Consider the following production function 𝑌 = 𝐹(𝐾, 𝐿) = 𝐾 0.75𝐿 0.75. The marginal rate of technical substitution (of K for L). a) ½ b) 𝐾 2 𝐿 2 c) 𝐾 𝐿 d) √ 𝐾 �

The production function given is:

\[
Y = F(K, L) = K^{0.75} L^{0.75}
\]

To find the **Marginal Rate of Technical Substitution (MRTS)** of capital (K) for labor (L), we use the ratio of the marginal products of capital and labor. The MRTS is defined as:

\[
\text{MRTS} = \frac{MP_K}{MP_L}
\]

where \( MP_K \) is the marginal product of capital and \( MP_L \) is the marginal product of labor.

### Step 1: Compute the Marginal Product of Capital (MP_K)
The marginal product of capital is the partial derivative of the production function with respect to \( K \):

\[
MP_K = \frac{\partial Y}{\partial K} = 0.75 K^{0.75 - 1} L^{0.75} = 0.75 K^{-0.25} L^{0.75}
\]

### Step 2: Compute the Marginal Product of Labor (MP_L)
Similarly, the marginal product of labor is the partial derivative of the production function with respect to \( L \):

\[
MP_L = \frac{\partial Y}{\partial L} = 0.75 K^{0.75} L^{0.75 - 1} = 0.75 K^{0.75} L^{-0.25}
\]

### Step 3: Find the MRTS (K for L)
The MRTS is the ratio of \( MP_K \) to \( MP_L \):

\[
\text{MRTS} = \frac{MP_K}{MP_L} = \frac{0.75 K^{-0.25} L^{0.75}}{0.75 K^{0.75} L^{-0.25}} = \frac{K^{-0.25} L^{0.75}}{K^{0.75} L^{-0.25}}
\]

Simplifying the expression:

\[
\text{MRTS} = \frac{L^{0.75} L^{0.25}}{K^{0.25} K^{0.75}} = \frac{L}{K}
\]

Thus, the MRTS of \( K \) for \( L \) is:

\[
\boxed{\frac{K}{L}}
\]

Therefore, the correct option is:

\[
\boxed{c) \frac{K}{L}}
\]

---

Do you have any questions or would you like more details?

Here are some follow-up questions for you to explore:

1. What is the economic interpretation of the Marginal Rate of Technical Substitution?
2. How does the MRTS change as the production inputs change in the context of diminishing returns?
3. What happens to output if we increase both capital and labor proportionally in this production function?
4. How would the MRTS differ if the exponents in the production function were different?
5. What role does MRTS play in optimizing production costs?

**Tip:** The Marginal Rate of Technical Substitution tells us how much of one input (e.g., labor) can be substituted by another (e.g., capital) while keeping output constant.

Consider the following production function Y = F(K, L) = K^0.75 L^0.75. The marginal rate of technical substitution (of K for L). a) 1/2 b) K^2/L^2 c) K/L d) √K/L

Explore the Marginal Rate of Technical Substitution (MRTS) for the production function Y = K^0.75 L^0.75. Learn how to calculate MRTS by finding the ratio of marginal products of capital and labor. Step-by-step solution provided, including key formulas and concepts.

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