Transcript Rational Expectations Lecture

Lecture 4
Evaluating Competitive Equilibrium
This lecture analyzes how well competitive
equilibrium predicts industry outcomes as a
function the of the production technology,
the number of firms and the aggregation of
information about supply and demand.
What happens as we increase the
number of firms in the industry?
Up until now we have concentrated on
situations where there is just on supplier
(or auctioneer) and many demanders (or
bidders), or where there is just one
demander and many suppliers.
Suppose there are just two firms in the
industry (and many demanders). We
shall see that their market value depends
on whether they compete on price, or on
quantity.
Demand and Technology
Consumer demand for a product is a linear
function of price, and that market pre-testing
has established:
q  13  p
We also suppose that the industry has constant
scale returns, and we set the average cost of
producing a unit at 1.
Price competition
When firms compete on price, the firm which
charges the lowest price captures all the market.
When both firms charge the same price, they share
the market equally.
These sharp predictions would be weakened if
there were capacity constraints, or if there was
some product differentiation (such as location rents
or market niches).
Profit to the first firm
As a function of (p1,p2), the net profit to the first firm
is:
1 ( p1 , p2 )  (13  p1 )( p1 1) if p1  p2
1
 (13  p1 )( p1  1) if
2
0 if
p1  p2
p2  p1
Net profit to the second firm is calculated in a similar
way.
Market games with price competition
We could try to solve
the problem
algebraically.
An alternative is to see
how human subjects
attack this problem
within an experiment.
We have substituted
some price pairs and
their corresponding
profits into the depicted
matrix.
Solving the price setting game
Setting price equals 7 is
dominated by a mixture of
setting price to 5 or 2, with most
of the probability on 5.
Eliminating price equals 7 for
both firms we are left with a 3 by
3 matrix.
Now setting price equals 5 is
dominated by a mixture of
setting price to 3 or 2.
In the resulting 2 by 2 matrix a
dominant strategy of charging 2
emerges for both players.
The algebraic solution
to the price setting game
Notice that neither firm will offer the product at
strictly less than cost, other wise they will make a
loss.
Suppose one firm offers the product at a price p1
greater than cost. The best response of the other firm
is to charge a price between cost and . This proves
there is no Nash equilibrium in which either firm
charges a price strictly more than cost.
Finally suppose one firm charges at price equals cost.
A best response of the second firm is to do the same
thing. This is the unique Nash equilibrium to this
pricing game.
Quantity competition
When firms compete on quantity, demanders set a
market price that clears inventories and fills every
customer order.
If firms have the same constant costs of
production, and hence the same markup, then their
profits are proportional to their market share.
This predictions might be violated if the price
setting mechanism was not efficient, or if the
assumptions about costs were invalid.
Calculating profits when there is
quantity competition
Letting q1 and q2 denote the quantities chosen by the
firms, the industry price is derived from the demand
curve as:
p = 13 - q1 – q2
When the second firm produces q2, as a function of
its choice q1, the profits to the first firm are
q1(13 - q1 – q2) - q1 = q1[12 - q1 – q2]
The profits of the second firm are calculated the
same way.
Market games with quantity competition
As in the price setting
game, we could try to solve
the game algebraically, or
set the model up as an
experiment.
If we can compute profits
as a function of the
quantity choices, using the
second approach, we can
easily vary the underlying
assumptions to investigate
the outcomes of alternative
formulations.
The first order for a firm in
to the quantity setting game
When the second firm produces q2, as a function of
its choice q1, the profits to the first firm are
q1(13 – q1 – q2) - q1 = q1[12 – q1 – q2]
Maximizing with respect to q1, we obtain the first
order condition
[12 – 2q1 – q2] = 0
or
[12 – q2] = 2q1
Solving the quantity setting game
In a symmetric equilibrium with both firms producing
the same quantity this yields a solution with each
firm producing 3q1 = 12 or q1 = 4.
More generally suppose there are N firms in the
industry and they each produce the same amount.
Let q1 denote the quantity produced by any given
firm and q2 denote the quantity produced by the
remaining N – 1 firms.
The first order condition is still
0 = 12 – 2q1 – q2 = 12 – 2q1 – (N - 1)q1
= 12 – (N + 1)q1
The competitive limit
Solving we have that
or
0 = 12 – (N + 1)q1
q1 = 12/ (N + 1)
Since there are N firms in the industry this implies
the amount supplied by the industry is
Nq1 = 12N/ (N + 1)
Compare this with the competitive equilibrium
quantity, 12, obtained be setting price equal to
marginal cost.
Marginal costs
Consider the following two
examples in which:
1. There are declining costs
for a single good
2. There are scale
complementarities in a
production mix.
Aggregating information
In the trading games described above, all traders had
the same information, and trade occurs because of
differences in endowments and preferences.
How is trading affected when there are differences
between traders in their information?
With respect to competitive equilibrium:
1. How much does competitive equilibrium reveal
of the knowledge demanders and suppliers
have?
2. Are the predictions of competitive equilibrium
theory accurate?
The no-trade theorem
Suppose people have differential information
about an asset they would all value the same way
if they were fully informed.
Will any trade take place?
Note that if one trader party benefits from the
trading then the other party must lose.
