Income pooling and the distribution of individual
consumption among couples in Switzerland
Aline Bütikofer, Michael Gerfin and Gabrielle Wanzenried∗
This version: October 2009
Abstract
Textbook models of household behavior assume that a household is a single decision unit that
allocates time and money by maximizing a common utility function with respect to a common budget
constraint. But households consist of individuals with distinct preferences. Household decisions are
the result of a bargaining process with usually unequal bargaining power. If this collective view of
household behavior is true the traditional tools used in the measurement of inequality, poverty and
welfare may produce misleading results. In this paper we test the so-called income pooling
assumption which postulates that redistributing income within the household does not change
individual welfare. We also provide estimates of a more general collective household model. We use
subjective income evaluation as measure for individual utility. Income pooling is rejected by our
results. We find evidence for an unequal distribution of the bargaining power within the household,
which depends on how much each spouse contributes to household income. Consequently, there is
also an unequal distribution of consumption within the household.
JEL Classification : D12, C14, C23
∗
Corresponding author: Michael Gerfin, Department of Economics, University of Bern, Schanzeneckstasse 1, CH-3001
Bern, Switzerland, and IZA, Bonn, Germany; email: [email protected]. Aline Bütikofer, Department of
Economics, University of Bern, Schanzeneckstasse 1, CH-3001 Bern, Switzerland; [email protected].
Gabrielle Wanzenried, Lucerne University of Applied Sciences and Arts, Institute of Financial Services Zug IFZ,
Grafenauweg 10, Postfach 4332, 6304 Zug, [email protected]. This study has been realized using the data
collected in the ''Living in Switzerland'' project, conducted by the Swiss Household Panel (SHP), which is based at the
Swiss Foundation for Research in the Social Sciences FORS, University of Lausanne. The project is financed by the
Swiss National Science Foundation.
1
I. INTRODUCTION
Textbook models of household behavior assume that a household is a single decision unit
that allocates time and money by maximizing a common utility function with respect to a
common budget constraint. These models are called unitary models of household behavior
(Becker, 1981, is a leading example). In these models, it is assumed that there is an equal
distribution of income and welfare within the household. Redistribution of income from one
member to another leaving total household income unchanged will not affect individual
welfare and will not change household behavior. This property of unitary models is called
income pooling. The problem with this approach is that individuals and not households have
preferences. If the partners in couples have different preferences, household behavior must
be the outcome of some decision making process, e.g. collective bargaining. In this case,
individual preferences and individual bargaining power determine household outcomes such
as consumption decisions. Within household inequality of welfare is likely in this setup, and
redistribution of income within the household will change household behavior. Models that
take account of these issues are called collective household models.
A prime example where this distinction may be important is the analysis of income
distribution and poverty. Traditionally, the incomes of households with different sizes are
made comparable with equivalence scales. These scales reflect economies of scale of living
together and adjust household incomes such that two households with the same equivalent
income are treated as having the same utility. Each member of the household is assumed to
receive this equivalent income. Only inter-household inequality matters in this view. If the
unitary model is wrong, the commonly used measures of inequality and poverty are biased.
These concerns are also relevant for the analysis and design of welfare benefits. In the
collective view it may matter who receives the transfers. A well-known example is Lundberg
et al. (1997) who show that in the UK a reform of shifting child allowances from fathers to
mothers lead to an increased share of expenditure on children and female clothing. These
results indicate that specific targeting of benefits may be important for the intended welfare
effects.
The standard collective household model assumes that the outcomes of decision making
within the household are Pareto efficient (Chiappori, 1988, 1992).1 Household decisions are
1
Alternative models include cooperative household models as proposed by McElroy and Horney (1981) or Manser and
Brown (1980), who describe household behavior as a 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
2
the result of a constrained maximization of the weighted sum of individual utilities, where
the Pareto weights can be seen as a measure of the influence of household members on the
decision process. To any Pareto weight there exists a unique sharing rule function η. The
household behavior can then be described as allocating the fraction η to member f and the
fraction (1 - η) to member m. The concept of a sharing rule is part of the standard collective
household model. Browning, Chiappori and Lewbel (2008, BCL hereafter) extend the
standard collective model by including a consumption technology function which generates
the returns to scale of living together. 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 difficult to estimate, and consequently several simplifications have been
proposed (Lewbel and Pendakur, 2008, Cherchye et al., 2008). 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 individual income satisfaction with
household finances to estimate the parameters of the individual utility functions, the sharing
rule and the consumption technology parameter. Already in the early 1970s van Praag and
co-authors estimated individual utility functions based on subjective income evaluation data
(e.g. van Praag and Kapteyn, 1973, for an early contribution, and van Praag and Frijters,
1999, for a survey). Nowadays, 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 and Browning, 2009, and Lührmann and Maurer, 2007). As is
common with virtually all empirical analyses of collective household models, we focus on
households without children. The incorporation of children in these models is the major
challenge for future research.
