John Maurice Clark on Marginal Productivity Theory. A note with some unpublished correspondence

In a recent and insightful paper on Institutionalism Between the Wars, Malcolm Rutherford has correctly included John Maurice Clark’s Studies on the Economics of Overhead Costs (Clark 1923a) among the works «most often cited as paradigms of institutional research in the period up to the late 1920’s» (Rutherford 2000a, p. 295; see also Rutherford 2000b). Yet, the relevance of Clark’s Magnus Opus should not be limited just within the narrow limits of the institutionalist camp. Clark’s monograph was in fact one of the most influential works produced by an American economist during the interwar years. In 1922, for example, anticipating some of the main themes of the book in the paper Some Social Aspects of Overhead Costs (Clark 1923b), Clark received a standing ovation at the annual meeting of the American Economic Association at which the paper was presented (Shute 1994; see also Shute 1997). After its publication, favorable reviews of the volume appeared both in the “Political Science Quarterly” (Copeland 1925) and in the “American Economic Review” (Hugh Jackson 1925), while across the Atlantic, Francis Y. Edgeworth, who reviewed the book for the “Economic Journal”, found it to be like «an atlas which together with the general map of a kingdom contains maps of counties and districts on a smaller scale» (Edgeworth 1925, p. 245). In one case, then, the book triggered a brief but significant controversy with one of the – if not the – harshest opponent of institutionalism, namely, Frank Knight. In his critique, the Chicago economist focused in particular on the last chapter of the book, the one dealing with the impact of overhead costs on the law of distribution. In a note which appeared in the “Journal of Political Economy”, Knight informed the reader that:

[i]n his brilliant and valuable book on the Economics of Overhead Costs, Professor J.M. Clark gives an arithmetical example to illustrate the theory of distribution by marginal productivity which seems to the writer involving a slip worth pointing out (Knight 1925a, p. 550).

In order to better understand Knight’s contention, it is necessary to briefly review Clark’s proof that if all productive agents are rewarded in accord with their marginal products, then the total product will be exactly exhausted. We can summarize his arithmetical example as follows. Let 100 acres of land and 1.000 days of labor yield 2.000 bushels of wheat. Suppose that by adding 50 days of labor the product is increased to 2.060 bushels, diminishing returns entering the scene. Now, if we add 5 acres of land, the proportion of the productive factors will be the same as in the original combination, and if the product is increased in the same ratio as both factors, the new output will be 2.100 bushels. Then the marginal contribution of the 50 days of labor is 60 bushels – or 1,20 bushels per unit – while the marginal contribution of the 5 acres of land is 40 bushels – or 8 bushels per unit of land. Multiplying the marginal productivity of a single unit of labor by the 1.000 units of labor used in the original combination gives 1.200 bushels for the total share of the labor, and performing the same operation for the land gives 800 bushels for its total share, which together make 2.000 bushels, exactly exhausting the total product. In Clark’s words:

this is a law and not a coincidence the reader may for himself, using any figures he likes so long as they abide by the crucial assumption of the problem (Clark 1923b, emphasis added).

In his note, Knight declared to appreciate Clark’s effort to present the marginal productivity theory without the use of higher mathematics – «a thing evidently to be desired for purposes of exposition» (Knight 1925a, p. 552) – but he nevertheless found his illustration to be a much more general relation than that of Euler’s theorem. We can follow Knight’s reasoning by use of simple mathematics. Let:

Which, as Knight put it, is «an obvious identity», in the sense that «the reasoning does hold for any figures not violating the restrictions as to proportions». But the interesting thing is that the P₂’s of the original equation disappear entirely. That is,

any assumption whatever may be made as to the product of the second or intermediate combination, without invalidating the result. The reader may work it out with the product of the second combination set equal to zero, or a negative quantity, or a magnitude greater than the product of the third combination or any negative or positive quantity whatever (ibid., p. 553).

