Edgeworth’s formalization of parametric external
economies as a germ of a game theoretic view
: What was the hard core of the British Marginal
Revolution?
Satoko Nakano
Faculty of Economics, Meiji Gakuin University
Discussion paper No.18-01
January, 20191

Abstract
While Marshall introduced the concept, ‘external economy’, it is said that
Chipman(1970) formalized ‘parametric economies of scale’ in order to reconcile themaximization behavior of individual firms with increasing return to scale of the economy.
Chipman’s idea promoted the next stage of the development, the endogenous growth theory
through Romer (1986) and in this, a new type of capital, knowledge or human capital, hasbeen focused in the driving forces of economic growth. But who is the true originator ofthis development and how can we appraise the development in the history of economics?
Though Chipman indicated that his parametric idea is originated from Edgeworth,and H. Cunynghame, their contributions are scarcely known and appraised in the historyof economic theory. This paper tries to show that the parametric formulation is intrinsicallyof Edgeworth and that his aim is not only showing how the individual’s maximization
behavior can be consistent with the external economy under the competitive system but
also trying to suggest the more general framework where individuals are partly consciousof other’s consumption and production behavior as seen in the modern game theoretic
models.

In this formulation, Edgeworth tried to base the micro theory on a kind ofstrategic interactive system and to suggest that the structure of the system should not be
arbitrarily determined by analyst’s preference but should be positively specified by using
statistical data.
There is an important message in this on the methodological and theoretical hardcore of the British Marginal Revolution. Marshall’s external economy and Edgeworth’parametric idea have been, in their understanding, presumed to be synthesized in a broader
approach toward market interactions, differing from the general equilibrium type ofapproach. They have assumed the whole economic structure where the productivity of theeconomy, therefore economic gains, would be marginalized by the frontier of enveloping
structure of each specific technology or preference.

Within the structure, individual

economic actions are cooperatively or conflictingly strategic interactive.

2

Edgeworth’s formalization of parametric external economies
as a germ of a game theoretic view
: What was the hard core of the British Marginal Revolution?
1.
2.
3.
4.
5.

Introduction
The historical appraisal of Chipman’s parametric formulation
H. Cunynghame and his contribution toward the modern externality
Cunynghame, Marshall and Edgeworth; what is their opposition?
Edgeworth and his formalization of ‘parametric economies of scale

1.

Introduction
While A. Marshall constructed the concept of ‘external economy’, J. Chipman

considered ‘parametric economies of scale’ in Chipman (1965), where he referred H.
Cunynghame and F.Y. Edgeworth as the source of the idea. Chipman analyzed how thecompetitive equilibrium can be derived from a kind of myopic optimization behaviors ofindividual firms under the external economy.

Chipman’s formalization has been

extended in the endogenous growth theory by P.Romer since 1980’s.

In this, humancapitals such as knowledge and education are focused in the literatures, in which various
endogenous economic growth paths beyond the physical constraints are analyzed in thedevelopment economics and international trade theory with some implications for the
economic analysis and policy. This paper investigates what kind of role Cunynghame andEdgeworth had played in the development of the basic ideas of ‘parametric economies ofscale’, in particular, in relation to Marshall.
The contributions of Edgeworth, and Cunynghame in these literatures are scarcely
known and appraised in the history of economic theory. This paper tries to show that theparametric formulation is intrinsically of Edgeworth and it is a very special case within his
scope. His aim is not only showing the consistency of the individual behavior under the
scale economy with the competitive system but also trying to suggest the more general casewhere individuals are partly conscious of other’s consumption and production behavior.
On the one hand, Cunynghame provided concrete descriptions and geometricalexplanations of externalities, in particular, snob effects or conspicuous consumptions,earlier than Veblen’s the theory of Leisure Class. Cunynghame (1904) gave a direct
influence on Leibenstein’s bandwagon effect analysis.

Edgeworth, on the other hand,interpreted that each individual’s myopia that is assumed in the parametric economy is aproperty of the special and limit case applicable to consumption side of externality. Thiscase is later, in 1980’s, extended to the idea of human capital of the endogenous economic
growth, the property of which provides the unique interpretive potential that market

3behaviors are sometimes negligibly small to count their own effects as seen in social tastes
or prevalent knowledge and education of the social standard. However,

Edgeworthsaw that the myopic behavior on the production side scarcely finds its concrete example
and that each producer being conscious of other’s economic action, their relations can beanalyzed generally in an interactive system rather than as a special case of ‘parametriceconomies of scale’

The parametric formulation is Edgeworth’s device as a connector

between his general scope of interactive markets and Marshall’s partial equilibrium
analysis under the competitive regime.
The above massage provides a fundamental question, “What was the methodologicaland theoretical hard core of the British Marginal Revolution, at least for Edgeworth andMarshall?”Marshall’s external economy and Edgeworth’ parametric idea have been, in

their understanding, presumed to be synthesized in a broader approach toward market
interactions, differing from the general equilibrium type of approach. They have assumed
the whole economic structure where the productivity of the economy, therefore economic
gains, would be marginalized by the frontier of enveloping structure of each specifictechnology or preference.

While Edgeworth saw that individual economic actions arecooperatively or conflictingly strategic interactive and the outcome of their action would
be cycle, jump or indeterminate, Marshall made the partial equilibrium analysis along histime structure, where his interpretative view for the organic and evolutionary growth ofindustrial organizations was developed.

Edgeworth’s parametric formulation connects

these two analytical scopes and what is to be remarked is the fact that the former vision
precedes the latter in the British Marginal Revolution period.
2.

The historical appraisal of Chipman’s parametric formulation"A parametric economy of scale “(Chipman (1970) p. 349)" is a term used by Chipman(1970).
"Entrepreneurs are assumed to believe that his firm is operating under constantreturn to scale, and any departures from this assumed output-factor relationship areinterpreted by him as brought about by a perturbation in his unit-homogeneous
production function, even if such departures are caused in part by changes in his own
level of output.”(Chipman (1970) p. 349)
Chipman explains this interpretation with the development of division of labor at Adam
Smith's pin factory. One company uses specialized labor as a result of expanding
production and progressing division of labor. Then, such specialized labor will be available

4to other companies in the industry. However, since the specialized labor is pooled only
when the whole industry is expanded, the contribution of one enterprise to this process is
negligibly small. Therefore, it is external to a company and is not recognized. (Chipman(1970) p. 349) In this way, Chipman (1970) has introduced the concept of parametriceconomies of scale, Chipman (1965) investigating the history of this idea considerably indetail. However, there are few researches in the history of economics to consider thehistorical background of Chipman (1965) 1.
Boblulescu (2007) and Gehrke (2015) have studies the significance of Chipman’s
formulation in relation to the modern analysis of increasing return to scale 2.

