Perfect Competition
and the
Nash Bargaining Solution
Reinhard John
Department of Economics
University of Bonn
Adenauerallee 24-42
53113 Bonn, Germany
email: [email protected]
May 2005
Abstract
For a linear exchange economy, it is proved that the utility allocation
of a Walrasian equilibrium with respect to a proportional division of total
endowments coincides with the corresponding asymmetric Nash bargaining
solution of the bargaining problem induced by the feasible utility allocations of the economy. In particular, the equal division Walrasian equilibrium yields the symmetric Nash solution. It is further shown that the proof
can be adapted to an extension of this result to exchange economies with
linear homogeneous utility functions.1
1
Introduction
Imagine that a bundle of divisible goods should be distributed ”fairly” among
some individuals whose preferences are described by known linear utility functions. Consider the following two alternative approaches to solve this problem.
The first one is to interpret it as a bargaining problem represented by the set
of feasible utility allocations and to apply one of the known bargaining solution
concepts. However, which one should be used?
1
Addendum (May 2013): Recently, I became aware of two contributions that already dealt
with the subject. The result for linear utility functions can be found in Gale (1960), Theorem
8.5. The extension to linear homogeneous utility functions is presented in Polterovich (1973),
Theorem 4.
1
The other one is to propose an equal division of the given bundle and to determine the final allocation by perfect competition, i.e. to choose a Walrasian
equilibrium allocation with respect to this endowment distribution as a solution.
It has been shown by Gale (1976) that the induced utility allocation is unique.
Thus, the natural question is: Does it coincide with one of the conceivable bargaining solutions in the first approach?
The answer given in this paper is affirmative. It will be shown that the second
approach yields the (symmetric) Nash bargaining solution. Actually, we prove in
Section 3 a slightly more general result that includes the asymmetric case. The
proof even reveals that it can be further generalized to the case that the utility
functions are linear homogeneous. We sketch this generalization in a concluding
remark.
This contribution is not the first one that relates the Nash bargaining solution
to Walrasian equilibrium. Trockel (1996) established a relationship in a very
different way. Given a bargaining problem, he defined a production economy for
which the agents’ utilities serve as commodities such that the consumption vector
of its unique equilibrium coincides with the Nash solution.
Although its message is different, the spirit of this paper is more akin to the
work by Ervig and Haake (2003) who employed the idea of perfect competition
in order to support the Perles-Maschler bargaining solution. At first sight, this
does not seem compatible with our result. However, the crucial difference is that,
in their approach, consumer demands are restricted by the total endowments of
the economy. This implies that in the linear case there may be many equilibrium
utility allocations, in particular the one corresponding to the Perles-Maschler
solution.
There is another contribution that seems to contradict our observation. Sertel and Yildiz (2003) claim the impossibility of a Walrasian bargaining solution.
The reason is their different model. They take an arbitrary endowment distribution as given and consider the bargaining problem defined by the feasible utility
allocations and the initial utility allocation as threat point.
2
Basic definitions
We consider an economy with m consumers i = 1, ..., m and n commodities
j = 1, ..., n. Each consumer is described by a linear utility function ui : Rn+ −→ R,
defined by
i
i
u (x ) =
n
X
aij xij
(1)
j=1
for xi = (xij )nj=1 . It is assumed that these functions are monotone, i.e. the
coefficients aij are nonnegative real numbers and for every i there is some j such
that aij > 0.
2
The total endowment of the economy is given by the commodity vector e =
(ej )nj=1 ∈ Rn++ . Thus, a feasible allocation x = (xi )m
i=1 has to satisfy the inequality
m
P
xi 6 e. Consequently, the set
i=1
i
U = {(u (x
i
))m
i=1 |
m
X
xi 6 e} ⊆ Rm
+
i=1
consists of all feasible utility vectors for this economy. It is obvious that U is
compact, convex, and satisfies the free disposal condition (U − Rn+ ) ∩ Rn+ ⊆ U , i.e.
(U, 0) has the usual properties required for a bargaining problem with status-quo
point 0 ∈ Rm
+.
A (not necessarily symmetric) Nash bargaining solution for (U, 0) defines a
unique utility vector in U induced by some (not necessarily unique) feasible allocation.
More precisely, let α = (αi )m
i=1 be a vector of positive weights, i.e. αi > 0 for
m
P
i = 1, ..., m and
αi = 1. Define for an allocation x = (xi )m
i=1
i=1
α
U (x) =
m
Y
ui (xi )αi .
