The Japanese Economic Review
Vol. 51, No. 4, December 2000
THE OPTIMAL TARIFF FORMULA IN A TWO-PERIOD
ECONOMY
By TADASHI INOUE
University of Tsukuba, Ibaraki, Japan
The optimal tariff formula is derived for a large country trading both consumption
goods and an investment good in a two-period economy. The formula greatly simpli®es
the results of the standard one-period economy where both consumption goods and real
capital are traded with or without a non-traded good; in particular, the results do not
depend on the relative intensities of the two goods.
JEL Classi®cation Numbers: D11, F11, F34.
1.
Introduction
This paper tries to investigate the formula of the optimal tariff/subsidy in a two-period
economy. This is a generalization of the optimal tariff and tax formula for a large
country trading both consumption commodities and real capital, obtained by Jones
(1967), Kemp (1969), Chipman (1972) and Inoue and Kiyono (1988), among others.
Re¯ecting the assumption that it is not capital itself but the investment good that is
traded in our two-period economy, optimal tariff con®gurations in the two-period
economy are different to some extent from those of the standard one-period economy.
To be more precise, some of the signs of the optimal tariff formula for investment
goods are different, though the signs of the optimal tariff formula for consumption
goods are the same. Also, these results simplify the classi®cations greatly. In particular,
they do not depend on the ranking of factor intensities of the two goods. Section 2 sets
out the assumptions of the model.
2.
The model
Let there exist two countries (the home country and the foreign country), two periods
(period 1 ˆ present period, and period 2 ˆ future period), one consumption good in
each period and one investment good only in period 1. Goods are produced employing
labour and capital in both periods with neoclassical technology (constant returns to
scale and concave production functions); l, the labour endowments of the home country,
are constant over time while a, the capital endowments, increase in period 2 by v, the
amount of investment good employed in period 2. Capital depreciation is assumed away
for simpli®cation.
Let y1 be the amount of the consumption good produced in period 1 (good 1) in
the home country and p1 be its domestic price, and let z be the amount of the
investment good produced in period 1 in the home country and q be its domestic
price. Each country is assumed to lend and borrow money at the common
international interest rate r, which is the discount rate for future factor and
commodity prices. But this is not a monetary model. Money is used only to exchange
present goods for future goods.
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T. Inoue: The Optimal Tariff Formula in a Two-period Economy
Let y2 be the amount of the consumption good produced in period 2 (good 2) in
the home country and p2 be its present domestic price; i.e., p ˆ (1 ‡ r)ÿ1 p^2 where p^2
is the current domestic price of good 2. Both consumption good and investment good
are assumed to be tradable.
Let v be the amount of the investment good used to produce the consumption good
in period 2 in the home country and k ˆ v ÿ z (resp., z ÿ v) be the amount of
imports (resp., exports) of the investment good in period 1 if k . 0 (resp., if
ÿk ˆ z ÿ v . 0).
Let ð be the present-value revenue function of the home country so that:
ð ˆ ð( p1 , p2 , q, ù) ˆ maxf p1 y1 ‡ p2 y2 ÿ q(v ÿ z)j y1 ˆ f 1 (l11 , a1 ), z ˆ f 2 (l21 , a2 ),
y2 ˆ f (l, a ‡ v), l11 ‡ l21 ˆ l, a1 ‡ a2 ˆ ag
where ù ˆ (l, a) is the endowment vector in period 1. Here f 1 and f 2 are, respectively,
the production functions of the consumption good and investment good in period 1, and
f is the production function of the consumption good in period 2. All production
functions are assumed to be twice continuously differentiable. Furthermore, l11 and a1
are, respectively, the amount of labour and capital employed to produce y1 units of the
consumption good in period 1; similarly, l21 and a2 are, respectively, the amount of
labour and capital employed to produce z units of the investment good in period 1.
Let w i and ri be the present domestic prices of labour and capital in period i in the
home country, i ˆ 1, 2. (Hence w2 ˆ (1 ‡ r)ÿ1 w^2 , and r2 ˆ (1 ‡ r)ÿ1 r^2 hold where
^2 and r^2 are, respectively, the current prices of labour and capital in period 2.)
w
Here it is in order to explain the present-value revenue function ð. Just as in a oneperiod static model, from national accounting identities we observe GNE ˆ GNP
ˆ GNI, i.e., p : c ˆ ð ˆ wl ‡ ra where p : c is the inner product of p ˆ ( p1 , p2 ),
c ˆ (c1 , c2 ), w ˆ w1 ‡ w2 and r ˆ r1 ‡ r2 . In a two-period model the present-value
revenue function is the sum of the revenue of consumption goods and the net export
of the investment good. Especially in autarky, ð is the sum of the revenue of
consumption goods only.1 Here
p : cÿðˆ0
implies that trade balance is attained not in each period but over the entire time horizon.
