Optimal unemployment insurance: When search takes effort and money

Labour Economics 36 (2015) 1–17

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Optimal unemployment insurance: When search takes effort and money☆ J. Schwartz ⁎ Loyola University Maryland, Department of Economics, 4501 North Charles Street, Sellinger 315, Baltimore, MD 21210, United States

H I G H L I G H T S • • • • •

I develop a model where finding work requires effort and monetary expenses. The model assumes search effort, savings, and search capital are hidden actions. I use the model to determine the optimal unemployment insurance (UI) policy. Without savings high upfront benefits is optimal so search expenses are affordable. With savings UI should be high for the long-term unemployed.

a r t i c l e

i n f o

Article history: Received 3 April 2013 Received in revised form 11 May 2015 Accepted 5 July 2015 Available online 16 July 2015 JEL classification: J64 J66 D82 Keywords: Unemployment insurance Moral hazard Hidden information Unemployment Search

a b s t r a c t Searching for work is costly. It involves finding available positions, completing applications, and attending interviews, to name but a few of the activities involved. The optimal unemployment insurance (UI) literature models the cost of these activities as either a reduction in leisure or an unpleasant bad that reduces utility, ignoring their associated monetary costs. If search requires out of pocket expenses on goods and services that improve the probability of a successful job search, a low UI benefit may make a job search unaffordable. This paper investigates the optimal structure of UI in an economy where job search is not only unpleasant, but also requires a monetary investment. Numerical experiments suggest that without access to capital markets, the optimal UI system should include a higher benefit for the newly unemployed than is implied by assuming a job search is free. This allows workers to purchase the stock of goods and services needed to find work. In contrast, when workers can accumulate savings, more benefits should be provided to the long-term unemployed, so they have the financial resources needed to conduct a job search even as they exhaust their own savings. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Searching for work is a costly endeavor. The process involves a set of fairly unpleasant activities, such as searching through help wanted advertisements, preparing resumes and applications, and attending interviews. Modelers interested in the search process, and in particular the optimal design of unemployment insurance (UI), incorporate these costs as either a loss of leisure time or an intangible bad that reduces

☆ I would like to thank the Loyola University of Maryland Sellinger Business School for their generous financial support. ⁎ Tel.: +1 410 617 2919. E-mail address: [email protected].

http://dx.doi.org/10.1016/j.labeco.2015.07.002 0927-5371/© 2015 Elsevier B.V. All rights reserved.

utility. To date, the literature largely ignores the out of pocket expenses that are required to perform a search. A job search not only requires some intangible effort, but a variety of goods and services ranging from transportation and professional attire for interviews to computing resources and paid recruiters. The degree to which monetary expenses are important to the job hunt raises a variety of new questions on how to best design a UI system. For instance, does assuming that job search is costly, rather than free, suggest that benefits should be higher early in an unemployment spell so job seekers can purchase an initial stock of the items that are required for a successful job search? Or, alternatively, should UI benefits be low so workers are incentivized to self-insure against an employment shock by maintaining a stock of these goods and services while employed? Finally, should benefits be higher for the long-term unemployed so they may continue to have the financial resources to find a job even after they

2

J. Schwartz / Labour Economics 36 (2015) 1–17

have exhausted their own savings? To answer these questions I investigate an economy where search reduces utility, both because it is unpleasant and because it diverts financial resources from consumption. To explore the optimal design of UI, this paper develops a search model where workers' actions may be hidden from the government along three dimensions: (1) the intangible search effort they exert, (2) the financial resources workers devote to search, and (3) the degree workers use precautionary savings to self-insure against an employment shock. The first hidden action, the non-monetary, intangible effort exerted by the unemployed, I refer to simply as search effort. Shavell and Weiss' (1979) and Hopenhayn and Nicolini (1997) seminal works on optimal unemployment insurance focus on the moral hazard problem that arises from not being able to observe search effort. Shavell and Weiss (1979) model maximizes worker utility subject to a fixed budget, while Hopenhayn and Nicolini (1997) use a recursive contract approach to determine the cost minimizing UI benefit and employment tax contingent on a worker's employment history. Both studies find that a UI benefit that declines with unemployment duration extracts a second best level of search effort from forward looking workers that wish to avoid a decline in their consumption. While these important papers, and many extensions to their models by other researchers, establish the optimal form of UI when workers must expend effort, they ignore the monetary costs of the search process. In this paper I argue that search effort alone is an incomplete picture of the job search process. Finding a job also requires additional goods and services that increase the probability of obtaining employment, but do not increase the utility of the unemployed. For instance, locating job opportunities is not just unpleasant, but may involve transportation costs to inquire if positions are available. Filling out on-line applications cannot be done with effort alone, but one needs computing resources. In addition, moving to where jobs are more plentiful and obtaining certifications of technical expertise that signals one's qualifications are large monetary expenditures that improve the chances of finding employment. These purchases may differ by socio-economic status. At the upper end of the wage distribution job searchers may pay for resume writing assistance, the help of recruiters, or networking opportunities, such as flying to professional conferences. At the lower end of the wage distribution the long-term unemployed may struggle to maintain professional attire, and email access, which are near necessities for a successful job search. Given the variety of goods and services needed for a successful job search, the associated monetary costs can be a substantial portion of the rather small average weekly UI benefit in the United States of just $291.1 Given these significant expenses I model search capital as a second action hidden from the government. While programs like Supplementary Nutritional Assistance Program (Food Stamps) suggest that governments may be able to monitor the purchases of beneficiaries, it is likely to be difficult to observe if an item is being used for a job search. For instance, a worker may fly to New York for a conference vital for her job search or simply for vacation, someone may move to North Dakota because jobs are more plentiful or to be close to family, and someone may use the internet to search help wanted ads or just for entertainment. Due to this ambiguity I assume that search capital is unobservable, but also explore cases where the government can monitor search capital in the sensitivity analysis sections of this paper. The idea of search capital has thus far been largely ignored by the literature.2 One exception is (Hassler and Mora, 2002), where workers may be short or long-term unemployed, search costs money, and workers may make an additional discrete investment to improve their 1 The average weekly benefit amount reported by the Department of Labor for the first quarter of 2011. 2 Note, Carrillo-Tudela and Smith (2012) recently use the term search capital to refer to the number of past contacts a job searcher has accumulated.

exit rate from unemployment. Workers vary in the cost of this investment which gives rise to an adverse selection problem. Workers with a high cost of investment desire greater insurance against long-term spells. In order to incentivize low cost workers to make these investments and provide insurance to high cost workers, benefits are low early in the spell and rise for the long-term unemployed. While this paper's focus is on the moral hazard problem, it also builds upon Hassler and Mora (2002) by allowing for monetary costs of search as well as search effort. Further, unlike Hassler and Mora (2002), I allow for more than two periods of unemployment, which allows for the possibility of a non-monotonic optimal benefit schedule. The last hidden action is private savings, which has long been shown to significantly affect the optimal UI policy. Flemming (1978) determines that an optimal replacement rate, provided for an infinite duration, goes from under 20% with perfect capital markets to over 70% when capital markets are nonexistent. Wang and Williamson (2002) find that it is optimal for benefits to be low in the first quarter of unemployment when workers' assets are high and then increase to provide insurance as assets are exhausted. Late in the unemployment spell UI should fall to provide incentives to search. Hansen and Imrohoroglu (1992) frame the moral hazard question in terms of the government's inability to monitor whether job offers and job continuations are accepted or rejected. Abdulkadiroglu et al. (2002) extend Hansen and Imrohoroglu (1992) model to allow for hidden savings. Abdulkadiroglu et al. (2002) find that it is optimal to pay large benefits upfront, which can be saved and used throughout the spell, and then keep benefits low while workers exhaust their savings. Benefits should then increase for the long-term unemployed to provide insurance for those with no remaining savings. Lentz (2009) estimates an empirical model of job search to determine the optimal UI replacement rate for Denmark where the duration of UI is nearly unlimited. The author estimates that the optimal replacement rate is between 43% and 80%. These studies suggest that savings can significantly influence the optimal UI schedule. Since each of these actions are unobservable, a moral hazard problem arises where workers may provide a sub-optimal level of search effort, purchase too little search investment and accumulate too little precautionary savings, than the government would wish under a full insurance system. As a result, the UI program must play the role of both insuring workers against an employment shock, as well as incentivizing these behaviors. The government can do this by varying the timing of benefits based on a worker's current unemployment history, subject to a balanced budget constraint and the optimizing behavior of workers. I develop, and numerically simulate, a search model under several scenarios: Scenario 1 allows for only search effort, Scenario 2 allows for search effort and search capital, Scenario 3 allows for search effort and private savings, and Scenario 4 allows for all three of these actions. The traditional wisdom of Shavell and Weiss (1979) and Hopenhayn and Nicolini (1997) holds for Scenarios 1 and 2 where it is optimal for benefits to fall throughout an unemployment spell. However, when relaxing the assumption that a job search is free, it is optimal to provide a much larger benefit at the beginning of an unemployment spell, so workers can purchase an initial stock of search capital. Allowing for precautionary savings suggests a different policy recommendation. Here, rather than more benefits for the newly unemployed, assuming that job search has monetary costs implies more benefits for the long-term unemployed, who are in danger of exhausting their savings. I also find that the welfare improvement that occurs from moving to an optimal UI system is substantially higher, when one assumes that searching for work requires financial resources. Sensitivity analysis shows that these conclusions are subject to how quickly search capital depreciates and the degree of risk aversion. If search capital depreciates slowly workers are willing to self-insure against unemployment by accumulating search capital while employed.

