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Journal of Economic Dynamics & Control

Journal of Economic Dynamics & Control 35 (2011) 1744–1768

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The (un)importance of unemployment fluctuationsfor the welfare cost of business cycles

Philip Jung a,1, Keith Kuester b,�

a Department of Economics, University of Mannheim, 68131 Mannheim, Germanyb Research Department, Federal Reserve Bank of Philadelphia, Ten Independence Mall, Philadelphia, PA 19106-1574, USA

a r t i c l e i n f o

Article history:

Received 21 July 2009

Accepted 22 May 2011Available online 12 June 2011

JEL classification:

E32

E24

J64

Keywords:

Cost of business cycles

Unemployment

Search and matching

89/$ - see front matter & 2011 Elsevier B.V. A

016/j.jedc.2011.05.008

responding author. Tel.: þ1 215 574 3415.

ail addresses: p.jung@vwl.uni-mannheim.de (

l.: þ49 621 1811854.

r example, 70% of the respondents in Shiller’s

more than 80% agree that smoothing out bo

ve well-being. He finds that eliminating unem

roughly equal to that from lowering the ave

oyment fluctuations even if agents are well i

a b s t r a c t

This paper studies the cost of business cycles within a real business cycle model with

search and matching frictions in the labor market. We endogenously link both the

cyclical fluctuations and the mean level of unemployment to the aggregate business

cycle risk. The key result of the paper is that business cycles are costly: fluctuations over

the cycle induce a higher average unemployment rate since employment is nonlinear in

the job-finding rate and the past unemployment rate. We show this analytically for a

special case of the model. We then calibrate the model to U.S. data. For the calibrated

model, too, business cycles cause higher average unemployment; the welfare cost of

business cycles can easily be an order of magnitude larger than Lucas’s (1987) estimate.

The cost of business cycles is the higher the lower the value of nonemployment is, or,

equivalently, the lower is the disutility of work. The ensuing cost of business cycles rises

further when workers’ skills depreciate during unemployment.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

There is a large literature that computes the cost of business cycles. Most of the studies focus on fluctuations around agiven average level of economic activity, and typically find small estimates of the aggregate cost of cyclical fluctuations;see Lucas (2003) for an overview. The current paper instead puts the effect of the business cycle on average employmentfront and center. In particular, the paper points out that models with labor market matching frictions along the lines ofinfluential papers by Hall (2005) and Shimer (2005) imply an endogenous link between the business cycle and both

fluctuations of unemployment risk and the mean of unemployment. When we calibrate the matching model in line withU.S. data, we find a higher average unemployment rate in the stochastic steady state than in the nonstochastic steadystate. This renders economic volatility costly and might rationalize why economic volatility ranks so high on the public’sagenda.2

The previous literature has largely assumed that the business cycle does not affect average unemployment. As a result,if equilibrium prices would not react to the business cycle, there would be no welfare cost of such business cycles, or only a

ll rights reserved.

P. Jung), keith.kuester@phil.frb.org (K. Kuester).

(1997) survey, economists and laymen alike, state that preventing recessions is important. Of these,

th recessions and booms is preferable to having a business cycle. Wolfers (2003) uses surveys on

ployment volatility, that is, both low and high unemployment rates, would raise well being by an

rage level of unemployment by a quarter of a percentage point. Our paper finds welfare costs of

nsured against the idiosyncratic risk associated with unemployment.

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–1768 1745

very small cost: the size of unemployment fluctuations would be unimportant for the welfare cost of business cycles; forexample, Atkeson and Phelan (1994) and Krusell and Smith (1999). The effect that we wish to stress, instead, mattersprecisely because it affects the average unemployment rate. Through the employment-flow equation, we first show thata negative covariance between job-finding and unemployment, as is found in the data, can imply higher meanunemployment. The reason for this is that the number of new matches between firms and workers is the product of thejob-finding rate and unemployment. A negative covariance between the two implies that unemployed workers are morelikely to find a job in a boom, when there are few unemployed workers to start with, than in a recession, when manyworkers are unemployed. To the extent that the business cycle leaves average job-finding rates untouched, the strongerthe business cycle fluctuations, the higher the average unemployment rate. Thus, a reduction in business cycle volatilityleads to higher employment and more consumption and higher welfare.

We describe a real business cycle model with matching frictions in the labor market. For a special case of our modelwith risk-neutral consumers and no capital accumulation, we can prove analytically that the business cycle indeed doesnot affect the mean job-finding rate. Using a second-order approximation, we derive an expression for the mean increasein unemployment that shows that the more volatile productivity is, the more persistent it is, and the more the job-findingrate reacts to the cycle, the more does mean unemployment exceed its steady-state level.

When we calibrate the model to U.S. data, in both the special case and in the general case involving risk aversion andcapital, we find a cost of business cycles that is an order of magnitude larger than Lucas’s (1987) estimate. For example,for log-utility in our calibrated model the average unemployment rate is 0.13 percentage point higher than in thenonstochastic steady state for the model without capital, and 0.11 percentage point for the model with capital. For thesecases, depending on the disutility of work, we obtain a cost of business cycles of up to 0.14% and 0.09% of steady-stateconsumption, respectively.

It is interesting to note that the higher mean unemployment in our model causes lower returns to capital. In turn, formodest degrees of risk aversion the capital stock in the stochastic economy is, therefore, lower than in the nonstochasticsteady state—by 0.05% for log-utility, for example. This effect contrasts with the previous literature that typically findsthat consumers can dissave along the transition path to the nonstochastic steady state; for example, Krusell and Smith(1999). In our model environment, for many degrees of risk aversion commonly used in the literature, the unemploymenteffect dominates the precautionary savings effect, so capital needs to be accumulated along the transition path. In addition,employment has to be increased along the transition path as well, which is a costly and time-consuming process. Takingthe transitional dynamics into account, therefore, leads to estimates of the welfare costs of business cycles that are lowerthan a steady-state comparison would suggest.

Beyond the effects described, the rise in mean unemployment has further repercussions, which we explore. In anextension, we allow for an interaction of the mean effect on unemployment with the average skills in the economy.In particular, we follow Ljungqvist and Sargent (2008) and allow human capital of workers to depend on their employmentand unemployment history. This is motivated by an extensive literature that documents that earnings after anunemployment spell on average tend to be lower than previous earnings; for example, Keane and Wolpin (1997) orJacobson et al. (1993). At the same time, it is well documented that wages rise with work experience; for example,Kambourov and Manovskii (2009). While some previous literature has identified these earnings losses as an importantcandidate for causing costs of business cycles, in our model the mechanism that induces these costs differs from previouswork. Krebs (2007), for example, assumes that business cycles induce a mean-preserving spread in individual income risk;that is, he emphasizes changes in the second moment of individual risk. In his case, this leads to a significant cost ofbusiness cycles. In our formulation we abstract from idiosyncratic risk by means of a family assumption. Rather the effectsare driven by a higher mean level of unemployment that translates into a lower mean level of skills due to an increase inskill losses off the job. We show that this composition effect is quantitatively important. It results in a cost of businesscycles that, for the case of log-utility, is about 40% larger than the cost that we find in the absence of skill transitions. Alsofor this case, the degree of risk aversion and the amount of precautionary saving are crucial for the size of the welfare coststhat we obtain.

The previous literature has typically abstracted from mean effects on employment; for example, Lucas (1987). In aseminal paper, in an economy with heterogeneous agents, Krusell and Smith (1999) highlight that the cost of businesscycles varies with employment and wealth status. They find that on average, however, the cost is small. The reason is thatcapital/savings typically allow consumers to self-insure against transitory income fluctuations. Recently, Krusell et al.(2009) have revisited the calculation and allowed for a greater reduction of idiosyncratic risk when the aggregate shock iseliminated, but they retained the assumption that average unemployment is not affected by the size of business cyclefluctuations. They find a welfare cost of business cycles that is on average an order of magnitude larger than Lucas’s (1987)estimate. Their later estimate is similar in size to ours. Interestingly, however, the sources of the cost of business cycles inour papers differ notably. In their model, consumers are imperfectly insured against idiosyncratic income risk, and thereduction in aggregate risk helps to alleviate the obstacles to consumption-smoothing. In our model, instead, as in Merz(1995) and Andolfatto (1996), we assume that workers perfectly insure each other’s consumption against idiosyncraticincome and unemployment risk. The cost of business cycles that we find arises from an increase in economy-wide averageunemployment that is caused by fluctuations in the job-finding rate. Krusell et al. (2009) also examine the cost of businesscycles when accounting for two states of unemployment: short-term and long-term. When they allow that the eliminationof business cycles significantly reduces the risk of long-term unemployment, a case we do not examine here, they find

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–17681746

welfare costs that are on average two orders of magnitude larger than Lucas’s number, namely, about 1% of consumption.Storesletten et al. (2001) and Krebs (2003) assume that the cross-sectional variation of idiosyncratic human capital riskincreases in recessions and shrinks in booms for all workers. In this paper, we do not account for this pattern and thecorresponding component of the cost of business cycles. Beaudry and Pages (2001) analyze the welfare cost of businesscycles when the contractual structure in the labor market insures existing workers against wage cuts, while workers whoare laid off in a recession enter the labor market at a lower wage level. In our paper as well the wages of re-entrants intothe labor market are persistently lower in recessions than in booms when we interact the business cycle with skilltransitions, due to a loss of skills off the job.

Fewer papers emphasize that mean effects can generate costly business cycles. In a cross-country study, Ramey andRamey (1995) document a negative relationship between economic volatility and growth. Barlevy (2004) points out that inan economy with endogenous growth and decreasing returns to investment, eliminating cycles increases average growthrates. Mean effects of business cycles are wider-spread in New Keynesian business cycle models in which real and nominalfrictions imply that fluctuations induce an inefficient utilization of resources; for example, Levin et al. (2005). Finally,Hairault et al. (2010) also exploit the nonlinearity of the employment-flow equation in matching models. In a modelwithout capital, they look at risk-neutral workers throughout and focus largely on constant wages, following Hall (2005).Their main focus is the dependence of the job-finding rate on the curvature of the matching function. This focus, the detailsof the bargaining – and, in parts, the calibration3 – differ from our paper. For our model, we derive an intuitive formula thatprovides a second-order approximation to the mean effects on unemployment in the special case without capital. We thenexplain the link between the mean effect, risk aversion and capital accumulation, and make the link between the cost ofcycles, the level of mean unemployment and the mean level of skills in the economy explicit. Barlevy (2005) provides abroader overview of the literature on the welfare cost of business cycles, concluding – as do we – that business cycles arelikely costly. Lucas’s (2003) survey touches only marginally on effects of the business cycle on the means of economicvariables and arrives at the opposite conclusion.

The remainder of the paper is organized as follows. In our model, the key to the welfare cost of business cycles is thatbusiness cycles cause a higher unemployment rate on average. Section 2 shows how the nonlinearity of the employment-flow equation can generate these mean effects. Section 3 describes a real business cycle (RBC) model with search andmatching frictions. Section 4 shows that in a special case with risk-neutral workers and in the absence of capital, the meanjob-finding rate is indeed not affected by the degree of cyclical fluctuations. It also shows analytically that the morevolatile the economy is, the more does mean unemployment exceed the steady-state level of unemployment. Section 5.1discusses the calibration of the model to U.S. data. Section 5.2 discusses how we measure the welfare cost of businesscycles. Section 5.3 presents estimates of the welfare cost of business cycles. Section 6 shows that the welfare costs canincrease notably when allowing for employment-dependent skill transitions, highlighting the effects of the business cycleon the average skill level. A final section concludes. The appendix provides background information and sensitivityanalysis.

2. The employment-flow equation and mean employment

The business cycle is costly in our model because cyclical fluctuations cause a higher average unemployment rate, andthus lower consumption, than would be observed in the absence of business cycles. Before going to the generalequilibrium model, we here seek to highlight the importance of the nonlinearity of the employment-flow equation and thebehavior of the job-finding rate. In the following, we assume that the mass of workers is normalized to one and thatworkers can either be employed or unemployed. There is a mass ut of unemployed workers who are looking for a job.An unemployed worker will be matched with a firm with probability ft : New matches are effective from tþ1 onward.The remaining mass of workers, et ¼ 1�ut , is employed. Each period, as in Shimer (2005) and Hall (2005), a constant shareW of employed workers becomes unemployed. As a result, employment evolves according to the following employment-flow equation

et ¼ ð1�WÞet�1þ ft�1ut�1: ð1Þ

It is important to note that unemployment and the job-finding rate enter nonlinearly in the final term of Eq. (1).4 We areinterested in comparing the unconditional average of unemployment, Efutg, in an economy with aggregate fluctuations(‘‘mean unemployment’’ henceforth) to the nonstochastic steady state of that same economy when aggregate fluctuationsare eliminated (the ‘‘steady state’’ henceforth). The steady-state value of any variable is denoted by dropping the timeindex; for example, u for steady-state unemployment. We have the following condition for the mean effects that wediscuss in this paper:

3 For example, Hairault et al. (2010) calibrate their model to a steady-state unemployment rate that is twice as big as in our calibration. As a result,

a higher welfare cost of business cycles is obtained. Our Proposition 3 explains the reason for this.4 An alternative representation of Eq. (1) that also makes the nonlinearity in unemployment apparent is

ut ¼ Wþut�1ð1�W�ft�1Þ:

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–1768 1747

Proposition 1. Suppose that the employment-flow equation (1) holds. Suppose further that all variables in the employment-

flow equation are covariance stationary. Then the effect of the business cycle on the mean of unemployment is given by

Efutg�u¼�1

Wþ fðCOVðft ,utÞþ½Efftg�f �EfutgÞ: ð2Þ

As a result, sufficient conditions for mean unemployment to exceed steady-state unemployment, that is, for Efutg4u are,therefore: (i) the job-finding rate and the unemployment rate are non-positively correlated, Covðut ,ftÞr0, (ii) the average job-

finding rate does not exceed the steady-state job-finding rate, Efftgr f , and at least one of the inequalities in (i) and (ii) holds

strictly.

