Liquidity and Insurance for the Unemployed*


 Jessica Farmer
 3 years ago
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1 Fedeal Reseve Bank of Minneapolis Reseach Depatment Staff Repot 366 Decembe 2005 Liquidity and Insuance fo the Unemployed* Robet Shime Univesity of Chicago and National Bueau of Economic Reseach Iván Wening Massachusetts Institute of Technology, National Bueau of Economic Reseach, and Fedeal Reseve Bank of Minneapolis ABSTRACT We study the optimal design of unemployment insuance fo wokes sampling job oppotunities ove time. We focus on the optimal timing of benefits and the desiability of allowing wokes to feely access a iskless asset. When wokes have constant absolute isk avesion pefeences, it is optimal to use a vey simple policy: a constant benefit duing unemployment, a constant tax duing employment that does not depend on the duation of the spell, and fee access to savings using a iskless asset. Away fom this benchmak, fo constant elative isk avesion pefeences, the welfae gains of moe elaboate policies ae minuscule. Ou esults highlight two lagely distinct oles fo policy towad the unemployed: (a) ensuing wokes have sufficient liquidity to smooth thei consumption; and (b) poviding unemployment benefits that seve as insuance against the uncetain duation of unemployment spells. * s: and We ae gateful fo comments fom Daon Acemoglu, GeogeMaios Angeletos, Hugo Hopenhayn, Naayana Kochelakota, Samuel Kotum, and Ellen McGattan, and numeous semina paticipants on this pape and an ealie vesion entitled Optimal Unemployment Insuance with Sequential Seach. Shime s eseach is suppoted by gants fom the National Science Foundation and the Sloan Foundation. Wening is gateful fo the hospitality fom the Fedeal Reseve Bank of Minneapolis duing which this pape was completed. The views expessed heein ae those of the authos and not necessaily those of the Fedeal Reseve Bank of Minneapolis o the Fedeal Reseve System.
2 1 Intoduction Thee is wide vaiation in the duation of unemployment benefits acoss OECD counties (Figue 1). In Italy, the United Kingdom, and the United States, benefits last fo six months. In Gemany benefits lapse afte one yea and in Fance afte five yeas. In Belgium they last foeve. Which county has the ight policy? A standad agument fo teminating benefits afte a few quates is that extending the duation of benefits lengthens the duation of jobless spells (Katz and Meye, 1990). But benefits also povide insuance and help wokes maintain smooth consumption while unemployed (Gube, 1997). Detemining which policy is best equies a dynamic model of optimal unemployment insuance. Shavell and Weiss (1979) and Hopenhayn and Nicolini (1997) develop such models and show that benefits should optimally decline duing a jobless spell. Many economists have intepeted these esults as boadly suppotive of the Italian, Bitish, and Ameican vesion of unemployment insuance. But they hinge on some subtle assumptions, notably a estiction that the unemployed can neithe boow no save and so consume thei benefits in each peiod. This means that unemployment benefits play a dual ole: they insue wokes against uncetainty in the pospect of finding a job, and they povide wokes with the ability to smooth consumption while unemployed. In this pape, we eexamine the optimal timing of benefits, distinguishing the two oles by allowing the woke to boow and save. Ou main conclusion is that when wokes have sufficient liquidity, in eithe assets o capacity to boow, a constant benefit schedule of unlimited duation is optimal o nealy optimal. 1 The constant benefit schedule insues against unemployment isk, while wokes ability to dissave o boow allows them to avoid tempoay dops in consumption. Ou esults suggest two conceptually distinct oles fo policy towad the unemployed. Fist, ensuing that wokes have sufficient liquidity to smooth thei consumption; and second, poviding constant unemployment benefits that seve as insuance against the uncetain duation of unemployment spells. This dichotomy is consistent with the spiit of Feldstein and Altman s (1998) ecent policy poposal fo unemployment insuance savings accounts (see also Feldstein, 2005). We epesent the unemployed woke s situation using McCall s (1970) model of sequential 1 This does not necessaily imply that the Belgian policy is optimal. Policies diffe along othe dimensions, notably in the maximum yealy benefit; see OECD (2002), Table 2.2. This pape focuses on the optimal duation of benefits. In ongoing wok fo a sepaate pape, we examine the deteminants of the optimal level. 1
3 60 50 Months If you ae eading this invisible text, you have too much time on you hands 0 Czech Rep. Italy U.K. U.S. S. Koea Switzeland Slovak Rep. Austia Japan Canada Gemany Geece Hungay Luxemboug Ieland Sweden Poland Spain Finland Potugal Noway Denmak Fance Iceland Nethelands Belgium Figue 1: Duation of unemployment benefits in select OECD counties. Souce: OECD Benefits and Wages 2002, Table 2.2. Duation is unlimited in Belgium. job seach. Each peiod, a iskavese, infinitely lived unemployed woke gets a wage offe fom a known distibution. If she accepts the offe, she keeps the job at a constant wage foeve. If she ejects it, she seaches again the following peiod. Ou main pupose is to compae two unemployment insuance policies. We begin by consideing a simple insuance policy, constant benefits, whee the woke eceives a constant benefit while she is unemployed and pays a constant tax once she is employed. The woke can boow and lend using a iskless bond. We show that the woke adopts a constant esevation wage although he assets and consumption decline duing a jobless spell. The esevation wage is inceasing in both the unemployment benefit and the employment tax, a fom of moal hazad. An insuance agency sets the level of benefits and taxes to minimize the cost of poviding the woke with a given level of utility. We then conside optimal unemployment insuance. An insuance agency dictates a duationdependent consumption level fo the unemployed, funded by an employment tax that depends on the length of the jobless spell. The woke has no access to capital makets and so must consume he aftetax income in each peiod. Absent diect monitoing of wage offes o andomization schemes, this is the best insuance system possible. The path of unemployment consumption and employment taxes detemines the woke s esevation wage, which the insuance agency cannot diectly contol. It sets this path to minimize the 2
