A Note on 'Sector Specific Capital, Interconnectedness in Production, and Welfare

A Note on 'Sector Specific Capital, Interconnectedness in Production, and Welfare'Author(s): Leslie YoungSource: The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 19, No. 4(Nov., 1986), pp. 678-684Published by: Wiley on behalf of the Canadian Economics AssociationStable URL: http://www.jstor.org/stable/135320 .

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A note on 'Sector specific capital, interconnectedness in production, and welfare' LESLIE YOUNG University of Texas at Austin

Abstract. We note some errors in a recent paper by Chaudhuri and Khan on the effects of international investment in a labour surplus country and identify a mechanism whereby comparative static changes apparently harmful to the investing country can end up benefiting it.

Note sur 'Le capital dont l'utilisation est specifique a un secteur, l'interconnexion de la production et le bien-etre' L'auteur identifie quelques erreurs dans l'article recent de Chaudhuri et Khan sur les effets de l'investissement international dans un pays qui a un surplus de main d'oeuvre et identifie un mecanisme par lequel des changements en statique comparative qui paraissent nuisibles pour le pays investisseur finissent par lui etre benefiques.

In a recent paper in this JOURNAL, Chaudhuri and Khan (1984) postulated an industrialized country 'Industria' which invests in the resource sector of a labour-surplus country 'Resourcia' producing food, manufactures, and minerals. They considered the effects of changes in world prices, endowments, and policy variables on factor rewards, employment, and the welfare of the two countries. However, their analysis of the welfare effects contains serious errors (specifically in their propositions x, xii, and xiv).1 This note reconsiders their welfare results and identifies an important mechanism that operates when there is international investment in a labour-surplus country. The benefits of comparative static changes which appear to favour the latter can then be entirely appropriated by the investing country via increases in its return to investment in the host country.

In the Chaudhuri-Khan (CK) model, Industria and Resourcia have

I In a subsequent paper Chaudhuri and Khan (1985) present an alternative model with no food sector and with capital flows between the two countries. In this model, unlike the paper under review, they include tax revenue in Resourcia's objective function and import payments in Industria's objective function.

Canadian Journal of Economics Revue canadienne d'Economique, XIX, No. 4 November novembre 1986. Printed in Canada Imprime au Canada

0008-4085 / 86 / 678-684 $1.50 ? Canadian Economics Association



A note on sector specific capital 679

endowments K* and K of capital and L and L of labour. Industria produces only manufactures X* using labor Lu*, capital Ku* and a mineral resource Mu . Resourcia produces: food Xf using labour Lf and capital Kf; manufactures Xu using labour Lu, capital Ku and the mineral resource Mu; and the mineral resource itself Xr using labour Lr and capital from Industria Kr . There are exogeneous world prices Pf and Pu for food and manufactures, but the price Pr of minerals is determined endogeneously.

Resourcia has surplus labour at the subsistence wage w and its employmernt is demand-determined. Industria has full employment at an endogenously determined wage w*. Let R and R* be the returns to capital in Resourcia and Industria, respectively. Under perfect competition and constant returns to scale, price equals unit cost in each industry, that is,

Pf= Cf(R,O) (1)

Pu Cu (R v,Pr) (2)

Pr = Cr(R* F) (3)

Pu = Cu (R ,w ,P) (4) where the right-hand sides are the unit cost functions of the corresponding goods.

Given w-, Pf and Pu*, equations (1)-(4) determine the equilibrium values of R, Pr, R, and w, and hence the factor intensities in the four industries. Given these intensities, outputs are determined by the market-clearing conditions for Industrian labour, Industrian capital, Resourcian capital, and the mineral resource. The output levels and labour intensities of the three Resourcian industries then determine Resourcian employment at the subsistence wage:

Le = Lr + Lu + Lf

We first consider CK's analysis of the effects of an ad valorem tariff t on Resourcian imports of manufactures. The Resourcian domestic price of manufacturers is then PuI(1 + t), which now replaces Pui on the left side of (2). CK assert the following.

