Recycling and International Trade Theory

Transcription

1 Recycling and International Trade Theory Kazunori Tanigaki aculty of Economics Ritsumeikan University Abstract Recently recycling and/or production of secondary materials have increased in many countries. However, there seldom be the analyses which examine the effect of recycling on comparative advantage, trade and welfare. Under the existing of recycling sector, we examine whether the Rybczynski theorem is valid or not, what is the effect of subsidy on recycling and welfare. We show that the recycling subsidy could change production structure and comparative advantage. We find that demand structure has an affect on final goods production and comparative advantage. It is shown that optimal recycling ratio depends on social waste cost, unit subsidy. Key words: recycling, secondary materials, comparative advantage, Rybczynski theorem, welfare JEL Classification Numbers: 11, 18, Q24 Address for correspondence: aculty of Economics, Ritsumeikan University Nojihigashi Kusatsu, Shiga Japan Acknowledgement: I thank M.Okawa, Hajime Sugeta, and M.C. Kemp for their helpful comments. 1

2 1. Introduction As the concern to environmental problem has been raised, the governments have encouraged the recycling. However, the economic implication of the recycling has not been sufficiently examined. Especially little attention to the recycling or the production of secondary material has been paid in international trade and trade theory. This paper tries to fill the gap in international trade theory. As an exceptional paper related to our paper, there is a paper of Grace, Turner and Walter (1978). Their analysis was partial equilibrium. They considered international trade and international market of secondary materials. They applied the model to the case of waste paper. We focus on the effects of recycling on production, comparative advantage and welfare. In our model, there are two sectors of recycling. To recycle, the factor inputs of labor and capital are necessary so that the expansion of recycling could change the structure of production and trade. In section 2 we show the model which introduces a recycling sector. In section 3, we examine how the Rybczynski theorem is altered by the existing of the sector. In section 4, the effect of factor endowment on intermediate import is analyzed. inally in section 5, welfare and optimal recycling ratio are examined. 2. The model We consider a small country. There are two traded final consumption goods. There is one intermediate goods which is imported or which is recycled. As long as the subsidy is admitted, we assume that the production sectors for secondary materials exist. In our paper, we don t consider import of recycled goods. Production functions for final consumption goods are homogenous of degree one and are given by X1 = X1 (L1, K1, M1), X2 = X2 (L2, K2, M2), where Xi is the production of final goods i(i = 1,2), Ki is capital input of final goods i, Li is labor input, Mi is intermediate input of final goods i. Production functions for recycling goods are given by R1 = R1(IR1, LR1, KR1), R2 = R2(IR2, LR2, KR2) where Ri is production of recycling goods i(i = 1,2), KRi is capital input, LRi is labor input, IDi is input of final consumption i. IRi is gathered and used to produce secondary materials. To transform from consumption goods to intermediate input, labor and 2

3 capital inputs are necessary. We assume these functions are also homogenous of degree one. Models are represented as follows: (1) D1= D1(Y, p1, p2) (2) D2= D2(Y, p1, p2) (3) Y = wl + rk - S (4) R = R1 + R2 (5) am1x1 + am2x2 = M M + R (6) al1x1 + al2x2 + alr1r1+ alr2r2= L (7) ak1x1 + ak2x2 + akr1r1 + akr2r2 = K (8) al1w + ak1r + am1pm = p1 (9) al2w + ak2r + am2pm = p2 (10) alr1w + akr1r - ad1r1s1 = pm (11) alr2w + akr2r - ad2r2s2 = pm (12) IR1 = 1 D1, (13) IR2 = 2 D2, (14) S = S1 + S2 = s1 1 D1 + s1 2 D2 where Di is demand for final goods i, Y is income with subsidy burden, S, pi is price of final goods i and it is fixed under the assumption of a small country, a is input-output coefficient of i input for j output, M M is import of intermediate goods (M M = M1 + M2), pm is price of intermediate goods and it is also fixed under the assumption of a small country, i is recycling ratio to final consumption goods i (0 < 1 < 1), si is subsidy to recycling sector i. We call s unit subsidy to distinguish usual subsidy. By government, i is exogenously determined and si is automatically decided for the profit to be zero. In our system, there are 14 equations, and 14 endogenous variables, D1, D2, Y, w, r, X1,X2, M, R, R1, R2, s1, s2, S, and there are 7 exogenous variables, p1, p2, pm, L, K, 1, 2. The policy variables are recovery rates, 1 and 2. rom p1, p2, pm, and (8), (9), factor prices, w and r, are determined. Income, Y, is determined by w, r and S 1. rom w, r, pm, and (10), (11), each unit subsidy to recycling production, s1 and s2, are determined. By governments the unit subsidy is determined 1 When there are intermediate import goods, M, the income or GDP is written as Y = p X + p X - p M. Using the assumption of homogenous of degree one, zero profit condition of pmri + Si = wlr + rkr and M +M = M D, we have Y= wl + rk + p M + wl + rk + p M - p (M D - R1 - R2) = wl + rk - S. 3

