1. The demand and cost function for a company are estimated to be as follows: P(Q)=100-8Q; C(Q)=50+80Q-10Q^2+0.6Q^3
(a) What price should the company charge if it wants to maximize profits in the short-tun?
(b) What price should ti charge it it wants to maximize it's revenue?
(c) Assume the company is a monopoly - what would expect their profits to be in the long-run, and why?
(d) Now, assume the company exists under monopolistic conditions - what you you expect their profits to be in the long-run, and why?
Profit is maximized when Marginal Revenue (MR) = Marginal Cost (MC)
To find MR, first find Total Revenue (TR)
TR = PQ
TR = (100 - 8Q)Q
TR = 100Q - 8Q^2
MR is the derivative of TR
MR = 100 - 16Q
MC is the derivative of Total Cost (TC)
TC = 50 + 80Q - 10Q^2 + 0.6Q^3
MC = 80 - 20Q + 1.8Q^2
To maximize profit, let MR = MC
100 - 16Q = 80 - 20Q + 1.8Q^2
0 = 1.8Q^2 - 4Q - 20
This solution gives a detailed, step-by-step solution of the math required to calculate a firm's profit-maximizing and revenue-maximizing price and quantity when its demand and cost functions are known. Solutions are given for a firm in a monopoly market and a monopolistic competition market.