Perfect competition is assumed to prevail in both the commodity and factor markets in
each country as well as in the world economy in each period.
The world markets of consumption goods in each period and of the investment good
in period 1, and the markets of labour and capital of each country in each period, are
all assumed to clear. Only the external balance is not assumed to clear in each period,
but rather to clear over time.
Since the investment good produced in period 1 is the same as the increase in the
capital endowment in period 2 in terms of production ef®ciency, perfect competition
in the investment good and capital markets leads to:
q ˆ r2
(1)
in equilibrium.
1
For the equality between p : c and ð in the two-period closed model, see e.g. Woodland (1982, pp.
469±470).
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Throughout the paper, the foreign country is assumed to retain free commodity
trade, so that
P : C ˆ P : Y ‡ Qk
(2)
holds where P ˆ (P1 , P2 ), C ˆ (C1 , C2 ) and Y ˆ (Y1 , Y2 ). Pi is the international
(foreign) present value of the price of consumption good in period i, i ˆ 1, 2. C i and Y i
are, respectively, the amount of the consumption and the amount of production of the
consumption good in the foreign country in period i, i ˆ 1, 2, and Q is the international
(foreign) price of the investment good. Here, P : C is the inner product of P and C.
Henceforth in general capital letters are used to express the variables and parameters of
the foreign country.
Let
e( p, u) ˆ minf p : cju(c) > ug
(3)
be the expenditure function of the home country, where p ˆ ( p1 , p2 ) is the price vector
of the consumption good, and c ˆ (c1 , c2 ) is the consumption vector of the home
country; u is the social utility function of the home country. The social utility function
is concave and strictly increasing in c in each country. Each consumption good is
assumed to be normal in each country. Let xi ˆ c i ÿ yi , i ˆ 1, 2, be the excess demand
for the consumption good in period i. xi . 0 (resp., , 0) means that the home country
imports (resp., exports) the consumption good in period i.
Throughout the paper, the consumption good in period 2 is the numeraire so that
p2 ˆ P2 ˆ 1
always holds. From this equation, the interest rate r is seen to equal (1 ÿ P1 )=P1 for
both countries. If the consumption good is imported (resp., exported), p1 ˆ (1 ‡ ô )P1
(resp., (1 ‡ ô) p1 ˆ P1 ) holds where ô (resp., ô) is the tariff on imports (resp., the
export tax). I adopt the convention that (1 ‡ ô)(1 ‡ ô ) ˆ 1 always holds so that ô , 0
means either that the export of good 1 is subsidized at a rate of ÿ100ô%, or that there
exists a duty of 100ô % on imports of good 1 where ô ˆ ÿô=(1 ‡ ô).
Similarly q ˆ (1 ‡ t )Q (resp., (1 ‡ t)q ˆ Q) holds if the investment good is the
home imported (resp., exported) good, with t (resp., t) being the import tariff (resp.,
export tax) on the investment good. I adopt the convention that (1 ‡ t)(1 ‡ t ) ˆ 1
always holds. (t , 0 means either a ÿ100t% rate of subsidy for the export of the
investment good, or a 100t % rate of tariff for its import where t ˆ ÿt=(1 ‡ t).) By
de®nition, we then observe
m ˆ p : c ˆ P : c ‡ ô P c ˆ ð( p , p , q, ù) ‡ ô P x ‡ t Qk,
(4)
1 1
1
2
1 1
where m is the national wealth which is composed of the revenue ð, the tariff revenue
of the consumption good ô P1 x1 and the tariff revenue of the investment good t Qk.
Here it is to be noted that the equilibrium conditions for good 1 and investment good
determine their prices P1 and Q.2
2
Let h i and H i be, respectively, the home and foreign demand functions for good i so that
c i ˆ h i ( p1 , p2 , q, m) and C i ˆ H i (P1 , P2 , Q, M), where M ˆ P : C ˆ Ð(P1 , P2 , Q, Ù). Then h i ( p1 ,
p2 , q, m) ‡ H i (P1 , P2 , Q, M) ˆ ð1 ( p1 , p2 , q, ù) ‡ Ð1 (P1 , P2 , Q, Ù), k ‡ K ˆ ð3 ( p1 , p2 , q, ù)
‡ Ð3 (P1 , P2 , Q, Ù) ˆ 0, equation (4) and M ˆ Ð(P1 , P2 , Q, Ù) de®ne implicitly P1 and Q.
