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The life-cycle labor supply of married women and its implications for household income inequality.

I.INTRODUCTION

There has been an abundance of papers on female labor supply over the years, but most do not rigorously test the life-cycle model of labor supply, and none focus on the potentially important role of nonlabor income in shaping labor supply.(1) Their primary focus is on the wage elasticity, and, most recently, on the intertemporal wage elasticity. The intertemporal wage elasticity, or the change in labor supply as wages change over the woman's life-cycle, is economically cleaner to interpret because it contains only a substitution effect (the income effect is constant over the life-cycle). But this advantage comes at a cost. We learn nothing about why labor supply varies across women. Such variability is extremely important for policy purposes, because, for example, it has implications for income inequality. The popular press often creates the impression that income inequality is worsening, as upper income men marry upper income women. The implication is that the rising labor force participation rate of women is increasing family income inequality. On the other hand, it is possible that the income of wealthy husbands exerts a negative income effect on their wives' labor supply, so income inequality is little changed or declines as participation rates of wives rise.

The objectives of this paper are twofold: to estimate a life-cycle labor supply model with panel data that emphasizes the husband's income effect on the wife's hours of work, and to draw implications for family income inequality. I provide new estimates of the income elasticity that nest the life-cycle hypothesis of labor supply with the current-income hypothesis. The life-cycle model suggests that it is the permanent income of the husband that matters, not his current income, yet nearly all models of female labor supply include only current income (the exception is Jakubson |1988~). I include both and add a varying-parameters specification for changes in tastes. The implications for income inequality arise as an application of the life-cycle labor supply model. A summary of the empirical results is contained in the conclusion. This paper is unusual in that it covers two relatively disparate topics--though their linkage is important and demands their integration in one paper.

II. SPECIFICATION OF THE LABOR SUPPLY MODEL

Following the standard life-cycle labor supply model, assume that an individual maximizes lifetime utility, V, where utility is separable over time and separable between consumption and leisure:(2)

|Mathematical Expression Omitted~

where C and L are consumption and hours of leisure for individual i, |Rho~ is the rate of time preference, ||Gamma~.sub.it~ reflects consumers' tastes (for home production, leisure, etc.), and |Omega~ |is less than~ 1 is fixed across individuals. The lifetime budget constraint is

|Mathematical Expression Omitted~

for interest rate r, wage rate |W.sub.it~, hours of work |H.sub.it~, and the present value of nonlabor income, |A.sub.i~. For an interior solution, the leisure equation is

(2) ln |L.sub.it~ = (1/(|Omega~-1)) |ln ||Lambda~.sub.i~ - ln ||Gamma~.sub.it~ - ln |Omega~ + ln |(1 + r).sup.t~ |(1 + |Rho~).sup.-t~ + ln |W.sub.it~~

where ||Lambda~.sub.i~ is the marginal utility of lifetime wealth.

The marginal utility of wealth, ||Lambda~.sub.i~, is a fixed effect that captures all the permanent-income effects that influence labor supply, including expected lifetime potential wages, and lifetime nonlabor income. For the labor supply of a married woman, the largest wealth effect is likely to be her husband's income as it enters through |Lambda~: an increase in the husband's lifetime earnings will decrease the marginal utility of lifetime wealth and decrease the wife's labor supply.(3) Incorporating this relationship in the parameterization of |Lambda~:

|Mathematical Expression Omitted~

where |Mathematical Expression Omitted~ is the husband's lifetime income, education |E.sub.i~ is a proxy for the wife's potential lifetime wages, and |e.sub.1i~ is i.i.d. across individuals. Making distributional assumptions about this fixed effect, ln ||Lambda~.sub.i~, is the only means of estimating the effect of nonlabor income: Differentiating the data would eliminate the fixed effect and thus the nonlabor income elasticity; and the assumption of a random-effects model that g() is independent of the right-hand-side variables in (2) is violated by the theory. Analogously, time-varying tastes for labor supply, ln ||Gamma~.sub.it~ in equation (2), are

(4) ln ||Gamma~.sub.it~ = h (|CH.sub.it~, |AGE.sub.it~, |E.sub.i~) + |e.sub.2it~

where C|H.sub.t~ is the number and ages of children, and |e.sub.2it~ is i.i.d. across individuals and time.

