5. NASH BARGAINING AND PERFECT COMPETITION the equivalent maximization problem
max
n
X
i=1
αiln (di) (5.3)
w.r.t. X
di = 1. (5.4)
Using the Lagrange approach we get that for a maximizer of this problem dˆ1, ...,dˆn there existsλ≥0 such that for all i
αi
dˆi
=λ and
Xdˆi = 1.
It is easy to see that λ= 1 and ˆdi =αi is the unique solution to this system of equations.
Hence, ˆdi =αi is the unique solution for this maximization problem.
Together with Theorems 3.8 and 3.9 from Chipman and Moore (1979) this completes the proof.
This result shows that under the given assumptions the function ˜Uα can be regarded as the utility function of a representative consumer. While Chipman and Moore (1979) denote their result in terms of Marshallian demand we use the notation in terms of Walrasian demand here. Furthermore, compared with the version of Chipman and Moore (1979) this version is more general as it includes the case that utility functions are homogeneous of degreek. Hereby, one should be aware of the fact that the utility functions of all agents have to be homogeneous of the same degree.
5.3. RESULTS Proof. 1. Assume that ¯x is a Walras allocation with respect to α. By the definition of a Walras allocation there exists a price vector p ∈ Rl+\ {0} such that (p,x) is an¯ α-Walrasian equilibrium. By definition this means that for each i the vector ¯xi maximizes ui xi
subject to xi ≥ 0 and p·xi ≤p·(αie). By Theorem 17 this implies that ¯x is a maximizer of the function
U˜α(x) =
n
Y
i=1
ui xiαi
on the set
Bp = (
x∈
×
i=1nXi
n
X
i=1
p·xi ≤p·e )
. Notice that
A⊆Bp and that
¯ x∈A.
Hence, ¯x also maximizes ˜Uα(x) on the set A. So, ¯x is a Nash allocation with respect toα.
2. Let ¯x= (¯x)ni=1 be a Nash allocation with respect to α. By the definition of Nash allocations the allocation ¯x maximizes ˜Uα(x) =
n
Q
i=1
ui xiαi
on the setA. Notice that the functionUα is quasiconcave and that for eachi∈I the utility functionui is concave.
Therefore the set T =n
z∈Rl+
∃xi ∈Xi, i= 1, ..., n, such that X
xi=z,U˜α x1, ..., xn
>U˜α(¯x)o is convex. Furthermore, we observe thate /∈T. By the separating
Hyper-plane Theorem there exists an vectorp∈Rlwithp6= 0 such thatp·y > p·e) for all y∈T.
As the utility functions are weakly increasing and locally nonsatiated and as the functionUαis strictly increasing , we observe that for allx∈Awith x >x¯ we have ˜Uα(x)>U˜α(¯x). It follows thatp≥0 holds.
Thus, for any allocation ˆx = (ˆx)ni=1 with ˜Uα(ˆx) > U˜α(¯x) we have that 139
5. NASH BARGAINING AND PERFECT COMPETITION
p· P ˆ xi
> p·e. Furthermore, we observep· n
P
i=1
¯ xi
≤p·e. Therefore, ¯x maximizes ˜Uα(x) on the set Bp. Applying Theorem 17 it follows that for allithe vector ¯xi maximizesui on
xi ∈Rl+
p·xi ≤p·(αie) . Hence, the tuple p, x¯in
i=1
is an α-Walrasian equilibrium and the allocation ¯x is a Walras allocation with respect toα.
There are some immediate consequences of this result that can be used to derive properties of competitive equilibria under the given assumptions:
• Fix some vector of weights α. It is well known that the α-asymmetric Nash bargaining solution yields a single point. Hence, each agent’s utility is constant on the set of of Nash allocations with respect to α. As Nash allocations coincide with Walras allocations the same holds true for Walras allocations with respect toα.
• As the utility functions are concave and as each agent’s utility is constant on the set of Nash allocations, it is easy to see that the set of Nash allocations with respect to α is convex. The same holds true for the set of Walras allocations with respect toα.
• It is easy to see that an allocation ¯xcan be at the same time Nash allocation with respect toα1 andα2 for different vectors of weightsα1, α2. Then, ¯xis also a Walras allocations with respect toα1 andα2. This implies that ¯x is an equilibrium allocation for different pricesp1 ∈Rl+ and p2 ∈Rl+. Thus, it can happen that an allocation is an equilibrium allocation for different prices (and different endowments).
