significantly amplify the fall in aggregate consumption, the latter also feeds back to adverse labor-market conditions). We find qualitatively similar, though quantitatively less important, amplification effects during the 1990–1991 recession. In contrast, we find no evidence of strong aggregate demand effects during the 2001 recession; if anything, the supply effects dominate the aggregate dynamics (that is, there is less aggregate volatility with the precautionary motive than without). Our analysis relates to several strands of the business-cycle literature. Sticky-price models emphasize the role of aggregate demand as a key driver of the business cycle (see, e.g., Christiano, Eichenbaum, and Evans (2005), Galí (2010), Smets and Wouters (2007), Woodford (2003)). These models have recently been extended to incorporate labor-market frictions; see Walsh (2005), Trigari (2009), Blanchard and Galí (2010),
Quantitative Economics 8 (2017) Precautionary saving and aggregate demand 439 Gertler, Sala, and Trigari (2008), Heer and Maussner (2010), Leduc and Liu (2014), and Galí (2010) for a survey. We relax the perfect-insurance assumption from this framework. Krusell, Mukoyama, and Sahin (2010), Nakajima (2012) and, more recently, Kehoe, Midrigan, and Pastorino (2014) analyze imperfect-insurance models with search frictions wherein the idiosyncratic unemployment risk faced by households is endogenized through firms’ job creation policy. These models assume flexible prices, implying that only the aggregate supply effect is operative. Other papers combine nominal frictions with imperfect insurance, but as in Krusell and Smith (1998), treat labor-market flows as exogenous constraints on labor supply. This, by construction, rules out any feedback from aggregate demand to unemployment risk, which is the key amplification mechanism in our model. This class of models includes Guerrieri and Lorenzoni (2011), who study the impact of a tightening of the borrowing constraint, Oh and Reis (2012)andMcKay and Reis (2013), who study the impact of fiscal and transfer policies, and McKay, Nakamura, and Steinsson (2015), who examine the effectiveness of “forward guidance” at the zero lower bound. Two papers consider the same frictions in goods, labor, and asset markets as we do: Gornemann, Kuester, and Nakajima (2012)andRavn and Sterk (2013). There are important differences between these papers and ours, both in terms of focus and method. Gornemann, Kuester, and Nakajima (2012) are concerned with the redistributive impact of monetary policy shocks. The authors thus construct an imperfect-insurance model with large-dimensional cross-sectional heterogeneity and show that an increase in the policy rate raises income and wealth inequalities, consistent with the empirical findings of Coibion, Gorodnichenko, Kueng, and Silvia (2012). Ravn and Sterk (2013) study how an exogenous shock to the job separation rate can explain the depth and length of the Great Recession. The latter paper illustrates the feedback loop between unemployment risk and aggregate demand, but it has no capital, and hence the aggregate supply effect of precautionary savings is shut down. In contrast to both contributions, we construct the likelihood function and estimate, rather than calibrate, our model.5 Our interest in the aggregate demand effect of time-varying precautionary savings is shared by several recent theoretical contributions, most notably Rendhal (2014), Beaudry, Galizia, and Portier (2014), and Heathcote and Perri (2014). Rendhal (2014) shows how a zero lower bound problem coupled with labor-market frictions and rigid nominal wages can cause the economy to fall into a liquidity trap. Beaudry, Galizia, and Portier (2014) show that when the economy has excess capital, precautionary savings against idiosyncratic unemployment risk may cause a demand shortage. The authors’ approach is closely related to Heathcote and Perri (2014), who show that the feedback loop between aggregate demand and idiosyncratic unemployment risk may lead to multiple equilibria. Like Beaudry, Galizia, and Portier (2014)andHeathcote and Perri (2014), our paper focuses on the interactions between households’ wealth and idiosyncratic unemployment risk, although the specific mechanism by which this occurs in our model is 5Den Haan, Rendahl, and Riegler (2015) identify an alternative mechanism generating a feedback loop between unemployment risk and aggregate demand—based on nominal wage rigidities (but flexible prices) and the possibility of hoarding (zero-interest) cash for precautionary purposes—in addition to real assets. Moreover, their model is calibrated and not estimated.
440 Challe, Matheron, Ragot, and Rubio-Ramirez Quantitative Economics 8 (2017) different from theirs and is embedded into the standard sticky-price framework. Moreover, in contrast to all three papers, we design our model to estimate it and extract the strength of the unemployment risk–aggregate demand feedback loop from the data. On the methodological side, we show how a first-order approximation to an incomplete-insurance model can capture time-varying precautionary savings and can be used for estimation. Our approach differs from the alternative approach of Reiter (2009) in that our framework endogenously generates a finite-dimensional state space, whereas the state space is infinite dimensional and approximated by a finite distribution in Reiter (2009). As a consequence, perturbation methods and likelihood-based estimation are particularly easy to use within our framework. The rest of the paper is organized as follows. Section 1describes the model, from agents’ behavior to the definition of the recursive equilibrium. Section 2shows how our assumptions lead to a collapse of the dimension of the state space, while preserving the precautionary motive. Section 3estimates the model and evaluates its empirical performance. Section 4discusses our counterfactual experiment and investigates the amplifying role of the precautionary motive during the last U.S. recessions. Section 5offers some concluding remarks. 1. The model 1.1 Model overview The model introduces imperfect insurance against time-varying idiosyncratic unemployment risk into a quantitative “New Keynesian” model with labor-market frictions. There are two household types: “workers” and “firm owners.” All households participate in a market for one-period nominal bonds, supply labor when employed, and transit between employment and unemployment. However, only firm owners own the capital stock as well as all firms. Idiosyncratic unemployment risk cannot be perfectly insured by workers, who also face a borrowing constraint (as in, e.g., Krusell and Smith (1998)). Such financial frictions will motivate employed workers’ accumulation of assets for precautionary reasons. Hence, to the extent that the idiosyncratic unemployment risk is time-varying, so will be the amount of assets in the economy for precautionary reasons. The production side has four types of firms, in the spirit of, for example, Trigari (2009)orHeer and Maussner (2010). Labor intermediaries hire labor from the households in a market with matching frictions (modeled as in Mortensen and Pissarides (1994)) and transform it into labor services. Competitive wholesale goods firms buy labor and capital services to produce wholesale goods that are then used as inputs by intermediate goods firms. Every intermediate good firm is the monopolistic supplier of the differentiated good it produces, but faces Calvo (1983)-type nominal frictions when setting nominal prices (as in, e.g., Christiano, Eichenbaum, and Evans (2005), Smets and Wouters (2007)). Finally, intermediate goods firms sell their goods to a competitive final goods sector, which aggregates them into a single final good that is ultimately used for consumption and investment, as well as utilization and vacancy posting costs. Even though intermediate goods firms set nominal prices, we express all prices in real terms
Quantitative Economics 8 (2017) Precautionary saving and aggregate demand 441 Figure 1. Model time line within a period. using the final good as the numeraire. A central bank determines the nominal interest rate via a Taylor-like rule. The timing of events within a period is as follows (see Figure 1). A period is divided into three stages: labor-market transitions, production, and consumption–saving. In the first stage, after the innovations to the exogenous aggregate state have been revealed, some existing employment relationships are destroyed, then hiring decisions are made, and new relationships are formed.6In the production stage, production takes place and total income is shared between households in the form of net wages (for employed households, whether workers or firm owners), capital service payments (for firm owners, whether employed or unemployed), unemployment benefits (for unemployed households, whether workers or firm owners), and monopolistic profits (for firm owners, whether employed or unemployed). Finally, households’ assets holdings are determined in the consumption–saving stage, after the nominal bonds issued in the previous period have paid out. We present the model recursively and use primes to denote the next period’s values. We call the aggregate state X, a vector containing all the relevant aggregate state variables in the model. We assume that all agents know the current value of Xas well as its law of motion X=(X),whereis the innovation to the exogenous aggregate state.7The exogenous aggregate state is Markovian and includes a stochastic productivity trend ez,wherezdrifts at rate μz≥0. For expositional clarity, we summarize the content of Xin Section 1.6 below, only after the presentation of the model has been completed. We first present the behavior of the households (Section 1.2), then present that of the firms (Section 1.3), and finally turn to the market-clearing conditions (Section 1.5) 6Our timing assumption allows a worker who is separated from the firm in the current period to be rematched within the same period, in which case the worker does not effectively experience unemployment. This timing is consistent with the fact that labor-market flows occur at a frequency that is higher than the quarterly frequency (see, e.g., Walsh (2005), Galí (2010)). 7We will use “exogenous aggregate state” and “aggregate shocks” as interchangeable expressions.
