Welfare Cost of Fluctuations When Labor Market Search Interacts with Financial Frictions Eleni Iliopulos∗
François Langot†
Thepthida Sopraseuth‡§
February 2016
Abstract We show that the interaction between labor search and financial frictions can account for the high responsiveness of the job finding rate to the business cycle through counter-cyclical opportunity costs of opening a vacancy and endogenous wage sluggishness. The matching process in the labor market leads positive shocks to reduce unemployment less than negative shocks increase it. The magnitude of this nonlinearity is magnified by financial frictions and lead to sizable welfare costs. By amplifying this asymmetric effect of the business cycle, financial frictions shift the distribution of welfare costs to regions characterized by more asymmetry and greater losses.
JEL Classification : E32, J64, G21 Keywords: Welfare, business cycle, financial friction, labor market search ∗
U. of Evry and CEPREMAP.
[email protected] U. of Le Mans (GAINS-TEPP & IRA), Paris School of Economics and IZA.
[email protected] ‡ U. of Cergy Pontoise (THEMA) and CEPREMAP.
[email protected] § We thank Guido Ascari, Paul Beaudry, Florin Bilbiie, Matteo Cacciatore, Daniel Cohen, Martin Ellison, Andrea Ferrero, Pietro Garibaldi, Céline Poilly, Nicolas Petrosky-Nadeau, Vincenzo Quadrini, Xavier Ragot, Federico Ravenna, Gilles Saint-Paul, Etienne Wasmer, Pierre-Olivier Weill, Francesco Zanetti for helpful comments as well as seminar participants at Paris School of Economics macro-workshop (2014), Cepremap (2014), University of Lille I (2014), University of Lyon - Gate (2014), HEC Montreal (2014), T2M Conference (2015, Berlin), Search and Matching Annual Conference (2015), Conference in honor of C. Pissarides (2015), TEPP conference (2015), Oxford macroeconomics seminar (2015). Thepthida Sopraseuth acknowledges the support of the Institut Universitaire de France †
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1
Introduction
The Great Recession was characterized by a large increase in unemployment and sudden deleveraging. This phenomenon is not new. Figure 1 shows that, starting from the mid-1970s, recessions have been characterized by both de-leveraging and increases in unemployment.1 It also shows that episodes of job creation (periods when the job-finding rate rises) are also times when firms accumulate debt. Data suggest the existence of an interaction between Figure 1: Cyclicality of unemployment rate U , job finding rate Ψ and debt stock B. 0.08
U Unemployment rate B Corporate Debt
0.06 0.04 0.02 0 -0.02 -0.04 -0.06 1975
1980
1985
1990
1995
2000
2005
2010
2015
2005
2010
2015
Ψ Job finding rate B Corporate Debt
0.06 0.04 0.02 0 -0.02 -0.04 -0.06 1975
1980
1985
1990
1995
2000
HP Filtered logged quarterly data. Shaded area shows recessions (NBER dates). HP filtered data on U and Ψ were divided by 4 for the purpose of scale consistency. Smoothing parameter: 1600. Source : See Appendix A.
developments in financial markets and labor markets. This paper aims to assess the welfare cost of fluctuations in a framework that allows for financial and labor market frictions. We argue that their interaction causes sizable welfare costs and costlier recessions, compared to welfare gains in expansion. We also show that the interplay between labor search and financial frictions helps predict the high volatility of financial and labor market aggregates. 1
Jermann & Quadrini (2012) also notice that debt repurchases (a reduction in outstanding debt) increase during or around recessions.
2
We build a DSGE model where labor markets are characterized by standard search-andmatching frictions, following Mortensen & Pissarides (1994) (hereafter DMP) and entrepreneurs’ access to credit is limited by a collateral constraint as in Kiyotaki & Moore (1997) because of enforcement limits. We also allow entrepreneurs to borrow so as to finance the intra-period costs associated with hiring. Our originality lies in focusing on the interplay between two nonlinearities that magnify the costs of fluctuations. The first nonlinearity comes from the intrinsic structure of labor market frictions. Indeed, it takes time to search and form a new match, whereas job separation is instantaneous. As a result, employment falls during recessions more than it increases during expansions.2 Canonical search-and-matching labor frictions therefore introduce a gap between the unemployment level at its deterministic steady state and its mean. We show how financial frictions introduce a second nonlinearity, which amplifies this gap and generates sizable business-cycle costs. Entrepreneurs must borrow to finance their activity and the creation of new jobs. Credit costs associated with financial frictions lead to (i) a lower equilibrium employment level through an increase in labor costs. This makes the economy more sensitive to the asymmetries of the labor market by enlarging the gap between employment (or the jobless rate) at the deterministic steady state and its mean (level effect). Moreover, financial frictions (ii) introduce a financial-accelerator multiplier, which magnifies aggregate shocks (business-cycle effect). We recover a strong propagation mechanism, which depends both on a credit-multiplier effect and an additional amplification mechanism. The latter affects the wage-bargaining process and is specific to financial frictions à la Kiyotaki & Moore (1997). The implied amplification of fluctuations is the second source of welfare costs in our model. Notice, finally, that both nonlinearities amplify each other simultaneously. Because of congestion effects on the labor market, employment gains during booms are more than offset during recessions. The introduction of financial frictions work in the same direction, magnifying this asymmetry. Indeed, credit costs increase during recessions, thus dampening further hiring. Moreover, this latter effect is not compensated during booms as credit frictions increase the costs of recessions without affecting the gains during booms. In fact, the inefficiency intrinsic to financial frictions à la Kiyotaki & Moore (1997) varies countercyclically (booms cause the equilibrium to move closer to that of an efficient and frictionless economy). This explains why the propagation of shocks is highly asymmetric. Our evaluation of welfare costs can be considered reasonable only if our model is able to 2 These asymmetries in employment dynamics are supported by empirical studies such as Acemoglu & Scott (1994). Collard et al. (2002) show that these asymmetries are also present at the level of job flows.
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reproduce the main features of the business cycle.3 We show that the mechanisms that generate both the level and the business-cycle effects on welfare can also help to solve the volatility puzzle emphasized by Shimer (2005a)4 , allowing our model to match the cycle. In our model, in response to a positive productivity shock, countercyclical credit costs associated with financial frictions give firms an incentive to post more vacancies, independently of the expected benefit of new workers. Moreover, financial and labor market frictions interact by affecting the wage-curve dynamics. As suggested by Chéron & Langot (2004) and Pissarides (2009), and echoing the old Keynes-Tarshis-Dunlop controversy, a solution to the puzzle must explain fluctuations in both unemployment and wages.5 Our analysis shows why financial constraints as in Kiyotaki & Moore (1997) can per se improve the performance of a DSGE matching model in generating fluctuations in labor-market aggregates and wages. Our quantitative results support the theoretical analysis. The model can replicate both the volatilities of labor-market aggregates (worker flows and unemployment stock) and the cyclicality of real wages6 in line with the data. Moreover, the fit of labor market volatility is not obtained at the cost of a poor fit in the other dimensions of the model. Indeed, fluctuations in debt and the price of collateral also remain consistent with the data.7 Notice finally that there is only a technological shock in our model.8 This restrictive view of the sources of fluctuations has a double advantage: first, it allows us to isolate the mechanism at work in the model; second, it facilitates the comparisons of our framework with the other contributions on the Shimer puzzle.9 The good fit of the cycle of our model is explained by the countercyclical cost of posting vacancies, which directly amplifies the volatility of labor-market tightness (i.e., the ratio of the number of vacancies to the number of unemployed workers). In the standard DMP model, unemployed workers are scarce during booms; thus, labor-market tightness increases. Because of congestion effects, hiring takes longer. Thus, workers’ outside option during negotiations goes up. This raises the workers’ threat point in the Nash bargaining process, 3
We focus in particular on second moments because welfare costs depend on the volatility of macroeconomic aggregates. 4 As stressed by Shimer (2005a), the textbook search and matching model cannot generate enough business-cycle-frequency fluctuations in unemployment, job vacancies, and thus job-finding rates (the "Shimer puzzle"). Indeed, a large wage increase in expansion tends to lower firms’ incentive to hire more workers, thus dampening hiring. 5 Since the work of Shimer (2005a), real-wage rigidity is known to solve the volatility puzzle in the Diamond-Mortensen-Pissarides model. This approach however neglects the observed wage volatility that is not zero. 6 Our calibration does not exogenously impose wage rigidity. 7 Hence, the discipline of the general equilibrium approach is ensured. 8 We incorporate financial shocks as a robustness check. 9 This approach differs from the method proposed by Liu, Wang & Zha (2013) and Christiano et al. (2016), but shares its feature of the small sample size.
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entailing an increase in wages. As higher wages absorb most of the productivity increase, firms’ incentive to hold open vacancies is dampened. To reduce the reaction of wages and ensure that firms’ incentive to hold open vacancies is not eliminated (a crowding-out effect), wage rigidity is needed. With rigid wages, labor market tightness is volatile, as in our model. In the standard Nash solution, wage results from the firm’s and the worker’s evaluation of the gains associated with a match. When financial markets are frictionless, the expected returns of the search activity are capitalized in the same way by the firm and the workers. This is no more the case in the presence of financial imperfections à la Kiyotaki & Moore (1997). As entrepreneurs must borrow to finance job creations (and incur credit costs), the value of a match is lower for them than for workers. This dampens wages in proportion to credit costs. Notice also that credit costs vary throughout the cycle: in booms, scarcity of unemployed workers delays the hiring process (congestion effect), thereby increasing the financial costs incurred by firms when they search for new hires. This reduces in turn firms’ surplus and the wage that they are ready to pay. In booms, this mechanism thus leads to endogenous wage moderation: the (highly pro-cyclical) increase in labor market tightness is dampened by congestion effects and by the (countercyclical) dynamics of the vacancy-filling rate, which in turn increases the firm’s financial costs. Financial frictions raise welfare costs of fluctuations several times those with only labor frictions. With respect to a DSGE model with labor market search but without financial frictions, the relative volatility of the job-finding rate is multiplied by two. This suggests a solution to the volatility puzzle in a framework that also reproduces the volatility of real wages. The business-cycle cost of fluctuations with financial and labor market frictions is 2.5% of workers’ permanent consumption. It drastically falls without financial frictions and labor market frictions only (at 0.12% for workers). These costs are far larger than the estimates by Lucas (1987, 2003), who reports a welfare cost of 0.05% in the case of logarithmic utility. We compute the time-varying welfare cost and report its empirical distribution for the model with financial and search frictions and the model with search frictions only. Whatever the model, welfare costs in recessions are larger than welfare gains in expansion. This result captures the asymmetric business cycles in our nonlinear environment. In addition, the presence of financial frictions shifts the distribution of welfare costs to regions showing more asymmetry and greater losses; not only are welfare costs larger whatever the state of the economy (welfare gains therefore appear only in case of large expansions), but welfare gains remain small in case of expansion (while they increase quasi-linearly without financial frictions). The paper is organized as follows. Section 2 outlines our original contribution to the litera-
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ture, and section 3 introduces the main mechanisms at work. Section 4 describes the model. Section 5 provides the analytical results on labor market adjustments. Our quantitative analysis is developed in section 6. Section 7 concludes the paper.
2
Related literature
Our contribution lies in bridging the gap between two strands of the literature, one studying the welfare costs of fluctuations and the other investigating the macroeconomic impact of financial frictions on labor market dynamics. Lucas (1987, 2003) shows that welfare costs associated with business cycles are negligible. In his framework, costs associated with recessions are indeed compensated by the gains during booms. In this paper, we challenge Lucas’s result by proposing that a nonlinear DSGE model, where the allocation is sub-optimal, can generate significant asymmetries at businesscycle frequency. In our model, the costs of recessions cannot be compensated by the gains of expansions because average employment and average consumption are significantly lower than their deterministic steady-state levels. Welfare costs of fluctuations can then be significantly greater than those found by Lucas.10 As explained by Hairault et al. (2010), Jung & Kuester (2011), and Petrosky-Nadeau & Zhang (2013), canonical search-and-matching labor frictions introduce a gap between the deterministic steady-state and mean unemployment levels. Using the DMP model, they show how this generates business-cycle costs ranging between 0.2% and 1.2% of permanent consumption in their calibrated models. However, these papers neglect financial frictions, which are also highly nonlinear because of borrowing constraints. As suggested by Pissarides (2011), the "equilibrium matching models are built on the assumption of perfect capital markets. ... But future work needs to explore other assumptions about capital markets, and integrate the financial sector with the labour market. This might suggest another amplification mechanism for shocks, independently from wage stickiness or fixed costs" (Pissarides (2011)).11 The link between the matching model and financial imperfection is in fact intuitive; search activity is costly for firms and needs to be financed with borrowing. However, this investment cannot be used as a collateral. In this paper, we study the interaction between nonlinearities that are associated with the matching process on the labor market and that are specific to financial frictions so as to 10
In case of a significant gap between average and deterministic steady-state values, such costs are of a first-order magnitude – as the costs of tax distortion also evaluated by Lucas (1987). 11 Dromel et al. (2010) show that Pissarides (2011)’ intuition is supported by significant empirical links between unemployment dynamics and financial frictions.
