Journal of Monetary Economics 16 (1985) 309-327. North-Holland - PDF document

Journal of Monetary Economics 16 (1985) 309-327. North-Holland INDIVISIBLE LABOR AND THE BUSINESS CYCLE Gary D. HANSEN* Unioersify of California, Santa Barbara, CA 93104, USA A growth model with shocks to technology is studied. Labor is indivisible, 1. Introduction Equilibrium theories of the business cycle, such Kydland and Prescott (1982) or Lucas (1977), have been criticized for failing to account for some important *This paper is part of my doctoral dissertation written while a student at the University of Minnesota. I have benefited from conversations with many people including Robert King, Thomas Sargent, Christopher Sims, Neil Wallace, 310 G. D. Hunsen. Indivisible lubor und the bminess qde uals can either work some given positive number of hours or not at all - they are unable to work an intermediate number of hours. This assumption is motivated by the observation that most people either work full time or not This follows ‘The fact that existing equilibrium models are inconsistent with this observation has been stressed by Heckman (1983) and Coleman (1984). ‘Kydland and Prescott (1982) attempt to explain the above fact by including past leisure as an argument in the individual’s utility function so as to enhance the intertemporal substitution response to a productivity shock. However, even after introducing G. D. Hansen, Indivisible hbor and the business cycle 311 The paper is divided as follows: The next section provides a more detailed explanation and motivation of the indivisible labor assumption. In section 3 the artificial economies to be studied are constructed. The first is a 2. Motivation Existing equilibrium theories of the business cycle analyze individuals who are free H, is total hours worked, h, is ‘The data used for this analysis is available from the Bureau of Labor Statistics’ Labstat data tape. The series I used were collected from households using the Current Population Survey. For a description of the detrending method, see footnote 18. 312 G. D. Hansen. Indioisihle l&or and rhe business cycle Using this decomposition, 55% of the variance of H, is due to variation in N,, while only 20% of this variance can be directly attributed to h,. The remainder is due to the covariance term.6 Most people either work full time or 6Coleman (1984) comes to a similar conclusion using establishment data. ‘One advantage of modeling the non-convexity as a feature of the technology is that it would likely explain why part-time workers are paid less than full-time workers, in addition to accounting for features of the data discussed in this paper. G. D. Hansen, Indivisible labor and rile business cycle 313 sort could probably be safely abstracted from when studying business cycle phenomena. However, it happens that the two models have very different implications 3. Two economies 3.1. A one-sector stochastic growth model with divisible labor The economy to be studied is populated by a continuum of identical infinitely lived households with names t decisions. The assump- tion of one firm is made for convenience. Since the technology displays constant returns to scale - implying that firms make zero profit in (3) where 6 is the rate of capital depreciation. The stock of capital is owned by the households who sell capital services to the firm. The technology shock is assumed to follow r+l =Yh,+Et+lr (4) where the E,‘S are iid with distribution function F. This distribution is assumed to have a positive support with a finite upper bound, which guarantees that output will 314 changes in the inputs (capital and labor). We follow Solow (1957) and Kydland and Prescott (1982) in interpreting this residual as reflecting shocks to technol- %Y* Households in economy �AO. (5) We now have a complete specification of the preferences, technology, and stochastic structure of a simple economy where individuals are able to supply any level of employment in the interval [O,l]. Each period three (6) c,+i,~ w,h,+r,k, and (3). Agents are assumed to make period t decisions based on all information available at time (which includes r, and w,). They have rational expectations in that their forecasts of future wages and rental rates are the same as those implied by the equilibrium laws of motion. The first-order conditions for the firm’s profit maximization problem imply that the wage k, and h,, r-0 subject to (l)-(4) and E, - c.d.f. F. (7) G. D. Hunsen. Indivisible bhor and the business qcle 315 The state of the economy in period t is described by k, and A,. The decision variables are h,, c,, and i,. This problem can be solved using dynamic programming techniques.’ This 03) where the maximization is over c and h and is subject to the same constraints as (7). The value function, V(k, A), is the maximum obtainable expected 3.2. An economy with indivisible labor The assumption of indivisible labor will now be to the above stochastic growth This will give rise to an economy where all variation in the labor input reflects adjustment along the extensive margin. This differs from the economy described above where all h,, or not at all9 “For a detailed presentation of dynamic programming methods, see Lucas, Prescott and Stokey (1984). ‘This is consistent with the interpretation given in section 2. An alternative interpretation of indivisible labor assumes that households can h, hours, and when both members work the household is working 11s hours. 