The Economies of Scale of Living Together and How They are Shared: Estimates Based on a Collective Household Model

How large are the economies of scale of living together? And how do partners share their resources? The first question is usually answered by equivalence scales. Traditional estimation and application of equivalence scales assumes equal sharing of income within the household. This paper uses data on financial satisfaction to simultaneously estimate the sharing rule and the economy of scale parameter in a collective household model. The estimates indicate substantial scale economies of living together, especially for couples who have lived together for some time. On average, wives receive almost 50% of household resources, but there is heterogeneity with respect to the wives' contribution to household income and the duration of the relationship.


Introduction
How large are the economies of scale associated with living together? And how do households allocate resources to their members? These questions are important for many economic topics such as designing welfare benefits, determining alimony and life insurance payments (Lewbel, 2003), measuring inequality and poverty, and measuring the willingness to pay for public goods (Munro, 2005). Traditionally, the analysis is based on equivalence scales which allow to compare well-being across households with different sizes. Within household distribution of well-being is not an issue, i.e. it is implicitly assumed to be equally distributed.
However, if there is unequal distribution of well-being within the household, traditional equivalence scales are biased and misleading in practice. In order to address this problem a richer model of household behaviour is necessary.
From a theoretical point of view, models of household behavior can be classified in two main groups. One strand is the general household utility framework, which goes back to Becker (1974Becker ( , 1981 and Samuelson (1956). This unitary framework is based on the assumptions that husband and wife have identical preferences 1 and household utility is maximized subject to a single budget constraint. Accordingly, it is irrelevant who earns the money in the household.
The second strand refers to the collective models of household behavior. The main assumption of this approach is that the individuals within a household have distinct preferences, which have to be considered separately. The standard collective approach assumes that the outcomes of decision making within the household are Pareto efficient (Chiappori, 1988(Chiappori, , 1992. 2 A standard result of welfare theory is that Pareto efficient decisions can be written as a constrained maximization of the weighted sum of individual utilities 1 Alternatively, there can also be an altruistic dictator who controls a larger share of the family income. 2 Alternative models include cooperative household models as proposed by McElroy and Horney (1981) or Manser and Brown (1979), who describe household behavior as result of a Nash-bargaining game, where the bargaining power depends on the emerging expenditure patterns on the options outside marriage. Non-cooperative household models describe household behavior as a non-cooperative game with no binding and enforceable contracts between the household members and resulting inefficiencies.
. The Pareto weight µ may depend on prices, total expenditures and on socalled distribution factors. These are defined as variables with no direct influence on preferences, technology or the budget constraint. From a bargaining perspective, the Pareto weight µ can be seen as a measure of the influence of household member f on the decision process. One difficulty with using µ as a measure of the weight given to member f is that the magnitude of µ will depend on arbitrary cardinalizations of the utility functions U.
Recently, Browning, Chiappori and Lewbel (2008, BCL hereafter) have shown that under specific assumptions there exists a unique Pareto weight corresponding to any sharing rule function η and vice versa. The sharing rule is defined as the fraction of household resources that are at the disposal (usually for consumption) of household member f. The household behaviour can be described as allocating the fraction η to member f and the fraction (1-η) to member m. The sharing rule η is invariant to cardinalizations of the utility function. This concept of a sharing rule is part of the standard collective household model. The BCL model is richer than the standard collective models due to the inclusion of a consumption technology function. BCL derive the conditions necessary to estimate the consumption technology function, the sharing rule, and individual preferences and estimate their model using functional form assumptions for these functions.
The BCL model is hard to estimate, and consequently several simplifications have been proposed, either in terms of theoretical restrictions (Lewbel and Pendakur, 2008) or in terms of estimation method (Cherchey et al., 2008). Lise and Seitz (2007) also follow the BCL approach, but focus on the demand for leisure and a composite consumption good. A different approach, also following the main ideas of BCL, has been suggested by Alessie, Crossley, and Hildenbrand (2006, ACH hereafter). They use subjective data on income satisfaction to estimate the parameters of the individual utility functions, the sharing rule and a consumption technology parameter. Subjective data are increasingly used in the happiness literature (see Frey and Stutzer, 2002, for an overview), but also in the collective household framework (e.g. Bonke, 2009, andLührmann andMaurer, 2007 share of almost 0.5 of total private consumption. If she contributes half of household income the consumption share is significantly larger than 0.5, and if she contributes only 25% of household income, her consumption share is significantly below 0.5.
The paper is structured as follows: Section 2 outlines the theoretical model. Section 3 describes the data, and the empirical model and the results are in section 4 and 5. Section 6 concludes.

