Takeup, Social Multipliers, and Optimal Social Insurance∗

Kory Kroft†

THIS DRAFT: October, 2006 FIRST DRAFT: February, 2006

Abstract

This paper examines optimal social insurance with endogenous takeup. A simple framework is presented that accounts for behavioral responses along both the intensive and extensive margins. Two formulations of takeup are considered: in the first, individuals face a takeup cost that is exogenous; in the second, the cost depends endogenously on the number of people who take up social benefits. The latter formu- lation captures the intuitive notion that taking up benefits is less costly, possibly due to lower stigma or learning, when others are also taking up benefits. Such endogenous costs to takeup lead to a social mul- tiplier. Formulas for the optimal replacement rate are derived directly using the behavioral elasticities and the social multiplier. These formulas are presented in the tradition of the optimal income taxation literature, facilitating intuition and interpretation, and leading to several novel insights: first, the optimal replacement rate depends on the consumption smoothing benefit- for the segment of the population that receives UI benefits- and the takeup elasticity; second, for a given aggregate takeup elasticity, the precise decomposition into direct effects and multiplier effects matters, implying that knowledge of the social multiplier is important for conducting normative analysis. This paper uses a reform-based strategy to estimate the social multiplier in unemployment insurance takeup. This leads to an estimate of 1.9 for the social multiplier. On the consumption side, a large consumption smoothing role is found when the full takeup assumption is relaxed: consumption for UI recipients, in the absent of unemployment insurance benefits, would fall by 4 times as much as consumption for non-UI recipients. Formulas for the optimal replacement rate are calibrated using the estimated social multiplier, along with the behavioral elasticities and consumption measures. Social multiplier effects significantly increase optimal replacement rates.

∗I would like to thank Alan Auerbach, David Card, Raj Chetty, Bryan Graham, and Emmanuel Saez for many helpful comments, corrections, and suggestions. Useful feedback from Madhav Chandrasekher, Paul Chen, Thomas Davidoff, Zack Grossman, Rafael Lalive, Day Manoli, Yooki Park, Devin Pope, and Justin Sydnor is also gratefully acknowledged. Finally, a special thank you to Raj Chetty and Adam Szeidl for PSID data and to Brian McCall for CPS data. †Department of Economics, University of California - Berkeley, 549 Evans Hall 3880, Berkeley, CA 94720. E-MAIL: [email protected]. WEB: http://korykroft.googlepages.com/

1 1 Introduction

One of the central assumptions in the theory of social insurance provision is that all agents who are eligible for benefits claim them. This assumption originates in the seminal paper by Baily (1978). It is also present in more recent analyses of optimal unemployment insurance (Lentz, 2005; Shimer and Werning, 2005; Chetty,

2006).

Empirically, the takeup rate, defined as the fraction of eligible individuals who claim benefits when unemployed, is well below one, with estimates ranging from .4 to .7 (Blank and Card, 1991; Vroman,

1991; McCall, 1995; Anderson and Meyer, 1997). As Currie (2004) remarks, “although people generally have means-tested programs in the United States in mind when they discuss takeup (where entitlement is conditional on income), low takeup is also a problem in many non means-tested social insurance programs and in other countries...takeup of Unemployment Insurance (UI) is generally similar to takeup of programs such as AFDC and Food Stamps.”1 In addition, takeup rates tend to correlate positively with the size of expected benefits.2 This evidence suggests that taking up benefits is costly and calls into question the “full takeup” assumption.

Moreover, a recent empirical literature has emerged suggesting the prominence of social interactions in takeup decisions (Borjas and Hilton, 1996; Bertrand et al., 2000; Sorensen, 2001; Duflo and Saez, 2003;

Aizer and Currie, 2004). This has motivated some to consider the theoretical implications of social inter- actions in social insurance programs (Lindbeck et al., 1999; Lindbeck and Persson, 2006; Lindbeck, 2006).

These studies are purely positive analyses however and as a result, do not cast light on the debate of whether it is important to decompose the general equilibrium effect from a policy change into partial equilibrium and social multiplier effects (Glaeser et al., 2003).

The goal of this paper is to derive elasticity-based formulas for the optimal replacement rate in a setting with endogenous takeup and social interactions in takeup. A simple framework is developed that encom-

1Low takeup is also a feature of health insurance programs like Medicaid and CHIP (Remler et al., 2001). 2The consensus elasticity of takeup with respect to benefits is around .4. Blank and Card (1991) find evidence of a takeup response using state-level data but not micro data; McCall (1995) and Anderson and Meyer (1997) estimate elasticities in the range .35-.5.

1 passes behavioral responses along both the intensive and extensive margin. Two formulations of takeup are considered, one where the takeup cost is exogenous and a second where the takeup cost endogenously de- pends on the number people who takeup social benefits.3 The latter formulation captures the intuitive notion that when others are taking up benefits, takeup might be less costly due to a reduction in social stigma or increased learning. Formulas for the optimal replacement rate are derived directly using behavioral elas- ticities. These formulas are presented in the tradition of the optimal income taxation literature, facilitating intuition and interpretation, and leading to several novel insights.

First, the optimal replacement rate depends on the sum of the behavioral elasticities and the consump- tion smoothing benefit– for the segment of the population that receives UI benefits. To see the intuition for this, consider the thought experiment of increasing benefits by $1. This change induces behavioral re- sponses along the intensive margin and extensive margin. On the intensive margin, individuals will choose to stay unemployed longer, an effect that is summarized by the duration elasticity. Next, when benefits are increased, costs will become outweighed for some, leading to a higher takeup rate. This latter response– one that has heretofore been ignored in the literature on optimal social insurance– is represented by the elasticity of takeup with respect to benefits. The finding that the formula depends on the consumption smoothing ben- efit for UI recipients is a primae facie one; nevertheless, since the decision to takeup benefits has not been formalized in models of unemployment insurance, the literature has not distinguished between UI recipients and non-UI recipients when estimating the consumption smoothing benefits of unemployment insurance and when calibrating optimal replacement rates. This paper fills this gap.

Second, when social interactions are present, the formula depends on the social multiplier. The formula shows that for a given aggregate takeup elasticity, the precise decomposition into direct effects and multiplier effects matters. Intuitively, this follows since agents do not internalize the effect of their takeup decision on others, generating an externality that social welfare policy seeks to correct. This paper shows that this externality is fully summarized by the social multiplier, along with private utility. The implication is that

3Davidson and Woodbury (1997) consider the impact of less than full eligibility on optimal unemployment insurance. They find that accounting properly for UI-ineligibles raises the optimal replacement rate by around 10 percentage points to 70-80%. To focus on the issue of takeup, this paper assumes that all individuals who become unemployed are eligible for benefits.

2 the social multiplier is a sufficient statistic, that can be used with the other reduced-form parameters of the model, to test whether existing unemployment insurance benefit levels are optimal, in the presence of social spillovers.

There is currently no evidence on the importance of spillover effects in unemployment insurance takeup.

This paper estimates the social multiplier using a reform-based strategy, one that exploits policy variation in benefit levels across states and years. It is shown that by removing a single layer of identification from a quasi-experimental design very standard to the social insurance literature, it is possible to recover an estimate for the social multiplier. The intuition underlying this identification strategy is related to the well-known idea that social interactions cause excess between-group variance, a result that has been formally established for models with continuous action spaces in Glaeser and Scheinkman (2002) and Graham (2005), and for the discrete action model considered in Glaeser et al. (1996). A contribution of this paper is to demonstrate that for a more general interactive process with discrete actions, based on individual optimization, the social multiplier is equal to the ratio of the ‘between-variance’ to the ‘within-variance’.4 Interestingly, this ratio is also the F-statistic for the equality of group means in a standard one-way analysis of variance (ANOVA), making it relatively simple to compute estimates. Importantly however, if there are omitted variable cor- related within groups, or which vary across groups, this statistic provides only suggestive evidence of a multiplier effect.

Finally, the formulas for the optimal replacement rate are implemented using the estimate for the mul- tiplier and elasticities estimated from empirical studies, along with the consumption drop for recipients.

It is shown that UI recipients, in the absence of an unemployment insurance system, would experience a drop in consumption that is 4 times as large as that for non-recipients, implying higher replacement rates.

With no social effects, optimal replacement rates remain very similar to those in Gruber (1997), once the takeup distortion is taken into account. Calibrations show that for low levels of risk aversion, optimal re- placement rates can be twice as high with social multiplier effects as they would be if takeup decisions were independent.

4The prediction that social interactions cause excess variance in takeup rates provides an explanation for the substantial cross- state variation in UI takeup rates that has puzzled applied researchers (Blank and Card, 1991).

3 The remainder of this paper is organized as follows. Section 2 models the optimal provision of benefits with endogenous takeup and no social interactions. Section 3 extends this to allow for social interactions.

Section 4 formally establishes the relationship between the excess between-group variance and the social multiplier. Section 5 discusses identification of the multiplier using a reform-based strategy and Section

6 contains results. Section 7 discusses generalizability of the theoretical results, in the context of a search model adapted from Lentz (2005). Section 8 calibrates the optimal replacement rate and Section 9 concludes.

2 The Baily Model with Takeup

This section outlines a simple unemployment insurance problem similar to Baily (1978). In the model, an individual fully controls his unemployment duration. Though unrealistic, it is shown below that similar results are obtained in a richer, dynamic setting that explicitly incorporates the uncertainty inherent in the unemployment problem. The main features of the model are as follows.

There is a population of individuals of measure of 1, with well-behaved preferences represented by u(c), where u0(c) > 0 and u00(c) < 0. Each individual begins his life employed (with probability p) or unemployed (with probability 1 p). If employed, an individual works for the entire period at a wage w and pays an − unemployment tax of τ. The assumption that labor supply is fixed focuses the discussion on the interaction between unemployment and unemployment insurance. Assume the interest rate and subjective discount rate are zero. Given initial assets A0, this implies that consumption during the period is ce = A0 + w τ. − An individual initially unemployed makes two decisions. First, he chooses whether to collect benefits b. If he collects, his utility is u(cu) ψi where ψi, the takeup cost, is known and is continuously distributed − across individuals with distribution function F and density function f . Assume that F and f are continuous and that the support of F is bounded below by ψ 0. An individual that does not collect benefits has ≥ utility u(cu). This formulation of takeup can reflect information barriers or stigma (Moffitt, 1983; Besley and Coate, 1992). The assumption that the takeup cost does not vary with the benefit level is for simplicity, although some evidence exists to support this. Moffitt (1983) finds in the context of AFDC that stigma arises

4 from the act of taking up welfare, and does not vary with the amount of the benefit once on welfare.

