# Family composition, parental time, and market goods: life cycle trade-offs.

Family Composition, Parental Time, and Market Goods: Life Cycle
Trade-Offs

Analyzing the joint determination of family consumption and time allocation has been the subject of many recent studies (1) that appeal to Becker's (1965) and Gronau's (1977) theoretical work. However, as Kooreman and Kapteyn (1987) note, the analysis of time spent by members in household production of goods and services and its relationship to the demand for either market goods or leisure is restricted by lack of detailed time use data. Studies designed to examine family expenditures usually collect labor supply data but little information regarding other time allocation decisions. Similarly, time budget studies usually collect little family expenditure information. Thus, previous studies of the joint determination of demand for time and goods examine expenditures among commodity groups omitting the time allocation (home production and leisure time) function (e.g., Blundell 1980; Blundell and Walker 1982) or examine time allocation decisions among activities omitting the consumption function (e.g., Kooreman and Kapteyn 1987). In this paper results are reported from a unique study that incorporates information about both family expenditures and time allocation.

Of particular interest is further examination of family composition's influence on full income allocated to leisure and home production activities as well as to market-purchased goods. The hypothesis of interest is one suggested by Deaton and Muellbauer (1986), that families with young children face large time (foregone leisure) relative to goods expenditures, and the trend reverses as children age.

The first step toward accomplishing the stated objectives was to estimate parameters of a nonlinear expenditure system using a linear logit specification that includes a continuous measure of family composition (Tyrrell and Mount 1982; Douthitt and Fedyk 1988 (2)). The present analysis differs from the authors' previous study where family composition influenced only goods consumption. The purpose of that study was to examine substitutions made by families within categories of current consumption in response to changes in family composition. In that model it was implicity assumed that earned income (leisure) and levels of home production were exogenous to current consumption decisions. The present study examines how the presence of children influences not only current consumption decisions, but also time allocation decisions and income generation.

Finally, pursuant to the objective of comparing family composition effects on time versus goods trade-offs, parameter estimates are used to calculate expenditure and family size elasticities and to conduct life cycle simulations of full income allocations to time and goods over the life cycle.

The remainder of the paper is organized as follows. Section two reviews the theoretical underpinnings of the model used. The third section explains the model specification. Section four discusses the data and statistical method used to estimate the model's parameters. Section five presents parameter estimates, and section six contains life cycle simulations of full income allocation decisions. The final section summarizes findings and compares the results to previous studies.

The present analysis embraces Becker's (1965) theory of time allocation. Relying on Gronau's (1977) adaptation of this theory, it is assumed that families maximize household utility through the consumption of combinations of goods and services (X) and leisure (R). Goods and services are either purchased in the market place ([X.sub.m]) or produced in the home ([X.sub.h]). A family's utility is thus:

U = U(X,R) (1)

where X = [X.sub.m] + [X.sub.h].

Expenditures for market-purchased goods and services are made subject to the budget constraint:

[piX.sub.m] = WN + V (2)

where [pi] is an index of market prices, W and N are, respectively, market wage and hours worked in the market, and V is unearned income.

As with Gronau's theory, it is assumed for this study that home-produced goods and services ([X.sub.h]) are primarily a function of the amoung of time spent working in the home (H), and any market goods that may be used in home production are ignored:

[X.sub.h] = f(H). (3)

Household production is constrained by the total time available:

T = N + H + R. (4)

Since each hour spent in nonmarket time (H + R) has an opportunity cost equal to W, full income ([M.sub.F]) then becomes:

[M.sub.F] = WT + V. (5)

Maximization of utility subject to the full income constraint makes the marginal rate of substitution between goods and services (X) and leisure (R) equal to the shadow wage (W*), which, for persons in the paid labor force, is also equal to W--the market after-tax wage rate. Thus, while market prices represent the cost of market-purchased goods, the shadow wage represents the implicit cost of home-produced goods.

Assuming input prices are constant permits specification of the dual problem minimizing costs subject to a given level of goods and services, [X.sub.m.sup.o] and [X.sub.h.sup.o], and of leisure, [R.sup.o] (Deaton and Muellbauer 1980, p. 37). The associated Lagrangian function for minimizing expenditures is

Min L = [[pi]X.sub.m] + [W*X.sub.h] + W*R + [lambda][[X.sub.m.sup.o] - (W*N + V)] + [theta][[X.sub.h.sup.o] - f(H)] + [Phi][[R.sup.o] - (T - H - N)].

The input demand functions derived from minimizing equation (6) are the basis of the expenditure functions analyzed in this study and are expressed as:

[f.sub.1.([pi],X.sub.m.sup.o]) = [e.sub.1.([pi],U.sup.o]), [f.sub.2(W*,X.sub.h.sup.o)] = [e.sub.2(W*,U.sup.o)], and [f.sub.3(W*,R.sup.o)] = [e.sub.3(W*,U.sup.o)].

The demands for inputs can be derived directly from the expenditure functions:

[[derivative]e.sub.1.([pi],U.sup.o]) / [derivative][pi] = [X.sub.m] = [g.sub.1.([pi],M.sub.F]), [[derivative]e.sub.2.(W*,U.sup.o]) / [derivative]W* = [X.sub.h] = [g.sub.2.(W*,M.sub.F])M and [[derivative]e.sub.3.(W*,U.sup.o] / [derivative]W* = R = [g.sub.3.(W*,M.sub.F]),

where [M.sub.F], defined as WN + V, is the full income necessary to maintain utility at [U.sup.o].

EMPIRICAL MODEL

The empirical model used in this analysis is an extension of a nonlinear expenditure system described by Tyrrell and Mount (1982). It incorporates both continuous measures of family composition and a flexible functional form.

A revealed preference approach is used to derive continuous measures of household composition effects on family expenditure decisions. Although reliance upon revealed preference is a commonly used strategy for deriving parameter estimates of consumer demand, (3) few studies also incorporate continuous measures of family size and structure effects on spending behavior. (4) Friedman (1957) first developed the concept of a continuous composition or equivalence scale measure. Its strengths include continuity over size or age range measures (i.e., measured effects scales do not "jump" between adjacent age categories) and fewer required parameters for estimation.

In their continuous measure, Tyrrell and Mount (1982) combine the effects of family size and a composition multiplier (specific to each of the [i.sup.th] commodity groups) by taking the log of family size multiplied by a cubic function of family member age variables. Because it is not possible to estimate these cubic functions directly, they approximate them by using a set of Almon lag structures referred to as Lagrangian interpolation polynomials (LIPs). (5) The LIPs ([L.sub.j]'s) are expressed in terms of member's age (a) as it deviates around four reference ages ([a.sub.1.. . . a.sub.4]) (6) and have the form:

[L.sub.j](a) = ([a - a.sub.2])([a - a.sub.3])([a - a.sub.4]) / ([a.sub.1 - a.sub.2])([a.sub.1 - a.sub.3])([a.sub.1 - a.sub.4])

The four reference ages were chosen to correspond with points in the life cycle where changes in consumption were expected to occur. The sum of the LIP parameters from each equation estimate the weight associated with an age-specific equivalence effect. In Tyrrell and Mount's specification, separate age LIP functions were estimated for males and females, thus eight LIPs were required.

