Female Labor Supply: Theory and Estimation

Female Labor Supply: Theory and Estimation


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Female Labor Supply: Theory and Estimation by James P. Smith, John F. Cogan

This collection of original essays brings econometric theory to bear on the problem of estimating the labor force participation of women. Five scholars here examine, both theoretically and empirically, the determinants of women's wages in the market, the value of their home time, and the factors that affect their employment.

Originally published in 1980.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Product Details

ISBN-13: 9780691042237
Publisher: Princeton University Press
Publication date: 04/21/1980
Series: Princeton Legacy Library Series
Pages: 400
Product dimensions: 6.46(w) x 9.54(h) x 1.13(d)

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Female Labor Supply: Theory and Estimation

By James P. Smith


Copyright © 1980 The Rand Corporation
All rights reserved.
ISBN: 978-0-691-04223-7




I. Introduction

There are ample reasons to refine nonexperimental cross-sectional estimates of labor supply parameters in light of continuing advances in conceptual and statistical methodology (Cain and Watts, 1973). First, from a policy standpoint, more precise parameters of long-run labor supply functions would help in the design and administration of welfare reform legislation.

Second, and central to the research strategy of this investigation, there is a growing awareness that many aspects of family decision-making over the life cycle are interrelated, implying that unexplained deviations or disturbances in individual behavior are related over time and across different behavioral outcomes (Nerlove and Schultz, 1970). Problems of simultaneous equation bias in disentangling behavioral relations underlying labor market behavior are undoubtedly more severe in the case of married women than of married men. For example, age at marriage, cumulative fertility, the presence of young children in the household, prior experience in the labor force, and market wage offers are all behaviorally interrelated with a wife's current labor supply decisions, and all are probably influenced by common exogenous factors. Inclusion of these relatively endogenous and in certain instances error-prone variables directly in the conditional analysis of labor market behavior is likely to have biased past estimates of the long-run responsiveness of married women's labor supply. To evaluate the consequences of proposed welfare reforms, for example, it is necessary to estimate not only short-run adjustments of labor supply conditioned on these endogenous variables, but also to consider long-run repercussions, allowing these other aspects of household behavior to adapt to new institutional and economic environments. Since the identification problems involved in this approach are formidable, it may be necessary to settle on estimating partially reduced form equations in which such outcomes as fertility or family composition are implicitly solved out of the estimation equation.

Third, many statistical problems in estimating labor supply functions call for special treatment, and though they have often been neglected in the study of men, they cannot be ignored in the study of married women. These problems occur because a large proportion of married women do not participate in the market labor force, and occupy a corner solution in their allocation of time between market and nonmarket activities. This volume addresses some of these issues: the discrete labor force entry decision, complicated by noncontinuous costs of entry; a distribution of market job options that is not independent over hours and wages; the time allocation process as essentially multidimensional in character, admitting to a variety of measurement conventions such as hours per day and week, and days and weeks per year. Nonlinear maximum likelihood estimation techniques seem more appropriate than ordinary linear regression to the analysis of dichotomous decisions, such as labor force participation, and limited dependent variables, such as hours worked. However, with these nonlinear models, simultaneous equation estimation techniques do not always have well-established statistical properties. Subsequent studies in this book address this problem at greater length, and develop estimation and computational techniques.

Where firm theoretical or convincing empirical guidance is now lacking, the specification and estimation of labor supply models will remain controversial. For example, how is one to choose among identifying restrictions? The choice of instruments to identify one relationship may merely introduce other endogenous variables resulting in biased and unstable parameter estimates. Where are nonlinearities and discontinuities important, and how should predictive errors be compared across methodologies? Which variables are appropriately treated in a particular context as endogenous and which as exogenous or at least predetermined? I suspect that the answers to these questions will emerge only as alternative approaches to new sources of survey data begin to evidence regularities that are insensitive to seemingly minor matters of variable definition, sample composition, and estimation technique. Much work remains to be done before this stage of consensus is achieved, but there has also been palpable progress in this direction, as documented in this volume.

To estimate labor supply parameters that could be useful to policymakers, a number of relatively strong assumptions are needed. If simple and sophisticated formulations of the labor supply model lead to essentially the same outcome, those sources of bias implicit in the simple approach can be pragmatically neglected, while refinement of model specification and estimation proceeds in those areas where different approaches imply substantial differences in policy-relevant parameter estimates.

Section II of this paper presents the time-allocation-labor supply framework and explores alternative specifications for the underlying structural equations determining market demand and individual supply functions. Section III discusses the empirical specification of the model and the data used to obtain the estimates reported in Section IV. The fifth and final section sums up how these empirical findings differ from earlier studies and discusses their implications for policy and further research.

