Labor Market Cycles, Unemployment Insurance Eligibility, and Moral Hazard

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1 Labor Market Cycles, Unemployment Insurance Eligibility, and Moral Hazard Min Zhang y Miquel Faig z May 4, 2011 Abstract If entitlement to UI bene ts must be earned with employment, generous UI is an additional bene t to working, so, by itself, it promotes job creation. If individuals are risk neutral, then there is a UI contribution scheme that eliminates any e ect of UI on employment decisions. As with Ricardian Equivalence, this result should be useful to pinpoint the e ects of UI to violations of its premises. Our baseline simulation shows that if the neutral contribution scheme derived in this contribution 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 bene ts. JEL classi cation: E24 E32 J64 Keywords: Search, Matching, UI Eligibility, Moral Hazard, Unemployment, Business Cycles, Labor Markets We are grateful for all the feedbacks we received, which are too extensive to list here. In particular, we bene ted from Shouyong Shi, Michael Reiter, and Giovanni L. Violante and two anonymous referees. Min Zhang thanks the nancial support from the Leading Academic Discipline Program, 211 Project for Shanghai University of Finance and Economics (the 3rd phase), and Miquel Faig thanks the nancial support of SSHRC of Canada. We are the only ones responsible for any remaining errors. y Mailing Address: Shanghai University of Finance and Economics, Department of Economics, 777 Guoding Road, Shanghai, China, zhang.min@mail.shufe.edu.cn. z Mailing Address: University of Toronto, Department of Economics, 150 St. George Street, Toronto, Canada, M5S 3G7. miquel.faig@utoronto.ca. 1

2 1 Introduction Most models of employment ows in the labor market assume that workers automatically qualify for unemployment insurance (UI) bene ts 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 always entitled to UI bene ts 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 o ers. If UI bene ts 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 bene ts reduce the expected pro ts of lling a vacancy, and hurt rms incentives for job creation and therefore employment. In contrast, if UI bene ts are conditional on prior employment and a worker cannot collect UI if bargaining with an employer breaks down, UI bene ts are no longer an opportunity cost but an indirect bene t of employment. Therefore, UI bene ts promote the value of lling a vacancy and stimulate job creations. This is the entitlement e ect 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 e ect operates through the bargaining positions of rms and workers. The UI bene ts, making the employment match more attractive to workers, enable rms to appropriate a larger fraction of the match surplus, which translates into a stronger incentive to post vacancies. Even if generous UI bene ts encourage job creations due to the entitlement e ect, they may hurt employment due to other e ects. 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 o er. As a result, UI bene ts have two detrimental e ects on employment. First, they increase the bargaining power of workers since they can now threat refusing a job to collect UI, which reduces rms incentives to create jobs. Second, they may actually trigger actual moral hazard quits or rejections, which directly increases unemployment. In addition 2

3 to these e ects, a generous UI system is also an expensive one, and the fees needed to nance it are an opportunity cost of employment. Taking into account all these e ects, 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 e ect on the determination of output and employment. Like Ricardian Equivalence, this irrelevance result should be a useful benchmark to pinpoint the economic e ects 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 uctuations 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 e ects of the UI bene ts do not cancel each other for some or all workers. Therefore, the UI system a ects the incentives of rms 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 e ects of UI by making it impossible to collect bene ts after rejecting a job, then the long-term average unemployment rate would fall from 5.7 to 4.5 percent. This e ect is actually stronger than the e ect 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 o ers the following insights on the current debate about the appropriateness of the Mortensen-Pissarides model in explaining the cyclical uctuations in the labor market. Even though, as in Hagedorn and Manovskii (2008), our model needs a relatively large opportunity cost of employment to generate realistic cycles for unemployment and vacancies. The model is able to simultaneously generate realistic responses to productivity shocks and to changes in UI bene ts. So, the criticism of Hornstein, Krusell, and Violante (2005) and Costain and Reiter (2008) that such simultaneous t is impossible applies with less force to our model. Intuitively, the entitlement e ect counteracts some of the detrimental e ects of UI on employment, which reduces the e ect of an increase in bene ts. Consequently, 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 bene ts. 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 3

