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Suppose on the x axis we have shelter and on the y axis we have composite goods. Now, if the price of shelter increases, the optimal bundle changes from point A to point D. Standard textbook tells me to draw a line parallel to the new budget line B1, which is tangent to the indifference curve I0. In this way we can get the substitution effect. The income effect follows. My question is, can we reverse the two procedures and measure the income effect first by drawing a line parallel to the original budget line and tangent to the new indifference line?image from Robert Frank's Microeconomics and Behavior Thanks!

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    $\begingroup$ Is the price of the shelter decreasing or increasing? The x intercept decreases from curve $B_0$ to curve $B_1$. So shouldn't it be increasing? $\endgroup$ – Dayne Oct 1 '20 at 16:19
  • $\begingroup$ @Dayne You are correct. Thank you for pointing that out! $\endgroup$ – Gracie Oct 1 '20 at 23:22
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You can definitely do so! There are two versions of the income effect. The one that you are referring to is called the Equivalent variation. The two ways of looking at the problem were introduced by Sir John Richard Hicks.
To see the impact of a price change in monetary terms, we ask how much money should have been taken before the price change to leave a consumer at the same utility level he attains after the price change.
This is the same thing as you had asked: can we reverse the two procedures and measure the income effect first by drawing a line parallel to the original budget line and tangent to the new indifference line?
I hope this has helped but if you wish to explore further, see the 'Consumer Surplus' chapter in Hal Varian's 'Intermediate Microeconomics'.

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