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

<p>If we rearrange the terms on the right hand side:</p>

<p>$$\frac{\Delta PCI}{PCI}=\left(\frac{\frac{\Delta Y}{Y}}{\frac{\Delta P}{P}}\right)$$</p>

<p>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.</p>

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

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

<p>in more formal notation used in macroeconomics:</p>

<p>$$\hat{PCI}=\hat{y}-\hat{p}$$</p>

<p>hope this helps.</p>

CC BY-SA 3.0
2017-11-19T19:42:00.127Z
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 $\frac{\Delta PCI}{PCI}$?
Giskard
CC BY-SA 3.0
2017-11-19T19:48:37.467Z
@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.
EconJohn
CC BY-SA 3.0
2017-11-19T19:50:08.843Z
@denesp I have no clue how else one would come to the result in the OP without using logs
EconJohn
CC BY-SA 3.0
2017-11-19T21:36:12.787Z
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)
Alecos Papadopoulos
CC BY-SA 3.0
2017-11-20T00:18:18.160Z
@AlecosPapadopoulos Now that you mention it, for $0 < \Delta PCI/PCI < 1$ the 'approximation' $\ln(\Delta PCI/PCI)$ does look a bit negative...
Giskard
CC BY-SA 3.0
2017-11-20T00:39:58.147Z
@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.
Alecos Papadopoulos
CC BY-SA 3.0
2017-11-20T21:56:45.207Z
hmmm, log approximation requires taking the log of (1+growth rate).
london
CC BY-SA 3.0
2017-11-19T19:36:50.683Z
2017-11-19T19:51:11.703Z
2017-11-19T19:51:11.703Z
EconJohn
<p>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 $g_x=\frac{dlog(x)}{dt}$</p>

<p>hence, we have $$g_y=g_Y-g_P$$</p>

CC BY-SA 3.0
2017-11-20T22:13:24.827Z
2017-11-20T22:51:27.523Z
2017-11-20T22:51:27.523Z
london
<p>We know: $PCI = \frac{Y(Income)}{P(population)}$ and that
$PCI$ growth rate is $\frac{\Delta PCI}{PCI}$.</p>

<p>How is this equal to the difference between income growth rate and population growth rate?</p>

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

<macroeconomics>
Per capita income growth rate is difference between income growth rate and population growth rate?
Shoaib Ashraf
# Per capita income growth rate is difference between income growth rate and population growth rate?

(Asked by: username_0 on 2017-11-19)

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 $\left(\frac{∆\frac{Y}{P}}{\frac{Y}P}\right) = (∆Y/Y) - (∆P/P)$ ?

**Tags:** macroeconomics

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## 2 answers

### Answer by username_1 on 2017-11-19. Score: 1

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.

> **Comments:**
> **username_2**: 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 $\frac{\Delta PCI}{PCI}$?  
> **username_1**: @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.  
> **username_1**: @denesp I have no clue how else one would come to the result in the OP without using logs  
> **username_3**: 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)  
> **username_2**: @AlecosPapadopoulos Now that you mention it, for $0 < \Delta PCI/PCI < 1$ the 'approximation' $\ln(\Delta PCI/PCI)$ does look a bit negative...  
> **username_3**: @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.  
> **username_4**: hmmm, log approximation requires taking the log of (1+growth rate).  

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### Answer by username_4 on 2017-11-20. Score: 0

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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