Labor market cycles, unemployment insurance eligibility, and moral hazard

Review of Economic Dynamics 15 (2012) 41–56

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Review of Economic Dynamics www.elsevier.com/locate/red

Labor market cycles, unemployment insurance eligibility, and moral hazard ✩ Min Zhang a , Miquel Faig b,∗ a b

Shanghai University of Finance and Economics, Department of Economics, 777 Guoding Road, Shanghai, 200433, China University of Toronto, Department of Economics, 150 St. George Street, Toronto, Canada, M5S 3G7

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 9 June 2010 Revised 5 September 2011 Available online 21 September 2011 JEL classification: E24 E32 J64 Keywords: Search Matching UI eligibility Moral hazard Unemployment Business cycles Labor markets

If entitlement to UI benefits must be earned with employment, generous UI is an additional benefit to working, so, by itself, it promotes job creation. If individuals are risk neutral, then there is a UI contribution scheme that eliminates any effect of UI on employment decisions. As with Ricardian Equivalence, this result should be useful to pinpoint the effects of UI to violations of its premises. Our baseline simulation shows that if the neutral contribution scheme derived in this paper were to be implemented, the average unemployment rate in the United States would fall from 5.7 to 4.7 percent. Also, the results show that with endogenous UI eligibility, one can simultaneously generate realistic productivity driven cycles and realistic responses of unemployment to changes in UI benefits. © 2011 Elsevier Inc. All rights reserved.

1. Introduction Most models of employment flows in the labor market assume that workers automatically qualify for unemployment insurance (UI) benefits while they are searching for a job. As pointed out by Mortensen (1977), Burdett (1979), and Hamermesh (1979), this simplistic view of how a UI system operates may lead to highly misleading conclusions about its impact on the labor market. To avoid this criticism, several papers taking into account more realistic features of the UI systems have emerged. However, because of the institutional complexities of actual UI systems, these models rely exclusively on numerical methods for their analyses, and, they either assume an exogenous distribution of real wages (Andolfatto and Gomme, 1996) or a non-standard mechanism for its determination (Brown and Ferrall, 2003). In this paper, we advance an analytically tractable version of the standard Mortensen–Pissarides search and matching model in which workers are not

✩ This contribution combines and extends the results from two previously circulated papers: Faig and Zhang (2009), which analyses a simpler version of the model without idiosyncratic match heterogeneity, and Zhang (2009), which extends the earlier paper with heterogeneous productivities. Both papers were part of Min Zhang’s dissertation. We are grateful for the many suggestions we have received in the various formats of this work. In particular, we have benefited from comments by Shouyong Shi, Michael Reiter, Giovanni L. Violante (editor), and two anonymous referees. Min Zhang thanks the financial support from the Leading Academic Discipline Program, 211 Project for Shanghai University of Finance and Economics (the 3rd phase), and Miquel Faig thanks the financial support of SSHRC of Canada. We are the only ones responsible for any remaining errors. Corresponding author. E-mail addresses: [email protected] (M. Zhang), [email protected] (M. Faig).

*

1094-2025/$ – see front matter doi:10.1016/j.red.2011.09.002

© 2011

Elsevier Inc. All rights reserved.

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M. Zhang, M. Faig / Review of Economic Dynamics 15 (2012) 41–56

always entitled to UI benefits because such an entitlement must be earned with prior and not too distant employment, and it can be lost if workers quit their jobs voluntarily or refuse job offers. If UI benefits are unconditionally received while searching for a job, they unequivocally represent an opportunity cost of employment, and improve the bargaining position of workers while negotiating over wages with their employers. As a result, UI benefits reduce the expected profits of filling a vacancy, and hurt firms’ incentives for job creation and therefore employment. In contrast, if UI benefits are conditional on prior employment and a worker cannot collect UI if bargaining with an employer breaks down, UI benefits are no longer an opportunity cost but an indirect benefit of employment. Therefore, UI benefits promote the value of filling a vacancy and stimulate job creation. This is the entitlement effect stressed by Mortensen (1977), Burdett (1979), and Hamermesh (1979) but operating through a new channel. In those papers, the desire to earn UI entitlement reduces the reservation wage of workers searching for jobs, which, in turn, reduces unemployment. In our model, the entitlement effect operates through the bargaining positions of firms and workers. The UI benefits, making the employment match more attractive to workers, enable firms to appropriate a larger fraction of the match surplus, which translates into a stronger incentive to post vacancies. Even if generous UI benefits encourage the creation of jobs due to the entitlement effect, they may hurt employment due to other effects. With the realistic assumption that the UI agency is not able to perfectly monitor the reason for a job loss, workers are able to collect UI with positive probability even if they quit a job voluntarily or reject a job offer. As a result, UI benefits have two detrimental effects on employment. First, they increase the bargaining power of workers since they can now threat to refuse a job to collect UI, which reduces firms’ incentives to create jobs. Second, they may actually trigger actual moral hazard quits or rejections, which directly increases unemployment. In addition to these effects, a generous UI system is also an expensive one, and the fees needed to finance it are an opportunity cost of employment. Taking into account all these effects, we obtain the following analog to Ricardian Equivalence: If the UI system is fully funded and workers have linear utilities, then contribution fees can be designed to prevent moral-hazard behavior and to render the UI system neutral in the sense that it has no effect on the determination of output and employment. Like Ricardian Equivalence, this irrelevance result should be a useful benchmark to pinpoint the economic effects of a UI system as violations of its premises. That is, the economic relevance of a UI system must be found on the risk aversion of workers or the “improper” pricing of UI services. If workers are risk averse, UI provides the valuable service of smoothing consumption fluctuations in the presence of employment shocks. The Mortensen–Pissarides model typically abstracts from this purpose by assuming linear utilities, and we follow this tradition in this paper. If UI contributions, or equivalently taxes that ultimately fall on employed workers, are not carefully crafted, the positive and negative effects of the UI benefits do not cancel each other for some or all workers. Therefore, the UI system affects the incentives of firms to post vacancies or the incentives of some workers to accept and continue employment relationships. The details of how workers earn or lose UI eligibility are quantitatively important for the predictions of the model. For example, in our baseline calibration, if a reform could eliminate the moral-hazard effects of UI by making it impossible to collect benefits after rejecting a job, then the long-term average unemployment rate would fall from 5.7 to 4.5 percent. This effect is actually stronger than the effect that would result from changing the scheme of contribution fees to achieve neutrality or from completely eliminating the UI system, in which case the average unemployment rate would fall to 4.7 percent. Making UI eligibility endogenous offers the following insights on the current debate about the appropriateness of the Mortensen–Pissarides model in explaining the cyclical fluctuations in the labor market. Even though, as in Hagedorn and Manovskii (2008), our model needs a large opportunity cost of employment to generate realistic cycles for unemployment and vacancies, it is able to simultaneously generate realistic responses to productivity shocks and to changes in UI benefits. In contrast, as emphasized by Hornstein et al. (2005) and Costain and Reiter (2008), this simultaneous fit is impossible in the standard Mortensen–Pissarides model. More precisely, in our model the response of unemployment to an increase in UI benefits is small and similar to the estimates in Costain and Reiter (2008), whereas the response of unemployment to an increase in productivity is large and of the order of magnitude needed to generate realistic cycles in the labor market. In the standard model, these two responses are similar. Intuitively, the two responses can differ in our model because the entitlement effect reduces the adverse effect on employment of an increase in UI benefits. This mechanism is an alternative to assuming real wage rigidity, as in Hall (2005), Kennan (2010), and Menzio and Moen (2010), to explain why unemployment responds strongly to productivity shocks but weakly to changes in benefits. The rest of the paper is organized as follows. Section 2 sets up our stochastic version of the Mortensen–Pissarides model with a UI system in which individuals need to earn their UI eligibility. In addition, it establishes conditions that make this system neutral. Section 3 calibrates the model to data in the United States and analyzes its quantitative predictions. In particular, it studies how far apart the UI system in the United States is from the neutral one derived in Section 2. Also, it reports the responses of the model to productivity shocks and changes in UI benefits. Finally, Section 4 concludes. 2. The baseline model Our model is a stochastic discrete time version of Pissarides (1985) search and matching model with the following two special features: (1) to collect UI benefits unemployed workers must have earned eligibility with a previous job, and (2) the quality of a match between a firm and a worker is heterogeneous.

M. Zhang, M. Faig / Review of Economic Dynamics 15 (2012) 41–56

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2.1. Environment In the economy, there is a continuum of measure one of workers, and a large measure of potential firms who can enter freely into the labor market. Both workers and firms are infinitely lived, risk neutral, maximize their expected utilities, and discount future utility flows at the common rate r .1 Production requires the cooperation of one worker and one firm. For this cooperation to take place, workers and firms must first enter the labor market and search for a suitable partner. Once a match has been formed, it produces output until it breaks down. Employment matches dissolve either exogenously as a result of separations which come at an arrival rate s, or endogenously when breaking the match is in the interest of at least one of the two parties. The surplus from a match is assumed to be split between the two parties according to the generalized axioms of Nash. There is a single labor market where firms and workers are matched. Search frictions in this market are characterized by a constant returns to scale matching technology: M ( v t , ut ). In a period, denoted by the t subscripts, the function M maps vacancies posted v t and unemployment ut onto the measure of successful matches formed. The constant returns to scale of M implies that the probability of a worker finding a firm is just a function of the vacancy-unemployment ratio θt = v t /ut : f (θt ) = M ( v t , ut )/ut = M (θt , 1). Likewise, the probability of a firm finding a worker is a function of θt that satisfies:

q(θt ) =

M ( v t , ut ) vt

=

M ( v t , ut ) ut ut

vt

=

f (θt )

θt

.

