Labor market search and optimal retirement policy.

Bhattacharya, Joydeep ; Mulligan, Casey B. ; Reed, Robert R., III 等

A popular and long-standing view is that social security is a means for young, unemployed people to "purchase" jobs from older workers. Can social security, by encouraging retirement and hence creating job vacancies for the young, improve the allocation of workers to jobs? Maybe, according to a standard model of labor market search, but public retirement programs currently pay the elderly substantially more than their jobs are worth. An important effect is that retirement reduces the value of other vacant jobs. Our results imply that recent reforms aimed at reducing retirement incentives are likely to improve labor market efficiency.

I. INTRODUCTION

The objective of this article is to formalize a popular argument about the economic rationale for social security and public pension programs in the United States and other countries. We construct a model in which there can be significant frictions slowing the process of matching workers with firms, and social security is a labor market policy designed to remove old workers from the labor market. In this sense, it can be said that social security is a means for young, unemployed individuals to "purchase" jobs from older, employed workers. President Franklin D. Roosevelt, in particular, in one of his fireside chats suggested that this would be an important goal for social security:

 The program for Social Security now pending
 before the Congress is a necessary part of the
 future unemployment policy of the Government.
 ... It proposes, by means of old-age pensions, to
 help those who have reached the age of retirement
 to give up their jobs and thus give to the younger
 generation greater opportunities for work and to
 give all a feeling of security as they look toward
 their old age. (Roosevelt 1938, 134-35)

This position on social security continues to receive some support, at least enough to receive serious consideration in recent social security debates. In recent discussion about the retirement earnings test in Congress, many agreed with the following perspective: "Social Security, when it was created in 1935, sought to achieve two goals--moving older workers out of the workforce to make way for younger workers, and to partially replace lost income due to retirement" (Testimony of Honorable John J. Rhodes III in Social Security Retirement Test 1991). These debates are not unique to the United States. As is well known, European countries have recently devoted much attention to youth unemployment. As a result, public pension programs have become more generous and increasingly used to encourage labor force withdrawal among older workers.

Although public pension programs have been used in many countries as age-targeted employment policies, there has been little work aimed at understanding their implications for the market at the aggregate level. In this article we begin to address whether policies aimed at discouraging work among the elderly, and thereby creating job vacancies for the young, can improve the allocation of workers to jobs in the labor market. In particular, the idea that continued employment by the old contributes to increased youth unemployment implies that the life cycle contributes to congestion problems in the labor market. Accordingly, an important aspect of a modeling approach aimed at addressing these issues should be a setting in which unemployment is an equilibrium phenomenon, and the labor market participation decisions of workers impose a negative externality on the pool of the unemployed.

We introduce a simple model of labor market search along the lines of Mortensen (1982) and Pissarides (2000) extended to include retirement decisions and show how the optimality of induced retirement depends on the model's parameter values, some of which imply that the social value of retirement exceeds the private value while other parameter values imply that the private value exceeds the social value. We then consider the parameter values implying the largest wedge between private and social retirement values and show that even in this extreme case, optimal retirement policy does much less to encourage retirement than do the policies used by governments around the world. To put it simply, labor market search theory provides at best only a partial justification for publicly induced retirement; public retirement programs pay the elderly substantially more than labor market search theory implies that their jobs are worth. An important effect, ignored in arguments for policy-induced retirement, is that the creation of a vacant job by a retirement reduces the value of other vacant jobs. In this regard, our results might be interpreted as either a critique of public retirement policy or as a puzzle to be explained by positive theories of the public sector.

Section II begins with an overview of retirement-inducing policies used around the world. Section III presents a standard model of search, simplified, amended, and reinterpreted to allow for a retirement decision. Section IV considers an extreme parameterization of the model to show that even if it is the case that the social value of retirement exceeds the private value, an optimal retirement program does much less to induce retirement than do observed programs. Section V provides some concluding comments and discusses the possible welfare effects of recent reforms and policy proposals aimed at reducing retirement incentives and thereby promoting labor market participation among older workers.

II. AN OVERVIEW OF RETIREMENT-INDUCING POLICIES AROUND THE WORLD

There is a growing body of literature comparing public pension systems and their retirement incentives across countries and over time. We report some of the main results from that work. The purpose of our article here is not to conduct a detailed statistical analysis but merely to highlight the empirical regularities relevant to a theoretical study of publicly induced retirement. The most conspicuous (and theoretically most relevant) regularity is that implicit earnings tax rates are highest for the elderly.

Public Policies Encourage Retirement

As of 1995, over 100 countries had public pension programs. (1) Among the 88 of those countries reporting to the U.S. Social Security Administration sufficient detail of their public pension benefit formulas, 75% pay pension benefits in such a way as to discourage work by its elderly citizens. The most typical means by which benefit formulas induce retirement is remarkably transparent: Retirement is a necessary condition for receiving public pension benefits, and no credit is given to those who decide to retire later and collect benefits for fewer years. Other countries had more complicated benefit formulas extending some less-than-actuarially fair credits to those who delay retirement, or allowing employed elderly to collect partial benefits, or both (the case, until the year 2000, for U.S. Social Security for elderly aged 65-69). But the more complicated formulas have much the same effect as the simple one: Elderly labor income is implicitly taxed.

