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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).

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Price competition.

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  1. Price competition.

  2. Firm Behavior under Profit Maximization • Monopoly • Bertrand Price Competition

  3. 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.

  4. 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?

  5. 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.

  6. 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).

  7. 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.

  8. Sample result: Bertrand Game Two Firms Fixed Partners Five Firms Random Partners Two Firms Random Partners

  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)*(q1+q2)-3(q1+q2).

  12. Graph of total profit:(15-price)(price-3) Maximum is price=9 With profit 36. Profit Price

  13. 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).

  14. 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.)

  15. Price-matching Policy

  16. Price-Beating Policy

  17. 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).

  18. 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?

  19. 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!

  20. 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?

  21. 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.

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