CSIS Discussion Paper No.55

A Theory of Optimal Tariffs under a Revenue
Constraint
Tatsuo Hatta a,∗, Yoshitomo Ogawa b
Center for Spatial Information Science and Center for International Research on the JapaneseaEconomy, Faculty of Economics, University of Tokyo.
School of Business and Economics, Kinki Universityb

Abstract
This paper examines the optimal tariff structure under a revenue constraint. When a fixed
level of tax revenue has to be collected from the tariff alone, no adjustment in tariff rates canachieve an efficient resource allocation, even in a small open economy. Hence, the optimal
tariff problem arises under a revenue constraint.

We show that the revenue-constrained

optimal tariff structure is characterized by the following two rules: (i) the optimal tariff rate islower for the import good that is a closer substitute for the export good; and (ii) the stronger the
cross-substitutability between imports, the closer the optimal tariff is to uniformity. This
theoretically explains why empirical studies have shown that the efficiency loss from a uniform∗

Corresponding authors. H. Hatta. Address: 705 Faculty of Economics Building, University of Tokyo, 7-3-1,

Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan. Tel.&Fax: +81-3-5841-5631. E-mail address: t@hatta.net. Y.
Ogawa. Address: School of Business and Economics, Kinki University, 3-4-1, Kowakae, Higashiosaka, Osaka
577-8502, Japan. E-mail address: ogawa@shok.kindai.ac.jp.

tariff structure is negligible.

Keywords: Revenue-Constrained Optimal Tariff; Corlett-Hague Rule; Cross-Substitutability Rule
JEL classification: F11; F13; H21.

1

1. Introduction
In many developing countries, tariffs are the main source of government revenue.1 However, if
a fixed level of tax revenue has to be collected from tariffs alone, no adjustment in tariff rates
can achieve efficient resource allocation, as first pointed out by Dasgupta and Stiglitz (1974).
Thus, we need to analyze the tariff structure that attains the second-best resource allocation.
We call this the optimal tariff problem under a revenue constraint. This problem does not arisein an economy with a lump-sum tax, but it does arise in an economy without a lump-sum tax
even when the economy is small.
This problem is entirely different from the more familiar optimal tariff problem in alarge economy, which was studied by Kaldor (1940) and Johnson (1954–55) among others. In alarge economy, the optimal tariff problem arises even if a lump-sum tax is available.
The optimal tariff problem under a revenue constraint is an extension of the optimal
taxation problem pioneered by Ramsey (1927) and Diamond and Mirrlees (1971).2 This iscaused strictly by an institutional framework within which a fixed level of tax revenue has to becollected by tariffs alone. The exportable good in the optimal tariff model plays the role of theleisure good in the optimal commodity tax model, as the untaxed good.3 Dasgupta and Stiglitz

1

See Rajaram (1994) for a survey on actual trade policies under a revenue constraint in several

developing countries.
2

In the R-D-M model, labor supply is endogenous, and the distortion is generated between goods

and leisure.

Commodity taxes and wage subsidies at a uniform rate remove this distortion.

However, tax revenue is zero. At this point, the optimal commodity tax problem is generated. In the
optimal tariff model, we make the labor supply constant, and hence we can disregard this distortion.
3

See Hatta (1994) for a comparison of the two theories in a simple context. In the optimalcommodity tax model, labor supply is endogenous and leisure is untaxable. Hence, a distortion is
generated between goods and leisure. In the revenue-constrained optimal tariff model, labor supply
is assumed to be constant, but the consumption of the exportable goods is untaxed. Hence, a2(1974) originated the study of optimal tariffs in a small open economy. Dahl, Devarajan andvan Wijnbergen (1986), Heady and Mitra (1987) and Mitra (1992) numerically analyzed
optimal tariff rates for a few developing countries. More recently, this problem has beenstudied by Hatta (1994), Dahl, Devarajan and van Wijnbergen (1994), Panagariya (1994),
Chambers (1994), Mitra (1994) and Keen and Ligthart (2002).4
This literature has shown non-uniformity of the optimal tariff structure in a number ofways.

