# Solow model response function

Consider a Solow model without technological progress so that the steady state $$k^*$$ occurs at $$sf(k^*) = (n+\delta)k^*$$ where $$n$$ is population growth rate, $$\delta$$ is capital depreciation rate and $$s$$ is the proportion of output that is saved and invested. Given that consumption per worker is $$c = f(k) - sf(k)$$, how does steady state consumption change if $$n$$ decreases?

Intuitively, output per worker rises so $$c = (1-s)f(k)$$ should also rise. However, if I try to solve for the change numerically I would do:

$$c^* = f(k^*(s, n, \delta)) - (n+\delta)k^*(s, n, \delta)$$ at the steady state.

So, $$\frac{\partial c^*}{\partial n} = f'(k^*)\frac{\partial k^*}{\partial n} - k^* -(n + \delta)\frac{\partial k^*}{\partial n}$$.

Re-arranging:

$$\frac{\partial c^*}{\partial n} = [f'(k^*) - (n+\delta)]\frac{\partial k^*}{\partial n} - k^*$$.

We know $$\frac{\partial k^*}{\partial n}$$ < 0 since a higher slope of the breakeven investment causes it to intersect with $$sf(k^*)$$ at a lower $$k^*$$. The sign of the term in the square bracket is ambiguous so $$\frac{\partial c^*}{\partial n}$$ itself is ambiguous.

There seems to be a contradiction between my intuition and the numerical analysis, so where have I gone wrong?

The direction of $$\frac{\partial c^{*}}{\partial n}$$ is not ambiguous.
An easy way to show this is taking derivative of $$c^*=(1-s)f(k^*)$$ so that $$\frac{\partial c^{*}}{\partial n}=(1-s)f'\frac{\partial k^{*}}{\partial n}$$ and because $$f'>0$$ and we can prove $$\frac{\partial k^{*}}{\partial n}<0$$ we thus have $$\frac{\partial c^{*}}{\partial n}<0$$.
Through this you can also easily see that $$c^{*}$$ will be monotone in $$n$$ and $$\delta$$ but it will not be monotone in $$s$$.
It is somehow harder to prove this from $$\frac{\partial c^{*}}{\partial n}=\left[f^{\prime}\left(k^{*}\right)-(n+\delta)\right] \frac{\partial k^{*}}{\partial n}-k^{*}$$
• Thank you, this makes perfect sense. For record: The sign in the square term of the bracket is actually not ambiguous. However, I notice that the response function $\frac{\partial c^*}{\partial s} = [f'(k^*) - (n + \delta)]\frac{\partial k^*}{\partial s}$. If we were to reason the same way, wouldn't steady state consumption be ever-increasing in savings, and so no such thing as the golden rule maximum? Sep 29 at 11:36
• @PanhabothK Sorry, you are correct. I made a mistake in my initial answer. I find it is harder than I thought to prove the sign from the equation you are dealing with. But I show that you can easily prove this in an alternative way. And it helps you see the difference between $n$ and $s$. Sep 29 at 13:40