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

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.

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

To find the firm's profit-maximizing price and quantity, we need to determine the quantity at which marginal revenue equals marginal cost.

a) The profit-maximizing quantity can be found by setting the marginal revenue (MR) equal to the marginal cost (MC) equation.

Given:
Cost function (C) = 160 + 16Q + 0.1Q^2
Price equation (P) = 96 - 0.4Q

To find the marginal cost (MC), we differentiate the cost function with respect to Q:
MC = dC/dQ = 16 + 0.2Q

To find the marginal revenue (MR), we differentiate the price equation with respect to Q:
MR = dP/dQ = -0.4

Set MR equal to MC and solve for Q:
-0.4 = 16 + 0.2Q
0.2Q = -16.4
Q = -82

Since the quantity cannot be negative, we discard this negative value.

To find the profit-maximizing price, substitute the found quantity into the price equation:
P = 96 - 0.4(-82)
P = 96 + 32.8
P = 128.8

Therefore, the profit-maximizing quantity is Q = 0 units, and the profit-maximizing price is P = $128.8.

To calculate the profit, we need to use the profit equation:
Profit = Total Revenue - Total Cost
Total Revenue = Price * Quantity
Total Cost = Cost function (C)

Substitute the profit-maximizing price and quantity into the equations:

Total Revenue = 128.8 * 0 = $0
Total Cost = 160 + 16(0) + 0.1(0)^2 = $160

Profit = Total Revenue - Total Cost
Profit = $0 - $160
Profit = -$160

Therefore, the firm's profit is -$160 (which indicates a loss).

b) The production manager's claim that the firm's average cost of production is minimized at an output of 40 units is not necessarily true. The average cost is minimized at the output level where the marginal cost equals the average cost. It is possible for the average cost to be minimized at a different level of output than the profit-maximizing level. Additionally, the claim that 40 units is the firm's profit-maximizing level of output is also incorrect, as we found earlier that the profit-maximizing quantity is 0 units.

c) The firm cannot increase its profit by using a second plant to produce the output in part (a) since the profit-maximizing quantity for the single plant is already 0 units. Adding a second plant with identical costs would double the cost, resulting in a larger loss.