CHAPTER XIII

SECOND APPROXIMATION IN TERMS OF FORMULAS§1. Introduction

The object of this chapter is to express in algebraicformulas the six principles comprising the second ap-proximation. 1 In Chapter XII we assumed that all in-come streams were unalterable, except as they could bemodified by borrowing and lending, or buying and sellingrights to specified portions of these income streams. Inthe second approximation now to be put into formulas,we substitute for this hypothesis of fixity of the incomestreams the hypothesis of a range of choice between dif-ferent income streams.

The income stream of Individual 1 no longer consistsof known and fixed elements, c x , c", c x ", etc., in suc-cessive periods but of unknown and variable elementswhich we shall designate by y x , y x ", y x ", etc. (y x andy x " are the coordinates of the Opportunity line).

This elastic income stream may now be modified intwo ways: by the variations in these y’s, as well as bythe method which we found applicable for rigid incomestreams, namely, the method of exchange, borrowing andlending, or buying and selling. The alterations effectedby the latter means we shall designate as before by thealgebraic addition of x Xt x x ", x x ", ...., x x m , for succes-

1 These have already been expressed in words in Chapters VI andIX and geometrically in Chapter XI.

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