Economic Growth, Intermediate Products, and the Terms of Trade: CommentAuthor(s): Neil BruceSource: The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 8, No. 1(Feb., 1975), pp. 112-117Published by: Wiley on behalf of the Canadian Economics AssociationStable URL: http://www.jstor.org/stable/134068 .

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Notes

Economic growth, intermediate products, and the terms of trade: comment

NEIL BRUCE / Queen's University

In a recent article, Batra and Pattanaik (1971) (henceforth BP) examined the effects of biased growth on the terms of trade when Vanek-type (1963)-inter- mediate goods are used in production. They demonstrated that when each good uses the other good as an input in fixed proportions to output, biased growth will be even more biased, in that the excess supplies and demands for the goods created before relative prices change will be larger. They went on to assert that as a result the change in the terms of trade resulting from biased growth must be greater in order to restore equilibrium (1971, 232).

This note will demonstrate that the BP conclusion cannot be made on the basis of the larger excess supplies and demands but is nevertheless valid under certain conditions. It will be demonstrated that a sufficient condition for its validity is that demand conditions are homothetic, and that in the absence of this assumption the conclusion may be invalid.

For simplicity, only growth in capital will be considered, so that dK > O while dL = 0, where Kand L represent the total supplies of capital and labour to the economy. Letting Xi and xi denote gross and net output of commodity i respectively,

x, = X1 - a12X2,

x2= X2 -a2lXl,

where aij is the amount of Xi used to produce a unit of Xj. The linear homo- geneous production functions can be represented as

Xi- Fi(Ki, Li, Xji), (2) This note was written while the author was a graduate student at the University of Chicago. I am indebted to A.C. Harberger and the members of the International Trade Workshop for valuable comments and suggestions. I would also like to thank the anonymous referees for correcting some errors in an earlier draft and for their careful comments.

Canadian Journal of Economics/Revue canadienne d'Economique, VIII, no. I February/f6vrier 1975. Printed in Canada/Imprim6 au Canada.

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Notes/ 113

where Ki and Li are the capital and labour used in producing Xi, while

Xii = ajiXi is the amount of the other good used in producing Xi. Because of linear homogeneity, (2) can be expressed as:

Xi = Lifi(ki, xji), (2')

where ki = Ki/Li and xji = XjiLi.

Full factor employment requires that

L1 + L2 =L

k1L, + k2L2 = K

By substituting (2') into (1) and differentiating along with (3) assuming relative prices fixed, we can derive the Rybczynski (1955) relations' between net outputs and K as

dxl f, + al 2f2 > o dK k,-k2

dx2 -(f2 + a21fl) < ,

(4) dK k,-k2

if k1 > k2.

Clearly Idxi/dKI is greater when aij > 0, assuming the other parameters are the same. Therefore, with given demand conditions and assuming x1 is capital-intensive, the augmentation of capital at fixed relative prices would generate a larger excess supply of x1 and excess demand of x2 than if inter- mediate goods were not involved,2 that is, if aij = 0.

This point alone is not sufficient to infer that a greater change in the rela- tive price of the goods is necessary to restore equilibrium because it can be shown that supply response of the net outputs to a given change in their relative price is also greater in the case of intermediate goods. Differentiating (1) with respect to p (the relative price of x1 in terms of x2) we get:

dxl/dp = dX,/dp - al2dX2/dp. (5)

Since dX,/dp > 0 while dX2/dp < 0, it is found that dxl/dp > dX,/dp aild conversely that dx2/dp < dX2/dp. This result agrees with intuitive reasoning,

1 The slope of the Rybczynski line with intermediate goods can be calculated as:

dX2 = -(f2 + a21f1) < o. dX, fi + al2f2

Note that it may be steeper or flatter than the corresponding locus in the non-inter- mediate goods case depending on whether

t2 >- gt2,

Vf2 ) < a21

2 The intuitive reasoning is as follows. The gross output of X1 must rise (and 2 must fall) by the same amount in both cases. Therefore the net output x1 must rise by more because the decline in X2 releases some of the commodity for final use. Similarly the net output x2 must fall by more because the increased output of X1 requires more X2.

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1 14 / Neil Bruce

X2 x X2

Tv

tv TA

0 T tv Tv 2

FIGURE 1

since as p rises, gross output of X1 increases in any case, but net output rises by more in the intermediate goods case because some X1 is released when X2 declines.

This is apparent in Figure 1 where t,t, is the net output transformation curve derived by subtracting intermediate inputs from the gross output transformation curve TV TV (the subscript v will denote the Vanek intermediate goods case). It is easy to show that t, t, will be convex like TV TV but will have less curvature (providing TV TV is strictly convex), a point which Vanek (1963) noted in his original analysis. By differentiating (1) twice we get

dx2 dX2/dXj - a21

dxl I - a12(dX2/dXj)

d2x2 (1- al2a2l)(d2X2/dX 12)

dx1 2 [1- a12a2j(dX2/dX 1)]2

Since 0 < a12a21 < 1 by the Hawkins-Simons condition and dX2/dXj < 0, then

[1 - a12a2j(dX2/dX1)]2

Therefore

d2X2/(dX1)2 < d2X2/(dx1)2 < 0.

