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?

  • $\begingroup$ Can you please include your attempt in the question? $\endgroup$
    – 1muflon1
    Jan 8 at 12:39
  • $\begingroup$ Is g same as q ? $\endgroup$
    – erik
    Jan 9 at 18:18
  • $\begingroup$ 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? $\endgroup$
    – erik
    Jan 9 at 18:21
  • $\begingroup$ @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: $u'(c_t)=\beta u'(c_{t+1})(MPK_t +1-\delta )$? $\endgroup$
    – guest
    Jan 10 at 4:07
  • $\begingroup$ @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. $\endgroup$
    – guest
    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. :)

  • $\begingroup$ 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}$. $\endgroup$
    – guest
    11 hours ago

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