Value & Capital, CHAPTER XII, Section 3

This somewhat lengthy section examines the simplified model of an economy with only "short lending" and describes the determination of the system of spot prices of all goods and services, as well as the single-period "short rate" of interest.  The prices in this system must be those that equate the supply and the demand for all goods and services.  If we assume there are n different kinds of exchangeable goods and services, then there must be n different prices to be determined.  As in previous models, we assume one of the goods is "taken as a standard of value (money)."  Then each of the n - 1 other goods must have a price in terms of the standard;  with the interest rate on single period loans, this makes a total of n prices to be determined.  To determine these prices there are n - 1 equations for the supply and demand of the goods and services, plus one equation for the supply and demand for money, plus one for the supply and demand for loans.  It would appear, then, that there are n + 1 equations.  But, as the discussion notes, "one of these n + 1 equations follows from the rest."  Therefore there are the same number of equations as unknowns.  "The system is neither over- nor under-determined."

The text goes on to explain (at length) how one of the n + 1 equations can be eliminated.  An individual cannot spend more than he receives from trading, unless he borrows.  Conversely, an individual who receives more from trading than he spends must end up with more cash on hand (unless he lends out more than his profits from trading).  Thus we have the following equation for any private individual:

Acquisition of cash by trading = Receipts - Expenditures - Lending

(keeping in mind that lending may be negative).  With firms, the situation is slightly more complicated.  In addition to the effects of borrowing and paying back loans, a firm "will reduce its cash balance by any acquisition it makes of factors of production, increase it by any sales of products.  Finally, it will diminish its cash balance by any dividend it pays out to entrepreneurs."  Thus the following equation holds for any firm:

Acquisition of cash by trading = Value of output - Value of input - Repayment of loans + New borrowing - Dividends

In considering what happens over the entire community of individuals and firms, taken in aggregate, the supply and demand equations for unfinished goods can be taken as canceled out.  "The input to be reckoned is simply the input of labour and material property provided by private persons;  the output is simply the output of finished goods sold to private persons."  If demand equals supply in the input markets, the net receipts of private persons equal the value of  (labor) inputs to firms, plus dividends, plus loan repayments.  If demand equals supply in the output markets, the value of the output of production equals the net expenditure of private individuals.  Also, borrowing will equal lending if supply equals demand in the loan market.
 
Therefore, the following equation holds for the community as a whole:
Net Acquisition of cash by trading
= (Value of Output - Net expenditures by private persons)
+ (Net receipts by private persons - Value of Input - Dividends - Repayment of loans)
+ (Borrowing - Lending)
= 0

Therefore, if there is equilibrium in the markets for all goods and services, and if there is equilibrium in the market for loans, then there must also be equilibrium in the market for money.  Thus there are only n independent equations to determine the n unknown prices.