A firm uses a single plant with costs C= 160 +16Q +.1Q2 and faces the price equation P= 96 – .4Q.

Question

a) Find the firm’s profit-maximizing price and quantity. What is the profit?

b) The firm’s production manager claims that the firm’s average cost of production is minimized at an output of 40 units. Furthermore, she claims that 40 units is the firm’s profit-maximizing level of output. Explain whether these claims are correct.

c) Could the firm increase its profit by using a second plant (with costs identical to the first) to produce the output in part (a)? Explain.

Answer 1

a) Given total cost function as C=160+16Q+0.1Q(2){squared} and price equation P=96-0.4Q.

Total Revenue function is R=PQ=96Q-0.4Q(2).

The profit function is F=R-C=(96Q-0.4Q(2))-(160+16Q+0.1Q(2))=80Q-0.5Q(2)-160.

Profit attains maximum when and is negative dF/dQ=0 and d(2)F/dQ(2) is negative.

dF/dQ(fraction)=0 -> 80-Q= 0 -> Q=80

d(2)F/dQ(2) (fraction)= -1

Now it is clear that profit attains maximum when Q = 80
Price when Q = 80 is P=96-0.4*80=64

The profit-maximizing price = 64 and quantity = 80

Maximum Profit = [80Q-0.5Q(2)-160]Q=80(subscript) = 3040

b) The average cost function is AC = C/Q= 160/Q{fraction}+16+0.1Q

The AC function attains minimum when dAC/dQ {fraction}=0 and d(2)AC/dQ(2){fraction} is positive

dAC/dq{fraction}=0 -> -160/q(2){fraction} +0.1=0 -> -160+0.1Q(2)=0

i.e., Q(2) -1600=0 -> (q+40)(Q-40)=0 -> 40 or -40

Since Q cannot be negative Q = 40

d(2)AC/dQ(2){fraction}=320/Q(3){fraction}

Clearly d(2)AC/dQ(2){fraction} is positive when Q = 40

It is clear that the firms average cost of production is minimized at an output of 40 units.

The firm’s production manager’s first claim is the firm’s average cost of production is minimized at an output of 40 units, is correct.

But, her second claim, the firms’ profit maximizing level of output is 40 units, is not correct.
Note that from (a) we got the firms’ profit maximizing level of output is 80 units.
When Q = 40 the firms profit is (F=80*40-0.5*40(2)-160) = 2240,
But, when Q = 80 we got profit = 3040

c) If the firm uses a second plant with costs identical to the first. Then the cost function becomes C=320+32Q+0.2Q(2).

But, the demand function remains the same; unfortunately the revenue function remains unchanged. i.e.,R=PQ=96Q-0.4Q(2).

Now the new profit function becomes

F=R-C=96Q-0.4Q(2))-(320+32Q+0.2Q(2))=64Q-0.6Q(2)-320

dF/dQ{fraction}=0 -> 64-1.2Q= 0->Q=53.33

d(2)F/dQ(2){fraction}=-1.2

That is profit attains maximum when Q = 53.33

The maximum profit is[64Q-0.6Q(2)-320]Q=53.33{subscript} = 1386.67
So the firm cannot increase its profit by adding a second plant.

Answer 2

1. Given a firm’s demand function, P = 24 - 0.5Q and the average cost function, AC = Q2 – 8Q + 36 + 3/Q,
calculate the level of output Q which
a) maximizes total revenue b) maximizes profits

Answer 3

what is the relationship of investment and interest rates