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Competitive equilibrium (also called: Walrasian equilibrium) is a concept of
economic equilibrium In economics, economic equilibrium is a situation in which economic forces such as supply and demand are balanced and in the absence of external influences the ( equilibrium) values of economic variables will not change. For example, in the st ...
introduced by
Kenneth Arrow Kenneth Joseph Arrow (23 August 1921 – 21 February 2017) was an American economist, mathematician, writer, and political theorist. He was the joint winner of the Nobel Memorial Prize in Economic Sciences with John Hicks in 1972. In economi ...
and
Gérard Debreu Gérard Debreu (; 4 July 1921 – 31 December 2004) was a French-born economist and mathematician. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize ...
in 1951 appropriate for the analysis of
commodity market A commodity market is a market that trades in the primary economic sector rather than manufactured products, such as cocoa, fruit and sugar. Hard commodities are mined, such as gold and oil. Futures contracts are the oldest way of investin ...
s with flexible prices and many traders, and serving as the benchmark of
efficiency Efficiency is the often measurable ability to avoid wasting materials, energy, efforts, money, and time in doing something or in producing a desired result. In a more general sense, it is the ability to do things well, successfully, and without ...
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: * A price function P. It takes as argument a vector representing a bundle of commodities, and returns a positive real number that represents its price. Usually the price function is linear - it is represented as a vector of prices, a price for each commodity type. * An allocation matrix X. For every i\in 1,\dots,n, X_i is the vector of commodities allotted to agent i. These elements should satisfy the following requirement: * Satisfaction (market-envy-freeness): Every agent weakly prefers his bundle to any other affordable bundle: ::\forall i\in 1,\dots,n, if P(Y) \leq P(X_i) then Y \preceq_i X_i. Often, there is an initial endowment matrix E: for every i\in 1,\dots,n, E_i is the initial endowment of agent i. Then, a CE should satisfy some additional requirements: * Market Clearance: the demand equals the supply, no items are created or destroyed: ::\sum_^n X_i = \sum_^n E_i. * Individual Rationality: all agents are better-off after the trade than before the trade: ::\forall i\in 1,\dots,n: X_i \succeq_i E_i. * Budget Balance: all agents can afford their allocation given their endowment: ::\forall i\in 1,\dots,n: P(X_i) \leq P(E_i).


Definition 2

This definition explicitly allows for the possibility that there may be multiple commodity arrays that are equally appealing. Also for zero prices. 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: 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: * The bundle allocated by X to each agent is in that agent's demand-set for the price-vector P; * Every good which has a positive price is fully allocated (i.e. every unallocated item has price 0).


Approximate equilibrium

In some cases it is useful to define an equilibrium in which the rationality condition is relaxed. Given a positive value \epsilon (measured in monetary units, e.g., dollars), a price vector P and a bundle x, define P^x_\epsilon 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 \epsilon more than their price in P. In a ''\epsilon-competitive-equilibrium'', the bundle x allocated to an agent should be in that agent's demand-set for the ''modified'' price vector, P^x_\epsilon. This approximation is realistic when there are buy/sell commissions. For example, suppose that an agent has to pay \epsilon 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 P^x_\epsilon. This makes the equilibrium more stable.


