Discussion Paper No.

483

A THEORY OF OPTIMUM TARIFF
UNDER REVENUE CONSTRAINT

Tatsuo Hatta
andYoshitomo Ogawa

July 1999

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanA theory of optimum tariff under revenue constraint

Tatsuo Hatta
Institute of Social and Economic Research
Osaka University
Yoshitomo Ogawa
Graduate School of Economics
Osaka University

Abstract
This paper analyzes the revenue-constrained optimum tariff problem.

When afixed level of tax revenue has to be collected only from tariffs, an efficient resource
allocation can not be achieved by any tariff structure.

Thus we need to find the

optimum tariff structure as the second best resource allocation.

We will consider asmall open economy with a non-tradable good, and with full technologicalsubstitutability between each good.
characterized by following:

Then, the optimal tariff structure is(i) the optimum tariff rate is lower for the importablethat is the closer substitute of the untariffed goods, and (ii) the stronger is the crosssubstitutability between importables, the closer is the optimal tariff to uniformity.
We also show that the Inverse Elasticity Rule no longer holds in this model.

Key words:
tradable,

Revenue-constrained optimum tariff,Corlett and Hague Rule,

Rule
JEL classification:

F1, H21

Optimal tax rules,

Cross Substitutability Rule,

Non-

Inverse Elasticity

1

1.

Introduction
The World Bank has often recommended reduction of highest tariff rates todeveloping countries.1

However, tariff is the main revenue source of many

developing countries, and hence reduction of highest tariffs forces these countries
to increase lower tariffs.

The World Bank has in effect recommended thesecountries to make the tariff structure closer to uniformity.
On the other hand, the literature of revenue-constrained optimum tariff
problem, such as Dahl, Devarajan, and van Wijnbergen (1994), Panagariya (1994)
and Mitra (1994), has pointed out that the optimum tax rules must be applied totariff rates in developing countries where tariff is the main source of thegovernment revenue.

In particular, this literature has emphasized theimportance of the Inverse Elasticity Rule as a conceptual guidance for tariffreform.
In the models employed in this literature, which have only tradable goods, achange in a tariff rate does not affect the prices of the untariffed goods.2

This isbecause the only uatariffed goods in these models are exportables, whose prices
are exogenously given from the assumption of a small country.

In an economy

with non-tradable goods, however, a change in a tariff rate affects the prices of theuntariffed goods, i.e., the non-tradables.

The present paper shows which rules

are robust and which rules are not in such an economy.
The purpose of the present paper is four fold.

The first is to derive optimal

tariff rules in a model with full technological substitutability.

In order to1

See Rajaram (1994) for a review of the World Bank’s tariff recommendation.

2

The non-tradable goods here include (i) intrinsically non-tradable goods due to thepreference of each country, (ii) goods with prohibitive transportation cost, and (iii) goods with

2

analyze the economy where prices of the non-tradables are flexible, we need toconsider a model with full technological substitutability.

The second is to showthat the Inverse Elasticity Rule does not hold in an economy with a non-tradable
good.

The third is to establish that the Corlett and Hague Rule continues to holdin an economy with a non-tradable good.

Thus the tariff rate has to be higher for

the imports that is more complementary with the compound of the non-tariffed
goods.

The fourth is to demonstrate that the so called Cross Substitutability

Rule also continues to hold even when a non-tradable good is introduced.

Thus

substitutability between the imports tends to make the tariffs more uniform in
the non-tradable good economy as well as in the economy with a non-traded good.
When a fixed level of tax revenue has to be collected only from tariffs, i.e.,
when a lump-sum tax is not available, an efficient resource allocation can not be
achieved by any tariff structure.3

Thus we need to find the tariff structure thatattains the second best resource allocation.

We will call this the revenue-

constrained optimal tariff problem in a small country.
This problem arises under the institutional framework that a fixed level ofrevenue has to be collected only by tariffs.

This problem is entirely different

prohibitive tariff rates or restrictions.
3

The reason is as follow: An efficient resource allocation would require that the domestic

prices of all the tradables be proportional to their world prices.

Given import tariffs, thedomestic prices of importables are higher than the world prices.

Thus the first best policy

would require that the domestic prices of exportables be higher than the world prices, implyingthat subsidies must be given to all the exportables.

Besides, the ad volorem rates of subsidies

on exportables and tariffs on importables must be exactly equal.
would be zero.

But then the tariff revenue

See Appendix 1 for this.

This would conflict with the original constraint of raising fixed revenue.
given tariff revenue, price distortions are inevitable.
structure of tariff is.

Hence, to raise a

Problem is what is the optimal distorted

3

from the more familiar optimal tariff problem in a large country, studied by Graaff(1949-50) and Johnson (1953-54) among others, where a lump-sum tax isavailable.

