Balanced growth path with specific technology

Question

Resource constraint: $Y_t =C_t +I_t $
CRS Production function: $Y_t =K_t^{\alpha} (N_t X_t )^{1-\alpha}$
Investment function: $I_t =\frac{1}{q_t}(K_{t+1} -(1-\delta)K_t )$
The labor-augmenting technological progress: $X_{t+1} =\gamma_X X_t $
The specific technological progress: $q_{t+1} =\gamma_q q_t$

Show that if $C_t$, $K_{t+1}$, $Y_t$ cannot increase with the same rate.

Answer 1

Assume there is a solution where $ Y, K, C$ all have constant growth rates.

Impose that solution on Euler and you will have $Y/K$ is constant i.e growth rates of $Y$ and $K$ are same.

Use the production function and you will have growth rate of $Y = \gamma_x + n $ where $n$ is the growth rate of $N$.

Then use the budget constraint (replacing $I$) and you will find that $C/K$ is not a constant because of $q$.

Thus if a solution exists where output, capital and consumption all have constant growth rates, in such a solution output and capital grow at the same rate which is different from that of consumption.

1. I deliberately did not show how to manipulate the production function and budget constraint - the first time I worked on this type of problem, figuring out the manipulation was fun; perhaps you will also enjoy it :)

2. Don’t take logs and differentiate - your notation suggests you have discrete time. :)