1. The demand function for a firm’s product is Q(P) = 50-P/10. The firm’s cost o
ID: 1192509 • Letter: 1
Question
1. The demand function for a firm’s product is Q(P) = 50-P/10. The firm’s cost of production is C(Q) = Q^3-20Q^2+125Q. The firm’s problem is to choose the value of Q0 that maximizes its profit. You may occasionally find an irrational number and in those cases simplify your answer as much as possible.
(j) Calculate the price(s) that would cause the firm to break even, meaning: earn exactly zero profit.
(k) For this part only, change the demand function by assuming that demand (at any given price) is
half of what it was before. In this new situation, calculate the firm’s inverse demand function,
profit-maximizing point, and maximized profit.
(l) For this part only, suppose that the problem is to maximize revenue instead of profit. Does this
problem satisfy the global SOC? Find all points (if any) that satisfy the FOC. Calculate the revenue-maximizing price and quantity. (Justify your answer carefully.) Calculate the firm’s maximized revenue. How much profit does the firm sacrifice by choosing to maximize revenue instead of profit?
Explanation / Answer
j.
Given the demand function, Q = 50 – (P/10)
Therefore, P = 500 – 10Q
PQ = 500Q – 10Q^2
Break-even is a situation where PQ – CQ = 0
Therefore, 500Q – 10Q^2 – (Q^3 – 20Q^2 + 125Q) = 0
Solving the equation, (Q + 15) (Q – 25) = 0
Either Q = -15, or Q = 25
Since quantity (Q) can’t be negative, Q would be 25, (Q = 25).
Putting the value of Q = 25 in the demand function, Q = 50 – (P/10)
50 – (P/10) = Q
P = 500 – (10 × Q)
= 500 – (10 × 25)
= 500 – 250
= 250
Answer: The price would be $250 for break-even.
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