The distribution of wealth and the efficiency of institutions.

Eaton, B. Curtis ; White, William D.

THE DISTRIBUTION OF WEALTH AND THE EFFICIENCY OF INSTITUTIONS

This paper explores the effect on economic efficiency of the distribution of wealth and systems for enforcing property rights. We construct a two-person, two-period economy in which each person can consume, plant, transfer or steal corn. We find circumstances in which redistribution of wealth is Pareto optimal and in which increasing sanctions against theft to their maximum level is not. These results suggest that it is not only important to consider distribution in the design of property institutions, but also that redistribution itself may serve to increase the efficiency of systems of property rights.

I. INTRODUCTION

A fundamental insight of economics is that property rights may foster creation of wealth. Nevertheless, property rights which can be freely violated serve no function, so that to be meaningful, they must be enforceable. Effective enforcement requires the existence of enforcement systems to monitor violations, impose sanctions and provide for restitution of stolen property.

An effective enforcement system produces obvious desirable effects. By deterring theft and providing for restitution, it assures that individuals will appropriate the fruits of their labors, and hence enforcement serves to increase efficiency. It is a standard observation in the literature on crime and punishment that the possibility of crime may lead potential victims to alter their behavior in efficiency-reducing ways. The possibility of street crime may, for example, empty the streets [Neher 1978]. Conversely, the establishment of an effective enforcement system can return people to the streets both by deterring theft, or reducing the probability that it will occur, and by providing for restitution in the event theft occurs and the thief is apprehended. We refer to these efficiency-enhancing effects of enforcement as deterrence and restitution effects.

However, the benefits of operating an enforcement system must be weighed against its costs. Such a system may be resource using, and may yield undesirable as well as desirable effects. Thus, maintaining a criminal justice system entails direct costs--resources to operate a police force, courts and prisons--and indirect costs in that potential criminals may modify their behavior to adjust to the prospect of sanctions in efficiency-reducing ways (see Becker 1968, Usher 1986). A criminal may, for instance, dissipate resources in the face of expected sanctions, or otherwise deploy them in an inefficient fashion to escape appropriation. Hence, the tendency of criminals to exhibit apparently high discount rates observed by Wilson and Herrnstein [1985] may reflect the prospect of sanction rather than indicate a strong preference for immediate gratification, as they suggest. Likewise, the reduced risk of appropriation may explain the willingness of criminals to pay large transactions costs to maintain secrecy in Swiss bank accounts. We refer to these efficiency-reducing effects of enforcement systems as sanction effects.

Cheung [1970] and others have observed that the costs of enforcing property rights may depend on their distribution.(1) Ehrlich's [1973] finding that there is a positive relation between the level of crime and the degree of income inequality is supportive of this view. Beyond this, it has frequently been suggested that redistribution may increase respect for property rights along with monitoring and sanctions. The notion that giving people more to lose can render them more law abiding is central to liberal theories of rehabilitation. It is also embedded in efficiency wage models, where high wages are shown to be economical to the extent they increase the efficacy of sanctions and extend an employer's basis for control. (See for example Eaton and White [1983], Yellen [1984]).

Any wealth transfers to individuals must, of course, come from other individuals. Redistribution may serve to control behavior not only because it gives those who potentially may violate property rights more to lose, but also because it reduces what is left for them to steal. For instance, Johnson [1986] draws on this notion to explore the role of ritual gift giving to guests during potlach ceremonies among the Kwakiutl Indians. He argues that in the Pacific Coast salmon fisheries, where yearly harvests were highly variable, gifts from tribal groups who had experienced a good year (typically potlach hosts) to those who had experienced a lean one (typically guests) served, through the resulting redistribution of wealth, to deter theft in both ways.

In this paper we focus on the role of wealth distribution in a very simple general equilibrium model in which deterence, restitution, and sanction effects are possible. Our model is a two-person, two-period, corn economy. Both individuals in our model have the option of consuming initial endowments of corn in the first period or planting them for harvest in the second period. The yield in the second period exceeds the amount planted. But whereas property rights are assumed inviolate in the first period, they are not in the second; corn is safe in hand, but not on the stalk. Attempts to enforce property rights may entail deterrence and restitution effects--individuals plant corn they otherwise would have consumed because they feared theft. However, enforcement may also entail sanction effects--those intending to steal anyway may consume corn instead of planting it since they anticipate its confiscation through sanctions.