At least in higher-income countries, the rates of implicit taxation are enormous. Although an exact calculation of marginal tax rates is complicated due to nonlinearities and other details of benefit formulas, the reason for the high rates is simple: The elderly must retire to obtain full benefits, and full benefits are typically a very large fraction of the earnings enjoyed if one does not retire. Gruber and Wise (1999, table 1, based on even more detailed computations of their coauthors) attempt to quantify the rates of implicit taxation for 11 countries. According to their calculations for the early 1990s, the "typical" implicit tax rate for "someone of retirement age" ranges from roughly 20% for Japan, United States, and Canada, to more than 80% for Belgium and the Netherlands. (2)

Another way to appreciate the quantitative significance of the implicit taxation of elderly labor income by public pension programs is to notice the prevalence of 100% (!) marginal tax rates. Mulligan (1998) discusses in some detail a number of examples, including U.S. Social Security benefit formulas between 1939 and 1971, under which retirees lost all of their Social Security benefits if their earnings exceeded a rather low earnings limit by even $1. Other American examples of 100% marginal tax rates can be found prior to the Social Security Act in U.S. state-administered old age assistance programs, which typically implicitly taxed earnings at a 100% rate (US Congress, Joint Economic Committee 1966, 26-27). Spain is one of several international examples where their elderly are not allowed to collect a government pension if they earn any labor income at all (Boldrin et al. 1999, 322; Social Security Administration 1995, 330), and those benefits are typically close to or more than what the pensioner would have earned after taxes if working (Boldrin et al. 1999).

Perhaps these implicit taxes are not distortionary because they are not enforced or because other government regulations prohibit people from changing their behavior in response to them? There are two reasons to be skeptical of such a claim. First, Gruber and Wise (1999) show that retirement behavior is highly correlated across countries and across age groups with the measured incentives. (3) Second, the stated purpose of the implicit tax provision is often to encourage retirement (Sala-i-Martin 1996; Gruber and Wise 1999, 31).

Pensions are not the only public programs encouraging retirement. "Disability insurance" and "unemployment insurance" programs "essentially provide early retirement benefits before the official Social Security early retirement age" (Gruber and Wise 1999, 9) in many countries. Tax-favored company pensions, mandatory defined benefit company pensions, and public health insurance are some other government policies that may substantially induce retirement.

Marginal (Implicit + Explicit) Tax Rates Are Highest for the Old

Perhaps it is unsurprising that public policies discourage work, because governments need to raise revenue or may want to assist the poor. But another feature of public pension programs, and government policy in general, is that elderly work is discouraged more than young work. Hence, although payroll tax rates are paid by young and old workers and can be large in many countries more than 10% in the United States and nearly 50% in Egypt, Italy, and the Netherlands--public pension benefit formulas in many countries substantially reduce the incentive to work beyond its reduction due to payroll and income taxation.

Income taxes, payroll taxes, and public pension benefits are not the only public policies discouraging work. Minimum wages, unemployment compensation, welfare payments, work week restrictions, and other policies have the effect of discouraging work, and a full analysis of public policy and work incentives would include detailed calculations of the effects of these programs. However, two observations strongly suggest that, taken together, the various public policies tax elderly labor income at much higher marginal rates. First of all, a number of these programs--such as unemployment and welfare--affect work incentives for both elderly and young people. Often unemployment and welfare payments are most generous for the elderly and implicitly tax elderly labor earnings at higher rates. Indeed, the unemployment insurance programs in Belgium, Finland, and other countries are hard to distinguish from public pension programs in terms of their intergenerational incidence and their age profile of marginal tax rates. (4) Second, it seems that because of public pension programs, the prevalence of 100% and near-100% marginal tax rates is much higher among the elderly than among the young (as a consequence of tax and other policies), and as a result work is so much more prevalent among the young.

Pensions Designed This Way Have Existed for Many Decades

For decades, social security benefit formulas have implicitly taxed labor income of the elderly. To prove this, Mulligan (2000) constructs an international data set for the years 1958, 1975, and 1995 based on Social Security Administration reports (various issues). It was somewhat more common in 1958 and 1975 for benefit formulas to induce retirement with the simpler formula making retirement a necessary condition for receiving public pension benefits (e.g., the United States did so in 1958 but not in 1995). Delayed retirement credits and gradual phase-out of benefits with earnings were more common in 1995, so it might be said that retirement was induced more dramatically in 1958 and 1975. However, the size of the benefit forgone by the elderly worker has grown over time relative to what a retiree would have earned, so in this sense benefit formulas induce retirement more in recent years. More research is required to determine exactly how the incentive to retire has changed over the years in various countries, but it is clear that for decades public pension benefits have provided an important incentive to retire.

