Transcript 
i
BY T H E S A M E AUTHOR
ECONOMIC CYCLES
LAWS OF WAGES
FORECASTING THE
OF COTTON
THEIR LAW AND
YIELD AND THE
GENERATING ECONOMIC CYCLES
CAUSE
PRICE
SYNTHETIC ECONOMICS
BY
HENRY LUDWELL MOORE
PROFESSOR OF POLITICAL ECONOMY IN COLUMBIA UNIVERSITY
" L'unification synthétique qui transforme une
pluralitê discontinue de faits en un réseau continu
de relations."
LÉON BKUNSCHVIOO
Mt\o gorfe
THE MACMILLAN COMPANY
1929
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COPYRIGHT, 1929,
BY T H E M A C M I L L A N COMPANY
Set up and electrotyped. Published May, 1929
All rights reserved, including the right of reproduction
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CONTENTS
CHAPTER I
INTRODUCTION
PAGE
Two Approaches to Economic Theory . . . . 1
The Scope of Synthetic Economics . . . . 5
CHAPTER II
FUNDAMENTAL NOTIONS
The Premise of Free Competition 11
Capitals and Services 17
Economic Equilibria 19
The Postulate of the Negligibility of Indirect Effects 23
Obsolescent Disabilities 28
CHAPTER III
THE LAW OF DEMAND
Elasticity of Demand and Flexibility of Prices . . 33
Typical Demand Functions of One Variable . . 38
Statistical Derivation of Typical Demand Functions
of One Variable 41
Partial Elasticity of Demand and Partial Flexibility
of Prices 52
Typical Demand Functions of More than One Variable 56
Statistical Derivation of Typical Demand Functions
of More than One Variable 61
V
vi Contents
CHAPTER IV
THE LAW OF SUPPLY
PAGE
Elasticity of Supply and Expansiveness of Supply
Price 65
Typical Supply Functions of One Variable . . . 66
Statistical Derivation of Typical Supply Functions
of One Variable 69
Partial Elasticity of Supply and Partial Expansiveness
of Supply Price 70
Typical Supply Functions of More than One Variable 71
Statistical Derivation of Typical Supply Functions
of More than One Variable 75
Relative Cost of Production (K) and Relative Efficiency
of Organization (w) 76
Laws of Relative Cost and Relative Return Contrasted
with Laws of Cost and Laws of Return . 78
Cost Curves and Supply Curves; Relative Cost
Curves and Relative Supply Curves . . . 82
Partial Relative Efficiencies of Organization . . 85
Typical Production Functions 88
CHAPTER V
MOVING EQUILIBRIA
A Moving Particular Equilibrium of Demand and
Supply 93
Walrasian Equations 100
The Premise of Free Competition: A Spurious Superfoetation
107
The Real Functions in a Moving General Equilibrium 111
Coefficients of Production 116
A Moving General Equilibrium: First Approximation 123
The Theory of Capitalization 127
A Moving General Equilibrium: Second Approximation
133
Contents vii
PAGE
Statistical Treatment of the Productivity Theory of
Distribution 143
CHAPTER VI
ECONOMIC OSCILLATIONS
Oscillations about Particular Equilibria . . . 147
The Mechanism of Oscillations about General Equilibria
152
An Index Number of the Oscillations of General
Prices 154
A Synthetic Theory of Economic Oscillations . . 163
CHAPTER VII
CONCLUSION
Economic Certainties 178
Economic Probabilities 180
Economic Dreams 182
SYNTHETIC ECONOMICS
SYNTHETIC ECONOMICS
CHAPTER I
INTRODUCTION
The mathematical approach to economic theory
bifurcated at an early point in its course. One road,
travelled chiefly by practical men intent upon the
exigent business of the day, has led through a dreary
region directly to practical, but unrelated, results;
the other road, followed mainly by philosophers with
primary interest in causes and relations, has ascended
to picturesque heights affording distant views of the
ensemble of economic activity, but has stopped short
amid the enchanting scene and left the explorers in
doubt as to what might be the real destination of so
promising a beginning.
Two Approaches to Economic Theory
In 1874 when Leon Walras had just published the
early instalment of his demonstration of the theory of
general equilibrium, he wrote a letter to the aged
Cournot which contains a sentence marking the
divergence of the two roads to economic theory:
"Notre methode est la même, car la mienne est la
votre, seulement vous vous placez immédiatement au
benefice de la hi des grands nombres et sur le chemin
qui mène aux applications numériques. Et moi, je
demeure en dega de cette loi sur le terrain des données
rigoureuses et de la pure theorie."
1
2 Synthetic Economics
Both Cournot and Walras employed the mathematical
method, but Cournot entered upon a course
which seemed adapted to lead, through the use of
, the theory of probabilities, to numerical applications;
; while Walras concentrated his strength upon the solu
, tion of a central problem in pure theory and made no
= effort whatever to obtain an empirical test of the
adequacy of his theoretical construction in the interpre
, tation of the world of economic realities.
The central problem to the solution of which Walras
gave his life was the interdependence of all economic
quantities and the necessity of expressing in simultaneous
mathematical equations the conditions of
their common determination. To facilitate his inquiry
he created an hypothetical static state having
the properties that are familiar to all students of
economic theory. His solution of the problem of the
mathematical conditions of equilibrium in a static
state inspired an ideal conception of the goal toward
which future investigators must work: A comprehensive
treatment of economic questions in a changing
society must take cognizance of the interdependence
of all types of economic change, and the only kind of
treatment that will lead to rational forecasting and
control is mathematical in character.
Cournot had a clear conception of the interdependence
of the parts of the economic system and he stated
very compactly the difficulties with which the economist
is confronted:
"So far we have studied how, for each commodity
by itself, the law of demand, in connection with the
conditions of production of that commodity, deter
Introduction 3
mines the price of it and regulates the incomes of its
producers. We considered as given and invariable
the prices of other commodities and the incomes of
other producers; but in reality the economic system is •
a whole of which all the parts are connected and react
on each other. An increase in the income of the
producers of commodity A will affect the demand for
the commodities B, C, etc., and the incomes of their
producers, and, by its reactions, will involve a change
in the demand for commodity A. It seems, therefore,
as if, for a complete and rigorous solution of the
problems relative to some parts of the economic
system, it were indispensable to take the entire system ;
into consideration. But this would surpass the powers
of mathematical analysis and of our practical methods
of calculation, even if the values of all the constants
could be assigned to them numerically." ' He abandoned
the forbidding task of determining concretely
the conditions of a general equilibrium and sought, by
other means, to reach approximate solutions of special
problems affecting the general economic system.
Cournot's results were published in 1838; Walras',
in 187477. Cournot's mastery of the theory of
probability was, in his day, so great that Poisson conceded
his priority in the conception and development
of ideas that subsequently appeared in the latter's
Probabilités des jugements; while Czuber, a careful
historian of the theory of probability, has ranked him
second only to Laplace in his grasp of the philosophy
of chance. But in the half century since Cournot's
' Cournot: Researches into the Mathematical Principles of the Theory
of Wealth. Bacon's translation, p. 127.
4 Synthetic Economics
death the development of the theory of probability
' has entered upon an entirely new phase, and the sup
, plying of statistical material affording the means of
I immediate application in economic questions has become
the special function of bureaus, public and
private, in many parts of the world. Walras' successors,
while extending his work, have for the most part
1 followed his example of remaining in the domain of
pure^ theory, and in certain cases have even deprecated
any attempt to pass to a statistical investigation of the
questions to which they have devoted such eminent
ability. Notwithstanding this attitude of the Ecole
de Lausanne, the idea that economic theory may be
fruitfully approached through the conception of the
equilibrium of the interdependent parts has made such
headway since the publication of Walras' epochal
essays that Pantaleoni, with perhaps pardonable
exaggeration, could say, in 1909, "I'economia si
presenta ora quale scienza delle leggi delVequilihrio
economico."
The thought that inevitably presents itself to an
investigator in touch with these collateral developments
is whether, equipped with the new tools and
supplied with the new material, he may not advisedly
reconsider the problem abandoned by Cournot in
1838 and left as a pure statical theory by Walras in
' 1877. Is it not possible to solve the problem dynami
I cally and to give, by means of recent statistical methods,
a concrete, practical form to the theoretical ideas of
moving equilibria, oscillations, and secular change?
Introduction 5
The Scope of Synthetic Economics
The title of this essay. Synthetic Economics, is t
intended to indicate a concrete, positive description of
moving equilibria, oscillations, and secular change, by
a method which presents all of the interrelated eco j
nomic quantities in a synthesis of simultaneous, real
equations.
As far as I am aware neither Walras nor Pareto used
the term Synthetic Economics. Pareto insists upon
the need, in every science, of following up the work of
analysis with synthesis; ^ he uses repeatedly the adjective
synthetic to describe the peculiar point of view
of Walras and himself in their use of mathematics in
the treatment of economic theory; ^ but it seems that
the definite collocation of the words synthetic economics
did not appear in his work. Barone and Sensini use
casually the exact expression synthetic economics in
describing the method of Walras and Pareto as a
presentation of the whole of statical economics in a
series of simultaneous equations.^
Another Italian economist, quite unconnected with
the School of Lausanne, Professor Loria, uses as the
title of one of his works La Sintesi Economica, which
" Pareto: Cours d'économie politique. Vol. 1, p. 13.
^ Pareto: "Nota sulle equazioni dell'equilibrio dinamico." Giornale
degli Economisti, Settembre 1901, p. 14.
* "Cournot, Walras, Pareto, le cui opere contengono piü che un'intera
biblioteca, sono i grandi maestri della nuova economia sintetica." Enrico
Barone: Principi di Economia Politica, 1908, p. 25 n.
E questo uno dei principali vantaggi della moderna economia
sintetica, che apparsa per la prima volta colle immortali ricerche di
Walras. . . . " Guide Sensini: La Teoria della Rendita, 1912, pp.
157158.
m
6 Synthetic Economics
approaches in form, but not in substance, the title
borne by the present essay.
There are three special characteristics which I should
like the name Synthetic Economics to imply: (1) the
use of simultaneous equations to express the consensus
of exchange, production, capitalization, and distribution;
(2) the extension of the use of this mathematical
synthesis into economic dynamics where all of the
variables in the constituent problems are treated as
functions of time; and (3) the still further extension
of the synthesis to the point of giving the equations
concrete, statistical forms. With these implications
Synthetic Economics is both deductive and inductive;
dynamic, positive, and concrete.
A first advantage of this method of treating as an
ensemble the totality of prices and their determinants
is the elimination of many controversies in economics
as to the causes of phenomena. Is the cause of the
value of a commodity its cost of production or its
marginal utility? Is the cause of the rate of interest
the productivity of capital or the discount of future
goods? Is the cause of the rate of wages the laborer's
standard of living or the marginal value of his product?
In the history of economic theory these questions
have been discussed to the point, not of the conversion
of the disputants, but of the exhaustion of their
faculties. Yet when a synthetic view of exchange,
production, capitalization, and distribution is taken,
we see at once that each of the alternatives of the preceding
questions contains a partial truth; that the
sum of the partial truths is not the whole truth; that
Introduction 7
the proper weight and place of each partial truth may •
be specified; and that the ensemble of the determiningconditions
may be mathematically expressed. )
A second advantage of the synthetic method is that
it enables one to know when an economic problem has
reached a solution. Here, a distinction should be
made in the meaning of the word "solution" according
as one sees it from the point of view of the mathematical,
or of the synthetic method. The problem is(
solved by the mathematical method when there are
as many independent equations as there are unknown
quantities in the problem. From the point of view
of the synthetic method, however, this is only half of /
the solution: over and above the presentation of the
abstract simultaneous equations, proof must be sup >
plied that the equations themselves may be empirically ?
derived and, consequently, that the problem admits of (
a real solution. For example, we know that Marshall :
regarded Note xxi in the "Mathematical Appendix"
of his Principles as embodying "a bird'seye view of
the problems of joint demand, composite demand, joint
supply and composite supply when they all arise
together." As a mathematical economist he seemed
to be satisfied, for he says: "however complex the
problem may become, we can see that it is theoretically
determinate, because the number of unknowns is
always exactly equal to the number of the equations
which we obtain." Marshall devoted his genius
throughout a long, laborious life to an endeavor to give
a realistic form to the abstract equations of Note xxi. /
Did he, from the point of view of the synthetic method, 
reach his goal? Every demand equation and every i
8 Synthetic Economics
supply equation that figures in Note xxi is a function
simply of one variable. The work of Marshall affords
no method of deriving concretely either a demand
equation or a supply equation; it affords no cogent
reason for believing that, if methods for deriving the
equations were devised, either demand or supply could,
as a function of one variable alone, be expressed with
sufficient accuracy to be useful in the treatment of real
problems. Note xxi may be a possible, abstract mathematical
solution of the problem Marshall had in
mind, but economists will differ as to whether it may
suggest a useful first step towards a concrete, real
solution, and no opinion can be more than a lucky
guess until methods have been devised for submitting
the mathematical formulation to an empirical test.
