Answer to Question #42181 in Economics of Enterprise for Bre

Question

In this module, it is demonstrated that some times extensive diseconomies of scale, say, due to high transportation costs, would require that the firm produce its product in a multiple of plants. Suppose a beer brewing company has determined that its total production cost is: TC = 1000 Q - 1.2 Q2 + 0 .004Q3 where Q is its annual output measured in metric tons.

A. The average hauling (freight) cost is $0.8Q; that is AFC = 0.8Q. Write the firm's average aggregated cost equation.
B. Now suppose the firm is facing the following market demand:
Q = 760,000 - 10 P
Determine the optimal number of plants that the firm should have to take full advantage of the market demand.
C. Calculate the firm's profit.

Answer 1

TC = 1000Q - 1.2Q2 + 0.004Q3
A. AFC = 0.8Q.

Average aggregated cost ATC = TC/Q = 1000 - 1.2Q + 0.004Q2
B. Qd = 760,000 - 10P, P = 76,000 - Q/10

To find optimal quantity produced, we should find the quantity, for which marginal revenue equals marginal cost MR = MC.
MR = TR' = (P*Q)' = (76,000Q - Q2/10)' = 76,000 - Q/5
MC = TC' = 1000 - 2.4Q + 0.012Q2
76000 - 0.2Q = 1000 - 2.4Q + 0.012Q2
0.012Q2 - 2.2Q - 75000 = 0
D = 2.2^2 - 4*0.012*75000 = -3595.16, so the equality has no solutions, so we can't calculate the optimal number of plants and firm's profit.