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We know: $PCI = \frac{Y(Income)}{P(population)}$ and that $PCI$ growth rate is $\frac{\Delta PCI}{PCI}$.

How is this equal to the difference between income growth rate and population growth rate?

In other words how is $\left(\frac{∆\frac{Y}{P}}{\frac{Y}P}\right) = (∆Y/Y) - (∆P/P)$ ?

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Knowing that: $$\frac{\Delta PCI}{PCI}= \left(\frac{∆\frac{Y}{P}}{\frac{Y}P}\right) $$

If we rearrange the terms on the right hand side:

$$\frac{\Delta PCI}{PCI}=\left(\frac{\frac{\Delta Y}{Y}}{\frac{\Delta P}{P}}\right)$$

In order to get your last equation we would have to log the variables, for logged growth rates. This makes things easier because we can think of the variables in terms of percentages.

Logging the variables we get $$\ln\left(\frac{\Delta PCI}{PCI}\right)=\ln\left(\frac{\frac{\Delta Y}{Y}}{\frac{\Delta P}{P}}\right)$$

$$\ln\left(\frac{\Delta PCI}{PCI}\right)=\ln\left(\frac{\Delta Y}{Y}\right)-\ln\left(\frac{\Delta P}{P}\right)$$

in more formal notation used in macroeconomics:

$$\hat{PCI}=\hat{y}-\hat{p}$$

hope this helps.

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    $\begingroup$ It seems you are arguing $$ \frac{\Delta PCI}{PCI} \approx \ln\left(\frac{\Delta PCI}{PCI}\right) = \ln\left(\frac{\Delta Y}{Y}\right)-\ln\left(\frac{\Delta P}{P}\right) \approx (∆Y/Y) - (∆P/P). $$ The only questions left unanswered is why $\ln\left(\frac{\Delta Y}{Y}\right)-\ln\left(\frac{\Delta P}{P}\right)$ is a good approximation for $(∆Y/Y) - (∆P/P)$ and why $\ln\left(\frac{\Delta PCI}{PCI}\right)$ is a a good approximation for $\frac{\Delta PCI}{PCI}$? $\endgroup$ – Giskard Nov 19 '17 at 19:42
  • $\begingroup$ @denesp the difference between logs can be interpreted as a percentage change. The difference between $(∆Y/Y) - (∆P/P)$ gives us a decimal, however it is difficult to interpret due to the non-comparability of units i.e population and income. $\endgroup$ – EconJohn Nov 19 '17 at 19:48
  • $\begingroup$ @denesp I have no clue how else one would come to the result in the OP without using logs $\endgroup$ – EconJohn Nov 19 '17 at 19:50
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    $\begingroup$ We don't take the logarithm of the growth rate, we take the logarithm of 1+ the growth rate, which for growth rates in $(-0.10, 0.10)$ has an approximation mistake less then half percentage point. (@denesp also) $\endgroup$ – Alecos Papadopoulos Nov 19 '17 at 21:36
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    $\begingroup$ @denesp Well yes, last time I checked numbers less than unity had negative logarithm, while negative numbers had no logarithm... I don't think the WMA (World Mathematical Authority) has changed this particular rule as of yet. $\endgroup$ – Alecos Papadopoulos Nov 20 '17 at 0:39
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Yes, there is a rule: the growth rate of the ratio of two variables is equal to the difference between the two growth rates. To show this, assume, $PCI=\frac{Y}{P}=y$, take logs of both sides $log(y)=log(Y)-log(P)$. Now take time deriveative$$\frac{dlog(y)}{dt}=\frac{dlog(Y)}{dt}-\frac{dlog(P)}{dt}$$ and note that the growth rate of variable $x$ is $g_x=\frac{dlog(x)}{dt}$

hence, we have $$g_y=g_Y-g_P$$

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