2.1. Single Households

We start with the maximization problem of a single female and use subscripts fs on variables to indicate her choices. In addition, we use superscripts to indicate the location of the activity, with a 1 for market activities and a 2 for home activities. Specifically, single females choose market and home consumption (c ¹_fs, c ²_fs), market and home hours (ℓ¹_fs, ℓ²_fs), and the amount of home-specific capital (k_fs) to solve

\begin{equation*} \max _{c_{fs}^{1},c_{fs}^{2},\ell _{fs},\ell _{fs}^{1},\ell _{fs}^{2},k_{fs}}\qquad \mu \log \left(c_{fs}^{1}\right)+\nu \log \left(c_{fs}^{2}\right)+(1-\mu -\nu )\log (\ell _{fs}) \end{equation*}

subject to

(1) \begin{eqnarray} &\!&c_{fs}^{1}+qk_{fs}\le (1-\tau _{f})w\ell _{fs}^{1}, \nonumber \\ &\!&c_{fs}^{2}\le Ak_{fs}^{\theta }\left(\ell _{fs}^{2}\right)^{1-\theta }, \nonumber \\ &\!&\ell _{fs}+\ell _{fs}^{1}+\ell _{fs}^{2}=1,\ \end{eqnarray}

where q is the price of home-specific capital, w is the wage rate, A is a home-specific productivity factor, θ is the capital share in home production, and τ_f is the wedge between actual productivity and income for the typical female.Footnote ²

The maximization problem for single men—identified with the subscript m instead of f—is similar, with the only difference being in the tax rate faced. As noted above, we will assume that 1 − τ_f = (1 − τ_d)(1 − τ_m), where τ_m represents the common labor income tax rate and τ_d represents the additional wedge faced by a female when there is discrimination in the market activity. This wedge is a proxy for either direct wage discrimination—a woman being paid less than her marginal product—or the shadow value on a constraint that restricts a woman’s job opportunities—for example, a glass ceiling. (See the appendix for a model with a glass ceiling policy.)

It is straightforward to generalize the problem to allow for sex-specific differences in productivities, allowing for a rich variety of potential differences in both absolute advantage and comparative advantage across the sexes. Since this will not change any of the results given below, we leave this generalization to the reader.

Let W_fs = (1 − τ_f)w. Then the solution to the single woman’s problem is

\begin{eqnarray*} \vspace{-10.0pt}\ell _{fs}^{1} &=&\mu +\theta \nu , \nonumber \\ \ell _{fs}^{2} &=&(1-\theta )\nu , \nonumber \\ c_{fs}^{1} &=&\mu W_{fs}, \nonumber \\ c_{fs}^{2} &=&Ak_{fs}^{\theta }\left( \ell _{fs}^{2}\right) ^{1-\theta }=A\left( \theta \nu \frac{W_{fs}}{q}\right) ^{\theta }((1-\theta )\nu )^{1-\theta }, \nonumber \\ k_{fs} &=&\theta \nu W_{fs}/q. \nonumber \end{eqnarray*}

Thus, as is standard in problems with log utility, expenditure shares on the different goods (market consumption, home consumption, and leisure) are constant fractions of wealth, W_fs. In this case, this implies that the expenditure on home investment goods is also a constant fraction of wealth and that time spent in the home is independent of prices. A similar set of equations holds for single men, with the only difference being that τ_m appears everywhere in place of τ_f. Otherwise, the solutions are identical.

Clearly, these equations show that hours used in both the market and the home are independent of w, A, q, and 1 − τ. These parameters do have an impact on both the level of consumption and the amount of the home capital good purchased. Thus, improvements in technologies do not alter the amount of labor supplied to the market by either single women or single men. Also, the market labor supply of single women and single men will be the same even if women face an additional tax wedge arising from discrimination.Footnote ³

In a dynamic setting in which w and A are endogenously determined by human capital formation decisions that may differ across the sexes (because of discrimination or natural productivity differences), analogs of these static first-order conditions will still apply, and hence, much of this reasoning will continue to hold. The main difference is that the levels of consumption and labor supply will enter the optimality conditions governing optimal capital accumulation, and hence, the effects will be more complex.