In other words, Knight pointed out that Clark’s proof would work not only in presence of diminishing returns but also with constant and even increasing returns, thus violating the main assumptions of the marginal productivity theory. Knight’s note was followed in turn by a Reply by Clark (1925a), a further Rejoinder by Knight (1925b), and a final Concluding Note by Clark (1925b). In none of his two replies, however, did Clark succeed in confuting the validity of Knight’s penetrating criticisms.

From our archival researches it emerges that the controversy hosted in the pages of the “Journal of Political Economy” was anticipated by an epistolary exchange between the two leading economists (the letters is reproduced in the appendix). What is relevant about the correspondence is that it contains a diagrammatic version of the illustration Clark provided in his 1923 volume. Interestingly, from Joseph Dorfman (1964, p. 294) we learn that Clark had informed Wicksteed as early as in 1915 that he had worked out a “diagrammatic geometric proof” of the equality of the sum of marginal products and total products². Clark also mentioned his “independent” discovery of the “exhaustion” problem in a letter he wrote to Paul Samuelson who had asked him about the “psychological genesis” of the accelerator principle.

Your question reminds me of a couple of other instances: the proposition about the sum of the marginally-imputed products absorbing the total product, and the conditions necessary to this, and the working out of a rough form of the R.F. Kahn-type multiplier. I remember where I got the answer to the first: namely in a hotel room where my father and I had gone to meet David Kinley. I had been wrestling with the problem, and figured out a geometrical solution. Later, I talked it over with my friend, Charley Cobb, and he converted it into a case of Euler’s theorem. Before trying to publish it, I did some investigating, and discovered Wicksteed’s monograph, and Flux’s review, which converted Wicksteed’s demonstration into a case of Euler’s theorem. So that had clearly been anticipated (John M. Clark to Paul A. Samuelson: April 21, 1953, John M. Clark Papers, Rare Book and Manuscript Library, Columbia University).

Clark’s diagrammatic exposition does not need any particular comment: as the reader will note it follows the traditional practice of assuming only two factors, in this case land and labor, each of which is homogeneous. There are two aspects, however, which deserve to be emphasized. Firstly, as showed by Knight in his letter of April 24, Clark’s graphical proof is subject to the same sort of criticisms of his simple mathematical example. Following Clark’s procedure, in fact, Knight ably demonstrated that his diagrammatic proof would “check” also with negative marginal products of the eleventh unit of the variable element (see Knight to Clark April 24, 1924 and related graph). Secondly, it should be noted that Sydney Chapman in 1906 had already presented an elegant diagrammatic proof that the residual share is equal to the marginal product of the factor receiving the residual. Differently from Clark’s, Chapman’s demonstration works exclusively under the assumption of diminishing returns and introduced explicitly the problem of external economies (Stigler 1941).

I am wondering what you would say if I were to submit to you something of the nature of the enclosed paragraphs, in your capacity as an editor of the J.P.E. anyway I hope you’ll let me know your reaction to the criticisms^*.

An additional point is considered important by a colleague here, Henry Simons, who I believe was in your class last summer. It is that in figuring the product of the marginal unit of labor by adding labor and that of land by subtracting land, you automatically give the labor too much and the land too little as compared wit what you would get by integrating between the same limits the productivity of infinitesimal increments. If you did not do this you would not of course have enough to pay the land and the labor, using finite increments as units and any productive function giving rise to diminishing returns. It seems to me that my point annuls this in a way, as I show that the actual marginal products may be opposite in sign without destroying the balance of your illustration.

You are quite right in saying that my proposition, as I have put it in the arithmetical illustration you are discussing, is an identity. I am not sure about Mr. Simons point: he seems to be proposing to convert the problem from one of finite increments to one of infinitesimals, in which case I think that Euler’s theorem would apply and the whole product would be divided without shortage of surplus.

Of course, I was using the illustration for the purpose of showing one large discrepancy between the whole product and the sum of the shares of separate contributors, due to the existence of constant factors with unused capacity. I was not concerned with minor discrepancies due to the presence of the variable factors in finite increments, and the nature of my book did not justify taking the space necessary to a full discussion of the logic of the thing in all its relations, such as would be necessary in a treatise on distribution for economists only. I was not even concerned to prove that the product is equal to the sum of the shares, but rather to show that the argument it says it does so, requires that the function be homogenous in the first degree. However, it is an essential part of my case to show a tendency for factors which are “homogenous” in their behavior to absorb approximately the whole product if they get their full marginal worth, so that there would be little or nothing left for “constant” factors.

