RAUL V. FABELLA
A CONTRACTARIAN APPROACH TO PARETO EFFICIENCY IN TEAMS: A NOTE
ABSTRACT. We show that if identical members first decide on the sharing technology (stage I) taking into account their subsequent effort supply (stage II) decisions, the resulting contractarian sharing technology (constitution) channels individual self-seeking towards team (Pareto) optimum. Voting with one’s feet and open entry can ensure symmetry and majoritarian decision making in the real world teams. The model helps explain the differential performance of the Israeli Kibbutz and the Russian Kolkhoz. KEY WORDS: Nash equilibrium, Individual optimum, Social optimum
1. INTRODUCTION
Team equilibrium is largely associated with Nash equilibrium (see, e.g., Moulin, 1995). Nash equilibria are notorious for being in general Pareto suboptimal (see, e.g., Cornes and Sandler, 1986 for a crisp discussion). The Nash equilibrium output in a regular duopoly exceeds the cooperative (Pareto efficient) output to the detriment of the firms. Overexploitation of common property is a regular feature of Nash equilibrium. The Nash equilibrium number of firms in an oligopolistic market with free entry exceeds the Pareto optimal number signalling ‘excessive competition’ (Okuno and Suzumura, 1985). The myopic Nash solutions to prisoner’s dilemma games are, in general, Pareto inferior. But the Nash equilibrium also exhibits the irreplaceable propery of being self-enforcing and uniquely consistent with a set of plausible rationality axioms (Johansen, 1982). Most facile and direct routes to Pareto efficiency are not self-enforcing or they violate strict agent rationality. The team game is no exception. Nash solutions in teams are generally Pareto inefficient. Even when effort is observable, Pareto efficiency is attained only for a very special and unlikely combination of production and allocation technology parameters (Sen, 1966). When effort is unobservable Holmstrom (1982) has shown that a budget balancing Pareto efTheory and Decision 48: 139–149, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.
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ficient Nash solution fails to exist. This formalized Alchian and Demsetz’ (1972) rejection of team structure due to moral hazard. Although moral hazard has since been associated with inefficiency in teams, the window closed by moral hazard is surprisingly quite small. Fabella (1988) showed that with observable effort and proportional sharing, Pareto efficiency of the Nash equilibrium is supported by a linear production function with identical marginal products. Closer to reality, Guttman and Schnytzer (1987) wondered why the (Israeli) Kibbutz remains a viable (though beseiged) institution while the (Russian) Kolkhoz is not. They zeroed in on the sharing technology as possible explanation for the performance divergence. The Kibbutz adheres to egalitarian sharing while the Koklhoz practices proportional sharing based on effort. They showed that egalitarian sharing supports Pareto efficiency under an independent two stage (Guttman, 1978) effort-matching mechanism. In stage I, members determine their matching rate and in stage II, their flat rate. If agents are identical and there is no income effect, the Nash equilibrium is Pareto optimal. Effort-based sharing fails to support Pareto efficiency under the same matching mechanism! This is interesting if counterintutitive. We maintain the view that in the universe of differences between the Kibbutz and the Kolkhoz, there are elements as or even more important than the sharing mechanism. In this paper, we harness the contractarian viewpoint (see, e.g., Buchanan and Tullock, 1964; Brennan and Buchanan, 1985) to mediate the conflict between individual and social optimum. Hobbes in ‘Leviathan’ viewed the state or governance as a social contract freely agreed upon by individuals facing the prospect of existence that is ‘nasty, brutish and short’. Buchanan and Tullock (1964), reinforcing Hobbes, prefer to call the outcome of the consensus a constitution which as an alternative to chaos, works to the benefit of all. In this paper, we employ a majoritarian process which obeys the will of the membership in the determination of the environment. We will stick with methodological rationality and, with David Hume (1741), suppose each man ‘a knave, and to have no other end, in all his actions, than private interest’. The heart of the constitutional process in this case is naturally the team sharing technology. Since membership symmetry is central to the results in this paper, we identify a real life mechanism by which symmetry may be
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attained. This mechanism is free entry and exit and the freedom to vote with one’s feet. In 2, we set down the bare ingredients of a team, showing first that even with membership symmetry, Pareto efficiency is only a happenstance with an exogenous sharing technology. We then proceed to endogenize sharing by resorting to an earlier stage where the sharing technology is agreed upon before effort supply is even begun. Allowing for interdependence in the decision nodes, we show that individuals will choose the environment or constitution that channels self-seeking towards the Pareto optimum. In 3, we define the contractarian team as one which adopts as sharing technology the one most preferred by a majority of the members. We then argue that the contractarian process and the symmetry assumption are, in the real world achieved by voting with one’s feet, i.e. with free mobility of members and, thus self-selection.
