Definitions and axioms based on engineering and economic concepts in production are presented, and then a conceptual model of production is constructed illustrating the methods which the manager uses in planning and scheduling production. Even under the assumed conditions of certainty in demand and prices, it is shown to be a difficult task and one in which optimal solutions are seldom achieved. In order then to indicate the relevance of the model and its eventual possible usefulness, it is compared with industrial practices and with other mathematical models of production.

ATT EMPTS by economists and statisticians to deal with the problem of the best method for mathematical formulation of the laws of economic behavior from statistical data go back at least as far as Ragnar earlier contributions.' According to Jacob Marschak, however, Frisch did not take full account of the random disturbances (shocks) in the economic relations, nor of the simultaneous character of these relations. 2 In addition to this, Frisch's hypotheses on random disturbances (errors) in variables were not specified in probability terms. 3 These shortcomings have been corrected, over the years, by Koopmans,4 Wald,' Mann,' and Haavelmo.7 Out of this body of contributions has grown the Cowles Commission's simultaneous-equation approach as a method for mathematical formulation of the laws of economic behavior from statistical data, and in I947 two papers were published making a practical application of this method to the marginal propensity to consume ' and to the demand for food.' simultaneous-equation approach draws attention, among other things, to what might be called the inconsistency bias inherent in the single-equation, least-squares method of estimating the parameters (constants) of a functional relationship between two (or more) variables-where the so-called independent variable (or variables) is not completely independent of the dependent variable.10 In this particular respect, the principal purpose of these articles has been to demonstrate that the reduced-form, simultaneous-equation method of estimating parameters is superior superior in the sense that it leads to estimates for the parameters of these functional relationships which are consistent with the estimates for the parameters of the other relationships embraced by the entire system of equations encompassed by the model. Apparently the inconsistency bias of the single-equation, least-squares method can lead to results widely divergent from those of the reduced-form, simultaneous-equation method. rLna possible significance of this to economicmodel building-whether for purposes of 1 Correlation and Scatter in Variables, Nordi7 Journal, Vol. I, I929, Pitfalls in the Construction of Demand and Supply Curves, Veroffentlichungen der Frankfurter Gesellschaft fur Konjunkturforschuing, Neue Folge, Heft 5, Leipzig, I933; and others. For a more complete list see Inference in Dynamic Economic Models, ed. Tjalling C. Koopmans (New York, I950), pp. 423-28. 2 Inference in Dynamic Economic Models, p. 4 of the Introduction by Marschak. 3Idem. 'Tjalling C. Koopmans, Linear Regression Analysis of Time Series (Haarlem, 1937). 5 Abraham Wald, The Fitting of Straight Lines If Both Variables Are Subject to Error, Annals of Mathematical Statistics, September, I940. 'H. B. Mann and A. Wald, On the Treatment of Linear Stochastic Difference Equations, Econometrica, ii (July-October 1943). 7Trygve Haavelmo, The Implications of a System of Simultaneous Equations, Econometrica, iI (January I943); The Probability Approach in Econometrics, Econometrica, Supplement, 12 (July I944). 'Trvgve Haavelmo, Methods of Measuring the Marginal Propensity to Consume, Journal of the American Association, XLII (March 1947). ' M. A. Girshick and Trygve Haavelmo, Statistical Analysis of the Demand for Food: Examples of Simultaneouls Estimation of Structural Equations, Economofetrica, I5 (April 1947). 10 From the statistician's viewpoint this is terribly imprecise phraseology, but it is not easy to phrase precisely without falling into the very terminology and modes of expression this paper seeks to avoid. Essentially -and this is much of the essence of this paperwhat is meant by the so-called independent variable not being completely independent of the dependent variable is that the dependent variable is neither functionally nor stochastically independent of the independent variable. Chart i, for example, illustrates the assumed stochastic behavior of consumption as a function of income: consumption is functionally dependent upon income, but stochastically independent of it in the sense that its random fluctuations about the functional relationship do not affect income. Chart iI, on the other hand, is illustrative of both functional and stochastic dependence: consumption is functionally dependent upon income, but stochastic dependence also exists in the sense that random fluctuations about the functional relationship do affect income. It is the fact that stochastic independence between consumption and income cannot logically be assumed, where both are assumed to be jointly determined by a third variable (as is the case in the model here under discussion), that leads to a biased estimate of the functional relationship between consumption and income using the single-equation, least-squares method.

E VER since the General Theory, continued experimentation with various sets of independent variables has produced numerous consumption functions. The bulk of this work was completed during or immediately following World War II, and therefore little of it has been tested with respect to the postwar period. Some may object to the use of the postwar years for testing purposes, on the grounds that the economic system had not returned to normal conditions, especially in the sphere of consumers' expenditures, owing to the existence of the deferred demand which had grown to immense proportions as the result of five years of war production. But, as we are now engaged in a staggering rearmament program coupled with a foreign war, these immediate postwar years of I946-50 appear as close an approach to the normal period as may legitimately be expected in the near future. Thus, it seems opportune to take stock of some of the existing contributions to see if we may find a consumption function suitable for use as a tool for prediction. The functions which we shall endeavor to test for the postwar period fall into three groups.

Abstract There are three relatively simple methods in common use for determining the rate of interest in installment payment plans. These are known as the Constant Ratio, Series of Payments and Interest at End methods. They have been developed from independent assumption, apparently without any thought that they might be approximations to the more complicated Compound Interest method. In a paper in the American Mathematical Monthly it is shown algebraically that the above methods are approximations to the Compound Interest Method. In order to obtain a better understanding, let us assume temporarily that we know the rate and form a schedule of the payment of the debt. The magnitudes of the rates for the approximate formulas may be compared by means of the areas of the trapezoids. The rate is inversely proportional to the area. The areas for the series of payments, constant ratio, and interest at end methods are in order. This solution is often approximated by means of interpolation in tables. It is the purpose however to compare the compound interest method with the above methods and to present a simple but more accurate approximate formula which can be used without knowledge of the theory of finance.

Abstract The approach to courses for all fields of accounting, public, private, cost, tax and governmental, represents something of a paradox today. Viewed by some as too practical for admission to the cultural-centered liberal arts curriculum, by others as a field of graduate study, these courses on many campuses have been sustained largely by the intellectual prowess of faculties and the ever-increasing demand of students for practical training in areas of employment arising from the war and post-war economics. Even the most cursory glance at recent college catalogues reveals that accounting courses appear under a variety of departments, most frequently under business administration or social science, but sometimes correlated with instruction in mathematics, statistics or secretarial studies, and less frequently as part of the pre-engineering or pre-law curriculum. Teachers of all phases of accountancy in liberal arts colleges, and indeed in other institutions of higher learning, have made a valiant, if at times a somewhat ineffectual, effort to train students to cope with the rapidly shifting economic climate, and to relate the complex technical knowledge of this age to human progress.