- Email: [email protected]
Durable Consumption Goods and the Pattern of International Trade and Indebtedness in a Dynamic Ricardian Model
Copyright © IFAC Modelling and Control of National and Regional Economies. Queensland. Australia. 1995
DURABLE CONSUMPTION GOODS AND THE PATTERN OF INTERNATIONAL TRADE AND INDEBTEDNESS IN A DYNAMIC RICARDIAN MODEL Koji Shimomura Research Institute for Economics and Business Administration, Kobe University Rokko, Nada, Kobe, Japan, 657
Abstract: Constructing a two-country and two-commodity dynamic general equilibrium model a la Ricardo in which tradeable goods consist of durable and nondurable consumption goods, that the pattern of trade depends on comparative cost advantage and the pattern of international indebtedness depend on absolute cost advantage.
we show
Keywords: duable consumption goods, the pattern of trade, international indebtedness, endogeneous time preference, absolute cost advantage, comparative cost advantage
1. INTRODUCTION
national lending and borrowing issues and trade pattern issues in a commondynamic framwork. That is, we can consider under what conditions a country is an international creditor as well as an exporter{resp. importer) of durable(resp. nondurable) goods , just like Japan during the last two decades. Shimomura(1993b) derive those conditions in a dynamic version of the two-country and two-commodity Heckscher-0hlin model in which the two commodities consist of a durable and a nondurable goods.
A considerable amount of commodities traded among developed countries are consumer durbles. For example, Japan is"a major exporter of automobiles and electric appliances. In view of this fact, it is surprising that there is almost no literature in trade theory which explicitly takes consumer durables into account. Dynamic trade models often assume duarability in production but never considers that in consumption. Thus, it may be worth constructing a dynamic trade model in which tradeable goods consist of durable and nondurable consumption goods and examining how the pattern of trade may be determined. Shimomura(1993a) formulated a dynamic version of the two-country and two-commodity Heckscher-Ohlin model under the above assumption and proved an intutively obvious proposition that, other things being equal, the country with higher (resp. lower) discount rate exports the durable (nondurable) consumption good.
The present paper takes up the same issue in another fundamental framework in trade theory, viz .• David Ricardo's trade model in Chapter 7 of his Principles(Ricardo(1817)). The main result in the subsequent argument is in the following: In the long run the pattern of trade is determined by comparative cost advantage, while the pattern of international indebtedness is determined by absolute cost advantage. The present paper is organized as follows. Section 2 presents a dynamic version of Section 3 characterizes Ricardo's model. the steady state of the dynamic model. Section 4 proves the main result and Section 5
There is an interesting aspect in dynamic trade models involving consumer durables. By assuming that consumerdurables can be internationally rented, we can argue inter133
gives a concluding re.ark.
F(p)
2. THE KODa
Consider a trading world which consists of two countries and asSUIIe that there are two tradeable consumption goods. We shall call the countries Japan and US and the goods car and rice. It is assumed~hat car is durable while rice is nondurable. The service flow of car, B, is assumed to be proportinate to the stock:One unit of car brings one unit of "service" to its userper unit of time. Let c be the rate of consumption of rice and let the instantaneous utility function of the representative household be u(c,B), which is assu.ad to be increasing, strictly quasiconcave and homogeneous of degree one in c and B.
Fig. 1. The Graph of GNP Function aA,x=r,c:the labour coefficient of rice(r) and car(c) sub. to
K= F(p) + R(M-B) y = - v(u(c,B»y
- hM - pc
M(O) = Ko given, y(O) = 1 [P) where p is the price of rice, F(p) the GNP function of Japan, Mthe stock of car owned by the Japanese household, R the rental rate and h the rate of depreciation which is assumed to be constant. Following Recardo, we assume that labour is a single factor of production and that production is carried out according to fixed technical coefficients. Then, the graph of F(p) can be depicted as in Figure 1. That K-B being positive (resP. negatve) means that Japan is a creditor (resp. debtor). Note that car serves as numeraire.
