Academic journal article
*Journal of Risk and Insurance*

# State Dependent Unemployment Benefits

## Article excerpt

ABSTRACT

Optimal design of unemployment insurance is considered in a search setting where the state of nature (business cycle) affects the unemployment risk and thus the return to search. The incentive effects or distortions of individual job search arising due to the unemployment insurance scheme are crucial for optimal policies, so is the scope for risk diversification that depends critically on whether the balanced budget requirement applies to each state of nature or across states of nature. In the former case a basic budget effect tends to cause optimal benefits to be procyclical. If risk diversification across states of nature is possible, the fact that incentives are more distorted in good than bad states of nature tends to make both benefits and contribution rates countercyclical. It is shown that countercyclical benefits exacerbate employment fluctuations but increase average employment by aligning benefits more with states of nature where the incentive costs are small.

INTRODUCTION

The design of unemployment insurance scheme is an important policy issue. Most of the recent empirical and theoretical literature has focused on the incentive effects of unemployment benefits, and many recent reforms have gone in the direction of tightening benefit systems (lower benefits, shorter duration, higher fees, etc.). However, these changes may impair the insurance properties of the schemes. An important question is whether a better balance between insurance and incentives can be attained by making benefits and their financing dependent on the state of nature (business cycle situation). Intuitively there is more need for benefits in a bad state of nature with a high unemployment risk at the same time as benefits are likely to be more distortionary in a good state of nature with good opportunities for job creation. These arguments both go in the direction of having countercyclical elements in unemployment insurance schemes, for example, more generosity in bad states of nature and vice versa.

Interestingly, some countries do have state of nature contingencies in the unemployment insurance scheme. In Canada, benefit levels, duration, and eligibility conditions are state of nature dependent, and in the United States, benefit duration is state of nature dependent. Many other countries have used semiautomatic adjustments of the unemployment insurance scheme, and even more have made discretionary changes depending on the state of nature.

There is a large literature on the design of unemployment insurance schemes. Since Bailey (1978), it is well known that the optimal benefit level trades off insurance and incentives. Recent work has extended these insights in various directions (for a survey see, e.g., Frederiksson and Holmlund, 2004). Surprisingly, there is neither a large theoretical literature on the design of state dependent unemployment insurance nor an empirical literature exploring the state dependencies in the effects of various labor market policies including the benefit level. Kiley (2003) and Sanchez (2008) argue within a search framework that the initial benefit level should be higher and its negative duration dependence weaker in a business cycle downturn compared to an upturn. Both models are partial in the sense that budget effects are disregarded and they both rely on the assumption that benefits are more distortionary in a boom.

The following develops a simple static search model to clarify the interaction between insurance and incentives in the determination of optimal unemployment benefits financed by an income tax. The model is cast in such a way as to yield insights on how the benefit level depends on the overall employment level (unemployment risk). The financing of the benefit scheme turns out to be very important, and in particular the ability to diversify risk not only between employed and unemployed but also across states of nature. We therefore consider, in turn, the cases where the budget balances for each state of nature and across states of nature. The possible diversification of shocks via the public budget is closely related to the macro discussion of the automatic budget effects and the automatic stabilizers (see, e.g., Rejda, 1966; van der Noord, 2000). The present article considers these mechanisms from the micro perspective of insurance and incentives, disregarding aggregate demand effects.

The article is organized as follows. The second section sets out the static state space model with contribution (tax) financed unemployment benefits. The third section considers the case where the budget balances for each state of nature, implying that risk diversification is restricted to be between employed and unemployed workers in a given state of nature. The fourth section allows for a balanced budget constraint across states of nature, which makes risk diversification across states possible, and considers the implications for optimal policies and employment performance. The fifth section offers a few concluding comments.

UNEMPLOYMENT INSURANCE: INSURANCE VERSUS INCENTIVES

The following model is a static one-period model. Denote the state of nature by [s.sub.i] where i = 1 ... N, and let the probability that state [s.sub.i] is realized be given by p([s.sub.i]) [member of] [0, 1] where [[summation].sup.N.sub.i=1] p([s.sub.i]) = 1. Assume an infinity of identical individuals; that is, they all have the same productivity and share the same preferences. Search effort in state i is denoted [e.sub.i], and the probability of finding a job is given by (1)

[pi]([s.sub.i], [e.sub.i]), [[pi].sub.e] ([s.sub.i], [e.sub.i]) > O, [[pi].sub.e3] ([s.sub.i], [e.sub.i]) < 0;

that is, the employment probability is concave in search effort.

The state of nature index i is ordered such that a higher index i is associated with a higher s value and a probability of finding a job, ceteris paribus, that is,

[pi]([s.sub.i], e) > [pi]([s.sub.j],e) for i > j.

The higher the index i, the better the state of nature in terms of job finding.

The wage rate is denoted w and it is assumed exogenous to simplify. There is a universal unemployment insurance scheme covering all workers, and it offers a benefit level denoted by [b.sub.i] to all unemployed. This is financed by a contribution rate or tax [[tau].sub.i] levied on the income of employed workers. Both of these policy instruments are allowed to depend on the state of nature. Without loss of generality unemployment benefits are assumed to be nontaxable income. The disposable income depending on labor market status is thus

w(1 - [[tau].sub.i]) if in job

[b.sub.i] if unemployed.