The second decision is choice of unemployment duration D. Following Chetty (2006), assume there are costs to search and/or utility from leisure, captured by a smooth, concave and increasing function of D, v(D). Given the choice for D, an individual is reemployed for the remainder of the period 1 D and earns wage w.5 − Consumption for those who are unemployed initially and receive UI benefits is cu = A0 + Db + (1 D)w = − A0 + D(b w) + w and for those who do not receive benefits is cu = A0 + (1 D)w. This completes the − − description of the individual’s problem.

There is an unemployment insurance agency whose objective is to maximize a weighted sum of individ- ual utilities, with the weights corresponding to population weights. The agency controls the benefit level, b, and the unemployment tax, τ. Benefits are paid in proportion to the length of the unemployment spell, and only to those who claim them. Assume that the distribution of ψi, F, is known to the insurer. The UI agency is required to balance its budget in expectation.

There are some features of the UI problem that are not address here. First, unemployment insurance spillovers that take place through the process of searching for jobs are ignored. Although individuals may learn from one another about the set of job openings that are available or how to search more effectively, the fact that there are only a finite number of job openings implies that competition between individuals for jobs could be important.6 There could be also spillover effects from individuals who receive UI to those who do not, if benefits cause UI recipients to prolong their unemployment spells. This possibility has been examined by Levine (1993) who finds that non-UI recipients have shorter UI spells when benefits are more generous. Future research that provides more evidence on the importance of spillovers effects and their relevance for optimal unemployment insurance would be very valuable. Second, assume that p, the probability of unemployment, does not respond to UI benefits. This assumption means that firms do not react to benefit changes, something that is likely to be true if there is an effective experience-rating provision.

Several studies have found that layoffs tend to increase when benefits are made more generous (Feldstein,

5The assumption that earnings are untaxed is made primarly for simplicity. It is discussed below how relaxing this assumption affects the results. 6Lalive (2003) studies a large benefit reform in Austria and finds evidence of positive spillovers in unemployment durations.

5 1978; Topel, 1983; Anderson and Meyer, 1994; Card and Levine, 1994). This increases the cost of providing

UI benefits and therefore would presumably lower the optimal replacement rate.

2.1 Elasticity Concepts

In this model, there are two types of unemployed agents, UI recipients (R) and non-UI recipients (N), that are defined by a critical value ψ∗. For UI recipients, ψi < ψ∗, and for non-UI recipients, ψi > ψ∗. Note that since individuals only differ in terms of ψi, this cutoff value will be the same for everyone and hence is not indexed by i. This is stated precisely as follows:

R N R N ψ∗ = u(c ) u(c ) + v(D ) v(D ) ψ∗ (1) i u − u − ≡

Moreover, this implies that DR(b w), the unemployment duration for a UI recipient, will be independent − of i. Denote P(b w), the fraction of the unemployed population that collects benefits in equilibrium. − The term pPDR represents the expenditure base. This is the number of unemployment weeks that the

UI system subsidizes. An increase in benefit levels can change the expenditure base for one of two reasons.

First, along the extensive margin, more individuals can take up benefits. The size of this behavioral response is captured by the elasticity of participation with respect to the benefit level. Since the decision to participate depends only on w b, this may be written as −

w b ∂P ν = − (2) P ∂(w b) −

This elasticity measures the percentage increase in the fraction of individuals who takeup UI benefits when the net wage is increased by 1 percent. Also, along the intensive margin, individuals may extend their unemployment durations. This behavioral response is summarized by

w b ∂DR ε = − (3) DR ∂(w b) −

6 This elasticity measures the percentage increase in the number of weeks unemployed for a UI recipient when the net wage is increased by 1 percent.

A few final points merit discussion. First, it is important to keep in mind throughout that this elasticity does not depend on i. In other words, the individual duration elasticity is also the average population duration elasticity. Second, the elasticities are defined with respect to the net wage, not the benefit level as in Baily (1978) and Gruber (1997). This is related to the approach taken in the optimal income tax literature, where the labor supply elasiticities are defined with respect to the net-of-tax wage (Saez, 2001).

Note that the elasticities here are well defined as the benefit level approaches zero. On the other hand, the elasticities defined with respect to the benefit level are not defined in this limiting case, imposing undesirable features on the nature of underlying preferences. Other advantages of this approach are that it allows for a clearer and more intuitive interpretation of the formula for the optimal replacement rate. Also important, the formula is well-defined when all of the reduced-form parameters are constant. The formulas for the optimal replacement rate in Baily and Gruber are not well-defined in this case. Last, note that the elasticities defined here are uncompensated elasticities.

2.2 Deriving the Optimal Replacement Rate

Consider a small increase of db in benefits. This change has two effects on unemployment insurance expen- ditures, pbPDR. First, there is a mechanical effect, which is the change in expenditures assuming there were no behavioral responses, and second, there is the change in expenditures due to behavioral responses. These are examined sequentially.

2.2.1 Mechanical Effect

The mechanical effect (denoted by M) represents the dollar amount of the change in benefits paid out to unemployment recipients, if they do not change their behavior. A UI recipient would receive an additional

DRdb of benefits. Therefore summing over the population of individuals who receive UI benefits, the total

7 mechanical effect M is

M = pPDRdb (4)

2.2.2 Behavioral Responses

The benefit increase induces two behavioral responses. First, some individuals who were not collecting benefits choose to collect them. This behavioral response is

∂P P dP = db = ν db (5) ∂b w b − using definition (2). This increases expenditures by pDRbdP. That the relevant duration in this expression is

DR rather than DN follows from the fact that an individual’s incentives to search are lower when collecting benefits. Second, an individual changes his duration DR by

∂DR DR dDR = db = ε db (6) ∂b w b − where definition (3) is used. The change in duration displayed in equation (6) implies an increase in expen- ditures for that individual equal to bdDR. For the population, the total change in expenditures is pPbdDR.

The total increase in expenditures due to the behavioral responses is the sum of the terms bpPdDR and bpDRdP and is written as

bdb bdb B = pPDRε + pPDRν (7) w b w b − −

Expressions (4) and (7) represent how much total expenditures change when benefits increase. Solving for the optimal replacement rate requires specifying a social welfare criterion. To make things simple, as- sume that the social welfare function (SWF) is utilitarian, with weights equal to population shares. Start by considering how this reform affects a UI recipient’s utility. Assuming an optimum, the envelope the-

8 R R R orem implies that the effect of a small change in benefits on utility is simply u0(cu )D db where D db is the mechanical increase in individual benefits. Next, summing over the population of UI recipients gives

R u0(cu )M, the total social welfare gain. Now consider the effect of the reform on individual taxpayers. This is given by u (ce)( dτ) where dτ is the total increase in individual taxes necessary to balance the budget. 0 − Summing over the population 1 p of employed taxpayers and noting that the implied total increase in taxes − 1 is 1 p (M + B) gives u0(ce)(M + B), the total social welfare loss. Social welfare is maximized when the − − R total welfare gain is equal to the total welfare loss. Defining g = u0(cu ) , this condition is (1 g)M + B = 0. u0(ce) − Thus the optimal benefit level must satisfy

b 1 = (g 1) (8) w b ε + ν − −

This may be further expressed as

r 1 = (g 1) (9) 1 r ε + ν − −

b where r = w is the replacement rate.

In deriving equation (9), it is implicitly assumed that the set of individuals who might jump discontinu- ously because of the small reform is negligible. In particular, this means that individuals who takeup benefits in response to the small reform do not gain since, on the margin, they are indifferent between taking up ben- efits and not taking up benefits. More formally, this requires that the c.d.f. F and the p.d.f. f are continuous as is shown in the appendix. As Saez (2001) points out, this property cannot be used when the objective of the UI agency is non-welfarist since in this case, the government has a different objective function than individuals and as a result, the envelope theorem no longer applies. Finally, note that g is endogenous to the benefit level.

9 2.3 Interpretation and Implications

It is worth analyzing the formula for the optimal benefit level that appears in equation (9). It appears that two elements determine the optimal replacement rate: elasticity effects, and the social marginal weight.

2.3.1 Elasticity Effects

The optimal social insurance literature has focused exclusively on how benefits affect the duration of un- employment spells. The formula derived here contains two new insights. First, the formula depends on the elasticity of duration with respect to the replacement rate- for the population of individuals that receive benefits. In principle, a change in benefits should have no effect on the unemployment spells of those indi- viduals who do not receive benefits. The second insight is that the behavioral response along the extensive margin also determines the replacement rate.7

2.3.2 Social marginal welfare weight

The social marginal weight g enters the formula for the optimal benefit level in the numerator. This weight summarizes the relative gain in social welfare, on average, when benefits are increased by $1 and taxes are raised to finance this increase, for a given (b,τ).8 Intuitively, the optimal replacement rate is an increasing function of the weight g.

Note that in this model, consumption for the unemployed rises one-for-one with the benefit level. That is, a $1 increase in benefits raises unemployment consumption by exactly $1, and hence utility for UI recipients

R by exactly u0(cu ). This is true whether an individual is at an optimum or not. In practice, unemployment

7Note that the elasticity of duration, ε, is the total effect of an increase in benefits and taxes. This is trivially true since these individuals never pay taxes. A behavioral response that one would think would enter the optimal benefit formula is the labor supply response to the employment tax. Although this could be important in reality, the model effectively assumes away this distortion since labor supply is fixed when employed. 8Note that if all individuals were required to pay taxes on earnings, the denominator in g would be a weighted-average of consumption during employment with the weights reflecting the population distribution. In practice, since individuals spend a small fraction of their lifetimes unemployed, the difference between these terms is likely to be small. To make things simple, g is calibrated using the consumption drop for UI recipients.

