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Weekly employee hours, weeks worked and intertemporal substitution
JOURNALOF
Monetary ELSEVIER
Journal of Monetary Economics 41 (1998) 185-199
ECONOMICS
Weekly employee hours, weeks worked and intertemporal substitution J a n g - O k C h o a'b, P h i l i p M e r r i g a n b'c, L o u i s P h a n e u f *'bx aDepartment of Economics, Sogang University, Seoul, 121-742, South Korea b Center for Research on Employment and Economic Fluctuations, Montreal, H3C 3P8, Canada CDepartment of Economics, Universitb du Quebec ~ Montreal, Montreal, H3C 3P8. Canada
Received 12 July 1995; accepted 25 September 1997
Abstract
We show that the representative consumer model fits the aggregate consumption and employment data well if a choice of work is allowed both at the intensive and extensive margins. The structural preference parameters recovered from the estimation of the Euler equations of the model are economically meaningful and the null hypothesis of the overidentifying restrictions implied by our model is far from being rejected. We find that the shares in preferences associated with leisure time in the weeks off and in the workweeks are quite large and about equal. Our estimates also uncover relatively large intertemporal substitution elasticities. © 1998 Elsevier Science B.V. All rights reserved.
JEL classification: E24; E32; J22 Keywords. Weekly employee hours; Weeks worked; Representative consumer model
1. Introduction
In neoclassical theories of the labor market, aggregate employment fluctuations are modeled as movements along a labor supply curve. Therefore, to account for observed variations in employment, the elasticity of labor supply with respect to changes in the relative return from working currently and in the
* Corresponding author. Tel.: + 1 514 987 3015; fax: + 1 514 987 4707; e-mail: phaneuf.louis@ uqam.ca. 0304-3932/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 39 3 2 ( 9 7 ) 0 0 0 6 9 - X
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near future ought to be large. However, the search for an elastic labor supply has been met with mixed success. For example, Altonji (1982) reports real wage elasticities which are insignificant and wrongly signed, whereas Mankiw, Rotemberg and Summers (1985) (hereafter, MRS) find implausible parameters of the intertemporal utility function. In contrast, Alogoskoufis (1987) obtains sizable labor supply elasticities measuring aggregate labor supply by the total number of employees, adjusted for the total population. Also, work by Angrist (1991) and Moffitt (1993), which better control for measurement error, yield quite large labor supply elasticities in panel or pseudo-panel data. But to support optimizing models of employment fluctuations, one must simultaneously account for variations in aggregate consumption. Until now the representative consumer model with complete markets has generally failed to fit the aggregate consumption and labor data. MRS have shown that the utility function is often not concave, implying that the representative agent is not at a maximum of utility, but at a saddle-point or a minimum, and that the overidentifying restrictions implied by the representative consumer theory are generally rejected, despite experimenting with different functional forms of the utility function, definitions of the variables, and frequency of the data. Moreover, even when the estimated utility function is found to be concave as in Eichenbaum, Hansen and Singleton (1988) (hereafter, EHS), the evidence still refutes the orthogonality conditions implied by the representative consumer theory. Based on these findings, one can only conclude that the representative consumer model has not been borne out by empirical evidence and that it does not offer, as it stands, a satisfactory explanation of aggregate consumption and employment fluctuations. In this paper, we try to overcome these difficulties by allowing the representative consumer a choice of work both at the intensive and extensive margins. Indeed, one of the striking features of US post-war data is that most of the variation in total hours worked has taken the form of movements in and out of the labor force rather than adjustments in the weekly average hours worked. In spite of this, the participation and worked hours decisions have not been modeled as distinct decisions in former intertemporal substitution studies based on the estimation of Euler equations. Our paper is related to the work of Rogerson and Rupert (1991) who have assumed a two-dimensional labor supply decision. Using micro data from the Panel Survey of Income Dynamics and restricting their analysis to labor supply, they show that prime-aged married males who are not at a corner solution for weeks worked engage in substantially more intertemporal substitution than found by other researchers. Here, while we also emphasize a choice of work at the two margins, our main objective is to explain aggregate consumption and employment data and to assess the validity of the representative consumer model as a whole. Our work is closer to Cho and Cooley (1994) who have allowed decisions at the intensive and extensive margins in a real business cycle
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model. However, in assigning values to the parameters governing the relative importance of both margins in preferences, they rely on calibration rather than on the estimation of the Euler equations, t Using quarterly post-war aggregate data for the US economy and a generalized method of moments procedure (GMM), we estimate and test a model of a representative optimizing agent designed to capture variations in consumption, weekly hours of work per worker, and weeks of work. This allows us to recover the preference parameters of a representative agent whose labor supply is two-dimensional. Our framework is built upon the idea that the marginal utility of consumption depends on the level of consumption and two types of leisure: leisure time in the workweek and leisure time in the weeks off. Former studies have considered only one standard measure of leisure time defined as the difference between total hours available to the representative agent in a period and the per capita total hours worked by the civilian labor force in the same period. 2 Therefore, these models typically generate two Euler equations, one for consumption and one for standard leisure time, from which the estimated utility function parameters are obtained. In contrast, our model generates three Euler equations since we incorporate two types of leisure in the utility function. We briefly summarize our main findings. First, we find that the estimated parameters are always consistent with a concave utility function, thus lending support to the hypothesis that the representative agent is at a maximum of utility. Second and more importantly, we find that the null hypothesis of the overidentifying restrictions imposed by our theoretical model is not rejected in several cases we study. In fact, the evidence in favor of the two-margin model is overwhelming when the three Euler equations are estimated jointly. Third, we find that the shares in preferences of leisure time in the nonworking weeks and in the workweeks are relatively large and about equal. Fourth, we compute the intertemporal substitution elasticities implied by our estimates using McLaughlin's (1995) 2-constant elasticity and find the intertemporal substitution of labor supply to be sizable and in the range of the estimates obtained by Angrist (1991) and Moffitt (1993) with panel or pseudo-panel data, and of Alogoskoufis (1987) with the aggregate work effort measured by the total number of employees. The paper is organized as follows. In Section 2, we present the two-dimensional representative agent model of labor supply and derive the system of Euler equations. In the third section, we address some issues related to model estimation and data. Our results are presented and discussed in the fourth section. The fifth section contains concluding remarks.
See also Bils and C h o (1994) for a real business cycle model where choice of work at the two margins is allowed. 2 See, for example, MRS, EHS and Cooley and Ogaki (1996).
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2. The two-dimensional labor supply model Consider a collection of households who derive utility, u, from consumption, ct, hours of leisure in the workweeks, lit, and leisure time in the weeks offin the quarter, 12,. Each household maximizes the following lifetime expected utility: of)
U = Eo ~ fltu(ct, lit, lzt),
(I)
t=O
where fl is the discount factor with 0 < fl < 1, expectations are conditioned upon the information set of the household at time O, and u(.) is a utility function which is concave. All variables are defined in end-of-period terms. The representative agent's quarterly time endowment is expressed as the product of a fixed number of weeks per quarter, E, and a fixed amount of time available in a week, T, which is allocated to the two types of leisure, work, and commuting, i.e. (2)
E x T = lit + 12t + et(ht + ~),
where et denotes the working weeks in the quarter, ht stands for the weekly hours of work per worker, and ~ is a fixed time cost associated with commuting. The amount of leisure time in the workweek and in the weeks off are, respectively, given by lit = et(T -- h, - "c),
(3)
12t = (E - et)T.