(1)

The function M is assumed to be continuously differentiable, increasing in both arguments, and concave. Furthermore, it satisfies the Inada conditions: M (1, 0) = M (0, 1) = 0, and M 1 (0, 1) = M 2 (1, 0) = ∞. Therefore, workers find it easier to find firms when vacancies are abundant relative to unemployment, while firms find it easier to match with workers when the reverse is true. The key feature we introduce to this standard environment is that workers do not always collect unemployment insurance (UI) benefits while they are searching for jobs. For workers to be eligible for UI, they must first be employed for a while, and benefits do not last forever. Furthermore, UI benefits are meant to be collected by workers who lose their jobs involuntarily, although, to be realistic, we allow for some workers who quit to successfully pretend to have lost their jobs involuntarily. More precisely, eligible workers can always collect UI if they suffer an exogenous separation from their jobs, but if they quit an employment relationship or reject a job offer they can collect UI with respective probabilities π and π˜ . We allow these two probabilities to be identical, but we differentiate between them to capture the fact that in most jurisdictions, workers eligible to collect UI are more likely to maintain eligibility if they reject a job offer than if they quit an ongoing employment relationship, so we assume π˜  π . To balance the trade-off between realism and tractability, earning and losing UI eligibility are introduced in the following stylized way. Let i be an indicator for a worker’s UI eligibility state. A value i = 1 denotes that the worker is eligible to collect UI in case of being unemployed, while a value i = 0 denotes ineligibility. Ineligible workers can only earn UI entitlement while employed, and the probability of a transition from i = 0 to i = 1 during one period is g . Eligible workers can lose UI entitlement either when they are “caught” voluntarily quitting or rejecting jobs, as mentioned above, or while they are unemployed and benefits expire. The probability in one period of running out of UI benefits while unemployed (transition from i = 1 to i = 0) is d. In our numerical simulations, the parameters g and d are chosen so that the average time required for an ineligible worker to gain UI entitlement and the average duration of benefits predicted from the model are the same as their empirical counterparts in the United States. Unemployment insurance benefits are provided by the government, which finances the UI system by charging employed workers a mandatory state dependent contribution fee, τxi , where the superscript i denotes the UI eligibility state, and the subscript x denotes the aggregate state of the economy (to be specified later). Since the government can borrow and save at the interest rate r , the UI program can run deficits or surpluses over time. To allow for the possibility that some matches break down endogenously while others do not, we assume that the productivity of a firm–worker pair has an idiosyncratic component. To model this, we assume that at the beginning of a period, after a new match is formed, workers draw a random value  that determines the match-specific productivity. The total productivity of the match is defined as: pt ( ) = pt +  . The component pt is common to all matches in the economy in period t. The other component  is assumed to be match specific, so it will be called the quality of the match. The common component of productivity pt is a stochastic variable that follows a Markov chain and takes values in a finite support P ⊂ R n+ . The idiosyncratic component  is randomly drawn from a distribution with finite support E ⊂ R m + . The density function of  is h( ), and its cumulative distribution function is H ( ). After the realization of the match-specific productivity, the newly matched workers decide whether to take their respective offers or not. Workers who turn down their offers are allowed to collect UI benefits with the probability π˜ as long as they were eligible for UI. In contrast, workers who accept their offers start an employment relationship and  stays constant during the spell of employment. Denote ˆti the critical value of  that determines if a worker of type i accepts 1 The reason for choosing risk neutrality is to ensure that the different results in the quantitative analysis are caused by the introduction of the UI eligibility. We conjecture that with risk aversion the main quantitative results should be even stronger since risk averse workers should care more about UI entitlement.

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the job (  ˆti ) or not ( < ˆti ) given the state of the economy in period t. The effective job-finding rate for workers of type i is the product of the rate at which they match with a firm and the probability of accepting the ensuing job offer: f (θt )[1 − H (ˆti+1 )]. Similar to job rejections, if workers eligible for UI choose to quit their jobs, they can collect UI benefits with probability π . For a given match-quality  , the only difference between an employed worker who becomes eligible for UI through working for a while and the one who is eligible for UI at the time of forming a match with a firm lies in the probability of collecting UI in case of turning down the job offer. Since π  π˜ , moral hazard quits can only happen either because a worker has gained UI eligibility while employed or because the state of the economy has changed since the time that hiring took place. Net of the UI contribution fee, employed workers earn a wage rate w ti ( ). The wage rate w ti ( ) depends on UI eligibility because UI benefits raise the opportunity cost of employment, so they improve the worker’s bargaining position in the negotiations to split the match surplus. Each period, unemployed workers receive utility from leisure , and UI benefits b if they are eligible for UI. Both  and b are assumed to be positive, and  is assumed to be smaller than the productivity in a match:  < pt ( ) for i ∈ {0, 1}, all t and  ∈ E. This assumption ensures that in the absence of UI there are no voluntary dissolutions of employment relationships. However, strategic job losses induced by the UI system are possible because the total opportunity cost of employing a worker includes the UI contribution fee and, for those who are eligible, the expected UI benefits to be received in case of rejecting an offer or quitting a job. Ex-ante, all firms possess the same production technology and preferences. Each one of them chooses to either stay idle or be active in the labor market. Each period, an active firm paired up with a worker obtains output pt ( ) and incurs a labor cost w ti ( ) + τti . In a period when an active firm is searching for a worker, it posts a vacancy at a cost c. Even though the probability of finding a job by an unemployed worker is independent from his or her eligibility, the probability that a firm matches with a worker of a particular type depends on the composition of unemployment. That is, the probability that a vacant firm matches with a worker of type i ∈ {0, 1} during a period is q(θt )uti /ut , where uti is the measure of unemployed workers of type i and ut is total unemployment. 2.2. Laws of motion of the distribution of workers In detail, the timing of the model inside a period is the following. At the beginning of the period, the state of aggregate productivity is realized and the qualities of the matches formed in the previous period become known. Soon after, matched workers decide whether they continue their matches or not. The workers that remain employed produce output, while those who quit search for jobs. During the period, employed workers may lose their jobs exogenously with probability s, and those who remain employed and are ineligible for UI may gain eligibility with probability g . Unemployed workers get matched with probability f t , f t = f (θt ), and the probability of each match quality is h( ). Therefore, the laws of motion for the measures of employed workers by eligibility type and match quality are:







et0 ( ) = (1 − s)(1 − g )et0−1 ( ) 1 − Q t0 ( ) + ut0−1 f t −1 h( ) A t0 ( ),

and

(2)

    ) = (1 − s)et1−1 ( ) + (1 − s) get0−1 ( ) 1 − Q t1 ( ) + ut1−1 f t −1 h( ) At1 ( ),

et1 (

(3)

where Q ti ( ) for i ∈ {0, 1} is an indicator function that a worker of type i and match quality  quits in period t; and A ti ( ) for i ∈ {0, 1} is an indicator function that an unemployed worker of type i accepts  a job with match quality  in period t. For i ∈ {0, 1}, let the probability of rejecting a job be: H ti = H (ˆti ) = 1 −  ∈ E A ti ( ). In period t, unemployed workers become employed if they get matched (probability f t −1 ) and accept the employment offer (probability 1 − H ti ). Unemployed workers eligible for UI lose eligibility either if they run out of benefits (probability (1 − f t −1 )d) or if they reject a job offer and are caught “cheating” by the UI agency (probability f t −1 H t1 (1 − π˜ )). Because of exogenous separations, employed workers of type i become unemployed of the same type with probability s. Workers also become unemployed if they quit an employment relationship. In such a case, ineligible workers cannot collect UI, but eligible workers collect UI with probability π and lose eligibility with probability (1 − π ). Consequently, the laws of motion for the measures of unemployed workers by eligibility type are:





ut0 = ut0−1 1 − f t −1 1 − H t0

+ (1 − s)(1 − π ) 

 ∈E





   + ut1−1 (1 − f t −1 )d + f t −1 H t1 (1 − π˜ ) + et0−1 s + (1 − s) (1 − g )et0−1 ( ) Q t0 ( )

et1−1 (

) +

  ) Q t1 ( ),

get0−1 (



∈E

and

(4)



ut1 = ut1−1 1 − f t −1 1 − H t1 − (1 − f t −1 )d − f t −1 H t1 (1 − π˜ ) + et1−1 s

+ (1 − s)π

 ∈E



et1−1 ( ) + get0−1 ( ) Q t1 ( ).

