suppose a firm's constant-returns to scale production function requires it to use capital and labor in a fixed ratio of two workers per machine to produce 10 units and that the rental rates for capital and labor are given by v=1, w=3.

a. calculate the firm's long run total and average cost curves.
b. Suppose K is fixed at 10 in the short run. Calculate the firms short run toatl and average cost curves. What is the marginal cost of the 10 th unit? The 25th unit? the 50th unit? the 100th unit?

take a shot, what do you think?

Hint. With the given constant returns to scale, and that all labor and capital can be purchased at constant prices, the TC 'curve' should be an upward sloping line, and the MC and AC curves should be flat lines.

A firm uses capital to produce revenue. The marginal revenue from the first 5 units of capital is as follows: 1st unit has MR 2.1, 2nd unit has MR 1.93, 3rd unit has MR 1.76, 4th unit has MR 1.55, and 5th unit has MR 1.35. If the interest rate is 51%, what is the optimal amount of capital for this firm to borrow?

To calculate the firm's long run total and average cost curves, we need to determine the total cost and average cost at each level of output.

a. In the long run, the firm can adjust both capital and labor inputs to minimize costs. Since the constant-returns to scale production function requires using a fixed ratio of two workers per machine, we have the following relationship:

10 units = 2 workers/machine × K machines

Therefore, K = 10/2 = 5 machines.

To calculate the long run total cost (C) at each level of output, we need to consider the cost of capital and labor inputs. The total cost can be expressed as:

C = vK + wL

where v represents the rental rate for capital (v = 1) and w represents the rental rate for labor (w = 3). Since we know the fixed ratio of labor to capital (2 workers per machine), we can substitute the values into the equation:

C = (1)(5) + (3)(2L)

where L represents the labor input.

To calculate the long run average cost (AC), we divide the total cost by the level of output (q):

AC = C/q

b. In the short run, the firm can only adjust the labor input while keeping capital fixed at a level of 10 machines. To calculate the short run total cost and average cost curves, we substitute the fixed capital value into the total cost equation:

C = (1)(10) + (3)(L)

To calculate the short run average cost (AC), divide the total cost by the level of output:

AC = C/q

To find the marginal cost of the 10th, 25th, 50th, and 100th units, we need to calculate the additional cost incurred when producing each additional unit. The marginal cost (MC) represents the change in total cost when output increases by one unit:

MC = ΔC/Δq

The marginal cost can be calculated using the short run total cost function:

MC = (C(q) - C(q-1))/(q - (q-1))

where C(q) represents the total cost at a certain level of output q, and C(q-1) represents the total cost at the previous level of output (q-1).

Therefore, using the short run total cost function, we can calculate the marginal cost for the 10th, 25th, 50th, and 100th units by substituting the corresponding values of q into the formula.