Brouwer's fixed point theorem cannot be constructively proved, so the existence of an equilibrium in a competitive economy also cannot be constructively proved. On the other hand, Sperner's lemma which is used to prove Brouwer's theorem is constructively proved. Some authors have presented a constructive (or an approximate) version of Brouwer's fixed point theorem using Sperner's lemma. In this paper, I prove the existence of an approximate equilibrium in a competitive economy directly by Sperner's lemma. Also I show that the existence of an approximate equilibrium leads to Sperner's lemma. I follow the Bishop style constructive mathematics according to Bishop and Bridges (1985), Bridges and Richman (1987), and Bridges and Vîţă (2006).
It is often demonstrated that Brouwer’s fixed point theorem cannot be constructively proved. Thus, the existence of an equilibrium in a competitive economy also cannot be constructively proved. On the other hand, however, Sperner’s lemma which is used to prove Brouwer’s theorem is constructively proved. Some authors have presented a constructive (or an approximate) version of Brouwer’s fixed point theorem using Sperner’s lemma. Thus, Brouwer’s fixed point theorem can be constructively proved in its constructive version. See [
Then, can we prove the existence of an approximate equilibrium directly by Sperner’s lemma?
This paper presents such a proof and also shows that the existence of an approximate equilibrium leads to Sperner’s lemma. An approximate equilibrium in a competitive economy is a state such that excess demand for each good is smaller than
In the next section, the proof of Sperner’s lemma will be presented. This proof is a standard constructive proof. In Section
To prove Sperner’s lemma, we use the following simple result of graph theory, the Handshaking lemma (For another constructive proof of Sperner’s lemma, see [
Example of graph.
Every undirected graph contains an even number of vertices of odd degree, that is, the number of vertices that have an odd number of incident edges must be even.
It is a simple lemma. But for completeness of arguments we provide a proof.
Prove this lemma by double counting. Let
Let
Partition and labeling of 2-dimensional simplex.
Let the vertices of if a vertex of if a vertex of a vertex contained inside of
A small simplex of
If one labels the vertices of
See the appendix.
Consider a competitive exchange economy. There are
We assume that the excess demand function
Consider the following function from the set of price vectors
Now, we show the following.
In a competitive exchange economy, if the excess demand functions for the goods are uniformly continuous about their prices, then there exists an approximate equilibrium and one can constructively find the prices at the approximate equilibrium.
(1) First, we show that we can partition
For example, let
Consider the case where
Next, consider the case where
The conditions for Sperner’s lemma are satisfied, and there exists an odd number of fully labeled simplices in
(2) Suppose that we partition
The uniform continuity of
(3) Denote a point which satisfies (
In this section, we will derive Sperner’s lemma from the existence of an approximate equilibrium in a competitive economy. Let us partition an
Now, using
Let
Since
In this paper, I have presented a proof of the existence of an approximate equilibrium in a competitive economy directly by Sperner’s lemma from a viewpoint of constructive mathematics. In another paper [
We prove this lemma by induction about the dimension of
Next, consider the case of 2 dimension. Assume that we have partitioned a 2-dimensional simplex (triangle)
Sperner’s lemma.
Now, assume that the theorem holds for dimensions up to
If the number (label) of a vertex other than vertices labeled with
We have completed the proof of Sperner’s lemma.
Since
The author thanks the anonymous referees for their very useful comments. This research was partially supported by the Ministry of Education, Science, Sports and Culture of Japan, Grant-in-Aid for Scientific Research (C), no. 20530165, and the Special Costs for Graduate Schools of the Special Expenses for Hitech Promotion by the Ministry of Education, Science, Sports and Culture of Japan in 2010.