4 cost of poviding the woke with a given level of utility. Ou main esult is that with constant absolute isk avesion (CARA) pefeences and no lowe bound on consumption, constant benefits and optimal unemployment insuance ae equivalent. That is, the cost of poviding the woke with a given level of utility is the same, he esevation wage is the same, and the path of he consumption is the same unde both insuance systems. In both cases consumption falls by a constant amount each peiod that the woke is unemployed, both duing and afte the unemployment spell. When the woke can boow and save, this is consistent with a constant benefit and tax. Ou esult that the optimal unemployment insuance can do no bette than constant benefits with boowing and savings contasts with a lage liteatue on the need fo savings constaints in dynamic moal hazad models. 2 Rogeson (1985) consides an envionment in which a iskavese woke must make a hidden effot decision that affects he iskneutal employe s pofits. He poves that optimal insuance is chaacteized by an invese Eule equation, ( 1 u (c t ) = E t 1 β(1 + )u (c t+1 ) whee E t is the expectation opeato conditional on infomation available at date t. In paticula, since the function 1/x is convex, ), u (c t ) < E t β(1 + )u (c t+1 ). An individual facing this path of consumption would consume less today and moe tomoow and hence is savingsconstained by optimal insuance. In contast, in ou model a woke confonted with the optimal unemployment insuance policy satisfies this Eule condition with equality. We also exploe optimal insuance with constant elative isk avesion (CRRA) pefeences. We do this fo two easons. Fist, CRRA pefeences ae theoetically moe appealing than CARA pefeences. We want to exploe the obustness of ou findings to this assumption. And second, this intoduces a nonnegativity constaint on consumption that limits a woke s debt to the amount that she can epay even in the wost possible state of the wold, Aiyagai s (1994) natual boowing limit. We highlight an inteesting inteaction between 2 A ecent example is Golosov, Kochelakota, and Tsyvinski (2003), who emphasize that capital taxation may discouage saving. Allen (1985) and Cole and Kochelakota (2001) povide a paticulaly stiking example of the cost of unobseved savings in a dynamic economy with asymmetic infomation. They pove that if a woke pivately obseves he income and has access to a hidden saving technology, then no insuance is possible. 3
5 a woke s ability to boow and optimal insuance. The pefect equivalence between optimal unemployment insuance and constant benefits beaks down with CRRA pefeences, but we find that ou esults with CARA povide an impotant benchmak. As in the CARA case, optimal unemployment insuance dictates a declining path of consumption fo unemployed wokes and an inceasing tax upon eemployment. Howeve, the implicit subsidy to unemployment, the amount that a woke s expected lifetime tansfe fom the insuance agency ises if she stays unemployed fo an additional peiod, inceases vey slowly duing a jobless spell. By its vey definition, constant benefits ae always at least as costly as optimal insuance. But if the woke has enough liquidity so as to have a minimal chance of appoaching any lowe bound on assets, the additional cost of constant benefits is minuscule, less than 10 7 weeks (o about 0.01 seconds) of income in ou leading example. The diffeence between the optimal timevaying and timeinvaiant subsidy is also vey small. If the woke is nea he debt limit, the diffeence between constant benefits and optimal unemployment insuance is lage. This is because benefits ae foced to play the dual ole of poviding insuance and smoothing consumption. Howeve, using benefits to ceate liquidity in this indiect way is likely to be less efficient than measues designed to addess the liquidity poblem diectly. The geneal message that emeges fom ou model is that unemployment insuance policy should be simple a constant benefit and tax, combined with measues to ensue that wokes have the liquidity to maintain thei consumption level duing a jobless spell. Ou intuition fo these esults is the following. With CARA utility the fall in assets and consumption that occus duing an unemployment spell does not affect attitudes towad isk; as a consequence, the optimal unemployment subsidy is constant. With CRRA utility, the woke becomes moe isk avese as consumption falls; this explains why the optimal subsidy inceases ove time. Howeve, this wealth effect is small duing a typical, o even elatively polonged, unemployment spell povided the woke is able to smooth he consumption. Befoe poceeding, we note that ou use of a sequential seach model depats fom Hopenhayn and Nicolini (1997) and many othes, which assumes that thee is only a job seach effot decision. 3 Thee ae thee easons fo this modeling choice. Fist, ou model poduces stak esults on optimal policy in a staightfowad way, which we believe is in 3 Shavell and Weiss (1979) allow fo both hidden seach effot and hidden wage daws. See also execise 21.3 in Ljungqvist and Sagent (2004). Howeve, both of these models assume that employed wokes cannot be taxed, and neithe examines optimal benefits when wokes have access to liquidity. 4
6 tinsically useful. On the othe hand, the sequential seach model is not citical fo these esults. Indeed, the pape most closely elated to ous is Wening (2002), which intoduces hidden boowing and savings into the Hopenhayn and Nicolini (1997) seach effot model, and some of his esults ae analogous to ones we epot hee. Fo example, he poves that constant benefits and taxes ae optimal unde CARA pefeences if the cost of seach is monetay. Despite this, and in contast to ou esults hee, in Wening (2002) constant benefits ae not equivalent to optimal unemployment insuance, even with CARA utility, since it is always desiable to exclude the woke fom the asset maket. Second, the sequential seach model is empiically elevant. Stating with the wok of Feldstein and Poteba (1984), a numbe of authos have documented that an incease in unemployment benefits aises wokes esevation wage and consequently educes the ate at which they find jobs. The sequential seach model is a natual one fo thinking about this fact. Thid, the sequential seach model is the backbone of most eseach on equilibium unemployment. At the heat of the Lucas and Pescott (1974) equilibium seach model and of vesions of the Pissaides (1985) matching model with heteogeneous fims ae individual sequential seach poblems. Moe ecently, Ljungqvist and Sagent (1998) examine a lage economy in which each individual engages in sequential job seach fom an exogenous wage distibution. This pape poceeds as follows. Section 2 descibes the model s envionment and the two policies we conside. Section 3 then establishes the equivalence between the two systems unde CARA pefeences. Section 4 quantitatively evaluates optimal unemployment insuance and optimal constant benefits with CRRA pefeences, highlighting the elationship between unemployment insuance and liquidity. Section 5 concludes. 