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Let's solve the problem step-by-step.

### Given:
The utility function is:
\[
u(x_1, x_2) = \sqrt{x_1} + \sqrt{x_2}.
\]

We are asked to:

1. **Compute the demand functions** \( x_i(p_1, p_2, y) \) for \( i = 1, 2 \).
2. **Compute the substitution term** in the Slutsky equation for the effects on \( x_1 \) of changes in \( p_2 \).

### Solution:

#### (a) Demand Functions

1. **Set up the utility maximization problem** with the budget constraint. We aim to maximize:
   \[
   u(x_1, x_2) = \sqrt{x_1} + \sqrt{x_2}
   \]
   subject to the budget constraint:
   \[
   p_1 x_1 + p_2 x_2 = y,
   \]
   where \( p_1 \) and \( p_2 \) are the prices of goods \( x_1 \) and \( x_2 \), respectively, and \( y \) is the income.

2. **Form the Lagrangian** for this constrained optimization:
   \[
   \mathcal{L} = \sqrt{x_1} + \sqrt{x_2} + \lambda (y - p_1 x_1 - p_2 x_2).
   \]

3. **Take the first-order conditions** by differentiating with respect to \( x_1 \), \( x_2 \), and \( \lambda \):
   - With respect to \( x_1 \):
     \[
     \frac{\partial \mathcal{L}}{\partial x_1} = \frac{1}{2\sqrt{x_1}} - \lambda p_1 = 0.
     \]
   - With respect to \( x_2 \):
     \[
     \frac{\partial \mathcal{L}}{\partial x_2} = \frac{1}{2\sqrt{x_2}} - \lambda p_2 = 0.
     \]
   - With respect to \( \lambda \):
     \[
     \frac{\partial \mathcal{L}}{\partial \lambda} = y - p_1 x_1 - p_2 x_2 = 0.
     \]

4. **Solve the system of equations**:
   - From the first two conditions, we have:
     \[
     \frac{1}{2\sqrt{x_1}} = \lambda p_1 \quad \text{and} \quad \frac{1}{2\sqrt{x_2}} = \lambda p_2.
     \]
   - Equate these to find a relationship between \( x_1 \) and \( x_2 \):
     \[
     \frac{1}{2\sqrt{x_1} p_1} = \frac{1}{2\sqrt{x_2} p_2},
     \]
     which simplifies to
     \[
     \frac{\sqrt{x_2}}{\sqrt{x_1}} = \frac{p_1}{p_2}.
     \]
   - Squaring both sides, we get:
     \[
     \frac{x_2}{x_1} = \frac{p_1^2}{p_2^2}.
     \]
   - Therefore,
     \[
     x_2 = x_1 \frac{p_1^2}{p_2^2}.
     \]

5. **Substitute \( x_2 \) into the budget constraint**:
   \[
   p_1 x_1 + p_2 \left( x_1 \frac{p_1^2}{p_2^2} \right) = y.
   \]
   Simplifying, we get:
   \[
   x_1 \left( p_1 + \frac{p_1^2}{p_2} \right) = y.
   \]


Consider the utility function, u(x1, x2) = (x1)^(1/2) + (x2)^(1/2).
(a) Compute the demand functions, xi(p1, p2, y), i = 1, 2.
(b) Compute the substitution term in the Slutsky equation for the effects on x1 of changes in p2.

Calculate the demand functions for a given utility function, u(x1, x2) = (x1)^(1/2) + (x2)^(1/2), and explore the Slutsky equation's substitution term effect on x1 from changes in p2. This solution includes utility maximization techniques and applies the Slutsky equation.

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