Job quality, search, and optimal unemployment contracts

da Costa, Carlos E.; Maestri, Lucas Jóver; Santos, Cézar Working Paper Job quality, search, and optimal unemployment contracts IDB Working Paper Series, No. IDB-WP-1667 Provided in Cooperation with: Inter-American Development Bank (IDB), Washington, DC Suggested Citation: da Costa, Carlos E.; Maestri, Lucas Jóver; Santos, Cézar (2025) : Job quality, search, and optimal unemployment contracts, IDB Working Paper Series, No. IDB-WP-1667, InterAmerican Development Bank (IDB), Washington, DC, https://doi.org/10.18235/0013396 This Version is available at: https://hdl.handle.net/10419/315920 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/3.0/igo/ J ob Quality, Search, and Optimal Unemployment Contracts Carlos da Costa Lucas Maestri Cezar Santos WORKING PAPER No IDB-WP-1667 InterA merican Development Bank Department of Research and Chief Economist January 2025 * FGV EPGE ** Inter-American Development Bank and CEPR J ob Quality, Search, and Optimal Unemployment Contracts Carlos da Costa* Lucas Maestri* Cezar Santos** InterA merican Development Bank Department of Research and Chief Economist January 2025 Cataloging-in-Publication data provided by the Inter-American Development Bank Felipe Herrera Library Da Costa, Carlos. Job quality, search, and optimal unemployment contracts/ Carlos da Costa, Lucas Maestri, Cezar Santos. p. cm. — (IDB Working Paper Series ; 1667) Includes bibliographical references. 1. Unemployment insurance-Econometric models-United States. 2. Job hunting-Econometric models-United States. 3. Labor market-Econometric models-United States. I. Maestri, Lucas. II. Santos, Cezar. III. Inter-American Development Bank. Department of Research and Chief Economist. IV. Title. V. Series. IDB-WP-1667 http://www.iadb.org Copyright ©2025 Inter-American Development Bank ("IDB"). This work is subject to a Creative Commons license CC BY 3.0 IGO (https://creativecommons.org/licenses/by/3.0/igo/legalcode). The terms and conditions indicated in the URL link must be met and the respective recognition must be granted to the IDB. Further to section 8 of the above license, any mediation relating to disputes arising under such license shall be conducted in accordance with the WIPO Mediation Rules. Any dispute related to the use of the works of the IDB that cannot be settled amicably shall be submitted to arbitration pursuant to the United Nations Commission on International Trade Law (UNCITRAL) rules. The use of the IDB's name for any purpose other than for attribution, and the use of IDB's logo shall be subject to a separate written license agreement between the IDB and the user and is not authorized as part of this license. Note that the URL link includes terms and conditions that are an integral part of this license. The opinions expressed in this work are those of the authors and do not necessarily reflect the views of the Inter-American Development Bank, its Board of Directors, or the countries they represent. Abstract When searching for employment, workers consider non-wage job characteristics, such as effort requirements or amenities. We study an environment where unemployed workers search for jobs of different quality in a labor market characterized by directed search. In equilibrium, firms are more likely to post vacancies for low-quality jobs, as these are more profitable. Hence, high-quality jobs are hard to come a cross. The non-observability of these employment contracts influences t he o ptimal unemployment insurance (UI) program, leading to distortionary taxation. Calibrating the model to the U.S. economy, we find that non-observability of employment contracts results in faster-declining UI benefits, steeper taxes upon re-employment, distortionary taxation, and a 10.5% costlier program than an observable contract scenario providing equal welfare. JEL classifications: H21, J64 Keywords: Unemployment insurance, Directed search, Intensive margin, Amenities, Hidden savings Filipe Fiedler, Pedro Guerra, Artur Rodrigues and Alejandra Torres provided excellent research assistance. The views expressed in this article are those of the authors and do not necessarily represent those of the Inter-American Development Bank. da Costa thanks CNPq project 304955/2022-1 for financial support. This study was financed in part by the Coordenac¸˜ao de Aperfeic¸oamento de Pessoal de N´ıvel Superior - Brasil (CAPES) - Finance Code 001. 1 Introduction The labor market encompasses more than just wage compensation, as jobs vary significantly in terms of amenities, perks, work hours, and effort requirements. When searching for jobs, workers consider these non-pecuniary factors that determine job quality, and firms tailor positions accordingly. Recognizing and understanding these complexities of the labor market is crucial for crafting effective economic policies, particularly in the context of unemployment insurance. Unemployment insurance programs must strike a balance between providing adequate insurance and avoiding disincentives for job search and reemployment. By considering different aspects of job quality, policymakers can develop unemployment insurance policies that better align with the complexities of the modern job market. In this paper, we study the problem of a government that offers unemployment insurance in a dynamic environment featuring directed search. We innovate by considering non-wage aspects of job quality, which firms can provide at different levels. Firms may expand the supply of vacancies of jobs of lower quality, such as jobs that require more effort and/or provide fewer amenities for the same level of earnings. As a prime example, many people prefer home office jobs due to the flexibility they provide in balancing personal and professional lives, such as eliminating long commutes and accommodating family responsibilities. Furthermore, remote work reduces expenses on commuting, dining out, and professional attire, making it a more economical choice, which motivates individuals to seek out such roles. A growing body of empirical research, discussed in the next section, strongly emphasizes the significance of job quality. These non-pecuniary dimensions of job quality are important for our analysis. From the workers’ perspective, they can reduce the expected unemployment spell if they look for lower-quality jobs. For the design of optimal policies they are important because these adjustments in job quality are typically not controlled by the planner. We characterize the optimum for general separable preferences when the planner controls the agent’s savings. At the optimum, unemployment benefits and net earnings decline with the length of the unemployment spell. The repeated moral hazard nature of the problem implies that, at the optimum, the stochastic process governing consumption satisfies the inverse Euler Equation. In the long run, unemployment benefits converge to zero. The optimal contract also prescribes a positive wedge on the marginal rate of substitution between consumption and job quality, i.e., distortionary taxation. This result materializes even though the planner can use non-distortionary instruments and there is no distributive motive. 