This post was updated on May 04, 2016; 9:41am.
Case 1 - (alpha<1)
Indifference curves are convex shaped. And since both goods get equal weights and prices are equal, x=y=(M/2p)
Case 2 - (alpha=1)
Perfect substitutes case. So any combination of x and y satisfying x+y=M
Case 3 - (alpha>1)
Indifference curves are concave to the origin (if that's the right terminology). And the indifference curve touches the budget line at the x and y axes. So either x=(M/p) or y=(M/p)
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