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The reciprocity theorem in a model with variable factor supply
Economics Letters 6 (1980) 153- 156 North-Holland Publishing Company
THE RECIPROCITY VARIABLE FACTOR Stephen
T. EASTON
153
THEOREM SUPPLY
IN A MODEL
WITH
*
Simon Fraser University, Burnaby, B.C. V5A IS, Canada University of Rochester, Rochester, NY 14627, USA Received
2 February
The reciprocity model.
1981
theorem
is extended
to include
variable
factor
supply
in a specific
factor
The reciprocity relationships were originally developed by Samuelson (1953) and describe the duality that exists between the Stolper- Samuelson by Rybczynski theorems. More generally they provide the link between the response of a commodity output to a change in the endowment of a particular factor, and the response of that factor’s rental to a change in the commodity’s price under a wide variety of assumptions about technology, numbers of goods and numbers of factors. ’ Formally these relationships may be written as ax,/av,
= aRi/ap,,
where x, and pj are the output and price of commodity j, and V, and R, are the endowment and rental of factor i. The reciprocity relationships are normally developed with given endowments of factors. In this note I discuss the reciprocity relationships appropriate to a model in which one
*
I am indebted
to Ronald
W. Jones for many helpful discussions of these issues. under very general conditions may be found, for example, in Samuelson (1953), Kemp (1976) or Jones and Scheinkman (1977). The relationships appear most explicitly in trade theoretic discussions of foreign investment. See Jones (1967) or Kemp (1966). Kemp and Ethier [in Kemp (1976, ch. 7)] develop a particularly broad interpretation of reciprocity.
’ Proofs of this relationship
0165-1765/81/0000-0000/$02.50
0 North-Holland
154
S. T. Easfon / The reclprociry theorem in a model with variable factor supply
of the factors of production is in elastic supply at a given rental price. The issue arises in the context of a two good three-factor model in which two factors, V, and V,, are specific to the production of X, and x2 respectively, and the third factor, V,, is used in both activities. 2 If one of the factors is in elastic supply, then we have what is commonly known as a staple or export-led growth model which has had wide application in economic history and development. 3 To show the appropriate reciprocity relationships, we begin by writing the value of output along the ‘transformation schedule’ as 4
and the value of factor rewards
as
Y = R,V, + R2V2 + RN&,. Assuming that factor V, is available in elastic supply at rental rate R,, we may rewrite (1) to expose the implicit relationships embedded in the transformation schedule:
Differentiate
(1’) with respect
top,
holdingp,,
V,, V, and R, constant:
Jones (1971) develops the properties of the fixed endowment three factor, two good specific factor model. Among those who use variants of the staple model, Chambers and Gordon (1966) consider sector specific capital as mobile internationally; North (1966) views sector specific land as available at a given price; and Lewis (1954) considers the supply of unskilled labor as elastic at a given wage. Strictly speaking a transformation schedule is drawn for given factor endowments. The appropriate maximal amounts of x, and x1 available depend upon the endowed factors, the rental rate on the factor available in elastic supply and commodity prices (in general). This is written as (1’) in the text. I shall retain the term transformation schedule for this locus for expositional convenience. Martin (1976) describes some of the other properties of the ‘transformation schedule’ when a factor is in elastic supply in the 2 X 2 HeckscherOhlin model.
S. T. Easion / The reciprocity theorem in a model with variable facior supply
Now
when
P&%/aP*) as
V, is held constant, we know that p,(ax,/CIp,)( ,,, + 5 Further, since p,(ax,/aV,) = R,, we may rewrite (3) = 0.
av,
i!lY
=x,-i-R-.
(3’)
’ aP,
aP1 P,,RI,VN,VZ Turning holding
our attention to (2) we differentiate Y with respect to V, commodity prices, R,, and factor endowment V, constant: =R,fR,-$.
‘t
To complete the proof, respect to p, :
a9
av?.‘apl*R, v2 p2
+,R N
9
a*y
Equating that
(4
N
iV2.P,#2,RI
aPlavN
155
aRN
P,,R,,V2
=-+ apI
differentiate
(3’) with respect
to V, and (4) with
a2v, ___
(5)
1avNap, 2
a*v,
(6)
Rlap,avN .
