Factor intensities and factor substitution in general equilibrium

Journal of International Economics 15 (1983) 65-99. North-Holland

FACTOR INTENSITIES AND FACTOR SUBSTITUTION IN GENERAL EQUILIBRIUM R o n a l d W. J O N E S * University of Rochester, Rochester, N Y 14627, USA

Stephen T. E A S T O N Simon Fraser University, Burnaby, B.C., Canada

Received February 1982 This article examines how factor intensity rankings between industries and the economy-wide asymmetry in the degree of factor substitution combine to influence the manner in which changes in relative commodity prices affect the factoral distribution of income. (The reciprocal influence of factor endowment.changes on the composition of outputs is also discussed.) The analysis is undertaken in a general three-factor, two-commodity framework, the minimal sized model that allows both influences to affect factor prices and admits of the possibility of complementarity between factors. Factors which are good substitutes find their returns behave somewhat similarly when commodity prices change while factors which are complements experience strongly asymmetrical fortunes. A crucial role is played by a comparison of the share of the 'middle' factor in each sector.

1. Introduction Two aspects of t e c h n o l o g y d o m i n a t e the s u p p l y side of general e q u i l i b r i u m models. O n e of them is c o n c e r n e d with the degree to which factors of p r o d u c t i o n can be substituted for each o t h e r in the event that d i s t u r b a n c e s in factor m a r k e t s suggest changes in techniques to minimize unit costs. The other details the m a n n e r in which industries differ from each o t h e r in the p r o p o r t i o n s in which they rely on v a r i o u s factors of p r o d u c t i o n . F o r s o m e issues, such as the effect of c o m m o d i t y price changes on c o m m o d i t y supplies in an e c o n o m y with a given resource base, factor intensities a n d the extent of factor substitution are clearly seen j o i n t l y to affect the results. 1 F o r o t h e r questions, such as those dealing with the effect of c o m m o d i t y price changes on the d i s t r i b u t i o n of income, we k n o w in general that these aspects of the *This research has been supported, in part, by National Science Foundation Grant SES - 78O6159. ~For example, consider the elasticity of relative output response along a transformation schedule in a two-factor, two-commodity world in which the elasticity of substitution between 0022-1996/83/$3.00 © 1983, Elsevier Science Publishers B.V. (North-Holland)

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R.W. Jones and S.T. Easton, Factor intensities and factor substitution

technology may interact but we have scarcely begun to probe the nature of such interaction in the smaller-scaled models in which comparative statics exercises are carried out in fields such as international trade, public finance, economic history, or economic growth. One reason for such neglect is the heavy reliance on the standard model with two mobile productive factors producing two commodities. In this 2 x 2 model and, indeed, in higher-dimensional 'even' models of general equilibrium with matching numbers of factors and commodities, factor prices generally are determined only by commodity prices, not by factor endowments. And the nature of this dependence is characterized solely by factor intensities, completely independent of the other aspect of technology, the extent of factor substitution.2 The last decade, however, has witnessed the revival of the 'specific-factor' model, so-called because each of two sectors of production makes use of a productive factor employed only in that sector as well as a factor that is mobile between sectors) This model specifies three factors and two commodities and thus opens up the possibility that factor intensities and the degree of factor substitution jointly determine the effect on factor rewards of changes in commodity prices. In a sense this is what happens, but the distributional consequences are very strict: a rise in the relative price of the factors in each industry has the same value, a. Then the elasticity of output response is

{

1--1~lloft,.~ I~lloI J

'

where the expression [2[ 10[ is a positive fraction, the value of which captures the extent of the difference in factor intensities between industries. For further details see Jones (1979, ch. 7). 2As described in Jones and Scheinkman (1977), the basic equilibrium dependence of factor prices (w) and outputs (x) on factor endowments (V) and commodity prices (p) is expressed in the matrix equation:

[_s.,,_q

q.

l oJ Ld~/ LdpJ The A matrix of input-output coefficients reveals factor-intensity rankings while the S matrix incorporates the substitution elasticities. If techniques are linearly independent and the number of factors is at least as great as the number of commodities, the matrix is invertible. Let

I-KI G1 [-S A-1-~ Then elements of the G-matrix reveal how a change in the jth commodity price affects the ith factor reward when endowments and other commodity prices are held constant. If the number of factors equals the number of commodities, the K matrix will be the zero matrix (endowment changes do not independently influence factor prices) and G will equal the inverse of A', and thus be independent of S. 3The basic references are to Jones (1971), and Samuelson (1971), as well as to work by Mayer (1974), Mussa (1974), and Near), (1978).

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

67

first commodity must raise the return to the factor used specifically in that sector in real terms, lower the real reward to the other specific factor, and trap the change in the mobile factor's return between the changes in the two commodity prices. 4 Despite the acknowledged usefulness of the specific-factors model, there are issues involving more complicated substitution relationships and the possibility of complementarity that are avoided since only two inputs are involved in any productive activity. In a pioneering article, Batra and Casas (1976) explored the more general three-factor, two-commodity model, the 'minimal' sized model in which substitution and intensity influences have an opportunity significantly to interact in explaining income distribution. Although successful in setting out the basic equilibrium conditions, their solutions were at times flawed and, perhaps more importantly, the economic rationale for their results seemed missing. Two dissertations at the University of Rochester, written a decade apart, have also attempted to analyze this more general model. T h a k k a r (1971) provided an exhaustive taxonomy of results and applied the model to issues in growth theory. Suzuki (1981) independently solved the open-economy comparative statics exercises of changing commodity prices and factor endowments to solve for factor prices and commodity outputs, and pointed out a key error in the Batra-Casas article [see Suzuki (1983)]. Ruffin (1981) published a paper providing a simple argument to show how endowment changes influence factor prices in the general model precisely in the same manner as in the sector-specific model, s In the present paper we analyze this general three-factor, two-commodity model with the twin purpose of bringing out the economic meaning behind the rather complicated technical relationships of the model and presenting new results. Section 2 deals in a preliminary fashion with the important role of the 'middle' factor in this model. In section 3 we turn briefly to a formal solution - - the same kind as obtained by Thakkar, Batra and Casas, and Suzuki. Section 4 introduces a diagrammatic technique that proves useful in revealing the interaction of intensity and substitution characteristics that is expounded in section 5. As is well understood in these models, commodity prices and factor prices are related in a manner that finds its dual counterpart in the factor e n d o w m e n t - c o m m o d i t y output relationships. A few *In the specific-factorsmodel it is possible to ask whether the return to the mobile factor rises more or less than the average of all factor returns. The criterion pits a ratio of intensities with which the mobile factor is used in each sector against the ratio of elasticities of demand for the mobile factor. [For details see Rutlin and Jones (1977).I A related question involves the same criterion: What happens to the ratio of outputs produced if the terms of trade are kept constant and the endowment of the mobile factor rises? SBurgess (1980) analyzes a related model - - one in which each sector makes use in part of the output of the other sector. If each sector also employs a factor specific to that sector it will, indirectly through its intermediate-good requirement, use both specificfactors as well as labor.

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

68

remarks about these output changes are made in section 6. In the concluding section (section 7) we summarize our results and suggest areas of applied interest in which this model may prove useful.

2. Extreme factors, the middle factor, and the influence of factor endowments on factor prices

In the specific-factor model there is a clear distinction between those factors that are used only in one sector and that factor which is mobile between sectors. In our general case all three factors are mobile. Nonetheless, a similar kind of asymmetry exists, and the influence of endowment changes on factor prices precisely parallels that in the specific-factor model. The concept of specific factor is blunted and transformed into that of extreme factor, while the role of mobile factor is, in the general model, undertaken by the middle factor. We follow the numbering scheme suggested by Suzuki (1983), whereby the first factor is used most intensively in the first industry, the second factor in the second industry, leaving factor 3 to adopt the position of middle factor. If a o denotes the input of factor i required to produce a unit of output in the jth sector, our assumption is that: a l l > a 3 t > a21. O12

a32

(1)

a22

Furthermore, although we allow flexibility in the technology so that each a~j responds to any change in factor prices, we assume that the factor-intensity ranking shown in (1) never reverses. Fig. 1 illustrates this ranking by the ordering of the slopes of each factor constraint line. For some initial set of factor prices and techniques all three factors are fully employed at the output levels shown at point A. Each constraint line shows the trade-off between outputs (x~ and x2) that would allow a particular factor to remain fully employed when techniques remain frozen at their initial values. Formally, eqs. (2) state these full-employment conditions: a l lXl 4- a l 2 x 2 = VI,

a21xt + a22x2 =

V2,

(2)

a 3 1 x l -~a32x 2 ~---V 3.

