Consider a market with demand given by Q = 400/p^2. To enter the market a firm m

Question

Consider a market with demand given by Q = 400/p^2. To enter the market a firm must first pay an entry cost of k, thereafter it can produce at a constant marginal cost of 2 with no other fixed costs. (a) If a government grants a firm an exclusive monopoly in this market, how low must k be for the declared monopolist to enter the market? (b) If instead there are a two potential entrants who will engage in symmetric Cournot competition if they both enter, how low must n be for them both to enter the market?

Explanation / Answer

a) A monopolist will enter the market if it can produce profitably (at least positive accounting profit and zero economic profits are needed for it to enter). Thus, it must be willing to pay an amount k for entering if that amount is at most as large as its profits.

To compute the profits, note that profits are the difference between total revenue and total cost. With the demand given by Q = 400/P2 or P = 20/(Q)1/2 , and total cost being 2Q, Profit function is:

TR – TC

= PQ – CQ

= [20/(Q)1/2]Q – 2Q

= 20(Q)1/2 – 2Q

To find the quantity that will maximize profits (as a monopolist would always do), find the derivative and set it equal to zero:

10/(Q)1/2 – 2 = 0

5 = (Q)1/2

Q = 25, P2 = 400/25 => P = $4

Profits are equal to 20(Q)1/2 – 2Q or $50.

This implies the lowest value of k should be 50. If k is higher than $50, monopolist will incur losses and would not dare to enter.

b)

For two Cournot duopolists, each firm’s marginal cost function is MC = 2 and the market demand function is P = 20(Q)1/2

Where Q is the sum of each firm’s output q1 and q2.

Find the best response functions for both firms:

Revenue for firm 1

R1 = P*q1 = 20q1(q1 + q2)1/2

Firm 1 has the following marginal revenue and marginal cost functions:

MR1 = 20(q1 + q2)-1/2 – 10q1(q1 + q2)-3/2

MC1 = 2

Profit maximization implies:

MR1 = MC1

20(q1 + q2)-1/2 – 10q1(q1 + q2)-3/2 = 2

10(q1 + q2)-1/2 – 5q1(q1 + q2)-3/2 = 1

The game is symmetric so the situation faced by firm 2 is:

10(q1 + q2)-1/2 – 5q2(q1 + q2)-3/2 = 1

Solving these gives the following: q1 = q2.

Substituting these relation in any of the response functions:

10(2q1)-1/2 – 5q1(2q1)-3/2 = 1

15q1 = (2q1)-3/2

q1 = 225/8, q2 = 225/8

Total quantity is 450/8 or 225/4

Price P = 20/(225/4)1/2 = 20*2/15 = 2.67

With that the profit each firm earns is

TR – TC

= 2.67*225/8 – 2*225/8

= $75 - $56.25

= $18.75

This implies the lowest value of k should be $18.75 for each firm. If k is higher than $50, duopolist will incur losses and would not dare to enter.

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