This paper provides the first empirical test of income pooling using Swiss data. In addition,
we present first estimates of a collective household model for Switzerland. Like most
comparable studies, we reject the income pooling assumption. We find substantial scale
economies of living together. Total private consumption exceeds household income by
almost 40%. The consumption share of a wife increases significantly with her contribution to
household income. On average, a wife has a share of almost 0.5 of total private consumption.
If she contributes half of household income the consumption share is significantly larger
household behavior as a non-cooperative game with no binding and enforceable contracts between the household members
and resulting inefficiencies.
3
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.
II. THEORETICAL BACKGROUND
1.
Income pooling
In this section we provide a definition of income pooling. Consider a household with two
members i, where i ∈ { f , m} . Total household income yh is the sum of the individual labor
income y f and y m and non-labor income of the household y0, and it is given by:
yh = y0 + y f + ym
(1)
The indirect utility of each household member is a function of household income. For
simplicity, all other arguments that typically enter the indirect utility function as well as
details about the bargaining and allocation process within households are suppressed for the
moment. Accordingly, the indirect utility function of individual i is as follows:
Vi = Vi ( y f , y m , y 0 ) .
(2)
In this setup, the income pooling hypothesis can be formulated as follows:
Income pooling holds for all i, y j , y 0 if the following condition holds for any ∆y :
Vi ( y 0 , y f − ∆y , y m + ∆y ) = Vi ( y 0 , y f , y m ) = Vi ( y h ) .
(3)
In words, indirect utility of household member i only depends on the total level of household
income, and the individual income contributions of each household member do not matter.
In the next section, we present a simple test of the income pooling hypothesis.
It should be noted that the term “income pooling” is unfortunate. Bargaining models of the
household may also assume that partners first pool their income and then divide the pool
according to their bargaining power. As Apps and Rees (2007) argue, “anonymity” would be
a much better term for what is meant by equation (3), namely that the identity of the income
contributor does not matter for individual welfare and household decisions.
4
2.
A simple collective household model
In order to make the approach useful for welfare analysis, we need a structural model. For
this next step we follow Alessie et al. (2006) and specify a collective household model which
captures both returns to scale in household consumption and unequal allocation within the
household.
We assume that individual indirect utility of person i can be written as
Vi = α ( zi ) + β ln xi + ε i ,
(4)
where Vi is utility, α(zi) is a linear function of observable characteristics zi, xi is individual
consumption, and εi is the error term. Throughout this section we assume that Vi 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. x = y h .2
The level of individual consumption in couple households, however, may depend on a
sharing rule and returns to scale. Returns to scale exist if total private consumption of both
household members f and m exceeds household income. This effect can be captured by
writing
( x m + x f ) = y h / A or A( x m + x f ) = y h ,
(5)
where xj denotes consumption of person j = f,m. The scalar A can be seen as representing a
household consumption technology that transforms household income yh into total household
consumption ( x m + x f ) . If A = 1 there are no returns to scale (all consumption is private).
The logical lower bound for A is 0.5 (all consumption is public). The specification adopted
in this paper is a simple version of the more complex consumption technology used by BCL,
who estimate a collective household model using expenditure data. By contrast to their
model we assume that A is a constant. The parameter A captures the fact that some goods are
2
Throughout this section we denote with y h the total household income.
5
at least partly public, e.g. housing, household operation, transportation or newspapers. In
other words, A-1 is an overall returns to scale factor. 3
Given (5) we can write individual consumption of wife and husband as
x f = η f y h A−1 and x m = (1 − η f ) y h A−1 ,
(6)
where η f is the sharing rule that determines which share of y h A-1 is allocated to the wife.
The sharing rule depends on so-called distribution factors d. A convenient specification for
η f is given by
ηf =
exp[γ 0 + γ d d ]
.
1 + exp[γ 0 + γ d d ]
(7)
As distribution factors, we use the female contribution to household income and the duration
of the relationship. While the first distribution factor is commonly used in the collective
household literature, the duration of the relationship has not been used yet as far as we
know.4
Combining equations (4), (5), (6) and (7) we can derive the indirect utility functions for
singles and both members of couples as follows (suppressing the error term).
Singles:
V = α ( z ) + β ln y h .