2. THE MODEL
Consider a team of n members indexed by i. Revenue net of other inputs and effort monitoring costs other than labor’s is F defined P over total effort L = n Li . F is nondecreasing, continuous, twice differentiable and homogenous of degree r > 0. We assume that an effort monitoring subcontract ferrets out true effort levels of members so that moral hazard is insignificant. F is exhaustively allocated among members according to the Sen sharing mechanism: Pmember i’s share si = [(1−α)(Li /L)+(α/n)], 0 6 α 6 1. Thus n si = 1 and no residual claimant exists. If α = 0, si = (Li /L) and sharing is purely effort-based; if α = 1, si = n−1 , and sharing is purely egalitarian. Holmstrom (1982) used si = n−1 to reflect complete unobservability of effort, i.e, as a moral hazard adjustment. If α > 0, one may say, that, true effort is only imperfectly measured by observed effort Li . In this paper, we will interpret α as a Rawlsian equalization parameter acting as a tax rather than as a moral hazard adjustment parameter since moral hazard is assumed addressed by a monitoring subcontract. Member i’s utility is defined as Ui = si F − Vi (Li ) where Vi (•), the effort disutility function, is increasing, strictly convex and twice differentiable. We assume team members to exhibit a Cournot con-
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jectural variation. The first order condition for the maximum of Ui , with respect to Li is: !2 n−1 n X X (1 − α) Lj / Lj (F /F 0 ) + si F 0 = Vi0 ,
i = 1, 2, . . . , n
(1)
where F 0 = dF /dL and Vi0 = dV /dLi . Imposing symmetry and suppressing subscripts, we have [((1 − α)(n − 1)/nr) + n−1 ]F 0 = V 0
(2)
where r = (F /LF 0 ) is the reciprocal of the effort elasticity of revenue (also the degree of homogeniety). This can be solved for the Cournot-Nash equilibrium effort level L∗ . Note that L∗ is a function of α, i.e., L∗ (α). Suppose theP team as a wholePcan deploy effort to maximize team welfare W = n Ui = F − n Vi (Li ). The first order condition for team maximum is, under symmetry, F0 = V0
(3)
which can be solved for the Pareto optimal effort level L∗∗ . Note that in (3), the allocation parameter α does not figure in contrast to its role in (2). In general, therefore, L∗ 6 = L∗∗ , and Nash equilibrium effort level is Pareto suboptimal. The circumstances where L∗ = L∗∗ is of importance to us here. Clearly L∗ = L∗∗ if and only if, in (2), we have [(1 − α)(n − 1)/nr) + n−1 ] = 1 which simplifies to α = 1 − r.
(4)
This is the well-known Sen condition for a Pareto optimal Nash equilibrium in the absence of other inputs. (4) then constitutes an environment where member self-seeking leads to the social optimum. A. Smith’s ‘invisible hand’ works in this constrained environment. In the 60’s milieu, this result is of interest for another reason. One
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can claim that some egalitarianism (equity) is compatible with (indeed, conduces towards) efficiency – an issue which seems dated at the close of this century. The drawback of (4) is that both α and r are exogenous and equality is only an unlikely happenstance – a problem it shares with, say, the ‘Harrod’Domar knife edge’. Thus, even with observable effort, member self-seeking will in general, block the social optimum. On the other hand, without the environment given by (4), the cooperative (Pareto optimal) solution given by F 0 = V 0 is not self-enforcing. Our task is to endogenize α.
3. CONTRACTARIAN MECHANISM
V. Pareto (1896) in another context remarked that the question of income distribution must first be resolved before the issue of resource allocation can be addressed. The degree of equity in income distribution we assume to be resolved within the team itself and therefore by the members themselves even before effort supply by members becomes an issue. The former is stage I in the decision tree for members and the latter, stage II. The important idea that we will exploit is that stage I decisions are not independent of stage II decisions – the so-called backward induction routine of the subgame perfect equilibrium. We assume that members are consulted on their preferred level of egalitarianism (α > 0) and the value preferred by a simple majority is adopted as the team α. The crucial issue is how a member i chooses his preferred α? We again assume that he chooses that α which maximizes his own individual utility subject to the constraint that α now affects not only his own effort supply Li in stage II but also the effort supply of other members in that stage, i.e., ∗0 that L∗j (α), j = 1, 2, . . . , n, L∗0 j = dLj /dα 6 = 0. This makes eminent sense because α constitutes with r the environment under which his effort supply decision will be made. This is the backward induction routine that leads to subgame perfection – very popular in modern industrial organization especially in regard the choice of R & D in oligopolistic markets (see, e.g., Bulow, Geanokoplos and Klemperer, 1984). Members absolutely ignore how others choose their own preferred α.