(4)
and
The representative agent faces the lifetime budget constraint (5)
RtPtct <~ ~, RtW,etht + Ao, t=O
t=O
where Ao is the initial stock of assets, Pt is the aggregate price level of period t, Wt is the aggregate after-tax hourly nominal wage rate of period t, and Rt is the discounting factor, 3
1 R t = ( 1 +ro)(1 + r l ) . - . ( 1 + r t
1)'
where r is the after-tax rate of return.
3Note that we assumeno frictionsin the loan market.
(6)
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For the purpose of estimation, we specify the utility as 1 u(G, lit, 12,) = - - 1f t ~- a ' la~~2t1 1 -"lt'2t ~'
~2)1-a - -
(7)
1},
where el, ~2 and a are preference parameters. Concavity requires that a > 0, that el and o~2 be positive parameters, and 0 < el + 0~2 < 1. This specific form of the utility function has been used by EHS, among others. However, EHS consider only one type of leisure. This utility function has a few desirable characteristics. It is quite general in that it accommodates preferences that are separable across consumption and the two types of leisure as a special case. More specifically when a = 1, Eq. (7) becomes the logarithmic specification, u(ct, lit, 12t) = gllogct + :¢21ogI1, + (1
-
:~1
-
(7')
~2)log/2t.
The Euler equations implied by Eq. (7) do not exhibit any time trends. 4 Moreover, to account for co-movements of wages and consumption, the marginal utility of consumption must be a decreasing function of leisure time, that is, consumption and leisure must be direct substitutes. The assumed specification, Eq. (7), admits this possibility by allowing 1 - ~ r to be negative. If 1 - a is indeed negative, then the two types of leisure lit and Izt are also direct substitutes: the marginal utility of a week off is lower the shorter the workweek. 5 Each agent chooses consumption, the weekly hours of work, and the number of working weeks in the quarter. More specifically, the agent maximizes expected total discounted utility, Eq. (7), subject to the budget constraint, Eq. (5), the definitions of lit and lzt in Eq. (3) and Eq. (4), and the nonnegativity constraints. Agents are assumed to make period t decisions based upon all the information available at time t. Their expectations are rational. Setting up the Lagrangian, we derive the following first-order conditions: fl' ~ l ~....... /1 -~t,- ~t2}l - aCt- 1 = 2tRtPt, t Iltl2t
(8)
flt~""21t~t ~..,,1~211 "ltt2t . . . .
(9)
fit( 1 -- ~1
--
~2J].t~t
}1 -al~,l = 2 , R , W , ,
tlt'2t
-~2}
-~lft 1 = AtRtWt
1 -
,
(10)
where 2, is the multiplier attached to the lifetime budget constraint in period t.
4 E H S s main motivation for the specification is to remove the time trend in the Euler equations under reasonable a s s u m p t i o n s so that the G M M estimation procedure be appropriate. 5 In fact, the sign of 1 - a determines the signs of the cross-partials, so if any two of the three commodities are direct substitutes (complements) then the other pairs m u s t also be direct substitutes lcomplements).
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Since 2t follows a martingale (see MaCurdy, 1985; and MRS), we can derive from Eqs. (8)-(10) the following Euler equations: E(M, It2,) = 0,
(I 1)
where t2, is the information set available in period t, for Mt = (Mat, Mzt, M3t)'. The elements of Mt are defined as
Mlt= ~(Ct+l l C,-~,]"(llt+l~It a"~a2(12t+l /[ ~ T J'~ \ l-al-~2}
x(C,+,']-'Pt(1 +r,) \ c~ / P,+I (/1,+ 1"~ - 1Wt(1 -I-rt) x\
11, ]
1,
W,+I
(12t+ 1~- 1Wt(1 ~- rt) x \ 12, J
1,
1.
(12)
(13)
(14)
W, + 1
Eqs. (12)-(14) are the Euler equations for consumption, the average weekly hours of work, and the working weeks, respectively. This set of Euler equations will serve for the purpose of estimation.
3. Econometric issues and data
3.1. Estimation method We estimate the model with the Euler equations, Eqs. (12)-(14), using the generalized method of moments proposed by Hansen (1982), and Hansen and Singleton (1982). The method involves choosing instrumental variables that belong to the information set and invoking the orthogonality conditions embodied in the Euler equations. If we let Z, be a vector of instrumental variables, we have the following: E[M,®Z,] = 0.
(15)
If Z, is stationary, we can use these moment restrictions to estimate the parameters with nonlinear optimization methods. The sample moment
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191
corresponding to the expected value in Eq. (15) is N
g = (l/N) ~ [Mr®Z,].
(16)
t=l
If Eq. (15) holds and the number of parameters is equal to the number of equations in Eq. (16), then the G M M estimates are the values of the unknown parameters that simultaneously set each equation in Eq. (16) equal to zero. However, in most cases, the number of equations is greater than the number of parameters, and the parameter values are estimated by solving the following minimization problem: min S = ,q'Vg,
(17)
0
where 0 = (3, ~1, ~z, o), and V is a consistent estimate of the covariance matrix of g. Hansen (1982) has shown that the minimized value S multiplied by N (the number of observations), denoted by J, has a x2-distribution with N1N2 - N3 degrees of freedom, with N1, Nz, and N3 representing the number of instruments, the number of equations in M,, and the number of parameters, respectively. To estimate the parameters of the model, we must choose a set of instruments and a method to estimate the matrix V. We use the following instruments: 6
Xt = (RG, Rltt, Rl2t, RctAPt, RIttAWt, RlzfAWt),
(18)
where Ryt = YJYt 1, AYt = Yt-1(1 + rt-1)/Yt and Zt = (1,
Xt, X , - 1).
(19)
To construct a covariance matrix V which is consistent and robust to heteroskedasticity and autocorrelation, we follow the estimation procedure proposed by Andrews (1991). First, we compute the lag truncation parameter to be 3. We then use the Parzen kernel to compute V. The parameters of the utility function can be recovered after the Euler equations have been estimated separately, or jointly with sets of Euler equations restricting the parameter values to be the same in all equations.
3.2. Data We use US seasonally adjusted aggregate data in the estimations (complete data sources are listed in an appendix). The per capita real consumption series, ct, is obtained by dividing the aggregate real consumption of nondurables and
6 See EHS and D u t k o w s k y and Dunsky (1996) for a similar choice of instruments.
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services by the total adult (age sixteen and over) population. The aggregate price level, Pt, is the implicit price deflator associated with our measure of consumption. The asset return is the value-weighted average of returns on the New York Stock Exchange and the American Exchange as in Hansen and Singleton (1982), EHS and Cooley and Ogaki (1996). Monthly returns for the three months in each quarter are compounded. Nominal wages are measured as the average hourly compensation in nonagricultural employment. The after-tax asset return, rt, and the after-tax nominal wage rate, Wt, are obtained using the effective marginal tax rate on capital income and the effective marginal tax rate on labor income, respectively, constructed by McGrattan (1994) following the method of Joines (1981). The representative consumer is given a weekly time endowment, T, of l l 2 h o u r s and a quarterly week endowment, E, of 13 weeks. As in Alogoskoufis (1987), the extensive margin, et, is approximated by the product of E and the ratio of civilian employees to the working population. The weekly average hours worked, ht, is the series of hours worked from the household survey.
