Transcript lecture 6 - ComLabGames

Lecture 5
Monopoly practice
and the competitive limit
The latter parts of the lecture analyze other
aspects of monopolistic practices. We discuss
mechanisms for setting prices and quantities,
the role of commitment, market segmentation,
and product bundling. Then we investigate the
effects of competition,the competitive limit and
the related concept of competitive equilibrium.
A dynamic inconsistency?
But what if the monopolist set price so that
marginal revenue equals marginal cost, and the
demanders whose valuations exceeded the price
immediately purchased the item at the beginning
of the game?
The static solution illustrated
Price in dollars
20
inverse demand curve
Uniform price
solution
unit cost
10
marginal revenue curve
quantity
0
Uniform quantity
solution
Residual demand
Price
20
New vertical axis for origin of
residual inverse demand
Uniform price
solution
Unit cost
10
New marginal revenue curve
Quantity
0
Will price fall to marginal cost?
The solution to this game is for the monopolist to let the
price decline to where marginal revenue equals marginal cost
at the end of the game, thus presenting each consumer
simultaneously with a take it or leave it offer at that price.
Equivalently, the monopolist can commit to a uniform
price policy by committing to everyone the lowest price he
offers to anyone.
Letting a firm split can also resolve the problem if each of
the new break off firms guarantee to match each other’s
discounts.
Durable goods monopoly
One way of avoiding the problem associated with a
random cut-off time is to rent the good for short
periods.
But this is not always possible, and it might be
desirable to attempt to price discriminate between
consumers.
There are two cases to focus on:
1.
Constant marginal cost
2.
Fixed supply
Discriminating monopolist
Price and product discrimination is more widespread
than in durable goods problems, where the monopolist
may be able to sort customers by their urgency.
Suppose the monopolist knows the valuations of the
players, and can commit to prices.
Make a take it or leave it offer to each person: multi
person ultimatum game.
Now imagine that it cannot prevent players from
buying in any market they like.
Now let monopolist condition on characteristics that
are related to their valuations which he cannot observe.
Multiple markets
Consider now another related method for
segmenting market demand to extract greater
economic rent.
The firm exploits the idea that customers who
demand several of the firm’s products might exhibit
more elastic demands (be more price sensitive) than
customers who only wish to purchase a smaller
subset of the firm’s products.
Perhaps the most common example of this
behavior is quantity discounting (sometimes enforced
through packaging).
Other examples
1. Firms sell assembled goods such as cars or other durables
for new car buyers and demand from previous buyers,
plus replacement parts arising from collision damage or
wear and tear.
2. Restaurants (furniture stores, car dealers) offer complete
dinners (suites, high performance and luxury packages)
with a limited (selected) range of items, and also offer
portions a la carte (set pieces, individual components).
3. Ski resorts (amusement parks, cellular phone companies
internet or cable operators) offer vacation packages for
lodging and tickets (entry or connection plus service
charges) as well as sell tickets (services) by themselves.
A product bundling monopolist
A resort owner has a monopoly over two products,
called accommodation on the mountain, and ski lift
tickets.
Some demanders visit the mountain resort to ski
downhill, while others come to cross country ski or
snowshoe (neither of which requires lift tickets).
Demanders also choose between commuting from the city
90 minutes by road, or by renting an apartments or a
hotel room at the resort.
What should the resort owner charge for ski tickets, for
accommodation, and for the holiday package of both?
The number of rivals
Now we investigate how the solution to trading
games is affected by relaxing the assumption that there
is only a single supplier (or more generally dealer) in
each market.
First we analyze how monopoly power breaks down
with competition from rival producers.
This leads us to define price taking behavior and a
definition of competitive equilibrium.
The competitive limit
We first consider two extensions of the multiunit
auction, where there is a constant marginal cost of
production.
In the first case we assume that entry occurs
sequentially until it is unprofitable to do so. This
corresponds to a competitive market where rival
suppliers compete for demanders.
In the second case we assume that the
monopolist or cartel maximizes producer surplus.
Duopoly
Considering the monopoly problem of the previous
lecture, let us now introduce a second seller with same
marginal cost schedule, and no fixed costs.
Three or more producers
Continuing in this vein, one could fragment the
organization of production even more.
For example consider how three or more producers would
compete against each other.
Can we endogenously determine the number of entrants?
Price competition
and capacity constraints
It seems that a remarkably small number of
competing firms suffice to drive the price down to
marginal cost.
But this result is partly driven by the cost
structure.
Now suppose there is a two stage game, where
firms construct capacity for production in the first
stage, and market their produce in a second stage.
Declining marginal cost
Now suppose unit costs fall with scale of
production. For example suppose there is a fixed
cost of entry (technological know how or plant
set up) as well as a constant marginal cost.
If there is only one producer, then the profit
maximizing quantity for the firm is
What happens in the case of two producers?
Is there convergence?
A natural question to ask is where this process
would converge, and whether there is an easy
way to model what would happen in the limit.
Do our experiments suggest that the limit
point depends on the cost structure?
Another question is how many firms are
required to reach this limit (that is when it exists).
Free entry
Consider first the uniform distribution. In a
second price auction
In the first case we assume that entry occurs
sequentially until it is unprofitable to do so. This
corresponds to a competitive market where rival
suppliers compete for demanders.
In the second case we assume that the
monopolist or cartel maximizes producer surplus.
Definition of competitive equilibrium
A competitive equilibrium is a single price, or a price
band (an interval on the real line), with two defining
properties:
1.
Traders treat each point in the competitive
equilibrium as a fixed price, seeking to buy or sell units
of the good that maximize their objective function at
that fixed price.
2.
At every price above those in the competitive
equilibrium, demand exceeds supply. At every price
below those in the competitive equilibrium, supply
exceeds demand.
Competitive equilibrium
as a tool for prediction
The key advantage from assuming that markets are in
competitive equilibrium is that models of competitive
equilibrium 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
same game.
In other words, using the tools of competitive
equilibrium we can sometimes make accurate predictions
with minimal effort.
An economy with one stock
Consider the following economy:
There is one stock, as well as money. The common
value of the asset is constant, and every one is fully
informed.
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).
Example 1
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 4 units of the good, and everybody
else has $12 to buy units of the good.
We also assume that everyone has the same
access to the market, and can place limit or
market orders.
The front page of a player’s folio.
Example 2
Now we modify the example a little.
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 4 units of the good, and everybody
else has $12 to buy units of the good.
As before we assume that everyone has the
same access to the market, and can place limit or
market orders.
The front page
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.
Then aggregate across players to obtain the
aggregate net demand.
The set of competitive equilibrium prices is found by
applying the second part of the definition: every price
below (above) prices in the set generate positive
(negative) aggregate net demand.
Individual optimization in a
competitive equilibrium
In a competitive equilibrium with price p the
objective of player i is to pick the quantity of stock
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
Aggregate 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:
mi
I
D p  i 11 vi  p
p
   

