Question

A firm has fixed costs of $4,000. 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 $4,000 per unit. Where y is the amount of output, the short-run total cost function is:____

Answer 1

The short-run total cost function of the firm, given a production function and fixed costs, is calculated by adding the fixed costs of $4,000 to the variable costs, which derive from the price of the variable factor and its relation to the output. The total cost function is TC = $4,000 + 1,000y^2.

Step-by-step explanation:

The student is asking for the short-run total cost function of a firm given a production function, fixed costs, and the price of a variable factor. The total cost (TC) in the short-run is comprised of fixed costs (FC) and variable costs (VC). Fixed costs are those costs that do not change with output, while variable costs vary with the level of output. According to the information provided, the fixed costs are $4,000.

To find the variable costs, we use the production function y = 4x1/2, where x is the amount of the variable factor used. Since the variable factor costs $4,000 per unit, the variable cost (VC) can be expressed as $4,000 * x. Substituting x for (y/4)2 from the production function, the VC becomes $4,000 * (y/4)2.

Adding the variable costs to the fixed costs gives us the total cost (TC): TC = FC + VC = $4,000 + $4,000 * (y/4)2. So, the short-run total cost function is TC = $4,000 + 1,000y2.

Answer 2

To determine the short-run total cost function for the given firm, we need to consider both the fixed costs and the variable costs associated with the production function.

Fixed Costs (FC): $4,000 (as given)

Variable Costs (VC): The variable cost is the cost of the variable factor, which is priced at $4,000 per unit. In the production function y = 4x^(1/2), x represents the amount of the variable factor used, and it's raised to the power of 1/2. So, VC can be calculated as:

VC = $4,000 * x^(1/2)

Total Cost (TC): Total cost is the sum of fixed and variable costs:

TC = FC + VC

TC = $4,000 + $4,000 * x^(1/2)

Therefore, the short-run total cost function is:

TC = $4,000 + $4,000 * x^(1/2)