4. Results 4.1. E s t i m a t i o n r e s u l t s
We estimate the parameters of the utility function using quarterly data for the sample period 1958:1-1992:111. To ensure convergence, the value of r is fixed throughout the estimations. Therefore, the parameters to be estimated are fl, Ctl, ~2 and a. To verify if the estimations are sensitive to variations in the value of z, we estimate the Euler equations for z = 2 .... ,10. Since they are not, we report only the results for z = 6. They are presented in Table 1. First, the Euler equations for consumption, the average weekly hours of work, and the working weeks are estimated separately. The structural parameters of the utility function recovered from these estimations are reported in the first three columns of Table 1. All the structural parameters are statistically significant and economically meaningful with the exception of the estimated value of fl, which is slightly larger than unity. The latter finding is common to several empirical studies of intertemporal Euler equations. 7 According to our estimates, the concavity conditions are easily satisfied and 1 - a is found to be negative. However, for each separate Euler equation, the p-value indicates that the null hypothesis of the over-identifying restrictions implied by the theory is rejected or very close to being rejected by the data. The results are more favorable to the model as the Euler equations are estimated jointly. The parameters recovered from the estimation of the three 7See for example EHS, and Singleton (1988).
13.48 0.142
1.01 (126.2) 0.478 (2.2) 0.193 (1.1) 0.189 (4.1) -7.20 (1.9)
hr
13.48 0.142
1.01 (162.2) 0.478 (2.2) 0.332 (2.2) 0.190 (1.6) -7.20 (1.9)
e,
30.26 0.112
1.01 (146.1) 0.295 (3.2) 0.368 (5.3) 0.336 (8.4) - 13.65 (4.3)
c,, ht
30.14 0.115
1.01 (170.0) 0.322 (3.1) 0.348 (4.4) 0.329 (7.1) - 11.43 (3.6)
Estimates ~,b
G, et
30.19 0.092
1.01 (183.6) 0.372 (1.9) 0.278 (1.7) 0.349 (4.5) -6.46 (1.9)
h~,et
37.17 0.369
1.01 (296.6) 0.278 (3.2) 0.359 (5.4) 0.361 (9.4) - 12.38 (4.7)
c,, ht, e,
0.995
40.38 0.542
0.279 (2.9) 0.374 (7.3) 0.345 (8.2) -6.32 (3.7)
--
c,, hr, ef c
28.91 0.067
0.552 (6.2) -22.97 (2.5)
1.04 (139.7) 0.448 (5.0) 0.00
ct, htd
~Asymptotic t-statitics in parentheses bThe first three columns contain estimates from the Euler equations estimated separately. Columns four to six present estimates from the joint estimation of couples of Euler equations, whereas the seventh column displays the results of the joint estimation of the three Euler equations. One finds in the eight column the estimates from the joint estimation of the three Euler equations constraining/~ to be equal to 0.995. Finally, estimates from the one-margin model are presented in the last column. eWe added the current and lagged interest rate to the list of instruments in Z, (see Eq. (19)). dThe instrument list does not contain instruments involving the extensive margin.
J-statistic p-value
I -a
1 - ~l - ~2
~2
15.40 0.080
1.01 (139.4) 0.345 (2.9) 0.330 (3.6) 0.325 (6.4) - 11.55 (3.1)
/3
~1
c~
Parameter
Table 1 Separate and joint estimation of the Euler equations (1958:1-1992:111)
~.
~
~,~
-"
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pairs of Euler equations are found in columns four to six of Table 1, whereas those from the joint estimation of the three Euler equations appear in column seven. Again, all the individual structural parameters are statistically significant at conventional confidence levels, the estimated utility function is concave and 1 a is negative. With pairs of Euler equations, the p-values show that the null hypothesis of the overidentifying restrictions is rejected or close to being rejected at any conventional critical value. In contrast, when the three Euler equations are jointly estimated, the model is very far from being rejected at any conventional critical value as the p-value increases to 0.40. Therefore, as the Euler equations are jointly estimated, the nonrejection of the two-dimensional labor supply model becomes increasingly significant. There are three more noticeable features of the results displayed in Table 1. First, the share parameter, (1 - 51 -- 0~2) , associated with leisure in the weeks off, which is the extra margin in our model, is always very significant and sizeable. Second, the joint estimation of the three Euler equations reveals that the shares in preferences associated with both types of leisure are quite large and about equal. Conversely, the share of consumption in preferences decreases significantly when the three equations are estimated jointly. Third, our estimates are quite stable and show much less variation than those reported in former intertemporal substitution studies, as in MRS for example. Since our previous estimations indicate that fl is slightly above unity, we reestimate the Euler equations separately and jointly, constraining the value of fl to be equal to 0.995. The results are almost unaffected, so for the sake of brevity we do report only those from the joint estimation of the three Euler equations. They are shown in the eight column of Table 1. Finally, to assess more clearly the role of separate margins in our findings, we estimate a version of our model with only the standard measure of leisure time in the utility function. This model yields two Euler equations, instead of three: one for consumption and one for leisure time defined as the difference between the representative consumer's quarterly time endowment and the ratio of the quarterly total hours worked by the civilian labor force to the total adult (age sixteen and over) population. We report only the estimates of the structural parameters from the joint estimation of the Euler equations since the results from the separate estimation are even less favorable to the representative consumer model. These findings are presented in the last column of Table 1. Although we find that the estimated parameters are statistically significant and the utility function is concave, we note a significant deterioration in the specification of the model based on the computed p-value. In fact, the null hypothesis of the over-identifying restrictions implied by the theoretical model is rejected at the 90% level of confidence. This is not surprising since we have found that the structural parameters attached to leisure in the workweek and in the weeks off in the more general model are both statistically significant. Therefore, the specification test simply reveals that the one-margin model is
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195
misspecified. Overall, these findings confirm that the e x t r a m a r g i n is the m a i n feature d r i v i n g o u r results.
4.2. Intertemporal substitution elasticities T h e q u e s t i o n we next a d d r e s s is t h a t of the size of the i n t e r t e m p o r a l elasticities. W e follow M c L a u g h l i n (1995), w h o argues t h a t the s h o r t - r u n u n c o m p e n sated l a b o r elasticity is likely to e m b e d small wealth effects in an e n v i r o n m e n t like ours where the shocks are a s s u m e d to be t e m p o r a r y , the h o r i z o n is infinite a n d the d i s c o u n t i n g is small. H e suggests a p p r o x i m a t i n g the u n c o m p e n s a t e d elasticity by the 2 - c o n s t a n t elasticity, s W e c o m p u t e the 2 - c o n s t a n t elasticity of i n t e r t e m p o r a l s u b s t i t u t i o n with the p a r a m e t e r estimates o b t a i n e d from the j o i n t e s t i m a t i o n of the three Euler e q u a t i o n s (seventh c o l u m n in T a b l e 1). 9 First, we t a k e the log of b o t h sides of Eqs. (8)-(10) to have the following: t ln(fl) + ln(~l) + [cq(1 - a) - 1] In(G) + ~¢2(1 - - 0")ln(ll,) + (1 - c~1 -- ~2)(1 - a)ln(12t) = In(At) + ln(Rt) + ln(Pt),
(20)
t ln(/3) + ln(cq) + el(1 - a)ln(c,) + [e2(1 - a) - 1] ln(la,) + (1 - :q - e2)(1 - a)ln(12t) = ln(2,) + ln(Rt) + ln(W,),
(21)
tln(/3) + ln(~l) + :q(1 - or)In(c,) + ~2(1 - or)In(/1,) + [(1 - ~1 - ~2) x(1 -- a) - 1] ln(12,) = In():,) + ln(R,) + ln(Wt) + ln(1 - r/T).