If the utility functions of the two sexes are identical but not logarithmic, the results given above need no longer hold. How they are changed depends on the elasticity of substitution between home and market goods. For example, if the utility function aggregates home and market goods using a constant elasticity of substitution aggregator, and home and market goods are substitutes, an increase in productivity in the home (A) causes both single women and single men to consume more home production and fewer market hours. If the goods are complements, the opposite occurs, causing market hours to increase for both sexes. Similarly, the effects of differences in sex-specific tax rates depends on whether home and market goods are substitutes or complements. For example, if home and market goods are substitutes, and women face higher effective tax rates than men, single women’s hours supplied to the market will be lower than those of single men. Correspondingly, single women will consume more home goods and fewer market goods than their male counterparts. This may account for the small but measurable difference in market hours between single women and single men seen in the data. (Single women work slightly less than single men do, and this difference has been relatively stable over time.) The size of these effects will depend both on the changes in relative productivities of the two activities (or the change in sex-specific tax rates) and on the degree to which preferences depart from the log specification.

2.2. Married Couples

We turn now to the problem of a married couple, or partnership, in this environment. We assume that the bargaining problem within the household is resolved efficiently, so that a weighted form of a planner’s problem describes the couple’s decisions. As before, we use subscripts to indicate gender and marital status. In this case, p indicates partnership, c ¹_fp and c ¹_mp are the consumption of the market good by the woman and the man of the pair, c ²_fp and c ²_mp are their consumption levels of the home good, ℓ¹_fp and ℓ¹_mp are the hours they work in the market, and ℓ²_fp and ℓ²_mp are the hours they work in the home. For the partnership, the maximization problem solved is

\begin{eqnarray*} &&\max \ \lambda _{f}\left[\mu \log \left(c_{fp}^{1}\right)+\nu \log \left(c_{fp}^{2}\right)+(1-\mu -\nu )\log (\ell _{fp})\right] \nonumber \\ &&\qquad \qquad +\lambda _{m}\left[\mu \log \left(c_{mp}^{1}\right)+\nu \log \left(c_{mp}^{2}\right)+(1-\mu -\nu )\log (\ell _{mp})\right], \nonumber \end{eqnarray*}

subject to

\begin{eqnarray*} &&c_{fp}^{1}+c_{mp}^{1}+qk_{p}\le (1-\tau _{f})w\ell _{fp}^{1}+ (1-\tau _{m})w\ell _{mp}^{1}, \nonumber \\ &&c_{fp}^{2}+c_{mp}^{2}\le Ak_{p}^{\theta }\left(\ell _{fp}^{2}\right)^{1-\theta }, \nonumber \\ &&\ell _{fp}+\ell _{fp}^{1}+\ell _{fp}^2=1, \nonumber \\ &&\ell _{mp}+\ell _{mp}^{1}+\ell _{mp}^2=1, \nonumber \end{eqnarray*}

where the remainder of the parameters are as discussed above. Note that we have maintained the assumption that tax rates are sex-specific and will, as above, interpret differences between τ_f and τ_m as arising from the effects of discrimination in the market activity.

Here, as in the work of Becker (Reference Becker1991), the solution to this problem is not interior in general, since men’s and women’s hours are perfect substitutes in both home and market activities. Because of this, there will be specialization within the household. In keeping with what is seen in the data, we will use the first-order conditions that result when ℓ²_mp = 0, but will assume that otherwise the solution to the problem is interior.

The solution to the married couple’s problem is

\begin{eqnarray*} \vspace{-10.0pt}\ell _{fp}^{1} &=&1-[\lambda _{f}(1-\mu -\nu )+\nu (1-\theta )] \frac{W_p}{(1-\tau _{f})w}, \nonumber \\ \ell _{fp}^{2} &=&(1-\theta )\nu \frac{W_p}{(1-\tau _{f})w}, \nonumber \\ \ell _{mp}^{1} &=&1-\lambda _{m}(1-\mu -\nu )\frac{W_p}{(1-\tau _{m})w}, \nonumber \end{eqnarray*}

where W_p ≡ (1 − τ_f)w + (1 − τ_m)w. We have also assumed, for simplicity, that there are no economies of scale in living as a couple. This could be reflected in the example in a variety of ways, but would not affect the results below.

For married as for single households, changes in A, q, and w do not affect the household’s allocation of hours to any of the activities—leisure, work in the home, or work in the market. As is the case with single agents, there are changes in quantities consumed and in k , however. The form of these quantity adjustments mirrors that for the single agents and will not be included here.