4 UI eligibility. In addition, it establishes conditions that make this system neutral. Section 3 calibrates the model to the 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 bene ts. 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 bene ts unemployed workers must have earned eligibility with a previous job, and (2) the quality of a match between a rm and a worker is heterogenous. 2.1 Environment In the economy, there is a continuum of measure one of workers, and a large measure of potential rms who can enter freely into the labor market. Both workers and rms are in nitely lived, risk neutral, maximize their expected utilities, and discount future utility ows at the common rate r: 1 Production requires the cooperation of one worker and one rm. For this cooperation to take place, workers and rms must rst 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 rms and workers are matched. Search frictions in this market are characterized by a constant returns to scale matching technology: M (v t ; u t ). In a period, denoted by the t subscripts, the function M maps vacancies posted v t and unemployment u t onto the measure of successful matches formed. The constant returns to scale of M implies that the probability of a worker nding a rm is just a function of the vacancy-unemployment ratio t = v t =u t : f ( t ) = M (v t ; u t ) =u t = M ( t ; 1) : Likewise, the probability of a rm nding a worker is a function of t that satis es: q ( t ) = M (v t; u t ) = M (v t; u t ) u t = f ( t) : (1) v t v t t The function M is assumed continuously di erentiable, increasing in both arguments, and 1 The reason for choosing risk neutrality is to ensure that the di erent 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. u t 4

5 concave. Furthermore, it satis es the terminal conditions: M (1; 0) = M (0; 1) = 0, and M 1 (0; 1) = M 2 (1; 0) = 1. Therefore, workers nd it easier to nd rms when vacancies are abundant relative to unemployment, while rms nd 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 bene ts (UI) while they are searching for jobs. For workers to be eligible for UI, they must rst be employed for a while, and bene ts do not last forever. Furthermore, UI bene ts are meant to be collected by workers who lose their jobs involuntarily, although, to be realistic, we allow some workers who quit to successfully pretend to have lost their jobs involuntarily. More precisely, eligible workers can always collect UI if they su er an exogenous separation from their jobs, but if they quit an employment relationship or reject a job o er they can collect UI with respective probabilities and ~. We allow these two probabilities to be identical, but we di erentiate 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 o er than if they quit an ongoing employment relationship, so we assume ~ : To balance the trade-o between realism and tractability, earning and losing UI eligibility is 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 bene ts expire. The probability in one period of running out of UI bene ts 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 bene ts predicted from the model are the same as their empirical counterparts in the United States. Unemployment insurance is provided by the government, which nances the UI system by charging employed workers a mandatory state dependent contribution fee, i x; where the superscript i denotes the UI eligibility state, and the subscript x denotes the aggregate state of the economy (to be speci ed later). Since the government can borrow and save at the interest rate r; the UI program can run de cits 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 rm-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-speci c productivity, and will be called quality of the match. The total productivity of the match is de ned as: p t () = p t + : The component p t is common to all matches in the economy in period 5

