<p>To partially answer this question, in Advanced Macroeconomics by Romer, question 1.5(c) asks </p>

<blockquote>
  <p>"What saving rate is needed to yield the golden-rule capital stock?"</p>
</blockquote>

<p>Answering this tells us what $s_{gold}$ is.</p>

<p>But before we can find this we must find the golden rule level of capital stock $(k_{gold})$.  Which we shall derive with the following equations: The intensive form of the production function, the dynamics of capital in the Solow Model, and consumption per unit of effective labour</p>

<p>\begin{cases}
Y_t = (A_t L_t)^{1-\alpha} K_t^{\alpha} = A_t L_t \left( \frac{K_t}{A_t L_t} \right)^{\alpha} = A L k^{\alpha} \\
\Rightarrow Y/AL = y = k^{\alpha} \\
\dot{k} = s y - (n+g+\delta) k \\
c = (1-s)y \\
\end{cases}</p>

<p>Under the balanced growth path, $\dot{k}=0$ so </p>

<p>$$ \displaystyle s k^{* \alpha} = (n+g+\delta)k^* \Rightarrow k^* = \left( \frac{n+g+\delta}{s} \right)^{\frac{1}{\alpha-1}} $$</p>

<p>We maximize $c^*$ with respect to $k^*$ to implicitly define the golden-rule level of capital per unit of effective labour</p>

<p>$$ \displaystyle \frac{\partial }{\partial k^*} c^* = \frac{\partial }{\partial k^*} (1-s)k^{* \alpha} = (1-s)(\alpha k^{* \alpha - 1}) $$</p>

<p>$$ \displaystyle = \alpha (1 - s) \left( \frac{n+g+\delta}{s} \right)^{\frac{\alpha-1}{\alpha-1}} $$</p>

<p>Next replace $s$ with $(n+g+\delta)k^{* 1-\alpha}$</p>

<p>$$ \displaystyle = \alpha (1 - (n+g+\delta)k^{* 1-\alpha}) \left( \frac{n+g+\delta}{(n+g+\delta)k^{* 1-\alpha}} \right) $$</p>

<p>$$ \displaystyle  0 = \alpha (k^{* (\alpha-1)} - (n+g-\delta)) \Rightarrow k^*_{gold} = \left( \frac{\alpha}{n+g+\delta} \right)^{1/(1-\alpha)} $$</p>

<p>We can finally compute, $s_{gold}$</p>

<p>$$ \displaystyle s_{gold} = (n+g+\delta) k_{gold}^{*1-\alpha} = (n+g+\delta) \left( \frac{\alpha}{n+g+\delta}\right)^{(1-\alpha)/(1-\alpha)} = \alpha $$</p>

<p>So for a Cobb-Douglas production function the golden-rule saving rate should be whatever $\alpha$ is.</p>

CC BY-SA 3.0
2016-03-16T15:06:09.433Z
2016-03-16T15:06:09.433Z
Sunhwa
<p>Consider Solow growth model. Let $s$ denote savings rate. </p>

<p>Suppose we estimate China invests 40% of it GDP rather than consuming. Assuming China is dynamically inefficient because they are a developing nation, how can I estimate what it's $s_{gold}$ is? Furthermore, how do I decide whether China is at this $s_{gold}$ versus some other $s$?</p>

<p>Furthermore, how does one determine this in general? Like are there distinct characteristic patterns that allow me to understand these rates? </p>

<p>Let our production function be $Y_t = (A_tL_t)^{1-\alpha} K_t^{\alpha}$.</p>

<p><strong>I am asking how one calculates $s$ and $s_{gold}$ in <em>practice</em>, as in, if I had to find realistic numbers for these how would I do it?</strong></p>

CC BY-SA 3.0
2016-03-16T15:09:51.450Z
For the purposes of calculating the golden-rule savings rate, why are you assuming that "China is dynamically inefficient"?
Alecos Papadopoulos
CC BY-SA 3.0
2016-03-16T17:00:29.817Z
I added a justification, see if u think its sufficient
Stan Shunpike
<macroeconomics>
Is China at the golden savings rate s? If so, what is it?
Stan Shunpike
# Is China at the golden savings rate s? If so, what is it?