442 Challe, Matheron, Ragot, and Rubio-Ramirez Quantitative Economics 8 (2017) and the definition of the equilibrium (Section 1.6). In the recursive representation of the model, all variables either belong to Xor are a function of X. To save on notation when presenting the model we will only make this explicit for the value and policy functions. The rest of the relationships will be clarified when describing the equilibrium.8 1.2 Households There is a unit mass of households, each of which is endowed with one unit of labor that is supplied inelastically during the production stage if the household is employed by the end of the labor-market transitions stage. All households are subject to idiosyncratic changes in their employment status: a share f(resp. s)∈[01]of the households that are unemployed (respectively, employed) before the labor-market transitions stage will be employed (respectively, unemployed) at the end of that stage. We refer to fand sas the job-finding and job-loss rates. There are two types of households: there is a measure Ω∈[01)of workers (indexed by Whenceforth) and a measure 1−Ωof firm owners (indexed by F). All households have the same period utility function u(c −hc)=lim˜σ→σ(c−hc)1−˜σ−1 1−˜σ,withσ>0,where cis consumption, cis the level of consumption habits, and h∈[01)is a constant habit parameter. Workers and firm owners have subjective discount factors βWand βF,respectively, and we assume that 0<β W<β F<e(σ−1)μz where the second inequality states that workers are more impatient than firm owners and the third inequality ensures that the intertemporal utility of all households remains bounded. Habits are external and defined as follows. We let cFbe the common consumption habit of firm owners in the current period, and it is assumed to be equal to the average consumption of firm owners in the previous period. Regarding workers, we let cW(N) denote the habit level of workers in the current period having been continuously unemployed for N∈Z+periods. It is assumed to be equal to the average consumption of workers having experienced the same number of consecutive periods of unemployment (=N) in the previous period. For example, cW(0)is the habit level of currently employed workers, and it is equal to the last period average consumption of employed workers. Similarly, cW(1)is the consumption habit of an unemployed worker who was employed in the previous period, and it is equal to the average consumption of those workers who had lost their jobs in the previous period, and so on. This implies that all workers with the same Nshare the same habit level, while two workers with different Ns in general have different habit levels.9 8In this section, we only describe the households’ and firms’ problems along with the aggregation and market-clearing conditions. All the optimality conditions are derived in the Technical Appendix, available in a supplementary file on the journal website, http://qeconomics.org/supp/714/code_and_data.zip. 9In our model, habits serve the usual purpose of producing an inertial response of aggregate consumption to aggregate shocks, which greatly improves the model’s empirical fit; we then assume external rather
Quantitative Economics 8 (2017) Precautionary saving and aggregate demand 449 technology ym=˘ kφ(ez˘ n)1−φφ∈(01),where ˘ nand ˘ kdenote labor and capital services. It solves max ˘ n ˘ kpm˘ kφez˘ n1−φ−Q˘ n−rk˘ k(14) where Qis the real unit price of labor services. The solution to (14) gives the optimal demands for factor services ˘ n=g˘ n(X) and ˘ k=g˘ k(X). 1.3.4 Labor intermediaries and labor-market flows Labor services are sold to wholesale goods firms by labor intermediaries, who hire labor from households in a market with search frictions. More specifically, at the beginning of the labor-market transition stage, a fraction ρ(ϕs)of existing employment relationships are destroyed, where ϕsis a job-destruction shock. The workers who loose their jobs on that occasion enter the unemployment pool, where they join the workers who were already unemployed at the end of the previous period. At the same time, labor intermediaries post vacancies, at the unit cost κvezin terms of the final good, where the term ezis included to ensure the existence of a balanced growth path. One employed worker provides one unit of labor services, but a firm owner provides ψ>1units of labor services, and we refer to ψas the skill premium.16 The values to the labor intermediary of a match with a worker and a firm owner are, respectively, JW=Q−w+EX1−ρMFJWJ F=ψQ −wF+EX1−ρMFJF(15) We assume that, when posting a vacancy, labor intermediaries cannot target a particular skill type. Labor intermediaries thus adjust vacancies until the expected payoff on a posted vacancy is equal to its cost, that is, λΩJW+(1−Ω)JF=κvez(16) where λis the economy-wide vacancy-filling rate. Let ˜ n=Ω˜ nW+(1−Ω) ˜ nFdenote the economy-wide employment rate before the labor-market transitions stage and n=ΩnW+(1−Ω)nFis the same rate after the labormarket transitions stage. These two definitions imply that ˜ n=n.Theunemployment pool is made up of workers who are unemployed at the beginning of the labor-market transitions stage (in number 1−˜ n)aswellasworkerswhowereemployedatthebeginning of that stage but lost their job after the job-destruction shock (in number ρ˜ n). The matching technology produces memployment relationships using as inputs the unemployment pool and the aggregate number of vacancies v. This technology has the form m=¯ m1−(1−ρ) ˜ nχv1−χ(17) where ¯ mis a scaling parameter and χ∈(01)the elasticity of mwith respect to (w.r.t.) the size of the unemployment pool. Accordingly, the economy-wide job-finding and 16The extent of consumption dispersion across U.S. households cannot be entirely accounted for by assets dispersion: dispersion in labor income is needed in addition to dispersion in asset income. This is adequately captured by a skill premium (see, e.g., Challe and Ragot (2014)).
