DSE 2019 Question 5: Demand theory and elasticity

Consider utility functions $u_{1}(x,y)=\begin{cases}2x & \text{if }y/x>2\\ \max\{x,y\} & \text{if }y/x\in[1/2,2]\\ 2y & \text{if }y/x<1/2\end{cases}$ and $u_{2}(x,y)=\begin{cases}2x & \text{if }y/x>2\\ x+y & \text{if }y/x\in[1/2,2]\\ 2y & \text{if }y/x<1/2\end{cases}$. Let $p_{x}>0$ and $p_{y}>0$ be the prices of goods $x$ and $y$, and $w>0$ wealth. For $i=1,2$, let $h_{i}(p_{x},p_{y},U)$ denote the set of solutions of: choose $x>0$ and $y>0$ to minimise $p_{x}x+p_{y}y$ subject to $u_{i}(x,y)\ge U$. Let $e_{i}(p_{x},p_{y},U)=p_{x}X+p_{y}Y$, where $(X,Y)\in h_{i}(p_{x},p_{y},U)$. Then