Question

Suppose that there are two firms in an industry and they face market demand y=400-0.5p where y=y₁+y₂ . The total cost functions of the firms are C₁(y₁)= 40y₁ and C₂(y₂)= 2y₂².

a) Assume initially that the firms enter into Cournot competition. Calculate the equilibrium market price and each firm’s equilibrium output. That is, find y₁^c, y₂^c, and p^c.

b) Calculate the equilibrium market price and each firm’s equilibrium output assuming that firm 2 is the Stackelberg leader and firm 1 is the follower. That is, find y₁^s, y₂^s, and p^s

c) Will the firms prefer collusion (cartel) equilibrium or Cournot game? Based on your calculations in parts above, explain clearly.  

Answer 1

a) The demand function in this duopoly industry faced by both the firms is given as y=400-0.5p. The total output y can be written as (y1+y2) where y1 and y2 represent the respective output produced by firm 1 and firm 2. The inverse demand function, in this case, can be written as 800-y/0.5=p or 800-(y1+y2)0./5=p

The total functions of firm 1 and firm 2 are C1(y1)=40y1 and C2(y2)=2y2^2. The total revenue of firm 1 or TR1=p*y1={800-(y1+y2)/0.5}*y1=800y1-(y1^2+y1y2)/0.5 and the total revenue of firm 2 or TR2=p*y2={800-(y1+y2)/5}*y2=800y2-(y1y2+y2^2)/0.5.

The total profit function of firm 1 or TP1=800y1-(y1^2+y1y2)/0.5-40y1 or (400y1-y1^2-y1y2-20y1)/0.5 or (380y1-y1^2-y1y2)/0.5 and the total profit function of firm 2 or TP2=800y2-(y1y2+y2^2)/0.5-2y2^2 or (400y2-y1y2-y2^2-2y2^2)/0.5 or (400y2-3y2^2-y1y2)/0.5. The marginal profit function of firm 1 or MP1=dTP1/dy1=(380-2y1-y2)/0.5. The marginal profit function of firm 2 or MP2=(400-6y2-y1)/0.5

Now, based on the profit-maximizing condition, the duopolist firms would produce the output level at which their respective marginal profits are equal to 0.

Therefore, based on the profit-maximizing condition, in this case, we can state for firm 1:-

MP1=0

(380-2y1-y2)/0.5=0

380-2y1-y2=0

-2y1=y2-380

y1=(y2-380)/-2

y1=190-y2/2

Now, using the profit-maximizing condition for firm 2, we can obtain:-

MP2=0

(400-6y2-y1)/0.5=0

400-6y2-y1=0

-6y2=y1-400

y2=(y1-400)/-6

y2=66.66-y1/6

Now, plugging the profit-maximizing value of y2 into the profit-maximizing function of firm 1, we can derive:-

y1=190-y2/2

y1=190-(66.66-y1/6)/2

y1=190-(33.33-y1/12)

y1=190-33.33+y1/12

y1=156.67+y1/12

y1-y1/12=156.67

11y1/12=156.67

11y1=1880.04

y1=1880.04/11

y1=170.91 approximately.

Therefore, the profit-maximizing equilibrium output produced by firm 1 is y1=170.91 units approximately.

Plugging the value of the profit-maximizing value of y1 into the profit-maximizing function of firm 2, we get:-

y2=66.66-y1/6

y2=66.66-170.91/6

y2=66.66-28.485

y2=38.17 approximately

Hence, the profit-maximizing output produced by firm 2 is y2=38.17 units approximately.

Therefore, the equilibrium output level y=(170.91+38.17)=209.08 units

Plugging the equilibrium output value into the demand function given, we get:-

p=800-y/0.5

p=800-209.08/0.5

p=800-418.16

P=381.84

Thus, the equilibrium price in the duopoly market would be $381.84

b) Now, under the Stackelberg competition, firm 2 is the leader and firm 1 is the follower in the market. Therefore, the MP1 has been derived as (380-2y1-y2)/0.5. By backward induction, we can calculate the profit-maximizing output of firm 1 as a follower in terms of the output of firm 2 or y2.

In part a) the profit-maximizing output of firm 1 in terms of y2 has been derived as y1=190-y2/2

Now, plugging the profit-maximizing value of y1 into the MP2 derived in part-a), we get:-

(400-6y2-y1)/0.5

={400-6y2-(190-y2/2)}/0.5

=(400-6y2-190+y2/2)/0.5

=(210-5.5y2)/0.5

Therefore, based on the profit-maximizing condition of a duopolist firm, we can state:-

MP2=0

(210-5.5y2)/0.5=0

210-5.5y2=0

-5.5y2=-210

y2=-210/-5.5

y2=38.18

Thus, the profit-maximizing equilibrium output produced by firm 2 under the Stackelberg competition as the leader would be 38.18 units.

Hence, plugging the value of profit-maximizing equilibrium level of y2 into the reaction function of y1, we obtain:-

y1=190-y2/2

y1=190-38.18/2

y1=190-19.09

y1=170.91

Hence, the profit-maximizing equilibrium output of firm 1 is 170.91 units. Therefore, the output price in the market would be $381.84 which is identical to the result obtained in part a).

c) Now, as both under the Cournot and Stackelberg competition, the respective output levels produced by each firm and the output price in the market remain identical, the profit level of firm 1 and 2 would also remain unchanged implying that both firms would be indifferent about preferring the cartel or collusive operation or the Cournot equilibrium outcome in the market.