# Intertemporal substitution and constraints on labor supply: evidence from panel data.

INTERTEMPORAL SUBSTITUTION AND CONSTRAINTS ON LABOR SUPPLY: EVIDENCE
FROM PANEL DATA

I. INTRODUCTION

Through much of the 1970s and 1980s, macroeconomic stabilization analysis has been dominated by equilibrium models in which fluctuations in employment result from "intertemporal substitution": workers voluntarily vary their labor supply over time in response to movements in wages and interest rates. A number of recent empirical tests of intertemporal substitution models (for example, Altonji [1982]; Mankiw et al. [1985]; Ham [1986]) strongly reject the intertemporal substitution approach. These results have led some economists to reject equilibrium macro models in favor of Keynesian models in which workers face quantity constraints on labor supply. In Keynesian models, employment fluctuations result from changes in the severity of constraints - that is, changes in involuntary unemployment - rather than intertemporal substitution.

On the other hand, many economists remain unconvinced by recent rejections of intertemporal substitution models because the results are consistent with many possible explanations. For tractability, empirical studies always impose restrictive assumptions, such as separability of utility over time and between goods, or such as the existence of a "representative consumer." Data also suffer from numerous problems, such as aggregation bias, measurement error, and the use of average rather than marginal wage rates. Consequently, even the authors of the studies seem hesitant to draw firm conclusions. They often simply conclude that workers' behavior is explained either by quantity constraints or by a more complicated intertemporal substitution model than the one they have tested.(1)

This paper attempts to determine which of these explanations is correct. Are changes in employment a sophisticated form of intertemporal substitution, or are they involuntary? The crucial data come from questions in the Panel Study of Income Dynamics about whether workers experience unemployment or are unable to work as many hours as they want. Following Ham [1982; 1986], this information is used to classify workers as quantity constrained in their labor supply or unconstrained. I then test a simple intertemporal substitution model for each group of workers. If the tests suffer from overly restrictive utility functions or data problems, then the model should be rejected for both samples. On the other hand, if some workers truly face quantity constraints, then the model should perform better for the sample of unconstrained workers.(2)

The tests yield two main results. First, the model indeed performs worse for the constrained workers. For this group, the data reject the model's overidentifying restrictions, and the parameter estimates contradict the model. In contrast, the results for the unconstrained workers are weakly consistent with the model. As Keynesian theories suggest, quantity constraints alter workers' behavior.

Second, the estimates of the intertemporal labor supply elasticity (for prime-age men) are close to zero, even for unconstrained workers. Even if workers are free to vary their hours, they apparently choose to respond little to movements in wages.

The paper addresses two other issues. First, do constraints on borrowing influence labor supply? In the intertemporal substitution model, workers borrow and lend to smooth consumption while varying hours of work. As this suggests, the model performs best for workers who not only face no labor supply constraints, but who also own substantial assets and thus are unlikely to face liquidity constraints.

Finally, the paper proposes a new approach to identifying intertemporal labor supply equations. Previous micro studies do not adequately address the problem that wages are correlated with unobservable characteristics that affect tastes for work. I propose two identifying restrictions: (1) changes in tastes are uncorrelated with levels of time-in-variant variables; and (2) changes in tastes have no aggregate component. This choice of identifying restrictions strongly influences the empirical results, and diagnostic tests support my approach.

Section II of the paper presents a simple intertemporal labor supply model. Section III discusses identification, and section IV describes the data and how workers are classified as constrained or unconstrained. Sections V and VI present the results concerning labor supply constraints and liquidity constraints respectively. Section VII summarizes the findings.

II. THE MODEL

This section presents a simple intertemporal labor supply model similar to the ones in MaCurdy [1981] and Altonji [1986]. A worker chooses labor supply and consumption to maximize expected lifetime utility subject to a budget constraint. I assume time-separable utility, so the worker's problem is

[MATHEMATICAL EXPRESSION OMITTED]

subject to

[MATHEMATICAL EXPRESSION OMITTED]

where E is the expectation operator at time tau, C is consumption, L is labor supply, [theta] is a taste shock, Upsilon is one-period utility, [delta] is the rate of time preference, w is the wage, A is the stock of assets, R is the gross interest rate, and [MATHEMATICAL EXPRESSION OMITTED] is the discount factor for year t [MATHEMATICAL EXPRESSION OMITTED]. All variables are real.

This problem yields two well-known first-order conditions: [MATHEMATICAL EXPRESSION OMITTED] (static first-order condition) and [MATHEMATICAL EXPRESSION OMITTED] (Euler equation)

The static condition equates the marginal rate of substitution (MRS) between consumption and leisure to the wage. Ignoring the error [u.sub.tau+1], the Euler equation equates the MRS between leisure at t and tau+1, adjusted for time preference, to the relative price of leisure in the two periods. The term [u.sub.t+1] is an expectational error; it appears because [omega.sub.t+1] is unknown when the worker chooses [L.sub.tau]. Under rational expectations, [u.sub.tau+1] is uncorrelated with any variable known at tau.

Assume the functional form for worker i's utility in year t is

[MATHEMATICAL EXPRESSION OMITTED]

where [eta] and [epsilon] are taste shocks and K is a scale factor. Concavity in consumption and leisure implies [alpha] > 0 and [beta] < 0. With this utility function, the two first-order conditions become

[MATHEMATICAL EXPRESSION OMITTED]

+ (1/[Beta])([epsilon].sub.[iota][tau] - [eta].sub.[iota][tau]), (static) and

[MATHEMATICAL EXPRESSION OMITTED]

+ (1/[Beta])([DELTA][epsilon].sub.[iota][tau]+1 - [u.sub.[iota][tau]+1)

(Euler) where [DELTA] ln[L.sub.[iota][tau]+1] is (ln[L.sub.[iota][tau]+1] -ln[L.sub.[iota][tau]).(3) In the empirical work, the static equation is estimated in first differences (this makes identification easier):

[MATHEMATICAL EXPRESSION OMITTED]

+ (1/[Beta])([DELTA][epsilon].sub.[iota][tau] +1 - [DELTA][pi].sub.[iota][tau] +1). (static)

The static and Euler equations have intuitive interpretations. Since [alpha] > 0 and [beta] < 0, the static equation states that hours of work move in the same direction as the wage and, for a given wage, in the opposite direction from consumption. The second prediction arises because consumption and leisure are both normal goods, and thus respond in the same direction to news about lifetime prospects. The static equation also includes changes in tastes for consumption and leisure. The Euler equation states that hours rise when the wage rises or when the interest rate is low (a low interest rate reduces the price of working at t+1 rather than [tau]). The Euler equation is also affected by changes in tastes for leisure and by new information, [u.sub.t+1]. In both equations, the coefficient on [DELTA][omega] is -1/[beta]. This coefficient is the intertemporal labor supply elasticity: the elasticity of current hours with respect to the current wage holding constant wages in other periods.(4)

The model is empirically tested in two ways. First, I test the model's predictions concerning the signs of coefficients, focusing on the negative coefficient on [delta c] in the static equation. For a given wage, a Keynesian consumption function implies a positive relation between hours, which determine income, and consumption (Hall [1984]). Mankiw et al. and Barro and King [1984] show that aggregate data support the Keynesian prediction. This paper's sample split will show whether labor supply constraints cause this failure of equilibrium models.

Second, I test the prediction that the only sources of changes in hours besides the observable variables in the equations are the taste changes and new information in the errors. If, contrary to the model, workers face quantity constraints, then changes in these constraints also affect hours. I ask whether variables that are uncorrelated with tastes and new information but correlated with constraints help to explain the equation residuals. Formally, this is a test of the overidentifying restrictions described below.

III. IDENTIFICATION

Identification of labor supply equations is difficult because wages, the crucial explanatory variable, is correlated with tastes for work, which influence the error term.[5] The reason is simply that the determinants of wages - whether observables like education and experience or unobservables like ability and motivation - influence tastes as well. This identification problem appears insoluble for equations explaining the level of labor supply, including traditional static labor supply models and the undifferenced static condition of intertemporal models. Researchers who estimate these equations generally add "taste shifter" variables such as age and number of children, but this is inadequate because unobservables strongly influence both wages and tastes. Furthermore, good instruments for wages do not exist because, again, any variable that effects wages (age, education, family background, health...) is probably correlated with tastes.[6]

Identification of equations for changes in hours, including this paper's Euler and static-in-differences equations, is less hopeless because differencing eliminates time-invariant components of tastes. Differencing does not, however, solve the identification problem by itself. Changes in wages are correlated with the changes in tastes in the errors, because they have many common sources, such as health, job changes, and - perhaps most important - the life cycle. Mincer [1974] shows that [DELTA][omega] falls with age: age-earnings profiles are concave. It is likely that age also systematically affects changes in tastes (although the direction of the relationship is less obvious).

In light of these problems, I propose a new approach to identifying labor supply equations. The variables age and age-squared and a constant (change in age) are added to the Euler and static-in-differences equations to remove life-cycle effects from the errors.[7] I then make the following identifying assumptions.