In April 2000, the earnings test was lifted for workers age 65 and over in the United States. Workers in this age bracket now have a greater incentive to continue working, but the social security program still discourages their work. Namely, even without penalties in the form of reduced benefits, older workers are still liable for the payroll tax, and the payment of this tax (more precisely, the accrual of payroll-taxable earnings) does not enhance their ultimate social security benefit to the degree that it would for younger workers. We also point out that the very recent lack of social security benefit penalties for elderly work in the United States is unusual in the context of the international history of social security.

III. THE BASIC MODEL

Tastes and Technology

With these observations of public policies in mind, we ask whether publicly induced retirement can alleviate congestion problems in the labor market induced by the life cycle. We consider a simple one-period extension of the standard Mortensen (1982) and Pissarides (2000) model. As in the standard search model of the labor market, there are two groups of agents: workers and firms. Both types of agents derive utility from income and are risk-neutral.

Workers are heterogeneous in exactly two dimensions. First, some workers are matched with an employer, and some are unmatched. We refer to workers that begin the period matched with employers as "old" workers and the initially unemployed as "young" workers, and their population shares as [lambda] and (1-[lambda]), respectively. (5) Second, workers also differ in their nonpecuniary costs of working. To be specific, we let [[gamma].sub.i] represent the cost of working for worker i. One may interpret this cost as the opportunity cost of working in terms of lost leisure time. These costs of working may vary across cohorts. The cost of working for old workers is described by the cumulative distribution function, [F.sub.1]([gamma]), whereas the cost for young workers is described by [F.sub.0]([gamma]). One interesting case has old workers having a higher value of leisure time than young workers, so that [F.sub.1]([gamma]) first-order stochastically dominates [F.sub.0]([gamma])or [F.sub.1]([gamma])<[F.sub.0]([gamma]). (6) As in the standard search model of the labor market, firms take the output from the match and sell it on the market at price p. (7) Given that both the firm and the worker have linear preferences over income, the total surplus created on filling a vacancy with a worker i is p - [[gamma].sub.i]. (8)

Technically, this is a departure from the standard Mortensen-Pissarides model (whose workers have homogeneous preferences for leisure) because one group of workers, those who are initially employed, will on average have a higher value of leisure time than those who are not endowed with jobs. The higher value of leisure time among the employed provides an economic motivation for a process like retirement in which older workers are replaced by younger workers. It also permits a straightforward calculation of a "retirement rate" in the interval (0,1). Furthermore, some studies have found that the value of leisure time (interpreted literally) is an important determinant of who and how many retire. (9)

Worker-firm matches are made in one of two ways. First, matches are part of the initial conditions for so-called old workers. Second, a young worker can be matched with a firm, or an unmatched firm with a young worker, by "search." Job search costs s for each worker, and for simplicity, worker search costs nothing for firms. Job searchers and firms with vacancies are brought together according to a matching technology M. The matching technology, M(U,V), denotes the aggregate number of matches as a function of the aggregate number of searchers U and aggregate number of vacancies V. The matching technology is stochastic and undiscriminating--namely, all searchers enjoy the same ex ante probability of a match m = M/U and all employers posting a vacancy enjoy the same ex ante probability of a match M/V. Because the matching technology exhibits constant returns to scale, we let m = m([theta]) = M(1,V/U) where [theta] = V/U is the number of vacancies per worker and therefore may be viewed as the degree of labor market tightness.

Initial worker firm matches can be dissolved, in which case there is no surplus associated with the initial worker. We interpret this situation as retirement, with the retiree consuming leisure and his or her former employer participating in the aforementioned matching process by posting the vacancy. (10) There are two possibilities in which young workers may consume leisure. The first, which might be called unemployment, occurs when a worker searches for a job but does not find one. The second, which might be called out of the labor force, occurs when a worker does not search at all.

Note that there is a direct link between the amount of retirement and the number of job vacancies in our model. We focus on a setting where there is a fixed potential stock of vacancies that is dictated by the number of old workers. As has been widely noted (see, for example, the discussion in Layard et al. 1991), the idea of removing old workers from the labor market to enhance the employment possibilities for the young is based in large part on the partial equilibrium idea of a fixed stock of vacancies. (11) More general equilibrium work needs to be done, but we expect that general equilibrium models would still allow for the important possibility that retirement makes search more difficult for other employers searching for employees.

Efficient Allocations

An efficient allocation is the aggregate surplus-maximizing list of retirees, job searchers, and firms posting vacancies, given the economy's matching technology and the costs of searching. An efficient allocation involves: (a) all unmatched employers posting their vacancies, (b) retirement for the initially matched workers with high nonpecuniary costs of work (relative to the others initially matched), and (c) job search among the initially unmatched workers with low nonpecuniary costs of work (relative to the others initially unmatched).

