F(A,K,L)=A(K^(1/2) + L^(1/2))^2
Find MPL and MPK

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.

F(A,K,L)=A(K^(1/2) + L^(1/2))^2 Find MPL and MPK

Learn how to calculate the Marginal Product of Labor (MPL) and Marginal Product of Capital (MPK) for the production function F(A,K,L)=A(K^(1/2) + L^(1/2))^2. This step-by-step guide includes the use of partial derivatives and the chain rule. Suitable for undergraduate students studying economics or mathematics.

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