In this module, it is demonstrated that some times extensive diseconomies of sca

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.

D. Compare the firm's profit with multiple plants with its profit with a single plant.

Hint: The firm's MC equation based on its aggregated total cost (including the freight cost) is:  MC = 1000 - 0.8 Q + 0 .012 Q2

Explanation / Answer

A.

The total Cost equation including the frieght cost will be (Add the freight cost which is given as 0.8Q for each unit and hence TC will be 0.8Q per unit * (Q units) = 0.8Q^2

TC = 1000Q-1.2Q^2+0.004Q^3 + Q(0.8Q) = 1000Q + Q^2(-1.2+0.8) + 0.004Q^3 = 1000Q -0.4 Q^2 + 0.004Q^3

B. The optimal profit happens when the marginal cost is equal to marginal revenue.

To obtain the marginal cost equation, just differentiate the Total Cost equation,

Then you get 1000 + 2*(0.4Q) + 3*(0.004Q^2) = 1000 + 0.8Q + 0.012Q^2

The total revenue is Price * Quantity = P*(760,000-10P) = 760,000 P -10 P^2

Hence Marginal revenue (differentiating Total Revemue) is 760,000-2(10P) = 760,000-20P

Hence the intersection of both the curves can be obtained and solved for Q.

You get 7961 as one postive root and that should be the optimal production.

Substituting that P = (7691 - 760000)/-10 = 75203.9

C. The firm's profit will be TR - TC

= 760,000 P -10 P^2 - (1000Q -0.4 Q^2 + 0.004Q^3)

D

Further information is needed to compare with multiple locations and single location to find out disadvantage of transportation costs.

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