Asked by Daniel Klucker on Jun 16, 2024
Verified
A firm has fixed costs of $2,000.Its short-run production function is y 4x1/2, 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.
Short-Run Production Function
A mathematical representation of the relationship between input changes and output levels in a production process, considering some inputs as fixed.
Total Cost Function
The total cost function describes the total cost of production as a function of the quantity of output, including both fixed and variable costs.
Variable Factor
In economics, it refers to any input whose quantity can be changed in the short term to affect production levels.
- Examine the characteristics of short-run and long-run cost curves in diverse scenarios.
- Evaluate the effect of fixed costs, variable costs, and total costs on firm decisions.
Verified Answer
MA
Muhammad AbubakarJun 20, 2024
Final Answer :
D
Explanation :
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 = ^(1/2) * ^(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.
(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 = ^(1/2) * ^(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.
Learning Objectives
- Examine the characteristics of short-run and long-run cost curves in diverse scenarios.
- Evaluate the effect of fixed costs, variable costs, and total costs on firm decisions.