( A third, and by far the chief, advantage of the
/ synthetic method is that it gives ground for the hope
I of introducing into economic life rational forecasting
) and enlightened control.
The synthetic method is concerned with the ensemble
of prices and their determinants. Suppose it
is assumed,
(i) that the laws of supply and demand in a
changing society may be empirically ascertained
;
(ii) that all of the conditions determining a moving
general equilibrium may be statistically ex
I pressed;
(iii) that it is possible to know concretely the results
of the solution of the simultaneous equations
expressing a moving general equilibrium.
Introduction 9
Under these conditions, it is possible to obtain a
rational forecast of the result of changing, in any
definite way, any factor in the moving general equilibrium.
But in social life, to foreknow the effect, \\
qualitative and quantitative, of specific change is to pos /
sess, precisely, the sine qua non of enlightened control.
The problem of the rational forecasting of oscillations
about the moving equilibrium introduces a
complexity similar to that of forecasting the demand l
for a single commodity from a knowledge of its demand ! j
function. A first approximation in the latter problem < \
is reached by considering the demand for the commod '
ity as a simple function of its price. A second approximation
in the accuracy of the predicted demand
is obtained by regarding the demand as a function of
many prices. In case of oscillations about the moving
general equilibrium a similar procedure may be
followed. From a study of the simultaneous equations
determining the moving general equilibrium the possible
sources of oscillations may be located, whether
these be in the amounts of consumable goods demanded,
the quantities of productive services supplied,
the degree of saving, the kinds and quantities of new
capital goods manufactured, or the quantity of money
in circulation. The complete theory of oscillations,
like the complete theory of the quantity demanded of
a single commodity, would be approached by successive
approximations. A first approximation would
take cognizance of the most important cause of perturbation,
and the successive approximations would be
made by combining the effects of the several perturbing
causes.
10 Synthetic Economics
But can these complex actions and reactions be
followed up in reality? In mathematical physics there
is a theory of oscillations developed by Daniel Bernouilli
which presents a curious and inspiriting parallel
to the problem before us. According to Bernouilli's
theorem, a system in equilibrium tends, under the
influence of a perturbing cause, to oscillate about its
position of normal stability, and the partial oscillations
that are due to different causes coexist in harmony
with the general conditions determining the equilibrium.
The successful application of this principle
in different branches of mathematical physics should
give courage to the synthetic economist.
CHAPTER II
FUNDAMENTAL NOTIONS
"En dépit de certaines apparences, I'ensemble de nos connaissances
et de notre science est surtout pratique, subjectif, provisoire
et convenu en sa definition, subordonné a son utilisation
sociale."
A L P B E D L O I SY
The pure theory of economics rests upon an obscure
premise; it employs certain concepts with a narrowly
technical meaning; it gives two interpretations of
facts according as a mathematical postulate is accepted
or rejected; it has reached a point in its growth
where, through an accumulation of hypotheses and i
postulates, the burden of presuppositions impedes its !
further progress. The object of the present chapter 
is to appraise these theoretical premises, concepts,
postulates, and disabilities for the purpose of gaining
a larger freedom in the constructive work of the
ensuing chapters.
The Premise of Free Competition
One of the chief difficulties of economic theory is the
bewildering vagueness of its fundamental premise.
Competition is the fundamental hypothesis of the
science in the sense that competition is postulated in
nearly every argument concerned with the determination
of prices. But what is the meaning of "perfect
competition"? In what respect is the idea of "competition"
changed by the addition of such modifiers as
11
12 Synthetic Economics
"perfect," "unlimited," "indefinite," "free," "pure,"
"absolute"? If by these additions there is a change
of meaning in the term, then, in cases in which the
state of industry admits only of "competition," what
is the nature of the limitation of propositions deduced
on the hypothesis of "perfect competition"? This
question is usually evaded by saying,—the imperfection
of competition is simply a form of friction, producing
for the most part a negligible variation from the
standards prevailing in a regime of perfect competition.
This complacent reply should be scrutinized. If
one is inclined to accept it, what shall be said to the
following statement by Pareto:
"La libre concurrence produit le maximum
d'ophélimité; la libre concurrence règne dans nos
sociétés: ce sont la deux propositions difïérentes.
La première est tres probablement vraie; la seconde
est certainement fausse." '
Here is an authoritative statement that "free competition"
does not reign in modern societies. Then where,
precisely, is the limit of the applicability of propositions
deduced by the Ecole de Lausanne on the hypothesis
of '' free competition " ? Is it allowable to infer, for
example, that actual wages deviate only in a negligible
way from the standard which would prevail under
"free competition"? That is not the idea of Pareto:
"II n'y a done nuUe contradiction entre la theorie
qui assigne comme effet le maximum d'utilité a
cette libre concurrence, et I'observation qui fait voir
qu'un régime essentiellement différent produit une
elïroyable misère." "M. Bodio, le savant directeur
' Pareto: Cours d'économie politique. Vol. II, p. 130, note (788)^.
Fundamental Notions 13
de la statistique italienne, dit que la misère des
classes agricoles italiennes atteint des limit es absolument
incroyables." ^
According to the theory of the Ecole de Lausanne
wages under "free competition" would reach a level at
which there would be a maximum of satisfaction
compatible with private property, whereas the actual
state of Italian agricultural laborers, at the time in
which Pareto wrote, was one of incredible wretchedness.
The word ''competition," as used in economic
theory, is a blanketterm covering more or less completely
the following implicit hypotheses:
(a) Every economic factor seeks and obtains
a maximum net income. This is the
essential meaning of the term.^ It is
made the basis of the definition of the
science given by Edge worth: "Economics
investigates the arrangements between
agents each tending to his own maximum
utihty." ^ This aspect of competition is
always explicitly emphasized in those
2 Pareto: Cours d'économie politique. Vol. II, pp. 166 and 166, note
(814)2. Cf. also ibid., Vol. II, p. 137, for a statement more in harmony
with the current conception: "Nous avons suppose dans nos theories la
concurrence parfaite, et nous avons insisté sur Ie fait que ce n'est lè,
qu'un 6tat limite. En réalité, la concurrence est souvent imparfaite, il
se produit un effet analogue a celui des frottements dans les machines."
^ "It is to Quesnay in his Dialogues sur les travaux des artisans that
we owe the first, and very categorical enunciation of the formula which
has been so famous under the name of the edonistic (?) principle, and
constitutes, in fact, the basis of economics: 'To obtain the greatest
possible increase of enjoyment with the greatest decrease of expense is
the perfection of economics.' It is no exaggeration to say that he who
enunciated this principle has indeed a right to the title of Founder of
Economic Science." Gide's review of Higgs' Physiocrats, Economic
Journal, June, 1897, p. 248.
* Edgeworth: Mathematical Psychics, p. 6.
14 Synthetic Economics
systems of economics using analytical
symbols, since it at once suggests, as
Malthus foresaw, that "many of the
questions, both in morals and politics,
seem to be of the nature of problems de
maximis et minimis in Fluxions." ^ It is
the prime hypothesis used in Cournot's
essay: "Nous n'invoquerons qu'un seul
axiome, ou, si I'on veut, nous n'employerons
qu'une seule hypothese, savoir quechacun
cherche a tirer de sa chose ou de son
travail la plus grande valeur possible." ^
It may be called the maximum hypothesis
of competition.
(b) There is but one price for commodities of
the same quality in the same market.
This is referred to by Jevons as the law of
indifference, and it is constantly used as a
premise in his theory of equilibrium. It
is also used by Cournot, notwithstanding
the above statement that he would invoke
but a single axiom: "II ne peut pas y
avoir dans une ordre de chose stable, et
sur une grande échelle, deux prix différents
pour une même quantité débitée." ^ As
an illustration of the identification of this
hypothesis with competition, a passage
from Jevons' Principles of Economics
^ Malthus: Observations on the Effects of the Corn Laws, 1814, p. 30.
' Cournot: Recherches sur les principes mathématiques de la theorie
des richesscs, p. 46.
' Ibid., p. 73.
Fundamental Notions 15
may be cited: "This law of indifference,
in fact, is but another name for the
principle of competition which underlies
the whole mechanism of society" (p. 60).
(c) The influence of the product of any one
producer upon the price per unit of the
total product is negligible. An illustration
is found in Pareto's Cours d'économie
politique, Vol. I, p. 20: "L'échangeursubit
les prix du marché sans essayer de les
modifier de propos délibéré. Ces prix
sont modifies effectivement par son offre et
sa demande, mais c'est a son insu. C'est
ce qui caractérise l'état que nous appelons de
libre concurrence. . . . En langage mathématique
nous dirons que pour établir les
conditions du maximum, on differentie en
supposant les prix constants." ^
{d) The output of any one producer is negligible
as compared with the total output.
Marshall has discussed this assumption
in Note xiv of the Appendix to his
Principles of Economics, particularly p.
801 of the 4th edition.
(e) Each producer regulates the amount of
his output without regard to the effect of
his act upon the conduct of his competitors.
Where (c) and {d) coexist, (e) is a
simple corollary; otherwise, it is an ^^'^
independent and inadmissible hypothesis.
* One sentence in the quotation I have italicized in order to draw , ^w*''^
attention to Pareto's having definitely regarded this feature as the . '' '
characteristic of "free competition."
16 Synthetic Economics
In most systems of economics, a theory of exchange,
production, and distribution is developed by reasoning
consciously from hypotheses (a) and (e). It is not by
any means always perceived, however, that the truth
of the theory is further limited by the implicit assumption
of hypotheses (c) and (d). This loose method of
procedure entails no necessary harm so long as the
investigation is confined to a simplified, hypothetical
static state, but great harm is done when, in approaching
the problems of actual industry—which, to a large
extent, is in a state intermediate between perfect
monopoly and perfect competition—the economist
flings the inquirer into the vague with the assurance
that theoretical standards will tend to prevail. In
this intermediate state between perfect monopoly and
perfect competition hypothesis (a) is, at best, only
approximately true; hypothesis (b) is frequently
untrue; and hypotheses (c) and (d), in many spheres,
are never true.
In pure economic theory, when the assumption of
"free competition" lies at the basis of the reasoning,
all five of the above enumerated hypotheses are implicitly
postulated. Here we come upon a principal
cause of the gross conflict of theoretical conclusions
and observed facts. "La libre concurrence produit le
maximum d'ophélimité." "La misère des classes
agricoles italiennes atteint des hmites absolument
incroyables " (Pareto). Obviously, some method must
be devised for bringing theory into closer harmony
with observed facts. In pure theory, the fundamental
laws, for example the laws of demand and
supply, are such as would obtain on the hypothesis of
Fundamental Notions 17
"free competition" in a static state. In Synthetic
Economics, the fundamental laws are obtained directly
from reality. The difference in the two superstructures I
is the difference between a purely formal science and a
positive science. J
Capitals and Services
A scientific classification, like a scientific method, is
to be judged by its fitness for its purpose. In the
historic treatment of capital and income much ability
has been directed toward a definition of terms and an
effort to bring them into a logical classification. But
the goal of the great talents thus employed has not
always been well defined, and without a distinct
specification of the goal such dialectical endeavors
might obviously be endlessly prolonged. The Ecole
de Lausanne has been exemplarily clear as to its
aim. Its ultimate object has been to describe the
ensemble of the interrelations in exchange, production,
and capitalization, and with this object always
in view it has defined and classified capitals and
services.
With the work of the Ecole de Lausanne as a point
of departure in Synthetic Economics it is desirable, in
the interest both of science and of personal loyalty, to
adhere as far as possible not only to Walras' terms but
also to his symbols. His terms are in some cases uncommon,
but the use of them has been continued by
his most distinguished disciples; his symbols, for the
most part, have been retained, and where substitutes
have been offered the innovations have seldom proved
to be betterments.
18 Synthetic Economics
Walras' classification of capitals and services may be
summarized in the following scheme:
!
Ca) Land
(b) Labor
(Personal faculties)
(c) Capitals in
narrow sense
Economic/
goods
(2) Goods used
only once
/ ( I ) Goods
(material)
(ii) Flow of
goods and
services
, (2) Services
C(a) Consumable goods
((b) Raw materials
!
