Link between Productivity and Labour Force?

Question

In this article from VoxEU, the author states:

By definition growth in output per capita equals growth in labour productivity times growth in hours per capita. The slowdown in productivity growth that began 40 years ago was partly offset between 1972 to 1996 by an increase in the labour-force participation rate of 0.4% per year, as females and baby-boom teenagers entered the labour force. In contrast during 2004-2014 the participation rate has declined at an annual rate of 0.5%, and over the shorter 2007-2014 interval at an annual rate of 0.8%. This transition from a 0.4% increase to a 0.8% decline accounts for a 1.2% reduction in the growth of per-capita real GDP for any given growth rate of labour productivity.

I don't understand how the participation rate influences the growth of GDP per capita. Is it because in the per capita, besides the labour force, we also take into consideration the rest of the population? And what's the link between productivity and labour force?

Any help would be appreciated.

Answer 1

Output per capita is $\dfrac{Y}{N}$. This can be decomposed as:

$$ \frac{Y}{N} \equiv \frac{Y}{h} \times \frac{h}{L} \times \frac{L}{A} \times\frac{A}{N} $$

where $h$ are aggregate hours of work, $L$ is number of employed, and $A$ is active population. $\frac{Y}{h}$ is aggregate labour productivity, $\frac{h}{L}$ is "work intensity", $\frac{L}{A}$ is the employment rate and $\frac{A}{N}$ is the participation rate.

The above equality can be turned into growth rates, which means in approximation, the growth of output per capita is the sum of the growth rates of each of the subcomponents.