2 The logic is as follows. Consider a firm that increases job quality (lower work requirements or increased amenities) for a fixed level of earnings to attract workers. This increase in the value of the job, which goes under the planner’s radar since only earnings are observed, leads to a higher probability of hiring. But, from a worker’s perspective, high-quality jobs are harder to find. Because agents searching for a job are entitled to unemployment benefits, high-quality jobs are expensive for the unemployment insurance program. The question is how the planner can discourage firms from offering these highquality jobs. Now, a worker who manages to get such a job has a higher utility of job quality than those who get lower-quality jobs. In an economy without distortions at the margin, these workers would like the firm to provide less job quality in exchange for a proportional increase in earnings. By taxing earnings at the margin, the planner discourages such a change and makes these high-quality jobs less attractive. This result relies on three realistic assumptions embedded in our framework: directed search, intensive margin adjustments, and unobservability of the details of the employment contract. We calibrate our model to the U.S. economy. The non-observability of employment contracts has a significant quantitative impact on the optimal unemployment insurance contract designed by the social planner. When contracts are unobservable, the unemployment insurance benefits decline faster, taxes upon re-employment increase more rapidly with unemployment duration, and distortionary tax rates are introduced. These factors lead to an unemployment insurance program that is 10.5% more expensive than one that provides the same level of welfare in a world in which contracts are observable. To implement the optimal allocation described above, the planner must control the agent’s savings, which may not be possible in practice. We take the possibility of hidden savings and borrowing in perfect capital markets into account. For this case, we restrict our analysis to preferences of the Greenwood et al. [1988] type specialized to the case of Constant Absolute Risk Aversion (GHH-CARA preferences). The optimal allocation can be implemented by a simple stationary contract: an upfront unemployment installment, constant gross earnings, and taxes when the agent finds a job. The pattern of declining consumption in both employment and unemployment is achieved by the worker’s (dis)savings along the unemployment spell. In this hidden-savings case too, a positive wedge at the margin characterizes the optimum. Government agencies deploy various strategies to oversee unemployed individuals during their job search to validate eligibility for unemployment benefits. Despite widespread reporting mandates, such as recording job applications and interviews, ensuring job seekers pursue suitable employment proves challenging. Confirming individuals do not solely 3 target highly competitive yet appealing positions that frequently draw numerous applicants presents a significant hurdle. Our research demonstrates that the benefits of establishing an effective unemployment insurance monitoring agency are substantial. The rest of the paper is organized as follows. After a brief literature review, in Section 2, we motivate empirically the interaction between receipt of unemployment insurance, the likelihood of finding employment, and certain job characteristics. In Section 3, we describe the environment and offer a one-period account of the forces explaining our findings. We derive the properties of an optimal system under the assumption that the planner controls agents’ savings in Section 4and analyze the optimal contract quantitatively. Section 5 describes the optimal contract for the case of hidden savings. Section 6concludes. Literature Review The modern treatment of unemployment insurance program design has its roots in Shavel and Weiss [1979] and found its first canonical treatment in Hopenhayn and Nicolini [1997]. We contribute by focusing on directed search and by introducing the possibility of selecting jobs according to their effort requirements. Acemoglu and Shimer [1999] consider a general equilibrium model of directed search with risk aversion. The static version of our model generalizes theirs by considering the possibility of adjusting the effort requirements of different jobs. Moreover, while their focus is on the general equilibrium aspects of unemployment insurance, we concentrate on the planner’s solution to the optimal policy. Shimer and Werning [2007,2008] evaluate the consequences of allowing agents to borrow and save in perfect capital markets using McCall’s (1970) model of sequential job search. Under CARA preferences, a policy comprised of a constant benefit during unemployment, a constant tax during employment, and free access to a riskless asset is optimal. In our directed search environment with the possibility of intensive margin adjustments once employed, simple stationary policies are also optimal under CARA. We add to the prescription by proving the optimality of introducing distortionary taxation to incentivize search towards easier-to-find jobs.2 A strand of the literature investigates redistributive policies in the presence of labor market frictions. Golosov et al. [2013] consider the redistribution of residual income. Under directed search, the optimal redistribution of residual income can be attained with positive unemployment benefits and a positive, increasing, and regressive income tax schedule. They do not consider an intensive margin of non-wage job quality as we do. 2We also contribute to the literature that studies the optimal path of UI benefits [e.g., Kolsrud et al.,2018, Lindner and Reizer,2020]. We add by considering non-observable aspects of job quality. 4 The presentation of our theory below is centered on variations of effort as the relevant intensive margin adjustment. In practice, workers may adjust their search not only by becoming more selective about wages and how much effort they must exert once employed but also about the quality of their prospective work environment, neither of which is within the reach of policy.3We show that the same logic leading to the wedge in effort implies a wedge in the supply of amenities. Recent research shows that job amenities are important for workers [e.g., Sockin,2022]. For instance, Morchio and Moser [2024] demonstrate that amenities play an important role in explaining the gender pay gap. Bagga et al. [2024] show that increased preferences for telework, a key job amenity, help explain the postpandemic labor market experience in the United States. We contribute to this literature by showing that these non-wage characteristics of job quality influence the design of the optimal unemployment insurance program. Kroft et al. [2020] find sufficient statistics for the optimal combination of income taxes and unemployment benefits but do not consider intensive margin adjustments as we do. da Costa et al. [2022] study optimal distributive policies in the presence of labor market frictions. While they emphasize intensive margin choices, their model is static and focused on the interaction between distributive motives and unemployment insurance design. Here, we abstract from redistribution while highlighting the dynamics of insurance when contracts are not observed and there is scope for adjustments in the intensive margin. 