so easily redirected at a moment’s notice, we cannot help thinking of them all the same. Under these conditions the practical value of the final result, at which we arrive nevertheless, is no doubt much reduced. It reads: both for a numéraire and for a money economy, Walras’ system of the economic process is determined and stable, though he did not quite succeed in proving this rigorously; for a process which is stationary except for positive or negative investment on traditional lines, it is hitchless in the sense defined above, and full employment of resources is in fact one of its properties; conclusions other than these can be arrived at only by introducing hypotheses at variance with those of Walras. 72 If in the last analysis Walras’ system is perhaps nothing but a huge research program, it still is, owing to its intellectual quality, the basis of practically all the best work of our own time. 8. THE PRODUCTION FUNCTION All that remains to be said about the period’s work on the higher levels of theoretical analysis may be grouped conveniently around the two sets of data that were the two pillars of the classical temple of 1900, the given tastes of consumers and the given technological possibilities within the horizon of producers. The former topic will be dealt with in the appendix to this chapter, the latter best fits in here. In both cases we shall be only supplementing what, for a lower level of analytic rigor, we have already learned before. In both 72 This may be illustrated by the question of the possibility of underemployment of labor in equilibrium that has played so conspicuous a role in the Keynesian controversy. In the Walrasian system such underemployment is possible only if the Walrasian supply conditions of labor are replaced by the hypothesis that wage rates are ‘rigid downwards’ at a figure higher than the Walrasian equilibrium figure. But we may add the further hypothesis that, if rigidity be removed, the fall in wages that would ensue fails to attain equilibrium because this fall may so reduce firms’ receipts or, even without doing so, create such pessimistic anticipations as to induce shrinkage of operations all round so that the falling wage rates would never catch up with the ever-falling equilibrium level. We may reach a similar result, given some wage rigidity, by assuming that capitalists, while bent on saving without any regard to returns, are unwilling to accept the current returns to investment and wish to hold whatever they have decided to save ‘in the form of immediate, liquid command (i.e. in money or its equivalent)’ (Keynes, General Theory, p. 166). Whatever we may think of the realistic virtues of such assumptions, the point to be kept in mind is that, even if accepted, they would not invalidate Walras’ theory within his assumptions. In particular, they would not prove that the Walrasian ‘condition of full employment’—which is not a postulate but a theorem—makes his system overdeterminate and, in this sense, self-contradictory. It should be added again that economists who wish to establish a tendency in the capitalist economy to produce perennial unemployment have nothing to fear from a proof that, on so high a level of abstraction, perfect equilibrium in perfect competition would involve full employment. Nor has this proof itself anything to fear from the ubiquity of unemployment in a world that is never in perfect equilibrium and never perfectly competitive. History of economic analysis 992 cases I shall carry the story to the present situation. In both cases I shall have to be sketchy to the point of incorrectness. 1 [(a) The Meaning of the Concept.] We begin by recalling the concept of a production function as it is commonly used today. Suppose that a business man A contemplates producing a well-defined commodity X at the rate of per unit of time in a single plant that is to be constructed for this purpose. This may require a unique set of rates of inputs per unit of time—such as of the equally well-defined services V 1 , V 2 , …V n —that are technologically fixed like the Walrasian coefficients of production and define for us economists the only ‘process’ or ‘method’ of production that is available. As a rule, however, there exist several or even infinitely many such processes or methods of production by which can be produced. Each of them is identified by a distinct set of time rates of inputs—again, for us economists: should it happen that two or more technologically different processes use exactly the same combination of rates of inputs in order to produce they would be the same process for us. Mr. A will choose between these possibilities with a view to minimizing the total cost of producing and hence reject from the outset all those processes that use more of all the (scarce) services V 1 , V 2 ,…V n than does another. Among the rest, which we may call the eligible choices, he will choose according to the price situation he expects to prevail in the factor markets during the period for which he plans. The complete list of all those eligible choices, with which A or his consulting engineer is fully familiar, defines A’s or his engineer’s technological horizon. 