J. Schwartz / Labour Economics 36 (2015) 1–17

As a result, to encourage this behavior, UI benefits should be lower than in cases where search capital depreciates at a moderate or significant rate. Additionally, lower risk aversion implies a lower value for insurance and, consequently UI generosity should decrease with the coefficient of risk aversion. The remainder of this paper is organized as follows. In the next section I describe an economy where job search involves both effort and money. This is followed a brief discussion of how UI influences the job finding rate, in Section 3. In Section 4 I discuss the calibration strategy. Section 5 presents the results for the economy without savings, and Section 6 presents results with savings. Section 7 examines the welfare effects of moving to an optimal UI policy and finally, Section 8 concludes. 2. The economy To understand how search capital influences the optimal UI program, I develop a dynamic discrete time model of the savings, purchasing, and effort behavior of workers. The model accommodates each of the four scenarios the introduction describes, depending on various parameter restrictions. The economy consists of a continuum of utility maximizing workers, normalized to a measure of one. Workers are infinitely lived and exante identical, but differ ex-post as a result of random employment shocks. Agents maximize the expected value of time separable discounted utility: E

∞ X

βt uðct ; et Þ

ð1Þ

t¼0

where β is a common subjective discount factor and ct and et are consumption and effort levels at time period t. Utility is given by constant relative risk aversion utility function over consumption, and e enters the utility function linearly: cð1−σ Þ −e if σ ≠1 uðc; eÞ ¼ 1−σ uðc; eÞ ¼ logðcÞ−e if σ ¼ 1:

borrow. Denoting the stock of savings as a, the law of motion for savings is as follows: 0

a ¼ a þ y−i−c

y¼



w−τ if d ¼ 0 : bðdÞ if d ¼ 1; …; D

ð4Þ

Employed workers produce, w, of which a portion, τ, is paid to the government to fund the UI program. When workers are unemployed they receive a benefit, b(d), that is conditional on the current unemployment state. While those that investigate optimal unemployment insurance using the recursive contract approach, such as Hopenhayn and Nicolini (1997) and Pavoni (2007), allow benefits to depend on an individual's entire employment history, the restriction that UI is contingent only on the current length of the unemployment, similar to Abdulkadiroglu et al. (2002), is required for tractability. Workers have perfect foresight and a stochastic process governs the transitions into and out of employment. The employed can exert effort in order to retain their position. The probability of a worker continuing employment is given by q(e) with qe N 0 and qee b 0. The unemployed can influence their probability of finding work, p(k′,e), through the accumulation of search capital, k′, and by exerting search effort. The impact of search effort on the probability of finding a job conforms to p e(k′,e) N 0 and p ee (k′,e) b 0. For the purposes of this paper search capital has the following primary characteristics: (1) it requires monetary expenditures, (2) search capital has a strictly positive impact on the job finding rate probability (pk′(k′,e) N 0), (3) it has no direct impact on utility, and (4) it has no impact on human capital dw ¼ 0. dk0 The law of motion for search capital is: 0

Similar to Wang and Williamson (2002), employed workers exert effort to maintain employment, and as is standard throughout much of the UI literature the unemployed's effort impacts the probability of finding work.3 For convenience, in the remainder of this section I drop the t subscripts in favor of indicating a variable x's value next period as x′. I categorize each worker into D + 1 states. The first, d = 0, indicates the worker is employed. States d = 1 to D − 1 correspond to workers unemployed for d periods, and state D workers are unemployed D or more periods. Production can be seamlessly transformed into consumption, search investment or stored. Saved production earns no interest, and can also be freely converted into search investment or consumption in future periods. Workers do not have access to capital markets and cannot

ð3Þ

where y denotes disposable income and i, search investment. Disposable income is:

k ¼ kð1−δÞ þ i ð2Þ

3

ð5Þ

where the depreciation rate of search capital is δ. The depreciation rate allows one to consider a range of types of search capital. For instance, durable items may include certifications that signal qualifications, professional attire, computing and telecommunications equipment, resume assistance, or even moving costs that put workers in a labor market with better job prospects. For groups in which these items are important a low value of δ may apply. In contrast, non-durable items, such as transportation costs, childcare services, and internet and phone service, may also assist in the job search. For individuals in which these items are important to the job search, a high depreciation rate may be appropriate. Intermittent values of δ might be applicable for groups that require a mixture of durable and non-durable items for their job search. Denoting Vu and Ve as the discounted utility for the unemployed and employed, the maximization problem facing the unemployed can be written as a dynamic programming problem. The bellman equation is uðc; eÞ V u ðd; a; kÞ ¼ max a0 ;k0 ;e   0   0   0  e 0 0  þ β p k ; e V 0; a0 ; k þ 1−p k ; e V u d ; a0 ; k

ð6Þ

3

Note that others have chosen to model search effort as a time cost which reduces leisure. However, using the American Time Use Survey, Aguiar and Hurst (2008) show that the actual time spent on the job search is quite small, just 1.4 h a week for nonemployed males. As a result, I view these search activities as unpleasant and a direct reduction of utility. Modeling the effort of the employed provides reasonable bounds on what the optimal UI benefit should be at the beginning of an unemployment spell. This is because of what Atkinson and Micklewright (1991) call the “entitlement effect”, the theoretical possibility that workers increase their job search effort when UI becomes more generous because greater insurance against unemployment makes employment more valuable. This effect alone suggests it is optimal to pay implausibly large benefits for the newly employed. Modeling the effort choice of the unemployed mitigates the entitlement effect since a large benefit early in the unemployment spell would decrease effort while employed, resulting in more unemployment.

subject to Eqs. (3) and (5) and non-negativity constraints, e ≥ 0, k′ ≥ 0 and c ≥ 0. I also assume that the value of UI arises from imperfect capital markets that require 0 ≤ a ≤ ā. The inability to borrow signifies that workers cannot perfectly self-insure, and suggests that government provided insurance will be welfare improving. While I assume that there is a maximum amount of accumulated savings, in scenarios where workers may save, ā is set high enough to be inconsequential. In scenarios where the possibility of precautionary savings is excluded ā is set to zero. Similarly, k is bounded above by k which is also chosen to be high enough to be immaterial to the results.

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J. Schwartz / Labour Economics 36 (2015) 1–17

Eq. (6) illustrates the timing of the model. Unemployed workers enter each period with a stock of search capital and savings. During the period they decide how much search effort to exert and how much search investment to purchase, which determines the probability of finding work. Simultaneously, workers choose the level of savings next period, a′, which, together with the choice of k′, and y, determine consumption. At the end of the period the uncertainty in the labor market resolves itself and the unemployed either transition to employment, with d′ = 0, or advance to the next unemployment state, with d′ = min(d + 1,D). Employed workers derive utility from consumption, disutility from effort devoted to work, and also decide whether or not to self-insure against a job loss by either accumulating precautionary savings or search capital which decreases the expected duration of future unemployment spells. The employed's problem is: uðc; eÞ V e ð0; a; kÞ ¼ max 0 a0 ;k ;e    0 0  þ β ð1−qðeÞÞV u 1; a0 ; k þ qðeÞV e 0; a0 ; k

ð7Þ

0

subject to Eqs. (3) and (5) and 0 b a′ b ā and 0bk bk and non-negativity constraints on c and i. With probability 1 − q(e), workers separate from productive employment and enter the first state of unemployment, d = 1, and with probability q(e) workers maintain employment. Definition: Collecting state variables, (d,a,k) into vector w, a stable equilibrium for a given benefit schedule b(d) is a set of policy functions c(w),a′(w), k′(w), and e(w) as well as a time invariant distribution of workers across states, λ(ω), and tax rate, τ, that satisfy: 1. A balanced budget constraint ∫ λð0; a; kÞτ ¼ ∫ d ≥ 1;a;k λðd; a; kÞbðdÞ 2. The goods market clearsa;k such that w∫ a,k λ(0, a, k) = ∫ωλ(ω)[c(ω) + i(ω) + a′(ω) − a(ω)] 3. A stable distribution across states that solves:

 0 0 0 d ;a ;k ¼

8Z > > > > > > Z > > > > > > > > >
ða;kÞ∈Ω

qð0; a; kÞλð0; a; kÞ þ

Z

Z d≥1

0

ða;kÞ∈Ω

pðd; a; kÞλðd; a; kÞ if d ¼0

0

ða;kÞ∈Ω

ð1−qð0; a; kÞÞλð0; a; kÞ if d ¼1   0  1−p d−1; a; k λðd−1; a; kÞ if d ¼ 2; …; D−1

> > Z ða;kÞ∈Ω > >    > 0 > > 1−p D−1; a; k ÞλðD−1; a; k > > > ð a;k Þ∈Ω > Z >   > > 0  > 1−p D; a; k λðD; a; kÞ if d ¼ D :þ ða;kÞ∈Ω

ð8Þ where Ω(a′, k′) = {(a, x) : a′ = a′(d, a, k) and k′ = k′(d, a, k)} and defines the transition rule from (a,k) to (a′,k′) and p(d,a,k′) = p(e(d,a,k),k′(d,a,k)). The first line of Eq. (8) states that q(0,a,k) of the employed last period will continue to be employed and p(e,k′) from each unemployment state transition to employment. The second line indicates that a proportion 1 − q(0,a,k′) of the employed will become newly unemployed. For states d = 2,…,D − 1 a proportion (1 − p(d,a,k)) will continue to the next state of unemployment. Finally, the measure in state d = D, which covers those unemployed D periods or more, is determined by those that were in state D and D − 1 last period, but did not find work. A benevolent government maximizes a utilitarian social welfare function of the form: Z Z Z Z Z λð0; a; kÞV e ða; kÞ þ λðd; a; kÞV U ðd; a; kÞ ð9Þ SW ¼ max bðdÞ;τ

a

k

dN0 a

k

subject to the definition of a stable equilibrium. The government's problem is to maximize Eq. (9) by choosing the level of benefits for each state of unemployment and the tax rate, τ. While the complexity of the model makes analytical results difficult, the next section provides

some intuition on how the choice of the benefit system affects the exit rate from unemployed in a somewhat simpler framework.

3. UI's influence on the job finding rate This section provides some basic intuition on how unemployment benefits affect the exit rate from unemployment. To accomplish this I need to use a simplified framework which uses the following additional assumptions: 1. All workers begin life unemployed and can search for work up to two periods. 2. Employment is permanent (q = 1) and consequently there is no motivation to accumulate search capital while employed. 3. Workers do not have access to credit markets, ā = 0. 4. The job finding rate is Cobb–Douglas p(k′, e) = k′γe1− γ. These assumptions imply that the bellman equation for d = 1 is given by Eq. (6), and the second period Bellman equation can be 0 written as V u2 ¼ maxk0 ;e uðcÞ−e þ βpðk ; eÞV e , where Ve can be taken as a parameter. I summarize the comparative statics in three propositions and the proofs for each can be found in Appendix A. I use the phrase “almost certain” to recognize the proofs require that the partial of the first order condition for the search effort in the first period with respect to search capital in the first period, be positive. In Appendix A I argue that this condition is very likely to hold for all possible parameter values and will hold with certain for some level of δ. Proposition 1. An increase in benefits in the first period of unemployment almost certainly leads to an increase in search effort, search capital and the job finding rates in periods one and two. The intuition behind Proposition 1 is straight forward. More resources in the first period allow one to purchase more search capital. The marginal benefit of search effort is made up of two parts, the marginal effect of increasing the probability of finding work next period, and the future gain one achieves when transitioning to employment. Since both of these components are unaffected by contemporaneous UI benefits, b(1) does not directly impact the search effort decision. However, the additional purchases of search capital does increase the marginal effect of search effort on the job finding rate, and, as a result, it is optimal for workers to increase search effort. The stock of search capital at the end of the second period also increases with more generous UI benefits in the first period, since workers enter the period with more search capital. Again, greater search capital increases the productivity of search effort, so it is optimal for workers to exert more search effort in the second period as well. Since inputs into the job finding rate all increase with greater benefits in the first period of unemployment, the government can positively influence the job finding rate throughout an unemployment spell by increasing benefits for the newly unemployed. Proposition 2. An increase in benefits in the second period almost certainly leads to a decrease in search effort and search capital for the unemployed in this state. Increasing UI benefits in the second period makes continuing to be unemployed less of a burden. As a result, forward-looking workers in the first period of unemployment will decrease their search effort, an effect that is common throughout the literature. Similarly, workers are willing to sacrifice less consumption in favor of search capital in the first period, since the cost of continuing to be unemployed is lower. Thus, to encourage a higher job finding rate in the first period the government may wish to lower UI benefits in subsequent periods of unemployment.