CK PROPOSITION X. Under conditions of laissez faire, that is t = 0, a tariff improves Resourcian welfare. If there is already an existing tariff the effect of a further increase depends on the compensated demand elasticity of imports. In such a case, a tariff increase is beneficial if and only if an improvement in domestic income more than compensates for the loss of tariff earnings.

CK'Is income-expenditure identity for Resourcia is:

g(Pf, Pu*(1 + t), U) = Le +RK + Pu* t(Zu - Xu), (CK16)

where g is the Resourcian expenditure function, U is its utility level, and Zu is its demand for manufactures. CK differentiate this to obtain



680 Leslie Young

go[ - mt/(l + t)(aU/at)] = w(aLe/at) + K(aR/at) + Pu[t(Zu - XU)/(1 + t) ]E, (CK17)

where go = ag/laU, m is the Resourcian marginal propensity to consume manufactures and cu is its compensated price elasticity of demand for manufacturing imports. (cK17) is incorrect. Differentiation of (cK16) yields

Pu*(ag/aPu) + go(dU/at) = (aLe/at) + K(aR/at) + P*(Z - XU) + Pu*t{ (aZu/apu) - (aXu/apu) (au/at)]. (5)

But ag/aPu = Zu and aZu/aU = mgo/Pu*(1 + t), so (5) simplifies to

go[l - mt/(l + t) ](aU/at) = w(dLe/at) + K(aR/at) + Pu [t(Zu - XU)/(1 + t) ]u - Pu U (6)

Thus, CK have omitted the term - X which captures the direct effect of the tariff on the value of its output of manufactures.2 Since aR/at = 0 and stability implies that 1 - mt/(l + t) > 0, a U/at has the sign of

PaLe/at - *X + tPu*[t(Zu - xu)/(l + t) ],E* (7)

Thus, Resourcia would be harmed by a tariff increase if and only if the increase in its labour income from increased employment is less than the increase in the internal value of current Resourcian manufacturing plus the (absolute value of the) loss in tariff revenue due to the substitution effects of the rise in the internal price of manufactures. In particular, it is clear that a rise in t could reduce Resourcian welfare even if t 0, contrary to CK proposition x.

CK PROPOSITION XII. 'Under conditions of laissez-faire, that is t = 0 and under the assumptions of proposition IX, (a) a decrease in the profit tax and an increase in the export tax increases Resourcian welfare. (b) An increase in the subsistence wage improves Resourcian welfare if and only if the improvement on the part of employed Resourcian labour is compensated for by the loss of employment and capital income. When t > 0, all the above assertions have to be modified to take account of the fall in tariff revenue.'

The proof offered is that if z denotes T, v, or w-, then differentiating (cK16) yields

goaU/az = WP(aLe/az) + K(aR/az) - Pu*t(dXu/az). (CK18)

However, if z = T then (cK 16) is not the income-expenditure equation for Resourcia: CK have omitted the term TRR*K,* representing the revenue from the profit tax which is, of course, part of Resourcian income. The derivative of this term needs to be added to (cK18) to yield

go(aU/WaT) = iv(dLe/dT) + Rr* + TRr (dr/dT),

2 The mislocation of a right-hand bracket behind dU/lt instead of in front of it on the left of (cK 17) and the omission of a star above Pu on the right side of (cK 17) are presumably print- ers' errors.




where we have used the fact that aR/aT = 0. Contrary to CK proposition xIia, a decrease in the profit tax need not increase Resourcian welfare; an increase in welfare would occur if and only if the increase in labour income due to increased employment outweighed the loss of tax revenue.

Similarly, if z = v then (cK16) omits the term vP,.M, representing the revenue from the export tax which is now part of Resourcian income. Since aR/av = 0, when t =- 0 the correct version of (cK18) is

go(aU/av) - i(dLe/dv) + P,Mu + VPP(dMu*/dV).

