<p>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.</p>

<p>First determine the steady state capital by setting <span class="math-container">$\Delta K_{t+1}=0$</span> and solving for <span class="math-container">$K^*$</span>. </p>

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

<p>The expression you have now can be shown to be the same as the growth rate of <span class="math-container">$y_t$</span> that can be found from the production function.</p>

CC BY-SA 4.0
2019-02-08T15:52:05.393Z
2019-02-08T15:52:05.393Z
Maarten Punt
<p>Conside the following Cobb-Douglas Production Function:
<span class="math-container">$$Y_t=\bar AK_t^{1/3}\bar L^{2/3}$$</span></p>

<p>The growth rate of per capita GDP for this equation in continuous time is <span class="math-container">$\frac{y'_t}{y_t}= \frac{1}{3K_t}$</span> where <span class="math-container">$y_t=\frac{Y_t}{\bar L}$</span></p>

<p>Using the Law of Motion of Capital:</p>

<p><span class="math-container">$$\Delta K_{t+1}=\bar sY_t-dK_t \implies \frac{\Delta K_{t+1}}{K_t}=\bar s \frac{Y_t}{K_t}-d$$</span></p>

<p>where <span class="math-container">$\bar s$</span> represents the savings rate.</p>

<p>And Capital-Output Ratio in Steady State:
<span class="math-container">$$\frac{K^*}{Y^*}=\frac{\bar s}{\bar d}$$</span></p>

<p>where <span class="math-container">$\bar d$</span> represents the depreciation rate</p>

<p>Prove that the growth rate of per capita GDP is equal to
<span class="math-container">$$\frac{1}{3} \bar s \frac{Y^*}{K^*} (\frac{(K^*)^{2/3}}{K_t^{2/3}} -1)$$</span></p>

<p>I have no idea how am I supposed to get the equation from <span class="math-container">$\frac{y'_t}{y_t}= \frac{1}{3K_t}$</span>, because it seems like this is not derived by substitution.</p>

<p>After some work, I realised that the following equation looks somewhat like:</p>

<p><span class="math-container">$$\frac{1}{3} \bar s \frac{Y^*}{K^*} (\frac{(K^*)^{2/3}}{K_t^{2/3}} -1) =\frac{1}{3} \bar d \frac{\Delta K_{t+1}}{K_t}$$</span></p>

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

<solow>
Growth Rate of Solow Model
Derp
# Growth Rate of Solow Model

(Asked by: username_0 on 2019-02-06)

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!

**Tags:** solow


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## 1 answers

### Answer by username_1 on 2019-02-08. Score: 1 | Accepted answer

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.


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