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[ The following memoir is in substance the writer's thesis for the degree of

Compliments of



[ Read April 27, 1892.)


[ The following memoir is in substance the writer's thesis for the degree of

Ph . D . at Yale University , 1891. ]


[ Read April 27, 1892.]

‫‪ ۱۹۰۵‬ر ‪ .‬ع‬
‫‪| 12 % 7۹۰‬‬

‫در یهبر تمر‬



‫رج ‪3‬‬



John Stuart Mill* asserted that he had left nothing in the laws of
value for any future economist to clear up. Until 1871 this state
ment doubtless had much the force of dogma. Even Jevonsmade
preliminary obeisance before proceeding to break the ground afresh
with the mathematical instrument. Jevons with characteristic can
dor expressly disclaimed finality ;t but few of his followers have

realized with his clearness and honesty the need of further analysis
along the lines which he laid down.

The truth is, most persons, not excepting professed economists,
are satisfied with very hazy notions. How few scholars of the lit
erary and historical type retain from their study of mechanics an

adequate notion of force ! Muscular experience supplies a concrete
and practical conception but gives no inkling of the complicated
dependence on space, time, and mass. Only patient mathematical
analysis can do that. This natural aversion to elaborate and intri
cate analysis exists in Economics and especially in the theory of
value. The very foundations of the subject require new analysis
and definition . The dependence of value on utility, disutility , and
commodity, the equality of utilities, the ratio of utilities, the utility
of a commodity as a function of the quantity of that commodity
solely , or of that commodity and others conjointly, are subjects, the
neglect of which is sure to leave value half understood , and the

mastery of which claims, therefore, the first and most patient effort
of the economic scientist.
These form the subject matter of the following memoir which is
a study by mathematicalmethods of the determination of value and

Much germane to the subject has been omitted because already
elaborated by others. Cases of discontinuity belong to almost

every step , to modify or extend the continuous case. But the appli
cation of this correction has been thoroughly worked out by Auspitz
und Lieben. Multiple equilibrium and monopoly value are omitted
for a similar reason .

The two books which have influenced memost are Jevons : “ The
ory of Political Economy," and Auspitz und Lieben : “ Untersuch
* Pol. Econ., Bk. III, Ch. I, $ 1.

+ Pol. Econ., Pref. 3rd ed .


ungen ueber die Theorie des Preises.” To the former I owe the idea
of marginal utility and of mathematical treatment in general, to the
latter the clear conception of the “ symmetry ” of supply and demand

and the use of rate of commodity in place of absolute commodity ,
and to both many minor obligations.
The equations in Chapter IV , § 10 , were found by me two years
ago, when I had read no mathematical economist except Jevons.

They were an appropriate extension of Jevons' determination of
exchange of two commodities between two trading bodies to the
exchange of any number of commodities between any number of
traders and were obtained as the interpretation of the mechanism
which I have described in Chapter IV . That is, the determinate
ness of the mechanism was expressed by writing as many equations
as unknowns. These equations are essentially those of Walras in
his Éléments d' économie politique pure. The only fundamental

differences are that I use marginal utility throughout and treat it as

a function of the quantities of commodity, whereas Professor Walras
makes the quantity of each commodity a function of the prices.

That similar results should be obtained independently and by sepa
rate paths is certainly an argument to be weighed by those skeptical
of the mathematicalmethod. It seemed best not to omit these ana

lytical portions of Part I, both because they contribute to an under

standing of the other portions of the work and because they were in
a proper sense my own .
Three days after Part II was finished I received and saw for the

first time Prof. Edgeworth 's Mathematical Psychics. I was much
interested to find a resemblance between his surface on page 21 and

the total utility surfaces* described by me. The resemblance , how
ever, does not extend far. It consists in the recognition that in an
exchange, utility is a function of both commodities (not of one only

as assumed by Jevons), the use of the surface referred to as an inter
pretation thereof and the single phrase (Math . Psych., p. 28) “ and
similarly for larger numbers in hyperspace” which connects with
Part II, Ch . II, $ 5 .
There is one point, however, in which, as it seems to me, the

writer of this very suggestive book has gone far astray. Mathe
* His result, which translated into my notation is
| dU

/ AU



(da ) (ab,) - (2B )(and) = 0,
laA .

becomes by transposition and division identical with part of the continuous pro
portion , Part 1, Ch . IV , $ 3.

matical economists have been taunted with the riddle : What is a
unit of pleasure or utility ? Edgeworth, following the Physiological
Psychologist Fechner, answers : “ Just perceivable increments of
pleasure are equatable ” (p. 99). I have always felt that utility
must be capable of a definition which shall connect it with its posi

tive or objective commodity relations. A physicist would certainly
err who defined the unit of force as the minimum sensible of mus
cular sensation . Prof. Edgeworth admits his perplexity : “ It must

be confessed that we are here leaving the terra firma of physical
analogy ” (p. 99). Yet he thinks it is “ a principle on which we are
agreed to act but for which it might be hard to give a reason ;"
and again : [ such equality ] “ it is contended , not without hesitation
is appropriate to our subject.”

This foisting of Psychology on Economics seems to me inappro
priate and vicious. Others besides Prof. Edgeworth have done it.
Gossen * and Jevons appeared to regard the “ calculus of Pleasure
and Pain ” t as part of the profundity of their theory. They doubt
less saw no escape from its use. The result has been that “ mathe

matics” has been blamed for “ restoring the metaphysical entities
previously discarded .” I

These writers with Cournot,$ Menger,|| and Marshall appear to
me to have contributed the most to the subject in hand. With the
exceptions noted I have endeavored not to repeat them but to add a
little to them , partly in the theory of the subject and partly in the

mode of representing that theory. Readers to whom the subject is
new will find the present memoir exceedingly condensed. In the
attempt to be brief, the possible uses of the diagrams and mechan

isms have been merely sketched , and elaborate explanations and
illustrations have been omitted. I have assumed that my readers are
already familiar with (say) Jevons, Walras,Menger or Wieser where

illustrations and explanations regarding “ final utility ” abound.
Much of Part II and Appendix Imay notbe thoroughly intelligible
to those not familiar with higher geometrical analysis. These parts
are made as brief as possible.
My especial thanks are due to Prof. Gibbs and Prof. Newton for
valuable criticism .

Yale University, May, 1892 .

* Menschlich Verhehr., Braunschweig , 1854.
# Dr. Ingram .

+ Jevons, p . 23, also pp. 8– 9.

S Théorie des Richesses, Paris, 1838.

| Volkswirthschaftslehre, Wien , 1871. [ Prin . of. Econ., Macmillan , 1890.


CHAPTER I. Utility as a Quantity.

§ 1. Psycho -economic postulate,




§ 2. Definitionsof equal and unequal utilities,





§ 3. Application of these definitions to an equation of



§ 5.
$ 6.
§ 7.
§ 8.
§ 9.
$ 10.





. . . . . .


§ 4. Definition of the ratio of two utilities,

Analogy with other mathematical definitions, .
Marginal utility , unit ofutility,
Exact meaning of a utility curve, . . .
Total utility, utility -value, gain or consumer's rent,
Element of time,
Utility curves for a time-period and for an instant,







$ 11. A yearly utility curve implies no exact foreknowledge
of amounts consumed , . . . . . .
$ 12. Theory of probabilities partially eliminates sporadic
influences except as regards “ gain ,” . . . .
$ 13. Infinitesimals appropriate though non -existent,
$ 14 . The preceding definitions ofutility absolve the econ

omist from psychologicaland metaphysical disputes,

$ 15. Summary of Chapter 1, ·

· · · ·

CHAPTER II. Mechanism .


Mechanicalanalogy common in economic thought, .
Ideal suppositions concerning themarket,
A new utility diagram . Its related cistern (Fig. 2),
A single commodity and a single consumer, . . .

§ 5. One commodity — one producer; disutility diagram


6. One commodity - many consumers (Fig. 4),


and cistern (Fig. 3),



§ 7. Analytical interpretation of S6 ,
$ 8. Aggregate commodity, . .





§ 9. One commodity - many producers (Fig . 5),
§ 10. Preceding simply gives exactitude to the common
imagery of thought,







CHAPTER III. One consumer (or producer)- many commodities.

$ 1. Distribution of income,
S2. Mechanism (Fig . 6), .







83. The three determining conditions of such equilibrium ,
$ 4 . Analytical interpretation of g3,






. .


§ 5 . Aggregate income, . . . . .
$ 6. One producer - many commodities (Fig. 7),


CHAPTER IV . m commodities— n consumers (or producers).

§ 1. Relative points of view of Chapters II, III, and IV, .
§ 2 . These points of view compared exactly and analyti
cally ,








§ 3. Equality of marginal utilities now gives way to their




§ 5. Equilibrium automatic , . . .
$ 6 . The various magnitudes represented ,


§ 7. Two simple relations of these magnitudes,
§ 8. Complex interdependence traced ,
1. Equal incomes, · · · · ·
2. One income increased , . . . .
3. One income increased , one decreased, .

4. One commodity increased , .


. . . . . .

proportionality ,

§ 4. Mechanism (Figs. 8, 9, 10, 11), .





5. One commodity increased with resulting in




7. All incomes increased , .



come-increase, .



8. Change in individual characteristics, .
§ 9. Aggregate cisterns impossible , . . . .
$ 10. Analytical,

$ 11. m commodities - n producers (Figs. 12, 13),


· · · · · ·


6 . All commodities increased , .





CHAPTER V . Production and consumption combined .

8 1. Interdependence of production and consumption must
not be overlooked , .

§ 2. Assumptions,










$ 3. One individual consuming just that quantity of a given

commodity which he himself produces (Fig. 14), .
$ 4. Analytical, .



$ 5 . n individuals- m commodities (Figs. 15 , 16 ),










. ..




83. Further subdivision of production, . . . .


$ 6. Analytical, .











CHAPTER VI. The component processes of production .

$ 1. Retailing (Fig. 17),


. . .

82. Analytical, .






§ 1. Introduction , . . . .
§ 2. Interdependence of utilities common ,
§ 3. Effect on the cistern-analysis, . . .
4. Competing and completing commodities defined,
5 . Cistern -analysis correct but incomplete, . .
§ 6. The third definition of utility (Part 1, Ch. I), . .

· · · · · · · · · · ·


CHAPTER I. Two commodities with interrelated utilities .



Two commodities,
§ 8. Impossible combinations of two commodities, . ..
$ 9 . Curves of utility ,
. .
$ 10. Indifference curves, . . . . . .
S 11. Partial income line tangent to indifference curve,


$ 12. Different sensitiveness of competing and completing
commodities, .








$ 13. Good and poor qualities of the same commodity,


S 14. “ Maximum directions,"






$ 15 . “ Maximum directions” of all consumers parallel,

. . . .
S 16 . Production ,
$ 17. Production and consumption , .
CHAPTER II. Three or more commodities.
§ 1. Construction in three dimensions,

.. .
. .

§ 2. Complicated dependence of utilities,

. ..
. .





§ 3. Equilibrium if only three commodities exist, .


8 4. Passage to two dimensions by plane sections, .
§ 5. m dimensions requisite , ·
$ 6 . m dimensions used , .


§ 7. Conditions of equilibrium ,
§ 8. Passage to lower dimensions by “ fat ” sections,
$ 9. Analytical (vector analysis ), . . .

$ 10 . Peculiar cases ofmarginalutility, .





$ 1. No need to compare two persons' utilities, .


$ 2. How such a comparison might be made, .


CHAPTER III. Mechanical analogies.

$ 1. Mechanical analogies discussed ,
82. Analogies in parallel columns, .
CHAPTER IV . Utility as a quantity .



S 3. How it could be applied to statistics, . . .
84. Definition (3 ) (Part I, Ch. I) unessential in Part II , .

$ 5 . “ Maximum directions” alone important. .



$ 6 . Integration of total utility probably impossible for in
terrelated utilities, . . . . . . .
87. Arbitrary constants unessential,
. . . .

$ 8. Four attributes of utility unessential in the present in
vestigation ,












I. Failure ofequations,

. . . . . . . . .


II. The cisterns and diagrams of Part I compared with the diagrams of

Jevons and of Auspitz und Lieben .

$ 1. Possible geometrical representations of commodity and



III. Gain a maximum ,
$ 1. For one individual, . . . . .
$ 2. For one commodity and in what sense true,





. . . . . .

$ 7. Total utility and gain ,

· · · · ·

$ 5. Properties essential to the cisterns, .
$ 6. Meaning of the abscissa, . . .

· ·

utility, . . . . . . . . .
$ 2. Scheme comparing the cistern -coördinates with those
of Jevons and of Auspitz und Lieben , . . .
$ 3. A linear assumption , . . . . . . .
$ 4. The relative value of the diagrams, . .


$ 3. For whole market and in what sense true,
$ 4. Under what conditions would the totalmarket gain be
maximum if we could obtain the “ true” equivalence
between two persons' utilities,





IV . Elimination of variables, . . . . . . . . .
Each price is the quotientof two determinants, and all equa


tions can be reduced to a single set involving commodities
§ 1. The suppositionswere ideal, . . . . .

§ 2. Utility a function of many variables,

§ 3. Articles not homogeneous nor infinitely divisible,





$ 4. Discontinuity in time,
§ 5 . Statics and Dynamics ,
. .
§ 6 . Population ,








$ 7. No perfect individual freedom to stop producing (or
consuming) at any point,


§ 8. No perfectknowledge of prices,



. . . .





§ 9. Production different from consumption in many im
portant respects, . . . . . . . .
$ 10. Marginal utility and disutilitymay occasionally vary
in a manner opposite to that which has been sup
posed , . . . . . . . . .



$ 11. Markets are not isolated and there is no perfect
market, .












§ 1. Its utility is of samekind as the utility of mathemat
ical method elsewhere, . . . . . .
§ 2. Mathematicalmethod distinguished from mathemat
ics, .










$ 3. Utility of mathematical method is relative to the
§ 4. Mathematicalmethod is for higher economics, . .
$ 5. Has mathematicalmethod come to stay ? . . .
§ 6 . What has it already accomplished ? . . . .
§ 7. Quotations in favor of mathematical method from



Whewell, Cournot, Gossen , Jevons, Walras, New
comb, Launhardt, Wicksteed , Foxwell , Auspitz und

Lieben , Edgeworth, Marshall, and Cunynghame, .
$ 8. Quotations opposed to mathematical method from


Anonymous, Cairnes, Wagner, Ingram , and Rab
berio , with comments , .
§ 9. A reason for opposition , .
§ 10. Conclusion ,
. . .








§ 1. Scope of bibliography,








82. List selected from Jevons' bibliography, .
3. Extension of Jevons'bibliography, . .






$ 1.
The laws of economics are framed to explain facts. The concep

tion of utility has its origin in the facts of human preference or
decision as observed in producing , consuming and exchanging goods
and services.

To fix the idea of utility the economist should go no farther than
is serviceable in explaining economic facts. It is not his province to
build a theory of psychology. It is not necessary for him to take
sides with those who wrangle to prove or disprove that pleasure and

pain alone determine conduct. These disputants have so mangled
the ideas of pleasure and pain that he who follows them and their

circular arguments finds himself using the words in forced senses.
Jevons makes utility synonymous with pleasure. Cairnes* objects
and claims that it leads to a circular definition of value. The circle
is however at the very beginning and vitiates psychology not eco
nomics ; the last dollar's worth of sugar (we are told ) represents the
same quantity of pleasurable feeling as the last dollar's worth of
dentistry. This may be true as a mere empty definition, but we
must beware of stating it, as a real “ synthetic proposition,” f or of

connecting it with the mathematics of sensations| as did Edgeworth.
The plane of contact between psychology and economics is desire.
It is difficult to see why so many theorists endeavor to obliterate the

distinction between pleasure and desire .|| No one ever denied that

economic acts have the invariable antecedent, desire. Whether the
necessary antecedent of desire is “ pleasure ” or whether indepen

dently of pleasure it may sometimes be “ duty ” or “ fear ” concerns
a phenomenon in the second remove from the economic act of choice

and is completely within the realm of psychology.
Wecontent ourselves therefore with the following simple psycho
economic postulate :
Each individual acts as he desires.
* Pol. Econ., p . 21.
+ Kant, Critique Pure Reason , Introduction .
# Ladd, Physiological Psychology, p. 361.
See above (Preface).

| See Sidgwick , Methods of Ethics, Chap. IV .


Irving Fisher - Mathematical investigations
$ 2.

The sense in which utility is a quantity is determined by three

(1) For a given individual at a given time, the utility of A units
of one commodity or service (a) is equal to the utility of B units of
another (6), if the individual has no desire for the one to the exclu
sion of the other.
A and B are here used as numbers. Thus if the first commodity
is sugar and the second calico and if the individual prizes 2 pounds

of sugar as much as 10 yards of calico , A is 2 and B is 10.
(2 ) For a given individual, at a given time, the utility of A units
of (a ) exceeds the utility of B units of (6) if the individual prefers
(has a desire for) A to the exclusion of B rather than for B to the
exclusion of A . In the same case the utility of B is said to be less
than that of A .

The third definition will be given in $ 4.
The two preceding definitions are exactly parallel to those of any
other mathematical magnitude.

Thus: two forces are equal if at the same time they alone act on
the same particle in opposite directions and no change of motion
results. One is greater when additional motion is produced in its
direction. Again : “ two masses are equal which if moving with

equal velocities along the same straight line in opposite directions
and impinging on each other are reduced to rest by the collision." *
Two geometricalmagnitudes are equal if they can be made to coin
cide, etc., etc.

Just as coincidence is the test of equality and inequality of geo
metrical figures, and the tip of the scales the test of equality and

inequality of weights, so is the desire of the individual, the test of
•the equality and inequality of utilities. It is to be noted that in
each definition of equality the word “ no ” or some equivalent
occurs. A standard mode of cancellation is thus designated .
$ 3.

Let us see how these definitions of utility apply to an act of pur
chase. An individual I enters a market with fixed prices to ex
change some of a commodity (a ) for another (6). We may sup
pose prices to be such that he gives one gallon of (a) and receives two
bushels of (6 ) , then a second gallon for two more bushels and so on

* Price , Calculus, vol. iii, p . 316 .

in the theory of value and prices.


until finally he has given A gallons and received B bushels. At
what point does he stop ?

Although the “ exchange values ” of A gallons of (a) and B
bushels of (6) are equal, their utilities (to I) are not. He prefers B
to the exclusion of A , for his act proves his preference (postulate).
Therefore by definition (2 ) the utility of B exceeds that of A .
Wemay write :

ut. of B > ut. of A .
Why then did he cease to buy (6) ? He sold exactly A gallons for
B bushels. By stopping here he has shown his preference to buy
no more (postulate). Ergo the utility of a small increment, say

another bushel of (6 ) is less than the utility of the corresponding
number of gallons of (a ) (Def. 2). Likewisehe prefers to buy no less.
Ergo the utility of a small decrement, say one less bushel is greater

than the gallons for buying it. Now by the mathematical principle
of continuity , if the small increment or decrement be made infinites
imal dB, the two above inequalities become indistinguishable, and

vanish in a common equation , viz :
ut. of dB = ut. of dA

dB and dA are here exchangeable increments. But the last incre
ment dB is exchanged for dA at the same rate as A was exchanged
for B ; that is


B - dᏴ

where each ratio is the ratio of exchange or the price of B in terms
of A .

B =- dAA


multiplying this by the first equation , wehave :
t. ooff dA
ut. of dB . B
. A

which may be written :*

ab . B == d
JAu A .

The differential coefficients here employed are called by Jevons
“ final degree of utility ,”! and by Marshall “ marginal utility.” I
Hence the equation just obtained may be expressed: For a given
* Cf. Jevons, Pol. Econ ., p . 99.
# Marshall, Prin . of Econ. , Preface, p . xiv.

Jevons, Ibid ., p. 51.


Irving Fisher - Mathematical investigations

purchaser at the time of purchase the quantity of the commodity
purchased multiplied by its marginal utility equals the like product

for the commodity sold . Or again : for a given purchaser the utili
ties of A and B , though actually unequal would be equal if every
portion of A (and also of B ) were rated at the same degree of utility
as the last infinitesimal. This hypothetical equality underlies, as
will subsequently appear, the notion of the equality of values of A
and B .

§ 4.

But the two definitions (1) and (2) do not fully determine the
sense in which utility is a quantity. To define when the “ grades ”
of two parts of a highway are equal or unequal (viz : when they

make equal or unequal angles with a horizontal), does not inform us
when one shall be twice as steep as the other. It does not oblige us
to measure the “ grade ” by the sine of the angle of elevation , or
by the tangent, or by the angle itself. If the two highways were
inclined at 10° and 20° respectively , the “ grades ” have a ratio of

1.97 if measured by sines, of 2:07 by tangents , and exactly 2 by
angles. For a long time philosophers could define and determine
when two bodies were equally or unequally hot. But not till the
middle of this century * did physicists attach a meaning to the phrase
" twice as hot.”
It is here especially that exactitude has been hitherto lacking in

mathematical economics.

Jevons freely confesses that “ We can

seldom or never affirm that one pleasure is an exact multiple of
another.” +

Now throughout Part I the assumption is made that the utility of

any one commodity (or service ) depends on the quantity of that
commodity or service, but is independent of the quantities of other
commodities and services. This assumption is preliminary to the
definition we seek .

Our first problem is to find the ratio of two infinitesimal utilities.
If an individual I consumes 100 loaves of bread in a year the utility

of the last infinitesimal, or to fix our ideas, the utility of the last
loaf is (presumably) greater than what it would be if he consumed

150 loaves. What is their ratio ? It is found by contrasting the
utilities of the 100th and 150th loaves with a third utility. This
* The first thermodynamic definition of one temperature as a multiple of

another was made by W . Thomson in 1848. See Maxwell, Theory of Heat, p .
155 .
7 p . 13.

in the theory of value and prices.


third utility is that of oil (say) of which let B gallons be consumed
· by I during the year. Let ß be that infinitesimal or small increment

of B whose utility shall equal that of the 100th loaf. Now in sub
stituting the hypothesis of 150 loaves let us not periit our individ

ual to alter B , his consumption of oil.* The utility of the 150th
loaf will be pronounced by him equal (say) to the utility of } B.
Then the utility of the 150th loaf is said to be half the utility of
the 100th .

That is, if :
ut. of 100th loaf = ut. of B ,
and ut. of 150th loaf = ut. of B2,
the ratio is defined :
ut. of 100th loaf

B being the total,
B being the total again ,

ut. of 150th loaf 8 /2
It is essential to observe that if the 100th loaf is twice as useful
as the 150th when their ratio is defined as above in terms of incre
ments of oil, it will also be twice as usefulwhen the ratio is defined

by any other commodity ; also that it matters not what total quan
tity ( B ) of oil or other commodity is employed .
This theorem may be thus stated :
Given (1) ut. of 100th loaf = ut. of B , . B
and (2 ) ut. of 150th loaf = ut. of B /2,
also (3) ut. of 100th loaf = ut. of y ,
To prove ut. of 150th loaf = ut. of y /2 ,

being total,
being total,
being total,
being total,

where C is the quantity of another commodity (c) consumed by I in
the same period and y is such an increment of C that its utility
shall equal that of the 100th loaf.

Wemay write from (1) and (3):
ut. of 100th loaf = ut. of B = ut. of y ,

( 100 loaves, B and C, being totals).
Now , if the first total (100 loaves) be changed to 150, B and C being

unchanged , the above equation, dropping the firstmember, will still
be true, viz :
ut. of B = ut. of y ,

( 150 loaves, B and C , totals),
for, by our preliminary assumption these utilities are independent of
the quantity of bread.
* As a matter of fact an individual who, if consuming 100 loaves of bread
would consume B gallons of oil might, if consuming 150 loaves , use also more
oil. But this fact in no wise hinders our inquiring how he would reckon utili
ties if he used the same amount.


Irving Fisher — Mathemat
ical nvestigations
Since B and y are infinitesimal it follows from the mere mathe

matical principle of continuity that :
ut. of B /2 = ut. of y /2 ,

(B , C , totals),

. : by (2 )

ut. of 150th loaf = ut. of y /2 ,
(150 loaves, C, totals)

Q. E, D.

Hence our definition becomes :
ut. of 100th loaf
ut. of 150th loafy 2
Likewise :

ut. of 100th loaf

ut. of 150th loaf = 92 = 2,
etc., etc.,

all of which results harmonize .
Since C is any arbitrary quantity it follows that the definition of
the above ratio is independent not only of the particular commodity
employed as a means of comparison but also of the total quantity of

that commodity .
It is to be noted here that if the utility of one commodity were
dependent on the quantities of others, two applications of the defini

tion would yield discordant results.*

Wemay state our definition in general terms as follows:
( 3 ) The ratio of two infinitesimal utilities is measured by the ratio

of two infinitesimal increments of the same commodity respectively
equal in utility to the two utilities whose ratio is required, provided

these increments are on the margin of equal finite quantities :

In general symbols this becomes :
ut. of dA -

= n :-- if ut. of dA = ut. of ndM

ut. of dB

(M total),
and ut. of dB = ut. of AM

(M also total),
where n is any finite number, positive or negative, whole or frac

This definition applies not only to infinitesimal utilities of the
same commodity (as of the 100th and 150th loaves of bread) but to
those of different commodities or services.

* We shall afterward see how this affects our notions of utility (Part II,
Ch. IV,

in the theory of value and prices.

$ 5.
Definition (3) is perfectly analogous to other mathematical defini

tions. To define equality of forces does not fix their proportionality.
This property is found in the additional definition : “ The ratio of

two forces is the ratio of their mass-accelerations.”

Before me

chanics was a science, “ force ” stood for a “ common sense ” notion
resolvable in the last analysis into a muscular sensation felt in push

ing and pulling.* But to construct a positive science, force must
be defined with respect to its connection with space, time and mass.
So also, while utility has an original “ common sense ” meaning
relating to feelings, when economics attempts to be a positive
science, it must seek a definition which connects it with objective

$ 6.

(4 ) The marginal utility of a commodity (as implied in § 3) is the
limiting ratio of the utility of the marginal increment to themagni
tude of that increment. Hence the ratio of two marginal utilities is
the ratio of the utilities of two marginal increments divided by the
ratio of these increments .
If the units of the commodity are small, the marginal utility is

practically the utility of the last unit - for bread, of the last loaf,
but if this loaf is sliced into 10 parts and these slices have different
utilities, the marginal utility of bread is more nearly the utility of
the last slice divided by it , and so on ad infinitum .

It is now an easy matter to find a unit of utility, the lack of
which has been the reproachſ of mathematical economists. The
utility of the 100th loaf per year may be regarded as the unit of

utility. Or in general:
* Spencer, First Principles, p . 169.

+ Jevons, Marshall, Gossen , and Launhardt, omit indicating in any way what
they mean by the ratio of utilities . Yet each of them enabody the idea in their

diagrams. Edgeworth (Math . Psych., p . 99) thinks “ just perceivable increments
[of pleasure ] are equatable ” and uses this “ minimum sensible ” as a unit in
terms of which any pleasure is to be measured (in thought at least). His defini

tion and mine show perhaps the very point of departure between psychology
and economics. To measure a sensation , theminimum sensible is perhaps the
only thinkable method (see Ladd, Physiological Psychology , p . 361). Here the

phenomenon is subjective and so is its measure ; while in economics the phe
nomena are objective and likewise their measure .

† Dr. Ingram , Article : Pol. Econ ., Ency . Brit., xix, 399.

