# Capital and Time: A Neo-Austrian Theory

By J. R. Hicks | Go to book overview

Appendix

The Mathematics of Traverse

1

There were several occasions, in Part II of this book, when I advanced propositions which were not intuitively obvious, though I hope they will have seemed to 'make sense'. Proofs can be given and this is the place to give them.

It is not my intention to give in this Appendix a continuous mathematical treatment, which can be read, in place of the text of Part II , by those who prefer such a version. The Appendix is complementary to the text, not a substitute for it. It will nevertheless be convenient to begin with a summary of the assumptions that underlie the analysis of Chapters VII-XI . This is chiefly for reference in what follows.

2

The model has been reduced to bare bones. There is just one original factor (labour) and one final product. A technique (at, bt) is a process, of unit size, extending from t = 0 to t = Ω, by which a stream of labour inputs (a) is converted into a stream of homogeneous outputs (b). We begin from a steady state, in which wage (w) and rate of interest (r) are constant, and in which just one such technique is used; it is a condition of equilibrium, in that steady state, that the rate of interest should equal the rate of return on the technique. The ex-ante capitalized value of the process, at wage w and rate of interest r, must therefore be zero;

(2.1) where

Then at a certain date (called T = 0) a new technique is introduced, which at the old steady state rate of wages has a higher rate of return. This new technique is adopted for new processes, begun in week 0 and subsequently; but meanwhile old processes are continued, as long as it is profitable to continue them. We have to determine the path of the economy from T = 0 onwards. We do so by comparing this actual path with a reference path—that which would have been followed if the technical change had not occurred, so that the old steady state had continued. Magnitudes which refer to the old technique, to the old steady state, or to the reference path, will henceforward be starred.

-185-

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Capital and Time: A Neo-Austrian Theory

• Capital and Time iii
• Preface v
• Contents ix
• Part I- Model 1
• I- Goods and Processes 3
• II- The Process and Its Profiles 14
• III- Social Accounting 27
• IV- Technique and Technology 37
• V- Full Performance and Full Employment 47
• II- Traverse 80
• VII- The Standard Case and the Simple Profile 81
• VIII- The Fixwage Path 89
• IX- The Full Employment Path 100
• X- Substitution 110
• XI- Shortening and Lengthening 125
• Part III- Controversy 149
• XIII- The Measurement of Capital—value and Volume 151
• XIV- The Accumulation of Capital 167
• XV- The Production Function 177
• Appendix 185
• Index 211
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