# Approximating the impact of inflation

The other day someone mentioned to me a rule of thumb that he was using to estimate the number of years \(n\) it would take for inflation to destroy half of the purchasing power of today's money:\[ n = \frac{70}{p}\]

Here \(p\) is the inflation in percent, e.g. if the inflation rate is \(2\%\) then today's money would buy only half of today's goods and services in 35 years. You can also think of a saving account with an interest rate of \(2\%\) that would double your money in 35 years.

It is not difficult to derive this formula. The starting point is:

\[

2K = K (1 + \frac{p}{100})^n

\]

This is equivalent to:

\[

2 = (1 + \frac{p}{100})^n

\]

Taking the log gives:

\[

\log(2) = n \log(1 + \frac{p}{100})

\]

The first term of the Taylor series approximation of \(\log(1+x)\) for small \(x\) is \(x\). Hence for small \(p\) I can set:

\[

\log(2) \doteq n \, \frac{p}{100}

\]

Next I have to estimate the value for \(\log(2)\). Writing it as an integral leads to:

\[

\log(2) = \int_1^2 \frac{1}{x} \,dx

\]

Using Simpson's rule I can approximate the integral with:

\[

\int_1^2 \frac{1}{x} \,dx \doteq \frac{2-1}{6} (1+4\frac{2}{1+2}+\frac{1}{2} )

= \frac{25}{36} \doteq 0.7

\]

Thus,

\[

n \doteq \frac{70}{p}

\]

Plotting the two formulas against each other reveals that the approximation works pretty well, even for inflation rates up to 10%.

### R Code

Here is the R code to reproduce the plot.curve(70/x, from=1, to=10,
xlab="Inflation rate p%",
ylab="Number of years for purchaing power to half",
main="Impact of inflation on purchasing power",
col="blue",
type="p", pch=16, cex=0.5)
curve(log(2)/(log(1+x/100)),
from=1, to=10, add=TRUE,
col="red")
legend("topright",
legend=c("70/p","log(2)/log(1+p/100)"),
bty="n",
col=c("blue", "red"),
pch=c(16,16), pt.cex=c(1,1))

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what's K?

ReplyDeleteK=Kapital (if you read Marx, or capital in English).

ReplyDeleteIt seems to be an interesting chart tool however I'm getting a "Failed to execute default browser. Failed to execute child process "chrome" (No such file or directory)" error when trying to plot using GoogleVis. Any idea how to correct this? Thanks!

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ReplyDeleteHello,

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I tryed the following code:

icons=paste0("{",

"'default': {'normal': 'http://icons.iconarchive.com/",

"icons/icons-land/vista-map-markers/48/",

"Map-Marker-Ball-Azure-icon.png',\n",

"'selected': 'http://icons.iconarchive.com/",

"icons/icons-land/vista-map-markers/48/",

"Map-Marker-Ball-Right-Azure-icon.png'",

"}}"))and it changed all the markers (as it is supposed to). Is there a way I can modify this so that my data identified with a "1" on some variable will be plotted with the icon 1, on the data with a "0" with the icon 0 ?

Thanks in advance

I am not sure. Perhaps the

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ReplyDelete