II. The General Framework and Problems of Estimation

The objective of any labor supply study is to explain the decision of the wife whether to participate in the market labor force, P, and if she does, the number of hours to work in this activity per year, H. The expected supply of labor, L, can then be expressed as the product of the expected probability of participation and the expected hours worked, conditional on participation. Consumer demand theory, when simplified to a single period framework without uncertainty, implies that an individual's demand for time to engage in nonmarket activities can be expressed as a function of his own market wage offer, market prices, nonearned market income, and peculiar tastes and talents that are initially assumed to be distributed randomly across populations. To extend this framework to a two-person household, in particular that of a husband and wife, only one additional market price variable, the wage offer available to the other spouse, must be added to the list of factors determining the reduced-form demand equations for nonmarket time if both spouses engage in some market activity (Heckman, 1971; Ashenfelter and Heckman, 1973).

This classical demand approach to labor supply functions yields several well-known predictions. The income-compensated own-wage effect of either spouse on the household's demand for his nonmarket time is negative (i.e., own-price effect). The influence of increased real income, holding household technology, market wage offers, and prices constant, is to increase or to decrease demand for nonmarket activities depending on whether the final (unobserved) composite product of nonmarket activities is a normal or inferior good. The income-compensated cross-wage effect, namely, the compensated effect of one spouse's market wage offer on the household's demand for the other spouse's nonmarket time, is positive or negative depending on whether nonmarket time of one spouse is a substitute or complement for the nonmarket time of the other spouse in the household. Regardless of the sign, symmetry conditions imply equality for both compensated cross-wage effects. When one estimates the mirror image of the demand for nonmarket time, or the supply of labor to the market, all of the above signs of the effects are, of course, reversed.


The labor supply behavior of each spouse is determined by two relationships: (1) the market wage offer or market demand function, W, and (2) the shadow value of time in nonmarket activities or the individual's supply function, S.

W = f(Z, ε1), (1.1)

S = g(X, ε2), (1.2)

where Z and X are vectors of possibly overlapping endogenous and exogenous variables that affect the market demand wage, W, and the individual supply wage, S, respectively, and ε1 and ε2 are two, not necessarily independent, normally distributed random disturbances that undoubtedly embody errors in measurement, the effects of many minor omitted variables, and purely stochastic variability. Both of these relationships are probably a function of the number of hours worked in the market. The inclusion of at least one variable among the Z's that is excluded from the X's, and vice versa, permits one to identify statistically the supply from the demand functions in the household labor market.

An interior equilibrium occurs if the schedules intersect and the slope of the individual supply wage function, with respect to hours worked, exceeds (positively) the slope of the market demand wage function.

If the two wage functions are assumed linear, as illustrated in Figure 1.1, we have the following

Wi = α0 + α1Hi + α2Zi + ε1i i = 1, ..., n (1.3)

Si = β0 + β1Hi + β2Xi + ε2i (1.4)

where i refers to individuals and the α's and β's are parameters to be estimated.

When no market work is undertaken, the reservation or supply wage for entry into market work is denoted S0, where H = 0. The market offer provided for the first hour worked in the market is similarly denoted W0. The individual participates in the labor force if W0 exceeds S0, and works some amount, say H*, in Figure 1.1, where the marginal value of time in market and nonmarket activities is equal. For these interior equilibria W* = W = S for the observed H*. The individual does not participate if W0< S0, and in this case the levels of both the supply and the demand functions are not directly observed.

Solving equations 1.3 and 1.4 for the labor supply, one obtains


if α0 - β0 + α2Zi - β2Xi > ε2i - ε1i, (1.6)

and otherwise,

Hi = 0.

Market participation occurs if inequality 1.6 obtains, in which case the number of hours worked in the market is determined by equation 1.5. Otherwise, hours worked is definitionally zero for nonparticipants.

Expected hours worked is proportional to the difference between the systematic components of the individual's demand and supply functions, if this difference is positive, where the factor of proportionality, 1 / (β1 - α1), is the slope of this excess labor demand schedule with respect to hours worked in the market. Illustrated in Figure 1.2, this framework lends itself to maximum likelihood estimators proposed by Tobin (1955, 1958).

Unexplained individual deviations in the hours worked decision (about the Tobit Index, T) are reduced to the composite disturbance

υi = ε1i - ε2i/β1 - α1,

that is assumed normally distributed with zero mean, constant variance, and, of course, distributed independently of X and Z.

The expected value locus of H given Y, when there is no residual variance, i.e., σ2υ = 0, is two line segments abc. But allowing disturbances, i.e., σ2υ > 0, the expected value locus of hours worked per person, which is the market labor supply, becomes a nonlinear function of Y, approximately a'b'c in Figure 1.2. Individuals do not enter the market labor force if Ti > - υi. At point b, where the Tobit index equals zero, half of the population participates in the market labor force, and participants work, on average, twice as many hours as expected for the entire population.

This general framework is used here for interpreting labor market behavior. Though not always expressed in precisely these terms, most previous attempts to estimate "labor supply functions" can be viewed as imposing particular restrictions on this general model in order to obtain parameter estimates that describe one or both dimensions of the labor supply phenomenon.


