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1 V. On the Use and Misuse of Mathematics in Presenting Economic Theory Author(s): D. G. Champernowne Source: The Review of Economics and Statistics, Vol. 36, No. 4 (Nov., 1954), pp Published by: The MIT Press Stable URL: Accessed: 26/05/ :34 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. The MIT Press is collaborating with JSTOR to digitize, preserve and extend access to The Review of Economics and Statistics.

2 between those more interested in qualitative and descriptive research and those more interested in quantitative and analytic work. Both MATHEMATICS IN ECONOMICS have their places in economic science and the second group will need more mathematical tools than the first. V. ON THE USE AND MISUSE OF MATHEMATICS IN PRESENTING ECONOMIC THEORY D. G. Champernowne Articles on economic theory are usually intended to explain to the reader what results may be expected to follow various policies or disturbances in given circumstances. The task of explanation is made difficult by the varying capacities and demands of the readers. If the theory is to be realistic, the given circumstances must reflect a great many of the complications of a real-world situation: an exhaustive account of these, whilst necessary for logical completeness, would be intolerably lengthy and boring for the reader who is well informed about the situation to which the theory is intended to apply. He can supply for himself, as obvious, many of the underlying assumptions about institutions and psychology which justify the reasoning in good non-mathematical articles on economic theory. But some readers will lack the experience or the docility to do this, and will either attempt to apply the theory to inappropriate circumstances, or will attack the writer for not giving an exhaustive account of his assumptions. Since proof is not always the most effective method of explanation, many writers are content to sketch their ideas in a manner which will enlighten sympathetic readers without attempting to convey complete conviction to anybody. The cautious economic theorist, whose overriding ambition is never to appear wrong and yet to appear in print at all, has little scope beyond the discussion of economic models. These are shadows of the real situations so drastically simplified that they can be completely described and many of their workings exactly portrayed within the compass of a single article. The logic leading from the assumptions about the model to the conclusions about how it will behave can now be made rigorous and independent of the reader's knowledge of the real world. Such models serve the useful purpose of making it possible to reach complete agreement within reasonable time: occasionally a study of them will reveal that earlier disagreements have been entirely due to a difference of intuitive assumptions about the real world. Unfortunately for the cautious theorist, his economic models will be judged according to the degree in which they appear to be relevant to the real world; so that in avoiding the appearance of being wrong, he may yet appear silly by publishing a long article whose relevance to any practical issue seems to be superficial. This danger of manufacturing mere "toys" is especially great since the assumptions which are most convenient for modelbuilding are seldom those which are most appropriate to the real world. Many would regard as the best and most important articles on economic theory those which reveal keen observation and judgment in choosing assumptions which accord well with facts and which are yet able to demonstrate fairly convincingly powerful conclusions of a simple nature, which suggest important analogues in the real world. They would regard the rigor of the logic and the exactness of the descriptions of the assumptions as secondary matters relating to their style rather than to their importance. The ability to judge the relevance of an economic theory and its conclusions to the real world is but rarely associated with the ability to understand advanced mathematics. An important article on economic theory is therefore likely to be wasted unless it can be set out in prose supported by the most elementary mathematics. "It is obvious that there is no room in economics for long trains of deductive reasoning," 4 so that those economic models which are realistic and yet sufficiently simple to allow ' Marshall, Principles of Economics, 8th ed., Appendix D, p. 78I, I.9.

3 370 THE REVIEW OF ECONOMICS AND STATISTICS useful conclusions to be suggested can usually be set out in prose: their simplicity makes the drawing of the conclusions straightforward and obvious in terms of diagrams or elementary mathematics, whose translation into descriptive prose is straightforward. Even where the argument is not so simple, there are some advantages to be gained from translating the mathematical steps into prose. This may draw attention to a step which is obviously incongruous in relation to the real world, but whose implication would have been entirely lost so long as the argument was left as a sequence of algebraic manipulations. These and other considerations can be advanced to show that it would "be very helpful if those who use mathematics as a language would apply the additional time and effort to restate both the theory and the data requirements in a literary form so that a larger group could study and benefit from the work which they are doing." But the following points should be borne in mind by those editors and readers who accept this view. There is a danger that a prose translation of a mathematical argument will render it inexact and slipshod. Economic theory which is not rigorously set out can suggest false conclusions and indirectly persuade a wide public into accepting them. The mathematical presentation of axioms, reasoning, and deductions is a discipline which, strictly followed, will pinpoint assumptions, expose weak logic to expert scrutiny, and confine conclusions to their proper limits. When the reasoning has withstood the scrutiny of qualified mathematical economists the non-mathematical reader has a good assurance that at any rate that part of the article is correct. At the same time the mathematical presentation will achieve an economy of space whose degree probably few non-mathematicians realize. The economic model has one further advantage to be weighed against its neglect of so many of the particular circumstances of the real world: that is its generality, in the sense that it can deal simultaneously with a whole family of conceivable situations. This is a direct result of the employment of algebraic 'Novick, p. 358, above. notation and roughly corresponds to the fact that a statement such as x2- J = (x-y) (x+y) can stand simultaneously for (32-22) = (3-2) (3 + 2) ( II2 _ 992) = (IoI - 99) (IoI + 99) and for an endless series of other such statements. The advantages of such generality in an economic theory should not however be rated too highly. There is seldom much to be learned from it unless one can conveniently select and interpret those particular cases which are most relevant to actual situations. Despite Marshall's devastating comment about long trains of deductive reasoning, some theoretical developments in economics fail to exclude them altogether: they are concerned with models whose behavior is complicated enough to require quite sophisticated mathematical argument. To express their argument shortly and reveal its structure may demand the use of a condensed notation and an appeal to known general theorems at an advanced level. To translate axioms, proof, and results all into prose would, at a guess, often increase the length of such an article tenfold, whilst blunting the precision considerably: moreover, unless diagrams were substituted for much of the mathematics, the whole would become almost unintelligible even to a mathematician until he translated it back into symbols. In addition to saving space, the elegant use of mathematics will simplify the understanding of a proof by revealing it as a special case of a more general line of argument with which mathematicians are already familiar. It is a common experience to find, after hacking out a tortuous proof of the properties of an economic model, that the whole thing can indeed be done far more briefly and directly by a method of proof with which mathematicians are already familiar in another context. Occasionally the tortuous proof appears in print and it is left to the reader to discover the more direct line of reasoning. Whilst elegant presentation and appeal to familiar results and lines of argument will

4 MATHEMATICS IN ECONOMICS greatly assist the mathematical reader, this is cold comfort to the non-mathematician. The author appears to have the choice of being completely unintelligible to the non-mathematician, or else making his article very longwinded, imprecise, and unnecessarily obscure to those with mathematical training. It is to guide him in this choice that the Editors of the Economic Journal have recently been giving helpful advice.6 We may consider in conclusion these and any further precepts which editors may reasonably require their contributors to observe or, at least, to strive toward observing, as an ideal. Three essential stages in an article about an economic model are: i. To set out the axioms determining the working of the model and the set of initial conditions or disturbances whose effects are to be investigated; 2. To set out the results which, it is claimed, follow from the axioms; 3. To give a complete and rigorous proof of the results on the basis of the axioms. Many non-mathematical readers may be especially qualified to judge the relevance of the model to the real world. Hence it is important that, in addition to the mathematical statement of the axioms, a fairly accurate translation of them into prose should be attempted. Similarly the non-mathematician may be interested in tracing analogies between the results claimed for the model and possible consequences in the real world: therefore these results also should be translated if possible into a set of conclusions expressed in prose. If it is impossible to make an exact translation the reader's attention should be directed to any points where the prose translation is only suggestive of what has been proved, or where it is more general than the proven results. On the other hand it should not be a primary concern to make the proofs themselves understandable by non-mathematicians: it is more important that these should be put in the form in which they can be easily grasped and scrutinized by those who have mathematical training. This, however, does not rule out the employment of diagrams to illustrate the reasoning, nor the use of elementary notations and algebraic processes in cases where these can express the argument almost as well as more advanced ones, and without increasing the length of the proof more than twoor threefold. For example, there is a strong case for writing out matrix equations 7 such as Y = AX + B.'. AY =AA in the expanded form nn Y= NA jxj + Bi. A Y = ja_axj j=l I, 2... m I, 2... n or in certain journals as Y, = A,Xi + A12X2 + --AlnXn + B1 1 Y2 = A21X1 + A22X2+.A2nXn + B2 Ym = A mrxi + A m2x2 + j=l.a mnxn + Bm J therefore if X1X2... Xn change to X11X2' *.. XnJ, then Y1Y2... Ym change to YliY2... Ymnl where y11 = Y1 + Ali(Xll - X1) + A12 (X21 - X2) + * Aln(Xnl - Xm) Y2= Y2 + A21(X1- X1) + A22 (X21 - X2) + + A2n(Xnl - Xn) Ym1 = Ym + Aml(Xl1 - X1) + Am2 (X21 - X2) +* + A mn(xn1 - Xn) But there is little point in such concessions at any one stage of the proof unless the whole argument can be reduced to equally simple terms. The semi-mathematical reader will have little cause for gratitude if he is lured by such means to read the first few pages and is then abandoned out of his depth in a quagmire of unfamiliar notation. That the use of elementary notation may not always be an advantage is suggested by the following example: Suppose y (income) must satisfy the relation s y = b + i (investment), where s and b are constants. Then with the aid of a diagram there should not be much 37I 'Economic Journal, LXIII (March I954), Notice by the Editors, p. i. 'In these and the following equations the symbol A denotes "change in."

