<p>The formula for cross price elasticity of demand is:</p>
<p><span class="math-container">$e_{x,p_y} = \frac{\partial x}{\partial p_y} \cdot \frac{p_y}{x}$</span></p>
<p>In my notation it’s measuring the percentage change in demand for good <span class="math-container">$x$</span> per <span class="math-container">$1\%$</span> change in the price of good <span class="math-container">$y$</span> on the margin, <span class="math-container">$\textbf{keeping everything else constant}$</span>.</p>
<p>Here good <span class="math-container">$x$</span> would be beef and good <span class="math-container">$y$</span> would be lamb.</p>
<p>You’re right, price elasticity formulas are for price changes along the demand curve.</p>
<p>Shifts in demand curves don’t satisfy <span class="math-container">$\textbf{keeping everything else constant}$</span>.</p>
<p>The logic for substitute goods being defined as <span class="math-container">$e_{x,p_y} &gt; 0$</span> is as follows:</p>
<p><span class="math-container">$p_y$</span> increases <span class="math-container">$\implies$</span> moving along <span class="math-container">$y$</span>’s demand curve, less units of <span class="math-container">$y$</span> will be demanded.</p>
<p><span class="math-container">$e_{x,p_y} &gt; 0 \implies$</span> as <span class="math-container">$p_y$</span> increases, demand for good <span class="math-container">$x$</span> increases.</p>
<p>Combining these statements,</p>
<p>When <span class="math-container">$e_{x,p_y} &gt; 0$</span>, as <span class="math-container">$p_y$</span> increases, the consumer will demand more <span class="math-container">$x$</span> and less <span class="math-container">$y$</span>, i.e. the consumer is substituting good <span class="math-container">$y$</span> for good <span class="math-container">$x$</span> as a result of its price increase.</p>
<p>The substitution would happen the other way around if <span class="math-container">$p_y$</span> falls.</p>
<p>Here the substitution is a direct consequence of <span class="math-container">$p_y$</span> increasing.</p>
<p>It is possible that the shock causing the scenario your peers discuss is a change of preferences towards lamb.</p>
<p>If it was caused by let’s say a famous nutritionist publicly states that lamb is healthier than beef for whatever reason, then this event/shock would also cause the demand curve for beef to shift left.</p>
<p>The overall result of this event is that <span class="math-container">$p_y$</span> increases while demand for <span class="math-container">$x$</span> decreases and demand for <span class="math-container">$y$</span> increases.</p>

