Chapter 23Competitive Equilibrium

1.Partial Equilibrium2.General Equilibrium3.Pareto Optimality

1. Partial Equilibrium

This segment of the course introduces the concept of competitive equilibrium. First we define competitive equilibrium, price taking behavior and market clearing, within a single market, and compare the predictions of competitive equilibrium with the solution to a limit order market.

Then we demonstrate with several examples how competitive equilibrium can be applied to multiple markets. We discuss why, under fairly wide ranging conditions, that competitive equilibrium yields an optimal allocation of resources, and last, illustrate by way of examples, the distorting effects of taxes and subsidies.

Definition of competitive equilibrium

In microeconomics you learned that a competitive equilibrium describing market behavior has two defining properties:

1. Traders take the price as given. Each trader chooses only the quantity he or she wishes to trade at that price.

2. Markets clear. There are neither unfilled orders by demanders nor unanticipated inventory accumulation by suppliers.

Partial equilibrium Consider the following market for a stock.

There are a finite number of player types, say I. Every player belonging to a given player type has the same asset and money endowment, and the same private valuation.

Players belonging to type i are distinguished by their initial endowment of money mi and the stock si, as well as their private valuation of the stock vi. Thus a player type i is defined by the triplet (mi, si, vi).

Using supply and demand curves to derive competitive equilibrium

To derive the competitive equilibrium, compute the demand for the asset minus the supply of the asset (both as a function of price), otherwise known as the net demand for the asset.

This is found by computing the individual net demands for the asset, and then aggregating across players.

The competitive equilibrium price is determined by setting the net demand for the asset to zero.

Individual optimization in a competitive equilibrium

In a competitive equilibrium with price p the objective of player i is to pick the quantity of good traded, denoted qi, to maximize the value of his or her portfolio subject to constraints that prevent short sales (selling more stock than the the seller holds) or bankruptcy (not having enough liquidity to cover purchases).

The value of the portfolio of player i is: iiiii qsvpqm

Constraints in the optimization problem

The short sale constraint is:

ii pqm

These constraints can be combined as:

iii sqp

m

0 ii qs

The solvency constraint is

Solution to the individual’soptimization problem

The solution to this linear problem is to specialize the stock if vi exceeds p, specialize in money if p exceeds vi, and choose any feasible quantity q if vi = p. That is:

pvifs

pvifp

m

q

ii

ii

i

and:

pvifp

mqs i

iii

Market demand

Summing across the individual demands of players we obtain the demand across players curve D(p).

Let 1{ . . .} be an indicator function, taking a value of 1 if the statement inside the parentheses is true, and 0 if false. Then, the demand from those players who wish to increase their holding of the stock is: I

ii

i p

mpvpD

11

As p falls the number of players with valuations exceeding p increases. Thus D(p) declines in slanting steps. The top step has height max{v1,v2, . . . ,vI} and a typical step length is mi/p.

Market supply

Summing over the individual supply of each player we obtain the aggregate supply curve S(p), the total supply of the asset from those players who want to sell their shares, as a function of price :

I

i ii spvpS11

Following the same reasoning as on the previous slide, the supply curve is actually a step function which increases from min{v1,v2, . . . ,vI}, where the steps have variable length of si.

Indifferent traders

This only leaves stockholders whose valuation vi = p, who are indifferent about how much they trade. They are equally well off selling up to their endowment si versus buying up to their budget constraint mi/p:

The next step is to those prices for which there is excess supply, which we denote by p+. Then we derive those prices for which there is excess demand, denoted p-.

The set of competitive equilibrium are the remaining prices.

p

mqs iii

Solving for competitive equilibrium

This only leaves stockholders whose valuation vi = p, who are indifferent about what they trade:

I

i ii qpvpSpD11

The competitive equilibrium price pe is the unique solution in p found by equating the difference between demand and supply to the quantity traded by those who are indifferent about how much they trade:

p

mq ii 0

Example

To make matters more concrete, suppose there are 10 players, with private valuations that take on the integer values from $1 to $10.

Suppose the third player (with valuation $3) is endowed with 2 units of the stock, the first and second have one unit, and everybody else has $12 to buy units of the asset.