Examples


Divisible resources

The new solution method (Riley 022is to solve not for a single out come but for all possible outcomes. Solving for the graph of Equilibrium outcomes. CHOICE Preferences are represented by an individuals marginal rate of substitution MRS(X,Y). This is the marginal willingness to trade away y for x. Alex has a MRS of Ay(a)/x(a). Bev's MRS is By(b)/x(b). Below the case in which (A,B) = (2.1) is solved. DEMAND Given a price p for commodity x and 1 for commodity y Alex and Bev choose to consumer where MRS(X,Y) =p The Walrasian equilibrium WE price ratio P is the price ratio that clears the market Maximizing equations p = 2y(a)/x(a) = y(b)/x(b). SUPPLY Define units so that the total supply of each commodity is 1. Then x(b) = 1 - x(a) and y(b) = 1 - x(a). Sub into the maximizing equations P = 2Y/X (*) and P = (1 -Y)/(1-X) (**) Cross multiplying and rearranging yields the following result (X-2)(Y+1) = - 2. Then PX =2Y and P- PX = 1-Y Adding these equations, P=1-Y. Therefore Y=P-1. From (*) PX=2Y =2(P-1) RESULTS SPENDING EQUATIONS PX = 2(P-1). Y=P-1, BUDGET EQUATION: PX+Y= 3P - 3 and the WE outcomes lie on the graph of the hyperbola (X-2)(Y+1)=-2 BUDGET EQUATION. PX + 1Y = 3P - 3 = P(3) + 1(-3) The economy therefore has a very special fixed point F = (3, -3). ALL WALRASIAN EQ BUDGET LINES PASS THROUGH THE FIXED POINT. THE SOLUTION Pick any endowments. For example, (1,1) The prices are P and 1. The value of the endowment is therefore P ( 1/2 ) +1 ( 1/2 ) =P(3) + 1(-3) Then 2P = 4 and so P = 7/5.. From the spending equations you can solve for the WE outcome. If the endowment is (0,1) show that the WE price ratio is 3/2 INTRODUCTORY EXAMPLES The following examples involve an
exchange economy Exchange economy is technical term used in microeconomics research to describe interaction between several agents. In the market, the agent is the subject of exchange and the good is the object of exchange. Each agent brings his/her own endowment, ...
with two agents, Jane and Kelvin, two
goods In economics, goods are items that satisfy human wants and provide utility, for example, to a consumer making a purchase of a satisfying product. A common distinction is made between goods which are transferable, and services, which are not ...
e.g. bananas (x) and apples (y), and no money. 1. Graphical example: Suppose that the initial allocation is at point X, where Jane has more apples than Kelvin does and Kelvin has more bananas than Jane does. By looking at their
indifference curve In economics, an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is ''indifferent''. That is, any combinations of two products indicated by the curve will provide the c ...
s J_1 of Jane and K_1 of Kelvin, we can see that this is not an equilibrium - both agents are willing to trade with each other at the prices P_x and P_y. After trading, both Jane and Kelvin move to an indifference curve which depicts a higher level of utility, J_2 and K_2. The new indifference curves intersect at point E. The slope of the tangent of both curves equals -P_x / P_y. And the MRS_ = P_x / P_y; MRS_ = P_x / P_y. The marginal rate of substitution (MRS) of Jane equals that of Kelvin. Therefore, the 2 individuals society reaches
Pareto efficiency Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engi ...
, 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: :u_J(x,y) = x^a y^ :u_K(x,y) = x^b y^ where a,b are constants. Suppose the initial endowment is E=
1,0), (0,1) Onekama ( ) is a village in Manistee County in the U.S. state of Michigan. The population was 411 at the 2010 census. The village is located on the shores of Portage Lake and is surrounded by Onekama Township. The town's name is derived from "On ...
/math>. The demand function of Jane for x is: :x_J(p_x,p_y,I_J) = \frac = \frac = a The demand function of Kelvin for x is: :x_K(p_x,p_y,I_K) = \frac = \frac The market clearance condition for x is: :x_J + x_K = E_ + E_ = 1 This equation yields the equilibrium price ratio: :\frac = \frac We could do a similar calculation for y, but this is not 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 p_1+p_2=1. Then we get p_1=\frac, p_1=\frac. But any other normalization will also work. 3. Non-existence example: Suppose the agents' utilities are: :u_J(x,y)=u_K(x,y) = \max(x,y) 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) The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
Since the agents have the same income, necessarily p_y = 2 p_x. 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 In economics and consumer theory, quasilinear utility functions are linear in one argument, generally the numeraire. Quasilinear preferences can be represented by the utility function u(x_1, x_2, \ldots, x_n) = x_1 + \theta (x_2, \ldots, x_n) wh ...
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 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 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 A double auction is a process of buying and selling goods with multiple sellers and multiple buyers. Potential buyers submit their bids and potential sellers submit their ask prices to the market institution, and then the market institution choose ...
. 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 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 Below may refer to: *Earth * Ground (disambiguation) *Soil *Floor * Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Ernst von Below (1863–1955), German World War I general *Fred 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 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 v_ 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 - he receives a utility of v_ from holding one of them and the same utility for holding both of them. Here, when v_ < 2 v_, 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): * The sum is less than v_. Then, AND wants both items. Since the price of at least one item is less than v_, OR wants that item, so there is excess demand. * The sum is exactly v_. Then, AND is indifferent between buying both items and not buying any item. But OR still wants exactly one item, so there is either excess demand or excess supply. * The sum is more than v_. Then, AND wants no item and OR still wants at most a single item, so there is excess supply. D. Unit-demand consumers: There are ''n'' consumers. Each consumer has an index i=1,...,n. There is a single type of good. Each consumer i wants at most a single unit of the good, which gives him a utility of u(i). The consumers are ordered such that u is a weakly increasing function of i. If the supply is k\leq n units, then any price p satisfying u(n-k)\leq p\leq u(n-k+1) 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 In mathematical economics, the Arrow–Debreu model suggests that under certain economic assumptions (convex preferences, perfect competition, and demand independence) there must be a set of prices such that aggregate supplies will equal aggregat ...