2 Two Policies fo the Unemployed We begin by descibing the common physical envionment of the model. We then discuss the two policies we conside, constant benefits and optimal unemployment insuance. 2.1 The Unemployed Woke Thee is a single iskavese woke who maximizes the expected pesent value of utility fom consumption, E 1 β t u(c t ), t=0 5
7 whee β < 1 epesents the discount facto and u(c) is the inceasing, concave peiod utility function. At the stat of each peiod, a woke can be employed at a wage w o unemployed. A woke employed at w poduces w units of the consumption good in each peiod; she neve leaves he job. An unemployed woke eceives a single independent wage daw fom the cumulative distibution function F. 4 Let w 0 denote the lowe bound of the wage distibution. The woke obseves the wage and decides whethe to accept o eject it. If she accepts w, she is employed and poduces w units of the consumption good in the cuent and all futue peiods. If she ejects w, she poduces nothing and is unemployed at the stat of the next peiod. In eithe case, the woke decides how much to consume at the end of the peiod, afte obseving the wage daw. The woke cannot ecall past wage offes. We assume that an unemployment agency only obseves whethe the woke is employed o unemployed. In paticula, it does not obseve the woke s wage, even afte she decides to take a job. 5 The objective of the unemployment insuance agency is to minimize the cost of poviding the woke with a given level of utility. We assume costs ae discounted at ate = β Policy I: Constant Benefits The policy we call constant benefits is defined by a constant unemployment benefit b, a constant employment tax τ, and pefect access to a iskless asset with net etun = β 1 1, subject to a noponzigame condition. 6 Since the woke s poblem is stationay, we pesent it ecusively. Stat by consideing a woke who is employed at wage w and has assets a with budget constaint a = (1+)a+ w τ c. Since β(1 + ) = 1, she consumes he afte taxincome plus the inteest on he assets c e (a,w) = a + w τ, so that assets ae kept constant, a = a. This means that he 4 We assume that F is continuous and has finite expectation and that thee is some chance of dawing a positive wage, so F(w) < 1 fo some w > 0. 5 If the wage wee obsevable, an unemployment insuance agency could tax employed wokes 100 pecent and edistibute the poceeds as a lumpsum tansfe. Wokes would be indiffeent about taking a job and hence would follow any instuctions on which wages to accept o eject. This makes it feasible to obtain the fist best, complete insuance with the maximum possible income. Pivate infomation is a simple way to pevent the fist best, but othe modeling assumptions could also make the fist best unattainable, e.g., moal hazad among employed wokes. 6 That is, debt must gow slowe than the inteest ate, lim t (1 + ) t a t 0, with pobability one, whee a t denotes asset holdings. Togethe with the sequence of intetempoal budget constaints, this is equivalent to imposing a pesentvalue lifetime budget constaint, with pobability one. 6
8 lifetime utility is V e (a,w; τ) = u(a + w τ). (1) 1 β Next conside an unemployed woke with assets a and let V u (a; b, τ) denote he expected lifetime utility, given policy paametes b and τ. This must satisfy the Bellman equation V u (a; b, τ) = w { ( max max u(c) + βvu (a ; b, τ) ), c } u(a + w τ) df(w), (2) 1 β whee a = (1 + )a + b c. An unemployed woke chooses whethe to accept a job o not. If she does take the job, he utility is given by V e (a,w; τ) in equation (1). Othewise, she collects unemployment benefits b, consumes c and saves a in the cuent peiod, and emains unemployed into the next peiod, obtaining expected continuation utility V u (a ; b, τ). The solution to the Bellman equation defines the woke s unemployment consumption c u (a), esevation wage w(a), and next peiod s assets a (a), conditional on this peiod s assets. Given these objects, the cost of the unemployment insuance system is defined ecusively by ( ) S(a S(a; b, τ) = b (a); b, τ) + F( w(a)) (1 + ) τ( 1 F( w(a)) ). (3) 1 + A woke with assets a fails to find a job with pobability F( w(a)). In this event, the cost of the unemployment insuance system is the unemployment benefit b plus the discounted continuation cost S(a ). If she finds a job, the pesent value of he tax payments is (1+) τ. An unemployment insuance agency chooses b and τ to maximize the woke s utility given some available esouces and an initial asset level. Equivalently, we conside the dual poblem of minimizing the total esouce cost of deliveing a cetain utility fo the woke. The optimal constant benefit policy solves C c (v 0,a) min b, τ S(a; b, τ) + (1 + )a subject to V u (a; b, τ) = v 0. Since thee ae no ad hoc constaints on boowing, a standad Ricadian equivalence agument implies that V (a; b, τ) = V (a+x; b x, τ +x). The same is tue with total cost, so it follows that C c (v 0,a) is independent of a. Abusing notation we wite C c (v 0 ). 7
