The following post is by Daniel Rigney, Professor Emeritus of sociology at St. Mary’s University in San Antonio and currently a Complimentary Visiting Scholar in the humanities at Rice University in Houston. He is the author of The Matthew Effect: How Advantage Begets Further Advantage. He can be reached at drigney3@gmail.com. (You can also read Daniel Rigney’s previous post The Matthew Effect as Social Spiral):

Evidence from across several academic disciplines suggests that advantage tends to beget further advantage (in popular parlance, “the rich get richer”) while disadvantage tends to beget further disadvantage (“the poor get poorer”) in the social world, whether the “riches” in question are economic, political, cultural or personal. The late sociologist Robert Merton called this phenomenon the “Matthew effect,” from a passage in biblical scripture pertaining to the development of talents and spiritual understanding. My book of the same title The Matthew Effect (hereafter referred to as TME) has sought to integrate the scattered fragments of research on Matthew effects that have appeared in psychology and the social sciences since Merton first coined the term in 1968.

The process that underlies Matthew effects resembles the accumulation of compound interest. If money deposited into an account accrues a 10% annual rate of interest (optimistically and for ease of calculation), and if this interest is returned to the initial principle to gather further interest compounded annually, the principle will grow with time in a curvilinear manner. The shape of a given growth curve will depend, of course, on the size of the initial principle and the rate of interest, whether constant or variable. Thus, as I note in an example in the book (TME, p. 11), an investor who begins with a deposit of $1,000 will accumulate$2,594 over ten years, while an investor with an initial deposit of $100 will accumulate $259. The proportional or percentage increase in the two amounts remains constant over time, but the dollar gap between the two widens substantially.

Why is this growing gap important? Here is the crucial point: People do not survive and pay their expenses with proportions or percentages, but with money. (See TME, pp. 41; 119-20). If we measure inequality between investors monetarily, the gap between investors increases dramatically over time even when their respective assets remain proportional to each other.

This first example illustrates what I have called a relative Matthew effect. Two individuals or groups both improve their position, but one gains more than the other. To make the point even more dramatically, suppose that you begin with $10 million and I begin with $10. After only a year, you will have gained an additional $1,000,000 while I will have gained only an additional $1. Both of us will have gained relative to where we began, but the gap between us will have widened profoundly.

Surprisingly, economists who measure inequality using conventional indices such as the Lorenz curve and its derivative, the Gini coefficient, do not recognize this as an instance of growing inequality. Because our respective growth rates are the same and the growth is proportional, they see no real increase in inequality. Measures such as the Gini coefficient have their virtues (see TME. 119-121), but they miss an important aspect of inequality, which Glenn Firebaugh, in The New Geography of Global Income Inequaity, somewhat reluctantly acknowledges and calls “gap inequality.” While the Gini index and similar measures help us to understand what is happening toward the middle of income distributions – for example, the growth of China and India in the total world economy – it draws attention away from the widening gaps at the extremes, such as the gap between the richest and poorest nations, or between the richest and the poorest within a nation such as China or the United States.

Such inequities are even more apparent when we consider not just relative but also absolute Matthew effects. In our first compound interest example, a relative Matthew effect occurs when both investors gain the benefits of compound interest. But what about the case of a debtor whose debt is simultaneously compounded. In compound debt, initial disadvantage (e.g., a debt of $1,000 compounded by fees and penalties due to inability to pay) is magnified over time in a manner that mirrors the magnified gains of the investor. Indeed, the investor’s gains may even have been won at the expense of the debtor, especially if the investor is a banker. In the case of absolute Matthew effects, gap inequality grows even more rapidly than in the case of relative Matthew effects. Investors pull away from debtors, not just in relative but in absolute terms.

Now consider a second example. Imagine that the investor who began with $1,000 receives not a 10% but rather a 20% annual rate of return, while the investor who began with $100 receives only a 10% rate and the debtor goes deeper into debt by 10% per year as in the previous example. We know that in real life, not everyone receives the same return on investment. Those who begin with more and receive higher rates of return obviously gain more than other investors in relative terms, and do ever better than debtors in absolute terms as well.

Finally, consider a third example. Imagine that the first investor’s initial advantage increases by 20% this year, by 30% next year, by 40% the year after that, and so on. Now both the proportional and gap inequalities are rising with the first investor’s accelerating rate of return. Something like this has happened in the United States in recent decades as the super-rich have pulled away from the merely rich, the middlers, and those at the bottom, who are falling still further behind both relatively and absolutely. Meanwhile the super-rich accumulate fortunes beyond our imagining. As energy tycoon T. Boone Pickens puts it in his autobiography of the same title, “the first billion is the hardest.”

I propose that we refer to gap increases in our second and third illustrations as “Matthew hypereffects.” I allude to such effects in TME (p. 9) when I write that in our initial example , compound interest causes growing inequalities because the first investor’s dollar gains on a much larger base greatly exceed the second investor’s, so that the dollar gap between the two widens dramatically over time. “The gap widens even more rapidly when, as often happens in the real world of finance, those who begin with more receive a higher rate of return on their investments than those who begin with less.” I could have added that their gains are even more dramatic when their rates of return accelerate over time. In certain instances they enter a kind of economic hyperspace, and their gains are almost boundless.

The basic compound interest model proposed in TME can be adapted flexibly to apply to situations in which social actors begin with varying initial advantages and receive varying and fluctuating (including decelerating or accelerating) rates of return (or penalty) on these initial conditions over time. Originally I intended the compound interest model merely as an example or illustration of Matthew effects, but it may prove useful as a simple base model as well , to be adjusted and elaborated mathematically in response to specific circumstances by those so inclined.

As a friend has noted, Bill Gates didn’t get where he is just through compound interest. That is certainly true, and nothing in this book implies that Matthew effects are the only game in town where inequalities are concerned. Yet the compound interest model, suitably refined, freely acknowledges that some social actors enjoy far larger rates of return than others on their initial investments. Those who enjoy higher, and especially accelerating, rates of return are, for good or ill, the beneficiaries of Matthew hypereffects. I thank my friend Paul Ingmundson for stimulating this elaboration of a point made too fleetingly in TME.