The Tyrrell and Mount model also incorporates a flexible functional form. Many econometric models of demand a priori restrict estimated parameter values to be consistent with postulates of economic theory. For example, many expenditure allocation models assume functions homogeneous of degree one in income and family size (e.g., Prais and Houthakker 1955).

However, it can be demonstrated that the assumption of homogeneity can generate nonsensical results when applied to actual behavior and that homogeneity, coupled with an equivalence scale specification, implies constant returns to scale. Blundell and Walker (1982), for example, apply these assumptions to their work and "conclude" no economies of scale exist in rearing children. Numerous illustrations can be cited to refute this restriction. The purchase of food in larger quantities sold at lower per unit prices and the reuse of clothing are standard examples of economies of scale that could occur upon the addition of a family member.

The Tyrrell and Mount model ensures the theoretical restrictions of adding-up while allowing for nonhomogeneous demand functions and economies of scale. Thus, the model provides an effective balance between the concern for theoretical plausibility and the applied need to maximize explained variance. Their theoretical model, expressed in the logistic form of the budget shares ([c.sub.i]), is

[Mathematical Expressions Omitted]

where:

[c.sub.i] = expenditures for [i.sup.th] commodity group as a share of total expenditures, M = income (WN + V), S = family size, and L = a vector of family composition terms.

In this adaptation of the Tyrrell and Mount model, budget shares ([Q.sub.i]) are defined as commodity outlays divided by total expenditures (E) for market goods and services ([[pi] X.sub.m]), spouses' household production ([[Sigma]w.sub.i.*h.sub.i]), and leisure ([[Sigma]w.sub.i.*r.sub.i]). Specifically, shares are defined as:

[Q.sub.1] = [[pi]X.sub.m./E]; share of total expenditures allocated to family expenditures on market purchased goods, [Q.sub.2] = [w.sub.1.*h.sub.1./E]; share of total expenditures allocated to wife's home production, where [w.sub.1.*] is wife's shadow wage, and [h.sub.1] is the amount of time the wife spends in home production, [Q.sub.3] = [w.sub.2.*h.sub.2./E]; share of total expenditures allocated to husband's home production, where [w.sub.2.*] is husband's shadow wage and [h.sub.2] is husband's home production time, and [Q.sub.4] = ([w.sub.1.*r.sub.1] + [w.sub.2.*r.sub.2])/E; share of total expenditures allocated to leisure time by both parents, where [r.sub.1] is wife's leisure time, and [r.sub.2] is husband's leisure time.

Empirically, then, each budget share is expressed as a function of full income ([M.sub.F]) and family composition:

[Mathematical Expression Omitted]

where:

[B.sub.i]'s = parameter estimates associated with the [i.sup.th] expenditure group.

Tyrrell and Mount estimate a linear approximation of the logistic share model (equation 11) by expressing each of the n - 1 commodity shares as a proportion of the share for the [n.sub.th] commodity group. The same technique is employed in this study where the [Q.sub.i.sup.th] share, i.e., the share of total expenditures allocated to market goods ([Q.sub.1]), to wife's home production ([Q.sub.2]), and husband's home production ([Q.sub.3]) are divided by the budget share for combined spouse leisure ([Q.sub.4]). Thus, the final estimating equations are:

[Mathematical Expression Omitted]

where estimated coefficients measure the differences in effects of each independent variable on the n - 1 budget shares relative to its effect on leisure.

The effects of family composition (L) are approximated by using Tyrrell and Mount's LIP functions with associated reference ages of 1, 14, 20, and 64. The ages were chosen to correspond with points in the life cycle where changes in consumption have previously been found to occur (Tedford, Capps, and Havlicek 1986).

DATA AND METHOD DESCRIPTIONS

The data used in this study were collected as part of Statistics Canada's 1982 Survey of Family Expenditures (FES). They include both expenditure and demographic information for a random sample of over 10,000 Canadian households. The scope of the study was limited in several respects because Statistics Canada did not release the entire survey for public use. For example, children's gender was not revealed, meaning that adjustments in family budget allocations attributable to such differences (Olson 1983) could not be examined.

Since prices and preferences are unobservable, the analysis was limited to a single region, the Canadian prairie provinces of Manitoba, Saskatchewan, and Alberta. The final sample includes only nonfarm families where all family members are present in the household for the entire year. Extended families, or families with more than two adults (where the third adult is not a parent's child) are eliminated from the sample. Further selection criteria are that neither spouse was older than 64 years of age, not more than one-third of family income comes from income-tested government sources, and children are not employed full-time in the paid labor force. Mean sample characteristics are presented in Table 1.

Using equation (12), the system was jointly estimated with generalized least squares (GLS). Because all equations contain the same independent variables and no across-equation constraints are imposed, the GLS estimators are equivalent to ordinary least squares.

Independent variables in the analysis are full income, family size, and four transformations of each family member's age (LIPs). For each family in the sample, full income ([M.sub.F]) (Equation 5) was calculated. The FES data contain information regarding total money expenditures ([X.sub.m]), as well as parental earnings, weeks worked, and unearned income. Data for time spent in household production (H) and a value for earnings per hour of work ([W.sub.p]) for those parents who were not paid labor force participants were not available.

With regard to estimating home production time, fortunately, in the same year that the FES data were collected (see Kinsley and O'Donnell 1983), Employment and Immigration, Canada conducted a national time use study (NTBS). After defining a NTBS sample and independent variables analogous to the FES prairie sample/data, leisure hours equations were estimated for both men and women. (7) The NTBS parameter estimates were then used to predict time spent in leisure for the FES families. Time spent in home production (H) was calculated as a residual of total time (T) available in the year (52 weeks) minus predicted leisure time (R) and FES reported time spent in labor force activities (N): H = T - N - R.

Measures of the value of parental earnings for those who did not report a market wage were derived using a technique outlined by Heckman (1979) to derive "shadow" wages (W*). As implied by the first order conditions of the theoretical model, individuals with observable wages were assigned a value of earnings equal to their actual wages, while those with unobserved wages were assigned a shadow wage. Appendix B presents intermediate results from the sample selection procedure for women who were not paid labor force participants. Table 2 presents descriptive statistics regarding both actual (for employed persons) and imputed shadow (for all) wages for men and women. Home production and leisure time were each assigned a value equal to the shadow or market after-tax wage, which was multiplied by hours spent in each activity to derive estimates of expenditures of full income allocated to spouses' home production and leisure.

As mentioned in the empirical section, the effects of family composition were estimated by using Tyrrell and Mount's LIP functions. However, because there was no information regarding children's gender, it was necessary to estimate separate LIPs for parents and children. The equation was normalized by omitting the third adult LIP, making the reference household a twenty-year-old childless couple. In total, then, seven LIP parameters (8) were estimated.