10 consumption rises less than one-for-one with benefits as shown in Gruber (1997). This happens because

UI benefits crowd out other private insurance arrangements. Thus, a potential criticism of this model is that it omits economic realities like private insurance arrangements. Interestingly, Chetty (2006) shows that Baily’s reduced-form formula is infact valid in a much broader context that explicitly accounts for these features. This result follows from an application of the envelope theorem and continues to hold with endogenous takeup. There is one important difference however that affects the implementation of formula

(9). Although at an optimum, a $1 increase in benefits will be spent solely on consumption, this does not imply that in general, a $1 increase in benefits raises consumption by $1. In particular, with other sources of income, a change in benefits will be partially offset by behavioral responses. This issue will be given careful consideration when calibrating g below.

3 Social Interactions in Takeup

This section considers the optimal unemployment insurance problem when the individual propensity to takeup benefits is a function of the overall takeup rate. There are a few empirical studies that provide support for this model of takeup. Lemieux and MacLeod (2000) find that first time receipt of unemployment insurance increases future receipt. They interpret this primarly as evidence of individual learning, but note that some of their findings are consistent with ‘social’ learning. Bjorn et al. (2000) attempt to distinguish between social learning and individual learning effects by considering the role of family background in the determination of UI receipt. Using Canadian data, they find that a parent’s use of UI benefits affects both

first and subsequent use of UI benefits, evidence of social learning. While difficult to establish empirically, in theory, social interactions can reflect social learning (Bikhchandani et al. (1992); Banerjee (1992); Ellison and Fudenberg (1993)), conformity (Bernheim (1994)) and social stigma (Moffitt (1983); Besley and Coate

(1992)).

The key insight is that in all of these cases, there is a ‘social multiplier’ effect: the aggregate impact of an intervention on a group is larger than the sum of its effects on each individual decision. While some

11 have argued less formally that it is important for policy purposes to separate these effects, there has been no systematic effort to show why this might be the case (Glaeser et al., 2003). The next section attempts to shed light on this debate by considering the importance of social spillovers in takeup in the context of an unemployment insurance program.

3.1 Preferences

To study how social interactions affect the optimal provision of unemployment insurance, I reconsider the

fixed utility cost model considered in section (2). An individual who takes up unemployment benefits has utility

u(cu) (ψi + S(P)) (10) −

P is the aggregate takeup rate and ψi is distributed across individuals with distribution F and density f . Assume that S, F and f are continuous and bounded, and that f is positive on the whole real line.9 Agents take S(P) as given, not internalizing the effect that their behavior has on the takeup rate. In the parlance of the social interactions literature, the dependence of agent i0s takeup cost on the action of others reflects

10 endogenous social interactions. Throughout this discussion, it is assumed that S0(P) < 0.

3.2 Solution Concept

Let ωi be the indicator variable for whether individual i collects benefits when unemployed. Define V(ωi,P,ψi)

R N 11 as individual i0s lifetime utility when the choices of cu and cu are at an optimum. This satisfies

R R N N V(ωi,P,ψi) ωi u(c ) + v(D ) (ψi + S(P)) + (1 ωi) u(c ) + v(D ) (11) ≡ u − − u 9The model developed in this sectionclosely follows Brock and Durlauf(2001) and Glaeser and Scheinkman (2002). 10It is worth pointing out that since social interactions only affect the takeup cost in this model, the choices of unemployment R N R N durations, D and D , and therefore consumption, cu and cu will be unaffected. 11To simplify notation, I have suppressed the dependence of V on the benefit level, b.

12 Now define h(P,ψi) V(1,P,ψi) V (0,P,ψi) and z(P,ψi) V (0,P,ψi). Notice that this implies V(ωi,P,ψi) = ≡ − ≡ hωi + z. Using this change of variables, the individual’s decision problem can be re-expressed

ωBR ψ ω i (P, i) = argmaxh i + z (12) ωi 0,1 ∈{ } where the superscript BR is used to express the fact that this function is a best response strategy to mean peer group behavior, P. It should be emphasized that since ωi is binary, equation (12) is non-parametric and does not rely on any functional form assumptions. An individual’s decision rule is to collects benefits if h > 0.

Assume that agent i observes ψi and then chooses ωi to maximize utility conditional on ψi. The solution concept follows.

Definition 1 A social equilibrium12 is defined as a takeup rate Pe and a cutoff value λψ that satisfy:

(1) h(Pe,λψ) = 0 (13)

ωBR ψ λ (2) i = 1 i < ψ (14)

(3) Pe =F(λψ) (15)

It is shown in the appendix that an equilibrium exists in this model. To ensure that the unemployment insurance agency’s problem is well-defined, the following assumption is made.

dh Pe λψ Assumption 1 For a given takeup rate Pe, ( , ) < 0. dψi

This assumption guarantees that the equilibrium is unique.13 Figure (1) in the appendix shows that if dh(Pe,λψ) dh(λψ) ψ 0, there are atleast two other equilibria. Since ψ = hP(λψ) f (λψ) + hψ , two key elements d i ≥ d i i determine whether there are multiple equilibria: first, the density of agents on the margin; second, the marginal impact of the population takeup rate on the individual preference for taking up to not taking up.

12Glaeser and Scheinkman (2002) refer to this as a Mean Field Equilibrium (MFE). 13A discussion of the conditions necessary for stability of equilibria is provided in the appendix.

13 Since f > 0 by assumption and since hψ = 1, for there to exist multiple equilibria, it must be the case that i − hP > 0. When this true, preferences are said to exhibit strategic complementarities in the sense of Cooper and John (1988).14

3.3 Optimal Replacement Rate with Social Interactions

When social interactions are present, a change in b elicits two takeup responses: there is a change in takeup since utility depends directly on the benefit level through consumption in the unemployed state, and there is a change in takeup since utility depends indirectly on the benefit level through the takeup rate P. This is

0 illustrated in Figure (2). The movement from the initial equilibrium to E0 represents the partial equilibrium

0 response of the benefit increase and the movement from E0 to P = E1 represents the social spillover response. Since the UI agency cares about the total or general equilibrium response, this requires introducing the following elasticity concept.

w b ∂Pe νe = − (16) Pe ∂(w b) −

This elasticity measures the percentage increase in the fraction of individuals who takeup UI benefits when they are increased by 1 percent, taking into account spillover effects. There is a close relationship between

νe and ν, the partial equilibrium elasticity, that is examined below.

The other key insight is that a change in benefits causes individuals to claim them, thereby generating an externality. Whereas before, the takeup response lead only to deadweight loss, with social interactions, the takeup response can actually be welfare-enhancing. Intuitively, this depends on the importance of social

∂S spillovers, ∂Pe , leading to the following concept.

R 1 ∂S ∂Pe u0(c ) + R ∂ e ∂ ge = u D P b (17) u0(ce)

14Glaeser and Scheinkman (2002) contains an interesting discussion about the role of heterogeneity in the population in producing multiple equilibria. They show that if F is symmetric and monotonic, then the model in certain situations must exhibit multiple equilibria.

14 It is now a straightforward exercise to derive the formula for the optimal replacement rate with social inter- actions.

Proposition 1 The optimal replacement rate with endogenous takeup and social spillovers is defined im- plicitly as

r 1 = (ge 1) (18) 1 r ε + νe − −

It is difficult to implement formula (18) empirically since the primitive function S(P) and hence its first derivative are not directly observable. As a consequence, ge is not directly estimable. The next section tries to overcome this difficulty.

3.4 The Social Multiplier

To link (18) to a parameter that can potentially be estimated, it is useful to introduce the social multiplier

(Cooper and John, 1988; Glaeser et al., 2003). Similar to the concept of an elasticity, the social multiplier is a reduced-form parameter that depends on the primitives of the model. It measures the discrepancy between the initial takeup response to a change in b, holding social spillovers constant, and the full equilibrium re- sponse which occurs once the change in takeup rates have been fully capitalized into takeup costs. Formally, the social multiplier for a change in the benefit level is defined as follows:

Definition 2 For a stable equilibirum, the social multiplier is defined locally as

Pe m(b) = b (19) Pb

where Pb Fb. ≡

The next lemma gives a simple characterization for m(b).

15 Lemma 1 Suppose FPe = 0. The social multiplier is 6

1 m(b) = (20) 1 FPe −

e e e Proof: The equilibrium takeup rate, P , solves the equation P = F(P ,b). Suppose that FPe = 0. Then, by 6 the Implicit Function Theorem, Pe is (locally) a function of b; that is, Pe = Pe(b). Moreover,

dPe F = b (21) db 1 FPe −

To see the intuition for this, suppose that Fb > 0 so that a higher replacement rate implies a higher takeup rate. In any stable equilibrium, FPe < 1, and because of the assumed complimentarity between own takeup

dF and the aggregate takeup rate, FPe > 0. Therefore, db > Fb, whereas in the absence of the complimentarities, dF 1 db = Fb. The term m = 1 F e represents a (local) social multiplier effect. − P The concepts introduced in (16) and (17) can now be expressed in terms of m, thus allowing the optimal policy in (18) to be more easily identified. First, letting ν denote the denote the partial equilibrium elasticity holding spillover effects constant, νe = mν. Thus, a social multiplier above 1 will amplify the individual- level takeup response to an increase in benefits. This may explain why the level of aggregation has proved important in estimating the takeup elasticity (Blank and Card, 1991). It may also explain why the short-run response to a policy change is smaller than the long-run response (Lindbeck and Nyberg, 2006). The next

1 ∂S ∂Pe proposition formally establishes the link between m and DR ∂Pe ∂b .

Lemma 2 The social marginal weight ge with social interactions satisfies

ge = mg (22) where g is defined in equation (9).

16 Proof: First note that by definition (2),

∂Pe = mF ∂b b

Next, consider the direct effect Fb. I will show that this is related to the marginal utility of consumption. First,

∂ψ F = f ∗ b ∂b

Using definition (1), this can be further expressed as

∂cT ∂DR F = f u (cT ) u + v (DR) b 0 u ∂b 0 ∂b  

∂ R ∂ R Since cu = DR + (b w) D , this becomes ∂b − ∂b

∂ R R ∂ R R R D v0(D ) D Fb = f u0(c ) D + (b w) + u − ∂b u (cR) ∂b  0 u 

Agent optimization implies v (DR) = (w b)u (cR). Substitution gives the result that 0 − 0 u

R R Fb = f u0(cu )D (23)

Finally, using the fact that FPe = f SPe allows me to write ·

∂ ∂ e 1 S P R = (m 1)u0(c ) (24) DR ∂Pe ∂b − u thus concluding the proof.

