Estimating a model of inefficient cooperation and consumption in collective households

Abstract

Lewbel and Pendakur (2021) propose a model of consumption inefficiency in collective households, based on “cooperation factors". We simplify that model to make it empirically tractable, and apply it to identify and estimate household member resource shares, and to measure the dollar cost of inefficient levels of cooperation. Using data from Bangladesh, we find that increased cooperation among household members yields the equivalent of a 13% gain in total expenditures, with most of the benefit of this gain going towards men.

Appendices

Appendix

See Table 6–8

Formal assumptions and proofs

Here we formally derive our model, and prove that it is semiparametrically point identified. To simplify the derivations and assumptions, we first prove results without unobserved random utility parameters (as would apply if, e.g., our data consisted of many observations of a single household, or of many households with no unobserved variation in tastes). We then later add unobserved error terms to the model, corresponding to unobserved preference heterogeneity.

Let f, r, y, p, π, and z be as defined in the main text. Note that the first few Lemmas below will not impose the restriction that f only equal two values.

Assumption 1

Conditional on f, r, y, p, π, and z, the household chooses quantities to consume using the program given by equation (6) in the main text.

Assumption A1 describes the collective household’s conditionally efficient behavior. For each household member j, U_j is that member’s utility function over consumption goods, u_j is that members additional utility or disutility associated with f, and ω_j is that member’s Pareto weight.

As can be seen by Eq. (6) in the main text, the way that private assignable goods q_j differ from other goods g is that each q_j only appears in the utility function of individual j (which makes it assignable to that member) and these goods are unaffected by the matrix A_f in the budget constraints, meaning that they are not shared or consumed jointly (which makes them private goods).

We next assume some regularity conditions. These assumptions ensure sensible and convenient restrictions on economic behavior like no money illusion, preferring larger consumption bundles to smaller ones, and the absence of corner solutions in the household’s maximization problem.

Assumption 2

Each ${\omega }_{j}\left(f,z,p,\pi ,y\right)$ function is differentiable and homogeneous of degree zero in $\left(p,\pi ,y\right)$. Each ${U}_{j}\left({q}_{j},{g}_{j},z\right)$ function is concave, strictly increasing, and twice continously differentiable in g_j and q_j. For each f, the matrix A_f is nonsingular with all nonnegative elements and a strictly positive diagonal. The variable y and each element of p and π are all strictly positive, and the maximizing values of g₁, q₁, . . . g_J, q_J in Assumption A1 are all strictly positive.

Lemma 1

Let Assumptions A1 and A2 hold. Then there exist positive resource share functions ${\eta }_{j}\left(p,\pi ,y,f,z\right)$ such that $\mathop{\sum }\nolimits_{j = 1}^{J}{\eta }_{j}\left(p,\pi ,y,f,z\right)=1$, and the household’s demand function for goods is given by each member j solving the program

$$\mathop{\max }\limits_{{g}_{j},{q}_{j}}{U}_{j}\left({q}_{j},{g}_{j},z\right)$$

(13)

$${{{\rm{such}}}}\,{{{\rm{that}}}}\,{p}^{{\prime} }{A}_{f}{g}_{j}+{\pi }_{j}{q}_{j}={\eta }_{j}\left(p,\pi ,y,z,f\right)y\,{{{\rm{and}}}}\,g={A}_{f}\mathop{\sum }\limits_{j=1}^{J}{g}_{j}.$$

To prove Lemma 1, first observe that the values of g₁, q₁, . . . g_J, q_J that maximize Eq. (6) in the main text are equivalent to the values that maximize

$$\mathop{\max }\limits_{{g}_{1},{q}_{1},...{g}_{J},{q}_{J}}\mathop{\sum }\limits_{j=1}^{J}{U}_{j}\left({q}_{j},{g}_{j},z\right){\omega }_{j}\left(p,\pi ,y,f\right)$$

(14)

given the same budget constraint. because the terms in Eq. (6) in the main text that are not in (14) do not depend on g₁, q₁, . . . g_J, q_J. With that replacement, the proof of Lemma 1 then follows immediately from the results derived in BCL. BCL only considered J = 2, but the extension of this Lemma to more than two household members, and to carrying the additional covariates, is straightforward. Note that the resource share functions η_j in Lemma 1 do not depend on r, because r, including the component v, does not appear in either Eq. (14) or in the budget constraint, and so cannot affect the outcome quantities.

Our empirical work will make use of cross section data, where price variation is not observed. Most of the remaining assumptions we make about resource shares and about the U_j component of utility are the same, or similar, to those made by DLP, and for the same reason: to ensure identification of the model without requiring price variation.

Assumption 3

The resource share functions ${\eta }_{j}\left(p,\pi ,y,f,z\right)$ do not depend on y.

DLP give many arguments, both theoretical and empirical, supporting the assumption that resource shares do not vary with y. Given Assumption A3, we hereafter write the resource share function as ${\eta }_{j}\left(\pi ,p,f,z\right)$.

For the next assumption, recall that an indirect utility function is defined as the function of prices and the budget that is obtained when one substitutes an individual’s demand functions into their direct utility function.

Assumption 4

For each household member j, the direct utility function ${U}_{j}\left({g}_{j},{q}_{j},z\right)$, when facing prices p and π and having the budget y, has the associated indirect utility function

$${V}_{j}\left({\pi }_{j},p,y,z\right)=\left[\ln y-\ln {S}_{j}\left({\pi }_{j},p,z\right)\right]{M}_{j}\left({\pi }_{j},p,z\right)$$

(15)

For some functions S_j and M_j.