Let [[gamma]*.sub.1] and [[gamma]*.sub.0] represent the planner's choices for the critical values of the costs of working for the initially matched and unmatched respectively. Let [[PHI]*.sub.1] = [F.sub.1]([[gamma]*.sub.1]) and [PHI]*.sub.0] = [F.sub.0]([[gamma]*.sub.0]) represent the planner's choices for the fractions of the initially matched who do not retire and the unmatched who choose to search, respectively. The social surplus W in the economy is, as functions of the number of old workers [[PHI].sub.1] and young searchers [[PHI].sub.0],

given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The first term is the aggregate surplus resulting from matches initially filled with old workers, calculated by adding p - [gamma] for each old person who works (i.e., who has [gamma] < [F.sup.-1.sub.1] [[[PHI].sub.1]]). The second term is the average surplus of matches with successful young searchers, whose quantity is M and whose average surplus is in square brackets. The final term is the total lost surplus incurred by all young workers who incur the costs of search.

From the description of aggregate surplus, we can derive conditions for the efficient amount of retirement and job search. Efficient retirement is described by

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The left-hand side represents the marginal social surplus of less retirement, and the right-hand side is the social marginal cost of less retirement. Of particular importance for our analysis is the last term, which is the effect of retirement by the marginal worker on the surplus of inframarginal workers because, as we will show, it will be ignored by a person choosing retirement solely to maximize the joint surplus of himself and his employer.

The efficient amount of labor market participation by the young is given by

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The efficient choice of job search recognizes that the marginal searcher has probability m of finding a job and hence probability m([theta]) of enhancing surplus by p minus [[gamma]*.sub.0]. But there are two social costs of search: the search cost s and the marginal searcher's effect on the surplus of others. This second cost is the product of the inframarginal searchers' potential surplus (p - [E.sub.0][gamma]) and the effect [theta]m'([theta]) of additional search on job finding by the inframarginal searchers.

Equilibrium Retirement, with Public Policy

In this section, we seek to determine the equilibrium allocations of workers to jobs and the unemployment rate and model some simple public policies that might affect these decisions. We suppose that the government can observe whether a worker is matched and producing, and levies a tax [T.sub.1] on (or, if [T.sub.1] < 0, pays a subsidy to) each old person, regardless of his or her retirement status, and a tax To on each young person. The government pays [B.sub.1] to (or, if [B.sub.1] < 0, taxes) old nonworkers (aka, retirees) and [B.sub.0] to young nonworkers.

We follow those in the literature and suppose that the net postfisc surplus, (p - [gamma]), derived from work is split between workers and firms with shares [beta] and (1 - [beta]), respectively. For simplicity, we assume that workers' outside options do not affect the bargaining between workers and firms. That is, we assume that workers have zero threat points and that the outside options of retirement or unemployment benefits do not affect the determination of wages. This is equivalent to assuming that workers cannot actively engage in on-the-job search and is adopted so that there are no wage distortions resulting from either the public pension or unemployment programs. As we will describe, the only margin in which public policy affects labor market activity is through the labor market participation decision of each group of workers.

Each initially matched old worker is assumed to decide jointly with his or her employer whether or not to retire. Each young agent, initially unemployed, is assumed to decide on her own whether or not to search, based only on her expected costs and benefits. In other words, the search decision is not made jointly by the young, old, and the young's ultimate employer, because the essence of the search friction is that searchers and employers do not know who are their ultimate match partners. As we shall show, distortionary public policy can in principle help coordinate some of these decisions.

Given matching probabilities, the cost of searching, distributions of working costs, government policy ([B.sub.1], [T.sub.1], [B.sub.0], [T.sub.0]), and a sharing parameter [beta], an equilibrium allocation is a list of retirees, job searchers, and firms posting vacancies, so that (i) the government budget balances; (ii) a job searcher's match probability is M (U,V)/U and an employee searcher's match probability is M(U, V)/V, where U and V are the population measures of job searchers and employee searchers, respectively; (iii) a young person cannot improve his or her expected surplus by changing the decision to search or not; and (iv) an old person and his or her employer cannot improve their joint expected surplus by changing the person's retirement status.

An equilibrium can be characterized algebraically as a pair of scalars [[PHI].sub.1] and [[PHI].sub.0] satisfying:

(i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(iv) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [E.sub.0][gamma] is the average cost of work among those young people searching for a job.

Condition (i) is the government budget constraint, where taxes are collected from all persons and benefits are paid to retirees and young nonworkers. Second, condition (ii) accounts for U and V, the number of job searchers and job vacancies. Third, condition (iii) defines the marginal retiree. The left-hand side represents the total surplus from maintaining the match between the firm and the marginal retiree. In contrast, the right-hand side provides the total surplus to the firm and the worker if the individual retires. The first term, [B.sub.1], is the retirement benefit received by the retiree, whereas the second term is the expected net surplus for the firm from searching for a young worker in the labor market. In equilibrium, the firm and the worker jointly determine the retirement decision so that the worker, together with the original employer, is indifferent between remaining employed and retiring (in which case benefit [B.sub.1] is collected and a vacancy is created).

