BLS WORKING PAPERS 

 

U.S. Department of Labor 

U.S. Bureau of Labor Statistics 

Office of Employment and Unemployment Statistics 

How Does Household Production Affect Earnings Inequality? 

Evidence from the American Time Use Survey 

Harley Frazis, U.S. Bureau of Labor Statistics 

Jay Stewart, U.S. Bureau of Labor Statistics 

Working Paper 393

May 2006

All views expressed in this paper are those of the authors and do not necessarily reflect the views or policies of the 

U.S. Bureau of Labor Statistics. 
How Does Household Production Affect Earnings Inequality? 

Evidence from the American Time Use Survey 

April 2006 

Harley Frazis and Jay Stewart 

U.S. Bureau of Labor Statistics 

2 Massachusetts Avenue, NE 

Frazis.Harley@BLS.GOV 

Stewart.Jay@BLS.GOV 

The authors thank Peter Gottschalk, Jim Spletzer, Frank Stafford, and participants in sessions at 

the 2004 meetings of the International Association of Research on Income and Wealth, the 2005 

Allied Social Science Associations meetings, and the Levy Economics Institute conference 

“Time Use and Economic Well-Being.” The views expressed here are those of the authors and 

do not necessarily reflect the views of the U.S. Department of Labor or the Bureau of Labor 

Statistics. 
How Does Household Production Affect Earnings Inequality? 

Evidence from the American Time Use Survey 

April 2006 

Harley Frazis and Jay Stewart 

U.S. Bureau of Labor Statistics 

2 Massachusetts Avenue, NE 

Frazis.Harley@BLS.GOV 

Stewart.Jay@BLS.GOV 

Abstract 

Although income inequality has been studied extensively, relatively little attention has been paid 

to the role of household production. Economic theory predicts that households with less money 

income will produce more goods at home. Thus extended income, which includes the value of 

household production, should be more equally distributed than money income. We find this to 

be true, but not for the reason predicted by theory. Virtually all of the decline in measured 

inequality when moving from money income to extended income is due to the addition of a large 

constant--the average value of household production--to money income. This result is robust to 

alternative assumptions that one might make when estimating the value of household production. 
I. Introduction 

Inequality of earnings and inequality of household income have increased over recent 

decades, both in the United States and to a lesser extent in other industrialized countries (see 

Gottschalk and Smeeding 1997 for a review). In response to these developments, there has been 

an outpouring of research describing and trying to explain these trends. Most of these studies 

concern inequality in money income, where data are more readily available. However, money 

income is not a complete measure of economic welfare. Recent studies have attempted to 

describe inequality along other dimensions--for example, inequality in consumption (Johnson 

and Shipp 1995, 1997, and 2005; Krueger and Perri 2002) and in total compensation including 

fringe benefits (Pierce 2001). 

Another branch of the inequality literature has incorporated the value of household 

production--goods and services produced at home--into measures of income to arrive at what is 

referred to as “extended income.”1

 This measure is sensible, because home production 

represents additional “income” that is available for consumption but is not included in money 

income. Models of household production (for example, Gronau 1986) predict that high-wage 

workers will spend less time on nonmarket work than low-wage workers (assuming that all 

individuals have identical preferences and are equally productive in nonmarket work). 

Extending this model to two-person households generates analogous predictions for married 

couples.2

 These predictions lead us to expect a negative correlation between money income and 

 

1

 The definition of extended income varies somewhat across studies. Our definition is similar to that used in Jenkins 

and O’Leary (1996). 

2

 For more details, please see the Appendix, which presents the Gronau (1986) model, extends the model to twoperson households, and discusses the assumptions that drive these results. 
- 2 -

time spent in household production, which implies that extended income will be more equally 

distributed than money income. 

Several studies have empirically compared inequality measures using extended income 

and money income, and all but one have found that, consistent with theory, extended income is 

more equally distributed than money income.3

 However, the data that have been available to 

examine this question are far from perfect, and it is possible that the procedures used to address 

the deficiencies may be driving the results. 

The ideal data would include information on household production, as well as income, 

for every member of the household. However, time-use surveys, which are the main data source 

for household production, typically collect data for only one or two days, and many collect data 

for only one household member. This results in an incomplete picture of household production 

at both the individual and household level. 

The most common approach to addressing this deficiency is to fill in the missing data 

using regression techniques to impute the time spent in household production (Bonke, 1992; 

Jenkins and O’Leary, 1996; and Wolff, Zacharias, and Caner, 2004). However, these imputation 

procedures smooth over the individual variation in the data and reduce measured inequality in 

household production. The Jenkins and O’Leary study perturbed estimated household 

production by adding a random term to the imputed values. A more recent study by Bonke, 

Deding, and Lausten (2004)--the only study to find that extended income is less equally 

distributed than money income--multiplied each respondent’s average hours per day spent in 

household production by 365. This approach tends to magnify the individual variation, because 

 

3

 Bonke (1992) found that when taxes are incorporated, extended income is less equally distributed than money 

income. 
- 3 -

in addition to the household's average time spent in production it also includes day-to-day 

variation given that the data contain only one or two diaries per person. 

