Donald J. Boudreaux
Sometimes simple arithmetic reveals realities that are both astonishing and astonishingly important. Such is the case for the arithmetic performed by this little button on every scientific calculator – including, no doubt, on the calculator you have on your smartphone: x!
Called “factorial,” it’s a mathematical formula as straightforward as they come. Although comprehending its formal definition might require a high-school education – it’s the product of all positive integers less than or equal to x – fourth graders can understand how a factorial works. Simply pick a whole number (x) and then multiply that number by the whole number just below it; keep doing this multiplication until you get down to the number 1.
For example, suppose you choose the number four. 4! = 4 x 3 x 2 x 1, which equals 24. Five factorial – written as 5! – is 5 x 4 x 3 x 2 x 1, which equals 120.
Simple Arithmetic with Big Implications
Pretty simple. Equally simple is what the answer to any factorial operation reveals: it’s the number of possible ways to arrange, in a single dimension, any given number of items. For example, the factorial operation tells you how many different ways you can arrange the seating of you and three dinner guests around your dining-room table. To your immediate right you can seat Mike, to Mike’s right you can seat Chloe, and to Chloe’s right you can seat Max. Or, instead, to your immediate right you can seat Chloe, to her right Max, and to his right Mike.
With four people dining, the number of different ways to arrange the seating order is 24. This is the answer to 4! (4 x 3 x 2 x 1 = 24)
If you had one more dinner guest – a total of five people to seat at your table – the number of different ways to arrange the seating of you and your four guests is 120. (5! = 5 x 4 x 3 x 2 x 1 = 120)
“So what?” you ask. So plenty. As the number of items increases, the number of different ways to arrange them skyrockets.
When I was young, the number of family members gathered for our family’s Thanksgiving dinner was about 20 – a large dinner gathering, yes, but nothing out of the ordinary. Yet the number of different ways to arrange the seating of a mere 20 people around a dinner table is – drumroll! – 2,432,902,008,176,640,000.
A number this large has no non-scientific name. It is inconceivably high and unimaginably humongous. It’s just a bit larger, by a mere several thousand trillion, but roughly equivalent to this number: 2,428,272,000,000,000,000 – which is the number of seconds (as in “60 seconds in a minute”) there are in 77 billion years. Note that astronomers estimate the universe’s age to be 13.8 billion years.
I didn’t write the above to impress you with my knowledge of advanced arithmetic or my ability to google “How old is the universe?” I wrote the above to make a point about the economy.
Arranging Economic Inputs Productively
Solving economic problems – creating value – requires discovering how to arrange inputs productively. The number of inputs available for use is not 20 or even 20,000; it’s in the billions. Therefore the factorial operation means that the number of possible ways to arrange these billions of inputs is indescribably far beyond human comprehension. Yet only a minuscule fraction of these ways has any prospect of being productive. Almost all possible arrangements are useless or even dangerous.
One among these countless arrangements is gin mixed with turpentine and anchovies, but the resulting martini would be terrible. How best to discover the vanishingly small number of productive arrangements out of the sprawling and gigantic number of possible arrangements?
Obviously, relying on random chance will not do. Equally futile would be reliance on what on the surface appears to be the opposite of random chance: central planning. No human being or committee could possibly even survey and list all of the possible different arrangements of billions or inputs. Much less could any individual genius, or agency of geniuses, discover from this vast list of possibilities which of these arrangements are most useful compared to the uncountably large number of other possible arrangements.
Given the puniness of the human mind relative to the incalculable number of possible different arrangements of an economy’s inputs, to turn over to government the responsibility for choosing how resources should be allocated is, in effect, to rely upon random chance.
The only way to ‘solve’ this problem – the only feasible method of discovering from among the practically infinite number of ways to arrange inputs those precious few arrangements that improve the living standards of ordinary people – is to divide decision-making authority among the multitude of creative human minds. Let each individual, occupying her own unique place in the world, survey the relatively few input-arrangement options that are, or might become, visible to her. Give that person freedom to experiment with her – but only with her – tiny sliver of economic reality. Then allow that person to enjoy benefits if she figures out how to rearrange inputs in ways that improve the lives of fellow human beings, and allow her to suffer losses if she rearranges inputs in ways that prove to be useless or wasteful.
With each among the earth’s billions of individuals attending to his or her own unique part of the economy, and responsible for experimenting with different arrangements of his or her relatively small set of inputs, ‘the economic problem’ becomes manageable.
How can such a division of responsibility be attained in practice? And how can each person know if his or her experiments with different input arrangements are useful or not to fellow human beings? For answers, tune in to my next column.
Epilogue: On Today’s Pandemic
Before closing, however, here’s a word about COVID-19. If a vaccine or cures (or both) for COVID-19 are possible, these necessarily involve discovering arrangements of different inputs – mostly, I presume, of different chemicals – that destroy the virus or prevent it from harmfully invading human bodies. We today know of about 50 million different chemicals. If a unique arrangement of some subset of these chemicals can treat COVID-19, we do not now – in mid-May 2020 – know what this arrangement is. If it exists, how best to find it?
The realities of factorials – and of combinations and permutations – tell me that we want as many people as possible experimenting with different arrangements and combinations of chemicals.
To rely only on select and officially approved researchers, or to ignore ideas from all or even some foreigners, is to unnecessarily reduce the number of human beings working to discover from among an incalculably vast number of possible chemical arrangements the one or a few that might render the coronavirus harmless. That policy – that artificial arrangement of human decision-making authority – would evince madness.