CBMR

76261986214

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RESEARCH MEMORANDUM

ILBURG. UNIVERSITY

EPARTMENT OF ECONOMICSistbus 90153 - 5000 LE Tilburgstherlands

~~~c.u.e.Bii~LIOTHE~~:Til~BURG

'.he theo:y of wage differenti.a]s:a correction

A..1.W. van c~e Gevel

1

'fhe authoritative article by Bhagwati and Srinivasan (1971)

tried to prove that when thPre is a distortionary wage differential

between sectors the production possibility curve might tiave both convex

and cencave stretches. This was based on the sign of the second derivative.

However, their complex equation (15) and their next ones as special cases

contain a rtiistake. This paper presents the correct outcomes. The Bwagwati-

`:r-nivasan conclu~-on~ are af:'ected in the following way.

1. 'fhe condit:on~ unZe~ w.,;c'z ~ne f-on:~er -.~ convex a':, one speciali~at'or.

,„-~.t and concave at the other are somewhat more intricats.~ than chose

stated by Bhagwati and Srinivasan. A general classificati~n of the

conditions leading to different combinations of curvatures at the

sper..ialisation points is presented.

2. In the special case of CES production functions, the production possibi-

lit.y frontier wil] be convex under less stringent condition than those

stat,ed by Bhagwati and Srinivasan.

:'he cerrect equat;or. `o. tre cecend derivative .~.

d2Q1 -~N(R2 - R1)2 ~ N(YR1 - R2) ( Y - 1)Flp2flflt -dQ2 D2 (w t R1)(y w t R2) D

X j~(R2 - R)a1R1 t(R - R1)a2R2}~al(R2 - R) t a2(R - R1)}

r da do- w(R2 - R1)(R2 - R)(R - R1)ja2

1-Q1 2

L dw dw

-(~r - 1)flfi r .- {a 1R1(R2 - R) t a2R2(R - R1)~ al a2(R2 - R1)(R1R2 - wR)

D ((( (15

2

'Phis re~~ilc influences the outcomes for the cases of complete spe-

cialisation. In the case of complete specialisation in Q1 the

result is:

d`i~ 1

dP2

-w(R1 - R)? r N(YR - R2)

D2 (w t R)(yw t R2)

t (Y - ~) (fifiai(R2 - R)2 R2i R2(1 - a2) t Q2w~

D `

r ~ ~ ~where N --fi J(w t R2)(R2 - R) a1R ~ 0 as R1 ~ R ~ R2

and D- fi I(Yw t R2)(R2 - R) alRl ~ O as R1 ~ R~ R2

For the case of complete specialisation in Q2 the result is:

d2QI -w(R - R1)2 r N(YR1 - R)

dQ2 p2 L(w t R1)(Yw t R)

t (Y - 1) j lfifio2 (R - R1)2 R2 {R1(1 - Q1) } Qlw ~ JD l

where N--fi 1(w t R1)(R - R1) a2R ~~ 0 as R1 ~ R ~ R2

~ ~ ~and D- fi ~(Yw t RI)(R - R1) a2R ~~0 as R1~ R ~ R2

These revised outcomes have certain consequences for the conditions

under which the second derivative in the neighbourhood of the points

of specialisation is negative or positive. These conditions differ

3

f'rom those of Fiha~wat~ and Srinivaoan especially with respect to

a.íi - ~~)iIn case RI ~ R~ R2, so that N ~ 0 and D ~ 0,

the second derivative for complete specialisation in QI is negative,

i.e. concavity, if both terms in square brackets are positive. This

holds if YR ~ R2, what is certain if Y~ 1 and is possible even if

Y ~ 1, and either if Y~ 1 and o 2~ 1 or if Y ~ 1 and a 2 ~ 1.

For complete specialisation in Q2 the second derivative is positive,

i.e. convexity, if both terms in square brackets are negative.

This holds if YR1 ~ R, that requires that Y~ l, and either if Y~ 1

and al ~ 1 or if Y ~ 1 and al ~ 1. Thus there is a concavity for

complete specialisation in Q1 and convexity for complete speciali-

sation in Q2 if Y ~ 1, Y R ~ R2, 02 ~ 1, YRl ~ R and o 1 ~ 1.

In case R2 ~ R~ R1, so that N ~ o and D~ o, the second

derivative for complete specialisation in Q1 is negative if both

terms in square brackets are positive. This holds if YR ~ R2, what

is certain if Y ~ 1 and is possible even if Y~ 1, and either if Y~ 1

and a2 ~ 1 or if Y~ 1 and a2 ~ 1. For complete specialisation in Q2

the second derivative is positive if both terms in square brackets

are negative. This holds if YR1 ~ R1 what requires that Y~ 1,

and either if Y~ 1 and aI ~ 1 or if Y ~ 1 and ol ~ 1. Thus due to

the requirement that Y~1. There is a possibility of concavity for

complete specialisation in Q1 and convexity for complete speciali-

sation in Q2 if Y~ 1, Y R ~ R2, 02 ~ 1, Y R1 ~ R and ol ~ 1.

In order to save space we summarize the different possibilities

by presenting next table.

4

Concavity in Q1 and

Convexity in Q2

Convexity in Q1 and

Concavity in Q2

Cc~ncavity in Q1 and

Concavity in Q2

Convexity in Q1 and

Convexity in Q2

Table 1

R1 ~ R ~ R2

(N ~ o, D ~ o)

y ~ 1 YR ~ R2 02 ~ 1

YR1 ~ R al ~ 1

y ~ 1 YR ~ R2 a2 ~ 1

yRl ~ R al ~ 1

y~ 1 YR ~ R2 02 ~ 1

YR1 ~ R ol ~ 1

Y ~ 1 YR ~ R2 a2 ~ 1

yRl ~ R al ~ 1

R2 ~ R ~ Rl

(N ~ o, D ~ o)

y~ 1 yR ~ R2 a2 ~ 1

YR1 ~ R ol ~ 1

Y~ 1 YR ~ R2 a2 ~ 1

YR1 ~ R al ~ 1

Y ~ 1 YR ~ R2 a2 ~ 1

YR1 ~ R al ~ 1

Y~ 1 YR ~ R2 a2 ~ 1

yRl ~ R al ~ 1

J

Finally Bhagwati and Srinivasan consider the case in wich the

elasticities of substitution in both sectors are equal and constant.

The revised second derivative should read as:

d2Q1 -w(R2 - RI)2 r N(YR1 - R2)

d Q2 D2 ~, (w t R1)(Yw t R2)

}(Y D 1) fifi ( RZ - R2)2 a R~ R1 R2a(1 - a) } o2wRl

where N--fi a~(R2 - RI)(R1R2 t wR)} ~ o as R1 ~ R2

and D- fl a{(R2 - R1 )(R1R2 t Y wR)1 ~ o as R1 ~ R2

In case R1 ~ R ~ R2 throughout convexity is possible if Y R1~ R2,

what requires that Y ~ 1, and if a~l. In case R2 ~ R~ Rlthroughout

convexity is possible if YR1~ R2, what requires that Y~ 1, and if

a ~ 1.r 1

For the CES function fl - LaiRiE t ( 1 - a i)1 - e the revised

second derivative becomes:

d2Q1 -w ( n- 1) 3R13 f 1 r( n-Y) Rl a ( wR t n Ri )

d Q2 -

t

D2 ~, ( w f Rl )( Yw t n Rl )

(Y-. 1) R{Ri n a(1 -a ) ta2 w R~

( Y w R t n 1)

If a 1- a 2 and a ~ 1 the second derivative is positive because

either 1 ~ n~ Y or Y~ n~ 1. Thus the production possibility

curve is indeed convex throughout, although the condition on the

R2 J(16)

6

elasticity of substitution is less stringent than suggested byBhagwati and Srinivasan.

References:

J.N. Bhagwati and T.N. Srinivasan, 1971, The theory of wagedifferentials: production response and factor price equalisation,Journal of International Economics, 1, 19-35.