2 Empirical Motivation This section explores data from the United States to check whether there are discernible differences in labor market outcomes between individuals receiving unemployment insurance (UI) and those without such coverage. We rely on data extracted from the March supplement of the Current Population Survey (CPS). This supplement provides data on UI receipts among the unemployed, as well as key characteristics of their job for those currently working. Our analysis encompasses data from 2009 to 2022. We run linear probability regressions to draw comparisons between the labor market trajectories of unemployed individuals benefiting from UI and those without such benefits. Figure 1provides the main estimates (see Appendix Afor the full regressions). Figure 1(a) plots the difference in the likelihood of being unemployed one year ahead for unemployed workers who receive UI versus those who do not. UI recipients are about 3These equalizing differences, surveyed by Rosen [1987], have been shown to be quantitatively important in recent work by Mas and Pallais [2017], Sorkin [2018], Hall and Mueller [2018]. 5 The planner cannot force the agent to find a job if φ(cu)> φ(ce)−η(ne). Hence, the program that the planner solves is C(W) = max p,ce,cu,ne,˜ W p 1−βne−ce−κϱ(p) p+ (1 −p)h−cu+βC(˜ W)i, subject to the promise-keeping, W=p 1−β[φ(ce)−η(ne)] + (1 −p)hφ(cu) + β˜ Wi,(5) and the incentive constraint, φ(ce)−η(ne) 1−β≥φ(cu) + β˜ W. (6) We can write the program above as the following Kuhn-Tucker problem,6 C(W) = max p,ce,cu,ne,˜ W p 1−βne−ce−κϱ(p) p+ (1 −p)h−cu+βC(˜ W)i+ µp 1−β[φ(ce)−η(ne)] + (1 −p)hφ(cu) + β˜ Wi−W+ λφ(ce)−η(ne) 1−β−φ(cu)−β˜ W To proceed, we first assess whether the moral hazard and the promise-keeping constraints bind at the optimum. Lemma 4.1 below states that whenever agents search for a job, they are indifferent between doing so and remaining unemployed for another period. Lemma 4.1 The promise-keeping constraint (5) binds in every period, and µ > 0.In any period in which there is positive search, the moral-hazard constraint binds, φ(ce)−η(ne) = [1 −β]hφ(cu) + β˜ Wi, and λ > 0. For every period tin which the moral hazard constraint binds we have µt+1 =µt−λt 1−pt , 6We can rely on Lemma B.3 to write the problem as such. This lemma refers to the case in which contracts are not observed, but the argument is easily adapted to the case with observed contracts. 12 which implies that unemployment consumption decreases over time, cu t−1= (φ′)−1µ−1 t>(φ′)−1µ−1 t+1=cu t. Moreover, the consumption process is described by an inverse Euler equation, 1 φ′cu t−1=µt=ptµt+λtp−1 t+ (1 −pt)µt+λtp−1 t=pt φ′(ce t)+1−pt φ′(cu t). Also, the first-order conditions with respect to ceand nimply that, in contrast to our one-period model with non-observed contracts, the effort is not distorted at the optimum in the dynamic model with observable contracts. We gather these findings in Proposition 4.1. Proposition 4.1 The solution for the planner’s problem when contracts are observable has the following properties: 1. It entails a zero marginal income tax rate. 2. The unemployment insurance is decreasing over time. Moreover, if the agent searches in period t, then the unemployment insurance is strictly lower than the one from the previous period. 3. The consumption process is described by an inverse Euler equation. To understand 1, note that the incentive-compatibility constraint (6) only depends on the agent’s utility when employed, not on how it is generated. Since the government observes contracts, it can choose them to minimize the cost of providing this utility. That is, given any utility level, there is no reason for the government to distort effort, which implies 1. Second, unemployment insurance should decrease over time to make it more costly to turn down employment opportunities, which is the content of 2. Finally, similar to several dynamic moral-hazard models—e.g., Rogerson [1985] —the consumption process is described by an inverse Euler equation. 4.2 Non-observable Contracts Section 4.1 adopted the strong assumption that the government observes the contracts chosen by workers and hence the disutility of effort from a particular job. We now consider optimal policies under non-observable labor contracts. In this setup, the optimal policy 13 must be based only on whether or not the agent is employed, on their earnings, and the length of the unemployment spell. If the agent is promised a sufficiently high utility, then there is no search in equilibrium at the solution of the planner’s program; it is cheaper to deliver the promised utility if the agent remains unemployed forever; we show this in Lemma B.1 in the Appendix. This is an uninteresting case, and we instead focus on the case in which utility is not too high. To characterize the optimal unemployment insurance program in this case, we rely on a first-order approach. Lemma B.2 shows that the solution for this relaxed problem is the solution to the original program. Hence, the planner’s problem has a recursive structure and can be written as follows, C(W) = max p,ce,cu,ye,˜ W p 1−β(ye−ce) + (1 −p)h−cu+βC ˜ Wi, subject to a promise-keeping constraint p 1−βφ(ce)−ηye+κϱ(p) p+ (1 −p)hφ(cu) + β˜ Wi−W≥0,(7) and an incentive compatibility constraint 1 1−βφ(ce)−ηye+κϱ(p) p−φ(cu)−β˜ W =pκ 1−βη′ye+κϱ(p) pϱ(p) p′ .(8) Lemma B.3 shows that the planner’s problem is differentiable, and hence the optimum must satisfy a constrained optimization in which we write µand λfor the multipliers relative to the constraints (42) and (43). Both multipliers are strictly positive. If µwere not strictly positive, the planner would be able to save resources by lowering the utility promised to the agent in both states with no consequences for incentives. λis strictly positive because the worker does not internalize the fiscal externality when unemployed. Combining the first order conditions with respect to yeand ce, one obtains φ′(ce)−η′ye+κϱ(p) p=λpκ µp +λη′′ ye+κϱ(p) pϱ(p) p′ >0.