450 Challe, Matheron, Ragot, and Rubio-Ramirez Quantitative Economics 8 (2017) vacancy-filling rates are f=m 1−(1−ρ) ˜ nand λ=m v(18) Since the workers who are separated from the firms can be rematched within the period, the period-to-period job-loss rate sis given by s=ρ(1−f) (19) As usual, there are two equivalent ways to view labor-market flows. From the point of view of the households, employment dynamics are determined by the flows of job losers and job finders, that is, n=f(1−˜ n)+(1−s)˜ n. From the point of view of the labor intermediaries, it follows from the natural process of job destruction and the intensity of vacancy postings, that is, n=(1−ρ) ˜ n+λv. 1.3.5 Wages The presence of labor-market frictions implies that there may exist a full bargaining set over which a labor intermediary and an employee (whether worker or firm owner) find it mutually profitable to be matched. Following Hall (2005), we assume that there are some rigidities in nominal-wage adjustment and we use the implied expression for the base real wage w. More specifically, the latter is given by17 w=w−1 1+πγw¯ wez+ϕwn ¯ nψn1−γw (20) In equation (20), w−1denotes last period’s real wage rate, ¯ wdenotes a scale factor, γwdenotes the degree of indexation to past wages, and ψndenotes the sensitivity of wages to the business cycle, here measured as the ratio of aggregate employment nto its steady-state value ¯ n. The wage equation is also perturbed by a wage shock ϕw,and is appropriately scaled by the technology shock ezto ensure the existence of a balanced growth path. We assume that the wage premium wF/w paid to firm owners is equal to the skill premium ψ(as would be the case in a competitive labor market). Finally, we assume that wlies within the appropriate bargaining set (this implies that wFdoes too), and will verify that this condition holds over our sample once the model has been estimated.18 1.4 Central bank The central bank is assumed to set the nominal interest rate Raccording to the rule (see, e.g., Christiano, Motto, and Rostagno (2014), Gust, Lopez-Salido, and Smith (2012), Guerrón-Quintana, Fernández-Villaverde, and Rubio-Ramírez (2010)) log1+R 1+¯ R=ρRlog1+R−1 1+¯ R+(1−ρR)aπlog1+π 1+¯π+aylog1+g 1+¯ g+ϕR(21) 17The corresponding nominal wage dynamics is recovered by multiplying both sides of (20) by nominal final goods prices and rearranging to eliminate 1+π. 18See the Technical Appendix for details. Note that with wF=ψw,wehaveJF=ψJW, so that the freeentry condition reduces to λJW=[Ωψ +(1−Ω)]−1κvez.
Quantitative Economics 8 (2017) Precautionary saving and aggregate demand 451 where ¯ Ris the steady-state nominal interest rate, ρR∈(01)is an interest rate smoothing parameter, (aπay)are the reaction coefficients to inflation and output growth, g=y/y−1−1isthegrowthrateoffinaloutput,wherey−1is the last-period final output, and ϕRis a monetary-policy shock. 1.5 Market clearing 1.5.1 Labor services Recall from Section 1.2 that all households face the same labormarket transition rates (fs). Hence, in the steady state, the employment rates in every family of workers and firm owners are the same. Assuming that employment is symmetric at the beginning of the date-0labor-market transition stage, by the law of large numbers they remain symmetric at every point in time, that is, ˜ nW=˜ nF=˜ nW=˜ nF≡˜ nn W=nF=nW=nF≡n(22) Because a matched firm owner provides ψtimes more units of labor services than a worker, the total supply of labor services is ΩnW+(1−Ω)ψnF=(Ω +(1−Ω)ψ)n. Denoting by ˘ nfirms’ demand for labor services, market clearing requires Ω+(1−Ω)ψn=˘ n (23) 1.5.2 Assets markets Recall that firm owners are symmetric and in measure 1−Ω. Since each of them supplies υk units of capital services, the total supply of capital services is (1−Ω)υk. Thus, market clearing gives (1−Ω)υk =˘ k (24) All the households may participate in the market for nominal bonds, which are in zero net supply. In symmetric equilibrium, at the end of the consumption–saving stage, all firm owners hold the same amount of assets aF, while workers hold different levels of assets depending on their individual state (aN). Clearing of the market for bonds requires (1−Ω)aF+Ω Na aWdμ=0(25) 1.5.3 Goods markets The aggregate demand for final goods is made of total investment (by firm owners), the consumption of all households, as well as capital utilization costs (directly paid by firm owners) and vacancy costs (paid by the labor intermediaries). Again, taking into account workers’ heterogeneity, we write the market-clearing condition as (1−Ω)cF+i+η(υ)k+Ω Na cWdμ+κvezv=y (26) The intermediate goods sector demands one unit of wholesale goods for any unit of intermediate goods. Hence, the market-clearing condition for the wholesale goods
452 Challe, Matheron, Ragot, and Rubio-Ramirez Quantitative Economics 8 (2017) sector is 1 0 xςdς=ym=˘ kφez˘ n1−φ(27) The total demand for intermediate goods by the final goods sector is 1 0yς(X pς)dς=Λy,whereΛevolves as shown in equation (13). The total supply of wholesale goods is equal to 1 0xςdς−κyez. Hence, the clearing of the market for intermediate goods requires, using equation (27), Λy =˘ kφez˘ n1−φ−κyez(28) 1.6 Aggregate state and equilibrium We are now in a position to summarize the content of the aggregate state. Again, we are focusing on a symmetric equilibrium, where family-level variables are identical across families of workers and families of firm owners (e.g., μ=μetc.). The aggregate state is then given by X=˜μ(·)kaFicFcW(N)N∈Z+aeR−1Λ−1π−1y−1w−1(29) where ≡{zϕiϕcϕsϕRϕwϕp}is the exogenous aggregate state. Definition 1. A symmetric recursive equilibrium is a set of value and policy functions, a set of prices, and labor-market flows such that the following statements hold: (i) Workers:Givenr(X),w(X),τ(X),buez,cW(N)N∈N,f(X),ands(X), the value and policy functions VW(μX),gaW(aNX),andgcW(aNX)solve the workers’ problem. (ii) Firm owners:Givenr(X),rk(X),wF(X),cF,Υ(X),f(X),ands(X), the value and policy functions VF(nFkaFiX),gaF(X),gcF(X),gi(X),gυ(X),andgk(X) solve the firm owners’ problem. (iii) Final goods firms:Givenpς,ς∈[01], the demand for intermediate goods yς(pςX)is optimal from the point of view of final goods firms. (iv) Intermediate goods firms:Givenpm(X),yς(pςX),andMF(X X), the value functions VR(X) and VN(pς−1X) and the reset price p∗(X) solve the problem of intermediate goods firms. (v) Wholesale goods firms:Givenpm(X),Q(X),andrk(X), the demand for labor and capital services ˘ n(X) and ˘ k(X) solves the problem of wholesale goods firms. (vi) Labor intermediaries:GivenQ(X),w(X),andMF(XX),thejobvaluesJW(X) and JF(X) are given by (15), the free-entry condition (16) determines the vacancy-filling rate λ(X),andm(X),f(X),v(X),ands(X) are determined according to (17), (18), and (19). (vii) Profits:TheprofitfunctionΥ(X)results from the optimal decision of the intermediate goods firms and the labor intermediaries.