(8)
Females in couples: V = α ( z ) + β {(γ 0 + γ d d ) − ln[1 + exp(γ 0 + γ d d )] + ln( y h / A)} (9)
Males in couples:
V = α ( z ) + β {− ln[1 + exp(γ 0 + γ d d )] + ln( y h / A)}
(10)
The term in curly brackets is individual consumption determined by household income, the
returns to scale and the sharing rule. Combining equations (8), (9) , and (10) we can express
indirect utility for each individual. Identification is obtained by restricting α and β to be
identical in the three equations, i.e. by assuming that preferences do not change when the
3
BCL specify z j = A ( q jf + q mj ) + a , where zj is an observable vector of household j’s consumption of n goods, q jf and
q mj are unobservable vectors of private consumption of both spouses. A is an n x n nonsingular matrix and a is a n-
vector. This linear consumption technology nest familiar cases, e.g. the well-known Barten scales if A is diagonal and a
= 0. By contrast to BCL we can only identify an aggregate consumption technology, i.e. A is diagonal with identical
elements A. Assuming the budget constraint holds with equality we have y j = p ' z j = ( p ' Aq jf + p ' Aq mj ) . Given the
restriction on A this simplifies to A( x jf + x mj ) = y j .
4
Recently, Bonke and Uldall-Poulsen (2006) have shown that the probability of income pooling increases with 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
6
living situation changes. This identification assumption has been made by all papers based
on the BCL approach.
Obviously, indirect utility for individuals in couples is nonlinear in the parameters of the
sharing rule and the consumption technology. In order to estimate these parameters using
standard ordered response models, simplifying assumptions are necessary. These
assumptions are described in the technical appendix. Using this simplification, we can
estimate a reduced form ordered response model and derive the structural parameters from
the reduced form parameters. This is described in the next section.
If we define C = 1 if the individual is living in a couple household, and F = 1 if the
individual is female we can write a reduced-form individual-level model of indirect utility as
V = θ 0 ( z ) + θ 1 ln y h + θ 2C + θ 3 F ⋅ C + θ 4 F ⋅ C ⋅ d1 + θ 5 F ⋅ C ⋅ d 2 + θ 6 (1 − F ) ⋅ C ⋅ d1
+ θ 7 (1 − F ) ⋅ C ⋅ d 2 + ε
(11)
where d1 and d2 are the two distribution factors. This model can easily be estimated by an
ordered discrete response model if we measure indirect utility as the individual satisfaction
level with the household’s financial situation. If the female’s contribution to household
income is one of the distribution factors, say d1, this model provides a convenient test of the
income pooling hypothesis described above. Under income pooling it must hold that
θ 4 = θ 6 = 0 , i.e. conditional on household income the individual contributions must not
influence individual utility.
The structural parameters α, β, γ and A are functions of the reduced-form parameters θ. The
exact relationships are given in the appendix; here we only show the parameters central to
our analysis:
θ3 1 θ4
θ5
0
2
;
γ
=
1
+
exp(
γ
)
;
γ
=
1 + exp(γ 0 ) )
)
1
1 (
1 (
θ
θ
θ
2
⎛θ
⎞
ln A = − ⎜ 1 + ln (1 + exp(γ 0 ) ) ⎟
⎝θ
⎠
γ0 =
(12)
Note that both A and γ are identified even if the reduced form parameters are only estimated
up to scale. In order to estimate the structural parameters we use a minimum distance step
which is also described in the appendix.
Typically, (indirect) utility is a theoretical and ordinal concept and cannot be observed. This
view has been challenged by Van Praag and several co-authors since the early 1970s. They
7
claim that subjective income evaluation questions can be used to estimate utility functions,
i.e. the answers to these questions are valid proxies of utility. Over the years, this so-called
“Leyden-school” produced a series of credible evidence for the usefulness of the approach
for empirical welfare analysis. The happiness literature is also grounded on the assumption
that answers to subjective satisfaction questions provide valid proxies for happiness (or
utility). Easterlin (1974) initiated this research, a recent overview is given in Frey and
Stutzer (2002). Bonke and Browning (2009) suggest using information on self-reported
satisfaction with the financial situation of the household to approximate the indirect utility
level of the individual household members. Subjective data are also increasingly used in the
empirical analysis of collective household models (Lührmann and Maurer, 2006, Bonke and
Browning, 2009).
III. DATA
The data source is the Living in Switzerland Survey conducted by the Swiss Household
Panel (SHP).5 This survey is based on a representative sample of the population of
permanent Swiss residents. It has been conducted annually since 1999 and covers objective
questions on resources, social position, and participation as well as subjective questions on
satisfaction, values and evaluations.6
Our analysis is based on data from the years 2000 to 2006.7 The main selection rules we
impose are that all single persons are employed, that all men in couples are employed, and
that all persons are aged between 20 and 60. We only consider households without children.
After eliminating missing observations, the sample used for our analysis consists of 3’361
single households and 1357 couple households (of which 863 are married). The data set is a
panel but panel attrition is severe. More than 50% of the individuals have only one
observation, and 75% have at most two observations. For this reason, we prefer pooled
estimators to a random effects estimator.
We use the answer to the following survey question to measure individual satisfaction with
income:
5
6
7
See also www.swisspanel.ch.
