key: cord-0973562-uuognidf
authors: Mitman, Kurt; Rabinovich, Stanislav
title: Whether, when and how to extend unemployment benefits: Theory and application to COVID-19()
date: 2021-05-24
journal: J Public Econ
DOI: 10.1016/j.jpubeco.2021.104447
sha: 1a3a087edbc695884489e9c6d311f0ae479a9332
doc_id: 973562
cord_uid: uuognidf

We investigate the optimal response of unemployment insurance to economic shocks, both with and without commitment. The optimal policy with commitment follows a modified Baily-Chetty formula that accounts for job search responses to future UI benefit changes. As a result, the optimal policy with commitment tends to front-load UI, unlike the optimal discretionary policy. In response to shocks intended to mimic those that induced the COVID-19 recession, we find that a large and transitory increase in UI is optimal; and that a policy rule contingent on the change in unemployment, rather than its level, is a good approximation to the optimal policy.

Extending unemployment insurance (UI) in economic downturns is common policy practice in the United States, as exemplified by the Great Recession and the COVID-19 crisis. The academic and policy debate over the desirability of such a policy continues. At the heart of the discussion is the insuranceincentive tradeoff involved in UI design: it is arguably optimal to extend UI during downturns if its moral hazard cost is lower during those periods. Despite the wealth of literature on this topic, several important questions remain unresolved.

To start with, the standard insurance-incentive tradeoff is complicated because workers' job search behavior responds to their expectations of future UI benefits as well as current UI benefits. Further, search behavior depends on future economic conditions in addition to current ones. Therefore, the optimal policy may contain substantial history-dependence, and its shape may depend on the government's ability to commit to future policy actions. More-over, the general prescription of extending UI in recessions is somewhat vague when it comes to applying it in practice. Should UI be indexed to the unemployment level-commonly the caseor to another economic indicator? Does a crisis, such as COVID-19, call for a large but short-lived UI extension, as implemented under the CARES Act, or a moderate but prolonged one? Does the nature of the economic shock that induced the recession change the policy prescription?.

Using a simple dynamic framework, we theoretically and numerically address these questions. We consider a parsimonious environment in which risk-averse workers search for jobs at a cost, facing potentially stochastic search efficiency and separation rates. Our model's two-period version is analytically tractable and allows for a transparent characterization of the optimal policy, both with and without commitment. We use this framework to examine how optimal UI depends on search efficiency and the unemployment level, both current and future.

Our framework provides a dynamic generalization of the standard Baily-Chetty logic (Baily (1978) , Chetty (2006) ), whereby optimal policy balances the consumption smoothing benefit of UI against its moral hazard cost. Our analysis leads to three insights about optimal UI. First, the moral hazard cost depends most directly on search efficiency rather than the unemployment rate. Thus, it is optimal on theoretical grounds-at least from a purely static perspective-to index UI to the primitive shock, not the unemployment rate. Second, the dynamic nature of the environment complicates the standard Baily-Chetty formula because future unemployment benefits distort current search effort and hence have a moral hazard cost today. The optimal policy under https://doi.org/10.1016/j.jpubeco.2021.104447 0047-2727/Ó 2021 Elsevier B.V. All rights reserved. q An earlier (and different) version of this paper was circulated under the title ''Optimal Unemployment Benefits in the Pandemic." We are grateful to the editor, Johannes Spinnewijn, and two anonymous referees for very helpful constructive comments. We thank seminar participants at the Third ICEF Conference on Applied Economics for useful suggestions. Support from the European Research Council grant No. 759482 under the European Union's Horizon 2020 research and innovation programme and the Ragnar Söderbergs stiftelse is gratefully acknowledged. commitment accounts for this cross-period moral hazard cost. The optimal policy under discretion does not: as we show, it effectively follows a sequence of static Baily-Chetty formulas, which trade off consumption smoothing against the contemporaneous moral hazard cost only. Third, while the level of unemployment does not matter per se, its path over time does. Suppose unemployment is expected to fall in the future. In that case, future UI has a low consumption smoothing benefit, which accrues to the small number of future unemployed, but a high moral hazard cost, since it distorts the behavior of all the currently unemployed. If the government can commit to future UI, then promising to cut it in states of low unemployment is a cheap way of providing incentives today.

The second and third insights have important implications for the optimal policy's shape and how it varies with the government's commitment ability. A government with commitment power can use both current and future UI to provide current incentives. This commitment power is particularly advantageous in scenarios of rapidly falling unemployment or rapidly rising job-finding ability. In such cases, the optimal policy provides incentives primarily by promising low future UI, leading to a steeply declining UI profile over time. Without such commitment, a government cannot credibly promise to lower UI in the future, worsening the moral hazard problem today. In turn, the government provides less UI today. The no-commitment government, therefore, implements a flatter UI profile in an economy recovering from a crisis.

We illustrate these results by extending the framework to an infinite horizon and computing both the optimal Ramsey policy (with commitment) and the Markov equilibrium policy (without commitment). In response to a negative shock to search efficiency or a destruction of job matches, the policy responses of the two types of government differ: the Ramsey government raises the level of UI benefits more initially but keeps them elevated for a shorter period of time than the Markov government. We also ask how to implement the Ramsey policy-which features complicated history-dependence-if in practice the government can only commit to a simple policy rule. We find that a policy rule conditioning UI on the growth rate in unemployment across periods performs remarkably well in approximating the optimal policy. In contrast, a policy rule conditioning on the level of unemployment would raise UI benefits insufficiently and keep them elevated for an excessively long period of time.

