Paternalistic preferences, interpersonal transfers and reciprocity.

I. 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.

II. 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, which again only alter consumption via income effects. I consider subgame-perfect Nash equilibria, which in the present instance means that each individual chooses how much to give in the first stage on the basis of what both they and the other person will choose optimally to consume in the second stage. This implies that the model must be solved "backwards"; first, find the optimal consumption choices as functions of the gifts given and received, and then find each person's optimal gift to give, taking the gift received as given, conditional on those functions.

Let [B.sup.T] and [Y.sup.T] denote the transfers of B received and Y given by H in the first stage, and [P.sub.A], [P.sub.B], and [P.sub.Y] denote the prices of A, B, and Y respectively. Then H's second-stage optimization problem is;

[Mathematical Expression Omitted]

subject to:

[P.sub.A][A.sup.H] + [P.sub.B]([B.sup.H] - [B.sup.T]) [is less than or equal to] [I.sup.H] - [P.sub.Y][Y.sup.T]

[B.sup.H] [is greater than or equal to] [B.sup.T]

taking [Y.sup.W], [B.sup.T], and [Y.sup.T] as given. The second constraint arises from the no-resale assumption, and may hold as a strict inequality, particularly if [B.sub.T] is small. W's second-stage problem is symmetric.

Solutions of the second-stage problems yields optimal consumption patterns conditional on the levels of transfers, denoted(6) [A.sup.H*]([Y.sup.T], [B.sup.T]), [B.sup.H*]([Y.sup.T], [B.sup.T]), [X.sup.W*]([Y.sup.T], [B.sup.T]) and [Y.sup.W*]([Y.sup.T], [B.sup.T]). Then, in the first stage, H's problem is to

[Mathematical Expression Omitted]

taking [B.sup.T] as given; W's problem is again symmetric. A Nash equilibrium occurs when H's choice of [Y.sup.T] is an optimal response to W's choice of [B.sup.T] and vice versa.

To make the example concrete, consider the case of log-linear (Cobb-Douglas) utility functions;

[U.sup.H] = [U.sup.H]([A.sup.H], [B.sup.H], [Y.sup.W]) = [[Alpha].sub.1]ln[A.sup.H] + [[Alpha].sub.2]ln[B.sup.H] + [[Alpha].sub.3]ln[Y.sup.W],

[U.sup.W] = [U.sup.W]([X.sup.W], [Y.sup.W], [B.sup.H]) = [[Beta].sub.1]ln[X.sup.W] + [[Beta].sub.2]ln[Y.sup.W] + [[Beta].sub.3]ln[B.sup.H].

Under this assumption, it is straightforward to show that in the second stage, H's optimal consumption choices are given by:

[A.sup.H*] = [([I.sup.H] - [P.sub.Y][Y.sup.T] + [P.sub.B][B.sup.T])/[P.sub.A]][[[Alpha].sub.1]/([[Alpha].sub.1] + [[Alpha].sub.2])],

[B.sup.H*] = [([I.sup.H] - [P.sub.Y][Y.sup.T] + [P.sub.B][B.sup.T])/[P.sub.B]][[[Alpha].sub.2]/([[Alpha].sub.1] + [[Alpha].sub.2])],

if [([I.sup.H] - [P.sub.Y][Y.sup.T] + [P.sub.B][B.sup.T])/[P.sub.B]][[[Alpha].sub.2]/([[Alpha].sub.1] + [[Alpha].sub.2])] [is greater than or equal to] [B.sup.T], and are given by:

[A.sup.H*] = ([I.sup.H] - [P.sub.Y][Y.sup.T])/[P.sub.A], [B.sup.H*], = [B.sup.T],

if not. Similar conditions hold for [X.sup.W*] and [Y.sup.W*].