A simple structural collective household model
We specify a collective household model which attempts to capture both returns to scale in household consumption and unequal allocation of resources within the household. As stated in the introduction collective household models are based on individual preferences that are aggregated into household utility according to some rule. Hence we first have to specify individual preferences. Following ACH we assume that individual indirect utility can be described by the PIGLOG specification where p denotes prices and x total consumption expenditure. Because we do not observe prices in the data we treat β as a constant parameter. Furthermore, α is specified as a function of observable individual characteristics. The empirical indirect utility function of person i is specified as where V i is utility, z i are observable characteristics, and ε i is the error term. Throughout this section we assume that V is observable. To simplify notation we drop the individual subscript i unless it is necessary for clarity.
This specification implies that preferences are egoistic, that is people only care about their own consumption. Single individuals are assumed to consume their income in each period, i.e.  (1 ) where f η is the sharing rule that determines which share of A h y -1 is allocated to the wife.
The sharing rule depends on so-called distribution factors d (see e.g. Browning, Chiappori and Lechene, 2003). We specify a simple linear sharing rule given by where γ d is a vector of unknown parameters. As distribution factors, we may use the ratio of female income to household income, the age difference between the spouses, and the duration of the relationship. While the first two distribution factors are commonly used in the Combining equations (2), (5), and (6) gives the indirect utility function for singles and both members of couples as follows. For singles, we get as in (2) For females in couples we get while for males we have The term in square brackets is individual consumption determined by household income, the returns to scale and the sharing rule. The model is estimated by nonlinear least squares using equation (7) for singles, equation (8) for women in couples, and equation (9) for men in couples. Identification is obtained by restricting α and β to be identical in the three equations.

Equivalence and indifference scales
Traditionally, an equivalence scale is defined as the ratio of the expenditures (or income) of two different household types with the same standard of living. Formally, this corresponds to the ratio of the cost functions of two household types evaluated at the same utility level. This requires comparability of the utility levels of different households. The impossibility of interhousehold utility comparison lies at the heart of the well-known identification problem of equivalence scales. 5 As BCL state, the notion of household utility is flawed because individuals have utility, not households.
If we cast our model in this framework then household indirect utility may be written as where C is a dummy equal to one for couples, and δ is the utility effect of being a couple. This expression reduces to individual utility for single households given in eq. (2). The corresponding log cost function is given by The resulting log equivalence scale if we take the couple as the reference household is ln ln / The small literature on estimating equivalence scales using satisfaction with income data generally provides this estimate (see e.g. van Praag and van der Sar, 1988, Charlier, 2002, Schwarze, 2003, Stewart, 2009. While this approach does not suffer from the usual identification problem that arises when equivalence scales are estimated from consumption data (provided utility is adequately measured), the problem of having specified a household utility function remains.
Let us reinterpret the results of estimating eq. (10) in terms of the individual utility model The indifference scales are can only be identified if we explicitly model individual preferences as described above.

