<p>The good <strong>is not a giffen good</strong>. These goods have a positive elasticity of demand. Yours has a negative one, by assumption. It cannot yield a giffen good at all.</p>

<p>The error is in the negative sign that you (and also Thomas's answer) assumed. </p>

<p>The formula for the elasticity is:</p>

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

<p>The negativeness of the elasticity is given by the <strong>sign</strong> of the derivative (a price increase leads to a fall in quantity demanded; the contrary in a Giffen good).</p>

<p>In your example, assuming as starting point $Q_1$ and $P_1$ (you could alternative use the end point, or the arc elasticity; see <a href="https://en.wikipedia.org/wiki/Price_elasticity_of_demand#Definition" rel="nofollow noreferrer">here</a>), you get:</p>

<p>$$ -8 = \frac{Q_2 - 800,000}{12}\frac{24}{800,000}$$</p>

<p>Which yields </p>

<p>$$Q_2 = -2,400,000$$</p>

<p>This is, in effect, a negative quantity. To see why this is the case, rewrite the elasticity function as follows:</p>

<p>$$ e_d \frac{\Delta P}{P_1} = \frac{\Delta Q}{Q_1}$$</p>

<p>This is, the <strong>percentage</strong> change in the quantity is equivalent to the <strong>percentage</strong> change in prices times the elasticity.</p>

<p>Replacing the numbers, you get:</p>

<p>$$ -4 = \frac{\Delta Q}{Q_1}$$</p>

<p>This is, <strong>the quantity falls by 400% percent!</strong>. That is why it becomes negative (from 800,000 to -2,400,000).</p>

<p>The origin of the problem is the massive change in price (50%) together with the massive elasticity of demand (-8), which gives the total change in quantity ($50\% \times -8 = -400\%$). A less dramatic combination would yield a positive $Q_2$.</p>

CC BY-SA 3.0
2017-09-13T09:59:31.580Z
2017-09-13T09:59:31.580Z
luchonacho
<p>You have that price elasticity is defined by:</p>

<p>$$\varepsilon_{d} = -\frac{\partial Q}{\partial P}\frac{P}{Q} \approx -\frac{P}{Q}\frac{\Delta Q}{\Delta P}$$</p>

<p>Initially we have $P = P_{1}$ and $Q=Q_{1}$ and after the perturbation we have $\Delta P = (P_{2} - P_{1})$ and $\Delta Q = (Q_{2} - Q_{1})$, therefore:</p>

<p>$$-\frac{P_{1}}{Q_{1}}\frac{Q_{2}-Q_{1}}{P_{2}-P_{1}}=-8 \implies Q_{2}-Q_{1}=\frac{8Q_{1}(P_{2}-P_{1})}{P_{1}}$$</p>

<p>And therefore:</p>

<p>$$ Q_{2}=Q_{1}+\frac{8Q_{1}(P_{2}-P_{1})}{P_{1}}=\frac{8800000}{3} $$</p>

<p>This is so large, partly because the price elasticity is so large, but also probably because of the error in the finite difference approximation method over such large changes in $P$.</p>

CC BY-SA 3.0
2015-03-09T10:31:52.253Z
Thank you. But does it make sense that the quantity increases even when the price increases? My textbook problem is about potatoes.
Jamgreen
CC BY-SA 3.0
2015-03-09T10:34:40.260Z
@Jamgreen Well in the case of potatoes it doesn't seem to make sense; although as $\varepsilon_{d} < 0$ it does make sense mathematically. Does your textbook give the formula for $\varepsilon_{d}$, sometimes they define it without the negative coefficient? Usually this sort of behaviour occurs with [Giffen Goods](http://en.wikipedia.org/wiki/Giffen_good).
Thomas Russell
CC BY-SA 3.0
2015-03-09T10:12:03.900Z
2017-09-13T10:01:43.380Z
2017-09-13T10:01:43.380Z
Thomas Russell
<p>I have $P_1 = 24$, $Q_1 = 800000$, $P_2 = 32$ and price elasticity $e = -8$ and need to find the function $P(Q)$.</p>

<p>I guess I need to find $Q_2$ by</p>

<p>$$
\frac{(Q_2-Q_1)/Q_1}{-(P_2-P_1)/P_1} = -8 \Leftrightarrow Q_2 = \frac{8800000}{3}
$$</p>

<p>But does it make sense that the quantity will be so much higher by a price increment and thus not follow the law of demand? I know that $e = -8$ is a high price elasticity but this is quite extreme I think.</p>

CC BY-SA 3.0
2015-03-09T13:18:21.157Z
You probably have an error in your equation. It's hard to be sure without seeing the context in your textbook (this could be an example intented to illustrate the concept of Giffen goods). But if that's not the case then the left hand side should be $[(Q_2-Q_1)/Q_1] / [(P_2-P_1)/P_1]$ (i.e. with no negative sign).
Ubiquitous
CC BY-SA 3.0
2015-03-09T13:51:41.820Z
But without the negative sign I get $Q_2 = -4000000/3$. Wouldn't it be a bad example if the quantity becomes negative?
Jamgreen
CC BY-SA 3.0
2015-03-09T18:34:45.413Z
Yes, and I now notice that you said the example concerns the potato market, which is the classic example of a Giffen good. I'd recommend that you read about such goods (if you haven't already) to see why price and quantity move in the same direction.
Ubiquitous
<microeconomics><elasticity>
Find quantity from price elasticity
Jamgreen
# Find quantity from price elasticity

(Asked by: username_0 on 2015-03-09)

I have $P\_1 = 24$, $Q\_1 = 800000$, $P\_2 = 32$ and price elasticity $e = -8$ and need to find the function $P(Q)$.

