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
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.