Abstract The single strongest predictor of changes in the Fed Funds rate in the period 1982–2007 was the level of the layoff rate (initial unemployment claims divided by total employment). That fact is puzzling from the perspective of representative-agent models of the economy, which typically imply that the welfare gains of stabilizing employment fluctuations are small. It is now well known, though, that accounting for the heterogeneous effects of business cycles can substantially increase their welfare costs. We therefore augment a standard New Keynesian model with a labor market featuring endogenous countercyclical layoffs that lead to large and persistent wage declines. In our benchmark calibration, welfare may be increased by 0.5 percent of lifetime consumption when the central bank’s policy rule responds to the layoff rate instead of purely targeting inflation. The theory provides a theoretical rationale for the Federal Reserve’s dual mandate and its apparent responsiveness to changes in the number of layoffs.

1

Introduction

To the average person, a recession is a relatively minor event. But to those who lose their jobs in recessions, the effects are large and long-lasting. While the average person’s income might fall by a percentage point or two, a large empirical literature shows that people who are laid off in recessions can see declines in lifetime income of as much as 30 percent.1 Idiosyncratic risk varies over the business cycle (Storesletten, Telmer, and Yaron (2001); Guvenen, Ozkan, and Song (2014)), and welfare calculations imply that such variation can be a source of large losses (Imrohoroglu (1989)?; Krebs (2007)). So a key potential reason that stabilization of the business cycle might raise welfare it that it would stabilize idiosyncratic consumption risk.2 ∗

Berger, Dew-Becker, Milbradt, and Takahashi: Northwestern University. Schmidt: University of Chicago. We thank Gadi Barlevy for helpful comments 1 Notable examples are von Wachter, Song, and Manchester (2009) (finding long-term wage losses of 30 percent); Jacobson, LaLonde, and Sullivan (1993) (25 percent long-term wage loss); Davis and von Wachter (2011) (10 to 20 percent loss); Ruhm (1991) (11–15 percent); and Kletzer and Fairlie (2003) (9–13 percent losses for young workers); among many others. 2 See Mukoyama and Sahin (2006), Krebs (2007), Krusell, Mukoyama, Sahin, and Smith (2009) and Krueger, Mitman and Perri (2016).

1

If time-variation in idiosyncratic risk is the source of welfare costs in recessions, then we might expect government policy to respond to it. In particular, given the evidence on the cost of job loss, a natural reading of the Federal Reserve’s mandate to promote “maximum employment” might be to attempt to minimize layoffs. This paper begins by documenting a novel fact about monetary policy in the US: in the period since 1982, the aggregate layoff rate – initial unemployment claims divided by aggregate employment – has been the single strongest predictor of changes in interest rates. That result holds even after controlling for a wide range of measures of inflation and real activity and examining both forward- and backward-looking policy rules. In the context of many models of the business cycle and monetary policy, that empirical result is puzzling. A common implication of workhorse models is that policy should purely target inflation and that setting interest rates to respond to real activity in any way leads to measurable welfare losses (Schmitt-Grohe and Uribe (2007); though Erceg, Henderson, and Levin (2000) find that wage stickiness can motivate attention to the output gap). Motivated by the evidence on the welfare costs of idiosyncratic risk and the behavior of the Federal Reserve, we develop a tractable New Keynesian model of the business cycle featuring uninsurable risk associated with job losses.3 The fundamental source of heterogeneity in the model is in human capital. Human capital risk is modeled as being purely idiosyncratic, so it is potentially insurable.4 Following Harris and Holmstrom (1982), firms in the model provide such insurance by promising not to reduce a worker’s wages if their human capital declines. But if, due to market forces, the firm must lay off some workers, that insurance contract is effectively broken (as in Gamber (1988) and Berk, Stanton, and Zechner (2010)). The loss of the insurance is extremely costly, leading to large and persistent wage declines. But that loss is borne exclusively by the workers who are laid off. Those who keep their jobs are largely unaffected by recessions. While simple, the model is able to quantitatively match a range of patterns in income, including the paths of wages both within jobs and following job loss. To study optimal policy, the paper derives a novel expression for welfare under consumption heterogeneity. As in past calibrations of endowment economies, the key feature of the model that generates welfare losses around business cycles is that idiosyncratic consumption risk is higher when the layoff rate is higher. The particular source of that risk (in our case, wage contracts) is not directly relevant for the policy conclusions. In fact, our results are equivalent to a setting in which human capital is simply assumed to permanently decline following a layoff, thus generating observed consumption and income declines. The model allows us to revisit well known results on optimal policy. We find that the model 3

The importance of idiosyncratic risk for policy analysis has been understood for some time. Clarida, Gali and Gertler (1999) write: “while the widely used representative agent approach may be a reasonable way to motivate behavioral relationships, it could be highly misleading as a guide to welfare analysis. If some groups suffer more in recessions than others (e.g. steel workers versus professors) and there are incomplete insurance and credit markets, then the utility of a hypothetical representative agent might not provide an accurate barometer of cyclical fluctuations in welfare.” 4 There is a large literature on wage insurance. See Guiso, Pistaferri, and Schivardi (2005) for a recent review of the theory and for empirical evidence that firms do provide such insurance.

2

rationalizes the observed behavior of the Federal Reserve: welfare is in general improved when the central bank in the model sets interest rates to respond to the level of layoffs. Furthermore, since layoffs are the direct source of welfare losses, it is optimal to use layoffs as the relevant measure of real activity, as opposed to, say, the output gap. Interestingly, this is consistent with both our empirical evidence and the Federal Reserve’s statutory mandate, which requires stabilizing employment rather than output. The reason that pure inflation targeting is optimal in representative agent models is that realistic fluctuations in output have extremely small welfare costs in those settings (Lucas (1987)), while variation in inflation leads to inefficiencies in production. Schmitt-Grohe and Uribe (2007) find therefore that it is optimal to stabilize inflation, which maximizes productivity and output. In that setting, the optimal policy, which responds only to inflation, can lead to welfare gains of approximately 0.3 percent of lifetime consumption compared to a rule that also responds to the output gap. Our results are therefore notable both for their direction – the Federal Reserve should pay attention to the state of the business cycle – and for their magnitude: we obtain welfare gains two-thirds larger those that appear in Schmitt-Grohe and Uribe (2007). It is important to note that it is optimal in the model for the central bank to intervene in markets even at the flexible-price equilibrium. That is due to the simple fact that there is uninsurable risk, and by stabilizing the level of employment, the central bank also stabilizes risk. Because insurance is incomplete, the central bank’s behavior reduces the welfare cost of market incompleteness. In the end, the paper is easily summarized: when a New Keynesian model is augmented so as to generate large wage losses following layoffs, it implies that optimal monetary policy will attempt to minimize fluctuations in the layoff rate. Empirically, the Federal Reserve appears to do just that. Our work builds on a number of important and influential areas of past research. Lucas (1987, 2003) discusses the welfare cost of business cycles in settings with both representative and heterogeneous agents. Storesletten, Telmer, and Yaron (2004) and Guvenen, Ozkan, and Song (2014)), among others, provide evidence on the magnitude and coutercyclicality of idiosyncratic income risk.5 The theoretical model builds on a large literature that examines monetary policy in microfounded New Keynesian settings, including Rotemberg and Woodford (1999), Woodford (2003), Christiano, Eichenbaum, and Evans (2005), Schmitt-Grohe and Uribe (2007), and Coibion, Gorodnichenko, and Wieland (2012), among many others. Krause and Lubik (2007) and Braun and Nakajima (2012), also study optimal policy in the presence of heterogeneity. We differ from Krause and Lubik (2010) in having a larger cost of job loss, and from Nakajima (2010) for making idiosyncratic risk endogenous to the business cycle. In considering idiosyncratic labor income risk in a model with sticky prices, our work is closely related to that of Ravn and Sterk (2015), Challe and Ragot (2015), Challe et al. (2015), den 5

See also Lucas (2003) for a discussion of how the Lucas (1987) welfare calculation can be substantially magnified when heterogeneity is taken into account and the variance of idiosyncratic shocks is countercyclical, as in Storesletten, Telmer, and Yaron (2004) and Guvenen, Ozkan, and Song (2014).

3

Haan et al. (2015) and Werning (2015). This paper differs from those in two important respects. First, our primary focus is on monetary policy while those papers focus on the specifics of labor matching, wage setting, and consumption demand. Second, our model is sufficiently simple that it can be easily linearized, which suggests that it will be relatively simple to estimate or modify in future work. Ravn and Sterk (2015) deserves special mention. This paper focuses on how labor market risks can be amplified in a world with incomplete markets and sticky prices and argues that aggressively responding to inflation is close to optimal because it also mitigates the aggregate demand amplification. One difference between our papers is one of emphasis: we focus on optimal policy and welfare while they focus on the amplification mechanism. Our policy perscriptions are also different. In contrast to their paper, we find strong empirical and theoretical support for stabilizing variation in the layoff rate at the expense of stabilizing prices. From a modeling perspective, we build on Mankiw (1986) and the recent work of Constantinides and Ghosh (2015) and Schmidt (2015), who examine asset prices in models with uninsurable idiosyncratic risk.6 Those papers use a modeling approach in the optimization problems of heterogeneous agents scale in such a way that aggregation still takes place and Euler equations can be formed as though there is a representative agent, but with extra terms accounting for idiosyncratic risk. The ability of the model to account for idiosyncratic risk in both the Euler equations and the welfare calculations represents a methodological contribution. Variation in idiosyncratic risk over time passes through to affect precautionary savings and hence consumption demand. And the welfare and demand effects are obtained in a purely linear approximation that can easily be solved, simulated, and estimated. It is widely understood that models can generate much larger welfare costs when they account for the possibility that the pain of business cycles is focused on only a fraction of the population. But because such models are usually very difficult to work with, they are rarely used for policy analysis. So an important contribution of this paper to the literature on optimal monetary policy is to extend a standard New Keynesian model of the business cycle to account for heterogeneous effects of business cycles on workers The remainder of the paper is organized as follows. In section 2 we briefly review the evidence on the long term earnings losses following a job displacement event. Section 3 estimates models of the interest rate rule in the U.S. since 1982 and shows that the layoff rate has been the most important driver of interest rate changes. We then proceed to build the model of the economy. Section 4 discusses the preferences and provides a simple calibration to illustrate how different types of shocks affect welfare. We then build the remainder of the model in section 5 and examine its implications for monetary policy in section 6. Finally, section 7 concludes.

2

Empirical estimates of the cost of job loss

Jacobson, Lalonde, and Sullivan (1993) examine the income of workers in Pennsylvania between 1974 and 1986. They find large and persistent effects from job loss: the average initial drop in 6

See also Constantinides and Duffie (1996), and Storesletten, Telmer, and Yaron (2007).

4

earnings relative to pre-displacement earnings in their data is 50 percent, and six years after the displacement event earnings are still 25 percent below their pre-displacement level. More recently von Wachter, Song, and Manchester (2009) and Davis and von Wachter (2011) use Social Security records to document economy-wide earnings consequences from displacement over a more representative sample and a longer time-span. Both studies find large and highly persistent earnings losses for both low and high tenure workers. Davis and von Wachter (2011) report that the present discounted value of earnings losses for men with three or more years of tenure after a the displacement event are 11.9 percent relative to pre-displacement earnings trends. The average effect masks significant heterogeneity in the cost of job loss across time and across different workers. The present discounted value of earnings losses is 9.9 percent of pre-displacement earnings in expansions and 19.8 percent in recessions. Davis and von Wachter (2011) also document that while there is significant cross-sectional heterogeneity in the cost of job loss, the size and persistence of the loss is always sizable. Broadly speaking, the costs are smallest for men between the ages of 31–40 (7.7 percent) and largest for men above 50 (24 percent) and the mean cost of job loss is slightly smaller (10.9 percent) for women but still sizable. Kletzer and Fairlie (2003) examine the National Longitudinal Survey of Youth (NLSY) and find that the long-term earnings losses of male young adults with low tenure are approximately 10 percent, similar to the decline for high-tenure workers found by Davis and von Wachter (2011). Recent work by Couch and Placzek (2010) also establishes that the cost of job loss is large irrespective of which industry the worker is employed. Using administrative data from Connecticut during the 1990s and 2000s, they show that the losses are large – the earnings losses six years after a displacement event are 13-15 percent of pre-displacement earnings no matter which industry the job displacement event happens in. The smallest losses occurred in the education and health sector and the largest losses occurred in the financial sector with the losses in the manufacturing sector in the middle. Similarly, Phelan (2014) finds large effects in most sectors. This suggests that these earnings losses are not solely an artifact of depreciating human capital that is only valued in a declining industry, but rather an inherent consequence of job loss. Recent work by Guvenen, Ozkan, and Song (2014), which uses income information from Social Security records, supports this conclusion. They find that both within and across industries, the probability of an extreme decline in income rises significantly in recessions. A natural explanation for that finding is that the negative income changes are the result of layoffs. Given the size of the reduction in lifetime income that people face due to being laid off, and given that layoffs are countercyclical, it is natural to think that variation in the layoff rate could be a major source of welfare costs due to business cycles.

