Private versus public consumption within groups: testing the nature of goods from aggregate data

Abstract

We study the testability implications of public versus private consumption in collective models of group consumption. The distinguishing feature of our approach is that we start from a revealed preference characterization of collectively rational behavior. Remarkably, we find that assumptions regarding the public or private nature of specific goods do have testability implications, even if one only observes the aggregate group consumption. In fact, these testability implications apply as soon as the analysis includes three goods and four observations. This stands in sharp contrast with existing results that start from a differential characterization of collectively rational behavior.

Appendix

1.1 Example 1

There exists a general-CR of S. Consider the following personalized quantities and prices:

$$\begin{aligned} \widehat{\mathbf{q }}_{1}&= (\mathbf q _{1}, \mathbf 0 , \mathbf 0 ), \widehat{ \mathbf q }_{2}=\left(\frac{1}{2} \mathbf q _{2}, \frac{1}{2}\mathbf q _{2}, \mathbf 0 \right), \widehat{\mathbf{q }}_{3}=\left(\mathbf 0 , \mathbf q _{3}, \mathbf 0 \right); \\ \mathfrak p _{t}^{1}&= \mathbf p _{1}, \mathfrak p _{t}^{2}=\mathbf 0 \quad \text{ for}\; t=1,2,3. \end{aligned}$$

Then one can easily verify that the GARP conditions in Proposition 1 are satisfied for both members. This implies that there exists a general-CR of $S$.

There exists an egoistic-CR of S. By Proposition 3 we can conclude that the above construction also shows that there exists an egoistic-CR of $S$.

There does not exist a public-CR of S. Let us prove this ad absurdum and assume that we have a construction of feasible prices that satisfy condition (ii) in Proposition 2.

Observe that for the given set of observations we have for any $t,s \in \{1,2,3\}$, with $t\not = s$, that $\mathbf p _{t}\mathbf q _{t} > \mathbf p _{t} \mathbf q _{s}$. Therefore, we must have for our solution of feasible prices that either $\mathfrak p ^{h}_{t}\mathbf q _{t} > \mathfrak p ^{h}_{t} \mathbf q _{s}$ or $(\mathbf p _{t}-\mathfrak p ^{h}_{t})\mathbf q _{t} > ( \mathbf p _{t}-\mathfrak p ^{h}_{t})\mathbf q _{s}$. As a result, the GARP conditions in Proposition 2 require that if $\mathfrak p ^{h}_{t}\mathbf q _{t} \ge \mathfrak p ^{h}_{t}\mathbf q _{s}$, we must have that $\mathfrak p ^{h}_{s}\mathbf q _{s} \le \mathfrak p ^{h}_{s} \mathbf q _{t}$ and thus $(\mathbf p _{s}-\mathfrak p _{s}^{h})\mathbf q _{s} > (\mathbf p _{s}-\mathfrak p _{s}^{h})\mathbf q _{t}$. Or, alternatively, if $\widehat{\mathbf{q }}_{t} R^{1}_{0} \widehat{\mathbf{q }} _{s}$, then we must have $\widehat{\mathbf{q }}_{s} R^{2}_{0} \widehat{ \mathbf q }_{t}$. Given that this holds for any $t,s \in \{1,2,3\}$, with $t\not = s$, we may therefore conclude that, without losing generality, the solution of feasible prices leads to (i) $\widehat{\mathbf{q }}_{1} R^{1}_{0} \widehat{\mathbf{q }}_{2}$ and $\widehat{\mathbf{q }}_{2} R^{1}_{0} \widehat{ \mathbf q }_{3}$ for member 1 and (ii) $\widehat{\mathbf{q }}_{3} R^{2}_{0} \widehat{\mathbf{q }}_{2}$ and $\widehat{\mathbf{q }}_{2} R^{2}_{0} \widehat{ \mathbf q }_{1}$ for member 2.

