The cost function for an enterprise has the following form:
TC = 10L + 20K
The production function has the form: 4 ln L + 10 K 1/2
a. Derive the mathematical forms of the marginal and average products of capital and
labor. How is the marginal product of labor affected by changes in capital?
b. Describe the production position and combinations of inputs for when total costs are
equal to 300 and 700.
a. To derive the mathematical forms of the marginal and average products of capital and labor, we need to differentiate the production function with respect to each input separately.
Let's start with the marginal product of labor (MPL). We need to take the derivative of the production function with respect to labor (L):
MPL = d(4 ln L + 10 K^(1/2)) / dL
To differentiate the natural logarithm, we use the chain rule: d(ln(u))/du = 1/u. Applying this, we get:
MPL = 4 (1/L) + 0 (because K is treated as a constant)
Simplifying this:
MPL = 4/L
Now let's find the marginal product of capital (MPK). We differentiate the production function with respect to capital(K):
MPK = d(4 ln L + 10 K^(1/2)) / dK
Using the power rule, we differentiate K^(1/2) to get:
MPK = 0 (because the derivative of a constant is zero) + 1/2 * 10 * K^(-1/2)
Simplifying this:
MPK = 5 / √K
The average product of labor (APL) is the total output divided by the amount of labor used:
APL = (4 ln L + 10 K^(1/2)) / L
The average product of capital (APK) is the total output divided by the amount of capital used:
APK = (4 ln L + 10 K^(1/2)) / K
Regarding the second part of the question, the marginal product of labor (MPL) is not directly affected by changes in capital (K). This means that increasing or decreasing capital will not change the rate at which additional units of labor contribute to output. The MPL only depends on the amount of labor used (L) and not on the amount of capital (K).
b. To find the production position and combinations of inputs for when total costs are equal to 300 and 700, we can equate the total cost function (TC) with the given costs and solve for the inputs.
When TC = 300:
10L + 20K = 300
You would need to determine specific numerical values for this equation by either having more information or assuming some particular values. By solving this equation, you can find corresponding combinations of labor and capital inputs that result in a total cost of 300.
When TC = 700:
10L + 20K = 700
Similarly, you would need additional information or assumptions to solve this equation. By finding solutions for L and K, you can identify the combinations of inputs that result in a total cost of 700.