Denote the utility from consumption, c = w(1 - [[tau].sub.i]), when in job by U((1 - [[tau].sub.i])w), where U((1- [[tau].sub.i])w) is concave, and utility from consumption when unemployed by V([b.sub.i]), where V([b.sub.i]) is concave. This formulation of the utility function is thus sufficiently general to capture various assumptions concerning the disutility from work and stigmatization from unemployment.

The expected utility for a representative individual is

[pi]([s.sub.i], [e.sub.i])U((1 - [[tau].sub.i])w) + (1 - [pi]([s.sub.i], [e.sub.i]))V([b.sub.i]) - [e.sub.i], (1)

where the last term denotes the utility costs of search effort. (2) The sequencing structure is such that agents choose search effort ([e.sub.i]) with knowledge of the current state of nature ([s.sub.i]) but under risk with respect to whether a job is found or not. Hence, there is no information problem with respect to the prevailing state of nature, but agents face a state of nature dependent job finding rate.

It is assumed throughout that employed workers are better off than unemployed workers in all states of nature, that is, the participation

Individual Search Effort

Since all agents are ex ante identical, they make the same effort choice in a given state, and it is characterized by the following first-order condition (3)

[[pi].sub.e] ([s.sub.i], [e.sub.i]) [U((1 - [[tau].sub.i])w) - V([b.sub.i])] = 1. (2)

The left-hand side gives the marginal benefit from increasing search as the product of the marginal efficiency of search ([[pi].sub.e]([s.sub.i], [e.sub.i])) times the utility gain from becoming employed (U((1 - [[tau].sub.i])w)- V([b.sub.i])), and the right-hand side gives the marginal costs of search. The optimal search effort in a given state can now be summarized by the implicit function

[e.sub.i] = e([s.sub.i], [[tau].sub.i], [b.sub.i]).

The behavioral response to benefits and taxes are important in the following and we have the standard effects that higher benefits and taxes tend to lower job search, that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is easily verified by use of (2) that comparing search effort across states for the same benefits and tax level, we have

e([s.sub.i], [[tau].sub.i, [bi.sub.i]] > e([s.sub.j], [[tau].sub.i, [bi.sub.i] > [s.sub.j] if [[tau].sub.e]([s.sub.i], e)] > [tau].sub.e]([s.sub.j], e) e([s.sub.i], [[tau].sub.i, [bi.sub.i]] < e([s.sub.j], [[tau].sub.i, [bi.sub.i] < [s.sub.j] if [[tau].sub.e]([s.sub.i], e)] > [tau].sub.e]([s.sub.j], e).

That is, other things being equal, agents search more in a good state of nature if the marginal efficiency of job search ([[pi].sub.e]) is high, and vice versa.

An important question is whether the incentive effects of benefits and taxes are state dependent. We have that the elasticity of search effort with respect to benefits is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

and with respect to the tax rate it is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

These expressions feature the elasticity

[eta]([s.sub.i], [e.sub.i]) [equivalent to [partial derivative][[pi].sub.e]/[partial derivative][e.sub.ei] [e.sub.i/[[pi].sub.e < O,

which measures the elasticity of the marginal search efficiency with respect to the level of search, that is, how more search affects the search efficiency measured in terms of the job finding probability. In the following we term this the "elasticity of search efficiency."

[FIGURE 1 OMITTED]

A key issue is how the elasticity of search efficiency depends on the state of nature since it determines how the incentive effects of benefits and taxes varies across states of nature. It is immediately clear from (3) and (4) that benefits and taxes reduce search effort more in a good than a bad state of nature, other things being equal, that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

if

0 > [eta]([s.sub.i], e) > [eta]([s.sub.j], e) for i > j, (5)

and oppositely for [eta]([s.sub.i], e) < [eta]([s.sub.j], e).

Although (5) does not hold generally, it is satisfied for a wide class of reasonable job finding probability functions. Figure 1 gives as an illustration a case that implies that (5) holds. (4)

The specific additional assumptions on the job finding probability [pi] ([s.sub.i, [e.sub.i]) underlying the figure are that (1) [pi] ([s.sub.i], 0) = 0 for all i, (2) [[pi].sub.e]([s.sub.i], e) > [[pi].sub.e] ([s.sub.j], e) for i > j, and (3) [lim.sub.e[right arrow][infinity]] [pi] ([s.sub.i], [e.sub.i]) = [alpha]([s.sub.i]), where [alpha]([s.sub.i]) > [alpha]([s.sub.i]) for i > j. (5)

UNEMPLOYMENT INSURANCE UNDER A BALANCED BUDGET IN EACH STATE OF NATURE

We consider first the determination of benefits and taxes when there is a balanced budget constraint applying to each state of nature. We compare the decentralized outcome to the social optimum. The policymaker is assumed to be utilitarian and the sequential structure is such that policy is determined before individuals decide on job search. The social optimum has the planner to decide on both the unemployment insurance scheme and job search simultaneously.