6 t; and is assumed to be a stochastic variable that follows a Markov chain and takes values in a nite support P R+: n The idiosyncratic component is randomly drawn from a distribution with nite support E R+ m : The density function of is h () ; and its cumulative distribution function is H (). After the realization of the match-speci c productivity, the newly matched workers decide whether to take their respective o ers or not. Workers who turn down their o ers are allowed to collect UI bene ts with the probability ~ as long as they are entitled to UI. In contrast, workers who accept their o ers start an employment relationship and stays constant during the spell of employment. Denote ^ i t the critical value of that determines if a worker of type i accepts the job ( ^ i t) or not ( < ^ i t) given the state of the economy in period t. The e ective job- nding rate for workers of type i is the product of the rate at which they match with a rm and the probability of accepting the ensuing job o er: f ( t ) 1 H ^ t+1 i : Similar to job rejections, if workers eligible for UI choose to quit their jobs, they can collect UI bene ts with probability : For a given match-quality ; the only di erence 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 rm lies in the probability of collecting UI in case of turning down the job. Since ~; 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 of employment. Net of the UI contribution fee, employed workers earn a wage rate wt i (). The wage rate wt i () depends on UI eligibility because UI bene ts 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 bene ts 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: ` < p t () for i 2 f0; 1g ; all t and 2 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 bene ts to be received in case of rejecting an o er or quitting a job. Ex-ante, all rms 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 rm paired up with a worker obtains output p t () and incurs a labor cost wt i () + i t: In a period when an active rm is searching for a worker, it posts a vacancy at a cost c. Even though the probability of nding a job by an unemployed worker is independent from his or her eligibility, the probability that a rm matches with a worker of a particular type depends on the composition of unemployment. That is, the probability that a vacant rm matches with a worker of type i 2 f0; 1g during a period is: q ( t ) u i t=u t ; where u i t is 6

7 the measure of unemployed workers of type i and u t is total unemployment. 2.2 Laws of Motion of the Distribution of Workers In detail, the timing of a model inside a period is the following. At the beginning of a 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. In a 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 ( t ) ; 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: e 0 t () = (1 s) (1 g) e 0 t 1 () 1 Q 0 t () + u 0 t 1f t 1 h () A 0 t () ; and (2) e 1 t () = (1 s) e 1 t 1 () + (1 s) ge 0 t 1 () 1 Q 1 t () + u 1 t 1f t 1 h () A 1 t () ; (3) where Q i t () for i 2 f0; 1g is an indicator function that a worker of type i and match quality quits in period t; and A i t () for i 2 f0; 1g is an indicator function that an unemployed worker of type i accepts a job with match quality in period t. For i 2 f0; 1g ; let the probability of rejecting a job be: Ht i = H ^ i P t = 1 2E Ai t () : In period t, unemployed workers become employed if they get matched (probability f t 1 ) and accept the job ensuing o er (probability 1 H i t). Unemployed workers eligible for UI lose eligibility either if they run out of bene ts (probability (1 f t 1 ) d) or if they reject a job o er and are caught "cheating" by the UI agency (probability f t 1 H 1 t (1 ~)). 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 and lose eligibility with probability (1 ) : Consequently, the laws of motion for the measures of unemployed workers by eligibility type are: u 0 t = u 0 t 1 1 ft 1 1 Ht 0 + u 1 t 1 (1 ft 1 ) d + f t 1 Ht 1 (1 ~) + e 0 t 1s + (1 s) X (1 g) e 0 t 1 () Q 0 t () (4) 2E + (1 s) (1 ) X 2E e 1 t 1 () + ge 0 t 1 () Q 1 t () ; and 7

8 u 1 t = u 1 t 1 1 ft 1 (1 Ht 1 ) (1 f t 1 ) d f t 1 Ht 1 (1 ~) + e 1 t 1s (5) + (1 s) X e 1 t 1 () + ge 0 t 1 () Q 1 t () : 2E 2.3 Bellman Equations The aggregate state of the economy, to be denoted by a vector x t ; includes the common component of labor productivity p t 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 a ects the incentives to post vacancies because it determines the type of workers who are going to ll 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 ; u 0 t ; u 1 t ; e 0 t () and e 1 t () for all 2 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 x 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 bene ts or not. Analogously, rms 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 rm or not in aggregate state x, respectively, be Wx i () and Ux. i Similarly, let the values of a rm matched with a worker with UI eligibility state i be Jx i (). Finally, let a prime denote next period, E calculate averages over, and E x calculate averages over x 0 conditional on x: Using this notation, the utility values of workers and rms 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 rm next period and continuing being unemployed. These events happen with probabilities f x = f ( x ) and 1 f x ; respectively. Hence, Ux 0 = ` + f x E x E Wx 0 () + (1 f 0 x ) E x Ux 0 ; (6) 0 8