Given the nature of the demand functions, it is difficult to calculate analytical expressions for the reaction functions [Y.sup.T]([B.sup.T]) and [B.sup.T]([Y.sup.T]). It is easy, however, to compute them numerically. The reaction function of each individual depends on both persons' incomes and on the parameters of both persons' utility functions, and as a result, the model is capable of producing a variety of equilibrium outcomes. The results that emerge are quite reasonable. For example, if the paternalistic aspect of preferences is relatively weak (i.e., [[Alpha].sub.3] and [[Beta].sub.3] are sufficiently small relative to [[Alpha].sub.1] + [[Alpha].sub.2] and [[Beta].sub.1] + [[Beta].sub.2] respectively), then each person's optimal response to any gift received is to give no gift. In that event, no transfers will occur in equilibrium. Likewise, if one person has more strongly paternalistic preferences than the other, or if they both have strongly paternalistic preferences but have incomes that differ greatly, the unique equilibrium transfer is unilateral; in the first case, from the more paternalistic person, in the second from the wealthier person, but not vice versa.

Since my interest here is in the notion of reciprocity, I will consider a case in which bilateral (indeed, symmetric) transfers are an equilibrium.(7) Figure 1 shows the reaction functions that arise when [[Alpha].sub.1] = [[Beta].sub.1] = 0.4, [[Alpha].sub.2] = [[Beta].sub.2] = 0.2 and [[Alpha].sub.3] = [[Beta].sub.3] = 0.4, and [I.sup.H] = [I.sup.W] = 6.(8) The kinked and discontinuous nature of the reaction functions deserves some discussion. Recall that H's transfer can alter W's consumption of Y in two ways; small transfers lead to increased consumption via the income effect by displacing her own purchases of Y, and large transfers have more direct effects by making the no-resale constraint bind. At low levels of [B.sup.T], H finds it optimal not to give a gift; the cost in terms of goods foregone for either a small gift or a large gift is greater than the benefit derived from W's additional consumption of Y. At intermediate levels of [B.sup.T], it becomes optimal to give a small gift; W is reducing consumption of Y (as well as X) in order to give [B.sup.T], and H finds it worthwhile to give some Y in return to increase W's consumption of Y through the income effect. The cost of giving enough Y to make the no-resale constraint bind is still too high to justify the benefits. At high enough levels of [B.sup.T], however, it becomes optimal for H to make a transfer so big that the no-resale constraint binds. This occurs for two reasons. First, when W is giving a large gift, she is consuming relatively little Y, which reduces the size of the gift necessary to make the constraint bind. Second, when H is receiving a large gift, the goods foregone in order to give the gift have low marginal utility relative to that of W's consumption of Y.

There are two Nash equilibria in Figure 1, one involving no giving and one involving positive levels of giving, where the reaction functions intersect at [B.sup.T] = 3 and [Y.sup.T] = 3. The intuition behind this result is straightforward. If H believes that he will receive no gift, then the cost of giving a gift outweighs the benefits he expects to receive. The consumption he would have to give up to afford a large gift is too great, since W is expected to spend all her own income herself and it would take a lot to alter her consumption via the no-resale constraint. Trying to alter her consumption via the income effect from a small gift is expensive, too, because W spends only [[Beta].sub.2]/([[Beta].sub.1] + [[Beta].sub.2]) = 1/3 of each additional dollar on Y. Thus H will find it optimal to give no gift. The same holds for W, and so these expectations are self-fulfilling. On the other hand, if H believes that he will receive a large gift of B, he will find it optimal to give a large gift of Y. Since he expects W to reduce her consumption in order to give him a large gift, it won't take as much to move her past the no-resale constraint, and since he is expecting a large gift himself, the utility of the consumption foregone should be small relative to the benefit derived from W's additional consumption of Y. Again, the same holds for W, and so these beliefs are also self-fulfilling.

