To find the technical rate of substitution (TRS) for this firm, we start by identifying the marginal products of labor and capital. The TRS is given by the negative ratio of the marginal product of labor to the marginal product of capital:

$\text{TRS} = -\frac{MP_L}{MP_K}$

Step 1: Define the Production Function

The production function is: $f(L, K) = 50.00 L^{0.25} K^{0.75}$

Step 2: Find the Marginal Product of Labor (MP_L)

The marginal product of labor is the partial derivative of $f(L, K)$ with respect to $L$:

$MP_L = \frac{\partial f(L, K)}{\partial L} = 50.00 \cdot 0.25 \cdot L^{0.25 - 1} \cdot K^{0.75}$ $= 50.00 \cdot 0.25 \cdot L^{-0.75} \cdot K^{0.75}$ $= 12.5 \cdot \frac{K^{0.75}}{L^{0.75}}$

Step 3: Find the Marginal Product of Capital (MP_K)

The marginal product of capital is the partial derivative of $f(L, K)$ with respect to $K$:

$MP_K = \frac{\partial f(L, K)}{\partial K} = 50.00 \cdot 0.75 \cdot L^{0.25} \cdot K^{0.75 - 1}$ $= 50.00 \cdot 0.75 \cdot L^{0.25} \cdot K^{-0.25}$ $= 37.5 \cdot \frac{L^{0.25}}{K^{0.25}}$

Step 4: Calculate the Technical Rate of Substitution (TRS)

Now, we can calculate the TRS as follows:

$\text{TRS} = -\frac{MP_L}{MP_K} = -\frac{12.5 \cdot \frac{K^{0.75}}{L^{0.75}}}{37.5 \cdot \frac{L^{0.25}}{K^{0.25}}}$

Simplifying the expression by dividing the constants and the terms with $L$ and $K$:

$= -\frac{12.5}{37.5} \cdot \frac{K^{0.75} \cdot K^{0.25}}{L^{0.75} \cdot L^{0.25}}$ $= -\frac{1}{3} \cdot \frac{K}{L}$

Conclusion

The technical rate of substitution (TRS) for this firm is:

$\text{TRS} = -\frac{1}{3} \cdot \frac{K}{L}$

Answer

The correct choice is: D. $-\frac{1}{3} \times \frac{K}{L}$

Would you like any further explanation on this solution?

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Here are some related questions for further understanding:

1. How would changes in the exponents of $L$ and $K$ in the production function affect the TRS?
2. What does the TRS signify in terms of substitution between labor and capital?
3. How would you find the TRS if the production function was Cobb-Douglas with different coefficients?
4. How can the TRS be used to determine the cost-minimizing combination of inputs?
5. How does a change in input prices affect the TRS in a firm's production decision?

Tip: Remember, in economics, the TRS represents the rate at which a firm can substitute labor for capital while keeping output constant.