Abstract
We consider a two-stage voluntary provision model where individuals in a family contribute to a pure public good and/or a household public good, and the parent makes private transfers to her own child. We show not only that Warr’s neutrality holds, regardless of the different timings of parent-to-child transfers, but also that there is a continuum of Nash equilibria which individuals’ contributions and parental transfers are indeterminate, although the allocation of each’s private consumption and total public good provision is uniquely determined. Furthermore, impure altruism or productivity difference in supplying public goods may not break our results above.
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Notes
Lundberg and Pollak (1993) consider a threat point to be a “separate sphere” contribution equilibrium where socially prescribed gender roles assign the primary responsibility for certain public goods to the wife and others to the husband, while Konrad and Lommerud (2000) and Chen and Woolley (2001) define a threat point as the utility levels obtained from the spouses’ non-cooperative decisions of human capital investments and contributions to a household public good prior to the bargaining, respectively.
Although the form of the utility function given by Eq. 1 has been commonly used in the literature on family economics, it is easy to show that the results obtained in the present paper remain valid in the more general utility function such as \( U_{p}^{i}(c_{p}^{i},G,u_{k}^{i}(c_{k}^{i},G))\).
Since it can be confirmed that child’s payoff function \( U_{k}^{i}\left(g_{p}^{1},g_{k}^{1},g_{p}^{2},g_{k}^{2}\right)\) in its own contribution is concave (i.e., its second derivative is equal to zero), the second-order condition is certainly satisfied. Moreover, it is well documented that when the strategy set of each player is compact and convex and when each player’s payoff function is quasi-concave in its own strategy, there exists a Nash equilibrium in this game (see, e.g., Glicksberg 1952).
When the child has another choice variable such as a labor supply (i.e., “a lazy kid” in the sense of Bergstrom 1989), the optimizing problem of the child becomes
$$ \underset{\left\{ g_{k}^{i}\right\} }{\max } U_{k}^{i}=u_{k}^{i}\left(y_{k}^{i}\left(a^{i}\right)+\hat{\pi} ^{i}\left(g_{k}^{i},g_{k}^{j},a^{i}\right)-g_{k}^{i},\hat{G},a^{i}\right), $$where \(y_{k}^{i}(a^{i})\) represents the labor income earned by the child which depends positively on the child’s labor supply a i, and is assumed to be a C 2-class function. The optimal transfer function \(\hat{\pi} ^{i}(g_{k}^{i},g_{k}^{j},a^{i})\) has been obtained by solving the first-order condition for the parent’s optimizing problem given by Eqs. 6 and 7, given that \(g_{k}^{i}\), \(g_{k}^{j}\), and a i have been determined at stage 1. Assuming Eqs. 10 and 11, we can show that neutrality as well as indeterminacy hold true in this model as well.
When all individuals make positive contributions to the public good but the parents stop the transfers to their children, the resulting case precisely corresponds to the standard Nash provision game (see, e.g., Bergstrom et al. 1986).
As evident in Definition 1, the concept of linked individuals is not restricted in the case involving a single public good and two families, each consisting of a single parent and a single child. It is straightforward to demonstrate the validity of partial neutrality in the context of many public goods, many families and many children.
The concept of “double link” can more carefully be defined using the terminology of graph theory or the analysis of networks as follows (see, e.g., Jackson 2008). Since all nodes including the node labeled G in Figs. 1, 2, 3, and 4 are tied by several links, all graphs or networks depicted in Figs. 1, 2, 3, and 4 are termed as “the network or graph is connected” since every two nodes in the network are connected by some path (i.e., a sequence of links) in the network. Hence, if the network is connected, neutrality emerges as shown in Proposition 1. Moreover, if one of those links is to be dropped in Figs. 2, 3, and 4, then the network becomes “disconnected”, then the original network is called a “tree”; alternatively, if a connected network consisting of n nodes has n − 1 links, it is a tree. Therefore, if the network is a tree, these links conform “the minimum set of links” (which is called as “minimality of a network” in the analysis of networks), indeterminacy never arises. Thus the concept of “double link” means that if there are more links than the number of the minimum set of links, there exist extra links at some node and we call them “double links”.
We can address the efficiency problem. To do this, combining Eqs. 6 and 7, and rearranging yields
$$ \frac{\partial u_{p}^{i}(c_{p}^{i},G)/\partial G}{\partial u_{p}^{i}(c_{p}^{i},G)/\partial c_{p}^{i}}+\frac{\partial u_{k}^{i}(c_{k}^{i},G)/\partial G}{\partial u_{k}^{i}(c_{k}^{i},G)/\partial c_{k}^{i}}=1\text{, }i=1,2. $$This implies that the Samuelson’s efficiency condition for provision holds within a family. This is because by making use of the ex-post transfers, the parent of each family can internalize the externality of the public good so that the Rotten-kid theorem (see, e.g., Bruce and Waldman 1990) in terms of private provision of public goods holds true. However, it should be noted that it is not fully Pareto efficient in the whole society.
Note that the two expressions in Eq. 28 simplify to \(\partial \hat{g}_{p}^{1}/\partial g_{p}^{2}=\partial \hat{g} _{p}^{1}/\partial \pi ^{2}\).
In the presence of impure altruism, moreover, there are several other possibilities which undermine the neutrality property. In addition to impure altruism, for example, either when imperfect substitutability (or cost/productivity differentials) across individual contributions to interfamily public goods as in Ihori (1996) or imperfect information regarding other individuals’ preferences introduced by Abel and Bernheim (1991) emerges, neutrality fails. This non-neutrality result once again arises from the fact that the number of links is less than the minimum number of links which connect individuals when a pure income redistribution policy is carried out among them.
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Acknowledgments
Earlier versions of this paper have been presented at the seminars of Hokkaido University, Chukyo University, and Kobe University. We have greatly benefited from helpful comments and suggestions from Professors Midori Hirokawa, Kimiyoshi Kamada, Takashi Sato, Hideki Konishi, Jun Iritani, Eiichi Miyagawa, and other seminar participants. We also thank the editor, Alessandro Cigno, and two anonymous referees of this journal for their helpful comments. All remaining errors are our own responsibility.
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Cornes, R., Itaya, Ji. & Tanaka, A. Private provision of public goods between families. J Popul Econ 25, 1451–1480 (2012). https://doi.org/10.1007/s00148-011-0388-2
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DOI: https://doi.org/10.1007/s00148-011-0388-2