A farm equipment firm produces small tractors at an estimated total cost: C = 37
ID: 2441693 • Letter: A
Question
A farm equipment firm produces small tractors at an estimated total cost: C = 37,500,000 + 5,000Q + 1.5Q2, where Q denotes annual tractor output. Regional demand for small tractors is given by: P = 30,000 – Q.
a. Determine the firm’s optimal output and price and its annual profit from tractors.
b. Because of increased foreign competition, the demand for the firm’s tractors falls permanently to: P = 20,000 – Q. The firm is considering closing its plant immediately. By doing so, it can probably save $18 million of its annual $37.5 million in fixed costs. Alternatively, it can continue to operate the plant for 12 months after which if it shuts down it can walk away from its labor contracts and plant lease and incur no continuing costs. Recommend its most profitable operating strategy and justify.
Explanation / Answer
C = 37,500,000 + 5,000Q + 1.5Q2
Marginal cost (MC) = dC/dQ = 5,000 + 3Q
(a) Profit is maximized when Marginal revenue (MR) equals MC.
Total revenue (TR) = P x Q = 30,000Q - Q2
MR = dTR/dQ = 30,000 - 2Q
Equating with MC,
30,000 - 2Q = 5,000 + 3Q
5Q = 25,000
Q = 5,000
P = 30,000 - 5,000 = $25,000
TR = $25,000 x 5,000 = $125,000,000
C ($) = 37,500,000 + (5,000 x 5,000) + (1.5 x 5,000 x 5,000) = 37,500,000 + 25,000,000 + 37,500,000
= 100,000,000
Profit ($) = 125,000,000 - 100,000,000 = 25,000,000
(b)
New TR = 20,000Q - Q2
New MR = dTR/dQ = 20,000 - 2Q
Equating with MC,
20,000 - 2Q = 5,000 + 3Q
5Q = 15,000
Q = 3,000
P = 20,000 - 3,000 = $17,000
If firm closes immediately, Q = 0, TR = 0 and Loss = $37,500,000 - $18,000,000 = $19,500,000
If firm operates for 12 months,
TR = $17,000 x 3,000 = $51,000,000
TC ($) = 37,500,000 + (3,000 x 3,000) + (1.5 x 3,000 x 3,000) = 37,500,000 + 9,000,000 + 13,500,000
= 60,000,000
Profit ($) = 51,000,000 - 60,000,000 = -9,000,000 (Loss of $9 million)
Since continuing production for next 12 months will decrease the loss (-$9,000,000 < -$19,500,000), the most profitable strategy is to continue production for next 12 months and then shut down.
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