6
focus on their impact on welfare throughout the cycle. This analysis constitutes our original contribution to the literature on the welfare costs of fluctuations.12 In our model, the financial contract concerns both firms (i.e., the borrowers) and workers (i.e., the lenders). We thus depart from the DMP models, too, where the borrowing constraint is supported by households, with access to incomplete financial markets. Indeed, Krusell et al. (2010) show that the nonlinearities linked to financial constraints à la Aiyagari (1994) do not change the gaps of endogenous variables between bad and good states, in contrast to linear-utility models where this type of financial constraints do not matter.13 Recent research has studied how evolving conditions on credit markets affect the dynamics of labor markets and improve the ability of standard search models to match data.14 Regarding these studies, we consider a streamlined model à la Kiyotaki & Moore (1997) and show how the interaction between financial and credit frictions generates a strong propagation mechanism, which depends both on a credit-multiplier effect15 and an additional amplification mechanism. This latter mechanism affects the wage-bargaining process and is specific to financial frictions as in Kiyotaki & Moore (1997). Our quantitative results suggest that these mechanisms are sufficient to solve the volatility puzzle of the DMP model. Hence, we show that it is not necessary to combine financial constraints à la Kiyotaki & Moore (1997) with exogenous fluctuations in collateral requirements to close the gap between the model and the data, as suggested by Liu, Miao & Zha (2013) or Garin (2015).16 Liu, Miao & Zha (2013) also include house-land holdings in the utility function, allowing generation of countercyclical movements in workers’ outside options. However, their model yields limited interaction between labor and financial frictions as labor market variables do not enter the collateral constraint: these limited interactions can lead them to overestimate the weight of the financial shock, necessary to fit the data. In Petrosky-Nadeau (2013), for the mechanism associated with canonical financial frictions à la Bernanke et al. (1999) to match data, an ad hoc countercyclical monitoring cost is 12
This study differs from previous works on business-cycle costs, such as Beaudry & Pages (2001), Storesletten et al. (2001) or Krebs (2003), in that it regards interactions between the labor and financial frictions as the root of the welfare cost of business cycles. 13 See table 5, p.1492, of the Krusell et al. (2010) paper and section 5.11.1 for a discussion on the implications of business cycles. Notice that that financial constraints à la Aiyagari (1994) (i.e., exogenous labor market transitions) do not per se entail positive welfare effects of eliminating business cycles (see Krusell & Smith (1999)). 14 See Wasmer & Weil (2004), Petrosky-Nadeau (2013), Petrosky-Nadau & Wasmer (2013), or Zanetti (2015) for an analysis based on real business-cycle models and Zanetti & Mumtaz (2013) and Christiano et al. (2016) for one based on New-Keynesian DSGE models. 15 Wasmer & Weil (2004), Petrosky-Nadeau (2013), and Petrosky-Nadau & Wasmer (2013) also share this feature. 16 We also depart from Garin (2015) as we do not introduce fixed training costs and time-varying vacancy, already known to change the prediction of the basic DMP model (even without financial constraints).
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incorporated into the financial contract. The impact of financial frictions on job creation and on the bargained wage proposed in our work is then an original mechanism to reconcile search-and-matching theory with business-cycle data. Our mechanism also differs from those used by Hairault et al. (2010), Jung & Kuester (2011), and Petrosky-Nadeau & Zhang (2013), where real wages are exogenously fixed (for the former) or rigid (for the other two, because of bargaining à la Hall & Milgrom (2008)).17 Notice finally that Petrosky-Nadau & Wasmer (2013), Petrosky-Nadeau (2013), Liu, Miao & Zha (2013), and Garin (2015) do not discuss the implications of their models with respect to real wage dynamics. As mentioned above, we replicate the volatilities of both labor market aggregates and real wages (the KeynesTarshis-Dunlop controversy) in our work, ensuring that the volatility of financial variables, such as debt and the price of collateral, is not exessive.18 We also contribute to this strand of literature by looking at business-cycle costs, amplified by the interaction between the nonlinearities linked to financial and labor market frictions.
3
Welfare costs of fluctuations: why do search frictions and financial imperfections matter?
In this section, we introduce the main mechanisms that are at the root of our results. Another way to understand why we obtain welfare costs of cycles that are much higher than the results of Lucas (1987, 2003) is to focus on the fundamental difference between the two frameworks. In fact, while the Lucas economy is "linear" (in the sense that the business cycle does not affect average unemployment and output), our model is characterized by nonlinearities due to the combination of labor market frictions and credit imperfections. In what follows, we show how these nonlinearities have important implications for welfare both via level effects, which are intrinsic to the structure of the model (section 3.1) and business-cycle effects – i.e., business-cycle fluctuations that are amplified by the financial accelerator (section 3.2). 17
In this respect, our work also differs from the DMP model, without the financial frictions of Hagendorn & Manovskii (2008), who calibrate the worker bargaining power so as to achieve a small value with a regression coefficient between the HP-filtered log of wage and the HP-filtered log of productivity. In equilibrium, this calibration strategy leads to "over-employment." 18 Notice that Garin (2015) does not control for the implications of shocks on financial aggregates. Garin (2015) argues that credit shocks are critical to understanding labor market dynamics. However, according to Figure 3, p.122, of his paper, debt is five times as volatile as output, following a credit shock. This volatility of debt is counterfactual (see Table 2 in this paper). In Liu, Miao & Zha (2013), the estimation includes the real land price (but not corporate debt).
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3.1
Nonlinearities and the level effect
As in Lucas (1987, 2003), our work consists of a comparison between welfare in a stabilized economy (i.e., an economy that is always at the deterministic steady state) and welfare in a stochastic economy (i.e., its mean). As emphasized by Hairault et al. (2010), nonlinearities inherent in the standard search and matching model imply that the level of unemployment at its deterministic steady state is smaller than its mean. Due to congestion effects, average employment and therefore average consumption are lowered by the mere process of alternate expansions and contractions. Therefore, the welfare loss due to fluctuations in an economy characterized by labor friction is not negligible. At the steady state, unemployment is a convex function of the job-finding rate Ψ because ΨU = sN leads to U =
s , s+Ψ
with U the
number of unemployed workers, s the exogenous job destruction rate, and N = 1 − U the number of employed workers, given that the population size is normalized to 1. Consider now a stochastic environment. Suppose for simplicity that Ψ19 follows a Markov stochastic process defined over states i and that unemployment converges instantaneously from one steady state to another, depending on the value of Ψ. Formally, conditional steady-state Moreover, stabilized unemployment is u¯ = s+Ps πi Ψi , where πi i P is the probability that Ψ takes the value Ψi and i πi Ψi is the mean of the job-finding rate. unemployment is u˜i =
s . s+Ψi
Because of convexity, s+
s P
i
πi Ψi
0 a scale parameter measuring the efficiency of the matching function, and Vt the number of vacancies. Following Blanchard & Gali (2010), we suppose that a pool of jobless individuals, St , is available for hire at the beginning of period t. This implies that the pool of jobless agents is larger than the number of unemployed workers. Indeed, individuals are either employed or willing to work (full participation) at all times so that St is given by St = Ut−1 + sNt−1 = 1 − (1 − s)Nt−1 20
(3)
We present in section 6.3.3 an extension with capital. Following Merz (1995), Langot (1995) and Andolfatto (1996), we assume complete insurance against the unemployment risk. Hence, no wealth heterogeneity exists among workers at the equilibrium. 21
11
where Ut = 1 − Nt is the stock of unemployed workers when the size of the population is normalized to 1 and full participation is assumed. Ut thus measures the fraction of the population left without jobs after hiring takes place in period t. Among agents looking for jobs at the beginning of period t, a certain number Mt is hired, and they start working in the same period. Only workers in the unemployment pool St at the beginning of the period can be hired (Mt ≤ St ). The ratio of aggregate hires to the unemployment pool is the rate at which jobless people in the pool find a job, Ψt ≡ Mt /St , whereas Φt ≡ Mt /Vt is the rate at which vacancies are filled. Labor market tightness θt equals
Vt 22 . St
4.2
Households
Households maximize the utility function of consumption and labor. In each period, an agent can engage in only one of two activities, working or enjoying leisure. Employment lotteries ensure that the individual idiosyncratic risks faced by agents in their jobs match. Hence, the representative household’s preferences are represented by E0
"∞ X
# µt {Nt U n (Ctn ) + (1 − Nt )U u (Ctu + Γ)}
(4)
t=0
where 0 < µ < 1 is the discount factor and Γ the utility of leisure. Ctz stands for the consumption of employed (z = n) and unemployed agents (z = u). We assume U (Ctn )
=
U (Ctu + Γ) =
(Ctn )1−σ 1−σ (Ctu +Γ)1−σ 1−σ
≡ U˜tn for employed workers ≡ U˜tu for unemployed workers
with σ > 0 the coefficient of relative risk aversion.23 . The budget constraint is [Nt Ctn + (1 − Nt )Ctu ] + Bt ≤ Rt−1 Bt−1 + Nt wt + (1 − Nt )bt + Tt
(5)
where w is the real wage and b the unemployment benefit. T is a lump-sum transfer from the government. Moreover, B represents private bond-financing firms and R is the gross investment return associated with these loans. Households’ labor opportunities evolve as s The jobless rate is thus a convex function of the job-finding rate, S = s+(1−s)Ψ . As shown in section 6.3.1, our welfare measure indirectly depends on the employment level of this utility function. 22
23
12
follows: Nt = (1 − s)Nt−1 + Ψt St
(6)
Each household chooses {Ctn , Ctu , Bt } to maximize (4) subject to (5) and (6).24
4.3
Entrepreneurs and Firms
The economy includes many identical firms. Entrepreneurs maximize the following sum of expected utilities: " E0
∞ X
# F
β t U Ct
(7)
t=0
where β denotes the entrepreneurs’ discount factor, and µ > β, implying that workers are more patient than firms25 . Their budget constraint is CtF + Rt−1 Bt−1 + qt [Lt − Lt−1 ] + wt Nt + ω ¯ Vt ≤ Yt + Bt + πt
(8)
where B is private debt, L productive land or infrastructure, and q its price. Moreover, wN denotes total wages, with N the number of employees, whereas Y is the final output and π lump-sum dividends. Each firm has access to a Cobb-Douglas constant-return-to-scale production technology combining workers and infrastructure (land): α Yt = At L1−α t−1 Nt
(9)
where At represents the global productivity of factors in the economy, assumed to evolve stochastically as follows: log At = ρa log At−1 +(1−ρa ) log A+εat , with εat the iid innovations. Firms’s activity can be financed by funds lent by households under imperfect debt contracts. Enforcement limits à la Kiyotaki & Moore (1997) imply that entrepreneurs are subject to collateral constraints. As Quadrini (2011) and Jermann & Quadrini (2012), we assume that, at the beginning of the period, firms can access financial markets to finance both their expenditures (i.e., consumption by entrepreneurs and land investments) as well as the costs associated with working capital within the period. Moreover, following Petrosky-Nadeau (2013) and Wasmer & Weil (2004), we suppose for simplicity that the costs associated with working labor represent hiring costs. Entrepreneurs can thus borrow from agents subject to 24
See Appendix B for a complete description of the model. This assumption is needed to ensure that firms are debt constrained in equilibrium. We will discuss this point in the following. 25
13
the following collateral constraint: Bt + ωVt ≤ mEt [qt+1 Lt ]
(10)
This constraint shows that if the capitalist fails to repay the loan, the lender can seize the collateral. Given that liquidation is costly, the lender can recover up to a fraction, m, of the value of collateral assets. m is the (exogenous) loan-to-debt ratio. The firms’ constraint associated with the evolution of vacancies is Nt = (1 − s)Nt−1 + Φt Vt
(11)
Each entrepreneur chooses {CtF , Lt , Bt , Vt , Nt } to maximize (7) subject to (8), (10), and (11), where Yt is given by (9). The Job Creation (JC) curve is F λt+1 ω (1 + ϕt ) ∂Yt ¯ ω ¯ = − wt + (1 − s) βEt (1 + ϕt+1 ) Φt ∂Nt λFt Φt+1
(12)
where ϕt is the Lagrangian multiplier associated with the credit constraint (Equation (10)) and represents the "credit multiplier" of this model, and λFt is the Lagrangian multiplier associated with the budget constraint (Equation (8)). The credit multiplier (ϕ) appears both on the LHS and RHS of Equation (12). Nevertheless, there is almost no persistence in the adjustment of ϕt ; after a jump at the time of the shock, it comes back to its steadystate value. We thus shift our attention to its impact on the LHS of (12). In recession, tight credit conditions (large values of ϕt ) drive up the opportunity costs associated with vacancy posting, ω ¯ (1 + ϕt ). This introduces a countercyclical and time-varying wedge that has the potential to magnify productivity shocks. If the real wage is sufficiently rigid, the adjustments of quantities on the labor market can thus be large.
