Determinants of voluntary overtime decisions.

Robins, Philip K. ; Idson, Todd L.

DETERMINANTS OF VOLUNTARY OVERTIME DECISIONS

I. INTRODUCTION

Since the passage of the Fair Labor Standards Act in 1938, the minimum legal overtime pay premium has remained time and one-half for persons working more than forty hours per week. In 1978, approximately 59 percent of the labor force was subject to this provision, and coverage has not changed since then. (1) The labor market effects of overtime provisions, including the decision on the part of both employers and employees to utilize overtime, has generated a great deal of intellectual debate. For example, union groups have long stressed the importance of raising the legal minimum premium in order to stimulat employment through a substitution of additional employees for longer (e.g. overtime) hours. Previous research by Ehrenberg and schumann [1982! has estimated that these effects are likely to be quite small.

To date, studies of overtime behavior, such as Ehrenberg [1971a!, have focused on the decisions of employers. Implicit in these studies is the assumption of a perfectly elastic supply response by the employee to employer overtime policy. However, just as employers attempt to minimize costs in demanding overtime hours (taking into account quasi-fixed labor costs in addition to overtime premiums), employees attempt to maximize utility in supplying overtime hours. From the worker's perspective, the decision to work overtime (and perhaps more generally, to take a job where overtime is regularly available or required) can be thought of as resulting from a comparison of the utility derived from an overtime versus a no-overtime state. Within this framework, the voluntary overtime decision depends on relative wages (overtime to straight time) and nonlabor income and will be strongly conditioned on such personal factors as the worker's marital status and age.

This study attempts to advance our knowledge of the labor market effects of overtime policies by explicitly modelling employee responses to variations in the legal parameters of the overtime system. (2) Our model allows us to estimate substitution and income elasticities for overtime, and thereby enables us to evaluate the effects of a variety of policies on a worker's voluntary overtime decision. Policy simulations suggest that legislative changes to increase the overtime premium, or reduce the number of hours after which the mandatory premium takes effect, would induce greater voluntary overtime on the part of worker. Yet, because of offsetting income and substitution effects, the net induced effects of such policies are not quantitatively large. We also find that mandatory overtime provisions are most likely to be found in settings where quasi-fixed employment costs are highest.

The remainder of this paper is organized as follows. Section II develops a theoretical model of the voluntary overtime decision. Section III presents the empirical model to be estimated. Section IV discusses the data used in estimation and the construction of key variables. Section V presents the empirical findings. Section VI reports the results of a simulation analysis of the effects of a number of policy alternatives on the voluntary overtime decision. Section VII investigates the sensitivity of the policy simulations to the empirical specification. Finally, section VIII summarizes the major conclusions and gives suggestions for further research.

II. THE VOLUNTARY OVERTIME DECISION

The voluntary overtime decision can be though of as resulting from a straightforward maximization of utility subject to a budget constraint. Consider a worker who possesses a monotonic, strictly quasi-concave utility function in market goods (income) and labor supply, U(Y,H), where Y is income and H is hours of work. The budget constraint facing the worker is Y = wH + n, where w is the net (of taxes) wage rate and n is net nonwage income. To make the problem empirically tractable, we utilize the Stone-Geary utility function in this paper. (3)

(1) U = [alpha!ln(Y - a) + (1 - [alpha!)ln(b - H),

where a is subsistence income, b is total time available for work, and 0

[alpha! [is less than! 1, where [alpha! is the Stone-Geary parameter.

Maximization of the utility function subject to the budget constraint yields equations determining income, labor supply, and the marginal utility of income. Substituting these solution equations into the direct utility function (1) yields the indirect utility function:

(2) V = [alpha!ln[alpha! + (1 - [alpha!)ln(1 - [alpha!), + lnz - (1 - [alpha!)lnw

where z = bw + n - a, termed "supernumerary" income by Goldberger [1967!. (4)

Using the indirect utility function (2), one can derive the conditions that determine whether or not an individual chooses to work overtime. The decision to work overtime depends on a comparison of utility between two segments of the kinked budget constraint one at the straight-time wage and one at the overtime wage. Figure 1 depicts the budget constraint for a worker subject to overtime provisions. In this diagram, [w.sub.1! is the net wage rate under overtime ([w.sub.1!=WP(1-[t.sub.1!)) and [w.sub.0! is the net wage under straight time ([w.sub.0! = W(1 - [t.sub.0!)), where W is the gross straight-time wage, P is the overtime premium, [t.sub.0! is the average marginal tax rate on the straight-time segment of the constraint and [t.sub.1! is the rate when overtime wages are received. In addition, [z.sub.1! is net supernumerary income under overtime ([z.sub.1! = [bw.sub.1! + [n.sub.1! - a, where [n.sub.1! is virtual nonwage income under overtime) (5) and [z.sub.0! is virtual supernumerary income under straight time ([z.sub.0! = [bw.sub.0!+[n.sub.0! - a, where [n.sub.0! is actual nonwage income). The threshold level of hours, after which the overtime premium comes into effect, is H (*1).