Quantitative Economics 8 (2017) Precautionary saving and aggregate demand 453 (viii) Social contribution rate, real interest rate, stochastic discount factor, wages, and nominal interest rate:Giveny(X),π(X),andbuez, the social contribution rate τ(X) is such that (1) holds; the real return on nominal bond holdings r(X) follows (2); the stochastic discount factor MF(XX)is given by (5); firm owners’ wage wF(X) is equal to ψw(X),wherew(X) is given by (20); the nominal interest rate R(X) is given by (21). (ix) Market clearing: The market-clearing conditions (23)–(28) hold. (x) Laws of motion:Givenp∗(X), inflation π(X) and price dispersion Λ(X) evolve according to (12)and(13), respectively; given f(X),s(X),andgaW(·), the laws of motion from ˜μto μand then from μto ˜μare given by ˜μto μ:⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ μ(a0)=f(X) N≥1 ˜μ(aN) +1−s(X)˜μ(a0) μ(a1)=s(X) ˜μ(a0) μ(aN) =1−f(X)˜μ(aN −1) for N≥2 μto ˜μ:˜μ(ˆ aN) =1gaW(aNX)≤ˆ adμ(aN) for N≥0 (xi) Habits:GivengcF(X) and gcW(·), tomorrow’s habit level of a particular household type is equal to the average consumption of this type today, that is, cF=gcF(X) and cW(N) =gcW(aNX)dμ(aN) The solution of the model will rely on a linear approximation of its dynamics around a balanced growth path (BGP henceforth). We now provide a formal definition of the BGP and derive its key theoretical properties in the next section. Definition 2. A BGP is a symmetric recursive equilibrium where the following statements hold: (i) Innovations to the exogenous aggregate state () are zero at every point in time; therefore, aggregate shocks are absent. (ii) The variables w(X),cW(N)N∈N,wF(X),cF,Q(X),Υ(X),and ˘ k(X) all grow at rate μz. (iii) The variables r(X),rk(X),f(X),s(X),λ(X),m(X),ν(X),˘ n(X),R(X),pm(X), and π(X) are constant. 2. Equilibrium dynamics We now study the equilibrium dynamics of the symmetric recursive equilibrium defined above. We will first show that under some assumptions, the cross-sectional distributions of workers over individual assets and length of unemployment spell in a BGP have finite support. We will then use this result to show that around the BGP the symmetric recursive equilibrium is summarized by a finite number of equilibrium conditions. Finally, we
454 Challe, Matheron, Ragot, and Rubio-Ramirez Quantitative Economics 8 (2017) will isolate the key determinants of workers’ precautionary savings and show how they survive when the nonlinear system that characterizes the equilibrium is approximated at the first order. 2.1 A cross-sectional distribution of workers with finite support We now derive our main theoretical results regarding the properties of the crosssectional distributions of workers (over assets and length of unemployment spell) and workers’ consumption–saving choices. We focus on the symmetric recursive equilibrium characterized in Section 1.6, so that the family-level distributions coincide with their aggregate counterparts, that is, ˜μ=˜μand μ=μ(at the beginning of time by assumption, and in every period by implication). We proceed in three steps. First, we show that, under Assumption 1, the cross-sectional distributions ˜μand μasymptotically tend toward distributions with countable supports. We then show that, as a consequence, workers’ consumption–saving choices are summarized by a countable number of Euler conditions, which have exactly the same form as in standard heterogenous-agent models. Finally, we show that, under the additional assumption that the borrowing limit is tighter than the natural limit, the supports of ˜μand μare not only countable but also finite in and around any BGP. In what follow we state the relevant propositions and leave their proofs for the Appendix at the end of the paper. Proposition 1. Under Assumption 1,(a)if the distribution ˜μ(aN) has a unique mass point in afor all N,then both ˜μ(aN) and μ(a N) have a unique mass point in afor all Nin the following periods;it follows that all workers with the same N=01have the same levels of consumption cW(NX) and end-of-period assets aW(NX).Additionally, (b) if ˜μ(aN) does not have this property,then for any ˇ N∈Z+,both ˜μ(aN) and μ(aN) have a unique mass point in afor all N≤ˇ Nafter ˇ N+1periods. Proposition 1(a) states that, under Assumption 1, the countability of the support of the cross-sectional distribution of workers over individual assets and length of unemployment spell is preserved over time; that is, the distributions ˜μand μcan be characterized by a countable set of pairs (aN). By implication, all workers with the same Nare indistinguishable: they enter the consumption–saving stage with identical assets aW(N), consume the same amount cW(N), and end the consumption–saving stage with the same assets aW(N). The intuition for this result is best conveyed in the context of an economy without aggregate shocks, with a zero-debt limit, and where workers are so impatient that in equilibrium they decide to hold at most very little precautionary wealth. The fact that all employed workers pool their assets at the beginning of the consumption–saving stage implies that they all hold the same end-of-period asset wealth; in this sense, all employed workers are alike, despite the fact that they have different individual employment histories. When such an employed worker falls into unemployment, the worker faces a binding debt limit, and hence liquidates his or her entire asset wealth (again, under the assumption that workers are so impatient that they hold at most very little wealth). It follows that the end-of-period wealth of all workers falling into unemployment is zero (i.e.,
Quantitative Economics 8 (2017) Precautionary saving and aggregate demand 455 the amount afforded by the debt limit). It the next period, such a worker will either find a job or remain unemployed. If the worker finds a job, he or she becomes again identical to all other employed workers (by virtue of the risk-pooling assumption). If the worker remains unemployed, then he or she again faces a binding debt limit and holds zero end-of-period wealth. This precisely makes this unemployed worker, and ultimately all unemployed workers (regardless of how long they have been unemployed), identical to those who are currently falling into unemployment. In this sense, all unemployed workers are alike, just like all employed workers were alike. So in this simple economy we have exactly two types of workers: employed workers, who all have the same positive end-of-period asset wealth, and unemployed workers, who all have the same (zero) endof-period asset wealth. The same reasoning can be extended to construct an equilibrium with three, rather than two, wealth states/types of workers. Suppose that workers are slightly less impatient than previously considered, so that those who fall into unemployment choose to keep some wealth (rather than fully liquidating it) as a precautionary buffer against an additional period of unemployment. All workers enter their first unemployment period with the same asset wealth—that inherited from the previous period, when they were employed and hence “identical.” Consequently, they all make the same asset holding choice and remain symmetric at the end of that first unemployment period. The period after, they will either find a job (in which case they will become identical to the existing employed workers) or they will remain unemployed and liquidate what is left of their wealth. In this configuration, there are exactly three worker types: employed workers (with “high” end-of-period wealth), unemployed workers at their first period of unemployment (with “low” end-of-period wealth), and all the other unemployed workers (with zero asset wealth). And again, we can go from three to four wealth states, then from four to five, and so on, by gradually making workers more and more patient. Proposition 1(b) clarifies the sense in which starting from any ˜μ0the distribution asymptotically “converges” toward a distribution with countable support (in the sense that the measure of workers whose individual state (aN) does not belong to a countable subset of R×Zvanishes asymptotically). When all workers with the same Nare indistinguishable, we call any worker with length of unemployment spell equal to Nperiods a type-Nworker. Proposition 2. Assume that μ(aN) has a unique mass point in afor all N.Then workers’ intertemporal marginal rates of substitution (IMRS)are given by MW(0) =βW1−succW(0)−hcW(0)+succW(1)−hcW(1) uccW(0)−hcW(0)for N=0(30) MW(N) =βW1−fuccW(N +1)−hcW(N +1)+fuccW(0)−hWcW(0) uccW(N) −hcW(N) for N≥1(31)