(1) Levels of time-invariant individual characteristics are uncorrelated with changes in tastes. With this assumption, variables such as education and family background are valid instruments for labor supply equations in differences. The assumption means that highly educated people (for example) may like work more or less than others, but that education does not systematically alter the way that tastes change over time. One can imagine reasons that the assumption fails - for example, people with certain education levels suffer job burnout more quickly than others. But such problems seem much less important than the correlation between education and levels of tastes. The data below do not reject the assumption.

Variables like education and background are correlated with [delta omega], and thus are useful instruments, because they affect the slopes of age-earnings profiles. Greater education, for example, causes wages to grow more quickly (Mincer [1974]).

Previous studies of intertemporal labor supply (MaCurdy [1981]; Altonji [1986]) use instruments such as education and background for the Euler equation. But they also use age, which is invalid by my criteria because it varies over time. As shown below, this difference matters for empirical results.

(2) Taste shocks have no aggregate component. This assumption implies that all aggregate variables are uncorrelated with changes in tastes. Any aggregate variable is a linear combination of year dummies, and so the efficient way to use aggregate information for estimation is to use year dummies as instruments.

Again, this assumption might fail, but the failures do not appear important. Aggregate tastes may contain long-term trends, but aggregate taste shifts are not important for the year-to-year changes in hours studied here. Aggregate tastes for work may have changed between the counter-culture 60s and the current age of yuppies. But surely shocks such as the sharp decline in hours in 1982 and the recovery in 1983 had little to do with taste changes.

(3) Any variable known at t is uncorrelated with the expectational error [u.sub.[tau]+1. This assumption follows from rational expectations.

These three assumptions determine the proper instruments for the static-in-differences and Euler equations. The static errors are changes in tastes, and so both year dummies and time-invariant micro variables are valid. The Euler errors include both expectational errors and taste changes, and the former are probably correlated with year dummies: while I rule out aggregate taste shocks, aggregate information shocks, such as news that a recession has begun, are allowed. I generally add year dummies to the Euler equation to capture aggregate news and use only micro variables as excluded instruments.

In section V the (over) identifying restrictions described here are tested. The tests of assumption (2) are particularly important because they provide evidence on the basic model of unconstrained optimization. I test (2) by adding aggregate variables, such as the change in the unemployment rate, to the labor supply equations. Given (2), the model implies that these variables have zero coefficients. The alternative hypothesis of quantity constraints suggests that aggregate variables matter, because fluctuations in constraints have an aggregate component. (Constraints are tightest, and so hours of work are lowest, during a recession.) Quantity constraints could cause the data to reject assumption (2).

IV. DATA

Data are drawn from the first fourteen waves (1968-1981) of the Michigan Panel Study of Income Dynamics. The sample is restricted to men ages 25-59 to avoid the econometric complications that arise when many individuals work zero hours.(8)

I use the standard PSID measures of hours, wages, and consumption. Labor supply is annual hours of work and the wage is annual labor earnings divided by hours. Since the wage is an average, it differs from the marginal wage in the model's first-order condition if a worker holds a second job or receives an overtime premium. Consumption is measured by food consumption; one can justify this rigorously only if utility is separable in food consumption and other consumption (see Zeldes [1989] for details).(9) The data problems increase the value of comparing results for constrained and unconstrained workers. One could blame negative results for the entire sample on poor data, but differences between the two groups cannot be dismissed as easily.

The consumer price index and PSID data on marginal tax rates are used to construct real after-tax wages.(10) Food consumption is deflated by the CPI for food. The interest rate at t is the annual average of the after-tax Treasury bill rate at t deflated by the change in the CPI between [tau] and [tau]+1. Finally, the micro instruments for the labor supply equations are dummies for education level and economic status of parents, interactions among these variables, and dummies for race, region, residence in an SMSA, union status, and industry.(11)

Following Ham [1982; 1986], I classify a worker as constrained in a given year if he either experiences a spell of unemployment or cannot work as many hours as he wants. He meets the latter criterion if he reports that no more hours were available on his job and that he wanted to work more.(12) A worker is assigned to the constrained group if he is constrained in any year.(13)

This procedure for determining which workers face quantity constraints is imperfect. Some workers may say they are constrained because they cannot work at overtime wages even though they work as much as they want at their actual wages (Ham [1982]). Similarly, if workers receive fixed salaries, the wage that they consider when saying whether they want more work might be unrelated to their current earnings. Given these problems, the empirical work is partly a test of the data. If the model performs best for the unconstrained sample, this suggests that the PSID questions are reliable guides to who is constrained.

In a given year, most workers are unconstrained, but most are constrained in at least one year: the sample contains 9290 annual observations on the constrained group and 3975 on the unconstrained group. Table I presents sample statistics for the two groups. Not surprisingly, the constrained workers receive lower wages and are concentrated among disadvantaged segments of the labor market: they are younger, less educated and more likely to be minorities and from the South.

Table : TABLE I Sample Means (Standard Deviations in Parentheses)

Dummy Variables:

a. Base Group [is equal to] High School Graduate

b. Base Group [is equal to] Parents Average

c. Base Group [is equal to] North Central

V. RESULTS: HOURS CONSTRAINTS

Part A of this section presents estimates of the Euler and static equations for all workers and for the constrained and unconstrained sub-samples. Part B tests the model for each sample by adding aggregate instruments to the equations. Part C addresses econometric issues, focusing on alternative identifying restrictions.

A. The Basic Estimates

Table II presents two-stage least squares estimates of the Euler and static-in-differences equations. The endogenous variables are [DELTA][epsilon], [DELTA]c, and (after-tax) R. As described in section III, age and age-squared are included as taste shifters; the excluded instruments are time-invariant micro variables for the Euler equation and micro variables plus year dummies for the static equation. In the Euler equation, the restriction is imposed that the coefficients on [DELTA][epsilon] and R are equal in absolute value.(14)

Table : TABLE II Estimates of Euler and Static-in-Differences Equations

Standard errors are in parentheses.

a. Endogenous variables. Instruments are time-invariant micro variables.

b. Endogenous variables. Instruments are time-invariant micro variables and year dummies.

The results have two important features. First, the coefficients on [DELTA] w are close to zero and very insignificant statistically for both equations and all samples. Recall that these coefficients measure the intertemporal labor supply elasticity (again, the elasticity with respect to the current wage holding constant wages in other periods). The data contain no evidence of intertemporal substitution by anyone.

Second, the static equation results imply that the model performs better for unconstrained workers than for constrained workers. The [DELTA]c coefficient for all workers is .216 ([tau] = 3.3). This positive coefficient contradicts the model and, as noted above, is consistent with a Keynesian consumption function. When the sample is split, the [DELTA]c coefficient for the constrained group is .265 ([tau] = 3.2). The coefficient for the unconstrained group, while positive, is only .069, and it is very insignificant ([tau] = .8). For unconstrained workers, one cannot reject the model's prediction of a negative coefficient.

How should we interpret these results? We can clearly reject the model for the constrained group. The data for the unconstrained group provide no evidence in favor of the model, since the key variables in both equations have insignificant coefficients. On the other hand, the results for unconstrained workers are at least consistent with the model. The model predicts a near-zero coefficient on [DELTA][epsilon] if the intertemporal labor supply elasticity is small. In other words, workers may be unconstrained but nonetheless choose to respond little to changes in wages. The model predicts a near-zero coefficient on [DELTA]c if labor supply is inelastic and workers are not too risk averse in consumption. To see this, recall that the labor supply elasticity and the [DELTA]c coefficient are -1/[beta] and [alpha]/[beta] respectively. The parameter [alpha] is the coefficient of relative risk aversion. If the labor supply elasticity is .05 and [alpha] is one (the common assumption of log utility), then the [DELTA]c coefficient is -.05. This value lies within the 90 percent confidence interval implied by the point estimate of .07 and standard error of .08 for the unconstrained sample.

B. Tests of the Model

This section tests the overidentifying restriction that the taste shocks in the errors have no aggregate component. As explained above, aggregate variables are added to the equations and the hypothesis tested that their coefficients are zero. The data are likely to reject this restriction if workers face quantity constraints, because fluctuations in constraints have an aggregate component.

For the static equation, two versions of the test are performed. First, I add the full set of year dummies to the equation (while continuing to use micro variables as excluded instruments). The results are inconclusive. For the constrained group, the hypothesis that the dummies have zero coefficients is rejected at the 90 percent but not the 95 percent level ([X.sup.2.sub.10] = 17). For the unconstrained group, the hypothesis is not quite rejected at the 90 percent level ([X.sup.2.sub.10] = 15).

Second, I add the change in the aggregate unemployment rate for men ages 20 and older. This yields a more powerful test against the alternative hypothesis of quantity constraints if the unemployment rate is a good aggregate measure of constraints. The results, reported in Table III, are clear-cut. The coefficient on the change in unemployment is -.008 ([tau] = 3.3) for the constrained group and -.001 ([tau] = .4 for the unconstrained group. Thus the behavior of constrained workers violates the model's restrictions while that of unconstrained workers does not. The coefficient of -.008 means that a one-percentage point rise in unemployment reduces the average hours of constrained workers by eight-tenths of a percent.