(22)
Then, by solving this system, we derive the following ):-constant d e m a n d s for c o n s u m p t i o n a n d the two types of leisure: ln/,~=Cl+(~(--0~)l!lnW,+
(O0~l)llnP,+(O _ )
+ wl .n,) : ' ' ~ l (23)
lnl2t = C2 + ( +1,7, 0~ O "~ ) l n W t
( ~'n(O°q)' -1) ~
l
1 (24)
8 M aCurdy (1981) distinguishes three types of wage rate changes. The first captures changes along the given (or expected) lifetime wage profile, the second captures (unexpected) temporary changes while the third captures (unexpected) permanent changes. There is no income effect involved in the first type. However, wealth effects are involved in the second and third types of wage changes. McLaughlin argues that since the income effect in the second case is negligible, the total effects of wage changes in the first and in the second case are similar. 9 With Our preference specification, the compensated elasticity of intratemporal substitution of leisure is the same as the 2-constant elasticity of intertemporal substitution.
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196
lnC,=Csq-
0(1 - ~q)ln (0-1~ W,+
1 - 0(1 -
1
1
i~--~')lnP,+(--O-~_l)lnR,+(~_l)lnA,, (25)
where the Ci's are constants composed of parameters of the utility function and 0 = 1 - a. The 2-constant elasticities of intertemporal substitution with respect to output price and wage rate changes are given by the coefficients of the variables. Substituting the estimated parameters in these expressions we obtain the elasticities found in the first three columns of Table 2. The 2-constant elasticity of intertemporal substitution of consumption with respect to price changes is about - 0.81 and those of the two types of leisure are about 0.26. On the other hand, the elasticities with respect to wage rate changes are 0.73, - 0.33 and - 0.33, respectively. One notable fact is that the two types of leisure have the same elasticity of intertemporal substitution. This can be explained as follows. Although they may have different share parameters in preferences, both types of leisure are governed by the same intertemporal substitution parameter, namely a. In addition, from the first-order conditions, we observe that the ratio llt/12, is equal to a constant. In other words, the representative agent values the two types of leisure differently and hence the amount of the two types of leisure is different. However, the rate of changes due to the changes in the price level or the wage is the same across the two types of leisure. To obtain the elasticities of hours and weeks worked with respect to wages and prices, we totally differentiate llt and 12, and use the definition of elasticity. They are presented in columns four and five of Table 2. These elasticities are computed at the means of the explanatory variables. The results show that the weekly hours of work are about four times more sensitive to changes in price and wage than the working weeks. The elasticity of intertemporal substitution of the total hours with respect to the wage rate is about 1.22. We consider this estimate to be large compared to those obtained in the literature. It is much larger than the early estimates reported by MaCurdy (1981), Browning et al. (1985) or Ham (1986), but in the range of those most recently obtained by Alogoskoufis (1987), Angrist (1991) or Moffitt (1993). Table 2 Elasticities o f the c h o i c e v a r i a b l e s w i t h r e s p e c t to w a g e s a n d prices a
ct W,
Pt
0.731 -- 0.806
lit - 0.332 0.258
lzt - 0.332 0.258
ht 0.238 -- 0.185
et 0.978 - 0.758
et × ht 1.22
aThe elasticities a r e c o m p u t e d f o r the v a r i a b l e s in the first r o w w i t h r e s p e c t to t h o s e in the first column.
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197
5. Conclusions In recent years, the attempts that have been made to investigate whether the representative consumer model with complete markets can fit the aggregate consumption and labor data well have not been entirely successful. Therefore, one alternative has been to take into account heterogeneity, and to correct compensation series for compositional effects. Here, we have taken a different approach. We have shown that the representative consumer model is consistent with the data if a choice of work is allowed both at the intensive and extensive margins. More specifically, when the system of Euler equations is estimated jointly, the null hypothesis of the over-identifying restrictions imposed by our two-margin model is very far from being rejected at any conventional confidence level. Whether these findings mean that composition is just a bell or whistle is a question that remains to be studied more thoroughly in the future.
Acknowledgements We thank Mark Bils, Bob Amano, and an anonymous referee for their useful comments. We also thank Ali Dib and Marianne Sauthier for excellent research assistance. Financial support from FCAR is gratefully acknowledged. All errors are, of course, our own.
Appendix A. Data appendix The data used for the estimations are US seasonally adjusted aggregate data for the sample period 1958:I-1992:III. They are: (i) Pz: implicit price deflator of personal consumption expenditures of nondurable goods and services [Citibase variables (GCN + GCS)/(GCNQ + GCSQ)]. (ii) c,: real personal consumption expenditures of nondurable goods and services [Citibase variables (GCN + GCS)/P,] divided by the civilian noninstitutional population, 16 years and over (Citibase variable PO16; monthly data transformed into quarterly data). (iii) r,: after-tax asset return which is the product of the value-weighted average of returns on the New York Stock Exchange and the American Exchange (Center for Research in Security Prices) and one minus the effective marginal tax rate on capital income [constructed by McGrattan (1994)]. (iv) W,: after-tax hourly nominal wage rate which is the product of the average hourly compensation ( G C O M P (total compensation to the civilian labor force)/LHOURS (total hours worked by the civilian labor force)) and one minus the effective marginal tax rate on labor income [constructed by McGrattan (1994)].
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(v) ht (two-margin model): weekly average hours worked (Citibase variable LHCH; monthly data transformed into quarterly data). (vi) ht (one-margin model): ratio of total hours worked by the civilian labor force (Citibase variable LHOURS; monthly data transformed into quarterly data) to the total adult (age sixteen and over) population (Citibase variable PO16). (vii) et: weeks worked per quarter which is the product of thirteen and the ratio of civilian employment to total population of working age (Citibase variable LHEMPA; monthly data transformed into quarterly data).
References Alogoskoufis, G., 1987. On the intertemporal substitution and aggregate labor supply. Journal of Political Economy 95, 938-960. Altonji, J., 1982. The intertemporal substitution model of labour market fluctuations: an empirical analysis. Review of Economic Studies 69, 783-824. Andrews, D.W.K., 1991. Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817-858. Angrist, J.A., 1991. Grouped-data estimation and testing in simple labor supply models. Journal of Econometrics 47, 227 242. Bils, M., Cho, J., 1994. Cyclical factor utilization. Journal of Monetary Economics 33, 319-354. Browning, M., Deaton, A., Irish, M., 1985. A profitable approach to labour supply and commodity demands over the life cycle. Econometrica 53, 503-544. Cho, J., Cooley, T.F., 1994. Employment and hours over the business cycle. Journal of Economic Dynamics and Control 18, 411-432. Cooley, T., Ogaki, M., 1996. A time series analysis of real wages, consumption, and asset returns: a cointegration-Euler equation approach. Journal of Applied Econometrics 11, 119-134. Dutkowski, D.H., Dunsky, R.M., 1996. Intertemporal substitution, money and labor supply. Journal of Money, Credit and Banking 28, 216-232. Eichenbaum, M., Hansen, L.P., Singleton, K.J., 1988. A time-series analysis of representative agent models of consumption and leisure choice under uncertainty. Quarterly Journal of Economics 103, 51-78. Ham, J., 1986. Testing whether unemployment represents intertemporal labor supply. Review of Economic Studies 53, 579-588. Hansen, L.P., 1982. Sample properties of generalized method of moment estimators. Econometrica 50, 1029 1054. Hansen, L.P., Singleton, K.J., 1982. Generalized instrumental variables estimation of nonlinear rational expectations models. Econometrica 50, 1269 1286. Joines, D.H., 1981. Estimates of effective marginal tax rates on factor incomes. Journal of Business 54, 191-226. MaCurdy, T.E., 1981. An empirical model of labor supply in a life-cycle setting. Journal of Political Economy 89, 1059-1085. MaCurdy, T.E., 1985. Interpreting empirical models of labor supply in an intertemporal framework with uncertainty. In: Heckman, J.J., Singer, B. (Eds.), Longitudal Analysis of Labor Market Data, Econometric Society Monograph Series, Vol. 10. Cambridge University Press, Cambridge. Mankiw, N.G., Rotemberg, J., Summers, L.H., 1985. Intertemporal substitution in macroeconomics. Quarterly Journal of Economics 100, 225-251. McGrattan, E., 1994. The macroeconomic effects of distortionary taxation. Journal of Monetary Economics 33, 573-601.