J. Schwartz / Labour Economics 36 (2015) 1–17

Proposition 3. An increase in benefits in the second period has an ambiguous effect on search effort and capital in the second period. Both ∂k1 1 Nð− ∂b Þ. will almost certainly increase if ð1−δÞ 2

The effect of period two UI benefits depends on the degree that search capital purchased in the first period falls is offset by the higher purchases of new search capital in the second period. If the negative effect of b(2) on purchases of search capital in the first period is sufficiently large, then additional purchases of search capital made in period two will not lead to higher levels of search capital at the end of period two. In addition, if depreciation of search capital is high, then search capital in period two is more independent of search capital purchased in period one. As a result, the increase in new purchases of search capital in period two is more likely to offset any reductions of purchases in the first period. Search effort in the second period moves in the same direction as search capital, again since additions to search capital makes search effort more productive. In this simple framework increasing benefits early in an unemployment spell will lead to a higher probability of finding work throughout an unemployment spell. More benefits late in the unemployment spell will decrease the probability of finding work for the newly unemployed, but could potentially increase the job finding rate for the long-term unemployed. Using numerical methods the remainder of the paper explores the full model the last section describes. 4. Methodology I calibrate the model using a period length of one quarter and set β = 0.951/4, a fairly standard value in the literature. I consider ten unemployment states, D = 10. In contrast to last section's simple framework, for the numerical experiments I ensure p(k′, e) ∈ [0, 1] by using a function similar to that of Hopenhayn and Nicolini (1997) who use the form:    0   p ¼ 1− exp −υ k ; e

ð10Þ

 0  0γ υ k ; e ¼ rk e1−γ

ð11Þ

where υ can be thought of search units produced by inputs search effort and capital. The model accommodates scenarios without search capital by setting γ = 0. Similarly, the functional form for the job retention rate is q = 1 − exp(−ρe). Parameterizing the model involves choosing values for r, σ, γ, ρ, w, δ, ā and k. I start by normalizing the value of production, w, to one. Next, I set σ = 1, consistent with the values Fredriksson and Holmlund (2001) and Cahuc and Lehmann (2000) use. In addition, I set the upper bounds on personal savings and search capital such that no worker in the simulations reaches these bounds. The procedure to accomplish this is discussed in more detail in Appendix B. The parameters that directly relate to search capital, δ and γ are particularly uncertain since estimates of how much search capital is purchased by the unemployed are limited. Additionally, there is likely a large range of plausible figures that may apply to individuals depending on their socio-economic status and occupation. One of the only estimates available is from Stephenson (1976)), who finds that white youth use 25% of their income to find work. The estimate encompasses expenditures on transportation, applications costs, employment agency fees and interview attire. Unfortunately, a review of the literature did not uncover more recent rigorous estimates of the costs of a job search. The consumer expenditure survey (CEX) provides some expenditure figures for the unemployed on basic items one would expect to be used in a job search. According to the CEX, telephone service accounts for 6% of the average quarterly UI benefit of $4030 that is reported by the Department of Labor. Additionally, apparel accounts for 2%, internet service 7%, transportation 43%, and education 4%. Thus, spending on the basic goods

5

and services needed for a job search could potentially be as high as 62% of a quarterly UI benefit. However, it is unlikely 100% of these expenditures, particularly transportation, relate solely to a job search. Assuming that, or 31%, do relate to a job search provides an estimate close to Stephenson (1976). While these basic expenses may provide a rough idea of average spending, it may vary greatly by individual. For instance, individuals in regions where jobs are scarce may be more likely to move. Ludwig and Raphael (2010) reports the average cost of moving is $2600, or 65% of a quarter's UI benefit. Others may need a certificate attesting a specific qualification. Porter (2013) reports an average certificate program could amount to 169% of the quarterly UI benefit. Further, services like resume consultation may be more important to those seeking management level occupations and a session with a resume consultant may amount to 6% of a quarter's benefit (CNNMoney, 2001). With this large range of possibilities for spending on search capital, it is extremely difficult to pin down exact values for δ and γ. Consequently, I begin by using intermediate values. I set δ to 0.5, and γ to 0.25, and refer to these as the baseline parameters. These values result in a plausible percentage of average search capital purchased by the unemployed of 30% for Scenario 2 when calibrated to the current U.S. system, consistent with Stephenson (1976) estimates and my own calculation based on expenditure data. For consistency, I use the same parameters for the case with savings, where the amount of search capital purchased as a percentage of the UI benefits is considerably higher, 56%. Individuals that have meaningful savings likely purchase more expensive items, such as resume assistance, recruiting services, or conduct a national search, and use some of their own wealth to fund consumption and search capital while unemployed. Consequently, one would expect this group to devote more resources to search capital. Given the uncertainty surrounding these parameters, I also conduct an extensive sensitivity analysis that includes three different values for δ and setting γ = 0.10, which results in much lower search capital expenditures.4 The remaining parameters, r and ρ are calibrated to meet two targets. The first is an unemployment rate of 6.3%. This is the average unemployment rate for the United States as reported by the Bureau of Labor Statistics for the 2003–2010 period. The second is the percentage unemployed greater than twelve months, which the OECD reports as 14.0% for the United States over the same period. This period is chosen because it approximates about one business cycle and a longer period for U.S. data is made difficult by a change in the Current Population Survey in 1994. I ensure the model replicates these targets with the current U.S. unemployment insurance system, which has a UI replacement rate of 0.462 for the first six months5 and a social assistance (SA) replacement rate of 0.17 thereafter, as Wang and Williamson (1996) report. I then adjust r and ρ until the model reaches the two calibration targets. I use this procedure for each scenario, along with the baseline parameters, such that the calibration targets hold, but r and ρ are allowed to vary across scenarios. Table 1 presents the parameter values. To determine the benefit schedule that maximizes the social welfare function, I use value function iteration to approximate continuous functions of Ve(d, a, k) and Vu(d, a, k) and policy functions x(d, a, k), k′(d, a, k) and a′(d, a, k). I then simulate the behavior of a sample of agents over several periods until the distribution, λ(d, a, k), is stable. Given this stable distribution SW can be determined as the sum of the discounted utility for each agent in the sample. I ensure a balanced budget by adjusting τ and repeating these steps until the government's budget is balanced. Given this procedure to determine SW, I use a combination of a standard

4 A sensitivity analysis, available from the author, was also conducted for the case with γ = 0.50. However, this results in search capital expenditures as a proportion of UI income that are probably at the edge of what one would think of as realistic and the effect of increasing γ can be inferred from comparing the γ = 0.10 and γ = 0.25 cases. As a result, I do not present results with γ = 0.50. 5 Department of Labor Data, 2010.

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J. Schwartz / Labour Economics 36 (2015) 1–17

Table 1 Calibrated parameters. Scenario

(1)

(2)

(3)

(4)

Search capital:

No

Yes

No

Yes

Private savings:

No

No

Yes

Yes

γ δ Calibrated parameters r ρ

0.000 –

0.250 0.500

0.000 –

0.250 0.500

0.189 5.628

0.503 6.175

0.199 6.526

0.403 6.977

gradient approach and pattern search algorithm to determine the benefit schedule that maximizes SW. More details can be found in Appendix B. 5. Results: UI policy without hidden savings 5.1. Baseline parameters In this section I examine two scenarios that do not allow for savings, Scenario 1, which is the standard assumption that a job search is free, and Scenario 2, which assumes search capital influences the job finding rate. Both cases use the baseline parameters and Table 2 presents the key results. The first column indicates the income for the U.S. UI system the calibration uses, which is also the consumption choice under the U.S. UI system for Scenario 1. The second set of columns presents the average end of period search capital and average consumption choices under the current UI system for Scenario 2. The third and fourth sets of columns present the same data for the optimal UI systems. Fig. 1 illustrates the job finding rates and UI benefits graphically. Under the U.S. UI system, in Scenario 2 workers use their income either for consumption or search investment. While it is plausible that the employed may choose positive amounts of search investment as way of insuring against future lengthy unemployment spells, this is not the case with this set of parameters. Since the average worker enters an unemployment spell with k near zero, workers use a substantial portion of the UI benefit to build a stock of search. After benefit exhaustion, the limited available financial resources results in a declining stock of search capital. On average, workers spend 30% of the initial UI benefit on search capital or 14% of the quarterly wage. Fig. 1 presents the job finding rate by period of unemployment for the two scenarios under the U.S. UI system. Standard models of unemployment insurance, such as Mortensen (1976), often look similar to

those for Scenario 1. The job finding rate increases up to the point of benefit exhaustion as workers increase search effort in order to avoid further loss in income. Katz and Meyer (1990) and Meyer (1990) provide evidence for such a pattern empirically and many of the models the optimal UI literature uses, such as those in Shavell and Weiss (1979) and Hopenhayn and Nicolini (1997), imply a similar pattern. Under the U.S. UI system, Scenario 2 represents a departure from the more common pattern of a non-decreasing job finding rate. While the job finding rate increases prior to UI exhaustion, it falls as workers cannot maintain the same level of search capital. Declining unemployment exit rates for the long-term unemployed are not unique to this paper. Imbens and Lynch (2006), Shimer (2008) and Kroft et al. (2012) find empirical evidence of such negative duration dependence. Reasons for this phenomenon include depreciation of human capital (Pissarides, 1992; Ljungqvist and Sargent, 1998), and statistical discrimination by employers (Kroft et al., 2012). A contribution this paper offers is a new mechanism, search capital, that both explains declining exit rates for the long-term unemployed and captures the empirical fact of increasing exit rates until UI exhaustion. In this context, if the government wishes to use UI benefits to increase the job finding rate they must consider the consequences the UI system has on a worker's stock of search capital. The optimal benefit for Scenario 1 (see Fig. 1(c)), which assumes search is free, mimics the results elsewhere in the literature. It is welfare maximizing for benefits to decline throughout the unemployment spell. The continual reduction in consumption incentivizes workers to exert search effort, which results in a continually increasing job finding rate, see Fig. 1(d). The optimal benefit schedule under Scenario 2 can be thought of as consisting of two parts. The first is a benefit schedule, nearly identical to that of Scenario 1, which falls with the duration of unemployment in order to incentivize search effort, and to ensure workers devote resources to maintaining their search capital. The second is a one time payment upon becoming unemployed, in excess of the benefit indicated in Scenario 1. Under the optimal UI schedule the stock of search capital in quarter one is twice that under the current UI system, which results in a higher job finding rate for almost every quarter of the unemployment spell than under the current UI benefit schedule. The one exception is in the second quarter where search effort and, consequently, the probability of finding a job, spikes prior to benefit exhaustion under the current UI system (see Figs. 2(b) and 1(d)). This is consistent with the analytical results presented in Section 3, which suggest that increasing benefits for the newly unemployed will increase the job finding rate during an unemployment spell. This result also calls into question the conventional

Table 2 Optimal UI without hidden savings: baseline parameters.

