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Consider the utility function $u(x,y)$ and two budget constraints $B_1: px + qy = 1$ and $B_2 : p_1x + qy = 1$ where $p_1 < p$. If $(x^*, y^*)$ and $(x_1^*, y_1^*)$ maximize $u$ subject to $B_1$ and $B_2$ respectively, will it mean that $y_1^* \leq y^*$?

When the price of good $X$ falls, it is evident that $X$ will be consumed more and so $x_1^* > x^*$. I was wondering how $y^*$ is influenced and it seems $y_1^* \leq y^*$ but I am not sure. Any help will be appreciated. ^_^

Assume rationality and convexity.

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  • $\begingroup$ "it is evident that $X$ will be consumed more" While evident, the statement is not true, see Giffen-goods. $\endgroup$
    – Giskard
    Commented Jul 6, 2022 at 12:35
  • $\begingroup$ What happens to the consumed quantity of $Y$ when the price of $X$ increases depends on their relationship, see substitutes vs. complements. $\endgroup$
    – Giskard
    Commented Jul 6, 2022 at 12:36
  • $\begingroup$ @Giskard If this helps, I drew it. The red point belongs to the new budget constraint and its $y$ coordinate is more than that of the blue point. share.sketchpad.app/22/ab5-b2b4-924de0.png I am asking if this is possible. $\endgroup$
    – Isa
    Commented Jul 6, 2022 at 12:45
  • $\begingroup$ This is indeed possible, in fact this is what happens if the utility function is e.g., $U(x,y) = \sqrt{x} + \sqrt{y}$. $\endgroup$
    – Giskard
    Commented Jul 6, 2022 at 12:48
  • $\begingroup$ @Giskard I read about perfect complements. Apart from $\sqrt{x} + \sqrt{y}$, I think perfect complements $\min(x,y)$ will also satisfy it. Are complements Gidden goods then? $\endgroup$
    – Isa
    Commented Jul 6, 2022 at 13:04

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