5 372 THE REVIEW OF ECONOMICS AND STATISTICS difficulty in explaining and establishing the multiplier argument s y = b + i.. Ay =- Ai (provided s 7 o) giving the relation between an autonomous increase Ai in investment and the corresponding increase Ay in income. This can be algebraically generalized to the case where Y is a vector of n different economic variables which must satisfy n relations given by SY= B+I where B is a vector of constants and I is a vector of parameters reflecting autonomous economic conditions and S is an n X n matrix. The changes AY may then be obtained in terms of autonomous changes AI by the argument S Y = B + I.. AY = S-l{AI} (provided ISI 7# o). This argument would only be confused by expanding the notation into one involving separate terms and fractions involving determinants. Diagrams cannot help here and a prose translation would presumably have to cover an explanation about the inversion of matrices and the meaning of determinants. This argument, that AY = S-1(AJ) or something even more elaborate, is in principle involved whenever one wishes to find the effect of a given disturbance on a position of equilibrium involving many simultaneous condi- tions. Marshall himself insisted on the fact that economic variables are interconnected and determined in just this way.7a The difficulty in making AY = S-1(AI) intelligible to persons who have not found time to learn about matrices is therefore a main obstacle in getting them to understand the ramifications of any change involving three or more interdependent variables. A set of numerical examples is probably the best available substitute for matrix algebra in cases of this kind, but it is a very poor substitute indeed. Whilst the axioms and the results should be fully translated into prose, only minor concessions should be made to the semi-mathematician in the exposition of the proof itself. But there remains one further subject that is very much the concern of the non-mathematical reader. This is the possible effects of relaxing the simplifying assumptions. Usually the writer can only guess at such effects by using his mathematical judgment. Only the mathematical readers will be able to pronounce an informed verdict on the worth of such guesses. But the non-mathematical reader will be very interested in precisely these effects and it is therefore essential that any such guesses should be accurately translated into language which both he and the mathematicians can understand. Editors may ask their contributors to bear such precepts as these in mind when compiling articles on economic theory. But they are no more than a guide, and the editor who insisted on literal obedience to them would be unlikely to publish much in this field. VI. THE SURVIVAL OF MATHEMATICAL ECONOMICS Robert Solow I have a hard time imagining what a contributor to this discussion is supposed to say. I am certainly not going to insist that I have stopped beating my wife or that I shortly will. Surely research (of any kind) is a more useful activity than bickering. In Edgeworth's words "the margin of profitableness in the intellectual as in the external world will differ with the personality of individuals. No general rule is available, except that, like cultivated Athenians, we should eschew the invidious dispar- agement of each other's pursuits." For some reason Novick chooses to base his complaint on the proposition that literary economists are too simple-minded for words (or rather for symbols). Apparently the literary economist, in his native habitat, is alert and clear-eyed; he can talk about profit maximiza- "' See his analogy of the equilibrium of a number of balls resting against one another in a basin. Principles of Economics, Book V, Ch. i and i6 and App. I? 3 para. 4 and 5.