Proof. Using the assumed stationarity, (1) implies WEf1�utg ¼ COVðft ,utÞþEfftgEfutg. Subtracting the steady-state versionof (1) from both sides of the above equation, we have that �W½Efutg�u� ¼ COVðft ,utÞþEfftgEfutg�fu, or equivalently�W½Efutg�u� ¼ COVðft ,utÞþ½Efftg�f �Efutgþ f ½Efutg�u�, so

Efutg�u¼�1

Wþ fðCOVðft ,utÞþ½Efftg�f �EfutgÞ:

The final part of the proposition follows by recognizing that Efutg40. &

Proposition 1 highlights that mean unemployment can be affected by business cycle fluctuations. Condition (i) in theproposition is not particularly demanding. Empirically, it holds with a strict inequality. Indeed, the sign of the covariancein (i) is at the core of the search and matching model with exogenous separation of which employment-flow equation (1)forms a part; see, for example, Hagedorn and Manovskii (2008). Namely, unemployment fluctuations arise in the modelsince unemployed workers are more likely to find a job in a boom – when there are fewer unemployed workers to startwith – than in a recession, when many workers are unemployed. Condition (ii) is less general. While we will presentspecial cases below in which the job-finding rate is exactly linear in the technology shock, so Efftg ¼ f , and highlight casesin which the inequality in (ii) holds strictly, one cannot make a model-free claim for this condition. This is the reason whywe build a general equilibrium model in Section 3 to quantitatively evaluate the effect of the business cycle on meanunemployment, and to subsequently assess the welfare cost of business cycles. In this sense, Proposition 1 serves anillustrative purpose: The business cycle can cause higher mean unemployment.

3. The model

The precise interplay of the job-finding rate, unemployment, and aggregate cyclical risk is important for the cost ofbusiness cycles in this paper. This section lays out a real business cycle model with matching frictions along the lines ofPissarides (1985) and Shimer (2005) that generates this interplay endogenously. The wage bargaining follows Hall andMilgrom (2008). Section 4 proves that for a special case of this model, with linear utility and in the absence of capital, thejob-finding rate is indeed linear in productivity, so that Proposition 1 implies that business cycle fluctuations lead to higheraverage unemployment. We derive an intuitive expression for the effect of the business cycle on the mean ofunemployment for this case.

To be able to focus most clearly on the mean effects that we wish to stress, we introduce a family structure as in Merz(1995) and Andolfatto (1996). Family members pool their assets and incomes from market work. They thereby perfectlyinsure individual consumption streams against individual unemployment risk.

3.1. Consumption and investment

There is a measure of size one of identical families in the economy. Each family consists of a unit measure of members.In period t, a measure et of these are employed and a measure ut ¼ 1�et of family members are unemployed. The family’sutility is an equal-weighted average of the utility of individual family members, i 2 ½0,1�:

Z 1

0Et

X1j ¼ 0

bj½uðci,tþ jÞ�zIði is employed in tþ jÞ�

8<:

9=;di, ð3Þ

where

uðctÞ ¼

logðctÞ if s¼ 1,

c1�st

1�s if sZ0, sa1:

8><>:

Above, zZ0 indexes the disutility of work, and Ið�Þ denotes the indicator function. We later seek to illustrate that, throughthe effects on mean (un)employment, the welfare cost of business cycles depends crucially on the society’s value ofnonemployment relative to the value of market work. We use z to trace out the gap between the utility from market workand nonemployment. Alternatively, we could have modeled the gap between employment and nonemployment through

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–17681748

introducing home production.5 The family collects and distributes all income, maximizing the sum of expected utilities ofits individual members. As a result, the family equates the marginal utility of consumption of each member, and as aconsequence of additive separability of the utility function all members consume the same amount. The family’s budgetconstraint is

ctþ it ¼ etwtþrtktþCt : ð4Þ

Here it marks real investment. The terms etwt are the real wages earned by employed household members. kt is theamount of physical capital owned by the family at the beginning of the period. The real rental rate of capital is rt. Ct

denotes income arising from the firms’ profits, described below in Eq. (9). Capital evolves according to

ktþ1 ¼ ktð1�dÞþ it , ð5Þ

where d 2 ð0,1Þ is the rate of depreciation. The family maximizes its objective (3) by choosing investment, it, andconsumption, ct , subject to (4) and (5). The first-order condition for investment is

1¼ Etfbt,tþ1½ð1�dÞþrtþ1�g,

where bt,tþ j ¼ bltþ j=lt is the stochastic discount factor, and lt :¼ c�st is the family’s marginal utility of consumption.The optimal consumption plan satisfies the transversality condition

limj-1

Etfbt,tþ jktþ jg ¼ 0, 8t:

3.2. Goods markets

There are two competitive sectors of production. One sector produces a homogeneous ‘‘labor good.’’ The other, finalsector uses the labor good and physical capital to produce ‘‘output,’’ yt. Output is produced according to

yt ¼ Atkat l1�at , a 2 ð0,1Þ:

Technology, At, is given by

At�A¼ rðAt�1�AÞþeAt , r 2 ½0,1Þ,

where eAt �

iidNð0,s2

AÞ.6 Final good firms can rent capital and the labor good in competitive markets at rates rt and xt,

respectively. The demand functions for capital and the labor good are, respectively, kt ¼ ða=rtÞyt , and lt ¼ ðð1�aÞ=xtÞyt . Eachfirm-worker match constitutes a one-worker labor firm that produces one unit of the labor good. As a result, the totalsupply of the labor good is given by the number of employed workers

lt ¼ et :

3.3. Labor market

The timing of the labor market is as follows. For workers who are already matched with firms, the families bargainabout wages. Production takes place. New matches are determined. Separations occur. As a result, employment evolvesaccording to employment-flow equation (1). Period profits from production of a labor firm, Ut , are given by

Ut ¼ xt�wt ,

where xt denotes the price of the labor good and wt is the wage paid to the firm’s worker. At the end of the period, afterproduction has taken place, the match is severed with probability W. As a result, the value of a labor firm in period t, Jt , isgiven by

Jt ¼ Utþð1�WÞEtfbt,tþ1Jtþ1g:

Firms and the families bargain about their share of the overall match surplus. In this paper, we adopt a bargainingmechanism analyzed by Hall and Milgrom (2008), who assume that the outside option in the bargaining process is to delaythe bargaining by one period, a ‘‘strike.’’ A worker, whose bargaining is delayed, faces the same disutility, z, from striking

5 For the exercises that we will show below, varying the disutility of work, z, changes neither the steady state of the economy nor the dynamics.

We believe that this is the cleanest way to illustrate the dependence of the welfare cost of business cycles on the gap between the value of market work

and nonemployment.6 Here we present results for neutral technology only. Results for labor-augmenting technology are available upon request. In any case, this choice

does not affect the results with labor as the only factor of production ða¼ 0Þ. In addition, for the variants with capital the main mechanisms discussed in

this paper would remain at work if technology were labor-augmenting. For the case with capital, with labor-augmenting technology the welfare costs of

business cycles would be about 20%–30% larger than the costs reported here. This is due to a different response of labor productivity. With neutral

technology business cycle fluctuations induce higher average labor productivity. With labor-augmenting technology, instead, average labor productivity

falls for low to moderate degrees of risk aversion.

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–1768 1749

than if he were working productively. In addition, the worker receives a stream of income in the periods in which thebargaining is delayed, labeled pZ0. In equilibrium, under complete information, rational firms and families would neverdelay the bargaining, but instead they would agree on a wage immediately. A strike thus would never actually occur.7 Wefollow den Haan et al. (2000) in assuming that the family bargains on behalf of its workers. The surplus for the family ofhaving a marginal member employed rather than on strike, Dt , is

Dt ¼wt�p: ð6Þ

When working, the worker earns the wage but loses the strike payment. The firm’s surplus from settling the bargaining inperiod t rather than deferring it to tþ1 is given by period profits Ut . Each period, wages are determined by means ofbargaining over the match surplus, where Z 2 ð0,1Þ denotes the family’s bargaining power

maxwt

ðDtÞZðUtÞ

1�Z:

The first-order condition for wages yields that earnings are a convex combination of the firm’s revenue and the termsdetermining the bargaining position (remuneration when delaying the bargaining):

wt ¼ Zxtþð1�ZÞp: ð7Þ

This wage equation resembles the standard Nash bargaining solution when the outside option is unemployment ratherthan delaying the bargaining, except for two differences. First, labor-market tightness would enter the wage equation.Second, a term in disutility of work, z=lt , would enter instead of the strike value, p. Parameter p captures a shift in thebargaining position of the worker not related to utility flows in equilibrium. There are two reasons for this way ofmodeling. First, as we will discuss further below, the fluctuation of the job-finding rate is key for the mean effect onunemployment. In order to achieve sufficient fluctuations of unemployment, we will rely on the mechanism stressed inHagedorn and Manovskii (2008) and set a high strike value. Yet, in contrast to their paper, this does not mean the family isclose to indifferent between having a worker employed or unemployed. Second, in particular, the disconnect between theoutside option in the bargaining and the value of unemployment allows us to trace out the welfare cost for different levelsof disutility of work, z, without affecting the cyclical or steady-state properties of the model. We consider this importantbecause, as stressed repeatedly, in our model business cycles cause higher mean unemployment. Therefore, the gapbetween the value of market work and the value of nonemployment (determined here by the disutility of work) is key forthe size of the welfare cost of business cycles. New matches arise according to

mt ¼ wuxt v1�x

t , w40, x 2 ð0,1Þ: ð8Þ

Here mt is the number of new matches and vt is the number of vacancies posted. With probability qt ¼mt=vt , a firm with avacant position finds a worker in period t. Unemployed workers always search for a job. With probability ft ¼mt=ut , anunemployed worker will find a job. In order to find a worker, firms need to post a vacancy. As a result of free entry into thevacancy posting market, in equilibrium the cost of posting a vacancy, k40, equals the discounted expected value of alabor firm

k¼ qtEtfbt,tþ1Jtþ1g:

3.4. Market clearing and equilibrium

In equilibrium, the final goods market and the labor and capital markets clear. The final good is used for consumptionand investment. Vacancy posting activity requires resources as well, so output is used according to

yt ¼ ctþ itþkvt :

Finally, total period profits that accrue to the family are given by

Ct ¼ Utet�kvt : ð9Þ

4. The mean effect on unemployment: a special case of the model

Section 2 argued that the business cycle can induce a higher mean unemployment rate. In particular, Proposition 1showed that in the matching model the mean unemployment rate exceeds its steady-state level if cyclical fluctuations leadto a negative correlation of job-finding and unemployment, and if the mean job-finding rate does not increase due to the

7 Our calibration strategy here differs from that used by Hall and Milgrom (2008) in the following dimensions: below, we use the strike value to

reproduce the comovement of unemployment and the job-finding rate. The strike value thus is larger in our calibration than in theirs. Our modeling

further – on purpose – disentangles the strike value from the disutility of work, contrary to their calibration. Finally, their model allows for an exogenous

probability that the bargaining breaks down when it is delayed, which we do not allow for here with a view toward keeping the analytical results

tractable.

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–17681750

fluctuations. This section looks at a special case of the model that we just laid out. Proposition 2 below shows that the job-finding rate is indeed exactly linear in the technology shock, At , if there is no capital, workers are risk-neutral, and thematching function gives equal weight to vacancies and unemployment ðx¼ 0:5Þ. For this case, Proposition 3 further belowin this section gives a closed-form approximation for the mean of unemployment, showing the factors that determine thesize of the effect that the business cycle has on mean unemployment.

Proposition 2. Suppose that the economy is described by the model shown in Section 3. Suppose further that labor is the only

factor of production; that labor enters the production function linearly ða¼ 0Þ; that consumers are risk-neutral, s¼ 0; and that

the matching function gives equal weight to vacancies and unemployment (the elasticity of the matching function is x¼ 0:5).Then

ft ¼ f þff ðAt�AÞ,

where f ¼ ð1�ZÞðw2=kÞðb=ð1�ð1�WÞbÞÞðA�pÞ and ff ¼ ð1�ZÞðw2=kÞbr=ð1�ð1�WÞbrÞ. Hence, the job-finding rate is linear in the

technology shock; so EðftÞ ¼ f .