In a large country, the optimal tariffs have to be non-uniform even if alump-sum tax is available.
The revenue-constrained optimal tariff problem in a small country is anextension of the optimal taxation problem pioneered by Ramsey (1927) andDaimond and Mirrlees (1971).4

The exportable in the optimal tariff model plays

the role of the leisure in the optimal commodity tax model as the untaxed good.5
Dasgupta and Stiglitz (1974) started the field of optimal tariff in a small economy.
Devarajan et al(1986)，Heady and Mitra (1987) numerically analyzed optimal
tariff rates for a few developing countries.

More recently, this problem has beenstudied by Devarajan et al (1994), Panagariya (1994), Chambers(1994), Mitra(1994) and Hatta (1994).
The most prominent among the optimal tariff rule is the Inverse Elasticity
Rule.

For example, Dusgupta and Stiglitz (1974)， Devarajan et al (1994), andPanagariya (1994) discussed this rule.

The Inverse Elasticity holds, however,

only under the stringent condition that the imports (i.e., tariffed goods) are
independent of each other in consumption and production, and its practical value
is limited.

The present paper shows that in an economy where a non-tradable

good exists, the Inverse Elasticity Rule no longer holds even under the condition

4

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

goods and leisure.

Commodity taxes and wage subsidy at a uniform rate get rid of this

distortion, and tax revenue, however, is zero.
problem is generated.

At this point, the optimal commodity tax

In the optimal tariff model, we will make labor supply constant, andhence can disregard this distortion.
5

See Hatta (1994) for a simple comparison of the two theories.

4

that import goods are independent of each other.6
The main message of the present paper is that the Corlett and Hague Ruleand the Cross-Substitutability Rule are more relevant than the Inverse Elasticity
Rule as the conceptual guidance for the revenue-constrained optimal tariff designin the realistic situation where non-tradables are abundant.
Section 2 presents the model that has only tradable goods.
analyzes the optimal tariff problem with a non-tradable.

Section 3 will

Section 4 proves themain theorem.

6

Chambers (1994) showed the sufficient condition of uniform tariff structure.

implication is similar to that of Sadka (1977) in the optimal commodity tax problem.
(1994) derived the Samuelson rule in the optimal tariff problem.

Its
Mitra

5

2.

The T-Economy

2.1.

The model
We consider an economy that satisfies the following assumptions.

Assumption 1.

The economy is small and open.

It has perfectly competitivemarkets for goods and factors.

Assumption 2.

The economy produces three goods, one exportable good and two

importable goods.

The only inputs of the economy are endowed factors.

We willdenote the exportable good by 0 and the importable goods by 1 and 2.

Assumption 3.

There is only one consumer.

factors, whose endowments are fixed.

Initially he possesses all of the

All of his income is obtained from factor

markets.

Assumption 4.

The consumer consumes all of the three final goods.

He has awell-behaved utility function u (x) , where x ′ = ( x0 , x1 , x 2 ) is the demand vectorof the final goods, and chooses the commodity bundle that maximizes his utility
level under given prices and income.7

The budget constraint of the consumer is given by

7

Since the level of the public good provision is fixed throughout the analysis, it does not enter
the utility function as an explicit argument.

A function is well behaved if it is (i) increasing

6

q ′x = m(1)

where q ′ = ( q 0 , q1 , q 2 ) is the domestic-price vector and m is the consumer‘s
income.
His compensated demand function for the i -th good is given by

xi = xi (q, u ) ,(i = 0, 1, 2)(2)

where u is utility level.

Assumption 5.

A producer maximizes his profit, taking prices as given.

The aggregate of the net revenue of the all firms, and hence of all industries
is equal to the income of the consumer.

Thus the aggregate budget equation of

the producers is given by

q ′y = m ,(3)where y ′ = ( y 0 , y1 , y 2 ) is the output vector.
The supply function of the economy gives the commodity bundle thatmaximizes the total revenue q ′y , of the production sector, under the givenin each argument, (ii) strictly quasi-concave, and (iii) twice continuously differentiable.

7

technology and prices.

y i = y i (q)Assumption 6.

Its i -th element is given by8(i = 0, 1, 2) .

(4)

Tariffs are imposed on the two importables.

The relationship among the world prices, the domestic prices, and import tariffs isgiven byq = p+t,(5)

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

Assumption 7.

The only revenue source of the government is import tariffs.

particular, the government can not levy commodity taxes and income taxes.

In
The

government spends all of the tariff revenue on the purchase of the public goodwhich is imported from a foreign country.9

Thus, the budget equation of the government can be written as

8

Since the domestic factors are fully employed by production sector and its supply level isfixed, it does not enter the supply function as an explicit argument.
9Note that the world price of the public good that the government imports is fixed, though we8t ′(x − y ) = r(6)

where r is the government spending in the public good and x − y represents thenet import vector of the private goods.

Assumption 8.