316 G. D. Hunsetr. lodivisible lubor atrd the business cycle In order to guarantee [using Theorem 2 of Debreu (1954)] that the solution of the representative agent’s problem can be supported as a competitive equi- librium, it is necessary that the consumption possibilities set be convex. However, if one of the commodities traded is hours X[O,ll+R U(c,,ol,)=logc,+Aa,log(l-ho). (9) “In Rogerson’s paper, a static economy with indivisible labor is studied and lotteries are introduced to solve the problem introduced by this non-convexity. Readers may wish to consult Rogerson’s paper for a rigorous general equilibrium formulation of this type of model. “Adding lotteries to i E [0, l] is now ‘renamed’ according to the following rule: x,(i,z)=i+r, if i+z,Sl, = i + z, - 1 otherwise. The amount worked by agent x in period t is equal to h,(x) =o if x,(i.z)ll-a,, = ha if �s,(i,z) l-a,. This provides a mechanism for dividing the continuum of agents into two subsets, one where each individual works zero hours and another where ho. The first will have measure (1 - a,) and the other measure a,. This follows from the easily verified fact that Prob[s,( i, z) 5 1 - a,] is equal to 1 - a, for each i. “This uses the fact that, since preferences are separable in consumption and leisure, the consumption level chosen in equilibrium is independent of whether the individual works or not. G. D. Hansen, Indivisible labor and Ihe business qcle 317 Since a fraction OL, of households will work h, and the rest will work zero, per capita hours worked in period t k,, h,) = w,. However, due to the fact that lottery contracts are being traded, households are not paid for the time they actually spend working, c, + i, 4 wp,h, + r,k,. (11) Thus, the problem solved by a typical household is maxEfP’U(c,,a,), given k, and A,,, r-0 02) subject to The following is the representative agent’s problem that must be solved to derive the equilibrium decision rules and laws of motion: m maxEx/3’U(c,,(Y,), given k, and A,, r-o subject to (l)-(4), (10) and aggregate economy is infinite and independent of the willingness of individuals to substitute leisure across time.14 4. Solution method and calibration The problems (7) and (13) are not in i, and h, and state variables X, and k,. 14The fact that in this type of model the representative agent’s utility function is linear in leisure was originally shown by Rogerson (1984) for his model. This result depends, however, on the utility function being additively separable across time. G. D. Hansen. Indivisible labor and the bwiness cycle 319 objective functions are non-linear. For each of these problems, Kydland and Prescott’s procedure is used to construct a quadratic approximation of the objective function to be accurate in a neighborhood of the steady state for the appropriate model after the technology shock has been set equal to its unconditional mean of 0ne.l’ The reader may F, and specific parameter values for 19, 6, /3, A, y, and he. Kydland and Prescott (1982,1984) follow a methodology for choosing parame- ter values based on evidence from growth observations and micro studies. This methodology will also 15Let the steady states for the certainty version of these models be denoted by the variable’s symbol without any subscript. Eq. (3) implies that investment in the steady state is given by i = 6k. Expressions for k and h can be determined by deriving the Euler equations for the appropriate representative agent h, = h, k, = k, and i, = i= Sk for all f. For both economies, the steady state capital stock is given by k= [(p+S)/O]““-” h where p=(l/p)-1. Hours worked in the steady state for the economy with divisible labor is given by h = (1 - 6) X (p + S)/(3(p + 6) - f?(p + 36)]; and for the economy with indivisible labor, h = (1 - O)(p + S)/ h,)]/h,. l6 Kydland and Prescott’s method for approximating this problem requires choosing a vector of average deviations, z E R4, which determines the size of the neighborhood around the steady state within which the approximation is accurate. The four components of z are average deviations from trend of the four variables, x, = (X,, k,, i,, h,). as found in U.S. time series data. This implies that along those dimensions 'i/jzt I:‘- 1 = (0.012.0.006,0.08,0.017), reflecting the average standard deviations of these series as reported in the next section. Although attention was paid to specifying this vector in a reasonable way, it turns out that the results are not altered when the zi components are decreased by a factor of 320 G. D. Hansen, Indivisible labor and fire bminess cycle their time engaged in market activities and 2/3 of their time in non-market activities. To determine the parameter h,, I set the expressions for hours of work in the steady state for the two models equal to each F along with the parameter y determine the properties of the 5. Results For the purposes of this study, the statistical properties of the economies studied are summarized by a set of standard deviations and correlations with output that are reported in table 1. The statistics for the U.S. economy are reported in the first two columns of the table. Before these statistics “The production, function residual is measured, using U.S. time series, by logX,-logy,-elogk,-(1-e)logh,, where data on GNP, capital stock (nonresidential equipment and structures). G. D. Hansen, Indivisible labor and the business cycle 321 Table 1 Standard deviations in percent (a) and correlations with output (II) for U.S. and artificial economies. Quarterly U.S. rime series’ Economy with Economy with (55.3-84.1) divisible labo? indivisible laborh Scrics (a) (a trend. The ‘detrending’ procedure used is the method employed by Hodrick and Prescott (1980).