altruism and time consistency: the economics of fait accompli

Altruism and Time Consistency: The Economics of Fait AccompliAuthor(s): Assar Lindbeck and Jörgen W. WeibullSource: The Journal of Political Economy, Vol. 96, No. 6 (Dec., 1988), pp. 1165-1182Published by: The University of Chicago PressStable URL: http://www.jstor.org/stable/1831946Accessed: 03/06/2010 05:38

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Altruism and Time Consistency: The Economics of Fait Accompli

Assar Lindbeck and Jdrgen W. Weibull Institute for International Economic Studies

This paper analyzes the strategic and intertemporal interaction between two economic agents who have "overlapping" concerns, such as altruistic concerns for each other's welfare. The agents may be two individuals, a social bureau and a client, or two units in an organization. We show how the presence of such common concerns may lead to socially inefficient outcomes, in which one economic agent "free- rides" on the other's concern. We also briefly discuss how this inefficiency and free-riding, in the context of interaction between individuals, might be mitigated by compulsory social security systems. As another example we interpret the inefficiency in terms of Kornai's "soft budget constraints" within organizations.

I. Introduction

The presence of altruism, in the sense of concern for others' welfare, may easily lead to socially inefficient outcomes in an intertemporal setting with strategic behavior, despite the fact that both the donor and the recipient are rational and well informed about each other's preferences and endowments, and all choices are voluntary.' The source of the inefficiency is the recipient's strategic incentive to

An earlier version of this paper was presented at the Institute for International Economic Studies, December 1986, and at the conference of the European Economic Association in Copenhagen, August 1987 (Lindbeck and Weibull 1987). We want to thank Wolfgang Leininger, Torsten Persson, Lars E. 0. Svensson, and an anonymous referee for helpful comments. The research was supported by the Bank of Sweden Tercentenary Foundation.

1'"True" altruism, which is the topic of this paper, should be distinguished from "cooperative egoism," i.e., help to others in the expectation of being helped back in the future (see, e.g., Hammond 1975; Kurz 1978; Wintrobe 1981; Axelrod 1984).

[Journal of Political Economy, 1988, vol. 96, no. 6] ( 1988 by The University of Chicago. All rights reserved. 0022-3808/88/9606-0010$01.50

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1 i66 JOURNAL OF POLITICAL ECONOMY

"squander" in an early period in order to subsequently receive more resources from the other agent, that is, to "free-ride" on his concern. A "threat" by a potential donor not to give additional support to an agent because he squanders is not credible if the recipient knows that, ex post, it will be in the donor's (altruistic) interest to give such additional support. A rational donor anticipates this possibility of "fait accompli" and may hence reserve some additional resources for this purpose.

The sense in which this outcome is inefficient is that both individuals' welfare could be improved if at the outset the donor could make a binding commitment to the given support. The reason why the recipient would gain by such an arrangement is that, although his total resources would not be enhanced by the commitment, he could then allocate them more efficiently over time. By doing so, he in- creases not only his own welfare but also that of the donor, via the donor's "true" altruism. In other words, there is a problem of time consistency: intertemporal equilibrium may be in conflict with (Pareto) optimality.2 The same problem may arise also for types of common concern other than altruism.

The general, multiperiod and multiperson problem of strategic interaction between altruistic economic agents is quite complex. It is possible, however, to highlight the basic incentive mechanisms already in a model with two periods and two agents, who may give transfers in the second period only. We therefore restrict the analysis to this simple case. The two agents may be individuals who have more or less "altruistic" preferences toward each other. Alternatively, they may be agents who have some other type of common concern, as in the case of a social bureau that cares about the well-being of a client or two units in an organization with overlapping objectives. For the sake of definiteness, however, we will develop the model mainly in terms of altruism between two individuals.

The main result in our study is that an intertemporal (subgame perfect) equilibrium is (Pareto) inefficient whenever a transfer is given and the recipient does some saving for himself. Moreover, the recipient then is a "free-rider" in the sense that his strategic behavior induces the donor to give larger support than he would have given if he had had the possibility to precommit himself to a support of his own choice.