EMPIRICAL RESULTS

The parameter estimates of equation (12) are presented in Table 3. Over half of the independent variables in the system are significant at the .05 level or above. All three estimated share equations explain a significant amount of share variance vis-a-vis the reference leisure share of full income. The greatest difference in total explained variance is found in the women's home production equation. Consistent with the specification of the final estimating equation (12), all parameters represent effects on the dependent variable of a unit change in the independent variables vis-a-vis parameters of the leisure equation (4), by which the remaining three equations were normalized. Thus, interpretation of each coefficient is the difference between the effect of a change in the independent variable on the dependent variable vis-a-vis its effect on family leisure allocations. For example, the parameter on full income (-.238) in the market goods equation is equal to the difference ([B.sub.11.-B.sub.14]) and implies that the effect of a one percent change in full income on market good expenditures is -.238 less than that of a comparable change in full income on leisure expenditures.

Table 4 presents predicted budget shares and estimates of both income and size elasticities calculated from the model parameters for two family types. The top half of Table 4 shows estimates for a childless couple where the adult male is 32 and the adult female is 29 years of age. The bottom half of the table gives findings for a similar pair of adults with two children ages four and two.

Results shown in Table 4 indicate that childless couples allocate larger shares of full income to both market goods and leisure compared with their counterparts with children. In other words, if wages are assumed constant, the childless couple spends less time in home production activities than does the couple with children. For both families, men's time in home production and combined spouse leisure shares have an income elasticity greater than one. This means that an increase in full income generates a greater increase in husband's home production share and that both husband and wife spend an increasing share of time on leisure when full income rises.

An increase in family size results in an increase in the share of full income allocated to market goods, men's and women's time in home production, and a decrease in the share of full income allocated to leisure. It is noteworthy that the effect of family size on women's home work is equal to that for market goods. That is, a one percent increase in family size increases both shares of market goods and women's home production by .13 percent. This result strengthens claims by Espenshade (1984) and others that the opportunity costs of children are comparable to their direct costs. Further, as evidenced by the negative size elasticity, increases in family size cause substitutions to occur primarily out of leisure.

SIMULATION RESULTS

From these parameter estimates, simulations were conducted to illustrate how an "average" family reallocates full income over the life cycle. The average family was identified by using provincial statistics regarding prairie family economic, fertility, and marriage behavior. Given the 1982 population average age at first marriage, it is assumed that the hypothetical couple marries when the man is 26 and the woman is 23. In accordance with the national figures for median age at first and second birth, the first child is assumed to be born when the father is 28 and the mother is 25, and the second child is born two years later when the mother is 27 and the father is 30. Children are assigned an age of 0 for the year of their birth since the sample does not provide information on expenditures for families with members present for only part of the year.

In order to simulate results for the average Canadian prairie family, it was necessary to select a flow of full income values over the life cycle. To do this a full income equation (9) was estimated for the sample and used to generate predicted values for the average family over the life cycle.

Predicted budget shares are generated for each year with every family member aging accordingly. It is assumed that children leave home after their eighteenth year. For comparative purposes, simulations of the allocations of a childless couple with the same demographic characteristics and who marry at the same age as the couple with children were conducted. Differences between these two simulations permit comparisons of shifts in budget allocations attributable to the presence of children.

Figures 1 through 4 present the effects of children on family budget shares over the life cycle. Separate full income streams were generated for families with one and two children. To analyze the first child's effects on full income allocation decisions, the one-child family full income streams were used in the simulations comparing childless and one-child allocation decisions. Alternatively stated, the expenditures of an average one-child family are compared with how that couple would have otherwise allocated its same full income in the absence of a child. Similarly, the two-child full income stream was used for generating budget share comparisons between the one-and two-child families. In this way, one effectively holds full income constant in the simulations to focus the analysis on the effects of changes in family composition (the addition of a child) on full income allocation.

As demonstrated in Figure 1, prairie families reduce the share of full income allocated to market goods upon the birth of children and only surpass what they would have spent on market goods in the absence of the marginal child once the first (only) child is nine and when the oldest (of two) is thirteen years of age.

The share of full income allocated to father's home production increases in the year of each child's birth and subsequently declines (see Figure 2), although the allocation is larger than it would have been in the absence of the child.

The share of full income allocated to the mother's household production increases steadily until the youngest child reaches the age of two (see Figure 3). After that point the share declines, but remains greater than it would have been in the absence of the marginal child.

It is important to remember that comparisons of spousal home production shares do not reveal differences in actual time spent in home production by parents. A gross comparison of actual hours spent by parents over the life cycle would require information about married men's and women's age earning profiles. Thus, for example, if both mothers' and fathers' shares of home production each amounted to ten percent of full income (twenty percent in total) and if on average women's hourly wages are half that of men, it would indicate that households allocate twice the number of women's hours to home production than that of men.

The first child has the effect of reducing the share of full income devoted to leisure activities. The second child has a somewhat smaller but negative marginal effect on share of leisure as well.

Overall, the effect of children appears to cause (1) a steady increase in the market good's share over the life cycle, (2) an increased but diminishing effect on allocations to home production over the life cycle, and (3) a reduction in share of full income devoted to leisure activities over the life cycle.

DISCUSSION

Simulations from this study provide a descriptive account of the influence of children on family full income allocations over the life cycle. As posited by Deaton and Muellbauer (1986), the simulations confirm that the opportunity costs associated with child rearing (women's home production) peak when children are young and gradually decline over the life cycle as goods' costs are increasing. Taken together with the simulation result that consumption shares increase over the life cycle, results confirm the findings of Browning, Deaton, and Irish (1985) that home production time and market purchased goods are substitutes, at least in the production of child services, over the life cycle.

Blundell and Walker (1982) examine the relationship between consumption over the life cycle and husbands' and wives' leisure. However, in employing Becker's general model of household time allocation by aggregating all time spent in nonmarket activities, they neglect to provide insight regarding the important distinction made by Mincer (1962) and Gronau (1977) between women's home production time and true leisure (consumption) time. For example, Blundell and Walker find that women's "leisure" time is complementary with consumption of clothing, food, energy, and durable goods. Men's leisure time, however, proves complementary only to consumption of services and energy. Although the present results do not permit empirical testing of the Blundell and Walker findings, descriptive simulations accent the important distinctions to be made regarding the relationship between the purchase of market goods and time allocations to each of home production and leisure (consumption) activities.

Kooreman and Kapteyn (1987) found that husband's time allocation is affected little by the presence of children. The present results indicate that the effect of family size on men's share of home production is significantly greater than its effect on his leisure. Further, it was found that families' allocation of fathers' share of home production is sensitive to life cycle stage, with allocations peaking during the first year of the child's life.