9 where is the discount factor: 1=(1 + r): To capture in a tractable way that workers are not forced to collect UI bene ts, 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 bene ts, and the expected present values of being matched with a rm next period or not. In the latter case, the worker may lose UI eligibility with probability d : n Ux 1 = maxfux; 0 ` + b + f x E x he max ~W 1 x () ; ~U 1 0 x + (1 0 + (1 f x ) de x U 0 x 0 + (1 d) E xu 1 x 0 g: ~) U 0 x 0 oi (7) To clarify the expected value of being matched (expression after E ), note that an eligible unemployed worker meeting with a rm will have to decide whether to accept the o er or not. In case of rejecting the o er, the worker will continue receiving UI with probability ~: 2 The value of this option is ~U x 1 + (1 ~) Ux: 0 In case of accepting the o er, the utility value attained is a value we denote as W ~ x 1 () : In general, this value di ers from the value of an employed worker, Wx 1 () ; because of the di erent probabilities of collecting UI after a job quit and a job rejection, which give di erent outside values to workers before and after accepting a job o er. Consequently, if Nash bargaining takes place at all times, as we assume, at the moment of accepting a job o er there will be one time transfer between the rm and the worker. This transfer captures 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 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 Ux: 0 If this option is not taken, the worker is employed in 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 0 x () = max U 0 x; w 0 x () + se x U 0 x 0 + (1 s) ge x W 1 x 0 () + (1 g) E xw 0 x 0 () : A matched worker eligible for UI can choose to quit a job to collect UI with probability. The value of this option is U 1 x + (1 ) U 0 x: The value of continuing employment is the sum of the current wage and the expected present values of exogenously losing the 2 With our de nitions (chosen to minimize algebraic expressions), the probability ~ combines the probability that UI bene ts have not expired and the probability that the UI agency does not catch the worker rejecting an o er. (8) 9

10 job next period and continuing employment: W 1 x () = max U 1 x + (1 ) U 0 x; w 1 x () + se x U 1 x 0 + (1 s) E xw 1 x 0 () : (9) The value of a rm employing a worker is the sum of current pro ts plus the expected present value of the rm 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 rm can terminate the match. Consequently, J 0 x () = max 0; p x () 0 x w 0 x () + (1 s) ge x J 1 x 0 () + (1 g) E xj 0 x 0 () : (10) Jx 1 () = max 0; p x () 1 x wx 1 () + (1 s) E x Jx 1 0 () : (11) Unmatched rms 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 ( x ) (u i x=u x ) : Because of free entry, the value of an unmatched rm is driven to zero in equilibrium: c = q ( x ) E h u 0 x=u x Ex Jx 0 () + n oi 0 u1 x=u x Ex max ~J 1 x () ; 0 : (12) 0 As with the worker, the utility value of starting an employment relationship with a UI eligible worker, ~ J 1 x 0 () ; di ers from the ongoing value of maintaining such employment relationship, Jx 1 0 () ; because of possible one time transfers between rms and workers at the moment of signing an employment contract. 2.4 Nash Bargaining and the Decision to Start an Employment Relationship Workers and rms who are matched split the gains from continuing the match according to the generalized axioms of Nash. This Nash bargaining takes place as soon a worker and a rm rst 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 di er after a rejection and a quit, the conditions that determine these two actions di er. 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 has established or continued 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 rm are respectively Wx 0 () Ux 0 and Jx 0 () : Therefore, the total match surplus 10