The representative household maXlmlzes the Uzawa-type of intertemporal welfare functional: 00
U = J u(c(t) ,B(t»y(t)dt
o
where t
yet) - exp[- J v(u(c(s),B(s»)ds]
o
Let r=R-h. Using this, the first dynamic equation of [P] can be rewritten as
and the time discount rate v(u) satisfies the restrictions originally imposed by Uzawa
(l968)l oo>v(O»O, v(u»O, v' (u»O, v· (u»O, v(u)-uv' (u»O
K = F(p) + rK - (r+h)B - pc (1)
Then, associated with [P) is the Hamiltonian H = Bg(x)y + A. [F(p) + rM - (r+h+px)B] - v (Bg (x) ) y,
We normalize so that the number of households is the same between two countries and is equal to be unity. The dynamic behaviour of the household in Japan is described by the optimization problem
and, resorting to the Maximum Principle(See, for example, Leonard and Long(1992», we obtain the dynamic system
max U
( c (t) }
[1 - v' (Bg(x»]g(x) = (px+r+h)q
(3a)
v' (Bg(x) )Jg' (x) = pq
(3b)
[l -
IThe reason for assuming endogenous time preferences is that by doing so we can single out the steady state. I t is wellknown that if the rate of time prefernce is constant then there are infinite steady states and it depends on the initial conditions of state variables to which steady state the dynamic system converges. See Epstein(1987, Pp. 70-71).
q = q[v(Bg(x» ~ = -Bg (x) +
134
- r] If> v (Bg (x) )
(3c) (3d)
X
K= F(p)
+ rH - (px+r+h)B
~
~
[1 -
xv x' (BXgX(x"»]g"(x") = (pxx+r+h)qX
~X
= -B"'gX(x") + ~Xv"'(B"'gX(x"'»
M" = FX(p) + rH'" - (px"'+r+h) B"
(8)
Second, cons idering (3c) and (4c) , ,we have r
= v(Bg(x» = v(B"'.g(x X»
(9)
in the steady state. Since v(.) is increasing in u, (8) and (9) together imply that B and BX are uniquely determined and
(4a)
(IO)
xv x' (B"'gX (XX» ]g"" (XX) = pq'"
XX
(3e)
where x=c/B, g(x)=u(x, 1) and q= ')./y. Bymaking a parallel calculation, we obtain the dynamic system concerning the household in US. [1 -
=
B = B'"
(4b) Third, suppose for the time being that the steady-state price is within two relative labour costs of two countries, which implies that both countries completely specializes to the production of either good. Since it is assumed in Ricardo's model that labour is a single factor of production, internationally immobile and inelastically supplied, world production of car and rice, say Yc and Yr , are positive constant 2 • Thus, combin-in g (3e) and (4e) together and consideringth e m. c. c. 's, (5) and (6), we have, in thest eady state,
(4d) (4e)
where the asterisk (*) indicates the variables and functions belonging to US. In what follws we assume that there is no difference in preference between US and Japan. That is, v(.)=v x (.) and g(.)=g"(.).
o = pVc + Yr + reM + MX - B -B") Next, let us consider market-clearing conditions (m.c.c.). First, we havethe m.c.c. for rice. F' (p)
+ F (p) X
=
'
- (px+h)B - (px"+h)B X
c + cX
= Bx + B"x X
=
Yc
-
h(B + BX)
.=
Yc
-
heM
+ MX)
(ll)
(5)
Therefore, Second, the m.c.c. for renatal-car services, which should be H + If"
=B+ B
X
B = BX = Yc/2h
(12)
and M+ MX is also uniquely determined. Moreover, from (5) and (8)
(6)
(3)-(6) constitute the dynamic Ricardian model in this paper.