How do we apply our results to the COVID-19 pandemic? In the context of the current crisis, our results suggest that the CARES Act's transitory nature was desirable: not only because the fall in search efficiency itself was transitory but also because frontloading UI is a cheaper way of providing a given amount of incentives. In other words, the optimal policy intertemporally substitutes UI from periods with few unemployed to periods with many unemployed. With credible commitment, intertemporal substitution can be executed at minimal incentive cost.

This paper contributes to an already rich existing literature on optimal UI over the business cycle, including Jung and Kuester (2015) , Mitman and Rabinovich (2015) and Landais et al. (2018) with government commitment, and Pei and Xie (2020) without commitment. These papers undertake thorough quantitative analyses of optimal policy, in particular incorporating general equilibrium effects through labor demand. We have examined the same question in a deliberately simplified framework, which allows us to transparently isolate the key insurance-incentive tradeoff and the bearing of dynamics on it.

This leads to several contributions. We demonstrate the effect of worker expectations and government commitment (or lack thereof) on the standard Baily-Chetty formula, thereby connecting the Baily-Chetty literature with the literature on time-inconsistency. Moreover, our findings -which are robust and likely to emerge in richer models -shed new light on the results found in the aforementioned papers. For example, our results are consistent with the cuts in UI benefits during economic recoveries advocated in Mitman and Rabinovich (2015) under commitment, and the lack thereof in Pei and Xie (2020) under discretion. Finally, the tendency to front-load UI under commitment is clearly reminiscent of the optimally declining UI profile for an individual unemployed worker in Shavell and Weiss (1979) , Hopenhayn and Nicolini (1997) , stemming from the ability to use future UI for incentives today. 1 The re-emergence of the same front-loading tendency in designing economy-wide UI in response to aggregate shocks is an important and, to our knowledge, novel insight.

Our analysis also contributes to understanding the appropriate policy response to the COVID-19 crisis. In this regard, our stylized framework complements the growing literature incorporating the more unique features of the pandemic, such as the distinction between temporary and permanent separations (as examined in Gregory et al. (2020) and Birinci et al. (2020) ) and the epidemiological side of the crisis (as applied to a search model by e.g. Kapicka and Rupert (2020) , Birinci et al. (2020) and Fang et al. (2020) ). The purpose of our paper is less to provide a comprehensive and quantitative analysis of the recent policy response, and more to provide qualitative insights on the key tradeoffs involved. 2 While our results are of relevance for the COVID-19 crisis, the mechanisms underlying them are applicable more generally, for understanding the optimal UI response in recessions and the central role of expectations in its design.

The paper is organized as follows. In Section 2, we consider a simple two-period environment, which allows for analytical characterization of optimal policy, both with and without commitment. Section 3 extends the analysis to an infinite horizon, showing that the main insights remain intact. In Section 4 we numerically illustrate the main results, by simulating policy responses to shocks intended to mimic the COVID-19 crisis. Section 5 concludes.

In this section we consider a two-period model. The simplified environment allows us to analytically characterize the relationship between search efficiency and the moral hazard cost of UI -a key statistic for the optimal policy -and to precisely illustrate the role of government commitment.

The economy is populated by a continuum of risk-averse workers, who, in each period, can be either employed or unemployed. The economy is subject to aggregate shocks to search efficiency f t P 0. Workers have utility

where x t denotes period-t consumption, and S t denotes period-t search effort, incurred only when unemployed and restricted to be between 0 and 1. The utility function v x ð Þ satisfies 1 As shown by Kolsrud et al. (2018) , however, the optimal UI profile may no longer be declining in the presence of worker heterogeneity or duration dependence. We discuss how heterogeneity impacts the results in our environment in Section 4.1.1. 2 We have also abstracted from two general equilibrium feedback mechanisms.

First, we have ignored potential aggregate demand effects induced by providing transfers to the unemployed that could speed the recovery (Kekre (2019 (Kekre ( , 2016 (Kekre ( , 2018 ). Our view is that the COVID-19 pandemic (and ensuing policy response with lockdown orders) represents a supply shock and thus that normal demand channels will be muted (see Guerrieri et al. (2020) for an alternative view). Second, we have abstracted from firm labor demand and the response of wages and labor force participation to benefit policy (see, e.g., Hagedorn et al. (2013 Hagedorn et al. ( , 2015 ).

v 0 > 0; v 00 < 0. The cost of search c S ð Þ satisfies c 0 > 0; c 00 > 0; c 0 0 ð Þ ¼ 0 and c 0 1 ð Þ ¼ 1. In the full-fledged model of the next section, we will assume that f t is stochastic. For the moment, we assume that f 1 and f 2 are known deterministically before any decisions are made in period 1.

The economy begins with an initial fraction u 0 of workers who are unemployed and 1 À u 0 who are employed. Unemployment subsequently evolves as follows. At the start of each period, a fraction d of the employed workers lose their job. We assume that a job loser joins the pool of unemployed job searchers immediately and can find re-employment within the period. The probability that a job searcher finds a job when exerting search effort S is simply S. Thus, end-of-period unemployment follows the law of motion

When employed at the end of period t, workers receive exogenous income w and pay a tax s. When unemployed, they receive a government-provided unemployment benefit b t ; t ¼ 1; 2. Unemployed workers choose S t at each point in time to maximize expected utility, taking as given the path of search efficiency f t and the government policy b t , for t ¼ 1; 2. Unemployment benefits are chosen by a benevolent government, who faces an exogenous cost of public funds g. The timing

is such that the government announces b t in each period before workers choose S t . Below, we will consider both a version where the government can commit to the sequence b 1 ; b 2 at the beginning of time, as well as one where it re-optimizes b 2 at the start of period 2. We refer to these types of government throughout the paper as Ramsey and Markov, respectively.