Substituting (3) and (4) into (2), the equation to be estimated is

|Mathematical Expression Omitted~

where ||Epsilon~.sub.it~ = |e.sub.1i~ + |e.sub.2it~, |B.sub.0~ is an intercept containing (-ln |Omega~) and unobserved factors in |Lambda~ orthogonal to |e.sub.1i~, and ||Tau~.sub.t~ is a set of time dummies to control for ln |(1 + r).sup.t~ |(1 + |Rho~).sup.-t~.(4)

The important feature of this hours specification is that the husband's current income does not influence female labor supply. Current income diverges from permanent income due to transitory changes in wages and hours and due to life-cycle variation in investment. In a perfect certainty life-cycle model, the husband's permanent income operates entirely as a wealth effect, influencing the overall level of the wife's labor supply, but not its intertemporal variability. However, there are several reasons why the husband's current income may add predictive power to the life-cycle model. If uncertainty is introduced, current income may be a proxy for changes in permanent income. Second, if the household is liquidity constrained (i.e., one can't borrow against future income), there may be an "added worker" effect--the wife may increase her hours of work to supplement family income during periods of reduced male earnings (if she cannot find employment, it creates a discouraged worker effect). Last, if female hours of work are constrained on the downside, associated with employer restrictions on hours or with fixed costs of employment, hours cannot adjust at the margin within subperiods and may instead adjust intertemporally.(5)

To characterize the model econometrically, note that women not working are at a corner solution when reservation wages exceed market wages:

|Mathematical Expression Omitted~

where |Mathematical Expression Omitted~ is maximum hours of leisure and the transformation of the leisure equation to |Mathematical Expression Omitted~ sets the mass point at 0 for zero hours of work, following Jakubson |1988~, and the equation for |L.sub.it~ is (5). Our inability to observe wages for nonworking women suggests a comparable wage equation

|Mathematical Expression Omitted~

where X is a standard set of variables determining productivity and |u.sub.it~ is distributed i.i.d. over i and t.

Two methods of estimating the labor supply parameters arise from (6) and (7). The first method of estimating labor supply is the Tobit estimation of (6). The drawback of this method is that the structure of (6) assumes continuity between the decision to work and the hours of work by restricting the |Beta~ coefficients to be the same for both decisions. The fixed costs of working suggest that the decisions are not continuous. The second method is the "Heckit" (or generalized Tobit) estimation of (6), in which (6) is estimated for working wives with the Heckman |1979~ sample selection correction that inserts the inverse of Mills' ratio to control for the conditional error term structure E(|u.sub.it~/|H.sub.it~|is greater than~0). Heckit and Tobit methods are contrasted below. Due to the likely endogeneity of wages in the hours equation, wages are replaced by |Mathematical Expression Omitted~ from (7).

III. ESTIMATES OF THE LABOR SUPPLY MODEL

Fourteen years, 1968-1981, of the Panel Study of Income Dynamics (PSID) are used to estimate the model of the labor supply of married women. In the process of surveying households, the PSID introduces new households into the sample as children age and form their own households. Thus, unlike most panels, the PSID panel contains young households and new entrants. The wives that are used in the sample below are only those who have been married to the same spouse for at least four years and are in the 18-64 age range. After dropping the low-income subsample, there are a total of 19103 family-years of data.(6)

The dependent variable for the labor supply model is |Mathematical Expression Omitted~, where H is annual hours of work and |Mathematical Expression Omitted~ is 365 x 24=8760 hours. The equation for the fixed effect, or the marginal utility of lifetime wealth in equation (3), is the most important component of the model. It is assumed that the predominant variables that affect this marginal utility are the husband's permanent income and the wife's education (perhaps representing personal tastes for income). As described in footnote 3 above, other forms of nonlabor income (returns on assets, income subsidies) may be endogenous to labor supply and so are omitted from the equation for lifetime utility. The husband's permanent income is defined as

|Mathematical Expression Omitted~

where T varies from a minimum of four years to a maximum of fourteen years for complete panel data (about 40 percent of the sample has complete data; the average T is 9.6). To introduce uncertainty in the model, the husband's current income is added as the deviation from the fixed effect:

|Mathematical Expression Omitted~

This is a variant of a fixed-effects model. Fixed-effects models are increasingly used with panel data now that the number of panel years have increased for many data sets (e.g., see Keane |1989~). Other variables in the labor supply equation are listed in the footnote to Table I.