2 Allowing the to vary continuously and smoothly 3 and letting vary continuously and smoothly also, we may express a man’s technological horizon by setting up a transformation function of the form, x=f(ν 1 , ν 2 ,…v n ), which we call the production function and which associates with any given set of the v i ’s(i=1, 2,…n) a definite maximum value of x which it is possible for him to produce with the given set. Any change in the technological horizon, for example, caused by the discovery of a new process or even by some known process becoming commercially available which was not so before, destroys this production function and replaces it by another. All this is quite simple, and it should be fairly obvious which properties we are to assign to the production function on the various levels of abstraction that are prescribed 1 My inability to present an account of either topic that would be at the same time brief, elementary, and correct—an inability of which I was never fully aware before I put that appendix and this section into their final shapes—had to be stressed because it illustrates so tellingly the conditions in both fields, in which faltering advance was incessantly being undone by mutual misunderstandings between workers—attended with unnecessary peevishness—and all but universal unwillingness to pull the same way. Confusion went so far as to make it difficult, sometimes, to make sure what writers really meant e.g. by the marginal productivity theory. 2 From this must be distinguished the firm’s (or anybody’s) time horizon, i.e. the time span over which it plans. The concept of a time horizon has been introduced by Tinbergen. 3 A continuous function has no jumps, a smooth function has no kinks. Equilibrium analysis 993 to us by the requirements of the particular problems we wish to investigate. Thus, when we are high up in thin air and hunting for the ‘purest’ features of the logic of production, we shall assume, as we have just done, that production functions are continuous and also that they are differentiable twice in all directions. 4 Reality very frequently fails to correspond to these assumptions. But this is no objection so long as we are concerned with the pure logic of production. It becomes an objection only when we apply results, derived by means of them, to patterns and problems for which discontinuity and nonexistence of partial derivatives of the first and second order are relevant: there is no sense whatever in either asserting the presence of continuity and differentiability or denying it for all patterns and all problems. Neglect of this trivial truth has been an unbelievably fertile source of futile controversy to this day and has impeded analytic advance in a manner that is most interesting from the standpoint of the student of ‘scientific progress’ and of the ‘ways of the human mind.’ In order to bring out this aspect it will be convenient to touch first upon a number of points as they present themselves today in order to clear the ground (or part of it) for our story of the historical development and in order to supply to the reader information that may help him to appreciate it. Some modern expositions of the theory of production and cost (mainly of static aspects) are listed for reference in the footnote below. 5 (1) We have come to distrust the idea of any well-defined commodity or service. Moreover, firms do not as a rule produce just one commodity in one quality but many commodities in many qualities, and ability to shift their production from one to the other is obviously an important consideration in the choice of a productive set-up. 6 Finally, a change in the combination of productive services will frequently affect of itself the quality or even kind of the commodity a firm produces. To some extent, this may be taken into account by admitting many commodities (x 1 , x 2 ,…x m ) into the production function and by writing the latter in the implicit form, φ(x 1 , x 2 ,…x m ; v 1, 4 That is, we shall assume that, v i and ν j being representative productive services (i, j=1, 2,…n), all expressions of the forms and exist and are continuous. 5 R.G.D.Allen, Mathematical Analysis for Economists (1938); on the production function and constant product curves, see especially pp. 284–9, which are readily understandable for non mathematicians and reading of which would greatly facilitate the perusal of this section. J.R.Hicks, Value and Capital (2nd ed., 1946), especially Part II. P.A.Samuelson, Foundations of Economic Analysis (1947), particularly ch. 4, perusal of which, strongly recommended, requires some mathematics, but very little. E.Schneider, Theorie der Produktion (1934). Gerhard Tintner, ‘A Contribution to the Nonstatic Theory of Production’ (with an excellent summary of the static theory) in Studies in Mathematical Economics and Econometrics (H.Schultz memorial vol.), 1942. 