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Anghela Arrieta Guinoza. Industrial Organization-EUS
Problem set 4
Two steel mills produce steel bars of equal quality. They compete with one homogeneous product a la Cournot.
The inverse of the daily market demand is the following
Both companies have the same total cost function: Company 1, and firm
2,.
Calculate the following results of the Cournot Nash equilibrium in quantities, the collusion solution and the outcome from deviating with respect to a collusive agreement:
1. The amount of the Cournot Nash equilibrium in quantities for each company.
The firm 1 will maximize its profits the following way:
Max (^) Q1 1 = (130-Q 1 - Q 2 ) Q 1 - 10Q 1 Max (^) Q1 1 = 130Q 1 - Q 12 - Q 1 *Q 2 - 10Q 1 Max (^) Q1 1 = 120Q 1 - Q 12 - Q 1 *Q 2
According to the optimum condition of first order 1 /Q 1 = 120-2Q 1 - Q 2 =
We obtain the reaction function: 120-Q 2 =2Q 1 60 - 0.5Q 2 =Q 1 Firm 1 best response
The firm 2 will maximize its profits the following way:
Max (^) Q2 2 = (130-Q 1 - Q 2 ) Q 2 - 10Q 2 Max (^) Q2 2 = 130Q 2 - Q 22 - Q 1 *Q 2 - 10Q 2 Max (^) Q2 2 = 120Q 2 - Q 22 - Q 1 *Q 2
According to the optimum condition of first order 2 /Q= 120-2Q 2 - Q 1 =
We obtain the reaction function: 120-Q 1 =2Q 2 60 - 0.5Q 1 =Q 2 Firm 2 best response
60-0.5(60-0.5Q 1 )=Q 1
60-30+0.25Q 1 = Q 1
30= Q 1 -0.25Q 1
30=0.75Qe 1 40=Qe 1 Quantity of equilibrium for firm 1
60-0.5(60-0.5Q 2 )=Q 2
60-30+0.25Q 2 = Q 2
30= Q 2 -0.25Q 2
30=0.75Qe 2 40=Qe 2 Quantity of equilibrium for firm 2
The quantity of the competitive equilibrium of Nash in quantities, under Cournot model for each firm is:
Qe 1 COMP = 40
2. The price of the Cournot Nash equilibrium.
Substituting the equilibrium quantity in the demand function, we obtain the equilibrium price common:
P 130 Q1 Q2 130 40 40 50
PeCOMP = 50
3. The profits of each firm in this equilibrium.
1 = PQ 1 - TC (Q 1 ) =50Q 1 - 10Q 1 = (50-10) Q 1 = (50-10)*40 = 1600
2 = PQ 2 - TC (Q 2 ) =50Q 2 - 10Q 2 = (50-10) Q 2 = (50-10)*40 = 1600
eCOMP 1600 Profits for each firm
5. The price of the collusive solution.
Substituting in the demand function we obtain the price:
P= 130 - QTOTAL = 130-60 = 70
PeCOL = 70
6. The profits obtained by each company in this collusive solution.
1 = PQ 1 - TC (Q 1 ) = 70Q 1 - 10Q 1 = (70- 10 ) Q 1 = (70-10)30 = 1800 (FIRM1)
2 = PQ 2 - TC (Q 2 ) = 70Q 2 - 10Q 2 = (70-10) Q 2 = (70-10)30 = 1800 (FIRM2)
7. The amount that would cause firm 1 to maximize its profits if firm 2 continues to produce the perfect amount of the collusion. Remember that firm 1 will want to maximize profits given that its residual demand is the following residual
In order to maximize its profits:
Max (^) Q1 1 = (100-Q 1 ) Q 1 - 10Q 1
Max (^) Q1 1 = 100Q 1 - Q 12 - 10Q 1
When two firms decide to collude (join profits), the price offered in collusion is higher than when these firms compete independently.
1 / Q 1 = 100-2Q 1 - 10 = 0
90 = 2Q 1
45 = Q 1 Qe 1 DEV
8. The price in this equilibrium in which firm 1 deviates from the collusive agreement, and firm 2 continues to produce the amount of collusion.
P = 100-Q 1 = 100-45 = 55
PeDEV = 55
9. The profit that obtains firm 1 when it deviates and breaks the collusive agreement.
1 = P*Q 1 -TC (Q 1 )
1 = 55* Q 1 -10 Q 1
1 = (55-10)*Q 1
e 1 DEV = 2025
10. The profit of colluding company that keeps producing the amount of perfect collusion.
2 =PQ 2 -TC (Q 2 )= 55Q 2 -10Q 2 = (55-10) Q 2 = 4530 = 1350
2 e^1 DEV 1350
With the deviation option firm one obtains more profits, in the other hand firm 2(collusion) is less profitable.
11. Compare the present value of the profit stream to maintain in the current period and in all subsequent periods collusion with respect to the present value of the profit stream to break the collusive agreement in the current period and compete a la Cournot in all the following periods when the interest rate is 5%. Will firms collude or compete?
m= 1/1- = 1/1-0.9524 = 21
AV (COL) = ^0 COL+^1 COL+^2 ^2 COL+^3 ^3 COL+… = 1/1-COL= mCOL AV (DEV, COMP) = ^0 DEV+ (^1 COMP+^2 COMP+^2 ^3 COMP+...) =DEV+/1-COMP
14. What if the interest rate is extremely high, such as 95%, as in a super innovative industry in which firms are very short lived?
m= 1/1- = 1/1-0.5128 = 2.
The firm 1 collude today and if and only if:
AV (COL) AV (DEV, COMP)
m (COL) DEV+ m COMP
2.05(1800) 2025+0.51282.05
3694.583706.
He will compete! If they collude, profits will be less than in a competitive situation.