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T. Inoue: The Optimal Tariff Formula in a Two-period Economy
3.
Optimal tariff con®gurations
I deal with a large home country which can manipulate terms of trade by tariff/subsidy policy.
In particular, I investigate the optimal tariff con®gurations for the home country,
which is large enough to change the international price P1 and the level of imports of
the investment good by manipulating the level of tariff rates ô and t on the
consumption good and investment good in period 1. (Henceforth I follow essentially
the arguments of Chipman, 1972, and Inoue and Kiyono, 1988.)
By totally differentiating the social utility function of the home country
u ˆ u(c1 , c2 ), assuming that utility is maximized subject to the budget constraint
m ˆ p : c,
I obtain
du ˆ p1 dc1 ‡ p2 dc2 ,
where the marginal utility of national wealth m is assumed to be unity without loss of
generality. By substituting c i ˆ xi ‡ yi ˆ ÿX i ‡ yi , i ˆ 1, 2, into the above equation, it
follows that
du ˆ p1 d y1 ‡ p2 d y2 ÿ p1 dX 1 ÿ p2 dX 2 :
Furthermore, since
p1 d y1 ‡ p2 d y2 ÿ qdk ˆ 0
holds for a given ù, the above is reduced to
du ˆ qdk ‡ ( p1 ÿ P1 )dx1 ÿ P1 dC1 ÿ P2 dC2 ‡ P1 dY1 ‡ P2 dY2 :
(5)
Here, by totally differentiating (2), while observing that
P1 dY1 ‡ P2 dY2 ‡ Qdk ˆ 0
holds for a given foreign factor endowment vector Ù ˆ (L, A), I obtain
P1 dC1 ‡ P2 dC2 ˆ ÿX 1 dP1 ÿ X 2 dP2 ‡ kdQ:
(6)
Substituting this into (5) and noting that dP2 ˆ 0, I observe
du ˆ [(q ÿ Q)dk ÿ ( p1 ÿ P1 )dX 1 ] ‡ (X 1 dP1 ÿ kdQ):
(7)
Equation (7) is Jones's (1967) equation (6) and Inoue and Kiyono's (1988) equation (15).
Next, I note that
(8)
X ˆ C (P , P , u ) ÿ Y (P , Q)
1
1
1
2
1
1
holds, where C1 ˆ C1 (P1 , P2 , u ) is the compensated demand function for the
consumption good in period 1 in the foreign country, u ˆ U (C1 , C2 ) being the
foreign social utility function, with du ˆ P1 dC1 ‡ P2 dC2 , Y1 ˆ Y1 (P1 , Q) being the
foreign supply function of that good, and
k ˆ Z(P1 , Q) ÿ V (P2 , Q)
(9)
where Z ˆ Z(P1 , Q) is the foreign supply function of the investment good and V is its
demand function (which is de®ned indirectly by the foreign counterpart of (1) and
P2 @ f (L, A ‡ V )=@(A ‡ V ) ˆ Q). I therefore observe that (9) implicitly de®nes Q
to be a function of P1 and k, and hence by (8) so does X 1 with X 11 ˆ
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@ X 1 =@ P1 ˆ S11 ÿ (Y11 ‡ Y12 Q1 ) ÿ H 1 M (X 1 ÿ kQ1 ), where S11 , 0 is the Slutsky
substitution term (i.e., S11 ˆ @C1 (P1 , P2 , u )=@ P1 ), Y11 ˆ @Y1 =@ P1 . 0, Y12 ˆ
@Y1 =@Q , 0, Q1 ˆ @Q=@ P1 ˆ Z 1 =(V2 ÿ Z 2 ) . 0 and H 1 M ˆ @ H 1 (P1 , P2 , M)=@ M . 0,
Q k ˆ @Q=@ k ˆ ÿ1=(V2 ÿ Z 2 ) ˆ ÿQ1 = Z 1 . 0,
X 1 k ˆ @ X 1 =@ k ˆ ( H 1 M k ÿ Z 1 )Q k ,
Z 1 ˆ @ Z=@ P1 , 0, V2 ˆ @V =@Q , 0 and Z 2 ˆ @ Z=@Q . 0. (For the derivation of