Studies consistently find that increased age of youngest child has a significant negative relationship with woman's time spent in home production (e.g., Gronau 1977; Kooreman and Kapteyn 1987). The present results support both these previous studies and others evidencing a concomitant positive relationship between female labor force participation and age of youngest child (e.g., Gronau 1977; Nakamura and Nakamura 1981). Results (Appendix B) indicate that the number of children less than seven years of age has a significantly negative effect on both labor force participation rates and hours spent in paid work.

Deaton and Muellbauer (1986) explain that one effect of adding a child to the household is to increase the price of consumption (market goods). Blundell and Walker (1982) make a similar observation with regard to the relationship between children's age and consumption. One would, thus, expect that as an increase in the price of consumption reduces real income, the amount of time spent in leisure would also decrease (assuming leisure is a normal good). Results in Table 4 indicate that the relationship between family size and share of full income allocated to leisure is indeed negative. In fact, leisure is the only share with a negative size elasticity. This result further confirms Gronau's (1977) finding that leisure activities of both spouses will decline as the number of children in the family increases.

CONCLUSIONS

In summary, through analysis of a model using consumer expenditure survey data supplemented with data from a unique time budget study, one is able to tie together results from previous studies and shed new light on the manner in which family composition influences the trade-off between demand for market purchased goods, household production, and leisure over the life cycle. The analysis further contributes to a better understanding of the manner in which families reallocate resources in order to meet the costs of raising children. Of particular note is the result that home production is at least an important method of meeting those costs as is increasing consumption of market goods, with substitutions occuring out of leisure.

Finally, three caveats must be noted. The first relates to the limitations of augmenting expenditure data with external time use estimates. In estimating parameters of a leisure equation from the time use data, extreme care must be exercised not to make a choice of variables that will have the consequence of ensuring a particular outcome in the final estimation. That is, one wants to avoid generating predicted NTBS time use values that will a priori result in well-fitting final parameter estimates. Two strategies were followed to avoid this dilemma. First, leisure time was chosen over home production time as the external measure of time use since it is less volatile over the life cycle. Second, variables in the leisure equation were specified in a functional form different from that used in the final expenditure system.

Next, it was implicitly assumed in simulating life cycle budget allocations from cross-sectional data that there were no cohort effects. For example, it was assumed that a 29-year-old husband and wife will in twenty years behave like their current 49-year-old counterparts.

Finally, the population from which inferences can be drawn in this study is relatively limited. Specifically excluded from the sample, and thus from the inferential population, are single-parent households, low-income households, and households who have children born (or adopted) to them when the husband is over age 46.

APPENDIX A

The continuous age variables ([L.sub.j]) are expressed in terms of an individual's actual age as deviations around the four reference ages and have the form:

[Mathematical Expression Omitted]

where:

[a.sub.s] = ln(age of person s) [a.sub.1] = ln(1), [a.sub.2] = ln(14), [a.sub.3] = ln(20), [a.sub.4] = ln(64), and [L.sub.i] = the [i.sup.th] LIP.

The LIP functions have the feature that if one of the reference ages is the person's actual age, the LIP corresponding to that reference age is equal to one while all the other LIPs equal zero. By taking the natural log of a family member's age, the estimated LIP parameters minimize the effects that higher ages would otherwise have on consumption in a linear specification.

APPENDIX B

The technique used to estimate shadow wages for nonemployed women in this sample was originally outlined by Heckman (1979). For an application of Heckman's technique to estimate shadow wages using United States data see Zick and Bryant (1983). Basically, the model involves the estimation of three equations: (1) a labor force participation equation (labelled probit below), (2) a market wage equation, and (3) an hours worked (labor supply) equation. The shadow wage is assumed to be a function of home productivity characteristics and the number of hours worked in the market and can be derived from the estimated coefficients of equations 1 and 3.

(1) See Abbott and Ashenfelter 1976, 1979; Blundell 1980; Blundell and Walker 1982; Browning, Deaton, and Irish 1985; Kooreman and Kapteyn 1987.

(2) In the previous study the authors applied a multinomial budget share allocation model specification rather than the linear logit specification.

(3) See, for example, Barten 1964; Blokland 1976; Buse and Salathe 1978; Espenshade 1984; Henderson 1950; Lazear and Michael 1980; Muellbauer 1980; Olson 1983; Prais and Houthakker 1955; Price 1971; Singh and Nagar 1973; Sydenstricker and King 1921; Tedford, Capps, and Havlicek 1986; Van der Gaag and Smolensky 1982.

(4) See, for example, Blokland 1976; Blundell and Walker 1982; Buse and Salathe 1978; Douthitt and Fedyk 1988; Tedford, Capps, and Havlicek 1986.

(5) See Appendix A and Tyrrell 1983 for a more complete exposition of the LIP functions.

(6) In addition to Tyrrell and Mount, see also Tyrrell (1983) and Douthitt and Fedyk (1988) for a more complete specification of the LIP functions.

(7) Simple linear regressions of leisure hours on independent variables that included dummy variables measuring respondent's education, age, and income and continuous variables measuring time spent in market work and family size were conducted separately for men and women. Complete details are available from the authors upon request.

(8) See Douthitt and Fedyk 1988 for a more complete exposition of the LIP specification.

(9) A simple linear regression of full income on age of head, age of head squared, and the logged values of age of youngest child and number of children was estimated to generate values of full income for each year in the simulated family life cycle.

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Lazear, F. P. and R. T. Michael (1980), "Family Size and the Distribution of Real per Capita Income," American Economic Review, 70: 91-107.

Mincer, J. (1962), "Labor Force Participants of Married Women: A Study of Labor Supply," Aspects of Labor Economics, A Conference of the Universities, National Bureau of Economic Research, Princeton, NJ: Princeton University Press.

Muellbauer, J. (1980), "The Estimation of the Prais-Houthakker Model of Equivalence Scales," Econometrica, 48: 153-176.

Nakamura, A. and M. Nakamura (1981), "A Comparison of the Labor Force Behavior of Married Women in the United States and Canada, With Special Attention to the Impact of Income Taxes," Econometrica, 49: 451-489.

Olson, L. (1983), Costs of Children. Toronto, Canada: Lexington Books.

Prais, S. J. and H. S. Houthakker (1955), The Analysis of Family Budgets, Cambridge: Cambridge University Press.

Price, D. W. (1971), "Unit Equivalence Scales for Specific Food Commodities," American Journal of Agricultural Economics, 52: 224-233.

Singh, B. and A. L. Nagar (1973), "Determination of Consumer Unit Scales," Econometrica, 41: 347-355.

Sydenstricker, F. and W. I. King (1921), "The Measurement of the Relative Economic Status of Families," Quarterly Publication of the American Statistical Association, 17: 842-857.

Tedford, J. R., O. Capps, and J. Havlicek (1986), "Adult Equivalence Scales Once More--A Developmental Approach," American Journal of Agricultural Economics, 68(2): 321-333.

Tyrrell, T. J. (1983), "The Use of Polynomials to Shift Coefficients in Linear Regression Models," Journal of Business and Economic Statistics, 1: 249-252.