• In this context it is individual rational to apply the α-asymmetric Nash bargaining solution with status quo point 0. This follows from the fact that Nash allocations coincide with Walras allocation together with the fact that Walras allocations are individual rational. This point is not a priori clear as individual rationality has to be seen in relation to the utility of the endowments vectors. Individual rationality as given in the axiomatization
5.3. RESULTS of theα-asymmetric Nash bargaining solution with status quo point 0 just shows that the each agent receives at least 0.
The implications of Proposition 19 also hold if all the utility functions are homogeneous of the same degree k with 0 < k < 1 instead of homogeneous of degree 1. Hereby, one should be aware of the fact if a function is homogeneous of degree k >1 then this function can not be concave.1
Corollary 1. Suppose for some 0 < k < 1 the utility functions of all agents i∈ I are homogeneous of degree k and that in addition all assumptions (except homogeneity of degree 1) from Section 2 hold. Then, an allocation x¯ = ¯xim
i=1
is a Nash allocation with respect toα if and only if it is a Walras allocation with respect to α.
Proof. This Proposition can be shown in the same way like Proposition 19.
Hereby, it is important to see that the homogeneity of degree 1 enters in the proof of Proposition 19 only indirectly via Theorem 17. But Theorem 17 is also valid for utility functions that are homogeneous of degreek with 0< k <1.
To prove Corollary 1 it is important that all the utility functions are homo-geneous of the same degree. If the utility functions are homohomo-geneous of different degrees the implication of the result does not have to hold. This point will be discussed more precisely in subsection 5.5.3.
5.3.2 Non-proportional endowments
If the initial endowments are not proportionally distributed there is the problem that it is not clear for which α the α-symmetric Nash bargaining solution could be applied.
If the initial endowments are ω= ωin
i=1 one can compute an equilibrium price
1To see this suppose that a functionf is homogeneous of degreek >0 and concave and suppose that for some x we have f(x) > 0. Then we have f(x) = f 120 +122x
1 ≥
2f(0) + 12f(2x) = 0 +12f(2x) = 2k−1f(x)⇒1 ≥2k−1. Ifk > 1 this inequality is not satisfied.
141
5. NASH BARGAINING AND PERFECT COMPETITION
vectorp(ω). As seen before, this price vector does not have to be unique. Now, one can compute the vector of weightsα(p(ω), ω)∈Rn+ given by
αi(p(ω), ω) :=
t∈R+
p(ω)· teN
=p·ωi
fori∈N. Using this definition, it is easy to see that Walras allocations (coming from the vector of initial endowmentsω) are also Nash allocations with respect to α(p(ω), ω). Hereby, one can not just look on the data of the model and knows for which α the α-bargaining solution is related to competitive equilibria (givenω).
One first has to computeα(p(ω), ω) via the equilibrium prices and then is able to construct α. Moreover, we observe that the vectorα(p(ω), ω) depends on the choice of the equilibrium price vector and the equilibrium price vector does not have to be unique. In particular, consider the situation thatω is already a Nash allocation with respect to someα. Then, it can happen that ω is an equilibrium allocations for different prices which leads to different choices ofα.
On the other hand consider that some allocation ¯x is a Nash allocation for some α. Then, by Proposition 19 there exists a price p ∈ Rl+ such that (p,x) is an¯ α-Walrasian equilibrium. Then, ¯x is obviously an equilibrium allocation for all
ωin
i=1satisfyingp·ωi=p·(αie) fori= 1, ..., n. Moreover, one should recall that
¯
x can also be a Nash allocation with respect to some α′ 6= α. In this way, one could find even more vectors of endowments such that ¯xis a Walras allocation with respect to these endowments. Again, we observe the problem that the relation of α andω is only indirect via the equilibrium prices.
To summarize, these result illustrate that it is mathematically possible to relate Walras allocation and Nash allocations even if the vector of initial endow-ments is not proportionally distributed. Nevertheless, there is a problem with the economic interpretation. The relation of ω and α is in both cases indirectly via equilibrium prices. Thus, the concepts of cooperative game theory and general equilibrium theory are mixed. We do no more analyze distinct concepts and hence a comparison of those is not that meaningful.