Unemployment Insurance Design: inducing moving and retraining

Transcription

1 Unemployment Insuance Design: inducing moving and etaining John Hassle y José V. Rodíguez Moa z Septembe 8, 2006 Abstact Evidence suggests that unemployed individuals can sometimes a ect thei job pospects by undetaking a costly action like deciding to move o etain. Realistically, such an oppotunity only aises fo some individuals and the identity of those may be unobsevable ex-ante. The poblem of chaacteizing constained optimal unemployment insuance in this case has been neglected in pevious liteatue. We constuct a model of optimal unemployment insuance whee multiple incentive constaints ae easily handled. The model is used to analyze the case when an incentive constaint involving moving costs must be espected in addition to the standad constaint involving costly unobsevable job-seach. In paticula, we deive closed-fom solutions showing that when the moving/etaining incentive constaint binds, unemployment bene- ts should incease ove the unemployment spell, with an initial peiod with low bene ts and an incease afte this peiod has expied. JEL Classi cation: J65, J64, E24 Keywods: Unemployment bene ts, seach, moal hazad, advese selection This is a substantial evision of a pevious vesion that ciculated unde the title Should UI bene ts eally fall ove time?. We thank Chistina Lönnblad fo editoial assistance and Kjetil Stoesletten, Jaume Ventua and Nicola Pavoni fo valuable comments. JVRM acknowledges nancial suppot fom the Spanish Ministy of Education, poject numbe SEJ JH thanks the Swedish Reseach Council fo nancial suppot. y Institute fo Intenational Economic Studies, Stockholm Univesity, S-06 9 Stockholm, Sweden, IZA, CEPR and CESIfo. John@Hassle.se. Telephone: z Univesity of Southampton (SO7 BJ, Southampton, United Kingdom), CREA, Univesitat Pompeu Faba and CEPR. sevi@soton.ac.uk, Telephone:

2 Intoduction An impotant featue of the moden welfae state is the existence of an extensive unemployment insuance (UI) system. It is now well established that the design of the unemployment insuance a ects the incidence of unemployment by distoting the incentives of unemployed to seach fo a job (see, e.g., Holmlund (998) fo a suvey). This has motivated a gowing liteatue on how the UI system should be designed to make an optimal tade-o between poviding good insuance, on the one hand, and not distoting the incentives too much, on the othe. The key infomational fiction in this liteatue is that seach activity cannot be monitoed, so su cient seach incentives must be povided. The main contibution of this pape is that we cast the focus on anothe impotant infomational fiction, lagely neglected in the liteatue. We will conside the case when individuals who become unemployed have di eent oppotunities to nd a new job. Howeve, we assume that the insue cannot (pefectly) obseve these di eences. Speci cally, we assume that some, but not all, unemployed can incease the pobability of being hied by undetaking a costly investment, e.g., by etaining o moving to a location with bette employment pospects. Unde the ealistic assumption that the insue is unable to obseve who has this option, an incentive poblem aises and failue to take this into account may lead to sub-optimal UI-design. One diect way of mitigating the poblem would be to o e subsidies to moving o etaining. While we will discuss this case at the end of the pape, ou main case is when full cost-compensation is not feasible, fo example because the insue cannot fully distinguish voluntay and involuntay job-sepaations. Although an empiical investigation is outside the scope of this pape, we ague that the consequences of not poviding easonable incentives fo people to move o etain may be of substantial quantitative impotance. Fo instance, Batel (979) documents that the popotion of geogaphical mobility in the U.S. caused by the decision to change jobs is one-half of all migation decisions fo young wokes and one thid of all migation decisions fo wokes aged above 45. Futhemoe, geogaph- 2

3 ical mobility is substantially lowe in continental Euope, and Hassle, Rodíguez Moa, Stoesletten and Zilibotti (2004) document in panel-data a negative coelation between geogaphical mobility and UI-geneosity as well as between mobility and aggegate unemployment ates. Othe empiical documentations of the link between unemployment and geogaphical mobility ae DaVanzo (978), Pissaides and Wadswoth (989) and McComick (997). Seach incentives and incentives to move ae geneally not independent and should theefoe be jointly analyzed. The eason why moving incentives ae not included in the standad analysis is that multiple incentive constaints with di eent chaacteistics ae di cult to analyze. Including both seach and moving/etaining incentive constaints complicates the analysis, since it is di cult to evaluate which of many constaints ae binding, in paticula when unemployment bene ts ae allowed to be non-constant. Suppose, fo example, that the bene t schedule contains x ties, so that the bene t level b is an element of B fb ; b 2 ; ::b x g. The incentive constaint fo an individual at a paticula tie then depends on bene ts in all ties that the individual could eventually end up, in geneal all elements of B. The methodological contibution of the pape is to show that the poblem of nding the optimal bene t stuctue can be fomulated in such a way that that all incentive constaints ae linea and paallel o independent of each othe. It is then immediate to check which constaints ae binding and optimal bene ts can easily be chaacteized, both gaphically and analytically. Ou model easily lends itself to allowing multiple incentive poblems and adding, fo example, a moal hazad poblem in job-etention e ot as in Wang and Williamson (996) should be staightfowad. Thee is empiical evidence indicating that pecautionay saving is used to selfinsue against unemployment isk. Using PSID, Gube (997) nds that, in the absence of UI, consumption falls by 22% when an individual becomes unemployed, showing that individuals ae able to smooth consumption also when thee is no UI. Similaly, Engen and Gube (200) show that UI cowds out nancial savings, indicating that households use nancial makets to self-insue against unemploy- 3

