Growth Rate of Solow Model

Question

Conside the following Cobb-Douglas Production Function: $$Y_t=\bar AK_t^{1/3}\bar L^{2/3}$$

The growth rate of per capita GDP for this equation in continuous time is $\frac{y'_t}{y_t}= \frac{1}{3K_t}$ where $y_t=\frac{Y_t}{\bar L}$

Using the Law of Motion of Capital:

$$\Delta K_{t+1}=\bar sY_t-dK_t \implies \frac{\Delta K_{t+1}}{K_t}=\bar s \frac{Y_t}{K_t}-d$$

where $\bar s$ represents the savings rate.

And Capital-Output Ratio in Steady State: $$\frac{K^*}{Y^*}=\frac{\bar s}{\bar d}$$

where $\bar d$ represents the depreciation rate

Prove that the growth rate of per capita GDP is equal to $$\frac{1}{3} \bar s \frac{Y^*}{K^*} (\frac{(K^*)^{2/3}}{K_t^{2/3}} -1)$$

I have no idea how am I supposed to get the equation from $\frac{y'_t}{y_t}= \frac{1}{3K_t}$, because it seems like this is not derived by substitution.

After some work, I realised that the following equation looks somewhat like:

$$\frac{1}{3} \bar s \frac{Y^*}{K^*} (\frac{(K^*)^{2/3}}{K_t^{2/3}} -1) =\frac{1}{3} \bar d \frac{\Delta K_{t+1}}{K_t}$$

Appreciate it if anyone can provide me with some advice on how I should go on further! Many thanks!

Answer 1

I do not directly see what intuition you are supposed to get from this growth rate equation, and the resulting math exercise but here is a sketch of the proof.

First determine the steady state capital by setting $\Delta K_{t+1}=0$ and solving for $K^*$.

Substitute this value for the $(K^*)^{(2/3)}$ in the fraction. Replace the $ \frac{Y^*}{K^*}$ in the same expression with the value that you get capital-output steady state ratio, and work the $d$ into the parentheses.

The expression you have now can be shown to be the same as the growth rate of $y_t$ that can be found from the production function.