Decentralized Equilibrium

The balanced budget constraint applying to each state of nature reads

[[tau].sub.i]w[pi]([s.sub.i], [e.sub.i]) = (1 - [pi]([s.sub.i], [e.sub.i]))[b.sub.i] for all i. (6)

In the following, the benefit level [b.sub.i] is taken as the policy instrument, and hence the tax rate [[tau].sub.i] follows from (6) as

[[tau].sub.i] ([s.sub.i], [e.sub.i], [b.sub.i]) = 1 - [pi]([s.sub.i], [e.sub.i])/[pi]([s.sub.i], [e.sub.i]) [b.sub.i]/w.

The tax rate is thus proportional to the ratio of benefits to wages (replacement ratio) with a proportionality factor given by the ratio of benefit recipients (1 - [pi]) to taxpayers ([pi]). The response of the tax rate to a change in the benefit rate when taking the behavioral response into account is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the elasticity of the ratio of benefit recipients (1 - [pi]) to tax payers ([pi]) with respect to job search. The tax response to a benefit change is made up of the direct cost effect of changing the benefit level plus the effect this change has on the ratio of benefit recipients to tax payers.

Using (2) and (6), we have that equilibrium search under the balanced budget requirement is determined by

[[pi].sub.e] ([s.sub.i], [e.sub.i]) [U(w - [[tau].sub.i] ([s.sub.i], [e.sub.i], [b.sub.i])w)- V([b.sub.i])] = 1. (7)

Job search can in this case be summarized by the implicit function (6)

[e.sub.i] = [PHI]([b.sub.i], [s.sub.i]),

and it is readily verified that the higher the benefit level, the lower the equilibrium level of job search, that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that this job search effect is the combination of the direct benefit effect and the implied tax effect (cf. (3) and (4)) arising via the requirement that the budget balances (6). Since the effect of both the benefit level and the tax rate is to decrease search, it is no surprise to find that equilibrium search under a balanced budget rule is decreasing in the benefit level. It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Social Optimum

As a point of reference it is useful first to work out the social optimum. A utilitarian criterion is imposed and the social planner jointly determines search effort and benefits (the tax rate follows endogenously from the budget constraint). This is equivalent to maximization of (1) with respect to [e.sub.i] and [b.sub.i]. The first-order conditions to this problem read

[U.sub.c] (w - (1 - [pi]([s.sub.i], [e.sup.*.sub.i]/[pi] ([s.sub.i], [e.sub.*.sub.i] [b.sub.i]) = [V.sub.b] ([b.sub.i]) (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

The social optimum has two important implications. The first condition (8) implies that the marginal utility of income to employed and unemployed workers is the same in any state of nature i. This is known in the literature as the full insurance (Botch) condition since it ensures that income in a given state of nature is redistributed--here between employed and unemployed--up to the point where the two groups have the same marginal utility of income/consumption. Note that it is an implication of (8) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

That is, the benefit level is higher in states of nature where the job finding and thus the employment rate is higher. This effect is driven by the basic budget mechanism that a high employment rate implies few on benefit recipients and hence the tax rate can be low other things being equal. This will increase consumption for the employed, lowering their marginal utility, which in turns calls for a higher benefit level to ensure that the two groups have the same marginal utilities. Hence, a basic budget effect will make optimal benefits procyclical.

The second condition (9) gives the social optimal effort level ([e.sup.*.sub.i]). Comparing this to the effort level chosen by the decentralized decision makers ([e.sub.i]; cf. (7)), we find that the determination of the social optimal effort level ([e.sup.*.sub.i]) includes an additional marginal benefit term

[U.sub.c] (w - (1 - [pi]([s.sub.i], [e.sup.*.sub.i]/[pi] ([s.sub.i], [e.sup.*.sub.i] [b.sub.i]) = [[pi].sub.e]([s.sub.i], [e.sup.*.sub.i]/[pi]([s.sub.i], [e.sup.*.sub.i] > 0. (11)

This term arises from the fact that the planner takes into account that more job search increases the employment rate, which in turn lowers the tax rate. This effect thus arises because the planner internalizes the budget constraint, whereas individual decision makers do not (the common pool problem). It is a straightforward implication that the social optimal search level exceeds the level arising in decentralized equilibrium,

[e.sup.*.sub.i] > [e.sub.i].

Optimal Benefits in Decentralized Equilibrium

Return now to the case where the policymakers decide on the properties of the unemployment insurance scheme and individuals choose effort (cf. (7)). Since the planner is assumed utilitarian, individual utilities are respected and the criterion is therefore to maximize expected utility for the representative agent (here equivalent to the weighted sum of utility to employed and unemployed), that is,

[pi](s.sub.i]([e.sub.i] U (w - (1 - [pi]([s.sub.i], [e.sub.i]/[pi] ([s.sub.i], [e.sub.i] [b.sub.i]) + (1 - [pi] [s.sub.i]([s.sub.i])) V([b.sub.i]) - [e.sub.i], (12)

Note that it has been used that the policymaker takes the budget constraint (6) into account in setting the benefit level.

The optimal benefit level is now found as the value of bi maximizing (12) subject to (7), and the optimum is characterized by the following first-order condition (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or using (2) as

[V.sub.c]([b.sub.i]) - [U.sub.c] (w - (1 - [pi]([s.sub.i], [e.sub.i])/[pi] ([s.sub.i], [e.sub.i]) [b.sub.i]) [1 + D ([s.sub.i], [e.sub.i], (13)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This term captures the behavioral response of job search to the benefit level, and therefore how job search is distorted.