(9) The optimal allocation now displays a positive wedge at the intensive margin. The dynamic model inherits the finding from our one-period model. If a firm offers a better job, 14 i.e., one requiring less effort for the same earnings, then it will attract more job candidates. Workers, in turn, will find it harder to land such a job, thus remaining unemployed for a longer horizon. Conditional on getting one of these jobs a worker would have a higher willingness to exert effort compared to someone who got one of the jobs offered by firms along the equilibrium path. To make these deviations less attractive, the planner distorts effort downwards by taxing earnings at the margin. Since preferences are separable in consumption and effort, it is always feasible to vary the unemployment consumption utility in a period and compensate for it by varying the consumption utility in all states of nature in subsequent periods. Such a strategy changes neither incentives nor expected utility. Thus, these perturbations cannot save resources at the optimum. Because the marginal cost of delivering utility is 1/φ′, the inverse Euler equation ensues. These findings are summarized in Theorem 4.1, which is proved in the Appendix. Theorem 4.1 At the optimum, in every period in which the worker searches, 1. the marginal income tax rate is always positive; 2. the moral-hazard constraint (43) binds, and the government benefits from strictly increasing p, and; 3. conditional on not finding a job at period t, the worker’s marginal utility of consumption satisfies the inverse Euler equation, 1 φ′(cu t)=E1 φ′(ct+1). The planner can avoid distorting the effort margin. Taxes may be based on employment, independently of earnings. Moreover, the utility conditional on finding a job depends on φ(ce)−η(ne), regardless of whether ceand neare efficiently chosen. What is then the rationale for distorting the intensive margin prescribed in Theorem 4.1? It is the same as in the static setting. Consider a worker deciding whether to apply for a job in a sub-market that is slightly less tight than what the planner has prescribed ˆp<p. The planner controls yeand ce, but not the amount of effort the agent must make to earn ye. Upon landing a job in a less tight market, the worker is required to supply effort, ˆn=ye+κϱ(ˆp)/ˆp<ye+κϱ(p)/p =n while receiving the same ce. This worker, therefore, has a lower marginal disutility of effort than agents who followed the optimal policy. To make this downward deviation less valuable, which is the relevant deviation according to 1, the planner distorts effort 15 downward by introducing a positive wedge. A little less surprising is the fact that, as in Rogerson [1985], Atkeson and Lucas [1995], the Inverse Euler equation characterizes the dynamics of consumption for the unemployed. When is search optimal? Theorem 4.1 describes the efficient allocation in periods in which there is search. But when is it optimal to search? Proposition 4.2 The unemployment benefit is decreasing over time with cu t> cu t+1 whenever the worker searches in period t+ 1. Moreover, whenever the worker searches in period t+ 1, their consumption from employment is strictly greater than the unemployment benefit from any period τ≥t. When the promised utility is very high, the optimal contract provides constant benefits and asks the worker never to search for a job. On the other hand, job search must be incentivized when the government promises a sufficiently low utility to the worker. These two possibilities render the government’s cost of providing utility Wto the worker not convex in W, in general. As a consequence, we cannot rule out the possibility that the worker does not search for a job in the first period of the optimal contract. To better understand when it is optimal to search in every period, define z(W)by z(W)≡argminzzs.t. maxye[φ(ye+z)−η(ye+ϕ)] ≥W, where, as we recall, ϕ= limp↓0ϱ(p)/p > 0. Intuitively, z(W)is the minimum amount of resources that would cost the government to motivate the worker to search for employment if their unemployment continuation utility were W, assuming that the labor market was competitive. To see this, we use the fact that ϱ(p)/p is increasing in p. Hence, to find a job with probability p the worker would have to pay ϱ(p)/p > ϕ to the firm upon landing a job. Let also cu(W) by φ(cu(W)) = W, the cost of providing utility Wfor a worker who never searches for a job. We show in Lemma B.1, in the Appendix, that there is a level of utility, W∗, above which z(W)> cu(W)and below which z(W)< cu(W). Lemma 4.2 below shows that, if the initial unemployment insurance provides less utility than W∗, then the worker must search for a job in every period. Lemma 4.2 Assume that φ(cu 0)< W∗. Then, pt>0in every period, t. When the initial utility, W0, is smaller than W∗, the initial contract must induce search in some period. Moreover, whenever the worker searches in some period the unemployment benefits eventually fall so that φ(cu t)< W∗for some period t. Hence, the worker searches in every period, τ > t, which is the content of Lemma 4.3, below. 16 Lemma 4.3 The following conditions hold in any optimal contract: a) Assume that W0< W∗. Then, there is t > 0such that φ(cu t)< W∗. Hence, the worker who is unemployed in any period τ > t searches for a job. b) Assume that the worker searches for a job in some period t. Then there is T > t such that the unemployed worker searches in any period τ > T. In this case, according to Proposition 4.2,cu t> cu t+1 for all t. Therefore, the unemployment benefit converges to a non-negative number. Proposition 4.3 shows that this number is 0. Proposition 4.3 Assume that W0< W∗, then unemployment benefits converge to zero. We have focused thus far on the case of separable preferences between consumption and effort. This has been the most frequently studied case in the literature. In Appendix C, we study non-separability for the case of GHH-CARA utility U(c, n) = −exp −αc−η(n).7 These preferences will be the focus of our analysis when we assume that savings cannot be controlled by the planner. The results of this section carry over to the GHH-CARA case. In particular, the optimal policy for this case also prescribes a positive wedge between effort and consumption. 4.3 Quantitative Analysis In this subsection, we analyze quantitatively the impact of implementing the optimal unemployment insurance (UI) contract derived above. We use the United States as our benchmark. Our initial step involves the calibration of model parameters based on the prevailing policy framework. To do this, we first write the problem of the agent under such a policy. An unemployed worker is entitled to UI for a fixed duration of Tperiods, with a constant benefit payout of b. After T, if the individual remains unemployed, they receive a guaranteed minimum consumption floor of f. The worker chooses in which market to search; that is, they choose the job-finding rate p. Plus, they choose their preferred consumption and savings bundle, (c, a′). 7The constant absolute risk aversion (CARA) case is the only one for which Shimer and Werning [2007] have theoretical results for the non-observable savings scenario. They offer numerical explorations for the constant relative risk aversion (CRRA) case. Because we are also interested in understanding choices at the intensive margin, we suppress income effects at this margin through the assumption of quasi-linearity, as in Greenwood et al. [1988]. 