Tyrrell, T. and T. Mount (1982), "A Nonlinear Expenditure System Using a Linear Logit Specification," American Journal of Agricultural Economics, 64: 539-546.

Van der Gaag, J. and E. Smolensky (1982), "True Household Equivalence Scales and Characteristics of the Poor in the United States," Review of Income and Wealth, 28: 17-28.

Zick, C. D. and W. K. Bryant (1983), "Alternative Strategies for Pricing Home Work Time," Home Economics Research Journal, 12: 133-144.

Robin A. Douthitt is an Assistant Professor at the University of Wisconsin, Madison, and Joanne M. Fedyk is an Assistant Professor at the University of Saskatchewan, Canada.

Special thanks are due Susan Bruns for running what must have seemed like thousands of simulations and Brian Gould for reading what must have seemed like endless revisions. Anonymous reviewers also made suggestions that substantively improved the manuscript's exposition. None of the aforementioned persons, however, is responsible for any remaining errors or omissions.

Analyzing the joint determination of family consumption and time allocation has been the subject of many recent studies (1) that appeal to Becker's (1965) and Gronau's (1977) theoretical work. However, as Kooreman and Kapteyn (1987) note, the analysis of time spent by members in household production of goods and services and its relationship to the demand for either market goods or leisure is restricted by lack of detailed time use data. Studies designed to examine family expenditures usually collect labor supply data but little information regarding other time allocation decisions. Similarly, time budget studies usually collect little family expenditure information. Thus, previous studies of the joint determination of demand for time and goods examine expenditures among commodity groups omitting the time allocation (home production and leisure time) function (e.g., Blundell 1980; Blundell and Walker 1982) or examine time allocation decisions among activities omitting the consumption function (e.g., Kooreman and Kapteyn 1987). In this paper results are reported from a unique study that incorporates information about both family expenditures and time allocation.

Of particular interest is further examination of family composition's influence on full income allocated to leisure and home production activities as well as to market-purchased goods. The hypothesis of interest is one suggested by Deaton and Muellbauer (1986), that families with young children face large time (foregone leisure) relative to goods expenditures, and the trend reverses as children age.

The first step toward accomplishing the stated objectives was to estimate parameters of a nonlinear expenditure system using a linear logit specification that includes a continuous measure of family composition (Tyrrell and Mount 1982; Douthitt and Fedyk 1988 (2)). The present analysis differs from the authors' previous study where family composition influenced only goods consumption. The purpose of that study was to examine substitutions made by families within categories of current consumption in response to changes in family composition. In that model it was implicity assumed that earned income (leisure) and levels of home production were exogenous to current consumption decisions. The present study examines how the presence of children influences not only current consumption decisions, but also time allocation decisions and income generation.

Finally, pursuant to the objective of comparing family composition effects on time versus goods trade-offs, parameter estimates are used to calculate expenditure and family size elasticities and to conduct life cycle simulations of full income allocations to time and goods over the life cycle.

The remainder of the paper is organized as follows. Section two reviews the theoretical underpinnings of the model used. The third section explains the model specification. Section four discusses the data and statistical method used to estimate the model's parameters. Section five presents parameter estimates, and section six contains life cycle simulations of full income allocation decisions. The final section summarizes findings and compares the results to previous studies.

The present analysis embraces Becker's (1965) theory of time allocation. Relying on Gronau's (1977) adaptation of this theory, it is assumed that families maximize household utility through the consumption of combinations of goods and services (X) and leisure (R). Goods and services are either purchased in the market place ([X.sub.m]) or produced in the home ([X.sub.h]). A family's utility is thus:

U = U(X,R) (1)

where X = [X.sub.m] + [X.sub.h].

Expenditures for market-purchased goods and services are made subject to the budget constraint:

[piX.sub.m] = WN + V (2)

where [pi] is an index of market prices, W and N are, respectively, market wage and hours worked in the market, and V is unearned income.

As with Gronau's theory, it is assumed for this study that home-produced goods and services ([X.sub.h]) are primarily a function of the amoung of time spent working in the home (H), and any market goods that may be used in home production are ignored:

[X.sub.h] = f(H). (3)

Household production is constrained by the total time available:

T = N + H + R. (4)

Since each hour spent in nonmarket time (H + R) has an opportunity cost equal to W, full income ([M.sub.F]) then becomes:

[M.sub.F] = WT + V. (5)

Maximization of utility subject to the full income constraint makes the marginal rate of substitution between goods and services (X) and leisure (R) equal to the shadow wage (W*), which, for persons in the paid labor force, is also equal to W--the market after-tax wage rate. Thus, while market prices represent the cost of market-purchased goods, the shadow wage represents the implicit cost of home-produced goods.

Assuming input prices are constant permits specification of the dual problem minimizing costs subject to a given level of goods and services, [X.sub.m.sup.o] and [X.sub.h.sup.o], and of leisure, [R.sup.o] (Deaton and Muellbauer 1980, p. 37). The associated Lagrangian function for minimizing expenditures is

Min L = [[pi]X.sub.m] + [W*X.sub.h] + W*R + [lambda][[X.sub.m.sup.o] - (W*N + V)] + [theta][[X.sub.h.sup.o] - f(H)] + [Phi][[R.sup.o] - (T - H - N)].

The input demand functions derived from minimizing equation (6) are the basis of the expenditure functions analyzed in this study and are expressed as:

[f.sub.1.([pi],X.sub.m.sup.o]) = [e.sub.1.([pi],U.sup.o]), [f.sub.2(W*,X.sub.h.sup.o)] = [e.sub.2(W*,U.sup.o)], and [f.sub.3(W*,R.sup.o)] = [e.sub.3(W*,U.sup.o)].

The demands for inputs can be derived directly from the expenditure functions:

[[derivative]e.sub.1.([pi],U.sup.o]) / [derivative][pi] = [X.sub.m] = [g.sub.1.([pi],M.sub.F]), [[derivative]e.sub.2.(W*,U.sup.o]) / [derivative]W* = [X.sub.h] = [g.sub.2.(W*,M.sub.F])M and [[derivative]e.sub.3.(W*,U.sup.o] / [derivative]W* = R = [g.sub.3.(W*,M.sub.F]),

where [M.sub.F], defined as WN + V, is the full income necessary to maintain utility at [U.sup.o].

EMPIRICAL MODEL

The empirical model used in this analysis is an extension of a nonlinear expenditure system described by Tyrrell and Mount (1982). It incorporates both continuous measures of family composition and a flexible functional form.

A revealed preference approach is used to derive continuous measures of household composition effects on family expenditure decisions. Although reliance upon revealed preference is a commonly used strategy for deriving parameter estimates of consumer demand, (3) few studies also incorporate continuous measures of family size and structure effects on spending behavior. (4) Friedman (1957) first developed the concept of a continuous composition or equivalence scale measure. Its strengths include continuity over size or age range measures (i.e., measured effects scales do not "jump" between adjacent age categories) and fewer required parameters for estimation.

