Correct Fixed-capital Replacement in Input-output Growth Models

Abstract

Under constant technique defined by constant current 3 and capital-capacity and inventory-output coefficients and constant lengths of fixed-capital lives, then, under steady growth, fixed capital comprises a set of balanced stocks which undergo apparent radioactive decays defined by the lengths of lives and the growth rate. The principle behind this fact was elucidated some time ago by R. Eisner [3], and almost as explicitly by J. von Neumann ([14] p. 2, note (e)), but it has been unfortunately ignored in subsequent literature. This article endeavours to put matters straight with particular reference to steady-state and non-steady-state n-industry models employing square input-output matrices, n> 0. The matter at issue is this: providing an economy's history is entirely known, this year's 4 replacements of fixed capital may be listed by industry of origin, as a vector d, or by industry of origin and use, as a matrix D; also, since last year's capacity outputs of industries and extensions thereto are known, so are this year's capacity outputs, and the respective columns of D may be divided by them giving a matrix of ratios U of fixed-capital replacements by industry of origin divided by capacity output of industry of use. Moreover, if the economy has long grown, with capacity operation of its industries, at a common growth rate r, with unchanging technique, and continues to grow so, both the vector d and the matrix D will do likewise from this year to next year and so on; the matrix of ratios U will then be invariant. However, in general, previous growth may not have been steady growth, and in this case (as well as for steady growth) both d, D and U are always a function of everything that has happened previously up to and including last year . Except for the case of prior and continuing (exponential) growth at the common rate r, one cannot dictate steady future growth of the elements of fixedcapital replacements in d and D, nor can one state accurately, using D (before solving for industries' outputs), the proper ratios of fixed-capital replacements to this year's industries' outputs. (The latter are not necessarily the capacity outputs obtained from last year's information; see Concluding Remarks below. If solved-for outputs q do equal capacity outputs q*, U is a correct matrix; otherwise it is arbitrary and both q, and d and D (obtained from U and q) are incorrect). Stone and Brown, in [13], want to solve the problem of the conditions for sustained growth; commodities given in final consumption are to grow at their own specific rates, or at a common rate, for this [outputs' solution] year and all future years. For reasons just given, they are wrong in dictatorially including the vector of fixed-capital replacements d in their final-consumption vector e, the first possibility that they mention, and they may also be wrong in trying to make a modification of [their] matrix [of current 5