Competitive equilibrium
Competitive equilibrium (also called: Walrasian equilibrium) is a concept of economic equilibrium introduced by Kenneth Arrow and Gérard Debreu in 1951 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. Definitions . A competitive equilibrium (CE) consists of two elements: These elements should satisfy the following requirement: Often, there is an initial endowment matrix formula_9: for every formula_3, formula_11 is the initial endowment of agent formula_5. Then, a CE should satisfy some additional requirements: Alternative definition. 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: formula_16 for every other bundle y. A "competitive equilibrium" is a price function P and an allocation  matrix X such that: Approximate equilibrium. In some cases it is useful to define an equilibrium in which the rationality condition is relaxed. Given a positive value formula_17 (measured in monetary units, e.g., dollars), a price vector formula_1 and a bundle formula_19, define formula_20 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 formula_17 more than their price in P. In a "formula_17-competitive-equilibrium", the bundle x allocated to an agent should be in that agent's demand-set for the "modified  price vector, formula_20. This approximation is realistic when there are buy/sell commissions. For example, suppose that an agent has to pay formula_17 dollars for buying a unit of an item, in addition to that item's price. That agent will keep his current bundle as long as it is in the demand-set for price vector formula_20. This makes the equilibrium more stable. Examples. Divisible resources. 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. 1. Graphical example: Suppose that the initial allocation is 	Competitive equilibrium
15436686	at point X, where Jane has more apples than Kelvin does and Kelvin has more bananas than Jane does. By looking at their indifference curves formula_26 of Jane and formula_27 of Kelvin, we can see that this is not an equilibrium - both agents are willing to trade with each other at the prices formula_28 and formula_29. After trading, both Jane and Kelvin move to an indifference curve which depicts a higher level of utility, formula_30 and formula_31. The new indifference curves intersect at point E. The slope of the tangent of both curves equals -formula_32. And the formula_33; formula_34. The 	Competitive equilibrium
15436687	marginal rate of substitution (MRS) of Jane equals that of Kelvin. Therefore, the 2 individuals society reaches Pareto efficiency, where there is no way to make Jane or Kelvin better off without making the other worse off. 2. Arithmetic example: suppose that both agents have Cobb–Douglas utilities: where formula_37 are constants. Suppose the initial endowment is formula_38. 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 	Competitive equilibrium
15436688	needed, since Walras' law guarantees that the results will be the same. Note that in CE, only relative prices are determined; we can normalize the prices, e.g, by requiring that formula_43. Then we get formula_44. But any other normalization will also work. 3. Non-existence example: Suppose the agents' utilities are: and the initial endowment is [(2,1),(2,1)]. In CE, every agent must have either only x or only y (the other product does not contribute anything to the utility so the agent would like to exchange it away). Hence, the only possible CE allocations are [(4,0),(0,2)] and [(0,2),(4,0)]. Since the 	Competitive equilibrium
15436689	agents have the same income, necessarily formula_46. But then, the agent holding 2 units of y will want to exchange them for 4 units of x. 4. For existence and non-existence examples involving linear utilities, see Linear utility#Examples. Indivisible items. When there are indivisible items in the economy, it is common to assume that there is also money, which is divisible. The agents have quasilinear utility functions: their utility is the amount of money they have plus the utility from the bundle of items they hold. A. Single item: Alice has a car which she values as 10. Bob 	Competitive equilibrium
15436690	has no car, and he values Alice's car as 20. A possible CE is: the price of the car is 15, Bob gets the car and pays 15 to Alice. This is an equilibrium because the market is cleared and both agents prefer their final bundle to their initial bundle. In fact, every price between 10 and 20 will be a CE price, with the same allocation. 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 	Competitive equilibrium
15436691	and the price will be anywhere between 10 and 20. On the other hand, any price below 10 is not an equilibrium price because there is an excess demand (both Alice and Bob want the car at that price), and any price above 20 is not an equilibrium price because there is an excess supply (neither Alice nor Bob want the car at that price). This example is a special case of a double auction. B. Substitutes: A car and a horse are sold in an auction. Alice only cares about transportation, so for her these are perfect substitutes: she 	Competitive equilibrium
15436692	gets utility 8 from the horse, 9 from the car, and if she has both of them then she uses only the car so her utility is 9. Bob gets a utility of 5 from the horse and 7 from the car, but if he has both of them then his utility is 11 since he also likes the horse as a pet. In this case it is more difficult to find an equilibrium (see below). A possible equilibrium is that Alice buys the horse for 5 and Bob buys the car for 7. This is an equilibrium since Bob 	Competitive equilibrium