9 2.3 Policy II: Optimal Unemployment Insuance Unde optimal unemployment insuance, a woke who is unemployed in peiod t consumes b t, while a woke who finds a job in peiod t pays a tax τ t, depending on when she finds a job, fo the emainde of he life. One can conceive of moe complicated insuance policies whee the agency asks the woke to epot he wage daws, advises he on whethe to take the job, and makes payments conditional on the woke s entie histoy of epots. That is, one can model unemployment insuance as a evelation mechanism in a pincipalagent poblem. We pove in Appendix A that the policy we conside hee does as well as any deteministic mechanism as long as absolute isk avesion is noninceasing. Given {b t } and {τ t }, conside a woke who chooses a sequence of esevation wages { w t }. He lifetime utility is U ( { w t,b t,τ t } ) = ( t 1 ( β t F( w s )) u(b t )F( w t ) + t=0 s=0 ) u(w τ t ) w t 1 β df(w) The woke is unemployed at the stat of peiod t with pobability t 1 s=0 F( w s). If she daws a wage below w t, she ejects it and he peiod utility is u(b t ). If she daws a wage above w t, she takes the job and gets utility u(w τ t ) each peiod, foeve. Now conside an unemployment insuance agency that sets the sequence of unemployment consumption and employment taxes {b t,τ t } to minimize the cost of poviding the woke with utility v 0 : C (v 0 ) min { w t,b t,τ t} ( t 1 ) ( (1 + ) t F( w s ) b t F( w t ) 1 + t=0 s=0 (4) ) τ t (1 F( w t )), (5) subject to two constaints. Fist, the woke s utility must equal v 0 if she uses the ecommended esevation wage sequence, v 0 = U ( { w t,b t,τ t } ). And second, she must do at least as well using the ecommended esevation wage sequence as any othe sequence { ˆ w t }, U ( { w t,b t,τ t } ) U ( { ˆ w t,b t,τ t } ). That is, the agency ecognizes that the woke will choose he esevation wage sequence { w t } to maximize he utility given {b t,τ t }. The solution to this poblem descibes optimal unemployment benefits. It is useful to expess this poblem ecusively. The cost function defined above must 8
10 solve the Bellman equation (( ) C (v) = min b + C (v ) F( w) 1 + ) τ(1 F( w)) w,b,v,τ 1 + (6) subject to v = (u(b) + βv )F( w) + u(w τ) df(w) w 1 β (7) u(b) + βv u( w τ) = 1 β. (8) Moeove, the optimal sequence { w t,b t,τ t } must be geneated by the Bellman equation s policy functions. An unemployed woke stats the peiod with some pomised utility v. The agency chooses consumption fo the unemployed b, the tax τ it will collect on wokes who become employed in the cuent peiod, the woke s continuation utility if she emains unemployed v, and the esevation wage w in ode to minimize its cost. If the woke gets an offe below the esevation wage, then the cost is the unemployment consumption b plus the discounted cost of deliveing continuation utility v in the next peiod. If instead the woke finds a job above the esevation wage, then the agency s costs ae educed by the pesent value of taxes. Equation (7) imposes that the policy must delive utility v to the woke. Finally, equation (8) is the incentive constaint, which incopoates the fact that the woke sets he esevation wage at the point of indiffeence between accepting and ejecting the wage. 3 Equivalence fo a Benchmak: CARA Utility Thee ae two disadvantages to constant benefits elative to optimal unemployment insuance. Fist, thee is a estiction on the time path of unemployment benefits and taxes, so b t and τ t ae constant. Second, the planne does not diectly contol the woke s consumption and so is constained by he savings choices. This can be thought of as an additional dimension of moal hazad. In geneal, constant benefits ae moe costly than optimal unemployment insuance: C c (v) C (v); howeve, in this section we pove analytically that constant benefits achieve the same outcome as optimal unemployment insuance fo the case with CARA pefeences, u(c) = exp( ρc) with c R. A key featue is that thee is no limit on the amount of debt that wokes can accue and all wokes have the same attitude 9
11 towads lotteies ove futue wages, which makes the model paticulaly tactable. We late show that these esults povide a good benchmak fo othe pefeence specifications. Fo the esults in this section, it is convenient to define CE( w) u 1 ( w u ( max{ w,w} ) ) df(w), (9) the cetainty equivalent fo a woke of a lottey offeing the maximum of w and w F. The CARA utility function has a convenient popety that we exploit thoughout this section, u(c 1 + c 2 ) = u(c 1 )u(c 2 ) fo any c 1 and c Constant Benefits We chaacteize constant benefits in two steps. Fist, we chaacteize individual behavio given unemployment benefits b, employment taxes τ, and assets a. Then we discuss how to choose these paametes optimally. It is convenient to define the net benefit o unemployment subsidy by B b + τ. The fist step follows fom solving the Bellman equation (2). Poposition 1 Assume CARA pefeences. The esevation wage, consumption and utility of the unemployed satisfy Poof. In Appendix B. (1 + ) w = CE( w) + B. (10) c u (a) = a + w τ (11) V u (a) = u(a τ + CE( w)) 1 β Equation (10) genealizes a standad equation fo a iskneutal woke s esevation wage, e.g., equation (6.3.3) in Ljungqvist and Sagent (2004), to an envionment with isk avesion and savings. It can be eexpessed as the condition that a woke is indiffeent between accepting he net wage w τ today and ejecting it, getting he unemployment benefit today, and then eaning the cetainty equivalent CE( w) τ theeafte: w τ 1 β = b CE( w) τ + β. 1 β Equation (10) indicates that the esevation wage w is inceasing in the net unemployment 10 (12)
12 subsidy B. This is the essence of the moal hazad poblem in ou model the moe one ties to potect the woke against unemployment by aising unemployment benefits and funding the benefits by an employment tax, the moe selective she becomes. The equation also shows that a woke s assets a do not affect he esevation wage, so it is constant duing a spell of unemployment. Consumption in equation (11) has a pemanent income fom with a constant pecautionay savings component. Assets fall by w B > 0 as long as thee is some chance of getting a wage in excess of the unemployment subsidy, F( B) < 1. 7 Consumption falls by CE( w) w each peiod that the woke emains unemployed. Unemployed wokes face uncetainty: a wage daw above CE( w) is good news leading to an incease in consumption, while a wage daw below CE( w) is bad news leading to a decline in consumption. The next step is to minimize the cost of poviding the woke with utility v 0. Using the esult that the esevation wage is constant, equation (3) becomes S(a; b, τ) = 1 + ( bf( w) (1 + ) τ(1 F( w)) 1 + F( w) ), (13) which is independent of a. Optimal constant benefit policy minimizes S(a; b, τ) + (1 + )a subject to (10) and (12). Poposition 2 Assume CARA pefeences. Then the optimal constant benefits policy is independent of v 0. The esevation wage satisfies w ag max w Φ( w), whee Φ( w) CE( w) wf( w), (14) 1 + F( w) and b and τ ae then detemined by equations (10) and (12). The minimum cost is independent of a. C c (v 0 ) = 1 + ( u 1 ((1 β)v 0 ) (1 + )Φ( w ) ), (15) Poof. Use equations (10) and (12) to solve fo b and τ as functions of w and v = V u (a; b, τ). Substituting into the cost (13) delives the desied esult. 7 Substitute equation (11) into the unemployed woke s budget constaint to get a = a + B w. If w B, condition (10) implies w CE( w). But the definition of the cetainty equivalent (9) implies this is possible only if F( w) = 1, a contadiction. 11