For given techniques and factor endowments (V~), only point A satisfies eqs. (2). Following the ranking in (1) the V~ constraint is the steepest, the V2

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

69

A

Xl Fig. 1

constraint the flattest, with the V3 constraint line in the middle.6 In the specific-factor model the V~-line is vertical since V1 is only used to produce xl, and the Vz-line is horizontal. The transformation schedule (not drawn) passes through point A and is trapped between the steepest (V1) and flattest (V2) constraint lines. As we point out below, a crucial question involves a comparison of its slope with the middle (V3) constraint line. Further properties of extreme factors and the middle factor are revealed in two alternative expressions for the ranking in (1). First, let 0~ denote factor i's distributive share in the jth sector. Multiply the numerator and denominator in the first ratio of (1) by w, (the return earned by factor 1), in the second ratio by w3, and in the third by w2, and divide each of the numerators by the price of commodity 1, p,, and each denominator by P2 to obtain the alternative ranking:

01L 031 021 0,

(3)

Since the sum of the numerators, unity, equals the sum of the denominators, the first fraction must exceed unity and the last must be smaller than one. That is, the share of each 'extreme' factor in the industry using it intensively 6This diagram plays a key role in Rullin's (1981) analysis, and is based on a similar diagram in Jones and Scheinkman (1977).

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R.W. Jones and S.T. Easton, Factor intensities and factor substitution

must exceed the share of that same factor in the other sectorfl (In the specific-factor model, the first ratio is infinite and the last is zero.) Of crucial importance to much of what follows is the value of the middle ratio. If it equals unity, neither sector is relatively 'intensive' in its use of middle factor 3. By contrast, if 031 exceeds 032, the middle factor is used relatively intensively in the first industry, s The ranking of factors according to the ratios in the two industries, (1) or (3), can also be expressed by the statement:

~11>~31>~2x,

(4)

where 20 refers to the fraction of the total supply of factor i used in the jth industry. (In the specific-factor model 211 is unity and 221 is zero.) The relative position of 231 in the range 211 to 221 could be defined by ~+:

~31--~21 ~1~----211__221'

(5) ~11--~31 O~2 ~ 211--221 Thus, if the first sector uses almost as high a fraction of middle factor 3 as it does of extreme factor 1, gl approaches unity. The sizes of the ~ turn out to be crucial in revealing the relative effect each extreme factor has on factor prices. But a straightforward comparison of ~1 and ~2 does not by itself yield a factor intensity ranking for middle factor 3. The ~{s also reflect the importance of each extreme factor in the national income. To pursue our analysis of the significance of the ~, suppose factor prices remain at their initial levels, but that factor endowments change. If there were as many commodities produced as factors required to produce them, any (small) variation in factor endowments could be matched by an appropriate alteration in the composition of outputs that would clear all factor markets without requiring any change in factor prices. This is the hallmark of 'even' models of general equilibrium. But with more factors than commodities, such as clearing of all factor markets by appropriate output changes is generally not feasible; outputs can adjust to clear only as many factor markets as there are outputs. Thus, suppose all endowments change, 7Note that nothing can be assumed, a priori, about the share of different factors in the same industry. Thus, 9tl might be smaller than 02t if factor 1 is relatively unimportant in producing either commodity. 8The importance of factor-intensity rankings using distributive shares in the specific-factor model is pointed out in Jones (1971), while in the Heckscher-Ohlin 2 x 2 model it is synonymous with the ranking of factor proportions.

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

71

that factor prices are kept frozen at their initial values, and that outputs Xl and x2 adjust so as to clear extreme factor markets V~ and V2. At these new output levels we ask whether the market for middle factor 3 is cleared, or whether, instead, there exists excess demand or supply of factor 3 that would put pressure on factor prices to change. Proceeding formally, differentiate the first two equations in (2) (keeping the ai~'s constant since we are examining the state of the market at initial factor prices):

211~1+212X2=171, 221X1 +222X2 = 172. The 2's are the factor allocation fractions previously defined and a hat (') over a variable denotes a relative change. (Thus ~2=dx/x.) Solve these two equations for output changes that clear markets for the two extreme factors: .

1

^

Xl = ~ (/~22Vl -- ~12 172), 1

:~2 = ~ ( 2 t 1172--221 171),

where 121"~--(,~11,~22--,~12,~21)=(211--221)>0. The demand for factor 3 is shown by (a31x I -I-a32x2) in the third equation of (2), so that at initial factor prices the relative change in demand for the middle factor is (23x~1-1-)-32-~2). Thus, substituting for the output changes that clear extreme factor markets, the relative change in the economy's demand for middle factor 3 is:

{~1 171+ ~2 172}" The only way all factor markets can clear without any change in factor prices is if 173 exactly matches (~t 171+~2172)Fig. 1 illustrates this result. If the VI constraint alone shifts out, extreme factor markets could clear if outputs move from A to B. By itself this creates excess demand for middle factor 3. A downward shift in the V2 constraint line so that it intersects the new Vl-line at C would serve exactly to clear the market for middle factor 3 in the event that the Va-constraint line does not shift. The relationship between ~1 and ~2 shows how much one extreme factor must change to balance the other in terms of demand for middle factor 3. The relationship the ~'s bear to the comparison in the intensity with which the middle factor is used in the two industries can be brought out by

72

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

considering, in fig. 1, the position of the transformation schedule through point A. The transformation curve must be flatter than the V1-1ine and steeper than the V2 line. Suppose it is flatter than the V3-line. This implies (i) that the first industry is intensive in its use of middle factor 3 (031 >032); and (ii) that the ratio ~1/~2 exceeds the ratio of distributive shares of extreme factors 1 and 2 in the national income, 01/02. Since the absolute value of the slope of the transformation curve is the price ratio, Pl/P2, and that of the V3 line is a31/a32 , contention (i) follows immediately by multiplying each a3j by w3. To derive (ii) consider the price line that is tangent to the transformation curve at A. Outputs south-east of A along this line could be obtained (at constant factor and commodity prices) by increasing V1 and decreasing V z by appropriate amounts. Endowment changes (at constant prices) in general alter the national income, Y, by amounts proportional to factor shares:

011~'1--1.-02~'2 .-~-03V3 = ~" All points on the tangent line to the transformation curve display the same value of Y. Therefore since point C lies below this line the increase in V1 and reduction in V2 that leads to C along the unchanged V3-1ine must reduce Y: 01 02 01+02 9 1 + ~

P2
This same change in the (V1, V2) mix, however, leaves the economy's demand for middle-factor 3 unchanged, so that ux lPl + c~2lP2--O. Therefore ~1 must exceed 01/(OX+O 2) or, in relative terms, ~1/~2 exceeds 01/02. A comparison of the ~'s in relation to the importance of each extreme factor in the national income is therefore equivalent to an intensity ranking for middle factor 3. Formally, 9 9By definition0q],,,2=(231-221)/0,tt-~.30. But since ~ . 0 = 1, this ratio also equals (231222 --221232)/().t1232--2312xal. Directlyfrom the definitionof the 2's and O's. 20 00 where 0j refers to output xj's share of the national incomeand 0~,as before, to factor i's share of the national income.By substitution:

~! 01 I (031022--021032)~ and the expression in brackets exceedsunity if and only if Osl > 032.

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

~1 0(2 "-A-T> "A-'5-, iff O" tl-

031>032.

