Value & Capital, CHAPTER XII, Section 5

This section explores the significance of eliminating the extra equation from the systems of equations discussed earlier in this chapter.  The author explains the meaning of this elimination as follows:

It means that if a system of prices is established which equates the demand and supply for each of the n - 1 goods and services, and equates the demand and supply for securities (or loans), then the demand and supply for money must be equal, so that that equation has nothing further to tell us. 

Mathematically what we have is simply a system of equations and unknowns, and any one of the equations could be chosen for elimination.  Because one equation is superfluous, the same values of the unknowns would result from solving the remaining system of equations, no matter which equation is the one eliminated.

The author focuses on two special cases for the choice of which equation to eliminate.
In the first case, the money equation is eliminated, and prices and interest are determined through the markets for goods and services, along with the market for loans.  In this case the author describes the money equation as "otiose," which, according to the dictionary, can mean "lacking use or effect."

When the money equation is included in the system of equations, it can determine the money values of the goods and services, but some other, independent means of determining their relative values will be needed.  As the author explains, "it is impossible to determine even relative prices except in terms of some standard.  Thus the prices of goods and services must first be fixed in terms of some auxiliary standard commodity."  He goes on to note that writers of "the classics" used unskilled labor as their auxiliary standard, whereas more modern writers used a representative consumption good.  This leads to the other special case the author highlights:  the elimination of the equation for supply and demand of the auxiliary standard commodity.

As the author notes, "this is a perfectly legitimate line of approach," but he emphatically warns that it is subject to a "great danger."  This danger has to do with confusion about "what happens to the rate of interest."  An economist who is not careful "will find himself determining, not the true rate of interest," but instead "a rate indicating the value of future deliveries of the auxiliary standard commodity in terms of current deliveries of the same auxiliary standard."  There is no reason why this rate of interest in terms of the auxiliary standard commodity should be the same as the money interest rate.  We saw in the previous chapter that the two rates were the same only if the futures price of the commodity is the same as the spot market price.  This condition will be satisfied if the money price of the standard commodity is expected to remain constant - and if there is no risk that it will not.  Although the author concedes that the difficulties with using this approach could be overcome, he concludes the section by noting that, "It looks as if it will be better to eliminate a different equation."