# Converting per capita growth rates for subgroups into contribution to total per capita growth rate

You are given total debt for the second quarter of 2003 and the second quarter of 2004 for the overall population and four age groups. The four age groups collectively account for the entire population and their debt holdings account for overall total debt.

Quarter, Age_Group, Population, Total_Debt, Per_Capita_Debt
Q2 2003,0,1000,10000,10
Q2 2003,1,250,3000,12
Q2 2003,2,400,5000,12.5
Q2 2003,3,250,1500,6
Q2 2003,4,100,500,5

Q2 2004,0,1300,15000,11.5
Q2 2004,1,380,4000,10.5
Q2 2004,2,450,7000,15.6
Q2 2004,3,270,2500,9.3
Q2 2004,4,200,1500,7.5


The per capita growth rate for age group 0, representative of the total per capita growth rate is 15 percent. But how does one convert the individual per capita growth rates to the contribution to the total rate? Perhaps weight by the population shares? Taking a weighted average of the per capita growth rates yields approximately 6 percent. Dividing the change in per capita debt for each of the groups over the per capita debt level for the population doesn't work as it would when using total debt. The result should be r1 + r2 +r3 + r4 = r0 Where rn is the growth rate for age group n.

If the initial populations are $p_1,p_2,p_3,p_4$ and the population growth rates are $g_1,g_2,g_3,g_4$ and the initial per capita debts are $d_1,d_2,d_3,d_4$ and the growth in per capita debts are $r_1,r_2,r_3,r_4$ then if you insist on some kind of weighted sum then you could use something like:

$r_0= \displaystyle \sum _i \tfrac{ \left(\sum_j p_j\right)p_i(1+g_i)d_i}{\left(\sum_j p_j d_j\right) \left(\sum_j p_j(1+g_j)\right)}(1+r_i) - 1$

but you would probably find it easier to calculate the intermediate population totals and debt totals

The problem is that you've got two inconsistent definitions of growth rate.

One is driven by per capita debt across the whole population.

The other by per capita debt within each subgroup.

These are inconsistent, because different subgroups have different baseline debts in Q2, and different population sizes. Because both of those things are different, and both are divisors in the calculation of your statistic of interest, then simply re-weighting by population size can't fix the problem.

So you can calculate a meaningful contribution of each to overall growth rate, so that together they satisfy r1 + r2 +r3 + r4 = r0. But then the individual r1 to r4 won't mean much within their own groups - only in terms of their contribution to the whole. Or you can calculate growth rates for each sub group, as you've done in your question, but then they won't mean much in the overall context.

First, compute total growth in dollars for each year.

Then, divide this value by total debt.