73

(6)

Even before turning to the formal solution, we are now in a position to discern how factor endowment changes alter factor prices? ° With outputs assumed to adjust to clear markets for extreme factors 1 and 2, an increase in V3 by itself must create excess supply of factor 3 (and lead to a lowering of w3). By contrast, an increase in either extreme factor leads to excess demand for factor 3 (e.g. at point B in fig. 1 when VI rises) and a subsequent rise in w3. If commodity prices are held constant, any such rise in w3 must be balanced by a fall both in w~ and w2. This natural pairing of extreme factor returns is a reflection of the fact that all factor price changes depend upon the sign of excess supply for middle-factor 3, which pits ~'3 against the combination in demand, ~t17"~+~2122. These asymmetric properties of the general model precisely parallel those of the specific-factor model, x~ 3. The determinants of factor prices: The formal solution Three relationships are required to obtain a solution for the three factor prices. Competitive profit conditions in each sector provide two such relations: al 1wl + a21w2 "1-a31w3 ----Pl, (7) a12wl + a22w2 + a32w3 --'--/92. Assuming the economy continues to produce both goods, small changes in the Pi and w~ are related by: 011Wx "{-021W 2"JF 031W 3 = r i d

(8) 012W1 +022W2"1- 032W3 = P 2 ,

where these relationships have been simplified by the envelope property in each sector (reflective of the minimization of unit costs): 01jdlj + 02~d2~+ 03j~3~= 0.

(9)

A third relationship is required and, as in the special case of the specific-factor model, this must build upon the full-employment conditions set out in (2). '°These results and the graphical technique are stressed by Ruffin (1981). The findings are also obtained by Thakkar (1971), Suzuki (1981), and Batra and Casas (1976). l~In the specific-factor model ~ reduces to 23t and ~2 to 232.

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

74

With more factors employed than outputs produced, factor endowments exercise an influence on factor prices independently of commodity prices. Techniques of production in each sector depend upon factor prices. For a given technology, a 0 = ao(wl, w2, wa),

(10)

where each such function is zero homogeneous. In elasticity terms we take Ekj to reflect the impact on a~i of a rise only in Wk, with the other two factor prices held constant. Thus: _

Wk

Oao.

EiJ=~w k ~ ,

(11)

so that 3

% " = ~

~ w" k. E~j

(12)

k=l

The zero homogeneity of each aij function implies that: 3

Eij = 0.

(13)

k=l

Furthermore, the general envelope property stated in (9) must hold in the special case in which each factor price changes separately, so that 3

y, O~jE~j=O, k for all j,k.

(14)

i=l

The sign of E~k/ reflects the substitution or complementarity relationship between factors i and k. Of course 'own' substitution terms, Eli, must all be negative. If i and k are substitutes in producing commodity j, E~k is positive; a rise in wk would induce producers to use factor i more intensively. We also examine the possibility of complementarity, where Ekj is negative. Indeed, we shall even consider the special case in which two factors, say i and k, are perfect complements, implying that a rise in Wk not only lowers aki, it lowers a u as well by the same relative amount. Finally, the substitution matrix in 'slope' terms is symmetric, ~aij = ~akj

dwk

awl'

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

75

so that in elasticity terms,

(15)

Ek _ Ok1 wi

ij ---~-- "-'kj" vU

With these basic concepts describing factor substitutions defined, we can proceed in a fashion analogous to the previous section to derive a third relationship involving factor prices. Differentiate the full employment conditions for extreme factors in (2) to obtain: /11IX1 +/112X2 =

P1--{/111a11+/112a12},

/121X1+/122X2=~2--{/121a21+/122a22}.

(16)

Clearly, the substitution terms in the two industries are always averaged together. With this in mind we define the term a~ to denote the economywide substitution towards or away from the use of factor i when the kth factor becomes more expensive, under the assumption that each industry's output is kept constant: -/1,~ E~I +/l,2E~2.

(17)

That is, an economy-wide isoquant 'super' bowl can be constructed for any f i x e d set of industry outputs. The effect of changes in outputs on factor

demands is picked up on the left-hand side of eqs. (16). 12 Substituting these aggregate ~'s into (16) yields (18): /11

+/112 2 =

+

+

'2"21"~1"q"/122"~2 = ~2 -- {O'Iw1 + 0"2W2 "~ 0"23W3}•

(18)

As before, we assume outputs adjust to clear the two markets for e x t r e m e factors, where these markets are disturbed not only by endowment changes but also by changes in the intensity with which factors are used to produce commodities. The relative change in the economy's demand for middle factor 3 is {/131X1 "1"/~32X2} "1- {~31031 -~ ~32a32}. 12The concept of an isoquant 'super' bowl is analogous in demand theory to that of a Scitovsky community indifference curve, whereby a locus is constructed of minimal combinations of commodities that keep, in the background, each individual on the same indifference curve. See Seitovsky (1942), Samuelson (1956), and Jones (1972) for further discussion.

76

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

The relative change in supply is I7'3. Decompose the intensity terms, dai, substitute for the output changes that clear eqs. (18) for the extreme factors. and equate demand and supply for middle factor 3 to obtain the third equation in set (19) [for convenience the competitive profit equations of change, (8), have been repeated]:

012W1 -F 022~ 2 "1-032 ~,(~3=/72,

(19)

~1~'1 + ¢_,~'_,+ ~3~'3 = ~'3- {a, 17~+ ~ } , and ~ = a ~ 3 - { ~ a ~ + ~ 2 a ~ } . Note the role played by the ct; in bringing together (i) extensive changes in the endowments of extreme factors, 17"1 and ~'2, and (ii) intensive changes in the use of extreme factors, expressed in the ~,coefficients. In each case the average of extreme factor response (at the extensive or intensive margin) is weighed against the response in middle factor 3. This basic equation set, or equivalent variations of it, has been developed in the literature) 3 The challenge lies in deriving economically meaningful results. Proceeding a bit further in a formal fashion consider the solution for ~'3: 1 f

~'3=7 ~[012C,-02,_~t]:t + [02,¢1- 01,~2]:2 (20) where 011

021

031

A ~ 012

022

032 <0,

101-0. °0:;1>0 = 011

The A determinant of coefficients is negative. A heuristic proof focuses on our previous question about the impact of an increase in V3 on the return to factor 3 at constant commodity prices. This, we argued, must be negative so taSoe, for example, eq. (17) in Batra and Casas (1976). Thakkar (1971) and Suzuki (1981) also derive equations sumcientto determinethe ~(,~.

R.W. dones and S.T. Easton, Factor imensities and factor substitution

77

that, with Iol positive, A must be negative, t'* We could, in similar fashion, solve explicitly for ~'~ and ~,_,. Before proceeding instead to exploit a simple diagrammatic device whereby the solutions for the ~,~ may be analyzed, note two features of the solution for ~'3. (i) The impact of commodity price changes on factor prices depends both on factor intensity terms, as captured by the 0;j's, attd upon the degree of factor substitutability or complementarity, as expressed in the ~. This reflects the recurrent theme of our analysis. (ii) The sign of the impact of factor endowment changes on factor prices (at constant commodity prices) depends only upon factor intensities. Indeed, it depends only upon the specification of which factor plays a middle role, rather than upon more detailed knowledge of the relative intensity with which each sector uses the middle factor. An increase in V3 that is greater, relatively, than the s-weighted sum of changes in extreme factor endowments must lower the return to middle factor 3 and raise the return to both extreme factors. Factor substitutability, working through the denominator, A, serves merely to dampen or exacerbate these results.

4. A diagrammatic technique In this section we develop a diagram to help explore the question: How do changes in relative commodity prices (at constant endowments) affect all factor prices? With three factors and two commodities, a diagram ~howing factor prices explicitly on the axes would lead into three dimensions. But there is a trick that can usefully be employed. As in most general equilibrium models, an inflation of all commodity prices at the same rate would force up all factor prices to the same extent; only relative prices matter. This common observation usually precedes the assumption that one of the commodities is to be chosen as numeraire. Instead, we select one of the factors for this role. Indeed, we have stressed the asymmetry between extreme factors 1 and 2, on the one hand, and middle factor 3, on the other. We therefore choose the single middle factor, 3, as numeraire. Fig. 2 illustrates the technique. At the initial equilibrium w3 has a certain value (say unity), which will be kept constant throughout, and w t and w2 are shown by point A. Four loci are drawn through A. Consider them in turn. (i) The pl-locus shows combinations of w I and w2 that (together with the given, constant, value of w3) keep costs of producing xl constant at the value pl achieves at A. t4Ruffin (1981)makes use of this argument. More directly, the negativityof A followsfrom the required negative semjdefinite nature of the factor substitution matrix, S, as described in Jones and Scheinkman {1977, p. 926). Suzuki (1981) provides the elements of the u vector such that u'Su (which must be negative if it is not proportional to the factor price vector)equals A.