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McLaughlin, K.J., 1995. Intertemporal substitution and )~-constant comparative statics. Journal of Monetary Economics 35, 193-213. Moffitt, R.A., 1993. Identification and estimation of dynamic models with a time series of repeated cross-sections. Journal of Econometrics 59, 99 123. Rogerson, R., Rupert, P., 1991. New estimates of intertemporal substitution. Journal of Monetary Economics 27, 255 269. Singleton, K.J., 1988. Econometric issues in the analysis of equilibrium business cycle models. Journal of Monetary Economics 21,361 386.
Monetary ELSEVIER
Journal of Monetary Economics 41 (1998) 185-199
ECONOMICS
Weekly employee hours, weeks worked and intertemporal substitution J a n g - O k C h o a'b, P h i l i p M e r r i g a n b'c, L o u i s P h a n e u f *'bx aDepartment of Economics, Sogang University, Seoul, 121-742, South Korea b Center for Research on Employment and Economic Fluctuations, Montreal, H3C 3P8, Canada CDepartment of Economics, Universitb du Quebec ~ Montreal, Montreal, H3C 3P8. Canada
Received 12 July 1995; accepted 25 September 1997
Abstract
We show that the representative consumer model fits the aggregate consumption and employment data well if a choice of work is allowed both at the intensive and extensive margins. The structural preference parameters recovered from the estimation of the Euler equations of the model are economically meaningful and the null hypothesis of the overidentifying restrictions implied by our model is far from being rejected. We find that the shares in preferences associated with leisure time in the weeks off and in the workweeks are quite large and about equal. Our estimates also uncover relatively large intertemporal substitution elasticities. © 1998 Elsevier Science B.V. All rights reserved.
JEL classification: E24; E32; J22 Keywords. Weekly employee hours; Weeks worked; Representative consumer model
1. Introduction
In neoclassical theories of the labor market, aggregate employment fluctuations are modeled as movements along a labor supply curve. Therefore, to account for observed variations in employment, the elasticity of labor supply with respect to changes in the relative return from working currently and in the
* Corresponding author. Tel.: + 1 514 987 3015; fax: + 1 514 987 4707; e-mail: phaneuf.louis@ uqam.ca. 0304-3932/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 39 3 2 ( 9 7 ) 0 0 0 6 9 - X
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near future ought to be large. However, the search for an elastic labor supply has been met with mixed success. For example, Altonji (1982) reports real wage elasticities which are insignificant and wrongly signed, whereas Mankiw, Rotemberg and Summers (1985) (hereafter, MRS) find implausible parameters of the intertemporal utility function. In contrast, Alogoskoufis (1987) obtains sizable labor supply elasticities measuring aggregate labor supply by the total number of employees, adjusted for the total population. Also, work by Angrist (1991) and Moffitt (1993), which better control for measurement error, yield quite large labor supply elasticities in panel or pseudo-panel data. But to support optimizing models of employment fluctuations, one must simultaneously account for variations in aggregate consumption. Until now the representative consumer model with complete markets has generally failed to fit the aggregate consumption and labor data. MRS have shown that the utility function is often not concave, implying that the representative agent is not at a maximum of utility, but at a saddle-point or a minimum, and that the overidentifying restrictions implied by the representative consumer theory are generally rejected, despite experimenting with different functional forms of the utility function, definitions of the variables, and frequency of the data. Moreover, even when the estimated utility function is found to be concave as in Eichenbaum, Hansen and Singleton (1988) (hereafter, EHS), the evidence still refutes the orthogonality conditions implied by the representative consumer theory. Based on these findings, one can only conclude that the representative consumer model has not been borne out by empirical evidence and that it does not offer, as it stands, a satisfactory explanation of aggregate consumption and employment fluctuations. In this paper, we try to overcome these difficulties by allowing the representative consumer a choice of work both at the intensive and extensive margins. Indeed, one of the striking features of US post-war data is that most of the variation in total hours worked has taken the form of movements in and out of the labor force rather than adjustments in the weekly average hours worked. In spite of this, the participation and worked hours decisions have not been modeled as distinct decisions in former intertemporal substitution studies based on the estimation of Euler equations. Our paper is related to the work of Rogerson and Rupert (1991) who have assumed a two-dimensional labor supply decision. Using micro data from the Panel Survey of Income Dynamics and restricting their analysis to labor supply, they show that prime-aged married males who are not at a corner solution for weeks worked engage in substantially more intertemporal substitution than found by other researchers. Here, while we also emphasize a choice of work at the two margins, our main objective is to explain aggregate consumption and employment data and to assess the validity of the representative consumer model as a whole. Our work is closer to Cho and Cooley (1994) who have allowed decisions at the intensive and extensive margins in a real business cycle
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model. However, in assigning values to the parameters governing the relative importance of both margins in preferences, they rely on calibration rather than on the estimation of the Euler equations, t Using quarterly post-war aggregate data for the US economy and a generalized method of moments procedure (GMM), we estimate and test a model of a representative optimizing agent designed to capture variations in consumption, weekly hours of work per worker, and weeks of work. This allows us to recover the preference parameters of a representative agent whose labor supply is two-dimensional. Our framework is built upon the idea that the marginal utility of consumption depends on the level of consumption and two types of leisure: leisure time in the workweek and leisure time in the weeks off. Former studies have considered only one standard measure of leisure time defined as the difference between total hours available to the representative agent in a period and the per capita total hours worked by the civilian labor force in the same period. 2 Therefore, these models typically generate two Euler equations, one for consumption and one for standard leisure time, from which the estimated utility function parameters are obtained. In contrast, our model generates three Euler equations since we incorporate two types of leisure in the utility function. We briefly summarize our main findings. First, we find that the estimated parameters are always consistent with a concave utility function, thus lending support to the hypothesis that the representative agent is at a maximum of utility. Second and more importantly, we find that the null hypothesis of the overidentifying restrictions imposed by our theoretical model is not rejected in several cases we study. In fact, the evidence in favor of the two-margin model is overwhelming when the three Euler equations are estimated jointly. Third, we find that the shares in preferences of leisure time in the nonworking weeks and in the workweeks are relatively large and about equal. Fourth, we compute the intertemporal substitution elasticities implied by our estimates using McLaughlin's (1995) 2-constant elasticity and find the intertemporal substitution of labor supply to be sizable and in the range of the estimates obtained by Angrist (1991) and Moffitt (1993) with panel or pseudo-panel data, and of Alogoskoufis (1987) with the aggregate work effort measured by the total number of employees. The paper is organized as follows. In Section 2, we present the two-dimensional representative agent model of labor supply and derive the system of Euler equations. In the third section, we address some issues related to model estimation and data. Our results are presented and discussed in the fourth section. The fifth section contains concluding remarks.