Thus D(p) declines in steps, for two reasons. As p falls
the number of players with valuations exceeding p
increases, and demanders who are willing to buy at higher
prices can now afford to buy more units.
Aggregate 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 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
We find 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.
By definition the p+ prices are defined by the
inequality that:
 
Dp




I
1
v
i 1 i

 
 p  mi p   S p 
Similarly the p- prices are defined by:
 
D p


 
S p




I
1
v
i
i 1

 p  si
Aggregate supply in the first example
At prices above $3, the third player will supply 4
units, and at price $3, the player is indifferent
between supplying quantity between 0 and 4. No one
supplies anything to the market at less than $3.
Define q as any quantity satisfying the inequalities:
0q4
Then the supply function is:
S  p  1p  34  1p  3q
Aggregate demand in the first example
To make the problem more manageable we
will assume that traders can buy fractions of
units, rather than just whole ones.
Then we can write the demand schedule as:
D p 
12
 i 11i  p
p
10
Graph of supply and demand curves
Competitive Equilibrium
in the first example
In this example, there is a unique equilibrium
price at Note that at prices above , demand shrinks
quite markedly because infra marginal demanders
can no longer afford more than one unit. Similarly
at prices below , demanders want considerably
more than what producers can supply.
However at the unique equilibrium price not all
demanders are able to fulfill their plans. In limit
order markets those demanders who enter their
orders first receive priority over those who
recognize the equilibrium price later.
Aggregate supply in the second example
Recall that aggregate demand in the both examples is
the same. We now derive supply as a function of price.
At prices above $3, the third player will supply 4 units,
and at price $3, the player is indifferent between
supplying quantity between 0 and 4. No one supplies
anything to the market at less than $3.
Define q as any quantity satisfying the inequalities:
0q4
Then the supply function is:
S  p  1p  34  1p  3q
Supply and demand in second example
Competitive equilibrium
in the second example
In this example, there is a band of equilibrium
prices. At every price between and suppliers and
demanders wish to trade units between them. At all
these prices both demanders and suppliers are able
to fulfill their plans.
However the price is not determined uniquely by
the theory of competitive equilibrium. Whereas in
the previous example demanders competed with
each other for the limited supplier, here demanders
and suppliers can bargain over who should receive
the most gains from trading.
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.
But is competitive equilibrium realistic?
The short answer is maybe. Whether or not this is
true depends on the:
1. Cost structure
2. Durability and nature of product demand
3. Number of firms in the industry
4. Threat of entry by new firms
•
Clearly strategy consultants search for situations
where these factors are not conducive to the
existence of a competitive equilibrium.