Question

Suppose that a firm estimates that the demand curve for its product is Q= 120,000 - 10,000P.  Suppose that the firm has fixed costs of $12,000 and variable costs per unit (AVC) is $1.50.

1. Write an equation for total profits in terms of Q.  At what level of output (Q) are total profits maximized?  What price will be charged?  What are total profits at this output level?
2. Check your answers in part e) by equating marginal revenue and marginal cost functions and solving for Q.
3. What market model has been assumed in this problem?

Answer 1

Q= 120,000 - 10,000P
So, 10,000P = 120,000 - Q
So, P = (120,000/10,000) - (Q/10,000) = 12 - 0.0001Q
Total revenue = TR = P*Q = (12 - 0.0001Q)Q = 12Q - 0.0001Q²
Total cost, TC 1.= Fixed cost + Total variable cost = 12,000 + 1.5Q

Profit = Total revenue - Total cost = (12Q - 0.0001Q²) - (12,000 + 1.5Q) = 12Q - 0.0001Q² - 12,000 - 1.5Q = 10.5Q - 0.0001Q² - 12,000
So, Profit = 10.5Q - 0.0001Q² - 12,000

d(Profit)/dQ = 10.5 - 2(0.0001Q) = 0
So, 0.0002Q = 10.5
So, Q = 10.5/0.0002 = 52,500
so, Q = 52,500 maximizes profits.

P = 12 - 0.0001Q = 12 - 0.0001(52,500) = 12 - 5.25 = 6.75
So, P = 6.75

Profits = 10.5Q - 0.0001Q² - 12,000 = 10.5(52,500) - 0.0001(52,500)² - 12,000 = 551250 - 275625 - 12000 = 263,625
So, Profit = 263,625

MR = d(TR)/dQ = 12 - 2(0.0001Q) = 12 - 0.0002Q
MC= d(TC)/dQ = 1.5
So, MR = MC gives,
12 - 0.0002Q = 1.5
So, 0.0002Q = 12 - 1.5 = 10.5
So, Q = 10.5/0.0002
So, Q = 52,500

Monopolist market model has been assumed.