x
= XX = hYr/Yc
(13)
Sustituting (12) and (13) to (9), the steady state r can be uniquely derived, viz. ,
3. THE EXISTENCE,UNIQUENESS AND SADDLEPOINT-STABILITY OF THE STEADY STATE
(14)
Let us characterize the steady state of the dynamic Ricardian model (3)-(6). First, from (3a) , (3b) , (4a) and (4b) , we see that (r+h)/p = z(x) = z(x*)
Therefore, p is also derived from (7) p = (r+h)/z(x) (u(g(hYr/Yc)Yc/2h)+h)/z(hYr/Yc)
(7)
(15)
where z(x) = (g(x)-xg' (x»/g' (x). Considering the assumptions with regard to the function u(.), one may verify that z' (x»O for any x) 0, that z(oo)=oo and that z(O)=O. It follows that x and XX are uniquely determined for any positive (r+h)/p and that we always have
Now, we arrive at the proposition. (2) Since we assume that there is one household in each country, Ye (resp. Yr ) xis either 1/ac or l/a c x (resp. l/a r or l/a r ). 135
-v' v /2v· .p ,
PROPOSITION 1: Denote the relative labour costs of Japan and US by ar/a c and arx/acx respectively, where ax, x=c,r, is the unit cost of production of car and rice in teras of labour. The steady state of the dynaaic" Ricardian lOdel uniquely exists if
v'v/2v· .p ,
o
y - v
£
0 0
0 0
v' v/2q, y - (r+h)
= y(y-v) (r-(r+h)} [y(y-v) - (I-uv /v)vv Iv 4>]
=0
ain [ar/a c , arX/aCX] <: (U(g(hYr/Yc)Yc/2h)+h}/z(hYr/Yc) <: .ax [ar/a c , arX/aCE] (16)
where
o
Next let us check saddlepoint-stability of the steady state of the dynamic Ricardian model. Adding (3e) to (4e) and noting the m. c.c.'s (5) and (6), we have
PROPOSITION 2:The steady state of the Ricardian dynamic model is locally saddlepointstable. o 4. MAIN RESULTS
(17)
Proposition 1 already includes the statement concerning the pattern of international trade. Thus in this section we shall concentrate on international indebtedness. In the steady state we have, from (3e) and (4e) ,
which is clearly stable by itself. Since x = xx, (5) and (6) together imply that x = XX = Yr/(M+M X) also asymptotically converges to (13) by itself. Therefore, considering that the number of backward variables is two (M and MX), it suffices for the saddlepointstability to show that the Jacobian matrix concerning the following system has just one negative and three positive eigen values.
F(p) + r(M-B) = pxB + hB
As we have already shown, we always have x = xE and in the steady state B = BX. Since M-B = -(MZ-B"), it is clear that in the steady state international indebtedness, i.e., M-B ~ 0 ~ M"-Bx is completely determined by F(i» ; FX(p), where p is the steady state price of rice, viz., (15).
[1 - .p v' (Bg (x) ) ][g (x) -ig' (x))
q(r+h)
=
(1 -
.p xv' (B g (x» ][g (x) -ig' (x)] = qX (r+h)
B + BX
X
=
M
q = q[v(Bg(x»
- rJ
qX = qX(v(BXg(x»
PROPOSITION 3: Japan is a creditor in the long run if its GNP is smaller than that of US.
- rJ
~ = -Bg(x) + .pv(Bg(x» ~
x = -BXg(x) +
4>
== g(x)-ig' (x) and it is assumed
without loss that g(x)=I. Since (1) ensures us that (I-uv' /v)vv' Iv·.p is posi tive, the characteristic equation e(y) = 0 has one negative and four positive real roots, as was to be proved. Therefore we now have the stability proposition.
In the steady-state Japan exports car if ar/a c > a r x/ac".
(H+H X) = pY r + Yc - p(xB+xxB X) - h(B+B") = Yc - h(M+M"),
£
o
Xv(B"g(x»
Suppose that US has absolute cost advantage in both goods and comparative cost advantage in the production of rice, i. e. , ar/a e is greater than ar"/a e x • Then F(p) and F"(p) can be depicted as in Figure 2. In that case Japan is the exporter of car and a creditor country in the long run.
where M and x are positive constants. Transforming (18) to another system of differential equations whose variables are B. B", f.L, f.L x and r. we can derive the following characteristic equation which is evaluated at the steady state. (Calculations are available from the author on request.) y 0, v'v/2v· .p o y, -v'v/2v· .p e(y)== q(r+h)/ £, 0 Y - v o q(r+h)/ £, 0 o 0 E v'v/2q
PROPOSITION 4:In the dynamic Ricardian model comparative cost advantage determines the pattern of trade and absolute cost advantage determines the pattern of interna-tional 136
F(p) F(p)
1/ac "1.1-_ _ _'" I
I; :
~ 1/acl--~~~-~~ ,,'
/
,,'
;,,;.,.,,/' I
... " ,
J,,' I.:
~----~---------- p
Fig. 3. ar"l. lac x
Fig. 2. ar"l./ac"l. ( ar/ac ar"l. ( a r and ac"l. ( lac
ar
indebteness.
(
"I.
