Full text of Economic Quarterly (Federal Reserve Bank of Richmond) : March-April 1992, Vol. 78

The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.

Marshallian

Cross Diagrams

and

Their Uses before Alfred Marshall:
The Origins of Supply and Demand

Geometry

ThonaasM. Humphrey

Undoubtedly the simplest. and most frequently
used tool of microeconomic analysis is the conventional partial equilibrium demand-and-supply-curve
diagram of the textbooks. Economics professors
and their students put the diagram to at least six
main uses. They use it to depict the equilibrium or
market-clearing price and quantity of any particular
good or factor input. They employ it to show how
(Walrasian) price or (Marshallian) quantity adjustments ensure this equilibrium: the first by eliminating
excess supply and demand, the second by eradicating
disparities between supply price and demand price.
They use it to illustrate how parametric shifts in
demand and supply curves induced by changes in
tastes, incomes, technology, factor prices, and prices
of related goods operate to alter a good’s equilibrium
price and quantity. They apply it to show how the
shifting and incidence of a tax or tariff on buyers
and sellers depends on elasticities of demand and
supply. With it they demonstrate that price ceilings
and price floors generate shortages and surpluses,
respectively. Finally, they employ it to compare the
allocative effects of competitive versus monopoly
pricing and to indicate the welfare costs of market
imperfections.

And Karl Rau (184 l), Jules Dupuit (1844), Hans von
Mangoldt (1863), and Fleeming Jenkin (1870) thoroughly developed it years before Marshall presented
it in his Pzm Theory of Domestic Vafues (1879) and
later in his Pnitciples offionomics (1890). Far from
merely introducing the diagram, these writers applied
it to derive many of the concepts and theories often
attributed to Marshall or his followers. The notions
of price elasticity of demand and supply, of stability
of equilibrium, of the possibility of multiple equilibria,
of comparative statics analyses involving shifts in the
curves, of consumers’ and producers’ surplus, of constant, increasing and decreasing costs, of pricing of
joint and composite products, of potential benefits
of price discrimination, of tax incidence analysis, of
deadweight-welfare-loss
triangles and the allocative
inefficiency of monopoly: all find expression in
early expositions of the diagram.

The diagram’s applications are of course well
known. Not so well known, however, are its origins
and early history. Economists typically tend to
associate the diagram with Alfred Marshall, its
most persuasive and influential nineteenth century
expositor. So strong is the association that economists
have christened the diagram the Marshallian cross
or Marshallian scissors after Marshall’s analogy comparing the price-determining properties of a brace of
demand and supply curves with the cutting properties of the blades of a pair of scissors.

These expositions, however, have not always been
fully appreciated. John Maynard Keynes ([ 192.51
1956: 24), for example, dismissed them as vastly
inferior to “Marshall’s diagrammatic exercises” which
“went so far beyond the ‘bright ideas’ of his
predecessors that we may justly claim him as the
founder of modern diagrammatic economics.” In the
same vein, Michael Parkin (1990: 85) recently has
claimed that of early graphical treatments of supply
and demand only Marshall’s is sufficiently modern
to be recognized by textbook readers today. Such
discounting of the pre-Marshall work has contributed
to what Joseph Schumpeter (1954: 839 n. 13) complained of as economists’ “uncritical habit of
attributing to Marshall what should, in the ‘objective’
sense, be attributed to others (even the ‘Marshallian’
demand curve!).” Seen this way, Marshall’s definitive
contribution
emerges as the cdmination of the
diagram’s development.
Economists, Schumpeter
thought, misperceived it as the origin.

The diagram itself, however, long predates Marshall. Antoine-Augustin Cournot originated it in 1838.

In an effort to correct such misperceptions and to
give the earlier writers their due, this paper traces

FEDERAL RESERVE BANK OF RICHMOND

3

their pioneering contributions so as to counter the
notion that the Marshallian cross diagram begins with
Marshall. To paraphrase Frederick Lavington’s
famous remark that “it’s all in Marshall, if you’ll only
take the trouble to dig it out,” (Wright 1927: 504)
the following paragraphs attempt to show, with
respect to the diagram and its applications, that it’s
all in Marshall’s predecessors too.

ANTOINE-AUGUSTINCOURNOT

COURNOT’S

1

DEMAND

CURVE

[quantity]

0

(1801-1877)

Though hardly the first to state that supply and
demand determine price, Antoine-Augustin Cournot,
in his 1838 Recherchs sur l’esprincipes mathtfmatiques
ak b t&wriedes tihesses (Reseamk into tk Mathematial
Principbs of the Thor-y of Wealth), was the first to draw
market demand and supply curves for a particular
good.’ Of the two curves, Cournot analyzed demand
before introducing supply. Figure 1 shows his diagram
of the demand function D = F@) where D is quantity sold at different prices p, which Cournot
measured along the horizontal rather than the vertical axis as is customary today.z Cournot ([1838]
197 1: 52-3) noted that corresponding to each price
on the demand curve ab is a price-quantity rectangle
whose area R = pD represents total revenue R
associated with that price. He sought to determine
the particular price at which revenue is at a maximum.
A profit-maximizing monopolist would charge this
price if his costs were either zero or a fixed sum independent of quantity produced.
To find the revenue-maximizing
price, Cournot
differentiated the revenue function R = pD = pF@)
with respect to price. He obtained the expression
pF’(p) + F(p), where F’ is the derivative of the
demand function F. Setting this expression equal to
zero as required for a maximum and rearranging, he
gotp = -F(p)/F’@), which says that the revenuemaximizing price must equal the ratio of the quantity demanded to the slope of the demand curve at
that price. To depict this solution diagrammatically, he rewrote the expression as F@)lp = -F’@).
’ That supply and demand determine price must have been
known for thousands of vears. But the phrase “sutmlv and demans
itself is of more recent vintage; Sir James Steuart
originated it in 1767 (Thweatt 1983). Adam Smith (117761
19307:56) in his weo/t/r‘ofNationsused ;he notion of sup$y and
demand to explain deviations of actual market price from natural
price determined by long-run cost of production. And David
Ricardo ([ 18 17-Z 1] 195 1) gave the idea added prominence when
he incorporated it into the title “On the Influence of Demand
and Supply on Price” of Chapter 30 of his Principlesof Political
Economy and Taxation.
.

.

I

* On Cournot see Theocharis (1983: 138-46) and Ekelund,
Furubotn, and Gramm (1972: 15-19).

4

Figure

Source:

Cournot ([11338] 1971: Fig. 1 at end of book)

The left-hand side he represented, for any point n
on the demand curve, by the slope qn/Oq of the ray
On (see Figure I). The right-hand side he portrayed
by the slope qnlqt of the line segment nt. He noted
that these slopes are equal only when Oq equals qt,
which uniquely determines the revenue-maximizing
price Oq as one-half of Ot.
Price Elasticity of Demand

Cournot (53-4) also anticipated Marshall’s concept
of price elasticity of demand, defined as the percentage change in quantity demanded divided by
percentage
change in price: (dD/D)/(dp/p)
or
pdD/Ddp. True, William Whewell had foreseen the
idea before him in 1829. But Whewell, whose work
was unknown to Cournot, did not use diagrams or
draw a demand curve. Cournot, in drawing the curve,
argued that one can determine the effect of a small
change in price Ap on total revenuepF@) by comparing the curve’s slope AD/Ap with the ratio of
quantity demanded to price D/p. A small rise in
price, he claimed, will cause total revenue to rise,
fall, or remain unchanged at its peak level as
ADlAp is less than, greater than, or equal to D/p.
Setting ADlAp = D/p and dividing both sides by D/p
yields (p/D) (AD/Ap) = 1. Now the left-hand side
of this expression is Marshall’s measure of elasticity. Hence Cournot’s statement that total revenue

