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Until recently ‘unitary’ models, which assume that household members act as if they maximize a unique utility function under a budget constraint, were largely predominant in the literature on household behaviour. There is increasing agreement, however, that economists cannot ignore the fact that most households are composed of several individuals who take part in the decision process. Consequently, the ‘collective’ models, which postulate that (a) each household member has specific, generally different preferences and (b) the decision process results in Pareto-efficient outcomes, have attracted considerable attention from the profession during recent years.

To examine the properties of collective models, let us consider a household consisting of two persons, A and B, who make decisions about consumption. These persons are characterized by well-behaved utility functions of the form: ui(xA, xB, X), where xi denotes a vector of private goods consumed by member i and X a vector of public goods (i = A, B). This specification of preferences is very general; it allows for altruism but also for externalities or any other preference interaction. We denote the vector of prices for private goods by p, the vector of prices for public goods by P and the household total expenditure by y. Finally, we suppose that there exists a vector of distribution factors, that is, a set of exogenous variables which influence the intra-household allocation of resources without affecting preferences or the budget constraint. Examples are given by the respective contribution of each member to the exogenous household income, the state of the marriage market or divorce legislation. These variables, which are often assigned a crucial role in the derivation of the results, are denoted by s.

To simplify notation, let π′ = (p′, P′) be the vector of prices. Then, efficiency essentially means that household behaviour can be described by the maximization of a utilitarian social welfare function, that is,

$$ \underset{{\mathbf{x}}_A,{\mathbf{x}}_B,\mathbf{X}}{\max}\mu \left(\pi, y,\mathbf{s}\right){u}_A\left({\mathbf{x}}_A,{\mathbf{x}}_B,\mathbf{X}\right)+\left(1-\mu \left(\pi, y,\mathbf{s}\right)\right){u}_B\left({\mathbf{x}}_A,{\mathbf{x}}_B,\mathbf{X}\right) $$
(1)

subject to p′ (xA + xB) + P′X= y. In this programme, the function μ determines the location of the household equilibrium along the Pareto frontier. If μ = 1, then the household behaves as though member A always gets her way whereas, if μ = 0, it is as if member B is the effective dictator. We denote the solutions to Eq. (1) by xA (π, y, s), xB (π, y, s) and X(π, y, s).

Characterization

The first objective of the theory of collective models is to investigate the properties of the household demands derived from Eq. (1). These properties can either be tested statistically or be imposed a priori for simplifying the estimation task. From this perspective, one crucial point is that individual demands for private goods, xA and xB, are generally unobservable by the outside econometrician; demands for these goods are observed only at the household level, x = xA + xB To be useful, the restrictions derived from the collective setting have thus to characterize household demands, x or X, instead of individual demands, xA and xB.

Let ξ = (x′, X′) be the vector of household demands. We define the Pseudo–Slutsky matrix as follows:

$$ \mathbf{S}=\frac{\partial \boldsymbol{\upxi}}{\partial {\pi}^{\prime }}+\frac{\partial \boldsymbol{\upxi}}{\partial y}{\boldsymbol{\upxi}}^{\prime } $$

There exist at least three different sets of testable restrictions that characterize household behaviour.

SR1 Condition

Browning and Chiappori (1998) and Chiappori and Ekeland (2006) show that household demands compatible with Eq. (1) have to satisfy the following condition:

$$ \mathbf{S}=\sum +{\mathbf{R}}_1, $$

where Σ is a symmetric, semi-definite matrix and R1 is a rank one matrix. The interpretation is the following. For any given pair of utility functions, (a) the budget constraint determines the Pareto frontier as a function of π and y, and (b) the value of μ determines the location of the household equilibrium on this frontier. Consequently, a change in π implies a shift of the Pareto frontier. The latter entails the modification of household demands described by Σ. However, the value of μ varies as well, hence the location of the equilibrium moves along the Pareto frontier. Since the frontier is of dimension one, this effect is very restricted and defined by R1.

Proportionality Condition

The particular structure of Eq. (1) leads to further restrictions on behaviour. To make things simple, let us suppose that the vector of distribution factors is twodimensional: s = (s1, s2). Then, Bourguignon et al. (1993) demonstrate the following result:

$$ \frac{\partial \boldsymbol{\upxi}}{\partial {s}_1}=\theta \frac{\partial \boldsymbol{\upxi}}{\partial {s}_2}, $$

where θ is a scalar. Thus, the response to different distribution factors is co-linear. The interpretation is that distribution factors can only change the location of the outcome on the frontier (through function μ), and the latter is of dimension one.