4 ment isk. The assumption in most of the ealy liteatue on optimal UI (e.g., the seminal papes by Shavell and Weiss (979) and Hopenhayn and Nicolini (997)), namely that the insue can pefectly contol individual consumption, is thus not entiely ealistic. Following the emeging tadition in a ecent line of papes (e.g., Pavoni (2006), Apad and Pavoni (2005) and Wening (2002)), we will theefoe allow the individual to make he own consumption decisions, allowing access to a maket fo saving and boowing. We will povide analytical expessions fo the (constained) optimal bene t schedule and, in paticula, focus on the issue of whethe bene ts should incease o decease ove time. Two impotant assumptions ae key to analytical tactability. Fist, individuals have access to a pefect maket fo boowing and lending. Second, we assume constant absolute isk-avesion. These assumptions imply seach incentives to be independent of asset holdings. This allows us to focus on simple bene t schemes not contingent on the full employment histoy of the agent and with a limited numbe of bene t levels. Neithe of the key assumptions is pefectly ealistic and a quantitative analysis might equie wealth e ects, eithe because of non-constant absolute isk avesion and/o because of vaiations in the bite of liquidity constaints. Nevetheless, we hope that illustating a mechanism not peviously exploed in the liteatue might povide guidance fo futue quantitative wok. The pape is stuctued in the following way. The model is pesented in section 2, whee the elevant value functions ae deived in subsection 2.. The fomal optimality poblem is de ned and solved in section 3. In subsection 3., we show the methodology in the simplest case with a constant bene t level and in subsection 3.2, we allow time vaying bene ts. In section 4, the optimal insuance scheme is chaacteized unde di eent assumptions on seach and moving costs. The section ends with some extensions and section 5 concludes. Some poofs ae given in the Also if access to the fomal capital maket is limited, altenative means of smoothing consumption may exist, see e.g., Cullen and Gube (2000). 4

5 main text, othes in the appendix and the emaining ones ae available fom the authos upon equest. 2 The model Conside an economy in continuous time whee individuals can eithe be employed o unemployed. They have access to a maket fo safe saving and boowing with an exogenous etun, equal to the subjective discount ate (possibly including a positive pobability of dying). Unemployed individuals can a ect thei chances of nding a job. As noted in the intoduction, we will focus on the case whee some, but not necessaily all, individuals can make a costly investment inceasing thei chances of becoming employed. Allowing unobsevable heteogeneity in this espect ceates an infomational poblem simila to an advese selection poblem. 2 In addition, we will allow a moe standad moal hazad poblem whee seach activity entails a ow cost. Speci cally, we assume that an employed individual, who is said to be in state, loses he job at the exogenous ate q. A shae p 2 [0; ] of those who lose thei job can undetake a costly investment. We will intepet this as epesenting a cost of moving, denoted m > 0 (fo example between geogaphical locations o between occupations equiing some etaining). Fo simplicity, we assume that if the unemployed pays this cost ( moves ), she is immediately ehied. 3 Unemployed who cannot move o decide not to move and who seach fo a job nd one at ate h. Seaching has a cost of s 0 pe unit of time. We may conside this cost as epesenting the oppotunity cost of seaching, aising fom, fo example, some altenative economic activity. Whethe the agent actually seaches o not 2 Thee ae few papes on UI which deal with advese selection. One ecent pape is Hagedon, Kaul and Memmel (2003), whee individuals with di eent hiing ates ae sepaated by being o eed di eent bene t menus. 3 This assumption educes the numbe of states, since thee ae no unemployed moves, which makes an easy gaphical epesentation of the esults possible. Howeve, it would be staightfowad to allow a highe but nite hiing ate of moves. 5

6 and whethe she has the oppotunity to move ae assumed to be he own pivate infomation. To make the poblem inteesting, we assume that it is socially optimal to induce individuals to seach and move (if they have the oppotunity). It is easily shown that unde this assumption, agents with the option of moving should be induced to do so immediately. Theefoe, in the optimal solution, no mass of agents should be unemployed while having the oppotunity to move. A key question we want to analyze is if and how UI bene ts should change ove the duation of the unemployment spell. To answe this question, we make two assumptions that will simplify the analysis and make gaphical epesentations of ou esults possible. Fist, we assume the bene t schedule to be a ladde with a nite numbe of steps. In fact, we only allow two bene t levels, b 2 and b 3 ; but the extension to any a nite numbe of bene t levels is staightfowad. Moeove, we can show that ou main esults would not change by allowing moe than two bene t ties with x bene t ties, only the st should have a unique value, all latte bene t ties should be identical. 4 Second, we assume tansition between the steps in the bene t schedule to occu with a constant hazad ate f: Individuals who lose thei jobs ente state 2 and eceive bene ts b 2 : In state 2, they face a constant hazad ate f of enteing state 3 and then eceiving bene ts b 3 : 5 Motivated by eal-wold pactical consideations, and in contast to, e.g., Hopenhayn and Nicolini (997), we assume that bene t levels can only be given conditional on cuent unemployment status (2 o 3), not conditional on employment histoy o asset holdings. 6 Given the multiple incentive constaints, an extended unemployment insuance, whee individuals can choose between di eent menus, may be bette than a simple twotie system. In subsection 4.3, we allow such a scheme, showing that ou esults 4 Poof available upon equest. 5 This assumption implies that seach incentives emain constant as long as the individual emains in state 2. An altenative would be to use discete time and assume that shot-tem UI bene ts ae paid fo one peiod only, as done by e.g., Cahuc and Lehmann (2000). Assuming that UI bene ts change afte some xed peiod of time would make seach incentives depend on the emaining time of cuent bene ts and consideably complicate the analysis with little gain. 6 In fact, unde CARA utility, also this assumption is innocuous. 6