4 for the secondary materials sectors to exist under the assumption of alriw + akrir > pm. All input-output coefficients are determined by w, r, pm, s1 and s2. D1 and D2 are determined by Y, p1 and p2. Since firms receive subsidy by using the input of secondary materials and the other input prices and output price are constant, the firms want to use all the inputs which can be available. The government determines the policy variables, 1 and 2. Since the inputs are shown by 1D1, 2D2, each production function of secondary materials is rewritten by Ri = i( i Di, LRi, KRi). The government can determine the output of the secondary materials, Ri. Then X1 and X2 are determined by (6) and (7). inally the volume of intermediate import, M M, is given by (5). Rybczynski theorem In this section we examine Rybczynski theorem in the presence of recycle sector under constant prices. In the subsection 3-1 we examine the relationship between the recycling policy and income. In the subsection 3-2 we analyze the theorem for dl, and for dk in the subsection The relationship between the recycling policy and income Before analyzing the Rybczynski theorem, we examine the relationship between the recycling policy and income. Letting Y0= wl + rk be the basic income, and i(y) = pidi/y be consumption ratio of goods I to income, we can rewrite (3) by using (1), (2), (12), (13), (14), as follows: Y = Y0 (s1 1 D1 + s1 2 D2) = Y0 (s1 1 1(Y)/p1 + s2 2 2 (Y)/p2)Y. Thus for constant price we obtain: Y {1 + (s1 1 1(Y)/p1 + s2 2 2 (Y)/p2)} = Y0, dy/dy0 = 1/{1 + s1 1 ( 1/p1)(1 + 1) + s2 2 ( 2/p2)(1 + 2)}, (15) = 1/ (1 + s s2 2 2) = 1/ (1 + S/ Y) where i = (d i/dy)(y/ i) is the income elasticity of consumption ratio, i is defined as the income propensity to consume goods i. rom the stability condition, we have: 1 > S/ Y = s1 1 ( 1/p1)(1 + 1) + s2 2 ( 2/p2)(1 + 2). 4

5 Then, we obtain dy/dy0 > (1/2). We find that the effect depends on the change in subsidy for income. We also find that the smaller the income elasticity is, the larger dy/dy0 is. or example, when the recycling is only conducted in the first goods, that is, 2 = 0, and the income elasticity of goods 1 is minus one, then dy/dy0 becomes 1 since the demand of goods 1 is constant and the subsidy also constant in this case. 3-2 Labor endowment and the Rybczynski theorem How about the Rybczynski theorem for the increase in labor endowment? In our model, the factor prices are determined by goods prices. Then dy = (1/{1 + s1 1 ( 1/p1)(1 + 1) + s2 2 ( 2/p2)(1 + 2)})dY0 = (1/{1 + s1 1 ( 1/p1)(1 + 1) + s2 2 ( 2/p2)(1 + 2)})wdL, dd1 = 1dY. If the government intends to keep the recycling ratio, the input, IRi, is increased under the assumption of normal goods, that is, 1> 0 or (1 + 1) > 0. Then government subsidy expenditure is raised. The volume of diri is given by iwdl. In the equilibrium, we have = i/ LRi, r= i/ KRi. Then, the effects of dl on dlri and dkri are given by ill ikl ilk ikk dl dk Ri Ri ilir = di ikir Ri. Thus we obtain: 1 dl dk Ri Ri ill ilk ilir = di ikl ikk ikir Ri = ill ikl ilri illri + ilk ikk iklr iklr iwdl, where 1 = illikk - iklikl > 0 under the assumption of concavity. rom the assumptions of > 0 (m, j = L, K) and normal goods, we find that dlri > 0, dkri > 0. Then the recycling sectors are expanded from dri = iiridiri + ilridlri + ikridkri > 0. rom (6) and (7), we obtain: a a L1 K1 a a L2 K 2 dx 1 dl a LR1 alr2 dr dx 2 dk 1 a dr2. KR1 akr2 5

