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


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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