Balanced growth path in the Hicks neutral technology and CES function

Question

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!

Answer 1

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.