Question

Consider an inverse demand curve for a monopolist: P = 200 - 0.05Q. The Marginal Cost function is MC = 50 + 0.2Q; Fixed Cost (FC) =17,500.

What is the total cost (TC) function and the value of TC at pmax Q? {Hint: Think “integration” of the MC function and then add fixed cost (FC).}

Intercept or Q0 coefficient?

Q1 coefficient?

Q2 coefficient?

Answer 1

P = 200 - 0.05Q

TR= 200Q-0.05Q²

MR= Differentiation of TR wrt Q= 200-0.1Q

TC= Variable cost(VC) + Fixed cost(FC)

Variable cost is the cost which vary with level of output.

MC= Differentiation of VC

So if we integrate both sides with respect to Q we get value of VC:

VC= Integration of MC

VC= 50Q+ 0.2Q² /2

FC= 17500

TC=0.2Q² /2 +  50Q+17500 Total cost function

Optimal condition for monopoly: MR=MC

200-0.1Q= 50+0.2Q

150= 0.3Q

Q= 150/0.3= 500 profit maximizing optimal quantity

TC= 0.2(500)(500)/2 +50(500)+17500= 67500

Intercept coefficient of TC(Q=0)= 17500

Q1 coefficient= 50

Q2 coefficient = 0.2/2= 0.1