<p><span class="math-container">$E = PQ \\
dE = PdQ + QdP
$</span></p>
<p>Given: <span class="math-container">$dE =0 $</span></p>
<p><span class="math-container">$\implies PdQ = - QdP \\ 
\implies \frac{dQ}{dP}\frac{P}{Q} =-1$</span></p>

CC BY-SA 4.0
2024-11-16T05:16:14.977Z
2024-11-16T05:16:14.977Z
shgwani2804
<p>The question is apparently trying to test the concept of total expenditure test. You can read about it <a href="https://www.studysmarter.co.uk/explanations/microeconomics/supply-and-demand/total-expenditure-test/" rel="nofollow noreferrer">here</a> or <a href="https://www.amosweb.com/cgi-bin/awb_nav.pl?s=wpd&amp;c=dsp&amp;k=demand%20elasticity%20and%20total%20expenditure" rel="nofollow noreferrer">here</a>. Using the total expenditure test  the demand is inelastic when total expenditure increases, unit elastic when it stays same, and elastic when it decreases.</p>
<p>I would particularly refer you to the graph on AmosWeb:</p>
<p><a href="https://i.sstatic.net/oflK3.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/oflK3.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2023-10-22T22:08:59.983Z
If $P = 25$ and $\Delta P = -0.2$, then $P^{\prime} = 25-0.2= 24.8$. Similarly, $Q^{\prime} = 20.25$. Multiplying them, $P^{\prime}Q^{\prime}  = 502.2$, which is not the same as $PQ = 25 * 20$. So, even though $|e_d| = 1$, then isn't total expenditure is no longer the same?
asdf
CC BY-SA 4.0
2023-10-22T22:13:50.180Z
Also, I calculated the percent change in $Q$ needed to offset the change in $P$, not $\Delta Q$.
asdf
CC BY-SA 4.0
2023-10-22T22:16:46.570Z
@asdf 1. you just calculated two changes for which total expenditure is same. That alone does not tell you what elasticity is as my answer explains. 2. By definition any change whether change to offset something or not is $\Delta Q$ so your change to offset change in P is $\Delta Q$ by definition
1muflon1
CC BY-SA 4.0
2023-10-22T22:21:33.297Z
the change I calculated is a percent change, while $\Delta Q$ is just the change in the value of $Q$. Also, isn't elasticity just defined as the percent change in demand divided by the percent change in price? In that case, finding the percent changes for $P$ and $Q$ are sufficient to determine the elasticity of demand.
asdf
CC BY-SA 4.0
2023-10-22T22:29:50.733Z
@asdf oh sorry, I did not noticed you were talking about percentages my bad. In any case the first part of my answer still applies to you.
1muflon1
CC BY-SA 4.0
2023-10-22T09:17:25.033Z
2023-10-22T22:38:41.377Z
2023-10-22T22:38:41.377Z
1muflon1
<p>I was recently given the following question: assuming that a consumer's total expenditure for a good does not change when the price of that good declines, what is the price elasticity of that consumer's demand?</p>
<p>Here's my initial thought process: Let their total expenditure <span class="math-container">$E = PQ$</span>, where <span class="math-container">$P$</span> is the price and <span class="math-container">$Q$</span> is the quantity of that good. Suppose that <span class="math-container">$P$</span> decreases by <span class="math-container">$20\%$</span>. Denote <span class="math-container">$P^{\prime}$</span> and <span class="math-container">$Q^{\prime}$</span> as the new price and quantity after the price decline. <span class="math-container">$P^{\prime} = 0.8P$</span>, so <span class="math-container">$Q^{\prime} = \frac{1}{0.8} Q$</span> to make sure the final expenditure is the same. <span class="math-container">$\frac{1}{0.8} = 1.25$</span>, which means quantity demanded increases by <span class="math-container">$25\%$</span>. Thus, the price elasticity for his demand is relatively elastic.</p>
<p>However, the correct answer is that his demand is unit elastic. Could someone explain why this is true?
Thank you!</p>

<microeconomics><elasticity><demand>
Price elasticity of demand when total expenditure is constant
asdf
# Price elasticity of demand when total expenditure is constant

(Asked by: username_0 on 2023-10-22)

I was recently given the following question: assuming that a consumer's total expenditure for a good does not change when the price of that good declines, what is the price elasticity of that consumer's demand?

Here's my initial thought process: Let their total expenditure $E = PQ$, where $P$ is the price and $Q$ is the quantity of that good. Suppose that $P$ decreases by $20\%$. Denote $P^{\prime}$ and $Q^{\prime}$ as the new price and quantity after the price decline. $P^{\prime} = 0.8P$, so $Q^{\prime} = \frac{1}{0.8} Q$ to make sure the final expenditure is the same. $\frac{1}{0.8} = 1.25$, which means quantity demanded increases by $25\%$. Thus, the price elasticity for his demand is relatively elastic.

However, the correct answer is that his demand is unit elastic. Could someone explain why this is true?
Thank you!

**Tags:** microeconomics, elasticity, demand

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## 2 answers

### Answer by username_1 on 2024-11-16. Score: 0

$E = PQ \\
dE = PdQ + QdP
$

Given: $dE =0 $

$\implies PdQ = - QdP \\
\implies \frac{dQ}{dP}\frac{P}{Q} =-1$


---

### Answer by username_2 on 2023-10-22. Score: 0

The question is apparently trying to test the concept of total expenditure test. You can read about it [here](https://www.studysmarter.co.uk/explanations/microeconomics/supply-and-demand/total-expenditure-test/) or [here](https://www.amosweb.com/cgi-bin/awb_nav.pl?s=wpd&c=dsp&k=demand%20elasticity%20and%20total%20expenditure). Using the total expenditure test the demand is inelastic when total expenditure increases, unit elastic when it stays same, and elastic when it decreases.

I would particularly refer you to the graph on AmosWeb:

[![enter image description here](https://i.sstatic.net/oflK3.png)](https://i.sstatic.net/oflK3.png)

> **Comments:**
> **username_0**: If $P = 25$ and $\Delta P = -0.2$, then $P^{\prime} = 25-0.2= 24.8$. Similarly, $Q^{\prime} = 20.25$. Multiplying them, $P^{\prime}Q^{\prime}  = 502.2$, which is not the same as $PQ = 25 * 20$. So, even though $|e_d| = 1$, then isn't total expenditure is no longer the same?  
> **username_0**: Also, I calculated the percent change in $Q$ needed to offset the change in $P$, not $\Delta Q$.  
> **username_2**: @asdf 1. you just calculated two changes for which total expenditure is same. That alone does not tell you what elasticity is as my answer explains. 2. By definition any change whether change to offset something or not is $\Delta Q$ so your change to offset change in P is $\Delta Q$ by definition  
> **username_0**: the change I calculated is a percent change, while $\Delta Q$ is just the change in the value of $Q$. Also, isn't elasticity just defined as the percent change in demand divided by the percent change in price? In that case, finding the percent changes for $P$ and $Q$ are sufficient to determine the elasticity of demand.  
> **username_2**: @asdf oh sorry, I did not noticed you were talking about percentages my bad. In any case the first part of my answer still applies to you.  

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