(5)

M. Zhang, M. Faig / Review of Economic Dynamics 15 (2012) 41–56

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2.3. Bellman equations The aggregate state of the economy, to be denoted by a vector xt , includes the common component of labor productivity pt and the distribution of workers by employment status, eligibility for UI, and match quality. To see why this distribution matters for individual decisions, note that the composition of unemployment affects the incentives to post vacancies because it determines the type of workers who are going to fill them. Moreover, the future composition of unemployment depends not only on its current composition, but also on the UI eligibility of employed workers and their match qualities, which determine their quitting rates and their type once they become unemployed. Therefore, the whole distribution of employed workers is part of the aggregate state of the economy. As a result, a complete set of aggregate state variables includes p¯ t , ut0 , ut1 , et0 ( ) and et1 ( ) for all  ∈ E . (One of these measures can be dropped because the total measure of workers is one.) The set of all possible aggregate states is denoted as X ⊂ R n+2m+1 . The dynamics of xt inside this set are dictated by the Markov chain followed by the common component of productivity and the laws of motion of the distribution of workers. In the recursive formulation of an equilibrium that follows, all endogenous variables and value functions are functions of the state of the economy, so x subscripts will replace the t subscripts used up to this point. Workers may be in four possible individual states depending on whether they are employed or not and whether they are eligible for UI benefits or not. Analogously, firms paired with a worker may be in two possible states depending on the worker’s UI eligibility state. Let the utility values of a worker of type i who is matched with a firm or not in aggregate state x and with match quality  , respectively, be W xi ( ) and U xi . Similarly, let the values of a firm matched with a worker with UI eligibility state i and match quality  be J xi ( ). Finally, let a prime denote next period, E  calculate averages over  , and E x calculate averages over x conditional on x. Using this notation, the utility values of workers and firms are recursively determined by the following Bellman equations. The value of an unemployed worker who does not collect UI is the utility of enjoying leisure during the current period plus the sum of the expected present values of being matched with a firm next period and continuing being unemployed. These events happen with probabilities f x = f (θx ) and 1 − f x , respectively. Hence,









U x0 =  + ρ f x E x E  W x0 ( ) + (1 − f x ) E x U x0 ,

(6)

where ρ is the discount factor: 1/(1 + r ). To capture in a tractable way that workers are not forced to collect UI benefits, an unemployed eligible worker is allowed to renounce such eligibility. If eligibility is maintained, the value of an unemployed worker collecting UI includes the current utilities from leisure and UI benefits, and the expected present values of being matched with a firm next period or not. In the latter case, the worker may lose UI eligibility with probability d:







˜ 1 ( ), π˜ U 1 + (1 − π˜ )U 0 U x1 = max U x0 ,  + b + ρ f x E x E  max W x x x



  + ρ (1 − f x ) dE x U x0 + (1 − d) E x U x1 .

(7)

To clarify the expected value of being matched (expression after with a firm will have to decide whether to accept the job offer continue receiving UI with probability π˜ .2 The value of this option

E  ), note that an eligible unemployed worker meeting or not. In case of rejecting the offer, the worker will is π˜ U x1 + (1 − π˜ )U x0 . In case of accepting the offer, the

˜ 1 ( ). In general, this value differs from the value of an employed worker utility value attained is a value we denote as W x

eligible for UI, W x1 ( ), because of the different probabilities of collecting UI after a job quit and a job rejection, which give different outside values to workers before and after accepting a job offer. Consequently, if Nash bargaining takes place at all times, as we assume, at the moment of accepting a job offer there will be one time transfers between firms and workers. These transfers capture in a tractable way the common practice of bargaining for concessions before signing a job contract (e.g. teaching reductions in academia), which is a phenomenon that occurs precisely because workers recognize that their bargaining power typically drops at the moment when a job contract is signed. At the beginning of a period, a matched worker ineligible for UI may choose to quit the current job to become unemployed. The value of this option is U x0 . If this option is not exercised, the worker is employed during the period. In this case, the value of the worker is the utility from the current wage plus the sum of the expected present values of next period losing the job exogenously, gaining UI eligibility, and continuing with the same status. The probability of losing the job exogenously is s, and the probability of gaining UI eligibility conditional on continuing employment is g:









W x0 ( ) = max U x0 , w 0x ( ) + ρ sE x U x0 + (1 − s) g E x W x1 ( ) + (1 − g ) E x W x0 ( )

.

(8)

A matched worker eligible for UI can choose to quit a job to collect UI with probability π . The value of this option is π U x1 + (1 − π )U x0 . The value of continuing employment is the sum of the current wage and the expected present values of exogenously losing the job next period and continuing employment:

W x1 ( ) = max







π U x1 + (1 − π )U x0 , w 1x ( ) + ρ sE x U x1 + (1 − s) E x W x1 ( ) .

(9)

2 With our definitions (chosen to minimize algebraic expressions), the probability π˜ combines the probability that UI benefits have not expired and the probability that the UI agency does not catch the worker rejecting an offer.

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M. Zhang, M. Faig / Review of Economic Dynamics 15 (2012) 41–56

The value of a firm employing a worker is the sum of current profits plus the expected present value of the firm next period. The probability of the current employment match surviving exogenous separations is (1 − s), and if the match survives the probability that a worker gains UI eligibility is g . At any time, a firm can terminate the match. Consequently,







J x0 ( ) = max 0, p x ( ) − τx0 − w 0x ( ) + ρ (1 − s) g E x J x1 ( ) + (1 − g ) E x J x0 ( ) ,

(10)

J x1 (

(11)



1 x

 ) = max 0, p x ( ) − τ −

w 1x (

 ) + ρ (1 −

) .

s) E x J x1 (

Unmatched firms post vacancies until the cost of posting a vacancy is equal to the expected present value of having the vacancy matched with a worker of type i, which occurs with probability q(θx )(u ix /u x ). Because of free entry, the value of an unmatched firm is driven to zero in equilibrium. Therefore, the following free entry condition must hold:



c = ρ q(θx ) E 











u 0x /u x E x J x0 ( ) + u 1x /u x E x max ˜J x1 ( ), 0 .

As with the worker, the utility value of starting an employment relationship with a UI eligible worker,

(12)

˜J 1 ( x

 ), differs from

the ongoing value of maintaining such employment relationship, J x1 ( ), because of possible one time transfers between firms and workers at the moment of signing an employment contract. 2.4. Nash bargaining and the decision to start an employment relationship Workers and firms who are matched split the gains from continuing the match according to the generalized axioms of Nash. This Nash bargaining takes place as soon as a worker and a firm first meet. If the surplus of the match at that point is negative, the match is immediately dissolved, and we say that the worker has rejected the job. Likewise, an ongoing employment relationship is dissolved if the surplus of the match turns negative. In this case, we say that the worker has quit the job. Because we assume that the probabilities of collecting UI differ after a rejection and a quit, the conditions that determine these two actions differ. Therefore, the surplus of a match depends not only on the worker’s entitlement to receive UI in case the match were dissolved, but also on if the worker is in an ongoing employment relationship or not. If the worker is not eligible for UI, the values of continuing the match by the worker and the firm are respectively W x0 ( ) − U x0 and J x0 ( ). Therefore, the total match surplus is:

S 0x ( ) = W x0 ( ) − U x0 + J x0 ( ).

(13)

If the worker is eligible for UI, the match surplus depends on if the UI agency would maintain eligibility after a potential dissolution. Because the agency imperfectly monitors why employment separations occur, we assume that if a match were to break down while bargaining, the eligible worker would be able to collect UI with probability π˜ (rejections) and π (quits). These are the same as the probabilities of collecting UI after a voluntary rejection or a quit because a worker who takes these actions should be considered as one who cannot successfully negotiate suitable terms with the matched firm. Consequently, the worker’s outside value from a match before and after an employment relationship begins are respectively π˜ U x1 + (1 − π˜ )U x0 and π U x1 + (1 − π )U x0 , and the match surpluses that correspond with these two values are:

˜ x1 ( ) − π˜ U x1 − (1 − π˜ )U x0 + ˜J x1 ( ), S˜ 1x ( ) = W

(14)

S 1x (

(15)

) =

W x1 (

) − π

U x1

− (1 − π

)U x0

+

J x1 (

 ).

For future reference, we denote as V xi for i ∈ {0, 1} the expected match surplus of a newly formed match, which is:

V x0 = E  S 0x ( )

and

V x1 = E  S˜ 1x ( ).

(16)

The generalized Nash solution to the bargaining problem maximizes the weighted product of the match surpluses of the two parties: [ J xi ( )]1−β [ S ix ( ) − J xi ( )]β for i ∈ {0, 1} and [ ˜J x1 ( )]1−β [ S˜ 1x ( ) − ˜J x1 ( )]β , where β denotes the worker’s bargaining power. The solution to this problem leads to the familiar sharing rule:

J xi ( ) = (1 − β) S ix ( ),

for i ∈ {0, 1},

and

˜J x1 ( ) = (1 − β) S˜ 1x ( ).