4.4
Wages
The wage is the solution of the maximization of the generalized Nash product maxwt with VtF =
∂W(ΩF t ) ∂Nt−1
the marginal value of a match for a firm and VtH =
∂W(ΩH t ) ∂Nt−1
VtF λF t
VtH λt
the marginal
household’s surplus from an established employment relationship. denotes the firm’s share
14
1−
,
of a job’s value, i.e., firms’ bargaining power. The wage curve (W C) is ∂Yt wt = (b + Γ) + (1 − ) | {z } ∂Nt | {z } (a) (b)
λF +(1 − ) (1 − s)βEt (1 + ϕt+1 ) t+1 | {z } λFt (3) |
F λt+1 λ t+1 λ t+1 ω β − µ µ F ¯ λ λt t + λλFt ω ¯ θt+1 (13) λF t+1 t+1 Φ t+1 β λF β λF t t {z } | {z } | (1) (2) {z } (Σ)
where (a) represents the weight of the reservation wage in total wage and (b) + (Σ) is the workers’ gain from the match. This gain can be decomposed into the marginal productivity of the new employed worker, (b), and the saving on search costs if the job is not destroyed in the next period (Σ).26 In the presence of a borrowing constraint on the vacancies, the bargaining process is influenced by the fact that workers and firms evaluate the aggregate surplus differently. As firms borrow to finance their vacancies and pay a premium on the interest rate to compensate for their impatience, they underestimate the value of the intertemporal surplus. In practice, the value of the agreement after a match is lower for them than for workers. Thus, term (1) in (Σ) reduces the bargained wage by an amount proportional to the vacancy costs and the premium on the interest rate. The latter corresponds in equilibrium to the gap between the firm’s and the workers’ price kernels – i.e., the gap in impatience rates. In addition, during the bargaining process, the firm-worker pair shares the returns on the search process. For the worker, this is equal to the discounted time duration to find a job offer; for the firm, returns are instead equivalent to the discounted time duration to find a worker. Note however that these relative time spans cannot be proxied by the ratio of the average duration for these two search processes – as would be the case without discounting heterogeneity. Given that firms are less patient than workers, the“subjective” duration of the search process is also underestimated by the firm. Thus, term (2) of (Σ) pushes up the wage through the ratio of discount rates. Indeed, entrepreneurs are impatient; this state pushes them to quicker negotiations and to accept demands for higher wages. Note that this second mechanism has been emphasized by Rubinstein (1982): once an agent enters a bargaining 26
When firms and workers have the same discount factor (let µ = β, ϕ = 0, thus λt = the standard Blanchard & Gali (2010) wage curve: wt = (1 −
λF to t ), equation (13) collapses h i λt+1 ∂Yt ) ∂Nt + ω ¯ (1 − s)βEt λt θt+1
+ (b + Γ).
15
process, the most impatient one eventually gains less from the match.27 Moreover, the outcome of the two mechanisms is amplified by the "credit channel," term (3), (1 + ϕt+1 ), which is also at work in Petrosky-Nadeau (2013) and Wasmer & Weil (2004) and discussed in Monacelli et al. (2011).
4.5
Markets clearing
In order to close the model, we assume that the government does not accumulate debt and pays unemployment benefits using a lump-sum tax; i.e., Tt = (1 − Nt )bt . It is possible to show (see Becker (1982), Becker & Foias (1982)) that, in presence of standard levels of uncertainty28 , firms are collateral constrained in each period. Thus, the debt limit eventually determines the equilibrium level of corporate debt and workers savings. The private-bond market clears. Good market equilibrium requires the condition Yt = Nt Ctn + (1 − Nt )Ctu + ω ¯ Vt + CtF . Finally, we assume that land supply is fixed and that the land market clears in each period; i.e., Lt = 1.
5
Interactions between labor and financial frictions
In what follows, we analyze in detail the mechanisms that are at the root of the level effect (section 5.1) and the business-cycle effect (section 5.2), shown in Figure 3.
5.1
The level effect
Firms incur relatively higher costs in the presence of financial frictions. With access to credit becoming increasingly costly, open vacancies become more expensive. The incentive to hold open vacancies falls, thereby lowering the job-finding rate Ψ. This is at the root of our level effect: the equilibrium shifts towards the more convex part of the curve linking unemployment and the job-finding rate. To characterize the long-term equilibrium in the presence of financial frictions à la Kiyotaki & Moore (1997), one need only study the system (JC) − (W C), given that ϕ = 1 −
β µ
at
the steady state. By analyzing the equilibrium of this system, we find that, as in Acemoglu (2001), Wasmer & Weil (2004), Petrosky-Nadeau (2013), and Petrosky-Nadau & Wasmer 27 28
Notice that, in Rubinstein (1982), term (1) has no effect because of the static approach of the game. For a discussion see, among others, Iacoviello (2005).
16
(2013), financial frictions entail lower levels of employment. Proposition 1. The steady-state level of labor market tightness in presence of discounting heterogeneity is smaller than in the baseline search and matching model. Proof. See Appendix B.4. This result provides theoretical foundations for the shift of Ψ to the left in Figure 3 and thus, to our level effect.
5.2
The business-cycle effect
In what follows, we discuss the impact of a technological shock on the labor market equilibrium. We provide analytical results that are easily comparable to those presented in the literature, following Shimer (2005b). We first develop our analysis in a partial equilibrium environment. Given that our argument is based on the transitory impact of the tightness of financial frictions, we cannot use the standard methods of simply studying conditional steady states. Hence, we solve the dynamic solution of the labor market equilibrium for a given exogenous process of the tightness of financial contracts ϕ. We then provide the set of restrictions that allow our model to solve the volatility puzzle of the DMP model. Finally, by using the quantitative predictions of our model, we show that the set of restrictions is satisfied at the general equilibrium. Assumptions 1. 1. Entrepreneurs’ utility function is linear,
λF t+1 λF t
= 1.
2. The labor market is hit by one shock. Let yt denote the marginal return of employment that follows the exogenous process ybt = ρb yt−1 + εt where ybt denotes log(yt /y). β 3. The tightness of the collateral constraint ϕt is given by the reduced form ϕ bt = − µ−β Λεt .
Assumptions 1.1 and 1.3 imply λt = λ(1 + Λεt ).29 At the general equilibrium, Λ is deduced from agents’ interaction on all other markets. 29
1 By combining assumption 1.1 with the worker’s Euler equation, we obtain µ Etλλtt+1 = β 1−ϕ . The t
ϕt impatience gap is now given by β−µ Etλλtt+1 = −β 1−ϕ . Countercyclical movements in ϕt lead this impatience t
gap to be pro-cyclical. Assumption 1.3 is deduced from (1 − ϕt ) Etλλtt+1 = βµ with the reduced form for the solution of λt = λ(1 + Λεt ). Using assumption 1.3, the Lagrange multiplier associated with the debt limit declines for one period only and is driven only by the innovation of the technological shock.
17
Proposition 2. Under Assumptions 1, and for reasonable restrictions on parameter values, e financial frictions magnify labor-market-tightness fluctuations if Λ > Λ. Proof. See Appendix B.5. This result provides theoretical foundations for the larger volatility of Ψ in Figure 3. To study the mechanisms at work, we log-linearize both (JC) and (W C) curves. The log-linearized (JC) curve is ϕ Φy Φw ϕ b b ϕ bt + (1 − ψ)θt = ybt − w bt + (1 − s) βEt ϕ bt+1 + (1 − ψ)θt+1 1+ϕ ω ¯ (1 + ϕ) ω ¯ (1 + ϕ) 1+ϕ Given that Et [ϕ bt+1 ] = 0 (Assumption 1.3), countercyclical movements in ϕt amplify the response of labor-market tightness θt , for a given level of wages (w bt = 0). This is the first mechanism at work. Wages do fluctuate pro-cyclically. Higher wages tend to absorb the increase in productivity, thereby dampening the hiring incentive, and hence the responsiveness of θbt . 30 Therefore, for labor-market tightness to be really responsive, we need an additional component entering the wage equation so as to counterbalance the increase in workers’ outside options during booms. The log-linearized (W C) curve is w bt = with
(1 − )y (1 − )Σ b ybt + Σt w w ω ¯ (β − µ) ϕ Φ b b t+1 ] Σt = Et [ϕ bt+1 ] − ω¯ Et [Φ 1+ϕ (β − µ) + µ¯ ω θ Φ ω µ + µ¯ ωθ µ¯ ωθ β − µ −¯ Φ Et [θbt+1 ] − ϕ bt + ω¯ ω ¯ (β − µ) + µ¯ ωθ β Φ (β − µ) + µ¯ ωθ Φ
Since Et [ϕ bt+1 ] = 0 (Assumption 1.3), it is possible to isolate the new mechanism affecting b t ) with respect to the usual DMP model. It the dynamics of workers’ outside options (Σ b t ) and tightness of the collateral is associated with the dynamics of the job-filling rate (Φ constraint (ϕ bt ) (in addition to the new values of the multipliers). The wage equation is indeed key to our model, providing an original method to account for endogenous real wage stickiness – despite Nash bargained wages and the high volatility of the labor-market tightness.31 One 30
Indeed, the Nash solution of the wage bargaining process depends on the worker’s outside options – characterized by large fluctuations (the dynamics of θbt ). 31 From the (W C) and our timing à la Blanchard & Gali (2010), the expected value of ϕt+1 associated with the vacancy costs disappears. Therefore, wage stickiness cannot come from the same argument as in Petrosky-Nadeau (2013). In Petrosky-Nadeau (2013), the interaction between the labor market demand and the financial-frictions mechanism is different. The credit multiplier à la Bernanke et al. (1999) is a function of the productivity threshold leading firms to default. Notice also that, for the mechanism associated with financial frictions à laBernanke et al. (1999) to match data, an ad hoc countercyclical monitoring cost is
18
can identify two effects of financial frictions: • Fluctuations in the time duration to fill a vacancy: the expected credit cost of a vacancy. In expansion, the time until another worker is found increases (the probability of b t+1 = (ψ − 1)θb < 0). Hence, the time until the firm is filling a vacancy is such that Φ compensated by a profit flow for the credit costs associated with an open vacancy rises. Given that entrepreneurs are less patient (β < µ), these first periods of search count more for them. This leads them to discount the match surplus less and to become reluctant to pay higher wages. This component is a countercyclical force in the wage ω ¯
(β−µ)
Φ equation if − ω¯ (β−µ)+µ¯ > 0, which is always satisfied for a β value sufficiently close ωθ Φ
to µ. • Fluctuations of the impatience gap: the divergence in evaluating the future. These fluctuations are given by ϕ bt , which is countercyclical.32 In booms, the gap between workers’ and firms’ discount factors falls from the steady state. Hence, the expected credit cost of a vacancy is reduced (ϕ bt < 0), dampening the countercyclical fluctuations in the time until a vacancy is filled. However, the same force acts in the opposite direction during the bargaining process; entrepreneurs are more patient in expansions, leading them to delay negotiations in order to pay lower wages. The total impact is however unambiguous; the impact of heterogenous discounting dynamics on wages is pro-cyclical, because − β−µ β
−ω ¯ µ+µ¯ ωθ Φ ω ¯ (β−µ)+µ¯ ωθ Φ
< 0, for a β value sufficiently close to µ.