The enforcement mechanisms we consider involve a probability of apprehension and wealth-constrained sanctions. We discover that whether a particular mechanism produces an efficient or an inefficient outcome is very much dependent on the distribution of the economy's initial endowment of corn. Indeed, when the mechanism produces an inefficient equilibrium, Pareto-optimal redistribution is a distinct possibility.

Our model captures the externalities which are associated with theft and with attempts to control it. Since theft uses resources, it raises other economic issues which we have ignored. Benoit and Osborne [1987] examine the role of distribution in a model where the only economic problem raised by the possibility of theft is that it is resource using. Hence their analysis is complementary to ours, as are their results. They too discover that distribution plays a central role in determining the efficiency of any system of property rights.

The balance of this paper is organized as follows. In section II we present our model. In sections III and IV, we explore equilibria of the model. In section V we examine comparative statics with respect to the distribution of wealth. In section VI we close with a brief discussion of our results.

II. A TWO-PERSON, TWO-PERIOD CORN ECONOMY

We begin our examination of the role which the distribution of wealth plays in supporting property institutions by developing a primitive two-person, two-period, corn economy. The preferences of the two individuals, One and Two, are captured in the following linear utility functions: (1) [U.sub.i]([x.sub.i],[y.sub.i]) = [x.sub.i] + [y.sub.i], i = 1,2, where [x.sub.i] is corn consumed in period 1 and [y.sub.i] is corn consumed in period 2.(2) The aggregate initial endowment of corn is W units, and [W.sub.i] denotes i's initial endowment. Thus (2) [W.sub.1] + [W.sub.2] = W. We assume throughout that [W.sub.1] [is greater than or equal to] [W.sub.2]. A unit of corn planted in period 1 yields G > 1 units in period 2. Since utilities are linear in aggregate corn consumption and G exceeds unity, efficiency or Pareto-optimality requires that all the corn be planted in period 1 and the harvest consumed in period 2.

The Problem

We assume that either individual can eat some or all of his endowment in period 1, where there is an inviolate property right in corn for one period. However, once corn is planted and ready for harvest a question arises. How much of the total harvest will One appropriate and how much will Two appropriate? To answer this question, we need an assumption regarding the technology of corn harvesting. It seems natural to assume that each individual will harvest half of the corn in the field. That is, in the absence of an institution of property, corn in the field is common property.

The equilibrium of the common property model depends on the magnitude of G. Let [z.sub.i] denote the quantity of corn planted by individual i. Thus (3) [z.sub.i] = [W.sub.i] - [x.sub.i], i = 1,2, that is, the individual plants what he does not consume in period 1. Objective functions, [V.sub.i](z), can then be written as (4) [V.sub.i](z) = ([W.sub.i] - [z.sub.i]) + G([z.sub.1] + [z.sub.2])/2, where z = ([z.sub.1], [z.sub.2]). The first term is [x.sub.i], equal to [W.sub.i] - [z.sub.i], and the second is [y.sub.i], equal to one-half the total harvest.

The partial derivative of [V.sub.i](z) with respect to [z.sub.i] is -1 + G/2. This derivative is positive if and only if G exceeds two. Hence in a Nash equilibrium, each individual will plant all of his initial endowment if G exceeds two, and none of it if G is less than two. In the former case there is no interesting economic problem since the economy is on its utility possibilities frontier. Thus we assume that G is less than two. For simplicity, we let (5) G = 3/2 In this case the sum of utilities in the common property equilibrium is W, which is less than the maximum potential utility, 3W/2.

Now we have a problem--the economy is inside its utility possibilities frontier--that an institution of property rights might solve. Before we proceed, we should be clear about the nature of the economic problem our model captures. Given an institution of property, harvesting the other person's corn is theft. As noted above, there are two distinct economic problems associated with theft: it wastes resources, and it creates an externality. We are ignoring the first of these--theft is not resource using in our model. It does however create a negative externality--theft reduces the private return to planting corn and hence may induce either or both individuals to plant no corn, forcing the economy inside its utility possibilities frontier.

This externality is quite similar to the externality associated with common property. If, for example, we were to simply promulgate property rights by announcing that "in period 2, [Gz.sub.i] is the property of individual i," but provided no enforcement mechanism, then each (amoral) individual would steal half of the other's harvest. Consequently, neither would capture all the returns to the corn he plants. Anticipating this externality, neither would plant any corn, and the economy would be inside its utility possibilities frontier just as it was when corn in the field was common property.

