Discretionary Extensions to Unemployment Insurance Compensation and Some Potential Costs for a McCall Worker

Abstract

Unemployment insurance provides temporary cash benefits to eligible unemployed workers. Benefits are sometimes extended by discretion during economic slumps. In a model that features temporary benefits and sequential job opportunities, a worker’s reservation wages are studied when policymakers can make discretionary extensions to benefits. A worker’s optimal labor-supply choice is characterized by a sequence of reservation wages that increases with weeks of remaining benefits. The possibility of an extension raises the entire sequence of reservation wages, meaning a worker is more selective when accepting job offers throughout their spell of unemployment. The welfare consequences of misperceiving the probability and length of an extension are investigated. Properties of the model can help policymakers interpret data on reservation wages, which may be important if extended benefits are used more often in response to economic slumps, virus pandemics, extreme heat, and natural disasters.

1. Introduction

Each month, millions of workers lose their jobs. Unemployed workers search for jobs by answering online adverts, contacting employers, using the services of employment agencies, and asking friends and relatives about employment opportunities. Workers understand the the types and frequency of job offers they expect to receive. Depending on how long they are unemployed, workers are entitled to unemployment insurance compensation, typically 26 weeks. Upon receiving a job offer, a worker has the option to reject the offer and continue their search or accept the job. A sequence of reservation wages describes optimal choices: any wage offer above the reservation level should be accepted. As more weeks of unemployment are endured, and fewer weeks of unemployment insurance remain, the reservation wage will fall, indicating that workers are less selective as benefits expire. Less is known about optimal decisions, however, when policymakers can extend benefits by discretion.

In brief, here is the main issue: Unemployed workers are often entitled to claim 26 weeks of UI compensation. In periods of economic distress, policymakers sometimes extend the number of weeks an individual might claim UI compensation by discretion. When a worker misperceives the probability and length of an extension, what are the costs?

I answer the question from a worker’s perspective. Imagine that Sonia is unemployed and submitting her resume to online adverts. Each week, Sonia receives a job offer that she can accept or reject. Meanwhile, her unemployment compensation benefits are expiring. Between job offers, however, there is a chance that benefits are extended. An extension would allow Sonia to claim additional weeks of UI compensation. If an extension is likely, then Sonia can reject low-wage offers, knowing that her search will likely be supported by extended UI compensation. Sonia’s reservation wage increases. Sonia, though, might not know the true probability that benefits are extended and the length of the extension.

I am interested in Sonia’s welfare costs when she misperceives the probability and length of an extension. If Sonia believes an extension is unlikely to happen, so the true probability of an extension is higher than what Sonia believes, then Sonia risks accepting job offers she would like to reject. In this scenario, if Sonia knew the true probability, then, on average, she would like to spend more time searching in order to find a higher-paying job. To understand these costs, I develop a formal dynamic model of sequential job search with expiring UI benefits, which allows for the possibility that benefits are extended. Burdett (1979) studied expiring benefits in McCall’s (1970) environment of sequential job search. I add to this environment the possibility that benefits are extended. I show how perceptions about the possibility of an extension affect reservation wages, which determine a worker’s job-acceptance criteria.

The decision-making environment I consider captures some features of UI extensions made during and after the Great Recession and the COVID-19 pandemic. For example, extensions to UI compensation benefits created by Congress during the Great Recession under the Emergency Unemployment Compensation program expired three times, and each time Congress had to reauthorize the program to the previous expiration date (Rothstein 2011). In fact, temporary additional UI benefits have been created by Congress nine times: in 1958, 1961, 1971, 1974, 1982, 1991, 2002, 2008, and 2020. These policies have extended the number of weeks a worker could claim UI compensation anywhere from 6 to 53 weeks (Whittaker and Isaacs 2022). As I document in Section 2, there is much uncertainty and risk from a worker’s perspective when extensions are made by discretion.

Because discretionary extensions made to UI compensation benefits are endogenous by design, making econometric work challenging, there is scope for investigating a theoretical model. I present the model in Section 3 and Section 4. Section 5 presents a numerical example. The exercise is not meant to be definitive. Instead, it is meant to demonstrate the theoretical properties established in Section 3 and Section 4. Nevertheless, it is worth noting that the welfare calculations imply small costs to misperceiving benefit extensions. Finding small costs is discussed in the context of the literature in Section 6. Section 7 concludes.

2. Two Episodes of Extended Benefits

There are two ways for UI compensation benefits to be extended: automatically and by discretion. Benefits are automatically extended through the Extended Benefits (EB) program of the UI system. The UI system is a joint federal–state partnership, essentially with 53 different systems (the 50 states, Puerto Rico, the District of Columbia, and the US Virgin Islands). Regular UI in most states provides 26 weeks of UI compensation. The EB program extends the number of weeks a worker can claim UI compensation by 13 or 20 weeks when state-specific unemployment-rate triggers are reached. But, “in practice, the required EB trigger is set to such a high level of unemployment that the majority of states do not trigger onto EB in most recessions” (Whittaker and Isaacs 2022, pp. 1–2). In response, Congress often temporarily extends UI benefits, like they recently did under the Coronavirus Aid, Relief, and Economic Security Act (CARES Act) during the COVID-19 pandemic and the Emergency Unemployment Compensation (EUC) program during the Great Recession (Fujita 2010). From a worker’s perspective, both types of extensions involve uncertainty and risk: Automatic extensions require a forecast of unemployment-rate statistics and discretionary extensions require a forecast of legislative action.1

Details of the EUC program and CARES Act justify investigating the costs associated with misperceiving the probability and length of an extension. Section 2.1 considers a thought experiment about how difficult it was for a worker to perceive that they would be entitled to claim up to 99 weeks of UI benefits after the Great Recession. Section 2.2 shares an index of Google searches for “unemployment benefits extension,” which spikes around potential cutoffs to extended benefits after the COVID-19 pandemic. Both episodes document the significant uncertainty a worker must manage when making decisions about accepting job offers when benefits can be extended.

