Effect of population growth on Solow steady state My textbook says that:


  Ratio of capital per capita to income per capita in the steady state is a positive function of s and an inverse function of η and δ.  Thus, k*/y* is a constant.  This means when saving increase, the ratio does not change as both capital per capita and income per capita increase at the same rate.  However, both capital per capital and income per capital decrease at the same rate if the depreciation and population growth rate are higher.


It also says that:


  When you get to the steady state of output per worker, it means that your economy will still grow, just that it will only grow at the rate of growth of the labour force i.e. equals to the rate of population growth)


Doesn't both statement contradict each other?  I mean if the population growth rate grows, why could the capital and income per capita decreases which the first statement seems to state.  Am I reading the statements wrongly?
if the population growth rate grows, why could the capital and income per capita decreases


This is basically asking "why does higher population growth lower the steady state capital per worker"?

Mathematics

Let us assume a Cobb-Douglas production function with constant returns to scale. This is,

$$Y=K^{\alpha}L^{1-\alpha}$$

It can be shown that in the steady state optimal capital per capita is:

$$k^* = \left(\frac{s}{\delta+n}\right)^\frac{1}{1-\alpha}$$

(for example, see here)

Therefore, an increase in $n$ lowers $k^*$.

Similarly, in the steady state, output per capita is $y^*=k^*{^{\alpha}}$. Thus, a fall in $k^*$ leads to a fall in $y^*$, given that $\alpha \in (0,1)$.

Intuition

To start simple, think of an economy with a fixed level of workers ($n=0$). This economy has an exogenous saving rate $s$. Thus, for an initial level of output $Y$, workers save $s$ percentage of their income $Y$. These savings become investment, which then contributes to expanding capital. Meanwhile, there is a positive depreciation rate $\delta$, which in a given year renders a $\delta$% of the capital stock unusable. These two forces go in the opposite direction, moving capital stock up or down, depending on which force is stronger. Will there be a point in which these forces are in equilibrium? Necessarily there is one. Why? Because $\alpha \in (0,1)$ implies there is decreasing marginal returns to capital, case in which the Inada Conditions hold. In other words, if capital is very low, adding more capital will expand output and investment by very much. Conversely, when capital is very low, an identical increase in capital expands output and investment by very little. In the middle, there is a point where such expansion is equal to the depreciation (which is constant). In the Solow diagram with axes $K$ and $Y$ (in capital letters, not in per capita terms), that point is represented by the intersection between the depreciation function (a straight line) and the $sY$ line (concave, in line with the Inada Conditions).

Now, let us move to the case of positive population growth ($n&gt;0$). Here, employment is increasing every period. Therefore, $Y$ is also increasing. In consequence, savings, investment and therefore capital is also increasing. In this economy, there is no steady state in terms of $K$ (unlike before), because $K$ is growing every period with the expansion of the population size. In terms of the same Solow diagram of above (with axes $K$ and $Y$, in capital letters, not in per capita terms), the function $sY$ is shifting upwards every period, pushing the never-reachable point where $sY$ intersects $\delta K$ further to the right.

Since there is no steady state in terms of $K$, the next question to ask is, is there a steady state in terms of $K/L$? Because of the same logic than before, necessarily there is one. To see this, notice that capital per capita terms has decreasing marginal returns in the production function (see the function above). This is, an increase in $K/L$ by 1% leads to an increase in $Y/L$ by less than 1%. Therefore, for a very low $K/L$, an increase in $K/L$ means a relatively large increase in $Y/L$ and therefore in savings $sY$ and in investment, and therefore in capital. Conversely, for a very high $K/L$, an increase in $K/L$ means a relatively small increase in $Y/L$ and therefore in savings $sY$ and in investment, and therefore in capital. The Inada Conditions ensure there is a middle point by which the forces that lower $K/L$ (in this case, depreciation and population growth) equilibrate the expansion of capital.

Finally, why does an increase in $n$ lowers that optimal $K/L$? On the one hand, the first force in the equilibrium (pushing $K/L$ up) is given by both the savings rate $s$ and the degree of marginal returns ($\alpha$). Neither of these have changed. On the other hand, the second force in the equilibrium (pushing $K/L$ down) is given by both the depreciation rate $\delta$ (which makes a certain proportion of $K$ obsolete) and by the exogenous increase in $L$, which itself depends on $n$. In consequence, an increase in $n$ makes the second force stronger. This pushes the equilibrium to a lower level of $K/L$.

Notice that a way to compensate this is to make the first force stronger, for instance, by increasing the savings rate. This can be easily seen in the Solow diagram, where it is possible to revert to the original $K/L$ steady state by increasing $s$.