International Trade with a Public Intermediate Good and the Gains from Trade

Transcription

1 International Trade with a Public Intermediate Good and the Gains from Trade Nobuhito Suga Graduate School of Economics, Nagoya University Makoto Tawada Graduate School of Economics, Nagoya University ugust, 003 bstract We develo a one-rimary factor, two-consumer good and two-country model of international trade where a country-secific ublic intermediate good is sulied efficiently through the indahl ricing rule in each country. Then it is shown that the country with larger factor endowment exorts the good whose roductivity is more sensitive to the ublic intermediate good. s for the gains from trade, we show that an incomletely secializing country necessarily loses from trade. We also resents the necessary and sufficient condition for a comletely secializing country to gain from trade. Corresonding author: Makoto Tawada. Graduate School of Economics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya , Jaan. Tel/Fax: mtawada@soec.nagoya-u.ac.j

2 . Introduction In recent decades, many trade theorists have studied the influences of ublic intermediate goods on the fundamental theorems in the traditional trade theory. (See Manning and McMillan (979), Kahn (980), Tawada and Okamoto (983), Tawada and be (984), Okamoto (985), ltenburg (987) and Ishizawa (988).) But all of these studies aid attention to a small oen economy and did not give any exlicit analysis in a two-country framework. nother lack in these studies is the analysis of a welfare asect. Therefore, whether a country can gain from trade still remains as an oen question in an economy with ublic intermediate goods. n excetion is the study made by Manning and McMillan (979) who considered an economy with one rimary factor, two consumer goods and one ure ublic intermediate good and showed by a simle Ricardian tye of argument that a country necessarily gains from trade in a small oen economy. But their assertion is not so robust because, as we will show in our analysis, a henomenon that a country will lose from trade is not eculiar but sufficiently lausible in a two-country economy. Since Manning and McMillan concentrated on a small oen economy, they did not argue how to determine the roduction attern and the terms of trade both of which are crucial to the argument of the gains from trade. In this aer, in order to deal with these subjects, we consider an economy where two tradable goods, one rimary factor, one ure ublic intermediate good and two countries. nd we assume that the ublic intermediate good is sulied efficiently by means of the indahl ricing rule. Moreover, we introduce a simle Marshallian adjustment rocess into the model in order to determine the atterns of trade. Then we discuss the gains from trade, according to the attern of trade realized by this dynamic rocess. Three main conclusions emerge from the analysis. The first is that the country with larger factor endowment exorts the good whose roductivity is more sensitive to the ublic intermediate good. The second is that an incomletely secializing country necessarily becomes worse off by free trade. Finally, concerning a comletely secializing country, we roose the necessary and sufficient condition for the country to have a gain from trade.

3 i This aer is organized as follows. Section describes the model and exhibits some basic results regarding the roduction side. Section 3 examines the autarkic economy and characterizes the adjustment rocess. Then Section 4 investigates the atterns of trade attained at the trading steady-state equilibrium and Section 5 deals with the question of the gains from trade. Concluding remarks are resented in the last section.. The Model We consider an economy with two consumer goods called good and good, one rimary factor called labor, and one ubic intermediate good which is collectively used in the roduction of all consumer goods.. roduction technologies and the labor endowment We suose that the roduction technologies of two consumer goods and ublic intermediate good are described as, () Q = ( R), ( ) > 0, ( ) < 0, i =,, i i () R = f ), f ( ) > 0, f ( ) > 0, resectively, where ( R i i Q and R are the oututs of good i and the ublic intermediate i good, resectively, and are the amounts of labor used in the roduction of good i R i and the ublic intermediate good, resectively, f ( ) is the roduction function of the ublic intermediate good, and i th industry for a given R. (R) i is the marginal roductivity of labor in the It is assumed that the elasticity of marginal labor roductivity with resect to the rovision of ublic inut, η ( R) ( R) R / ( R), is always greater in the first industry than the other, namely, (3) R) η ( ) ( R i i η > for all R > 0. This suggests that the roductivity of the first industry is more sensitive to the ublic inut than that of the second industry. The labor endowment is assumed to be given and constant. The full emloyment condition of labor is (4) + = R +, 3