(
ar/a c
a r and a c
(
ac x
REFERENCES
o Epstein,L.G. (1987). A simple dynamic general equilibrium model, Journal of Economic Theory 41, pp.68-95. Leonard,D. and N. V.Long(1992). Optimal Control Theory and Static Optimization in Economics, Cambridge University Press. Ricardo,D. (1817). On the Principles of Political Economy and Taxation, London,J. Murray. Reprinted as Vol.1 of The Works and Correspondences of David Ricardo (ed. by P.Sraffa), Cambridge University Press for the Royal Economic Society, 1951. Shimomura, K. (1993a). Durable consumption goods and the pattern of international trade, in H. Herberg and N. V.Long eds., Trade, Welfare, and Policies:Essays in Honor of Murray C. Kemp, Michigan Universi ty press. Shimomura,K. (1993b). Durable consumption goods, endogenous discount rate and the pattern of inteernational trade and indebtedness, mimeo. Uzawa,H. (1968). Time preference, the consumPtion function, and the optimal assets holdings, in J.N.Wolfe ed., Capital and Growth:Papers in Honor of Sir John Hicks AIdine, Chicago.
When absolute cost advantage differs between two commodities, the pattern of international indebtedness depends on the shapes of u(.) and g(.) as well. See Figure 3 in which we assume that ar/a c :> ar"l./a c"', a r :> a rx but ac <: acx. Then, the pattern of international indebtedness is determined by whether the steady state price p is greater or smaller than acx/a r . Considering (15), we have the following proposition. PROPOSITION 5: If ar/a c :> arx/a cx , a r :> a r '" but ac <: acx, i.e., if US has absolute cost advantage only in rice, then Japan is a creditor country if and only if (u(g(hacl arX)/2hac)+h}/z(hac/arX) is greater than a r '" lac.
o
5. A CONCLUDING REMARK There may be several theoretical issues which we can analyzes by making use of the dynamic model in the present paper. One of them, ' which is one of our next research agenda, is to seek an optimal dynamic tariff reform. Suppose that the government of Japan initially imposes heavy tariff on imported rice and that it faces the problem that it has to reduces the rate of tariff to zero within a certain period. Then, if the objective of the government is the welfare of Japanese household then what is the optimal schedule of tariff reform like? This is the problem which Japan in the real world has been facing since the end of GATT negociations on December 15, 1993.
137
DURABLE CONSUMPTION GOODS AND THE PATTERN OF INTERNATIONAL TRADE AND INDEBTEDNESS IN A DYNAMIC RICARDIAN MODEL Koji Shimomura Research Institute for Economics and Business Administration, Kobe University Rokko, Nada, Kobe, Japan, 657
Abstract: Constructing a two-country and two-commodity dynamic general equilibrium model a la Ricardo in which tradeable goods consist of durable and nondurable consumption goods, that the pattern of trade depends on comparative cost advantage and the pattern of international indebtedness depend on absolute cost advantage.
we show
Keywords: duable consumption goods, the pattern of trade, international indebtedness, endogeneous time preference, absolute cost advantage, comparative cost advantage
1. INTRODUCTION
national lending and borrowing issues and trade pattern issues in a commondynamic framwork. That is, we can consider under what conditions a country is an international creditor as well as an exporter{resp. importer) of durable(resp. nondurable) goods , just like Japan during the last two decades. Shimomura(1993b) derive those conditions in a dynamic version of the two-country and two-commodity Heckscher-0hlin model in which the two commodities consist of a durable and a nondurable goods.
A considerable amount of commodities traded among developed countries are consumer durbles. For example, Japan is"a major exporter of automobiles and electric appliances. In view of this fact, it is surprising that there is almost no literature in trade theory which explicitly takes consumer durables into account. Dynamic trade models often assume duarability in production but never considers that in consumption. Thus, it may be worth constructing a dynamic trade model in which tradeable goods consist of durable and nondurable consumption goods and examining how the pattern of trade may be determined. Shimomura(1993a) formulated a dynamic version of the two-country and two-commodity Heckscher-Ohlin model under the above assumption and proved an intutively obvious proposition that, other things being equal, the country with higher (resp. lower) discount rate exports the durable (nondurable) consumption good.