11 is: S 0 x () = W 0 x () U 0 x + J 0 x () : (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 ~ (rejections) and (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 rm. Consequently, the worker s outside value from a match before and after an employment relationship begins are respectively ~U x 1 + (1 ~) Ux 0 and Ux 1 + (1 ) Ux; 0 and the match surpluses that correspond with these two values are: ~S 1 x () = ~ W 1 x () ~U 1 x (1 ~) U 0 x + ~ J 1 x () : (14) S 1 x () = W 1 x () U 1 x (1 ) U 0 x + J 1 x () : (15) For future reference, we denote as V i x for i 2 f0; 1g the expected match surplus of a newly formed match, which is: V 0 x = E S 0 x () ; and V 1 x = E ~ S 1 x () : (16) The generalized Nash solution to the bargaining problem maximizes the weighted product of the match surpluses of the two parties: [Jx i ()] 1 [Sx i () Jx i ()]) for i 2 f0; 1g h i 1 h i and ~J 1 x () ~S 1 x () J ~ 1 x () ), where denotes the worker s bargaining power. The solution to this problem leads to the familiar sharing rule: J i x () = (1 ) S i x () ; for i 2 f0; 1g ; and ~ J 1 x () = (1 ) ~ S 1 x () : (17) A job will be rejected if and only if S ~ x 1 () < 0: In case the job is rejected the rm gets value zero, and the worker gets value ~U x 1 + (1 ~) Ux: 0 If the job is accepted, the total value of the rm-worker pair is W ~ x 1 () + J ~ x 1 () = Wx 1 () + Jx 1 () : As explained above, the values W ~ x 1 () and Wx 1 (), and J ~ x 1 () and Jx 1 () may di er because of the possibility of side payments at the moment when an employment relationship starts. 2.5 Equilibrium A stochastic recursive equilibrium is a set of functions fu i x; e i x () ; x ; wx i () ; Ux; i Wx i () ; Jx i () ; W ~ 1 x () ; and J ~ x 1 ()g for i 2 f0; 1g ; 2 E; and x 2 X that satisfy the laws of motion (2) to (5), the Bellman equations (6) to (11), the free entry condition (12), the 11

12 match surplus de nitions (13) to (15), and the Nash bargaining rules (17). To establish the existence of an equilibrium and 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 rm-worker pair. The value of UI eligibility for an unemployed worker, de ned as ^U x U 1 x U 0 x; satis es the following equation: n ^U x = max b + [1 f x (1 ~) (1 f x ) d] E x ^Ux 0 + f x Ex Vx 1 E o xv 0 0 x ; 0 : (18) 0 Equation (18) is obtained by substituting the value functions U 0 x and U 1 x from (6) and (7) into the de nition of ^U x ; and simplifying with the help of the Nash bargaining rules (17), and the de nitions of S 0 x () ; ~ S 1 x () ; and V i x. Intuitively, eligibility for UI gives unemployed workers current UI bene ts plus the expected present value of being unemployed and eligible for UI next period plus the di erence in expected utility values attained by eligible and ineligible workers in case they get matched. The value of ^U x cannot be negative because unemployed workers are not forced to collect UI. The total value of UI eligibility for a rm-worker pair when the worker becomes eligible for UI is ^B x () = J 1 x () + W 1 x () J 0 x () W 0 x () : Substituting (8) to (11) into this de nition and simplifying, we obtain: n ^B x () = max ^U h io x Sx 0 () ; 0 x 1 x + se x ^Ux 0 + (1 s)(1 g) E x ^Bx 0 () : (19) Intuitively, if the match is dissolved, the rm-worker pair loses the surplus of the match S 0 x (), but eligibility for UI gives the worker the extra value ^U x with probability : If the match continues, gaining UI eligibility implies di erent contribution fees for the employing rm, 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 ^U x and ^B x () to simplify, the surplus of an employment match that involves an ineligible worker satis es the following equation: n h Sx 0 () = max p x () ` 0 x + (1 s) E x Sx 0 () f xe 0 x Vx 0 + (1 0 i o s) ge ^B x x 0 () ; 0 : The existence of the UI system a ects this match surplus in two ways: the rm has to pay the current contribution fee 0 x and the worker gains UI eligibility next period with probability g as long as employment continues by then. Combining the de nitions of ^U x and ^B x () with those of S i x () for i 2 f0; 1g and ~S 1 x (), and using ~ W 1 x () + ~ J 1 x () = W 1 x () + J 1 x (), we obtain: (20) S 1 x () = S 0 x () + ^B x () ^U x ; and ~ S 1 x () = S 0 x () + ^B x () ~ ^U x : (21) Eligibility for UI by an employed worker has two opposing e ects on the match surplus: 12