JULY, 1892.

Irving Fisher — Mathematical investigations


(5 ) The marginalutility of any arbitrarily chosen commodity on
the margin of some arbitrarily chosen quantity of that commodity
may serve as the unit of utility for a given individual at a given

This unit may be named a util.
Any unit in mathematics is valuable only as a divisor for a

second quantity and constant only in the sense that the quotient is
constant, that is independent of a third quantity. If we should
awaken to -morrow with every line in the universe doubled, we
should never detect the change, if indeed such can be called a

change, nor would it disturb our sciences or formulæ .

$ 7.
With these definitions it is now possible to give a meaning to
Jevons' utility curve, whose abscissas represent the amounts of a
commodity (say bread) which a given individual might consume

during a given period and the ordinates, the utilities of the last (i. e.
the least useful) loaf. For if corresponding to the abscissa 100
loaves an ordinate of arbitrary length (say one inch ) be drawn to
stand for the utility of the 100th loaf, we may use this as a unit
(util. For any other abscissa as 85 loaves whose marginal utility
is (say) twice the former, the ordinate must be two inches, and so

on. For any other commodity as oil the marginal utility of A

gallons being contrasted with the utility of the 100th loaf of bread
and this ratio being (say) three, an ordinate of three inches must be
drawn. In all the curves thus constructed only one ordinate is
arbitrarily selected , viz : that representing the utility of the 100th

$ 8.

Only differentials of utilities have hitherto been accounted for.

To get the total utility of a given amount of bread we sum up the
utilities for the separate loaves. Or in general:

(6 ) The total utility of a given quantity of a commodity at a
given time and for a given individual is the integral of the mar

ginal utility times the differential of that commodity.
That is :

ut. of (x ) = ut. (dx,) + ut. (dxn) + . .. . . + ut. (dan)

= S.*ut.(dove)

in the theory of value and prices.


(7 ) The name UTILITY-VALUE of a commodity may be given to the

product of the quantity of that commodity by its marginal utility or
2 . doc

The name is suggested from money-value which is quantity of
commodity times its price. (Cf. § 3).
(8 ) The GAIN or consumer's rent is total utility minus utility value.
That is :

Gain = / dU ,


Gain = % de dx – X . die
It is the actual total utility diminished by that total utility which
the commodity would have if it were all rated at the same degree of
utility as the last or least useful increment.

It is to be observed that total utility and gain are not experiences
in time but the sum of increments of utility substitutionally succes
sive. The individual is to assign the marginal utility for the 90th

loaf on thehypothesis that hewere consuming 90 loaves per year, and
then abandoning this supposition to substitute successively the hy

pothesis of 91 loves, 92, 93, etc., all for the same year. That is, a
number of mutually exclusive hypotheses for the same period are
thought of.
$ 9.

The preceding definitions have been expressed relative to a par
ticular instant of time. This was because in actual life purchases
are made by separate instantaneous acts. But the important com
modity-magnitudes in economics are “ tons per year,” “ yards per

day,” etc., bought, sold , produced , consumed . In order to make our
definitions applicable to such quantities the element of timemust be

introduced. Hence the following supposition :

During the given period of time (that is, the period for which
commodity -magnitudes are considered) the marginal utility to a
given individual of a given commodity is the same at all instants at

which he buys or consumes it or sells or produces it.
This involves supposing that prices do not vary, for prices (as we
shall see) are proportional to marginal utilities.

A housewife buys (say) 10 lbs. of sugar at 10 cts. per pound. As
she closes the bargain she roughly estimates that the last or tenth

pound is about worth its price. She did not stop at five pounds for

she wanted a sixth more than the 10 cts. it cost her. She may not
buy sugar again for a fortnight. When she does, we shall suppose


Irving Fisher - Mathematical investigations

the price to be the same, so that the last pound she then buys has
the same utility as the last pound she previously bought. She may

buy fifteen pounds. A fortnight later only five, all depending on
her plans for using it. The whole yearly purchase may be 250 lbs.
and wemay write :

ut. ( 10th 1b .) Jan. 1 = ut. (15th lb .) Jan. 15 .
= ut. ( 5th lb .) Jan. 30.
= etc.

== ut. (250th lb.) for whole year.
Thus : The marginal utility of a certain quantity of a commodity
for a given period (say a year) is defined to be the marginal utility
of that commodity on all occasions during that yeur at which it is
bought or consumed , the sum of the individual purchases being the

given yearly purchase and consumption .
§ 10.

In the hypothetical case the marginal utility of 250 pounds per
year equalled the marginal utility of 10 cts. In the same manner

wemay practically estimate the marginal utility of 200 pounds by
supposing the price to be such that our housewife would buy 200
pounds. Thus a number of alternative suppositions are made for
the same period. By means of these a utility curve can be con
structed, one of the coördinates of which is the yearly consumption
of sugar. To do this statistically is of course quite a different and

more difficult though by no means hopeless proceeding.
Curves of this nature are the only ones to be here considered .

But it is clear that there also exist utility curves for each time of
purchase.* These would differ both from the “ yearly ” curve as
well as from each other .
$ 11.

To meet a possible objection it must be pointed out that the use
of a “ yearly ” utility curve assumes no nice calculation on the part
of the individual as to his future incomeand receipts. Hemay even

be and generally is totally ignorant of the number of pounds of
butter he consumes per year. He creeps along from purchase to
purchase and only at these individual acts does he estimate his needs

and his abilities. Yet if he always completes his purchase with the
sameestimate of marginal utility as measured against other com
* They would be the curves of Fleeming Jenkin : Graphic Representation of
Supply and Demand . Grant's Recess Studies, p . 151, Edinburgh , 1870.

in the theory of value and prices.


modities , this must be the marginal utility for the year and the total

yearly purchase is the quantity which bears this marginal utility.

This marginal utility or “ final” degree of utility of the commodity
for the year is clearly not the utility of the last amount chrono
logically (that is Dec. 31), but the utility of the least useful part
of any and each of the separate purchases.
$ 12 .

It may further be objected that there is a fitful element in the

problem which the above supposition ignores. We have supposed
prices do not vary during the given period and also that the indi

vidual's utility -estimate does not vary . It may justly be claimed

that not only do prices vary from day to day, but even if they did
not, the individual's estimate of utility is fitful and, although at
the instant he closes a bargain his estimate of utility must be
regarded as corresponding to the given price, yet he is likely gen

erally and certain sometimes to regret his action so that if he were

to live the year over again he would act very differently .
This objection is a good illustration that a microscopic view often
obscures the general broad facts. As a matter of fact the use of a
period of time tends to eliminate those very sporadic elements
objected to. First, though prices vary from hour to hour under the

influence of excitement and changing rumors, and from season to

season under causesmeteorological and otherwise, yet these fluctua
tions are self-corrective. The general price through the year is the
only price which is independent of sporadic and accidental influences.

This general price is not the arithmetical mean of the daily prices
but a mean defined as such that had it been the constant price during

the period the amounts bought and sold would have been just what
they actually are. Secondly, the individual caprice is self-correc

tive. If a man lays in too large a stock of provisions this week he
will buy less next.

The theory of probabilities therefore substan

. tially harmonizes the theoretical and the actual. The apparently
arbitrary suppositions regarding constancy of price, etc., may be

looked upon as convenient definitions of an ideal average as just
described .

One observation however must not be overlooked . Although
accidental variations of price or choices of caprice afford both posi
tive and negative errors and thus largely cancel each other, yet the

effect on the total utility and the gain is always to diminish them .
To buy too much or too little, to sell too cheap or too dear will be


Irving Fisher - Mathemati


equally sure to diminish gain.



Herein lies the virtue of insur

ance and the vice of gambling.

Neither alters (directly) the

amount of wealth. But insurance modifies and gambling intensi

fies its fluctuations. Hence the one increases the other decreases
gain .
$ 13.

Again it may be objected to the foregoing definitions that the use

of infinitesimals is inappropriate since an individual does not and

cannot reckon infinitesimals. The same apparent objection attaches
to any application of the calculus. We test forces by weights but

cannot weigh infinitesimal masses nor do they probably exist; yet
the theory of forces begins in infinitesimals. We apply fluxions to the
varying density of the earth , though we know that if we actually
take the infinitesimal ratio of mass to volume we shall generally get

zero since matter is discontinuous. The pressure of a confined gas
is due to collisions of its molecules against the containing vessel.
As each molecule rebounds the change of momentum divided by the
infinitesimal time is the pressure. Yet at any actual instant the

value of this fluxion is quite illusory. But these facts do not mil

itate against the use of fluxions for a thinkable theory of forces,
density and gaseous pressure . In cases of discontinuity fluxions

have important applications though infinitesimals may not exist .
The rate of increase of population at a point in time is an impor

tant idea, but what does it mean ? It is convenient to define it as
infinitesimal increase of population divided by the infinitesimal time
of that increase though we know that population increases discon

tinuously by the birth of whole individuals and not of infinitesimals.
Practically we can find the approximate

marginal utility of a commodity just as

we approximately find the rate of increase
of population by taking small increments
in place of infinitesimals.

In actual fact inequality of utilities is
the rule and absolutely equal utilities never
exist. Instead of a curve of utility we
should draw a belt (fig . 1) whose limits

are ill-defined and whose width in general
depends on the amount of antecedent atten

tion which the individual has bestowed on the alternative amounts

and modes of consumption .

in the theory of value and prices.


$ 14.

Utility as defined in the preceding sections does not involve the
economist in controversy as to the laws of the subjective states of
pleasure and pain , the influence of their anticipation as connected
with their probabilities,* the vexed questions whether they differ in

quality as well as in intensity and duration, whether duty can or
cannot exist as a motive independently of pleasure, etc .

It does not follow that these discussions have no meaning or im
portance . Doubtless pleasure and pain are connected with desire
and doubtless they have an important biological and sociological

function as registering “ healthful” or “ pathological” conditions.
But the economist need not envelop his own science in the hazes of

ethics, psychology, biology and metaphysics.

Perhaps utility is an unfortunate word to express the magnitude
intended. Desirability|| would be less misleading, and its opposite ,
undesirability is certainly preferable to dis -utility. “ Utility ” is the

heritage of Bentham and his theory of pleasures and pains. For us
his word is the more acceptable, the less it is entangled with his
$ 15.

This chapter may be thus summarized :

Postulate : Each individual acts as he desires.
Definitions of utility .
(2) and (1) ut. of A ? ut. of B
if the given indiv. at the given time
prefers A to B or neither.
ut. of dA

ut. of dB =

if ut.ofdA = ut. of ndM ( M total)
and ut. of dB = ut. of dM (M also total).
JA = Marginal utility .

= Unit of utility (util.) (A being given).
+ Jevons, p . 28, etc.
* Jevons, p . 72.
† Darwin , Descent Man , I, p . 76 , Sidgwick , Methods Ethics, Ch . IV .
§ Marshall, Prin . of Econ., p. 181, Spencer, Data of Ethics, p. 79 , L . Stephen ,
Science of Ethics, p. 366 .

| Marshall, Prin . of Econ., p. 306.


Irving Fisher - Mathematical investigations
CA du

- . dA = Total utility.

A du
pAdU dA - A
dAdA - A
Assumption :
Corrollaries : From

. JA

= Utility-value.
= Gain .


= Function of A only.

(1) and (2) and postulate , when B is ex

changed for A
B : B , DdU

CdA · A .
From (3) and assumption , in the equation : ut. of

dA /ut. of dB = n , the value of n is independent
of the particular commodity and of its quantity
M used in the definition.

$ 1.
Scarcely a writer on economics omits to make some comparison
between economics and mechanics . One speaks of a “ rough cor

respondence” between the play of “ economic forces ” and mechan
ical equilibrium . Another compares uniformity of price to the level
seeking of water. Another (Jevons) compares his law of exchange
to that of the lever. Another (Edgeworth ) figures his economic
“ system ” as that of connected lakes of various levels. Another

compares society to a plastic mass such that a “ pressure” in one
region is dissipated in all “ directions.” In fact the economist bor
rowsmuch of his vocabulary from mechanics. Instances are : Equi

librium , stability, elasticity, expansion, inflation, contraction, flow ,
efflux, force, pressure, resistance, reaction, distribution (price ), levels,
movement, friction .

The student of economics thinks in terms of mechanics far more
than geometry, and a mechanical illustration corresponds more fully

to his antecedent notions than a graphical one. Yet so far as I know ,
no one has undertaken a systematic representation in terms of me
chanical interaction of that beautiful and intricate equilibrium which
manifests itself on the “ exchanges ” of a great city but of which the
causes and effects lie far outside.

in the theory of value and prices.


$ 2.

In order to simplify our discussion the following preliminary sup
positions* are made :
(1) A single isolated market large enough to prevent one man's

consciously influencing prices.
(2) A given period , say a year.
(3) During this period the rate of production and consumption

are equal and such that stocks left over from last year and stocks
held over for next may have an influence which is unvarying or
which is not a function of quantities produced and consumed during
the year. Their influence is accounted for in the form of the curves

to be employed just as is the influence of climate, population, polit
ical conditions, etc.
(4 ) Each individual in themarket knows all prices, acts freely and
independently and preserves the same characteristics during the
period , so that the forms of his utility curves do not change.
(5 ) All articles considered are infinitely divisible and each man
free to stop producing and consuming at any point.
(6 ) The marginal utility of consuming each commodity decreases

as the amount consumed increases, and themarginal disutility of pro
ducing each commodity increases as the amount produced increases.
(7) As stated in Chapter 1, § 4, the utility of each commodity is

independent of the quantities of other commodities and likewise for
disutility .
$ 3.

In fig. 2 let the curve MN be drawn with axes OE and OA. This
curve is such that the shaded area represents

any amount of the given commodity consumed
by the given individual in the given period of

time, and the ordinate (drawn downward) from
OtoR represents its marginal utility. The figure

evidently interprets the fact thatas the quantity
of commodity increases its marginal utility de
creases and vice versa . t OA indicates what the

marginalutility would be if only an infinitesimal
quantity of the commodity were consumed.
Furthermore let a glass cistern (fig. 2) be
formed having the figure OAMN for its front Af





-- -

* These are (essentially) those of Auspitz und Lieben .
+ For the further properties of the curve MN and its relation to the curves of

Jevons, Auspitz und Lieben and Fleeming Jenkin , see Appendix I, Division II .



Irving Fisher - Mathemati



face and a uniform thickness of unity so that the volume of liquid
contained is always equal numerically to the area on the face. Hence
the amount of liquid in the cistern may represent commodity and the

distance of its surface from 0 , its marginal utility .


Let fig. 2 represent the utility cistern for I relative to A . Let
us select as a unit of utility themarginal utility of money supposing
this to be constant. Thus the cistern is (say) one inch in thickness ;
the number of cubic inches of water represents the number of units
of the commodity ( yards, gallons, or pounds, etc.) consumed by the
individual during a given period (say a year) and the ordinate OR

(in inches ) represents the number of dollars at which the individual
prizes the last yard or gallon ( say) of the commodity.
Since the market is large enough to prevent any conscious influ
ence on the price by the individual I, he acts with reference to a
fixed price ( p dollars ). He will therefore consume such an amount

of A that its marginal utility in dollars equals that of the price P,
that is, the cistern will be filled till OR = p . This is evident, for
if less should be consumed OR would be greater than p , that is, a
little more commodity would be valued more highly than the dollars
exchanged for it and so would be purchased , and if more should be
consumed , reverse considerations hold .

If the price rises OR will increase and less be consumed but if it
falls, more. If the price falls to zero as is the case for water and air
the quantity consumed fills the whole cistern up to the horizontal
axis. This volume is therefore the quantity ofmaximum satisfaction .

If the price rises to OA the individual will cease consuming. This
price is therefore the limiting maximum price at which he will buy.

The liquid contents of the cistern may be regarded as made up of
successive horizontal infinitesimal layers each representing an incre
ment of commodity.

The height or distance of each layer from the

origin represents the degree of utility of that layer. The last or top
layer is on the margin of the whole and its vertical distance from

the origin is the degree of utility of that marginal layer or incre
ment of the commodity or briefly its marginal utility . Thus the
margin of consumption has in the cistern an actual physical analogue.

in the theory of value and prices.
$ 5.


The definitions of utility in Chapter I apply also to negative util
ity or disutility . Corresponding to all that has been said relative to

consumption are analagous remarks for production.


Thus wemay construct a disutility curve and cisterna
(fig. 3) marginal disutility (O R ) being measured a
upward from the origin . If utility be measured in
money as in the last section , O A represents the
minimum price at which the individual will produce

the commodity, O R the current price and the shaded
area (or the cubic contents behind it ) the output.


The marginal disutility of production is here represented as de
creasing as the amount of the product increases.

“ law of diminishing returns.”

This assumes a

It is true that this law is seldom if

ever rigorously true when applied to small amounts ; that is, the
cost or disutility of producing the first unit is not less but greater
than that of producing the second. But the marginal disutility con
tinues to decrease only up to a certain point, after which it increases.
This is usually true even of manufacturing. American bicycle fac

tories are now running behind their orders. If they attempted to
run their factories at a higher velocity the cost of the additional
product would become greater than its price. In general at the
actual rate at which a concern produces, the law of increase of dis
utility applies.

It would be possible by looping the curve MN near the bottom to
make a cistern of such a form as to represent correctly both the law

of decrease and increase , but as we are chiefly concerned with the

point of equilibrium and as at equilibrium the law of increase usually
applies such complicated curves are not here drawn .
If a producer has such a productive capacity as to consciously in

fluence prices by a variation of his product, he may find his maxi
mum gain by restricting his output even at a point where the law of
decreasing disutility applies ; for if he should extend his production,

his price might decrease faster than his cost.
These considerations together with the important one that in a
productive enterprise the expenses are classified as " fixed ” and

“ running,” make many interesting cases of instability and indeter
minateness and lead to the discussion of monopolies, combinations,

rate wars, etc., etc. These each require special analysis. In the


Irving Fisher - Mathematical investigations

present memoir, however, attention is confined to those features of

production which are strictly analogous to consumption. (See Ap
pendix II, § 8 .)
$ 6.
Let fig. 4 represent the utility cisterns for all individuals I, II,


III, IV, . . . N , in the market and let utility be measured in money

as before,the marginal utility of money being considered constant
(say 1 util.).
The water in the connecting tubes (represented by oblique shad
ing) does not stand for commodity.
The water will seek its own level. This is exactly what happens

in the economic world and may be stated in the theorem : A given
amount of commodity to be consumed by a market during a given
period will be so distributed among the individuals thatthe marginal
utilities measured in money will be equal. Furthermore the margi

nal utility thus determined will be the price.
This follows, for there can be but one price , and each individual
will make his marginal utility equal to it, as shown in $ 4.
If the stopper,* S, be pressed; more liquid (commodity ) flows into
the cisterns, there is an inevitable change in level and the price de

creases. When it cheapens to 2, IIbegins to indulge. It is for the
first time “ within his reach .”

It is to be noted that from the standpoint of a single individual
the existence of the general price level is an unalterable fact and the
amount which he consumes is accommodated to it, just as the gen
eral water level in several hundred cisterns may be said to determine
* A rubber compression ball would be used in practice. Throughout the de
scriptions, the mechanisms are those simplest to delineate and in many cases

not those which might be actually employed.

in the theory of value and prices.


the amount in any particular cistern . But, for the system as a
whole, the price level is a consequence of the amount of commodity
marketed. What appears as cause in relation to effect to an indi
vidual is effect in relation to cause for the whole market.
The quantities of commodity and the marginal utilities mutually

limit and adjust themselves, subject to three conditions, (1) that due

to the forms of cisterns, (2) that due to the total amount of commod
ity marketed, (3) uniformity of price, or of marginal utility .
$ 7.


The algebraic interpretation of the preceding mechanism or of the
economic phenomena themselves is as follows :
Let A ,, A ,, A ,, . . . An be the (as yet unknown) quantities of the

commodity consumed by I, II, III, . . . N . Let


dA ' MA,'


.. DA,

be their (unknown) marginal utilities. Then the three conditions
mentioned in § 6 become:
(The unit of utility (util.) is that of themarginal dollar.)

IN = F,(A,) ?
JA = F,(A,). |n equations.
2 n unknowns.

au =F.(A.) J
on .
K ſnoequati
A , + A , + A , + . . .. + A , =- K11
new unknowns.
(Unit of utility is that of marginal dollar.)



dU in - 1 independent equations

dA = A JA, .. . . A, S no new unknowns.
Hence the number of equations is :
n + 1 + (n - 1) = 2n

and of unknowns :
2n + 0 + 0 = 2n .

Therefore the numbers of equations and unknowns are equal and
all quantities and utilities are determinate .


Irving Fisher-- Mathematical investigations


$ 8.

Let C , fig . 4,be the average curve* of all the individual curves, I,

II, III, . . . . N , and let the new cistern have a thickness equal to the
sum of the thicknesses of the individual cisterns. Then asmuch water
will be in the aggregate cistern as in all the others.* The water in

the aggregate cistern may be regarded as a repetition of the con
tents of the individual cisterns. It represents no new commodity.
In cistern C it is almost too evident to require mention that an in
creased supply of this commodity (indicated by pressing the stopper)
reduces the price while a diminished supply increases it. This fact

is all that is usually exhibited in “ demand curves” such as of

Fleeming Jenkin .
$ 9.
Fig. 5 and completely analogous explanations apply to production


* Formed as follows : Select pts. of like price on the individual curves, that is,
pts. of like ordinates (as Yı, ya, y3, . . . Yn ) and using the same ordinate for the
new ordinate, take the average of their abscissas for the new abscissa and make
the thickness of the new cistern equal to the sum of the thicknesses of all the in
dividual cisterns.

Then if in such a cistern liquid be allowed to flow to the

level of the individual cisterns the amount of liquid contained in it will equal
all that contained in the individual cisterns. For evidently the free surface of
the water in the large cistern equals in area the total free surfaces in the small,
and as such equality of horizontal infinitesimal layers or laminæ holds true at

all successive levels, it holds true of the sums of the layers.
+ The Graphic Representation of Supply and Demand . Grant's recess studies ,
p . 151.

in the theory of value and prices.


$ 10 .

The mechanism above described simply gives exactness to a com
mon imagery in economics, such as “ margin ," " price levels,"

“ planes” of demand (and supply) and : “ a plentiful supply brings
the commodity within reach ' of consumers."

The notion of a cistern is also natural. Says Adam Smith : “ The
demand for food is limited by the capacity of a man's stomach .”
Not only is there a “ limit,” but the demand for food has varying

intensities according to the degree in which the stomach is filled.
The economic man is to be regarded as a number of cisterns or
stomachs, each relative to a particular commodity.


$ 1.
The next problem is that of the distribution of an individual's in
come over all the commodities in the market .
The income-spender considers not only the price of a given article
in determining how much of that article he will take but also the
relative advantages of using the samemoney for other things.
The manner in which this consideration affects the mechanism de

scribed in Chapter II is through the utility of money.*
In the last chapter, while the price varied in relation to the quan
tity of commodity, each individual's valuation or marginal utility

ofmoney was regarded as constant. This is nearly true when only
one commodity is considered. In the present chapter, on the other
hand , the individual valuation of money varies in relation to the

quantity ofmoney income, butthe prices of all commodities are re
garded as constant. This is nearly true when only one individual
is considered .
* This sort of interaction, especially when extended to several consumers and
several commodities (as in the next chapter), presented the most difficulties to

the Auspitz und Lieben Analysis ; on p . 63 in $ 16 they say : “ Welche Aende


rung eine Einzelkurve erleidet wenn sich die Vermögensverhältnisse des be
treffenden Individuumsändert, lässt sich im allgemeinen nicht verfolgen . Wenn
auch in der Regel die Ordinaten der kurven länger werden , wenn das Individuum
wolhabender wird , so wird dies doch keineswegs gleichmässig der Fall sein, vol
· lends nicht, wenn wir verschiedene Artikel betrachten ."

Irving Fisher — Mathematical investigations
$ 2.

Let the individual I distribute his income over the commodities
A , B , C , . . . . . M . Let the thickness of each cistern in fig . 6 be

proportional to the price of the commodity it contains. Thus if A
bears a price of $2 per yard, B $ 1 per gallon and C $ t per pound, the

thickness of cistern B is 2, of B 1, and of Ct.


Let the unit of area on the front surface of each cistern represent

a unit of commodity , yards for A , gallons for B , etc .
Then the volume of liquid will evidently indicate themoney value

of the commodity , for it equals the front area times the thickness,
that is, the quantity of commodity times its price. Moreover the
sum of all the water will indicate the whole * income in dollars.
The unit of volume thus represents not a yard, gallon, pound, etc .,
but a dollar's worth in each case. For A it would be } yard , for B
1 gallon, for C 2 lbs., etc .
Accordingly let the curves which limit the cisterns be so con

structed that the ordinates shall represent marginal utility per
dollar's worth not per yard , gallon , etc.

$ 3.

The liquid will seek its own level corresponding to the economic
proposition : A consumer will so arrange his consumption that the

marginal utility per dollar 's worth of each commodity shall be the

* Saving is here regarded as a form of spending, the commodity purchased
being capital. The analysis implies that the marginal utility of saving a dollar
equals the marginal utility of the dollar spent in other ways. This would be

elaborated from another standpoint in a theory of distribution . Cf. Launhardt ;
Volkswirthschaftslehre; Böhm -Bawerk ; Kapital und Kapitalzins.

33 •

in the theory of value and prices.

This follows because if the individual should vary his consumption .

from such a distribution, by expending an extra dollar on A he
would divert that amount from another article or articles, say B .
Then the level in the A cistern would be higher than in the B , which
interpreted , is the dollar spent on A had less utility than if it had
been devoted to B .
If the stopper be pressed , i. e. if the individual had had a larger
income, the valuation of the last dollar's worth of each commodity

decreases, or themarginal utility of money decreases. If it becomes
at the maximum marginal utility of B he begins to spend on B. As

it is in the figure he “ cannot afford it.”

The amount spent on any particular commodity depends on the
general water level, i.e. the valuation of a dollar, while reversely
the valuation of money depends on the total amountto be spent on
all commodities.

Three conditions suffice to make the distribution determinate :
(1) that due to the forms of the cisterns, (2) the condition that the
total income equals a specified amount, (3 ) uniformity of marginal

utility (per dollar's worth ) of each commodity .
$ 4.


Let A , B , C , . . . M be the (unknown) quantities of various com


modities consumed by 1, and ini . . . . JM their (unknown) mar

ginal utilities. Let Pa, Po, . . . Pm be their (known ) prices.
Then the above three conditions become :
( The unit of commodity is the dollar's worth .)


dA = F(A)


= F (B ) | m equations.

· · :

| 2m unknowns .

AM = F(M )

Ape+ Bpo + . . . . +Mpm = K —no new unknowns.

Trans. Conn. ACAD., Vol. VIII. ,

- JULY, 1892.


Irving Fisher - Mathematical investigations

(Unit of commodity is dollar's worth.)
NU Im - 1 independent equation .
QA = dB = . . . = M no new unknowns.
Number of equations = m + 1 + m - 1 = 2m .

66 - unknowns = 2m + 0 + 0
Hence the system is determinate.

= 2m .


Let I, fig. 6, be the average curve* of all the separate commodity
curves A , B , C , . . , M , and let the new cistern have a thickness
equal to the sum of the thicknesses of the original cisterns. Then
the water in the resultant cistern equals the sum of that in the com

The liquid in the new cistern represents the money collectively
considered and the ordinate the utility of the last dollar.

