From Individual to Aggregate Labor Supply

Transcription

1 From Individual to Aggregate Labor Supply Yongsung Chang Department of Economics University of Pennsylvania 3718 Locust Walk Philadelphia, PA Sun-Bin Kim Department of Economics Concordia University 1455 de Maisonneuve Blvd, W. Montreal, QC, H3G 1M8 March 27, 2002 Abstract Macroeconomists often assume an elastic labor supply, despite the low estimates from various micro studies. The extensive margin (entry-exit margin) has been recognized as a potential resolution. We construct a model economy in which heterogeneous agents decide on labor-market participation and the capital market is incomplete. Heterogeneity of the workforce is designed such that the evolution of wages, worker flows between employment and nonemployment, and cross-sectional earnings distribution are consistent with micro data. We find the aggregate labor-supply elasticity of such an economy less than 1, smaller than those assumed in aggregate models. Keywords: Aggregate Labor-Supply Elasticity, Heterogeneity, Indivisible Labor JEL Classifications: E24, E32, J21, J22 We thank Olivier Blanchard for sharing the worker-flow data with us. We thank Jose-Victor Rios-Rull for helpful comments. Chang gratefully acknowledges the financial support of the University of Pennsylvania Research Foundation.

2 1 Introduction Despite enormous heterogeneity, both in productivity and preferences in the workforce, economists often postulate and analyze an economy populated by identical agents for its simplicity and tractability. A fully specified representative-agent model has become a workhorse in macroeconomics, since Lucas and Rapping (1969) s pioneering work on aggregate labor-market fluctuations. It is now a common practice to rely on micro evidence to pin down the key parameters of highly aggregated models (e.g., Kydland and Prescott, 1982; Prescott, 1986; King, Plosser, and Rebelo, 1988). Unfortunately, however, this practice often creates a conflict between micro and macro observations. One of the stylized facts in economic fluctuations is that total hours vary greatly over the business cycle without much variation in wages. Among many, one explanation is substitution of leisure over time by workers, requiring an elastic labor supply beyond the admissible estimates based on micro data (e.g., Ghez and Becker, 1975; MaCurdy, 1981; Altonji, 1986; Abowd and Card, 1989). 1 The participation margin the so-called extensive margin has been recognized as a potential resolution, as the variation in the number of employees is the dominant source of fluctuations in total hours worked (e.g., Coleman, 1984; Heckman, 1984). For example, Prescott (1986) refers to this margin as a rationale for an elasticity higher than micro estimates. Another prominent example is Hansen (1985) s a linear utility in leisure, derived from the indivisible labor and employment lottery of Rogerson (1988). An economy with indivisibility at the micro level may be approximated by a representativeagent economy with divisible labor, as the indivisibility is smoothed by an aggregation over hetero- 1 See Pencavel (1986) for a survey on labor-supply elasticity. See also Mulligan (1998) on how the current micro estimates may underestimate the workers willingness to substitute leisure over time. 1

3 geneous agents. While this point is well illustrated in Mulligan (2001), we have yet to investigate its quantitative implications, because the mapping from the micro to the macro function depends crucially on the nature of heterogeneity in the workforce. 2 Heckman (1984, p. 210), in his comments on Kydland (1984), argues A fictional macro function must be created which is related to the true micro behavior by an aggregation process that Kydland and others for some reason feel they can safely ignore. The literature, so far, has not echoed to this concern. 3 In this paper we examine the aggregate labor-supply of an economy in which population possesses a serious heterogeneity in earnings and preference. Our strategy and main findings can be illustrated as follows. We construct a dynamic general equilibrium model economy populated by many workers. A worker faces idiosyncratic shocks to his productivity and preference, and the capital market is incomplete. Workers decide on labor supply in a discrete manner labor is indivisible. Heterogeneity in the workforce is disciplined by the two sets of micro data. The stochastic process of idiosyncratic productivity shocks is estimated by the wage process from the Panel Study of Income Dynamics (PSID). Conditional on the idiosyncratic productivity process, the stochastic process of individual preference shocks is specified to match the average worker flows between employment and nonemployment in the Blanchard and Diamond (1990) data based on the Current Population Survey (CPS) for In order to assess the validity of the heterogeneity in the model, we compare the cross-sectional distributions of earnings and wealth to those in Diaz-Gimenez, Quadrini, and Rios-Rull (1997), 2 For example, suppose there is a mass of workers, almost indifferent about either working or not working at the current wage. A small wage increase induces a large increase in participation. Alternatively, if the reservation-wage distribution is scattered, the same wage increase ends up with a small variation in total hours. 3 See also Browning, Hansen, and Heckman (1999) that raises warning flags about the current use of micro evidence in calibrating macro models. 2