We also assume that everyone has the same access to the market, and can place limit or market orders.

Aggregate supply

At prices above $1, the first player will supply a unit, at prices above $2 the second player will supply a unit, and at prices above $3, the third player will supply 2 units.

Define q as any (integer) quantity between 0 and 4, and p as a positive real number, the supply function is:

Sp 1p 1 1p 2 1p 32

Aggregate demandIf the price p exceeds 5, the players with valuations greater than p will demand one unit, if p lies between 3.3 and 5, the players with valuations above p will demand 2 units each, if p is between 2.5 and 3.3 the players with valuations above p will demand 4 units each, and so on:

Therefore the demand for the good as a function of prices above 2.5 is:

Dp 110 p 5 15 p 3. 33 13. 33 p 2. 5 i 110 v i p

Competitive equilibrium price

In this example, there is a competitive equilibrium at any price between $6 and $7.

Note that at prices above $7, demand shrinks by one unit since only 3 buyers wish to purchase a unit, But at any price below $6, five units are demanded in aggregate.

Competitive equilibrium as a tool for prediction

An advantage from assuming that markets are in competitive equilibrium is that they are relatively straightforward to analyze.

For example, deriving the properties of a Nash equilibrium solution to a trading game is typically more complex than deriving the competitive equilibrium for the game.

Comparing competitive equilibrium with the solution to trading games

In limit order markets players choose prices and quantities, not just quantities.

Moreover there is no presumption in limit order markets that every trade will take place at the same price, whereas in models of competitive equilibrium this is a premise.

When does competitive equilibrium approximate the outcome of the Nash equilibrium solution to a trading game?

Similarities with competitive equilibrium

The trading mechanism we discussed last lecture is self financing, and satisfies the participation and incentive compatibility constraints.

If the number of players is odd, the mechanism mimics the price and resource allocation of a competitive equilibrium!

If the number of players is even, this mechanism approaches the competitive equilibrium price and quantities as the market share of the trades made by each player declines.

2. General Equilibrium

The definition of competitive equilibrium we provided for a single market for a single market can readily be extended to multiple markets. After giving a definition of general equilibrium we demonstrate with point wih several examples.

General equilibrium

The definition of competitive equilibrium extends to multiple markets.

Suppose traders live in an economy where there are a total of I markets. A competitive equilibrium is a price an I-1 dimensional price vector, such that when the traders take this price vector as given and choose their respective allocations to individually maximize their objective functions subject to their budget constraints that limits their expenditures, all markets clear.

Production in the goods and services sector

Consider an economy where firm owners maximize their value by selling a good called housing, which is produced using two inputs, wood and, clay and labor.

The firms adopt the same production technology. Denote the output of firm j by yj and the three inputs by x1j, x2j and x3j.

There is a fixed amount of raw materials distributed throughout the population that are traded on the three markets.

Differential products

In the previous examples several suppliers competed with each other by selling exactly the same product.

We now consider how well competitive equilibrium predicts outcomes in assignment and matching problems, where goods are not perfect substitutes for each other.

Some examples include the labor market, where companies recruit job applicants, professional sportsmen with their teams, graduating high school students with their teams, and the housing market.

Valuations

We consider two ways of calculating the value of the match to the buyer. It is:

1. the product of the indexed values of the market.

2. The sum of the two markets

3. Pareto Optimality

The final section in this chapter discusses why, under fairly wide ranging conditions, that competitive equilibrium yields an optimal allocation of resources, and concludes by illustrating the distorting effects of taxes and subsidies.

Optimality of competitive equilibrium

The prisoner’s dilemma illustrates why games reach outcomes in which all players are worse off than they would be in one of the other outcomes.

Notice that in a competitive equilibrium is a single the potential trading surplus is used up by the traders. It is impossible to make one or more players better off without making someone else worse off.

This important result explains why many economists recommend markets as a way of allocating resources.

Distortions from taxation and regulation

The last part of this segment segment analyzes the role of the government in affecting market outcomes.

We focus on one areas of government intervention: through taxes and subsidies on trade.

We modify our previous example of trade. For example, how would a sales tax levied on consumers affect market supply and demand?

What about a production tax levied on suppliers?