Agency business cycles

Golosov, Michail Ju.; Menzio, Guido Article Agency business cycles Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Golosov, Michail Ju.; Menzio, Guido (2020) : Agency business cycles, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 15, Iss. 1, pp. 123-158, https://doi.org/10.3982/TE3379 This Version is available at: https://hdl.handle.net/10419/217101 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 15 (2020), 123–158 1555-7561/20200123 Agency business cycles Mikhail Golosov Department of Economics, University of Chicago and NBER Guido Menzio Department of Economics, New York University and NBER We develop a theory of endogenous and stochastic fluctuations in economic activity. Individual firms choose to randomize over firing or keeping workers who performed poorly in the past to give them an ex ante incentive to exert effort. Different firms choose to correlate the outcome of their randomization to reduce the probability with which they fire nonperforming workers. Correlated randomization leads to aggregate fluctuations. Aggregate fluctuations are endogenous— they emerge because firms choose to randomize and they choose to randomize in a correlated fashion—and they are stochastic—they are the manifestation of a randomization process. The hallmark of a theory of endogenous and stochastic fluctuations is that the stochastic process for aggregate “shocks” is an equilibrium object. Keywords. Endogenous and stochastic cycles, coordinated randomization, unemployment fluctuations. JEL classification. D86, E24, E32. 1. Introduction What causes cyclical fluctuations in economic activity? This is a central question in macroeconomics and, naturally, it has attracted a great deal of attention from both theorists and empiricists. One explanation for cyclical fluctuations is that the economy is subject to aggregate shocks to its fundamentals (see, e.g., Kydland and Prescott 1982). A second explanation for cyclical fluctuations is that the economic system admits multiple equilibria and there are switches in the equilibrium played by market participants Mikhail Golosov: [email protected] Guido Menzio: [email protected] We are grateful to Gadi Barlevy, Paul Beaudry, Jess Benhabib, Katarina Borovickova, Veronica Guerrieri, Christian Haefke, Boyan Jovanovic, John Kennan, Narayana Kocherlakota, Ricardo Lagos, Rasmus Lentz, Igor Livschits, Paolo Martellini, Nicola Pavoni, Thijs van Rens, Guillaume Rocheteau, Tom Sargent, Karl Shell, Ben Schoefer, Rob Shimer, Mathieu Taschereau-Dumouchel, and Randy Wright for comments on earlier drafts of the paper. We are also grateful to participants at seminars at the University of Wisconsin Madison, New York University, University of Chicago, University of California Berkeley, CREI, LSE, New York FRB, Minneapolis FRB, and at the Search and Matching European Conference in Aix-en-Provence, the Search and Matching Workshop at the Philadelphia FRB, the conference in honor of Chris Pissarides at Sciences Po, the Econometric Society World Congress in Montreal, and the NBER EFG Meeting at the New York FRB. ©2020 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://econtheory.org.https://doi.org/10.3982/TE3379 124 Golosov and Menzio Theoretical Economics 15 (2020) (see, e.g., Benhabib and Farmer 1994). According to these two theories, cyclical fluctuations are exogenous and stochastic, in the sense that they are driven by an exogenously given stochastic process for the shocks to fundamentals or to the selection of equilibrium. The main criticism to these theories is that they leave the driving force of business cycles completely unexplained. A different view of cyclical fluctuations is that the economic system does not tend toward stasis—where the extent of economic activity remains constant over time—but it naturally oscillates between periods of high and low activity (see, e.g., Benhabib and Day 1982). According to this theory, cyclical fluctuations are endogenous and deterministic. The main criticism to this theory is that business cycles do not appear to follow a deterministic pattern. In this paper, we develop a theory of endogenous and stochastic business cycles. The structure of our theory is simple: individual agents find it optimal to randomize over some choice in order to overcome a nonconvexity in their decision problem, and different agents find it optimal to correlate the outcome of their randomization. Aggregate fluctuations are endogenous because they are an equilibrium outcome: individual agents choose to randomize over some decision (which endogenously creates individual uncertainty) and they choose to correlate the outcome of their randomization (which endogenously generates aggregate uncertainty from individual uncertainty). Aggregate fluctuations are stochastic because they are the manifestation of aggregate uncertainty. The distinguishing feature of our theory—and more generally the hallmark of a theory of endogenous and stochastic fluctuations—is that the stochastic process for aggregate “shocks” is endogenous. Therefore, our theory makes predictions about the economies in which shocks will be frequent or infrequent, large or small. While the structure of our theory is fairly general, we exemplify it in the context of a search-theoretic model of the labor market in the spirit of Pissarides (1985) and Mortensen and Pissarides (1994). We consider a market populated by risk-averse workers and risk-neutral firms. Unemployed workers and vacant firms come together through a frictional search process. Once matched, a firm-worker pair bargains over the terms of an employment contract and starts producing output. Production is subject to moral hazard—in the sense that the firm does not observe the effort of the worker but only output, which is a noisy measure of effort. The employment contract allocates the gains from trade between the firm and the worker and tries to overcome the moral hazard problem. In particular, the contract specifies the level of effort recommended to the worker, the wage paid to the worker, and the probability with which the worker is fired conditional on the realization of the worker’s output and possibly on the realization of a public sunspot. Two features are necessary to develop our theory of endogenous and stochastic fluctuations. First, we need firms to find it optimal to use a firing lottery. In the model, we obtain this feature by assuming that the firm pays the wage before it observes the worker’s output and that the firm and the worker renegotiate the terms of the employment contract every period. Under these assumptions, the firm can provide the worker with the incentive to exert effort only by randomizing over firing or keeping him in case he produces low output. Second, we need firms to find it optimal to correlate the outcomes of their firing lotteries. In the model, we obtain this feature by assuming that the Theoretical Economics 15 (2020) Agency business cycles 125 firm’s vacancy cost is convex, which in turn, implies that the cost of losing a job to a worker is decreasing in the unemployment rate. In the first part of the paper, we characterize the properties of the optimal employment contract. We show that the optimal contract is such that the worker is fired only when the realization of output is low and the state of the world (i.e., the realization of the sunspot) is one in which the gains from continued trade accruing to the worker are highest relative to those accruing to the firm. The result is intuitive. Firing is costly—as it destroys a valuable firm-worker relationship—but necessary—as it is the only way for the firm to give the worker an incentive to exert effort. When firing takes place, however, it is only the value of the relationship that would have accrued to the worker that provides incentives. The value that would have accrued to the firm is collateral damage. The optimal contract minimizes the collateral damage by loading the firing probability on the states of the world in which the worker’s gains from continued trade are highest relative to the firm’s. In other words, the optimal contract