State Employed Quarter unemp. 1 2 3 4 5 6 7 8 9 10+ τ ī/b(1) ī/w a

Income

Search capital: Yes Savings: No Scenario (2)

(All scenarios)

Search capitala

Cons.

1−τ

0.01

0.98

0.46 0.46 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.02 – –

0.19 0.30 0.16 0.12 0.11 0.11 0.11 0.11 0.11 0.11

0.27 0.26 0.16 0.13 0.12 0.12 0.12 0.12 0.12 0.12

Measured as end of period stock.

0.02 0.30 0.14

Search capital: No Storage: No Scenario (1)

Search capital: Yes Storage: No Scenario (2)

Income = cons

Income

Search capitala

Cons.

0.96

0.94

0.01

0.94

0.69 0.50 0.38 0.31 0.24 0.20 0.16 0.13 0.08 0.04 0.04 – –

1.02 0.53 0.38 0.29 0.26 0.21 0.15 0.12 0.10 0.03

0.43 0.34 0.27 0.21 0.19 0.17 0.14 0.12 0.11 0.05 0.06 0.23 0.23

0.60 0.40 0.29 0.21 0.17 0.13 0.10 0.07 0.05 0.02

J. Schwartz / Labour Economics 36 (2015) 1–17 Scenario 1: Search Capital No, Savings No Scenario 2: Search Capital Yes, Savings No

UI Benefit

1.00 0.80

7

Job Finding Rate 0.60

Scenario 1: Search Capital No, Savings No Scenario 2: Search Capital Yes, Savings No

0.50

0.60 0.40 0.40 0.30

0.20 0.00

0.20 1

2

3

4

5

6

7

8

9

10+

1

2

3

4

Quarter of Unemployment

(a)UI Benefit: U.S. UI System

UI Benefit

5

6

7

8

9

10+

Quarter of Unemployment

(b) Job Finding Rate: U.S. UI System

Scenario 1: Search Capital No, Savings No Scenario 2: Search Capital Yes, Savings No Job Finding Rate 0.60

1.00

Scenario 1: Search Capital No, Savings No Scenario 2: Search Capital Yes, Savings No

0.80 0.50

0.60 0.40

0.40

0.30

0.20 0.00 1

2

3

4

5

6

7

8

9

10+

Quarter of Unemployment

(c) UI Benefit: Optimal UI System

0.20 1

2

3

4

5

6

7

8

9

10+

Quarter of Unemployment

(d) Job Finding Rate: Optimal UI System

Fig. 1. UI benefits and job finding rates without hidden savings: baseline parameters.

wisdom that reducing UI benefits will always lead to greater unemployment exit rates. The job finding rate does fall in the final unemployment period. This is because the low benefit for the D = 10 state, while encouraging workers to exert search effort in earlier periods of unemployment, leaves workers unable to maintain the same level of search capital. Consequently, the job finding rate falls for extremely long unemployment spells. 5.2. Sensitivity to δ and γ Since the most uncertainty surrounds γ and δ, this section provides results for six alternative sets of parameters. Specifically, I explore how sensitive the results are to the importance of search capital to a job search by using a lower value for γ, 0.10. I also look at alternatives to the durability of the search capital that might be purchased. The choices for different values of δ are in part motivated by desire to explore the circumstances in which workers may self-insure against long-unemployment spells by purchasing search capital while employed. Individuals make such decisions by examining the return to search capital relative to the subjective discount rate. As a result, for a low level of depreciation I use δ = 1 − β and for symmetry, for a high level of depreciation case, I set δ = β. A moderate level of depreciation is given by the baseline value of 0.50. I present all six combinations of these two parameters in Table 3, and Fig. 2. While the different combinations can be thought of as alternate possibilities for economy wide parameters, one may also think of them as all applicable, but for different demographic and occupational groups. These groups may differ in the importance of search capital to their job search and the durability of the search capital that they purchase.

The second section of Table 3 and Fig. 2 displays the sensitivity of the results to different levels of search capital depreciation, while holding γ at 0.25. For the high level of deprecation (δ = β), search capital must almost be completely replaced each period. This may be applicable to workers for whom non-durable goods and services, such as cell-phone service, transportation, daycare and internet access, are critical to the job search. Similar to δ = 0.50, workers are not willing to accumulate search capital when employed, but since search capital is less valuable, due to the high level of depreciation, average expenditures on search capital by the unemployed is just 16% of wages, 7 percentage points less than the baseline parameters. In the case where search capital depreciates quickly, it is optimal to provide a somewhat lower benefit in the first quarter, but greater benefits over the next several quarters. Large benefits early in a spell are less desirable because purchases of search capital will last just one quarter. Rather it is important to continue to provide relatively higher benefits in subsequent quarters so workers can replenish these non-durable goods and services. The case where depreciation is very low applies to groups that primarily require durable items for their job search, such as formal certifications of one's skills, proper interview attire, or moving to locations where jobs are plentiful. In this case, accumulating search capital while employed is a reasonable precaution to an employment shock, since it is likely to still be available if a worker suddenly loses their job. To incentivize this behavior, it is optimal to have lower UI benefits throughout an unemployment spell. Since the employed realize unemployment will mean a significant reduction in consumption, they are willing to accumulate search capital in the amount of 1.53 times quarterly wages to ensure a short unemployment spell and then not devote UI income to search capital while unemployed.

8

J. Schwartz / Labour Economics 36 (2015) 1–17 UI Benefit 1.00

Job Finding Rate 0.80

0.80

0.70 0.60

0.60

0.50 0.40

0.40

0.30 0.20

0.20

0.10 0.00

0.00 1

2

3

4

5

6

7

8

9

1

10+

2

3

Quarter of Unemployment

4

5

6

7

8

9

9

10+

10+

Quarter of Unemployment

(a) UI Benefit

(b) Job Finding Rate Job Finding

UI Benefit 1.00

Rate

0.80 0.70

0.80

0.60 0.60

0.50 0.40

0.40

0.30 0.20

0.20

0.10 0.00

0.00 1

2

3

4

5

6

7

8

9

10+

Quarter of Unemployment

(c) UI Benefit

1

2

3

4

5

6

7

8

Quarter of Unemployment

(d) Job Finding Rate Fig. 2. UI benefits and job finding rates sensitivity to γ and δ.

The first set of rows in Table 3 and graphs in Fig. 2 present results for the same values of δ, but a value of γ = 0.10. This may correspond to groups for which the role of search capital in a job search is more limited. The more minor role of search capital has a few main effects on the optimal UI benefit schedule. First, the optimal benefit schedules for the different depreciation rates converge, since the design of the unemployment insurance system is more driven by the need to address the moral hazard issues surrounding search effort. Additionally, the effect of a lower γ differs by depreciation level. Further, the level of benefits is much lower, 24% and 22% lower in the first period, for δ = 0.5 and δ = β, and the percentage of income that workers use to purchase search capital on average falls by about onethird. For δ = 1 − β, the benefit schedule is slightly more generous than with γ = 0.25. This is because the cost of greater benefits, in terms of a disincentivizing purchasing search capital while employed, is lower when search capital is less important to a job search. Indeed the optimal benefit schedule elicits an average stock of search capital that is about one-third of what it is when γ = 0.25. The sensitivity analysis in this section suggests a somewhat complex relationship between the optimal benefit schedule and δ and γ. For very low values of depreciation, an increase in γ indicates that lower benefits are optimal in order to encourage accumulating search capital while employed. For moderate and high levels of depreciation, higher levels of γ suggest that the government should provide more UI income to the unemployed so, upon losing one's job, they could afford to purchase more search capital.

5.3. Sensitivity to risk aversion This subsection explores how the results may vary with the level of risk aversion. Table 4 and Fig. 3 display the same set of statistics as in the

other subsections using the baseline parameters, where σ is set to 1.00, along with a lower level of risk aversion, σ = 0.50. The effect of the level of risk aversion on the optimal UI system is very intuitive. Less risk averse workers find insurance less valuable and consequently it is optimal to provide lower UI benefits as σ falls. The amount of search capital bought at the beginning of a UI spell falls somewhat with σ as smaller UI benefits are available. Also, as UI benefits fall with the length of unemployment, the stock of search capital falls less dramatically for the lower coefficient of risk aversion. This is because less risk averse workers are more willing to sacrifice current consumption for search capital as UI income falls. Finally, panel (b) of Fig. 3 shows that the less generous UI system for σ = 0.50, corresponds to lower probabilities of finding work. This is because less risk averse individuals will exert less effort to avoid future loss of income despite less generous UI benefits. 5.4. Sensitivity to ability to observe search effort Since, whether or not worker's search capital can be reasonably monitored by the government is debatable, exploring how the optimal UI system may change if search effort is observable is important. Table 5 and Fig. 4 present the results with the baseline parameters both assuming that the government cannot observe search capital purchases and that the government can monitor search capital and proscribe the level of search investment in each employment state. In comparison to the unobservable search capital assumption, when search capital is observable the benefit system should include slightly lower benefits for the newly unemployed, greater benefits for the long-term unemployed, and a search investment requirement that ensures greater search capital throughout the unemployment spell. The slightly lower UI benefit in the first

J. Schwartz / Labour Economics 36 (2015) 1–17

9

Table 3 Optimal UI without hidden savings: sensitivity to γ and δ. γ = 0.10, δ = 1 − β State Employed Quarter unemp. 1 2 3 4 5 6 7 8 9 10+ τ ī/b(1) ī/w

γ = 0.10, δ = 0.50

Search capital*

Consump.

Income

Search capital*

Consump.

Income

Search capital*

Consump.

