Econ 460: Industrial Organization
0101, Fall 2008
Problem set 3
Due: Tuesday November 18, 11am
1. Hotelling with fixed locations.
Consider again the Hotelling model, but this time rather than fixing prices we fix
locations. Assume that there is a 1 mile long beach, with n customers spread uniformly
along the beach. Denote any point on the beach as being a value x. For simplicity,
assume that n is a continuous variable (so we can have fractions of customers). Suppose
that there are two ice-cream trucks, one at each end of the beach (ie firm 1 is at position
x=0, firm 2 is at position x=1). Each customer gets a utility of A – ti – pi from buying an
ice-cream from firm i, where ti is the distance between a customer and the ice-cream they
purchase from, and pi is the price charge by firm i. Assume the marginal cost to a firm of
selling an additional ice-cream is zero.
Both firms simultaneously choose prices p1 and p2 to maximize profit, and each customer
buys from whichever firm leaves them with higher utility. You may assume that all
customers will choose to purchase an ice-cream from one or other firm.
a) Suppose that firm 1 chooses some price p1 and firm 2 chooses some price p2. Derive
an expression for the location x* of a customer who is indifferent which firm they buy
from.
Utility from buying from firm1 = A – x* - p1 = A – (1 – x*) – p2 = Utility from buying
from firm2
p2 - p1 = x* - 1 + x*
x* = (p2 – p1 + 1)/2
b) Noting that firm 1 will sell to a quantity of customers = nx*, and that firm 2 will sell to
a quantity of customers = n(1-x*), derive expressions for the profits of firm 1 and firm 2
in terms of p1, p2, and n.
Firm 1 profit:
π1 = p1nx*
= p1n(p2 – p1 + 1)/2
π2 = p1n(1-x*)
= p2n(p1 – p2 + 1)/2
c) Suppose that n=1. Write down each firm’s profit maximization problem, and solve to
find the unique Nash equilibrium for this model.
Firm 1 solves:
Maxp1 : p1(p2 – p1 + 1)/2
FOC: (p2 – 2p1 + 1)/2 = 0
Best response: p1 = (p2+1)/2
By symmetry: p2 = (p1+1)/2
Solve simultaneously to find: p1 = p2 = 1. Unique Nash equilibrium.
2. Subgame perfection in dynamic games
Consider the following dynamic game:
1
B
A
2 2
C D E F
0,1 3,2
2, 3 1
G H
4,2 3,5
Players 1 and 2 sequentially choose actions from A, B, C, D, E, F, G and H, as shown in
the game tree. Payoffs are as given in the game tree, with payoffs to player 1 listed first.
a) Write down all possible strategies for each player [Recall that a strategy must specify
an action for a player at every node in which that player can make a decision)
Player 1: AG, AH, BG, BH
Player 2: CE, CF, DE, DF
b) Find the unique SPNE for this game.
Player 1 will never play H, G is better. So Player 2 will never play D, C is better. Player
2 will never play E, F is better. So player 1 will never play A, B is better.
So the unique SPNE is: BG, CF.
3. Industry collusion characteristics.
Choose industry OTHER than one discussed in class (oil, electricity, shipping, diamonds)
where collusion might be a feasible possibility. Briefly describe what features of this
industry might help facilitate collusion, and what features of the industry might
discourage collusion.
4. Repeating a joint project
Suppose there are two firms who are involved in a joint venture. The value of the venture
is increasing in the investment e of each player, but making an investment is costly for
i
each player. Each player gets a profit equal to half the value of the project, less the cost of
their own effort. Specifically, the value of the project is 5eiej and the cost to each player
2
of investment is 2ei2, so the profit to firm i is (5e e )/2 – 2e
i j i
Each firm simultaneously chooses an investment level from the interval [0, 1] (ie any
number between zero and one inclusive). Then the value of the project is realized, and
firms receive their profits.
NOTE: This is not saying that they can only choose e = 0 or e = 1. They can choose
those, or ANY number between 0 and 1. They can choose e = 0.45389 or e = 2/3.
a) Show that the unique Nash equilibrium if this game is played once is for each player to
choose e = 0.
i
Player i solves:
2
Max : (5e e )/2 – 2e
ei i j i
FOC: 5e /2 – 4e = 0
j i
e = 5e /8
i j
This is the best response function for player i.
Similarly,
e = 5e /8
j i
Solving simultaneously:
e = 25e /64
i i
which implies e = e = 0.
i j
b) Show that the players maximize the sum of their payoffs by choosing e = 1.
i
(Hint: you may assume that e = e = e*, and maximize the joint payoff by choosing e*.)