In the Euler equation, the error term includes the expectational error [u.sub.[tau] +1, which has an aggregate component. Thus the model does not imply that the errors are uncorrelated with all aggregate variables. But since expectational errors are unforecastable (the third identifying restriction), the Euler errors should be uncorrelated with any aggregate variable known at time t. Adding the unemployment rate at time t to the Euler equation produces t-statistics of 3.7 for the constrained sample and .7 for the unconstrained sample; again, only the results for constrained workers contradict the model.

To summarize, the test results, like the parameter estimates in part A, are consistent with the model for the unconstrained sample and inconsistent for the constrained sample. To understand the rejection for the constrained group, note that the model suggests a negative simple correlation between changes in an individual's hours and changes in aggregate unemployment. But this correlation works through the wage: if wages are generally low, individuals choose fewer hours and (since some choose zero hours) measured unemployment rises. The model is rejected because aggregate unemployment helps to explain the part of hours' movements that are not explained by wages, and that the model thus ascribes to taste shifts. Of course the results are consistent with the basic model if, contrary to the identifying restrictions, changes in

Table : TABLE III

Standard errors are in parentheses.

a. Endogenous variables. Instruments are time-invariant micro variables.

b. Endogenous variables. Instruments are time-invariant micro variables and year dummies.

aggregate unemployment capture aggregate taste shifts. It seems unlikely, however, that such taste shifts would cause rejection for one sample of workers but not the other. In contrast, if the rejection for the constrained sample is indeed caused by quantity constraints, it makes sense that the results are better for the unconstrained sample.

C. Alternative Identifying Restrictions and Diagnostics

Here I consider econometric issues, focusing once more on identification. As described in section III, previous micro studies of intertemporal substitution use age and age-squared as excluded instruments, while the micro instruments in this study are time-invariant. Table IV shows that this difference matters for the results. Adding age and age-squared to the excluded instruments for the Euler equation produces significantly positive estimates of the intertemporal labor supply elasticity. The point estimates of .4 to .6 are at the high end of the range of previous estimates. Recall that the elasticity estimates are essentially zero when age and age-squared are treated as taste shifters.[15]

Why is the treatment of age important? The coefficients on age and age-squared in Table II imply that [Delta] L falls with age for most ages in the sample - that is, hours are a concave function of age. As noted above, wages are also a concave function of age. Since both [Delta] L and [Delta] w fall with age, they are positively correlated. This correlation produces positive elasticity estimates if one does not control for age in the labor supply equation.[16]

When age and age-squared are added to the Euler equation, the hypothesis that they have zero coefficients, and thus are valid instruments, is rejected at the 99 percent level [Mathematical Expression Omitted]. In contrast, the validity of the time-invariant instruments - education, family background, and so on - cannot be rejected. I test the overidentifying restrictions on the basic Euler equation by regressing the equation residuals on the instruments and the exogenous variables in the equation. The [Mathematical Expression Omitted] statistic (R.sup.2 times the sample size) is a tiny 7.9. Thus the data support this study's approach to identification.

Finally, I consider the potential econometric problem of serial correlation in the errors for a given individual. Taste changes need not be white noise. In addition, white noise measurement error in the level of hours creates negative serial correlation in the errors of the differenced equations. To gauge the importance of these problems, I compute the correlations between errors for the same individual at various lags. For the unconstrained sample, the correlation between a worker's Euler equation errors at t and t-1 is -.3. The correlations at all lags greater than one are very close to zero. These results suggest substantial measurement error in the level of hours. In any case, since the serial correlation dies out quickly, it has little effect on the results. This is confirmed by experimenting with Altonji's [1986] procedure for calculating standard errors robust to serial correlation. As in Altonji's work, the correction reduces the standard errors, but only slightly; the OLS results are reliable.

Table : TABLE IV

Standard errors are in parentheses.

a. Endogenous variables. Instruments are time-invariant micro variables, age, and [age.sup.2].

VI. LIQUIDITY CONSTRAINTS

In intertemporal labor supply models, workers borrow and lend to smooth consumption as they vary hours of work. Thus liquidity constraints, as well as constraints on hours, can cause violations of the models. This section investigates the effects of liquidity constraints by splitting the sample of workers who are unconstrained in hours into liquidity constrained and unconstrained subsamples. Following Zeldes [1989], I place a worker in the liquidity constrained group if his asset income is zero or negative in any year. Zeldes presents evidence from consumption data that individuals without assets are liquidity constrained while those with assets are not.

Liquidity constraints imply a violation of the model's Euler equation, which concerns the tradeoff between leisure in different periods. Intuitively, if a worker cannot borrow then he does not reduce his hours when his wage falls, because his consumption would fall considerably. The static first-order condition is robust to liquidity constraints, because it states that workers optimally balance leisure and consumption within a period, which does not require borrowing. Thus the hypothesis that asset income measures liquidity constraints has rich implications: the Euler equation should hold only for the high-asset group, while the static equation should hold for both groups.

Table V reports estimates of the Euler and static equations for the two subsamples. The estimates support the predicted effects of liquidity constraints. In the Euler equation, the [Delta] w coefficient for the high-asset group is .042 (t [ equal to] .3), which is consistent with unconstrained optimization and a small labor supply elasticity. The corresponding coefficient for the low-asset group is -.453 (t [ equal to ] 2.5); this unambiguously contradicts the model's prediction of a positive coefficient. Finally, as predicted by liquidity constraints, the difference between the subsamples disappears for the static equation: there, the [Delta] w coefficient is close to zero for both groups.

VII. CONCLUSIONS

This paper has three findings. First, an intertemporal labor supply model performs better for workers who say they do not face constraints on hours than for workers who say they do. This suggests that the first group really does face quantity constraints, and that these constraints contribute to the failures of intertemporal substitution models in previous studies. Given the empirical evidence of quantity constraints, macroeconomists should focus more attention on Keynesian models that include these constraints.

Second, the intertemporal substitution model performs best for workers with significant assets, who are unlikely to face liquidity constraints. Future labor supply models should incorporate these constraints as well.[17]

Finally, the paper presents new evidence on the intertemporal labor supply elasticities of workers who do not face hours or liquidity constraints. Elasticity estimates for prime-age men are close to zero and statistically insignificant. Even when workers can choose their hours, intertemporal substitution appears close to non-existent. Along with the other results, this suggests that employment fluctuations result primarily from changes in involuntary unemployment.

The estimates of labor supply elasticities differ from estimates in previous studies, which are often positive and significant. Differences in identifying restrictions explain the different results, and tests of over-identifying restrictions support this paper's approach. Similar approaches might be useful for other studies in which error terms include tastes, such as panel studies of consumption.

Table : TABLE V

Standard errors are in parentheses.

a. Endogenous variables. Instruments are time-invariant micro variables.

b. Endogenous variables. Instruments are time-invariant micro variables and year dummies.

( 1.) Mankiw et al. conclude that "the abundance of plausible explanations for the results we obtained... leads us to be somewhat skeptical of the power of aggregate time series data in distinguishing alternative macroeconomic hypotheses." And Ham (who uses micro data) cautions that his result "does not indicate which model of the labour market is appropriate.... [O]ne could turn to models which allow for the possibility that unemployed workers are off their labour supply functions. Alternatively, one could turn to more complex models of interporal substitution. Unfortunately, the estimation and testing of these alternative or more complex models is likely to be extremely difficult."

( 2.) Ham [1986] tests an interporal labor supply model by adding PSID measures of quantity constraints to the estimated first-order conditions. The most important departure of his study from Ham is that the model is tested separately for constrained and unconstrained workers. As argued below, the crucial evidence that constraints are the true reason for rejection of the model is the relatively good results for the unconstrained sample. The sample split also allows one to estimate the interporal labor supply elasticity for unconstrained workers. Other departures of this study from Ham and other earlier papers, such as the approach to identification and the attention to liquidity constraints, described below. For another recent study that tests an interporal labor supply model for constrained and unconstrained workers, see Biddle [1988].

( 3.) The Euler equation uses the approximations ln(1 + [Delta]) [is equal to] and ln(1 + [upsilon ]) [is equal to] [upsilon].

( 4.) More precisely, the coefficient is the elasticity of hours with respect to the current wage holding constant the marginal utility of wealth. This definition differs slightly from the one in the text, because changing the current wage while holding other wages constant has a slight effect on the marginal utility of wealth.

( 5.) There are two other identification problems. First, consumption and interest rates are endogenous. (After-tax interest rates are endogenous because labor supply affects tax rates.) And second, there is severe measurement error in micro data on wages, hours, and consumption. Under plausible assumptions, the instrumental variables estimators proposed below are robust to these problems as well as to the problem stressed in the text.