Another drawback to using regression techniques to impute household production is that 

most time-use surveys have little or no income data. Imputed values of household production are 

matched to a second dataset that contains income using variables common to both datasets. This 

means that income cannot be used as a covariate in the imputation procedure, and that any 

relationship between income and time spent in household production may be lost.4

 

A study by Gottschalk and Mayer (2002) avoided both of these problems by using data 

from Panel Study of Income Dynamics (PSID), which contains information on earned and 

unearned income as well as a measure of the usual amount of time spent doing household work. 

The main drawback to this approach, as they acknowledge, is that the PSID measure of 

household production leaves much to be desired. The question does not define household 

production, which could result in biased estimates if there are systematic differences in how 

respondents report. And because the question asks about time usually spent doing housework, it 

may be subject to recall and social desirability biases. 

In this paper, we use data for 2003 from the American Time Use Survey (ATUS) to 

construct measures of earnings inequality that include the value of household production and 

examine how this inclusion affects measured inequality. Although, as in previous studies, it is 

still necessary to impute the value of household production to fill in missing data, our 

imputations are an improvement over those in earlier studies because ATUS makes it possible to 

include earnings as a covariate in the imputation process. We also perform a number of 

 

4

 Bonke (1992) was able to use income data from the register of income taxation for the respondents in the time-use 

survey, but this alternative is usually not available to researchers. 
- 4 -

sensitivity analyses to determine the extent to which these imputation procedures are likely to 

matter. 

We show that the greater equality of extended income compared to money income is not 

caused by the procedures used to address data deficiencies. But nor is it due to the predicted 

negative relationship between money income and time spent doing household production. In 

fact, we find, as did Jenkins and O’Leary (1996), that the correlation between money income and 

household production is close to zero. Instead, the greater equality of extended income is almost 

entirely due to the addition of a large constant--the average value of household production--to 

money income. 

II. Methods 

Ideally, we would have a long-run measure of household production corresponding to the 

time period of our income measure (in this case, annual). In contrast to other time-use surveys, 

the ATUS can be directly linked to data on income. But like many other time-use surveys, the 

ATUS interviews only one person per household and collects only one diary per person. Thus, 

we can only estimate means of household production conditional on observable characteristics. 

We use a variation of Bonke's (1992) regression method of predicting household production. We 

regress the value of household production on the log of annual family income, the log of weekly 

earnings, the log of non-labor income, the log of the hourly wage, dummies for employment 

status (2 categories), education level (4 categories), age, and the number of children zero to 5, 6 

to 12, and 13 to 17. We run separate regressions by marital status and sex. For married 

respondents, we also include the log of spouse's weekly earnings, log of the spouse’s wage, and 

dummies for spouse’s employment status, education level, and age. 

Because of our interest in household production by income, we use a flexible 
- 5 -

specification for the log of family income. Specifically, we use Gallant's (1981) Fourier series 

expansion. Transforming the log of family income into the variable Z∈(0,2π) and letting X 

denote the vector of regressors listed above, our Fourier specification is: 

(1) ( , ) ( cos( ) sin( )) , 2

1

1

2 = + + +∑ β + β + β =

f Z X a bZ cZ jZ j jZ X

J

j

j

A function’s Fourier expansion has the property that the differences between the true value of a 

function g and the value of its Fourier expansion f and between the derivatives of g and the 

derivatives of f can be minimized to an arbitrary degree over the range of the function by 

choosing J to be sufficiently large. It thus provides a global approximation to the true function, 

rather than a local approximation (as in a Taylor series expansion). We selected J by crossvalidation, minimizing the sum of the squared prediction errors ∑ − −

2 ( ˆ ) i i y y , where i y−ˆ is the 

leave-one-out prediction generated by omitting observation i from the regression.5 

Using the flexible functional form in (1), we estimate the following equation to impute 

household production: 

(2) d

i i i

d Pit = f ( ( Z , X ) + u d = D,E), 

where separate equations were estimated for weekdays (D) and weekends (E) for each sex ×

marital status cell (8 regressions total). For each cell, we combine the predicted values from the 

weekday and weekend equations to generate imputed weekly value of household production for 

person i as follows: 

(3) ( , ) ˆ ( , ) 2 ˆ 5 ˆ

i D i i E Zi Xi P = f Z X + f , 

where X is appropriately defined. For married households, total household production is simply 
- 6 -

the sum of the husband’s and wife’s predicted values. 

