# Showing Growth rates are equal in BGP?

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?

• 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
Dec 13 '21 at 18:17
• The law of motion of capital appears wrong. Where is investment in it? Dec 13 '21 at 20:47

## 1 Answer

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?