13 Ou next esult chaacteizes the woke s allocation given optimal policy. We take cuent unemployment utility v as a state vaiable, expess the allocation as a function of v, and descibe the evolution of v. Poposition 3 Assume CARA pefeences. Let v denote the utility pomised to the unemployed at the beginning of a peiod. Then if the agent emains unemployed, she consumes c u (v) = w CE( w ) + u 1( (1 β)v ) (16) and he utility evolves to v (v) = u( w CE( w ))v. (17) If she accepts a job at wage w, she foeve afte consumes c e (v,w) = w CE( w ) + u 1( (1 β)v ). (18) Poof. This follows diectly by changing vaiables fom a to v = V u (a; b, τ) using equations (11) (12), c e (a,w) = a + w τ, and the budget constaint a = (1 + )a + b c. This poposition will be useful when compaing constant benefits with optimal unemployment insuance, which we tun to now. 3.2 Optimal Unemployment Insuance We chaacteize optimal unemployment insuance using the Bellman equation (6) (8). To do so, it is convenient to fist deduce the shape of the cost function diectly fom the sequence poblem. Lemma 1 Assume CARA pefeences. The cost function satisfies C (v 0 ) = 1 + ( ) u(0) u 1 ((1 β)v 0 ) + C. (19) 1 β Moeove, let { w t,b t,τ t } denote the optimum fo initial pomised utility u(0)/(1 β). Then { w t,b t + x,τ t x} with x u 1 ((1 β)v 0 ) is optimal fo any othe initial pomise v 0. Poof. Let ˆb t b t + x and ˆτ t τ t x fo all t. Use equation (4) and CARA pefeences to show that adding a constant x to unemployment consumption in each peiod and subtacting 12
14 the same constant x fom the employment tax simply multiplies lifetime utility by the positive constant u(x), that is U ( { w t,b t,τ t } ) = u(x)u ( { w,ˆb, ˆτ} ). The esult follows immediately. The optimal path fo consumption shifts in paallel with pomised utility, while the path fo the esevation wage is unchanged. The cost function eflects these two featues. Indeed, since pomised utility is a state vaiable fo the poblem, the lemma implies that the optimal esevation wage path will be constant. These esults ae implications of the absence of wealth effects with CARA pefeences. To solve the agency s poblem futhe, we substitute the cost function fom (19) into (6) and use the incentive constaint (8) to eliminate the employment tax τ. The Bellman equation at v = u(0)/(1 β) then becomes ( ) (( u(0) C = min b + u 1 ((1 β)v ) 1 β w,b,v + 1 ( )) u(0) 1 + C F( w) 1 β 1 + ( w u 1( (1 β)(u(b) + βv ) )) (1 F( w)) ) (20) subject to u(0) 1 β = (u(b) + βv )u(ce( w) w). (21) The solution to this cost minimization poblem must solve the subpoblem of minimizing the cost b+u 1 ((1 β)v )/ of poviding a given level of utility u(b)+βv to those emaining unemployed. The fist ode condition fo this poblem yields o equivalently u(b) + βv (1 β)v = u(b) (22) = u(b)/(1 β). The pomise keeping constaint (21) is then equivalent to b = w CE( w). Substitute these conditions into the Bellman equation to eliminate b and v, and solve fo C (u(0)/(1 β)) to obtain ( ) u(0) C = 1 β (1 + )2 min w F( w) w CE( w) 1 + F( w) (1 + )2 = max Φ( w), (23) w whee Φ( w) is defined by equation (14). The optimal esevation wage w is independent of pomised utility and hence constant ove time. Substituting equation (23) into equation (19) poves that the cost to the agency of poviding a woke with utility v is identical to the cost with constant benefits C c (v) in equation (15). 13
15 Once we have found the optimal esevation wage w, the associated unemployment consumption, employment tax and continuation utility fall out using equations (7), (8), and (22) along with Lemma 1. The next poposition summaizes the main esult of this section. Poposition 4 Assume CARA pefeences. Unde optimal unemployment insuance, the esevation wage is constant ove time and maximizes Φ( w), given by (14). If an agent has expected utility v and emains unemployed, she consumes c u (v) (equation 16) and has continuation utility v (v) (equation 17). If she accepts a job at wage w, she consumes c e (v,w) (equation 18) foeve. This is the same allocation as unde an optimal constant benefit and the cost is the same, C c (v 0 ) = C (v 0 ). Thus, when the woke can boow and lend at the same ate as the agency, a vey simple policy attains the same allocation as optimal unemployment insuance. Of couse, Ricadian equivalence implies that the timing of tansfes is not pinned down, only the net subsidy to unemployment. If a woke takes a job in peiod t, she must pay taxes equal to (1+)τt in pesent value tems. If she emains unemployed fo one moe peiod, she eceives a benefit b t and then pays taxes τ t+1 in pesent value tems. The sum of these is the unemployment subsidy, a measue of insuance: B t b t + (1 + )τ t τ t+1 (24) Using b t = c u (v t ) and τ t = w c e (v t,w) with equations (16) (18), we find that B t = ((1 + ) w CE( w))/. The unemployment subsidy is constant and the same as B in the poblem with constant benefits, given by equation (10). At the othe end of the spectum fom Ricadian equivalence, imagine a woke who can neithe boow no save and so lives handtomouth consuming cuent income. In this exteme case, benefits and taxes ae uniquely pinned down, as in Shavell and Weiss (1979) and Hopenhayn and Nicolini (1997). One intepetation of this exteme case is that it calls fo deceasing benefits and inceasing taxes. Howeve, it is equivalent to think of the insuance agency simultaneously lending to the woke and poviding he with a constant unemployment subsidy. Conceptually, even in this case, it emains useful to distinguish between these two components of policy. 14
