Cant find equations for solving for MP and AP.

Suppose firms A and B each make T-shirts. Firm A's production function is q = L0.5 K0.5. Firm B's production function is q = 1.2 L0.5 K0.5.

a.If the two firms each hire the same amounts of capital and labor, compare the two firms in terms of their APL.

b.Given their MPL of 0.5 L-0.5 K0.5 and 0.6 L-0.5 K0.5 respectively, state which one can increase productivity of their labor force faster by increasing capital the same amount? .

c.Suppose Firm A has 125 units of K and 5 units of L. Firm B has 625 units of K and 25 units of L. Find out their APL.’s and MPL’s. What sets the difference between the firms in terms of average and marginal product of labor, their scale, or their technology?

To solve for MP (marginal product) and AP (average product), we need to understand the production functions and the relationship between inputs (labor and capital) and output (quantity produced).

a. To compare the firms in terms of their APL (average product of labor), we need to calculate the average product of labor for each firm. The formula for APL is APL = q/L.

For Firm A:
APL = q/L = (L^0.5 * K^0.5) / L = (L^0.5 * K^0.5) * L^-1 = K^0.5.

For Firm B:
APL = q/L = (1.2 * L^0.5 * K^0.5) / L = (1.2 * L^0.5 * K^0.5) * L^-1 = 1.2 * K^0.5.

Comparing the two firms in terms of APL, we can see that Firm A has an APL of K^0.5, while Firm B has an APL of 1.2 * K^0.5. Firm B has a higher APL, indicating that, on average, Firm B is more productive in terms of output per unit of labor.

b. To determine which firm can increase productivity of their labor force faster by increasing capital, we need to compare their MPL (marginal product of labor) values. The MPL represents the additional output produced by employing an additional unit of labor while keeping capital constant.

For Firm A:
MPL = 0.5 * L^(-0.5) * K^0.5.

For Firm B:
MPL = 0.6 * L^(-0.5) * K^0.5.

To compare the impact of increasing capital, we can ignore the constant values (0.5 and 0.6) and focus on the exponents of L and K. Both firms have the same exponent for K^0.5, so increasing capital will have the same effect on their MPL. Therefore, the rate at which the productivity of their labor force increases by increasing capital will be the same for both firms.

c. To calculate the APL and MPL for each firm given their specific quantities of capital and labor, substitute the values of K and L into the production functions.

For Firm A:
APL = q/L = (5^0.5 * 125^0.5) / 5 = 5 * 5^0.5 = 11.18.
MPL = 0.5 * 5^(-0.5) * 125^0.5 = 0.5 * 0.447 * 11.18 = 2.49.

For Firm B:
APL = q/L = (1.2 * 25^0.5 * 625^0.5) / 25 = 1.2 * 25 * 25^0.5 = 30.
MPL = 0.6 * 25^(-0.5) * 625^0.5 = 0.6 * 0.2 * 25 = 3.

The APL for Firm A is 11.18 and for Firm B is 30. Therefore, Firm B has a higher APL, indicating it has a higher average output per unit of labor.

The MPL for Firm A is 2.49 and for Firm B is 3. Therefore, Firm B has a higher MPL, indicating that an additional unit of labor in Firm B will produce more output compared to Firm A.

In terms of the difference in average and marginal product of labor between the two firms, the difference is primarily due to their technology. Firm B's production function includes a higher constant (1.2) compared to Firm A's production function (which has an implicit constant of 1), contributing to higher average and marginal product values. Additionally, the scale of production, as indicated by the quantities of capital and labor employed, also plays a role in the difference. Firm B has higher quantities of capital and labor, which can contribute to higher productivity.