2.1. Extensions Created in Response to the Great Recession

Imagine that Vernon lived in California and became unemployed in early January 2010 when the United States was dealing with the fallout from the Great Recession. The EB program in California had been triggered on since 22 February 2009, so Vernon could expect to receive 26 regular weeks of UI coverage and 20 EB weeks, for a total of 46 weeks—as long as California did not trigger off its extended-benefits thresholds, requiring Vernon to forecast unemployment-rate statistics. At that time, Vernon could reasonably expect to receive no temporary benefits because the EUC program was set to expire in late February 2010 (Rothstein 2011, p. 150, Table 1). Around 25 weeks later, or around 27 June 2010, California’s EB program had triggered off and the federal EUC program had also expired.2

What Vernon expected then is unknowable, but by 25 July 2010, the EB program had triggered on in California and the EUC program had been reauthorized. Vernon, at that time, could reasonably expect—in addition to the 26 weeks of regular UI compensation just received—20 EB weeks plus $20+14+13+6=53$ weeks of UI compensation under the four tiers of the reauthorized EUC program (where some of the compensation would be paid retroactively) (Rothstein 2011, p. 153). In total, Vernon could have received up to $26+20+53=99$ weeks of UI compensation. Yet, the 99 weeks belies the uncertainty and risk faced by Vernon.3

Vernon’s predicament was pointed out by Kahn (2011) in the context of UI extensions in the aftermath of the Great Recession. After the EUC program was authorized in June 2008, temporary extensions were enacted and reauthorized by Congress in “fits and starts” (Rothstein 2011, p. 149). “For much of the program’s history, the expiration date was quite close. Indeed, on three occasions … Congress allowed the program to expire. Each time, Congress eventually reauthorized it retroactive to the previous expiration date” (Rothstein 2011, p. 150).

2.2. Extensions Created in Response to the COVID-19 Pandemic

Uncertainty similarly surrounded CARES Act extensions. The CARES Act was created by Congress in response to the recession caused by the COVID-19 pandemic. Two CARES Act programs extended UI compensation benefits. The Pandemic Emergency Unemployment Compensation (PEUC) program provided additional weeks of federally funded benefits similar to the EUC program in the Great Recession. And the Pandemic Unemployment Assistance (PUA) program expanded coverage to individuals who would be ineligible for UI benefits (the self-employed, independent contractors, gig-economy workers) and to individuals unemployed due to COVID-19-related reasons. Both programs were reauthorized multiple times. Data on internet searches suggest that workers were concerned about access to extended benefits throughout 2020 and 2021.

Figure 1 depicts a weekly index of Google searches for “unemployment benefits extension.” In early 2020, before the COVID-19 pandemic, the index was near zero. Google searches began rising by mid-March and peaked locally in the week after 27 March 2020, which coincided with the signing of the CARES Act into law, creating “several temporary, now-expired UI programs” (Whittaker and Isaacs 2022).

During the third week of June 2020, the index was around 25. But, when the additional USD 600-per-week benefit created by the Federal Pandemic Unemployment Compensation (FPUC) program expired in the week ending 25 July 2020, the indexed neared its global peak. The 100 peak was reached in the week of 8 August 2020, when President Donald Trump authorized the USD 300-per-week Lost Wages Assistance benefit.

A little over 21 weeks passed between 13 March 2020 (when a nationwide emergency was declared) and 8 August 2020.4 So, up to this point, many workers would have been covered by 26 weeks of regular UI benefits. The local peaks that follow have to do with benefit extensions. In the week of 27 December 2020, the Continued Assistance Act was signed into law, which increased the maximum number of PEUC weeks from 13 to 24 and increased the maximum number of PUA weeks from 39 to 50. In the week of 11 March 2021, the American Rescue Plan increased the maximum number of PEUC weeks to 53 and increased the maximum number of PUA weeks to 79. Both the PEUC and PUA programs were extended through 4 September 2021, when temporary UI programs ended (Spadafora 2023, p. 4). As documented in Figure 1, each of these policy milestones coincided with increased Google searches for “unemployment benefits extension”.5

3. Job Search with Expiring Benefits

Before turning to a decision-making environment where UI benefits can be extended, I start with a canonical model of sequential search. In the environment, a worker is entitled to N periods of UI compensation benefits. The worker searches for a job, taking market conditions as given. Market conditions are summarized by a wage-offer distribution. The worker receives a wage offer each period. Upon receiving an offer, the worker decides whether to accept the job or continue search. After N rejected job offers, the worker will no longer receive UI benefits.

This sequential-search problem, where remaining periods of UI compensation is a state variable, was studied by Burdett (1979). The optimal solution is a sequence of reservation wages that is increasing in the remaining periods of UI compensation. To maximize their expected income, a worker rejects any job offer with a wage below the reservation wage and accepts any job offer with a wage above the reservation wage. Burdett (1979) proves the result using Bellman equations for employment and unemployment. In contrast, I provide a proof that expresses the model in terms of reservation wages, using techniques described by Rogerson et al. (2005). I believe the optimal policy is more transparent. In addition, my proof provides an algorithm for directly computing the sequence of reservation wages.

