0

I am running into a real world problem. But I think this is more like a math problem. So here it is. Suppose I have $$A = B + C.$$

The $A, B, C$ in this period are called $A_{1}, B_{1}, C_{1}$. Similarly, $A, B, C$ in previous period are given by $A_{0}, B_{0}, C_{0}$. The changes in $A$ compared to the previous period is give by $$\Delta A = \frac{A_{1}}{A_{0}}-1.$$

The changes in $B$ and $C$ are defined in a similar way. I want to decompose the changes in $A$ in to changes in $B$ and $C$. Obviously, I cannot do the following $$\Delta A = \Delta B + \Delta C$$ since mathematically it would be wrong.

I wonder if there is a smarter way to decompose $\Delta A$ into changes in $B$ and changes in $C$, which makes sense regarding the restriction $A = B + C$ and is also mathematically correct. Maybe there's not such a way. But just curious.

JasonSmith
  • 103
  • 2
  • Do you want to calculate $\Delta A$ given $\Delta B$ and $\Delta C$? – Ragnar Aug 22 '14 at 21:11
  • I think the simplest way to describe this would be: How does one construct the percent change in the sum of two quantities in terms of the percent changes in these quantities? – Semiclassical Aug 22 '14 at 21:16
  • I have all $A, B, C, A_{1}, B_{1}, C_{1}, A_{0}, B_{0}, C_{0}$. I want to keep $\Delta A = \frac{A_{1}}{A_{0}}-1$ as it is. Just want to calculate $\Delta B$ and $\Delta C$ differently so that I can have change in $A$ is equal to change in $B$ + change in $C$ – JasonSmith Aug 22 '14 at 21:17

1 Answers1

2

Using your notation for $\Delta A$ (which most people would call $\frac{\delta A}{A}$), the formula you are looking for is

$$ \Delta A = \frac{1}{A_0}\left( B_0 \Delta B + C_0 \Delta C \right) $$

Mark Fischler
  • 41,743
  • This points out the fact that it is not possible to compute the relative change in $A$ from just the relative changes in $B$ and $C$. You must also know the actual values of $B$ and $C$ to recover the absolute changes in them and to compute the relative change in $A$. For example: If the number of men in Smallville increased by 5% and the number of women there increased by 13%, by what percent did the population there increase? Answer: impossible to tell without more information. What if there were 10 men and 100,000 women? Or the other way around? Or if they were equal? +1 for you. – MPW Aug 22 '14 at 21:31
  • Thank you. This is brilliant! – JasonSmith Aug 22 '14 at 22:08