Question

The following equations represent the function of total cost of a monopoly and the demand for a product made by the monopoly.

c = (1/12)q³ - (5/2)q² + 30q + 100

q = 25 - p

a. Obtain the equation for revenue (r) coming from the sales of the product expressed in fonction of the quantities produced (q)

b. Obtain the equation for profits (?)

c. Obtain the level of production q* that would maximize the profits of the monopoly. Verify that at the level of production q = q*, the profits are maximized. What is the highest level of profits under which this business can aspire?

Answer 1

Cost function is c = (1/12)q3 - (5/2)q2 + 30q + 100.

Inverse demand function is p = 25 - q

a. Revenue function r = pq = (25 - q)q = 25q - q^2.

b. For the profit function we use ? = r - c (revenue - cost)

= 25q - q^2 - (1/12)q3 - (5/2)q2 + 30q + 100)

c. Profit is maximized when its derivative with respect to q is zero

d?/dq = 0

25 - 2q - (3/12)q^2 + (10/2)q - 30 = 0

-5 + 3q -q^2/4 = 0

q^2 -12q + 20 = 0

This gives q = 10 or q = 2. At q = 10, profit/loss = -83.33 and at q = 2, it is -104.67

Hence loss is minimized when q = 10. It is equal to -83.33