This imputation procedure eliminates deviations from the conditional mean of household 

production. Because, as we noted in the Introduction, the absence of these residuals could bias 

estimates of income inequality, we assess the potential bias by adding a random perturbation and 

recomputing the inequality measures. The first step is to derive an estimate of the upper bound 

for the variance of the long-term household production that was eliminated in the imputation 

procedure. It is useful to decompose the residual in (2) into two components as follows: 

(4) ) ( , ) ( d

it

d

d i i i

d

it P = f Z X + m + e (d = D,E), 

where the residual is equal to the sum of a person-specific fixed effect ( d

mi ) and a term denoting 

day-to-day variation ( d

it e ).6

 If the Var( dmi ) = 0, the residual consists entirely of day-to-day 

variation in household production, and our imputation procedure will generate consistent 

estimates of long-run household production for each observation in the sample, which in turn 

will result in consistent estimates of inequality measures for extended income. However, if 

Var( d

mi ) > 0, our procedure will underestimate the variability of long-run household production 

across households, and usually generate downwardly biased inequality measures. 

We can use the residuals from (2) to place an upper bound on the variance of long-run 

household production. Letting σ d (d = D,E) denote the standard deviation of the residual in (2) 

for weekdays and weekends, the maximum possible variance for long-run weekly household 

production (i.e., assuming Var( d

it e ) = 0) is 2 (5 2 ) M = σ D + σ E . Thus Var( d

mi ) ∈ [0,M]. We 

 

5

 Andrews (1991) shows this criterion is asymptotically optimal in the sense that the probability of choosing the J

that minimizes the expected sum of squared errors converges to 1 as the sample size increases, even in the presence 

of heteroscedasticity. 

6

 More precisely, mi is the long-run average of Pit - f(Xi, Zi). We do not assume that the eit are independent across 

time. 
- 7 -

generate perturbed values of imputed household production for single-person households by 

using: 

(5) D i E i i

S

i P = f X + f (X ) + ks ˆ ( ) 2 ˆ 5 , 

where 0 < k ≤ 1 and si is drawn from N(0,M ) . 

For married-couple households, the maximum possible variance for long-run production 

across married couple households occurs when the residuals for spouses' production are perfectly 

positively correlated. Extending the definition of the maximum residual variance M to include 

spouses, we have: 2 ' (5[ ] 2[ ]) M = σ Dw +σ Dh + σ Ew +σ Eh where σ ds is the standard deviation of the 

residuals from the married-couple versions of (2) and subscripts denote day of week (d = D,E) 

and spouse (s = w,h). Total production is 

 (6) = M Pi ( , ) ' ˆ ( , ) 2 ˆ ( , ) 5 ˆ ( , ) 2 ˆ 5 Dw i i Ew i i Dh i i Eh i i i f Z X + f Z X + f Z X + f Z X + ks , 

where si' is drawn from N(0,M '). We recomputed our inequality measures assuming k = 0.25, k

= 0.50, and k = 1.0 for both singles and married couples. 

III. Data and Definitions 

 Our data come from the 2003 ATUS, which is a stratified random sample drawn from 

households that have completed their participation in the Current Population Survey (CPS) and is 

representative of the U.S. civilian population. The ATUS collects information on the amount of 

time spent in over 400 detailed activities. The ATUS time diary does not collect information 

about what else the respondent was doing at the time of each episode (secondary activities), but 

several summary questions asked at the end of the diary collect information about times and 

activities during which children under 13 were in their care. 
- 8 -

ATUS also contains labor force information that is comparable to that in the CPS, 

including employment status, usual hours worked per week, and earnings on the main job.7

 For 

the respondent’s spouse or unmarried partner, the ATUS collects basic labor force information--

employment status (employed or not employed) and total hours usually worked per week. 

Earnings are available from the CPS if the spouse was employed at the time of the last CPS 

interview. ATUS does not collect any labor force information for other household members. 

We divide the sample into single-adult and married-couple households. Our sample of 

single-adult households includes respondents age 25-64 who had no spouse or unmarried partner 

present. Our married-couple sample includes households where both spouses are between 25 

and 64. We excluded households with other adult (18+) family members in order to avoid the 

need to estimate the contribution of the other adult to household production. 

Finally, to obtain data on unearned income, we matched the ATUS data to the March 

supplement. Because of the sample rotation scheme used in CPS, only about one-third of 

respondents--those whose final CPS interviews were in March-June--were interviewed in 

March.8

 Family income and non-labor earnings variables for the remaining two-thirds of the 

sample are predicted by regression using variables common to both the ATUS and the March 

CPS. Households with allocated family income data were excluded from the sample, and family 

incomes below the 1st percentile were replaced with the 1st percentile value. Households where 

other (minor) family members contributed more than 10 percent of income were also excluded. 

 

7

 The earnings data are carried over from the final CPS interview. The earnings questions are asked in ATUS if the 

respondent had a new job in ATUS (either changed jobs or made a nonemployment-to-employment transition) or 

earnings were allocated in the last CPS interview. 

