<p>We want to prove that </p>

<p>$$\frac{n+g+\delta}{s} &gt; f'(k^*)$$</p>

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

<p>$$ \frac{f(k^*)}{k^*} &gt; f'(k^*) $$</p>

<p><strong>Cobb-Douglas case</strong></p>

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

<p>Then, the above inequality is:</p>

<p>$$ {k^*}^{\alpha-1} &gt; \alpha{k^*}^{\alpha-1} $$</p>

<p>This is:</p>

<p>$$ 1&gt; \alpha $$</p>

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

<p><strong>General case</strong></p>

<p>A concave function has the <a href="https://en.wikipedia.org/wiki/Convex_function#Properties" rel="nofollow noreferrer">following property</a>, for any $x$ and $y$ in the domain:</p>

<p>$$f(y) \leq f(x) + f'(x)(y-x) $$</p>

<p>Rearranging, this is:</p>

<p>$$ \frac{f(y)- f(x)}{y-x} \leq f'(x) \hspace{1cm}, \text{for } y&gt;x $$</p>

<p>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 ($&lt;$), except in the trivial case of $x = y$. </p>

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

<p>$$ \frac{f(k^*)}{k^*} &gt; f'(k^*) $$</p>

<p>which is what we wanted to demonstrate.</p>

CC BY-SA 3.0
2017-10-01T22:51:28.260Z
Very nice. Everything makes sense; I just have one lingering concern. Why can you assume WLOG that the function takes on some Cobb-Douglas form? I had heard from someone that given the Inada conditions the production function basically has to be "asymptotically Cobb-Douglas", perhaps it has to do with that? Would love some clarification! Many thanks
spacetime
CC BY-SA 3.0
2017-10-02T07:16:26.927Z
@spacetime Well, you mentioned the "regular Solow growth model", which is based on a Cobb-Douglas. Is that not the case? A general CES also holds the [Inada Conditions](https://en.wikipedia.org/wiki/Inada_conditions). The proof is more complex then.
luchonacho
CC BY-SA 3.0
2017-10-02T11:09:19.523Z
@luchonacho That is definitely not a WLOG situation. You have to make that assumption in the beginning. You can explain that it is customary, but it does come with loss of generality.
Giskard
CC BY-SA 3.0
2017-10-02T11:16:05.447Z
@denesp it is WLOG within Cobb-Douglas. Personally, I have never seen a Solow model with a PF other than CD; hence my interpretation of "regular Solow growth model". If the OP would have been clearer, we could get what s/he meant in the first place.
luchonacho
CC BY-SA 3.0
2017-10-02T16:46:04.860Z
It is true, I was thinking of the "regular" Solow growth model as one with a CD production function. However my broader interest, especially after @luchonacho's answer, was in taking any arbitrary function with constant returns to scale that satisfies the Inada conditions (these https://en.wikipedia.org/wiki/Inada_conditions). Apologies for the confusion and thank you for the thorough answer.
spacetime
CC BY-SA 3.0
2017-10-01T21:07:07.100Z
2017-10-02T11:42:18.280Z
2017-10-02T11:42:18.280Z
luchonacho
<p>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 $\frac{\partial k^*}{\partial s}&gt;0$. 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 $\frac{n+g+\delta}{s} &gt; f'(k^*)$, and that this is a fact implied by the Inada assumptions.</p>

<p>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).</p>

CC BY-SA 3.0
2017-09-30T20:41:25.027Z
Do you make the assumption that $f$ is concave?
Giskard
CC BY-SA 3.0
2017-10-01T22:48:14.763Z
Yes, it is one of the Inada conditions.
spacetime
<macroeconomics><mathematical-economics><solow>
Solow growth model - analytic proof that Inada conditions imply steady state capital is increasing in the savings rate
spacetime
# Solow growth model - analytic proof that Inada conditions imply steady state capital is increasing in the savings rate

(Asked by: username_0 on 2017-09-30)

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).

**Tags:** macroeconomics, mathematical-economics, solow

> **Comments:**
> **username_1**: Do you make the assumption that $f$ is concave?  
> **username_0**: Yes, it is one of the Inada conditions.  

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## 1 answers

### Answer by username_2 on 2017-10-01. Score: 3 | Accepted answer

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](https://en.wikipedia.org/wiki/Convex_function#Properties), 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

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

which is what we wanted to demonstrate.

> **Comments:**
> **username_0**: Very nice. Everything makes sense; I just have one lingering concern. Why can you assume WLOG that the function takes on some Cobb-Douglas form? I had heard from someone that given the Inada conditions the production function basically has to be "asymptotically Cobb-Douglas", perhaps it has to do with that? Would love some clarification! Many thanks  
> **username_2**: @username_0 Well, you mentioned the "regular Solow growth model", which is based on a Cobb-Douglas. Is that not the case? A general CES also holds the [Inada Conditions](https://en.wikipedia.org/wiki/Inada_conditions). The proof is more complex then.  
> **username_1**: @username_2 That is definitely not a WLOG situation. You have to make that assumption in the beginning. You can explain that it is customary, but it does come with loss of generality.  
> **username_2**: @denesp it is WLOG within Cobb-Douglas. Personally, I have never seen a Solow model with a PF other than CD; hence my interpretation of "regular Solow growth model". If the OP would have been clearer, we could get what s/he meant in the first place.  
> **username_0**: It is true, I was thinking of the "regular" Solow growth model as one with a CD production function. However my broader interest, especially after @username_2's answer, was in taking any arbitrary function with constant returns to scale that satisfies the Inada conditions (these https://en.wikipedia.org/wiki/Inada_conditions). Apologies for the confusion and thank you for the thorough answer.  

---

