Abstract
In this note we identify and clarify a confusion that has arisen in the literature about the exact relationship between unitary and collective models and what enters the Pareto weight and the sharing function. We suggest that we should denote as ‘unitary’ any model that leads to outcomes that satisfy the Slutsky conditions whether or not these outcomes depend on distribution factors. In particular, income pooling is neither necessary nor sufficient for a unitary model. We also show that the presence of prices or total expenditure in the sharing rule cannot be used as a test for a unitary model.
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Notes
This is conceptual in the sense that we doubt that any household uses this for all private goods (the latter including much food that is consumed within the household). On the other hand, the concept may be used for some goods in the form of an ‘allowance’ or ‘pocket money’.
If we do have public goods and preferences over private goods are not separable from them, we can decentralise any allocation by giving each agent money to spend on both types of goods and then using Lindahl prices for the public goods (see Donni, 2002).
In this note we shall always assume adding up and homogeneity so these conditions are symmetry and negative semidefiniteness of the Slutsky matrix.
Browning and Chiappori (1998) show that in general the demands from a non-trivial collective model will fail the Slutsky conditions and are hence non-unitary, by this definition. In a two person household, collective demands will satisfy the condition that the Slutsky matrix is the sum of a symmetric and negative semidefinite matrix and a rank one matrix.
The decision about how much to spend may itself be non-unitary in the sense that the two partners may have different preferences over intertemporal allocation; see Browning (2000) for such a model. An unexplored area is the interactions between intertemporal and intratemporal allocations within the household. For example, it might be that one partner agrees to extra total expenditure if the extra is spent in a particular way.
To ease notation we assume that preference factors are disjoint from distribution factors, but we can easily accommodate the case in which they overlap.
By definition, any unitary model is also a collective model but we shall implicitly exclude the latter when we talk of a collective model.
‘For most’, but not for all. For example, household income or gender specific windfall incomes would not usually be considered preference factors and hence their significant presence in an empirical demand equation would signal a failure of DFI.
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We thank the Danish National Research Foundation for support through its grant to the Centre for Applied Microeconometrics. We thank three referees for comments and suggestions.
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Browning, M., Chiappori, PA. & Lechene, V. Collective and Unitary Models: A Clarification. Rev Econ Household 4, 5–14 (2006). https://doi.org/10.1007/s11150-005-6694-2
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DOI: https://doi.org/10.1007/s11150-005-6694-2