ECONOMIC REVIEW. MARCtiIAPRIL

1992

Figure

achieves its stationary maximum when ALI/& equals
D/p is equivalent to saying that it does so at the point
of unitary elasticity on the demand curve. Similarly,
his contention that revenue rises or falls with a rise
in price as ADlAp is less than or greater than D/p
is tantamount to declaring that it does so as elasticity is above or below one in value.
Supply Curve

COURNOT’S
TAX

SCISSORS
INCIDENCE

2

DIAGRAM

AND

ANALYSIS

[price]

2D

Introduced

After specifying the elasticity-revenue relationship,
Cournot incorporated supply curves into his diagram.
Assuming a regime of perfect competition, he (90-Z)
argued that profit-maximizing firms equate price with
marginal cost. Since marginal cost after a certain point
rises with the level of output, each firm produces
along a schedule showing output as an increasing
function of price. Summing over the individual firms’
supply functions, Cournot obtained the total market
supply function S@) expressing a schedule of
quantities offered at all possible prices. Finally, he
equated market supply and demand S@) = F@) to
determine equilibrium price and output. All this he
depicted in Figure ‘2 in which the downward-sloping
demand curve MN is combined with the upwardsloping supply curve PQ to yield the equilibrium
price-quantity combination S.3

T

I

1r-

-7
[quantity]
Source:

Cournot ([1838] 1971: Fig. 6 at end of book).

Effects of a Tax

Having depicted supply-demand
equilibrium,
Cournot (92-3) next used his curves to show the
effect of a per-unit tax of amount VS’ levied on a
particular good. He argued that the tax, by adding
L’S’ francs to the cost of supplying each unit of
output, would shift the supply curve upward by that
same amount. The result is a rise in the equilibrium
price from OT to OT’ and a fall in the equilibrium
quantity from TS to T’S’. He noted, however, that
with an upward-sloping supply curve the price increase m’ is always less that the tax VS’ provided
the demand curve is not vertical. Indeed, if the
demand curve were horizontal the tax would not
increase price at all. Cournot was thus the first to
show that given a positively sloped supply curve the
portion of an excise tax shifted to buyers in the form
of higher prices varies inversely with demand elasticity, being nil at infinite and complete at zero values
of that parameter. Cournot, not Marshall, invented
geometrical tax-incidence analysis.

This is Cournot’s Figure 6 with the axes switched to conform
to current practice.

3

KARL HEINRICH RAU

(1792-1870)

Cournot did not use his scissors diagram to expound the stability of market-clearing equilibrium.
Although he employed stability analysis in his famous
reaction-function model of duopoly, he never did so
in his competitive supply and demand model. In
184 1, three years after the publication of Cournot’s
Researc/res but with no knowledge of its contents,
the German economist Karl Heinrich Rau rectified
this oversight (Hennings 1979).4 In so doing, he
became the first to employ a Marshallian cross
diagram to investigate the stability of market
equilibrium and to indicate the forces that restore that
equilibrium once it is disturbed.
Rau’s diagram, which owed nothing to Cournot’s
and indeed differed from it by putting price on the
vertical axis and quantity on the horizontal, first
4 Hennings (1979: 6) argues that Rau never saw a copy of
Cournot’s Researchesand thus was ignorant of its diagrams. He
points out that neither Rau’s library nor that of his university
possessed Cournot’s book.

FEDERAL RESERVE BANK OF RICHMOND

5

appeared in a note in the 1841 volume of the Bultin
de C’Acadhie des Sciences et des Bekdettres
de
Bmxefhs.~ He published a more elaborate version
in the appendix to section 154 of the fourth (1841)
and all subsequent editions of his textbook &wz~s&ze der lWh-witihscha~slehm (see Figure 3). There
he argued that, given the linear demand curve g/l and
the vertical supply curve a& a below-equilibrium price
ec would produce an excess demand cb. Suppliers
would take advantage of the resulting shortage of the
good to raise price until the shortage disappeared and
equilibrium m was restored.6
Rau thought this result important enough to state
algebraically. His formula, cm = (ab -ac) tan tw,
expresses the price rise cm as the product of the
excess demand (ab -ac) times the slope of the
demand curve tan w where w is the angle formed
by lines cb and bm. This trigonometric expression,

of course, holds only for the linear demand-vertical
supply case. Alternatively, if the demand curve were
instead the concave schedule j, the same excess
demand would induce sellers to raise price top. And
if the supply schedule were the positively sloped
curve ek, then excess demand would spur sellers to
boost price to I or n depending on which demand
curve they faced. Conversely, an above-equilibrium
price would generate an excess supply that would
put downward pressure on price until equilibrium
was restored.
Rau’s diagram seems to have influenced only one
of his contemporaries,
Hans von Mangoldt, who
used it as the starting point for his own demand-andsupply-curve analysis in 1863, twenty-two years after
Rau first drew it.’ In the meantime, other writers,
including the French engineer Jules Dupuit who
published demand-curve diagrams in 1844, either
ignored it or were unaware of its existence.

5 On Rau’s diagram and its applications see Hennings (1979)
who provides an English translation of the relevant passages.
6 Of course demanders too could take the initiative and bid up
the price. But Rau did not mention this possibility.

Figure 3
RAU’S

DIAGRAM

[price]

Despite their originality, neither Cournot nor Rau
derived welfare propositions from their diagrams.
Because they saw demand curves as empirical sales
schedules rather than as replicas of theoretical
marginal utility functions, they said nothing about
the welfare implications of monopoly pricing, public
utility rate setting, discriminatory pricing, or commodity taxation.8
Jules Dupuit, however, was not so hampered.
Explicitly identifying demand curves with marginal
utility schedules, he became, in 1844, the first to
derive welfare theorems with the aid of a Marshallian
diagram. True, he drew no supply curves. He merely
assumed a constant supply price or one that varies
independently of the level of output. But he made
path-breaking use of the demand curve to define such
Marshallian concepts as total utility, consumer
surplus, and deadweight-welfare-loss
triangles, not
to mention Laffer-curve relationships between tax
rates and revenues. In so doing he advanced demand
theory far beyond Cournot and Rau, whose work was
unfamiliar to him.9

B
45 a#
40--o

33 'OH
30 -

25-a”
-a’

.20 -

I5 -

-a

10 _

’ Hennings (1980: 670) notes than Mangoldt, in advertising his
book, explicitly cited Rau as a precedent for employing diagrams.
h
e

Source:

JULESDLJPUIT(1804-1866)

Rau (1841b: 527).

x

[quantity]

8 On Cournot’s and Rau’s view of demand functions as purely
empirical schedules, see Ekelund, Furubotn, and Gramm (1972:
17) and Hennings (1979: 2).
9 Ekelund and Htbert (1983: 260) note that Dupuit was ignorant
of Cournot’s work even though both writers once lived and
worked in Paris at the same time.

ECONOMIC REVIEW, MARCH/APRIL

1992

Laws

of Demand

prices and responding to lower prices, give the curve
its characteristically convex shape.