An informal discussion of this type of incentive problem, in the

2 The concept of time consistency, as currently used in macroeconomics, can be interpreted in (at least) three different ways, depending on the stringency of the under- lying equilibrium concept. While the pioneering study by Kydland and Prescott (1977) requires only Nash equilibrium, later studies usually require (at least) subgame perfect equilibrium (see McTaggart and Salant 1986).

ALTRUISM AND TIME CONSISTENCY 1167

context of one altruistic and one selfish individual, has been pursued earlier by Buchanan (1975b).3 However, in contrast to his verbal discussion, the present paper provides a formal, game-theoretic analysis. In fact, this analysis shows that inefficiency can arise even when both individuals are equally altruistic toward each other, a case not consid- ered by Buchanan.

Our result is related to, but different from, Becker's (1974, 1976) "rotten kid theorem." Becker assumes that one agent (the "father") is altruistic, while another (the "rotten kid") is purely selfish, and that the former can transfer resources to the latter after the rotten kid has taken his actions, which affect family income. In this asymmetric setting, Becker argues that the rotten kid, despite his selfishness, will act in the interest of the whole family." The present analysis, by contrast, establishes inefficiency in an intertemporally symmetric setting with or without altruistic symmetry between the agents. Since our analysis does not require the kid to be rotten, only rational, one may baptize it the "smart kid theorem."

The paper is organized as follows. The basic assumptions of the model are given in Section II. Section III is restricted to the special case of Cobb-Douglas preferences, while Section IV provides a more rigorous and general game-theoretic setup. Questions of efficiency and free-riding are analyzed in Section V, while Section VI draws some general conclusions from the analysis and suggests some reme- dies to the inefficiency and free-rider problem under consideration. (Mathematical derivations and some extensions of the model are given in Lindbeck and Weibull [1987].)

II. The Model

Suppose that there are two individuals, 1 and 2, who both live in periods t = 1 and t = 2 and whose (total) welfare, U1 and U2, can be written in the following separable but nested form:

U, = u-(c1) + a, U1 for i = 1, 2,j1 i. (1)

Here ci = (C11, Cr2), where c1, is the consumption of individual i in period t. The "individualistic utility" from consumption, u1(c1), is assumed to be additively separable,

U1(CZ) = U11(C11) + 1112(C12), i = 1, 2, (2)

where the four functions ui have the usual mathematical properties

3Varian (1982) has discussed a simple numerical example along the same lines as Buchanan.

4 See Becker (1974, p. 1076). For further discussion, analysis, and criticism of his result, see Becker (1976), Hirshleifer (1977), Wintrobe (1981), and Bergstrom (1987a).

i i68 JOURNAL OF POLITICAL ECONOMY

of utility functions.5 Each parameter at, : 0 represents the altruistic concern of individual i for the welfare of the other.

Hence, individual i cares not only about j's individualistic utility from consumption (uj) but about the latter's total welfare (Uj), which also comprises j's altruistic concern for i. Thus there is an infinite sequence of mutual concerns. However, if a1a2 #& 1, then one may solve equation (1) for U1 and U2' hence obtaining (1 - I a2)Ut =

u-(c,) + atiuj(Cj). Thus equation (1) then in fact defines UC and U2 as real-valued functions of the consumption vector c = (cl, c2). To avoid "bizarre" behavior, we assume that

a1a2 < 1.6 (3)

Consequently, in the present model, the behavior of individuals who care about the total welfare of others, including also others' altruistic concerns, is, under condition (3), identical to the behavior of individuals who are concerned only about other citizens' individualistic utility. Since the positive and constant factor (1 - aX1aX2) is behaviorally irrelevant, we will subsequently represent the individual's preferences as utility functions U. defined over consumption allocations:

U.(c) = u-(c.) + atiuj(cj), (4)

where c = (C11, c12, c21, c22). Note that altruism, as represented in this equation, is a special case

of common concerns, in the sense that both individuals have preferences over the same vectors (in this case vectors c), and each of them associates positive marginal utilities with every component. It is this "overlap" of concern that is the basic reason for inefficiency.

