Knowledge that Transforms

IN RECENT YEARS a number of theoretical and statistical investigations' in the field of business-cycle analysis have made use of differeince equations, or mixed difference and differential equations. These equations serve to study the possible endogenous movements of a schematized economic system governed by a set of as many structural relations as there are variables, the movements of which are considered. In the investigations referred to, the treatment of such a set of equations has been to eliminate successively all variables but one, which leaves one from which the possible movements of the system under consideration are studied. If the final equation in the variable Zt is a linear homogeneous difference equation or mixed difference and differential equation, possible movements of Zt are found by substituting for it in the final equation an expression of the form2

IN MODERN business-cycle research the following proceeding is being commonly used: First, a mathematical model (a determinate dynamic system) is set up, as an attempt to describe approximately the interconnections between a set of economic variables, their time derivatives, lagged values, and so on, in terms of strict functional relations. By some statistical procedure the constants in the system are estimated from the corresponding observed time series. Then the system is solved, i.e., the variables are expressed as explicit functions of time, involving the estimated parameters. The degree of conformity between these theoretical solutions and the corresponding observed time series is used as a test of the validity of the model. In particular, since most economic time series show cyclical movements, one is led to consider only mathematical models the solutions of which are cycles corresponding approximately to those appearing in the data.' This means that one restricts the class of admissible hypotheses by inspecting the apparent form of the observed time series. This condition for a good theory is of course not a sufficient one, since there are in general many different a priori setups of theory which are capable of reproducing approximately the observed cycles. But, what is more important, it may not even be a necessary condition, and its application may result in a dangerous and misleading discrimination between theories. The whole question is connected with the type of errors we have to introduce as a bridge between pure theory and actual observations. Compared with actual observations, each equation in a dynamic model splits the observed variations into two parts, one part which is explained by the equation, and another part which is not accounted for, and which is ascribed to external factors. This kind of splitting is common to all theory. We usually consider such equations as good and useful theories if, in order to get full agreement between theory and observations, it is and continues to be sufficient to allow for only relatively small and random external factors. There are two main ways in which such external factors may be

WHEN TO REPLACE individual units of durable equipment by similar or improved units is one of the main problems, upon which the success of industrial enterprise depends. Nevertheless, no unified presentation of its many aspects appears to have been published up to the present. The principal writers refer to replacement merely incidentally, when discussing the subject of depreciation. From the theoretical point of view, such an approach really amounts to putting the cart before the horse.' Replacement is the basic problem, because it actually affects the composition and productivity of a plant. Calculations of depreciation are mere figures entered into books, the significance of which depends entirely on the use to which they are put. The concept of depreciation does not enter into the theory of capital value at all. In practice, on the other hand, differences in depreciation methods do to some extent influence the judgment of traders in the negotiable symbols of composite capital goods. This anomaly is due partly to defective accounting methods. A study of the replacement problem by itself must precede attempts to correct the situation. The value aspect of replacement or arises from the familiar phenomenon that many types of machines outlive their usefulness. The income stream derived from their operation gradually declines, until a more attractive alternative becomes available. The theory that the economic life of a machine is a period which makes the unit cost (plus interest) of the product a minimum, appears to have been originated by Professor J. S. Taylor.2 His algebraic presentation was simplified and refined by Professor Harold Hotelling,3 who employs continuous functions for the purpose. The basic formula given by the latter writer is:4