The nature of capital and income . Fig. 43. the direction OQ of the curve for the continuous case. Thisdirection Oq is the diagonal of a parallelogram formed byproducing Oh to meet the ordinate from W in IT, producingOq to Q and completing the parallelogram. Since HQ is twice as far from 0 as hq { RR =2 (rR)by hypothesis), it follows, by similar triangles ( Ohq andOHQ), that it is also twice as long. But hq represents ahalf years installment of income. Hence H^Q represents two APPENDIX TO CHAPTER XIII 385 such installments, or the annual rate of income. ThereforeOK, being of the same l
The nature of capital and income . Fig. 43. the direction OQ of the curve for the continuous case. Thisdirection Oq is the diagonal of a parallelogram formed byproducing Oh to meet the ordinate from W in IT, producingOq to Q and completing the parallelogram. Since HQ is twice as far from 0 as hq { RR =2 (rR)by hypothesis), it follows, by similar triangles ( Ohq andOHQ), that it is also twice as long. But hq represents ahalf years installment of income. Hence H^Q represents two APPENDIX TO CHAPTER XIII 385 such installments, or the annual rate of income. ThereforeOK, being of the same length, also represents this annual rateof income. In other words, the direction from O to g lies along a paral-lelogram of which the side OK represents the annual rateof income and the side OIF a chord of the discount curveOP. Now it is evident that if, instead of a semi-annual installment,we assume greater frequency, the same statement will applyexcept that the point h will be nearer 0. By proceeding in this. Fio. 44. manner the chord OhlT approaches the tangent OH as itslimit, and the parallelogram OITQK becomes at the limitthe parallelogram OHQK as originally described. That is, itssides are OK, the annual rate of income, and OH, the tangentto the discount curve drawn from 0 to a vertical line one yearto the left. That this construction and its demonstration bear a strikinganalogy to the construction and demonstration which apply to2c 386 NATURE OF CAPITAL AND INCOME the composition of forces or motions is very evident. Availingourselves of this analogy, we may say that in the case of dis-continuous income, the point 0 traces the capital curve (back-ward) by obeying alternately two tendencies, — one, to followthe discount curve at times when no income occurs, and theother, to rise vertically whenever income occurs. In the caseof continuous income these motions occur simultaneouslyinstead of alternately, and the resultant is a smooth curveinstead of a series of teeth. The s
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