Bobulescu(2007) evaluates Chipman’s "parametric economy of scale" as a logical development under
Marshall's tradition.

In other words, Chipman’s formulation is a logically naturaldevelopment based on Marshall’s original external economy including the cost controversy
and the subsequent criticism by Sraffa etc. in the 1920s.

However, Bobulescu (2007) istrying to differentiate some of the development in 1920’s, such as, J. Viner's pecuniary
externality, the monopolistic competition of J. Robinson and E. Chambelin from Chipman's formulation.
Gehrke (2015) took the same theme as Boblulescu (2002, 2007) and gave anotherevaluation, comparing the formulation of Viner, Chipman and Krugman (Gehrke (2015)
P.23.note 2). Gehrke (2015) opposed to positioning the modern formulation on theextension of the logical development of Marshall's tradition (Gehrke (2015) P.25.note 31).
Gehrke (2015) emphasizes that on the one hand, the formulation of Chipman is just a matterof technological externality in the general equilibrium while on the other Viner deals withpecuniary type of externality in the partial equilibrium. It appraised that Viner's intention
is to convert the context of partial equilibrium into the framework of monopolisticcompetition theory. (Gehrke (2015) P.3) And it is said that Krugman’s formulation is on
the extension of Viner’s trial, the development of which is interpreted as a positive
contribution. On the other hand, the formulation of Chipman is not evaluated as a promising
revision of Marshall's framework.
It is written in footnote 27 of Gehrke (2015) as follows.
"Chippman’s concept was adopted, among others, by Panagariya (1980), Inoue(1981), Suzuki (1996) and Silvestre (1999), but it cannot be said to have triggered a

1

Suzuki (2009) quoted Chipman (1965) and mentioned that Edgeworth is one of souces of the parametricidea.
2 Bobulescu (2007) does not refer to Chipman (1965) and therefore does not mention Edgeworth andCunyngham. Gehrke (2015) refers to Edgeworth and Cunyngham with reference to Chipman (1965), but does
not consider their content.

5particularly broad stream of literature. It was also mentioned, and immediately
dismissed, in the opening paragraphs-that is, in the so-called “literature survey” – ofRomer's seminal contribution to the new growth theory: Subsequent work
demonstrated that it is possible to construct a consistent, general equilibrium models
with perfect competition, increasing returns and externalities (see e.g., Chipman(1970)) ... Following Smith, Marshall and Young, most authors justified the existenceof increasing returns on the basis of increasing specialization and division of labor. It
is now clear that these changes in organization of production cannot be rigorously
treated as technical externality. Formally, increased specialization opens new markets
and introduces new goods.

All producers in the industry may benefit from theintroduction of these goods, but they are goods, not technological externalities.
(Romer (1986) p. 1005) ” (Gehrke (2015) p.25.)
On the one hand, Gehrke interprets that Chipman's concept covers only technologicalexternality with little economic implication to its application.

On the other hand, Gehrke(2015) positively focuses on the later development of increasing return to scale phenomena
such as spillover effects of knowledge, linkages of industries, in the international
economics and the spatial economics.
Together with these evaluations, there are two types of developments in the approachof increasing returns to the modern era. One is (1) Chipman type = technologicalexternalities under the perfect competition, the other is (2) Viner-Krugman type, =pecuniary externalities under the imperfect competition. (1) Chipman type is compatible
with the competitive equilibrium. (2) Viner-Krugman type considers some processes inwhich various consumers and companies and capital goods interactively spread throughcumulative propagation under incomplete competition. The problem that this paper
considers is how and in what context the original concept of “parametric economy of scale"started in the history of economics. In addition to it, why does Romer type come out from
Chipman type?

Romer type considers the externality of knowledge / human capital whereindividual contribution in the economy is negligibly small.
3.

H. Cunynghame and his contribution toward the modern externality
According to Cunynghame’s biography (Ward and Spencer (1938)), he was equippedwith a little academic education, but he was a lawyer who worked vigorously on realistic
issues and did the job of a member of the government. He worked for the enactment of theregulation law for workers working in coal mines, traveled to the coal mine himself,
gathered various opinions and discussions, and led them. He was a key person who tackles

6the reality of Britain after the Industrial Revolution from all angles. Among scientists or
artists in London at the turn of the century, probably there is no one who has never met him.
For example, Painters Holman Hunt, Whistler, Herbert Spencer, Huxley, William Morris,
Wilde and others. He was a "Brain of the town", which connects in various way people
and stories that he encountered in society and parties. Also, he surveyed colonial financegoing abroad, gave an advice on construction of London's arts school, and mediated Alaska
whale ownership problems. It can be seen in the photograph of Cunynghame at the
beginning of the biographies, that he is an active person who has a very big head on asquatting petite body. However, in this biography, economic contributions are hardly told.
Marshall is mentioned as one of the teachers of the university era and an economic writing
1904 is slightly mentioned. Perhaps there was a lot of involvement with economics, but
the share of that would have been negligible in the whole.
In economics related literature, mentioning Cunynghame’s person is the comment by
Whitaker who wrote Cunynghame’s section of Palgrave Dictionary and Keynes'
Cunynghame memorial service (Keynes (1935)). The following Times died article quoted
by Keynes is interesting.
“It may be doubted whether any important official position has ever been held byan odder or, in some ways, a more remarkable personality than Cunynghame. Of his
cleverness there never could be any doubt, though the routine of a public department
was hardly the best field for its display. There appeared to be no subject of which hehad not at least a working knowledge, and certainly none of which he was notprepared to talk at large. He was a practical electrician, and at one time had been
vice-president of the Institution of Electrical Engineers; in his house were installeda laboratory and workshop; and official interviews would be enlivened withdisquisitions upon Hegelian philosophy and analogies drawn from the novels of
Balzac (the plots of which he seemed to know by heart), the medieval Schoolmen,
or indeed any other branch of literature or learning upon which his nimble
intelligence had recently been engaged. The story goes that when " phossy-jaw,"
contracted in the manufacture of matches, was engaging the attention of the Home
Office, Cunynghame announced that he had himself discovered and manufactured a
non-phosphoric safety match. A meeting in the Home Secretary's room wasaccordingly arranged for the testing of this invention, but the future prospects of the
Cunynghame match went up in the smoke of the explosion that ensued. On anotheroccasion, when the prevention of miners' phthisis was the subject of inquiry by theRoyal Commission on Mines, he astonished his colleagues by the production of a7

