Adam produces pencils and is a monopolist. The demand for pencils is P(q) = 10 0

Question

Adam produces pencils and is a monopolist. The demand for pencils is P(q) = 10 0.5q, where P is the price and q is the quantity. Assume that Adam’s total cost function is T C(q) = 3q. Assume that Adam chooses how much to produce in order to maximize his profits.

a) (15 points) Compute the Lerner index.

Next, assume that Adam is a third-degree discriminating monopolist operating in two markets. In market 1, the demand for pencils is P1(q1) = 10 0.5q1 (as above). We know that, at optimum, the price elasticity of demand in market 2 is E2 = 13/9 .

c) (15 points) Compute the optimal price in market 2, P*2 .

Explanation / Answer

Question a). Solution :- P = 10 - 0.5q (Demand function).

Marginal revenue (MR) = 10 - q. (The slope of marginal revenue function is twice that of the slope of inverse demand curve function.)

TC(q) = 3q (Total cost function)

MC = 3. (MC denotes marginal cost. It is the first derivative of the total cost function.)

With a view to maximise the profit, a monopolist will produce upto that quantity at which marginal revenue (MR) equals to the marginal cost (MC).

Accordingly, equating marginal revenue (MR) and marginal cost (MC) in the given question,

10 - q = 3

10 - 3 = q

q = 7

P = 10 - 0.5 * 7 (Put the value of q = 7 in demand function given in the question)

P = 10 - 3.50

P = $ 6.50.

Lerner index = (P - MC) / P

= (6.50 - 3.00) / 6.50

= 3.50 / 6.50

= 0.54 (approx).

Conclusion :- Lerner index = 0.54 (approx).

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