Assumption A4 says that household members each have utility functions in the class that Muellbauer (1974) called PIGLOG (price independent, generalized logarithmic) preferences. As noted in the main text, this is a class of functional forms that is widely known to fit empirical continuous consumer demand data well. Examples of popular models in this class include the Christensen et al. (1975) Translog demand system and Deaton and Muellbauer’s (1980) AIDS (Almost Ideal Demand System) model.¹⁰

Lemma 2

Let Assumptions A1, A2, A3, and A4 hold. Then the value of ${U}_{j}\left({q}_{j},{g}_{j},z\right)$ attained by household member j is given by

$${U}_{j}=\left[\ln {\eta }_{j}\left(\pi ,{A}_{f}p,f,z\right)+\ln y-\ln {S}_{j}\left({\pi }_{j},{A}_{f}p,z\right)\right]{M}_{j}\left({\pi }_{j},{A}_{f}p,z\right)$$

(16)

and the household’s demand functions for the private assignable goods q_j are

$${q}_{j}={\eta }_{j}\left(\pi ,{A}_{f}p,f,z\right)y\left(\frac{\partial \ln {S}_{j}\left({\pi }_{j},{A}_{f}p,z\right)}{\partial {\pi }_{j}}-\frac{\partial \ln {M}_{j}\left({\pi }_{j},{A}_{f}p,z\right)}{\partial {\pi }_{j}}\ln \left(\frac{{\eta }_{j}\left(\pi ,{A}_{f}p,f,z\right)y}{{S}_{j}\left({\pi }_{j},{A}_{f}p,z\right)}\right)\right)$$

(17)

To prove Lemma 2, observe that by Lemma 1, household member j maximizes the utility function ${U}_{j}\left({q}_{j},{g}_{j},z\right)$ facing shadow prices ${A}_{f}^{{\prime} }p$ and π_j and having the shadow budget ${\eta }_{j}\left(\pi ,{A}_{f}p,f,z\right)y$. Therefore, using the definition of indirect utility, member j’s attained utility level ${U}_{j}\left({q}_{j},{g}_{j},z\right)$ is given by ${V}_{j}\left({\pi }_{j},{A}_{f}^{{\prime} }p,{\eta }_{j}\left(\pi ,{A}_{f}p,f\right)y\right)$, which by Assumption A4 equals Eq. (16). Next, a property of regular indirect utility functions is that the corresponding demand functions can be obtained by Roy’s identity. Equation (17) is obtained by applying Roy’s identity to Eq. (15) for the private assignable goods q_j, and then replacing p and y in the result with ${A}_{f}^{{\prime} }p$ and ${\eta }_{j}\left(\pi ,{A}_{f}p,f\right)y$.

We could similarly obtain the demand functions for other goods g, as in BCL, but these will be more complicated due to the sharing, with Roy’s identity being applied to each member to obtain each g_j demand function, and substituting the results into $g={A}_{f}\mathop{\sum }\nolimits_{j = 1}^{J}{g}_{j}$. However, our empirical analyses will only make use of the private assignable goods q_j with demands given by Eq. (17).

Assumption 5

Let $\ln {M}_{j}\left({\pi }_{j},{A}_{f}p,z\right)={m}_{j}\left({A}_{f}p,z\right)-\beta \left(z\right)\ln {\pi }_{j}$ for some functions m_j and β.

There are two restrictions embodied in Assumption A5. One is that the functional form of $\ln {M}_{j}$ in terms of prices is linear and additive in $\ln {\pi }_{j}$, and the other is that the function $\beta \left(z\right)$ does not vary by j. The functional form restriction of log linearity in log prices is a common one in consumer demand models, e.g., the function M_j in Deaton and Muellbauer’s (1980) AIDS (Almost Ideal Demand System) satisfies this restriction. Assumption A5 could be further relaxed by letting β depend on p (though not on A_f) without affecting later results.

To identify their model, DLP define and use a property of preferences called similarity across people (SAP), and provide empirical evidence in support of SAP. The restriction that β not vary by j suffices to make SAP hold for the private assignable goods (but not necessarily for other goods).

Assumption 6

Let $\ln {S}_{j}\left({\pi }_{j},{A}_{f}p,z\right)=\ln {s}_{j}\left({\pi }_{j},p,z\right)-\ln \delta \left({A}_{f}p,z\right)$ for some functions s_j and δ. Without loss of generality, let $\ln \delta \left({A}_{0}p,z\right)=0$.

Assumption A6 assumes separability of the effects of π_j and f on the function S_j. DLP discuss various ways in which the matrix A_f can drop out of a function of prices, as required in the function s_j.¹¹ This assumption is not vital, but will be helpful for making the cost of an inefficient choice of f identifiable. Assuming $\ln \delta \left({A}_{0}p,z\right)=0$ in Assumption A6 is without loss of generality, because if it does not hold then one can make it hold if one redefines δ and s_j by subtracting $\ln \delta \left({A}_{0}p,z\right)$ from both $\ln \delta \left(f,p,z\right)$ and $\ln {s}_{j}\left({\pi }_{j},p,z\right)$.

It will be convenient to express our demand functions in budget share form. Define w_j = q_jπ_j/y. This budget share is the fraction of the household’s budget y that is spent on buying person j’s assignable good q_j.