Krueger and Mueller (2016), using a model discussed by Mortensen (1977), consider a similar environment—where workers can claim UI compensation for a finite number of weeks—but allow for jobs to end. A worker who holds a job that does not last 6 months does not qualify for UI compensation. The added complexity, however, requires them to rely on numerical results. Unlike the environment I study, they do not allow for the possibility that benefits are extended. Boar and Mongey (2020) consider a related problem where workers consider expiring additional benefits added by the CARES Act. They are interested in whether workers will reject return-to-work offers in order to claim the USD 600-per-week additional UI benefit. Benefits expire with a fixed probability, though, so optimal decisions are not characterized by a sequence of reservation wages, a feature that is essential to the search environment I consider. Thus, the results of Krueger and Mueller (2016) and Boar and Mongey (2020) are complementary to my analysis.

The environment considered in this section is generalized in Section 4, where extensions to UI benefits are considered. Additional references to the literature on unemployment insurance are shared in Appendix A.

3.1. The Sequential Search Environment with Expiring Benefits

Time is indexed by t. A worker searches for a job in discrete time. They seek to maximize

$${\mathrm{E}}_0\sum _{t=0}^{\infty }{\beta }^t{x}_t,$$

(1)

where $\beta \in \left(0,1\right)$ is the discount factor such that $\beta ={\left(1+r\right)}^{-1}$, $x_t$ is income at time t, and ${\mathrm{E}}_0$ denotes the expectation taken with respect to income given available information at time 0. A worker’s income is their wage if employed and the value of nonwork if unemployed. The value of nonwork includes unemployment insurance compensation, the value of leisure, and the value of home production. Justification for modeling a risk-neutral worker with no savings includes the fact that “a majority of unemployed workers have only a trivial amount of savings” and “extending the model to include savings is an unnecessary complication for a large portion of the unemployed” (Krueger and Mueller 2016, pp. 146–47).6

A job is fully characterized by the wage the job pays, but a worker has imperfect information about job possibilities. The imperfect information is characterized as a distribution of possible wage offers. While unemployed, a worker samples one independent and identically distributed offer each period from a known cumulative distribution function F. I assume F is continuous, has finite expectation, and has support $\left[\underline{w},\overline{w}\right]$ or $\left[\underline{w},\infty \right)$. While I refer to w as the wage, “more generally it could capture some measure of the desirability of the job, depending on benefits, location, prestige” (Rogerson et al. 2005, p. 962).

If a worker rejects a wage offer, they remain unemployed. If a worker accepts a wage offer, they keep the job forever. An accepted job entitles the worker to a wage payment each period. I let $W\left(w\right)$ denote the payoff from accepting a job offering wage w. The Bellman equation for W satisfies

$$W\left(w\right)=w+\beta W\left(w\right)\mathrm{or}W\left(w\right)=\frac{w}{1-\beta }.$$

(2)

While unemployed, a worker’s income includes the value of leisure and home production, $z>0$. In addition, a worker may be entitled to UI compensation benefits, $c>0$. I let n denote the remaining periods of benefits, typically weeks, where $n\in \left\{0,\dots ,N\right\}$ and N is the maximum number of periods a worker is entitled to claim UI, typically 26 weeks. While unemployed, a worker’s income is $x=z+c$ if $n>0$ and $x=z$ if $n=0$. For positive n, the payoff of rejecting a wage offer, earning $z+c$, and sampling a wage offer the following period is

$$U\left(n\right)=z+b+\beta \mathrm{E}\left[max\left\{U\left(n-1\right),W\left(w\right)\right\}\right].$$

(3)

The expectation is taken with respect to potential wage draws. When $n=0$, the payoff of rejecting a wage offer, earning z, and sampling a wage offer the following period is

$$U\left(0\right)=z+\beta \mathrm{E}\left[max\left\{U\left(0\right),W\left(w\right)\right\}\right].$$

(4)

The Bellman equations in (2)–(4) will be used to write the problem in terms of reservation wages in the next section.

3.2. Optimal Search

Optimal search implies a reservation-wage policy.7 The payoff of accepting a job, $W\left(w\right)=w/\left(1-\beta \right)$, starts from 0 and strictly increases. This implies there is a unique reservation wage, $w_R\left(n\right)$, which satisfies $W\left(w_R\left(n\right)\right)=U\left(n\right)$ and depends on the remaining periods of UI compensation. With $n\in \left\{0,\dots ,N\right\}$ remaining periods of UI compensation, under the convention that the worker accepts a job when they are indifferent between the job and unemployment, any wage offer $w<w_R\left(n\right)$ should be rejected and any wage offer $w\ge w_R\left(n\right)$ should be accepted.

Using $W\left(w\right)=w/\left(1-\beta \right)$ and $U\left(w_R\left(n\right)\right)=w_R\left(n\right)/\left(1-\beta \right)$ in the expressions for W and U in Equations (2)–(4) implies, for $n\in \left\{1,\dots ,N\right\}$,

$$w_R\left(n\right)=\left(z+c\right)\left(1-\beta \right)+\beta \left\{{\int }_0^{w_R\left(n-1\right)}{w}_R\left(n-1\right)dF\left(w\right)+{\int }_{w_R\left(n-1\right)}^{\overline{w}}wdF\left(w\right)\right\}$$

(5)

and, for $n=0$,

$$w_R\left(0\right)=z\left(1-\beta \right)+\beta \left\{{\int }_0^{w_R\left(0\right)}{w}_R\left(0\right)dF\left(w\right)+{\int }_{w_R\left(0\right)}^{\overline{w}}wdF\left(w\right)\right\}.$$

(6)

Additional details are provided in Appendix B.