4 documented from the Survey of Consumer Finance (SCF). The model economy reproduces the earnings and wealth distributions in the SCF reasonably well. We then introduce exogenous shifts in labor demand through the variation of aggregate productivity. We find the response of total hours from our heterogeneous-agent economy comparable to those from the economies with laborsupply elasticity between 0.6 and 0.8, which is much smaller than those often assumed in aggregate models. Among others, earlier important works on the labor-market heterogeneity in the context of a stochastic growth model include Kydland (1984), Cho (1995), Andolfatto and Gomme (1996), Castaneda, Diaz-Gimenez, and Rios-Rull (1998), Merz (1999), and Gomes, Greenwood, and Rebelo (2001). Kydland and Cho are the closest to our work as they focus on labor supply. Kydland constructs an economy with two types of workers, skilled and unskilled, and reproduces some labor-market regularity in relative wages and hours. However, this approach does not reflect the participation margin, a dominant source of variation in total hours. Cho first incorporates the extensive margin into the real-business-cycle model in which workers are ex ante identical and ex post heterogeneous. This considerably simplifies the computation since the consumption is shared among workers. It is, however, clear in the data that persons with greater hours or greater earnings per hour consume more. We argue that realistic ex ante heterogeneity is important because workers take into account their ability and preference in making decisions on consumption, savings, and labor supply. The recent advance made by Krusell and Smith (1998), also adopted in Castaneda et al. and Gomes et al., allows us to characterize the equilibrium fluctuations in incomplete market. Gomes et al. examine the cyclical behavior of unemployment rates; Andolfatto and Gomme the Canadian unemployment insurance policy; Castaneda et al. the income distribution and unemployment spells; Merz the cyclical behavior of labor turnover. 3

5 The paper is organized as follows. Section 2 presents the model. Section 3 calibrates the parameters of the model, including the heterogeneity in productivity and preference. In Section 4, we characterize the steady state and fluctuations of the model, including the response of aggregate labor supply to exogenous shifts in (labor) demand. Section 5 is the conclusion. 2 The Model 2.1 Environment The model economy is a version of the stochastic-growth model with a large (measure one) population of infinitely lived workers. Individual workers differ from each other in productivity and preference for leisure. Each worker maximizes the expected discounted lifetime utility: U = max {c t,h t} t=0 E 0 { t=0 } β t u(c t, h t ; z t ), with (1 h t ) 1 1/γ u(c t, h t ; z t ) = ln c t + Bz t, 1 1/γ where E 0 [ ] denotes the expectation operator conditional on information available at time 0, β is the discount factor, c t consumption, and h t hours worked at time t. The utility is separable between consumption and leisure and across times. The assumption about the form of utility is popular in both business-cycle analysis (e.g., Hansen, 1985; King, Plosser, and Rebelo, 1988) and the empirical labor-supply literature (e.g., MaCurdy, 1981; Altonji, 1986). The parameter γ denotes the intertemporal substitution elasticity of leisure. Log utility in consumption supports a balanced growth path. The individual preference shock z t, affecting the marginal rate of substitution between goods and leisure, varies exogenously over time according to a stochastic process with a transition probability distribution function, π z (z z) = Pr(z t+1 z z t = z). 4

6 According to our production technology, which will be specified below, labor input enters simply as an effective unit. Thus, a worker who supplies h t units of time earns w t x t h t, where w t is the market wage rate for the efficiency unit of labor, and x t represents the worker s productivity. We assume that individual productivity x t exogenously varies over time according to a stochastic process with a transition probability distribution function π x (x x) = Pr(x t+1 x x t = x). Following Hansen and Rogerson, we abstract from an intensive margin and assume that labor supply is indivisible; i.e., h t takes either zero or h(< 1). A worker can save and borrow by trading a claim for physical capital, which yields the rate of return r t and depreciates at rate δ. Workers face a borrowing constraint; the level of asset holdings, a t, cannot be negative at any time. The capital market is incomplete; the physical capital is the only asset available to insure against idiosyncratic risks in x and z. A worker s budget constraint is: c t = w t x t h t + (1 + r t )a t a t+1, and a t+1 0. Firms produce output according to constant-returns Cobb-Douglas technology in capital, K t, and effective labor, L t : Y t = F (L t, K t, λ t ) = λ t L α t K 1 α t, where λ t is aggregate productivity, following a stochastic process with a transition probability distribution function, π λ (λ λ) = Pr(λ t+1 λ λ t = λ). It is useful to consider a recursive equilibrium definition, including the law of motion for aggregate state of the economy. Suppose µ(a, x, z) denotes the distribution (measure) of workers Let A, X and Z denote sets of all possible realizations of a, x and z, respectively. The measure µ(a, x, z) is defined over a σ-algebra of A X Z. 5

7 Let V E and V N denote the values of employed and nonemployed. If a worker decides to work, she solves the following Bellman equation by choosing the next period asset holdings a : { V E (a, x, z; λ, µ) = max u(c, 1 h; z) a A [ + βe max { V E (a, x, z ; λ, µ ), V N (a, x, z ; λ, µ ) } ] } (1) x, z, λ subject to c = wx h + (1 + r)a a, a 0, and µ = T(λ, µ). where T denotes a transition operator for µ. If the worker decides not to work, her Bellman equation is: { u(c, 1; z) V N (a, x, z; λ, µ) = max a A subject to [ + βe max { V E (a, x, z ; λ, µ ), V N (a, x, z ; λ, µ ) } ] } (2) x, z, λ c = (1 + r)a a, a 0, and µ = T(λ, µ). Having solved (1) and (2), it is straightforward to deal with worker s labor-supply decision: { } V (a, x, z; λ, µ) = max h { h,0} V E (a, x, z; λ, µ), V N (a, x, z; λ, µ). (3) 6

