Balanced growth path with investment-specific technology case

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 investment-specific technological progress: $$q_{t+1} =\gamma_q q_t$$

Show that why $$C_t$$, $$K_{t+1}$$, $$Y_t$$ do not increase the same rate and we can not detrend them with the same variable.

I tried to solve this using Kaldor's stylized facts but too hard for me.
Does anyone can help me to solve this problem?

– 1muflon1
Jan 8 at 12:39
• Is g same as q ?
– erik
Jan 9 at 18:18
• Anyway these proofs usually require showing that Y/K is constant on the Balanced Growth Path - which requires looking at the Euler. What does your Euler say?
– erik
Jan 9 at 18:21
• @erik You're right. That's typo and $g_t=q_t$. I edited it. And according to your second reply, euler means consumption's euler equation? How can I show $Y/K$ raito in euler equation: $u'(c_t)=\beta u'(c_{t+1})(MPK_t +1-\delta )$? Jan 10 at 4:07
• @1muflon1 I used capital's law of motion and derive $\gamma_q * \gamma_I= \gamma_X$ where $\gamma_I$ means investment's growth rate. I stuck here. Jan 10 at 4:11

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. :)

• Euler equation of this problem: $(C_t)^{-1} =\beta (C_{t+1})^{-1} [q_t * MPK_{t+1} +\frac{q_t}{q_{t+1}}(1-\delta)$]. In here how can I derive $Y/K$ ratio is constant? $\gamma_C =\frac{C_{t+1}}{C_t} =\beta [q_t * \alpha \frac{Y}{K} +{\gamma_q}^{-1} (1-\delta)$] : We have $q_t$ in front of $\frac{Y}{K}$. 11 hours ago