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Please read the lecture notes and do the questions with detailed process.
1. Differentiated Bertrand competition versus price leadership. The demand for two brands of
laundry detergent, Wave (W) and Rah (R), are given by the following demands:
Qw = 80 – 2pW + pR QR = 80 – 2pR + pW
The firms have identical cost functions, with a constant marginal cost of 10.
The firms compete in prices.
(a) What is the best response function for each firm? (that is, what is firm W’s optimal price
as a function of firm R’s price, and vice-versa?) What is the equilibrium to the one-shot
pricing game? What are the profits of each firm?
(b) Suppose the manufacturer of Wave could commit to setting pw before the manufacturer of
Rah could set pR. How would this change the equilibrium? What are the profits of each
firm in this case? Should Wave take advantage of this commitment possibility? Why or
why not?
(c) Is there a first or second-mover advantage in this game? First-mover advantage is like the
conventional Stackelberg quantity-leadership story, while second-mover advantage is
reversed. Explain the intuition for your answer, and compare / contrast with the Stackelberg
quantity-setting story.
2. Entry deterrence via quantity pre-commitment. The U.S. market for hand sanitizer is controlled
by a monopoly (firm I, for incumbent) that has a total cost given by TC(qI) = 0.025
2
I
q and
MC(qI) = 0.05qI. The market demand for hand sanitizer is given by P = 50 – 0.1Q. Under
monopoly, Q = QI.
(a) What is the monopolist’s optimal price and output?
(b) Now let there be a foreign firm (firm E, for entrant) that is considering entry into the
market. Because the entrant must ship hand sanitizer all the way across the ocean, its costs
are higher. Specifically, the entrant’s costs are given by TC(qE) = 10qE + 0.025
2
E
q and
MC(qE) = 10 + 0.05qE. Suppose that the incumbent monopolist has committed to the
monopoly output level. What is the residual demand faced by the entrant? How much
output will the entrant export to the U.S.? What will be the U.S. price of hand sanitizer?
(c) Show that the monopolist would need to commit to produce 400 units in order to deter
entry of the foreign firm. (Hint: figure out the monopolist’s output level q* such that the
entrant loses money if it exports anything other than zero.) What are the incumbent’s profits
if it commits to this output level and deters entry?
(d) If the incumbent decides to accommodate entry, what quantity will it commit to?
(e) Will the incumbent deter or accommodate entry in this market?
3. Collusion and punishment. Suppose the market demand for lumber is given by:
( ) 100 / 2 P Q Q
There are two symmetric producers in the market, each with a constant marginal cost of 10.
(a) What are the monopoly price, quantity, and profits in this market?
(b) What are the Cournot price, quantities, and profits in this market?
(c) Suppose the two firms compete in the following infinitely repeated
game:
(i) Each firm produces qi = q
*
(ii) If any firm produces q>q*, then each firm believes that both will revert to the one-
shot Cournot quantity qc, forever.
What is the critical value of the firms’ discount factor δ such that q* = 0.5Qm (where Qm is
the monopoly output) is the equilibrium outcome to this game?
(d) Suppose the firms instead set price, given the cost functions above and no capacity
constraints. What is the equilibrium price and quantity to this one-shot stage game?
(e) Let the firms in part (d) compete repeatedly in the following infinitely repeated Bertrand
game:
(i) Each firm sets pi = p
*
(ii) If any firm produces p shot Bertrand price pB, forever.
What is the critical value of the firms’ discount factor δ such that p* = pm is the equilibrium
outcome to this game? Which type of competition, price or quantities, is more likely to
sustain the monopoly outcome? Why? 4. Factors affecting the sustainability of collusion. Consider an infinitely repeated Bertrand trigger
pricing game (for example, question 3(e) above). Describe how each of the following
conditions would affect the sustainability of a collusive outcome, if at all.
(a) The government’s Competition Commission announces plans to publish a monthly list of
all transactions prices and volumes in this market, in an effort to improve “market
transparency” for consumers.
(b) Recent regulations require users of the product to convert to less environmentally-
hazardous substitutes over the next five years. At that point, production and sales of this
product will be banned.
Slides 6
(Chapter 11) 2
Review: Cournot competition
Two firms that compete in quantities (BP is firm 1
and Shell is firm 2).