16 4 Liquidity and Wealth Effects: CRRA Utility The shap closed fom esults obtained so fa wee deived unde an assumption of CARA pefeences and in paticula allowed consumption to be negative. We now tun to wokes with constant elative isk avesion (CRRA) pefeences with nonnegative constaint on consumption. Let σ > 0 denote the coefficient of elative isk avesion. Then the peiod utility function is u(c) = c1 σ 1 σ one. fo σ 1, with u(c) = log(c) coesponding to isk avesion of We again conside ou two altenative policies: optimal unemployment insuance and constant benefits. The equivalence between these two policies beaks down with CRRA pefeences. Nevetheless, we find little welfae gain in moving fom constant benefits to optimal unemployment insuance. Moeove, we find that the optimal policy and allocations obtained analytically with CARA povide an excellent appoximation fo ou CRRA specifications. Fo both easons, we conclude that the CARA case is indeed a vey useful benchmak. 4.1 Optimal Unemployment Insuance Optimal unemployment insuance solves the Bellman equation (6) (8). The fom of the utility function is diffeent with CRRA pefeences than with CARA pefeences, and so the analytical expession fo the cost function in equation (19) no longe holds. We theefoe use numeical simulations to examine the economy. To poceed we need to make choices fo the discount facto β = (1 + ) 1, the coefficient of elative isk avesion σ, and the wage distibution F(w). As in Hopenhayn and Nicolini (1997), we view a peiod as epesenting a week and set β = 0.999, equivalent to an annual discount facto of We fix the coefficient of elative isk avesion at σ = 1.5 but late conside the obustness of ou esults to a highe value, σ = 6. 8 We adopt a Fechet wage distibution, F(w) = exp( zw θ ) with suppot (0, ), and paametes z,θ > 0. 9 With CRRA the paamete z acts as an uninteesting scaling facto, 8 Hopenhayn and Nicolini (1997) use the much lowe value of σ = 1/2 in thei baseline calibation. They ague that ove shot hoizons, a high intetempoal elasticity of substitution may be appopiate. In ou view, this emak esonates intospectively, but is at the same time misleading since it confounds attitudes egading consumption and net income paths. In thei model, consumption and net income wee equivalent; but ou model allows saving and boowing, and as a esult a woke displays an infinite elasticity of substitution with espect to the timing of tansfes. 9 A Fechet distibution has some desiable popeties in this envionment. Fist, it displays positive skewness. Second, suppose a woke eceives n wage daws within a peiod fom a Fechet distibution with paametes (ẑ,θ), and must decide whethe to accept the maximum of these daws. The distibution of the maximum wage daw is also Fechet with the same θ and z = nẑ. Thus thee is no loss of geneality in ou 15
17 40 30 Density F (w) Wage w Figue 2: Wage density: F (w) = θw θ 1 e w θ, θ = so we nomalize by setting z = 1. The mean log wage daw is then γ, whee γ is θ π Eule s constant, and the standad deviation of log wages is 6θ θ Following Hopenhayn and Nicolini (1997), we set θ so that the mean duation of an unemployment spell is about ten weeks, consistent with evidence in Meye (1990) on a weekly job finding pobability of 10 pecent fo the United States. This equies setting θ = , giving a standad deviation of log wages of about 1.2 pecent. 10 Figue 2 plots the density function F (w). We also conside the obustness of ou esults to changes in the wage distibution, in paticula to a substantial decease in θ, which inceases the dispesion in wages, aising the option value of job seach and the expected duation of unemployment. It will be useful to have a way of compaing the cost o policy functions obtained fom ou CRRA specification with those obtained fom the CARA case. To this end, note that a woke with a constant coefficient of elative isk avesion σ who consumes c has local coefficient of absolute isk avesion equal to σ/c. This suggests compaing the cost o policy functions obtained fo a CRRA woke with the appoximation povided by those of a fictitious CARA woke with coefficient of absolute isk avesion ρ = σ/u 1 ((1 β)v), whee assumption that the woke gets one wage daw pe peiod. 10 Specifically, we chose θ so that a iskneutal woke without unemployment insuance would have exactly F( w) = 0.90, making use of equation (10). 16
18 C (v) B t 1 F( w) Diffeence Utility (consumption equivalent) u 1 ((1 β)v) Figue 3: Diffeence between actual values and CARA appoximations. we take the consumption equivalent utility as a poxy fo consumption. The appoximations can be computed analytically using ou esults fom Section 3. We begin by discussing ou esults fo the minimum cost of poviding a woke with a given level of utility. Recall that with CARA utility, this cost is linea when utility is measued in consumption equivalent units with a slope of 1+ ; see equation (19). Fo ou CRRA specification, we find that the cost is nealy linea with almost the same slope. Fo this eason, we do not gaph the cost function. Instead, we compae the cost obtained fom the CRRA specification with an appoximation povided by the CARA execise. The solid black line in Figue 3 shows that the diffeence between C (v) and this CARA appoximation is small, less than in absolute value when utility exceeds a cetainty equivalent of 0.3. Tuning to the optimal allocation and policy, the left panel in Figue 4 shows that, as a function of the woke s pomised utility, unemployment consumption b is inceasing (solid bown line) while employment taxes τ ae deceasing (dashed oange line). The ight panel shows how b and τ evolve ove an unemployment spell stating with initial pomised value v 0 = u(1.1)/(1 β). Although initially unemployment consumption is high and the employment tax is slightly negative, afte a sufficiently long unemployment spell the employment tax ises to a high level and unemployment consumption falls to nealy zeo. Putting these togethe, a woke s expected utility v t declines ove time. This line is not gaphed because 17
