We are given a production function $Q = K^{0.5}L^{0.5}$ and asked to determine its returns to scale and the marginal productivities of capital (K) and labor (L).

1. Problem Description

We are given a production function $Q = K^{0.5}L^{0.5}$ and asked to determine its returns to scale and the marginal productivities of capital (K) and labor (L).

2. Solution Steps

First, let's determine the returns to scale. Returns to scale refer to what happens to output when we increase all inputs by the same proportion. Let's multiply both inputs $K$ and $L$ by a constant $\lambda > 0$. The new output $Q'$ will be:

$Q' = (\lambda K)^{0.5} (\lambda L)^{0.5}$

$Q' = \lambda^{0.5} K^{0.5} \lambda^{0.5} L^{0.5}$

$Q' = \lambda^{0.5+0.5} K^{0.5} L^{0.5}$

$Q' = \lambda K^{0.5} L^{0.5}$

$Q' = \lambda Q$

Since $Q' = \lambda Q$, this production function exhibits constant returns to scale.

Next, let's examine the marginal productivities of capital (MPK) and labor (MPL). The marginal product of capital is the partial derivative of the production function with respect to capital:

$MPK = \frac{\partial Q}{\partial K} = \frac{\partial (K^{0.5}L^{0.5})}{\partial K} = 0.5 K^{-0.5} L^{0.5} = 0.5 (\frac{L}{K})^{0.5}$

As K increases, $MPK$ decreases, so we have diminishing marginal productivity of capital.

The marginal product of labor is the partial derivative of the production function with respect to labor:

$MPL = \frac{\partial Q}{\partial L} = \frac{\partial (K^{0.5}L^{0.5})}{\partial L} = 0.5 K^{0.5} L^{-0.5} = 0.5 (\frac{K}{L})^{0.5}$

As L increases, $MPL$ decreases, so we have diminishing marginal productivity of labor.

3. Final Answer

The production function exhibits constant returns to scale and diminishing marginal productivities for K and L. Therefore, the answer is c. exhibits constant returns to scale and diminishing marginal productivities for K and L.