Let's go through each question step-by-step.

Given:

The production function is $f(x_1, x_2) = x_1^{1/2} x_2^{3/2}$.

1.1. Marginal Product of Input 1

The marginal product of input $x_1$ (MP1) is the partial derivative of $f(x_1, x_2)$ with respect to $x_1$: $\text{MP}_1 = \frac{\partial f(x_1, x_2)}{\partial x_1} = \frac{1}{2} x_1^{-\frac{1}{2}} x_2^{\frac{3}{2}} = \frac{x_2^{3/2}}{2\sqrt{x_1}}$

1.2. Is the Marginal Product of Input 1 Increasing or Decreasing with $x_1$?

To determine if MP1 is increasing or decreasing with respect to $x_1$, we need to examine the sign of the second partial derivative of $f$ with respect to $x_1$: $\frac{\partial^2 f}{\partial x_1^2} = \frac{-1}{4} x_1^{-\frac{3}{2}} x_2^{\frac{3}{2}} = -\frac{x_2^{3/2}}{4 x_1^{3/2}}$ Since this derivative is negative, MP1 is decreasing with $x_1$.

1.3. Marginal Product of Input 2

The marginal product of input $x_2$ (MP2) is the partial derivative of $f(x_1, x_2)$ with respect to $x_2$: $\text{MP}_2 = \frac{\partial f(x_1, x_2)}{\partial x_2} = \frac{3}{2} x_1^{\frac{1}{2}} x_2^{\frac{1}{2}} = \frac{3 x_1^{1/2} x_2^{1/2}}{2}$

1.4. Is the Marginal Product of Input 2 Increasing or Decreasing with $x_2$?

To determine if MP2 is increasing or decreasing with respect to $x_2$, we look at the sign of the second partial derivative of $f$ with respect to $x_2$: $\frac{\partial^2 f}{\partial x_2^2} = \frac{3}{4} x_1^{\frac{1}{2}} x_2^{-\frac{1}{2}} = \frac{3 x_1^{1/2}}{4 x_2^{1/2}}$ Since this derivative is positive, MP2 is decreasing with $x_2$.

1.5. Technical Rate of Substitution (TRS) and Diminishing TRS

The technical rate of substitution (TRS) between $x_1$ and $x_2$ is given by the ratio of the marginal products: $\text{TRS} = \frac{\text{MP}_1}{\text{MP}_2} = \frac{\frac{x_2^{3/2}}{2 \sqrt{x_1}}}{\frac{3 x_1^{1/2} x_2^{1/2}}{2}} = \frac{x_2}{3 x_1}$ Since $\text{TRS}$ decreases as $x_1$ increases, the production function features diminishing TRS.

1.6. Returns to Scale

To determine returns to scale, examine what happens when both inputs are scaled by a factor $t$: $f(t x_1, t x_2) = (t x_1)^{1/2} (t x_2)^{3/2} = t^{1/2 + 3/2} x_1^{1/2} x_2^{3/2} = t^2 f(x_1, x_2)$ Since scaling both inputs by $t$ results in output being scaled by $t^2$, this production function exhibits constant returns to scale.

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Would you like further details or explanations on any of these points? Here are a few related questions for further exploration:

1. How would the TRS change if the exponents on $x_1$ and $x_2$ were different?
2. What impact does decreasing marginal product have on production efficiency?
3. How does diminishing TRS affect input choices in production?
4. Can you illustrate the concept of returns to scale using a graphical representation?
5. How do returns to scale differ in production functions with different forms, such as Cobb-Douglas?

Tip: To understand marginal productivity, try exploring how small changes in each input affect total production, as it provides insights into resource allocation efficiency.