8 2.2 Equilibrium Equilibrium consists of a set of decision rules of an individual for consumption, asset holding, and labor supply, {c(a, x, z; λ, µ), a (a, x, z; λ, µ), h(a, x, z; λ, µ)}, the value function, V (a, x, z; λ, µ), functions for aggregate inputs, {K(λ, µ), L(λ, µ)}, functions for factor prices, {w(λ, µ), r(λ, µ)}, and a law of motion for the distribution µ = T(λ, µ) such that: 1. Individual optimization: Given w(λ, µ) and r(λ, µ), the individual decision rules c(a, x, z; λ, µ), a (a, x, z; λ, µ), and h(a, x, z; λ, µ) solve (1), (2), and (3). 2. The firm s profit maximization: w(λ, µ) = F 1 ( L(λ, µ), K(λ, µ), λ ) (4) r(λ, µ) = F 2 ( L(λ, µ), K(λ, µ), λ ) δ (5) for all (λ, µ). 3. The goods market clears: {a (a, x, z; λ, µ) + c(a, x, z; λ, µ) } dµ = F ( L(λ, µ), K(λ, µ), λ ) + (1 δ)k (6) for all (λ, µ). 4. Factor markets clear: L(λ, µ) = xh(a, x, z; λ, µ)dµ (7) K(λ, µ) = adµ (8) for all (λ, µ). 7

9 5. Consistency of individual and aggregate behavior: µ (A 0, X 0, Z 0 ) = A 0,X 0,Z 0 { A,X,Z for all A 0 A, X 0 X and Z 0 Z. } 1 a =a (a,x,z;λ,µ) dπ x (x x)dπ z (z z)dµ da dx dz (9) 3 Calibration A key element in the mapping from individual to aggregate labor supply is the nature of heterogeneity in the workforce. In our model this amounts to properly specifying the stochastic process of x and z. We assume that the individual productivity follows an AR(1) process in logs: ln x = ρ x ln x + ε x, ε x N(0, σ 2 x). (10) Panel data from the PSID for the period of are used to estimate this process. Appendix A.1 describes the PSID data we use in detail. According to our model, the log wage for individual i at time t, denoted by ln wt, i can be written as ln wt i = ln w t + ln x i t, where w t and x i t denote the market wage rate for an effective unit of labor and individual i s productivity, respectively. When quasi-differenced, individual wage evolves as: ln w i t = ρ x ln w i t 1 + (ln w t ρ x ln w t 1 ) + ε i x,t. (11) Equation (11) is estimated with year dummies which capture aggregate effects including ln w t ρ x ln w t 1. The OLS estimate for ρ x is with a standard error of (R 2 = 0.69). 5 This is an annual estimate, and we uncover the quarterly values ρ x = 0.95 and σ x = as described in Appendix A.2. 5 When we include age, sex, and years of schooling in the regression to control for individual characteristics, the persistence decreases from to Both estimates are somewhat lower than the persistence of idiosyncratic labor-income risks reported in Storesletten, Telmer, and Yaron (1999). 8

10 Labor economists and econometricians (e.g., Heckman 1978; Burtless and Hausman, 1978; MaCurdy, Green, and Paarsch, 1990) have found enormous heterogeneity in the marginal rate of substitution between goods and leisure. While this rate tends to depend on observed characteristics, such as marital status and number of children, the magnitude of unobserved component is not only large but also persistent over time. We assume that the preference shock z follows an AR(1) process in logs and is independent of productivity x. ln z = ρ z ln z + ε z, ε z N(0, σ 2 z). (12) We do not see a practical way to estimate this directly. Thus, we simply set ρ z = ρ x. Concerning the standard deviation σ z, we use a model-based approach. In steady state, there are constant flows of workers between employment and nonemployment. Conditional on the stochastic process for productivity, we search for a magnitude of preference shock σ z such that the steady-state worker flows from the model match the average gross-worker flows in the data. According to our calculation based on Blanchard and Diamond (1990) data, which we describe in Appendix A.3, during , on average 5.15 percent of the population moved from employment to nonemployment each quarter; however, 5.35 percent of the population moved in the opposite direction, from nonemployment to employment. 6 With σ z = 0.45 we obtain 5.25 percent for both flows in the steady state. Other parameters of the model are in accord with business-cycle analysis and empirical laborsupply literature. According to the Michigan Time-Use Survey, a typical married couple allocates about 33 percent of its discretionary time for paid compensation (e.g., Hill, 1984; Juster and Stafford, 1991): we set h = 1/3. Regarding the intertemporal substitution elasticity of leisure 6 Nonemployment includes both unemployment and non-labor-force. According to Blanchard and Diamond, the flows between employment and non-labor-force are as big as as those between employment and unemployment. 9