Dasgupta and Stiglitz (1974), Dahl, Devarajan and van Wijnbergen (1986) andPanagariya (1994) have derived the inverse elasticity rule in the trade context under theassumption of zero cross-substitutability between importable goods. In addition, simulations
have been conducted by assuming Leontif-type production functions in part.
However, the literature has not demonstrated an explicit, general optimal tariff
formula for a three-good trade model that allows full technological substitutability. This is incontrast with the literature on optimal tax, where the Harberger expression (1964) gives a fully
general optimal tax formula for a three-good economy. A Harberger-like formula has been
derived for an optimal tariff in Hatta (1994), but only under an extremely simple assumption
about technology.
In the present paper, we derive an explicit formula for a revenue-constrained optimal
import tariff in a trade model with full technological substitutability. Our optimal tariff
formula is expressed in terms of the elasticity of the excess demand function, i.e., thedistortion is generated between the exportable and the importable goods. As Hatta (1994) explains,the exportable good in the optimal tariff model plays the role of the leisure good in the optimalcommodity tax model, as the untaxed goods.
4Among the literature cited here, Keen and Ligthart (2002) analyzed the welfare effect of asimultaneous change in tariffs and commodity taxes under a revenue constraint. A similar analysis
was undertaken by Panagariya (1992), who considered an economy where only tariffs can be
adjusted and there is an imported input subject to a tariff.

3

compensated demand function minus the supply function. Thus, the optimal tariff structure
depends on the supply elasticities as well as the demand elasticities. This contrasts with theHarberger expression for optimal tax, which is expressed in terms of demand elasticities alone,even if the production possibility frontier is strictly convex, as shown by Auerbach (1985). Thisdifference is caused by a variation in the consumer’s budget constraints in the two models: theproducer’s behavior does not affect the consumer’s income in the optimal tax model, while itdoes in the optimal tariff models.
As special cases of this general formula, we obtain various optimal tariff rules, such as
the inverse elasticity rule and the Corlett and Hague-type tariff rule. In particular, we observethat the stronger the cross-substitutability between imports, the closer the optimal tariff is touniformity. These rules are expressed in terms of the elasticities of excess demand functions.
This means that the optimal tariff rules depend upon both producer’s and consumer’s responses
to price changes. This is in contrast with the optimal tax rules, which are expressed in terms of
demand elasticities alone.
We investigate two features of the optimal tariff rules through this formula. First, theformula enables us to untangle the puzzle shown by Dahl et al. (1994, p. 222) and Mitra (1992,
p. 246), which is that the welfare loss caused by a uniform tariff rather than the optimal tariff
is negligible. Indeed, the welfare loss reported by Dahl et al. (1994) is 0.005% of the welfare
level under the optimal tariff, and that reported by Mitra is 0.05%. The reason for thisnegligible welfare loss from a uniform tariff has not been explained. Our formula shows thatthe virtual optimum is attained by any tariff structure close to uniformity when imports areclosely substitutable for each other in consumption and production.
Second, our formula is generalized to complex trade settings.
extended to the case where a non-tradable good exists.

For example, it is

The non-tradable good has an

endogenous price, which is adjusted so as to keep the amount of demand equal to that of supply.

4

Even in this case, the implication of our basic formula, that the optimal tariff structure is close
to uniformity when imports are closely substitutable, is robust. In addition, our formula isinterpretable in the case where one of the imports is an imported input (intermediate good). An
imported input is used in the production of final goods, so that imposing a tax on it distorts
resource allocation in the production sector.
The imported input and the non-tradable good are peculiar to the optimal tariff
framework, and do not appear in the optimal tax framework analyzed in a closed economy
model.

Our optimal tariff formulae expressed in terms of elasticities of net demand areessential in analyzing the imported input case.
In Section 2, we present the model. Section 3 proves that the first-best tariff (subsidy)
policies would yield zero tariff revenue. In Section 4, a formula that characterizes the optimal
tariff structure is derived and we examine the optimal tariff structure. Section 5 analyzes the
optimal tariff problems in the presence of the non-tradable good. Section 6 extends the optimal
tariff rules to the cases with the export tax (or subsidy) and with an imported input.

2. The Model
We consider an economy that satisfies the following assumptions:

Assumption 1: The economy is small and open. It has perfectly competitive markets for goods
and factors. The economy produces three goods: one export good and two import goods. The
only inputs of the economy are endowed factors. We denote the export good by 0 and the import
goods by 1 and 2.