These two components affect wages in the opposite directions. Proposition 2 states that, under "reasonable" parameter restrictions33 , when the elasticity of the Lagrange multiplier associated with the debt constraint, φ, with respect to the technology shock is sufficiently e the volatility of quantities in the labor market are magnified by financial high (i.e., Λ > Λ), frictions. In this case, labor market aggregates overshoot. Figure 4 shows the impulse response function of market tightness θ and jobless people S in response to a productivity shock, in the models both with and without financial frictions incorporated into the financial contract. 32 We use the following approximation of the Euler equations on consumption: h i bt+1 bt bt + Et λ R = λ Rt µEt[λt+1] = λt h i ≈ F F Rt βEt λt+1 = (1 − ϕt )λt bF bt + Et λ R = − t+1
bt+1 − λ b t − Et λ bF + λ bF ⇒ Et λ t+1 t
=
ϕ bt 1−ϕ ϕ
bF +λ t
ϕ ϕ bt 1−ϕ
33
These restrictions are "reasonable" in the sense that they ensure that the search value is positive and that the equilibrium is a saddle path, as in the usual DMP model (See Appendix B.5)
19
(benchmark calibration at the general equilibrium). In response to the shock, financial frictions trigger a strong instantaneous reaction of market tightness. This leads unemployment on a longer trajectory towards the steady state. Figure 4 thus confirms that the volatility of labor market aggregates is amplified by financial frictions, as stated in proposition 2. Figure 4: Financial frictions amplify the response of labor market aggregates (business-cycle effect)
0.2
No Financial Friction Financial Frictions
0.18 0.16 0.14
3ˆ
0.12 0.1 0.08 0.06 0.04 0.02 0 -0.04 -0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 Sˆ
0
0.005
Impulse Response Function (IRF) in the model with and without financial frictions. Both IRFs start at 0, the period of the shock. Benchmark calibration. hat-variable denotes deviation from the steady state.
Notice finally that beyond these arguments based on the log-linearized version of the model, our quantitative analysis is based on a second-order approximation method. Indeed, the impact of financial frictions is asymmetric. During booms, both the "credit multiplier" and the impatience gap fall, taking the economy closer to a (financial) frictionless environment. The costs associated with financial frictions arise in recessions, where the impatience gap and the credit multiplier are greater; the deeper the recession (in case of very negative shocks), the more significant is their role (i.e., their size is greater). This asymmetric mechanism has important implications for welfare.
6
Quantitative analysis
In this section, we present the calibration and show that our model is able to match the magnitude of cycles. Indeed, for our quantitative calculation of business-cycle costs to be relevant, we need to match the volatility of data. Overcoming the "Shimer puzzle" is thus a necessary condition for our exercise. We then measure the welfare costs of business cycles.
20
6.1
Calibration
Preference and technology parameters: The calibration is based on quarterly US data.34 The discount factor for patient agents is consistent with a 4% annual real interest rate. For the impatient consumer, we set β = 0.99, which is within the range of values chosen by Iacoviello (2005)35 . The risk aversion is set to 1 for firms and 2 for workers. Both values lie within a standard interval in the literature. In addition, the firm is characterized by a lower risk aversion because, as shown by Iacoviello (2005), such a calibration ensures that the borrowing constraint is binding for a wide range of volatility shocks, impatience levels, and loan-to-value ratios (m values).
Financial parameters: The corporate debt-to-GDP ratio pins down the value of m in the collateral constraint. To this end, we use the average corporate debt over GDP for 20012009 (debt outstanding, annual data, corporate sector, Flow of Funds Accounts tables of the Federal Reserve Board).
Labor market parameters: Employment level N is consistent with the average unemployment rate (N = 0.88) estimates of Hall (2005). According to Hall (2005), the observed high transition rate from “out of the labor force” directly to employment suggests that a fraction of those classified as out of the labor force are nonetheless effectively job-seekers. Hall (2005) adjusts the US unemployment rate to include individuals out of the labor force who are actually looking for a job. As in Shimer (2005a), the quarterly separation rate s is 0.10, so jobs last for about 2.5 years on average. Using steady-state labor-market flows, we infer Ψ given s and N. This leads to Ψ = 0.423. This value is lower than in the usual Mortensen-Pissarides model. Indeed, the pool of job seekers is larger in Blanchard & Gali (2010) than in the standard MP model. The elasticity of the matching function with respect to the number of job seekers is ψ = 0.5, which lies within the range estimated in Petrongolo & Pissarides (2001). This value is also chosen so as to illustrate the pure effect of the nonlinearities in the unemployment dynamics of the model without financial frictions (see section 3.1), as opposed to the nonlinearities in the job-finding rate (see Hairault et al. (2010)). The efficiency of matching, χ, is set such that firms with a vacancy find a worker within a quarter with a 95% probability, which is consistent with Andolfatto (1996). 34 35
Data are described in Appendix A. Notice that an extremely low degree of impatience heterogeneity is sufficient for debt limits to hold.
21
As stressed by Hagendorn & Manovskii (2008), the parameters that determine the responsiveness of job creation to business cycles are the ratio of unemployment benefits (or home production without policy) to wages and firms’ bargaining power. The utility of leisure parameter, Γ, is pinned down so as to match an unemployment benefit equal to 0.7 at the steady state. This gives b/w = 0.720 (which is consistent with Hall & Milgrom (2008)). The cost of posting a vacancy, ω, is set to 0.17 as in Barron & Bishop (1985) and Barron et al. (1997). We obtain
ωV Y
= 0.0179, which is within the range found in the literature (0.005 in
Chéron & Langot (2004) or 0.05 in Krause & Lubik (2007)). Notice finally that, in order to reproduce the volatility of the job finding rate, Hagendorn & Manovskii (2008) need to calibrate b/w = 0.955 and = 0.9480, which implies that the share of wage that can fluctuate is negligible (equal to 1 − (b)/w = 0.10). In this paper, with b/w = 0.720 and = 0.5, half of the wage can fluctuate. Indeed, in our case,1 − (b + Γ)/w = 0.55. Thus, there is room for economic mechanisms to endogenously lead wages to fluctuate (which is consistent with the data). In addition, the model must endogenously generate limited fluctuations in w so as to preserve firms’ incentives to hire in booms.
Shocks: The technological shock is calibrated as in Hairault et al. (2010). We choose the standard deviation of technological shock to reproduce the observed GDP standard deviation. Table 1 summarizes the calibration.
6.2
Business-cycle properties
In this section, we document the unconditional business cycles facts on financial variables and labor market adjustments. Our contribution lies also in bringing together financial data (from Jermann & Quadrini (2012)) and data from the labor market literature (Shimer (2012)). In both markets, we focus on fluctuations in quantities (debt, unemployment) as well as equilibrium prices (interest rate, wage).36 Table 2, column 1, reports business-cycle properties found in the data. The volatility of real wages is not close to zero. Moreover, it is larger than that of labor productivity. This clearly suggests that real wage rigidity (implying a zero standard deviation for fluctuations in w) is not a realistic explanation for the strong cyclicality of labor market 36 All data have been recomputed and updated so that our sample covers five recession episodes from 1976 January through January 2013 (see Appendix A for a complete description of the data set). Previous works that study the interaction between financial and real variables in DSGE models such as Monacelli et al. (2011) and Christiano et al. (2010) summarize labor market adjustments using only fluctuations in employment and unemployment.
22
Table 1: Calibration (a) External information Notation Label β discount factor (impatient) α production function σW risk aversion, worker σF risk aversion, firm s Job separation rate N Employment ψ Elasticity of the matching function ω cost of job posting b replacement ratio w A average TFP ρA Persistence (b) Empirical target Notation Label µ discount factor (patient) χ scale parameter of matching function m collateral constraint σA Standard deviation (c) Derived parameter values Notation Label Ψ Job finding rate Γ preference
value 0.99 0.99 2 1 0.1 0.88 0.5 0.17 0.72 1 0.95
Reference Iacoviello (2005) Iacoviello (2005)
value 1 1/(1.04 4 ) 0.634 0.61 0.0031
Empirical target Annual real rate of 0.04 Probability of filling a vacancy Φ = 0.95 corporate debt to GDP ratio B/Y = 0.595 Observed σY
Iacoviello (2005) Shimer (2005a) Hall (2005) Petrongolo & Pissarides (2001) Barron et al. (1997) and Barron & Bishop (1985) Hall and Milgrom (2008) Normalization Hairault et al. (2010)
value 0.423 0.19
aggregates. corr(Ut , Vt ) summarizes the dynamics around the Beveridge curve. The negative covariance is consistent with the view that aggregate shocks have a more important weight than reallocation shocks at business-cycle frequency. As expected, corr(Ut , Ψt ) is negative. Jung & Kuester (2011) point out that mean unemployment exceeds steady-state unemployment when the job-finding rate and the unemployment rate are non-positively correlated and the average job-finding rate is lower than the steady-state job-finding rate.37 This is the case in our model. The second moments from simulated data are reported in Table 2, column 2. Comparing columns 1 and 2 of Table 2, we note that the model generates volatile employment, vacancies, unemployment, and job-finding rates. The simulated volatilities are even a bit higher than the observed volatilities, as we tend to slightly underestimate wage volatility. Notably, 37
This can be inferred from the employment-flow equation taken at the steady state sNt = ΨUt where Nt = 1 − Ut . Hence, sE(1t − Ut ) = cov(Ut , Ψt ) + E(Ut )E(Ψt ). Subtracting the steady-state from both sides 1 of the above equation, leads to E(Ut ) − ut = − s+Ψ [cov(Ut , Ψt ) + (E(Ψt ) − Ψt ) E(Ut )]. We deduce that if (i) E(Ψt ) − Ψt < 0 and (ii) cov(Ut , Ψt ) < 0, then necessarily, E(Ut ) − ut > 0. The correlation at the bottom of Table 2 suggests that ii) holds in the data.
23
Table 2: Business-cycle volatility : Models versus data (1) Data
(2) Benchmark Model
std(.)
std(.)
(3) Without Financial Frictions std(.)
Y
1.44
**
1.44
**
1.44
**
C N Y /N w U Ψ V B q R
0.81 0.72 0.54 0.62 7.90 5.46 9.96 1.68 3.21 0.92
* * * * * * * * * *
0.88 0.74 0.28 0.49 5.45 6.26 12.7 1.35 2.59 0.32
* * * * * * * * * *
0.94 0.46 0.56 0.49 3.71 2.79 4.60
* * * * * * *
corr(U, Ψ) -0.91 -0.86 -0.92 corr(U, V ) -0.97 -0.71 -0.76 ** std (in percentage); * relative to GDP std σA = 0.0031 in column (2); σA = 0.0063 in column (3)
the predicted volatile adjustments on the labor market are not obtained under unrealistic fluctuations on financial markets. The model also reproduces fluctuations of corporate debt, nearly as volatile as in the data. Same considerations apply to the dynamics of land prices q. Simulations confirm that financial frictions do generate wage stickiness. At business-cycle frequencies, the decomposition of the wage equation confirms that countercyclical components dominate pro-cyclical elements. As a result, the wage increase in expansions is dampened. Firms then have a stronger incentive to create jobs, which raises the job-finding rate. Table 2 shows that the endogenous wage sluggishness featuring in the model is consistent with the wage dynamics in the data.38 This allows the model to generate sufficiently large movements in job-finding rates. Our model thus solves Shimer’s volatility puzzle without introducing the counterfactual assumption of a constant real wage. Finally, note that when the model is simulated without financial frictions (Table 2, column 3) and with a TFP process adjusted to match the volatility of output39 , the standard deviation of the job-finding rate relative to GDP is twice lower. This result underlines the significant interaction between labor search 38
Notice that the wage relative standard deviation is actually a little lower than that found in the data. In fact, the model’s job-finding rate is slightly more volatile than in the data. 39 Tho obtain this level of volatility the standard deviation of the technological shock must be multiplied by two.
24
and financial frictions in the propagation mechanism. Moreover, the relative standard deviation of the wage is the same as in the model with financial frictions, whereas the volatility b > Ψ) b is more than twice as small. This clearly of the outside option (given by θb = ψ1 Ψ confirms that financial frictions dampen large movements in workers’ outside options in the Nash bargained wage rule. Despite large movements in θ, wages are endogenously rigid.
6.3
Welfare cost of fluctuations: the crucial role of nonlinearities
To measure implied business-cycle costs, we compute simulated paths using a second-order approximation of decision rules. This allows us to take into account nonlinearities and business-cycle asymmetries.
