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.
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.