This paper provides a formal proof of the Factor Price Equalization Theorem within the Heckscher Ohlin model derived by Ronald W. Jones in The Structure of Simple General Equilibrium Models" (1965), where formal proof is provided for the Heckscher Ohlin, Stolper Samuelson and Rybczynski Theorems."International Trade

Hernán Vallejo

English

Research Papers in Economics

 
 
 
 
A FORMAL PROOF OF  
THE FACTOR PRICE EQUALIZATION THEOREM 
 
 
HERNÁN VALLEJO G1. 
 
 
 
Abstract 
 
This paper provides a formal proof of the Factor Price Equalization Theorem within 
the Heckscher Ohlin model derived by Ronald W. Jones in “The Structure of Simple 
General Equilibrium Models” (1965), where formal proof is provided for the 
Heckscher Ohlin, Stolper Samuelson and Rybczynski Theorems. 
 
 
Key  Words:  International Trade, Heckscher Ohlin, Rybczynski, Stolper 
Samuelson, Factor Price Equalization. 
 
 
JEL Classification:  F1. 
 
 
 
 
                                                           
1   Department of Economics, CEDE, Universidad de los Andes, hvallejo@uniandes.edu.co.  The author 
gratefully acknowledges useful comments made by Leopoldo Fergusson. Comments on this working 
paper are welcomed. 
 
 CEDE 
DOCUMENTO CEDE 2004-05 
ISSN-1657- 7191(Edición Electrónica) 
FEBRERO DE 2004 
   2
 
 
 
 
UNA PRUEBA FORMAL DEL 
TEOREMA DE IGUALACIÓN DE PRECIOS DE LOS FACTORES 
 
 
 
Resumen 
 
Este documento provee una prueba formal del Teorema de Igualación de Precios 
de los Factores dentro del modelo de Heckscher Ohlin desarrollado por Ronald W. 
Jones en el artículo “The Structure of Simple General Equilibrium Models” (1965), 
en el cual se presentan pruebas formales de los teoremas de Heckscher Ohlin, 
Stolper Samuelson y Rybczynski. 
 
 
 
Palabras clave:  Comercio Internacional, Heckscher Ohlin, Rybczynski, Stolper 
Samuelson, igualación de precios de los factores. 
 
 
Clasificación JEL:  F1. 
 
 
 
   3
 
I.   INTRODUCTION 
 
This paper provides a formal proof of the Factor Price Equalization Theorem within 
the Heckscher Ohlin model derived by Ronald W. Jones in “The Structure of 
Simple General Equilibrium Models” (1965), where formal proof is provided for the 
Heckscher Ohlin, Stolper Samuelson and Rybczynski Theorems. 
 
The basic structure of the model is: 
 
a.  Two countries:    (A and B). 
b.  Two  goods:      Manufactures  and  food. 
c.  Two factors of production:  Labor and land. 
 
The assumptions of the model are: 
 
a.  Technologies of production are homogeneous of degree one, (i.e., they are 
homothetic and have constant returns to scale) and are identical in both 
countries. 
 
b.  There are different factor intensities in the two sectors (and there are no 
factor intensity reversals for all ranges of factor prices).  Manufactures is 
labor intensive while food is land intensive. 
 
c.  Consumer preferences are identical in both countries. 
 
d.  All markets are competitive, there are no transportation costs within and 
between countries and there is no government (and no trade policy). 
   4
e.  Both countries are identical in everything except in their factor endowments, 
which are different but sufficiently similar in order to avoid complete 
specialization in production (factors lie within the diversification cone). 
 
Notation can be expressed as follows: 
 
L  =   Labor  endowment. 
T  =   Land  endowment. 
M   =   Manufactures sub-index. 
F   =   Food sub-index. 
Qj =  Output  of  sector  j, where j = M or F. 
Lj  =  Labor used in sector j, where j = M or F. 
Tj  =  Land used in sector j, where j = M or F. 
w =   Wages. 
r =  Rent. 
PM  =   Unitary price of manufactures under free trade. 
PF  =   Unitary price of food under free trade. 
aij =  Amount  of  factor  i required to produce one unit of good j. 
α  = Positive  technological  parameter. 
β  = Positive  technological  parameter. 
 