7 egading when bene ts should be inceasing and when they should be deceasing emain valid in the case of menu-based insuance. The simplest and most obvious way of intepeting the unemployment states is as an indication of the passage of time: individuals in state 3 have, on aveage, been unemployed longe than individuals in state 2. Theefoe, we label state 2 as shot-tem unemployment and state 3 as long-tem unemployment. Ou pefeed intepetation of the thid state is that it is a puely administative state and we may allow the insuance povide to choose f: In this case, it is natual to assume that seach costs (s) and hiing pobability (h) ae the same in both states. We may also intepet the thid state as epesenting loss of skills duing unemployment in the sense of job- nding ates and seach costs developing disadvantageously ove the unemployment spell. As an extension, we modify the model so that with a constant instantaneous pobability f, unemployed individuals su e a shock, and thei seach costs incease (s 2 < s 3 ) and/o thei hiing pobabilities decease (h 2 > h 3 ). Although this intepetation aises issues about obsevability, we abstain fom these and assume bene ts to be paid contingent on whethe the individual is in state 2 o 3. Individuals maximize thei intetempoal utility, given by E Z 0 e t U (c t ) dt; whee c t is consumption at time t and is the subjective discount ate. To facilitate analytical solutions when individuals have access to makets fo saving and boowing, we choose the CARA utility function U (c t ) e ct ; whee is the coe cient of absolute isk avesion. All individuals ae bon (ente the labo maket) as employed without assets and ae identical at that point. The pupose of this pape is to discuss how an unemployment insuance system should be constucted when thee ae incentive poblems. To this end, we want to emove othe motives fo unemployment bene ts than poviding insuance. 7

8 In paticula, in this pape, we ae not inteested in motives fo using the UI system to ceate non-actuaial tansfes between individuals with di eent chaacteistics. 7 Theefoe, we assume that individuals face an actuaially fai insuance. This means that when an individual entes the labo foce, the expected pesent discounted value of the bene ts she will eceive duing he life-time exactly balances the expected pesent discounted value of he contibutions. An altenative intepetation of actuaial fainess is that in a decentalized equilibium, whee individuals can sign binding insuance contacts with competitive insuance companies when enteing thei st job, actuaial fainess is identical to a beak-even condition fo the insuance companies, which would be satis ed unde pefect competition. 8 Without loss of geneality, we let individuals pay lump-sum taxes, denoted, implying that _ A t = A t + y c t ; () except at the points in time when the cost of moving is paid, and whee y 2 fw; b s; b 2 sg, depending on the employment state. We de ne the aveage discounted pobabilities (ADP s) of being in state 2 and 3, espectively, by 2 3 Z 0 Z 0 e t 2;t dt; e t 3;t dt; whee 2;t and 3;t ae the pobabilities of being shot-tem and long-tem unemployed at time t, espectively, conditional on being employed at time zeo, povided that individuals who can move do so and that unemployed seach fo a job. 9 actuaial fainess equiement of the UI system is now a simple linea function of the bene ts The = 2 b b 3 : (2) 7 Fo positive implications, the edistibutive elements of unemployment insuance ae, howeve, likely to be cental. See e.g., Wight (986). 8 Since we use the CARA speci cation, individual assets do not a ect pefeence ove insuance, so that olde employed agents with non-zeo asset holdings would not want to enegotiate thei contact. 9 It is staightfowad to calculate that 8

9 2. Value functions and consumption It is well known that unde constant absolute isk avesion and stationay income uncetainty, the value functions fo the thee states j 2 f; 2; 3g can be sepaated V (A t ; j) = W (A t ) ~ V j (; b 2 ; b 3 ) ; (3) whee W (A t ) e At ~V j e c j ; (4) and j ae state-dependent consumption constants and whee the state dependent consumption functions ae c j (A t ) = A t + j : (5) The consumption constants j ae nonlinea functions of income in all states and thus, depend on the planne choice vaiables ; b 2 and b 3 : The constants ae found as the unique solutions to the Bellman equations fo each state: 0 q e 2 = w (6) 2 = b 2 s + h e 2 f e ( 3 2 ) 3 = b 3 s + h e 3 ; whee 2 2 ; (7) 3 3 ; q ( p) (h + ) 2 ( + h + q ( p)) ( + h + f) ; f 3 2 h + : 0 See the appendix fo poof. 9

10 ae the consumption di eences between state and 2 and between state and 3, espectively. 3 Optimal Insuance Given the discussion above, the poblem we set out to solve is to maximize the ex-ante value of unemployment insuance, that is, we want to maximize the welfae of an individual upon enteing the economy. This welfae is given by V (0; ) ; since we assume that agents ente the economy as employed with no assets. Due to the sepaability and the fact that W (A t ) is independent of the insuance system, we immediately see that this is equivalent to maximizing V ~ ove f; b 2 ; b 3 g : Using the budget constaint = 2 b b 3 ; ou objective is theefoe to solve max V ~ ( 2 b b 3 ; b 2 ; b 3 ) (8) b 2 ;b 3 s.t. IC2, IC3, and ICM, whee IC2 and IC3 ae the incentive constaints that unemployed individuals voluntaily seach fo a job and ICM is the constaint that individuals with the oppotunity to move to get a job voluntaily do so. In the diect fomulation of the poblem, the incentive constaints ae highly non-linea functions of the choice vaiables b 2 and b 3. This makes it had to nd the binding constaints, which is necessay to nd the solution. Howeve, it tuns out that we can fomulate the poblem so that the incentive constaints ae linea and eithe paallel o othogonal. Finding out which is binding is then tivial. Futhemoe, adding moe states and incentive constaints is also vey simple. We egad this as the methodological contibution of the pape. Finding the constained optimal insuance now involves the following steps: Obviously, we could equally well have chosen any othe initial condition. Note also that the sepaability implies that the insuance system that maximizes the ex-ante utility also maximizes the utility of all employed, egadless of thei histoy. 0