As for the resource constraints of the two individuals, let wx > 0 be the initial wealth holding, or endowment, of individual i at the beginning of the first period.7 Abstracting from interest earnings, we take the resources available at the beginning of the second period, a-, to be the amount saved:

ai = wi - Ci I (5)

The strategic interaction between the two individuals is supposed to take place as follows. At the first stage both individuals independently choose their levels of first-period consumption and plan their second-

5 More precisely, each function u-,: R + R is continuous and twice continuously differentiable for positive arguments, with ui', > 0, u" < 0, and ui,(ci,) -m as i -* 0.

6 Drazen (1978, pp. 511-12) analyzes altruistic gifts between generations and uses a similar constraint on the degree of mutual altruism. If instead o1C2 > 1, then each individual i strives to minimize ui + otu_. Bergstrom (1987b) analyzes cardinal represen- tations of a more general class of intertwined altruistic preferences, including this bizarre case.

7 For an extension of the analysis to the case in which the first-period income is endogenous and depends on labor supply, see Lindbeck and Weibull (1987).

ALTRUISM AND TIME CONSISTENCY 1 169

period consumption and transfer, with each other's preferences and initial endowments as common knowledge. At the second stage, they observe the resulting "state" vector a = (a,, a2), and each individual i now decides how much, t,, of his current resources (ai) to give to the other (where 0 < t- S a-). The individuals also make this second decision independently of each other. Consumption in the second period is then

C,2 = a,+t-t, j i. (6)

III. Cobb-Douglas Preferences

Before we go to a more general and rigorous game-theoretic treat- ment, let us examine the special case in which the two individuals have identical Cobb-Douglas preferences for consumption: u,(c-) = log(c.i) + log(cI2).

Looking at the second stage first, we note that the pattern of transfers between the two individuals depends on their assets, a, and a2, at the beginning of that period. Before any transfer is made, each individual i derives marginal utility 1/ a, from his own assets and ai/aj from those of the other. Hence, if tIa I > a2, then individual 1 can increase his (total) welfare by giving a transfer to individual 2, and vice versa if cx2a2 > a,. The "state space" A can accordingly be partitioned into three regions, Al, A2, and AO, where AO is the residual set (see fig. 1).

a2

W2

A2

A A

WF

FIG. 1.-The subsets AO, AI, and A2 in the Cobb-Douglas case

1170 JOURNAL OF POLITICAL ECONOMY

If a = (a,, a2) lies in A, (i 7$ 0), then it turns out to be optimal for individual i to make such a transfer toj that their joint resources, a1 + a2, are split into shares 1/(1 + axi) and ot-/(I + (xi), the first for himself and the second for the other.8 Moreover, such an optimal transfer can be shown to constitute the unique Nash equilibrium in the second- period interaction, for any given state a in A (including zero transfers in AO). Hence, if the two individuals save a, and a2, respectively, for the second period and the corresponding equilibrium transfers are given in the second period, then the utility of individual i is

'r(aj, a2) = log(w, - a,) + aX log(wj - aj)

log< a + a2) + a, logy .<1a+ a2)1 for a E A,

+ log(a,) + ax log(aj) for a E AO (7)

log[ o(aI + ) + a, log(1 a, + 2) for a E A

for i = 1, 2 andj $ i; compare equations (5) and (6). If, in period 1, the two (rational) individuals anticipate the corre-

sponding equilibrium transfers in the second period, then each player i should choose an a- that is optimal against the other's choice. Hence such Nash equilibria (a,, a2) can be obtained as the intersections between the two individuals' "best-reply" correspondences. In fact, it can be shown that, in the present Cobb-Douglas setting, these correspondences always intersect at least once.9 Hence existence of (subgame perfect) equilibrium is guaranteed here.

A. Equal Endowments

To examine the nature of such equilibria, we will first consider the special case of one altruistic and one selfish individual with equal endowments: now assume w1 = = 1 and oa1 > 0 = a2. Then the set A2 is clearly empty: individual 2 never wants to give a transfer since he is strictly selfish. It is straightforward to show that then the best- reply correspondence of the altruistic individual is the graph of a continuous, nonincreasing, and piecewise linear function as in figure 2a. Since in the present Cobb-Douglas case his utility function for consumption is the same in each period and future consumption by

' This follows directly from first-order conditions, just as in intertemporal Cobb- Douglas allocation of resources al + a2 over two time periods, with a discount factor ai.