The coefficients and elasticities of nonlabor income in the life-cycle labor supply model are presented in Table I. In these nonlinear models, elasticities must be evaluated at mean values of hours of work. A one-percent increase in the husband's permanent income causes the wife's labor supply to fall by .48 percent, at mean hours of 800.(7) This is a very sizable elasticity, as will be shown in section TABULAR DATA OMITTED IV below. Moreover, the permanent-income variable is extremely significant statistically--its t-statistic is 15.0, far exceeding the significance of all other variables except children. This life-cycle model is compared to the current-income model in row 2 by adding deviations of current income from permanent income. Current income does have a significant effect (the reduction in the log-likelihood is extremely significant), yet the significance of permanent income is little changed.(8) Referring back to the interpretation of the current-income effect in section II, its negative elasticity suggests that either it induces reductions in labor supply because it provides new information on the husband's permanent income, or it causes an "added worker" effect--the wife works more when the husband's income declines because they are liquidity constrained in borrowing against his future permanent income (rejecting the discouraged worker effect).

Another way of incorporating permanent-income effects is to include lags in the husband's income in the wife's labor supply equation. The advantage of doing so is that if future income is highly uncertain, YP, the husband's permanent income, may be mismeasured. The drawback of doing so is the loss of degrees of freedom--the panel years must be shortened to include lags in the husband's income (in this case, from fourteen years to eight years for six lags). The information on permanent income then covers a shorter time period and thus may be less representative of permanent income. This would be especially true for younger men, whose earnings are growing, and whose families can readily anticipate such growth. The coefficient on mean lagged husband's income is in row 3.(9) It produces almost the same elasticity as the fixed-effect permanent-income variable (though with a larger standard error)--suggesting that six lags in income may be a sufficient approximation of permanent income relative to the variable YP, the average over a longer period.

As is often reported by other researchers, such as Schultz |1980~ and Mroz |1987~, female labor supply elasticities calculated from Tobit models are larger than those from Heckit models which condition the sample on the wife working. The elasticities of female labor supply with respect to husband's permanent income are -.51 in the Tobit model and -.19 in the Heckit model (rows 2 and 4).(10) The model of section II implies that the Heckit and Tobit models should produce the same elasticities, because both assume hours are a linear function of the difference between market wages and reservation wages. In practice, Tobit elasticities are often larger because the probability of participation is more sensitive to the husband's income,(11) or there are discontinuities in labor supply, as in Cogan |1980b~ and Mroz |1987~. Cogan estimates a model of labor supply with the discontinuity associated with the fixed costs of labor supply (the discontinuity could also arise from fixed costs of labor demand in a constrained system). He finds a pronounced discontinuity at 1151 hours, implying that women tend to enter the labor market by working more than half-time, but once they are working, their hours are not very responsive to wage rates. The result is that the Tobit model will tend to underpredict hours of work at the low end of the hours distribution and overpredict them at the high end. Unfortunately, there is no standard solution to the discontinuity problem--Cogan estimates a maximum likelihood function with an assumed functional form for the minimum hours equation. The problem is that women are making their labor supply decisions at two different margins--the extensive margin of whether to participate and the intensive margin of how many hours to work given participation. The Heckit model estimates only the intensive margin and underpredicts labor supply variability. At the extensive margin, the probability of participation can be modeled as a probit equation, assuming a normal distribution for errors. It is difficult to combine these in one estimate of the labor supply elasticity, since minimum hours constraints that determine participation vary across women as a function of unobservables. Schultz |1980~ finds that the Tobit husband's wage elasticity is very similar to the sum of the separate elasticities for the participation and hours decisions. The approach pursued here is to report probit and Heckit results, but to focus on the Tobit model for several reasons: the nonlinear probit model cannot produce elasticities; the Tobit provides a summary elasticity of labor supply for both extensive and intensive margins; and, for the purposes of comparison, the Tobit is still the most frequently used methodology, as in Jakubson |1988~.

Before turning to the income distribution application, let us consider how well the model explains the dramatic aggregate trends in female labor supply. The average annual hours of work for all women ages 30-34 rose from 627 in 1956 to 1081 in 1980 (Killingsworth and Heckman |1986, 118~). Though large increases in participation occurred prior to 1968, in the PSID data the participation rate for married women rose from .54 to .64 from 1968 to 1981. Any formal model of female labor supply should be able to explain these increases. Possible reasons for the increases are divided into changes in right-hand-side variables (wages, income, children), and changes in the sensitivity to these variables (tastes). I will focus on changes in tastes, since changes in the distributions of wages, income, and children appear to be too small to explain the large changes in labor supply using the functional forms estimated in Table I.(12)