Lemma 3

Given Assumptions A1 to A6, the value of ${U}_{j}\left({q}_{j},{g}_{j},z\right)$ attained by household member j is given by

$$\left[\ln {\eta }_{j}\left(\pi ,{A}_{f}p,f,z\right)+\ln y-\ln {s}_{j}\left({\pi }_{j},p,z\right)+\ln \delta \left({A}_{f}p,z\right)\right]\left[{m}_{j}\left({A}_{f}p,z\right)-\beta \left(z\right)\ln {\pi }_{j}\right]$$

(18)

and the budget share demand functions for each private assignable good are given by

$${w}_{j}={\eta }_{j}\left(\pi ,{A}_{f}p,f,z\right)\left[{\gamma }_{j}\left({\pi }_{j},p,z\right)+\beta \left(z\right)\left(\ln y+\ln {\eta }_{j}\left(\pi ,{A}_{f}p,f,z\right)+\ln \delta \left({A}_{f}p,z\right)\right)\right].$$

(19)

where the function γ_j is defined by

$${\gamma }_{j}\left({\pi }_{j},p,z\right)=\frac{\partial \ln {s}_{j}\left({\pi }_{j},p,z\right)}{\partial \ln {\pi }_{j}}-\beta \left(z\right)\ln {s}_{j}\left({\pi }_{j},p,z\right)$$

The proof of Lemma 3 consists of substituting the expressions for M_j and S_j given by Assumptions A5 and A6 into the equations given by Lemma 2, and converting the quantity q_j into the budget share w_j.

Assumption 7

Market prices p and π are the same for all households.

Our data come from a single time period, which (assuming the law of one price) justifies assuming p and π are the same across all households. This assumption makes our demand functions reduce to Engel curves. For simplicity, we abuse notation here and redefine objects that were functions of A_fp as just functions of f, since with fixed prices the only source of variation of A_fp is just variation in f).

Lemma 4

Given Assumptions A1 to A7, the value of ${U}_{j}\left({q}_{j},{g}_{j},z\right)$ attained by household member j is given by

$$\left[\ln {\eta }_{j}\left(f,z\right)+\ln y-\ln {s}_{j}\left(z\right)+\ln \delta \left(f,z\right)\right]{M}_{j}\left(f,z\right)$$

(20)

and the budget share Engel curve functions ${w}_{j}={W}_{j}\left(f,z,y\right)$ for each private assignable good are given by

$${W}_{j}\left(f,z,y\right)={\eta }_{j}\left(f,z\right)\left[{\gamma }_{j}\left(z\right)+\beta \left(z\right)\left(\ln y+\ln {\eta }_{j}\left(f,z\right)+\ln \delta \left(f,z\right)\right)\right].$$

(21)

Lemma 4 entails a small abuse of notation, where we have absorbed the values of p and π into the definitions of all of our functions, noting that any function of A_fp remains a function of f even if p is a constant. Lemma 4 is just rewriting Lemma 3 after dropping the prices.

Lemma 5

Let Assumptions A1 to A7 hold. Let ${W}_{j}\left(f,z,y\right)$ be defined by Eq. (21) for j = 1, . . . , J. Given functions ${W}_{j}\left(f,z,y\right)$, the functions ${\eta }_{j}\left(f,z\right)$, $\delta \left(f,z\right)$,${\gamma }_{j}\left(z\right)$, and $\beta \left(z\right)$ are all point identified.

To prove Lemma 5, observe first by Eq. (21) that ${\eta }_{j}\left(f,z\right)\beta \left(z\right)=\partial {W}_{j}\left(f,z,y\right)/\partial \ln y$. Next, since resource shares sum to one, we can identify $\beta \left(z\right)$ and ${\eta }_{j}\left(f,z\right)$ by

$$\beta \left(z\right)=\mathop{\sum }\limits_{j=1}^{J}\frac{\partial {W}_{j}\left(f,z,y\right)}{\partial \ln y}\,{{{\rm{and}}}}\,{\eta }_{j}\left(f,z\right)=\frac{1}{\beta \left(z\right)}\frac{\partial {W}_{j}\left(f,z,y\right)}{\partial \ln y}$$

Next, define ${\rho }_{j}\left(f,z,y\right)$ by

$${\rho }_{j}\left(f,z,y\right)=\frac{{W}_{j}\left(f,z,y\right)}{{\eta }_{j}\left(f,z\right)}-\beta \left(z\right)\left(\ln y+\ln {\eta }_{j}\left(f,z\right)\right)$$

The function ${\rho }_{j}\left(f,z,y\right)$ is identified because it is defined entirely in terms of identified functions. By Eq. (21), ${\rho }_{j}\left(f,z,y\right)={\gamma }_{j}\left(z\right)-\beta \left(z\right)\ln \delta \left(f,z\right)$. It follows from Assumption A6 that $\ln \delta \left(0,z\right)=0$, so ${\gamma }_{j}\left(z\right)$ and $\delta \left(f,z\right)$ are identified by

$${\gamma }_{j}\left(z\right)={\rho }_{j}\left(0,z,y\right)\,{{{\rm{and}}}}\,\ln \delta \left(f,z\right)=\frac{{\rho }_{j}\left(f,z,y\right)-{\rho }_{j}\left(0,z,y\right)}{\beta \left(z\right)}$$

evaluated at any value of y (or, e.g., averaged over y).