11 γ, most estimates based on micro labor-supply data fall between 0 and 0.5: we use The labor share, α, is 0.64, and the quarterly depreciation rate, δ, is 2.5 percent. When we search for the value of σ z that matches the average worker flows, we jointly search for B such that the steady-state employment rate is 65 percent, the sample average in Blanchard-Diamond data. The discount factor β is chosen so that the real rate of return to capital is 4 percent annually. 8 Table 1 summarizes the parameter values. Finally, when we investigate the response of hours with fluctuations, we introduce exogenous shifts in aggregate productivity that resembles the total factor productivity from the post-war U.S. data: λ can take two values, ln λ { , }, and its transition matrix between the two states is π λ = [ ]. 4 Results We solve the worker s problem by computing an approximation to value functions (1), (2), and (3) on a discrete-state space. Tauchen s (1986) algorithm is used to approximate the transition probabilities when evaluating the conditional expectations in value functions. For the steady state, we use the algorithm described in Rios-Rull (1999) to find the invariant distribution of workers. When we introduce aggregate fluctuations, we adopt the bounded rationality method developed by Krusell and Smith, in which agents are assumed to make use of a finite set of moments of 7 With discrete choice of hours of work, the value of γ by itself is not so important. Given the level of consumption, the utilities of the employed and the non-employed differ by a constant term. 8 The discount factor is lower than that in the representative-agent model, because market incompleteness increases savings as noted in Aiyagari (1994). 10

12 the distribution µ. The justification of this method is that by using partial information about µ, households do almost as well as by using all the information in µ when predicting future prices. In fact, Krusell and Smith show that use of the first moment only provides a remarkably good approximation in a stochastic-growth model. We show that this is the case in our economy as well. A detailed description of the computational procedures is provided in Appendix A The Steady State We first characterize the steady state of the model economy in which there is no aggregate uncertainty, and the distribution of workers is invariant. As described in the previous section, we have calibrated B and σ z such that employment rate and worker flows replicate the data in the steady state. Table 2 confirms this. Employment rate, gross worker flows into and out of employment, and hazard rates of worker flows are very close to those in the data. The data statistics are quarterly averages for 1968 to 1986 computed from the Blanchard-Diamond data. We ask whether the model generates a reasonable cross-sectional distribution of wealth and earnings. While our primary objective is not to explain the income inequality, we view this as an indirect way of assessing our calibration of idiosyncratic productivity and preference shocks. Figure 1 shows the Lorenz curves of the wealth distribution for the U.S. and the model. The statistics for the U.S. are from Diaz-Gimenez, Quadrini, and Rios-Rull, which is based on the 1992 SCF. The wealth is much more concentrated in the data than our economy. In terms of earnings, however, both economies exhibit similar inequality, as shown in Figure 2. Table 3 reports the Gini indices. For earnings, it is 0.56 in the model, close to 0.63 in the U.S. data; for wealth, it is 0.49, much lower than 0.78 in the data. Table 4 shows the detailed comparison between the model and 1992 SCF, again, documented 11

13 by Diaz-Gimenez et al. For each quintile group of the wealth distribution, we calculate the wealth share, ratio of group average to economy-wide average, and earnings share. In 1992 households in the 4th and 5th quintile of the wealth distribution own and percent of total wealth, respectively. According to the model, they own and percent, respectively. The average asset holdings of the households in the 5th quintile is 3.97 times larger than that of a typical household in the data, while this ratio is 2.64 according to our model. For earnings, households in the 5th quintile of the wealth distribution earn percent of total earnings in the data, whereas they earn percent in the model. Overall, the wealth is more concentrated in the data. In particular, the model fails to match the highly concentrated wealth in the right tail of the distribution. According to Diaz-Gimenez et. al, about half of total wealth is held by the top 5 percent of the population in the U.S., whereas only 18.5 percent of total wealth is held by them in our model (not shown in the Table). However, our primary objective is not to explain the income inequality or the behavior of the top 1 or 5 percent of population. The model economy seems to possess a reasonably large heterogeneity for our purpose, to study aggregate labor supply; that is, the stochastic process of wages is directly estimated from the panel data and the cross-sectional earnings distribution is, by and large, consistent with micro data. Before we introduce aggregate uncertainty, we illustrate the labor-market participation and consumption decisions of workers in the model. Figure 3 shows the combination of ln x and ln z that makes the worker indifferent between working and not-working. Each line corresponds to the workers at the 1st, 2nd, 3rd, and 4th quintile of the wealth distribution. At a given asset level, workers with productivity above the cutoff line choose to work. Equivalently, workers with preference below the cutoff choose to work. As the asset increases, from the 1st quintile to the 2nd, and so forth, the line shifts to the right; the richer individuals are, the more selective in working. To illustrate, consider a worker on the 4th quintile of the wealth distribution with average 12