6.3.1
Decomposing the welfare cost of fluctuations: the level effect matters more than the business-cycle effect
P t The expected lifetime utility of a worker is U˜ w = E0 ∞ t=0 µ U (Ct + (1 − Nt )Γ), because Ct ≡ Nt Ctn + (1 − Nt )Ctu and the FOC on consumption imply Ctn = Ctu + Γ. We define the welfare costs associated with business cycles, τ , as the fraction of steady-state consumption that workers would give up to be indifferent between the steady state and the fluctuating P t ¯ + (1 − N ¯ )Γ](1 − τ ) = µ U [ C economy. The welfare cost of fluctuations τ is such that ∞ t=0 U˜ w , where variables marked with an overbar denote their steady-state values.40 We deduce
(1 − µ) (1 − σ) τ = 1 − U˜ w ¯ ¯ )Γ)1−σ (C + (1 − N
1 1−σ
The result is reported in Table 3, line 1, column A. The business-cycle cost of fluctuations with financial frictions is 2.50% of workers’ permanent consumption. This number is far larger than the estimates found by Lucas (1987, 2003), who reports a welfare cost of τ = 0.05% with log utility. Notably, welfare costs are large even though agents can save by lending to firms; workers can actually smooth business cycles with savings. One way to understand these quantitative results is to decompose the cost. We use a Taylor expansion of welfare in the volatile economy. The crucial point at this stage is to consider the Taylor expansion around the mean of the stochastic economy, and not around P∞ t 40 F ¯F ˜F = A similar computation is possible for the firm’s owner: = U t=0 β U C (1 − τ ) P∞ t F t=0 β U Ct . When introducing financial frictions, firms’ welfare costs of fluctuations can also be taken into account. In this case, aggregate welfare costs would then be greater. We choose to focus on workers’ welfare costs only to compare our results to the existing literature.
25
the deterministic steady state. This ensures that the computation takes into account the gap between the mean (i.e., the stochastic steady state) and the deterministic steady state. We approximate welfare in the volatile economy as the sum of a level effect and the business-cycle effect. Indeed, we have U˜ w ≈
1 1 U (E0 [C + (1 − N )Γ]) 1 − σ(1 − σ) (γc V ar(b c) + γu V ar(b u) + γcu Cov(b c, u b)) 1−µ 2
where we denote41 x b= and γcu =
2ΓE0 [C(1−N )] . E0 [(C+(1−N )Γ)2 ]
(1 − τ ) ≈
Xt −E0 [X] , E0 [X]
for x = C, U and γc =
E0 [C 2 ] , E0 [(C+(1−N )Γ)2 ]
γu =
Γ2 E0 [(1−N )2 ] E0 [(C+(1−N )Γ)2 ]
This leads to
E0 [C + (1 − N )Γ] ¯ )Γ C¯ + (1 − N
1 1−σ 1 1 − σ(1 − σ) (γc V ar(b c) + γu V ar(b u) + γcu Cov(b c, u b)) 2
¯ )Γ beIf we neglect the level effect, then we have U (E0 [C + (1 − N )Γ]) ≈ U C¯ + (1 − N ¯ )Γ, then cause we assume that E0 [C + (1 − N )Γ] ≈ C¯ + (1 − N 1 1−σ 1 c) + γu V ar(b u) + γcu Cov(b c, u b)) = 1 − σ(1 − σ) (γc V ar(b 2
1 − τBC
where τBC denotes the welfare costs of the BC computed in the spirit of Lucas (1987, 2003) . In contrast, if we neglect the business-cycle effect (γc V ar(b c) + γu V ar(b u) + γcu Cov(b c, u b) ≈ 0), we have (1 − τL ) =
E0 [C + (1 − N )Γ] ¯ )Γ C¯ + (1 − N
where τL denotes the welfare costs of the business-cycle effect linked to the level effect. We deduce that (1 − τ ) = (1 − τBC )(1 − τL ) ⇒ τ ≈ τBC + τL . Numerical computations give τ ¯ E0 [U ] and U¯ . The previous formula then gives τBC .42 and τL given E0 [C] and C, Welfare costs due to business-cycle fluctuations alone, τBC , are reported in Table 3, line 3, column A, and equal 0.27% of permanent consumption for workers. They are five times as large as in Lucas (1987, 2003) (0.05%). This first result comes from labor-market fluctuations (which are magnified by financial frictions). The latter are neglected by Lucas. In our model, τBC is a function of not only V ar(b c), as in Lucas, but also V ar(b u) and Cov(b c, u b), which, for a given "level effect", clearly magnify the costs of cycles. The most striking result is the measure of τL = 2.23% in Table 3, line 2, column A. It accounts for the great increase 2
2
2
∂U ∂U ∂ U ∂ U 00 ∂ U 2 00 Indeed, given that ∂C = U 0 , ∂N = −ΓU 0 , ∂C and ∂C∂N = −ΓU 00 , we obtain, with 2 = U , ∂N 2 = Γ U 1−σ x1−σ 0 −σ 00 −σ−1 = −σ(1 − σ) x1−σ x12 . the usual functional form U (x) = 1−σ , U = x and U = −σx 42 The same considerations apply to firms’ owners, with E0 [C F ], C¯F and τ F . 41
26
Table 3: Decomposition of welfare costs of business cycle Worker with financial frictions A Total welfare cost 1. τ × 100 2.50 Decomposing the welfare cost 2. τL × 100 2.23 3. τBC × 100 0.27
Worker without financial frictions B 0.12
0.06 0.06
line 1 = line 2 + line 3
in business-cycle costs: 90% of welfare costs come from this level effect. In fact, in Lucas (1987, 2003), τL = 0. By assumption, there is no gap between average and steady-state consumption. Our model shows that this approximation is not acceptable because businesscycle volatility significantly affects the gap between average and steady-state employment and consumption. Thus, business-cycle costs are sizable: they are 50 times the amount estimated by Lucas. Without financial frictions, workers’ welfare cost falls drastically (100 × τ = 0.12, line 1, column B, Table 3).43 The magnitude of business-cycle costs is reduced to 2.4 times Lucas’s evaluation. This estimation is slightly lower than the value reported in Hairault et al. (2010) or Jung & Kuester (2011). In our model, wages are set by using a Nash bargaining solution while Hairault et al. (2010) and Jung & Kuester (2011) assume exogenous wage rigidity or sluggishness. If we apply our computations to the data, the contrast with Lucas’s results is straightforward. Our calculations show that, with financial frictions, the mean of employment is E(N ) = ¯ = 0.88. This implies that the level effect associated 0.85227, while its steady-state value is N with employment only is 100 ×
¯ −E(N ) N ¯ N
= 3.15. If we apply this percentage to US civilian
employment stock in 2015, the job losses associated with fluctuations represent more than 4.7 millions jobs. Analogous considerations apply to GDP per capita in 2015:Q4. A comparison 43
This estimate is computed by our model, with discounting heterogeneity and collateral constraints eliminated but the parameter values in Table 1 (panels (b) and (c)) retained. The rationale behind this approach is as follows. The model with financial frictions is considered the "true" model of the economy. By calibrating the model to match key financial and labor market targets, we uncover the "true" parameter values. The business-cycle costs without financial frictions are then computed with these parameter values, including the standard deviation of technological shock. If, in the model without financial frictions, we adjust the standard deviation of technological shock to match output volatility, the welfare costs amount to 0.41% of permanent consumption, which remains far below the values reported in the model with financial frictions. Finally, in the model without financial frictions, under a calibration that mimics the first-order allocation (i.e., b = 0, no unemployment allocation) and the Hosios condition = ψ, welfare costs are negligible (0.02%).
27
between E(Y ) and Y¯ leads to the following estimate: the level effect entails a GDP per capita loss of approximately 1520 dollars a year. Without financial frictions, the level effect entails a loss of about 100.000 jobs, and each household loses 38 dollars per year. 6.3.2
Asymmetric welfare costs of business cycles
As stressed by Hairault et al. (2010), Jung & Kuester (2011) and Petrosky-Nadeau & Zhang (2013), search and matching models feature asymmetric responses to business-cycle shocks; recessions (expansions) are characterized by severe and rapid rises (gradual decline) in unemployment. These asymmetric fluctuations are supported by empirical evidence (MacKay & Reis (2008) and Petrosky-Nadeau & Zhang (2013), among others). The IRFs in model B (without financial frictions) are consistent with these features. Figure 5, panel (b), displays the employment response to the calibrated positive productivity shock when the economy is in boom – i.e., starting from a point where the economy is already hit by the same shock (dotted line)– and to a negative shock of the same magnitude when the economy is in recession (solid line). Employment falls more in a recession than it increases in a boom. Figure 5, panel (a), shows the IRFs from model A (with financial frictions) in a recession versus economic boom. The asymmetric response of employment is even larger than in model B. In order to measure this increased asymmetry due to the presence of financial frictions, we report, in panel (c), the gaps between IRFs in recession versus boom in both models. The gap between IRFs is twice as large in the economy with financial frictions (model A) as in the economy without financial frictions (model B); the maximum gap is around 15% in model A, but only 7.5% in model B.44 . In addition, since labor market adjustments are connected to credit market conditions, the immediate response of credit market conditions directly affect employment dynamics. The maximum gap is reached after two quarters in model A and seven quarters in model B. In order to measure the implied business-cycle costs of these nonlinearities, we compute the time-varying welfare cost τ as in Petrosky-Nadeau & Zhang (2013) for model A (with financial frictions) and model B (without financial frictions).45 Figure 5, panel (d), plots the welfare cost τ × 100 against the technological shock. First, in model B, the welfare cost is countercyclical. In addition, welfare gains in expansion are much lower than welfare 44
This last case summarizes the results reported in Hairault et al. (2010), Jung & Kuester (2011) and Petrosky-Nadeau & Zhang (2013). 45 The welfare cost at each date is based on a comparison between the deterministic worker’s welfare and the fluctuating worker’s welfare summarized by his value function. This value function is time-varying. Its expected value is computed, for each current state of the economy, using the second-order Dynare approximation of endogenous variables (see Adjemian et al. (2014)). We consider the decision rules around the mean, rather than the steady state, which is the default setting in Dynare
28
Figure 5: The Welfare cost of business cycles and asymmetry: Why do financial frictions matter? (a) N and -N, Model A with Fin. Fric.
(b) N and -N, Model B without Fin. Fric.
1.2
0.8 Expansion Recession
1
0.7 0.6
0.8
0.5 0.4
0.6
0.3 0.4 0.2
0.2 0
5
10 quarter
15
0.1 0
20
(c) Comparing recession to expansion in models A vs. B 15 A with Fin. Fric B no Fin. Fric.
5
Model A. With Fin. Fric. Model B. No Fin. Fric. Model A. With Fin. Fric. withτ q). Thus, wealth increases, on average, raising consumption relative to the case with technological shocks only.47 Financial shocks improve the match between debt and land-price volatilities (see Column (2) of Table 7 in Appendix C.4), but this shock induces excess volatility in vacancies. Figure 6 also shows that the asymmetries intrinsic to our model are not affected by the introduction of an additional shock. Indeed, the structure of our framework remains unaffected. Column (3) of Table 4 incorporates capital into the model along the lines of Liu, Wang & Zha (2013).48 Notice that welfare costs are greater than in the benchmark model. Indeed, capital amplifies our basic mechanism. As employment is lower, on average, than in steady state, the marginal productivity of capital in the stochastic economy, and thus the incentive to save, decreases. Notice also that when capital is included in the collateral constraint, 46
See Appendix C.1 for more details. See Appendix D.1 for a formal analysis of this mechanism using a simplified version of our model. 48 See Appendix C.2 for more details. 47
30
Figure 6: The Welfare cost of business cycles and asymmetry Benchmark With credit shocks Capital Capital and wage bill Capital with credit shock
14 12 10
100 × τ
8 6 4 2 0 -2
0.94
0.96
0.98 1 1.02 Technological shock
1.04
1.06
The steady-state value of technological shock is 1. Curves are quadratic fit over 30,000 simulations for each model.
models with only technological shocks are not able to replicate the volatilities of debt and land price (see column (3) of Table 7 in Appendix C.4) – making the assessment of the welfare cost of fluctuations less convincing.49 Welfare costs increase further when the collateral constraint also includes the wage bill; see column (4) of Table 4. As costs associated with hiring are greater in equilibrium, the incentive to hold open vacancies is lower. In this modified framework, the financial wedge in the (JC) curve magnifies only the marginal product of labor (not net of the real wage, as in the benchmark model). In equilibrium, the economy thus shifts towards the more convex part of labor market equilibrium (on Figure 3) where welfare costs are higher. Finally, the introduction of both capital and financial shocks can be viewed as a solution to fit the financial indicators of the business cycle as well as those of investment (column (5) of Table 4). Indeed, this proceeds in the right direction, but at the cost of excess volatility on labor market fluctuations. This comes from the very low sensitivity of wages to the business cycle in this model; financial frictions are too strong and thus overestimate the wage moderation induced by the credit channel.50 Notice also that the role of financial shocks in our model depends on whether we include capital. In economies without capital (columns (1) and (2) of Table 4), financial shocks reduce welfare costs. On the contrary, 49
In Figure 6, the introduction of capital exacerbates the nonlinearities of the model (welfare gains and losses are larger than in the benchmark). As large gains are, on average, compensated by large losses, new asymmetries are not the main reason behind the change in average welfare costs when capital is introduced. 50 A solution for this shortcoming of this extension would be to consider a more complex modeling of the collateral constraint as in Liu, Wang & Zha (2013), where the collateral is not the simple sum of the value of capital and land, but a weighted sum of these two assets. Given that these weights are unknown, this is left for future research.