11 . Note that V ~ e c is a monotone tansfomation of. Fo convenience, we theefoe use as the objective function and expess it as a function of the consumption di eences, using the budget constaint (2) to eplace. 2. Expess the incentive constaints in tems of consumption di eences j : 3. Maximize ove the consumption di eences, subject to the incentive constaints. 4. Veify that the optimal consumption di eences 2 can be implemented by some combination of b 0 j s: 3. Two states Fo illustative puposes, we stat with the simplest case of two states, i.e., we assume that f = 0 so unemployment bene ts ae constant foeve. The st step is now to deive an expession fo in tems of 2 whee the budget constaint (2) is used to eplace the tax ate: Fo this pupose, we subtact the second line of (6) fom the st and solve fo b 2 : Then, we use this expession in the budget constaint = 2 b 2 and substitute fo in the st line of (6). This yields = he 2 ( 2 ) q ( p) e 2 ; (9) whee is a constant, independent of the choice vaiables. Staightfowad calculus shows that (9) de nes as a concave function of 2 with a unique maximum at 0. The eason fo being maximized at 2 = 0 is obvious when actuaial insuance is available, full insuance maximizes utility. Howeve, 2 = 0 is not incentive compatible. Neithe seaching no moving will occu voluntaily unde full insuance. Theefoe, we tun to step 2 whee we nd the incentive constaints. The ICM constaint implies that a peson who has lost he job and has the oppotunity to move must be induced to do so. We st note that if he assets upon sepaation

12 wee A t, he value immediately afte moving is V (A t m; ) = e (At m) e ; since she has paid the moving cost, m: We compae this to the value of a one-peiod deviation, i.e., the value if the individual does not move duing this unemployment spell. Immediately afte being laid o, he assets ae A t and she is unemployed, i.e., in state 2, since she did not take the oppotunity to move to get a job. He value is theefoe, To induce moving, we need V (A t that this equies We label (0) the ICM-condition. V (A t ; 2) = e At e 2 : m; ) V (A t ; 2) : It immediately follows 2 m: (0) Now, conside the incentive to seach. Remembe that fo now, we assume unemployment bene ts to be at (the assumption f = 0 implies that b 3 is ielevant). If the individual does not seach, she theefoe gets an income b 2 fo eve, since she will not nd a new job without seaching. Without uncetainty, she consumes exactly he total income A t + b 2 ate) and he utility is theefoe (since coincides with the subjective discount e At e (b 2 ) : The utility if the individual instead seaches is e At e 2 seach, we clealy need 2 b 2 : so to induce Note that the consumption of the unemployed who seach is A t + 2 : Futhemoe, he total income net of seach costs is A t + b 2 s: Theefoe, the seach condition implies consumption to be stictly highe than income. Ove time, the unemployed depletes he assets and consumption theefoe falls, despite the bene ts being constant. The celebated esult by Shavell and Weiss (979) and Hopenhayn 2

13 and Nicolini (997) that consumption should optimally fall ove the unemployment spell when the insue can fully contol consumption (no hidden savings) is theefoe mimicked in this case, whee hidden savings ae allowed. The nal pat of step 2 is to expess the seach constaint in tems of the consumption di eence 2 : Using the second line of (6) and setting f = 0; the seach constaint can be witten 2 = ln s h ; () which we label the IC2-condition. As can be seen, the incentive constaints ae simply constants and it is immediate to see which one is binding. The poblem is now simply depicted in Figue, whee we note that the two constaints ae paallel. σ IC2 ICM 0 ln( γ s / h) γ m 2 Figue : Objective function and constaints in a two-state case. In the depicted case, it is the ICM-constaint that binds and step 3 is tivial. 3

14 Maximizing ove 2 subject to the ICM constaint implies 2 = m: Finally, we want to implement this. This is easily done using (6); set the di eence between the st and the second line equal to m and solve fo b 2, giving b 2 = w + s m q (em ) + h ( e m ) : In the altenative case, whee the IC2 constaint binds, we instead get b 2 = w + ln s h sq h s ; (2) whee both expessions ae unique and easily lend themselves to compaative statics. 3.2 Thee states The pocedue in the case of thee states is exactly analogous to the two-state case and simply extends to any numbe of nite states. We use (6) and the budget constaint (2) to expess as a function of the consumption di eences, now 2 and 3 3 (step ). Then, we expess the incentive constaints in tems of 2 and 3, check which ae binding (step 2), maximize ove f 2 ; 3 g subject to the binding constaints (step 3) and nd the implementing b 2 ; b 3 (step 4) Objective and constaints Using the equations fo the consumption constants (6) and the budget constaint (2), the objective becomes = ( 2 3 )! 2 h e 2 + f e( 3 2 ) q ( p) e 2 (3) 3 h e 3 ; whee 2 is an unimpotant constant. In gue 2, we make a gaphical epesentation of the objective function by dawing indi eence cuves in a gue with 3 on the x axis and 2 on the y axis. 2 The bliss point is at full insuance, when 2 The indi eence cuves in gue 2-6 ae dawn fo fh; f; q; ; ; pg = f; ; :; :05; :5:g but the esults below hold fo all paamete values. 4

15 f 3 ; 2 g = f0; 0g, again, fo the eason that the insuance is actuaially fai. The indi eence cuves have elliptical shapes aound the bliss point, of which we ae only inteested in the segment in the positive quadant, since incentive compatibility cetainly equies 3 ; 2 0: Fo the late analysis, we should note that the slope of an indi eence cuve is stictly positive if 3 = 0 and 2 > 0 and that it is downwad sloping at 2 = 3, egadless of the paamete choice Figue 2: Indi eence cuves. Regading the thee incentive constaints, it is staightfowad to see that they ae identical to the case of two states, 4 i.e., the ICM is 2 m and the IC2 and IC3 constaints ae, 2 ; 3 ln s : (4) h The intuition fo the fact that IC2 and IC3 ae identical is simple. In ou base line case, hiing pobabilities and seach costs of seaching individuals ae the 3 Di eentiating the objective function, we nd the deivative of the indi eence cuve to be fe 2 +(h+f)(+e 2 ) 2 (0; ) at 3 = 0 and 4 See the appendix fo a fomal poof. e 2 + h + (h+)2 fh + (h+)(e 2 ) f 2 ( ; 0) at 2 = 3: 5