Based on the above definitions, the voluntary overtime decision depends on whether utility is higher under overtime or under straight time. Depending on this the worker will choose to locate on the overtime segment of the constraint ([z.sub.1!x) or the straight-time segment ([xn.sub.0!). Define [V.sub.1! as the value of utility if the worker works overtime, [V.sub.0! as the value of utility if the worker does not work overtime, and [Delta!V = [V.sub.1! - [V.sub.0!. The overtime decision may be formally written as

(3) Work overtime if [Delta!V = [Delta!lnz - (1 - [alpha!)[Delta!lnw 0

Do not work overtime if [Delta!V = [Delta!lnz - (1 - [alpha!)[Delta!lnw [is less than or equal to! 0

where [Delta!lnz = [1nz.sub.1! - 1[nz.sub.0 and [Delta!lnw = [lnw.sub.1! - [lnw.sub.0!. Hence, the overtime decision depends on the sign of [Delta!V, which in turn depends on [alpha!, [Delta!lns and [Delta!lnw.

III. EMPIRICAL SPECIFICATION

Some workers are in jobs where voluntary overtime decisions are constrained by the existence of mandatory overtime provisions. For workers in these jobs, voluntary overtime decisions are unobservable, (6) thus our analysis must be restricted to workers in jobs without mandatory overtime provisions. (7) However, if job choice and overtime decisions are correlated, estimates based on a subsample of workers not facing mandatory overtime provisions would be biased. Hence, we must correct our estimates for possible sample selection bias, e.g. we only observe voluntary overtime decisions for those individuals who first locate in jobs without required overtime, so that the observed data are non-randomly selected from the universe of all possible workers overtime responses. Therefore, we modify equation (3) to get the following empirical model:

(4) [Delta!V = [Delta!lnz - (1 - [alpha!)[Delta!lnw

+ [Theta![X.sub.1! + [epsilon.sub.1! [epsilon.sub.1! N(0,[o.sup.2),

(5) NROVT = [BETA!'[X.sub.2! + [epsilon.sub.2! [epsilon.sub.2!

N(0,1), corr([epsilon.sub.1,[epsilon.sub.2!) = p,

OT = 1 IF [Delta!V [is greater than' 0 NR=1 if (6) NROVT 0

OT is only

observed when NR = 1,

where [X.sub.1! is a vector of personal characteristics reflecting observed heterogeneity in labor supply behavior, [epsilon.sub1! is a random error term reflecting measurement error, unobserved heterogeneity, and optimization error, (8) NROVT is a latent variable representing the absence of mandatory overtime provisions, [X.sub.2! is a vector of employer and demographic variables reflecting factors associated with mandatory overtime provisions, and [epsilon.sub.2! is a random error term. In (4), [Delta!V is only observed when

NROVT 0 or, equivalently, the indicator of whether the individual works overtime, OT, is only observed when NR, the indicator of whether there is no mandatory overtime, equals one. Hence, equations (4)-(6) comprise a bivariate probit model with sample selection. To estimate the model, we use the technique of full information maximum likelihood (FIML). (9)

Having obtained estimates of the Stone-Geary parameter [alpha!, it is possible to calculate income and substitution elasticities pertinent to the overtime decision. From

Roy's identity were derive H = ([derivative!V/[derivative!w/ ([derivative!V/[derivative!n) using this it can be shown that the total income elasticity is -(1 - [alpha!), the uncompensated wage elasticity is (1 - [alpha!) (n - a)/wH, and the compensated wage elasticity is (1 - [alpha!){[(n -a)/wH! + 1}. (10)

The probit model implied by equation (4) differs slightly from the conventional probit model in that a normalization of the error variance is not required because of the unitary constraint imposed by the Stone-Geary specification on the coefficient of [LAMBDA!lnz. Given this constraint on the coefficient of [LAMBDA!lnz, it is possible to estimate the variance of [epsilon.sub.1!.

IV. DATA AND VARIABLE SPECIFICATION

The empirical analysis is based on a sample of male workers from the 1977 Quality of Employment Survey (QES). (11) While the QES suffers from a relatively small sample size, it has the critical information needed to estimate equations (4) and (5) namely information on the respondent's actual overtime rate for those working overtime, what respondent's who are not working overtime say they would receive if they did work overtime, and whether or not they are subject to mandatory overtime provisions. The QES also has a rich vector of information on both job and respondent characteristics. Specifically, we include in the [X.sub.1! vector, for the overtime equation (4), the age of the respondent (AGE), his education level (ED), and dummy variables for race (RACE) and employment in the south (SOUTH). The [X.sub.2! vector, in the required overtime equation (5), includes information on job and personal characteristcs namely the logarithm of employer size (LFS) and dummy variables for the availability of certain fringe benefits (FB), being in a blue-collar occupation (BCOCC), whether the worker is salaried (SALARY), and union status (UC). Following Mellow [1983!, Oi [1983!, and Barron, Black and Lowenstein [1987!, the firm size and fringe benefit variables are used as proxies for quasi-fixed employment costs, which previous research (e.g., Rosen [1968! Ehrenberg [1971b!) has identified as a key factor in firms' decisions concerning their optimal mix of hours and number of employees, and hence whether they are likely to require overtime.

In order to increase the size of our sample, we merged the 1977 QES cross-section with the 1977 wave of the 1973/77 QES panel. (12) To further increase the amount of usable information, and to avoid biases arising from any nonrandom factors associated with missing wage values, wages are imputed for those respondents with missing wage data. (13) Excluded from the analysis sample are the self-employed and respondents employed in public service or agriculture. (14) The resulting analysis sample has complete data for 612 observations.

The overtime wage rate is constructed by taking the actual reported overtime wage for those who responded to the question concerning what they would make if they worked more than their usual hours during a week. For those who responded time and one-half or double time, we assign a 50 percent and 100 percent markup, respectively, of the straight-time wage. Both straight-time and overtime wages are adjusted to account for federal income taxes (i.e., net wages are used). Marginal tax rates are derived from a regression function estimated using data from the U.S. Department of the Treasury [1977!. A cubic function relating adjusted gross income to taxes is estimated for three groups: married (filing jointly,) single heads of households (with children), and single (no children). For each sample member, the average marginal tax rate along each budget segment is used. (15) Hence, we follow the literature is specifying a two-segment, linearized budget constraint.

Direct measures of nonwage income are not available in the QES. Our measure of nonwage income is constructed indirectly by taking the difference between reported total family income (from all sources) and the respondents' total earnings from employment. (16) The virtual full income measure for the straight-time segment of the budget constraint ([z.sub.0!) is calculated as the maximum number of weekly hours in the sample (eighty-four) times the wage rate plus the defined nonwage income measure ([n.sub.0!) minus the official poverty level (our proxy for the subsistence income level (17) term in the Stone-Geary function). For the overtime segment of the constraint the same formula is used to calculate [z.sub.1!, except that the nonwage virtual income level, [n.sub.1!, is calculated as [n.sub.1! =H* [w.sub.0! - [w.sub.1!) + [n.sub.0!, where H* is the kink point in the constraint (set equal to forty). As with the wage rate measures, the nonwage income measures used in the empirical work are after-tax values.

V. EMPIRICAL RESULTS

Table I presents definitions and descriptive statistics for the variables used in the empirical analysis. As Table I indicates, approximately 21 percent of the respondents state that they are subject to mandatory overtime provisions. The overtime measure indicates that about 35 percent of the sample not subject to mandatory overtime provisions reports some overtime hours.

Table II reports the FIML estimates of the bivariate probit model. Referring to the results in column 1, it is seem from the statistically significant and negative coefficients for the firm size and fringe benefit variables that the higher the percentage of labor costs due to quasi-fixed employment costs, the higher the demand for overtime on the part of the employer. The results also indicate that blue-collar workers (BCOCC) are more likely to be in jobs with required overtime, while workers covered by union contracts (UC) are less likely to be subject to these overtime requirements.