Lemma 5 shows that, given the household demand functions, the resource share functions ${\eta }_{j}\left(f,z\right)$ are identified, so our model, like DLP, overcomes the problem in the earlier collective household literature of (the levels of) resource shares not being identified. Lemma 5 also shows identification of the preference related functions ${\gamma }_{j}\left(z\right)$ and $\beta \left(z\right)$, and identification of our new cost of inefficiency function $\delta \left(f,z\right)$.

Lemma 6

Let Assumptions A1 to A7 hold. Assume f is determined by maximizing ${{\Psi }}\left({U}_{1}+{u}_{1},...,{U}_{J}+{u}_{J}\right)$ for some function Ψ. Then $f=\arg \max {{\Psi }}\left({R}_{1}\left(p,y,f,v\right),...{R}_{J}\left(p,y,f,v\right)\right)$ where ${R}_{j}\left(f,y,v,z\right)$ is given by

$${R}_{j}\left(f,y,v,z\right)=\left(\ln {\eta }_{j}\left(f,z\right)+\ln y-\ln {s}_{j}\left(z\right)+\ln \delta \left(f,z\right)\right){M}_{j}\left(f,z\right)+{u}_{j}\left(f,v,z\right)$$

The proof of Lemma 6 is then that, by Eq. (20) and the definition of u_j, for any f the level of U_j + u_j attained by member j is given by the function ${R}_{j}\left(f,y,v,z\right)$.

The above analyses apply to a single household. Our data will actually consist of a cross section of households, each only observed once. To allow for unobserved variation in tastes across households in a conveniently tractible form, replace the function $\ln {S}_{j}\left({\pi }_{j},{A}_{f}p,z\right)$ with $\ln {S}_{j}\left({\pi }_{j},{A}_{f}p,z\right)-{\widetilde{\varepsilon }}_{j}$ where ${\widetilde{\varepsilon }}_{j}$ is a random utility parameter representing unobserved variation in preferences for goods. This means that ${\widetilde{\varepsilon }}_{j}$ appears in member j’s utility function U_j. We assume these taste parameters vary randomly across households, so $E\left({\widetilde{\varepsilon }}_{j}| r,z\right)=0$. Similarly, replace ${u}_{j}\left(f,r,z\right)$ with ${u}_{j}\left(f,r,z\right)+{\widetilde{e}}_{jf}$ where ${\widetilde{e}}_{jf}$ represents variation in the utility or disutility associated with the choice of f. The errors ${\widetilde{e}}_{jf}$ and ${\widetilde{\varepsilon }}_{j}$ can be correlated with each other and across household members.

Substituting these definitions into the above equations, we get

$${w}_{j}={\eta }_{j}\left(f,z\right)\left[{\gamma }_{j}\left(z\right)+\beta \left(z\right)\left(\ln y+\ln {\eta }_{j}\left(f,z\right)+\ln \delta \left(f,z\right)\right)+{\varepsilon }_{j}\right]$$

(22)

where ${\varepsilon }_{j}=\beta \left(z\right)\,{\widetilde{\varepsilon }}_{j}$ so $E\left({\varepsilon }_{j}| r,z\right)=0$, and f is now determined by

$$f=\arg \max {{\Psi }}\left({\widetilde{R}}_{1f},...{\widetilde{R}}_{Jf}\right),{{{\rm{where}}}}\,{\widetilde{R}}_{jf}={R}_{j}\left(f,y,r,z\right)+\left({M}_{j}\left(f,z\right)/\beta \left(z\right)\right){\varepsilon }_{j}+{\widetilde{e}}_{jf}$$

(23)

We will want to estimate the Engel curve Eq. (22) for j = 1, . . . , J. Equation (23) shows that f is an endogenous regressor in these equations, because f depends on both ε_j and ${\widetilde{e}}_{jf}$. As discussed in the main text, we do not try to empirically identify or estimate Eq. (23), because both the functions R_j and errors ${\widetilde{e}}_{1f}$ depend on u_j, and there may be important determinents of u_j (the direct utility or disutility from cooperation) that we cannot observe. However, we will require at least one instrument for f.

Another source of error in our model is that, in our data, y is a constructed variable (including imputations from home production), and so may suffer from measurement error. We will therefore require instruments for y. Our current collective household model is static. This is justified by a standard two stage budgeting (time separability) assumption, in which households first decide how much of their income and assets to save versus how much to spend in each time period, and then allocate their expenditures to the various goods they purchase. The total they spend in the time period is y, and the household’s allocation of y to the goods they purchase is given by Eq. (6) in the main text. These means that variables associated with household income and wealth will correlate with y and so are potential instruments for y.

This time separability applies to the utility functions over goods, ${U}_{j}\left({q}_{j},{g}_{j},z\right)$ for each member j, but need not apply to the utility or disutility associated with f, that is, ${u}_{j}\left(f,v,z\right)$. So at least some of these income and wealth variables could be components of v. Let $\widetilde{r}$ denote a vector of potential instruments for y. These are measures related to income or wealth that are not already included in v.