Looking at (6), the expression can be written as $\mathcal{T}\left(w_R\left(0\right)\right)=w_R\left(0\right)$, where

$$\begin{array}{c}\hfill \mathcal{T}\left(x\right)\equiv z\left(1-\beta \right)+\beta \left\{{\int }_0^{x}xdF\left(w\right)+{\int }_x^{\overline{w}}wdF\left(w\right)\right\}.\end{array}$$

In other words, $w_R\left(0\right)$ is a fixed point of the function $\mathcal{T}$.

Appendix C establishes that $\mathcal{T}$ is a self-map and $0<{\mathcal{T}}^{\prime }\left(x\right)=\beta F\left(x\right)<1$ for $x\in \left(\underline{w},\overline{w}\right)$. Thus, $\mathcal{T}$ is a contraction. The contraction mapping theorem implies that $\mathcal{T}$ admits one and only one fixed point $w_R\left(0\right)\in \left[\underline{w},\overline{w}\right]$.8 Given $w_R\left(0\right)$, an induction argument implies that the reservation wages $w_R\left(n\right)$ are increasing in n. A worker is more selective when there is a single remaining week of UI benefits than when there are no remaining weeks, $w_R\left(1\right)>w_R\left(0\right)$. And, using an induction argument, because $\mathcal{T}$ is increasing, $w_R\left(n\right)>w_R\left(n-1\right)$ implies $w_R\left(n+1\right)>w_R\left(n\right)$. Proposition 1 summarizes these results, and Appendix C provides the details.

Proposition 1.

Assume a risk-neutral, infinitely lived worker searching for a job. The worker perceives no possibility that benefits are extended. The worker is entitled to N periods of UI benefits and regularly receives wage offers from the known offer distribution F with support $\left[\underline{w},\overline{w}\right]$ and mean ${\mu }_w$. A single independently and identically distributed offer is received each period. The solution to the worker’s sequential-search problem is a sequence of reservation wages that increases in the number of remaining periods of UI compensation benefits:

$$\overline{w}>w_R\left(N\right)>\cdots >w_R\left(n+1\right)>w_R\left(n\right)>\cdots >w_R\left(0\right)>\underline{w}.$$

(7)

In state n, the worker accepts any wage $w\ge w_R\left(n\right)$. For offer distributions where $\underline{w}<\left(1-\beta \right)z+\beta {\mu }_w$, the reservation wages exist and are unique. The support of wages can extend to $\left[0,\infty \right)$.

When the future means more to workers, then they are more selective throughout their entire unemployment spells. In other words, a higher $\beta$ shifts the entire sequence of reservation wages upward. Workers are also more selective throughout their unemployment spells when z or c is higher. A higher value of nonwork means they can reject some jobs they would not have otherwise. Appendix G provides more details.

4. Extending Benefits by Discretion

In many situations, a worker must make decisions about accepting a job or continuing their search when there is a chance that UI benefits are extended. Policymakers extend UI compensation benefits by discretion. For example, temporary additional UI benefits were created by Congress after the Great Recession and COVID-19 pandemic. Or extended benefits can be triggered automatically, but this feature of the UI system requires a worker to forecast a state triggering on and off. From the worker’s perspective, there is uncertainty over the following:

• Whether benefits are extended;
• The length of the extension.

I investigate theoretically how the the possibility of extending UI benefits affects job-acceptance decisions. I allow workers to form a fixed belief about an extension—summarized by two parameters for items 1 and 2—and compute their welfare conditional on that belief. My formulation abstracts from how beliefs are formed and how beliefs are updated. While central to decision-making, there is no agreement on how to specify beliefs (Caplin and Leahy 2019, provide a recent take). In addition, my formulation avoids specifying how legislators enact extensions. These features allow me to develop a formal, tractable dynamic model that investigates the costs of misperceiving the probability and length of an extension.

The economic environment is described in Section 4.1. The environment generalizes the environment described in Section 3. Section 5 then considers the costs to a worker who cannot know the true probability that benefits are extended and the length of the extension.

4.1. Environment

I analyze optimal sequential search when UI compensation benefits can possibly be extended. To make the model tractable, once benefits are extended, there is no chance that benefits are extended subsequently.

An extension affects how the state variable n evolves. When there are n remaining periods, a worker is entitled to collect UI compensation and while the worker is deciding to accept or reject a job offer, the worker understands that benefits will be extended by $\Delta$ periods with probability $\delta$. If benefits are extended, then in the following period, there are $n-1+\Delta$ remaining periods of UI compensation. If benefits are not extended, then in the following period, there are $n-1$ remaining periods of benefits. From the worker’s perspective, the probability that benefits are extended, $\delta$, and the length of the extension, $\Delta$, remain constant but potentially unknown.

