How to calculate a proportion

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likesomehelp

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I am doing a growth decomposition for my city's economic growth. I have gotten the data and processed it and calculated the growth rates.
I have total economic growth and the growth rates of the four factors that contributed to it. So when I calculate (growth of x/total economic growth) I get the proportion that factor x contributed to total growth.
My problem is that one of the factors changed more in magnitude than total growth (say -6% vs -1% growth) so the absolute value of the share I get is greater than 1.
Basically, using that formula is saying that factor x contributed to 500% of total growth, which is not possible.
How do I adapt my formulae to fix that?
 
I'm not convinced that what you want to do even makes sense. The growth rate of a given factor that contributes to overall growth does not necessarily represent how much it actually contributed. It depends very much on how each "factor" is involved; in particular, are they actually "factors" in a literal sense (multiplied), or do they add together or combine in a more complicated way?

This is all the more true when some of the factors decreased (negative "growth").

At the very least, I would expect that you might need to work not with percent growth (like +6% or -6%) but with factors (106%, 94%), which combine more appropriately.

Perhaps the people with more financial expertise here will have more to say, especially if you can tell us a little more specifically what your factors are, and how they affect overall growth. We may also want to see what the numbers look like, in order to experiment with them to see what results you get.
 
You're right! I'm not sure it makes sense at this point.

They are multiplied in a function. like Y=zKaL(1-a).

where the growth in Y is the total economic growth. So, I though it makes sense that %∆Y=%∆z+%∆(aK)+%∆(1-a)L.
 
If you actually have a formula that relates these factors (rather than just using "factor" in the broad sense of "something the influences the result"), and it has this form, then we can be somewhat more precise.

If we differentiate your formula (and I'll just write differentials rather than dY/dt), we get by the product rule

[MATH]dY = (K^aL^{1-a})dz + (zL^{1-a})aK^{a-1}dK + (zK^a)(1-a)L^{-a}dL[/MATH]​

Dividing this by Y = zK^aL^{1-a} in order to obtain relative change (percent increase), we get

[MATH]\frac{dY}{Y} = \frac{dz}{z} + a\frac{dK}{K} + (1-a)\frac{dL}{L}[/MATH]​

In terms of percent change, this is essentially what you wrote, though we can change that slightly to

[MATH]\%∆Y=\%∆z+a\%∆K+(1-a)\%∆L[/MATH]​

Since we now have an actual sum of parts, it may make sense to get the percentage of the percent increase due to each; and if any are actually decreasing, they will be counteracted by others, so you might end up with, say, 30%, -10%, and 80% of the change.

On the other hand, differentials are only an approximation to deltas, for small changes, so for non-infinitesimal changes, it may be nonsense. The sum of the parts will not really be the whole. So I'm not positive that this will make any sense in reality (which tends to make a fool of mathematicians), but it's an idea.
 
That makes sense.

I'm using real data. Could that be why the proportions are adding up to numbers that aren't 1 (100%)?

Because when %Δz=-6 and %ΔY=-1. I'm %Δz/%ΔY ends up being greater than 1. and the sum of all the proportions ends up being about 5 (500%) rather than 1.
 
I think I'd divide each delta not by delta Y, but by the sum of the individual deltas.

Or not.
 
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