these equations, see Appendix A.) Hence (7) is rewritten as
du ˆ (@u=@ P1 )dP1 ‡ (@u=@ k)dk,
(10)
with @u=@ P1 ˆ X 1 ÿ ( p1 ÿ P1 )X 11 ÿ kQ1 , and @u=@ k ˆ ÿ[(Q ÿ q) ‡ ( p1 ÿ P1 )X 1 K ‡
kQ K ], and hence
(11)
@u=@ P ˆ X [1 ÿ ( p ÿ P )ç =P ‡ ì ã ],
1
1
1
1
1
1
1
where ç1 ˆ X 11 P1 =X 1 , ì ˆ ÿQk=PX 1 and ã1 ˆ Q1 P1 =Q and
"
#
( p1 ÿ P1 )ã1
p
1
‡ ä
m ‡ m2 ,
@u=@ k ˆ q ÿ Q 1 ‡
P1
P1 1
(12)
where ä ˆ kQ K =Q and mi ˆ Pi H iM (the foreign marginal propensity to consume the
consumption good in period i.) Equation (10) is a generalization of Jones's (1967)
equation (9), Chipman's (1972) equations (5.5), (5.6) and (5.7), and Inoue and Kiyono's
(1988) equation (18). Equation (11) is the same as Inoue and Kiyono's (1988) equation
(19-1), while (12) is different from their equation (20-1).
Denote å1 ˆ ì ã1 ˆ ÿ(k=X 1 )Q1 , åk ˆ ÿ Z 1 P1 =k. Then ã1 ˆ åk ä holds so that
å1 ˆ ì åk ä follows. By letting @u=@ P1 ˆ 0 and @u=@ k ˆ 0, I obtain the optimal
taxes ô for the consumption good and t for the investment good when the home
country exports both goods in period 1:
(13)
ô ˆ ÿ(1 ‡ å )=(ç ‡ 1 ‡ å )
1
1
1
and
ÿ[ç ‡ (m1 ‡ åk )(1 ‡ å1 )]ä
:
tˆ 1 ç1 ‡ [ç1 ‡ (m1 ‡ åk )(1 ‡ å1 )]ä
(14)
When the home country imports the consumption good in period 1, the optimal import
tariff ô is:
(15)
ô ˆ ÿô=(ô ‡ 1) ˆ (1 ‡ å1 )=ç1 ,
and similarly, the optimal import tariff for the investment good t is:
(16)
t ˆ ÿt=(1 ‡ t) ˆ [ç1 ‡ (m1 ‡ åk )(1 ‡ å1 )]ä =ç1 :
The optimal tariff formulae ô and ô for the consumption good are the same as the
standard case, while those for the investment good, t and t , are different.3 By letting
@u=@ P1 ˆ 0, I obtain
3
In the standard case (i.e. the one-period economy), when the home country imports (resp., exports)
capital, with ^t (resp., ^t) being the tax on rental price of capital, Q ˆ (1 ÿ ^t )q (resp., q ˆ (1 ÿ ^t )Q)
holds so that t (resp., t) in my notation equals ^t =(1 ÿ ^t ) (resp., ^t=(1 ÿ ^t )). However, t and t in
(14) and (16) are different from ^t=(1 ÿ ^t ) and ^t =(1 ÿ ^t ). The latter in the standard case equal,
respectively, ÿã1 (1 ‡ å1 )=[ç1 ‡ ã1 (1 ‡ å1 )] and [ã1 (1 ‡ å1 )]=ç in my notation. (For this, see e.g.
Inoue and Kiyono's (1988) equations (22-1), (23-1) and (24-1).)
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T. Inoue: The Optimal Tariff Formula in a Two-period Economy
p1 X 11 =X 1 ˆ 1 ‡ ç1 ‡ å1 :
(17)
Furthermore, from the de®nition of X11 and (17), I see that
p1 m1 ‡ m2 , 0
X 11 ˆ [S11 ÿ (Y11 ‡ Y12 )Q1 ]=
P1
(18)
follows.4 Hence I obtain
X 1 . 0 , ç1 , 0:
Furthermore, by de®nition, I observe that
㠈 å ä . 0,
1
k
åk . 0 , k . 0,
(19)
ì . 0 , kX 1 , 0 , å1 . 0,
(20)
ç1 , 0 , X 1 : X 11 , 0 , 1 ‡ ç1 ‡ å1 , 0 , X 1 . 0:
(21)
and
Now I obtain the following con®gurations of the optimal tariff and tax formula for both
the consumption good and the investment good in period 1.