3

Does the Federal Reserve react to the layoff rate?

This section studies how the Federal reserve has historically set interest rates. Standard implementations of the monetary policy rules do not include the layoff rate. We provide evidence, though, 5

that the Federal Reserve has in fact historically responded to the layoff rate, consistent with the view that layoffs are a major source of welfare losses over the business cycle. Moreover, in the period since 1982, the layoff rate has been the single strongest predictor of changes in the Fed Funds rate among the standard variables that we examine.

3.1

Data

The policy interest rate that we study is the target Fed Funds rate. The analysis is conducted at the quarterly frequency to allow us to include data on the output gap. All data is obtained from the Federal Reserve Bank of St. Louis’s FRED database. We consider three measures of inflation: the personal consumption expenditures (PCE) deflator, the core PCE deflator that excludes food and energy purchases, and the gross domestic product (GDP) deflator. We measure output using real GDP either in growth rates or detrended using a Hodrick–Prescott (HP) filter with a smoothing parameter of 1600. We use the CBO’s definition of potential output to construct our baseline measure of the output gap. We also examine the unemployment rate, the HP-filtered unemployment rate (using a smoothing parameter of 12800) and the change in the unemployment rate. We measure layoffs using weekly initial claims. Initial claims are then used either in their raw level, scaled by total employment, HP detrended with a smoothing parameter of 12800, or smoothed using an exponentially weighted moving average with a decay rate of 0.025. We perform all detrending of the variables on the 1967–2014 sample, but the analysis of the determinants of interest rates below uses the period 1982–2008 to avoid endpoint problems, major changes in monetary policy prior to the tenure of Paul Volcker, and the zero lower bound. That said, the results are highly similar in both the full (1967–2014) and post-Volcker (1982–2014) sample.

3.2

Analysis

We begin by examining simple pairwise correlations between innovations in the interest rate and the various explanatory variables. The innovation in the interest rate is measured as the residual in an AR(1) regression (the estimated autoregressive coefficient is 0.970). Table 1 lists the pairwise correlations sorted by their magnitude. The four largest correlations are for various measures of layoffs, with HP-detrended initial claims performing best with a correlation of -0.63. Of the other variables, the strongest is the change in the unemployment rate, with a correlation of -0.55 (and the change in the unemployment is obviously mechanically closely related to the layoff rate). The other measures of unemployment and output have substantially smaller correlations, almost all below 0.4 in absolute value. While the pairwise correlations do not represent a fully specified policy rule, they provide a simple first indication that the layoff rate is a major determinant of changes in interest rates. Next we estimate both backward and forward looking monetary policy rules and show that the layoff rate continues to have strong explanatory power for the federal funds rate. The top

6

panel of table 2 reports results of regressions of the fed funds rate on its own lag and the various explanatory variables. Our measure of the layoff rate is HP-detrended initial claims, and across all 10 specifications, this measure always has the largest t-statistic, indicating that they have the highest marginal explanatory power of any of the variables excluding the lagged fed funds rate.7 The t-statistics do not account for autocorrelation in the residuals; we use them here as measures of explanatory power – since they are monotonically related to the marginal R2 of each variable – rather than as indicators of statistical significance (but using Newey–West (1987) standard errors does not change our conclusions). Next we estimate forward looking policy rules of the following form:

rt = (1 − ρ) (α + βEt [inf lationt,t+k ] + γEt [outputgapt,t+q ] + δEt [layof f st,t+q ] + ρrt−1 ) + εt (1) where the parameters k and q determine the forecast horizon used in setting interest rates and εt is a residual. It is well known that forward looking rules have a strong theoretical appeal since in standard New Keynesian models the optimal policy function is typically forward-looking. We closely follow the approach outlined in Clarida, Gali and Gertler (2000) and our baseline policy function is identical to theirs. Our innovation is that we also estimate a policy rule that includes both the expected output gap and expected initial claims. We estimate our policy rules by GMM using the following 21 instruments: four lags of inflation: output gap, the federal funds rate, the 10/1-year Treasury yield spread, and commodity price inflation.8 The bottom panel shows results for five different sets of policy rules where each set of rules differs first in whether expected layoffs are included and second in what horizon the central bank is considering for expected inflation, output gaps and layoffs. No matter which horizon we consider, whenever expected layoffs are included it is always the most significant predictor of federal funds rate other than the lagged policy rate. Moreover, since we standardize all of the regressors, we can formally test whether the central bank puts more weight on layoffs rather than the output gap or inflation. In all five specifications we find that expected layoffs are the strongest predictor of the federal funds rate and the estimated weight on expected layoffs is greater than that on either inflation and the output gap. So across a range of specifications, involving three different measures of layoffs, a range of measures of inflation and the output gap, and both backward and forward looking policy rules, the layoff rate is the dominant driver of changes in interest rates. The Federal Reserve thus appears to respond strongly to the layoff rate. The remainder of the paper builds an analyzes a model that can rationalize that result. 7 8

In the appendix we show this result is robust to alternative definitions of the layoff rate. This is the exact instrument set used by Clarida, Gali and Gertler (2000).

7

4

Preferences and welfare

4.1

Utility function

We model people as having generalized recursive preferences over consumption streams (Epstein and Zin (1989), Weil (1989)). The lifetime utility of person i, denoted vi,t , follows the recursion

vi,t = (1 − β) ci,t +

β log Et exp ((1 − α) vi,t+1 ) 1−α

where ci,t = log Ci,t

(2) (3)

where vi,t is person i’s lifetime utility, β is the effective time discount factor, α the coefficient of relative risk aversion, and Ci,t is person i’s consumption.9 For the sake of simplicity, we assume a unit elasticity of intertemporal substitution.10 The preferences in (2) differ from the standard recursive representation of time-separable preferences in that they involve a certainty equivalent over the continuation value, (1 − α)−1 log Et exp ((1 − α) vi,t+1 ), instead of a simple mathematical expectation. The parameter α thus adds extra curvature to utility over continuation values compared to what is caused by the finite elasticity of intertemporal substitution, making people particularly averse to shocks that reduce lifetime utility, rather than just consumption in a single period. For our purposes, the key features of the preferences are that they separate the elasticity of intertemporal substitution (EIS) from the coefficient of relative risk aversion and that they admit closed-form expressions for welfare in log-linear equilibria. As to the EIS, while there is evidence that people are relatively willing to substitute consumption over time, which is at least part of the reason that the business cycle literature often focuses on the case of log utility, that need not imply that the average person’s coefficient of relative risk aversion is only 1. For example, it is well known that it is very difficult to rationalize the behavior of asset prices under any specification of power utility (Mehra and Prescott (1985)), but Epstein–Zin preferences with a coefficient of relative risk aversion greater than 1 are consistent with major features of asset prices (Tallarini (2000) and Bansal and Yaron (2004), among many others). Hansen and Sargent (2007) also show that the recursion (2) can be obtained if people face model uncertainty and choose decision rules to be robust against unfavorable outcomes. Our desire is to provide a reasonable description of how people view uncertainty while at the same time modeling them as being willing to substitute consumption relatively freely over time. 9 When people die stochastically with a constant probability, as we will discuss later, β incorporates both pure time preference and the effect of the death rate. 10 It is straightforward to extend the analysis to a case where the elasticity of substitution differs from 1, it just requires an additional approximation. The assumption of a unit EIS is also common in the business cycle literature (e.g. Christiano, Eichenbaum, and Evans (2005)), and micro evidence suggests that 1 is a reasonable calibration for the EIS (Attanasio and Weber (1993); Beaudry and van Wincoop (1996); Vissing-Jorgensen (2002)).

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4.2

The determinants of welfare

Before turning to the remainder of the full general equilibrium model, we first briefly discuss the determinants of welfare under Epstein–Zin preferences. Since utility depends only on consumption, the production side of the economy (i.e. the source of consumption dynamics) is irrelevant at this point. This subsection therefore just examines an exogenously fixed consumption process. Consumption will then be endogenized in the full equilibrium in the following section. 4.2.1

Consumption

Person i’s consumption has both an aggregate component, ct , and an idiosyncratic component, η i,t ci,t = η i,t + ct

(4)

We think of layoffs as representing idiosyncratic disasters – they lead to large declines in the net present value of income, which then also cause declines in consumption. In reduced-form, then, there is a small probability in each period that a person’s consumption may decline by a large amount. η i,t is therefore modeled in this section following the process ∆η i,t = −1 layoffi,t gL,t + 1 − 1 layoffi,t gH,t Pr layoffi,t = φt

(5) (6)

where 1 layoffi,t is an indicator function equal to one if person i is laid off in period t and zero otherwise, ∆ is the first-difference operator and gL,t is an exogenous process determining the size of income losses in each period. The probability that a particular person is laid off in period t is denoted φt . gH,t is set to ensure that E exp ∆η i,t |φt , gL,t = 1 in each period and the cross-sectional mean of the person-specific part of consumption is constant. This specification for consumption growth is similar to that studied in the asset pricing literature on rare disasters (e.g. Barro (2006, 2009) and Wachter (2010)) but here the disasters are isolated to individual people, instead of hitting the entire population simultaneously. The welfare effects are identical, though, regardless of whether disasters are common or idiosyncratic, as long as they are uninsurable. Subtracting η i,t from both sides of equation (2) to obtain utility relative to human capital, we obtain vη t ≡ vi,t − η i,t = (1 − β) ct + where kt+1 ≡

(7) β log Et exp (1 − α) vη t+1 + kt+1 1−α

1 log Et+1 exp (1 − α) ∆η i,t+1 1−α

9

(8)

(9)

and where (8) follows from the fact that shocks to ∆η i,t+1 are conditionally independent. Utility depends on current consumption, scaled utility on date t + 1 (vη t+1 ), and a certainty equivalent over the shocks to lifetime utility, which we denote kt+1 . The certainty equivalent k incorporates information about the layoff rate, the consumption loss following a layoff, and the coefficient of relative risk aversion. Higher layoff risk, higher losses in layoffs, and higher risk aversion all reduce the certainty equivalent k. This variable will appear throughout our analysis as the key driver of welfare. We model aggregate consumption and the certainty equivalent kt as following AR(1) processes,11

4.2.2

ct = ρ∆c ct−1 + εc,t

(10)

kt = (1 − ρk ) k¯ + ρk kt−1 + εk,t ε∆c,t ∼ N 0, σ 2c εk,t ∼ N 0, σ 2k

(11) (12)

Expression for welfare

Given the above process, the appendix then shows that the average level of lifetime utility scaled by human capital is " β 1−α E [vη t ] = k¯ + 1−β 2

σk 1 − βρk

2

+

1−β σc 1 − βρ

2 !# (13)

Average scaled utility depends on three terms. k¯ is the utility cost of the steady-state level of idiosyncratic risk. The second and third terms measure the effect on welfare of fluctuations in the state of the economy. Aggregate consumption risk affects welfare through the term term becomes small, since

1−β 1−βρ

σk 1−βρk

2

. When β ≈ 1, that

≈ 0. It is possible to show that under Epstein–Zin preferences,

welfare depends on the magnitude of the shocks to the long-run level of consumption (Bidder and Dew-Becker (2016)). Here, the aggregate shock to consumption affects only fluctuations around the mean – it has no effect on the long-run level. So its effect on welfare is approximately zero (and this result holds for essentially any stationary process for the level of aggregate consumption). Temporary fluctuations in the level of consumption due to, e.g. monetary policy or government spending shocks, have quantitatively trivial effects on welfare. For the disasters, through the parameters gL,t and φt , though, the results are completely different. Because the idiosyncratic shocks have permanent effects on the level of consumption, they σk 1−βρk

measures the cumulative effect of a shock εk,t on 2 σk idiosyncratic risk (discounted at the rate of pure time preference). When β ≈ 1, 1−ρ is known

have non-trivial welfare effects. The term

k

as the long-run variance, since it measured the variance of the cumulative innovation to kt . When 11

It would obviously be preferable to directly model processes for φ and gL , but in order for there to be an analytic expression for welfare, k must follow an affine process.