The present paper takes up the same issue in another fundamental framework in trade theory, viz .• David Ricardo's trade model in Chapter 7 of his Principles(Ricardo(1817)). The main result in the subsequent argument is in the following: In the long run the pattern of trade is determined by comparative cost advantage, while the pattern of international indebtedness is determined by absolute cost advantage. The present paper is organized as follows. Section 2 presents a dynamic version of Section 3 characterizes Ricardo's model. the steady state of the dynamic model. Section 4 proves the main result and Section 5
There is an interesting aspect in dynamic trade models involving consumer durables. By assuming that consumerdurables can be internationally rented, we can argue inter133
gives a concluding re.ark.
F(p)
2. THE KODa
Consider a trading world which consists of two countries and asSUIIe that there are two tradeable consumption goods. We shall call the countries Japan and US and the goods car and rice. It is assumed~hat car is durable while rice is nondurable. The service flow of car, B, is assumed to be proportinate to the stock:One unit of car brings one unit of "service" to its userper unit of time. Let c be the rate of consumption of rice and let the instantaneous utility function of the representative household be u(c,B), which is assu.ad to be increasing, strictly quasiconcave and homogeneous of degree one in c and B.
Fig. 1. The Graph of GNP Function aA,x=r,c:the labour coefficient of rice(r) and car(c) sub. to
K= F(p) + R(M-B) y = - v(u(c,B»y
- hM - pc
M(O) = Ko given, y(O) = 1 [P) where p is the price of rice, F(p) the GNP function of Japan, Mthe stock of car owned by the Japanese household, R the rental rate and h the rate of depreciation which is assumed to be constant. Following Recardo, we assume that labour is a single factor of production and that production is carried out according to fixed technical coefficients. Then, the graph of F(p) can be depicted as in Figure 1. That K-B being positive (resP. negatve) means that Japan is a creditor (resp. debtor). Note that car serves as numeraire.
The representative household maXlmlzes the Uzawa-type of intertemporal welfare functional: 00
U = J u(c(t) ,B(t»y(t)dt
o
where t
yet) - exp[- J v(u(c(s),B(s»)ds]
o
Let r=R-h. Using this, the first dynamic equation of [P] can be rewritten as
and the time discount rate v(u) satisfies the restrictions originally imposed by Uzawa
(l968)l oo>v(O»O, v(u»O, v' (u»O, v· (u»O, v(u)-uv' (u»O
K = F(p) + rK - (r+h)B - pc (1)
Then, associated with [P) is the Hamiltonian H = Bg(x)y + A. [F(p) + rM - (r+h+px)B] - v (Bg (x) ) y,
We normalize so that the number of households is the same between two countries and is equal to be unity. The dynamic behaviour of the household in Japan is described by the optimization problem
and, resorting to the Maximum Principle(See, for example, Leonard and Long(1992», we obtain the dynamic system
max U
( c (t) }
[1 - v' (Bg(x»]g(x) = (px+r+h)q
(3a)
v' (Bg(x) )Jg' (x) = pq
(3b)
[l -
IThe reason for assuming endogenous time preferences is that by doing so we can single out the steady state. I t is wellknown that if the rate of time prefernce is constant then there are infinite steady states and it depends on the initial conditions of state variables to which steady state the dynamic system converges. See Epstein(1987, Pp. 70-71).
q = q[v(Bg(x» ~ = -Bg (x) +
134
- r] If> v (Bg (x) )
(3c) (3d)
X
K= F(p)
+ rH - (px+r+h)B
~
~
[1 -
xv x' (BXgX(x"»]g"(x") = (pxx+r+h)qX
~X
= -B"'gX(x") + ~Xv"'(B"'gX(x"'»
M" = FX(p) + rH'" - (px"'+r+h) B"
(8)
Second, cons idering (3c) and (4c) , ,we have r
= v(Bg(x» = v(B"'.g(x X»
(9)
in the steady state. Since v(.) is increasing in u, (8) and (9) together imply that B and BX are uniquely determined and
(4a)
(IO)
xv x' (B"'gX (XX» ]g"" (XX) = pq'"
XX
(3e)
where x=c/B, g(x)=u(x, 1) and q= ')./y. Bymaking a parallel calculation, we obtain the dynamic system concerning the household in US. [1 -
=
B = B'"
(4b) Third, suppose for the time being that the steady-state price is within two relative labour costs of two countries, which implies that both countries completely specializes to the production of either good. Since it is assumed in Ricardo's model that labour is a single factor of production, internationally immobile and inelastically supplied, world production of car and rice, say Yc and Yr , are positive constant 2 • Thus, combin-in g (3e) and (4e) together and consideringth e m. c. c. 's, (5) and (6), we have, in thest eady state,
(4d) (4e)
where the asterisk (*) indicates the variables and functions belonging to US. In what follws we assume that there is no difference in preference between US and Japan. That is, v(.)=v x (.) and g(.)=g"(.).