Changes in tastes for work enter the formal model as time-varying individual differences in the power on leisure, |Omega~.(13) In equation (2), this produces a constrained nonlinear model in which variation in |Omega~ shifts the coefficients on all right-hand-side variables. I hypothesize that |Omega~ varies by age group and time period, so that younger cohorts may have greater tastes for work than older ones. To conserve degrees of freedom, I try two time periods, 1968-1974 and 1975-1981, interacted with two age groups, ages 18-32 and over 32. Estimates of the constrained nonlinear model (2) are rejected in favor of an unconstrained model in which age and period effects interact separately with right-hand-side variables.(14)

The varying-parameters model results indicate very pronounced demographic differences in the sensitivity of labor supply to the husband's permanent income. Looking at cohort effects in Table II, the elasticity at the mean for young women (age 18-32) falls from -.49 in 1968-74 to -.29 in 1975-81, while the elasticity for older women (age |is greater than~ 32) also declines very substantially over time, though it remains at higher levels than that of young women, going from -.78 to -.45. Thus, the elasticity weakens by 42 percent for both age groups, so the increasing labor supply of women over time reflects, in part, a substantially decreased sensitivity to their husband's income.(15) Comparing the probit and Heckit results, it is clear that most of the change is associated with greater labor force participation.

These estimated demographic differences in the income elasticities are ascribed to variation in tastes, but may represent structural changes as well as changes in tastes. Based on aggregate statistics, a number of structural changes in the economy occurred from 1965 to 1981: the divorce rate of women doubled; husbands' real income growth rates plummeted; birth rates fell by about 35 percent;(16) and far more women entered the upper tail of the educational distribution--more entered college and the percentage of law and M.D. degrees obtained by women rose from about 7 percent to 30 percent.(17) Some of these structural changes are surely endogenous to changes in labor supply, but they also suggest that the changes in tastes may represent increased incentives for women to develop permanent attachments to the labor force that are little influenced by their husbands' incomes. What has changed is that wives are more likely to lose their husbands' incomes via divorce, there are fewer children to care for, women are investing in skills that have higher wage rates and higher depreciation rates, and male incomes are more variable. In general, changes in "tastes" may represent less need for home production and greater returns to market work. These considerations suggest several testable hypotheses:

Women with young children may be more sensitive to their husband's income because the value of their home production is greater. Interactions between the number of children and the husband's permanent income are highly significant: young mothers are much less TABULAR DATA OMITTED likely to work when their husbands earn more.(18) However, the decline in the income elasticities over time is even more pronounced after controlling for interactions with children: income effects today are less sensitive to the presence of children.

The sensitivity of the wife's labor supply to the husband's income may decline with her education, because more educated wives are investing more in market skills. In fact, there are no significant educational differences in the husband's income elasticity. The interaction is surprisingly negative--more educated women have larger income effects--but insignificant across ages and vintage (not shown).

As was true above, these interactions are more sizable in the participation probit and are smaller for the hours Heckit. In general, the declining income elasticities with age and time are likely to be a reflection of changing tastes and changing structural factors, and one cannot readily disentangle the differences.(19)

In summary, I have found strong evidence supporting the role of permanent-income effects in the life-cycle model of labor supply, but the possibilities of liquidity constraints or imperfect information also introduce a lesser role for current income in female labor supply. Even more dramatic is the evidence that the wife's response to the husband's permanent income has shifted down over time: the income elasticity fell by 42 percent from 1968-74 to 1975-81.

IV. IMPLICATIONS FOR INCOME INEQUALITY

Many have been concerned that family income is becoming less equally distributed as more women enter the labor force, based on the notion that similarly able men and women marry ("assortative mating"). Previous empirical results suggest that women have had little effect on family income distribution, or if anything, cause it to be more equally distributed.(20) The crucial determinant of the extent of family income inequality is the elasticity of female labor supply with respect to the husband's income. If there is a very small husband's income effect on the wife's labor supply, the incomes of husbands and wives will be highly correlated (if there is assortative mating), and the rise in female labor force participation will increase inequality. On the other hand, a sizable husband's income effect would lessen married households income inequality, because the wives of men with high incomes would work less. This section applies the results of the labor supply model to issues of income inequality.

The magnitudes of the supply elasticities estimated here are large enough that, when combined with the very large cross-sectional differences in husbands' incomes, they can produce very dramatic cross-sectional differences in wives' labor supply.(21) An illustrative example of potential labor supply outcomes emphasizes this point.