6 or dl > 0, dk = 0, we have: 2 dx 1 ak 2 al2 1 alr 1dR1 / dl alr2dr2 / dl dl, (16) dx 2 ak1 a L1 akr1dr1 / dl akr2dr2 / dl where 2 = al1al2(k2 k1), k1 = K1/L1, k2 = K2/L2. The effects of dl on production X1 and X are given by 2dX1/dL = ak2(1 alr1dr1/dl alr2dr2/dl) + al2(akr1dr1/dl + akr2dr2/dl) = ak2 + (al2akr1 ak2alr1)dr1/dl+(al2akr2 ak2alr2)dr2/dl = ak2 + al2alr1(kr1 k2)dr1/dl+ al2alr2(kr2 k2)dr2/dl, 2dX2/dL = ak1 + al1alr1(k1 kr1)dr1/dl + al1alr2(k1 kr2)dr2/dl, where kri = KRi/LRi. Then we can say that dx1/dl > 0 under the conditions of k2 > k1 and kr1, kr2 > k2, normal goods. Similarly we obtain that dx2/dl < 0 under the conditions of k2 > k1 and kr1, kr2 > k1, normal goods. 2 rom the above results we can state Proposition 1. The production structure is affected by the recycling and income propensity. Under the conditions of normal goods, and kr1, kr2 > k2, k1, the gross Rybczynski theorem for dl becomes consistent with net Rybczynski theorem. In the proposition, net Rybczynski theorem is related to (k2-k1) and /or dxi/dl when there is no recycling sector. or the increase in labor endowment, the productions of secondary materials are raised under the assumption of normal goods. If the both sector in the recycling sector are more capital intensive than the final goods sector, the increase in the recycling production demands capital more than labor. Then since the capital-labor ratio in final output sector is lower than recycling sector, the final output of labor intensive goods increases. 2 2dX2/dL = ak1(1 alr1dr /dl alr2dr /dl) al1(akr1dr /dl + akr2dr /dl) = ak1 + ( al1akr1 + ak1alr1)dr /dl+ ( al1akr2 + ak1alr2)dr /dl = ak1 + al1alr1(k1 kr1)dr /dl + al1alr2(k1 kr2)dr /dl 6

7 3-3 Capital endowment and the Rybczynski theorem How about the Rybczynski theorem for the increase in capital endowment? Eq. (16) is rewritten as 2 dx 1 ak 2 al2 alr1dr1 / dk alr2dr2 / dk dk. (16) dx 2 ak1 a L1 1 akr 1dR1 / dk akr2dr2 / dk Similarly dx1/dk and dx2/dk are obtained as follows 2dX1/dK = al2 + al2alr1(kr1 k2)dr1/dk + al2alr2(kr2 k2)dr2/dk, 2dX2/dK = al1 + al1alr1(k1 kr1)dr1/dk + al1alr2(k1 kr2)dr2/dk. Then we have: dx1/dk < 0, if k2 > k1 and kr1, kr2 < k2, and both goods are normal goods, dx2/dk > 0, if k2 > k1 and kr1, kr2 < k, and both goods are normal goods. rom the above results, we can state Proposition 2. The production structure is affected by the recycling and income propensity. Under the conditions of normal goods, and kr1, kr2 < k2, k1, the gross Rybczynski theorem for dk is consistent with net Rybczynski theorem. Note that the above two propositions are symmetric and the gross Rybczynski theorem and the net Rybczynski theorem are not always consistent for both dk > 0 and dl > Import of intermediate goods What is the effect of factor endowments on the volume of intermediate import? Is the import increased or reduced by factor endowments? 4-1 dm M /dl irst we consider the effect of labor endowment. rom dm M = dm1 + dm2 dr1 dr2 = am1dx1 + am2 dx2 (dr1 dr2), we have: 2dM M /dl = am1{ak2 + al2alr1(kr1 k2)dr1/dl+ al2alr2(kr2 k2)dr2/dl} + am2{ ak1 + al1alr1(k1 kr1)dr1/dl + al1alr2(k1 kr2)dr2/dl} 2 ( dr1/dl + dr2/dl) 7