10

kt is more persistent or more volatile, due to greater persistence and volatility in the layoff rate or consumption loss, the long-run variance rises and welfare declines. Putting together the results on aggregate consumption and idiosyncratic risk then, this section derives a key result that will drive our analysis: any government policy directed at stabilizing the economy and minimizing business cycles matters almost exclusively to the extent that it stabilizes idiosyncratic risk, kt , rather than aggregate consumption. 4.2.3

A calibration

We now consider a simple calibration of the consumption process to measure the welfare effects of fluctuations in idiosyncratic risk. The details of the sources for the calibrated parameters are discussed below in section 5. Here we briefly outline the main parameters. The average layoff rate is set to 1.3 percent per quarter, and the average consumption decline following a layoff is 20.3 ¯ The discounted long-run standard deviation of idiosyncratic risk, σk , is set percent, yielding k. 1−βρ k

to 0.0053, which is based on various data sources on job transitions.12 For the level of consumption, we set σ c = 0.01 and ρc = 0.95. The rate of pure time preference is set to 2 percent per year. Finally, we set the coefficient of relative risk aversion, α, to 5, as in Lucas (1987). The three components of welfare are then Average idiosyncratic risk Idiosyncratic risk fluctuations

β ¯ 1−β k β 1−α 1−β 2 β 1−α 1−β 2

Aggregate cons. fluctuations

-0.3348

σk 1−βρk

2

1−β 1−βρc σ c

2

-0.0109 -0.0004

By far the most important is the average level of idiosyncratic risk. If idiosyncratic risk were set to zero on average, then welfare would improve by the equivalent of 33.5 percent of lifetime consumption (which we find to be intuitively plausible). The contribution of fluctuations to risk is far smaller. Variation in aggregate consumption around its trend has quantitatively trivial effects reducing utility by the equivalent of 0.04 percent of lifetime consumption. Variation in idiosyncratic risk has effects approximately 30 times larger: if the layoff rate and consumption loss from layoffs could be fully stabilized, we find that welfare could be increased by 1.1 percent of lifetime consumption. While that number is not enormous, it is within the range of estimates that other work has found in similar experiments, and an order of magnitude larger than the loss calculated by Lucas (1987).

5

Full model

We now develop the remainder of the model. Except for the description of the labor market, it follows the typical New Keynesian framework. The goal of the labor market component of the 12

The calibration of the dynamics of kt here is set to match what is obtained in the full model below.

11

model is to obtain a specification that captures the idiosyncratic risk that we observe empirically in a setting that remains tractable.

5.1

Production/employment sector

There is a competitive sector of the economy that produces undifferentiated intermediate goods using labor. It is the only part of the economy that uses labor, and it is where our model deviates from past work. 5.1.1

Labor flows

Intermediate-good producing firms are indexed by m. Each firm’s total employment in terms of units of human capital follows the law of motion Nm,t = (1 − qt ) Nm,t−1 − Fm,t + Hm,t

(14)

where Fm,t ≥ 0 is firm m’s firing in period t, Hm,t ≥ 0 is its hiring, and qt is the quit rate in period t. Each firm faces the same quit rate, losing a fraction qt of its workers at the beginning of the period. qt will be determined endogenously, but it is exogenous to the decisions of the individual firms. Each firm produces an identical output which sells at price St . The firms have no intertemporal decisions, and simply maximize revenue net of costs in each period, γ − Wt Nm,t max St At Am,t Nm,t Nm,t

(15)

where At is the aggregate level of technology, Am,t is firm m’s level of technology, and Wt is the wage paid per unit of human capital on date t. Firms have decreasing returns to scale determined by the parameter γ. We assume that Am,t is a martingale with log-normal innovations, ∆am,t

1 1 2 2 ∼N − σ ,σ 21−γ a a

(16)

where the drift is chosen to ensure that labor demand is stationary. As above, lower-case letters denote logs and ∆ denotes the first difference operator.13 Optimization by the firms implies that log labor demand follows exp (∆nm,t ) = exp

1 (−∆wt + ∆st + ∆at + ∆am,t ) 1−γ

13

(17)

As a technical matter, this law of motion for productivity implies that the distribution of firm-specific productivity is undefined (or non-stationary). To account for this issue, one could allow firms to die with some fixed probability d, being replaced by new firms (whose initial level of productivity may depend on d). Our model represents the limit as d → 0.

12

Firms hire if the optimal change in employment is greater than the decline induced by firings. That is,

1 exp (−∆wt + ∆st + ∆at + ∆am,t ) > 1 − qt =⇒ Hm,t > 0 and Fm,t = 0 1−γ 1 (−∆wt + ∆st + ∆at + ∆am,t ) < 1 − qt =⇒ Hm,t = 0 and Fm,t > 0 exp 1−γ

(18) (19)

We then calculate aggregate hiring and firing in the economy as their sums across all firms (see the appendix for the details of that calculation; it is a straightforward integration over the density of ∆am,t ). Firms that receive sufficiently negative shocks to their productivity fire workers, while firms that receive positive shocks hire workers. There is thus both hiring and firing in all periods. If the quit rate in the model were constant, then in any period when employment rises – even if it is still below its steady-state – firing would be below average. But in the data, firing is countercyclical – it is high when output and employment are low, even if they are rising. That is due to the fact that the quit rate is procyclical – when aggregate employment is below average, fewer workers quit, so firms must fire more workers. In order to accurately capture the behavior of layoffs, then, it is important for us to account for variation in the quit rate. Recall that there are always firms who receive positive idiosyncratic productivity shocks and desire to hire more workers. The people they hire may be currently unemployed or employed. We denote the fraction of workers are willing to leave their jobs for a new offer as ϕ0,t−1 (which will be derived endogenously below). Firms hire workers proportionally from the two sources of available workers. Aggregate quits, Qt , and the quit rate, qt , are then ϕ0,t−1 (1 − Nt−1 ) + λNt−1 ϕ0,t−1 /Nt−1 ≡ Qt /Nt−1 = Ht (1 − Nt−1 ) + λNt−1

Qt = Ht qt

(20) (21)

where aggregate labor supply is normalized to 1. In periods when employment is low, currently employed workers account for a smaller fraction of the available work force to hire than do the unemployed. If that effect is sufficiently large, the quit rate is procyclical. Equation (21) is important for giving a simple explanation for why quits might be procyclical. We are not forced to assume any kind of exogenous rate of failure for employment relationships. 5.1.2

Wages and disequilibrium

All people in the economy inelastically supply a unit of labor, so if wages were perfectly flexible then they would all be employed at all times. Following Blanchard and Gali (2010), we instead assume that the aggregate real wage (which will be different from what individual workers are paid)

13

imperfectly tracks aggregate productivity, Wt = ΞAt1−ζ

(22)

The level of employment is then set to satisfy demand for intermediate goods (consistent with the observation that the unemployment rate is always positive, we assume that firm labor demand is always lower than labor supply). We use a reduced-form specification for the aggregate part of the wage because our primary focus is on idiosyncratic risk, and hence the determination of the worker-specific part of wages. Modeling aggregate wage determination and unemployment is generally complicated. There is important recent work in this area, including Hall and Milgrom (2008), Challe et al. (2015), and Christiano, Eichenbaum, and Trabandt (2015). Our contribution is in modeling idiosyncratic risk, which we now proceed to discuss. 5.1.3

Human capital and risk sharing

Each worker has log human capital xi,t that follows a random walk on a binomial tree, ( ∆xi,t = where pup =

+d with prob. pup −d with prob. 1 − pup

1 − exp (−d) exp (d) − exp (d)

(23) (24)

For the sake of parsimony, we assume that the probability of an increase, pup , is selected so that the level of human capital, exp (xi,t ), is a martingale. Workers face substantial idiosyncratic income risk due to shocks to their human capital. Since that risk is idiosyncratic, it should be possible for firms to provide insurance against it. But in reality wage insurance appears to be far from perfect: there is substantial evidence that workers see large and persistent declines in income following layoffs. It is well known that such behavior, along with many other features of labor markets, can be generated by models with downward rigidity in wages.14 We follow Beaudry and Pages (2001)? and others in modeling wages as being downwardly rigid but upwardly flexible. Harris and Holmstrom (1982), MacLeod and Malcolmson (1993)?, and Beaudry and Pages (2001) provide theoretical derivations of optimal downwardly rigid contracts, but proving optimality in a general setting with both aggregate and idiosyncratic risk is beyond the scope of this paper (we are also not aware of a derivation of an optimal wage contract in a setting with generalized recursive preferences). Our model is in that sense similar to that of Bernanke, Gertler, and Gilchrist (1999), who also study a risk-sharing contract that abstracts from the effects of aggregate risk and is optimal in a specific special case. 14

For empirical evidence, see Chiappori, Salanie, and Valentin (1999)?, Holzer and Montgomery (1990)?, Bils (1995)?, Beaudry and DiNardo (1991, 1995)??, and Jacobson, LaLonde, and Sullivan (1993).

14

The model of downward rigidity that we use is a version of the simple contract derived by Harris and Holmstrom (1982). Firms are able to commit to contracts, but workers are not. The contract the firm offers workers is therefore able to insure the workers against declines in productivity, but if productivity rises, wages must rise, since the worker cannot commit not to quit for a better paying job. Because human capital shocks are purely idiosyncratic, the firm is able to insure those shocks costlessly on average. They do not, though, offer insurance against aggregate shocks, which are priced. To be more specific, we model wages as having both an aggregate and an idiosyncratic component. The aggregate component is denoted Wt , and worker i’s idiosyncratic component is denoted ˜ i,t . The idiosyncratic component of the ˜ i,t , so that the total wage that worker i earns is Wt W W wage follows ˜ i,t = exp x∗ − w ¯ W i,t ( ∗ max x , x if job continues i,t i,t−1 x∗i,t = xi,t in a new job

(25) (26)

x∗i,t is the running maximum of the worker’s productivity during the life of the current job. The contract states that firms do not cut workers’ wages due to declines in human capital. But if a worker’s human capital rises, the firm will raise their wage. Obviously complete insurance would imply constant wages, whereas in this setting there is only insurance against negative shocks. In Harris and Holmstrom (1982), that result follows from the fact that firms can commit to contracts but workers cannot. When the worker’s human capital rises to a new peak, the firm must increase the worker’s pay, otherwise they will quit and move to a new job. The constant w ¯ is set so that the present value of the worker’s idiosyncratic wage is equal to the present value of their marginal products when x∗i,t = xi,t (i.e. at the beginning of an employment relationship). That constraint follows from the assumption that firms are competitive, and it implies that contracts are actuarially fair. Since the contracts are actuarially fair, the firms make hiring and firing decisions as though all workers are paid their marginal products. That is, all actors in the model understand that firms will hire and fire workers from time to time, but the firm commits to only condition that decision on its own productivity and demand shocks, not on xi,t or x∗i,t (see equation (15)).15 The firm does not hire or fire workers in response to changes in either the average idiosyncratic productivity or the average tenure of its workers. Specifically, firms fire workers when profit maximization requires it to, treating workers as though they are all paid Wt multiplied by their human capital. In other words, the model says that a firm’s primary duty is to its shareholders, but to the extent that it can provide insurance across workers, it does so. The insurance that firms provide implies that when xi,t < x∗i,t , a worker is effectively overpaid in 15

It is not formally optimal for firms to ignore the average tenure of their workers – firms with higher tenure workers should in general fire them and hire newer workers. We assume that they commit not to do so.