Start this example with a 60 percent increase in the husband's permanent income, which is approximately equivalent to a one standard deviation change, or to the difference between median income and the income of the 90th percentile. This increase results in a drop in the wife's average hours from 800 (at the mean) to 554 based on the estimated elasticity of the wife's labor supply with respect to the husband's permanent income.(22) There are two ways of looking at the effect on female incomes. If husbands' and wives' wages are uncorrelated, the average wife's income falls 31 percent when she reduces her labor supply, falling from $6029 to $4175 in 1989 dollars. On the other hand, the wages of husbands and wives are actually highly correlated--the simple correlation between them is .24.(23) Using simple regression analysis to predict wives' wages as a function of husbands' wages, a 60 percent increase (cross-sectionally) in the husband's wage is likely to be associated with a 14 percent increase in the wife's wage. Therefore, when we compare the labor supply of the average wife to that of the wife whose husband's income is 60 percent greater and her wage is 14 percent above average, the husband's income effect is more sizable: the wealthier wife's income falls from $6849 to $4743, a decline of $2106 relative to the decline of $1854 in the example above (all else constant--there is also the wife's own wage effect, but it is generally estimated to be very small). Thus, the husband's income exerts a substantial income equalizing effect, relative to the distribution of family income had there been no husband's income effect. As a result, the correlation between the actual incomes of husbands and wives is only .085, compared to a correlation of .24 between their expected wages.

The wife's labor supply estimates above show that her elasticity with respect to her husband's permanent income has weakened by 42 percent from 1968-74 to 1975-81--going from -.782 to -.449 for women older than 33. The implication is that family income inequality should worsen as the importance of the husband's negative income effect declines. The standard measure of income inequality used in this literature is the coefficient of variation (CV = |Sigma~ / mean). The reason is that the coefficient of variation is scale free and the family's coefficient can be written as a function of the coefficient of variation of the husband and wife, C|V.sup.h~ and C|V.sup.w~, to examine the wife's contribution to inequality:(24)

C|V.sup.2~ = |(1 - |Alpha~).sup.2~ |(C|V.sup.h~).sup.2~ + ||Alpha~.sup.2~|(C|V.sup.w~).sup.2~ + 2|Alpha~(1-|Alpha~)|Rho~C|V.sup.h~C|V.sup.w~

where |Alpha~ is the wife's share of family income and |Rho~ is the correlation between the spouses' incomes. Each component of this equation is displayed in Table III, including a breakdown by time period and age group. On average, from 1968 to 1981, wives have had an equalizing effect on household income, resulting in a family coefficient of variation of .68 relative to the husband's coefficient of .75. But this average hides a move towards greater inequality that is consistent with the changing female labor supply elasticity.(25)

TABULAR DATA OMITTED

Family income inequality rose a small amount from 1968-74 to 1975-81.(26) Inequality worsened because the correlation between husbands' and wives' incomes increased dramatically from .037 in the first period to .112 in the second, with a more pronounced increase for younger women (.024 to .145).(27) The increased income correlation must reflect the evidence presented above that wives are becoming less sensitive to their husbands' incomes.

Family income inequality would have worsened further due to the rising correlation between husbands' and wives' incomes, but this effect is partially offset by a decline in the variance of female earnings. Greater entry of women into the labor force has tended to equalize their wages and hours, resulting in a decline in their coefficient of variation from 1.38 to 1.36.(28) As younger vintages of women, such as the baby-boom generation, increasingly dominate the labor force, their weaker husband's income effect will increase inequality through the higher income correlation |Rho~, unless it is offset by further equalization of women's wages and hours.

IV. CONCLUSION

The life-cycle labor supply model implies that the husband's permanent income will shape the wife's labor supply; his current income will alter labor supply only if it provides new information on his permanent income or if the household is liquidity constrained. Models of consumption behavior have emphasized the distinction between permanent and current income, but it has received little attention in estimates of female labor supply--previous labor supply models focus on current income. Using fourteen years of the Panel Study of Income Dynamics, I estimate a life-cycle labor supply model in which the husband's permanent income is his lifetime average for the fourteen observed years. His permanent income has a very significant and large negative effect on the wife's labor supply, dominating current income. Current income does have an additional, though weaker, negative effect.

Next, I amend the life-cycle model by adding a nonlinear varying-parameters specification that introduces changes in tastes for work. The hypothesis is that the rising participation rates of women reflect a greater preference for work among younger cohorts. In fact, the labor supply of younger, more recent vintage women is far less sensitive to their husbands' incomes; their income elasticity falls to -.29, relative to -.78 for the older cohort.

