Optimal Unemployment Bene ts Policy and the Firm Productivity Distribution Tomer Blumkin and Leif Danziger, y Ben-Gurion University Eran Yashiv, z Tel Aviv University January 10, 2014 Abstract This paper provides a novel justi cation for a declining time pro- le of unemployment bene ts, without resorting to moral hazard and consumption-smoothing considerations. We consider a simple economy with homogeneous workers and low- and high-productivity rms. In a search environment with random matching, there is a tradeo between employment and average worker productivity. By introducing unemployment bene ts, the government is able to a ect the equilibrium wage pro le in a manner that enhances the sorting of workers across low- and high-productivity rms. A decreasing time pro le of bene ts may be optimal in this context, whereby employment and average worker productivity fall between what they would be with no unemployment bene ts and with high, constant unemployment bene- ts. We demonstrate that optimal government policy depends on the dispersion and skewness of the rms productivity distribution. JEL Classi cations: J64, J65 Keywords: Unemployment bene ts policy, declining unemployment bene ts, productivity distribution, skewness, dispersion. tomerblu@bgu.ac.il. y danziger@bgu.ac.il. z Corresponding author. yashiv@post.tau.ac.il; 972-3-640-9233.
Optimal Unemployment Bene ts Policy and the Firm Productivity Distribution 1 1 Introduction A prevalent feature of unemployment bene ts policy in most OECD countries is a declining bene ts schedule. Typically, high unemployment bene ts are o ered for a limited period and are subsequently replaced by lower bene ts, often labeled social or income assistance. Thus, in OECD countries (except for Belgium), the duration of the initially high unemployment bene- ts ranges between 6 and 48 months, averaging 16 months (OECD, 2007). 2 The existing literature on the optimal design of unemployment bene ts policy has focused on the tradeo between the consumption-smoothing role of bene ts and the associated moral hazard due to reduced incentives for job search. A key result is the desirability of setting a declining time-pro le of unemployment bene ts as a means to mitigate the moral hazard e ect. For a prominent example, see Hopenhayn and Nicolini (1997). This paper revisits the case for a declining time pro le of unemployment bene ts and provides a novel justi cation. Our approach follows Mortensen (1977), Diamond (1981), Albrecht and Axell (1984) and Marimon and Zilibotti (1999) in viewing unemployment bene ts as a search subsidy, and we study the role policy may play in attaining a better match between jobs and workers. The crucial insight is that in a search environment where the assignment of unemployed workers to heterogenous rms is uncoordinated, there is a tradeo between employment and average worker productivity. As this tradeo is not adequately re ected in the equilibrium wage pro le, unemployment bene ts may help internalize matching externalities. We consider a simple economy with homogeneous workers and low- and high-productivity rms. In the benchmark case with no unemployment bene ts, both types of rms will be active and will o er the same wage rate, which will be accepted by any unemployed worker. In this case, no sorting of 1 We thank Yoram Weiss and numerous seminar participants for helpful comments. Any errors are our own. 2 For updated statistics see http://www.oecd.org/els/bene tsandwagesstatistics.htm. 1
workers across rms takes place. However, since random matching implies an identical matching probability of workers to low- and high-productivity rms, average worker productivity will be relatively low. By introducing unemployment bene ts, the government may be able to a ect the equilibrium wage pro le in a manner that enhances the sorting of workers across low- and high-productivity rms. The main result of this paper is that a decreasing time pro le of bene ts may be optimal in this context. This would be the case when there are moderate di erences between rms productivity levels. The logic is as follows: low-productivity rms o er lower wages than high-productivity rms. 3 Workers at the beginning of their unemployment spell, when bene- ts are high, turn down o ers to work in low-productivity rms. This leads to voluntary unemployment and to more workers who search, shifting employment to high-productivity rms. Partial sorting of workers across lowand high-productivity rms ensues. In this case, employment and average worker productivity fall between what they would be with no unemployment bene ts and with high, constant unemployment bene ts. We demonstrate