17 Denote by Vu(t, a)the value function for an unemployed worker that still has tperiods of UI and owns assets a. If the worker is still eligible for UI (i.e., t≥0), their value function reads: Vu(t, a) = max p,c,a′pVe(a, p) + (1 −p) [φ(c) + βVu(t−1, a′)] s.t. c+a′= (1 + r)a+b, where Ve(a, p)denotes the value of being employed in a type-pjob with asset level a. The continuation value Vu(t−1, a′)reflects the fact that the worker will have one fewer period of UI next period if they do not find a job in the current period. The value function for an unemployed worker without UI (i.e., after Tperiods of unemployment) reads: Vu(0, a) = max p,c,a′pVe(a, p) + (1 −p) [φ(c) + βVu(0, a)] s.t. c+a′= (1 + r)a+f. The value function for an employed worker is given by: Ve(a, p) = max c,a′φ(c)−ηy(p) + κϱ(p) p+βVe(a′, p) s.t. c+a′= (1 + r)a+y(p), where the income y(p)is determined by: η′y(p) + κϱ(p) p=φ′(c) We must now set functional forms and parameter values to perform counterfactuals. We assume the utility function for consumption is logarithmic: φ(c) = log c. The disutility for effort is given by: η(n) = η1nη2. Moreover, the labor market tightness is determined by the function: ϱ(p) = 1/(1/p −1). Each model period corresponds to one week. Accordingly, we set the discount factor β= 0.961/52, a standard value. We set the UI in the benchmark, b, to 40% of the average income, the same ratio as in Shimer [2005]. We assume this benefit lasts for T= 26 weeks, as it does in the United States. Additionally, the consumption floor is fixed at 10% of the average income. Three parameters are chosen internally so that the benchmark model matches certain 18 data targets: the parameters that control the disutility of effort, η1and η2, and the vacancy posting cost κ. These parameters are jointly chosen to match three data targets. The first is the mass of unemployed workers who find a job before the UI expires: 86.8% according to Shimer [2008]. The second data target is the wage markdown; that is, how much lower is the wage relative to the worker’s productivity. Berger et al. [2022] report an average wage markdown between 11% and 22%. We target the intermediate value of 16.5%. Finally, we match the relative search effort spent by an unemployed worker at week 26 of unemployment (right before losing the UI benefit) versus week 1. We target 50%, the number reported by Marinescu and Skandalis [2020]. Table 1reports the parameter values and the model fit. Table 1: Calibrated Parameters and Model Fit Parameters κ η1η2 0.1352 0.200 5 Moments Markdown % Reemployed Rel. search effort Model 0.165 0.867 1.549 Data 0.165 0.868 1.500 The fit of the model is quite good, as reported in Table 1. From this benchmark, we take the value function of a worker that still has all of their UI payments to receive and owns the average level of assets as in the data: Vu(26,¯a).8We set this value as the baseline utility that the planner will promise the worker in the optimal unemployment insurance contracts: W0. We solve for these optimal contracts under two scenarios: one in which the contracts are observable (the contract characterized in Section 4.1) and another in which they are not (Section 4.2). Under observable contracts, the planner can provide the corresponding contract more cheaply, though the promised utility is the same. Quantitatively, with non-observable contracts, the cost of the program is 10.5% higher, a substantial increase. Figure 2reports the comparison of different outcomes under observable versus nonobservable contracts. UI declines with the duration of unemployment in both cases. However, the decline is steeper under non-observability (Panel a). This reflects a decreasing promised utility for an unemployed worker throughout the unemployment spell (Panel b). 8The level for ¯ais calibrated by targeting the level of liquid assets for the median individual in the United States as reported in Kaplan and Violante [2014]. We take their number and divide it by real GDP per worker (USARGDPE from the St. Louis FRED database). The corresponding ratio is 1.59. 19 Figure 2: Outcomes under the Optimal UI Contract, Observable versus Non-observable (a) Unemployment Insurance 20 40 60 80 100 Week (t) 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1Obs. Non obs. (b) Welfare of Workers 20 40 60 80 100 Week (t) -580 -560 -540 -520 -500 -480 -460 -440 Observable Unemployed - Non obs. Employed - Non obs. (c) Probability of Finding a Job 20 40 60 80 100 Week (t) 5.85 5.86 5.87 5.88 5.89 5.9 5.91 5.92 Obs. Non obs. (d) Production of Employed Worker 20 40 60 80 100 Week (t) 1.03 1.035 1.04 1.045 1.05 Obs. Non obs. (e) Income of Employed Worker 20 40 60 80 100 Week (t) 1 1.005 1.01 1.015 1.02 Obs. Non obs. (f) Consumption of Employed Worker 20 40 60 80 100 Week (t) 0.94 0.95 0.96 0.97 0.98 0.99 1 1.01 Obs. Non obs. Notes: The outcomes for an unemployed worker are reported for each week tduring the unemployment spell. The outcomes for an employed worker are those for an individual who found a job at exactly week t. 20 Welfare upon employment also goes down with the duration of unemployment, as the planner wants to incentivize the agent to search harder earlier on. The planner achieves this by increasing taxes with the duration of the unemployment spell, as we will see momentarily. In the observable contract case, due to a binding incentive compatibility constraint, the welfare of the unemployed and the employed coincide (so that only one line is displayed in the figure for the observable case). Lower unemployment insurance and welfare over time incentivize the worker to search harder for a job (Panel c). Therefore, with non-observable contracts, the search effort the worker engages in increases faster over time. This higher search effort (higher probability of finding a job) materializes because the individual is searching for jobs in which they have to work harder and produce more (Panel d). Consequently, the worker is compensated for this higher effort, and their income as an employed worker is higher when they find a job later (Panel e). This happens because search effort increases with time and the worker is compensated for this. However, the consumption of the employed worker is lower for those who find a job later (Panel f), implying the tax increases with the duration of unemployment. As seen in the previous sections, the planner finds it optimal to increase taxes over the duration of unemployment to incentivize search. Again, under non-observable contracts, the variation throughout unemployment is steeper. With non-observable contracts, the tax collected by the planner increases with the unemployment spell. This transpires by putting together Panels (e) and (f) in Figure 2. In the previous section, we proved that, in this non-observable case, the planner imposes a distortionary tax rate to incentivize the effort supplied by the worker. We can compute such a tax rate in our numerical exercise: 0.03%. So, though the planner does distort the intensive margin of the worker, it does so with a somewhat low tax rate. This low level of the distortionary tax materializes because, in our model, once employed, the worker never loses their job. Hence, the planner must take into account that, by imposing this distortion, the worker will face it forever. Were the worker at risk of losing their job and searching again, this tax rate would have been higher. This distortionary tax rate also increases with the duration of the unemployment spell. In sum, the non-observability of the employment contract has an important quantitative effect on the optimal UI contract offered by the planner. With non-observable contracts, the planner decreases UI faster and increases taxation upon employment faster with the duration of unemployment (including adding a distortionary component to the tax). These changes all add up to a considerably more expensive UI program. 