# Competitive equilibrium

(Redirected from Walrasian equilibrium)

Competitive equilibrium (also called: Walrasian equilibrium)[1] 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.

## Definitions

A competitive equilibrium consists of two elements:

• A vector of prices - a price for each different type of commodity;
• For each agent, an allocation vector - the quantity of each commodity allocated to this agent.

These vectors should satisfy the following requirements:

• Feasibility - the total demand of each good equals the total supply of that good (i.e. the market is cleared);
• Rationality - every agent weakly prefers their allocation to any other possible allocation they might receive given their budget. In other words, if an agent strongly prefers another combination of goods, the agent can't afford it in the given prices.

An alternative definition[2] relies on the concept of a demand-set. Given a price vector 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 vector P and an allocation vector 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.[3] 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

### Indivisible item assignment

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 competitive equilibrium 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 competitive equilibrium price. 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.

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). 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 him only 1 additional utility.

C. Complements: A horse and a carriage are sold in an auction. Alice wants only the horse and the carriage together - she receives a utility of 100 from holding both of them but a utility of 0 for holding only one of them. Bob wants either the horse or the carriage but doesn't need both - he receives a utility of 60 from holding one of them and the same utility of 60 for holding both of them. Here there is no competitive equilibrium, i.e. no price will clear the market. To see this, consider the following options for the sum of the prices (horse-price + carriage-price):

• The sum is less than 100. Then Alice wants both items. Since the price of at least one item is smaller than 60, Bob wants that item, so there is an excess demand.
• The sum is exactly 100. Then Alice is indifferent between buying both items and not buying any item. But Bob still wants exactly one item, so there is either an excess demand or excess supply.
• The sum is more than 100. Then Alice wants no item and Bob still wants at most a single item, so there is an 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.

### Resource allocation

There are two kinds of products: bananas and apples, and 2 individuals: Jane and Kelvin. The price of bananas is $P_b$, and the price of apples is $P_a$.

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 curves $J_1$ 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 $P_b$ and $P_a$. 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 -$P_b / P_a$.

And the $MRS_{Jane} = P_b / P_a$; $MRS_{Kelvin} = P_b / P_a$. The marginal rate of substitution 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.

## Existence of a competitive equilibrium

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 $\frac{\Delta \text{demand}(X)}{\Delta \text{price}(Y)}\geq 0$. 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.[3][4] 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.[5]

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.[6]

## The competitive equilibrium and allocative efficiency

By the Fundamental theorems of welfare economics, any competitive equilibrium leads to a Pareto efficient allocation of resources, and any efficient allocation can be sustainable by a competitive equilibrium.

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.[7]

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). This 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:[2]

1. 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.

2. 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.[3][5][8] Such mechanisms are often called Walrasian auction, Walrasian tâtonnement or English auction. One such mechanism is explained below.

### Main procedure

The mechanism keeps a price vector P, and a CurrentBundle for every agent.

1. Initialize P to 0; initialize the CurrentBundle of every agent to the set of all items.
2. For each item, calculate its total demand - the number agents that have this item in their CurrentBundle.
3. If the total demand for each item equals its supply, then assign the CurrentBundles to their agents and finish.
4. Otherwise, select a good x for which the demand is larger than the supply.
5. Increase the price of x continuously, updating the CurrentBundles as necessary (see below). This eventually decreases the demand for x, and possibly increases the demand of other goods, but does not decrease the demand for other goods (thanks to the GS property). Hence, eventually the demand of x equals the supply, and there is no demand-shortage in any other product.
6. Go back to step 3.

Since eventually the demand of each good must drop, the algorithm must terminate. Since there cannot be a shortage of demand, the algorithm must terminate when the demand of every good equals its supply, which means a competitive equilibrium.

### Increasing the price of item x

In principle, the price of x must increase continuously. Otherwise, if the price 'jumps' too high, it is possible that two or more agents will simultaneously decrease their demand for x, leading to a demand shortage.

In practice, we can assume that the values are all whole dollars (or whole cents), and increase the prices in jumps of one dollar (or one cent). In general, the price increase step should be a common divisor of all the utilites. In the sequel, we assume it is a cent.

After each cent of increase in the price of x, each agent can make one or both of these declarations:

1. "I don't want x anymore". This means that x should be removed from that agent's CurrentBundle, since its marginal utility for the agent is strictly less than its price.
2. "I want y instead of x". This means that x should be removed and y should be added to that agent's CurrentBundle, since the marginal utility of y minus its price is strictly more than the marginal utility of x minus its price.