(a) Consumable
goods
(«Raw
materials
Ub) Used more than
once (Capitals)
^ (a) Directly Consumable
(Consumable services)
(b) Indirectly Consumable
(Productive services)
Anything that has rareté (marginal utility) is by that
fact an economic good. These goods may be regarded
from the point of view of a given epoch, or from the
point of view of a flow of time. At any given epoch
the stock of goods will be made up of those goods which
Fundamental Notions 19
afford more than one use and of those which are used
only once. The former are called Capitals and are
subdivided into land, labor (personal faculties), and
capitals in the narrow sense; the latter are made up of
consumers' goods and raw materials. Capitals in the
narrow sense include all those goods that are used
more than once and are neither land nor labor. This
category embraces savings and money as well as commodities
usually called capital goods.
With the passing of time there is a flow of material
goods and services. The flow of material goods takes
the form either of capitals or of consumers' goods and
raw materials. The capital goods are either of the
current types or of new types. The services are either
productive services or directly consumable services.
The object of the classification, as has been said, is
to describe the ensemble of interrelations in exchange,
production, and capitalization. This problem of
description is approached by successive approximations.
In the theory of exchange those interrelations
are considered which determine the prices of consumable
material goods and directly consumable services.
In the theory of production those supplementary interrelations
which are required to determine the prices of
raw materials and productive services are added; in
the theory of capitalization the remaining conditions
which are necessary to determine the prices of capital
goods are introduced into the general system of equations.
Economic Equilibria
The idea of a social equilibrium is found, in germ,
wherever the study of social science has been ap
20 Synthetic Economics
preached systematically. It appears in Greek speculation,
in the thought of the schoolmen of the Middle
Ages, in the theory of the Physiocrats, in the treatises
of the classical economists. The germ idea of an
economic equilibrium is simply that of a balance of the
many forces operative upon a price configuration,
which configuration, with respect to the balanced
forces, remains in a state of rest. It received a
marked and explicit development in the work of
Cournot,' and has been made a fundamental notion
in the treatises of the Ecole de Lausanne. In the
hands of these mathematical economists the notion is
so carefully elaborated and so definitely linked with the
whole of economic theory that it is virtually the means
of a new conception of the science.^"
This new conception of economic science we have
sought to develop, starting with the following classification
of equilibria:
' Cournot: Researches into the Mathematical Principles of the Theory
of Wealth. Bacon's translation, p. 127. Cf. also Professor Umberto
Ricci: "Non è che al Cournot mancasse una visione deU'equilibrio
generale. Egli ebbe una nozione deU'equilibrio, per i suoi tempi assai
ragguardevole." Giornale degli Economisti, GennaioFebbraio, 1924,
p. 35.
" "Una delle scoperte piü profonde a cui gli economisti letterari
siano pervenuti in fatto di teorie concernenti I'equilibrio economico,
consiste appunto nel rilievo che il concetto di equilibrio e di interdipendenza
dei fatti economic! non è poi tanto nuovo, esso ritrovandosi in fondo
anche presso molti economisti classici, specialmente in J. B. Say.
Quasichè sia la stessa cosa accennare piü o meno di sfuggita, e male, ad un
concetto, ovvero precisarlo matematicamente, trasformandolo poi in
una base nuova ed estremamente feconda di studio di tutto un gruppo di
fatti!" Guido Sensini: La Teoria delta Rendita, p. 311, note.
Fundamental Notions 21
(1) Static
' ( i ) Particular
Economic
Equilibria
( i i ) General
(2) Moving
(1) Static
(2) Moving
(a) Tentative
(b) Final
(a) Tentative
(b) Final
The equilibrium may refer either to a single commodity
or to the whole economic system. The former is a
particular equilibrium; the latter, a general equilibrium.
Either of these may be either static or moving,
and a moving equilibrium, whether particular or
general, may be either tentative or final.
The key to the distinction between tentative and
final moving equilibria lies in the flow of time. In case
of a particular commodity the distinction is similar to
that between market price and normal price. The
classical discrimination between market price and
normal price is directed toward an understanding of
the effect upon price of forces which take time to be
brought into play. The market price is not directly
affected by the cost of production but is determined by
the law of demand and the quantity of commodity
that is put upon the market. There is a tentative
equilibrium of demand and supply which determines
the market price. But if the market price exceeds, or
falls short of, the cost of production, forces are brought
into play which, theoretically, will lead to such a readjustment
of the supply that the resultant price, the
normal price, will afford a final equilibrium. Demand
22 Synthetic Economics
and supply will then be equated at a price which will
ofïer no inducement to change any factor in the equilibrium.
In case of the general economic system the distinction
between tentative and final moving equilibria is
also analogous to that between market price and normal
price, and is likewise traceable to the flow of time. In
a definite time, say a particular year, the economic
organism brings to fruition many consumable commodities
and consumable services. For the time being
the prices of these are determined by the existing laws
of demand for the commodities and services, and the
quantities of services and commodities put upon the
markets. In the determination of the prices, cognizance
is taken of the whole complex of commodities and
services and their respective laws of demand, but little
or no attention is paid to what the costs of production
may have been. The resulting prices are prices at
tentative equilibria. These tentative equilibria involve
profits or losses arising out of the differences
between prices and costs of production, and these profits
and losses become spurs to economic readjustments.
The magnitude of the readjustments is again a
question of the flow of time. It takes time to make a
readjustment with the existing machinery of production,
and it takes a still longer time to make a readjustment
involving the saving of capital and the creation
of new machinery. If the readjustment is made with
the existing technical equipment, there may be other
tentative equilibria where demand and supply are
balanced but where prices and costs are at variance.
Readjustment will now be made by altering the instru
Fundamental Notions 23
ments of production, and, if there are no new perturbations,
will proceed until a stage is reached in which not
only demand and supply with reference to all commodities
and services are balanced, but costs of production
and market prices are, likewise, equated. Equilibrium
will then exist throughout the whole economic system,
and the equilibrium will be final because there will be
no inducement to alter any factor in the equilibrium.
This series of readjustments has been described on
the supposition that no new perturbation occurs.
But new perturbations always do occur, and, consequently,
the final equilibria become shifting ideal goals
whose lines of motion trace out the trends of the
system of economic quantities. Perturbations of
final equilibria start the changes whose immediate
goals are tentative equilibria; but the occurrence, or
prospect, of new perturbations disturbs the immediate
adjustment of the results of earlier changes, with the
consequence that the tentative equilibria likewise become
ideal goals, whose lines of motion trace out the
oscillations about the trends of final equilibria.
The Postulate of the Negligibility of Indirect Effects
In an early stage of economic speculation there was
a dim perception that a clue to the understanding of
the oscillatory character of economic changes could be
found in demand and supply. Indeed, the terms demand
and supply were vaguely used as summary names
for the groups of forces whose opposed working produced
the oscillatory course of economic changes.
With the advance of economic theory it became
necessary to give, first, an abstract, then, a concrete
24 Synthetic Economics
formulation of the laws of demand and supply. For
theoretical purposes it does not suffice to refer vaguely
to the laws of demand and supply as being responsible
for observed changes; one must know the general
mathematical character of the function descriptive of
demand and the function descriptive of supply. For
more exacting, concrete, practical purposes, the parameters
in the functions descriptive of the laws must be
deduced in numerical form from actual observations.
But how shall the investigator proceed to formulate
these laws abstractly, then concretely?
Looking back over the history of economic theory,
we see that the formulation has varied according as
the investigator has, or has not, made use of the
hypothesis of the negligibility of indirect effects.
The root idea of this postulate has been described in
general terms by Marshall:
" I t was not till the seventeenth century that the
physical sciences appreciated the full importance of
the fact that when several causes act together and
mutually affect one another, then each cause produces
I two classes of effects: those which are direct, and those
which result indirectly from the influence exerted by
it on other causes: for indeed these direct and indirect
effects are apt to become so intricately interwoven
that they can by no means be disentangled. So far
the results are negative: they seem to indicate that the
task of following out and understanding the combined
action of several causes, which are in various degrees
mutually interdependent, is beyond the power of
human faculty. But a way out of the difficulty was
found, chiefly under the guidance of Leibnitz and
Fundamental Notions 25
Newton. An epochmaking process of reasoning
showed that, though the indirect effects might grow
cumulatively, and ere long become considerable, yet
at first they would be very small indeed relatively to
the direct effects. Hence it was concluded that a
study of the tendency to change, resulting from each
several disturbing cause, might be made the starting
point for a broad study of the influence of several
causes acting together. This principle is the foundation
of the victory of analytical methods in many '
fields of science. Its best known triumph is that of the
Nautical Almanack which takes into account the
disturbing influences exerted by any two planets on
one another directly, and also indirectly as the result i
of their disturbances of other planets." "
It would not be difficult to show that Marshall constructed
his Principles of Economics, from beginning to
end, upon this method of analysis. Our misfortune is
that he followed the method only through the first
stage. His laws of demand and supply are laws of the
variation of demand price and supply price with the
variation of the quantity of commodity, on the
hypothesis that all other things remain equal. The indirect
effects of the variation of any one commodity
upon the prices of other commodities are neglected.
There is a principle of research in chemistry the
purport of which is in this precept—if you want to
make discoveries, look to your residues. Marshall's
practice in his Principles has been to invoke systematically
the postulate of the negligibihty of indirect
effects. What discovery shall we make if we examine
" Marshall: Industry and Trade, pp. 677678.
26 Synthetic Economics
the discarded residue? Nothing less, as we shall see,
j than the explanation of general economic oscillations.
'' The postulate described above in general terms by
Marshall has also been employed by Cournot, whose
description of his particular problem shows the truth
of the statement which' has just been made, that the
discarded residue of indirect effects contains the explanation
of general economic oscillations:
"In general . . . it must be the case, that a perturbation
experienced by one element of the system (of
prices) makes itself felt from that to the next, and by
reaction throughout the entire system. Nevertheless,
since the variation occurring in the price of commodity
A, and in the income of its producers, leaves intact the
sum total of the funds applicable to the demand for the
other commodities, B, C, D, E, etc., it is evident that
the sum diverted, by hypothesis, from commodity B,
by reason of the new direction of demands, will
necessarily be applied to the demand for one or several
of the commodities C, D, E, etc. Strictly speaking,
this perturbation of the second degree, which occurs in the
incomes of the producers of B, C, D, etc., would react on
the system in turn until a new equilibrium is established;
but, although we are unable to calculate this series of
'reactions, the general principles of analysis will show us
\that they must go on with gradually decreasing ampli
• tude, so that it may be admitted, as an approximation,
that a variation occurring in the incomes of the producers
of A, while modifying the distribution of the
remainder of the social income among the producers of
B, C, D, E, etc., does not alter the total value of it, or
only alters it by a quantity which is negligible in
Fundamental Notions 27
comparison with the variation which is experienced
by the incomes of producers of A."/^
In the preceding quotation the words I have italicized
contain the expression "a new equilibrium" ("un
nou vel équilibre"); also the germidea of a general
equilibrium, the conception of oscillations about the
equilibrium, and the explicit statement that "the
general principles of analysis will show us that they
(the reactions) must go on with gradually decreasing
amplitude." When an economic system is in equilibrium
any perturbation, according to Cournot, will
set up primary or direct effects, which are limited to
the immediate object disturbed, and secondary or indirect
effects, which result from the liaisons between
all the elements in the system. In consequence of
these functional liaisons, the indirect effects are
diffused throughout the entire system in the form
of oscillations which "the general principles of analysis
will show us must go on with gradually decreasing
amplitude." Neglect of the indirect effects obviously
involves the neglect of the phenomena and mechanism
of general economic oscillations.
If one employs the postulate of the negligibility of
indirect effects, a first approximation to the laws of
demand and supply may be obtained by representing
both demand and supply as functions of a single
variable. This is the course followed by Cournot and
Marshall. If, on the other hand, one aspires to explain
general economic equilibria and to follow out the
oscillations about the general equilibria, the liaisons
" Cournot: Researches into the Mathematical Principles of the Theory
of Wealth. Bacon's translation, § 76, pp. 130131
28 Synthetic Economics
among all the elements of the systems must be known,
and the indirect effects of perturbations become the
conditions of the explanation of oscillations. The
point of departure for this undertaking is to represent
demand and supply not as functions of a single price
but as functions of all prices. This is the course
followed by Leon Walras and his disciples of the
1 Ecole de Lausanne.
Obsolescent Disabilities
A discerning critic, Professor Umberto Ricci, has
summarily appraised the work of the Ecole de Lausanne.