1 2
To maximize the joint payoffs of the players, we solve:
2 2
Max : (5e*e*)/2 – 2e* + (5e*e*)/2 – 2e*
e*
2 2
Max : 5e* – 4e*
e*
2
Max : e*
e*
FOC: 2e*
This is always positive for any positive e*, so we maximize the joint payoffs by choosing
the highest possible e that we can, ie e = 1.
i
c) Suppose now that this game is repeated an infinite number of times, and that players
discount payoffs in future periods by a discount factor δ every period. Suppose that
players play the following trigger strategy:
Choose e = 1 in the first period, and choose e = 1 in every period as long as every player
i i
has chosen e = 1 in all prior periods.
If any player has ever chosen any effort level other than 1, then choose e = 0 in every
i
period forever after.
Find the optimal deviation from this strategy, and show that this strategy will constitute a
subgame perfect Nash equilibrium as long as δ ≥ 9/25.
2
Payoff from both players choosing 1 = 5(1)(1)/2 – 2(1 ) = 1/2.
Optimal deviation when the other player chooses e = 1 comes from the best response
j
function: choose e = 5(1)/8 = 5/8
i
2
Payoff from optimal deviation = 5(1)(5/8)/2 – 2(5/8) = 25/16 – 50/64 = 100/64 – 50/64
= 25/32.
2
Payoff from both choosing zero effort = 5(0)(0)/2 – 2(0) = 0
2
So: payoff along the equilibrium path = 1/2 + δ/2 + δ /2 + … = (1/2)/(1 – δ)
2
Payoff from optimal deviation = 25/32 + (0)δ + (0)δ + … = 25/32
So, we have an equilibrium when: (1/2)/(1 – δ) ≥ 25/32
Ie 1/(1 – δ) ≥ 50/32
1 ≥ 50/32 – 50δ/32
50δ/32 ≥ 50/32 – 32/32
50δ/32 ≥ (50 – 32)/32
50δ/32 ≥ 18/32
50δ ≥ 18
δ ≥ 18/50
δ ≥ 9/25
5. Repeated entry game.
Consider the following entry game:
A challenger may enter a market (E) or stay out (O). If they enter, the incumbent may fight
(F) or accommodate. Payoffs are as specified below (first payoff for the challenger, second
payoff for the incumbent).
C
O
E
I 0,3
3
A F
2, 2 -1, 1
a) Suppose the game is played only once. Find all Nash equilibria of this game. Indicate
which ones are subgame perfect.
The game has two NE, EA and OF. But OF is not subgame perfect, it relies on a non-credible
threat, since once the challenger enters, the incumbent will accommodate.
A F
E 2,2 -1,1
O 0,3 0,3
For parts b) and c), suppose that this occurs in an industry with two firms, firm 1 and firm 2,
both large chain stores. Suppose that the firms are playing this game repeatedly (and
sequentially) in T different markets, but alternating roles. In the first market, firm 1 is the
challenger and firm 2 is the incumbent. In the second market, firm 2 is the challenger and
firm 1 is the incumbent. Then in the third market, firm 1 is the challenger again, and so on.
Each market is resolve sequentially: firms choose to enter or not in market 1, and to
accommodate or fight in market 1, then observing this whether to enter or not in market 2,
and to accommodate or fight in market 2, etc.
b) First, suppose that 1 < T < ∞, so there are a finite number of markets. Firms preferences
are to maximize the sum of their payoffs across all T markets (there is no discounting).
Find the unique subgame perfect Nash equilibria of this repeated game.
The unique subgame perfect equilibrium is for all challengers to enter and all incumbents to
accommodate. This can be found from backward induction: in the last market, the incumbent
has no incentive to fight if the challenger entered, so the challenger will enter. The same
holds true in the T-1 market, and the T-2 and so on, back to the first round.
c) Now suppose that T = ∞, so that the firms are considering entry over an infinite number of
markets. Each round of entry takes a month, so payoffs from the game in second market are
less than those from the first market, and less again in the third market, and so on.
2
Specifically, assume that we put weights δ on payoffs from the second market, δ from the
3
third market, δ from the fourth market and so on.
Consider a collusive agreement (where firms do not enter each others’ markets) supported by
a grim trigger strategy.
In this strategy, firm i will not enter at their first opportunity to do so, and will continue to not
enter in any market where are they are the challenger as long as firm j does not enter any
market when firm 2 is the challenger. However, if a challenger enters a market, then the
incumbent will accommodate the challenger that round, both firms will enter all future
markets (where they are the challenger) and will accommodate all future entry.
Can this supportive equilibrium, where no firm enters any market, be supported in this
infinitely repeated game? If so, for what ranges of δ?
[Hint: it is sufficient to consider only deviations by firm 1, who in equilibrium stays out in
the first market.]
