<p>The problem is that, as in most cases for the Solow model, <strong>we don't know an explicit analytical solution of the differential equation for <span class="math-container">$k_t$</span>,</strong> the so-called <em>fundamental equation of growth</em>, and in general one must rely on a qualitative analysis, as exemplified by the usual graphs of the Solow model.</p>
<p>The fundamental equation of the Solow model with the <em>total factor productivity</em>  <span class="math-container">$A_t$</span> in general is:</p>
<p><span class="math-container">$$\dot{k}_t = sA_tf(k_t) - (n+\delta)k_t \tag{1}.$$</span></p>
<p>Substitute into <span class="math-container">$(1)$</span> the expression of <span class="math-container">$f(k)$</span> for the <span class="math-container">$CES$</span> function, <span class="math-container">$f(k_t) = (ak_t^\rho + b)^{\frac{1}{\rho}}$</span>,
to obtain:</p>
<p><span class="math-container">$$\dot{k}_t = sA_t( (ak_t^\rho + b)^{\frac{1}{\rho}}) - (n+\delta)k_t \tag{2}.$$</span></p>
<p><strong>How to solve this differential equation in the unknown function <span class="math-container">$k_t$</span>, in order to know explicitly <span class="math-container">$k_t$</span>?</strong></p>
<p>Observe that:</p>
<ol>
<li><p>Equation <span class="math-container">$(2)$</span> depends on the function <span class="math-container">$A_t$</span>, which we don't know.</p>
</li>
<li><p>Even if we knew the function <span class="math-container">$A(t)$</span>, it is highly probable that we are not able to solve analytically the differential equation <span class="math-container">$(2)$</span>, because, unfortunately, in mathematics the class of differential equation we know how to solve is not very large, we don’t know how to solve very many differential equations.</p>
</li>
</ol>
<p>As a consequence, we can’t know <span class="math-container">$k^*$</span>.</p>
<p>As far as I know, the only case (of economic relevance) in which we know how to  solve explicitly, analytically, the Solow model is when  <strong>the production function is a Cobb-Douglas.</strong></p>
<p>In this case, it can be shown that the fundamental equation reduces to a type of differential equation, the so-called <em>Bernoulli equation</em>, for which we have an explicit method of solution.</p>
<p>For an explanation, if you are interested,  you can  see my answer in this thread  about the Cobb-Douglas production function:
<a href="https://economics.stackexchange.com/questions/18998/why-is-the-cobb-douglas-production-function-so-popular/55958#55958">Why is the Cobb-Douglas production function so popular?</a></p>
<p><span class="math-container">$$***$$</span></p>
<p><strong>[edit]</strong>  It came to mind   that a discussion of the <span class="math-container">$CES$</span> production function case is in the classical paper by <strong>Solow, <a href="http://piketty.pse.ens.fr/files/Solow1956.pdf" rel="nofollow noreferrer">A contribution to the Theory of growth, 1956</a>,</strong> pp. 77-78.</p>
<p>Solow examines the case <span class="math-container">$p={1\over {2}}$</span>, so that the <span class="math-container">$CES$</span> production function becomes:</p>
<p><span class="math-container">$$Y=(a\sqrt{K}+\sqrt {L})^2$$</span></p>
<p>And the fundamental equation becomes (he writes <span class="math-container">$r$</span> instead of <span class="math-container">$k$</span>):</p>
<p><span class="math-container">$$\dot r = s(a\sqrt r+1)^2-nr$$</span>.</p>
<p>Then he points out that:</p>
<blockquote>
<p>The solution has to be given implicitly: <span class="math-container">$$\left(\frac{A\sqrt r+1}{
 A\sqrt r_0+1}\right)^{1/A}\left(\frac{B\sqrt r+1}{ B\sqrt
 r_0+1}\right)^{1/B}= e^{\sqrt {nst}}$$</span></p>
<p>Once again it is easier to refer to a diagram.</p>
</blockquote>
<p>Therefore, the problem is that we haven’t an explicit solution for <span class="math-container">$r$</span>.</p>
<p>Moreover, here the total factor productivity <span class="math-container">$A_t$</span> is absent, and its eventual presence can evidently make things even more complicated.</p>

CC BY-SA 4.0
2024-02-04T08:40:30.757Z
I was afraid of that... thanks for your answer BakerStreet, I have been struggling with this for some weeks now...
Veronica
CC BY-SA 4.0
2024-02-04T09:15:51.337Z
You are welcome!
BakerStreet
CC BY-SA 4.0
2024-02-05T15:35:08.213Z
@Veronica I added a discussion of the CES case in the Solow's 1956 paper.
BakerStreet
CC BY-SA 4.0
2024-02-02T16:41:21.193Z
2024-02-05T15:21:40.887Z
2024-02-05T15:21:40.887Z
BakerStreet
<p>In an another post (Balanced growth path definition in the Hicks neutral steady state with technology), the problem of having a not stationary steady state in a Solow model with Hicks neutral technological shock was addressed.</p>
<p>It was proven that in the case of <span class="math-container">$Y_t = A_tF(K_t,L_t)$</span>, the rate of growth of Y is :</p>
<p><span class="math-container">$g_Y = g_A + g_L + \frac{f'(k^*)}{f(k^*)}\dot{k^*}$</span></p>
<p>with <span class="math-container">$k^*$</span> being the &quot;steady state&quot; of <span class="math-container">$k_t$</span> which comes by assuming that:</p>
<p><span class="math-container">$\dot{k}_t = sA_tf(k_t) - (n+\delta)k_t = 0$</span></p>
<p>Then <span class="math-container">$g_Y$</span> is only known if <span class="math-container">$f$</span> is known.</p>
<p>Let's assume that:</p>
<p><span class="math-container">$Y_t = A_t(aK_t^\rho + bL_t^\rho)^{\frac{1}{\rho}}$</span></p>
<p>then we can show that:</p>
<p><span class="math-container">$Y_t = A_tf(k_t)$</span></p>
<p>with <span class="math-container">$k_t = \frac{K_t}{L_t}$</span> and:</p>
<p><span class="math-container">$f(k_t) = (ak_t^\rho + b)^{\frac{1}{\rho}}$</span>.</p>
<p>Can anyone help me with the expression for:</p>
<p><span class="math-container">$\frac{f'(k^*)}{f(k^*)}\dot{k^*}$</span></p>
<p>Thanks!</p>