We first solve for the workers' optimal choices of S 1 and S 2 , given f 1 ; f 2 and b 1 ; b 2 (the detailed derivations are in Appendix A.1). The optimal levels of search effort satisfy:

Note that period-1 search effort depends on both period-1 and period-2 unemployment benefits, since the worker anticipates the possibility of remaining unemployed in period 2. Define the elastic- (4) and (5), we obtain

These elasticities will be play a key role in the subsequent optimal policy analysis, as they capture the moral hazard cost of UI. Note that, in addition to the standard distortionary effects of UI on contemporaneous search effort captured by (6) and (8), there is also an inter-period distortion captured by (7): future UI benefits distort current search effort.

Next, we examine how the moral hazard cost of UI depends on search efficiency. The sign of this dependence is theoretically ambiguous and depends on the shape of the cost function c S ð Þ. Differentiating ()()()(6)-(8) with respect to f 1 and f 2 -and taking into account that S 1 and S 2 depend on f 1 ; f 2 via (4) and (5) -yields the following result.

Lemma 1. All else equal, a necessary condition for the elasticities e 1ÀS 1 ;b 1 and e 1ÀS 1 ;b 2 to be increasing in f 1 , and for the elasticity e 1ÀS 2 ;b 2 is increasing in f 2 , is

All else equal, a necessary condition for the elasticities e 1ÀS 1 ;b 1 and e 1ÀS 1 ;b 2 to be increasing in f 2 is

Condition (9) provides a restriction on the cost function under which the moral hazard cost of UI is increasing in current search efficiency. Similarly, condition (10) provides a restriction under which the moral hazard cost of UI is increasing in future search efficiency. Note that, higher f 1 and higher f 2 move the level of 1 À S 1 in opposite directions. A higher f 1 makes search effort less costly and thus raises the optimal level of S 1 , all else equal. A higher f 2 makes future search less costly-lowering the opportunity cost of remaining unemployed today-hence lowers the marginal benefit of finding a job today. Condition (10) says that a higher S 1 , all else equal, lowers the elasticity of 1 À S 1 . Condition (9) says that this effect does not outweigh the direct effect of a higher f 1 , which amplifies this elasticity. 3 In the subsequent numerical analysis we will assume a functional form for c S ð Þ such that (9)-(10) hold; it turns out that standard functional forms from the literature, e.g. the one used by Mitman and Rabinovich (2015) , naturally satisfy these conditions. In doing so, we are motivated by the empirical findings of Kroft and Notowidigdo (2016) , who find that the elasticity of unemployment duration with respect to UI does in fact co-vary positively with the unemployment rate. It is worth emphasizing that this empirical result is not theoretically obvious, as discussed here. The derivation in Lemma 1 is, to our knowledge, new in the literature.

In a static (i.e. one-period) model, the conditions (9) and (10) would be sufficient for optimal UI to be decreasing in current f. This is not necessarily the case here, precisely because of the additional anticipation effects emphasized in our dynamic setting.

In this section we assume that the government can commit up front to the policy path. The government is choosing b 1 and b 2 to maximize social welfare using a utilitarian welfare metric

3 To understand these conditions mathematically, observe that we can write the optimal search condition in either period (t ¼ 1; 2) as ln c 0 St ð Þ ¼ ln f t þ ln Dt, where Dt is the marginal benefit of job search (i.e. the right-hand side of Eq. (4) or (5)). All else equal, a one-percent increase in the marginal benefit Dt translates into a one-percent increase in the marginal cost c 0 St ð Þ: this is simply a consequence of the worker's optimal search behavior. But the resulting percent change in 1 À St depends on the mapping from ln c 0 S ð Þ to ln 1 À S ð Þ ; the business cycle dependence of these elasticities therefore depends on how this mapping depends on the level of S.

with S 1 and S 2 determined as functions of b 1 and b 2 through the optimal search conditions (4) and (5), and u 1 and u 2 determined by the laws of motion (2) and (3).

The following result characterizes the optimal levels of unemployment benefits, leading to a two-period version of the standard Baily-Chetty formula.

Then the allocation under the optimal policy with commitment satisfies

Proof. See Appendix A.3.

We draw three lessons from this result. First, as is standard in the Baily-Chetty formula, each expression relates the consumption smoothing benefit of unemployment benefits in each period to its moral hazard cost. In turn, the moral hazard cost is captured by the respective elasticity, i.e. the distortionary effects of benefits on search effort. The terms K 1 and K 2 are fiscal externality measures, which capture the extent to which a decrease in search effort represents a budgetary loss for the government. The expression for K 1 in particular indicates that a period-1 reduction in search effort results in fiscal losses in both periods, since a worker who fails to find a job in period 1 might remain unemployed in period 2.