19 1.25 Optimal Unemployment Insuance b t τ t B t Utility u 1 ((1 β)v) Time (weeks) Figue 4: Optimal unemployment consumption b t, employment taxes τ t, and unemployment subsidy B t = b t + (1+)τt τ t+1. it is only slightly highe than, and would be scacely distinguishable fom, unemployment consumption b t. We also look at the subsidy to unemployment, the additional esouces that a woke gets by emaining unemployed fo one moe peiod, as peviously defined in equation (24). The dashdot blue line in Figue 4 shows that this unemployment subsidy is small when utility is high at the stat of an unemployment spell and then inceases gadually as pomised utility falls and the spell continues. The dashdot blue line in Figue 3 illustates the high accuacy fo B t of the CARA appoximation with ρ = σ/u 1 ((1 β)v). The unemployment subsidy B t paints a vey diffeent pictue of optimal unemployment insuance than do unemployment consumption b t o employment taxes τ t in isolation. The pictue fo b t and τ t in Hopenhayn and Nicolini (1997) is qualitatively simila. Wening (2002) computes the net subsidy to unemployment fom thei allocation and finds that it is nealy constant, stating quite low and ising vey slowly. This distinction between unemployment consumption and subsidies is cucial in undestanding the diffeence between the esults of this pape on the one hand, and Shavell and Weiss (1979) and Hopenhayn and Nicolini (1997) on the othe. Finally, Figue 5 shows the pobability that an unemployed woke accepts a job, 1 18
20 Job Finding Pobability 1 F( wt) Utility u 1 ((1 β)v) Time (weeks) Figue 5: Optimal Job Finding Pobability 1 F( w t ). F( w t ), unde optimal unemployment insuance. This stats just above 10 pecent pe week when pomised utility is high, then initially ises befoe falling when pomised utility is vey low. This nonmonotonicity illustates two opposing foces at play as the woke gets pooe: the incease in absolute isk avesion and the incease in unemployment subsidies. The fist effect encouages the woke to accept moe jobs, while the second effect, which is patly an endogenous esponse to the fist, encouages he to become moe selective. The dashed ed line in Figue 3 shows the high accuacy of the CARA appoximation. It follows that the nonmonotonicity of the job finding ate with espect to pomised utility v found in ou CRRA specification eflects a nonmonotonicity with espect to isk avesion ρ in the CARA case. The next subsection exploes this notion futhe by studying the constant benefits policy with CRRA utility. Befoe closing, it is impotant to emphasize that in this CRRA specification, the pobability that a woke emains unemployed fo 100 weeks o moe is emote, on the ode of Ove the elevant time peiod, the unemployment subsidy and the job finding pobability ae vitually constant. In this sense, ou esults with CARA povide an excellent benchmak fo the CRRA specification. 19
21 4.2 Constant Benefits The Bellman equation (2) descibes the poblem of an unemployed woke with assets a facing a constant benefit b and a constant employment tax τ. In addition, since consumption is nonnegative, assets a cannot fall below some, possibly negative, level a, Aiyagai s (1994) natual boowing limit. A woke can boow as long as she can pay the inteest on he debt following any sequence of wage daws. The natual boowing limit is a = max{ b,w τ} = max{ B,w} τ. Thus, the details of the natual boowing limit depend on whethe the smallest possible wage, w, is bigge o smalle than the net unemployment benefit, B b + τ. In the fist case, w > B, and the natual boowing limit is w τ, since a woke with assets above this level could always have positive consumption and pay the inteest on he debt by taking the next job offe. This implies that a change in the left tail of the wage distibution can substantially affect a woke s debt limit, and potentially he behavio, even if she is extemely unlikely eve to accept a wage fom this pat of the distibution. 11 If w B, the natual boowing limit is detemined by a woke s ability to use he unemployment income to pay the inteest on he debt, so a = b. An incease in the net unemployment benefit, obtained by an incease in b and a budget balance change in τ, then has two distinct effects. It tansfes income to states in which the woke does not find a job (insuance) and it allows the woke to go futhe into debt while she is unemployed (liquidity). The liquidity effect is absent fom the model with CARA pefeences because thee is no boowing limit. Appendix C discusses an efficient method of solving the woke s Bellman equation (2) fo V u (a; b; τ) and equation (3) S(a; b, τ). It is then simple to choose b and τ to minimize the total esouce cost (1 + )a + S(a; b, τ) of poviding a woke with utility v = V u (a; b, τ). We paameteize the economy as befoe: β = 0.999, σ = 1.5, F(w) = exp( w θ ), and θ = Thus, we begin with a specification whee w = 0 B so that we ae in the case whee benefits affect the boowing constaint. In the next subsection we tun to the othe case. To stat, we examine the cost of poviding the woke with a given level of utility. Theoetically, this is highe than the cost unde optimal unemployment insuance. Rathe than 11 With CARA pefeences, Poposition 1 shows that the distibution of wages below the esevation wage is ielevant fo equilibium behavio. This is because thee is no boowing limit with CARA pefeences. 20
22 Diffeence in Cost, Logaithmic Scale w = 0 w = 0.95 Ad Hoc Utility (consumption equivalent) u 1 ((1 β)v) Figue 6: Additional cost of constant benefits. showing the cost diectly, the solid puple line in Figue 6 plots the additional cost of constant benefits on a logaithmic scale. At small values of utility, the cost of constant benefits is easonably lage, equal to a few weeks of consumption. But at high levels of utility, nea those coesponding to a net esouce cost of zeo, aound v = u(1.03), the additional cost of 1 β using a constant benefits system is vey small, about 0.01 weeks of consumption. On the othe hand, Figue 7 shows that the optimal constant unemployment subsidy B fo a woke who stats an unemployment spell with a given level of utility (solid puple line) is typically much highe than the optimal timevaying unemployment subsidy B t fo a woke with the same level of utility (dashdot blue line). Wokes with lowe utility demand highe benefits fo two easons. Fist, they have highe absolute isk avesion, and value insuance moe. Second, elaxing the boowing constaint is moe impotant to them because they ae close to it. 4.3 Liquidity The goal of this section is to isolate the insuance ole of unemployment benefits. To do this, fist obseve that unde the Fechet wage distibution F(w) = exp( w ), the pobability that a single wage daw is less than 0.95 is minuscule, appoximately