o = pVc + Yr + reM + MX - B -B") Next, let us consider market-clearing conditions (m.c.c.). First, we havethe m.c.c. for rice. F' (p)
+ F (p) X
=
'
- (px+h)B - (px"+h)B X
c + cX
= Bx + B"x X
=
Yc
-
h(B + BX)
.=
Yc
-
heM
+ MX)
(ll)
(5)
Therefore, Second, the m.c.c. for renatal-car services, which should be H + If"
=B+ B
X
B = BX = Yc/2h
(12)
and M+ MX is also uniquely determined. Moreover, from (5) and (8)
(6)
(3)-(6) constitute the dynamic Ricardian model in this paper.
x
= XX = hYr/Yc
(13)
Sustituting (12) and (13) to (9), the steady state r can be uniquely derived, viz. ,
3. THE EXISTENCE,UNIQUENESS AND SADDLEPOINT-STABILITY OF THE STEADY STATE
(14)
Let us characterize the steady state of the dynamic Ricardian model (3)-(6). First, from (3a) , (3b) , (4a) and (4b) , we see that (r+h)/p = z(x) = z(x*)
Therefore, p is also derived from (7) p = (r+h)/z(x) (u(g(hYr/Yc)Yc/2h)+h)/z(hYr/Yc)
(7)
(15)
where z(x) = (g(x)-xg' (x»/g' (x). Considering the assumptions with regard to the function u(.), one may verify that z' (x»O for any x) 0, that z(oo)=oo and that z(O)=O. It follows that x and XX are uniquely determined for any positive (r+h)/p and that we always have
Now, we arrive at the proposition. (2) Since we assume that there is one household in each country, Ye (resp. Yr ) xis either 1/ac or l/a c x (resp. l/a r or l/a r ). 135
-v' v /2v· .p ,
PROPOSITION 1: Denote the relative labour costs of Japan and US by ar/a c and arx/acx respectively, where ax, x=c,r, is the unit cost of production of car and rice in teras of labour. The steady state of the dynaaic" Ricardian lOdel uniquely exists if
v'v/2v· .p ,
o
y - v
£
0 0
0 0
v' v/2q, y - (r+h)
= y(y-v) (r-(r+h)} [y(y-v) - (I-uv /v)vv Iv 4>]
=0
ain [ar/a c , arX/aCX] <: (U(g(hYr/Yc)Yc/2h)+h}/z(hYr/Yc) <: .ax [ar/a c , arX/aCE] (16)
where
o
Next let us check saddlepoint-stability of the steady state of the dynamic Ricardian model. Adding (3e) to (4e) and noting the m. c.c.'s (5) and (6), we have
PROPOSITION 2:The steady state of the Ricardian dynamic model is locally saddlepointstable. o 4. MAIN RESULTS
(17)
Proposition 1 already includes the statement concerning the pattern of international trade. Thus in this section we shall concentrate on international indebtedness. In the steady state we have, from (3e) and (4e) ,
which is clearly stable by itself. Since x = xx, (5) and (6) together imply that x = XX = Yr/(M+M X) also asymptotically converges to (13) by itself. Therefore, considering that the number of backward variables is two (M and MX), it suffices for the saddlepointstability to show that the Jacobian matrix concerning the following system has just one negative and three positive eigen values.
F(p) + r(M-B) = pxB + hB
As we have already shown, we always have x = xE and in the steady state B = BX. Since M-B = -(MZ-B"), it is clear that in the steady state international indebtedness, i.e., M-B ~ 0 ~ M"-Bx is completely determined by F(i» ; FX(p), where p is the steady state price of rice, viz., (15).
[1 - .p v' (Bg (x) ) ][g (x) -ig' (x))
q(r+h)
=
(1 -
.p xv' (B g (x» ][g (x) -ig' (x)] = qX (r+h)
B + BX
X
=
M
q = q[v(Bg(x»
- rJ
qX = qX(v(BXg(x»
PROPOSITION 3: Japan is a creditor in the long run if its GNP is smaller than that of US.