9 It turns out that each correspondence is the closure of the graph of a nonincreasing function that has at most one discontinuity.

a2

a1

A /" 0~

- A1 A

a 1 - rlo | ( a

-a" I\

-/ I \

- I \ a

o 1 IA-az

2+a1

a2

a1o

A0

.01~~~~~~ 2 i > 1 |(b)

Is ,I

A~~~~~~~~ / A I

0 al

FIG. 2.-The "best-reply" correspondences of individuals 1 (in a) and 2 (in b) in the case of equal endowments.


assumption is not discounted, it is evidently optimal for him to consume half his endowment in each period when no transfers are given. This is the explanation behind the vertical part (in the region AO) of his best-reply correspondence. Moreover, this individual knows that he will want to support individual 2 if the latter has little left in period 2, so in such states he saves more than what would be optimal from a purely selfish viewpoint-and more so the less individual 2 has in the second period. This is the rationale behind the sloping part of his best-reply correspondence (in Al).

The best-reply correspondence of the potential recipient is some- what more complex. It turns out that the value of this correspondence is a one-point set for all a, 7# al, where

1 1/2

al= (1+ ) -1. (8)

At the critical point a, = al, each of two distinct values of a2 is an optimal reply: individual 2 can then maximize his welfare either by saving half his endowment for period 2, a2 = 1/2, or by saving only a2 = (1 - dj)/2. If a1 < al, then it is best for him to save "prudently" (a2 = 1/2), while if a1 > a,, then a certain amount of "undersaving" is optimal, a2 = (1 - a1)/2. In other words, the more resources individual 2 expects 1 to save beyond 1/2, the less 2 should save, and the shift from the prudent saving regime to the undersaving regime takes the form of a "jump," as in figure 2b.

Note also that equation (8) confirms the intuition that the more altruistic individual 1 is, the wider is the range of values of a, at which undersaving is optimal. In particular, al > 1 for all (xl below 1/3; that is, when the altruism of the potential donor is sufficiently small, then the potential recipient does not have an incentive to undersave, irre- spective of his expectation concerning the altruistic individual's saving. The reason is, of course, that the anticipated transfer will then be too low to make any level of undersaving advantageous. This actually proves that a* = (1/2, 1/2) is the unique Nash equilibrium in G, for (xl < 1/3 (see fig. 3a).

Conversely, da < 1/2 for all (x1 above 4/5; then the potential recipient has an incentive to undersave whenever the potential donor saves at least half his endowment for the second period, which he always does. Hence, there exists a unique Nash equilibrium in G when (xl > 4/5, and a transfer is then given (see fig. 3b). In this equilibrium, the altruistic individual's expectation about the other's undersaving is fulfilled, as is the selfish individual's expectation about the support to be received. For some intermediary values of (x1 there are two equilibria: one prudent saving and one undersaving equilibrium.

a2

(a)

1 1+a1 1 2+a1

a2

2 __e _____________ (b)

a 1+ 1C 1

1+ 1

FIG. 3.-Nash equilibrium in C, for al < 1/3 and ot, > 4/5, respectively, in the case of equal endowments.


B. Equal Altruism

Let us turn to the polar case of equal altruism but differing endowments: ot, = CL2 = ot > 0 and w1 > W2 > 0. It turns out that if

OtLW I > W12, (9)

that is, if the difference in wealth is sufficiently large and the degree of altruism is sufficiently high, then a transfer is given in equilibrium, from the wealthier to the poorer individual. Moreover, for some combinations of parameters, the poorer individual will save some of his initial endowment, while for other combinations he will not save at all.