In their continuous measure, Tyrrell and Mount (1982) combine the effects of family size and a composition multiplier (specific to each of the [i.sup.th] commodity groups) by taking the log of family size multiplied by a cubic function of family member age variables. Because it is not possible to estimate these cubic functions directly, they approximate them by using a set of Almon lag structures referred to as Lagrangian interpolation polynomials (LIPs). (5) The LIPs ([L.sub.j]'s) are expressed in terms of member's age (a) as it deviates around four reference ages ([a.sub.1.. . . a.sub.4]) (6) and have the form:

[L.sub.j](a) = ([a - a.sub.2])([a - a.sub.3])([a - a.sub.4]) / ([a.sub.1 - a.sub.2])([a.sub.1 - a.sub.3])([a.sub.1 - a.sub.4])

The four reference ages were chosen to correspond with points in the life cycle where changes in consumption were expected to occur. The sum of the LIP parameters from each equation estimate the weight associated with an age-specific equivalence effect. In Tyrrell and Mount's specification, separate age LIP functions were estimated for males and females, thus eight LIPs were required.

The Tyrrell and Mount model also incorporates a flexible functional form. Many econometric models of demand a priori restrict estimated parameter values to be consistent with postulates of economic theory. For example, many expenditure allocation models assume functions homogeneous of degree one in income and family size (e.g., Prais and Houthakker 1955).

However, it can be demonstrated that the assumption of homogeneity can generate nonsensical results when applied to actual behavior and that homogeneity, coupled with an equivalence scale specification, implies constant returns to scale. Blundell and Walker (1982), for example, apply these assumptions to their work and "conclude" no economies of scale exist in rearing children. Numerous illustrations can be cited to refute this restriction. The purchase of food in larger quantities sold at lower per unit prices and the reuse of clothing are standard examples of economies of scale that could occur upon the addition of a family member.

The Tyrrell and Mount model ensures the theoretical restrictions of adding-up while allowing for nonhomogeneous demand functions and economies of scale. Thus, the model provides an effective balance between the concern for theoretical plausibility and the applied need to maximize explained variance. Their theoretical model, expressed in the logistic form of the budget shares ([c.sub.i]), is

[Mathematical Expressions Omitted]

where:

[c.sub.i] = expenditures for [i.sup.th] commodity group as a share of total expenditures, M = income (WN + V), S = family size, and L = a vector of family composition terms.

In this adaptation of the Tyrrell and Mount model, budget shares ([Q.sub.i]) are defined as commodity outlays divided by total expenditures (E) for market goods and services ([[pi] X.sub.m]), spouses' household production ([[Sigma]w.sub.i.*h.sub.i]), and leisure ([[Sigma]w.sub.i.*r.sub.i]). Specifically, shares are defined as:

[Q.sub.1] = [[pi]X.sub.m./E]; share of total expenditures allocated to family expenditures on market purchased goods, [Q.sub.2] = [w.sub.1.*h.sub.1./E]; share of total expenditures allocated to wife's home production, where [w.sub.1.*] is wife's shadow wage, and [h.sub.1] is the amount of time the wife spends in home production, [Q.sub.3] = [w.sub.2.*h.sub.2./E]; share of total expenditures allocated to husband's home production, where [w.sub.2.*] is husband's shadow wage and [h.sub.2] is husband's home production time, and [Q.sub.4] = ([w.sub.1.*r.sub.1] + [w.sub.2.*r.sub.2])/E; share of total expenditures allocated to leisure time by both parents, where [r.sub.1] is wife's leisure time, and [r.sub.2] is husband's leisure time.

Empirically, then, each budget share is expressed as a function of full income ([M.sub.F]) and family composition:

[Mathematical Expression Omitted]

where:

[B.sub.i]'s = parameter estimates associated with the [i.sup.th] expenditure group.

Tyrrell and Mount estimate a linear approximation of the logistic share model (equation 11) by expressing each of the n - 1 commodity shares as a proportion of the share for the [n.sub.th] commodity group. The same technique is employed in this study where the [Q.sub.i.sup.th] share, i.e., the share of total expenditures allocated to market goods ([Q.sub.1]), to wife's home production ([Q.sub.2]), and husband's home production ([Q.sub.3]) are divided by the budget share for combined spouse leisure ([Q.sub.4]). Thus, the final estimating equations are:

[Mathematical Expression Omitted]

where estimated coefficients measure the differences in effects of each independent variable on the n - 1 budget shares relative to its effect on leisure.

The effects of family composition (L) are approximated by using Tyrrell and Mount's LIP functions with associated reference ages of 1, 14, 20, and 64. The ages were chosen to correspond with points in the life cycle where changes in consumption have previously been found to occur (Tedford, Capps, and Havlicek 1986).

DATA AND METHOD DESCRIPTIONS

The data used in this study were collected as part of Statistics Canada's 1982 Survey of Family Expenditures (FES). They include both expenditure and demographic information for a random sample of over 10,000 Canadian households. The scope of the study was limited in several respects because Statistics Canada did not release the entire survey for public use. For example, children's gender was not revealed, meaning that adjustments in family budget allocations attributable to such differences (Olson 1983) could not be examined.

Since prices and preferences are unobservable, the analysis was limited to a single region, the Canadian prairie provinces of Manitoba, Saskatchewan, and Alberta. The final sample includes only nonfarm families where all family members are present in the household for the entire year. Extended families, or families with more than two adults (where the third adult is not a parent's child) are eliminated from the sample. Further selection criteria are that neither spouse was older than 64 years of age, not more than one-third of family income comes from income-tested government sources, and children are not employed full-time in the paid labor force. Mean sample characteristics are presented in Table 1.

Using equation (12), the system was jointly estimated with generalized least squares (GLS). Because all equations contain the same independent variables and no across-equation constraints are imposed, the GLS estimators are equivalent to ordinary least squares.

Independent variables in the analysis are full income, family size, and four transformations of each family member's age (LIPs). For each family in the sample, full income ([M.sub.F]) (Equation 5) was calculated. The FES data contain information regarding total money expenditures ([X.sub.m]), as well as parental earnings, weeks worked, and unearned income. Data for time spent in household production (H) and a value for earnings per hour of work ([W.sub.p]) for those parents who were not paid labor force participants were not available.

With regard to estimating home production time, fortunately, in the same year that the FES data were collected (see Kinsley and O'Donnell 1983), Employment and Immigration, Canada conducted a national time use study (NTBS). After defining a NTBS sample and independent variables analogous to the FES prairie sample/data, leisure hours equations were estimated for both men and women. (7) The NTBS parameter estimates were then used to predict time spent in leisure for the FES families. Time spent in home production (H) was calculated as a residual of total time (T) available in the year (52 weeks) minus predicted leisure time (R) and FES reported time spent in labor force activities (N): H = T - N - R.