456 Challe, Matheron, Ragot, and Rubio-Ramirez Quantitative Economics 8 (2017) where cW(N) satisfies cW(N) =cW(N) (i.e., current consumption determines next period’s habits for any N). As usual, those IMRS consist of ratios of next-period to current marginal utilities. For example, the marginal utility of an employed worker (the denominator of MW(0))isuc(cW(0)−hcW(0)). Next-period marginal utility (the numerator) must be broken into two idiosyncratic states, because the worker will either stay employed in the next period (with probability 1−s), in which case he will enjoy marginal utility uc(cW(0)−hcW(0)), or fall into unemployment (with probability s) and enjoy marginal utility uc(cW(1)−hcW(1)). The IMRS of currently unemployed workers follows from the fact that a worker with N≥1today may stay unemployed in the next period (with probability 1−f), and thus become N+1with marginal utility uc(cW(N +1)−hcW(N +1)), or find a job (with probability f) and obtain marginal utility uc(cW(0)−hWcW(0)).By making future marginal utility a weighted average of marginal utilities in each idiosyncratic state, these IMRS tie current and future marginal utilities for both employed and unemployed workers exactly as any model with incomplete insurance and encompass the same precautionary motive as they do. We now make an additional assumption that will further simplify the cross-sectional distribution of workers over individual assets and length of unemployment spell. Assumption 2. We have a>a nat ≡βFbu βF−e(σ−1)μz. Under Assumption 2, the exogenous debt limit aezis strictly tighter than the “natural” debt limit anatezin any BGP, where the natural limit is defined as the maximum amount that a household can borrow while still being able to repay in the worst possible individual history (Aiyagari (1994)). In our model, this worst possible history corresponds to a history of permanent unemployment, and the present value of the corresponding income stream is buez+μz 1+¯ r+buez+2μz (1+¯ r)2+buez+3μz (1+¯ r)3+···= βFbu βF−e(σ−1)μzez=−anatez(≥0) wherewehaveusedthefactthatthegrossrealinterestrateis1+¯ r=eσμz/βFin any BGP. We then have the following proposition. Proposition 3. Under Assumption 2,in a BGP,workers face a binding debt limit after a finite number of unemployment periods.Formally, ∃ˆ N∈Z+ˆ N<∞:∀N< ˆ N MW(N)1+r=1 ∀N≥ˆ N MW(N)1+r<1 The intuition for this result is as follows. We know from Proposition 1that for any ¯ N∈Z+,after ¯ N+1periods,allunemployedtype-Nworkers, for all N≤¯ N, are indistinguishable. Those workers keep on decumulating assets while remaining unemployed, and in so doing they gradually approach the debt limit aez. A worker who would remain
Quantitative Economics 8 (2017) Precautionary saving and aggregate demand 457 indefinitely unemployed would never actually reach the natural limit because borrowing up to that limit would force zero consumption in every following period, which would be suboptimal. However, whenever the debt limit is tighter than the natural limit (as Assumption 2states), then that tighter limit can be reached in finite time (and will be, due to worker’s impatience) while still allowing strictly positive consumption. This implies the following key corollary. Corollary 1. In a BGP,under Assumptions 1and 2,the distribution μhas a finite support. This follows directly from the fact that ˆ Nin Proposition 3is finite, which implies that the support of μhas at most ˆ N+1points. Moreover, we can state another corollary. Corollary 2. Under Assumptions 1and 2,the number of points in the support of μis the same in a BGP and in the vicinity of this BGP. This corollary comes from the fact that, along the BGP, unemployed workers who face a binding debt limit have their Euler equation holding with strict inequality, and this will also be the case when aggregate shocks are sufficiently small. 2.2 Equilibrium conditions We now characterize the workers’ problem in the symmetric recursive equilibrium in the vicinity of a BGP. From Corollary 2, in the vicinity of a BGP the distribution of workers has finite support, which will allow us to characterize the equilibrium by a finite set of equilibrium conditions. In every period, workers’ end-of-period wealth and consumption are given by the sequences {aW(N) cW(N)}N≥0. Since workers face a binding debt limit after ˆ Nconsecutive periods of unemployment, we have aW(N) =aezfor N≥ˆ N (32) From the workers’ budget constraint, we have cW(N) =bu−aez+(1+r)a(N) for all N> ˆ N (33) while cW(ˆ N)=bu−aez+(1+r)a( ˆ N) (34) The remaining elements of {aW(N)cW(N)}N=0··· ˆ N−1are determined as follows. First, workers with unemployment spells of length 1to ˆ N−1have the budget constraints aW(N) +cW(N) =buez+(1+r)a(N) N =1 ˆ N−1(35) where the wealth of a type-Nworker at the beginning of the consumption–saving stage, a(N), is his accumulated assets at the end of the consumption–saving stage of the previous period, when he was an N−1worker: a(N) =aW(N −1) (36)
458 Challe, Matheron, Ragot, and Rubio-Ramirez Quantitative Economics 8 (2017) Second, the budget constraint of employed workers is cW(0)+aW(0)=(1−τ)w +(1+r)A (37) where A=1−snWaW(0)+f ˆ N N=1 n(N)aW(N) +f1−nW− ˆ N N=1 n(N)aez nW(38) Third, there are ˆ Nfirst-order conditions, corresponding to the interior asset holding choices of type-0to ˆ N−1workers: EXMW(N)1+r=1for N=01 ˆ N−1(39) Given the processes that workers take as given (see point (i) of Definition 1), equations (35)–(39) form a system of 2ˆ N+1equations in the 2ˆ N+1variables (A{aW(N) cW(N)}N=0 ˆ N−1}. From the solution of the system, we can find cW(ˆ N) =(bu−a)ez+ (1+r)a( ˆ N), since equation (36)impliesthata( ˆ N)=aW(ˆ N−1). Equations (32)–(39) form the finite set of conditions that characterize the symmetric recursive equilibrium in the vicinity of a BGP. Note that while ˆ Nis constant in and around a BGP, it is an endogenous variable that depends on the underlying parameters of the model. Thus, to construct the equilibrium, we must first conjecture a particular value of ˆ N, then check that the existence conditions for this equilibrium to exist are verified. 2.2.1 Existence conditions Equations (32)–(39) above determine the dynamics of aW(N) and cW(N) as functions of the aggregate state, under the conjecture that the debt limit is binding for all type-Nworkers such that N≥ˆ Nand only for them. This requires that equation (32) hold for all workers of type N< ˆ N, while EXMW(N)1+r<1for N≥ˆ N (40) Those conditions can be checked empirically for a specific joint distribution of the structural parameters. Given a value of the aggregate shocks, we will provide a posterior probability that the existence conditions (32)and(40) hold. 2.2.2 The case ˆ N=1While Proposition 3establishes the existence of an ˆ Nunder Assumptions 1and 2, it does not give the exact value of ˆ Nin a particular model economy. However, in quantitative applications, the data impose additional discipline on the equilibrium structure. For example, the amount of wealth that workers hold (hence the initial wealth of a worker falling into unemployment) must be consistent with the broad features of the empirical cross-sectional distribution of wealth, and the job transition rates (which are a key determinant of both initial bond holdings and the pace of asset decumulation) must be consistent with their empirical counterparts. We argue in Section 3below that, given our quantitative exercise and our focus on liquid (rather than
Quantitative Economics 8 (2017) Precautionary saving and aggregate demand 465 Table 1. Estimation results. Parameter Prior Shape Prior Mean Prior S.D. Post. Mean Post. S.D. Low High σGamma 150 020 072 010 056 089 hBeta 050 010 066 004 060 072 νiGamma 200 020 190 019 159 221 νuBeta 050 010 058 009 044 073 αBeta 050 010 073 004 066 079 γpBeta 050 010 034 009 018 048 γwBeta 050 010 082 004 076 088 ψnGamma 100 020 186 032 132 238 ρBeta 075 010 046 006 036 055 aπGamma 150 010 201 010 185 218 ayGamma 013 010 052 016 027 076 ρzBeta 020 010 042 006 032 052 ρcBeta 050 010 058 004 052 065 ρwBeta 050 010 079 006 070 089 ρiBeta 050 010 086 005 079 093 ρpBeta 050 010 091 003 086 096 ρsBeta 050 010 066 005 057 075 ρRBeta 050 010 049 006 039 059 ρuBeta 050 010 083 004 077 089 σcInverted gamma 100 020 067 009 053 081 σwInverted gamma 100 020 052 004 045 058 σiInverted gamma 100 020 280 037 220 337 σpInverted gamma 100 020 142 023 107 175 σzInverted gamma 100 020 117 008 104 129 σRInverted gamma 100 020 043 003 037 048 σsInverted gamma 100 020 695 047 617 769 σuInverted gamma 100 020 080 006 071 089 Note: “Low” and “high” stand for the lower and upper boundaries of the 90 percent HPD interval, respectively. standard error of the markup shock σpis not directly comparable to results discussed in the literature. The reason is that, in general, the markup shock is rescaled by the slope of the New Keynesian Phillips curve, resulting in a relatively low standard error of markup shocks. In the present paper, σpis not rescaled. If it were, our estimate would broadly fall in the ballpark of available estimates. 3.5 Verification of the existence conditions We may now check empirically that the the conditions for the existence of an equilibrium with ˆ N=1are satisfied. From (32)and(40), with a zero debt limit, this requires aW(N) =0⇔EX[MW(1)(1+r)]<1for all N≥1. However, all workers with N≥2 are indistinguishable, so it is enough to check the latter inequalities for N=12.The equilibrium also requires positive precautionary savings for employed workers, that is, aW(0)>0(⇔EX[MW(0)(1+r)]=1). From left to right and top to bottom, the panels in Figure 2report the posterior mean (thick line) of 1−EX[MW(1)(1+r)],1−EX[MW(2)(1+r)],andaW(0)(each appropriately normalized), respectively, over the estimation sample, as implied by the smoothed
466 Challe, Matheron, Ragot, and Rubio-Ramirez Quantitative Economics 8 (2017) Figure 2. Existence conditions. Note: The thick line is the posterior mean path; the grey area is the 90 percent HPD interval. values of the state variables. In each panel, we also report the associated 90 percent HPD interval (the grey area delineated by the thin, black dashed lines). Figure 2makes clear that the posterior probability that the existence conditions are indeed satisfied is very close to 1. 3.6 Empirical performance In this section, we show that our baseline imperfect-insurance model empirically outperforms the perfect-insurance benchmark, so that taking into account time-varying precautionary savings improves the fit to the data. The perfect-insurance benchmark is structurally identical to our baseline model, except that all workers enjoy perfect insurance (and not only firm owners). Because workers are impatient relative to firm owners, they then borrow up to the borrowing limit in every period, as in, for example, Kiyotaki and Moore (1997)orIacoviello (2005). This model does not completely eliminate household heterogeneity (only heterogeneity among workers), so that the cross-sectional distribution of consumption is not degenerated (workers and firm owners consume different amounts), and we can compare the two models using the observable variables described above, that is, including the consumption shares. However, in the perfect-insurance model, workers no longer hold any precautionary wealth in excess of the borrowing limit (by construction), so the aggregate demand and supply effects of time-varying precautionary savings are absent.24 Let MII and MPI denote our imperfect-insurance model and its perfect-insurance counterpart, respectively. Both model versions are estimated using the (i) same data, 24See the Technical Appendix for a complete formal description of the perfect-insurance benchmark.
Quantitative Economics 8 (2017) Precautionary saving and aggregate demand 467 (ii) the same set of calibration restrictions, and (iii) the same prior distributions on the estimated parameters. We compare the fit of the two specifications by comparing their marginal likelihoods. Let log(p(O1:T|Mj)) denote the log marginal likelihood of model Mjfor j∈{IIPI},whereO1:Tdenotes the sample observations of the data vector O.We obtain log(p(O1:T|MII)) =−8705and log(p(O1:T|MPI)) =−8779.25 Although the evidence is overwhelming in favor of MII, it is the case that this exercise alone does not tell us where these empirical gains are coming from. In other words, what features of the data is the partial-insurance model fitting better and why? This is a well known drawback of this approach. It is also the case that these figures show that model MII is to be preferred to model MPI, but they do not tell us by how much. To provide an answer to this question, we endow each model with its own prior, p(Mj),forj∈{IIPI}. Here, we adopt a noninformative choice by setting p(MII)=p(MPI)=05. Armed with these priors, we can then compute the posterior probability of each model specification. Given the above results, we obtain a posterior probability on specification MII,p(MII|O1:T), that is very close to 1. Hence, almost all the probability mass is shifted toward the imperfect-insurance model. Note that the imperfect-insurance model still outperforms the perfect-insurance model when we exclude the consumption share of the 60 percent poorest households from the set of observable variables log(p(O∗ 1:T|MII)) =−7436and log(p(O∗ 1:T|MPI)) = −7553,whereO∗ 1:Tdenotes the history of observable variables when the consumption share of the 60 percent poorest households is excluded. Once again, under the noninformative prior on models p(MII)=p(MPI)=05, we obtain a posterior probability p(MII|O∗ 1:T)very close to 1. As an additional conclusion, these results show that considering the unemployment risk is important for the macroeconomy, which justifies the focus of this paper. 4. Precautionary savings during post-Volcker recessions 4.1 Measuring the contribution of time-varying precautionary savings We now use our estimated imperfect-insurance model to measure the contribution of time-varying precautionary savings in the propagation of the recent U.S. recessions (the 1990–1991 recession, the 2001 recession, and the Great Recession). From a theoretical point of view, our model embodies the two aggregate effects of time-varying precautionary savings discussed in the introduction: the aggregate demand effect is operative, because the three basic frictions that we have assumed generate a mutually reinforcing feedback between idiosyncratic unemployment risk and aggregate consumption demand; but the aggregate supply effect is also operative, because our model has capital and, thus, the traditional smoothing effect of imperfect-insurance models (e.g., Krusell and Smith (1998)). In the presence of both effects, the question naturally arises as to which effect dominates, that is, whether time-varying precautionary savings ultimately 25This difference is larger than 7, a bound for DNA testing in forensic science, often accepted by courts of law as evidence beyond reasonable doubt (see Evett (1991)).
