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).

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.

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.

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.

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?

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?

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.

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).

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.

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<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

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.

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).

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!

17.88 17.88

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.

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.

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

5/4M=1

S=3/4

9/85/4

15/169/8

115/16 1

M=1

S=3/4

1/2

9/169/8

3/4

9/8 3/49/16 1/2

00

L=3/2

L=3/2

Cournot Duopoly: Normal Form GameProfit1=(3-q1-q2)q1 and Profit 2=(3-q1-q2)q2

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)?

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.

Stackelberg Game.

A

B

L

L

S

S

(1.13,.56)

(1.13,1.13)

(0,0)

(.56,1.13)

B

B

A B

L

S (1.25,.94)

(.5,.75)

B

S

M

L

M

M (1,1)

(.75,.5)

(.94,1.25)M

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.

Stackelberg Game.

A

B

L

L

S

S

(1.13,.56)

(1.13,1.13)

(0,0)

(.56,1.13)

B

B

A B

L

S (1.25,.94)

(.5,.75)

B

S

M

L

M

M (1,1)

(.75,.5)

(.94,1.25)M

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.

Stackelberg Game.

A

B

L

L

S

S

(1.13,.56)

(1.13,1.13)

(0,0)

(.56,1.13)

B

B

A B

L

S (1.25,.94)

(.5,.75)

B

S

M

L

M

M (1,1)

(.75,.5)

(.94,1.25)M

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.