The same is not true for changes in taxes. If either τ_m or τ_f is changed, with the other held fixed, then hours adjust. For example, if τ_m is unchanged but τ_f falls, or, equivalently, discrimination is reduced so that τ_d falls, it follows that ℓ¹_fp increases while ℓ²_fp falls (as does ℓ_fp)—the woman works more in the market and less in the home (and consumes less leisure). At the same time, ℓ¹_mp falls (and ℓ_mp increases). Thus, in response to a reduction in market discrimination, the woman works more in the market; the man works less. In contrast, if τ_m and τ_f are both changed proportionally with 1 − τ_d = (1 − τ_f)/(1 − τ_m) fixed, no change in hours takes place.

In sum, we see that if utility is logarithmic, changes in technology are neutral for labor supply decisions of both singles and married couples, whereas reductions in discrimination leave the decisions of singles unchanged but increase married women’s market hours. For utility specifications that differ from logarithmic, there will be effects on all agents of changes in technology, even if preferences are homothetic, but the direction of the effects will depend on the substitutability between home and market goods. By continuity, the effects are likely to be small unless the changes are very large or the utility structure deviates greatly from the unit elasticity of substitution. In this case, the effects will be present for all agents, single and married, women and men.

2.3. Inferiority of the Home Good

Technological change will also have effects on labor supply even if preferences are not homothetic. Since these changes are substantial for some specifications when home goods are inferior, we present a simple version of this phenomenon here. We consider a perturbation on the model above in which households become satiated in c ² once it is equal to c*. Beyond that, the formulation is identical.

We restrict attention to the problem of a single household. If parameters are such that the solution to the original problem satisfies c ²_fs ⩽ c*, the solution is that presented above. This will hold as long as A[θν(1 − τ_f)w/q]^θ[(1 − θ)ν]^{1 − θ} ⩽ c*. This requires a relatively low A, w, and 1 − τ_f and a relatively high q. If this does not hold, the solution is given by c ²_fs = c* with k_fs and ℓ²_sf chosen to minimize the cost of producing c*. Let C(c*; q, (1 − τ_f)w) denote the minimum total cost of producing c ²_fs = c*.

The solution to the household optimization problem is

\begin{eqnarray*} \ell _{fs}^{1} &=&\frac{1-\mu -\nu }{1-\nu }+\frac{c^{\ast }}{A}\left[ \frac{ 1-\theta }{\theta }\frac{q}{(1-\tau _{f})w}\right] ^{\theta }\left[ \frac{ 1-\mu -\nu }{1-\nu }\frac{1}{1-\theta }-1\right], \nonumber \\ \ell _{fs}^{2} &=&\frac{c^{\ast }}{A}\left[ \frac{\theta }{1-\theta }\frac{ (1-\tau )w}{q}\right] ^{\theta }, \nonumber \\ c_{fs}^{1} =\frac{\mu }{1-\nu } \left[(1-\tau _{f})w-C(c^{\ast };q,(1-\tau _f)w)\right], \nonumber \\ k_{fs} =\frac{\theta }{1-\theta }\frac{(1-\tau )w}{q}\ell _{fs}^{2}, \nonumber \\ c_{fs}^{2} &=&c^{\ast }. \nonumber \end{eqnarray*}

In this case, increases in w and decreases in τ_f decrease ℓ²_fs but increase ℓ_fs. Whether ℓ¹_fs increases or decreases depends on which is larger, 1 − θ or (1 − μ − ν)/(1 − ν). If 1 − θ is larger, ℓ¹_fs rises with increases in (1 − τ_f)w, but the opposite holds if 1 − θ is smaller. Similar results hold for changes in both A and q; if 1 − θ is larger than (1 − μ − ν)/(1 − ν), increases in A and decreases in q increase ℓ¹_fs. Thus, as is intuitive, what is important is the share of ℓ in the production of c ² relative to its share in the reduced form utility, (1 − μ − ν)/(1 − ν).

Thus, in some cases, satiation provides an alternative route to changes in ℓ¹_fs, but as we can see from the example, this effect is present in single households as well as in those of married couples. Note, however, that as A or w rises, or q falls, qk falls as a fraction of income.