31
in presence of capital (columns (3) and (5)), financial shocks increase welfare costs. This is because financial shocks make banks’ return on the collateral more risky. This uncertainty generates a premium on the borrowing constraint (i.e., an over-accumulation). This in turn generates a new motive to increase debt (B increases in the volatile economy relative to the stabilized economy), entailing a parallel increase in workers’ wealth. As a result, in our benchmark model, consumption is, on average, greater than at the steady state. This is not the case when we include capital accumulation. This latter setting is characterized by decreasing returns to scale. As greater leveraging in the volatile economy induce additional capital, over-accumulation is then costly in terms of consumption.51
7
Conclusion
This paper provides a quantitative assessment of welfare costs of fluctuations in a labor market search model with financial frictions à la Kiyotaki & Moore (1997). Because of labor market search frictions, fluctuations generate a higher average unemployment rate relative to its steady-state value, increasing the welfare cost of fluctuations. Financial frictions amplify this mechanism, together with the associated welfare costs. We show that business-cycle costs are sizable: they are 50 times the amount estimated by Lucas. Without financial constraints, the magnitude of business-cycle costs is reduced to 2.4 times Lucas’s evaluation. Our model also allows the job-finding rate a high degree of responsiveness to the business cycle. Indeed, financial frictions entail wage sluggishness that helps the model match the large changes in job-finding rates observed in the data; at the same time, it preserves the real wage volatility observed in the data. We have shown that the presence of financial frictions sharply increases the welfare cost of fluctuations. Moreover, our paper reveals significant asymmetries in the welfare response to business cycles. These results suggest that structural policies aiming to remove financial frictions, per se, could have significant stabilizing macroeconomic effects. This is left for future research.
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See Appendix D.2 for an analysis of this mechanism in a simplified Mickey-Mouse model.
32
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Krusell, P. & Smith, A. (1999), ‘On the welfare effects of eliminating business cycles’, Review of Economic Dynamics 2, 245–272. Langot, F. (1995), Unemployment and real business cycle : A matching model, in P. Henin, ed., ‘Advances in Economic Business Cycles Theory’, Springer Verlag. Liu, Z., Miao, J. & Zha, T. (2013), Land prices and unemployment, NBER Working Paper 19382. Liu, Z., Wang, P. & Zha, T. (2013), ‘Land-price dynamics and macroeconomic fluctuations’, Econometrica 81(3), 1147–1184. Lucas, R. (1987), Models of Business Cycles, Blackwell Publishing. Lucas, R. (2003), ‘Macroeconomic priorities’, American Economic Review 93(1), 1–14. MacKay, A. & Reis, R. (2008), ‘The brevity and violence of contractions and expansions’, Journal of Monetary Economics 55, 738–751. Merz, M. (1995), ‘Search in the labor market and the real business cycle’, Journal of Monetary Economics 36(2), 269–300. Monacelli, T., Quadrini, V. & Trigari, A. (2011), Financial markets and unemployment, NBER Working Paper Series 17389, NBER. Mortensen, D. & Pissarides, C. (1994), ‘Job creation and job destruction in the theory of unemployment’, The Review of Economic Studies 61(3), 397–415. Petrongolo, B. & Pissarides, C. (2001), ‘Looking into the black box : A survey of the matching function’, Journal of Economic Literature 39, 390–431. Petrosky-Nadau, N. & Wasmer, E. (2013), ‘The cyclical volatility of labor markets under frictional financial markets’, American Economic Journal: Macroeconomics 5(1), 193–221. Petrosky-Nadeau, N. (2013), ‘Credit, vacancies and unemployment fluctuations’, Review of Economic Dynamics . Petrosky-Nadeau, N. & Zhang, L. (2013), Unemployment crises, NBER Working Paper 19207, NBER. Pissarides, C. (2009), ‘The unemployment volatility puzzle: Is wage stickiness the answer?’, Econometrica 77(5), 1339–1369.
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Pissarides, C. (2011), ‘Equilibrium in the labor market with search frictions’, American Economic Review 101(4), 1092–1105. Quadrini, V. (2011), ‘Financial frictions in macroeconomic fluctuations’, Economic Quarterly 97(3), 209–254. Rubinstein, A. (1982), ‘Perfect equilibrium in a bargaining model’, Econometrica 5(1), 97– 109. Shimer, R. (2005a), ‘The cyclical behavior of equilibrium unemployment and vacancies’, American Economic Review 95(1), 25–49. Shimer, R. (2005b), ‘The cyclical behavior of equilibrium unemployment and vacancies’, American Economic Review 95(1), 25–49. Shimer, R. (2012), ‘Reassessing the ins and outs of unemployment’, Review of Economic Dynamics 15, 127–148. Storesletten, K., Telmer, C. & Yaron, A. (2001), ‘The welfare cost of business cycles revisited: Finite lives and cyclical variation in idiosyncratic risk’, European Economic Review 45, 1311–1339. Wasmer, E. & Weil, P. (2004), ‘The macroeconomics of labor and credit market imperfections’, American Economic Review 94(4), 944–963. Zanetti, F. (2015), Financial shocks and labor market fluctuations, Working Paper 746, University of Oxford. Zanetti, F. & Mumtaz, H. (2013), The effect of labor and financial frictions on aggregate fluctuations, Working Paper 690, University of Oxford.
Appendix For Online Publication A
Data
Aggregate data. The following quarterly time series come FRED database, the Federal Reserve Bank of Saint Louis’ website (1976Q1-2013Q1). y is Real Gross Domestic Product from the FRED database (mnemonicsGDPC96) divided by the Civilian Non institutional
36
Population from the FRED database (mnemonics CNP16OV). c is Real Personal Consumption Expenditures from the FRED data-base (mnemonics PCECC96) divided by the Civilian Non institutional Population from the FRED database (mnemonics CNP16OV)
Labor market data. w is Compensation of Employees: Wages & Salary Accruals from the FRED database (mnemonics WASCUR) divided by Civilian Employment (CE16OV). N is Civilian Employment (CE16OV) divided by Civilian Non institutional Population. U is FRED, Civilian Unemployment Rate (UNRATE), Percent, quarterly, Seasonally Adjusted. The latter time series are taken from the FRED database. As for the time series of the job finding rate, we use monthly CPS data from January 1976 to March 2013. We follow all the steps described in Shimer (2012). As in Shimer (2012), we correct for time aggregation and take quarterly averages of monthly observations. V are vacancies Total Nonfarm, Total US Job Openings JTS00000000JOL, Seasonally Adjusted Monthly data from BLS. We take quarterly averages of this time series that is available only from December 2000 onwards.
Debt, loan Rate and land price. We follow Jermann & Quadrini (2012). Financial data come from the Flow of Funds Accounts of the Federal Reserve Board. The debt stock is constructed by using the cumulative sum of net new borrowing measured by the ‘Net increase in credit markets instruments of non financial business’52 . Since the constructed stock of debt is measured in nominal terms, it is deflated by the price index for business value added from NIPA. The initial (nominal) stock of debt is set to 94.12, which is the value reported in the balance sheet data from the Flow of Funds in 1952.I for the nonfarm non financial business. The cumulative sum starts in 1952, which, as in Jermann & Quadrini (2012), is not likely to affect our data starting on January 1976. R is the log of 1+ the Bank Prime Loan Rate (MPRIME) (used as a reference for short-term business loan) from the FRED database. Finally, we use as a proxy for q the price index for residential land as computed by Liu, Wang & Zha (2013).
Cyclical components of the data: All data are quarterly (from 1976:Q1 through 2013:Q1), in logs, HP (λ = 1600) filtered and multiplied by 100 in order to express them in percent deviation from steady state. Ψ is the job finding rate computed from Monthly CPS data from January 1976 to March 2013 using Shimer (2012)’s methodology. It measures the probability for an unemployed worker to find a job. As for financial data on debt and interest rate, we 52
Nonfinancial business; credit market instruments; liability; Net increase in credit markets instruments of non financial business, millions of dollars (nominal). FA144104005.Q, F.101 Line 28.
37
follow Jermann & Quadrini (2012). We finally check that our financial and labor market time series are consistent with the data available on line for Shimer (2012) and Jermann & Quadrini (2012).
B B.1
Model Household
Each household knows that the evolution of S follows (3), so that (6) can be written as: Nt = (1 − s)Nt−1 + Ψt (1 − (1 − s)Nt−1 )
(14)
The dynamic problem of a typical household can be written as follows W(ΩH t ) = nmax u
Ct ,Ct ,Bt
Nt U (Ctn ) + (1 − Nt )U (Ctu + Γ) + µEt W(ΩH t+1 )
subject to (14) and (5), given the initial conditions on state variables (N0 , B0 ) and ΩH t = {Nt−1 , Ψt , wt , bt , Tt , Bt−1 }, the vector of variables taken as given by households. Let λt be the shadow price of the budget constraint. The first order conditions associated with consumption choices are (Ctn )−σ = (Ctu + Γ)−σ = λt Hence Ut (Ctn ) = Ut (Ctu + Γ). The first order condition associated to bond holdings reads:
B.2
−λt + µEt [Rt λt+1 ] = 0
(15)
U CtF + βEt W(ΩFt+1 )
(16)
Entrepreneur
The firm’s program is W(ΩFt ) =
s.t.
max
CtF ,Lt ,Bt ,Vt ,Nt
−CtF − Rt−1 Bt−1 − qt [Lt − Lt−1 ] − wt Nt − ω ¯ Vt +Yt (At , Lt−1 , Nt ) + Bt = 0 (λFt ) −Bt − ω ¯ Vt + mEt [qt+1 Lt ] = 0 (λFt ϕt ) −Nt + (1 − s)Nt−1 + Φt Vt = 0 (ξt )
38
given the initial conditions N0 , B0 , where ΩFt = {Nt−1 , Ψt , wt , bt , πt , Tt , Bt−1 , Lt−1 } is the vector of variables taken as given by firms. Letting λFt , λFt ϕt , and ξt be the Lagrange multipliers associated to (8), (10) and (11) the first order conditions of problem (29) read: U 0 CtF
= λFt
λFt qt = (1 − ϕt )λFt = ξt = ξt =
(17)
∂Yt+1 F βEt λt+1 qt+1 + + λFt ϕt mEt [qt+1 ] ∂Lt βEt λFt+1 Rt (1 + ϕt ) ¯ λFt ω Φ t ∂Yt F λt − wt + (1 − s) βEt [ξt+1 ] ∂Nt
(18) (19) (20) (21)
where (17) is the condition associated to consumption and (20) the one on vacancy posting53 . Equation (C.2) is the one associated to land accumulation. It implies that, in equilibrium, the value of current marginal utility of consumption needs to equal the indirect value of utility deriving from land accumulation, i.e.:
i) the value of future consumption utility
deriving from reselling land in the next period, βEt λFt+1 qt+1 ; ii) the future consumption ; iii) the additional utility arising utility arising from the product of land, βEt λFt+1 ∂Y∂Lt+1 t from current consumption related to the effect of land in loosening the collateral constraint, ϕt mλFt Et [qt+1 ]. Equation (19) is a modified Euler equation. When the collateral constraint is not binding, ϕt is equal to zero and we recover the standard Euler equation. When the debt limit is binding, ϕt > 0 and ϕt = 1 − β
Et λF t+1 Rt λF t
implying that firms’ marginal utility of current consumption
is greater than their discounted marginal utility of future consumption. Impatient firms choose thus to increase consumption up to the limit imposed by (10).