(17)

A job will be rejected if and only if S˜ 1x ( ) < 0. In case the job is rejected the firm gets value zero, and the worker gets ˜ x1 ( ) + ˜J x1 ( ) = W x1 ( ) + J x1 ( ). As value π˜ U x1 + (1 − π˜ )U x0 . If the job is accepted, the total value of the firm–worker pair is W ˜ x1 ( ) and W x1 ( ), and ˜J x1 ( ) and J x1 ( ) may differ because of the possibility of side payments explained above, the values W at the moment when an employment relationship starts. 2.5. Equilibrium

˜ x1 ( ), and ˜J x1 ( )} for A stochastic recursive equilibrium is a set of functions {u ix , e ix ( ), θx , w ix ( ), U xi , W xi ( ), J xi ( ), W i ∈ {0, 1},  ∈ E , and x ∈ X that satisfy the laws of motion (2) to (5), the Bellman equations (6) to (11), the free entry condition (12), the match surplus definitions (13) to (15), and the Nash bargaining rules (17).

M. Zhang, M. Faig / Review of Economic Dynamics 15 (2012) 41–56

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To establish the existence of an equilibrium and to study the conditions that a UI system must satisfy to be neutral, it is useful to calculate the values of UI eligibility for an unemployed worker and for a matched firm–worker pair. The value of UI eligibility for an unemployed worker, defined as Uˆ x ≡ U x1 − U x0 , satisfies the following equation:











Uˆ x = max b + ρ 1 − f x (1 − π˜ ) − (1 − f x )d E x Uˆ x + ρ β f x E x V x1 − E x V x0 , 0 .

(18)

Eq. (18) is obtained by substituting the value functions and from (6) and (7) into the definition of Uˆ x , and simplifying with the help of the Nash bargaining rules (17), and the definitions of S 0x ( ), S˜ 1x ( ), and V xi . Intuitively, eligibility for UI gives unemployed workers current UI benefits plus the expected present value of being unemployed and eligible for UI next period plus the difference in expected utility values attained by eligible and ineligible workers in case they get matched. The value of Uˆ x cannot be negative because unemployed workers are not forced to collect UI. The total value of UI eligibility for a firm–worker pair when the worker becomes eligible for UI is Bˆ x ( ) = J x1 ( ) + W x1 ( ) − J x0 ( ) − W x0 ( ). Substituting (8) to (11) into this definition and simplifying, we obtain: U x0

Bˆ x ( ) = max



U x1





π Uˆ x − S 0x ( ), τx0 − τx1 + ρ sE x Uˆ x + (1 − s)(1 − g ) E x Bˆ x ( ) .

(19) S 0x (

Intuitively, if the match is dissolved, the firm–worker pair loses the surplus of the match  ), but eligibility for UI gives the worker the extra value Uˆ x with probability π . If the match continues, gaining UI eligibility implies different contribution fees for the employing firm, and the prospect of collecting UI by the worker in case of an exogenous separation. Substituting the value functions from (6) to (11) into (13) and using Uˆ x and Bˆ x ( ) to simplify, the surplus of an employment match that involves an ineligible worker satisfies the following equation:







S 0x ( ) = max p x ( ) −  − τx0 + ρ (1 − s) E x S 0x ( ) − β f x E x V x0 + (1 − s) g E x Bˆ x ( ) , 0 .

(20)

The existence of the UI system affects this match surplus in two ways: the firm has to pay the current contribution fee τx0 and the worker gains UI eligibility next period with probability g as long as employment continues by then. ˜ x1 ( ) + ˜J x1 ( ) = Combining the definitions of Uˆ x and Bˆ x ( ) with those of S ix ( ) for i ∈ {0, 1} and S˜ 1x ( ), and using W W x1 ( ) + J x1 ( ), we obtain:

S 1x ( ) = S 0x ( ) + Bˆ x ( ) − π Uˆ x

and

S˜ 1x ( ) = S 0x ( ) + Bˆ x ( ) − π˜ Uˆ x .

(21)

Eligibility for UI by an employed worker has two opposing effects on the match surplus: eligibility for UI brings total gains Bˆ x ( ) to the firm–worker pair involved in the match. However, UI eligibility tends to reduce the match surplus because the worker’s outside value increases if the pair were to break down while bargaining. In case an ongoing relationship is voluntarily dissolved, the expected value of UI eligibility is π Uˆ x . In contrast, the expected value of UI eligibility in case of rejecting a job offer is π˜ Uˆ x . Since we assume π˜  π , (21) implies S 1x ( )  S˜ 1x ( ) for all x ∈ X and  ∈ E . A stochastic recursive equilibrium is fully characterized by Eqs. (18)–(21), together with the laws of motion of x, the definitions in (16), and the following restatement of the condition for free entry:











c θx = ρ f (θx )(1 − β) u 0x /u x E x V x0 + u 1x /u x E x V x1 .

(22)

The following proposition establishes the existence of a solution to this system of equations and some basic properties of an equilibrium. Proposition 1 (Existence). There is a set of functions θx , Uˆ x , Bˆ x ( ), S ix ( ) for i ∈ {0, 1} and S˜ 1x ( ) that solves the system of Eqs. (18)– (22), so an equilibrium exists. In this equilibrium, the unemployed will not voluntarily give up eligibility ( Uˆ x > 0 for all x ∈ X ) if one of the following two conditions hold: (i) τx0  τx1 for all x ∈ X and (ii) contribution fees are such that Bˆ x ( )  π˜ Uˆ x for all x ∈ X and  ∈ E . Furthermore, in the absence of UI, S 0x ( ) = S 1x ( ) = S˜ 1x ( ) > 0 for all x ∈ X and  ∈ E , so, in such a case, workers never quit or reject a job. Although in principle, unemployed workers may voluntarily renounce UI eligibility, they will not do so under the two important conditions stated above. The first one is that contribution fees are not raised because of eligibility (τx0  τx1 ). The rationale for this condition is that if the reverse were true, then gaining UI eligibility could become painful due to the heavily increased fees, which would reduce the value of finding a job if eligibility is maintained. A special case of this condition is identical contribution fees for all employed workers, which we will use in our baseline numerical simulations. The second condition is that eligibility does not reduce the match surplus, or equivalently, Bˆ x ( )  π˜ Uˆ x for all x ∈ X and  ∈ E . If the match surplus were to decrease upon a worker gaining eligibility, then both the firm and the worker would suffer losses because they split the surplus according to the Nash bargaining rule. If these losses were sufficiently large, the worker would be willing to give up eligibility. As we will see next, this second condition with equality is satisfied if the UI system is fully funded and neutral.

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2.6. Neutrality of the UI system An interesting issue for the design of a UI system is the conditions that ensure that the UI system is neutral in the sense of not changing economic incentives for job creation or job destruction. For this inquiry, we will be focusing on UI systems that are neither subsidized from other revenue sources, nor used to raise revenue to finance other public services. For this purpose, we will center our analysis on fully funded UI systems defined as follows: Definition. A fully funded UI system is one in which the expected present value of net benefits from the UI system for a worker who is newly hired but not yet entitled to collect UI is zero. The three key actions in the present model are the posting of vacancies, the creation of employment relationships, and their possible dissolution once they are formed. All these actions depend on the surplus of a match. The incentive for firms to post vacancies is the expected present value of the profits extracted from the employment matches once the vacancies are filled, and, because of Nash bargaining, they are proportional to the surplus of the matches formed. Consequently, a UI system will not affect the posting of vacancies if it does not affect the expected match surpluses of newly created jobs by ineligible and eligible workers, which respectively are S 0x ( ) and S˜ 1x ( ). The sign of the match surplus S 0x ( ) also determines the creation and the dissolution of employment relationships involving ineligible workers, while the acceptance of job offers by eligible workers depends on the sign of S˜ 1x ( ). Therefore, if a UI system has no effect on S 0x ( ) and S˜ 1x ( ), it will be neutral with respect to the posting of vacancies, the creation and dissolution of employment relationships involving workers ineligible for UI, and the acceptance of job offers by eligible workers. Such UI system may still affect the quitting decisions of workers eligible for UI. However, if workers never quit an employment relationship in the absence of UI because p x ( ) −  > 0 for all x ∈ X and  ∈ E , as we have assumed, they will not quit any job either with a UI system which has no effect on S 0x ( ) and S˜ 1x ( ). The reason for this is that π˜  π implies S 1x ( )  S˜ 1x ( ), so S 1x ( ) cannot be negative if S˜ 1x ( ) is positive.3 Inspection of (20) reveals that the condition τx0 = ρ (1 − s) g E x Bˆ x ( ) ensures that the match surplus S 0x ( ) is not affected by the UI system because it makes the firms employing ineligible workers pay the expected present value of gaining from eligibility that would bring to the firm–worker pair. Also, Eq. (21) implies that UI eligibility does not change the match surplus S˜ 1x ( ) if Bˆ x ( ) = π˜ Uˆ x . This equality implies that the gain from eligibility by a firm–worker pair is equal to the worker’s outside value brought by eligibility in case the worker in the pair were to reject the job. Therefore, together, these two conditions ensure the neutrality of the UI system. The following proposition provides contribution fees that achieve these conditions in a fully funded UI system. Proposition 2 (Neutrality of the UI system). Contribution fees can be designed to make the UI system both fully funded and neutral. In such a case, the level of UI benefits, the duration of these benefits and the time it takes to become eligible for UI are all irrelevant for the determination of output, vacancies, and unemployment. In particular, if the UI contribution fees are such that τx0 = π˜ (1 − s) g ρ E x Uˆ x and τx1 = π˜ [ρ (1 − s) E x Uˆ x − Uˆ x ] + ρ sE x Uˆ x , then Uˆ x is equal to the expected present value of the UI benefits to be received by an unemployed worker eligible for UI, and the UI system is fully funded and neutral. Proposition 2 provides a set of conditions that makes a UI system irrelevant. Like other irrelevance results, such as Ricardian Equivalence, this proposition should be useful to pinpoint the economic effects of a UI system as violations from its stated premises. In this vein, if risk neutrality is maintained, the effects of a UI system have to be found in a poor structure of contribution fees that is either distorting the posting of vacancies or not preventing strategic behavior such as quitting once eligibility is achieved or rejecting jobs while benefits last. It should be noted here that the neutral financial structure in Proposition 2 involves fees that depend on the productivity of a match, but not on how the output of the match is distributed between the worker and the firm. Contribution fees proportional to wages would affect the Nash bargaining rule, and therefore, prevent the neutrality of the UI system. With risk aversion, a UI system will reduce income uncertainty, which will affect the willingness to work and save in ways that are beyond the scope of the present contribution. Because we assume risk neutrality, our analysis cannot address the deeper question of what is the optimal size and configuration of a UI system when it plays a fundamental role in insuring employment risks. However, we hope that a generalized version of Proposition 2 will be useful in building and understanding the optimal system with risk aversion. In this regard, Proposition 2 shows that differentiating contribution fees by eligibility types is a good mechanism to discourage moral hazard with respect to the acceptance or dissolution of employment relationships. This mechanism should enhance the schemes advanced by Hopenhayn and Nicolini (2009) based on raising the contributions of workers who have experienced prior unemployment spells.

3 Without the assumption p x ( ) −  > 0 for all x ∈ X and  ∈ E , a neutral UI system requires a more sophisticated contribution scheme than the one in Proposition 2. For example, one way of ensuring neutrality without this assumption would be to complement the contribution scheme in Proposition 2 with a one time subsidy for workers eligible for UI when they accept a job.

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Unfortunately, the contribution fees of Proposition 2 cannot be reduced to simple functions of the parameters of the model. However, in the absence of productivity shocks, the contribution fees are functions of a few parameters and the endogenous finding rate. Therefore, in this case, we obtain the following corollary. Corollary. In the absence of productivity shocks, the contribution fees of the neutral UI system are the following:

τ0 =

π˜ g (1 − s) r + f x (1 − π˜ ) + (1 − f x )d

b

and

τ1 =

s − π˜ (s + r ) r + f x (1 − π˜ ) + (1 − f x )d

b.

The contribution fees that achieve neutrality depend crucially on how likely it is for a worker to collect UI after a moralhazard rejection of an employment offer. In the extreme case that such “cheating” is impossible (π˜ is zero), no fees should be charged to firms employing ineligible workers. In this case, the fee charged to firms employing eligible workers is equal to the expected present value of the benefits that such eligibility entitles times the probability of an exogenous separation in one period. In contrast, if workers have no difficulty in collecting UI after they reject a job offer (π˜ is close to one), then ineligible workers must pay large fees that finance not only all the UI benefits that eventually they will be entitled to receive, but also cover the cost of a subsidy to eligible workers since in this case τ 1 becomes negative. Such a subsidy is necessary to avoid the strategic rejections that the high value of π˜ entices. Intermediate values of π˜ bring more balanced contribution structures. In particular, the two types of workers must pay the same fees if π˜ = s/[r + s + g (1 − s)]. 3. Numerical analysis This section calibrates the model laid out in Section 2 using data from the United States and analyzes its quantitative predictions. One objective of this section is to show that, when eligibility for UI has to be earned, one can calibrate the model to realistically predict that unemployment responds strongly to productivity shocks and weakly to changes in UI benefits. The other objective of this numerical analysis is to inquire how far apart the UI system in the United States is from the neutrality of Proposition 2. Finally, our analysis unveils that institutional features, such as the ease with which workers can collect UI after voluntarily rejecting an offer of employment, are crucial for the effects of a UI system on employment. The numerical analysis of this section adopts the following specializations. The matching function is assumed to be Cobb–Douglas: M ( v , u ) = μ v 1−η u η for i ∈ {0, 1}.4 With this functional form, the finding rate with respect to the vacancy– unemployment ratio is: f (θ) = μθ 1−η , so it has constant elasticity 1 − η. Following Shimer (2005), the common part of aggregate labor productivity is assumed to be a stochastic process that satisfies p x =  + e y ( p ∗ − ), where p ∗ is a positive parameter, and y is a zero mean random variable that follows a symmetric 51-states Markov process in which transitions only occur between contiguous states. The transition matrix governing this process is fully determined by two parameters: the step size of a transition δ and the probability that a transition occurs λ. Zhang (2008) provides further details on modeling of this stochastic process. The distribution of match qualities is assumed to be uniform with a discrete support of 201 states evenly spread in an interval [−¯ , ¯ ]. As for the UI contribution fees, we assume that they are uniform in our baseline simulations. However, when we simulate a neutral system, the contribution fees are contingent on eligibility and the aggregate state of the economy. 3.1. Parameterization The calibration targets that we adopt aim to replicate the main rates and flows in the labor market, and, in a stylized way, the key features of the UI system in the United States and the effects of changes in this system on the labor market. The model period in the simulations is set to be one week, and consecutive periods are aggregated to construct quarterly series to match the implications of the model for variables observed at this frequency. The model is calibrated in two stages. In the first stage, the four top parameters in Table 1 are determined independently from the rest. The interest rate (r ) is set to correspond to the typical annual rate of 4 percent. The probabilities to earn UI eligibility ( g ) and to stop collecting UI benefits (d) in one period are set to respectively match the average time it takes for a worker to gain UI eligibility (20 weeks) and to exhaust UI benefits (24 weeks) in the United States.5 Finally, the value of c is normalized to one. By doing so, we are just defining the units in which vacancies are measured.6 In the second stage, the remaining thirteen parameters of the model are jointly calibrated to match the eleven targets in the bottom part of Table 1, in addition to the Hosios condition, and a zero average budget deficit for the UI system. The first seven of these targets are empirical moments from the United States that describe the main features of the business cycle in

4 Strictly speaking a Cobb–Douglas matching function must be truncated to ensure that the finding and the filling rates are probabilities. In our simulations, this is never an issue because their values are always between zero and one without any truncation. 5 See Card and Riddell (1992) and Osberg and Phipps (1995) for the weeks needed to gain eligibility. The number of weeks eligibility lasts is an average over the period 1951–2003 reported by annual report and financial data from the U.S. Department of Labor Employment and Training Administration (column 27). It is available at http://workforcesecurity.doleta.gov/unemploy/hb394.asp. 6 The normalization adopted by Shimer (2005) of setting average θ equal to one yields identical results except for the calibrated values of μ.

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Table 1 Calibration targets. Annual real interest rate (r ) Average weeks of employment needed for UI eligibility (1/ g ) Average weeks before UI benefits expire (1/d) Cost of posting a vacancy (c ) Average labor productivity Standard deviation of labor productivity (quarterly in logs) Autocorrelation of labor productivity (quarterly in logs) Average unemployment rate Average short-term unemployment rate Standard deviation of unemployment rate conditional or not on p (quarterly in logs) Standard deviation of θ conditional or not on p (quarterly in logs) Effective replacement rate of UI benefits (b/ w ) Effect of small changes in b on the separation rate Increased average unemployment duration (in weeks) if (b/ w ) increases by 0.1 Increase in the elasticity of duration of unemployment with respect to b due to eligibility

0.04 20 24 1 1 0.020 0.878 0.0567 0.0244 0.0775/0.190 0.151/0.382 0.25 0 1 0. 3

Note: These are the targets that our calibration of the model aims to reproduce. Most of these targets correspond to empirical moments in the United States. Each one of the first four targets pins down one parameter, specified in parenthesis. The rest collectively pin down the rest of parameters. We conduct two different simulations depending on if the targeted standard deviations of unemployment and θ are conditional on p (conditional moments) or not (unconditional moments).