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.
Since not all necessary variables are available for the first wave in 1999, this wave is excluded from this study.
8
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.
Figure 1: Frequencies of reported levels of satisfaction with financial situation
Satisfaction by gender and marital status
Single men
0
0
.1
.1
Fre qu ency
.2
.3
Frequency
.2
.3
Single women
0
2
4
6
8
10
Satisfaction level with financial si tuation
0
Married men
0
0
.1
Fre qu ency
.2
.3
Frequency
.1
.2
.3
.4
Married women
2
4
6
8
10
Satisfaction level with financial situation
0
2
4
6
8
10
Satisfaction level with financial si tuation
0
2
4
6
8
10
Satisfaction level with financial situation
Data source: Swiss Household Panel, 2000-2006.
Table 1 reports descriptive statistics of the person- and household-specific characteristics
included in the regression analysis. Net annual income is defined as labor as well as nonlabor income net of taxes. Single women earn 17% less than single men but are on average
more satisfied with their financial situation. Furthermore, there is a significant difference in
the average satisfaction level between couples and singles. The gender difference is not
affected by household type.
9
Table 1: Descriptive statistics of person- and household specific characteristics
Singles
Couples
Men
Women
Net annual household income (in SF)
Income contribution of woman (in%)
Duration of relationship
Age
Low education
High education
French speaking
Swiss
Financial satisfaction
71’297 (37’484)
59’357 (26’563)
Men
39.54 (9.13)
0.08
0.34
0.23
0.88
6.86
40.63 (10.48)
0.12
0.22
0.27
0.88
7.05
125’768 (84’211)
0.35 (.19)
14.27 (11.04)
39.03 (11.02)
41.25 (10.65)
0.04
.14
0.36
.22
0.25
.26
0.87
0.91
7.31
7.47
Women
Number of households
1592
1769
1357
Notes: Data source: Swiss Household Panel, 2000-2006.
Own calculations
Sample means (standard deviations in parentheses)
On average, the female contribution to household income is 0.35 and the duration of the
relationship is around 14 years. In roughly 13% of the households, the female contribution is
zero. The distribution of positive female contributions has a mode around 0.5, but it is
asymmetric with much more mass below 0.5.
IV. EMPIRICAL MODEL
Ordered response models are traditionally analyzed in a latent variable framework. Denote
this latent variable as y * and write
yi* = xi β + ε i ,
(13)
for i = 1, 2, …, N, where x is a vector of explanatory variables, β is a vector of unknown
parameters, and ε is a random error with distribution function F. We do not observe y * but
the discrete variable y, taking one of the values {1,2,…, J}. y * and y are assumed to be
related as follows
⎧1
⎪
⎪2
yi = ⎨
⎪#
⎪J
⎩
if yi* < α1
if α1 ≤ yi* < α 2
if α J −1 ≤ yi*
10
with αj being additional parameters such that α1 < α 2 < ... < α J −1 . The dependent variable y is
ordinal, and the probability of a particular outcome yj is given by
P[ yi = j ] = P[α j −1 ≤ yi* < α j ]
= P[α j −1 ≤ xi β + ε < α j ]
(14)
= F (α j − xi β ) − F (α j −1 − xi β )
with α 0 = −∞ and α J = ∞ .
Assuming the standard normal distribution for ε gives the standard ordered probit model.
The model is only identified up to location and scale, so it is assumed that the variance of ε
is unity and that x does not contain a constant term. Estimation with maximum likelihood is
straightforward.
A semiparametric alternative to the ordered probit model is based on the semi-nonparametric
series estimator proposed by Gallant and Nychka (1987). It approximates the unknown
density of ε by a Hermite form and can be written as the product of a squared polynomial
and a normal density. The class of densities that can be approximated by this form is very
general. Estimation is done with maximum likelihood. Details are provided in the Appendix.
We do not present results using the random effects ordered probit estimator. It imposes two
strong statistical restrictions: a) the unobserved time-varying determinants of utility cannot
be serially correlated, and b) strict exogeneity of all explanatory variables. The pooled
estimators provide consistent estimates of the reduced form parameters up to scale under
fairly mild conditions (see Wooldridge, 2002). Furthermore, there is a high degree of panel
attrition in the data. For these reasons, we believe that the pooled ordered probit estimator
and its semi-nonparametric version are superior to the random effects estimator in the
present case.
V. RESULTS
In this section we discuss the estimation results. We estimate equation (11) using pooled
ordered probit and the semi-nonparametric ordered probit estimator (column 2,). Table 4
displays the estimation results. Standard errors are corrected to take account of multiple
observations per person.