B.3
The wage curve
From the household’s intertemporal program, one gets: VtH
53
∂W(ΩH ∂W(ΩH t ) ∂Nt t ) = = + µEt ∂Nt−1 ∂Nt ∂Nt−1
∂W(ΩH t+1 ) ∂Nt
∂Nt ∂Nt−1 ∂W(ΩH ∂Nt ∂Nt t+1 ) n u n u = [Ut (Ct ) = Ut (Ct + Γ) + λt wt − λt bt − λt (Ct − Ct )] + µEt ∂Nt−1 ∂Nt ∂Nt−1
Note that, entrepreneurs are not risk neutral. By letting λF t = 1 we recover the canonical search model.
39
With Ut (Ctn ) = Ut (Ctu + Γ), we have VtH Where, from (14)
∂Nt + µEt = [λt wt − λt (bt + Γ)] ∂Nt−1
∂Nt ∂Nt−1
∂W(ΩH t+1 ) ∂Nt
∂Nt ∂Nt−1
= (1 − s) (1 − Ψt ), so that
VtH 1 ∂W(ΩH t+1 ) = (1 − s) (1 − Ψt ) wt − (bt + Γ) + µEt λt λt ∂Nt From the firms’ program VtF =
∂W(ΩF t ) ∂Nt−1
(22)
t) ¯ (1+ϕ = ξt (1 − s) where ξt = λFt ω , thus: Φt
∂W(ΩFt+1 ) ω ¯ F = (1 − s) λ (1 + ϕt+1 ) ∂Nt Φt+1 t+1 ω ¯ VtF = (1 − s) (1 + ϕt ) F Φt λt Then, using (12) we obtain: ∂Yt VtF 1 ∂W(ΩFt+1 ) = (1 − s) − wt (1 + ϕt ) + βEt ∂Nt ∂Nt λFt λFt Therefore, the surpluses are, respectively: VtF λFt VtH λt
F F λt+1 Vt+1 ∂Yt = (1 − s) − wt + βEt ∂Nt λFt λFt+1 H λt+1 Vt+1 = (1 − s) (1 − Ψt ) wt − (bt + Γ) + µEt λt+1 λt
By maximizing the Nash product with respect to the wage, we obtain
VtH λt
=
(23) (24)
VtF λF t
(1−)(1−Ψt ) .
By substituting for (23) and (24), and rewriting it, we obtain the wage curve
B.4
Proof of proposition 1
In equilibrium, the job creation JC and the wage curve W C need to intersect, i.e.: β MP L + θ 1+1− [(1 − s) β − 1] (25) χ µ 1−ψ β θ = O + (1 − ) M P L + (1 − s) (1 − ) ω ¯µ 1 + 1 − (β − µ) + µθ µ χ ¯ 1−ψ ω
40
where we
∂Y ∂N
≡ M P L and we denote by O = b + Γ the constant component of the outside
option. For simplicity, let as proxy the impatience gap with the steady-state level of the Lagrange multiplier associated to the collateral constraint, ϕ¯ = 1 − βµ . Indeed, ϕ¯ is increasing in the impatience gap. Equation (25) can be rewritten as: θ1−ψ (M P L − O) = (1 + ϕ) ¯ ω ¯ (1 − s) (1 − ) µθ + (1 − (1 − s) µ (1 − ϕ)) ¯ χ | {z }
(26)
≡g(θ,ϕ) ¯
Note that gθ0 (θ, ϕ) ¯ > 0 and gθ00 (θ, ϕ) ¯ < 0. In addition, ∀ϕ, ¯ we have lim g(θ, ϕ) ¯ = +∞ and θ→+∞
g(0, ϕ) ¯ = 0. Finally, g(θ, ϕ) ¯ is steeper in an economy with financial frictions, indeed: ∂g 0 (θ, ϕ) ¯ =ω ¯ ∂ ϕ¯
"
−ψ
(1 − s) (1 − ) µ + (1 − ψ) θ χ (1 − (1 − s) µ (1 − ϕ)) ¯ −ψ
+ (1 + ϕ) ¯ (1 − ψ) θ χ (1 − s) µ
# >0
Hence, labor market tightness is lower in case of financial frictions (θ2 < θ1 with ϕ2 > ϕ1 , see figure 7). As no restrictions on parameter values are needed, this result result is not ambiguous. This means that the equilibrium level of θ is eventually driven by the steeper JC curve. Figure 7: Steady state labor market tightness
0
Labor market tightness is lower in case of financial frictions : θ2 < θ1 with ϕ2 > ϕ1
B.5
The labor market in response to a productivity shock : analytical results
We characterize analytically the dynamics of the labor-market equilibrium in response to a productivity shock, under assumptions 1. More precisely, we restrict our analysis to an
41
equilibrium where the value of search in the bargaining process is positive, and where the dynamics of θ are determined (saddle path). The values of parameters are to be considered "reasonable" as they imply usual properties of the labor market. Within these restrictions, our results are directly comparable to the standard DMP framework. The system of equations of this problem is thus: (1 + ϕt ) (1 + ϕt+1 ) ω ¯ = yt − wt + (1 − s) βEt ω ¯ Φt Φt+1 wt = (b + Γ) + (1 − ) [yt + Σt ] ω ¯ 1 + ϕt+1 β − ϕt + ω ¯ θt+1 Σt = (1 − s)Et 1 − ϕt Φt+1 Φt = χθtψ−1 β ϕt = 1 − (1 + Λεt ) µ while its log-linearized counterpart is: ϕ Φy Φw ϕ bt = b t+1 ϕ bt − Φ ybt − w bt + (1 − s) βEt ϕ bt+1 − Φ 1+ϕ ω ¯ (1 + ϕ) ω ¯ (1 + ϕ) 1+ϕ (1 − )y (1 − )Σ ˆ w bt = yˆt + Σt w w ω ¯ (β − µ) ϕ bt = b t+1 ] Σ Et [ϕ bt+1 ] − ω¯ Φ Et [Φ 1+ϕ (β − µ) + µ¯ ω θ Φ ω µ + µ¯ ωθ µ¯ ωθ β − µ −¯ Φ b + ω¯ Et [θt+1 ] − ϕ bt ω ¯ β Φ (β − µ) + µ¯ (β − µ) + µ¯ ωθ ωθ Φ ybt = ρb yt−1 + εt β ϕ bt = − Λεt µ−β b t = (ψ − 1)θbt Φ where, using assumption, 1.3 Et [ϕ bt+1 ] = 0. Notice that for β sufficient close to µ, we obtain ω ¯
(β−µ) Φ b t < 0 in • − ω¯ (β−µ)+µ¯ > 0 ⇒ a countercyclical component into the wage (because Φ ωθ Φ
booms). • − β−µ β
−ω ¯ µ+µ¯ ωθ Φ ω ¯ (β−µ)+µ¯ ωθ Φ
< 0 ⇒ the pro-cyclical component into the wage (because ϕ bt < 0 in
booms). In order to gauge the relative weight of these two opposite mechanism, it is necessary to analyze them at the equilibrium. Hence, we solve the system so as to compute the effect of 42
shocks on market tightness, θ: h i ϕ Φy ϕ bt + (1 − ψ)θbt = ybt + (1 − s) β(1 − ψ)Et θbt+1 1+ϕ ω ¯ (1 + ϕ) " # (1 − ψ) Φω¯ (β − µ)Et [θbt+1 ] (1 − )ΦΣ − −¯ ω ω ¯ (1 + ϕ) Φω¯ (β − µ) + µ¯ ωθ +µ¯ ω θEt [θbt+1 ] − β−µ µ + µ¯ ω θ ϕ bt β Φ This equation can be rewritten as: h i A1 θbt = A2 εt + A3 ybt + A4 Et θbt+1 where the values of coefficients Ai are given in Table 5, whereas steady-state restrictions are reported in table 6.
Restriction 1: A positive value of search Σ in w. Without financial frictions, Σ is always larger than zero. With financial friction, one must restrict the analysis for β sufficiently close to µ, ie.
ω ¯ (β Φ
− µ) + µ¯ ω θ > 0. Hence, we have
ω ω ¯ ϕ 1 Σ ¯ = (1 − s)β − +ω ¯θ (β − µ) + µ¯ ωθ > 0 = (1 − s) 1+ϕ Φ1−ϕ 1−ϕ Φ Restriction 2: The saddle path. The dynamics of the model with financial friction are a saddle path, as the one without of financial friction iff h i A1 θbt = A2 εt + A3 ybt + A4 Et θbt+1 ⇒ θbt = a1 εt +
a2 ybt 1 − a3 ρ
where a1 = A2 /A1 , a2 = A3 /A1 , a3 = A4 /A1 and a3 < 1. Given the steady-state values for Σ, term A4 can be rewritten as ( A4 =
(1 − s) [β(1 − ψ) + (1 − )µ((1 − ψ) − Φθ)]
with financial frictions
(1 − s)β [(1 − ψ) − (1 − )Φθ]
without financial frictions
Table 5: Model coefficients: with versus without financial frictions Coeff. A1 A2 A3 A4
β − µ−β Λ
With financial frictions 1−ψ h −ω ¯ µ+µ¯ ωθ (1−)ΦΣ β−µ Φ ω ¯ ω ¯ (1+ϕ) β Φ (β−µ)+µ¯ ωθ Φy ω ¯ (1+ϕ)
(1 − s) β(1 − ψ) −
(1−)ΦΣ ω ¯ (1+ϕ)
No financial frictions 1−ψ −
ϕ 1+ϕ
i
(1−ψ)ω ¯ (β−µ)+µ¯ ωθ Φ ω ¯ (β−µ)+µ¯ ωθ Φ
43
0 Φy ω ¯
(1 − s) β(1 − ψ) −
(1−)ΦΣ ω ¯
Table 6: steady state values: with versus without financial frictions Variable y−w w Σ Φ ϕ
With financial frictions No financial frictions ω ¯ ω ¯ (1 + ϕ) [1 − β(1 − s)] [1 − β(1 − s)] Φ Φ (b + Γ) + (1 − ) [y + Σ] 1+ϕ ¯θ (1 − s)β ω ¯θ (1 − s) 1−ϕ β − Φω¯ ϕ + ω ψ−1 ψ−1 χθ χθ µ−β 0 µ
Therefore, if (1 − ψ) > Φθ = Ψ, then A4 > 0, with or without financial frictions. This condition is always satisfied for our calibration where ψ = 0.5 and Ψ ≈ 0.4. Moreover, the solution is a saddle path iff A4 /A1 < 1 ⇔ a3 < 1, ie.
a3
h i (1 − s) β + (1 − )µ 1 − Φθ (A2 + A3 )|without financial frictions µ−β Φ y ⇒Λ> β 2µ − β ω ¯ (1 − s)(1 − )µ (−1 + Φθ) + 2µ−β h
where, using Tables 5 and 6, we deduce that A2 = Λ (1 − s)(1 − )µ (−1 + Φθ) + iff β is sufficiently close to µ, so that
β 2µ−β
(27)
β 2µ−β
i
> 0,
is sufficiently closed to one so as to compensate
(1 − s)(1 − )µ (−1 + Φθ) ∈ (−1; 0). e for Λ : for any Λ > Λ, e a model The RHS of the inequality (27) gives the threshold value Λ with financial frictions amplifies the short run impact of a technological shock. The solution 44
for Λ is given by the general equilibrium model.
C
Sensitivity analysis
C.1
Financial shocks
As in Liu, Wang & Zha (2013) and Jermann & Quadrini (2012), the financial shock is captured as a shock on m. This is interpreted as shocks on the tightness of the enforcement constraint and, therefore, the borrowing capacity of the firm. The financial shock follows the stochastic process log(mt ) = (1 − ρm )log(m) + ρm log(mt−1 ) + m t with the calibrated values from Liu, Wang & Zha (2013)’s estimation results (ρm = 0.9804 m
and σ
= 0.0112). The standard deviation of technological innovation σA is adjusted to
match the observed standard deviation of output. This calibration is used in column (2) Table 7 (section C.4).
C.2
Model with capital
Model.
