Solow growth model - analytic proof that Inada conditions imply steady state capital is increasing in the savings rate

Question

Let's take the example of a generic Harrod-neutral (labor-augmenting) production function $f(k)$; all letters denote the growth rates they usually would. In the regular Solow growth model with the Inada assumptions, it is a result of its comparative statics that, since $sf(k^*)=(n+\delta+g)k^*$ is the steady-state condition, if we assume $k^*$ is a function of the savings rate then $\frac{\partial k^*}{\partial s}>0$. Total differentiation gives us that $$ \frac{\partial k^*}{\partial s} = \frac{f(k^*)}{(n+g+\delta)-sf'(k^*)},$$ and in my class the professor claimed that this last expression is strictly positive because the numerator is (fine) and the denominator is too. Basically, it boils down to the fact that $\frac{n+g+\delta}{s} > f'(k^*)$, and that this is a fact implied by the Inada assumptions.

I have tried messing around with the statement about the limiting behavior of the derivative at 0 (infinity) and as k tends to infinity (zero); all to no avail. If someone could please offer some hints or an explanation or a proof of this fact I would deeply appreciate. This was not a question posed for an assignment or anything of the sort, just personal interest (having some background in analysis).

Answer 1

We want to prove that

$$\frac{n+g+\delta}{s} > f'(k^*)$$

Replace the left hand side with the equivalent from the expression $sf(k^*)=(n+\delta+g)k^*$, and you get:

$$ \frac{f(k^*)}{k^*} > f'(k^*) $$

Cobb-Douglas case

Without loss of generality, assume $$f(k^*) = {k^*}^{\alpha}$$

Then, the above inequality is:

$$ {k^*}^{\alpha-1} > \alpha{k^*}^{\alpha-1} $$

This is:

$$ 1> \alpha $$

This condition represents a production function with decreasing marginal returns to capital, necessary for the Inada conditions to hold.

General case

A concave function has the following property, for any $x$ and $y$ in the domain:

$$f(y) \leq f(x) + f'(x)(y-x) $$

Rearranging, this is:

$$ \frac{f(y)- f(x)}{y-x} \leq f'(x) \hspace{1cm}, \text{for } y>x $$

This is, the derivative of $f(x)$ at point $x$ is no smaller than the slope of the segment between $x$ and $y$. For the case of a strictly concave function (which is needed for the Inada conditions to hold), the inequality is strict ($<$), except in the trivial case of $x = y$.

For further confirmation, apply the above when $y=0$ and $x=k^*$ (notice here $y<x$ so the sign of the above inequality reverses). Since $f(0)=0$, you get:

$$ \frac{f(k^*)}{k^*} > f'(k^*) $$

which is what we wanted to demonstrate.