As late as 1924 he expressed the following opinion:
"After due acknowledgment has been made to the
authors of one of the most wonderful creations of
human thought, one cannot but circumscribe the field
of its application. The whole construction gives the
effect of an enchanted palace which delights the
fantasy but does not help to solve problems of housing.
Or, to drop the metaphor, the theory remains abstract
and intangible." '^ In particular. Professor Ricci
remarks that there is no bridge between the pure
theory of Pareto and ninetenths of the problems with
which the economist is usually concerned. He asks
whether the bridge between theory and fact may yet
be built after mathematical knowledge shall have
progressed beyond its present stage and statistical
" " Ma tutto questo riconosciuto, e tributata la dovuta riconoscenza
agli autori di una delle piü meravigliose creazioni del pensiero umano,
non si puo non circoscrivere di questa il campo di applicazione.
"Tutta la costruzione fa un po' I'effetto di un castello incantata che
bea la fantasia, ma non aiuta a risolvere il problema degli alloggi.
Ossia, per uscir di metafora, la teoria rimane astratta e inafferrabile."
Giornale degli Economisti, GennaioFebbraio, 1924, p. 43.
Fundamental Notions 29
data shall have become more numerous, and he replies
to his own question with the noncommittal
words: Let us hope so. It cannot be denied that his
views are shared by many wellinformed scholars.
But what is the source of the sense of unreality so many
experience after having heroically struggled through
the writings of Walras and Pareto? What is the
prospect not only of removing from the theory of
equilibrium the sources of unreality but of utilizing, in
a practical manner, a theoretical scheme of thought
which embraces the ensemble of interrelated phenomena?
Foremost among the causes of the sense of unreality
are these: the method of proceeding by successive
approximations in the approach to a theory of general
equilibrium, which gives a feeling of an indefinitelj''
postponed real solution; the use of the hypothesis of
perfect competition with a meaning which does not
accord with reality; the limitation of all conclusions
to a static state, when, as a matter of fact, all economic
phenomena are in a perpetual flux; the assumption of
an immediate adjustment of changes, when in reality
there are always lags and leads; the complexity of the
functions that must be derived from reality and the
absence of any known method of making the derivation;
the assumption that the simultaneous equations
can never be solved, first, because their empirical forms
can never be known, and secondly, if they were known,
their great number would preclude the possibility of
their solution. There may be other reasons, but those
enumerated are certainly sufficient to account for the
sense of unreality.
30 Synthetic Economics
Every one of these disabilities may be mitigated, if
not entirely removed.
With regard to the method of successive approximations,
a distinction should be made between the successive
approximations which have been traversed in
the historic development of a theory and the successive
approximations in the accuracy with which a known
theory may be made to describe reality. Walras
began his work with the theory of utility and made the
conduct of the individual his point of departure.
Working in an uncharted region he observed all the
caution necessary to safeguard every step, being content
if each stage seemed to bring him nearer his goal of
a comprehensive view of social conduct. When at
last he reached the abstract simultaneous equations
descriptive of social behavior, the long laborious
process by which the results were achieved became,
primarily, of historic interest, but the abstract equations
may now be given a real form and may be made
to yield, by successive approximations, an increasingly
accurate description of reality.
The manner of circumventing the difficulties of the
hypothesis of perfect competition will be treated in
Chapter V, on "Moving Equilibria."
Two other causes of the sense of unreality in
the work of the Ecole de Lausanne—the limitation
of the results to an hypothetical static state,
and the absence of any known method of deriving
the necessary empirical equations from reality—disappear
with the actual derivation of these equations
for a perpetually changing state. Moreover, once we
are in possession of the concrete functions, it is not
Fundamental Notions 31
only unnecessary to make the assumption of immediate
adjustment of economic changes, but the real lags and
leads themselves become the foundation of a realistic
theory of economic oscillations about the general
equilibria.
Of the enumerated sources of the sense of unreality
there remains the assumption that the simultaneous
equations can never be solved, first, because their
empirical forms can never be known, and secondly,
if they were known, their great number would preclude
the possibility of their solution." The first part of this
assumption, that the empirical forms of the equations
'*"Les conditions que nous avons ónumérées pour I'cquilibre économique
nous donnent une notion générale de eet équilibre. Pour
savoir ce qu'étaient certains phénomènes nous avons dü étudier leur
manifestations; pour savoir ce que c'était que I'cquilibre économique,
nous avons dü rechercher comment il était determine. Remarquons,
d'ailleurs, que cette determination n'a nullement pour but d'arriver è,
un calcul numérique des prix. Faisons l'hypothèse la plus favorable a
un tel calcul; supposons que nous avons triomphé de toutes les difficultés
pour arriver a connaitre les données du problème, et que nous connaissions
les ophélimités de toutes les marchandises pour chaque individu,
toutes les circonstances de la production des marchandises, etc.
C'est \k déja une hypothese absurde, et pourtant elle ne nous donne pas
encore la possibilité pratique de résoudre ce problème. Nous avons vu
que dans Ie cas de 100 individus et de 700 marchandises il y aurait
70,699 conditions (en réalité un grand nombre de circonstances, que
nous avons jusqu'ici negligees, augmenteraient encore ce nombre);
nous aurons done a résoudre un système de 70,699 equations. Cela
dépasse pratiquement la puissance de l'analyse de l'analyse algébrique,
et cela la dépasserait encore da vantage si l'on prenait en consideration Ie
nombre fabuleux d'équations que donnerait irne population de quarante
millions d'individus, et quelques milliers de marchandises. Dans ce
cas les róles seraient changes: et ce ne seraient plus les mathématiques
qui viendraient en aide è l'économie politique, mais l'économie politique
qui viendrait en aide aux mathématiques. En d'autres termes, si on
pouvait vraiment connaitre toutes ces equations, Ie seul moven accessible
aux forces humaines pour les résoudre, ce serait d'observer la solution
pratique que donne Ie marché." Pareto: Manuel d'économie politique,
pp. 233234.
32 Synthetic Economics
can never be known, is, of course, erroneous if we
actually do derive the necessary concrete functions.
The second part of the assumption—that the great
number of the equations, even if known, would preclude
the possibility of their solution—seems to be a
serious, if not a fatal, disability. If the equations can
not be solved it might seem necessary to infer that the
equilibria values can never be known, and in this case
the whole theory of economic equilibria must forever
remain an unverifiable speculation. But, fortunately,
it would be an error to make such an inference. The
equilibria values may be known without solving the
equations. The method of deriving the empirical
functions suggests a way of guessing the equilibria
values, and the guessed values satisfy the system of
simultaneous equations.
The ensemble of economic interrelations may be
described concretely in their perpetual flux.
CHAPTER III
THE LAW OF DEMAND
"Kann man nicht die Nachfragefunktion genauer festellen, so
genau, dass wir nicht bloss ein eindeutiges, sondern ein konkretes
Resultat gewinnen? Ich glaube die Antwort zu horen: Welch'
ein phantastisches Unterfangen—Unberechenbarkeit der wirtschaftlichen
Vorgange—steter Wechsel—u.s.w.!"
JOSEPH SCHUMPETER
Effective use of mathematical methods in the elucidation
of economic theory was begun by Cournot in
his treatment of the law of demand. The same law
has been more amply investigated in the researches of
his mathematical successors, and, because of its critical
importance, will, with the law of supply, occupy a
principal place in Synthetic Economics. The statistical
methods which prove adequate to the concrete
presentation of the law of demand will serve also to
clothe with reality the law of supply. The theory of
demand will be presented in this chapter and will be
followed in the next with the theory of supply.
Elasticity of Demand and Flexibility of Prices
The quantitative treatment of the conception that
lies at the basis of elasticity of demand originated with
Cournot. Marshall gave the conception a name and
simplified and extended its presentation.
According to Cournot, if D = F(p) is the symbolic
expression of the relation between the quantity of
commodity demanded, D, and the price per unit of
33
34 Synthetic Economics
commodity, p, then for many problems in economics
it is of importance to know for what value of p the
product pF{p) is a maximum. One of Cournot's
concrete illustrations is that of a monopolist, owning
a mineral spring where the cost of production is
negligible, who wishes to know what price of the commodity
will yield him the largest monopoly return.
The mathematical condition of a maximum return is
& m = ^ ( , ) _, , ^ . ( , ) = 0. (1)
The root of equation (1) is the price that will afford
the maximum profit. In order to carry this problem
to a concrete solution we must know the empirical
form of D = F{p). Upon this statistical problem
Cournot makes the following comment:
"We may admit that it is impossible to determine
the function F(p) empirically for each article, but it is
by no means the case that the same obstacles prevent
the approximate determination of the value of p which
satisfies equation (1) or which renders the product
pF{p) a maximum. The construction of a table, where
these values could be found, would be the work best
calculated for preparing for the practical and rigorous
solution of questions relating to the theory of wealth.
"But even if it were impossible to obtain from statistics
the value of p which should render pF{p) a
maximum, it would be easy to learn, at least for all
articles to which the attempt has been made to extend
commercial statistics, whether current prices are
above or below this value. Suppose that when the
price becomes p + Ap the annual consumption as
The Law of Demand 35
shown by statistics . . , becomes D — AD. According
as AD/Ap < or > D/p, the increase in price, Ap, will
increase or diminish the product pF{p); and, consequently,
it will be known whether the two values p
and p \ Ap (assuming Ap to be a small fraction of p)
fall above or below the value which makes the product
under consideration a maximum.'
"Commercial statistics should therefore be required
to separate articles of high economic importance into
two categories, according as their current prices are
above or below the value which makes a maximum of
pF(p). We shall see that many economic problems
have different solutions, according as the article in question
belongs to one or the other of these two categories." ^
Some of Cournot's statements in the preceding
quotation have been italicized to indicate that, in his
opinion, we must know the empirical laws of demand
1 The method of reaching the inequality discussed in the tejd; may
be indicated:
The increase in price will increase the gross receipts if
{p + Ap){D  AD) > pD,
or
yD  P  A D + flAp  ApAZ) > pD,
or
 pAD + DAp  ApAD > 0,
or
AZ)(p + Ap) < DAp,
or
AD D
Ap p + Ap '
or, when Ap is small as compared with p,
AD ^ D
Ap p '
^ Cournot: Researches into the Mathematical Principles of the Theory
of Wealth, Bacon's translation, pp. 5364.
36 Synthetic Economics
for commodities if we are to pass to the practical and
rigorous solution of questions relating to the theory of
wealth.
The classification that Cournot makes between
commodities according as AD/Ap < or > D/p is essentially
the classification which has subsequently been
made to distinguish between inelastic and elastic
commodities, flexible and inflexible prices. To show
this let us pass from Cournot's symbols to those that
have been rendered more familiar by Marshall. In
Marshall's notation, x stands for the quantity demanded,
and y, for the price per unit of commodity.
If we substitute these symbols for those of Cournot,
we may write Cournot's statement in this way:
According as AxjAy < or > xjy, the increase in price.
Ay, will increase or diminish
the product xy.
So far Cournot carried
the problem. But
his statement may be
easily simplified and its
economic significance
rendered much clearer.
In Figure 1 we have
the familiar graph of
the law of demand,
where x — the quantity
of commodity demanded; y = the price per unit of
commodity; and DD' is the demand curve. Let the
quantity actually demanded be x = OM, and the
resulting actual price he y = MP. Then, in agreement
with Cournot's formula, it would be profitable
0
\ p'
\ p
\ X"
\
M' M M"
FIGURE 1.
The Law of Demand 37
to the monopolist to raise or lower his price according
as
Ax X .,.
— < o r >   (2)
Ay y
The inequality (2) may be divided through either
by x/y or by Ax/Ay. If we divide through by x/y, we
get
^ • ^ < o r > l , (3)
Ay X
which is Marshall's form of statement. He regards
Ax/Ay • y/x as the measure of the elasticity of demand
and describes the demand for the commodity as
inelastic or elastic according as Ax/Ay • y/x is numerically
less or greater than unity. If we call dx/dy • y/x
the coefficient of elasticity of demand and indicate it
by 7), we may state Cournot's proposition as follows:
The gross receipts of a monopolist will be increased or
diminished by an increase of price according as
— ?? < or > 1; that is, according as the demand for
the commodity is inelastic or elastic. This form of
expression we have reached by dividing Cournot's
inequality (2) by x/y.
But we might have divided it by Ax/Ay. If we do
so now, we shall have
l < o r > ^ . ^  (4)
y Ax
Let us call dy/dx • x/y the coefficient of flexibility of
price and indicate it by ^. Cournot's proposition
would then be as follows: The gross receipts of a
monopolist will be increased or diminished by an in
38 Synthetic Economics
crease in price according as — ^ > or < 1; that is,
according as the price is flexible or inflexible.
Returning now to the use of Cournot's symbols, we
may give formal definitions of elasticity of demand
and flexibility of price. If ?? be taken as the symbol to
represent elasticity of demand, and \l p = the price
per unit of commodity and D = the quantity of commodity
demanded at price p, the elasticity of demand is
dD I dp p dD ,^.
Elasticity of demand is then defined as the ratio of the
relative change in the quantity of commodity demanded
to the relative change in the price per unit of
commodity. When ?; is numerically greater than
unity, the demand for the commodity is said to be
elastic; when rj is numerically less than unity, the
demand is inelastic.
Similarly, if ^ is taken as the symbol to represent
flexibility of price, the definition of flexibility becomes
dp I dD D dp
That is to say, flexibility of price is the ratio of the
relative change in price to the relative change in the
quantity of commodity demanded. When
'a, or
a H a'D, or
a\ a'D + a"D^.
In these equations, 4> is the flexibility of price, and,
according to the preceding paragraph,
dp IdD
where D = the quantity of commodity demanded,
and p is the price per unit of commodity. The first
assumption made in the above differential equations is
that the flexibility of price is a constant; the next
assumption is that the flexibility of price is a linear
function of the quantity of commodity demanded;
the third assumption is that flexibility is a quadratic
function of the quantity demanded.
40 Synthetic Economics
If the flexibility of price is assumed to be constant,
we have
dp IdD
or
dp dD
—p = «D77 
Integrating, we get
log. p = a log, D + loge A.
Or, passing from logarithms to absolute numbers, we
obtain as the typical demand function
p = AD". (7)
In this formula, A is the constant of integration to be
determined from the observations.
If the flexibility of price is assumed to be a simple
linear function of the quantity of commodity demanded,
the differential equation is
dp IdD
or
dp dD
— = a=+ a dD,
p D
which, when integrated, yields the typical demand
function
p = ADe"'". (8)
In this formula, e is the base of natural logarithms.
The third assumption,
dp IdD , ,nL "n2
0 = — / ;r = a + a D f a D',
p I D
The Law of Demand 41
leads to the typical demand function,
p = AD"e"'^+^""^\ (9)
The typical demand functions (7, 8, 9) have been
obtained by taking the quantity of commodity demanded
as the independent variable, and starting
with three simple assumptions with regard to the
character of flexibility of price. If price is taken as
the independent variable, we shall obtain other typical
demand functions by starting with corresponding
assumptions with regard to elasticity of demand. We
may put
>, or
dD/dp
Dl p
p + fi'p, or
j8 + /3'p + /3'y.
From these equations we obtain, by integrating, the
typical demand functions,
D = Bp^; (10)
D = 5pV^• (11)
D = Bp^e^'''+'^^"^\ (12)
In these equations, the B's are constants of integration,
and e is the base of natural logarithms.
Statistical Derivation of Typical Demand Functions of
One Variable
The preceding section gives useful forms of demand
functions, and the present problem is to derive from
observations the values of the parameters in those
typical equations. The great difficulty in the under
42 Synthetic Economics
taking lies in the fact that both prices and quantities
of commodities are in a state of constant secular
change. Several methods for overcoming this difficulty
have been devised,^ but in the subsequent reasoning
use will be made only of the method of trendratios.
In preparing the data of prices and of quantities of
commodities so that they may be used to deduce the
law of demand by the method of trendratios, the
prices of the commodity at successive points in time
are expressed as ratios to the corresponding trends at
the respective points in time. Similarly the quantities
of commodity at the points in time are expressed as
ratios to the corresponding commodity trends.
The general formula for the law of demand according
to the method of trendratios is either
where D is the trend of the quantity demanded at the
particular time, and p is the trend of the corresponding
price at the same time; or, if the quantity of commodity
is taken as the independent variable,
To find the concrete laws of demand, the general
equation (13) may be taken in any one of the three
forms (10, 11, 12), or the general equation (14) may
be taken in any one of the forms (7, 8, 9).
^Henry Schultz: "The Statistical Law of Demand," Journal of
Political Economy, October and December, 1925.
The Law of Demand 43
The steps in the statistical procedure will be illustrated
by fitting typical equation (8) to data.
The Yearbook of the Department of Agriculture gives
statistics of the production of agricultural commodities
and of their respective prices throughout a long interval.
The data referring to the production and prices
of potatoes from 1881 to 1913 will be used to illustrate
the derivation of the empirical law of demand for
potatoes and the law of the flexibility of potato prices.
1 1 1 I I I M I I I I I I I I I I I I I I I I r I I I [ I I I I [ I
1880 1890 1900 1910
FiGUEE 2. The secular trend in the production of potatoes in
the United States.
y = 222.3 t 5.711«  .1758^2  .004363^', origin at 1897.
In Figure 2 the secular trend of the production of
potatoes from 1881 to 1913 is graphed, and a similar
curve for the secular trend of prices of potatoes might
be given. The equation to the trend of production is
printed on Figure 2; the equation to the trend of
prices is, with origin at 1897.
V = 48.86 + 0.775i f 0.0443^^ _ o.002935«'.
44 Synthetic Economics
TABLE I
THE ANNUAL PRODUCTION OF POTATOES AND THEIR DECEMBER FARM
PRICES IN THE UNITED STATES. PRODUCTION RATIOS AND
PRICE RATIOS
Year
1881
82
83
84
85
86
87
88
89
1890
91
92.
93
94
95
96
97
98
99 .
1900...
01
02
03. .
04
05
06 .
07
08
09
1910
11
12
13
Total
Mean
Production;
millions of
bushels
D
109
171
208
191
175
168
134
202
218
148
254
157
183
171
297
252
164
192
273
211
188
285
247
333
261
308
298
279
389
349
293
421
332
Price:
cents per
bushel
P
91.0
55.7
42.2
39.6
44.7
46.7
68.2
40.2
35.4
75.8
35.8
66.1
59.4
53.6
26.6
28.6
54.7
41.4
39.0
43.1
76.7
47.1
61.4
45.3
61.7
51.1
61.8
70.6
54.1
55.7
79.9
50.5
68.7
Production
ratio
DID
.690
1.062
1.261
1.137
1.023
.960
.753
1.110
1.172
.783
1.316
.793
.906
.826
1.401
1.161
.739
.842
1.162
.876
.758
1.113
.936
1.224
.929
1.058
.990
.894
1.201
1.039
.840
1.163
.883
33.001
1.00
Price
ratio
viv
1.52
.97
.77
.74
.88
.94
1.41
.85
.75
1.63
.76
1.41
1.27
1.13
.56
.59
1.12
.84
.77
.84
1.46
.88
1.13
.82
1.09
.89
1.06
1.21
.90
.93
1.33
.83
1.14
33.42
1.01
The Law of Demand 45
After the trends of the two series of figures—production
and prices—have been computed, the next step in
the problem is to express the observed values of the
quantities as ratios to the corresponding respective
trends. The ratios relating to production and prices
we shall refer to, respectively, as productionratios and
priceratios. In Table I these ratios are tabulated.
The correlation of the priceratios and the production
ratios of potatoes is r = — 0.84, which is sufficient
evidence of a very high relation between the two series.
Suppose, now, that the law of demand is of type (8).
We should then be required to fit to the trendratios
the formula
f = ^ (§)>'"'"'• (15)
Before proceeding to fit (15) to the data, we may effect
a simplification by observing that the value of A is
already known. When the priceratio p/p is 1.0,
the production ratio is likewise 1.0; and if these values ^
are substituted in (15), the value of A is found to be
e~"', and consequently (15) becomes
H.()V(i). •
To fit equation (16) to the observations, let us first
take logarithms of both sides of the equation. We
have
l o g (  ) = a l o g (  ) + « ' (   l ) l o g e . (17)
* In the particular case of potatoes the mean values of DjD, pfp are,
practically, 1.0. See Table I. There may be cases where it would be
better not to use this hypothesis but to determine A from the observations.
This could be done by following the same method as that
described in the text.
46 Synthetic Economics
If the method of least squares is used to fit (17) to the
n observations, we have as the observation equations
[lo.()«.o.()„(l),„,.J=.,,
The sum of the squares of the errors, SC^^), is a function
of a, a', and, in order to find their most probable
values, we have the normal equations
öSi)2 „ / , Ï ) , D\ „ / , DV
= 2 ( l o g  . l o g  )  « 2 ( l o g § ) '
0.4343a's(log£)= 0;
 0.4343a':
D'
dZv' KM)('l)(§'i)
+ «2 ( l o g  )  0.4343a'2 ( I J
+ 0M86a'^(^\  0.4343na' = 0.
By means of these two normal equations the most
probable values of a and a' may be determined from
the observations. When the calculation is carried
out for the productionratios and priceratios of
The Law of Demand 47
potatoes, the equation connecting the two is found to
be
1.376 it'). (18)
Its graph
Y
1.6
1.5
1.4
1.3
CO 1.2 o
fe 1.1
OC
Ö 1.0
is traced in Figure 3

D
\
N
L _!_ _!_ 1 I 1 L  L .
.7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5
PRODUCTION RATIOS
FiGUKE 3. The law of demand for potatoes.
p ^ /Z)\oi43
e1.37G[(£i//))!]_
Equation (18) gives the relation between the priceratios
and productionratios for the interval 18811913.
It may be called the law of demand for potatoes in the
ratio form. By an obvious transformation of (18),
the law of demand for any one year may be obtained
48 Synthetic Economics
in terms of absolute quantities. It is
„ _ [ ^ ^1376 1 r)0.143.1.376(0/5)
P  J 0 ) o . u 3 e ji> e
= AD'''^h'^'^^'"'^\ (19)
where
1(^)0.143 6 I
If, for example, the law of demand for potatoes in 1913
is to be ascertained, find the trend of prices for 1913,
f, and the trend of production for 1913, D, and substitute
these values in (19). The values of f, D may
be obtained for any one year from the equations to
the trends which have already been given.'^
The preceding derivation of the demand function
also solves the problem of the law of variation of
flexibility of price. By comparing the typical equation
of demand (15) with the concrete equation of demand
in ratio form (18) and with the demand equation in
absolute form (19), we see that, in this particular case,
the flexibility of price is
<^ = a + a ' ( ^ £ j = 0.143  1.376(^£V
^ The objection may be made that, since the trends are described by
parabolas of the type y = uo + ait + a2<^ + asP, if we extrapolate for
many years beyond the observations the curves may give impossible
results. But the subject of the present discussion is not what will occur
many years beyond the limits of observation. To a similar objection
on the part of an unfriendly critic Pareto once replied: Quando vi si
da una formula valevole entro certi limiti, chi vi insegna ad applicarla
fuori di quei limiti?
I hope I have also made clear that the statistical data of quantities of
commodity and of prices used in the problem of the text have been
taken merely to illustrate a statistical conception, and not to indicate
what data should be selected to yield the most accurate demand curve.
The Law of Demand 49
Its graph is traced in Figure 4, and we see at a glance
how the flexibility of prices of potatoes varies from
point to point in the demand curve.
1.0
UJ
CJI.I
cc
^1.2
>1.3
S1.4
X
!±j1.5
I—^
1 .. 5 .
PRODUCTION RATIOS
3 .7 .8 .9 1.0 1.1 1.2 1.3 1.4
>
<
N \.
\
\
\
\ ,
N s
\
\
s k.
\ 0'
1.6
1.7
FIGURE 4. The flexibility of the prices of potatoes.
4> = 0.143  1.376(Z)/5).
These results have been reached by starting with
the typical equation of demand (8) in which quantity
of commodity is taken as the independent variable.
If the start had been made with the typical equation
(11), in which price is the independent variable, the
typical demand function in the ratio form would be
D \v)
g3'(p/p)_ (20)
By following the same method as the one used in
50 Synthetic Economics
fitting (15) to the data of observation, the above function
(20) could be fitted to the same material. After
the constants /3 and /S' should have been determined
concretely, we should have the law of variation of
elasticity of demand, since, by hypothesis, the
elasticity of demand is
This concrete derivation of the law of variation of
elasticity of demand would clear up the following
perplexing statement by Marshall in his classic
chapter, "The Elasticity of Wants": "The elasticity
of demand is great for high prices, and great, or at least
considerable, for medium prices; but it declines as
the price falls; and gradually fades away if the fall
goes so far that satiety level is reached. This rule
appears to hold with regard to nearly all commodities
and with regard to the demand of every class." ® This
statement is perplexing, because we have had no means
of deriving empirical laws of demand and because it
has been impossible to picture concretely what is
meant by elasticity of demand being "great" or
"considerable." We know when a coefiicient of correlation
is "great" or "considerable," but what is
"considerable" or "great" elasticity of demand?
Graphs of the values of q for various commodities,
after the manner of the graph of (f> in Figure 4, would
answer the question.
The statistical methods that have just been described
make possible the mastery of the problems to
* Marshall: Principles of Economics, 4th edition, p. 178.
The Law of Demand 51
which Cournot drew attention. He saw that many
economic problems have different solutions according
as the product of pF(p) is increasing or decreasing for
increasing values of p; and he recommended, as the
first step toward a practical and rigorous solution of
these problems, that commodities should be separated
into two categories according as their current prices
are above or below the value which makes a maximum
of pF(p). In the first section of this chapter we found
that the distinction with which Cournot was concerned
is the same as the modern distinction between commodities
of elastic and of inelastic demand. Demand
curves of type (20)
D \pj
give summary descriptions of data within the limits
in which p actually oscillates.^ Moreover, since
, = . + .'(1),
' Criticism is apt to degenerate into discussion of what might be
the course of the curves beyond the limits of actual observation. It
may not, therefore, be amiss to quote two authorities who would have
been content to know what actually occurs within the limits of observation:
Cournot: "If we cease considering the question from an exclusively
abstract standpoint, it will be instantly recognized how improbable it is
that the function pF{p) should pass through several intermediate
maxima or minima inside of the limits between which the value of p can
vary; and as it is unnecessary to consider maxima which fall beyond
these limits, if any such exist, all problems are the same as if the function
pF(p) admitted only a single maximum. The essential question is
always whether, for the extent of the limits of oscillation of p, the
function pF{p) is increasing or decreasing for increasing values of p."
Bacon's translation of Cournot's Researches, p. 55, ^ ,j) i_
52 Synthetic Economics
this type, of course, likewise gives the law of the variation
of elasticity with the variation of p.
Partial Elasticity of Demand and Partial Flexibility
of Prices
The preceding sections of this chapter have presented
the theory of demand as a function of a single
variable and have described methods by which the
theory may be given a concrete, statistical form. The
remainder of the chapter will be devoted to a consideration
of the more difficult problems of the theory and
technique of demand when demand for any one commodity
is regarded as a function of the prices of all
commodities. To treat this more complex question
there is need of a more ample notation, and to meet this
requirement we shall take as our point of departure the
notation of Walras.
Walras' symbols are as follows: The commodities
produced in a unit of time are m in number and are
represented by {A), (B), (C), • • •. The factors of
production fall into three classes: services of land,
services of persons, and services of capital. The total
number of services, for the unit of time, is assumed to
be n, and these are designated as
Services of land (terre), (T), (T'), (T"), • • •
Services of persons, (P), {P'), {P"), • • •
Services of capital, (K), (K'), (K"), •••.
Marshall: "The general demand curve for a commodity cannot be
drawn with confidence except in the immediate neighbourhood of the
current price, until we are able to piece it together out of the fragmentary
demand curves of different classes of society." Principles, 4th
edition, p. 189.
The Law of Demand 53
If the commodity {A) is taken as a standard of value
{numéraire) in terms of which the prices of commodities
and services are expressed, the respective prices for
the commodities may be represented as pb, pc, Pd, • • •
and the respective prices for the services as pi • • •,
PP ••, Pk •••,•• •• Since the prices of the (m — 1)
commodities are expressed in terms of the standard
of value—commodity (A)—there are in Walras' system
(m — 1) demand functions, which he represents
with these symbols:
Db = Fiipt, PP, Pk, ••• Pb, Pc, Pd, •••),
Do = Fcipt, PP, Pk, ••• Pb, Pc, Pd, •••),
Dd = Fdipt, PP, pk, ••• Pb, Pc, Pd, •••),
(21)
In these expressions the demand for any one commodity
is regarded as a function not only of its own
price but of the prices of all other commodities.
Walras' reasoning is limited to a static state. His
quantities of commodities and prices are such as would
prevail in his hypothetical construction, in a state of
equilibrium, under a régime of perfect competition.
To carry his reasoning into the sphere of economic
realities, a method must be found for obtaining the
concrete, complex functions of demand in a constantly
changing society where perfect competition does not
generally exist, and where equilibrium is incessantly
perturbed.
In order to approach the real problem, let us recognize
that all prices and all quantities of commodities
are subjected to such forces as give to each of them
an individual secular trend. Let us suppose that the
54 Synthetic Economics
secular trend of each price and of each commodity is
determined statistically by fitting to the data, through
the use of the method of least squares, a curve of type
2/ = ao + «iJ + dil? + cizt^ + • • •.
If the D's and p's are now taken as the actual quantities
of commodities and the actual prices, their trendvalues
at a given time may be represented by putting
a bar over these symbols, so that, for example, Dc will
indicate the trendvalue of the quantity of commodity
(C) demanded at a time when the actual quantity of
commodity demanded is Dc. Similarly pc will indicate
the trendvalue of the price of (C) at the time the
actual price is pc The new type of demand function
will then be represented as
B^^pfPl 'EL PH ...Pi 'El EÉ. ...\. (22)
Dc \pt' PP' Pk' Pb' Pc' Pd' J
This formula expresses the simple hypothesis that the
trendratio of the quantity demanded is a function of
the trendratios of all prices. While this hypothesis
is simple it is the means of making the transition from
a purely rational construction to a real situation. The
introduction of the conception of a function of trendratios
makes possible the statistical evaluation of the
general demand function just as soon as its algebraic
form is known.
When dealing with demand as a function of a single
variable we reached appropriate typical demand
curves by starting with the conceptions of elasticity
of demand and flexibility of prices. Elasticity of
demand, rj, was defined as the ratio of the relative
The Law of Demand 55
change in the quantity demanded to the relative
change in price. Its formula is
dD Idp p dD
'^D/J^Dd^ (^^^
Flexibility of price, 0, is the ratio of the relative change
in the price to the relative change in the quantity demanded.
Symbolically
dp IdD D dp
'^^j/'D^^dD ^^^^
If now demand, for example of commodity (C), is
regarded as a function of all prices, the demand function
is
Dc = Fcipt, PP, Pk, •" Pb, Pc, Pd, • • •)• (25)
If price is regarded as a function of all quantities of
commodities demanded, the price function is
Pa = MDt, D^, Dk, ••• Dt, Dc, Da, •••). (26)
As a means of deriving concrete demand functions
and concrete price functions, in place of (25) and (26),
which are only abstract representations, the above
procedure in case of a single variable suggests the
wisdom of defining the conception of partial elasticity
of demand and of partial flexibility of price.
If the demand function is in the form of (25), the
partial elasticity of demand for commodity (C) with
respect to pt may be written, by following the analogy
of (23),
_ pj dDa ,^^ .
J>c opt
In this notation the primary subscripts of i? are sepa
56 Synthetic Economics
rated by a single dot from the secondary subscripts,
and the whole symbol indicates the partial elasticity
of demand for commodity (C), with respect to price
Pt, when the demand for (C) is a function of pt, Pp, Pk,
• • • Pb, Vc Pd, • • •• In a similar manner the partial
elasticity of demand for the commodity with respect
to every other price may be indicated.
If the price function is in the form
Po = f{Dt, Dp, Dk, '•' Dt, Do, Da, • • •),
the partial flexibility of price for commodity (C) with
respect to Dt may be written, by following the analogy
of (24),
Pc OJJt
Here, again, the primary subscripts are separated by
a single dot from the secondary subscripts, and the
whole symbol indicates the partial flexibility of the
price of commodity (C), with respect to Dt, when the
price of (C) is regarded as a function of Dt, Dp, Z)&,
• • • Dh, Do, Dd, • • •.
Typical Demand Functions of More than One Variable
After developing general conceptions of partial
elasticity of demand and partial flexibility of prices,
we are confronted with the problem of determining
the partial elasticities and partial flexibihties in
particular cases. But a prerequisite to obtaining
concrete results is the knowledge of appropriate
types of demand functions and price functions. What
steps may be taken toward finding these appropriate
typical functions of demand and of price?
The Law of Demand 57
Obviously it is wise to go forward in the direction in
which positive conclusions have already been attained.
Progress in the treatment of elasticity of demand has
been made (a) by using the method of trendratios in
the preparation of the statistical data, and (6) by
deriving appropriate demand curves from one of the
hypotheses
•/3, or
^ = • /3 h ^'p, or
/3 + ;8V + ^"v\
The suggestions from this experience that occur with
reference to the problem of the typical forms to be
given to the representative demand function
Do = Fcivt, PP, Pk, ••• Pb, Pc, Pd, •••)
are (a) to retain the method of trendratios in the
preparation of the data, and (b) to derive increasingly
complex typical functions that will give the partial
elasticities with increasing accuracy. In case of the
above representative demand function we know from
equation (26a) that a representative partial elasticity
of demand is
VcpfPpPliPbPcPd ~ rj • ^
The increasingly complex demand functions, which give
the partial elasticities with increasing accuracy, would
obviously be obtained if functions could be found in
which each partial elasticity of demand is given as a
constant, or as varying in a linear function of the
corresponding price, or as a quadratic function of the
corresponding price. Supposing the functions dis
58 Synthetic Economics
covered, we should then have, for the representative
commodity (C), the partial elasticity of demand with
respect to Pi in one of these three forms,
Vcp, fpppf •PbPcPi
Vt
dpt
l3ot, or
At + P'ctPt, or
At + P'ctPt + P'c'tPl
(28)
In this notation the proper subscripts for the /3's are
the same as those of t], but abbreviated symbols have
been used in order to avoid unwieldy formulas for
the demand function in which, as we shall see, the
/3's are the parameters.
If, now, the first hypothesis in (28) is followed and
the partial elasticities are assumed to be constant, the
typical demand function for (C) is
Do = Constant {piY"{pj,Y'"'{pkY"'{• •)
X {p,Y''{poY"{vdY'\). (29)
If the second hypothesis is followed, in which the
partial elasticities are assumed to be simple linear
functions of the corresponding prices, the typical
demand function for (C) is
Dc = Constant {ptY^'iPvY'^iPkY^i • •)
X {vbY*i.PcY"{pdY'\• •)
\y pWclPt +P'cpPp +/SctM H Htepi +P'ccPc+KdPd\ ) f ^ 0 1
If the third hypothesis is followed, in which the
partial elasticities are assumed to be quadratic func
The Law of Demand 59
tions of the corresponding prices, the typical demand
function for (C) is
D, = Constant {vty''iVpY'''{,PkY"'{ • •)
X {vbY^ivcY^iVdY'^• •)
In formulas (30) and (31) the symbol e is the base of
natural logarithms.
That these three functions fulfil the conditions
expected of them may be proved by showing that
any one of the partial elasticities, for example,
VcPfPpPk • PbPcPd ••>
is, in (29), a constant; in (30), a simple linear function
of the corresponding price; in (31), a quadratic function
of the corresponding price. To carry out the
proof with regard to (29), take the logarithm of both
sides of the equation; differentiate with respect to pt',
and then find the above representative partial elasticity.
By following these directions we have
loge Dc = loge Constant + Pct loge Pt + Pep loge Pp
+ ••• + Pcb loge Pb + Pec loge Pc
+ 0cdlogePd + '••
Consequently,
1 dPc _^ Pot
Dc dpt Pi'
whence
_Pt dPq _
VcpfPpPkPbPcPd" TJ ' )„ "<:'•
60 Synthetic Economics
By following the same directions with regard to (30),
we have
loge Dc = loge Constant + Pot loge ft + Pep loge PP
+ ••• + iP'ctPt + P'CPPP + • • • ) •
Consequently,
1 dDo Pot , „,
IT ' ."a— ~ 1 Pat,
Dc dpt pt
whence
_ Pt dDo _ ^ , or
VcpfPpPfpbPcPd T\ ' J ~ Pet "r PctPt'
In a similar manner, proof may be given that, by
following the same directions as in the preceding cases,
equation (31) yields as the value of a representative
partial elasticity
VcpfPjiVfPiPcPd ^^ "T^" * ^ ^^ Pet ~r PctPt ~r PetPt
Equations (29), (30), (31) are typical equations of
demand, appropriate not only for connecting demand
with prices but for supplying the derivative laws of
the variation of partial elasticities of demand. If it is
desired to regard price as the dependent variable, to
take quantities of commodities demanded as the
independent variables, and to arrive at laws of the
variation of partial flexibilities of prices, the start
would be made with the general price function
Pc = fe(Di, Dp, Djc, ' • ' Db, Dc, Dd, '••).
Partial flexibilities of prices could then be obtained
with increasing accuracy if functions could be found
in which the partial flexibihties are constants, or vary
The Law of Demand 61
in simple linear functions of corresponding quantities
of commodities demanded, or vary as quadratic
functions of the corresponding quantities of commodities
demanded. Supposing these price functions
to be known, we should have for a representative commodity
(C) the partial fliexibiUty of price pc with
respect to Dt,
Dt dpc
Pc OUt
act, or
act + a'ctDt, or •. (32)
act + a'ctDt + a'c'tD^
If the first hypothesis is adopted, the typical price
function is
Pc = Constant {PtT'KD^TKDkT"'{ • •)
X {DiY'\DcT"{DiY'%). (33)
The second hypothesis yields as the typical price
function
Po = Constant {DtT'^D;)"''{• • •)(i)6)''"'(A)""( • •)
The third hypothesis gives
Pc = Constant {DtY''%D^Y'\DuY"'{ • •)
X {DiY'KDcT'ii^dT'^• •)
S/ p(.oi'ciDt+a'cpDp+a'ckDk{ ) +Ha!:iD'i+a'cpDl+ailJ)k\ ) (^R)
Statistical Derivation of Typical Demand Functions of
more than One Variable
With the typical demand functions (29), (30), (31)
and the typical price functions (33), (34), (35) before
62 Synthetic Economics
us, the next step is to fit the functions to the statistical
observations. The observations themselves, however,
refer to a constantly changing society: all of the
variables that enter into the demand functions and
price functions are themselves functions of time; and
these typical functions must be adjusted so as to
bring out the interrelations of prices and quantities of
commodities not only at a particular time, but throughout
a flow of time. To meet this need the device of
trendratios, which was explained in detail in an earlier
section of this chapter, is as applicable to functions of
many variables as to functions of a single variable.
Repeating the notation used in an earlier section of
this chapter, where symbols with bars placed over them
refer to the trends of the variables, we may indicate the
transformation which demand functions (29) and (30)
undergo when trendratios are introduced. Demand
function (29) becomes
g=c—()(ir(ir()
or, in the logarithmic form,
log ( ^^ j = log Constant 1 i3c« log ( t r )
+ ,.,i„g(a)+...+ft.,„,()
+ ft,log(?:)+. (37)
The Law of Demand 63
Demand function (30) becomes
x (  y ' (  ƒ(...)o'«(g)«='(I)*, (38)
or, in the logarithmic form,
log ( =^ j = log Constant + pet log f rr j
\p'ck(P^+ •••'\\oge. (39)
Equations (37) and (39) are linear functions of the /3's,
and hence the /3's and the constants of integration may
be directly ascertained by fitting these functions to
the observations, either by the method of least squares
or by the method which has become familiar in the
theory of partial correlation. With the /3's and the
constants of integration empirically determined, we
have the complex demand functions not only in concrete
forms, but in such forms as give the partial
elasticities of demand either as constants or as simple
linear functions of the corresponding prices. If (37)
is fitted to the data, a representative partial elasticity
of demand is
VcpippPkpiPcPd /'«<•
If (39) is used, a representative partial elasticity of
64 Synthetic Economics
demand, described as a simple linear function of the
corresponding price, is
 R M R' {Pl\
Vcpipppic• pbPcPd• • "c( " r Pet \ — I"
\Pt /
Reasoning similar to the above with regard to the
demand functions (29) and (30) and the adjusted
trendratio forms (37) and (39) will hold with regard
to the price functions (33) and (34) and the corresponding
adjusted trendratio forms. These two types
of demand functions and two types of price functions
will probably suffice for most practical purposes, but
if still more complex forms are required, resort may
be had to types (31) and (35).
The prices and quantities of commodities to be used
as raw data will be determined by the problem in
hand: they may be of as many individual commodities
as desired; or they may be of representative commodities
of various categories; or they may be price
indexnumbers and corresponding quantity indexnumbers
of whole classes of commodities.
CHAPTER IV
THE LAW OF SUPPLY
"That fundamental symmetry of the general relations in
which demand and supply stand to value, which coexists with
striking differences in the details of those relations."
ALFRED MARSHALL
The fundamental symmetry with which demand
and supply cooperate in the determination of price
suggests the possibility, and indicates the desirability,
that the typical functions descriptive of supply may
be of the same general forms as those which have been
found useful when dealing with demand. In the
development of this chapter we shall see that, in fact,
the actual practice of business and the exigences of
economic theory concur in leading to the conclusion
that the same typical functions reproduce the essential
characteristics of both demand and supply.
Elasticity of Supply and Expansiveness of Supply Price
In the foregoing chapter we have seen that, if the
demand for the representative commodity (C) is
indicated as
Dc = F,(pc),
where D^ is the quantity of the commodity demanded
at price pc per unit of commodity, then elasticity of
demand and flexibility of price may be expressed,
65
66 Synthetic Economics
dDc idpe _ Pc dDc
De I Pe De dpc '
dpc idDc _ De dpc_
Pc I De Pc dDc
These conceptions in the theory of demand are paralleled
in the theory of supply with corresponding
notions of elasticity of supply and of expansiveness of
the supply price. Elasticity of supply is the ratio of
the relative change of the quantity supplied to the
relative change in the supply price. If Sc is adopted
to indicate the quantity of commodity (C) supplied at
price Pc and ijs is used to represent the elasticity of
supply, the above definition becomes
dSe idpc Pc dSc ,.„
Se I Pc Sc dpc
Expansiveness of supply price is the ratio of the
relative change in supply price to the corresponding
relative change in quantity of commodity supplied.
If (/) s is used as the symbol for expansiveness of supply
price, the above definition becomes
dpc idSc Sc dpe ,..,
Pc I Sc Pc dSc
Typical Supply Functions of One Variable
Elasticity of supply and expansiveness of supply
price are conceptions that enter into the exact solution
of many economic problems, and for this reason
it is desirable to describe laws of supply by typical
functions in which the characteristics of these two
conceptions are revealed through the parameters of
respectively, as
VD =
4>D =
The Law of Supply 67
the functions. By following the method employed in
the preceding chapter on "The Law of Demand," we
may find typical functions of supply in which elasticity
of supply is described as a constant, or as a simple
linear function of the supply price, or as a quadratic
function of the supply price. Increasingly complex
supply functions are obtained by integrating the
differential equations
' Ic, or
•ns = • yc + I'cVc or
. 7c + 7'Po + Tc'Pc.
If the elasticity of supply is assumed to be a constant
and, in case of the representative commodity (C), equal
to 7c, we have
or
ns =  5  / — = 7o,
dSa _ épc
Sa ~^' Pc'
Integrating, we get
loge Sc = 7= log. Pc + r.
Or, passing from logarithms to absolute numbers, we
have
So = VpV. (42)
In this formula r is the constant of integration to be
determined from the observations.
If the elasticity of supply is assumed to be a simple
linear function of the supply price, the differential
equation is
dSc Idpc . ,
vs =  0  / — = 7c + ycPc
o I Pc
68 Synthetic Economics
which, when integrated, yields the typical supply
function
Sc = Vvl'e''^'. (43)
If the elasticity of supply is regarded as a quadratic
function of the price,
dSJdpo , , „ 2
VS = ^ — = 7c + IcPc + loVc
JOc ' j/c
and the typical supply function is
Sc = rpj«e^«^'+*^"''l (44)
These typical supply functions have been obtained
by taking supply price as the independent variable
and starting with three simple assumptions as to the
character of elasticity of supply. If quantity of commodity
is taken as the independent variable, we may
derive corresponding typical functions by making
similar assumptions as to the variation of expansiveness
of supply price. We may put for the representative
commodity (C)
dpc
Sc
(57)
The Law of Supply 75
If the first hypothesis is adopted, the typical function
is
pt = Constant {StY"{S^y'{S,y"'{ • •)
X is,y"(Soy"(s,y'\). (ss)
The second hypothesis yields as the typical function
Pt = Constant {Sd'"(Sj,y'( • •)(S,y"{Scy"{ • •)
The third hypothesis leads to
Pt = Constant {Sty"(Spy''{ • •)
Statistical Derivation of Typical Supply Functions of
More than One Variable
To fit to the observations the typical functions
(54), (55), (56), (58), (59), (60), they are first transformed
so that they relate to trendratios instead of
absolute quantities; these adjusted functions are then
put into logarithmic forms, which are fitted to the
trendratio observations by the method of least
squares. An illustration of the procedure may be
given by indicating the steps in case of (55). For
practical purposes, where all prices and all quantities
of commodities are in a state of flux, (55) becomes, by
substitution of trendratios for absolute quantities,
1=—(gr(ir()
X e* ^P'' ^pp' '. (61)
This typical trendratio function is now put into the
76 Synthetic Economics
logarithmic form
log ( "^ ) = log Constant + ju loglzr)
+ r ; . (  ) +    ) l o g « . (62)
As (62) is a linear function of the 7's it may be easily
fitted to the trendratio observations by the method of
least squares.
Relative Cost of Production (K) and Relative Efficiency
of Organization (w)
In consequence of the ceaseless changes in the conditions
of business, a representative entrepreneur is
constantly asking, and constantly answering, the
question whether he shall increase or diminish the
quantity of his physical output. His decisions are
made from the point of view of the probable movement
of demand, which is beyond his control, and
from the point of view of the efficiency of his own
organization, which he is capable of modifying. This
latter phase of the business problem relates to supply,
and the criterion upon which his decision turns should
have a technical name.
The quantities that are compared by the entrepreneur
are total cost and total physical output.
If total physical output is regarded as the independent
variable and total cost as the dependent variable, the
proposed criterion may be called the coefficient of
relative cost of production and may be represented
The Law of Supply 77
by K. Relative cost of production, K, may then be defined
as the ratio of the relative change of the total cost
to the relative change in total production. If y equals
the total cost of production, and x, the total quantity f
produced, the symbolic representation of relative cost'
of production is
or, at the limit.
Ay ILx
y I X
X dy ^. . . xé'(x)
y dx !^ ^^ '^ 0(a;)
This criterion gives the information desired by the
entrepreneur. He wishes to know, if he increase his
output, whether the relative increase in total cost will
be greater than, equal to, or less than the relative
increase in the output. That is to say, he wishes to
know whether K = 1.
If total cost is regarded as the independent variable
and total output as the dependent variable, the criterion
may be called the coefficient of relative efficiency
of organization, and be represented by the symbol co.
Relative efficiency of organization, co, is then defined as
the ratio of the relative change in total production to
the relative change in total cost. Symbolically,
dx Idy (f>(x)
X I y x4)'{x)
The information desired by the entrepreneur is
whether w = 1.
These criteria, K and co, not only will facilitate the
theoretical treatment of the laws of cost, but, in the
next chapter, will be of critical importance in statistical
studies of coefficients of production.
78 Synthetic Economics
Laws of Relative Cost and Relative Return Contrasted
with Laws of Cost and Laws of Return
A description in mathematical terms of laws of
economic return was first given by Cournot.^ Cournot
shows that ii y = (l>{x) is the expression for total cost
of production, then there are three types of laws of
cost or return, according as "{x) = 0. He avoided
many difficulties by contenting himself with a mathematical
definition of the laws without passing on to
identify them by name as the law of diminishing return,
the law of constant return, and the law of increasing
j return. Without using the customary unprecise desig
' nations, he amply made good his claim that the condition
0"(x) = 0 is of great importance in the solution
'of the principal problems of economic science.^ It
would be conducive to clearness and accuracy if the
Cournot criterion ^"(x) were regarded as the criterion
of laws of cost or laws of return.
In the preceding section the coefficient of relative
return, or of relative cost, was defined as
_x dy _ X(j>'{x)
y dx 4>{x)
^ Cournot: Recherches sur les principes mathémaliques de la theorie
des richesses, p. 66, §§ 2930. He considers, in addition to the above
three cases, a fourth, where {x) is a constant.
^Cournot: Recherches, p. 65: "Dans la suite de nos recherches,
nous aurons rarement occasion de considérer directement la function
0(D) [the 4>{x) of the text], mais seulement son coefficient différentiel
d{D)ldD que nous désignerons par la characteristique '{D). Ce coefficient
différentiel est une nouvelle function de D, dont la forme exerce
la plus grande influence sur la solution des principaux problèmes de la
science économique. La fonetion '{D) est, selon la nature des forces
productrices et des denrées produites, susceptible de croitre ou de
décroitre quand D augmente."
The Law of Supply 79
According as K = 1 we have to do with the law of increasing
relative cost, or of diminishing relative return;
the law of constant relative cost, or of constant relative
return; the law of decreasing relative cost, or of increasing
relative return. Just as Cournot's criterion
"{x) > 0, (63)
/c > 1. . (64)
But, by definition,
_ x4>'(x)
and, as a result of the inequality (64), the law of
diminishing relative return gives the information that
'{x) > ^ • (65)
That is to say, in stating the law of diminishing relative
return we assume that the marginal cost of production
is greater than the average cost of production.
The inequality in (65) may be written
xcl>'(x) > 4,{x). (66)
This states the wellknown proposition in economic
theory that where the law of diminishing relative re
80 Synthetic Economics
turn prevails, the ordinate of the integral supply curve
is greater than the corresponding ordinate of the total
cost curve. Economic theory has also led to the
conclusion that under these conditions the slope of the
supply curve is greater than the slope of the cost
curve. This means that
£{x0'(a;)i >4>\x). (67)
That is,
[0'(x) + x4>"{x)'} > '{x), or 4>"{x) > 0. (68)
Here we reach a conclusion in case of diminishing
return and diminishing relative return which, by
similar reasoning with regard to constant relative
return and to increasing relative return, may be proved
to be general, namely, where a given condition of K
holds, so likewise does the corresponding condition
with reference to 4>"{x) hold. In this particular case,
we started in (64) with K > 1 and we reached, in (68),
4>"{x) > 0. Where the law of diminishing relative
return exists, there likewise does the law of diminishing
return exist. The latter conception is at least as broad
as the former: we shall now proceed to show that it is
broader.
Auspitz and Lieben have described how a total cost
curve is made up of bits of individual expense curves.
Following their method of exposition,^ let us suppose
that an individual producer has such a plant that he
can produce, with moderate variations of outlay, a
quantity of commodity ranging between mi and m/
' Auspitz und Lieben: Untersuchungen über die Theorie des Preises,
p. 112, Fig. 27 a.
The Law of Supply 81
FIGURE 5.
(see Figure 5), and let us assume, further, that the
total cost of producing quantities between Wi and m/
is given by the respective ordinates of the curve Cic/,
which, by hypothesis, is assumed to be, throughout its
extent, convex to the y
axis of X. If 0i(x) is
put for the integral cost
curve of this producer
(1), the convexity of
the curve requires that
^\"{x) > 0, and, consequently,
that the business
be subject to the
law of diminishing return
between the limits
X = Ml and X = m/. Suppose, now, that the price
of the commodity is given by the price line oPi. At
the price tan Piox, producer (1) could not afford to
produce less nor more than oxi, for which quantity
the total cost would be yi. But when oxi units of the
commodity are produced we have
dx Xi
For any point on the cost curve between Ci and p, we
have
ax X
and, consequently, by the criterion K, the industry is
subject between these limits to the law of increasing
relative return. For any point on the cost curve
82 Synthetic Economics
between p and c/, we have
ax X
and, consequently, by the criterion K, the business between
these limits is subject to the law of diminishing
relative return.^
The result in this particular case of diminishing return
is general: The criterion 4>"{x) is more inclusive
than the criterion K: where K occurs in a particular
form the corresponding form of 4>"{^) always occurs;
but where 4>"{x) exists in any particular form, the
corresponding condition of K may or may not be fulfilled.
Cost Curves and Supply Curves; Relative Cost Curves and
Relative Supply Curves
The supply functions thus far described in this
chapter are derivable immediately from the data of
prices and quantities of commodities supplied. When
the supply curves are known, it is possible to deduce
from principles of economic theory the corresponding
cost curves. There are, however, many cases where
it is desirable to begin with cost data, to derive therefrom
cost curves, and then to deduce, by means of
principles of economic theory, the corresponding supply
curves. If this need is to be met it is clearly desirable,
i in view of the critical importance of laws of cost in
i 1 the solution of practical problems, to choose typical
1 cost functions that shall reveal the nature either of
I 4>"{x) or of K.
* This point, as far as I am aware, was first made by Edgeworth.
Economic Journal, June, 1899, p. 294.
The Law of Supply 83
We shall derive the cost functions and supply functions
first by means of 4>"{x). The simplest possible
assumptions as to the nature of "(a;) are summarized
in (69):
TO, or
(/>"(a;) =  m + m'x, or • (69)
. TO + m'x + 'm"x'^,
If {x)lx. When the law of increasing return prevails,
the supply price ^ is likewise ps = 4>{x)lx. More
complex cost functions and corresponding supply
functions could be deduced from the other assumptions
in (69).
The criterion K leads to other useful types of cost
functions and supply functions. The simplest possible
assumptions as to the nature of K are summarized in
(70):
' k, or
K =  A; + k'x, or
k + k'x + Wx" \
(70)
Suppose that K = k, a constant. Since by definition
^ Marshall: Principles of Economics, 4th edition, p. 539, note 1,
Tig. 36.
84 Synthetic Economics
the coefHcient K = {xjy) • (dy/dx), the hypothesis becomes
X dy , dy , dx
 • r = k, or — = k
y dx y X
Integrating this last equation, we have
y — Constant x'' = 4>{x), (71)
which is the law of the variation of total cost with the
quantity of commodity that is produced.
The derivation of the equation to the supply curve
from the equation to the cost curve will vary according
as, K%\, that is, according as the business is subject to
increasing, constant, or diminishing relative cost.
When the industry is subject to increasing relative
cost, K is greater than unity; the supply price p, is
equal to the marginal cost of production, ^'(x); and
the equation to the supply curve is
p, = (i)'{x) = Constant kx'''^. (72)
When the industry is subject to constant relative cost,
K = 1, and the supply price per unit of commodity is
equal to the mean cost of production, that is,
Ps = = Constant x^~^ = Constant,
X
since k = 1. (73)
When the industry is subject to decreasing relative
cost, K < 1, and the supply price per unit of commodity
is equal to the mean cost of production, that is,
V, = ^ = Constant x''~\ (74)
X
In the preceding discussion, laws of supply have
been deduced from laws of cost, and the equations
The Law of Supply 85
(72), (73), (74) show that an expression of the type
y = Constant x* is an appropriate form for the law of
supply whatever may be the constant value of K.
It will, therefore, be allowable to take this type of
function to describe the law of supply directly and
then deduce from it the corresponding law of cost.
This characteristic is valuable in treating supply and
cost concretely: When it is impossible to find statistical
data to serve as material for an empirical cost
curve, it is sometimes ^ practicable to obtain directly
from statistical data the curve of supply as a function
of one variable from which, according to the above
reasoning, the law of cost may be immediately de [
duced. 
Just as the typical equation, y — Constant z^, has
been derived from the simplest hypothesis regarding
the value of K, SO, by similar reasoning, more complex
functions may be derived from other hypotheses in
(70). If, for example, it be assumed that the variation
of K is linear, we have
and the typical equation to the cost curve, from which
the supply curve may be deduced, is
y = 4>{x) = Constant x^e^'"". (75)
Partial Relative Efficiencies of Organization
There are three classes of general functions that
appear in the theory of moving general equilibria:
functions of demand, functions of supply, and pro
* An instance is given in the following chapter under the section
"A Moving Equilibrium of Supply and Demand."
86 Synthetic Economics
duction functions. In the concrete, practical theory
of moving equilibria these three classes of functions
must be known not only in their algebraic mathematical
forms, but in the numerical definiteness of
their parameters. The studies of the foregoing and
present chapters have met these desiderata for the
demand functions and the supply functions. The
means of arriving at a knowledge of the algebraic
forms of the demand functions and supply functions
was, in the former case, the theory of partial elasticity
of demand; and in the latter case, the theory of partial
elasticity of supply. The method of fitting the general
algebraic functions to the statistical data was in both
cases the method of trendratios.
Of these three classes of general functions there
remains to be dealt with only the third, the production
functions. We have to find appropriate algebraic
forms of these functions and, then, to fit them to
empirical data. The clue to the solution of the first
problem is the theory of partial relative efficiency of
organization.
The conception, relative efficiency of organization,
which is symbolically represented by w, we have
already defined as the ratio of the relative change in
total production to the relative change in total cost.
If we suppose that Qc represents the quantity of commodity
(C) which is produced and we assume, as a
first approximation, that the cost consists only of
services of persons, which are represented in Walras'
notation by (Pc), the relation between quantity produced
and cost of services may be indicated by
Qc = *c(Po), (76)
The Law of Supply 87
and the relative efficiency of organization, according to
the above definition, would be
dQc IdP, Pe dQ, .^„^
''^'Ql/'K^Q/dP: ^^^^
If, now, total cost is made up of services of land,
services of persons, and services of capital, and the
respective quantities of these are represented by Tc,
Pc, Kc, • • •, we have as the expression for the general
production function
Q, = ^o{To, Pc, Kc, • • •). (78)
In treating this more complex function we are led, by
analogy with the reasoning employed in the study of
demand functions and supply functions, to the conception
of partial efficiencies of organization, which
may be symbolically indicated as follows:
(79)
The meaning of these coefficients is clear from their
analogy with the coefficients of partial elasticity of
demand and of partial elasticity of supply. For
example, the first coefficient in (79) is the partial
efficiency of organization with respect to the service
(Tc) when the factors employed in the production of
(C) are Tc, Pc, Kc, ••••
Qc ' dTc
^Pc dQ^
••• Qc ' dP,
^Kc dOc
'••• 0. • dK.
88 Synthetic Economics
Typical Production Functions
One of the greatest difficulties in dealing with moving
general equilibria is the problem of the determination
of the coefficients of production. In the works both of
Walras and of Pareto, these coefficients are purely
hypothetical, and both economists assume, in their
mathematical syntheses, that the coefficients are
constants. To treat moving equilibria concretely and
practically, typical functions must be found by means
of which the coefficients may be derived from statistical
data, and, moreover, the coefficients must be given in
forms that approach reality with increasing accuracy.
In the following chapter on "Moving Equilibria,"
the coefficients of production are shown to be functions
of partial efficiencies of organization; in the remainder
of this chapter, these coefficients of partial efficiency
of organization are shown to be determinable as
constants, or as simple linear functions of the corresponding
factors of production, or as quadratic functions
of the factors of production.
Typical functions of production may be derived in
exactly the same way in which foregoing studies have
derived typical functions of demand and typical
functions of supply. Partial efficiencies of organization
play the same role in production functions as
partial elasticities of demand play in demand functions,
and partial elasticities of supply, in supply functions.
Suppose, for example, that the production functions
for commodity (C) are to be determined and that the
factors of production are (Tc), (Pc), {Kc), • • •. Let a
representative partial efficiency of organization be
^ct.pk... We may then impose upon the required
The Law of Supply 89
functions the conditions that they shall give the partial
efficiencies of organization as constants, or as linear
functions of the factors, or as quadratic functions.
These conditions, in case of the representative coefficient
of partial efficiency of organization, are
(^ctpk —
€ct, or
€c( + i'ctTc, or (80)
If the partial efficiencies are to be constants, the
first of the above conditions is followed, and we obtain
as the typical production function
a = Constant {TcT%PoT''{KcT''{ • •)• (81)
If the partial efficiencies are to be given as simple
linear functions of the corresponding factors of production,
the second of the above conditions is imposed,
and the typical production function is
a = Constant {T,y'\Pcy'''{K,y'\ • •)
y g^tï'c+ecp^c+eóü^cH (82)
If the partial efficiencies are to be given as quadratic
functions of the factors of production, the third of the
above conditions is chosen, and the typical production
function is
Qo = Constant {T,y'\PcT'{K,Y\ • •)
To fit (81), (82), (83) to statistical data, the variables,
independent and dependent, are taken as trendratios
and, in this transformed shape, are fitted to the
90 Synthetic Economics
observations by the method of least squares. When
the trendratios are used as variables the three transformed
functions are
I—(tnsr(ir() <>
Xe''(v>''(r)*''(i)*, (86)
i=c—(ini^nin..)
« [ • " © •   ( & ) ' « ( t ) '  ] . ,85,
As a preliminary to the use of the method of least
squares in the evaluation of the e's, logarithms are
taken of both sides of equations (84), (85), (86).
For example, (85) becomes
log \d) = log Constant + ed log f =? j
+ . , l o g ( g ) + . * l o g (  ) + . . .
+ e ^ . ( § ) + •••]loge, (87)
which is a linear function of the e's and may, therefore,
be easily fitted to the statistical data by the method of
least squares.
The Law of Supply 91
Since the partial efficiencies may be determined
empirically as constants, or as linear functions of the
factors of production, or as quadratic functions of
t 