Question

1. You are the manager of a large firm that has obtained a patent on its unique low-carbohydrate energy bars. As such, you are the only firm that sells this unique type of energy bar in the United States. An economist that you have hired estimated that the demand function for energy bars is Q = 200 – 2P and you have estimated your cost function is C(Q) = 50 + 5Q2. This implies that MC(Q) = 10Q. a. What is the inverse demand function? b. What is the marginal revenue function? c. What is the profit-maximizing output for your firm? d. What is the profit-maximizing price for your firm? e. What are profits at the profit-maximizing output? f. What is the own-price elasticity of demand at the profit-maximizing output? g. Will the firm be able to sustain profits over time?

Answer 1

Ans: Here we are talking of monopoly context.

a.

Inverse demand function => P = f(Q)

=> Q = 200 - 2P

=> P = 100 - Q/2.

b.

Marginal Revenue = ?

Total revenue = P* Q

=> (100-Q/2)*Q

differentiating wrt Q,

MR = 100 - Q.

c. Profit maximizing output is when MR=MC

=> 100 - Q = 10Q

=> 100/11 = Q

or, Q = 9.09 = 9 energy bars.

d. Price=?

putting this Q in demand function,

P = 100 - Q/2 = 100 - 9.09/2 = $95.455 .

e. Profit = TR - TC

= PQ - TC

= 95.455*9 - (50+5*81)

=$404.095.

f. own price elasticity = delta(Q)/Q / delta(P)/P = -2 * P/(200-2P) = -2*95.455/9 = - 21.212.

So the price elasticity of demand is elastic, this means that if revenue for the firm can be increased if prices went down (but just because revenue would rise, doesn’t mean profit would, because we have positive costs).

g. The existence of high barriers to entry prevents firms from entering the market even in the long?run. Since patented product is there, it is possible for the monopolist to avoid competition and continue making positive economic profits in the long?run.