8 = am1am2(km2 km1)+ {al2alr1(kr1 k2)dr1/dl+ al2alr2(kr2 k2)dr2/dl} + { al1alr1(k1 kr1)dr1/dl + al1alr2(k1 kr2)dr2/dl} al1al2(k2 k1)( dr1/dl + dr2/dl) (17) where ami= Mi/Xi, kmi=ki/mi (i = 1, 2). Then we obtain: dm M /dl < 0, If km2 < km1, kr1 < k2, kr2 < k2, k1 < kr1, k1 < kr2, k2 > k1, that is, If km2 < km1, k1 < kr1 < k2, k1< kr2 < k dm M /dk What is the effect of capital endowment on the volume of intermediate import? Similarly we can obtain the effect as 2dM M /dk = am1{ al2 + al2alr1(kr1 k2)dr1/dk + al2alr2(kr2 k2)dr2/dk } + am2{ al1 + al1alr1(k1 kr1)dr1/dk + al1alr2(k1 kr2)dr2/dk } 2 ( dr1/dl + dr2/dl) Letting lm1=l1/m1 be labor intermediate input ratio, we have ( am1al2 + am2 al1) = am1am2(lm1 lm2). Then we obtain: dm M /dk < 0, If lm1 < lm2, kr1 < k2, kr2 < k2, k1 < kr1, k1 < kr2, k2 > k1, that is, If lm1 < lm2, k1 < kr1, kr2 < k2.. That is, we can say Proposition 3. When factor endowment is increased, the intermediate import is reduced if the capital-labor ratio of the recycling sector is middle between the two capital-labor ratios of final production sector and if a capital (or labor) intermediate input ratio condition is satisfied. The results in this section mean that under some conditions, the economic growth by the increase in factor endowments reduces the primary production of the rest of the 8

9 world through the decrease in the import of intermediate goods. When the recycling sector exists, the import of intermediate goods is not always increased by the economic growth. 5. Welfare and Policy 5-1 The effect of subsidy on production In this section we consider the effect of subsidy on production, trade and welfare. Now the government promotes to recycle and increases the recycling ratio 1, keeping 2 constant. In our model, the increase in the volume of subsidy ( it is unit subsidy, s1) raises R1, that is, dr1/d 1 > 0. The effect on final production is given by 2 dx dx 1 2 ak a 2 K1 a a L2 L1 a a LR1 KR1 dr1, Then we have: 2dX1/dR1 = al2alr1(kr1 k2), 2dX2/dR1 = al1alr1(k1 k R1). Thus, under k2 > k1 we have: kr1> k2 > k1 dx1 > 0, dx2 < 0, 2 > kr1 > k1 dx1 < 0, dx2 < 0, 2 > k1 > kr1 dx1 < 0, dx2 > 0. If the recycle sector 1 is most capital intensive than the other two final production sectors, capital is more necessary by the increase in production of R1 and it causes the reduction of capital intensive final production. If the country imports the goods 1 and exports goods 2 and 2 > k1 > kr1, the government policy (d 1 > 0) which promotes recycling expands the volume of trade. Then we can say Proposition 4. If the capital-labor ratio of the recycling sector is middle between the capital-labor ratios of final two production sectors, the production of the two goods are both reduced by recycling. If it is not middle, the production of the one final goods is increased and the other is reduced. The policy of promotion of recycling affects also the volume of final 9

10 consumption trade. 5-2 The effect on welfare Since (1-1)D1 and C2((1 2) are the volume of wastes, we define social waste cost as C1((1-1)D1) + C2((1 2)D2). We assume C i > 0 and C i > 0. Then social welfare, W, is defined as = wl + rk S C1((1 1)D1) C2((1 2)D2) Similarly to (15), the real income effect of subsidy on the recycling sector 1 is given by dy/d 1 = s1d1/{1 + s1 1 ( 1/p1)(1 + 1) + s2 2 ( 2/p2)(1 + 2)} = s1d1/(1 + S/ Y) As the denominator is negative from the stability condition, the income is decreased by recycling. We find that the income effect depends on unit subsidy (s ), demand of the recycled goods (D1) and the partial subsidy effect of income ( S/ Y). rom (3), welfare effect is given by d /d 1= dy/d 1+ {C (1 1) +C (1 2) }( dy/d 1)+ D1C1. (18) The welfare effect is divided into direct subsidy burden, indirect subsidy burden and the reduction of social waste cost. The second term is positive and it shows the reduction in waste cost due to negative income effect. If the marginal waste cost is sufficiently large so that {C (1 1) + C (1 2) } 1, welfare is always improved. In this case the reduction of waste cost through negative income effect always overweights the distortion by subsidy. If subsidy is initially expended and the marginal waste cost is sufficiently small ({C (1 1) +C (1 2) } < 1), the following condition is necessary for welfare improving for 1= 2= 0: {1 C (1 1) C (1 1) } < C /s1. Since the term, {1 C (1 1) C (1 1) }, is given by (Y C C )/ Y, it can be interpreted that the term is welfare effect except for the effect on subsidy. Thus as long as the partial welfare effect, {1 C (1 1) C (1 1) }, is larger than the 10