Proof. Without capital, xt¼At. By (7), wt ¼ ZAtþð1�ZÞp: Since s¼ 0, bt,tþ1 ¼ b8t. This means that the value of the firm islinear in At as well. Guessing and verifying yields

Jt ¼1�Z

1�ð1�WÞbðA�pÞþ 1�Z

1�ð1�WÞbrðAt�AÞ: ð10Þ

The vacancy posting condition reads k=qt ¼ bEtfJtþ1g. Using matching function (8), and the worker and job-finding rates,qt ¼mt=vt ¼ wðvt=utÞ

�x and ft ¼mt=ut ¼ wðvt=utÞ1�x, yields that

ft ¼ wwk

� �ð1�xÞ=x½bEtfJtþ1g�

ð1�xÞ=x: ð11Þ

If x¼ 0:5, we have that ft ¼ wðw=kÞbEtfJtþ1g. Using Eq. (10) to substitute for EtfJtþ1g, and rearranging, proves theproposition. &

Proposition 2 illustrates that for a special case of the model, the job-finding rate will be exactly linear in the technologyshock. As a result, by Proposition 1, if the calibration of the model accounts for the negative correlation of unemploymentand the job-finding rate, mean unemployment is higher in the economy with business cycles than in the steady state,making business cycles costly. The proof of the proposition relied on two elements that show why more generally theeffect of the business cycle on the mean of the job-finding rate, and on mean unemployment, is difficult to prove. First,even with the bargaining that we entertain, once capital is present, wages will typically not be linear in the technologyshock, At, since the price of the labor good, xt, ceases to be linear in the shock. Second, even if labor is the only factor ofproduction with a¼ 0, once consumers are risk-averse, the stochastic discount factor varies over time, so in generalbt,tþ1ab. This means that the value of the firm, Jt, would cease to be linear in productivity, too. A third element in theproof was that x¼ 0:5. Hairault et al. (2010) have previously discussed the dependence of the mean job-finding rate on theelasticity of the matching function with respect to unemployment, x, for other bargaining schemes. In particular, for Hall’s(2005) bargaining scheme with constant wages, they show that the job-finding rate is convex in productivity for xo0:5and concave for x40:5. The bargaining in this paper instead is based on Hall and Milgrom (2008). Still, the proof ofProposition 2, in particular Eq. (11), suggests that the same cut-off applies in our model as in the aforementioned paper.

While, more generally, the effect of the business cycle on mean unemployment, and the welfare cost of business cyclesneeds to be computed numerically, for the case underlying Proposition 2 we can derive the following approximation forthe mean of unemployment:

Proposition 3. Under the conditions of Proposition 2, the unemployment rate, up to a second-order approximation, has a

mean of

Efutg ¼ uþf2

f

1�ð1�W�f Þru

Wþ f

r1�r2

s2A: ð12Þ

Proof. See Appendix A.1.

Proposition 3 shows for the assumptions underlying Proposition 2 (namely, risk-neutrality, no capital, and x¼ 0:5) thatwhenever there is persistence in productivity shocks, r40, the mean unemployment rate in the cyclical economy exceedsthe steady-state level, and increasingly so the more volatile innovations to productivity are (the higher sA).8 In addition,the more the job-finding rate reacts to the technology shock, captured by the term ff , and the higher the steady-stateunemployment rate, u, the stronger the effect of business cycles on mean unemployment. When we fill Proposition 3with the parameter values implied by our calibration with linear utility and labor as the only factor of production in

8 Note that if productivity was not persistent, that is, r¼ 0, due to the one-period lag between vacancy posting and production, the job-finding rate

and unemployment would not respond to the shock but would stay at their steady-state levels indefinitely. r¼ 0 is, therefore, not sensible.

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–1768 1751

Section 5.3.1, we find that the business cycle makes mean unemployment rise by 0.106 percentage point above its steady-state level.9

5. The welfare cost of business cycles

We first present the calibration strategy that we use for the different variants of the model laid out in Section 3.We then present the welfare measures that we use to estimate the welfare cost of business cycles for these variants.The appendix presents a sensitivity analysis.

5.1. Calibration of the baseline

One period in the model is one month. We match the model’s unconditional second moments to the second momentsof hp-filtered quarterly U.S. data from 1951Q1 to 2007Q1.10 As in Shimer (2005), we use the Hodrick–Prescott filter with aweight of 105 to separate fluctuations and trends. All data are in logs and seasonally adjusted. Nominal variables aredeflated by the GDP deflator. Consumption is from the national accounts. Our measure of output is GDP net of governmentspending. We use the civilian unemployment rate among those 16 years of age and older. Vacancies are measured by theConference Board’s index of help-wanted advertising. The job-finding rate in our model is the hazard rate of transitionfrom unemployment to employment in any given month. A measure of this time series is taken from Shimer (2007). Wageand labor productivity data are from the BLS private nonfarm business series.

Table 1 reports the second moments and autocorrelations in the hp-filtered data.To compare the results for different degrees of risk aversion and to explore the role of capital for the mean effect on

unemployment, we fix a set of targets that we want our model to replicate across different variants of the model. Thisimplies that some parameters will remain constant, while others will need to be adjusted in each comparison. Table 2summarizes the set of parameters that are the same in each of the variants of the model or are altered only whenconsidering the model with or without capital.

The time-discount factor targets an annual rate of return of 4%. The elasticity of the matching function with respect tounemployment is set to x¼ 0:5, the lower bound of what Petrongolo and Pissarides (2001) suggest as reasonable. Thisvalue is chosen so as to preserve the linearity of the job-finding rate in the case of risk-neutrality; compare Proposition 2.When labor is the only factor of production in the model, we assume that output is constant-returns-to-scale in the laborgood, a¼ 0. When capital is present, the elasticity of output with respect to capital is set to a¼ 0:33. The depreciation rateof d¼ 0:0052 targets an investment to output ratio of 20%; recall that we measure output as GDP net of governmentconsumption.

Finally, some parameters will be adjusted jointly to put the different model variants on a similar footing whencomputing the mean effects on unemployment and the welfare cost of business cycles. Proposition 1 suggests using thecovariance between the unemployment rate and the job-finding rate as a calibration target. The same proposition alsosuggests keeping the mean job-finding rate and the mean unemployment rate the same across model variants. These aresummarized in Table 3.

The vacancy posting costs, k, are set so as to ensure that the mean unemployment rate in each variant is 5.6%, equal tothe average in the data. The job-separation rate, W, is adjusted so as to ensure that the average job-finding rate per month is45%, which is the mean value in the data. The covariance between unemployment and the job-finding rate is equal to�5.06, measuring both rates in percentage points. As Proposition 1 highlights, this covariance is key to the mechanismthat we wish to stress. Each variant of the model replicates the value of the covariance in the data by appropriately settingthe strike position, parameter p. The efficiency of matching, w, is set such that firms with a vacancy find a worker with a33% probability within a month’s time, as in den Haan et al. (2000). Given that the strike value mainly determines theequilibrium profits of firms, we can then use the bargaining power, Z, to match the elasticity of the wage with respect tolabor productivity to the value of 0.446 reported in Hagedorn and Manovskii (2008). The autocorrelation of theproductivity shock, r, is set so as to replicate the autocorrelation of labor productivity in the data. Finally, in order tomake the welfare costs comparable across model variants, in each variant we scale the standard deviation of theproductivity shock, sA, so as to match the standard deviation of consumption in the data.

Two parameters are left undetermined in the previous tables. First, in the analysis that follows the risk-aversionparameter will be set to several values in the range s 2 ½0,4�, as in the previous literature. Second, the disutility of work

9 As it should be, this is the same result that we obtain later in Section 5.3.1. More precisely, the first entry for the mean effect on unemployment in

Table 4 (column s¼ 0) is 0.106 percentage point, rounded up to 0.11 percentage point. The underlying parameters for this example imply ff ¼ 4:51.

The separation rate is W¼ 0:0265. The steady-state unemployment rate is u¼0.055. The steady-state job-finding rate is f¼0.45. The persistence of the

shock is r¼ 0:946, and the standard deviation of the innovation is sA ¼ 0:505=100.10 An alternative calibration strategy, which is also commonly used in the literature, would have simulated time series from the model, hp-filtered

these series, and matched the moments of these hp-filtered series to the moments of the hp-filtered data. The sensitivity analysis in Appendix A.3

documents for the model variants with capital that using this alternative approach would have resulted in a mean effect on unemployment that would

have been three times as large as we find with the calibration approach that we follow here. The current calibration approach has the advantage that it

allows us to most clearly show the economic effects that are at work and to link these back to Proposition 1, which is formulated in terms of

unconditional moments of the model.

Table 1Second moments in the data—1951Q1–2007Q1.

Variable Std. deviation Corr. with dy=et AR(1)

Output and consumptionbyt 2.77 0.61 0.91bct 2.00 0.66 0.93

Labor market: Wages and labor productivitybwt 1.62 0.63 0.93dy=et1.96 1.00 0.89

Labor market: Job-finding, unemployment, and vacanciesbf t11.56 0.38 0.91

but 18.55 �0.40 0.94bvt 19.65 0.38 0.95

Notes: The table reports second moments of the data. All data are quarterly, in logs,

HP(105) filtered and multiplied by 100 in order to express them in percent deviation from

steady state. Labor productivity is real output per person employed, and wages are real

compensation per employee in the nonfarm business sector. The first column reports the

standard deviation. The next column reports the correlation with labor productivity. The

final column reports first-order autocorrelation coefficients.

Table 2Parameters that are the same across variants.

Preferences

b 0.997 Annual real rate of 4%

Labor marketFmatching and separation

x 0.5 Petrongolo and Pissarides (2001)

If no capital

a 0 Production of output constant-returns-to-scale

If capital

a 0.33 Conventional configuration

d 0.0052 Steady-state investment/output ratio of 20%

Notes: This table presents the parameters that are the same across versions of the model,

and the targets for these parameters.

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–17681752

parameter z has not been specified. We chose the setup of the model such that different values of this parameter do nothave any bearing on the equilibrium dynamics or the steady state. Rather, we later vary the value of z to illustrate how theestimate of the welfare cost of business cycles depends on the gap between the value of market work and nonemployment.

5.2. Measuring the welfare cost of business cycles

In this paper, removing business cycles means setting the innovation to the technology shock, eAt , to zero. The welfare

cost of business cycles is defined as the share, g, of steady-state consumption that would leave the family indifferentbetween the stochastic economy and the nonstochastic economy. The family’s welfare in the stochastic economy isgiven by

Wt ¼ uðctÞ�etzþbEtfWtþ1g: ð13Þ

Let fW sðgÞ be the welfare of the family when, instead, eA

t equals zero in the current and all future periods so that there areno business cycles and when a share, g, is deducted from actual consumption in that economy in all periods. Superscript s

indexes the current state of the economy in the period in which the switchover to eAt ¼ 0 occurs. The welfare cost of

business cycles is computed ex ante: we average over the states of the stochastic economy.In the following, we report two different measures of the cost of business cycles. The first measure (subsequently

labeled ‘‘static’’ welfare cost) calculates the cost of business cycles, neglecting the transition path to the new steady state.This measure computes the welfare cost, g, by equating

EfWtg �fW sðgÞ,

where fW sðgÞ is the same as fW s

ðgÞ but evaluated at the steady-state values of the states of the economy.The second measure of the welfare cost of business cycles that we report, however, does include the transition

dynamics to the new steady state. Subsequently, we refer to this measure as the welfare costs with ‘‘transition.’’ In order to

Table 3Common targets for parameters.

k Targets an average unemployment rate of E(ut)¼0.056

W Targets an average job-finding rate of E(ft)¼0.45

p Covariance between unemployment and job-finding, cov(ut,ft)¼�5.06/1002

w Targets q¼0.33, den Haan et al. (2000)

Z Targets an elasticity of aggregate wages with respect to labor productivity of 0.446

Hagedorn and Manovskii (2008)

r Targets the autocorrelation of aggregate labor productivity

sA Adjusted to match the standard deviation of consumption

A Normalizes steady-state output, y, to unity

Notes: This table presents the targets for parameters that are not necessarily the same

across modeling variants. In each of the model variants analyzed below, the targets are

the same.

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–1768 1753

obtain this measure, we first draw initial states from the stochastic steady state of the model economy. More precisely,we simulate an initial 100,000 periods for the model without capital and 1,000,000 periods for the model variants withcapital so as to initialize the system. Starting from there, we then simulate another S¼100,000 periods for the modelvariants without capital and S¼1,000,000 periods for the variants with capital. This second set of draws represents thedraws from the stochastic steady state of the economy. We compute welfare, fW s

, in the nonstochastic economy,initializing the economy at the states corresponding to each of those draws and withdrawing a share g from consumptionin all periods, including for the periods on the transition path to the nonstochastic steady state. We then compute thevalue of g that solves

EfWtg �1

S

XS

s ¼ 1

fW sðgÞ:

That is, we average over the initial states of the economy using the nonstochastic steady state as the starting point. Tosolve the model, to compute the mean effects and the welfare cost of business cycles, we rely on second-orderapproximations as in Schmitt-Grohe and Uribe (2004). When simulating series, we apply pruning; see Kim et al. (2007).