It is readily seen that the term D([s.sub.i], [e.sub.i]) determines how the condition for the optimal benefit level differs from the Borch condition (8) that marginal utilities should be equalized. Specifically, we have

[V.sub.c]([b.sub.i]) > [U.sub.c] (w - (1 - [pi]([s.sub.i], [e.sub.i])/[pi] ([s.sub.i], [e.sub.i]) [b.sub.i]) for D([s.sub.i], [e.sub.i]) > 0.

That is, the optimal benefit level implies that the marginal utility for unemployed exceeds the marginal utility for the employed. This difference arises because benefits affect incentives. This is most easily seen by considering the special case where there is no incentive issue; that is, the search effort is constant or unaffected by the benefit level ([[pi].sub.e] = 0 or [partial derivative][e.sub.i]/[partial derivative][b.sub.i]|bb = 0). In this case there is no distortion term (D([s.sub.i], [e.sub.i]) = 0) and we recover the Borch condition for full insurance, (8) that is, [V.sub.b]([b.sub.i]) = [U.sub.c] (w - (1 - [pi]([s.sub.i], [e.sub.i])/[pi] ([s.sub.i], [e.sub.i]) [b.sub.i]). In this case we have that benefits move procylical (cf. (10)).

Returning to the general case where incentives are distorted (D([s.sub.i], [e.sub.i]) > 0), we have that a divergence from complete insurance arises because the optimal benefit level trades off incentives and insurance. This is seen most clearly by rewriting (13) as

[V.sub.b]([b.sub.i]) > [U.sub.c] (w - (1 - [pi]([s.sub.i], [e.sub.i])/[pi] ([s.sub.i], [e.sub.i]) [b.sub.i]) = [U.sub.c] (w - (1 - [pi]([s.sub.i], [e.sub.i])/[pi] ([s.sub.i], [e.sub.i]) [b.sub.i]) D([s.sub.i], [e.sub.i]),

where the left-hand side gives the value of insurance measured by the difference in marginal utilities of income, and the right-hand side gives the incentive or distortions from the benefit level. The optimal benefit level balances the value of insurance to the incentive effects. To interpret this condition further, first note that the distortions arising from the benefits are driven by the effects of benefits on job search. As shown earlier, individual job search is inefficiently low, and since higher benefits lead to lower job search there is an incentive cost of providing benefits. The costs of providing benefits are thus higher, the higher the distortion. The incentive effect implies that it is not optimal to provide full insurance. Therefore, the marginal utility of income of the unemployed is larger than that of the employed in optimum.

The interesting question here is how the incentive effect affects how optimal benefits vary with the state of nature. The distortion expression D([s.sub.i], [e.sub.i]) can be written

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In considering the variation of the distortion terms across states of nature, we have two effects at play: (1) the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] term capturing how a change in benefits affect the budget via a change in the ratio between benefit recipients and tax payers, and (2) the responsiveness of job search to the benefit level

([b.sub.i]/[e.sub.i] [partial derivative][e.sub.i]/[partial derivative][b.sub.i]|bb). We have already considered these terms and the conditions under which the distortion term is procyclical.

If the distortion term is procyclical this tends to make optimal benefits counter-cyclical. To see this compare two states i and j, we have from (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The relation of benefits in the two states thus depends on two factors. The first part is related to the distortions, and this goes in the direction of having benefits to be countercyclical ([b.sub.i] < [b.sub.j]) since

[1 + D([s.sub.i], [e.sub.i])/[1 + D([s.sub.ij, [e.sub.j]) > 1 if D([s.sub.i], [e.sub.i]) > D([s.sub.j], e) for i > j.

The second part relates to the budget effect discussed earlier, and it goes in the direction of making benefits procyclical ([b.sub.i] > [b.sub.j]) since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, even for D([s.sub.i],e) > D([s.sub.j], e) for i > j it cannot be concluded unambiguously that [b.sub.i] < [b.sub.j]. This shows that the balanced budget requirement has an important role for the determination of the benefit level.

THE PUBLIC BUDGET AS A BUFFER

The preceding analysis assumed a balanced budget requirement for each state of nature. This clearly precludes risk diversification across states, and it is well known that the public budget can offer some scope for risk smoothing (see Gordon and Varian, 1988; Andersen and Dogonowski, 2002). In the macro literature, this is associated with the automatic budget effects and automatic stabilizer effects (see, e.g., van der Noord, 2000).

To see the role of such a smoothing device, consider the design of a tax-benefit scheme under the constraint that the expected expenses due to state dependent benefits are covered by the expected revenue from state dependent tax revenues. This may be interpreted as reflecting either a small open economy with access to a complete international capital market, or as a situation where the budget is always balanced via lump sum tax levied on some other groups (capitalist) that are risk neutral, i.e., the constraint is

[N.summation over (i = 1)] p ([s.sub.i]) [(1 - [pi] ([s.sub.i], [e.sub.i]))[b.sub.i] - [pi] ([s.sub.i], [e.sub.i]) [[tau].sub.i] w] = 0. (14)

The important difference to the previous section with a balanced budget requirement is that the benefit level and the tax rate in a given state of nature are not tied via the balanced budget requirement, and therefore they are separate instruments.

As in the preceding section, we consider both the social optimum and the optimal policies for the decentralized equilibrium.