14 productivity and preference (ln x = 0 and ln z = 0). According to Figure 3, he is indifferent between working and not-working. Suppose, for example, his hourly wage was 20 dollars. A worker on the 1st quintile with the same preference is indifferent in working at hourly wage of 5 dollars 50 cents (= $20 exp 1.3 ). Figure 4 shows the average consumption of employed and nonemployed individual for each asset holdings. Given the separable utility and no risk-sharing, the consumption of employed is greater than that of nonemployed at all asset levels. Also, note that the consumption, so is savings, is close to linear in assets in most range except for those near the borrowing constraint Fluctuations In the model, exogenous shifts in productivity generate aggregate fluctuations. We do not take a stand on the sources of the business cycle here, but we intentionally exclude other types of aggregate disturbances, especially those that shift the labor-supply curve. Aggregate productivity shocks act like an instrument, shifting the labor-demand curve (marginal product of labor), to identify the slope of the aggregate labor-supply curve. Table 5 summarizes the cyclicality and volatility of three labor-market variables: employment and worker flows into and out of employment. Since we assume a single aggregate disturbance, we do not expect our economy to replicate the labor-market fluctuation in the data. However, it is still of interest to see the response of key labor-market variables to aggregate disturbances. The model statistics are calculated from 5,000 periods of simulated data. As a reference, we report the comparable statistics for the post-war U.S. economy. Employment is less volatile than the data; the standard deviation relative to output is 0.36 in our model, whereas it is 0.46 in the data. 9 This partially justifies our approximation procedure which uses the first moment only in computing equilibria under aggregate fluctuations below. 13

15 Employment is highly correlated with output (0.84) as in the data (0.77). The cyclical behavior of the flow out of employment is somewhat close to that in the data; the relative standard deviation and the correlation with output are 2.71 and in the model, respectively, while they are 2.39 and -0.21, respectively, in the data. The flow into employment is almost a mirror image to the flow out-of employment in the model, whereas it is somewhat less volatile and virtually uncorrelated with output in the data. We now investigate the elasticity of aggregate labor supply. We do so by comparing with the divisible-labor representative-agent economies under the same economic environment. For a representative-agent economy, the Frisch labor-supply equation around the steady state is ĥ t = ψ(ŵ t ĉ t ), ψ = 1 h γ, (13) h where the circumflexes denote the variable s percentage deviation from its steady-state value. The Frisch labor-supply elasticity ψ, also known as lamda-constant-labor-supply elasticity or the compensated labor-supply elasticity, represents the response of hours with respect to wages holding consumption (or wealth) constant. We consider four economies with ψ equaling to 0.4, 0.6, 0.8, and 1. With h = 1/3, these correspond to γ of 0.2, 0.3, 0.4, and 0.5, respectively. 10 The representativeagent economies have the same parameter values in Table 1 except for γ. 11 We compute the equilibria of all model economies by the same method. Table 6 summarizes the main findings of the paper. It displays the statistics of five model economies (our indivisible-labor heterogeneous-agent economy and four representative-agent economies) and the U.S economy. The upshot is that the response of hours in the heterogenous-agent economy is similar to that from the representative-agent economies with labor-supply elasticity between For example, Prescott (1986) uses ψ = 2 by supposing γ = 1 and h = 1/3. 11 For each representative economy, the parameter B is also adjusted to yield h = 1/3 in the steady state. 14

16 and 0.8. Regarding the labor-market variables in the 4th row, the volatility of aggregate hours for our model is 0.53; they are 0.47 and 0.59 for the economies with labor-supply elasticity of 0.6 and 0.8, respectively. The volatility of hours relative to labor productivity is 0.46, falls between those with ψ = 0.6 (0.43). The volatility of hours relative to output is 0.34, close to that with ψ = 0.8 (0.36). 12 A similar comparison applies to other aggregate variables. The volatility of output (1.56) is close to that with ψ=0.6 (1.56), the volatility of consumption (0.42) is close to that of the economy with ψ=0.8 (0.41), the volatility of investment (5.16) is close to that of the economy with ψ=0.6 (5.20). Aggregate labor supply is not sensitive to fluctuations in productivity, because of the large heterogeneity in the workforce. The dispersion of individual productivity (measured by the crosssectional standard deviation of log wages in the PSID) is larger than that of aggregate productivity (measured by the time-series standard deviation of aggregate TFP) by a factor of nearly 27. The standard deviation of individual preference shocks is at least twice as big as that of individual productivity. The implied reservation-wage distribution is scattered. As a result, a temporary high wage, say, a few percents above the steady state, does not generate a big increase in participation. In sum, under the heterogeneity we have, designed to match the certain aspects of the labor market, the aggregate labor-supply elasticity is slightly bigger than micro estimates, but smaller than those often assumed in macro models. 13 A highly elastic labor supply resulting from the indivisible labor 12 Given the single aggregate disturbance which shifts the labor demand, one might be tempted to determine the aggregate labor-supply elasticity of the model simply by estimating Equation (13) using the model-generated aggregate time-series data. However, with an incomplete market, the aggregation theorem does not hold. When we actually estimate (13) based on model-generated data, the value for ψ is 0.4 by OLS (with R 2 of only 0.45) and 0.27 by IV with lagged consumption and wages as instruments. 13 Since the variation of hours in the model stems sorely from the extensive margin, allowing for an intensive margin may generate a bigger response of aggregate labor supply. However, under the small intertemporal substitution 15

