Answer to Question #42176 in Economics for Emily

Question

2. In this module, it is demonstrated that sometimes 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= 1000Q-1.2Q^2 +0.004Q^3 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=760,000-10P
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.
For 5 extra bonus points:
D. Compare the firm’s profit with multiple plants with its profit with a single plant.
Hint: The firms MC equation based on its aggregated total cost (including the freight cost) is MC=1000-0.8Q+0.012Q^2

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.