468 Challe, Matheron, Ragot, and Rubio-Ramirez Quantitative Economics 8 (2017) amplify or dampen recessions. Answering this question requires the use of a counterfactual economy; to this purpose we use the perfect-insurance benchmark discussed in the previous section, which by construction has constant (zero) precautionary savings.26 We run our counterfactual experiments as follows. First, using the posterior mean of the distribution of estimated parameters ϑ2computed in Section 3, we run the Kalman smoother to extract the sequences of aggregate shocks experienced by the U.S. economy during recession episodes. To that end, we extend the estimation sample to the period covering the Great Recession.27 Second, we feed these shocks into the perfectinsurance counterpart of our model. The perfect-insurance model is the one described in Section 3.6, but with all the structural parameters maintained at the posterior mean of the estimated imperfect-insurance model. We also make sure that the calibrated parameters ϑ1have the exact same values as those used in the above smoothing step. Third, we compare the time series generated by the perfect-insurance model (grey dashed line of Figures 3and 4) with the historical time series (solid lines) for consumption and investment, as well as the job-finding, job-loss, and employment rates (the first two variables are expressed as log deviations from their value at the National Bureau of Economic Research (NBER) peak, while the last four are in level deviations from their value at the peak). Because the precautionary motive is shut down in the perfect-insurance benchmark, this comparison gives us a measurement of the propagating role of time-varying precautionary savings in the last three recessions. Figures 3analyzes the Great Recession, while Figure 4considers the 1990 and the 2001 recessions.28 4.2 The great recession Figure 3compares the actual and counterfactual paths of the abovementioned variables during the Great Recession. The amplification generated by the precautionary motive is striking. The fall in consumption from peak to trough would have been 175 times smaller without the precautionary motive than it actually was in the data (it is −35% in the data but about −2% in our counterfactual experiment). This fall in consumption 26The perfect-insurance benchmark is a natural counterfactual model because it is identical to the baseline model in every respect except for the assumption of perfect insurance, and differs from it only in that dimension. Despite their proximity, the two models have their own internal structure and parameter distributions, so they transmit structural shocks differently, not only because of the precautionary motive. For example, the workers’ marginal propensity to consume is different across the two models. 27At the end of this extended sample, the zero lower bound is binding. We adopt the method advocated by Ireland (2011) to deal with the latter. The procedure is basically as follows. Since we work with a fully linear model, sometimes agents expect the nominal interest rate to become negative because of the Taylor rule. Then, because in the data the interest rate always remains nonnegative, we need to find combinations of shocks that also allow the interest rate to always remain nonnegative in the model. These shocks mostly show up as contractionary monetary policy shocks. 28When running these counterfactual experiments, each model version is initialized with its own historical state variables. Thus the dynamics reported in Figures 3and 4reflect that the two model versions (i) have different initial values and (ii) react in different ways to the same shocks. As a robustness exercise, we also considered running the same experiments while forcing each model’s state variables to zero at the onset of each of the recessions under study. The results, reported in the Technical Appendix, are quantitatively similar.
Quantitative Economics 8 (2017) Precautionary saving and aggregate demand 469 Figure 3. The Great Recession. Note: The solid lines correspond to the actual paths of consumption, investment, the job-finding rate, the job-loss rate, and the employment rate. The dashed lines correspond to the counterfactual sample paths. Consumption and investment are reported in proportional deviation from their level at the beginning of the recession. All the other variables are expressed in level deviation from their values at the beginning of the recession. The grey areas indicates the recession dates. reflects the rational response of imperfectly insured workers to the huge increase in idiosyncratic labor-market risk that they have faced, as is illustrated by the sharp rise in the job-loss rate over the period (about 25standard deviations of the job-loss rate in the pre-recession sample). Moreover, there is evidence of a strong feedback from aggregate demand to idiosyncratic labor-market risk. This can be inferred from the responses of the job-finding and job-loss rates between the data and the prefect-insurance counterfactual. Let us recall that, on the eve of the Great Recession, the quarter-to-quarter job-finding and job-loss rates were 76 percent and 4percent, respectively, while during the Great Recession the former crashed to 50 percent and the latter went up to 6 percent. Since the model allows for within-period labor-market transitions, the job-loss rate s=ρ(1−f)combines an exogenous separation component ρand an endogenous job-finding component fthat responds to aggregate demand. During the Great Recession, all of the increase in sis explained by the latter component.29 Our counterfactual analysis indicates that without this feedback, the rise in the job-loss rate would have 29More specifically, having computed quarterly values for fand son the basis of the monthly rates (Section 3), we then calculated the value taken by the separation shock ρ=s/(1−f)as a residual. It turns out
470 Challe, Matheron, Ragot, and Rubio-Ramirez Quantitative Economics 8 (2017) Figure 4. The 1990Q3 and 2001Q1 Recessions. Note: The red lines correspond to the actual paths of consumption, investment, the job-finding rate, the job-loss rate, and the employment rate. The dashed, grey lines correspond to the counterfactual sample paths. Consumption and investment are reported in proportional deviation from their level at the beginning of the recession. All the other variables are expressed in level deviation from their values at the beginning of the recession. The grey area indicates the recession dates.
Quantitative Economics 8 (2017) Precautionary saving and aggregate demand 471 been half as large as it actually was. But, as explained above, since sin turn drives timevarying precautionary savings and thereby consumption demand, this closes the feedback loop. Combined with the greater fall in the job-finding rate, this manifested itself as a large drop in the employment rate. Hence, as far as consumption and labor-market risk are concerned, the aggregate demand effect of the precautionary motive dominated the supply effect during the Great Recession. Our counterfactual analysis also indicates that the fall in investment was not significantly affected by the precautionary motive. But this is to be expected, because the precautionary motive has two contradicting effects on investment. On the one hand, larger precautionary wealth in a recession takes down the real interest rate. This drop is transmitted to the market for capital claims by firm owners (who participate in both asset markets) and ultimately stimulates investment. This is precisely the aggregate supply effect of the precautionary motive, which tends to smooth the fall in investment relative to the perfect-insurance case. On the other hand, because of the aggregate demand effect, consumption and output are depressed, which tends to discourage investment. The overall impact of these two forces on investment is a priori ambiguous, and in the present case they roughly offset each other. Similarly, inflation is moderately affected by time-varying precautionary savings. This reflects the fact that the New Keynesian Phillips curve flattens in the post-Volcker period, that is, there are significant nominal price rigidities (see, e.g., Sbordone (2007), Coibion and Gorodnichenko (2013), Del Negro, Giannoni, and Schorfheide (2015)). Hence, large variations in aggregate demand, including those due to the precautionary motive, are associated with limited movements in inflation. 4.3 The 1990–1991 and 2001 recessions We now turn to the other two recessions in our sample. Figure 4compares the actual and counterfactual paths of consumption, investment, the job-finding rate, the job-loss rate, and the employment rate over the 1990–1991 and 2001 recessions. The impact of the precautionary motive on the propagation of the 1990–1991 recession shares several features of the Great Recession, except, of course, for the size of the effect. More specifically, over the duration of the recession, and also over the three quarters after the recession ended, consumption stagnated. Our counterfactual experiment indicates that it would have kept growing at a moderate pace during this time had the precautionary motive not been active. The feedback from stagnating consumption to depressed labor-market conditions is also apparent from the comparison between the paths of actual labor-market transition rates and those that would have prevailed without the precautionary motive; in particular, the job-loss rate would have roughly stayed at prerecession level during most of the year 1991. In other words, just as in the Great Recession, the aggregate demand effect of the precautionary motive dominated the supply effect during the 1990–1991 recession. that ρhas slightly fallen (from 15 percent to 12 percent) during the Great Recession, so the rise in sentirely comes from the fall in f.