78

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

w2

!

Wl Fig. 2

(ii) The p2-10cus shows a comparable set of (wl, w2) combinations that keep P2 constant. By the profit equations in (19) each of these loci is negatively sloped, but the pl-locus is steeper at A since 011/021, the absolute value of its elasticity, exceeds 012/022 , the comparable value along the p2-1ocus. (iii) The V-locus is derived from the third, factor-market clearing relationship in (19) once if3 is set equal to zero. Thus, the elasticity of the Vcurve is (-~1/~2)- The curve shows combinations of w~ and w2 that clear factor markets for an unchanging set of factor endowments. In fig. 2 the Vlocus is drawn with a negative slope, suggesting that a drop in w2 creates excess supply of middle factor 3 which must be matched by a rise in w~ to create excess demand for factor 3 and thus clear its market. Recall that outputs are, in the background, adjusting to clear markets for extreme factors 1 and 2. Furthermore, fig. 2 illustrates the V-locus as trapped between the Pl and P2 loci. Thus, this construction is special in two respects: the Vlocus could break out of the P ~ - P 2 spread, and it might even become positively sloped. But we postpone both the possibility of these cases and a consideration of the circumstances under which they might appear. (iv) The positively sloped p-locus through point A shows all combinations of wt and w2 (together with the fixed value of w3) which, although causing both commodity prices to change, keeps the price ratio constant. In the first two competitive profit equations of (19) set/~1 =/~2 equal to a common value,

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

79

/~, and keep if3 = 0. Subtraction reveals that if2 (011-012) WI (022--021) '

(21)

along the p-locus. The p-loci in fig. 2 are special in that they aim towards the origin. This illustrates the case in which 03~ equals 032, i.e. middle factor 3 is as intensively used in one industry as in another. By contrast, should the first industry use middle factor 3 intensively, 031 would exceed 032 and the ratio in eq. (21) would be less than unity; the p-locus would cut a ray from the origin from above. From point A in fig. 2 suppose that commodity l's relative price is raised to a point on the p'-locus. In particular, if Pl were to rise to p'~ and P2 remain unchanged, w 1 and w2 would settle at point B. But our choice of factor 3 as numeraire (instead of commodity 2) implies that the absolute commodity price level must be adjusted along the p'-locus (to keep the relative price at its new level) until point C is reached. At point C factor markets are cleared and: (i) (ii) (iii) (iv) (v) (vi)

commodity l's relative price is higher than at A, commodity l's absolute price is higher than at A, commodity 2's absolute price is lower than at A, the return to factor 3, w3, has been held constant, the return to factor 1, wl, has risen compared to A, the return to factor 2, w2, has fallen compared to A.

These changes are summarized by the ranking: (22) By the magnification effect, some factor return must rise by more than either commodity price, and w I is the only candidate. 15 Similarly, if2 must anchor the bottom end. The ranking in factor returns shown in (22) follows from our locating the V-locus between the p~-loci in fig. 2. This position corresponds to the necessary ranking in the specific-factors model. As already mentioned, the latter has a vertical px-locus, and a horizontal p2-1ocus. And, as we prove below, the V-locus must be negatively sloped. In the general model, whose analysis we now pursue, the p~-loci remain negatively sloped but the V-locus can break out of the bounds suggested in fig. 2. ~SThe magnificatiorieffect, discussed in Jones (1965) and Jones and Scheinkman (1977), is based upon the presumed lack of joint production.

80

R.W.. Jones and S.T. Easton, Factor intensities and factor substitution W2

.p,

0

W~ Fig. 3

Fig. 3 illustrates the technique for the case in which the V-curve is steeper, at A, than the p~-locus. In the next section we discuss what underlying substitution and/or intensity possibilities this might reflect. From initial point A suppose the relative price of commodity 1 rises to some point on the p'locus. Factor markets are cleared (keeping w3 constant) only at the intersection of the if-locus with the V-schedule at C. But at C both p~ and P2 have fallen (compared with an unchanged value of w3). Comparing C with A: (23) The rise in the relative price of commodity 1 has driven up the real return both of extreme factor 1 and middle factor 3. Extreme factor 2 unambiguously loses. From the construction in fig. 3, and from a similar diagram which could illustrate a negatively-sloped V-curve flatter than the p2-1ocus, it is possible to conclude that if the V-curve is negatively sloped, an increase in commodity l's relative price must lead to

wt > wa > w2-

(24)

The return to the middle factor is trapped between the returns to the extreme factors, although it may break out of the (P~,P2) range.

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

81

w2 V

//,

\ -,q

0

Wl Fig. 4

The V-curve may become positively sloped. Fig. 4 illustrates a case in which ~2 has become negative and ~1 is positive; that is, from A a cut in w2 would create excess demand for middle factor 3, requiring a fall in w I to clear factor markets. Following our graphical technique, compare point C, at which factor markets clear at the higher relative price for commodity 1, with point A. Fig. 4 is consistent with any of the following three alternatives: (i)

~3(= 0) > ~q >/~1 >/~2 > ~,2,

(ii)

~V3(= 0) > p l > W1 >/92 > W2,

(iii)

W a ( = 0 ) > p l >/92 > WI > W2.

In all three cases a rise in commodity l's relative price unambiguously increases the real return to middle factor 3, and by more than the return to extreme factor one. Although wx rises relative to w2, in real terms factor 1 might unambiguously lose, as in case (iii). In the next section we see what distinguishes these three cases and show that they may reflect either complementarity between factors 2 and 3 or a combination of extreme factors 1 and 2 being very good substitutes and middle factor 3 being intensively used in the commodity (1) whose relative price has increased. The symmetry of this situation rightly suggests that the return to middle factor 3 might fall relative to both commodity prices as well as to the returns to extreme factors 1 and 2.

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R.W. Jones and S.T. Easton, Factor intensities and factor substitution

5. Substitutability and complementarity: The interaction with factor intensities

The diagrammatic techniques just decribed point to the importance of the factor-market-clearing V-locus. The elasticity of this curve is given by (minus) the ratio of ~.~ to ~2, and our first task is to ascertain the meaning of the ¢;. Each ~ is the answer to the following question. If w~ increases by 1 percent and other factor prices and all factor endowments are kept constant, by how much will demand for middle factor 3 rise if, in the background, commodity outputs are adjusting to clear markets for extreme factors 1 and 2? Fig. 5 is used to illustrate the component of ~2. It resembles fig. 1 in showing an initial output combination (A) at which the three factor constraint lines intersect. These constraint lines are now pictured as shifting not because factor endowments change, but because we assume w2 rises and this encourages changes in techniques. By eq. (19) the expression for ~2 is:

(25) Leave complementarity aside for the moment so that tr~ and a~ are positive, with a 2, the 'own' elasticity, negative. The tr32-term captures the extent to which the rise in w2 causes middle factor 3 to be used more intensively. That is, both a31 and a32 increase, causing the Va-constraint line in fig. 5 to move inwards to position V~. Thus, if outputs were unchanged (at A), excess demand for middle factor 3 would be created. But the rise in w2 also shifts the extreme V1- and V2-constraint lines, and outputs are assumed to adjust to clear their markets.