0.95

0.56

0.94

0.94

0.01

0.94

0.95

0.00

0.95

0.58 0.44 0.32 0.24 0.17 0.14 0.11 0.07 0.03 0.01

0.55 0.55 0.54 0.54 0.53 0.52 0.51 0.51 0.51 0.49 0.06 0.00 0.00

0.58 0.44 0.32 0.24 0.17 0.14 0.11 0.07 0.03 0.01

0.78 0.47 0.32 0.26 0.21 0.17 0.12 0.09 0.06 0.02

0.15 0.12 0.08 0.07 0.06 0.05 0.04 0.03 0.03 0.02 0.06 0.14 0.11

0.63 0.42 0.29 0.23 0.19 0.15 0.11 0.08 0.05 0.02

0.73 0.51 0.36 0.28 0.22 0.18 0.14 0.09 0.07 0.03

0.09 0.08 0.06 0.05 0.04 0.04 0.03 0.02 0.02 0.01 0.06 0.11 0.08

0.64 0.43 0.30 0.23 0.17 0.14 0.11 0.07 0.05 0.02

γ = 0.25, δ = 1 − β State Employed Quarter 1 2 3 4 5 6 7 8 9 10+ τ ī/b(1) ī/w *

γ = 0.25, δ = 0.50

γ = 0.25, δ = β

Income

Search capital*

Consump.

Income

Search capital*

Consump.

Income

Search capital*

Consump.

0.94

1.53

0.92

0.94

0.01

0.94

0.94

0.00

0.94

0.56 0.43 0.31 0.23 0.18 0.12 0.09 0.06 0.03 0.00

1.51 1.49 1.47 1.45 1.43 1.41 1.38 1.37 1.36 1.32 0.06 0.00 0.00

0.56 0.43 0.31 0.23 0.18 0.12 0.09 0.06 0.03 0.00

1.02 0.53 0.38 0.29 0.26 0.21 0.15 0.12 0.10 0.03

0.43 0.34 0.27 0.21 0.19 0.17 0.14 0.12 0.11 0.05 0.06 0.23 0.23

0.60 0.40 0.29 0.21 0.17 0.13 0.10 0.07 0.05 0.02

0.93 0.70 0.48 0.37 0.27 0.22 0.16 0.11 0.06 0.00

0.27 0.25 0.19 0.17 0.13 0.12 0.10 0.08 0.04 0.00 0.06 0.17 0.16

0.66 0.45 0.29 0.20 0.14 0.10 0.06 0.03 0.01 0.00

Measured as end of period stock.

quarter of unemployment incentivizes effort while employed, which, unlike when search effort is unobservable, can be done without leading to a reduction in search capital that is purchased. After the first quarter it is optimal to provide greater benefits for those unemployed two quarters or more. Someone unemployed ten quarters would receive 15% greater benefits after the first quarter than the unobservable case. Finally, the proscribed spending on search capital results in a five percentage point increase in average spending on search capital, which is higher throughout the unemployment spell. This suggests that the optimal UI benefit schedule Table 4 Optimal UI without hidden savings: sensitivity to σ. σ = 0.50

Employed Quarter unemp. 1 2 3 4 5 6 7 8 9 10+ τ ī/b(1) ī/w

Search

6.1. Baseline parameters

Income

Capital*

Consump

Income

Capital*

Consump

0.95

0.01

0.95

0.94

0.01

0.94

0.85 0.41 0.25 0.22 0.18 0.16 0.10 0.11 0.08 0.04 0.06 0.24 0.20

0.40 0.34 0.26 0.24 0.22 0.21 0.16 0.16 0.14 0.08

0.46 0.26 0.16 0.11 0.08 0.06 0.04 0.03 0.02 0.01

1.02 0.53 0.38 0.29 0.26 0.21 0.15 0.12 0.10 0.03 0.06 0.23 0.23

0.43 0.34 0.27 0.21 0.19 0.17 0.14 0.12 0.11 0.05

0.60 0.40 0.29 0.21 0.17 0.13 0.10 0.07 0.05 0.02

Measured as end of period stock.

in the unobservable case only elicits a second best level of search capital. The job finding rates for the observable and unobservable search capital assumptions are very similar. This is because the additional benefits for the long-term unemployed are not entirely devoted to search capital. Without the additional moral hazard issue involving search capital, the government also provides for greater consumption for those unemployed four or more quarters. The higher search capital, which increases the job finding rate, is offset by somewhat lower search effort because of the higher level of consumption for the long-term unemployed. The result is a similar job finding rate to the unobservable case (see Fig. 5(b)), but greater insurance for the long-term unemployed. 6. Optimal UI policy with hidden savings

σ = 1.00

Search State

*

γ = 0.10, δ = β

Income

This section explores the impact of search capital on the optimal UI system when workers have access to capital markets. Similar to Table 2, Table 6 shows the results for Scenario 3 (savings, but no search capital), and Scenario 4 (savings and search capital). The first column indicates the income the U.S. UI system provides, which the calibration also uses. The second and third sets of columns present the results for Scenarios 3 and 4 under the U.S. UI system and the fourth and fifth sets of columns present the results for the optimal UI systems. Each of these sets of results indicate the average end of period savings and search capital, where relevant, as well as the consumption choices. Fig. 5 presents the UI benefit schedules and corresponding job finding rates graphically. In the context of Scenario 3, under the U.S. UI system, the model suggests that the average worker accumulates just over 70% of quarterly

10

J. Schwartz / Labour Economics 36 (2015) 1–17 UI Benefit

Job Finding Rate

1.00

0.80 0.80 0.60 0.60 0.40

0.40

0.20

0.20 0.00

0.00 1

2

3

4

5

6

7

8

9

10+

Quarter of Unemployment

1

2

3

4

5

6

7

8

9

10+

Quarter of Unemployment

(a)UI Benefit

(b)Job Finding Rate

Fig. 3. UI benefits and job finding rates sensitivity to σ.

wages as precautionary savings to prepare for the lower income available when unemployed. This savings is exhausted, on average, after four quarters of unemployment. Those unemployed longer must rely solely on government benefits. The use of savings allows workers to mitigate lost income and results in a smooth decline in consumption and a corresponding increase in the job finding rate (see Fig. 5(b)). Scenario 4 allows for both search capital and savings. In this scenario workers partially self-insure by accumulating 1.70 times quarterly wages while employed, significantly more than Scenario 3 since savings must also be used to fund search capital. Similar to Scenario 3, in Scenario 4 the unemployed on average exhaust their savings after quarter four. On average purchases of search capital amount to 57% of UI income or 23% of quarterly wages, substantially higher than Scenario 2. This conforms to the intuition that those with wealth can devote some of their savings towards a successful search. The job finding rate for Scenario 4 increases even after benefit exhaustion, but decreases for the long-term unemployed. In this scenario, the job finding rate starts to fall after four quarters of unemployment as workers near the exhaustion of their savings. In this setting, a quickly declining benefit schedule will enhance the incentives to exert search effort and adequately save, but may leave the long-term unemployed without the resources to conduct a job search. The optimal benefit schedule for Scenarios 3 and 4 can be thought of as being composed of two parts. The first is an up-front benefit for the newly unemployed which can be used or saved. The second is a stream of benefits that insures against long unemployment spells and occurs after a period of zero or very low benefits, when workers have exhausted much of their own resources.

Table 5 Optimal UI without hidden savings: sensitivity to the ability to monitor search effort. Search capital is

Search capital is

Unobservable State Employed Quarter unemp. 1 2 3 4 5 6 7 8 9 10+ τ ī/b(1) ī/w *

Observable

Income

Capital*

Consump.

Income

Capital*

Consump.

0.94

0.01

0.94

0.94

0.02

0.94

1.02 0.53 0.38 0.29 0.26 0.21 0.15 0.12 0.10 0.03 0.06 0.23 0.23

0.43 0.34 0.27 0.21 0.19 0.17 0.14 0.12 0.11 0.05

0.60 0.40 0.29 0.21 0.17 0.13 0.10 0.07 0.05 0.02

0.99 0.55 0.38 0.33 0.30 0.24 0.21 0.14 0.13 0.11 0.06 0.28 0.28

0.50 0.43 0.32 0.28 0.25 0.23 0.21 0.15 0.13 0.15

0.50 0.36 0.28 0.21 0.18 0.14 0.11 0.10 0.07 0.04

Measured as end of period stock.

Under the assumption that a job search is free, the benefit for the newly unemployed is 0.59 and the stream of benefits for the longterm unemployed begins in the fourth quarter of unemployment after workers have exhausted nearly all of the own savings. The two quarter gap between the initial benefit and benefits for the long-term unemployed incentivizes workers to accumulate precautionary savings, which is 71% higher for the unemployed than under the current U.S. UI system. The benefit in the fourth quarter of 0.42 declines thereafter to incentivize search effort. The benefit system results in a declining level of consumption throughout the unemployment spell and a rising job finding rate (see Fig. 5(d)). This is because workers begin to exert more effort to avoid the future decline in consumption that occurs as first savings are exhausted and UI income falls. The pattern of benefits is similar to that of Abdulkadiroglu et al. (2002). The authors' low monitoring scenarios suggest a relatively large upfront payment for the unemployed and followed by a decline in benefits to near zero. As in this paper, benefits increase in subsequent periods for the very long-term unemployed who exhaust their savings. These results do suggest that the benefits should further decline in later quarters of unemployment, which differs from Abdulkadiroglu et al. (2002). This is likely because they explore four separate periods of unemployment rather than ten, as is done in this paper. The results also differ from Lentz (2009), who estimates a single optimal replacement rate of between 60% and 80% for cases when the return to savings is zero, as it is in this paper. The initial UI benefit in this paper is at the lower end of the range in Lentz (2009) and benefits are lower than this range in subsequent quarters. The difference is likely due to the author's focus on the Danish economy and excluding UI's effect on the effort exerted while employed. The lower benefits in this paper are optimal, in part, to ensure that the employed workers exert adequate effort to retain their job. The optimal UI system for Scenario 4 suggests a similar UI benefit to Scenario 3 for the newly unemployed, but generally higher benefits for the long-term unemployed. This type of benefits encourages 32% higher savings than under the current UI system and close to twice the savings under the optimal UI system for Scenario 3. Scenario 3 and 4 differ in the fourth quarter where benefits are 28% lower in Scenario 4. This is because in Scenario 4 savings are five times that of Scenario 3 in this quarter. Additionally, the savings from a lower benefit in this quarter can be used to fund benefits that are, on average, 28% higher through quarters five to nine when workers need more financial resources to fund their job search. Also in Scenario 4, while search capital is somewhat lower under the optimal UI system early in a UI spell than the current UI system, it is significantly higher at longer durations of unemployment. The result is that the job finding rate increases through the first nine periods of unemployment in contrast to just the first four quarters, under the current U.S. system (see Fig. 6(b) and (d)). For those unemployed more than nine quarters the job finding rate does fall as it becomes difficult to maintain search capital. The results in this subsection indicate that

J. Schwartz / Labour Economics 36 (2015) 1–17 UI Benefit 1.00

Unobservable Search Capital Observable Search Capital

11 Unobservable Search Capital Observable Search Capital

Job Finding Rate 0.80

0.80 0.60

0.60 0.40

0.40

0.20

0.20

0.00

0.00 1

2

3

4

5

6

7

8

9

1

10+

2

3

4

5

6

7

8

9

10+

Quarter of Unemployment

Quarter of Unemployment

(a) UI Benefit

(b) Job Finding Rate

Fig. 4. Optimal UI without hidden savings: sensitivity to the ability to monitor search effort.

