Question

This question is borrowed from Exercise 3.5 in the book. Suppose the demand is q=D(p)=10200-100p. The cost of producing q units is q²/2 (note this implies that MC=q).

Hint: Write the demand as p=102-q/100 first

What is your total revenue schedule (or function)?

What is your marginal revenue schedule (MR)?

What is the profit-maximizing q and what is the corresponding price p? Given p and q, what is your profit?

Answer 1

Inverse demand function is as follows -

p = 102 - q/100

Calculate the Total revenue function -

TR = p*q = (102 - q/100) q = 102q - q²/100

The Total Revenue function is 102q - q²/100.

Calculate the marginal revenue function -

MR = dTR/dq = d(102q - q²/100)/dq = 102 - q/50

The Marginal Revenue function is 102 - q/50.

At profit-maximizing level,

MR = MC

102 - q/50 = q

q/50 + q = 102

51q/50 = 102

q = (102 * 50)/51 = 100

p = 102 - q/100 = 102 - (100/100) = 102 - 1 = 101

Thus,

The profit-maximizing q is 100.

The profit-maximizing p is 101.

Calculate the profit -

Profit = TR - TC = (102q - q²/100) - (q²/2)

Profit = [(102 * 100) - (100²/100)] - (100²/2)

Profit = [10200 - 100] - 5000

Profit = 10100 - 5000 = 5100

The profit is 5,100.