8

 Households are in the CPS for 4 consecutive months, out for 8, then back in for 4. Because of the lag between the 

final CPS interview and introduction into the ATUS, most of the ATUS respondents who were matched to March 

were interviewed for ATUS in June through September. 
- 9 -

Our sample consists of 10,048 observations from the 2003 ATUS data that fit our sample 

inclusion criteria. Of these, 3,329 observations had income data available from the March 

supplement and 2,639 of these had unallocated earnings. Our first step was to use these 2,639 

observations to estimate (2) for each of the sex × marital status × day cells, and determine the 

optimal value of J in (1).9

 Next, we reestimated (2) over the entire sample using the optimal 

value of J.10 For these regressions, we imputed income for observations that could not be 

matched to March or had allocated income in March.11 Coefficient estimates from these 

regressions were used to generate imputed values of household production in (3). Our extended 

income measure is computed for the 3,329 values matched to the March supplement as the sum 

of family income from March, including allocated values, and imputed household production. 

As in previous studies, we classify activities as household production if the same result 

could be achieved by hiring a third person (Reid, 1934). We used two alternative definitions of 

nonmarket work. The first definition includes household activities (including purchasing goods 

and services) and care of household members done as a primary activity.12 The second definition 

is the same, but adds child care done as a secondary activity. To avoid double counting in the 

second definition, we excluded secondary child care that was done at times when the respondent 

was engaged in household production as a primary activity.13 

 

9

 The number of observations for these regressions ranged from 180 (for single men on weekdays) to 468 (married 

women on weekends). 

10 For these regressions, the sample sizes ranged from 599 to 1,831. 

11 Family income was imputed using predicted values from a regression of income on covariates. As noted in 

Greene (2000, p. 363), including observations with imputed family income increases does not change the 

coefficient on family income, but it does increase the precision of the coefficients on the other variables. 12 We exclude volunteer work and care of non-household members from all of our measures. These activities could 

legitimately be classified as nonmarket work, but they do not contribute directly to the household’s income. In any 

case, the time spent in these activities is small, and their inclusion would have no effect on our results. 13 The implicit assumption is that it is possible to hire someone to do household chores and look after household 

children. Alternatively, one could assume that it would be necessary to hire two people--one to do the housework 
- 10 -

We use the replacement-cost approach to value household production. Under this 

approach, time spent in household production is valued at either a specialist wage that 

corresponds to the specific activity or a generalist wage. The specialist wages were generated 

using the Outgoing Rotation Group files from the CPS as follows. We computed the hoursweighted mean wage for each 3-digit occupation. The time spent in each nonmarket activity was 

valued at the wage for the occupation that most closely resembles the activity.14 For the 

generalist wage, we used the average wage for Maids and Housekeepers. We made no 

adjustments to account for differences in productivity in household production across 

households, although the lower productivity of non-specialists is a primary justification for using 

a generalist wage.15 

We considered using other approaches to valuing household production. The 

opportunity-cost approach, which uses the individual’s market wage to value the time spent in 

household production, has some conceptual and practical difficulties associated with it. On a 

conceptual level, the implicit assumption that hours of paid work are freely variable at the 

margin may not hold; workers, at least in the short run, may have no choice in their working 

hours. Perhaps more importantly, the opportunity cost approach assumes that people who are 

highly productive in market work are just as productive doing household work. It is hard to 

imagine that a lawyer is 5 times more productive building a deck than a carpenter. On a practical 

level, it would be necessary to impute a wage for nonworkers. Another approach, taken by 

Gronau (1980), is to specify a functional form for the marginal product of nonmarket work, 

 

and one to look after the children. Given that individuals routinely perform these tasks simultaneously, the former 

assumption makes more sense. 

14 This crosswalk is available from the authors upon request. 

15 For example, Wolff et al. (2004) multiplied this wage by a performance index that depends on household-level 

characteristics as well as characteristics of household members. 
- 11 -

estimate the parameters of the function using time-diary data, and integrate the function for each 

individual in the sample.16 This approach has the advantage of being based in theory, but it is 

sensitive to the functional form of the production function. 

Finally, because we are comparing extended income across households of different sizes, 

we adjusted extended income measures using two alternative equivalence scales. The first is the 

OECD equivalence scale (OECD, 2005), which is given by: E = I /(1.0 + 0.7(A −1) + 0.5C),

where E is equivalent income, I is the income measure, A is the number of adults in the 

household (either one or two in our case) and C is the number of children less than 18. The 

second is: E = I A + C . We run separate regressions for each equivalence scale. 

IV. Results 

Table 1 shows person-weighted estimates of mean household earnings and household 

production for the four definitions of household production.17 Under all measures, household 

production is a substantial fraction of household money earnings, from 31 percent (using the 

generalist wage and excluding secondary childcare) to 47 percent (specialist wages and including 

secondary childcare). Put differently, household production comprises 23-32 percent of 

combined labor earnings and household production. Married households do more household 

production than do single households, with household production comprising 32-49 percent of 

money income for married couples compared to 24-32 percent for singles. 

