The Two Commodities and Three Factors Model with Increasing Returns to Scale

1. INTRODUCTION

The purpose of this paper is to extend the works by Ide-Takayama (1991, 1993a, 1993b) to the discussion of the two commodities and three factors model with increasing returns to technology. Ide-Takayama discuss the topics on the variable returns to scale in the framework of Hechscher-Ohlin (H-O) model. The topics include the Stolper-Samuelson theorem, the Rybczynski theorem, Comparative Advantage, etc. They show that the Marshallian adjustment process plays an important role in order to obtain the useful results when we allow the variable returns to scale technology.

The H-O model considers an economy consisting of two commodities and two factors, in a competitive situation, under constant returns to scale technology.

Using such a 2×2 framework, the Hechscher-Ohlin theorem states that a country exports a commodity which uses intensively the country’s relatively abundant

The Two Commodities and Three Factors Model with Increasing Returns to Scale

Technology−Another Interpretation of the Leontief Paradox −

Toyonari Ide

＊Faculty of Economics, Fukuoka University, Fukuoka, Japan

−１６５−

（ １ ）

factor. Leontief (1953) tested the empirical plausibility of the above theorem, and found that the U.S. (which is presumably capital abundant) exports labor intensive commodities and imports capital intensive commodities, contrary to the assertion of the above theorem. This paradoxical findings, known as the Leontief paradox, lead the discussions of “demand bias” and “factor intensity reversals,” and so on.

In contrast with the H-O model, there has been an alternative specification, so- called “specific factor model” in the literature (e.g., Haberler 1936, Chapter 12, Harrod 1957, Samuelson 1971, Jones 1971, Mayer 1974, Mussa 1974, Amano 1977, Falvey 1979). Later, a general discussion of the two commodity, three factor model has been followed by Batra and Casas (1976), Suzuki (1983), Ruffin (1981). Also, for an excellent survey article on this topic, see Takayama (1982).

This 2×3 framework, is partly supported by the empirical finding that the U.S.

exports skilled labor intensive (relative to unskilled labor) commodities and imports capital intensive (relative to unskilled labor) commodities. Also, this finding leads to the importance of a three factor model of unskilled labor, (physical) capital, and skilled labor. And the 2×3 model casts a light on the Leontief paradox, and theoretically shows that the U.S., relatively abundant both in skilled labor and capital, exports skilled labor intensive commodities and imports capital intensive commodities.

As mentioned above, the 2×3 model succeeds to explain the Leontief paradox.

Needless to mention, the results above obtained under the 2×3 framework are carried under the assumption of constant returns to technology. Natural question is what can be said if we introduce increasing returns to scale technology into the 2×3 model. We show that the Leontief paradox can be explained by using the 2

×3 model with increasing returns to scale technology. Namely, we show that if skilled labor intensive commodity is produced under increasing returns to scale

−１６６−

（ ２ ）

technology, and capital intensive commodity is produced under constant returns to scale technology, then a larger country (the U.S.) exports skilled labor intensive commodity and imports capital intensive commodity. Thus, the existence of increasing returns to scale can explain the Leontief paradox.

A brief summary of the paper is now in order. Section 2 presents the basic model, in which increasing returns to scale technology is introduced in a simple form. Section 3 discusses the topics such as the endowment-factor price matrix, the Stolper-Samuelson matrix and the Rybczynski matrix. Section 4 discusses the comparative advantage, in which the another interpretation of the Leontief paradox is given. Section 5 provides the concluding remarks.

2. MODEL

We consider an economy consisting of two industries, X and Y, each using the same three factors of production, Vi, i＝1, 2, 3. If only two factors are used in each industry out of three factors, then the model is similar to the specific factor model used by Jones (1971). We write the production function as :

X＝X^T・f(V1X, V2X, V3X), 0＜T＜1 (1a)

Y＝g(V1Y, V2Y, V3Y) (1b)

where Vijdenotes the amounts of factor i used in the jth industry. It is assumed that f and g are homogeneous of degree one with respect to inputs, Vij. The argument X^Tcaptures a scale economy in a X industry while Y industry exhibits a constant returns to scale. The simple form of production functions will suffice for the discussion of this paper.

The Two Commodities and Three Factors

Model with Increasing Returns to Scale Technology（Ide） −１６７−

（ ３ ）

Let aij denote the quantity of factor i required to produce one unit of good j.

The requirement that three factors are fully employed in given by :

aiX・X＋aiY・Y＝Vi, i＝1, 2, 3 (2)

where Vi denotes the endowment of factor i. In addition, assuming an average cost pricing, we have the following zero-profit condition :

p＝w1・a1X＋w2・a2X＋w3・a3X (3a)

1＝w1・a1Y＋w2・a2Y＋w3・a3Y (3b)

where p≡px/py denotes the price of good X in terms of good Y, and where wi

denotes factor price of factor i in terns of good Y.

Differentiation of (2) and (3) yields :

aiXdX＋aiYdY＋XdaiX＋YdaiY＝dVi, i＝1, 2, 3 (4a) dp a_iX !