Formula (18) is now restated in terms of the social multiplier.

Proposition 2 The optimal replacement rate with endogenous takeup and social spillovers, summarized by

17 m, is defined implicitly as

r 1 = (mg 1) (25) 1 r ε + mν − −

Equation (2) provides some insight into the debate about whether it is important for policy purposes to separate individual level effects from social multiplier effects. Although the denominator in equation (2) shows that we only need an estimate for the total response, νe, the numerator shows that we also need an estimate for m, the social multiplier. Note that if m = 1 (i.e., there are no social interactions), then we get the simple formula in equation (9).

1 ∂S ∂Pe That the term DR ∂Pe ∂b can be expressed in a very simple way in terms of the marginal utility of con- sumption in the unemployed state and the social multiplier is not obvious and is worth commenting on.

To see the intuition for this, it is helpful to consider the marginal agent. The marginal agent is defined by

ψi = ψ∗ where ψ∗ solves the following equation:

R N u(c ) u(c ) ψ∗ = S(P) (26) u − u −

R N At an optimum, an individual sets the marginal direct gain of taking up benefits, u(c ) u(c ) ψi, equal to u − u − the marginal indirect cost of taking up benefits, S(P). Equation (26) states that for the marginal individual, these two terms are equal. Consider what happens when there is a small perturbation in benefits. This directly increases consumption, leading to increased takeup; however, since takeup has increased, there is a reduction in the indirect cost, since S(P) is decreasing in P. A new equilibrium is reached when the increase in the direct gain exactly offsets the reduction in the indirect cost.15 This illustrates the deeper connection between the direct and indirect terms that is implicit in equation (22). Interestingly, it implies that knowledge of the social multiplier, m, a term that is in principle estimable, and the social welfare weight, g, is sufficient

15Figure (4) in the appendix illustrates this idea. ∆U represents the direct gain and ∆S represents the indirect cost. One can see from this figure that ∆S is determined both by ∆U and the slope of the curve S(P).

18 to identify the optimal replacement rate.16

4 Excess Between-Group Variance

This section shows that social interactions cause the variance in average outcomes across social groups to be greater than what would be predicted if individuals’ decisions were independent. In this sense, social interactions can provide an explanation for the historically large variation in UI takeup rates across U.S. states that has puzzled applied researchers (Blank and Card, 1991). More specifically, it is shown that the ratio of the between-variance to the within-variance identifies the social multiplier, in the absence of unobserved group-level heterogeneity.

Consider a random sample of outcomes for j = 1,...,M social groups that are ex ante identical. Each group has N individuals. Horst and Scheinkman (2006) show that equilibria in a system with an infinite number of agents can be viewed as the limit of equilibria of a large finite system. When necessary, this section will appeal to some of the arguments in Horst and Scheinkman (2006) to make the connection to the model presented earlier, with an infinite number of agents. All of the basic elements of the model above N ∑ ωi ω i=1 are retained. Let N = N be the empirical average of a social group. Individual i0s utility function is

Vi = V(ωi,ωN,ψi). The takeup outcome for individual i in a group is denoted by ωi = ω(ωN,ψi).

A social equilibrium in a finite system is a situation where each individual plays a best response to the perceived average choice of their neighbors actions and where the forecast of the average choice coincides with the empirical average of the action profile.

16Note that if the unemployed were required to pay taxes when they became employed, the takeup response would also depend on the tax burden and as a result would make the problem less tractable. In particular, ψ∗ would depend additionally on how much taxes changed and the difference in marginal utility of consumption in the employed state for recipients and non-UI recipients.

19 4.1 Large Economy Behavior

For large group sizes, Horst and Scheinkman (2006) demonstrate that the equilibrium of the finite system is well approximated by the equilibrium of the infinite system, so that the mean action ωN converges (almost surely) to Pe as defined in (13).17 The next proposition formally relates the between-group variance to the social multiplier. This is a well known result in the case where an individual chooses a continuous action (Glaeser and Scheinkman, 2002; Graham, 2005). It is also related to expression (3) in Glaeser et al.

(1996) who model social interactions in binary outcomes, the difference being that instead of m, the relevant concept is the number of imitators, f (π). Both of these terms however capture the covariance across agents’ actions.18 The main contribution of this paper is in establishing that the result holds for a more general interactive process. Finally, note that the expressions below concern the variance across a fixed set of social groups. Since social groups are identical, double index notation is not introduced.

Proposition 3

N e ∑ (ωi P ) i=1 − lim Var   = Pe(1 Pe)m (27) N ∞ √ − → N       The way the proof to this proposition will proceed is to calculate the finite-sample variance and then evaluate the limit of this variance as the group size, N, tends to infinity.

e Proof: Consider Var √N(ωN P ) . This can be decomposed as −
N 1 e e e ∑Var (ωi P ) + 2 ∑ Cov(ωi P ,ω j P ) (28) N "i=1 − 1 i< j N − − # ≤ ≤ 17See Theorem 4.1 for a proof. 18More specifically, in Glaeser et al. (1996), there are a fixed number of agents who choose some pre-determined action. The remaining agents mechanically follow or imitate the others with some probability.

20 N 1 ∑ ω e e e It is simple to see that N Var ( i P ) = P (1 P ). Now consider the second term and note the following i=1 − − two things:

e e e e 2 e 2 Cov(ωi P ,ω j P ) = E[ωiω j P (ωi + ω j) + (P ) ] = E[ωiω j] (P ) (29) − − − −

e E[ωiω j] = Pr(ωi = 1,ω j = 1) = Pr(ωi = 1 ω j = 1)Pr(ω j = 1) = Pr(ωi = 1 ω j = 1)P (30) | |

Therefore,

e e e e Cov(ωi P ,ω j P ) = P (Pr(ωi = 1 ω j = 1) P ) (31) − − | −

e 2 ∑ ω ω e Since P factors out of the sum, it is only necessary to consider N (Pr( i = 1 j = 1) P ). Define 1 i< j N | − ≤ ≤ ∆Pi j Pr(ωi = 1 ω j = 1) Pr(ωi = 1 ω j = 0), the total increase in the probability that individual i takes , ≡ | − | e e e up, when individual j takes up. Using the fact that P = Pr(ωi = 1 ω j = 0)(1 P )+Pr(ωi = 1 ω j = 1)P , | − | this expression can be re-written as

1 e ∑ ∆Pi, j(1 P ) (32) N 1 i< j N − ≤ ≤

e 1 ∑ ∆ Since the term (1 P ) factors out of the summation, all that remains is to show that N Pi, j con- − 1 i< j N ≤ ≤ verges to m 1. First, note that since individual j s action only influences i s probability of takeup through − 0 0 the mean and since the ex ante probabilities should not depend on i, ∆Pi, j = ∆P. This exchangeability property implies that equation (32) can be expressed compactly as

N 1 − ∆P (33) 2

Given a population of size N, an additional individual taking up benefits changes the overall takeup rate ∆ by 1/N. Consider P . By definition of m, as N gets large, this approaches m 1. Intuitively, for a large 1/N −

21 population, an additional individual taking up benefits causes an initial increase in the mean takeup rate of

FPe . This is the partial equilibrium response. The total equilibrium response, taking into account social

FPe spillovers, is 1 F e . Formally, − P

1 lim ∑ ∆Pi, j = m 1 (34) N ∞ N − → 1 i< j N ≤ ≤

The result follows immediately.

For practical purposes, if the propensity to takeup benefits, Pe, is constant across groups, one can esti- mate the between variance using the variance of population-weighted takeup rates across groups. Further, the variance among all individual takeup decisions in the total population is Pe(1 Pe). Therefore, the − ratio of the ‘between-variance’ to the ‘within-variance’, the F-statistic in a standard one-way analysis of variance (ANOVA), identifies the social multiplier. This estimator will be implemented in the empirical section below. Of course, the mean takeup rate may differ across social groups for reasons other than social interactions. In particular, there may be omitted variables correlated within groups, or which vary across social groups. This points to the main disadvantage of using the excess variance approach to estimate the multiplier. The next section discusses a reform-based identification strategy, explicitly recognizing these potential confounds .

5 Identification of the Social Multiplier

As discussed above, a change in benefit levels elicits two responses, a direct response, and a spillover response. To identify the direct response, one needs exogenous variation in individual benefit levels. In practice, benefit levels vary almost one-for-one with previous earnings, as illustrated in Figure (5), making it difficult to disentangle the effect of previous earnings from the effect of benefits on takeup. Previous studies that estimate labor supply effects to social insurance programs overcome this problem by exploiting periodic changes in the maximum weekly benefit amount (WBA) (Blank and Card, 1991; Meyer, 1992;

22 McCall, 1995).19

An insight of this study is that changes in the maximum WBA can be exploited to estimate both the direct effect and the social spillover effect. An example illustrates the main idea. Consider the state of California.

In 1985, the maximum WBA was 154 and in 1986, it was 189. Thus, individuals eligible for the maximum

WBA experienced a benefit increase, while others did not. This benefit change causes a direct effect, and a spillover effect. If spillovers affect individuals uniformly, a simple “diff-in-diff” within states will identify the direct effect. Now consider another state, Texas. In 1985 and 1986, Texas had a maximum WBA of 189.

Essentially, the average weekly benefit level increased in 1986 in California, but not in Texas. The change in the average takeup rate in California during this time period represents the combination of the direct effect, spillover effects, and other shocks. Assuming that Texas is a valid counterfactual for California, a diff-in-diff across states will identify the total effect. The ratio of the total effect to the direct effect provides evidence on the importance of social spillovers. This reasoning also shows that studies that difference across and within states are implicitly assuming spillover effects are negligible.