15

the sense that the NPV of their wages going forward is higher than the NPV of their productivity. The insurance is therefore the cause of wage declines following layoffs: for any worker for whom xi,t < x∗i,t , their current wage is above what any other firm would offer them. When the firm is forced to end an employment relationship, then, x∗i,t falls down to xi,t , and the worker’s wage falls. This loss is a transfer to the firm in the sense that the NPV of the worker’s productivity was below that of their wages if xi was less than x∗i . Workers with xi,t = x∗i,t , though, face no cost from job loss and represent the pool of workers willing to quit their jobs – if they start a new job, they will still have xi,t = x∗i,t and hence earn the same wage. The fraction of people with xi,t = x∗i,t is denoted ϕ0,t , which was defined above as the fraction of employed workers willing to quit their current jobs. Note also that firms are indifferent to whether employees with xi,t = x∗i,t quit – they can simply be replaced by new hires, who will also have xi,t = x∗i,t . The model has a range of qualitative predictions that fit well with empirical data: wages increase rapidly for new hires and follow a concave path (Buchinsky et al. (2010)); there is a large mass of workers with zero wage change in any given period (Daly and Hobijn (2014)?); and there are large wage losses following layoffs (section 2). We will calibrate the model to match micro-level data on wage changes and show that it performs well quantitatively. Household insurance Firms in the model do not insure workers against all shocks – workers experience changes in wages due to both increases in productivity and job losses. Moreover, that risk changes over time. When xi,t for a worker is equal to or close to x∗i,t , the worker faces a substantial possibility of an increase in income, but relatively little risk of a wage decline (since if they are laid off, their wage will fall only by x∗i,t − xi,t ). On the other hand, when xi,t is much smaller than x∗i,t , the worker has little prospect of seeing a wage increase in the near future, and they will experience a substantial decline in wages if their job ends. The household provides two forms of insurance against that residual risk: it eliminates idiosyncratic variation in income losses following job loss and income gains during employment, and it provides consumption during unemployment. When workers lose jobs, the household reduces their consumption by the same proportion that the NPV of wages declines on average following job loss. The household insurance is thus of a very limited form. It does not eliminate the main risk that workers face, which is job loss. It simply eliminates uncertainty about what happens when a job is lost, helping make the model tractable.16 We assume that the household provides insurance so that each worker’s consumption follows 16 While beyond the motivation for this paper, it interesting to note that having a penalty for job loss helps ensure that workers do not voluntarily want to lose their jobs. This is an potential insure since the household pays its members consumption goods during unemployment, so it is important to properly incentivize them not to lose their jobs. Christiano, Eichenbaum, and Trabandt (2015) suggest a similar type of deal between the household and its members.

16

the process Ci,t = Ct exp (˜ xi,t ) ∆˜ xi,t = −1 layoffi,t gL,t + 1 − 1 layoffi,t gH,t

(27) (28)

(as in (5)) where Ct is the aggregate production of consumption goods. The variable gL,t represents the decline in consumption following a layoff. The consumption process smooths losses across laid off workers. The household calculates the average decline in expected income for workers laid off in period t and reduces the consumption of all laid off workers by that amount – that is, the household provides insurance against variation across workers in the size of wage declines.17 gH,t is then chosen to ensure that the allocation of consumption across workers exhausts all available goods, therefore satisfying the following equation that guarantees that the cross-sectional mean of x ˜ is constant, 1 = φt exp (−gL,t ) + (1 − φt ) exp (gH,t )

(29)

The appendix provides the details of the calculations of gL,t an ϕ0,t . As in section 4, we define a certainty equivalent, 1 E [exp ((1 − α) ∆˜ xi,t+1 )] 1−α 1 log (φt exp (− (1 − α) gL,t ) + (1 − φt ) exp ((1 − α) gH,t )) 1−α

kt = =

5.2

(30) (31)

Price setting

There is a set of monopolistically competitive firms that, employing no labor, buy the intermediate good and differentiate it. They then sell their output to competitive final good aggregators. This part of the model leads to the usual New Keynesian Phillips curve and is entirely standard, so we leave the derivation to the appendix.

5.3

Preferences

As in section 4, people have Epstein–Zin preferences over consumption. We now assume, though, that workers die with probability (1 − δ) , yielding the utility function ˜ ˜ log Ci,t + βδ log Et exp ((1 − α) vi,t+1 ) vi,t = 1 − βδ 1−α

(32)

Specifically, since human capital is on a discrete space, x∗i,t − xi,t also lies on a discrete space. We calculate the net present value of wages for a worker at each value of x∗i,t − xi,t (up to a bound) on each date and then calculate gL,t as the difference between the NPV of wages for a person who is currently unemployed and NPVs for workers calculated on date t and summed over the density of workers at each x∗i,t − xi,t state (that density is taken to be fixed over time for simplicity). 17

17

When workers die, they are replaced by workers with xi,t = 0.18 The key effect of worker death (which we interpret as stochastic retirement) is to reduce the NPV of wages and make layoffs relatively more consequential – to a person who plans to work into eternity, losing a few months of income is not particularly painful. The aggregate dynamics of the model are ultimately determined by the composite parameter ˜ β ≡ βδ

(33)

We show in the appendix that, due to the properties of the consumption process provided by the household, all agents in the economy agree on the pricing of aggregate risks, in the sense that they all agree on the prices of securities whose returns depend only on the aggregate state of the economy. There is thus a pricing kernel for aggregate risks (including riskless assets), which we denote Λagg,t+1 , Λagg,t+1

exp ((1 − α) vt+1 ) Et+1 −α−1 exp (−α∆˜ xi,t+1 ) = β exp (−∆ct+1 ) Et [exp ((1 − α) (vt+1 + kt+1 ))]

where vt = (1 − β) ct +

β log Et [exp ((1 − α) (vt+1 + kt+1 ))] 1−α

(34)

(35)

and kt is the certainty equivalent from (30). The only financial asset that is relevant for the model is a single-period nominal bond, which $ . The Euler equation for the nominal bond is has a gross interest rate of Rt,t+1

Pt $ 1 = Et Mt,t+1 Rt,t+1 Pt+1 where Mt,t+1 = Λagg,t+1 exp (ut )

(36) (37)

Pt is the aggregate price level. Mt,t+1 is the market pricing kernel, which can differ from the pricing kernels of individuals, Λagg,t+1 , through an exogenous wedge ut . ut plays the role played by shocks to the time discount factor in many other New Keynesian models (e.g. Smets and Wouters (2007)); it shifts the equilibrium real interest rate conditional on the path of real consumption.

5.4

Monetary policy

We assume that the central bank sets the nominal interest rate following a linear rule that allows it to potentially respond to the layoff rate, $ $ rt,t+1 − r¯$ = ρπ rt−1,t − r¯$ ¯ + (1 − ρπ ) bπ (π t − π ¯ ) + by (yt − y¯) − bφ φt − φ

+ εµ,t

(38)

18 For simplicity, the new worker inherits the dead worker’s job status, including the state of the wage contract xi,t − x∗i,t .

18

where µt is a shock to the policy rate and bars over variables represent steady-state values. Given the expressions for welfare, it is perhaps more natural for policy to respond to the certainty equivalent kt than to the layoff rate. But kt depends on risk aversion and the consumption loss following a layoff, neither of which is directly observable empirically. Furthermore, in simulations of the model we find that φt is nearly perfectly correlated with kt at 99.1 percent.

5.5

Exogenous processes

There are three persistent shock processes: technology (At ), markups (ψ t ), and discount rates (ut ). They all follow AR(1) processes with independent and normally distributed innovations,

5.6

log At = ρA log At−1 + εA,t

(39)

ψ t = ρψ ψ t−1 + εψ,t

(40)

ut = ρu ut−1 + εu,t

(41)

Solution

We solve the model with a standard first-order approximation around the non-stochastic steadystate. In the first-order approximation, the deviations from the non-stochastic steady-state of the variables in the model, here in a vector Xt , follow a first-order VAR, Xt = GX Xt−1 + Gε εt

(42)

where εt is a vector containing the four innovations to the exogenous processes. The matrices GX and Gε are determined by the solution of the model.

5.7

Calibration

Table 3 reports the benchmark calibration of the model. The majority of the parameters, e.g. those related to price stickiness and monopolistic competition, have been extensively discussed in the literature. We set the persistence of the markup, demand, and technology shocks to 0.95 to generate business-cycle frequency fluctuations. The shock volatilities are chosen to match the standard deviation of inflation since 1982 (0.31) and also its unconditional variance decomposition according to Smets and Wouters (2007) (where we assume that the variance accounted for by the price and wage markup shocks in their model are accounted for purely by the price markup shock in our model).19 The preference parameters are calibrated as above: β = 0.981/4 and α = 5 (as in Lucas (1987)). 19

The volatilities are chosen so that the fraction of the variance of inflation accounted for by the markup, demand, monetary, and technology shocks are 85, 4, 7, and 4 percent, respectively.

19

5.7.1

Idiosyncratic risk

We calibrate the idiosyncratic human capital process to have a standard deviation of 7 percent per quarter. That parameter is chosen to generate a path of wages following a layoff that generates the closest possible fit to the empirical path reported by Davis and von Wachter (2011) in terms of mean squared error. Figure 1 plots the average wage differential between workers who are fired at a random point in year 1 and a sample of otherwise identical workers who are not fired in that year. The blue line plots the model-implied path, red is the empirical path.20 The paths are qualitatively similar, though not identical. The initial wage decline following a layoff is much larger in our calibration than in the data, but wages also recover more quickly than observed empirically. The cumulative loss in income is 297 percent of initial annual earnings in Davis and von Wachter’s data and 314 percent of initial annual earnings in our model. In terms of raw risk, the standard deviation and skewness in 5-year wage growth in the model are both similar to what is observed by Guvenen et al. (2015). The cross-sectional standard deviation of 5-year log wage income growth in the model is 0.34, and Kelley’s skewness is -0.25. The standard deviation is substantially smaller than what Guvenen et al. report for workers at the median wage, approximately 0.65. The skewness is more similar – Guvenen et al. report cross-sectional Kelley measure of skewness (based on percentiles to control outliers) of 0.14 in expansions and 0.30 in recessions. Since the wage contract states that the idiosyncratic component of wages can only rise during a particular job, it implies that there should be an increasing profile of wages as tenure rises, even though on average there is no increase in skill with experience. As another check of the model’s parameters, then, we ask how realistic its implications for the relationship between wages and seniority are. Figure 2 plots the model-implied relationship between average wages and tenure over a 20-year employment period compared to the estimated relationship from Buchinsky et al. (2010). The model implies that wages rise on average by 35 percent (in log terms) over a 20-year period, approximately half the empirical increase of 74 percent measured by Buchinsky et al.. There thus remains substantial room for wage increases due to other factors like job-specific human capital. We calibrate the death rate, δ, so that workers have on average 20 years remaining in their working lives. The calibration of the parameters d and δ imply that at the steady-state, the net present value of wages declines by 31 percent following the average layoff. If consumers are able to insure 1/3 of human capital losses (e.g. if they have financial wealth equal to 1/3 percent of human wealth; see Blundell, Pistaferri, and Preston (2008) for empirical estimates supporting this magnitude) then that income loss corresponds to a 20.4 percent decline in consumption. Finally, the model of idiosyncratic risk also has implications for the quit rate. In any given quarter, 11.9 percent of workers have x∗i,t = xi,t . That is the set of workers who would be willing to potentially quit their jobs. At the model’s steady-state, then, assuming workers are hired 20

The empirical path is measured as 0.2 times the path Davis and von Wachter (2011) measure in recessions and 0.8 times the path in expansions.

20

proportionally from the pool of unemployed workers and those who are employed and willing to quit, the implied quit rate is 3.2 percent per quarter. In the JOLTS data from the BLS, the average quit rate is 5.5 percent. The model thus gives a plausible description of average quits (presumably sometimes workers quit even when xi,t < x∗i,t , e.g. due to non-pecuniary factors). 5.7.2

The layoff rate

Since, as discussed above (and again below in section 6.1), it is the long-run standard deviation of the layoff rate that determines welfare, that is a key moment to target in the calibration. Specifically, for a variable xt , the long-run standard deviation is21 std Et+1

∞ X

xt+j − Et

j=1

∞ X

xt+j

(43)

j=1

The two available time series closest to the concept of layoffs in our model are initial unemployment claims and the JOLTS layoffs and discharges series. Initial claims have a longer sample, and their quarterly long-run standard deviation estimated from an AR(1) model is 5.3 percent. The JOLTS time series is much shorter. In the available sample, though, it suggests that firings may be much less volatile. During the 2008–2009 recession, the layoff rate in the JOLTS data rises by 1.7 percentage points, while it rises by 3.0 percentage points in the initial claims data. We therefore scale the long-run standard deviation from the initial claims data down to match the magnitude of the increase in last recession in the JOLTS data, yielding a calibration target of 3.0 percent.22 Consistent with this calibration, the measure of job destruction in manufacturing from Davis et al. (2010) implies that the long-run standard deviation for the job destruction rate is 3.11 percent. The volatility of the shocks to firm-specific productivity, σ a , controls the dynamics of hiring and firing. We use it to control the volatility of the layoff rate to hit our target long-run standard deviation of 3.0 percent. That value also then implies a mean layoff rate of 5.2 percent per year. Davis and von Wachter (2011) report an average displacement rate (through mass layoffs only) for high-tenure workers of 3.5 percent per year, compared to rates of job destruction, initial unemployment claims, and layoffs from JOLTS that average 14 percent or more. Our mean layoff rate is thus closer to that of high tenure workers.