Table 2 shows results for five inequality measures: the coefficient of variation, the Gini 

coefficient, and the ratios of the 90th to the 50th percentile, the 50th to the 10th percentile, and the 

 

16 Gronau points out that an alternative approach in the same vein is to specify a functional form for the home 

production function, and then solve for the marginal product function. 

17 All of our estimates are person weighted, rather than household weighted. 
- 12 -

90th to the 10th. As expected, moving from money income to extended income (that is, from row 

1 to row 2 for each measure) substantially reduces measured inequality. Both the coefficient of 

variation and the Gini coefficient fall by about one-quarter regardless of whether the OECD or 

the square root scale is used. The effect on the 50/10 and 90/10 ratios are also quite dramatic, 

with the ratios falling by about one-third and one-half, respectively, under both scales. The 

effect on the 90/50 ratio is somewhat smaller, with the ratio falling by about one-fifth under both 

scales. (All of these differences are statistically significant at the 1 percent level.) The larger 

effects on the 50/10 and 90/10 ratios compared with the 90/50 ratio are not surprising, because 

we would expect household production to be a larger fraction of extended income for those 

lower in the money income distribution. 

Relative inequality measures such as the Gini coefficient will always fall if a positive 

constant is added to the income of all members of the population. Jenkins and O'Leary (1996) 

pointed out that inequality of extended income is positively related to the variance of household 

production and the correlation between money income and household production. In our data, 

this correlation ranges from -0.10 to 0.20 across extended income measures and equivalence 

scales. Using specialist wages increases the correlation, while including secondary child care 

reduces the correlation. Gottschalk and Mayer (2002) also found a weak relationship between 

money income and household production, with high-money-income households spending more 

time in household production. This would seem to imply that including household production 

should increase measured inequality. But the value of household production is a larger fraction 

of money income for low-income households than for high-income households, and this effect 

dwarfs differences in the amount of time spent in household production by income level. 
- 13 -

We investigated this further by adding the mean household production (normalized 

across households by the equivalence scale), rather than imputed household production, to 

household money income (row 3 of Table 2). Adding mean household production accounts for 

by far the greatest portion of the effect of household production on inequality measures. There 

are only two instances (both using the 90/50 ratio) where the predicted-household-production 

extended income (row 2) inequality measure is less than the corresponding mean-householdproduction measure (row 3) by a statistically significant amount. In most cases point estimates 

for the mean-household-production measure are less than the predicted-household-production 

estimates, in many cases by a statistically significant amount. 

Rows 4-6 of Table 2 show the effect of adding random disturbances to predicted 

household production as in (5) and (6). The results in row 6, which were generated assuming the 

maximum variance for the disturbance, are upper bounds on the effect of the perturbations. 

Hence, we will focus on the results in rows 4 and 5 (k = 0.25 and k = 0.50), which were 

generated under more realistic assumptions. Adding the disturbances increases the variance of 

household production but reduces the magnitude of the correlation with money income, so the 

direction of the effect is ambiguous if the correlation is negative. In almost all cases adding the 

disturbance increases the inequality of extended income, frequently to the extent that the 

inequality measures in rows 4-6 are statistically significantly greater than the constanthousehold-production measure in row 3. However, the differences for rows 4 and 5 are rather 

small and are not economically significant. 

The inclusion of secondary childcare and to a lesser extent the use of specialist rather 

than generalist wages both tend to reduce measured inequality. In both cases this is mostly due 

to greater mean levels of household production. For most comparisons, it makes little difference 
- 14 -

whether the square root or OECD equivalence scale is used. The most notable exception is the 

50/10 ratio when secondary childcare is included (both using generalist and specialist wages). 

The decline in inequality is larger when using the OECD scale than when using the square root 

scale. And the difference between the predicted-household-production and the mean-householdproduction measures is smaller for the OECD scale. 

Thus far we have shown that the lower measured inequality of extended income, 

compared to money income, is due to the addition of mean household production, rather than to 

the substitution of household production for market work. Given the small correlation between 

the value of household production and money income, it is natural to wonder by how much 

measured inequality would change if the correlation were large in magnitude. 

For convenience, we confine ourselves to the coefficient of variation (CV). Note that 

 2

2 2

2

2 2

2

2

2 2 (1 2 )

E E E

CV E Y YP P σY + β + σP = σ + σ + σ = σ =

where E is the mean of extended income; the subscripts on the variance and covariance terms 

denote money income (Y), household production (P), and extended income (E); and 2 β ≡ σ YP σ Y

is the coefficient on money income in a simple regression of household production on money 

income. 

The coefficient β is a convenient measure of the relationship between household 

production and market goods, as it represents the amount by which household production 

increases or decreases for a dollar increase in money income. It varies from -0.018 to 0.011 

across the various combinations of measures of household production and equivalence scales. 

The minimum occurs when using the OECD equivalence scale, the generalist wage, and 

including secondary care, while the maximum occurs when using the square root equivalence 

scale, the specialist wage, and excluding secondary care. 
- 15 -

Table 3 shows how the CV varies across values of β and 2 σ P using the specification that 

produces the minimum value of β (−0.018) as a reference point. The values of 2 σ P considered 

correspond to those in Table 2, with 2

P σ ) corresponding to the variance of predicted household 

production (row 2 in Table 2). 

