Regression theorem

Regression theorem is the theorem by which Mises applies the subjective theory of value to the objective-exchange value, or purchasing power of money.

Objective-exchange values of all other goods and services are explained by the subjective theory of value, whereby the values are traced to the ultimate subjective use values of the marginal consumers who value such goods and services for their objective-use values which they expect to consume. This is not true for money because (1) money is not consumed in its use and (2) the subjective and objective use values of money coincide and are equal to its objective-exchange value, the estimated value of the goods and services for which it can be exchanged. Mises explains the origin of the objective-use value of money by tracing it back step by step from the point at which it is being valued to the point where the monetary good served only non-monetary uses, an essential point preceding the first use of anything as money. At this point, its objective-exchange value is explained by the general theory of subjective value and marginal utility.

Workings
For many economists, a marginal utility explanation of money demand would simply be a circular argument: We need to explain why money has a certain exchange value on the market. It won't do (so these economists thought) to merely explain this by saying people have a marginal utility for money because of its purchasing power. After all, that's what we're trying to explain in the first place—why can people buy things with money?

Mises eluded this apparent circularity by his regression theorem. In the first place, yes, people trade away real goods for units of money, because they have a higher marginal utility for the money units than for the other commodities given away. It's also true that the economist cannot stop there; he must explain why people have a marginal utility for money. (This is not the case for other goods. The economist explains the exchange value for a Picasso by saying that the buyer derives utility from the painting, and at that point the explanation stops.)

People value units of money because of their expected purchasing power; money will allow people to receive real goods and services in the future, and hence people are willing to give up real goods and services now in order to attain cash balances. Thus the expected future purchasing power of money explains its current purchasing power.

But haven't we just run into the same problem of an alleged circularity? Aren't we merely explaining the purchasing power of money by reference to the purchasing power of money?

No, Mises pointed out, because of the time element. People today expect money to have a certain purchasing power tomorrow, because of their memory of its purchasing power yesterday. We then push the problem back one step. People yesterday anticipated today's purchasing power, because they remembered that money could be exchanged for other goods and services two days ago. And so on.

So far, Mises's explanation still seems dubious; it appears to involve an infinite regress. But this is not the case, because of Menger's explanation of the origin of money. We can trace the purchasing power of money back through time, until we reach the point at which money first emerged from a state of barter. And at that point, the purchasing power of the money commodity can be explained in just the same way that the exchange value of any commodity is explained. People valued gold for its own sake before it became a money, and thus a satisfactory theory of the current market value of gold must trace back its development until the point when gold was not a medium of exchange.