After all agents make their declarations, the demand for x is re-calculated. There are 3 cases:

• If there is still an excess demand in x, then make another step of price-increase.
• If the demand of x equals the supply, then return to the main procedure.
• If there is a demand shortage, this means that several agents decreased their demand for x simultaneously. Call these agents "Unlikers". We know that all Unlikers don't want to buy x in its new price. But, because the increase in price is only a cent and all utilities are whole cents, we also know that in the previous price, all Unlikers were indifferent between buying x and not buying x. Remove one cent from the price, select a random subset of the Unlikers and undo their declarations so that the demand equals the supply. Return to the main procedure.

### Examples

Consider the substitutes example (example B). The utility gained from (horse,car,both) is (8,9,9) for Alice and (5,7,11) for Bob.

• Initially, the prices of the (horse,car) are (0,0). Both Alice and Bob want (horse,car). The demand for each good is 2 while the supply is 1.
• The horse is picked for increase and its price rises to 1. Now Alice wants only (car), since the horse will not give her any additional utility. Bob still wants (horse,car). The demand of horses equals the supply (1), but the demand for cars is larger than the supply.
• The car price is rises to 1, then 2, then 3. At this point, Alice changes her CurrentBundle to (horse), since buying the horse for 1 will give her a net utility of 7 while buying the car for 3 will leave her with a net utility of only 6. Bob still wants (horse,car).
• The horse price rises to 2, then 3. Alice changes her demand to (car) but Bob still wants (horse,car).
• The car price rises to 4, then 5. Alice changes her demand to (horse) but Bob still wants (horse,car).
• The horse price rises to 4, then 5. Alice changes her demand to (car). Bob now also changes his demand to (car) only, since the horse will only give him 4 additional utility. We run into a demand shortage on the horse. To remedy this, we decrease the horse price back to 4 and undo Alice's declaration. So the CurrentBundle of Alice is (horse) and the CurrentBundle of Bob is (car), and the auction terminates.

We now have an equilibrium in which Alice buys the horse for 4 (her net utility is 4) and Bob buys the car for 5 (his net utility is 2). Alice would not switch since buying the car would give her the same net utility (4). Bob would not switch since buying the horse alone will give him a smaller net utility (1) and adding the horse to the car will add him no additional net utility.

Alternatively, it is possible to do a descending auction, in which prices start high and go down as long as there is excess supply. For example:

• Initially, the prices of the (horse,car) are (20,20). Both Alice and Bob want an empty bundle (), so there is excess supply.
• The horse price drops down to 7. At this point, Alice wants (horse) but Bob still wants (), so there is excess supply of one car.
• The car price drops to 7. At this point, Alice changes her CurrentBundle to (car). Bob still wants (), so there is excess supply of one horse.
• The horse price drops to 5. Alice changes her CurrentBundle to (horse) but Bob still wants (); an excess supply of a car.
• The car price drops to 6. At this point, Bob changes his CurrentBundle to (car). Alice still wants (horse), and the auction terminates.

The final outcome is that Alice buys the horse for 5 (her net utility is 3) and Bob buys the car for 6 (his net utility is 1). This is different than the result of the ascending auction, but still an equilibrium.

As a third example, consider what happens when we try to run an ascending auction on the complements example (example C):

• Initially, the prices of the (horse,carriage) are (0,0). Both Alice and Bob want (horse,carriage).
• We increase the price of the horse to 1. Now Bob wants only (carriage) because the horse gives him no additional utility. Alice still wants (horse,carriage).
• We increase the price of the carriage to 2. Now Bob wants only (horse). This goes on like this until the prices of (horse,carriage) are (49,50), at which point Bob only wants (horse) but Alice still wants (horse,carriage).
• We increase the price of the horse to 51. At this point, Bob wants only (carriage), but Alice does not want anything! Because the horse and the carriage are complements, she wants either both or none, and here, since their price is more than 100, she wants none.

As we can see, in this case the ascending auction does not terminate with an equilibrium. This could be expected, since in this case the utilities are not GS and there is no competitive equilibrium at all.

### Strategic considerations

The above ascending auction is not incentive compatible. In some cases, it may be worthwhile for agents to report a reduced demand in order to increase their net utility.[3] For example, consider the ascending auction in the horse-and-car example above. By playing truthfully, Bob gets the car for 7 and his net utility is 0. But if he says that he wants only the horse, we will get it after a single step for a price of 1, and his net utility will be 4.