that optimal government policy hinges on the properties of the rm productivity distribution. Sorting gains are closely associated with the technological dispersion given by the di erence in productivity between low- and high- productivity rms. As may be expected, the higher the di erence, the bigger is the gain from the enhanced sorting of workers across rms. More surprisingly, the asymmetric nature of the technological dispersion, captured by the proportion of high-productivity rms, also plays a key role. When the productivity distribution is su ciently right-skewed, i.e., the proportion of high-productivity rms is small enough, 4 the cost in 3 Mortensen (2003) provides evidence whereby wage dispersion for observationally equivalent workers can be explained by more productive rms paying higher wages. For detailed empirical work, see Davis, Haltiwanger and Schuh (1996), Abowd, Kramarz and Margolis (1999), Haltiwanger, Lane, and Spletzer (1999, 2007) and Bartelsman, Haltiwanger and Scarpetta (2013). 4 For empirical evidence on the skewness of the distribution of rm productivity, see Feng and Horrace (2012). The much discussed skewness of the distribution of rm size (see Luttmer (2007)) is consistent with our model, as declining unemployment bene ts imply that the expected number of workers per rm increases with rm productivity. 2
terms of reduced employment, implied by crowding out all low-productivity rms from the market, is high. Hence, partial sorting dominates full sorting of workers, warranting the implementation of a declining time pro le of bene ts. In contrast, when the productivity distribution is less right-skewed, i.e., the proportion of high-productivity rms is big enough, full sorting is optimal, which implies a constant ( at) bene t schedule. The paper is structured as follows: Section 2 outlines the set-up. Section 3 presents the benchmark of no unemployment bene ts while Section 4 shows two cases of government intervention a at unemployment bene ts schedule and a declining one. Section 5 outlines the government objective and derives the optimal policy. Section 6 concludes. 2 The Set-Up There is a continuum of homogeneous risk-neutral workers with a measure L > 0 and a continuum of rms with a measure M > 0. Firms di er in the technology they possess. A proportion p 2 (0; 1) of rms have high productivity and worker s output per period is x, while a proportion 1 p have low productivity and worker s output per period is x 2 (0; x). The mean of the rms productivities is px + (1 p)x, and the di erence in their productivities is x x. Firms maximize their discounted expected pro ts. Each rm can employ one worker and costlessly post a vacancy at a speci ed wage rate, and each unemployed worker can costlessly apply to one vacancy per period. There is no on-the-job search. The assignment of unemployed workers to vacancies occurs at the end of a period. It is random and governed by a constant-returns-to-scale matching function m(u; V ), where m denotes the measure of matches, U the measure of unemployed workers, and V the measure of vacancies. 5 Workers maximize their discounted expected income. A worker who is matched to a vacancy decides whether to accept the rm s o er of a wage rate which s/he will receive for the duration of employment in the rm. An unmatched worker or a matched worker who 5 For empirical evidence on random matching see Petrongolo and Pissarides (2001). The random matching assumption seems an appropriate way of capturing labor market frictions. These underlie the potential gain from government intervention discussed below. 3
rejects a rm s o er remains unemployed in the next period. The imputed value of leisure is normalized to zero, with no loss of generality. A successful match terminates with an exogenous probability s > 0. 3 The Benchmark Regime: No Unemployment Bene ts We start by characterizing the equilibrium in a benchmark regime without government provision of unemployment bene ts. As there is no on-the-job search, all rms will post the same wage o er coinciding with the workers common reservation wage given by the imputed value of leisure (as in Diamond (1971)). Let V N and U N denote, respectively, the measures of vacancies and unemployed workers, and 2 (0; 1) the workers discount factor. The continuation value for an unemployed worker is H = [n max(j; H) + (1 n)h]; (1) where J = w + [(1 s)j + sh] (2) is the continuation value for an employed worker, and n = m(u N ; V N ) U N (3) is the probability of being matched with a rm. We henceforth simplify by looking at the limit case where the workers discount factor converges to unity; namely, workers maximize their expected income ow. We also let the rms discount factor converge to unity; i.e., rms maximize their expected pro t ow. As employed workers do not search, a rm s posted wage o er must coincide with the workers reservation wage. In other words, unemployed workers must be indi erent between accepting and rejecting the wage o er, that is, J = H. This implies that the equilibrium wage rate will be equal to zero and therefore that all rms will be active. Unemployed workers will be randomly assigned to vacancies across 4