To see this, consider figure 4, which shows l's and 2's best-reply correspondences (drawn for the special case (XW2 < w1/2 < W2/kX)- In contrast to the case of only one altruistic individual, there now appears a region A2 in the state space A. It can be shown that, when (9) holds, then individual 2's switch from a prudent saving regime to an undersaving regime occurs at a value of a, that is lower than the individualistically optimal savings level, wI/2, of individual 1. Thus the two correspondences intersect only in A1. If, in addition to (9), we have (2 + 0)W2 > w1I, that is the difference in wealth is not too large, then there is a unique intersection, at a positive value of a2.

Suppose, in contrast, that the difference in wealth is so large that (2 + U)W2 - 1 w, but (9) still holds. As is seen in figure 4, the two correspondences then instead have a unique intersection on the a,-axis. In such an equilibrium, the wealthy individual's saving is as large as it can be in any equilibrium, and the less wealthy individual consumes all his initial endowment already in the first period, correctly anticipating the transfer he will receive in period 2.

IV. A More General and Rigorous Game-Theoretic Approach

In the two-stage game outlined in Section II, which we will denote G, a (pure) strategy for individual i is a pair s, = (ab, vi), where a, is his savings for the second period and v, a function that specifies the transfer t- to be given in every possible state a = (a,, a2): t a = ).'' The payoffs associated with any pair (sI, S2) of strategies for the two agents, T1I(SI, S2) and 'r2(S1, S2), are simply the corresponding (total) welfare levels defined in equation (4):

Tr1(SI, S2) = u[w, - a7, a, - v,(a) + Tj(a)] (10) + ot u,[w. - aya, + T,(a) - T(a)], i = 1, 2, j = i.

10 Formally, let A = [0, w1I] X [0, W2], and define the strategy set of individual i as S = {(ai, Ti); a, E [0, Wi], Ti: A -* R+ and Ti(a) S ai V a E A}.

a2

A A 0/

2 t A2 / A ? a / A

/ A

+~~~~ 1 2 + 1

2 - I."

/ A A

i2/ A AA a

// A

1 1+a)2 1

FIG. 4.-The "best-reply" correspondences of individuals I (in a) and 2 (in b) in the case of equal altruism.


The second stage of the game G, played after the state a has been revealed, constitutes a subgame, G(a), in which a (pure) strategy of individual i simply is his transfer t1. Hence, his pure strategy set in G(a) is the interval [0, aj, and his payoff function is

*1(tI, t2la) = u12(a- t, + tj) + xLuj2(aj + t - t) (11)

+ ui (w(- ai) + tuj I (wj - a), i = 1, 2,j $/ i.

Note that the last two terms are exogenous to the subgame and hence are strategically irrelevant in the subgame.

If a Nash equilibrium ((a,, TI), (a2, T2)) in the full two-stage game G satisfies the further requirement that, for every state a', the point (I (a'), T2(a')) be a Nash equilibrium in the corresponding subgame G(a'), then ((al, T1), (a2, T2)) is a subgame perfect equilibrium of G. This is the intert.mporal equilibrium concept that we used in the Cobb-Douglas example above and that we will subsequently use because it reflects the real-life difficulty of making commitments to future actions that will then not be optimal.

To get some handle on the set of subgame perfect equilibria, let us first examine the set of (pure) Nash equilibria in a subgame G(a), where a is a fixed but arbitrary point in A = [0, X 1I] X [0, W2]. The two (disjoint) sets AI and A2 are here defined by

A, = {a E A; a $ (0,0 ) arid u,'2(a) < 0x"u2(aJ)}, i = 1, 2,j $ i.

Let AO be the subset of points in A that belong to neither AI nor A2. Along with this partitioning of the state space A, a corresponding

pair of functions, Tl and T*, will be useful. Each such function v* defines, for every state a in A, the transfer that individual i would like to give toj, granted that gives nothing to i. Formally, T* (a) is defined to be zero for all a outside A1 and, for a in A?, as the (unique) solution of

u.2(ai- t U) - CtUj2(a- + ti). (12)

With the aid of these two "transfer functions" one can prove that there exists a unique Nash equilibrium in every subgame G(a) and that at most one of the individuals gives a transfer in equilibrium.