See also Bils and C h o (1994) for a real business cycle model where choice of work at the two margins is allowed. 2 See, for example, MRS, EHS and Cooley and Ogaki (1996).
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2. The two-dimensional labor supply model Consider a collection of households who derive utility, u, from consumption, ct, hours of leisure in the workweeks, lit, and leisure time in the weeks offin the quarter, 12,. Each household maximizes the following lifetime expected utility: of)
U = Eo ~ fltu(ct, lit, lzt),
(I)
t=O
where fl is the discount factor with 0 < fl < 1, expectations are conditioned upon the information set of the household at time O, and u(.) is a utility function which is concave. All variables are defined in end-of-period terms. The representative agent's quarterly time endowment is expressed as the product of a fixed number of weeks per quarter, E, and a fixed amount of time available in a week, T, which is allocated to the two types of leisure, work, and commuting, i.e. (2)
E x T = lit + 12t + et(ht + ~),
where et denotes the working weeks in the quarter, ht stands for the weekly hours of work per worker, and ~ is a fixed time cost associated with commuting. The amount of leisure time in the workweek and in the weeks off are, respectively, given by lit = et(T -- h, - "c),
(3)
12t = (E - et)T.
(4)
and
The representative agent faces the lifetime budget constraint (5)
RtPtct <~ ~, RtW,etht + Ao, t=O
t=O
where Ao is the initial stock of assets, Pt is the aggregate price level of period t, Wt is the aggregate after-tax hourly nominal wage rate of period t, and Rt is the discounting factor, 3
1 R t = ( 1 +ro)(1 + r l ) . - . ( 1 + r t
1)'
where r is the after-tax rate of return.
3Note that we assumeno frictionsin the loan market.
(6)
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For the purpose of estimation, we specify the utility as 1 u(G, lit, 12,) = - - 1f t ~- a ' la~~2t1 1 -"lt'2t ~'
~2)1-a - -
(7)
1},
where el, ~2 and a are preference parameters. Concavity requires that a > 0, that el and o~2 be positive parameters, and 0 < el + 0~2 < 1. This specific form of the utility function has been used by EHS, among others. However, EHS consider only one type of leisure. This utility function has a few desirable characteristics. It is quite general in that it accommodates preferences that are separable across consumption and the two types of leisure as a special case. More specifically when a = 1, Eq. (7) becomes the logarithmic specification, u(ct, lit, 12t) = gllogct + :¢21ogI1, + (1
-
:~1
-
(7')
~2)log/2t.
The Euler equations implied by Eq. (7) do not exhibit any time trends. 4 Moreover, to account for co-movements of wages and consumption, the marginal utility of consumption must be a decreasing function of leisure time, that is, consumption and leisure must be direct substitutes. The assumed specification, Eq. (7), admits this possibility by allowing 1 - ~ r to be negative. If 1 - a is indeed negative, then the two types of leisure lit and Izt are also direct substitutes: the marginal utility of a week off is lower the shorter the workweek. 5 Each agent chooses consumption, the weekly hours of work, and the number of working weeks in the quarter. More specifically, the agent maximizes expected total discounted utility, Eq. (7), subject to the budget constraint, Eq. (5), the definitions of lit and lzt in Eq. (3) and Eq. (4), and the nonnegativity constraints. Agents are assumed to make period t decisions based upon all the information available at time t. Their expectations are rational. Setting up the Lagrangian, we derive the following first-order conditions: fl' ~ l ~....... /1 -~t,- ~t2}l - aCt- 1 = 2tRtPt, t Iltl2t
(8)
flt~""21t~t ~..,,1~211 "ltt2t . . . .
(9)
fit( 1 -- ~1
--
~2J].t~t
}1 -al~,l = 2 , R , W , ,
tlt'2t
-~2}
-~lft 1 = AtRtWt
1 -
,
(10)
where 2, is the multiplier attached to the lifetime budget constraint in period t.
4 E H S s main motivation for the specification is to remove the time trend in the Euler equations under reasonable a s s u m p t i o n s so that the G M M estimation procedure be appropriate. 5 In fact, the sign of 1 - a determines the signs of the cross-partials, so if any two of the three commodities are direct substitutes (complements) then the other pairs m u s t also be direct substitutes lcomplements).
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Since 2t follows a martingale (see MaCurdy, 1985; and MRS), we can derive from Eqs. (8)-(10) the following Euler equations: E(M, It2,) = 0,
(I 1)
where t2, is the information set available in period t, for Mt = (Mat, Mzt, M3t)'. The elements of Mt are defined as
Mlt= ~(Ct+l l C,-~,]"(llt+l~It a"~a2(12t+l /[ ~ T J'~ \ l-al-~2}
x(C,+,']-'Pt(1 +r,) \ c~ / P,+I (/1,+ 1"~ - 1Wt(1 -I-rt) x\
11, ]
1,
W,+I
(12t+ 1~- 1Wt(1 ~- rt) x \ 12, J
1,
1.
(12)
(13)
(14)
W, + 1
Eqs. (12)-(14) are the Euler equations for consumption, the average weekly hours of work, and the working weeks, respectively. This set of Euler equations will serve for the purpose of estimation.
3. Econometric issues and data
3.1. Estimation method We estimate the model with the Euler equations, Eqs. (12)-(14), using the generalized method of moments proposed by Hansen (1982), and Hansen and Singleton (1982). The method involves choosing instrumental variables that belong to the information set and invoking the orthogonality conditions embodied in the Euler equations. If we let Z, be a vector of instrumental variables, we have the following: E[M,®Z,] = 0.
(15)
If Z, is stationary, we can use these moment restrictions to estimate the parameters with nonlinear optimization methods. The sample moment
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191
corresponding to the expected value in Eq. (15) is N
g = (l/N) ~ [Mr®Z,].
(16)
t=l
If Eq. (15) holds and the number of parameters is equal to the number of equations in Eq. (16), then the G M M estimates are the values of the unknown parameters that simultaneously set each equation in Eq. (16) equal to zero. However, in most cases, the number of equations is greater than the number of parameters, and the parameter values are estimated by solving the following minimization problem: min S = ,q'Vg,
(17)
0
where 0 = (3, ~1, ~z, o), and V is a consistent estimate of the covariance matrix of g. Hansen (1982) has shown that the minimized value S multiplied by N (the number of observations), denoted by J, has a x2-distribution with N1N2 - N3 degrees of freedom, with N1, Nz, and N3 representing the number of instruments, the number of equations in M,, and the number of parameters, respectively. To estimate the parameters of the model, we must choose a set of instruments and a method to estimate the matrix V. We use the following instruments: 6
Xt = (RG, Rltt, Rl2t, RctAPt, RIttAWt, RlzfAWt),
(18)
where Ryt = YJYt 1, AYt = Yt-1(1 + rt-1)/Yt and Zt = (1,
Xt, X , - 1).
(19)
To construct a covariance matrix V which is consistent and robust to heteroskedasticity and autocorrelation, we follow the estimation procedure proposed by Andrews (1991). First, we compute the lag truncation parameter to be 3. We then use the Parzen kernel to compute V. The parameters of the utility function can be recovered after the Euler equations have been estimated separately, or jointly with sets of Euler equations restricting the parameter values to be the same in all equations.