Specific Conditions

The econometrician is often inclined to put more structure on preferences. For example, let us suppose that agents have utility functions of the form: ui(xi,X). In that case, we say that agents are ‘egoistic’ in the sense that the utility does not depend on the partner’s consumption. This assumption implies, in particular, that the decision process can be decentralized. In a first step, household members agree on the level of public goods as well as on a particular distribution of the residual expenditure between them. In a second step, they maximize their utility function, taking into account the level of public goods and their own budget constraint. It means, formally, that there exists a pair of functions (ρA(p, X, y*, s), ρB(p, X, y*, s)), satisfying ρA+ ρB = y* where y* = y – P′X, such that the demand for private goods by member i is the solution to

$$ \underset{{\mathbf{x}}_i}{\max }{u}_i\left({\mathbf{x}}_i,\mathbf{X}\right)\;\mathrm{subject}\;\mathrm{to}\;{\mathbf{p}}^{\prime }{\mathbf{x}}_i={\rho}_i. $$

Hence, household demands for private goods, conditionally on the demands for public goods, can be written as:

$$ \mathbf{x}={\mathbf{x}}_A\left(\mathbf{p},\mathbf{X},\rho \left(\mathbf{p},\mathbf{X},{y}^{\ast },\mathbf{s}\right)\right)+{\mathbf{x}}_B\left(\mathbf{p},\mathbf{X},{y}^{\ast }-\rho \left(\mathbf{p},\mathbf{X},{y}^{\ast },\mathbf{s}\right)\right), $$

where ρ = ρA and y* − ρ = ρB. This structure generates strong testable restrictions because the same function ρ(p, X, y*, s) enters each demand for private goods. Bourguignon, Browning and Chiappori (1995) explicitly derive these restrictions under the form of partial differential equations, whereas Donni (2004) shows that the demands for public goods have a particular but different structure, which implies testable restrictions as well.

Welfare Analyses – Identification

One of the main sources of interest in collective models is to provide the theoretical background for performing welfare comparisons at the individual level. The key concept in that case is what Chiappori (1992) calls the ‘collective’ indirect utility function. Let us suppose again that agents are egoistic. If so, the collective indirect utility function is defined as follows:

$$ {v}_i\left(\pi, y,\mathbf{s}\right)={u}_i\left({\mathbf{x}}_i\left(\pi, y,\mathbf{s}\right),\mathbf{X}\left(\pi, y,\mathbf{s}\right)\right). $$
(2)

This expression describes the level of welfare that member i attains in the household when he or she faces the price-income bundle(π, y) and a set of distribution factors s. This representation of utility differs from the ‘unitary’ indirect utility function in that it implicitly includes the sharing function, and hence an outcome of the collective decision process. However, the knowledge of Eq. (2) is usually sufficient to evaluate the impact of economic policies on individual welfare.

In general, if agents are egoistic, the collective indirect utility functions can be retrieved. Nonetheless, the econometrician must observe the demand for some specific goods, referred to as ‘exclusive’, which benefit only one person in the household. More precisely, we say that good X (x) is exclusively consumed by member i if ∂uj/∂X = 0(∂uj/∂xj = 0) for j ≠ i. The intuition is that the household demand for ‘exclusive’ goods can be used as an indicator of the distribution of bargaining power within the household. Donni (2006) considers the case of purely private consumption (X = 0) and shows that, if there is a single exclusive good, the collective indirect utility functions can be identified up to composition by an increasing transformation. Similarly, Chiappori and Ekeland (2003) consider the opposite case of purely public consumption (x = 0) and show that, if there are two exclusive goods (one for each member), the identification is still possible. However, the general case with both private and public consumption has not been completely treated until now; see Blundell, Chiappori and Meghir (2005) for a first investigation.

Bibliographical Note

The main idea of collective models can be traced back to Leuthold (1968), who estimates a model of household labour supply based on non-cooperative game theory, where the individual is the basic decision-maker. However, this model differs from collective models in that the underlying decision process does not result in efficient outcomes. It actually belongs to the family of ‘strategic’ models (which are sometimes referred to as ‘collective’ models in a broad sense). Nevertheless, a significant advance towards the development of collective models is made by Manser and Brown (1980) and McElroy and Horney (1981) at the beginning of the 1980s. These authors study the properties of models based on bargaining theory, which implies Pareto-efficiency. In that case, the location along the Pareto frontier is determined by the Nash (or Kalai–Smorodinsky) solution. However, the first formal investigation of a model based on the sole efficiency assumption is due to Chiappori (1988, 1992) in the context of labour supply decisions. This model is not explicitly examined in this article because it can be seen as a particular case of the model of consumption. Note, however, that Apps and Rees (1997), Chiappori (1997), Donni (2003), and Fong and Zhang (2001) present theoretical extensions of Chiappori’s initial model, whereas Chiappori, Fortin and Lacroix (2002) exhibit empirical results. Finally, we must mention that several authors have generalized collective models to inter-temporal decisions and uncertain environment. One of the most representative examples of these studies is given by Mazzocco (2005).

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