5.3. Results: the welfare cost of business cycles

Next we discuss the mean effect on unemployment and the welfare cost of business cycles implied by the respectivevariants of the model. We report results separately for the cases with and without capital. Appendix A.2 reports theparameterizations that the calibration strategy implies and also the implied second moments.

5.3.1. Model with labor as the only factor of production

First, we focus on the model variant without capital. Table 4 reports summary statistics for selected variables fordifferent degrees of risk aversion. The first line reports the effect of the business cycle on the average rate ofunemployment. As Eq. (2) in Proposition 1 makes clear, this effect depends on both the covariance between the job-finding rate and the unemployment rate, and on how the business cycle affects the average job-finding rate. Thecalibration strategy that we follow in this paper keeps the first determinant, the covariance between unemployment andthe job-finding rate, the same across model variants. Table 4, therefore, can focus exclusively on the effect of the degree ofrisk aversion on the average job-finding rate.

Proposition 2 highlights that the job-finding rate is linear in productivity if workers are risk-neutral, so the averagejob-finding rate is not affected by business cycle fluctuations in that case. This is borne out by the table (second row,column s¼ 0). For the case of risk-neutral workers, business cycle fluctuations increase the average unemployment rate by0.11 percentage point. This in turn means that mean consumption falls so that welfare is negatively affected by businesscycle fluctuations.11 The proof to the same proposition also highlights, however, that risk aversion potentially changes theevolution of the job-finding rate. This effect enters through the stochastic discount factor, bt,tþ1, when s40. In particular,concave utility means that the owners of the labor firms discount good times, when profits tend to be high, moreintensively than bad times, when profits tend to be small. As a result, firms have less of an incentive to post vacancies, andthe average job-finding rate falls as risk aversion increases. Consequently, in Table 4 the average job-finding rate is abouthalf a percentage point lower for a degree of risk aversion of s¼ 4 than it is for linear utility or in the absence of cyclicalfluctuations. This implies that the effect of the business cycle on mean unemployment is more than one and a half times as

11 The increase in unemployment and the decrease in consumption do not line up perfectly. There are two reasons for this. First, with business cycle

fluctuations workers tend to be employed when working is most productive, so average output and consumption drop by less than the average increase

in unemployment suggests. Second, there are also changes in the average number of vacancies. The effect on average vacancy posting costs drives a

further, albeit small, wedge between the effect of the business cycle on average consumption and on average unemployment.

Table 4Mean effects and welfare costs in the baseline without capital.

Mean effects, welfare costs s¼ 0 s¼ 1 s¼ 2 s¼ 3 s¼ 4

100 � ðEfutg�uÞ 0.11 0.13 0.15 0.16 0.17

100 � ðEfftg�f Þ 0.00 �0.23 �0.40 �0.50 �0.54

100 � ðEfctg�cÞ=c �0.11 �0.13 �0.14 �0.14 �0.14

100 � ðEfwtg�wÞ=w 0.00 0.00 0.00 0.00 0.00

Welfare cost (static) 0.11 0.15 0.18 0.20 0.22

Welfare cost (transition) 0.10 0.14 0.17 0.19 0.21

Notes: The table compares the mean values of endogenous variables in the economy with business cycles to the steady-state values. The entries for the

unemployment rate and the job-finding rate are in percentage points. The entries for consumption and wages are in percent deviation from steady state.

The table further reports the welfare cost of business cycles with z¼ 0 using the static measure of the welfare costs and the measure that takes into

account the transition dynamics. From left to right: five different degrees of risk aversion.

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–17681754

large with s¼ 4 than in the case of risk-neutrality.12 Apart from the dislike of risk-averse workers for fluctuations per se

the mean effects thus add considerably to their cost of business cycles.The bottom rows of Table 4 report the two different measures of the welfare cost of business cycles. The first measure is

static in the sense that it computes the welfare costs ignoring the period of transition from the economy with aggregatefluctuations to the new steady state without fluctuations. The second measure, instead, takes into account that theeconomy does not immediately jump to the new steady state since additional workers need to be hired to bring theemployment stock to the steady-state level. This requires both time and resources. As a result, the welfare gains fromabolishing the cycle are somewhat lower if one accounts for the transition path.

For three degrees of risk aversion (risk-neutrality, log-utility, and s¼ 4), Fig. 1 plots the welfare cost of business cyclesin this model variant against alternative values for the disutility of work.

In terms of scale, the x-axis reports the disutility of work transformed into consumption units. The value of 0 on thex-axis pertains to z¼ 0, so the family does not suffer any disutility of work. The value of 100 on the x-axis in Fig. 1corresponds to the value of the disutility of work, z, that fully compensates the family for a one-unit drop in consumption.This value is given by z=l¼ 1, where l marks the steady-state value of the marginal utility of consumption.

Two observations are apparent. First, as discussed above, even risk-neutral consumers would be willing to pay slightlyabove 0.10% of steady-state consumption to abolish the business cycle. The reason is that the average unemployment raterises if there are business cycles. Second, the value of the disutility of work is crucial for the size of the welfare costs. In ourmodel environment business cycle fluctuations induce a higher average unemployment rate. The larger the disutility ofwork, the more are workers compensated for the resulting lower output and consumption through an increase in theutility derived from leisure. In the extreme, therefore, with linear utility the family would not suffer virtually any cost ofbusiness cycles when z=l¼ 1; see the solid line at the x-axis value of 100 in Fig. 1. Therefore, it is important to providesome indication with regard to the size of the disutility of work. Appendix B argues that microeconometric estimates ofthe labor supply elasticity at the intensive margin suggest that the disutility of work would be no larger thanz¼ ð1=3Þð1=ð1�uÞÞl, and possibly smaller than this. A vertical dotted line in Fig. 1 marks this cut-off value. Followingthis interpretation, risk-neutral consumers would be willing to pay at least 0.07% of steady-state consumption to eliminatethe business cycle, and possibly more. Naturally, also for the calibrations with higher degrees of risk aversion, the lowerthe disutility of work is, the higher are the costs of business cycles. For log-utility, the welfare cost ranges between 0.10% ofsteady-state consumption at the upper bound for z (an order of magnitude larger than estimated in Lucas, 1987) and avalue of 0.14% at z¼ 0. For a degree of risk aversion of s¼ 4, the costs rise to between 0.15% and 0.21% of steady-stateconsumption.

5.3.2. Model with labor and capital

Physical capital gives the family an instrument for smoothing consumption. This section analyzes the interaction of themean effect on unemployment and the mean effect on the job-finding rate with capital accumulation. Toward that end, wenow study the model variant that features both capital and labor as production factors. Table 5 provides the mean effect onunemployment and the mean effect on the job-finding rate for different values of risk aversion. The same table also reportsthe effect of the business cycle on the average capital stock. Accounting for capital accumulation turns out to be importantfor the inference regarding the interaction of the degree of risk aversion with the mean effect on unemployment and,therefore, for the welfare costs of business cycles.

Three effects shape the interaction of the mean effect on unemployment with the capital stock. First, since employmentis lower, the return to capital falls. The lower employment that results from business cycle fluctuations thus has a negativeeffect on the capital stock. As a result, the average capital stock in the economy falls below the steady-state capital stock

12 For higher risk aversion, an additional effect works to balance the reduction in the job-finding rate: The precautionary savings motive means that

the families seek to build buffers against bad times. In the absence of capital, creating more matches (and thus employment) is the only means to

generate that saving. If s is large enough, therefore, the average job-finding rate does not fall by as much as it does for lower degrees of risk aversion.

% ss

con

sum

ptio

n

0 20 40 60 80 1000

0.05

0.1

0.15

0.2

disutility of work, % of �/�

� = 4� = 1� = 0

Fig. 1. The cost of business cycles—no capital. Notes: Welfare cost of business cycles in percent of steady-state consumption (y-axis) for alternative

values of disutility of work, z (x-axis). The x-axis varies z between 0 and 100% of the equivalent of one unit of steady-state consumption as explained in

the main text. The vertical dotted line at 35% of that level marks the upper bound for the size of the disutility of work as derived in Appendix B.1. A thick

solid line shows the welfare cost for risk-neutral consumers, squares mark the case of log-utility, and circles show the case of higher risk aversion ðs¼ 4Þ.

The welfare costs shown here take into account the transition from the stochastic steady state to the nonstochastic steady state.

Table 5Mean effects and welfare costs in the baseline with capital.

Mean effects, welfare costs s¼ 0:1 s¼ 0:5 s¼ 1 s¼ 2 s¼ 3 s¼ 4

100 � ðEfutg�uÞ 0.105 0.108 0.110 0.109 0.105 0.099

100 � ðEfftg�f Þ 0.013 �0.017 �0.034 �0.028 0.008 0.058

100 � ðEfctg�cÞ=c �0.100 �0.095 �0.091 �0.080 �0.067 �0.052

100 � ðEfktg�kÞ=k �0.100 �0.083 �0.051 0.045 0.172 0.326

100 � ðEfwtg�wÞ=w 0.001 0.003 0.006 0.017 0.032 0.050

100 � ðEfxtg�xÞ=x 0.002 0.006 0.013 0.037 0.071 0.112

Welfare cost (static) 0.102 0.105 0.111 0.120 0.127 0.132

Welfare cost (dynamic) 0.080 0.081 0.087 0.098 0.112 0.125

Notes: The table compares the mean values of endogenous variables in the economy with business cycles to the steady-state values. The entries for the

unemployment rate and the job-finding rate are in percentage points. The entries for consumption, capital, wages and the price of the labor good are in

percent deviation from steady state. The table further reports the welfare cost of business cycles with z¼ 0, using the static measure of the welfare costs

and the measure that takes into account the transition dynamics. From left to right: six different degrees of risk aversion.

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–1768 1755

for lower degrees of risk aversion; for example by 0.1% for s¼ 0:1, see row ‘‘100 � ðEfktg�kÞ=k.’’ Second, all else equal theeffect of risk aversion on the mean job-finding rate that we described above is also present in the model economy withcapital. Namely, as the degree of risk aversion increases, good times are discounted more intensively than bad times, andall else equal, the job-finding rate therefore falls with the degree of risk aversion. Third, there is now a strong effect thatcomes from the precautionary savings motive. As the degree of risk aversion increases, the family increases the amount ofprecautionary savings, and thus the capital stock. As a result, all else being equal, the marginal product of labor rises andwith it the return to hiring. The precautionary savings effect, therefore, reduces the mean effect on unemployment.Therefore, for higher degrees of risk aversion, the job-finding rate rises with the degree of risk aversion, contrary to whatwas suggested by the model with labor as the only factor of production. This reduces the average unemployment rate;compare, for example, the entry in row ‘‘100 � ðEfftg�f Þ’’ for s¼ 1 to the one for s¼ 4. In sum, beyond a certain degree ofrisk aversion the precautionary savings effect starts to dominate for the job-finding rate and the unemployment rate.Therefore, the relationship between risk aversion and mean unemployment is not as strong here as in the model withoutcapital, and nonlinear.

For the lower degrees of risk aversion business cycle fluctuations tend to reduce the average capital stock. Nevertheless,comparing Tables 4 and 5 shows that accounting for capital accumulation lowers the welfare cost of business cyclessomewhat even for the static welfare measure. This is because the capital stock does decline, but in percentage terms, itdoes not decline by more than the labor input. In particular, row ‘‘100 � ðEfxtg�xÞ=x’’ in Table 5 illustrates that the businesscycle induces an increase in average labor productivity (see also Footnote 6).

The negative effect on the capital stock that Table 5 bears out for low to intermediate degrees of risk aversion is notpresent in the previous literature that abstracts from the effects that the business cycle has on mean unemployment.Krusell and Smith (1999), for example, find that with log-utility precautionary saving induces the mean level of capital toexceed its steady-state level. In contrast, in our economy, for lower degrees of risk aversion, the negative effect on thereturn to capital that is induced by lower employment can dominate the precautionary savings effect. In the presence of

% ss

con

sum

ptio

n

0 20 40 60 80

0

0.05

0.1

disutility of work, % of �/�

� = 4� = 2� = 1

Fig. 2. The cost of business cycles—labor and capital. Notes: welfare cost of business cycles in percent of steady-state consumption (y-axis) for alternative

values of disutility of work, z (x-axis). The x-axis varies z between 0 and 80% of the equivalent of one unit of steady-state consumption as explained in the

main text. The vertical dotted line at 23.5% of that level marks the upper bound for the size of the disutility of work as derived in Appendix B.2. Squares

mark the case of log-utility. Diamonds mark the case of a degree of risk aversion of s¼ 2, and circles show the case of s¼ 4. The welfare costs shown here

take into account the transition from the stochastic steady state to the nonstochastic steady state.

