So far our model cannot explain sustained economic growth: a higher saving rate leads to higher growth temporarily, but the economy eventually approaches a steady state in which ouptut is constant. To explain the sustained economic growth that we observe in many countries, we must incorporate population growth and technological progress. We start with population growth.

Instead of assuming that the labour force $L$ is fixed as we did previously, we suppose that it grows at a constant rate $n$. For example, the US population grows at around 1 percent per year, so $n=0.01$.

To understand how population growth affects the steady state, we must discuss how population growth (along with investment and depreciation) influences the accumulation of capital per worker. As we have seen, investment adds to the capital stock whilst depreciation reduces it. Now there is a third force which influences the amount of capital per worker: growth in the number of workers causes the amount of capital per worker to decrease.

The change in the capital stock per worker is now:

$\Delta k = s f(k) - (\delta + n) k$

The change in the capital stock in any period is the difference between the amount invested $sf(k)$ and break even investment $(\delta + n) k$: the amount of investment necessary to keep the capital stock per worker constant.

Break-even investment includes the amount of investment needed to replace depreciated capital $\delta k$, as well as the amount of investment necessary to ensure that new workers have as much capital as existing workers. Because there are $n$ new workers for each existing worker and because $k$ is the amount of capital for each worker, the amount of investment necessary to achieve this purpose is $n k$

Population growth influences the per-worker capital stock much the way depreciation does. Depreciation reduces $k$ by diminishing the capital stock, whereas population growth reduces $k$ by spreading it more thinly amoung more workers.

If $k$ is less than the steady state $k^*$, investment is greater than break-even investment so the capital stock rises (and vice versa). In the steady state, capital per worker $k$ is constant despite growth in the number of workers. This is because investment both replaces depreciated capital $\delta k$ and provides new workers with $n k$, to ensure that they work with the steady state amount of capital.