Question

A firm is a price-searcher; that is, it has some monopoly power. Its demand equation is given by P(q) = 10 – q. Its total cost of producing its output is given by the function TC(q) = (q²/8) + q + 16, and it can be shown that its marginal cost equation is MC(q) = (q/4) + 1.

3.You know from earlier in the course that if the firm has the linear demand equation P(q) = a –bq, then the price elasticity of demand at an output qis ε= (bq-a)/bq. Use this result to calculate the price elasticity of demand at the firm’s profit-maximizing point on the demand curve.

a. Based on your result in the last part, is the firm’s demand elastic or inelastic at the profit-maximizing point? Explain.

b.Using the price and marginal cost you found in 1(c) and 1(d), calculate the firm’s monopoly markup ratio (P –MC)/P. Is this equal to -1/ε, as we’d expect? Price= $6 MC=2

Answer 1

P(q) = 10 – q, Here a=10 and b=1

TC(q) = (q2/8) + q + 16

MC(q) = (q/4) + 1.

The price elasticity of demand= ε= (bq-a)/bq

For profit maximization:

P=10-q

TR=P*q= 10q-q2

MR= differentiation of TR with respect to q= 10-2q

Condition for profit maximization:

MR=MC

10-2q= q/4 + 1

10-1= q/4 + 2q

9 = 9q/4

36= 9q

q= 4

P= 6

ε= (bq-a)/bq= (1x4-10) / 1x4= -6/4= -3/2= -1.5

a) Demand is inelastic if value of elasticity of demand is less than 1 and elastic if value is more than 1. Here the value of elasticity of demand is 1.5 (ignore (-) sign) which is more than 1 so demand is elastic. It means % change in quantity is more than the % change in price.

b) The firm’s monopoly markup ratio:

(P-MC)/P = -1/ε

(6-2)/6 = -1/(-1.5)

4/6 = 10/15

2/3 = 2/3

Yes the equation satisfies. Markup ratio is equals to negative of reciprocal of elasticity of demand.