Figure 2 shows a particular instance of how the state variable n evolves. Starting from the left side of the figure, a worker is at the node with n remaining periods of UI compensation. Benefits are not extended, which occurs with probability $1-\delta$, and the worker does not accept a job. As indicated by the dark cyan path, in the following period, the worker makes a decision about accepting or rejecting a job offer when there are $n-1$ periods of UI compensation and the possibility of extension. Again, as indicated by the dark cyan path, benefits are not extended and the job offer is rejected, which places the worker at the node labeled $n-2$. Then, benefits are extended, which occurs with probability $\delta$. As indicated by the dark cyan path, the following period after the job offer is rejected, the worker makes a decision about accepting or rejecting a job offer when there are $n-3+\Delta$ periods of UI compensation. There is no longer a chance of extension.

The extension of benefits rules out benefits being extended again. The one-time occurrence of an extension implies that, upon extension, the worker solves a standard McCall model that includes the remaining periods of compensation as a state variable. In other words, once benefits are extended, the worker solves the problem described in Section 3.

Like in the sequential-search model with finite benefits considered in Section 3, preferences are the same as in (1). The value of a job is $W\left(w\right)=w/\left(1-\beta \right)$. In this environment, I need to distinguish between the value of unemployment in the upper and lower halves of Figure 2. In the upper half, there is no chance of an extension. The value of unemployment when there is no chance of extension is denoted by U. In the lower half of Figure 2, there is a chance—or a perceived chance—that benefits will be extended. The value of unemployment in this case is denoted by $U^{\delta }$. Both U and $U^{\delta }$ depend on the number of remaining periods of UI compensation.

The value of U corresponds to the basic model of job search in Section 3. The value of $U^{\delta }$ requires characterization. When there is a perceived chance that benefits can be extended, the value of unemployment is, for $n\in \left\{1,\dots ,N\right\}$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill U^{\delta }\left(n\right)& =z+c+\beta \delta \mathrm{E}\left[max\left\{U\left(n-1+\Delta \right),W\left(w\right)\right\}\right]\hfill \\ \hfill & +\beta \left(1-\delta \right)\mathrm{E}\left[max\left\{U^{\delta }\left(n-1\right),W\left(w\right)\right\}\right].\hfill \end{array}\end{array}$$

(8)

The first component of the expression on the right side, $z+c$, corresponds to the value of nonwork plus the flow UI compensation benefit. The following period, discounted by $\beta$, corresponds to choosing to accept or reject a job when benefits have or have not been extended, which occurs with probability $\delta$ and $1-\delta$. Likewise,

$$U^{\delta }\left(0\right)=z+\beta \left\{\delta \mathrm{E}\left[max\left\{U\left(\Delta \right),W\left(w\right)\right\}\right]\left(1-\delta \right)\mathrm{E}\left[max\left\{U^{\delta }\left(0\right),W\left(w\right)\right\}\right]\right\}.$$

(9)

When there is a perceived chance that benefits will be extended, the reservation wage differs from the case when there is no chance. I need to distinguish between the two cases. The reservation wage when there is a chance of extension is denoted by $w_R^{\delta }$, and the reservation wage upon an extension is denoted by $w_R$. Both depend on the remaining periods of UI benefits.

Using the techniques in Section 3, the value of a job, and the expressions for $U^{\delta }$ in (8) and (9), the reservation wages satisfy, for $n\in \left\{1,\dots ,N\right\}$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill w_R^{\delta }\left(n\right)& =\left(z+c\right)\left(1-\beta \right)+\beta \delta \left[{\int }_{\underline{w}}^{w_R\left(n-1+\Delta \right)}{w}_R\left(n-1+\Delta \right)dF\left(w\right)+{\int }_{w_R\left(n-1+\Delta \right)}^{\overline{w}}wdF\left(w\right)\right]\hfill \\ \hfill & +\beta \left(1-\delta \right)\left[{\int }_{\underline{w}}^{w_R^{\delta }\left(n-1\right)}{w}_R^{\delta }\left(d-1\right)dF\left(w\right)+{\int }_{w_R^{\delta }\left(d-1\right)}^{\overline{w}}wdF\left(w\right)\right].\hfill \end{array}\end{array}$$

(10)

and, when $n=0$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill w_R^{\delta }\left(0\right)& =z\left(1-\beta \right)+\beta \delta \left[{\int }_{\underline{w}}^{w_R\left(\Delta \right)}{w}_R\left(\Delta \right)dF\left(w\right)+{\int }_{w_R\left(\Delta \right)}^{\overline{w}}wdF\left(w\right)\right]\hfill \\ \hfill & +\beta \left(1-\delta \right)\left[{\int }_0^{w_R^{\delta }\left(0\right)}{w}_R^{\delta }\left(0\right)dF\left(w\right)+{\int }_{w_R^{\delta }\left(0\right)}^{\overline{w}}wdF\left(w\right)\right].\hfill \end{array}\end{array}$$

(11)

The expressions in (10) and (11) can be interpreted as the benefit of search when an offer $w_R^{\delta }$ is in hand. For example, subtracting $\left(1-\beta \right)z$ and $\left(1-\beta \right)w_R^{\delta }\left(0\right)$ from both sides of Equation (11) yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill w_R^{\delta }\left(0\right)-z& =\frac{\delta }{r}\left\{\underset{\underline{w}}{\overset{w_R\left(\Delta \right)}{\int }}\left[w_R\left(\Delta \right)-w_R^{\delta }\left(0\right)\right]dF\left(w\right)+\underset{w_R\left(\Delta \right)}{\overset{\overline{w}}{\int }}\left[w^{\prime }-w_R^{\delta }\left(0\right)\right]dF\left(w^{\prime }\right)\right\}\hfill \\ \hfill & +\frac{1-\delta }{r}\left\{\underset{w_R^{\delta }\left(0\right)}{\overset{\overline{w}}{\int }}\left[w^{″}-w_R^{\delta }\left(0\right)\right]dF\left(w^{″}\right)\right\}.\hfill \end{array}\end{array}$$

