The given production function is:

$F(A, K, L) = A(K^{1/2} + L^{1/2})^2$

To find the Marginal Product of Labor (MPL) and the Marginal Product of Capital (MPK), we need to take the partial derivatives of the production function with respect to $L$ and $K$, respectively.

Marginal Product of Labor (MPL)

The MPL is the partial derivative of the production function with respect to $L$.

1. Start with the production function: $F(A, K, L) = A(K^{1/2} + L^{1/2})^2$

2. Apply the chain rule to differentiate with respect to $L$: $\frac{\partial F}{\partial L} = A \cdot 2(K^{1/2} + L^{1/2}) \cdot \frac{\partial}{\partial L}(K^{1/2} + L^{1/2})$

3. Differentiate $K^{1/2} + L^{1/2}$ with respect to $L$, noting that $K^{1/2}$ is a constant: $\frac{\partial}{\partial L}(K^{1/2} + L^{1/2}) = \frac{1}{2L^{1/2}}$

4. So, the Marginal Product of Labor (MPL) is: $\text{MPL} = A \cdot 2(K^{1/2} + L^{1/2}) \cdot \frac{1}{2L^{1/2}} = A \frac{K^{1/2} + L^{1/2}}{L^{1/2}}$

   Simplifying: $\text{MPL} = A \left( \frac{K^{1/2}}{L^{1/2}} + 1 \right)$

Marginal Product of Capital (MPK)

The MPK is the partial derivative of the production function with respect to $K$.

1. Differentiate $F(A, K, L)$ with respect to $K$: $\frac{\partial F}{\partial K} = A \cdot 2(K^{1/2} + L^{1/2}) \cdot \frac{\partial}{\partial K}(K^{1/2} + L^{1/2})$

2. Differentiate $K^{1/2} + L^{1/2}$ with respect to $K$, noting that $L^{1/2}$ is a constant: $\frac{\partial}{\partial K}(K^{1/2} + L^{1/2}) = \frac{1}{2K^{1/2}}$

3. So, the Marginal Product of Capital (MPK) is: $\text{MPK} = A \cdot 2(K^{1/2} + L^{1/2}) \cdot \frac{1}{2K^{1/2}} = A \frac{K^{1/2} + L^{1/2}}{K^{1/2}}$

   Simplifying: $\text{MPK} = A \left( 1 + \frac{L^{1/2}}{K^{1/2}} \right)$

Final Results:

• MPL: $A \left( \frac{K^{1/2}}{L^{1/2}} + 1 \right)$
• MPK: $A \left( 1 + \frac{L^{1/2}}{K^{1/2}} \right)$

Would you like further details or clarifications?

Here are 5 related questions you may find interesting:

1. How would you interpret the significance of MPL and MPK in this model?
2. How do changes in $A$ (productivity) affect MPL and MPK?
3. What happens to MPL and MPK as $L$ or $K$ increases significantly?
4. How does the Cobb-Douglas production function compare to this form?
5. Can you calculate MPL and MPK for specific values of $A$, $K$, and $L$?

Tip: When analyzing marginal products, pay close attention to diminishing returns as inputs increase independently.