( 6.) Economist who estimate one-period labor supply models often ignore the identification problem (for example, Hausman, [1980; 1981]). In estimating the undifferenced static condition of interporal models, researchers generally use instrumental variables, but they choose invalid instruments like education and background (for example, MaCurdy [1983]). For other discussions of the identification problem, see Mankiw et al. [1985], Ham [1982], and Altonji [1986].

( 7.) These taste shifters can be added to the formal model by assuming that [Mathematical Expression Omitted] where X is the vector of taste shifters and [Mathematical Expressi on Omitted] is the unobservable part of tastes (with a similar equation for [eta]). The identifying assumptions about tastesneed hold only for [Mathematical Expression Omitted]. In the Euler equation, the constant term captures the constan t in the basic model (which depends on the rate of time preference) as well as life-cycle changes in tastes.

( 8.) There is no consumption data for Waves 1 and 6. Therefore, for the static-in-differences equation the change from Wave 1 to Wave 2 is deleted and the Wave 5 to Wave 7 change replaces the two one-year changes. (The two-year change is weighted by 1/[square root of 2], which is efficient in the absence of serial correlation.) Observations are deleted if a worker is out of the labor force (rather than employed or unemployed), receives food stamps (which makes the food consumption data difficult to interpret), or is not the head of his household. I also delete observations with missing data and extreme outliers: observations in which the wage falls by 60 percent or rises by 150 percent, or in which hours of work or consumption falls by 80 percent or rises by 400 percent (these criteria follow Altonji [1986]). Finally, I use only the random part of the PSID sample.

( 9.) Another problem is the timing of the food consumption question. PSID questions on hours and wages refer to the previous calendar year, but the consumption question is simply "how much do you spend on food in an average week?" (The question is asked separately for restaurants and food at home.) It is unclear what time period is relevant to an "average week," and it is likely that responses are affected by changes in consumption between the end of the previous year and the interview date (late March, on average). We can interpret the difference between the available data and data for the previous year (which would be consistent with the hours and wage data) as measurement error. Unfortunately, this error is probably correlated with the year dummies used as instruments, because aggregate shocks affect changes in consumption between different periods. To determine the importance of this problem, I compare the basic static equation estimates to estimates obtained when only micro variables are used as instruments. The estimates are similar, and so there is no evidence that the problem is serious. On the other hand, the standard errors rise when only micro variables are used, and this makes the results less conclusive. (With the restricted set of instruments, the crucial [Delta]c coefficient is .35 (t = 1.5) and .15 (t = .8) for the constrained and unconstrained groups.)

(10.) I take into account Federal income taxes and Social Security taxes. The PSID reports marginal income tax rates for Waves 9-14. For earlier waves, marginal rates were constructed from Federal tax tables and PSID data on income and total taxes.

(11.) Some of these variables, such as region and industry, change occasionally for a given individual, and therefore do not quite meet the requirement of time-invariance (identifying restriction (1)). It seems unlikely that changes are frequent enough to cause serious problems. Tests of overidentifying restrictions provide no evidence that the variables are invalid instruments.

(12.) The questions are "Was there more work available on your job [or "any of your jobs" if more than one] so that you could have worked more if you had wanted to?"; and "Would you have liked to work more if you could have found more work?"

(13.) If labor supply equations are estimated in levels, splitting workers into constrained and unconstrained groups creates sample selection bias. In particular, it is likely that the expectation of the error conditional on being unconstrained is negative, because workers who desire few hours are unlikely to face binding constraints. Ham [1982] uses a sophisticated econometric procedure to overcome this problem. Under plausible assumptions, the problem does not arise here, because the equations are differenced. Knowing that a worker is in the unconstrained group - that is, that he does not face a constraint in any year - suggests that his tastes for work are generally negative. But there is no reason that the expectation of his change in tastes, which appears in the error, is non-zero. That is, knowing that a worker is unconstrained from 1970 to 1975 reveals nothing about whether he liked work better in 1973 than in 1972.

(14.) This restriction is an implication of the model. However, relaxing this restriction or omitting the interest rate from the equation has little effect on the results.

(15.) Altonji [1986] sometimes adds age to the Euler equation and still obtains positive elasticity estimates. Even in these regressions, however, he uses age * schooling and age * schooling[2] as excluded instruments. He also uses year dummies, which are invalid instruments for the Euler equation by my criteria.

(16.) Browning et al. [1985] discover similar relationships among hours, wages, and age in synthetic cohort data, and they point out the resulting biases in elasticity estimates.

(17.) See Dau-Schmidt [1988] for some work along these lines.

REFERENCES

Altonji, Joseph G. "The Intertemporal Substitution Model of Labor Market Fluctuations:

An Empirical Analysis." The Review of Economic Studies, Supplement 1982, 783-824.

___."Intertemporal Substitution in Labor Supply: Evidence from Micro Data." Journal

of Political Economy, June 1986, S176-S215.

Barro, Robert J. and Robert G. King. "Time-Separable Preferences and Intertemporal

Substitution Models of Business Cycles." Quarterly Journal of Economics, November

1984, 817-39.

Biddle, Jeff E. "Intertemporal Substitution and Hours Restrictions." Review of Economics

and Statistics, May 1988, 347-51.

Browning, Martin, Angus Deaton, and Margaret Irish. "A Profitable Approach to Labor

Supply and Commodity Demands over the Life-Cycle." Econometrica, May 1985,

503-43.

Dau-Schmidt, Kenneth G. "An Empirical Study of the Effect of Consumption Commitments

and the Liquidity Constraint on Labor Supply." Working Paper, University of

Cincinnati, February 1988.

Hall, Robert E. "The Role of Consumption in Economic Fluctuations." Working Paper,

Stanford University, 1984.

Ham, John C. "Estimation of a Labor Supply Model with Censoring Due to Unemployment

and Underemployment." The Review of Economic Studies, July 1982, 335-54.

___."Testing Whether Unemployment Represents Intertemporal Labor Supply

Behavior." The Review of Economic Studies, August 1986, 559-78.

Hausman, Jerry A. "The Effects of Wages, Taxes, and Fixed Costs on Women's Labor Force

Participation." Journal of Public Economics, October 1980, 161-94.

___."Labor Supply," in How Taxes Affect Economic Behavior, edited by Henry Aaron and

Joseph Pechman. Washington, D.C.: Brookings Institution, 1981.

MaCurdy, Thomas. "An Empirical Model of Labor Supply in a Life Cycle Setting." Journal

of Political Economy, December 1981, 1059-86.

___."A Simple Scheme for Estimating an Intertemporal Model of Labor Supply and

Consumption in the Presence of Taxes and Uncertainty." International Economic Review,

June 1983, 265-90.

Mankiw, N. Gregory, Julio Rotemberg, and Lawrence H. Summers. "Intertemporal

Substitution in Macroeconomics." Quarterly Journal of Economics, February 1985,

225-51.

Mincer, Jacob. Schooling, Experience, and Earnings. Chicago: University of Chicago Press for

the National Bureau of Economic Research, 1974.

Zeldes, Stephen. "Consumption and Liquidity Constraints: An Empirical Investigation."

Journal of Political Economy, April 1989, 305-46.

LAURENCE BALL(*)

(*) Assistant Professor of Economics, Princeton University. This is a revised version of Chapter I of my M.I.T. dissertation (January 1986). I am grateful for many suggestions from Oliver Blanchard, Henry Farber, Lawrence Katz, N. Gregory Mankiw, James Poterba, David Romer, Robert Solow, Lawrence Summers, Frank Wykoff, Stephen Zeldes, the referees, and participants in workshops at M.I.T. and the 1985 N.B.E.R. Summer Institute.

I. INTRODUCTION

Through much of the 1970s and 1980s, macroeconomic stabilization analysis has been dominated by equilibrium models in which fluctuations in employment result from "intertemporal substitution": workers voluntarily vary their labor supply over time in response to movements in wages and interest rates. A number of recent empirical tests of intertemporal substitution models (for example, Altonji [1982]; Mankiw et al. [1985]; Ham [1986]) strongly reject the intertemporal substitution approach. These results have led some economists to reject equilibrium macro models in favor of Keynesian models in which workers face quantity constraints on labor supply. In Keynesian models, employment fluctuations result from changes in the severity of constraints - that is, changes in involuntary unemployment - rather than intertemporal substitution.

On the other hand, many economists remain unconvinced by recent rejections of intertemporal substitution models because the results are consistent with many possible explanations. For tractability, empirical studies always impose restrictive assumptions, such as separability of utility over time and between goods, or such as the existence of a "representative consumer." Data also suffer from numerous problems, such as aggregation bias, measurement error, and the use of average rather than marginal wage rates. Consequently, even the authors of the studies seem hesitant to draw firm conclusions. They often simply conclude that workers' behavior is explained either by quantity constraints or by a more complicated intertemporal substitution model than the one they have tested.(1)

This paper attempts to determine which of these explanations is correct. Are changes in employment a sophisticated form of intertemporal substitution, or are they involuntary? The crucial data come from questions in the Panel Study of Income Dynamics about whether workers experience unemployment or are unable to work as many hours as they want. Following Ham [1982; 1986], this information is used to classify workers as quantity constrained in their labor supply or unconstrained. I then test a simple intertemporal substitution model for each group of workers. If the tests suffer from overly restrictive utility functions or data problems, then the model should be rejected for both samples. On the other hand, if some workers truly face quantity constraints, then the model should perform better for the sample of unconstrained workers.(2)

The tests yield two main results. First, the model indeed performs worse for the constrained workers. For this group, the data reject the model's overidentifying restrictions, and the parameter estimates contradict the model. In contrast, the results for the unconstrained workers are weakly consistent with the model. As Keynesian theories suggest, quantity constraints alter workers' behavior.

