Question

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 𝑝̅̅₁ such that if 𝑝₁ > 𝑝̅̅₁ then the unique utility maximizing choice is to buy ONLY good 2. Then 𝑝̅̅₁ is _________ (in integer).

Answer 1

Correct Answer: 2

Given:
Utility \[ u(x_1,x_2)=\max\{0.5x_1,\,0.5x_2\}+\min\{x_1,x_2\}, \] income \(y>0\), prices \(p_1\) and \(p_2=1\). We seek the smallest \(\bar p_1\) such that for every \(p_1>\bar p_1\) the unique utility-maximiser buys only good 2. 
 
Step 1 — utility for corner bundles
If buy only good 2: \(x_1=0,\;x_2=y\) so \[ u(0,y)=\max\{0,\,0.5y\}+0 = 0.5y. \] If buy only good 1: \(x_1=\tfrac{y}{p_1},\;x_2=0\) so \[ u\Big(\tfrac{y}{p_1},0\Big)=0.5\cdot\frac{y}{p_1}. \] Clearly \(u(\tfrac{y}{p_1},0)1\), so the main competitor to the pure-good-2 bundle is an interior/mixed bundle. 

Step 2 — consider mixed bundles
From the budget \(p_1x_1+x_2=y\) we write \(x_2=y-p_1x_1\). There are two regions: - If \(x_1\le x_2\) then \(u=x_1+0.5x_2\). Substituting \(x_2\) gives \[ u(x_1)=0.5y + x_1\big(1-\tfrac{1}{2}p_1\big), \] which is linear in \(x_1\). Feasible \(x_1\) must also satisfy \(x_1\le x_2\), which (with the budget) implies \(x_1\le \dfrac{y}{1+p_1}.\) - If \(x_1\ge x_2\) then \(u=0.5x_1+x_2 = y + x_1\big(\tfrac12-p_1\big)\), also linear; feasibility gives \(x_1\ge \dfrac{y}{1+p_1}\). In both cases the candidate interior solution occurs at the boundary where \(x_1=x_2=\dfrac{y}{1+p_1}\). At that point \[ u_{\text{mix}} = 1.5\,x_1 = 1.5\cdot\frac{y}{1+p_1}. \] 

Step 3 — compare mixed bundle with only good 2
We require the pure good-2 utility to be at least as large as any mixed bundle, i.e. \[ 0.5y \ge 1.5\cdot\frac{y}{1+p_1}. \] Divide by \(y>0\) and solve: \[ 0.5 \ge \frac{1.5}{1+p_1} \quad\Rightarrow\quad 1+p_1 \ge 3 \quad\Rightarrow\quad p_1 \ge 2. \] For \(p_1>2\) the inequality is strict and the unique maximiser is to buy only good 2. At \(p_1=2\) the consumer is indifferent. 

Answer: \(\boxed{\bar p_1 = 2}\)