la E

Xl Fig. 5

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

83

The V2-constraint line must shift outwards (to V~) when w2 rises since both a2~ and a22 are reduced. If extreme factors are substitutes, as we are now assuming, the Vl-constraint line shifts inwards (just as the V3-1ine shifted in). If the degree of substitutability between extreme factors is slight, outputs would move to point B to clear extreme factor markets. This is the case in which the negative value of a22 dominates the sign of {~lat2+0t2022} in the ~2-term. A higher degree of substitutability between extreme factors would cause the new V~-constraint line to intersect the new V~-line at C in fig. 5, and this would turn {~la2+a2¢r22} positive, but still outweighed by the direct expression a]. Point D illustrates how an even greater degree of substitutability between extreme factors can turn ~2 negative; when outputs adjust from A to D to clear extreme factor markets 1 and 2, the increase in w2 would create excess supply of middle factor 3. (Point D lies below the new V~-constraint line.) The potential for a negative value of ~2 in this example comes from the presumed substitutability relationship between extreme factors. But a negative value of ~2 could also reflect a strong degree of complementarity between extreme factor 2 and middle factor 3. In such a case an increase in w2 would serve to shift the V3-constraint line outwards, and, when outputs adjust to clear markets for extreme factors (such as at B in fig. 5), excess supply for middle factor 3 may be created. (Should complementarity exist between extreme factors 1 and 2 instead, the Vt-constraint would shift outwards, with outputs adjusting to point E to clear factor markets. At E there exists excess demand for middle factor 3.) To recapitulate: each ~r captures the direct and indirect effects of a rise in w~ on the market for middle factor 3. The 'direct' effect is the term cry, while the 'indirect' effect refers to the altered demand for middle factor 3 when outputs adjust to clear extreme factor markets 1 and 2. Several further links among the ~ are known. Since a doubling of all factor prices does not disturb factor markets, ~..~=l~=0. Secondly, the determinant of coefficients in eq. (20), A, must, we have argued, be negative. Since ~3 equals - ( ~ t +~2), this condition can be written as --A={(0221021)¢1.-[-(011--012)~2}>0~

(26)

where the coefficients of ~1 and ~2 are positive since factor i is extreme in producing commodity i.t 6 trThus condition (26) guarantees that at most one of ~t or ~2 is negative.. If ~2 is negative, restriction (26) indicates that (~t/--~2)>(Oll--Ot2)/(O22--O2t). By eq. (21), the fight-hand side is the elasticity of the (relative price) p-locus. That is, the V-locus, which is now positively sloped, must cut the p-locus from below (i.e. be steeper at the point of intersection). If, instead, ~t is negative, (-~t/~2)<(0tl-0ti)/(02:l-02t), and the positively sloped V-locus must now cut the plocus from above.

84

R.W. Jones and S.T. Easton, Factor intensities andfactor substitution

Finally, consider the expression for ~3:

If all factors are substitutes, ~a clearly is negative. To restrict our discussion of complementarity, we assume 'perfect' complementarity to be the limiting case. That is, if w3 rises, we allow the possibility that not only factor 3, but also either factor 1 or factor 2 may be used less intensively (i.e. be complements with factor 3). In the case of perfect complementarity, the use of the complementary factor is reduced by as much, relatively, as factor 3. Since a~ is pre-multiplied by the fraction ~i, this restriction guarantees a negative value for ~3.17 These remarks lead directly to a powerful result. Suppose all factors are substitutes for each other: I f extreme factors are each better substitutes for the middle factor than for each other, an increase in commodity l's relative price must trap the middle factor's return between that of the extreme factors:

ffl > ffa > ff2. That is to say, if a~ is smaller than a32 (and a,~ smaller than a~), both ~1 and ~2 must be positive, the V-curve negatively sloped, thus (as illustrated in figs. 2 and 3) guaranteeing the chain of inequalities. This finding is, relative to our subsequent results, rather remarkable because it is based solely on assertions about substitutability and is independent of the factor-intensity ranking for the middle factor. But factor intensities for extreme factors have an obvious role to play in determining middle factor 3's real return. If the V-curve lies between the p~ and p2-10ci, as in fig. 2, w3 would be trapped between Pl and/~2 as well. This is the kind of result obtained in the specific-factors model, where V~ is vertical and Vz horizontal, but is 'less likely' to occur if extreme factor intensities are not significantly different. 5.1. A reformulation of ~ 1 and ~2

In proceeding to obtain more results it is useful to reformulate the expressions for the ~i. From the definitions in (19) nine separate elasticities are involved. Homogeneity has allowed us to concentrate just on ~1 and ~2,

17In terms of our diagrammatic technique, a negative value for Ca guarantees that an increase in w I and w2 along a ray from the origin in figs. 2 or 3 must create excess demand for middle factor 3.

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

85

which leaves six elasticities. With unit costs minimized in each sector, 010.,~+ 0 20"i2 + 0 30"3i =

(27)

0,

an application of eq. (14) to the aggregate economy. Solve (27) for the 'own' substitution terms tr~ and 0"2 and substitute, respectively, into ~1 and ~2 to obtain: 03

02

(29) This further reduces the number of separate elasticities to four, as well as revealing the crucial role of {(02/01)al-a2}, whose sign is, by (6), the same as the intensity difference (031-032). Since a~ and o~ each express the degree of substitutability between the two extreme factors, 01

1 2 0"2 ~--"0-2 0.1,

as in (15). Finally, it proves convenient to express the degree of substitutability between each extreme factor and the middle factor in terms of tx~ and 0"2 a instead of 0.a~ and tz2. Applying (15) once again, k __ Ok

0.3 - N 0 . L With these substitutions, the elasticity of the full-employment V-locus can be represented by (minus) the ratio between ~t and ~2:

~A.i= 0__1_ ~ .~o910.3-t-(co~- co2)0"~'[ ¢2 0 2 where co,

_

I

(x i

1

O(2

o~2-=~-+~-> 0, 0~2

@i-co~)=~-~>O,

iff 031>032.

(30)

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

86

The expression in brackets in (30) thus depends upon (i) three independent elasticities of factor substitution - - between each extreme factor and the middle factor, on the one hand, and between the two extreme factors directly, on the other. Also, the expression depends upon (ii) the relative intensity with which middle factor 3 is used in the two sectors. If the first sector uses the middle factor more intensively than the second sector, coefficient col exceeds co2 in (30). 18

5.2. Special cases 52.1. Same intensity of middle factor 3 (031 = 032) To highlight the role of substitution terms on their own, we begin by setting 031 equal to 0a2. This implies that co1 equals co2 so that the elasticity of the V-curve reduces to

~l= ~2

01 a3 02 a'3"

(31)

With intensity differences for the middle factor set aside, the V-curve is unaffected by the relative degree of substitutability between extreme factors 1 and 2. But the comparison between the extent to which the middle factor substitutes for each of the extreme factors is important. We start with the most neutral case, namely that of separability, in which a change in w3 would not affect the overall ratio in which extreme factors are used to produce a given output bundle, so that aia =~r2. 3 In this case the Vcurve must lie strictly between the pl-locus and the pj-locus, as in fig. 2, since

o,

Io,2

)•

That is, the elasticity of the full-employment V-locus becomes a strictly positive weighted average of the elasticities of the pi-loci. The expression in brackets in (30) becomes different from unity if asymmetry is introduced either in the degree to which each extreme factor substitutes for the middle factor or in the intensity with which the middle factor is used in the two sectors. Now leave the case of separability and let middle factor 3 become a better substitute for extreme factor 1 than for factor 2. The V-curve becomes leAn alternative expression for (col- co2)is (1/03)[(0at-032)/10rl, where [0[,positive, is defined in eq. (20).

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

87

steeper until it breaks out of the (Pt,P2) range, as in fig. 3. The change in the return to middle factor 3, trapped not only between fix and if2 but also between/~t and/~2 in the case of separability Eas in (22)], rises towards Wl as factors 3 and 1 become relatively better substitutes. Eventually if3 exceeds/~1 [as in (23) and fig. 31, so that the real returns of both factors 1 and 3 rise, but as long as the middle factor is also a substitute for factor 2, w3 cannot rise by as much as Wr

5.2.2. Extreme factors independent (a 2 = O) The most obvious example of such independence is the case in which neither industry uses both extreme factors, such. as the specific-factors model. However, even if all three factors are used, independence merely asserts that extreme factors 1 and 2 are on the borderline between being substitutes and complements. With extreme factors independent, the elasticity of the V-curve becomes:

+,

+i ++t,,-,++ =

-

02 to92++ j.

