Monopoly

1
Profit Maximization

• What is the goal of the firm?
• Expand, expand, expand Amazon.
• Earnings growth GE.
• Produce the highest possible quality this class.
• Many other goals happy customers, happy workers,
  good reputation, etc.
• It is to maximize profits that is, present value
  of all current and future profits (also known as
  net present value NPV).

2
Firm Behavior under Profit Maximization

• Monopoly
• Oligopoly
• Price Competition
• Quantity Competition
• Simultaneous
• Sequential

3
Monopoly

• Standard Profit Maximization is
• max r(y)-c(y).
• With Monopoly this is Max p(y)y-c(y) (the
  difference to competition is price now depends
  upon output).
• FOC yields p(y)p(y)yc(y). This is also
  Marginal RevenueMarginal Cost.

4
Example (from Experiment)

• We had quantity Q15-p. While we were choosing
  prices. This is equivalent (in the monopoly case)
  to choosing quantity.
• r(y) yp(y) where p(y)15-y. Marginal revenue
  was 15-2y.
• We had constant marginal cost of 3. Thus,
  c(y)3y.
• Profity(15-y)-3y
• What is the choice of y? What does this imply
  about p?

5
Rule of thumb prices

• Many shops use a rule of thumb to determine
  prices.
• Clothing stores may set price double their costs.
• Restaurants set menu prices roughly 4 times
  costs.
• Can this ever be optimal?
• qAp? (p(1/A) 1/?q1/?)
• Notice in this case that p(y)p(y)y(1/ ?)p(y).
• If marginal cost is constant, then p(y) ?mc for
  any price.
• There is a constant mark-up percentage!
• Notice that (dq/q)/(dp/p) ?. What does ?
  represent?

6
Bertrand (1883) price competition.

• Both firms choose prices simultaneously and have
  constant marginal cost c.
• Firm one chooses p1. Firm two chooses p2.
• Consumers buy from the lowest price firm. (If
  p1p2, each firm gets half the consumers.)
• An equilibrium is a choice of prices p1 and p2
  such that
• firm 1 wouldnt want to change his price given
  p2.
• firm 2 wouldnt want to change her price given p1.

7
Bertrand Equilibrium

• Take firm 1s decision if p2 is strictly bigger
  than c
• If he sets p1gtp2, then he earns 0.
• If he sets p1p2, then he earns 1/2D(p2)(p2-c).
• If he sets p1 such that cltp1ltp2 he earns
  D(p1)(p1-c).
• For a large enough p1 that is still less than p2,
  we have
• D(p1)(p1-c)gt1/2D(p2)(p2-c).
• Each has incentive to slightly undercut the
  other.
• Equilibrium is that both firms charge p1p2c.
• Not so famous Kaplan Wettstein (2000) paper
  shows that there may be other equilibria with
  positive profits if there arent restrictions on
  D(p).

8
Bertrand Game
Marginal cost 3, Demand is 15-p. The Bertrand
competition can be written as a game.
Firm B
9
8.50
35.75
18
9
18
0
Firm A
17.88
0
8.50
17.88
35.75
For any pricegt 3, there is this incentive to
undercut. Similar to the prisoners dilemma.
9
Cooperation in Bertrand Comp.

• A Case The New York Post v. the New York Daily
  News
• January 1994 40 40
• February 1994 50 40
• March 1994 25 (in Staten Island) 40
• July 1994 50 50

10
What happened?

• Until Feb 1994 both papers were sold at 40.
• Then the Post raised its price to 50 but the
  News held to 40 (since it was used to being the
  first mover).
• So in March the Post dropped its Staten Island
  price to 25 but kept its price elsewhere at 50,
• until News raised its price to 50 in July,
  having lost market share in Staten Island to the
  Post. No longer leader.
• So both were now priced at 50 everywhere in NYC.

11
Collusion

• If firms get together to set prices or limit
  quantities what would they choose. As in your
  experiment.
• D(p)15-p and c(q)3q.
• Price Maxp (p-3)(15-p)
• What is the choice of p.
• This is the monopoly price and quantity!
• Maxq1,q2 (15-q1-q2)(q1q2)-3(q1q2).

12
Anti-competitive practices.

• In the 80s, Crazy Eddie said that he will beat
  any price since he is insane.
• Today, many companies have price-beating and
  price-matching policies.
• A price-matching policy (just saw it in an ad for
  Nationwide) is simply if you (a customer) can
  find a price lower than ours, we will match it. A
  price beating policy is that we will beat any
  price that you can find. (It is NOT explicitly
  setting a price lower or equal to your
  competitors.)
• They seem very much in favor of competition
  consumers are able to get the lower price.
• In fact, they are not. By having such a policy a
  stores avoid loosing customers and thus are able
  to charge a high initial price (yet another
  paper by this Kaplan guy).

13
Price-matching

• Marginal cost is 3 and demand is 15-p.
• There are two firms A and B. Customers buy from
  the lowest price firm. Assume if both firms
  charge the same price customers go to the closest
  firm.
• What are profits if both charge 9?
• Without price matching policies, what happens if
  firm A charges a price of 8?
• Now if B has a price matching policy, then what
  will Bs net price be to customers?
• B has a price-matching policy. If B charges a
  price of 9, what is firm As best choice of a
  price.
• If both firms have price-matching policies and
  price of 9, does either have an incentive to
  undercut the other?