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–17681756

business cycles the average capital stock, therefore, can be lower in our economy than in the steady state, while theprecautionary savings effect alone would have meant more savings and thus more capital.13

Consequently, the welfare cost can be notably lower when one accounts for the transition relative to the static measureof the welfare cost. The reason is that not only employment but also capital may need to be accumulated during thetransition to the nonstochastic steady state. This weighs on consumption and makes the nonstochastic steady staterelatively less attractive. For log-utility, for example, we obtain welfare costs of 0.111% of steady-state consumption whenneglecting the transition path and of 0.087% when accounting for it. This gap starts to close for higher degrees of riskaversion since the precautionary savings effect pushes up the capital stock in the stochastic economy.

For three degrees of risk aversion, Fig. 2 shows how the welfare costs of business cycles (accounting for the transitionpath) depend on the value of the disutility of work. For the model with capital, Appendix B.2 argues that a reasonableupper bound for the disutility of work is z¼ ð1�aÞð1=3Þl=ð1�uÞ, a value indicated by a vertical line in Fig. 2. For log-utilitythe welfare costs range from 0.06% at this value of the disutility of work to 0.087% of steady-state consumption at z¼ 0.The welfare costs range between 0.07% and 0.10% for s¼ 2 and between 0.1% and 0.12% for s¼ 4.

In Appendix A.3 we report on a sensitivity analysis that explores alternative calibration strategies. The baselinecalibration used here is conservative in the sense that it tends to yield lower estimates of the welfare cost of businesscycles than would other calibration strategies.

6. Welfare cost when skills depend on employment

The previous sections illustrated that the business cycle can increase average unemployment and thereby the length ofthe average unemployment spell. Since it is well documented that the unemployed lose skills when off the job, while theemployed gain skills through work experience, this higher average unemployment can in addition negatively affect thedistribution of skills. This section shows that this can further exacerbate the welfare cost of business cycles.

6.1. Model with skill transitions

For comparison, we will henceforth refer to the model without skill transitions of Section 3 as the ‘‘baseline model.’’ Wenow augment the baseline model with a process for skill transitions that follows Ljungqvist and Sargent (2008). Thissection gives a verbal overview. Appendix C provides the detailed equations. There are H skill levels indexed by h¼ 1, . . . ,Hordered from the lowest to the highest level of human capital. Transitions across skill levels depend on the employmentstatus of the worker. Transitions are stochastic and – conditional on the individual employment state – independent acrosstime and aggregate states. A worker of skill h, who is matched with a firm, produces an amount of lh,t ¼ eh, h¼ 1, . . . ,H, ofthe labor good, where eh indexes the skill-dependent productivity level. These are equally spaced between e1 ¼ 1�o andeH ¼ 1þo.

13 The model environment presented here assumes perfect insurance against idiosyncratic risk. To the extent that the business cycle interacts with

idiosyncratic risk more so than a representative agent framework suggests, allowing for incomplete insurance against idiosyncratic risk would generate

an additional reason for precautionary saving that would be linked to business cycle fluctuations. We leave a quantitative exploration of this for future

research. In any case, our results imply that accounting for the mean effect on unemployment would reduce the average capital stock in an environment

with uninsurable idiosyncratic risk relative to a model environment in which the mean effect on unemployment is not present.

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–1768 1757

A worker who has reached the highest skill level, H, retains this skill for as long as he remains employed. All otheremployed workers, after a period of employment, move to the next higher skill level with probability pe, or otherwiseretain their current skill level—provided a separation does not occur. Match separation occurs for three reasons:retirement, layoff, or for another reason. The probability of retirement is t. The size of the labor force is normalized to one.For every worker who retires, a new worker enters the labor force as unemployed and at the lowest skill level. Layoffsoccur with probability l. Layoffs entail the risk of instantaneous human capital depreciation. When a worker is laid off, hisskill falls from the current level h to a new level s, srh,with probability pLðh,sÞ. Finally, with probability f a match issevered for some other reason, which does not entail the risk of an instantaneous depreciation of human capital. At the endof every month of unemployment, the skills of an unemployed worker deteriorate by one skill level with probability pu, orotherwise stay at their current level. An unemployed worker who has reached the lowest skill level remains at this level foras long as he remains unemployed.

The timing of events is as follows: retirement shocks and separations occur. Idiosyncratic skill shocks materialize.Knowing this period’s skill level, the family and the firm bargain about wages for the employed workers of the differentskill types. Production takes place. At the same time, unemployed workers search for a job, and firms without a match canpost vacancies. Matches are formed in a common matching market for all types. Therefore, regardless of their skill level,all workers have the same job-finding rate. At the end of the period, new matches are determined.

6.2. Mean effects and welfare costs

For this model, we next discuss the effect of the business cycle on mean (un)employment. We then discuss how theskill transitions can amplify the cost of business cycles. We do so by means of a number of propositions that we haverelegated to Appendix C.2. Proposition 4 in the appendix shows that the mean effect on unemployment here has the samedeterminants as in the baseline model, namely, the covariance of unemployment with the job-finding rate, and the effectof the business cycle on the average job-finding rate. Therefore, if the skill transitions do not (much) affect the mean of thejob-finding rate, the mean effect on aggregate unemployment will be very close to that in the baseline model.14

Nevertheless, the skill transitions can add to the welfare costs of business cycles: this will be the case if business cyclefluctuations mean that, relative to the respective steady-state level, the mass of employed workers who have a higher skilllevel falls by more than does the mass of employed workers in the lower skill groups. Proposition 5 in Appendix C.2 provesthis for a version of the model with H¼2 skill levels. Staying with the two-skill case, and restricting itself to setting t¼ 0,Proposition 6 in the same appendix shows that – changes in the average job-finding rate apart – the skill transitions implyhigher welfare costs of business cycles than in the baseline model whenever unemployment in the higher skill group isnegatively correlated with the job-finding rate, that is, if the correlation of unemployment in the higher skill group withthe job-finding rate is of the same sign as we have found for aggregate unemployment.15 Allowing for skill transitionsalone will, therefore, not automatically increase the welfare costs of business cycles. In our model, unemployed workerssearch at a fixed intensity and separations are exogenous, so labor market fluctuations in our model occur entirely throughthe hiring margin. Therefore, the interaction of the skills with the length of the unemployment spell is crucial for thewelfare costs, and so the probability of a skill loss when off the job, pu, is a key margin. In particular, the propositionillustrates that the skill dynamics do not add to the welfare cost of business cycles if workers do not gradually lose skills offthe job, regardless of the size of the turbulence. For example, the skill transitions do not entail additional welfare costs ofbusiness cycles if unemployed workers always lose all their skills off the job.16 We now assess numerically the importanceof the respective mean effects.

6.3. Calibration of the model with skill transitions

For the parameters that overlap with the baseline model, we retain the same calibration strategy as in Section 5.We restrict the strike values for the respective skill groups to ph=eh ¼ ps=es :¼ p for all h,s 2 f1, . . . ,Hg. This conditionensures that the strike payment relative to productivity is the same for all skills. As before, we set p so as to replicate the

14 The vacancy posting condition for the model with skill transitions, Eq. (17) in the appendix, shows that the value of hiring depends on the

composition of unemployment (term put ðhÞ), which will generally not be constant. Even for risk-neutral workers and if labor is the only factor of

production, we could not, therefore, show algebraically that the average job-finding rate is not affected by business cycle volatility. Our numerical results,

however, suggest that the composition effect on the job-finding rate is very small for this case; compare row ‘‘100 � ðEfftg�f Þ’’ in the column labeled

‘‘Labor only, s¼ 0’’ in Table 6.15 More generally, Proposition 7 in the Appendix shows that also for H42 skill groups the mean effect on the respective skills depends on an

interplay of the covariance of unemployment in the respective skill groups with the job-finding rate.16 Extreme turbulence does, therefore, not necessarily imply high welfare costs of business cycles in our environment. To see this, observe that in the

two-skill case, the case of extreme turbulence coincides with the case pu¼1, which Proposition 6 shows yields zero additional welfare costs. In our model

there is no interaction of turbulence with the (exogenous) separation margin or the search margin. Explicitly modeling these margins might make

turbulence more important for the welfare costs of business cycles. While the two-skill case captures much of the intuition for H42, it does not capture

the entire complexity. In particular, with H42, a high value of pu means that workers lose skills fast but gradually, namely, by at most one skill level per

month. A large value of pu, therefore, for H42, does not coincide with extreme turbulence. As a result, a high value of pu can imply larger welfare costs of

business cycles if H42. The sensitivity analysis in Appendix C.4 provides further discussion.

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–17681758

covariance of aggregate unemployment with the job-finding rate, thus anchoring the mean effect on unemployment;compare Proposition 4 in the appendix.

We do not have data on the incidence of (un)employment broken down by the respective skill groups and, therefore,cannot target the respective disaggregated covariances. Rather we borrow the calibration of the skill process fromLjungqvist and Sargent (2008), which we consider a reasonable starting point. There are H¼11 skill levels. We set theprobability to climb up one skill level after a month of employment to pe¼0.0975, and the probability of losing one level ofskill while unemployed to pu¼0.19.17 The average duration of a working life is 40 years, so the separation rate intoretirement is t¼ 1=ð40 � 12Þ. The workers with the best skills are twice as productive as workers with the worst skills; thisamounts to setting o¼ 1=3. The probability of a layoff is l¼ 0:012. Workers who are laid off draw new skills. Withprobability pLðh,sÞ they move from their current skill h to another skill level srh. The new skill level is drawn from anormal distribution with mean h and variance s2

l , which is truncated such that the support ranges from the lowest skilllevel up to the current skill level h. Therefore, the bigger s2

l (the bigger the ‘‘turbulence’’ upon being laid off), the higher theprobability that a worker who enters unemployment after a layoff loses a notable amount of skills. We set s2

l ¼ 0:089.18

The probability, f, that a separation occurs that does not entail the risk of an immediate depreciation of human capital isset so as to reproduce the average unemployment rate in the data.

The remaining parameters for which there is overlap between the model with skill losses and the earlier model variantsare set as laid out in Tables 2 and 3. The tables in Appendix C.3 report the parameter values for the model variants withskill transitions and the implied second moments.

Last, we put the calibration of the skill process into context. It implies that the skills of workers who experience anunemployment spell that lasts for a year depreciate by 20% on average. This is in line with Keane and Wolpin (1997),who estimate that the skills of white-collar workers depreciate by 30% after a year-long unemployment spell and that theskills of blue-collar workers fall by 10%. Our calibration furthermore implies that workers who have just completed anunemployment spell earn on average 11% less than on their previous job. This is at the upper end of earnings lossesreported in the literature when a broad definition of job loss is used.19 Stevens (1997), for example, uses the PSID and findsthat the earnings loss in a new job relative to the previous job is about 7% in the year of the job loss. Most closely related toour measure of separations are the earnings losses reported in Fujita and Rao (2009). They compare the earnings after anyunemployment spell to the earnings before that spell. That is, they look at all flows from employment into unemploymentregardless of the underlying reason. Their estimates, after correcting for the fact that their measures were taken in arecession, suggest an earnings loss of about 4%–5% for an average unemployment spell. Finally, in our baseline, the averagereturn to five years of continuous employment after an unemployment spell is 29%. Kambourov and Manovskii (2009)argue that there are significant returns to tenure, but that what matters is occupational tenure more than tenure with anemployer. The simple model in this section does not allow for job-to-job transitions and also cannot distinguish to whatextent workers switch occupations when they lose their jobs, so we add up the returns to occupational, industry, andemployer tenure in their paper. This yields an upper bound for the return to five years of continuous employment ofbetween 12% and 20%, suggesting that in our calibration workers may accumulate skills somewhat too rapidly. AppendixC.4 presents a sensitivity analysis that explores the welfare costs with alternative parameterizations of the skill process.

6.4. Results: welfare costs with skill transitions

Table 6 reports the mean effects and the welfare costs for the calibration laid out above. The table also reports thepercent increase in the welfare cost relative to the baseline model without skill transitions.

Allowing for the mean effect on the skill distribution turns out to be quantitatively important. For log-utility, forexample, the cost of business cycles is about 44% higher than in the baseline for the labor-only variant, and 37% higherthan in the baseline for the model variant with capital accumulation.

The same mechanisms that we explored in Section 5.3 are present here: the business cycle induces higherunemployment on average, and the mean effect on the unemployment rate and the job-finding rate shows a similardependence on risk aversion and the capital stock as discussed in Section 5.3. In addition, another effect is at work nowthat exacerbates the cost of business cycles: higher average unemployment means that the average length of anunemployment spell increases. As a result, workers tend to lose more of their skills in an average unemployment spell,which means that fewer of the employed workers have high skills. The skill effect is borne out by the sixth through eighthrow in Table 6. Focus, for example, on the case of risk-neutrality, s¼ 0, and on the model with labor as the only factor of

17 A period in Ljungqvist and Sargent (2008) lasts for two weeks. Our model runs at a monthly frequency. As a result, we have scaled their transition

probabilities to a monthly level.18 In Ljungqvist and Sargent (2008), workers’ skills are assigned to the interval [1,2]. We instead opt for having the skills distributed on ½1�o,1þo�.

