Answer to Question #215332 in Microeconomics for Krish

Question #215332
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.
1
Expert's answer
2021-07-12T11:48:08-0400

"xa=2ya\\\\\n\n2xb=yb\\\\\n\nxa+xb=1\\\\\n\nya+yb=1\\\\"

Putting

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

"2xa+2xb=2\\\\\n\n\\frac{3xa}{2}=1\\\\\n\nxa=\\frac{2}{3}\\\\\n\nya=\\frac{1}{3}\\\\\n\nxb=\\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


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