Question

Short Run Cost Curves: Consider a firm with the following production function: q=(KL)/20 a. For a short-run situation in which K=10, wage = 4 and cost of capital = 1, derive expressions for short run total cost and short run average cost for this production function. b. Plot the short-run total cost curve and label it “TC1”. Now suppose that the cost of capital goes up to 2. (Continue to assume we’re in the short run and K can not be altered). How would this alter your short run total cost curve? Plot the new cost curve as “TC2.” c. Alternatively, beginning from your original total cost function with cost of capital = 1, suppose that wages fall to 2. How would this alter your TC curve? Plot this new curve and label it “TC3.” d. Will either of the changes above – the change in capital costs or the change in wages – affect marginal costs for this firm? Explain.

Answer 1

Given:

Production function:

where the capital K is fixed at 10 units in the short run.

Wage rate: w = 4

cost of capital: r = 1

Now in the short the amount of labor employed are:

Putting K = 10 into the production function.

equation (1)

Therefore the amount of labor employed is equal to output units.

The total cost function is the cost as a function of output. here, the total cost is the sum of fixed cost (capital cost) and variable cost. (labor cost)

where, w = 4, r = 1

From the above finding, L= 2q. and K is fixed at 10. Putting the values, we get the total cost function:

Therefore, this is the expression of the short-run total cost function:

Now, the average cost function:

The average cost is the per-unit cost. It the ratio of total cost over the output produced.

Therefore this is the expression for the average cost function.

b.) Plotting the total cost curve: Putting quantity produced on the x-axis and total cost on the y-axis.

The y-intercept of the total cost curve is where the quantity produced is 0.

Therefore TC1 y-intercept is 10. It starts at 10 and increases with the cost rate as the wage rate is fixed 4.

Total cost curve: TC1

New total cost curve as the cost of capital increases to 2.

Cost of capital: r = 2

wage: w = 4

Capital units are fixed in the short run: K = 20

where, w = 4, r = 2

from the equation (1).   and as K = 10

Therefore,

Plotting new total cost curve TC₂:

The TC2 curve starts from point 20 as the fixed cost is 20. Then it rises with the constant rate. At each additional unit produced, the total cost increases by 8.

TC2:

Description of the curve:

As the cost of capital increases to 2. The total curve parallelly, shift upward. As of now at each level of output, the fixed cost is 20. So at each level, it increases by 20. The New TC2 is parallel upwards to the old TC1.

c.) The new total cost curve as the wages fall to 2.

Cost of capital: r = 1

wage: w = 2

Capital units are fixed in the short run: K = 20

where, w = 2, r = 1

from the equation (1).   and as K = 10

Therefore,

Plotting new total cost curve TC₃:

Description of the diagram:

The new TC after the fall of wage is flatter to the old TC1. As the wage rate falls to 2. Each additional cost also changes. And this changes the slope of the TC curve. While the y-intercept remains the same at 10 as the fixed cost, the cost of capital is the same.

ALL TC curves: As we can see that TC2 is parallelly upward to TC1 while TC3 is flatter to the TC1.

d.) Marginal cost in all the above cases.

Marginal cost is the additional cost that occurred on producing one more unit of the good. It is the derivative of the total cost function with respect to the output produced.

Case 1 without a change in the cost of capital and wage

Marginal cost:

The marginal cost in the initial case is 8.

Case 2 when the cost of capital changes to 2:

Marginal cost:

Therefore in this case, when the cost of capital increases to 2. The marginal cost remains the same as 8.

Case 3 when the wage changes to 2:

Marginal cost:

Therefore in this (3rd) case, when the wage falls to 2. The marginal cost changes and decreases to 4 and doesn't remain the same as the original case.

The explanation for the change in marginal cost:

As marginal cost is the additional cost, occurred on producing one more unit of the good. It only changes when the variable cost changes. As only variable factor units change in the short run, so additional cost occurred is only on the variable input. Whereas, the fixed factor doesn't have any additional cost. It remains the same for all levels of output in the short run. So marginal cost doesn't take into account the fixed cost.

Here, variable cost is the cost of labor, wage. The fixed cost is the cost of capital.

So when the cost of capital increases to 2, it increases the fixed cost. As capital is the fixed factor, so it only changes the initial cost occurred and additional cost remains the same at all levels of output. Whereas, the change in the wage to 2, it increases the variable cost. So at each additional unit of labor employed, the variable cost changes. It changes the marginal cost.