From an economist's perspective, the most important explanatory variable in a labor supply equation is the market wage rate. In this section, I discuss some problems in using available information on market wage rates. First, observed wage rates are derived in most data sources by dividing market earnings by hours worked in the labor force during the relevant period, either a year or a week. Errors in reporting hours worked are thereby transmitted inversely to the measured market wage rate, biasing down directly estimated effects of the measured wage rate on labor supplied to the market.

A second problem in obtaining an adequate measure of wages for secondary workers is the simultaneous equations bias. Past decisions regarding labor market participation and market- or nonmarket-specific training will be reflected over time in current market wage offers. But since these earlier decisions are jointly and simultaneously determined with current labor market behavior, the current market wage outcome is itself endogenous. To estimate long-run labor supply response parameters, an appropriate exogenous wage rate would be wage opportunities available to an individual at the outset of the marriage and adult life-cycle planning period. This source of parameter bias implies that for at least older married women, the direct relation between an error-free measure of current market wages and market labor supply would be positively biased (overstated) from the "true" relation that is sought here between life-cycle wage opportunities and labor supply.

A third and related source of parameter bias stressed by Mincer (1962) is the confounding effect of transitory variation in market wages, which would also appear to bias up (positively) compensated own-wage effects and bias down the wealth or pure income effects embodied in the effect of current wages on current labor supply.

Finally, variation in marginal taxes on market earnings and prices (the market purchasing power of money) may systematically affect the real market wage across the sample, adding additional potentially endogenous complications to measuring the appropriate market wage variable.

These problems of measuring, without systematic error, permanent (life-cycle) market wage opportunities may be resolved, or at least alleviated, by replacing the observed market wage rate with an instrumental variable estimator inferred from the sample of workers for whom market wages are observed (see Hall, 1973). This auxiliary instrumental variable equation might be interpreted, under certain assumptions, as the market demand structural equation 1.3, having been estimated by single equation techniques. Gronau (1974) has stressed that this interpretation neglects selectivity bias, for unexplained variation in market wages will tend to be associated with condition 1.6, and hence, with inclusion of the observation in the working sample. In this case, the parameter estimates of the instrumental variable market wage equation do not provide a consistent or asymptotically unbiased basis to infer market wage offers for the population of workers or nonworkers.


But how serious is this bias? If the residual variation in the correctly specified equation 1.3, namely, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], could be entirely attributed to errors in measuring wages, and therefore, were independent of the selection of the observation for the "working" sample, there would be no selectivity bias. Also, as labor force participation within a group became universal, the bias would vanish as the observed wage for any Z approached the expected value of the market wage offer for that Z. Other things being equal, the bias would therefore appear more serious in the analysis of secondary workers such as married women than for the study of market wages of males between the ages of 25 and 55.

More generally, if an instrumental variable in the market wage equation is correlated with the probability of participation, its coefficient in the market wage equation will tend to be biased by the systematic censorship of data. Moreover, since supply determinants, in addition to demand determinants, might plausibly influence observed market wages, it is not clear whether a market demand wage equation or a reduced form equation inclusive of both demand and supply factors should be estimated. Several examples of different sources of selectivity bias may illustrate the problem.

Some variable in Z might be omitted from the estimated demand function, such as ability (Griliches and Mason, 1972; Hause, 1972), quality of schooling (Welch, 1973), or taste for market work, and persons with this "ability" might be offered higher market wages and be more frequently found in the "working" sample, other things being equal. Even if this "ability" is uncorrelated with explanatory variables included in the market demand function, intercept estimates based on the censored "working" sample are biased because the criterion for censorship depends positively on "ability." In this instance, the "predicted" market wage offer for nonworkers would tend to exceed the "expected" market wage offer. Observed proxies for "ability" might be used, or information on individuals over time assembled, in order to infer the nature of the covariance structure between unobserved components in the residual disturbances.


Excerpted from Female Labor Supply: Theory and Estimation by James P. Smith. Copyright © 1980 The Rand Corporation. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents

  • FrontMatter, pg. i
  • Contents, pg. vii
  • Preface, pg. ix
  • Introduction, pg. 1
  • Chapter 1. Estimating Labor Supply Functions for Married Women, pg. 25
  • Chapter 2. Married Women's Labor Supply: A Comparison Of Alternative Estimation Procedures, pg. 90
  • Chapter 3. Hours And Weeks In The Theory Of Labor Supply, pg. 119
  • Chapter 4. Assets And Labor Supply, pg. 166
  • Chapter 5. Sample Selection Bias As A Specification Error With An Application To The Estimation Of Labor Supply Functions, pg. 206
  • Chapter 6 A Multivariate Model Of Labor Supply: Methodology And Estimation, pg. 249
  • Chapter 7 Labor Supply With Costs Of Labor Market Entry, pg. 327
  • Bibliography, pg. 365
  • Index, pg. 377

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