Notice that [T.sub.0] and [T.sub.1] do not enter (iii) because those particular taxes are collected (subsidies paid) regardless of labor market status, whereas [B.sub.1] is viewed as benefit of retirement from the point of view of a potential retiree and his or her employer because it is an outside source of income that unlike [T.sub.1] is paid only conditional on retirement. Potential retiree and employer are maximizing joint surplus, so [beta] enters (iii) only to the extent that it determines the employer's surplus after retirement. Finally, condition (iv) defines the marginal job searcher, who is indifferent between the employment benefit and the expected proceeds from search. The left-hand side represents the surplus received by choosing not to search for a job because the worker will definitely receive unemployment benefits. In contrast, the right-hand side is the total net surplus from choosing to search for a job in the labor market. The first term is the expected surplus obtained from working, and the second term is the surplus obtained if the worker cannot find employment. Finally, regardless of the labor market outcome resulting from participating in the labor market, the worker bears the costs of job search.

This algebraic characterization (i) (iv) of an equilibrium permits easy proofs of our two main propositions.

PROPOSITION 1. For any [beta], [gamma], s [greater than or equal to] 0, any distribution functions [F.sub.0](*) and [F.sub.1](*), and any homogeneous function M(*) defined on [[0,1].sub.2], there exists a government policy ([T.sub.0], [T.sub.1], [B.sub.0], [B.sub.1]) consistent with an efficient equilibrium.

Proof We prove Proposition 1 by constructing a government policy that makes the efficient [[PHI]*.sub.1] and [[PHI]*.sub.0] consistent with equilibrium. First, pick [B.sub.1] = [m'([theta]) - (1 - [beta])m([theta])(U/V)] (p - [E.sub.0][gamma]), calculating U, V, and [E.sub.0][gamma] from an efficient allocation. Plugging this into equilibrium condition (iii), we have that the equilibrium retirement margin is efficient. Second, pick [B.sub.0] - [beta]([[theta]m'([theta])]/m[[theta]])(p - [E.sub.0][gamma] - (1 - [beta])(s/m[[theta]]). Plugging this into equilibrium condition (iv), we have that the equilibrium search margin is efficient. Third, pick any ([T.sub.0], [T.sub.1] satisfying the government budget constraint for the ([B.sub.0], [B.sub.1]) calculated above.

PROPOSITION 2. If the elasticity of m ([theta]) is constant and equal to 1 - [beta], then a government policy consistent with an efficient equilibrium has [B.sub.1] = O, and [B.sub.0] [greater than or equal to] 0.

Proof Use Proposition 1's formula for efficient [B.sub.1] to show that it is zero when the elasticity of m([theta]) is constant and equal to 1 - [beta]. Use Proposition 1's formula for efficient [B.sub.0], and the equilibrium condition (iv), to show it is > ( = ) zero as the work cost of the marginal searcher is > ( = ) the work cost of the average searcher.

Proposition 1 says that efficient allocations are consistent with equilibrium with the right government policy. Proposition 2 considers a special case explored by Hosios (1990), where the sharing parameter [beta] is related to the elasticity of the matching function. Hosios (1990) finds that if the elasticity of the matching technology is the same as the worker's degree of bargaining power, the supply of job vacancies is efficient without any taxes or government transfers. In contrast to his results, we find that only the retirement decision will be efficient if the elasticity of the matching technology is the same as the firm's degree of bargaining power. However, due to the differing degrees of leisure costs and access to jobs, the efficient labor market participation decision among the young still requires an unemployment subsidy. In summary, Proposition 2 provides conditions where governments do not need a retirement policy to generate labor market efficiency, but an optimal unemployment policy is still required. In Hosios (1990), all workers have the same access to the labor market and disutility of work so that labor market efficiency is guaranteed under a particular matching function.

Propositions 1 and 2 are important for understanding the quantitative relationships between search and retirement, so we explore them further. Notice that from the social point of view, there are three distortions of the equilibrium retirement decision. The first is the retirement subsidy (or tax, if [B.sub.1] < 0) seen on the right-hand side of (iii). Second, the potential retiree and his or her employer put some value on creating a vacancy according to the average match rate M/V. When M is homogeneous, and the number of matches are at least somewhat elastic to the number unemployed, the average match rate exceeds the marginal match rate [delta]M/[delta]V that is relevant from the social point of view. This second distortion tends to cause employers (in agreement with their potential retirees) to excessively encourage retirement. Third, equilibrium retirement decisions do not consider the creation of surplus for the unemployed group, which is the product of the marginal match rate and the average surplus [beta](P - [E.sub.0][gamma]) for those matched. By itself, this third distortion means that an equilibrium has too little retirement. The second distortion can overwhelm the third, as is the case when [beta] is small and/or the gap between marginal and average match rates is large, or vice versa. The optimal retirement subsidy is positive when the third distortion dominates, and negative when the second dominates.

To isolate and quantitatively evaluate the third distortion, consider the limiting case of a linear matching function M(U,V) = V. Search is very efficient in this limiting case, because all vacancies costlessly find a match with probability one, although inefficient in an important sense which we will demonstrate. (12) Now equilibrium condition (iii) becomes (iii)':

(iii)' p - [F.sup.-1.sub.1]([[PHI].sub,1]) = [B.sub.1] + (1 - [beta])(p - [E.sub.o][gamma]).