16 same fo long- and shot-tem unemployed. The incentives in tems of utility and thus, in tems of consumption inceases upon successful seach, must theefoe be the same. Allowing di eent seach costs and/o hiing pobabilities in the two states is, howeve, vey simple by allowing s and h to be state dependent in the IC conditions; this is done in section 4.2. Theefoe, we each the key conclusion that the incentive constaints fo the two states (IC2 and IC3) ae identical and othogonal in the f 2 ; 3 g space. We emphasize that this does not mean that only b 2 (b 3 ) is of impotance fo seach incentives of the shot-tem (long-tem) unemployed. On the contay, both b 2 and b 3 a ect consumption and theefoe incentives in all states. Howeve, individual optimization and access to makets fo saving and boowing imply that the value function is a monotonous tansfomation of consumption. Thus, the wedge between consumption in the cuent state and duing employment is a su cient statistic to detemine whethe seach incentives ae su ciently stong. In the next subsection, we will use ou model to chaacteize the optimal UIscheme unde di eent assumptions on which the constaint is binding. As in the two-state case, the analysis is geatly simpli ed by the incentive constaints in f 3 ; 2 g space being linea and paallel o othogonal. When the optimal f 2 ; 3 g ae found, we nd the optimal bene ts fom the implementation mapping, which is deived by taking the di eence between lines and 2 and between and 3 in (6) and solving fo b 2 and b 3 : q e 2 + h e 2 b 2 = w + s 2 q e 2 + h e 3 b 3 = w + s 3 : f e ( 3 2 ) ; (5) 4 Chaacteization of optimal UI-schemes In this section, we use ou model to chaacteize (constained) optimal unemployment insuance unde thee di eent scenaios. In the st, the hiing pobability is 6

17 the same independent of unemployment duation. In the second scenaio, we look at the consequences fo the optimal UI scheme of having a hiing pobability that is deceasing with unemployment tenue. Finally, in the thid scenaio, we allow the insue to o e a menu of possibilities to the unemployed, including the choice of a lump-sum tansfe. 4. Constant hiing ates and seach costs 4.. Small seach costs We stat the analysis with the assumption that seach costs ae su ciently small to be ignoed, late they ae e-intoduced. Fist, we analyze the poblem gaphically by including the ICM constaint, i.e., 2 m in the indi eence cuve gaph (Figue 3), and then we povide analytical esults. The ICM constaint is satis ed fo all values of 2 above the ICM-cuve, which is hoizontal. The optimizing choice of 3 is whee the ICM constaint is tangent to an indi eence cuve. This occus fo the solid indi eence cuve in gue 3. As noted above, the indi eence cuve is positively sloped at 3 = 0 and negatively sloped at 2 = 3 implying that the tangency must be at a point whee 3 > 0 and 3 < 2. This means that state 2 should be "wose" than state 3 in the sense that, given assets, utility and consumption ae highe in state 3 than in state 2. It is intuitive (and easily poved) that 2 > 3 > 0 implies that b 2 s < b 3 s < w. The intuition fo this is that when b 2 s = b 3 s, the two unemployment states ae, by constuction, identical so that 2 = 3. Making 2 lage than 3 equies a eduction in bene ts fo shot-tem unemployed and/o an incease in bene ts fo long-tem unemployed. Result : If seach costs ae su ciently low, only the ICM constaint is binding and bene ts should optimally incease ove time. The economic eason fo ou esults can be phased in the following way. To sepaate individuals with the option of moving fom those who have not, a positive 2 is equied. Howeve, this does not call fo an ine cient stuctue of the bene t 7

18 ICM Figue 3: Indi eence cuves and Incentive Constaint fo Moving (ICM). schedule. Speci cally, stating fom a at bene t schedule (along the 45 degee line whee 2 = 3 ), the welfae in all states can be inceased, while maintaining the necessay wedge 2 = m, by inceasing bene ts fo long-tem unemployed and educing bene ts fo shot-tem unemployed. The eason fo this is that the expected maginal utility is highe fo individuals who have been unemployed fo a long time. The optimum is, howeve, eached befoe bene ts to long-tem unemployed ae su ciently high to make the latte indi eent between having a job and emaining unemployed. On the othe hand, when 3 = 0 while 2 = m, long-tem unemployed ae as well o as the employed (given assets) and thei expected maginal utility is elatively low. A eallocation fom long-tem to shot-tem bene ts theefoe inceases the value of the insuance so that the tax-cost of poviding a given insuance value can be educed. Now, let us deive closed-fom solutions to ou poblem. Using the binding ICM condition 2 = m to substitute fo 2, the objective function (3) simpli es and 8

19 the poblem can then be witten max 3 2R + ( 3 3 h e 3 2 f e( 3 m) ) ; (6) These tems have staightfowad intepetations. The st tem is due to the bene t of educing the tax-cost of long-tem bene ts. This tem is inceasing in 3, since highe 3 is achieved by lowe bene ts fo long-tem unemployed, which educe taxes in popotion to the ADP of long-tem unemployment, 3 : Note that this tax eduction comes fom two souces; thee is a diect e ect that is popotional to 3 but thee is also an indiect e ect, captued by the second tem inside the paenthesis. Long-tem unemployed nd jobs at a positive ate, h. The pospect of nding a job keeps up consumption, so that it falls less than popotionally to the eduction in bene ts. Convesely, given an incease in 3, bene ts can be educed moe than popotionally. The second tem in (6) is due to the bene t of educing the tax cost of shottem bene ts. It is deceasing in 3 since less consumption fo long-tem unemployed has a negative impact on consumption also of the shot-tem unemployed, popotional to f: As 3 inceases, bene ts to the shot-tem unemployed must theefoe incease to keep 2 = m: This has a tax-cost popotional to the ADP of shot-un unemployment 2 : The objective function in (6) is concave in 3. Thus, the unique solution to the poblem is obtained by the solution to the st-ode condition, given by q 2 ln 2h + e m h+ h 2h 3 = > 0: Using the implementation mapping (5), we can nd the optimal insuance scheme. In paticula, in optimum b 3 b 2 = m 3 + f + he 3 e (m 3) > 0: (7) Notice also that since the solution fo 3 is independent of f, the di eence b 3 b 2 should incease in f. It can be shown that the deivative of the objective function 9

