I'm studying the Lab Equipment Model (Barro-Sala-i-Martin, Chapter 6). I'm having trouble when trying to prove that every variables grows at the same rate as consumption.
I was able to prove that $\dot{C}/C = g_{c}$, which is constant. Also, I have that
$b \dot{N} = Y - X - C$ where $b$ is a parameter,
$Y/N$ is a constant related to the interest rate and
$X = a^{2}Y$, where $a$ is a parameter from the production function.
From the second and third equations above, it's clear that $Y,X$ and $N$ grow at the same rate. However, it's not clear that this rate should be $g_{c}$ or even constant at all. I'm trying to use the first equation to show it, but I couldn't do it.
The book forgoes this proof. It says that it's quite similar to the AK model since there's no transitional dynamics. Is there a simple way to show this result? I think it should follow from a simple manipulation of the first equation. Any ideas? thanks a lot in advance!