How to prove that GDP grows at the same rate as consumption on Lab Equipment Model?

Question

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!

Answer 1

The household side of this model is pretty standard. Denote $K(t)$ be household's assets at time $t$. Then the transversality condition (which is an optimizing condition, not a constraint), is

$$\lim_{t \to \infty} [e^{-\rho t}\lambda(t) K(t)] = 0$$

where $\lambda(t)$ is the current value multiplier on assets in the Hamiltonian. Given the assumed form of the utility function we have the f.o.c.

$$\lambda(t) = \frac {1}{c(t)^{\theta}}$$

So we want

$$\lim_{t \to \infty} [e^{-\rho t}\frac {1}{c(t)^{\theta}} K(t)] = 0$$

and since in the model we have a constant consumption growth rate, $g_c$ we have

$$c(t)^{\theta} = c(0)^{\theta}\cdot e^{g_c\theta t}$$

Ignoring inconsequential constants, we require

$$\lim_{t \to \infty} [\frac {1}{e^{(\rho+g_c\theta) t}} K(t)] = 0$$

Use the relation between household assets and the number of goods $N$, the relation between income (from labor and assets) to aggregate output, as well as the expression of aggregate output as a function of $N$, to obtain a first-order linear differential equation in $N$, and proceed as in pages 208-209 of the book.