loads the firing probability on the states of the world where the cost to the worker from losing the job is highest relative to the cost to the firm from losing the worker. In the second part of the paper, we characterize the equilibrium relationship between firing and relative gains from trade.1We show that there exists a correlated equilibrium in which all firms fire their nonperforming workers for some realizations of the sunspot, and they all keep their nonperforming workers for the other realizations. The measure of realizations of the sunspot for which there is firing is uniquely pinned down by the workers’ incentive compatibility constraint. In this equilibrium, individual firms correlate the outcomes of their firing lottery. There is a simple logic behind this finding. Suppose that firms load up the firing probability on some states of the world. In those states of the world, the unemployment rate is higher, and because the vacancy cost is convex, the labor market tightness is lower and so is the probability with which unemployed workers find jobs. Since the job-finding probability is lower, workers have a weaker outside option when bargaining with firms and their wage is lower. Since the wage is lower, the marginal utility of income of workers (who are risk-averse) relative to the marginal utility of income of firms (who are risk-neutral) is higher, and per the Nash bargaining solution, the gains from trade accruing to the workers are high relative to those accruing to the firms. Thus, if the other firms load the firing probability on some states of the world, an individual firm finds it optimal to load the firing probability on the very same states of the world. In other words, firms want to correlate the outcome of the randomization between firing and keeping their nonperforming workers, and the sunspot allows them to achieve correlation. Alongside the correlated equilibrium described above, there exists an uncorrelated equilibrium in which firms ignore the realization of the sunspot and randomize over firing or keeping their nonperforming workers in an independent fashion. The uncorrelated equilibrium, though, is not robust. The uncorrelated equilibrium only exists because firms randomize simultaneously, and hence, they must rely on an inherently 1The characterization is done under the conjecture that the worker’s and firm’s gains from trade are strictly increasing in unemployment and that the equilibrium wage is strictly decreasing in unemployment. The conjecture holds for the calibrated version of the model. 126 Golosov and Menzio Theoretical Economics 15 (2020) meaningless signal (the sunspot) to achieve correlation. When a signal is inherently meaningless, there is always an equilibrium in which the signal is ignored. We show that, in a version of the model where firms randomize sequentially, history can always be used to achieve correlation and the uncorrelated equilibrium disappears. Specifically, firms who act later always find it optimal to correlate the outcome of their randomization to the randomization outcome of the firms who move first.2 In the correlated equilibrium, the economy goes through aggregate fluctuations, which we dub Agency Business Cycles or ABC’s. ABC’s are endogenous. They are not caused by exogenous shocks to current or future fundamentals nor by exogenous shocks to the equilibrium played by market participants. ABC’s are caused by correlated randomization, i.e., individual firms find it optimal to randomize on firing or keeping their nonperforming workers, and different firms find it optimal to correlate the outcomes of their randomizations. Correlation is achieved either through the sunspot (in the simultaneous version of the model) or through history (in the sequential version of the model). ABCs are stochastic. The economy does not follow a deterministic cycle, but a random process in which the probability of a correlated firing episode, and hence, a recession is endogenous. We show that the probability of a recession depends positively on the worker’s cost of effort and negatively on the worker’s cost of losing a job (which rises with unemployment). In the last part of the paper, we calibrate the theory to the US labor market. We show that, for some parameter values, ABCs feature fluctuations in unemployment, unemployment-to-employment (UE), and employment-to-unemployment (EU) rates that have the same magnitude and pattern of comovement as in the data. Moreover, ABCs generate fluctuations in unemployment, UE and EU rates that are, as in the data, uncorrelated with fluctuations in labor productivity. However, we conclude that ABCs— at least in the formulation developed in this paper—are not a complete explanation of labor market fluctuations because they feature a correlation between unemployment and vacancies that is counterfactually positive. The nature of recessions in ABC’s is quite different than in theories of business cycles where fluctuations are driven by aggregate productivity shocks (such as Real Business Cycles, or RBC’s). In RBC’s, recessions are times when productivity is unusually low and so are the gains from trade in the labor market. For this reason, workers and firms have weaker incentives to trade and unemployment is high. In ABC’s, recessions are states of the world in which the value of being unemployed to a worker is unusually low, and hence, the gains from trade in the labor market are high. In these states of the world, firms find it optimal to fire their nonperforming workers, and hence, unemployment is high. Overall, in RBC’s, the gains from trade move in the opposite direction as unemployment. In ABC’s, the gains from trade move in the same direction as unemployment. We show that, at least when measured from the perspective of a worker, the gains from trade are countercyclical. Our theory of business cycles is not a mere intellectual curiosity. There is empirical evidence consistent with the view that firms use firing as an incentive device. 2The sequential version of the model also shows that a sunspot is not necessary for our theory of aggregate fluctuations. Theoretical Economics 15 (2020) Agency business cycles 127 Cappelli and Chauvin (1991) examine the internal records of a large car manufacturing company. Exploiting geographical variation across plants, they establish a negative relationship between the plant’s wage relative to the average manufacturing wage in the plant’s area (which is a measure of the cost to the worker of losing the job) and the frequency of disciplinary dismissals. The finding suggests that the firm uses the threat of firing as an incentive device, that workers understand the threat, and that they adjust their effort according to the strength of the threat. Ichino and Riphahn (2005) examine days of absence per week for white-collar workers at a large Italian bank. During the first 12 weeks of their tenure, workers are in a probationary period and can be fired at will. Afterwards, workers enjoy strong employment protection. Ichino and Riphahn (2005) find that days of absence per week triple right after the end of the probationary period. That is, workers’ effort (as measured by absenteeism) varies depending on whether they can or cannot be fired at will. The finding suggests that workers expect the bank to use firing as part of its incentive scheme. There is also empirical evidence consistent with the mechanism behind our theory. Agarwal and Kolev (2016) show that Fortune 500 companies tend to cluster mass layoffs within a few days of each other, even though mass layoffs are a relatively infrequent event. Interestingly, they find that an announcement of mass layoffs by one of the top 20 firms is positively related with announcements of mass layoffs by other Fortune 500 firms in the five following business days, while it is uncorrelated with mass layoffs in the five previous business days. The asymmetry suggests that firms are not being hit by a common shock. As in our theory, it could be that smaller firms use the layoff decisions of larger firms as a coordination device. Similar clustering seems to take place at the