Paternalistic preferences, interpersonal transfers and reciprocity.

• Author: Solow, John L.

1. Introduction

   Anthropologists since Bronislaw Malinowski [20] and Marcel Mauss [21] have observed that reciprocity is the primary defining characteristic of voluntary interpersonal transfers. While this perspective developed from observation of behaviors in exotic societies (e.g., the "kula ring" of Eastern New Guinea and the "potlatch" of the Kwakiutl tribe of the Pacific Northwest), contemporary observers find reciprocity to be an important feature of interpersonal transfers in modern societies(1) as well [7; 13; 14]. Sociologists such as Alvin Gouldner [17] and Claude Levi-Strauss [19] have given reciprocity the status of a social norm. Members of society are held to have a three-part obligation: to give gifts; to accept gifts; and to respond to a gift with a gift in return. In this literature, gift exchange serves to establish, perpetuate and define social relationships.

   Robert Sugden [25] suggests that social conventions and norms enable people to coordinate their behavior in the face of multiple Nash equilibria, where there is no uniquely rational choice and hence rationality alone is insufficient for decision-making. The purpose of this note is to demonstrate, by means of an example, how the norm of reciprocity can play this role in supporting transfers as the equilibrium of non-cooperative behavior. The example is based on the "paternalistic preferences" model that Robert Pollak [22] proposed as an extension of Gary Becker's [6] model of altruism in the family. Where Becker assumes that children's utility levels enter as arguments in their parents' utility function, Pollak assumes that in addition children's levels of consumption of certain goods directly affect parents' utility.(2) Thus, Becker assumes that parental utility has the form [U.sup.P] ([C.sup.P], [U.sup.i]([C.sup.i]), [U.sup.j]([C.sup.j])), where [U.sup.P], [U.sup.i], and [U.sub.j] denote the utility functions of the parent and children i and j respectively, and [C.sup.P], [C.sup.i] and [C.sup.j] denote their respective consumption vectors. Pollak assumes instead that parental utility is given by [U.sup.P] ([C.sup.P], [C.sup.i], [C.sup.j], [U.sup.i] ([C.sup.i]), [U.sup.j] ([C.sup.j])), where the derivatives of [U.sup.P] with respect to the various elements of [C.sup.i] and [C.sup.j] differ. This explains why intergenerational transfers within families are often tied to the children's consumption of particular goods or even take the form of in-kind transfers to the children.(3) For example, parents may be willing to pay for their children's college education but not to give their children the equivalent sum of money to spend as they please because parents value having college-educated children more than they value having children who have purchased, say, a new BMW.(4)

   The notion that the utility of one individual might depend on the quantities of goods consumed by other individuals has a long history; Harvey Leibenstein [18] traces it back to the 19th century. Melvin Reder [23, 64] suggested that such preferences could lead individuals to give gifts of the relevant goods to the appropriate people. It is easy to see how this might work in the intergenerational setting, since children typically have low incomes and have difficulty borrowing [1; 15].

   Things become more complicated with transfers between individuals with similar incomes. The problem is that recipients may desire to purchase some amount of the relevant good on their own, and since money is fungible, a gift that is less than what the recipient will desire to purchase simply frees up some money. This may lead to increased consumption of the relevant good via an income effect, but that will be less than the amount of the transfer, since the recipient will increase expenditures on other things as well. Hence, large transfers may be necessary in order to increase appreciably the recipient's consumption of the relevant good, and the benefit to the giver of doing so may not justify the cost in terms of foregone consumption. Another way to increase the recipient's consumption of the desired good is to make a transfer that exceeds the amount that recipient would buy, but this may also be too expensive to justify the benefits that the giver receives.

   Furthermore, the anticipated effect of a transfer on the recipient's consumption will depend on the giver's expectations about how the recipient will spend her own income, including importantly whether she is expected to be foregoing consumption in order to make a transfer herself. The opportunity cost of making a transfer will likewise depend on these expectations, since the consumption foregone will depend on the transfer received. Thus the question of whether transfers between people with similar incomes can be an equilibrium is not a trivial one.

2. An Example of Equilibrium Paternalistic Transfers

   Consider two individuals (H and W) who are faced with the problem of allocating their incomes ([I.sup.H] and [I.sup.W]) over the purchases of four goods, denoted A, B, X and Y. Assume that H and W have convex preferences that can be represented by utility functions of the following kind,(5)

   [U.sup.H] = [U.sup.H]([A.sup.H],[B.sup.H],[Y.sup.W]),

   [U.sup.W] = [U.sup.W]([X.sup.W], [Y.sup.W],[B.sup.H]), (1)

   where the superscripts on the goods denote who is doing the consuming. Thus H would receive no utility if he consumed Y himself, but he receives utility when W consumes Y.

   To examine the equilibrium level of transfers, I consider a two-stage game. In the first stage, in-kind transfers of the relevant goods are made, and in the second stage, H spends his remaining income on A and B so as to maximize his utility taking as given W's purchases of X and Y, and vice versa. As is common in this literature, I will assume that recipients of in-kind transfers are unable to resell them. Otherwise, in-kind transfers are equivalent to cash transfers...