0
$\begingroup$

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 :)

$\endgroup$
2
$\begingroup$

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, 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 explaining the rules of total differentials and showing another example of how to apply them to an equation.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.