top of organizations as well. Jenter and Kanaan (2015) find that CEOs who underperform the industry average are much more likely to be fired when the industry-wide performance is poor, even though one would imagine that only a CEO’s relative performance is informative about effort. The finding implies that the firing of nonperforming CEOs is clustered during downturns. The main contribution of the paper is to develop a new theory of endogenous and stochastic business cycles. We show that endogenous and stochastic cycles emerge in equilibrium when individual agents want to randomize over some economic decision and different agents find it optimal to correlate the outcome of their randomization. The hallmark of a theory of endogenous and stochastic business cycles is a stochastic process for aggregate “shocks” that is determined endogenously. Conceptually, a theory of endogenous and stochastic cycles is useful because it explains why the aggregate economy is subject to shocks, rather than simply assuming the existence of shocks. This implies that the theory has something to say about what determines the frequency of shocks, the magnitude of shocks, and what policies may affect the stochastic process of shocks. Empirically, a theory of endogenous and stochastic business cycles is useful because it helps explain why economic activity seems (at first blush) to be more volatile than its fundamentals, and it does so without resorting to unobserved shocks to equilibrium selection. Intellectually, a theory of endogenous and stochastic cycles adds to the class of theories that we can use to understand macroeconomic fluctuations. Some of the exist- 128 Golosov and Menzio Theoretical Economics 15 (2020) ing theories of aggregate fluctuations are based on exogenous shocks to fundamentals. These can be shocks to the current value of economy-wide fundamentals (e.g., Kydland and Prescott 1982 or Mortensen and Pissarides 1994), to the future value of fundamentals (e.g., Beaudry and Portier 2004 or Jaimovich and Rebelo 2009), to the stochastic process of fundamentals (e.g., Bloom 2009), or to higher-order beliefs (e.g., Angeletos and La’O 2013). Relatedly, there are granular theories of business cycles, in which aggregate fluctuations are driven by shocks to the fundamentals of individual agents who are large enough or connected enough to others to cause aggregate swings in economic activity (e.g., Jovanovic 1987 or Gabaix 2011). Other theories of business cycles are driven by exogenous shocks to equilibrium selection (e.g., Heller 1986,Cooper and John 1988, Benhabib and Farmer 1994,Kaplan and Menzio 2016). Finally, there are theories of endogenous and deterministic aggregate fluctuations, where the economy converges to a limit cycle (e.g., Benhabib and Nishimura 1979,Diamond 1982,Diamond and Fudenberg 1989,Benhabib and Rustichini 1990,Mortensen 1999,orBeaudry et al. 2015)or follows chaotic dynamics (e.g., Benhabib and Day 1982,Boldrin and Montrucchio 1986 or Boldrin and Woodford 1990). The only other theory of endogenous and stochastic business cycles of which we are aware is Benhabib et al. (2015). Their theory shares with ours the fact that the probability distribution of aggregate “shocks” is an equilibrium object. However, the mechanism leading to endogenous, stochastic cycles is different from ours, as it builds on a signal extraction problem. The particular illustration of our theory contributes to the literature on labor market fluctuations. Shimer (2005) showed that the basic search-theoretic model of the labor market implies very small fluctuations in unemployment in response to the observed fluctuations in labor productivity. Building on this observation, many papers have identified channels through which labor productivity shocks can lead to sizeable movements in unemployment (e.g., wage rigidity in Hall 2005,Menzio 2005,Kennan 2010,Menzio and Moen 2010, small gap between home productivity and market productivity in Hagedorn and Manovskii 2008, match heterogeneity in Menzio and Shi 2011). These papers typically generate a perfect negative correlation between labor productivity and unemployment. Yet, since 1984, this correlation has vanished. Recent work has thus focused on identifying different sources of unemployment fluctuations (e.g., Farmer 2013,Galí and van Rens 2014,Kaplan and Menzio 2016,Beaudry et al. 2015,Hall 2017). Our model offers a novel explanation for why unemployment is so volatile and why its volatility is uncorrelated with productivity. A distinguishing feature of our explanation relative to others is that it implies a positive correlation between the net value of employment and unemployment. We find evidence of this positive correlation in the data. Finally, let us briefly relate our paper to Shapiro and Stiglitz (1984). Shapiro and Stiglitz (1984) showed that the existence of a moral hazard problem between firms and workers generates unemployment in a frictionless labor market. Our paper also features a moral hazard problem between firms and workers, but the focus of our paper is not on explaining the existence of unemployment—which is caused by search frictions—but the volatility of unemployment. Theoretical Economics 15 (2020) Agency business cycles 129 2. Environment and equilibrium In this section, we describe the physical and contractual environment of the model and derive the conditions for a recursive equilibrium. 2.1 Environment Time is discrete and continues forever. The economy is populated by a measure 1of identical workers. Every worker has preferences described by the expected sum of current and future periodical utilities discounted at the factor β∈(01).Whenaworkeris unemployed, his periodical utility is given by υ(b) +ζ,whereυ(·)is a strictly increasing, strictly concave function of consumption, bis the worker’s unemployment income, and ζis the worker’s utility from leisure. When a worker is employed, his periodical utility is given by υ(wt)−ψet,wherewtis the worker’s labor income,3and ψetis the worker’s disutility from putting effort on the job, where ψ>0and et∈{01}. The economy is also populated by a measure 1of identical firms. Every firm has preferences described by the expected sum of current and future periodical profits, discounted at the factor β. Every firm operates a constant returns to scale production technology that transforms one unit of labor (i.e., one employee) into ytunits of output, where ytis a random variable that depends on the employee’s effort et.Inparticular, yttakes the value yhwith probability ph(e) and the value ywith probability p(e) =1−ph(e),withyh>y ≥0and 0<p h(0)<p h(1)<1. Production suffers from moral hazard, in the sense that the firm does not directly observe the effort of its employee, but only the output. Every period tis divided into five stages: sunspot, separation, matching, bargaining, and production. At the first stage, a random variable, zt, is drawn from a uniform distribution with support [01].4The random variable is aggregate, in the sense that it is publicly observed by all market participants. The random variable is a sunspot, in the sense that it does not directly affect technology, preferences, or any other fundamentals, although it may help correlate the outcome of the lotteries played by different market participants. At the separation stage, some employed workers become unemployed. An employed worker becomes unemployed for exogenous reasons with probability δ∈(01).Inaddition, an employed worker becomes unemployed because he is fired with probability s(yt−1zt),wheres(yt−1zt)is determined by the worker’s employment contract and it is 3As the reader can infer from the notation, we assume that workers are hand-to-mouth, in the sense that they consume their income in every period. We discuss the robustness of our theory to this assumption in the conclusions. 