when allowing for savings benefits should be more generous for the longer-term unemployed than what is implied by assuming a job search is free, rather than more generous for the newly unemployed as is implied by the case without savings. 6.2. Sensitivity to δ and γ Table 7 and Fig. 6 present the results for the same sensitivity analysis to δ and γ that was done for the case without savings. Similar to the prior section, the second section of Table 7 and Fig. 6 examine the sensitivity of the optimal benefit system keeping γ at 0.25, but adjusting the depreciation rate. For the high depreciation case, workers need sufficient resources in order to replenish the stock of search capital almost anew each quarter. As a result, it is optimal to provide a higher level of benefits in the first two quarters, after which there is period of zero benefits for two quarters. In subsequent quarters, as savings begins to be significantly depleted, it is again optimal to have a higher stream of UI Benefit 0.80

Scenario 3: Search Capital No, Savings Yes Scenario 4: Search Capital Yes, Savings Yes

benefits so workers can still fund a successful search. An individual who was unemployed for ten quarters would receive 20% more benefits between quarters five to ten under the benefits system with δ = β, in comparison to δ = 0.5 in order to keep replenishing search capital. The optimal UI benefit schedule when depreciation is higher results in very similar average purchases of search capital and similar job finding rates. The job finding rate rises slightly more sharply than in the baseline parameters, since as benefits become smaller and savings exhausted, consumption falls considerably. To avoid this possibility workers exert greater search effort. The job finding rate falls significantly in the last period of unemployment as workers with long unemployment spells cannot devote resources to search capital. When depreciation is very low, search capital is more valuable, and workers are willing to accumulate search capital while employed. Similar to Scenario 2, to encourage this, UI benefits are more limited than with moderate and high depreciation rates. The initial benefit when depreciation is low is about half what it is when setting δ = 0.50. Similarly, Job Finding Rate 0.80

Scenario 3: Search Capital No, Savings Yes Scenario 4: Search Capital Yes, Savings Yes

0.70

0.60

0.60 0.40

0.50 0.40

0.20 0.30 0.20

0.00 1

2

3

4

5

6

7

8

9

1

10+

2

3

Quarter of Unemployment

4

(a) UI Benefit: Current U.S. UI System

6

7

8

9

10+

(b) Job Finding Rate: Current U.S. UI System Job Finding Rate 0.80

UI Benefit 0.80

5

Quarter of Unemployment

Scenario 3: Search Capital Yes, Savings No Scenario 4: Search Capital Yes, Savings Yes

Scenario 3: Search Capital No, Savings Yes Scenario 4: Search Capital Yes, Savings Yes 0.60

0.60

0.40 0.40 0.20

0.00 1

2

3

4

5

6

7

8

9

10+

Quarter of Unemployment

(c) UI Benefit: Optimal UI System

0.20 1

2

3

4

5

6

7

8

9

10+

Quarter of Unemployment

(d) Job Finding Rate: Optimal UI System

Fig. 5. UI benefits and job finding rate with hidden savings.

12

J. Schwartz / Labour Economics 36 (2015) 1–17

Table 6 Unemployment insurance with hidden savings: baseline parameters. State

Employed Qrtr. unemp. 1 2 3 4 5 6 7 8 9 10+ τ ī/b(1) ī/w a

U.S. UI income

1−τ 0.46 0.46 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17

Search capital: No Savings: Yes Scenario (3)

Search capital: Yes Storage: Yes Scenario (4)

Savingsa

Cons.

Search capitala

Savingsa

Cons.

Income

Savingsa

Cons.

Income

Search capitala

Savingsa

Cons.

0.71

0.96

0.01

1.70

0.94

0.98

1.21

0.96

0.98

0.01

2.23

0.94

0.54 0.47 0.21 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.02 – –

0.62 0.54 0.43 0.33 0.22 0.17 0.17 0.17 0.17 0.17

0.40 0.44 0.44 0.40 0.27 0.16 0.13 0.12 0.12 0.12

1.17 0.89 0.42 0.10 0.00 0.00 0.00 0.00 0.00 0.00

0.60 0.51 0.41 0.31 0.20 0.14 0.12 0.11 0.11 0.11

0.59 0.00 0.00 0.42 0.34 0.26 0.24 0.20 0.14 0.09 0.02 – –

1.18 0.59 0.08 0.05 0.02 0.01 0.00 0.00 0.00 0.00

0.62 0.59 0.52 0.45 0.36 0.28 0.25 0.20 0.14 0.09

0.59 0.04 0.00 0.33 0.39 0.33 0.28 0.24 0.22 0.08 0.02 0.40 0.16

0.40 0.28 0.43 0.36 0.35 0.31 0.26 0.22 0.21 0.09

1.85 1.32 0.49 0.25 0.12 0.04 0.01 0.00 0.00 0.00

0.60 0.50 0.54 0.43 0.35 0.27 0.20 0.15 0.11 0.06

Search capital: No Storage: Yes Scenario (3)

0.02 0.57 0.23

Search capital: Yes Storage: Yes Scenario (4)

Measured as the stock at end of period.

the stream of benefits for the long-term unemployed is also significantly lower. An individual unemployed ten quarters would receive 45% less benefits in the δ = 1 − β case than when δ = 0.50. As a result of the limited level of UI benefits workers accumulate, an average stock of search capital of 1.20 times quarterly wages while employed and while unemployed purchase search capital in the amount of just 8% of quarterly wages. Additionally, as a result of low UI benefits, workers accumulate savings while employed greater than for the other two levels of depreciation. Finally, the job finding rate increases throughout the unemployment spell even in the D = 10 state. This is because even though financial resources to purchase new search capital are limited in this

UI Benefit 0.80

state the level of search capital remains high because of the low level of depreciation. Lowering the importance of search capital by setting γ = 0.10 impacts the results in a few ways. First, savings among the employed is lower for all depreciation levels. Since search capital is less important to a job search, it is not as necessary to have funds on hand to purchase search capital if one becomes unemployed. Correspondingly, the amount of search capital purchased is generally lower for all depreciation levels, including the amount accumulated while employed in the low depreciation case. For all depreciation levels, the proportion of UI income and wages used to purchase search capital is significantly lower.

Job Finding Rate 0.80

0.60 0.60 0.40

0.40

0.20

0.20

0.00

0.00 1

2

3

4

5

6

7

8

9

10+

1

2

3

Quarter of Unemployment

4

5

6

7

8

9

10+

9

10+

Quarter of Unemployment

(b) Job Finding Rate

(a) UI Benefit UI Benefit 0.80

UI Benefit 0.80

0.60

0.60

0.40

0.40

0.20

0.20

0.00

0.00 1

2

3

4

5

6

7

8

Quarter of Unemployment

(c) UI Benefit

9

10+

1

2

3

4

5

6

7

8

Quarter of Unemployment

(d) Job Finding Rate Fig. 6. UI benefits and job finding rates sensitivity to γ and δ.

J. Schwartz / Labour Economics 36 (2015) 1–17

13

Table 7 Optimal UI with hidden savings: sensitivity to γ and δ. γ = 0.10, δ = 1 − β State Employed Quarter unemp. 1 2 3 4 5 6 7 8 9 10+ τ ī/b(1) ī/w

γ = 0.10, δ = 0.50

Search capital*

Savings*

Consump.

Income

Search capital*

Savings*

Consump.

Income

Search capital*

Savings*

Consump.

0.98

0.46

1.93

0.94

0.98

0.00

1.57

0.95

0.97

0.00

1.42

0.94

0.38 0.07 0.00 0.34 0.31 0.26 0.20 0.15 0.11 0.00 0.02 0.11 0.04

0.57 0.56 0.57 0.56 0.54 0.53 0.51 0.51 0.51 0.51

1.56 1.06 0.41 0.25 0.14 0.08 0.04 0.02 0.01 0.00

0.66 0.59 0.64 0.51 0.42 0.33 0.25 0.17 0.11 0.01

0.43 0.00 0.31 0.32 0.29 0.24 0.22 0.15 0.12 0.07 0.06 0.22 0.10

0.15 0.16 0.14 0.13 0.11 0.09 0.09 0.06 0.05 0.03

1.22 0.52 0.28 0.14 0.06 0.02 0.01 0.00 0.00 0.00

0.65 0.61 0.49 0.41 0.32 0.25 0.19 0.14 0.10 0.06

0.77 0.00 0.00 0.37 0.36 0.27 0.26 0.16 0.13 0.03 0.06 0.12 0.09

0.11 0.09 0.09 0.08 0.07 0.06 0.06 0.04 0.04 0.01

1.43 0.74 0.13 0.04 0.02 0.00 0.00 0.00 0.00 0.00

0.66 0.60 0.52 0.38 0.31 0.23 0.20 0.12 0.09 0.02

γ = 0.25, δ = 1 − β State Employed Quarter unemp. 1 2 3 4 5 6 7 8 9 10+ τ ī/b(1) ī/w *

γ = 0.10, δ = β

Income

γ = 0.25, δ = 0.50

γ = 0.25, δ = β

Income

Search capital*

Savings*

Consump.

Income

Search capital*

Savings*

Consump.

Income

Search capital*

Savings*

Consump.

0.98

1.20

2.88

0.92

0.98

0.01

2.23

0.94

0.98

0.00

2.57

0.94

0.28 0.00 0.00 0.27 0.23 0.22 0.18 0.13 0.16 0.06 0.02 0.27 0.08

1.44 1.44 1.42 1.36 1.29 1.24 1.20 1.17 1.13 1.10

2.30 1.74 1.12 0.90 0.68 0.53 0.39 0.26 0.19 0.09

0.66 0.57 0.63 0.55 0.48 0.42 0.35 0.28 0.24 0.15

0.59 0.04 0.00 0.33 0.39 0.33 0.28 0.24 0.22 0.08 0.02 0.40 0.24

0.40 0.28 0.43 0.36 0.35 0.31 0.26 0.22 0.21 0.09

1.85 1.32 0.49 0.25 0.12 0.04 0.01 0.00 0.00 0.00

0.60 0.50 0.54 0.43 0.35 0.27 0.20 0.15 0.11 0.06

0.65 0.17 0.00 0.00 0.47 0.42 0.34 0.28 0.28 0.06 0.02 0.36 0.23

0.25 0.23 0.18 0.27 0.25 0.24 0.22 0.17 0.18 0.04

2.38 1.82 1.23 0.46 0.28 0.16 0.08 0.03 0.03 0.01

0.59 0.51 0.41 0.50 0.39 0.31 0.23 0.15 0.12 0.03

Measured as end of period stock.

