The problem states that the production function is given by $Q = K^2L^2$, where $Q$ is the quantity of output, $K$ is capital, and $L$ is labor. The question asks about the nature of the marginal product of capital (MP of capital). We need to determine whether the marginal product of capital is diminishing, increasing, constant, or undetermined.

1. Problem Description

The problem states that the production function is given by $Q = K^2L^2$, where $Q$ is the quantity of output, $K$ is capital, and $L$ is labor. The question asks about the nature of the marginal product of capital (MP of capital). We need to determine whether the marginal product of capital is diminishing, increasing, constant, or undetermined.

2. Solution Steps

The marginal product of capital (MPK) is the change in output resulting from a one-unit change in capital, holding labor constant. It is calculated as the partial derivative of the production function with respect to capital.

First, we find the marginal product of capital (MPK):

$MPK = \frac{\partial Q}{\partial K} = \frac{\partial (K^2L^2)}{\partial K} = 2KL^2$

Now, we need to determine how the marginal product of capital changes as the amount of capital increases. To do this, we take the derivative of the MPK with respect to K:

$\frac{\partial MPK}{\partial K} = \frac{\partial (2KL^2)}{\partial K} = 2L^2$

Since $L^2$ is always positive (assuming $L$ is non-zero), $2L^2$ is always positive. This means that as capital (K) increases, the marginal product of capital (MPK) also increases. Therefore, the marginal product of capital is increasing.

3. Final Answer

b. increasing