4 where is the labor endowment.. the efficient suly of the ublic intermediate good and the indahl ricing rule Q The roduction ossibility frontier is the locus of Q, Q ) satisfying ( = Γ( Q, ) for a given, where Γ ( Q, ) reresents the maximum outut of good when Q and are fixed; that is, Γ ( Q, max,,, ( R) R R subject to ( R) = Q, R = f ( R ) and (4). The otimal conditions of the above roblem are derived as (5) ( ) Q / ( ) + ( ) Q / ( ) = / f ( ), (6) Q / ( ) + Q / ( ) + f ( R) =. et Λ ( Q, ) be the otimal suly of the ublic intermediate good for given Q and. Then Γ( ) and Λ( ) constitute the solution to the simultaneous equations given by (5) and (6). Hence, by using (5) and (6) we can obtain the artial derivatives of Γ( ) and Λ( ) with resect to Q and as follows. (7) Λ Q = ( η η ) / R 0, / > (8) Γ Q = / 0, / < (9) Λ / = η / R > 0, (0) Γ / > where = Σ [ ( ) ] / /( = Q + f f ) 0. Partial differentiation of (8) i i i i i i < with resect to Q and, together with the use of (7) and (9), yields Γ ( η η ) Λ ( η η ) () = = > 0, Q R Q ( R) Γ ( η η ) Λ ( η η ) η () = = > 0. Q R R It is obvious from () that the roduction ossibility frontier is strictly convex to the origin, as shown in Figure. (Figure ) Suose that the government sets the rice of the ublic intermediate good by the 4

5 indahl ricing rule. Then the outut of the ublic intermediate good is determined by rofit maximization. The rofit maximizing conditions are exressed as, R = (, R f ( ) = w, ) + ( ) where and are the rices of good i and the ublic intermediate good, i R resectively, and w is the wage rate. We assume that the roduction of the ublic intermediate good is financed through the lum sum tax on total income (total wage bill lus rofit). ll firms within each industry are assumed to take the quantity of the ubic intermediate good as given and act as if they oerate under constant returns to scale. Perfect cometition and rofit maximization in all rivate industries, together with the otimal conditions of ublic good suly, bring forth of (3) / = / = Γ( Q, / Q. Thus, if both consumer goods are roduced in the market equilibrium, roduction takes lace at the oint of the roduction ossibility frontier where the budget line is tangent. It is shown in Figure..3 references Suose that social references are described by the homothetic utility function u = U C, C ) = φ[ v( C, )], φ ( ) > 0, ( C where Ci is the social consumtion for good i and v ( ) is monotonely increasing, strictly quasi-concave and linearly homogeneous in and C. C Normalizing the rice of good to unity and denoting the rice of good relative to good by, the exenditure function with the level of utility, E(, u) = e( ) φ ( u), e ( ) > 0, e ( ) < 0. u, is exressed as Thus the indirect utility level, V, and the consumtion ratio of good to good, C / C, are described, resectively, as (4) V = φ [ I / e( )] V (, I), (5) C C = γ /( γ ) Z( ), where / I Q + Q is national income and γ e ( ) / e( ) is the share of income sent on good. Then the consumtion ratio of good to good has a negative relation 5

6 to its relative rice since (6) dz / d = ε Z /( γ ) < 0, where ε e ( ) / e ( ) is the rice elasticity of comensated demand for good. 3. utarky Now we adot the following Marshallian adjustment rocess: The outut of good rises if the corresonding demand rice, D, exceeds the corresonding suly rice, S and vice versa. So the recise descrition of the adjustment rocess is Q& = D S, where dot denotes a time derivative. The demand rice is defined as the relative rice of good that clears the commodity markets for a given outut level of good. From (5) and the market-clearing condition, say C / C = Q / Q, the autarkic demand rice, D, is thus reresented as the solution to (7) Z D ) = Q / Γ( Q, ). ( Total differentiation of (7) with resect to D and Q and the use of (8) and (6), we obtain (8) χ = γ )( Q / Q + ) / ε 0 D Q ( <, where χ ( dx / dy ) /( Y / X ) denotes the elasticity of X with resect to Y. The above XY equation suggests that the autarkic demand rice is decreasing in the domestic outut level of good. The suly rice is defined as the relative average roduction cost of good. By using (3) we can exress it as S = Γ( Q, / Q. So, from (8) and () the S elasticity of with resect to Q is given by (9) χ η η ) χ 0. the outut-rice air S Q = ( ΛQ < In view of (7) and (9), we find that the suly rice is also decreasing in the domestic outut level of good. et us denote the autarkic demand rice and the suly rice by = D ( Q, D ) and S = S( Q,, resectively. Then the autarkic demand curve, showing the locus of Q, ) such that = D ( Q, ) for a given, is dislayed ( D D 6