6

Optimal policy

We now explore the implications of our model for optimal monetary policy. 21

Equivalently, it is proportional to the square root of the spectral density of xt at frequency zero. Since the welfare cost of business cycles is proportional to the long-run variance of the layoff rate (or kt ), using the long-run standard deviation from the initial claims data without rescaling it to match the JOLTS data would generate welfare costs of business cycles roughly four times larger than in our benchmark calibration. 22

21

6.1

Calculating welfare

In order to evaluate alternative policies, we calculate welfare for each policy choice. Define two vectors, ec and eφ , that select elements of the vector Xt so that ec Xt = ∆ct

(44)

ek Xt = kt − k¯

(45)

where k¯ is the non-stochastic steady-state of kt and the steady-state of aggregate consumption growth is zero. Denote the cross-sectional average scaled level of welfare as v¯ ≡ E [vi,t ]

(46)

The appendix then shows that 0 β 1−α −1 −1 0 ¯ v¯ = cmean + k+ (ec + ek ) (I − βGX ) Gε Σε Gε (ec + ek ) (I − βGX ) (47) 1−β 2 where GX and Gε are from the model solution (42), Σε is the variance matrix of the innovations, and cmean is the unconditional average level of aggregate consumption. This result is a simple extension of the expression for welfare in the linear Gaussian endowment economy analyzed in section 4. The welfare cost of uncertainty again depends on the discounted long-run variance of the shocks, (I − βGX )−1 Gε Σε G0ε (I − βGX )−10 . That is, the effect of each shock on welfare depends on the sum of its impulse response functions for consumption and layoffs, discounted by β.23 The relevant long-run variance is that of (ec + ek ) Xt , i.e. taking into account fluctuations in both consumption growth and the cross-sectional distribution of the innovations. Again, the variance is multiplied by

β 1−α 1−β 2 ,

which accounts for risk aversion and the fact that the shocks

appear on all future dates. We account for the cost of inflation volatility by calculating a second-order approximation for the dynamics of price dispersion, ∆t (similar to Erceg, Henderson, and Levin (2000)). Following Yun (1996), output in the economy is, Yt = At Ntγ /∆t

(48)

where ∆t measures price dispersion, with ∆t = (1 − ξ)

1 − ξΠε−1 t 1−ξ

ε ε−1

+ ξΠεt ∆t−1

(49)

where 1 − ξ is the Calvo probability that an intermediate good producing firm is able to change its 23 That is, the vector of impulse responses for consumption growth at horizonPj is ec Φj Ψ. The discounted long-run −1 j j Ψ. The effect of a unit innovation in the VAR on the level of consumption is then ec ∞ j=0 β Φ Ψ = ec (I − βΦ) term determining utility is the variance of these long-run innovations.

22

price; ε is the elasticity of substitution across intermediates, and Πt is gross price inflation. When inflation is more volatile, ∆ is higher on average, thus reducing average output. Since all output is consumed, a one percent increase in ∆ is associated with a one percent decline in consumption. The appendix shows that the second-order approximation for ∆t around a zero-inflation steadystate is24

∞ 1X log ∆t ≈ ε (1 − ξ)−1 ξ m (Πt−m − 1)2 2

(50)

m=1

Defining, ∞ 1X ε (1 − ξ)−1 ξ m (Πt−m − 1)2 ≡ E 2 m=1 ε = ξ (1 − ξ)−2 var (Πt ) 2

"

¯2 log ∆

# (51) (52)

we measure cmean as ¯2 cmean = cnss − log ∆

(53)

where cnss is average consumption in the non-stochastic steady state of the model (which is invariant ¯ 2 , is to policy parameters and shock volatilities). The welfare cost of inflation volatility, log ∆ precisely the same formula obtained by Erceg, Henderson, and Levin (2000). In this model, government policy does not affect the steady-state of the model; it only affects dynamics following shocks. In examining welfare across policies, then, we can ignore steady-state terms, which yields the final relative measure of welfare that we examine: 0 β 1−α ε (ec + ek ) (I − βGX )−1 Gε Σε G0ε (ec + ek ) (I − βGX )−1 v¯ = − ξ (1 − ξ)−2 var (Πt ) + 2 1−β 2 +terms independent of policy (54) The coefficient on inflation variance in our benchmark calibration is -19.4. Equation (54) is a key result of the model: taking the linearized dynamics (plus the second-order approximation to ∆t ), we are able to calculate welfare exactly, even though there is substantial heterogeneity in consumption. When we discuss optimal policy, what we mean is policy that increases v¯. The only way to do that is to reduce the discounted long-run variances of consumption growth and idiosyncratic risk or the unconditional variance of inflation. To the extent that monetary policy only affects transitory fluctuations in the level of consumption (i.e. it has no effect on the long-run expectation of the level of consumption), it will have only minimal ability to affect the discounted long-run variance. And in the limit as β → 1, monetary policy has precisely zero effect on the long-run variance of consumption growth. On the other hand, there is nothing stopping policy from being able to substantially affect the discounted long-run variance of the layoff rate, regardless of the discount rate. We should thus in fact expect that the quantitative motivation for activist policy 24 Adjusting the calculation to allow for a two-percent steady-state level of inflation has quantitatively trivial effects on the results

23

is purely idiosyncratic risk, rather than the fluctuations in aggregate consumption that have often been analyzed in the past. An obvious question is why there is a role for the central bank to play in raising welfare. That is, why is optimal policy active? There are two reasons. First, as usual, because of price stickiness, equilibrium output in general can deviate from what would be chosen by a central planner. More importantly, though, in this model layoffs are associated with uninsured idiosyncratic risk. A small number of workers lose their jobs and see large wage declines. A social planner would insure that risk across workers, but in the model there is no mechanism through which that can happen, so the natural policy response is to try to at least minimize fluctuations in layoff risk. By reducing layoff volatility, they stabilize the amount of insurance that firms provide to workers.

6.2

Welfare across policy rules

We begin by simply examining the performance in terms of welfare of various combinations of coefficients in the policy rule (38). Figure 3 displays contour plots for total welfare, the contribution of fluctuations in idiosyncratic risk, and the welfare loss due to inflation volatility lowering mean consumption (the contribution from variation in aggregate consumption is quantitatively trivial). We vary β π ∈ [1.1, 3] and bφ ∈ [0, 4]. The plots are normalized so that they report the difference between the level of welfare for the particular combination of coefficients and the maximal welfare attainable in the grid of coefficients explored in the plot. by is set to zero for the moment. The values reported for welfare correspond to equivalent percentage-point differences in lifetime consumption. Figure 3 shows that in our benchmark calibration there is a substantial benefit to the central bank responding to the layoff rate. When bπ = 2, increasing the coefficient on layoffs from 0 to 3 raises welfare by 0.52 percent of lifetime consumption. In the context of the optimal policy literature, that is a relatively large change – Schmitt-Grohe and Uribe (2007) find that variation in the coefficient on output in the Taylor rule (in a model without layoff costs) induces changes in welfare of approximately 0.3 percent of lifetime consumption. More importantly, though, they find that responding to measures of real activity is suboptimal, whereas here we find that there are substantial benefits to doing so. The contour plot shows clearly that the peak levels of welfare lie along a diagonal line. Conditional on responding more strongly to inflation, it is also optimal to respond more strongly to the firing rate. When bπ rises, policy will restrain inflation and allow higher layoffs in response to shocks. That is then optimally counteracted by a stronger response to control the layoff rate. The optimal combination in figure 3 is to use as low a coefficient on inflation as possible (while maintaining determinacy), which then also allows a relatively weak response to firings (we note below, though, that this result is not robust to changes in other parameters). The middle and right-hand panels of figure 3 show the two major sources of the welfare differences across the policy rules: their impacts on the long-run standard deviation of idiosyncratic risk (kt ) and their impact on average consumption. The middle panel shows that when bφ increases,

24

welfare costs from fluctuations in idiosyncratic risk decrease. Since the central bank is responding to layoffs more strongly, which are the major driver of idiosyncratic risk (and over 99 percent correlated with kt ), obviously the volatility of kt falls, raising welfare. On the right-hand side we see the cost of that more aggressive policy: inflation becomes more volatile, which reduces mean output and consumption due to the greater dispersion in output across intermediate good producers. One can also note in the two figures that the benefits of increasing bφ are concave, while the costs are convex, which leads to an interior solution. The cost of layoffs is closely related to the coefficient of relative risk aversion, as we saw in table 1. A natural question, then, is whether our result that the central bank should actively try to control the layoff rate is robust to variation in α. To see the effects of variation in α, figure 4 plots the level of welfare as we vary bφ for different values of α (setting bπ = 2.03, as in Smets and Wouters (2007)). As α falls, the optimal value of bφ also falls, until it reaches zero when α = 2. But even for relative risk aversion as low as 3, it is optimal for the central bank to at least respond weakly to the layoff rate. To understand the low risk aversion case better, figure 5 replicates figure 3, but setting α = 3. In this case, we see broadly similar results to figure 3: there is still a ridge in welfare that runs on a diagonal line: high values of bπ are optimally combined with high values of bφ . Compared to figure 3, though, that line is now steeper, indicating that it is generally optimal to use lower values of bφ than when α = 5. Furthermore, the globally optimal policy is no longer towards the bottom-left with low bπ and low bφ – it is reversed, and the central bank should optimally raise both bπ and bφ . So it is a robust finding of our results that bφ > 0 and that it increases with bπ , but whether the {bφ , bπ } pair should be both high or both low depends on the level of risk aversion. Monetary policy rules are typically stated in terms of responses to inflation and the output gap, rather than the layoff rate, A natural question, then, is how well policy works when it responds to output instead of the layoff rate. Figure 6 therefore replicates figure 3, but varying by instead of bπ . We hold bπ = 2.03. Figure 6 shows that there is a negative trade-off between responding to output and the layoff rate – they are complements to some extent. However, welfare is maximized when policy responds purely to the layoff rate and the loading on output is set to zero. Furthermore, unlike with bπ , that result holds even if we reduce α as low as 3. So a prediction of the model is that if the central bank sets its policy rule to maximize welfare, it should respond relatively more strongly to the layoff rate than to output.

6.3

Effects of activist policy on dynamics

Table 4 reports the unconditional and long-run standard deviations of φt , the standard deviation of quarterly inflation, and the standard deviation of output across a range of calibrations of the monetary policy rule. The first three columns show the effect of increasing the response to φt . When bφ rises from 0 to 4, both the unconditional and long-run standard deviations of the layoff rate fall substantially, by one-third to one-half. So increasing the responsiveness of interest rates

25

to the layoff rate can lead to very large increases in the stability of employment. The consequence of controlling employment is that inflation becomes more volatile (due to the fact that we have cost-push shocks in the model). When bφ rises from 0 to 4, the standard deviation of inflation rises as high as 1.31 percent per quarter, or 2.6 percent per year. The middle set of columns shows the effects of increasing by . Increasing by from 0 to 0.8 increases inflation by more than setting bπ to 4, but it reduces the long-run standard deviation of φt by less. Manipulating by is thus clearly suboptimal compared to manipulating bφ since it yields a smaller benefit in terms of layoff volatility at a larger cost in terms of inflation volatility. Finally, the last three columns explore varying bπ . As we would expect, a higher bπ reduces inflation volatility, but it does so at the cost of substantially increasing the long-run volatility of φt . Raising bπ from 1.5 to 2.5 raises the long-run standard deviation of the layoff rate from 2.1 to 2.6 percent, which explains why we find that optimal policy should have a relatively small response to inflation. To explore the effects of responding to the layoff rate in more detail, figure 7 plots impulse response functions for output, inflation, interest rates, and the layoff rate in response to a markup shock in the model for bφ = 0 and bφ = 4. A markup shock is useful to study here because it presents a stark trade-off for the central bank: it must either allow inflation to rise, or raise interest rates, hold inflation stable, and accept a decline in output. Differences in bφ represent two different responses to the shock. When bφ = 0, the central bank holds inflation relatively stable with a maximum response of less than 0.5 percent. When bφ = 4, though, inflation is allowed to rise far higher, by as much as 2.8 percent. The two policies then also obviously yield very different responses for output and layoffs – with bφ = 4, the cumulative output shortfall is only 2/3 as large as with bφ = 0. There is a similar decline in the cumulative sum of layoffs.