LEMMA. For every a E A, (T*i(a), T*-(a)) is the unique Nash equilibrium of G(a).

This result implies that the subgame perfect equilibria of the (infinite-dimensional) two-stage game G are in a one-to-one correspondence with the Nash equilibria of a (two-dimensional) one-shot game. For, by definition, any subgame perfect equilibrium in G induces a Nash equilibrium on every subgame G(a), so if ((al, T1), (a2, T2)) is such a subgame perfect equilibrium, then, by the lemma, the


functions Tv must be identical to the functions v*. Hence, just as in the Cobb-Douglas example above, one can model the full strategic interaction between the two agents in terms of a one-shot game, G, in which the strategies simply are the scalar variables a, and a2, and the payoff functions are defined by substitution of <* for vi in the payoff functions of the full two-stage game G:

-Ifr(a1, a2) = ui(w, - ai) + a- ujI (wj - aj)

Ui2-[a- T*(a)] + ci uj2[aj + T*(a)] for a E A-

+ u12(ai) + OL'Uj2(aj) for a E AO

tu,2[a- + Tj*(a)] + OUj2[aj - T* (a)] for a E A1. (This is a generalization of eq. [7].)

V. The "Smart Kid Theorem"

Now suppose that a* is a (pure-strategy) Nash equilibrium of G, in which individual i makes a transfer to j, that is, a* E Ai. Necessary first-order conditions then are

____(cl) 1 (14a)

uil(c71) |a X(14b)

uj2(c>) ____*

Uj'2 (Cj*2) ~~aT*

1 + (1 - O12)d if a* = 0. aa.

As can be expected, aT/laa, < 0; that is, the transfer is decreasing in the recipient's second-period resources. By hypothesis CLI L2 < 1, SO

condition (14b) implies that, whenever a* > 0, the recipient's marginal utility of first-period consumption is lower than his marginal utility of second-period consumption. From a nonstrategic viewpoint, this is clearly not optimal for individual j since in the absence of interest earnings, the return to first-period savings is one to one. In other words, whenever he does some saving in equilibrium, then the strategic motive induces him to undersave, not only in comparison with what would have been optimal in the absence of a transfer, but also as compared with what would have been optimal had the equilibrium transfer been exogenously fixed. The reason for the "distortionary wedge" in (14b) is the (rational) speculation in the first period for more support in the second: individuals knows that the transfer to be received is a decreasing function of his own savings. Hence, such an


"interior" equilibrium (i.e., in which the recipient does some saving himself) is Pareto inefficient. For if the donor would be able to commit himself to the equilibrium transfer, then his individualistic utility would be unchanged, whereas the recipient could increase his individualistic utility by consuming less in the first period and more in the second.' 1 This way, both individuals' total welfare could be increased via the donor's altruistic concern.

The corner solution a* = 0 is slightly more complicated because ifj has a sufficiently small initial endowment and/or i is sufficiently altruistic, then it may indeed be in both individuals' interest for j to consume his whole wealth already in period 1. It turns out that an equilibrium with a* E A, and a* = 0 is (Pareto) inefficient if and only if ai* < y where y= max1a E [0, Wxi; uj2[v*(a)] ? ujl (wj) when an = 0}.12 (With inefficiency, here we mean a deviation from what would be feasible under commitment, as discussed above.) Not surprisingly, this upper level on a* for inefficiency is decreasing in ',- and increas- ing in wo (but is functionally independent of stj and wx).

One may also say that the problem of inefficient equilibria in this model is an example of free-riding. The reason for borrowing this term from the theory of optimum supply of public goods is that altruism renders a public-good character to individual consumption since (at least) one individual's consumption then enters also others' utility functions.'3 In the public finance literature, an individual is said to be a free-rider if he does not contribute to the provision of a public good on the grounds that others will pay enough to cover its financing. But in the present context it is not clear what one should mean by an individual's "contribution" to his own welfare-when this is a public good.

Hence, we need a definition of free-riding for situations of strategic interaction between altruistic individuals. In such contexts, it seems natural to say that an individual is a free-rider if his strategic behavior induces others to contribute more to his welfare than they would like to, had they had the possibility to commit themselves to a support of their own choice. This intuitive notion can be formalized as follows.

