Answer :
The marginal product of capital in the given production function is represented by 13/√K, where K is the amount of physical capital employed.
To compute the marginal product of capital, we differentiate the production function with respect to capital (K) while holding the number of workers (L) constant. The resulting expression represents the marginal product of capital, indicating the additional output produced when one additional unit of capital is employed.
The production function is given as [tex]Y = 2K^{0.5}L^{0.5}[/tex] where Y represents output, K is physical capital, and L is the number of workers. To compute the marginal product of capital (MPK), we differentiate the production function with respect to K, while holding L constant.
Taking the partial derivative of Y with respect to K, we have:
∂Y/∂K = [tex]0.5 \cdot 2 \cdot K^{(0.5-1)} \cdot L^{0.5}[/tex]
Simplifying further, we have:
∂Y/∂K = [tex]K^{-0.5} * L^{0.5}[/tex]
Since we are interested in the marginal product of capital when L is fixed at 13 workers, we substitute L = 13 into the expression:
∂Y/∂K = [tex]K^{-0.5} \cdot 13^{0.5}[/tex]
Simplifying the equation further, we have:
∂Y/∂K = 13/√K. Therefore, the marginal product of capital is given by 13 divided by the square root of K, where K represents the amount of physical capital employed by the firm.
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