For the moderate and high levels of depreciation, UI benefits are generally less generous and the stream of benefits for the long-term unemployed occurs sooner. Similar to results for Scenario 2, since search capital is less important, there is less a need to provide funds for investment in search capital through the UI system. Benefits in the first two quarters of unemployment are 31% lower for the moderate depreciation case and 6% lower in the high depreciation case. In the high depreciation case search capital still needs to be replaced each period and, as a result, the sensitivity to a reduction in γ is smaller than in the moderate depreciation case. Since savings is more limited with γ = 0.10, it is exhausted quicker. As a result, it is optimal to begin the stream of benefits for the long-term unemployed a quarter sooner for both the moderate and high

depreciation cases. These benefits, however, are more limited, which is again because it is less critical for the UI system to provide the necessary funds for individuals to purchase search capital. In contrast, when depreciation is high, it is optimal to have higher UI benefits as γ falls. The total benefits for the first two quarters are almost 60% higher. Also, the stream of benefits for the long-term unemployed is generally higher. An individual unemployed ten quarters would receive 7% more benefits for quarters four through ten in comparison to when γ = 0.25. Fig. 7(b) presents the job finding rates for the different depreciation levels for when γ = 0.10. The job finding rate rises the fastest through an unemployment spell for the low depreciation case in comparison to higher levels of depreciation because search capital can be

UI Benefit 0.80

Job Finding Rate 0.80

0.60

0.60

0.40

0.40

0.20

0.20

0.00

0.00 1

2

3

4

5

6

7

8

Quarter of Unemployment

(a) UI Benefit

9

10+

1

2

3

4

5

6

7

8

Quarter of Unemployment

(b) Job Finding Rate

Fig. 7. UI benefits and job finding rates sensitivity to σ.

9

10+

14

J. Schwartz / Labour Economics 36 (2015) 1–17

Table 8 Optimal UI with hidden savings: sensitivity to σ. State

Employed Quarter unemp. 1 2 3 4 5 6 7 8 9 10+ τ ī/b(1) ī/w *

σ = 0.50

σ = 1.00

Income

Search capital*

Savings*

Consump.

Income

Search capital*

Savings*

Consump.

0.99

0.01

1.51

0.96

0.98

0.01

2.23

0.94

0.15 0.00 0.29 0.28 0.31 0.22 0.15 0.14 0.11 0.07 0.02 1.49 0.22

0.36 0.39 0.34 0.33 0.34 0.30 0.24 0.21 0.18 0.13

0.90 0.30 0.16 0.07 0.03 0.01 0.00 0.00 0.00 0.00

0.40 0.38 0.29 0.22 0.17 0.11 0.07 0.05 0.03 0.02

0.59 0.04 0.00 0.33 0.39 0.33 0.28 0.24 0.22 0.08 0.02 0.40 0.24

0.40 0.28 0.43 0.36 0.35 0.31 0.26 0.22 0.21 0.09

1.85 1.32 0.49 0.25 0.12 0.04 0.01 0.00 0.00 0.00

0.60 0.50 0.54 0.43 0.35 0.27 0.20 0.15 0.11 0.06

Measured as end of period stock.

maintained even as UI benefits fall. In addition, the job finding rate for the high depreciation case rises faster than with δ = 0.50. This is because for the high depreciation case more financial resources need to be devoted to search capital each period and consequently consumption is lower for the long-term unemployed which provides incentives for exerting search effort. When considering savings, increasing depreciation from δ = 0.50 to δ = β results in more generous benefits early in the unemployment spell and a stream of long-term benefits that begin later, but are more generous. In contrast, a low level of depreciation suggests more limited UI benefits to encourage accumulating search capital while employed. Similar to Scenario 2, lowering the importance of search capital to a job search generally decreases UI generosity for moderate and high levels of depreciation, and increases UI generosity for the low depreciation case. 6.3. Sensitivity to risk aversion Table 8 and Fig. 7 explores the sensitivity of the optimal UI benefit schedule to a lower degree of risk aversion. For the lower value of σ, workers are willing to accept more consumption volatility. As a result, workers save less and it is optimal to pay a lower UI benefit in the first period of unemployment. However, the combination of lower savings when becoming unemployed, and the lower initial UI benefit, means

savings is exhausted sooner. Consequently, UI benefits are significantly positive after just one quarter without benefits rather than two under the baseline parameters. In subsequent quarters, again because the value of insurance is lower, lower UI benefits are optimal. The job finding rate for the lower lever of risk aversion is much flatter than the baseline case. At σ = 0.50, since workers are willing to accept more consumption volatility, they exert less search effort. As a result, the job finding rate does not increase as much as when UI benefits fall for the long-term unemployed. Further, the job finding rate begins to fall slightly even in the seventh quarter of unemployment as workers' search capital begins to fall substantially when faced with less financial resources. 6.4. Sensitivity to observable search effort Table 9 and Fig. 8 present results using the baseline parameters under the assumption that search capital is observable as well as unobservable for comparison purposes. The initial UI benefit is not sensitive to this assumption. This is because although purchases of search capital can be set by the government, increasing the initial benefit could still result in less effort and savings while employed. After the initial quarter, benefits are more generous for the next six quarters. Since the government can mandate spending on search capital, it can ensure that a portion of the additional benefit is spent on the goods and services

Table 9 Optimal UI with hidden savings: sensitivity to unobservable effort assumption. State

Employed Quarter unemp. 1 2 3 4 5 6 7 8 9 10+ τ ī/b(1) ī/w *

Search capital: Yes

Search capital: No moral hazard

Storage: Yes Scenario (4)

Storage: Yes Scenario (4)

Income

Search capital*

Savings*

Cons.

Income

Search capital*

Savings*

Cons.

0.98

0.01

2.23

0.94

0.96

0.01

2.20

0.93

0.59 0.04 0.00 0.33 0.39 0.33 0.28 0.24 0.22 0.08 0.02 0.40 0.24

0.40 0.28 0.43 0.36 0.35 0.31 0.26 0.22 0.21 0.09

1.85 1.32 0.49 0.25 0.12 0.04 0.01 0.00 0.00 0.00

0.60 0.50 0.54 0.43 0.35 0.27 0.20 0.15 0.11 0.06

0.59 0.35 0.19 0.52 0.49 0.41 0.34 0.18 0.11 0.03 0.04 0.55 0.32

0.47 0.56 0.47 0.59 0.49 0.40 0.32 0.22 0.14 0.06

1.75 1.26 0.69 0.36 0.22 0.13 0.08 0.05 0.04 0.02

0.60 0.53 0.58 0.50 0.42 0.34 0.27 0.16 0.08 0.03

Measured as end of period stock.

J. Schwartz / Labour Economics 36 (2015) 1–17 Unobservable Search Capital Observable Search Capital

UI Benefit 0.80

15 Unobservable Search Capital Observable Search Capital

Job Finding Rate 0.80

0.60

0.60

0.40

0.40

0.20

0.20

0.00

0.00 1

2

3

4

5

6

7

8

9

1

10+

2

3

Quarter of Unemployment

4

5

6

7

8

9

10+

Quarter of Unemployment

(a) UI Benefit

(b) Job Finding Rate

Fig. 8. UI benefits and job finding rates sensitivity to unobservable search capital assumption.

necessary for a job search. While benefits are higher for quarters 2–8, the benefit schedule incentivizes search effort by having lower UI benefits for those unemployed more than eight quarters. Overall, expenditures on search capital are, on average 32% of wages, when mandated by the government, in comparison to just 24% when search capital is assumed to be unobservable.

Among the unemployed, consumption is far less volatile since some of the drop in income that occurs when benefits are exhausted can be mitigated by spending less on search capital. Under the optimal UI system, consumption volatility among all workers would fall by 0.03, as the drop in consumption upon becoming unemployed is just 36%. This is offset, however, by an increase in volatility among the unemployed of 0.10, since benefits fall more sharply than the current UI system. While in both Scenarios 1 and 2, the benefit of an optimal UI system is cushioning the consumption shock from losing one's job, in Scenario 2 the welfare improvement is nearly four times what one may conclude when assuming that a job search is free. As others in the literature find when workers have access to capital markets, as in Scenario 3, there is less of a role for UI to improve social welfare and assisting workers in smoothing consumption. The optimal UI system reduces consumption volatility among all workers and among the unemployed by very small amounts. These benefits are offset in part by the slightly lower employment rate. The result is an improvement in welfare of just 0.005. In contrast to Scenario 3, Scenario 4 improves welfare by reducing consumption volatility among the unemployed, since the optimal UI schedule includes a more generous stream of benefits for the longterm unemployed, which allows workers to continue to consume and purchase search capital after they have exhausted a significant amount of their own savings. In this case, the standard deviation falls by 0.02. This is offset in part by a small increase in the employment rate. The result is a welfare improvement that is almost three times as high as in Scenario 3, suggesting that the welfare improvements of UI are still significant even if workers do have access to capital markets when one assumes a job search involves monetary expenses.

7. Welfare improvements The improvement in welfare from moving to an optimal UI system can come from several sources. One is the ability to smooth consumption across states of employment, another is smoothing consumption during a long-unemployment spell, and finally increasing total employment. Table 10 summarizes these factors for the four scenarios using the baseline set of parameters. Table 10 presents the standard deviation of consumption for all agents and for just the unemployed. These figures are given under the current U.S. UI system and the optimal system. In addition, the table shows the employment rate under the optimal UI system and the increase, or decrease, from the calibration target. Finally, the last column indicates the welfare improvement from moving to the optimal system, which I measure as the percentage increase in consumption all agents would accept under the current U.S. UI system to be indifferent between it and the optimal system. The welfare improvement under Scenario 1 is primarily due to lower consumption volatility. Losing one's job reduces consumption by about one-half under the U.S. UI system, while under the optimal UI system the drop would be slightly more than 25%. As a result, the standard deviation of consumption among all workers falls from 0.16 to 0.14. This is offset by a slightly lower employment rate and more consumption volatility among the unemployed under the optimal UI system, for a total increase in welfare of 0.6%. Scenario 2 also improves welfare by lowering consumption volatility between the employed and unemployed states, but more dramatically than Scenario 1. In Scenario 2, under the U.S. system consumption drops by 72% as a proportion of the UI benefit is used for search capital.

