## Abstract

The collective household model is based upon the assumption that decision makers have achieved efficient outcomes. This paradigm, which has become one of the leading approaches in family economics, is seldom, if ever, rejected, raising doubt about its falsifiability. We show that the standard approach to test the collective model may yield misleading inferences. We develop a new test procedure to assess its validity. Our approach extends to households that potentially include more than two decision makers. We provide an informal meta-analysis that suggests that much of the evidence in favor of collective rationality in the empirical literature appears to be inconsistent with our test. We illustrate the latter using data from a survey we have conducted in Burkina Faso. Consumption efficiency within monogamous households is not rejected using the standard testing procedure while it is clearly rejected using our proposed test procedure. Furthermore, our test also rejects consumption efficiency for bigamous households. We conclude that intra-household efficiency does yield empirically falsifiable restrictions despite being scarcely rejected in the literature.

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## Notes

- 1.
Distribution factors are variables, such as the state of the marriage markets, that influence the decision process within the household but neither individual preferences nor the household budget set.

- 2.
See the first sentence of their Proposition 2.

- 3.
A distribution factor cannot influence a single demand since, by definition, it cannot affect the household budget constraint.

- 4.
In a somewhat complementary paper, Udry (1996) investigated

*production*outcomes of rural Burkina Faso households and strongly rejected efficiency. Our analysis of household efficiency in consumption can be interpreted as conditional on the household production choices. - 5.
Strictly speaking, testing the joint “all or nothing” hypothesis would require knowledge of the covariance between different parameter estimates. Since we do not have this information, and since we do dot compare the studies between themselves, we will refer to our discussion as an

*informal*meta-analysis. - 6.
Note that Dauphin et al. (2011) investigate the efficiency of households comprising three potential decision-makers (couples with an adult child) using price-based statistical tests.

- 7.
This assumes that the household does not produce any of these N goods, or that the goods produced within the household can be freely sold and purchased on the market.

- 8.
This notation is used to simplify the presentation. Note that if a good is purely private to member i, then the marginal utility of the other members for that good will simply be zero.

- 9.
Henceforth, we will omit

*m*from the argument to simplify the notation. - 10.
When

*N*≤ 2 or*K*= 1, collective rationality imposes no restrictions on the demand system. - 11.
- 12.
An earlier extension of

*Proposition 2.ii*of BBC2009 to multiple decision makers can also be found in Dauphin and Fortin (2001). - 13.
When

*N*≤*I*+ 1 or*K*<*I*+ 1, collective rationality imposes no restrictions on the demand. - 14.
- 15.
In the case where each Pareto weight depends on all distribution factors, one has \({\bf{z}}_L^* = {{\bf{z}}_{ - J}}\).

- 16.
Note that just as for Proposition 2, Proposition 3 does not require any of the distribution factors to affect all the demands.

- 17.
This analysis is based on

*point hypothesis*testing, that is, on inspection of the marginal p-values of various estimates of \({D_{\bf{z}}}{x_n}\left( {\bf{z}} \right),\forall n = 1, \ldots ,N.\) Of course, as noted earlier, to . test the “all or nothing”*joint hypothesis*would require information on the covariance between the various coefficient estimates, which is unavailable. However, in papers where one distribution factor is random (e.g., generated from a randomized experiment), the covariance between the two distribution factors is likely to be close to zero. - 18.
Rookhuizen (1986), p. 59, free translation.

- 19.
Lallemand (1977), p. 263, free translation.

- 20.
Consequently, we implicitly ignore the whole area of the household decisions that concern time allocation, and in particular leisure choices. Note that the original formulation of the collective model did the exact opposite (see Apps and Rees 1988; Chiappori 1988): It focused on household labor supply but omitted the allocation of consumption goods, thanks to the Hicksian composite-good theorem. In that framework, wage rates were the main distribution factors.

- 21.
According to the National 2006 Census.

- 22.
An alternative approach would be to implement a joint hypothesis test. The latter would lead to the same decision rule as long as the covariance between the estimates of our two distribution factors is negligible. Otherwise, a more complex test must be implemented.

- 23.
As long as the covariance between the estimates of our two distribution factors can be neglected

- 24.
Portier and Delyon (2014) developed a so-called

*constrained bootstrap method*which allows to compute the bootstrap distribution of three distinct rank-test statistics proposed in the literature under the null hypothesis that the rank of the matrix is of a given size. Among the three available statistics, we have chosen that of Li (1991) since the consistency of its associated constrained bootstrap test relies on less stringent conditions than the other two. The Li (1991) statistics is the following:\({\hat {\rm \Lambda }} = n\mathop {\sum}\nolimits_{p = m + 1}^P {{{\hat \lambda }_p}} ,\)

where \(n\) is the sample size, \(\left( {{{\hat \lambda }_1}, \ldots ,{{\hat \lambda }_P}} \right)\) are the singular values of the matrix \({D_{\bf{z}}}{\bf{x}}({\bf{z}})\)arranged in descending order and

*m*> 0 is the assumed rank of*D*_{ z }**x**(**z**). The null assumption is \({H_0}:rank\left[ {{D_{\bf{z}}}{\bf{x}}\left( {\bf{z}} \right)} \right] = m\) against \({H_1}:rank\left[ {{D_{\bf{z}}}{\bf{x}}\left( {\bf{z}} \right)} \right] >m\). Since this procedure cannot test whether \({H_0}:rank[{D_{\bf{z}}}{\bf{x}}({\bf{z}})] = 0\), we begin with an*F*test that all the distribution factors are simultaneously statistically significant in all the demand functions. If rejected, the next step is to test \({H_0}:rank[{D_{\bf{z}}}{\bf{x}}({\bf{z}})] = 1\) with a constrained bootstrap test of \({\hat {\rm \Lambda}}\), and so on until the maximum rank of \({D_{\bf{z}}}{\bf{x}}\left( {\bf{z}} \right)\)*(*i.e*., I)*is reached. - 25.
Few, if any, papers ever report tests of weak instruments in the empirical literature. For instance, no paper in Table 1 using a z-conditional demands approach reports such tests.

- 26.
We acknowledge that the approach we use may raise questions about the presence of a pretest bias, as the choice of which demand to invert depends on the statistical significance of the estimated coefficients. Note that this issue is present in all the studies that use the

*z*-conditional demand condition approach.

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## Acknowledgment

We thank two anonymous referees for their valuable comments which helped improve the paper. We are grateful to Pierre-André Chiappori, Jesse Naidoo, Idrissa Diagne, Marion Goussé, Olivier Donni, Nicolas jacquemet, Carolin Pflueger, and François Portier, as well as numerous seminar participants for useful discussions and comments.

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Dauphin, A., Fortin, B. & Lacroix, G. Is consumption efficiency within households falsifiable?.
*Rev Econ Household* **16, **737–766 (2018). https://doi.org/10.1007/s11150-016-9359-4

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### Keywords

- Collective model
- Distribution factors
- Rationality
- Efficiency
- Polygamy

### JEL Classification

- D1
- D7
- J12