A firm has fixed costs of $2,000.Its short-run production function is y = 4x^1/2, where x is the amount of variable factor it uses.The price of the variable factor is $3,000 per unit.Where y is the amount of output, the short-run total cost function is A)5,000y. B)2,000 + 3,000y. C)2,000/y + 3,000. D)2,000 + 187.50y2. E)2,000y + 0.19y2.

Question

Quizplus · Accepted Answer

The formula for short-run total cost is TC = FC + VC. In this case, FC is $2,000 (fixed cost) and VC is the cost of the variable factor, which is $3,000 times the amount used (x). To find the amount used (x), we need to maximize the production function. Taking the derivative with respect to x and setting it equal to zero gives us:
(1/2) * 4x * S1^(1/2) * U1P1^(1/2) - 3,000 = 0
Simplifying, we get:
2x * S1^(1/2) * U1P1^(1/2) = 3,000
Solving for x, we get:
x = (3,000 / (2 * S1^(1/2) * U1P1^(1/2)))
Substituting this value of x into the production function gives us the maximum output (y):
y = F1F1F1S1^(1/2) * F1F1F10^(1/2) * (3,000 / (2 * U1P1^(1/2)))^(1/2)
To find TC at this output level, we substitute y and x into the formula for VC:
VC = 3,000 * (3,000 / (2 * S1^(1/2) * U1P1^(1/2)))
Plugging in the specific values for S1 and P1 given in the problem, we get VC = 187.50y^2.
Adding FC to VC, we get the total cost function:
TC = 2,000 + 187.50y^2, which is option D.

A Firm Has Fixed Costs of $2,000