17 and employment lottery does not survive a serious heterogeneity and market incompleteness. 5 Conclusion The labor supply is at the heart of macroeconomic research. It is a cornerstone of the equilibrium approach that relies on intertemporal substitution of leisure. In a disequilibrium approach, in which the role of labor supply is dismissed in the short run, the labor supply is still crucial for the welfare loss of the economy departing from the equilibrium. Aggregate models often assume elastic labor supply, despite the low estimates from empirical studies based on individual data. The extensive margin, a major source of hours variation over the business cycle, has been proposed as a potential resolution. We construct and compute the equilibrium of an economy where heterogeneous agents decide on labor-market participation and the capital market is incomplete. Heterogeneity of workforce is designed such that evolution of wages, gross worker flows, and cross-sectional income distribution are consistent with the micro data. We find the responses of hours from such an economy comparable to those from an economy with labor-supply elasticity between 0.6 and 0.8, which is much lower than those often assumed in aggregate models. elasticity, this will not increase the aggregate labor-supply elasticity substantially. 16

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21 [30] Prescott, Edward, 1986, Theory Ahead of Business-Cycle Measurement, Carnegie-Rochester Conference Series of Public Policy, 25, [31] Rios-Rull, Jose-Victor, 1999, Computation of Equilibria in Heterogeneous-Agents Models, in Computational Methods for the Study of Dynamic Economies, Ramon Marimon and Andrew Scott eds., Oxford University Press. [32] Rogerson, Richard, 1988, Indivisible Labor, Lotteries and Equilibrium, Journal of Monetary Economics, 21, [33] Storesletten, Kjetil, Chris Telmer, and Amir Yaron, 1999, The Risk-Sharing Implications of Alternative Social Security Arrangement, Carnegie-Rochester Conference Series on Public Policy, 50, [34] Tauchen, George, 1986, Finite State Markov-Chain Approximations to Univariate and Vector Autoregressions, Economics Letters, 20,

22 A Appendix A.1 The PSID Data The PSID sample period used is The sample consists of heads of households and wives. We start in 1979 because the wage data for wives are available only since Wages are annual hourly earnings (annual labor incomes divided by annual hours). Nominal wages are deflated by the Consumer Price Index. The base year is Workers who worked less than 100 hours per year or whose hourly wage rate was below $1 (in 1983 dollars) are viewed as nonemployed even though their employment status is reported as employed in the survey. We use workers who were employed in non-agricultural sectors and not self-employed. We also restrict the sample to hourly earnings less than or equal to $500. The descriptive statistics for our PSID data are summarized in Table A.1. Table A.1: Summary Statistics for the PSID Variable Mean S.D. Obs. Real Wage Rate (in 1983 $) Log Real Wage Log Real Wage (net of aggregate effect) Annual Hours of work Age Years of schooling Sex (male =1) Note: Log real wages (net of aggregate effect) are the residuals from the regression of log wages on year dummies. 21

23 A.2 Conversion between Annual and Quarterly Variances After controlling for aggregate price of an effective unit of labor, the wage evolves according to x. Since the wages in the PSID are annual averages: 14 ln x τ = 1 4 { ln x (τ,1) + ln x (τ,2) + ln x (τ,3) + ln x (τ,4) }, where x τ is annual average and x (τ,q) denotes the wage of the qth quarter in year τ. According to a AR(1) process for x, the quarterly persistence ρ x is simply 0.95(= ) where is the annual estimate from the PSID. Furthermore, ln x τ = 1 4 {(1 + ρ x + ρ 2 x + ρ 3 x) ln x τ 1 + (1 + ρ x + ρ 2 x)ε x,(τ,2) + (1 + ρ x )ε x,(τ,3) + ε x,(τ,4) }. The standard deviation of annual average is: σ(ln x) = 1 4 σ x { (1 + ρx + ρ 2 x + ρ 3 x) 2 1 ρ 2 x + (1 + ρ x + ρ 2 x) 2 + (1 + ρ x ) 2 + 1} 1/2. When we replace σ(ln x) with 0.601, the standard deviation of log annual hourly earning (net of aggregate effects) from the PSID, we obtain σ x = A.3 The Blanchard-Diamond Data The Blanchard and Diamond (1990) data, which is preceded by Abowd and Zellner (1985), is based on monthly CPS. We convert the monthly worker-flow series into the quarterly data consistent with our model. We also normalize them by dividing by the noninstitutional population of age 65 or less. There are three possible labor-market status: employment, unemployed, and non-labor-force, denoted by e, u, and n, respectively. The following formula is used to construct the flow of workers from labor-market status i to j during the quarter, denoted by f ij : f ij = ī h 1 ik h2 kl h3 lj, i, j {e, u, n}, k,l {e,u,n} 14 Note that 1 4 P 4 q=1 ln x (τ,q) can be interpreted as a log-linear approximation of the arithmetic average ln ex τ = ln[ 1 4 P 4 q=1 x (τ,q)]. 22