model man on a large scale which he had made in his own workshop, showing theworking of the throat and lungs and how dust was absorbed into the system. These
incidents are typical of this unconventional public servant, who, with his wide
learning and practical knowledge of many departments of life, might, if he hadconcentrated more, have been one of the great departmental heads of his generation.
As it was, his official career was less successful than that of many of his intellectual
inferiors, and to his disappointment he was not made head of the Home Office when
Sir Mackenzie Chalmers retired in 1908.” (Keynes (1935) p.401, The Times (1935))

The explosion incident in the article is as if it is a caricature scene, reminiscent of an
eccentric inventor. Cunynghame is also inventing tools in economics.

He entered St.

John's College in Cambridge University in 1870 and was under the influence of Marshall,
who was appointed as a lecturer of the moral science. According to Mrs. Marshall’s letterto Keynes, Cunynghame was one of Marshall’s favorite disciples in the early 70's.

And

while attending Marshall's lecture, Cunynghame invented a creative instrument to draw aright-angle hyperbola, seeing that Marshall was in trouble drawing the curve on theblackboard. The tool was lost now, but it was exhibited at the Cambridge Science Society.
Marshall published a paper to explain its usage and analyzed its monopolistic value in thepaper 3.
Among Cunynghame's socializing with cultural people and key persons, there were also
mathematical economists in the marginal revolution, such as Edgeworth or P.H. Wicksteed.
It is known that Jevons and Edgeworth lived in Hampstead, a suburban residential area in
London, as a neighbor, and had interactions such as hiking, cycling, and study sessions.
Cunynghame’s unpublished thesis of the 1880’s was delivered at the study meeting ofHampstead. The meeting was called the Economic Circle of Wicksteed and was held atHamstead's Beaton house (Ramée Beeton) once every two weeks, and Cunynghame,
Edgeworth, Shaw (G.B. Shaw), Foxwell (H.S. Foxwell), Web (S. Webb) participated, andMarshall also occasionally participated4.

3“The machine, which Mrs. Marshall mentions, for drawing rectangular hyperbolas on the black-board, wasfamous thereafter to forty generations of undergraduates attending Marshall's lectures. It was communicated to
the Cambridge Philosophical Society along with Marshall's first original contribution to economics (and his
second appearance in print)-‘Graphic representation by aid of a series of Hyperbolas of some Economic
Problems having reference to Monopolies,’ Proceedings of the Cambridge Philosophical Society, Oct. 1873.”
Keynes (1935) p.400.
4 Refer to Whitaker (1996) vol. 1, p.255 Letter no.224 from Marshall to Foxwell. Whitaker (1987) p.739.
Howey (1960) pp.118-30. Barbé, L. (2010) p.116.

8

The economic contribution of Cunynghame can be seen from Keynes' biographicalmemoirs and Edgeworth's book review. One of them is to illustrate and generalize thedemand curve by ‘successive utility curves’, the generalized supply curve, what is called‘successive cost curves’. Discussions on this successive utility curve, successive cost curve
have been developed in Cunynghame (1892), but they have not been developed in
Cunynghame (1904). Cunynghame had withdrawn them from his main contribution in
Cunynghame (1904)
First, Cunynghame mentioned a case where the externalities of demand, in particular
the consumption of other consumers, influences the utility obtained from his consumption.
“It affords a man who is eating bread no satisfaction to know that his neighbour has
got none; but almost the whole value of strawberries in March, to those who like thistasteless mode of ostentation, is the fact that others cannot get them. As my landlady
once remarked, ' Surely, sir, you would not like anything so common and cheap as a
fresh herring?' The demand for diamonds, rubies, and sapphires is another exampleof this. As the number increases, not only does the price go down, but the very
pleasure of those who already have them is decreased by their becoming common.”(Cunynghame (1892), p. 37)
This discussion can be evaluated as a pioneering one of Veblen's conspicuous
consumption or snob effect as seen in "The Theory of the Leisure Class" in 1899. Moredirectly, it is positioned as a prior study of the bandwagon effect in Leibenstein (1950).
Leinenstein refers to Cunynghame (1892) at P.185 of Leibenstein (1950).

Both arecommon in that they try to show by means of a diagram that the aggregate demand function
composed of individual demand curves has a upward sloping part.
It is worthwhile to note what motivated Leibenstein (1950), “Bandwagon, Snob, andVeblen Effects in the theory of Consumers’ demand”. His enquiry was suggested by O.
Morgenstern as the “non-additivity” problem, where market demand curve is not the lateralsummation of the individual demand curve. This phenomenon is seen, for example, inthe collective demand curve with bandwagon, snobs and Veblen effects. And Leibensteinslightly mentions that the phenomenon also applies to the collective supply curves. While
Morgenstern imply that “since coalitions are bound to be important in this area only the“Theory of Games” (developed by Von Neumann and Morgenstern) is likely to give anadequate solution to this problem,” (Leibenstein (1950) p.183), Leibensteain tried toanalyze through the use of conventional method by graphical demand curves, because hethought he is not competent to judge whether the game theory is proper to the analytical9tool. It is notable that the “non-additivity” problem is originally related to the market
view based on strategic interactions of the game theory.
Furthermore, Leibenstein refers to Cunynghame and his 1892 paper as one of the past
literatures in this line of research. He explains the reason why the interpersonal effect inthe market has been ignored in the current texts as follows.
“One reason why the interpersonal effects on demand have been ignored incurrent texts may be the fact that Marshall did not consider the matter in his
Principles. We now, however, from Marshall’s correspondence (note: Pigou,
Memorials of Alfred Marshall, pp.433 and 450.

