You are the manager of a firm that produces output in two plants. The demand for your firm’s product is P = 80 – Q, where Q = Q1+ Q2. The marginal cost associated with producing in the two plants are MC1 = Q1 and MC2 = 8. How much output should be produced in plant 1 in order to maximize profits?
2 |
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4 |
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8 |
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14 |
Please help/ show work.
Thank you
The answer is C.
TR = P×Q
= (80- Q)×Q
= 80Q - Q2
MR = dTR/dQ = 80 - 2Q = 80 - 2(Q1 + Q2) = 80 - 2Q1 - 2Q2
For profit maximization of plant 1
MR = MC1
80 - 2Q1 - 2Q2 = Q1
80 - 2Q2 = 3Q1
(80/3) - (2/3)Q2 = Q1. .............(i)
For profit maximization of plant 2
MR = MC2
80 - 2Q1 - 2Q2 = 8
72 - 2Q1 = 2Q2
36 - Q1 = Q2
Putting the value of Q2 in equation 1.
(80/3) - (2/3)(36-Q1) =Q1
(80/3) - { 24 - (2/3)(Q1)} = Q1
(80/3) - 24 + (2/3)Q1 = Q1
(80/3) - 24 = Q1 - (2/3)Q1
(80/3) - 24 = (1/3)Q1
80 - 72 = Q1
8 = Q1.
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