Suppose that a* is a Nash equilibrium in 6, and assume that a* E Al for i ? 0. Then t* = v* (a*) > 0 is the transfer that i (voluntarily) gives to j in this equilibrium. If individual i instead were able to commit himself in advance to any transfer t in [0, wxl], while the intertemporal allocation of consumption were left at each individual's discretion,

Note that the commitment is to give exactly t*, neither less nor more. 12 Note that Uj2[T*(a)], when a, = 0, is continuous in ai > 0 and tends to plus infinity

as ai -* 0. 13 The public-good character of others' consumption, in the presence of altruism, is

discussed in an atemporal setting in Buchanan (1975a).


then the individualistic utilities associated with such a transfer would be

vi(t) = max u7(wi - a,, ai - t) (the donor) (1 5a)

subject to a, E [t, wj,

v>(t) = max uj(w, - ap, aj + t) (the recipient) (15b)

subject to a, E [0, wj].

The total welfare that i would derive from such a committed transfer t would thus be Vi(t) = vi(t) + otivj(t). It can be shown that the (indirect utility) function Vi: [0, wx -> R is continuous and strictly concave, and hence achieves its maximum at a unique level i. In the spirit of the intuitive definition above, we accordingly say that is a free-rider in a* if ? < t*.

If a* is "interior" (in the sense a* > 0), then V1'(t*) < 0 by the envelope theorem (and conditions [14]). Thus i < t*, by concavity of Vi, so the recipient indeed is a free-rider by the definition above. If instead a* = 0, then one can likewise show that Vi'(t*) < 0 if a* < -Y, and Vi'(t*) - 0 if a* y Hence, j is then a free-rider if and only if a*< y In sum, an equilibrium a* E A,, for i $ 0, is socially inefficient if and only if j is a free-rider.

Applying these conditions for inefficiency and free-riding to the example in Section IIIA of an altruistic and a selfish individual with equal endowments, we immediately find that when the altruism of the first individual is large (otl > 4/5), then the (unique) equilibrium is inefficient and the selfish individual is a free-rider (since then a* E AI and a*> 0).

In the example of equally altruistic but differently endowed individuals in Section IIIB, it was found that a transfer is given from the wealthy to the poor, in equilibrium, if the endowment of the less wealthy individual is less than the share a of the more wealthy individual, where cx is the (common) degree of altruism (condition [9]). This is the area below the diagonal in figure 5. For certain combinations of endowments and altruism (in this area) involving a relatively small difference in endowments, the poor individual will do some saving in equilibrium, which results in inefficiency and free-riding. This is area I, in the figure. For other combinations, he will save nothing, in which case we have to compare a* with -yI in order to know whether he is a free-rider or not. In this example, a* = (1 + ot)wI/(2 + ot) and -y= W2(1 + ot)/cx, so the equilibrium is inefficient if otwl < (2 + 00)W2 (see area I2 in the figure), while it is efficient if otw I (2 + 0)0w2 (this is area E). In the latter area, it is in both individuals' interest for the poor individual to consume all his wealth in the first period.

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FIG. 5.-Efficient (E) and inefficient (II and I2) equilibria in the Cobb-Douglas example of equally altruistic (ot) but differently endowed individuals (w1 > W2).

VI. Final Remarks

The binding agreements that are necessary to avoid inefficiency and free-riding are difficult to enforce in practice. Then what are the realistic possibilities to reduce or, ideally, eliminate the "strategic distortions" analyzed above?

In the case of life cycle savings, the analysis above suggests that compulsory savings systems could be welfare improving even when actuarially fair. This opens up the possibility for an explanation, or at least a rationalization, of social security systems along the lines of free- riding, as a complement to the "traditional" explanations based on market failure (for instance due to adverse selection or moral hazard), paternalistic concern for myopic citizens, or ambitions to redistribute


income within and between generations.'4 However, there are some limitations to this free-riding argument for social security.