Assumption 2: There is a representative consumer, who initially possesses all of the factors

5

(whose endowments are fixed), has an income m , consisting of factor incomes,5 and consumes
all of the three final goods. The consumer has a well-behaved utility function u ( x ) , wherex ′ = ( x0 , x1 , x 2 ) is the demand vector of the final goods, and he or she chooses the commodity
bundle that maximizes the utility level under given prices and income.6

The budget constraint of the consumer is given by:

q ′x = m ,(1)where q ′ = ( q 0 , q1 , q 2 ) is the domestic-price vector.
The consumer’s compensated demand function for the i -th good is given by:

x i = x i (q , u ) ,for i = 0,1,2,(2)

where u is the utility level.

Assumption 3: A producer maximizes profit, taking prices as given.

The aggregate of the net revenues of all firms, and hence of all industries, is equal to theincome of the consumer. Thus, the aggregate budget equation of the producers is given by:

5

The profit is zero, because free entry and exit are assumed.

6

Since the level of the public good provision is fixed in the analysis, it does not enter the utility

function as an explicit argument.

The function is well behaved if it is: (i) increasing in each6

q ′y = m ,(3)where y ′ = ( y 0 , y1 , y 2 ) is the output vector.
The supply function of the economy provides the commodity bundle that maximizes thetotal revenue q ′y of the production sector, under a given level of technology and prices. Itsi -th element is given by:7

y i = yi (q) ,for i = 0,1,2.

(4)

Assumption 4: Tariffs are imposed on the two imports.

The relationship between world prices, domestic prices, and import tariffs is given by:

q = p+t,(5)

where p ′ = ( p0 , p1 , p 2 ) is the world price vector, and t ′ = (t 0 , t1 , t 2 ) is the specific tariff
vector.

Assumption 5: The only revenue source of the government is import tariffs. In particular, thegovernment cannot levy commodity taxes or income taxes. The government spends all of theargument; (ii) strictly quasi-concave; and (iii) twice continuously differentiable.
7

Since the domestic factors are fully employed by the production sector and their supply levels

are fixed, they do not enter the supply function as explicit arguments.

7

tariff revenue on the purchase of the public good, which is imported from a foreign country.8

Thus, the budget equation of the government can be written as:t ′( x − y ) = r ,(6)where r is government spending on the public good and x − y represents the net import
vector of the private goods.

Assumption 6: The international balance of payments is in equilibrium, and is written as:p ′( x − y ) + r = 0 .

(7)

The left-hand side of the equation represents the sum of the international value of the net
imports of private goods and the public good.

Assumption 7: The world price of the export good is chosen as the numeraire: p 0 = 1 .

Equation (7) is the market equilibrium condition. Equations (1), (3) and (6) are thebudget equations of the economic agents. However, Equations (1) and (3) can be combined into:

8

Since the world price and the government revenue are constant, and since the government

expenditure is equal to revenue, the quantities imported by the government should also beconstants. The public good may be the good traded by the private sector of the home country. In
that case, the excess demand for the i -th tradable good is redefined as z i = x i − y i − g i where g i isthe amount the government imports of the i -th good.

8

q ′x = q ′y .

(8)This equation is the budget constraint of the private sector, and implies that the consumer’s
expenditure equals the producer’s revenue. Since Equations (7) and (8) immediately yield (6),
we represent this economy by Equations (2), (4), (5), (7) and (8) in the following.

Definition:

An economy satisfying Assumptions 1 through 7 is called a General Tariff

Economy (hereafter, GTE). When Equations (2), (4), (5), (7) and (8) are satisfied, it is said that
the GTE is in equilibrium.

A special case of the GTE is given a name, as stated below.

Definition. An economy satisfying Assumptions 1 through 7 and t 0 = 0 is called an Import
Tariff Economy (hereafter, ITE). When Equations (2), (4), (5), (7) and (8) are satisfied andt 0 = 0 , we say that the ITE is in full equilibrium.

In the ITE, q 0 = 1 holds from Assumption 7.
We define the excess demand functions as:z i ( q , u ) ≡ x i (q , u ) − y i (q ) ,for i = 0,1,2.

and

9z (q, u ) ≡ x (q, u ) − y (q ) ,where z ′(q, u ) = ( z 0 (q, u ), z1 (q, u ), z 2 (q, u )) . By substituting these functions for x − y in(7) and (8), we have:q ′z (q, u ) = 0 ,(9)p ′z (q, u ) + r = 0 .