15436693	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. C. Complements: A horse and a carriage are sold in an auction. There are two potential buyers: AND and OR. AND wants only the horse and the carriage together - she receives a utility of formula_47 from holding both of them but a utility of 0 for holding only one of them. OR wants either the horse or the carriage but doesn't need both 	Competitive equilibrium
15436694	- he receives a utility of formula_48 from holding one of them and the same utility for holding both of them. Here, when formula_49, a competitive equilibrium does NOT exist, i.e, no price will clear the market. Proof": consider the following options for the sum of the prices (horse-price + carriage-price): D. Unit-demand consumers: There are "n" consumers. Each consumer has an index formula_54. There is a single type of good. Each consumer formula_5 wants at most a single unit of the good, which gives him a utility of formula_56. The consumers are ordered such that formula_57 is  a weakly increasing function of formula_5. If the supply is formula_59 units, then any price formula_60 satisfying formula_61 is an equilibrium price, since there are "k" consumers that either want to buy the product or indifferent between buying and not buying it. Note that an increase in supply causes a decrease in price. Existence of a competitive equilibrium. Divisible resources. 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. A. For concreteness, assume that there are formula_64 agents and formula_65 divisible goods. Normalize  the prices such that their sum is 1: formula_66. Then, the space of all possible prices is the formula_67-dimensional unit simplex in formula_68. We call this simplex the "price simplex". B. Let formula_69 be the excess demand function. This is a function of the price vector formula_60 when the initial endowment formula_9 is kept constant: It is known that, when the agents have strictly convex preferences, the Marshallian demand function is continuous. Hence, formula_69 is also a continuous function of formula_60. C. Define the following function from the price simplex to itself: This is a continuous function, so by  the Brouwer fixed-point theorem there is a price vector formula_76 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, formula_81. But this implies that the inequality above must be an equality: This means that formula_76 is a price vector of a competitive equilibrium. 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. Algorithms for computing the market equilibrium are described in Market equilibrium computation. Indivisible items. In the examples above, a competitive equilibrium existed when the items were substitutes but not when the items were complements. This is not a coincidence. Given a utility function on two goods "X" and "Y", say that the goods are weakly gross-substitute (GS) if they are either Independent goods or gross substitute goods, but "not" Complementary goods. This means that formula_84. I.e., if the  price of "Y" increases, then the demand for "X" either remains constant or increases, but does "not" decrease. A utility function is called GS if, according to this utility function, all pairs of different goods are GS. With a GS utility function, if an agent has a demand set at a given price vector, and the prices of some items increase, then the agent has a demand set which includes all the items whose price remained constant. He may decide that he doesn't want an item which has become more expensive; he may also decide that he wants another  item instead (a substitute); but he may not decide that he doesn't want a third item whose price hasn't changed. When the utility functions of all agents are GS, a competitive equilibrium always exists. Moreover, the set of GS valuations is the largest set containing unit demand valuations for which the existence of competitive equilibrium is guaranteed: for any non-GS valuation, there exist unit-demand valuations such that a competitive equilibrium does not exist for these unit-demand valuations coupled with the given non-GS valuation. For the computational problem of finding a competitive equilibrium in a  special kind of a market, see Fisher market#indivisible. The competitive equilibrium and allocative efficiency. 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. Note that microeconomic analysis does not assume additive utility, nor does it assume any interpersonal utility tradeoffs. Efficiency, therefore, refers to the absence of Pareto improvements. 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. Welfare theorems for indivisible item assignment. In the case of indivisible items, we have the following strong versions of the two welfare theorems: Finding an equilibrium. 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 an ascending auction, the auctioneer publishes a price vector, initially zero, and the buyers declare their  favorite bundle under these prices. In case each item is desired by at most a single bidder, the items are divided and the auction is over. 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. Several different ascending-auction mechanisms have been suggested in the literature. Such mechanisms are often called Walrasian auction, "Walrasian tâtonnement" or English auction.