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A Solution to the Tragedy of the Commons and a Problem for Anti-Trust Authorities

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A Solution to the Tragedy of the Commons and a Problem for Anti-Trust Authorities
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  A Solution to the Tragedy of the Commons and a Problem forAnti-Trust Authorities Martin D. Heintzelman Stephen W. SalantFebruary 2, 2004 1 Introduction We examine here a clever solution proposed by Schott (2003) to the “tragedy of the com-mons.” It is well known that if N individuals independently choose their effort levels toextract from a common property resource, subject to congestion externalities, that aggre-gate effort will exceed the social optimum: over-harvesting will result. If, on the other hand,N individuals form a single “partnership” that  shares   the fruits of their efforts equally re-gardless of individual effort while each individual continues to pay the cost of his own effort,aggregate effort in the resulting Nash equilibrium would be socially insufficient because of excessive free-riding: under-harvesting would result. In both of these cases, every individualbears his own cost of effort, the partnership receives a share of aggregate output equal to itsshare of aggregate effort, and, within each partnership, output is shared equally regardlessof individual effort. In the first case, every individual belongs to a different partnershipwhile, in the second case, all  N   individuals belong to the same partnership. Since over-harvesting occurs in the first case and under-harvesting in the second, Schott hypothesizedand then proved that  socially optimal   effort could be induced (or approximated in case of 1  integer problems) if the N players are divided into an intermediate number of partnershipsof equal size. Intuitively, the free-riding induced by the partnerships then exactly offsetsthe inherent congestion externality.To illustrate Schott’s solution, consider the following example. Suppose 12 players earntheir livelihood working in an activity plagued by a congestion externality. Normalize theprice of output to unity and assume aggregate production equals 19 X   −  X  2 , where  X  represents their aggregate effort. Suppose that the cost per unit of effort is equal to 3.For concreteness, the players might be lobster men,  X   might be the aggregate number of traps they set, and 19 X   −  X  2 might be the dollar value of their aggregate catch. It isstraightforward to see that the socially optimal effort level (or number of traps) is 8 sincethat equates marginal benefit (19  −  2 X  ) and marginal cost (3).Assume that, given their exogenous partnership assignments, individuals choose effortsimultaneously and a Nash equilibrium results. If each individual belongs to his own solopartnership, then aggregate effort would be more than 75% larger than the social optimum(the excessive effort we associate with the tragedy of the commons). If, on the other hand,every individual belongs to the same grand partnership, then equilibrium aggregate effortis 0 (an extreme form of the insufficient effort we associate with free-riding). But if thereare  exactly   four groups, each with 3 individuals, then in the resulting Nash equilibrium eachof the 12 agents will choose an effort level of 2/3 and aggregate effort will be exactly thesocial optimum (12  ·  2 / 3 = 8). 1 In a sequel (Schott et. al., 2003), Schott has verified using controlled experiments thataggregate effort does in fact increase as the number of partnerships increases and thatsocially optimal effort can be approximated by grouping experimental subjects into theappropriate number of partnerships. 1 As we will show,  any   partition of the 12 individuals into four partnerships results in a pure-strategyNash equilibrium with the socially efficient level of effort. 2  Schott’s partnership scheme induces a Nash equilibrium in individual effort levels that issocially optimal: no individual has an incentive to alter his effort level unilaterally. But inboth his analytical and experimental work, Schott treats the group assignment as  exogenous  .In reality, however, there is nothing to prevent an agent from switching groups—or for thatmatter from going solo although that would entail forgoing any benefits of team production.For Schott’s solution to be workable in practice, no individual can have a strict incentiveto deviate from his assigned group unilaterally. The purpose of our paper is to explore thestability of Schott’s proposed solution when, in addition to deviating in effort, agents maydeviate