21 the margin under the non-observability of employment contracts. This distortion occurs because unobserved high-quality jobs are effectively subsidized by the unemployment insurance program. Introducing a marginal tax on earnings discourages such jobs, rendering them less attractive despite providing higher non-wage quality. This distortionary tax emerges whether or not the agent can hide their savings from the planner. We calibrate our model to the U.S. economy. When employment contracts are unobservable to policymakers, several consequences arise: unemployment benefits decline more rapidly, taxes upon re-employment increase more steeply with unemployment duration, and distortionary tax policies become necessary. Collectively, these adjustments result in an unemployment insurance program that, when compared to a scenario where contracts are fully observable, is 10.5% costlier while providing the same level of welfare. Our findings highlight the need for effective monitoring of the non-wage dimensions of jobs to validate benefit eligibility and deter job seekers from solely pursuing highly desirable yet improbable positions. Implementing robust oversight policies can help mitigate moral hazard and facilitate a more efficient unemployment insurance system aligned with the realities of today’s complex labor market. References Daron Acemoglu and Robert Shimer. Efficient unemployment insurance. 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American Economic Review, 98(2):268– 73, May 2008. doi: 10.1257/aer.98.2.268. 19 Robert Shimer and Iv´ an Werning. Reservation wages and unemployment insurance,. The Quarterly Journal of Economics, 112:1145–1185, 2007. 4,17,22 Robert Shimer and Iv´ an Werning. Liquidity and insurance for the unemployed. American Economic Review, 98(5):1922 – 1942, 2008. 4 Jason Sockin. Show me the amenity: Are higher-paying firms better all around? Working paper, 2022. 5 Isaac Sorkin. Ranking firms using revealed preference. The Quarterly Journal of Economics, 133(3):1331–1393, 2018. 5 A Data Appendix This appendix provides the full results for the regressions that yield the coefficients from Figure 1in Section 2. See Tables 2and 3. These regressions use U.S. data from the March Supplement of the Current Populations Surveys (CPS) between 2009 and 2022. The controls used in some of the regressions are age, gender and education. Table 2: Linear Probability Model, Probability of Being Unemployed One Year Later (1) (2) Unemployed Unemployed insurance 0.0820∗∗∗ 0.0691∗∗∗ (0.00995) (0.0101) Controls No Yes N 11804 11804 Standard errors in parentheses ∗p < 0.05,∗∗ p < 0.01,∗∗∗ p < 0.001 30 Table 3: Linear Probability Model, Probability of Having a Job with some Characteristics One Year Later (1) (2) (3) (4) Unionized Unionized Health Health insurance 0.0167∗∗∗ 0.0148∗∗∗ 0.0310 0.0149 (0.00431) (0.00439) (0.0366) (0.0373) Controls No Yes No Yes N 7422 7422 670 670 Standard errors in parentheses ∗p < 0.05,∗∗ p < 0.01,∗∗∗ p < 0.001 B Theoretical Appendix B.1 Proofs of Section 4.1 Proof of Lemma 4.1.First, we show that the constraint (5) binds. The first order condition with respect to cureads φ′(cu) = 1−p µ(1 −p)−λ>0. If µ≤0then λ < 0and thus p > 0and then using the first order condition w.r.t. cewe obtain φ′(ce) = p pµ +λ<0, a contradiction. Next, towards a contradiction, assume that, without loss of generality, the constraint (6) does not bind at t= 0,φ′(cu 0) = φ′(ce 0) = µ−1 0=η′(n0).In this case, φ(ce 0)−η(n0)< φ(cu 0).(13) The moral hazard constraint must bind for some t > 0, otherwise, φ(cu t) = µ−1 0,for every t. This means that getting a job in period zero is worse than being unemployed forever. Assume that the first period in which the constraint binds is t= 1 (the other case is analogous). We have µ1=µ0,λ1>0and, hence, φ′(ce 1) = p1 p1µ0+λ1 =η′(n1). 31 Therefore, φ(ce 0)−η(n0)< φ(ce 1)−η(n1)(14) Hence, using (13) and (14) we obtain φ(ce 0)−η(n0) 1−β< φ(cu 0) + βφ(ce 1)−η(n1) 1−β, which, using the fact that the moral hazard constraint was binding in the second period, implies that the worker strictly prefers being unemployed to getting a job at zero, a contradiction. B.2 Proofs of Section 4.2 Lemma B.1 Both mappings, z(·)and cu(·), are strictly increasing, twice differentiable, and strictly convex. Moreover, there exists W∗such that z(W∗) = cu(W∗),z(W)> cu(W), for all W > W∗, and z(W)< cu(W), for all W < W∗. Proof of Lemma B.1.Let ye(W)be given by argmax ye [φ(ye+z(W)) −η(ye+ϕ)] , and note that if φ′(z(W)) −η′(ϕ)≤0, then ye(W) = 0. Otherwise, ye(W)is given by φ′(ye+z(W)) −η′(ye+ϕ) = 0. Hence, because z(W) + ye(W)> cu(W), we have z′(W) = 1 φ′(z(W) + ye(W)) >1 φ′(cu(W)) =cu′(W). This implies that if z(·)and cu(·)cross at most once, and z(W)> cu(W)(resp. z(W)< cu(W)) for every utility greater (resp. lower) than this utility level. Since z(W)→ ∞ as W→ ∞, we have ye(W)=0for Wlarge enough, which implies z(W)> cu(W). The existence of a small Wsuch that z(W)< cu(W)holds by assumption. Therefore, W∗exists by continuity. It remains to show that both mappings are strictly convex. Since ce(W) := z(W) + 32 ye(W)is strictly increasing with positive derivative, we have z′′ (W) = −φ′′ (ce(W)) φ′(ce(W))2ce′(W)>0, and cu′′ (W) = −φ′′ (cu(W)) φ′(cu(W))2cu′(W)>0. Lemma B.2 Suppose that, if a worker gets a job, then they must earn ce+T, paying T to the government, to consume ce, whereas if the worker fails to get a job then they obtain the continuation utility W. Then this problem admits a unique solution. If the solution is interior, it is given by the associated first-order conditions. Proof. Consider the problem max pφ(ce)−ηce+T+κϱ(p) p−W This problem admits an interior solution if and only if φ(ce)−η(ce+T)> W. Assume that this is the case and consider pthat makes its derivative equal to zero: φ(ce)−ηce+T+κϱ(p) p−W−pη′ce+T+κϱ(p) pκd dp ϱ(p) p= 0 Differentiate the left-hand side again to obtain −2η′ce+T+κϱ(p) pκd dp ϱ(p) p−pη′ce+T+κϱ(p) pκd2 dp2ϱ(p) p −pη′′ ce+T+κϱ(p) pκd dp ϱ(p) p2 . To show that the expression above is negative, it suffices to show that −2d dp ϱ(p) p−pd2 dp2ϱ(p) p<0⇔2ϱ′(p)p−ϱ(p) p2+pd dp ϱ′(p)p−ϱ(p) p2>0 ⇔p2d dp [ϱ′(p)p−ϱ(p)] >0⇔ϱ′′(p) p>0. 