The Regression Theorem: Summary

In the late 1800s, three economists working independently radically transformed the way we think about value. Classical economists had struggled to explain why water is typically much cheaper than diamonds, despite being more useful on the whole and, indeed, crucial to one’s survival. Then, William Stanley Jevons, Carl Menger, and Léon Walras offered an elegant solution. They maintained that prices reflect an item’s marginal use, as valued by the user. Sure, water is more valuable on the whole. But an additional gallon of water is not worth much since many of our uses for water have already been satisfied. Diamonds, in contrast, are rare. Although the first gallon of water is surely more valuable to the typical consumer than the first carat of of diamonds, the marginal value of water is considerably lower than the marginal value of diamonds—and this is reflected in their prices.

Needless to say, the subjectivist-marginalist view caught on. Pretty soon, economists were applying the it to a host of topics. But they ran into a problem when they tried to apply it to money. How is the value of money determined? Classical economists had referenced the equation of exchange. And, although the marginalists did not reject that identity, they recognized that something more would be required to produce an explanation consistent with the subjectivist-marginalist theory of value.

Dollars and Donuts

To understand the problem, let’s first consider how individuals determine the value of some good they consume, like donuts. We can then compare that with how individuals value some item they might employ as a medium of exchange, like dollars.

It is relatively straightforward for me to determine how much I value a donut. (BTW: I love donuts…perhaps a bit too much). In assessing the worth of a donut to me, simple introspection will do the trick. I can survey all that I own and ask myself how much of it I am willing to give up in exchange for a donut. What am I willing to pay for a donut? I can also ask the question the other way around. I can consider something I own—my slightly used coffee mug, for example—and ask myself what I am willing to accept for it—perhaps a donut. Again, simple introspection will do. Since I consume donuts, I can determine its subjective value to me through introspection.

Mere introspection is of little use when trying to value a medium of exchange like the dollar. By definition, a medium of exchange is an item one accepts, not to consume, but rather to trade for some other good or service that one consumes. I can value the other good or service that I consume. And I can value the good or service I might give up for the dollars. But how do I value the dollars? Unlike the donut, it is not a question of what a dollar is worth to me. It is a question of what the dollar is worth to someone else, since that will determine how much of the stuff I really want (probably donuts) I ultimately get for the dollars.

Hopefully you can now see the problem valuing money presents for an economist working in the subjectivist-marginalist tradition. If value depends on how much utility one derives from consuming a good or service, it isn’t obvious how one values an item she intends to employ as a medium of exchange since employing an item as a medium of exchange means she will not consume it.

The Regression Theorem

Working in the tradition of Menger, Ludwig von Mises explained the value of money by what is known today as the regression theorem.

Mises (1934, p. 109) starts by reaffirming the subjectivist-marginalist position.

The price of money, like other prices, is determined in the last resort by the subjective valuations of buyers and sellers. But [..] the subjective use-value of money, which coincides with its subjective exchange-value, is nothing but the anticipated use-value of the things that are to be bought with it.

The question, then, concerns how those anticipations are formed.

Mises notes that there is a continuity in the purchasing power of money. “Once an exchange-ratio between money and commodities has been established in the market, it continues to exercise an influence beyond the period during which it is maintained; it provides the basis for the further valuation of money. Thus the past objective exchange-value of money has a certain significance for its present and future valuation. The money-prices of to-day are linked with those of yesterday and before, and with those of to-morrow and after.” Hence, the value of money today depends on the the value of money yesterday; the value of money yesterday depends on the value of money the day before that; and so on. But Mises does not propose an infinite regress. It’s not turtles all the way down. Rather, there is a clear stopping point. For Mises, any item valued as a medium of exchange today must have been valued for consumption by someone at some point in the past prior to its employment as a medium of exchange. It is that initial use value that enabled money to be exchanged initially. And that it is no longer employed for some end use is of little consequence now since an exchange value has already been established.

We might express the regression theorem formally as follows:

Let v_t be the exchange value of some item that has functioned as a medium of exchange for t periods.

An item functioning as a medium of exchange at time t Þ v_t-1 > 0.

An item functioning as a medium of exchange at time t-1 Þ v_t-2 > 0.

…

An item functioning as a medium of exchange at time 0 Þ v_0-1 > 0.

.: An item functioning as a medium of exchange at time t had some positive exchange value before it was initially employed as money.

By this point, you may be wondering why I have taken you on a short journey through the history of economic thought, only to end up explaining an old theory on how we might account for the positive exchange value of money with rejecting the subjectivist-marginalist approach. The short answer: I think new innovations in money, especially bitcoin, call into question the validity of the regression theorem.

The goal of this post is to make sure we are all on the same page with respect to what the regression theorem says and why it matters. In the next post, I will explore some practical applications of the regression theorem. Then, in two subsequent posts, I will turn to the case of bitcoin to identify some misconceptions of the regression theorem and how it might be amended given what we now know.