A FORMAL PROOF OF THE FACTOR PRICE EQUALIZATION THEOREM
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1 DOCUENO CEDE ISSN (Edición Electrónic EBRERO DE 2004 CEDE A ORA ROO O HE ACOR RICE EQUAIZAION HEORE HERNÁN VAEJO G. Abstrct his pper provides forml proof of the ctor rice Equliztion heorem within the Heckscher Ohlin model derived by Ronld W. Jones in he Structure of Simple Generl Equilibrium odels (965, where forml proof is provided for the Heckscher Ohlin, Stolper Smuelson nd Rybczynski heorems. Key Words: Interntionl rde, Heckscher Ohlin, Rybczynski, Stolper Smuelson, ctor rice Equliztion. JE Clssifiction:. Deprtment of Economics, CEDE, Universidd de los Andes, hvllejo@unindes.edu.co. he uthor grtefully cknowledges useful comments mde by eopoldo ergusson. Comments on this working pper re welcomed.
2 UNA RUEBA ORA DE EOREA DE IGUAACIÓN DE RECIOS DE OS ACORES Resumen Este documento provee un prueb forml del eorem de Igulción de recios de los ctores dentro del modelo de Heckscher Ohlin desrrolldo por Ronld W. Jones en el rtículo he Structure of Simple Generl Equilibrium odels (965, en el cul se presentn pruebs formles de los teorems de Heckscher Ohlin, Stolper Smuelson y Rybczynski. lbrs clve: Comercio Interncionl, Heckscher Ohlin, Rybczynski, Stolper Smuelson, igulción de precios de los fctores. Clsificción JE:. 2
3 I. INRODUCION his pper provides forml proof of the ctor rice Equliztion heorem within the Heckscher Ohlin model derived by Ronld W. Jones in he Structure of Simple Generl Equilibrium odels (965, where forml proof is provided for the Heckscher Ohlin, Stolper Smuelson nd Rybczynski heorems. he bsic structure of the model is:. wo countries: (A nd B. b. wo goods: nufctures nd food. c. wo fctors of production: bor nd lnd. he ssumptions of the model re:. echnologies of production re homogeneous of degree one, (i.e., they re homothetic nd hve constnt returns to scle nd re identicl in both countries. b. here re different fctor intensities in the two sectors (nd there re no fctor intensity reversls for ll rnges of fctor prices. nufctures is lbor intensive while food is lnd intensive. c. Consumer preferences re identicl in both countries. d. All mrkets re competitive, there re no trnsporttion costs within nd between countries nd there is no government (nd no trde policy. 3
4 e. Both countries re identicl in everything except in their fctor endowments, which re different but sufficiently similr in order to void complete speciliztion in production (fctors lie within the diversifiction cone. Nottion cn be expressed s follows: bor endowment. nd endowment. nufctures sub-index. ood sub-index. Q j Output of sector j, where j or. j bor used in sector j, where j or. j nd used in sector j, where j or. w Wges. r Rent. Unitry price of mnufctures under free trde. Unitry price of food under free trde. ij Amount of fctor i required to produce one unit of good j. ositive technologicl prmeter. ositive technologicl prmeter. II. A ORA ROO O HE ACOR RICE EQUAIZAION HEORE he ctor rice Equliztion heorem sttes tht under the ssumptions, free trde will mke fctor prices equl in the prticipting countries. Since the unitry fctor requirements re endogenous (they depend on fctor prices, it is importnt to provide production functions with functionl forms tht comply with the bsic Heckscher Ohlin ssumptions, in order to prove the ctor rice Equliztion heorem. 4
5 As noted erlier, technologies re identicl in ll countries nd re ssumed to be homothetic nd to hve constnt returns to scle. hey re lso different between sectors (mnufctures is lbor intensive nd food is lnd intensive. hus, the following Cobb Dougls production functions were ssumed in this pper: Q ( Q ( No fctor intensity reversls imply tht for ny fctor prices: > Homothetic production functions imply tht given ny fctor prices: he unitry production functions cn be represented by: ( ( ( (2 5
6 nd thus, for ny fctor prices2: > he zero profit conditions re: w + r C (3 w + r C (4 hus, unitry profits re: Π ( ( w ( r (5 Π ( ( w ( r (6 Equtions ( to (6 give us system of 6 equtions nd the following 6 endogenous vribles: w, r,,,,. his system hs four exogenous vribles:,, nd. nd re constnt technologicl prmeters tht re identicl in both countries by ssumption, while nd re lso identicl in the two countries under free 2 he ppendix of this pper shows tht for this condition to be fulfilled, ll tht is required is tht >. 6
7 trde becuse of competitive mrkets nd perfect rbitrge, given tht there re no trnsport costs nd no trde brriers. In wht follows, ll 6 endogenous vribles will be expressed in terms of the exogenous vribles. Differentiting (5 for profit mximiztion in mnufctures: Π ( ( ( w Π ( r irst order conditions stte tht: ( ( ( w ( r Dividing the lst two equtions: ( ( ( r w ( ( ( r w (7 Differentiting (6 for profit mximiztion in food: 7
8 Π ( ( ( w Π ( r irst order conditions stte tht: ( ( ( w ( r Dividing the lst two equtions: ( ( ( ( r w ( ( ( r w (8 rom (7, wges cn be expressed s: w ( r ( ( (9 Replcing wges into (3: ( r ( ( + r 8
9 ( r + r ( r ( + + ( ( r ( ( ( r r ( ( (0 Replcing (0 into (9: w ( ( ( ( ( ( w ( rom (8: ( w ( ( r (2 Replcing (2 into (4: 9
10 w ( ( w + r ( r w ( + ( + w ( w w (3 Replcing (3 into (2: ( w ( rw ( r (4 Replcing ( into (3: ( (5 Replcing (0 into (4: 0
11 (6 rom (: (7 Replcing (5 nd (6 into (2: (8 Replcing (7 in (8: +
12 2 (9 Defining: (20 nd replcing it into (, (5 nd (7: w (2 (22 (23 Defining:
13 ( ( ( (24 nd replcing it into (0 nd (6: ( r (25 ( ( (26 In the system defined by equtions ( to (6, ll six endogenous vribles hve now been expressed in terms of the exogenous vribles. ctor prices nd unitry fctor requirements re identicl between countries since they depend on prmeters tht re identicl in both countries. Goods prices re equl becuse of free trde, competitive mrkets nd perfect rbitrge due to the lck of trnsporttion costs nd trde brriers, while nd re identicl becuse of equl technologies. hus, wges nd rent will be the sme in both A nd B s shown in equtions (2 nd (25. QED. III. SOE URHER RESUS In wht follows, some of the results obtined so fr re used to find expressions for Q, Q,,, nd, bsed on the exogenous prmeters. he use of fctors in ech sector is defined s: Q (27 Q (28 3
14 Q (29 Q (30 Competitive mrkets nd full employment of fctors imply tht: Q + Q (3 Q + Q (32 Equtions (27 to (32 give us gin system of 6 equtions nd the following 6 endogenous vribles:,,,, nd. In this cse, there re six exogenous vribles which re:,,,. nd where nd re the fctor endowments. Solving for Q nd Q in (3 nd (32 by Crmer s Rule: Q (35 Q (36 4
15 5 Replcing (20, (22, (24 nd (26 into (35 nd (36: Q Q (37 Q (38 is defined s Q. hus: (39 is defined s Q. hus: (40 is defined s Q. hus:
16 6 (4 is defined s Q. hus: (42 Once more, ll the endogenous vribles hve been expressed in terms of the exogenous vribles. he results obtined show tht both the totl employment of ech fctor per sector nd the output of food nd mnufctures my differ between countries since these vribles depend on fctor endowments (which re ssumed to be different, s indicted by equtions (37, (38, (39, (40, (4 nd (42. hese results re in shrp contrst to fctor prices nd unitry fctor requirements, which hve lredy been shown to be identicl in both ntions.
17 IV. CONCUSIONS his pper hs shown tht it is possible to prove the ctor rice Equliztion heorem in the frmework developed by Jones (965 with endogenous unitry fctor requirements in production, by providing functionl form tht meets the Heckscher Ohlin ssumptions. It hs lso shown tht under the model s ssumptions, unitry fctor requirements per sector will lso be identicl between countries, while totl use of fctors in ech sector nd totl output of mnufctures nd food will be different in ech ntion. V. BIBIOGRAHY Jones, Ronld W. (965 he Structure of Simple Generl Equilibrium odels he Journl of oliticl Economy Vol XXIII December 965 No. 6 pp Krugmn. R, y. Obstfeld (200 Interncionl Economics, ifth Edition, cgrw-hill Appleyrd D. R. y A. J. ield. (997 Economí Interncionl, cgrw-hill, dird. VI. AENDIX: HE NO ACOR INENSIY REVERSA CONDIION or no fctor intensity reversls, the required condition is tht for ny fctor prices: > Replcing ech term with the expressions (20, (22, (24 nd (26: 7
18 ( ( ( ( ( ( Now, > ( ( if nd only if >. QED. 8
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