This change in tastes for labor supply has important implications for income distribution. The size of the husband's negative income effect is crucial in assessing the impact of female earnings on family income inequality: given that married spouses tend to be of similar ability, a pronounced negative income effect on the wife's earnings will depress the earned income of more able households and thus equalize the distribution of income. In the past, the negative income effect has clearly acted to equalize earnings--actual earnings of husbands and wives are much more weakly correlated than are their wage rates. The finding that the husband's income effect is diminishing implies the income inequality should worsen,(29) but this effect has been offset by a declining variance of wives' earnings. Previous researchers have not explicitly attributed changes in income inequality to causal factors uncovered in a formal life-cycle labor supply model.

1. See Killingsworth and Heckman |1986~ for a review and references.

2. This is a standard model for men--see MaCurdy |1981~. For women see Heckman and MaCurdy |1980~, and Jakubson |1988~. When the model is applied to women it rests on the crucial assumption that the birth of children and the husband's income are exogenous to the wife's labor supply decision. In very careful econometric tests, Mroz |1987~ could not reject these assumptions. Thus, while there are certainly elements of endogeneity, they were found to be very weak empirically.

3. Other forms of income--such as Aid to Families with Dependent Children--are omitted from the |Lambda~ equation because they are often conditional on labor earnings (note that they are only 3 percent of average family income in the data below). Income from wealth accumulated while working is also inappropriate in a life-cycle labor supply model because assets are a function of past labor supply. Smith |1980~ provides an excellent summary of the volatility of estimated income effects when income includes assets, and then develops a model of life-cycle asset accumulation that the estimation of asset effects requires.

4. Few studies of female labor supply have acknowledged the importance of a fixed effect in the labor supply equation. Jakubson |1988~ focuses on the fixed effect, estimating a conditional random-effects model and a fixed-effects model. The conditional random-effects model conditions on children and other income and assumes that all other determinants of |Lambda~ are uncorrelated with all observed variables. He finds that lifetime variables do matter, but that there are no significant differences between random-effects and fixed-effects models. MaCurdy |1981~ produced the original fixed-effects estimates for male labor supply and Heckman and MaCurdy |1980~ for female labor supply, using a fixed-effect tobit model. MaCurdy first-differences his labor supply equation to omit |Lambda~, then solves for |Lambda~ and utilizes this proxy of |Lambda~ to develop estimates of parametric wage elasticities. Mroz |1987~ omits fixed effects. The equation estimated herein, (5), imbeds the individual random effect |e.sub.1~ in error term |Epsilon~, so in estimation of (5) there is a loss of efficiency but no harm to consistency. Efficiency losses are likely to be small given the large data sets used below.

5. For evidence that fixed costs alter female labor supply, see Cogan |1980b~.

6. The low-income subsample is dropped because the weights supplied by the PSID are not valid when I restrict the sample to women married continuously for at least four years. If the low-income sample were included, weights would be necessary in the labor supply regressions because there may be omitted variables that explain the differential labor supply patterns of low-income women. The random portion of the PSID is a very large data set (N=19103) and contains low-income women. However, if sample attrition is higher for low- income women, the estimated coefficients may be subject to some sample censoring bias.

7. Mean hours of all wives is 795, mean hours of working wives is 1310. Due to the nonlinearity of the model, income elasticities of working wives are lower than those for all wives.

8. In this fixed-effects model, the coefficient on deviations from current income is the same as the coefficient on the current income level. Previous work on the effect of nonlabor income has predominantly used current income, the only exception being Jakubson |1988~, who includes lags in income. However, in Jakubson and almost all other models, the measure of nonlabor income includes income from dividends, other family members, and social welfare payments, which I argue above are endogenous to the current labor supply and would introduce an upward bias in the income elasticity. The appropriate asset measure would be initial assets, prior to labor market entry, but it is never available. Therefore, I prefer to focus only on the husband's earnings. One other researcher reporting the husband's income effects is Cogan |1980a; 1980b~ for current income. He finds current income elasticities in the Tobit and Heckman maximum likelihood models to be -.29 and -.30, but his sample includes only National Longitudinal Survey white women age 30-44, an age when income effects are likely to be stronger due to child-rearing. In Cogan |1980a~, he uses PSID data to get elasticities of -.23 and -.085 for the Heckman model and the fixed-cost model, respectively (the latter is discussed more below).

9. Six unconstrained separate lags in husband's income were also tested, but rejected by a likelihood ratio test in favor of the constrained mean lag income variable.

10. The probit equation used to calculate the increase of the Mill's ratio in the Heckit model differs from the one reported on line 5 of Table I, in that the former contains several variables not included in the strict theory of the life-cycle labor supply model (like husband's wage) to improve the predictive power of the probit model and identify it relative to the linear labor supply model.

