Posted by J on Sunday, April 11, 2010 at 11:14pm.
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.
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
what is the relationship of investment and interest rates