(5) and (6) by ignoring
the order of differentiation
establishes
ax,/av, = aRN/ap,. Similarly,
ax,/ay
we may prove that
= aR,/ac,
where the index j = 1,2 and i = 2, N. Were the factor that is used in both industries, V,, to be in elastic supply at rental rate RN, and specific factors V, and V, to be available as inelastic endowments, then the reciprocity relationships hold and the index i now runs i = 1,2. These two cases fully characterize the reciproc-
’ Recall that the slope of the fixed endowment (negative) of the relative price ratio.
transformation
schedule
is equal
to the
156
S. T. Eusfon / The reciprocity theorem in a model with vuriahle fuctor suppQ
ity conditions supply. 6
in the specific factor
model
when one factor
is in elastic
References Chambers, E.J. and D.F. Gordon, 1966, Primary products and economic growth: An empirical measurement, Journal of Political Economy 74, 3 15-332. Jones, R.W., 1967, International capital movements and the theory of tariffs and trade, Quarterly Journal of Economics 8 1, I- 38. Jones, R.W., 1971, A three-factor model in theory, trade and history, in: J.N. Bhagwati and R.W. Jones, et al., Trade balance of payments and growth: Papers in international economics in honor of Charles P. Kindleberger (North-Holland, Amsterdam) 49-65. Jones, R.W. and J.A. Scheinkman, 1977. The relevance of the two-sector production model in trade theory, Journal of Political Economy 85, 909-935. Kemp, M.C., 1966, The gain from international trade and investment: A neo-HeckscherOhlin approach, American Economic Review 56, 788- 809. Kemp, M.C., 1976, Three topics in the theory of international trade (North-Holland, Amsterdam). Lewis, W.A., 1954, Economic development with unlimited supplies of labour, Manchester School Economic and Social Studies 22, 139- 191. Martin, J.P., 1976, Variable factor supplies and the Heckscher-Ohlin-Samuelson model, Economic Journal 86, 820- 83 1. North, D.C., 1966, The economic growth of the United States 1790- 1860 (W.W. Norton and Company, New York). Samuelson, P.A., 1953, Prices of factors and goods in general equilibrium, Review of Economic Studies 2 1, 1- 20.
6 We should notice that the actual values of ax,/aK =aIt,/aP, the usual fixed endowment model, some values are zero.
are not stipulated.
Unlike
THE RECIPROCITY VARIABLE FACTOR Stephen
T. EASTON
153
THEOREM SUPPLY
IN A MODEL
WITH
*
Simon Fraser University, Burnaby, B.C. V5A IS, Canada University of Rochester, Rochester, NY 14627, USA Received
2 February
The reciprocity model.
1981
theorem
is extended
to include
variable
factor
supply
in a specific
factor
The reciprocity relationships were originally developed by Samuelson (1953) and describe the duality that exists between the Stolper- Samuelson by Rybczynski theorems. More generally they provide the link between the response of a commodity output to a change in the endowment of a particular factor, and the response of that factor’s rental to a change in the commodity’s price under a wide variety of assumptions about technology, numbers of goods and numbers of factors. ’ Formally these relationships may be written as ax,/av,
= aRi/ap,,
where x, and pj are the output and price of commodity j, and V, and R, are the endowment and rental of factor i. The reciprocity relationships are normally developed with given endowments of factors. In this note I discuss the reciprocity relationships appropriate to a model in which one
*
I am indebted
to Ronald
W. Jones for many helpful discussions of these issues. under very general conditions may be found, for example, in Samuelson (1953), Kemp (1976) or Jones and Scheinkman (1977). The relationships appear most explicitly in trade theoretic discussions of foreign investment. See Jones (1967) or Kemp (1966). Kemp and Ethier [in Kemp (1976, ch. 7)] develop a particularly broad interpretation of reciprocity.