24 where ī denotes the number of workers in status i in the beginning of the quarter, and h m kl the monthly hazard rate from status k to l in the m-th month of the quarter. This takes into account all possible paths, direct and indirect, from i to j during a quarter, and avoids a potential double counting in a simple sum of monthly flows. A.4 Computational Procedures A.4.1 Steady-State Equilibrium The distribution of workers is invariant in the steady state; so are the factor prices. We search for the discount factor β that clears the capital market given an quarterly rate of return of 1 percent. Computing the steady-state equilibrium amounts to finding the value functions, the associated decision rules, and the time-invariant measure of workers. Details are as follows: 1. Choose 936 grid points for asset holdings, a i, in the range of [0, 199], where average asset holding is The grid points of assets are not equally spaced. We assign more points on the lower asset range to better approximate savings decisions of workers with lower assets. For productivity, x j, and preference for leisure, z k, we construct grid vectors of 9 equally spaced points in which ln x j s lie on the range of ±3σ x / 1 ρ 2 x and ln z k s on the range of ±3σ z / 1 ρ 2 z. 2. Given β, we solve the individual optimization problem in (1), (2), and (3) at each grid point of the individual states. In this step, we also obtain the optimal decision rules for asset holdings a (a i, x j, z k ) and labor supply h(a i, x j, z k ). This step involves the following procedure: (a) Initialize value functions V E 0 (a i, x j, z k ), V N 0 (a i, x j, z k ), and V 0 (a i, x j, z k ). (b) Update value functions by evaluating the discretized versions of (1), (2), and (3): { V1 E (a i, x j, z k ) = max u ( w hx j + (1 + r)a i a, 1 h; ) z k (A.4.1) 23

25 + β 9 9 j =1 k =1 { V1 N (a i, x j, z k ) = max u ( (1 + r)a i a ), 1; z k and + β 9 9 j =1 k =1 [ ] } max V0 E (a, x j, z k ), V0 N (a, x j, z k ) π x (x j x j )π z (z k z k ), (A.4.2) [ ] } max V0 E (a, x j, z k ), V0 N (a, x j, z k ) π x (x j x j )π z (z k z k ), { } V 1 (a i, x j, z k ) = max V1 E (a i, x j, z k ), V1 N (a i, x j, z k ), (A.4.3) where π x (x j x j ) and π z (z k z k ) are the transition probabilities of x and z respectively, which are approximated using Tauchen s (1986) algorithm. (c) If V 1 and V 0 are close enough for all grid points, then we found the value functions. Otherwise, set V0 E = V 1 E and V0 N = V1 N, and go back to step 2-(b). 3. Using a (a i, x j, z k ), π x (x j x j ) and π z (z k z k ) obtained from step 2, we obtain time-invariant measures µ (a i, x j, z k ) as follows: (a) Initialize the measure µ 0 (a i, x j, z k ). (b) Update the measure by evaluating the discretized version of (9): µ 1 (a i, x j, z k ) = i=1 j=1 k=1 9 1 ai =a (a i,x j,z k )µ 0 (a i, x j, z k )π x (x j x j )π z (z k z k ) (A.4.4) (c) If µ 1 and µ 0 are close enough for all grid points, then we found the time-invariant measure. Otherwise, replace µ 0 with µ 1, and go back to step 3(b). 4. We calculate the real interest rate as a function of β, i.e., r(β) = α ( K(β)/L(β) ) 1 α δ, where K(β) = i=1 j=1 k=1 a iµ (a i, x j, z k ) and L(β) = i=1 j=1 k=1 x jh(a i, x j, z k )µ (a i, x j, z k ). Other aggregate variables of interest are calculated using µ and decision rules. If r(β) is close enough to the assumed value of the real interest rate, we found the steady state. Otherwise, choose a new β and go back to step 2. 24

26 A.4.2 Equilibrium with Aggregate Fluctuations Approximating the equilibrium in the presence of aggregate fluctuations requires us (i) to include the measure of workers and the aggregate productivity shock in the list of state variables, and (ii) to keep track of the evolution of the measure µ over time. Since µ is an infinite dimensional object, it is almost impossible to implement these tasks as they are. We follow the procedure suggested by Krusell and Smith (1998). We assume that agents make use of its first moment only. The law of motion for aggregate capital stock is restricted to the log-linear function. Therefore, computing the equilibrium with aggregate fluctuations amounts to finding the value functions, decision rules, and law of motion for the aggregate capital within the class of a parametric form of the log linear in K and λ. Details are as follows: 1. In addition to the grids for individual state variables specified above, we choose 7 grid points for the aggregate capital K in the range of [0.93K, 1.07K ], where K denotes the steadystate aggregate capital. In our numerous simulations, the capital stock has never reached the upper or lower bound. The stochastic process for the aggregate productivity shock π λ (λ λ) is described in the text. 2. Let the parametric law of motion for the aggregate capital take a log linear in K and λ: ln K t+1 = κ κ 0 1 ln K t + κ 0 2 ln λ t. (A.4.5) In order for individuals to make their decisions on savings and labor supply they have to know (or predict) the interest rate and wage rate for an effective unit of labor. While the factor prices depend on aggregate capital and labor aggregate labor input is not known to individuals at the moment when they make decisions. Thus, individuals need to predict the factor prices. These predictions on factor prices, in turn, must be consistent with the 25