13 Eligibility for UI brings total gains ^B x () to the rm-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 ^U x : In contrast, the expected value of UI eligibility in case of rejecting a job o er is ~ ^U x : Since we assume ~ ; (21) implies Sx 1 () S ~ x 1 () for all x 2 X and 2 E: A stochastic recursive equilibrium is fully characterized by equations (18) to (21), together with the laws of motion of x; the de nition (16), and the following restatement of the condition for free entry: c x = f ( x ) (1 ) u 0 x=u x Ex V 0 x 0 + u1 x=u x Ex V 1 x 0 : (22) The following proposition establishes the existence of a solution to this system of equations and some basic properties of the equilibrium. Proposition 1 (Existence): There is a set of functions x ; ^U x, ^Bx () ; S i x () for i 2 f0; 1g and ~ S 1 x () that solves the system of equations (18) to (22), so an equilibrium exists. In this equilibrium, the unemployed will not voluntarily give up eligibility ( ^U x > 0 for all x 2 X ) if one of the following two conditions hold: (i) 0 x 1 x for all x 2 X and (ii) contribution fees are such that ^B x ~ ^U x for all x 2 X: Furthermore, in the absence of UI, S 0 x () = S 1 x () = ~ S 1 x () > 0 for all x 2 X and 2 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 above two important conditions. The rst one is that contribution fees are not raised because of eligibility ( 0 x 1 x). 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 nding 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, ^B x ~ ^U x for all x 2 X: If the match surplus were to decrease upon a worker gaining eligibility, then both the rm and the worker would su er losses because they split the surplus according to the Nash bargaining rule. If these losses were su ciently large, the worker would be willing to give up eligibility. As we will see next, this second condition with equality is satis ed if the UI system is fully funded and neutral. 13

14 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 creations or job destructions. For this inquiry, we will be focusing on UI systems that are neither subsidized from other revenue sources, nor they are used to raise revenue to nance other public services. For this purpose, we will center our analysis on fully funded UI systems de ned as follows: De nition: A fully funded UI system is one in which the expected present value of net bene ts 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 rms to post vacancies is the expected present value of the pro ts extracted from the employment matches once the vacancies are lled, and, because of Nash bargaining, they are proportional to the surplus of the matches formed. Consequently, a UI system will not a ect the posting of vacancies if it does not a ect the expected match surpluses of newly created jobs by ineligible and eligible workers, which respectively are S 0 x () and ~ S 1 x (). The sign of the match surplus S 0 x () also determines the creation and the dissolution of employment relationships involving ineligible workers, while the acceptance of job o ers by eligible workers depends on the sign of ~ S 1 x (). Therefore, if a UI system has no e ect on S 0 x () and ~ S 1 x () ; 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 o ers by eligible workers. Such UI system may still a ect 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 2 X and 2 E; as we have assumed, they will not quit either with a UI system which has no e ect on S 0 x () and ~ S 1 x () : The reason for this is that ~ implies S 1 x () ~ S 1 x () ; so S 1 x () cannot be negative if ~ S 1 x () is positive. 3 Inspection of (20) reveals that the condition 0 x = (1 s) ge x ^Bx 0 () ensures that the match surplus S 0 x () is not a ected by the UI system because it makes the rms employing ineligible workers pay the expected present value of gains from eligibility that would bring to the rm-worker pair. Also, equation (21) implies that UI eligibility does not change the match surplus ~ S 1 x () if ^Bx = ~ ^U x : This equality implies that the gain 3 Without the assumption p x () ` > 0 for all x 2 X and 2 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. 14