the labor market, and were constructed from the data reported by Shimer (2005) or its original sources.7 Note that, following Mortensen and Nagypál (2007), we conduct a calibration using moments conditional on labor productivity p, which is the only variable with exogenous shocks in our simulations. So, we multiplied the unconditional standard deviations reported by Shimer with the respective correlations of each variable with p . For comparison purposes, we also conduct a calibration using unconditional moments. The remaining targets that we seek to replicate define key characteristics of the UI system in the United States and its effects on labor market flows. As for the size of the UI system, we target the effective replacement rate measured as the product of the take-up rate (fraction of eligible unemployed workers who actually collect UI) times the observed replacement rate conditional on receiving benefits. As documented by Blank and Card (1991), the take-up rate over the period 1977–1987 was fairly stable around 0.7, while the replacement rate conditional on receiving benefits averaged 0.357 over the period 1972–2003.8 Consistent with the effects of UI summarized in the review survey by Atkinson and Micklewright (1991), we aim that in our model increases in UI benefits affect the unemployment rate moderately through the average duration of unemployment spells, but not through their incidence. To this end, we target the effect of a 10 percentage point increase in the replacement rate to raise the average duration of unemployment by one week,9 but to leave the separation rate unchanged.10 Finally, we target the extra elasticity of duration of unemployment with respect to b for those who are eligible to collect UI relative to those who are not to be 0.3. That is, we aim at realistic responses of the likelihood that an eligible worker will reject a job offer when b increases. As the empirical counterpart of this response, we pick the 0.3 elasticity found by Meyer and Mok (2007). In their study, they compared the durations of unemployment for workers who got an increase in b in a reform by the State of New York in 1989 and other workers searching for jobs in the same markets but without receiving such a raise. As these authors argue, this analysis isolates the individual response to the increase in b for the workers who received it, controlling for other effects such as the changes the increase in b had on the distribution of wages and the posting of vacancies. Although all the parameters in the model jointly determine the targets in the bottom part of Table 1, each parameter can be associated to a few empirical moments that it affects most directly. The value of p ∗ determines, together with ¯ , the average aggregate labor productivity. The values of λ and δ control respectively the autocorrelation and the standard deviation of the aggregate productivity process. The value of s is a key determinant of the average short-term unemployment rate. The value of μ determines the average finding rate, and, for a given value of s, the average unemployment rate. The standard deviation of θ is primarily determined by how close the value of leisure is to productivity (see Mortensen and Nagypál, 2007). The values of η and β, which are equal to satisfy the Hosios condition, are inversely related to the elasticity

7 For these calculations, we used Table 1 in Shimer (2005). The average short-term unemployment rate from 1951(1) to 2003(12) was calculated using Shimer’s methodology from the following series of the Current Population Survey by the Bureau of Labor Statistics: (i) Number of Unemployed for Less than 5 Weeks (Series ID: LNS13008396) and (ii) Civilian Labor Force Level (Series ID: LNS11000000), both available at http://www.bls.gov/cps/. Finally, the average labor productivity being one is a normalization. 8 This ratio is reported in the annual report and financial data of the U.S. Department of Labor Employment and Training Administration (Column 33), which is available at http://workforcesecurity.doleta.gov/unemploy/hb394.asp. 9 Moffitt and Nicholson (1982) estimate that a 10 percentage point increase in the replacement rate leads to an increase in unemployment duration of up to one week. For the same change, Moffitt (1985) and Meyer (1990) offer estimates of around 0.5 weeks and 1.5 weeks, respectively. 10 The survey by Atkinson and Micklewright (1991) remarks the importance of the outflows from unemployment in explaining the effect of benefits on overall unemployment. Also, Sider (1985), Pissarides (1986), and Burda (1988), after examining a variety of countries and time periods, emphasize that variations in unemployment duration are the primary driving force of variations in unemployment.

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Table 2 Parameter values. Conditional moments p∗

¯  λ δ s

μ η β b

τ π π˜

0.9997 0.0069 0.8330 0.3787 0.0347 0.0079 0.3687 0.6916 0.6916 0.2460 0.0111  0.35 0.6117

Unconditional moments 0.9984 0.0006 0.9268 0.3771 0.0769 0.0079 0.6729 0.5852 0.5852 0.2463 0.0111  0.30 0.4468

Note: With these parameters the model replicates the moments in Table 1. In the column “conditional moments”, the model is calibrated to match the standard deviations of log of unemployment and the log of the vacancy/unemployment ratio conditional on productivity. In the column “unconditional moments”, these targeted standard deviations are unconditional.

of the finding rate with respect to θ (the vacancy–unemployment ratio). So the standard deviation of the unemployment rate falls with η for a given degree of variability of θ. The key parameters that determine the features of the UI system in the United States and its effects are the following. The value of b determines the UI benefits replacement rate. Also, given the other parameters of the UI system, the contribution fee τ determines the average budget deficit. The probability of collecting UI after quitting (π ) determines the effect of changes in b on the incidence of unemployment. This effect is guaranteed to be zero if the probability of collecting UI after a quit (π ) is sufficiently smaller than the probability of collecting UI after a rejection of an offer of employment (π˜ ). Finally, the parameters ¯ and π˜ are key determinants for the durations of unemployment spells and the effects of increasing b on such durations. To see this, note that the parameter ¯ determines the degree of heterogeneity in matches. Hence, everything else equal, the frequency of rejections to employment offers increases with ¯ . Therefore, large values of ¯ are associated with long unemployment spells of UI eligible workers and large effects of b on the duration of these spells. High values of π˜ are also associated with frequent rejections because with a high π˜ UI eligible workers are unlikely to lose UI eligibility if they reject a job. However, this is not the only effect of a high π˜ . A high π˜ also reduces the incentives to post vacancies because it increases the bargaining power of UI eligible workers when they first meet an employer. Consequently, since there is a single search market, high values of π˜ affect the expected duration of unemployment spells for all workers. This effect is amplified as b increases. As a result, ¯ and π˜ jointly determine how an increase in b affects both the average duration of unemployment and the differential effects on the durations of unemployment by eligible and ineligible workers. 3.2. Benchmark results The calibrated model can successfully replicate all the targets listed in Table 1. The parameter values obtained in the second stage of the calibration are reported in Table 2. The calibrated values for some of these parameters deserve some commentaries. First of all, even though we target average productivity to be one, the calibrated value of p ∗ is smaller than one because the workers with the poorest match qualities are more likely to reject jobs in bad times, so average productivity ends up being higher than p ∗ . However, the values of p ∗ are very close to one because Jensen inequalities work in the opposite direction, and in our calibrations not many workers end up rejecting jobs. (The fraction of matched workers who reject their offers of employment in the calibrations to conditional moments and unconditional moments are 2.8% and 5.3%, respectively.) Secondly, the values of leisure, 0.83 and 0.93, must be fairly large to generate the wide variability of the vacancy–unemployment rate observed over the business cycle. This is particularly true when unconditional moments are targeted. However, our values are lower than those of Hagedorn and Manovskii (2008) because neither UI benefits nor the UI contribution fees are included in . Curiously, in both of our simulations, the sum  + τ + b is larger than one. If this were the case in a model with universal UI eligibility, all workers would quit because this sum would be the opportunity cost of being employed in such a model. However, in our simulations workers do not quit because they realize that if they were to do it, it is likely for them to lose UI eligibility (π is low in both calibrations), and even if they did not lose eligibility, UI benefits would not last forever. Thirdly, the calibrated value of ¯ is small in the calibration to conditional moments, and tiny in the calibration to unconditional moments. This is just a manifestation that with high values of leisure small changes in productivity have strong effects on labor market flows. Fourthly, the calibrated values of the probability of collecting UI after a rejection of an offer of employment are around 1/2 (0.61 and 0.45). Finally, the precise value of π cannot be determined because as long as quits do not occur, the value of π is irrelevant. However, we can determine that the maximum values of π for which quits are insignificant (less than 10−6 ) are 0.35 in the calibration to conditional moments and 0.3 in the calibration to unconditional moments.

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Table 3 Effects of changes in UI benefits and productivity (in percent). Conditional moments

b = 0.01 (keeping τ unchanged) log average unemployment rate log average duration of UI eligible workers log average duration of UI ineligible workers b = 0.01 (rebalancing UI budget) log average unemployment rate log average duration of UI eligible workers log average duration of UI ineligible workers p = 0.01 (or ∇ τ = 0.01) ∇ log average unemployment rate ∇ log average duration of UI eligible workers ∇ log average duration of UI ineligible workers

Unconditional moments

1.34 1.69 0.53

1.64 1.73 0.76

1.57 1.95 0.68

2.46 2.56 1.26

4.75 5.80 2.54

14.06 14.87 8.07

Note: Even when contribution fees are adjusted to rebalance the UI budget, changes in UI benefits have a much weaker effect on unemployment than changes in labor productivity. In both changes, the effect on the duration of unemployment of eligible workers is larger than the effect on the duration of ineligible workers.

