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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!

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1 Answer 1

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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?

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[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, 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.

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  • $\begingroup$ I was afraid of that... thanks for your answer BakerStreet, I have been struggling with this for some weeks now... $\endgroup$
    – Veronica
    Commented Feb 4 at 8:40
  • $\begingroup$ You are welcome! $\endgroup$ Commented Feb 4 at 9:15
  • $\begingroup$ @Veronica I added a discussion of the CES case in the Solow's 1956 paper. $\endgroup$ Commented Feb 5 at 15:35

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