in group membership.Before we begin, it is important to recognize the darker side of Schott’s proposed so-lution. While it holds the promise of resolving an important environmental problem, hisproposal also poses the threat of exacerbating an important problem in industrial organi-zation. If Schott’s solution is stable, it can also be used to keep the homogeneous outputof a cartel stabilized at the monopoly level. To see this, reinterpret the foregoing exampleas follows. Let  X   denote aggregate output, let 19 X   − X  2 denote industry revenue, andassume that each of 12 players produces at a constant marginal cost of 3. Then, the inversedemand curve is  P   = 19 − X,  and an output of 8 units maximizes industry profit. If the12 players compete against each other as Cournot players, output is more than 75% abovethe monopoly level. This is the standard textbook story of overproduction by cartel mem-bers. But what if the 12 members are divided into four partnerships, each group sharingequally with its members the revenues received. In the resulting Nash equilibrium, themonopoly output will be produced and each of the 12 members will earn an equal shareof the monopoly profits without a need for sidepayments or complex, history-dependentstrategies over an infinite horizon. Note that law firms, architectural firms, accountingfirms, consulting firms, and medical practices often use this organizational form (revenues3  distributed according to fixed shares), as do some cooperatives. 2 In Section 2, we assume that each individual chooses a group and an effort level at thesame time. In that case, unilateral deviations in effort or group membership are met withno reaction from the other players since such deviations cannot be detected. In Section 3,we assume that each individual simultaneously selects a group in the first stage and then,after observing those choices, simultaneously selects an effort level in the second stage. Inthat case, everyone can observe a unilateral deviation in group membership chosen in thefirst stage and can  react   to it when selecting his effort level in the second stage. In Section4, we re-examine our conclusions about the stability of Schott’s proposal under alternativeassumptions. Section 5 concludes the paper. 2 Simultaneous-move Game First, we define the notation that will be used throughout this paper. m i  = number of members of group i  x ik = effort level of agent k in group i  Y   − ki  = aggregate effort level of members of group i other than agent k  X  − i = aggregate effort of other groups X = total effort level (sum of all agents’ efforts) f(X) = aggregate production function w = constant marginal cost of effort (in dollars per unit effort) A ( · ) =  f  ( X  ) X   =  average product (in dollars per unit effort) ¯ x i  = ( x i + Y   − ki  ) m i =  mean effort level in group i  2 See Farrell and Scotchmer (1988) for other examples and an analysis of partnerships using cooperativegame theory. 4  We also assume that (1)  A ( X  ) is strictly positive, strictly decreasing, and twice continuouslydifferentiable; (2)  A (0) − w >  0; and (3) the Novshek (1985) condition,  A  ( X  )+ XA  ( X  )  <  0 , holds for all  X >  0 . By definition, socially optimal effort ( X  ∗ ) maximizes X  ( A ( X  ) − w ) . Since A ( · ) is assumedsmooth and  A (0)  − w >  0 ,X  ∗ must satisfy the following first-order condition: A ( X  ∗ ) + X  ∗ A  ( X  ∗ )  − w  = 0 .  (1)Since the Novshek condition holds,  X  ∗ is unique. Suppose each agent simultaneouslychooses a group number and a level of effort. Since there are  N   agents, there can, inprinciple, be at most  N   groups. Let  n  denote the number of distinct groups specified bythe agents and index them  i  = 1 ,...,n . An individual in group  i  would choose his owneffort level ( x ik ) taking as given the aggregate effort level of his colleagues in partnership i  ( Y  − ki  =   l  = k x il ) as well as the aggregate effort levels of the other partnerships ( X  − i ).Hence, he would maximize π ik  =  Max x ik   1 m i   x ik  + Y  − ki x ik  + Y  − ki  + X  − i  · f  ( x ik  + Y  − ki  + X  − i )  − wx ik  which is equivalent to maximizing: m i π ik  =  x ik  + Y  − ki  · A  x ik  + Y  − ki  + X  − i  − m i wx ik .  (2)To find the equilibrium effort levels for a given partition of the players into groups  { m i } ni =1 ,differentiate the objective function (2) and substitute  X   =  x ik  + Y  − ki  + X  − i  to arrive at thefollowing first-order conditions: A ( X  ) +  x ik  + Y  − ki  · A  ( X  )  − wm i  = 0 for  i  = 1 ,...,n  and  k  = 1 ,...,m i . 5
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