3.3. Response to changes in benefits and productivity Our calibration succeeds in simultaneously matching the wide variability of the business cycles generated by productivity shocks and the modest responses of unemployment to changes in UI benefits. To further clarify this point, Table 3 presents the semi-elasticities of unemployment with respect to changes in UI benefits and productivity. More precisely, it records the effect of increasing b and p (for all P ) by 0.01 on the logarithm of the average unemployment rate. Also, it decomposes this effect by reporting the changes in the average durations of unemployment spells suffered by eligible and ineligible workers. As Table 3 shows, regardless of if contribution fees are adjusted to rebalance the UI budget or not, changes in UI benefits have much weaker effects than changes in productivity. In the calibration to conditional moments, a 0.01 increase in benefits raises the unemployment rate by 1.34 percent if contribution fees are kept constant. This increase is a little larger if τ is adjusted to rebalance the UI books (1.57 percent), but is still well below the 4.75 percent increase delivered by a 0.01 drop in productivity. In the calibration to unconditional moments, this contrast is even sharper. The semi-elasticities of unemployment with respect to b are 1.64 (keeping τ constant) and 2.46 (adjusting τ to balance the budget), while the semi-elasticity of unemployment with respect to productivity is (minus) 14.06. Table 3 also shows that changes in the duration of unemployment spells by eligible workers are responsible for the bulk of the adjustments that take place after changes in benefits or productivity. This implies that, when b rises or p drops, unemployment increases not only because firms post fewer vacancies as in the standard Mortensen and Pissarides model, but also because eligible workers reject a larger fraction of employment offers. In our model, unemployment reacts weakly to change in benefits, but strongly to changes in productivity. In contrast, in the standard model where all unemployed workers are automatically eligible for UI, the two reactions have equal strength, that is, a drop in productivity has the same effect as an increase in UI benefits. To illustrate this, we recalibrated a model similar to ours, but without different match qualities and with all the workers being exogenously entitled to UI. With these modifications, our model becomes indistinguishable from the standard one used, for example, in Shimer (2005). Once recalibrated, this model could reproduce all the moments in Table 1 except for the effects of changes in b on the duration of unemployment (last two targets in the table), which are now much larger. The absolute value of the semi-elasticities of unemployment with respect benefits and productivity become both equal to 4.69 in the calibration to conditional moments, and 12.25 in the calibration to unconditional moments. Consequently, our departures from the standard model strongly reduce the semi-elasticity with respect to benefits, while changing little the semi-elasticity with respect to productivity. The two modifications we introduce to the standard model are both important for our results. The endogenous UI eligibility is crucial for lowering the semi-elasticity with respect to benefits. Meanwhile, heterogeneous match quality is important to guarantee that most of the adjustments take place as changes in the rejection rate. To illustrate this, we eliminated different match qualities from the baseline model while maintaining endogenous UI eligibility. This variation of the model is then recalibrated to the targets in Table 1 except for the differential elasticity of duration with respect to b of eligible and ineligible workers. In the recalibrated model, the responses to changes in b and p are still of the same order of magnitude as our baseline model. For example in the conditional moments calibration, the semi-elasticity with respect to b marginally falls from 1.34 to 1.30 and (minus) the semi-elasticity with respected to p marginally falls from 4.75 to 4.49. However, in this calibration all the adjustments take place through changes in the posting of vacancies since no rejections take place in equilibrium.11

11 In the calibration to unconditional moments, some eligible workers reject jobs even if ¯ = 0, but the number is smaller than the one in the calibration reported in Table 2.

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Table 4 Effects of UI reforms (in percent).

Calibrated models Average unemployment rate Uniform contribution fees Neutral contribution fees Average unemployment rate Average fees paid by UI ineligible workers τx0 Average fees paid by UI eligible workers τx1 Elimination of “cheating” Average unemployment rate Uniform contribution fees Elimination of “cheating” controls for rejections Average unemployment rate Uniform contribution fees

Conditional moments

Unconditional moments

π˜ = 0.61

π˜ = 0.45

5.67 1.11

5.67 1.11

π˜ = 0.61

π˜ = 0.45

4.67 7.58 0.65

4.06 3.92 0.72

π˜ = π = 0

π˜ = π = 0

4.52 0.94

3.73 0.80

π˜ = 1 − d

π˜ = 1 − d

12.27 2.42

19.28 4.05

Note: Our calibration to conditional moments predicts that if the United States were to adopt the neutral contribution fees of Proposition 2, the average unemployment rate would fall from 5.67 to 4.67 percent. To discourage moral hazard, neutral fees are much higher for workers who are not eligible to collect UI than those who are eligible. Alternatively, if the UI system could prevent all workers who reject a job from collecting UI, the average rate of unemployment would fall to 4.52 percent. In contrast, if all workers who reject a job were certain of maintaining eligibility, the average unemployment rate would more than double and become 12.27 percent. These predicted effects are stronger in the calibration to unconditional moments, but still of the same order of magnitude.

3.4. The effect of UI reforms The objective of this subsection is to examine how the average unemployment rate would be affected if the UI system in the United States were to be reformed either to adopt the neutral contribution fees of Proposition 2 or to change the likelihood of collecting UI after a rejection of an offer of employment. With our calibrated parameter values, if the uniform contribution fee were to be replaced with the neutral contribution fees of Proposition 2, the average unemployment rate would fall by 1 percent in the calibration to conditional moments and by 1.6 percent in the calibration to unconditional moments. In principle, the neutral contribution fees are contingent on both the eligibility type of workers and the aggregate state of the economy. However, in practice, what really matters is to condition the fees to the UI eligibility of workers. In fact, replacing the state contingent contribution fees of the neutral system by their averages for each eligibility type has virtually no effect on the average unemployment rate (not shown in Table 4). To achieve neutrality, contribution fees must give the proper incentive to eligible workers to avoid strategic rejections of offers of employment or credible threats to reject such offers while bargaining with firms. In the calibration to conditional moments, Table 4 shows that in the neutral system, the contribution fees of eligible and ineligible workers are quite different. This is because our calibrated probability of collecting UI after rejecting a job offer is fairly high (π˜ is 61 percent). With a high π˜ , eligible workers have a high outside value to their employment match. Therefore, one needs a low contribution fee (0.65 percent of productivity) to induce them to accept their jobs. This implies that, to maintain the UI system fully funded, ineligible workers must pay large contribution fees (7.58 percent of productivity). Such large contribution fees are still neutral because the low fees charged to UI eligible workers make UI eligibility highly valuable. Therefore, the large fees charged to the ineligible workers essentially buy a fair draw to obtain the highly valuable UI eligibility. In the calibration to unconditional moments, the two contribution fees are more similar than in the calibration to conditional moments because the calibrated value of π˜ is lower (45 percent). The incentive for strategic job rejection is key to understand why the prevalent UI system delivers much higher unemployment than the neutral system. A reform that were to make it harder for workers to collect UI after rejecting or quitting a job would achieve similar reductions in the unemployment rate as the neutral system. In fact, if the probability of collecting UI after rejecting an offer or quitting a job were to drop to zero, the unemployment rate would fall below the rates achieved with the neutral system. With π˜ = π = 0, the UI system gives no incentive for eligible workers to reject or quit a job, yet it still makes employment more attractive to ineligible workers. As a result, the UI system creates employment instead of destroying it. In contrast, as Table 4 shows, if all “cheating” controls on job rejections were to be eliminated, so π˜ were to become 1 − d (the same as the probability of keeping UI eligibility in a period without a job offer), the unemployment rate would climb to 12.3 percent in the calibration to conditional moments and 19.3 percent in the calibration to unconditional moments.12 Consequently, changes in π˜ have a large predicted impact on the average unemployment rate. This result suggests that institutional differences in the design of UI systems determining how easy it is for a worker to collect UI after a strategic job rejection are probably more important in explaining the observed wide distribution of unemployment rates across countries than differences in the generosity of UI benefits.

12 In this simulated reform, we kept π = 0. We tried another reform were π = π˜ = 1 − d, and an equilibrium fails to exist because the fraction of workers employed becomes so small that they cannot support the expensive UI system.