The no-giving equilibrium is inefficient; a reallocation of H's and W's consumption bundles that marginally increases both H's consumption of B and W's consumption of Y (and reduces H's consumption of A and W's consumption of X so as not to violate their respective budget constraints) will increase the utility of each. To see this intuitively, suppose we start at the no-giving equilibrium and let a hypothetical benevolent central planner reallocate H's purchases so that he spends one more dollar on A and one less dollar on B. Since H had previously chosen his consumption bundle optimally, this change has no impact on his utility; he has already equated the marginal utility per dollar of expenditure across the two goods. But the reallocation increases W's utility, since her marginal utility of H's consumption of B is positive. A similar reallocation of W's purchases so that she spends one more dollar on Y and one less dollar on X similarly increases H's utility at no loss in utility to W.(9) Note that the central planner need not make any reallocations of goods between the two individuals; all he need do is reallocate each one's own consumption, and both will be better off as a result. But since each is better off as a result of the reallocation of the other's consumption, neither has a private incentive to reallocate their own purchases.

The giving equilibrium involves exactly this sort of reallocation; H consumes more B and less A than he otherwise would, and similarly for W. Of course, the changes involved are not strictly marginal reallocations, and for non-marginal reallocations there is a net reduction in utility that needs to be offset by the gain in utility from the reallocation of the other's consumption. In the example depicted in Figure 1, the giving equilibrium Pareto-dominates the no-giving equilibrium. It remains possible that excessively large transfers in the gift-giving equilibrium could make the no-giving equilibrium yield higher welfare. I have not been able to produce such a case as an example.(10)

III. Coordination, Welfare and Reciprocity

The transfer problem between people with similar incomes, then, is an example of the sort of coordination problem that Sugden [25] discusses. There is no uniquely rational action; what is a rational choice for one individual depends on what the other is expected to do, and either choice by the other is a rational response to something that the first individual might rationally do. In this setting, social conventions that prescribe behavior, not because it is rational but because that is how one ought to behave, can serve to coordinate behavior and select an equilibrium. The norm of reciprocity, as described in the sociological literature, achieves that result here. If both individuals expect that the other will obey the social obligation to give, they will respond in like fashion, thus leading to the preferable of the two equilibria. The social norm is both self-fulfilling and, in this case, beneficial.(11)

It is interesting to consider the functions of the three parts of the norm of reciprocity in this interpretation. The obligation to give is key here, while the obligation to accept is subsidiary and the obligation to reciprocate is redundant. If individuals expect that others will heed the obligation to give, it will be in their own best interest to reciprocate; no social obligation is required. The obligation to accept a gift has a role to play, too. It serves to enforce the no-resale constraint in situations where resale is economically feasible. This contrasts with the functions attributed to the various obligations in the sociological interpretation. There, the obligation to reciprocate is primary, and the obligation to give is superfluous. In the sociological model, individuals give gifts in order to accumulate reciprocal obligations from others, thereby establishing a relationship. Hence, if individuals expect that others with whom they desire a relationship will heed the obligation to repay gifts, it will be in their own interest to give them. Again, the obligation to accept gifts plays the subsidiary role of ensuring that gifts not be disposed of, which would defeat their presumed purpose.

IV. Some Cautious Concluding Remarks

The economist's instinctive reaction to social conventions and norms is to assert that upon closer inspection, they serve to enhance either private [5, 5, 14] or collective self-interest [2, 22]. Sugden [25] and Jon Elster [16] both argue persuasively that while this may be the case, it certainly need not be. In any event, I have followed such instincts (or, perhaps, fallen into that trap). Elster makes a stronger point about economic modeling;

It is only a slight exaggeration to say that any economist worth his salt could tell a story--produce a model, that is, resting on various simplifying assumptions--which proves the individual or collective benefits derived from the norm. The very ease with which such "just-so stories" can be told suggests that we should be skeptical about them [16, 113].

He goes on to argue that his skepticism about the economic benefits of the norm would be reduced if we could point to a mechanism linking those benefits to the emergence or perpetuation of the norm. In the present case, at least, there is a plausible feedback mechanism of this sort. Although the norm of reciprocity involves social interaction, the assumed setting is one governed by individualistic benefits, so that reinforcement through personal reward can serve to maintain the norm. Those who give, accept and reciprocate gifts are individually better off than those that do not.