33 Lemma B.3 For every W, let C(W)be the planner’s cost of providing utility W. The mapping C(·)is differentiable at Wtfor every t > 0. Proof. We prove that Cis differentiable at Wt. For that, we assume that pt>0as the other case is analogous. Consider any small ϵ∈Rand note that the following perturbation is feasible: ˜ut−1,˜ut,˜ce t=ut−1+ϵ, ut−ϵβ−1, φ−1(φ(ce t) + ϵ). One can thus apply the argument in Clausen and Strub to conclude that C′(Wt) = −c′(ut) = 1 φ′(ut). Lemma B.4 The multipliers, µand λ, are strictly positive if there is a search. Proof. First, notice that [µ(1 −p)−λ]φ′(cu) = 1 −p and pµ +λ 1−βφ′(ce) = p 1−β Hence, µ= 0 implies φ′(cu)φ′(ce)≤0, which is absurd. Hence assume towards a contradiction that λ0≤0.Clearly, there is a last period at which λt≤0and λt+1 >0. Otherwise, as we will verify below, cu t≥ce tfor every t, and hence there is no search. Assume that λ1>0(case in which λs≤0for all s < t and λt>0for some t > 1can be handled analogously). From the first-order condition with respect to p, we get φ′(cu) = 1 µ−λ(1 −p)−1≤1 µ+λp−1=φ′(ce). Hence, cu≥ce. Moreover, notice that from the first order condition we have C′(W1) = −µ0+λ0 (1 −p)=−µ1, 34 which implies µ1=µ0−λ0 (1 −p)≥µ0. This and λ0≤0< λ1imply φ′(ce 1) = 1 µ1+p−1 1λ1 <1 µ0+p−1 0λ0 =φ′(ce 0). Hence, ce 1> ce 0.(15) We can rearrange the first order condition with respect to yeto get µη′ye+κϱ(p) p= 1 −λη′′ ye+κϱ(p) pκd dp ϱ(p) p−λ pη′ye+κϱ(p) p. Therefore, λ0≤0< λ1imply η′ye 1+κϱ(p1) p1< µ−1 1. Similarly, η′ye 0+κϱ(p0) p0≥µ−1 0. Since µ1≥µ0, this implies ye 1+κϱ(p1) p1 < ye 0+κϱ(p0) p0 , and ηye 1+κϱ(p1) p1< η ye 0+κϱ(p0) p0, because ηis strictly convex. 35 Since p0>0, by the assumption of the lemma, we have 0<1 1−βφ(ce 0)−ηye 0+κϱ(p0) p0−[φ(cu 0) + βW1] =1 1−βφ(ce 0)−ηye 0+κϱ(p0) p0−φ(cu 0) −βp1 1 1−βφ(ce 1)−ηye 1+κϱ(p1) p1+ (1 −p1) [φ(cu 1) + βW2] =φ(ce 0)−ηye 0+κϱ(p0) p0−φ(cu 0) + β"1 1−βφ(ce 0)−ηye 0+κϱ(p0) p0 −p1 1 1−βφ(ce 1)−ηye 1+κϱ(p1) p1−(1 −p1) [φ(cu 1) + βW2]# =1 1−βφ(ce 0)−ηye 0+κϱ(p0) p0−φ(cu 0) −βp1 1 1−βφ(ce 1)−ηye 1+κϱ(p1) p1+ (1 −p1) [φ(cu 1) + βW2](16) Since p1>0, due to λ1>0, we have 1 1−βφ(ce 1)−ηye 1+κϱ(p1) p1> φ(cu 1) + βW2 Hence, φ(ce 0)−ηye 0+κϱ(p0) p0−φ(cu 0) + β(1 1−βφ(ce 0)−ηye 0+κϱ(p0) p0 −p1 1 1−βφ(ce 1)−ηye 1+κϱ(p1) p1−(1 −p1) [φ(cu 1) + βW2]) < φ(ce 0)−ηye 0+κϱ(p0) p0−φ(cu 0)+β1 1−βφ(ce 0)−ηye 0+κϱ(p0) p0−[φ(cu 1) + βW2] Since the first line from the last term is negative, the entire term is less than β1 1−βφ(ce 0)−ηye 0+κϱ(p0) p0−[φ(cu 1) + βW2], 36 which is less than 1 1−βφ(ce 0)−ηye 0+κϱ(p0) p0−[φ(cu 1) + βW2], since the term is positive. Since φ(ce 0)< φ(ce 1), and ηye 0+κϱ(p0) p0> η ye 1+κϱ(p1) p1, this is less than 1 1−βφ(ce 1)−ηye 1+κϱ(p1) p1−[φ(cu 1) + βW2]. Hence, using the first-order conditions with respect to p, the algebra just performed means that p1 1−βη′ye 1+κϱ(p1) p1κϱ(p1) p1>p0 1−βη′ye 0+κϱ(p0) p0κϱ(p0) p0.(17) Since ye 1+κϱ(p1) p1 < ye 0+κϱ(p0) p0 , If ye 1≥ye 0,we will have p1< p0which together contradict (17). We conclude that ye 1< ye 0. Finally, notice that λ1>0and the first order condition with respect to pand the fact that pis a local maximum imply ye 0−ce 0 1−β≤ −cu 0+βC (W1).(18) Analogously, in period 1, using λ0≤0, the first order condition with respect to p implies ye 1−ce 1 1−β≥ −cu 1+βC (W2). But notice that C(W1) = p1 ye 1−ce 1 1−β+ (1 −p1) [−cu+βC (W2)] ≤ye 1−ce 1 1−β(19) 37 Proof of Theorem 5.2.Let {(p∗ t, ye∗ t, cu∗ t, ce∗ t)}∞ t=0 be the optimal allocation. Notice that W∗ 0=p∗ 0We∗ 0+ (1 −p∗ 0)Wu∗ 0. If We∗ 0≤Wu∗ 0, then p∗ 0= 0. In this case, the optimal allocation can be implemented by assets a0=−α−1log (−(1 −β)W0) 1−β and a some pair (ye, Te)with ye=Te. The worker best responds by never searching for a job and consuming −(1 −β)α−1log (−(1 −β)W0)in every period. According to Lemma B.8 this is optimal. Next, assume that We∗ 0> Wu∗ 0. Consider the first order condition: −1 1−βexp −αce∗ 0−ηye∗ 0+κϱ(p∗ 0) p∗ 0−Wu∗ 0− αp∗ 0 1−βexp −αce∗ 0−ηye∗ 0+κϱ(p∗ 0) p∗ 0η′ye 0+κϱ(p∗ 0) p∗ 0κϱ(p∗ 0) p∗ 0′ = 0, and the following promise-keeping condition, W∗ 0=p∗ 0We∗ 0+ (1 −p∗ 0)Wu∗ 0. By solving these two equations we obtain: We∗ 0 W∗ 0 =1 + αp∗ 0(1 −p∗ 0)η′ye∗ 0+κϱ(p∗ 0) p∗ 0κϱ(p∗ 0) p∗ 0′−1 Wu∗ 0 W∗ 0 = 1 + αp∗2 0η′(ne∗ 0)κϱ(p∗ 0) p∗ 0 1 + αp∗ 0(1 −p∗ 0)αη′(ne∗ 0)κϱ(p∗ 0) p∗ 0. Next, notice that We∗ 0delivers ce∗ 0by −1 1−βexp −αce∗ 0−ηye∗ 0+κϱ(p∗ 0) p∗ 0=We∗ 0, which implies ce∗ 0=−α−1log (−(1 −β)We∗ 0) + ηye∗ 0+κϱ(p∗ 0) p∗ 0. 44 We claim that there exists (a∗ 0, Te∗)that solves the system: ce∗ 0= (1 −β)a0+ye∗ 0+Te(22) Wu∗ 0= max c−exp {−αc}+βU β−1(a0−c), ye∗ 0, Te,(23) where U(a, ye∗ 0, Te)is the utility of an agent who starts a period unemployed and faces a simple policy, (a, ye∗ 0, Te). If Te=yeand a0=ce∗ 0 1−β, then W∗ 1<max c−exp {−αc}+βU β−1(a0−c), ye∗ 0, Te,(24) as the agent can keep consumption constant at ce∗even without taking a job. The individual best responds to that contract by choosing p= 0 in every period. From this point, if we decrease a0by −ε 1−βand decrease Teby ε, the planner’s payoff is increased by ε 1−β1−p(a, ye∗ 0, Te) 1−(1 −p(a, ye∗ 0, Te)) β>0.(25) Next, notice that, by construction, −1 1−βexp −α(1 −β)a0+ye∗ 0+Te−ηye∗ 0+κϱ(p∗ 0) p∗ 0=We∗ 0. Recall that the inequality (24) implies that p(a, ye∗ 0, Te)< p∗. We claim that, if we keep decreasing a0by −ε 1−βand Teby ε, we can generate (a∗ 0, Te∗)satisfying (22) and (23). Otherwise, as we take a0to −∞, the planner’s revenue goes to infinity while the worker’s utility at the beginning remains above W∗ 0, a contradiction. From the first order condition, we know that premains bounded below p∗(and by lemma B.5, this holds in every future period) and the principal obtains infinite profits because of (25). At the same time, the worker’s utility remains greater than pWe∗ 0+ (1 −p)Wu∗ 0, a contradiction. The reasoning above shows that offering (a∗ 0, ye∗, Te∗)in the first period is optimal to generate utility W∗ 0. In this case, Lemma B.8 implies that (a∗ 1, ye∗, Te∗)is optimal to generate utility W∗ 1,where a∗ 1is the asset holdings chosen by the agent. Inductively, we conclude that (a∗ t, ye∗, Te∗)is optimal to generate utility W∗ tfor every tand hence the simple policy (a∗ 0, ye∗, Te∗)is optimal. Lemma B.9 We have ∂p ∂ce>0and ∂p ∂ye<0. Proof. We must calculate ∂p ∂ceand ∂p ∂ye. Let ce:= ye−Te, assume without a loss that the 45 agent starts with zero assets (Lemma B.5) and write W1for the payoff of an agent who starts a period of unemployment with zero assets. Start with the first order condition with respect to p: −1 1−βexp {−α[ce−η(ne)]} − max a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}] −αp 1−βexp {−α[ce−η(ne)]}η′(ne)κϱ(p) p′ = 0.(26) Next, we remark that the problem is strictly concave in p, and hence the derivative of (26) w.r.t. pis strictly negative. Differentiating this condition w.r.t. cewe obtain α 1−βexp {−α[ce−η(ne)]} − d dcehmax a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i +α2p 1−βexp {−α[ce−η(ne)]}η′(ne)κϱ(p) p′ . Now, notice that d dcehmax a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i< −αmax a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}],(27) where the last number is obtained by the derivative of an increase in cin every state of nature. 