- rJ
~ = -Bg(x) + .pv(Bg(x» ~
x = -BXg(x) +
4>
== g(x)-ig' (x) and it is assumed
without loss that g(x)=I. Since (1) ensures us that (I-uv' /v)vv' Iv·.p is posi tive, the characteristic equation e(y) = 0 has one negative and four positive real roots, as was to be proved. Therefore we now have the stability proposition.
In the steady-state Japan exports car if ar/a c > a r x/ac".
(H+H X) = pY r + Yc - p(xB+xxB X) - h(B+B") = Yc - h(M+M"),
£
o
Xv(B"g(x»
Suppose that US has absolute cost advantage in both goods and comparative cost advantage in the production of rice, i. e. , ar/a e is greater than ar"/a e x • Then F(p) and F"(p) can be depicted as in Figure 2. In that case Japan is the exporter of car and a creditor country in the long run.
where M and x are positive constants. Transforming (18) to another system of differential equations whose variables are B. B", f.L, f.L x and r. we can derive the following characteristic equation which is evaluated at the steady state. (Calculations are available from the author on request.) y 0, v'v/2v· .p o y, -v'v/2v· .p e(y)== q(r+h)/ £, 0 Y - v o q(r+h)/ £, 0 o 0 E v'v/2q
PROPOSITION 4:In the dynamic Ricardian model comparative cost advantage determines the pattern of trade and absolute cost advantage determines the pattern of interna-tional 136
F(p) F(p)
1/ac "1.1-_ _ _'" I
I; :
~ 1/acl--~~~-~~ ,,'
/
,,'
;,,;.,.,,/' I
... " ,
J,,' I.:
~----~---------- p
Fig. 3. ar"l. lac x
Fig. 2. ar"l./ac"l. ( ar/ac ar"l. ( a r and ac"l. ( lac
ar
indebteness.
(
"I.
(
ar/a c
a r and a c
(
ac x
REFERENCES
o Epstein,L.G. (1987). A simple dynamic general equilibrium model, Journal of Economic Theory 41, pp.68-95. Leonard,D. and N. V.Long(1992). Optimal Control Theory and Static Optimization in Economics, Cambridge University Press. Ricardo,D. (1817). On the Principles of Political Economy and Taxation, London,J. Murray. Reprinted as Vol.1 of The Works and Correspondences of David Ricardo (ed. by P.Sraffa), Cambridge University Press for the Royal Economic Society, 1951. Shimomura, K. (1993a). Durable consumption goods and the pattern of international trade, in H. Herberg and N. V.Long eds., Trade, Welfare, and Policies:Essays in Honor of Murray C. Kemp, Michigan Universi ty press. Shimomura,K. (1993b). Durable consumption goods, endogenous discount rate and the pattern of inteernational trade and indebtedness, mimeo. Uzawa,H. (1968). Time preference, the consumPtion function, and the optimal assets holdings, in J.N.Wolfe ed., Capital and Growth:Papers in Honor of Sir John Hicks AIdine, Chicago.
When absolute cost advantage differs between two commodities, the pattern of international indebtedness depends on the shapes of u(.) and g(.) as well. See Figure 3 in which we assume that ar/a c :> ar"l./a c"', a r :> a rx but ac <: acx. Then, the pattern of international indebtedness is determined by whether the steady state price p is greater or smaller than acx/a r . Considering (15), we have the following proposition. PROPOSITION 5: If ar/a c :> arx/a cx , a r :> a r '" but ac <: acx, i.e., if US has absolute cost advantage only in rice, then Japan is a creditor country if and only if (u(g(hacl arX)/2hac)+h}/z(hac/arX) is greater than a r '" lac.
o
5. A CONCLUDING REMARK There may be several theoretical issues which we can analyzes by making use of the dynamic model in the present paper. One of them, ' which is one of our next research agenda, is to seek an optimal dynamic tariff reform. Suppose that the government of Japan initially imposes heavy tariff on imported rice and that it faces the problem that it has to reduces the rate of tariff to zero within a certain period. Then, if the objective of the government is the welfare of Japanese household then what is the optimal schedule of tariff reform like? This is the problem which Japan in the real world has been facing since the end of GATT negociations on December 15, 1993.
137