Referring to the results in column 2, we see that the relative wage (LNW1W0) has the expected negative sign on the probability of working overtime (given the fact that the ratio of the virtual full income measures (LNZ1Z0) is held constant), but is only marginally significant at the 11 percent level. (18) AGE has the expected negative sign and is significant at the 9 percent level. The estimate of [rho!, while farily large, is not statistically significant, indicating that sample selectivity would not have produced serious biases in our estimates if we had simply deleted workers who are subject to mandatory overtime provisions and thereby ignored the possible correlation between the error terms in equations (4) and (5). In general, the results are consistent with the theory, but the relatively small sample size has produced somewhat high coefficient standard errors and, hence, imprecise results.

Table III reports elasticities, derived from the estimated Stone-Geary parameter, evaluated at the sample means. The elasticities are well within the range generally accepted in the labor supply literature, as reported by Killingsworth [1983!, (19) with the weakly negative uncompensated wage elasticity revealing a slightly backward-bending supply curve to overtime hours at the sample means. This result is consistent with the backward-bending labor supply curves found in most recent studies of male labor supply.

VI. POLICY SIMULATIONS

Table IV uses the estimates in Table II to predict the effects of various government policies regarding overtime. Three policies of interest are (a) increasing the minimum overtime premium, (b) reducing the number of weekly hours above which an overtime premium must be paid, and (c) varying the level of quasi-fixed employment costs. For (a) and (b) we use the results in Table II to predict the effects of increasing the minimum overtime premium to double and triple time, and reducing the kink point for overtime hours from forty to thirty-five and thirty hours. For these changes, we assume no response on the part of the employers that is, we examine the effects of these changes on the voluntary overtime decision holding constant the straight-time wage and any labor substitution effects made by the employer. It is worth noting that for both of these changes, the probability of voluntarily working overtime increases. This is because such changes increase utility along the overtime segment of the budget constraint but do not a ffect utility along the straight-time segment of the budget constraint. (20)

In response to an increase in the overtime premium, the results indicate that increasing the minimum required overtime premium from the present time and one-half to double time and triple time raises the probability of an employee offeering to work overtime from 0.2999 to 0.3032 (a 1.1 percent increase) and 0.3295 (a 9.9 percent increase), respectively. (21) A policy requiring that overtime be paid for weekly hours exceeding thirty and thirty-five, respectively, instead of the present forty, acts to increase the probability of desired overtime work from 0.2999 to 0.03103 (a 3.7 percent increase) and 0.3206 (a 6.9 percent increase), respectively. Finally, increasing (decreasing) the level of quasi-fixed employment costs acts to increase (decrease) the probability that employees would be subject to mandatory overtime provisions. If all employees had available paid vacation and medical and retirement plans, then the likelihood of not being subject to mandatory overtime falls from a probability of 0.7799 to a probability of 0.7636 (a 2.1 percent decrease). Similarly, if these fringe benefits were not available to any employees, the probability of not being subject to these provisions would increase to 0.9158 (an 17.4 percent rise). (22)

VII. SENSITIVITY OF THE POLICY

SIMULATIONS TO THE EMPIRICAL

SPECIFICATION

The empirical estimates and policy simulations reported above are based on an empirical specification derived from the Stone-Geary utility function. Because of this it is of considerable interest to determine the sensitivity of the policy simulations to the empirical specification. We performed sensitivity tests in two dimensions. First, we reestimated the empirical model using alternative values for the level of subsistence income. Specifically, we varied the level of subsistence income from zero to twice the poverty level. (23) Note that when zero is used the utility function becomes Cobb-Douglas. Second, we reestimated two very simple unrestricted models, using the actual poverty level for the level of subsistence income. The first model represents the probability of working overtime as a log-linear function of [w.sub.0!, [w.sub.1!, [z.sub.0!, and [z.sub.1!, while the second represents the probability of working overtime as a linear function of [w.sub.1!/[w.sub.0! and [z.sub.1!/[z.sub.0!. The full set of estimates are available on request from the authors.

When the level of subsistence income is varied, the estimated labor supply elasticities change in a systematic way. As the subsistence level increases, all three elasticities monotonically decrease. (24) The policy simulations also change systematically. When subsistence income is set at zero (the Cobb-Douglas case), the responses to increasing the overtime premium and reducing the overtime hours threshold are larger than those reported in Table IV. However, when subsistence income is set at twice the official poverty level, the responses to reducing the overtime hours threshold a similar, yet increases in the overtime premium slightly reduce the probability of an employee offering to work overtime. In contrast, the effects of fringe benefits on the probability of being subject to mandatory overtime are virtually unaffected by the value used for subsistence income. (25)

When the model is reestimated using the log-linear and ratio specifications, the coefficients are of the expected sign, (26) but the significance levels are somewhat lower. The policy simulations from both sets of results are very similar to those reported

in Table IV. Hence, we conclude that the results in Table IV are not due to the functional form imposed by the Stone-Geary specification.