i

IN 1985 REEDS VERSCHENEN

l68 T.M. lloup, A.J.J. TalmanA con[inuous deformation algorithm on the product space of unitsimplices

169 P.A. BekkerA note on the identification of restricted factor loading matrices

170 J.H.M. Donders, A.M, van NunenEconomische politíek in een twee-sectoren-model

171 L.H.M. Bosch, W.A.M, de LangeShift work in health care

172 B.B. van der GenugtenAsymptotic Normality of Least Squares Estimators in AutoregressiveLinear Regression Models

173 R.J, de GroofGeisoleerde versus gecoSrdíneerde economiache politiek in een twee-regiomodel

174 G. van der Laan, A.J.J. TalmanAdjustment processes for finding economic equilibria

175 B.R. MeijboomHorizontal mixed decomposition

176 F. van der Ploeg, A.J. de ZeeuwNon-cooperative strategies for dynamic policy games and the problemof time inconsistency: a comment

177 B.R. MeijboomA two-level planning procedure with respect to make-or-buy deci-sions, including cost allocations

178 N.J. de BeerVoorspelprestaties van het Centraal Planbureau in de periode 1953t~m 1980

178a N.J, de BeerBIJLAGEN bij Voorspelprestaties van het Centraal Planbureau in deperiode 1953 t~m 1980

179 R.J.M. Alessie, A. Kapteyn, W.H.J. de FreytasDe invloed van demografische factoren en inkomen op consumptieveuitgaven

180 P. Kooreman, A. KapteynEstimation of a game theoretic model of household labor aupply

181 A.J. de Zeeuw, A.C. MeijdamOn Expectations, Information and Dynamic Game Equilibria

ii

18'L Crístina PennavajaPeriodizatíon approaches of capitalist development.A critical survey

I8"i J.k'.C. Kleijnen, c;.L.J. Kloppenburg and F.L. Meeuwsen'festinR the mean of an asymmetric population: Johnson's modified Ttest revisited

184 M.O. Nijkamp, A.M. van NunenFreia versus Vintaf, een analyse

185 A.Ii.M. GerardsHomomorphisms of graphs to odd cycles

186 P. Bekker, A. Kapteyn, T. WansbeekConsistent sets of estimates foruncorrelated measurementvariables

errorsregressions with correlated orin arbitrary

187 F'. Bekker, J. de LeeuwTlie rank uf reduced dispersion matrices

subsets of all

188 A.J. de Zeeuw, F. van der PloegConsistency of conjectures and reactions: a critique

189 E.N. KertzmanBelastingstructuur en privatisering

190 J.P.C. KleijnenSimulation with too many factors: review of random and group-screening designe

191 J.P.C. KleijnenA Scenario for Sequential Experimentation

192 A. DortmansDe loonvergelijkingAfwenteling van collectieve lasten door loontrekkers?

193 R. Heuts, J. van Lieshout, K. BakenThe quality of some approximation formulas in a continuous reviewinventory model

194 J.P.C. KleijnenAnalyzing simulation experiments with common random numbers

195 P.M. KortOptimal dynamic investment policy under financial restrictions andadjustment costs

196 A.H. van den Elzen, G. van der Laan, A.J.J. TalmanAdjustment processes for finding equilibria on the simplotope

iii

197 J.Y.C. KleijnenVariance heterogeneity in experimental design

198 J.P.C. KleijnenSelecting random number seeds in practice

199 J.P.C. Kleijnenltegression analysis of simulation experimenta: functional softwarespecification

200 G. van der Laan and A.J.J. TalmanAn algorithm for the linear complementarity problem with upper andlower bounds

201 P. KooremanAlternative specification tests for Tobit and related models

iv

IN 1986 REEDS VERSCHENEN

2l)2 J.N.F. SchilderinckInterregional Structiire of the European Community. Part III

203 Antoon van den Elzen and Dolf TalmanA new strategy-adjustment process for computing a Nash equílibriumin a noncooperative more-person game

204 Jan VingerhoetsFabrication of copper and copper semis in developing countries.A review of evidence and opportunities.

205 R. Heuts, J. v. Líeshout, K. BakenAn inventory model: what is the influence of the ahape of the leadtime demand distribution?

206 A. v. Soest, P. KooremanA Microeconometric Analysis of Vacation Behavior

207 F. Boekema, A. NagelkerkeLabour Relations, Networks, Job-creation and Regional DevelopmentA view to the consequences of technological change

"208 R. Alessie, A. KapteynHabit Formation and Interdependent Preferences i n the Almost IdealDemand System

209 T. Wansbeek, A. KapteynEstimation of the error components model with i ncomplete panels

210 A.L. HempeniusThe relatiod between dividends and profits

211 J. Kriens, J.Th, van LieshoutA generalisation and some properties of Markowitz' portfolioselection method

212 Jack P.C. Kleijnen and Charles R. StandridgeExperimental design and regression analyeis in simulation: an FMScase study

213 T.M. Doup, A.H. van den Elzen and A.J.J. TalmanSimplicial algorithms for solving the non-linear complementarityproblem on the simplotope

Ai i ~ u~~~m i MU ~i wai~~u~u

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