Second, the elasticities in question, whose expressions are given earlier by ()()()(6)- (8), depend on f 1 and f 2 . However, and importantly, they do not directly depend on u 1 and u 2 . In fact, the optimal benefit formulas (12) and (13) do not directly feature u 1 or u 2 except for the term u 1 u 2 in (13), which we will discuss momentarily. The reason is that both the insurance benefit and the moral hazard cost are proportional to the number of potentially unemployed workers. If the unemployment level with which the economy entered the period increases, this raises both the consumption smoothing benefit and the moral hazard cost by the same factor, keeping the resulting optimal benefit level unchanged.

Third, the role of commitment is captured by the presence of e 1ÀS 1 ;b 2 in (13). Period-2 benefits affect period-1 search effort. The

Ramsey planner takes this into account when deciding on period-2 benefits. This also explains the presence of the u 1 u 2 term. The insurance benefit of period-2 unemployment benefits applies to u 2 unemployed workers. The moral hazard cost in period 1 applies to u 1 unemployed workers. This means that if u 1 is large and u 2 is small, there is a small ex post benefit and a large ex ante cost. In particular, this channel implies a larger moral hazard cost in faster-recovering economy, since such an economy would have a larger ratio u 1 =u 2 .

We now consider the UI benefits chosen by a government who re-optimizes b 2 in period 2, rather than committing to the path of benefits in period 1. We solve the problem by backward induction. 4

In period 2, the government takes u 1 as given and chooses b 2 to maximize

subject to the period-2 law of motion (3) and the period-2 optimal search condition (5). Denote by b 2 ¼ B 2 u 1 ð Þ the period-2 government's optimal policy function, and denote by W 2 u 1 ð Þ the resulting maximized value of (35). Then, the period-1 government takes the functions B 2 Á ð Þ and W 2 Á ð Þ as given and chooses b 1 to maximize

subject to laws of motion (2)-(3), the optimal search conditions (4)-(5), and the behavior of the period-2 government b 2 ¼ B 2 u 1 ð Þ. The following result characterizes the equilibrium choices of b 1 and b 2 .

Then the allocation under the optimal policy without commitment satisfies v 0 b 1 ð Þ À g g

Proof. See Appendix A.4.

We now contrast the choice of optimal policy under discretion (Markov) and under commitment (Ramsey). First, comparison of (17) with (13) clearly illustrates the role of commitment: the term e 1ÀS 1 ;b 2 is now absent from the optimal UI formula. The period-2 government trades off the consumption smoothing benefit against the contemporaneous moral hazard cost, but does not internalize the fact that period-2 UI benefits distort search effort in the previous period. Thus, each period's government effectively follows a static Baily-Chetty formula. From this, it is straightforward to show that the choice of b 2 is higher under discretion than under commitment.

Second, while the optimal benefit formula in (13) depends on u 1 u 2 , the optimal benefit formula (17) does not, precisely because u 1 u 2 only enters in the term through which period-2 benefits affect period-1 search. This shows that the level of unemployment matters for the wedge between the Markov policy and the Ramsey policy, but it matters via the Ramsey policy, not the Markov policy. In particular, there is no sense here in which the Markov government is tempted to raise benefits more when unemployment is high. Instead, the Ramsey government would like to promise low benefits in future states in which unemployment is low; the Markov government fails to credibly promise this. As a result, the Markov government will set UI too high (relative to the commitment case) particularly in states of rapidly recovering unemployment. Third, we inspect how inability to commit affects period-1 benefits. The form of (16) is exactly the same as that of (12). However, the magnitude of the elasticity e 1ÀS 1 ;b 1 (and, for that matter, K 1 ) changes due to lack of commitment. Since a higher b 2 leads to a lower S 2 , the elasticity e 1ÀS 1 ;b 1 rises, all else equal, by the assumptions made earlier. In turn, this means that the period-1 UI benefits are lower under discretion than under commitment. The inability to commit to future policy worsens the moral hazard problem today.

To summarize the lessons from the two-period model: the government with commitment uses both current and future UI benefits to provide current incentives, and therefore has a tendency to front-load UI benefits, which is absent under discretion. This tendency to front-load UI benefits -and hence the wedge between commitment and discretionary optimal policy -is stronger when economic conditions are rapidly improving, i.e. when search efficiency is rising and unemployment is falling. These results also have an important implication for implementing the optimal policy in practice, as they clearly indicate that the time path of unemployment, not its level, is of first-order importance for the optimal policy. We will confirm below that simple policy rules conditioning UI on the change in unemployment in fact perform better than those conditioning on the level of unemployment.

We now describe the full model, which differs from the simplified two-period model in two main respects. First, the time horizon is infinite. Second, the aggregate shocks to search efficiency are now stochastic and we add shocks to the separation rate. The economy is populated by a continuum of infinitely-lived risk-averse workers, with utility given by

where, as before, x t denotes period-t consumption, and S t denotes period-t search effort. We assume that f t and d t follow AR(1) processes

and denote the history of f t and d t shocks up to period t as Z t ¼ f 1 ; . . . f t ; d 1 ; . . . d t f g . Unemployment u t evolves according to the law of motion

When employed, workers receive exogenous income w and pay a tax s; when unemployed, they receive h þ b t , where h is an exogenous value of home production and b t is the government-provided unemployment benefit. The unemployment benefit b t , which is the policy choice of interest, can potentially be contingent on the entire past history of shocks, Z t . Unemployed workers choose S t at each point in time to maximize expected utility, taking as given the government policy b t Z t À Á . We show in Appendix B.1 that the worker's optimal search behavior leads to the Euler equation for search intensity,

The Euler equation equates the marginal cost of additional search to the marginal benefit; the latter is the combination of the consumption gain from becoming employed and the benefit of economizing on search costs in the future. Given a policy path b t Z t À Á and an initial condition for unemployment u 0 , the equilibrium is fully characterized by law of motion (21) and Euler Eq. (22).