In the model with capital, households’ behavior do not change. The introduction
of producing capital alters the entrepreneurs’ problem. As in Liu, Wang & Zha (2013), the production function is now
h i1−α 1−φ Yt = At Lφt Kt−1 Ntα
with K the stock of capital. Capital accumulation is subject to adjustment costs such that Ω Kt = (1 − δ) Kt−1 + It + 2
It It−1
2 − 1 It
λK t
with It the investment flow and Ω the scale parameter on adjustment costs. The entrepreneur’s budget constraint is now CtF +Rt−1 Bt−1 +qtk [Kt − (1 − δ) Kt−1 ]+qt [Lt − Lt−1 ]+wt Nt + ω ¯ Vt ≤ Yt +Bt +πt
45
(λt ) (28)
with λt the Lagrange multiplier on equation (28) and qtk the price of capital in consumption units. The collateral constraint now includes capital k Kt + Et [qt+1 ] Lt Bt + ωVt ≤ m Et qt+1 The firm’s program is W(ΩFt ) =
max
CtF ,Lt ,Bt ,Vt ,Nt ,Kt
U CtF + βEt W(ΩFt+1 )
¯ Vt −CtF − Rt−1 Bt−1 − qt [Lt − Lt−1 ] − wt Nt − ω −qtk [Kt − (1 − δ) Kt−1 ] + Yt (At , Kt−1 , Lt , Nt ) + Bt k −Bt − ω ¯ Vt + mEt qt+1 Kt + mEt [qt+1 ] Lt −Nt + (1 − s)Nt−1 + Φt Vt 2 It −Kt + (1 − δ) Kt−1 + It + Ω2 It−1 − 1 It
s.t.
(29)
= 0 (λFt ) = 0 (λFt ϕt ) = 0 (ξt ) = 0 (λK t )
Let us define the shadow price of capital in consumption units qtk =
λK t λFt
then the FOCs with respect to Kt is qtk
= βEt
λFt+1 λFt
k Yt+1 k + qt+1 (1 − δ) + ϕt mEt qt+1 (1 − α) (1 − φ) Kt
and the choice of investment is such that " " 2 # 2 # F λ Ω I I I I I t+1 t t t t+1 1 = qtk 1 − + βEt t+1 qk Ω −1 − Ω −1 −1 2 It−1 It−1 It−1 It It λFt t+1 Calibration.
The capital adjustment cost parameter Ω = 0.1881 and φ = 0.0695 are set
to the estimated values found in Liu, Wang & Zha (2013). δ = 0.025 as in Liu, Wang & Zha (2013). α = 0.805 is set so as to mimic the capital-output ratio (4.15 in the US data, as reported in Liu, Wang & Zha (2013)). m = 0.123 is chosen to match the benchmark value of B/Y = 0.59. Finally, the standard deviation of technological innovation σA is adjusted to match the observed standard deviation of output. This calibration is used in columns (3) and (5) of Table 7.
46
C.3
Model with capital and wage bill in the collateral constraint
Model.
As previously, the behaviors of the households do not change. But as in Liu,
Wang & Zha (2013), we introduce wage payment in the borrowing constraint. The wage bill needs to be financed by working capital such that the collateral constraint becomes k Bt + ωVt + wt Nt ≤ m Et qt+1 Kt + Et [qt+1 ] Lt The job creation curve becomes (1 + ϕt ) ∂Yt ω ¯ F (1 + ϕt+1 ) ω ¯ = − wt (1 + ϕt ) + (1 − s)β F Et λt+1 Φt ∂Nt Φt+1 λt The Nash bargaining is also altered such that the wage curve becomes:
wt = (bt + Γ) +
Calibration.
(1 − ) ∂Yt + (1 − s) ω ¯ (1 − ) (1 + ϕt ) ∂Nt
λF t+1 (1+ϕt+1 ) F λt Φt+1 (Ψt+1 −1) +µEt λλt+1 Φt+1 t
β E (1+ϕt ) t
We consider the same calibration as in section C.2, except for m = 0.28
that is adjusted again to match the benchmark value of B/Y = 0.59. α = 0.81 is set so as to mimic the capital-output ratio (4.15 in the US data, as reported in Liu, Wang & Zha (2013)). Finally, the standard deviation of technological innovation σA is adjusted to match the observed standard deviation of output. This calibration is used in column (4) of Table 7.
C.4
D
Business-cycle properties
Understanding the welfare effect of financial shocks using Mickey Mouse models
In this section, we show in Mickey Mouse models that financial shocks actually decrease welfare costs of fluctuations in a model with land only (section D.1) This provides a rationale for the quantitative results found in columns (1) and (2) in Table 4. Financial shocks actually increase business-cycle costs in a model with capital (section D.2), which explains the quantitative results found in columns (3) and (5) in Table 4
47
Table 7: Business-cycle volatility : Models versus data (0) Data
(1) Benchmark
(2) Financial shocks
(3) Capital
(4) Capital and wage
(5) Financial shocks and capital
0.0031
0.0024
0.0066
0.0071
0.0067
std(.)
std(.)
std(.)
std(.)
std(.)
σA std(.) Y
1.44
**
1.44
**
1.44
**
1.44
**
1.44
**
1.44
**
C N Y /N w U Ψ V B q R I
0.81 0.72 0.54 0.62 7.90 5.46 9.96 1.68 3.21 0.92 4.59
* * * * * * * * * * *
0.88 0.74 0.28 0.49 5.45 6.26 12.7 1.35 2.59 0.32
* * * * * * * * * * *
0.85 0.86 0.21 0.50 6.30 7.50 15.5 1.63 2.83 0.38
* * * * * * * * * * *
0.80 0.65 0.52 0.35 4.81 12.2 24.1 0.27 0.53 0.25 2.52
* * * * * * * * * * *
0.76 0.61 0.57 0.31 4.51 11.3 22.3 0.91 0.64 0.09 3.00
* * * * * * * * * * *
0.82 0.65 0.52 0.36 4.80 12.5 24.5 1.03 0.89 0.64 2.96
* * * * * * * * * * *
corr(U, Ψ) -0.91 -0.86 corr(U, V ) -0.97 -0.71 ** std (in percentage); * relative to GDP std
D.1
-0.86 -0.72
-0.54 -0.40
-0.55 -0.40
-0.54 -0.39
A simple model with land : Financial shocks decrease welfare costs
D.1.1
Household
The dynamic problem of a typical household can be written as follows H W(ΩH s.c. Ct + Bt ≤ Rt−1 Bt−1 + wt t ) = max U (Ct ) + µEt W(Ωt+1 ) Ct ,Bt
given the initial conditions on state variables B0 and ΩH t = {wt , Bt−1 }, the vector of variables taken as given by households. Let λt be the shadow price of the budget constraint. The FOC are (Ct )−σ = λt and −λt + µEt [Rt λt+1 ] = 0. The labor supply is Nt = 1.
48
D.1.2
Entrepreneur
The firm’s program is W(ΩFt ) = s.t.
max U CtF + βEt W(ΩFt+1 ) CtF ,Lt ,Bt ( −CtF − Rt−1 Bt−1 − qt [Lt − Lt−1 ] − wt Nt + Yt (At , Lt−1 , Nt ) + Bt = 0 (λFt ) −Bt + mt Et [qt+1 Lt ] = 0 (λFt ϕt )
given the initial conditions B0 , where ΩFt = {wt , Bt−1 , Lt−1 , mt } is the vector of variables taken as given by firms. Letting λFt and λFt ϕt be the Lagrange multipliers associated to each constraint,the first order conditions of problem read: U 0 CtF
λFt qt
= λFt = βEt
λFt+1
∂Yt+1 qt+1 + + λFt ϕt mt Et [qt+1 ] ∂Lt
(1 − ϕt )λFt = βEt λFt+1 Rt ∂Yt wt = ∂Nt D.1.3
Equilibrium
Given that Nt = 1 and Lt = 1 ∀t and assuming that Yt = At L1−α Ntα = At , we deduce t
1 = qt = 1 =
h
Dynamic system i
Rt µEt λλt+1 h Ft λt+1 βEt λF (qt+1 + (1 h Ft i λ Rt + ϕt βEt λt+1 F t
Steady state 1 = µR
i − α)At+1 ) + ϕt mt Et [qt+1 ]
wt = αAt
1 =
β µ
C =
At + Bt = CtF + Rt−1 Bt−1 + wt
+ϕ + αA 1−µ B µ
B = mq
How does B change with m? This depends on mq = 00 > 0 whereas Fmm =
1−µ B µ
C F = (1 − α)A −
Bt = mt Et [qt+1 ]
β(1−α)(1−β) (1−β−ϕm)2
β(1−α) A 1−β−ϕm
w = αA
Ct + Bt = Rt−1 Bt−1 + wt
Fm0 =
q =
2ϕβ(1−α)(1−β) (1−β−ϕm)3
mβ(1−α) A 1−β−ϕm
≡ F (m, A). We have
> 0. Hence, this function is convex
implying that E[mq] > mq, which means that expected debt is larger than steady state debt: E[B] > B. Given the worker’s budget constraint (C =
49
1−µ B µ
+ αA), we have E[C] >
C: expected consumption is larger than steady state consumption. The level effect on consumption actually implies that financial shocks in an economy with land are actually welfare-improving. Uncertainty generates a premium on the price of land as its return is risky for the bank. In this case, the value of worker’s wealth increases on average, which explains the larger consumption in the stochastic economy, with respect to the stabilized economy.
How does B change with A? Given that F is linear in A, we deduce that E[mq] = mq, hence E[B] = B and thus E[C] = C. Technological shocks are neutral on the level effect of the welfare costs of the business cycle. This last result shows that welfare costs provided by our benchmark model are the result of the interaction between labor market and financial frictions, given that, without labor market frictions, the technological shocks are not costly in an economy with only financial constraints.
D.2
A simple model with capital : Financial shocks increase welfare costs
D.2.1
Individual behaviors
The household’s program does not change. The firm’s program becomes W(ΩFt ) = s.t.
max U CtF + βEt W(ΩFt+1 ) CtF ,Bt ,Kt ( 1−α α −CtF − Rt−1 Bt−1 − wt Nt − Kt + At Kt−1 Nt + Bt = 0 (λFt ) −Bt + mt Kt = 0 (λFt ϕt )
The FOCs are U 0 CtF
λFt (1 − ϕt )λFt wt
= λFt F ∂Yt+1 + λFt ϕt mt = βEt λt+1 ∂Kt = βEt λFt+1 Rt ∂Yt = ∂Nt
50
D.2.2
Equilibrium
Given that Nt = 1 ∀t, we deduce Dynamic isystem h Rt 1 = µEt λλt+1 t h F i λt+1 −α 1 = βEt λF (1 − α)At+1 Kt + ϕt mt h Ft i λt+1 1 = βEt λF Rt + ϕt
Steady state 1 = µR α1 β(1−α) K = A 1−ϕm 1 =
t
wt = αAt Kt1−α
β µ
+ϕ
w = αAK 1−α
Ct + Bt = Rt−1 Bt−1 + wt
C =
1−α At Kt−1 + Bt = CtF + Rt−1 Bt−1 + Kt + wt
Bt = mt Kt
1−µ B µ
+ αAK 1−α
C F = (1 − α)AK 1−α −
1−µ B µ
−K
B = mK 1
1
The steady state, conditional to {A, m}, gives K = K(m, A) = (β(1−α)A) α (1−ϕm)− α and B = B(m, A) = mK(m, A). Assume a first restriction, which is satisfied in our calibration exercises, namely C F ≈ 0 ie.
1−µ B µ
≈ (1 − α)AK 1−α − K. We deduce that consumption C
is given by C ≈ AK(m, A)1−α − K(m, A), ( with
0 Km (m, A) = 00 (m, A) = Kmm
ϕ 1 K(m, A) > 0 α 1−ϕm 1+α ϕ K0 (m, A) > α 1−ϕm m
0
This leads to ϕ 1 0 C 00 = Km (m, A) 1−ϕm [(1 − α)AK(m, A)−α − (1 + α)] α ϕ 1 0 Given that Km (m, A) 1−ϕm > 0, C 00 has the same sign as the term between brackets, which α
consists of 2 terms. The first term (1 − α)AK(m, A)−α < 1 because it represents an interest rate, and the second term 1 + α > 1. This implies that C 00 < 0, hence E[C] < C when the uncertainty comes from financial shocks. This shows that fluctuations in m reduce welfare.
Why do changes in m increase welfare costs when the collateral includes capital? In presence of financial shocks, the return on the collateral becomes risky for banks. This uncertainty generates a premium on the borrowing constraint (an over-accumulation). This generates a new motive to increase leveraging (B increases in the volatile economy relative to the stabilized economy) and thus, capital. As capital in our economy is characterized by decreasing returns to scale, this over-accumulation is then costly in terms of consumption.
51