VIII. CONCLUSIONS AND SUGGESTIONS FOR

FUTURE RESEARCH

This paper develops and estimates a model of voluntary overtime decisions on the part of workers. The supply of overtime work is assumed to depend on a comparison of utility along the straight-time and overtime segments of the budget constraint. To account for the establishment of mandatory overtime provisions by employers, we estimate a bivariate probit model with sample selection. The results indicate that employee decisions to work overtime are, in fact, consistent with the principle of utility maximization.

Simulations indicate that an increase in the overtime premium or a decrease in the hours above which overtime must be paid would both lead to increases in desired overtime by workers. However, none of the estimated effects are quantitatively large. (For example, increasing the overtime premium from time and one-half to double time increases the probability of wanting to work overtime by two percentage points.) This suggests that workers are fairly insensitive to changes in supply factors when making overtime decisions. Although thos paper does not examine employer responses to increasing the overtime premium, previous research by Ehrenberg and Schumann [1982! suggests there would be a weak reduction in the demand for overtime work. Coupled with our finding of a weak supply response, the implication is that legislation to increase the overtime premium would not lead to any significant alteration in the amount of overtime work performed.

Our results also indicate that firms offering fringe benefits to their workers in the form of medical or retirement plans are more likely to have mandatory overtime provisions. This suggests that as fringe benefits become more pervasive throughout the economy, more workers are likely to become subject to mandatory overtime provisions. If certain groups (e.g., women with small children) do not desire overtime work, a significant change in the demographic composition of the labor force might occur in sectors where these quasi-fixed employment costs are most pronounced.

There are several issues that need to be addressed in future research on overtime behavior. First, the hours-of-work decision should be integrated with the basic overtime decision, perhaps using a methodology similar to that of Burtless and Hausman [1978!. In this context it might be fruitful to analyze the effects of alternative tax schemes on overtime decisions some work on this has already been done by Brown et al. [1974! and Wales and Woodland [1979!. Second, the use of panel data to examine the duration and frequency of overtime work appears to be a fruitful avenue of investigation. Third, incorporation of moonlighting as an alternative (or perhaps complement) to overtime work would provide a richer picture of the options facing the worker, with possible differential estimates of supply elasticities and policy effects. Preliminary studies of moonlighting include Perlman [1966! and Shishko and Rostker [1976!. Finally, a more thorough analysis is needed of how employer constraints influence the overtime decision. (27) Such an analysis might utilize a simultaneous equation framework in which the supply of and demand for overtime work are both endogenous.

(*1) Visiting Assistant Professor, Columbia University and Professor, University of Miami. We are grateful to David Blau, A. G. Holtmann and two anonymous reviewers for helpful comments, and to Adedeji Adebayo for competent research assistance.

(1.) U.S. Department of Labor [1979!. Excluded categories of workers are primarily administrative, executive, and professional personnel, outside salespersons, most state and local government employees, and agricultural workers. Our sample exludes the latter two groups.

(2.) The analysis takes explicit account of the fact that some workers are unable to maximize utility because of mandatory overtime provisions of employers.

(3.) For a discussion of the properties of the Stone-Geary utility function, see Goldberger [1967!. Numerous researchers have fruitfully employed the Stone-Geary specification in labor supply studies see, for example, Abbott and Ashenfelter [1976! and Johnson and Pencavel [1984!. In section 7 we discuss the sensitivity of our results to the Stone-Geary specification.

(4.) With a = 0, z is the familiar "full" income of Becker [1965!.

(5.) Virtual nonwage income is the intercept (at H = 0) of the projected budget line under overtime (see Burtless and Hausman [1978!) and is calculated by setting [z.sub.1! = [z.sub.0! at [H.sup.*!, which yields [n.sub.1! = [n.sub.0! + [H.sup.*! ([w.sub.0! - [w.sub.1!). Virtual supernumerary income has the same interpretation for H = b.