6.4

Precautionary saving effects

Thus far we have focused on the implications of layoffs for welfare. But variation in the risk that people also causes variation in how much they want to save. That is, fluctuations in the layoff rate induce fluctuations in demand and hence the natural rate of interest. To see the magnitude of that mechanism in our setting, figure 8 plots responses to a markup shock in the baseline model and in a setting where the baseline Euler equation has been replaced with the one that would be obtained if agents had log utility and there were a representative agent (i.e. a world with perfect insurance). In the baseline model, inflation, output, and interest rates are all lower following a markup shock than they are in the alternative model. In other words, the model endogenously generates a demand effect – a markup shock is associated with increased layoffs and hence low consumption demand. Output, prices, and interest rates then all decline.

26

6.5

The importance of endogenous quits

Recall that welfare in the model depends critically on the long-run standard deviation of the layoff rate. The long-run standard deviation depends on the cumulative sums of the impulse response functions of layoffs to shocks. A shock that raises layoffs above their mean by 1 percent and then reduces them below it by 1 percent makes no contribution to the long-run standard deviation. When the quit rate is constant, all shocks have that characteristic. The intuition is simple. Any shock that reduces the level of employment must increase layoffs. But then as employment recovers, firms must hire more workers and lay off fewer. In a calibration of our model with a constant quit rate, the long-run standard deviation of the layoff rate falls by two orders of magnitude, from 3.0 percent in the baseline to only 0.02 percent. The fact that the model is able to endogenously generate a procyclical quit rate is thus critical to its ability to generate non-trivial welfare costs.

7

Conclusion

Recent results on optimal monetary policy imply that central banks should not use policy rules that attempt to stabilize output. Yet statutory policy mandates and also the actual behavior of central banks suggests that policymakers believe that it is important to try to stabilize the business cycle, and in particular employment. We argue in this paper that the reason that such an emphasis is placed on employment is that there are large welfare losses associated with consumption declines following job loss. Not only are these welfare losses large under standard preference specifications, but we also provide evidence that in fact it is precisely the layoff rate, as opposed to other measures of the state of the business cycle, that the Federal Reserve has historically targeted. We then build an equilibrium model of the business cycle and use it to examine optimal policy, showing that is in fact optimal for the central bank to respond to the layoff rate, and that the welfare gains from such a policy rule can be quantitatively large – up to 0.5 percent of lifetime consumption. While our focus is on optimal policy, the modeling tools developed here are also important as a technical contribution to the literature. Building on work in the asset pricing literature (Constantinides and Ghosh (2013) and Schmidt (2015)) and on Blanchard and Gali’s (2010) model of unemployment, we develop a New Keynesian model with heterogeneous agents and endogenous layoffs that can still be solved and simulated using standard techniques. We also provide novel expressions for welfare that allow for straightforward policy analysis.

27

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31

A

Welfare in an endowment economy

Proposition 1 Suppose aggregate consumption growth and the disaster certainty equivalent kt follow ct kt

c = k

c

=

k

"

(ct

c) + "c;t

1

kt

1

(2)

2 c

N 0;

c;t

(1)

k + "k;t "k;t

N 0;

2 k

(3)

Lifetime utility is then vi;t

i;t

= c+ +

Proof: Subtracting

i;t

vi;t

k

1

kt

k +

2

1 1

k

1

k k

2

+

1

k

(4) 1 1

2 c c

!

(5)

from both sides of the recursion for lifetime utility yields, vi;t

=

(1

) ci;t +

i;t

=

(1

) ct +

log Et exp ((1

1

log Et exp (1

1

) vi;t+1 ) ) vi;t+1

(6) i;t+1

+

i;t+1

(7)

Now note that the law of iterated expectations log Et exp (1 =

) vi;t+1

log Et exp (1

i;t+1

) vi;t+1

+

(8)

i;t+1

Et+1 exp (1

i;t+1

)

(9)

i;t+1

where in a slight abuse of notation, Et+1 here denotes the set of information available on date t + 1 excluding the realization of i;t+1 (it may also be thought of as the cross-sectional mean on date t + 1). Inserting the de…nition of kt+1 yields log Et exp (1

) vi;t+1

i;t+1

+

i;t+1

= log Et exp (1

) vi;t+1

i;t+1

+ kt+1

(10)

The desired result then can be easily con…rmed (it can be derived more constructively by guessing that lifetime utility is a linear function of the state variables and then solving using the method of undetermined coe¢ cients).

A A.1

Intermediate goods …rms’optimization Law of motion and quits

Each …rm faces a quit rate qt . They can …re workers Fi;t size of each …rm’s labor force is Ni;t = (1 qt ) Ni;t 1

0 or hire Hi;t Fi;t + Hi;t

0. The law of motion for the (11)

The speci…c timing is 1. Firms choose the number of workers to hire. These workers are hired partly from unemployment and partly from the pool of current workers, so quits also happen in this step. 2. Firms decide the number of workers to …re. In the aggregate when we sum across all …rms, we have Nt Nt Nt 1

= Nt =

(1

Qt

1

qt ) 1

Ft + Ht Ft Ht + Nt 1 Nt 1 1

(12) (13)

We assume that hiring is taken randomly from the pool of available workers. The workers available to be hired are the unemployed from the end of the previous period, 1 Nt 1 , those who retired at the beginning of the current period, (1 ) Nt 1 , and those whose wage makes them indi¤erent to leaving the …rm, the measure of which we denote '0;t 1 (the dynamics of that variable are derived below). So the fraction of hiring from people employed at the end of period t 1 is then '0;t 1 Nt 1 + '0;t

1

(14)

Total hiring of previously employed workers, which is quits, Qt , is then Qt

'0;t 1 Nt 1 + '0;t

(15)

'0;t 1 = (Nt 1 ) Qt = Ht Nt 1 1 Nt 1 + '0;t

(16)

Ht

qt

1

The fraction of unemployed workers who are hired is Ht

A.2

(1 1

Nt Nt

1

1)

(17)

+ '0;t

Idiosyncratic shocks and optimization

Each individual …rm solves the problem max St At Am;t Nm;t

Wt Nm;t

Ni;t

) Wt = St At Am;t Nm;t1 Nm;t

1=(1

=

(St At Am;t =Wt )

(18) (19)

)

(20)

We assume that idiosyncratic productivity follows a random walk, with am;t

1 1 21

N

2 a;

2 a

Integrating Nm;t across all …rms yields aggregate employment, h 1=(1 1=(1 ) Nt = (St At =Wt ) E Am;t

(21)

)

i

:

(22)

h i h i 1=(1 ) 1=(1 ) is the cross-sectional mean. We can set E Am;t = 1 by suitably normalizing each where E Am;t …rm’s initial level of productivity (and recall the discussion from the text on death). Each individual …rm’s employment growth is nm;t

1

=

1 N

( st +

at

wt +

am;t )

2 n

n;

(23) (24)

where 1 n 2 n

st +

1 1 1

at

wt

1 1 21

2 a

(25)

2 2 a

(26)

2

Aggregate employment growth is Nt = Nt 1

Z

Ni;t di = Ni;t 1

Z

exp ( ni;t ) di

(27)

i

st +

= exp

at

wt

(28)

1

where the …rst equality follows from the fact that employment growth is uncorrelated across …rms with their initial level of employment. Firm i sets Hm;t and Fm;t to achieve the desired change in employment, Nm;t . Speci…cally, if nm;t < log ((1 qt )), …ms hire, and if nm;t < log ((1 qt )), they …re. Aggregate hiring and …ring are then calculated by integrating across …rms (again using to the independence of nm;t and Nm;t 1 ), Z Z Ht Hm;t = dm = 1 f nm;t > log (1 qt )g [exp ( nm;t ) (1 qt )] dm Nt 1 Nm;t 1 = exp

Ft = Nt 1

Z

= (1

n

+

1 2

n

2 n

log (1

qt )

(1

log (1

qt ) 1

qt )

n

Fm;t dm = (1 Nm;t 1

qt )

log (1

qt )

qt )

2 n

+

(log (1 n

n

n

qt )) exp

Z n

n

+

1 f nm;t 6 ln (1 1 2

2 n

1

qt )g exp ( nm;t ) dm n

+

2 n

log (1

qt )

:

n

where 1 f g denotes the indicator function.

B B.1

Idiosyncratic productivity and wages Productivity gap dynamics

We assume that each worker’s productivity follows the process xi;t =

xi;t + d with prob. pup xi;t d with prob. 1 pup

(29)

The parameter pup controls the mean of the income process. We set the mean growth in human capital, exp (xi;t ), to zero for simplicity. De…ning x ^i;t = xi;t xi;t (30) x ^i;t takes on values that are positive integer multiples of d. We denote the fraction of workers with x ^i;t = nd on date t to be 'n;t . 'n;t follows the law of motion 'n;t =

(

(1 Ht 1

(1 t) (1 Nt 1 ) Nt 1 +'0;t 1

pup ) 'n 1;t 1 + pup 'n+1;t 1 for n > 0 + (1 pup '0;t 1 + pup '1;t 1 for n = 0 t)

(31)

For n > 0, the set of workers in state n on date t is equal to the number of workers in the neighboring states (n 1 and n + 1) who are not …red (1 ft ) who move into state n. For n = 0, a worker can move into that state from state 1, remain there (if human capital rises). Furthermore, workers who are hired out of unemployment also enter the labor force with x ^ = 0. We can …nd a steady-state for this object. Guess that it is exponential with decay q. We then solve '0 sn

=

s =

(1 (1

) (1 ) 1

pup ) '0 sn

1 2

pup + pup s 3

+ pup '0 sn+1

(32) (33)

s= Ht

1 1

q

1

Nt

1

4pup (1

pup ) (1

2pup (1

Nt 1 + (1 1 + '0;t

)

2

(34)

) t)

pup (1 + s) '0

(35)

and '0

B.2

= H

H (1 ('0 + 1

N) N ) '0

=

('0 + 1

N ) '0

=

1 N + (1 '0 + 1 N

1

(1

1

(1

t ) pup

t ) pup

(1 + s) '0

(36)

(1 + s)

(37)

N) ) p up (1 + s) t

(38)

H (1

The present value of wages

It is possible to calculate the present value of wages for each possible value of x ^. Note that the wage of a worker with productivity gap x ^ is Wi;t

=

exp xi;t

w Wt

(39)

Wi;t = exp (xi;t )

=

exp (^ xi;t

w) Wt

(40)

The net present value of wages for a person in state n (^ x = nd) scaled by current productivity is denoted npv, 1 X m s N P Vn;t = npvn;t exp (xi;t ) Et Mt+m Wi;t+m (41) m=0

where the subscript n indicates the current state of x ^ and the superscript s indicates employment status. The NPV follows a recursion,

E npvn;t = exp (^ xn;t

npvtU

2

2

w) Wt + Et 4Mt+1 4

"

= Et Mt+1

"

Ht '

0;t

+ 1

+ 1 1

1 +1

Ht '

33

U 1 t+1 npvt+1 2

0;t

t+1

1 2

(exp (d) + exp ( d)) E npvmax(n 1;0);t+1 exp (d) E +npvn+1;1;t+1 exp ( d)

E Nt 1 npv0;t+1 1 U npvt+1 1 +1 Nt 1

#

1 (exp (d) + exp ( d)) 2

#

55

(42)

(43)

Finally, we need to calculate gL;t . We assume that the household reduces the consumption of workers who are laid o¤ in period t by the average decline in the NPV of wages for workers laid o¤ in that period. We calculate that as ! !! 1 E E X npvmax(n npvn+1;t 1 1;0);t gL;t = 'n;t 1 log + log (44) 2 npvtU npvtU n=0 That is, the average NPV loss that the household calculates takes the distribution of workers by x ^ state at the beginning of period t and averages the NPV loss that they would have received. As noted in the main text, we then also have 1 t exp ( gL;t ) gH;t = log (45) (1 t)

4

C

Price setting equations for the di¤erentiator …rms

None of the results here are novel to this paper; they can be found throughout the literature. We index the di¤erentiator …rms by i. Each di¤erentiator has the production function Yt (i) = Xt (i)

(46)

where Yt (i) is the output of …rm i, and Xt (i) is the quantity of the intermediate good that it buys. There is a competitive sector of …nal good …rms that combine the intermediate goods, Yt (i), into …nal output, Yt , according to the technology Z

Yt =

"

1

"

Yt (i)

"

1

1

(47)

di

"

0

Di¤erentiator i charges price Pt (i) for its goods. The price of …nal output is Pt =

Z

1

1

1 "

Pt (i)

1

"

di

(48)

0

and the demand curve for each di¤erentiated good is Yt (i) = Yt (Pt (i) =Pt )

"