If this income increases, its marginal utility decreases and de
creases in a law whose relation to the laws of utility for the separate

commodities is shown by the relation of the resultant cistern to the
* In this case the average is not a simple arithmetical mean but a weighted
average. Select points of like utility on the component curves, that is, points of

equal ordinates . Average their abscissas, multiplying each by the ratio of the
thickness of its cistern to that of the resultant cistern (viz : the sum of the thick
nesses of the original cisterns). Thus if the thicknesses are p , P ., . . . Pon and

the abscissas xq, xg,. . . xm, the resulting thickness and abscissa (P and X ) are :
P = P, + P8 + . . . + Pm
2 ,2 , + X, P, + . . . + xm P.
X = 1
Pc + Pr + . . . + P.m
If in a cistern thus formed liquid enters to the level of the component cisterns,
the liquid in the resultant cistern equals the total in the component. For the
sum of the free surfaces in the component cisterns is

X ,P , + X7P8 + . . . + xmp m
and the free surface in the resultant is
( PQ + P , + . . . + 2 ) . MaPa + 3 , P , + . . . + a in p m


Pe + P + . . . + Pm

Since these two expressions are equal and this equality holds of infinitesimal
layers at the free surface and so successively at all levels it must hold of the
sums of these layers.

in the theory of value and prices.


$ 6.

An analogous discussion applies to fig. 7. In place of a given in
come wemust suppose a given amount of expenses to bemet by the

production of various commodities.* It is at this point that an im
portant distinction between production and consumption enters, viz :

in civilized life men find it advantageous to consume many things
but to produce few . The discussion of this difference pertains to
Part II.

$ 1.

We have seen the laws of distribution of commodities from two
points of view , by first restricting our discussion to one commodity
among many consumers and afterward to one consumer among many
commodities. Our discussion is like a tourist's view of a great city ,
who glances up each east and west street while riding along the same
avenue and then takes a “ cross town ” course and sees each avenue
from a single street. We are now to seek a bird's-eye view .
The variables and their variations which have been described are

comparatively simple. But the possible variations in the more gen
eral case are so complicated that they can scarcely be seen or de
scribed without the aid of a mechanism .
* Borrowing capital is to be here regarded as a form of producing. The dis
utility of borrowing the last dollar equals the disutility of producing the last

dollar's worth of goods. See foot note to $ 2.

Irving Fisher - Mathematical investigations


$ 2.
First of all an analysis will serve to set the two preceding discus

sions in a common point of view .
In any purchase the last infinitesimal commodity bought has a

utility equal to that of the money given , that is :
ut. of dA = ut. of dm
or :

au da = a...Am

or :

(see Ch. I, § 3 .)

du dm

JA = dm .da
du _ dU

or :

dA = dm . Pa

where pa is the money price .

That is, the marginal utility of a commodity (per pound, yard,
etc.) equals the marginal utility of money (per dollar) times the
ratio of exchange of money for commodity :

This equation is fundamental. · In our first discussion (one com

modity, various consumers) the marginal utility of money was sup
posed constant so that

dA « Pa
or the marginal utility of a commodity is measured by it price.

In the second discussion the other factor, the price, was supposed
constant, and :


dA " dm

or the marginal utility is measured by the valuation of money.
§ 3.
In the present chapter we are restricted to neither of these special

suppositions. For the individual I, we may write



A = dm , .Pa
dB, – dm , .Po
. . . .. . .. . . .

d [ _ «U


dm = dm , Pm


in the theory of value and prices.

whence, since the marginal utility of money to I is the same in each




da, " ĀB, : .. . . īm . = PaiPoi. .. .Pm
Since this is true for every individual and the prices to all individ
uals are the same, wemay write :



. . . .

Pa : Po : . . . . Pm =

• ' dM


. ....


= da . : dB. : . .. .


These equations express in the most general way the theory of
marginal utilities in relation to prices.

This theory is rot, as some

times stated, “ the marginal utilities to the same individual of all
articles are equal,” much less is it “ the marginal utilities of the same

article to all consumers are equal,” but : The marginal utilities of
all articles consumed by a given individual are proportional to the

marginal utilities of the same series of articles for each other con
sumer, and this uniform continuous ratio is the scale of prices of
those articles.

The idea of equality is inadequate and must be replaced by the

idea of proportionality. The problem which confronts the individ
ual must be figured as to so adjust his consumption of all commodi
ties that the utilities of the last pound, yard , gallon , etc., shall bear
the ratio which he finds their prices do, while the market as a whole
must cause such prices to emerge as will enable each individual to

solve this problem and at the same time just take off the supply.
$ 4.

This notion of a ratio is introduced into the following more com
plicated mechanism (fig . 8 ). Fig. 9 (an elevation of fig . 8 ) shows
the various cisterns of various commodities for the individual I.

The ordinates represent marginal utility per unit of commodity .
It corresponds to fig . 6 , except that in the latter the utility is per

dollar'sworth of commodity. The tops of the cisterns are no longer
at the same level.

The cisterns are now to float like boats in a


Irving Fisher— Mathematical investigations





пс |


wun an do

in the theory of value and prices.












tank* and free to move only vertically (being so constrained by a
telescope arrangement beneath and not shown in the diagram ).
A glance at fig . 8 or fig .
10 .

10 (a plan of fig . 8 ) will
show that any right and
left row of cisterns is rela
tive to a single individual
and corresponds to fig . 6
and thatany front and back

row is relative to a single
commodity and corresponds
to fig. 4.

The water in these cis
terns must be subjected to
two sets of conditions, first:
the sum of all the contents

of IA , IIA , IIIA , etc., shall
be a given amount (viz :
the whole of the commod
ity A consumed during the

given period) with a like given sum for the B row , C row , etc.,
secondly : the sum of IA , IB , IC , etc., each multiplied by a coeffi

cient (the price of A , of B , of C , etc.), shall be given (viz: the
whole income of I during the period ),with a like given sum for the

II row , III row , etc.
* The level of water in each cistern is intended to be that of the level in the

tank. The only constant cause which will makethe levels different is the differ
ence between the weight of the whole cistern and the weight of the water dis
placed by its walls (partly wood) which difference is slight, may be plus or
minus, and is equal to the weight of the excess or deficit of water in the cistern
above or below the outside level.


Irving Fisher — Mathematical investigations
To realize these two sets of conditions each cistern is divided into

two by a vertical partition of wood. The frontcompartments are all

of unit thickness one inch (say). All front compartments belonging
to the same front-and -back row are mutually connected by tubes (in
the tank but not in connection with the water of the tank ) thus ful
filling the first set of conditions.
The thickness of the back compartments is adjustable but is (as

will soon appear) constrained to be always equal to the price, thus
if the price of A is $ 1, of B $ 3 and C $ 1.20, the thickness of all
cisterns in the A row will be 1, in the B row 3 and in the Crow
1.2 (inches).

Since the thickness of the front compartment is unity, the con
tents of each back compartment equals the contents of the front
multiplied by the number of inches of thickness of the back cistern,
that is the back compartment contains a volume of water equal to
the amount of the commodity multiplied by its price. . It contains

therefore the money value of the commodity. The double cistern
represents the double light in which each commodity is commonly

regarded — so many pounds, yards, etc. and so many dollar's worth .
All back compartments of the same right and left rows are

mutually connected by tubes — that is the sum of their contents is
given — thus fulfilling the second set of conditions.
The back compartments can change their thicknesses, as the walls
at the right, left and bottom are of flexible leather; the back plane

is kept parallel to the wood partition by two double “ parallel rules”
not diagramed .

There remains to be described the system of levers. The purpose
of these levers is to keep the continuous ratio of marginal utilities,

the same for all individuals and equal to the ratio of prices.

First there is a system of oblique* levers (F12, etc., fig. 9 ) con
nected by sliding pivots with the tops of the cisterns and having
their lower extremities hinged to wooden floats F , the hinges being
on the level of the water of the tank. These floats are free only to

shift laterally . It is evident from the similar triangles FR1 and
FR2 in fig . 9 that the ordinates of the two cisterns IA and IB are
proportional to the distances of the A and B rods R and K from
the hinge in the left float F . Likewise in the row behind , the ordi
nates are proportional to the same distances. Hence the four
ordinates are proportional to each other and in general all the

- -- - -- -- - -

- -





-- -

- - ---

* A convenient angle for each lever can be assured by a careful selection of
commodity units. Thus if the marginal utility per pound gives inconvenient
ordinates in the A row , reconstruct the cisterns in that row so that the ordinates

are lengthened to represent marginal utility per ton or shortened for the ounce .

in the theory of value and prices.


ordinates of the front row are proportional to those of the row next

behind , also of the second row behind and so on . Remembering

that each ordinate is a marginal utility we have: .
du du
du du
du du
da , ' dB, . . . = da . : dB . : . . . = A : B : . . . = . .

which is the required condition that marginal utilities must be pro
portional (83).

Secondly there are the horizontal levers (F34, etc., fig. 10 ) lying
on the surface of the water in the tank. These relate to prices.
The sliding pivots 3, 4, etc. are connected with rods RRR , which

in turn are connected by vertical pins with the rear walls of the

cisterns. A motion of one of these rods causes all back compart
ments in that row to expand or shrink in unison. The pivots 3, 4,

etc . are so situated on these rods that if the levers F34, etc. should

assume a right-and-left position along the dotted line FF , the back
compartment of every cistern would be completely closed. Hence

R3 equals the thickness of each back compartment in the A row ,
R4 the corresponding thickness in the B row and so on.
By the similar triangles FR3 and F :34 in fig . 10, it is clear that
the lines R3 and R4, and consequently the rear thickness in the A
and Brows are proportional to the distances of the A and B rods
R and R from the float F . But we have just seen that the ordinates

of IA and IB are proportional to these samedistances. Hence the

thicknesses of the back compartments of the cisterns are propor.
tional to the ordinates of those cisterns, that is to marginal utilities.

Hence we are free to call the thickness of each back compartment,
the money* price of the commodity to which that cistern relates.
- - - - -- -



- - - --

* Money is here used solely as a measure of value. It is not one of the com
modities in the market. The high or low price of commodities in terms of
this money is dependent entirely on the amount of it at which we agree to rate
the yearly consumption of the market , that is the amount of liquid originally in

the back cisterns. We are so accustomed to regard money as the medium of

exchange and therefore as a commodity that we may not observe that it is per
fectly possible to have a measure of value which is not a commodity at all. Thus

wemight agree to call the consumption of the United States for a year $ 10 ,000,
000,000, and this agreementwould immediately fix a measure of value, though the
new dollar need bave no equality to the gold or silver dollar. It would be easy

to translate between such an arbitrary standard and any commodity standard.
Thus if statistics showed that the consumption measured in gold dollars was
$ 12,000 ,000 ,000 , the agreed standard is at 120 compared with gold and by means

of this factor we can reduce the prices of all commodities. In the mechanism
the aggregate amount of liquid in the back cisterns corresponds to the $ 10 ,000 ,

000,000. If we take it so and if the amount of liquid in the I row is given at
$ 1,000, this means that (in whatever standard ) the consumption of I is one-ten
millionth in value the aggregate consumption .


Irving Fisher - Mathematical investigations

It is to be observed that the cisterns are free to move only verti
cally , the rods and rear cistern walls only forward and backward ,

the wooden floats can shift only sidewise right and left while the

levers assume such positions as the mechanism compels.

Let given quantities of water be introduced into each front-and
back -row of front cisterns and into each right-and-left row of back
cisterns. The system will attain a stable equilibrium and the level
of water in each cistern will be that of the tank .

The front cisterns of a front-and-back row must have a uniform
level on account of their mutual connection. The back cisterns of
a right-and-left row must preserve a uniform level for a similar rea
son . The movable rear walls allow the pressure of the outside
water in the tank to keep the back cisterns at the samelevel as the
front. Without taking account of the levers the cisterns would
thus all have the same level as the tank . But it would be possible

to arrange their vertical positions and their rear thicknesses in many
arbitrary ways. The levers simply specify or determine this arrange

$ 6.
It may be needful to restate carefully the magnitudes, their units
and the conditions which determine them . The magnitudes are :

1. The quantities of each commodity consumed by each individual

during the year. These are represented by the quantities of water
in each front compartment.
2. The given total quantities of each commodity consumed by the

whole market — represented by the fixed amount of water in each
front and back row of front comparments and registered on scales*

A , B , C , at the rear of the tank. Each commodity-water may have
a distinguishing color.

3. Themoney paid for each commodity by each individual— rep
resented by the water in each back compartment.
4. The total money income of each individual— represented by
the fixed amount of water in each right-and-left row of back com
partments, and registered on scalest I, II, III, at the right of the
tank .
* The stoppers A B , C regulate this amount of water.

The stoppers are each

directly connected with the pointers on the scales A , B, C , and so arranged that
when the stopper is withdrawn so that the scale reads zero, the water entirely
disappears from the cisterns.
+ The stoppers I, II, III are also directly connected with pointers on the scales


in the theory of value and prices.

5. The marginal utility of each commodity to each individual
— represented by the ordinate of each cistern, i. e. by the distance
from its top to the water level.

6 . The money price of each commodity — represented (in any
cistern in the same front-and-back -row ) by the thickness of the back

compartment, and registered on scales* Pa, Po, P. at the rear. (The

relation of price to marginal utility will recur.)
7. The prices of commodities in terms of each other — represented
by the ratios of their ordinates.
8. The marginal utility of money to each individual — repre
sented (in any cistern in the same right-and -left-row ) by the ratiot
of the ordinate of that cistern to the
thickness of its back compartment


and registered on scalesI UI, U JI,

U III at the right.
The units of these magnitudes
are :

1. The unit of commodity is a

ton, yard, gallon , etc., and is repre
sented by (say) a cubic inch of

2. The unit of money is (say) a
dollar and is represented by (say) a
cubic inch of water .
3. The unit of price is one dollar

per ton , yard, gallon, etc., and is
represented by one inch .

4. The unit of marginal utility
for each individual is the marginal
utility of (say ) 100 tons of A . It
may be called a util and by a proper
* The rodsRRR are each connected by a cord and pulley with the pointers of
the scales p . , P . , P . .

+ This ratio is evidently the marginal utility of money (“ valuation ofmoney ?)

because as seen in chapter IV, 82,


da = dm • Pa


whence :


dm – P

ordinate of cistern

thickness of its back compartment

Fig . 11 (which views the outside of the right wall of the tank) shows the

device by which this is accomplished. Evidently from the labels


ī = p or x = dm
The pointer obviously varies with x. It is so arranged as to register zero when
c = 0.

Irving Fisher - Mathematical investigations
adjustment of the breadth of each cistern may be represented by one
inch . That is, if 100 cu. in . of water are put in each A cistern the
ordinate must be one inch. This applies as well to the utility of
money, so that the scale U at the left indicates the number of utils
at which the individual values the last dollar of his income. It
should , however , be noted that the variation of utils is only valuable

in the same register , that is, for the same individual. There is no

important meaning attached to the ratio of the scale readings U for
two individuals. If that of I is 1 and of II 2 it means simply that

II values his last dollar twice asmuch as his 100th ton of A , while
I values his last dollar just asmuch as his 100th ton of A . It is in
teresting to observe that analogously the price registers are not to
be compared, for while one indicates price per ton the other indicates

price per yard, etc. Thus the mechanism is independent of any
common measure of utility for different individuals and any common
measure of prices for different commodities.
$ 7.

It will be observed that the numbers on the various registers are
so connected that the product of the register of A by that of its

price added to the like products for B, C, etc., will equal the sum
: of all the income registers.
Moreover if each cistern is provided with a graduation to show

marginal utility, this number will be found to be the product of the

number for price in its front-and-back row , by that for valuation of
money in its right-and -left row .
$ 8.
The mechanism just described is the physical analogue of the

ideal economic market. The elements which contribute to the
determination of prices are represented each with its appropriate
rôle and open to the scrutiny of the eye. We are thus enabled not
only to obtain a clear and analytical picture of the interdependence
of the many elements in the causation of prices, but also to employ

the mechanism as an instrument of investigation and by it, study
some complicated variations which could scarcely be successfully

followed without its aid . Its chief uses may be briefly classified as
follows :

1. Arrange the stoppers I, II, III, etc ., so that the money incomes

of I, II, III, are all equal. The differences of distribution of the
commodities will depend on individual characteristics, that is, on

in the theory of value and prices.


the character of the cisterns. If all the A cisterns are alike and

also all B cisterns, all C cisterns, etc., then each commodity will be
distributed in equal parts among the individuals.

2. Press stopper I. This amounts to increasing the income of I.
It does not increase the amount of commodities in the market but
gives a larger share to I.

The total money value of the same aggre

gate commodities in the whole market has increased by the amount

of liquid added by depressing the stopper.
The added water in the back cisterns of the I row will make the
back compartments in this row fuller than the front. The back level
will be temporarily above the water level of the tank and (as the

cisterns will sink) the front level will be temporarily below . The
effect of the former is to bulge out the movable rear wall in the I
row , to extend the rods and to cause the same expansion in the back
compartments of the II, III, etc. rows. This makes the back liquids

in these rows lower and the front liquids higher than the tank level.
Hence the front cisterns of the II, III, etc . rows pour part of their

contents into the I row whose level as we have seen is below that of
the tank .

In economic language to give a greater money value to one indi

vidual causes for him smaller marginal utilities (cisterns sink ), a
lower marginal utility of money, and increased consumption of com

modities. For other individuals it increases marginal utilities (cis
terns rise), decreases consumption , increases prices (back cisterns
'expand), and may increase or decrease their marginal utility of

money-income according as marginal utilities (ordinates) increase
faster than prices (back thicknesses) or the reverse.
So much for the effect on different individuals. Now as to the
effect on the various commodities. Prices in general have risen but

not necessarily of all articles. Suppose article C is consumed little
or not at all (cistern narrow ) by the enriched individual I but is ex
tensively used by those whose valuation of money has increased .
Then since the valuation of money to II is equal to the quotient of
the ordinate of IIC divided by the thickness of the back cistern of

IIC, and since this ordinate has not lengthened by any appreciable
loss of commodity C from II to I,the thickness must have lessened,
that is, the price has been reduced.

Not only may there be such exceptional commodities but there
may be exceptional individuals. Thus a man may be the principal
consumer of just those commodities and those only whose price has

fallen. His consumption will increase,his marginal utility ofmoney


Irving Fisher - Mathematical investigations

decrease. He is benefited not injured by the increase of income of

his neighbor I.
3. Press stopper I and raise III. I, II, III now represent a wealthy
middle class and poor man respectively . We observe first that this
change causes the poor man to relinquish entirely some things

(luxuries) as C while decreasing his necessaries slightly ; second that
the rich man increases his luxuries enormously and his necessaries
slightly, and thirdly that slight modifications will appear in the
prices and hence in the middle-class consumption.

The nature of the effect on prices depends on the character of the
cisterns of I and III, and on themagnitude of the changes in their

incomes. In order that prices may not change, one condition (neces
sary but not sufficient) is that the amount of money income added
to I must equal that taken from III, for if the amounts of com
modities are not to change, nor their prices, their totalvalues cannot.

If all prices rise it proves a net increase of money income in the
whole system .
If the increase of income of I equals the decrease of that of III,

so that the total money value in the market is unchanged , and if
furthermore all the cisterns of I and III have straight walls on the

right and have their breadths* proportional, there will be no change
in price. For if the cistern breadths of the III row are each, (say)
half the corresponding ones in the I row , equilibrium will clearly be
satisfied by shortening each ordinate of the I row by a uniform per
centage ( say 10 %), and lengthening those of the III row by just
twice the amount of shortening in the corresponding I ordinates.

This will evidently cause the lengthening of the III ordinates to be
uniform (say 15% ). The ratio of marginal utilities has thus been
preserved and hence the prices. Obviously the contents added to

IA equals that taken from IIIA and equilibrium is reëstablished by a
simple transfer from III to I. In this case there is no effect on II

or any individual save I and III.
* The breadth of a cistern is evidently the differential of its area divided by
the differential of the ordinate that is the fluxion of commodity in reference to its
marginal utility. It is a magnitude important in the discussion of distribution
of commodities. Involving as it does the second differential of utility it has no
perfectly distinct recognition in popular language. A narrow cistern means that

a slight reduction of its contents causes its ordinate to increase much , i. e. causes
it to be greatly desired .

The individual is very sensitive to a change in that

commodity. He misses a little less of it and appreciates a little more. Reversely
a broad cistern signifies that it is hard to satisfy the man by increase and hard
to annoy him by decrease. These two sorts of cisterns may be called “ sensitive ”
and “ callous ” (see Appendix I).

in the theory of value and prices.


More generally in a redistribution of incomes without altering
their aggregate , in order that no prices may change (1) no condition

is necessary for those whose incomes have not changed ; ( 2) for
those whose incomes have changed the geometrical character of the

cisternsmust be such that a proportional shortening of the ordinates

for each and every richer man will absorb in the aggregate,the
same additional commodity of each sort as is lost in the aggregate
by the poorer through a proportional lengthening in the ordinates
of each of them .

If the enriched man or men absorb more of a given commodity
than this requirement its price will rise, if less it will fall.

If the increase of income of I equals the decrease of III effects on
prices must be compensatory. If one rises some other or others
must fall. If IA is much broader than IIIA but IB is much nar
rower than IIIB , the price of A may rise and of B fall unless
counteractions come from other commodities. For if we were to

suppose prices unaltered, the cistern IA would absorb from IIIA
so much and IB from IIIB so little that the ordinate of IIIA
would be too long and of IIIB too short for equilibrium . In order

to partially permit this lengthening and shortening there must be a

corresponding lengthening and shortening in the whole A and B
rows respectively and prices must be proportioned to these ordinates.
In this case it is to be noted furthermore that a change in prices
causes a change in the distribution of the income of II and all other
individuals .

The marginal utility of money for I decreases, for III increases,
and for II may slightly rise or fall, owing to the change of prices.
With the breadths of the cisterns properly adapted to the changes
in prices there may be no change* in the valuation of money for II.
- -

* If the prices of only two commodities A and B change and AII and BII are
straight walled , and if their breadths are inversely proportional to the difference

of the squares of the old and the new prices, there will be no change in the valu
ation of money. For, let p and p ' be the old and new prices, let a , and x , be
the breadths (for II) of the A and B cisterns and let y ., y , and y ', y , be their

old and new ordinates. Since the marginal utility of money is not to change
nor the prices of C , D , etc., their ordinates cannot and therefore their quanti
ties (for II) cannot change. Hence the added expenditure (by II) on A must

equal that taken from B , i. e. :

x,yPC - X,90Pa = x,y,Po — X;y "Pó
But since the valuation of money is to be kept constant,

Ya _ Y _ Y _= Y , _=
Po Po Pó


Irving Fisher — Mathematical investigations


If the price of A rises slightly and of B falls relatively more while

the breadth of IIA is less than of IIB , the valuation of money to II
will fall. For if not then the ordinates of IIA and IIB must change
pari passu with the thickness of the back cisterns. The thickness
and ordinate for IIA are , say each increased 10 % and for IIB
reduced 50 % . There is clearly not room in IIA for all the money

poured out of IIB . This surplus will spread over all A, B , C , etc.,
and reduce the ordinates and reduce the money valuation of II.
These artificially exact cases obviously stand for more general and

approximate economic theorems. There are no such delicate adjust
ments in the actual world as here presented , but through ideal cases

we study real tendencies .
4. Depress stopper A. The chief effect will be to lower the price

of A . If it is a necessary* a relatively large share of the increase
will go to the poor. It will probably occur that while the total
money expenditure by the poor for this commodity will increase,

that for the rich will decrease. Themarginal utility of money in
general decreases especially for the poor man.
Most other commodities will rise in price if A decreases in price
faster than it increases in quantity. For there will be a saving in
the expenditure for A which must be made up elsewhere . But an

exceptional commodity may fall in price. Thus if B happens to be
extensively used (cisterns deep and broad) only by those who use A
slightly, these persons will not save materially in the expenditure

x,kp." — kpo = x,kp,? — ,kp,"?.

X ,: x,:: p.9- P ”? : Po - Po ,

which is the condition required . More generally in order that the valuation of
money to an individual shall not change, the cisterns of II must be so formed
that when the money saved on some articles equals the extra spent on the others
the ordinates may all change proportionally with the prices. If the ordinates
increase more than this requirement or decrease less, the valuation of money
will rise. In the reverse cases it will fall.

* A necessary may be defined as a commodity whose cistern is relatively deep
and narrow . I. e. a very small quantity has a very great utility and a slight
addition gives satisfaction very rapidly .

A luxury has the reverse properties .

+ When commodity begins to flow into a cistern its money value (the contents
of the back cistern ) increases in about the same rate as the commodity — itmatters
little how much the price (thickness) falls . Contrariwise when the cistern is
nearly full a fall of price decreases the money value at about the same rate - the

increase of commodity matters little . The dividing point is where the commod
ity increases at the same rate as the price decreases.

These characters are more

plainly shown in the diagrams of Auspitz und Lieben , p. 48, etc.

in the theory of'valueand prices.
for A , but will be compelled to pour much money into C , D , etc. of
which the prices have risen. This will cause a rise in their valuation

of money and as the quantity of B does not decrease its price must.
Moreover there will be slight changes in all other quantities IB ,
IIC , etc. If (say) IIC decreases, it is due to one or both of two
causes, a rise in price of C or a rise in valuation of money of II.
In general the valuation of money will decrease. The decrease will
be relatively great for the poor as compared with the rich , but (as
just seen ) will not necessarily decrease for all persons.
If A is a “ luxury ” the fall in its price will be small relatively to

the foregoing case. Most of the increase of A will go to the rich .
The total amount of money spenton it will probably increase which
will in general decrease the price of other articles. Exceptions can

be found analogous to that in the former case. The valuation of
money will in general decrease, most perhaps for the middle class

and more for the rich than the poor, but not necessarily for all.
5. The cases just discussed assume that the additional production
of A is such that the incomes of I, II, III, etc. are not disturbed .
To represent the case in which I produces all of A , after depressing
A a given amount, slowly depress I until the difference of income as
registered on the I scale shall equal the final reading on the A scale
multiplied by the price of A minus the former A by its former price.
The chief change to any one article will be in the price of A
which will decrease. The chief change to any one person will be

to I whose income is increased (especially if the commodity is a
luxury), whose expenditure for most other articles will increase

though not necessarily for all, and whose valuation of money will

decrease, owing both to an increase of income and to a decrease in
price of other articles consequent on the withdrawal of money from
them to be spent on A . *Only exceptional articles will increase in

price if their chief consumers sufficiently decrease their expenditure
for A .

But it may be that the increase of A will so greatly depress the

price that the value of the total will decrease. This is generally
true of necessaries. The producer I will lose income, that is stopper
I must be raised instead of depressed . His valuation ofmoney will
increase doubly, owing to the contraction of his income and the rise

in price of other articles. The money* return to such a benefactor
* Monopoly price is not treated here . It is interesting to note that the Dutch
East India Co . used to destroy a part of their spices to prevent a great fall of

price . The same thing has been done by the Japanese in silk -worm eggs.

JULY, 1892.


Irving Fisher - Mathematical investigations

is therefore not even roughly proportioned to his benefaction . If
the exact shares among I, II, III, etc . in the old and new produc
tion of A are known, the proper combination of stopper-positions
may be made and the reactions, now exceedingly complicated,may
be watched.

6. Depress each stopper A , B , C , etc. There will be a general fall
in prices. But it will not be true that if the quantity of each com

modity is doubled its price will be halved , and the price of one
commodity in terms of another unaltered as Mill* apparently
thought, for the ratios of exchange are not the ratios of the con

tents of the cisterns but of their ordinates. Nor will the ratios of
distribution of commodities remain the same. If however all cis
terns in each front and back row are geometrically similar and their
filled portions also similar (a most unreal condition ), the ratios of
distribution of commodities will be unaffectedt and if furthermore

all cisterns are similar, the ratios of prices will be unaltered.I
In the actual world aside from differences in the shapes of cisterns
there are more important differences in the way in which they are
filled. Those for necessaries are relatively full as compared with

those for luxuries and those for the rich as compared with those for
the poor. Hence the effect of a proportionate increase of produc
tion in all commodities will depress the price of necessaries much
more than of luxuries.

The effects on the valuation or marginal utility of money will be

more complicated. If we suppose the depression of the stoppers to

begin when they are far extended, the effects may be roughly
described as follows. At first the valuation of money increases
since the prices decrease fasters than the marginal utilities, reaches
a maximum (which is different for each individual and depends on
the initial distribution ), and decreases when the decrease of ordin

ates is faster than that of the thickness of the back cisterns. These
- --

- - -- - -


* Pol. Econ., Bk. III, Ch. XIV , $ 2 .
+ For proportional increase of the contents of the cisterns in the same front
and back row will reduce their ordinates proportionally and shrink the back com
partments alike, thus restoring equilibrium .
# For in addition to the above consideration the reduction of ordinates in all
rows will be alike.

$ Because when a cistern is relatively empty , a rise in the surface of its con
tents diminishes the long ordinate by only a slight percentagebut very materially
contracts the back compartment.

in the theory of value and prices.


changes in the valuation of money are of course subject to the con

dition that each incomemeasured in money remains the same.

7. Depress all income stoppers proportionally , i. e. increase all
incomes in the same ratio. Then will all prices increase and the
valuation of money decrease exactly in this ratio. There will be
no change in the distribution of commodities. There is merely a

depreciated standard of money. Formerly the whole marketed
commodity was valued at a given number of dollars, now this
number is increased .

We have seen under number 1, that an increase in the money
income of a single individual without an increase in commodities
is a benefit to him , but such an increase when universal is bene
ficial to no one.

8. Remove cistern IA and replace it with a shallower one, i. e.
suppose a change in the taste of I for A , making the article less

It is as if we raise the bottom of the original cistern IA. More
of A will flow to other consumers and more of l's money will flow
to the purchase of other commodities.

A will fall in price, most

other articles will rise. I's valuation of money will fall. For those
who consume A extensively the valuation of money will fall. For
others it may rise .

If all of the I cisterns grow shallower there will be a fall in the
valuation of money for I, but either prices will not change or their

changes must be compensatory, for the quantities of commodities
have not been altered nor their aggregate value. If all of the I row
cisterns change so as to admit of a uniform percentage shortening

of ordinates without any commodity flowing out of any cistern , no
commodity will flow out, no prices will change and there will be
no change whatsoever in the distribution of commodities nor in the
valuation of money to other people. If one cistern shortens more
than this requirement, the effects will be analogous to those just
described for a single cistern.

• If all the cisterns of the A row are made shallower the price of
A will decrease.* That of other articles will in general increase.
In order that the distribution of commodities may not change, the
A cisterns must be so changed as to admit of a shortening of ordin
- - --

* Otherwise while the A ordinates shorten and their ratio to other ordinates
lessens, the back cisterns would have a relatively too great thickness compared
with the other thicknesses.


Irving Fisher - Mathemat

ates in a uniform percentage without loss or gain of commodity. In
this case the price of A will decrease while that of all other articles

will increase exactly alike.* The valuation of money will be re
duced since the ordinate of a B cistern (say) has not changed while
its back thickness has increased. The changes just considered may
be brought about if A suddenly goes out of fashion .
Perfectly analogous changes occur if a cistern or cisterns become


The individual is then more keenly “ sensitive ” to

changes of quantities. This change may occur through a discovery
by which a little of the commodity is made to “ go farther ” than

Reverse changes occur if cisterns are broadened or deepened .
$ 9.

It is impossible to combine all the A cisterns into a single demand

cistern for A as was done in Ch. II or to combine all the I cisterns
into an income cistern as in Ch. III, for we can no longer overlook

the influence of other commodities and other individuals.


analysis therefore which treats of but one commodity at a time and
constructs a demand curve for it is a superficial one for it does not

reach all the independent variables .

Suppose there are n individuals and m commodities in our given
isolated market during the given period and suppose the amounts

of the commodities A , B, C , etc., are given Ka, K , K ., etc., and the

given incomes of I, II, III, etc. are K ,, K ,, Ky, etc. Then the con
dition that the commodity -sums are given is:

+ A + A , + . . . . . . . . + An = K , )

+ B, =- KK ,.
B , + B , ++ BC ., ++ .. .. .. .. . . . .. AC.

C . + C


-- -- --- --- M ,+ M , + M , + . . . . . . .

= K. Į

- -+ M , K

m equations.

í mn unknowns.

* For their mutual ratios cannot change since the ordinates to which they are
proportional do not.



. in the theory of value and prices.
The condition that the incomes are given is :

A , · pa+ B, · pot . . . . . + M , - Pw = K , ?
A , · Pa + B , . Pot . . . . . + M , · Pm = K , ?į

n equations.
m new unknowns

. . . . . . . .. . . .. . . . .. . . .. . . . . . . . . . . . . .


An · Pa + B , · Pot . . . . . + M , · Pm = K ,
The utility functions (the cistern-forms) are:

dU E


= F (B ) ; : . . . ;


mn equations.


= F (À ,); je = F (B .); . . . . ; IN = F (M )

mn new

--- . .

(marg. ut.).

ax.=F(A.); 28.= F(B.);.... ;MN. F(M.)
The principle of proportion is :
du du du

da, dB,- 70, . . . am ,
du du

n (m — 1)


DA, -dB, . . . . . :

no new

- - -



AU _ m .mm

dA, - dB.: . . . . .


M . = Pa: Po:Pc:...: P.

Total number of equations : m + n + mn + n (m - 1) = 2mn + m


unknowns mn + m + mn + 0

= 2mn + m

Therefore allmagnitudes are determinate and the number of these
magnitudes as well as the number of the equations is twice the num
ber of commodities times the number of individuals plus the num
ber of commodities.
The valuation of money for each individual can be found from
the equations:


IV , $ 2.)

= etc. .


Irving Fisher - Mathemat
ical investigation


$ 11
For production the treatment is precisely parallel to the foregoing
(figs. 12, 13).






- =








= =



$ 1.

Hitherto it has been assumed that the quantities of commodities
and incomes (or expenditures) have been given. But these quanti
ties have themselves been determined by economic causes. Jevons*
arranges the sequence as follows:
“ Cost of production determines supply,
Supply determines final degree of utility,
Final degree of utility determines value.”

* Pol. Econ., Ch. IV, p. 165.

in the theory of value and prices .


This represents the chronological order but only part of the causa


Cost of production is not the sole determinator of supply .

Production is prophetic . When prices are steady the certain future
price is an unquestionable regulator of supply . Auspitz und Lieben

appear to me to deserve much credit for showing how all these facts
harmonize . Price, production , and consumption are determined by
the equality of marginal utility and marginal cost of production.*
Their clear exposition of this theory not only exbibits the “ funda
mental symmetry of supply and demand,” but reconciles in a
captivating manner the old one-sided and seemingly contradictory
theories of value making them fall in place as opposite facets of the

same gem . It is discouraging to find the old fight still going on.
Dietzelf attempts to play the peacemaker by the makeshift of
dividing the field between the contesting theories.
The apparent conflict grows out of an inadequate conception of
mathematical ileterminateness. As the quantity of any commodity
increases its marginal utility to consumers decreases while its mar

ginal disutility to producers increases. If the latter exceeds the
former the price which consumers will give is less than what pro

ducers will accept. Production is contracted and the utility and

disutility approach each other. If the quantity is too small the
machinery acts in the reverse way. The equilibrium though always

miscalculated is constantly sought and its more delicate and rapid

deflections are corrected by a special functionary, the speculator.
$ 2.

It is assumed that the rate of production during the given period
is exactly equal to the rate of consumption. This is asserting an
ideal equilibrium .
The expenses of transportation and retailing are included in “ pro

The principle of proportion previously explained is now extended.
The marginal utilities of consuming and the marginal disutilities of
producing are in the same continuous ratio for each individual- the

ratio of prices.
$ 3.
As the simplest case of combining production and consumption ,

suppose an individual to consume himself just that quantity of a
given commodity which he produces.
- - --

* Auspitz und Lieben , S5, p . 17.
+ Die Klassische Werttheorie und die Theorie vom Grenznutzen . Conrad 's Jahr
buch , 20 .


Irving Fisher - Mathematical investigations
14 .


In Fig. 14 the stoppers * or pistons S and S' which regulate the
quantities in the cisterns for production and consumption, respect
ively, are so connected as to move together, keeping the quantities
in the two cisterns equal. Furthermore the water pressure on them

from the tank keeps the level of all three liquids the same— that in

the tank and those in the two cisterns. The lever keeps the mar
ginal utility equal to the marginal disutility, for its pivot is a fixed
one and is placed midway between the axes of ordinates. The
resulting determinate equilibrium is subject to three sets of condi
tions :
(1) The quantity consumed equals that produced - a condition pro
vided for by the duplicate pistons.

(2) There must be a relation between the quantity produced and
its marginal disutility and between the quantity consumed
and its marginal utility — the character of the cisterns.

(3) Marginal utility and disutility are equal— the lever.
If A , and A , be the quantities of A produced and consumed ,
respectively, the conditions of equilibrium are :

A . = A { 1 equation .
* | 2 unknowns.


dA = F (A.)' | 2 equations.
40 = F (A ) !? 2 new unknowns.

dA. =

dU 11equation.
MA, S no new unknown.

No. equations: 1 + 2 + 1 = 4 .

No.unknowns : 2 + 2 + 0 = 4.
* In practice a more intricate frictionless bellows would be used .


in the theory of value and prices.


In the more general case there are n individuals and m com



Fig. 15 simply connects fig . 9 and fig . 12 by a series of new levers
like that in fig . 14, so that for each individual the ordinates of the
production cistern and its consumption cistern shall be equal. There

are also analogous horizontal levers (fig. 16) to keep the price for
16 .




- -




consumers equal to that for producers. The stoppers are all duplicate
as in fig. 14 for each commodity. Moreover there are analogous

duplicate pistons to keep each individual's incomes and expenditures

The industrial machinery is now seen to be self-regulative. There
is no arbitrary assignment of incomes or of commodities. The only


ical nvestigations

Irving Fisher - Mathemat

changes possible are effected by change in the forms of the cisterns
or by changing their number, that is by changing the “ cost” of pro

duction or the utility of consumption, or by changing the population
(which changes, we may remark , go together). By making the
cisterns removable and replaceable the effects of varied conditions

can be studied as in the preceding chapter.
However, this equilibrium is indeterminate in one respect. Unlike
the former it does not fix the unit of value. The sum of the

income-cistern -contents is arbitrary. If all duplicate income-and
expenditure -pistons are simultaneously depressed so as to increase
all incomes proportionately, the equilibrium will not be upset nor

will the distribution of commodities be affected. The rear cisterns
will simply dilate in uniform * ratio . The money standard has alone
changed .
This may be remedied by making the thicknesses of all back cis
terns for the commodity A equal to unity . A thus becomes the stand
ard of value, and henceforth all prices are in terms of this com
modity. This is what is done in the actual world.

A.1,1 + An, + ... + A.,n = Ax,1 + Ax,2 + . .. + Axon
B.7,1 + B1, 2 + . . . + B .,n = Bx,1 + Bx2 + .. . + Bkv m equations.
2mn unknowns.

. .. . . . .. . . . . . . . . .. . . . .

M .,it M .,2 + ... + Mnn = M 1 + Mx2 + . .. + Moni

) (n — 1) inde
A 1,1•Pat .. .. + M .. . Pm = Axıl. Pat ... + M «,1• Pm ! pendent equa
---- . .. .... .. .. .- -- --- - ---- ---- --- - ---- - --. į

A ,,n•Pat .. . . + M .

Pm = Arn : Pat . . . + Myn Poi m new un
ſ knowns (prices).

. du




TM 1 = F

F'(A7,1) ; . . .

TAK, = F ( A / I) ; . . .

- - -

= FC

U Mko1

- - - -

- - - -

- - - -

2mn equations.

- - -

. - -

. - -

- - - - - - - - - - -

DAF, = F (A ,,n) ; . .
= F (Akın) ; . . .


d M , = F (M ,,n)

= F (Mx,n)

* Cf. Ch . IV , § 8, number 7.

2 mn new un

(marg. ut.).

' in the theory of value and prices.







da , dB .. M , JA : dB ., . .-2M , =
du du
du du
dA. , AB , , . . . an ... : JA : dB . . . JM
. . .

. . . . . . - - - -

- - - . . . - . . - - - . . - . . . . . . . . . . . . . .








dB. .. .2M ndAconi a Be ...AM =

e onations



dᏌ .

- Pai - Po : . . : - Pm : + Pa : + Po : . . : + Pm
No. equations: m + (n

| (2 m - 1) n

no new


1) + 2mn + (2m – 1 ) n = tmn + m - 1

No. unknowns: 2mn + m + 2mn + 0

= 4mn + m .

There are just one too few equations. It may not be evident at
first why the second set does not contain n independent equations

instead of (n - 1). The point is that any one of these equations can
be derived from the others together with the equations of the first
set. Thus multiply the equations of the first set by Pa Po . . . Pm
respectively and add the resulting equations arranging as follows:

A7,1 · Pa + B .,1. Put . . . + M1,1· Pmt ?

+ A7,2 · Pa + B7,2 • Po + . . . + M7, · P. t.

-- -. .- .- .- .- .- .- . -. - . -. - .- -. -- -. . .. ..

+ A .,n . Pa + Brenº pot . . . + Mn,n · Pm )
[ A ,1 · Pa + Bk 2• po + . . . + Mk,* ·

} + Ax,2 • Pa + Bk,2 • Pi + . . . + M & • P'm +
. - . - . - -.

. . . . . . . . . .. . . . . . . . . . . . .. . . .

+ Arn : Pa + Bxr · Pot . . . + Man: Pm .
Subtracting from this equation the sum of all but the first (say )
of the second set, our result is :

A. ,1 · Pat B7,1 · Pot . . . + M .,1 . Pm =
Ak,1 . Pat Bk1 . Pot . . . + Mx, 2· Pm

which is the first equation of the second set. This equation is there
fore dependent on the others, or there is one less independent equa
tion than appears at first glance. Hence we need one more equa
tion . Wemay let :
Pa = 1.

This makes A the standard of value (cf. $ 5 ).
No such limitation applies to the equations in Chapter IV .


ical nvestigations

Irving Fisher — Mathemat


$ 1.
Without dwelling on the economic applications of the mechanism

just described we hasten on to the description of a more complicated
mechanism .

Production usually consists of a number of successive processes .

The last of these is retailing. Let us group all other processes
under the head of production. The price for production and con
sumption are no longer equal.

Hitherto we have had two sets of cisterns, the production set and the

consumption set. Separate now , these sets far enough to introduce
a third set for exchange or retailing as in fig. 17.

The exchange set is a series of double cisterns each related to a
particular commodity,and a particular person . Consider the cistern

IA for instance (the sub-letter for exchange or retailing). In the
front compartment is the quantity of A which I buys and sells or
transfers from producer to consumer. The back compartment con
tains the money pay for doing it.
These exchange cisterns are connected with each other and with
the production set by levers precisely as if they were so many new

commodities produced.


in the theory of value and prices.

So also the rodsmaintain a constantmoney rate for exchange; in
stead, however, of the former simple relation between the producers
and consumers there is now the following that the sum of the ordin

ates of A I., and A I , equals the ordinate of Alk, and likewise for
II, III, etc., also that the thickness of the back cisterns of A , plus
that of A , equals that of An. These results are effected by parallel
rulers, those for the former purpose being represented in fig . 17.

The new machinery required for the exchange process consists

then (1) of triplicate pistons* which necessitate that the same
quantity of A shall be produced , exchanged , and consumed ; (2) the
additional rods and levers (horizontal and inclined ) to make the

marginal disutilities of producing and exchanging proportional to
the recompense and which also maintain a constant price for exchang

ing the same thing ; and (3) the special contrivance to add the

marginal disutilities of producing and exchanging for any individ
ual so as to equal that of consuming, and also equate the sum of
the prices of producing and exchanging to that of consuming.

A..,1 t . .. + A1.n = A6,1 + .. . + Ac, = A \,1 + . .. + Akon ) 2 m
- - - - - - - - - - - - - - - - . . . . . . . . . . . . . . . . . . .. . . . . . .



| 3 min
M . , + . . . + Mon - Mc,1 + . . . + Mon = Mx, + . . . + Mkn )


A7, Part + . .. + M .,1P.,7 + A ., Page + . .. + M .,1 Pm,e= Ak, Pok + ... + Mxi Pmore
-- - - - -- - - -- -- -- - - - - - -- - - -- -- - - -- - - . -- . - -. - - -- - -- - - -- - - -- - -- -

A.,n Patt.. . + M ,1,n Pm,8 + Ac,nPave to.. + MenPm,e = Axın Park + .. . + Men Pink
n - 1 independent equations. 3 m new unknowns (prices) .


_ F (1 731)

- = F (A ,,1) ; . . . ;

DA ,


dA . =

F (A )) ; . . . ;


DAC, =

F (AX,1) ; . . . ;

- = F (M )
dU - =

F (M ., ) )

3 mn equations.
į 3 mn new unknowns


dA = F (A ,,n); . . . ;

- = F (M _ ) 1 1 (marg. ut.).

TT ,



dAC, = F (Ac,n) ; . . . ; M

= F (M .) |


dAkin = F (A ..») ; . . .;

= F (M «,n)

* The income and expenditure-pistons are merely duplicate as before.

Irving Fisher — Mathematical investigations





dA. .. UM , JA,







) n (3m - 1)

-AM = | independ


ent equa

dA . MM., da ,:..-JM , JA... .--dM ..=


- - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - . . =

no new








dAFT .-- MandACXdM ., d . ..-AM ,na

.. .:Pm,

wns ,

:Pare i. . :Pm,e : – Paki.. : – Pm,
Pa,y + Pane = Paix ) m equations.
-- -. . .-. .. . . ...

P .,g + Pm,e = Pmin ) no new unknowns.
No. equations : 2m

+ (n - 1) + 3mn + n (3m - 1) + m = 6mn + 3m – 1.

No.unknowns: 3mn + 3m

+ 3mn + 0 .

+ 0 = 6mn + 3m .

The second set apparently contains n equations instead of n - 1

as above recorded . But, by multiplication of the first line of the
first set, we have :

(AT + . . . + A ,,n) Pat = (Ax,1 + . . . + Axın) Pa,
(Ac,1 + . . . + Ac,n) Page = (Ax,1 + . . . + Akin ) Pare
adding and remembering that Pak = Pat + Pare we get :

A ., 1 · Part . . . + Anni Part Ac, 1 • Pase + .. . + Ac,n. Pane =
Ak,1• Parkt . . . + Axon • Pa,K

Writing the similar equations from the second, third , etc. lines of the
first set and adding we get (rearranging terms):

A7,1•Part .. + M .,1• Pm,n + A ,1.Page + .. + M ,1. Pm,et ?
+ A .,2. Part . . + M7,2• Pm,it - - -

.. . . . .. ..


; :

- - - - -- - - - - - - - - - - -

.. . . .. .. .

- --- .. .. + Men : Pm,ej


+ 4 ,1,n · Part -- --. .

r Æg,1• Pax + . . . + Mx,1• Pmx +
+ Ak,2 · Pa,x + -- -- - -- -- - -- -- --- - ---

- - - . .. - - - - - - - - - - - - -

+ Ann · Part . . . + Mane Pmir
If from this equation the sum of all but one of the second set be
subtracted the result will evidently be the remaining one.
Weare therefore at liberty to write
Park = 1
to determine a standard of value.

in the theory of value and prices.


$ 3.

An analogous mechanism and discussion applies to the separation
of production into retailing, wholesaling, transportation and even the
various technical processes distinctive of the production of each
commodity. In making worsted for instance there are some 16

processes having this sort of dependence.

The reactions and equilibrium in the real world are still more
complicated than those here presented . Notonly is there equilibrium

in one market as New York city, but a mutual dependence of vari
ous markets. The rate of transportation determines in part the
amount of dependence and the amount of communication deter
mines in part the rate of transportation . As Cournot* says, “ * *

le système économique est un ensemble dont toutes les parties se
tiennent et réagissent les unes sur les autres."
End of Part I.
* Principes mathématiques, Ch. XI, p . 146.


Irving Fisher - Mathematical investigations




§ 1.

Hitherto it has been assumed that the utility of a commodity is a
function of the quantity of that commodity alone. It is true that
it depends upon that quantity more than any other and the analysis
of Part I is a necessary first approximation . In astronomy the

attraction of the sun on the earth is first studied alone to determine
the earth 's motion ; next the moon 's influence is admitted, then the
occasional “ perturbations ” due to planets and comets. Absolute

accuracy is never attained for the earth's motion is a function of
the mass and position of every body in the universe.

So also the utility of the 100th lb . of butter (100 lbs. per year)
depends mostly on that 100 lbs. It would not be perceptibly in
fluenced by a change in the quantity of clothing, but it would be

perceptibly reduced if the amount of bread consumed were reduced
from 300 loaves to 200, for bread and butter go together.
It is needful here to distinguish carefully between two ways in

which the quantity of one commodity can affect the utility of others.
Even under the supposition of Part I, a change in the price of
clothes effected a change in the individual valuation of money and

so changed the quantity of bread consumed and so in turn changed
the marginal utility and price of bread . But under our new sup

position, a change in the price of butter directly changes the utility ·
of the same quantity of bread . In the first case marginal utility of
bread can change only after a change in its quantity . In the second
the marginal utility of the same amount of bread changes; the first
contemplates a variation in the quantity of water in a cistern , the
second contemplates a variation in the cistern wall itself.

. assumed
. .: dU _= FN ( A ) ; but now we must write:
In Part I we
we dusuncu
" Nil

= F (A ,, B , C , . . . M ).

in the theory of value and prices.


$ 2.
It will be seen that this sort of dependence of particular commodi
ties is very common. Articles are 'bought with reference to each
other, oil with reference to the number of lamps used, bed linen to
the number of beds, bureaus to the quantity of clothes to be stored ,
carpets to the amount of floor rented or built, bookcases to the

number of books owned ; the demand for steel rails is connected
with that for railroad ties, that for locomotives with that for cars ,
etc .

Again in production , the “ peculiar cases of value ” of which
Mill* speaks and which Jevonst treats come under the samehead ;
coke and coal gas; mutton and wool; beef, hides, and tallow , etc .
The cases above instanced are cases of “ completing” ! articles.

Under the head of “ competing ” # articles, come, mineral oil and
other oils, various “ qualities ” of any article as meats, grades of
flour, etc., while under production almost every two articles are
competing. A man in one business does not wish to meddle with
another or, otherwise expressed, the marginal disutility of produc
ing 1,000 tons per year of coal is increased if the producer attempts

to run a paper mill or trade in jewelry.
$ 3.
Introducing this new dependence of utilities, it is seen that, if the
cisterns contain at one point of equilibrium the proper amount of

water and have as ordinates the proper marginal utilities , as soon as
any income or commodity stopper is pressed , not only does the
water redistribute but the shapes of the cisterns change. If the
quantity of bread is increased, the cisterns for biscuit may shrink
and those for butter widen. That is the ordinate (marginal utility)

for the same quantity of biscuit decreases, and of butter increases.
The general effect is to keep the ratio of marginal utilities of bread

and biscuit and so also their prices nearly constant, while the

cheapening of bread may directly increase the marginal utility and
price of butter irrespective of its quantity .
The essential quality of substitutes or competing articles is that
the marginal utilities or the prices of the quantities actually pro

duced and consumed tend to maintain a constant ratio . Wemay
* Bk . III, Ch . XVI.

# Auspitz und Lieben , p . 170.

+ Page 197.

JULY, 1892.

cal nvestigations

Irving Fisher - Mathemati


define perfect substitutes as such that this ratio is absolutely constant.

The essential attribute of completing articles is that the ratio of

the quantities actually produced and consumed tends to be constant
(as many shoe-strings as shoes for instance, irrespective of cost).
Wemay define perfect completing articles as such that this ratio is

absolutely constant.

If we suppose each set of competing and completing articles to
be “ perfect,” it is possible to arrange the cisterns so that the change

of form of some cisterns as due to change in the contents of other
cisterns shall be small or nothing. Thus if four grades of flour be
“ perfect ” competing, so that their marginal utilities are always in

the ratio 8, 9, 11, 17 ,we may form a joint cistern for individual I
whose contents shall be “ flour,” the quality unspecified .


cubic unit of liquid shall represent equivalent quantities of each
grade, i. e. barrel of the first quality, \ of the second, 1 of the
third or 1 of the fourth, while the ordinate shall represent the com

mon utility of any one of these equivalent quantities.
If four completing articles as the parts of a coat, sleeves, pockets,

buttons, and coat proper are always produced and consumed in num
bers proportional respectively to 2, 4, 3 and 1, we may form a joint

cistern for individual I whose contents shall be “ coats,” parts un
With such combinations as these, the cistern analysis of Part I
will represent the economic relations fairly well and almost per
fectly if the deviations from equilibrium are not followed too far.

But few articles are absolutely perfect representatives of either
the competing or the completing group, and a member of one group
may also belong to another. Thus butter is completing to bread
and biscuit, and although a cheapening of bread directly increases

the utility of butter it indirectly increases it by decreasing the use
of biscuit.

It is readily seen that the interrelations of the shapes of the cis
terns — if we now treat each quality of meat, etc. and each part of a
utensil as a separate commodity — are too complicated even to be
mentally representable without some new mode of analysis.

$ 5.

The former analysis is incomplete, not incorrect. All the inter
dependence described in Part I exists , but there also exist other

connections between the shapes of the cisterns which could not be
mechanically exhibited . For any one position of equilibrium the


in the theory of value and prices.

cistern mechanism may represent accurately the quantities, utilities ,
and prices, but the shape of each cistern is a function of the whole
state of equilibrium and differs as soon as that differs. However in

general the interdependence in the shapes of the cisterns is very
slight. That is, the utility of a commodity usually varies so much
more under a variation in the quantity of that commodity than
under variations of other commodities that the relations discussed

in Part Imay be regarded as good first approximations. Especially
is this true if the interdependent commodities are grouped as in § 4,
so as to eliminate all the really important influences of commodities

on each other.* It will subsequently appear that the analysis of
Part II is also incomplete and so will it ever be. Neither economics

nor any other science can expect an exhaustive analysis.
$ 6.

Recurring to the definitions of utility as a quantity (Part I, Ch. 1),
it will be noted that the third definition which indicated the ratio of
two utilities was based on the assumption that the utility of each

commodity was independent of the quantity of any other com
modity. This assumption was necessary to prove that two applica
tions of def. (3 ) led to harmonious results (Part 1, Ch. I, § 4 ). To

abandon this assumption as we have now done is to forego the use
of that third definition. At the close of Part II a further discus

sion of “ utility as a quantity ” will be given . At present we con
tent ourselves by assuming the marginal utility of a given amount

of some one article as our unit of utility. Of course if we should
use some other marginal utility as a unit, themeasurements will not
now agree. This, however, is no calamity . It will presently appear

that the meaning of the phrase “ one utility is twice another ” is of

no real importance for the subject in hand.
$ 7.
Confine attention first to two commodities (a ) and (6 ) consumed by
one individual. Let this individual first arrange his whole consump
tion combination to suit himself. Then in order to partially analyze
this equilibrium of choice let us metaphorically experiment on him
* Marshall, Prin . Econ ., Math , note xii, p. 756, says: “ Prof. Edgeworth 's plan

of representing U and V as general functions of x and y [see preface to this
memoir ] has great attractions to the mathematician ; but it seems less adapted
to express the every day facts of economic life than of regarding, as Jevons did ,

themarginal utilities of apples as functions of x [the quantity of apples ] simply.”,

Irving Fisher — Mathematical investigations


as follows. He is directed to alter this consumption combination
by arranging bis quantities A and B of the two selected commodities
(a ) and (b) in all possible ways, but without changing the quantities

C , D , etc. of other commodities. The marginal utility of each will
vary not only in relation to its own quantity but also the quantity
of the other commodity.


A = F (A , B,)

dB = F (B , A ,)
These may be regarded as derivatives with respect to A and B of

U , = P ( A ,B ,)
where U , is the total utility to I of the consumption combination
A , and B .

In fig. 18 let the abscissa OX represent the quantities B , of (6) and

theordinates (OY) thequantities A ,of (a ).

Any point P by its co -ordinates represents

a possible combination of quantities A ,
and B , consumed by I. By varying point

P all possible combinations of A , and B ,
are represented . At P erect à perpen
dicular to the plane of the page whose

length shall representthe marginal utility
of A , for the combination , that is, the


degree of utility of a small addition of
A , ( B , remaining the same). If Pas

sumes all possible positions, the locus of the extremity of this per

pendicular will be a surface.

Again at P erect a different perpendicular for the marginal utility
of B,; its extremity will generate another surface. The first sur
face takes the place of a utility curve for (a ), the second for (6).
These two surfaces may be regarded as the derivative surfaces
(with respect to the variation of A , and of B,), from a primitive
whose ordinate (perpendicular at P ), is the total utility of the com

bination of A , and B , represented by the point P. This surface is
usually convex like a dome with a single maximum part ,but it need

not always be. There may be twomaxima as will presently appear.
In such a case it cannot be everywhere convex.

If a plane be drawn tangent to this last surface at a point over P,
the slope of the plane parallel to the A direction will be the ordinate

in the theory of value and prices.


of the first derived surface ; i. e., will be the marginal utility of A ,,
while the right and left slope will be the marginal utility of B , or
theordinate of the second derivative surface . The primitive surface
thus supplies a convenient way of uniting in thought the two mar
ginal utilities. Its absolute height* above the plane of the paper is
of no consequence ; it may be lowered or heightened without dis

turbing tangential directions or affecting its two derivatives.
$ 8.

The three surfaces thus constructed need not extend indefinitely

over the plane. They may approach vertical plane or cylindrical
asymptotes so that for some points in the plane there may be no
surface vertically over or under

Mathematically the total utility and marginal utilities at these
points are imaginary. Economically it is impossible that the indi
vidual should consume quantities of (a ) and (b ) indicated by the co
ordinates of such points. Those parts of the plane where such
points are may be called “ empty.”

If (fig. 18 ) the point P moves vertically (up and down on the page)
the extremity of the perpendicular for the total utility describes
one of Auspitz und Lieben 's curves for An, it being understood how
ever that the quantities of other commodities do not change.I
The perpendicular for the marginal utility of A , generates in the
first derivative surface a Jevonians curve of utility for A , it being
understood that B , C , etc. are constant. This curve will usually
descend but it may not and cannot in certain regions if the surface
is derived from a primitive with two maxima, or any concave primi
tive. The other perpendicular, however, traces a curve which has

never been used , viz : one which shows the relation between the
quantities A , and themarginal utility of B , while B , remains con

stant. This curve will in general descend or ascend according as
the articles (a ) and (b) are competing or completing. For instance,
* It is in fact the arbitrary constant of integration .

+ This “ asymptote ” and “ imaginary ” interpretation appears to cover the
class of difficulties which led Marshall to say his curves failed to have meaning

at points at which the individual could not live.

# It is rather, then , an “ Elementarkurve ” of a “ Lebensgenusskurve ” there
being an “ anfangsordinate .”

Jevons' curve is evidently the derivative of Auspitz und Lieben's. See table

Appendix I, Diyision II, S 2.


ical nvestigations

Irving Fisher — Mathemat

suppose (a) and (6) are two brands of flour. If I consumes during
the period X units of one brand and 20 units of the other his desire
for a 21st unit of the latter will depend on how much he has of
the former (how large X is). If he has much of the first kind his

desire is small.

A similar pair of curves may be found by moving P horizontally .
If the supposition in Part I were true the two strange curves (viz:
connecting marginal utilities of A and B with quantities of B and

A , respectively), would reduce to straight lines parallel to the plane
ofthe paper.
$ 10.

The relations indicated by these three surfaces are really all
included in one of them — the primitive. Consequently , to avoid
troublesome transitions from one mode of representation to another

we shall hereafter confine ourselves to this primitive surface .

Consider horizontal sections of this surface, that is sections par
allel to the plane of the A and B axes. Each section forms a curve

which may be called an indifference curve. It is the locus of points
representing all consumption-combinations of A and B which have a
given total utility. In fig . 18 the attached number to each curve

represents the amount of this utility. They in general form a
family of concentric curves vanishing finally at the point M of max

imum satisfaction . M is the point at which the individual would
arrange his consumption -combination of A and B if they cost noth

ing. There may be two or more maxima. For competing articles
these maxima may lie in the axes (fig . 19), for one may prefer not to
The ordinates may of course have any units of length. Suppose

this unit to be indefinitely reduced from an
inch to a millimeter, etc. Then our surface

becomes a layer. Its thickness may be fig
ured as a density (rather than an ordinate),
distributed over the plane of the paper as
electricity over à conductor. Each indif
ference curve is the locus of points where
the density (formerly ordinate ), is a given
amount. This idea of density will behence

forth used though the necessity for its use
does not come till the next chapter .
Fig . 20 shows the curves for competing articles and fig . 22 for
completing. For “ perfect” substitutes the curves (fig. 21) reduce


in the theory of value and prices.

to parallel straight lines whose intercepts on the A and B axes are
inversely proportional to the fixed ratio of their marginal utilities.

The point M is indeterminate on the line 99. “ Lehigh ” and


20 .







“ Lackawanna ” anthracite coal are nearly perfect substitutes. If it

cost nothing the individual would indifferently consume the quan




B "

tity 09 (vertical) of one or 09 (horizontal) of the other or any

combination of the two on the straight line 99 inclined in this case
at 45°.

For perfect completing articles the whole family of curves

reduces to a straight line passing through the origin (fig. 23). Let
us regard a pair of shoes as two distinct commodities : right shoes
and left shoes. For any point in the line OM (fig. 23), the desire

for right shoes vanishes as long as no new left shoes are admitted ,
and yet the desire for a new pair may exist. The idea of marginal
utility for right shoes has no application though that for pairs of
shoes has.
$ 11.

There are endless points of view from which the primitive and its
derivatives may be approached and made to yield the economic

relations we seek .* Descriptions will be confined chiefly to the
* For instance we might take curves corresponding to the sections of the deriv
ative surfaces at various heights, or curves orthogonal to the indifference curves
these will be again referred to ), or curves representing the locus of points at
which the marginal utilities of the two commodities have a given ratio .

Irving Fisher - Mathematical investigations
indifference curves, the tangents and normals to which play an
important rôle .

When our individual fixed his whole consumption combination to

suit himself, let us suppose that he spent $25 per year on the two
articles fa and b) under consideration. Wemay metaphorically
compel him , while not altering in the least his purchases of other
articles and hence having the same $ 25 to spend on (a ) and (6 ), to

contemplate spending it in a different way. If the price of (a) is
$0. 25 and of (6) is $0.50, the two simplest methods of spending his
$ 25 is to spend it all on (a ) and purchase 100 units, or to spend all
on (6 ) and purchase 50 units.
In fig . 18 lay off OA = 100 units and OB = 50 units. Then any
point on the straight line AB will represent a consumption combina

tion of A and B purchasable for $ 25.* AB may be called a partial

incomeline. Our individual is therefore left free only to select his
combination somewhere on this line. The combination 5 or 5 pre
sent equal inducements but not as great as 6 or 6 on an arc of

greater utility, nor there asmuch as at I. Hewill select his combtha
tion in such a manner as to obtain the maximum total utility, which

is evidently at the point I where AB is tangent to an indifference
curve.f At this point “ he gets the most for his money."

His selection I is of course just what it was before we began our
analysis. But we have advanced one step. Wehave partially anal
yzed this equilibrium , that is we see the equilibrium for A and B
while the prices and quantities of other articles remain the same. It
is as if a pendulum free to swing in any vertical plane is found at

rest and a scientist attempts to analyze its equilibrium . He forth
with confines its motion to a single plane and discusses its equilib
rium there. The analogy suggested may be extended . The prin
ciple underlying the equilibrium of a pendulum or any mechanical
equilibrium (as of a mill pond or of a suspension bridge) is : that

configuration will be assumed which will minimize the potential. So
also the supreme principle in economic equilibrium is : that arrange
ment will be assumed which will maximize utility 1.
-- --- -

* Proof: Equation of AB is


+ 20 = 1 where x and y are the co-ordinates

of any point on AB. This becomes y .


+ x . 00 = 25 ; that is, ac times

its price + y times its price equals $ 25 .
+ When AB is tangent to two indifference curves that one will be selected
which has the greater utility.
# See interesting remarks, Edgeworth : Mathematical Psychics. Also in his

address as Pres. section Econ. Sci. and Statistics Brit. Asso., Nature, Sept. 19 ,
1889, p. 496 .


in the theory of value and prices.
$ 12.

Since OA and OB represent quantities A and B of commodities
( a) and (6) purchasable for the same sum ($ 25 ), they are inversely

proportional to the prices of (a ) and (6).
If prices remain the same but the individual grows richer and the

sum he can afford to spend on (a ) and (6) is no longer $ 25 but $50,

the line AB simply recedes twice as far remaining parallel to itself.
As it changes, its varying point of tangency follows a tortuous line
the locus of all points at which the individual would arrange his
combination of A and B at the given prices.
If the price of (a ) increases, O A relatively diminishes and a new

point of tangency is found. If the articles are completing (fig . 22)
a change of price will not cause the tangent line to very greatly

alter the proportion of consumption of the articles for it will merely
change the position of I to (say) I', and it is clear that the coördin
ates of l' have nearly the same ratio as those of I ; if substitutes
( fig . 20) a slight relative change in price will cause an enormous

change in the proportions used (I and I'). This was found to be the
case in 1889 when a copper syndicate attempted to raise the price
of copper. Hardly any article exists which has not some substi
tute . This sort of dependence keeps manufacturers watchful. It

is because of this dependence that some “ useful” articles go out of
use .

$ 13.
Fig . 24 represents two “ grades ” of the same commodity, as brown

and granulated sugar. The superior grade is laid off on the B (hor

izontal) axis, and the inferior on the

24 .

A (vertical) axis. The point of maxi

mum satisfaction is in or near the B
axis. If the individual is poor and can

afford to spend little on the article he
will buy the poorer quality. The line
AB is tangent to an indifference curve

in or near the A axis at I. If he grows

richer the line AB recedes from the
origin and he purchases the combina

tion I' containing considerably more of



B ; he uses this superiorquality on Sun
days (say) while consuming A on week days. If he grows richer

still, he changes the position to I" using none of A or only a little.

Irving Fisher — Mathematical investigations

The inclination of the line AB is such that ( A > OB that is A is
cheaper than B, for OA and OB are the quantites of A and B pur
chasable for the samemoney. If the prices of A and B were equal
so that OA = OB , it would not be tangent to an indifference curve
unless on the B axis and A would go out of use.

Moreover it is evident that a slight variation in the relative prices

of A and B will change greatly the position of I for a poor man but
will not change materially that of l" for a rich man.
If the poor consumers predominate the line AB will follow the
general trend of the curves near the origin . If the rich consumers
predominate the line AB will become steeper (as in the dotted posi

tions). That is the two prices of the two qualities separate widely .
This interprets the fact that in a rich market like New York City

a slight difference in quality will make an enormous divergence in
price while in some country towns different grades either do not exist
or sell for nearly the same price. In the country districts of “ the
west ” all cuts of beef sell for the same price (about 10 cts. per lb .).

In the cities of the west two or three qualities are commonly dis
tinguished , while in New York a grocer will enumerate over a dozen

prices in the same beef varying from 10 to 25 cts. per lb .
$ 14.

In fig . 25 if the individual III attempts to change the position of

III he may do so in many
different “ directions.” If he
changes in the direction III
&, he will increase his con
sumption of A without alter

ing that of B or if toward B ,
III B , without altering A , if

in an intermediate direction,
III O , he will increase both A
and B and in the ratios of

the components of that direc

tion (III a and III B). The
direction of maximum in
crease of utility is perpendicular to the indifference curve.* We
may figure III S as a force. If III were in any other position the
force would evidently have a component along the line A , B , and
would move III back to the position of equilibrium III.
* For between two infinitesimally distant indifference curves the shortest route
is on their perpendicular.

in the theory of value and prices.


We may call the perpendicular direction III Ở the “ maximum


It has the important property that its components III a

and III B are proportional to the marginal utilities of A and B .
This follows from a theorem * of vector calculus or thus : III a and
III B are inversely proportional to 0A , and OB,, that is directly
proportional to the prices of A and B and therefore proportional to

their marginal utilities.
$ 15 .

If (fig . 25) the separate curve systems of all individuals I, II, etc.
are drawn, and the lines AB drawn in each case , they will be paral

lel. For the prices are uniform among all individuals and OA and
OB in each case are inversely as the prices.
Since the normals to these lines will also be parallel, this theorem
may be stated : The “ maximum directions ” of all are alike.

$ 16.
These methods apply to the comparison of any two commodities

and afford a means of graphically representing statistical relations
connecting the demands for two articles so far as the variations in
the quantities of other articles can be eliminated

The same principles apply to the production of two articles. Hides
and tallow are completing articles from a producer's standpoint.
Likewise coke and coal gas, mutton and wool, and in general any
article and its “ secondary product.”
On the other hand most articles are competing or substitutes from

a producer's point of view .

The difficulty of producing cloth is

greatly increased if the same individual produces books. This is
the root of the principle of division of labor and leads to that im
portant contrast between production and consumption once before

alluded to. This and other contrasts will be mentioned in Appen

dix II, § 8. Marshall and others are fond of using the expression
“ fundamental symmetry of supply and demand."

This notion must

be supplemented by that of a “ fundamental asymmetry.” As social
organization progresses each man (and each community or nation)
tendsto become producer of fewer things but consumer of more.
* Gibbs, Vector Analysis , 88 50 -53.


| For by similar triangles : il 3 = 11L 3 = OB
0 , =



Irving Fisher - Mathe




Fig . 26 shows the usual sort of indifference production curves.
26 .


B is here laid off to the left and A downward ;
the line AB is the locus of production combina

tions of A and B which can be sold for the
same money, say $ 1,000. The point of tan
gency * I is the point at which the individual can
produce the required $ 1,000 worth of A and B

with the minimum disutility. The curves are
such that the points of tangency will be gener
ally at or near the axes, especially if the amount
of production is large i. e. if the line AB is far

from the origin. If B becomes cheaper (OB longer) the point of
tangency will change but slowly until presently there are two points
of tangency and if B becomes still cheaper the individual will change

his profession suddenly from the position I to a position in or near
the A axis.

The numbers on the indifference curves for production increase in
definitely negatively .

There is usually no maximum or minimum


$ 17.
Finally an article consumed may be competing or completing to
another produced. A blacksmith finds small utility in dumb bells;

the production of horseshoes " competes ” with the consumption of
dumb-bells .

The relations between competing articles and completing articles
are not always so simple, for articles may be competing at some
combinations and completing at others . Statistical inquiries along

these lines might be made with profit, and have apparently attracted
little attention .t

$ 1.
The foregoing methods extend very readily to three dimensions.

Suppose the whole market to attain equilibrium . As before, let us
as it were,freeze this equilibrium except for three commodities A , B ,
and C . Then as before,we obtain a fixed sum of money disposable
* The tangency must be such that the curve is on that side of the straight line
toward the origin . The other kind of tangency represents an unstable equilib
+ See Jevons, p. 135 .
rium .


in the theory of value and prices.

for the purchase of A , B, and C , by each individual. Construct
three mutually perpendicular axes (OA, OB, OC,) in space. Con
ceive this space to be filled with matter whose density distribution

is the total utility for A , B, and C ,relative to a particular individual
I. There may be " empty ” portions of space. The locus of points
representing combinations of A , B , and C , possessing a given utility

will be an indifference surface. All such loci will form a “ family "
of concentric surfaces like the coats of an onion around one or more
points of maxima.
Lay off on the A axis OA , equal to as many units of A as can be
bought for the sum of money disposable by I for the purchase of
A , B , and C. Lay off OB and OG similarly defined . Draw the
plane A B C . This is the locus* of all consumption-combinations
of A , B , and C, purchasable with the given sum of money. It is a

“ partial income plane.” Its point of tangency with an indifference
surface will mark the chosen combination. A normal at this point

indicates the “ maximum direction ” and its A , B , and C components
are the marginal utilities, proportional to the prices of A , B , and C.
§ 2.

The utility distributions may be very complicated. If the three
articles are substitutes like oats, corn , and rye, the indifference sur
faces may be almost plane and will allow but little change in the
orientation of the partial income plane, while each slight change
shifts the point of tangency greatly (cf. fig . 20 for two dimensions).

If they are completing articles as cuffs, collars, and ties the indiffer
ence surfaces are arranged like concentric cocoons directed toward
the origin (cf. fig . 22 for two dimensions).
But the three articles may be more intricately related in utility.
Of tea, coffee and sugar, the first two are substitutes while the last
is completing to both . If this triple completing and competing rela

tion of articles were “ perfect,” the utility distribution would reduce

to a plane passing through the origin and cutting between the
“ sugar ” and “ tea ” axes, also between the “ sugar ” and “ coffee ”
axes. Several characteristics of such an ideal utility dependence
would exist. If the triple dependence is not “ perfect ” the plane

referred to swells out into a flat disk or rather a “ family ” of con
centric disks. The triple variation of prices and its effects on the



ñ = 1, whence : A .
* For its equa. is a + OB5 + oc

or Apa + Bpo + Cpc = 50.



+ B .00
OR ++ C .OC

= 50


Irving Fisher — Mathematical investigations

relative amounts of the three articles (that is on the position of

the point of contact) can readily be discerned by its aid . Far more
complicated cases are supposable and exist in reality.
$ 3.

If we suppose for an instant that there are but three commodities
in the market, the preceding analysis yields a complete account of
the equilibrium in that market.
To sketch this briefly let us suppose the space to be filled with a

utility density for I, another superposed but different distribution
for II, and so on . Let us include production . If one man should

be both a consumer and producer of the same article,the net con
sumption or production is now to be taken , and the totalutility or dis
utility of this net amount is the density. The planes before referred

to as partial income planes may now be called “ total income and
expenditure planes,” and they must each pass through the origin *

(OI, fig. 27 for two dimensions). Since the “ maximum directions”
(normal to their planes) are parallel, these planes must all coincide.
The point in this plane selected by I will be that of tangency to an
indifference surface for I. Likewise for II , III, etc. Such points
* For since income balances expenditure, if A1, B , C , represent the (net)
amounts consumed or produced by I, those consumed being treated as positive,
and those produced as negative, the whole money value must be zero : i. e.

A . Pa + B . Po + C . P . = 0 ,
which is the equation of a plane passing through the origin .

in the theory of value and prices.
could be found whatever the position of the plane. But the plane
must assume such an orientation that the center of gravity of these

points shall be the origin . That is the algebraic sum of all the A
coördinates consumed must equal the sum produced . Likewise the
algebraic sum of the B and C coördinates must each be zero.

Hence with the geometrical analysis just described the equilib
rium for a market of three commodities is determined when :
(1) All individuals' combinations lie in a common plane through
the origin (each individual's sales and purchases cancel).
(2 ) Each individual's combination is at the point where this plane

is tangent to an indifference surface for that individual (the point of
maximum net utility).
( 3) The points in the plane are so distributed as to make the origin

their centre of gravity (the production and consumption of each com
modity balance).
Whence it follows geometrically that the “ maximum directions "
are parallel, their components (marginal utilities) proportional as

between different individuals and that this proportion is that of the
orientation of the plane (the ratio of prices).

When this equilibrium is attained, let us, through the point of

tangency I, representing the consumption combination for I, pass a
section parallel to the plane of the A and B axes. The section of
this plane with the total income plane gives a straight line which is

none other than the partial income line of Ch . I, § 11 and its section
with the indifference surfaces gives back the indifference curves of

Ch. I, $ 10.
$ 5.

We have temporarily assumed only three commodities for we have
only three dimensions wherewith to represent them .

A complete

presentation of the interdependence of utilities would require m

dimensions, for the utility of any one commodity A , is subject to
m independent variations according to a change in any one of the

m commodities, though (in general) the change of the quantity A
itself is most important.
There is a curious glamour over “ the fourth dimension.” The
popular interest is all to prove that it “ exists.” Its origin histor
ically and its present usefulness is in the interpretation of a fourth

independent variation, i. e. in representing just such relations as now


Irving Fisher — Mathematical investigations

concern us. It seems unfortunate that only mathematicians should

be acquainted with this fact.
$ 6.

In this m dimensional space make m mutually perpendicular axes
for the commodities A , B , C , . . . M . Fill the space with a total

utility density. Pass an m - 1 flat* through the origin giving it the
proper orientation in view of the prices. The indifference loci will
be (m - 1) spaces (curved). The point of tangency of the (m - 1)
flat with an (m - 1) indifference locus will indicate the total con
sumption and production combination for an individual. A normal

· to the (m — 1) flat and (m - 1) indifference locus at their tangency
shows his “ maximum direction ” and its components the marginal
utilities of all articles.

These ideas are not so unfamiliar as they appear. This space is
simply the “ economic world ” in which we live. We often speak of

spending an income in this or that “ direction,” to express the rela

tive amounts of commodities. When one speaks of the “ point”
which a consumer or producer reaches, the use of the word is a
natural attempt to group in thought m different magnitudes. This
is accomplished by regarding them as coördinates of a “ point” in

the “ economic world.” It is an application to economics of those
ideas of “ multiple algebra ” which have addedt so to the beauty

and simplicity of geometry and mathematical physics.

These conceptions will tend to a more compact comprehension of
the nature of economic equilibrium . In order to have equilibrium

in the whole system including production :
(1) The utility distribution must be given for each individual.
(2) The “ maximum directions ” must be alike among all indi

viduals and between production and consumption.
(3) The origin must be the centre of gravity of all the individ
ual points : that is the sum of all A coördinates for consumption

must equal the sum for production and likewise for B, C, etc.
(4 ) The common income and expenditure flat must pass through
the origin : that is the money values of each man's production and

consumption must cancel.
* I. e. a Euclidean space of (m — 1) dimensions related to the m -dimensional
space as a plane is to our space.

+ See J. W . Gibbs, Multiple Algebra, Proceedings Amer. Asso, Adv. Sci., vol.

in the theory of value and prices.
$ 8.


By passing sections successively through the point I, we may nar- ,
row the discussion to as few variables as we choose. Wemay thus
select any three and discuss them as before in real space (cf. § 4 ).

For those familiar with multiple algebra, that is with the quater
nion analysis of Hamilton, the “ ausdehnungslehre ” of Grassman, or
the vector analysis of Prof. J. Willard Gibbs, the foregoing geo
metrical simplification will lead to a striking analytical simplifica
tion .*

Let I, II, . . . N , be vectors to the points I, II, . . . N from the
origin. Let U ,, U ,, etc., represent the total utility at the points
I, II, etc. Let vU , U , etc ., be vectors to represent in magnitude
and direction the maximum rate of increase of utility at the points
I, II, etc. (i. e. in tke “ maximum directions ” ).

The conditions of equilibrium expressed in $ 7 become :
(1) TU , = F (I) ; •U, = F ( II ) ; . . .

(2) VU, «

Un = F (N )

U , a v U , a . . . 0v Un

(3) I + II + III + . . . + N = 0

(4 ) I . VU, = II . VU, = . . . = N . VU, = 0
The first equation represents the several utility distributions.
The second means that the “ maximum directions ” are alike ; the

third that the amount of each commodity produced and consumed
cancel, and the fourth that for each individual the values of produc

tion and consumption cancel. t

- - --


- -

* See J. W . Gibbs' Vector Analysis, p . 16 , $ 50.

+ The scalar equations which the preceding vector equations replace can
readily be deduced from them . Let a , b , c, etc., be unit vectors along the
A , B, C, etc. axes . Multiply v U = F (I) by a , b , c , etc . respectively . Weob
tain m equations of the form U , . a = F (I) . a or:
' dU
JĀ. = F (A1, B1, C1, . . . . M ).
Likewise m scalar equations are contained in U , = F ( II), etc.
Again from (2) since v U , a v U2,
VU , . a : VU , . b = VU , . a : VU , . b or:




dA , - dB, - dA , AB.Z.
Likewise for C ., D ., . . . M . Likewise for v U3, etc.
Again (3 ) yields I . a + II , a + III , at . . . + N . a = () or

Ai + A2+ A3 + . . . + Ar = 0 .

Likewise for B , C , . . . M , making m equations.

JULY, 1892.


Irving Fisher - Mathematical investigations

$ 10 .

It is seen that analytically the treatment of interdependent com

modities differ from that of independent commodities only in this,
that the equations which represent the functions have more letters ;
i. e. we have

N = F (A ,, B , . .. N ,) instead of = F (A ,).

All other equations are just as in Part I. In fact these function
equations are, so to speak, the residuary formulæ ; they contain all
the unanalyzed conditions of the problem .
The marginal utilities are (as in Part I) in a continuous ratio

which is the ratio of prices. Yet there are some peculiar cases
which could not occur under the suppositions of Part I, viz : those

cases arising when the marginal utility of one or more articles has
no meaning .

If two articles are perfect completing articles, as gun and trigger,
there is no such quantity as the marginal utility of triggers alone.

There is, however, a marginalutility of a combined gun and trigger.
Now there are separate marginal disutilities for producing the gun
and trigger. How are all these quantities to be introduced into our

continuous proportion of marginal utilities ?
Suppose for a moment there were no difficulty of this sort.


proportion for each individual would be just as before (Part I,
Ch. IV , $ 10) and might be expressed as follows [G & g for gun T & t

for trigger] :

- dG



i * dU


dG .

1 -

I dT )

I P + P

į dū


i det dT,

P, + p

MG& T),

Finally : It is clear that
au a dUBib
I = Aa + Bb + . . . + Mm and vU, = GA
+ (ID ) + . . . + dU
. M . Mm .
Substituting these values in 1 .VU, = () we have after performing the multipli
cation and remembering that a , a = 1 and a . b = a . c = . . . = b . c = . . . = 0 ,

AiJA, + B , B , + . . . + M , MM , = 0
or since prices are proportional to marginal utilities:

Alpe + Bip: + . . . M , Pm = 0.
Likewise for II, III, etc. making n equations.
Conversely we could derive the vector equations from the scalar.

in the theory of value and prices.


The last two members of this equation are new and require a

word of explanation. The next to the last is an obvious conse,
quence of the principles of composition and division. Its denom
inator represents the marginal utility of Gun and Trigger combined
and is written Teen

in the last member.

But the quantities which are starred are those which can under
our supposition no longer be said to exist. Hence all members of

the equation containing a star drop out and we have left the first,
second and last members. In other words, if two articles are
perfectly completing their joint marginal utility is in the ratio to

their joint price as the marginal disutility of producing either
article is to its price (negatively ) or as every other marginal utility.
is to its price.

In like manner if two articles are perfect completing articles from
the producer's standpoint, as beef-hides and beef-meat, their joint
marginal disutility is to their joint price as the marginal utility of

either is to its price (negatively ) or as any marginal utility is to its

If two articles are such that they are perfect completing both as
to production and consumption and in the same ratios, they not

only have no separate utilities or disutilities but they can have no

separate prices. Thus, the head, limbs, tail and other parts of a
horse are produced together and consumed (used) together ; they
have no separate price.

It is impossible for articles to exist which are perfect completing
articles both for consumption and production but are produced in
one ratio and consumed in another.
Suppose two articles are such that the production of one is per

fectly completing to the consumption of the other. Suppose, for
instance, that the production of a ton of iron involves the consump

tion of a ton of coal, and that the consumption of the ton of coal
also implies the production of a ton of iron. The iron producer in
this case could not be said to have utility for more coal so long as

he does not produce more iron, nor can he be said to have disutility
of producing more iron without consuming more coal. What
utilities or disutilities then does he have ? He may be said to have

a joint marginal disutility of producing iron and consuming coal.
This “ joint ” disutility is to the difference of the prices of iron and
coal as themarginal utility of any commodity to him is to its price.
Like principles apply to three or more perfectly completing
articles. As long as articles are not perfectly completing there is


tical nvestigations

Irving Fisher - Mathema

no need for the substitution of joint utilities for single ones. As a

matter of fact the number of really perfectly completing articles is
relatively small.
If two articles are “ perfect ” substitutes for consumption and the

ratio of their marginal utilities is the same for all consumers, while
from a producers standpoint they are not “ perfect” substitutes, the
consumers fix the ratio of their prices (viz : that of their marg. ut.)
and the producers produce quantities accordingly . But the quan

tities of each consumed by different individuals is entirely indeter
minate. · Thus the milk from each cow may be regarded as a sep
arate commodity.

The consumer is indifferent to which milk he

drinks, and purely accidental causes determine how much of each he
gets ; the producer, however, milks determinate amounts from each
cow .

If two articles are perfect substitutes both for production and
consumption and the ratio of their marginal utilities and of their

marginal disutilities are all alike their prices will have this ratio , but
the relative quantities of each produced and consumed is entirely
indeterminate ; (e. g. the colors in the binding of a book ).
If two articles are perfect substitutes and the ratio of their mar

ginal utility of the first to the second is for every consumer greater
than the ratio of their marginal disutilities to all producers, the
first commodity alone will be produced and consumed and its price
will be determined as for any commodity.
In general if two articles are perfect substitutes, but the ratio of
their marginal utilities and the ratio of their marginal disutilities is

different for different individuals, those to whom the ratio of mar
ginal utilities of the first to the second is greater than the ratio of
their prices will consume only the first, those whose utility ratio is
less than the price ratio will consume only the second, those whose

disutility ratio is greater than the price ratio will produce only the
second ; those for whom it is less, only the first.* In this case the
price of each article is determined just as usual, but for each indi
vidual who does not consume or produce one or the other, its mar

ginal utility or disutility simply fails to have meaning and drops
out of the equations ; just as in Part I, occasionally a cistern may

be entirely out of the tank water.




- - - -- - - - -

* If some producers and consumers should have their utility or disutility ratio
identical with the price ratio the relative amounts produced and consumed are
indeterminate to the extent of this coincidence.

in the theory of value and prices.


$ 1.

For each individual situated in the “ economic world ,” suppose

a vector drawn along each axis to indicate the marginal utility in
that “ direction .”

The marginal utility of consuming (a ) is a vector

positive along the A axis, the marginal disutility of producing (a )
(or the disutility of paying money for a) is an equal vector in the
opposite direction. In like manner the marginal utilities and
disutilities along all axes are equal and opposite.
This corresponds to the mechanical equilibrium of a particle the

condition of which is that the component forces along all perpen

dicular axes should be equal and opposite.
Moreover wemay combine all the marginal utilities and obtain a
vector whose direction signifies the direction in which an individual

would most increase his utility. The disutility vector which indi
cates the direction in which an individual would most increase the

disutility of producing. These two vectors are (by evident geo
metry ) equal and opposite.

The above is completely analogous to the laws of composition and
resolution of forces.

If marginal utilities and disutilities are thus in equilibrium “ gain”
must be a maximum . This is the mere application of the calculus
and corresponds exactly to the physical application of the calculus

which shows that at equilibrium the balancing of forces implies that
energy is a maximum . Now energy is force times space, just as
gain is marginal utility times commodity .
$ 2.
In Mechanics.
A particle

corresponds to


Work or Energy = force x space.
Force is a vector (directed in space).
Forces are added by vector addition .

(“ parallelogram of forces." )
Work and Energy are scalars.

In Economics.
An individual.
Commodity .
Marg. ut. or disutility.
Utility .

Disut, or Ut. = marg . ut. x commod .
Marg. ut. is a vector (directed in com .)

Marg. ut. are added by vector addition .
(parallelogram of marg. ut.)
Disut. and ut, are scalars.


Irviny Fisher - Mathematical investigations

The total work done by a particle in | The total disutility suffered by an indi
moving from the origin to a given po - vidual in assuming a given position in

sition is the integral of the resisting

the “ economic world ” is the integral

forces along all space axes (resisting
forces are those directed toward the

axes (marg. disut. are directed to

origin ) multiplied by the distances

ward the origin ) multiplied by the

moved along those axes.

distances moved along those axes.

of themarg. disut. along all commod .

The “ total energy ” (the work done | The total utility enjoyed by theindivid
upon the particle) may be defined as ual is the like integral with respect
the like integral with respect to im
to marg . utilities.

pelling forces.
The net energy of the particle may be | The net ut. or gain of the individual is
defined as the “ total energy” less the the “ total utility ” less the " total
“ total work ."

Equilibrium will be where net energy Equilibrium will be where gain is max
is maximum ; or equilibrium will be

imum ; or equilibrium will be where

where the impel. and resist. forces

themarg . ut, and marg . disut. along

along each axis will be equal.
each axis will be equal.
“ total ut.” be subtracted from
(If “ total energy ” be
“ total work " instead of vice versa
total disut." instead of vice versa

the difference is “ potential” and is
minimum ).

the difference may be called “ loss”

and is minimum ).



$ 1.
In Part I, Chap. I, Utility was defined with reference to a single

individual. In order to study prices and distribution it is not neces

sary to give any meaning to the ratio of twomen 's utilities. Jevons
apparently did not observe this. Auspitz und Lieben did . So did
George Darwin.*
§ 2.

It would doubtless be of service in ethical investigations and pos
sibly in certain economic problems to determine how to compare the

utilities of two individuals. It is not incumbent on us to do this.

When it is done the comparison will doubtless be by objective stand
ards. If persons alike in most respects show to each other their sat
--- -- - - -

- --

-- - - - - -

- - -

* The Theory of Exchange Value.

- - -- ---

- - - - -- -- -

Fortnightly Review , new series, xvii, 243.

in the theory of value and prices.


isfaction by similar gestures, language, facial expression , and gen
eral conduct we speak of their satisfaction as very much the same.
What however this may mean in the “ noumenal ” world is a mys
tery. If on the other hand differences of age, sex, temperament, etc .
enter, comparison becomes relatively difficult and inappropriate .
Very little could be meant by comparing the desire of a Fuegian for

a shell-fish with that of a college conchologist for the same object
and surely nothing is meant by comparing the desires of the shellfish
itself with that of either of its tormentors.
§ 3.
When statistics becomes a developed science it may be that the

wealth of one age or country willbe compared with that of another
as “ gain ” not money value.

If the annual commercial product of

the U . S. was in 1880 $ 9,000,000,000 * and by increased facilities for
production prices are lowered so much that the product in 1890 is
only valued at (say ) $8,000,000,000 it proves a gain not a loss. The
country would be the richest possible when all things were as plenti
ful as water, bore no price, and had a total valuation of zero. Now
money value simply measures utility by a marginal standard which
is constantly changing. Statistical comparison must always be rough
but it can be better than that. The statistician might begin with
those utilities in which men aremost alike- food utilities , and those
disutilities in which they are most alike — as the disutilities of de
finite sorts of manual labor. By these standards he could measure
and correct the money standardt and if the utility curves for vari

ous classes of articles were constructed he could make rough sta
tistics of total utility, total disutility, gain , and utility-value which
would have considerable meaning. Men are much alike in their di
gestion and fatigue. If a food or a labor standard is established it
can be easily applied to the utilities in regard to which men are

unlike as of clothes, houses, furniture, books, works of art, etc.
§ 4.

These inquiries however do not belong here. Let us instead of add
ing to the meaning of utility do the very opposite and strip it of all
attributes unessential to our purpose of determining objective prices
- - - - --

* Edward Atkinson , Distribution of products , p . 141.

+ Cf. Edgeworth, On the method of ascertaining and measuring variations in
the value of the monetary standard , Report of the British Association for the
Advancement of Science, 1887.

Irving Fisher - Mathematical investigations


and distribution. Definition 3, Part I, Chap. I, § 4 yielded uniform
results only on the assumption that the utility of each commodity was
independent of the quantity of others. Similar assumptions are nec
essary in geometry. A unit of length is a yard. A yard is the length
of a standard bar in London. To be used it must be assumed that its
length is not a function of its position nor dependent on the changes
in length of other bodies. If the earth shrinks we can measure the
shrinkage by the yard stick provided it has not also shrunk as a nec
essary feature of the earth 's change. Definition 3 was essential in
Part I to give meaning to the cisterns used. Such a definition is essen

tial to the analyses of Gossen, Jevons, Launhardt, Marshall, and all
writers who employ coördinates.
analysis of Part II.

Yet it is not necessary in the

$ 5.
In fig. 28 the “ lines of force ” are drawn perpendicular to the in
difference loci. The directions of these lines of
28 .
force are alone used in the formulæ in Ch . II, § 9
which determine equilibrium .

Therefore the

directions alone are important. It makes abso
lutely no difference so far as the objective de
termination of prices and distribution is con

cerned what the length of the arrow is at one

point compared with another. The ratios of the
components at any point are important but these
ratios are the same whatever the length of the
arrow . Thus we may dispense with the total
utility density and conceive the “ economic world ” to be filled
merely with lines of force or “ maximum directions.”
§ 6.

Even if we should give exact meanings to the length of these ar
rows (so that the equation v U , = F (I) should signify not only that
for each position in the economic world a definite “ maximum direc

tion ” exists but also that the rate of increase of utility or the length

of the vector along this line is given ) — even then there would not be
a complete primitive U = P (I) unless certain conditions were ful
filled .*

These conditions are ( 1) that the lines of force are so ar

ranged that loci (surfaces in two dimensions , m - 1 spaces in m di

mensions) perpendicular to them can be constructed , and (2) that
* Osborne, Differential Equations, p. 12 . .

in the theory of value and prices.


the rate of passing from one locus to the next along a line of force

shall for all positionsbetween the two loci be inversely proportional
to the value of v U , already assigned to these positions. If v U , is
not distributed in the above manner integration is impossible and
there is no such quantity as total utility or gain .

$ 7.

Even if the integration were possible there would still be an arbi
trary constant. We could even claim that total disutility exceeds
total utility and all man can do is to minimize the disagreeable in

stead of maximize the agreeable. In other words, if we embrace
hedonism , there is nothing in economic investigation to cause us to
choose between optimism and pessimism .

$ 8.

Thus if we seek only the causation of the objective facts of prices

and commodity distribution four attributes of utility as a quantity
are entirely unessential, (1 ) that one man 's utility can be compared to
another's, (2) that for the same individual the marginal utilities at

one consumption-combination can be compared with those at another ,
or at one time with another, ( 3) even if they could , total utility and
gain might not be integratable, (4 ) even if they were, there would
be no need of determining the constants of integration .


Irving Fisher - Mathematical investigations


Jevons (p. 118 ) discusses the failure of equations for simple ex
change. It is clear that such failure must frequently occur in com
plex exchanges butno one has apparently commented on it. It would
seem at first sight that this would introduce an indeterminate element
into our results. Such however is not the case unless we take account

of articles neither produced nor consumed ; then the highest price
which any consumer will pay for the first infinitesimal is less than
the lowest price at which any one will produce it ; there is no pro
duction nor consumption and the term price has no determinate
meaning. As soon as changes in industrial conditions, that is in the

shape of the cisterns or their numbermakes this inequality into an
equality, the article enters into our calculations.
Suppose A is produced by n , people, consumed by now , and ex

changed or retailed by nic, where N . , Ny and n are each less than n
(the number of individuals.) Moreover from the nature of our
former suppositions if any of the three are greater than zero all
must be, for anything once in the system is supposed to be produced ,

exchanged and consumed within the given period of time.
The number of people who do not

produce A is n - n.
exchange A is n - ne,
consume A is n - nr.
The number of unknowns dropped out of the equations in Ch. VI,
§ 2 , is

3n - (n. +ne+ nr) of the type A , Ale, Ak, etc.,
and 3n - (n. + na + nx) of the type ZA ,
or 6n — 2 (n + n + nn) altogether. .
The failing equations in the first set are none,













31 - (n + n + n ),






3n - (notnetnx),

or 6n — 2(n + ne+ n .) altogether.
From the above agreement it appears that there can be no indeter
minate case under the suppositions which were first made. Let us
look at this somewhat more closely.

in the theory of value and prices.


In the fourth set of equations there are really n (3m - 1) (0 2 )
separate equations but only n3m – 1) are independent. Which shall
be selected is a matter of convenience. Wemay make every equa
tion contain Pan for instance and write
Par : P6,1 =


• OR



: 2D - - = etc .
T 2

Par : Per = etc.
. .. .. .. .. .

Paso : Pax = , . .
. . . . . . . . . . . . .

Pan : Page = , . . . . . .
. . . . . . . . . . . . . .

Now from the first two equations wemay derive by division

Por :Pes = dB

. 2C , '

but we might wish to use this last as one of the n (3m - 1 )3mindepen
dent equations, if A1, should “ fail.” From the n ( 3m - 1) -6 sepa
rate equations we are at liberty to select for use any n (3m – 1) inde

pendentones; and if in this selection there occur any which by some
change of quantities fail, we are compelled to change our selection so
that the new n (3m - 1) equations shall avoid the “ failing ” magni

This is interpreted in the mechanism as follows : when a cistern
is wholly above the surface of the tank (as IIIC fig . 8) and so con

tains nothing, the quantity of commodity and its utility “ fail.” The
levers which keep the ordinates in proportion to the corresponding
ordinates of other individuals may be far more numerous than the
levers pictured in former diagrams. Thus for four cisterns there
may be six levers (by joining each pair) but only three are neces

sary . The “ failure ” of any magnitudes will not invalidate any
system of levers originally selected ; it will simply make their num
ber greater than necessary.


$ 1.

In order to represent geometrically the relations between quantity

of commodity ,marginal utility , total utility, and gain (any two of
which four magnitudes are determined by a specified relation between


Irving Fisher - Mathematical investigations

the other two) it is only necessary to have a plane curve of appro
priate form and to represent any two of the above economic magni
tudes by any two geometrical magnitudes determined by the posi
tion of points in the curve.

Out of the numerous possible methods thus included , the one se
lected for the preceding discussion was to representmarginal utility
by the Cartesian ordinate and commodity by the area included be
tween the curve, the axes of coördinates, and the abscissa drawn
from the point.
$ 2.

In order to show the connection between this system of coördi
nates and those of Jevons and of Auspitz und Lieben , the following
scheme is presented :

Auspitz & Liehen .

The new curves.

= fædy

Commodity - -- - = x ;
Marginal .... = y;
utility Ś -Total ,

= ya

= Syædy

Gain .- .-

= Ya — Xa

= fyxdy - yfædy

utility s

- = Sy;də ; — X;Y ;

aya = tan 0

= y





in the theory of value and prices.
34 .

33 .


: 93


lo x


These curves are shown in figs. 29 and 30 (Jevons), 31, 32 ( Aus

pitz und Lieben ), and 33, 34 (new ). The first in each case is for
consumption the second for production.*
$ 3.

If Jevons' curve for consumption becomes a straight line, fig. 35 ,
its equation is:t

X ;+ 99; = m .
Using the preceding table substituting for x, and Y; we get in
Auspitz und Lieben coördinates :
36 .



dya = in ,
3 , - dan

which integrated gives
299. = 2mx, -

+ C.

Since the curve must evidently pass through the origin, C = 0,
and using new constants wemay write :

Ya = « X — Bxa",
which is a parabola (fig. 36 ).
* Jevons used no production curve. The one drawn is inserted to complete

the comparison. Fleeming Jenkins? “ Demand and Supply” curves are the same
as Jevons save that price replaces marginal utility .
+ Gossen , Launhardt, Whewell, and Tozer (the last two use no geometric analy

sis) employ such a linear supposition, though the meanings of their variables are
not identical.

# Launhardt's equation .

Irving Fisher - Mathematical investigations


For the new coördinates the substitutions from the table give :
Fædy + ay = m ,
2C == - 9,
which reduces to
a straight line parallel to the axis of ordinates (fig . 37).

The Auspitz und Lieben curve does not reveal to the eye the spe
cial supposition (that commodity and marginal utility change pro

portionally ). If we suppose that marginal utility decreases at a con
stant rate in relation to constant second differences of commodity,
the new diagram reduces to a straight line :
2 - 17 - M = 0,
while the other curves would be :

(ya + Ax, + B )' = C (D - x,)*
<; = E - Fy; - Gy;'.

$ 4.
The value of Jevons' diagram consists in the use of a simple and
familiar system of coördinates (the Cartesian ) as representing the

two chief economic quantities, and is probably the best for elemen
tary purposes.
The value of Auspitz und Lieben ’s diagram together with a

“ derivative ” curve* not shown above consists chiefly in the ease with
which maxima are discovered and the clear association of maxima

with equality of marginal utilities. It is believed that the third
method will, by means of its applicability to the mechanisms of
Part I, more clearly reveal the interdependence of the many com
modities of many individuals and of their many utilities.
$ 5.

The properties which are essential to the curve we have adopted
are :

First. That the curve shall never admit of being intersected twice
by a horizontal line (i. e. that it shall not cease to run in a general
up and down manner), to express the fact for consumption thatmar
ginal utility decreases as quantity of commodity increases and for

production thatmarginal disutility increases as the quantity of com
modity increases.

-- -


and yyaa ayxaa
* Whose Cartesian coördinates are x , and


in the theory of value and prices.


Second. That the curve shall approach the axis of ordinates asymptotically and in such a manner that the whole area between
it and the axis is finite, to express the fact that marginal utility

becomes infinitely minus for consumption of, and infinitely plus for
production of finite limiting quantities of commodity .*

Third. The curves begin (have commodity equal to zero ) at a

finite vertical distance from the origin . (These assumptions are less
generally true of production than of consumption , but they have
been here employed throughout.)
$ 6.

It is evident that in comparing the forms of curves for different
articles their differences and peculiarities are determined in a most

delicate fashion by the form of the curve . . . far more delicately
than, with our present statistical knowledge, is necessary.
Observe, then , what the abscissa of our curve stands for. An
infinitely thin layer ædy is the amount additional demanded (or

supplied) in response to an infinitesimal decrease (or increase) dy in
marginal utility . The abscissa x is the ratio of the infinitesimal
layer ædy to the infinitesimal change of price, dy. It is therefore
the rate of increase of quantity demandedt (or supplied ) in relation
to change of marginal utility. AM (figs. 2 and 3) is the initial rate.

Consulting II, § 2 of this appendix, we see that
ae; = f xdy
Hence ,

da; = wdy
y = y; and dy = dy;
dæ ; – X .

That is the abscissa of our curve is the tangential direction in
Jevons' curve, considered with respect to the axis of ordinates.

Hence if Jevons' curve be subjected to the condition of being
convex, the new curve must have the simple condition that succes
sive abscissas diminish , etc., etc .
$ 7.

Hitherto nothing has been said as to the mode of representing
total utility and gain ,

If y, is the marginal utility (which may be figured in money) at
which a consumer actually ceases to buy, y, that at which he would
* Cf. Auspitz und Lieben . pp. 7 and 11.
+ Cf. foot note Ch . IV , § 8 , div . 3 .


Irving Fisher - Mathematical investigations
m just begin to buy, then his consumer's rent or gain is (see Ch. I, § 8)

G = Syk % yæ dy –y, S % zdy
ormeasuring this gain in the given commodity as valued at y, cents

(say) per unit,
PYiy x dy


ry, a dy .

y Jyry " " Yx
This may be interpreted by a simple geometrical construction .

In fig. 38 OA = Yx and OR = y .

. -



Selecting the point 3 make the evident dotted construction deter
mining a point 5.
Evidently: 7375 = 04
= OR = 5 . 70 = 17, . ]

Let 3 assume all positions from M to 6. Then 5 will trace a curve
26 .

The area described by the moving line 73 is evidently / 91xdy.
The area described by the moving line 75 is evidently

- 2 ) dy.

Ya ly

Hence area described by themoving line 35 is the difference of
these expressions or G /y ,.

That is the area M62 represents the gain measured in commodity.
Thus suppose a person buys corn measured by RAM6 and let corn

at the valuation RO be the unit of utility. It is only the last layer
R6 on which no gain is felt. For any preceding layer 75 the price
really paid is OR while the price which it is worth to him is 07.
The layer 73 may be considered as lengthened in his eyes by that

in the theory of value and prices.


ratio 07/0R so that by getting it at less than he was willing to pay,

he has gained the element 35 measured in corn. His gain is maxi
mum when he purchases such a quantity that its final utility equals
its price.

Fig. 39 applies to “ producer's rent ” or “ gain,” substituting

“ sale ” for “ purchase ” ; “ sell ” for“ buy.”
To express the gain in money the area M26 must be multiplied by
the price. On each cistern construct the curve 62 (fig. 38) and con
sider the area RA26 to move front and back one unit (say one inch )
so as to trace a volume (fig . 40) adjacent to the front cistern and



again to move pinches further back so as to trace a volume adjacent
to the back cistern .

The front volume gives again the total utility measured in com
modity . The back volume gives the totalutility measured in money.

That is the whole back cistern and its adjacent volume represent the
money which if the individual paid he would neither gain nor lose,
provided his marginal valuation of it is unchanged by the operation.
The cistern portion is the money he actually pays and the outside
volume 7 , 10, 12, 9, 8 , 11 is his “ gain .” Likewise for the producer.

$ 1.
In the case of a single individual distributing a fixed income over
various commodities under fixed prices the distribution actually

TRANS. Conn. ACAD., Vol. IX .

JULY, 1892.


Irving Fisher - - Mathema




attained and specified in Part 1 yields the maximum total utility ,

for, since (Ch. IV , $ 2 ):



da, = dm , PaidB, = dm , Poz.. .
therefore :




dA, - dB, _

_ AM ,

Pa - Po = . . . = Pm
The numerators are the marginal utilities per unit of commodity.
To divide by the price is to make the unit of commodity the dollar's

worth. Each fraction is thus the marginal utility per dollar's worth .
The equation expresses the fact that the rate of increase of utility
from spending more money on any one commodity equals the rate

of increase for any other . Hence by a familiar theorem of the cal
culus the total utility must be the maximum attainable by any dis
tribution of a fixed income. In like manner the individual dis
tributes his production so that the marginal disutilities in all modes
of producing dollar's worth of commodity are equal so that his total

disutility is a minimum . Hence the difference between his total
utility and total disutility or his economic gain is a maximum .
$ 2.

In the distribution of a single commodity over many individuals
since :






dA, – dm , · Pai QA = dm . · Pa; . . . ; JA . = im Pa,






dA .

DƯ = ZU = . . . . = DU '
dm , dm ,
that is, themarginal utilities (when the unit of utility is the marginal
utility of money for each individual) are equal and the total utility
is maximum . In like manner the total disutility is a minimum and
gain therefore a maximum .

$ 3.
The first continuous equation may be divided by


and the

in the theory of value and prices.


second by P, and since the first members will become identical we
have a common continuous equation :
dA ,



dA ,


dm , ·

- = ... =
- = etc.
dm , · Pa
dm , . ?


that is, the marginal utilities of all commodities to all consumers are
equal when the unit of utility is the marginal utility of money and

the unit of commodity the dollar 's worth . Hence the total utility
in whole market thus measured is a maximum .*
§ 4.

However it may justly be objected that the marginal utility of

money to one person is not equatable to that of another, that is that
it is unfair to use the unit of utility for the poor man the high mar

ginal utility of his small income and add the smallnumber of such
large units in a poor man's gain to the corresponding rich man ’s gain
in which the unit of measure is small and the number of units large.
If we suppose by some mysterious knowledge an exact equiv

alence of utilities were possible between different individuals (see
Part II, Ch. IV , $ 2 ) and by some equally mysterious device of
socialism we could without changing the aggregate commodities alter

their distribution so as to make the whole market utility a maximum
our condition would be

[ dU _ dU
dA = dd = etc .


This could be brought about by a change in the relative incomes ,
taking from the rich and giving to the poor until
dm , — dm ,

which applied to equation (3) will evidently afford the required (4 ),
or by breaking down the condition of uniformity of price and mak

ing each man's price inversely as his marginal utility of money,
which applied to (3) will evidently yield (4).
To interpret equation 4 in the mechanism we may alter the posi

tion of the stoppers in fig . 8 until the ordinates in each front and
back row are equal. (This will not be when “ incomes are equally
divided ” nor when “ gains ” are equal, for persons differ in their

power of enjoyment, and it would still be true that those whose
* Cf. Auspitz und Lieben , p . 23 and 435 .


Irving Fisher - Mathematical investigations

capacities for pleasure were great would consume the most in order

to make the aggregate gain in the whole market a maximum ). Or
wemay destroy all the levers and re-arrange the rear thicknesses
until the front and back ordinates are made equal.
In like manner the minimum disutility would be attained if all

marginal disutilities were equal. The maximum gain would then
result. This is the maximum gain obtainable when the amounts of

each commodity consumed and produced are fixed and given . If
we are permitted to rearrange these amounts also , we shall secure
the maximum gain when the marginal utilities equal the marginal
disutilities ; i. e .



A ,*


Under such a socialistic regime more “ necessaries ” and less

“ luxuries ” would be consumed and produced than previously .
The “ rich ” or powerful would produce more and consume less than
previously ; the poor or weak would consume more and produce

less. Yet for each the marginal utilities and disutilities would be
It is needless to say that these considerations are no plea for

socialism , but they serve to clear up a subject sometimes discussed
by mathematical economists and reconcile Launhardt's contention*
that utility is not a maximum with Auspitz und Lieben 's that it is.
The former unconsciously has reference to equation (4) which is not

true, the latter to equation (3) which is.f

The four sets of equations, Part I, Ch. IV , $ 10 , can be reduced.
TU .

Wemay substitute for

itsvalue F (A,) and thus eliminate allmar

ginal utilities. Moreover we can get an expression for Pa, Po, etc.,

in terms of commodities. First, if m = n the second set of equa
tions are easily solved by determinantst giving ::
* Volkswirthschaftslehre under “ Widerholte Tausch . "

+ Auspitz und Lieben appear to overlook this difference of standpoint.
Preface, p . xxv .

Burnside and Panton . Theory of Equations, p. 251. .
S This equation does not mean that any arbitrary values can be assigned to

A2, B ,, etc., and the resulting price of A be so simply expressed ; only when A1,

Bı, etc. satisfy all the conditions of Ch .'IV , $ 10 will the price be expressible as
the quotient of the two determinants .


in the theory of value and prices.


( KK, B, . .. M ) ( A , B, . . . M ) .
K , B, . . ..D
M , }L , A , B, . . . M , Į
. -- - -. ... . ...... 'y

į A , BB,e . . . My


| K, B , . . . M ,


Pa =

in which obviously in general a change in A will produce a greater
influence on pathen an equal change in B , etc. But it shows clearly
that pa is not a function of A alone.

Usually n > m . Hence we may use the first m equations of the
second set, or in fact any m equations. The resulting determinant
quotients must be equal and must equal also the several like
determinates for production.

The corresponding values of Po, po, ctc., may be found and may
be substituted in the fourth set.

If wherever A , now occurs in the fourth set, we substitute

K . - A , - A , - . . . - A , from the first set, and likewise for B ,, etc.,
the resulting fourth set is self-sufficient. Wehave thus eliminated
; dU
the variables
T , etc., Pa, etc., A , B ,, etc., and have gotten rid of
the first, second and third set of equations. We can proceed no
further, however, until the explicit forms of the functions F ( A ),

etc., are given.

$ 1.

No pretense is made that the preceding analysis is perfect or ex
haustive. There is no such analysis of any phenomena whatever

even in physics. The suppositions in Ch. II, § 2 of Part I, are of
course ideal. They only imperfectly apply to New York City or
Chicago. Ideal suppositions are unavoidable in any science . In fact
it is an evidence of progress when the distinction between the ideal

and the actual arises.* Even in hydrostatics the assumption of per
fect fluidity is never fully realized . The physicist has never fully
explained a single fact in the universe.

He approximates only .

The economist cannot hope to do better. Somewriters, especially
those of the historical school are disposed to carp at the introduc
tion of a refined mathematical analysis. It is the old story of the
---- - -- - - -


* See Prof. Simon Newcomb . The Method and Province of Pol. Econ. , N . Am .

Rev ., CCXI, IX .


Irving Fisher - Mathematical investigations

“ practical” man versus the scientist. A sea -captain can sail his
vessel and laugh at the college professor in his elaborate explanation
of the process. What to him is all this resolution of forces and

velocities which takes no account of the varying gusts of wind, the
drifting of the keel, the pitching and tossing, the suppositions which
makes of the sail an ideal plane and overlook the effect of the wind
on the hull ?

There is no need to point the moral.

Until the

economist is reconciled to a refined ideal analysis he cannot profess
to be scientific. After an ideal statical analysis the scientist may

go further and reintroduce one by one the considerations at first

This is not the object at present in view . But it may

be well to merely enumerate the chief of these limitations.
$ 2.

In Part I the utility of A was assumed to be a sole function of
the quantity of A , and in Part II a function of all commodities con
sumed by a given individual. We could go on and treat it as a

function of all commodities produced and consumed, treating not

net production for each article, but the actual amounts separately
produced and consumed by the given individual.
Again we could treat it as a function of the quantities of each
commodity produced or consumed by all persons in the market.
This becomes important when we consider a man in relation to
the members of his family or consider articles of fashion as dia
monds,* also when we account for that (never thoroughly studied )
interdependence, the division of labor.

This limitation has many analogies in physics. The attraction of
gravity is a function of the distance from the center of the earth.

A more exact analysis makes it a function of the revolution of the
earth, of the position and mass of the moon (theory of tides) and

finally of the position, and mass of every heavenly body.
$ 3.

Articles are not always homogeneous or infinitely divisible. To
introduce this limitation is to replace each equation involving mar

ginal utilities by two inequalities and to admit an equilibrium inde
terminate between limits.† As an extreme case we may imagine an
article of which no one desires more than a single copy as of a book .

The utility of (say) Mill's Pol. Econ. is considerably greater than
* See David Wells, Recent Economic Changes, on Diamonds.
+ Auspitz und Lieben , 117 – 136 and 467.

in the theory of value and prices.


its cost, but the utility of a second copy is considerably less than

its cost. In the aggregate market, however, there will be a mar.
ginal person whose utility is very close to the price. A change in

price will not alter the amount purchased by everyone, but will
alter the number of purchasers.*
§ 4.

Producing, consuming and exchanging are discontinuous in time.
The theory of utility when applied to a single act of production or

consumption or of sale or purchase, is independent of time, or rather

the time element is all accounted for in the form of the utility
function.f But an analysis of a number of such acts must take
account of their frequency. The manner in which the time element
enters has puzzled not a few economists.

An example from physics may not be amiss. In the kinetic
theory of gases the pressure on the walls of the containing vessel

is explained by its continual bombardment by molecules. But an
apparent difficulty must be observed.

A rebound of a molecule

involves the idea of momentum only while that which we wish to

explain is pressure or force which is not by any means momentum ,
but momentum divided by time. How does this time enter ? By
regarding not one but many molecules and taking account of the
frequency of their collision. The average momentum of each blow
divided by the average interval between the blows is the pressure
So a produce exchange is a channel connecting production and con
sumption . Instead of an even flow of one bushel per second, the
machinery of the exchange is such that by an instantaneous blow of a

bat, so to speak, a thousand bushes are knocked along. Time is in
appropriate to explain the single blow but necessary to explain the
§ 5.
The ideal statical condition assumed in our analysis is never satis
fied in fact.

No commodity has a constant yearly rate of production or con
sumption. Industrial methods do not remain stationary. Tastes
and fashions change. Panics show a lack of equilibrium . Their
explanation belongs to the dynamics of economics. But we have
* The analysis of H . Cunynghame in the Ec. Jour., March ’92 , applies to this
case .

+ Cf. Jevons, 63 – 68 .


Irving Fisher , Mathematical investigations

again a physical analogue. Water seeks its level, but this law does
not fully explain Niagara. A great deal of special data are here
necessary and the physicist is as unfit to advise the captain of the
Maid of the Mist as an economist to direct a Wall street speculator.

The failure to separate statics from dynamics appears historically *
to explain the great confusion in early physical ideas. To make
this separation required the reluctant transition from the actual
world to the ideal. The actual world both physical and economic

has no equilibrium . “ Normal” + price, production and consumption
are sufficiently intricate without the complication of changes in social
structure. Some economists object to the notion of “ normal” as an
ideal but unattainable state. They might with equal reason object
to the ideal and unattainable equilibrium of the sea.

The dynamical side of economics has never yet received system
atic treatment. When it has, it will reconcile much of the present
apparent contradiction , e. g. if a market is out of equilibrium , things
may sell for “ more than they are worth ,” as every practical man
knows, that is the proper ratios of marginal utilities and prices are
not preserved .

$ 6.

We have assumed a constant population. But population does
change and with it all utility functions change. An analysis whose
independent variable is populationſ leads to another department of
economics . In the foregoing investigation the influence of popula
tion was included in the form of the utility function. So also with
all causes physical, mental and social not dependent on the quantities
of commodities or services.

Individuals are not free to stop consuming or producing at any
point. Factory operatives must have uniform working hours. The
marginal undesirability of the last hour may for some workmen

equal, for others exceed or fall short of the utility of its wages.
$ 8.

No one is fully acquainted with all prices nor can he adjust his
actions to them with the nicety supposed ; both these considerations

are starting points for separate discussion .
* Whewell, Hist. Induct. Sci., I, 72 –3 and 186 .
+ Marshall, p . 84 .
See article of Prof. J . B . Clark . Distribution as determined by a law of
rent. Quart. Jour. Econ., Apr. '91, p . 289.

in the theory of'value and prices.


The “ fundamental symmetry of supply and demand worked out
by Auspitz und Lieben should not bind us to the fundamental

asymmetry. The symmetry enables us to investigate the general
dependence of consumption and production but special investigation
of production , e. g. of railroad rates should be independently pur
sued .

(1.) Production of a commodity always precedes its consumption .
(2 .) The maximum advantage in production involves few com

modities for each individual, in consumption many.
(3.) Increasing social organization intensifies the former fact not
the latter.

(4.) There are more successive steps in production than consump
tion .

(5 ) Social organization intensifies this distinction .
(6) Owing to (4) and (5 ) service rather than commodity becomes
increasingly the unit in production.

(7.) Freedom to leave off consuming at any point is greater than
for producing.

(8.) Social organization intensifies this.
(9.) Combination and monopoly are more feasible and frequent in

production than in consumption .
( 10 .) In production the distinction of fixed charges and running

expenses often plays an important rôle. This deserves a separate

The transportation charges on a steamship are not what

it costs to transport an extra ton but it is this quantity plus the pro
portionate share of that ton in the fixed charges (interest, insurance ,
etc ). That is, the marginal cost of service involves the margin of
capital invested as well as the marginal cost of running the ship )

(which is purely nominal). This is so in theory of railroad rates
but the railroad investor cannot foresee the results of his enterprise

as well nor can he change his road when built from one route to
another as a steamship can do. To apply the theory to railroads
assumes that railroad projectors know what the traffic will be. Con
sequently the proper discussion of railroad rates, assuming that the
railroads are already built, takes no account of fixed charges but

becomes formulated as “ what the traffic will bear." *

A complete theory of the relation of cost of production to price
in its varying and peculiar ramifications is too vast a subject to be
treated here.
* See Hadley, Railroad Transportation .


Irving Fisher - Mathematical investigations

§ 10.

It has been assumed throughout this investigation that marginal
utility decreases as quantity of commodity increases. This is not

always true, e. g. it is obviously not true of intoxicating liquors.
A study of the liquor traffic would require a somewhat different

treatment from that of most other commodities. Still less is it
always true that marginal cost of production always increases as the
quantity produced increases. It is clearly not true that it costs
more in a shoe factory to produce the second shoe than it costs to
produce the first . Yet it is probably quite generally true that at

the actual margin reached in business the disutility of extending the

business grows greater. When this is not true and when it is not
true that marginal utility decreases as quantity of commodity in .

creasess an instability is theresult. The matter of instability is one
element at the bottom of the present industrial tendency toward
trusts and pools.
$ 11.

There is no isolated market. Not only this but a “ market ” itself
is an ideal thing. The stalls in the same city meat market may be

far enough apart to prevent a purchaser from behaving precisely as
if he stood before two counters at once . The relation of the counters

ten feet apart differs in degree rather than in kind from the relation
of London to New York.

$ 1.

Mathematics possesses the same kind though not the same degree
of value in every inquiry. Prof. B . Peirce,* in his memorable
Linear Associative Algebra , says : “ Mathematics is the science
which draws necessary conclusions. * * * * * Mathematics is not

the discoverer of laws, for it is not induction , neither is it the
framer of theories for it is not hypothesis, but it is the judge over
both. * * * * * It deduces from a law all its consequences.

Mathematics under this definition belongs to every inquiry , moral
as well as physical. Even the rules of logic by which it is rigidly
bound could not be deduced without its aid. The laws of argu
* Amer. Jour. Math ., IV., p. 97.

in the theory of value and prices.


ments admit of simple statement, but they must be curiously trans
posed before they can be applied to the living speech and verified
by observation.
In its pure and simple form the syllogism cannot be directly com
pared with all experience, or it would not have required an Aristotle
to discover it. It must be transmuted into all the possible shapes

in which reasoning loves to clothe itself. The transmutation is the
mathematical process in the establishment of the law ." *
I make this quotation for I believe many persons, especially econo
mists, do not understand the character of mathematics in general,
They imagine that a physicist can sit in his study and with the cal

culus as a talisman spin out some law of physics. Some economists
have hoped for a similar mysterious use of mathematics in their own
science .

$ 2.

· We must distinguish carefully between what may be designated
as mathematics and mathematical method. The former belongs, as
Prof. Peirce says, to every science. In this sense economics has
always been mathematical. The latter has reference to the use of

symbols and their operations. It is this which is to be discussed
here. A symbol may be a letter, a diagram , or a model. All three
are used in geometry and physics.f
By an operation on symbols is meant a rule the formulation of
which depends on the mention of those symbols (as the operation of
differentiation). To employ mathematical method is to pass from
what is given to what is required by the aid of such a rule. To
avoid mathematical method is to do it without the rule. Symbols

and their operations are aids to the human memory and imagina
tion .
$ 3.

The utility of mathematical method is purely relative, as is all
utility. It helps greatly some persons, slightly others, is even a
hindrance to some.
Before a schoolboy studies “ mechanics ” he is usually given in
his arithmetic problems of uniform motion . It would sorely puzzle

him if he were compelled to use the formula s = ut. The employ
ment of symbols has for him

only disutility.

But when in

* Cf. Grassmann, Ausdehnungslehre, Introduction.
+ Few are aware how important models sometimes are in the treatment of
these sciences. Maxwell's model to represent the relations of volume, entropy
and energy in thermodynamics is an excellent example.


Irving Fisher — Mathematical investigations

“ mechanics ” proper a few years later the sameboy studies “ falling
bodies " he finds it helpful to use the formula v = gt which contrasts.
with the preceding formula only in that space (s) is replaced by
space per unit of time (v) and velocity (1 ) by velocity acquired per
unit of time (g). The increased complexity of the magnitudes
makes a formula relatively desirable. Yet for someminds the latter
formula is of no use. Experience in teaching this very subject has
convinced me that there are a few who understand it better without
the aid of the formula , but they are just those individuals whose
comprehension of the relations involved is the vaguest and the
The formulæ , diagrams and models are the instruments of higher

study. The trained mathematician uses them to clarify and extend

his previous unsymbolic knowledge. When he reviews the mathe
matics of his childhood , the elementary mechanics is to him
illumined by the conceptions and notation of the calculus and qua

ternions. To think of velocity, acceleration , force,as fluxions is not
to abandon but to supplement the old notions and to think of
momentum , work, energy, as integrals is greatly to extend them .
Yet he is well aware or ought to be that to load all this on the

beginner is to impede his progress and produce disgust. So also the
beginner in economics might be mystified , while the advanced

student is enlightened by the mathematicalmethod .
$ 4.

The utility of a mathematical treatment varies then according to
the characteristics of the user, according to the degree of his mathe
matical development and according to the intricacy of the subject

handled .

There is a higher economics just as there is a higher

physics, to both of which a mathematical treatment is appropriate.

It is said that mathematics has given no new theorems to economics.
This is true and untrue according to the elasticity of our terms.

The challenge of Cairnes might be answered by a counter challenge
to show the contents of Cournot, Walras, or Auspitz und Lieben in
any non -mathematical writer.
If I may venture a speculation, those who frown on the mathe
matical economistbecause he “ wraps up his mysterious conclusions
in symbols ” seem to me in some cases to point their finger at those

“ conclusions ” which when “ unwrapt ” of symbols they recognize as
old friends and lustily complain that they are not new ; at the same
time they seem to ignore completely those “ mysterious ” conclu

in the theory of value and prices.


sions which are new because they think the former and admitted

theorems exhaust all that is important on the subject. Why should
the mathematician be obliged to vindicate the exercise of his science

by overturning economics or by deducing some“ laws” more funda
mental than those already admitted ?

Elementary physics is the fundamental physics and it can be
taught with little or no mathematical symbols. Advanced physics
is relatively less popular while more mathematical. By actual count

Ganot's elementary physics of 986 pages contains a formula for
every three and one-third pages. The chapter on electricity and
magnetism of 320 pages, a formula for every 44 pages, while the
profound treatise.of Mascart and Joubert on Electricity and Mag
netism , vol. I, of 640 pages, contains 31 formulæ for each page or 15

times as many per page as the same subject in Ganot.
Similarly in economics, mathematical treatment is relatively use

ful as the relations becomerelatively complicated . The introduction
of mathematical method marks a stage of growth - perhaps it is not
too extravagant to say, the entrance of political economy on a scien
tific era .

$ 5.
Has the mathematical method attained a firm footing ? Before

Jevons all the many attempts at mathematical treatment fell flat.
Every writer suffered complete oblivion until Jevons unearthed their
volumes in his bibliography. One chief reason for this is that these
writers misconceived the application of mathematics. I think this

was true even of the distinguished Whewell. Jevons thinks it is so
of Canard though his work was crowned by the French institute.
The second reason for this oblivion is that the world was not pre
pared for it. The movement was too advanced and premature.
Cournot certainly ,Gossen possibly, now exert considerable influence

on economic thought. Marshall,whose recentbook is acknowledged
to be to modern economics what Mill's was to the economics of a
generation ago repeatedly expresses his admiration for and obliga
tion to Cournot.

Thus the mathematical method really began with Jevons in 1871.
Up to this time pol. econ. had been the favorite field for those per
sons whose tastes were semi-scientific and semi-literary or historical.

But the scientific and literary temper are seldom equally balanced
and as might have been expected after once beginning to divide

they have steadily differentiated. On the one extreme is the histori


Irving Fisher - Mathematical investigations

cal school of Roscher and Leslie, on the other the mathematical,
deductive, or so-called Austrian school of Jevons, Menger and
Walras, while the “ orthodox ” economists the legitimate successors

of Adam Smith , Ricardo andMill constitute the central body from
which both have split. This cleavage is, however, largely a division
of the field of research rather than opposed theories or methods on
the same field .

The mathematical economics apparently has its warmest adherents
in Austria, Italy and Denmark. France occupies the next position ,

while England, America and Germany have their individual enthu
siasts but are still restrained largely by classic traditions. Prof.

Pantaleoni thinks “ the most active movement in Italian pol.
econ, is that of the new school styled rather inexactly the “ Aus
trian,"* while Graziani says that the utility theory of value “ seems

to close the evolutionary cycle of Italian thought.” *
In England, Prof. Edgeworth, noted for his enthusiasm on mathe
matical economics, has recently been elected to the chair of pol.

econ. at Oxford, while Prof. Marshall is carrying forward the same
movement at Cambridge.
There has been a great increase in mathematico-economic litera

ture since 1871. Just two decades have passed by since Jevons'
epoch-making books appeared.

Of the mathematico-economic

writingst appearing in this period which here come to my notice,

the number in the first decade was 30, representing 12 writers,
while in the second decade it was 66, representing 23 writers. From
all apparent evidence the mathematicalmethod has come to stay.
$ 6.

We can see why this is so if we glance at the work which the

mathematicalmethod has already accomplished. It is perhaps fair
to credit the idea of marginal utility to mathematicalmethod. This
idea had five independent origins with Dupuit, Gossen, Jevons,
Menger, and Walras. All except Menger presented this idea and
presumably attained it by mathematical methods. No idea has been

more fruitful in the history of the science. This one achievement
is a sufficient vindication of the mathematical method .
* Article on Economics in Italy , by Prof. Ugo Rabberio , Pol. Sci. Quart., Sept.,
1891, pp. 439-473 .
+ I have not even included here Menger, Böhm -Bawerk and other writers of the
Austrian school, who in spite of a mathematical tone have omitted to use math

ematical symbols.

in the theory of value and prices.


To pass in review all that has been done in expanding and apply
ing the idea of marginal utility (and most of this expansion has
been purely mathematical) would not be possible here, nor would it
be possible to state all the other notions which have grown out of a

mathematical treatment. It has corrected numerous errors and con
fusion of thought. This correcting function has really been the
chiefmission of mathematics in the field of physics though few not

themselves physicists are aware of the fact.
In fact the ideas of marginal utility and disutility may be re
garded as corrections of two old and apparently inconsistent theories

of value — the utility theory and the cost of production theory.
Utility was first thought of as proportional to commodity. (That
this was never explicitly assumed is a splendid illustration of how

without a careful mathematical analysis in which every magnitude
has definite meaning, tacit assumptions creep in and confuse the

mind ). It was next pointed out that utility could not explain price
since water was useful. So “ utility ” and “ scarcity ” were jointly

privileged to determine price. It was Jevons clear and mathemat
ical exposition of utility which showed the shallowness of the for

mer discussion and brought to light the preposterous tacit assump
tion , unchallenged because unseen , that each glass of water has an
inherent utility independent of the number of glasses already drunk.
Jevons laid emphasis on demand . Many who accepted his work
were still applying the analogous errors to supply. Ricardo* had

indicated the idea of marginal cost. But even Mill did not perceive
its extension beyond agricultural produce. Considerable credit

belongs to Auspitz und Lieben for working out the legitimate con

sequences and showing by a beautiful mathematical presentation
that the marginal utility theory and the marginal cost theory are

not opposed but supplementary. In fact the “ margin ” itself is
determined by the condition that the utility and the cost of final
increments shall be equal (when measured in money).

Mathematical method is to be credited with the development of
the ideas of consumers' and producers' rent or gain so ingeniously
applied by Auspitz und Lieben and so conspicuous in the orig
inal article of Prof. J . B . Clark on the law of the three rents. t

The intimate and mathematically necessary relation between the
equality of marginal utilities and disutilities and the maximum sum

of consumers' and producers' rent, a theorem emphasized by Auspitz
und Lieben, and Edgeworth , is of course due to the mathematical
* Pol. Econ., Ch. 2.

+ Quart. Jour. Econ ., April, 1891.


Irving Fisher - Mathematical investigations

Mathematical method is making a new set of classifications based
on mathematical properties.

Thus the classification by Auspitz und

Lieben of all commodities into three groups* is, I believe, a new
one, and one suggested by, and readily discussed by the use of their
diagrams. The classification of capital into free and sunk is one

which is emphasized by the mathematicalwriters, as Marshall, and
is bearing fruit.t
I believe therefore that mathematical method has made several
real contributions to economics, and that it is destined to makemore.

To verify this statement I would refer the reader to the books men
tioned in the bibliography among recent writers, especially Walras,
Auspitz & Lieben , Marshall, Edgeworth , Wicksteed and Cunyng

hame ; also , if it is proper to include those writers, who while

avoiding mathematical language are interpreting and extending
the same ideas, Menger, Wieser, Böhm -Bawerk, Clark and Hobson .

$ 7.
It may not be amiss to present a list of quotations from those who
have pursued or admired the mathematical path :
Whewellf says : [Mathematical method in mechanics saves scien

tists three errors, viz :] “ They might have assumed their principles
wrongly, they might have reasoned falsely from them in conse
quence of the complexity of the problem , or they might have

neglected the disturbing causes which interfered with the effect of
the principal forces. * * * It appears, I think, that the sciences of
mechanics and political economy are so far analogous that some
thing of the same advantage may be looked for from the application

of mathematics in the case of political economy.” Again :S “ This
mode of treatment might be expected to show more clearly than
any other within what limits and under what conditions propositions
in political economy are true.”
Cournot: | L' emploi des signes mathématiques , est chose naturelle

toutes les fois qu'il s'agit de discuter des relations entre des gran

deurs ; et lors même qu'ils ne seraient pas rigoureusement nécess
aires,s'ils peuvent faciliter l'exposition , la rendre plus concise,mettre ·
sur la voie de développements plus étendus, prévenir les écarts
* Page 46 .

+ See Cunynghame, Geom . Meth . of treating exchange value, monopoly , and
rent. Econ . Jour. , March , ’92, p . 35 .
| Cambridge Philosophical Transactions, 1830, p . 194 .

S Cambridge Philosophical Transactions, 1856, p . 1 .
|| Principes math . de la théorie des richesses, 1838.

Preface, p . viii.

in the theory of value and prices.


d'une vague argumentation , il serait peu philosophique de les
rebuter, paree qu 'ils ne sont pas également familiers à tous les
lecteurs et qu 'on s'en est quelquefois servi à faux.”

Gossen :* Was einem Kopernikus zur Erklärung des Zusammen
seins der Welten im Raum zu leisten gelang, das glaube ich für die

Erklärung des Zusammenseins der Menschen auf der Erdoberfläche

zu leisten. * * * Darum ist es denn eben so unmöglich, die wahre
Nationalökonomie ohne Hülfe der Mathematik vorzutragen , wie
dieses bei der wahren Astronomie, der wahren Physik, Mechanik
u . S. W .”

Jevons:f “ I have long thought that as it deals throughout with

quantities, it must be a mathematical science in matter if not in
language. I have endeavored to arrive at accurate quantitative
notions concerning utility, value, labor, capital, etc., and I have

often been surprised to find how clearly someof the most difficult
notions, especially that most puzzling of notions value, admits of
mathematical analysis and expression.”
Walras:f “ Je crois bien que les notations qui y sont employées
paraitrout tout d'abord un pen compliquées; mais je prie le lecteur

de ne ponit se rebuter de cette complication qui est inhérents au
sejet et qui en constitue d'ailleurs le seule difficulté mathématique.
Le système de ces notations une fois compris le systeme des phé

nomènes économiques est en quelque sorte compris par cela même.”
Newcomb :S “ To ultimately expect from pol. econ. results of such
certainty and exactness, that it can present the legislator with

numerical predictions like those we have described is by no means
hopeless.” * * * * “ Mathematical analysis is simply the application

to logical deduction of a language more unambiguous,more precise,
and for this particular purpose, more powerful than ordinary lan

Launhardt:|| “ Es ist ja die Mathematik nichts anderes als eine
Sprache, welche in strenger Folgerichtigkeit die Beziehungen mess
bare Dinge zu einander darstellt, was durch die gewohnliche
Sprache entweder gar nicht oder doch nur in weitschweifiger

ungenauer Weise erreicht werden kann."
* Menschlicher Verkehr. Preface, p. v.
+ Preface to first edition , p . vii.

# Econ. pol. pure, 1874 , Preface, p. VI.

$ Themethod and province of pol. econ . [Review of Cairne's logical method
in pol. econ .), N . Amer . Rev., No. CCXLIX , '1875 , p . 259.

| Volkswirthschaftslehre : Preface, p. v.
TRANS. Conn . ACAD., VOL. IX .

JULY, 1892.


Irving Fisher - Mathematical investigations

Wicksteed :* “ The diagrammatic method of studying economics
may be regarded from three points of view : (I)many teachers find

in it a stimulating and helpful appeal to the eye and use it as a
short and telling way of making statements and registering results.
( II) a few students treat it as a potent instrument for giving pre

cision to hypotheses in the first instance and then for rigorously
analysing and investigating the results that flow from them . (III)

a very few investigators (among whom I think we must rank
Jevons), have hoped ultimately to pass beyond the field of pure
hypotheses and analysis and to build up constructive results upon

empirical curves of economic phenomena established by observa
tion .”

Foxwellt [speaking of the mathematics of Jevons and Marshall ] :
“ It hasmade it impossible for the educated economist to mistake

the limits of theory and practice or to repeat the confusion which
brought the study into discredit and almost arrested its growth .”
Auspitz und Lieben : f “ Wir haben uns bei unseren Untersuch

ungen der analytischen Methode und namentlich der graphischen

Darstellung bediehnt, nicht nur weil sich diese Behandlungsweise
überall, wo sie überhaupt anwendbar ist, und namentlich in den
naturwissenschaftlichen Fächern glänzend bewährt hat, sondern

hauptsächlich auch darum weil sie eine Prazision mit sich bringt,
welche alle aus vieldeutigen Wort-definitionen entspringender Miss
verständnisse ausschliest.”
Edgeworth : $ * * * “ the various effects of a tax or other impedi

ment, which most students find it so difficult to trace in Mill's labori

ous chapters, are visible almost at a glance by the aid of the mathe
matical instrument. It takes Prof. Sidgwick a good many words to

convey by way of a particular instance that it is possible for a
nation by a judiciously regulated tariff, to benefit itself at the

expense of the foreigner. The truth in its generality is more clearly
contemplated by the aid of diagrams. * * * * There seems to be a

natural affinity between the phenomena of supply and demand, and
some of the fundamental conceptions of mathematics, such as the
relation between function and variable * * * and the first principle
* On certain passages in Jevons' “ Theory of pol. econ.” Quart. Jour. Econ.,
April, '89, p . 293.

+ The Economic Movement in England , Quart. Jour. Econ ., Oct., '88.
† Untersuchungen. Preface, p . xiii.
S Address before Brit . Assoc, as president of the section on economic science

and statistics. Published in Nature, Sept. 19 , '89, p . 197.

in the theory of value and prices.


of the differential calculus ; especially in its application to the
determination of maxima and minima.” [It seems to ] “ supply to

political economy what Whewell would have called appropriate
and clear ' conceptions. * * * Algebra and geometry are to ordinary

language in political economy somewhat as quaternions are to
ordinary algebraic geometry in mathematical physics ” (Quotes
Maxwell on quaternions : “ I am convinced that the introduction of
the ideas as distinguished from the operations and methods * * *

will be of great use.” )
Again :* “ I do not mean that the mathematical method should
form part of the curriculum as we make Greek obligatory for the
students of philosophy. But may we not hope that the higher path
will sometimes be pursued by those candidates who offer special
subjects for examination .”

Marshall :f “ It is not easy to get a clear full view of continuity
in this aspect without the aid either of mathematical symbols or
diagrams. * * * * experience seems to show that they give a firmer

grasp of many important principles than can be got without their
aid ; and there are many problems of pure theory, which no one
who has once learnt to use diagrams will willingly handle in any
other way.

The chief use of pure mathematics in economic questions seems

to be in helping a person to write down quickly, shortly and exactly ,
some of his thoughts for his own use : and to make sure that he has
enough , and only enough , premises for his conclusions (i. e. that his
equations are neither more nor less in number than his unknowns).

But when a great many symbols have to be used, they become very
laborious to any one but the writer himself. And though Cournot's
genius must give a new mental activity to everyone who passes

through his hands, and mathematicians of calibre similar to his
may use their favorite weapons in clearing a way for themselves to
the center of some of those difficult problems of economic theory ,
of which only the outer fringe has yet been touched ; yet it is
doubtful whether any one spends his time well in reading lengthy
translations of economic doctrines into mathematics, that have not

been made by himself. A few specimens of those applications of
mathematical language which have proved most useful for my own
purpose have, however, been added in an Appendix.”
* An introductory lecture on pol. econ . delivered before the University of

Oxford , Oct. 230, 1891, published in Economic Journal, Vol. i, No. 4, p. 629 .
+ Prin . of Econ . Preface to first Ed., p . xiv ; in 2d ed .


Irving Fisher - Mathematical investigations

Cunynghame:* “ But curves play in the study of pol. econ.

much the samepart as the moods and figures play in logic. They
do not perhaps assist in original thought, but they afford a
system by means of which error can be promptly and certainly
detected and demonstrated . And as in logic so in graphic pol.
econ. the chief difficulty is not to solve the problem , but to state it

in geometrical language.”
Contrast with the preceding the following statements from a few
who can see nothing good in mathematicalmethod :

A writer in the “ Saturday Review ” (Nov. 11, 1871), quoted by
Prof. Edgewortht says of Jevons : “ The equations, * * * assum

ing them to be legitimate, seem to us to be simply useless so long as
the functions are obviously indeterminable.” [Mathematics studies
relations as well as calculations. Numerical indeterminability is

common even in mathematical physics.]
Cairnes:f “ Having weighed Prof. Jevons's argument to the best
of my ability, and so far as this is possible for one unversed in

mathematics, I still adhere to my original view . So far as I can see,

economic truths are not discoverable through the instrumentality of
mathematics. If this view be unsound , there is at hand an easy
means of refutation - the production of an economic truth , not
before known, which has been thus arrived at ; but I am not aware

that up to the present any such evidence has been furnished of the
efficiency of the mathematical method . In taking this ground, I

have no desire to deny that it may be possible to employ geometrical
diagrams or mathematical formulæ for the purpose of exhibiting
economic doctrines reached by other paths, and it may be that there
are minds for which this mode of presenting the subject has advan

tages. What I venture to deny is the doctrine which Prof. Jevons
and others have advanced — that economic knowledge can be ex

tended by such means ; that mathematics can be applied to the
development of economic truth , as it has been applied to the devel

opment of mechanical and physical truth ; and, unless it can be

- -

-- -

* Geometrical methods of treating exchange value, monopoly and rent. H .

Cunynghame. Econ . Jour., March, '92 , p. 35 .
+ Math. -Psychics, p . 119.

# The Character and Logical Method of pol. econ . New York, 1875 . Preface.
See also, p. 122 ; also : Some leading principles of pol. econ . newly expounded.
Preface .

in the theory of value and prices .


shown either that mental feelings admit of being expressed in pre

cise quantitative forms, or, on the other hand, that economic phe
nomena do not depend upon mental feelings, I am unable to see how
this conclusion can be avoided.” [ There are examples in Cournot,

Walras, Auspitz und Lieben , Marshall, etc., which I think are fair
instances of the “ production of an economic truth, not before

known.” It is admitted, however, that each of these truths could
have been discovered without “ mathematical method ” by some

remarkably clear headed reasoner. But the same is true in physics.
The deduction used in every physical truth could be reasoned out

without diagrams or formulæ . A railway will best convey a man
from New York to San Francisco though it is perfectly possible to
walk .

Cairnes certainly has an erroneous idea of the use of mathe

matical method in physical investigations. Mathematics afford the
physicist a complete and precise view of his subject, and this con
dition of mind permits and facilitates his discovery.

The discovery

is only indirectly due to mathematics though it might never have

been madewithout it. Cairnes apparently thinks that physical truth
has been discovered by the manipulation of equations. The history
of physics will not bear him out. So far as I know only one physi
cal discovery was made in that way — a discovery in light. See the

quotation from Peirce at the beginning of this appendix .]
Wagner * [in reviewing Marshall's Prin. of Econ. ]: “ I do not
believe that this mode of treating the subject has an independent

value of its own for solving our problems. Indeed Marshall himself
admits as much [does he ? Cf. preceding statement of Marshall.]

* * * He has used diagrams and formulæ only for purposes of

illustration and for greater precision of statement." [Diagrams and
formulæ are never used for any other purpose yet they surely have
an independent value in (say) physics. Cf. $ 1.]
Ingram :f “ There is not much encouragement to pursue such
researches, which will in fact never be anything more than academic

playthings, and which involve the very real evil of restoring the
metaphysical entities previously discarded .” Also , “ Units of animal
or moral satisfaction, of utility and the like are as foreign to positive

science as a unit of dormative faculty would be.” [See Part 1, Ch. I].
Also :8 “ Mathematics can indeed formulate ratios of exchange when
* Quart. Jour. Ec., April. '91, p. 327.
| Ency. Brit., 9th ed . Vol. xix, p. 399.

Ency. Brit., 9th ed. Vol. xix , p . 386 .
& Hist. Pol. Econ., New York, 1888, p. 182.


Irving Fisher - Mathematical investigations

they have once been observed ; but it cannot by any process of its

own determine those ratios; for quantitativeconclusions imply quan
titative premises and these are wanting. There is then no future
for this kind of study, and it is only waste of intellectual power to

pursue it.”

[What a “ therefore ” ! Why require mathematics to

predict prices in order to be admitted into good society with the his
torical school ? No mathematical economist has ever tried to do
this. Dr. Ingram does not discuss what mathematics has done or
attempted, but complains loudly that it cannot do everything and

therefore has no future.]
Rabberio* in speaking of Prof. Pantaleoni's Principi di Economia
Pura says: “ As a monument of abstract logic, it bears fresh witness
to the unusual qualities of the author's genius; but it is based on a
method which , frankly speaking, I consider dangerous. In the face
of pressing practical problems of every kind, both in production and
in distribution , economic thought is drawn off into the field of bar

ren abstractions. Under an attractive semblance of mathematical
accuracy these abstractions conceal much that is really false ; for
they do not correspond in the least to the complexity of concrete

facts. While they distract the student with an imaginary logical
construction, they lessen his interest in that positive study which
tells us what is, whereas logic by itself gives us only what is
thought. Thus in last result they deprive economic science of that
great practical importance which it should have in society.” [ I am
not acquainted with Prof. Pantaleoni's book nor with any Italian

writer. As to the criticism on mathematical method, however, I
may say that experience in other sciences shows that “ in face of

many practical problems” it is wisest to “ draw off thought ” for a
time to pure theory. Before solving the problems of cannon pro
jectiles it is best to solve the problem of projectiles in general.
Before an engineer is fit to build the Brooklyn bridge or to pro

nounce on it after it is built it is necessary to study mathematics,
mechanics, the theory of stress and of the natural curve of a hang

ing rope, etc., etc. So also before applying political economy to
railway rates, to the problems of trusts, to the explanation of some
current crisis, it is best to develop the theory of pol. econ, in general.
When these special “ practical problems” are examined the mathe
matical instrument will, I believe, often be the one to get the best

I am far from denying, however, that some mathematical econo
mists have exhibited a “ false accuracy.” It has been due to
* Economics in Italy, Prof. Ugo Rabberio , Pol. Sci. Quart., Sept. 1891, p . 462.

in the theory of value and prices.


making special assumptions not with the purpose of facilitating
economic investigation but for permitting algebraic transformation.

A writer who intentionally parades his mathematics really does the
cause of mathematical economics much harm . I venture to think
that Launhardt's Volkswirtschaftslehre which contains some excel
lent things would have exhibited these excellencies better if the
author had contented himself with solving problems in all their

generality ].
$ 9.

I cannot refrain from venturing an opinion the application of
which may not apply to all of those writers just quoted but which
certainly applies to many : Mathematics is looked upon as an
intruder by those students of economics who have not had the
mathematical education to understand and make use of them , and

who are unwilling to believe that others enjoy a point of view
unattainable by themselves. A friend of mine much interested in
economics asked mewhat was the service of mathematics in the
subject. On hearing my reply be said : “ Well, I don't like to
admit that I can 't understand economics as well as those who have

studied highermathematics.”
Thus part at least of the opposition to mathematical method is a
mere incident to its novelty.

It must be remembered that the

character of economists is itself a variable and from generation to
generation those choose or reject the pursuit of economics according
to what it is at the time of choice. It may not be rash to expect
that the next generation of the theoretical (as distinct from histori
cal) economists will have fitted themselves by mathematical training
for this mode of treating their theme, and that they will be such
men as by natural aptitude can so fit themselves.
$ 10 .

The effort of the economist is to see, to picture the interplay of
economic elements. The more clearly cut these elements appear in
his vision , the better; the more elements he can grasp and hold in
mind at once, the better. The economic world is a misty region .
The first explorers used unaided vision . Mathematics is the lantern

by which what before was dimly visible now loomsup in firm , bold
outlines. The old phantasmagoria disappear. We see better. We
also see further.


Irving Fisher - Mathematical investigations


$ 1.

A bibliography of mathematico-economic writings was constructed
by Jevons and extended * by his wife up to 1888. This list con

tains a number of works mathematical in tone only . I have selected
out of the whole number (196 ), those 50 which are either undoubt

edly mathematical or are closely associated logically or historically
with the mathematical method . Thus Menger, though his writings
are not explicitly mathematical, is included for he founded the
“ Austrian School” which has ever since been allied with the mathe
matical method. In this selected list the references are much abbre
viated and only the first edition of each work is cited.
The second list is intended to be an extension of that of Jevons up
to the present date. I shall be indebted for information as to inac

curacies and omissions. A star has been placed opposite those writ
ings in which mathematical method is employed only occasionally or

whose mathematical character is not explicitly expressed in symbols
or diagrams. In the case of Italian and Danish writings, with which

I am wholly unacquainted and in thecase of a large number of others
which I have not been able to see and examine, I have been guided
by book notices or the wording of the title.
The list in Jevons' appendix and the second list here given may
be taken as a reasonably complete bibliography of mathematico

economic writings in the broadest sense, while the unstarred writings

in the abridged list of Jevons here quoted together with the un
starred writings in the second list represent the economic literature

which is strictly and avowedly mathematical. The distinction
between these two classes is tolerably wellmarked.

$ 2.


1711 CEVA - De re nummaria quoad fieri potuit geometrice Nactata .

1765 BECCARIA — Tentativo analitico sui contrabandi. Etc.
1801 CANARD — Principes d'economie politique.
1824 THOMPSON - Instrument of Exchange.
1826 von THÜNEN — Der isolirte Staat, etc.

* Pol. Econ ., Appendix I to third edition , 1888.

in the theory of value and prices.


WHEWELL — Mathematical Exposition of some Doctrines of Pol. Econ.
COURNOT- Recherches sur les principes math . de la théorie des richesses.
TOZER — Math. Investigation of the Effect of Machinery, etc.
ANONYMOUS — On Currency.
TOZER — On the Effect of the Non -Residence of Landlords, etc.
DUPUIT — De la mesure de l'utilité des travaux publics.

1844 HAGEN - Die Nothwendigkeit der Handelsfreiheit, etc.

- 1850

BORDAS — De la mesure de l'utilité des travaux publics.
DUPUIT - De l'influence des péages sur l'utilité des voies de communication.
LARDNER — Railway Economy (chapter xiii).
WHEWELL — Mathematical Exposition of Certain Doctrines of Pol. Econ.

1854 GOSSEN - Entwickelung derGesetze des menschlichen Verkehrs, etc.

BENNER - Théorie mathématique de l'economie politique.
MANGOLDT- Grundriss der Volkswirthschaftslehre.
FAUVEAU _ Considérations math . sur la théorie de l' impôt.
FAUVEAU — Considérationsmath , sur la théorie de la valeur.
JENKIN — The Graphic Representation of the lawsof Sup. and Demand , etc.
JEVONS — The Theory of Political Economy.
MENGER – Grundsätze der Volkswirthschaftslehre.
LAUNHARDT– Kommerzielle Trassirung der Verkehrswege.
POCHET - Géométrie des jeux de Bourse.

1874 WALRAS - Principe d'une théorie math . de l'échange .

1874 WALRAS - Éléments d 'économie politique pure.
1874 * LETORT—- De l'application des math , à l'étude de l'econ . pol.

1875 * DARWIN — The Theory of Exchange Value.
1875 * BOCCARDO — Dell' applicazione deimetodi quantitativi, etc.
1876 WALRAS - Equations de l'échange, etc.
1876 WALRAS - Équations de la capitalisation .

1876 WESTERGAARD - Den moralske Formue og detmoralske Haab.
1878 * WEISZ — Die mathematische Methode in der Nationalökonomie.
1879 WALRAS - Théorie math . du billet de banque.
1881 EDGEWORTH ---Mathematical Psychics.
1881 WALRAS — Théorie math. du bimétallisme.
1883 LAUNHARDT— Wirthschaftliche Fragen des Eisenbahnwesens.
1884 * WIESER - Hauptgesetze des wirthschaftlichen Werthes.
1885 LAUNHARDT— Math. Begründung der Volkswirthshaftslehre.
1886 GROSSMAN — Die Math . im Dienste der Nationalökonomie. I Lieferung.
1886 * NEWCOMB— Principles of Political Economy.
1886 * BÖHM -BAWERK — Theorie des wirtschaftlichen Güterwerts.
1886 ANTONELLI Teoria math . della economica politica.
1886 GROSSMAN - Die Math . im Dienste der Nationalökonomie. II Lieferung.
1887 VAN DORSTEN - Math , onderzoekingen op het-gebied Staathuishoudkunde.
1887 WESTERGAARD — Math . i Nationalökonomiens Tjeneste.
1887 PANTALEONI— Teoria della pressione tributaria, etc.
1888 WICKSTEED — The Alphabet of Economic Science.

TRANS. Conn. ACAD., Vol. IX .

JULY, 1892.

Irving Fisher - Mathematical investigations


$ 3.
1867 WITTSTEIN -- Mathem . Statistik . Hanover .
1882 PANTALEONI (M .)/ La Traslazione dei Tributi. Rome. Paolini .
1884 SCHROEDER (E . A .)— Das Unternehmen und der Unternehmergewinn vom

historischen , theoretischen und praktischen Standpunkte. Wien . 92 pp.
1884* SAX (E .)- -Das Wesen und die Aufgabe der Nationalökonomie.
1887* Sax (E .) - Grundlegung der theoretischen Staatswithschaft.
1887 PIOARD (A .) Traité des Chemins de fer. 4 vols. Paris : Rothschild .
1888 EDGEWORTH ( F . Y .)- New method of measuring variations in general

prices. Jour. Stat. Soc. London , p . 347.
1888* Sax (E .) Die neusten Fortschritte der nationalökonomischen Theorie.
Vortrag gehalten in Dresden märz. Leipzig : Duncker & Humblot. 8vo.
38 pp.

1888 * MENGER (C .) — Contribution à la théorie du Capital. [ Trans, from Jahrb .

für Nat. Oek., by C. Secrétan .] Rev. d' Écon . Pol., Dec. '88.
1888* SALERNO (Ricca)— Manuale di Scienza delle Finanze. Florence. Barbera .
1888 HADLEY (A . T.)— Railroad Transportation, its History and its Laws. New
York and London . 269 pp . (Appendix II.)

1888 GOSSEN (H . F .)- Entwickelung der Gesetze des menschichen Verkehrs.
New edition .] Berlin : Prager. 8vo . 286 pp.

1888 * MENGER (C .)— Zur Theorie des Kapitals. Jahrb. Nat. Oek ., 17 Heft 1.
1889 PANTALEONI (M .)— Principi di Economia Pura. Florence. Barbera .

1889 AUSPITZ UND LIEBEN - Untersuchungen über die Theorie des Preises.
Leipzig : Duncker & Humblot. 555 pp.
1889* ZUCKERKANDL (R .)- Zur Theorie des Preises mit besonderer Berück

sichtigung der geschichtlichen entwicklung der Lehre. Leipzig. 348 pp.
1889* WIESER (F . von )- Der natürliche Werth . Wien . 237 pp.

1889* BÖHM -BAWERK (E.)- Kapital und Kapitalzins. Translated into English
by Wm. Smart. 1890 . London and New York : Macmillan .

1889* LEHR (J.)— Wert, Grenzwert, und Preise. Jahrb. Nat. Oek., 19 Heft 1.
1889 SUPINO (C .)— La Teoria del Valore e la Legge del minimo mezzo . Giorn .
degli Econ . Aug. '89.

1889 WALRAS (L .)- Théorème de l'Utilité maxima des Capitaux Neufs. Rev.
d'Econ . Polit., June '89.

1889* MACLEOD (H . D .)— The Theory of Credit . Vol. I. London : Longmans &
Co. 8vo. 342 pp.

1889 ST. MARC (H .)— Les Procédés d ’Analyse Graphique à l'Exposition Uni
verselle. Rev. d'Écon . Polit., Aug. '89.
1889 VIRGILII (F .)- La Statistica Storica e Mathematica . Giorn . degli Econ .,

Aug. '89, concluded Oct. '89.
1889 * HEARN (W . E .) – Plutology ; or, The Theory of the Efforts to satisfy Hu
man Wants. [New edition .] Melbourne : Robertson . 8vo. 486 pp .
1889* KOMARZYNSKI (J.)— Der Werth in der isolirten Wirthschaft. Wien .
Manz. 8vo . 105 pp.

1889 Rossi (G .) - La Mathematica applicata alla Teoria della Ricchezza Sociale :
Studî Bibliografici. Storici, e Critici. Vol. I, fasc. 1. Reggio Emilia
Artegianelli. 8vo . 103 pp., 4 charts.

in the theory of value and prices.


1889 * BöHM -BAWERK (E . von )-- Une Nouvelle Théorie sur le Capital. Rev.
d ' Écon . Pol., April, 1889.
1889* Böhm -BAWERK (E . von )— Kapital und Kapitalzins. Zweite Abteilung :
Positive Theorie des Kapitals. Innsbrück . 8vo.
1889* CLARK (J. B .) - Possibility of a Scientific law of Wages. [Publ. of Am .

Econ. Assoc.] Baltimore. 8vo. 32 pp.
1889 WICKSTEED (P. H .)— On certain Passages in Jevons' “ Theory of Political

Economy.” Quart. Jour. Econ., April, '89, p. 293.
1889 EDGEWORTH (F . Y .)— On the application of Mathematics to Pol. Econ .
Journ . Stat. Soc. London , Dec. '89.
1890 * DIETZEL - Die Klassische Werttheorie und die Theorie vom Grenznutzen .
Conrad 's Jahrbuch N . F . Band 20. pp . 561-606 .
1890 * MACLEOD (H . D .)— The Theory of Credit. Vol. II, Part I. London :
Longmans. 8vo.
1890 MARSHALL ( A .)— Principles of Economics. Vol. 1, 1st ed . London :
Macmillan . 770 pp . Mathematical Footnotes and Appendix. ]
1890 PANTALEONI (M .)- Principidi Economia Pura. Florence : Barbèra . 16mo.
376 pp.

1890 JURISCH ( K . W .) — Mathematische Diskussion des Eutwickelungsgesetzes
der Werterzengung durch industrielle Produktionsgruppen . Viertelj. f.
Volksw . 27 Band 3, 1. Second paper, same title, 27 Band 3, 2 .
1890 VAUTHIER (L . L .) Quelques Considérations Élémentaires sur les Construc
tions Graphiques et leur Emploi en Statistique. Journ . de la Soc. Sta
tist., June, ’90 .

1890 * AUSPITZ (R .)- Die klassische Werttheorie und die Lehre vom Grenznutzen .
Jahrb. Nat. Oek . 21 Heft 3 ; reply to Dietzel, same journal, 20 Heft 6 .

1890* ZUCKERKANDLE (R .)— Die klassische Werttheorie und die Theorie vom
Grenznutzen . Jahrb. Nat. Oek . 21 Heft 5. Reply to Dietzel.
1890 Colson (G .)- Transports et Tarifs . Précis du Régime, Lois Économiques
de la Détermination des Prix de Transport, Tarifs de Chemins de Fer,
etc. Paris : Rothschild . 8vo. 479 pp .

1890 LAUNHARDT (W .) - Theorie der Tarifbildung der Eisenbahnen. Berlin :
Springer. 8vo . 84 pp .
1890 WESTERGAARD (H .) - Die Grundzüge der Theorie der Statistik . Jena :
Fischer. 8vo . 286 pp .
1890 Cossa (E .) — Le Forme Naturali della Economia Sociale. Milan : Hoepli.

1890 MARSHALL (A .)- Principles of Economics. Vol. I, 2nd ed. London :
Macmillan . 770 pp. [Mathematical Footnotes and Appendix.]
1891* HOBSON ( J. A .)- The law of the three rents. Quart, Jour. Econ ., April,
1891, p . 263.

1891* CLARK (J. B.)— Distribution determined by a law of rent. Quart. Jour.
Econ ., April, 1891, p . 289 .

1891 EDGEWORTH (F . Y .)- Osservarioni sulla Teoria matematica dell' Economia

Politica con riguardo speciale ai Principi di Economia di Alfredo Mar
shall. Giorn . degli Econ ., March , '91.

1891 * SMART (W .)- An Introduction to the Theory of Value on the lines of
Menger, Wieser and Böhm -Bawerk . London and New York : Macmillan.

16mo. 88 pp.


Irving Fisher - Mathematical investigations, etc.

1891* CLARK (J. B.) — The statics and the dynamics of Distribution. Quart,
Jour. Econ., Oct. '91, p. 111.
1891* WIESER (F .)- The Austrian School and the Theory of Value. Economic
Journal, March , '91.

1891* BöHM -BAWERK (E. von)— The Austrian Economist. Annals of Am . Acad .
of Polit . Sci., Jan . '91.
1891 EDGEWORTH (F . Y .)— La ThéorieMathematique de l'Offre et de la Demande
et le Coût de Production . Rev. d 'Écon . Polit., Jan. '91.
1892* BÖHM -BAWERK (E . von )- Wert, Kosten und Grenznutzen. Jahrbücher
für Nationalökonomie und Statistik , Dritte Folge, Dritter Band, Drittes

Heft, pp. 321–378.
1892 BILGRAM (H .) — Comments on the “ Positive Theory of Capital ” ſof
Böhm -Bawerk ). Quart. Jour. Econ ., Jan . ’92, pp. 190 – 206 .

1892 GROSSMAN (L .)- Die Mathematik in Dienste der Nationalökonomie unter
Berücksichtigung auf die praktische Handhabung der Finanzwissenschaft

und der Versicherungstechnik (schluss Lieferung). Vienna .

1892* WIESER (F. von)— The Theory of Value. A reply to Prof. Macvane. An
nals of Am . Acad . of Pol. and Soc. Sci., March, '92.
1892* SELIGMAN (E . R . A .)- On the Shifting and Incidence of Taxation. Publ.
of Amer. Econ . Asso., Vol. VII, Nos. 2 and 3.
1892* PATTEN (S. N .) — The Theory of Dynamic Economics. Publ. of Univ. of
Penn ., Pol. Econ , and Public Law Series, Vol. III, No. 2 . Phila . 8vo.
153 pp.

1892* BÖHM -BAWERK (E.)- Wert, Kosten , und Grenznutzen . Jahrb . Nat. Oek .,
3, Heft 3.
1892 CUNYNGHAME (H .)— Geometricalmethods of treating Exchange-value, Mo

nopoly, and Rent. Econ . Journ., March , '92.
1892 PARETO (V .)— Considerazione sui Concipi Fondamentali dell' Economia
Politica Pura. Giorn . degli Econ ., May, '92 .
1892 PARETO (V .).- La Teoria dei Prezzi dei Signori Auspitz e Lieben e la Osser
vazioni del Professore Walras. Giorn . degli Econ ., March , ’92.
1892* VOIGT (A .)— Der Oekonomische Wert derGüter. Zeitschr. f. Ges. Staatsw .,

48, Heft 2.
1892 WALRAS (L .)— Geometrical Theory of the Determination of Prices. Annals
Amer. Acad . Polit. and Social Sci., Phila., July, '92. Translated under
the supervision of Irving Fisher. Part I was published in French in
the Bulletin of Soc. of Civil Eng. of Paris, Jan. 1891, and Parts II and
III in the Recueil inaugural of Univ. of Lausanne, July ( ) ’92.
1892 FISHER (I.)— Mathematical Investigations in the Theory of Value and
Prices Transactions of the Connecticut Academy of Arts and Sciences.
Vol. IX , pp. 1- 124.