Assume there exists values v₀ and v₁ such that ${u}_{j}\left(f,{v}_{0},z\right)\,\ne \,{u}_{j}\left(f,{v}_{1},z\right)$ for some member j who’s utility appears in Ψ. Then it follows from Eq. (23) that f varies with v, so v can serve as an instrument for f. Similarly, assume that $\ln y$ correlates with $\widetilde{r}$, which can serve as instruments for $\ln y$ (elements of v could also be instruments for y). Based on Eq. (22), we then have conditional moments

$$E\left[\left(\frac{{w}_{j}}{{\eta }_{j}\left(f,z\right)}-{\gamma }_{j}\left(z\right)-\beta \left(z\right)\left(\ln y+\ln {\eta }_{j}\left(f,z\right)+\ln \delta \left(f,z\right)\right)\right)| \widetilde{r},v,z\right]=0$$

(24)

Later in this Appendix we consider nonparametric identification of the functions in this expression based on these moments, but for now consider using these moments parametrically. If we parameterize each of the unknown functions using a parameter vector θ, then Eq. (24) implies unconditional moments

$$E\left[\left(\frac{{w}_{j}}{{\eta }_{j}\left(f,z,\theta \right)}-{\gamma }_{j}\left(z,\theta \right)-\beta \left(z,\theta \right)\left(\ln y+\ln {\eta }_{j}\left(f,z,\theta \right)+\ln \delta \left(f,z,\theta \right)\right)\right)\phi\, \left(\widetilde{r},v,z\right)\right]=0$$

(25)

for any suitably bounded functions $\phi \left(\widetilde{r},v,z\right)$. Our actual estimator will consist of parameterizing the unknown functions in this expression, choosing a set of functions $\phi \left(\widetilde{r},v,z\right)$, and estimating the parameters by GMM (the generalized method of moments) based on these moments. At the end of this Appendix we discuss choice of the ϕ functions.

Equation (25) can suffice for parametric identification and estimation, but is it still possible to nonparametrically identify the functions in this model in the presence of unobserved heterogeneity? The following Theorem shows that the answer is yes, if we make some additional assumptions. Theorem 1 shows these additional assumptions are sufficient for nonparametric identification of these functions, These additional assumptions, which are not required for parametric identification, are listed in Assumption A8.

Assumption 8

Add unobservable heterogeneity terms ${\widetilde{\varepsilon }}_{j}$ and ${\widetilde{e}}_{jf}$ to the model by replacing the function $\ln {S}_{j}\left({\pi }_{j},{A}_{f}p,z\right)$ with $\ln {S}_{j}\left({\pi }_{j},{A}_{f}p,z\right)-{\widetilde{\varepsilon }}_{j}$ and ${u}_{j}\left(f,v,z\right)$ with ${u}_{j}\left(f,v,z\right)+{\widetilde{e}}_{jf}$, for j = 1, . . , J. Assume f is determined by maximizing Ψ, where Ψ is linear, so ${{\Psi }}\left({\widetilde{R}}_{1f},...{\widetilde{R}}_{Jf}\right)=\mathop{\sum }\nolimits_{j = 1}^{J}{\widetilde{c}}_{j}{\widetilde{R}}_{jf}$ for some constants ${\widetilde{c}}_{1}$,...,${\widetilde{c}}_{J}$. Let $\widetilde{e}=\mathop{\sum }\nolimits_{j = 1}^{J}{\widetilde{c}}_{j}\left({\widetilde{e}}_{j1}-{\widetilde{e}}_{j0}\right)$. Define $\widetilde{y}\left(\widetilde{r},v,z\right)$ by $\ln \widetilde{y}\left(\widetilde{r},v,z\right)=E\left(\ln y| \widetilde{r},v,z\right)$. Assume the following: The function $\widetilde{y}\left(\widetilde{r},v,z\right)$ is differentiable in a scalar $\widetilde{r}$ with a nonzero derivative. The error $\widetilde{e}$ is independent of $y,\widetilde{r},v,z$ and $\left({\varepsilon }_{j},\widetilde{e}\right)$ is independent of $\widetilde{r}$ conditional on $\left(v,z\right)$. $E\left({\varepsilon }_{j}| \widetilde{r},v,z\right)=0$. The functions ${M}_{j}\left(f,z\right)$ do not depend on f. There exist values v₁ and v₀ of v such that $\mathop{\sum }\nolimits_{j = 1}^{J}{\widetilde{c}}_{j}{u}_{j}\left(f,{v}_{1},z\right)\ne \mathop{\sum }\nolimits_{j = 1}^{J}{\widetilde{c}}_{j}{u}_{j}\left(f,{v}_{0},z\right)$.

Theorem 1

Let Assumptions A1 to A8 hold. Then the functions ${\eta }_{j}\left(f,z\right)$, $\delta \left(f,z\right)$,${\gamma }_{j}\left(z\right)$, and $\beta \left(z\right)$ are identified.

To prove Theorem 1, first observe that, with f binary, it follows from Eq. (23) that f = 1 if $\mathop{\sum }\nolimits_{j = 1}^{J}{\widetilde{c}}_{j}\left[{R}_{j}\left(1,y,r,z\right)+\left({M}_{j}\left(1,z\right)/\beta \left(z\right)\right){\varepsilon }_{j}+{\widetilde{e}}_{j1}\right]$ is greater than