(49)

The probability that a di¤erentiator can change its prices in any given period is 1 . If it cannot change its price, the price remains at the same level as in the previous period (there is no indexation). Firms maximize the present value of pro…ts over the period that the price is …xed, discounting using the market pricing kernel, Mt;t+1 . We denote the price that the …rms that optimally choose on date t as Pt . The equilibrium conditions that come out of the maximization are Pt =Pt x ~1;t

"

=

"

x ~1;t exp ( 1x ~2;t

t)

= St + Et Mt;t+1 "

x ~2;t

=

1 + Et Mt;t+1

1 " t

=

(1

(50) Pt+1 Pt Pt+1 Pt

1 "

1 " t

) (Pt =Pt )

"

Yt+1 x ~1;t+1 Yt

" 1

(51)

Yt+1 x ~2;t+1 Yt

#

+

(52) (53)

where x ~1;t and x ~2;t are endogenous variables that arise as part of the optimization problem. Following Yun (1996)?, we can write the aggregate production function as Yt = At Nt = where

t

(54)

t

measures price dispersion, with t

(1

)

1 X

j

Pt

j=0

Pt j j Pt

"

(55)

The …rm’s optimization problem in setting prices is max Et

1 X

Mt;t+j

j=0 1 X

= max Et

j=0

Pt Pt+j

Mt;t+j

j

Pt Pt+j

[Pt j

5

[Pt

PI;t+j ] Yt+jjt PI;t+j ]

Pt Pt+j

"

Yt+j

(56)

where the second line follows from the fact that the …rm’s sales in each period given a price Pt set in period "

P

t Yt+j . t are Yt+jjt = Pt+j The …rst-order condition yields

"Et

1 X

Mt;t+j

j=0 1 X

"Et

Pt Pt+j

Mt;t+j

j=0

" 1

j

Pt Pt+j

PI;t+j j

(Pt ) Pt+j"

Yt+j

PI;t+j Yt+jjt (Pt )

=

1

(1

=

") Et

(1

") Et

1 X j=0 1 X

Mt;t+j

Pt Pt+j

Mt;t+j

Pt Pt+j

j=0

Pt =Pt

Et

"

=

"

1

j

j

Pt Pt+j

"

Yt+j

Yt+jjt

(57)

(58)

P1

j PI;t+j Pt j=0 Mt;t+j Pt+j Pt Yt+jjt P1 j Pt Et j=0 Mt;t+j Pt+j Yt+jjt

(59)

The numerator and denominator of the above are recursive, so we obtain x1;t

= Et

1 X

Mt;t+j

j=0

Pt Pt+j

PI;t Yt Pt

Pt Pt

"

PI;t Yt Pt

Pt Pt

"

=

PI;t Yt Pt

Pt Pt

"

=

=

j

PI;t+j Yt+jjt Pt

(60)

Pt+1 pt

+ Et Mt;t+1

1 "X

Mt+1;t+j

j=0

Pt+1 pt

"

+ Et Mt;t+1

Pt+1 Pt

"

+ Et Mt;t+1

Pt+1 Pt+j

j

PI;t+j Yt+j+1jt+1 Pt+1

x1;t+1

(61)

(62)

Pt+1 =Pt+1 Pt =Pt

"

x1;t+1

(63)

where the second line uses the fact that yt+jjt+1 =

Pt+1 Pt+j

"

Yt+j =

Pt+1 pt

"

Yt+jjt

(64)

For the denominator, x2;t

=

Et

1 X

Mt;t+j

j=0

=

=

Pt Pt

"

Pt Pt

"

Pt Pt+j

j

Yt+jjt

Yt + Et Mt;t+1 "

Yt + Et Mt;t+1

(65) Pt+1 Pt Pt+1 Pt

" 1

Pt+1 =Pt+1 Pt =Pt

" 1

We …nally have Pt =Pt =

Pt+1 =Pt+1 Pt =Pt "

"

Note also that we can rescale x1;t and x2;t , de…ning x ~j;t

1 "X

Mt+1;t+j

j=0 "

x2;t+1

#

x1;t 1 x2;t xj;t

Pt+1 Pt+j

j

Yt+j+1jt+1

(66)

(67)

(68) pt Pt

"

=Yt . We augment the model with a

shock to markups, exp ( t ), "

x ~1;t

=

x ~2;t

=

PI;t Pt+1 Yt+1 exp ( t ) + Et Mt;t+1 x ~1;t+1 Pt Pt Yt " # " 1 Pt+1 Yt+1 1 + Et Mt;t+1 x ~2;t+1 Pt Yt 6

(69) (70)

The aggregate price index follows 1 " t

= (1

) (Pt =Pt )

1 "

1 " t

+

(71)

The conditions for price setting that go into the model are then Pt =Pt x ~1;t

where st

C.1

=

" "

x ~1;t 1x ~2;t

= st exp (

(72)

t)

Pt+1 Pt

+ Et Mt;t+1

"

x ~2;t

=

1t + Et Mt;t+1

1 " t

=

(1

) (Pt =Pt )

" 1

Pt+1 Pt

1 "

1 " t

"

Yt+1 x ~1;t+1 Yt #

(73)

Yt+1 x ~2;t+1 Yt

(74)

+

(75)

PI;t =Pt .

Welfare in log-linear equilibria

This section derives the welfare criterion for the full model. Lifetime utility is vi;t = (1

) ci;t +

log Et exp ((1

1

) vi;t+1 )

(76)

De…ne vci;t

vi;t

ci;t

(77)

We then have vci;t =

log Et exp ((1 ) (vxi;t+1 + ct+1 + x ~i;t+1 )) 1 The deviations of the state variables from their non-stochastic steady-states follows Xt = GX Xt

1

(78)

+ G" "t

(79)

for a vector of normally distributed innovations "t with covariance matrix We guess that vci;t = v + ev Xt

".

Xt may include a constant. (80)

for some unknown vector ev . De…ne ec and ek such that

(noting that the steady-state of the right-hand side of (??) as 1 = =

1 1

log Et exp ((1

ec Xt

=

ek Xt

=

ct kt

(81) k

(82)

ct is equal to zero). Using the law of iterated expectations, we can write

) (v + ev Xt+1 + ec Xt+1 +

x ~i;t+1 ))

log Et exp ((1

) (v + ev Xt+1 + ec Xt+1 )) E exp ((1

log Et [exp ((1

) (v + ev Xt+1 + ec Xt+1 + kt+1 ))]

(83) )( x ~i;t+1 )) j

t+1 ; gL;t+1

(84) (85)

We then guess that there is a linear solution to the recursion, v + ev Xt

= =

1

log Et exp (1

) v + ev Xt+1 + ec Xt+1 + ek Xt+1 + k

v + (ev + ec + ek ) GX Xt + k + 7

1 2

(ev + ec + ek ) G" G0" (ev + ec + ek )

(86) 0

(87)

Matching coe¢ cients yields v=

1

k+

1

ev

(ev + ec + ek ) G" G0" (ev + ec + ek )

2 =

0

(88)

(ev + ec + ek ) GX

(89)

(ec + ek ) GX (I

GX )

1

ev + ec + ek = (ec + ek ) (I

GX )

1

=

(90) (91)

So then …nally, vi;t

ci;t

=

(ec + ek ) GX (I k+

1

1

GX )

1

Xt +

(ec + ek ) (I

2

(92) 1

GX )

G" G0" (I

10

GX )

(ec + ek )

0

0

(93)

Since E [Xt ] = 0, and since E [~ xi;t ] = 0, we have E [vi;t ] = E [ct ] +

k+

1

1

(ec + ek ) (I

2

1

GX )

G" G0" (I

10

GX )

(ec + ek )

0

0

(94)

which is the result from the text.

D

In‡ation costs

The distortion is expressed as follows: = (1

t

)

"

" 1 t

1

"

1

" t

+

1

t 1:

Thus the scaled distortion is bt ,

t

=

1

"

" 1 t

1

"

1

+

1

1 X

j

j=1

jY1

" 1 t j

1

" t k

1

k=0

!""1

:

(95)

We approximate (95) in second order around the zero-in‡ation steady state: bt =

1 X @ bt ( @ t j j=0

1) +

t j

1 1 X X 1 j=0 k=0

@2 b t 2 @ t j@ t

(

1) (

t j

1) :

t k

k

It is easy to show

@

@ b t = t

" "

="

" 1 t

1 1

1 "

("

1 " 1 t

1

" 2 t

1

1)

1 "

1

1

" 2 t

1

+

1 +

1 X

j

jY1

1

"

t

j=1

1 X

j

"

k=0

jY1

1 t

j=1

" t k

1 " ! " 1 1

!""1

" t j

1

" t k

" 1 t j

1

1

k=0

Also @ @

t l

bt =

l

=

l

lY 1

k=0 lY 1

k=0

" t k

"

" t k"

" 1 t l

1

" 1

1 " 1 t l

1 1

!"

!"11

(" 1) 1

1 1

1

" 2 t l

+

" 2 t l

+

j

"

1 t l

j=l+1 1 X

j=l+1

8

1 X

j

"

1 t l

jY1

" t k

k=0 jY1

k=0

" t k

" 1 t j

1 1 " 1 t j

1 1

!""1

:

!""1

At the zero-in‡ation steady state, these …rst order e¤ect is zero.

@

@ b t ="

1

t

="

@ @

t l

bt = =

+

l

" "

j

"

j=1

+

1 l

1 X

+

1

"=0

1

1 X

j

"

j=l+1

+

1

"

l+1

1

= 0: As by-products, we know that at the steady states, 0 j 1 1 @ @X j Y " t @ t l j=m

" 1 t j

1 k

k=0

1

!""1 1

The following function, F1 ; is prepared for notational convenience.

F1 ( ) = =

@

1

t l

1 "

" 1 t l

1

@

" 1

1 1

=

(" 1

1 1

1 "

1

1 1

1

" 2

" 2

1

2)

("

(96)

" 2 t l

(" 1) 1

1 2

F1 (1) =

!"11

A = 0:

2) :

9

+

" 1

1 1

1 "

1

(" 1

2)

" 3

The second order terms are 0 @ @ 1 @ b " t = @ 2t @ t 1 0 = "F1 (

@ @ t@

t l

@2 @

2 t l

bt =

@ @ t 0

B B = "B B @ bt = =

t)

l

@ l

t l lY 1

@ |

0 @

l

2 t

l

lY 1

j

=

k=0

{z

1

" t k"

@ (

" t k "F

(

+

" 1 t l

1

t l

t l)

+

@ @

t l)

!"11

!"11 1 t l

t l

2 t l

" 1 t j

1

" 1 t l

1

@

j 2

"

j=l+1

jY1

1 X

j

1

1 X

j

2 t

j=1

1

jY1

jY1

" t k

1 X

" 1 t j

1

j

@

"

1 t l

t l

1 t l

1 X

j

"

j=l+1

!""1 1

1 " 1 t j

1

jY1

" t j

1

k=0 " t k

1

!""1

"

1

:

Then for m > l (wlog),

@

t

@2 m@

t l

bt = +

=

=

@ @

t m

@ @

t m

@ @

t m

@ @

t m

l

lY 1

" t k"

k=0 1 X

j

"

1 t l

j=l+1 1 X

" 1 t l

1 1 jY1

" t k

!"11

j

"

1 t l

jY1

j=l+1

k=0

1 X

jY1

j

"

j=l+1

1 t l

k=0

10

" t k

" t k

1 " 1 t j

1 1

k=0

" 1 t j

1

1

!""1 1

A:

!""1 1 A

" 1 t j

1 1

" 1 t j

1 1

" t k

" 2 t l

!""1 !""1 !""1

= 0:

jY1

k=0

" 1 t j

1 1

k=0

@

" t k

" t k

1

j=l+1

jY1

A

(* 96)

+

+

!""1 1

k=0

k=0

1

" 2 t l

"

jY1

!"