Appendix

I show here that a reallocation of H's and W's consumption bundles that marginally increases both H's consumption of B and W's consumption of Y from the no-giving equilibrium quantities without violating their respective budget constraints leads to a strict Pareto improvement. To see this, first totally differentiate the utility functions in (1) to get;

[Mathematical Expression Omitted],

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] denotes [[Delta]U.sup.H]/[[Delta]A.sup.H], and so forth. Consider the effect of simultaneously increasing H's purchase of B and decreasing H's purchase of A in small amounts such that H remains on his budget constraint, and similarly increasing W's purchase of Y and decreasing W's purchase of X in small amounts such that W remains on her budget constraint. From the budget constraints, we know that;

d[A.sup.H] = -([P.sub.B]/[P.sub.A])d[B.sup.H],

d[X.sup.W] = -([P.sub.Y]/[P.sub.X])d[Y.sup.W], (A2)

and from the first-order conditions in the no-giving equilibrium, where each spends their entire income to maximize their own utility, we know that;

[Mathematical Expression Omitted],

[Mathematical Expression Omitted].

Substituting (A2) for d[A.sup.H] and d[X.sup.W] in (A1), and solving (A3) for [Mathematical Expression Omitted] and [Mathematical Expression Omitted] and substituting in (A1) yields;

[Mathematical Expression Omitted],

[Mathematical Expression Omitted],

which are both positive since marginal utilities are positive and we have increased both [B.sup.H] and [Y.sup.W].

1. While intergenerational transfers are often viewed as being unilateral (from grandparents and parents to children but not vice versa), the reciprocation may take less tangible forms [2; 8].

2. Note that this use of "paternalism" differs from its use in the discussion of governmental transfers. There, paternalism refers to the notion that someone (e.g., the government) understands what is beneficial to recipients (frequently, the poor) better than the recipients do [24, 94]. Hence, government transfers of "merit goods" are made for the recipients' own good. This also seems plausible in the family setting, particularly between parents and children.

3. The same argument was put forth by James Buchanan [12] and others to explain why it may be optimal for government transfers to be in kind, rather than in cash. Another explanation is that in-kind transfers deter the non-deserving from masquerading as deserving recipients, which is valuable if the government cannot easily distinguish the two [10]. This would seem to be less important in the family setting.

4. Neil Bruce and Michael Waldman [11] offer another explanation for conditional or in-kind intergenerational transfers. They point out that if parents leave bequests that equalize the well-being of their children, then children have an incentive to squander any inter vivos transfers; the cost of doing so is borne partly by their siblings. Thus, parents will prefer to make inter vivos transfers in the form of investments (e.g., college educations) that are less likely to be dissipated.

5. For simplicity, I have assumed away the purely altruistic interactions, so that neither individual derives utility from the utility of the other.

6. For simplicity, the price and income arguments of the second-stage demand functions are suppressed.

7. This example requires the a priori assumption of a relationship that involves mutually paternalistic preferences. While Becker [4; 6] argues that mutually altruistic relationships should arise endogenously, Bernheim and Oded Stark [9] cast doubt on this result. Hence, the example should be interpreted as describing paternalistic transfers conditional on the existence of a mutually paternalistic relationship.

8. For these calculations, all prices equal unity.

9. This is shown more formally in a brief Appendix.

10. It is possible to find parameter values for which the unique Nash equilibrium involves large gift exchanges, and for which both individuals would be better off if there were no gift exchange. This arises, for example, if [[Alpha].sub.1] = [[Beta].sub.1] = 0.1, [[Alpha].sub.2] = [[Beta].sub.2] = 0.1 and [[Alpha].sub.3] = [[Beta].sub.3] = 0.8, so that the external effects of consumption dominate strongly. In these cases, the gift-giving equilibrium is inefficient, but the no-gift point is not an equilibrium.