Condition (iii)' allows us to compute the efficient retirement subsidy (namely, that for which equilibrium and planned retirement coincide), in this case [B.sub.1] = [beta] (p - [E.sub.0][gamma]). In words, the efficient retirement subsidy is that surplus received by the employee who takes over the job of the retiree because that future employee is not at the table when employer and potential retiree make the retirement decision. Or, using the optimal retirement condition for the planner facing a linear matching function, we see that the optimal retirement subsidy equates the disutility of work of the marginal retiree ([F.sup.-1.sub.1] [[[PHI]*.sub.1]]) with the average disutility of work among the unemployed ([E.sub.0][gamma]).

Furthermore, in this special case, it is quite clear that the optimal labor market policy should be age-targeted. That is, the optimal retirement policy definitely affects labor market participation among both the young and the old so that the unemployment and pension policies are interdependent. To see this, suppose that [E.sub.0][gamma] = 0. On substituting the optimal pension subsidy [B.sub.1] into the retirement condition (iii), we find that all workers who incur any loss of utility from working will choose to retire. That is, in this case, [[gamma]*.sub.1] = 0. If old workers experience any additional desire for leisure relative to their youth, the optimal retirement policy encourages all of the old workers to retire.

We can also examine how the optimal pension and unemployment policies affect labor market participation among young workers. In the case of the linear matching technology, [B.sub.0] = [beta]p - (1 - [beta])s(U/V). On substitution into the labor market participation condition for the young, we find that [[gamma]*.sub.0] = -(s/m[[theta]]). Because all workers must incur the costs of search, each worker imposes a congestion externality on all other young individuals who choose to look for jobs. Therefore, in an effort to conserve on search costs in the labor market, the optimal unemployment policy would encourage some of the young (even though they derive utility from working) to withdraw from the labor market. Thus, despite our use of a static model for tractability, our results demonstrate that the optimal labor market policy requires that public pension programs and unemployment policies are dynamically connected. (13)

The special linear matching function permits an intuitive calculation of the magnitude of the optimal retirement subsidy, although we point out that the optimal subsidy is especially big in the linear case because there is no gap between average and marginal match rates (see our previous discussion of the second distortion). A precise calculation requires precise estimates of the sharing parameter [beta] and the average disutility of work among the unemployed (LOT), but notice how [B.sub.1] = [beta](p - [E.sub.0][gamma]) implies that an average young worker with a job who suffered a reduction in his or her after-tax earnings in the amount [B.sub.1] would still have a nonnegative surplus from working. (14) We believe that in reality, 50-80% reductions in earnings would eliminate the surplus and then some--from working for a great many young workers (e.g., practically all women workers, if it is indeed the case that their labor supply is wage elastic), so the formula [B.sub.1] = [beta] (p - [E.sub.0][gamma]) implies that optimal elderly earnings tax rates would be no greater than 25-50% in the linear case. Given that the optimal elderly earnings tax rates are even smaller in the nonlinear cases, the retirement subsidies of 50% and higher seen in several European countries are excessive.

IV. LIMITS ON THE OPTIMAL RETIREMENT INCENTIVE

As long as the matching function is nonlinear, the formula [B.sub.1] = [beta](p - [F.sub.0][gamma]) overstates the optimal retirement subsidy. This effect occurs because it ignores the possibility that the creation of a job vacancy by a retirement lowers the probability that other job vacancies get filled. The optimal retirement subsidy for nonlinear matching functions is shown in the proof of Proposition 1, which we rewrite below:

(4) [ILLUSTRATION OMITTED]

The first two terms show exactly how the optimal retirement subsidy is less in the nonlinear case. First, subtracted from [beta] in parentheses is the elasticity of M with respect to V, which is proportional to the gap between the average and marginal match rates (M/V and [delta]M/[delta]V). This first reduction can be substantial, because [alpha] has been estimated in the range of 0.4 to 0.7. (15) In other words, the worker's share [beta] has to exceed 0.4 if the optimal retirement subsidy were even to be positive! With [beta] = 0.5, and the lower estimate of [alpha] = 0.4, the ([beta] - [alpha]) alone is 80% smaller than [beta] ([0.5-0.4]/ 0.5 = 0.2). In other words, even if M/V were arbitrarily close to one, and [E.sub.0][gamma] were zero (so that [beta][p - [E.sub.0][gamma]] were as large as employee earnings), the optimal implicit tax on elderly earnings is only 20%. Most of the countries studied by Gruber and Wise have implicit taxes in excess of this amount.

