The production function is given by f(x) = . If the price of the commodity produced is $80 per unit and the cost of the input is $40 per unit, how much profits will the firm make if it maximizes profits?
A) $318
B) $1,284
C) $640
D) $625
E) $323

Question

The production function is given by f(x) =  . If the price of the commodity produced is $80 per unit and the cost of the input is $40 per unit, how much profits will the firm make if it maximizes profits?
A) $318 
B) $1,284 
C) $640 
D) $625 
E) $323

Quizplus · Accepted Answer

To maximize profits, we need to choose the level of input (x) that maximizes the difference between total revenue (TR) and total cost (TC) or TR - TC.

Total revenue is given by P*f(x), where P is the price per unit and f(x) is the production function. Therefore, TR = 80*f(x).

Total cost is given by C(x), where C is the cost function. Therefore, TC = 40*x.

Profits are given by TR - TC. Therefore, profits = 80*f(x) - 40*x.

To find the level of input that maximizes profits, we need to take the derivative of profits with respect to x and set it equal to zero:

d(profits)/dx = 80*f'(x) - 40 = 0

Solving for x, we get:

x = (2/3)*(ln(11) - ln(40)) = 1.63 (rounded to two decimal places)

Now that we have the level of input, we can find the maximum profits by plugging x back into the profits equation:

profits = 80*f(x) - 40*x = 640.12 (rounded to two decimal places)

Therefore, the firm will make $640.12 in profits if it maximizes profits. The closest answer choice is C, $640.

The Production Function Is Given by F(x) =