4Assuming that the sunspot is an i.i.d. draw from a uniform with support [01]is without loss of generality. In fact, as the sunspot does not directly affect preferences or technology, an equilibrium of a model where the sunspot is drawn from some arbitrary CDF can always be represented as an equilibrium of the model where the sunspot is drawn from a uniform distribution. Moreover, in any equilibrium in which firms perfectly correlate their randomization (which is the “robust” equilibrium of the model), the probability of the different aggregate events is uniquely pinned down. Therefore, the same distribution of aggregates would emerge if the sunspot was i.i.d. or autocorrelated. 130 Golosov and Menzio Theoretical Economics 15 (2020) allowed to depend on the output of the worker in the previous period, yt−1, and on realization of the sunspot in the current period, zt. For the sake of simplicity, we assume that a worker who becomes unemployed in period tcan search for a new job only starting in period t+1. At the matching stage, some unemployed workers become employed. Firms decide how many vacancies vtto create at the unit cost k(vt),wherek(·)is a strictly increasing function such that k(0)=0. Then the ut−1workers who were unemployed at the beginning of the period search for the vtvacancies created by firms. The outcome of the search process is described by a constant return to scale matching function, M(ut−1vt), which gives the measure of bilateral matches formed between unemployed workers and vacancies. Hence, the probability that an unemployed worker meets a vacancy is λ(θt)≡M(1θt),whereθt≡vt/ut−1is the tightness of the labor market and λ(·) is a strictly increasing and concave function such that λ(0)=0. The probability that a vacancy meets an unemployment worker is η(θt)≡M(1/θt1),whereη(·)is a strictly decreasing function such that η(θ) =λ(θ)/θ. At the bargaining stage, new and continuing firm-worker pairs negotiate the terms of a one-period employment contract xt. The contract xtspecifies the effort etrecommended to the worker in the current period, the wage wtpaid by the firm to the worker in the current period, and the probability s(ytzt+1)with which the firm fires the worker at the next separation stage, conditional on the output of the worker in the current period and on the realization of the sunspot at the beginning of next period. We assume that the outcome of the bargain between the firm and the worker is the axiomatic Nash bargaining solution. At the production stage, an unemployed worker home-produces and consumes b units of output. An employed worker chooses an effort level, et, and consumes wtunits of output. Then the output of the worker, yt, is realized and observed by both the firm and the worker. A few comments about the environment are in order. We assume that employment contracts are short-term—in the sense that they can only specify effort, wages, and separation probabilities for one period before being renegotiated—and incomplete—in the sense that they cannot specify a wage that depends on the contemporaneous realization of output. Short-term, incomplete contracts imply that firms must use firing lotteries to provide incentives to their workers.5We assume that vacancy costs are convex, in the sense that the cost of an additional vacancy is increasing in the number of vacancies opened by the firm. Convex vacancy costs imply that the value of unemployment to an individual worker is decreasing in aggregate unemployment.6 5In the conclusions, we discuss the robustness of our theory to relaxing the assumption of short-term, incomplete employment contracts. 6In the textbook search-theoretic model of the labor market (see, e.g., Pissarides 1985 or Mortensen and Pissarides 1994), the vacancy cost is assumed to be linear. For this reason, the equilibrium is such that the value of unemployment to an individual worker is independent from aggregate unemployment. Empirical studies of the hiring behavior of firms (see, e.g., Gavazza et al. 2018) suggest that vacancy costs are convex. Theoretical Economics 15 (2020) Agency business cycles 137 optimal to correlate the randomization to economize on agency costs. The uncorrelated equilibrium exists because the only instrument that firms have to achieve correlation is an inherently meaningless sunspot that has meaning to an individual firm only to the extent that it has meaning to others. In the second part of this section, we argue that the only robust equilibrium is the correlated equilibrium. Specifically, in a version of the model in which firms fire sequentially (and thus history can act as a correlation device), the unique equilibrium is one with perfect correlation. In the last part of this section, we describe the key properties of the equilibrium dynamics. 4.1 Stage equilibrium In equilibrium, the firms’ firing probability s∗(yˆ z) and the workers’ relative gains from trade φ(ˆ z) must simultaneously satisfy two conditions. For any realization ˆ zof the sunspot, the firing probability s(ˆ z) =s∗(yˆ z) must be part of the optimal employment contract given that the workers’ relative gains from trade are φ(ˆ z) and the firing cutoff is φ∗-cutoff which depends on the whole distribution of the workers’ relative gains from trade across realizations of the sunspot. Moreover, for any realization ˆ zof the sunspot, the workers’ relative gains from trade φ(ˆ z) must be consistent with the optimal employment contracts that are negotiated next period given the firing probability s(ˆ z).Formally, in any equilibrium, the functions φ(ˆ z) and s(ˆ z) must be a fixed point of the mapping we just described. Borrowing language from game theory, we refer to such a fixed point as the stage equilibrium, as it describes the key equilibrium outcomes between the bargaining stage of the current period and the bargaining stage of the next period. We characterize the stage equilibrium under the conjectures that the optimal wage w∗(u) is strictly decreasing in unemployment, and that the worker’s and firm’s gains from trade V(u)and J(u) are strictly increasing in unemployment. These conjectures are natural. If unemployment is higher, the convexity of the vacancy cost implies that the job-finding probability of unemployed workers is lower, and so is the lifetime utility of unemployed workers. In turn, this implies that the gains from trade between workers and firms are higher, and through Nash bargaining, so are the gains from trade accruing to the workers and the firms, Vand J. Since the output of a firm-worker match is independent of unemployment, a higher Jrequires a lower wage. In Section 5,weverifythat the conjectures hold for the calibrated version of the model. We can now characterize the effect of the firms’ firing probability s(ˆ z) on the worker’s relative gains from trade φ(ˆ z). Given that the unemployment at the bargaining stage of the current period is uand that the firing probability at the separation stage of next period is s(ˆ z) for a realization of the sunspot ˆ z, it follows that unemployment at the bargaining stage of the next period is ˆ u(s(ˆ z)) given by ˆ us(ˆ z)=u−uλθJˆ us(ˆ z)u+(1−u)δ+(1−δ)p(1)s(ˆ z)(13) Under the conjecture that Jis a strictly increasing function, there exists a unique ˆ u(s(ˆ z)) that solves (13)and ˆ u(s(ˆ z)) is strictly increasing in s(ˆ z). At the bargaining stage of the next period, the optimal employment contract signed by workers and firms is such that φ(ˆ z) =υw∗ˆ us(ˆ z) 138 Golosov and Menzio Theoretical Economics 15 (2020) Figure 2. Stage equilibrium. Under the conjecture that w∗is a strictly decreasing function, the workers’ relative gains from trade φ(ˆ z) are strictly increasing in ˆ u(s(ˆ z)), and hence, strictly increasing in s(ˆ z). The solid red line in Figure 2 illustrates the effect of the firing probability s(ˆ z) on the worker’s relative gains from trade φ(ˆ z). Next, we characterize the effect of the workers’ relative gains from trade φ(ˆ z) on the firms’ firing probability s(ˆ z). From the characterization of the optimal contract in Theorem 1, it follows that s(ˆ z) is such that s(ˆ z) =⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0∀ˆ z∈[01]s.t. φ(ˆ z) < φ∗ ∈[01]∀ˆ z∈[01]s.t. φ(ˆ z) =φ∗ 1∀ˆ z∈[01]s.t. φ(ˆ z) > φ∗ where φ∗is implicitly defined by the worker’s incentive compatibility constraint (11). The dashed green line in Figure 2 illustrates the effect of the worker’s relative gains