This is inconsequential for the welfare costs, but has the advantage that the baseline model is a limiting case for o¼ 0. Our value for the variance is in line

with their ‘‘T20’’ turbulence level.19 The literature reports higher estimates when it focuses on a smaller subset of the flows into unemployment. Jacobson et al. (1993), for example,

find that the earnings of a sample of high-tenure workers in Pennsylvania that lost their jobs in mass layoffs were roughly 25% lower even six years after

they had lost their jobs. Farber (2005) uses data from the displaced workers survey to document that displaced full-time job losers who find new

full-time jobs on average earn about 13% less on their new jobs than in their previous positions.

Table 6Mean effects and welfare costs when there are skill transitions.

Labor only With capital

Mean effects, welfare costs s¼ 0 s¼ 1 s¼ 4 s¼ 0:1 s¼ 1 s¼ 2 s¼ 4

100 � ðEfutg�uÞ 0.11 0.14 0.18 0.10 0.11 0.11 0.09

100 � ðEfftg�f Þ 0.00 �0.26 �0.63 0.01 �0.04 �0.02 0.10

100 � ðEfctg�cÞ=c �0.16 �0.20 �0.25 �0.16 �0.15 �0.14 �0.11

100 � ðEfktg�kÞ=k – – – �0.16 �0.12 �0.04 0.18

100 � ðEfxtg�xÞ=x 0.00 0.00 0.00 0.00 0.01 0.03 0.08

100 �P11

h ¼ 8ðEfeh,tg�ehÞ�0.18 �0.23 �0.30 �0.18 �0.19 �0.19 �0.16

100 �P7

h ¼ 5ðEfeh,tg�ehÞ0.02 0.03 0.03 0.02 0.02 0.02 0.02

100 �P4

h ¼ 1ðEfeh,tg�ehÞ0.06 0.07 0.10 0.06 0.06 0.06 0.05

Welfare cost (static) 0.16 0.22 0.33 0.16 0.17 0.18 0.19

Increase relative to baseline 51% 52% 52% 54% 56% 54% 44%

Welfare cost (transition) 0.15 0.20 0.30 0.11 0.12 0.13 0.14

Increase relative to baseline 42% 44% 44% 42% 37% 30% 14%

Notes: The table compares the mean values of endogenous variables in the economy with business cycles to the steady-state values. The sixth through

eighth rows show the difference between means and steady-state values for the number of employed workers in the top four skill categories, in the

middle three categories and in the lowest four skill categories. The final four rows report the welfare cost of business cycles (in percent of steady-state

consumption) for z¼ 0 both with and without accounting for the transition path. Also shown is the percent increase in the welfare costs relative to the

baseline without skill transitions. From left to right: different values for risk aversion in the model without/with capital.

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–1768 1759

production: the share of workers who are employed in the top four skill categories falls by 0.18 percentage point due to thebusiness cycle fluctuations, which is larger than the overall drop in employment (0.11 percentage point). Employmentwith good skills is, therefore, affected more than proportionally by the business cycle fluctuations and the associated meaneffect on employment.20

Interestingly, there is a nontrivial interaction with capital accumulation in the model variants with capital. The loss ofskills implies that the average labor input measured in efficiency units declines by more than the average unemploymenteffect suggests. Therefore, the incentives to accumulate capital decline relative to the baseline. As a result, the averagecapital stock here is lower than in the baseline model. Therefore, accounting for the transition to the nonstochastic steadystate, the skill transitions induce a smaller increase of the welfare costs of business cycles in the model variants withcapital than in the labor-only variants. In addition, as discussed earlier, the mean effect on unemployment is lesspronounced for higher levels of risk aversion in the model with capital due to the precautionary savings motive; comparealso Table 5. As a consequence, for higher degrees of risk aversion the mean effect on the skill distribution is weaker aswell, which further reduces the relative attractiveness of the nonstochastic steady state. Indeed, for s¼ 4, and in the modelwith capital, the welfare costs turn out to be only 14% higher when accounting for the skill transitions than in theirabsence, whereas they add considerably to the welfare costs for lower degrees of risk aversion.

7. Conclusions

This paper developed a real business cycle model with search and matching frictions in the labor market. We calibratedthe model to U.S. data and used it to compute the welfare cost of business cycles. We computed the cost for differentvalues of the disutility of work, which we understood as a catch-all term for determining the value of market work relativeto nonemployment. Importantly, we let the model govern how both the fluctuations and the levels of labor market riskchange when the business cycle risk is eliminated. In a second step, we extended the model to allow for employment-dependent human capital levels of workers.

General equilibrium effects apart, in the model unemployment fluctuations by themselves would have only minorimplications for the welfare cost of business cycles. Nevertheless, our estimates for the cost of business cycles are easily anorder of magnitude larger than the estimates provided by Lucas (1987). This is due to the fact that besides fluctuations inunemployment and consumption, which have been the focus of the previous literature, the model also impliessignificantly higher unemployment rates on average in the presence of a business cycle. These mean effects arise as adirect consequence of the nonlinearity between unemployment and the job-finding probability in the employment-flowequation. Therefore, business cycles are costly even for workers who are well insured against idiosyncratic fluctuations inincome and unemployment risk. Reducing business cycle fluctuations reduces average unemployment risk and increases

20 As in the baseline model, the business cycle also affects the mean job-finding rate for most specifications. The effect on the mean job-finding rate is

somewhat bigger here than in the baseline. For the relative increase in welfare costs in each variant with skill losses, Proposition 4 in the Appendix can be

used to ascertain that the effect on the skill distribution is more important than is the difference in mean job-finding rates between the baseline and the

model with skill transitions.

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–17681760

welfare. In the economy without skill transitions, we find that for log-utility, for example, mean unemployment is between0.11 and 0.13 percentage point higher than in the steady state and that the potential gain from eliminating business cyclesis up to 0.14% of steady-state consumption. We explored the sensitivity of our results with respect to the parameterizationof the model and the calibration strategy. The cost of business cycles tended to be bigger still in alternative calibrations.

We then assessed the cost of business cycles when unemployment spells increase the risk of losing skills acquiredthrough previous work experience. In our calibrated model, using the calibration of the skill transitions in Ljungqvist andSargent (2008) as our baseline, the interaction of skills with the mean effect on unemployment caused by the businesscycle is quantitatively important. Again for log-utility, the welfare cost rises by about 40% relative to a model without skilltransitions.

Our estimates of the cost of business cycles focused on the business cycle’s effect on mean unemployment and mean skillswhile we clearly have omitted further sources for costly business cycles. Most important to us, a number of authors have pointedout that the risk of infrequent disasters linked to cyclical phenomena significantly raises the cost of business cycles. These authorstypically appeal to a (once in a lifetime) great depression scenario; see Chatterjee and Corbae (2007) and Salyer (2007). In thecurrent paper, not only do we abstract from such aggregate disasters, but also in the same vein we limit the damage thatunemployment can do to skills. In particular, regardless of the length of the unemployment spell, skills in the paper never fallbelow a certain level. Business cycles would be more costly if very long-term unemployment – which is much more likely tooccur when there are lasting deep recessions – was associated with a very deep (disastrous) loss of skills or with the absence ofunemployment insurance. Needless to say that this would point to an even higher cost of business cycles.

Another omission is that, in the data, the probability that an individual finds a job tends to fall with the length of hisunemployment spell; see, for example, Shimer (2008). Indeed, several studies find that the job-finding rate declinesrapidly over the length of an individual’s unemployment spell even after controlling for unobserved heterogeneity; see, forexample, van den Berg and van Ours (1996). The current paper, instead, assumes that two workers who have beenunemployed for a different length of time still face the same probability of finding a job. Allowing for duration-dependentjob-finding rates would reinforce the mean effect on unemployment and would increase the welfare costs of businesscycles. We leave a quantitative exploration of this for future research.

In sum, in a model with labor market search and matching frictions, we found that business cycles increase the averageunemployment risk and that cycles reduce the skill level of the workforce. Accordingly, business cycles are considerablymore costly than the mere degree of aggregate fluctuations suggests, and this cost affects even consumers who are wellinsured against the idiosyncratic risk of fluctuations in income and employment.

Acknowledgments

We would like to thank two anonymous referees, as well as Gadi Barlevy, Wouter den Haan, Shigeru Fujita, MarcusHagedorn, Michalis Haliassos, Tom Krebs, Dirk Krueger, Lars Ljungqvist, and Makoto Nakajima for their comments.We would also like to thank Gisle Natvik, Franck Portier, and Pedro Silos for their discussions of the paper. Earlier versionsof this paper circulated under the titles ‘‘The (Un)importance of Unemployment Fluctuations for Welfare’’ and ‘‘The Cost ofUnemployment Fluctuations Revisited.’’ Comments from participants at the following conferences and seminars aregratefully acknowledged. 2009: ‘‘Ensuring Economic and Employment Stability’’ at the Kiel Institute, Search and MatchingWorkshop Philadelphia; 2008: New York/Philadelphia Workshop on Quantitative Macroeconomics, Oslo Workshop onMonetary Policy, Bank of Canada, Tilburg University; 2007: Norges Bank, ‘‘Using Dynamic Economic Models to Make PolicyRecommendations’’ in San Sebastian, and the North American Summer Meeting of the Econometric Society. The viewsexpressed in this paper are those of the authors. They do not necessarily coincide with the views of the Federal ReserveBank of Philadelphia or the Federal Reserve System.

Appendix A. Baseline model without skill transitions

A.1. Proofs for the baseline model

Proof of Proposition 3. The employment-flow equation (1) can be written as ut ¼ Wþð1�W�ft�1Þut�1. Usingft ¼ f þff ðAt�AÞ from Proposition 2, and rewriting gives

�ut ¼ ð1�W�f Þ �ut�1�ff�At�1 �ut�1�ff u �At�1: ð14Þ

Here a check marks deviations from steady state, for example, �ut ¼ ut�u: Take unconditional expectations of (14), use thecovariance stationarity of the model, and that Ef �Atg ¼ 0. This gives

Ef �utg ¼�1

Wþ fff Ef �At�1 �ut�1g: ð15Þ

In order to obtain an expression for Ef �ut�1�At�1g, multiply (14) by �At , and expand the right-hand side by using �At ¼ r �At�1þeA

t . Asecond-order approximation of the resulting terms yields �ut

�At � ð1�W�f Þ½r �ut�1�At�1þ �ut�1eA

t ��ff ur �A2

t�1�ff u �At�1eAt : Taking

Table 7Parameters in the respective variants without skill transitions.

Labor only With capital

Parameters s¼0 s¼1 s¼2 s¼3 s¼4 s¼0.1 s¼0.5 s¼1 s¼2 s¼3 s¼4

A 1.059 1.058 1.058 1.058 1.058 0.310 0.310 0.310 0.310 0.310 0.310

Z 0.438 0.437 0.436 0.434 0.432 0.434 0.425 0.420 0.413 0.408 0.403

W 0.026 0.026 0.026 0.026 0.026 0.026 0.026 0.026 0.026 0.026 0.026

k 0.213 0.239 0.271 0.311 0.365 0.208 0.370 0.456 0.578 0.676 0.763

w 0.387 0.388 0.388 0.388 0.389 0.387 0.387 0.387 0.387 0.387 0.387

p 1.024 1.020 1.015 1.008 1.000 0.676 0.651 0.638 0.620 0.606 0.594

r 0.946 0.946 0.946 0.946 0.946 0.936 0.935 0.933 0.931 0.929 0.928

100 � sA 0.505 0.518 0.532 0.549 0.571 0.201 0.323 0.375 0.440 0.486 0.524

p=ðy=eÞ 0.967 0.964 0.959 0.952 0.945 0.639 0.615 0.603 0.586 0.573 0.561

p=ð@y=@eÞ 0.967 0.964 0.959 0.952 0.945 0.953 0.918 0.900 0.875 0.855 0.838

Notes: This table presents the parameters for the respective model variants in the baseline model without skill transitions. For comparison with the

literature, the last two lines also report the strike value as a fraction of steady-state labor productivity and as a fraction of the steady-state marginal

product of labor, @y=@e¼ x.

Table 8Standard deviations in the respective variants without skill transitions.

Labor only With capital Data

Variable s ¼0 s ¼1 s ¼2 s ¼3 s ¼4 s ¼0.1 s ¼0.5 s ¼1 s ¼2 s ¼3 s ¼4

Output and consumptionbyt 2.22 2.26 2.29 2.34 2.40 2.60 3.76 4.25 4.85 5.29 5.65 2.77bct 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00

Wages and labor productivitybwt 0.64 0.66 0.67 0.69 0.72 0.81 1.33 1.56 1.83 2.03 2.20 1.62dy=et1.43 1.47 1.51 1.56 1.62 1.82 3.00 3.50 4.11 4.56 4.92 1.96

Job-finding, unemployment, and vacanciesbf t15.14 15.11 15.07 15.04 15.01 15.03 14.97 14.94 14.89 14.87 14.84 11.56

but 13.98 13.97 13.96 13.95 13.94 13.94 13.92 13.92 13.90 13.89 13.89 18.55bvt 17.43 17.30 17.18 17.07 16.95 17.00 16.80 16.66 16.48 16.37 16.29 19.65

Notes: The table reports second moments implied by the different calibrations of the baseline model without skill transitions. The second moments are

the population moments from the model. All data are quarterly aggregates, in logs, and multiplied by 100 in order to express them in percent deviation

from steady state. The final column reports the second moments in the corresponding hp-filtered data.

