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# Short run and long run cost functions: Profit maximization

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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?

https://brainmass.com/economics/short-and-long-run-cost-functions/short-run-long-run-cost-functions-profit-maximization-621145

#### Solution Preview

1a)
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

Solving ...

#### Solution Summary

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.

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