(10)

In terms of this notation, the GTE is in full equilibrium if and only if it satisfies Equations (5),(9) and (10). The set of Equations (9) and (10) contains four variables, q0 , q1 , q2 and u , sincer is fixed by assumption. When two of the four variables are exogenously given, the two
equations determine the remaining variables. For example, if q0 and u are given, the model
determines the remaining variables (q1 , q 2 ) . Therefore, from Equation (5), the tariff vector t ,which maximizes the utility level u , can be found. The following definition is now required:

Definition: The tariff combination (t1 , t 2 ) that maximizes the utility level u in the model of(5), (9) and (10) for a fixed level of r and for the zero level of t 0 is called the optimum tariff ofthe ITE.

3. Tariff Revenue and Efficient Resource Allocation
A Pareto optimal resource allocation is attained only if the international and domestic prices of
the traded goods are proportional. At first, it may seem that the optimal tariff of the ITE must
be a proportional tariff structure, which makes the international and domestic prices of the10

traded goods proportional. It turns out that a proportional tariff yields zero revenue, as isshown below.

Hence, any tariff structure that yields positive revenue has to be

non-proportional and is inefficient in resource allocation. In other words, the optimal tariff ofthe ITE is the one that minimizes the inevitable distortion.
We now show that a proportional tax yields zero revenue. To this end, we define the advalorem equivalent rate of ti by:

τi =

ti
,
qi

for i = 0,1,2.

(11)This and (5) yield:9qi = pi + τ i qi ,for i = 0,1,2,(12a)

for i = 0,1,2.

(12b)

for i = 0,1,2.

(12c)

and

qi = pi +

τi
pi ,
1−τi

From (12a), we also have:10

qi =

9

1
pi ,
1−τi

Let t i = σ i p i .

Then σ i p i = τ i q i , and hence σ i = τ i q i p i .

This and p i = (1 − τ i )q i yield

σ i = τ i (1 − τ i ) , implying (12b).
10

The definition of the ad valorem tariff rate, as shown by (11), ensures that τ i < 1 . Hence,11

A tariff structure is called proportional if all tradable (including exportable) goods sharean identical ad valorem tariff rate, i.e., if:

τi = ε ,for i = 0,1,2,(13)

holds for some scalar ε . Equation (12c) implies that, under a proportional tariff structure, thedomestic prices of both exports and imports are proportionally higher than their world prices.
Hence, efficient resource allocation is attained with a proportional tariff. The fact that thedomestic price of the exports is higher than the world price implies that a subsidy is given to
the export goods at the same rate as the import tariffs. Thus, the “proportional tariff”, defined
as above, implies a combination of positive tariffs on the imports and a subsidy of equal rate to
the export. This means that a proportional tariff structure yields zero revenue, and hence wehave the following:

Theorem: In the GTE, proportional tariff structures achieving efficient resource allocation

yield zero revenue.

Proof. Under proportional tariffs, it follows from (12c) and (13) that:

qi =

1
pi .
1− ε

(14)

Substituting (14) for qi in (9) yields:

1−τ i > 0 .

12

p ′z = 0 .

From this and (10), we obtain r = 0 .

Q.E.D.

This implies that under a proportional tariff structure, all of the revenue collected by importtariffs is spent on the export subsidy. Therefore, only a non-proportional tariff can raise apositive tariff revenue.

In other words, a positive tariff revenue necessarily generates a

distortion.11
A tariff structure is called uniform if all import goods share an identical ad valorem
tariff rate, i.e., if τ 0 = 0 , τ 1 = ε , τ 2 = ε for some scalar ε . A uniform tariff structure can raisea positive tariff revenue. However, under a uniform tariff structure, the vector of the domestic
prices of the three goods is not proportional to that of the world prices, even though the vector ofthe domestic prices of the two imports is proportional to that of the world prices. Hence, theresource allocation under a uniform tariff is inefficient. In the next section, we examine how
the optimal tariff is different from uniformity under a positive revenue constraint.

4. Optimal Tariff Structure in the ITE
4-1. The Basic Lemma
In this section, we examine the optimal import tariff structure of the ITE, i.e., the economy thathas a zero export tariff and subsidy.
Let z ij = ∂z i ∂q j . We define the import elasticity of the i -th good with respect to theprice q j by η ij = q j z ij z i . Then we have the following:

13

Lemma. In the ITE, the optimal tariff structure (τ 1 , τ 2 ) satisfies the following:(i)τ 1 η 12 − η 22=,
τ 2 η 21 − η 11(ii)

τ 1 η 20 + η 12 + η 21,=
τ 2 η 10 + η 12 + η 21and

τ 1 = (η 20 + η12 + η 21 )θ ,(iii)

τ 2 = (η10 + η12 + η 21 )θ ,where θ > 0 .

Proof. We must first choose a level of q that maximizes the utility level in the model of (9) and(10) for the fixed level of r :12

max

u,

s.t

q ′z (q, u ) = 0,

t1 , t 2 , up′z (q, u ) + r = 0.

The Lagrangian of this maximization problem is:

11

Sandmo (1974) discussed this problem in the optimal tax context.

14

L = u − λ (q ′z (q, u )) − δ ( p ′z (q, u ) + r ) ,where λ and δ are Lagrange multipliers. The first-order conditions with respect to qi are

λz i + δp′z i = 0 , for i = 1, 2 , where z i′ = (∂z ∂qi ) = ( z 0i , z1i , z 2i ) . By using the homogeneity
condition, q ′z i = 0 , of the compensated demand function, this equation can be rewritten as:

 z11
−
 z12z 21   t1  z1   = ν   ,z 22  t 2  z 2 (15)

where ν = − λ δ .13 Solving (15) for t1 and noting z12 = z 21 and τ i = ti qi , we obtain:

τ 1 = (η12 − η 22 ) ⋅ θ ,(16a)

θ = νz1 z 2 q1 q 2 ( z11 z 22 − ( z12 ) 2 ) .

(17)

where

Similarly, we have:τ 2 = (η 21 − η11 ) ⋅ θ .

(16b)

12

See Mirrlees (1976) and Hatta (1993) for this formulation of the maximization problem.

13

The proof following this expression adapts the proof of Diamond and Mirrlees (1971, pp.

15

From (16), we obtain Formula (i).
Since:

η10 + η11 + η12 = 0 ,(18)

η 20 + η 21 + η 22 = 0 ,Formula (i) immediately yields (ii).
From (16) and (18), we obtain two equations in Formula (iii). Multiplying (15) by(t1 , t 2 ) gives:z− [t1 , t 2 ] 11
 z12

z 21   t1 = ν (t1 z1 + t 2 z 2 )z 22  t 2 

= νr .

Since the left-hand matrix is negative semi-definite, ν has the same sign as government
revenue r . From this and (17), we obtain θ > 0 since the denominator of (17) is positive.
Q.E.D.

A formula similar to (ii) was first obtained by Harberger (1964) for an optimalcommodity tax.14 However, his formula is expressed in terms of demand elasticities. Even in amodel with a fully substitutable production possibility surface, the supply elasticities have no262-263) for optimal taxation.
14

This paper was later published as Chapter 2 in Harberger (1974). The Harberger formula is16

place in the optimal tax formula, as demonstrated by Auerbach (1985).
On the other hand, our formula is expressed in terms of the elasticities of the excess
demand function, i.e., the compensated demand function minus the supply function. Hence,unlike the optimal tax structure, the optimal tariff structure is affected by supply elasticities.15
The consumer’s budget equations are different in the two models. In the optimal tariffmodel, the budget equation is given by q ′x = q ′y as in (8); in the optimal tax model, it is given
by q ′x = q ′x where x represents the initial endowments that the consumer possesses. Hence,the producer’s response to a price change affects the consumer’s budget in the optimal tariffmodel, but not in the optimal tax model.
Note that in the optimal tax model, the producer’s profit is assumed to be zero either
because of free entry or due to a 100% tax on profit, while in the model of the optimal tariff, nosuch assumptions are made.

4-2. The Inverse Elasticity Rule
We say that import goods are independent of each other if (∂z1 ∂q 2 ) = (∂z 2 ∂q1 ) = 0 , i.e.,found on p. 49 in Harberger (1974).
15

We define the compensated demand elasticity as φ ij = q j x ij x i and the supply elasticity as

ε ij = q j y ij y i . From the definition of the excess demand elasticity, we have:η ij ===

q j z ijzi
q j xij − q j y ij
zi
xi
zi

⋅ φ ij −yizi⋅ ε ij .

By substituting these for η ij s in the formulae in the Lemma, we find that the optimal tariff rates
depend on the compensated demand elasticities and the supply elasticities.

17η12 = η 21 = 0 .

Inverse Elasticity Rule: The optimal tariff rate is inversely proportional to the own elasticity

of excess demand if the imports are independent of each other.

Proof.

Independence among the imports implies η12 = η 21 = 0 , and hence Formula (i)degenerates into:τ 1 η 22=,
τ 2 η 11which proves this rule.

(19)

Q.E.D.

This rule has been widely used in the literature on optimal tariffs under a revenue constraint.
The seminal paper by Dasgupta and Stiglitz (1974, p. 21) showed that the revenue-constrained
optimal tariff is not uniform, by directly establishing the inverse elasticity rule. The inverse
elasticity rule was also derived by Dahl at al. (1994, p. 217) and Panagariya (1994, p. 234).
Formula (i) of the Lemma immediately implies that if cross-elasticities between the two
imports, η12 and η 21 , are not equal to zero, the ratio τ 1 τ 2 would diverge from the inverse
elasticity rule. In fact, when η12 ≠ η 21 , the optimal tariff rate of the import with the lower own
elasticity may become lower than the rate of the other import.
Note, however, that Formula (i) does not imply that the tariff system necessarily
approaches uniformity as cross-elasticities become large when η12 ≠ η 21 .

18

4-3. The Corlett–Hague Rule

Definition: We say that the i -th good is a closer substitute for the k -th good than the j -thgood, if η ik > η jk .

We are in a position to state and prove the following proposition.

Proposition 1: The optimal tariff rate is lower for the import good that is the closer substitutefor the export good.

Proof.

Taking the difference of the two equations in Formula (iii), we have

τ 1 − τ 2 = (η 20 − η10 )θ . Since θ > 0 , this expression shows that τ 1 > (<)τ 2 if and only if
η 20 > (<)η10 .

Q.E.D.

This implies that the optimal tariff rate is higher for the good that is more complementary withthe export good. The export is the untaxed good, and hence it is over-consumed. Taxation on
the good that is more complementary with the export partially offsets this over-consumption.
This shows that the ranking of tariff rates depends upon the relative degree ofcomplementarity between the taxed goods (import goods) and the untaxed good (the exportgood). Since this was first shown by Corlett and Hague (1953), in the context of commodity
taxation, we call this the Corlett–Hague rule.

4-4. The Cross-Substitutability Rule
The Corlett–Hague rule shows that optimal tariff structure is non-uniform, while the following

19

proposition shows that the degree of non-uniformity is limited by the degree ofcross-substitutability between imports.

Proposition 2:

Assume that τ 1 > 0 and τ 2 > 0 .

The stronger the cross-substitutability

between imports, the closer is the optimal tariff to uniformity if all the cross-elasticities
involving the export good are kept constant.

Proof.

In case τ 1 > 0 and τ 2 > 0 , we obtain η 20 + η12 + η 21 > 0 and η10 + η12 + η 21 > 0 ,respectively, from Formula (iii). The cross-elasticity of imports η12 + η 21 is common in both thenumerator and the denominator of Formula (ii). When η10 and η 20 are kept constant, thelarger is the cross-substitutability, the closer are the values of the numerator and denominator,and hence the closer is the ratio of the tariff to uniformity.

Q.E.D.

Proposition 2 may be called the Cross-Substitutability Rule. The non-uniformity of the tariff
rates will create distortions in the choice among the imports.

In particular, whencross-elasticities among imports are high, a non-uniform tariff structure creates strong
distortionary effects in the choice among the imports,16 and hence the optimum tariff structure
tends to be close to uniform.
Although the inverse elasticity rule is the most well-known optimal tax rule, its
applicability is extremely limited. A good usually has high demand elasticity because it has
close substitutes. For example, beef has high demand elasticity because chicken and pork areavailable as substitutes. However, the inverse elasticity rule assumes that a good has no
substitutes among imported goods. In the context of our trade model, this assumption implies

16

For the case that generalizes the number of taxed goods, see Hatta (1986).

20

that an imported good has high demand elasticity if and only if it is a close substitute for theexported good, and a good has low demand elasticity if and only if it is independent of theexported good.
There is an intriguing puzzle in this literature. Empirical simulations by Dahl at al.
(1994, p. 222) and Mitra (1992, p. 246) show that the welfare loss caused by a uniform tariff,
rather than the optimal tariff, is negligible.
As was pointed out earlier, the inverse elasticity rule appears contradictory to thefinding of Dahl et al. (1986, 1994) and Mitra (1992) that the welfare loss caused by a uniform
tariff structure is negligible. This is because their findings suggest that virtual optimality is
attained by a uniform tariff structure.

However, the puzzle is solved if we realize theunrealistic nature of the assumption in the inverse elasticity rule that the imports are
independent of each other in both consumption and production.

It is likely that thecross-substitutability term among the imports is dominating, undermining the basic
assumption of the inverse elasticity rule.
We assume that import tariffs are positive in Proposition 2. Note that the optimal tariff
rate can be negative for an import if the import is a substitute of the export and if it is acomplement of the other import.

5. Optimal Tariff Rules under the Presence of a Non-Tradable Good.
In this section, we introduce a non-tradable good to the ITE and examine the optimal tariffstructure in such an economy.
We consider the model obtained by adding the non-tradable good to the ITE as the
fourth good. We continue to adopt the notations of the ITE, and denote the non-tradable goodby n . The market-clearing condition of the non-tradable good is added to the equilibrium
conditions in the ITE, and it is expressed as:21xn = yn ,(20)

where x n and y n denote the demand and supply level of the non-tradable good, respectively.
The excess demand function in this economy can be represented by:

z i = z i (q, q n , u ) ,for i = 0,1,2, n ,(21)

where q n is the price of the non-tradable good. From (20) and (21), we obtain:z n (q , q n , u ) = 0 .

This equation determines the price q n as a function q and u . Suppose that this equation canbe solved for q n . The resulting function may be written as q n = q n (q, u ) . Substituting thisfor q n in (21) yields:z i∗ (q, u ) = z i (q, q n (q, u ), u ) ,for i = 0,1,2.

∗

We call z i (q, u ) the reduced form of the excess demand function (hereafter, the reduced form).
The reduced form has the same properties as the excess demand function used in the ITE.17

17

This function is (i) differentiable, (ii) homogenous of degree zero, and (iii) concave in q . For thedetail, see Dixit and Norman (1982, p. 91).

22

Note that Equations (9) and (10) also hold in this economy.18 Then, replacing the∗

reduced form z i (q, u ) for the excess demand function z i (q, u ) in (9) and (10), we find thatthe equilibrium equations in this economy can be expressed as:

p ′z * (q, u ) + r = 0 ,q ′z * (q, u ) = 0 .

Solving the maximizing problem in the same way as in the proof of the Lemma, we obtain theHarberger expression for the presence of a non-tradable good:∗∗+ η 12∗ + η 21
τ 1 η 20= ∗,∗
τ 2 η 10 + η 12∗ + η 21

(22)

which is the same as Formula (ii) except that each η ij is replaced by η ij = (q j z i ) ⋅ (∂z i ∂q j ) .
∗

∗

∗

∗

∗

Formula (22) has implications similar to those of Propositions 1 and 2. Note that (∂z i ∂q j )involves the indirect effect through the price change of the non-tradable good.

6. Other Extensions.
6-1. An Export Tax or Subsidy
In this section, we analyze the optimal tariff problem in the GTE, where a tax or subsidy is
imposed on the export good.
In the GTE, if all the tariffs (t 0 , t1 , t 2 ) are simultaneously adjusted, the tariff vector

18

The budget equation of the private sector in this economy can be written as q ′x + q n x n23

that maximizes the utility level is not unique.19 Fixing a tariff level on one of the three tradable
goods, we can find the optimal tariff levels on the two remaining goods. Here, we will fix theexport tax (or subsidy) at a given level. In this case, we obtain the following:τ 1 − τ 0 η 20 + η12 + η 21.
=
τ 2 − τ 0 η10 + η12 + η 21

(23)

