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

Firm B

Firm A

£9

1835.75

018

0 17.88

£8.50

£9

£8.50

Marginal cost= £3, Demand is 15-p.

The Bertrand competition can be written as a game.

For any price> £3, there is this incentive to undercut. Similar to the prisoners’ dilemma.

35.75 17.88

Sample result: Bertrand Game

0

1

2

3

4

5

6

7

8

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Time

Pri

ceAverage Price Average Selling Price

Marginal Cost

Two Firms

Fixed Partners

Two Firms

Random Partners

Five Firms

Random Partners

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)

4 6 8 10 12 14

5

10

15

20

25

30

35

40

Profit

Price

Maximum is price=9With profit 36.

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

Firm B

Firm A

£9

1817.88

17.8818

17.88 17.88

£8.50

£9

£8.50

Marginal cost= £3, Demand is 15-p. If both firms have price-matching policies, they split the demand at the lower price.

The monopoly price is now an equilibrium!

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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.