Select production to maximize profits, taking as
given the decision of the other firm
Linear demand 𝑃 𝑞1,𝑞2 = 𝑎 − 𝑏(𝑞1 + 𝑞2)
Same linear cost 𝐶 𝑞𝑖 = 𝑐𝑞𝑖 for 𝑖 = 1,2 Equilibrium and Profits
𝑞 1
∗ = 𝑞2
∗= 𝑎−𝑐
3𝑏
𝑄𝐶 = 2(a − c)/3b and 𝑃𝐶 = 𝑎/3 + 2c/ 3
The profits of the firms are
𝜋1 = 𝜋2 = (𝑎 − 𝑐) 4
Stackelberg competition
Two firms (BP is firm 1 and Shell is firm 2)
BP is the leader and Shell is the follower
Select production to maximize profits
BP chooses 𝑞1 first
Shell sees the choice of BP, then chooses 𝑞2
Linear demand 𝑃 𝑞1,𝑞2 = 𝑎 − 𝑏(𝑞1 + 𝑞2) 5
How do we solve this problem?
We solve the problem by backward induction
We start with the decision of the follower
We then figure out the choice of the leader
What firm makes more profits? 6 Problem of the follower
The profits of Shell (the follower) are
𝜋2 𝑞1, 𝑞2 = 𝑎 − 𝑏 𝑞1 + 𝑞2 𝑞2 − 𝑐𝑞2
The First Order Condition (FOC) is 𝜕𝑞2 Thus, 𝑞2 𝑞1 = (𝑎 − 𝑐 − 𝑏𝑞1)/2𝑏 7
Problem of the leader
The profits of BP (the leader) are
𝜋1 𝑞1, 𝑞2 = 𝑎 − 𝑏 𝑞1 + 𝑞2 𝑞1 𝑞1 − 𝑐𝑞1
Using the fact that 𝑞2 𝑞1 = (𝑎 − 𝑐 − 𝑏𝑞1)/ 2𝑏
𝜋1 𝑞1,𝑞2 = 𝑎 + 𝑐 − 𝑏𝑞1 /2 𝑞1 − 𝑐𝑞1
Thus, the FOC is 𝜕𝑞1 𝑎 − 𝑐
2 Equilibrium and profits
BP will produce 𝑞1 𝑎−𝑐 Shell will produce 𝑞2 𝑎−𝑐−𝑏𝑞1 2𝑏 𝑎−𝑐 4𝑏
𝑄𝑆 = 3(a − c)/4b and 𝑃𝑆 = 𝑎/4 + 3c/4
The profits of the firms are 𝜋1 = (𝑎 − 𝑐) (𝑎 + 𝑐)/4𝑏 𝜋2 = (𝑎 − 𝑐) (𝑎 + 𝑐)/8𝑏 Equilibrium and profits BP makes more profits than Shell
There is a first mover advantage
Commitment (Credible? Investment capacity!)
BP makes more profits with Stackelberg than
with Cournot competition Slides 7
(Chapter 12) 2 Entry
What drives entry in markets?
What are barriers to entry?
Scale economies
One firm is “better”
Network effects
Predation
Limit pricing: Set low price to prevent entry
Predatory pricing: Set low price to force exit
Capacity choice 3 Two-stage game:
Bresnahan and Reiss 1991 whether to enter or not
Upon entry, they compete in price or quantity
Demand is 𝐷 𝑝 𝑆 where 𝑆 is market size
Entry (or fixed) cost 𝐸
If 𝑁 firms enter, variable profit per custom is 𝑉 𝑁 4 How many firms?
The profit of each firm 𝑖 = 1,2,…,𝑁 is To sustain 𝑁 firms we need 𝑆 ≥ 𝐸/𝑉 𝑁
If 𝑆 is large we get lot of firms with small 𝑉 𝑁
Big markets have a lot of firms
Lot of restaurants in SF, a few less in SC
But we know many big markets with small 𝑁
NYC newspapers
Cars Example:
Entry with BP/Shell Cournot
Demand is 𝑃 𝑄 = 𝑎 − 𝑏𝑄
Same linear cost 𝐶 𝑞𝑖 = 𝑐𝑞𝑖 for 𝑖 = 1,2 Example: Entry with BP/Shell Cournot 𝜋1 = 𝜋2 = (𝑎 − 𝑐) If only BP enters (monopoly)
𝜋1 = (𝑎 − 𝑐) Example: Entry with BP/Shell Cournot If both enter and play Cournot
𝜋1 = 𝜋2 = 7 ൗ 9
If only BP enters (monopoly) 𝜋1 = 16
If entry cost 𝐸 = 10 only BP enters Example: Suppose 𝐸 = 6. What can BP do?
Threatening monopoly quantity
NOT credible without commitment (Why?)
Suppose BP can commit to 𝑞1 = 𝑞1 If Shell enters: 𝑞2 = 2, 𝑃 = 4, and 𝜋2 = 4
Shell does not enter! Example: If 𝐸 < 4, BP needs 𝑞1
𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦
> 4 to deter entry!
This goes beyond the optimal monopoly level!!