The international balance of payment is in equilibrium, i.e.,p ′(x − y ) + r = 0 .

(7)

The left hand side represents the sum of the international value of the net importof private goods and that of the public good.

Assumption 9.

The exportable good is chosen as the numerair good: q 0 = 1 .

Equation (7) is the market equilibrium condition.
are the budget equations of the economic agents.

Equations (1), (3) and (6)However, equations (1) and (3)can be combined intoq ′x = q ′y .

(8)This equation is the budget constraint of the private sector, and implies that theconsumers’ expenditure equals the producers’ revenue.

Since equations (7) and(8) immediately yield (6), we will represent this economy by equations (2), (4), (5),do not denote it explicitly.

9(7) and (8) in the following.

Definition 1.
Economy.

The economy satisfying Assumptions 1 through 9 is called the T-

When equation (2), (4), (5), (7) and (8) are all satisfied, we say that the

T-Economy is in the full equilibrium.

Define the excess demand functionsz i (q, u ) ≡ xi (q, u ) − yi (q)(i = 0, 1, 2)

(9)

and

z (q, u ) ≡ x(q, u ) − y (q) ,(10)

where z ′(q, u ) = ( z 0 (q, u ), z1 (q, u ), z 2 (q, u )) .

By substituting these functions

for and x − y in (7) and (8), we haveq ′z (q, u ) = 0 ,(11)p ′z (q, u ) + r = 0 .

(12)

In term of this notation, the T-Economy is in full equilibrium if and only if itsatisfies equations (5), (11) and (12).

Two equations (11) and (12) contain threevariables, q1 , q 2 , u , since r is fixed by assumption.

When one of the threevariables are exogenously given, the two equations determine the remaining

10

For example, if u is given, the model determines the remaining

variables.

variables (q1 , q 2 ) .

Therefore, from equation (5) we can find the tariff vector tthat maximizes the utility level u .

Definition 2.

We now define the following.

The tariff combination (t1 , t 2 ) that maximizes the utility level u

in the model of (5), (11) and (12) for a fixed level of r is called the optimum tariffof the T-Economy.

2.2.

The optimal tariff in the T-Economy
We will derive the optimal tariff in the T-Economy.

definitions.

Let z ij = ∂z i ∂q j .

Let us make a few

Define the import elasticity of the i -th goodwith respect to the price q j by η ij = q j z ij z i and ad valorem tariff rate of the i th good by τ i = t i qi .

Theorem 1.

Then we have following:

In the T-Economy, the optimal tariff rate is expressed asτ 1 = (η 20 + η12 + η 21 )θ ,
τ 2 = (η10 + η12 + η 21 )θ ,(13)

for some scalar θ > 0 .

Proof.

We must first choose q that maximizes the utility level in the model of11(11) and (12) for the fixed level of r 10:

max

u

s.t

q ′z (q, u ) = 0,

q, u

(14)

p ′z (q, u ) + r = 0.

The Lagrangian of this maximization problem is

L = u − λ (q ′z (q, u )) − δ (p ′z (q, u ) + r )where λ and δ are Lagrangian multipliers.

Its first-order conditions withrespect to qi are− λz i − δp ′z i = 0 ,

where(i = 1, 2) ,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− λz i + δt ′z i = 0(i = 1, 2) .

To derive equation (13) from equation (15), see Auerbach (1985, p.92).11

(15)

As to

10

See Hatta (1993) for this formation of maximization problem.

11

Alternatively (13) is derived as a special case of (25), for which a full proof is given in12

θ > 0 , see Daimond-Mirrlees (1971, p.262).

Q.E.D.

The term η12 +η 21 is called the cross-elasticity between the importables.
We will say that the i -th good is more substitutable for the k -th good than thej -th good is, if η ik > η jk , and importable goods are independent of each other, if(∂z1 ∂q 2 ) = (∂z 2 ∂q1 ) = 0 , i.e., η12 = η 21 = 0 . We are in a position to state andprove the following Proposition.

Proposition 1.
(a)

The following holds in the T-Economy:

The optimal tariff rate is lower for the importable good that is the closer

substitute of the exportable good .
(b)

The stronger is the cross substitutability between importables, the closer isthe optimal tariff to uniformity when all of the cross elasticities involving theexpert good are kept constant.
(c)

The optimal tariff rate is inversely proportional to the own elasticity ofexcess demand if the importables are independent of each other.

Proof.

From equation (13), we immediately obtainτ 1 − τ 2 = (η 20 − η10 )θ ,Appendix 2 of the present paper.

(16)

13

τ 1 η 20 + η12 + η 21 12.
=τ 2 η10 + η12 + η 21

(17)

Proposition 1-(a) is obvious from equation (16).
equation (17).

Proposition 1-(b) is derived fromThe cross elasticity η12 +η 21 is common in both numerator anddominator of equation (17).

