Question #42181

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.

Expert's answer

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.

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.

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