The Engel Curves of Noncooperative Households: Director’s Cut

A parametric example, cut from Chiappori and Naidoo (2020), in which collective and noncooperative demand systems differ.

Proposition 2 of Chiappori and Naidoo (2020) states certain restrictions on a household’s $z$-conditional demands that are true of a noncooperative model, but not in general true of collective models. Testing whether those restrictions hold is then a way to discriminate between the two classes of models. Below is a parametric example of a household model where we can see how this would work.

For ease of reference, here is the proposition in question:

Proposition 2.

Order the public goods such that $a$ contributes to goods $1,...,L$ and $b$ contributes to $L+1,...,N$. Let ${\chi }_j$ represent the $z$ -conditional demand for the $j$-th private good, and let ${\mathrm{\Xi }}_k$ represent the demand for the $k$-th public good. In a neighborhood of any point $(\overline{y},\overline{z})$ such that $\partial Q_1/\partial z(\overline{y},\overline{z})\ne 0$ for some $j$, the $z$-conditional demand functions generated by the noncooperative model must satisfy the following partial differential equations: for all private goods $i,j$,

$$\begin{array}{rcl}(1)\frac{{\partial }^2{\chi }_i/\partial y\partial Q_1}{{\partial }^2{\chi }_i/\partial y^2}-\frac{{\partial }^2{\chi }_j/\partial y\partial Q_1}{{\partial }^2{\chi }_j/\partial y^2}& =& 0\text{ and }\\ (2)\frac{\partial }{\partial y}\left(\frac{{\partial }^2{\chi }_i/\partial y\partial Q_1}{{\partial }^2{\chi }_i/\partial y^2}\right)& =& 0;\end{array}$$

for all public goods contributed by $a$, $2\le j\le L$, $$\frac{\partial {\mathrm{\Xi }}_j(y,Q_1)}{\partial y}=0;$$ and for all public goods $L+1\le k,l\le N$ contributed by $b$,

$$\begin{array}{rcl}\frac{\partial {\mathrm{\Xi }}_k(y,Q_1)/\partial Q_1}{\partial {\mathrm{\Xi }}_k(y,Q_1)/\partial y}& =& \frac{\partial {\mathrm{\Xi }}_l(y,Q_1)/\partial Q_1}{\partial {\mathrm{\Xi }}_l(y,Q_1)/\partial y}\\ \text{ and }\frac{\partial }{\partial y}\left(\frac{\partial {\mathrm{\Xi }}_k(y,Q_1)/\partial Q_1}{\partial {\mathrm{\Xi }}_k(y,Q_1)/\partial y}\right)& =& 0.\end{array}$$

Lastly, the effective division of wealth $\rho$ is identified up to an additive constant; for any value of that constant, individual Engel curves are identified up to one common constant each. ◼️

Note that it is necessary that the aggregate Engel curves of the household are nonlinear, because otherwise the ratios of partial derivatives in (1) and (2) above (equations (8) and (9) in the published version) are undefined.

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The example.

Suppose that there is one public good, $Q$, and three private goods. Let $\epsilon >0$, and suppose the preferences of $a$ and $b$ are of the following additively separable, isoelastic form:

$$U^a(Q,q_A)=\frac{Q^{1-{\epsilon }^{-1}}}{1-{\epsilon }^{-1}}+\frac{q_{A1}^{1-(2\epsilon {)}^{-1}}}{1-(2\epsilon {)}^{-1}}+\frac{q_{A2}^{1-{\epsilon }^{-1}}}{1-{\epsilon }^{-1}}+\frac{{\varphi }_a{q}_{A3}^{1-{\epsilon }^{-1}}}{1-{\epsilon }^{-1}}$$

$$U^b(Q,q_B)=\frac{Q^{1-{\epsilon }^{-1}}}{1-{\epsilon }^{-1}}+\frac{q_{B1}^{1-{\epsilon }^{-1}}}{1-{\epsilon }^{-1}}+\frac{q_{B2}^{1-(2\epsilon {)}^{-1}}}{1-(2\epsilon {)}^{-1}}+\frac{{\varphi }_b{q}_{B3}^{1-{\epsilon }^{-1}}}{1-{\epsilon }^{-1}}$$

for some ${\varphi }_a,{\varphi }_b\ge 0$. In what follows, we use two stars in the superscript to denote efficient quantities, and one star to denote equilibrium quantities in the noncooperative model.

If $\theta \in [0,1]$ is $a$’s Pareto weight - so that the social planner maximizes $\theta U^a+(1-\theta )U^b$ subject to the household’s budget constraint - then the collective aggregate demands satisfy $$\begin{array}{rcl}{Q}^{\ast \ast }(y,\theta )& =& {\lambda }^{-\epsilon }\\ q_1^{\ast \ast }(y,\theta )& =& {\theta }^{2\epsilon }{\lambda }^{-2\epsilon }+(1-\theta {)}^{\epsilon }{\lambda }^{-\epsilon }\\ q_2^{\ast \ast }(y,\theta )& =& {\theta }^{\epsilon }{\lambda }^{-\epsilon }+(1-\theta {)}^{2\epsilon }{\lambda }^{-2\epsilon }\\ q_3^{\ast \ast }(y,\theta )& =& [(\theta {\varphi }_a{)}^{\epsilon }+((1-\theta ){\varphi }_b{)}^{\epsilon }]{\lambda }^{-\epsilon }\\ y& =& [{\theta }^{2\epsilon }+(1-\theta {)}^{2\epsilon }]{\lambda }^{-2\epsilon }+[1+(1+{\varphi }_a^{\epsilon }){\theta }^{\epsilon }+(1+{\varphi }_b^{\epsilon })(1-\theta {)}^{\epsilon }]{\lambda }^{-\epsilon }\end{array}$$ where $\lambda$ is a Lagrange multiplier. Consider the special case where $\epsilon =1/2$, ${\varphi }_a=1$, and ${\varphi }_b$ = 0, and let $$s(y,Q)=\frac{y-Q^2}{Q}-1.$$ Notice that $s$ is observable. Now, this is a case in which it is easy to solve for $\theta (y,Q)$, as implicitly defined by $$s(y,Q)=2{\theta }^{1/2}+(1-\theta {)}^{1/2}$$ and hence to compute the ratios of second derivatives in Proposition 2. The solution is $$\theta (s)={\left[\frac{1}{5}(2s+\sqrt{5-s^2})\right]}^2.$$

