Showing Growth rates are equal in BGP?

Question

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?

Answer 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?