Firm A and Firm B are the only two firms in the housing market. Both these firms

Question

Firm A and Firm B are the only two firms in the housing market. Both these firms produce the same type of houses. They face the following market demand curve:

where Q=QA + QB

a) Write the demand equation in terms of Q.

b) Write down the total revenue functions for both firms in terms of QA and QB.  

c) Write down the marginal revenue (MR) functions for both firms in terms of QA and QB

Marginal cost (MC) for both firms is equal to $20. When maximizing profits, both these firms take their rival’s output as given (i.e., the firms behave as Cournot oligopolists).

d) Set MR = MC for both firms and solve for QA and QB. These are the reaction curves of these firms.

e) Solve the two equations of part d. simultaneously to calculate the Cournot equilibrium. How much will each firm produce individually? What is the total output produced by the two firms collectively?

f) The total cost function is given by the following equation: TC=20Q. What are each firm’s total profits?

g) Suppose these firms collude and decide to split profits evenly. How much will each firm produce in such a case? What will be the total output produced by the two firms? What will be each firm’s total profit?

h) Suppose these firms produce competitively without taking the other firm’s quantity into consideration. How much will each firm produce now? What will be the total output produced by the two firms? What will be each firm’s total profit?

i) Draw the reaction curves of both these firms and show the Cournot equilibrium. On your graph also show the collusive equilibrium and the competitive equilibrium. Compare these three equilibriums.

Explanation / Answer

Given demand function: Q = 28-(1/5)P

a)

The required demand function in terms of Q will be:

Q = 28-(1/5)P

5Q = 140-P

P = 140-5Q or, P = 140-5(Q1+Q2)

b)

Total revenue functions are computed by multiplying price with quantity.

The TR function for both the firms will be: PQ1 and PQ2 calculated as below:

Firm 1:

P = 140-5(Q1+Q2)

TR1 = PQ1 = 140Q1 – 5Q12 – 5Q1Q2

Firm 2:

P = 140-5(Q1+Q2)

TR2 = PQ2 = 140Q2 – 5Q22 – 5Q1Q2

c)

MR is calculated by derivating the TR function with respect to Q.

That is,

Firm 1:

TR1 = 140Q1 – 5Q12 – 5Q1Q2

MR1 = dTR1/dQ1 = 140-10Q1-5Q2

Firm 2:

TR2 = 140Q2 – 5Q22 – 5Q1Q2

MR2 = dTR2/dQ2 = 140-10Q2-5Q1

d)

MC1 = MC2 = 20

Calculate reaction functions of both the firms by setting MR1 = MC1 and MR2 = MC2

Firm 1:

            MR1 = MC1

            140-10Q1-5Q2 = 20

            Q1 = 12-0.5Q2

Firm 2:

            MR2 = MC2

            140-10Q2-5Q1 = 20

            Q2 = 12-0.5Q1

Substitute the reaction functions in each other to get the final values of output as Q1 = Q2 = 8 units.

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