Lecture 5 Monopoly practice and the competitive limit

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Lecture 5Monopoly 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.
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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?
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The static solution illustrated
Price in dollars
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inverse demand curve
Uniform price solution
unit cost
10
marginal revenue curve
quantity
0
Uniform quantity solution
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Residual demand
Price
New vertical axis for origin of residual inverse
demand
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Uniform price solution
Unit cost
10
New marginal revenue curve
Quantity
0
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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 others discounts.

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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

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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.

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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 firms products might
  exhibit more elastic demands (be more price
  sensitive) than customers who only wish to
  purchase a smaller subset of the firms products.
• Perhaps the most common example of this
  behavior is quantity discounting (sometimes
  enforced through packaging).

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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.

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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?

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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.

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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.
  

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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.
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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?

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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.

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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?
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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).

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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.
  

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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.
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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.

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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).

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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.

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The front page of a players folio.
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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.

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The front page
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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.

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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

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Constraints in the optimization problem
The short sale constraint is
The solvency constraint is
These constraints can be combined as
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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
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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

• 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.

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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
  

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

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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.

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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

• Similarly the p- prices are defined by

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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

• Then the supply function is

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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

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Graph of supply and demand curves
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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.

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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

• Then the supply function is

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Supply and demand in second example
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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.

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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.

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But is competitive equilibrium realistic?

• The short answer is maybe. Whether or not this is
  true depends on the
• Cost structure
• Durability and nature of product demand
• Number of firms in the industry
• 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.