These are Marshall’s letters to

Pigou and Cunynghame which indicate that Marshall had read the articles (EJ. 1892and E.J. 1903), where Pigou and Cunynghame consider the matter.), that he wasaware of the problem.” (Leibenstein (1950) p.186.)
The above indicates that Leibenstein’s theme is placed on the extension of Cunynghame(1892). At the same time, Leibenstein remarked that though Marshall knew the content
of the paper, he did not consider the interpersonal effects on demand in his Principles.
Leinbenstein’s bandwagon analysis could be placed on the half way to the strategicinteractive market view in the history of the concept, one of origins of which is
Cunynghame (1892). Marshall may have intentionally avoided the way.
The following is a diagram of the demand curve that is derived from a group of
successive utility curves, one of which is designated, for example, as Δδ and is negative tothe axis OX.

Cunynghame explains that the demand curve will be in general negative,

but that in rare instances it might at the commencement be positive as in the Figure 4(Cunynghame (1892) p.40.)

10

This case is, according to Cunynghame, as when a particular edition of hymns became
useful in direct proportion as it was widely used.

It is the case that there is positive

externality among users in the society as Leibenstein’s bandwagon effects or Romer’s
knowledge capital.

Edgeworth explains the relationship between the individual demand

curve and the aggregate demand curve in a form to be seen later. Though Cunynghame's
description has ambiguity and not necessarily given a consistent explanation as to therelation of groups of successive utility curves with the demand curve.
However, Cunynghame is clear in asserting that a group of successive curves does not
represent states of some lapse of time but a state at one time.
“But a group of successive curves is the expression of a state of facts existingat one time, viz. a set of hypothetical consumers’ values, and not a group of
successive time phenomena.” (Cunynghame (1892) p.39.)
Individual demand curves depend on some social conditions, for example, the state
where "strawberry is scarce", the state where "bread is common", etc. Though the social
condition can be resolved into individual demand curves as Leibenstein analyzed,
Cunynghame does not provide consistent explanations. 5

5

Leibenstein tried to construct aggregate demand curves with external consumption effect, such as

bandwagon effects, snob effects and Veblen effects. He indicates that under the diminishing marginalexternal consumption effect, there is a stable collective demand curves.

11

Cunynghame keeps track of the decline of cost reduction of the supply side in the similar
structure to the demand side.
“Just as the demand value varies with the amount purchased, so also will the supplycost, or, in other words, the cost of the manufacture of an article estimated in money
vary with the amount supplied, and sometimes this cost will rise, sometimes it will
fall. The phenomenon of an increasing cost of production is illustrated generally byagricultural produce, and raw articles, of which the difficulty of production increases
with the amount produced. On the other hand, the cost of production of manufactured
articles generally diminishes with the amount produced on account of the greater
facilities for the use of machinery and other economy of production which is afforded
by a large output. "(Cunynghame (1892), p. 39-40)

12

The production cost curve (dotted line), which depends on the conditions of economy
available for production, such as "the workers of related work are exhausted" and "there isa limit to the necessary means of transportation". 6 A series of points on the cost curves is
a supply curve that slopes downward to the right.
Overall, Cunynghame’s achievement can be evaluated as intuitively showing how to
configure sequentially the demand curve that depends on state as the state changes. Whilehis graphical explanation is too naïve to hold the consistent analysis, the framework of thestate dependent demand curve or supply curve is an important step toward the insight ofthe strategic interactive view for the market. Based on this intuitive insight, it can be said
that Leibenstein considered the problem of constructing the aggregate demand curve fromthe individual demand curve when there is consumer’s externality under some necessary
conditions.
4.

Cunynghame, Marshall and Edgeworth; what is their opposition?
In the background that Cunynghame (1905) book review was written, what kind of

discussion had Marshall, Edgeworth and Cunynghame developed on their analysis of
6

Cunynghame states as follows in Cunynghame (1904) p.90.”Various causes prevent the full effect of theinstability of manufacturing prices being felt. The necessity of carriage from one place to another is an
obstacle. The impossibility of suddenly creating the necessary skilled labour is another; timidity inenterprise, and the persistence of old habits is another; and of course protective tariffs would also help. Theintervention of a sort of arbitration board, to retard sudden transformations, would possibly have some effect,
if it were not probable that the Problem is so complicated that the medicine might produce worse effects thatthe disease.”13‘successive utility curves’ and ‘successive cost curves’?

Why had Cunynghame

withdrawn them from his main contribution in Cunynghame (1904).
It is known that there were serious conflicts between Edgeworth and Marshall and that
Cunyghame (1892) was a trigger to the conflict.

In a letter to Edgeworth, Marshallreckons Edgeworth’s discussion as "Murder" while discussing Cunningham's analysis.
Concerning this letter, contemporaries like Keynes pointed out, suggesting that there wasa serious confrontation between them. (Keynes (1935) p.404)
Marshall wrote to Edgeworth on 26 April 1892.
“I cannot refrain when I see you plunging it into a medium in wh it cannot breathe,from calling MURDER!” (Whitaker (1996) p.69, Letter 406)
Marshall took up the discussion of Cunynghame (1892), develops an objection, andfinally writes to shout this sentence. In this letter, Marshall is asking the following question.
”Let y=f(x) be the equation to one of Cunynghame’s successive cost curves: What
does the y mean, & what does the x mean?”
Cunynghame may have withdrawn his successive cost curves from Cunynghame (1904),
probably confronted with Marshall’s sever criticism.

However, despite being a bookreview to Cunynghame (1904), Edgeworth has reconstructed this issue taking up the
successive utility curve and the successive cost curve of Cunynghame (1892).

It isinferred that Edgeworth's aim of the book review is to discuss these parts and answer
Marshall’s question, “What does the y mean, & what does the x mean?”
Apart from Edgeworth’s interpretation, how Marshall himself treated this problem in
the Principles as the edition goes from the first to the eighth edition.

Guillebaud’s

comparison of the editions elucidated in the eighth edition does not necessarily let us know
how Marshall had changed.

The following is the comparison through editions only

concerning with Marshall’s mention to Cunynghame.
There are three mentions to Cunynghame in the Principles. (1) Cunynghame’s
instrument for drawing hyperbolas (2) Cunynghame’s private circulated paper
Cunynghame (1892) in Economic Journal.
published are the following.

(3)

The years in which each edition was

The first edition in 1890, the second edition in 1891, third

edition I 1895, the fourth edition in 1898, fifth edition in 1907, sixth edition in 1910,
seventh edition in 1916 and eighth edition in 1920.

14

As to (1), in the preface for the first edition, there was an acknowledgement expressing
Marshall’s thanks to Cunynghame for his making the instrument for drawing hyperbolas.
This mention reflects their close and friendly relationship.

This preface for the firstedition has been inserted from the second to fourth edition in the same form (p. xv n.1 forthe second and third edition and p. xiii for the 4th edition).

However, though the preface

for the first edition is printed in the following edition (from fifth edition to eighth edition),
the mention to Cunynghame was omitted.
As to (2), from the first edition, the following statement has been inserted.
“If the change is gradual, the supply curve will assume in succession a series of
positions, each of which is a little below the preceding one; and in this way we might
have represented the effects of that gradual improvement of industrial organization
which arises from an increase in the scale of production, and which we haverepresented by assigning to it an influence upon the supply price for long-period curves.
In an ingenious paper privately printed by Mr H. Cunynghame, a suggestion is made,which seems to come in effect to proposing that a long-period supply curve should be
regarded as in some manner representing a series of short-period curves; each of thesecurves would assume throughout its whole length that development of industrialorganization which properly belongs to the scale of production represented by the
distance from Oy of the point in which that curve cuts the long-period supply curve(compare Appendix H, 3) and similarly with regard to demand.” (Marshall (1997)
p.383-384)
The above is Marshall’s interpretation of Cunynghame’s successive cost curve
which expresses the effects of increasing return to scale as the development ofindustrial organizations.

Marshall clearly states that the effects are graphicallyshowed as the relation between a series of short-period supply curves and the log
period curves.

This interpretation is based on Marshall’s consistent modeling bypartial equilibrium analysis with time period differentiation, mainly short-run andlong-run.
As to (3), from the fourth edition there appears the following statement in the
Appendix H.
“One difficulty arises from the fact that a suitable time to allow for the
introduction of the economies appertaining to one increase in the scale of production
is not long enough for another and larger increase, so we must fix on some fairly

15

long time ahead, which is likely to be indicated by the special problem in hand, andadjust the whole series of supply prices to it.
We could get much nearer to nature if we allowed ourselves a more complex
illustration. We might take a series of curves, of which the first allowed for the
economies one year, a second curve doing the same for two years, a third for three
years, and so on. Cutting them out of cardboard and standing them up side by side,
we should obtain a surface, of which the three dimensions represented amount, price,and time respectively. If we had marked on each curve the point corresponding tothat amount which, so far as can be foreseen, seems likely to be the normal amountfor the year to that curve would be a fairly true long-period normal supply curve for
a commodity obeying the law of increasing return.

Compare an article by Mr.

Cunynghame, in the Economic Journal for 1892.” (Marshall (1997) P.667 n.)
Marshall subdivides the time structure by introducing economies of one year, the samefor two years, for three years etc. in this statement. What is remarkable is that in spite of
the clear assertion by Cunynghame indicated above, “a group of successive curves is theexpression of a state of facts existing at one time”, Marshall never concede that the scaleeconomy can be analyzed in the model of a state of facts existing at one time.

He

presupposes that each individual firm or consumer behave within the time structure, while
effects of increasing return to scale can be introduced within some time period enoughappertaining to it. The interactions of behaviors firms and consumers at one time were
not considered.

Why?

What is the reason of Marshall’s excitement when he used theword, ‘Murder’, in his letter to Edgeworth?
It should be clarified that Marshall and Edgeworth were conflicting in their analysis ofmarket equilibrium under increasing returns with an opposition between the two methodsof economic analysis.

Nakano (2012) (2013) (2015) elucidated the opposition between

Marshall and Edgeworth, whereas Edgeworth has been interpreted under the Marshall’s
umbrella except for the technical arguments on the barter process, called the barter
controversy. Their opposition will be explained in the following as long as it makes it
easier to understand the background of their discussion of this paper.
The following figure is inserted on a page of Nature, Edgeworth (1890), which is his bookreview for the first edition of Marshall’s Principle. Edgeworth indicates Marshall’s
misrepresentation of the situation where there are multiple equilibria under the increasing
return to scale on the supply side. Edgeworth states that an individual firm acting on the
declining supply curve would not maximize his profit.

16

Edgeworth (1890) Nature p.363
When Edgeworth criticizes the above Marshall’s partial equilibrium analysis in his
Principle, he has the offer curve analysis in Marshall (1879) in his mind. Particularly
diagrams (figures 5, 9) concerning the movement of trade equilibrium when shifting fromlow production efficiency technology to high technology in domestic industry. This is ananalysis of how the economic equilibrium using high technology appears in the economic
environment potentially having increasing production technology 7.
In other words, the analysis concerns how a trade equilibrium shifts from A through B
to C in figure 5 when increasing return on the scale is brought about through expansion of
the domestic market or expansion of foreign markets. Marshall refers to the concept ofSmith's "the extent of the market" and explains how the phenomenon that the market size
reduces the relative price of imported goods and export goods while expanding tradetransactions. Therefore, we assume crossover of offer curves as shown in figure 5 by thesetting "Decrease in the value of export goods, the amount of imported goods to beexchanged for export goods will decrease". When OE is the home offering curve and thepartner country is OG, the horizontal axis shows the export amount x and the vertical axis
shows the import amount y from the partner country. 8
7Marshall explained that the contents of figures 5 and 9 are the "Class II" problem of the exceptional case of
trade problems. “give the name of ‘Class I.’ to the second exceptional case to which reference has been made;the case namely, in which an increase in the amount of wares which a country produces for exportation effects
a very great diminution in the expenses at which she can produce them; so that the consequent fall in their value
diminishes the total amount of the imports that she receives in exchange for them. (Marshall (1879) p.5-6.)“The case has its origin in the fact that the wares which a country exports maybe such that the difficulty of
producing them diminishes very rapidly when their amount increases. It is indeed true, as has been said, that ingeneral the production of a commodity on a large scale for home consumption precedes the development of
any considerable foreign trade in it. Still the extent to which division of labour in the production of it can be
carried, is onlarged by every extension of the foreign markets for it.” (Marshall (1879) p.12-13.)
8 Marshall does not write P and Q in the figure, but according to Marshall's explanation, the point where astraight line going vertically from M crosses OE is called P and the point vertically rising from R is confronted
with OE can be regarded as Q . Marshall says as follows. "Continue to interpret the problem of Class II in the
figure, P and Q (figure 5) are two points on O and E, PM and QR are perpendicular to Ox, QR is larger than17

Marshall (1879) figure 5
Edgeworth highly appraises the analysis of Marshall (1879) figure 5, because it
presupposes envelop structures of various technologies with increasing return to scale anddepicts a dynamic competitive process through which traders or firms compete each other todominate the market by watching their competitor’s actions.

There is a possibility that

Edgeworth could have been influenced by this analysis when he developed his game theoretic
view in his Mathematical Psychics, which will be discussed in another paper in detail.
However, Marshall omitted this analysis from his Principles and replaced it with the partial
equilibrium analysis by the figure above (Edgeworth (1890) Nature p.363). Accordingly,
Edgeworth trial in Mathematical Psychics based on the analysis of Marshall (1879) has been
lost importance in Principles.

This is the background of their opposition that lies in theparametric economies idea.

5.

Edgeworth and his formalization of ‘parametric economies of scale’
Edgeworth puts a book review on Cunynghame's 1904 book in the Economic Journal in1905. However, rather than mentioning only 1904 books, the mention of 1892 papers is
PM In case and Class I case, the ratio of QR to OR must be greater than the ratio of PM to OM (Proposition II)
However in Class II, the increase in the amount of fabric produced in the UK is 1 As the production cost per
yard is greatly reduced, the increase in imported linen will reduce the value of 1 yard linen in the UK, as well
as allowing more linen to be purchased at each yard of linen than before. Therefore, the increase in the exportof dough will increase by a large proportion rather than the smaller proportion than the import of linen.
"(Marshall (1879) p.14.) In other words, because economies of scale work, the situation that the price ofexport goods for imported goods become cheap when trade expands is caught in the shape of the offer curve18

the main object of his review. On the one hand, it is pointed out that using real economic
case and the metaphor, Cunynghame's explanation is attractive to the scholars of beginners.
On the other hand, Edgeworth confirms that it is insufficient by merely explaining with thefigure, citing Cunynghame’s statement, "The figure cannot determine the issue of free tradeversus tariffs, the figure only shows the status of the problem" (Cunynghame, (1904) p.
102), at the beginning of his review. Furthermore, indicating it is difficult to combine
properly both concrete explanations and scientific explorations, Edgeworth evaluates thatthe balance between them is not well enough in Cunynghame’s discussion.
“We had hoped that the path struck out by him in his original article on ''Exchange
Value" in an early number of the Economic Journal (1892) would have been now
converted by him into a high-road accessible to the wayfarers of science, even though
not specialists. But this hope has not been fully gratified. There still, as it appears
to us, remains some difficulty, which we shall endeavor to smooth over by a freerestatement.” (Edgeworth (1905) pp.136-137)
As is stated above, the free restatement is one of the objects of Edgeworth's book review.
In other words, although Edgeworth was hoping Cunynghame to reformulate the paper in
1892, as it was not yet done in the 1904 book, Edgeworth embarked on it in the review
paper.
Edgeworth draws the following quote from Cunynghame’s book at the beginning of thebook review.
“Two manufacturing rivals are like men pulling against one another on a rope, one
on each side of the summit of a hill. When one of them is once pulled over the summit
the other can run away with him. But if the men were pulling on each side of a hollow,
as in rival production of the agricultural order, then when one got pulled down a little
his opponent's task would become harder; so they would come at last like a marble ina bowl to a position of equilibrium.” (Edgeworth (1905) p.136. Cunynghame (1904)
p.89)
In the book of Cunynghame, the above quotation is followed by the statement, "Thepoints on the downward supply curve is unstable, theoretically, it will be produced by thecheapest producers and countries ".

In other words, monopoly will occur if there is anincreasing return to scale, but if there is some friction in reality, there is a possibility ofantagonizing the competitors.

For example, some cases are enumerated, for example,

19

where industrial leaders retire, where competitors are exhausted to make money, where theworkers of related work are exhausted, and where there is a limit to the necessarytransportation means. Some of these are concerned with the internal economy of eachenterprise and some can be interpreted as external economies. However, Cunynghame
did not develop the theory to explain how these frictional factors derive the downward
sloping supply curve.
Why did Edgeworth quote a metaphor comparing to the situation where the above rival
enterprises have tug-of-war across the summit of the mountain at the beginning of his
review? It is inferred that the view that captures unstable competition under increasing
returns is the key for their opposition among Marshall, Cunynghame and Edgeworth.
According to Guillebaud, a relative of Marshall, who edited Marshall's Principles, states ina retrospective review of Marshall that Marshall often says about Edgeworth as follows.
“Apropos of Edgeworth, for whom he had a strong personal liking and whose
ability as an economics he rated highly, he said, ‘I would illustrate his way of dealing
with an economic problem as follows: “let us assume” says Edgeworth “that we have
two elephants suspended from the end of a rope. Give the elephants a push and then,disregarding the weight of the elephants, work out what happens to the rope.’ ”(Guillebaud (1971) p.7)
In other words, Marshall cynically told Guillebaud that the characteristics ofEdgeworth's economic analysis lie in instability of the firm's competition under increasing
returns.

It can be interpreted that Marshall, at least after publishing Principle in1890, had

not base his analysis on the micro-instability of the competitive process between firms.
Marshall thought that the time period for economic activities are basically divided into theshort-run and the long-run, and that the economic process in the long run can betheoretically traced through the partial equilibrium in the short run. Marshall interpretsthat the increasing return to scale is due to the external economy uncontrollable forindividual firm, which develops through the complication of industrial organizations.
Marshall, at least in his Principles, opposed to Edgeworth, who thinks that competition
between firms under increasing returns is unstable, as shown below by indicating the logic
to handle the parametric economy to handle the external economy as a special case,
Edgeworth tried to take the case of the micro instability actively.
In Edgeworth (1905), Edgeworth picks up the parametric economy as a case of
externalities in consumption at first. After that, Edgeworth explains the logical structure
equivalent to the externality of consumption as the externality of production. However,20

as an interpretation for the real economy, Edgeworth considers that there are rare cases inwhich the externality of production can be interpreted through the myopic behaviors ofindividual firm, rather considers that in the most cases where the oligopolistic market ormonopoly progresses, each firm is conscious of their behaviors in the market.
Edgeworth starts with the theory of demand and lists the case of "orchid". Whenorchids are rare, demand increases with rising price of orchids, but as orchids becomecommon flowers, demand will not increase with price rise, only with price decline demand
will increase. In this way, whether or not the consumption of orchids of other people issmall or large affects the consumption behavior of individuals. Within the normal partial
equilibrium analysis, Edgeworth explains Cunynghame’s illustration as follows.
The demand function of r-th consumers within n persons is set as follows.
𝝃𝝃𝒓𝒓 ＝𝑨𝑨𝒓𝒓 − 𝑩𝑩𝒓𝒓 𝒑𝒑 − 𝑪𝑪𝒓𝒓 𝑿𝑿′𝟐𝟐
𝑨𝑨𝒓𝒓 , 𝑩𝑩𝒓𝒓 , 𝑪𝑪𝒓𝒓 > 𝟎𝟎, 𝒑𝒑 is the price and X is the general consumption level.

As for r＝1, 2, ‥, n The aggregate demand function that adds up to all n individuals is asfollows.
𝒏𝒏

𝒏𝒏

𝒏𝒏

𝒏𝒏

𝒓𝒓＝１

𝒓𝒓=𝟏𝟏

𝒓𝒓=𝟏𝟏

𝒓𝒓=𝟏𝟏

� 𝝃𝝃𝒓𝒓 = � 𝒂𝒂𝒓𝒓 − � 𝒃𝒃𝒓𝒓 ∙ 𝒑𝒑 − � 𝒄𝒄𝒓𝒓

When 𝐴𝐴 = ∑𝑛𝑛𝑟𝑟=1 𝑎𝑎𝑟𝑟 , 𝐵𝐵 = ∑𝑛𝑛𝑟𝑟=1 𝑏𝑏𝑟𝑟 , 𝐶𝐶 = ∑𝑛𝑛𝑟𝑟=1 𝑐𝑐𝑟𝑟 ,∙ 𝑿𝑿′𝟐𝟐the aggregate demand function is asfollows.

𝒙𝒙 = 𝑨𝑨 − 𝑩𝑩𝑩𝑩 − 𝑪𝑪𝒙𝒙′𝟐𝟐

𝐼𝐼𝐼𝐼 𝑋𝑋 ′ is not the constant but the same variable as X,

𝑿𝑿＝𝑨𝑨 − 𝑩𝑩𝑩𝑩 − 𝑪𝑪𝑿𝑿𝟐𝟐
𝟏𝟏
𝟏𝟏
�
�
𝟐𝟐
𝟐𝟐
𝑩𝑩𝑩𝑩＝ − 𝑪𝑪(𝑿𝑿 + 𝑪𝑪) + 𝑨𝑨 + 𝟒𝟒 𝒄𝒄

The aggregate demand function is in the form of a parabola as above, and X and P areplaced in the positive quadrant according to the way of giving constants of A, B, C. In
the positive quadrant, (1) a case where P goes down along X goes the right and (2) Theremay be a case where both the rising part and the descending part exist. Edgeworth thinksthat the case (2) is a case where demand rises with price rise and external economy ofconsumption is occurring as the Figure 4 in Cunynghame (1904).

Individual consumers

consider the demand volume of the market as a whole as a given amount, if their influenceis negligibly small with respect to the aggregate demand in the competitive market.
Therefore, X' is considered as a constant. This myopia of individual consumer is an idea
of parametric economy inherited by Chipman. It is pointed out that Edgeworth is actually21

reluctant to apply this idea to the supply side.

Edgeworth says that this parametricexternality concept can be logically developed in a similar way for the external economy
of production, but Edgeworth also points out that there are few cases where production
actually applies to production.
“The liability of an industry to be monopolized when it obeys the law of increasing
returns creates peculiar difficulty in the application of the geometrical method to
supply. In order that the theory which has been above set forth with reference todemand should be extended to supply, it must be postulated that the output of eachproducer is small in comparison with the collective output of all his competitors. But
this postulate is apt not to be adequately fulfilled in modern manufacturing industry;
as Mr. Cunynghame reminds us in many a striking passage (pp. 79, 86-69).”(Edgeworth (1905) p.141)
In the corresponding part, a monopoly case is compared to the beggar-my neighbor ofplaying cards, where a monopolist eradicates the market (Cunynghame (1904) p.79).
Cunynghame depicted an unstable process of market competition. (Cunynghame (1904)
p.89.)
Edgeworth said that, either in case of externality of consumers or that of producer, thecase can be logically treated as a parametric economy as its individual demand or supply
is negligibly small compared to the total as a whole in a competitive market. However,
Edgeworth does not explicitly lead the competitive equilibrium in the case where such an
external economy exists.

It can be said that the development of this part is thecontribution of Chipman (1970).

Edgeworth only refers to the shape of the demand curve,

on the supposition that the demand and supply curves in the partial equilibrium intersect.
However, Edgeworth is reluctant to apply the parametric economy on the supply side as
seen above, would rather keep in mind the process in which individual suppliers reactaccording to aggregate production volume. Edgeworth insists that demand curves andsupply curves dealt with in discussion of parametric externality are short-term curves interms of short-term and long-term classification of Marshall usage as follows.
“They are “short-period” curves because when there occurs a change in theconditions of supply, and accordingly a new point of intersection between the new
supply curve and the old " successive utility curve, then—the correspondence
between our x and x' being disturbed—the successive utility curve mast be conceived
as changing its form until x and x' once more coincide. No such change of form is22

suffered by the proper, or, as we should like to say, “long-period” demand curve. We
should have said so if Mr. Cunynghame had not expressly repudiated thisnomenclature.” (Edgeworth (1905) p.142)
It can be seen that Edgeworth presupposes the process where individual behavior given
parameters respond to changes in the total amount.

Taking micro instability and its

concrete description by Cunningham into consideration, Edgeworth mostly has theoligopolistic market in mind.

For that reason, he insists that the supply curve based on

the successive cost curves is of the ‘short-period’.

Considering the case where thebehaviors among firms are strategically interdependent, the interdependence can occur inthe short-period not in the long-period. This point is crucial in interpreting what is the
opposition between Marshall and Edgeworth. Marshall never takes into the analyticalframework the situations where individual firms are strategically interdepend in the shortperiod or at a time.
The extension of Leibenstein and the development of Romer are basically concerned withexternalities of social levels of consumption and available standards of living including
knowledge and education.

