A consumer has utility function u(x1, x2) = max {0.5 x1, 0.5 x2, min{x1, x2}}. She has some positive income y, and faces positive prices p1, p2 for goods 1 and 2 respectively. Supp... A consumer has utility function u(x1, x2) = max {0.5 x1, 0.5 x2, min{x1, x2}}. She has some positive income y, and faces positive prices p1, p2 for goods 1 and 2 respectively. Suppose p2 = 1. There exists a lowest price p1 such that if p1 > p1 then the unique utility maximizing choice is to buy ONLY good 2. Then p1 is __________ (in integer). Read more

Steps to Solve

1. Identify the Utility Function

   The utility function given is $$ u(x_1, x_2) = \max {0.5 x_1, 0.5 x_2, \min {x_1, x_2}} $$ This function suggests that the consumer derives utility from the amounts of goods $x_1$ and $x_2$, with diminishing marginal utility for each.

2. Determine the Budget Constraint

   The consumer has a positive income $y$ and faces prices $p_1$ and $p_2$. With $p_2 = 1$, the budget constraint is given by: $$ p_1 x_1 + x_2 \leq y $$

3. Analyze Pricing Conditions

   To find the price $p_1$ where the consumer would choose only good 2, we need to analyze the situation when the utility from $x_2$ exceeds that of $x_1$. This occurs when: $$ 0.5 x_2 > 0.5 x_1 ; \text{and} ; 0.5 x_2 > \min {x_1, x_2} $$

4. Setting Up the Equations

   Since we are focusing on purchasing only good 2, the utility simplifies to: $$ u(0, x_2) = 0.5 x_2 $$ with $x_1 = 0$.

5. Determine When to Buy Only Good 2

   The consumer will buy only good 2 when the entire budget can be spent on it: $$ x_2 = y $$ leading back to the budget constraint: $$ p_1 \cdot 0 + y \cdot 1 \leq y $$

6. Equating Prices for Conditions

   To check the threshold for $p_1$: If the consumer buys both goods, we have: $$ y = p_1 x_1 + x_2 $$ To maximize with $x_1 = 0$, the price at which the consumer switches to good 2 alone satisfies the inequality: $$ p_1 > 2 \Rightarrow p_1 = 2 $$ Hence, if $p_1 > 2$, the utility maximization favors purchasing only good 2.