Note that even fairly large negative values of β produce only moderate differences in the 

CV compared to including the mean value of household production in the income measure. 

(Also note that, as shown in the last column of Table 3, the possible values of β are limited by 

2 σ P , as large magnitudes of β violate the Cauchy-Schwarz inequality.) For example, using the 

variance of household production equal to that in row 5 of Table 2 and increasing the magnitude 

of β from −0.018 to −0.1 (i.e., increasing the magnitude of the correlation from −0.10 to −0.58) 

decreases the CV from .689 to .626. The implied elasticity of the CV with respect to β is very 

small--about 0.05. Thus changes in β over time, unless they are very large, should have 

negligible effects on measured inequality. 

V. Conclusion 

In this study, we used data from the ATUS to further examine how measures of 

inequality of earnings are affected by using extended income, which includes household 

production, instead of money income. The ATUS has the advantage over other time-use surveys 

of having data on income, but, like many time-use surveys, it covers one day for one household 

member. We extended previous authors’ imputation procedures to take advantage of the income 

data in ATUS using a flexible functional form. 

Consistent with our expectations and with most of the previous literature, we found that 

adding household production to money income reduces measured income inequality 
- 16 -

significantly. But little of this effect comes from variation in household production across 

households--virtually all of it is due to the addition of mean household production to earnings. 

Perturbing imputed values of household production to restore variation in household production 

across households causes inequality to increase only slightly. It makes virtually no difference 

whether generalist or the specialist wages are used or which equivalence scale is used, although 

the inclusion of child care as a secondary activity matters--but only to the extent that mean 

household production is greater when secondary childcare is included. We also found that 

measured inequality is not very sensitive to the relationship between money income and the 

value of household production. Thus, we conclude that measures of extended income inequality 

are robust to alternative assumptions one could make when estimating the value of household 

production. 

The practical implication of these results is that one can simply use the mean value of 

household production when estimating trends in extended income, and that there are virtually no 

gains to using more sophisticated imputation techniques. Any changes in the variance of 

household production across households or in the relationship between money income and the 

value of household production will have such a small effect on measured trends that they can 

safely be ignored. 
- 17 -

Appendix 

In order to analyze how the inclusion of household production in income would be 

expected to affect measures of inequality, we present the Gronau (1986) model of household 

production, extend it to two-person households, and discuss its implications. 

We begin with a single-person household. Using Gronau’s nomenclature, the utility 

function is given as: 

U = U(X,L,H,N), 

where X is the quantity of goods and services purchased in the market plus those produced at 

home, L is time spent in leisure, H is time spent in nonmarket production, and N is time spent 

working for pay. The individual maximizes utility subject to the following constraints: 

 X = XM + f(H) = W⋅N + V + f(H) 

 T = L + H + N, 

where XM represents goods and services purchased in the market, W is the individual’s market 

wage, V is unearned income, and f(H) is the home production function (fH > 0 and fHH < 0). 

There are several features of this model that are worth pointing out. First, as is evident 

from the first constraint, home-produced goods are perfect substitutes for market goods. This 

may seem unrealistic, because households clearly do not produce most of the goods that they 

consume. An alternative way to specify the model would be to allow goods and services to enter 

into the production function separately, and to assume that home production is a perfect 

substitute only for services. Under this specification, the qualitative results are the same, so we 

opted for the simpler specification. Second, the time spent in market and nonmarket work enters 

directly into the utility function. This allows individuals to obtain utility or disutility from these 

activities. We assume that, at the margin, that the marginal utility of time spent in these 
- 18 -

activities is negative, and that the disutility of work is concave (UH,UN,UHH,UNN < 0). Third, 

market goods do not enter into the production function. This is consistent with our notion that 

home production mostly is a substitute for services, but abstracts somewhat from reality in that 

much of this production would involve the use of household capital (vacuum cleaners, stoves, 

dishwashers, etc.). 

Solving the above maximization problem yields the following equilibrium conditions: 

(1) 

X

L N

U

U U W − =

(2) 

X

H N

H U

U U W f − = + , 

for individuals who are employed. For nonemployed individuals, the condition is: 

(3) 

X

L

U

U

X

H

H U

U = f +

Differentiating equations (1) and (2) and simplifying the expressions obtains the following 

comparative static results: 

(4) ( )( ) LN NN X L N XN

X

U U U U U U

U

dW

dN

− − − =

2

(5) ( )( ) X HH HH NH X H N XH

X

U f U U U U U U

U

dW

dH

+ − − − =

2

(6) ( )( ) LL NL X L N XL

X

U U U U U U

U

dW

dL

− − − =

2

To sign equations (4) and (5), it is sufficient to assume that the utility function takes the 

following form: U = U(X,L,H,N) = U(X,L) − C(H,N), where C(H,N)= C(αH) + C(βN) or 

C(H,N)= C(H+N). Under these assumptions, the derivatives in equations (4) and (5) will be 
- 19 -

positive (dN/dW,dH/dW > 0). The derivative in equation (6) is negative if, in addition, UXL > 0 

(or is not too negative). Equations (4) - (6) give the standard result that an increase in the wage 

will result in an increase in the amount of time spent in market work and decreases in time spent 

in nonmarket work and time spent in leisure. 