This development is considered to be consistent with

Edgeworth's parametric idea applicable for the common goods augmented by negligibly
small agent behaviors. But this extension is the special case in which Edgeworth tried tofind a point of contact between Marshall’s approach and Edgeworth own.
Edgeworth, on the one hand, tries to clarify in what cases the external economy iscompatible with the competitive market, but on the other hand, he mainly targeted in
Edgeworth (1905) to reveal a new methodological stance how to analyze not necessarily
competitive markets. The following two points concerns the methodology. Firstly, as aalternative to Marshallian method, the partial equilibrium analysis with time-period
structure, Edgeworth pointed out a method called Collocation, which is J.S. Mill's
discussion. Secondly, while referring to G. Barkley's epistemology, Edgeworth pointed out
the importance of constructing abstract mathematical models with checking on positive
data.
First of all, Edgeworth insists that Cunynghame type of supply and demand curves, thatis, the state dependent supply and demand curve, are reasonable as the usual type of supply
and demand curves are conceived as price dependent functions.

Cunynghame

presupposes that there are two periods when orchids are rare and common, and thenconsumers are exposed to different periods.

Edgeworth emphasizes Cunynghame’s pointthat the change in these two periods is not a historical event and that these universal states
can be introduced into economic analysis by using the way of thinking, ‘collocation’.

23

“He demands, as we understand, that the new collocation should itself be
explicable by conditions which are pre-existent and co-existent in much the same
sense as the dispositions represented by the ordinary demand curve. It is thus that weinterpret his doctrine: "a group of successive curves is the expression of a state offacts existing at one time, and is not a group of successive time phenomena"(Economic Journal II., p. 39). We understand that there is here predicted the same
sort of permanence as that which belongs to the state of facts designated by anindividual's demand for a commodity, say tea: the law of demand does not change
when the price changes.” (Edgeworth (1905) p.137)
Edgeworth claims that a model can be constructed as individual demand behavior andsupply behaviors are dependent on aggregate behaviors as well as price. In other words,
this discussion describes a methodological basis for modeling where individual player
behaves dependent on other player’s behaviors.
The original idea of "collocation” in Mill's A System of Logic is as follows.
“It is necessary here to remark, that in this resolution of the law of a complex effect,
the laws of which it is compounded are not the only elements. It is resolved into thelaws of the separate causes, together with the fact of their coexistence. The one is as
essential an ingredient as the other whether the object be to discover the law of theeffect, or only to explain it. To deduce the laws of the heavenly motions, we require
not only to know the law of a rectilinear and that of a gravitative force, but theexistence of both these forces in the celestial regions, and even their relative amount.
The complex laws of causation are thus resolved into two distinct kinds of elements:
the one, simpler laws of causation, the other (in the aptly selected language of Dr.
Chalmers) collocations; the collocations consisting in the existence of certain agentsor powers, in certain circumstances of place and time.” (Mill (1862) p.320-321)
In the above, collocation is explained in the laws of the heavenly motions.

As thephysical mechanics are resolved into complex combinations of certain agents or powers incertain circumstances of place and time, the complex law of markets, “new collocation”, is
not only explained by the causal relation with price but also the interdependence ofindividual demand and supply behaviors in the circumstances of the place and time. Thiscan be understood as a discussion about the methodological basis for making a market

24model. In other words, it can be interpreted as an expression for the idea of a model building
in a way that economic agents are strategic interdependent".
In addition, Edgeworth expresses that mathematical formulation is more suitable for thecomplicated modeling than the graphical method, which Cunynghame and Marshall prefer.
Because mathematical formulation makes it possible to check in what extent the model fitssome various discrete data that are fragmental reflection of the real market. Edgeworth
presupposes the statistical analysis, some of which he had contributed. He also interpretsthat the diagram method can be arbitrary because it depends on freehand writing of analyst.
“Nor are we convinced by the following objection:“To express an experimental

supply curve, as, for instance, Fig. 27, or price of getting coal. Pig. 36, or of producing
a book, Pig. 27, or still more the curves of demand for corn or sugar by such an
expression asy=f(x)
is to invest these curves with an apparently simple law-determined character thatthey do not really possess.” But as all that is knowable—much more than is usuallyknown—is a set of discrete data, so much commodity corresponding to such a price,
whether is it more arbitrary to draw a freehand curve through points representing thosedata, or to use a form which stands for any one of an indefinite number of equations
each representing a curve passing through the given points? [In particular, a rational
algebraical function𝑦𝑦 − 𝐴𝐴0 + 𝐴𝐴1 𝑥𝑥 +𝐴𝐴2 𝑥𝑥 2 + ⋯ + 𝐴𝐴𝑚𝑚 𝑥𝑥 ′′ ; whether the constants are
considered as numerous as the observations and so fitting them exactly, or lessnumerous, fitting the observations as well as possible.] ” (Edgeworth (1905) p.144)
On that basis, Edgeworth insists that formulating a problem with mathematical
functional form is of lower arbitrariness and that it can prove more general properties thandrawing a concrete curve. He explains using Euclid's geometric proof as an example.
Edgeworth says that if you draw a diagram of a specific triangle and prove it using thespecific properties of that figure, it is hard to say that it is a general proof. And quoted fromBerkeley’s philosophical writings as follows. “it is true that the diagram I have in view
includes all these particulars, but then there is not the least mention made of them in the
proof of the proposition.” (Berkeley (1994), p. 248.) Edgeworth concludes as follows. “Itseems to us quite tenable that the indefinite symbol "f" obtrudes particularity even less thana concrete curve-line." (Edgeworth (1905) p.144) These methodical discussions can beinterpreted as answering Marshall’s question in the his letter, ”Let y=f(x) be the equation
to one of Cunynghame’s successive cost curves: What does the y mean, & what does the x25mean?”In this way, Edgeworth presented the parametric economy mathematically formulated.
Its methodological view is trying to open up a way to mathematical analysis and empiricalanalysis toward a kind of strategic interactive market view. In Edgeworth's argument, weneed to understand not only the idea of the consistency of the competitive equilibrium withthe externality, but also the different approaches between Marshall and Edgeworth. The
possibility that the game theoretic view precedes the Marshallian partial equilibrium
analysis would provide a new interpretation for the marginal revolution in the history of
economics.

26

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