<p>First, note that the growth rate of <span class="math-container">$\mu$</span> is defined as <span class="math-container">$\dot\mu = \frac{ d\mu }{ \mu }$</span>. Therefore, you will have to</p>

<ol>
<li>take the total differential of the equation; and</li>
<li>divide by <span class="math-container">$\mu$</span>.</li>
</ol>

<p>For the first step, as in <a href="https://economics.stackexchange.com/questions/30635/hep-with-total-differentiation-of-an-ad-function">your previous question</a>, simply use the rules of total differentials, specifically the product rule that states that for variables <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>, the total differential of <span class="math-container">$XY$</span> is given by</p>

<p><span class="math-container">$$d(XY) = Y \, dX + X \, dY$$</span></p>

<p>Assuming that <span class="math-container">$s_\pi$</span>, <span class="math-container">$s_W$</span> and <span class="math-container">$\zeta$</span> are (constant) parameters (so that by the constant rule, <span class="math-container">$ds_\pi = ds_W = d\zeta = 0$</span>), we can apply this to your equation:</p>

<p><span class="math-container">$$\begin{align}
d\mu &amp; = d \left[ s_\pi - v (s_\pi - s_W) + \zeta \right] \\
&amp; = ds_\pi - d \left[ v (s_\pi - s_W) \right] + d\zeta \\
&amp; = -\left[ (s_\pi - s_W) \, dv + v \, d(s_\pi - s_W) \right] \\
&amp; = -(s_\pi - s_W) \, dv
\end{align}$$</span></p>

<p>Now divide both sides by <span class="math-container">$\mu$</span>, noting that since <span class="math-container">$\dot v = \frac{ dv }v$</span>, we also have <span class="math-container">$dv = v \dot v$</span>:</p>

<p><span class="math-container">$$\begin{align}
\dot\mu = \frac{ d\mu }\mu &amp; = -\frac{ dv }\mu (s_\pi - s_W) \\
&amp; = -\frac v\mu (s_\pi - s_W) \dot v
\end{align}$$</span></p>

<p>which is the desired result. As before, see  chapter 8 of Chiang and Wainwright, <em>Fundamental Methods of Mathematical Economics</em>, 4th ed., McGraw-Hill 2005, for more on total differentials. Alternatively, see <a href="https://economics.stackexchange.com/questions/30639/rules-of-total-differentials">this Q&amp;A post of mine</a> explaining the rules of total differentials and showing another example of how to apply them to an equation.</p>

CC BY-SA 4.0
2019-08-24T16:14:54.033Z
2019-08-24T16:14:54.033Z
chsk
<p>I have the following equation:</p>

<p><span class="math-container">$$\mu =\left [s_{\pi }-v(s_{\pi }-s_{W})+\zeta   \right ]$$</span></p>

<p>And I have to derive its growth rate, which is:</p>

<p><span class="math-container">$$\dot \mu =-\frac{v}{\mu } (s_{\pi }-s_{W})\dot v$$</span></p>

<p>Do you have any idea about the approach to follow? Thank you so much :)</p>

<macroeconomics><mathematical-economics><economic-growth><linear-algebra>
Derive the growth rate of an equation
Alessandro
# Derive the growth rate of an equation

(Asked by: username_0 on 2019-08-24)

I have the following equation:

$$\mu =\left [s\_{\pi }-v(s\_{\pi }-s\_{W})+\zeta \right ]$$

And I have to derive its growth rate, which is:

$$\dot \mu =-\frac{v}{\mu } (s\_{\pi }-s\_{W})\dot v$$

Do you have any idea about the approach to follow? Thank you so much :)

**Tags:** macroeconomics, mathematical-economics, economic-growth, linear-algebra


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## 1 answers

### Answer by username_1 on 2019-08-24. Score: 2

First, note that the growth rate of $\mu$ is defined as $\dot\mu = \frac{ d\mu }{ \mu }$. Therefore, you will have to

1. take the total differential of the equation; and
2. divide by $\mu$.

For the first step, as in [your previous question](https://economics.stackexchange.com/questions/30635/hep-with-total-differentiation-of-an-ad-function), simply use the rules of total differentials, specifically the product rule that states that for variables $X$ and $Y$, the total differential of $XY$ is given by

$$d(XY) = Y \, dX + X \, dY$$

Assuming that $s\_\pi$, $s\_W$ and $\zeta$ are (constant) parameters (so that by the constant rule, $ds\_\pi = ds\_W = d\zeta = 0$), we can apply this to your equation:

$$\begin{align}
d\mu & = d \left[ s\_\pi - v (s\_\pi - s\_W) + \zeta \right] \\
& = ds\_\pi - d \left[ v (s\_\pi - s\_W) \right] + d\zeta \\
& = -\left[ (s\_\pi - s\_W) \, dv + v \, d(s\_\pi - s\_W) \right] \\
& = -(s\_\pi - s\_W) \, dv
\end{align}$$

Now divide both sides by $\mu$, noting that since $\dot v = \frac{ dv }v$, we also have $dv = v \dot v$:

$$\begin{align}
\dot\mu = \frac{ d\mu }\mu & = -\frac{ dv }\mu (s\_\pi - s\_W) \\
& = -\frac v\mu (s\_\pi - s\_W) \dot v
\end{align}$$

which is the desired result. As before, see chapter 8 of Chiang and Wainwright, *Fundamental Methods of Mathematical Economics*, 4th ed., McGraw-Hill 2005, for more on total differentials. Alternatively, see [this Q&A post of mine](https://economics.stackexchange.com/questions/30639/rules-of-total-differentials) explaining the rules of total differentials and showing another example of how to apply them to an equation.


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