8. Conclusion A long list of papers examines the optimal design of unemployment insurance. For the most part these studies focus on the need to exert search effort to find work, where the costs of this effort are either a direct reduction in utility or a loss of leisure. In this paper, I argue that in

Table 10 Welfare measures. Standard Deviation of Consump. All

Standard Deviation of Consump. Unemp

Scenario

Search Capital

Savings

Calibrated

Optimal

Calibrated

Optimal

Employment Rate

Change from Calibrated

Welfare Improvement

1 2 3 4

No Yes No Yes

No No Yes Yes

0.1552 0.1832 0.1243 0.1268

0.1421 0.1549 0.1227 0.1311

0.1419 0.0637 0.1517 0.1624

0.1789 0.1661 0.1458 0.1433

0.9181 0.9117 0.9301 0.9296

−0.0189 −0.0253 −0.0069 −0.0074

0.0057 0.0209 0.0045 0.0160

16

J. Schwartz / Labour Economics 36 (2015) 1–17

addition to this intangible search effort, workers may, and sometimes must, also purchase goods and services that assist in their job search. To capture the effect of these purchases I develop a simple, partial equilibrium search model where three actions of workers are hidden from the government: search effort, search capital and savings. In the first set of scenarios this paper explores workers do not have access to capital markets and, the standard recommendation of the literature, that benefits should decline during an unemployment spell, is upheld. Additionally, for workers whose search capital depreciates at a moderate level an additional benefit should be made to workers in the first quarter of unemployment in excess of what is suggested when one assumes a job search is free. For workers with savings, the UI system can be thought of as consisting of two parts regardless of whether a job search is free or not. The first is a one-time payment when becoming newly unemployed and the second is a stream of benefits that occurs after a few periods of near zero benefits which serves to insure individuals against unemployment spells that are long enough to significantly deplete one's personal savings. In the case where workers can save, assuming that there are monetary costs to a job search, the optimal UI benefit system should be more generous for the long-term unemployed than what is suggested by assuming a job search is free. Finally, while the welfare improvement from moving to the optimal UI system is often thought of as low when workers can access capital markets, when one assumes that a job search has monetary costs I find that the welfare improvements can still be significant. I find that the results are sensitive to assumptions on how quickly search capital depreciates and how important it is to the job search. For workers that require mostly durable goods for their job search, UI generosity should be more limited than when depreciation is higher in order to encourage accumulating search capital while employed. Additionally, when search capital is less important to a job search, benefits should be less generous for moderate and high levels of depreciation since it is less important for UI to fund purchases of search capital. However, when depreciation is low, a decline in the importance of search capital implies greater benefits, since the need to limit benefits to encourage accumulating search capital while employed is lower. Given the many different scenarios this paper evaluates, it is natural to ask which one might be the most appropriate policy to implement. This hinges on the role of savings, the importance of search capital and how quickly it depreciates. In terms of the prevalence of savings among the unemployed, Gruber (2001) finds that a substantial portion of the workforce (one-third) has just enough savings to cover 10% of their lost income during unemployment. This may suggest that in the absence of being able to observe who faces financial constraints, Scenario 2 might provide the best policy guidance. In regard to the importance of search capital and how quickly it depreciates, more research still needs to be done to determine the extent that search capital may influence the optimal UI benefit schedule. In particular, with little to no empirical work to rely upon, this paper was unable to pin down precise values for γ and δ, but instead provides a range of possibilities for these parameters. Future empirical studies may be able to shed light on which set of parameters this paper explores are closer to the truth and lead to better policy guidance. However, it may also be the case that each set of parameters may be plausible for different groups of workers. As a result, policymakers may consider alternative schemes for different groups of workers. Additionally, this paper explores a partial equilibrium model of the labor market with a fixed wage and ignores unemployment insurance's effect on job creation. Acemoglu and Shimer (1999) and Acemoglu and Shimer (2000) demonstrate one such effect where UI entices workers to take jobs with higher unemployment risk. As a result, firms are willing to make greater investments in these jobs, which result in higher overall output. This may suggest that the UI schemes this paper suggests are not generous enough to maximize output. In contrast, in

Fredriksson and Holmlund (2001) and Cahuc and Lehmann (2000), more generous UI, increases the opportunity costs of employment. Consequently, wages must be higher to attract workers to a firm. The resulting higher labor costs decrease job creation and suggest that a lower UI benefit, particularly at the beginning of an unemployment spell, might be optimal. If the increased labor costs were incorporated in this paper, it may imply less generous UI benefits. Unfortunately, including these effects was not computationally feasible and, as a result, is left to future work. Appendix A. Proofs of Propositions 1, 2 and 3 Proving propositions 1, 2 and 3 involves developing simple comparative statics using first order conditions for e1, k1, e2, and k2, where subscripts indicate periods one and two and in the case of search capital, k1 and k2 are end of period values. The first order conditions for each are given below:   ψe1 ¼ −1 þ βpk1 ðk1 ; e1 Þ V e −V u2 ðk1 Þ ¼ 0

ðA  1Þ

ψk1 ¼ −u0 ðc1 Þ     þ β pk1 ðk1 ; e1 Þ V e2 −V u2 ðk1 Þ þ ð1−pðk1 ; e1 ÞÞð1−δÞu0 ðc2 Þ ¼0 ðA  2Þ ψe2 ¼ −1 þ βpe2 ðk2 ; e2 ÞV e ¼ 0

ðA  3Þ

ψk2 ¼ −u0 ðc2 Þ þ βpk2 ðk2 ; e2 ÞV e ¼ 0:

ðA  4Þ

Using Eqs. (A-3) and (A-4), e2 and k2 can be expressed as follows:  1 e2 ¼ V e βð1−γÞ γ k2

ðA  5Þ

h 1−γ  1 i−1 σ k2 ¼ k1 ð1−δÞ þ b2 − γð1−γ Þ γ βV e γ :

ðA  6Þ 10

The propositions all rely on the condition that ψk1 e1 ¼ βp21 ðk ; e1 Þ 1

10

ðV e −V u2 ðk ÞÞ−βp2 ðk ; e1 Þu0 ðc2 Þð1−δÞN0. Using Eq. (A-2) to substitute for ðV e −V u2 ðk1 Þ one can show that this condition is met if:   0      1 u0 c1 N−β 1−2p k ; e1 ð1−δÞu0 c2 

ðA  7Þ

Note that k1 ∈ [0, b1]. The left hand side of Eq. (A-7) is increasing and convex ranging from a positive finite value when k1 = 0 to infinity when k1 = b1. The right side is an increasing function that ranges from a finite negative to a finite positive value. However, one cannot show that the right hand side is convex. As a result, while it is clear that at the extremes of the possible values for k1 this condition holds, one cannot eliminate the possibility that the right hand side does exceed the left for some range of k1. For Eq. (A-7) to be violated requires p(k1, e1) ≥ 0.5 which is not the case for the newly unemployed in any of the results this paper presents, further suggesting that Eq. (A-7) likely holds for reasonable parameter values. Finally, one could ensure this condition holds for all k1 by restricting the minimum level of possible δs. Proof of Proposition 1. Determining the comparative statics for b1 can be derived by applying Cramer's rule to the following:



3 2

∂e1 ψe e1 ψe1 k1 6 ∂b1 7 0 7¼ 6 1 : ″ 5 4 u ðc1 Þ ψk1 e1 ψk1 k1 ∂k1 ∂b1

ðA  8Þ

∂k1 ∂e1 Assuming concavity of the objective function, this gives ∂b N0 and ∂b N0. 1 1 ∂e2 ∂k2 N0 and ∂b N0 from Eqs. (A-5) and (A-6). It is straightforward that ∂b 1

1

J. Schwartz / Labour Economics 36 (2015) 1–17

Proof of Proposition 2. Apply Cramer's rule to the equation below ∂e1 ∂k1 b0 and ∂b b0. indicates that ∂b 2

2

3 2

∂e1 7 ψe1 e1 ψe1 k1 6 6 ∂b2 7 ψk1 e1 ψk1 k1 4 ∂k1 5 ∂b2  

βp k1;e1 u0 ðc2 Þ       ¼ : β p1 ðk1 ; e1 Þu0 ðc2 Þ− 1−p k1;e1 ð1−δÞu″ ðc2 Þ

ðA  9Þ

∂k1 1 Nð− ∂b Þ can be derived Proof of Proposition 3. The condition that ð1−δÞ 2 from take the derivative of k2 with respect to b2 using Eq. (A-6). The re∂e2 ∂k2 has the same sign as ∂b follows from Eq. (A-5). sult that ∂b 2

2

Appendix B. Numerical model solution and optimization algorithm Computing the optimal UI benefit schedule requires a means to calculate the social welfare function for a given set of benefits, and an algorithm for determining the set of UI benefits that maximize the social welfare function. First, to calculate the social welfare function, SW, I do the following: 1. I create a grid across k and a and choose an initial guess for the tax rate, τ that balances the government's budget constraint.6 2. Next, I approximate functions, Ve(0, a, k), and Vu(1, a, k), …, Vu(D, a, k) ga;kÞ; …; V u ðD; ga;kÞ by first initializing the apga;kÞV u ð1; k) with V e ð0; proximations with a guess at each grid point. Then I form continuous function across the state space using a cubic–spline interpolation. These functions are updated at each grid point using Eqs. (6) and (7). This process is repeated until convergence is reached.7 3. Once value functions converge I determine policy functions k ' (d, a, k), a ' (d, a, k) and e(d, a, k), by taking the optimal levels of k ', a ' and e at each grid point and interpolating using a cubic spline. 4. I then take a simulation approach to determine λ(d, a, k). I start by creating a data set of 200,000 workers with initial states drawn from a uniform distribution. Using policy functions k′(d, a, k), a ′(d, a, k) and e(d, a, k), I simulate search effort, search capital, savings and work history of these agents until the distribution is not informed by the initial state distribution.8 5. SW is then calculated using the simulated distribution λ(d, a, k), ga;kÞ and V u ð1; ga;kÞ… along with the interpolated value functions V e ðd; g u V ðD; a;kÞ. 6. Along with the value for SW, I determine the government's surplus using the simulated distribution λ(d ', a ', k '), the tax rate and the benefit schedule. 7. If the government's surplus is positive, τ is reduced, if it is negative, τ is increased, and the steps 2 and 7 are repeated. If the government's surplus is approximately zero, SW is determined. Given the steps above, I determine SW by alternating between gradient and pattern search algorithms to ensure a global maximum is reached. After each pattern search algorithm is run, the upper bounds of a and k are adjusted to ensure they are at least 25% higher than the values reached by any agent in the simulated sample.9

6

I use five grid points for k and a. To speed convergence I alternate between value function evaluations, where new optimal choices of a′, k′, and e are determined, and policy iterations where these choices are fixed. 8 I simulate for 35 quarters. 9 I run the algorithm first using less than the full 200,000 sample and allow for the government to choose UI benefits at a subset of the ten unemployment states, linearly interpolating between the chosen UI benefits to determine the whole UI benefit schedule. The algorithm is run again using the optimal solution as a starting point, but increasing the simulated sample, allowing the government to choose benefits at additional unemployment states and adjusting the upper bounds of a and k. The algorithm is repeated until benefits at all ten unemployment states are chosen and the simulated sample reaches 200,000. 7

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