<p>Consider the utility function <span class="math-container">$u(x,y)$</span> and two budget constraints <span class="math-container">$B_1: px + qy = 1$</span> and <span class="math-container">$B_2 : p_1x + qy = 1$</span> where <span class="math-container">$p_1 &lt; p$</span>. If <span class="math-container">$(x^*, y^*)$</span> and <span class="math-container">$(x_1^*, y_1^*)$</span>  maximize <span class="math-container">$u$</span> subject to <span class="math-container">$B_1$</span> and <span class="math-container">$B_2$</span> respectively, will it mean that <span class="math-container">$y_1^* \leq y^*$</span>?</p>
<p>When the price of good <span class="math-container">$X$</span> falls, it is evident that <span class="math-container">$X$</span> will be consumed more and so <span class="math-container">$x_1^* &gt; x^*$</span>. I was wondering how <span class="math-container">$y^*$</span> is influenced and it seems <span class="math-container">$y_1^* \leq y^*$</span> but I am not sure. Any help will be appreciated. ^_^</p>
<p>Assume <a href="https://economics.stackexchange.com/a/17853">rationality</a> and convexity.</p>

CC BY-SA 4.0
2022-07-06T12:35:45.987Z
"*it is evident that $X$ will be consumed more*" While evident, the statement is not true, see Giffen-goods.
Giskard
CC BY-SA 4.0
2022-07-06T12:36:41.547Z
What happens to the consumed quantity of $Y$ when the price of $X$ increases depends on their relationship, see substitutes vs. complements.
Giskard
CC BY-SA 4.0
2022-07-06T12:45:35.397Z
@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. https://share.sketchpad.app/22/ab5-b2b4-924de0.png I am asking if this is possible.
Isa
CC BY-SA 4.0
2022-07-06T12:48:04.147Z
This is indeed *possible*, in fact this is what happens if the utility function is e.g., $U(x,y) = \sqrt{x} + \sqrt{y}$.
Giskard
CC BY-SA 4.0
2022-07-06T13:04:16.097Z
@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?
Isa
<microeconomics><consumer-theory><choice-theory>
Influence on the consumption of $Y$ when the price of $X$ falls
Isa
# Influence on the consumption of $Y$ when the price of $X$ falls

(Asked by: username_0 on 2022-07-06)

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](https://economics.stackexchange.com/a/17853) and convexity.

**Tags:** microeconomics, consumer-theory, choice-theory

> **Comments:**
> **username_1**: "*it is evident that $X$ will be consumed more*" While evident, the statement is not true, see Giffen-goods.  
> **username_1**: What happens to the consumed quantity of $Y$ when the price of $X$ increases depends on their relationship, see substitutes vs. complements.  
> **username_0**: @username_1 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. https://share.sketchpad.app/22/ab5-b2b4-924de0.png I am asking if this is possible.  
> **username_1**: This is indeed *possible*, in fact this is what happens if the utility function is e.g., $U(x,y) = \sqrt{x} + \sqrt{y}$.  
> **username_0**: @username_1 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?  

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## 0 answers