’ Proofs of this relationship
0165-1765/81/0000-0000/$02.50
0 North-Holland
154
S. T. Easfon / The reclprociry theorem in a model with variable factor supply
of the factors of production is in elastic supply at a given rental price. The issue arises in the context of a two good three-factor model in which two factors, V, and V,, are specific to the production of X, and x2 respectively, and the third factor, V,, is used in both activities. 2 If one of the factors is in elastic supply, then we have what is commonly known as a staple or export-led growth model which has had wide application in economic history and development. 3 To show the appropriate reciprocity relationships, we begin by writing the value of output along the ‘transformation schedule’ as 4
and the value of factor rewards
as
Y = R,V, + R2V2 + RN&,. Assuming that factor V, is available in elastic supply at rental rate R,, we may rewrite (1) to expose the implicit relationships embedded in the transformation schedule:
Differentiate
(1’) with respect
top,
holdingp,,
V,, V, and R, constant:
Jones (1971) develops the properties of the fixed endowment three factor, two good specific factor model. Among those who use variants of the staple model, Chambers and Gordon (1966) consider sector specific capital as mobile internationally; North (1966) views sector specific land as available at a given price; and Lewis (1954) considers the supply of unskilled labor as elastic at a given wage. Strictly speaking a transformation schedule is drawn for given factor endowments. The appropriate maximal amounts of x, and x1 available depend upon the endowed factors, the rental rate on the factor available in elastic supply and commodity prices (in general). This is written as (1’) in the text. I shall retain the term transformation schedule for this locus for expositional convenience. Martin (1976) describes some of the other properties of the ‘transformation schedule’ when a factor is in elastic supply in the 2 X 2 HeckscherOhlin model.
S. T. Easion / The reciprocity theorem in a model with variable facior supply
Now
when
P&%/aP*) as
V, is held constant, we know that p,(ax,/CIp,)( ,,, + 5 Further, since p,(ax,/aV,) = R,, we may rewrite (3) = 0.
av,
i!lY
=x,-i-R-.
(3’)
’ aP,
aP1 P,,RI,VN,VZ Turning holding
our attention to (2) we differentiate Y with respect to V, commodity prices, R,, and factor endowment V, constant: =R,fR,-$.
‘t
To complete the proof, respect to p, :
a9
av?.‘apl*R, v2 p2
+,R N
9
a*y
Equating that
(4
N
iV2.P,#2,RI
aPlavN
155
aRN
P,,R,,V2
=-+ apI
differentiate
(3’) with respect
to V, and (4) with
a2v, ___
(5)
1avNap, 2
a*v,
(6)
Rlap,avN .
(5) and (6) by ignoring
the order of differentiation
establishes
ax,/av, = aRN/ap,. Similarly,
ax,/ay
we may prove that
= aR,/ac,
where the index j = 1,2 and i = 2, N. Were the factor that is used in both industries, V,, to be in elastic supply at rental rate RN, and specific factors V, and V, to be available as inelastic endowments, then the reciprocity relationships hold and the index i now runs i = 1,2. These two cases fully characterize the reciproc-
’ Recall that the slope of the fixed endowment (negative) of the relative price ratio.
transformation
schedule
is equal
to the
156
S. T. Eusfon / The reciprocity theorem in a model with vuriahle fuctor suppQ
ity conditions supply. 6
in the specific factor
model
when one factor
is in elastic
References Chambers, E.J. and D.F. Gordon, 1966, Primary products and economic growth: An empirical measurement, Journal of Political Economy 74, 3 15-332. Jones, R.W., 1967, International capital movements and the theory of tariffs and trade, Quarterly Journal of Economics 8 1, I- 38. Jones, R.W., 1971, A three-factor model in theory, trade and history, in: J.N. Bhagwati and R.W. Jones, et al., Trade balance of payments and growth: Papers in international economics in honor of Charles P. Kindleberger (North-Holland, Amsterdam) 49-65. Jones, R.W. and J.A. Scheinkman, 1977. The relevance of the two-sector production model in trade theory, Journal of Political Economy 85, 909-935. Kemp, M.C., 1966, The gain from international trade and investment: A neo-HeckscherOhlin approach, American Economic Review 56, 788- 809. Kemp, M.C., 1976, Three topics in the theory of international trade (North-Holland, Amsterdam). Lewis, W.A., 1954, Economic development with unlimited supplies of labour, Manchester School Economic and Social Studies 22, 139- 191. Martin, J.P., 1976, Variable factor supplies and the Heckscher-Ohlin-Samuelson model, Economic Journal 86, 820- 83 1. North, D.C., 1966, The economic growth of the United States 1790- 1860 (W.W. Norton and Company, New York). Samuelson, P.A., 1953, Prices of factors and goods in general equilibrium, Review of Economic Studies 2 1, 1- 20.
6 We should notice that the actual values of ax,/aK =aIt,/aP, the usual fixed endowment model, some values are zero.
are not stipulated.
Unlike