Question

Consider an economy with two individuals A and B, with utility functions UA = min[x^A, 2y^A] for A and UB= min[2x^B, y^B] for B and initial endowments given by WA= (1,0) and WB=(0,1). Check if goods are gross substitute and using this information comment whether the Walrasian equilibrium will be unique in this context.

Answer 1

$xa=2ya\\ 2xb=yb\\ xa+xb=1\\ ya+yb=1\\$

Putting

$\frac{xa}{2}+2xb=1$ as well by putting above two in y constraint

$2xa+2xb=2\\ \frac{3xa}{2}=1\\ xa=\frac{2}{3}\\ ya=\frac{1}{3}\\ xb=\frac{1}{3}\\$

$yb=\frac{2}{3}$ at competitive equilibrium

This shows $p1=p2=1$

Walsarian equilibrium is unique.

Goods are gross substitute as price of good 1 rises leads to demand for good 2 falls as we have to consume in fixed proportion