(2)
i=1
Definition 1
A feasible allocation x̄ is called a Nash allocation with respect to α if it maximizes
U α on the set of all feasible allocations, i.e. if x̄ is a solution to
max U α (x)
subject to
xij > 0 (i = 1, ..., m and j = 1, ..., n)
m
X
xij − ej 6 0 (j = 1, ..., n).
i=1
We want to relate Nash allocations to particular competitive allocations that
are introduced in
Definition 2
An allocation x̄ is called a Walras allocation with respect to (the ownership or
income distribution) α if there exists a price vector p = (pj )nj=1 ∈ Rn+ \ {0} such
3
that
(i)
x̄i maximizes ui (xi ) subject to
xij > 0 (j = 1, ..., n)
For i = 1, ..., m :
pxi 6 αi pe.
m
m
X
X
i
i
x̄ 6 e and p
x̄ − e = 0.
(ii)
i=1
3
i=1
Results
It will be shown that Nash allocations and Walras allocations coincide. The
main step of the proof is to observe that both can be characterized by first order
conditions.
Lemma 1
An allocation x̄ = (x̄i )m
i=1 is a Nash allocation with respect to α if and only if
there exist λ1 , ..., λn > 0 such that for i = 1, ..., m and j = 1, ..., n
αi aij U α (x̄) 6 λj ui (x̄i )
αi aij x̄ij U α (x̄)
m
X
x̄ij
i=1
m
X
λj x̄ij
i=1
=
(3)
λj x̄ij ui (x̄i )
(4)
6 ej
(5)
= λj ej
(6)
Proof: By Definition 1, x̄ is a solution to a constrained maximization problem
with a concave objective function U α and linear constraints. Hence, x̄ is characterized by the Kuhn-Tucker first order conditions for this problem, i.e. by the
existence of λ1 , ..., λn > 0 such that for i = 1, ..., m and j = 1, ..., n
∂U α (x̄)
− λj 6 0
∂xij
m
X
x̄ij − ej 6 0
∂U α (x̄)
and
− λj x̄ij = 0,
∂xij
m
X
x̄ij − ej λj = 0.
and
i=1
i=1
The definitions of ui and U α in (1) and (2) imply
α
Y
∂U α (x̄)
i i i αi −1
k k αk
i U (x̄)
u (x̄ ) = αi aj i i ,
= αi aj u (x̄ )
∂xij
u (x̄ )
k6=i
4
where we used the fact that all ui (x̄i ) are positive if x̄ solves the maximization
problem (the feasible allocation x with xi = m1 e yields positive utilities). Thus,
(3) - (6) follow immediately.
Lemma 2
An allocation x̄ = (x̄i )m
i=1 is a Walras allocation with respect to α if and only if
there is a price vector p = (pj )nj=1 ∈ Rn+ \ {0} and there exist µ1 , ..., µm > 0 such
that for i = 1, ..., m and j = 1, ..., n
aij 6 µi pj
(7)
µi pj x̄ij
(8)
px̄ 6 αi pe
µi px̄i = µi αi pe
m
X
x̄i 6 e
(9)
(10)
aij x̄ij
i
=
(11)
i=1
m
X
px̄i = pe
(12)
i=1
Proof: By Definition 2 (i), x̄i is a solution to a linear programming problem
and is characterized by the first order conditions (7) - (10). (11) and (12) are a
restatement of (ii) in Definition 2.
Proposition
An allocation x̄ = (x̄i )m
i=1 is a Nash allocation with respect to α if and only if it
is a Walras allocation with respect to α.
Proof: If x̄ = (x̄i )m
i=1 is a Nash allocation with respect to α then, by Lemma 1,
there are λ1 , ..., λn > 0 such that (3) - (6) are satisfied for all i and j. Since there
is at least a consumer i with aij > 0 for some j, (3) and U α (x̄) > 0 imply λj > 0
for some j, i.e. λ = (λj )nj=1 ∈ Rn+ \ {0}.
Define p = λ and µi = ui (x̄i )/αi U α (x̄) for i = 1, ..., m. In order to prove that
x̄ is a Walras allocation with respect to α, it remains to show that (7) - (12) are
satisfied.
Obviously, (3) and (4) imply (7) and (8). Summation of (4) with respect to j
yields
X
X
aij x̄ij = αi U α (x̄)ui (x̄i )
αi aij x̄ij U α (x̄) = αi U α (x̄)
j
j
equal to
X
λj x̄ij ui (x̄i )
=
X
j
j
5
λj x̄ij
ui (x̄i ).
Since ui (x̄i ) > 0, αi U α (x̄) =
P
λj x̄ij . This implies
j
U α (x̄) =
X
αi U α (x̄) =
i
and, by (6), U α (x̄) =
P
XX
i
λj x̄ij =
XX
j
j
λj x̄ij
i
λj ej .
j
Since λj = pj for j = 1, ..., n, we obtain
X
X
px̄i =
λj x̄ij = αi U α (x̄) = αi
λj ej = αi pe,
j
j
i.e. (9) and (10) are verified. Finally, (5) is equivalent to (11) and (6) implies
X
XX
XX
X
px̄i =
λj x̄ij =
λj x̄ij =
λj ej = pe,
i
i
j
j
i
j
i.e. (12) holds.
Assume now that x̄ is a Walras allocation with respect to α. By Lemma 2,
(7) - (12) hold for some p ∈ Rn+ \ {0} and some µ1 , ..., µm > 0.
From (8) it follows by summing over j that
X
X
ui (x̄i ) =
aij x̄ij = µi
pj x̄ij = µi px̄i .
(13)
j
j
Since ui (x̄i ) is positive if x̄i maximizes utility on the budget set, µi > 0 for
every i. Hence, by (10), px̄i = αi pe and, by (13), µi = ui (x̄i )/αi pe.
U α (x̄)
pj . By Lemma 1, it remains to show that (3) - (6) are
Define λj :=
pe
satisfied.
Inserting µi = ui (x̄i )/αi pe into (7) yields
αi aij U α (x̄) 6
ui (x̄i )
pj U α (x̄) = λj ui (x̄i ),
pe
i.e. (3) holds.
Analogously, by (8),
αi aij x̄ij U α (x̄) =
ui (x̄i )
pj x̄ij U α (x̄) = λj x̄ij ui (x̄i ),
pe
i.e. (4) is verified.
Trivially, (11) implies (5). From (5) it follows that
X
λj x̄ij 6 λj ej for all j.
i
6
Actually, these are equalities as required by (6) since otherwise
X
XX
X
λx̄i =
λj x̄ij <
λj ej = λe,
i
j
contradicting (12) by using λ =
4
i
j
U α (x̄)
p.
pe
Concluding Remark
The results in the previous section remain valid if the linear utility functions are
replaced by functions ui with the following properties:
(1) ui is linear homogeneous and concave.
(2) ui (0) = 0 and ui (x) > 0 if x ∈ Rn++ .
(3) ui is continuously differentiable on {x|ui (x) > 0}.
Under these assumptions, the set U of feasible utility vectors is compact, convex,
and satisfies the free disposal condition. Furthermore, the Nash and Walras
allocations are still characterized by first order conditions since the corresponding
optimization problems are concave programs. Hence, in the statements and proofs
of Section 3 the coefficients aij have to be replaced by the partial derivatives
∂ui (x̄i )/∂xij . By linear homogeneity, it follows that
ui (x̄i ) =
X ∂ui (x̄i )
∂xij
j
i.e. the essential identity ui (x̄i ) =
P
x̄ij ,
aij x̄ij remains valid after the replacement.
j
References
[1] Ervig, U. and C.J. Haake: ”Trading Bargaining Weights”, Journal of Mathematical Economics 41 (2005), 983-993.
[2] Gale, D.: The Theory of Linear Economic Models, McGraw-Hill 1960.
[3] Gale, D.: ”The Linear Exchange Model”, Journal of Mathematical Economics
3 (1976), 205-209.
[4] Polterovich, V. ”Economic Equilibrium and the Optimum”, Economics and
Mathematical Methods 9 (1973), 835-845 (in Russian, Engl. transl.: Matekon
XI(2), 3-20).
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[5] Sertel, M.R. and M. Yildiz: ”Impossibility of a Walrasian Bargaining Solution”, in Koray, S. and M.R. Sertel (eds.), Advances in Economic Design,
Springer 2003.
[6] Trockel, W.: ”A Walrasian Approach to Bargaining Games”, Economics Letters 51 (1996), 295-301.
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