(Asked by: username_0 on 2016-03-16)

Consider Solow growth model. Let $s$ denote savings rate.

Suppose we estimate China invests 40% of it GDP rather than consuming. Assuming China is dynamically inefficient because they are a developing nation, how can I estimate what it's $s\_{gold}$ is? Furthermore, how do I decide whether China is at this $s\_{gold}$ versus some other $s$?

Furthermore, how does one determine this in general? Like are there distinct characteristic patterns that allow me to understand these rates?

Let our production function be $Y\_t = (A\_tL\_t)^{1-\alpha} K\_t^{\alpha}$.

**I am asking how one calculates $s$ and $s\_{gold}$ in *practice*, as in, if I had to find realistic numbers for these how would I do it?**

**Tags:** macroeconomics
> **Comments:**
> **username_1**: For the purposes of calculating the golden-rule savings rate, why are you assuming that "China is dynamically inefficient"?  
> **username_0**: I added a justification, see if u think its sufficient  

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## 1 answers

### Answer by username_2 on 2016-03-16. Score: 6

To partially answer this question, in Advanced Macroeconomics by Romer, question 1.5(c) asks

> "What saving rate is needed to yield the golden-rule capital stock?"

Answering this tells us what $s\_{gold}$ is.

But before we can find this we must find the golden rule level of capital stock $(k\_{gold})$. Which we shall derive with the following equations: The intensive form of the production function, the dynamics of capital in the Solow Model, and consumption per unit of effective labour

\begin{cases}
Y\_t = (A\_t L\_t)^{1-\alpha} K\_t^{\alpha} = A\_t L\_t \left( \frac{K\_t}{A\_t L\_t} \right)^{\alpha} = A L k^{\alpha} \\
\Rightarrow Y/AL = y = k^{\alpha} \\
\dot{k} = s y - (n+g+\delta) k \\
c = (1-s)y \\
\end{cases}

Under the balanced growth path, $\dot{k}=0$ so

$$ \displaystyle s k^{\* \alpha} = (n+g+\delta)k^\* \Rightarrow k^\* = \left( \frac{n+g+\delta}{s} \right)^{\frac{1}{\alpha-1}} $$

We maximize $c^\*$ with respect to $k^\*$ to implicitly define the golden-rule level of capital per unit of effective labour

$$ \displaystyle \frac{\partial }{\partial k^\*} c^\* = \frac{\partial }{\partial k^\*} (1-s)k^{\* \alpha} = (1-s)(\alpha k^{\* \alpha - 1}) $$

$$ \displaystyle = \alpha (1 - s) \left( \frac{n+g+\delta}{s} \right)^{\frac{\alpha-1}{\alpha-1}} $$

Next replace $s$ with $(n+g+\delta)k^{\* 1-\alpha}$

$$ \displaystyle = \alpha (1 - (n+g+\delta)k^{\* 1-\alpha}) \left( \frac{n+g+\delta}{(n+g+\delta)k^{\* 1-\alpha}} \right) $$

$$ \displaystyle 0 = \alpha (k^{\* (\alpha-1)} - (n+g-\delta)) \Rightarrow k^\*\_{gold} = \left( \frac{\alpha}{n+g+\delta} \right)^{1/(1-\alpha)} $$

We can finally compute, $s\_{gold}$

$$ \displaystyle s\_{gold} = (n+g+\delta) k\_{gold}^{\*1-\alpha} = (n+g+\delta) \left( \frac{\alpha}{n+g+\delta}\right)^{(1-\alpha)/(1-\alpha)} = \alpha $$

So for a Cobb-Douglas production function the golden-rule saving rate should be whatever $\alpha$ is.


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