This formula is derived from Lemma 4 in Hatta and Ogawa (2002, p. 12).20 Assume thatτ i − τ 0 > 0 (for i = 1,2) and ηi 0 + η12 + η 21 > 0 (for i = 1,2). Noting that the export tax (orsubsidy) rate τ 0 is fixed, we find that the cross-substitution rule is satisfied since thecross-elasticity of imports η12 + η 21 is common in both the numerator and the denominator of= q ′y + q n y n . However, this equation is reduced to (8) by (20).
19

Since x (q, u ) and y (q ) are homogenous of degree zero with respect to q , a proportional

increase in q dose not affect the value of x (q, u ) and y (q ) , keeping the level of utility and the tariff
revenue of the GTE intact. If q ∗ is an equilibrium domestic price vector of the GTE, then the vectorκq ∗ is also an equilibrium price vector of the same model for any positive scalar κ . The vectors ofq ∗ and κq ∗ have the identical equilibrium allocation and hence attain the same levels of utility andtariff revenue.

Let t ∗ = q ∗ − p , and t ∗∗ = κq ∗ − p .

It is readily found that t ∗ and t ∗∗ have an

identical equilibrium allocation. Since κ is any positive scalar, t ∗∗ is not unique. This result is notaffected even if q ∗ is the equilibrium price vector that maximizes the utility level. Therefore, theoptimal tariff vector in the GTE is not unique. Hence, the optimal tariff rate vector is also not
unique.
20

Equation (19) in Hatta and Ogawa (2002, p. 12) is:ττ 1  η 20 + η12 + η 21   τ 0  ⋅ 1 −  + 1 ⋅ 0 .
=
τ 2  η10 + η12 + η 21   τ 2 
τ2Multiplying both sides by τ 2 (τ 2 − τ 0 ) and subtracting τ 0 (τ 2 − τ 0 ) from both sides, we obtain (23).

24

(23). We also obtain the Corlett-Hague rule: if and only if η10 < (>)η 20 then τ 1 > (<)τ 2 holds.

6-2. An Imported Input
We can analyze an imported input in our model simply by assuming that the consumer
neither consumes nor supplies one of the imported goods. Then all the optimal tariff rules for
the ITE hold as such.
It is generally considered that cross-substitution dominates in consumption. Thus, it isnatural to assume that two imports are substitutable if they are consumption goods. However,
the assumption of substitutable imports may not hold when one of the imports is an imported
input.

As Lopez and Panagariya (1992) pointed out, an imported input is necessarily

complementary with a produced good in certain models.

The first model of Lopez andPanagariya had a fixed coefficient technology with respect to the imported input, and theirsecond model was the Heckscher-Ohlin model, where the number of primary factors is equal tothat of produced goods.

In both models, an increase in the price of an imported input

necessarily increases the output level of the good that uses this input less intensively, throughthe Rybczynski effect.
In models where the Rybczynski effect does not work, the imported input can besubstitutable for all goods.

As was pointed out by Jones and Scheinkman (1977), theRybczynski effect does not work even in a Heckscher-Ohlin model when the number of inputs ismore than the number of outputs. In addition, in the case where each sector employs a specificfactor and an imported input, for example, all goods are necessarily substitutable for theimported input. The cross-substitutability rule can then be applicable even in the case with thepure imported input.

7. Conclusion

25

In the present paper, we derived an explicit formula for the revenue-constrained optimal
import tariff in a general trade model. This formula is expressed in terms of the elasticities ofsupply as well as those of demand.
This formula indicates that the optimal tariff is close to uniformity when imported
goods are close substitutes either in consumption or in production. This theoretically explains
why empirical studies show that the efficiency loss from a uniform tariff structure is negligible.
Our formula reveals that the existence of close substitutes in production makes the
optimal tariff structure more uniform than otherwise. Our explicit formula was generalized tocomplex trade settings that incorporate a non-tradable good and an imported input. In the caseof the imported input, some elasticities in our formula become purely supply elasticities. Evenin these cases, our basic formula reveals the conditions on the structure of production underwhich the optimal tariff structure is close to uniformity.

Acknowledgments
We are grateful to Kenzo Abe, Takashi Fukushima, Michael Keen, Jenny E. Ligthart,
Yoshiyasu Ono, Koji Shimomura, Katsuhiko Suzuki, David Tarr and Makoto Tawada for
helpful comments and suggestions. We would like to thank Shantayanan Devarajan andArvind Panagariya for introducing us to the topic of the optimal tariff problem under a revenue
constraint.

26

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