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–1768 1761

unconditional expectations and using the stationarity gives that up to second order Ef �ut�Atg ��1=ð1�ð1�W�f ÞrÞff ur=ð1�r2Þs2

A.Using this with (15) yields expression (12). This proves the proposition. &

A.2. Parameters and second moments in the baseline

This appendix, in Tables 7 and 8, reports the parameter values used for the versions of the model without skilltransitions described in Section 5.3 and the implied standard deviations.

A.3. Sensitivity analysis for the baseline model

This section summarizes a sensitivity analysis for the baseline model, presenting values for the variant with capital andlog-utility. The welfare costs reported here refer to the welfare costs that include the transition dynamics to the newsteady state. Anticipating the results, we document that the calibration strategy in the main text is conservative in thesense that it tends to yield lower estimates of the welfare cost of business cycles than would other calibration strategies.The sensitivity analysis varies the value of one parameter at a time. Unless stated otherwise, the other parameters areadjusted to continue to match the target values laid out in Section 5.1.

As a first robustness check, the results above followed Hagedorn and Manovskii (2008) by using a target for the wage-elasticity for determining the bargaining power of workers, Z. To check the sensitivity, we dropped the target for the wage-elasticity, setting Z¼ 0:1 and 0.9 instead. The welfare costs and mean effects were barely affected. As a second robustness

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–17681762

check, the results in Hairault et al. (2010) and the discussion of Proposition 2 suggest that the welfare cost of businesscycles depends on the value of the elasticity of the matching function with respect to unemployment, x. The resultingwelfare costs would range from 0.06% of steady-state consumption for x¼ 0:3, to 0.15% of steady-state consumption forx¼ 0:7.

We next turn to alternative targets. We note that our baseline calibration targeted the covariance of the job-finding rateand the unemployment rate, using the strike parameter p to match that target. Since the model abstracts from fluctuationsin the separation rate, this leaves unemployment about 25% less volatile than it is in the data; compare Table 8. If, instead,we use the strike parameter p to target the fluctuations of unemployment, the mean effect on unemployment rises to 0.19percentage point. As a consequence, the welfare cost of business cycles also increases: to 0.15% of steady-stateconsumption. This value is about 68% higher than the welfare cost in the baseline. Another alternative regards themapping between the moments in the model and the hp-filtered moments in the data. The calibration strategy abovematched hp-filtered moments in the data to population moments in the model. If, instead, we match the momentsfrom the hp-filtered data to moments from hp-filtered series obtained from simulating the model, the mean effects andthe welfare costs are larger. For log-utility and the model with capital, the mean effect on unemployment is 0.32percentage point and thus three times as large as in the baseline. This is because second moments after hp-filtering andunconditional moments in the model do not coincide. Using this alternative calibration strategy, for example, results in anunconditional covariance of unemployment with the job-finding rate that is three times as large as in the baselinecalibration, while the approach fits the covariance of the hp-filtered unemployment rate with the hp-filtered job-findingrate by construction. The welfare costs are 0.26% of steady-state consumption, or about three times the costs that weobtained in the baseline.

Appendix B. Size of the disutility of work, f

In our model environment, business cycles induce a higher average unemployment rate. The extent to which thistranslates into higher welfare costs of business cycles depends crucially on the value of the disutility of work, z. Inparticular, if z is large, workers could gain sufficient utility through leisure to compensate them for the lower output andthus lower consumption. This appendix provides an upper bound for the value of the disutility of work by resorting tomicroeconometric estimates of the labor supply elasticity. To do so, we use a thought experiment that infers the total sizeof the disutility of work from a putative intensive (hours worked) margin of labor adjustment. In the subsequentderivations, we first concentrate on the case in which labor is the only factor of production. Thereafter we focus on the casewith capital. Throughout, in this section, we focus on the steady state of the economy.

B.1. Model without capital

Assume that next to the extensive margin, each match, that is, each labor firm, would also have an intensive margin oflabor adjustment. Assume that output of each labor firm is given by the hours worked at that firm, h. Revenue of each laborfirm is given by xh, where x is the price of the labor good. Assuming separable utility and customary functional forms,an individual worker’s period utility function would be given by

uðcÞ�khh1þf

1þf, kh40, f40,

where 1=f is the elasticity of labor supply at the intensive margin. khðh1þf=ð1þfÞÞ is the equivalent of the disutility of

work-term z in our model that we wish to obtain an estimate for; compare Eq. (3). Suppose that the hours worked decisionin a match is bilaterally efficient, so that the firm’s marginal product of labor, @xh=@h, equals the worker’s marginal rate ofsubstitution between consumption and leisure, so

x¼ khhf

l, ð16Þ

where l :¼ u0ðcÞ. Note that the price of the labor good is x¼A in steady state. In equilibrium, there is a mass of ð1�uÞ

employed workers, all of whom work the same hours. Total output in the economy is therefore given by y¼ ð1�uÞAh. Sinceour calibration normalized steady-state output to y¼1, we have that

ð1�uÞAh¼ ð1�uÞkhh1þf

l¼ 1,

or, alternatively, z :¼ khðh1þf=ð1þfÞÞ ¼ l=ðð1�uÞð1þfÞÞ.

The elasticity of intertemporal substitution of labor, 1=f, tends to be small in most microeconometric studies (between0 and 0.5); for a survey of the literature, see Blundell and MaCurdy (1999). This implies fZ2, which allows us to boundthe disutility of work from above. For the model with only labor, an upper bound for the consumption value of thedisutility of work is given by z¼ l=ðð1�uÞð1þ2ÞÞ ¼ l=ðð1�uÞ3Þ:

Note that this number of, at most, about one-third of steady-state consumption seems to square well with an upperbound derived from cross-country data for unemployment benefit replacement ratios as collected by the OECD. Christoffel

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–1768 1763

et al. (2009), for example, summarize evidence on replacement rates for the euro area, the UK and the US.21 For bothlonger-term unemployment and short-term unemployment replacement rates can exceed 70% (for some countries theyexceed 80%) of previous wages. Since transitions from unemployment to employment are nevertheless observed, thissuggests that the disutility of work would not exceed 30% of wages, in line with the upper bound derived above.

B.2. Model with capital

Next, we derive an upper bound for the disutility of work in the model variants with capital. Revenue for the labor firmis xh. Total labor input in the economy is l¼eh. Output is given by y¼ kaðAlÞ1�a. The price of the labor good in the economyis given by x¼ ð1�aÞy=l.

As for the labor-only case, an efficient bargain over hours worked equates the marginal product of labor for the laborfirm ð@xh=@hÞ with the worker’s marginal rate of substitution between consumption and leisure, yielding Eq. (16).Substituting x¼ ð1�aÞy=l, using l¼ ð1�uÞh, and using y¼1 yields

z :¼ khh1þf

1þf¼ ð1�aÞ l

ð1�uÞð1þfÞ,

which (with 1�a¼ 2=3) gives as an upper bound for the disutility of work in the model with capital according to

z¼2

3

lð1�uÞ 3

:

Appendix C. Model with skill transitions

This appendix presents details for the model with skill transitions. Section C.1 reports the necessary adjustments to thebaseline model when we allow for skill losses during unemployment and skill accumulation when employed. Section C.2collects propositions as regards the mean effect on unemployment and the welfare costs of business cycles for that modelvariant. Section C.3 presents the parameterization of this model variant and the implied second moments.

C.1. Adjustments to the model of Section 3

We discuss the adjustments to the model equations in Section 3 that become necessary when we allow for H skillgroups; see also Section 6.1. Let h 2 f1, . . . ,Hg be the individual’s current skill level. pLðh,sÞ marks the probability to movefrom skill level h to skill level s immediately after a layoff. There continues to be a unit mass of workers. Employment inthe respective skill groups evolves according to

eH,t ¼ ð1�f�lÞð1�tÞeH,t�1þð1�f�lÞð1�tÞpeeH�1,t�1þð1�puÞð1�tÞft�1uH,t�1,

eh,t ¼ ð1�f�lÞð1�tÞð1�peÞeh,t�1þð1�f�lÞð1�tÞpeeh�1,t�1

þð1�puÞð1�tÞft�1uh,t�1þpuð1�tÞft�1uhþ1,t�1, h¼ 2, . . . ,H�1,

e1,t ¼ 1�XH

h ¼ 2

eh,t�XH

h ¼ 1

uh,t :

The laws of motion for unemployment in the respective groups are correspondingly given by

uH,t ¼ ð1�tÞ½fþlpLðH,HÞ�eH,t�1þð1�puÞð1�tÞð1�ft�1ÞuH,t�1,

uh,t ¼ ð1�tÞ½fþlpLðh,hÞ�eh,t�1þð1�tÞlXS

s ¼ hþ1

pLðs,hÞes,t�1

þpuð1�tÞð1�ft�1Þuhþ1,t�1þð1�puÞð1�tÞð1�ft�1Þuh,t�1, h¼ 2, . . . ,H�1,

u1,t ¼ ðtþð1�tÞ½fþlpLð1,1Þ�Þe1,t�1þXH

h ¼ 2

½tþð1�tÞlpLðh,1Þ�eh,t�1

þtXH

h ¼ 1

uh,t�1þpuð1�tÞð1�ft�1Þu2,t�1þð1�tÞð1�ft�1Þu1,t�1:

21 See Table 7 in Christoffel et al. (2009) for the euro area evidence. Their 2009 working paper version also reports the evidence for the US and the UK.

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–17681764

Period profits of firms depend on the skill level of the worker:

Uh,t ¼ xteh�wh,t , h 2 f1, . . . ,Hg:

The value of a firm that has a worker of type h is

Jh,t ¼ Uh,tþpeðh,hÞð1�tÞð1�f�lÞEtfbt,tþ1Jh,tþ1gþpeðh,hþ1Þð1�tÞð1�f�lÞEtfbt,tþ1Jhþ1,tþ1g,

where peðh,hÞ ¼ ð1�peÞ and peðh,hþ1Þ ¼ pe for hoH, whereas peðH,HÞ ¼ 1 and peðH,Hþ1Þ :¼ 0 for the highest skill level.Let ph be the strike value for workers of skill type h. The wage equation for these workers that results from bargaining isgiven by

wh,t ¼ Zxtehþð1�ZÞph:

The matching function is as in Eq. (8), but now the pool of unemployed workers needs to be aggregated up from thedifferent skill types, so ut :¼

PHh ¼ 1 uh,t . Job-finding and matching probabilities are defined the same way as before. The

composition of the pool of unemployed workers can matter for the incentives of firms to post vacancies. Let the fraction ofunemployed workers who will have skills of the respective skill group h at the end of the period, after skill transitions havetaken place, be pu

t ðhÞ. So in particular, put ðHÞ ¼ ð1�puÞuH,t=ut , pu

t ðhÞ ¼ ðð1�puÞuh,tþpuuhþ1,tÞ=ut for h¼ 2, . . . ,H�1, andpu

t ð1Þ ¼ ðu1,tþpuu2,tÞ=ut . The free-entry condition for vacancies becomes

kqt¼XH

h ¼ 1

put ðhÞEtfbt,tþ1Jh,tþ1g, ð17Þ

which accounts for the skill transitions. The total amount of the labor good produced is given by

lt ¼XH

h ¼ 1eheh,t :

C.2. Propositions and proofs for variants with skill transitions

Proposition 4. Suppose that the economy is given by the model with skill transitions as described in Section 6.1 and Appendix

C.1. Then the effect of the business cycle on the mean of unemployment is given by

Efutg�u¼�1

t1�tþfþlþ f

ðCOVðft ,utÞþ½Efftg�f �EfutgÞ, ð18Þ

where ut ¼PH

j ¼ 1 uj,t is the aggregate unemployment rate.