(6.) If we assume that workers knew that the jobs that they were taking required overtime, or else if the costs of changing jobs are not prohibitive, then the decision to locate in a constraining job is clearly endogenous. In future work we hope to simultaneously estimate overtime decisions with job choice.

(7.) The probability of observing a respondent working overtime is actually the product of the probability that the employer wants the employee to work overtime and the probability that the employee agrees to work overtime. Letting [E.sub.ij! denote the employer i's preference for employee j to work overtime, and [W.sub.j! denote employee j's willingness to work overtime, then the probability of observing an employee working overtime (WKOVT) is given by the following joint probability: P([WKOVT.sub.j! = 1) = P([E.sub.ij! [is greatern than! 0), where the indicator variables (E, W) are scaled to exceed zero if the employer (employee) offers (accepts) or requires overtime. The labor market outcome, WKOVT, is thereby a single binary random variable that is the product of E and W, the distribution of which has been derived by Poirer [1980!. As in the case of Ehrenberg and Schumann [1984!, rather than estimate the above joint probability we have approximated the decision rule by a single labor market outcome, where WKOVT = 1 if both the employer and the employee agree to overtime work.

(8.) In typical studies of labor supply behavior in the presence of kinked budget constraints, it is conventional to append an error term to the equation determining choice of hours rather than choice of budget segment (see Moffitt [1986! or Hausman [1985!). Because our study focuses on choice of budget segment, we append the error term to this equation. In general the various components of the error term can only be identified through direct estimation of the hours equation, or through a more elaborately specified budget selection equation in which partime, fulltime, and overtime work are distinguished.

(9.) For a general discussion of this model, along with the appropriate likelihood function, see Maddala [1983, ch. 9!

(10.) At first glance it would appear from equation (4) that an increase in the overtime wage (and hence [Lambda!lnw) would decrease utility. However w appears in z as well. The effect of an increase in the overtime wage, holding z constant, decreases utility since it involves rotating the budget constraint inward rather than outward. Such a change, though, is of no interest from a policy perspective.

(11.) For a description of the QES, see Quinn and Staines [1979!.

(12.) The 1973 wave of the QES panel is not included in the merged file because weeks worked was not asked in 1973. This omission made the computation of a wage variable either impratical or subject to such a high degree of measurement error as to make the efficiency gains from a larger sample insufficient to justify the inclusion of these observations.

(13.) The imputation method is based on the Heckman [1979! two-stage procedure. The resulting wage variable uses the imputed values for those respondents who didn't report wages and the actual wage values for those that did. Only 16.5 percent of the sample were missing wage values.

(14.) We do not select on eligibility for overtime, e.g. on covered occupations, so that sample selectivity along this dimension should not cause biases in our estimates.

(15.) The regression function for married males is T = 1,243 + 0.363[Y.sup.2! - 0.775[Y.sup.3!, for heads of households it is T = 2,309 + 0.386[Y.sup.2! - 0.847[Y.sup.3!, and for single males it is T = 1,448 + 0.419[Y.sup.2! -

0.931[Y.sup.3!, where T is total taxes and Y is adjusted gross income (the coefficient of [Y.sup.2! is multiplied by [10.sup.-3! and the coefficient of [Y.sup.3! is multiplied by [10.sup.-11!). This tax function was arrived at after much experimentation with various functional forms. By excluding the linear term the specification captures the fact that the marginal tax rate is zero at low levels of income. The empirical results are generally insensitive to changes in the specification of the tax function.

(16.) The approach used in the paper implicitly includes the earnings of the wife and any other family members, if present, in the husband's nonwage income.

(17.) The poverty level used is based on money income and is allowed to vary with family size (see U.S. Department of Commerce [1982! for a detailed discussion of how the poverty level is constructed). Johnson and Pencavel [1984! were able to estimate the subsistence level term of the utility function as part of their empirical analysis because they specified a dynamic utility model and had panel data to estimate the model.

(18.) The coefficient of the ratio of the virtual full income measures is an estimate of the inverse of [sigma! and has the expected positive sign. It is significant at the 15 percent level.

(19.) The estimated income elasticity is somewhat larger than those found in the literature for straighttime work, yet these elasticity estimates are not directly comparable since they are estimates of overtime responses.

(20.) However, among those working overtime, hours of work might fall.