<solow><ces-function><balanced-growth>
Balanced growth path in the Hicks neutral technology and CES function
Veronica
# Balanced growth path in the Hicks neutral technology and CES function

(Asked by: username_0 on 2024-02-02)

In an another post (Balanced growth path definition in the Hicks neutral steady state with technology), the problem of having a not stationary steady state in a Solow model with Hicks neutral technological shock was addressed.

It was proven that in the case of $Y\_t = A\_tF(K\_t,L\_t)$, the rate of growth of Y is :

$g\_Y = g\_A + g\_L + \frac{f'(k^\*)}{f(k^\*)}\dot{k^\*}$

with $k^\*$ being the "steady state" of $k\_t$ which comes by assuming that:

$\dot{k}\_t = sA\_tf(k\_t) - (n+\delta)k\_t = 0$

Then $g\_Y$ is only known if $f$ is known.

Let's assume that:

$Y\_t = A\_t(aK\_t^\rho + bL\_t^\rho)^{\frac{1}{\rho}}$

then we can show that:

$Y\_t = A\_tf(k\_t)$

with $k\_t = \frac{K\_t}{L\_t}$ and:

$f(k\_t) = (ak\_t^\rho + b)^{\frac{1}{\rho}}$.

Can anyone help me with the expression for:

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

Thanks!

**Tags:** solow, ces-function, balanced-growth


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## 1 answers

### Answer by username_1 on 2024-02-02. Score: 3

The problem is that, as in most cases for the Solow model, **we don't know an explicit analytical solution of the differential equation for $k\_t$,** the so-called *fundamental equation of growth*, and in general one must rely on a qualitative analysis, as exemplified by the usual graphs of the Solow model.

The fundamental equation of the Solow model with the *total factor productivity* $A\_t$ in general is:

$$\dot{k}\_t = sA\_tf(k\_t) - (n+\delta)k\_t \tag{1}.$$

Substitute into $(1)$ the expression of $f(k)$ for the $CES$ function, $f(k\_t) = (ak\_t^\rho + b)^{\frac{1}{\rho}}$,
to obtain:

$$\dot{k}\_t = sA\_t( (ak\_t^\rho + b)^{\frac{1}{\rho}}) - (n+\delta)k\_t \tag{2}.$$

**How to solve this differential equation in the unknown function $k\_t$, in order to know explicitly $k\_t$?**

Observe that:

1. Equation $(2)$ depends on the function $A\_t$, which we don't know.
2. Even if we knew the function $A(t)$, it is highly probable that we are not able to solve analytically the differential equation $(2)$, because, unfortunately, in mathematics the class of differential equation we know how to solve is not very large, we don’t know how to solve very many differential equations.

As a consequence, we can’t know $k^\*$.

As far as I know, the only case (of economic relevance) in which we know how to solve explicitly, analytically, the Solow model is when **the production function is a Cobb-Douglas.**

In this case, it can be shown that the fundamental equation reduces to a type of differential equation, the so-called *Bernoulli equation*, for which we have an explicit method of solution.

For an explanation, if you are interested, you can see my answer in this thread about the Cobb-Douglas production function:
[Why is the Cobb-Douglas production function so popular?](https://economics.stackexchange.com/questions/18998/why-is-the-cobb-douglas-production-function-so-popular/55958#55958)

$$\*\*\*$$

**[edit]** It came to mind that a discussion of the $CES$ production function case is in the classical paper by **Solow, [A contribution to the Theory of growth, 1956](http://piketty.pse.ens.fr/files/Solow1956.pdf),** pp. 77-78.

Solow examines the case $p={1\over {2}}$, so that the $CES$ production function becomes:

$$Y=(a\sqrt{K}+\sqrt {L})^2$$

And the fundamental equation becomes (he writes $r$ instead of $k$):

$$\dot r = s(a\sqrt r+1)^2-nr$$.

Then he points out that:

> The solution has to be given implicitly: $$\left(\frac{A\sqrt r+1}{
> A\sqrt r\_0+1}\right)^{1/A}\left(\frac{B\sqrt r+1}{ B\sqrt
> r\_0+1}\right)^{1/B}= e^{\sqrt {nst}}$$
>
> Once again it is easier to refer to a diagram.

Therefore, the problem is that we haven’t an explicit solution for $r$.

Moreover, here the total factor productivity $A\_t$ is absent, and its eventual presence can evidently make things even more complicated.

> **Comments:**
> **username_0**: I was afraid of that... thanks for your answer username_1, I have been struggling with this for some weeks now...  
> **username_1**: You are welcome!  
> **username_1**: @username_0 I added a discussion of the CES case in the Solow's 1956 paper.  

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