An enforcement mechanism that uses sanctions to deter theft and pays restitution to victims of theft may solve this problem. However, it may also create a negative externality borne by the thief. Suppose, for example, that the enforcement mechanism does not fully deter theft, so that one individual steals from the other in equilibrium. Sanctions then imply that the thief cannot capture all the returns to the corn he plants; that is, there are sanction effects. In our model, the sanction effect sometimes leads the thief to plant none of his corn. This in turn implies that maximal sanctions are not necessarily optimal.

These two externalities, the first associated with theft, and borne by the victim of theft, and the second associated with the attempt to control theft, and borne by the thief, imply that the design of efficient property institutions is a delicate matter. Increasing the effective sanction for theft mitigates the externality which the potential victim of theft bears, but it increases the externality which the potential thief bears.

Institutions of Property

There are many possible systems of property rights. To produce a tractable model, we choose to examine a system which can be characterized by two parameters. Each individual is endowed with a nominal property right to the corn he plants--if individual i plants [z.sub.i], then [Gz.sub.i] is nominally i's property in period 2. An enforcement mechanism exists, characterized by the parameters P and [Lambda]: 0 [is less than or equal to] P [is less than or equal to] 1, 0 [is less than or equal to] [Lambda] [is less than or equal to] 1. P is a probability of apprehension and 1 characterizes the sanction imposed on the (apprehended) thief. The enforcer is an external policeman--not One or Two--who is presumed to be incorruptible.

If individual i steals from j, he will be apprehended with probability P. In the event that he is apprehended, i is forced to return all of the stolen corn to j and, in addition, is forced to pay a sanction to j. The sanction paid to j is [Lambda][Gz.sub.i]; that is, the thief pays a proportion [Lambda] of his corn harvest to the victim.(3)

If both individuals steal, the situation is more complex. A thief may choose not to report the theft of his own corn if by doing so he increases the probability that his own transgression will be discovered. Hence the enforcement mechanism may or may not come into play. And if it does come into play, the probability that either thief is caught will be less than P, given a fixed effort from the enforcer. For these reasons it is not clear what we ought to assume regarding the probability of apprehension in this case. Guided again by tractability, we assume that when both individuals steal, the probability of apprehension is zero.

Note that while the magnitude of sanctions is here bounded by the thief's tangible wealth, it is straightforward to extend our analysis to consider other sorts of sanctions: for example, sanctions involving loss of life or liberty. It is sufficient for our purposes that the utility cost of sanctions be finite. In this regard, observe that even the death penalty has not always proved adequate to deter theft. In any case, society routinely sets limits on the magnitude of sanctions--the U. S. Constitution, for instance, prohibits cruel and unusual punishments. Such limitations may reflect humanitarian considerations or concerns about misapprehension, the perverse incentives that can be created when multiple types of offenses are involved (see Stigler [1970] on this sanction effect), or other sanction effects.

Equilibrium

The structure of the game we analyze is this: at the beginning of period 1 both individuals simultaneously choose [x.sub.i] (or [z.sub.i]); at the beginning of period 2 both individuals simultaneously choose how much corn, if any, to steal from the other; decisions with respect to theft having been made, thieves are instantaneously apprehended, and restitution and sanctions instantaneously paid.

Rational choice with respect to [x.sub.i] (or [z.sub.i]) in period 1 requires individual i to anticipate his own choices and j's choices in period 2. Hence the appropriate equilibrium concept is perfect Nash equilibrium, which ensures that individuals correctly anticipate both their own and each others actions in period 2. That is, perfect Nash equilibrium ensures that decisions are perfectly informed and individually utility maximizing. The common property equilibrium we found above is a perfect Nash equilibrium since, when making his own planting decision in period 1, each individual correctly anticipated that he would appropriate half of the total harvest in period 2.

