The problem provided asks for an analysis of a specific preference relation $\succeq^*$ on $\mathbb{R}^2_+$, defined as:

$\begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \succeq^* \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} \quad \text{if} \quad |x_1 - x_2| \geq |y_1 - y_2|,$ where $|a|$ is the absolute value of $a$.

(a) Is $\succeq^*$ convex?

To determine whether $\succeq^*$ is convex, recall that a preference relation is convex if, for any two bundles $\mathbf{x}, \mathbf{y} \in \mathbb{R}^2_+$ such that $\mathbf{x} \succeq^* \mathbf{y}$, any convex combination of $\mathbf{x}$ and $\mathbf{y}$ (e.g., $\mathbf{z} = \lambda \mathbf{x} + (1-\lambda) \mathbf{y}, \lambda \in [0, 1]$) satisfies $\mathbf{z} \succeq^* \mathbf{y}$.

Here, the inequality condition $|x_1 - x_2| \geq |y_1 - y_2|$ must hold for all intermediate combinations. For convexity to hold, the absolute difference of any weighted average of $(x_1, x_2)$ should remain greater than or equal to the absolute difference of $(y_1, y_2)$. However, the non-linear nature of the absolute value function typically breaks convexity. For example, consider extreme cases where the components switch dominance. Thus, $\succeq^*$ is not convex.

(b) Walrasian Demand $x(p_1, p_2, w)$:

The Walrasian demand function represents the utility-maximizing bundle given prices $p_1, p_2$ and wealth $w$, under the preference $\succeq^*$. Since the utility depends only on $|x_1 - x_2|$, the consumer's optimal bundle will maximize the absolute difference between the goods, subject to the budget constraint:

$p_1 x_1 + p_2 x_2 = w.$

1. When $p_1 = p_2$: Any combination $(x_1, x_2)$ such that $x_1 + x_2 = \frac{w}{p_1}$ maximizes utility, as all pairs yield the same $|x_1 - x_2|$.

2. When $p_1 \neq p_2$: The consumer will prioritize spending on the good with the smaller price $p_i$, maximizing the absolute difference $|x_1 - x_2|$. For example, if $p_1 < p_2$, the optimal choice is to spend as much as possible on $x_1$, yielding: $x_1 = \frac{w}{p_1}, \quad x_2 = 0.$

Final Answer:

(a) $\succeq^*$ is not convex.
(b) Walrasian demand depends on prices $p_1, p_2$ and wealth $w$, as outlined above.

Would you like me to elaborate further on any part or solve an example for clarification?

Follow-up Questions:

1. How does non-convexity affect consumer behavior under this preference?
2. Can we graphically interpret the preference relation $\succeq^*$ on $\mathbb{R}^2_+$?
3. What happens to the Walrasian demand when wealth $w$ changes?
4. How would the solution differ if $|x_1 - x_2|$ were squared instead of absolute?
5. Can this preference be linked to any real-world scenarios or models?

Tip:

For non-convex preferences, always check corner solutions in utility maximization problems, as intermediate solutions may not satisfy the preference condition.