Measures of the value of parental earnings for those who did not report a market wage were derived using a technique outlined by Heckman (1979) to derive "shadow" wages (W*). As implied by the first order conditions of the theoretical model, individuals with observable wages were assigned a value of earnings equal to their actual wages, while those with unobserved wages were assigned a shadow wage. Appendix B presents intermediate results from the sample selection procedure for women who were not paid labor force participants. Table 2 presents descriptive statistics regarding both actual (for employed persons) and imputed shadow (for all) wages for men and women. Home production and leisure time were each assigned a value equal to the shadow or market after-tax wage, which was multiplied by hours spent in each activity to derive estimates of expenditures of full income allocated to spouses' home production and leisure.

As mentioned in the empirical section, the effects of family composition were estimated by using Tyrrell and Mount's LIP functions. However, because there was no information regarding children's gender, it was necessary to estimate separate LIPs for parents and children. The equation was normalized by omitting the third adult LIP, making the reference household a twenty-year-old childless couple. In total, then, seven LIP parameters (8) were estimated.

EMPIRICAL RESULTS

The parameter estimates of equation (12) are presented in Table 3. Over half of the independent variables in the system are significant at the .05 level or above. All three estimated share equations explain a significant amount of share variance vis-a-vis the reference leisure share of full income. The greatest difference in total explained variance is found in the women's home production equation. Consistent with the specification of the final estimating equation (12), all parameters represent effects on the dependent variable of a unit change in the independent variables vis-a-vis parameters of the leisure equation (4), by which the remaining three equations were normalized. Thus, interpretation of each coefficient is the difference between the effect of a change in the independent variable on the dependent variable vis-a-vis its effect on family leisure allocations. For example, the parameter on full income (-.238) in the market goods equation is equal to the difference ([B.sub.11.-B.sub.14]) and implies that the effect of a one percent change in full income on market good expenditures is -.238 less than that of a comparable change in full income on leisure expenditures.

Table 4 presents predicted budget shares and estimates of both income and size elasticities calculated from the model parameters for two family types. The top half of Table 4 shows estimates for a childless couple where the adult male is 32 and the adult female is 29 years of age. The bottom half of the table gives findings for a similar pair of adults with two children ages four and two.

Results shown in Table 4 indicate that childless couples allocate larger shares of full income to both market goods and leisure compared with their counterparts with children. In other words, if wages are assumed constant, the childless couple spends less time in home production activities than does the couple with children. For both families, men's time in home production and combined spouse leisure shares have an income elasticity greater than one. This means that an increase in full income generates a greater increase in husband's home production share and that both husband and wife spend an increasing share of time on leisure when full income rises.

An increase in family size results in an increase in the share of full income allocated to market goods, men's and women's time in home production, and a decrease in the share of full income allocated to leisure. It is noteworthy that the effect of family size on women's home work is equal to that for market goods. That is, a one percent increase in family size increases both shares of market goods and women's home production by .13 percent. This result strengthens claims by Espenshade (1984) and others that the opportunity costs of children are comparable to their direct costs. Further, as evidenced by the negative size elasticity, increases in family size cause substitutions to occur primarily out of leisure.

SIMULATION RESULTS

From these parameter estimates, simulations were conducted to illustrate how an "average" family reallocates full income over the life cycle. The average family was identified by using provincial statistics regarding prairie family economic, fertility, and marriage behavior. Given the 1982 population average age at first marriage, it is assumed that the hypothetical couple marries when the man is 26 and the woman is 23. In accordance with the national figures for median age at first and second birth, the first child is assumed to be born when the father is 28 and the mother is 25, and the second child is born two years later when the mother is 27 and the father is 30. Children are assigned an age of 0 for the year of their birth since the sample does not provide information on expenditures for families with members present for only part of the year.

In order to simulate results for the average Canadian prairie family, it was necessary to select a flow of full income values over the life cycle. To do this a full income equation (9) was estimated for the sample and used to generate predicted values for the average family over the life cycle.

Predicted budget shares are generated for each year with every family member aging accordingly. It is assumed that children leave home after their eighteenth year. For comparative purposes, simulations of the allocations of a childless couple with the same demographic characteristics and who marry at the same age as the couple with children were conducted. Differences between these two simulations permit comparisons of shifts in budget allocations attributable to the presence of children.

Figures 1 through 4 present the effects of children on family budget shares over the life cycle. Separate full income streams were generated for families with one and two children. To analyze the first child's effects on full income allocation decisions, the one-child family full income streams were used in the simulations comparing childless and one-child allocation decisions. Alternatively stated, the expenditures of an average one-child family are compared with how that couple would have otherwise allocated its same full income in the absence of a child. Similarly, the two-child full income stream was used for generating budget share comparisons between the one-and two-child families. In this way, one effectively holds full income constant in the simulations to focus the analysis on the effects of changes in family composition (the addition of a child) on full income allocation.

As demonstrated in Figure 1, prairie families reduce the share of full income allocated to market goods upon the birth of children and only surpass what they would have spent on market goods in the absence of the marginal child once the first (only) child is nine and when the oldest (of two) is thirteen years of age.

The share of full income allocated to father's home production increases in the year of each child's birth and subsequently declines (see Figure 2), although the allocation is larger than it would have been in the absence of the child.

The share of full income allocated to the mother's household production increases steadily until the youngest child reaches the age of two (see Figure 3). After that point the share declines, but remains greater than it would have been in the absence of the marginal child.

It is important to remember that comparisons of spousal home production shares do not reveal differences in actual time spent in home production by parents. A gross comparison of actual hours spent by parents over the life cycle would require information about married men's and women's age earning profiles. Thus, for example, if both mothers' and fathers' shares of home production each amounted to ten percent of full income (twenty percent in total) and if on average women's hourly wages are half that of men, it would indicate that households allocate twice the number of women's hours to home production than that of men.

The first child has the effect of reducing the share of full income devoted to leisure activities. The second child has a somewhat smaller but negative marginal effect on share of leisure as well.

Overall, the effect of children appears to cause (1) a steady increase in the market good's share over the life cycle, (2) an increased but diminishing effect on allocations to home production over the life cycle, and (3) a reduction in share of full income devoted to leisure activities over the life cycle.

DISCUSSION

Simulations from this study provide a descriptive account of the influence of children on family full income allocations over the life cycle. As posited by Deaton and Muellbauer (1986), the simulations confirm that the opportunity costs associated with child rearing (women's home production) peak when children are young and gradually decline over the life cycle as goods' costs are increasing. Taken together with the simulation result that consumption shares increase over the life cycle, results confirm the findings of Browning, Deaton, and Irish (1985) that home production time and market purchased goods are substitutes, at least in the production of child services, over the life cycle.

