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: $x_1 \left( p_1 + \frac{p_1^2}{p_2} \right) = y.$