Question

The demand function for a monopolist's product is p=1300-7q and the average cost per unit for producing q units is c=0.004q2-1.6q+100+5000/q

-Find the quantity that minimizes the average cost function and the corresponding price. Interpret your results.

-What are the quantity and the price that maximize the profit? What is the maximum profit? Interpret your result.

Answer 1

c = 0.004q^²- 1.6q + 100 + (5,000 / q)

p = 1,300 - 7q

a) Average cost = (c / q) = 0.004q - 1.6 + (100 / q) + (5,000 / q^²)

Average cost is minimum when first derivative of average cost with respect to q is 0.

First derivative of Average cost = 0.004 - (100 / q^²) + (-10,000 / q^³)

0.004 - (100 / q^²) + (-10,000 / q^³) = 0

q = 195

At q = 195, p = 1,300 - 7 * 195 = -65

b) Profit = Total Revenue - Total Cost

Total Revenue = p * q = 1,300q - 7q^²

Total Cost = 0.004q^²- 1.6q + 100 + (5,000 / q)

Profit = 1,300q - 7q^² - 0.004q^² + 1.6q - 100 - (5,000 / q)

First derivative of profit = 1,300 - 14q - 0.008q + 1.6 - (5,000 / q^²)

1,300 - 14q - 0.008q + 1.6 - (5,000 / q^²) = 0

q = 9

Profit = 1,113.8 at this q level.