shows that a CE exists in every
exchange economy Exchange economy is technical term used in microeconomics research to describe interaction between several agents. In the market, the agent is the subject of exchange and the good is the object of exchange. Each agent brings his/her own endowment, ...
with divisible goods satisfying the following conditions: * All agents have strictly
convex preferences In economics, convex preferences are an individual's ordering of various outcomes, typically with regard to the amounts of various goods consumed, with the property that, roughly speaking, "averages are better than the extremes". The concept roughly ...
; * All goods are desirable. This means that, if any good j is given for free (p_j=0), then all agents want as much as possible from that good. The proof proceeds in several steps. A. For concreteness, assume that there are n agents and k divisible goods. Normalize the prices such that their sum is 1: \sum_^k p_j = 1. Then, the space of all possible prices is the k-1-dimensional unit simplex in \mathbb^k. We call this simplex the ''price simplex''. B. Let z be the
excess demand function In microeconomics, excess demand is a phenomenon where the demand for goods and services exceeds that which the firms can produce. In microeconomics, an excess demand function is a function expressing excess demand for a product—the excess ...
. This is a function of the price vector p when the initial endowment E is kept constant: :z(p) = \sum_^n It is known that, when the agents have strictly
convex preferences In economics, convex preferences are an individual's ordering of various outcomes, typically with regard to the amounts of various goods consumed, with the property that, roughly speaking, "averages are better than the extremes". The concept roughly ...
, the Marshallian demand function is continuous. Hence, z is also a continuous function of p. C. Define the following function from the price simplex to itself: :g_i(p) = \frac, \forall i\in 1,\dots,k This is a continuous function, so by the
Brouwer fixed-point theorem Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simples ...
there is a price vector p^* such that: :p^* = g(p^*) so, :p^*_i = \frac, \forall i\in 1,\dots,k 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: :z_j(p^*) \leq 0, \forall j\in 1,\dots,k E. The desirability assumption implies that all products have strictly positive prices: :p_j > 0, \forall j\in 1,\dots,k By Walras' law, p^* \cdot z(p^*) = 0. But this implies that the inequality above must be an equality: :z_j(p^*) = 0, \forall j\in 1,\dots,k This means that p^* is a price vector of a competitive equilibrium. Note that
Linear utilities In economics and consumer theory, a linear utility function is a function of the form: ::u(x_1,x_2,\dots,x_m) = w_1 x_1 + w_2 x_2 + \dots w_m x_m or, in vector form: ::u(\overrightarrow) = \overrightarrow \cdot \overrightarrow where: * m is the n ...
are only weakly convex, so they do not qualify for the
Arrow–Debreu model In mathematical economics, the Arrow–Debreu model suggests that under certain economic assumptions (convex preferences, perfect competition, and demand independence) there must be a set of prices such that aggregate supplies will equal aggregat ...
. 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 good In microeconomics, two goods are substitutes if the products could be used for the same purpose by the consumers. That is, a consumer perceives both goods as similar or comparable, so that having more of one good causes the consumer to desire less ...
s, but ''not'' Complementary goods. This means that \frac\geq 0. 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 There are two fundamental theorems of welfare economics. The first states that in economic equilibrium, a set of complete markets, with complete information, and in perfect competition, will be Pareto optimal (in the sense that no further exch ...
, 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 In economics, the marginal cost is the change in the total cost that arises when the quantity produced is incremented, the cost of producing additional quantity. In some contexts, it refers to an increment of one unit of output, and in others it ...
). 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.Callan, S.J & Thomas, J.M. (2007). 'Modelling the Market Process: A Review of the Basics', Chapter 2 in ''Environmental Economics and Management: Theory, Politics and Applications'', 4th ed., Thompson Southwestern, Mason, OH, USA 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 Distributive justice concerns the socially just allocation of resources. Often contrasted with just process, which is concerned with the administration of law, distributive justice concentrates on outcomes. This subject has been given considera ...
or
equity Equity may refer to: Finance, accounting and ownership *Equity (finance), ownership of assets that have liabilities attached to them ** Stock, equity based on original contributions of cash or other value to a business ** Home equity, the diff ...
). 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: # Any competitive equilibrium maximizes the social welfare (the sum of utilities), not only over all realistic assignments of items, but also over all ''fractional'' assignments of items. I.e., even if we could assign fractions of an item to different people, we couldn't do better than a competitive equilibrium in which only whole items are assigned. # If there is an integral assignment (with no fractional assignments) that maximizes the social welfare, then there is a competitive equilibrium with that assignment.


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.


See also

* Envy-free pricing - a relaxation of Walrasian equilibrium in which some items may remain unallocated. * Fisher market - a simplified market model, with a single seller and many buyers, in which a CE can be computed efficiently. * Allocative efficiency *
Economic equilibrium In economics, economic equilibrium is a situation in which economic forces such as supply and demand are balanced and in the absence of external influences the ( equilibrium) values of economic variables will not change. For example, in the st ...
*
General equilibrium theory In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an ov ...
* Walrasian auction


References

* {{Cite journal , doi = 10.1007/s001990050281, title = Non-computability of competitive equilibrium, journal = Economic Theory, volume = 14, pages = 1–27, year = 1999, last1 = Richter , first1 = M. K. , last2 = Wong , first2 = K. C.


External links


Competitive equilibrium, Walrasian equilibrium and Walrasian auction
in Economics Stack Exchange. Market (economics) Competition (economics) Auction theory