In the common property equilibrium, both individuals plant all of their corn when G > 2 and none of it when G < 2. These corner solutions are obviously driven by the linearity of the basic model--utility is linear in [x.sub.i] and [y.sub.i], and corn harvested is linear in corn planted. Not surprisingly, this linearity, in conjunction with the linearity of the property institution, produces corner solutions in the property rights model as well--in any pure strategy equilibrium each individual plants all of his corn or none of it. Hence we can greatly simplify the analysis by restricting individual strategies to these extreme cases. That is, we assume that [z.sub.i] (or [x.sub.i]) is either zero or [W.sub.i]. A version of the paper in which we do not use this simplifying assumption is available from the authors.(4)

III. PERIOD 2 EQUILIBRIA

Our assumption of a perfect Nash equilibrium requires that we begin the analysis by characterizing the equilibrium of the period 2 game of theft, taking period 1 planting decisions as given. Since there are four possible combinations of planting decisions in period 1, we must consider four possible subgames in period 2: Subgame I, in which z = (0,0); Subgame II, in which z = ([W.sub.1] ,0); Subgame III, in which z = (0,[W.sub.2]); Subgame IV, in which z = ([W.sub.1],[W.sub.2]). In Subgame I, there is no corn to steal and hence no theft. In Subgame II, Two will steal half of One's corn since Two has no corn of his own and hence nothing to lose in the event that he is apprehended.

Similarly, in Subgame III, One will steal half of Two's corn.

All of the interesting possibilities arise in Subgame IV where each individual is both a potential thief and a potential victim of theft. If i chooses to steal from j, he will steal as much as possible, half of j's harvest, since both the probability of apprehension and the sanction are independent of the amount of corn stolen. The payoffs in this game of theft are presented in Table I. An S denotes the strategy "steal" and a D the strategy "don't steal," and the first entry in each strategy combination is One's strategy and the second Two's.

With strategy combination (D,D) each individual consumes his own harvest, (3/2)[W.sub.1], and with combination (S,S) each individual eats half of his own harvest and half of the other's. With strategy combination (D,S), Two steals and One does not. In this case realized payoffs depend on whether Two is apprehended. If he is, One's consumption is (3/2)([W.sub.1] + [Lambda][W.sub.2]) and Two's is (3/2)(1-[Lambda])[W.sub.2] since there is full restitution and Two pays sanction (3/2) [Lambda][W.sub.2] to One. If Two is not caught, One's consumption is just (3/2)[W.sub.1]/2 and Two's is (3/2)[ [W.sub.2]+[W.sub.1]/2] since Two has successfully stolen half of One's corn. The payoffs associated with strategy combination (D,S) are obviously symmetric to those associated with (S,D).

The equilibrium of this two by two matrix game depends upon the nature of the enforcement mechanism, on (P,[Lambda]). The circumstances in which both property rights are honored--(D,D) is an equilibrium--are of central interest. Given strategy D for Two, One will prefer strategy D if (6.1) (3/2)[W.sub.1] [is greater than or equal to] (3/2)[P(1-[Lambda])[W.sub.1] + (1-P)([W.sub.1]+[W.sub.2]/2)] and, given strategy D for One, Two will prefer D if (6.2) (3/2[W.sub.2] [is greater than or equal to] (3/2)[P(1-[Lambda])[W.sub.2] + (1-P)([W.sub.2]+[W.sub.1]/2)] When both of these inequalities are satisfied, (D,D) is an equilibrium strategy combination in Subgame IV.

These inequalities can be written in a more compact way. (7.1) P [is greater than or equal to] R/(R+2[Lambda]), (7.2) P [is greater than or equal to] 1/(1+2[Lambda]R), where R is defined to be [W.sub.2]/[W.sub.1]. R parameterizes relative wealth and is bounded by zero and one since, by assumption, [W.sub.1] [is greater than or equal to] [W.sub.2]. Notice that if (7.2) is satisfied, then so is (7.1), since R [is less than or equal to] 1. Hence (D,D) is the equilibrium strategy combination whenever (7.2) is satisfied. From (7.2) we see that large values of P, [Lambda] and R tend to produce the equilibrium in which property rights are respected in Subgame IV.

Strategy combinations (D,S) and (S,S) are also equilibria of Subgame IV for certain values of the parameters P, [Lambda] and R. However, strategy combination (S,D) is never an equilibrium combination--that is, in Subgame IV if just one individual steals in equilibrium, the poorer individual is the thief. We record results for Subgame IV in proposition 1. In each of the three cases the first inequality pertains to One's choice and the second to Two's. Binding inequalities are indicated by asterisks.