II.  A FORMAL PROOF OF THE FACTOR PRICE EQUALIZATION THEOREM 
 
The Factor Price Equalization Theorem states that under the assumptions, free 
trade will make factor prices equal in the participating countries. 
 
Since the unitary factor requirements are endogenous (they depend on factor 
prices), it is important to provide production functions with functional forms that 
comply with the basic Heckscher Ohlin assumptions, in order to prove the Factor 
Price Equalization Theorem. 
   5
As noted earlier, technologies are identical in all countries and are assumed to be 
homothetic and to have constant returns to scale.  They are also different between 
sectors (manufactures is labor intensive and food is land intensive).  Thus, the 
following Cobb Douglas production functions were assumed in this paper: 
 
() ()
() α α − =
1
M M M T L Q  
 
() ()
() β β − =
1
F F F T L Q  
 
No factor intensity reversals imply that for any factor prices: 
 
F
F
M
M
T
L
T
L
>  
 
Homothetic production functions imply that given any factor prices: 
 
 
TM
LM
M
M
a
a
T
L
=  
 
 
TF
LF
F
F
a
a
T
L
=  
 
The unitary production functions can be represented by: 
 
() ()
() α α − =
1 1 TM LM a a          ( 1 )  
 
() ()
() β β − =
1 1 TF LF a a          ( 2 )  
   6
and thus, for any factor prices2: 
 
TF
LF
TM
LM
a
a
a
a
>  
 
The zero profit conditions are: 
 
M M TM LM P C r a w a = = +         ( 3 )  
 
F F TF LF P C r a w a = = +         ( 4 )  
 
Thus, unitary profits are: 
 
() ()
() ()() r a w a a a P TM LM TM LM M M − − = Π
− α α 1     (5) 
 
() ()
() ()() r a w a a a P TF LF TF LF F F − − = Π
− β β 1       ( 6 )  
 
Equations (1) to (6) give us a system of 6 equations and the following 6 
endogenous variables: 
 
w, r, aLM, aLF, aTM, aTF. 
 
This system has four exogenous variables: 
 
α , β , PM and PF.  
 
α  and β  are constant technological parameters that are identical in both countries 
by assumption, while PM and PF are also identical in the two countries under free 
                                                           
2   The appendix of this paper shows that for this condition to be fulfilled, all that is required is that  
α  > β .   7
trade because of competitive markets and perfect arbitrage, given that there are 
no transport costs and no trade barriers. 
 
In what follows, all 6 endogenous variables will be expressed in terms of the 
exogenous variables. 
 
Differentiating (5) for profit maximization in manufactures: 
 
() ()
() w a a P
a
TM LM M
LM
M − =
∂
Π ∂ − − α α α
1 1  
 
() ( ) ( )
() r a a P
a
TM LM M
TM
M − − =
∂
Π ∂ − α α α 1  
 
First order conditions state that: 
 
() ()
() w a a P TM LM M =
− − α α α
1 1  
 
() ( ) ( )
() r a a P TM LM M = −
− α α α 1  
 
Dividing the last two equations: 
 
() ()
()
() ( ) ( )
() r
w
a a P
a a P
TM LM M
TM LM M =
−
−
− −
α α
α α
α
α
1
1 1
 
 
()
() ( ) r
w
a
a
LM
TM =
− α
α
1
         ( 7 )  
 
Differentiating (6) for profit maximization in food: 
   8
()
() ()
() w a a P
a
TF LF F
LF
F − =
∂
Π − − β β β
1 1  
 
() ( ) ( )
() r a a P
a
TF LF F
TF
F − − =
∂
Π − β β β 1  
 
First order conditions state that: 
 
()
() ()
() w a a P TF LF F =
− − β β β
1 1  
 
() ( ) ( )
() r a a P TF LF F = −
− β β β 1  
 
Dividing the last two equations: 
 