from trade φ(ˆ z) on the firing probability s(ˆ z). The two equilibrium conditions relating the firms’ firing probability s(ˆ z) and the workers’ relative gains from trade φ(ˆ z) are intuitive. The higher is s(ˆ z), the higher is unemployment at the next bargaining stage, the lower is the workers’ outside option when bargaining, and the lower is the wage. Since Nash bargaining equates the workers’ relative gains from trade to the worker’s relative marginal utility of the wage and workers are risk averse, it follows that a higher s(ˆ z) implies a higher φ(ˆ z). Conversely, the higher is φ(ˆ z), the lower is the firms’ cost of firing workers in state of the world ˆ z.Hence,a higher φ(ˆ z) implies a higher s(ˆ z). For any realization ˆ zof the sunspot, the firms’ firing probability s(ˆ z) must be optimal given the workers’ relative gains from trade φ(ˆ z) (i.e., we must be on the dashed green line) and the workers’ relative gains from trade must be consistent with the firms’ firing probability (i.e., we must be on the solid red line). As it is clear from Figure 2,forany realization of ˆ z, only three outcomes are possible: points A, B, and C. The first outcome, Theoretical Economics 15 (2020) Agency business cycles 139 point A, is such that the firms’ firing probability s(ˆ z) is 0and the workers’ relative gains from trade φ(ˆ z) are smaller than φ∗. The second outcome, point B, is such that the firms’ firing probability s(ˆ z) is interior and the workers’ relative gains from trade φ(ˆ z) are equal to φ∗. The third outcome, point C, is such that the firms’ firing probability s(ˆ z) is 1and the workers’ relative gains from trade φ(ˆ z) are greater than φ∗. Let ZAdenote the realizations of the sunspot for which firms fire nonperforming workers with probability 0,andletπAdenote the measure of ZA.LetZBdenote the realizations of the sunspot for which firms fire nonperforming workers with a probability s∗(yˆ z) =sB∈(01),andletπBdenote the measure of ZB. Similarly, let ZCdenote the realizations of the sunspot for which firms fire nonperforming workers with probability 1,andletπCdenote the measure of ZC. Depending on πB, we can identify three qualitatively different types of equilibria. If πB=1,wehaveanuncorrelated equilibrium, in which firms fire their nonperforming workers with probability sB∈(01)for all ˆ z∈[01]. That is, in an uncorrelated equilibrium, firms ignore the realization of the sunspot and randomize over keeping or firing their non-performing workers independently of each other. The firms’ firing probability sBis such that the workers’ incentive compatibility constraint holds with equality, i.e., sB=ψ βph(1)−ph(0)(1−δ)V ˆ u(sB)(14) If πB=0,wehaveacorrelated equilibrium. In this equilibrium, every firm fires its non-performing workers with probability 0whenever the realization of the sunspot is ˆ z∈ZA, and every firm fires its nonperforming workers with probability 1whenever the realization of the sunspot is ˆ z∈ZC,withZA∪ZC=[01]. That is, in a correlated Eequilibrium, firms use the sunspot to randomize over firing or keeping their nonperforming workers in a correlated fashion. The measures of ZAand ZCare not free, but must be such that the workers’ incentive compatibility constraint holds with equality. Specifically, πA=1−πCand πC=ψ βph(1)−ph(0)(1−δ)V ˆ u(1)(15) If πB∈(01),wehaveapartially correlated equilibrium. In this equilibrium, firms fire workers with probability sB∈(01)for all ˆ z∈ZB.Thatis,whenˆ z∈ZB,firmsrandomize over firing or keeping their nonperforming workers independently from each other. However, if ˆ z/∈ZB, firms fire their workers with probability 0if ˆ z∈ZAand with probability 1if ˆ z∈ZC.Thatis,whenˆ z/∈ZB, firms use the sunspot to randomize over firing or keeping their nonperforming workers in a correlated fashion. A partially correlated equilibrium is a mixture of an uncorrelated and a correlated equilibrium. Given πB and πC, the firms’ firing probability sBis such that the workers’ incentive compatibility constraint holds with equality, i.e., sB=ψ−βph(1)−ph(0)(1−δ)πCVˆ u(1) βph(1)−ph(0)(1−δ)πBVˆ u(sB)(16) The above results are summarized in Theorem 2. 140 Golosov and Menzio Theoretical Economics 15 (2020) Theorem 2 (Stage equilibrium). There exist 3types of stage equilibria: (i) A unique uncorrelated equilibrium: s(ˆ z) =sBfor all ˆ z∈ZB,withπB=1and sB given by (14); (ii) A unique correlated equilibrium: s(ˆ z) =0for all ˆ z∈ZAand s(ˆ z) =1for all ˆ z∈ZC, with πA=1−πCand πCgiven by (15); (iii) A double continuum of partially correlated equilibrium: s(ˆ z) =0for all ˆ z∈ZA, s(ˆ z) =sBfor all ˆ z∈ZB,ands(ˆ z) =1for all ˆ z∈ZC,withπA=1−πB−πC, πB∈(01),πC∈(01−πB],andsBgiven by (16). 4.2 Stage equilibrium refinement The correlated equilibrium exists because firms want to correlate the outcomes of the randomization over keeping and firing nonperforming workers because doing so allows them to minimize the agency cost of moral hazard. Moreover, firm are able to correlate the outcome of the randomization because they can all observe the sunspot. The uncorrelated equilibrium (as well as the partially correlated equilibria) exists because the sunspot is inherently meaningless, and hence, there is always an equilibrium in which firms ignore it. If firms do not need to rely on an inherently meaningless signal, the uncorrelated equilibrium (as well as the partially correlated equilibria) should disappear. In this subsection, we show that—if firms fire sequentially, and hence, can use history as a correlation device—then the unique equilibrium of the stage game is the correlated one. Here is a formal description of the stage game. Let 1−udenote the measure of employed workers at the bargaining stage of the current period. The measure of employed workers is equally divided into a large number N·Kfirms, each employing 1worker of “measure” (1−u)/NK. Firms are clustered into a large number Kof groups, each that comprise a large number Nof firms. Firms and workers bargain over the terms of the one-period employment contract knowing the group to which they belong. At the separation stage of next period, firm-worker pairs in different groups decide to break up or stay together sequentially. First, the firm-worker pairs in group 1 decide whether to separate or not. After observing the outcomes of group 1, the firm-worker pairs in group 2 decide whether to separate or not. The process continues until the firm-worker pairs in group Kdecide to separate or not. Let Tidenote the measure of workers who separate from firms in groups 1 through i. We assume that each firm in group itakes as given the probability distribution of Ti conditional on Ti−1that we denote as Pi(Ti|Ti−1). The assumption implies that each firm views itself as small compared to its group, which is reasonable for Nlarge. We also assume that Pi(Ti|Ti−1)is strictly increasing—in the sense of first-order stochastic dominance—with respect to Ti−1. The assumption implies that firms in group iview themselves as small compared to the whole economy, which is reasonable for Klarge. To keep the analysis simple, we approximate the worker’s gains from trade, V(u),and the firm’s gains from trade, J(u), with linear functions. The approximation implies that Theoretical Economics 15 (2020) Agency business cycles 141 the worker’s expected gains from trade relative to the firm’s are equal to the worker’s relative gains from trade evaluated at the expectation of next period’s unemployment, E[V(ˆ u)]/E[J(ˆ u)]=φ(E[ˆ u]). We now characterize the optimal firing probability for firms in different groups. For firms in groups i=23K, the firing probability si(y Ti−1)depends on the realization of the worker’s output yand on the measure Ti−1of workers separating from firms in groups 1 through i−1.AsinSection 3, we can show that the optimal firing probability is such that: (i) the worker’s incentive compatibility constraint holds with equality; (ii) when the realization of output is high, the worker is fired with probability 0, i.e., si(yhTi−1)=0for all Ti−1; (iii) when the realization of output is low, the worker is fired with probability 0if the relative gains from trade are below a cutoff φ∗ i,andwith probability 1if they are above the cutoff, i.e., si(yTi−1)=0if φ(E[ˆ u|Ti−1])<φ ∗ i,and si(yTi−1)=1if φ(E[ˆ u|Ti−1])>φ ∗ i. Under the same conjectures about V,J,andw made in Section 4.1,φ(E[ˆ u|Ti−1])is strictly increasing in E[ˆ u|Ti−1]. For firms in group 1, the firing probability can only depend on the realization of the worker’s output y.Hence, the optimal contract is such that s1(yh)=0and s1(y)=s1,wheres1is such that the worker’s incentive compatibility constraint holds with equality. Firms in group Kunderstand that E[ˆ u|TK−1]is strictly increasing in TK−1,as PK(TK|TK−1)is strictly increasing in TK−1and unemployment is strictly increasing in TK. Therefore, there exists a cutoff T∗ K−1such that firms in group Kfire their nonperforming workers with probability 0if TK−1<T∗ K−1and with probability 1if TK−1> T∗ K−1. Firms in group K−1understand that E[ˆ u|TK−2]is strictly increasing in TK−2,as PK(TK−1|TK−2)is strictly increasing in TK−2and the firing probability of firms in group Kis increasing in TK−1. Hence, there exists a cutoff T∗ K−2such that firms in group K−1 fire their nonperforming workers with probability 0if TK−2<T∗ K−2and with probability 1if TK−2>T∗ K−2. The same reasoning implies that there exists a cutoff T∗ i−1for firms in all groups i=23K. Next, we compute the probability distribution of the measure tiof workers who separate from firms in group iconditional on Ti−1. A worker separates from a firm in group 1 with probability τ1=δ+(1−δ)p(1)s1. Since Nis large, we can apply the central limit theorem to approximate t1as a Normal with mean τ1(1−u)/K and variance τ1(1−τ1)[(i −u)/K]2/N. Conditional on Ti−1, workers separate from firms in group i= 23Kwith probability τi=δif Ti−1<T∗ i−1and with probability τi=δ+(1−δ)p(1) if Ti−1>T∗ i−1. Again, we can approximate tias a Normal with mean τi(1−u)/K and variance τi(1−τi)[(i −u)/K]2/N. We find it convenient to define tas δ(1−u)/K,and thas [δ+(1−δ)p(1)](1−u)/K. Lastly, we compare the incentive compatibility constraint for workers employed by firms in groups 2 and 3 to characterize the equilibrium for N→∞. The incentive compatibility constraint for workers in firms of group 2 is ψ=β(1−δ)ph(1)−p(1)PrT1>T∗ 1VEˆ u|T1>T∗ 1(17) The incentive compatibility constraint for workers in firms of group 3 is ψ=β(1−δ)ph(1)−p(1)PrT2>T∗ 2VEˆ u|T2>T∗ 2(18) 142 Golosov and Menzio Theoretical Economics 15 (2020) For N→∞,T2is approximately equal to τ1(1−u)/K +tif T1<T∗ 1and approximately equal to τ1(1−u)/K +thif T1>T∗ 1. Suppose that Pr(T2>T∗ 2)>Pr(T1>T∗ 1). By the reservation property of the firing probability of firms in group 3, Pr(T2>T∗ 2)>Pr(T1>T∗ 1)implies that s3(yT2)=1for all realizations of T1such that T1>T∗ 1, realizations that induce T2=τ1(1−u)/K +th, and also for some of the realizations of T1such that T1<T∗ 1, realizations that induce T2=τ1(1−u)/K +t. In light of these observations, we can rewrite (18)as ψ=β(1−δ)ph(1)−ph(0) ·PrT1>T∗ 1VEˆ u|T2=τ1(1−u)/K +th +PrT1<T∗ 1∨T2>T∗ 2VEˆ u|T2=τ1(1−u)/K +t(19) However, the first term on the right-hand side of (19)ispreciselythebenefitofexerting effort to a worker employed by a firm in group 2 and, by (17), it must equal to the cost ψ. Therefore, (17)and(19) can simultaneously hold only if Pr(T2>T∗ 2)≤Pr(T1>T∗ 1). Now, suppose that Pr(T2>T ∗ 2)<Pr(T1>T ∗ 1). By the reservation property of the firing probability of firms in group 3, Pr(T2>T∗ 2)<Pr(T1>T∗ 1)implies that s3(yT2)= 0for all realizations of T1such that T1<T∗ 1, realizations that induce T2=τ1(1−u)/K+t, and also for some realizations of T1such that T1>T ∗ 1, realizations that induce T2= τ1(1−u)/K +th. In light of these observations, we can rewrite (17)as ψ=β(1−δ)ph(1)−ph(0) ·PrT1>T∗ 1∨T2>T∗ 2VEˆ u|T2=τ1(1−u)/K +th +PrT1>T∗ 1∨T2<T∗ 2VEˆ u|T2=τ1(1−u)/K +th(20) However, the first term on the right-hand side of (20)ispreciselythebenefitofexerting effort to a worker employed by a firm in group 3 and, by (18), it must equal to the cost ψ. Therefore, (17)and(18) can simultaneously hold only if Pr(T2>T∗ 2)≥Pr(T1>T∗ 1). Combining the above observations, it follows that any equilibrium is such that Pr(T2>T∗ 2)=Pr(T1>T∗ 1). And, by repeating the same argument, it follows that any equilibrium is such that Pr(Ti>T ∗ i)=Pr(T1>T ∗ 1)for i=23K.Hence,inany equilibrium, if T1>T∗ 1, then all firms in the following groups fire their nonperforming workers with probability 1.IfT1<T∗ 1, then all firms in the following groups fire with probability 0. It is then straightforward to show that such an equilibrium does exist. We have thus established the following. Theorem 3 (Equilibrium refinement). For N→∞and K→∞, the unique equilibrium of the stage game with Kgroups of Nfirms firing sequentially is the correlated equilibrium. Theorem 3 is relevant for three reasons. First, as we have already discussed, the theorem shows that—if firms can use history and not only an inherently meaningless signal to coordinate their behavior—the unique equilibrium of the stage game is the correlated equilibrium. Second, the theorem shows that our theory does not rely on the Theoretical Economics 15 (2020) Agency business cycles 143 existence of a sunspot (in contrast with business cycle theories based on equilibrium indeterminacy). Third, the theorem exemplifies how firms may coordinate their behavior in practice. While a sunspot is a convenient theoretical construct to allow coordination, it does not have a clear empirical counterpart. What is the sunspot that firms use in the real world? The theorem suggests that, in practice, firms may achieve coordination by looking at the firing decisions of focal firms (e.g., industry leaders, large firms, etc.). 4.3 Agency business cycles Having established that the unique robust equilibrium of the stage game is the correlated equilibrium, we can now proceed to characterize some of the key properties of the dynamic equilibrium. A recursive equilibrium features aggregate uncertainty.8Given a current unemployment rate of u, firms fire their nonperforming workers for all realizations of the sunspot ˆ z∈ZC, an event which occurs with probability πCgiven by (15), and firms keep their nonperforming workers for realizations of the sunspot ˆ z∈ZA,anevent which occurs with probability πA=1−πC. In the first case, the fraction of employed workers who become unemployed is δ+(1−δ)p(1)and the unemployment rate goes to h(uZC). In the second case, the fraction of employed workers who become unemployed is δand the unemployment rate goes to h(uZA). Since h(uZC)>h(uZ A),the equilibrium features aggregate uncertainty. Aggregate uncertainty causes aggregate unemployment fluctuations, which we dub Agency Business Cycles (or ABC’s). These unemployment cycles are illustrated in Figure 3. Imagine an economy in which the unemployment rate is u0. As long as the realization of the sunspot falls in ZA, firms keep their nonperforming workers, the rate at which employed workers lose their job is δ, and the unemployment rate falls toward u∗.When the realization of the sunspot falls in ZC, firms fire their nonperforming workers, the rate at which employed workers lose their job is δ+(1−δ)p(1), and the unemployment rate goes back up. The unemployment rate starts falling again toward u∗when the realization of the sunspot returns in ZA. Agency business cycles are endogenous. ABC’s are not caused by changes in fundamentals, which remain constant over the cycle. ABC’s are not caused by changes in expectations about fundamentals in the future, as these expectations remain constant over the cycle. ABC’s are not caused by changes in the selection of equilibrium. In fact, the same stage equilibrium is played throughout the cycle. Moreover, while the unique stage equilibrium features randomization over two possible outcomes (e.g., firing and keeping nonperforming workers), these two outcomes are not equilibria on their own but only if properly mixed. Indeed, keeping nonperforming workers with probability 0 is not an equilibrium as this violates the workers’ incentive compatibility constraint. Firing nonperforming workers with probability 1is not an equilibrium as doing so would 8In an earlier version of the paper Golosov and Menzio (2015), we established the existence of a recursive equilibrium that satisfies the conjectures that we made in Section 4.1. The sufficient conditions for existence were quite stringent and not particularly informative, and for these reasons, here we simply check the existence of a recursive equilibrium and verify the conjectures for the calibrated version of the model. 