Question

You are the manager of a wine company that produces output in two plants. The demand for your company's product is P = 78 − 15Q, where Q = Q1 + Q2. The marginal costs associated with producing in the two plants are MC1 = 3Q1 and MC2 = 2Q2. How much output should be produced in plant 1 in order to maximize profits?

a. 1 b. 2 c. 3 d. 4

Answer 1

Solution:

The demand for your company's product is, P = 78 − 15Q

where Q = Q1 + Q2

The marginal costs associated with producing in the two plants are

MC1 = 3Q1 and MC2 = 2Q2

Calculate thye total revenue and then differentiate the total revenue function with respects to Q to get the marginal revenue function.

TR = PQ

TR = (78-15Q)Q

TR = 8Q-15Q²

MR = dTR/dQ

MR = 78-30Q

MR= 78-30Q1 - 30Q2

To maximize profits ,the firm should be in such a manner so that the following two conditions are satisfied

MR = MC1 ..........(1)

MR =MC2 ..........(2)

From (1) ,we have

78- 30Q1-30Q2 = 3Q1

33Q1=78-30Q2 ...............(3)

From (2) ,we have  

78 - 30Q1 -30Q2 = 2Q2

Q2 = (78-30Q1)/32 ..............(4)

Substitute (4) in (3)

33Q1 = 78-30[(78-30Q1)/32]

33Q1 = 78-[(2340-900Q1)/32]

1056Q1 = 2496 - 2340 + 900Q1

156Q1 = 156

  Q1 = 1

Therefore . the firm should produce only one unit at plan 1 to maximize profits