Question

The (inverse) demand for vitamin D dietary supplement is p = 85−2q and the cost function for any firm producing vitamin D is TC = (5q+F),where F is fixed cost.

(a) What is a marginal cost? How much of vitamin D would a monopolist produce? What price would it charge? How much profit would it earn?

(b) For which of the following values of F is a market for vitamin D a natural monopoly? F= $100, F= $200, F= $300, F= $400.

Answer 1

a) Marginal cost is the additional cost incurred in the production of one more unit of output. We have TC = 5q +F

Marginal cost = dTC/dq = 5

The monopolist will produce at a point where marginal revenue = marginal cost

We have, revenue = p*q = (85-2q)*q = 85q -2q2

Marginal revenue = dR/dq = 85 - 4q

Thus, at equilibrium we have 85 -4q = 5 => 4q = 80 => q =20

Price = 85 -2*20 = 45

Profits = pq - total cost = 45*20 - 5*20 - F = 900 -100 -F = 800 - F

b) Natural monopoliy will occur if the cost if production is higher with multi firms then with a single firm i.e ewhen average cost of production for 2 firms is higher than average cost of production for a single firm.

The average cost of 2 firs producing equal output = 20/2 =10 = Total cost / output = (5*10 + F)/10 = 5 + F/10

average cost of monoploist = (5*20 + F)/20 = 5 + F/20

since F/10 > F/20, there is natura monopoly at all F >0