<p>Adopting your notations, $E = P \times Q$. Taking natural log for both sides, we have $\log E = \log P + \log Q$.</p>

<p>When the percent change of some quantity, $\Delta x / x$, is small, we can approximate this percent change with $\Delta \log x$,  the change in $\log x$. This is because when $\Delta x / x$ is small,</p>

<p>$$
\log (x + \Delta x) - \log x = \log \frac{x + \Delta x}{x} = \log (1 + \frac{\Delta x}{x}) \approx \frac{\Delta x}{x}
$$</p>

<p>Now we come back to the equation $\log E = \log P + \log Q$. When the change is small, we can loosely say that the percent change in E is the sum of the percent changes in P and Q. But you are right, this relationship is not exact, and may not be appropriate when the percent change is large.</p>

CC BY-SA 3.0
2016-10-30T20:14:45.217Z
2016-10-30T21:58:01.263Z
2016-10-30T21:58:01.263Z
Paul
<p>I am learning the Economic class and I have a very specific question regarding a basic economic concept: change of expenditure in terms of change of price and change of quantity.</p>

<p>I'v learnt below:</p>

<pre><code>1. Expenditure = Price x Quantity (by definition)
2. change of Expenditure = change of Price + change of Quantity
</code></pre>

<p>I am really confused about how #2 above is inferred. I thought change of Expenditure is defined as</p>

<pre><code>E - Expenditure, P - Price, Q - Quantity

(E2 - E1) / E1 = (P2 x Q2 - P1 x Q1) / P1 x Q1
</code></pre>

<p>But this is really not equal to</p>

<pre><code>change of Price + change of Quantity = (P2 - P1) / P1 + (Q2 - Q1) / Q1
</code></pre>

<p>Can someone show me how does the formula "change of Expenditure = change of Price + change of Quantity" hold please?</p>

<p>Many thanks</p>

<elasticity>
study question: change of Expenditure in terms of price and quantity
Bob
# study question: change of Expenditure in terms of price and quantity

(Asked by: username_0 on 2016-10-30)

I am learning the Economic class and I have a very specific question regarding a basic economic concept: change of expenditure in terms of change of price and change of quantity.
I'v learnt below:
1. Expenditure = Price x Quantity (by definition)
2. change of Expenditure = change of Price + change of Quantity

I am really confused about how #2 above is inferred. I thought change of Expenditure is defined as
E - Expenditure, P - Price, Q - Quantity

(E2 - E1) / E1 = (P2 x Q2 - P1 x Q1) / P1 x Q1

But this is really not equal to
change of Price + change of Quantity = (P2 - P1) / P1 + (Q2 - Q1) / Q1

Can someone show me how does the formula "change of Expenditure = change of Price + change of Quantity" hold please?
Many thanks

**Tags:** elasticity

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## 1 answers

### Answer by username_1 on 2016-10-30. Score: 1

Adopting your notations, $E = P \times Q$. Taking natural log for both sides, we have $\log E = \log P + \log Q$.

When the percent change of some quantity, $\Delta x / x$, is small, we can approximate this percent change with $\Delta \log x$, the change in $\log x$. This is because when $\Delta x / x$ is small,

$$
\log (x + \Delta x) - \log x = \log \frac{x + \Delta x}{x} = \log (1 + \frac{\Delta x}{x}) \approx \frac{\Delta x}{x}
$$

Now we come back to the equation $\log E = \log P + \log Q$. When the change is small, we can loosely say that the percent change in E is the sum of the percent changes in P and Q. But you are right, this relationship is not exact, and may not be appropriate when the percent change is large.


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