14
Price-Matching Policy Game
Marginal cost 3, Demand is 15-p. If both firms
have price-matching policies, they split the
demand at the lower price.
Firm B
9
8.50
17.88
18
9
18
17.88
Firm A
17.88
17.88
8.50
17.88
17.88
The monopoly price is now an equilibrium!
15
Quantity competition (Cournot 1838)

• ?1p(q1q2)q1-c(q1)
• ?2 p(q1q2)q2-c(q2)
• Firm 1 chooses quantity q1 while firm 2 chooses
  quantity q2.
• Say these are chosen simultaneously. An
  equilibrium is where
• Firm 1s choice of q1 is optimal given q2.
• Firm 2s choice of q2 is optimal given q1.
• If D(p)13-p and c(q)q, what the equilibrium
  quantities and prices.
• Take FOCs and solve simultaneous equations.
• Can also use intersection of reaction curves.

16
FOCs of Cournot

• ?1(15-(q1q2))q1-3q1(12-(q1q2))q1
• Take derivative w/ respect to q1.
• Show that you get q16-q2/2.
• This is also called a reaction curve (q1s
  reaction to q2).
• ?2 (15-(q1q2))q2-3q2 (12-(q1q2))q2
• Take derivative w/ respect to q2.
• Symmetry should help you guess the other
  equation.
• Solution is where these two reaction curves
  intersect. It is also the soln to the two
  equations.
• Plugging the first equation into the second,
  yields an equation w/ just q2.

17
Cournot Simplified

• We can write the Cournot Duopoly in terms of our
  Normal Form game (boxes).
• Take D(p)4-p and c(q)q.
• Price is then p4-q1-q2.
• The quantity chosen are either S3/4, M1, L3/2.
• The payoff to player 1 is (3-q1-q2)q1
• The payoff to player 2 is (3-q1-q2)q2

18
Cournot Duopoly Normal Form Game Profit1(3-q1-q
2)q1 and Profit 2(3-q1-q2)q2
S3/4
M1
L3/2
9/8
9/8
5/4
S3/4
9/8
15/16
9/16
15/16
1
3/4
M1
5/4
1
1/2
1/2
9/16
0
L3/2
9/8
3/4
0
19
Cournot

• What is the Nash equilibrium of the game?
• What is the highest joint payoffs? This is the
  collusive outcome.
• Notice that a monopolist would set mr4-2q equal
  to mc1.
• What is the Bertrand equilibrium (pmc)?

20
Quantity competition (Stackelberg 1934)

• ?1p(q1q2)q1-c(q1)
• ?2 p(q1q2)q2-c(q2)
• Firm 1 chooses quantity q1. AFTERWARDS, firm 2
  chooses quantity q2.
• An equilibrium now is where
• Firm 2s choice of q2 is optimal given q1.
• Firm 1s choice of q1 is optimal given q2(q1).
• That is, firm 1 takes into account the reaction
  of firm 2 to his decision.

21
Stackelberg solution

• If D(p)15-p and c(q)3q, what the equilibrium
  quantities and prices.
• Must first solve for firm 2s decision given q1.
• Maxq2 (15-q1-q2)-3q2
• Must then use this solution to solve for firm 1s
  decision given q2(q1) (this is a function!)
• Maxq1 15-q1-q2(q1)-3q1
• This is the same as subgame perfection.
• We can now write the game in a tree form.

22
Stackelberg Game.
(0,0)
L
M
(.75,.5)
B
S
L
(1.13,.56)
(.5,.75)
L
M
A
M
A
B
B
(1,1)
S
(1.25,.94)
L
(.56,1.13)
S
M
B
B
(.94,1.25)
(1.13,1.13)
S
23
Stackelberg game

• How would you solve for the subgame-perfect
  equilibrium?
• As before, start at the last nodes and see what
  the follower firm B is doing.

24
Stackelberg Game.
(0,0)
L
M
(.75,.5)
B
S
L
(1.13,.56)
(.5,.75)
L
M
A
M
A
B
B
(1,1)
S
(1.25,.94)
L
(.56,1.13)
S
M
B
B
(.94,1.25)
(1.13,1.13)
S
25
Stackelberg Game

• Now see which of these branches have the highest
  payoff for the leader firm (A).
• The branches that lead to this is the equilibrium.

26
Stackelberg Game.
(0,0)
L
M
(.75,.5)
B
S
L
(1.13,.56)
(.5,.75)
L
M
A
M
A
B
B
(1,1)
S
(1.25,.94)
L
(.56,1.13)
S
M
B
B
(.94,1.25)
(1.13,1.13)
S
27
Stackelberg Game Results

• We find that the leader chooses a large quantity
  which crowds out the follower.
• Collusion would have them both choosing a small
  output.
• Perhaps, leader would like to demonstrate
  collusion but cant trust the follower.
• Firms want to be the market leader since there is
  an advantage.
• One way could be to commit to strategy ahead of
  time.
• An example of this is strategic delegation.
• Choose a lunatic CEO that just wants to expand
  the business.