Social Optimum

The social optimum is found as the solution to the problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to (14).

The first-order conditions read

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [psi] is the Lagrange multiplier associated with (14). It is seen that the job search is determined by the same condition as in (9), and the finding that individuals search is below the social optimal level also holds here. Risk diversification via the budget implies that

[V.sub.b]([b.sub.i]) = [V.sub.b] ([b.sub.j]) for i [not equal to] j

[U.sub.c]((1 - [[tau].sub.i])w) = [U.sub.c] ((1 - [[tau].sub.j])w) for i [not equal to] j.

Hence, the optimal policy has state independent benefit and tax rates, that is,

[b.sub.i] = [b.sub.j] for i [not equal to] j

[[tau].sub.i] = [[tau].sub.j] for i [not equal to] j.

The risk diversification made possible by not having to balance the budget in each state of nature but across states of nature thus has the important implication that all risk is absorbed via the budget leaving the consumption/income of both employed and unemployed invariant to the state of nature.

Optimal Policies in the Decentralized Case

In the decentralized equilibrium we have that the optimal tax-benefit scheme solves

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

Individual search effort (7)

[N.summation over (i = 1)] p([s.sub.i]) [[1 - [pi] ([s.sub.i], [e.sub.i])] [b.sub.i] - [pi] ([s.sub.i], [e.sub.i]) [[tau].sub.i] w] = O.

The first-order condition can be written (9)

(1 - [pi] ([s.sub.i], [e.sub.i])] [V.sub.b] ([b.sub.i]) - [lambda] [[1 - [pi] ([s.sub.i], [e.sub.i])] - [partial derivative][pi] ([s.sub.i], [e.sub.i])/[partial derivative][b.sub.i] ([b.sub.i] + [[tau].sub.i] w)] = 0, (15)

- [pi] ([s.sub.i], [e.sub.i])] [U.sub.c]((1 - [[tau].sub.i])w)w - [lambda] [- [pi] ([s.sub.i], [e.sub.i])] w - [partial derivative][pi] ([s.sub.i], [e.sub.i])/[partial derivative][[tau].sub.i] ([b.sub.i] + [[tau].sub.i] w)] = 0, (16)

where [lambda] is the Lagrange multiplier associated with the budget constraint. From the first-order conditions (15) and (16), we have that the optimal determination of the benefit level and the tax rate implies.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

This shows how the optimum balance the insurance and incentive effects. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

it follows that

[V.sub.c]([b.sub.i]) > [U.sub.c]((1 - [[tau].sub.i])w) for all i.

That is, the marginal utility of income for the unemployed is larger than the marginal utility of income for the employed in all states of nature. That is, although risk smoothing via the budget is now possible, there remains a trade-off between insurance and incentives, implying that the optimal policy does not equalize marginal utilities across the two groups.

The relation of optimal benefits in two states of nature (i [not equal to] j) are from (15) given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Similarly the relation of the optimal tax rates in two states of nature are from (16) given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

When risk diversification across states of nature is possible via the budget, we find that the question of state contingencies in optimal benefit levels and contribution rates only depends on whether the incentive effects or distortions differ across states of nature. To see this, consider first the case where benefits and taxes are equally distortive in all states of nature, that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this case it follows directly from (18) and (19) that the optimal benefit level and tax rate are state independent, that is,

[b.sub.i] = b for all i

[[tau].sub.i] = [tau] for all i.

It is a straightforward implication that the budget balance (net revenue)

r ([s.sub.i]) = [tau]w[pi] ([s.sub.i], [e.sub.i]) - [1 - [pi] ([s.sub.i], [e.sub.i])] b,

moves procyclically if [pi] ([s.sub.i], [e.sub.i] ([s.sub.i], b, [tau])) > [pi] ([s.sub.j], [e.sub.j] ([s.sub.j], b, [tau])) for i > j.

Proceeding to the more interesting case where distortions vary with the state of nature, observe first that the condition (17) relates the marginal utilities for unemployed and employed. Hence, for given distortions (10) there is an underlying tendency that the benefit level ([b.sub.i]) and the contribution rate ([[tau].sub.i]) should move in the same direction to maintain the ratio of marginal utilities. This is made possible by the balanced budget requirement applying across states of nature and thus the possibility of treating benefits and taxes as separate instruments in a given state of nature. In the following the interesting question is whether distortions in themselves give an argument for state of nature dependent benefit levels and contribution rates. Countercyclical benefits and taxes require that distortions are larger in good than in bad states. To see this, consider first the benefit level. The crucial factor here is

[partial derivative] (1 - [pi] ([s.sub.i], [e.sub.i]))/[partial derivative][b.sub.i] [b.sub.i]/1 - [pi] ([s.sub.i], [e.sub.i]) > 0,

measuring how benefits affect unemployment in a given state of nature. For state independent taxes (11) ([[tau].sub.i] b = b[tau] for all i), we have that if

[partial derivative] (1 - [pi] ([s.sub.j], [e.sub.j]))/[partial derivative][b.sub.j] [b.sub.j]/1 - [pi] ([s.sub.j], [e.sub.j]) < [partial derivative] (1 - [pi] ([s.sub.i], [e.sub.i]))/[partial derivative][b.sub.i] [b.sub.i]/1 - [pi] ([s.sub.i], [e.sub.i])

then it follows

[V.sub.b]([b.sub.i])/[V.sub.b]([b.sub.j]) > 1,

and hence optimal benefits imply [b.sub.i] < [b.sub.j]. The intuition is that if the benefit level is less distortionary in state j, it is possible to offer a better insurance in this state of nature in the form of a higher benefit level compared to state i. Hence, if i > j, it follows that benefits should be countercyclical.