Might be good to accommodate if 𝐸 is too low Example: Entry with BP/Shell Cournot What 𝑞1 is needed to deter entry?
Need to compute 𝜋2 given 𝑞1 = 𝑞1 𝜋2 = 4 − 2 2 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦 Example: Entry with BP/Shell Cournot 𝑞1 = 6 and 𝜋1 = 12
If BP accommodates and play Stackelberg
𝜋1 = 8
BP commits to 𝑞1 = 6 and deters entry! Slides 8
(Chapter 14) 2 What did we learn from static
oligopoly models? firms competing “non-cooperatively”
They only look out for their own interests
The game involves a Prisoners’ Dilemma
It is possible to improve both firms’ profits
But, the incentive to maximize own profits causes
both firms to “over produce” (or “under price”)
relative to monopoly behavior 3 Can the firms do better?
We know collusion exists
We will study how firms are able to support a
“better” equilibrium, compared to the Cournot or
Bertrand Nash equilibria
We will show that supporting collusive behavior
requires that the firms compete more than once
Another reason to know dynamic game theory! 4 Collusion and cartels
A cartel is an explicit attempt to enforce market
discipline and reduce competition between a
group of suppliers
Cartel members agree to coordinate their actions
Examples: prices, market shares, exclusive
territories
Collusion is an agreement to raise prices 5 Some cartels are explicit and difficult to prevent
OPEC: cartel members are sovereign nations
Some are “hidden”
Lysine (animal feed additive) 6 Collusion and cartels – the law
Laws make cartels illegal in the US and Europe
US: Section 1 of the Sherman Act says that it is
illegal to form any agreement in restraint of trade
Cartels are per se illegal
No defense for being caught in a price fixing scheme
Different from merger review or vertical restraints
These claims are examined under a rule of
reason analysis: Is there a legitimate, welfare-
enhancing reason to permit the activity? 7 Authorities continually search for cartels
Cartel investigations fall under the Department of
Justice (DOJ)
FBI wiretaps, etc.
Prison sentences for executives found guilty
It is illegal to talk about prices with your
competitor!!! 8 Some cartels are never formed
Putnam, CEO Braniff Air: Do you have a suggestion for me?
Crandall, CEO American Airlines: Yes, I have a suggestion
for you. Raise your damn fares 20%. I’ll raise mine the
next morning.
Putnam: Robert, we…
Crandall: You’ll make more money and I will too…
Putnam: We can’t talk about pricing!
Crandall: Oh (expletive deleted), Howard. We can talk
about any damn thing we want to talk about.
Conversation taped by DOJ, 2/21/82 9 What makes cartels and collusion difficult?
They are illegal
Thus, a cartel has to be covert
Enforced by non-legally binding threats or self- Cannot be enforced by legally binding contracts
There is always an incentive to cheat
Think about the Prisoner’s dilemma
High profits from collusion might attract entry
These often make cartels unstable 10
Collusion and cartels Other less explicit attempts to control competition, Formation of producer associations
Though they are not allowed to talk about pricing
Publication of price sheets
Repeated interaction that leads to higher prices
These are known as tacit collusion
No explicit discussion of price or collusion 11
Thinking about the economics
Do repeated interactions facilitate collusion?
What is the incentive to collude?
What is the incentive to cheat?
How to punish cheaters? (enforcement of the
collusive agreement) 12 An example before we dive into
theory: the lysine cartel
Lysine: amino acid and feed additive
During the mid-1990s, several international firms
conspired to fix worldwide lysine prices
Largest conspirator was Archer Daniels Midland
(ADM). Huge grain and seed company.
FBI learned of the conspiracy via an ADM
executive (Mark Whitacre) who was involved in a
different case
Whitacre became an informant, secretly recording
cartel meetings 13
Now a major
motion picture! 14
An example before we dive into Firms settled with the DOJ in 1996
Record (at the time) $100m in fines
Prison sentences for some executives
Whitacre wound up being convicted on separate
embezzlement charges
Got more prison time than the cartel conspirators 15
FBI Recordings of the lysine cartel
Segment 1: The firms joke about the FBI listening
in to their meeting, and about their biggest
customer (Tyson Foods) sitting in.
“Mr. Whitacre” is the snitch from ADM.
Segment 2: Phone call—one of the Japanese
firms doesn’t want to meet in Hawaii because it’s
in the U.S.