Second, the estimates of the intertemporal labor supply elasticity (for prime-age men) are close to zero, even for unconstrained workers. Even if workers are free to vary their hours, they apparently choose to respond little to movements in wages.

The paper addresses two other issues. First, do constraints on borrowing influence labor supply? In the intertemporal substitution model, workers borrow and lend to smooth consumption while varying hours of work. As this suggests, the model performs best for workers who not only face no labor supply constraints, but who also own substantial assets and thus are unlikely to face liquidity constraints.

Finally, the paper proposes a new approach to identifying intertemporal labor supply equations. Previous micro studies do not adequately address the problem that wages are correlated with unobservable characteristics that affect tastes for work. I propose two identifying restrictions: (1) changes in tastes are uncorrelated with levels of time-in-variant variables; and (2) changes in tastes have no aggregate component. This choice of identifying restrictions strongly influences the empirical results, and diagnostic tests support my approach.

Section II of the paper presents a simple intertemporal labor supply model. Section III discusses identification, and section IV describes the data and how workers are classified as constrained or unconstrained. Sections V and VI present the results concerning labor supply constraints and liquidity constraints respectively. Section VII summarizes the findings.

II. THE MODEL

This section presents a simple intertemporal labor supply model similar to the ones in MaCurdy [1981] and Altonji [1986]. A worker chooses labor supply and consumption to maximize expected lifetime utility subject to a budget constraint. I assume time-separable utility, so the worker's problem is

[MATHEMATICAL EXPRESSION OMITTED]

subject to

[MATHEMATICAL EXPRESSION OMITTED]

where E is the expectation operator at time tau, C is consumption, L is labor supply, [theta] is a taste shock, Upsilon is one-period utility, [delta] is the rate of time preference, w is the wage, A is the stock of assets, R is the gross interest rate, and [MATHEMATICAL EXPRESSION OMITTED] is the discount factor for year t [MATHEMATICAL EXPRESSION OMITTED]. All variables are real.

This problem yields two well-known first-order conditions: [MATHEMATICAL EXPRESSION OMITTED] (static first-order condition) and [MATHEMATICAL EXPRESSION OMITTED] (Euler equation)

The static condition equates the marginal rate of substitution (MRS) between consumption and leisure to the wage. Ignoring the error [u.sub.tau+1], the Euler equation equates the MRS between leisure at t and tau+1, adjusted for time preference, to the relative price of leisure in the two periods. The term [u.sub.t+1] is an expectational error; it appears because [omega.sub.t+1] is unknown when the worker chooses [L.sub.tau]. Under rational expectations, [u.sub.tau+1] is uncorrelated with any variable known at tau.

Assume the functional form for worker i's utility in year t is

[MATHEMATICAL EXPRESSION OMITTED]

where [eta] and [epsilon] are taste shocks and K is a scale factor. Concavity in consumption and leisure implies [alpha] > 0 and [beta] < 0. With this utility function, the two first-order conditions become

[MATHEMATICAL EXPRESSION OMITTED]

+ (1/[Beta])([epsilon].sub.[iota][tau] - [eta].sub.[iota][tau]), (static) and

[MATHEMATICAL EXPRESSION OMITTED]

+ (1/[Beta])([DELTA][epsilon].sub.[iota][tau]+1 - [u.sub.[iota][tau]+1)

(Euler) where [DELTA] ln[L.sub.[iota][tau]+1] is (ln[L.sub.[iota][tau]+1] -ln[L.sub.[iota][tau]).(3) In the empirical work, the static equation is estimated in first differences (this makes identification easier):

[MATHEMATICAL EXPRESSION OMITTED]

+ (1/[Beta])([DELTA][epsilon].sub.[iota][tau] +1 - [DELTA][pi].sub.[iota][tau] +1). (static)

The static and Euler equations have intuitive interpretations. Since [alpha] > 0 and [beta] < 0, the static equation states that hours of work move in the same direction as the wage and, for a given wage, in the opposite direction from consumption. The second prediction arises because consumption and leisure are both normal goods, and thus respond in the same direction to news about lifetime prospects. The static equation also includes changes in tastes for consumption and leisure. The Euler equation states that hours rise when the wage rises or when the interest rate is low (a low interest rate reduces the price of working at t+1 rather than [tau]). The Euler equation is also affected by changes in tastes for leisure and by new information, [u.sub.t+1]. In both equations, the coefficient on [DELTA][omega] is -1/[beta]. This coefficient is the intertemporal labor supply elasticity: the elasticity of current hours with respect to the current wage holding constant wages in other periods.(4)

The model is empirically tested in two ways. First, I test the model's predictions concerning the signs of coefficients, focusing on the negative coefficient on [delta c] in the static equation. For a given wage, a Keynesian consumption function implies a positive relation between hours, which determine income, and consumption (Hall [1984]). Mankiw et al. and Barro and King [1984] show that aggregate data support the Keynesian prediction. This paper's sample split will show whether labor supply constraints cause this failure of equilibrium models.

Second, I test the prediction that the only sources of changes in hours besides the observable variables in the equations are the taste changes and new information in the errors. If, contrary to the model, workers face quantity constraints, then changes in these constraints also affect hours. I ask whether variables that are uncorrelated with tastes and new information but correlated with constraints help to explain the equation residuals. Formally, this is a test of the overidentifying restrictions described below.

III. IDENTIFICATION

Identification of labor supply equations is difficult because wages, the crucial explanatory variable, is correlated with tastes for work, which influence the error term.[5] The reason is simply that the determinants of wages - whether observables like education and experience or unobservables like ability and motivation - influence tastes as well. This identification problem appears insoluble for equations explaining the level of labor supply, including traditional static labor supply models and the undifferenced static condition of intertemporal models. Researchers who estimate these equations generally add "taste shifter" variables such as age and number of children, but this is inadequate because unobservables strongly influence both wages and tastes. Furthermore, good instruments for wages do not exist because, again, any variable that effects wages (age, education, family background, health...) is probably correlated with tastes.[6]

Identification of equations for changes in hours, including this paper's Euler and static-in-differences equations, is less hopeless because differencing eliminates time-invariant components of tastes. Differencing does not, however, solve the identification problem by itself. Changes in wages are correlated with the changes in tastes in the errors, because they have many common sources, such as health, job changes, and - perhaps most important - the life cycle. Mincer [1974] shows that [DELTA][omega] falls with age: age-earnings profiles are concave. It is likely that age also systematically affects changes in tastes (although the direction of the relationship is less obvious).

In light of these problems, I propose a new approach to identifying labor supply equations. The variables age and age-squared and a constant (change in age) are added to the Euler and static-in-differences equations to remove life-cycle effects from the errors.[7] I then make the following identifying assumptions.

(1) Levels of time-invariant individual characteristics are uncorrelated with changes in tastes. With this assumption, variables such as education and family background are valid instruments for labor supply equations in differences. The assumption means that highly educated people (for example) may like work more or less than others, but that education does not systematically alter the way that tastes change over time. One can imagine reasons that the assumption fails - for example, people with certain education levels suffer job burnout more quickly than others. But such problems seem much less important than the correlation between education and levels of tastes. The data below do not reject the assumption.

Variables like education and background are correlated with [delta omega], and thus are useful instruments, because they affect the slopes of age-earnings profiles. Greater education, for example, causes wages to grow more quickly (Mincer [1974]).

Previous studies of intertemporal labor supply (MaCurdy [1981]; Altonji [1986]) use instruments such as education and background for the Euler equation. But they also use age, which is invalid by my criteria because it varies over time. As shown below, this difference matters for empirical results.

(2) Taste shocks have no aggregate component. This assumption implies that all aggregate variables are uncorrelated with changes in tastes. Any aggregate variable is a linear combination of year dummies, and so the efficient way to use aggregate information for estimation is to use year dummies as instruments.

Again, this assumption might fail, but the failures do not appear important. Aggregate tastes may contain long-term trends, but aggregate taste shifts are not important for the year-to-year changes in hours studied here. Aggregate tastes for work may have changed between the counter-culture 60s and the current age of yuppies. But surely shocks such as the sharp decline in hours in 1982 and the recovery in 1983 had little to do with taste changes.

(3) Any variable known at t is uncorrelated with the expectational error [u.sub.[tau]+1. This assumption follows from rational expectations.

