# the relationship between the growth rate of GDP, GDP per capita and the population

I was reading Capital in the Twenty-First Century by Thomas Piketty and it says the growth rate of GDP is equal to the growth rate of GDP per capita plus the growth rate of the population, and I wonder why this is the case.
Mathematically, $$Y = y * N$$ by definition, and I have no idea how we can get $$Y'-1=(y'-1)+(N'-1)$$ from the former eqution. Or am I getting this wrong?

If we are considering a long period in which GDP per capita and population both grow considerably, then the growth in GDP over the period will be significantly more than the sum of growth in the former quantities. Suppose that over a decade, GDP per capita and population both grow by 20%. Then GDP will have grown by:

$$\Big[\Big(1 + \frac{20}{100}\Big)\Big(1 + \frac{20}{100}\Big) - 1\Big]\times 100 = 44\% > 20\% + 20\% = 40\%$$

However, when considering short periods with small changes, summing the growth of GDP per capita and population gives a good approximation to growth of GDP. Making a similar calculation with growth in GDP per capita and population of just 2%, we find that GDP grows by:

$$\Big[\Big(1 + \frac{2}{100}\Big)\Big(1 + \frac{2}{100}\Big) - 1\Big]\times 100 = 4.04\%$$

In this case therefore the degree of approximation is only 0.04%.

When growth rates are measured in continuous time rather than over a period, then (as explained in 1muflon1's answer) summing the growth rates of GDP per capita and population gives the growth rate of GDP exactly.

Start by definition of GDP per capita:

$$\text{GDPpc}= \frac{Y}{N} \tag{1}$$

where $$Y$$ is GDP and $$N$$ population. Take log of both sides:

$$\ln \text{GDPpc}= \ln Y - \ln N \tag{2}$$

Take the time derivative of both sides (note all variables are implicitly functions of $$t$$ so this is allowed):

$$\frac{\dot{ \text{GDPpc} }}{\text{GDPpc} } = \frac{\dot{ Y} }{Y } - \frac{\dot{N}}{N} \tag{3}$$

Now solve for growth rate of $$Y$$

$$\underbrace{\frac{\dot{ Y} }{Y } }_{\text{GDP Growth}}= \underbrace{\frac{\dot{ \text{GDPpc} }}{\text{GDPpc} }}_{\text{GDPpc Growth}} + \underbrace{\frac{\dot{N}}{N}}_{\text{pop growth}} \tag{4}$$

The derivation above is expressed in continuous time, but note $$\frac{\dot{X}}{X}$$ is just a direct continuous analogue of discrete growth $$\frac{X_t-X_{t-1}}{X_t}= \frac{\Delta X_t}{X_t}$$ (for some infinitesimal $$\Delta$$ they are equal for large delta it would only be approximation) so if you want to switch to discrete time you get:

$$\underbrace{\frac{\Delta Y }{Y } }_{\text{GDP Growth}}= \underbrace{\frac{\Delta \text{GDPpc} }{\text{GDPpc} }}_{\text{GDPpc Growth}} + \underbrace{\frac{\Delta N}{N}}_{\text{pop growth}} \tag{5}$$

In discrete time (in which GDP is released), the statement is only true in approximation. Economists often use this kind of approximation implicitly.

Write GDP per capita as $$y = \frac{Y}{N}$$.

Then, growth of GDP per capita is related to GDP and population growth: $$\frac{y_t}{y_{t-1}} = \frac{Y_t}{N_t}\frac{N_{t-1}}{Y_{t-1}}$$ So, the gross growth rate of GDP per capita is actually the product of the gross growth rates of GDP and population. However, we can use $$\log(1+r)\approx r$$ where $$r$$ is the (net) growth rate and relatively small (say 0.025 year-on-year). So, for a net growth rate $$r$$ of $$x$$, $$1+r = 1+\frac{x_t - x_{t-1}}{x_{t-1}} = \frac{x_t}{x_{t-1}}$$ we obtain $$r \approx \log(1+r) = \log\biggl(\frac{x_t}{x_{t-1}}\biggr)$$

Applying this to the relationship between the growth rates gives $$\biggl(\frac{y_t}{y_{t-1}}-1\biggr) \approx \biggl( \frac{Y_t}{Y_{t-1}}-1\biggr) - \biggl(\frac{N_t}{N_{t-1}}-1 \biggr)$$ where the terms in brackets are growth rate of per-capita GDP, GDP, population, respectively. Rearranging the terms gives you the statement (in approximation).