Thus an increase in the export tax will increase welfare if and only if the increase in tax revenue outweighs the loss of labour income due to decreased unemployment.

In connection with a rise in the price of food, CK state: 'We cannot determine, without additional information whether improvements in the terms of trade (in the case at hand the price of its imports) help or harm Industria' (501). In fact, we shall show that a rise in the price of food always harms Industria. The reason for the discrepancy is as follows: CK write the income-expenditure identity for Industria as

g (Pf, P*, U) - PuXu + R Kr = P X* + R*X/.C,j. (CKl9)

and differentiate this to obtain

g*o(a */apf) = Zf + Pu*(aX*/aPf) + Xr*Crl[I + (R*Crl /C) ](aR*/aPf). (cK20)

CK's negative conclusion comes from their inability to sign this expression. Both (cK19) and (CK20) are incorrect, because income is taken to equal the value of Industrian output of manufactures plus factor payments from abroad. CK have thereby omitted to subtract Industrian payments for imports of minerals to use in manufacturing.3 In fact, Industrian income is domestic value-added Pu Xu - Pr Mu* plus factor payments from abroad R,.* Kr, leading to the income-expenditure identity

g*(Pf, PU1 U ) = P Xu* - PrMu* + R*K,r*. (8)

Differentiation with respect to Pf yields

g (Ppf 1<, U*) - Zf* + Pu*(aXu*/aPf) - PraMu*/apf) - Mu*(apr/aPf) * I *)

z * 11u ra

+ Xr Cr (aR laPf) + XrCr (R Cl. /Cr1)(WR/aPf) + R*Crl(aXr/aPf). (9) This expression differs from (CK20) not only by including the impact of Pf on the payments for mineral imports but also by including a term R*Crl(aXr/aPf),

reflecting the effect of Pf on the output of minerals, which CK ignored. Equation (9) can be simplified in the following way. A rise in Pf affects XU

and Mu* only by altering the level of capital Ku* invested in Resourcia. The

3 This would not have been necessary if CK had expressed the income of Industria as the sum of payments to its factors w*L* + R*K*. However, if income is expressed as the value of output, then payments for imports must be netted out.



682 Leslie Young

Envelope Theorem implies that if the capital in Industria were increased by one unit then the maximum value of Industrian value-added would increase by the marginal revenue product R* of Industrian capital. Combining the last two statements we have

Pu*(aXu*/aPf) - Pr(aMu*/aPf) = R*(dKu*/dPf) = -R*(dKr*/aPf). (10)

Since XrCr' = Kr,

R*Crl(aXr/aPf) + XrCrl(R*Crll/Crl)(aR */aPf) = R *(dKr /aPf). (11)

Substituting (10) and (11) into (9) yields:

g0*(au*aPf) L -Zj + K*aR*/af - M* apr/aPf (12)

Applying the Implicit Function Theorem to (3) and using the Shepherd Samuelson Relations:

aR*/aPr = Xr/Kr*

Since

Xr- Mu = Mu,

equation (12) implies that

go*(aU*/aPf) = Zf + Mu(aPr/aPf). (13)

Applying the Implicit Function Theorem to (1) and (2) and using the Shepherd-Samuelson relations:

aR/aPf = Xf/Kf

aPr/aR = -Ku/Mu.

Hence,

aPr/aPf = -XfKu/KfMu. (14)

By (13) and (14)

go* (a U*/ P) =-Zf - (XfKu/Kf). Hence a rise in Pf always harms Industria.

In stating the next proposition, CK defined ar to be the elasticity of substitution and 0rL as the share of labour in the resource sector.

CK PROPOSITION XIV. (a) An increase in the Resourcian tariff or (b) a decrease in the profits tax is harmfulfor Industria. (c) An increase in the export tax always harms Industria. (d) If Kj/Lf- Ku/Lu > 0 and U,/OrL - 1, an increase in the Resourcian subsistence wage benefits Industria.

The CK analysis is as follows. Letting z denote t, T, v, or 0, they obtain from (cKl9)




go (a U /aZ) P* (X,,/lz) + XC,r'[1 - ar/rLI,aR/aZ. (CK22)

They previously showed that

1 + R*Cr/ICr l 1 ar/OrL. (CK21)

In differentiating (CK19) CK ignore the effect of z on Xr, omitting the term R*CrIaXr/az. Moreover, just as (cK19) omits the term -PrM, M, (CK22) omits the term -PraM*/ az - Mu* (aPr/az). Adding these terms to (cK22) and using the Envelope Theorem as above, the corrected version of (CK22)

simplifies to

go*(aU*/az) = Kr*(QR*/az) - Mu*(aPr/aZ). (15)

For z = t, the right side of (15) reduces to P* so Industria always benefits from a rise in t, contradicting CK proposition xiva.

For z = -T, in (15) R* should be replaced by the after-tax return. Rr (1 - Tr), where Rr* is the return to Industrian capital in Resourcia. Since Rr and Pr are independent of z, the right side of (15) reduces to Kr Rr Hence, Industria always gains from a rise in z - - contradicting CK proposition xivb.

For z = -, 3R*/az = 0, but in (3) Pr should be replaced by Pr(l + v). Since a(Pr(l + i) )/a3 = Pr, the right side of (15) reduces to -M Pr; that is, the export tax always harms Industria. Although CK proposition xivc is valid, the analysis supporting it is invalid, since it is based on the incorrect equation (CK22) rather than on the correct equation (15).

For z = wv-, we can calculate aR*/3i7v and aPr/ai7 by differentiating (1), (2), and (3) with respect to wv and using the Shepherd-Samuelson relations to obtain

dR/dw75 = Lf/ Kf (16)

o -Lu + Mu(dPr/dW) + Ku(dR/diiw) (17)

Xu(dPr/di3) =Kr*(dR*/dw) + Lr. (18)

Equations (16) and (17) imply that

Mu(dPr/dv) KuLf/ Kf - Lu- (19)

Substituting from (18) and (19) into (15) yields

go (aU /av) (KtLf/Kf) - Lu .

Hence, contrary to CK proposition xivd, Industria benefits from a rise in w7 provided that LfJKf > (Lu + Lr)/Ku. To summarize, we have established the following.

PROPOSITION. In the Chaudhuri-Khan model (a) A rise in the price offood always benefits Industria.



684 Leslie Young

(b) A Resourcian tariff on manufacturing imports from Industria always benefits Industria but even a small tariff can harm Resourcia.

(c) A reduction in the Resourcian tax on resource exports to Industria or on the return to Industrian capital invested in Resourcia always benefits Industria but benefits Resourcia only if the increase in labour income from increased employment outweighs the loss of tax revenue. -

(d) A rise in the Resourcian subsistence wage benefits Industria if and only if Lj/Kf > (Lu + Lr)/Ku

The striking aspect of these conclusions is that Industria benefits from comparative static changes which at first sight are harmful to it (rises in the price of food, the Resourcian tariff or the Resourcian subsistence wage). The mechanism is that these changes drive up the price of Resourcian exports of the resource. Since Resourcian labour works at an exogeneously given wage, all the benefits from this are skimmed off via a higher return to Industrian capital. This benefit to Industria can outweigh the negative direct effect of the original parameter change.

REFERENCES

Chaudhuri, T.D. and M.A. Khan (1984) 'Sector specific capital, interconnectedness in production, and welfare.' This JOURNAL 17, 489-507

Chaudhuri, T. and M.A. Khan (1985) 'Commercial policy in an asymmetric world economy.' Working Paper, College of Commerce and Business Administration, University of Illinois at Champaign-Urbana



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A Note on 'Sector Specific Capital, Interconnectedness in Production, and Welfare