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DEFINITION 1. We call the team contractarian if the team sharing parameter α ∗ takes the value most preferred by at least a simple majority of members. THEOREM 1. A contractarian team achieves a Pareto optimal symmetric Nash equilibrium. Proof. We first substitute L∗j (α) from (1) for Lj , j = 1, 2, . . . , n, in Ui = si F − Vi (Li ). The first order condition for a maximum with respect to α is: " ! ! n n n X X X 0 ∗ −L∗i / L∗j + (1 − α) L∗j L∗0 L∗j i − Li X
L∗j
+ si F 0
−2
n X
−1
+n
(5)
F
0 ∗0 L∗0 j − Vi Li = 0,
i, j = 1, 2, . . . n
where L∗0 = dL∗j /dα = 1, 2, . . . , n and L∗j = L∗j (α). In (5), there will be n different α’s for n different member since Vi 6 = Vj . We now impose symmetry to get: [−n−1 + n−1 + (1 − α)(nL∗ L∗0 − L∗ nL∗0 )]F + F 0 L∗0 − V 0 L∗0 = 0,
(6)
which simplifies finally into F 0 = V 0.
(7)
Every member prefers the α that sets marginal product of effort equal to marginal disutility (the Pareto condition) in a contractarian team. At symmetric equilibrium this α ∗ is, from (4), α ∗ = 1 − r. Since symmetry is assumed, α ∗ will be preferred by all members and so becomes contractarian team α. Thus a symmetric contractarian team achieves a Pareto optimal Nash equilibrium. Remark 1. If F is not homogenous, then (7) implies that every member’s most preferred α is the one that sets [(1 − α)(n − 1)F (F 0 Ln)−1 + n−1 ] = 1 in (2) above. This does not readily yield a closed form solution.
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It is instructive to delve further into why this two-stage process works. With Cournot conjectures, agents act as if their effort decisions are independent of others’ effort decisions in the second stage. This is normally generative of Pareto suboptimality. If, in addition, agents are confronted with the decision on α, they are forced to confront the fact that others react to α which is public and known to everyone from the beginning of stage II (in contrast, Lj may be known to i only at the end of stage II from a monitoring report) just as they themselves react to α from (2). Thus, stage I forces the interdependence of agents’ decisions. How important this is is revealed by the following game. Suppose in lieu of Cournot conjectures in stage II (i.e., (dLj / dLi = L0j = 0)), we have perfect matching behavior (if effort is perfectly observable and at once). That is, dLj /dLi = L0j = 1. This may also be called ‘tit-for-tat’, the continuous rather than discrete binary version (Kreps, Milgrom, Roberts and Wilson, 1982; Axelrod, 1984) or ‘reciprocal altruism’ (Nowak, May and Sigmund, 1995; Lien, 1987). Then (1) becomes !−2 n n n X X X 0 0 (1−α) F +si F Lj −Li Lj L0j −Vi0 = 0. P P Under symmetry, ( n Lj − Li n L0j ) = (nL − nL) = 0 and P si F 0 n L0j = Vi0 . Thus (7) becomes F 0 = V 0,
(8)
and perfect matching delivers the Pareto optimum for an α and r in stage II. But perfect matching. like perfect altruism, is subject to strategic game playing. The individual who shirks will avoid additional effort but share in additional bounty at least before the onset of the next cycle. Nonetheless, the symmetric contractarian team exploits one of the perfect matching game’s steps, i.e., from (7) we see that (1 − α)(nL∗ L∗0 − L∗ nL∗0 ) = 0 ∗0 since L∗0 j = Li ’. The second stage contractarian process with symmetry operates like perfect matching ((dLj /dLi ) = 1) and appears
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to undercut the zero conjectural variation in stage II. Note that the contractarian mechanism works only if the budget balance is maintained in stage II. For example, if in lieu of Sen sharing, si = (1 − α)(Li /L) is used in stage II, F 0 = V 0 will not be attained because P si = (1 − α) < 1 showing a budget surplus. This results in the expression (n−1 −n−1 ) in (6) being replaced by another one unequal to zero. The stage II symmetry assumption does more. It sees to it that the α ∗ that maximizes i’s utility is the same α ∗ that maximizes j ’s. This ensures that a simple majority will prefer α ∗ . Now this, on the face of it, is heroic because the symmetry assumption is normally adopted to facilitate analysis but not to fundamentally affect the outcome. In this case, it does. This, therefore, calls for a practical motivation for symmetry.
