Profit Maximization

1. Profit Maximization • What is the goal of the firm? • 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).

2. Firm Behavior under Profit Maximization • Monopoly • Oligopoly • Price Competition • Quantity Competition • Simultaneous • Sequential

3. 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.

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(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?

5. 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) • 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?

6. 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.

7. 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).

8. 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.

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. 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 (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).

13. 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?

14. 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!

15. 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.

16. 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.

17. 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

18. 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

19. 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)?

20. 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.

21. 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.

22. Stackelberg Game. (0,0) L M (.75,.5) B S L (1.13,.56) (.5,.75) L M A M A B B (1,1) S (1.25,.94) L (.56,1.13) S M B B (.94,1.25) (1.13,1.13) S

23. Stackelberg game • How would you solve for the subgame-perfect equilibrium? • As before, start at the last nodes and see what the follower firm B is doing.

24. Stackelberg Game. (0,0) L M (.75,.5) B S L (1.13,.56) (.5,.75) L M A M A B B (1,1) S (1.25,.94) L (.56,1.13) S M B B (.94,1.25) (1.13,1.13) S

25. 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.

26. Stackelberg Game. (0,0) L M (.75,.5) B S L (1.13,.56) (.5,.75) L M A M A B B (1,1) S (1.25,.94) L (.56,1.13) S M B B (.94,1.25) (1.13,1.13) S

27. 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.