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2023-04-15T12:06:45.590Z
thank you very much i thought i was going insane for a second my own teacher was telling me that the price change can be due to shifts in demand.
j jose
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2023-04-15T12:10:05.903Z
@jjose That’s correct. Don’t forget you can accept my answer if you found it useful :)
Nicolas Torres
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2023-04-15T12:22:47.527Z
How would you answer this question [link](https://ibb.co/BNdQdH3) .Also im not sure what you mean by accept answer ,is that a function that i can do here ?
j jose
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2023-04-15T12:24:15.210Z
i know my answer was wrong but i simply didnt know what to put so i put anything , however there answer cant be right either, because the price change is due to demand atleast according to the picture
j jose
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2023-04-15T12:37:14.787Z
@jjose you accept an answer by clicking on the check mark
Nicolas Torres
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2023-04-15T12:44:30.550Z
wait you said complements shift in the direction, demand for strawberries went up, and the demand for cream fall? thats not the same direction?
j jose
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2023-04-15T12:52:55.893Z
@jjose Sorry, I got confused there, will delete that comment. My intuition says complements should shift in the same direction, giving your same wrong answer: demand for cream will rise.
Nicolas Torres
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2023-04-15T12:53:39.513Z
@jjose Do you know about utility functions?
Nicolas Torres
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2023-04-15T12:58:09.897Z
@jjose The utility function for perfect complement goods is $U(x,y) = \min(\alpha x, \beta y)$. A rational consumer would demand the quantities $x,y$ that maximize their utility subject to their budget ($I$ for income): $x p_x + y p_y = I$
Nicolas Torres
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2023-04-15T13:00:23.047Z
@jjose Let’s assume strawberries and cream are perfect complements. For simplicity, let’s assume $\alpha = \beta = 1$. $U(s,c) = \min(s,c)$
Nicolas Torres
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2023-04-15T13:07:35.877Z
no unfrotunaltey .I find this very strange i asked some other univerity economist and they said that PED price changes are movements along , but for xed it can be shifts along or movements in demand
j jose
CC BY-SA 4.0
2023-04-15T13:08:03.800Z
@jjose Here the consumer will pick a consumptions bundle where $s = c$. Substituting this equation into the budget constraint, $s p_s + c p_c = I \implies s p_s + s p_c = I \implies s = \frac{I}{p_s + p_c} \implies c = \frac{I}{p_s + p_c}$
Nicolas Torres
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2023-04-15T13:12:39.033Z
For $U(s,c) = \min(s,c), p_s = 3, p_c = 2, I = 10$, the solution would be $s=c=5$. To make it intuitive since you haven’t gotten to utility functions: try plotting a table of values for other combinations of $(s,c)$ on the budget equation/line. Check if you are getting higher or lower values of $U(s,c)$
Nicolas Torres
CC BY-SA 4.0
2023-04-15T13:18:17.703Z
Since here the consumer always picks $s = c$, if demand for $s$ shifts right, for the above equation to hold, then demand for $c$ must also shift right. Similarly, if demand for $s$ shifts left, or is $c$ the one which shifts first. We get here (perfect complements) that they should shift in the same direction.
Nicolas Torres
CC BY-SA 4.0
2023-04-15T14:02:20.100Z
but wouldn't this contradict the statement you made in your first post where you said price changes due to demand arent what the XED formulas refers to , so the price change of good x due to the demand of it increasing will not affect the demand changes of good y ?
j jose
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2023-04-15T11:37:39.357Z
2023-04-15T12:15:24.043Z
2023-04-15T12:15:24.043Z
Nicolas Torres
<p>I have been told by my peers that if lamb and beef were close substitutes, such that if the demand for lamb rises the price for lamb rises which causes the demand for beef to rise , this makes no sense to me one bit . I always referred to the price change in the XED formula as price changes along the demand curve thats the only way the arguments make sense if the price change was also due to demand shifts then we cant use the argument that PED depends on how many subs there.The reason we can say that is because price changes in both ped and xed is due to price changing along the demand curve. Please confirm if im right or not.</p>

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2023-04-15T17:37:57.703Z
You could have a curved surface in 4D space which jointly gave demand for beef and demand for lamb as a function of the price of beef and the price of lamb.  You can then take partial derivatives to help calculate price elasticities of demand and cross price elasticities of demand; these would be properties of the surface
Henry
<cross-price-elasticity>
Does the price change in the XED formula refer to price changes along the demand curve only?
j jose
# Does the price change in the XED formula refer to price changes along the demand curve only?

(Asked by: username_0 on 2023-04-15)

I have been told by my peers that if lamb and beef were close substitutes, such that if the demand for lamb rises the price for lamb rises which causes the demand for beef to rise , this makes no sense to me one bit . I always referred to the price change in the XED formula as price changes along the demand curve thats the only way the arguments make sense if the price change was also due to demand shifts then we cant use the argument that PED depends on how many subs there.The reason we can say that is because price changes in both ped and xed is due to price changing along the demand curve. Please confirm if im right or not.

**Tags:** cross-price-elasticity

> **Comments:**
> **username_1**: You could have a curved surface in 4D space which jointly gave demand for beef and demand for lamb as a function of the price of beef and the price of lamb.  You can then take partial derivatives to help calculate price elasticities of demand and cross price elasticities of demand; these would be properties of the surface  

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## 1 answers

### Answer by username_2 on 2023-04-15. Score: 1 | Accepted answer

The formula for cross price elasticity of demand is:

$e\_{x,p\_y} = \frac{\partial x}{\partial p\_y} \cdot \frac{p\_y}{x}$

In my notation it’s measuring the percentage change in demand for good $x$ per $1\%$ change in the price of good $y$ on the margin, $\textbf{keeping everything else constant}$.