These three assumptions determine the proper instruments for the static-in-differences and Euler equations. The static errors are changes in tastes, and so both year dummies and time-invariant micro variables are valid. The Euler errors include both expectational errors and taste changes, and the former are probably correlated with year dummies: while I rule out aggregate taste shocks, aggregate information shocks, such as news that a recession has begun, are allowed. I generally add year dummies to the Euler equation to capture aggregate news and use only micro variables as excluded instruments.

In section V the (over) identifying restrictions described here are tested. The tests of assumption (2) are particularly important because they provide evidence on the basic model of unconstrained optimization. I test (2) by adding aggregate variables, such as the change in the unemployment rate, to the labor supply equations. Given (2), the model implies that these variables have zero coefficients. The alternative hypothesis of quantity constraints suggests that aggregate variables matter, because fluctuations in constraints have an aggregate component. (Constraints are tightest, and so hours of work are lowest, during a recession.) Quantity constraints could cause the data to reject assumption (2).

IV. DATA

Data are drawn from the first fourteen waves (1968-1981) of the Michigan Panel Study of Income Dynamics. The sample is restricted to men ages 25-59 to avoid the econometric complications that arise when many individuals work zero hours.(8)

I use the standard PSID measures of hours, wages, and consumption. Labor supply is annual hours of work and the wage is annual labor earnings divided by hours. Since the wage is an average, it differs from the marginal wage in the model's first-order condition if a worker holds a second job or receives an overtime premium. Consumption is measured by food consumption; one can justify this rigorously only if utility is separable in food consumption and other consumption (see Zeldes [1989] for details).(9) The data problems increase the value of comparing results for constrained and unconstrained workers. One could blame negative results for the entire sample on poor data, but differences between the two groups cannot be dismissed as easily.

The consumer price index and PSID data on marginal tax rates are used to construct real after-tax wages.(10) Food consumption is deflated by the CPI for food. The interest rate at t is the annual average of the after-tax Treasury bill rate at t deflated by the change in the CPI between [tau] and [tau]+1. Finally, the micro instruments for the labor supply equations are dummies for education level and economic status of parents, interactions among these variables, and dummies for race, region, residence in an SMSA, union status, and industry.(11)

Following Ham [1982; 1986], I classify a worker as constrained in a given year if he either experiences a spell of unemployment or cannot work as many hours as he wants. He meets the latter criterion if he reports that no more hours were available on his job and that he wanted to work more.(12) A worker is assigned to the constrained group if he is constrained in any year.(13)

This procedure for determining which workers face quantity constraints is imperfect. Some workers may say they are constrained because they cannot work at overtime wages even though they work as much as they want at their actual wages (Ham [1982]). Similarly, if workers receive fixed salaries, the wage that they consider when saying whether they want more work might be unrelated to their current earnings. Given these problems, the empirical work is partly a test of the data. If the model performs best for the unconstrained sample, this suggests that the PSID questions are reliable guides to who is constrained.

In a given year, most workers are unconstrained, but most are constrained in at least one year: the sample contains 9290 annual observations on the constrained group and 3975 on the unconstrained group. Table I presents sample statistics for the two groups. Not surprisingly, the constrained workers receive lower wages and are concentrated among disadvantaged segments of the labor market: they are younger, less educated and more likely to be minorities and from the South.

Table : TABLE I Sample Means (Standard Deviations in Parentheses)

Unconstrained Constrained Workers Workers (3975 annual (9290 annual observations) observations) Annual Hours 2459 2291 (598) (596) Real After-tax Wage ($/hour) 3.33 2.93 (1.66) (1.41) Real Food Consumption ($/year) 1849 1776 (836) (806)

Dummy Variables:

No High School [Degree.sup.a] .11 .25 Some [College.sup.a] .30 .33 College [Degree.sup.a] .40 .20 Parents [Poor.sup.b] .36 .42 Parents [Rich.sup.b] .20 .17 Nonwhite .04 .09 Deep [South.sup.c] .05 .05 Other [South.sup.c] .20 .25 [Northeast.sup.c] .21 .22 [West.sup.c] .15 .17 Residence in SMSA .64 .65 Union Member .13 .33

a. Base Group [is equal to] High School Graduate

b. Base Group [is equal to] Parents Average

c. Base Group [is equal to] North Central

V. RESULTS: HOURS CONSTRAINTS

Part A of this section presents estimates of the Euler and static equations for all workers and for the constrained and unconstrained sub-samples. Part B tests the model for each sample by adding aggregate instruments to the equations. Part C addresses econometric issues, focusing on alternative identifying restrictions.

A. The Basic Estimates

Table II presents two-stage least squares estimates of the Euler and static-in-differences equations. The endogenous variables are [DELTA][epsilon], [DELTA]c, and (after-tax) R. As described in section III, age and age-squared are included as taste shifters; the excluded instruments are time-invariant micro variables for the Euler equation and micro variables plus year dummies for the static equation. In the Euler equation, the restriction is imposed that the coefficients on [DELTA][epsilon] and R are equal in absolute value.(14)

Table : TABLE II Estimates of Euler and Static-in-Differences Equations

Sample All Unconstr. Constr. Euler Equation Intercept .103 .168 .073 (.062) (.075) (.074) [DELTA] In[omega]a .070 -.124 .169 -InR)a (.201) (.205) (.222) age/100 -.519 -.774 -.386 (.295) (.348) (.364) [age.sup.2]/[10.sup.4] .508 .786 .351 (.342) (.401) (.432) other year year year variables dummies dummies dummies Static-in-Differences Equation Intercept .110 .118 .111 .043) .064) (.054) [DELTA] ln[omega.sup.b] .0047 .011 -.0085 (.080) (.108) (.097) [DELTA] ln[C.sup.b] .216 .069 .265 (.065) (.082) (.083) age/100 -.514 -.506 -.529 (.219) (.325) (.279) [age.sup.2]/[10.sup.4] .554 .498 .584 (.269) (.395) (.345)

Standard errors are in parentheses.

a. Endogenous variables. Instruments are time-invariant micro variables.

b. Endogenous variables. Instruments are time-invariant micro variables and year dummies.

The results have two important features. First, the coefficients on [DELTA] w are close to zero and very insignificant statistically for both equations and all samples. Recall that these coefficients measure the intertemporal labor supply elasticity (again, the elasticity with respect to the current wage holding constant wages in other periods). The data contain no evidence of intertemporal substitution by anyone.

Second, the static equation results imply that the model performs better for unconstrained workers than for constrained workers. The [DELTA]c coefficient for all workers is .216 ([tau] = 3.3). This positive coefficient contradicts the model and, as noted above, is consistent with a Keynesian consumption function. When the sample is split, the [DELTA]c coefficient for the constrained group is .265 ([tau] = 3.2). The coefficient for the unconstrained group, while positive, is only .069, and it is very insignificant ([tau] = .8). For unconstrained workers, one cannot reject the model's prediction of a negative coefficient.

How should we interpret these results? We can clearly reject the model for the constrained group. The data for the unconstrained group provide no evidence in favor of the model, since the key variables in both equations have insignificant coefficients. On the other hand, the results for unconstrained workers are at least consistent with the model. The model predicts a near-zero coefficient on [DELTA][epsilon] if the intertemporal labor supply elasticity is small. In other words, workers may be unconstrained but nonetheless choose to respond little to changes in wages. The model predicts a near-zero coefficient on [DELTA]c if labor supply is inelastic and workers are not too risk averse in consumption. To see this, recall that the labor supply elasticity and the [DELTA]c coefficient are -1/[beta] and [alpha]/[beta] respectively. The parameter [alpha] is the coefficient of relative risk aversion. If the labor supply elasticity is .05 and [alpha] is one (the common assumption of log utility), then the [DELTA]c coefficient is -.05. This value lies within the 90 percent confidence interval implied by the point estimate of .07 and standard error of .08 for the unconstrained sample.

B. Tests of the Model

This section tests the overidentifying restriction that the taste shocks in the errors have no aggregate component. As explained above, aggregate variables are added to the equations and the hypothesis tested that their coefficients are zero. The data are likely to reject this restriction if workers face quantity constraints, because fluctuations in constraints have an aggregate component.

For the static equation, two versions of the test are performed. First, I add the full set of year dummies to the equation (while continuing to use micro variables as excluded instruments). The results are inconclusive. For the constrained group, the hypothesis that the dummies have zero coefficients is rejected at the 90 percent but not the 95 percent level ([X.sup.2.sub.10] = 17). For the unconstrained group, the hypothesis is not quite rejected at the 90 percent level ([X.sup.2.sub.10] = 15).

Second, I add the change in the aggregate unemployment rate for men ages 20 and older. This yields a more powerful test against the alternative hypothesis of quantity constraints if the unemployment rate is a good aggregate measure of constraints. The results, reported in Table III, are clear-cut. The coefficient on the change in unemployment is -.008 ([tau] = 3.3) for the constrained group and -.001 ([tau] = .4 for the unconstrained group. Thus the behavior of constrained workers violates the model's restrictions while that of unconstrained workers does not. The coefficient of -.008 means that a one-percentage point rise in unemployment reduces the average hours of constrained workers by eight-tenths of a percent.