Although the example we have considered in this section is special, the qualitative nature of the results can be considerably generalized. For example, including quality choices for home production (cf. Mokyr Reference Mokyr2000) and the presence of glass ceilings for women does not change the conclusions.

3.1. Functional Forms and Parameter Choices

We start with the specification of the functional forms we will use in our quantitative analysis. We consider the class of preferences given by U^f = U^m = U, where

\begin{equation*} U={\frac{1}{1-\sigma }}\left[ \left( \psi _{1}(c^{1})^{\psi _{2}}+(1-\psi _{1})(c^{2})^{\psi _{2}}\right) ^{(1-\psi _{3})/\psi _{2}}(1-\ell ^{1}-\ell ^{2})^{\psi _{3}}\right] ^{1-\sigma }. \end{equation*}

The production function of both types of goods (market and nonmarket) are assumed to be Cobb–Douglas, with the same coefficients for market and nonmarket goods: Fⁱ(k, z) = Aⁱk ^θz ^{1 − θ}, i = 1, 2. We assume that the production functions of specific human capital are identical across all categories (sex and marital status) and are given by Φⁱ(h, η, ℓⁱ) = (h)^κ_i(η)^ζ_iℓⁱ, i = 1, 2.

The parameter choices for our benchmark case are shown in Table 1. We set n_p to match the fact that roughly 60% of the relevant U.S. population was married during the period we study. Values for government spending and tax rates on labor and capital are average U.S. postwar values. The annual growth rates γ_p and γ_A are long-run U.S. trend levels. The discount factor is chosen so that the trend interest rate is 4%.

Table 1. Benchmark parameter values

${}^\dagger \ U(c^1,c^2,\ell ) = {1\over 1-\sigma }\Big[(\psi _1(c^1)^{\psi _2}+(1-\psi _1) (c^2)^{\psi _2})^{(1-\psi _3)\over \psi _2} \ell ^{\psi _3}\Big]^{1-\sigma }$

Values for the capital share, θ, the rate of physical depreciation, δ_k, and two critical preference parameters, ψ₂ and σ, are the maximum likelihood estimates of McGrattan, Rogerson and Wright (Reference McGrattan, Rogerson and Wright1997) for a model with home production. We set the depreciation rates for human capital, δ_h and δ_η, equal to the depreciation rate for physical capital, δ_k, for our benchmark example. Since good estimates for human capital rates are not readily available, these parameter choices will be one focus of our sensitivity analysis.

We choose the remaining preference parameters (ψ₁, ψ₃, λ_f), two of the elasticities for effective labor (κ₁, κ₂), and the paths of technology and discrimination taxes to achieve several objectives. First, with no change in technology or discrimination, we want the benchmark parameters to yield initial hours of work that match the 1950 hours in Figure 1A for three groups—married women, married men, and single women—and to yield a relative wage of 51%—which is the value we obtain from extrapolating back the time series in Figure 1B.Footnote ⁶ Second, we assume that the initial leisure of married men is equal to the initial leisure of married women. This determines a value for the weight on married women’s utility, λ_f. This weight turns out to be very low, only 0.062. Because this value is so low, it will be one of the parameters that we focus on when we do sensitivity analysis.

A third objective is to match the U.S. time series on relative wages (Figure 1B) in the benchmark simulation with a change in discrimination. To do this and achieve the initial conditions above, we set the initial discrimination tax, $\tau _{d\,\scriptstyle {1950}}$, at 22% and set subsequent rates so that the model yields the same time path for relative wages as in Blau (Reference Blau1998) (see Figure 1B).

For the benchmark parameterization, we do not distinguish between type-h and type-η human capital; therefore, we assume that κ_i = ζ_i in both sectors, i = 1, 2. We experiment later by assuming no human capital and assuming sector-specific human capital.Footnote ⁷

We assume that the government purchases 20% of market goods and services and redistributes, in a lump-sum fashion, any remaining revenue generated. We interpret τ_ℓmt as the governmentally specified tax rate on labor income and assume that any difference arising from discrimination is completely used for redistributive purposes. For simplicity, we assume that this redistribution is equally divided among all agents in the economy. This is consistent with our assumption that although we have modeled discrimination as a tax, it is not being used for revenue generation. Later, we look at alternative specifications of the distribution of revenue.