To formalize this intuition and to interpret the results, a simple linear model is considered. Suppose that the takeup outcomes of S states (s = 1,...,S) where Ms individuals are sampled in the sth state in each year t = 1,...,T. Individuals are unemployed and are eligible for unemployment insurance benefits. An individual’s takeup decision, ysit, is a function of the weekly benefit amount, bsit, and the takeup rate in state s and year t, yst, so that: α η β α α ε ysit = ∗ + bsit + yst + s∗ + t∗ + sit (35)

α α ε where s∗ and t∗ capture unobserved state and year effects and sit is an idiosyncratic error. An exogenous increase in the mean takeup rate of a state translates into a final increased takeup rate for that state of

1/(1 β) percent through the social multiplier effect. − 19The maximum WBA is indexed to state average weekly earnings. Due to nominal wage growth, states automatically adjust the maximum benefit amount. As Meyer (1992) points out, high nominal wage growth in the early 1980s, the time period covered by this analysis, led to large benefit increases, averaging between 9 and 10 percent. The changes in the maximum weekly benefit amount provide the necessary variation to identify the direct and indirect effects. Note that very few people receive the minimum weekly benefit amount and it is not raised very often. Hence the empirical analysis will focus mainly on changes in the maximum weekly benefit amount.

23 Assume that outcomes observed by a researcher correspond to a social equilibrium in the sense that each individual’s action in a state and year is consistent with the average action in that state and year.

This assumption is analagous to the market equilibrium assumption that is employed in the estimation of supply and demand econometric models (Angrist et al., 2000). To achieve identification in these models, an exogenous source of variation is needed. This paper relies on the fact that the benefit level for a high wage earner in a state, but not a low wage earner, is exogenously increased over time. The identifying assumption is that an observed increase in takeup among low earners in a state, at the time of a benefit increase, is the effect of the exogenous increase in takeup among high earners within the state. To illustrate this, let the group of high wage earners be denoted by superscript E20:

E α η E β α α εE ysit = ∗ + bst + yst + s∗ + t∗ + sit (36)

Let the individuals within state s for whom the change in benefits has no direct impact be given by:

NE α η NE β α α εNE ysit = ∗ + bsi + yst + s∗ + t∗ + sit (37)

Note the following relationship between the overall mean outcome and the mean outcomes of the subgroups:

E E NE NE yst = mstyst + mst yst (38) where mE is the share of high wage earners in state s at time t and mNE = 1 mE . Substituting equation (38) − into equations (36) and (37), the social equilibrium of this system is given by the reduced-form model:

E α η E βR E E NE NE α α E ysit = + bst + (mstbst + mst bs ) + s + t + usit (39) NE α η NE βR E E NE NE α α NE ysit = + bsi + (mstbst + mst bs ) + s + t + usit (40)

20Note that the benefit level is essentially constant for high earners and so will not be indexed by i. Also, the benefit level for low earners rarely changes over time and so is not indexed by t.

24 α ηβ α α α α ∗ βR α α s∗ α α t∗ where = ∗ + 1 β , = 1 β , s = s∗ + 1 β and t = t∗ + 1 β . Assume for the present discussion − − − − E E E that mst = m . It is clear that bs gets absorbed by the state fixed effects. Given the assumption that bst is E uncorrelated with ust, an OLS regression of the takeup rate of the eligibles on the maximum weekly benefit amount, conditional on state and year effects, will identify η+βRmE . Similarly, an OLS regression of takeup of the non-eligibles on the maximum weekly benefit, controlling for the individual benefit level, amount will identify βRmE .21

There are atleast two important issues that this reduced-form approach cannot address. First, it is not possible to identify the underlying reason for why an individual’s propensity to takeup benefits might be increasing in the mean takeup rate. An individual might be more likely to takeup benefits if his neighbor is claiming them, not because his utility cost at the margin was reduced, but rather because he was ignorant of the program or his eligibility status.22 Though uncovering the underlying source of the relationship between own and mean takeup is clearly important and interesting, this paper will be unable to separate these effects.

A second and related point is that if there is an information campaign following an increase in benefits, this might have an independent effect on takeup outcomes of the non-eligibles. Thus, the estimates might not be capturing an effect that operates through the change in the overall takeup rate per se, but rather a correlated effect that is due to a shock that commonly affects both groups.

Lastly, it should be made clear that this is not the first study to consider how benefit levels affect takeup at different levels of aggregation. Blank and Card (1991) estimate takeup models and find large effects at the state level, but statistically insignificant effects at the individual level. Since they are not concerned with estimation of a multiplier effect however, they do not formally address identification of the multiplier.

21Note that the individual benefit level within a state is a function of earnings, the regressions will control directly for earnings, along with other individual characteristics. 22A way to think of ignorance is having an infinite takeup cost.

25 6 Results

To estimate the multiplier, data is collected from the 1984, 1986, 1988, 1990, and 1992 CPS Displaced

Worker Surveys. The sample includes individuals aged 20 to 61, displaced in the two years preceding the survey from nonagricultural, full-time jobs. To focus on individuals eligible for UI benefits, only those with atleast 1 year of previous job tenure and with reported weekly earnings in the lost job greater than the state minimum weekly benefit level are included in the sample.23 UI benefits are imputed from the U.S.

Department of Labor, using previous weekly earnings and state of residence. Since benefit entitlement is conditional on high quarter earnings, there might be measurement error in the estimates. Individuals who report 0 weeks of unemployment are also included in the sample. The final sample contains years 1982-

1991. Table (2) presents summary statistics for the sample and sub-samples of eligibles and non-eligibles.

Individuals are classified as eligibles if the weekly benefit amount exceeds the maximum weekly benefit amount in the previous year.24 Note that all figures are denominated in 1985 dollars.

6.1 Excess Variance Estimates

Table (3) provides estimates of the F-statistic from a one-way ANOVA for each year of the sample. These estimates are all statistically significant and range in size from 1.34 to 2.57. Overall, one cannot reject a positive social multiplier based on these “naive” estimates.

6.2 Reduced Form Differences

The first set of results provide evidence the importance of the direct effect. The coefficient estimate from a regression of the maximum WBA on takeup, where the sample consists of only eligibles, is reported in Panel

A in Table (4). The point estimate from this regression, controlling for year and state effects, is .035 and is precisely estimated: a 10% increase in UI benefits leads to a 6 percentage point increase in the probability

23The sample selection criteria follows McCall (1995). 24As illustrated in figure (5), when benefits increase, there is not a verticial shift in the top part of the schedule. Thus, in year t, an individual might not be observed to be receiving the maximum WBA, but might still have had an increase in benefits.

26 of takeup. This estimate is robust to individual and state level controls, though becomes less precise when state controls are included. It should be noted that the elasticity corresponding to this estimate (.95) is much larger than the estimates reported in McCall (1995). One reason for this discrepancy may be that McCall does not directly estimate the effect of changes in the maximum WBA on the population of eligibles. Infact, a regression of takeup on the maximum WBA using the pooled sample yields a coefficient estimate of .0011 and an elasticity of .25, in line with McCall’s estimates.

Next, the coefficient estimate from a regression of the maximum WBA on takeup, only including those with low previous earnings, is reported in Panel B in Table (4). The coefficient estimate for the indirect effect is .0006: a 10% increase in the maximum WBA is associated with a .5 percentage point increase in the probability of taking up benefits.

In the sample mE is on average .22 and is reasonably stable over time. Taken together, these estimates imply that ηˆ = .0029 and mˆ = 1/(1 βˆ) = 1.92. This suggests that an exogenous increase in the mean − takeup rate of a state translates into a final increased takeup rate for that state of 1.92 percent through the social multiplier effect. Put another way, an increase in benefits that causes 10 individuals to takeup because their benefits were increased, will induce through a spillover effect, on average, an additional 10 individuals to takeup.

7 Search Model with Takeup

To show that a formula similar to equation (9) can be derived in the context of a search model of unemploy- ment, a very simple extension of the discrete-time model in Lentz (2005) is considered. In the interests of space, the setup and solution of the model are briefly described here. Complete details are provided in the appendix.

There are a continuum of individuals that are indexed by a takeup cost. An individual’s preferences are represented by a utility function that is separable in consumption and search effort. When unemployed, an individual chooses his search effort, facing uncertainty in the prospects of finding a job. Individuals choose

27 whether to takeup benefits and how much to consume. If an individual finds a job, he earns a fixed wage and faces a constant, exogenous risk of becoming unemployed.

Individuals accumulate assets and can borrow or lend at the risk-free interest rate, x. Individuals choose a consumption and a search profile to maximize expected lifetime utility. The unemployment insurance agency, has a social weighting that corresponds exactly to the population distribution. Benefits paid out in expectation are required to be exactly offset by the expected present value of taxes. Under standard regularity conditions, the agency’s problem is well-defined, and the optimal replacement rate is characterized using the envelope theorem. The following proposition gives the optimal replacement rate.

Proposition 4 The optimal replacement rate with endogenous take up must satisfy

r 1 = (g 1) (41) 1 r ν + δε − −

δ β 1 where = 1 + 1 DR . −

Formula (41) is closely related to the simply Baily formula defined in equation (9). Two key differences emerge: first, ε is the elasticity of the expected fraction of an individual’s lifetime spent unemployed, with respect to benefits. This follows since individuals cannot deterministically control their durations. Second, g is defined similarly as above, but the marginal utilities of consumption are averaged over an individual’s life- time.25 Nevertheless, the basic intuition underlying the formula holds: the optimal replacement rate trades off the marginal benefit of raising average lifetime consumption of UI recipients by $1 in the unemployed state against the marginal cost of raising the UI tax in the employed state to cover the added benefits. The marginal cost of increasing benefits is given by the direct cost and some added terms arising from the be- havioral response along the intensive margin (agents extending their unemployment duration) and extensive margin (more agents collecting benefits). The main idea is that agents choose to stay unemployed longer and more individuals take up when benefits are higher implying that the unemployment insurance agency

25For precise definitions of these parameters, the reader is referred to the appendix.

28 needs to raise more revenue to satisfy its budget constraint.26

A consequence that follows from agents changing their behavior is a reduction in the tax revenue that is β 1 collected. The term 1 DR in the denominator of equation (41) reflects the amount by which the tax revenue, − or equivalently the tax base, has shrunk. To see this, if there were no change in the tax base, the expression on the right hand side would simply be the sum of the elasticities, ν+ ε. However the tax base is reduced as agents spend more time unemployed when benefits are higher. Therefore the duration elasticity is adjusted β 1 β by 1 + 1 DR , where is the fraction of time UI recipients spend employed relative to non-recipients. −

8 Empirical Calibrations

The modified Baily formulas derived in this paper show that the parameters needed to evaluate the optimal replacement rate are the social welfare weight (g), the elasticity of takeup with respect to the benefit level

(ν), the elasticity of duration with respect to the benefit level (ε), and the social multiplier (m).