11. Turning to a probit model for the probability of participating (row 5), at mean values of all right-hand-side variables, participation falls from .61 to .47 when the husband's permanent income rises 60 percent (one standard deviation), for an elasticity of -.38. The nonlinearity of the probit model makes it impossible to calculate one elasticity that is comparable to those of the Heckit and Tobit models.

12. The mean values (standard deviations) are (in 1974 dollars):
 1968 1981
Wife's wage 2.45(1.63) 2.51(1.92)
Husband's income 8811(4530) 9028(5490)
Number of children 1.54(1.34) 1.38(1.47)


The mean of the predicted values for the entire data set produces a small 3 percent increase in wives' mean hours.

13. Changes in tastes also enter the model in equation (4) as ln ||Gamma~.sub.it~. For example, education may increase preferences for labor supply; as women become more educated, tastes for work rise. Variation in ln ||Gamma~.sub.it~ was included in the estimates in Table I.

14. Likelihood ratio tests have been used to significantly constrain coefficients. There were no significant differences in labor supply coefficients for two older cohorts, ages 33-40 and over 40, so they are combined. There were also no significant differences in the sensitivity of different cohorts or vintages to current husband's income, wife's wage, or children. However, the participation probit was more sensitive to wages and husband's current income in the earlier period, 1968-74, so in Table II these elasticities are also permitted to vary by time and cohort, with only the arithmetic mean of the elasticities presented in rows 5 and 6 for the probit.

15. Labor supply is also more sensitive to education: the Tobit education coefficient rises from .0058 to .0089 over time for younger women.

16. The small estimated labor supply effects of the husband's income and number of children in Table I show that these variables explain little of the change in wives' labor supply over time using that functional form. However, these variables may interact with all the other structural changes in the economy in a very complicated fashion that is very difficult to model econometrically.

17. The percent of college freshman who were women rose from 42 percent to 51 percent--see various issues of the Statistical Abstracts of the United States.

18. The interaction between children and husband's income is likely to be larger for young women because they are more likely to have young children at home than are older women. Because the interactions are estimated by age group, the income interaction with the number of children is more significant than the interaction with a dummy variable for youngest child under age 6.

19. The permanent-income elasticities for younger age groups could reflect measurement error, because average sample income is a poorer proxy for lifetime income for younger workers. However, measurement error cannot explain why the income elasticity declines over time for the young age group when measurement error should be unchanged.

20. See Lehrer and Nerlove |1984~, Blau |1984~, and the references cited therein. Though these researchers have examined the impact of female income on income inequality and recognized the importance of the husband's income effect, they have not explicitly examined changes in inequality, nor attributed them to causal factors via a formal labor supply model.

21. Intertemporal wage elasticities can be larger, such as that of 1.14 to 1.73 estimated by Jakubson |1988~, but they have a very small total impact on female labor supply because intertemporal wage variability is very small (averaging 2 to 3 percent a year), though there is a cumulative effect over time if wage increases have a large time covariance. Because cross-sectional differences in the husbands' incomes are very large, female labor supply outcomes vary much more.

22. These examples are based on the Tobit elasticities, but note how similar the combined Heckit and probit elasticities are to the Tobit--so they produce similar results.

23. This correlation is between the actual wages of husbands and potential wages of wives, where the potential wage for non-working women is the predicted wage from the Heckit wage regression plus a random draw from the error distribution. Without the random error term, the correlation rises to .54.

24. For derivation of this equation see any of the papers listed in footnote 20.

25. These results must be interpreted with care. The coefficient of variation has the convexity property, such that the |(CV).sup.2~ of a weighted sum is always less than the squared weighted sum of the spouses' coefficients. The only time that adding the wife's income to the husband's will not be equalizing is when the incomes of the spouses are perfectly correlated.

26. The PSID data is often used to study income inequality because of its panel nature, but the randomness of the sample may be questioned. The entry of new families into the sample bodes well for its use. But missing data problems are greater for panels, suggesting potential selectivity bias, plus I condition on women married at least four years to estimate the labor supply model. The directions of the potential biases are unknown, but it is instructive that the male inequality results differ little from Current Population Survey results (see next footnote).