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4. Conclusions Once workers have to earn their entitlement to UI benefits with prior employment, a generous UI system is an additional benefit to an employment relationship and as such promotes job creation. This positive entitlement effect counteracts the negative effects arising from moral-hazard behavior and from the high cost of financing a generous UI system. If individuals are risk neutral, then there is a fully funded UI contribution scheme that eliminates moral hazard and any effects of the UI system on employment decisions. So, the UI system is rendered irrelevant for the determination of output and employment. As with Ricardian Equivalence, this irrelevance result should be useful to pinpoint the effects of a UI system to violation of its premises. That is, the economic effects of a UI system arise from three sources: The insurance it provides to smooth the income fluctuations experienced by risk-averse workers. The improper provisions for financing the UI system which ends up subsidizing or overly taxing segments of the labor force. And, the potential moral-hazard effects on search behavior, acceptance of job offers, and quitting decisions if incentives are not carefully crafted. In our numerical simulations, we find that the current UI system in the United States is substantially different from a neutral one. If the current contribution scheme were replaced by one that achieves neutrality, the average unemployment rate would drop from 5.7 percent to 4.7 percent in our preferred calibration. This gain is attained by reducing the contributions of UI eligible workers, who are tempted to reject an employment match, and by increasing the contributions of ineligible ones. In terms of cyclical fluctuations, endogenizing UI eligibility allows us to simultaneously generate realistic cycles for unemployment and vacancies resulting from productivity shocks and realistic responses to changes in UI benefits, which proves to be a major challenge for the standard Mortensen–Pissarides model with unconditional UI eligibility. A final lesson we learn from our inquiry is that institutional details of a UI system are more important for its economic effects than the generosity of UI benefits. In fact, in our preferred calibration, we calculated that changing the probability of collecting UI because of an employment rejection from 0 to 1 increases the average unemployment rate from 4.5 percent to 12.3 percent. Therefore, this is potentially an important factor in explaining the international differences in unemployment rates. Appendix A A.1. Proof of Proposition 1 Define Θ( V ) to be the real function V → θ that satisfies: (1 + r )c θ = f (θ)(1 − β) V . The assumed properties of the matching function imply that θ/ f (θ) is a strictly increasing function of θ such that limθ→0 [θ/ f (θ)] = 0, so Θ( V ) is well defined, continuous and increasing, and Θ(0) = 0. The set of possible state variables X ⊂ R n+2m+1 is bounded because there are a finite number of productivities and the measures describing the distribution of workers are bounded below by 0 and above by 1. Let C ( X ) be the set of continuous and bounded functions {Uˆ x , Bˆ x ( ), S 0x ( )} defined on X , with the sup norm. Define the mapping T as follows. For each element in C ( X ), use (21) and (16) to find S 1x ( ), S˜ 1x ( ), V x0 , and V x1 . With these functions, calculate (u 0x E x V x0 + u 1x E x V x1 )/u x for each x ∈ X , and use the function Θ( V ) to find θx and f x = f (θx ). Finally, use these functions on the right-hand side of (18) to (20) to define T (C ) as the set of values of Uˆ x , Bˆ x ( ) and S 0x ( ) on the left-hand side of (18)–(20). Let

 (τ 0 − τx1 )(1 + r ) Υ¯ = max max x ,0 and x r + s + g (1 − s) V¯ 0 =

maxx ( p x ( ) −  − τxi )(1 + r )

+

r+s

(1 − s) g B¯ , r+s

where B¯ = Υ¯ + U¯ and U¯ = (b(1 + r ) + Υ¯ )/r. Define the subset C¯ ⊂ C ( X ) by imposing the following bounds to its elements: 0  Uˆ x  U¯ , − V¯ 0  Bˆ x ( )  B¯ , and 0  S 0x ( )  V¯ 0 . The set C¯ is non-empty, closed, bounded, and convex. The mapping T : C¯ −→ C¯ is continuous and maps C¯ onto itself. The family of functions one obtains by repeated application of T (C¯ ) is equicontinuous because the number of productivity states is finite. Consequently, as a result of Schauder’s fixed point theorem, C¯ has a fixed point in X . If Bˆ x ( )  π˜ Uˆ x for all  ∈ E and x ∈ X , Eqs. (21) and (16) imply that V x1  V x0 for all x ∈ X . Consequently, Eq. (18) implies that Uˆ x > 0 for all x ∈ X since b > 0. If τx0  τx1 , Eq. (19) implies that Bˆ x ( )  0 for all  ∈ E and x ∈ X . Using (16) and (21), Eq. (18) can be rewritten as:











Uˆ x = max b + ρ β f x E x E  Bˆ x ( )

   + 1 − f x (1 − π˜ ) − β f x π˜ − (1 − f x )d E x Uˆ x , 0 .

Therefore, Bˆ x ( )  0 implies that Uˆ x  b > 0. In the absence of UI, Uˆ x = Bˆ x ( ) = τxi = 0 for all  ∈ E , x ∈ X , and i ∈ {0, 1}; so, Eqs. (20) and (21), together with p x ( ) > , imply S 1x ( ) = S˜ 1x ( ) = S 0x ( ) > 0 for all  ∈ E and x ∈ X . 2

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A.2. Proof of Proposition 2

τx0 and τx1 in the proposition into (19), we obtain:    Bˆ x ( ) = π˜ Uˆ x + max ρ (1 − s)(1 − g ) E x Bˆ x ( ) − π˜ Uˆ x , − S 0x ( ) − (π˜ − π )Uˆ x .

Substituting the definitions of

(23)

For a given pair of functions S 0x ( ) and Uˆ x in the set C¯ defined in the proof of Proposition 1, let T be the mapping of the function Bˆ x ( ) on the right-hand side of (23) onto its value of the left-hand side. Since ρ (1 − s)(1 − g ) ∈ (0, 1), T is a contraction (see Lemma 1 below). Also, given the definition of C¯ , the second argument of the max operator in (23) can never be positive. Therefore, T has a single fixed point, which must be Bˆ x ( ) = π˜ Uˆ x for all x ∈ X and  ∈ E. This equality, combined with (21), implies that S˜ 1x ( ) = S 0x ( ) for all x ∈ X and  ∈ E . Also, it implies that τx0 = ρ (1 − s) g E x Bˆ x ( ). Therefore, Eq. (20) simplifies into:







S 0x ( ) = max p x ( ) −  + ρ (1 − s) E x S 0x ( ) − β f x E x V x0 , 0 . Since this is the match surplus without a UI system, the proposed tax system does not affect S 0x ( ) and S˜ 1x ( ). Therefore, it is neutral with respect to the posting of vacancies, the creation and dissolution of employment offers involving workers ineligible for UI, and the acceptance of job offers by workers eligible for UI. Furthermore, S 1x ( )  S˜ 1x ( ) > 0 because of π˜  π and Proposition 1. Consequently, the proposed UI system is also neutral with respect to quitting decisions by workers eligible for UI. Using Uˆ x > 0 and S˜ 1x ( ) = S 0x ( ) for all  ∈ E and x ∈ X , Eq. (18) simplifies into:





Uˆ x = b + ρ 1 − f x (1 − π˜ ) − (1 − f x )d E x Uˆ x . Hence, Uˆ x is the expected present value of the UI benefits that an unemployed eligible worker expects to receive. Let B ix ( ) be the liabilities incurred by the UI system towards an employed worker of eligibility i ∈ {0, 1} and match quality  . For the system to be fully-funded, B 0x ( ) must be zero. To check that this is the case, given their definitions and the results from the proposition that with the proposed UI system workers never quit a job voluntarily, the liabilities B 0x ( ) and B 1x ( ) must obey:





B 1x ( ) = −τx1 + ρ sE x Uˆ x + (1 − s) E x B 1x ( ) B 0x (



0 x

 ) = −τ + ρ (1 −



s) g E x B 1x (

) −

and

   ) + (1 − s) E x B 0x ( ) .

B 0x (

(24) (25)

Substracting (25) from (24), a comparison with (19) reveals that Bˆ x ( ) = B 1x ( ) − B 0x ( ) because Bˆ x ( ) is equal to the second argument of the max operator when workers do not quit. Therefore, with the proposed system of contribution fees, Eq. (25) implies that B 0x ( ) = 0 for all  ∈ E and x ∈ X because as shown above it implies τx0 = ρ (1 − s) g E x Bˆ x ( ). 2 A.3. Lemma 1 (The notation of this lemma is not consistent with the rest of the paper.) Consider a compact set X ⊂ R n . Let C ( X ) be the space of continuous and bounded functions defined on X with the sup norm. Also, let S be a contraction mapping of modulus β in C ( X ). Define T : C ( X ) −→ C ( X ) as T f (x) = h1 (x) + max{ S f (x), h2 (x)} where h1 and h2 are functions in C ( X ). The mapping T is a contraction of modulus β. Proof.



T f − T g = sup T f (x) − T g (x) x   = sup max S f (x), h2 (x) − max S g (x), h2 (x) x  sup S f (x) − S g (x)  β f − g . 2 x

A.4. Sketch of the numerical algorithm Since the state vector x contains the whole distribution of workers by eligibility type, match quality, and employment status, the equilibrium functions have to be approximated in some way. The reason why the state vector x has more elements than p is that the weights u 0x /u x and u 1x /u x in (22) depend not only on p but also on the distribution of workers. Since transitions are fast, it turns out that these weights are well approximated by the “steady-state” weights, which are only conditional on p . Therefore, in our numerical simulations, we replaced u 0x /u x and u 1x /u x in (22) with their “steady-state” counterparts. As a result, θx , Uˆ x , V x0 , and V x1 are approximated with functions that depend only on p , and Bˆ x ( ), S 0x ( ), S 1x ( ), and S˜ 1x ( ) with functions that depend on p and  . We checked the accuracy of this approximation by comparing the

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simulated histories of the weighted sum inside the square brackets in (22) using the “steady-state” weights with the actual simulated weights. We found that the two weighted sums are almost identical: their correlation is 0.997 in the calibration to conditional moments and 0.989 in the calibration to unconditional moments. Our algorithm to calculate an equilibrium starts with a guess for the equilibrium functions (e.g. their steady state values in the absence of productivity shocks). Given these guesses, we used (18)–(22) (modified as explained in the previous paragraph) to calculate a new set of functions, and repeated this process until convergence was achieved. The model was then simulated for 52,000 periods (weeks) by using Eqs. (2)–(5) and the behavioral rules implied by our calculated value functions. From this artificial history we calculated our simulated moments by aggregating sets of 13 contiguous weeks into quarters to calculate moments observed at that frequency. 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