As noted, with no takeup margin, g is the ratio of the marginal utility of consumption in the unemployed and employed states. Previous calibrations have approximated this with the value of the consumption drop

ce cu ∆c due to unemployment, γ − , and have estimated the relationship between and replacement rates using · ce ce longitudinal data (Gruber, 1997). Since a similar approximation for g in formula (9) will be used, it is useful to review the methodology. Gruber (1997) is interested in estimating the reduced-form of the following two-equation model:

∆Ci = β0 + β1 BENi + εi (42) ·

BENi = π0 + π1 UIi + ηi (43) ·

where ∆C is the change in (log) consumption when an individual becomes unemployed, BENi measures ben-

26Note that while an increase in benefits affects durations and take up directly, since taxes must increase, this increase also affects them indirectly. The elasticities in the expressions above should be interpreted as the sum of the direct and indirect effects. As well, the elasticities expressed in terms of the total derivatives of D and P with respect to b. Thus these implicitly account for changes in other behaviors such as savings which indirectly affect the choice of D and P.

29 27 efits received and UIi measures potential benefits, the level of benefits that an individual is eligible for. The structural parameter in this model, β1, represents the amount unemployment consumption would increase if an extra dollar was given to an individual randomly selected from the population of unemployed individu- als. Assume that the disturbances (ε,η) are jointly normal and independent of the exogenous variable UIi. Solving for the reduced-form of this system gives

∆Ci = β0 + β1 π0 + β1 π1 UIi + β1 ηi + εi · · · ·

= δ0 + δ1 UIi + νi (44) ·

Calling the composite term νi = β1 ηi + εi, one can see that independence of both of the error terms of UIi · implies independence of νi and UIi. Hence it is valid to estimate δ1 by least squares giving an estimate of

β1 π1. A regression of BENi on UIi yields an estimate of π1. This represents the amount that UI benefits · received would increase if there were a marginal one-dollar increase in UI benefits eligibility. Hence the ˆ ratio is an estimate of β1. Gruber estimates δ1 = .28 and πˆ1 = .48 giving an estimate for β1 of -.58. The − implied consumption drop at unemployment, β0, reflects how much consumption would fall in the absence ˆ ˆ of any insurance arrangements. Gruber estimates δ0 = .24 = β0, where the last equality follows since π0 = 0.

Taken together, the estimates of β0 and β1 suggest that individuals have substantial resources to smooth their consumption. Finally, note that in practice, b/w 0.5, giving a mean consumption drop of 0.1 for the full ≈ sample.

Gruber’s objective is to consistently estimate the consumption drop for the full population of unem- ployed individuals. To compute the optimal replacement rate with takeup however, the formulas derived in this paper highlight the importance of estimating the consumption drop for the self-selected sample of indi- viduals who receive UI benefits. Under the identifying assumption that consumption of non-UI recipients does not respond to changes in the benefit level, it is possible to recover this. This assumption is likely to be violated if the option value of having UI benefits is large. For instance, if benefits are very generous for

27In Gruber’s specification, the fraction of benefits to the previous wage, b/w, is actually used.

30 some individuals, this might cause them to consume more at the beginning of an unemployment spell than if benefits were less generous, even if they were never observed to have received benefits. In practice, 70% of those who collect benefits do so within the first month of unemployment (Chetty, 2005). This suggests that the dynamic aspect of the takeup problem is not too important and can be taken as evidence that the option value of UI is small. Given this assumption, a baseline takeup rate of .54 in Gruber’s sample suggests that the consumption smoothing effect for those actually receiving UI is approximately twice as large in magnitude as the effect for the full sample, .28 1.85 = .52. − × − A potential problem that arises is the marginal individuals who select into the sample when benefits increase might experience less of a drop in consumption than the average individual. This would bias the estimate of the consumption smoothing effect downwards. It is possible however to guage the extent of this selection bias. This can be done by assuming that individuals are randomly drawn from the pool of non- recipients when benefits increase. Table (5) shows that the mean consumption drop for non-UI recipients is around 9%.28 For reasons described above, this is approximately the implied consumption drop. Given a takeup rate of 54%, the implied consumption for UI recipients is 0.37. Further, a 10% increase in the benefit rate is associated with a 2% increase in the probability of takeup.29 Starting from a takeup rate of 54%, this implies that a 10% increase in benefits will cause the average consumption drop to fall through this selection effect by 2% (.37 .09) = 0.1%. It follows that a 10% increase in the benefit rate causes a 5.1% reduction 54% × − in the consumption drop for UI recipients, compared to the estimate of 5.2% that is obtained when selection bias is ignored. Moreover, since the marginal individuals who take up benefits are likely to experience a larger drop in consumption than the average individual who does not takeup, the 0.1 is an upper bound for the selection bias. The full consumption drop for UI recipients can now be written as

∆cR .37 .52r (45) c ≈ −

It is noteworthy that at a mean replacement rate of .5, the mean consumption drop for UI recipients is 0.11,

28See appendix for data description. 29This estimate comes from Anderson and Meyer (1997).

31 which is very close to the estimate reported in Table (5). A final concern that must be addressed is that the results presented in this section are at odds with the model, which assumes that individuals differ only in the cost of taking up benefits. Taken literally, the model predicts that the implied consumption drop due to unemployment for non-recipients is the same as the implied consumption drop for recipients. The results however, suggest that in practice, only the truly needy– those with low assets– avail themselves of benefits.

Nevertheless, it can be shown that with heterogeneity in wealth and a fixed takeup cost, the formula for the ∆c ε optimal replacement rate is unaffected if ones uses population averages for c and in (9). Modeling why some individuals have low wealth and others do not would require a dynamic approach and is beyond the scope of this paper.

The other key elements that determine the optimal replacement rate are the elasticities ν and ε. Following

Gruber (1997) and Chetty (2004), assume that these elasticities are constant. The consensus estimate for both the elasticity of duration and takeup with respect to the benefit level is 0.5.30 These elasticities are ν ε 1 related to and by a scaling factor of 1 r . Given a mean replacement rate of .5, this yields elasticities of − 1. Table (1) reports the optimal replacement rate for a range of γ and m. For comparison, the first column reports the optimal replacement rate in the benchmark case with no takeup and no social interactions, using the consumption drop estimated from the full sample of unemployed.

30These numbers are from Krueger and Meyer (2002) in a review of the social insurance literature.

32 Table 1: Optimal Replacement Rates Benchmark Case m = 1 m = 1.5 m = 2 γ

1 .16 .13 .26 .32

2 .25 .21 .31 .37

3 .31 .26 .35 .40

4 .36 .30 .38 .42

5 .39 .33 .40 .44

10 .50 .43 .48 .51

Notes: Optimal replacement rates are computed according to formula (25) and using 0.5 as an estimate for both ν and ε.

9 Conclusions

This paper has incorporated the takeup decision into a standard model of unemployment insurance, in a setting with and without social interactions. Formulas for the optimal replacement rate were derived di- rectly using behavioral elasticities. A key contribution is to show that multiplier effects have normative implications.

There are several possible interesting directions for future research. First, it would be interesting to see how incorporating the extensive margin affects the optimal time path of benefits. This paper assumes that all individuals who receive benefits are alike and considers the simplifying case of a constant replacement rate. One can think of modeling the participation margin more generally with heterogeneous durations, similar to the approach taken by Saez (2002) who studies the optimal shape of an income transfer system.

Second, this analysis has assumed that individuals who become unemployed are all alike. In practice, unemployment insurance redistributes income from those who experience few spells over their lifetimes to those who experience many spells. It would be interesting to compare the potential role for redistribution in a social insurance setting with the role in a more standard optimal income taxation setting. Going in this

33 direction would require incorporating the labor supply decision into a search model. A final point is that while this paper has focused on unemployment insurance, the main points apply more broadly to other types of social insurance such as health insurance, disability insurance and worker’s compensation.

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38 A Derivation of Optimal Replacement Rate in Baily Model

R R R N N This section provides a formal derivation for equation (9). Let V = u(c ) + v(D ) ψi and V = u(c ) + i u − i u N τ λR λN v(D ). Denote the social welfare function by V(b, ). Letting u and u denote the lagrange multipliers of the individual’s optimization problem, this can be written as

ψi=ψ∗ τ R ψ N ψ V(b, ) = max (1 p)u(ce) + p Vi dF( i) + Vi dF( i) R N R N λ λR λN   ce,cu ,cu ,D ,D , e, u , u − ψiZ=ψ ψi>Zψ  ∗  R R R R N N N +λe[A0 + (w τ) ce] + λ [A0 + bD + w(1 D ) c ] + λ [A0 + w(1 D ) c] − − u − − u u − − u

∂V(b∗,τ) Assume an interior solution so that b∗ satisfies ∂b = 0. Under the assumption that F and f are continu- ous, Leibniz’s Rule for differentiation under the integral sign applies and leads to the following:

∂ τ ∂τ V(b∗, ) R R = λe + λ D = 0 ∂b − ∂b u ∂τ R R λe = λ D ⇒ ∂b u

39 Agent optimization further implies that

λe = (1 p)u0(ce) − ∂ψ λR = pP(b)u (cR) + p f (ψ ) ∗ (u (cR) + v(DR)) u 0 u ∗ ∂ R 0 u cu ∂ψ ∂ψ pψ f (ψ ) ∗ p f (ψ ) ∗ (u (cN) + ψ(DN)) ∗ ∗ ∂ R ∗ ∂ R 0 u − cu − cu λR R ψ R R ψ R u = pP(b)u0(cu ) + p f ( ∗)u0(cu )(u0(cu ) + (D ))