PROPOSITION 1: Equilibria of Subgame IV

A. (D,D) is an equilibrium if

P [is greater than or equal to] R/(R+2[Lambda]),

[.sup.*P] [is greater than or equal to] 1/(1+2[Lambda]R). B. (D,S) is an equilibrium if

[.sup.*P] [is greater than or equal to] R/(1+2[Lambda]R),

[.sup.*P] [is less than or equal to] 1/(1+2[Lambda]R). C. (S,S) is an equilibrium if

[.sup.*P] [is less than or equal to] R/(1+2[Lambda]R),

P [is less than or equal to] 1/(R+2[Lambda]).

These results are illustrated in Figure 1. In the upper-right portion of the figure neither individual steals, in the middle of the figure, Two steals but One does not, and in the lower-left portion, both of them steal. The dashed lines indicate how the partition of the parameter space responds to an increase in R--that is, a reduction in inequality.

IV. PERFECT EQUILIBRIA

The fact that Subgame IV, in which both individuals planted their corn in period 1, has three types of equilibrium in period 2 means that we must consider three different games in period 1. The first game we consider is the one in which there would be no theft if both One and Two planted their corn, which requires that the institution, (P, [Lambda]), be in the upper-right portion of Figure 1. This game seems to be the most interesting of the three, and consequently we discuss it at some length. We simply present results for the other two with minimal commentary.

Game One: P [is greater than or equal to] 1/(1+2[Lambda]R)

In period 1 an individual's strategy is either to plant all his corn, denoted by A, or none of it, denoted by N. Observe that in Game One, P is always greater than 1/3. In Table II we present the payoffs or utilities associated with the four possible strategy combinations. The payoffs associated with strategy combinations (A,A) and (N,N) are straightforward. With strategy combination (A,N), in the first period One plants his corn and Two eats his, and in the next period, Two steals from One. Hence Two's expected consumption is [W.sub.2], what he eats in the first period, plus the expected value of his theft in the second period, (1-P)(3/2)([W.sub.1]/2). One's consumption is (3/2)[W.sub.1] if Two is caught and (3/2)[W.sub.1]/2 if he is not, and his expected consumption is therefore (3/2)[W.sub.1][P+ (1-P)/2]. The payoffs associated with (N,A) are, of course, symmetric to those associated with (A,N).

The first thing to note about Game One is that (N,N) is never an equilibrium--at least one individual will plant his corn. Suppose, for example, that Two chose N. One would then anticipate that Two would steal from him in period 2 if he chose A. Whether One then chooses A or N depends upon the probability of apprehension, or more insightfully, the probability of restitution, since Two will pay full restitution if he's caught. As you can easily verify, when P exceeds 1/3, which is always the case in Game One, the restitution effect is strong enough to induce one individual to choose A if the other chooses N. Hence, (N,N) is never an equilibrium when P > 1/3. (On the other hand, (N,N) is always an equilibrium when P < 1/3.)

Now suppose that i does plant his corn, and consider j's strategy choice. If j also chooses A, he will anticipate that there will be no theft in period 2 and his utility will therefore be (3/2)[W.sub.j]. If j chooses N--to eat his corn in period 1--he will anticipate successfully stealing (1/2) of i's harvest with probability 1-P, and his expected utility will therefore be [W.sub.j] + (1-P) (3/4)[W.sub.i]. Hence i choosing A and j choosing N is a subgame perfect equilibrium if (8.1) [W.sub.j] + (1-P)(3/4)[W.sub.i] [is greater than or equal to] (3/2)[W.sub.j], or if (8.2) (3/2)(1-P) [is greater than or equal to] [W.sub.j]/[W.sub.i]. Because P [is greater than or equal to] 1/3 in Game One, it's clear that if someone does choose N it won't be the richer individual since (8.2) can never hold when [W.sub.j] > [W.sub.i]. That is, One will never choose N in Game One.

Letting i = 1 and j = 2, condition (8.2) can be written as (8.3) 3 [is greater than or equal to] 3P + 2R. If this inequality is satisfied then (A,N) is the subgame perfect equilibrium, and if it is not satisfied (A,A) is the equilibrium. (Both are equilibria if (8.3) holds with equality.) Note that large values of P and significant equality in the distribution of wealth (large values of R) work to produce the efficient equilibrium in which all the corn is planted. In particular, the inefficient equilibrium in which Two does not plant his corn is impossible if the distribution of wealth is equal (if R = 1), or if P = 1.

It makes sense that it will be the poorer individual who inefficiently eats his corn in the first period and steals in the second. Suppose that i does plant his corn. Then for individual j, the opportunity cost of not planting the corn is the increased quantity of corn that is foregone, equal to (1/2)[W.sub.j], and the inducement is the corn that would be stolen in period 2, equal to (3/4)[W.sub.i] (1-P). Thus the opportunity cost is proportional to one's own wealth, and the inducement is proportional to the other individual's wealth.