Data
The data source is the Living in Switzerland Survey conducted by the Swiss Household Panel 6 See also www.swisspanel.ch. 7 There is a household and an individual questionnaire. The household questionnaire includes questions about housing, living standard, financial situation, the household structure and family organization, whereas the individual questionnaire covers topics such as household and family, health and life events, social origin, education, work, income, integration and social networks, politics and values, as well as leisure and internet use. 8 Since not all necessary variables are available for the first wave in 1999, this wave is excluded from this study. 8 We use the answer to the following survey question to measure individual satisfaction with income: Overall, how satisfied are you with your financial situation, if 0 means "not at all satisfied" and 10 "completely satisfied?
The satisfaction levels are categorized in eleven groups, ranging from zero (not at all satisfied) to ten (completely satisfied). Figure 1 displays the distribution of reported financial satisfaction levels by sex and marital status. By and large, the distributions are rather similar across these cases. Women more often report the highest satisfaction levels 9 and 10, whereas men have a larger fraction reporting the levels 7 and 8. This is reflected in the higher means of income satisfaction reported in Table 1.  Table 1 reports descriptive statistics of the person-and household-specific characteristics included in the regression analysis. We differentiate between 6 types of individuals: single women and men, husbands and wives in legally married couples, and partners in cohabiting couples that are not married. Net annual income is defined as labor as well as non-labor income net of taxes. Single women earn 17% less than single men but are on average more 9 satisfied with their financial situation. Married couples on average have a somewhat higher income than non-married cohabiting couples. Married couples are older, but on average less educated than non-married couples. There is a significant difference in the average satisfaction level between married and cohabiting couples, who are similar to singles with respect to financial satisfaction. Regarding the distribution factors we observe that married couples on average have a much larger duration of their relationship, but also smaller female contributions to total household income. This is partly due to a higher share of non-working wives, but also to the higher share of part-time working wives. The data also contain a variable indicating who is mainly responsible for household finances.
This question is only answered by one person per household, hence it is not available for both partners in a couple. This information is not used in estimation, but we analyze ex post whether the estimated sharing rules are correlated with the way the household finances are managed. Table 2 presents a descriptive analysis of the financial responsibility variable.  identical results compared to ordered probit.
As far as we know the performance of POLS in a nonlinear setting has not been analyzed yet.
We conducted some Monte Carlo simulations 9 and found that the estimates of the utility parameters α and β of course depend on the scaling of the dependent variable. However, in all simulations we obtained unbiased estimates of γ and A. Replicating the reduced form approach of ACH both in the Monte Carlo analysis and with our data also yields almost identical estimates of the structural parameters regardless of estimation strategy (Bütikofer et al., 2009). This makes intuitive sense because monotone transformations of the dependent variable will change the intercept and slope of the estimated utility function, but not the transformation of household income into individual consumption. The question whether this is true for only modest monotone transformation is left to future research.

Results
In this section we present and discuss the empirical results. We proceed in two steps. First, we present a descriptive nonparametric analysis in the spirit of the literature of semiparametric estimation of equivalence scales. This provides a non-formal test of our identification assumptions. Second, we estimate the structural parameters directly by nonlinear least squares. Based on this we discuss the properties of the estimated sharing rules and equivalence scales.

Nonparametric analysis
In this section we provide a descriptive nonparametric analysis. We estimate the relationship between reported income satisfaction and log of household income by local linear regression, separately for single females, single males, females in couple and males in couples. Figure 2 shows the estimated regression lines.
By and large, in all four cases the regression lines are almost linear with similar slopes (except at the boundaries of the support of the independent variable). This is important for the following analysis because identification of A and γ hinges on a constant slope.  The results of the estimated shift parameters and the implied equivalence scales are displayed in Table 3. For both genders we report three sets of results: the OLS based and the semiparametric estimates 11 without further control variables, and OLS based results with further control variables. There is a difference between the female and male equivalence scales, which is significant at the 10% level if no further control variables are used. This is true both for the OLS based and the semiparametric estimates. With further controls, however, 14 the difference becomes insignificant. The simple test therefore is inconclusive with respect to the equal sharing hypothesis. However, as the results in the next section suggest this test result may be caused by the fact that on average the sharing rule is in fact almost 0.5. In order to gain more information we now turn to the estimation of the structural model described in section 2.