144 Golosov and Menzio Theoretical Economics 15 (2020) Figure 3. Agency business cycles. be suboptimal. ABC’s are endogenous because the unique equilibrium of the stage game features correlated randomization. Agency business cycles are stochastic. ABC’s are driven by correlated firing episodes, which occur randomly. Since ABC’s are endogenous, the probability of a correlated firing episode is not determined outside of the model by some free parameter, but it is determined in equilibrium. Specifically, the probability of a firing episode is given by πC=ψ β(1−δ)ph(1)−ph(0)Vh(uZC)(21) The probability πCof a firing episode (and hence, of a recession) depends on the worker’s cost of exerting effort, ψ, the difference between the probability of a positive realization of output when the worker does and does not exert effort, ph(1)−ph(0),and the worker’s cost of losing a job, V(h(uZ C)). Since the worker’s cost of losing a job is increasing in unemployment, it follows that the probability πCof a firing episode is higher when the unemployment rate is lower. Hence, going back to Figure 3,astheunemployment rate falls from u0to u∗, the probability of a firing episode gets larger and larger. When the firing episode eventually takes place, the unemployment rate increases and the probability of another firing episode falls. The endogeneity of the probability distribution of aggregate “shocks” is a genuinely distinctive feature of our business cycle theory and, more generally, it would be a distinctive feature of any theory of endogenous and stochastic fluctuations. Existing theories of business cycles take the probability distribution of shocks as exogenous, and hence, have nothing to say about the magnitude, persistence, and determinants of the distribution of shocks. In contrast, our theory has something to say about the relationship between the structure of aggregate shocks and the fundamentals of the economy. For instance, it says that negative shocks become more likely as the unemployment rate Theoretical Economics 15 (2020) Agency business cycles 145 falls. Similarly, it says that negative shocks are less frequent in economies or sectors where agency problems are less severe because either the cost of unobserved effort ψis lower or the ability to detect low effort, as captured by ph(1)−ph(0),ishigher. Lastly, we wish to point out that the nature of recessions in ABC’s is very different from the nature of recessions in business cycle theories driven by aggregate productivity shocks (e.g., the Real Business Cycles (RBC’s) of Kydland and Prescott 1982 and Mortensen and Pissarides 1994). In RBC’s, recessions are times when productivity is unusually low and so are the gains from trade between workers and firms in the labor market. As the gains from trade are small, workers and firms search less intensely and the unemployment rate is higher. In ABC’s, recessions are states of the world where the workers’ value of unemployment is unusually low, and hence, the gains from trade between workers and firms are large. For this reason, recessions are states of the world in which firms find it optimal to fire their nonperforming workers, in which unemployment is high and, because of convex vacancy costs, in which the workers’ value of unemployment is low. Thus, the correlation between unemployment and gains from trade is negative in RBC’s and positive in ABC’s. To paint a picture, in RBC’s, recessions are times when it is raining on the marketplace. In ABC’s, recessions are times when the TV set at home is broken. The above observations are summarized in the following theorem. Theorem 4 (Recursive equilibrium). A recursive equilibrium features: (i) Aggregate uncertainty: For any u∈[01], the next period’s unemployment is h(uZC)with probability πCand h(u ZA)with probability 1−πC,with h(uZC)>h(uZ A)as long as u<1. (ii) Endogenous probability of a recession: The probability of a correlated firing episode πCis given by (21), and it is increasing in ψand decreasing in ph(1)− ph(0)and in u. (iii) Countercyclical gains from trade: V(u)and J(u) are increasing in u. 5. Quantifying the theory In this section, we calibrate our theory of endogenous and stochastic fluctuations in order to assess quantitatively the features of agency business cycles and compare them with the data. We find that there are parameter values for which ABC’s display the same volatility and the same pattern of comovement of unemployment, UE and EU rates as in the data. Moreover, ABC’s display fluctuations in unemployment, UE and EU rates that are, as in the data, large relative to and uncorrelated with fluctuations in labor productivity. We find that the main counterfactual prediction of ABC’s is that the correlation between unemployment and vacancies is positive rather than negative. Lastly, we show that, in the data, the gains from trade in the labor market are countercyclical, as predicted by ABC’s, and not procyclical, as predicted by RBC’s. 146 Golosov and Menzio Theoretical Economics 15 (2020) 5.1 Calibration We calibrate the model to the US labor market between 1951 and 2014. We want to understand whether our theory of endogenous and stochastic cycles can possibly match the cyclical features of the US labor market. For this reason, we use as targets in the calibration not only average values of key labor market variables, but also their cyclical volatility. Let us start by reviewing the parameters of the model. Preferences are described by the discount factor βand by the worker’s periodical utility. When the worker is unemployed, his periodical utility is given by υ(b)+ζ,whereυ(·)is the utility of consumption, bis unemployment income, and ζis the value of leisure. When the worker is employed, his periodical utility is given by υ(wt)−ψet. We specialize the worker’s utility function υ(c) to be log(c). The production process is described by the possible realizations of the worker’s output, yhand y, by the probability distribution over realizations of the worker’s output conditional on effort, ph(1)and ph(0), and by the probability of exogenous job destruction, δ. The search and matching process is described by the vacancy cost function, k(v), and by the matching function, M(uv). We specialize the vacancy cost function to be of the form k(v) =k0vρ,wherek0>0is a scale parameter and ρ>0 is the elasticity of the vacancy cost with respect to vacancies. We specialize the matching function to be of the form M(uv)=uv(uγ+vγ)−1/γ, which is a constant returns to scale function with an elasticity of substitution γbetween uand v. We calibrate the model to the US labor market between 1951 and 2014. The calibration strategy for some of the parameters of the model is standard. We choose the model period to be 1month. We set the discount factor, β, so that the annual real interest rate is 5%. We set the scale coefficient in the vacancy cost function, k0, and the probability of exogenous job destruction, δ, so that the average unemployment rate and the average EU rate are the same in the model as in the data.9We normalize the expected labor productivity of a worker to 1, i.e., y≡ph(1)yh+p(1)y=1.Wesettheunemployment income bto be 40% of expected labor productivity, so as to reflect the typical replacement rate of US unemployment benefits.10 We set the value of leisure ζso that the flow value of unemployment expressed in units of output, b+ζ/υ(b),is70% of expected labor productivity, which Hall and Milgrom (2008) argue is a reasonable estimate for the US economy. We choose the elasticity of substitution γbetween unemployment and vacancies in the matching function to 124, which is the value estimated by Menzio and Shi (2011). We calibrate the remaining parameters, which are novel to our theory, to match some key cyclical properties of the US labor market. First, note that the probability that the realization of the worker’s output is low, p(1)=1−ph(1), determines the rate at which employed workers become unemployed during a firing episode, and hence, the 9We measure the UE and the EU rates using the civilian unemployment and short-term unemployment rates from the CPS, following the methodology in Shimer (2005). We measure labor productivity as output per worker in the nonfarm sector. 