Chapter 23
Competitive Equilibrium
1. Partial Equilibrium
2. General Equilibrium
3. 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:
mi  pqi  vi si  qi 
Constraints in the
optimization problem
The short sale constraint is:
si  qi  0
The solvency constraint is
mi  pqi
These constraints can be combined as:
mi
 qi   si
p
Solution to the individual’s
optimization 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:
 mi
if vi  p

qi   p
s if v  p
i
 i
and:
mi
 si  qi 
p
if
vi  p
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:
D p   i 11vi  p
I
mi
p
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 :
S  p  

I
1
v
i 1 i
 psi
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:
mi
 si  qi 
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.
Solving for competitive equilibrium
This only leaves stockholders whose valuation vi = p,
who are indifferent about what they trade:
mi
0  qi 
p
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:
D p   S  p   
vi
I
1
i 1
 pqi
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:
S
p1p 11p 21p 32
Aggregate demand
If 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:
D
p
110 p 515 p 3. 33 13. 33  p 2. 5
  i1 v i p
10
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?