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

(a) Profit is maximized when Marginal revenue (MR) equals Marginal cost (MC).

P = 10 - 0.5q

Total revenue (TR) = P x q = 10q - 0.5q2

MR = dTR/dq = 10 - q

MC = dTC/dq = 3

10 - q = 3

q = 7

P = 10 - (0.5 x 7) = 10 - 3.5 = 6.5

Lerner Index = (P - MC) / P = (6.5 - 3) / 6.5 = 3.5 / 6.5 = 0.54

(c) Price elasticity of demand (E) = - 13.9

Lerner index = - 1 / E = - 1 / -13.9 = 7.48

Again, Lerner Index = (P - MC) / P

7.48 = (P2 - 3) / P2

7.48P2 = P2 - 3

6.48P2 = - 3

P2 = - 0.46

NOTE: Since price cannot be negative, data seems incorrect. Check for correct value of price elasticity in market 2.

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