11
Table 4: Reduced form estimates
log household income (θ 1)
couple (θ 2)
couple x female (θ 3)
couple x female x female contribution(θ 4)
couple x female x years of relation (θ 5)
couple x male x female share (θ 6)
couple x male x years of relation (θ 7)
female
age
age squared/100
low education
high education
french speaking
foreign
Log Likelihood
Observations
Ordered Probit
0.688***
(0.047)
-0.191***
(0.056)
-0.063
(0.072)
(0.215)
0.424*
***
(0.041)
-0.197
(0.193)
-0.730***
-0.065
(0.039)
(0.051)
0.219***
(0.015)
-0.033*
(0.019)
0.047*
-0.089
(0.065)
(0.043)
0.087*
(0.040)
-0.249***
-0.207***
(0.054)
-11431.56
6075
SNP Ordered Probit
0.620***
(0.025)
-0.182***
(0.033)
-0.062
(0.043)
0.366**
(0.135)
***
-0.170
(0.026)
-0.659***
(0.127)
(0.024)
-0.055*
0.179***
(0.029)
-0.021*
(0.010)
0.028*
(0.012)
-0.075
(0.038)
0.060*
(0.025)
-0.195***
(0.026)
-0.163***
(0.035)
-11407.39
6075
Standard errors in parentheses
Estimation also used time dummies. Standard errors are adjusted for clustering by persons.
* p<0.10, ** p<0.05, *** p<0.01
SNP Ordered Probit: Semi-nonparametric ordered probit, order of Hermite polynomial = 4
Likelihood ratio test of ordered probit against SNP ordered probit: 50.39 (p-value = 0.00)
As expected, there are differences in the estimated parameters across models, mostly due to
the different scaling of the error terms. The ordered probit model is rejected against the seminonparametric model by a likelihood ratio test. It is difficult to interpret the reduced form
parameters directly, but one point is noteworthy. The results show a clear rejection of the
income pooling hypothesis. Conditional on household income there is a variation in
individual well-being with respect to the female contribution to household income.
Husbands’ well-being is reduced if their wives contribute more to household income (θ 6 <
0), and wives’ well-being is increasing (θ 4 > 0). This suggests that redistributing income
within the household will affect individual well-being, which is ruled out by the income
pooling assumption. Previous rejections of income pooling are provided by e.g. Attanasio
and Lechene (2002), Browning et al. (1994), Donni (2007), Fortin and Lacroix (1997),
Lundberg et al. (1997), Phipps and Burton (1998), Thomas (1990) and Ward-Batts (2008).
Table 5 displays the estimates of the household consumption technology parameter A and the
implied sharing rules. They are obtained from the reduced-form parameters by a minimum
distance step described in the Appendix. Note that there are hardly any differences across the
two models. This is due to the fact that the structural parameters are all functions of ratios of
the reduced form parameters, so all proportional scale factors cancel out. We focus on the
12
estimates of the semi-nonparametric model because the ordered probit model is rejected
against it.
The estimate of A is about 0.72, which implies that technologies of scale increase the sum of
individual consumption to 1.39 (= A-1) times household income. This is a plausible estimate
which conforms to previous estimates of the consumption returns to scale for a couple. The
γ-parameters of the sharing rule are displayed in the next three lines. The estimate of γ0 is not
significant which implies that at the mean of the distribution factors the sharing rule is not
significantly different from 0.5.8 The sharing rule varies significantly with the distribution
factors “female contribution to household income” and “duration of relationship”. However,
given the logistic specification of the sharing rule it is difficult to interpret the estimates of γ1
and γ2.
Table 5: Structural parameters of household behavior
Consumption technology A
Scale factor = A-1
Parameters of sharing rule
γ0
γ (female income contribution)
γ 2 (duration of relationship)
Sharing rule η
1
mean of income contribution
income contribution =0
income contribution =0.25
income contribution =0.50
income contribution =0.60
Ordered Probit
.724
(.040)
1.38
SNP Ordered Probit
.722
(.027)
1.39
-.063
1.646**
.019**
(.102)
(.423)
(.008)
-.068
1.597**
.019**
(.068)
(.299)
(.005)
.484
.347
.445*
.547*
.588**
(.025)
(.042)
(.028)
(.029)
(.034)
.483
.350
.445**
.544**
.584**
(.016)
(.029)
(.019)
(.020)
(.024)
Standard errors in parentheses
Structural parameters estimated by minimum distance from reduced form parameters. See Alessie et
al (2006) for details
All sharing rules are evaluated at the mean of the years of relationship
Mean of income contribution = 0.34; mean of years of relationship = 14.3
*
denotes significant difference from 0.5 at 10% level, ** denotes significant difference at 5% level
For this reason, the following rows display estimates of the share of household consumption
allocated to the wife, all evaluated at the mean duration of the relationship. At the mean of
the female contribution to household income, the female share is about 0.48, but not
significantly different from 0.5. This share increases to almost 0.55 when the wife
contributes half of household income. This estimate is significantly larger than 0.5 at the
10% level. Increasing the wife’s income contribution to 60% increases her consumption
8
Recall that the sharing rule is specified as η = exp(γ 0 + γ d ) / (1 + exp(γ 0 + γ d )) and that the distribution factors are
normalized to mean zero
13
share to 0.59, whereas reducing her income contribution to 25% reduces her share to 0.45.