4. VOTING WITH ONE’S FEET AND SYMMETRY
The membership symmetry assumption must be given an economic interpretation so as to affect economic outcomes. Actual symmetry in teams as opposed to methodological symmetry may be realized by ‘voting with one’s feet’ (Tiebout, 1956; Cornes and Sandler, 1986). Tiebout suggests that taxpayers who are free to move will relocate to areas whose tax structures agree with their priors. If the team has voluntary membership and there is an outside environment of sufficient diversity, then a team will attract members who subscribe to its α ∗ (its constitution in effect). Exit will render nonsubscribers scarce. Voting with one’s feet merely reverses the contractarian steps: the constitution first-membership later vs. membership first-constitution later. It does not change the contractarian substance. It has, however, the attractive property of effective symmetry and, via self-selection, a majoritarian decision process. Voting with ones feet gives indeed the empirical content of this exercise: We hypothesize that teams or partnership with free entry from a large pool of potential members will be more efficient than ones with forced entry (governed say by geography). The former teams (with free entry) will be so much more likely to be contractarian, i.e., a symmetric membership in agreement with preannounced rules.
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This is our account of the Kibbutz-Kolkhoz paradox. The Kibbutz subscribe to open entry and attracts members from far and wide. In contrast, the Kolkhoz was largely imposed on Russian peasant farmers bound only by geography and shirking naturally follows. There are of course other differences on interest, viz., Kibbutz oranges have to be competitive in price and quality in order to sell in Europe while the Kolkhoz survival was not subject to market discipline. It seems that for the superiority of the Kibbutz, egalitarian sharing is only incidental. The contractarian viewpoint is one way that the capitalist labor market may have contributed as well to the well-being of market economies, whose vitality has most of the time been granted other attributions (say, entrepreneurship or innovation). The role of homogeneity here is to facilitate the political process towards a Pareto-inducing constitution. When such homogeneity is absent, Pareto efficiency may escape the team’s grasp. This may explain, apart from pure prejudice, the persistence and attraction for some people of homogeneous societies (e.g. Japan). The results here differ from Fabella (1988) where the concern was the production technology that supports Pareto efficiency under proportional sharing. It also differs from Fabella (1998) which explores the efficiency possibility from the sharing technology angle and suggests one generalization that forces Pareto efficiency for certain circumstances.
5. CONCLUSION
In this paper, we deal with a team (or a partnership) where effort is observable and sharing is Sen type and exhaustive. Team members are strictly rational and supply effort voluntarily towards revenue production. It is well-known that only the environment given by α = 1 − r, where α is the egalitarian bias and r is the degree of homogeneity, supports a Pareto efficient symmetric Nash equilibrium. Thus, team Pareto optimality is a rare exception. What could make a rule of the exception? We show that if agents are asked to reveal their preferences for that degree of egalitarianism (in stage I) which maximizes individual utility subject to the effort supply response of everyone in the effort stage (II), each will choose that α which equates marginal product of
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effort with its marginal disutility. Under membership symmetry, the α’s for all n agents will be identical and becomes the team α if the team is contractarian. The contractarian team embracing the sharing technology most preferred by the membership adopts precisely the environment (constitution) that sustains Pareto efficiency. The prior contractarian constitutional process in stage I appears to disarm the Nash zero conjectural variation without robbing its outcome of attractive features (self-enforcing and other Johansen (1982) desiderata) in the effort supply stage. Voting with one’s feet appears to be the mechanism for symmetry realization and contractarian group choice in the real world and appears to partly explain the differential performance between the Israeli Kibbutz and the Russian Kolkhoz. It is also in order to exploit the Pareto-facilitating effect of symmetry that homogenous societies persist and continue to attract adherents.
ACKNOWLEDGMENT
I thank the Philippine Center for Economic Development (PCED) for financial support. I am grateful to Emmanuel de Dios, Joy Abrenica and Emmanuel Esguerra for helpful discussions. I also thank Millet Villanueva for excellent secretarial help. Errors are mine alone.
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Address for correspondence: School of Economics, University of the Philippines, Diliman, Quezon City 1101, Phone: (632) 927-9686 to 92 & 920-5463, Fax: (632) 920-5462 & 921-3359; e-mail:
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