20 with espect to f is always positive. Low values of f is an ine cient way of inducing sepaation between those who can move and those who cannot, as agents expect to spend a longe stochastic time su eing the low shot-un bene ts. Without fomally showing this, we conjectue that if lump-sum bene ts wee allowed, the best policy would be to punish unemployment by a lump-sum unemployment tax when an individual becomes unemployed. In eality, howeve, it may be politically di cult o even infeasible to implement a lump-sum punishment on those who lose thei jobs. Futhemoe, a lowe bound on b 2, fo example zeo, might be imposed fo political easons, in which case this would pin down f fom (7). As is clea fom the above analysis, a eduction in m educes 2 and allows a moe geneous unemployment insuance. Such a eduction could be achieved by subsidies to moving o etaining. Howeve, full compensation is unlikely to be optimal in eality. Suppose, ealistically, that individuals with a job sometimes expeience a pefeence o poductivity shock, making anothe job o a job in anothe location moe attactive than the cuent one. Suppose also that these shocks ae not su ciently lage to induce voluntay sepaation and moving if the individual must pay the moving cost heself. Clealy, such moves ae then not socially optimal. The insue would like to fully subsidize the moving cost of individuals who ae involuntaily sepaated fom thei job, but not subsidize it fo individuals who voluntay sepaate to claim the subsidy. Howeve, this is is infeasible if the insue cannot distinguish voluntay and involuntay sepaations. Theefoe, we ague that although patial subsidies may be feasible and, in fact, optimal, full subsidization may not. Moe speci cally, it seems clea that subsidies should be as lage as possible, without inducing ine cient voluntay sepaation. Thus, we could intepet m as the cost of moving o etaining, net the optimal subsidy. Futhemoe, a lage subsidy to moving might lead unemployed individuals to claim the subsidy, which is likely to be ine cient. This issue is analyzed below. 20

21 2 0.3 IC ICM IC Figue 4: Low seach costs 4..2 Lage seach costs We can now easily analyze the conditions such that IC2 and IC3 ae satis ed, despite positive seach costs. Gaphically, the constaints ae simply hoizontal and vetical lines and all values of 2 ( 3 ) above (to the ight of) these lines imply that the espective constaints ae satis ed. If seach costs ae su ciently small, none of the seach constaints bind, as shown in gue 4, whee IC2 is slack while IC3 almost binds at the tangency between ICM and an indi eence cuve. This occus at a point indicated by the aow on the solid indi eence cuve. Inceasing seach costs shift out IC2 and IC3 since fom (4) we see that the RHS is inceasing in s. Eventually (fo a seach cost which is su ciently lage) IC3 is no longe satis ed at the point whee the ICM constaint is tangent to the indi eence cuve. This situation is depicted in gue 5. Hee, the point whee the ICM is tangent to the most outwad dotted indi eence cuve satis es the IC2 constaint, but not the IC3 constaint. Thus, 3 must be inceased but since the IC3 and the ICM constaint ae othogonal, 2 need not be changed. The optimal 2

22 2 0.3 IC ICM IC Figue 5: Modeate seach costs point is whee the ICM and the IC3 constaint coss. This point is indicated by the aow and on the solid indi eence cuve. Clealy, 3 emains smalle than 2 implying an upwad sloping bene t po le, i.e., b 2 < b 3 : Speci cally, 2 should be set equal to m and 3 equal to ln. This means that individuals will be indi eent in the choice of moving and that long-tem unemployed ae indi eent to seaching, while the shot-tem unemployed stictly pefe to seach. Result 2: Fo an intemediate ange of seach costs, the ICM and the IC3 constaints ae binding and bene ts should optimally incease ove time. A futhe incease in seach costs will eventually call fo a situation like that in gue 6. s h Hee, both seach constaints bind, while the moving constaint is slack. Once moe, the optimum is indicated by the aow and on the solid indi eence cuve. Bene ts ae constant and given by expession (2) since 2 = 3 = ln : 5 We conclude: s h 5 This is a special case of the esult in Wening (2002) which shows that constant bene ts ae optimal unde CARA utility in a geneal class of UI-schemes. 22

23 2 0.3 IC IC2 ICM Figue 6: High seach costs. Result 3: Fo su ciently high seach costs, the IC2 and the IC3 constaints ae binding and bene ts should optimally be constant ove time. The conclusion so fa is that when the moving cost is lage elative to the seach costs, then the optimal unemployment insuance scheme involves an inceasing bene t po le in ode to, on the one hand, geneate incentives to move fo those agents who can and, on the othe hand, not too much limiting insuance fo the possibility that an unemployment peiod becomes long-lasting. If the seach costs ae su ciently high elative to the moving cost, stong seach incentives ae needed and the moving constaint is slack. In this case, the optimal bene t po le is at. The intuition behind this esult is that, one the one hand, seach incentives ae stengthened by falling bene ts. On the othe hand, when pivate savings ae allowed, bu e stock savings povide a good substitute fo shot but not fo long unemployment spells, calling fo an upwad sloping bene t po le. 6 6 See Hassle and Rodíguez Moa (999) fo an analysis of the elative value of insuance against long and shot unemployment spells unde CARA and CRRA utility. 23