These estimates are significantly different from 0.5. If the woman does not contribute at all
to household income, her consumption share drops to 0.35.
Compared to the results in Alessie et al. (2006) our estimates of A are rather large. They
estimate the model for 10 European countries, and the majority of their estimates is 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 Alessie et al.).
Let us illustrate these results with an example using the estimates from the seminonparametric ordered probit model. Consider a couple household with total income of SFr
100’000. The sum of private consumption of both spouses is then SFr 139’000. If both
spouses contribute half of household income, the wife has a consumption share of SFr
75’616, and her husband receives 63’384. In other words, if they were to separate the women
would need an income of 75’616 to be as well off as before, whereas the man would only
need 63’384. There is no unique equivalence scale that equates a couple’s utility with a
single’s utility. By contrast, we have estimated what Browning et al. (2008) call an
indifference scale. It is based on the concept of comparing two living situations for the same
individual. It answers the question how much income a person living in a couple would need
to be as well off when living alone, e.g. by separation or death of the partner. In the example,
the female indifference scale is 0.76 and the male indifference scale is 0.63. It is important to
note that these scales are not constant, but depend on the sharing rule. If all income is
contributed by the man, his indifference scale is 0.90, while the wife’s scale is only 0.49.
The concept of indifference scales can be important for determining alimony and life
insurance payments. Of course, these scales only refer to consumption utility and not overall
welfare, which would include the emotional dimensions associated with changing the living
situation.
Indifference scales may also be important for the analysis of income inequality and poverty.
As an illustration, we present the Gini coefficients computed using the traditional
equivalence scale and the estimated indifference scales for our data. Using the traditional
scale, the Gini in our sample is 0.25. Inequality is almost the same in the two subgroups
(singles and couples). Using the indifference scales, overall inequality increases to 0.26,
which is of course entirely driven by in the increase of inequality among couples (from 0.23
14
to 0.26). Inequality among individuals in couples is 10% larger when taking account of
unequal sharing rules.
Table 5: Income and consumption inequality
Income adjusted with constant equivalence scale
Income adjusted with indifference scale
All
0.247
0.260
Singles
0.235
0.235
Couples
0.233
0.258
Individual sharing rules are evaluated at the individual values of the distribution factors using
the estimated sharing rule parameters in column 2 of Table 4.
The constant equivalence scale is 1.39 x 0.5
The indifference scale is computed as 1.39 x sharing rule
VI. CONCLUSIONS
This paper provides first empirical evidence for Switzerland that the textbook model of
household behavior is not an adequate description of intra-household decision making. The
so-called income pooling hypothesis, which states that only total household income is
relevant for household behavior, independent of the individual contributions to household
income, is rejected. Based on a collective household model, this paper further provides
estimates of the returns to scale of living together and of the rule of sharing consumption
among spouses in Switzerland. 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.
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 1.39 times household income. The estimated sharing rule is somewhat
smaller than 0.5 at the mean, but significantly larger than 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.
The analysis can be extended in several ways. The most important extension is the inclusion
of children into the analysis. This is a difficult task, but absolutely necessary to make the
results of this kind of analysis relevant for policy applications. Furthermore, extending the
model to include household production is important. Overall, the results of this paper
indicate that the subjective evaluation approach yields additional and complementary
insights to the understanding of household behavior.
15
REFERENCES
Alessie, Rob, Thomas F. Crossley, and Vincent A. Hildebrand (2006), “Estimating a Collective Household
Model with Survey Data on Financial Satisfaction”, SEDAP Research Paper 161, McMaster University.
Apps, Patricia F. and Ray Rees (2007), “Household Models: An Historical Perspective”, CESifo Working
paper 2172.
Attanasio, Orazio and Valérie Lechene (2002), “Tests of income pooling in household decisions”, Review of
Economic Dynamics, 5, 720-748.
Becker, Gary S. (1981), A Treatise on the Familiy. Cambridge, Mass.: Harvard University Press.
Bonke, Jens and Hans Uldall-Poulsen (2006), “Why do families actually pool their income? Evidence from
Denmark”, Review of Economics of the Household, 5, 113-128.
Bonke, Jens and Martin Browning (2009), “The distribution of financial well-being and income within the
household”, Review of Economics of the Household, 7, 31-42.
Browning Martin, François Bourguignon, Pierre A. Chiappori, and Valérie Lechène (1994), “Incomes and
outcomes: a structural model of intra-household allocation”, Journal of Political Economcy, 102, 10671096.