(12)

The left side is the cost of searching another period when offer $w_R^{\delta }\left(0\right)$ is available. The right side is the expected benefit. When benefits are extended, the worker gains the expected net benefit of adopting reservation wage $w_R\left(\Delta \right)$. Adopting $w_R\left(\Delta \right)$ as opposed to $w_R^{\delta }\left(0\right)$ yields two values. First, a worker will reject wage offers below $w_R\left(\Delta \right)$, yielding net benefit $w_R\left(\Delta \right)-w_R^{\delta }\left(0\right)>0$. (The inequality is established in Lemma A5 in Appendix D.) Second, a worker will accept wage offers $w^{\prime }$ above $w_R\left(\Delta \right)$. When benefits are not extended, the worker gains the expected value of searching one more time, which equals the expected value of drawing $w^{″}$ above $w_R^{\delta }\left(0\right)$. Jobs are kept indefinitely. Their values, like perpetuities that pay off starting from the following period, equal the flow value divided by the interest rate, r, where $\beta ={\left(1-r\right)}^{-1}$.9

4.2. Optimal Search

Optimal decision rules are characterized by a sequence of reservation wages that are increasing in n. To establish this characterization, I first use the feature that once benefits are extended the environment coincides with the environment considered in Section 3. In other words, the upper part of Figure 2 is well defined and solved. This allows me to define the function ${\mathcal{T}}^{\delta }$ on $\left[\underline{w},\overline{w}\right]$ as

$$\begin{array}{cc}\hfill {\mathcal{T}}^{\delta }\left(x\right)& \equiv c\left(1-\beta \right)+\beta \delta \left[{\int }_0^{w_R\left(\Delta \right)}{w}_R\left(\Delta \right)dF\left(w\right)+{\int }_{w_R\left(\Delta \right)}^{\overline{w}}wdF\left(w\right)\right]\hfill \\ \hfill & +\beta \left(1-\delta \right)\left[{\int }_0^{x}xdF\left(w\right)+{\int }_x^{\overline{w}}wdF\left(w\right)\right].\hfill \end{array}$$

Appendix E establishes that ${\mathcal{T}}^{\delta }$ is a self map and $0<{\left({\mathcal{T}}^{\delta }\right)}^{\prime }\left(x\right)=\beta \left(1-\delta \right)F\left(x\right)<1$ for all $x\in \left(\underline{w},\overline{w}\right)$. This fact implies that ${\mathcal{T}}^{\delta }$ is a contraction that admits one and only one solution, $w_R^{\delta }\left(0\right)$.10

Given $w_R^{\delta }\left(0\right)$, an induction argument establishes that optimal sequential search is a sequence of reservation wages that is increasing in n. This is summarized in Proposition 2.

Proposition 2.

Assume a risk-neutral, infinitely lived worker searching for a job. The worker is initially entitled to N periods of UI benefits, and regularly receives wage offers from the known offer distribution F with support $\left[\underline{w},\overline{w}\right]$ and mean ${\mu }_w$. A single independently and identically distributed offer is received each period. In addition, between each job offer, there is a chance that benefits are extended by Δ periods with probability δ.

The solution to the worker’s sequential-search problem is a sequence of reservation wages that increases in the number of remaining periods of UI compensation benefits:

$$\overline{w}>w_R^{\delta }\left(N\right)>\cdots >w_R^{\delta }\left(n+1\right)>w_R^{\delta }\left(n\right)>\cdots >w_R^{\delta }\left(0\right)>\underline{w}.$$

(13)

In state n, when benefits have not been extended, the worker accepts any wage $w\ge w_R^{\delta }\left(n\right)$. In state n, when benefits have been extended, the worker accepts any wage $w\ge w_R\left(n\right)$. For offer distributions where $\underline{w}<\left(1-\beta \right)z+\beta {\mu }_w$, the reservation wages exist and are unique. The support of wages can extend to $\left[0,\infty \right)$.

4.3. How Beliefs Affect Optimal Search

As is often the case, a worker does not know whether benefits will be extended or the extension’s length. A worker’s subjective beliefs about a future extension are summarized by two parameters: $\delta$ and $\Delta$. The parameter $\delta$ summarizes beliefs about the probability that benefits are extended. The parameter $\Delta$ summarizes beliefs about the length of an extension. Both $\delta$ and $\Delta$ affect search behavior in an intuitive way. Extending benefits through $\delta$ or $\Delta$ provides relief from earning only the flow benefit of nonwork, which allows the worker to be more selective about the jobs they are willing to accept. This idea is summarized in Proposition 3.

Proposition 3.

Assume a worker searching for a job described in Proposition 2. Each reservation wage $w_R^{\delta }\left(n\right)$ is increasing in δ and Δ. That is, the worker is more selective when accepting job offers for at least two reasons. First, the worker is more selective when they think there is a greater chance that benefits will be extended. Second, the worker is more selective when they think benefits will be extended by more periods.

Appendix F provides a detailed proof.

Propositions 1–3 are illustrated by Figure 3, which shows sequences of reservation wages for different parameter values. The horizontal axis shows the remaining periods of UI compensation. As established, optimal choices imply a worker is less selective as benefits expire—all the sequences are increasing in remaining periods of UI compensation. The solid, dark blue line depicts the sequence of reservation wages after an extension has occurred. Alternatively, because extensions are made only once, this sequence could be interpreted as the sequence of reservation wages when there is no chance of an extension.