10To be more precise, the replacement ratio is a ratio between unemployment benefits and wages and not between unemployment benefits and output. However, because the UE rate is so high, the average wage is close to the average output. Theoretical Economics 15 (2020) Agency business cycles 153 wage, the worker has no incentive to exert any effort. The value to the firm of keeping a worker who exerts no effort and is paid the minimum wage may be negative. Hence, the value of the contract to the firm as a function of the value of the contract to the worker (i.e., the Pareto frontier) is hump-shaped. If the value of breaking the match lies above the Pareto frontier, the optimal contract should involve—for low values of the contract to the worker—a firing lottery. Clementi and Hopenhayn (2006) show, in the context of a contract between a lender and a borrower, that the lottery is played after a sufficiently long sequence of low realizations of output. We were able to show the same in a simple two-period version of the model. Second, we assumed that workers are hand-to-mouth, in the sense that they consume their income in every period. The assumption guarantees that the workers’ relative gains from trade are decreasing in the equilibrium wage. Note that, because of Nash bargaining, the workers’ relative gains from trade are decreasing in the equilibrium wage as long as the workers’ value function is concave with respect to cash-onhand (i.e., wealth plus wage). Even when workers are allowed to borrow and save, the value function is typically concave in cash-on-hand. Third, we assumed that the vacancy costs are convex. The assumption guarantees a negative relationship between the value of unemployment to a worker and the unemployment rate. This negative relationship implies that the wage is decreasing and, in turn, the workers’ relative gains from trade are increasing in the unemployment rate. This is why firms want to correlate the outcomes of their firing lotteries. There are several alternative assumptions that make the value of unemployment decreasing in the unemployment rate. In an earlier version of the paper, we assumed that the matching function has decreasing returns to scale. Inspired by Chodorow-Reich and Karabarbounis (2016), we also considered a version of the model in which the unemployment income bis decreasing with the unemployment rate. Finally, we assumed that the economy is not subject to aggregate shocks to fundamentals. We made this assumption in order to develop—as cleanly as possible—our theory of endogenous and stochastic aggregate fluctuations. In reality, though, there may be aggregate shocks, as the growth rate of technological progress, the generosity of unemployment benefits, and the tax code may vary unexpectedly over time. In a version of our model with aggregate shocks, firms may use the realization of these shocks to correlate the outcome of their firing lotteries. Consequently, correlated firing would act as a mechanism that amplifies exogenous shocks, rather than replacing them as the root cause of aggregate fluctuations. Such a version of our model would not only be more realistic than the one presented in this paper, but also, as explained in Section 5,would probably provide a better fit of the cyclical behavior of the US labor market. Appendix A: Proof of Lemma 1 (i) Let ρ≥0denote the Lagrange multiplier on the worker’s incentive compatibility constraint, let ν(y ˆ z) ≥0denote the multiplier on the constraint 1−s(y ˆ z) ≥0and let ν(y ˆ z) denote the multiplier on the constraint s(y ˆ z) ≥0. 154 Golosov and Menzio Theoretical Economics 15 (2020) The first-order condition with respect to the firing probability s(yˆ z) is given by (1−δ)F(x)V (ˆ z)+W(x)J(ˆ z)=ρβ(1−δ)ph(1)−ph(0)V(ˆ z)+ν(yˆ z)−ν(yˆ z) (23) together with the complementary slackness conditions ν(yˆ z) ·(1−s(yˆ z)) =0and ν(yˆ z) ·s(yˆ z) =0. The left-hand side of (23) is the marginal cost of increasing s(yˆ z). This cost is given by the decline in the product of the worker’s and firm’s gains from trade caused by a marginal increase in the firing probability s(yˆ z). The right-hand side of (23) is the marginal benefit of increasing s(yˆ z). This benefit is given by the value of relaxing the worker’s incentive compatibility and the s(yˆ z) ≥0constraints net of the cost of tightening the s(yˆ z) ≤1constraint by marginally increasing the firing probability s(yˆ z). Similarly, the first-order condition with respect to the firing probability s(yhˆ z) is given by (1−δ)F(x)V (ˆ z) +W(x)J(ˆ z) +ρβph(1)−ph(0)V(ˆ z)=ν(yhˆ z) −ν(yhˆ z) (24) together with the complementary slackness conditions ν(yhˆ z) ·(1−s(yhˆ z)) =0and ν(yhˆ z) ·s(yhˆ z) =0. The left-hand side of (24) represents the marginal cost of increasing s(yhˆ z). The right-hand side of (24) represents the marginal benefit of increasing s(yhˆ z). Notice that increasing the firing probability s(yhˆ z) tightens the worker’s incentive compatibility constraint, and hence, the term in ρis now on the left-hand side of (24). Suppose ρ=0. First, notice that the left-hand side of (23) is strictly positive as V(ˆ z) > 0,J(ˆ z) > 0by assumption, and W(x)>0F(x)>0at the optimum x∗.Therighthand side of (23) is strictly positive only if ν(yˆ z) > 0.Hence,ifρ=0, the only solution to the first order condition with respect to the firing probability s(yˆ z) is 0.Next,notice that the left-hand side of (24) is strictly positive and the right-hand side is strictly positive only if ν(yhˆ z) > 0.Hence,ifρ=0, the only solution to the first-order condition with respect to the firing probability s(yhˆ z) is 0. However, if s(yˆ z) =s(yhˆ z) =0,the worker’s incentive compatibility constraint is violated. Therefore, ρ>0and the worker’s incentive compatibility constraint holds with equality. (ii) The first-order condition with respect to s(yhˆ z) is given by (24) together with the complementary slackness conditions ν(yhˆ z)(1−s(yhˆ z)) =0and ν(yhˆ z)s(yhˆ z) =0. The left-hand side of (24) is strictly positive. The right-hand side of (24) is strictly positive only if ν(yhˆ z) > 0. Therefore, the first-order condition is satisfied only if ν(yhˆ z) > 0 and hence, only if s(yhˆ z) =0. (iii) Using the definition of φ(ˆ z), we can rewrite the first-order condition with respect to the firing probability s(yˆ z) as (1−δ)V (ˆ z)F(x)+W(x)/φ(ˆ z) −ρβph(1)−ph(0)=ν(yˆ z) −ν(yˆ z) (25) together with ν(yˆ z) ·(1−s(yˆ z)) =0and ν(yˆ z) ·s(yˆ z) =0. The left-hand side of (25) is strictly decreasing in φ(ˆ z). The right-hand side of (25) is strictly positive if ν(yˆ z) is strictly positive and it is strictly negative if ν(yˆ z) is strictly positive. Therefore, there Theoretical Economics 15 (2020) Agency business cycles 155 exists a φ∗such that if φ(ˆ z) > φ∗, the left-hand side is strictly negative and the solution to (25)requiresν(yˆ z) > 0. In this case, the solution to the first-order condition for s(yˆ z) is 1.Ifφ(ˆ z) < φ∗, the left-hand side is strictly positive and the solution to (25) requires ν(yˆ z) > 0. In this case, the solution to the first-order condition for s(yˆ z) is 0. Appendix B: Proof of Lemma 2 The first-order condition with respect to the wage wis given by F(x)υ(w) −W(x)=0(26) The left-hand side of (26) is the increase in the product of the worker’s and firm’s gains caused by a marginal increase in the worker’s wage w. A marginal increase in w, increases the worker’s gains from trade by υ(w) and decreases the firm’s gains from trade by 1. Therefore, a marginal increase in w, increases the product of the worker’s and firm’s gains from trade by F(x)υ(w) −W(x). 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