Browning, Martin, Pierre A. Chiappori, and Arthur Lewbel (2008), “Estimating Consumption Economies of
Scale, Adult Equivalence Scales, and Household Bargaining Power”, Boston College Working Paper 588,
revised 2008.
Cherchye, Laurens, Bram De Rock and Frederic Vermeulen (2008), "Economic well-being and poverty among
the elderly: an analysis based on a collective consumption model", CentER Discussion Paper, 2008-15,
Tilburg, CentER.
Chiappori, Pierre A. (1988), “Rational household labor supply”, Econometrica, 56, 63-89.
Chiappori, Pierre A. (1992), “Collective labor supply and welfare”, Journal of Political Economy, 102, 437467.
Donni, Olivier (2007), “Collective labor supply”, Economic Journal 117, 94-119.
Easterlin, Richard (1974), “Does economic growth improve the human lot? Some empirical evidence”, in: Paul
A. David and Melvin W. Reder (eds.), Nations and Households in Economic Growth: essays in honor of
Moses Abramovitz, New York: Academic Press, 89-125.
Fortin, Bernard and Guy Lacroix (1997), “A test of unitary and collective models of household labor supply”,
Economic Journal 107, 933-955.
Frey, Bruno S. and Alois Stutzer (2002), “What can economists learn from happiness research?”, Journal of
Economic Literature, 40, 402-435.
Gallant, A. Ronald and Douglas W. Nychka (1987), “Semi-nonparametric Maximum Likelihood Estimation“,
Econometrica, 55, 363-390
Lewbel, Arthur and Krishna Pendakur (2008), “Estimation of Collective Household Models with Engel
curves”, Journal of Econometrics, 147, 350-358.
Lundberg, Shelley J., Robert A. Pollak and Terrence J. Wales (1997), “Do Husbands and Wives Pool Their
Resources? Evidence from the United Kingdom Child Benefit”, Journal of Human Resources, 32, 463-480.
Lührmann, Melanie and Jürgen Maurer (2007), “Who wears the trousers? A semiparametric analysis of
decision power in couples”, cemmap working paper 25/07.
Manser, Marilyn and Murray Brown (1980), “Marriage and Household Decision-Making: A Bargaining
Analysis” International Economic Review 21, 31-44.
McElroy, Marjorie B. and Mary J. Horney (1981), “Nash-bargained household decisions”, International
Economic Review, 22, 333-350.
Phipps, Shelley A. and Peter S. Burton (1998), “What’s Mine is Yours? The Influence of Male and Female
Incomes on Patterns of Household Expenditure”, Economica, 65, 599-613.
Stewart, Mark (2005), "A Comparison of Semiparametric Estimators for the Ordered Response Model",
Computational Statistics and Data Analysis, 49, 555-573.
Thomas, Duncan (1990), “Intra Household Allocation, an Inferential Approach,” Journal of Human Resources,
25, 599-634.
16
Van Praag, Bernard M. S. and Arie Kapteyn (1973), “Further Evidence on the Individual Welfare Function of
Income: an Empirical Investigation in The Netherlands”, European Economic Review, 4, 33-62
Van Praag, Bernard M. S. and P. Frijters (1999), “The Measurement of Welfare and Well-Being; the Leyden
Approach”, in: Well-Being: the Foundations of Hedonic Psychology, D. Kahneman, E. Diener, N. Schwarz
(eds.), Russell Sage Foundation, New York, pp. 413-433
Ward-Batts, Jennifer (2008), “Out of the wallet and into the purse: Using micro data to test income pooling”,
Journal of Human Resources, 18, 325-351.