Assume that $\mathfrak p ^{h}_{2}=(\pi _{1},\pi _{2},\pi _{3})^{\prime }$. The GARP condition for member 1 in Proposition 2 requires that

$$\begin{aligned} \mathfrak p ^{h}_{2}\mathbf q _{2}\le \mathfrak p ^{h}_{2} \mathbf q _{1}\;\Leftrightarrow\;2\pi _{1}+5\pi _{2}+2\pi _{3} \le 5\pi _{1}+2\pi _{2}+2\pi _{3}\\\;\Leftrightarrow\;0\le \pi _{1}-\pi _{2}. \end{aligned}$$

The GARP condition for member 2 in Proposition 2 requires that

$$\begin{aligned} (\mathbf p _{2}-\mathfrak p _{2}^{h})\mathbf q _{2}\le (\mathbf p _{2}- \mathfrak p _{2}^{h})\mathbf q _{3}\;\Leftrightarrow\;2(1-\pi _{1})+5(4-\pi _{2})+2(1-\pi _{3}) \\&\le 2(1-\pi _{1})+2(4-\pi _{2})+5(1-\pi _{3}) \\\;\Leftrightarrow\;3\le \pi _{2}-\pi _{3}. \end{aligned}$$

Together this implies that $3\le \pi _{2}\le \pi _{1}$, which gives us the wanted contradiction since by construction $\pi _{1}\le 1$. We thus conclude that there cannot exists a public-CR of the data set in Example 1.

1.2 Example 2

There exists a general-CR of S. Consider the following personalized quantities and prices:

$$\begin{aligned}&\widehat{\mathbf{q }}_{1}=( \mathbf 0 , \mathbf 0 , \mathbf q _{1}),\quad \widehat{\mathbf{q }}_{2}=(\mathbf 0 , \mathbf 0 , \mathbf q _{2}),\quad \widehat{ \mathbf q }_{3}=(\mathbf 0 , \mathbf 0 , \mathbf q _{3}),\quad \widehat{\mathbf{q }}_{4}=(\mathbf 0 , \mathbf 0 , \mathbf q _{4}); \\&\mathfrak p ^{h}_{1}=(6, 2 , 2, 2)^{\prime }, \mathfrak p ^{h}_{2}=(4,3.5,0,0)^{\prime },\quad \mathfrak p ^{h}_{3}=(4,4,3.5,0)^{\prime },\quad \mathfrak p ^{h}_{4}=(2,2,2,1)^{\prime }. \end{aligned}$$

Then one can easily verify that the GARP conditions in Proposition 1 are satisfied for both members. This implies that there exists a general-CR of $S$.

There exists a public-CR of S. By Proposition 2 we can conclude that the above construction also shows that there exists a public-CR of $S$.

There does not exist an egoistic-CR of S. Let us prove this ad absurdum and assume that we have a construction of feasible prices that satisfy condition (ii) in Proposition 3.

Again we observe that for the given set of observations, we have for any $t,s \in \{1,2,3,4\}$, with $t\not = s$, that $\mathbf p _{t}\mathbf q _{t} > \mathbf p _{t} \mathbf q _{s}$. Therefore, without losing generality, we can as before assume that the solution of feasible prices leads to (i) $\widehat{ \mathbf q }_{1} R^{1}_{0} \widehat{\mathbf{q }}_{2}, \widehat{\mathbf{q }}_{2} R^{1}_{0} \widehat{\mathbf{q }}_{3}$ and $\widehat{\mathbf{q }}_{3} R^{1}_{0} \widehat{\mathbf{q }}_{4}$ for member 1; and (ii) $\widehat{\mathbf{q }}_{4} R^{2}_{0} \widehat{\mathbf{q }}_{3}, \widehat{\mathbf{q }}_{3} R^{1}_{0} \widehat{\mathbf{q }}_{2}$ and $\widehat{\mathbf{q }}_{2} R^{2}_{0} \widehat{\mathbf{q }}_{1}$ for member 2.