Here good $x$ would be beef and good $y$ would be lamb.

You’re right, price elasticity formulas are for price changes along the demand curve.

Shifts in demand curves don’t satisfy $\textbf{keeping everything else constant}$.

The logic for substitute goods being defined as $e\_{x,p\_y} > 0$ is as follows:

$p\_y$ increases $\implies$ moving along $y$’s demand curve, less units of $y$ will be demanded.

$e\_{x,p\_y} > 0 \implies$ as $p\_y$ increases, demand for good $x$ increases.

Combining these statements,

When $e\_{x,p\_y} > 0$, as $p\_y$ increases, the consumer will demand more $x$ and less $y$, i.e. the consumer is substituting good $y$ for good $x$ as a result of its price increase.

The substitution would happen the other way around if $p\_y$ falls.

Here the substitution is a direct consequence of $p\_y$ increasing.

It is possible that the shock causing the scenario your peers discuss is a change of preferences towards lamb.

If it was caused by let’s say a famous nutritionist publicly states that lamb is healthier than beef for whatever reason, then this event/shock would also cause the demand curve for beef to shift left.

The overall result of this event is that $p\_y$ increases while demand for $x$ decreases and demand for $y$ increases.

> **Comments:**
> **username_0**: thank you very much i thought i was going insane for a second my own teacher was telling me that the price change can be due to shifts in demand.  
> **username_2**: @username_0 That’s correct. Don’t forget you can accept my answer if you found it useful :)  
> **username_0**: How would you answer this question [link](https://ibb.co/BNdQdH3) .Also im not sure what you mean by accept answer ,is that a function that i can do here ?  
> **username_0**: i know my answer was wrong but i simply didnt know what to put so i put anything , however there answer cant be right either, because the price change is due to demand atleast according to the picture  
> **username_2**: @username_0 you accept an answer by clicking on the check mark  
> **username_0**: wait you said complements shift in the direction, demand for strawberries went up, and the demand for cream fall? thats not the same direction?  
> **username_2**: @username_0 Sorry, I got confused there, will delete that comment. My intuition says complements should shift in the same direction, giving your same wrong answer: demand for cream will rise.  
> **username_2**: @username_0 Do you know about utility functions?  
> **username_2**: @username_0 The utility function for perfect complement goods is $U(x,y) = \min(\alpha x, \beta y)$. A rational consumer would demand the quantities $x,y$ that maximize their utility subject to their budget ($I$ for income): $x p_x + y p_y = I$  
> **username_2**: @username_0 Let’s assume strawberries and cream are perfect complements. For simplicity, let’s assume $\alpha = \beta = 1$. $U(s,c) = \min(s,c)$  
> **username_0**: no unfrotunaltey .I find this very strange i asked some other univerity economist and they said that PED price changes are movements along , but for xed it can be shifts along or movements in demand  
> **username_2**: @username_0 Here the consumer will pick a consumptions bundle where $s = c$. Substituting this equation into the budget constraint, $s p_s + c p_c = I \implies s p_s + s p_c = I \implies s = \frac{I}{p_s + p_c} \implies c = \frac{I}{p_s + p_c}$  
> **username_2**: For $U(s,c) = \min(s,c), p_s = 3, p_c = 2, I = 10$, the solution would be $s=c=5$. To make it intuitive since you haven’t gotten to utility functions: try plotting a table of values for other combinations of $(s,c)$ on the budget equation/line. Check if you are getting higher or lower values of $U(s,c)$  
> **username_2**: Since here the consumer always picks $s = c$, if demand for $s$ shifts right, for the above equation to hold, then demand for $c$ must also shift right. Similarly, if demand for $s$ shifts left, or is $c$ the one which shifts first. We get here (perfect complements) that they should shift in the same direction.  
> **username_0**: but wouldn't this contradict the statement you made in your first post where you said price changes due to demand arent what the XED formulas refers to , so the price change of good x due to the demand of it increasing will not affect the demand changes of good y ?  

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