Now we will compare two economies which are identical except that one requires Vanek intermediate goods in its production processes. For the same total factor supplies, the intermediate goods economy will be poorer in the sense that tv tv lies inside the gross transformation curve; therefore we endow

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Notes / 115

p

S

S v

I D

'_X / X ? (X/X2 1 -2

FIGURE 2

the non-intermediate goods economy with a smaller factor endowment such that its transformation curve TT is tangent to t, t, at point A, the pre-growth equilibrium. This situation is also pictured in Figure 2 in terms of general equilibrium supply and demand3 curves between p and x1/x2. S, is the supply curve with intermediate goods and is flatter than S, the supply curve without them.

The effect of augmenting the capital stock is shown by shifting S, and S to the right - the horizontal shift of S, being greater according to BP and equations (4). It should be noted that the demand curve is invariant to capital accumulation if the social indifference curves are homothetic. Although Figure 2 is consistent with the BP analysis, it is clear that their assertion does not follow, since the opposite result is illustrated with the relative price falling by more in the non-intermediate goods case.

Nevertheless, the BP conclusion would follow if it could be shown that S- and S,' intersect at a lower value of x1/x2 than (xl/x2),* that is, that point D lies to the left A. This requires proving that p,' < p' in Figure 3. In the non- intermediate goods economy (where Xi = xi) the change in x1/x2 resulting from capital augmentation is

d(xl/x2) _ x2(dx1/dK) - xl(dx2/dK) dK X22

(6) x2f1 + XJf2

(k -k2)X22

Moreover,

d(x1/x2) _ d(X1/X2) (7) dp dp

3 This paper introduces only domestic demand conditions because the foreign offer curve facing the country will be the same in both cases.

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116 / Neil Bruce

s s' p S~

Sv

p p

X X ? (X X/X2) 2

FIGURE 3

For the interinediate goods economy:

(d(xl/x2)\ a(1-a2a2l)(fAX2 + f2X1) ,

dK v (k x - (6')

d(x(dp2) =(l-al 2a2) d(X1/X2) (7') dp dp

By setting each

d(x/X2)/dK -d(X1/X2)/dP,

we can calculate the dp < 0 needed for xl/x2 = (x/x2)* after capital aug- mentation in each case. We find that the ratio

(dp)vr dp = fA 2 + f2X , 1. (8) f1X2 +.f2X1 8

Since X, > xi (because net outputs were equal in both cases at the initial point) and all other parameters are identical, the right-hand side of (8) clearly exceeds unity. Therefore (dp), < dp < 0 andpv' < p' < p* in Figure 3. Thus, in Figure 3 the equilibrium price in the intermediate goods case will be lower than in the non-intermediate goods case, so the BP conclusion is correct.

However, the possibility that the indifference map is not homothetic would qualify their conclusion. In this case, the demand curve would shift with capital augmentation. If the income elasticity of x, were greater than unity, D would shift to the right in Figure 3 and the BP conclusion would hold a fortiori. If the income elasticity were smaller than unity (or especially when negative), D would shift to the left in Figure 3, making possible the example illustrated with D' where the BP conclusion is incorrect.

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Notes / 1 17

REFERENCES

Batra, R. N. and P. K. Pattanaik (1971) 'Economic growth, intermediate products, and the terms of trade.' This Journal 4, 225-37

Rybczynski, T. M. (1955) 'Factor endowment and relative commodity prices.' Economica 22, 336-41

Vanek, J. (1963) 'Variable factor proportions and inter-industry flows in the theory of international trade.' Quarterly Journal of Economics 78, 129-42

A note on welfare-increasing trade diversion

HARRY G. JOHNSON / London School of Economics and

Political Science and University of Chicago

One of the relics of criticism of Viner's distinction between trade creation and trade diversion in the analysis of the welfare-increasing and welfare-reducing effects of customs union is the possibility that trade diversion may be welfare- increasing, in the sense that the welfare loss from diversion to a higher-cost source of supply may be more than outweighed by the gains from substitution induced by the reduction of the price to consumers due to the elimination of the tariff on imports.' I have always taken the view that this possibility is a needless analytical complication, since any such gain must involve an increase in the volume of trade and hence 'trade creation' like any other trade-creation source of gain.2 The present note, however, accepts the conceptualization that permits the possibility of welfare-increasing trade diversion and examines for some simple cases the empirical conditions required for it to occur. The simple cases involve complete specialization of the customs-union-forming country on its exportable good and two alternative specializations of its utility func- tion: the Cobb-Douglas form, involving unitary income and price elasticity

1 The possibility was produced independently by Lipsey (1957) and Gehrels (1956), though it was implicit in Meade (1955) and subsequently became a standard part of customs unions theory as developed at the London School of Economics. For references, see Lipsey (1970) and the survey by Krauss (1972).

2 For references to the views of myself and others on this question, and a general endorse- ment of my approach, see the survey by Krauss mentioned in n. 1. More recently, what may be called "The London School of Economics line" has been presented by Bhagwait (1971) and Batra (1973). For a comment on Bhagwati and his subsequent controversy with Kirman, together with references, see Johnson (1974). In the context of that comment, it is worth pointing out that Lipsey (1970, 20) quotes the statement from Viner quoted in my comment, which clearly refers to new trade and shifts of existing trade, but then identifies trade creation and diversion with shifts of production location without reference to the volume of trade. Clearly, any increase in total importation of the goods whose production location is shifted from foreign to union sources, caused by the untaxed price of partner supplies being lower than the taxed price of foreign supplies, involves trade creation and not trade diversion.

Canadian Journal of Economics/Revue canadienne d'Economique, VIII, no. 1 February/f6vrier 1975. Printed in Canada/Imprime au Canada.

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