Assume that $\mathfrak q ^{1}_{2}=(0,\alpha ,0, 0)$ and $\mathfrak q ^{1}_{3}=(0, 0, \beta , 0)$. The GARP conditions for the two members in Proposition 3 require that the following holds:

$$\begin{aligned} \widehat{\mathbf{p }}^{1}_{2}\widehat{\mathbf{q }}_{2} \le \widehat{\mathbf{p }}^{1}_{2}\widehat{\mathbf{q }}_{1}\;\Leftrightarrow\;7\alpha \le 4; \\ \widehat{\mathbf{p }}^{1}_{3}\widehat{\mathbf{q }}_{3} \le \widehat{\mathbf{p }}^{1}_{3} \widehat{\mathbf{q }}_{2}\;\Leftrightarrow\;7\beta \le 4\alpha \le 4; \\ \widehat{\mathbf{p }}^{2}_{2}\widehat{\mathbf{q }}_{2} \le \widehat{\mathbf{p }}^{2}_{2}\widehat{\mathbf{q }}_{3}\;\Leftrightarrow\;7(1-\alpha )\le 4(1-\beta )\le 4; \\ \widehat{\mathbf{p }}^{2}_{3}\widehat{\mathbf{q }}_{3} \le \widehat{\mathbf{p }}^{2}_{3}\widehat{\mathbf{q }}_{4}\;\Leftrightarrow\;7(1-\beta )\le 4. \end{aligned}$$

This implies that $\frac{3}{7}\le \alpha \le \frac{4}{7}$, $\frac{3}{7} \le \beta \le \frac{4}{7}$ and $\frac{7\beta }{4} \le \alpha$ and thus also that $\alpha \ge \frac{3}{4}$. As such, we obtain the wanted contradiction, and we conclude that there cannot exist an egoistic-CR of the data set in Example 2.

1.3 Proof of Proposition 4

Example 1 of Cherchye et al. (2007) shows that there cannot exist a general-CR of $S$ if we observe that $\mathbf p _{1}\mathbf q _{1}\ge \mathbf p _{1}(\mathbf q _{2}+\mathbf q _{3})$, $\mathbf p _{2}\mathbf q _{2}\ge \mathbf p _{2}(\mathbf q _{1}+\mathbf q _{3})$ and $\mathbf p _{3} \mathbf q _{3}\ge \mathbf p _{3}(\mathbf q _{1}+\mathbf q _{2})$ holds simultaneously. Without losing generality, we assume that $\mathbf p _{2} \mathbf q _{2}< \mathbf p _{2}(\mathbf q _{1}+\mathbf q _{3})$.

Consider the following personalized quantities and prices for an $\alpha \in [0,1]$:

$$\begin{aligned} \widehat{\mathbf{q }}_{1}&= (\mathbf q _{1}, \mathbf 0 , \mathbf 0 ), \widehat{ \mathbf q }_{2}=(\alpha \mathbf q _{2}, (1-\alpha )\mathbf q _{2}, \mathbf 0 ), \widehat{\mathbf{q }}_{3}=(\mathbf 0 , \mathbf q _{3}, \mathbf 0 ); \\ \mathfrak p _{t}^{1}&= \mathbf p _{1}, \mathfrak p _{t}^{2}=\mathbf 0 \quad \text{ for} \; t=1,2,3. \end{aligned}$$

These feasible prices and quantities are consistent with the collective model with only private goods (i.e., $\mathfrak q _{t}^{h}=\mathbf 0$) and egoistic preferences (i.e., $\mathfrak p _{t}^{1}=\mathbf p _{t}$ and $\mathfrak p _{t}^{2}=\mathbf 0$).

Given that $\mathbf p _{2}\mathbf q _{2}< \mathbf p _{2}(\mathbf q _{1}+ \mathbf q _{3})$, there must exist an $\alpha \in [0,1]$ such that $\alpha \mathbf p _{2}\mathbf q _{2}< \mathbf p _{2}\mathbf q _{1}$ and $(1-\alpha ) \mathbf p _{2}\mathbf q _{2}< \mathbf p _{2}\mathbf q _{3}$. One can then easily verify that for such an $\alpha$ the GARP conditions in Proposition 3 are satisfied for both members. This implies that there exists an egoistic-CR of $S$.$\square$