" 1 t j

+"

t

k=0

1

k=0

j

" t k

!""1 C C C=0 C A }

1

"

j

" 2 t l

j=l+1

1

" t k

1 X

1

1 X

j=l+1 1 X

+

" 1 t j

1

k=0

!""1 1

j=1

1

k=0 2 t l

" 2 t

1

" t k

k=0

" t k "F

lY 1

1

jY1

1 t

1

1

k=0 l

" t k

1 "

j

jY1

" t j

1

k=0

1

1 X

+

j=1

=0

" t k"

lY 1

j

t j j=1

k=0

=

" 2 t

jY1

j=1

@

1

@

1 X

1 X

1

" 1 t

1 "@

t

"

1

+ "@

0

1

" 1 t

" t k

!""1 1 " 1 t j

1 1

A

!""1

Therefore 1

X et = 1 ( l) ( 2

1)

t j

2

()

l=0

t

where l

We can show that

=

1 (1 2

)

1 X

2

( l) (

1) ;

t l

l=0

0

= @ l "F (1)

1 X

j

"+

j=l+1

1 X

j=l+1

1

j 2A

"

:

l+1 l

=

2 ":

(1

)

Finally the cost is expressed as E

t

=E

1

(1 2

)X

( l) (

2

t l

1) =

l=0

1 " = 2 var ( 2 (1 )

t) :

11

1

(1 2

)X l=0

l+1

(1

2

)

(

2

t l

1)

Table 1: Pairwise correlations with Fed Funds rate innovations Variable Exponentially filtered initial claims Initial claims Change in unemployment rate Initial claims/total employment HP-filtered log output Output growth HP-filtered unemployment rate Unemployment rate PCE inflation Core PCE inflation

Correlation

R2

-0.60 -0.59 -0.55 -0.45 0.42 0.39 -0.35 -0.28 0.07 -0.07

0.36 0.35 0.30 0.20 0.18 0.15 0.12 0.08 0.00 0.00

Notes: The innovation in the Fed Funds rate is measured with an AR(1) regression.

1

Table 2: Estimated Policy Rules Panel a. Backward looking rules Lagged interest rate rt−1

0.928***

0.944***

0.941***

0.930***

0.936***

0.923***

0.939***

0.932***

0.921***

0.928***

(45.17)

(47.29)

(47.24)

(48.55)

(49.10)

(38.35)

(36.39)

(37.11)

(39.51)

(39.67)

0.212

0.227

0.239

0.317

0.266

(1.22)

(1.36)

(1.43)

(1.96)

(1.65)

-2.751***

-3.245***

-3.210***

-2.157***

-1.833***

-2.758***

-3.238***

-3.190***

-2.283***

-1.952***

(-7.09)

(-6.60)

(-5.80)

(-6.49)

(-4.74)

(-6.99)

(-6.44)

(-5.69)

(-6.87)

(-5.00)

PCE inflation

HP IC 0.055

0.042

Unemployment rate (1.23)

(0.80) 0.152*

0.137

(2.02)

(1.70)

HP unemployment rate -9.709

-8.437

(-1.60)

(-1.34)

HP log output

2

0.246**

0.224*

(2.85)

(2.59)

Output growth -0.630**

-0.593**

Δunemployment rate (-2.86)

(-2.66) 0.247

0.223

0.277

0.346

0.292

(0.94)

(0.98)

(1.24)

(1.66)

(1.39)

Core PCE inflation -0.202

0.020

0.025

-0.148

0.035

-0.122

0.050

0.047

-0.103

0.059

(-0.85)

(0.16)

(0.20)

(-1.06)

(0.29)

(-0.48)

(0.42)

(0.39)

(-0.77)

(0.50)

Constant Observations

103

103

103

103

103

103

103

103

103

103

Adjusted R2

0.967

0.968

0.968

0.969

0.969

0.967

0.968

0.968

0.969

0.969

Panel b. Forward looking rules Specification

k = 0, q = 0

k = 1, q = 1

k = 4, q = 1

k = 1, q = 2

k = 4, q = 2

0.571***

0.391***

0.597***

0.302

0.450***

0.107

0.634***

0.205

0.435***

0.032

(4.67)

(3.37)

(4.46)

(1.91)

(4.16)

(0.92)

(4.42)

(1.33)

(3.78)

(0.26)

0.365***

0.214***

0.424***

0.232*

0.242**

0.114

0.481***

0.289**

0.254**

0.174

Constant

Inflation πt,t+k

Output gap yt,t+q

Lagged interest rate rt−1

(4.44)

(3.43)

(4.87)

(2.28)

(3.15)

(1.37)

(5.46)

(3.02)

(3.16)

(1.92)

0.140*

-0.255***

0.189**

-0.114

0.223***

-0.186*

0.290***

-0.030

0.288***

-0.090

(2.51)

(-5.04)

(3.26)

(-1.13)

(3.45)

(-1.98)

(4.15)

(-0.30)

(3.72)

(-1.00)

0.894***

0.923***

0.884***

0.930***

0.914***

0.974***

0.871***

0.948***

0.913***

0.982***

(38.40)

(45.53)

(35.11)

(31.20)

(42.20)

(43.79)

(32.78)

(33.61)

(40.50)

(43.75)

Layoff st,t+q Observations

99

0.607***

0.591***

0.685***

0.520***

0.592***

(9.55)

(4.91)

(6.09)

(4.71)

(6.06)

99

98

98

95

95

97

97

95

95

P-value: layoff coef > Output gap coef

0.000

0.000

0.000

0.002

0.000

P-value: payoff coef > Infl. coef

0.000

0.027

0.000

0.087

0.003

3

Notes: The top panel estimates a backward looking monetary policy rule by OLS. The bottom panel estimates a forward looking policy rule using GMM. The set of instruments includes four lags of inflation: output gap, the federal funds rate, the short-long spread, and commodity price inflation as in Clarida, Gali and Gertler (2000). k and q refer to the number of future quarters considered for expected inflation and the output gap respectively. Whenever expected initial claims are included as a regressor we always include it with the same time horizon as the output gap. All regressors are standardized so that their coefficients can be compared. Furthermore, HP-IC was multiplied by negative one so that it would have the same expected sign as the output gap in the regressions. In all panels, numbers in parentheses are t-statistics. *** indicates significance at the 1 percent level, ** the 5 percent level, and * the 10 percent level. The dependent variable is the target Fed funds rate. All data is quarterly and averaged within the period and the time period is 1982q1-2007q4.

Table 3: Parameter values for structural model Parameter

Value

β ξ α γ ζ d pup σa ρA ρψ ρu σA σψ σu ρπ

0.998 0.6 5 .6 .5 .07 .48

Description

Time discounting Calvo parameter Risk aversion Power for the production Power for wage schedule Standard deviation for log human capital Probability of an increase of log human capital Standard deviation of ∆ai,t 0.95 Persistence of productivity .95 Persistence of markups 0.95 Persistence of discounting 0.00095 Volatility of productivity 0.00535 Volatility of markups 0.00142 Volatility of discounting 0.22 Interest rate smoothing parameter

Table 4: Moments under different policy rules std (πt ) 0.12 0.85 1.5 0.37 1.35 2.47 1.27 0.85 0.65 LRSD (φt ) 3.55 2.78 2.35 3.33 2.82 2.33 2.48 2.78 2.93 std (φt ) 1.4 0.62 0.49 1.18 0.78 0.59 0.55 0.62 0.69 std (Yt ) 4.41 3.34 2.93 4.13 3.63 3.26 3.06 3.34 3.5 bφ 0 2 4 0 0 0 2 2 2 bπ 2 2 2 2 2 2 1.5 2 2.5 by 0 0 0 0.1 0.4 0.8 0 0 0 Notes: The various moments are all in percentage terms. Inflation is quarterly. The other coefficients in the policy rule, with the exception of persistence, are all set to zero.

4

Figure 1: Wage losses following layoffs -0.05

Wage loss relative to starting wage

-0.1 -0.15 -0.2 -0.25 -0.3 -0.35 -0.4 -0.45

Data Model

-0.5 0

2

4

6

8

10

12

14

16

18

20

Year

Notes: Wage losses following layoffs compared to a control group of workers who are not laid off. Average wage differentials include workers with zero earnings and are scaled by initial earnings.

Figure 2: Wage profiles with seniority 0.8 0.7

Log wages

0.6 0.5 0.4 0.3 0.2 0.1

Model Buchinsky et al. estimates

0 0

10

20

30

40

50

Tenure

Notes: relationship between average log wages and years of tenure.

5

60

70

80

Figure 3: Welfare across policy rules Idiosyncratic risk fluctuations Mean consumption loss 2.8

-1.076 6

2.6

2.2

2.2

2.2 8

1.8

26

34

1.2

1.2 0

2 bφ

4

-0.

1.4

-1.615

1.4

1.6

53 83 2

1.6 -0 -0.0 .1 65 30 26 54 8

1.6

1.8

1.4

16

.

-0

0

2 bφ

73

03

9 26

-0.11

1.8

-0.8

-0.0

-0 .

2

074

68

2

652

2

bπ

2.4

bπ

2.4 -0.78322 -0.65268 -0.52215 -0.39161 -0.2610 7

2.4

-0.2 346 -0 -0.5-0.46 0.351 1 .7 8 9 03 65 22 91 83 2

2.6

-1.3458

-0.13

2.6

bπ

2.8

054

2.8

3

-1.615

3

073315 -0.-0.0 0146 63 -0.02932 6 -0.0586 52

Total

3

1.2 4

0

2 bφ

4

Notes: Contour plots of levels of welfare across policy rules. ’Total’ is overall welfare, Idiosyncratic risk fluctuations’ is the contribution from fluctuations in the idiosyncratic risk, and ’Mean’ is the contribution from the mean of price dispersion. Values of welfare reported in the figures are relative to maximum welfare on the parameter grid.

Figure 4: Optimal policy 0 -0.01

Welfare

-0.02 -0.03 -0.04 -0.05

α α α α α

-0.06

=2 =4 =5 =6 =8

-0.07 0

1

2

3

4

5

6

firing rates coefficient

6

7

8

9

10

Figure 5: Welfare gains for reduced risk aversion (3) Total

3

2.8

2.8 2.6

2.4

2.4

2.2

2.2

2

2

553 -0.2

bπ

bπ

-0.1

9

032

5

-0.01

2.6

-0.0079809 -0.015962 -0.031924 -0.0638 47

-0.20

996

3

2

1.2

0

2 bφ

769 -0.12

7 574

8

-0.1

49

1.4

-0

-0 .

258

1.6

.10

1.4

-0.0

1.2

1.6

61

12

1.4

1.8 -0.31493 -0.2624 4

1.6

1.8

-0 .41 29 94 8 8

-0.012906

1.8

-0. 20

64

-0 -0 -0 .7 .6 .5 66 38 10 0.383 17 47 78 08

9

bπ

2.2

-0.051623 -0.025812

2.4

-0.

2.8 2.6

Idiosyncratic risk fluctuations Mean consumption loss

00 12 90 6 2906 -0.02 5812 -0.0 516 23

3

4

1.2 0

2 bφ

4

0

2 bφ

4

Notes: Levels of welfare for varying levels of bπ and α. The red circles indicate maximal welfare for each value of α.

Figure 6: Welfare across policy rules (bφ and by ) Total

0.5 0.45

Idiosyncratic risk fluctuations Mean consumption loss

-0

0.4

0.5

0.45

0.45 -0.107

.3 56 07

0.5

0.4

0.4

0.05

by

by 9

0.1

0.15

4 -1.073

-0.1 186

0.15

0.1

0.05

0

0.05

0 0

2 bφ

4

0.25 0.2

71

0.1

-0.35607 476 -0.47

0.15

0.2

-0.858

0.2

0.25

4 987 798 -0.4 -0.39 199 92 -0.3 9 3 5 -0.2 599 -0.1 74 799 -0.0

0.25

0.3

35

-0.23738

0.3

68 -0.214

0.3

0.35

03 -0.644

by

34

0.35 -0.429

0.35

0 0

2 bφ

Notes: See figure 3.

7

4

0

2 bφ

4

Figure 7: Impulse responses to a markup shock with and without policy response to firings Response of inflation (annualized percent)

Response of layoff rate (percent) 1

2 0.5 1 0

0 5 0

10

15

20

Response of output (percent)

5

10

15

20

Response of interest rate (annualized percent) 1

-0.5 0.5 -1 0 5

10

15

20

5

10

15

20

Notes: The solid lines plot impulse responses when monetary policy places no weight on the layoff rate. The dotted lines plot responses when the weight on layoffs is set to 4.

Figure 8: Impulse responses to a markup shock with and without demand effects Response of inflation (annualized percent)

Response of layoff rate (percent) 1

0.6 0.4

0.5 0.2 0

0 5

0

10

15

20

Response of output (percent)

5

10

15

20

Response of interest rate (annualized percent) 0.4 0.3

-0.5

0.2 0.1

-1

0 5

10

15

20

5

10

15

20

Notes: The solid lines are for the baseline model. The dotted lines are generated by setting the pricing kernel to take the value it would in a representative-agent economy with log utility (i.e. ignoring the effects of layoff risk on consumption demand).

8