DSE 2019 Question 50: Budget constraint and consumer equilibrium

Scenario 1: Consider utility functions $u_1(x,y)=2x$ if $y/x>2$, $\max\{x,y\}$ if $y/x\in[1/2,2]$, and $2y$ if $y/x<1/2$; and $u_2(x,y)=2x$ if $y/x>2$, $x+y$ if $y/x\in[1/2,2]$, and $2y$ if $y/x<1/2$. Let $p_x>0$ and $p_y>0$ be the prices of goods $x$ and $y$, and let $w>0$ denote wealth (income). Let $m_i(p_x,p_y,w)$ denote the set of Marshallian demands for utility $u_i$ and let $v_i(p_x,p_y,w)=u_i\circ m_i(p_x,p_y,w)$. Then: