Knowledge that Transforms

AT least since Mandeville's Fable of the Bees (I728), there have been underconsumptionists who have ascribed trade depressions to deficiency in consumption expenditures.' Underconsumptionist thought may further be subdivided into two schools. Monetary underconsumptionism, which does not concern us here, blames underconsumption upon flaws in the processes of creation and circulation of money and credit. Social Credit movement in Great Britain and the Greenback movement in the United States may serve as illustrations. Since Marx and Rodbertus, however, the deficiency of consumption expenditures (and purchasing power) has been ascribed more commonly to maldistribution of real income. This we shall call maldistributionist or real underconsumption. During prosperity, income is concentrated in the higher brackets, where a large fraction is saved. If the savings are hoarded, there arises an immediate deficiency in consumption. If the savings are invested, the deficiency is only postponed until the day when additional consumption goods are produced because of the new investment, and come on to market without additional purchasing power to absorb them. Such, in briefest outline, is the position of the late John A. Hobson, the leading English-language representative of real underconsumptionism in the twentieth century.2 In this view, the principal means to prevent and to remedy depressions is substantial redistribution of income in the direction of greater eaualitv. In addition to Rodbertus, Marx, Hobson, and other leaders of the economic underworld, Keynes has given this position an indirect accolade in the General Theory,3 and it has been adopted by a substantial fraction of the neo-Keynesian school. Virtually all of the discussion, however, has been carried on in a quantitative semi-vacuum, which is to say, without any precise ideas as to the quantitative importance of possible income redistributions. It was of course recognized from the outset that personal savings rise faster than personal income, or in current jargon, that individuals' average propensities to save rise with their incomes. What was not recognized, however, was that for redistribution problems the relative marginal propensities to save of different income classes were likewise important, since redistribution involves shifts between income classes at the margin. To cite an extreme case, if all individuals' marginal propensities to save were identical, equalization of incomes would have no effect whatever on aggregate consumption and saving, however great might be the disparities in average propensities between rich and poor.4 Keynes himself, it would appear, was guilty of some inconsistency on this subject. He considered his consumption function relatively stable (which presumably means stable with respect to changes in income distribution), and at the same time he advocated income equalization in the interest of increased aggregate consumption. One of the first studies to apply modern aggregative analysis in estimating the quantitative effect of income redistribution on aggregate consumption was carried out by Harold Lubell at the Board of Governors of the Federal Reserve System.5 His study, which has been un* This study was financed by a grant from the Social Science Research Committee of the University of Wisconsin. 1 Harry G. Johnson cites the French Physiocrat Boisguillebert a century earlier as maintaining that trade would be more active if taxation fell on the rich than if it fell on the poor, which comes closer than Mandeville to a maldistributionist position. The Macro-Economics of Income Redistribution, in Alan T. Peacock (ed.), Income Redistribution and Social Policy (London, I954), p. I9. 2 For a full presentation of Hobson's views, see Erwin E. Nemmers, Hobson and Underconsumption (unpublished Ph.D. dissertation, Wisconsin, I953). We have called Hobson a twentieth-century writer, :but the initial presentations of his views appeared before the turn of the century. 'Keynes, General Theory, pp. 369-74. ' average propensities are important in this case only if ioo per cent of one individual's income is being taken away, or in a case where income is being given to individuals who had none before. 5 Harold Lubell, Effects of Income Redistribution on Consumers' Expenditures, American Economic Review, xxxvii (March 1947), 157-70, corrected in part, ibid., xxxvii (December 1947), 930. Lubell results appear to have furnished statistical

THE purpose of this paper is to analyze the formal structure of the matrix as an accounting tool (I to 7), to review actual inputoutput matrices under the aspect of their formal structure (8), and finally to construct an income-product matrix (g to I 2 ). i. A matrix, as used in macro-accounting, is a form of presentation of accounting material. It is distinguished from other forms of presentation mainly by four technical features: economy in figures, consolidation of interaccount flows, absence of narrations, and specific grouping of the accounting material. The economy in figures is obtained by making one figure serve two purposes simultaneously. According to the direction in which it is read, either vertically or horizontally, a figure is either a debit entry or a credit entry. This feature has the advantage that the number of figures required to communicate a given volume of information is in a matrix roughly one-half of that required in any other form of accounting statement. On the other hand, difficulties arise if the volume of information becomes large. The usual device of making statements manageable in size, which is to relegate part of the information to subsidiary schedules, is not available in the case of a matrix, because, in consequence of the double purpose served by each figure, figures cannot be taken out of the context within the matrix. A matrix grows therefore in direct proportion to the volume of information and, after a certain point, becomes an unwieldy instrument. The second feature is the necessity of consolidating inter-account flows. Transactions affecting any two accounts have to be contracted into two figures, because the framework of a matrix has only two places available for the record of flows affecting two accounts. One is the place of intersection of the row representing the credit entries to the first account with the column representing the debit entries to the second account; in this place, the flows from the first to the second account are recorded. The other place is the intersection of the row of the second account with the column of the first account, which is the place for the flows from the second to the first account. The matrix thus allows an expression of the direction in which transactions flow between any two accounts. But no more. If the transactions between two accounts are for instance composed of a number of heterogeneous flows and that occurs quite regularly in macro-accounting the social accountant faces an inconvenient choice. One alternative is to consolidate different transactions among the same transactors. This entails a loss of information or even of meaning, though the macro-accountant may help himself to some extent by cumbersome devices such as double rows 1 or footnotes. The other alternative is to split one transactor into as many accounts as different intertransactor-relations are to be recorded. This entails a loss of institutional or other transactor unity. In any case, the number of meaningful flows between two transactors which can be shown in a matrix is technically limited by the number of available accounts. The third feature of a matrix is the absence of what accountants call narration, that is a short explanation of the meaning of an entry. An economy in text is thus added to the economy in figures, making the matrix a still more concise form of presentation. Without the guidance of narrations, the reader of a matrix can derive the meaning of an entry solely from the captions of the row and column at whose intersection the figure stands. In this way, captions in a matrix are charged with two functions. They have to describe not only the account as a point of reference, but also the kind of transactions entered in the account. To call a personal account by the name of the transactor, or the impersonal account by the subject-matter assembled in it, is not sufficient. A way must also be found to describe the transactions entered in the account; otherwise

IT all began when that Schumpeterian innovator, John Maynard Keynes, introduced into a model the idea that savings and investment are brought to equality not (merely) by changes in the rate of interest but by changes in the level of income. Business cycle theorists -some of them -had assumed for decades that fluctuations in the rate of capital formation caused fluctuations in the level of income. But because the then foundations of economic theory included the axiom that the flow of saving and that of investment are brought to equality by the interest mechanism, without aid from change in the income level, cycle theorists were inhibited from pursuing the implications of their assumption, and schizophrenia divided cycle theory from the central body of economic theory. So it was not until Keynes broke the cake of custom and elevated the determinants of saving to a new importance that the search for knowledge of those determinants began. In a limited sense this statement is incorrect, for studies of the relationship between family income and components of family expenditure go back a century. But these were not directed toward an understanding of the determinants of aggregate consumer expenditure in an economy, and so at least in purpose they are not forerunners of the studies discussed here. Not many topics in the history of the development of economic theory have occasioned more discussion within a period of some two decades than has the form of the consumption function since the publication of the General Theory in I936. The discussion continues, among other places, in many of the papers in Savings in the Modern Economy,* the point of departure for this essay. And the end is not yet. On the contrary, the present state of the discussion invites further research. This is natural. The maturation period of an idea is often longer than that of a man.

A YEAR ago and for another audience, my ET friend David Novick wrote a short piece on the misuse of mathematics in which piece has rather unexpectedly been made the center of controversy.1 Under the circumstances, a rejoinder, and perhaps a more specific restatement of some of Novick's criticisms, is in order. As I share many of his misgivings regarding some economics, 2 and because, as colleagues, we could readily confer on this rejoinder, the following comments can be considered a joint expression of our views. Because some have incorrectly inferred that Novick is opposed to any use of mathematics in the following list of more specific complaints, together with some examples from a recent work of Samuelson, is offered. i. A few writers make a travesty of mathematics by using its notation, but neglecting its operations. Instead of using the powerful techniques of mathematics to make implicit relations explicit, they merely paraphrase with symbols, and sometimes with their own peculiar and unexplained notation, ideas that could readily be stated also in prose. A very few may use mathematics as a form of lifemanship in the best Stephen Potter manner. It is probable, though, that most of these writers have no wish to win arguments through intimidating unintelligibility.3 However, the result is that many able economists, especially the older and more experienced ones, cannot comment on some published ideas because they cannot read them. Not only do many competent economists thus become frustrated, but the constructive criticisms they might make are unnecessarily precluded. This is unhealthy and wasteful.4 2. Economic life is so rich in detail that only a few of its features can ever be described in a set of equations. Hence all the rest necessarily becomes lost from any succeeding analysis if it is limited to mathematical operations. Such caricatures of economic behavior can lead to logically correct conclusions granted the assumptions, the form of the equations, and the input magnitudes if there be any and yet the results may really be useless. This spinning of theories, with little reference to the real world, can retard progress within economics and bring the profession into disrepute among those who must apply economic theories to problem areas. 3. The widespread use of mathematics has often tended to transfer attention and inquiry from the substance of economic life to the formal properties of some of the mathematical expressions used to describe it. Part of Keynes's contribution was to suggest, from his knowledge of the environment, that aggregate income was somehow related to intended savings and intended investment, and that the equilibrium income would equate them. The repeated enunciation of this idea by others, in the form of graphs and equations, has mesmerized some economists into forgetting that many of the Keynesian functions may not exist precisely.5 What are now really needed are empirical investigations to determine the existence and nature of, say, consumption and investment as functions of income. Somewhat analogously a few economists have become so hypnotized by the delightful fluctuations that can be generated by some dynamic equations, chosen supposedly to simulate business cycles, that they forget that the validity of these models remains unknown, and, in fact, may not even be susceptible to test. For example, do all firms, if one can sensibly think of so general a class, really adjust their cyclical inventory position in some unique mechanical fashion? Certainly we need models but surely the cost need not be learning much less about economic behavior. 'This REVIEW, XXXVI (November 1954). 2 Especially the high deductive and exclusive use of algebraic symbols, in extremely abstract models, without ref erence to data and with little thought to the real world. 3 The atheist Diderot is supposed to have been routed in confusion from Catherine the Great's court when confronted by an equation that, Euler told him, demonstrated the existence of God. 'I am indebted, though, to Samuelson for the pains he has taken subsequently to explain his ideas in a language I can better understand. 'In the sense that observed behavior, among classes, times, and countries, indicate no central tendencies.

T HE recent development of linear programming has made possible a new type of economic analysis.1 M1ethods have been developed to allow the computation of numerical solutions for theoretical problems involving economic choice. Linear programming provides a general mathematical model which is applicable to problems that can be expressed as the maximization (or minimization) of a linear form subject to a system of linear inequalities. Many economic problems can be placed in this general format, and linear programming can be used for the analysis of a plant, a firm, an industry, a national economy, or the world economy. The general methods of linear programming are used here in the formulation and solution of a short-run model for the coal industry. The present application is formally similar to the transportation problems formulated by Hitchcock and Koopmans.2 The model for the coal industry contains two separate, but interrelated, programming problems. The data of the model are spatially distributed demands for coal, the capacities of spatially distributed deposits of coal, and the unit costs of deliveries from the deposits to the demand locations. The levels of the deliveries are the variables of the first of the programming problems; they are selected to minimize the cost of meeting the demands subject to the capacity restrictions for the coal deposits. The variables of the second programming problem are the delivered prices of coal at the demand locations and the unit royalties earned by the various deposits. The values of these variables are selected to maximize total revenue net of royalty payments subject to the condition that every possible delivery must yield a non-positive profit. The optimum solutions of these two problems provide a complete description of perfectly competitive short-run equilibrium for the coal industry. Once the model has been presented, it is implemented with factual information for the demands, capacities, and costs; and numerical solutions are computed for both problems for

IN a recent article in this REVIEW,' Professor Samuelson proves anew assertion that 'no decentralized pricing system can serve to determine optimally levels of collective (page 388). It is possible, he further states, that utopian voting and signalling schemes can be imagined. But there is still this fundamental technical difference going to heart of whole problem of social economy: by departing from his indoctrinated rules, any one person can hope to snatch some selfish benefit in a way not possible under self-policing competitive pricing of private goods; and 'external economies' or 'jointness of demand' intrinsic to very concept of collective goods and governmental activities makes it impossible for grand ensemble of optimizing equations to have that special pattern of zeros makes laissezfaire competition even theoretically possible as an analogue computer (page 389). conclusions and reasoning are familiar but never as cogently put. Samuelson divides budget into two sectors. first sector presents no difficulties to him. It consists of algebraic taxes and transfers (of an income redistribution type) would have to be varied until society is swung to ethical observer's optimum (page 388). It is in second sector, provision of collective goods, that difficulties arise. contention of this note is that a decentralized pricing system for all public services is technically difficult for reasons other than those stated by Samuelson; but more important is contention that his approach to a theory of public is unfruitful. heart of Samuelson's argument against a decentralized pricing system is definition of collective goods. They are goods which all enjoy in common in sense that each individual's consumption of such a good leads to no subtractions from any other individual's consumption of that good . (page 387). It follows from this that if both A and B desire an additional unit of a 6ollective good, it would be advantageous to both that other person pay, since each will receive full benefits of good no matter who pays. good, because of its technological characteristics, cannot become private property. Since Samuelson claims that he is analyzing the theory of optimal public expenditures we must assume that he is asserting that there are public on services if and only if these services are collective consumption goods. A more sympathetic reading of note might suggest that author was only discussing possibility of sole reliance on benefits taxation as a pricing mechanism for public services and was not explaining their genesis, or that he was even more modest and was suggesting technical difficulties of benefits taxation in case of a restricted set of public expenditures. Are there collective consumption goods? Are they typical public services? In defense of Samuelson there are a host of theorists who have begun their arguments same way. Against Samuelson are facts. He claims that collective goods are not rationed that use of a good by A does not involve any costs to B. Clearly this is not case in such common public services as education, hospitals, and highways, where capacity limitations and congestion are topics of daily press. Would it be true of more sovereign functions of justice and police? crowded calendar of courts certainly implies that use of this function by A makes it less available to B. Similarly a complaint to police ties up officers in a maze of arguments, forms to be completed, and hearings to be attended, reducing their availability to others. Possibly only goods * I am indebted to K. J. Arrow and R. A. Musgrave for helpful discussions on this note. This paper was prepared with support of Office of Naval Research under contract N6onr-2 5I33 (NR-o47-oo4). 1 Paul A. Samuelson, The Pure Theory of Public Expenditure, this REVIEW, XXXVI (November I954).