27. A very slight increase in husbands' income inequality also contributed to rising family inequality: the mens' coefficient of variation rises from .720 to .727. Another way of measuring income inequality is to look at the ratio of the income of the 90th percentile to the median income. In contrast to the coefficient of variation results, it fell from 2.02 to 2.00 for men (for women, it fell far more, from 10.83 to 8.22, consistent with the CV results). Underlying these opposing results, male hours became a little less equally distributed and wages a little more equally distributed. These wage results differ from those of Juhn, Murphy, and Pierce |1989~ who found male wage inequality worsening using Current Population Survey data. Using their measure of wage inequality, the variance of log wages, husbands' wage inequality also rises in the PSID data, by 8 percent. Overall, researchers have tended to conclude that male income inequality rose a small amount in the 1970s, but the cyclical variability of male hours makes income inequality trends less discernible, so the results are somewhat sensitive to the time period and the index used to measure inequality.

28. The declining income inequality for women results from evenly declining coefficients of variation of hours and of wages, for both younger and older women: the wage coefficient fell from .774 to .721 for women under 33, and fell .795 to .746 for women 33 and older; the hours coefficient of variation fell from .986 to .955 and 1.206 to 1.107 for the respective ages. The returns to education rose a significant 35 percent for women age 33 and older, and were unchanged for younger women (based on standard wage regressions). Because the womens' earnings distribution is skewed towards the lower tail, increasing returns to education tend to reduce inequality. This has been documented elsewhere using alternative measures of inequality--see Levy |1987~ and the footnote above.

29. Smith |1979~ points out that the correlation between the husband's income and the wife's hours is smaller for younger women, suggesting that the weaker correlation arises because actual income diverges from potential income most when young due to skill investment. He is indirectly making the same point that I am, that the husband's income effect should reflect permanent income not current income. This raises another point, that my income elasticities may be lower for younger cohorts because my measure of permanent income is limited to a fourteen-year period that may be too short for younger workers to attain their potential income. However, this limitation would not account for the decline in the elasticity over time.

REFERENCES

Blau, David. "Family Earnings and Wage Inequality Early in the Life Cycle." Review of Economics and Statistics, May 1984, 200-7.

Cogan, John. "Married Women's Labor Supply: A Comparison of Alternative Estimation Procedures," in Female Labor Supply: Theory and Estimation, edited by J. P. Smith. Princeton: Princeton University Press, 1980a, 90-118.

-----. "Labor Supply With Costs of Labor Market Entry," in Female Labor Supply: Theory and Estimation, edited by J. P. Smith. Princeton: Princeton University Press, 1980b, 327-64.

Heckman, James. "Sample Selection Bias as a Specification Error." Econometrica, January 1979, 153-63.

Heckman, James and Thomas MaCurdy. "A Life-Cycle Model of Female Labour Supply." Review of Economic Studies, January 1980, 47-74.

Jakubson, George. "The Sensitivity of Labor-Supply Parameter Estimates to Unobserved Individual Effects: Fixed- and Random-Effects Estimates in a Nonlinear Model Using Panel Data." Journal of Labor Economics, July 1988, 302-29.

Juhn, Chinhui, Kevin M. Murphy, and Brooks Pierce. "Wage Inequality and the Rise in Returns to Skill." Manuscript, University of Chicago, October 1989.

Keane, Michael. "Individual Heterogeneity, Interindustry Wage Differentials, and Unemployment." University of Minnesota, September 1989.

Killingsworth, Mark, and James J. Heckman. "Female Labor Supply: A Survey," in Handbook of Labor Economics, edited by O. Ashenfelter and R. Layard. Amsterdam: Elsevier Science Publishers BV, 1986, 103-204.

Lehrer, Evelyn, and Marc Nerlove. "A Life-Cycle Analysis of Family Income Distribution." Economic Inquiry, July 1984, 360-74.

Levy, Frank. "Dollars and Dreams: The Changing American Income Distribution." New York: Norton and Company, 1987.

MaCurdy, Thomas. "An Empirical Model of Labor Supply in a Life-Cycle Setting." Journal of Political Economy, December 1981, 1059-89.

Mroz, Thomas. "The Sensitivity of An Empirical Model of Married Women's Hours of Work to Economic and Statistical Assumptions." Econometrica, July 1987, 765-99.

Schultz, T. Paul. "Estimate Labor Supply Functions for Married Women," in Female Labor Supply: Theory and Estimation, edited by J. P. Smith. Princeton: Princeton University Press, 1980, 25-89.

Smith, James P. "Assets and Labor Supply," in Female Labor Supply: Theory and Estimation, edited by J. P. Smith. Princeton: Princeton University Press, 1980, 166-205.

-----. "The Distribution of Family Earnings." Journal of Political Economy, October 1979 (Supplement), S163-S192.
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