11. As Sugden points out, conventions are self-fulfilling, but not necessarily mutually beneficial. To see this, note that the convention that one is obliged not to give is also self-fulfilling. The norm of reciprocity is only one of many possible norms that could govern this situation; it happens to be the one that leads to a mutually beneficial outcome.

References

1. Altig, David and Steve Davis, "Government Debt, Redistributive Fiscal Policies, and the Interaction between Borrowing Constraints and Intergenerational Altruism." Journal of Monetary Economics, July 1989, 3-29.

2. Arrow, Kenneth. "Political and Economic Evaluation of Social Effects and Externalities," in Frontiers of Quantitative Economics, edited by Michael Intriligator. Amsterdam: North-Holland, 1971, pp. 3-35.

3. -----. "Gifts and Exchanges," in Altruism, Morality and Economic Theory, edited by Edmund Phelps. New York: Russell Sage, 1975, pp. 13-28.

4. Becker, Gary S., "A Theory of Marriage: Part II." Journal of Political Economy, March/April 1974, S11-S26.

5. -----. The Economic Approach to Human Behavior. Chicago: University of Chicago Press, 1976.

6. -----. A Treatise on the Family. Cambridge, Mass.: Harvard University Press, 1981.

7. Belk, Russell. "Gift Giving Behavior," in Research in Marketing, edited by Jagdish N. Sheth. Greenwich, Conn.: JAI Press, 1979, pp. 95-126.

8. Bernheim, B. Douglas, Andrei Shleifer, and Lawrence Summers, "The Strategic Bequest Motive." Journal of Political Economy, December 1985, 1045-76.

9. Bernheim, B. Douglas and Oded Stark, "Altruism Within the Family Reconsidered: Do Nice Guys Finish Last?" American Economic Review, December 1988, 1034-45.

10. Blackorby, Charles and David Donaldson, "Cash versus Kind, Self-Selection, and Efficient Transfers." American Economic Review, September 1988, 691-700.

11. Bruce, Neil and Michael Waldman, "Transfers in Kind: Why They Can Be Efficient and Nonpaternalistic." American Economic Review, December 1991, 1345-51.

12. Buchanan, James M., "What Kind of Redistribution Do We Want?" Economica, May 1968, 185-90.

13. Caplow, Thomas, "Christmas Gifts and Kin Networks." American Sociological Review, June 1982, 383-92.

14. Cheal, David J., "The Social Dimensions of Gift Behaviour." Journal of Social and Personal Relationships, December 1986, 423-39.

15. Cox, Donald. "Intergenerational Transfers and Liquidity Constraints." Quarterly Journal of Economics, February 1990, 187-217.

16. Elster, Jon, "Social Norms and Economic Theory." Journal of Economic Perspectives, Fall 1989, 99-117.

17. Gouldner, Alvin, "The Norm of Reciprocity." American Sociological Review, April 1960, 161-78.

18. Leibenstein, Harvey, "Bandwagon, Snob and Veblen Effects in the Theory of Consumers' Demand." Quarterly Journal of Economics, May 1950, 183-207.

19. Levi-Strauss, Claude. "The Principle of Reciprocity," in Sociological Theory, edited by Lewis A. Coser and Bernard Rosenberg. New York: Macmillan, 1965, pp. 84-94.

20. Malinowski, Bronislaw, "Kula: the Circulating Exchange of Valuables in the Archipelagoes of Eastern New Guinea." Man, July 1920, 97-105.

21. Mauss, Marcel. Essay on the Gift, translated (1954) by Ian Cunnison. London: Routledge, 1925.

22. Pollak, Robert A., "Tied Transfers and Paternalistic Preferences." American Economic Review, May 1988, 240-44.

23. Reder, Melvin. Studies in the Theory of Welfare Economics. New York: Columbia University Press, 1947.

24. Stiglitz, Joseph E. Economics of the Public Sector. New York: W.W. Norton & Co., 1986.

25. Sugden, Robert, "Spontaneous Order." Journal of Economic Perspectives, Fall 1989, 85-97.