Question

Consider an industry consisting of two firms which produce a homogeneous commodity. The industry demand function is Q= 100 − P, where Q is the quantity demanded and P is its price. The total cost functions are given as C₁ = 50q₁ for firm 1, and C₂ = 60q₂ for firm 2, where Q =q₁ + q₂. a. Suppose both firms are Cournot duopolists. Find and graph each firm's reaction function. What would be the equilibrium price, quantity supplied by each firm and their profits? b. Now suppose the firms start acting as Stackelberg duopolists. Suppose in particular that firm 1 acts as the leader, whereas firm 2 acts as the follower. Find the new equilibrium price, quantity produced by each firm and the profits. Comment on equilibrium price, total output and total surplus in comparison to the Cournot equilibrium. c. Suppose the two firms merge into one firm. Compare the profits in this case with those under the Cournot conditions assumed in (a) above. d. If the firms got into a Bertrand competition, what is likely to happen? You do not need to solve the problem; just explain verbally.

Answer 1

Given the industry demand Function: Q = 100 – P, we can find the inverse demand function i.e price as a function of the quantities,

Inverse Demand Function:        P = 100 – Q     {Q = q1 + q2}

Thus, P = 100 – [q1 + q2]

q1: Quantity produced by firm 1

q2: Quantity produced by firm 2

C (q1) = 50q1: Total cost function of firm 1

C (q2) = 60q2: Total cost function of firm 2

MC (q1) = dC(q1)/dq1 = 50 : Marginal cost of firm 1

MC (q2) = dC(q2)/dq2 = 60 : Marginal cost of firm 2

a) If both firms are Cournot Duopolists, then each firm sets their quantities independently but simultaneously

· For finding best response function for Firm 1 we find the profits for firm 1 and equate that with zero :

Firm 1 Profit Function = TR of Firm 1 – Total Cost of Firm 1

Π1 = P*q1 - C (q1)

Π1 = [100 – q1 - q2]q1 - 50q1

Π1 =   100q1 – q12 – q1q2 – 50q1   

Π1= 50q1 – q12 – q1q2              ------------------------- {Firm1 Profit function}

Differentiating the above profit function with respect to q1:

dΠ1/dq1 = 50 – 2q1 – q2

Let dΠ1/dq1 = 50 – 2q1 – q2 = 0

50 – q2 = 2q1

q1= 25 – 0.5q2       ---------------------- {Firm1 Reaction function}

· For finding best response function for Firm 2 we find the profits for firm 2 and equate that with zero :

Firm 2 Profit Function = TR of Firm 2 – Total Cost of Firm 2

Π2 = P*q2 - C (q2)

Π2 = [100 – q1 - q2]q2 – 60q2

Π2 = 100q2 – q1q2 – q22- 60q2

Π2 = 40q2 – q1q2 – q22    ------------------------- {Firm 2 Profit function}

Differentiating the above profit function with respect to q2:

dΠ2/dq2 = 40 – q1 – 2q2

Let dΠ1/dq1 = 0

40 – q1 – 2q2 = 0

40 – q1 = 2q2

  q2 = 20 -0.5q1     ----------------- {Firm 2 Reaction function}

Using the value of q2 and putting that in the Firm 1 reaction function, we get:

q1= 25 – 0.5q2      

q1 = 25 – 0.5 (20 -0.5q1)

q1 = 25 – 10 + 0.25q1

q1-0.25q1 = 15

0.75q1 = 15

q1=15/0.75

q1=20

Thus using the value of q1 in firms2’s reaction function we get :

q2 = 20 -0.5q1

q2= 20 -0.5(20)

q2 = 20 – 10

q2 = 10

Using the value of q1 and q2 we find the profits for each of the firms :

Π1= 50q1 – q12 – q1q2

Π1 = 50(20) – (20)2 – 20*10

Π1 = 1000 – 400 – 200

Π1 = 400

And, Π2 = 40q2 – q1q2 – q22    

Π2 = 40(10) – 10*20 – (10)2

Π2 = 400 – 200 – 100

Π2 = 100

Thus equilibrium price is P = 100 – [q1 + q2]

P = 100 – 20 – 10

P = 70

Refer to the below diagram, where the reaction function of both the firms are with q1 and q2 on x-axis and y-axis respectively. The lines are drawn by keeping q1 and q2 equal to zero one by one giving the x and y intercept. Both the reaction functions intersect at equilibrium point E.

b) If the firms start acting as a Stackleberg Duopolies’: Here firm 1 acts as leader then firm 1 sets its output first and firm to responds to its output. So we first find the best response function /reaction for firm 2.

Firm 2 keep choses to produce that level of output that maximizes its profits by keeping the marginal revenue equals to marginal cost:

MR(q2) = MC(q2)

Total revenue for firm 2 = p*q2

[100 – q1 - q2]q2 = 100q2 – q1q2 – q22

Differentiating TR(q1) with respect to q2 and equating it with MC(q2) we get :

dMR(q2)/dq2 = 100 – q1 – 2q2

Let, dMR(q2)/dq2 = MC (q2)

100 – q1 – 2q2 = 60

40 – q1 = 2q2

20 – 0.5q1 = q2    {Firm 1 reaction function, which we have already proved in part a}

Using the above reaction function for firm to and sing that to find the marginal revenue of firm 1 we get:

TR(q1) = p*q1

= [100 – q1 –q2]q1

= [100 – q1 – (20 – 0.5q1)]q1

= [100- q1 -20 +0.5q1]q1

= 100q1 – q12- 20q1+ 0.5q12

= 80q1 -0.5q12

MR(q1) = d[TR(q1)]/dq1 = 80 – q1

At equilibrium, MR(q1) = MC(q1)

80 – q1 = 60

q1= 20

Using q1 in the reaction function for firm 2 we get,

20 – 0.5q1 = q2   

20 – 0.5(20) = q2

20 – 10 = q2

q2 = 10

If use these quantities in the profit function of each firm derived in part a, we get the same results. Hence, the equilibrium prices, quantities and profits for each firm is same if the firm follow a cornot model or a stackleberg model.

c) If we assume that both the firms now merge, then then the total cost function is given by:

TC = C (q1) + C(q2)

TC(Q) = 50q1 + 60q2

Thus, merged profits can be given by :

Π(M) = Total Revenue - Total Cost

Π(M) = p*Q – TC(Q)

Π(M) = (100 – q1 – q2 ) (q1+q2) – [50q1 + 60q2]

Π(M) = 100q1 –q12-q1q2 + 100q2-q1q2- q22 – 50q1 – 60q2

Π(M) = 50q1 + 40q2 –q1² - q2² – 2q1q2 ------------------ (A)

Differentiation Equation A with respect to q1 and q2 and the setting them to zero we get :

dΠ(M)/dq1 = 0

50 – 2q1 – 2q2 = 0 ----------------- ( B )

dΠ(M)/dq2 = 0

40 – 2q2 – 2q1 = 0 ---------------- ( C )

In equation B & C we have, 2 equations and 2 unknowns, so we solve for q1 in B and put it in C. But solving for that gives us no solution. Which means inefficient outcomes. Thus, while merger is inefficient, profits are negative in this case as compared to Cournot model in part a.

d) If the firms get into Bertrand competition, then rather than competing via setting quantities, both the firms compete by setting prices. Here equilibrium condition is set by keeping their prices with the marginal cost. Thus, effort to increase the prices above the marginal cost by one of the firm , will shift the market demand towards other firm.