7 by the curve DD in Figure, and the suly curve, showing the locus of the outut-rice air Q, ) such that S = S( Q, for a given, is as deicted by ( S the curve SS. If the demand curve is steeer than the suly curve and there is at least one intersection, there exists a globally stable and unique autarkic steady-state equilibrium E. It is shown in Figure. Evaluating (8) at the autarkic steady-state equilibrium we have χ D Q = / ε stability condition is exressed as (0) χ, which, together with (9), suggests that the S Q < / ε, for Q such that D( Q, = S( Q, Hereafter we assume that this stability condition holds. (Figure ) 4. Trade in a Two-Country World We now consider free trade between two countries, say the home and foreign countries. The two countries are assumed to be identical in the roduction technologies and references but not necessarily in the labor endowment. The ublic intermediate good is suosed to be country-secific and have no international sillovers. In what follows we attach a suerscrit * to each variable ertaining to the foreign country. 4. trading equilibrium To see what the equilibrium will be after oening trade, we will begin with the descrition of the Marshallian dynamic rocess under free trade. It is easily shown that, under free trade, the world consumtion ratio of good to good, C C *) + C*) /( C +, is also exressed as Z( ) = γ /( γ ) for a given world ( relative rice,. From this fact and the world market-clearing condition exressed by the world demand rice, ( C + C *) /( C + C*) = ( Q + Q D, should be the solution to *) /( Q *) +, Q () Z( D ) = ( Q + Q*) /[ Γ ( Q, ) + Γ * ( Q*, *)]. ~ et us denote the world demand rice by D = D( Q, Q *,, *). Then the Marshallian adjustment rocess under free trade is described as follows. ~ () Q & = F(Q, Q*,, *) D( Q, Q*,, *) S( Q,, 7

8 (3) &Q ~ = F * ( Q *, Q, *, D( Q, Q *,, *) S * ( Q *, *), * where S ( ) and S * ( ) is the country-secific suly rices. Differentiating () totally with resect to, Q and Q *, and using (8) and D (6), the world demand rice is shown to be decreasing in Q and Q *, namely, ~ ~ (4) D / Q 0, D / Q * < 0. < This immediately leads to the fact that F ( ) and F * ( ) are decreasing in * and Q, resectively, that is, (5) F / Q * < 0, F * / Q < 0. s for the country-secific suly rices, S ( ) and S * ( ), it is assured that (6) S / Q < 0, S * / Q * 0 <. For the time being, in order to simlify the discussion we assume that the two countries are the same in regard to their labor endowments. Then, since two countries are the same in any resect, the existence of the ublic intermediate good itself lays a ivotal role in trade creation if trade occurs between countries. Figure 3 reresents the hase diagram of the dynamic system exressed by () and (3). Following Ethier (979, 98), we call the locus of outut air ( Q, Q *) satisfying Q & = F( ) = 0 the home allocation curve. Similarly we call the locus of outut air Q, *) satisfying Q &* = F * ( ) 0 the foreign allocation curve. Under ( Q = the assumtion that the two countries are identical in labor endowment, the home and foreign allocation curves can be drawn as the curves EB and CED, resectively. The two curves are symmetric about the 45 line, on which they intersect at a oint E, and the intersection corresonds to the autarkic steady-state outut level of good. (Figure 3) We further roceed to investigate the two allocation curves in more details. Because of (6) and the assumtion that the two countries are comletely identical, (7) S = S( Q, > (res. <) S ( Q *, *) = *, as * > (res. <) Q. * S Q Q From (), (3) and (7), it follows that, in the uer area of the 45 line, Q & = F( ) < 0 for any outut air Q, *) satisfying Q &* = F * ( ) 0. Taking this fact into ( Q = consideration and using the decreasingness roerty of F ( ) with resect *, we find that the home allocation curve lies below the foreign allocation curve in the uer Q 8

