<p>Think about what is <span class="math-container">$g_i$</span>. How does <span class="math-container">$g_k = (1-\delta) + \frac{i_t}{k_t}$</span> relate to the parts of <span class="math-container">$g_i$</span>? Notice that the left hand side of this equation is constant for all <span class="math-container">$t$</span>.</p>
<p>The same logic applies for <span class="math-container">$g_c$</span> and the corresponding equation. What can you say about <span class="math-container">$\frac{i_t}{k_t}$</span> and <span class="math-container">$\frac{i_{t+1}}{k_{t+1}}$</span> given previous derivations?</p>

CC BY-SA 4.0
2021-12-14T12:28:13.383Z
2021-12-14T12:28:13.383Z
LudwigNagasena
<p>One feature of Balanced Growth Path is that growth rates are equal along it:</p>
<p>Consider the following conditions:</p>
<p><span class="math-container">$y_t= Xk_t$</span></p>
<p><span class="math-container">$y_t = c_t + i_t$</span></p>
<p><span class="math-container">$k_{t+1} = (1-\delta)k_t$</span></p>
<p>From first equation it is easy enough to show <span class="math-container">$\frac{y_{t+1}}{y_t} = g_y = \frac{Xk_{t+1}}{Xk_t} = g_k$</span></p>
<p>But how do I go about showing other growth rates are equal too. My lecture notes claim:</p>
<p><span class="math-container">$g_k = (1-\delta) + \frac{i_t}{k_t}$</span> <span class="math-container">$\implies$</span> <span class="math-container">$g_k = g_i$</span>, and</p>
<p><span class="math-container">$X = \frac{c_t}{k_t} + \frac{i_t}{k_t} $</span> <span class="math-container">$\implies$</span> <span class="math-container">$g_k = g_c$</span>.</p>
<p>So along BGP: <span class="math-container">$g_k = g_c = g_y = g_i$</span>.</p>
<p>How do we arrive at the conclusion for the last two?</p>

CC BY-SA 4.0
2021-12-13T18:17:47.860Z
Can you please share link to the notes? It would help to see all assumptions and set up of the model. There are infinite ways of setting an endogenous growth model
1muflon1
CC BY-SA 4.0
2021-12-13T20:47:27.547Z
The law of motion of capital appears wrong. Where is investment in it?
Alecos Papadopoulos
<macroeconomics><economic-growth><endogenous-growth>
Showing Growth rates are equal in BGP?
Kinno
# Showing Growth rates are equal in BGP?

(Asked by: username_0 on 2021-12-13)

One feature of Balanced Growth Path is that growth rates are equal along it:

Consider the following conditions:

$y\_t= Xk\_t$

$y\_t = c\_t + i\_t$

$k\_{t+1} = (1-\delta)k\_t$

From first equation it is easy enough to show $\frac{y\_{t+1}}{y\_t} = g\_y = \frac{Xk\_{t+1}}{Xk\_t} = g\_k$

But how do I go about showing other growth rates are equal too. My lecture notes claim:

$g\_k = (1-\delta) + \frac{i\_t}{k\_t}$ $\implies$ $g\_k = g\_i$, and

$X = \frac{c\_t}{k\_t} + \frac{i\_t}{k\_t} $ $\implies$ $g\_k = g\_c$.

So along BGP: $g\_k = g\_c = g\_y = g\_i$.

How do we arrive at the conclusion for the last two?

**Tags:** macroeconomics, economic-growth, endogenous-growth
> **Comments:**
> **username_1**: Can you please share link to the notes? It would help to see all assumptions and set up of the model. There are infinite ways of setting an endogenous growth model  
> **username_2**: The law of motion of capital appears wrong. Where is investment in it?  

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## 1 answers

### Answer by username_3 on 2021-12-14. Score: 1

Think about what is $g\_i$. How does $g\_k = (1-\delta) + \frac{i\_t}{k\_t}$ relate to the parts of $g\_i$? Notice that the left hand side of this equation is constant for all $t$.

The same logic applies for $g\_c$ and the corresponding equation. What can you say about $\frac{i\_t}{k\_t}$ and $\frac{i\_{t+1}}{k\_{t+1}}$ given previous derivations?


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