Blundell and Walker (1982) examine the relationship between consumption over the life cycle and husbands' and wives' leisure. However, in employing Becker's general model of household time allocation by aggregating all time spent in nonmarket activities, they neglect to provide insight regarding the important distinction made by Mincer (1962) and Gronau (1977) between women's home production time and true leisure (consumption) time. For example, Blundell and Walker find that women's "leisure" time is complementary with consumption of clothing, food, energy, and durable goods. Men's leisure time, however, proves complementary only to consumption of services and energy. Although the present results do not permit empirical testing of the Blundell and Walker findings, descriptive simulations accent the important distinctions to be made regarding the relationship between the purchase of market goods and time allocations to each of home production and leisure (consumption) activities.

Kooreman and Kapteyn (1987) found that husband's time allocation is affected little by the presence of children. The present results indicate that the effect of family size on men's share of home production is significantly greater than its effect on his leisure. Further, it was found that families' allocation of fathers' share of home production is sensitive to life cycle stage, with allocations peaking during the first year of the child's life.

Studies consistently find that increased age of youngest child has a significant negative relationship with woman's time spent in home production (e.g., Gronau 1977; Kooreman and Kapteyn 1987). The present results support both these previous studies and others evidencing a concomitant positive relationship between female labor force participation and age of youngest child (e.g., Gronau 1977; Nakamura and Nakamura 1981). Results (Appendix B) indicate that the number of children less than seven years of age has a significantly negative effect on both labor force participation rates and hours spent in paid work.

Deaton and Muellbauer (1986) explain that one effect of adding a child to the household is to increase the price of consumption (market goods). Blundell and Walker (1982) make a similar observation with regard to the relationship between children's age and consumption. One would, thus, expect that as an increase in the price of consumption reduces real income, the amount of time spent in leisure would also decrease (assuming leisure is a normal good). Results in Table 4 indicate that the relationship between family size and share of full income allocated to leisure is indeed negative. In fact, leisure is the only share with a negative size elasticity. This result further confirms Gronau's (1977) finding that leisure activities of both spouses will decline as the number of children in the family increases.

CONCLUSIONS

In summary, through analysis of a model using consumer expenditure survey data supplemented with data from a unique time budget study, one is able to tie together results from previous studies and shed new light on the manner in which family composition influences the trade-off between demand for market purchased goods, household production, and leisure over the life cycle. The analysis further contributes to a better understanding of the manner in which families reallocate resources in order to meet the costs of raising children. Of particular note is the result that home production is at least an important method of meeting those costs as is increasing consumption of market goods, with substitutions occuring out of leisure.

Finally, three caveats must be noted. The first relates to the limitations of augmenting expenditure data with external time use estimates. In estimating parameters of a leisure equation from the time use data, extreme care must be exercised not to make a choice of variables that will have the consequence of ensuring a particular outcome in the final estimation. That is, one wants to avoid generating predicted NTBS time use values that will a priori result in well-fitting final parameter estimates. Two strategies were followed to avoid this dilemma. First, leisure time was chosen over home production time as the external measure of time use since it is less volatile over the life cycle. Second, variables in the leisure equation were specified in a functional form different from that used in the final expenditure system.

Next, it was implicitly assumed in simulating life cycle budget allocations from cross-sectional data that there were no cohort effects. For example, it was assumed that a 29-year-old husband and wife will in twenty years behave like their current 49-year-old counterparts.

Finally, the population from which inferences can be drawn in this study is relatively limited. Specifically excluded from the sample, and thus from the inferential population, are single-parent households, low-income households, and households who have children born (or adopted) to them when the husband is over age 46.

APPENDIX A

The continuous age variables ([L.sub.j]) are expressed in terms of an individual's actual age as deviations around the four reference ages and have the form:

[Mathematical Expression Omitted]

where:

[a.sub.s] = ln(age of person s) [a.sub.1] = ln(1), [a.sub.2] = ln(14), [a.sub.3] = ln(20), [a.sub.4] = ln(64), and [L.sub.i] = the [i.sup.th] LIP.

The LIP functions have the feature that if one of the reference ages is the person's actual age, the LIP corresponding to that reference age is equal to one while all the other LIPs equal zero. By taking the natural log of a family member's age, the estimated LIP parameters minimize the effects that higher ages would otherwise have on consumption in a linear specification.

APPENDIX B

The technique used to estimate shadow wages for nonemployed women in this sample was originally outlined by Heckman (1979). For an application of Heckman's technique to estimate shadow wages using United States data see Zick and Bryant (1983). Basically, the model involves the estimation of three equations: (1) a labor force participation equation (labelled probit below), (2) a market wage equation, and (3) an hours worked (labor supply) equation. The shadow wage is assumed to be a function of home productivity characteristics and the number of hours worked in the market and can be derived from the estimated coefficients of equations 1 and 3.

(1) See Abbott and Ashenfelter 1976, 1979; Blundell 1980; Blundell and Walker 1982; Browning, Deaton, and Irish 1985; Kooreman and Kapteyn 1987.

(2) In the previous study the authors applied a multinomial budget share allocation model specification rather than the linear logit specification.

(3) See, for example, Barten 1964; Blokland 1976; Buse and Salathe 1978; Espenshade 1984; Henderson 1950; Lazear and Michael 1980; Muellbauer 1980; Olson 1983; Prais and Houthakker 1955; Price 1971; Singh and Nagar 1973; Sydenstricker and King 1921; Tedford, Capps, and Havlicek 1986; Van der Gaag and Smolensky 1982.

(4) See, for example, Blokland 1976; Blundell and Walker 1982; Buse and Salathe 1978; Douthitt and Fedyk 1988; Tedford, Capps, and Havlicek 1986.

(5) See Appendix A and Tyrrell 1983 for a more complete exposition of the LIP functions.

(6) In addition to Tyrrell and Mount, see also Tyrrell (1983) and Douthitt and Fedyk (1988) for a more complete specification of the LIP functions.

(7) Simple linear regressions of leisure hours on independent variables that included dummy variables measuring respondent's education, age, and income and continuous variables measuring time spent in market work and family size were conducted separately for men and women. Complete details are available from the authors upon request.

(8) See Douthitt and Fedyk 1988 for a more complete exposition of the LIP specification.

(9) A simple linear regression of full income on age of head, age of head squared, and the logged values of age of youngest child and number of children was estimated to generate values of full income for each year in the simulated family life cycle.

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Abbott, M. and O. Ashenfelter (1976), "Labor Supply, Commodity Demand and the Allocation of Time," Review of Economic Studies, 43: 389-411.

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Robin A. Douthitt is an Assistant Professor at the University of Wisconsin, Madison, and Joanne M. Fedyk is an Assistant Professor at the University of Saskatchewan, Canada.

Special thanks are due Susan Bruns for running what must have seemed like thousands of simulations and Brian Gould for reading what must have seemed like endless revisions. Anonymous reviewers also made suggestions that substantively improved the manuscript's exposition. None of the aforementioned persons, however, is responsible for any remaining errors or omissions.

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Author: | Douthitt, Robin A.; Fedyk, Joanne M. |
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Publication: | Journal of Consumer Affairs |

Date: | Jun 22, 1990 |

Words: | 6112 |

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