15 from eligibility by the rm-worker pair is equal to the worker s outside value brought by eligibility in case the worker 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 them 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 bene ts, the duration of these bene ts 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 i fees are such that 0 x = ~ (1 s) ge x ^Ux 0 and 1 x = ~ h (1 s) E x ^Ux 0 ^Ux + se x ^Ux 0, then ^U x is equal to the expected present value of the UI bene ts 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 e ects of a UI system as violations from its stated premises. In this vein, if risk neutrality is maintained, the e ects 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 bene ts last. It should be noted here that the neutral nancial 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 rm. Contribution fees proportional to wages would a ect 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 a ect 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 con guration 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 di erentiating 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. 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, 15

16 the contribution fees are functions of a few parameters and the endogenous nding 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: 0 = ~g (1 s) r + f x (1 ~) + (1 f x ) d b; and 1 = s ~ (s + r) r + f x (1 ~) + (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 moral-hazard rejection of an employment o er. In the extreme case that such "cheating" is impossible, ~ is zero, no fees should be charged to rm employing ineligible workers. In this case, the fee charged for hiring an eligible workers is equal to the expected present value of the bene ts that such eligibility entitles times the probability of an exogenous separation in one period. In contrast, if workers have no di culty in collecting UI after they reject a job o er, ~ is close to one, then ineligible workers must pay large fees that nance not only all the UI bene ts that eventually they will be entitled to receive, but also cover the cost of a subsidy to eligible workers since 1 becomes negative. Such a subsidy is necessary to avoid the strategic rejections that the high value of ~ entices. Intermediate values of ~ bring more balanced contribution structures. In particular, the two types of workers must pay the same fees if ~ = s= [r + s + g (1 s)] : 3 Numerical Analysis This section calibrates the model laid out in Section 2 using United States data and analyzes its quantitative predictions. One objective 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 bene ts. 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 o er of employment, are crucial for the e ects of a UI system on unemployment. The numerical analysis of this section adopts the following specializations. The matching function is assumed to be Cobb-Douglas: M (v; u) = v 1 The u for i 2 f0; 1g : 4 With this functional form, the nding rate with respect to the vacancy-unemployment ratio is: f () = 1 ; so it has constant elasticity 1 : Following Shimer (2005), the common 4 Strictly speaking a Cobb-Douglas matching function must be truncated to ensure that the nding and the lling rates are probabilities. In our simulations, this is never an issue because their values are always between zero and one without any truncation. 16

17 part of aggregate labor productivity is assumed to be a stochastic process that satis es 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 and the probability that a transition occurs : Zhang (2008) provides further details on modelling 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 ows in the labor market, and, in a stylized way, the key features of the UI system in the United States and the e ects 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 rst 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 bene ts (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 bene ts (24 weeks) in the United States. 5 Finally, the value of c is normalized to one. By doing so, we are just de ning 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 de cit for the UI system. The rst seven of these targets are empirical moments from the United States that describe the main features of the business cycle in the labor market, and were constructed from the data reported by Shimer (2005) or its original sources. 7 Note that, following Mortensen and 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 reported by annual report and nancial data from the U.S. Department of Labor Employment and Training Administration (column 27). It is available at 6 The normalization adopted by Shimer (2005) of setting average equal to one yields identical results except for the calibrated values of : 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 17