27 outcome of individual actions the factor market clearing in (7) and (8). We also assume that individuals predict the market wage and the interest rate using a log-linear function of K and λ: ln w t = b b 0 1 ln K t + b 0 2 ln λ t. (A.4.6) and ln r t = d d 0 1 ln K t + d 0 2 ln λ t (A.4.7) 3. We choose the initial values for the coefficients κ 0 s, b 0 s and d 0 s. Good initial values may come from a representative-agent model. 4. Given the law of motion for the aggregate capital and the prediction functions for factor prices, we solve the individual optimization problem in (1), (2), and (3). This step is analogous to step 2 in the steady-state computation: (a) We have to solve for the value functions and the decision rules over a bigger state space. Now the state variables are (a, x, z, K, λ). (b) Computation of the conditional expectation involves the evaluation of the value functions not on the grid points along the K dimension since K predicted by (A.4.5) need not be a grid point. We polynomial-interpolate the value functions along K dimension when necessary. 5. Using a (a i, x j, z k, K l, λ m ), h(a i, x j, z k, K l, λ m ), π x (x j x j ), π z (z k z k ), π λ (λ m λ m ), and the assumed law of motion for the aggregate capital, we generate a set of artificial time series data {K t, w t, r t } of the length of 5,000 periods. Each period, {K t, w t, r t } is calculated by aggregating labor supply and asset holdings of 50,000 individuals. 6. We obtain new values for coefficients κ 1 s, b 1 s and d 1 s by the OLS from the simulated data. 26

28 If κ 1 s, b 1 s and d 1 s are close enough to κ 0 s, b 0 s, and d 0 s, respectively, we found the law of motion. Otherwise, update coefficients by setting κ 0 = κ 1, b 0 = b 1 s and d 0 = d 1 s, and go back to step 4. The estimated law of motion for capital and prediction functions and their accuracy, measured by R 2 for the prediction equations are as follows. the law of motion for aggregate capital in equation (A.4.5): ln K t+1 = ln K t ln λ t, R 2 = the market wage rate in equation (A.4.6): ln w t = ln K t ln λ t, R 2 = the interest rate in equation (A.4.7): ln r t = ln K t ln λ t, R 2 = The law of motion for aggregate capital provides the highest accuracy. The wage function is more accurate than the interest rate function. Overall, predictions functions are fairly precise as R 2 s are close to 1. 27

29 Table 1: Parameters of the Benchmark Economy Parameter α = 0.64 β = γ = 0.2 B = 0.65 h = 1/3 ρ x = 0.95 σ x = ρ z = 0.95 σ z = 0.45 Description Labor share in production function Discount factor Intertemporal substitution elasticity of leisure Utility parameter Steady-state hour Persistence of idiosyncratic productivity shock Standard deviation of innovation to idiosyncratic productivity Persistence of idiosyncratic preference shock Standard deviation of innovation to idiosyncratic preference Table 2: Labor Market Steady States Variable Data Model Employment rate Flow out of employment Flow into employment Hazard rate out of nonemployment Hazard rate into employment Note: All variables are in percentage. Data are quarterly averages for based on Blanchard and Diamond (1990) as described in Appendix A.3. 28

30 Table 3: Gini Indices for Wealth and Earnings Variable Data Model Wealth Earnings Note: Data statistics are from Diaz-Gimenez, Quadrini, and Rios-Rull (1997) based on 1992 Survey of Consumer Finance. Table 4: Characteristics of Wealth Distribution Quintile 1st 2nd 3rd 4th 5th Total U.S. Data Share of wealth Group average/population average Share of earnings Model Share of wealth Group average/population average Share of earnings Note: Data statistics are from Diaz-Gimenez, Quadrini, and Rios-Rull (1997) based on 1992 Survey of Consumer Finance. 29

31 Table 5: Cyclical Behavior of Labor Market Variable Volatility Cyclicality Data Model Data Model Employment rate Flow out of employment Flow into employment Hazard rate out of employment Hazard rate into employment Note: Variables are in percentage and detrended by the H-P filter. Volatility denotes the standard deviation of the variables divided by the standard deviation of output. Cyclicality denotes the correlation with output. Table 6: Comparison with Divisible Labor Economies Indivisible Divisible U.S. Data ψ = 0.4 ψ = 0.6 ψ = 0.8 ψ = :I-1998:4 σ(y ) σ(c) σ(i) σ(n) σ(n)/σ(y/n) σ(n)/σ(y ) Note: ψ denotes the Frisch labor-supply elasticity. Data are from the DRI data base. Y = GDP-government spending; C = consumption of non durables and services; I = nonresidential fixed investment; N = total employed hours in non-agricultural private sector from the establishment survey. All variables are divided by civilian noninstitutional population over 16. All variables are detrended by the H-P filter. 30

32 Figure 1: Lorenz Curves for Wealth o line Model U.S Figure 2: Lorenz Curves for Earnings o line Model U.S

33 Figure 3: Cutoff (x, z) for Work Decision for Selected Asset Levels ln z st Quintile 2nd Quintile 3rd Quintile 4th Quintile ln x Figure 4: Consumption by Employment Status Consumption Employed Non Employed Asset Holdings 32