Description

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

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.

Related Documents

Share

Transcript

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

Search

Similar documents

Related Search

Finding a resourceful solution to the problemFinding solution to the current problems in hSolution to the Problem on Stock Exchange MarEffect of Geogebra to the Achievement and AttGeneral Solution to the Unidimensional HamiltSolution to the problem that hinders infrastrsolution manual The Science and Engineering oEngineering Related to the Management and ProLetter to the Hebrews and Christian TheologyThe teaching and learning process for Zero wa

We Need Your Support

Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks