Social interactions and spillovers.

*(English)*Zbl 1229.91248This paper analyzes a model where agents choose socialization and production efforts simultaneously. The two main ingredients of the model are as follows. First, the model has local complementarities in productive investment. More precisely, spillovers are generated by paired agents and are multiplicative in own’s and other’s productive effort. There are two different sources of heterogeneity. On the one hand, agents can differ in their marginal returns to own productive effort. On the other hand, for identical levels of productive efforts, spillovers can vary with the strength of the synergistic linkage across different pairs of agents.

Second, it is assumed that agents devote an amount of resources to building synergies with others. The resulting collection of socialization efforts determines the meeting possibilities across pairs of agents and results in a pattern of bilateral interactions. Therefore, socializing is not equivalent to elaborating a nominal list of intended relationships, as in the literature on network formation.

The authors characterize the equilibria of the induced game. They first show that there is one (partial) corner equilibrium where agents do not invest at all in building synergies (Lemma 2). Next, they show that there are two interior equilibria as long as 1) there is a sufficiently large number of agents, and 2) the level of cross synergies as well as the heterogeneity in individual traits are not too large (Theorem 1). When there is a sufficiently large number of agents, the (partial) corner equilibrium is unstable, whereas the two interior equilibria are stable (Proposition 1).

When the number of agents tends to infinity, the two interior equilibria can be approximated by solving a rather simple system of equations. Its two solutions are hence approximate equilibria that are convenient for analyzing the two types of equilibria. Proposition 2 shows that the (approximate) equilibrium actions are ranked component-wisely and the equilibrium payoffs are Pareto-ranked accordingly. The socially efficient outcome lies in between the two equilibria. Thus, there is a too-high and a too-low equilibrium.

Next, the authors carry out a comparative statics analysis. They consider a compound index that is the product of a parameter indicating the synergistic returns and a quotient that measures the population heterogeneity in private returns to productive investment. In Proposition 3 it is shown that if the compound index increases then in both approximate equilibria, the percentage change in socialization effort is higher than that of productive investment for all agents (both increase at the too-low equilibrium and both decrease at the too-high equilibrium).

The authors interpret their results in terms of education where a parent (agent) exerts two types of costly effort: productive effort with the child and socialization effort related to education (e.g., parental meetings). They find that: (i) High-educated parents exert more productive effort educating their children than low-educated parents. (ii) High-educated parents are more prone to socialize with other parents than low-educated parents. (iii) In different locales, children whose parents have similar characteristics or are similarly talented as other children can end up having different educational outcomes or different levels of parental educational efforts. (iv) Different levels of parental education affect proportionally more the socialization effort of those parents than their direct effort with children.

Finally, in Section 6, the authors demonstrate that the analysis is robust to some specifications that generalize the model.

Second, it is assumed that agents devote an amount of resources to building synergies with others. The resulting collection of socialization efforts determines the meeting possibilities across pairs of agents and results in a pattern of bilateral interactions. Therefore, socializing is not equivalent to elaborating a nominal list of intended relationships, as in the literature on network formation.

The authors characterize the equilibria of the induced game. They first show that there is one (partial) corner equilibrium where agents do not invest at all in building synergies (Lemma 2). Next, they show that there are two interior equilibria as long as 1) there is a sufficiently large number of agents, and 2) the level of cross synergies as well as the heterogeneity in individual traits are not too large (Theorem 1). When there is a sufficiently large number of agents, the (partial) corner equilibrium is unstable, whereas the two interior equilibria are stable (Proposition 1).

When the number of agents tends to infinity, the two interior equilibria can be approximated by solving a rather simple system of equations. Its two solutions are hence approximate equilibria that are convenient for analyzing the two types of equilibria. Proposition 2 shows that the (approximate) equilibrium actions are ranked component-wisely and the equilibrium payoffs are Pareto-ranked accordingly. The socially efficient outcome lies in between the two equilibria. Thus, there is a too-high and a too-low equilibrium.

Next, the authors carry out a comparative statics analysis. They consider a compound index that is the product of a parameter indicating the synergistic returns and a quotient that measures the population heterogeneity in private returns to productive investment. In Proposition 3 it is shown that if the compound index increases then in both approximate equilibria, the percentage change in socialization effort is higher than that of productive investment for all agents (both increase at the too-low equilibrium and both decrease at the too-high equilibrium).

The authors interpret their results in terms of education where a parent (agent) exerts two types of costly effort: productive effort with the child and socialization effort related to education (e.g., parental meetings). They find that: (i) High-educated parents exert more productive effort educating their children than low-educated parents. (ii) High-educated parents are more prone to socialize with other parents than low-educated parents. (iii) In different locales, children whose parents have similar characteristics or are similarly talented as other children can end up having different educational outcomes or different levels of parental educational efforts. (iv) Different levels of parental education affect proportionally more the socialization effort of those parents than their direct effort with children.

Finally, in Section 6, the authors demonstrate that the analysis is robust to some specifications that generalize the model.

Reviewer: Flip Klijn (Barcelona)

##### MSC:

91B69 | Heterogeneous agent models |

91D30 | Social networks; opinion dynamics |

91B15 | Welfare economics |

91A80 | Applications of game theory |

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\textit{A. Cabrales} et al., Games Econ. Behav. 72, No. 2, 339--360 (2011; Zbl 1229.91248)

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