Inflationary math
Photo: Girish Gupta

Inflationary math
Girish Gupta
| 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.