If I were stating the proposition with reference to the grade of criticism which you are applying to it, I would add some things (which I do add in class-room discussion) about as follows.

The example in the book is an identity because, under the assumptions of our problem (homogenous functions) increments of capital and labor are not separate facts: a positive increment of capital (changing the proportions so that there is more capital per unit of labor than before) is in itself a negative increment of labor (in terms of the amount per unit of capital). There are no two things happening: one change has both aspects.

The question of distribution, however, cannot be answered by one such experiment. It starts with a given proportion and the share either factor can actually get depends in practice on a possible finite change in either direction, addition or subtraction, according to conditions of supply or bargaining situation. These make it possible for the sum of the shares to be either greater or less than the whole product, for any given producer. And even a marginal or “bunk-line” producer (as I understand the Tariff Commission people call him) might pay one factor on the more favorable basis, and another on the less favorable. Now the case is no longer an identity. It is in this form that I have put it in my classes. Discrepancies are limited by the smallness of the increments and the continuity of the curve of diminishing return, but some discrepancies are possible. This point did not seem to me important enough in connection with my overhead cost theme to justify taking up the reader’s attention, but it might be worth making in an economic journal, for economists who might want to follow the argument further than my book does.

Moreover, the chief force acting to bring this is the semi-natural selection which weeds out the worst discrepancies because they tend to raise the cost of production. Hence, more room for discrepancies!

So my arithmetical illustration unduly limits the choice of increments, but the essential point seems to me to be that the result, with its absolute equality, could be gotten by the device of infinitesimals, that any reader knows that bargaining power and other things introduce discrepancies (hence I did not feel responsible for taking up space to show just what discrepancies were involved in the abstract logic of my figures); and that my device for simplifying Euler’s theorem expresses the same central truth that Euler’s theorem does, without concealing the essential assumption involved. I knew what I was doing, and it seemed to me legitimate. I’m ready and anxious to state it more fully, now or later.

As for the fact that productivity may be zero or a minus quantity, that is to be taken in connection with the points noted in the footnote to p. 86. Constant return to factor A implies zero return to factor B, and increasing return to factor B implies less than zero return to factor B. It is a stock proposition that when added labor on a given land area would yield “increasing return”, you’re cultivating too much land and would get more crop by letting some of your land lie idle. In other words, land has a marginal product of less than zero and unless you paid less than nothing for it, it wouldn’t pay you to use it to that extent. In other words, unless someone paid you to use that much land, as Uncle Sam did by offering settlers title to it with a speculative future value attached. And the marginal product of labor is more than the whole crop, for it includes the speculative value of proving the claim. Such cases are illuminating as limiting cases, not economical on economic principles. I don’t see the fact that Euler’s theorem applies to them also proves its unreality. When it applies to them, it makes sense and not nonsense, even though the cases are “rare and peculiar”. And constant return to A and zero return to B is by no means rare and peculiar: it constitutes the very paradox I am discussing in that twenty-third chapter, sections 4 to 8. A’s share absorbs the whole product, and B either gets nothing or takes part of A’s share.

This is my reaction as writer and not as an editor. As an editor, whatever you care to say goes into print, and also I’d want to make some such explanation as I have made to you here, indicating something of what the larger story is, of which that arithmetical illustration is a part. Then I wonder if we can’t somehow telescope these first two moves and go on to your reaction to this statement. I suppose what we could do about that would depend on whether this is a separate note or part of a review. If it is a separate note we can do about what we please. I am not particular about having the last word, but I’d rather make this a cooperative putting forth of some considerations on a neglected point in the marginal-productivity theory, rather than a controversy of the usual sort.

You have certainly dug into the logic of the argument! However, I don’t think that the thing loses its meaning by being an identity. And if it does, I can simply take two increments, each yielding the same marginal product (a finite substitute for what infinitesimals would give you) start at the mid-point, after the first increment and before the second, and then go ahead with any possible combination of the four resulting marginal products. Then you’d get an equality that wouldn’t be an identity. It would be as valid as the infinitesimal assumption is: no more so. But isn’t all pure deduction an identity if you chase it enough? The conclusion is contained in the premises, but it may be worthwhile to find out what was in that premise by way of implications that weren’t obvious in the surface.

As to how I hit on the relation, it was in graphic form, long before I ever came to Chicago. I’d never read the Wieser-Menger controversy nor heard of Euler’s theorem. I was trying to prove that the sum of the marginal shares had to equal the whole product, and I hit on this relation.

Let OABC be the product with OC units of the variable factor yielding OA product per unit. Let ODFH be the product with, say, a ten percent increase of the variable factor, subject to diminishing return, so that yield per unit is less than before but total yield is more. Let CJGH be equal to this increase of total product. The easiest way to construct it is to construct EFGJ equal to ABED. Then CJGH is the marginal product of the variable factor (say labor). But at the same time, ABED is the marginal loss of product suffered by the labor already there, through having less capital to work with, owing to having to give the new unit of labor its share. (You remember my father talks in those terms: I think in one place he explains diminishing return by this reduction of the per-unit share of the constant factor). Then ABED (or JEFG by construction) are equal to the marginal decrement due to the loss of one eleventh of the original supply of capital. Eleven times JEFG is equal to IDFG. This is the marginal share of capital, and OIGH is the marginal share of labor. Check. Here you have the same identity you have pointed out, though it took me some time to see that it was an identity. You saw that fact quicker than I did.

That was the original form of the thing, I showed it to Charley Cobb, of the math. dept. of Amherst and he worked out the Euler’s theorem part. Later I found it had been done before, so I didn’t try to publish it. (I don’t mean that the graphic form of the thing was not new: I have never seen that in print).

Well, that’s that! Let me know what you are doing on it and I‘ll govern my actions according.

Incidentally, I’ve wanted to congratulate you on that article of yours on the ethics of competition. It’s one of the best thing you have done. I hope things are going all right with you and your wife and family – is there one?

Some other matters kept me from getting at once at the interesting task of studying out just what was involved and where it connects up with my algebra. I think I see it now, and have written up a few paragraph such as I might write if we should get out something of the nature of a colloquy and you should make a “Reply” along the lines of your letter.

Referring to this copy of your graph, you will see that the point F representing the product per unit of labor will not descend very far toward the base line until the area ADBE will become equal to and then exceed the whole area EFCH. After this point, (supposing the point F to be chosen at successively lower levels) the marginal product of your eleventh element becomes negative.

When this happens it becomes necessary to build an area CHCIHI below the X axis, as indicated. Your addition will still check, however, just as my algebra does, if you complete the rectangles and regard the negative sign of areas below the line.

On the other hand the point F may be placed above the line AB, showing increasing returns. You will still get a check by building up the marginal strip to E^IF^I, completing the rectangle and adding the area algebraically.

It would be entirely up to your editorial discretion whether such stuff as this is worth the pages. The fun of it has paid me for the trouble, in any case. You will see that I feel rather unscathed in my critical position by the considerations brot [sic] forward in your letter. Maybe you can still produce something that will disintegrate this complacent attitude. Or, it may be as such things so often are, a question of ultimate difference of opinion at some point. Anyway, I shall be glad to hear from you, and appreciate the attention you give to my animadversions. (It has been my “guess” that even if you publish the notes you probably would not care to give up the space and expense of engraving my discussion of the graph; hence the separation of it from the “paragraphs”). A reasonably clear exposition of the various “cases” would be rather long and for most readers “tedious”, as I state. it would go without saying that your graph and explanation should be given the light. I think in fact that it is enormously clever, and that I shall use the whole thing, as I suggest. I am sure, however, to date, that it is really a much more general relation than that of Euler’s theorem or the actual economic situation.

I have finally gotten around to the notes we exchanged last year and put in most of yesterday working over my thunder. You will see that I have re-written my first contribution with only verbal changes, except that I speak of share in distribution and factors instead of product and factors, in the last paragraph. I am still sure that the resemblance between your figures and the relation shown by the homogeneous functions of variables is the purest coincidence, but do not see anyway to make that point very conclusive.

Your reply carries the argument into the use of the illustration in your book. I have written a few pages showing what I would feel like saying on that point.
I am disappointed with the result as to length and in every other way, though I think my contents are sound. It seems clear to me that our conventional categories of competition and monopoly need only to be reasonably applied to cover most of the argument brought forward in your book and also in that of Professor Commons to which I refer^*. However, I would not want to raise questions as to the general argument of your book, at least that was not my intention at first.

Of course I believe in what I am saying and shall be glad enough to say it in print if the editors of the Journal think it worth the space and not too far from the canons of good taste. I should in any case be very glad, if the stuff goes into print, if it can be reorganized so as to put my two pieces into one and give you the last word. I am more interested in understanding the issues myself than I am in any debating or publicity aspects of the case and should be very glad to have any expression of opinion you may care to make.

^* Commons J.R. (1924), The Legal Foundations of Capitalism, University of Madison Press, Madison.

Chapman S.J. (1906), The Remuneration of Employers, in “Economic Journal”, vol. XVI, pp. 523-528.

Clark J.M. (1923a), Studies in the Economics of Overhead Costs, The University of Chicago Press, Chicago.

__________ (1923b), Some Social Aspects of Overhead Costs: An Application of Overhead Costs to Social Accounting, with Special Reference to the Business Cycle, in “American Economic Review”, n. 13, supplement, pp. 50-59.

__________ (1925a), Reply to Professor’s Knight Remarks, in “Journal of Political Economy”, n. 33, pp. 555-557.

__________ (1925b), Concluding Note, in “Journal of Political Economy”, n. 33, pp. 561-562.

Copeland M. (1925), Review of Studies in the Economics of Overhead Costs, in “Political Science Quarterly”, n. 40, pp. 296-299.

Dorfman J. (1964), Wicksteed’s Recantation of the Marginal Productivity Theory, in “Economica”, n.s., n. 31, pp. 294-295.

Edgeworth F.Y. (1925), Studies in the Economics of Overhead Costs. By John Maurice Clark, in “Economic Journal”, n. 35, pp. 245-251.

Hugh Jackson J. (1925), The Economics of Overhead Costs. By J. Maurice Clark, in “American Economic Review”, n. 15, pp. 82-84.

Knight F.H. (1925a), A Note on Professor Clark’s Illustration of Marginal Productivity”, in “Journal of Political Economy”, n. 33, pp. 550-553.

__________ (1925b), Rejoinder, in “Journal of Political Economy”, n. 33, pp. 557-561.

Rutherford M. (2000a), Understanding Institutional Economics: 1918-1929, in “Journal of the History of Economic Thought”, n. 22, pp. 277-308.

__________ (2000b), Institutionalism Between the Wars, in “Journal of Economic Issues”, n. 34, pp. 291-303.

Shute L. (1994), John Maurice Clark (1884-1963), in G.M. Hodgson, W.J. Samuels, and M. Tool (eds.), The Elgar Companion to Institutional and Evolutionary Economics, 2 vols., Elgar, London, pp. 50-54.

__________ (1997), John Maurice Clark: A Social Economics for the Twenty-First Century, St. Martin’s Press, New York.

Stigler G.J. (1941), Production and Distribution Theories, Macmillan, New York.

1* The Clark-Knight correspondence is published below with the permission of the University of Chicago Archives. The relevant material was found among Knight Papers, Folder C. I am indebted to Warren J. Samuels for encouraging publication and to Debra Levine, the assistant librarian, for much friendly cooperation during my research.

21 Weeksteed response to Clark, dated 14 February 1916, indicated that the English economist no longer considered the marginal productivity a sufficient explanation of distribution. See Dorfman (1964).