()
() ()
()
() ( ) ( )
() r
w
a a P
a a P
TF LF F
TF LF F =
−
−
− −
β β
β β
β
β
1
1 1
 
 
()
() ( ) r
w
a
a
LF
TF =
− β
β
1
         ( 8 )  
 
From (7), wages can be expressed as: 
 
  ()
() ( ) LM
TM
a
r a
w
α
α
−
=
1
           ( 9 )  
 
Replacing wages into (3): 
 
  ()
() ( )
M TM
LM
TM
LM P r a
a
r a
a = +  


 


− α
α
1
 
   9
  ()
()
M TM
TM P r a
r a
= +  


 


− α
α
1
 
 
  ()()
M TM P a r = 





+  


 


−
1
1 α
α
 
 
  () ()
()
M TM P a r =  


 


−
− +
α
α α
1
1
 
 
  ()
()
M
TM P
a
r =  


 


− α 1
 
 
  ()
() TM
M
a
P
r
α −
=
1
          ( 1 0 )  
 
Replacing (10) into (9): 
 
  ()
() ( )
()
() TM
M
LM
TM
a
P
a
a
w
α
α
α −
−
=
1
1
 
 
  () LM
M
a
P
w
α
=           ( 1 1 )  
 
From (8): 
 
  () () ( )
β
β
r
a w
a
LF
TF
−
=
1
         ( 1 2 )  
 
Replacing (12) into (4): 
  10
  () ( )
F
LF
LF P
r
r a w
w a =  


 

 −
+
β
β 1
 
 
  () ()
F LF P a w = 




 −
+
β
β 1
1  
 
  () ()
F LF P a w = 




 − +
β
β β 1
 
 
  F LF P
w
a = 





β
 
 
    

 
 =
w
P
a
F
LF
β
          ( 1 3 )  
 
Replacing (13) into (12): 
 
  () ()
β
β β
rw
P w
a
F
TF
−
=
1
 
 
  () ()
r
P
a
F
TF
β −
=
1
         ( 1 4 )  
 
Replacing (11) into (13): 
 
  ()






=
M
LM F
LF P
a P
a
α
β
         ( 1 5 )  
 
Replacing (10) into (14): 
  11
   () () ( )
() 





−
−
=
α
β
1
1
M
TM F
TF P
a P
a         ( 1 6 )  
 
From (1): 
 
  () () 







−
− −
=  


 


=
α
α
α
α
1 1 1
LM
LM
TM a
a
a        ( 1 7 )  
 
Replacing (15) and (16) into (2): 
 
  () () ( )
()
() β β
α
β
α
β
−






−
−






=
1
1
1
1
M
TM F
M
LM F
P
a P
P
a P
      ( 1 8 )  
 
Replacing (17) in (18): 
 
  ()
() ( )
()
() β
β
α
β
α
β
α
α −








−














−








−






=
−
1
1
1
1
1
M
LM F
M
LM F
P
a P
P
a P
 
 
  () ()
() ()
()
() β β
α
β α
β
α
β
α
β
−






−
−
−






−
−
 

 
 =
1
1
1
1
1
1
M
F
LM LM P
P
a a  
 
  ()
() () ()
()
() β β
α
β α α β
α
β
α
β
−
−
− − −






−
−
 

 
 =
1
1
1 1
1
1
1
M
F
LM P
P
a  
 
  () ()
()
() β β
α
αβ α αβ β
α
β
α
β
−
−
+ − −






−
−
 

 
 =
1
1
1
1
1
M
F
LM P
P
a  
  12
  () ()
()
() β β
α
α β
α
β
α
β
−
−
−






−
−
 

 
 =
1
1
1
1
1
M
F
LM P
P
a  
 
  ()
()
()
() β β
α
α β
α
β
α
β
−
−
−






−
−
 

 

= 1
1
1
1
1
M
F
LM
P
P
a  
 
 
()
()
()
α β
α
β β
α
β
α
β
−
−
−




















−
−
 

 

=
1
1
1
1
1
M
F
LM
P
P
a        ( 1 9 )  
 
Defining: 
 
