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. Profit Maximization • The environment is competitive: no one firm can influence the price. • We can write profit maximization of a competitive firm in terms of a cost function (cost of producing y units of output) • Maxy p*y-c(y) • What is the FOC? • What is profits and choice of y if c(y)=y2? • With general c(y), when would a firm shut down? When average cost is always above p.

3. Past, Present and Future • What happens if some decisions are already made in the past? • Remember one can’t change the past. • Euro-tunnel: spend billions to build it. Does this mean that prices have to be higher for tickets? • Similar for Airwave Auctions, Iridium and many other cases.

4. 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). • Maximization implies Marginal Revenue=Marginal Cost.

5. Example (from tutorial) • 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?

6. Example • Price is p(y)=120-2y, this implies marginal revenue is 120-4y. • Total cost is c(y)=y2. This implies marginal cost is 2y. • What is the monopoly’s choice of y (mr=mc)? • What is the competitive equilibrium y (price=mc)? • Why is a monopoly inefficient? Someone values a good above its marginal cost. • In a diagram, what is the welfare loss?

7. Why Monopolies? • What causes monopolies? • a legal fiat; e.g. US Postal Service • a patent or trade secret; e.g. a new drug • sole ownership of a resource; e.g. a toll highway • formation of a cartel/collusion; e.g. OPEC • large economies of scale; e.g. local utility companies.

8. Patents • A patent is a monopoly right granted to an inventor. It lasts about 17 years. • For the government: there is a trade-off between • loss due to monopoly rights. • incentive to innovate. • For the company • Must decide between patent and trade secret. • Minus side of patent is that it expires and is no longer secret (competitors can perhaps go around it). • Minus side of trade secret is that there is no legal protection, but lasts forever. For example, Coca Cola. • Strategy – protective, delay or shelve? License (temporarily remove competition).

9. Natural Monopoly • When is a monopoly natural such as in certain public utilities? • C(y)=1+y2. P(y)=3-y. • Notice the c entails a fixed cost of 1. • Where does p=mc (mc is 2y)? • What is profits at this point for a single firm that meets the whole demand? • What happens when another firm enters? They can’t charge a price close to competitive equilibrium and survive. • monopoly (mr=3-2y)? Y=3/4. If two firms try to split this output, they still lose money. • Government should allow a monopoly but force a price cap.

10. 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.

11. 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<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).

12. 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.

13. 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¢

14. 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.

15. 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. • 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).

16. 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!

17. Oligopoly • A monopoly is when there is only one firm. • An oligopoly is when there is a limited number of firms where each firm’s decisions influence the profits of the other firms. • We can model the competition between the firms’ price and quantity, simultaneously or sequentially.

18. Quantity competition (Cournot 1838) • Profit1=p(q1+q2)q1-c(q1) • Profit2= 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. • This is a Nash equilibrium! • Take FOCs and solve simultaneous equations. • Can also use intersection of reaction curves.

19. 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

20. 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

21. 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)?

22. Quantity competition (Stackelberg 1934) • 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 firm 2’s reaction. • This is the same as subgame perfection. • We can now write the game in a tree form.

23. 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

24. 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.

25. 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

26. 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.

27. 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

28. 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.