In this section, we describe the Ramsey problem, i.e. the problem of a benevolent government with full commitment, which can condition UI benefits on the entire history of aggregate states. 5 Such a government chooses the path b t Z t À Á together with S t Z t À Á

subject to (21) and (22). We show in Appendix B.2 that the optimal policy path satisfies the recursive equation

where K t is a measure of the fiscal externality, satisfying

The Ramsey equilibrium, for a given cost of funds g, thus consists of

and u t Z t À Á , and K t Z t À Á satisfying (21), (22), (24) and (25).

As already anticipated in the analysis of the two-period model, expression (24) also sheds light on what kinds of simple policy rules are likely to achieve desirable outcomes in practice. Policies that condition on the change in unemployment, rather than its level, are likely to perform well for two reasons. First, the change in unemployment directly enters in the optimal benefit formula with commitment. Second, the change in unemployment is also a relatively good proxy for f t , which likewise directly affects the optimal benefits. In contrast, the level of unemploymentper sedoes not enter the optimal benefit formula.

To characterize the optimal policy under discretion, we follow the approach of Klein et al. (2008) , solving for the Markov equilibrium of a sequential move game between successive governments. In a Markov equilibrium, the government's policy functions and the resulting actions of the private agents are functions only of the aggregate state u; f; d. Let Z ¼ f; d f g denote the exogenous aggregate states. Policy functions constitute an equilibrium if the current-period government finds it optimal to follow the policy functions given that future governments will follow them.

Defined recursively, the Markov equilibrium is described by a value function V u; Z ð Þ together with policy functions b ¼ B u; Z ð Þ; S ¼ S u; Z ð Þ and u 0 u; Z ð Þ, such that. 

The first condition states that the policy functions are optimal given that the future value function of the government is given by V. The second states that today's value is in fact given by V if the policy functions are followed. This is a system of functional equations, which is a complicated fixed-point problem in general. Given the structure of our model, we show in Appendix B that the problem simplifies further. Specifically, there is a Markov equilibrium such that the policy functions b ¼ B u; Z ð Þ; S ¼ S u; Z ð Þ and u0 u; Z ð Þdepend on Z only, not on u. The feature that enables this is precisely the one leading to the absence of u in Eqs. (16) and (17), characterizing the optimal (discretionary) policy in the twoperiod model: u affects symmetrically both the consumption smoothing benefit and the moral hazard cost of UI. In addition to the substantive implications discussed earlier, this simplifies the computation of the solution considerably. We show in Appendix B that the path of Markov optimal policy allocations satisfies the difference equation

with K t again given by (25). The Markov equilibrium is thus characterized by sequences b t Z t ð Þ; S t Z t ð Þ and u t u tÀ1 ; Z t ð Þ , and K t Z t ð Þ satisfying (21), (22), (30) and (25). Note that (30) differs from (24) due to the absence of the backward-looking term reflecting the time inconsistency of the Ramsey policy. In words, the Ramsey (commitment) policy calls for higher distortions today if higher distortions were required yesterday. The Markov (discretionary) policy faces no such constraint; as a result, the formula (30) looks very much like a static Baily-Chetty formula, just as the formulas (16) and (17) in the two-period model.

In this section we parameterize the full model and characterize the behavior of the optimal policy under commitment (Ramsey) and discretion (Markov), given various paths of economic shocks. The model period is one week. We assume that utility of consumption takes the functional form v x ð Þ ¼ ln x, and that the cost of search has the functional form

which satisfies all the assumptions made above; in particular it satisfies (9)-(10) and ensures that the optimal search effort is always strictly between 0 and 1. We set the discount factor equal to b ¼ 0:99 1 2 to match a 4% annual discount rate. We set d ¼ 0:0081 to match the weekly job separation rate from the CPS. We jointly estimate the disutility parameters in the search cost function A ¼ 3 and w ¼ 1:9 so that the model is consistent with the average unemployment rate and an elasticity of unemployment duration with respect to unemployment benefits of 0:12. 6 Finally, we normalize the wage to 1, so that b is interpreted as a replacement rate. 7 We set the value of home production h ¼ 0:2 to match estimates of the drop in consumption upon unemployment (e.g. Cox et al. (2020) ).

In all the experiments below, we consider the optimal policy response to an unexpected shock starting from steady state, which we assume has an unemployment benefit replacement rate of b ¼ 0:45 and unemployment rate of u ¼ 0:04 (to match the pre-COVID unemployment rate 8 ). This raises the question of whether to interpret the steady state of the US economy as optimal. We choose the following strategy. For the Ramsey experiment, we choose a cost of funds g so that the US allocation is the steady state Ramsey optimum. For the Markov experiment, we re-compute the cost of funds g so that the US allocation is the steady state of the Markov equilibrium. We make this assumption to facilitate comparison between the Markov and Ramsey policies. As we consider mean reverting shocks from steady state, under both policy scenarios the economy begins and returns to the same steady state. 9 The difference between the two policy paths can thus be entirely attributed to the commitment ability of the government along the transition, rather than its preference for a higher/lower UI on average.