24 These two e ects cancel exactly unde CARA utility. With othe utility functions both e ects ae pesent but will in geneal not cancel each othe. 4.2 Loss of skills and long-tem unemployment So fa, we have consideed the thid state as an administative state, used as a poxy fo the unemployment duation of the agent. Unemployment was assumed to have no othe e ect than depleting the nancial assets of the agent; hiing ates and seach costs emained constant. Howeve, it is easy to elax this assumption and analyze how the path of bene ts should be constucted if the unemployment duation also has eal diect e ects on, e.g., seach costs and hiing pobabilities. 7 Speci cally, let s 2 and s 3 denote the seach costs in states 2 and 3 and, coespondingly, h 2 and h 3 denote the state dependent hiing pobabilities. The idea that the human capital of the unemployed depeciates duing the unemployment spell (o that the individual "leans how to be unemployed") is captued by the assumption h 2 > h 3 and/o s 2 < s 3 ; implying s 2 h 2 < s 3 h 3 : It is staightfowad to show that the IC2 and IC3 constaints now become espectively, whee 2 3 ln s 2 ln h 2 s 3 h 3 ; ; ln s 2 h 2 < ln s 3 h 3 so that the IC3 constaint cosses the IC2 condition below the 45 degee line. If the binding constaints ae IC2 and IC3 (small moving costs), we must then 3 > 2 : Using the implementation equations (using the di eent seach costs and hiing ates), we nd that in this case, the optimal bene t schedule should be downwad sloping (b 2 > b 3 ). If the ICM constaint binds, athe than IC2, the possibility that the optimal bene t po le should be upwad sloping emains. 7 Similaily, we could easily analyze the case when the pospective wage depends on unemployment duation. 24

25 4.3 A menu of contacts Finally let us note that ou model can also easily handle moe complicated UI schemes, e.g., menus. 8 In paticula, let us conside the case when the insue allows individuals losing thei job to eithe get a lump-sum tansfe M, o a possibly non-constant UI-bene t steam. 9 Since the e ective cost of moving is now m M; the incentive constaint fo individuals with the oppotunity to move now becomes, 2 = (m M) ; i.e., a positive M slackens the constaint (moves it down in the gues). Inceasing M to a su ciently lage extent leads to a situation like that in gue 6, whee IC2 and IC3 bind. Potentially, its optimal to set M = m full subsidization. This is the case if unemployed without moving oppotunities pefe UI bene ts ove M, so that a sepaation between the goups is achieved also when the moving cost is fully insued. If such sepaation is not achieved unde full insuance but should be in optimum, M must be educed so that unemployed individuals choose UI bene ts. We note that we cannot incease the elative attactiveness of UI-bene ts by aising the latte, since this would violate the IC2 and IC3 conditions, which continue to bind. To analyze whethe sepaation is achieved, we need to add anothe state to the analysis, namely to be unemployed without bene t, which makes a two-dimensional gaphical analysis impactical. The analytical analysis emains simple, howeve. Setting the income of unemployed to zeo, the consumption constant associated with being unemployed without bene ts is given by u = s + h e ( u) so that u is a function of only. The incentive constaint implying that unemployed do not choose the lump-sum tansfe is then 2 ; u M; and it is easily 8 Some UI schemes o e this type of menus; in paticula, in the peiod of lage unemployment (end of the 80 s and beginning of the 90 s) the Spanish Unemployment agency o eed the option of a lump-sum tansfe o standad UI payments. 9 Fo simplicity, let us disegad the case of voluntay sepaations as discussed above. 25

26 checked if this is satis ed in the equilibium. If not, M must be educed. If the ICM condition is slack, bene ts should be constant. Howeve, as M is educed, the ICM condition might eventually bind, once moe calling fo an upwad sloping bene t schedule. 5 Conclusion In this pape, we have agued that thee ae easons to believe that an impotant infomation poblem associated with unemployment insuance has been neglected in the pevious liteatue. This poblem stems fom the fact that unemployed individuals sometimes have the option of making an investment that could incease thei chances of nding a job. Examples of such investments ae etaining and moving to anothe location. Since it is easonable to assume that it is di cult o impossible to obseve who has these options, the UI system should give incentives fo people to take advantage of any easonable option to incease thei labo maket pospects. By deiving gaphical and analytical closed-fom solutions fo how a simple UI system should be constucted to povide su cient incentives without excessively educing the value of the unemployment insuance. Unless the hiing ates of long-tem unemployed ae vey low and seach costs high, this equies an initial peiod of elatively low bene ts. The intuition hee is staightfowad, by setting initial bene ts at a low level, individuals with good oppotunities to get new jobs ae induced to exploit these. On the othe hand, individuals with wose oppotunities value insuance against long-tem unemployment moe than insuance against shot-tem unemployment. The value of the UI system can theefoe be maintained by poviding moe geneous bene ts fo long-tem unemployment, calling fo an upwad sloping bene t po le. We have assumed that individuals can self-insue via unobsevable savings, i.e., that individual consumption is unobsevable o, fo some othe eason, uncontactable. If, in contast, the insue has contol ove the consumption of the individual, it is well known that thee would be a tendency to povide a downwad 26

27 sloping path of consumption (and bene ts, if the individual has no othe income) to povide good seach incentives. Nevetheless, the point of this pape, that a peiod of low initial UI bene ts is an e cient way of sepaating individuals who can move fom those who cannot, is likely to still be tue. We have also assumed constant absolute isk-avesion, even if this epesentation of individual pefeences is not necessaily the most ealistic. Unde the moe standad assumption of constant elative isk-avesion, the analysis is geatly complicated by the fact that seach incentives would depend on asset holdings. Theefoe, incentive compatibility would not in geneal be consistent with a nite numbe of bene ts that ae independent of individual asset holdings. Howeve, the intuition fo the esults in this pape does not appea to be elated to such e ects. In ou model, the pefeence fo inceasing bene ts aises fom the need to sepaate between the two types of wokes and the fact that individual assets ae depleted duing unemployment, (which is tue fo geneal speci cations of utility, in paticula fo CRRA, as shown in e.g., Hassle and Rodíguez Moa (999)). Both mechanisms ae likely to be pesent also unde moe geneal pefeence speci cations. Howeve, since seach incentives in geneal depend on asset holdings and the duation of unemployment is likely to be coelated with the individual s asset holdings, unobsevability of the latte may have consequences fo optimal bene t time po les. Fo example, if the seach incentives ae einfoced as wealth decumulates and individuals with long unemployment spells ae likely to have less wealth, this might stengthen the case fo inceasing bene ts. The analysis of optimal UI design with hidden savings when individual behavio depends on asset holdings is likely to demand numeical models. This is left to futue eseach. Refeences Apad, A. and Pavoni, N.: 2005, The e cient allocation of consumption unde moal hazad and hidden access to the cedit maket, Jounal of the Euopean Economic Association 3,