I guess I need to find $Q\_2$ by

$$
\frac{(Q\_2-Q\_1)/Q\_1}{-(P\_2-P\_1)/P\_1} = -8 \Leftrightarrow Q\_2 = \frac{8800000}{3}
$$

But does it make sense that the quantity will be so much higher by a price increment and thus not follow the law of demand? I know that $e = -8$ is a high price elasticity but this is quite extreme I think.

**Tags:** microeconomics, elasticity
> **Comments:**
> **username_1**: You probably have an error in your equation. It's hard to be sure without seeing the context in your textbook (this could be an example intented to illustrate the concept of Giffen goods). But if that's not the case then the left hand side should be $[(Q_2-Q_1)/Q_1] / [(P_2-P_1)/P_1]$ (i.e. with no negative sign).  
> **username_0**: But without the negative sign I get $Q_2 = -4000000/3$. Wouldn't it be a bad example if the quantity becomes negative?  
> **username_1**: Yes, and I now notice that you said the example concerns the potato market, which is the classic example of a Giffen good. I'd recommend that you read about such goods (if you haven't already) to see why price and quantity move in the same direction.  

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## 2 answers

### Answer by username_2 on 2017-09-13. Score: 0 | Accepted answer

The good **is not a giffen good**. These goods have a positive elasticity of demand. Yours has a negative one, by assumption. It cannot yield a giffen good at all.

The error is in the negative sign that you (and also Thomas's answer) assumed.

The formula for the elasticity is:

$$ e\_d = \frac{\Delta Q}{\Delta P}\frac{P}{Q}$$

The negativeness of the elasticity is given by the **sign** of the derivative (a price increase leads to a fall in quantity demanded; the contrary in a Giffen good).

In your example, assuming as starting point $Q\_1$ and $P\_1$ (you could alternative use the end point, or the arc elasticity; see [here](https://en.wikipedia.org/wiki/Price_elasticity_of_demand#Definition)), you get:

$$ -8 = \frac{Q\_2 - 800,000}{12}\frac{24}{800,000}$$

Which yields

$$Q\_2 = -2,400,000$$

This is, in effect, a negative quantity. To see why this is the case, rewrite the elasticity function as follows:

$$ e\_d \frac{\Delta P}{P\_1} = \frac{\Delta Q}{Q\_1}$$

This is, the **percentage** change in the quantity is equivalent to the **percentage** change in prices times the elasticity.

Replacing the numbers, you get:

$$ -4 = \frac{\Delta Q}{Q\_1}$$

This is, **the quantity falls by 400% percent!**. That is why it becomes negative (from 800,000 to -2,400,000).

The origin of the problem is the massive change in price (50%) together with the massive elasticity of demand (-8), which gives the total change in quantity ($50\% \times -8 = -400\%$). A less dramatic combination would yield a positive $Q\_2$.


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### Answer by username_3 on 2015-03-09. Score: 0

You have that price elasticity is defined by:

$$\varepsilon\_{d} = -\frac{\partial Q}{\partial P}\frac{P}{Q} \approx -\frac{P}{Q}\frac{\Delta Q}{\Delta P}$$

Initially we have $P = P\_{1}$ and $Q=Q\_{1}$ and after the perturbation we have $\Delta P = (P\_{2} - P\_{1})$ and $\Delta Q = (Q\_{2} - Q\_{1})$, therefore:

$$-\frac{P\_{1}}{Q\_{1}}\frac{Q\_{2}-Q\_{1}}{P\_{2}-P\_{1}}=-8 \implies Q\_{2}-Q\_{1}=\frac{8Q\_{1}(P\_{2}-P\_{1})}{P\_{1}}$$

And therefore:

$$ Q\_{2}=Q\_{1}+\frac{8Q\_{1}(P\_{2}-P\_{1})}{P\_{1}}=\frac{8800000}{3} $$

This is so large, partly because the price elasticity is so large, but also probably because of the error in the finite difference approximation method over such large changes in $P$.

> **Comments:**
> **username_0**: Thank you. But does it make sense that the quantity increases even when the price increases? My textbook problem is about potatoes.  
> **username_3**: @Jamgreen Well in the case of potatoes it doesn't seem to make sense; although as $\varepsilon_{d} < 0$ it does make sense mathematically. Does your textbook give the formula for $\varepsilon_{d}$, sometimes they define it without the negative coefficient? Usually this sort of behaviour occurs with [Giffen Goods](http://en.wikipedia.org/wiki/Giffen_good).  

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