9 area of the 45 line, as shown in Figure 3. In a similar way, it is exhibited that the foreign allocation curve lies below the home allocation curve in the lower area of the 45 line. Moreover, The sloe of the home (res. foreign) allocation curve is negative at the intercet on the horizontal (res. vertical) axis such as in Figure 3, due to the decreasigness roerty of F ( ) (res. F * ( ) ) with resect to * (res. Q ). Q Therefore, one tyical relationshi between two allocation curves is ossibly shown as in Figure 3. The arrows in Figure 3 indicate the qualitative laws of motion of Q and Q *. From (5) we find that Q & = F( ) area of the home allocation curve. Thus is ositive (res. negative) in the lower (res. uer) Q rises (res. falls) over time in the lower (res. uer) area of the curve EB, as deicted by the horizontal arrows in the figure. Similarly we can denote the dynamic behavior of Q * by the vertical arrows. Q is defined as the maximum outut level of good in home when Q 0. So is Q * in foreign. Then the whole of each allocation curve is accommodated into the box OGJI in Figure 3. Therefore, if the oint reresenting the country-outut combination of good slightly moves from the oint E to somewhere in the uer (res. lower) region of the 45 line, the oint D (res. B) will eventually become the trading steady-state equilibrium. In this case, the country exorting good is diversified and the other country is secialized in good and exorts it. = 4. atterns of roduction and references We now consider the relationshi between the atterns of roduction and references. In view of (5), it is lausible to think that the greater the taste for good, the larger shares of income sent on good, for a given. Thus we suose Therefore, Q ( Q *) is the outut level of good in the home (foreign) roduction ossibility frontier when Q ( * ) is equal to zero. Q 9

10 references to satisfy this roerty. Suose that the economy is in equilibrium initially. Then, the world excess demand for good aears if the taste for good becomes greater. In this case, the relative rice of good has to rise in order to restore the market equilibrium. This imlies that the greater the taste for good, the higher the world demand rice, ~ D = D( Q, Q*,, *), for given ( Q, Q*,, *). Hence Q& (res. Q & * ) becomes ositive in sign for any outut air, *) that assures Q& (res. Q & * ) to be zero ( Q Q initially. This, together with (5), imlies that in Figure 3 the home (res. foreign) allocation curve shifts u (res. rightward) according to an intensification in the taste for good. Reminding the assumtion that = *, we consider the case where the taste for good is sufficiently strong, so that the segment E (res. CE) of the curve ED (res. CED) intersects the line GJ (res. IJ). Then, according to the qualitative laws of motion of Q and * indicated by arrows in Figure 3, the good exorting country will Q secialize in its exorting good and the other country will diversify at the trading steady-state equilibrium. Conversely, as for the case where the taste for good is not so strong, the segment E (res. CE) of the curve EB (res. CED) is within the box OGJI but the segment BE (res. DE) intersects it. Then each country is exected to secialize in its exorting good at the trading equilibrium. If the taste for good is sufficiently weak, the whole allocation curves are located within the box OGJI just as Figure 3 shows. Then, as stated earlier, the country exorting good is diversified and the other country is secialized in good and exorts it. 4.3 comarative advantage and country size Suose that references are described by the following CES function: ρ ρ / ρ U ( C, C ) = [ αc + ( α) C, ρ <, ρ 0, 0 < α <, ] where α reresents the extent to which good is referred, comared to good. Then we can see that, holding rices of consumer good constant, an increase in α raises the share of income sent on good. 0