Our experiment is intended to capture, in a stylized way, an economic crisis akin to the one triggered by COVID-19. We treat the COVID-19 recession as a combination of a temporary rise in d t and a prolonged fall in f t . We can think of these as encompassing policy responses and the decline in economic activity resulting from the spread of the virus. For example, the adverse shocks reflect NPI's, such as orders to limit restaurants to take-out only and stay-at-home orders, as well as reluctance or inability to search due to the fear of becoming infected, along with a shortage of newly posted job openings. We choose the paths for d and f to roughly correspond to the separation rate from the Job Openings and Labor Turnover survey by the BLS, and the decline in economic activity from Bognanni et al. (2020) , respectively. We assume that the economy experiences this sequence of adverse d t and f t shocks starting from a steady state at t ¼ 0. Agents have perfect foresight of the entire future path of f t and d t .

Panels 1a and 1b of Fig. 1 plots the time paths of the f t and d t shocks. Panel 1c shows the policy responses under the Ramsey and Markov policy; panel 1d shows the corresponding time path of unemployment under each policy response and under a constant-UI policy for reference purposes.

The figures clearly illustrate the contrast in policy responses between the Ramsey and Markov governments. The government with commitment power responds to the adverse shock by substantially raising unemployment benefits at the onset of the reces- 6 We note that there is an ongoing and active debate regarding the effects of unemployment benefits (levels and duration) on worker search effort (micro effects) and firm vacancy creation (macro effects). In innovative work using administrative data from Missouri, Johnston and Mas (2018) find significant affects of potential benefit duration on worker search effort, as measured through exits into employment (though, (Karahan et al., 2019) , also looking at Missouri find a smaller role for search effort). However, other work during the Great Recession by Rothstein (2011) finds an elasticity of 0. Thus, we see our calibration choice as on the lower side of the range of recent estimates. We show in Appendix C that our results are robust to different calibrations. 7 We abstract from the impact of unemployment benefits on re-employment wages, which is also the subject of an ongoing and active debate. 8 Our results are not sensitive to the choice of steady state unemployment in the ranges experienced in the post-War U.S. 9 In other words, in thinking about the Ramsey-optimal response to shocks, we adopt the ''timeless perspective" advocated by Woodford (1999) . Our Ramsey government at t ¼ 0 is thus one that has been already following the Ramsey policy for a long time prior to period 0.

sion, and then quickly cutting them back. This is both because the Ramsey policy responds more on impact to the decline in f, and because the Ramsey policy responds more to the high unemployment triggered by the increase in d. 10 The Markov policy response features a smaller but more prolonged rise in benefits, reflecting exactly the mechanism discussed earlier: the inability to commit to lower benefits upon the future recovery worsens the moral hazard problem today. Note that, while the Ramsey government appears to simply respond to high unemployment by raising benefits, this is not the mechanism here. Instead, the Ramsey optimal policy correctly anticipates that unemployment will subsequently fall, calling for a declining time profile of UI rather than a high level per se. It is also apparent that the Markov policy tracks very closely the simulated path of f t : in particular it does not lower benefits until the contemporaneous f t recovers itself, whereas the Ramsey policy lowers benefits preemptively, while search efficiency is still stagnant, in order to provide forward-looking incentives.

Importantly, the unemployment rate is very similar across policies (Ramsey, Markov, and constant benefits). This does not, by any means, imply that there are small welfare gains from commitment. On the contrary, the welfare gains from the Ramsey policy come about because the Ramsey planner reallocates UI from periods with few unemployed to periods with many unemployed. The latter still search just as intensively despite higher benefits today: in other words, the moral hazard effects of the initial rise in benefits are mitigated by the commitment to lower them in the future. By contrast, the Markov government does not raise benefits as much in the crisis because it cannot commit to lowering them in the recovery. The decline in welfare in consumption equivalent variation (CEV) terms as a result of the pandemic is 0:44% of lifetime consumption under the Ramsey policy, compared to 0:57% CEV under the Markov policy. Commitment, therefore, reduces the welfare cost by approximately one quarter.

Our analysis thus far has assumed an environment with ex-ante homogeneous workers, and economy-wide shocks that hit all the workers identically. However, the literature is replete with evidence of duration dependence in unemployment, whether it is due to human capital depreciation, dynamic selection on ability, or employer discrimination. More specific to the COVID-19 crisis, there is ample evidence of heterogeneous effects of the pandemic, which disproportionately affected service sectors and workers unable to work from home. Motivated by these considerations, we examine how heterogeneity in worker search efficiency affects our results.

Specifically, we assume that a fraction u of workers have high search efficiency f t a h , and 1 À u of workers have search efficiency f t a l , with 0 < a l < a h , where both a l and a h are time-invariant. The economy-wide moral hazard cost of UI now depends not only on aggregate f t , but also on the fraction of low types among the unemployed. We calibrate u ¼ 0:3; a h =a l ¼ 10 and a h ¼ 2:7, to match the same steady-state unemployment rate as in the homogeneous worker case. In steady state, in particular, the pool of unemployed consists disproportionately of low-type workers (whose unemployment rate is three times higher than the high-type workers), for whom the moral hazard cost of UI is lower than for hightypes, by the arguments made earlier.