the two kinds of rms, and any job o er will be accepted. Consequently, the assignment of workers across jobs will be random. In an equilibrium, the ow of successful matches between unemployed workers and rms equals the ow into unemployment of workers due to job separations, i.e., m(u N ; V N ) = s M V N : (4) In addition, the measure of lled vacancies is equal to the measure of employed workers, i.e., M V N = L U N : (5) 4 Government Bene ts Policy We now allow the government to provide unemployment bene ts. We assume that government expenditures on unemployment bene ts are nanced by lump-sum taxation and hence do not a ect the choices of workers and rms. 4.1 A Flat Regime Suppose unemployment bene ts, denoted by, are constant over time. Then, there are two possibilities to consider. If < x, all rms will be active, and the assignment of workers and aggregate output in equilibrium will be as in the benchmark case without government intervention. Furthermore, the equilibrium wage rate will coincide with the workers reservation wage, which is now. In contrast, if x < < x, only high-productivity rms will be active, unemployed workers will be randomly assigned to vacancies posted by these rms, and all wage o ers will be accepted. Hence, the equilibrium measures of unemployment, U F, and of vacancies, V F, will be determined by the ow condition m(u F ; V F ) = s pm V F (6) together with the condition that the measure of jobs lled is equal to the measure of employed workers, i.e., pm V F = L U F : (7) 5
Thus, the equilibrium wage rate will again be equal to the workers reservation wage which is. 4.2 A Two-Tiered Regime Now suppose newly unemployed workers receive two periods of high unemployment bene ts followed by an inde nite period of lower bene ts. In such a regime, short-term unemployed workers (one or two periods of unemployment) get unemployment bene t z, whereas long-term unemployed workers get bene t a, where a < z. We will derive an equilibrium in which the low-productivity rms o er a low wage rate, w, and the high-productivity rms o er a high wage rate, w, where w < w. We rst consider unemployed workers. These can be divided into those who are (i) in their rst period of an unemployment spell; (ii) in their second period of an unemployment spell; and (iii) unemployed for more than two periods during their current unemployment spell. Let V and V denote the measures of vacancies posted by rms o ering the low and high wage rates, respectively, V P V + V the measure of all vacancies, and U P the measure of unemployed workers. The continuation values associated with workers in each of the three possible states of unemployment are where H 1 = z + [n max(j; H 2 ) + n max(j; H 2 ) + (1 n n)h 2 ] H 2 = z + [n max(j; H 3 ) + n max(j; H 3 ) + (1 n n)h 3 ] H 3 = a + [n max(j; H 3 ) + n max(j; H 3 ) + (1 n n)h 3 ]; (8) J = w + [(1 s)j + sh 1 ] J = w + [(1 s)j + sh 1 ] (9) denote the continuation values associated with holding low and high wage jobs, and n = m(u P ; V P ) U P n = m(u P ; V P ) U P 6 V V P V V P ; (10)
denote the probabilities of being matched with rms o ering low and high wage rates. We again look at the limit case where the workers discount factor converges to unity. We next consider rms with a job vacancy. As employed workers do not search, a rm s posted wage o er must coincide with one of the reservation wage rates. That is, workers unemployed for more than one period must be indi erent between accepting and rejecting the low wage o er, i.e., J = H 3, whereas workers unemployed in the rst period must be indi erent between accepting and rejecting the high wage o er, i.e., J = H 2. Solving for the equilibrium wage rates and again letting converge to unity, we get w = (z a)[ s + n s(1 n)] + a (11) w = (z a)[ 1 + n s(1 n)] + z: (12) We can now describe the equilibrium. Given the behavior of unemployed workers and of rms with a job vacancy, the equilibrium must satisfy where U 1 s(l m(u P ; V P ) V V P = s pm V (13) m(u P ; V P ) V V P (U P U 1 ) U P = s[(1 p)m V ]; (14) U P ) denotes the measure of unemployed workers in their rst period of unemployment. Condition (13) states that the ow of successful matches between unemployed workers and high-productivity rms (the left-hand side) equals the ow into unemployment of workers due to separations from high-productivity rms (the right-hand side). Similarly, Condition (14) states that the ow of successful matches between unemployed workers and low-productivity rms equals the ow into unemployment of workers due to separations from low-productivity rms. Notice that the term (U P U 1 )=U P on the left-hand-side of Condition (14) is less than unity and captures the fact that only a proportion of the matches with low-productivity rms are successful as rst-period unemployment-bene t recipients turn down o ers from those rms. 7
In addition, in equilibrium the measure of lled vacancies is equal to the measure of employed workers, M V P = L U P : (15) Furthermore, low-productivity rms nd it optimal to post a low wage o er, i.e., the expected pro t ow associated with paying a matched worker the low wage rate weakly exceeds the pro t ow associated with paying a matched worker the high wage rate, (U P U 1 ) U P (x w) x w: (16) Similarly high-productivity rms nd it optimal to post a high wage o er, x w (U P U 1 ) U P (x w) : (17) By properly choosing the policy parameters a and z, we can ensure that there exists a two-tiered equilibrium. Formally, let w = x, and w = x+, where 2 (0; U 1 =(2U P U 1 )). Inequality (16) is satis ed, and since U 1 2U P U 1, (U P U 1 ) U P ( + ) ; inequality (17) is also satis ed. By substituting w = x (18) and w = x+ into equations (11) and (12), they can then be solved for the policy parameters a and z. As a consequence, there exists a two-wage equilibrium with two-tiered unemployment bene ts. In such equilibrium, conditions (13), (14), and (15) determine the measures of unemployment, U P, and of vacancies posted by rms o ering high and low wage rates, V and V. 5 The Government Objective and Optimal Policy We assume that the government seeks to maximize aggregate output. Due to matching frictions, however, the allocation of workers obtained under the benchmark setting does not generally achieve this aim. 8 The equilibrium
wage o ered by both types of rms will be the same and therefore will not re ect rms productivities. Firms are unable to signal their productivities via wage posting with the result that sorting externalities emerge. There is no sorting of workers across low- and high-productivity rms with the random matching of workers, although some sorting may be desirable when productivities are su ciently dispersed. As we show below, unemployment bene ts may serve to internalize these sorting externalities and be optimal. In the benchmark case both types of rms are active and all unemployment is involuntary. There is no sorting of workers across the two types of rms, so aggregate output is given by W N = (L U N )[px + (1 p)x] = (L U N ): (19) In light of the characterization of possible equilibria delineated above, there are two alternative con gurations of unemployment bene ts that need to be considered. One possibility is a at regime whereby the bene t level is set high enough so that only the high-productivity rms are active, all unemployment is involuntary and any wage o er is accepted. 6 Workers are fully sorted across low- and high-productivity rms, and as a consequence aggregate output is given by W F = (L U F )x (20) = (L U F )[ + (1 p)]: A second possibility is a two-tiered regime of unemployment bene ts that supports a two-wage equilibrium in which both types of rms are active and voluntary unemployment emerges with low wage o ers rejected by the short-term unemployed. Workers are partially sorted over low- and highproductivity rms, and the associated aggregate output is given by W P = (L U P )[qx + (1 q)x] (21) = (L U P )[ + (q p)]; 6 Given that the government objective is to maximize aggregate output, and hence sets aside redistributive concerns, a at regime whereby the bene t level is set so low such that all rms are active, coincides with the benchmark regime without unemployment bene ts. 9