Payoff along the equilibrium path:
3 2
Firm 1: 0 + 3δ + 0 + 3δ + 0 + …. = 3δ/(1 – δ )
2 4 2 2
Firm 2: 3 + 0 + 3δ + 0 + 3δ + … = 3 + 3δ /(1 – δ )
Payoffs from deviating:
2
Firm 1: 2 + 2δ + 2δ + ….= 2/(1-δ)
2
Firm 2: 3 + 2δ + 2δ + … = 3 + 2δ/(1 – δ)
When is there no incentive to deviate?
Firm 1:
2
3δ/(1 – δ ) ≥ 2/(1-δ)
3δ/[(1 + δ)(1 – δ)] ≥ 2/(1-δ)
3δ/(1 + δ) ≥ 2
3δ ≥ 2 + 2δ
δ≥2
This can only happen if we put more weight on the future than on the present; to be
impatient, we require δ to be ≤ 1. So for an impatient firm that discounts the future, we can
never support this equilibrium.
Firm 2:
2 2
3 + 3δ /(1 – δ ) ≥ 3 + 2δ/(1 – δ)
2
3δ/(1 – δ ) ≥ 2/(1-δ)
3δ/[(1 + δ)(1 – δ)] ≥ 2/(1-δ)
3δ/(1 + δ) ≥ 2
3δ ≥ 2 + 2δ
δ≥2
6. Consider Dixit’s 1980 model of capacity investment (from class, discussed in chapter
12). Explain why the incumbent might choose to invest in capacity in stage 1, when it
could wait and invest in capacity at the same cost in stage 2. Suppose that it was actually
cheaper to invest in capacity in stage 2 than in stage 1. Might the incumbent still choose
to invest in capacity in stage 1 rather than in stage 2?
incumbent has no incentive to fight if the challenger entered, so the challenger will enter. The
same holds true in the T-1 market, and the T-2 and so on, back to the first round.
c) Now suppose that T = ∞, so that the firms are considering entry over an infinite number of
markets. Each round of entry takes a month, so payoffs from the game in second market are
less than those from the first market, and less again in the third market, and so on.
2
Specifically, assume that we put weights δ on payoffs from the second market, δ from the
3
third market, δ from the fourth market and so on.
Consider a collusive agreement (where firms do not enter each others’ markets) supported by
a grim trigger strategy.
In this strategy, firm i will not enter at their first opportunity to do so, and will continue to not
enter in any market where are they are the challenger as long as firm j does not enter any
market when firm 2 is the challenger. However, if a challenger enters a market, then the
incumbent will accommodate the challenger that round, both firms will enter all future
markets (where they are the challenger) and will accommodate all future entry.
Can this supportive equilibrium, where no firm enters any market, be supported in this
infinitely repeated game? If so, for what ranges of δ?
[Hint: it is sufficient to consider only deviations by firm 1, who in equilibrium stays out in
the first market.]
Payoff along the equilibrium path:
3 2
Firm 1: 0 + 3δ + 0 + 3δ + 0 + …. = 3δ/(1 – δ )
2 4 2 2
Firm 2: 3 + 0 + 3δ + 0 + 3δ + … = 3 + 3δ /(1 – δ )
Payoffs from deviating:
2
Firm 1: 2 + 2δ + 2δ + ….= 2/(1-δ)
2
Firm 2: 3 + 2δ + 2δ + … = 3 + 2δ/(1 – δ)
When is there no incentive to deviate?
Firm 1:
2
3δ/(1 – δ ) ≥ 2/(1-δ)
3δ/[(1 + δ)(1 – δ)] ≥ 2/(1-δ)
3δ/(1 + δ) ≥ 2
3δ ≥ 2 + 2δ
δ≥2
This can only happen if we put more weight on the future than on the present; to be
impatient, we require δ to be ≤ 1. So for an impatient firm that discounts the future, we can
never support this equilibrium.
Firm 2:
2 2
3 + 3δ /(1 – δ ) ≥ 3 + 2δ/(1 – δ)
2
3δ/(1 – δ ) ≥ 2/(1-δ)
3δ/[(1 + δ)(1 – δ)] ≥ 2/(1-δ)
3δ/(1 + δ) ≥ 2
3δ ≥ 2 + 2δ
δ≥2
6. Consider Dixit’s 1980 model of capacity investment (from class, discussed in chapter
12). Explain why the incumbent might choose to invest in capacity in stage 1, when it
could wait and invest in capacity at the same cost in stage 2. Suppose that it was actually
cheaper to invest in capacity in stage 2 than in stage 1. Might the incumbent still choose
to invest in capacity in stage 1 rather than in stage 2?