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Price competition.
Firm Behavior under Profit
Maximization
• Monopoly
• Bertrand Price Competition
Monopoly
• A monopoly solves Max p(q)q-c(q)
– q is quantity.
– c(q) is cost of producing quantity q.
– p(q) is price (price depends upon output).
• FOC yields p(q)+p’(q)q=c’(q). 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(q)= q*p(q) where p(q)=15-q. Marginal revenue
was 15-2q.
• We had constant marginal cost of 3. Thus,
c(q)=3*q.
• Profit=q*(15-q)-3*q
• What is the choice of q? What does this imply
about p?
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.
Sample result: Bertrand Game
Average Price
Average Selling Price
8
7
6
Price
5
4
Marginal Cost
3
2
Two Firms
1
Five Firms
Two Firms
Fixed Partners Random Partners
Random Partners
3
21
0
1
5
7
9
11
13
15
Time
17
19
23
25
27
29
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.
•
•
•
•
•
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).
Graph of total profit:
(15-price)(price-3)
Maximum is price=9
With profit 36.
40
35
Profit
30
25
20
15
10
5
4
6
8
Price
10
12
14
Collusion by Repeated Interaction
• Let us say that firms have a discount factor of B.
• If each make 18 each period. How much is the
present value?
• The one period undercutting gains is close to 18.
• The other firm can punish under-cutters by
causing zero profit from then on.
• A firm will not cheat only if the punishment is
worse than the gains.
• For what values of B will the firm not cheat?
• 18B/(1-B)>=18 (or B>=1/2).
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
price-matching policies.
• A price-matching policy 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.)
Price-matching Policy
Price-Beating Policy
Price Matching/Price Beating
• 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!
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?
q=Apє (p=(1/A) 1/єq1/є) where -1> є
Notice in this case that p(q)+p’(q)q=((1+є)/ є)p(q).
If marginal cost is constant, then p= є/(1+є)mc for
any mc.
• There is a constant mark-up percentage!
• Notice that (dq/q)/(dp/p)= є. What does є represent?
Homework
• El Al and British Air are competing for passengers
on the Tel Aviv- Heathrow route. Assume marginal
cost is 4 and demand is Q = 18 − P.
– If they choose prices simultaneously, what will be the
Bertrand equilibrium?
– If they can collude together and fix prices, what would
they charge.
– In practice with such competition under what conditions
would you expect collusion to be strong and under
what conditions would you expect it to be weak.
– Under what conditions should the introduction of BMI
(another airline) affect prices?
– If the game is infinitely repeated, under what discount
factor B would full collusion be obtainable according to
standard game theory.