where Conditions (13) and (14) imply that p < q < 1; q lies between p and unity, which are the fractions of workers assigned to high productivity rms under no sorting and full sorting. Thus, the equilibrium associated with a two-tiered regime (declining bene ts) features partial sorting of workers across rms. In attempting to maximize aggregate output, there is a tradeo between employment and average worker productivity. There are three possible results, depending on the value of the di erence in the rms productivities: Under no sorting of workers, all rms are active and there is no voluntary unemployment; therefore, employment is the highest possible. However, the quality of matches is relatively poor due to the random nature of the matching process, which implies the same matching probability in low- and high-productivity rms. For a given mean of the rms productivities, when the di erence in the rms productivities is small, much is to be gained from increased employment and little to be lost from a reduction in average worker productivity due to there being no sorting of workers. In this case, W N exceeds both W F and W P so that no sorting maximizes aggregate output. Thus, no intervention is called for. In contrast, under full sorting of workers, low-productivity rms are inactive. Therefore, employment is low while average worker productivity is the highest possible, as all matches involve high-productivity rms. When the di erence in the rms productivities is large, the sorting consideration prevails. In this case, W F exceeds both W N and W P so that full sorting of workers maximizes aggregate output. When the di erence in the rms productivities is in an intermediate range, the gain from high employment with no sorting of workers is not large enough to justify not increasing average worker productivity by enhanced sorting, and the gain from full sorting of workers is not large enough to justify crowding out the low-productivity rms with its associated reduction in employment. Partial sorting of workers then constitutes a tting compromise between no and full sorting. It implies that W P exceeds W N and W F so that partial sorting maximizes aggregate output. Low-productivity rms remain active, but have a lower probability of lling a vacancy than their high-productivity counterparts. This would result in less employment than 10
under no sorting but more than under full sorting of workers. As an example, consider the following numerical exercise, which demonstrates that there exists a wide range of parameter values for which partial sorting of workers maximizes aggregate output. We set M = 100, L = 70, s = 0:01, and = 10, and let the matching function take the form 7 m(u; V ) = 0:1 U 0:5 V 0:5. Figure 1 depicts the optimal sorting con- guration for various combinations of the proportion p of high-productivity rms and the productivity di erence between high- and low- productivity rms. The feasible combinations of p and lie below the dashed curve = 10=p. 8 Figure 1 It is clear from the gure that there exist combinations of p and for which aggregate output is maximized under partial sorting of workers. As we have shown that a regime with declining unemployment bene ts is needed in order to obtain such partial-sorting equilibrium, the implication is that an unemployment bene ts policy with a decreasing time pro le is optimal in this case. Thus, we obtain the result that declining bene ts may be optimal without having to invoke the standard argument in the literature that declining bene ts serve as a means to mitigate the tradeo between consumption smoothing and moral hazard. Note the following key implications: (i) Dispersion. The gure illustrates that the optimal sorting of workers depends on the dispersion of productivities as measured by. Given the proportion p of high-productivity rms, the optimal degree of sorting increases in. That is, if is low enough, no sorting is called for, and hence, no 7 The constant-returns-to-scale Cobb-Douglas matching function has wide empirical support (see Petrongolo and Pissarides (2001)). Yashiv (2000) and Borowczyk-Martins at al. (2013) provide detailed estimates and discussion. 8 The numerical example is restricted to the range where p < 1. This parametric 2 assumption is consistent with the empirical evidence that the wage distribution is right skewed so that the median wage is lower than the average wage (see the survey in Neal and Rosen (2000)). Given the commonly observed two-tiered declining time pro le of unemployment bene ts, the proportion of employed workers who have high-wage jobs is given by q. Consequently, with a two-wage distribution, it follows that q < 1, and as 2 p < q, that p < 1. 2 11