‘* Since much of the discussion in section centers on the variability of hours worked and productivity (output divided by hours worked), some discussion of the hours series is appropriate. the Current Population Survey, which is a survey of households. This series was chosen in preference to the other available hours series which is derived from the establishment stiey. The hours series based on the household survey is more comprehensive than “This method involves choosing -_ min (l/r~~(r.-.,)‘+(h/r)~~~[(2*I-~,)-(+-4-~)12), I t-1 where X � 0 is the penalty on variation, where variation is measured by the average second difference. A larger value of X implies that the resulting (3,) series is smoother. FollowGig Prescott (1983). I choose d, - x, -s,. This method is used in order to filter out low frequency fluctuations. Although other methods (spectral techniques, for example) are available. this method was chosen because of its simplicity and the fact that other methods lead to basically the same results [see Prescott (1983)]. 322 G. D. Hanserl, Indivisible lubar and the business cycle the establishment series since self-employed workers and unpaid workers in family-operated enterprises are included. Another advantage is that the household series takes into account only hours actually worked rather than all hours paid for. That is, it doesn’t “The work referred to is a chapter of my dissertation. Copies will soon be available upon request. G.D. Hansen, Indivisible labor and the business cycle 323 Perhaps the most significant discovery made by examining table 1 is that the amount of variability in hours worked relative to variability in productivity is very different for the two model economies. This relative *‘This ratio is still not significantly different from one even when uC is increased to 0.00929. 324 G. D. Hmsen, Indioisible labor and the business qwzle in hours worked over the business cycle since most of the variability in total hours is unambiguously due to variation in the number employed rather than hours per employed worker. An important aspect of this economy is Appendix: A market for unemployment insurance The purpose of this appendix is to show that the equilibrium of the economy presented in section 3.2 is equivalent to the equilibrium of an economy where labor is G.D. Hansen, Indivisible labor arld the business cycle 325 paid according to the probability that it works, not according to the work it actually does. In other words, the firm is automatically providing full unem- ployment insurance to the households. i,, (s = 1,2). These are chosen to solve the following dynamic programming problem (primes denote next period values) : (A-1) subject to cl + i, I w(h, K)h,+ r(h, K)k (1 - S)k + i,, s= 1,2. (A.4 The function V(A, K, k) is the value function which depends on the house- hold’s state. The state vector includes the capital owned by the household, plus the economy wide state K, where K is the per capita capital stock.21 The functions w(h, K) and r(h, K) are the wage rate and rental rate “Since we are allowing households to choose any level of unemployment insurance they wish, we have to allow for the heterogeneity that may come K, and the households accumulated capital stock, k. However, this heterogeneity will disappear in equilibrium since all households will choose full insurance, so K = k in equilibrium. of capital respectively, and p(a) is the price of insurance, which is a function of the probability that the household works. Also, since individuals’ prefer- ences are the same as for the original model, u(c) = log c and v(I) = i,v and c,~ (s = 1,2), one can write the following first-order necessary conditions for k; and .y: 64.6) Eq. (A.6) implies, given the strict concavity of U, that c, = cz. This plus eq. (A.5) imply that k{ = k$. This, in turn, implies that i, = i,. Therefore, the left-hand sides of eqs. (A.2) and k. Substituting these results into the household’s optimization problem (A.l) yields the following problem: Households choose c, i, k’, and a! maxV(h,k)=u(c)+av(l-h,)+(l-ol)v(l)+pEV(X,k’), (A.7) subject to k’ = (1 - 6)k + i. This problem is identical to problem (12). Therefore, the equilibrium G. D. Honsen, Indivisible lubor and rite busiuess cycle 327 would not hold if cr depended some underlying choice variable like effort that was not directly observed by the insurance company. References Altonji, J.G.. 1984, Intertemporal substitution in labor supply: Evidence from micro data, Unpublished manuscript (Columbia University, New York). . Asbenfelter. 0.. 1984, Macroeconomic analyses and microeconomic analyses of labor supply, Carnegie-Rochester Conference Series