When η10 and η 20 are constant, the lager is the

cross elasticity, the closer is the value of numerator and dominator, that is, theratio of tariff to uniformity.13
Finally, consider the special case where importable goods are independent ofeach other.

Then, since η12 = η 21 = 0 , η i 0 = −η ii holds for i = 1, 2 .14

Therefore,(17) reduces toτ 1 η 22.
=
τ 2 η11(18)

From (18), we have Proposition 1-(c).

Q.E.D.

Proposition 1-(a) implies that the optimal tariff rate is higher for the goodthat is more complementary with the exportable.

12
13

The exportable is the untaxed

Harberger (1964) first indicates this formation.
This equation also yields Proposition 1-(a). If η10 = η 20 , τ 1 τ 2 = 1 , that is, the optimal

tariff rate is uniformity.

This corresponds to Sadka (1977) in the optimal commodity tax

problem.
14

Homogeneity condition yields η i 0 + η i1 + η i 2 = 0 .

ηi 0 = −ηii .

If η ij = 0 for i ≠ j and i, j ≠ 0 , we have

14

good, and hence it is over-consumed.

Taxation on the good that is more

complementary with the exportable partially offsets the over-consumption of theexportable.

Namely, Proposition 1-(a) shows that the ranking of tariff rates

depends upon the relative degree of complementarity between the taxed goods(importable goods) and the untaxed good (exportable good).

Since this was firstshown by Corllet and Hague (1953) in the context of commodity taxation, we willcall this the Corllet and Hague rule.
Proposition 1-(b) is called the Cross Substitutability Rule.

The stronger iscross substitutability between the taxed goods (i.e. the importable goods) createsthe stronger is the distortion.
Proposition 1-(c) is called the Inverse Elasticity Rule.

This rule has beenthem widely used in empirical estimates in the literature of optimal tariffs underrevenue constraints.

15

3.

The N-Economy

3.1.

The model
The

T-Economy

consumes

andproducesincorporate a non-tradable good in the T-Economy.

only-tradables.

We

now

To this end, we substitute

Assumptions 2’ and 4’ listed below for Assumptions 2 and 4, respectively.

Wealso add Assumption 10 also listed below.

Assumption 2’.

The economy produces four goods: one exportable good, twoimportable goods and one non-tradable good, which is not traded with foreigncountry.

The only inputs of the economy are endowed factors.

We will denotethe non-tradable by n , while other goods have same indexes as T-Economy.

Assumption 4’.

The consumer consumes all of the four final goods.

He has awell-behaved utility function u ( x, x n ) , where xn is the demand of non-tradable
and chooses the commodity bundle that maximizes his utility level under given
prices and income.

Assumption 10.

The market for the non-tradable good is in equilibrium:

xn = y n .

(19)

From Assumptions 3, 4’ and 5, the compensated demand and supply

16

functions for the i -th good are given by15:

xi = xi (q, q n , u )y i = y i (q, q n )(i = 0, 1, 2, n) ,(2’)

(i = 0, 1, 2, n) ,where q n is the price of the non-tradable.

(4’)

Since Assumptions 6 and 8 aresatisfied,q = p + t , and(5)p ′(x − y ) + r = 0(7)continues to hold.

Assumptions 2’, 3, 4’ and 5 yield the budget constraint of theprivate sector:

q ′x + q n x n = q ′y + q n y n .

(8’)

The N-Economy consists of (2’), (4’), (5), (7), (8’) and (19).
Equations (8’) and (19) yield the equation (8).

Equation (8) in this economy

implies that the revenue from the sale of the tradable goods equal the consumer’s
spending on tradable goods.

15

Thus the set of (8’) and (19) is equivalent to the set

Note that the utility function u ( x, x n ) is different from u (x) in T-Economy.

we will use the same notation to simplify the analysis.

This is also applied to thecompensated demand, supply and excess demand function defined below.
variable in this economy expresses the tradable good.

However,

The vector of each17

of (8) and (19).

We will therefore represent this economy by equations (2’), (4’),(5), (7) (8) and (19).

Definition 3.

The economy satisfying Assumptions 1, 2’, 3, 4’, 5-9 and 10 is called

the N-Economy.

When equation (2’), (4’), (5), (7), (8) and (19) are all satisfied, we

say that the N-Economy is in the full equilibrium.

The excess demand function in this economy is rewritten asz i (q, q n , u ) ≡ xi (q, q n , u ) − y i (q, q n ) ,(i = 0, 1, 2, n) .

(9’)

By substituting these functions for xi − yi and x − y in (7), (8) and (19), we havez n (q, q n , u ) = 0 ,(20)

q ′z (q, q n , u ) = 0 ,(21)

p ′z (q, q n , u ) + r = 0 .

(22)Note that z ′ = ( z 0 , z1 , z 2 ) is a vector of tradables, and do not contain z n .

Interm of this notation, the N-Economy is in full equilibrium if and only if itsatisfies equations (5), (20), (21) and (22).
Equations (20), (21) and (22) contain four variables q1 , q 2 , q n and u sincer is fixed by assumption. When one of the four variables are exogenously given,
the three equations determine the remaining variables.

For example, if u is

18

given, the model determines the remaining variables (q1 , q 2 , q n ) .

Therefore,

from equation (5) we can find the tariff vector t that maximizes the utility level

u.

3.2.

Optimal tariff in the N-Economy(I) : the general formula
We now define the following.

Definition 4.

The tariff combination (t1 , t 2 ) that maximizes the utility level u

in the model of (5), (20), (21) and (22) for a fixed level of r is called the optimum

tariff of the N-Economy.

The optimal tariffs in the N-Economy will be different from those in the TEconomy because of the following differences between the models.
First, the N-Economy has two untaxed goods, i.e., the exportable and thenon-tradable, while the T-Economy has only one, i.e., the exportable good.
Second, a change in a tariff rate does not affect the price of the untaxed goodin the T-Economy, while it affects the price of an untaxed good, i.e., the nontradable, in tariffs in the N-Economy.
In view of the second difference, the optimal tax formula has to contain theterm representing the impact of a tariff change upon the price of the non-tradable.
To this end, we have to express q n as a function of q1 and q 2 .

Equation (20)determines the price q n of the non-tradable good when q and u are exogenously
given.

If equation (20) satisfies the conditions of the Implicit Function Theorem,

it can be solved for q n .

The resulting function may be written as

19

q n = q n (q, u ) .

(23)

Define the elasticity of the price of non-tradable with respect to the price of i -theimportable byqˆ ni =

qi ∂q n,q n ∂qi(24)

and, the elasticity of the importable good i with respect to the price of nontradable byη in =

q n z in.
zi

Then the optimal tariff in the N-Economy can be stated in terms of this notation.

Theorem 2.

In the N-Economy, the optimal tariff rate is expressed asτ 1 = (η 20 + η 21 + η12 + η 2 n + (η1n − η 2 n )qˆ n 2 )θ ∗ ,τ 2 = (η10 + η 21 + η12 + η1n + (η 2 n − η1n )qˆ n1 )θ ∗ ,(25)

for some scalar θ > 0 .
∗Proof.

See Appendix 2

Q.E.D.

20

One difference between (13) and (25) is that (25) contains the terms (η jn − η in ) q̂ ni
representing the effect a change in a tariff rate upon the price of the untaxed good
through the change in the price of the non-tradable.
Taking the difference between the optimal tariff rates of the two goods in(25), we have

τ 1 − τ 2 = [{(η 20 + η 2 n ) − (η10 + η1n )}+ (η1n − η 2n )(qˆ n1 + qˆ n 2 )]θ * .

3.3.

(26)

Optimal tariff in the N-Economy (II) : independence between theimportables and the non-tradable
Let us first consider the special case where price changes of the importables
do not affect the prices of the non-tradables.

Thenqˆ n1 = qˆ n 2 = 0 ,(27)

τ 1 − τ 2 = ((η 20 + η 2n ) − (η10 + η1n ))θ

(28)

and hence

holds.
The expression η 20 + η 2 n in the RHS of (28) represents the percentage
increase in the excess demand for the second importable when the prices of both of
the untariffed goods (i.e., 0 and n ) is simultaneous increased by one percent.
Thus it indicates the closeness of substitutability between the second importable

21

and the compound good consisting of the two untariffed goods.
similar interpretation to η10 + η1n .

We can give a

Equation (28) implies, therefore, that when(27) holds in the N-Economy, the ranking of tariffs for the importables depends
upon their relative closeness of substitutability with the compound of the twountariffed goods in this economy.16

This is a natural extension of Proposition 1-(a), which shows that in the T-Economy, the ranking of tariffs for the importables
depends upon their relative closeness of substitutability with the untaxed good.
Equation (28) motivates the following definition.

Definition 5.

The first importable good is the closer direct-substitute of theuntaxed goods than the second importable is, if

η10 + η1n > η 20 + η 2 n ,For equation (28) to hold, however, (27) is unnecessarily strong.

qˆ n1 + qˆ n 2 = 0

16

It holds if(29)Consider an economy where an additional exportable good is added to the T-Economy

without adding the non-tradable.

If we denote the second exportable in this economy by n ,then it can be readily shown that the counterpart of the optimal tariff formula of (13)becomesτ1 = (η20 + η21 + η12 + η2 n )θ ,
τ 2 = (η10 + η21 + η12 + η1n )θ ,for some scalar θ > 0 .

The only change from (13) is that the terms η1n and η 2 n are added.

22

is satisfied.

Totally differentiating (20) in q n and q i , while keeping othervariables constant, we obtain∂q nz= − ni .
∂qi
z nn

(30)

Thus we have

q1 z n1 + q 2 z n 2 = 0(31)

if and only if (29) holds.

Definition 6.

It is said that the composite of the importable goods is independentof the non-tradable good if (31) holds.

It is needless to say that if (31) holds, a proportional increase in the prices of
the importable goods does not affect the net demand for the non-tradable good.
In terms of this definition, the following proposition holds:

Proposition 2.

If the composite of the importable goods is independent of the

non-tradable, the following holds in the N-Economy:(a)

The optimal tariff rate is lower for the importable good that is the closer

From these, (28) holds also in this economy, as the counterpart of (15).

23

direct-substitute of the untaxed goods.
(b)

The stronger is the cross-substitutability between importables, the closer isthe optimal tariff to uniformity when other elasticities are kept constant.
(c)

The optimal tariff rate is inversely proportional to the own elasticity ofexcess demand if the importables are independent of each other.

Proof.

Proposition 2-(a) follows directly from (26) and (29).

From (25) we have

τ 1 η 20 + η 2 n + η12 + η 21 + (η1n − η 2 n )qˆ n 2,=
τ 2 η10 + η1n + η12 + η 21 + (η 2 n − η1n )qˆ n1

which proves (b).

(32)

Next, consider the special case where importable goods are

independent of each other.

Then, since η12 = η 21 = 0 , η i 0 + η in + η ii + 0 holds for

i = 1, 2 . Substituting this and (29) into (32), we have

τ 1 η 22 + (η1n − η 2 n )qˆ n 2,=
τ 2 η11 + (η1n − η 2 n )qˆ n 2

which proves (c).

Q.E.D.

It is readily seen that if the independence between the composite of theimportable and the non-tradable is not assumed, Proposition 2-(a) and (c) nolonger hold.

3.4.

Optimal tariff in the N-Economy (III) : the main proposition

24

Equation (26) may be rewritten as

τ 1 − τ 2 = [(η 20 − η10 ) + (η 2 n − η1n ) + (η1n − η 2 n )(qˆ n1 + qˆ n 2 )]θ *〔the original effect〕〔direct effect〕(33)〔indirect effect〕

Comparing (16) and (33), we find that the introduction of non-tradable brings
about two new terms: the direct effect,(η1n − η 2 n )(qˆ n1 + qˆ n 2 ) .

(η 2 n − η1n ) and indirect effect,

These terms represent the difference in substitutability

between importable goods and non-tradable goods.

The problem is that the signof the sum of the two new terms is not apparently clear, since the two terms canhave opposite signs.

Indeed, we havethe direct effect

0 < (η 2 n − η1n ) andthe indirect effect

0 > (qˆ n1 + qˆ n 2 )(η1n − η 2 n ) ,if q̂ ni are positive and η 2 n > η1n .

Does the indirect effect upset the direct effect?

The answer can be derived from the following.

Theorem 3.

In the N-Economy, the optimal tariff rate is expressed asτ 1 = (η 20 + η12 + η 21 + η 2 n qˆ n 0 + η1n qˆ n 2 + η 2 n qˆ n1 )θ ∗ ,τ 2 = (η10 + η12 + η 21 + η1n qˆ n 0 + η1n qˆ n 2 + η 2 n qˆ n1 )θ ∗ ,

(34)

25

for some scalar θ > 0 .
∗Proof.

First apply Euler‘s Theorem to (20) to find

1 = qˆ n 0 + qˆ n1 + qˆ n 2 .

From (this) we have

η 2 n qˆ n 0 + η1n qˆ n 2 + η 2 n qˆ n1= η1n qˆ n 2 + η 2 n (1 − qˆ n 2 )= η 2 n + (η1n − η 2 n )qˆ n 2 .

Applying this to (25) yields the theorem.

Q.E.D.

From the proof (34) it is clear that (25) and (34) are equivalent.

By taking the difference between τ 1 and τ 2 in (34), we have

τ 1 − τ 2 = [(η 20 + η 2 n qˆ n 0 ) − (η10 + η1n qˆ n 0 )]θ ∗A comparison between (35) and (33) shows that(η 2 n − η1n ) + (η1n − η 2 n )(qˆ n1 + qˆ n 2 ) = q n 0 (η 2 n − η1n ) .

(35)

26

From this, it follows that if qˆ n 0 > 0 , the sum of the direct and indirect effect has
the same as the direct effect, and hence the indirect effect does not quite upset thesign of the direct effect.
Note that qˆ n 0 > 0 holds if the non-tradable is substitutable for theexportable good, as is clear from (24) and (30).
Equation (35) motivates the following definition

Definition 7.

The first importable good is the closer substitute of the untaxed

goods than the second importable is ifη10 + η1n qˆ n 0 > η 20 + η 2 n qˆ n 0 .

(36)Hence optimal tariff rates can be ranked in terms of Definition 7.
We are in a position to state and prove the following Proposition.

Proposition 3.
(a)

The following holds in the N-Economy:

The optimal tariff rate is lower for the importable that is the closer

substitute of the untaxed goods .
(b)

The stronger is the cross-substitutability between importables, the closer isthe optimal tariff to uniformity when other cross elasticities are kept constant.
(b’)

The stronger is the cross-substitutability between the importable and thenon-tradable, the closer is the optimal tariff to uniformity, when other crosselasticities are kept constant, provided that untaxed goods are independent ofeach other

27

(c)

The optimal tariff rate is inversely proportional to the own elasticity ofexcess demand if and only if the importables are independent of each other, i.e.,η12 = η 21 = 0 , and the importables are equally substitutable for the non-tradable,i.e., η1n = η 2 n .

Proof.

Equation (35) and Definition 7 immediately yield (a).

From equation(34), we also obtainτ 1 η 20 + η12 + η 21 + η 2 n qˆ n 0 + η 2 n qˆ n1 + η n1 q n 2.
=
τ 2 η10 + η12 + η 21 + η1n qˆ n 0 + η 2 n qˆ n1 + η n1 q n 2

Since the expressions in parentheses are common is both the numerator and thedenominator, the larger this term, the closer is the optimal tariff rates touniformity.

This proves (b) and (b’).

If we assume that the importables are independent of each other, then

η12 = η 21 = 0 hold, and hence we have η ii + η i 0 + η in = 0 .

By substituting theseequations into (25), we have

τ 1 η 22 + (η1n − η 2 n )qˆ n 2.
=
τ 2 η11 + (η 2 n − η1n )qˆ n1

Equation (37) proves (c).

(37)

Q.E.D.

Proposition 3-(a) is in line with the spirit of the Corlett and Hague Theorem.

Theranking of the optimal tariff rates of the two importable is again determined by

28

the ranking of substitutability between the two importables and the untaxed
goods(i.e., the exportable and the non-tradable).

Proposition 3-(b) and (b’) showthat strong cross substitutability between the importables and between
theuntariffed goods tends to make the import tariffs to become closer touniformity.

Proposition 3-(c) shows that if η 2 n ≠ η1n , the Inverse Elasticity Ruledoes not hold.17

17

See Hatta (1986) and (1994).

29

5.

Concluding remark
The optimal tariff problem in small open economy has been studied formodels with one untaxed good.

The present paper introduced the second

untaxed good whose domestic price changes endogenously.

It was observed thatthe celebrated Inverse Elasticity Rule is no longer valid in this situation.

Often

this rule is used as practical guide to obtain a rough estimate of the optimal tariff
rates.

In the economy with endogeneous price change, however, the rule fails togive such a guide.
On the other hand, it was observed that the Cross Substitutability Rule andthe Corlett and Hague Rule are quite robust in the new situation.

These rules

seem to give qualitative and quantitative insights into the optimal tariff rates inpractical situation.

30

Acknowledgments
We would like to thank Professors Kenzo Abe, Takashi Fukushima,
Yoshiyasu Ono, Koji Shimomura, Katsuhiko Suzuki and Makoto Tawada for theirinvaluable comments on several earlier drafts of this paper.

31

Appendix 1:
1 : uniform and proportional
proportional tariffs
In this appendix, we will prove that no tariff structure for a given
government revenue can achieve an efficient resource allocation.
A tariff stricture is called uniform, if all importable goods share an identicalad valorem tariff rate, i.e., if t 0 = 0 , and if

ti
=β,
qi

(i = 1, 2) ,(A1-1)

holds for some scalar β .
Under uniform tariff structure, domestic prices of the exportable goods areequal to world prices, while those of the importable goods are proportionallyhigher than the world prices.

Thus a uniform tariff structure generates

distortions, and the resource allocation is not efficient.
can raise a given tariff revenue.

However, a uniform tariff

Its resource allocation is, however, not efficient,because it is not proportional.
On the other hand, a tariff structure is called proportional if all tradable
goods i.e., including exportables share an identical ad valorem tariff rate, i.e., if

ti
=β,
qi

holds for some scalar β .

(i = 0, 1, 2) ,(A1-2)

Under a proportional tariff structure, domestic prices

of both exportables and importables are proportional to their world prices.

32

Subsidies are given to the export of the exportable goods at the same rate asimport tariffs.
world prices.

Thus the domestic prices of the exportables are higher than the
The proportional tariff attains an efficient resource allocation.

A proportional tariff structure, however, yields zero revenue.

Under theproportional tariff, from (A1-2) and (5), we haveq = ψpwhere ψ = 1 1 + β .

(A1-3)

Substituting (A1-3) into (7) yields

p ′z = 0 .

(A1-4)

From (A1-4), we obtain r = 0 .
This implies that the revenue collected by import tariffs is all spent on export
subsidies.

Namely, collecting a given tariff revenue necessarily generates a

distortion.

It is the revenue-constrained optimal tariff structure that minimizes

this distortion.

33

Appendix 2:
2 : the proof of Theorem 2
In this Appendix 2, we prove Theorem 2.

To this end, we must find thesolution of (q, qn ) in the following maximization problem:

max

u

s.t

zn (q, qn , u ) = 0,q, qn , u

(A2-1)

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

We can directly obtain Theorem 2 by solving this problem.

But we will take ashort cut by proving Theorem 3, the equivalent of Theorem 2, after transforming
the problem (A2-1) into a manageable form.
Substituting (23) for q n in the compensated excess demand function yields
the excess demand function which does not depend on the price of non-tradablegood:z ∗ (q, u ) = z (q, q n (q, u ), u ) ,where z∗′= ( z 0∗ (q, u ), z1∗ (q, u ), z 2∗ (q, u )) .

(A2-2)∗

We will call z i (q, u ) the reduced-

form of excess demand function, or more simply, the reduced form.
In terms of the reduced form, the market equilibrium condition (20)-(22) canbe rewritten as

34

q ′z ∗ (q, u ) = 0 ,(A2-3)

p ′z ∗ (q, u ) + r = 0 .

(A2-4)

These two equations contain three variables, q1 , q 2 and u .

The market

equilibrium conditions (A2-3) and (A2-4) in terms of the reduced form areequivalent to those in terms of (20), (21) and (22).
Thus (A2-1) is transformed into

max

u

s.t

q ′z ∗ (q, u) = 0,q, u

(A2-5)

p ′z ∗ (q, u ) + r = 0.

This is formally identical to (14), and hence we immediately obtain the following
optimal tariff rates exactly in the same manner as in Theorem 1.

Lemma 1.

In the N-Economy, the optimal tariff rate is expressed as∗∗τ 1 = (η 20+ η12∗ + η 21)θ ∗ ,(A2-6)

τ 2 = (η + η + η )θ ,
∗
10

∗
12

∗
21

∗

∗ ∗
where θ ∗ = − α ∗ z1∗ z 2∗ q1 q 2 ( z11z 22 − ( z12∗ ) ) and η ij is the elasticity of the i -th2

reduced with respect to the price q j .

∗35

Let us now decompose η ij into terms involving η ij and η in .
∗By partiallydifferentiating (A2-2) with respect to q j , we have∂z i∗ ∂z i ∂z i ∂q n.
=+⋅
∂q j ∂q j ∂q n ∂q j

(A2-7)

The first term in the RHS represents the effect of the price change of j -th goodupon the excess demand of i -th good.

We will call this the direct effect.

The

second term represents the effect of the price change of j -th good upon i -th good
through the price change of the non-tradable.

We will call this the indirect effect.

Thus, the LHS of (A2-7) represents total effect including direct and indirect effect.
By rewriting (A2-7), we obtainη ij∗ = η ij + η in q̂ nj .

(A2-8)The term η ij and η in q̂ nj correspond to the direct and indirect effect, respectively.
Substituting equation (A2-8) into (A2-6), we have equation (34), and hence(25).

This result and the following Lemma prove Theorem 2.

Lemma 2.

18

θ ∗ > 0 . 18

We are grateful to Professor Suzuki of Kwansei Gakuin University for suggesting thisproof.

36

The term θProof.

∗

has the same sign as government revenue

 z11∗ z12∗  is negative semi-definite.
Z =z ∗ z ∗  21 22 *

p.262).

r , ifSee Daimond and Mirrlees (1971,

From equation (A2-7),z1n z n1

 z11 − znnZ∗ = 
z
z n1
2
n
z −
21

z nn

z 1n z n 2 z nn 
.
z 2n z n2 
z 22 −
z nn 
z12 −From the property of matrix, we findZ 1∗ = z11 −z1n z n1
1 z11
=
z nn
z nn z n1

z1n z n1z nn
Z 2∗ =z z
z 21 − 2 n n1
z nn
z11 −

z1n
, and
z nn

z1n z n 2
z
1 11
z nn
z 21
=
z z
z nn zz 22 − 2 n n 2
n1
z nn
z12 −

z12
z 22
zn2

z1n
z2n .
z nn

If z i (q, q n , u ) is concave with respect to (q, q n ) , Z 1∗ < 0 and Z 2∗ > 0 .
since Z is the negative semi-definite, θ is positive.
∗∗

Incidentally, inequality (36) can be equivalently written as∗η10∗ > η 20
.

Thus,

Q.E.D.

37

From (A2-8), In Definition 7, therefore, (36) can be replaced by the above
inequality.

38

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