Proof. In terms of notation, let ext :¼ Efxtg�x, and let gxtyt :¼ Efxtytg�xy. Unemployment is given by ut ¼ 1�PH

j ¼ 1 ej,t . Theemployment-flow equations are given by

eH,t ¼ ð1�f�lÞð1�tÞeH,t�1þð1�f�lÞð1�tÞpeeH�1,t�1þð1�puÞð1�tÞft�1uH,t�1,

eh,t ¼ ð1�f�lÞð1�tÞð1�peÞeh,t�1þð1�f�lÞð1�tÞpeeh�1,t�1

þð1�puÞð1�tÞft�1uh,t�1þpuð1�tÞft�1uhþ1,t�1, h¼ 2, . . . ,H�1,

e1,t ¼ ð1�f�lÞð1�tÞð1�peÞe1,t�1þð1�tÞft�1u1,t�1þð1�tÞpuft�1u2,t�1:

Take expectations and subtract the steady-state versions from each equation. This gives

eeH,t ¼ ð1�f�lÞð1�tÞeeH,t�1þð1�f�lÞð1�tÞpeeeH�1,t�1þð1�puÞð1�tÞ gft�1uH,t�1 ,

eeh,t ¼ ð1�f�lÞð1�tÞð1�peÞeeh,t�1þð1�f�lÞð1�tÞpeeeh�1,t�1

þð1�puÞð1�tÞ gft�1uh,t�1þpuð1�tÞ gft�1uhþ1,t�1 , h¼ 2, . . . ,H�1,

ee1,t ¼ ð1�f�lÞð1�tÞð1�peÞee1,t�1þð1�tÞ gft�1u1,t�1þð1�tÞpugft�1u2,t�1 :

Sum up the left-hand sides and right-hand sides of the H employment-flow equations, and make use of the fact that theseries are covariance stationary. Observe that

PHh ¼ 1

eeh,t ¼ eet , and thatPH

h ¼ 1gftuh,t ¼

gftut . Collecting terms yields

eet ¼ ð1�f�lÞð1�tÞeetþð1�tÞgftut :

Now, eet ¼�eut and gftut ¼ Efftutg�fu¼ COVðft ,utÞþEfutgðEfftg�f Þþ f ðEfutg�uÞ. Making the substitutions and rearrangingyields (18). &

Proposition 5. Consider the conditions in Proposition 4. In addition, suppose that there are only two skill levels. Holding

constant the effect of the business cycle on aggregate (un)employment, the effect of the business cycle on the output of the

labor good (measured relative to its steady-state value) will be stronger in absolute terms with skill heterogeneity, o40, than

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–1768 1765

without if

Efe2,tg�e2

e2

��������4 Efe1,tg�e1

e1

��������: ð19Þ

In particular, if average employment is negatively affected by business cycle fluctuations, the effect that the business cycle has on

output of the labor good will be more negative with skill transitions if – relative to the respective steady state – the average

number of workers that are employed and of the higher skill level, e2,t , is more negatively affected by the business cycle than the

average number of workers, e1,t , that are employed and of the lower skill level.

Proof. If there are only two skill levels, output of the labor good is given by lt ¼ ð1�oÞe1,tþð1þoÞe2,t ¼ etþoðe2,t�e1,tÞ.The effect of the business cycle on the output of the labor good (relative to the steady state) is given by ðEfltg�lÞ=l. If o¼ 0,the effect of the business cycle on the output of the labor good would be given by ðEfetg�eÞ=e, where et ¼ e1,tþe2,t . We thusrequire that jEfltg�lÞ=ljo jðEfetg�eÞ=ej: This can be shown to be guaranteed by Eq. (19) through a sequence of algebraicsteps. &

Proposition 6. Consider the assumptions of Proposition 5. In addition, suppose that there is no retirement, t¼ 0. Comparing

two model variants that feature the same mean effects on aggregate unemployment and on the job-finding rate, and that have

the same calibration of the separation rate, of steady-state aggregate unemployment and of the steady-state job-finding rate, the

following holds: business cycle fluctuations make output of the labor good fall by more relative to the steady state if there are skill

differences, o40, than if skills are homogeneous, whenever (i) the skill distribution is non-degenerate, namely, pu 2 ð0,1Þ, and

(ii) COVðu2,t ,ftÞþEðu2,tÞðEfftg�f Þo0.

Proof. The proof proceeds through a series of algebraic transformations along the same lines as the previous proofs.We provide the key finding here. Let lt be the output of the labor good in the model with heterogeneous skills and let lhom

t

be the output of the labor good in a variant with homogeneous skills that has the properties that are postulated in theproposition. One can show that

Efltg�l

l�

Eflhomt g�lhom

lhom¼ ½Covðu2,t ,ftÞþEfu2,tgðEfftg�f Þ�

�Wþ f

f

2oð1�puÞpu

ð1þoÞð1�WÞpe½puþð1�puÞf �þð1�oÞ½Wpuþð1�puÞflpL�

with W :¼ fþl, and pL :¼ pLð2,1Þ being the probability of a skill loss after a layoff. Both fractions are strictly positive ifpu 2 ð0,1Þ. This equation, therefore, yields the conditions in the proposition. &

Proposition 7. Suppose that the model with skill transitions described in Section 6.1 and Appendix C holds. Then the effect that

the business cycle has on the average mass of employed and unemployed workers in each respective skill group is given by

Efxtg�x¼ ðI2 H�FÞ�1�G

CovðuH,t ,ftÞ

^

Covðu1,t ,ftÞ

264

375�D½Efftg�f �

0B@

1CA, ð20Þ

where xt :¼ ½eH,t , . . . ,e1,t ,uH,t , . . . ,u1,t�0, I2H denotes the identity matrix of dimension 2H. The mean effect on unemployment is

given by Efutg�u¼ ½01,H ,11,H�½Efxtg�x�, where 01,H , and 11,H are row vectors of length H of zeros and ones, respectively. The

output of the labor good lost due to the business cycle is given by ½ð1þoÞ, . . . ,ð1�oÞ,01,H�½Efxtg�x�.

Proof. The transition equations for employment and unemployment in the H skill groups are displayed in Appendix C.1.Apart from extending the number of transition equations, the proof follows the same algebraic steps as the proof ofProposition 1. Matrices F, G, and D are as follows: F ¼ ½F1,F2�, where

P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–17681766

&

C.3. Parameters and second moments with skill transitions

Parameters and standard deviation in the variants with skill transition are shown in Tables 9 and 10.

C.4. Sensitivity analysis for model with skill transitions

The welfare costs that arise from the mean effect on skills depend on the skill transition process. This section presents asensitivity analysis in that regard. As in the sensitivity analysis for the baseline model in Appendix A.3, we focus on themodel with capital and the case of log-utility. The first column of Table 11 repeats the welfare cost estimates that werereported in Table 6 and the adjacent discussion. The subsequent columns conduct sensitivity checks. First, the effect ofprobabilities pe and pu is illustrated in columns I–III. Starting from the baseline parameters the welfare costs will be largerif workers acquire skills less rapidly in an employment spell (see Column I, which halves the probability of gaining skillsafter a period in employment).22 The cost of business cycles will also be larger if workers have a higher chance to lose skillsin an unemployment spell (as long as the skill loss is gradual). To show this, Column II doubles the probability of losing onelevel of skills after a month of unemployment. Column III halves both probabilities, thereby leaving the ratio pu=pe as in thebaseline. The welfare costs are similar to the baseline.

Second, in columns IV and V, we report the sensitivity with respect to the turbulence parameters, namely, s2l and l.

Column IV doubles the probability of a layoff (and thus reduces the probability of a separation for other reasons, f,accordingly). Column V doubles sl relative to the baseline. These changes have a limited effect on the welfare costs;compare also the discussion in Footnote 16 and at the end of Section 6.2.

Third, the welfare costs associated with the loss of skills quite obviously will be the larger, the larger the skill difference,o. Column VI reports the welfare costs when halving o to o¼ 1=6. The welfare costs are still 19% larger than in thebaseline.

Fourth, as we have discussed in the main text, our baseline calibration implies somewhat steep returns to tenure andsomewhat large earnings losses after an unemployment spell. The final column of Table 11 chooses parameters pe, pu,and o so as to match a 10% loss of skills after a year of unemployment, a 10% return to 5 years of tenure, and a 5% earningsloss after the average unemployment spell, which are at the lower range of values that the literature finds; compareSection 6.4. The parameters are pe¼0.12, pu¼0.60, o¼ 0:095. Even for these moderate skill losses, the welfare costs ofbusiness cycles remain 22% larger than in the baseline, and thus higher than the pure costs originating from the meaneffect on unemployment.

22 There are two extremes: first, if workers do not acquire skills at all (pe¼0), there are no skill dynamics and so no additional welfare costs. If pe¼1,

in turn, the effects of any skill loss during unemployment are short-lived, and the welfare losses are small. For intermediate ranges, such as the one that

we have in our calibration, reducing pe means that the skill losses from unemployment are more persistent, which increases the cost of business cycles.

Table 10Standard deviations in variants with skill transitions.

Labor only With capital Data

Variable s¼ 0 s¼ 1 s¼ 4 s¼ 0:1 s¼ 1 s¼ 2 s¼ 4

Output and consumptionbyt 2.15 2.17 2.24 2.54 3.72 4.06 4.43 2.77bct 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00

Wages and labor productivitybwt 0.61 0.63 0.67 0.80 1.33 1.48 1.65 1.62dy=et1.36 1.38 1.47 1.76 2.96 3.30 3.69 1.96

Job-finding, unemployment, and vacanciesbf t15.23 15.19 15.08 15.10 14.97 14.92 14.85 11.56

but 14.00 13.99 13.96 13.96 13.92 13.91 13.89 18.55bvt 17.76 17.61 17.21 17.23 16.77 16.56 16.33 19.65

Notes: The table reports second moments implied by different degrees of risk aversion in model variants with skill transitions, corresponding to Table 6.

The second moments are the population moments from the model. All data are quarterly aggregates, in logs, and multiplied by 100 in order to express

them in percent deviation from steady state. The final column reports the second moments in the corresponding hp-filtered data.

Table 9Parameters in the respective variants with skill transitions.

Labor only With capital

Parameters s¼ 0 s¼ 1 s¼ 4 s¼ 0:1 s¼ 1 s¼ 2 s¼ 4

A 0.926 0.925 0.925 0.283 0.283 0.283 0.283

e 1.144 1.144 1.144 1.144 1.144 1.144 1.144

Z 0.418 0.418 0.416 0.411 0.406 0.401 0.394

f 0.012 0.012 0.012 0.012 0.012 0.012 0.012

k 0.141 0.155 0.225 0.143 0.279 0.337 0.414

w 0.386 0.386 0.389 0.386 0.386 0.386 0.386

p 0.900 0.897 0.884 0.594 0.571 0.562 0.549

p 1.029 1.026 1.012 0.680 0.653 0.642 0.628

r 0.937 0.938 0.939 0.930 0.929 0.925 0.921

100 � sA 0.441 0.447 0.470 0.182 0.296 0.328 0.365

p=ðy=eÞ 0.973 0.970 0.957 0.643 0.617 0.607 0.593

p=ð@y=@eÞ 0.973 0.970 0.957 0.959 0.921 0.906 0.885

Notes: This table presents the parameters for the model variants with skill transitions corresponding to Table 6. For comparison with the literature, the

last two lines also report the strike value as a fraction of steady-state labor productivity and as a fraction of the steady-state marginal product of labor, xt.

Above, e :¼PH

h ¼ 1 eheh=e is the employment-weighted average productivity of a worker in producing the labor good. p :¼PH

h ¼ 1 pheh=e is the

employment-weighted average of strike values. @y=@e :¼ Atkat ð1�aÞl�at e ¼ ð1�aÞy=e is the average marginal product that an employed worker produces.

Table 11Sensitivity: welfare costs with skill transitions.

Characteristics Table 6 I II III IV V VI VII

Skill loss (%), one-year u spell 19.5 16.5 26.0 17.2 22.4 20.3 10.8 10.0

Earnings loss (%) avg. u spell 11.4 9.4 13.0 8.8 15.8 12.6 6.8 5.2

Avg. return to 5 years tenure (%) 28.5 21.2 32.4 20.1 40.1 32.5 14.35 10.3

Time h¼1 to h¼H (yrs) 8.5 17.1 8.5 17.1 8.5 8.5 8.5 6.9

Welfare cost (static) 0.17 0.22 0.23 0.17 0.17 0.17 0.14 0.15

Increase from baseline w/o skill loss 56% 98% 107% 57% 52% 51% 27% 31%

Welfare cost (transition) 0.12 0.13 0.14 0.11 0.12 0.12 0.15 0.11

Increase from baseline w/o skill loss 37% 54% 62% 33% 34% 34% 19% 22%

Notes: The table compares eight variants of the model with skill transitions. The column labeled ‘‘Table 6’’ repeats the estimates for the calibration in

Section 6.4; cf. Table 6. Column I: pe ¼ 0:0975=2. Column II: pu ¼ 2 � 0:19. Column III: pe ¼ 0:0975=2, pu ¼ 0:19=2. Column IV: l¼ 2 � 0:012. Column V:

s2l ¼ 22

� 0:089. Column VI: o¼ 1=6. Column VII: pe¼0.12, pu¼0.60, o¼ 0:095. The remaining parameters are set as in the calibration laid out in Section

6.3. From top to bottom: average skill loss after an unemployment spell that lasted for one year (in percent), average earnings loss after an

unemployment spell (in percent), average wage increase for five years of tenure (in percent), and the average time it takes to climb from the lowest to the

highest skill level (in years). The final four rows report the welfare cost of business cycles (in percent of steady-state consumption) for z¼ 0 both with and

without accounting for the transition path to the nonstochastic steady state. Also shown is the percent increase in the welfare costs relative to the

baseline without skill transitions. The numbers reported refer to the model with capital and log-utility.

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P. Jung, K. Kuester / Journal of Economic Dynamics & Control 35 (2011) 1744–17681768

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