$\mathop{\sum }\nolimits_{j = 1}^{J}{\widetilde{c}}_{j}\left[{R}_{j}\left(0,y,r,z\right)+\left({M}_{j}\left(0,z\right)/\beta \left(z\right)\right){\varepsilon }_{j}+{\widetilde{e}}_{j0}\right]$, where the function R_j is given by Lemma 6. Taking the difference in these expressions, and using the assumption that ${M}_{j}\left(f,z\right)$ doesn’t depend on f, we get that f = 1 if and only if

$$\begin{array}{ll}&\mathop{\sum }\limits_{j=1}^{J}{\widetilde{c}}_{j}\left[\left(\ln {\eta }_{j}\left(1,z\right)+\ln \delta \left(1,z\right)\right){M}_{j}\left(z\right)+{\mu }_{j}\left(1,v,z\right)\right.\\ &\left.-\left(\ln {\eta }_{j}\left(0,z\right)+\ln \delta \left(0,z\right)\right){M}_{j}\left(z\right)-{\mu }_{j}\left(0,v,z\right)\right]+\widetilde{e}\end{array}$$

is positive. This means that $f=\widetilde{f}\left(v,z,\widetilde{e}\right)$ for some function $\widetilde{f}$. More precisely, f obeys a threshold crossing model where f is one if a function of v and z given by the above expression is greater than $-\widetilde{e}$, otherwise f is zero.

Now, again exploiting that f is binary,

$$E\left({w}_{j}| \widetilde{r},v,z,y\right)=E\left[{W}_{j}\left(f,z,y\right)+\beta \left(z\right)\ln \delta \left(f,z\right)\,{\widetilde{\varepsilon }}_{j}\,| \widetilde{r},v,z,y\right]$$

$$=E[{W}_{j}\left(1,z,y\right)f+\beta \left(z\right)\ln \delta \left(1,z\right)\,f{\widetilde{\varepsilon }}_{j}+{W}_{j}\left(0,z,y\right)\left(1-f\right)+\beta \left(z\right)\ln \delta \left(0,z\right)\left(1-f\right){\widetilde{\varepsilon }}_{j}| \widetilde{r},v,z,y]\,$$

$$\begin{array}{ll}={W}_{j}\left(0,z,y\right)+\left[{W}_{j}\left(1,z,y\right)-{W}_{j}\left(0,z,y\right)\right]E\left(f| \widetilde{r},v,z,y\right)\\ \quad+\,\beta \left(z\right)\left[\ln \delta \left(1,z\right)-\ln \delta \left(0,z\right)\right]E\left(f{\widetilde{\varepsilon }}_{j}| \widetilde{r},v,z,y\right).\end{array}$$

Next, observe that, since ${W}_{j}\left(f,z,y\right)$ is linear in $\ln y$, $E\left[{W}_{j}\left(0,z,y\right)| \widetilde{r},v,z\right]={W}_{j}\left(0,z,\widetilde{y}\right)$ and $E\left[{W}_{j}\left(1,z,y\right)| \widetilde{r},v,z\right]={W}_{j}\left(1,z,\widetilde{y}\right)$ where $\widetilde{y}=\widetilde{y}\left(\widetilde{r},v,z\right)$. Averaging the above expression over y, and noting that $f=\widetilde{f}\left(v,z,{\widetilde{e}}_{1}\right)$, we get

$$\begin{array}{ll}E\left({w}_{j}| \widetilde{r},v,z\right)={W}_{j}\left(0,z,\widetilde{y}\right)+\left[{W}_{j}\left(1,z,\widetilde{y}\right)-{W}_{j}\left(0,z,\widetilde{y}\right)\right]E\left(f| \widetilde{r},v,z\right)\\ \qquad\qquad\qquad\quad+\,\beta \left(z\right)\left[\ln \delta \left(1,z\right)-\ln \delta \left(0,z\right)\right]E\left(f{\widetilde{\varepsilon }}_{j}| \widetilde{r},v,z\right).\end{array}$$

and by the conditional independence assumptions regarding ${\widetilde{\varepsilon }}_{j}$ and ${\widetilde{e}}_{1}$,

$$\begin{array}{rc}E\left({w}_{j}| \,\widetilde{r},v,z\right)&={W}_{j}\left(0,z,\widetilde{y}\right)+\left[{W}_{j}\left(1,z,\widetilde{y}\right)-{W}_{j}\left(0,z,\widetilde{y}\right)\right]E\left(f| v,z\right)\\ &+\beta \left(z\right)\left[\ln \delta \left(1,z\right)-\ln \delta \left(0,z\right)\right]E\left(f{\widetilde{\varepsilon }}_{j}| v,z\right).\end{array}$$

Now the functions $E\left({w}_{j}| \widetilde{r},v,z\right)$ and $\widetilde{y}\left(\widetilde{r},v,z\right)$ (the latter defined by $\ln \widetilde{y}\left(\widetilde{r},v,z\right)=E\left(\ln y| \widetilde{r},v,z\right)$) are both identified from data (and could, e.g., be consistently estimated by nonparametric regressions. So the derivatives of these expressions with respect to $\widetilde{r}$ are identified. This means that the following expression is identified.

$$\frac{\partial E\left({w}_{j}| \widetilde{r},v,z\right)}{\partial \ln \widetilde{r}}/\frac{\partial \ln \widetilde{y}\left(\widetilde{r},v,z\right)}{\partial \ln \widetilde{r}}=\frac{\partial {W}_{j}\left(0,z,\widetilde{y}\right)}{\partial \ln \widetilde{y}}+\frac{\partial \left[{W}_{j}\left(1,z,\widetilde{y}\right)-{W}_{j}\left(0,z,\widetilde{y}\right)\right]}{\partial \ln \widetilde{y}}E\left(f| v,z\right)$$

(26)

