Some Unresolved Issues In Multiple Interest Rate Analysis

1.0 Introduction

Come October, as I understand it, the Review of Political Economy will publish, in hardcopy, my article The Choice of Technique with Multiple and Complex Interest Rates. I discuss in this post questions I do not understand.

2.0 Non-Standard Investments and Fixed Capital

Consider a point-input, flow-output model. In the first year, unassisted labor produces a long-lived machine. In successive years, labor and a machine of a specific history are used to produce outputs of a consumption good and a one-year older machine. The efficiency of the machine may vary over the course of its physical lifetime. When the machine should be junked is a choice variable in some economic models.

I am aware that in this, or closely related models, the price of a machine of a specific date can be negative. The total value of outputs at such a year in which the price of the machine of the machine is negative, however, is the difference between the sum of the price of the machine & the consumption good and the price of any inputs, like labor, that are hired in that year. Can one create a numerical example of such a case in which the net value, in a given year, is negative, where that negative value is preceded and followed by years with a positive net value?

If so, this would an example of a non-standard investment. A standard investment is one in which all negative cash flows precede all positive cash flows. In a non-standard investment at least one positive cash flow precedes a negative cash flow, and vice-versa. Non-standard investments create the possibility that all roots of the polynomial used to define the Internal Rate of Return (IRR) are complex. Can one create an example with fixed capital or, more generally, joint production in which this possibility arises?

Does corporate finance theory reach the same conclusions about the economic life of a machine as Sraffian analysis in such a case? Can one express Net Present Value (NPV) as a function combining the difference between the interest rate and each IRR in this case, even though all IRRs are complex? (I call such a function an Osborne expression for the NPV.)

3.0 Generalizing the Composite Interest Rate to the Production of Commodities by Means of Commodities

In my article, I follow Michael Osborne in deriving what he calls a composite interest rate, that combines all roots of the polynomial defining the IRR. I disagree with him, in that I do not think this composite interest rate is useful in analyzing the choice of technique. But we both obtain, in a flow-input, point output model, an equation I find interesting.

This equation states that the difference between the labor commanded by a commodity and the labor embodied in that commodity is the product of the first input of labor per unit output and the composite interest rate. Can you give an intuitive, theoretical explanation of this result? (I am aware that Osborne and Davidson give an explanation, that I can sort of understand when concentrating, in terms of the Austrian average period of production.)

A model of the production of commodities by means of commodities can be approximated by a model of a finite sequence of labor inputs. The model becomes exact as the number of dated labor inputs increases without bound. In the limit, the labor command by each commodity is a finite value. So is the labor embodied. And the quantity for the first labor input decrease to zero. Thus, the composite interest rate increases without bound. How, then, can the concept of the composite interest rate be extended to a model of the production of commodities by means of commodities?

4.0 Further Comments on Multiple Interest Rates with the Production of Commodities by Means of Commodities

In models of the production of commodities by means of commodities, various polynomials arise in which one root is the rate of profits. I have considered, for example, the characteristic equation for a certain matrix related to real wages, labor inputs, and the Leontief input-output matrix associated with a technique of production. Are all roots of such polynomials useful for some analysis? How so?

Luigi Pasinetti, in the context of a theory of Structural Economic Dynamics, has described what he calls the natural system. In the price system associated with the natural system, multiple interest rates arise, one for each produced commodity. Can these multiple interest rates be connected to Osborne's natural multiple interest rates?

5.0 Conclusion

I would not mind reading attempts to answer the above questions.