In the Euler equation, the error term includes the expectational error [u.sub.[tau] +1, which has an aggregate component. Thus the model does not imply that the errors are uncorrelated with all aggregate variables. But since expectational errors are unforecastable (the third identifying restriction), the Euler errors should be uncorrelated with any aggregate variable known at time t. Adding the unemployment rate at time t to the Euler equation produces t-statistics of 3.7 for the constrained sample and .7 for the unconstrained sample; again, only the results for constrained workers contradict the model.

To summarize, the test results, like the parameter estimates in part A, are consistent with the model for the unconstrained sample and inconsistent for the constrained sample. To understand the rejection for the constrained group, note that the model suggests a negative simple correlation between changes in an individual's hours and changes in aggregate unemployment. But this correlation works through the wage: if wages are generally low, individuals choose fewer hours and (since some choose zero hours) measured unemployment rises. The model is rejected because aggregate unemployment helps to explain the part of hours' movements that are not explained by wages, and that the model thus ascribes to taste shifts. Of course the results are consistent with the basic model if, contrary to the identifying restrictions, changes in

Table : TABLE III

Tests of the Model Sample All Unconstr. Constr. Static-in-Differences Equation Intercept .129 .121 .135 (.041) (.064) (.052) [Delta] ln [w.sup.b] -.041 .0043 -.067 (.078) (.109) (.094) [Delta] ln [C.sup.b] .156 .061 .178 (.065) (.084) (.083) [Delta] U -.0058 -.0011 -.0081 (.0019) (.0028) (.0025) other age, age, age, variables [age.sup.2] [age.sup.2] [age.sup.2] Euler Equation Intercept .086 .163 .050 (.065) (.079) (.076) [Mathematical Expression .057 -.123 .153 Omitted] (.198) (.203) (.218) [U.sub.t] .0057 .0014 .0075 (.0015) (.0022) (.0020) other age, age, age, variables [age.sup.2] [age.sup.2] [age.sup.2]

Standard errors are in parentheses.

a. Endogenous variables. Instruments are time-invariant micro variables.

b. Endogenous variables. Instruments are time-invariant micro variables and year dummies.

aggregate unemployment capture aggregate taste shifts. It seems unlikely, however, that such taste shifts would cause rejection for one sample of workers but not the other. In contrast, if the rejection for the constrained sample is indeed caused by quantity constraints, it makes sense that the results are better for the unconstrained sample.

C. Alternative Identifying Restrictions and Diagnostics

Here I consider econometric issues, focusing once more on identification. As described in section III, previous micro studies of intertemporal substitution use age and age-squared as excluded instruments, while the micro instruments in this study are time-invariant. Table IV shows that this difference matters for the results. Adding age and age-squared to the excluded instruments for the Euler equation produces significantly positive estimates of the intertemporal labor supply elasticity. The point estimates of .4 to .6 are at the high end of the range of previous estimates. Recall that the elasticity estimates are essentially zero when age and age-squared are treated as taste shifters.[15]

Why is the treatment of age important? The coefficients on age and age-squared in Table II imply that [Delta] L falls with age for most ages in the sample - that is, hours are a concave function of age. As noted above, wages are also a concave function of age. Since both [Delta] L and [Delta] w fall with age, they are positively correlated. This correlation produces positive elasticity estimates if one does not control for age in the labor supply equation.[16]

When age and age-squared are added to the Euler equation, the hypothesis that they have zero coefficients, and thus are valid instruments, is rejected at the 99 percent level [Mathematical Expression Omitted]. In contrast, the validity of the time-invariant instruments - education, family background, and so on - cannot be rejected. I test the overidentifying restrictions on the basic Euler equation by regressing the equation residuals on the instruments and the exogenous variables in the equation. The [Mathematical Expression Omitted] statistic (R.sup.2 times the sample size) is a tiny 7.9. Thus the data support this study's approach to identification.

Finally, I consider the potential econometric problem of serial correlation in the errors for a given individual. Taste changes need not be white noise. In addition, white noise measurement error in the level of hours creates negative serial correlation in the errors of the differenced equations. To gauge the importance of these problems, I compute the correlations between errors for the same individual at various lags. For the unconstrained sample, the correlation between a worker's Euler equation errors at t and t-1 is -.3. The correlations at all lags greater than one are very close to zero. These results suggest substantial measurement error in the level of hours. In any case, since the serial correlation dies out quickly, it has little effect on the results. This is confirmed by experimenting with Altonji's [1986] procedure for calculating standard errors robust to serial correlation. As in Altonji's work, the correction reduces the standard errors, but only slightly; the OLS results are reliable.

Table : TABLE IV

Alternative Identifying Restrictions Sample All Unconstr. Constr. Euler Equation Intercept -.023 -.016 -.024 (.009) (.014) (.011) [Mathematical Expression .561 .391 .532 Omitted] (.154) (.177) (.185) Other variables year year year dummies dummies dummies

Standard errors are in parentheses.

a. Endogenous variables. Instruments are time-invariant micro variables, age, and [age.sup.2].

VI. LIQUIDITY CONSTRAINTS

In intertemporal labor supply models, workers borrow and lend to smooth consumption as they vary hours of work. Thus liquidity constraints, as well as constraints on hours, can cause violations of the models. This section investigates the effects of liquidity constraints by splitting the sample of workers who are unconstrained in hours into liquidity constrained and unconstrained subsamples. Following Zeldes [1989], I place a worker in the liquidity constrained group if his asset income is zero or negative in any year. Zeldes presents evidence from consumption data that individuals without assets are liquidity constrained while those with assets are not.

Liquidity constraints imply a violation of the model's Euler equation, which concerns the tradeoff between leisure in different periods. Intuitively, if a worker cannot borrow then he does not reduce his hours when his wage falls, because his consumption would fall considerably. The static first-order condition is robust to liquidity constraints, because it states that workers optimally balance leisure and consumption within a period, which does not require borrowing. Thus the hypothesis that asset income measures liquidity constraints has rich implications: the Euler equation should hold only for the high-asset group, while the static equation should hold for both groups.

Table V reports estimates of the Euler and static equations for the two subsamples. The estimates support the predicted effects of liquidity constraints. In the Euler equation, the [Delta] w coefficient for the high-asset group is .042 (t [ equal to] .3), which is consistent with unconstrained optimization and a small labor supply elasticity. The corresponding coefficient for the low-asset group is -.453 (t [ equal to ] 2.5); this unambiguously contradicts the model's prediction of a positive coefficient. Finally, as predicted by liquidity constraints, the difference between the subsamples disappears for the static equation: there, the [Delta] w coefficient is close to zero for both groups.

VII. CONCLUSIONS

This paper has three findings. First, an intertemporal labor supply model performs better for workers who say they do not face constraints on hours than for workers who say they do. This suggests that the first group really does face quantity constraints, and that these constraints contribute to the failures of intertemporal substitution models in previous studies. Given the empirical evidence of quantity constraints, macroeconomists should focus more attention on Keynesian models that include these constraints.

Second, the intertemporal substitution model performs best for workers with significant assets, who are unlikely to face liquidity constraints. Future labor supply models should incorporate these constraints as well.[17]

Finally, the paper presents new evidence on the intertemporal labor supply elasticities of workers who do not face hours or liquidity constraints. Elasticity estimates for prime-age men are close to zero and statistically insignificant. Even when workers can choose their hours, intertemporal substitution appears close to non-existent. Along with the other results, this suggests that employment fluctuations result primarily from changes in involuntary unemployment.

The estimates of labor supply elasticities differ from estimates in previous studies, which are often positive and significant. Differences in identifying restrictions explain the different results, and tests of over-identifying restrictions support this paper's approach. Similar approaches might be useful for other studies in which error terms include tastes, such as panel studies of consumption.

Table : TABLE V

Results for High Asset and Low Asset Groups Sample High Asset Low Asset Euler Equation Intercept .051 .294 (.104) (.079) [Mathematical Expression .043 -.453 Omitted] (.160) (.184) other age, age, variables [age.sup.2] [age.sup.2] years years Static-in-Differences Equation Intercept .030 .185 (.099) (.079) [Delta] ln [w.sup.b] .015 -.043 (.120) (.122) [Delta] ln [C.sup.b] .037 .077 (.102) (.092) other age, age, variables [age.sup.2] [age.sup.2]

Standard errors are in parentheses.

a. Endogenous variables. Instruments are time-invariant micro variables.

b. Endogenous variables. Instruments are time-invariant micro variables and year dummies.

( 1.) Mankiw et al. conclude that "the abundance of plausible explanations for the results we obtained... leads us to be somewhat skeptical of the power of aggregate time series data in distinguishing alternative macroeconomic hypotheses." And Ham (who uses micro data) cautions that his result "does not indicate which model of the labour market is appropriate.... [O]ne could turn to models which allow for the possibility that unemployed workers are off their labour supply functions. Alternatively, one could turn to more complex models of interporal substitution. Unfortunately, the estimation and testing of these alternative or more complex models is likely to be extremely difficult."

