<p>Assume there is a solution where <span class="math-container">$ Y, K, C$</span> all have constant growth rates.</p>
<p>Impose that solution on Euler and you will have <span class="math-container">$Y/K$</span> is constant i.e growth rates of <span class="math-container">$Y$</span> and <span class="math-container">$K$</span> are same.</p>
<p>Use the production function and you will have growth rate of <span class="math-container">$Y = \gamma_x + n $</span> where <span class="math-container">$n$</span> is the growth rate of <span class="math-container">$N$</span>.</p>
<p>Then use the budget constraint (replacing <span class="math-container">$I$</span>) and you will find that <span class="math-container">$C/K$</span> is not a constant because of <span class="math-container">$q$</span>.</p>
<p>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.</p>
<ol>
<li><p>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 :)</p>
</li>
<li><p>Don’t take logs and differentiate - your notation suggests you have discrete time. :)</p>
</li>
</ol>

CC BY-SA 4.0
2022-01-11T20:43:25.130Z
2022-01-11T20:43:25.130Z
erik
<p>Resource constraint: <span class="math-container">$Y_t =C_t +I_t $</span><br />
CRS Production function: <span class="math-container">$Y_t =K_t^{\alpha} (N_t X_t )^{1-\alpha}$</span><br />
Investment function: <span class="math-container">$I_t =\frac{1}{q_t}(K_{t+1} -(1-\delta)K_t )$</span><br />
The labor-augmenting technological progress: <span class="math-container">$X_{t+1} =\gamma_X X_t $</span><br />
The specific technological progress: <span class="math-container">$q_{t+1} =\gamma_q q_t$</span></p>
<p>Show that if <span class="math-container">$C_t$</span>, <span class="math-container">$K_{t+1}$</span>, <span class="math-container">$Y_t$</span> cannot increase with the same rate.</p>

CC BY-SA 4.0
2022-01-08T12:39:37.547Z
Can you please include your attempt in the question?
1muflon1
CC BY-SA 4.0
2022-01-09T18:18:20.477Z
Is g same as q ?
erik
CC BY-SA 4.0
2022-01-09T18:21:46.597Z
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
CC BY-SA 4.0
2022-01-10T04:11:09.433Z
@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.
guest
<macroeconomics><balanced-growth>
Balanced growth path with specific technology
guest
# Balanced growth path with specific technology

(Asked by: username_0 on 2022-01-08)

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.

**Tags:** macroeconomics, balanced-growth
> **Comments:**
> **username_1**: Can you please include your attempt in the question?  
> **username_2**: Is g same as q ?  
> **username_2**: 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?  
> **username_0**: @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.  

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## 1 answers

### Answer by username_2 on 2022-01-11. Score: 2 | Accepted answer

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. :)


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