11 llowing the home and foreign countries to have different labor endowments, we investigate what attern of trade will be attained after oening trade. Hence we first consider the autarkic economy again and suose an increase in the labor endowment of one country. Then it is seen from (0) that the outut of good increases for a given outut level of good in that country. So the right-hand side of (7) becomes less than its left-hand side. By (6), however, the market equilibrium can be restored with a rise in the autarkic demand rice while the outut level of good is constant. Thus the autarkic demand curve shifts u corresonding to an increase in the labor endowment, which is shown by the uward arrow in Figure. By contrast, from (8), () and (3) it is clear that the home suly curve shifts downward in resonse to an increase in the labor endowment. This is shown by the downward arrow in the figure. By the use of these facts we can obtain the following lemma. emma Under the assumtion that η R) > η ( ) for all R > 0, a country with larger ( R labor endowment has a lower relative autarkic rice of good and a higher autarkic outut ratio of good to good. Proof. Suose that the relative autarkic rice of good is in Figure. Then an increase in the labor endowment shifts u the demand curve from DD to D D and it shifts down the suly curve from SS to S S, so that the relative autarkic rice declines from to '. This imlies that the country with larger labor endowment faces a lower relative autarkic rice of good. Furthermore, from (6) and (7), it is easily verified that a lower autarkic relative rice leads to a higher autarkic outut ratio of good to good. Thus we can conclude that the larger country has a higher autarkic outut ratio of good to good. Q.E.D. et us turn our attention to the trading equilibrium. Suose that the home and foreign allocation curves are, resectively, reresented by the curves EB and CED in Figure 3 initially. et the foreign labor endowment, *, increase. s already

12 mentioned, an increase in labor endowment lowers the suly rice of good. So an increase in * leads to a fall in the foreign suly rice, S * = S * ( Q*, *). From ~ (0), (6) and (), we find that the world demand rice, D = D( Q, Q*,, *), rises in resonse to an increase in *. Therefore, as a result of an increase in *, the sign of Q & (res. Q & * ) becomes ositive for any outut air Q, Q *) that lies on the ( curve EB (res. CED). This, together with (5), asserts that the home and foreign allocation curves shift outward in resonse to an increase in * 3. Bearing these shifts of the allocation curves in mind and using emma, we can obtain the following roosition with resect to the atterns of trade. Proosition Suose that roduction techno logies and references are the sam e between two countries. Then, under the assumtion that η ( R ) > η ( R) for all R, a country with larger labor endowment exorts goo d and the other country exorts good at the trading equilibrium. Proof. Without loss of generality we can assume that * >. Then the home and foreign allocation curves can be deicted by the curves E B and C E D in Figure 3. By the assumtion that * >, emma imlies that (8) Q Γ(Q Q + Q * Q * < <, Γ(, + Γ * ( Q *, *) Γ * ( Q *, *), Q where Q and Q * are, resectively, the home and foreign autarkic outut levels of good. From (6), (7), () and (8), it follows that ~ (9) D * ( Q *, *) < D( Q, Q *,, *) < D( Q,, where D (Q, ) and D * ( Q*, *) are the country-secific autarkic demand rices. Noticing that the autarkic demand rice becomes equivalent to the suly rice at the 3 n increase in * moves the intersection of the two allocation curves into the lower area of the 45 line. We understand the reason by noticing that a fall in the foreign suly rice due to an increase in * enlarges the value of Q that equalizes the home and foreign suly rices for any secific value of *. Q

13 autarkic equilibrium, and using (), (3) and (9), we can obtain (30) Q & = F Q, Q *,, *) 0, Q & = F * ( Q, Q *,, *) 0. ( < From (30) it is evident that an autarkic outut air * > ( Q, Q locates somewhere in the region D E in Figure 3. et F be such a oint. Then, according the adjustment rocess, the trading equilibrium attained at G in Figure 3, where the foreign country exorts good. Thus we can conclude that a larger country with larger labor endowment exorts good and the other country exorts good at the trading equilibrium. Q.E.D. *) n intuitive exlanation for Proosition is as follows. The larger country can take advantage of an abundant suly of labor to roduce the ublic intermediate good, so that it faces a lower relative autarkic rice of good. This is because good is more sensitive to the rovision of ubic inut than good. Hence, if the two countries commerce free trade, the larger country exorts good and the smaller country exorts good. 5. Gains from Trade Now we are in a osition to investigate the welfare effects of trade. In this section we first focus our analysis on the relation shi between the roduction atterns and the gains from trade in a country, and then we exlore the welfare effects of trasde to each country in a two-country framework. To start with, we shall derive some roerties of the home national income, I = Q + Q. Consider home country roducing both consumer goods and let us denote its national income by where Q i (, g, Q (, + Q (, ), ( is the amount of good i roduced by cometitive firms within the i th industry when the relative rice of good and the labor endowment are given by and, resectively. Because for given and the cometitive outut level of good is exressed as Q, = Γ[ Q (,, ], we can rewrite the national income ( 3