Estimation of structural model
In this section we discuss the estimation of the structural model. The dependent variable is the transformed income satisfaction v . The estimated model consists of eq. (7) for singles, eq. (8) for females in couples, and eq. (9) for males in couples. The model is estimated by nonlinear least squares with standard errors adjusted for clustering due to the panel structure of the data. Table 4 displays the estimation results. As distribution factors we use the female contribution to household income and the duration of the relationship. We also experimented with total household income and the age difference between partners as distribution factors, but both turned out to be completely insignificant in the sharing rule. The female contribution to household income enters the sharing rule linearly. We tested this against a spline function of the contribution and could not reject the linear effect. The distribution factors are both normalized to mean zero. Hence, 0 γ is an estimate of the share of total household consumption a wife with mean of female contribution to income and mean duration of the relationship receives. The first set of results refers to all couples. In the second and third column, couples are differentiated according to whether they are legally married or cohabiting without being married. 12 In column 1 of Table 4    Column 2 of Table 4 displays the results for legally married couples. Compared to column 1 there is a smaller estimate of A which corresponds to a larger return to scale factor of 1.61. By contrast, for cohabiting couples the estimated A is much larger implying a return to scale factor of only 1.16 (column 3). This finding may be explained by the fact that married couples, on average, have lived together much longer which allowed them to improve their consumption technology. This is also reflected by the estimate for the distribution factor "duration of relationship", which is important for married couples, but completely irrelevant for cohabiting couples. The effect of the female partner's contribution to household income on the sharing rule, on the other hand, is stronger for cohabiting couples.

The estimated sharing rule
In Table 5 we present calculations of how the sharing rule changes with the distribution factors as well as the distribution of the estimated sharing in our sample. We compute the estimated shares at income contributions of 0, 25%, 50% and 60%. 13 If the female contribution to household income is 50% her consumption share is 0.54 and significantly larger than 0.5 at the 10% level. It drops to 0.44 if the woman only contributes 25% of household income and increases to 58% if the contributes 60% of household income. If she does not contribute to household income at all her estimated consumption share is 0.35. These estimates are significantly different from 0.5 at the 5% level.
The differences by civil status are also reflected in the estimated sharing rules displayed in Table 5. The consumption shares of the females are larger in married couples at all levels of contribution to household income. A married woman contributing 50% to household income has a consumption share of 0.56, and this estimate is significantly larger than 0.5. If a married woman contributes 25% to household income her consumption share of 0.47 is not significantly smaller than 0.5. Without contributing to household income a married woman's consumption share is 0.39.
By contrast, a woman in a cohabiting couple has a consumption share of almost exactly 0.5 if she contributes 50%. to household income. It does not increase significantly if her income contribution increases to 60%. On the other hand, with small income contributions the female consumption share is rather small, dropping to 0.26 if she does not contribute at all. However, this estimate has a rather large standard error.
selections of singles (excluding divorced and widowed individuals, excluding those who are in a relationship but do not live together), but again results are robust with respect to the definition of a single person. 13 We did not use a contribution of 75% because only 1% of the sample have a contribution of at least 75%.  Figure 5 shows Kernel density estimates of the predicted sharing rules. The density of the female share in married couples lies to the right of the corresponding density for cohabiting couples. Further analysis reveals that almost 75% of all females in cohabiting couples have an income share less than 0.5. Overall, the distribution of estimated income shares mimics the female contribution to household income much closer in cohabiting couples than in married couples. This may be explained by the result that married couples generate much larger returns to scale which allows them to redistribute more consumption from males to females. In Table 6 we analyze how the estimated shares correlate with the reported responsibility for household income. We regress the predicted female shares on the indicators for financial responsibilities, controlling for household income and the age difference of the partners. None 18 of these variables have been used to estimate the sharing rule. Recall that only one person per household responded to the financial responsibility question.
For married couples we find that if there is only one partner responsible for household finances she or he is able to shift about one percent of household consumption to her or his own use compared to joint financial responsibility. This result makes intuitive sense and gives some credibility to the estimated sharing rules. In cohabiting couples, separate management of individual incomes reduces the female consumption share by 0.03, which probably is caused by the fact that couples with separate financial responsibilities also have the smallest female income contribution (see Table 2).    Table 4 Standard errors in parentheses a indifference scales are evaluated at the mean of the years of relationship The indifference scale is proportion of household income an individual needs if she or he moves from living in a couple to living alone * denotes significant difference at 10% level, ** denotes significant difference at 5% level