We record our results in proposition 2 and illustrate them in Figure 2. The first restriction in each case pertains to One's choice and the second to Two's. The restrictions pertaining to One's choice are always satisfied in Game One, and hence those pertaining to Two's choice are binding, and therefore marked by an asterisk. PROPOSITION 2: Perfect Equilibria of Game One Game One Restriction: P [is greater than or equal to] 1/(1+2[Lambda]R) A. (A,A) is an equilibrium if

3PR + 2 [is greater than or equal to] 3R,

[.sup.*3]P + 2R [is greater than or equal to] 3.

No theft in period 2. B. (A,N) is an equilibrium if

3P [is greater than or equal to] 1,

[.sup.*3]P + 2R [is less than or equal to]3.

Two steals in period 2.

When (8.3) holds with equality, Two is indifferent between strategies A and N. If Two chooses A, the economy is on its utility possibility frontier, and if he chooses N, the economy is inside the frontier. Since Two is indifferent between these options, we see that One bears the full utility cost, equal to [W.sub.2]/2, if Two chooses strategy N instead of strategy A. This result implies that Pareto-optimal redistribution of wealth prior to planting decisions in period 1 is a possibility in Game One. Given the initial distribution of wealth, if (P,[Lambda]) is at a point like A in Figure 2, then there is a redistribution of wealth from One to Two which transforms the inefficient (A,N) equilibrium into an efficient (A,A) equilibrium in which both individuals are better off.

Two is, of course, always better off if One redistributes wealth to him. Hence to discover the conditions under which Pareto-optimal redistribution is possible, we can focus on One's utility. Suppose that the initial distribution, [Mathematical Expression Omitted], is such that (8.3) holds with strict inequality, so that the equilibrium without redistribution is (A,N). Then One's utility is [Mathematical Expression Omitted]. Now consider the subsequent distribution, [Mathematical Expression Omitted], such that (8.3) holds with equality. [R.sup.1] is the most unequal distribution which produces an efficient equilibrium. Given [R.sup.1], One's utility in the efficient equilibrium is [Mathematical Expression Omitted]. Pareto-optimal redistribution is possible when (9.1) [Mathematical Expression Omitted] Using the (implicit) definition of [R.sup.1] and the identity [Mathematical Expression Omitted], condition (9.1) can be reduced to (9.2) 3[P.sup.2] - 2P + 4[R.sup.0] - 1 [is greater than or equal to] 0. For any (P,[Lambda]) in Game One, if [R.sup.0] [is greater than or equal to] 1/4, then (9.2) is satisfied. Hence, if the equilibrium of Game One without redistribution is (A,N) and if [Mathematical Expression Omitted], then it is always Pareto-optimal for One to redistribute just enough corn to Two so that the equilibrium with redistribution is (A,A). If [R.sup.0] < 1/4, then redistribution may or may not be Pareto-optimal. We've illustrated this case in Figure 2. If (P,[Lambda]) lies above the dashed line (3[P.sup.2]-2P+4R-1=0) and below the solid line (3P+2R=3), then redistribution is Pareto-optimal. Game Two: 1/(1+2[Lambda]R) [is greater than or equal to] P [is greater than or equal to] R/(1+2[Lambda]R)

In Game Two if both individuals plant their corn in period 1, the equilibrium in period 2 is (D,S)--One respects Two's property right, but Two does not respect One's. Hence payoffs in Game Two will differ from those in Game One (presented above in Table II) for strategy combination (A,A). One's payoff is (3/2)[P([W.sub.1]+[Lambda][W.sub.2]) + (1-P)[W.sub.1]/2], since One eats all of his own corn and a proportion [Lambda] of Two's when Two is caught, and eats just half of his own corn when Two is not caught. Two's payoff is (3/2)[P(1-[Lambda])[W.sub.2] + (1-P)([W.sub.2]+([W.sub.1]/2)], since Two eats a proportion 1-[Lambda] of his own corn if he is caught, and all of his own corn and half of One's if he is not.