unemployment bene ts. Increasing rst shifts the economy into a region where partial sorting is desirable, hence declining bene ts. As becomes high enough, full sorting, and hence a at regime, becomes optimal. (ii) Skewness. The gure highlights the key role played by the asymmetry or skewness of the productivity distribution as captured by p. Notably, given a for which at least some sorting is called for, there exists a threshold level of p below which partial sorting is desirable and above which full sorting is the optimal choice. The reason is that for a su ciently right-skewed productivity distribution (that is, for a su ciently small p), shifting from partial to full sorting by crowding out all low-productivity rms is too costly in terms of the associated reduction in employment. In contrast, with a less right-skewed productivity distribution, the sorting-employment tradeo tilts in the other direction and calls for implementation of the full-sorting regime. (iii) Skewness-dispersion tradeo. The curve separating the regions where partial and full sorting constitute optimal policy consists of the combinations of p and for which the government is indi erent between implementing the partial and the full sorting regimes. The curve is upward sloping, re ecting the fundamental tradeo between dispersion and skewness of the productivity distribution: the larger is the dispersion of productivities, the stronger is the case for enhancing the sorting by shifting from a partial to a full sorting regime; in contrast, a more right-skewed productivity distribution works in the direction of shifting policy from full to partial sorting. 6 Conclusions This paper has shown that a policy of declining unemployment bene ts may be an e cient way of internalizing externalities that are generated by the sorting of workers across rms in a labor market with matching frictions. Accordingly, declining unemployment bene ts may be optimal even in the absence of consumption-smoothing and moral-hazard considerations. For a wide range of realistic parameter values, declining unemployment bene ts will be preferred to at unemployment bene ts as the former maximize aggregate output by striking an optimal balance between employment and 12
average worker productivity. Our analysis highlights the close link between the properties of the productivity distribution and optimal policy choice. We emphasize the key role played by the asymmetric nature of technological dispersion, namely the extent to which the productivity distribution is skewed to the right, on the sorting-employment tradeo faced by the government. In particular, when the productivity distribution is su ciently skewed to the right, choosing full rather than partial sorting is too costly in terms of reduced employment, with the result that a declining unemployment bene ts policy is to be preferred. 13
References Abowd, J.M., Kramarz, F., Margolis, D.N., 1999. High wage workers and high wage rms. Econometrica 67, 251-233. Albrecht, J., Axell, B., 1984. An equilibrium model of search unemployment. Journal of Political Economy 92, 824-840. Bartelsman, E., J. C. Haltiwanger, S.Scarpetta, 2013. Cross-country di erences in productivity: the role of allocation and selection. American Economic Review 103,1, 305-334. Davis, S.J, Haltiwanger, J.C., Schuh, S., 1996. Job creation and destruction. MIT Press, Cambridge. Diamond, P., 1971. A model of price adjustment. Journal of Economic Theory 3, 156-168. Diamond, P., 1981. Mobility costs, frictional unemployment and e ciency. Journal of Political Economy 89, 798-812. Feng, Q., Horrace, W.C., 2012. Alternative technical e ciency measures: Skew, bias and scale. Journal of Applied Econometrics 27, 253-268. Haltiwanger, J.C., Lane, J.I., Spletzer, J.R., 1999. Productivity di erences across employers: The role of employer size, age, and human capital. American Economic Review Papers and Proceedings 99, 94 98. Haltiwanger, J.C., Lane, J.I., Spletzer, J.R., 2007. Wages, productivity, and the dynamic interaction of businesses and workers. Labour Economics 14, 575-602. Hopenhayn, H., Nicolini, J., 1997. Optimal unemployment insurance. Journal of Political Economy 105, 2, 412-438. Luttmer E.G.J., 2007. Selection, growth, and the size distribution of rms. Quarterly Journal of Economics 122, 1103-1144. Marimon, R., Zilibotti, F., 1999. Unemployment vs. mismatch of talents: Reconsidering unemployment bene ts. The Economic Journal 109, 266-291. Mortensen, D., 1977. Unemployment insurance and job search decisions. Industrial and Labor Relations Review 30, 505-517. Mortensen, D.T., 2003. Wage dispersion. MIT Press, Cambridge. Neal, D., Rosen. S., 2000. Theories of the distribution of earnings. In 14
A.B.Atkinson and F. Bourguignon (eds). Handbook of Income Distribution, Vol. 1, North Holland. OECD, 2007. Bene ts and wages. OECD, Paris. Petrongolo B., Pissarides, C.A., 2001. Looking into the black box: A survey of the matching function. Journal of Economic Literature 39, 390-431. Yashiv, E., 2000. The determinants of equilibrium unemployment. American Economic Review 90, 1297-1322. 15
θ 100 90 80 70 60 50 40 Full 30 20 Partial 10 0 No p 0.1 0.2 0.3 0.4 0.5 Figure 1: The Optimal Sorting