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CLASSICAL AND NEOCLASSICAL ROOTS OF
THE THEORY OF OPTIMUM TARIFFS
Thomas M. Hump/my
In current debates with protectionists, pure or
unilateral free traders insist that unrestricted com-
merce is optimally advantageous not only for the
world as a whole but for any individual nation, even
if it practices it alone. From this idea stems the cor-
ollary that a country automatically benefits from the
unilateral as well as reciprocal elimination of tariffs.
If true, it follows that, far from erecting tariffs, a coun-
try should immediately dismantle them and enjoy the
benefits of international specialization and division
of labor even if other nations do not.
In 1940, however, the British economist Nicholas
Kaldor challenged these notions by asserting that a
tariff always benefits the levying country provided
that the duty is not too large, that the country has
monopoly power in world markets, and that other
countries do not retaliate with tariffs of their own.’
Kaldor was here advancing the terms-of-trade or
optimum tariff argument according to which trade
taxes improve the levying country’s welfare by
turning the commodity terms of trade (relative price
at which exports exchange for imports or the quan-
tity of imports bought by a unit of exports) in its favor,
thus giving it a better bargain in world markets. By
taxing its imports, the country reduces its demand
for those goods thus driving down their world price.
Similarly, by taxing its exports it lowers the quan-
tity of those goods supplied on the world market thus
raising their price. In other words, it acts as a
monopolist exploiting an imperfectly elastic foreign
supply of its imports or demand for its exports. In
so doing it renders its imports cheaper and its ex-
ports dearer such that it obtains a larger quantity of
imports per unit of exports given up. Of course this
terms-of-trade gain comes at the expense of a loss
in real trade volume. The optimum rate of the duty
is that which maximizes the excess of the gain from
terms-of-trade improvement over the loss from lower
trade volume and reduced international division of
labor.
1 N. Kaldor, “A Note on Tariffs and the Terms of Trade,”
htwmka, n.s. 7 (November 1940): 377.
Kaldor demonstrated these propositions with a
geometrical diagram showing the tariff-imposing
country choosing to exchange the combination of ex-
ports for imports that allows it to reach its highest
attainable trade indifference curve given the offer
curve of the foreign country (see Figure 1).2
Shortly after, in 1944, Abba Lerner in his Economics
of COHTVI described how the same propositions could
be illustrated with conventional demand and supply
curves (see Figure 2). 3 Both diagrams quickly
worked their way into international trade textbooks
2 Kaldor, p. 379.
3 A.P. Lerner, Th Economics of Contmi(New York, Macmillan,
1944), pp. 357-59.
FEDERAL RESERVE BANK OF RICHMOND 17
where they became the standard model employed
in explaining the theory of the optimum tariff. Little
was said about earlier work on the subject. From the
point of view of the textbooks, the theory to all in-
tents and purposes largely dates from Kaldor’s
demonstration.4
To set the record straight, one must take issue with
this view. For, contrary to the impression conveyed
by textbooks, optimum tariff theory hardly originated
with Kaldor’s model but rather long predated it. It
can be documented that rudimentary statements of
4 Texts employing versions of the Kaldor-Lerner model with
no mention of its nineteenth and early twentieth century
predecessors include R.E. Caves and R.W. Jones, WorM Trade
and Payments, 3rd ed. (Boston: Little, Brown, and Co., 1981),
pp. 212-13; H.R. Heller, Inzmationai Trade: Thory and
&pi&al Etxhzce (Englewood Cliffs: Prentice-Hall, 1968)
pp. 145-47; CF. Kindleberger, In~emationa~Econotni, rev. ed.
(Homewood, 111.: R.D. Irwin, 19.58), pp. 617-20; D.B. Marsh,
WorM Tra& andlnvestmenr (New York: Harcourt Brace, and Co.,
1951). pp. 316-20; and J. Vanek, Intematimal Tmdk: Thoryand
Economic PO& (Homewood, Ill.: R.D. Irwin, 1962), pp. 294-97.
the theory go back at least to the 1830s and 184Os,
that these statements were embodied in formal
economic models rather than in mere casual remarks,
and that virtually all the elements of optimal tariff
theory were in place by 1907. In short, the origins
of optimum tariff theory are to be found in an earl&
vintage of models neglected by the textbooks. A
systematic survey of these models helps clarify what
economist Murray C. Kemp calls the “confusing and
little known early history” of the terms-of-trade argu-
ment.5 It also dispels the notion that all leading
classical and neoclassical trade theorists were doc-
trinaire free traders. True, of the six discussed below,
at least four thought that free trade was the best policy
from a practical standpoint. On a purely abstract
plane, however, all saw the terms-of-trade argument
as a valid theoretical qualification to the doctrine that
free trade is the best of all possible worlds for each
country.
HISTORICAL EVOLUTION
Early optimum tariff models evolved through five
distinct stages. First came the demonstration that im-
port duties improve the terms of trade either through
gold flows and their effects on relative national price
levels or by restricting import demand. Next came
the showing that export taxes accomplish the same
result by restricting export supply and that the ex-
tent of terms-of-trade improvement depends crucially
upon the size of certain demand elasticities. There
followed a geometrical restatement of these results
using the newly developed tool of offer curve analysis.
Next appeared indifference curve and consumer
surplus models measuring the gain from terms-of-
trade improvement and specifying the tariff rate that
maximizes the gain. Finally came a mathematical
statement of the theory including a rigorous
demonstration that a tariff can improve national
welfare and a derivation of the formula for the
optimum tariff. Each stage saw at least one different
innovator-Torrens, Mill, Marshall, Sidgwick,
Edgeworth, and Bickerdike being the key names
here-advance the theory.
ROBERTTORRENS
Priority for being the first to publish a formal
optimum tariff model goes to Robert Torrens in
5 M.C. Kemp, ‘“The Gain from International Trade and Invest-
ment: A Neo-Heckscher-Ohlin Approach,” American Eh~mic
Revim 56 (September 1966): 788.
18 ECONOMIC REVIEW, JULY/AUGUST 1987
1844. Long before then, however, he had per-
ceived that tariffs can turn the terms of trade in favor
of the levying country. He stated that idea as early
as 18 2 4 in his &says on the Prvdzhon of Wealth and
subsequently elaborated it in a series of letters
published in the BoGton Chnicle in 1832-33 and
reprinted in his 1833 LRtters on Commercial Policy.
Finally, in Letter II and the Postscript to Letter IX
of his 1844 Th Budget, he presented the idea in the
form of a hypothetical two-country, two-good
model-his famous Cuba case-in which he
showed that a 100 percent tariff, via its effect on
reciprocal demands, produces an equivalent 100
percent improvement in the terms of trade.6 This
result he depicted in two versions of his model: a
monetary version involving specie flows and their
effects on local prices and incomes and a pure barter
version involving trade in commodities. In the
monetary version, terms-of-trade improvement
comes from tariff-induced gold movements that raise
the price of the protecting country’s exports relative
to the price of its imports. In the barter version, the
same improvement comes from a reduced real de-
mand for imports. Of the two, the monetary version
provoked the stronger criticism from Torrens’s free
trade contemporaries. For that reason, it is de-
scribed in some detail below.
Torrens’s Cuba Model
In the monetary version of his model, Torrens
assumed that Cuba specializes in producing sugar and
England specializes in cloth, both goods being pro-
duced under conditions of constant real costs. He
further assumed that each good bears the same duty-
exclusive price wherever sold, that the prices of
home-produced goods vary directly with the quan-
tity of money in each country, and-of crucial im-
portance to the particular quantitative results he
obtained-that each country’s demand for the other’s
export good is of unit elasticity.
Employing these assumptions, he traced a chain
of causation from tariff to reduced quantity of
imports bought to trade balance surplus to specie
inflow and thence to a rise in the price of the pro-
tecting country’s exports relative to the price of its
imports. More precisely, he supposed that, starting
from a situation of balanced free trade with England,
6 On Torrens’s Cuba model, see D.P. O’Brien’s The CZ&aZ
Economir~ (London: Oxford University Press, 1975), pp. 191-94;
L.C. Robbins’s Robert Torrnrc and the Evolution of Classical
Ecwokcs (London: Macmillan, 1958), pp. 199-203; and J.
Viner’s Studies in the Thor-y of Znrmational Trade (New York.
Harper, 1937), pp. 298-99, 322, 463.
Cuba imposes a 100 percent ad valorem duty on im-
ports of English cloth. That good being produced
at constant cost, the immediate result is to double
its price in Cuba causing the quantity demanded to
fall by half, the Cuban demand for cloth being
assumed by Torrens to be of unit elasticity. In other
words, Cubans’ total expenditure (price-times-
quantity) on taxed cloth remains unchanged; but only
half that outlay goes to English exporters, the other
half being intercepted by the Cuban government at
the customs house.
But these are only proximate or first-round effects.
Later-round effects ensue. For, given the volume of
Cuban exports, the halving of her import bill pro-
duces a favorable trade balance with England and a
compensating specie flow from that country lower-
ing general prices in England and raising them in
Cuba. Since the price of each country’s exportable
commodity moves with its general price level and
since identical exportable goods bear the same
(duty adjusted) price in all markets, the price of sugar
rises in Cuba (and England) while the price of cloth
falls in England (and Cuba).
The fall in the price of cloth together with the rise
in Cuban money incomes occasioned by the specie
flow raises the quantity of cloth demanded in Cuba.
Conversely, the rise in sugar prices combined with
the fall in English incomes reduces the quantity of
sugar demanded in England. Gold continues to flow
from England to Cuba, lowering incomes in the one
and raising them in the other and likewise lowering
cloth prices and raising sugar prices, until the resulting
stimulus to cloth sales and check to sugar sales
restores trade balance equilibrium.
In the new equilibrium, Cuba imports the original
quantity of English cloth at two-thirds the original
unit price (four-thirds including duty) but exports only
half the initial quantity of sugar at four-thirds the
initial unit price. In barter terms, Cuba purchases the
same real quantity of imports at the cost of only half
the initial quantity of exports given up, her com-
modity terms of trade having improved 100 percent.
England’s terms of trade of course deteriorate by the
same amount.
Barter Version of Torrens’s Model
Torrens derived exactly the same results in the
pure barter version of his model, which he elaborated
with great precision in his Postscript to Letter IX of
Th Budget. There he.argued (1) that the equilibrium
terms of trade must lie between the comparative cost
ratios in the two countries, (2) that the precise
location of that equilibrium depends upon each coun-
try’s reciprocal demand for the product of the other,
FEDERAL RESERVE BANK OF RICHMOND 19
(3) that the resulting equilibrium lies most in favor
of the country with the weakest reciprocal demand,
and (4) that a tariff, by reducing the levying coun-
try’s reciprocal demand, turns the terms of trade in
its favor. Although he drew no diagrams himself, the
essentials of his analysis can be depicted with the
aid of Marshallian reciprocal demand or offer curves
showing the determination of the equilibrium terms
of trade by the intersection of the two curves (see
Figure 3).
As drawn, the curves differ from offer curves found
in standard textbooks in two respects. First, they
bend toward equilibrium only at the points on the
respective internal comparative cost ratio lines at
which the countries would operate in the absence
of trade. Second, within the range at which trade
occurs they take the form of horizontal and vertical
straight lines reflecting Torrens’s assumption of unit
elastic reciprocal demands. Given these elasticities
and starting from free trade equilibrium, Cuba’s tariff
shifts her effective offer curve down to half its initial
level thus producing at the original terms of trade
an excess world demand for sugar and a corre-
sponding excess supply of cloth. To eliminate these
excess supplies and demands England’s terms of trade
deteriorate by 100 percent. In the new equilibrium
England imports half the initial quantity of sugar at
the cost of the same initial amount of cloth given up.
Here is the key idea of optimum tariff models; namely
that trade taxes influence reciprocal demands which
determine the terms of trade thus allowing govern-
ments to manipulate those terms.
Money Stock Implications
The foregoing terms-of-trade effects were im-
portant. To Torrens, however, they were over-
shadowed by the impact of Cuba’s tariff on England’s
money stock. In the monetary version of his model
he explains how the redistribution of specie occa-
sioned by the tariff produces a one-third expansion
of Cuba’s money stock and a corresponding one-third
contraction of England’s No country, he thought,
could endure a monetary contraction of such
magnitude. For the resulting collapse of product
prices would bring ruinous rises in the real burden
of debts, wages, taxes, and other fixed charges whose
nominal values are sticky and thus respond sluggishly
to deflationary pressure. Economic stagnation,
“national bankruptcy, and revolution would be the
probable results.“’
Reciprocity in Commercial Policy
Having shown how England might lose from
foreign tariffs, Torrens next used his analysis to argue
for reciprocity in tariff removal. He pointed out (1)
that a unilateral abolition of tariffs would, like their
foreign imposition, worsen the home country’s terms
of trade and reduce its money stock, (2) that equal
retaliatory duties would cancel the unfavorable terms-
of-trade and monetary effects of foreign levies, and
(3) that the simultaneous removal of duties by all
countries tends to leave money stocks and the terms
of trade unchanged (see Figure 4). On these grounds
7 R. Torrens, Letter II of Tk Budget. On Commernal and
Cohial Policy. (London: Smith, Elder and Co., No. 65,
Cornhill, 1844), p. 37.
20 ECONOMIC REVIEW, JULY/AUGUST 1987
he proposed that Britain counter foreign tariffs with
equal duties of her own, that she trade freely only
with countries admitting her goods duty free, and that
she drop her tariffs only insofar as her trading part-
ners abolish theirs.
Criticisms
Torrens’s analysis was unsympathetically re-
ceived by his contemporaries who feared it would
undermine the case for free trade. His critics
refused to accept policy conclusions drawn from a
two-by-two model regarded by them as an inaccurate
representation of a world economy characterized by
many goods and many countries. Herman Merivale
argued that competition from third countries pro-
ducing sugar for export would limit Cuba’s power to
manipulate the terms of trade.8 Also England could
s H. Merivale, L.ectures w1 Cobtzization and Colonies, II, 1842, pp.
308ff. On Merivale’s criticisms see Viner, Studies, p. 322 and
Robbins, Robert Torren, pp. 209-11.
avoid Cuba’s tariff by selling to third countries and
exporting goods other than taxed cloth, such alter-
natives being possible in a multi-good, multi-country
model. This point was made by Nassau Senior who
also noted that what Cuba gains through terms-of-
trade improvement might be outweighed by her loss
of productivity and competitiveness due to reduced
international specialization and division of labor.9
The most cogent criticism, however, came from
George Warde Norman. He noted that England’s
terms of trade would hardly deteriorate to the
extent claimed by Torrens if one dropped the
assumption of unit elastic demands. He also argued
that the logic of Torrens’s model implied that England
should levy not equal but higher tariffs than those
levied abroad to improve the terms of trade and that
such action would intensify the danger of a trade war
with all parties losing. lo These criticisms were
telling. For Torrens indeed had overlooked the
possibility of trade warfare and the likelihood that
highly elastic reciprocal demand schedules would in
the long run severely limit the effectiveness of tariffs.
JOHN STUART MILL
Although Torrens’s Cuba case was the first
optimum tariff model to appear in print, it was
hardly the first formulated. Already in 1829-30, some
fifteen years earlier, John Stuart Mill had constructed
a similar model which he subsequently presented in
the first of his Essays on Some Unsettld Questions in
PoliticalEconomy, a volume he published in 1844 in
response to Torrens’s Th Budget.
Mill’s model possessed most of the features of
the monetary version of Torrens’s Cuba model,
namely two countries, two goods, complete
specialization, constant costs, law of one price,
Hume’s price-specie-flow mechanism, and quantity
theory of money. But Mill greatly enriched the model
by permitting demand elasticities to range from zero
to infinity and by incorporating export as well as im-
port taxes into the analysis. In so doing, he ex-
panded the model’s explanatory power thus en-
abling it to cover a greater variety of cases than con-
9 N. Senior, “Free Trade and Retaliation,” E&hrg/r Review
88 (July 1843): 12-15, 29-35. On Senior’s analysis see O’Brien,
The Ckzssicaal Economists, pp. 194-95.
lo G.W. Norman, Remarks on the Incidence of Import Duties with
special Refirence to the &gland and Cuba Case contained in “T/re
Budget, “privately printed (London: T. and W. Boone, 29 New
Bond Street, 1860), pp. 8, 12-19. On Norman’s criticisms, see
O’Brien, The C/as&a/ Economists, pp. 19596.
FEDERAL RESERVE BANK OF RICHMOND 21
sidered by Torrens. In particular, he showed how
different elasticities affect the degree of terms-of-
trade improvement.
Export Taxes, Foreign Demand Elasticities,
and the Terms of Trade
Mill applied his model first to export taxes, con-
cluding that such taxes tend to improve the taxing
country’s terms of trade by an amount equal to, more
than, or less than the tax as the elasticity of the foreign
demand for exports is equal to, less than, or greater
than one.” To demonstrate, he employed an ex-
ample in which England exports cloth to and imports
linen from Germany. In his example, he as-
sumed that England levies a tax on her exports of
cloth to Germany. Cloth being produced in England
at constant real cost, its price to Germans rises
initially by the amount of the tax. Provided the
German demand for cloth is of unit elasticity such
that her import expenditure on that good remains
unchanged after the tax raises its price, there results
no disturbance to the balance of payments requiring
equilibrating specie flows and further adjustments in
the prices of the traded goods. Cloth prices paid to
England consequently remain above their pre-duty
levels by exactly the amount of the tax. And there
being no change in the price of England’s import good
(linen), her terms of trade-that is, the ratio of the
price of cloth to the price of linen-improves
exactly by the amount of the tax. In short, unit elastic
German demand ensures a terms-of-trade improve-
ment equiproportional to the tax.
On the other hand, if Germany’s demand for
English cloth is inelastic such that she spends more
on that good when the tax boosts its price, her im-
port bill will rise producing a deficit in her trade
balance. The resulting flow of specie from Germany
to England will, via the operation of the quantity
theory of money and the law of one price, raise fur-
ther the price of cloth and lower the price of linen
in both countries. England will purchase more of the
cheaper linen and sell less of the dearer cloth, these
demand readjustments acting to restore trade balance
equilibrium. In the new equilibrium, England receives
a price for her cloth raised by more than the tax. As
she will also be paying a lower price for German lien,
her terms of trade-the relative price of cloth exports
to linen imports-will have improved by more than
the tax.
11 J.S. Mii, EcMys~~sMne(Ins~~~ti~inPo/itica/Econmny
(1844), (London: London School of Economics and Political
Science, 1948), pp. 21-24.
Finally, if Germany’s demand for English cloth is
elastic such that she spends less on it when the tax
raises its price, her import bill will shrink producing
a surplus in her trade balance and a corresponding
specie flow from England. The result of this money
flow is to lower the world price of cloth and to raise
the world price of linen-these price changes con-
tinuing until cloth sales are stimulated and linen sales
checked sufficiently to restore trade balance
equilibrium. With England’s export prices somewhat
lower than they were immediately after the impo-
sition of the tax and her import prices somewhat
higher, her terms of trade have improved but by less
than the amount of the tax.
Mill’s Model in Barter Terms
The foregoing conclusions can be presented in
barter terms, although why Mill himself did not do
so is something of a mystery since he applied barr.er
analysis involving his notion of reciprocal demand
schedules to other problems of trade theory. In any
case, Figure 5 shows England’s terms of trade im-
proving in greater, equal, or lesser proportion to the
export tax as the German offer curve is backward
bending (i.e., inelastic), vertical (of unit elasticicy),
or upward sloping (elastic), respectively-just as Mill’s
monetary model predicts.
22 ECONOMIC RNIEW, JULY/AUGUST 1987
An Exception
Mill admitted but one exception to the rule that
export taxes improve the taxing country’s terms of
trade: the case of an elastic German demand for cloth
combined with an inelastic English demand for linen.
Here the specie flow from England caused by the
tax-induced decline in Germany’s spending on cloth
is not self-correcting but rather is self-reinforcing. For
the faster gold flows abroad to raise the price of
German linen, the more England spends on that com-
modity. And the more she spends, the greater her
loss of gold and the greater the resulting fall in the
price of her cloth. To restore equilibrium, cloth may
have to fall so low in price relative to linen that the
terms of trade turn against England by more than the
amount of the tax. Such would be the case, Mill
thought, should Germany’s expenditure on cloth be
so insensitive to changes in income that prices alone
had to bear the full burden of adjustment.
Import Tariffs and the Terms of Trade
Having examined the terms-of-trade effects of
England’s export taxes, Mill next turned his atten-
tion to her import tariffs.12 He concluded that they
invariably improve her terms of trade except in the
singular case of a totally inelastic English demand for
German linen. But as long as England’s demand is
of greater than zero elasticity, quantity of imports
demanded falls as the tariff raises price. Since Ger-
man exporters producing under conditions of con-
stant cost receive a sum equal to the lower (post-
tariff) quantity times the old (pre-tariff) price, it
follows that England’s import bill falls. The resulting
gold flow from Germany to England lowers linen’s
supply price and raises the price of cloth, thus
improving England’s terms of trade. No such
improvement would occur, however, if England’s
import demand were perfectly inelastic such that the
quantity of linen demanded by that country re-
mained unchanged when the tariff raised its price.
With no shrinkage in quantity demanded, the price-
times-quantity sum paid to German exporters would
be the same as before, which means that there would
be no disturbance to the balance of payments re-
quiring gold flows and hence no changes in the
absolute and relative prices of cloth and linen. In
other words, England’s import bill, and hence her
terms of trade, would remain unchanged in this case.
12 Mill, pp. 26-27.
Views on Tariff Policy
To summarize, Mill, like Torrens, had clearly
established the theoretical possibility of a country
improving its terms of trade and its welfare through
trade restriction. Interestingly enough, however, Mill
opposed the application of his optimum tariff theory
to commercial policy on practical and moral grounds.
Tariffs, he said, invite retaliatory duties that not
only nullify the initial terms-of-trade improvement,
but also bring costly reductions in the volume of
world trade.13 Even in the absence of retaliation,
tariffs are unjust because one country’s gain is
another’s loss. Moreover, as the rest of the world’s
loss exceeds the dutying country’s gain the tariff is
inimical to global welfare and cannot be justified from
a cosmopolitan point of view. In his words, “if inter-
national morality . . . were rightly understood and
acted upon, such taxes, as being contrary to the
universal weal, would not exist.“14 He did, however,
agree with Torrens that reciprocity was a prime con-
sideration in the decision to remove tariffs. “A coun-
try,” he said, “cannot be expected to renounce the
power of taxing foreigners, unless foreigners will in
return practice towards itself the same forbearance.
The only mode in which a country can save itself
from being a loser by the revenue duties imposed
by other countries on its commodities, is to impose
corresponding revenue duties on theirs.“‘5
ALFRED MARSHALL
AND
HENRY SIDGWICK
In the 1870s and 1880s Alfred Marshall and Henry
Sidgwick constructed optimum tariff models. Mar-
shall’s innovation was to transform Mill’s model into
geometry, expressing his results in terms of reciprocal
demand or offer curves showing each nation’s desired
quantity of exports and imports as a function of the
terms of trade. Sidgwick too expressed some of Mill’s
conclusions in purely barter terms, but without
adding much to his analysis.
Marshall, in an unpublished manuscript which Pro-
fessor John Whitaker dates at 1872-74, employed his
reciprocal demand curves to show that when both
curves are elastic (provided the foreign curve is not
infinitely so) a tax on imports or exports always im-
I3 Mill, pp. 28-29.
I4 Mill, p. 25.
‘5 Mill, p. 29.
FEDERAL RESERVE BANK OF RICHMOND 23
proves the terms of trade of the levying country.16
He also showed that when the foreign curve is
inelastic-meaning that the foreign country bffers a
greater total quantity of its exports as its terms of
trade deteriorate-then the dutying country enjoys
a two-fold gain. l7 Not only do its terms of trade
improve, but, by obtaining a larger total quantity of
imports and sacrificing a smaller total quantity of its
exports, it has more of both goods to consume at
home (see Figure 6). A country lucky enough to face
an inelastic foreign offer curve, said Marshall, has
nothing to lose and everything to gain by exploiting it.
In general, however, Marshall thought that the
ability of the taxed country in a multi-country, multi-
commodity world to switch its production to non-
taxed exports and to trade its goods in nontaxed
markets rendered its offer curve so highly elastic as
to leave the dutying country little scope for tariff-
induced improvements in the terms of trade. He also
feared that the pressure of special interests would
push tariff rates far above the optimum level such
‘6 J.K. Whitaker (ed.), Th Eony Ecommk Witings of&edMar-
shall, 1867-1890, Vol. 1 (New York: Free Press, 1973, p, 270.
I7 Whitaker, pp. 275-76.
that the dutying country as well as the whole world
would lose.
Sidgwick’s analysis closely followed that of Mar-
shall, from whose abandoned 1873-77 manuscript
on trade theory Sidgwick had printed for private cir-
culation selected chapters under the title Th Put-e
Thoty of Foreign Truth (1879). In particular, Sidgwick
stressed three points previously made by Marshall.
First is the importance of monopsony power in
achieving terms-of-trade improvement. No country,
he said, could expect to improve its terms of trade
by means of tariff unless it “supplied a considerable
part of the whole demand for the [taxed] foreign prod-
ucts.“18 Second, a tariff affects the terms of trade
through its impact on reciprocal demands. Specifi-
cally, A’s tariff reduces her demand for B’s good, thus
producing an excess world supply of that good. This
excess supply is only eliminated by a deterioration
in B’s terms of trade.
Supposing trade to be in equilibrium at the time that the
demand in A for B’s commodities is artificially restricted
by import duties raising their price, and supposing that
other things-including the demand in B for A’s commodi-
ties-remain unchanged, one obvious result will be that B
will import more than she exports; hence in order to
restore the balance of trade, a certain readjustment of
prices will be necessary by which B will in most cases
tend to obtain a somewhat smaller aggregate of imports
on somewhat less advantageous terms.19
Third, the effectiveness of A’s tariff depends upon
the elasticity of B’s offer curve. If that curve is almost
totally inelastic, as when B urgently requires A’s good
at any price, the terms-of-trade gain realized by A
comes at the cost of little or no shrinkage in her ex-
port volume. But if B’s offer curve is perfectly elastic,
as when she can readily substitute third-country goods
for A’s good in her consumption mix, A’s tariff will
have no effect other than diminishing her (A’s) real
trade volume. Said Sidgwick:
This restriction on B’s import trade may possibly not
reduce materially the amount of her imports from A, if
the commodities supplied by A are strongly demanded in
B . . . . On the other hand . . . if the products of A are
closely pressed in the markets of B by the competition of
other countries, the protection given by A to . . . her
industry may very likely have the secondary effect of
inflicting a blow upon . . . the exports from A to B.zo
Here is Sidgwick’s recognition of one point stressed
by optimum tariff theory, namely that a tariff is
powerless to improve the terms of trade when the
foreign offer curve is perfectly elastic.
1s H. Sidgwick, T/l PrinGipl of Poktical Economy, 2nd edition
(London: Macmillan and Company, Ltd., 1887), p. 492.
19 Sidgwick, pp. 494-95.
z” Sidgwick, p. 495.
24 ECONOMIC REVIEW, JULY/AUGUST 1987
FRANCIS Y. EDGEWORTH
Although Torrens, Mill, Marshall, and Sidgwick
had shown that tariffs could benefit the dutying
country by turning the terms of trade in its favor they
did not provide a measure of this benefit nor did they
specify the precise tariff rate that would maximize
it. Not until 1894 did these ideas make their first
appearance with the publication of F. Y. Edgeworth’s
famous Economic Jounu1 article on “The Pure Theory
of International Values.” There in a demonstration
that anticipated Kaldor’s in all essential respects, he
employed the now-standard curves of trade geometry
to identify the optimum tariff (see Figure 7). In so
doing he advanced the theory in at least four ways.
First, he superimposed on Marshall’s reciprocal de-
mand or offer curves trade indifference curves essen-
tial to the demonstration of welfare gains from trade
restriction. His diagram shows the home country’s
trade indifference curve i, passing through the free
trade point P at which the offer curves intersect the
(free-trade) terms-of-trade line.21 This particular
z1 F.Y. Edgeworth, “The Theory of International Values, II,”
EconotnicJowna~4 (September 1894): 432. The same diagram
appears in his Papers Rdating to Political Economy, Vol. 2, (Lon-
don: Macmillan, 1925), p. 39.
indifference curve, he said, indicates the level of
welfare or satisfaction the home country enjoys under
free trade. It provides a benchmark against which
to compare alternative welfare levels yielded by
different degrees of trade restriction.
Second, he specified the range of tariff rates
beneficial to the home country. To do so, he noted
that the same indifference curve that passes through
the free-trade point P also cuts the foreign offer curve
at point M, which, by virtue of being on the same
indifference curve, yields the same level of welfare
as the free trade point. Since all points on the foreign
offer curve between these two extremes lie on higher
indifference curves, it follows that any movement to
a position between points P and M will result in the
home country being better off than under free trade.
In other words, points P and M mark the range of
terms-of-trade improvement beneficial to the home
country. Somewhere within this range benefit is at
a maximum.
Third, he identified the point Q at which the home
country reaches its highest possible trade indifference
curve given the foreign offer curve. The optimum
tariff, said Edgeworth, is that which distorts the home
country’s offer curve such that it intersects the foreign
offer curve at this point of tangency with the highest
attainable indifference curve. Here, almost fifty years
before Kaldor himself presented it, is the famous
tangency solution to the determination of the op-
timum tariff.
Fourth, Edgeworth showed that if the tariff is raised
too much it reduces rather than increases welfare.
For as the tariff is raised from point P to Q to M,
welfare at first rises, reaches a maximum, and starts
to fall. And if the tariff is raised beyond point M,
welfare falls below the level attained at the free trade
position P. It follows that the tariff must not be too
large if the nation is to benefit.
Finally, he noted some pitfalls in the practical
application of the model. For one thing, the optimum
point, though precisely identifiable in theory,
cannot be ascertained with any accuracy in practice.
Another consideration is the strong political pres-
sure exerted by protectionists. These factors make
it all too likely that policymakers would raise
tariffs far beyond the optimum point thus lowering
welfare. Then too there was the likelihood of retalia-
tion which would nullify any gains generated by the
tariff. Above all was the immorality of tariffs from
the cosmopolitan point of view; there is little to be
said for restrictions that cause other countries to lose
more than the dutying country gains.22 Taking all
22 Edgeworth, Papers II, pp. 17 (n.S), 18.
FEDERAL RESERVE BANK OF RICHMOND 2.5
these factors into account, free trade, he thought,
remains hands down the best and most practical
policy for a nation to follow.
C. F. BICKERDIKE
The last economist to be considered is C. F.
Bickerdike, who in his 1906 Economic Journal
article on “The Theory of Incipient Taxes” and his
1907 Review of A. C. Pigou’s Pmtective and Prefer-
ential Import D.&s contributed at least four innova-
tions to optimum tariff theory. First, he emphasized
the similarity between the theory of monopoly and
the theory of tariffs. He noted that when an individual
exporter expands his sales he drives down the price
received by other exporters. An export tax, he
claimed, corrects this tendency for competition
among exporters to lower the price obtained by all.
It does so by extracting from the gross price re-
ceived by exporters the amount by which an extra
unit sold lowers the price on all previous units. In
so doing, the duty forces exporters to behave as if
they take account of their collective influence on
prices paid by foreigners. The result is that the coun-
try acts as a single monopoly unit that fully exploits
its bargaining power to improve the terms of trade.a3
In effect, the export tax acts to form competing
exporters into a cartel.
Second, he specified anew the welfare gain from
trade restriction. As an alternative to Edgeworth’s in-
difference curve measure, he defined the net benefit
of an import duty as the sum of the tax revenue
collected from foreigners through lower import prices
less the deadweight loss in consumers’ surplus
caused by the shrinkage in trade volume. This welfare
gain he illustrated in a Marshallian demand-and-
supply curve diagram (see Figure 8) in which
crosshatched rectangular area 2 measures tariff
revenue collected from foreigners and shaded
triangular area 3 is the deadweight loss in consumers’
surplus.*4 To avoid Torrens-Mill type specie flow
and price level movements-complications that could
shift the demand and supply curves in Figure 8-he
assumed that each country operated with an incon-
vertible paper currency of constant purchasing power.
As noted by John Chipman, this assumption effec-
tively transformed a partial equilibrium diagram
into a consistent general equilibrium model.z5
23 C.F. Bickerdike, “The Theory of Incipient Taxes,” Ibn~nzi~
Journal 16 (December 1906): 530-3 1.
z4 Bickerdike, p. 533-34.
25 J.S. Chipman, “Bickerdike’s Theory of Incipient and Optimal
Tariffs,” unpublished paper, 1987.
In any case, Bickerdike concluded from his diagram
that a tariff benefits the dutying country whenever
rectangle 2 exceeds triangle 3 in size, which will be
the case provided the tariff is small enough, the
demand curve is of greater-than-zero elasticity, and
the import supply curve is not infinitely elastic. He
also concluded that the tariff is more beneficial the
more elastic the levying country’s demand for im-
ports. This is true because the more elastic the
demand curve the larger the foreigner’s tariff-burden
rectangle relative to the deadweight loss triangle. In
the limiting case of infinitely elastic demand,
foreigners would bear the entire burden of the tariff
and deadweight loss would be zero.
Third, he provided the frst mathematical proof rhat
a country could gain from a tariff. To obtain his
proof he constructed a two-country, two-commodity
algebraic model consisting of five groups of equations.
These included (1) export and import demand and
26 ECONOMIC REVIEW, JULY/AUGUST 1987
supply functions, (2) a trade-balance equilibrium con-
dition, (3) a law-of-one-price ,equation stating that the
foreign exchange rate must be such as to equalize
the common currency price (tariff-adjusted) of each
good across countries, (4) a tariff equation defining
the percentage tariff wedge inserted between the
prices domestic importers pay and foreign suppliers
receive, and (5) a collective utility function defining
national welfare as the excess of the total utility from
consuming import goods over the cost of obtaining
that utility through the production of exports.26
Having constructed his model, he then had to
demonstrate that national welfare increases upon a
small increase in the tariff. This he accomplished by
substituting equations (1) through (4) into the
utility function, differentiating that function with
respect to the tariff, and then showing that the
resulting first derivative is positive. His expression
reveals the welfare gain as depending critically upon
export supply and import demand elasticities at home
and abroad.
Last but not least he expressed the optimum tariff
rate in terms of a mathematical formula, being the
first to do so. To derive his optimum tariff formula
he set the foregoing first derivative of utility with
respect to the tariff rate equal to zero as required for
a maximum and solved for the tariff rate (or more
precisely for the reciprocal of one plus the tariff
rate-this term being his measure of the tariff wedge).
The result was
T = 1 -(h*)
1 -W%)
where T is the reciprocal of one plus the optimum
tariff rate t, or l/( 1 + t), and qa and q0 denote the ex-
port demand and import supply elasticities of the
foreign country. z7 Solving this formula for the tariff
rate t yields the expression
t = l/y+l/c
1 -l/c
where q = -q. and E = ye. Here is the classic formula
for the optimum tariff later made famous by R. F.
Kahn, J. de V. Graaff, and Harry G. Johnson in the
1940s and 195Os.28
z6 Bickerdike, Review of Protective and Pqbntial Itnpoti Duties
by A.C. Pigou, Economic JoumaZ 17 (March 1907): 100.
2’ Bickerdike, Review, p. 101.
28 See R.F. Kahn, “Tariffs and the Terms of Trade,” Rewiew
of Economic Studies 15, no. 1 (1947): 16; J. de V. Graaff, “On
Optimum Tariff Structures,” Review of Economic Studies 17.
nd. 1 (1949): 53; and H. G. Johnson,“Alternative Optimum
Tariff Formulae.” in his Intmrational Trade andEconomic Ghwth
(London: George Allen and Unwin, 1958), p. 60.
CONCLUSION
The impression conveyed by textbooks not-
withstanding, economists hardly had to wait until the
1940s to obtain theoretical models of the optimum
tariff. On the contrary, the key components of such
models already had been assembled long before.
Robert Torrens in the 1840s supplied two elements,
namely the notions that reciprocal demands deter-
mine the terms of trade and that tariffs affect those
reciprocal demands thus giving policymakers a means
of manipulating the terms of trade. John Stuart Mill
showed in an essay published in 1844 that an ex-
port tax works as well as an import tariff to improve
the terms of trade and that the extent of the improve-
ment depends crucially on the size of the coefficients
of elasticity of demand. Alfred Marshall in the 1870s
translated the Torrens-Mill analysis into graphical
form thus establishing the reciprocal demand or
offer curves used in modern models of the optimum
tariff. To Marshall’s reciprocal demand schedules
Edgeworth in 1894 added trade indifference curves
thus allowing one to identify in principle the particular
tariff rate that maximizes national gain. Finally,
C. F. Bickerdike in the early 1900s added three more
components to the theory: he proved mathematically
that a tariff could improve national welfare, he
presented alternative measures of the resulting gain,
and he derived the algebraic formula for the optimum
tariff rate. He also showed that the optimum tariff
restrains competition among individual importers and
exporters so that the dutying country acts as a cartel
exploiting its market power to improve the terms of
trade.
Except for Torrens and Bickerdike, these same
economists also specified the basic shortcomings of
optimum tariff theory. The theory, they noted,
assumes unrealistically (1) that foreign countries will
not retaliate with tariffs of their own, (2) that
elasticities of supply and demand in foreign trade are
not so large in the long run as to render the tariff
ineffective, (3) that the optimum tariff rate can be
precisely identified and skillfully administered, and
(4) that politicians can resist pressures to raise tariff
rates above the optimum level. None of these
assumed conditions, they felt, were likely to be
realized in practice. They further pointed out that
a tariff can benefit no nation except at the cost of
greater injury to others and is thus unacceptable
from a cosmopolitan point of view. For these reasons
they remained convinced that, despite the theoretical
FEDERAL RESERVE BANK OF RICHMOND 27
case that could be made for an optimum tariff, free optimum tariff long before modern analysts redis-
trade was the best policy from a practical and moral covered the issue. Here is a prime example of
standpoint. In other words, they established virtu- classical and neoclassical economists formulating
ally all the arguments for and against the use of an theories relevant to current trade policy analysis.
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I 28 ECONOMIC REVIEW, JULY/AUGUST 1987