3.2. Data We use US seasonally adjusted aggregate data in the estimations (complete data sources are listed in an appendix). The per capita real consumption series, ct, is obtained by dividing the aggregate real consumption of nondurables and
6 See EHS and D u t k o w s k y and Dunsky (1996) for a similar choice of instruments.
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services by the total adult (age sixteen and over) population. The aggregate price level, Pt, is the implicit price deflator associated with our measure of consumption. The asset return is the value-weighted average of returns on the New York Stock Exchange and the American Exchange as in Hansen and Singleton (1982), EHS and Cooley and Ogaki (1996). Monthly returns for the three months in each quarter are compounded. Nominal wages are measured as the average hourly compensation in nonagricultural employment. The after-tax asset return, rt, and the after-tax nominal wage rate, Wt, are obtained using the effective marginal tax rate on capital income and the effective marginal tax rate on labor income, respectively, constructed by McGrattan (1994) following the method of Joines (1981). The representative consumer is given a weekly time endowment, T, of l l 2 h o u r s and a quarterly week endowment, E, of 13 weeks. As in Alogoskoufis (1987), the extensive margin, et, is approximated by the product of E and the ratio of civilian employees to the working population. The weekly average hours worked, ht, is the series of hours worked from the household survey.
4. Results 4.1. E s t i m a t i o n r e s u l t s
We estimate the parameters of the utility function using quarterly data for the sample period 1958:1-1992:111. To ensure convergence, the value of r is fixed throughout the estimations. Therefore, the parameters to be estimated are fl, Ctl, ~2 and a. To verify if the estimations are sensitive to variations in the value of z, we estimate the Euler equations for z = 2 .... ,10. Since they are not, we report only the results for z = 6. They are presented in Table 1. First, the Euler equations for consumption, the average weekly hours of work, and the working weeks are estimated separately. The structural parameters of the utility function recovered from these estimations are reported in the first three columns of Table 1. All the structural parameters are statistically significant and economically meaningful with the exception of the estimated value of fl, which is slightly larger than unity. The latter finding is common to several empirical studies of intertemporal Euler equations. 7 According to our estimates, the concavity conditions are easily satisfied and 1 - a is found to be negative. However, for each separate Euler equation, the p-value indicates that the null hypothesis of the over-identifying restrictions implied by the theory is rejected or very close to being rejected by the data. The results are more favorable to the model as the Euler equations are estimated jointly. The parameters recovered from the estimation of the three 7See for example EHS, and Singleton (1988).
13.48 0.142
1.01 (126.2) 0.478 (2.2) 0.193 (1.1) 0.189 (4.1) -7.20 (1.9)
hr
13.48 0.142
1.01 (162.2) 0.478 (2.2) 0.332 (2.2) 0.190 (1.6) -7.20 (1.9)
e,
30.26 0.112
1.01 (146.1) 0.295 (3.2) 0.368 (5.3) 0.336 (8.4) - 13.65 (4.3)
c,, ht
30.14 0.115
1.01 (170.0) 0.322 (3.1) 0.348 (4.4) 0.329 (7.1) - 11.43 (3.6)
Estimates ~,b
G, et
30.19 0.092
1.01 (183.6) 0.372 (1.9) 0.278 (1.7) 0.349 (4.5) -6.46 (1.9)
h~,et
37.17 0.369
1.01 (296.6) 0.278 (3.2) 0.359 (5.4) 0.361 (9.4) - 12.38 (4.7)
c,, ht, e,
0.995
40.38 0.542
0.279 (2.9) 0.374 (7.3) 0.345 (8.2) -6.32 (3.7)
--
c,, hr, ef c
28.91 0.067
0.552 (6.2) -22.97 (2.5)
1.04 (139.7) 0.448 (5.0) 0.00
ct, htd
~Asymptotic t-statitics in parentheses bThe first three columns contain estimates from the Euler equations estimated separately. Columns four to six present estimates from the joint estimation of couples of Euler equations, whereas the seventh column displays the results of the joint estimation of the three Euler equations. One finds in the eight column the estimates from the joint estimation of the three Euler equations constraining/~ to be equal to 0.995. Finally, estimates from the one-margin model are presented in the last column. eWe added the current and lagged interest rate to the list of instruments in Z, (see Eq. (19)). dThe instrument list does not contain instruments involving the extensive margin.
J-statistic p-value
I -a
1 - ~l - ~2
~2
15.40 0.080
1.01 (139.4) 0.345 (2.9) 0.330 (3.6) 0.325 (6.4) - 11.55 (3.1)
/3
~1
c~
Parameter
Table 1 Separate and joint estimation of the Euler equations (1958:1-1992:111)
~.
~
~,~
-"
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pairs of Euler equations are found in columns four to six of Table 1, whereas those from the joint estimation of the three Euler equations appear in column seven. Again, all the individual structural parameters are statistically significant at conventional confidence levels, the estimated utility function is concave and 1 a is negative. With pairs of Euler equations, the p-values show that the null hypothesis of the overidentifying restrictions is rejected or close to being rejected at any conventional critical value. In contrast, when the three Euler equations are jointly estimated, the model is very far from being rejected at any conventional critical value as the p-value increases to 0.40. Therefore, as the Euler equations are jointly estimated, the nonrejection of the two-dimensional labor supply model becomes increasingly significant. There are three more noticeable features of the results displayed in Table 1. First, the share parameter, (1 - 51 -- 0~2) , associated with leisure in the weeks off, which is the extra margin in our model, is always very significant and sizeable. Second, the joint estimation of the three Euler equations reveals that the shares in preferences associated with both types of leisure are quite large and about equal. Conversely, the share of consumption in preferences decreases significantly when the three equations are estimated jointly. Third, our estimates are quite stable and show much less variation than those reported in former intertemporal substitution studies, as in MRS for example. Since our previous estimations indicate that fl is slightly above unity, we reestimate the Euler equations separately and jointly, constraining the value of fl to be equal to 0.995. The results are almost unaffected, so for the sake of brevity we do report only those from the joint estimation of the three Euler equations. They are shown in the eight column of Table 1. Finally, to assess more clearly the role of separate margins in our findings, we estimate a version of our model with only the standard measure of leisure time in the utility function. This model yields two Euler equations, instead of three: one for consumption and one for leisure time defined as the difference between the representative consumer's quarterly time endowment and the ratio of the quarterly total hours worked by the civilian labor force to the total adult (age sixteen and over) population. We report only the estimates of the structural parameters from the joint estimation of the Euler equations since the results from the separate estimation are even less favorable to the representative consumer model. These findings are presented in the last column of Table 1. Although we find that the estimated parameters are statistically significant and the utility function is concave, we note a significant deterioration in the specification of the model based on the computed p-value. In fact, the null hypothesis of the over-identifying restrictions implied by the theoretical model is rejected at the 90% level of confidence. This is not surprising since we have found that the structural parameters attached to leisure in the workweek and in the weeks off in the more general model are both statistically significant. Therefore, the specification test simply reveals that the one-margin model is
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195
misspecified. Overall, these findings confirm that the e x t r a m a r g i n is the m a i n feature d r i v i n g o u r results.
