Análise de Competição e Colusão em Duopólios

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1. Firm 1 and firm 2 are producing energy drinks in the market, which has a market
demand equal to P = 130−Q1−Q2. Each firm has a marginal cost of 10 and no fixed
cost.
(a) Suppose Firm 1 and Firm 2 compete by choosing quantities Q1 and Q2, respec-
tively. Solve for Firm 1’s and Firm 2’s reaction functions.
Choosing Qi to maximize profit = revenue - costs for each firm generates:
Qi = 60− (1/2)Q∗j for i = 1, 2 here is how:
Profits = Rev − Costs
Profits1 = [P ] ∗Q1 − 10 ∗Q1
Profits1 = [130−Q1 −Q∗2] ∗Q1 − 10 ∗Q1
Profits1 = 120Q1 −Q21 −Q∗2 ∗Q1
Taking a derivative with respect to Q(1):
120− 2Q1 −Q∗2 = 0
Q1 = 60−
1
2
Q∗2
Firm 2 faces the same demand function and marginal cost so Q2 is the same with
just the indexes reversed:
(b) Suppose the two firms choose quantities simultaneously. What are the equilibrium
price, firm outputs, and firm profits?
Plugging the equation for Qj into Qi = 60−(1/2)Q∗j yields: Q1 = Q2 = 40, P = 50
and profits = 1600 each.
(c) Suppose now Firm 1 has the first-mover advantage. What are the equilibrium
price, firm outputs and firm profits?
With the first mover advantage firm 1 can take the reaction by firm 2 as a fixed
responde to their choice of output. Thus firm 1’s profit function becomes:
Profits = Rev − Costs
Profits1 = [P ] ∗Q1 − 10 ∗Q1
Profits1 = [130−Q1 − 60 +
1
2
Q1] ∗Q1 − 10 ∗Q1
Profits1 = 60Q1 −
1
2
Q21
1
Taking a derivative with respect to Q(1):
60−Q1 = 0
Q1 = 60
⇒ Q2 = 30
To prove the reaction function of firm 2 still holds and results in Q2 = 30:
Profits = Rev − Costs
Profits2 = [P ] ∗Q1 − 10 ∗Q1
Profits2 = [130−Q2 − 60] ∗Q2 − 10 ∗Q2
Profits2 = 60Q2 −Q22
Taking a derivative with respect to Q2:
60− 2Q2 = 0
Q2 = 30
Q.E.D.
With Q1 = 60 and Q2 = 30 price = 130 − 60 − 30 = 40. Finally, given those
prices and quantities profits for firm one and two are 1800 and 900 respectively.
(d) Suppose the firms cooperate to jointly maximize profits. Find the resulting equi-
librium price, output, and total profits.
Profits = Rev − Costs
Profits2 = [P ] ∗QC − 10 ∗QC
Profits2 = [130−QC ] ∗QC − 10 ∗QC
Profits2 = 120QC −Q2C
Taking a derivative with respect to QC:
120− 2QC = 0
QC = 60
⇒ P = 70
⇒ Joint profits = 3600
2
Assuming the firm splits the output, we get that each firm produces 30 and has a
profit of 1800.
(e) What is the discount factor (δ) that would allow the firms to collude if they punish
each other by using the Cournot equilibrium?
When a firm collude, they get 1800 forever, that is: 1800
1−δ . If they deviate, the best
response to the other firm producing 30 is 60 − 1/2 ∗ 30 = 45. This generates
a price that is P = 130 − 30 − 45 = 55. The profits from deviation are thus
(55− 10) ∗ 45 = 2025. Thus, if deviating, the expected discounted future earnings
are: 2025 + δ1600
1−δ . The firm will be willing to cooperate if:
1800
1− δ
> 2025 +
δ1600
1− δ
1800 > 2025 ∗ (1− δ) + δ ∗ 1600
425δ > 225
δ >
9
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2. Consider two firms producing a homogeneous good in the market. They face a market
demand curve P = 8−Q. Each firm has a cost function equal to C (Qi) = 2Qi.
(a) Assume the two competing firms choose prices simultaneously, i.e. Bertrand com-
petition. Find the equilibrium price and firm outputs.
Both firms set price equal to marginal cost = 2. This results in market demand
equal to 6. It doesn’t really matter how the 6 units are split between the two firms
both firms achieve zero profits regardless.
(b) Suppose now that the firms sell differentiated products but have the same cost
functions and still choose prices simultaneously. The demands for each firm’s
output are P1 = 4 − Q12 +
P2
2
and P2 = 4 − Q22 +
P1
2
. Find the reaction functions
of Firm 1 and Firm 2, respectively. Solve for the Nash equilibrium.
First, it helps to rewrite the demand function with respect to Q1 and Q2.
Q1 = 8− 2P1 + P2
Q2 = 8− 2P2 + P1
Now solving to max profits for firm 1:
Profits1 = P1 ∗Q1 − 2 ∗Q1
Profits1 = P1 ∗ [8− 2P1 + P2]− 2 ∗ [8− 2P1 + P2]
Profits1 = 8P1 − 2P 21 + P2P1 − 16 + 4P1 − 2P2
Profits1 = 12P1 − 2P 21 + P2P1 − 16− 2P2
3
Taking a derivative with respect to P1:
12− 4P1 + P2 = 0
P1 = 3 +
1
4
P2
By symmetry, P2 = 3 +
1
4
P1. Plugging the reaction function of firm two into the
equation for firm one:
P1 = 3 +
1
4
[3 +
1
4
P1]
P1 = 3 +
3
4
+
1
16
P1
15
16
P1 =
15
4
P1 =
16
4
P1 = 4
⇒ P2 = 4
(c) What would happen if firms colluded?
If firms maximize colluded, they would maximize the sum of their profits, that is:
ProfitsT = P1 ∗Q1 − 2 ∗Q1 + P2 ∗Q2 − 2 ∗Q2
ProfitsT = (P1 − 2) ∗ [8− 2P1 + P2] + (P2 − 2) ∗ [8− 2P2 + P1]
ProfitsT = 8P1 − 2P 21 + P2P1 − 16 + 4P1 − 2P2 + 8P2 − 2P 22 + P1P2 − 16 + 4P2 − 2P1
ProfitsT = 10P1 + 10P2 − 2P 21 − 2P 22 + 2P1P2 − 32
Maximizing this by picking the best prices implies
∂πT
∂p1
= 0
10 + 2P2 − 4P1 = 0
P1 = 2.5 + 0.5P2
∂πT
∂p2
= 0
10 + 2P1 − 4P2 = 0
P2 = 2.5 + 0.5P1
4
Putting the two conditions together, we get:P1 = 2.5 + 1.25 + 0.25P1 or 0.75P1 =
3.75 or P1 = 5 and P2 = 5. Firms would produce each 3 units and profits would
be 18 in total.
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