Download Monopoly

Profit Maximization
• What is the goal of the firm?
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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).
Firm Behavior under Profit
Maximization
• Monopoly
• Oligopoly
– Price Competition
– Quantity Competition
• Simultaneous
• Sequential
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)y=c’(y). This is also
Marginal Revenue=Marginal Cost.
Example (from Experiment)
• We had quantity Q=15-p. While we were choosing
prices. This is equivalent (in the monopoly case)
to choosing quantity.
• r(y)= y*p(y) where p(y)=15-y. Marginal revenue
was 15-2y.
• We had constant marginal cost of 3. Thus,
c(y)=3*y.
• Profit=y*(15-y)-3*y
• What is the choice of y? What does this imply
about p?
Rule of thumb prices
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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?
q=Apє (p=(1/A) 1/єq1/є)
Notice in this case that p(y)+p’(y)y=(1/ є)p(y).
If marginal cost is constant, then p(y)= є·mc for any price.
There is a constant mark-up percentage!
Notice that (dq/q)/(dp/p)= є. What does є represent?
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
p1=p2, each firm gets half the consumers.)
• An equilibrium is a choice of prices p1 and p2
such that
– firm 1 wouldn’t want to change his price given p2.
– firm 2 wouldn’t want to change her price given p1.
Bertrand Equilibrium
• Take firm 1’s decision if p2 is strictly bigger than
c:
– If he sets p1>p2, then he earns 0.
– If he sets p1=p2, then he earns 1/2*D(p2)*(p2-c).
– If he sets p1 such that c<p1<p2 he earns D(p1)*(p1-c).
• For a large enough p1 that is still less than p2, we
have:
– D(p1)*(p1-c)>1/2*D(p2)*(p2-c).
• Each has incentive to slightly undercut the other.
• Equilibrium is that both firms charge p1=p2=c.
• Not so famous Kaplan & Wettstein (2000) paper shows
that there may be other equilibria with positive profits if
there aren’t restrictions on D(p).
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 price> £3, there is this incentive to undercut.
Similar to the prisoners’ dilemma.
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¢
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.
Collusion
• If firms get together to set prices or limit
quantities what would they choose. As in
your experiment.
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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)*(q1+q2)-3(q1+q2).
Anti-competitive practices.
• In the 80’s, Crazy Eddie said that he will beat any price
since he is insane.
• Today, many companies have price-beating and pricematching 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).
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 B’s net price be to customers?
• B has a price-matching policy. If B charges a price
of 9, what is firm A’s 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?
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!
Quantity competition (Cournot 1838)
• Л1=p(q1+q2)q1-c(q1)
• Л2= p(q1+q2)q2-c(q2)
• Firm 1 chooses quantity q1 while firm 2 chooses
quantity q2.
• Say these are chosen simultaneously. An
equilibrium is where
– Firm 1’s choice of q1 is optimal given q2.
– Firm 2’s 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.
FOCs of Cournot
• Л1=(15-(q1+q2))q1-3q1=(12-(q1+q2))q1
– Take derivative w/ respect to q1.
– Show that you get q1=6-q2/2.
– This is also called a reaction curve (q1’s reaction to q2).
• Л2= (15-(q1+q2))q2-3q2= (12-(q1+q2))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.
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 p=4-q1-q2.
• The quantity chosen are either S=3/4, M=1,
L=3/2.
• The payoff to player 1 is (3-q1-q2)q1
• The payoff to player 2 is (3-q1-q2)q2
Cournot Duopoly: Normal Form Game
Profit1=(3-q1-q2)q1 and Profit 2=(3-q1-q2)q2
S=3/4
M=1
L=3/2
9/8
9/8
5/4
S=3/4
9/8
15/16
9/16
15/16
1
3/4
M=1
5/4
1
1/2
1/2
9/16
0
L=3/2
9/8
3/4
0
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 mr=4-2q
equal to mc=1.
• What is the Bertrand equilibrium (p=mc)?
Quantity competition
(Stackelberg 1934)
• Л1=p(q1+q2)q1-c(q1)
• Л2= p(q1+q2)q2-c(q2)
• Firm 1 chooses quantity q1. AFTERWARDS,
firm 2 chooses quantity q2.
• An equilibrium now is where
– Firm 2’s choice of q2 is optimal given q1.
– Firm 1’s choice of q1 is optimal given q2(q1).
– That is, firm 1 takes into account the reaction of firm 2
to his decision.
Stackelberg solution
• If D(p)=15-p and c(q)=3q, what the equilibrium
quantities and prices.
• Must first solve for firm 2’s decision given q1.
– Maxq2 [(15-q1-q2)-3]q2
• Must then use this solution to solve for firm
1’s decision given q2(q1) (this is a
function!)
– Maxq1 [15-q1-q2(q1)-3]q1
• This is the same as subgame perfection.
• We can now write the game in a tree form.
Stackelberg Game.
L
M
S
B
L
(0,0)
(.75,.5)
(1.13,.56)
M
AA
M
B
B
S
L
S
(.56,1.13)
M
B
B
S
(.5,.75)
L
(.94,1.25)
(1.13,1.13)
(1,1)
(1.25,.94)
Stackelberg game
• How would you solve for the subgameperfect equilibrium?
• As before, start at the last nodes and see
what the follower firm B is doing.
Stackelberg Game.
L
M
S
B
L
(0,0)
(.75,.5)
(1.13,.56)
M
AA
M
B
B
S
L
S
(.56,1.13)
M
B
B
S
(.5,.75)
L
(.94,1.25)
(1.13,1.13)
(1,1)
(1.25,.94)
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.
Stackelberg Game.
L
M
S
B
L
(0,0)
(.75,.5)
(1.13,.56)
M
AA
M
B
B
S
L
S
(.56,1.13)
M
B
B
S
(.5,.75)
L
(.94,1.25)
(1.13,1.13)
(1,1)
(1.25,.94)
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 can’t 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.