Value & Capital, CHAPTER IV, Section 3

In this brief section, Hicks goes into a bit more detail about how we know that the number of prices that must be determined in order to define an equilibrium of exchange is always one less than the number of goods.  His argument runs as follows.

If prices are given, we know (using the methods described earlier) how to determine any individual's demand for each commodity, as well as the quantities of any commodities he already possesses that he will be willing to supply in exchange for these demands.  If we can sum up these demands and supplies for all the consumers, then we can determine the total demand and supply for each commodity.  Then, as Hicks puts it, "If the price-system is such as to make these demands and supplies equal, we have a position of equilibrium.  If not, some prices at least will be bid up or down."

Hicks then goes on to argue that the equation of supply and demand for the standard commodity (that is, the one assumed in the previous section to have some of the qualities of money) follows from the demand-and-supply equations for the rest of the goods.  In Hicks's words:

Once any particular individual has decided how much of each non-standard commodity he will sell or he will buy, he will automatically have decided how much of the standard commodity he will buy or sell.  Thus 

Demand for standard = Receipts from sale of other goods - Expenditure on purchase of others

Supply of standard = Expenditure on purchase of others - Receipts from sale of others 

Therefore for the whole community, 

Demand for - Supply of standard commodity = Total receipt from sale of others

- Total expenditure on purchase of others 

and, once the demand for every non-standard commodity equals the supply, this must = 0.

There are thus n-1 independent equations to determine the n-1 independent prices.