Comparative Statics on Balanced Growth Path

Question

It would be a silly question. In a model, I have found the BGP (balanced growth path) for all key variables.

As expected, these variables are constant variables ;

$$\mu^{BGP}=\frac{\alpha+\rho+\pi}{\epsilon\left(1+\pi\right)^{2}}$$

All paramaters on the right hand side are exogeneous constant parameters. Let's say $\mu$ is a variable for technological progress level.

For another variable, let's say capital accumulation $k$,I find a constant growth rate which is ;

$$g_{k}=\frac{\alpha\left(\rho+\pi-\delta\left(\frac{1-\alpha}{\alpha}\right)\right)}{1-\alpha}$$

The parameters on the RHS are again exogenous parameters.

My question is : Is it possible to see the effect of the variable $\mu$ on the growth rate of capital $g_k$ ?

My way of doing seems to me a little bit weird. I firstly put

$$\rho=\mu^{BGP}\epsilon\left(1+\pi\right)^{2}-\alpha-\pi$$

and replace it in $g_k$. So, I have

$$g_{k}=\frac{\alpha\left(\mu^{BGP}\epsilon\left(1+\pi\right)^{2}-\alpha-\delta\left(\frac{1-\alpha}{\alpha}\right)\right)}{1-\alpha}$$

After, I say that technological progress level at BGP affects positevly the growth rate of capital.

Do you think that it is correct to say that ? Making this kind of comparative statics analysis ?

Or any other way to do it in a more appropriate way ?

Answer 1

The approach is not correct, because $\mu^{BGP}$ is a result of underlying structural parameters. So to say "when $\mu^{BGP}$ increases..." immediately begs the question why it increases, which underlying parameter(s) has changed to cause such an increase... assume it was $\epsilon$ that decreased. But in this case $g_k$ is not affected at all, so you see that you cannot claim what you claim about $\mu^{BGP}$ affecting positively $g_k$.

What is meaningful to do is to perform comparative statics for the exogenous parameters, and record whether $\mu^{BGP}$ and $g_k$ move in the same or the opposite direction (or don't move) in each case. It may not be a causal link, but co-movement is also a useful result, and testable.