Taking the difference between the above expression evaluated at v = v₁ and at v = v₀ then gives (and so identifies)

$$\frac{\partial \left[{W}_{j}\left(1,z,\widetilde{y}\right)-{W}_{j}\left(0,z,\widetilde{y}\right)\right]}{\partial \ln \widetilde{y}}\left[E\left(f| {v}_{1},z\right)-E\left(f| {v}_{0},z\right)\right]$$

and, since $E\left(f| v,z\right)$ is also identified, this identifies $\partial \left[{W}_{j}\left(1,z,\widetilde{y}\right)-{W}_{j}\left(0,z,\widetilde{y}\right)\right]/\partial \ln \widetilde{y}$. We can then solve Eq. (26) for $\partial {W}_{j}\left(0,z,\widetilde{y}\right)/\partial \ln \widetilde{y}$ where all the terms defining this derivative are identified. Taken together, the last two steps identify $\partial {W}_{j}\left(f,z,\widetilde{y}\right)/\partial \ln \widetilde{y}$ for f = 0 and for f = 1.

Given these identified functions and derivatives, we may then duplicate the proof of Lemma 5, (replacing y with $\widetilde{y}$, to show that the functions $\beta \left(z\right)$, ${\eta }_{j}\left(f,z\right)$, ${\gamma }_{j}\left(z\right)$, and $\delta \left(f,z\right)$ are identified.

2.1 Instrument validity

To more formally define conditions under which village-level average f is a valid instrument, assume that the household h random utility parameters ${\widetilde{e}}_{1fh}$ and ${\widetilde{\varepsilon }}_{jh}$ defined in the Appendix are independent across households. Let ${\overline{f}}_{h}$ equal the expected value of f_h conditional on being a household other than h in the village. Then ${\overline{f}}_{h}$ is the probability that a randomly chosen household in the village, other than household h, cooperates. Assume that we include ${\overline{f}}_{h}$ in the function R_j (equals U_j + u_j, formally defined in the Appendix). Taking the conditional mean of Eq. (23) across households other than household h in the village then shows that ${\overline{f}}_{h}$ equals a function of the joint distribution of ${y}_{{h}^{{\prime} }}$, ${{{{\boldsymbol{r}}}}}_{{h}^{{\prime} }}$, ${{{{\boldsymbol{z}}}}}_{{h}^{{\prime} }}$, ${\widetilde{e}}_{1f{h}^{{\prime} }}$ and ${\widetilde{\varepsilon }}_{1{h}^{{\prime} }}$ across all households ${h}^{{\prime} }$ other than h in the village. It follows that ${\overline{f}}_{h}$ is a relevent instrument in that it affects the choice of f (by being in R_j) and that it is a valid instrument in the quantity demand equations because ${\overline{f}}_{h}$ is independent of household h’s specific value of ${\widetilde{\varepsilon }}_{jh}$ and hence of ε_jh.

2.2 Instruments

Our model has two endogenous regressors: the log of household total expenditures, $\ln$y_h, and the cooperation factor f_h. As discussed earlier, if we assume that the consumption allocation decision in our model is separable from the decision of how to allocate household income between total consumption and savings, then functions of household wealth are valid instruments for $\ln$y_h. This time separability is a standard assumption in the consumer demand literature, including in collective household models (see, e.g., Lewbel & Pendakur, 2008). We discuss time separability formally in the Appendix. Another reason y_h could potentially be endogenous is measurement error, stemming from, e.g., purchase mismeasurement, or infrequency of expenditures on some consumption items. Functions of wealth are also valid instruments for dealing with expenditure measurement issues (see, e.g., Banks et al., 1997).

Now consider instruments for f_h. We do not attempt to specify and estimate this equation, so we need an instrument v_h for f_h. This instrument does not need to be randomly assigned, but it does need to correlate with the choice of f_h, while not (after conditioning on other covariates) directly affecting the household’s food consumption decisions (in terms of the model, v_h must appear in one or more of the u_j functions, but not appear in the functions U_j and ω_j for j = 1, . . . , J).

Our primary instrument for f_h is the leave-one-out village level average value of f (the average excluding household h). The idea is that variation in the local prevalence of families whose members cooperate on consumption decisions is likely to correlate with an individual’s own decision to likewise cooperate. Roughly, village level average f (leaving out household h) is a valid instrument in our model if the choice of f in households other than household h is unrelated to the unobserved preference heterogeneity in member’s demand functions for food in household h. See the Appendix for a formal definition of conditions under which this instrument is is valid.

For estimation, we do not need to distinguish which elements of the instrument list r_h are intended to be specifically instruments for f_h vs for y_h (i.e., elements of v vs elements of $\widetilde{{{{\boldsymbol{r}}}}}$ in the Appendix). In particular, though we argue that ${\overline{f}}_{h}$ should primarily correlate with f_h and wealth should primarily correlate with y_h, either or both could affect both. Moreover, since we do not know the functional forms by which f_h and y_h depend on ${\overline{f}}_{h}$ and wealth, we let our instrument list r_h consist of r_1h and r_2h, where r_1h consists of the first through fourth powers of ${\overline{f}}_{h}$ and r_2h consists of the first through fourth powers of log wealth. We use these powers to flexibly capture how f_h and y_h might depend on these instruments. Descriptive statistics for our instruments are given at the bottom of Table 1b.