(21.) Ehrenberg and Schumann [1982! have shown that increasing the overtime rate will weakly reduce an employer's demand for overtime hours, while our results indicate a weak increase in the supply of overtime hours. Taken together these results imply that a increase in the overtime premium will create a slight shortage of overtime hours, leading possibly to conflicts among employees, and between employees and management, with regard to the decision rules adopted to allocate these scarce hours. Nevertheless, these effects are not likely to be terribly pronounced given our relatively inelastic supply estimates and Ehrenberg and Schumann's inelastic demand estimates. In future work we hope to develop a general equilibrium model and jointly identify the overtime demand and supply curves.

(22.) A number of our key variables, especially the relative wage rate, have somewhat low significance levels, so caution should be exercised in drawing any definitive policy inferences from these estimates.

(23.) When the official poverty levels is used, both z values are always positive in our sample, yet when we set the poverty level at twice the official rate the z's became negative for three cases which were omitted from the analysis.

(24.) As the subsistence level is varied from zero to twice the official level, the uncompensated wage elasticity varies from .1269 to -.4316, the total income elasticity varies from -.5028 to -.8940, and the compensated wage elasticity varies from .6297 to .4624.

(25.) When the subsistence level is set at zero, the probabilities reported in Table IV change to .3019, .3362, .4591, .3254, .3493, .7898, .7741, and .9205 while when subsistence is set at twice the official rate the probabilities are .2783, .2720, .2697, .2879, .2975, .7844, .7668, and .9303.

(26.) In particular, in the loglinear specification the signs are positive for [lnw.sub.0!, negative for [lnw.sub.1!, negative for [lnz.sub.0!, and positive for [lnz.sub.1!. In the ratio specification, the signs are negative for [w.sub.1!/[w.Sub.0! and positive for [z.sub.1!/[z.sub.0!

(27.) See Altonji and Paxson [1988! for an analysis of the effects of hours constraints on labor supply and hours variability.

REFERENCES

Abbott, M. and O. Ashenfelter. "Labor Supply, Commodity demand, and the Allocation of Time." Review of Economic Studies 43, October 1976, 389-411.

Altonji, J. G. and C. H. Paxson. "Labor Supply Preferences, Hours Constraints, and Hours-Wage Trade-offs." Journal of Labor Economics, 6(2), April 1988, 254-76.

Barron, J. M., D. Black, and M. Lowenstein. "Employer Size: The Implications for Search, Training, Capital Investment, Starting Wages, and Wage Growth." Journal of Labor Economics, January 1987, 76-89.

Becker, G. "A Theory of the Allocation of Time." Economic Journal, September 1965, 493-517.

Brown, C. V. and E. Levin. "The Effects of Income Taxation on Overtime: The Results of a National Survey." Economic Journal 84, December 1974, 833-50.

Burtless, G. and J. Hausman. "The Effect of Taxation on Labor Supply: Evaluating the Gary Income Maintenance Experiment." Journal of Political Economy, December 1978, 1101-30.

Ehrenberg, R. "Absenteeism and the Overtime Decision." American Economic Review, June 1970, 352-57.

_____. "The Impact of the Overtime Premium on Employment Hours in U.S. Industry." Western Economic Journal, June 1971a, 199-207.

_____. "Heterogeneous Labor, the Internal Labor Market, and the Dynamics of the Employment-Hours Decision." Journal of Economic Theory, 3, March 1971b, 85-104.

Ehrenberg, R. and Schumann, P. Longer Hours or More Jobs. Ithaca, NY: New York State School of Industrial and Labor Relations Publications Division, 1982.

_____. "Compensating Wage Differentials for Mandatory Overtime." Economic Inquiry, October 1984, 460-78.

Goldberger, Arthur S. "Functional Form and Utility: A Review of Consumer Demand Theory." Systems Formulation, Methodology, and Policy Workshop Paper No. 6703, Social Systems Research Institute, University of Wisconsin, Madison, October 1967.

Hausmann, Jerry A. "The Econometrics of Nonlinear Budget Sets." Econometrica, November 1985, 1255-82.

Heckman, J. "Sample Selection Bias as a Specification Error." Econometrica, January 1979, 153-61.

Johnson, T. R. and J. H. Pencavel. "Dynamic Hours of Work Functions for Husband's, Wives, and Single Females." Econometrica, March 1984, 363-90.

Killingsworth, M. Labor supply, Cambridge: Cambridge University Press, 1983.

Maddala, G. S. Limited-dependent and Qualitative Variables in Econometrics. Cambridge: Cambridge University Press, 1983.