Competitive equilibrium

Competitive equilibrium (also called: Walrasian equilibrium) is the traditional concept of economic equilibrium, appropriate for the analysis of commodity markets with flexible prices and many traders, and serving as the benchmark of efficiency in economic analysis. It relies crucially on the assumption of a competitive environment where each trader decides upon a quantity that is so small compared to the total quantity traded in the market that their individual transactions have no influence on the prices. Competitive markets are an ideal standard by which other market structures are evaluated.

A competitive equilibrium (CE) consists of two elements:

These elements should satisfy the following requirement:

Often, there is an initial endowment matrix ${\displaystyle E}$: for every ${\displaystyle i\in 1,\dots ,n}$, ${\displaystyle E_{i}}$ is the initial endowment of agent ${\displaystyle i}$. Then, a CE should satisfy some additional requirements:

An alternative definition relies on the concept of a demand-set. Given a price function P and an agent with a utility function U, a certain bundle of goods x is in the demand-set of the agent if: ${\displaystyle U(x)-P(x)\geq U(y)-P(y)}$ for every other bundle y. A competitive equilibrium is a price function P and an allocation matrix X such that:

In some cases it is useful to define an equilibrium in which the rationality condition is relaxed. Given a positive value ${\displaystyle \epsilon }$ (measured in monetary units, e.g., dollars), a price vector ${\displaystyle P}$ and a bundle ${\displaystyle x}$, define ${\displaystyle P_{\epsilon }^{x}}$ as a price vector in which all items in x have the same price they have in P, and all items not in x are priced ${\displaystyle \epsilon }$ more than their price in P.

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