I should have put the following in my working paper, on *Basic Commodities and Multiple Interest Rate Analysis*. This would go somewhere after Equation 10.

Let a technique of production be specified by a row vector, **a**_{0}, of labor coefficients and a square Leontief input-output matrix, **A**. The *j*th labor coefficient, *a*_{0,j}, and the *j*th column, **a**_{.,j}, of **A** represent the process for producing the *j* commodity when this technique is in use.

Consider a firm producing the *j*th commodity with this process. Suppose the firm faces prices of inputs and outputs, as represented by the row vector **p**. Let *w* be the given wage and *r* be the given rate of profits. Then the Net Present Value (NPV) for using this process, per unit output of the *j* commodity is:

NPV_{j}(r) =p_{j}- (pa_{.,j}+wa_{0,j})(1 +r)

Let *r*_{1} be the Internal Rate of Return (IRR) for this process. By definition, the NPV, evaluated for the IRR, is zero:

NPV_{j}(r_{1}) = 0

As the appendix proves, one can derive:

NPV_{j}(r) = - (pa_{.,j}+wa_{0,j})(r-r_{1})

In words, when an investment project consists of one payout and one expenditure, with the payout coming one period after the expenditure, the Net Present Value of the investment is the additive inverse of the (first) expenditure, accumulated for one period at the difference between the given rate of profits and the Internal Rate of Return for the investment. Notice that NPV is only positive if the rate of profits used for accumulating costs falls below the internal rate of returns.

This is a trivial application of multiple interest rate analysis because it applies when the multiplicity is one. The above formulation of NPV was suggested to me, however, by first considering a non-trivial application.

**Appendix**

By the definition of the IRR:

r_{1}= [p_{j}/(pa_{.,j}+wa_{0,j})] - 1

Substitute:

- (pa_{.,j}+wa_{0,j})(r-r_{1}) = - (pa_{.,j}+wa_{0,j})r+p_{j}- (pa_{.,j}+wa_{0,j})

Or:

- (pa_{.,j}+wa_{0,j})(r-r_{1}) = -(pa_{.,j}+wa_{0,j})(r+ 1) +p_{j}

Which is to say:

- (pa_{.,j}+wa_{0,j})(r-r_{1}) =p_{j}- (pa_{.,j}+wa_{0,j})(1 +r)

But the term on the right is the definition of NPV. So the two expressions for NPV in the main text are equivalent.

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