Turning to the relative tax rates, the key factor is

[partial derivative] [pi] ([s.sub.i], [e.sub.i])/[partial derivative][[tau].sub.i]/[[tau].sub.i] - [pi] ([s.sub.i], [e.sub.i]) < 0,

measuring how the tax rate distorts employment. For state independent benefits ([b.sub.i] = b for all i), we have that if

[partial derivative] [pi] ([s.sub.i], [e.sub.i])/[partial derivative][[tau].sub.i] [[tau].sub.i]/[pi] ([s.sub.i], [e.sub.i]) < [partial derivative] [pi] ([s.sub.j], [e.sub.j])/[partial derivative][[tau].sub.j] [[tau].sub.j]/[pi] ([s.sub.j], [e.sub.j]),

then it follows that

[U.sub.c] ((1 - [[tau].sub.i])w)/[U.sub.c] ((1 - [[tau].sub.j])w) < 1,

and thus optimal taxation implies [[tau].sub.i] < [[tau].sub.j]. That is, other things being equal, less taxes are levied in states of nature where the distortion is the largest. Hence, if i > j, the tax rate or contribution rate should move countercyclically. (12)

Finally, note that the elasticity of unemployment with respect to the benefit level can be expressed in terms of the elasticity of employment with respect to benefits as follows

[partial derivative] (1 - [pi] ([s.sub.i], [e.sub.i]))/[partial derivative][b.sub.i] [b.sub.i]/1 - [pi] ([s.sub.i], [e.sub.i]) = - [partial derivative][pi] ([s.sub.i], [e.sub.i]))/[partial derivative][b.sub.i] [b.sub.i]/[pi] ([s.sub.i], [e.sub.i]) [pi] ([s.sub.i], [e.sub.i])/(1 - [pi] ([s.sub.i], [e.sub.i]).

Hence, if the benefit level and the tax rate have the same distortions across states measured by their effect on the employment rate, (13) it follows that benefits affect unemployment more in a good than a bad state since

- [partial derivative] [pi] ([s.sub.i], [e.sub.i])/[partial derivative][b.sub.i] [b.sub.i]/[pi] ([s.sub.i], [e.sub.i]) [pi] ([s.sub.i], e)/(1 - [pi] ([s.sub.i], e) > - [partial derivative] [pi] ([s.sub.j], [e.sub.i])/[partial derivative][b.sub.j] [b.sub.j]/[pi] ([s.sub.j], [e.sub.j]) [pi] ([s.sub.j], e)/(1 - [pi] ([s.sub.j], e).

It follows from (18) and (19) that the optimal policy implies that [b.sub.i] < [b.sub.j] and (14) [[tau].sub.i] < [[tau].sub.j] where i > j; that is, benefit levels and taxes are lower in good states with higher distortions and higher in bad states with lower distortions.

Employment Effects

Since the optimal policy trades off insurance and incentives and since risk diversification via the budget plays an important role, it is an interesting question whether the strengthening of automatic budget reactions via state contingent policies also leads to a stabilization of employment. If the comparison is between a scheme with state independent taxes and benefits and a scheme with state dependent policies, then the latter will display the largest employment fluctuations across states of nature. To see this, note that employment in a given state is determined by

[pi] ([s.sub.i],e([s.sub.i], [[tau].sub.i], [b.sub.i])),

where

[partial derivative] [pi] ([s.sub.i], e ([s.sub.i], [[tau].sub.i], b.sub.i]))/[partial derivative][b.sub.i] = [[pi].sub.e] [partial derivative]e ([s.sub.i], [[tau].sub.i], b.sub.i])/[partial derivative][b.sub.i] < 0

[partial derivative] [pi] ([s.sub.i], e ([s.sub.i], [[tau].sub.i], b.sub.i]))/[partial derivative][[tau].sub.i] = [[pi].sub.e] [partial derivative]e ([s.sub.i], [[tau].sub.i], b.sub.i])/[partial derivative][[tau].sub.i] < 0.

Hence, we have that

[pi] ([s.sub.i], e ([s.sub.i], [tau], b.sub.i])) > [pi] ([s.sub.i], e ([s.sub.i], [tau], b)) if [b.sub.i] < b

[pi] ([s.sub.j], e ([s.sub.j], [tau], b.sub.j])) > [pi] ([s.sub.j], e ([s.sub.j], [tau], b)) if [b.sub.j] < b

and

[pi] ([s.sub.i], e ([s.sub.i], [tau.sub.i], b)) > [pi] ([s.sub.i], e ([s.sub.i], [tau], b)) if [[tau].sub.i] < [tau]

[pi] ([s.sub.j], e ([s.sub.j], [tau.sub.j], b)) > [pi] ([s.sub.j], e ([s.sub.j], [tau], b)) if [[tau].sub.j] < [tau],

It therefore follows that if the state contingent policy raises benefits and taxes in bad states of nature and lowers them in good states of nature, then this leads to lower employment in bad states and increase employment in good states. To put it differently, although the optimal state contingent policy contributes to insurance, it does not necessarily contribute to stabilizing employment.