The possibility—or the perception—that benefits are extended raises the entire sequence of reservation wages. These types of upward shifts are depicted in Figure 3 by the pink, dashed lines in the upper-left corner. These sequences are computed for different values of $\delta$. For example, when $\delta =0.1$, an extension is perceived to be unlikely relative to the case where $\delta =0.9$. Reservation wages associated with $\delta =0.9$ lie above reservation wages associated with $\delta =0.1$.

Besides illustrating Propositions 1–3, Figure 3 foreshadows the welfare results. Consider a worker who perceives the probability of extension to be 0.1 when the true probability of an extension is 0.5. In order for the worker to make suboptimal decisions, they need to receive a wage offer between the long-dashed sequence and the dash–dot sequence in Figure 3. There is not much room to make suboptimal decisions. Initially, at $n=N$, the worker is rejecting similar offers, as the two lines converge towards the case where benefits are available indefinitely. And even at $n=0$, a suboptimal decision will occur only when a wage offer falls between 0.864 and 0.825. Put another way, Figure 3 suggests that the welfare costs of misperceiving an extension are potentially small.

Nevertheless, policymakers often extend UI benefits temporarily by discretion. Workers are forced to make crucial forecasts about $\delta$ and $\Delta$. The next section considers the welfare consequences of misperceiving the probability and length of an extension.

5. Welfare

In this section, I explore a particular calibration of the model. The calibration is not meant to be definitive. Rather, it is meant to exhibit the features of Propositions 2 and 3.

To do so, I adopt a uniform wage distribution with support $\left[0,1\right]$.11 For the simulations, the discount factor, $\beta$, is set to 0.95. In the absence of UI compensation and the possibility of an extension, I find the value of nonwork so that the worker expects to be unemployed for 10 periods. I then take both the value of nonwork, z, and the value of UI compensation, c, to be half that value. I set $N=10$. The true probability of extension is set to 0.5, and the true length of an extension is set to 25. I consider what happens when $\delta$ and $\Delta$ vary from these values.

Figure 4 compares the expected welfare of misperceiving the probability that benefits are extended. I first compute the expected welfare associated with the baseline case where benefits are extended by $\Delta =25$ periods with probability 0.5. To perform this computation, I simulate the model 500 million times and take the average of the computed welfares.12 The welfare calculations add up and discount the flow values of nonwork, any UI benefits, and the value of accepting a job offer.

A worker is then allowed to hold different beliefs about the possibility of extension. In this scenario, $\Delta$ does not vary. While the true probability of extension is held at 0.5, the worker believes benefits are extended with probability $\delta$. This misperception causes the worker to compute a sequence of reservation wages that differs from the optimal sequence they would compute if they knew the true probability of extension. When $\delta =0.1$, for example, the worker believes the extension is unlikely and they are therefore more likely to accept offers they should reject. On average, the worker should remain unemployed more often to search for a better wage. In contrast, when $\delta =0.9$, the worker believes the extension is likely and they are therefore more likely to reject offers they should accept. On average, the worker should accept more job offers instead of searching for job offers that do not arrive and remaining unemployed.

These losses are depicted in Figure 4. When the worker’s belief coincides with the true probability of extension, which corresponds to 0 along the horizontal axis, the welfare loss is also 0 (theoretically). This is noted in Figure 4 by a horizontal gray, dotted line. As a worker misperceives the probability, they mistakenly reject offers they would optimally accept and accept jobs they would optimally reject. The vertical axis shows the relative welfare loss, computed as the percent away from welfare associated with the true probability that an extension is made.

The loss is asymmetric. A pessimistic worker, on average, experiences more loss than an equally optimistic worker. This feature can be seen by comparing the blue region, where the perceived probability of extension is less than 0.5, to the yellow region, where the perceived probability of extension is greater than 0.5.

Yet, as the vertical axis in Figure 4 indicates, the loss is very small. What accounts for the small costs of misperception? As suggested by Figure 3, reservation wages when $\delta =0.5$ are close to reservation wages when $\delta =0.9$ or even when $\delta =0.1$. Receiving wage offers in the gap between the two sequences is how suboptimal decisions are costly. But, the reservation-wage sequences show that even when a worker misperceives the probability of an extension by +0.4 or −0.4, they are nearly optimally making job-acceptance decisions. In addition, most spells of unemployment are short. Because sequences of reservation wags quickly converge to the reservation wage that would prevail if benefits were paid indefinitely (seen in Figure 3, when the remaining periods of UI compensation equals 15); again, many job acceptance decisions are made nearly optimally.

A comparison of reservation wages when $\delta =0.5$ to cases when $\delta =0.1$ and $\delta =0.9$ also reveals the source of the asymmetry. When $\delta =0.9$, reservation wages are nearly indistinguishable in Figure 3 to the case where $\delta =0.5$. In contrast, there is a distinguishable gap when $\delta =0.1$. To be clear, though, welfare loss is small whether a worker is optimistic or pessimistic about an extension.

The same pattern holds when workers misperceive the length of extension, which can be seen by comparing Figure 4 to Figure 5. To create Figure 5, I first compute the expected welfare associated with a baseline. In the baseline, compensation benefits are extended with probability $\delta =0.5$ for 25 periods. While the true length of an extension is held at 25, the worker holds a different belief about the length of extension, which they use to compute a sequence of reservation wages. Based on this sequence, the worker accepts and rejects offers—suboptimally, as the worker would have computed a different sequence of reservation wages to maximize their expected present value of welfare if they knew the true length of extension. In this scenario, $\delta$ does not vary.