 
()
()
()
() ()
µ
α
β
α
β α β
β α
α β
α β
α β
α
= 





−
−
 

 

 


 


=
 


 


−
− −
−  


 


−
−
−  


 


−
−
− 1 1 1 1
1
1
M
F
LM P
P
a     ( 2 0 )  
 
and replacing it into (11), (15) and (17): 
 
 
µ
α M P
w =           ( 2 1 )  
 
  





=
M
F
LF P
P
a
α
βµ
        ( 2 2 )  
 
  ()
α
α
µ
−
 


 


=
1 1
TM a          ( 2 3 )  
 
Defining:  13
 
  ()
()
()
η
α
β
α
β α β
β α
α β
βα
α β
α
= 





−
−
 

 

 


 


=
 


 


−
−
 


 


−
 


 


−
1
1
1
M
F
TM P
P
a      ( 2 4 )  
 
and replacing it into (10) and (16): 
 
  ()
η
α −
=
1 M P
r           ( 2 5 )  
 
  () () ( )
() 





−
−
=
α
η β
1
1
M
F
TF P
P
a         ( 2 6 )  
 
In the system defined by equations (1) to (6), all six endogenous variables have 
now been expressed in terms of the exogenous variables.  Factor prices and 
unitary factor requirements are identical between countries since they depend on 
parameters that are identical in both countries.  Goods prices are equal because 
of free trade, competitive markets and perfect arbitrage due to the lack of 
transportation costs and trade barriers, while α  and β  are identical because of 
equal technologies.  Thus, wages and rent will be the same in both A and B as 
shown in equations (21) and (25).  QED. 
 
III. SOME  FURTHER  RESULTS 
 
In what follows, some of the results obtained so far are used to find expressions 
for QM, QF, LM, LF, TM and TF, based on the exogenous parameters. 
 
The use of factors in each sector is defined as: 
 
  M LM M L a Q =           ( 2 7 )  
  F LF F L a Q =           ( 2 8 )   14
 
M TM M T a Q =           ( 2 9 )  
 
F TF F T a Q =           ( 3 0 )  
 
Competitive markets and full employment of factors imply that: 
 
L a Q a Q LF F LM M = +          ( 3 1 )  
 
T a Q a Q TF F TM M = +          ( 3 2 )  
 
Equations (27) to (32) give us again a system of 6 equations and the following 6 
endogenous variables: 
 
LM, LF, TM, TF, M and F. 
 
In this case, there are six exogenous variables which are: 
 
L , T , PM, PF. α  and β  
 
where  L  and T  are the factor endowments. 
 
Solving for QM and QF in (31) and (32) by Cramer´s Rule: 
 
  
LF TM TF LM
LF TF
M a a a a
a T a L
Q
−
−
=         ( 3 5 )  
 
 
LF TM TF LM
LM TM
F a a a a
a L a T
Q
−
−
=         ( 3 6 )  
  15
Replacing (20), (22), (24) and (26) into (35) and (36): 
 
 
() ( )
()
() ( )
() 





− 





−
−






− 





−
−
=
M
F
M
F
M
F
M
F
M
P
P
P
P
P
P
T
P
P
L
Q
α
βµ
η
α
η β
µ
α
βµ
α
η β
1
1
1
1
 
 
 
() ( )
()
() ( )
()  

 
 − 





−
−
 

 
 − 





−
−
=
α
βµ
η
α
η β
µ
α
βµ
α
η β
1
1
1
1
T L
QM        ( 3 7 )  
 
 
() ( )
() 





− 





−
−
−
=
M
F
M
F
F
P
P
P
P
L T
Q
α
βµ
η
α
η β
µ
µ η
1
1
       ( 3 8 )  
 
LM is defined as QMaLM.  Thus: 
 
 
() ( )
()
() ( )
()
µ
α
βµ
η
α
η β
µ
α
βµ
α
η β












 

 
 − 





−
−
 

 
 − 





−
−
=
1
1
1
1
T L
LM        ( 3 9 )  
 
TM is defined as QMaTM.  Thus: 
 