The second term is M/V, which of course is less than one with the nonlinear matching function and reflects the fact that some retirements will create vacancies that go unmatched. To quantify this term, we need to further parameterize our model and carefully distinguish stocks from flows. In particular, "additional" matches in our single-period model are more realistically interpreted as matches that occur more rapidly in the presence of an additional vacancy than they would otherwise. In other words, all vacancies find a worker with very high probability if they wait long enough. M/V< 1 is a way to capture, in our static model, the fact that a new vacancy can expect to go unused for some period of time. If we let r denote the interest rate, t time, and [delta] the instantaneous hazard at which a vacancy finds a match, a continuous-time expression for M/V is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For a vacancy with productive capacity one, the numerator is the expected present value of production, accounting for the fact that a match may take some time to occur. The denominator is the present value of production if the match were instantaneous. Because the expected duration of a vacancy (1/[delta]) is measured in months, [delta] is much larger than r and, for quantitative purposes, we can treat M/V as one. (16)

Table 1 presents calculations of the optimal retirement subsidy using formula (4) and various values for the parameters [alpha], [beta], and [E.sub.0][gamma]. To facilitate comparisons with quantitative studies of taxes and labor supply, we express the optimal subsidy and work disutility as fractions of worker earnings. Worker earnings are computed in our model by subtracting the employer surplus from worker productivity (i.e., earnings are p - [1 - [beta]][p - [E.sub.0][gamma]]).

The upper left cell reports the calculation discussed: The optimal implicit tax rate is 20% when [alpha] = 0.4, [beta] = 0.5, and [E.sub.0][gamma] = 0. The top half of the table shows how the optimal rate falls with [alpha] and [E.sub.0][gamma]. The bottom half fixes [alpha] = 0.5 and varies the employee's share [beta]. Optimal tax rates are small unless we use [beta] well in excess of 0.5. Even [beta] = 1 cannot justify elderly implicit tax rates in excess of 50%, as observed in a number of European countries.

V. CONCLUSIONS

Many countries around the world use public policy, especially Social Security programs, to induce their elder citizens to retire. An important motive behind such policy is to create job vacancies that can be occupied by the young. In this article we seek to evaluate the strength of the foundations underlying this rhetoric. To that end, we produce a relatively standard search model with young and old workers and firms. That model allows for the possibility of significant frictions in the process of matching workers with firms. We ask whether policy-induced retirement can be part of an efficient labor market search and matching equilibrium. In other words, is there really any reason for governments to intervene in elderly labor markets on behalf of the young and the unemployed?

Even when the model exhibits significant frictions, and inefficient labor market allocations without government intervention, it may be the case that there is too much retirement, in which case the optimal policy discourages retirement rather than encouraging it. Other cases imply that an efficient equilibrium can be supported with a positive subsidy to the old, but that its size is much smaller than what real-world governments routinely provide. This is primarily because the planner takes into account the negative effect on aggregate matching possibilities of an additional vacancy, an effect the empirical literature suggests to be strong. The Social Security rhetoric, on the other hand, ignores this effect, overemphasizing the beneficial effects of an additional vacancy on those searching for jobs. In addition, we find that when the Hosios (1990) condition is applied to our model, the amount of retirement is optimal without any government intervention, but the amount of labor market participation by the young is inefficient. In short, many societies excessively induce retirement by the elderly, at least from the standpoint of efficiency as understood in standard search models. Thus, our results imply that recent public pension and other policy reforms aimed at reducing retirement incentives (such as the elimination of the earnings test) are likely to improve labor market welfare.

There are a number of possible extensions of our framework to consider. The present analysis studies the impact of public pension programs on labor market activity assuming that the value of each match with an old worker is known and the worker and the firm jointly determine the retirement decision. Another important aspect of public pension reform involves the impact of Social Security on labor market contracts and aggregate labor market activity. Notably, old workers may experience health shocks that are privately known and adversely affect the firm's surplus more than for a given worker. In this regard, publicly induced retirement policies may be important because they help avoid adverse selection problems from retaining older workers. However, such an explanation for Social Security must also involve a rationale for why such contracts must be publicly rather than privately provided. This remains a challenging but important area for future research on the labor market implications of public pension programs. (17)

TABLE A-1
Summary Statistics

 Mean SD

Reliance on EIG 0.014 0.010
taxation
Reliance on general 0.311 0.135
sales taxation
Reliance on personal 0.248 0.156
income taxation
Percent of population 0.011 0.004
85 years or older
Percent of population 0.116 0.021
65 years or older
State per capita income 13231 7673
State per capita federal 408.6 284
transfers
State per capita debt 974 1167
State unemployment rate 1.8 18
Same party-Democrat 0.322 0.467
(Yes = 1)
Same party-Republican 0.152 0.359
(Yes = 1)
Election year dummy 0.277 0.448
(Yes = 1)

Note: N = 1536, annual data (1968-99) for all 48
contiguous states.