The exercise is repeated for different beliefs over the length of the extension. Relative losses are depicted in Figure 5. Welfare losses are asymmetric and small in magnitude. The magnitudes do suggest, however, that misperceptions about the length of extension may lead to greater welfare loss than misperceptions about the probability of an extension.

Figure 6 and Figure 7 illustrate the mechanisms through which welfare is lost. Figure 6 illustrates why welfare is lost when a worker misperceives the probability of an extension. In the gray region of both panels, looking at the horizontal axes reveals that in these cases the worker perceives that an extension is unlikely relative to the truth. The top panel shows that, on average, workers who are too pessimistic about an extension spend too little time unemployed and searching. If they knew the true probability of extension was higher, then they would reject wage offers they had accepted based on a worse forecast. This dynamic is shown in the bottom panel of the figure, which shows the relative accepted wage. In contrast, in the white region, workers are too optimistic about an extension. They spend too much time searching for a high wage, which costs them through experienced unemployment.

Both panels of Figure 6 show that misperceptions lead to suboptimal job-acceptance decisions, but these decisions are nearly optimal. The statistics on time spent unemployed and accepted wages corroborate the welfare losses depicted in Figure 4. Figure 7 illustrates that the same mechanisms reduce welfare when a worker misperceives the length of an extension. Overly optimistic workers spend too much time unemployed searching for a high wage. Overly pessimistic workers spend too little time unemployed, believing their benefits will soon run out.

6. Discussion

Propositions 2 and 3 link reservation wages to UI compensation benefits: reservation wages decline as UI compensation expires. This theoretical result is mildly consistent with survey evidence presented by Krueger and Mueller (2016). They asked 6,025 respondents in New Jersey who were unemployed as of 28 September 2009: “Suppose someone offered you a job today. What is the lowest wage or salary you would accept (before deductions) for the type of work you are looking for?” Respondents’ answers across 39,201 interviews indicate that there is “little tendency for the reservation wage to decline over the spell of unemployment” (Krueger and Mueller 2016, p. 158 and Figure 4).

What could account for a flat sequence of reservation wages? My results suggest that beliefs about extensions may be an important factor. The New Jersey respondents were unemployed as of 28 September 2009, so many were confronted with the possibility that benefits would be extended and they could claim up to 99 weeks of UI compensation. Many, in other words, faced a situation like Vernon’s, as discussed in Section 2.1. Two years later, in September 2011, the unemployment rate was around 9 percent. A worker could interpret these data as evidence that additional extensions would be made available. In addition, the model shows that reservation wages converge somewhat quickly to the level where UI compensation will be available indefinitely. Optimism about an extension and quick convergence both suggest a rule of thumb that computes reservation wages that are close to the reservation wage in the case where benefits are available indefinitely. If workers adopt this rule of thumb on average, then there would indeed be “little tendency for the reservation wage to decline over the spell of unemployment”.13

Reservation wages may well be more informative than what can be learned in a partial-equilibrium setting like the one considered here. Shimer and Werning (2007) point out that a reservation wage makes a worker indifferent between work and nonwork (as in $W\left(w_R\right)=U$). In addition, a worker’s take-home pay is directly related to consumption and, therefore, utility. Thus, the reservation wage (benefits do not expire in their model) measures the utility of unemployed workers. Any policy that raises the single reservation wage will increase workers’ well-being. In particular, if a marginal increase in UI compensation raises a worker’s reservation wage, then the change improves welfare.14 Krueger and Mueller’s (2016) finding that reservation wages do not respond to benefits suggests that increasing UI benefits would not be optimal. But, as pointed out here, that conclusion may be complicated by the uncertainty that surrounded extended benefits after the Great Recession. This perspective raises a number of questions for future research. In particular: what beliefs do people hold about extensions?

7. Conclusions

My goal was to identify a worker’s costs of misperceiving the probability and length of an extension to UI benefits. As discussed in Section 2, navigating possible extensions was an unavoidable feature of job searches after the two most recent recessions. The model I presented in Section 4 establishes channels through which a worker’s welfare can suffer. When policymakers like congressional representatives assure constituents that a significant extension is coming without a doubt, then workers will believe an extension is likely and the length of the extension will provide meaningful support. Workers will adjust their reservation wages upward. If the extension does not come, then workers will have rejected offers they would have liked to accept. On average, workers will experience too much unemployment. The mirror channel operates when a policymaker communicates their disapproval of UI benefits, despite wider support among policymakers that an extension will be granted. The numerical example, however, found small costs to misperceiving the probability and length of an extension. While the calibration is not definitive, the qualitative properties of the model may help explain some limited data on reservation wages.

Going beyond this paper, the numerical example suggests that extending benefits when the unluckiest and most unfortunate need them to avoid misery will alleviate some worst cases. Kahn (2011, p. 208) points out that searching for work in economic slumps is “particularly damaging” and notes that the damage extends long into careers. Small dithering costs borne by workers may be worth undertaking in order for policymakers to extend benefits for a preferred length and amount. Clear communication will help. Going forward, extensions may be relied upon more frequently as policymakers are forced to manage not only business cycles, but also natural events like viruses, wildfires, extreme heat, and hurricanes.