R R N N pψ∗ f (ψ∗)u0(c ) p f (ψ∗)u0(c )(u0(c ) + v(D )) − u − u u R R R R R N N λ = pP(b)u0(c ) + p f (ψ∗)u0(c )(u0(c ) + ψ(D ) (u0(c ) + v(D ))) u u u u − u R pψ∗ f (ψ∗)u0(c ) − u R R R R λ = pP(b)u0(c ) + p f (ψ∗)u0(c )ψ∗ pψ∗ f (ψ∗)u0(c ) u u u − u R = pP(b)u0(cu )

The government’s balance budget constraint implies that

∂τ p ∂P ∂DR = P(b)DR + bDR + P(b)b ∂b 1 p ∂b ∂b −  

This implies that

λR p ∂P ∂DR u DR = P(b)DR + bDR + P(b)b λe 1 p ∂b ∂b −   p P(b)u (cR) p ∂P ∂DR 0 u DR = P(b)DR + bDR + P(b)b ⇒ 1 p u (ce) 1 p ∂b ∂b − 0 −   ∂ ∂ R R b P b D P(b)u0(c ) = u0(ce)P(b) 1 + + ⇒ u P(b) ∂b DR ∂b   ∂ ∂ R R b P b D u0(c ) = u0(ce) 1 + + . ⇒ u P(b) ∂b DR ∂b  

R Substituting definitions (2), (3) and g = u0(cu ) yields (9). u0(ce)

40 B Existence of Equilibria

Lemma 3 For an arbitrary λ, let P = F(λ). There exists an m and M such that

λ m h(F(λ),λ) > 0 (46) ≤ ⇒ λ M h(F(λ),λ) < 0 (47) ≥ ⇒

Proposition 5 Consider λ such that λ (m,M) with m and M defined as in Lemma (3). Then there exists a ∈ social equilibrium.

Proof: Observe that h(F(m),m) > 0 and h(F(M),M) < 0. Since h(F(λ),λ) is continuous in λ over the interval (m,M), by the Weierstrass Intermediate Value Theorem, there is a λψ such that h(F(λψ),λψ) = 0.

41 C Stability of Equilibria

To consider the dynamics of the system, it is convenient work with the distribution function F rather than h.

Note that h = ψ (b) S(Pe) and define F(Pe,b) F(ψ (b) S(Pe)). Holding b fixed and letting Pe vary ∗ − ≡ ∗ − defines a univariate function of Pe. Call this function G. It is clear that G inherits the properties of F. In

∂G particular, G is a distribution function that has a well-defined density, ∂Pe . Following Brock and Durlauf (2001), subscript the variables by time and and assume that individuals condition their best responses on the takeup rate in the previous period. This implicitly assumes that agents are myopic and do not take into account how their action affects the takeup rate in subsequent periods. When this iterative process stops, a social equilibrium is reached. The dynamics of this process are described formally as a first-order difference equation in the per-period takeup rate.

e e Pt+1 = G(Pt ) (48)

A fixed point of this system, Pe = G(Pe) is a social equilibrium. Let me use Taylor’s formula to construct a linear approximation to (48) in some neighborhood of a social equilibrium Pe:

∂G Pe G(Pe) + (Pe Pe) (49) t+1 ≈ ∂Pe t −

The next result states that the system described by (48) is (locally) stable if and only if its linearization (49) is stable.

e ∂G ∂G Proposition 6 A social equilibrium P is asymptotically stable if ∂Pe < 1 and unstable if ∂Pe < 1.

Proof: This follows from Theorem 4.2 of de la Fuente (2000).

Thus, for an equilibrium to be stable, it must be the case that in a local neighborhood, individuals do not

∂G e respond too strongly to changes in average peer behavior. Since ∂Pe = FP , a stable equilibrium is unique if

42 FPe < 1 for all P. This is illustrated in Figure (3). Points A and C where FPe < 1 are stable equilibria and point B where FPe > 1 is an unstable equilibrium.

43 h(m)

ψ m λψ M

h(M)

Figure 1: Multiple Equilibria.

44 E1

E00

E0

F( ) ·

45o P

Figure 2: An Illustration of an Increase in Benefits with Social Interactions.

45 C

B

F( ) A ·

450 P

Figure 3: Stable and Unstable Equilibria.

46 R N u(c ) u(c ) ψi u − u −

∆U

S(P) ∆S

∆b > 0

ψi ψ∗ ψ∗∗

Figure 4: Connection Between Direct and Indirect Utility Gains.

47 Weekly benefit amount

1 Bmax Post Reform

0 Bmax Pre Reform

1 Bmax

0 Bmax

High Quarter Earnings

Figure 5: Illustration of Identification Strategy.

48 Table 2: Summary Statistics for CPS Sample Pooled Eligibles Non-Eligibles Mean Std Dev Mean Std Dev Mean Std Dev

Fraction receiving UI benefits .65 .48 .64 .48 .65 .48

Previous Weekly Earnings 386 228 538 236 344 206

Weekly Benefit Amount (WBA) 143 42 175 31 134 40

Maximum WBA 173 32 175 31 172 32

Tenure in Lost Job (yrs) 5.46 6.02 6.53 6.83 5.2 5.75

Female .35 .48 .22 .42 .38 .48

Reside in S.M.S.A .71 .45 .76 .43 .70 .46

Married .62 .48 .70 .46 .60 .49

Age 36.5 10.43 38.24 10.03 36 10.5

Head of Household .65 .48 .77 .42 .62 .49

Children .46 .50 .48 .50 .46 .50

Blue Collar .57 .50 .50 .50 .60 .49

More than 12 years of educ. .35 .48 .46 .49 .32 .47

Observations 8459 1831 6628

Notes: Data used is from McCall (1995) and includes the 1984, 1986, 1988, 1990, and 1992 CPS Displaced Worker Survey. The sample includes individuals aged 20 to 61, displaced in the two years preceding the survey from nonagri- cultural, full-time jobs, due to plant closure, slack work, abolishment of position, or the end of seasonal work. To focus on individuals eligible for UI benefits, only those with atleast 1 year of previous job tenure and with reported weekly earnings in the lost job greater than the state minimum weekly benefit level are included. UI benefits are imputed from the U.S. Department of Labor using previous weekly earnings and state of residence. Earnings and benefit levels are denominated in 1985 dollars.

49 Table 3: Excess Variance Estimates of the Social Multiplier F-statistic in One-way ANOVA P-value Observations 1982 1.69 .0024 1019

1983 1.57 .0082 990

1984 1.37 .0498 750

1985 1.62 .0048 931

1986 1.31 .0766 770

1987 1.63 .0047 748

1988 1.34 .0680 560

1989 1.46 .0242 709

1990 1.77 .0011 833

1991 2.57 .0000 1194

Notes: Standard errors corrected for state year clustering are reported in parentheses. *** significant at 1%. ** significant at 5%. * significant at 10%. Data is from the CPS Displaced Worker Surveys. A p-value of X indicates that if all the means were the same, then there would be a X probability of the sample means varying by as much as they do with samples of these sizes.

50 Table 4: Reduced Form Differences (1) (2) (3)

A. Effect of Benefit Changes for Eligibles

Max WBA .0035 .0042 .0028 (.0012)*** (.0012)*** (.0014)*

Year Effects Y Y Y

State Effects Y Y Y

Individual Controls N Y Y

State Controls N N Y

Observations 1831 1831 1831

B. Effect of Benefit Changes for Non-Eligibles

Max WBA .0006 .0006 .0004 (.0005) (.0005) (.0005)

Year Effects Y Y Y

State Effects Y Y Y

Individual Controls N Y Y

State Controls N N Y Observations 6628 6628 6628

Notes: Standard errors are reported in parentheses. *** significant at 1%. ** significant at 5%. * significant at 10%. Data is from the CPS Displaced Workers Survey. The takeup rate for the high group is .64. For the low group, the takeup rate is .65. The individual level control variables include tenure, reason for displacement, expected recall, head of household, marital status, kids, industry, urban/rural, blue collar, education, age, gender. The state level controls include the unemployment rate, revenue and the deficit.

51 D Search Model with Takeup

This section outlines a very simple extension of the discrete-time search model in Lentz (2005). In this model, agents are either employed or unemployed at any time t. Let θt denote an agent’s employment status at time t. If θt = 1, the individual is employed, works at a fixed wage w, and pays an unemployment tax of

τ. If θt = 0, the individual is unemployed and chooses whether to collect benefits b. Let ωi be the binary indicator for the individual’s takeup decision. The disutility of collecting benefits is given by the function

ψ(ωi), where ψ(0) = 0.

Transitions between states are modeled as follows. An employed worker faces an exogenous probability,

δ, of becoming unemployed. If unemployed at time t, individuals choose a search intensity, st. Searching is costly with costs given by the smooth, convex and increasing function e(st). The probability of finding a job is specified as an increasing function of st, µ(λst), where λ is an offer arrival rate. Note that in the simple static model, individuals have full control over their unemployment durations should they become unemployed. In this sense, the model resembles a standard labor supply problem. In this context, agents face uncertainty when searching for a job. For this reason, it is important to distinguish between the expected length of an unemployment spell, which individuals can control, and the realized length of an unemployment spell, which individuals cannot control. The expected duration of an unemployment spell is given by the inverse of the hazard rate

dt = 1/µ(λst) (50)

The financial environment is described as follows. Individuals accumulate assets, kt, and can borrow and lend at a (constant) risk-free interest rate x. Assets evolve according to the following equation of motion:

kt 1 kt = xkt + yt ct (51) + − −

where yt = w τ if employed, yt = b ψi if unemployed and collecting benefits and yt = 0 if unemployed − −

52 and not collecting benefits. The only requirement is that a worker’s wealth is bounded at time t so that kt k,k . The budget constraint for a UI recipient can be written as ∈  

kt 1 kt = xkt + θt(w τ) + (1 θt )(b ψi) ct (52) + − − − − −

For a non-UI recipient, this is simply

kt 1 kt = xkt + θt(w τ) ct (53) + − − −

R N Let µt and µt be the corresponding lagrange multipliers for these constraints. This represents the marginal value of relaxing the constraint at the optimum in period t.