28 Batel, A. P.: 979, The migation decision: What ole does job mobility play?, Ameican Economic Review 69(5), Cahuc, P. and Lehmann, E.: 2000, Should unemployment bene ts decease with the unemployment spell?, Jounal of Public Economics 77(), Cullen, J. B. and Gube, J.: 2000, Does unemployment insuance cowd out spousal labo supply?, Jounal of Labo Economics 8(3), DaVanzo, J.: 978, Does unemployment a ect migation?-evidence fom mico data, Review of Economics and Statistics 6(4), Engen, E. M. and Gube, J.: 200, Unemployment insuance and pecautionay saving, Jounal of Monetay Economics 47(3), Gube, J.: 997, The consumption smoothing e ect of unemployment insuance, Ameican Economic Review 87(), Hagedon, M., Kaul, A. and Memmel, T.: 2003, An advese selection model of optimal unemployment insuance. Mimeo, Univesity Pompeu Faba. Hassle, J. and Rodíguez Moa, J. V.: 999, Employment tunove and the public allocation of unemployment insuance, Jounal of Public Economics 73(), Hassle, J., Rodíguez Moa, J. V., Stoesletten, K. and Zilibotti, F.: 2004, A positive theoy of geogaphic mobility and social insuance, Intenational Economic Review. fothcoming. Holmlund, B.: 998, Unemployment insuance in theoy and pactice, Scandinavian Jounal of economics 00(), 3 4. Hopenhayn, H. A. and Nicolini, J. P.: 997, Optimal unemployment insuance, Jounal of Political Economy 05(2),

29 McComick, B.: 997, Regional unemployment and labou mobility in the uk, Euopean Economic Review 4(3-5), Pavoni, N.: 2006, On optimal unemployment compensation, Jounal of Monetay Economics. fothcoming. Pissaides, C. A. and Wadswoth, J.: 989, Unemployment and the inte-egional mobility of labou, Economic Jounal 99(397), Shavell, S. and Weiss, L.: 979, The optimal payment of unemployment insuance bene s ove time, Jounal of Political Economy 87(6), Wang, C. and Williamson, S. D.: 996, Unemployment insuance with moal hazad in a dynamic economy, Canegie-Rocheste Confeence Seies on Public Policy 44, 4. Wening, I.: 2002, Optimal unemployment insuance with unobsevable savings. Mimeo, Univesity of Chicago. Wight, R.: 986, The edistibutive oles of unemployment insuance and the dynamics of voting, Jounal of Public Economics 3(3), Appendix 6. Bellman equations and consumption constants Guessing that the value function is equation fo the employed is, e (At+ j) fo j 2 f; 2; 3g; the Bellman e (At+) = max e (At+) dt ( dt) ( qdt) e (A t+dt+ ) + qdt e (A t+dt+ 2 ) : 29

30 Using st-ode linea appoximations and dividing by e At ; this becomes e = max e dt ( dt) ( qdt) e ( (w ) dt) + qdt e 2 ( (w ) dt) Adding e to both sides, dividing by dt and letting dt appoach zeo, yields n o 0 = max e ( ) + + (w ) + q e ( 2 ) : (8) Similaly, fo the shot-tem and long-un unemployed, we obtain n 0 = max e ( 2) + + (b 2 s ) + h + f he ( 2 ) n 0 = max o e ( 3) + + (b 3 s ) + h e ( 3 ) : fe ( 3 2 ) o ; (9) Equations (8) and (9) ae maximized at = j, implying that fo the Bellman equation to be satis ed, the constants j ; must satisfy (6). Taking the di eence between line and 2 and between and 3 in (6) and solving fo b 2 and b 3 ; we obtain the implementation mapping (5). 6.2 The IC2 and IC3 conditions Let us now conside the incentives fo seaching duing unemployment. We st note if a long-tem unemployed does not seach, she gets an income b 3 foeve, implying a utility e At e (b 3 ) ; while she gets e At e 3 if she seaches. Theefoe, we need 3 b 3 (6), this implies which is the IC3-condition. to induce seach of the long-tem unemployed. Using 3 ln s ; (20) h Fo the shot-tem unemployed, we compute the value associated with a onepeiod deviation, i.e., no seach in the cuent employment state, conditional on e seaching in futue states. This value is A te c 2;n, whee 2;n satis es 2;n = b 2 + f e ( 3 2;n ) : 30

31 The IC2 constaint is given by 2 2;n 0: Futhemoe, 2 2;n = s + h e 2 = f e ( 3 2 ) e ( 3 2;n )! s + h e 2 f e( 3 2 ) e ( 2 2;n ) (2) R ( 2 2;n ) Clealy, R is a monotonously deceasing function with a hoizontal asymptote at s + h e 2 f e( 3 2 ) (achieved as 2 2;n appoaches in- nity), appoaches in nity as 2 2;n appoaches minus in nity and R (0) = s + h e 2. The solution to (2) is the unique xed-point of R. This value is non-negative if and only if s + h e 2 0. So 2 2;n, 2 s ln : h 7 Poofs not intended fo publication 7. Poof that esults extend to n unemployment states Suppose we have n states, then the consumption constants ae = w q pem + ( p) e 2 ; (22) 2 = b 2 s + h e 2 3 = b 3 s + h e 3 f e( 3 2 ) ; f 3 e ( 4 3 ) (23) ::: (24) n = b n s + h e n n = b n s + h e n : f n e (n n ) (25) 3