( 2.) Ham [1986] tests an interporal labor supply model by adding PSID measures of quantity constraints to the estimated first-order conditions. The most important departure of his study from Ham is that the model is tested separately for constrained and unconstrained workers. As argued below, the crucial evidence that constraints are the true reason for rejection of the model is the relatively good results for the unconstrained sample. The sample split also allows one to estimate the interporal labor supply elasticity for unconstrained workers. Other departures of this study from Ham and other earlier papers, such as the approach to identification and the attention to liquidity constraints, described below. For another recent study that tests an interporal labor supply model for constrained and unconstrained workers, see Biddle [1988].

( 3.) The Euler equation uses the approximations ln(1 + [Delta]) [is equal to] and ln(1 + [upsilon ]) [is equal to] [upsilon].

( 4.) More precisely, the coefficient is the elasticity of hours with respect to the current wage holding constant the marginal utility of wealth. This definition differs slightly from the one in the text, because changing the current wage while holding other wages constant has a slight effect on the marginal utility of wealth.

( 5.) There are two other identification problems. First, consumption and interest rates are endogenous. (After-tax interest rates are endogenous because labor supply affects tax rates.) And second, there is severe measurement error in micro data on wages, hours, and consumption. Under plausible assumptions, the instrumental variables estimators proposed below are robust to these problems as well as to the problem stressed in the text.

( 6.) Economist who estimate one-period labor supply models often ignore the identification problem (for example, Hausman, [1980; 1981]). In estimating the undifferenced static condition of interporal models, researchers generally use instrumental variables, but they choose invalid instruments like education and background (for example, MaCurdy [1983]). For other discussions of the identification problem, see Mankiw et al. [1985], Ham [1982], and Altonji [1986].

( 7.) These taste shifters can be added to the formal model by assuming that [Mathematical Expression Omitted] where X is the vector of taste shifters and [Mathematical Expressi on Omitted] is the unobservable part of tastes (with a similar equation for [eta]). The identifying assumptions about tastesneed hold only for [Mathematical Expression Omitted]. In the Euler equation, the constant term captures the constan t in the basic model (which depends on the rate of time preference) as well as life-cycle changes in tastes.

( 8.) There is no consumption data for Waves 1 and 6. Therefore, for the static-in-differences equation the change from Wave 1 to Wave 2 is deleted and the Wave 5 to Wave 7 change replaces the two one-year changes. (The two-year change is weighted by 1/[square root of 2], which is efficient in the absence of serial correlation.) Observations are deleted if a worker is out of the labor force (rather than employed or unemployed), receives food stamps (which makes the food consumption data difficult to interpret), or is not the head of his household. I also delete observations with missing data and extreme outliers: observations in which the wage falls by 60 percent or rises by 150 percent, or in which hours of work or consumption falls by 80 percent or rises by 400 percent (these criteria follow Altonji [1986]). Finally, I use only the random part of the PSID sample.

( 9.) Another problem is the timing of the food consumption question. PSID questions on hours and wages refer to the previous calendar year, but the consumption question is simply "how much do you spend on food in an average week?" (The question is asked separately for restaurants and food at home.) It is unclear what time period is relevant to an "average week," and it is likely that responses are affected by changes in consumption between the end of the previous year and the interview date (late March, on average). We can interpret the difference between the available data and data for the previous year (which would be consistent with the hours and wage data) as measurement error. Unfortunately, this error is probably correlated with the year dummies used as instruments, because aggregate shocks affect changes in consumption between different periods. To determine the importance of this problem, I compare the basic static equation estimates to estimates obtained when only micro variables are used as instruments. The estimates are similar, and so there is no evidence that the problem is serious. On the other hand, the standard errors rise when only micro variables are used, and this makes the results less conclusive. (With the restricted set of instruments, the crucial [Delta]c coefficient is .35 (t = 1.5) and .15 (t = .8) for the constrained and unconstrained groups.)

(10.) I take into account Federal income taxes and Social Security taxes. The PSID reports marginal income tax rates for Waves 9-14. For earlier waves, marginal rates were constructed from Federal tax tables and PSID data on income and total taxes.

(11.) Some of these variables, such as region and industry, change occasionally for a given individual, and therefore do not quite meet the requirement of time-invariance (identifying restriction (1)). It seems unlikely that changes are frequent enough to cause serious problems. Tests of overidentifying restrictions provide no evidence that the variables are invalid instruments.

(12.) The questions are "Was there more work available on your job [or "any of your jobs" if more than one] so that you could have worked more if you had wanted to?"; and "Would you have liked to work more if you could have found more work?"

(13.) If labor supply equations are estimated in levels, splitting workers into constrained and unconstrained groups creates sample selection bias. In particular, it is likely that the expectation of the error conditional on being unconstrained is negative, because workers who desire few hours are unlikely to face binding constraints. Ham [1982] uses a sophisticated econometric procedure to overcome this problem. Under plausible assumptions, the problem does not arise here, because the equations are differenced. Knowing that a worker is in the unconstrained group - that is, that he does not face a constraint in any year - suggests that his tastes for work are generally negative. But there is no reason that the expectation of his change in tastes, which appears in the error, is non-zero. That is, knowing that a worker is unconstrained from 1970 to 1975 reveals nothing about whether he liked work better in 1973 than in 1972.

(14.) This restriction is an implication of the model. However, relaxing this restriction or omitting the interest rate from the equation has little effect on the results.

(15.) Altonji [1986] sometimes adds age to the Euler equation and still obtains positive elasticity estimates. Even in these regressions, however, he uses age * schooling and age * schooling[2] as excluded instruments. He also uses year dummies, which are invalid instruments for the Euler equation by my criteria.

(16.) Browning et al. [1985] discover similar relationships among hours, wages, and age in synthetic cohort data, and they point out the resulting biases in elasticity estimates.

(17.) See Dau-Schmidt [1988] for some work along these lines.

REFERENCES

Altonji, Joseph G. "The Intertemporal Substitution Model of Labor Market Fluctuations:

An Empirical Analysis." The Review of Economic Studies, Supplement 1982, 783-824.

___."Intertemporal Substitution in Labor Supply: Evidence from Micro Data." Journal

of Political Economy, June 1986, S176-S215.

Barro, Robert J. and Robert G. King. "Time-Separable Preferences and Intertemporal

Substitution Models of Business Cycles." Quarterly Journal of Economics, November

1984, 817-39.

Biddle, Jeff E. "Intertemporal Substitution and Hours Restrictions." Review of Economics

and Statistics, May 1988, 347-51.

Browning, Martin, Angus Deaton, and Margaret Irish. "A Profitable Approach to Labor

Supply and Commodity Demands over the Life-Cycle." Econometrica, May 1985,

503-43.

Dau-Schmidt, Kenneth G. "An Empirical Study of the Effect of Consumption Commitments

and the Liquidity Constraint on Labor Supply." Working Paper, University of

Cincinnati, February 1988.

Hall, Robert E. "The Role of Consumption in Economic Fluctuations." Working Paper,

Stanford University, 1984.

Ham, John C. "Estimation of a Labor Supply Model with Censoring Due to Unemployment

and Underemployment." The Review of Economic Studies, July 1982, 335-54.

___."Testing Whether Unemployment Represents Intertemporal Labor Supply

Behavior." The Review of Economic Studies, August 1986, 559-78.

Hausman, Jerry A. "The Effects of Wages, Taxes, and Fixed Costs on Women's Labor Force

Participation." Journal of Public Economics, October 1980, 161-94.

___."Labor Supply," in How Taxes Affect Economic Behavior, edited by Henry Aaron and

Joseph Pechman. Washington, D.C.: Brookings Institution, 1981.

MaCurdy, Thomas. "An Empirical Model of Labor Supply in a Life Cycle Setting." Journal

of Political Economy, December 1981, 1059-86.

___."A Simple Scheme for Estimating an Intertemporal Model of Labor Supply and

Consumption in the Presence of Taxes and Uncertainty." International Economic Review,

June 1983, 265-90.

Mankiw, N. Gregory, Julio Rotemberg, and Lawrence H. Summers. "Intertemporal

Substitution in Macroeconomics." Quarterly Journal of Economics, February 1985,

225-51.

Mincer, Jacob. Schooling, Experience, and Earnings. Chicago: University of Chicago Press for

the National Bureau of Economic Research, 1974.

Zeldes, Stephen. "Consumption and Liquidity Constraints: An Empirical Investigation."

Journal of Political Economy, April 1989, 305-46.

LAURENCE BALL(*)

(*) Assistant Professor of Economics, Princeton University. This is a revised version of Chapter I of my M.I.T. dissertation (January 1986). I am grateful for many suggestions from Oliver Blanchard, Henry Farber, Lawrence Katz, N. Gregory Mankiw, James Poterba, David Romer, Robert Solow, Lawrence Summers, Frank Wykoff, Stephen Zeldes, the referees, and participants in workshops at M.I.T. and the 1985 N.B.E.R. Summer Institute.

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Publication: | Economic Inquiry |

Date: | Oct 1, 1990 |

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