6 Louis M.Court, ‘Entrepreneurial and Consumer Demand Theories for Commodity Spectra’ (Econometrica, April and July–October 1941) has considered the case of infinitely many commodities, an idea of great importance. History of economic analysis 994 v 2 ,…v n )=0. This has been done by Allen, Hicks, Leontief, Tintner, and others. (2) If we wish to base our theory of production on the Jevons-Böhm-Taussig theory of the ‘roundabout’ process, we may introduce time explicitly into the production function, that is by writing: x=ψ(ν 1 , ν 2 ,…v n ; t). This practice is strongly suggested by Wicksell’s treatment of capital problems and has been adopted by several modern authors. (See e.g. Allen, op. cit. p. 362.) 7 Evidently, however, there are other characteristics of a firm’s technological pattern besides rates of inputs and time: the rates of change of these rates, lags in some of them, cumulation (integrals) of others, outputs that are expected not for the immediate but the more distant future, may all be significant. Without going into these problems, we will advert to the practice of inserting shift parameters (α, β,…) into the production function, which then looks like this: x=f(ν 1 , ν 2 ,…v n ; α, β,…). This amounts to not more than a purely formal recognition of the fact that production functions do change in time. The practice may justify itself any day, of course, but so far it seems to me that this fact is equally well expressed by saying, as we said above, that an innovation destroys a production function and sets up a new one. 8 (3) For the economist a process or method of production is defined by the independent variables in the production function, even though this may amount to throwing together what are very different processes or methods to the engineer: this practice simply means that technological differences per se are without interest for us. But it follows that we must include all the productive services that may be required for any of the eligible methods of producing a commodity, although some of these methods may require services that are not required for others. This creates a difficulty that has induced some theorists (see, e.g., Schneider, op. cit. p. 1) to include in the production function only those processes or methods that use the same services (though in different proportions) and to define the technological horizon not by one production function but by many. More important, however, is another point. As defined, our production function refers only to a single firm—strictly, only to a single unit of production 7 The treatment of time as an independent variable fits well also into the Marshallian system, although for other reasons. But though Marshall so treated it in his verbal statements, he did not do so in his mathematical formulations, except of course for value problems—which is a different matter. 8 If, in the manner of Marshall and Hicks, we form a separate category of those innovations that are ‘induced’ by mere expansion of production—which must not be confused with simple changes to methods of production that are within the firm’s technological horizon from the first but do not pay until a certain output figure has been reached—we do in fact recognize an intermediate class of cases which it is useful to separate out for some purposes. But so long as it is not possible to foresee the effects of induced innovations exactly, there is no point in introducing them into the production function or into cost curves. If it is possible to foresee these effects exactly, then induced innovations must be already within the horizon and need not be ‘introduced.’ Equilibrium analysis 995 or ‘plant’ 9 —and not to the economy as a whole. But throughout that period and even today, it was and is common practice to reason as if there were such a thing as a social production function, 10 and it is not difficult to see the reason why: we obviously wish to speak of a ‘social’ marginal productivity when expounding the theory of distributive shares. And so most of the leaders of that period, among them Böhm-Bawerk, J.B.Clark, Wicksteed, and Wicksell, took the existence of an aggregative (social) production function for granted, at least by implication, without realizing that the logical right to use this concept must be acquired by proof. 11 Many modern authors, especially those of the Keynesian type, are just as careless. (4) Mathematically, the production function enters the theoretical set-up—in order to yield demand functions for productive services, see for example Allen, op. cit. pp. 369 et seq.—as a restriction upon firms’ behavior: these strive to maximize net profits subject to the possibilities listed in the production function. We might try to crowd into a single expression the whole of the technological facts that, for any purpose in hand, seem relevant to us. But even where this is possible, it is much more convenient to make a single relation basic—we shall of course choose one that has some primary economic significance; of this presently—and then to introduce the other facts (hypotheses) that are to be taken into account as further restrictions or, as we may say, as restrictions upon the restriction that we regard as fundamental. The best way of making this clear is as follows. Suppose we have n services 12 which define a ‘production surface’ in (n+1) dimensional