When workers are heterogeneous, the optimal UI profile in response to an adverse shock is potentially flattened through a dynamic selection effect, consistent with Kolsrud et al. (2018) : the initial destruction shock temporarily shifts the composition of the unemployed towards the high-efficiency workers and, as the recession progresses, the composition shifts back towards low-efficiency workers. 11 An additional layer of complexity arises if the aggregate shock itself hits the two types unequally. 12 These considerations are illustrated in panels 1e and 1f of Fig. 1 . The curve labeled ''Benchmark" illustrates the Ramsey optimal policy without heterogeneity. The curve labeled ''Heterogeneity" illustrates the Ramsey optimal policy with heterogeneity, assuming that the negative shock hits both types symmetrically. The flatter path of optimal benefits in this case is a manifestation of the dynamic selection effect alluded to earlier. Finally, ''Unequal incidence" illustrates the Ramsey optimal policy when low types bear the brunt of the pandemic and are subject to both higher job destruction and a larger decline in search efficiency. The dynamic selection effect is still present here, but initially muted, as the initial composition of the unemployment pool changes by less than when there is equal incidence of the destruction shock on the two types. However, as the high-types experience a milder shock to search effort the dynamic selection manifests itself more quickly as the high types find jobs. At that stage the original mechanism kicks in, whereby the optimal policy tracks the (more severe) f shock experienced by the low types. So benefits stay elevated, though at a lower level, and slowly decline providing the same dynamic incentives discussed in the benchmark. Thus, the presence of heterogeneity suggests a milder slope for the decline in benefits, all else equal. Nonetheless, our conclusion that optimal benefits should be indexed to the change in unemployment rather than the level continues to hold, for much the same reasons.

Finally, we return to the question of implementing the Ramsey policy in practice. The preceding discussion already suggests that a simple policy rule conditioning UI on the change in unemployment is likely to perform better than a policy rule conditioning on level of unemployment. We confirm this in our numerical examples. First, we find that a policy rule contingent on the unemployment level results in keeping UI benefits high for too long. 13 Second, we find that a policy rule conditioning UI on the change in three-month moving average of the unemployment rate between t and t À 1 performs remarkably well in terms of approximating the optimal Ramsey policy. 14 We obtain this by running the following numerical experiments. First, we simulate the optimal Ramsey policy in the COVID-19 experiment, in which f t and d t are assumed to evolve as Section 4.1. Having obtained the model-generated series for the optimal path of b t and the resulting unemployment rate, we then use the series to estimate a policy rule of the form b t ¼ b ss þ c level u tÀ1 À u ss

where b ss and u ss are the steady-state levels of UI benefits and unemployment. We term this policy rule the level rule. In the second experiment, we estimate an alternative policy rule contingent on the change in the moving-average of unemployment, u, rather than the level, b t ¼ b ss þ c change u tÀ1 À u tÀ2 ð Þ þ m t ;

terming this policy rule the change rule. We obtain c level ¼ 1:8 and c change ¼ 13:7. While the path of unemployment under the change rule closely follows that of the optimal Ramsey policy, the level rule, by keeping unemployment benefits high after search efficiency has recovered (because unemployment is still high) propagates the high unemployment further, generating hysteresis (see, e.g., Mitman and Rabinovich (2019)). These results are illustrate in Fig. 2 .

We have revisited the question of whether, when, and how to extend UI in recessions. The broad lesson is that expectations matter. People's job search behavior depends on future UI benefits as well as future labor market conditions. A government with commitment power takes advantage of this by front-loading UI, i.e., back-loading incentive provision when the labor market is recovering from a crisis. Inability to commit may manifest itself as a positive correlation between UI and unemployment, even though unemployment levels per se do not dictate the government's policy response.

The specific policy recommendation is for UI to track the growth rate of unemployment rather than its level. The change in unemployment between periods is a good proxy for search effi- 11 The size of this effect depends on both the ability distribution and the turnover post-initial shock. In particular, since regular layoffs continue to occur after the initial shock, newly laid-off high-type workers continue to enter the pool of unemployed even as the pool of legacy unemployed shifts toward low types. Because the fraction of the former is significant, this tempers the dynamic selection effect. 12 It is quite plausible, particularly in the COVID-19 episode, that the workers hit the most by the adverse shocks are the very workers with low search efficiency, e.g. service sector workers with few work-from-home opportunities. Indeed, Gregory et al. (2020) provide evidence that the separation shocks induced by the pandemic may have disproportionately affected workers that take significantly longer to find stable jobs in the future. 13 An implementation that depends on the three-month moving average of the unemployment rate yields similar conclusions. 14 Note that the current federal-state extended benefits (EB) program has triggers based on the level and change in the three-month moving average of the unemployment rate, so we believe that these are within the class of feasible simple rules to implement. cacy, which governs the moral hazard cost of unemployment insurance. Further, committing to lower UI when unemployment is falling provides proper search incentives for the unemployed workers in previous periods. It does so at minimal cost in terms of consumption insurance: it provides insurance to the many unemployed workers at the onset of the crisis while promising to cut it for the few unemployed workers at its conclusion.

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This section provides the derivations leading to the optimal search conditions (4) and (5). We solve the worker's problem backwards. A worker entering period 2 unemployed has the value U 2 ¼ maxand a worker entering period 2 employed has the valueThe optimal S 2 clearly satisfies (5). Then, a worker unemployed in period 1 solves A worker entering period 1 unemployed therefore has the value max s 1The optimal S 1 then solves 1which leads to (4).

