Chapter 23 Competitive Equilibrium

1
Chapter 23Competitive Equilibrium

1. Partial Equilibrium
2. General Equilibrium
3. Pareto Optimality

2
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.
3
Definition of competitive equilibrium

• In microeconomics you learned that a competitive
  equilibrium describing market behavior has two
  defining properties
• Traders take the price as given. Each trader
  chooses only the quantity he or she wishes to
  trade at that price.
• Markets clear. There are neither unfilled orders
  by demanders nor unanticipated inventory
  accumulation by suppliers.

4
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).

5
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.

6
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

7
Constraints in the optimization problem
The short sale constraint is
The solvency constraint is
These constraints can be combined as
8
Solution to the individualsoptimization 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
and
9
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

• As p falls the number of players with valuations
  exceeding p increases. Thus D(p) declines in
  slanting steps. The top step has height
  maxv1,v2, . . . ,vI and a typical step length
  is mi/p.

10
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
  

• Following the same reasoning as on the previous
  slide, the supply curve is actually a step
  function which increases from minv1,v2, . . .
  ,vI, where the steps have variable length of si.
  

11
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.

12
Solving for competitive equilibrium
This only leaves stockholders whose valuation vi
p, who are indifferent about what they trade
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
13
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.

14
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

15
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

16
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.

17
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.

18
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?

19
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.

20
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.
21
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.

22
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.

23
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.

24
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
25
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.
26
Optimality of competitive equilibrium

• The prisoners 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.

27
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?