Question

You are a manager of a monopoly, and your demand and cost functions are given by P =220 -2Q and C(Q) = 2000 + 10 Q2 What Price maximizes your profit

Answer 1

Given,the inverse demand function where, P = 220 - 2Q and cost function C(Q) = 2000 + 10Q^2.

In a monopoly, the profit is maximized at the stage where Marginal Revenue (MR) = Marginal Cost (MC).

In order to find MR, we need to calculate Total Revenue:

Total Revenue = P(Q)

= (220 - 2Q)Q

= 220Q - 2Q²

Marginal Revenue = derivative of Total Revenue / derivative of Q

= dTR / dQ

d(220Q - 2Q²) / dQ = 220 - 4Q since, [d(CQⁿ) / dQ = C*n*Q^n-1]

therefore, MR = 220 - 4Q

Now calculating MC from the cost function C(Q) = 2000 + 10Q²

Marginal Cost = derivative of Total Cost / derivative of Q

= dTC / dQ

d(2000 + 10Q²) / dQ = 20Q since, [d(CQⁿ) / dQ = C*n*Q^n-1]

therefore, MC = 20Q

Now, equating MR = MC

  220 - 4Q = 20Q

Q = 220/24

To find the price we must substitute the value of Q in the function P = 220 - 2Q

By substituting, we get the value,

P = 220 - 2(220/24)

P = 220 - 18.33

P = 201.67

Therefore, the profit is maximized when the price is $201.67.