18 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. Table 1 Calibration Targets Annual real interest rate (r) 0:04 Average weeks of employment needed for UI eligibility (1=g) 20 Average weeks before UI bene ts expire (1=d) 24 Cost of posting a vacancy (c) 1 Average labor productivity 1 Standard deviation of labor productivity (quarterly in logs) 0:020 Autocorrelation of labor productivity (quarterly in logs) 0:878 Average unemployment rate 0:0567 Average short-term unemployment rate 0:0244 Standard deviation of unemployment rate conditional or not on p (quarterly in logs) 0:0775=0:190 Standard deviation of conditional or not on p (quarterly in logs) 0:151=0:382 E ective replacement rate of UI bene ts (b=w) 0:25 E ect of small changes in b on the separation rate 0 Increased average unemployment duration (in weeks) if (b=w) increases by Increase in the elasticity of duration of unemployment with respect to b due to eligibility 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 rst four targets pins down one parameter, speci ed in parenthesis. The rest collectively pin down the rest of parameters. We conduct two di erent simulations depending on if the targeted standard deviations of unemployment and are conditional on p (conditional moments) or not (unconditional moments). The remaining targets that we seek to replicate de ne key characteristics of the UI system in the United States and its e ects on labor market ows. As for the size of the UI system, we target the e ective 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 bene ts. As documented by Blank and Card (1991), the take-up rate over the period was fairly stable around 0.7, while the than 5 Weeks (Series ID: LNS ) and (ii) Civilian Labor Force Level (Series ID: LNS ), both available at Finally, the average labour productivity being one is a normalization. 18

19 replacement rate conditional on receiving bene ts averaged over the period Consistent with the e ects of UI summarized in the review survey by Atkinson and Micklewright (1991), we aim that in our model increases in UI bene ts a ect the unemployment rate moderately through the average duration of unemployment spells, but not through their incidence. To this end, we target the e ect 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 o er 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 e ects 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 a ects most directly. The value of p determines, together with ; the average aggregate labor productivity. The values of and 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 determines the average nding rate, and, for a given value of s; the average unemployment rate. The standard deviation of is primarily determined by how close the value of leisure is to productivity (see Mortensen and Nagypál, 2007). The values of and ; which are equal to satisfy the Hosios condition, are inversely related to the elasticity of the nding rate with respect to (the vacancy-unemployment ratio). So the standard deviation of the unemployment rate falls with for a given degree of variability of : The key parameters that determine the features of the UI system in the United States and its e ects are the following. The value of b determines the UI bene ts replacement rate. Also, given the other parameters of the UI system, the contribution fee determines 8 This ratio is reported in the annual report and nancial data of the U.S. Department of Labor Employment and Training Administration (column 33) 9 Mo tt 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, Mo t (1985) and Meyer (1990) o er estimates of around 0.5 weeks and 1.5 weeks, respectively. 10 The survey by Atkinson and Micklewright (1991) remarks the importance of the out ows from unemployment in explaining the e ect of bene ts 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. 19

20 the average budget de cit. The probability of collecting UI after quitting () determines the e ect of changes in b on the incidence of unemployment. This e ect is guaranteed to be zero if the probability of collecting UI after a quit () is su ciently smaller than the probability of collecting UI after a rejection of an o er of employment (~). Finally, the parameters and ~ are key determinants for the durations of unemployment spells and the e ects 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 o ers increases with : Therefore, large values of are associated with long unemployment spells of UI eligible workers and large e ects of b on the duration of these spells. High values of ~ are also associated with frequent rejections because with a high ~ UI eligible workers are unlikely to lose UI eligibility if they reject a job. However, this is not the only e ect of a high ~: A high ~ also reduces the incentives to post vacancies because it increases the bargaining power of UI eligible workers when they rst meet an employer. Consequently, since there is a single search market, high values of ~ a ect the expected duration of unemployment spells for all workers. This e ect is ampli ed as b increases. As a result, and ~ jointly determine how an increase in b a ects both the average duration of unemployment and the di erential e ects 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 o ers 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 bene ts nor the UI contribution fees are included in `: Curiously, in both of our simulations, the sum ` + + 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 20