In Game Two anything is possible in the sense that there are permissible parameter values such that any of the four possible strategy combinations is a perfect equilibrium. The possibilities are summarized in proposition 3, and illustrated in Figure 3. Asterisks again indicate binding restrictions, and the first restriction is applicable to One and the second to Two. PROPOSITION 3: Perfect Equilibria of Game Two Game Two restrictions:

1/(1+2[Lambda]R) [is greater than or equal to] P [is greater than or equal to] R/(1+2[Lambda]R) A. (A,A) is an equilibrium if

[.sup.*P][1+(1+2[Lambda])R] [is greater than or equal to] 1+3R,

[.sup.*3][Lambda]P [is less than or equal to] 1.

Two steals in period 2. B. (A,N) is an equilibrium if

3P [is greater than or equal to] 1,

[.sup.*3][Lambda]P [is greater than or equal to] 1.

Two steals in period 2. C. (N,A) is an equilibrium if

[.sup.*3]P[1+(1+2[Lambda])R] [is less than or equal to] 1+3R,

[.sup.*3]P [is greater than or equal to] 1.

One steals in period 2. D. (N,N) is an equilibrium if

[.sup.*3]P [is less than or equal to] 1,

[.sup.*3]P [is less than or equal to] 1.

Two points are worth making: the first concerns sanction effects and the second concerns the possibility of Pareto-optimal redistribution from Two, the poorer individual, to One. If the institution is described by point V in Figure 3, then a small increase in P or [Lambda] transforms the efficient equilibrium, in which both individuals plant their corn and Two steals from One in period 2, into the inefficient equilibrium in which Two does not plant his corn and continues to steal from One. This result illustrates how the property institution can seriously distort the thief's incentives. At V the thief plants his corn just as efficiency demands, but a small increase in P or [Lambda] induces him to eat his corn in period 1 so that he can avoid the sanction in the event that he's caught in period 2. Interestingly, it is One who bears the burden of inefficiency. On the boundary separating these two types of equilibria, Two is indifferent between strategies A and N. If he chooses N instead of A, total utility is smaller by [W.sub.2]/2, which is identical to the (expected) sanction which Two pays to One if he chooses A. Hence One bears the entire burden of foregone utility if Two chooses N.

Now consider point Z in Figure 3. At Z, One is indifferent between A and N. If One chooses N, the economy is inside its utility possibility frontier, and if he chooses A, the economy is on the frontier. Since One is indifferent, Two bears the entire burden of the inefficiency. Because the boundary separating these two regions shifts downward as the distribution of wealth becomes more unequal, it's now possible that redistribution from Two to One is Pareto-optimal. That is, if the property institution is in region (N,A) but close enough to the boundary with region (A,A), Two will transfer just enough wealth to One to transform the inefficient (N,A) equilibrium into an efficient (A,A) equilibrium. Game Three: P [is less than or equal to] R/(1+2[Lambda]R)

In Game Three if both individuals plant their corn in period 1, the equilibrium in period 2 is (S,S)--property rights are not respected. For both individuals, the payoffs in Game Three associated with strategy combination (A,A) are then 3([W.sub.1]+[W.sub.2])/4 since total utility is 3([W.sub.1] +[W.sub.2])/2 and each gets half the total. (The payoffs for the other strategy combinations are those presented in Table II above.)

In Game Three there are three possible equilibria, (N,N), (N,A), and (A,A). The possibilities are summarized in proposition 4. The first inequality pertains to One's choice and the second to Two's, and an asterisk indicates a binding restriction. The equilibria in parts B and C entail a third restriction on R. If this restriction is not satisfied, then the set of parameter values that satisfy all relevant restrictions is empty. PROPOSITION 4: Perfect Equilibria of Game Three Game Three restriction:

P [is less than or equal to] R/(1+2[Lambda]R) A. (N,N) is an equilibrium if

[.sup.*P] [is less than or equal to] 1/3,

[.sup.*P] [is less than or equal to] 1/3. B. (N,A) is an equilibrium if

[.sup.*P] [is less than or equal to] 1/3R,

P [is greater than or equal to] 1/3,

[.sup.*R] [is greater than or equal to] 1/3. C. (A,A) is an equilibrium if

[.sup.*P] [is greater than or equal to] 1/3R,

P [is greater than or equal to] R/3,

[R.sup.*] [is greater than or equal to] 1/[Square root 3]. As in Game Two, Pareto-optimal redistribution from Two to One is again a possibility.