They agree to meet under the cover of a new
“trade association” 16 Segment 3: The firms set prices—to the penny
per pound—for the US and Canada
Segment 4: ADM makes threat against potential
cheaters. It will use its excess capacity
Segment 5: Cartel enforcement: end-of-year
compensation scheme
If a firm has more sales than agreed to, it must
buy from another firm
Tape features a great “pep talk” Slides 9
(Chapter 14) 2 Bertrand competition
Let’s stick with our example of BP and Shell
Demand is 𝑄 𝑃 = 10 − 𝑃
Same linear cost 𝐶 𝑞𝑖 = 2𝑞𝑖 for 𝑖 = 1,2
In the Bertrand one-shot game
𝑃∗ = 2 and 𝜋∗ = 0 for both of them!
In the monopolist case
𝑃𝑀 = 6 and 𝜋∗ = 8
But splitting is not a NE —incentives to deviate 3 Do repeated interactions help?
Suppose firms interact two periods
Potential collusive strategy
Play 𝑃𝑀 in period 1
Play 𝑃𝑀 in period 2 if both played 𝑃𝑀 in 1
This strategy incorporates punishment!
But it is not a Subgame Perfect NE (SPNE)
𝑃𝑀 in period 2 is not credible. It is just a one-shot ∗ = 2
Threat to reward/punish in period 1 is not credible.
So period 1 NE is also Bertrand 4 Continues…
We know Cartels exist!
What if we let firms to interact more periods?
Same problem!
Last period is one-shot Bertrand! Then, last but
one is also Bertrand! ….
What is the problem of the model?
There is a T after which the game is over! 5 Infinitely repeated game
Let T go infinity
Each time there is probability α the game will
continue to the next period!
With probability 1 – α the game will end
Regulatory intervention
Introduction of a new product
Competitor entry
Interest rate r and discount rate 1 / (1 + r)
The effective discount rare is δ = α / (1 + r) Continues… Suppose each firm makes 𝜋𝑖𝑡 per-period
The value of this profit stream is
𝑉𝑖 = Each firm goal is to maximize 𝑉𝑖
Question: Is 𝑃1𝑡 = 𝑃2𝑡 = 𝑃 Continues… Consider the next “grim trigger” strategies
t = 0, play 𝑃 = 𝑃𝑀
t > 0, play 𝑃 = 𝑃𝑀 if 𝑃1𝑠 = 𝑃2𝑠 = 𝑃 t > 0, play 𝑃 = 2 otherwise (punish if deviates!)
Is this a SPNE?
Need to check all subgames to see if a firm can be
better off by deviating Continues… There are two types of subgames here
The punishment subgame 𝑃1 = 𝑃2 = 2
The cooperative subgame 𝑃1 = 𝑃2 = 𝑃 Is this a SPNE? Collusive subgame
Will a firm deviate?
The firm might set price at 𝑃𝑀 − 𝜀
Gets almost full monopoly profits 𝜋𝑀 now
Gets punished afterwards!
The value of cooperating is
𝑉𝑖 = 2 1 1 − δ
1 2 If the firm deviates it gets 𝜋𝑀 Collusive subgame Cooperating is a SPNE if 1 − δ 2 In the example, if δ ≥ 2 To sustain collusive equilibrium we need firms to
be very patient —large δ
2/9𝑏
Same linear cost 𝐶 𝑞𝑖 = 𝑐𝑞𝑖 for 𝑖 = 1,2
𝜕𝜋2 𝑞1,𝑞2
= 𝑎 − 2𝑏𝑞2 − b𝑞1 − c = 0
𝜕𝜋1 𝑞1, 𝑞2
=
− 𝑏𝑞1 = 0
∗ =
2𝑏
∗=
∗
=
Potential entrants choose (simultaneously)
with 𝜕𝑉 𝑁 /𝜕𝑁 ≤ 0
𝜋𝑖 𝑆,𝑁 = 𝑆𝑉 𝑁 − 𝐸
If both firms enter and play Cournot
2/9𝑏
2/4𝑏
Assume 𝑎 = 10, 𝑏 = 1, and 𝑐 = 2
1
Entry with BP/Shell Cournot
𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦
𝑀 = 4
Entry with BP/Shell Cournot
Let 𝐸 = 1
𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦
𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦
1
𝑞1
𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦
≤ 1 if 𝑞1
≥ 6
If BP deters entry
𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦
𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦
We studied the equilibrium of a market with two
Collusion and cartels
Collusion and cartels – the law
Collusion and cartels
interest
which are legal
theory: the lysine cartel
FBI Recordings of the lysine cartel
Bertrand, so 𝑃2 = 𝑃
𝑡=0
δ𝑡𝜋𝑖𝑡
∗ = 2 t a SPNE?
𝑀 for all s < t
𝑀
Need to check all subgames to see if a firm can be
better off by deviating
𝑡=0
δ𝑡
1
𝜋𝑀 =
𝜋𝑀
1
1
𝜋𝑀 ≥ 𝜋𝑀
1