14 as (3) g, = Q (, + Γ[ Q (,, ]. ( Partial differentiation of (3) with resect to and use of (3) yield (3) g / = Q (,. We can make the analysis of foreign national income in a similar manner. Thus we can establish the following lemma. emma Consider home country and let Q i be the maximum outut level of good i ( i =,) when the othe r consumer good is not roduced. Then, the home national income in the case where both consumer goods are r oduced, g(,, satisfies the following: g (, < Q (res. Q ), for all > l (res. < u ), where l Γ( Q, / Q and u Γ( 0, / Q are, resectively, the lower and uer limits of such that both consumer goods can be roduced. The foreign national income also satisfies this roerty. Proof. We treat the home country case only, since the foreign country case can be shown in the same way. We should recall that the roduction ossibility frontier is strictly convex to the origin and that the budget line is tangent to the roduction ossibility frontier when both consumer goods are roduced, as deicted in Figure. From the figure we can see that fact, together with (3), says that follows that for any ' Q (, ) is decreasing in, say Q / 0. This < g / is negative for all l, ]. Thus it g(, < g( ', + [ g( ', / ]( ' ) = ' Q ( ', + Q ( ', + Q ( ', ( ' ) = Q ', + Q ( ', ). ( It is clear from Figure that (res. ) is the value of Q (, = Q (res. (, Q l u [ u such that Q = ). Hence, relacing ' with (or ) in the above equation, we can obtain emma. l u 4

15 Q.E.D. et us turn to the analysis of the gains from trade. For the time being we consider the case where both consumer goods are roduced and we denote the indirect trade utility function as V (,. Then, from (4) it follows that (33) V (, = φ[ g(, / e( )]. Concerning the roerties of V (,, we can show the following lemma. emma 3 Consider home country and let be a relative home autarkic rice of good. Then the home indirect trade utility in the case where both consumer goods are r oduced, V (,, has the following roerties: (i) V (, is strictly decreasing (res. increasing) in if is higher (res. lower) than. Hence V (, is maximized at. (ii) V (, is smaller than φ [ Q / e( )] (res. φ [ Q / e( )] ) for all > (res. < u ). The foreign indirect trade utility also satisfies the above roerty. l Proof. Consider the case of home countrypartially differentiating (33) with resect to and using (3) yield (34) V / = [ φ ( ) / e( )][ Q ( ) γg(, / ]. The last term in (34), γ g (, /, reresents the consumtion level of good when the relative rice and the labor endowment are given by and, resectively. Therefore V / = 0 if and only if =. Therefore, to show that V / is negative at suffices to rove emma 3 (i). Deriving the second order derivative of (33) with resect to and evaluating it at, we have V (, φ ( ) Q (, Q (, (35) = ε +, e(, Q (, from (3). Making use of (3), we find that Q (, ) is the solution to 5