4.2. Intertemporal substitution elasticities T h e q u e s t i o n we next a d d r e s s is t h a t of the size of the i n t e r t e m p o r a l elasticities. W e follow M c L a u g h l i n (1995), w h o argues t h a t the s h o r t - r u n u n c o m p e n sated l a b o r elasticity is likely to e m b e d small wealth effects in an e n v i r o n m e n t like ours where the shocks are a s s u m e d to be t e m p o r a r y , the h o r i z o n is infinite a n d the d i s c o u n t i n g is small. H e suggests a p p r o x i m a t i n g the u n c o m p e n s a t e d elasticity by the 2 - c o n s t a n t elasticity, s W e c o m p u t e the 2 - c o n s t a n t elasticity of i n t e r t e m p o r a l s u b s t i t u t i o n with the p a r a m e t e r estimates o b t a i n e d from the j o i n t e s t i m a t i o n of the three Euler e q u a t i o n s (seventh c o l u m n in T a b l e 1). 9 First, we t a k e the log of b o t h sides of Eqs. (8)-(10) to have the following: t ln(fl) + ln(~l) + [cq(1 - a) - 1] In(G) + ~¢2(1 - - 0")ln(ll,) + (1 - c~1 -- ~2)(1 - a)ln(12t) = In(At) + ln(Rt) + ln(Pt),
(20)
t ln(/3) + ln(cq) + el(1 - a)ln(c,) + [e2(1 - a) - 1] ln(la,) + (1 - :q - e2)(1 - a)ln(12t) = ln(2,) + ln(Rt) + ln(W,),
(21)
tln(/3) + ln(~l) + :q(1 - or)In(c,) + ~2(1 - or)In(/1,) + [(1 - ~1 - ~2) x(1 -- a) - 1] ln(12,) = In():,) + ln(R,) + ln(Wt) + ln(1 - r/T).
(22)
Then, by solving this system, we derive the following ):-constant d e m a n d s for c o n s u m p t i o n a n d the two types of leisure: ln/,~=Cl+(~(--0~)l!lnW,+
(O0~l)llnP,+(O _ )
+ wl .n,) : ' ' ~ l (23)
lnl2t = C2 + ( +1,7, 0~ O "~ ) l n W t
( ~'n(O°q)' -1) ~
l
1 (24)
8 M aCurdy (1981) distinguishes three types of wage rate changes. The first captures changes along the given (or expected) lifetime wage profile, the second captures (unexpected) temporary changes while the third captures (unexpected) permanent changes. There is no income effect involved in the first type. However, wealth effects are involved in the second and third types of wage changes. McLaughlin argues that since the income effect in the second case is negligible, the total effects of wage changes in the first and in the second case are similar. 9 With Our preference specification, the compensated elasticity of intratemporal substitution of leisure is the same as the 2-constant elasticity of intertemporal substitution.
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lnC,=Csq-
0(1 - ~q)ln (0-1~ W,+
1 - 0(1 -
1
1
i~--~')lnP,+(--O-~_l)lnR,+(~_l)lnA,, (25)
where the Ci's are constants composed of parameters of the utility function and 0 = 1 - a. The 2-constant elasticities of intertemporal substitution with respect to output price and wage rate changes are given by the coefficients of the variables. Substituting the estimated parameters in these expressions we obtain the elasticities found in the first three columns of Table 2. The 2-constant elasticity of intertemporal substitution of consumption with respect to price changes is about - 0.81 and those of the two types of leisure are about 0.26. On the other hand, the elasticities with respect to wage rate changes are 0.73, - 0.33 and - 0.33, respectively. One notable fact is that the two types of leisure have the same elasticity of intertemporal substitution. This can be explained as follows. Although they may have different share parameters in preferences, both types of leisure are governed by the same intertemporal substitution parameter, namely a. In addition, from the first-order conditions, we observe that the ratio llt/12, is equal to a constant. In other words, the representative agent values the two types of leisure differently and hence the amount of the two types of leisure is different. However, the rate of changes due to the changes in the price level or the wage is the same across the two types of leisure. To obtain the elasticities of hours and weeks worked with respect to wages and prices, we totally differentiate llt and 12, and use the definition of elasticity. They are presented in columns four and five of Table 2. These elasticities are computed at the means of the explanatory variables. The results show that the weekly hours of work are about four times more sensitive to changes in price and wage than the working weeks. The elasticity of intertemporal substitution of the total hours with respect to the wage rate is about 1.22. We consider this estimate to be large compared to those obtained in the literature. It is much larger than the early estimates reported by MaCurdy (1981), Browning et al. (1985) or Ham (1986), but in the range of those most recently obtained by Alogoskoufis (1987), Angrist (1991) or Moffitt (1993). Table 2 Elasticities o f the c h o i c e v a r i a b l e s w i t h r e s p e c t to w a g e s a n d prices a
ct W,
Pt
0.731 -- 0.806
lit - 0.332 0.258
lzt - 0.332 0.258
ht 0.238 -- 0.185
et 0.978 - 0.758
et × ht 1.22
aThe elasticities a r e c o m p u t e d f o r the v a r i a b l e s in the first r o w w i t h r e s p e c t to t h o s e in the first column.
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5. Conclusions In recent years, the attempts that have been made to investigate whether the representative consumer model with complete markets can fit the aggregate consumption and labor data well have not been entirely successful. Therefore, one alternative has been to take into account heterogeneity, and to correct compensation series for compositional effects. Here, we have taken a different approach. We have shown that the representative consumer model is consistent with the data if a choice of work is allowed both at the intensive and extensive margins. More specifically, when the system of Euler equations is estimated jointly, the null hypothesis of the over-identifying restrictions imposed by our two-margin model is very far from being rejected at any conventional confidence level. Whether these findings mean that composition is just a bell or whistle is a question that remains to be studied more thoroughly in the future.
Acknowledgements We thank Mark Bils, Bob Amano, and an anonymous referee for their useful comments. We also thank Ali Dib and Marianne Sauthier for excellent research assistance. Financial support from FCAR is gratefully acknowledged. All errors are, of course, our own.
Appendix A. Data appendix The data used for the estimations are US seasonally adjusted aggregate data for the sample period 1958:I-1992:III. They are: (i) Pz: implicit price deflator of personal consumption expenditures of nondurable goods and services [Citibase variables (GCN + GCS)/(GCNQ + GCSQ)]. (ii) c,: real personal consumption expenditures of nondurable goods and services [Citibase variables (GCN + GCS)/P,] divided by the civilian noninstitutional population, 16 years and over (Citibase variable PO16; monthly data transformed into quarterly data). (iii) r,: after-tax asset return which is the product of the value-weighted average of returns on the New York Stock Exchange and the American Exchange (Center for Research in Security Prices) and one minus the effective marginal tax rate on capital income [constructed by McGrattan (1994)]. (iv) W,: after-tax hourly nominal wage rate which is the product of the average hourly compensation ( G C O M P (total compensation to the civilian labor force)/LHOURS (total hours worked by the civilian labor force)) and one minus the effective marginal tax rate on labor income [constructed by McGrattan (1994)].
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(v) ht (two-margin model): weekly average hours worked (Citibase variable LHCH; monthly data transformed into quarterly data). (vi) ht (one-margin model): ratio of total hours worked by the civilian labor force (Citibase variable LHOURS; monthly data transformed into quarterly data) to the total adult (age sixteen and over) population (Citibase variable PO16). (vii) et: weeks worked per quarter which is the product of thirteen and the ratio of civilian employment to total population of working age (Citibase variable LHEMPA; monthly data transformed into quarterly data).
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