First, for an actuarially fair compulsory savings system to be effec- tive at all, it is evidently necessary that claims to future pensions cannot be fully used as collateral in the credit market. Second, even in the more realistic case in which claims to pensions cannot be effectively used as collateral for loans in the first, active, period of life, a compulsory savings system has to be binding for an individual that would otherwise be a free-rider. Moreover, the net welfare effect of such a compulsory savings system will be positive in equilibrium only if the welfare gain from the reduction of the "strategic" distortion exceeds possible welfare losses due to distortions caused by the compulsory savings system in combination with the assumed imperfection in the credit market. Hence, an actuarially fair compulsory savings system may very well turn out to be ineffective in many situations. An inter- esting question for future research then is whether there exists some, possibly redistributional, social security system that can overcome these difficulties.'5

In situations in which common concern, as modeled above, is oper- ative within or between organizations-for example, between the state and state-owned firms or an international aid organization and "client" countries-costs due to "strategic" distortions may clearly arise. This is a second line for further research. In particular, suppose that individual 1 in the model is one unit of an organization and individual 2 another unit of the same organization. Suppose that the first unit strives to maximize some overall "performance index" ul + u2, where ul is the component associated with its own activities and u2 the component associated with the second unit. In contrast, the second unit strives to maximize its performance index (u2) only. The analysis in this paper then shows that, for certain combinations of endowments and preferences, the second unit will use up the bulk of its resources already in the first period, in anticipation that the first unit will then transfer additional resources in the second period, even if this leads to an (ex ante) inefficient outcome. This phenomenon may be interpreted as a "soft" budget constraint in the sense of Kor- nai (1980a, 1980b).

References

Axelrod, Robert M. The Evolution of Cooperation. New York: Basic Books, 1984.

14 See, e.g., Diamond (1977) for a discussion of motives for social security systems. 15 For this to be politically realistic, the group of potential free-riders must be a

minority.

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Becker, Gary S. "A Theory of Social Interaction." J.P.E. 82 (November/ December 1974): 1063-93.

. "Altruism, Egoism, and Genetic Fitness: Economics and Sociobiol- ogy."J. Econ. Literature 14 (September 1976): 817-26.

Bergstrom, Theodore. "A Fresh Look at the Rotten Kid Theorem-and Other Household Mysteries." Manuscript. Ann Arbor: Univ. Michigan, January 1987. (a)

. "Systems of Benevolent Utility Dependence." Manuscript. Ann Ar- bor: Univ. Michigan, May 1987. (b)

Buchanan, James M. "The Political Economy of Franchise in the Welfare State." In Capitalism and Freedom: Problems and Perspectives, edited by Richard T. Selden. Charlottesville: Univ. Press Virginia, 1975. (a)

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Drazen, Allan. "Government Debt, Human Capital, and Bequests in a Life Cycle Model."J.P.E. 86 (June 1978): 505-16.

Hammond, Peter. "Charity: Altruism or Cooperative Egoism?" In Altruism, Morality, and Economic Theory, edited by Edmund S. Phelps. New York: Sage Found., 1975.

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. " 'Hard' and 'Soft' Budget Constraint." Acta Oeconomica 25, nos. 3, 4 (1980): 231-45. (b)

Kurz, Mordecai. "Altruism as an Outcome of Social Interaction." A.E.R. Pa- pers and Proc. 68 (May 1978): 216-22.

Kydland, Finn E., and Prescott, Edward C. "Rules Rather than Discretion: The Inconsistency of Optimal Plans."J.P.E. 85 (August 1977): 473-91.

Lindbeck, Assar, and Weibull, Jorgen W. "Strategic Interaction with Al- truism: The Economics of Fait Accompli." Seminar Paper no. 376. Stock- holm: Inst. Internat. Econ. Studies, February 1987.

McTaggart, D., and Salant, D. "Time Consistency and Subgame Perfect Equilibria." Manuscript. Blacksburg: Virginia Polytechnic Inst. and State Univ., November 1986.

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Wintrobe, Ronald. "It Pays to Do Good, but Not to Do More Good than It Pays." J. Econ. Behavior and Organization 2 (1981): 201-13.

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