16 = S( Q Q in regard to Q. Thus [ Q, / ][ / Q (, )] is rewritten as (36), Γ( Q, / Q (, Q (, = Q (, ( (, ) S( Q, Q = Q S( Q, = χ. By the use of (0), (35) and (36), we can show that V / is negative at. The roerty (ii) is easily derived from emma. In Figure 4, the curve BC reresents the locus of the rice-utility air (,V ) such that V = V (, for a given. Since a country exands the roduction of its S Q, we can confirm (, V ) (, V ) satisfying V = φ[ Q / e( )] and V = φ Q / e( )], resectively. That is, the curve Q The case of foreign country can be shown in the same way. exorting good by oening trade and Q (, is decreasing in that the segment B (res. CB) corresonds to the locus of the rice-utility air Q.E.D for which a country is incomletely secialized and exorts good (res. good ). On the other hand, the curves E and CD reresent the loci of the rice-utility air [ E(res. CD) indicates the utility level of a country secializing comletely in good (res. good ). Thus we can establish the following roosition as to the relationshi between the roduction atterns and the gains from trade. (Figure 4) Thus we can establish the following roosition as to the gains from trade. Proosition Suose that η R) > η ( ) for all R > 0. Then, as for the relationshi between ( R the roduction atterns and the gains from trade, the following hold. (i) The country that is incomletely secialized after trade becomes worse off. (ii) The country secializing in good gains from trade if and only if international rice of good exceeds the level such that φ [ Q / e( )] = V, where V reresents 6

17 the autarkic utility level. (iii) The country secializing in good gains from trade if and only if international rice of good falls short of such that φ Q / e( )] = V. [ Proof: Proosition (i) immediately follows from emma 3 (i). Next let us rove Proosition (ii) by the use of Figure 4. In the figure, the utility level of the good exorting country is indicated by the curve BFE and that of the good exorting country is denoted by the curve BCFD. Moreover, the rice levels and in the figure reresent the values of satisfying V = φ[ Q / e( )] and V = φ[ Q / e( )], resectively. Thus the country secializing in good (res. good ) loses from trade if and only if the relative rice of good falls short of (res. exceeds ). Q.E.D. et us give an intuitive exlanation of Proosition. First of all, we should remember that the autarkic demand curve is assumed to be steeer than the suly curve at the intersection. This assumtion, together with homothetic references, says that on the roduction ossibility frontier the marginal rate of substitution exceeds (res. falls short of) the marginal rate of transformation if than the autarkic outut level Q Q is smaller (res. larger) 4. n indifference curve corresonding to the autarkic utility level can be reresented by the curve UU and the curve FF indicates the roduction ossibility frontier in Figure 5. In this case, the budget line of the incomletely secializing country is as deicted by the line II that is tangent to the roduction ossibility frontier and lies below the autarkic indifference curve. Therefore, the country that incomletely secializes after trade becomes worse off. Concerning the interretation of (ii) and (iii) of Proosition, let the lower (res. uer) threshold rice (res. ) be given by the sloe of the line that starts from the oint F (res. F) and is tangent to the autarkic indifference curve UU in Figure 6. Thus, we can see that the 4 Note that the autarkic demand rice is equal to the marginal rate of substitution at the consumtion oint and D S = Γ( Q Q., / 7

18 country secializing comletely in good (res. good ) becomes better off by trade if and only if international rice exceeds (falls short of ). e (Figures 5 and 6) Finally, we consider an economy that consists of the two countries, say the home and foreign countries. Suose that they are identical in all resects initially and that the foreign labor endowment has increased. Then, by (0) and (), the foreign roduction ossibility frontier shifts uward and becomes flatter. So the foreign country sees its national income rise for given and it faces the lower l, which is the lower limit of such that both consumer goods can be roduced. Moreover, it follows from emma that > *. These facts, together with Proosition, imly that if the foreign labor endowment exceeds the home labor endowment, the home and foreign utility levels under free trade are, resectively, indicated by the curves BCFD and B FE in Figure 7. Proosition shows that trade can be welfare-decreasing for one country, but from Figure 7 we find that it is not ossible for both countries to lose from trade. For examle, suose that an international rice of good is given by, ) in the figure. Then, the foreign country comletely secializes in good ( u and enjoys a rise in its welfare level although the home welfare level falls by trade for any roduction atterns. (Figure 7) 6. Concluding Remarks We have considered trade between two countries in a simle general equilibrium model with a ubic intermediate good. By the introduction of Marshallian dynamic adjustment rocess, we have develoed a rigorous argument on the gains from trade as well as the atterns of trade in the two-country model. Three significant results have derived from our analysis. The first is that under free trade the larger country exorts the good whose roductivity is more sensitive to the ubic intermediate good. We should notice that the result bears a close resemblance to the result shown in the similar framework but dealing with the case of increasing returns to scale. For examle, Ethier (98) and 8