11 marginal waste ratio to unit subsidy (C /s1), the starting of recycling is always welfare improving. Then we can say Proposition 5. Initially when the recycling is introduced, that is, 1= 2 = 0, welfare is improved if the marginal waste ratio to unit subsidy (C /si) is larger than welfare effect except of the effect on subsidy. If the marginal waste cost is large so that {C (1 1) +C (1 2) } > 1, welfare is always improved by recycling. If the assumption of (d /d 1)/ 1< 0 is satisfied, optimal 1 could exist. Let 1 denote parameter of waste cost of final goods 1. By rewriting C to 1C in (18), we have: (d /d 1)/ 1= C (1 1) (dy/d 1) + D1C1 > 0. Let op1 denote optimal recycling ratio in sector 1. Then from d /d op1= 0 and the assumption of (d /d 1)/ 1< 0, we have: d op1//d 1 > 0. We find that the optimal recovery rate is increased by exogenous increase in waste cost. This result would be coincident with our intuition. rom (18) we also obtain (d /d 1)/ s1 = {1 C (1 1) C (1 1) } (dy/d 1)/ s1 + {C1 (1 1) 2 + C2 (1 2) 2 + C + C1 (1 1) }dy/ds1. rom (dy/d 1)/ s1 < 0 and dy/ds1 < 0, we obtain under the assumption of {1 C (1 1) C (1 1) } > 0 3, d op1/ds1 >0. 3 Noting that ( i/pi)(1 + i) = i, we obtain (dy/d 1)/ s1= D {1 + s1 1 + s2 2 } + s D 1/{1 + s1 1 + s2 2 } + dy/ds = [D /{1 + s1 1 + s2 2 }][ (1 + s2 2 )] + dy/ds < 0. 11

12 We find that the optimal recycling ratio is increased by exogenous increase in unit subsidy. This result would also be coincident with our intuition. Then we can say: Proposition 6. The optimal recovery rate is raised by the waste cost and reduced by unit-subsidy. The optimal rate also depends on social waste cost, unit-subsidy and recovery rate of another recycling sector. In this section we can simultaneously consider optimal recycling ratios of both recycling sectors. However, we don t examine the analysis since the results are ambiguous. Concluding remarks We examine the relationship between the recycling and international trade theory. It is shown that the export goods and comparative advantage could be changed by the subsidy to recycling. We examine welfare effect of recycling and the optimal recovery rate. We show that the optimal rate depends on waste cost and unit subsidy. These are analyzed by using a simple static two final goods and two factor models. There could be many other trade models which take the recycling into consideration. Some Ricardian and Specific factor models could be available. We may consider overlapping or some dynamic models. Instead of subsidy to recycling sectors, we could examine a sort of recycle tax which is put when consumers buy the final goods. Secondary materials are sometimes generated in the production sector. Then what are the results? We don t examine international factor movements. In the models in which factor prices are affected by its quantity, our results could be modified by introducing international factor movements. Although the recycling is closely related to environment problems, we don t treat it much in our paper. We don t consider the international trade of secondary materials. It is known that trade from developed countries to developing countries, secondary materials has grown. However, to introduce it, many production goods models may be useful. Of course in our model, we can treat the case in which the secondary materials are produced in foreign country. But as long as we use our model, the results would be obvious and not interesting. 12

13 References Grace, R., K. Turner and I. Walter (1978), Secondary Materials and International Trade, Journal of Environmental Economics and Management 5, Grant, D. (1999), Recycling and market power: A more general model and re-evaluation of the evidence, International Journal of Industrial Organization 17, P. J.H. van Beukering (2001), Recycling, International Trade and the Environment: an Empirical Analysis, Kluwer Academic Publishers. P. J.H. van Beukering and M.N. Bouman (2001), Empirical Evidence on Recycling and Trade of Papers and Lead in Developed and Developing countries, World development 29,