 It is straightforward to extend Gronau’s model to examine two-person households. 

Assuming that both household members share a common utility function, the maximization 

problem faced by the two-person household is: 

Max U = U(X,LH,LW,HH,HW,NH,NW) s.t. 

 X= WH⋅NH + WH⋅NH + V + f(HH,HW) 

 TH = LH + HH + NH

 TW = LW + HW + NW, 

where the subscripts indicate the husband (H) and wife (W). To derive comparative static results 

for changes in Li, Hi, and Ni with respect to own wage, Wi, the equilibrium conditions in (1) and 

(2) apply so that the expressions for the derivatives are the same as in the single-person 

household case. At first blush, it might seem that the solution is no different than putting two 

single-person households together. But the common consumption of X results in interactions 

between time use and spouse’s wage. Thus, although the derivatives are the same, the actual 

values they take on will differ from the single-person case. Combining equation (1) for i,j = 

H,W yields: 

(7) 

j j

i i

L N

L N

j

i

U U

U U

W

W

−

− = , 

and combining equation (2) for i,j = H,W yields: 

(8) ( )

( )

j j j

i i i

X H H N

X H H N

j

i

U f U U

U f U U

W

W

+ −

+ − = , 
- 20 -

for households in which both spouses work. Differentiating these conditions and arranging terms 

yields the following comparative static results: 

(9) =

j

i

dW

dH

{[( ) ( )][ ] [( ) ( )][ ]}

[ ]

2

XH H X H H H H N H j XH H X H H H H N H i

j

W U f U f U U R U f U f U U R

R

j i i i i i i i i i j j i j i j i + + − − + + −

(10) {[ ( )][ ] [ ( )][ ]}

[ ]

2

XN H H N N N j XN H H N N N i

j

j

i

W U f U U R U f U U R

R

dW

dN

j i i i i i i i i j i j i + − − + − =

where Ri = ) ( X Hi Hi Ni U f + U −U . If we maintain the assumptions above, and make the additional 

assumptions that the household utility function is separable in husband’s and wife’s disutility of 

work and that additional time spent in household production by one spouse does not reduce the 

marginal product of the oher spouse ( ≥ 0 Hi H j f ), these derivatives have the expected signs 

(dHi/dWj,dNi/dWj < 0). Under the same assumptions, the derivative of own leisure with respect 

to spouse’s wage has the expected positive sign: 

(11) 

L L N L j L L N L i

L N

j

i

U U W U U W

U U

dW

dL

i i i i j i j i

i i

− ⋅ − − ⋅

− = −

( ) ( )

( ) > 0 

To derive comparative static results for the case where one spouse does not work, we combine 

the condition in equation (2) (for the working spouse) with the following equilibrium condition: 

X

L H

U

U U f − ′ =

Differentiating and arranging terms yields the following: 
- 21 -

(12) =

i

j

dW

dH

0

( )( ) ( ) }

{[( ) ( ) ]( )

( )

2

2

<

− − − ⋅ + − ⋅

− ⋅ + − ⋅ −

−

j j j j i i i i i j j

i j i j j i j j j j j

j j

L H H H H N H H N H H

H H N H H H N H H L H

L H

U U U U f U U f

U U f U U f U U

U U

 (13) =

i

j

dW

dL

0

( ) ( ) ( )( )

( )

2

> − ⋅ ⋅ − − − − ⋅

−

i j i j j j j j j j j i i j

j j

H L N L H L H L L H L H N H

L H

U U f U U U U U U f

U U
- 22 -

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24

Table 1: Mean Annual Household Earnings and Household Production for Different Production Measures

Household Production 

Generalist Wage Specialist Wage 

 Secondary 

Childcare 

Excluded 

Secondary 

Childcare 

Included 

Secondary 

Childcare 

Excluded 

Secondary 

Childcare 

Included 

 Household 

Money 

Earnings 

All Households 70,553 21,542 31,517 23,385 32,969 Single-person Households 38,689 9,220 11,600 10,155 12,440 Married-couple Households 81,863 25,915 38,587 28,080 40,256 

Note: Household production estimated using the OECD equivalence scale. The square-root equivalence scale gives similar results. Data are from the 2003  ATUS. 
25