Theorem 1
Case 1: X 1 . 0 (The home country exports
(a)
å1 . 0
(b)
ÿ1 , å1 , 0
(i) 0 , ÿç1 =(1 ‡ å1 ) ÿ m1 , åk
(ii) åk , ÿç1 =(1 ‡ å1 ) ÿ m1
(c)
å , ÿ1
the consumption good in period 1)
Case 2: X 1 , 0 (The home country imports
(a)
å1 . 0
(b)
ÿ1 , å1 , 0
(i) ÿç1 =(1 ‡ å1 ) ÿ m1 , åk
(ii) åk , ÿç1 =(1 ‡ å1 ) ÿ m1 , 0
(c)
å , ÿ1
the consumption good in period 1)
k . 0, ô . 0, t . 0
k , 0, ô . 0
1
1
k , 0, ô . 0, t . 0
k . 0, ô . 0
t . 0
t , 0
k . 0, ô , 0, t . 0
t,0
t.0
k , 0, ô , 0, t . 0
(For the derivation of Cases 1 and 2, see Appendix B.)
4
Equation (18) follows from X 11 ˆ S11 ÿ (Y11 ‡ Y12 Q1 ) ÿ H 1 M X 1 (1 ‡ å1 ), equation (15) and
Y11 ‡ Y12 Q1 . 0. The last inequality is obtained from Q ˆ Q1 P1 ‡ Q2 P2 , as explained below. Here,
the homogeneity of degree 1 of Q in (P1 , P2 ) follows from the homogeneity of degree 1 of the revenue
function Ð(P1 , P2 , Q, Ù) and the homogeneity of degree 0 of its derivative Ð Q ˆ
Z(P1 , Q) ÿ V (P2 , Q) ˆ k in (P1 , P2 , Q). The latter equation implies that Q1 ˆ Z 1 =(V2 ÿ Z 2 ) . 0
and Q2 ˆ @Q=@ P2 ˆ ÿV1 =(V2 ÿ Z 2 ) . 0 and hence both Q1 and Q2 are positive. Noting Y12 , 0, it
follows that Y11 ‡ Y12 Q1 . Y11 ‡ Y12 Q=P1 ‡ (Y11 P1 ‡ Y12 Q)=P1 ˆ 0. (Let Ð(P1 , Q, Ù) ˆ maxfP1 Y1 ‡
QZjY1 ‡ F1 (L11 , A), Z ˆ F2 (L21 , A2 ), L11 ‡ L21 < L, A1 ‡ A2 < Ag. Then Ð1 ˆ Y1 is homogeneous
of degree 0 in (P1 , Q), which implies Y11 P1 ‡ Y12 Q ˆ 0, Y11 . 0 and Y12 , 0:)
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These results coincide with the standard case (see e.g. Chipman, 1972, and Inoue
and Kiyono, 1988), except for (i) and (ii) of (b) where ÿ1 , å1 , 0 for both X 1 . 0
and X 1 , 0. In that case, for X 1 . 0, t . 0 for (b)(i) and t , 0 for (b)(ii), and for
X 1 , 0, t , 0 for (b)(i) and t . 0 for (b)(ii), while in the standard case signs of t and
t are opposite. (The signs of ô and ô are always the same.)
However, as is seen from (20), since the signs of k and å1 are the opposite (resp.,
the same) for X 1 . 0 (resp., X 1 , 0), the above results greatly simplify the
classi®cation of the optimal tariff combinations in comparison with the standard
case, where, for three cases of the values of å1 , the sign of k can be positive as well
as negative depending on the ranking of the factor intensities.
4.
Concluding remarks
Of course, the results obtained so far have some limitations; for example, the number of
commodities, factors and periods should be more generalized. I do not have to assume
that the foreign country retains free trade: it can also adopt tariff policies to maximize
its utility. These generalizations should be followed up as a next step.
Appendix A
It is easy to obtain from equation (9) that
Q1 ˆ Z 1 =(V2 ÿ Z 2 ) . 0
with Z 1 ˆ @ Z=@ P1 , 0, Z 2 ˆ @ Z=@Q . 0, V2 ˆ @V =@Q , 0 and Q k ˆ ÿ1=(V2 ÿ Z 2 )
ˆ ÿQ1 = Z 1 . 0. Noting from (6) that
du ˆ P1 dC1 ‡ P2 dC2 ˆ ÿX 1 dP1 ‡ kdQ,
(A1)
then, by partially differentiating (8), it follows that
X 11 ˆ S11 ‡ C1u uP1 ÿ Y11 ÿ Y12 Q1 ,
where C1u ˆ @C1 =@u and uP1 ˆ @u =@ P1 . Here, from the two identities
E(P1 , P2 , u ) ˆ P1 C1 (P1 , P2 , u ) ‡ P2 C2 (P1 , P2 , u ) and H i (P1 , P2 , E(P1 , P2 , u ))
ˆ C i (P1 , P2 , u), i ˆ 1, 2, while utilizing (A1), I observe that E u ˆ @ E=@u ˆ 1
and H iM ˆ @ H i =@ M ˆ C iu, i ˆ 1, 2, uP1 ˆ ÿX 1 ‡ kQ1 from (A1), and hence
X 11 ˆ S11 ÿ (Y11 ‡ Y12 Q1 ) ÿ H 1 M (X 1 ÿ kQ1 ). Similarly, by partially differentiating
(8) with respect to k, we obtain:
X 1 k ˆ @ X 1 =@ k ˆ C1u uk ÿ Y12 Q k ˆ ( H 1 M k ÿ Z 1 )Q k
(A2)
where uk ˆ @u =@ k ˆ kQ k from (A1), and Y12 ˆ @Y1 =@Q ˆ @ 2 Ð(P1 , P2 , Q)=
@ P1 @Q ˆ @ 2 Ð(P1 , P2 , Q)=@Q@ P1 ˆ @( Z ÿ V )=@ P1 ˆ @ Z=@ P1 ˆ Z 1 , 0 from the
property of the revenue function Ð.
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T. Inoue: The Optimal Tariff Formula in a Two-period Economy
Appendix B
For Case 1 (X 1 . 0), ç1 , 0 always holds from (18). Furthermore, from (20) I obtain
å1 . 0 , k , 0. Hence it follows that
(a) å1 . 0,
k ,0
k .0
(b) ÿ1 , å1 , 0,
k . 0:
(c) å , ÿ1,
1
And similarly, for Case
(a) å1 . 0,
(b) ÿ1 , å1 , 0,
(c) å , ÿ1,
1
2 (X 1 , 0),
k .0
k ,0
k , 0:
The signs of ô and ô follow from (13), (15) and (21). As for the signs of t and t , I
observe from (12) with @u=@ k ˆ 0 and (A2) that
p1 m ‡ m2 k ÿ ( p1 ÿ P1 ) Z 1 Q k :
Qÿqˆÿ
P1 1
Noting that 0 , m , 1, i ˆ 1, 2, Z , 0 and Q . 0, I observe that
1
i
k
k.0
and
p1 . P1 ) Q , q
(i:e:, t . 0)
k ,0
and
p1 , P1 ) Q . q
(i:e:, t . 0):
and
In other words, it follows that
k.0
and
ô , 0 (resp:, ô . 0) in X 1 . 0 (resp:, X 1 , 0) ) t . 0,
(A3)
ô . 0 (resp:, ô , 0) in X 1 . 0 (resp:, X 1 , 0) ) t . 0:
(A4)
and
k ,0
and
The signs of t and t for (a) and (c) for Cases 1 and 2 are obtained from (A3) and
(A4). For (b), ÿ1 , å1 , 0 of Cases 1 and 2, I observe that
t . 0 , t , 0 , å . ÿç =(1 ‡ å ) ÿ m
k
1
1
1
from (15), noting 1 ‡ t . 0 by assumption.
Here it is to be noted that ÿç1 =(1 ‡ å1 ) ÿ m1 . 0 (resp., , 0) for Case 1 (resp. 2)
hold always.
Final version accepted 10 December 1998.
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Inoue, T. and K. Kiyono (1988) ``Optimal Restriction on Foreign Trade and Investment with a Nontraded
Good'', Economic Studies Quarterly, Vol. 39, pp. 246±257.
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# Japanese Economic Association 2000.
The Japanese Economic Review
Jones, R. W. (1967) ``International Capital Movements and the Theory of Tariff and Trade: Comment'',
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Kemp, M. C. (1969) The Pure Theory of International Trade and Investment, Englewood Cliffs, NJ:
Prentice-Hall.
Woodland, A. D. (1982) International Trade and Resource Allocation, Amsterdam: North-Holland.
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# Japanese Economic Association 2000.