Competitive equilibrium

Competitive equilibrium (also called: Walrasian equilibrium) is a concept of economic equilibrium, introduced by Kenneth Arrow and Gérard Debreu in 1951,[1] 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.

Then, a CE should satisfy some additional requirements: This definition explicitly allows for the possibility that there may be multiple commodity arrays that are equally appealing.

(measured in monetary units, e.g., dollars), a price vector

That agent will keep his current bundle as long as it is in the demand-set for price vector

The following examples involve an exchange economy with two agents, Jane and Kelvin, two goods e.g. bananas (x) and apples (y), and no money.

After trading, both Jane and Kelvin move to an indifference curve which depicts a higher level of utility,

The marginal rate of substitution (MRS) of Jane equals that of Kelvin.

The demand function of Jane for x is: The demand function of Kelvin for x is: The market clearance condition for x is: This equation yields the equilibrium price ratio: We could do a similar calculation for y, but this is not needed, since Walras' law guarantees that the results will be the same.

Non-existence example: Suppose the agents' utilities are: and the initial endowment is [(2,1),(2,1)].

When there are indivisible items in the economy, it is common to assume that there is also money, which is divisible.

The same situation holds when the car is not initially held by Alice but rather in an auction in which both Alice and Bob are buyers: the car will go to Bob and the price will be anywhere between 10 and 20.

This is an equilibrium since Bob wouldn't like to pay 5 for the horse which will give him only 4 additional utility, and Alice wouldn't like to pay 7 for the car which will give her only 1 additional utility.

XOR wants either the horse or the carriage but doesn't need both - they receive a utility of

, a competitive equilibrium does NOT exist, i.e, no price will clear the market.

The Arrow–Debreu model shows that a CE exists in every exchange economy with divisible goods satisfying the following conditions: The proof proceeds in several steps.

is kept constant: It is known that, when the agents have strictly convex preferences, the Marshallian demand function is continuous.

such that: so, D. Using Walras' law and some algebra, it is possible to show that for this price vector, there is no excess demand in any product, i.e: E. The desirability assumption implies that all products have strictly positive prices: By Walras' law,

Note that Linear utilities are only weakly convex, so they do not qualify for the Arrow–Debreu model.

However, David Gale proved that a CE exists in every linear exchange economy satisfying certain conditions.

For details see Linear utilities#Existence of competitive equilibrium.

When the utility functions of all agents are GS, a competitive equilibrium always exists.

By the fundamental theorems of welfare economics, any CE allocation is Pareto efficient, and any efficient allocation can be sustainable by a competitive equilibrium.

Furthermore, by Varian's theorems, a CE allocation in which all agents have the same income is also envy-free.

At the competitive equilibrium, the value society places on a good is equivalent to the value of the resources given up to produce it (marginal benefit equals marginal cost).

This ensures allocative efficiency: the additional value society places on another unit of the good is equal to what society must give up in resources to produce it.

It does not in any way opine on the fairness of the allocation (in the sense of distributive justice or equity).

An efficient equilibrium could be one where one player has all the goods and other players have none (in an extreme example), which is efficient in the sense that one may not be able to find a Pareto improvement - which makes all players (including the one with everything in this case) better off (for a strict Pareto improvement), or not worse off.

In the case of indivisible items, we have the following strong versions of the two welfare theorems:[2] In the case of indivisible item assignment, when the utility functions of all agents are GS (and thus an equilibrium exists), it is possible to find a competitive equilibrium using an ascending auction.

In case there is an excess demand on one or more items, the auctioneer increases the price of an over-demanded item by a small amount (e.g. a dollar), and the buyers bid again.