∞ A given individual chooses a program ct,st taking (b,τ) as given to maximize { }t=0

∞ ∞ t t E ∑(1 + ρ)− [u(ct) e(st)] + ∑(1 + ρ)− µt(kt+1 kt) "t=0 − t=0 − #

Let the maximal value of this problem be given by Vi(b,τ). The solution to the individual’s maximization ∞ ∞ t t problem defines the functions Li = E ∑ (1 + ρ) θt and Di = 1 L = E ∑ (1 + ρ) (1 θt) . These − − − − t=0  t=0  variables represent the fraction of time an individual spends employed and unemployed, respectively. It will also be useful to define the individual average marginal utility of consumption during employment and unemployment respectively as

∞ t E ∑ (1+ρ)− u0(ct )θt t=0 Eiu0(ce) =  ∞  ρ t θ E ∑ (1+ )− t t=0  ∞ t E ∑ (1+ρ)− u0(ct )(1 θt ) t=0 − Eiu0(cu) =  ∞  t E ∑ (1+ρ) (1 θt ) − − t=0 

These terms will be indexed by R and N when it necessary to distinguish between the subpopulations of UI recipients and non-UI recipients. Finally, the takeup rate is defined as P(b,τ) = F(ψi < b).

53 D.1 Elasticity Concepts

The dynamic setup requires redefining the behavioral inputs that enter the unemployment insurance agency’s problem. First,

w b ∂DR ε = − i (54) i DR ∂(w b) i − is the elasticity that measures the percentage increase in the fraction of agent i0s life spent unemployed, when benefits are increased by 1%. Note that a monetary cost of takeup affects the relative terms of trade and as a result, expected duration is indexed by i. Next, the takeup elasticity is given by

w b ∂P ν = − (55) P ∂(w b) −

D.2 Deriving the Optimal Replacement Rate

Assume that the agency has a social weighting function that corresponds exactly to the distribution of ψi. The UI agency has full control over the benefit level, b, and the unemployment tax, τ, which is collected from individuals that are employed. Benefits are paid out in proportion to the length of the unemployment spell and only to those who claim them. The assumption that the potential benefit duration is infinite is made for modeling convenience although the finding by Davidson and Woodbury (1997) that optimal benefits ought to last indefinitely suggest this is reasonable. Assume that the distribution of ψi, F, is known to the insurer. All individuals, regardless of whether they claim benefits when unemployed, pay taxes. Given a constant policy (b,τ), the agency’s utility function is

b τ R τ ψ N τ ψ V(b, ) = Vi (b, )dF( i) + Vi ( )dF( i) (56) ψ ψ iZ= ψiZ b ≥

54 The UI agency is required to balance its budget in expectation:31

R D b = Lτ (57)

b b R R ψ R ψ N ψ where D = Di dF( i) and L = Li dF( i) + Li dF( i). This takes into account the fact that ψ ψ ψ ψ i= i= ψi b the fraction of Rtime spent employed mayR depend on whetherR≥ the worker collects benefits when unemployed.

The agency’s full problem can now be stated as32

maxV(b,τ) b,τ R s.t.D b = Lτ (58)

To obtain a solution for the optimal replacement rate from first-order conditions, assume that individual life- ∞ ∞ time utility is smooth, increasing and strictly quasi-concave in ci(t) and the set of choices, ci(t) , { }t=0 { }t=0 that satisfy the constraints is convex. This guarantees that the individual problem has a unique global con- strained maximum and allows for the characterization of the optimal replacement rate using the envelope theorem. The next proposition characterizes the optimal replacement rate.

Proposition 7 The optimal replacement rate with endogenous take up must satisfy

r 1 = (g 1) (59) 1 r ν + δε − − where,

R Eu0(cu ) g = R N Eu0(ce )+Eu0(ce ) β 1 DR = −L ν PD R ν R∗ ≡ D 31This assumes that the UI agency commits to the benefit scheme which is constant over time. 32Although borrowing constraints are not formally included, the formula that is derived is valid if agents are infact liquidity constrained. This is a result that carries over from Chetty (2006). Thus, in the model, unemployment insurance can play the dual role of providing insurance and liquidity to the unemployed.

55 b R ε ε Di ψ i R dF( i) D ≡ ψi=ψ δ R β 1 = 1 + 1 DR −

Proof: To prove this result, consider a small revenue-neutral increase in benefits of db is considered. As before, the optimal replacement rate will be derived by exploiting the basic logic of the envelope theorem. A change in benefits has a mechanical and a behavioral effect. Since individuals are optimizing, the envelope theorem tells us that we should evaluate the effect on welfare as if their (average lifetime) consumptions were increased by $1; that is, the behavioral effect can be safely ignored.

D.2.1 Mechanical Effect

R A UI recipient receives an additional Di db of benefits. Summing over the population of UI recipients, the total mechanical effect M is equal to

R M = D db (60)

D.2.2 Behavioral Responses

Consider the behavioral responses. First, along the extensive margin, more individuals takeup benefits. ∂P ν P R This response is given by dP = ∂b db = w b db, implying an increase in expenditures equal to D∗ bdP = − Rν bdb ν PD R ν R D w b , where R∗ and D∗ is the fraction of time spent unemployed for the marginal individual. − ≡ D Second, there is a behavioral response along the intensive margin, as individuals who are receiving benefits b R R ε bdb ε ε Di ψ adjust their durations. The cost due to this response is given by D , where i R dF( i) is w b D − ≡ ψi=ψ a weighted average of the duration elasticity. While behavioral responses affect expenditures,R they also affect revenues since the tax base is changed. Therefore, the amount by which taxes must rise to finance the increase in expenditures must take this into account. To compute this effect, first note that L can be written

56 as

b R N L = (1 D )dF(ψi) + (1 D )dF(ψi) (61) − i − i ψ ψ iZ= ψiZ b ≥

Differentiating with respect to b, the reduction in the tax base is

b b ∂ ∂ R R L Di ψ ε R db ψ ε db dL = ∂b db = ∂b dbdF( i) = iDi w b dF( i) = D w b −ψi=ψ −ψi=ψ − − − R R This shows that the change in the tax base arises solely from individuals adjusting their behavior along the intensive margin. Define

1 DR β = − (62) L

R It follows that b dL = β D ε bdb . The total behavioral cost can now be written as L 1 DR w b − − −

R R bdb R bdb D bdb B = D ν + D ε + β ε (63) w b w b 1 DR w b − − − − To solve for the optimal replacement rate, define the population average marginal utility of consumption when employed and unemployed respectively as

b R R R Eu0(c ) (1 D )Eiu0(c )dF(ψi) e ≡ − i e ψiZ=ψ

N N N Eu0(c ) (1 D )Eiu0(c )dF(ψi) e ≡ − i e ψiZ=b b R R R Eu0(c ) D Eiu0(c )dF(ψi) u ≡ i e ψiZ=ψ

57 Define the social marginal welfare weight as

R Eu0(cu ) g = R N (64) Eu0(ce ) + Eu0(ce )

The first-order condition for the optimal replacement rate is (1 g)M + B = 033. Substitution leads directly − to formula (59).

33The fact that the solution can be written purely in terms of consumption follows from the separability of the utility function.

58 E Data From PSID

The mean consumption drops for UI and non-UI recipients are estimated using food expenditure data from the PSID.34 The PSID tracks roughly 6000 households and their splitoffs over time, and is the primary source of longitudinal data on consumption in the US. The data for this analysis spans the years 1977 to

1997. The key variables are food consumption, employment status and unemployment receipt.

Food consumption is obtained from the family files and is defined as the sum of household consumption at home and away, and paid for by food stamps, following Zeldes (1989). The food stamp variable reports the amount an individual received in the previous month and so is scaled up to an annual value. This is deflated using the CPI to obtain real growth rates. All values reported are in 2003 dollars.

The PSID records whether an individual is unemployed in the year of the survey. As well, the PSID records the number of weeks unemployed in the previous year. For the purpose of the analysis, an individual is defined as being in an unemployment spell if two conditions are satisfied: first, the individual reports being unemployed at the time of the current interview and working at the time of the previous interview; second, the individual reports a positive number for “weeks unemployed in previous year”, in the year following the survey. Data on unemployment compensation receipt in year t come from the survey in year t + 1. Prior to 1993, only the amount of unemployment insurance compensation was recorded. Following 1993, individuals were specifically asked whether they received benefits. UI receipt is defined as having received more than $100 in benefits prior to 1993 and having received benefits after 1993. Eligibility for UI benefits is defined as working in the private sector and having a weekly wage above the minimum weekly benefit amount in a state. This follows Gruber (1997).35

Three exclusion restrictions are made to arrive at the core sample. First, only observations for household heads who are between the age of 20 and 65 are included. Second, observations where there was a change in the number of people in the household are excluded. This is done to eliminate changes in total consumption

34Since food is a necessity, households are less likely to cut back on their food consumption during an unemployment spell, relative to other goods. This implies that the calibrated ratio used understates the true ratio. 35Thank you to Rick Evans and Tom Koch for providing the code to impute eligibility.

59 measures due to changes in the size of a household unit. Finally, only individuals who report becoming unemployed atleast once during the sample are included.

60 F Mean Consumption Drops

The key specification for the average consumption drop of individual i in period t is given by the following equation

git = α + β unempit + εi (65) ·

where gt = log(ct) log(ct 1) denotes the growth rate of food, unempit = 1 if individual i is unemployed − − in year t and 0 otherwise. The first column in Table (5) reports β for UI for the full sample. The next two columns report the consumption drop due to unemployment for those unemployment spells where benefits are received and for those spells where benefits are not received.

Table 5: Effect of Unemployment on Consumption Growth– UI Recipients vs. Non-UI Recipients Full Sample Recipients Non-UI Recipients

(Mean) Consumption Drop -.082 -.086 -.088 (.011)*** (.018)*** (.017)***

(Mean) Consumption Drop -.085 -.085 -.095 (Individual Effects) (.013)*** (.021)*** (.020)***

Observations 30468 28274 30468

Notes: Robust standard errors clustered on the individual are reported in parentheses. *** significant at 1%. ** significant at 5%. * significant at 10%. Data is from the PSID. Individuals are classified as unemployed if they are currently unemployed at the time of the interview and report a positive number for “weeks unemployed in previous year” in the following interview. An individual is defined as a UI recipient for the current UI spell if he reports receiving unemployment compensation in year t at time t + 1. All individuals are eligible for UI benefits. According to this definition, for a typical unemployment spell, the takeup rate is 38%.

61