However, the optimal state contingent policy may improve average employment compared to a state independent policy. To see this, note that expected employment across states of nature is given as

E = [N.summation over (i = 1)] p([s.sub.i])[pi]([s.sub.i], e([s.sub.i], [[tau].sub.i], [b.sub.i])).

Hence, the difference between expected employment under a state contingent policy ([E.sub.SD]) and a state independent policy ([E.sub.SID]) is given as

[E.sub.SD] - [E.sub.SID] = [N.summation over (i = 1)] [p([s.sub.i])[[pi]([s.sub.i], ([[tau].sub.i], [b.sub.i])) - [pi]([s.sub.i], e([s.sub.i], [tau], b))]].

Employment in any given state of nature under the state contingent policy can, by a first-order Taylor approximation, be written

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

Consider two states of nature, k and h, and assume that [b.sub.i] = b for all i [not equal to] k, h, and [[tau].sub.i] = [tau] for all i. Assume that

[partial derivative] [pi] ([s.sub.k], e ([s.sub.k], [tau], b))/[partial derivative]b b/[pi] ([s.sub.k], e ([s.sub.k], [tau], b)) < [partial derivative] [pi] ([s.sub.h], e ([s.sub.h], [tau], b))/[partial derivative]b b/[pi] ([s.sub.h], e ([s.sub.h], [tau], b)) < 0.

That is, benefits are more distortionary in state k than h. Consider now a situation where benefits in the two states are made state dependent. Given the result earlier, the optimal policy would have [b.sub.k] > b > [b.sub.h]; that is, the benefit is increased in state k and lowered in state h. For this policy shift to be consistent with the budget constraint, it is required that

p([s.sub.h] [pi] ([s.sub.h], e([s.sub.h], [tau], b)) ([b.sub.h] - b) + p([s.sub.k] [pi] ([s.sub.k], e([s.sub.k], [tau], b)) ([b.sub.k] - b) = 0.

It now follows from (20) that

[E.sub.SD] - [E.sub.SID] > O.

That is, this policy shift to a state dependent policy increases expected employment. The intuition is that by changing the benefit structure such that benefits become lower when the distortion is high and higher when the distortion is low, the overall distortion from benefits is reduced, and therefore the expected or average employment over the two states of nature increases. Similar reasoning clearly holds when changing from state independent to state dependent taxes. It can therefore be concluded that if there is a case for making benefits and tax rates state contingent, it follows that this contributes to increase expected employment. The effect of state dependent policies on the variability of employment across states of nature should thus be seen relative to the effect on the mean or average employment level.

So far it has been assumed that both the benefit level and the tax rate can be made dependent on the state of nature. Suppose that there are different costs of imposing such contingencies and that the costs of making benefits state dependent are less than the costs of making tax rates state dependent. One argument is that increasing benefit generosity can simply be made by linking benefits to unemployment duration, whereas it is administratively much more difficult to introduce contingencies in the tax system beyond those arising from taxes depending on income. In this case, it is easily seen that there is still a case for making benefits state contingent provided that the distortions vary across states of nature.

CONCLUDING REMARKS

The question of whether unemployment benefits should be state contingent has been analyzed in a stylized search framework. The optimal policy is critically affected by the fact that individual job search is distorted by the presence of an unemployment insurance scheme (both the benefit and the tax rate), implying that it falls short of the social optimal level of job search. When the budget for the unemployment insurance scheme has to balance for each state of nature, the first best tends to imply a procyclical benefit level (countercyclical tax rate). The reason is the budget effect arising from the fact that it is a lower burden to finance given benefits when unemployment is low than when it is large. In decentralized equilibrium the state dependency of the optimal benefits depends on the balance between the procyclical effect running via the budget and the countercyclical effect arising if benefits are less distortionary in bad than a good state of nature. In the case where risk diversification is possible via the budget (the budget balance requirement applies across states of nature), the benefit level and the contribution (tax) rate are independent instruments in a given state of nature, and the first best policy is to have state independent benefits and tax rates. In the decentralized case there is thus an unambiguous argument to have benefits and contribution rates be countercyclical if incentives are more distorted in good than bad states of nature.

The role of unemployment insurance for labor market performance is often assessed in terms of employment. In the case where the budget requirement applies across states of nature we find that the optimal state dependent policy tends to make employment more sensitive to the business cycle situation. The reason being that benefits are lowered in a good state, which further contributes to job search and thus employment, and vice versa in a bad state. However, this increased variability arises because the scheme is better aligned to distortions; that is, benefits are high when distortions are low, and vice versa. A consequence of this is that the "average" distortions are reduced and therefore the average employment level is increased. Hence, state dependent benefits tends to produce a higher average level of employment at the cost of more variability.

The present article has considered a static model, implying that the business cycle variations are modeled in a very stylized way. An interesting extension of the present work is thus to model business cycle fluctuations in an explicit intertemporal setting that will allow for a more realistic modeling of fluctuations but also how the budget can be a buffer by diversifying shocks over time. This article has disregarded aggregate demand effects, which in turn implies that possible stabilizing effects of state contingent unemployment insurance do not arise. It is an interesting issue whether an explicit consideration of this mechanism will strengthen the case for state of nature contingencies in the unemployment insurance scheme.