(32)

We have already examined the consequences of different degrees of substitutability between the middle factor and the two extreme factors, so let a 3 equal a23. The V-curve will be negatively sloped. If 031 is very close to 032, the elasticity of the V-curve will be close to the value 01/02, and the curve will thus be trapped between the p+-loci. Clearly, the more intensively does the first industry use middle factor 3, the steeper will be the V-locus and as Pl/P2 rises, the greater will be the gain to middle factor 3. But can differences in the intensity with which middle factor 3 is used in the two industries by itself allow the V-curve to break out of the bounds set by the pi-loci, as in fig. 3? If so, the return to middle factor 3 would change unambiguously in real terms. Perhaps surprisingly, the answer is yes. A rise in Pl/P2 could cause w 3 to rise relative to both commodity prices even if extreme factors are independent with respect to each other and of the same degree of substitutability with respect to the middle factor. 19

t9A numerical example can be designed to illustrate this point. First note that if a~=a3,, the expression for ~t/~2 in eq. (32) can be rewritten (with the aid of footnote 9) as: 01 (022-- 021)

02 (011--012)" Now assume that distributive shares in the first industry are [0t l, 021,03t] = [0.2, 0.3, 0.5] and in the second industry are [012,022,032]=[0.1,0.6,0.3]. Furthermore, assume that industry 1 bulks

88

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

5.2.3. Extreme factors substitutes (a 2 > O) We have already established that regardless of factor intensity rankings, if all factors are substitutes but the extreme factors are better substitutes with the middle factor than with each other, the V-curve must be negatively sloped and the change in w 3 trapped between that of w~ and w 2. But once extreme factors become better substitutes for each other than for the middle factor, ~'3 can break out of these bounds. And the direction 6'3 takes depends u p o n the factor intensity ranking for middle factor 3. To fix the framework, assume that the middle factor is used more intensively in x~ than in x2. Consider the following experiment: let a 3 and a23 have given (positive) values and ask how the impact of a rise in Pl/P2 on all factor prices would differ in a sequence of economies in which a 2, the elasticity of substitution between extreme factors, becomes higher and higher. 2° Fig. 6 is used to trace out the consequences of increasing try. N o t e that to leave the diagram relatively uncluttered, we have not drawn in the p~ and P2 loci. Instead, several alternative V-curves (differing in the value a 2 adopts) are drawn, with equilibrium points shown along the dashed p-curve along which Pl/P2 is a constant a m o u n t higher than at initial point A. Consistent with our assumption that 03~ exceeds 032, the p-curve is always cut from below by a ray from the origin. The V-curve in fig. 6 is consistent with values of tr 2 sufficiently small (relative to aa~ and a 3) so that ~1 and ~2 are each positive. Thus, although a cut in w2 from A to C' would be sufficient to raise c o m m o d i t y l's relative price to the level shown along the p-curve, it would create excess supply of middle factor 3. This necessitates a balancing increase in wl (and w2 along the p-curve) to deflect d e m a n d towards factor 3 and thus allow all factor markets to clear at C. Higher values of a 2 cause the curve to rotate clockwise a r o u n d point A because 031 exceeds 032. Vertical curve V' corresponds to a zero value for 42. That is, a drop of w2 from A to C' effects the required increase in P~/P2 without disturbing i'actor 3's market (and with

large in the national income in the sense that its share, 01, is 0.8 (and 02 is 0.2). Since, in general,

O~=OtOlt+ 020~z, we compute a value of 0.18 for 0 t and 0.36 for 02 so that the ratio of aggregate factor shares, 01/02, is 1/2. (Note, of course, that this ratio lies between 011/021, equal to 2/3, and 0t~0_,2, equal to 1/6, the elasticities of the pr-Ioci.) Now (022- 02~)/(0t ~-0~ 2) must exceed unity since 03t exceeds 032. Indeed, this ratio has a value equal to (0.3)/(0.1)=3. Therefore, ¢t/~2 equals 3/2, which exceeds the value of 011/021 of 2/3. The V-curve is steeper than the pt-locus so that an increase in Pt/P2 lets if3 exceed/~t2°As ~ (and a~) increase, we allow 'own' substitution terms a~ and tr_~to increase (in absolute value), in order to keep cross-terms a~ and o_a, constant.

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

IV1

89

V"

/P

,

/ /// 5 " / ¢ "

I

----]

WI Fig. 6

extreme factors 1 and 2 clearing in the background). For such an economy,

Compare point C' with point C to observe that for the same rise in Pl/P2, and with w3 constant in both cases, an economy with the V' curve (higher tr2) would, in comparison with an economy with a lower a 2, (i) have a sharper fall in w2; (ii) no longer have a rise in wl (it stays constant along with w3), (iii) have wa unambiguously rise in real terms (since both pl and P2 must fall), and (iv) experience a reduction in the spread between k~ and ~2. 2~ This latter point is important in revealing that the greater the degree of substitutability between extreme factors 1 and 2, the less can any given relative commodity price change drive their returns apart. But some element of costs must pick up the slack to ensure that P~/P2 goes up by the same amount. The burden falls on w3. Since xl is intensive in its use of middle factor 3, a smaller rise in wl/w2 (because of higher a 2) must be balanced by a greater (relative) rise in w3. The positively sloped V" curve in fig. 6 is appropriate for an economy with an even greater degree of extreme-factor substitutability than shown along the V or V' curves. Relative to the constant value of w3, Wx as well as w2 now falls. Indeed, the spread between ~vl and ~2 has been reduced (since 2~That is, the angle of the cone AOC' is smaller than the angle of cone AOC.

90

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

extreme factors are more substitutable for each other), as evidenced by the narrower cone AOC". To allow the relative cost of producing commodity 1 to rise (to its level along the p-locus) an even greater reliance must now be placed on a relative increase in w3. That is, comparing C" with C', w3 is unchanged in nominal magnitude, but both ~'1 and *~'2 (and/~i and iO2) have larger negative values. What is left unclear from the diagrammatic apparatus illustrated in fig. 6 is the change in factor l's real return (~'i compared to i61 and/~z) if the V-curve becomes positively sloped. Return t o t h e basic equilibrium set 119) and put if3=0. As ~2 adopts larger negative values, swinging the V-curve clockwise from the vertical V'-position in fig. 6, ffl becomes equal to i61 (both negative) when -

¢1 021+031 - = - -

(-¢2)

021

(33)

For slightly higher values of cry,6,1 will be trapped between/~1 and/~2. And, if

¢1 022+032 (-- ~'2)< 022 '

(34)

the rise in Pl/P2 will unambiguously lower w x relative to both commodity prices. In this extreme situation,

~3(= 0) >/~1 >/~2 > ~1 > ~2Extreme factors 1 and 2 are such close substitutes that almost the entire burden of raising the relative cost of commodity 1 falls upon middle factor 3 which, by assumption, is used more intensively in producing x x than in producing x 2.

5.2.4. Extreme factors perfect complements (4 =o'~2) Just as a high degree of substitutability between extreme factors prevents their relative rewards from changing much, high degrees of complementarity push factor rewards apart. Two cases of perfect complementarity illustrate this general point. First we assume that the two extreme factors are perfect complements in the sense that any factor price change does not alter the ratio of the intensities of their use ( 4 =o~2)• Consider the strong implications of such complementarity for the values of ¢1 and ¢2 and, thus, for the position of the V-curve. By definition,

Cx= ~ -

E~M + ~o~.].

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

91

But, since extreme factors are perfect complements, tr~ equals a~, so that ¢~ equals (try-try). The cost-minimization condition linking elasticities in eq. (27) reveals that a~ in this case equals -[(01 +02)/03]a~, so that 1

~I ~-" - - - - 0 " I 03 2"

Similarly, 1 =

-

o-

Combining these we find that when extreme factors are perfect complements, ~t 01 ~-2 =~-"

(35)

This is the type of result previously shown in eq. (31) (with a 3 = a 3) when middle factor 3 is used with the same intensity in each sector. Here, regardless of factor intensities, perfect complementarity between extreme factors drives their returns far apart, leaving the middle ground for if3:

5.2.5. Perfect complementarity between the middle factor and an extreme factor

Now let an extreme factor, say factor 2, become a perfect complement with middle factor 3. We achieve an analogous result, but with factor 3 and extreme factor 1 changing places in the ranking just shown. In the basic definition for ~l replace a~ by a21 since factors 2 and 3 are now assumed perfect complements. Thus, ~1 becomes 0q (tr~-a~). Use eq. (27) again, this time to solve for al, so that ~t converts to ~_ a...L_~ _ ctZ _2 --01 02--02 O1" 2 A similar procedure changes ~2 to 0q ( a23 - t~l), and with (27) revealing that 01

~ =

02 + 0 3 ~ ,

~2 becomes -[ct,/(02 + 03)]o-2. Taking their ratios,

~x . ~2

02 + 03 02

(36)

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R.W. Jones and S.T. Easton, Factor intensities and factor substitution

That is, the V-curve is positively sloped. But

021+031 02+03 022+032

021

'

so that now it is ffl that is trapped between/~1 and/~2. That !s, (37) as commodity l's relative price rises. Perfect complementarity between extreme factor 2 and middle factor 3 has caused their returns to pull as far away as possible in the ranking. This is accomplished by having ~i play the 'middle' role in factor spreads, thus freeing up ~3 to go to the extreme. Complementarity repels; substitutability attracts. These are the unifying underlying relationships at work in the link between commodity price and factor price changes. An increase, in commodity l's relative price raises wl relative to w2. But a high degree of substitutability between extreme factors 1 and 2 limits their price spread, forcing w3 to move upwards if good 1 is intensive in its use of middle factor 3 (or downwards if 032 exceeds 03x) in order to accommodate the required increase in relative unit costs of producing the first commodity. The extent of these moves depends on the relative degrees of intensity differences and asymmetries in substitution elasticities.

6. Outputs and factor endowments One of the relationships in general equilibrium theory that has proved most useful in trade theory is Samuelson's reciprocity theorem 22 whereby

Oxj aw~ OV, = Op--'~"

(38)

The right-hand side asks about the effect of a rise in commodity price j on the return to the ith factor when endowments and all other commodity prices are being held constant. This is the question that we have been analyzing in previous sections. The reciprocity theorem states that an answer to this question also reveals how a change in the endowment of factor i with all other endowments and all commodity prices constant affects the output of commodity j. Therefore answers to the effect which endowment changes at constant commodity prices has on the composition of outputs must be contained in our previous solutions. 22See Samuelson(1953),Jones and Schein.kman(1977),and Kemp (1976, ch. 7).

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

93

We choose to explore two questions about the possible effects of endowment changes on outputs. (1) If the endowment of the middle factor rises (at constant commodity prices), under what circumstances will both outputs expand? And, if both outputs do increase, what features of the technology are crucial in determining the change in relative outputs? (2) If the endowment of one of the extreme factors should rise (at constant commodity prices), is it possible (a) that the commodity in whose production that factor is extreme might nonetheless be reduced in output, or (b) that both outputs might rise? The specific-factors model has strong answers to each of these questions. An increase in the endowment of the mobile factor (i.e. middle factor) must, at constant commodity prices, always increase both outputs. And, a comparison of which output expands relatively more involves a mixture of the factor intensity rankings between commodities of the mobile factor and a comparison of substitution elasticities. As for question (2), in the specificfactor model an increase in V~ must always increase x~ and reduce x2 .23 In our general model this need not be the ease. Commodity outputs are unaffected by a rise in all commodity prices in the same degree. Therefore asking about Ox~/OV3 at constant commodity prices is equivalent to asking how V3 affects xl at constant relative prices. This observation suggests that we can once again use the diagrammatic device in which the absolute price level was adjusted to keep w3 constant. Making use of reciprocity relationship (38) it is clear that Oxl/OV3 and Ox2/OV3 are both positive if and only if the V-curve is trapped between the p : and p2-1oci, as in fig. 2, for in this case a rise in P~/P2 shows ~'1 >/~1 > ~'3 >,02 > ~'2.

(39)

Suppose, on the contrary, that the V-curve were a bit steeper than the Pllocus, as in fig. 3. Then

so that, relative to Px, a fall in P2 would raise w3. That is, the reciprocity theorem would reveal that Ox2/OV3 would be negative. In the specific-factors model the V-curve must be trapped between the Pl- and p2-10ci since the Vcurve is negatively sloped and the p : l o c u s is vertical while the p2-curve is horizontal. 2aWe find the question: Does .~ exceed P'I? generally less interesting than comparisons of the ~j with each other and the /~j. In the specific-factor model xl does not expand relatively as much as l/~ at constant commodity prices since factor prices are also changing. J.I,E,--D

94

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

Now suppose that the V-curve is trapped between the p~- and p2-10ci so that both outputs would expand if V3 increases (at constant relative commodity prices). Compare ~1/~'3 with ~2/P 3 by observing that the reciprocity theorem when translated into 'hat' notation requires:

X1 V3

03 1~3 01 i~1'

x2 03 if3 --~3 =~-. ~-~-2.

(40)

Eq. (20) shows solutions for ff3//3t and 1~3/P2 separately. Therefore if I23 rises, (40) and (20) reveal that Xl will exceed ~2 only if

02EO12~2--O22~l"1
~1 0t ~-2> ~ .

(41)

The reasons we previously cited why (41) might be satisfied - - industry 1 being intensive in its use of middle factor 3, or factor 3 being a better substitute for factor 1 than for factor 2, bear now on the way the transformation curve shifts. In section 5 we encountered two different situations in which equality held in (41) so that, in the present context, an increase in V3 would shift the transformation curve out uniformly. The first of these, discussed in relation to eq. (31), assumed that the middle factor was a substitute for each extreme factor - - to the same degree - - and that each sector used middle factor 3 with the same intensity. But in the second I-see eq. (35)], no condition was attached to factor intensity but we assumed extreme factors were perfect complements. The implications of such perfect complementarity for the behavior of outputs as V3 expands and commodity prices are held constant are readily detected with the aid of fig. 5. The increase in V3, as earlier argued, lowers w3 and raises Wl and w2. Since extreme factors 1 and 2 are perfect complements, they each substitute to the same e x t e n t with middle factor 3 and, given these factor price changes, both extreme factors are used less intensively. In fig. 5, the V1- and V2-constraint lines would shift out by the same relative amount, so that their new intersection point must be at a point such as G, on a ray

R.W. Jones and S.T. Easton, Factor intensities and factor substitution

95

from the origin through initial output bundle, A. Perfect complementarity for extreme factors locks output bundles in strict proportion if the endowment of the middle factor increases. Turning now to question (2) we ask how outputs respond (at a constant relative price ratio) to an increase in the endowment of extreme factor 1. Once again applying the reciprocity theorem, the only instance in which output x~ could be lowered is the same as that in which an increase in P~/P2 would unambiguously lower the real return to factor 1 so that:

In section 5 we identified the kind of situation in which this might occur: extreme factors are highly substitutable with each other, and 031 exceeds 032. Higher values for tr2 rotate the full-employment 1/-curve clockwise as in fig. 6, and if ~2 becomes sufficiently negative that (34) is satisfied, so also will be this ranking. This result, wherein at constant commodity prices an increase in the endowment of extreme factor 1 would actually lower production of commodity 1, can be interpreted along lines familiar from the Rybczynski argument in two-sector Heckscher-Ohlin theory. Factors 1 and 2 are highly substitutable, so think of them (almost) as a 'composite' input, with factor 3 the other input in a 2 x 2 world. If the endowment of the 'composite' factor rises, as it would if 1/1 increases, according to the Rybczynski argument the sector using the other factor (factor 3) intensively will suffer a reduction in output. But industry 1 uses factor 3 intensively. If this argument is to be persuasive, it must also indicate t h a t a rise in I,"2 (the other factor in the 'composite') would lower output xl as well. By reciprocity this would entail a negative value for Ow2/Opl, which is, indeed, indicated in our ranking. Finally, could an increase in 1/1 raise both outputs? Yes - - if the 1/-curve were positively sloped such that ranking (37) holds. Suppose this reflects high substitutability between extreme factors and 031 exceeding 032. As the 1/curve swings clockwise (with increases in tr2), it passes through ranking (37) before reaching the ranking described just previously in which an increase in V~ causes x~ actually to fall. With (37), both xl and x2 rise. But this situation could also reflect a relationship of complementarity - say perfect complementarity - - between middle factor 3 and extreme factor 1. Eq. (37) shows factors 1 and 3 completely reversing the positions they held in the 'neutral' case in which an increase in V3 raises both outputs uniformly. Thus, in this situation an increase in 1/I would raise xl and x2 in proportion. 7. Conclusions