Figures 4, 5 and 6 depict Dupuit’s diagrams as
presented in the appendix to his 1844 article “On
the Measurement of the Utility of Public Works” in
the Annales des Ponts et Chausstfes.‘OThe diagrams as
shown illustrate Dupuit’s two “laws” of demand.”
Law number one says that demand curvesexpressed by Dupuit as y = f(x), where y denotes
quantity demanded and x price-slope
downward
because of diminishing marginal utility: extra quantities of a good or service add less and less to total
satisfaction and thus command lower demand prices.
Law number two says that a given fall in price induces larger increases in quantity demanded the
lower the price at which it occurs. This law Dupuit
attributed to the pyramidal distribution of income;
each price decrement activates the demands of a new
group of buyers larger and poorer than the group
above it on the income scale. The resulting new
demands, added to those already existing at higher
10These are Dupuit’s Figures
switched to facilitate inspection.

Marginal Utility, Total Utility;
Consumers’ Surplus (Figure 4)

Having deduced the shape of the demand curve,
Dupuit used his first diagram to refute J. B. Say’s
contention that a good’s market price measures the
utility of each unit consumed. Not so, said Dupuit
([1844] 1969: 280-l).
Price Up measures the
(marginal) utility nr of the last unit purchased only.
Preceding or inframarginal units such as r’ and T”
yield higher marginal utilities as indicated by the
higher demand prices p ’and p ” buyers would pay
for those units rather than go without. Summing over
these successive demand prices as one moves up the
demand curve gives a measure of the total utility of
the entire quantity consumed. This measure is
represented by the roughly trapezoidal area OPnr
under the demand curve and not, as Say implied,
by the price-times-quantity or total-expenditure rectangle Opnr. Say’s measure understates utility by the
amount of the consumers’ surplus triangle pPncalled utifitb relative by Dupuit. Taken to its extreme, Say’s analysis erroneously implies that total

1, 3 and 4 with the axes

11On Dupuit’s two laws of demand, see Houghton (1958: 50)
and Ekelund and Thornton (1991).

Figure 4

DUPUIT
MARGINAL

ON TOTAL

UTILITY,

UTILITY,

AND CONSUMERS’

SURPLUS

[price]

r
Source:

Dupuit ([1844] 1969: 280).

FEDERAL RESERVE BANK OF RICHMOND

N
[quantity]

utility is zero when price is zero. In fact, total utility
would then be at its maximum equal to the entire
area under the demand curve.
As for the triangular area Nm in the lower righthand corner of the diagram, Dupuit defined it as the
utility lost when a positive market price Cp constrains
consumption short of the satiation point N. l2 Under
competitive conditions this loss is the natural result
of resource scarcity and cannot be avoided. Here
price measures the satisfaction forgone on other
goods whose production falls so that resources can
be freed to produce an extra unit of the good in question. That the last rN units of this good possess
marginal utilities falling below price indicates that they
are not worth producing; their opportunity cost
exceeds the extra satisfaction they would bring. Thus
units r through N are forgone so that resources can
be put to higher-valued uses elsewhere.
Under monopoly pricing, however, the loss may
be due to contrived restriction of output and is a
true measure of social harm. A costless monopoly
charging price Op, for example, needlessly deprives
consumers of satisfaction equivalent to the area Nm.
Here is the first diagram in the history of economic
thought to depict a deadweight-loss-from-monopoly
triangle and a total utility trapezoid under the demand
curve. And it is the first to partition the trapezoid
into a price-quantity rectangle showing buyer expenditure on the good and a consumers’ surplus triangle
showing the excess of what consumers would pay
over what they actually do pay. In short, Dupuit, not
Marshall, was the first to demonstrate diagrammatically that consumers get more utility than they
pay for when they buy a good or service at a single
market price. He was also the first to extend this insight to the evaluation of public works. In particular,
he noted that the potential benefits of such projects
cannot be measured by their costs. One needs to
estimate the area under the demand curve.
Dupuit’s Tax Theorems

(Figure

5)

Armed with the utility concepts developed in
his first diagram, Dupuit (281-2) used them in his
second diagram to derive key propositions concerning the welfare effects of commodity taxes. His first
proposition states that the imposition of a tax results
in a loss in consumers’ surplus that exceeds the yield
of the levy. His diagram shows how a per unit tax
12On Dupuit’s concept of lost utility (uhfh!pe&e)
(1970: 271-3).

8

see Ekelund

Figure

DUPUIT’S

5

TAX THEOREMS

[price]

P

K

M
p”
P”
P’
P

C

N
[quantity]

Source:

Oupuit (I18441 1969: 282)

of qn ’raises price by pp ‘, thus reducing purchases
top ‘n ’and consumers’ surplus top ‘Pn ‘. On thep ‘n ’
units still bought, consumers pay a total tax ofpp ‘n ‘q
to the government, which Dupuit assumed puts it
to socially productive uses. But, on the qn units no
longer bought, buyers lose consumers’ surplus qnn ’
with no corresponding gain to the government.
Hence the tax causes a loss of consumers’ surplus
that exceeds the tax yield by the roughly triangular
area n ‘qn: the deadweight loss of the tax. This loss,
consisting of the tax-induced distortion of relative
prices and consumption patterns, persists even if the
government returns the proceeds to the taxpayers.
Dupuit’s second theorem states that the deadweight loss is proportional to the square of the
tax rate. As mentioned above, the welfare loss AU
is the area of the triangle n ‘qn whose height is the

ECONOMIC REVIEW. MARCH/APRIL

1992

tax rate t and whose base is the reduction in quantity bought AQ caused by the duty. Since the area
of a triangle is half its height times its base, one sees
that the net loss in utility is half the tax rate times
the fall in amount purchased or AU = %tAQ. Now
AQ, the change in amount bought, can by definition
be expressed as AQ = kt, where k is the inverse
AQ/t of the slope of the triangle’s hypotenuse. Substituting this expression into its predecessor yields
Dupuit’s taxation theorem: AU = ‘/zkt2 or, in his
(28 1) words, “the loss of utility increases as the square
of the tax.“r3 It follows that the government minimizes the welfare burden of a given total tax collection by charging a low rate on a great many goods
rather than a high rate on a few. For the welfare
loss, which shrinks as the square of a lowered rate,
approaches zero as the tax becomes general or
diffused and its rate correspondingly small.
Dupuit’s third theorem posits an inverted U-shaped
or Laffer-curve relationship between tax rates and
tax revenue. Like Arthur Laffer in the 198Os,
Dupuit in 1844 saw tax revenues rising from zero
with small increases in the rate, reaching a maximum
pMTQ at rate PM, then falling with further rate
increases, and eventually returning to zero when the
rate becomes prohibitive. This rate-revenue relationship together with his second tax theorem led him
(282) to conclude “that the yield of a tax is no
measure of the loss which it causes society to
suffer.” For the same yield can be obtained from two
different rates entailing markedly different deadweight
losses. RatespK andpp “, for example, yield the same
revenue; yet the first rate’s welfare-loss triangle is
more than ten times the size of the seconds. Likewise, zero and prohibitive tax rates both yield zero
revenue. The zero rate, however, produces no
welfare loss while the prohibitive rate produces a loss
equal to the whole area under the demand curve.
Pricing Policies
(Figure 6)

would be larger and deadweight loss smaller by the
amounts pMTn and 71(m, respectively.
Dupuit also analyzed price discrimination-the
practice of charging separate customers different
prices for the same product-with
the aid of his third
diagram.r4 He argued that discriminatory pricing
could render profitable a firm that would suffer losses
if it charged a single price. Dupuit examined the case
of a monopolist whose fixed costs exceed his receipts
OMTR at the revenue-maximizing
price UM. The
monopolist, by dividing his market MT into two
groups, one paying price Op ’for quantity p ‘n ’and
the other price OM for quantity q’T, could expand
I4 On Dupuit’s analysis of price discrimination
(1970: 271-S).

see Ekelund

Figure 6

DUPUIT

ON PRICING
PRICE

POLICIES

AND

DISCRIMINATION

[price]

P

and Price Discrimination
p’

Dupuit (282-3) employed his last diagram to
specify appropriate pricing policies for private and
public monopolies having identical fixed costs Upnr.
A private monopoly would charge the price OM that
maximizes its receipts OMTR and, with costs given
and independent of output, its profits too. By contrast, a public utility would charge the lowest price
Op that maximizes consumer satisfaction subject to
meeting the cost constraint. Consumers’ surplus

M
P

0
[quantity]

13 See Htbert
and Ekelund (1984: 6’2) for an alternative derivation of this formula.

Source:

Dupuit ([18441 1969: 283).

