# Breaking down components of GDP per capita

GDP per capita can be broken down into 4 smaller components: Labor productivity, Average hours worked, Employment rate and Participation rate as can be deduced from the following formula:

What I'd like to ask is if there is a way to accurately measure the change in one component/factor if given the change in other components/factors?

Something akin to this: Suppose that everyone in the population of Country A is employed and equally productive. If the population grows by a%, hours of work fall by b%, and output per hour grows by c%, then output per person, or output per capita increases/decreases by how many percent?

For the aforementioned example, I tried to do the following calculation but the result came out wrong. $$\Delta \text{GDP per capita}=1-(1+c)\times \frac {1-b}{1+a}$$

For a more concrete illustration, let's assign 5,3,4 to a,b,c respectively, I would get $$\Delta \text{GDP per capita}=1-(1+0.04)\times \frac {1-0.03}{1+0.05}\approx 0.039$$ which means GDP per capita decreases approximately 4 percent but the result I got from the book is a 1% increase. I'm quite stuck here ...

• Your increases are wrong. You can answer your own question by writing down how much GDP, hours worked, employment and total population grow given your increases. Not all of these is simply one of a%,b%,c%. Apr 30 '17 at 18:22
• I tried to go with b+c-a but it didn't work either ...
– Rei
Apr 30 '17 at 22:48
• Brute force rarely works. Perhaps try by doing what I suggested in my previous comment. What is the rate of change of GDP? What is the rate of change of hours worked? What is the rate of change of employment? What is the rate of change of total population? May 1 '17 at 7:18
• @denesp I also get a 4% fall. I can't see where the possible mistake is. If GDP per hour grows by c%, and hours fall by b%, then GDP grew by (c-b)%. If population grows by a%, then GDP per capita grows by [(c-b)-a]%. Given the numbers provided, that is 4% fall. May 1 '17 at 9:20
• @denesp Well, let's see what the OP says. In any case, how would you compute the final solution if b% where hours per employment? Are you assuming employment and population growth together? So that (c-b)%=1%? If not, I'm not sure you can get a final number. We might not be just having an incorrect statement from the OP, but an omitted assumption too. May 1 '17 at 13:30