Sport Tee Corporation manufactures T-shirts bearing the logos of professional fo

Question

Sport Tee Corporation manufactures T-shirts bearing the logos of professional football teams. The wholesale market for sport T-shirts is perfectly competitive. The manager forecasts the wholesale price of T-shirts next year to be $8.00. The firm's estimated marginal cost is 1. SMC = 13-0.0060 + .000000902 where Q is the number of T-shirts produced and sold each month. Sport Tee Corporation will have a fixed cost of $4,000 per month. a. Given the estimated marginal cost function above, write the equation for the average variable cost function and the total variable cost function. What is the profit-maximizing number of T-shirts that Sport Tee should produce and sell each month? (Note: you will find 2 output levels that are solutions to your optimization problem. Test which one is actually the optimal level of output. Round to the nearest integer when calculating your answer.) b. c. How much monthly profit (or loss) will Sport Tee Corporation earn?

Explanation / Answer

We have SMC = 13 - 0.006Q + 0.0000009Q^2, fixed cost = 4000 and price = $8,

a) Find that Total cost = 13Q - 0.003Q^2 + 0.0000003Q^3 + 4000

TVC = 13Q - 0.003Q^2 + 0.0000003Q^3

Then we have AVC = TVC/Q = (13Q - 0.003Q^2 + 0.0000003Q^3)/Q = 13 - 0.003Q + 0.0000003Q^2

b) We must have MC = P = 8

13 - 0.006Q + 0.0000009Q^2 = 8

0.0000009Q^2 - 0.006Q + 5 = 0

We have two solutions, Q = 5690 and Q = 976. Find the minimum of AVC which is find by

dAVC/dQ = 0

0.003 = 0.0000006Q

Q = 5000

Since P = MC must be higher than minimum of AVC which is attained when Q = 5000, We select Q = 5690

c) Profits = TR - TC = 5690*8 - (13*5690 - 0.003*(5690^2) + 0.0000003*(5690^3) + 4000) = $9412

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