TABLE 1
The Optimal Tax Rate on Elderly Earnings
(B1/[p - (1 - [beta])(p-[E.sub.0][gamma])], Assuming (M/V
[approximately equal to] 1)

 Young Work Disutility as a
 Fraction of Earnings,
 [E.sub.0][gamma]/p -
 (1 - [beta])(p - [E.sub.0]
 [gamma])]

Alpha Beta 0 0.2 0.4 0.6

0.4 0.5 0.20 0.16 0.12 0.08
0.5 0.5 0 0 0 0
0.6 0.5 -0.20 -0.16 -0.12 -0.08
0.7 0.5 -0.40 -0.32 -0.24 -0.16
0.5 0.6 0.17 0.13 0.10 0.07
0.5 0.7 0.29 0.23 0.17 0.11
0.5 0.8 0.38 0.30 0.22 0.15
0.5 0.9 0.44 0.36 0.27 0.18
0.5 1.0 0.50 0.40 0.30 0.20

(1.) Data in this paragraph are reported and described in more detail by Mulligan and Sala-i-Martin (1999, 2004) and Sala-i-Martin (1996).

(2.) Gruber and Wise (1999) point out that in any one country, marginal implicit rates vary with earnings, age, calendar year, and other variables. For a person of age t in the early 1990s, where t is between the early retirement age (age 60 in 9 of the 11 countries they study) and 69, they compute for a worker of median earnings the present value of public pension benefits forgone by delaying retirement one year and express it as a fraction of earnings (after income and payroll taxes) for that year, a fraction [[tau].sub.t] that can be interpreted as an implicit tax rate. They sum [[tau].sub.t], between the early retirement age and t = 69, then divide their sum by the number of years in the sum (10 years are in the sum for 9 of the 11 countries they study) to arrive at the "typical" implicit tax rate for "someone of retirement age" reported in the text.

(3.) For a discussion on the implications of the earnings test for the labor supply behavior of older workers, see Baker and Benjamin (1999), Burtless and Moffit (1984), Disney and Tanner (2000), Friedberg (2000), and Gruber and Orszag (2003).

(4.) For international examples, see Social Security Administration (1995) and Gruber and Wise (1999). Leimer (1998, 16-17) reports results for the American disability insurance program.

(5.) In Shimer (2001), all workers are infinitely lived, but in each period a new generation of workers are born. "Young" workers are those who were born in the recent past. Because young workers have had less time to participate in the labor market, they are more likely to be unemployed than "older" workers.

(6.) See Hadar and Russell (1969) and Tesfatsion (1976) for definitions and discussion.

(7.) Because the price of output is exogenous, the model is a partial equilibrium setting in which workers and consumers are viewed as separate groups of agents.

(8.) An alternative interpretation is that [[gamma].sub.i] represents the adverse health consequences of aging, thereby lowering a worker's level of productivity.

(9.) See Costa (1998), Robinson et al. (1982), Parnes and Nestel (1981), and Schulz (2001) for discussion.

(10.) An alternative (but more complicated) model would have employer and potential retiree jointly making a contingency plan for search and retirement. Namely, the firm would (at some cost) begin searching for a young worker, and retirement could be conditional on the firm's success in the search. This would be a form of "on-the-job search" on the part of employers.

(11.) Although retirement programs in many countries have been introduced to alleviate problems with unemployment, Layard et al. (1991) argue that instead these policies have in fact contributed to more unemployment.

(12.) For the sake of simplicity, we do not consider the case V > U, in which case the more reasonable linear matching function would be M(U, V) = U.

(13.) Our modeling choice of a static model implies that we are do not capture some dynamic aspects of labor market behavior, such as the accumulation of firm-specific human capital. The use of a static model provides additional tractability to allow for closed-form solutions of the optimal retirement and unemployment subsidies. Bhattacharya and Reed (2001, 2002) study some of these issues in dynamic, general equilibrium frameworks, but they do not solve for the optimal labor market policies analytically.

(14.) Remember that a worker's surplus is less than his or her earnings; earnings are computed in our model by subtracting the employer surplus from worker productivity (i.e., p - [1 - [beta]][p - [E.sub.0][gamma]]).

(15.) Blanchard and Diamond (1989) find a to be 0.4 for the U.S. and Layard et al. (1991) find it to be 0.7 for the United Kingdom.

(16.) Van Ours and Ridder's (1992) calibration implies an average vacancy duration of about 45 days, whereas Blanchard and Diamond's (1989) implies 1 month.

(17.) We thank an anonymous referee for this suggestion.

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JOYDEEP BHATTACHARYA, CASEY B. MULLIGAN, and ROBERT R. REED III *

* We thank participants of the Macro Lunch Group at Chicago, an anonymous referee for helpful comments, and Kevin Tsui for research assistance.

Bhattacharya: Associate Professor of Economics, Department of Economics, Iowa State University, Ames, IA 50011. Phone 1-515-294-5886, Fax 1-515-294-0221, E-mail [email protected]

Mulligan: Professor of Economics and Research Associate, University of Chicago and National Bureau of Economic Research, Department of Economics, University of Chicago, 1126 East 59th Street, Chicago, IL 60637. Phone 1-773-702-9017, Fax 1-773-702-8490, E-mail [email protected]

Reed: Assistant Professor of Economics, Department of Economics, Gatton College of Business and Economics, University of Kentucky, Lexington, KY 40506. Phone 1-859-257-5975, Fax 1-859-323-1920, E-mail [email protected]