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 $6.00. The firm’s estimated marginal cost is

SMC = 12 - 0.008Q + .0000012Q^2

where Q is the number of T-shirts produced and sold each month. Sport Tee Corporation will have a fixed cost of $7,000 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.

b ) 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.)

c ) How much monthly profit (or loss) will Sport Tee Corporation earn?

Answer 1

Given that fixed cost is $7000 per month and an estimated price is $6.00, we have SMC = 12 - 0.008Q + 0.0000012Q^2.

a ) Given the estimated marginal cost function above, write the equation for the average variable cost function and the total variable cost function.

Total variable cost = integration of MC = 12Q - 0.004Q^2 + 0.0000004Q^3

AVC = TVC/Q = 12 - 0.004Q + 0.0000004Q^2

b ) What is the profit-maximizing number of T-shirts that Sport Tee should produce and sell each month?  

This is determined by P = MC

6 = 12 - 0.008Q + 0.0000012Q^2.

0.0000012Q^2 - 0.008Q + 6 = 0

This becomes a quadratic equation whose roots are

Q = (0.008 +/- (0.008^2 - 4*6*0.0000012)^0.5)/(2*0.0000012)

Q = 5806 or 861 (approx)

Profit = TR - TC = 6Q - (7000 + 12Q - 0.004Q^2 + 0.0000004Q^3) = -6Q + 0.004Q^2 - 0.0000004Q^3 - 7000

Profits when Q = 5806 , = -6*5806 + 0.004*(5806^2) - 0.0000004*(5806^3) - 7000 = 14715.29

Profits when Q = 861 =  -6*861 + 0.004*(861^2) - 0.0000004*(861^3) - 7000 = -$9456

Since Q = 861 gives us a loss, we select Q = 5806,

c ) How much monthly profit (or loss) will Sport Tee Corporation earn?

It earns a profit of $14715.29 per month.