Wooldridge, Jeffery (2002), The Econometric Analysis of Cross Section and Panel Data. MIT Press,
Cambridge
17
TECHNICAL APPENDIX
A.1
Minimum Distance step to estimate structural parameters
In this section we show how the model and the structural parameters are estimated. First, in
order to simplify the estimation problem, we linearize ln[1 + exp(γ 0 + γ d d )] around zero
(see also Alessie et al., 2006). The linearization is given by
ln[1 + exp(γ 0 + γ d d )] ≈ ln[1 + exp(γ 0 )] +
exp(γ 0 )
(γ d d )
1 + exp(γ 0 )
(A.1)
Based on this linearization the indirect utility of women living couples can be written as
⎧
⎫
exp(γ 0 )
(γ d d ) + ln y h − ln A⎬ + ε
V = α ( z ) + β ⎨(γ 0 + γ d d ) − ln[1 + exp(γ 0 )] −
0
1 + exp(γ )
⎩
⎭
(A.2)
The corresponding indirect utility of men living in couples is
⎧
⎫
exp(γ 0 )
0
(γ d d ) + ln y h − ln A⎬ + ε
V = α ( z ) + β ⎨ − ln[1 + exp(γ )] −
0
1 + exp(γ )
⎩
⎭
(A.3)
The estimated reduced-form parameters θ (see equation 11) are related to the structural
parameters α, β, γ and A as follows:
θ 0 ( zt ) = α ( z t )
θ1 = β
θ 2 = − β (ln A + ln[1 + exp(γ 0 )])
θ 3 = βγ 0
⎛
θ 4 = β ⎜1 −
⎝
⎛
θt5 = β ⎜ 1 −
⎝
exp(γ 0 ) ⎞ 1
βγ 1
γ
=
⎟
1 + exp(γ 0 ) ⎠
1 + exp(γ 0 )
exp(γ 0 ) ⎞ 2
βγ 2
=
γ
⎟
1 + exp(γ 0 ) ⎠
1 + exp(γ 0 )
⎛
exp(γ 0 ) ⎞ 1 − βγ 1 exp(γ 0 )
θ = β ⎜−
γ =
0 ⎟ t
1 + exp(γ 0 )
⎝ 1 + exp(γ ) ⎠
6
exp(γ 0 ) ⎞ 2 − βγ 2 exp(γ 0 )
γ =
0 ⎟ t
γ
p(
)
1 + exp(γ 0 )
1
ex
+
⎝
⎠
⎛
θ7 = β ⎜−
Obviously, the structural parameters are nonlinear functions of the reduced form parameters.
These can be written as
18
β = θ1 = π 1
⎛θ2
⎞
ln A = − ⎜ 1 + ln (1 + exp(θ 3 / θ 1 ) ) ⎟ = π 2
⎝θ
⎠
γ0 =
γ1 =
γ =
2
γ =
1
γ =
2
θ3
= π3
θ1
θ 4 (1 + exp(θ 3 / θ 1 ) )
θ1
θ 5 (1 + exp(θ 3 / θ 1 ) )
θ1
θ 6 (1 + exp(θ 3 / θ 1 ) )
θ 1 exp(θ 3 / θ 1 )
θ 7 (1 + exp(θ 3 / θ 1 ) )
θ 1 exp(θ 3 / θ 1 )
= π4
= π5
= π6
= π7
Using these expressions it is easy to compute the estimates of the vector π , πˆ , given the
estimates of the reduced form parameters. Alessie et al. show how the covariance matrix of
(πˆ ),Vˆ (πˆ ) , can be estimated. We refer to their paper for this step. The unknown structural
parameters π * = ( β ,ln( A), γ 0 , γ 1 , γ 2 ) can then be estimated by feasible GLS, i.e.
(
πˆ * = X 'Vˆ (πˆ ) −1 X
)
−1
X 'Vˆ (πˆ ) −1πˆ
with
⎛1
⎜0
⎜
⎜0
⎜
X = ⎜0
⎜0
⎜
⎜0
⎜0
⎝
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
1
0
0⎞
0 ⎟⎟
0⎟
⎟
0⎟
1⎟
⎟
0⎟
1 ⎟⎠
19
A.2
Semi-nonparametric ordered response model
A semiparametric alternative to the ordered probit model is based on the semi-nonparametric
series estimator proposed by Gallant and Nychka (1987). It approximates the unknown
density of ε by a Hermite form and can be written as the product of a squared polynomial
and a normal density. This gives a polynomial expansion with a Gaussian leading term. To
ensure that the density approximation is proper (i.e. that it integrates to unity) it is written as
2
1⎛ K
⎞
f K (ε ) = ⎜ ∑ γ k ε k ⎟ φ (ε )
θ ⎝ k =0
⎠
where φ (ε ) is the standard normal density of ε and where
2
⎛ K
⎞
θ = ∫ ⎜ ∑ γ k ε k ⎟ φ (ε )d ε
−∞
⎝ k =0
⎠
∞
This general specification is invariant to multiplication of γ = (γ 0 , γ 1 ,..., γ K ) by a scalar, so
normalization is required, which is obtained by setting γ0 = 1. The distribution function is
then given by
FK (u ) =
∫
∫
u
−∞
∞
−∞
(1 + ∑
(1 + ∑
K
)
γ ε ) φ (ε )d ε
2
γ ε k φ (ε )d ε
k =1 k
K
k =1
k
2
k
This defines a family of “semi-nonparametric” (SNP) distributions for increasing values of
K. The class of densities that can be approximated by this form is very general. See Stewart
(2005) for more details.
One location normalization is necessary for semiparametric identification. Following Stewart
(2005) we fix the first threshold α1 at the ordered probit estimate of this threshold. The
ordered probit estimator is a special case of the SNP ordered probit estimator with K = 2.
Estimation is done with maximum likelihood conditional on K. The optimal K can be
determined by likelihood ratio tests or the AIC.
20