This analysis has proved lengthy both because of the wide range of possible results and because new concepts and techniques are required to

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understand the essential complexities of a general model with three factors and two commodities. Two principal conclusions deserve to be highlighted: (i) the results of the sector-specific model lie in the 'neutral' range of possibilities evident in the more general model, despite the strong asymmetric assumption that extreme factors are each used only in one industry (i.e. are specific), and (ii) there are two types of interaction between factor intensities and factor substitutability in the general model. The first type combines the intensity with which the middle factor is used in each sector with the degree of substitutability between the middle factor and each extreme factor. Thus, an increase in commodity l's relative price is more apt to favor the return to the middle factor if (a) the middle factor is used intensively in the first industry, and/or (b) the middle factor is more substitutable with extreme factor 1 than with extreme factor 2 throughout the economy. This feature is also evident in the specific-factor model, except that in this case the return to factor 3 must remain trapped between the price changes. Also, the general model allows for complementarity between the middle factor and an extreme factor, which could let the return to the middle factor rise (or fall) by more than the return to the other extreme factor. The second type of interaction depends upon the degree of substitutability or complementarity between extreme factors, a feature totally absent in the specific-factors model. An increase in Pl/Pz raises wl relative to w2, but not by very much if extreme factors are highly substitutable. The implication of such high substitutability for rewards to the middle factor depends upon the factor intensity ranking of the middle factor in the two sectors. Thus, if the first industry uses the middle factor relatively intensively, the return to the middle factor might be forced to rise by more than extreme factor 1 if P~/Pz rises, in order to allow the relative cost of commodity 1 to increase by the specified amount when high substitutability prevents much spread between the returns to extreme factors. Indeed, the real return to extreme factor 1 might unambiguously fall. This model almost literally plays both 'extremes' against the 'middle'. Underlying the analysis are several key ways in which the two extreme factors might differ from each other in their relationship to the middle factor. (i) ~t and ~2, as defined in section 2, reveal how a change in the endowment of extreme factors 1 and 2 affects the market demand for middle factor 3 at initial factor prices, when the two outputs adjust to clear markets for extreme factors. (ii) tr~ and tr2, the substitution elasticities introduced in section 3, reveal how a change in the rewards to each extreme factor affects the market demand for middle factor 3 when outputs are kept constant.

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(iii) ~1 and ~2, defined in eq. (19), modify the direct effects of a3x and a32 in (ii) by allowing outputs to change to clear markets for the two extreme factors when w~ or w 2 changes. These indirect effects weight the elasticities (al and a~ for a change in w~ and a~z and a~ for a change in w2) by the ~, fractions in (i). In no case are we tempted to measure ~1 directly against ~2, a~ against a~, or ~1 against ~2. Each of these parameters is infected by the relative size or importance of extreme factors 1 and 2 in the national income. Instead, asymmetries between extreme factors are measured by the discrepancy between {~l~2,a~la~, and ~1/~2} and the ratio of extreme factor distributive shares in the national income, 0~/02. This latter ratio compares the impact on aggregate national income of both quantity and price changes for extreme factors 1 and 2. The most 'neutral' case is one in which: (a) both sectors use the middle factor with the same intensity (so that 031 =032); this implies el/e2 equals 01/02; and (b) the middle factor is separable from extreme factors in the sense that a change in w 3 does not disturb the ratio in which extreme factors are used at constant outputs (so that el3 =a2), 3. this implies that tr3/a ~ az equals 01/02. Conditions (a) and (b) together imply that ¢a/~2 is equal to 0x/02 as well. The diagrammatic technique we use to analyze the income-distribution effects of price changes introduces the full-employment V-locus, the combinations of extreme factor returns which, relative to a fixed nominal w3, would clear the market for factors (with outputs adjusting accordingly). If conditions (a) and (b) are satisfied, the V-locus is trapped strictly between the p~-loci so that the change in the return to middle factor 3 falls between /~t and iO2, the result which always obtains in the specific-factors model, z4 Departures from this neutral case involve the two types of interactions between factor intensities and factor substitutabilities we have described, and are analyzed diagrammatically by changing the position of the V-curve. If extreme factors are independent (¢r~=0), the V-curve rotates in a clockwise direction if, for the first type of interaction, ¢r3 rises compared to a 3 or if 031 becomes larger than 032. For the second type of interaction, as extreme factors become more substitutable the direction in which the V-curve rotates depends upon the intensity ranking for the middle factor. Solutions are

24More can be said. If we let ,6 represent ed~+02/~2, an output-weighted index of price changes, in the case described by (a) and (b) an increase in Pt/P2 changes w3 by the same relative amount as P. If the economy were closed to world trade, P would represent a consumer's price index and w3/P the real wage of middle factor 3 if all factors had identical (homothetic) taste patterns. For a discussion of how trade affects these terms in the context of a specific-factors model, see Ruffin and Jones (1977). In diagrams such as figs. 2--4, a curve showing a constant price level can be drawn. I.t has elasticity equal to (minus) e~/02. Compare with the elasticity of the full employment V-locusshown by eq. (30).

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shown by the intersection of the V-curve with the p-curve, the locus of w 1 and w2 consistent with a given, higher, relative price of commodity 1.25 Section 6 discusses the relationship between endowment changes and outputs at constant commodity prices, a discussion that could be kept brief because Samuelson's reciprocity theorem ties these changes into the commodity price-factor price links analyzed in sections 4 and 5. Among the potential applications of our model outside the traditional uses of small size general equilibrium models in trade, public finance, regional, and growth theory, we focus in conclusion on three diverse fields of economics in which the issues we have raised have already begun to surface in one form or another, and in which our results may be particularly useful. Economic historians have long chafed under the restrictions of the twogood, two-factor modeL, As early as 1967, Robert Fogel argued in the context of the labor scarcity debate for a characterization of the nineteenth-century American economy that included three factors of production [Fogel (1967)1. After the development of the specific factors model, 26 recent work by Clarke and Summers (1980) has revived the debate and included all factors in each production process albeit in a simplified (Cobb-Douglas) fashion. In questions concerning the optimal utilization of capital, e.g. in Winston and McCoy (1974), the possibility of using different shifts of labor to cooperate with the same capital equipment suggests the application of a three-factor model. Although the specific-factors framework would prove too stark for this issue, the possibility that two factors are 'independent' (day labor and night labor) considerably simplifies the analysis, as we argued in section 5. Finally, the rapidly growing energy literature has begun to turn toward simple general equilibrium models. A major controversy that limits the application of two-factor models concerns the complementarity of energy with cooperating factors [see Berndt and Wood (1975) and Griffen and Gregory (1976)]. For example, Solow (1980) traces the implications of complementarity in a two-sector model in which produced energy enters into production of the other sector. For economic theory, and trade theory in particular, our analysis represents the minimal model in which the interaction of factor intensity, substitutability, and complementarity in the production process may be studied together in the context of general equilibrium.

ZSSection 2 revealed how an increase in middle factor endowment V3 would, at unchanged prices, raise w~ and w2 and lower w3. Alternatively, if w3 is kept constant, our diagrammatic technique would illustrate a V-curveshifting outwards along a given p-locus, thus raising both w I and w 2 (and, by a lesser magnitude, pl and P2 in proportion to each other). 26Indeed the specific-factorsmodel was developed partly in response to the issues raised by economic historians in the labor scarcity debate, as the title in Jones (1971) attests. commodity

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