APPENDIX

This appendix shows that condition (5) holds for the job finding probability function depicted in figure holds. The specific additional assumptions for this function are: (1) [pi] ([s.sub.i], 0) = 0 for all i, (2) [[pi].sub.e] ([s.sub.i],e) > [[pi].sub.e] ([s.sub.j],e) for i > j, and (3) [lim.sub.e[right arrow][infinity]] [pi] ([s.sub.i], [e.sub.i]) = [alpha] ([s.sub.i]), where [alpha]([s.sub.i]) > [alpha]([s.sub.i]) for i > j.

We have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Next, observe that [pi]([s.sub.i],e) - [pi]([s.sub.j],e) = 0 for e = 0 and [lim.sub.e] [right arrow] [[pi]([s.sub.i],e) - [pi]([s.sub.j],e)] = [alpha]([s.sub.i]) > [alpha]([s.sub.j]) > 0, which given the monotonicity assumptions made implies that [pi]([s.sub.i],e) - [pi]([s.sub.j],e) is increasing in e, that is, [[pi].sub.e]([S.sub.i],e) - [[pi].sub.e]([s.sub.j],e) > 0.

Making a first-order Taylor approximation, we have

[[pi].sub.e]([s.sub.i],[e.sub.1]) [equivalent] [[pi].sub.e] ([s.sub.i],[e.sub.1]) + [[pi].sub.ee] ([s.sub.i],[e.sub.1]) [[e.sub.1] - [e.sub.0]]

and hence [[pi].sub.e]([s.sub.i],[e.sub.1]) - [[pi].sub.e]([s.sub.j],[e.sub.1]) = [[pi].sub.e]([s.sub.i],[e.sub.0]) - [[pi].sub.e]([s.sub.j],[e.sub.0]) + [[[pi].sub.ee]([s.sub.i],[e.sub.0]) - [[pi].sub.ee]([s.sub.j],[e.sub.0])] [[e.sub.1] - [e.sub.0]] and hence for [[pi].sub.e]([s.sub.i],[e.sub.1]) - [[pi].sub.e]([s.sub.j],[e.sub.1]) > [[pi].sub.e]([s.sub.i],[e.sub.0]) - [[pi].sub.e]([s.sub.j],[e.sub.0]) > 0 for [e.sub.1] - [e.sub.0] > 0 is ensured if [[pi].sub.ee]([s.sub.i],[e.sub.0]) - [[pi].sub.ee]([s.sub.j],[e.sub.0]) > 0. Hence, we have [[pi].sub.e]([s.sub.i],e) / [[pi].sub.e]([s.sub.j],e) > 1 and [[pi].sub.ee]([s.sub.i],[e.sub.0]) / [[pi].sub.ee]([s.sub.j],[e.sub.0]) > 1 and it follows that [eta]([s.sub.i],e) - [eta]([s.sub.j],e) > 0.

REFERENCES

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(1) It is assumed that 0 < [pi]([s.sub.i], [e.sub.i]) < 1 for all i and [e.sub.i] [member of] [0, [infinity]].

(2) Assuming linear costs simplifies the model but does not affect the results qualitatively.

(3) The second-order condition is fulfilled since [[pi].sub.ee] < 0.

(4) If [pi] ([s.sub.i], [e.sub.i]) = 1 - exp(-[s.sub.i] [e.sub.i]), as is the case considered by Kiley (2003) and Sanchez (2008), we have [eta] ([s.sub.i], [e.sub.i]) = (1 - [s.sup.2.sub.i]) exp(- [s.sub.i] [e.sub.i]) and hence (5) holds.

(5) See the Appendix for further details.

(6) Using [e.sub.i] = e([s.sub.i], [r.sub.i], [b.sub.i]), this may alternatively be written as [e.sub.i] = e([s.sub.i], 1 - [pi] ([s.sub.i], [e.sub.i])/[pi] ([s.sub.i], [e.sub.i]) [b.sub.i]/w, [b.sub.i]).

(7) The second-order condition is assumed fulfilled.

(8) Note that if U() = V(), the condition implies that [b.sub.i] = (1 - [[tau].sub.i])w = [pi]([s.sub.i], [e.sub.i])w; that is, there is complete income sharing.

(9) The second-order condition is assumed fulfilled.

(10) That is where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is constant.

(11) State independent taxes serve to remove the tendency for benefits and tax rates to move in the same direction induced by considerations of marginal utilities (cf. discussion earlier).

(12) Varian (1980) shows that tax progression can be justified on insurance grounds. A progressive tax system tends to imply procyclical tax rates contrary to the finding here. Notice, however, that in Varian the employed individual can realize different income levels whereas this is not

the case here. Hence, the present article does not include income risk but only employment risk.

(13) That is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all i, j.

(14) Note that taxes become state dependent although the distortion is state independent because benefits differ across states (cf. (19)).

DOI: 10.1111/j.1539-6975.2010.01379.x

Torben M. Andersen is at the School of Economics and Management, Aarhus University, CEPR, CESifo, and IZA. Michael Svarer is at the School of Economics and Management, Aarhus University, CAM, IZA. We gratefully acknowledge constructive comments from the referees. The usual disclaimer applies.

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