exponential – Pertinent Observations

Exponential need not mean explosive

Earlier on this blog I’ve written about the misuse of the term “exponential” when it is used to describe explosive increase in a particular number. My suspicion is that this misuse of the word “exponential” comes from Computer Science and complexity theory – where the hardest problems to crack are those which require time/space that is exponential in the size of the data. In fact, the entire definition of P, NP and NP-completeness have to do with the distinction between problems that take exponential resources versus those that take resources that are a polynomial function of the size of data.

Earlier today, I shared this blog post by Bryan Caplan on Puerto Rican immigration into the United States with a comment “exponential immigration”. I won’t rule out drawing some flak for this particular description, for Caplan’s thesis is that Puerto Rican immigration took a long time indeed to “explode”. However, I would expect that the flak I get for describing this variable as “exponential” would come from people who mistake “exponential” for “explosive”.

Caplan’s theory in the above linked blog post is that immigration from Puerto Rico to the United states was extremely slow for a very long time. It was in the late 1890s that a US Supreme Court ruling allowed free access to Puerto Ricans to the United States. However, it took close to a hundred years for this immigration to “explode”. Caplan’s theory is that the number of people moving to the US per year is a function of the number of Puerto Ricans who are already there!

In other words, the immigration process can be described by our favourite equation: dX/dt = kX, solving which we will get an equation of the form X = a exp(kt), which means that the growth is indeed exponential in time! Yet, given a rather small value of X_0 (the number of Puerto Ricans in the United States at the time the law was passed), and given a small value of k, the increase has been anything but explosive, despite it being exponential!

The point of this post is worth reiterating: the word “exponential”, in its common use, has been taken to be synonymous with “explosive”, and this is wrong. Exponential growth need not be explosive, and explosive growth need not be exponential! The two concepts are unrelated and people would do well to not confuse one with the other.

Exponential increase

“Increasing exponentially” is a common phrase used by analysts and commentators to describe a certain kind of growth. However, more often than not, this phrase is misused and any fast growth is termed as “exponential”.

If f(x) is a function of x, f(x) is said to be exponential if and only if it can be written in the form:

$f(x) = K alpha ^x$

So if your growth is quick but linear, then it is not exponential. While on the topic, I want to point you to two posts I’ve written on my public policy blog for Takshashila: one on exponential growth in bank transfers that I wrote earlier today and another on exponential growth in toilet ownership. The former shows you how to detect exponential growth and in the latter I use what I think is an innovative model to model toilet growth as exponential.

Exponential increase in uptake of IMPS

We had dealt with exponential increases on this blog once before. We revisit the topic, and this time this is in the context of the inter bank mobile payment system that came into place sometime last year. I’ve never used it so I’m not sure how it works, but going by the data put out by the National Payments Corporation of India, the volume of transactions is increasing at an exponential rate.

How do we determine this is an exponential rate? First, let us look at the time series of total volumes of transactions:

Source: http://www.npci.org.in/impsVolumes.aspx

Notice that after remaining flat for a couple of months (maybe even decreasing) the number of transactions has really taken off (March is probably an aberration – but given that it’s the month of financial closure the higher volumes can be expected). Increased exponentially, you say? How can we test that?

We can test that by using a logarithmic scale for the y-axis. Here is the same plot again, except that this time the Y-axis is logarithmic.

Notice that apart from the part with the aberration and the initial two months, the graph is now linear. In other words, we can describe this graph by a line of the form

log y = a + b x

or y = exp (a + bx)

Thus, exponential!

Coming back from the geekery, it is really good to note that IMPS has taken off. However, this should not be taken as proof of the fact that mobile payments are easy, for IMPS is anything but easy. New RBI Governor Raghuram Rajan has said in his inaugural speech that he hopes to make it simpler to make payments via mobile. Hopefully this will take off soon. Till then all we can do is to contribute to the exponential growth in the update of the IMPS!