23 Net Unemployment Subsidy, Bt and B Optimal w = 0 w = 0.95 Ad Hoc Utility (consumption equivalent) u 1 ((1 β)v) Figue 7: Net Unemployment Subsidy B t = b t + (1+)τt τ t+1 and B = b + τ. Conside an economy vey simila to this one but with the wage distibution F(w) = F(w) if w w 0.95 and F(w) = 0 othewise, i.e., with a (small) mass point at It tuns out that the optimal esevation wage always exceeds 0.95 and this change has no effect on optimal unemployment insuance policy. But if in the oiginal economy we had B <.95, then the natual boowing limit was b/ while in the modified economy it is (w τ)/. Thus, this slight change in the wage distibution may lead to a significant incease in the boowing limit, i.e., in the availability of liquidity. The geen dashed lines in Figue 6 and Figue 7 show how this mattes. When w > B, unemployment subsidies no longe play a ole in inceasing a woke s liquidity. The unemployment subsidy tuns into a pue insuance mechanism and is much lowe than with w = 0. In fact, the optimal subsidy is only slightly highe than the optimal timevaying unemployment subsidy at the same level of utility (dashdot blue line), at least when utility is high. Fo example, at v = u(0.5), the optimal timevaying subsidy is , ising to 1 β duing a tenweek unemployment spell. The optimal constant subsidy is if w = 0.95 but if w = 0. Thee is little need fo timevaying unemployment subsidies when w is high, and hence little additional cost of poviding utility though a constant subsidy (Figue 6). A slight modification in policy has an effect that is simila to this change in technology. 22
24 Suppose that afte the woke daws a wage, she has the option of exiting the labo maket and collecting w τ theeafte. She accepts this option if he esevation wage w falls below w. 12 (1+)( w τ)f( w) When this happens, the cost is, since w must be paid in all futue peiods. Like the lowe bound on the wage distibution, this option can substantially affect a woke s debt limit and he behavio even if she is extemely unlikely eve to execise it. The only diffeence is that the policy involves a cost to the planne, while the altenative distibution does not. Howeve, since the odds of getting a single wage daw below 0.95 ae negligible, the odds of the woke eve accepting w and hence the cost of the policy ae infinitesimal. This means that the cost and optimal unemployment subsidy unde constant benefits ae indistinguishable with w = 0.95 o with w = In summay, when the distibution of wages is such that wokes have liquidity poblems, optimal unemployment insuance is wellmimicked by a twopat policy: a subsidy to unemployment, which insues wokes against the failue to find a good job; and measues to ensue that wokes ae able to smooth thei consumption ove unusually long sequences of bad wage daws. Insuing wokes against the small pobability of a vey bad shock povides liquidity. Togethe the two policies mimic optimal unemployment insuance, which involves a nealy constant unemployment subsidy fo a long peiod of time, followed by a shap incease in the subsidy when wokes ae sufficiently poo (Figue 4). An open question is how to intepet the finding that aising the lowe bound on the wage distibution fom 0 to 0.95 can have a significant effect on the constant benefit policy even if F(0.95) In ou view, it is a shotcoming of exogenous incomplete makets models that vanishingly small pobability events can significantly affect boowing. On the othe hand, we view the simplicity and tanspaency of the exogenous incomplete maket model as a vitue. 4.4 Ad Hoc Boowing Constaints Although ou analysis focuses on the natual boowing limit, it is useful to note that policy can easily cicumvent any tighte ad hoc limit. To be concete, suppose boowing is pohibited but that the natual boowing limit is negative, a < 0. Conside giving the woke a lumpsum tansfe (1 + )a at the stat of the initial peiod, while loweing he unemployment benefit to b + a and aising he employment tax to τ a. This simply 12 In ou numeical examples, a woke only accepts w if doing so is the only way she can pay the inteest on he debt. 23
25 changes the timing of payments, and is equivalent to poviding the woke with a iskfee loan, but is an ideal instument fo cicumventing any ad hoc boowing constaint. Optimal unemployment insuance, taken liteally as a policy geaed towads a handtomouth consume, is also a loan. It pays out the duationdependent sequence b t and collects taxes τ t (as in Hopenhayn and Nicolini, 1997). Figue 4 shows that the net subsidy B t may be much lowe than consumption while unemployed, eflecting the accumulating employment tax liability ove the jobless spell. Thus, an impotant component of the agency s goss tansfes ae not net pesent value tansfes; the agency pays out ealy on and collects late, much as a loan. Fo completeness, we also conside biefly the case whee, fo some unspecified and abitay eason, the insuance agency does not cicumvent the ad hoc boowing constaint. This may significantly affect both the cost and level of optimal constant benefits. To take an exteme case, suppose a woke has no assets and no ability to boow, so she must consume he benefit each peiod she is unemployed. We compute the optimal constant benefit and taxes fo this case. The black dotted line in Figue 6 shows that this aises the cost of poviding the woke with a paticula level of utility by about thee to six weeks of income, a substantial amount given that unemployment spells last fo only ten weeks. The need to povide both insuance and consumption smoothing makes the optimal unemployment subsidy much highe, in excess of 0.5 ove the usual ange of utility (Figue 7). 4.5 Robustness This section asks the extent to which ou esults depend on the wage distibution, in paticula on the assumption that a woke finds a job in ten weeks on aveage. Thee ae a few easons to exploe this assumption. Fist, ou esults indicate that constant unemployment benefits and constant employment taxes do almost as well as a fully optimal unemployment insuance policy. It could be that this esult would go away if unemployment spells tended to last longe and theefoe pesented a bigge isk to individuals. Second, in many counties, notably much of Euope, unemployment duation is substantially longe, although this is at least in pat a esponse to unemployment benefits that ae high compaed to wokes income pospects (Ljungqvist and Sagent, 1998; Blanchad and Wolfes, 2000). And thid, wokes typically expeience multiple spells of unemployment befoe locating a longtem job (Hall, 1995). Although modeling this explicitly would go beyond the scope of this pape, aising unemployment duation may captue some aspect of this longe job seach pocess. 24
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