Transcript Price competition.

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 (1883) price competition.
• Two firms choose prices simultaneously and act
non-cooperatively.
• Firm one chooses p1. Firm two chooses p2.
• Firms have constant marginal cost c, and
products are homogeneous (no difference).
• 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.
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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).
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
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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) 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 Easyjet affect prices?