Finally, since we want to abstract from business cycle frequency effects, we consider a time period in our model to be five years. Thus, t = 0 corresponds to the year 1950, and t = 10 corresponds to the year 2000. The calculations that we perform assume that all agents perfectly anticipate the changes that are forthcoming.

For each experiment, we include sensitivity analyses on our results. The parameters used for these alternatives are included in Table 2.

Table 2. Numerical experiments

5.2. Sensitivity Analysis

Table 3 displays the numerical results for the example discussed above (in the second row of numbers) along with the results of several other related experiments. The third row of Table 3 shows the results for an example with even less substitution, ψ₂ = −4. In this case, married women’s hours increase more, by 4.7 hours per week between 1950 and 2000, but still significantly less than the 16.7 hours per week in the data. For married men, the change is 1 hour per week, similar in magnitude to the 0.7 hours seen above, whereas in the data the corresponding change is 2.9 hours. Not shown in the table are the results of experiments based on more inclusive notions of home capital goods. Since in those experiments, the corresponding reduction in q is smaller, even smaller changes in hours result.

Table 3. Effects of changes in home technology‡

‡ MF = married females, MM = married males, SF = single females, SM = single males.

An alternative way of studying the impact of improvements in home technologies is to study the effects of increases in A ² over and above any general technical change. In order for this alternative to have a chance at being successful, however, preferences must deviate substantially from the power utility case. In the power utility case, an increase in A ²_t, to say $\hat{A}^2_t=(1+\gamma )A^2_t$, raises home consumption by a factor γ but leaves all other variables, including hours of work, unchanged.Footnote ¹⁰ This theoretical result implies that, by continuity, changes in the home technology for any specification of preferences near unitary elasticity of substitution between home and market goods must necessarily result in a small effect on hours. As such, it serves as a useful benchmark for what follows.

Here, as above, we investigate what happens only when home and market goods are complements, since if they are substitutes, market hours actually fall. To generate the large changes in hours worked by married women observed in U.S. data, we are forced to examine very large changes in the value of A ². The results are displayed in Table 3. Again, with ψ₂ = −0.75, the size of the change in A ² that is needed to match the data is to increase it from A ² = 1.0 to A ² = 5.0, over and above our benchmark level of technological change. With our benchmark growth rate in the market sector of γ_A = 2%, this corresponds to a growth rate in home productivity of over 5% per year, whereas market productivity grows at only 2% per year.

Although this simulation matches the hours data well, with only a small change by singles, it has three problems. First, for this story to be successful, home and market goods must be complements, contrary to best estimates. Second, very large changes in technology are required changes, over and above those measured in market productivities. Finally, even in these cases, we see only small effects on the observed wage gap. This last point is important, since it is directly related to changes in human capital formation decisions, and as pointed out above, a dramatic shift in the schooling decisions of men and women seems to have taken place during the last 60 years.

Our results contrast with those of Greenwood, Seshadri and Yorukoglu (Reference Greenwood, Seshadri and Yorukoglu2005). There are two key differences. First, Greenwood, Seshadri and Yorukoglu (Reference Greenwood, Seshadri and Yorukoglu2005) assume that the labor supply decision is indivisible. Thus, married women are prevented from working part-time. This assumption implies that the elasticity of substitution between home and market goods plays no role in their model. If a household is sufficiently productive, then a decrease in the price of a durable that results in adoption on the part of a household frees up time—the technology is Leontief—that can only be used to produce either market goods or leisure.

Second, since Greenwood, Seshadri and Yorukoglu (Reference Greenwood, Seshadri and Yorukoglu2005) assume that the home technology is Leontief, and there are only two options for producing in the home, utility effectively exhibits satiation in the home good in their formulation. This seems to be why their model predicts that married women’s participation will eventually begin to fall and durables expenditures decline as a fraction of GDP. As we shall see in the next section, this is the driving force behind their results.

6.2. Sensitivity Analysis

We conduct numerous sensitivity analyses of the examples described above in an attempt to isolate the relative contributions of different sources of income growth and preference specification. We find that any of the sources of income growth produce only small effects when the utility function is of the type in V ₁. In contrast, when utility is given by the form in V ₂, durables prices alone produce almost no effect without overall growth in productivity. Similarly, when durables price reductions are added to the model with productivity growth, again, the marginal effect is quite small. Thus, we conclude that any effect that is present with this specification is present only when we have both strict satiation and overall productivity growth. The declining price of durables seems to play only a minor role.