V. DISTRIBUTION AND EFFICIENCY

We have already seen that the distribution of wealth plays a crucial role in determining the equilibrium of our model and that Pareto-optimal redistribution is a possibility. This, of course, means that distribution and efficiency cannot be disentangled--efficiency demands that both individuals plant their corn, but whether any property institution induces them to do so depends on the distribution of wealth. In this section we briefly present some comparative static results that illustrate just how central the role of distribution is in this model.

When P is less than 1/3, neither individual ever plants his corn and distribution is therefore irrelevant. Hence we ignore this portion of the parameter space in Figure 4. (The last panel in the figure depicts the portion of the parameter space on which we focus.) The shaded areas in each of the first eleven panels indicate the institutions which produce inefficient results. In the upper-right shaded area, Two fails to plant his corn, and in the lower-left shaded area One fails to plant his. One has the entire initial endowment of corn in panel 1 (R = 0), and distribution gets progressively more equal in panels 2 through 11. As the distribution gets more equal, the upper-right shaded area gets progressively smaller, while the lower-left shaded area initially increases and subsequently decreases. Interestingly, when the initial endowment is equally distributed, inefficiency is never a possibility, and when it is unequally distributed, there are institutions which induce either individual to behave inefficiently.

VII. DISCUSSION

As we observed in the introduction, it is evident that the allocation of property rights can affect their efficiency. Who has what can make a difference. It is also evident that when there are externalities associated with both theft and the use of sanctions, the effects of changes in distribution on efficiency cannot be predicted a priori. Society is faced with a balancing act between deterrence and restitution effects on the one hand, and sanction effects on the other. Because the relative importance of the two types of externalities, as well as the direct costs of operating enforcement systems, may vary, they must be evaluated on a case by case basis.

From a policy perspective, our analysis suggests two broad conclusions. First, it is important to consider the underlying distribution of wealth in designing enforcement systems. For example, we discover circumstances in which increasing the magnitude of sanctions and/or the probability of apprehension may reduce the efficiency of property institutions so that output falls. Second, it is evident from our analysis that redistribution itself may increase efficiency. That is, holding sanctions and the probability of apprehension fixed, circumstances exist in which it is optimal to redistribute wealth.

A key empirical question is whether in practice the magnitude of distribution effects is large enough to merit attention. That is, does (or could) distribution really matter in determining the costs and benefits of operating enforcement systems? We offered a number of empirical examples to motivate the discussion in our introduction. But we view these as suggestive, rather than conclusive, and see a challenging research agenda emerging out of the analysis presented here.

One promising area for research is the impact of changes in the distribution of wealth on economic efficiency when the structure of enforcement systems remains fixed. Extending the type of analysis performed by Ehrlich [1973] to consider changes in wealth distribution over time is one possibility. Another promising area for research is to explore the use of redistribution as a control variable in enforcement systems as Johnson [1986] does in his analysis of potlatching. There have been a number of studies of the development of systems of regulation to address common property problems in resource management, for example in oil fields [Libecap and Wiggins 1985] or fisheries [Karpoff 1987]. A common observation in these studies is that small operators receive favorable treatment. The standard interpretation is that favorable treatment reflects equity concerns and political lobbying. While not ruling out this possibility, we note a second possible explanation, namely that enforcement costs may be lower when small operators are given more to lose. [Tabular Data 1 and 2 Omitted] [Figures 1 to 4 Omitted]

(1)See also Bush's [1972] discussion of the formation of property right systems and Umbeck [1981] on the development of early property right systems in the California gold fields. (2)One might object that if [x.sub.i] were zero, the individual would starve. To blunt this objection we could assume that there are many goods in the economy (berries, nuts and game as well as corn), but that corn is the only scarce good. (3)We have analyzed the model with a more conventional variant of the property institution in which the sanction paid by the thief is some absolute amount S if [Gz.sub.i] > S, and [Gz.sub.i] otherwise. That is, the sanction for theft is a fixed, corn-constrained, sanction. This version of the model is considerably more complex, but produces the same general insights. For this reason we consider here the less conventional but more tractable proportional sanction. (4)Without this restriction on strategies, there are cases in which the model has no equilibrium in pure strategies. In these cases, the equilibrium of the simplified model is different from the equilibrium of the unrestricted model.

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B. CURTIS EATON and WILLIAM D. WHITE, Simon Fraser University and the University of Illinois at Chicago. We would like to thank the participants of more than a dozen workshops, two anonymous referees, and Tom Borcherding for valuable suggestions.