FEDERAL RESERVEBANK OF RICHMOND

9

his receipts by an amount Mp ‘n ‘q’sufficient to defray
his costs. Further discrimination would yield even
larger receipts. For example, were the monopolist
able to charge the maximum price for each successive
unit of the MT quantity sold, he could effectively
redistribute consumers’ surplus to himself and capture revenue equal to the entire area OP7R.

Figure

MANGOLDT’S

7

CROSS

[price]

To Dupuit, however, price discrimination could
accomplish more than merely redistributing a fixed
sum of welfare from buyers to sellers. It could increase total welfare if it led to increased output. Let
a monopolist initially charging price OM on output
OR find it profitable to sell extra output Rr at
discriminatory price Op. Then total utility, Dupuit
claimed, would rise by RTnr at the expense of a
corresponding shrinkage in deadweight loss to nrN.
Discriminatory pricing, in other words, would yield
a net social benefit. Later, in the 1920s and 193Os,
Marshall’s students A.C. Pigou and Joan Robinson
would echo Dupuit’s declaration of the welfare
superiority of output-increasing price discrimination
over simple monopoly pricing.

[quantity]
Source:

HANS

VON MANGOLDT

Taking his cue from Karl Rau, whose work he
cited, Mangoldt began his chapter on “The Exchange
Ratio of Goods” with a scissors diagram similar to
Rau’s (Figure 7). Like Rau, he identified the marketclearing equilibrium price P and described the
adjustment process that restores it once it is disturbed. His stability analysis, like Rau’s, highlights
the price-equilibrating
role of excess demand or
supply. Let price fall below equilibrium, he ([ 18631
1962: 32) said, and the resulting excess of demand
over supply bids it back to equilibrium. Likewise,
an above-equilibrium price activates an excess of
supply over demand that puts downward pressure
on price until it returns to equilibrium.
Curve

(Figure

8)

Following his stability analysis, Mangoldt (33-S)
proceeded to examine the demand curve in great
detail. He argued that (1) the height of each point
on the curve represents the marginal utility of the
10

Mangoldt ([1863] 1962: 33).

(1824-1868)

Dupuit had shown how much one could accomplish working with the demand curve alone. It was
time, however, to reintroduce the supply curve into
the diagram and to examine the role of cost in price
determination. Hans von Mangoldt took this step in
his 1863 Ghndtis der Voikswbm-hajislehre (Outline of
Pol’itical Economy).

Demand

DIAGRAM

corresponding quantity, (2) the curve slopes downward because of diminishing utility of additional units
and the resulting reduction in prices buyers are willing to pay, and (3) a rise in price induces buyers to
cut back their purchases until the marginal utility of
the last unit bought rises to match the higher price.
The demand curve cuts the vertical axis, he said,
at a price which just exceeds the marginal utility of
the good’s first unit (point Dm on Figure 8a). Conversely, demand reaches its satiation point D on the
quantity axis when price is zero.
As for shifts in the curve, Mangoldt attributed them
to population growth, to changes in tastes and
knowledge, and to economic development and the
resulting rise in income and wealth. Unlike Dupuit,
who believed that demand curves must be of convex shape, Mangoldt held that they could be either
convex or concave (see Figure 8b) depending on the
type of good (luxuries or necessities), on the degree
of inequality of income distribution, and on the
availability of close substitutes.
Finally, he noted certain exceptions to the law of
demand. Demand curves, he argued, could possess
upward-sloping segments (see Figure 8c) if tastes for
conspicuous consumption cvanity”) or expectations
of further price hikes (“fear”) motivated consumers

ECONOMIC REVIEW, MARCH/APRIL

1992

Figure 8

MANGOLDT’S

DEMAND

CURVES

(b)

(a)
[price]

(c)

[price]

[price]

L7m

i

I,,,,,,

I,,,,,,,,,

IIll
n

[quantity]

Source:

[quantity]

[quantity]

Mangoldt (118631 1962: 34-35)

to buy larger quantities at higher prices. And rising
demand curves, he realized, could intersect supply
curves more than once, giving rise to the possibility
of multiple equilibria.
Supply Curves

(Figure

9)

Turning his attention to supply curves, Mangoldt
(3.57) made his most enduring contribution. He was
the first to draw such curves with different shapes
depending on the behavior of costs of production.
Constant unit costs yield a horizontal or perfectly
elastic curve (Figure 9a). Constant costs up to the
limit of a rigidly fixed capacity yield a reverse
L-shaped curve possessing horizontal and vertical
segments (Figure 9b). Constant costs that give way
to increasing costs and then to rigidly fared limits yield
a curve with perfectly elastic, relatively elastic, and
perfectly inelastic components (Figure SC) . Finally,
decreasing costs owing to economies of scale over
a certain range of output followed by increasing costs
due to diseconomies of scale yield a roughly U-shaped
curve that falls before it subsequently rises (Figure
9d). As for outward secular shifts in supply curves,
Mangoldt ascribed them to technological progress and
resource discovery-forces
tending to lower the cost
of producing any level of output.

Comparative

Statics Exercises

(Figure

10)

Today Marshall’s name is associated with the
partial equilibrium, comparative statics method. But
it was Mangoldt, not Marshall, who pioneered the
technique. Having presented curves of demand and
supply, Mangoldt (38-40) put them through a series
of exercises designed to show how shifts in the curves
affect equilibrium price and quantity. Rightward shifts
of the demand curve along a perfectly elastic supply
curve raise quantity but not price (Figure lOa). Price,
that is, is supply-determined in the constant cost case.
The same demand shifts occurring along a vertical
or perfectly inelastic segment of the supply curve raise
price but not quantity (Figure lob). Price is demanddetermined in this case. Similarly, leftward shifts in
demand produce only price falls when supply is
perfectly inelastic (vertical curve PS,,, in Figure 10~)
and only quantity reductions when supply is perfectly
elastic (curve S&J. In the typical case of relatively
elastic supply, however, demand-curve shifts change
both price and quantity (Figure 1Od). Finally,
simultaneous rightward shifts in both demand and
supply curves can cause equilibrium price to rise, fall,
or remain unchanged depending upon which shift,
if either, predominates (Figure 1Oe).

FEDERAL RESERVEBANK OF RICHMOND

11

Figure 9

MANCOLDT’S

[price]

SUPPLY

CURVES

[price]

(4

(b)

[quantity]

[price]

f

Sm
Si

Source:

Mangoldt ([18631 1962:36-37)

These exercises alone are sufficient to ensure
Mangoldt’s place in the history of economic thought.
But he went beyond them to describe a three-step
adjustment process by which price and quantity move
from one equilibrium position to another. Years
before Marshall he (51) posited a Marshallian
mechanism. First comes an outward shift in either
the demand curve or supply curve. This produces
a positive gap between demand price and supply price
at the existing level of output. The resulting rise in
profits induces producers to expand output until the
price differential is eliminated and the new equilibrium is attained. All this, Mangoldt noted, refers
to unanticipated shifts in the curves. Should the shifts
be anticipated, price immediately jumps to its new
12

equilibrium, thus avoiding the sequential adjustment
process.
Multiple

Equilibria

Anticipating Marshall, Mangoldt (SO) discussed the
possibility of multiple equilibria of demand and
supply. He noted that such phenomena cannot
occur when the two curves slope in opposite directions and so intersect no more than once. But they
can occur when both curves slope in the same direction. Here Mangoldt cited demand curves that rise
with price because of desires for conspicuous consumption or expectations of even higher future prices.
Likewise he cited supply curves that fall because of

ECONOMIC REVIEW, MARCH/APRIL

1992

Figure 10

COMPARATIVE

[price]
T

STATICS

EXERCISES

b-4

[quantity]

[price]

[quantity]

[price]

(cl

(d)

.Sm

Sm

n
[quantity]

[price]
f-+Dm,------

[quantity]

(e)

--__

Sm

I

J
I

[quantity]
Source:

Mangoldt ([l&53]

1962: 37-38, 40).

FEDERAL RESERVE BANK OF RICHMOND

13

increasing returns to scale. In such cases, multiple
intersections are possible and there may be several
equilibrium prices. Having said this, however,
Mangoldt said nothing about which equilibria are
stable and which unstable. He failed to apply the
analysis he had used before in the case of a single
unique equilibrium.
With respect to multiple
equilibria, he did not recognize the stability problems
involved.
Pricing of Joint Products

(Figure

11)

Nevertheless, Mangoldt’s analysis of intersecting
supply and demand curves must be judged an
outstanding performance that in many respects
exceeded those of his predecessors.
Equally impressive was his application of the diagram to the
problem of price determination when goods are
jointly demanded or supplied in fixed proportions.
True, John Stuart Mill had briefly discussed this
problem in his 1848 Pritzc$des of PohicaL Economy.
But Mangoldt’s geometric and algebraic analysis
eclipsed Mill’s purely verbal treatment and was not
superseded until Marshall’s Principl’es. A “great
achievement” and “Mangoldt’s most significant contribution to price theory,” Eric Schneider (1960: 380,
384) called it. A “brilliant theoretical contribution”
concurred Jiirg Niehans (1990: 128). What follows
sketches the geometric part of Mangoldt’s contribution. Readers can find treatments of the algebraic
portion in the appendix to this article as well as in
Schneider (1960) and Creedy (1992: 38-46).
Mangoldt first examined the “joint demand” case
of a pair of goods A and B purchased in fixed proportions under a given spending constraint.r5 He
(41-4) showed how demand and supply determine
the equilibrium price and quantity of both goods
consistent with the constraint. He also showed that
a fall in A’s supply price-i.e.,
a rightward shift in
its supply curve (curvefl; in Figure 1 la)-raises
B’s
price and explained why. The cheapening of A
induces buyers to take more of it and, because of
fixed proportions, to demand more B too. The
resulting upward shift in the demand curve for B
(curve ggs) raises its price. In this way, a cost reduction that increases the supply of A raises the price
of B.
Turning to the “joint supply” case of a pair of goods
such as beef and cowhides produced in fixed proportions subject to a given cost constraint, Mangoldt
showed how equilibrium is established in both
I5Examples include (1) scotch and soda, and (2) copper and
zinc used in making brass.
14

markets. He (46-8) also showed that a rise in the
demand for beef must lower the price of hides and
gave the rationale. With fixed proportions, the increased demand for beef leads to a rise in its output
as well as that of hides and so, given the demand
for hides, to a fall in their price. In this way an
upward shift in the demand curve for beef (curve ee,
in Figure 1 lb) produces a downward shift in the
supply curve of hides (curve dd,) that lowers their
price.
Having examined joint demand and supply,
Mangoldt for completeness considered composite
demand and supply. Composite demand refers to the
case where two competing uses (e.g., furniture and
firewood) vie for one fixed input (timber). Here
Mangoldt (48-50) showed that a fall in the demand
for furniture would, by making more timber available
for firewood, increase the latter’s supply and lower
its price. Composite supply refers to the case where
two substitute goods (e.g., flax and cotton) satisfy
a single need (for cloth). Here he (44-6) showed that
a rise in supply and hence fall in the price of flax
lowers the demand for and so the price of cotton.
These topics were not further developed until Marshall took them up in his Principles.
Mangoldt’s

Influence

Mangoldt’s diagrammatic analysis should have
become common property to all economists by the
1870s. That it did not is attributable to one Friedrich
Kleinwachter who, upon publishing a reprint of
Mangoldt’s book in 187 1 shortly after his death,
deleted the diagrams on the grounds that “it is
utterly inconceivable to me that graphs or mathematical formulae could facilitate the understanding
of economic laws” (Creedy 1992: 46; see also
Schneider 1960: 392). Mangoldt’s contribution fell
into oblivion for twenty-three years until Francis
Edgeworth ([ 18941 1925: 53) rediscovered it in 1894
and proclaimed its author “one of the independent
discoverers of the mathematical theory of Demand
and Supply.” Edgeworth might well have said the
same thing about Henry Charles Fleeming Jenkin,
the distinguished electrical engineer and inventor,
who, with no formal training in economics and no
acquaintance with the work of Mangoldt or his
predecessors,
introduced
demand and supply
curves-indeed
the technique
of diagrammatic
analysis-into
the English economic literature circa
1870.16
I6 On these points see Brownlie and Lloyd Prichard (1963: ‘211,
2 16) who note that Jenkin had read little economics other than
J. S. Mill’s Principftx

ECONOMIC REVIEW. MARCH/APRIL

1992

Figure 11

PRICING

OF JOINT

PRODUCTS

(a)

[price]

__ ______

/

/

+-Ii

[quantity]

(b)

\

J’

.,es

I

I
1

t-

C

[quantity]
Source:

Mangoldt ((18631 1962: 43, 47).

FEDERAL RESERVE BANK OF RICHMOND

15

FLEEMINGJENKIN (1833-1885)
Jenkin presented his analysis in three papers: his
1868 Nmh British Rfx.zkv article “Trade-Unions: How
Far Legitimate ?,” his 1870 Recess Stzdies piece “The
Graphic Representation of the Laws of Supply and
Demand, and Their Application to Labour,” and his
1872 contribution to the Pmeed’ngs of th Royal Society
of Edinbz@
1871-Z “On the Principles Which
Regulate the Incidence of Taxes.” In his 1870 paper
he ([ 193 11: 77) drew intersecting curves representing equations which he (I 193 11: 17-8) had stated in
his 1868 piece, namely b-=
f(A+ I/x) and S = F(B+x)
where D and S denote
quantities demanded and
supplied, respectively,
x
denotes price, and A and B
denote shift parameters that
determine the location or
height of the curves on the
[price]
(a)
diagram.

equals the stock on hand times the fraction offered
for sale. This fraction varies directly with the differential between actual market price and traders’ reservation prices, i.e., prices below which stocks are held
for future sale and above which they are marketed
immediately. Different traders possess different reservation prices stemming from their expectations of
future prices (99, 109). Those expecting low future
prices have low reservation prices. Those expecting
high future prices have high reservation prices. A
rising market price surpasses a growing number of
reservation prices, thus enlarging the fraction of the

Figure 12

JENKIN
PRICE

ON MARKET-PERIOD
DETERMINATION

[price]

(b)

SM 4s

He (17-8) noted that the
curves’ intersection point
depicts
the equilibrium
price and quantity that solve
the market-clearing equation D =S or f(A+ I/x) =
F(B +x). To ensure stability of equilibrium he relied
on excess supply or demand
triggered by price deviations
from equilibrium. These excess supplies or demands,
he said, act immediately to
restore price to its marketclearing level.
Market-Period
Price
Determination
(Figures 12 and 13)

Anticipating
Marshall’s
assumption
of separate
operational time periods,
Jenkin (78, 89) conceived
two hypothetical intervalsmarket period and long
run-to which his analysis
applied.
In the market
period the stock of goods is
fured and cost of production
plays no role in price determination. Quantity supplied
16

90
80
70

60

50

40

SO

20

IO

0

Source:

Quarters

Quarters

[quantity]

[quantity]

Jenkin ([1870] 1931: 77, 79).

ECONOMIC REVIEW, MARCH/APRIL 1992

stock marketed. Accordingly,
quantity supplied rises with
price and the supply curve
slopes upward until it turns
vertical when the entire stock
or “whole supply” (Figure 12a)
is marketed.17
As for market-period
demand curves, Jenkin drew
them with a negative slope
indicating that lower prices are
required to compensate for
diminishing marginal utility of
additional units bought. Intersection of demand and supply
curves yields an equilibrium
(Figure 12a) characterized by
zero excess supply or demand
as the market clears (Figure
12b). This equilibrium, however, is extremely volatile. It
changes with every event, real
or imagined, that shifts the
curves. Demand curves shift
with variations
in buyers’
whims, desires, and expectations (Figure 13a) as well as
with changes
in incomes
(Figure 13b). Supply curves
shift with variations in the size
of stocks on hand (Figure 13~)
and with changes in traders’
expectations
and thus the
reservation prices they set
(Figure 13d).

Figure

SHIFTS

IN MARKET-PERIOD

[price]
Shillings

13

CURVES

[price]
(a)

Shillings

(b)

I
Qualtsrs

[quantity]

Quarters

[quantity]

[price]

[price]

Shlllin~s

h-0

Long-Run Price
Determination
(Figure 14)

Turning to the long period
when output can vary, Jenkin
(89-93) showed that the latter
adjusts to equilibrate demand
and supply prices. Long-period
supply price consists of average
cost of production, which includes the sum of the costs of
factor inputs per unit of output
I7 Jenkin’s diagrams which, like Cournot’s and Dupuit’s, measured price
horizontally and quantity vertically
are shown here with their axes
transposed.

Quarters

[quantity]

[quantity]

Source: Jenkin ([1870] 1931: 80-82)

FEDERAL RESERVE BANK OF RICHMOND

17

Figure

LONG-PERIOD

PRICE

14

DETERMINATION

[price]

production approximately determines price while demand approximately determines output. Here is
Jenkin’s version of the Marshallian cases of long-run
increasing and constant cost industries, respectively.
Application

Shillings

to Labor

Unions

(Figure

15)

As is evident from the titles of his papers, Jenkin
developed his supply and demand diagrams for two
main purposes: to examine the impact of trade unions
on wage determination and to elucidate the welfare
effects of taxes. With respect to trade unions, he
(94-106) argued that they could, by accumulating a
strike fund to support workers during walkouts, raise
equilibrium wage rates in both the market period and
the long run.
In the market period the labor force is given and,
if non-unionized, will work for whatever wage it can
get. Labor’s supply curve is a vertical line whose
intersection with employers’ demand-for-labor curve
determines the equilibrium wage (Figure 15a). Enter
the trade union. With its strike fund the union allows
its members to enjoy a reservation wage below which
they withhold labor from employers rather than
selling it for what it will fetch. The resulting labor
supply curve, instead of being a vertical straight line,
becomes a right-angled or reverse Lshaped curve
at the reservation wage set by the union (Figure 1Sa).
The labor demand curve will cut the supply curve
in its horizontal segment such that equilibrium wages
will be higher and equilibrium employment lower
than in the non-union case. Unions raise wages at
the expense of employment.

Millions of Bales

Source:

[quantity]

Jenkin ([11370] 1931: 90).

plus normal profits of producers. Unit or average cost
generally rises with output. The reason: productive
factors are in limited supply and must be paid higher
prices to bid them away in greater quantities from
alternative uses. Consequently, if a good requires
specialized inputs that are extremely scarce and thus
increasingly costly to obtain, unit costs rise rapidly
with output and the supply curve is steeply sloped
(curve 2 of Figure 14). Conversely, if inputs are so
plentiful that their prices are virtually invariant to
increases in the demand for them, unit costs will be
nearly constant and the supply curve relatively flat
(curve 1 of Figure 14). In this latter case, cost of
18

So much for the market period. In the long run
the workforce is variable and labor’s supply curve
horizontal. Labor is produced at constant cost consisting of the expense of rearing and maintaining
workers at some expected standard of comfort. Trade
unions, by setting a reservation wage and so raising
the standard of comfort, act to raise labor’s cost of
production. The resulting upward shift in the labor
supply curve causes it to cut the demand curve at
a higher equilibrium wage and a lower equilibrium
labor force (Figure 15b). Unions, by raising the
standard of comfort, influence the equilibrium size
of the population. Here is Jenkin’s most original
contribution: his extension of the scissors diagram
to the analysis of the labor market.
Welfare

Effects of Taxes

(Figure

16)

Jenkin also employed his diagrams to examine the
welfare effects of excise taxes, exhibiting originality
in conception if not establishing temporal priority in

ECONOMIC REVIEW, MARCH/APRIL 1992

Figure 15

JENKIN

ON WAGE

DETERMINATION

(a)
supply
+dow-

I

QuadiCy oj labour
(b)

Wa 8
rciL

Quantity 9 labour
Source:

Jenkin (I18701 1931: 83)

publication in doing so. First, without having seen
Cournot’s tax incidence analysis, he showed that who
bears the tax depends on the slopes of the demand
and supply curves. According to Jenkin (114), the
steeper the demand curve or the flatter the supply
curve the greater the share of the tax shifted to
demanders. Conversely, the steeper the supply curve
and the flatter the demand curve the greater the share
borne by suppliers. In the limiting case of a perfectly vertical demand curve or perfectly horizontal

supply curve, all of the tax is shifted to
demanders. But when supply is perfectly inelastic or demand perfectly elastic, the entire
burden falls on suppliers.
Second, Jenkin, in his tax analysis, derived
the Marshallian concepts of consumers’ and producers’ surplus. Being unaware of Dupuit’s invention of the former idea, he (110) thought
both were novel. Only the latter concept,
however, was new with him. Like Marshall, he
(109) defined it as the excess of sellers’ actual
receipts from supplying a good over the
minimum necessary to induce them to do so.
And, like Marshall, he measured it by the
roughly triangular area lying between the price
line and the supply curve (area 2 of Figure 16a).
He (110) also pointed out that consumers’ and
producers’ surplus triangles 2 and 2 are larger
the more steeply sloped the demand and supply
curves, respectively.
Finally, Jenkin (113-4) used these triangles
to show that (1) the welfare losses of consumers
and producers always exceed the tax they pay
and (2) these losses rise with the rate of the
tax. Thus, starting from pre-tax equilibrium D
in Figure 16b, a tax of C’C or MM’ per unit of
output drives a wedge between sellers’ supply
price OM and buyers’ demand price OM’thus
reducing output by C’D. The government
obtains tax revenue equal to the sum of rectangular areas 2 and 2, of which buyers pay area
2 and sellers area 2. But buyers lose consumers’
surplus equal to the sum of areas 2 and 3, this
loss exceeds their tax payment by the amount
of area 3. Similarly, sellers lose producers’
surplus equal to the sum of areas 2 and 4, which
exceeds their tax liability by the amount of area
4. These excess-burden triangles form constituent parts of the deadweight-loss triangle
3 + 4 whose size increases with the tax wedge.
At very high tax rates the deadweight-loss
triangle dwarfs the tax-yield rectangle and is a
powerful argument for keeping rates low.
Influence and Recognition

Jenkin’s work, especially his 1870 paper, fully
foreshadowed Marshall’s. Nobody saw this more
clearly than Marshall himself. Marshall had lectured
on demand and supply curves since 1868, but, at
the time of Jenkin’s writings, had published nothing.
He realized that Jenkin had “scooped” him, as the
following passage (Whitaker 1975: 45) from H. S.

FEDERAL RESERVE BANK OF RICHMOND

19

Figure 16

CONSUMERS’
AND PRODUCERS’
SURPLUS
THE WELFARE
EFFECTS OF A TAX

AND

[price]

[price]

(b)

(4

A

A4

P

F

v

0

Y

[quantity]
Source:

Jenkin ([18721 1931: 108, 113)

Foxwell’s letter of 24 April 1925 to J. M. Keynes
confirms: “I happened to come across Uenkin’s 1870
article] in the Easter vacation of 1870, when I was
attending
Marshall’s lectures on diagrammatic
economics, & I shall never forget his chagrin as he
glanced through the article when I showed it to him.
There was nothing in Cournot which so closely
agreed with Marshall’s general approach to the
Theory of Value & particularly to his statement of
the equation of supply & demand.”
Nevertheless,
Marshall continued to insist that
he, partly under the tutelage of Cournot, had invented
the scissors diagram and pioneered its applications
independently of Jenkin. William Stanley Jevons
20

[quantity]

likewise dismissed Jenkin’s contribution with the
claim that he Uevons) had used intersecting curves
to depict price determination in lectures at Owens
College as early as 1863, seven years before Jenkin
published his diagrams. Such disparaging comments,
together with Jenkin’s lack of formal training in
economics, caused his innovations to go unnoticed
(Brownlie and Lloyd Prichard, 1963: 21.5-6). Even
today his name is unfamiliar to most economists who
instinctively think of Marshall when supply-anddemand analysis is mentioned.18
18Thus Blaug and Sturges (1983: 186) remark that Jenkin’s work
“was little noticed . . . and had little effect on the subsequent
course of economic thought despite its striking quality and
originality.”

ECONOMIC REVIEW. MARCH/APRIL

1992

CONCLUSION

extended it to the labor market to explain
wage determination. Cournot formulated the
elasticity concept and Dupuit analyzed price
discrimination. There is little in Marshall’s
use of the diagram that was not anticipated
by his predecessors.

Economists typically consider Alfred Marshall the
father of the Marshallian cross or scissors diagram.
The pre-Marshall literature, however, reveals a
somewhat different picture. In particular:
1. The Marshallian cross diagram did not originate with Alfred Marshall. At least five economists-Cournot,
Rau, Dupuit, Mangoldt, and
Jenkin-employed
it in print before Marshall
published it. And the first four did so before
Marshall began his career as an economist.
2. Of the diagram’s five originators, all but
Mangoldt, who knew of Rau’s contribution,
were ignorant of the work of the others. The
diagram was a multiple independent discovery.
3. Besides conceiving the diagram itself, its
originators supplied all its components and
pioneered most of its applications. Cournot
contributed the original curves. Rau and Mangoldt provided the stability analysis. Mangoldt
furnished the comparative statics exercises
which Jenkin applied to the analysis of price
determination in the market period and long
run, respectively. The tax incidence analysis
stems from Cournot, Dupuit, and Jenkin. The
consumers’ surplus and deadweight-loss triangles are Dupuit’s and Jenkin’s. Jenkin devised the idea of producers’ surplus. Mangoldt
applied the diagram to the problem of the
pricing of joint and composite goods. Jenkin

4. The diagram thus illustrates Stigler’s (1980)
Law of Eponymy according to which no
scientific discovery is named for its original
discoverer. The Marshallian cross diagram
bears Marshall’s name because he gave it its
most complete, systematic, and persuasive
statement, not because he was the first to
invent it. His account was definitive, not
pathbreaking.
For this he received-and
deserved-credit.
5. Later economists could have obtained their
demand-and-supply
analysis from Marshall’s
predecessors as well as from his Prim-tjdes.
Had they done so, today’s microeconomics
textbooks probably would be little changed.
Given the existence of a well-formulated
geometry
of supply and demand before
Marshall, it follows that his contribution was
a sufficient but hardly a necessary condition
for the diagram’s subsequent dissemination.
Had he never published, later economists
probably would have discovered the work of
his predecessors
or invented the diagram
themselves.

REFERENCES
Blaug, M. and Sturges, P. 1983. %40’s Who in Economics.
Cambridge: MIT Press.
Brownlie, A.D. and Lloyd Prichard, M.F. 1963. Professor
Fleeming Jenkin, 1833-1885, pioneer in engineering and
political economy. OxfordEconomicPapers, NS 15: 204-16.
Cournot, A.A. 1838. ResearchesInto the MathematikalPrinc@es
of the Theoryof Weak/r. Trans. by N.T. Bacon, New York:
Macmillan, 1929; reprinted, New York: Augustus Kelley,
1971.

Ekelund, R.B., Jr. 1970. Price discrimination and product
differentiation in economic theory: an early analysis.
QuarterZyJournal of Economics84, May, 268-78.
Ekelund, R.B., Jr., Furubotn, E.G., and Gramm, W.P. 1972.
The evolution and state of contemporary demand theory.
In Evolution of Modenr Demand Theory, ed. R.B. Ekelund,
Jr., E.G. Furubotn, and W.P. Gramm, Lexington: D.C.
Heath.

Creedy, J. 1992. Demand and Exchange in Economic Analysti.
Aldershot: Edward Elgar.

Ekelund, R.B., Jr. and Htbert, R.F. 1983. A Historyof Economic Theoryand Method. 2nd edn., New York: McGrawHill.

Dupuit, J. 1844. On the measurement of the utility of public
works. Trans. by R.H. Barback in Readings in Welfare
Economics,ed. K.J. Arrow and T. Scitovsky. Homewood,
IL: Richard D. Irwin, 1969, 255-83.

Ekelund, R.B., Jr. and Thornton,
M. 1991. Geometric
analogies and market demand estimation: Dupuit and the
French connection.
History of Political Economy 23:
397-418.

Edgeworth, F.Y. 1894. The pure theory of international values.
Economic Journal 4. Reprinted in his Papers Relating to
PoliticalEconomy, Vol. 2, London: Macmillan, 1925.

Htbert, R.F. and Ekelund, R.B., Jr. 1984. Welfare economics.
In EconomicAnalysisin Historicaol
Penpective, ed. J. Creedy
and D.P. O’Brien, London: Butterworths.

FEDERAL RESERVEBANK OF RICHMOND

21

Hennings, K. 1979. Karl Heinrich Rau and the graphic representation of supply and demand. Diskussionspapier Serie
C, Nr. 35, Fachbereich Wirtschaftswissenschaften,
Universitat Hannover.
Hennings, K. 1980. The transition from classical to neoclassical
economic theory: Hans von Mangoldt. Kyklos33: 658-81.
Houghton, R.W. 1958. A note on the early history of consumer’s surplus. Economic0 2.5: 49-57.
Jenkin, F. 1868. Trade-unions:
how far legitimate? Norr/r
Britirh Rex&o, March. In Papen, Literaty, Scient& 6c,
ed. SC. Colvin and J.A. Ewing, London: Longmans,
Green & Co., vol 2, 1877; reprinted, London: The
London School of Economics, 1931.
Jenkin, F. 1870. The graphic representation of the laws of
supply and demand, and their application to labour. In
Recess Studies, ed. A. Grant, Edinburgh. Reprinted in
Papers, Literary, Scientific, 6c, ed. SC. Colvin and J.A.
Ewing, London: Longmans, Green & Co., vol 2, 1877;
reprinted, London: The London School of Economics,
1931.
Jenkin, F. 1872. On the principles which regulate the incidence
of taxes. Prvceedingsof the Royal SocietyofEdinburgh 18722.
In Papetx, Litmary, scientific, &c, ed. S.C. Colvin and J.A.
Ewing, London: Longmans, Green & Co., vol 2, 1877;
reprinted, London: The London School of Economics,
1931.
Keynes, J.M. 1925. Alfred Marshall, 1842-1924. In Memotials
of A&FedMa&a& ed. A.C. Pigou, London: Macmillan;
reprinted, New York: Kelley and Millman, 19.56.
Mangoldt, H. von. 1863. The exchange ratio of goods.
Trans. by E. Henderson from GNndriss der Voh%wirtschufifrs/e/lre,Stuttgart: Engelhorn. In IntenaatibnalEconomic
Papers, No. 11, 32-59, London: Macmillan, 1962.
Marshall, A. 1879. The Purz Thory of Domestic Values. Privately
printed; reprinted, London: London School of Economics,
Scarce Works in Political Economy No. 1, 1930; and in
amplified form in Whitaker, 1975.

Marshall, A. 1890. Ptinca>ks of Economics,
Vol. I. London:
Macmillan.
Niehans, J. 1990. A Historyof Economic Theory. Baltimore: The
Johns Hopkins University Press.
Parkin, M. 1990. Economics. Reading, Mass.: Addison-Wesley.
Rau, K.H. 1841a. (Communications:)
Economic politique.
Bulletinde l’Acadhie des scienceset Be//es-L.&% de Bruxebs
viii, pt. I: 148-52.
Rau, K.H. 1841b. Gtundsiitzeder Volkswirthschaj?slehm.
4th ed,
Heidelberg.
Ricardo, D. 18 17-Z 1. On t/e Principlesof Politico/Economyand
Taxation. Ed. P. Sraffa, Cambridge: Cambridge University
Press, 195 1.
Schneider, E. 1960. Hans von Mangoldt on price theory: a
contribution to the history of mathematical economics.
Economehico 28, 380-92.
Schumpeter, J.A. 1954. HistoryofEconomic Analysti. London:
George Allen & Unwin.
Smith, A. 1776. Th wea/h of Nations. Ed. E. Cannan, New
York: Modern Library, 1937.
Stigler, S.M. 1980. Stigleis law of eponymy. Transactionsof
tke NenmYo& Academy of Sciences
Series II, 39: 147-58.
Theocharis, R.D. 1983. Eurly Developmentsin Mathematiol
Economics. 2nd edn. Philadelphia: Porcupine Press.
Thweatt, W.O. 1983. Origins of the terminology, supply and
demand. &ottbh Jouma/ ofPoliticalEconomy30, November,
287-94.
Whitaker, J.K. (ed.) 1975. The E&y Economic Whitings of
Alfi-edMa&all, 2867-1890. Vol. 1. London: Macmillan.
Wright, H. 1927. Frederick

Lavington.

Economic JoumaL 37,

503-S.

APPENDIX
Mangoldt’s

Algebraic

Model

of Joint Demand

As stated in the article, Mangoldt ([ 18631 1962:
43-50) analyzed price determination of interrelated
goods algebraically as well as graphically. His first
algebraic model refers to two goods A and B jointly
demanded in fixed proportions under a given expenditure constraint.
Joint Demand

Model

Letting D denote quantity demanded, S quantity
supplied, p price, E total expenditure, f a functional
relationship between two variables, 1zthe fixed ratio
in which the two goods (identified by subscript) are
jointly demanded, Mangoldt’s first model consists of
the following equations:
22

and Supply

(1)
(2)
(3)

DA = nDj3
E = PADA + PBDB
PA
= fA(s,l

(4)

PB

(5)
(6)

= SA
De = SB.

=

fB(sB)

DA

Equation (1) states the fixed-proportion assumption and equation (2) the expenditure or budget
constraint. Equations (3) and (4) are supply functions,
while equations (5) and (6) are market-clearing
conditions.

(1)

To solve the system, Mangoldt first substituted
into (2) and solved for pB to obtain

ECONOMIC REVIEW, MARCH/APRIL

1992

(7)

PB = E/D,

- np~.

This expression he interpreted as good B’s demand
function containing good A’s price as a shift
parameter. He then eliminatedpA from the expression by using (3), (5), and (1) to obtain
(8)

PB

=

IX&

-

ny-ah&~,

which he equated with supply function (4)to determine the market-clearing price and quantity of good
B and thus equilibrium of the system as a whole.
Finally, from equation (7) he demonstrated that
a fall in good A’s supply pricepA raises the demand
price% of good B. He explained why: As good A
becomes less expensive, buyers take more of it.
Because the two goods are consumed in fiied proportions, however, buyers necessarily take more of
good B too. The resulting increased demand for B
bids up its price. In short, a movement down the
demand curve for A is accompanied by an outward
shift in the demand curve for B. In this way cost
reductions that increase the supply of A raise the price
of B.

Equation (9) states the fixed-proportions assumption that the goods are supplied in the ratio tl. Equation (10) says that revenues must cover total cost.
Equation (11) defines total cost K as the product of
the unit cost c of a complex unit of output SA +&
= (2 +n)SA times the number of units produced, or
K = c(2 +n)SA = kSA where k = ~(2 +n). Equations
(12) and (13) constitute the goods’ demand functions,
while equations (14) and (15) state the marketclearing condition that supplies equal demands.
To solve this system, Mangoldt substituted equations (9), (ll), (13), (14), and (15) into (10) to
obtain an expression for B’s price in terms of A’s:
(16)
Further
(17)

pB = (k-PA/n.
substitution

yielded the expression

SA = (Z/n)f/(k-PA/n/

which Mangoldt interpreted as A’s supply function
given equilibrium in the market for good B. Equating
(17) with (12) allowed him to solve for A’s marketclearing price and quantity and thus for equilibrium
of the system as a whole.

Joint Supply Model

Mangoldt’s second model refers to two goods A
and B jointly supplied in fixed proportions under a
given cost constraint. Letting K and R denote total
and unit costs, respectively, and using the same
variables defined above, his model appears as follows:
(9)

SB

?

nSA

(10)

PADA

(11)

K

+

(12)

DA

(13)
(14)
(15)

De = -L&t,)
SB = DB
SA = DA.

=

PBDB

=

He concluded that a rise in the demand for A
reduces B’s price and explained why. As A’s output
rises to match the increased demand so too, via the
fixed-proportions assumption, does B’s output. With
the demand for B given, however, the extra output
of that good constitutes an excess supply that puts
downward pressure on its price. The price of B varies
inversely with the demand for A.

K

k&
=

j-t&d

FEDERAL RESERVE BANK OF RICHMOND

23