Also, let $$\begin{array}{rcl}f(s)& =& \theta (s{)}^{\epsilon }+(1-\theta (s){)}^{\epsilon }\\ g(s)& =& 1-\theta (s)\end{array}$$ and notice that the $z$-conditional demands for this collective model satisfy $$\begin{array}{rcl}{\chi }_1^{\ast \ast }(y,Q)+{\chi }_2^{\ast \ast }(y,Q)& =& Q^2+f(s(y,Q))Q\\ {\chi }_2^{\ast \ast }(y,Q)-{\chi }_3^{\ast \ast }(y,Q)& =& g(s(y,Q))Q.\end{array}$$

Finally, notice that if (1) holds, then it must also be true that

$$\frac{{\partial }^2{\chi }_1/\partial y\partial Q+{\partial }^2{\chi }_2/\partial y\partial Q}{{\partial }^2{\chi }_1/\partial y^2+{\partial }^2{\chi }_2/\partial y^2}=\frac{{\partial }^2{\chi }_2/\partial y\partial Q-{\partial }^2{\chi }_3/\partial y\partial Q}{{\partial }^2{\chi }_2/\partial y^2-{\partial }^2{\chi }_3/\partial y^2}$$

The left-hand side of the above is $$\frac{\partial s/\partial Q}{\partial s/\partial y}=-y{Q}^{-1}-Q$$ while the right-hand side is $$\frac{{\partial }^2{\chi }_2^{\ast \ast }/\partial y\partial Q-{\partial }^2{\chi }_3^{\ast \ast }/\partial y\partial Q}{{\partial }^2{\chi }_2^{\ast \ast }/\partial y^2-{\partial }^2{\chi }_3^{\ast \ast }/\partial y^2}=\frac{\partial s/\partial Q}{\partial s/\partial y}+\frac{g^{\prime }(s)}{g^{″}(s)}$$ and since $g^{\prime }(s)$ is not zero, the restrictions of Proposition 2 do not hold for this collective model.

By way of comparison, when $a$ is the only contributor to the public good, the $z$-conditional demands arising from the noncooperative equilibria are given by $$\begin{array}{rcl}{\chi }_1^{\ast }(y,Q)& =& Q^2+\frac{1}{2}[-1+\sqrt{1+4(y-3Q-Q^2)}]\\ {\chi }_2^{\ast }(y,Q)& =& Q+{\left\{\frac{1}{2}[-1+\sqrt{1+4(y-3Q-Q^2)}]\right\}}^2\\ {\chi }_3^{\ast }(y,Q)& =& Q\end{array}$$

and the aggregate demand for the public good is $$Q^{\ast }(\rho )=\frac{1}{2}[-3+\sqrt{9+4\rho }].$$

Notice that $a$’s endowment $\rho$ can be recovered (up to a constant of integration) from the second derivatives of the $z$-conditional demands: inverting $a$’s Engel curve gives $\rho (Q)=Q^2+3Q$. But $$\frac{{\partial }^2{\chi }_i^{\ast }/\partial y\partial Q}{{\partial }^2{\chi }_i^{\ast }/\partial y^2}={\rho }^{\prime }(Q)$$ for any $i$ such that the ratio on the left-hand side is defined.

Under the collective model, the Pareto weight, say $\theta (y,z)$, is irrelevant for computing the extent to which the $z$-conditional demands violate the conditions of Proposition 2. Preferences alone determine the local properties of $z$-conditional demands, in both the noncooperative and the collective models. This is because, conditional on $y$, the set of equilibria in each model is one-dimensional. In collective models, the set of equilibria is parameterized by the Pareto weight $\theta$, and in the noncooperative model, it is parameterized by $a$’s endowment $\rho$. The $z$-conditional form of demands amounts to a change of variables, using the consumption of the public good rather than the unobservable $\theta$ or $\rho$ to parameterize the set of equilibria.

What is the empirical role of $\rho (y,z)$ then? It determines the strength of the “first stage”, where the distribution factors are used to predict the consumption of the public good. The properties of the first stage will also depend on preferences; in the noncooperative model, for example, the slope of the contributing spouse’s Engel curve for the public good will partially determine the strength of the distribution factors as instruments.

References

Chiappori, Pierre-Andre, and Jesse Naidoo. 2020. “The Engel Curves of Non-Cooperative Households.” The Economic Journal 130 (627): 653–74. https://doi.org/10.1093/ej/uez069.