Comparison to previous research
In this section we briefly discuss the results from other papers based on the BCL approach.
BCL use Canadian expenditure data from 1974 -1992. They specify a richer consumption technology that differs across consumption goods, i.e. they estimate good-specific Barten scales. As distribution factors, BCL use the wife's contribution to household income, the age 20 difference between the spouses, a home-ownership dummy, and total household expenditure.
They compute an overall measure of the economies of scale in consumption that varies between 1.27 and 1.41 (see Table 4 in BCL). 14 Our estimate is 1.47 (= 1/A). BCL's benchmark estimate of the sharing rule is 0.65, which appears to be quite large. BCL argue that this may be explained by the fact that household demand functions look more like women's demand functions than men's demand functions (p. 31). The estimated indifference scales vary between 0.58 and 0.74 for women, and 0.50 and 0.70 for men, depending on the restrictions imposed on the model. ACH is closest to our paper in term of methodology. They estimate a reduced form version of the model described in section 2 for 10 European countries. 15 Compared to our results the estimates of A in ACH are rather small, often below 0.6, in some cases even below 0.5. On the other hand, their estimate of the sharing rule evaluated at the mean of the distribution factors is above 0.5 in almost all cases. The estimates which are most similar to ours refer to the UK with estimates of A = 0.69 and a mean sharing rule of 0.49 (Table 6 in ACH).

Conclusions
Based on a collective household model, this paper provides estimates of the returns to scale of living together and of the rule of sharing consumption among spouses. Household income is transformed into individual consumption by a consumption technology (the returns of scale) and the rule that determines how much each member receives. An individual living alone has the same utility from his income as an individual living in a couple who receives individual income according to the above transformation. Assuming that preferences do not change by living together, it is possible to identify the returns to scale and the sharing function from data on singles and couples. This identification result is one of the major contributions of Browning, Chiappori, and Lewbel (2008). In this setup, it is possible to identify so-called indifference scales which allow to make welfare comparisons between the same person in different living conditions. We use data on financial satisfaction as a measure of indirect utility received from individual consumption. The estimated consumption technology parameter in our preferred specification implies that scale economies increase the sum of individual consumption of both members to 1.47 times household income. The estimated sharing rule is somewhat smaller than 0.5 at the mean, but above 0.5 if the women contributes exactly 50% to household income. It varies significantly both with the wife's contribution to household income and with the duration of the relationship. At the mean of the estimated sharing rule the female indifference scale is 0.71, while the male indifference scale is 0.76. These numbers measure which proportion of the couple's total income each member would need to be as well off when living alone.
There is heterogeneity of the results with respect to the civil status of couples. Married couples have a better consumption technology and a more equalizing sharing rule than couples that are not (yet) married. This leads to indifference scales that are much larger for married couples than for cohabiting couples. Both partners in married couples require about 80% of total household income in order to be as well off when living alone. This percentage is much smaller for cohabiting couples, where the female partners only need about 52% and male partners about 63% of total household income. These numbers are evaluated at the means of the estimated sharing rule.
The analysis can be extended in several ways. More work needs to be done with respect to estimation of models with satisfaction data, especially nonlinear models with panel data.
Another extension would be to consider more flexible specifications for individual utility. If 22 the data are available a very promising extension would be a combination of subjective satisfaction data with expenditure data. 23