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Consider a simple utility function U(x,y) = x+y such that Px < Py (which means only x is being consumed at optimal point, a corner solution).

In this case, assume Px falls further. Now won't the consumption of x increase further, and won't that be due to income effect entirely?

In the book i am studying from(Hal Varian), the author assumes Py reduces so that consumption shifts entirely from x to y, and concludes that perfect substitutes have no income effect.

But clearly, doesn't it depend on which commodity's price decreases?

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You are perfectly right. The analysis in Varian's Intermediate Microeconomics is incomplete. If the consumed good doesn't switch, then the entire effect is a pure income effect. This has also been noted in the literature.

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