 
() ( )
()
() ( )
()
η
α
βµ
η
α
η β
µ
α
βµ
α
η β












 

 
 − 





−
−
 

 
 − 





−
−
=
1
1
1
1
T L
TM        ( 4 0 )  
 
LF is defined as QFaLF.  Thus:  16
 
 
() ( )
()


























− 





−
−
−
=
M
F
M
F
M
F
F P
P
P
P
P
P
L T
L
α
βµ
α
βµ
η
α
η β
µ
µ η
1
1
 
 
 
() ( )
()
 

 













 

 
 − 





−
−
−
=
α
βµ
α
βµ
η
α
η β
µ
µ η
1
1
L T
LF        ( 4 1 )  
 
TF is defined as QFaTF.  Thus: 
 
 
() ( )
()
() ( )
() 





−
−




















− 





−
−
−
=
α
η β
α
βµ
η
α
η β
µ
µ η
1
1
1
1 M
F
M
F
M
F
F P
P
P
P
P
P
L T
T  
 
 
() ( )
()
() ( )
() 





−
−












 

 
 − 





−
−
−
=
α
η β
α
βµ
η
α
η β
µ
µ η
1
1
1
1
L T
TF       ( 4 2 )  
 
Once more, all the endogenous variables have been expressed in terms of the 
exogenous variables.  The results obtained show that both the total employment of 
each factor per sector and the output of food and manufactures may differ 
between countries since these variables depend on factor endowments (which are 
assumed to be different), as indicated by equations (37), (38), (39), (40), (41) and 
(42).  These results are in sharp contrast to factor prices and unitary factor 
requirements, which have already been shown to be identical in both nations.  17
IV. CONCLUSIONS 
 
This paper has shown that it is possible to prove the Factor Price Equalization 
Theorem in the framework developed by Jones (1965) with endogenous unitary 
factor requirements in production, by providing a functional form that meets the 
Heckscher Ohlin assumptions. It has also shown that under the model´s 
assumptions, unitary factor requirements per sector will also be identical between 
countries, while total use of factors in each sector and total output of manufactures 
and food will be different in each nation. 
 
V. BIBLIOGRAPHY 
 
Jones, Ronald W. (1965) “The Structure of Simple General Equilibrium Models” 
The Journal of Political Economy Vol LXXIII December 1965 No. 6 pp. 557-
572. 
Krugman P. R, y M. Obstfeld (2001) “Internacional Economics”, Fifth Edition, 
McGraw-Hill 
Appleyard D. R. y A. J. Field. (1997) “Economía Internacional”, McGraw-Hill, 
Madird. 
 
 
VI.  APPENDIX:  THE NO FACTOR INTENSITY REVERSAL CONDITION 
 
For no factor intensity reversals, the required condition is that for any factor prices: 
 
TF
LF
TM
LM
a
a
a
a
>  
 
Replacing each term with the expressions (20), (22), (24) and (26): 
  18
η
µ
=
TM
LM
a
a
 
 
() ( )
() 





−
−






=
α
η β
α
βµ
1
1
M
F
M
F
TF
LF
P
P
P
P
a
a
 
 
 
() ( )
() 





−
−
 

 

=
α
η β
α
βµ
1
1 TF
LF
a
a
 
 
  ()
() η αβ α
µ αβ β
−
−
=
TF
LF
a
a
 
 
Now, 
 
  ()
() TF
LF
TM
LM
a
a
a
a
=
−
−
> =
η αβ α
µ αβ β
η
µ
 
 
if and only if α  > β .  QED. 
 

APPENDIX: THE NO FACTOR INTENSITY REVERSAL CONDITION For no factor intensity reversals, the required condition is that for any factor prices: TF LF TM LM a a a a > Replacing each term with the expressions (20), (22), (24) and (26): η µ =

The Structure of Simple General Equilibrium Models”

A FORMAL PROOF OF THE FACTOR PRICE EQUALIZATION THEOREM

http://economia.uniandes.edu.co/publicaciones/D2004-05.pdf

A FORMAL PROOF OF THE FACTOR PRICE EQUALIZATION THEOREM

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