Since all traders anticipate this, we thus establish
that no trade occurs in competitive equilibrium, or
for that matter any other solution to a (voluntary)
trading mechanism, because of differences in
information alone.
Competitive equilibrium and information
Competitive equilibrium economizes on
the amount of information traders need to
optimize their portfolios.
Indeed a peculiar feature of competitive
equilibrium is that in some situations it
fully reveals private information to those
who are less informed about market
conditions.
Private valuations
in an endowment economy
A simple example illustrating how competitive
equilibrium aggregates information is in a market
where consumer valuations are identically and
independently distributed, and aggregate supply
is fixed.
Suppose no demander wants to consume more
than one unit of the good, and each demander
draws an
identically
and independently
distributed random variable that determines their
valuation for the first unit.
Demanders are said to have private valuations.
Competitive equilibrium
in endowment economies
The competitive equilibrium price does not
depend on whether each trader observes the
valuations of the others.
Hence every trader acts the same way as he
would if he were fully informed about
aggregate demand.
We compare two markets, one in which
demanders know the private valuations of
everyone, and the other in which they don’t.
Fully revealing prices
about a common shock
Suppose there are N traders, and the nth trader
receives a signal sn about the the state of the
economy, where n = 1,2, . . ., N.
In general, the competitive equilibrium price vector
p depends on all the signals the traders receive.
If, however, p is an invertible mapping of all the
relevant information s available to traders, then
every trader acts the same way as he would if he
were fully informed.
In this case p(s) has an inverse which we call f(p).
Each trader realizes that seeing p is as good as
seeing s.
In the example above s is aggregate demand, and
p is monotone increasing in s.
Differential information
about product quality
Now suppose a component of each demander
valuation is common, and traders have differential
information about that component.
The more favorable the signal to the informed traders,
the greater is their demand, and hence the higher is
the market clearing price.
As in the previous example, uninformed traders
compute their demands, deducing that if the market
clearing price is p, then the common component is
f(p). Thus informed traders cannot benefit from their
superior information in competitive equilibrium.
Implications for trading mechanisms
Some economists have used this theoretical result to
argue that markets are good at aggregating the
information that traders have about the preferences
of demanders and the technologies of suppliers.
Other economists have argued there is limited
investment in acquiring new information relevant to
suppliers and demanders, because those who use up
resources to become better informed cannot recoup
the benefit from their private information.
Both arguments implicitly assume that a competitive
equilibrium accurately predicts price and resource
allocations from trading.
When do competitive equilibrium
prices hide information?
There are two scenarios when competitive equilibrium
prices are not fully revealing:
1. The mapping from signals to prices p(s) is not
invertible. That is, two or more values of a signal,
s1 and s2, would yield the same fully revealing
competitive equilibrium price if everyone observed
the signal’s value, meaning pfr(s1) = pfr(s2).
2. Different units of the product are not identical,
although they are traded on the same market,
and these differences are observed by some but
not all the traders.
A comparative study
We start out with two shocks and explore what
happens if:
1. both shocks are observed by everyone
2. only one shock is observed by some of
the traders
3. both shocks are observed by somebody
but nobody observes both shocks
4. someone observes both shocks and the
others observe nothing.
Adding dimensions to uncertainty
The uninformed segment of the population can infer
the true state in the previous example because a
mapping exists from the competitive equilibrium price
to the shock defining the product quality.
In our next example we introduce a second shock.
Those traders who observe one shock can infer the
other from the competitive equilibrium price.
Those who observe neither can only form estimates
of what both shocks are from the competitive
equilibrium price.
Uncertain supply and quality
Now suppose that product quality is only known by
some of the demanders, and the aggregate quantity
supplied is also a random variable.
In this case an uninformed demander cannot infer
product quality from the competitive equilibrium
price, because a high price could indicate high
demand from informed traders or low supply.
Informed demanders benefit from the fact that
demand by uninformed traders is less than it would
be if they were fully informed when product quality is
high, depressing the price for high quality goods, and
vice versa.
Differential information about
heterogeneity across units
Suppose the quality of the individual units varies,
and that traders are differentially informed it.
What would competitive equilibrium theory
predict about the price and quantity traded?
Since traders condition their individual demand
and supply on their information, more informed
traders gain at the expense of the less informed.
The prospect of being exploited by a well
informed trader discourages a poorly informed
player from trading.
The market for lemons
For example, consider a used car market.
Suppose there are less cars than commuter traders, and
no one demands more than one car.
The valuation of a trader for owning one car is identically
and independently distributed across the population.
The quality of each car is independently and identically
distributed across the population. Each owner, but no
one, else knows the quality of his car.
The amenity value from car travel is the product of the
commuter’s valuation and the quality of the car he owns.
Lecture summary
As the number of firms in a constant cost industry
increases, there is convergence to the competitive limit
regardless of whether firms compete on price or quantity.
When there are declining unit costs or scope economies,
competitive behavior is not sustainable, and price is not
equated with marginal cost in equilibrium.
We competitive equilibrium with market outcomes when:
1. private valuations affect a subject’s willingness to
trade but not the distribution of demand.
2. differences in individual valuations are solely
attributable to the information about the product
3. heterogeneity in product quality is not observed by
both sides of the market