If our model were linear, then our nonlinear GMM estimator would (apart from weighting matrix) reduce to a linear two stage least squares. The first stage of that two stage least squares would consist of regressing the endogenous f and $\ln y$ on the instruments and exogenous regressors.

To assess the strength of our instruments, we ran those first stage linear regressions. In Table 2 we give regression estimates and associated standard errors from a linear regression of our endogenous regressors, f_h and $\ln {y}_{h}$ on our 18 demographic variables z_h and our 8 instruments r_h. Standard errors are clustered at the village (i.e., the Upazila) level. Table 2 shows that f_h is difficult to predict, with an R² of just 0.17, but the instruments collectively appear strong, in that the F-statistic for the relevance of the instruments (conditional on covariates) is 62. As expected, the village-level average instruments do most of the work here, with an F-statistic of 121, and the log-wealth instruments are also jointly insignificant in this equation. The low R² of this regression emphasizes the point that we can’t (and don’t try to) actually model the decision to cooperate. All we need are sufficiently strong instruments, which our F-statistic indicates is the case (being, e.g., much larger than the rule of thumb level of 10).

Although we can’t treat this regression as a formal model of cooperation, it is suggestive regarding covariates. The regression shows that village level average cooperation is positively correlated with a household’s individual decision to cooperate f_h, as expected. It is also positively correlated with the education of women and age of children, and negatively correlated with the age of women, suggesting that it may respond to women’s bargaining power.

The household log budget $\ln {y}_{h}$ is fitted with an R² of 0.43 and an F-statistic of the instruments of 101. Here, the log-wealth instruments do most of the work, with an F-statistic of 186. But, the cooperation instruments are also relevant in this equation, with an F-statistic of 9.

The above results provide evidence for the relevance our instruments. For further reassurance that the instruments are valid for our model, we later test the exogeneity of the instruments via overidentification tests.

Table 6 GMM estimates, varying covariates

Table 7 ▓

2.3 Discussion of Table 8

In Table 8, we consider three alternatives regarding data construction in our baseline model. In the leftmost column, labeled (7), we retain the previously dropped 238 households that had zero food intake for any member type (adult males, adult females or children). We dropped these households because they likely indicate measurement issues. However, if these zeroes result instead from infrequency that is correlated with regressors (e.g., if significant numbers of households are so poor that some members don’t eat every day), then excluding these households could lead to bias. In comparison with the baseline estimates, we see slightly smaller estimates of resource shares for men in the reference household type, and slightly larger estimates of resource shares for women and children. However, the estimated marginal effect of cooperation f on resource shares is very similar to the baseline estimates: cooperation increases male resource shares by about 3 percentage points, has roughly no effect on female resource shares and reduces children’s resource shares by 3 percentage points. The estimated efficiency gain from cooperation is also very similar to baseline, with $\delta =\exp (0.135)$, or about 14 percent. The resulting money metric utility gains Δ_j from efficiency are 28 percent for men, 14 percent for women, and 5 percent for children, compared to the baseline estimates of 23, 11, and 6 percent.

Table 8 Estimated efficiency and resource shares, varying samples

In the middle panel, we consider some additional sample restrictions that may be sensible. In this model we exclude: a) households where the female respondent is not married; b) households in the top or bottom percentile of the distribution of budgets; and c) households that report zero wealth. The restriction a) is relevant because our cooperation indicator specifically refers to husbands, and respondents in households where the female respondent is not married may not consider the response “self and husband" to be valid. The restriction b) is used because outliers in the budget may have excessive influence on the slopes of estimated Engel curves. The restriction c) is because reported zero wealth may actually be mismeasured wealth. These restrictions result in the loss of 302 observations (roughly 10 percent of the sample).

The resulting estimates in column (8) are somewhat different from the baseline. The estimated value of $\ln \delta$ is 0.173, which is a bit larger than that in the baseline. The associated efficiency gain δ is about 19 percent. Cooperation now increases male and female resource shares by roughly 4 and 1 percentage points, respectively, and decreases children’s resource shares by roughly 5 percentage points. At a gross level, these results are qualitatively the same as the baseline (men gain a lot, women a little and children’s money metric change is insignificant), but the estimated magnitudes are somewhat larger.

The nuclear households in our data have 1 adult man and 1 adult woman and one to four children. We also have 1325 non-nuclear households, having either more than 1 adult man or more than 1 adult woman. These non-nuclear households are a mix of polygamous and multi-generational households. Our model and data might be less appropriate for these non-nuclear households, since our cooperation factor f is only reported by “the main" adult female in the household, and primarily refers to joint decision making by the main adult woman and her husband. Column (9) in Table 4 reports result from estimating the model just with nuclear households, which greatly reduces the sample size.

Again, we see similar patterns as in the baseline case. Cooperation increases efficiency, and induces a shift in resource shares from children towards adult men, with a statistically insigificant impact on the resource shares of women. However, the estimated marginal effect of cooperation f on $\ln \delta$ is much smaller in these nuclear households than in the baseline model, with an estimated value of 0.078. This means that cooperation induces an efficiency gain of only ($\exp 0.078$) 8 percent in nuclear households, compared with 13 percent for all households.

The main difference between nuclear households and the full sample is that the nuclear subsample has smaller households on average. This suggests that the efficiency gains δ may depend on household size. Having more people in a household means goods can be jointly consumed by more household members, leading to greater efficiency. In our model, this can arise because more people sharing a good means a smaller element of A for that good, and hence a lower shadow price.