19 Tawada (989) dealt with a one-factor, two-good and two-country model in which one industry is subject to increasing returns to scale and verified that under free trade the larger country exorts the good of the industry subject to such externalities. The second is that, in site of the efficient sulies of the ublic intermediate good, the country that roduces both consumer goods under free trade necessarily becomes worse off. This result seems to be imortant when comared with that of the case of external economies in roduction. In the case of external economies, there exists a market distortion that the rice line cuts the roduction ossibility frontier at an equilibrium, which gives a negative effect to the gains from trade theorem. Nevertheless, it is ossible for an incomletely secializing country to have a gain from trade in this case. In the case of the ublic intermediate good, however, the source of the result that trade is harmful to a country is only the shae of the roduction ossibility frontier, which is convex to the origin, under the assumtion of efficient suly of the ublic intermediate good. Therefore, it is inevitable for an incomletely secializing country to have a loss from trade. In this reason, we can easily infer that the traditional gains from trade must hold in the case of semi-ublic intermediate goods since the roduction ossibility frontier becomes concave to the origin in this case. (See Tawada(980).) Thirdly, we roosed a necessary and sufficient condition for an comletely secializing country to have a gain from trade. ccording to this condition, if the international equilibrium rice is far away from the autarkic equilibrium rices, one of two countries must lose from trade. ccording to Figure 7, if the international equilibrium rice falls in between two autarkic equilibrium rices, both countries secialize and have a benefit from trade. Finally, we can consider two extensions of our analysis. In this aer we have assumed that the ublic intermediate good has no international sillovers. But there are many ublic intermediate goods that serve in roduction internationally. The infrastructure of information networks, satellite communication systems, international research and develoments etc. are the tyical examles. Therefore our resent analysis needs to be extended to the case of international ublic intermediate goods. nother imortant extension is to suose ublic intermediate goods to be durable, 9

20 since the ublic intermediate goods lay a role of infrastructure for rivate roduction. Then the ublic intermediate should be characterized as a caital stock and the analysis should be dynamic. 0

21 References [] ltenburg,., Production Possibilities with a Public Intermediate Good, Canadian Journal of Economics 0 (987), [] Ethier, W. J., Internationally Decreasing Costs and World Trade, Journal of International Economics 9 (979), -4. [3], Decreasing Costs in International Trade and Frank Graham s rgument for Protection, Econometrica 50 (98), [4] Ishizawa, S., Increasing Returns, Public Inuts, and International Trade, merican Economic Review 78 (988), [5] Khan, M.., Factor Price and Public Inut Equalization Theorem, Economic etters 5 (980), -5. [6] Manning, R. and J. McMillan, Public Intermediate Goods, Production Possibilities, and International Trade, Canadian Journal of Economics (979), [7] Okamoto, H., Production Possibilities and International Trade with Public Intermediate Good: Generalization, Economic Studies Quarterly 36 (985), [8] Tawada, M., The Production Possibility Set with Public Intermediate Goods, Econometrica 48 (980), [9], Production Structure and International Trade (Berlin: Sringer-Verlag, 989). [0], and K. be, Production Possibilities and International Trade with a Public Intermediate Good, Canadian Journal of Economics 7 (984), [], and H. Okamoto, International Trade with a Pubic Intermediate Good, Journal of International Economics 7 (983),

22 Q Q The grah of Q = Γ( Q, ) ; that is, the roduction ossibility frontier The budget line The roduction oint h / 0 Q Q Figure

23 , D S D D S S arger arger ' S D D S 0 Q Q' Q Figure

24 Q * D J h The 45 line Q * G D h J F h E h E O C C B I Q B Q Figure 3

25 V D V = φ[ Q / e( )] E V = φ[ Q / e( )] F V B V = V (, C 0 l u Figure 4

26 Q Q F U I U 0 I F Q Q Figure 5

27 Q Q F U U F 0 Q Q Figure 6

28 V,V * D E V* = φ[ Q * / e( )] F V = φ[ Q / e( )] V * B V B V = V (, V * = V (, *) C 0 * l * * u Figure 7