| Dec. 12, 2017 | Kabul, Afghanistan

A s Venezuela enters a period of hyperinflation (where monthly inflation is greater than 50%), I wanted to derive the basic math of inflation from first principles.

Where \(a\) is annual inflation and \(m\) is monthly inflation, the two are easily related by:

$$a = 100\times{({{(1+ {{m} \over 100})}^{12}-1})}$$

Rearranging this gives:

$$m = 100\times{({{(1+ {{a} \over 100})}^{1 \over 12}-1})}$$

But what if you want to work in a general unit of time such as years, \(t\), and want to know how long it will take for prices to, say, double?

Well, we know that if prices double in one month, then monthly inflation is 100%, so putting \(m = 100\) into the formula above and replacing the \(1 \over 12\) (the fraction of a year taken up by one month) with \(t_2\) (the \(_2\) signifying "double") gives:

$$100 = 100\times{({{(1+ {{a} \over 100})}^{t_2}-1})}$$

Rearranging for \(t_2\) (the time for prices to double) gives:

$$t_2 = {{\ln (2)} \over {\ln ({{a \over 100}+1)}}}$$

This means that with annual inflation of \(a\), prices will double every \(t_2\) years.

What if we wanted to generalize, to see how long it would take, \(t_x\) in years, for prices to change by a factor of \(x\)?

Well, the 100% on the left side of the third formula came from \(100(x-1)\), where \(x = 2\) (for doubling). So, we can generalize so that

$$100(x-1) = 100\times{({{(1+ {{a} \over 100})}^{t_x}-1})}$$

Rearranging this gives:

$$t_x = {{\ln (x)} \over {\ln ({{a \over 100}+1)}}}$$

This means that it would take \(t_x\) (in years) for prices to change by a factor of \(x\).

So, for prices to increase ten-fold (\(x = 10\)) in an environment with annual inflation \(a\), it would take

$$t_{10} = {{\ln (10)} \over {\ln ({{a \over 100}+1)}}} = {{1} \over {\log_{10} ({{a \over 100}+1)}}}$$

years.

According to the National Assembly, Venezuela's monthly inflation rate in November was 56.7%. That works out (as per the first equation) to an annual rate of 21,918%.

Using the above formulae, that means prices are doubling every 0.13 years, or every six weeks or so. And prices are going up ten-fold every 0.43 years, or just over every five months.