Question

Suppose there are two firms, 1 and 2, competing in quantity. The market demand is

p = 15-(q1 +q2), where q1 and q2 are the quantities produced by rms 1 and 2. Both

rms have constant marginal cost c1 = c2 = 3.

(a) [10] Find the Cournot equilibrium of this market. Compute the consumer surplus

in equilibrium.

b) Now suppose firms 1 and 2 merge, so that they become a monopolist with

demand function p = 15 ? q, where q is the total output they produce. Their

marginal cost remains to be 3. How much output do they produce, and what

is the price? Compute the consumer surplus after the merger. Has the merger

increased or decreased consumer surplus?

Answer 1

(a) For calculating cournot equilibrium of this market we need to calculate the best response function of individual firms. The best response functions are the profit maximizing condition for individual firms.

Let's calculate the best response function for firm 1.

Profit maximizing condition for firm 1,

= MR = MC

Revenue of firm 1, pq1= [15 - (q1 + q2)] q1

Partially differentiating with respect to q1 we get marginal revenue for firm 1.

MR = 15 - (q1 + q2) + q1(-)

= 15 - 2q1 - q2

From profit maximizing condition, MR=MC

= 15 - 2q1 - q2 = 3

= 12 - 2q1 - q2. - - - - - - - - (1)

Similarly, the best response function for firm 2 can be calculated.

Revenue for firm 2, pq2= [15 - (q1+q2)] q2

Differenting with respect to q2 we get marginal revenue for firm 2,

MR = 15 - (q1+q2) + q2(-)

= 15 - 2q2 - q1.

And from profit maximizing condition MR=MC

= 15 - 2q2 - q1 = 3

= 12 - 2q2- q1 - - - - - - - - (2)

Solving equation and two simultaneously we get,

= 12 - 2q2 - (12 - q2) /2

Using q1= (12 - q2)/2

= 24 - 4q2 - 12 + q2

= 12 - 3q2

= q2 = 4.

Putting q2 in equation (1) we get

= 12 - 2q1 - 4

= 8 - 2q1

q1= 4.

Putting q1 and q2 in market demand

P = 15 - 4 - 4

P = 7.

Consumer surplus is calculated by the area of triangle created by below the demand curve and above the price line. Maximum willingness to pay is 15 and market price is 7.

Consumer surplus = 1/2 (15-7)(8)

= 4×8 = 32.

So the consumer surplus is 32.

(b) The monopoly produces up to the point where marginal revenue equals marginal cost. MR=MC

Revenue of monopoly = pQ

= (15 - Q) Q

Differentiating with respect to Q

= (15 - Q) + Q(-)

MR = 15 - 2Q.

And the marginal cost is 3.

Putting MR = MC

= 15 - 2Q = 3

= 12 - 2Q

= Q = 6

Putting Q in market demand function, P = 15 - 6 = 9.

So the quantity produce is 6 and price is 9.

The consumer surplus here will be,

= 1/2(maximum willingness to pay - price) × (quantity)

And maximum willingness to pay is 15, it is the price at 0 level of Q.

Consumers surplus = 1/2 × (15-9) × 6

​​​​​ = 1/2×6×6 = 18.

The consumer surplus in this case is 18.

The consumer surplus in the case of monopoly decreases.

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