46 Therefore, we have α 1−βexp {−α[ce−η(ne)]} − d dcehmax a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i +α2p 1−βexp {−α[ce−η(ne)]}η′(ne)κϱ(p) p′ = α 1−βexp {−α[ce−η(ne)]}+αhmax a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i +α2p 1−βexp {−α[ce−η(ne)]}η′(ne)κϱ(p) p′ −αhmax a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i− d dcehmax a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i= −αhmax a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i− d dcehhmax a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]ii>0, where we have used (26) and (27). Therefore, ∂p/∂ce>0. Next, differentiating the first order condition with respect to ye, we get −αη′(ne) 1−βexp {−α[ce−η(ne)]}− d dyehmax a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i −α2p 1−βexp {−α[ce−η(ne)]}η′(ne)2κϱ(p) p′ −αpη′′(ne) 1−βexp {−α[ce−η(ne)]}η′(ne)κϱ(p) p′ . Notice that d dyehmax a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i> αη′(ne)hmax a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i.(28) 47 Hence, −αη′(ne) 1−βexp −αce−ηye+κϱ(p) p− d dyehmax a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i −α2pη′(ne) 1−βexp {−α[ce−η(ne)]}η′(ne)κϱ(p) p′ −αpη′′(ne) 1−βexp {−α[ce−η(ne)]}η′(ne)κϱ(p) p′ =−αη′(ne) 1−βexp {−α[ce−η(ne)]} − αη′(ne)hmax a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i −α2pη′(ne) 1−βexp {−α[ce−η(ne)]}η′(ne)κϱ(p) p′ αη′(ne)hmax a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i− d dyehmax a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i −αpη′′ (ne)1−βexp {−α[ce−η(ne)]}η′(ne)κϱ(p) p′ = αη′(ne)hmax a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i− d dyehmax a′[−exp {αa′β}+βW1exp {−αa′(1 −β)}]i −αp 1−βexp {−α[ce−η(ne)]}η′′ (ne)η′(ne)κϱ(p) p′ <0, where we have used (26) and (28). Lemma B.10 We have ∞ X t=0 pβt(1 −p)t−11 + exp {−α(1 −β)at} W0(1 −β)exp {−α[ce−η(ne)]}>0. 48 Proof. We have ∞ X t=0 pβt(1 −p)t−1−1 1−β−exp {−α(1 −β)at} (1 −β)2W0 exp {−α[ce−η(ne)]}<0 ⇔ ∞ X t=0 pβt(1 −p)t−1"−exp {−α(1 −β)at} (1 −β)P∞ t=0 pβt(1 −p)t−1exp {−α[ce−η(ne)]}#> W0, since z→ − exp{−αz}is strictly increasing. Notice that U0is the mixture of the distribution Feover employed payoffs defined above and the distribution over −exp{−αcu t}, which we call Fu. It follows that if Fefirst order stochastic dominates Fu. Hence for any λ∈(0,1), Zxd [λFe(x) + (1 −λ)Fe(x)] <ZxdFe(x). It, therefore, suffices to show that W0<RxdFe(x). We have W0(1 −β) = p(1 −β)W0 e+ (1 −p) (1 −β)h−exp{−αcu 0}+ β[pWe 1+ (1 −p) [−exp{−αcu 1}+βWu 2]] i. Using W0 e>−exp{−αcu 0} 1−βand −exp{−αcu 0} 1−β=pWe 1+ (1 −p) [−exp{−αcu 1}+βWu 2], we have W0<pW0 e+β(1 −p) [pWe 1+ (1 −p) [exp{−αcu 1}+βWu 2]] 1−(1 −p) (1 −β). Proceeding analogously, it follows that the last expression is less than pW0 e+β(1 −p) [pWe 1+ (1 −p)βWu 2] 1−(1 −p) (1 −β)−(1 −p)2β2. Proceeding analogously and taking the limit, we obtain the desired inequality. Proof of Theorem 5.3. Part (i). Recall from (11) ∂ ∂p p 1−(1 −p)βye−ce 1−β∂p ∂ce=p(ye, ce) 1−(1 −p(ye, ce)) β+Uce(ye, ce) eα(1−β)a0α(1 −β)W0 . Since ∂ ∂p p 1−(1 −p)β>0and ∂p ∂ce>0, 49 ye−cehas the same sign as − ∞ X t=0 pβt(1 −p)t−1"−1 1−β−exp −αce−ηye+κϱ(p)/p (1 −β)2W0#, by Lemma B.9, which is strictly positive by Lemma B.10. Part (ii). Consider the problem C(W0) = max W1,ce,ye pye−ce 1−β+ (1 −p)βC(e−αa(1−β)W1), subject to −p 1−βexp −αce−ηye+κϱ(p) p+ (1 −p) max a′−exp{αa′β}+ exp{−αa′(1 −β)}βW1−W0= 0 and −1 1−βexp −αce−ηye+κϱ(p) p −max a′−exp{αa′β}+ exp{−αa′(1 −β)}βW1 −αp 1 1−βexp −αce−ηye+κϱ(p) pη′ye+κϱ(p) pκϱ(p) p′ = 0. Plugging the last constraint into the problem, one obtains the following Lagrangian C(W0) = max W1,ce,ye pye−ce 1−β+ (1 −p)βC(e−αa(1−β)W1)+ µ"−1 1−βexp −αce−ηye+κϱ(p) p− α(1 −p)p1 1−βexp −αce−ηye+κϱ(p) pη′ye+κϱ(p) pκϱ(p) p′ −W0#. 50 Therefore, we have the first order conditions with respect to ce, p=µα exp −αce−ηye+κϱ(p) p1 + α(1 −p)pη′ye+κϱ(p) pκϱ(p) p′, and with respect to ye, p=µα exp −αce−ηye+κϱ(p) p"η′ye+κϱ(p) p+ α(1 −p)pη′ye+κϱ(p) p2 κϱ(p) p′# +µ(1 −p)pexp −αce−ηye+κϱ(p) pη′′ ye+κϱ(p) pκϱ(p) p′ . Therefore, we have η′ye+κϱ(p) p= 1 − (1 −p)pη′′ye+κϱ(p)/pκϱ(p) p′ α1 + α(1 −p)pη′ye+κϱ(p)/pκϱ(p) p′. Part (iii). Notice that W∗ t< We∗ tand hence it suffices to show that lim We∗ t=−∞. We have lim(1 −β)We∗ t= −lim exp −αc∗ 0+ (1 −β) ¯a0−ηye∗+κϱ(p∗) p∗α(t−1) ∆c=−∞. 51 C Extension: GHH-CARA type and Observable Savings In this section, we consider the case of observable savings with period utility of the form U(c, n) = −exp {−α[c−η(n)]}. We can write the Lagrangean as C(W0) = max p 1−β(ye−ce) + (1 −p) [−cu+βC (W1)] , subject to p 1−β[−exp {−α[ce−η(ne)]}] + (1 −p) [−exp {−α[cu]}+βW1]−W0≥0, and 1 1−β−exp −αc−ηye+κϱ(p) p+ exp {−α[cu]} − βW1 =1 1−βαη′ye+κϱ(p) pκϱ(p) p′ exp −αc−ηye+κϱ(p) p. Let Ue:= exp −αc−ηye+κϱ(p) p Uu:= exp {−αcu}. The first order condition for ce 0is −p+µpαUe+λαUe= 0. The first order condition for cu 0is −(1 −p) + (1 −p)µUu−λαUu= 0. 52 The first order condition for yeis p−pµαUeη′(ne)−λαUeη′(ne) −λUeαη′′(ne)κϱ(p) p′ +α2Ue[η′(ne)]2κϱ(p) p′= 0 From these, we have Ue=p µpα +λα Uu=1−p µ(1 −p)α−λα p−η′(ne)Ue[pµα +λα] = λUeαη′′(ne)κϱ(p) p′ +α2Ue[η′(ne)]2κϱ(p) p′ 1−η′(ne) = λUe pαη′′(ne)κϱ(p) p′ +α2[η′(ne)]2κϱ(p) p′(29) C′(W1) = −µ0+λ0 (1 −p0)=−µ1, which implies µ1=µ0−λ0 (1 −p). Moreover, the derivative with respect to pimplies p 1−β(ye−ce) = λαUe 1−β"η′(ne)κϱ(p) p′ +η′′(ne)κϱ(p) p′ +η′(ne)κϱ(p) p′′ +η′(ne)κϱ(p) p′2#.(30) Lemma C.1 The multipliers µand λare strictly positive if there is search. Proof. First notice that Uu 0=1−p0 µ0(1 −p0)α−λ0α Ue 0=p0 µ0p0α+λ0α hence µ0= 0 implies Uu 0Ue 0≤0, which is an absurd. Now, assume towards a contradiction that λ0≤0.Clearly, there is a last period at which 53 which is not possible, a contradiction. But then, by a continuity argument, for every ε > 0, there exists a period t∗such that t≥t∗implies that the planner’s utility is εaway from −cu ∞/(1 −β), while the worker’s utility is εaway from φ(cu ∞)/(1 −β). It follows by Assumption DS that there is ε > 0such that, if the planner demands production y∗in exchange for consumption η(y∗+κϕ) + χ(φ(cu ∞)/(1 −β)) + ε, then the worker searches with probability bounded away from some p > 0for every tlarge enough. Moreover, this εcan be chosen to make both players better off, a contradiction. D Variable Effort and Amenities Thus far we have talked about amenities suggesting that their supply plays an analogous role to effort requirements. In reality both dimensions will simultaneously help define what a desirable job is. In this extension, we add amenities to the one-period model explicitly connecting it to the effort model we have presented. Assume that amenities cost ato the firm and lead to a benefit ϕ(a)by making the working environment more pleasant. We can write the problem as C(W0) = max p 1−β(ye−ce)−(1 −p)cu subject to12 pφ(ce)−ηye−ϕ(a)+(κ+a)ϱ(p) p+ (1 −p)c(u)≥0,(42) and φ(ce)−ηye−ϕ(a)+(κ+a)ϱ(p) p−c(u) = p(κ+a)η′ye−ϕ(a)+(κ+a)ϱ(p) pϱ(p) p′ .(43) The efficient level of amenities is the solution for afb := argmax aϕ(a) + aϱ(p) p. 12We are assuming that the cost of amenities must be paid regardless of whether the vacancy is filled. 60 This implies afb (p) := (ϕ′)−1ϱ(p) p, which is decreasing in p. Suppose the government can choose a. The first order condition implies "(µ+pλ)η′ye−ϕ(a)+(κ+a)ϱ(p) p +λp (κ+a)η′′ ye−ϕ(a)+(κ+a)ϱ(p) pϱ(p) p′#×"ϕ′(a)−ϱ(p) p# =pλη′ye−ϕ(a)+(κ+a)ϱ(p) pϱ(p) p′ Therefore, the optimal policy implies ϕ′(a)> ϱ(p)/p, a positive wedge on the optimal level of amenities. The positive wedge on amenities arises whether the effort is another intensive adjustment margin or not. 61