Utility Function: $U(x_1, x_2) = \max\{x_1, 2x_2\}$

1. Interpretation of the Utility Function:

This utility function represents preferences where the consumer values $x_1$ and $x_2$ asymmetrically. Specifically, the utility is the maximum of $x_1$ and $2x_2$, meaning that:

• $x_1$ represents the consumption of good 1.
• $2x_2$ represents the consumption of good 2, but doubled in value compared to $x_1$.

Thus, for any bundle $(x_1, x_2)$, the utility is determined by the larger of $x_1$ and $2x_2$. This implies that the consumer is indifferent between bundles where $x_1 = 2x_2$, and the utility increases as either $x_1$ or $2x_2$ increases.

2. Indifference Curves:

The indifference curves for this utility function are determined by solving for $U = \max\{x_1, 2x_2\} = k$, where $k$ is a constant utility level. Therefore, for each level $k$:

• When $x_1 \geq 2x_2$, the utility is $x_1 = k$, so the indifference curve is a vertical line where $x_1 = k$.
• When $2x_2 \geq x_1$, the utility is $2x_2 = k$, so the indifference curve is a horizontal line where $2x_2 = k$, or equivalently $x_2 = \frac{k}{2}$.

Thus, the indifference curves are composed of two segments:

• A vertical line for $x_1 = k$, when $x_1 \geq 2x_2$.
• A horizontal line for $x_2 = \frac{k}{2}$, when $2x_2 \geq x_1$.

3. Direction of Preferred Bundles:

The consumer prefers bundles where either $x_1$ or $x_2$ is larger, with $2x_2$ contributing equally to utility as $x_1$. Therefore:

• Above the indifference curve, where either $x_1$ or $x_2$ is larger, utility is higher, and these bundles are preferred.
• The arrows indicating preferred directions would point upward and to the right.

4. Strict Convexity:

Strict convexity of preferences implies that the consumer strictly prefers a convex combination (i.e., a weighted average) of two bundles over the individual bundles themselves. However, in this case:

• The utility function $U(x_1, x_2) = \max\{x_1, 2x_2\}$ leads to "kinked" indifference curves. Along the line $x_1 = 2x_2$, preferences switch from valuing $x_1$ to valuing $2x_2$.
• Because of these kinks, the preferences are not strictly convex. For example, the consumer is indifferent between bundles on the line where $x_1 = 2x_2$, but a convex combination of two such bundles will lie on the same indifference curve and not be strictly preferred.

Thus, the preferences are not strictly convex due to the presence of "corner" solutions where either $x_1$ or $2x_2$ dominates utility.

Graph Explanation:

• Each indifference curve has two segments:
  • A vertical part at $x_1 = k$ (where $x_1 \geq 2x_2$).
  • A horizontal part at $x_2 = \frac{k}{2}$ (where $2x_2 \geq x_1$).
• The curves are kinked at the line $x_1 = 2x_2$.
• Arrows should point toward regions where either $x_1$ or $2x_2$ is increasing.

Would you like me to generate a visual representation of these indifference curves with arrows?

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Follow-up Questions:

1. What happens if the utility function treats $x_1$ and $x_2$ symmetrically, i.e., $U(x_1, x_2) = \max\{x_1, x_2\}$?
2. How do indifference curves look for $U(x_1, x_2) = x_1 + x_2$?
3. Can preferences be non-convex but still satisfy monotonicity?
4. What economic interpretation can you give for the non-convexity of these preferences?
5. How does strict convexity relate to the concept of diminishing marginal rates of substitution?

Tip:

Strict convexity of preferences typically means that a consumer prefers averages of bundles over extremes, implying smooth and strictly convex indifference curves.