7.2. Sensitivity Analysis

We turn now to the sensitivity of our results. Some of the modeling choices that we have made are standard; they have counterparts in all dynamic models (for example, discount factors and preference and production parameters). Others—the choice of welfare weights within a couple, the specification of effective labor including human capital, the nature of the transfer scheme for distributing the revenue raised by the discrimination tax, and the assumption of equal discrimination against married and single women—are unique to the questions that we are addressing. We find that generally speaking, our results are robust to alternative specifications. The one exception is the specification in which discrimination affects married women only; this example is difficult to match with the observed wage series.

Table 4 reports hours of work, relative wages, and home investment shares for several alternative versions of the model. The first row of numbers contains the U.S. data, and the remaining rows display statistics for the different model parameterizations, the first being the benchmark parameterization discussed above.

Table 4. Effects of fall in discrimination‡

‡ MF = married females, MM = married males, SF = single females, SM = single males.

In our benchmark case, the weight on married women’s utility is low (λ_f = 0.062). As indicated before, this is necessary in the benchmark example to keep total hours by the woman, home plus market, at the same level as those of her partner. The first experiment reported in Table 4 assumes a higher weight on women’s utility—the same, in fact, as that on married men’s. In order to match the wage series for this parameterization, a larger change in τ_d is required (see Table 2). In this experiment, we assume that $\tau _{d\,\scriptstyle {2000}}=0.12$; hence, the reduction is still a relatively modest 10%. Given this assumption, however, the changes in hours are similar to those in the benchmark model.

In a second experiment, we consider a variation of the model with no human capital. Here, we set κ_i and ζ_i, i = 1, 2, equal to 0.001 so that the return to human capital is negligible. All other parameters are as in the benchmark example except for the path of τ_d, which begins at $\tau _{d\,\scriptstyle {1950}}=0.48$ and falls to $\tau _{d\,\scriptstyle {2000}}=0.22$, as is required to match the wage series. In this case, it follows that the gap is simply the wedge introduced by the discrimination tax. No additional wedge is introduced by human capital differences. The smaller gap means that larger changes in discrimination are required to match the relative wage series, but beyond this, the results of the experiment are quite similar to the benchmark case. Thus, the role of human capital is quantitatively significant, in that with human capital included, only a 6% drop in τ_d is required, whereas without it, a drop of 26% is needed.

Also included in Table 4 are two further experiments on the role of human capital. In one, we assume that human capital is useful only in the market sector. In the other, we assume that the two types of human capital are sector-specific. The results are similar for all variations, with the exception of the path τ_d required to mimic the gender wage gap.

For the simulations discussed so far, we have assumed that revenues from the discrimination tax are lump-sum rebated in an equal per capita fashion. An alternative hypothesis is that the revenue raised from discrimination against women is used to subsidize some other group of agents. To analyze this possibility, we simulate a specification in which the revenues generated from the discrimination tax are used as a subsidy to married men [Goldin (Reference Goldin1990, p. 102)]. The parameters of the model are recalibrated to match initial hours and the time path of the wage gap. From this, we find that similar changes in the discrimination tax give nearly identical results, as our benchmark simulation does.

The benchmark simulation assumes equal discrimination against both married and single women. In fact, many of the discriminatory practices that have been documented [for example, marriage bars; see Goldin (Reference Goldin1990)] seem to exhibit more discrimination against married women. Because of this, we consider an example in which we set the discriminatory taxes for single women equal to zero, with the other parameters held fixed at their benchmark levels. This experiment is the last row in Table 4. Here, we can find no time path for τ_d that replicates the time series of the gender wage gap. Hence, we use the path from our benchmark parameterization. We find that this change significantly increases the relative wage of women in all periods and leads to more market hours in 1950 for all groups except married males. The reduction in discrimination, however, still creates a substantial increase in married women’s market hours, from 10.0 hours per week in 1950 to 26.3 in 2000.

In sum, the results reported in our benchmark example seem fairly robust to both changes in the parameters and the details of the treatment of the discrimination tax. As long as the levels and changes in τ_d are chosen to match the observed path of the gender wage gap, the pattern for hours is consistent with the data. The size and the change of τ_d that are needed for this depends critically on the role of human capital, however.