Table 2: Inequality Measures for Different Measures of Household Income

 Generalist Wage Specialist 

Secondary Childcare 

Excluded 

Secondary Childcare 

Included 

Secondary Childcare 

Excluded 

 OECD 

Equivalence 

Sq. Root 

Equivalence 

OECD 

Equivalence 

Sq. Root 

Equivalence 

OECD 

Equivalence 

Sq. Root 

Equivalence 

Coefficient of Variation (1) Family income 0.942** 0.917** 0.942** 0.917** 0.942** 0.917** 

(2) = (1) + Pred. HH production 0.741 0.720 0.679 0.663 0.730 0.710 

(3) = (1) + Mean HH prod. 0.738 0.714 0.684 0.657 0.723 0.700 

(4) = (2) + .25 S 0.745 0.725 0.684 0.669* 0.735 0.716 

(5) = (2) + .5 S 0.749 0.729** 0.689 0.674** 0.740 0.721* 

(6) = (2) + S 0.758** 0.740** 0.706** 0.696** 0.751* 0.735** Gini 

(1) Family income 0.416** 0.409** 0.416** 0.409** 0.416** 0.409** (2) = (1) + Pred. HH production 0.328 0.324** 0.299 0.301** 0.324** 0.320** (3) = (1) + Mean HH prod. 0.325 0.318 0.302 0.293 0.319 0.312 

(4) = (2) + .25 S 0.329* 0.325** 0.301 0.302** 0.326** 0.321** 

(5) = (2) + .5 S 0.332** 0.328** 0.305 0.306** 0.330** 0.326** (6) = (2) + S 0.340** 0.336** 0.317** 0.318** 0.339** 0.336** 90th percentile/50th percentile (1) Family income 2.416** 2.355** 2.416** 2.355** 2.416** 2.355** (2) = (1) + Pred. HH production 2.029 1.978 1.905* 1.879 2.002 1.956 

(3) = (1) + Mean HH prod. 2.046 1.998 1.955 1.904 2.021 1.975 (4) = (2) + .25 S 2.042 1.997 1.913* 1.881 2.014 1.980 (5) = (2) + .5 S 2.041 2.007 1.913* 1.883 2.018 1.990 (6) = (2) + S 2.042 2.014 1.928 1.911 2.029 2.007 50th percentile/10th percentile 

(1) Family income 3.231** 3.216** 3.231** 3.216** 3.231** 3.216** (2) = (1) + Pred. HH production 2.039 2.041 1.880 2.031** 2.027 2.024 (3) = (1) + Mean HH prod. 2.040 2.030 1.871 1.851 1.992 1.983 

(4) = (2) + .25 S 2.063 2.083 1.880 2.044** 2.065 2.077 

(5) = (2) + .5 S 2.110* 2.156** 1.943** 2.094** 2.132** 2.167** (6) = (2) + S 2.269** 2.319** 2.139** 2.255** 2.310** 2.346** 
26

Table 2: Inequality Measures for Different Measures of Household Income (continued)

 Generalist Wage Specialist 

Secondary Childcare 

Excluded 

Secondary Childcare 

Included 

Secondary Childcare 

Excluded 

 OECD 

Equivalence 

Sq. Root 

Equivalence 

OECD 

Equivalence 

Sq. Root 

Equivalence 

OECD 

Equivalence 

Sq. Root 

Equivalence 

90th percentile/10th percentile (1) Family income 7.807** 7.573** 7.807** 7.573** 7.807** 7.573** (2) = (1) + Pred. HH production 4.138 4.037 4.058 3.960** 4.058 3.960 (3) = (1) + Mean HH prod. 4.174 4.055 4.027 3.916 4.027 3.916 

(4) = (2) + .25 S 4.213 4.159 4.161 4.111** 4.161* 4.111 

(5) = (2) + .5 S 4.307* 4.327 4.302 4.311** 4.302** 4.311** (6) = (2) + S 4.631** 4.669 4.686** 4.706** 4.686** 4.706**  Data are from the 2003 ATUS. * Significantly different from Row (3) at 5 percent level. ** Significantly different from Row (3) at 1 percent level. 
27

Table 3: Coefficient of Variation For Differing Values of the Regression Coefficient of 

Household Production on Money Income and the Variance of Household Production

β

Actual 

(-0.018) -0.01 -0.02 -0.05 -0.1 -0.2

P P σ = σ ) CV 0.679 0.684 0.677 0.656 0.620 --

 Corr. -0.124 -0.069 -0.137 -0.344 -0.687 < -1

S P P σ = σ ) +.25 CV 0.684 0.685 0.678 0.657 0.621 --

 Corr. -0.119 -0.065 -0.131 -0.327 -0.655 < -1

S P P σ = σ ) +.5 CV 0.689 0.690 0.683 0.662 0.626 --

 Corr. -0.105 -0.058 -0.116 -0.290 -0.580 < -1

S P P σ = σ ) + CV 0.706 0.696 0.689 0.669 0.633 0.554

 Corr. -0.077 -0.042 -0.085 -0.212 -0.424 -0.849

Note: Household production measured using the OECD equivalence scale, the generalist wage, and including secondary childcare. Data are from the 2003 ATUS.