472 Challe, Matheron, Ragot, and Rubio-Ramirez Quantitative Economics 8 (2017) The 2001 recession was short and mild, even when compared to the 1990–1991 recession. For that reason, the precautionary motive may have been weak, as is reflected by the small difference between the actual paths and those implied by the counterfactual perfect-insurance model of all the variables of interest. If anything, the perfectinsurance model generates more volatility than the imperfect-insurance model, as is revealed by the dynamics of the labor market. This suggests that the precautionary motive has tended to stabilize the economy, that is, the aggregate supply effect of the precautionary motive dominated the demand effect during this period. 5. Concluding remarks In this paper, we have provided a general framework aimed at incorporating incomplete insurance and heterogenous agents in an estimatable New Keynesian dynamic stochastic general equilibrium model. Our theoretical framework relies on a minimal set of assumptions about the extent of risk sharing, under which the cross-sectional distribution of workers converges to a distribution with a potentially large, but always finite, support. Incorporating this distribution into the state-space representation of the model makes it possible to compute its likelihood function and thereby to recover the joint distribution of the structural parameters from the data. Our approach also expands the set of observables to times series of cross-sectional moments (e.g., consumption shares by income quantiles), which by construction cannot be used to discipline New Keynesian models with a representative agent. In the present paper, we have used our estimated baseline model to assess the relative strength of the aggregate demand and supply effects of time-varying precautionary savings. Our analysis shows that the demand effect has largely contributed to the amplification and propagation of the Great Recession, unlike the previous two recessions where this effect was weak and largely offset by the (stabilizing) supply effect of precautionary savings. Obviously, there are many dimensions other than the precautionary motive in which incomplete insurance and household heterogeneity matter, and thus where the approach that we propose can usefully be applied. One area that immediately comes to mind is the impact of transfer-based fiscal policy and its interactions with other macro policies. Kaplan and Violante (2014)andMcKay and Reis (2013) have taken the first steps in that direction, but in models that abstract from the unemployment risk/aggregate demand feedback loop that is likely to be activated following fiscal shocks. Note also that it is straightforward to introduce other assets into our framework, including public debt. Cyclical variations in the public debt cannot a priori be neglected when studying fiscal policy in heterogenous-agent environments, because the very reason why heterogeneity matters—imperfect insurance and borrowing constraints—makes the economy inherently non-Ricardian, causing the induced changes in public debt to have first-order effects on the equilibrium. We leave these themes for future research.
Quantitative Economics 8 (2017) Precautionary saving and aggregate demand 473 Appendix:Proofs Proof of Proposition 1. We prove (b) by induction, and then show that (a) is a special case of (b). First, recall that the pooling of assets among employed workers within every family implies that, at every point in time, the nWemployed workers in the economy all hold assets A= (1−s)R xd˜μ(x0)+f N≥1R xd˜μ(xN) nW at the beginning of the consumption–saving stage. Formally, μ(a0)=nW×1a≥Aat all dates. We now show that if μ(aN) has a unique mass point in afor all N≤ˇ ℵ,with0≤ˇ N≤ ∞, then it is also the case of ˜μ(aN) and, thus, of μ(aN) ∀N≤ˇ N+1. Suppose it is the case for μ(aN), and denote by a(N) and n(N) the wealth level of workers with N≤ˇ Nand their numbers, respectively (so that μ(aN) =n(N) ×1a≥a(N), while nW=n(0)). All these workers have the same individual state vector (aN)conditional on Nfor all N≤ˇ Nand, hence, they all save the same amount: aW(NX) = gaW(a(N) NX). This implies that the end-of-period wealth distribution ˜μ(aN) also has a unique mass point in aup to ˇ N,thatis, ˜μ(aN) =n(N) ×1a≥aW(N·)∀N≤ˇ N. The transition from ˜μto μ(see Section 1.6)givesμ(aN +1)=(1−f)˜μ(aN) = (1−f)nW(N) ×1a≥aW(N·)∀1≤xN ≤ˇ Nand μ(a1)=s˜μ(a0)=sn(0)×1a≥aW(0·), so that μ(aN) has a unique mass point in Nfor all N∈[1ˇ N+1].Fromthisrecursion, since μ(a0)necessarily has a unique mass point in aat the beginning of time (whatever ˜μ0), so do μ(a0)and μ(a 1)one period after, μ(a 0),μ(a1),andμ(a 2)two periods after, and so forth. The statement of part (b) of the proposition directly follows. The proof of (a) is a special case of the proof of (b): if ˜μ0has a unique mass point in afor all Nup to ∞, then so does μup to N=∞at the initial date, hence μ(by the induction argument above), and the same is true at every period. Proof of Proposition 2. Section A.2 of the Technical Appendix provides a detailed derivation of workers’ optimality conditions and how they lead to (30)–(31), so we only provide the main steps of the proof here. If μhas a unique mass point in a∀N≥0, then it is summarized by (a(N)n(N))N≥0(i.e., the wealth levels and numbers of family members for each N). We can thus rewrite the workers’ problem as ˆ VWa(N) n(N)N≥0X =max (aW(N)cW(N))N≥0 N n(N)ucW(N) −hcW(N)+βWEXˆ VWa(N) n(N)N≥0X such that aW(N) ≥aezand aW(N) +cW(N) =1N=0(1−τ)w +1N≥1buez+(1+r)a(N), N∈Z+. The solution to this problem is characterized by #Z+first-order and envelope
474 Challe, Matheron, Ragot, and Rubio-Ramirez Quantitative Economics 8 (2017) conditions. Combining those generates the #Z+Euler conditions uccW(0)−hcW(0) ≥βWEX1+r1−succW(0)−hcW(0)+succW(1)−hcW(1) for N=0 and uccW(N) −hcW(N) ≥βWEX1+rfuccW(0)−hcW(0)+1−fuccWN+1X −hcW(N +1) for N≥1 where the inequalities are strict any time the debt limit is binding. We can rearrange the previous expressions as EX[MW(N)(1+r)]≤1,N=012where the MW(N)sare stated in the proposition. Proof of Proposition 3. We provide the main steps of the proof and leave the full proof to the Technical Appendix. The proof is by contradiction. In the economy without aggregate shocks total factor productivity (TFP) grows deterministically at a rate μz, the real interest rate is given by 1+¯ r=eσμz/βF, and we denote by ˆ cW(N) =cW(N)e−z and ˆ aW(N) =aW(N)e−zthe detrended consumption and assets of a type-Nworker. If unemployed workers never faced a binding debt limit, their Euler equation (as written in the proof of Proposition 2above) would always hold with equality. Noting that in the absence of aggregate shocks we have uc(cW((N) −hcW(N)) =(ˆ cW(N)(eμz− h))−σe−σz+σμz, the Euler condition can be written as (after some calculations) ˆ cW(N)−σ=βW/βFfˆ cW(0)−σ+(1−f)ˆ cW(N +1)−σfor N=12 The latter expression defines a recursion on x(N) ≡ˆ cW(N)−σ, and it can be shown that x(N) →+∞as N→+∞,sothatˆ cW(N) →0as N→+∞. On the other hand, the budget constraint of a type-Nworker, expressed in detrended form, is given by ˆ aW(N) +ˆ cW(N) =bu+e(σ−1)μz/βFˆ aW(N) for N=12 In the limit, with ˆ cW(+∞)=0,itmustbethatˆ aW(+∞)=anat, which contradicts Assumption 2(according to which the debt limit is strictly tighter than the natural limit). References Aiyagari, S. (1994), “Uninsured idiosyncratic risk and aggregate saving.” The Quarterly Journal of Economics, 109, 659–684. [437,456] Ajello, A. (2014), “Financial intermediation, investment dynamics and business cycle fluctuations.” Working Papers 2014, Board of Governors of the Federal Reserve System. [437]