<p>Let's denote the elasticity of $X$ to $Y$ at $Y_0$ with $\eta_{X, Y}(Y_0)$ and solve for the demand ($Q$) elasticity in terms of the revenue elasticity ($R$).</p>

<p>We know that the elasticity is always given by</p>

<p>$$ \eta_{X,Y} = \frac{d X(Y)}{dY}\frac{Y}{X}$$ </p>

<p>or equivalently (showing the equivalence is a useful exercise),</p>

<p>$$ \eta_{X,Y} = \frac{d \log X(Y))}{d \log Y}$$ </p>

<p>Now, </p>

<p>$$ R(p) = p Q(p) \\
\eta_{R, p}(p) = R'(p) \frac{p}{pQ(p)}\\
\eta_{R, p}(p) = (Q(p) + pQ'(p_0) ) \frac{p}{pQ(p)} \\
\eta_{R, p}(p) = 1 + \frac{pQ'(p_0)}{Q} \\
\eta_{R, p}(p) = 1 + \eta_{Q, p}(p) \\
$$ </p>

<p>Given that the left-hand-side elasticity is $-1$ at the initial price $p_0$, the right-hand-side elasticity must be $-2$:</p>

<p>$$-1 = 1 + \eta_{Q, p}(p_0) \Rightarrow \eta_{Q, p}(p_0) = -2$$</p>

<h2>Your mistake</h2>

<p>You are being sloppy with the mathematics. First you state</p>

<p>$$ Q = \frac{TR}{P}$$ </p>

<p>But then you claim</p>

<p>$$ Q = \frac{0.8TR}{1.2P}$$ </p>

<p>These two equations just cannot be true at the same time.</p>

<p>Perhaps you want to look at initial $TR_0$, at total revenue after price change $TR_1$, and then state that $TR_1/TR_0 = 0.95$.</p>

CC BY-SA 3.0
2015-04-09T15:00:07.743Z
How did you derive the second line?
Frank Booth
CC BY-SA 3.0
2015-04-09T15:03:45.573Z
@orkhan471 I've added some general statements  about elasticities.
FooBar
CC BY-SA 3.0
2015-04-09T14:57:12.117Z
2015-04-09T15:00:35.700Z
2015-04-09T15:00:35.700Z
FooBar
<p>I have attempted to solve this problem and would be grateful if someone looked over my solution</p>

<p>5% increase in price  results in 5% decrease in total revenue. I need to determine price elasticity of demand. I have done calculations but I am not sure whether the way I have solved it is true.</p>

<p>$$TR=PQ\mapsto Q=\frac{TR}{P}\rightarrow Q=\frac{0.8TR}{1.2 P}=2/3$$
And then I wrote $$PE=\frac{\frac{8}{12}}{\frac{12}{10}}=5/9$$</p>

<p>But this solution doesn't appear in the test answer options. Can someone please check it and tell what is wrong?</p>

<p>Thanks in advance!</p>

CC BY-SA 3.0
2015-04-09T15:06:11.873Z
Perhaps a hint for future questions: You will get better answers when your questions are as general as possible. Also, then there are of more usefulness for future visitors. Here, the general question could've been on how to compute the demand-elasticity given the revenue-elasticity.
FooBar
<microeconomics><consumer-theory><elasticity>
Relationship between Revenue-Elasticity and Price-Elasticity
Frank Booth
# Relationship between Revenue-Elasticity and Price-Elasticity

(Asked by: username_0 on 2015-04-09)

I have attempted to solve this problem and would be grateful if someone looked over my solution

5% increase in price results in 5% decrease in total revenue. I need to determine price elasticity of demand. I have done calculations but I am not sure whether the way I have solved it is true.

$$TR=PQ\mapsto Q=\frac{TR}{P}\rightarrow Q=\frac{0.8TR}{1.2 P}=2/3$$
And then I wrote $$PE=\frac{\frac{8}{12}}{\frac{12}{10}}=5/9$$

But this solution doesn't appear in the test answer options. Can someone please check it and tell what is wrong?

Thanks in advance!

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

> **Comments:**
> **username_1**: Perhaps a hint for future questions: You will get better answers when your questions are as general as possible. Also, then there are of more usefulness for future visitors. Here, the general question could've been on how to compute the demand-elasticity given the revenue-elasticity.  

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## 1 answers

### Answer by username_1 on 2015-04-09. Score: 1 | Accepted answer

Let's denote the elasticity of $X$ to $Y$ at $Y\_0$ with $\eta\_{X, Y}(Y\_0)$ and solve for the demand ($Q$) elasticity in terms of the revenue elasticity ($R$).

We know that the elasticity is always given by

$$ \eta\_{X,Y} = \frac{d X(Y)}{dY}\frac{Y}{X}$$

or equivalently (showing the equivalence is a useful exercise),

$$ \eta\_{X,Y} = \frac{d \log X(Y))}{d \log Y}$$

Now,

$$ R(p) = p Q(p) \\
\eta\_{R, p}(p) = R'(p) \frac{p}{pQ(p)}\\
\eta\_{R, p}(p) = (Q(p) + pQ'(p\_0) ) \frac{p}{pQ(p)} \\
\eta\_{R, p}(p) = 1 + \frac{pQ'(p\_0)}{Q} \\
\eta\_{R, p}(p) = 1 + \eta\_{Q, p}(p) \\
$$

Given that the left-hand-side elasticity is $-1$ at the initial price $p\_0$, the right-hand-side elasticity must be $-2$:

$$-1 = 1 + \eta\_{Q, p}(p\_0) \Rightarrow \eta\_{Q, p}(p\_0) = -2$$

## Your mistake

You are being sloppy with the mathematics. First you state

$$ Q = \frac{TR}{P}$$

But then you claim

$$ Q = \frac{0.8TR}{1.2P}$$

These two equations just cannot be true at the same time.

Perhaps you want to look at initial $TR\_0$, at total revenue after price change $TR\_1$, and then state that $TR\_1/TR\_0 = 0.95$.

> **Comments:**
> **username_0**: How did you derive the second line?  
> **username_1**: @orkhan471 I've added some general statements  about elasticities.  

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