Proof (Proof of Lemma 1). Differentiation of e 1ÀS 1 ;b 1 , given by (6),with respect to f 1 (keeping policy and future variables fixed) giveswhere the second line follows from differentiating the worker's optimal search condition (4) with respect to f 1 . The above quantity is positive if and only if (9) holds. The same argument applies when differentiating e 1ÀS 2 ;b 2 , given by (6), with respect to f 2 . Next, differentiation of e 1ÀS 1 ;b 1 with respect to f 2 (again, keeping policy and future variables fixed) givesBy the worker's optimality condition (5), the derivative of the righthand side of (4) with respect to f 2 is À 1 f 2 2 c S 2 ð Þ < 0, so that @S 1 @f 2 < 0. This means that the expression in (40) is positive if and only if (10) holds.

Proof (Proof of Proposition 1). Let k 1 and bk 2 be the Lagrange multipliers on the laws of motion (2) and (3), and let l 1 and bl 2 be the Lagrange multipliers on the incentive constraints (4) and (5). Then the first-order conditions for b 1 ; b 2 ; S 1 ; S 2 ; u 1 , and u 2 , respectively, are: Finally, combining (47)-(48) with (6)- (8) gives us (12) and (13).

We solve the optimal policy problem backwards. We begin by solving the problem of the period-2 government, which maximizes (14) subject to (3) and (5), taking u 1 as given. Letting k 2 and l 2 be the multipliers on (3) and (5), we obtain the first-order conditions for b 2 ; S 2 and u 2 , respectively, asCombining (49)-(51) with (3) and (5) yieldswhich, from (8), is equivalent to (17). Note that the resulting solution b 2 ¼ B 2 u 1 ð Þ turns out to be independent of u 1 , since b 2 and S 2 are uniquely pinned down by (52) and (5).We next turn to the problem of the period-1 government. Since b 2 and S 2 do not depend on u 1 as established above, maximizing (15) is equivalent to maximizing (11) subject to (2)-(3), (4)-(5), and the additional constraint (52). The first-order conditions for b 1 ; S 1 , and u 1 , therefore, are still given by (41) where the period-t expectation is taken with respect to f tþ1 and dependence on Z t is suppressed for notational convenience. From (54), the first-order necessary condition for the optimal S ¼ S t isSubtracting (54) from (53) also givesCombining (55) with (56) gives (22).

The Ramsey problem consists of maximizing (23) subject to (21) and (22). Dependence on Z t is understood throughout. Letting b t k t and b t l t be the Lagrange multipliers on (21) and (22), we find the first-order conditions for b t ; S t and u t , respectively, to beThe term k t À 1 f t c 0 S t ð Þ is the fiscal externality from job search, and it is positive because the worker does not internalize that their job search affects future net government revenues. DefineCombining (59) with (22) it is easy to see that K t satisfies (25). Combining (57) with (58) to eliminate l t and l tÀ1 , and substituting for u tÀ1 þ d t 1 À u tÀ1 ð Þ ð Þ using (21), then gives the expression (24).

Consider the problem of maximizing (26) subject to (27) and (28). Let k and l be the Lagrange multipliers on (27) and (28). Then the first-order conditions for b; S, and u 0 , respectively, are:The envelope condition isWe now want to solve for a Markov equilibrium in which S u u; Z ð Þ¼0, i.e. the optimal policy functions depend on Z but not v 0 b ð Þ À g g Next, combining (62) with (63) and assuming future S u ¼ 0, we get that K satisfiesAnd so (64), (65) and (28) give us as system of functional equationswhich can be solved independently of u. In sequence form, (64) is (30).

In this section, we explore the robustness of our results to both the specification of the search technology and its precise parameterization. We first consider an alternative, commonly used, search technology:It can be verified that this search cost function likewise satisfies conditions 9 and 10 for sufficiently large / (which turns out to always be the case in our calibration). Below, we display the Ramsey policy in response to the ''COVID" shock with this alternative cost function. We set / ¼ 2 and A ¼ 95, which yields the same steady state unemployment as the benchmark and also a micro-elasticity of 0.11. The results are plotted in Fig. 3 . While the exact magnitudes are not identical, the qualitative conclusions are unchanged by this choice of functional form. Next, we return to our original specification of the search cost, but conduct sensitivity analysis with respect to the choice of parameters. In the benchmark we calibrated to a conservative value of the elasticity of search to benefits of 0:12. We recalibrate A and h to generate steady states with identical unemployment, but higher (0.16) or lower (0.09) elasticity. These correspond to h ¼ 0 and h ¼ 0:3, compared to a benchmark of h ¼ 0:2. We then re-run the benchmark ''COVID" shock and compute the Ramsey policy. We plot the path of benefits and unemployment under those three scenarios in Fig. 4 . While the dynamics of the unemployment rate are largely unaffected by the calibration of the search elasticity, the magnitude of the increase in benefits is sensitive to the calibration. However, the overall shape of the optimal benefit path is unchanged. Perhaps surprisingly, benefits are higher when the elasticity of search to benefits it higher. What underlies the result is that when the elasticity of search to benefits is higher, that elasticity itself is more sensitive to shocks to f, so that for the same size shock the moral hazard cost falls by more for the high elasticity calibration than the low elasticity one. Thus, the Ramsey planner optimally provides higher benefits when the search efficiency cost shock hits.