hyperspace. In general we shall find that firms cannot move about freely over the whole of this surface and that technological conditions permit choice only within the boundaries of a certain region. Thus there may be ‘limitational factors,’ which must, by technological necessity, be always used in strict proportion to the quantity of product or to the quantity of some other factor (R.Frisch); there 9 The problem of production functions for concerns that operate more than one plant will be excluded from consideration in order to save space. Some work has, however, been done on it of late. 10 Marshall and Walras were really the only authors whose argument, carefully scrutinized, turns out to be free from any implications of this kind. 11 To overlook this was perhaps natural for those literary economists who did not have any explicit concept of the production function at all. It was less natural for Wicksteed and Wicksell. But we must not forget that, under conditions of pure competition, the equilibrium relations between the marginal physical productivities realized in different firms and industries are easy to establish and that this is all that was required for their purposes: Marshall’s ‘marginal shepherd’ was well qualified to represent the marginal productivity of his kind of labor in any employment, hence the social productivity of this kind of labor in general. 12 Of course, if we consider scarce services only, account must be taken of cases in which it depends on the extent of firms’ demand whether a given service is scarce or not: water may cease to be ‘free’ in a given spot if firms need more than a certain amount of it. We have touched upon this point already in another connection. History of economic analysis 996 may be also restrictions of other types (A.Smithies). 13 We shall return to this in a moment, but must now advert to a particular short-run type of these additional restrictions, the importance of which for the theory of marginal productivity has been pointed out by Professor Smithies. I have emphasized the fact that the full logical meaning of the concept of production functions reveals itself only if we think of them as ‘planning’ functions in a world of blueprints, where every element that is technologically variable at all can be changed at will, without any loss of time, and without any expense. But whenever we apply the concept, as we certainly wish to do, to firms that own going concerns and are already committed to plant, equipment, and perhaps part of an existing administrative apparatus, then, according to the time we allow for adaptation, those elements of their existing set-up that are resistant to change will act upon technological choice as further restrictions. 14 To assume them away will bring us back into the sphere of pure logic and not alter the fact that reality will fail to correspond to the theoretical model and to the theorems, especially the marginal productivity theorems, that are derived from the model; and to allow time for full adaptation—Marshall’s method of dealing with this situation—will not help us either, because during the lapse of the necessary time other disturbances will occur that will prevent correspondence to the model from ever being brought about. It is as important to realize the inevitable discrepancies between theory and fact that must result from this as it is to realize that they do not constitute a valid objection to the former; it is no valid objection to the law of gravitation that my watch that lies on my table does not move toward the center of the earth, though economists who are not professionally theorists sometimes argue as if it were. (5) It is therefore under exceptionally favorable circumstances only that we can observe ‘logically pure’ production functions. This is the case particularly in agriculture, where we have not only observational but also experimental material with which to construct them. But whenever we try to do so from observations of going concerns alone, we meet difficulties similar to those that we meet in trying to construct statistical demand curves and cannot in general expect—not at any rate without taking special precautions— that we get the production functions of economic theory. Nevertheless and in spite of the errors in interpretation to which they may give rise, 15 ‘realistic’ production 13 R.Frisch, ‘Einige Punkte einer Preistheorie…’ Zeitschrift für Nationalökonomie, September 1931. A.Smithies, ‘The Boundaries of the Production Function and the Utility Function,’ Explorations in Economics, Notes and Essays Contributed in Honor of F.W.Taussig (1936). 14 Marshall was well aware of this fact and of its importance for the interpretation of actual business behavior. See his Pittsburgh gas case, Smithies op. cit. p. 328. 15 One of these errors stems from the observation, in itself quite correct, that a going plant, designed for a particular output and for a particular process of production, is often very rigid and leaves little room for adaptations to new situations of the productive combination it embodies, especially to changes in relative prices of services. Consideration of these relative prices, as foreseen at the time when the plant war erected, is embodied (often subconsciously) in the set-up of the plant itself. Equilibrium analysis 997 functions are of great importance. They help to destroy the layman’s impression that production functions and marginal productivity schedules are just theorists’ fictions. They confront us with new problems and shed light on the stretch of road before us. For examples I refer the reader to the report of a committee of the Econometric Society published in Econometrica (April 1936) by its chairman, Mr. E.H.Phelps Brown. [(b) The Evolution of the Concept.] As we have seen in the preceding chapter and in Parts II and III, schedules of marginal productivity, in terms of physical and of value products, have been in use ever since the times of Turgot and even before. The production function itself appeared in ‘classic’ times under the name of the State of the Arts—it being recognized that certain arguments hold only if technological knowledge is assumed to be constant. The most important of these arguments was the law of decreasing returns from land, but already Ricardo, by recognizing that ‘real values’ of commodities are ‘regulated’ by the ‘real difficulties’ encountered by the least-favored firm, pointed toward a wider generalization. And then there was what Marshall called Thünen’s ‘great law of substitution.’ All this had still to find its proper relation to the principle of marginal utility, but the rest looks to the backward glance—apart from the more difficult problems that lurk behind even the simplest case—like a fairly easy task of polishing, co-ordinating, and developing existing ideas, all of which were to be found, in one form or another, in J.S. Mill’s Principles or, at all events, in Mill plus Thünen. The Austrians accomplished this in their way and Marshall in his. 16 In Marshall’s Principles we find in fact, though he did not avail himself of the production function explicitly, a very complete and properly qualified marginal productivity theory of the firm and of distribution, and in addition many indications that he saw the problems beyond. 17 If we take in his treatment of the subject fully, even in the 16 Two other contributions should be mentioned here that seem to have remained almost unnoticed for a reason that is highly characteristic for the conditions prevailing in our field—their brevity: A.Berry in a paper on the ‘Pure Theory of Distribution,’ read before section F of the British Association for the Advancement of Science and published in its Report, 1890, presented ‘equations of marginal productivity’ that equated prices of productive services to marginal physical productivities multiplied by the prices of products. And Edgeworth, in 1889 and then again in 1894 (see Papers II, p. 298 and in, 54) did the same. Both use production functions explicitly and present the equalities referred to as elements of a comprehensive equilibrium system. Neither received much credit for what must be listed as a considerable achievement. Professor Stigler, however, noticed them both (Production and Distribution Theories, pp. 132 and 322). Owing to their close relation to Marshall and, especially in Edgeworth’s case, to all other builders in that field, it seems hopeless to try to appraise individual ‘rights.’ But their contributions help us to realize the breadth of the wave, at the crest of which stands Marshall’s work. 17 Professor Stigler (op. cit. ch. 12) has shown very well how Marshall, pushing his way slowly through traditional underbrush, ended up by accepting eventually the whole of the marginal productivity apparatus. If, however, he would never admit the full extent to which he actually did so, this is, I think, adequately explained by (1) his re- History of economic analysis 998 form given to it in the first edition of his Principles, we cannot help feeling some surprise at the statements at the beginning of Wicksteed’s Essay, to wit, that ‘in investigating the laws of distribution it has been usual to take each of the great factors of production…to inquire into…the special nature of the service that it renders and…to deduce a special law regulating [its] share of the product’ and to unify these laws on the basis of ‘the common fact of service rendered.’ 18 But Wicksteed, dropping Marshall’s wise hesitations and qualifications and writing down the production function explicitly, did set forth boldly the naked logic of the matter and also attempted a proof of the propositions—both of them guardedly affirmed but not proved by Marshall—that every ‘factor’s’ distributive share will under ideal conditions tend to equal its quantity multiplied by its marginal degree of productivity; and that those shares will tend to sum up to (to ‘exhaust’) the net product of every firm and, in the sphere of social aggregates, Marshall’s ‘national dividend.’ Now, both propositions are equilibrium propositions and need not hold outside of the point of equilibrium, assuming that one exists. Marshall was of course aware of this but it was left for Wicksell to state it explicitly. 19 Wicksteed, however, based his proof on the sufficient but not necessary postulate that the production function is homogeneous of the first order, in which case the ‘exhaustion theorem’ would hold identically, that is, all along the line and not only in equilibrium. 20 He did recant later on (see Common Sense of Po- luctance to throw in his lot with the non-English economists who did the same thing; (2) his justifiable aversion to assigning a ‘causal’ role to the partial coefficients of the production function; and (3) his awareness of conceptual difficulties, some of which were alluded to above. 18 P.H.Wicksteed, An Essay on the Co-ordination of the Laws of Distribution (1894), p. 7. If the statements above are hardly fair to Marshall, they are strikingly unfair to Walras and even to J.B.Say. The irritation that Walras displayed in his ‘Note sur la refutation de le théorie anglaise du fermage [rent of land] de M.Wicksteed’ (Recueil publié par la Faculté de Droit de l’Université de Lausanne, 1896, republ. as appendice III of the 3rd ed. of his Éléments, but left out in the 4th, which, however, contains the new no. 326 on marginal productivity) is therefore less unjustified than Professor Stigler declares it is. Moreover it is a misunderstanding to think that Walras claimed personal priority for the theory of marginal productivity as defined by himself. As far as this goes, the note on p. 376 of the Éléments is conclusive. 19 See his Lectures I, p. 129. Professor Stigler’s exposition of Wicksell’s share in the solution of the ‘modern’ marginal productivity theory—I call it ‘modern’ in order to distinguish it from that of Longfield and Thünen—is very interesting because it shows how difficult it is, even for first-class minds, to grasp and appreciate relatively new ideas that have already been displayed in broad daylight. Wicksell might have learned all or nearly all that was to be learned from Marshall and the Austrians. But it took him another decade, after having himself adumbrated the theory in 1893, to arrive at his final view of the matter (see Stigler pp. 373 et seq.), in part, as an acknowledgment shows, with the help of Professor D.Davidson. 20 A function of two or more independent variables is called homogeneous of the first order or ‘linear and homogeneous’ in all or some of these variables if, when these increase or decrease in a given common proportion—for instance when they are multiplied by a constant λ—the dependent variable increases or decreases also in the same Equilibrium analysis 999 litical Economy, p. 373n.) but without carrying out the alterations that this recantation would have called for. Before going on we had better see what happened at about the same time in Lausanne. Remember that Walras originally used what may be called a degenerate pro duction function, that is, a production function restricted to technologically fixed and constant coefficients of production. In 1894, Barone suggested to him the idea of turning these technological constants into economic variables and of introducing, for the determination of these, a new relation, the équation de fabrication, which was to express the fact that, if some coefficients are decreased, output may be maintained by an appropriate compensatory increase of others: the new ‘unknowns/ that is the new variable coefficients, were then to be determined by means of the condition that costs be minimized for any given output and any given factor prices. Barone himself started work on these lines and published two installments of a corresponding theory of distribution (‘Studi sulla distribuzione: la prima approssimazione sintetica’) in the Giornale degli Economisti, February and March 1896, 21 without however going on with it—we shall presently see why. Walras had already glanced at variability of coefficients of production in connection with his theory of ‘economic progress,’ which he defined (in contrast to ‘technological progress’) as progressive substitution of the services of capital goods for services of ‘land.’ He then reproduced Barone’s suggestion in his ‘Note’ of 1896 (mentioned above) and in the new no. 326 of the fourth edition (1900) of the Éléments. There he formulated ‘the theory of marginal productivity’ in three propositions of which the proportion. Call x the product as before, although now this x, standing for the total national dividend, would raise very delicate index-number problems, and (ν 1 , v 2 ,… v n ) the quantities of scarce factors used in producing x. Then the production function, x=f(ν 1 , ν 2 ,…ν n ) is said to be homogeneous of the first order if λx=f(λν 1 , λv 2 ,… λv n ) for any point (ν 1 , ν 2 ,…v n ) and any λ. In this particular case the relation, holds over the whole interval in which the x-func-tion exists. This is Euler’s theorem, or rather a special case of Euler’s theorem, on homogeneous functions. Identifying the with the various factors’ marginal degrees of physical productivity, we see that their shares exhaust the social product over the whole of that interval and whatever the amount of the product, although all that we can aver in cases not linear and homogeneous is that they do so in the equilibrium point. Translating this into economic terms, first-order homogeneity means that there are neither economies nor diseconomies of scale, or that large- and small-scale production is equally efficient, or that there are ‘constant returns to scale.’ In itself this implies nothing, of course, about what happens when only one of the ‘factors’ is increased, the others remaining constant, i.e. about the shape of each ‘factor’s’ marginal productivity curve. Note that, since λ is arbitrary, we may put it equal to the reciprocal of any of the v i ’s, e.g., to Then the production function reads: i.e. the average productivities of all ‘factors’ are functions of the proportions but not of the absolute amounts in which they are used. 21 See Stigler, op. cit. pp. 357 et seq. History of economic analysis 1000 last was omitted, without warning, or motivation, from the édition définitive (1926): (1) free competition brings about minimum average costs; (2) in equilibrium and if average cost equals price, the prices of productive services are proportional to the partial derivatives of any production function [that contains only substitutional (compensatory) services] or to the marginal productivities; (3) the whole amount of product is distributed among the productive services. 22 In 1897 (Cours II, 714–19) Pareto criticized the marginal productivity theory—mainly on the ground that it breaks down in the case of what are now called limitational factors—and blocked out a theory that covered all the more important possibilities and which was technically improved in the Manuel. But he looked upon this not as an improvement—especially not as an improvement on Walrasian lines—but as a renunciation of the marginal productivity theory, which in the Résumé of his Paris course (1901) he declared ‘erroneous.’ It was necessary to inflict these details upon the reader because they serve to clarify the situation in the late 1890’s. 23 By 1900, then, the production function had established itself, as a result of the efforts of many minds, 24 in its key position, alongside the utility func 22 In proposition (2) I have italicized the word ‘and’; and I have inserted the proviso that only substitutional factors are included because this was clearly Walras’ meaning, as a preceding sentence on the same page (375) shows, where he explicitly recognized the existence of other, non- substitutional ones. I think that both alterations only emphasize Walras’ true meaning. But I am unable to offer an explanation why, changing his careless (and meaningless) original statement that each service’s rate of remuneration is ‘equal’ to the partial derivative of the production function into the statement that it is proportional to it, he did not say what the factor of proportionality is, namely, in full equilibrium of pure competition, the price of the product. And I am also unable to say why, seeing that he imposed the condition that total receipts be equal to total cost, he dropped the exhaustion theorem which follows from this condition. Ob serve that, since firms will always try to minimize total cost, whatever their output, propositions (1) and (2) hold also for outputs other than the equilibrium output of pure competition. Then the factor of proportionality is no longer product price, but is still marginal cost. 23 The reader finds many further details in Stigler’s work (especially pp. 323 et seq.) and in H.Schultz, ‘Marginal Productivity and the General Pricing Process,’ Journal of Political Economy, October 1929. This paper contains much useful information and especially the simplest exposition in English of Pareto’s theory of production. Unfortunately it is also misleading not only in individual points but also in the total impression it conveys. In this respect, perusal of J.R.Hicks’s ‘Marginal Productivity and the Principle of Variation,’ Economica, February 1932, and of the subsequent controversy between Hicks and Schultz (ibid. August 1932) would provide an antidote. 24 It is hardly possible to be more specific than that. The names of Berry, Edgeworth, Marshall, Barone, Walras, and Wicksteed all enter in some way or another when we discuss this difficult case of paternity. Remember, we are now discussing the birth of the production function as such and not the older or newer marginal productivity ideas that had more or less definitely pointed toward it for a century or more. The Walras-Barone équation de fabrication is of course nothing but a particular form of the production function. Equilibrium analysis 1001 . reading of which would greatly facilitate the perusal of this section. J.R.Hicks, Value and Capital (2nd ed., 1946), especially Part II. P.A.Samuelson, Foundations of Economic Analysis (1947), particularly. History of economic analysis 996 may be also restrictions of other types (A.Smithies). 13 We shall return to this in a moment, but must now advert to a particular short-run type of these. April and July–October 1941) has considered the case of infinitely many commodities, an idea of great importance. History of economic analysis 994 v 2 ,…v n )=0. This has been done by