3

¦

¦

¦

3

¦

i 1 i 1 i 1 i !

(4b)

The aij values are chosen so as to minimize the cost in the usual fashion, We may obtain, aiX＝aiX(w1, w2, w3, X) and aiY＝aiY(w1, w2, w3). Differentiation of these, we have :

da_iX wa_iX ww_h i 1

3

¦

da_iY wa_iY ww_h i 1

3

¦

−１６８−

（ ４ ）

! dV₁

dV₂ dV₃ dp

0 ª

¬

««

««

«« º

¼

»»

»»

»»

S₁₁ S₁₂ S₁₃ a_1Xc a_1Y S₂₁ S₂₂ S₂₃ ac_2X a_2Y S₃₁ S₃₂ S₃₃ ac_3X a_3Y a_1X a_2X a_3X pR_X 0 a_1Y a_2Y a_3Y 0 0 ª

¬

««

««

««

º

¼

»»

»»

»» dw₁ dw₂ dw₃ dX dY ª

¬

««

««

«« º

¼

»»

»»

»» H˜

dw₁ dw₂ dw₃ dX dY ª

¬

««

««

«« º

¼

»»

»»

»»

a_3Xa_2Ya_2Xa_3Y!0, a_1Xa_3Ya_3Xa_1Y!0 Substituting (5) into (4), we may obtain :

! ac_iXdXa_iYdY S_ih

h 1 3

¦

dp a_iX i 1

3

¦ dw_ipR_XdX, 0 a_iY i 1

3

¦ dw_i (6b)

S_ih{wa_iX

ww_hXwa_iY

ww_hY, i,h 1,2, 3! (6c)

where a’iX＝(1−T)aiX.

The matrix S＝[Sih] is termed as the substitution matrix of the economy. The following properties of matrix S are fundamental ; S is symmetric and negative semidefinite with Sw＝w’S＝0, Sii＜0 and the rank of S is 2. A degree of scale economy for a X industry is obtained as ; 1/(1−T)＞1.

In a matrix form, (6) can be written as :

(7)

Following Suzuki (1980) and Ruffin (1981), we assume that the following factor intensity relations hold :

(8)

If a1Y, a2Y, a3Y＞0, then we may equivalently rewrite this as : The Two Commodities and Three Factors

Model with Increasing Returns to Scale Technology（Ide） −１６９−

（ ５ ）

a_1X a_1Y !a_3X

a_3Y!a_2X a_2Y

a_1X a_2Y 0, a_1X!0, a_2Y!0, a_3X!0, a_3Y!0

D1{a_3Xa_2Ya_2Xa_3Y, D2{a_1Xa_3Ya_3Xa_1Y, D3{a_2Xa_1Ya_1Xa_2Y!

E_i{S_i1D₁S_i2D₂S_i3D₃, i 1,2, 3, E { D₁E₁ D₂E₂ D₃E₃!

(8’)

Factors 1 and 2 may then be called extreme factors, and factor 3 may be called the middle factor. Note that the specific factor model amounts to assuming :

(9)

Defineα¹,α², andα³as :

(10)

Due to (8), we may observe :α¹＞0,α²＞0, andα³＜0. Also, we defineβi, i

＝1, 2, 3, andβas :

(11)

Recalling that factors i and h are substitutes if ∂aij/∂wh＞0, i≠h, and that factors i and h are complements if ∂aij/∂wh＜0, i≠h in industry j, and also recalling (6c), we may then define (for i≠h) :

Factor i and h are aggregate substitutes if Sih＞0 (12a)

Factor i and h are aggregate complements if Sih＜0 (12b)

Suppose that the extreme factors are aggregate complements for each other and the middle factors are aggregate substitutes in every industry. We then have :

−１７０−

（ ６ ）

! S₁₁ S₁₂ S₁₃

S₂₁ S₂₂ S₂₃ S₃₁ S₃₂ S₃₃ ª

¬

««

«

º

¼

»»

»

ª

¬

««

«

º

¼

»»

»

E10, E20, E3!0, E 0

H (1T)E pR_X

w₁w₂w₃˜K{ '

(13)

Note that S21＝S12＝0 for the specific factor model. Due to (13), we have :

(14)

By strightforward computation (e.g., use Cramer’s rule), we may obtain the following result from (7) :

(15)

where K＝w1S12S13＋w2S12S23＋w3S13S23.

Note that if T＝0, then (15) becomesΔ＝(1−T)β＜0 due to (14). However, for T≠0, the sign of K is important. So we have the following lemma.

LEMMAK＞0 always.

Proof : Using the homogeniety of Sih’s, i.e., w1Si1＋w2Si2＋w3Si3＝0, rewrite K as K＝w1(S12S13−S11S23). Since the matrix S is negative semidefinite, the matrix formed by deleting the 3rd row and 3rd column is negative definite.

Therefore, we have S11＜0 and S11S22−S12S21＞0. But again by the homogeniety of Sij’s, we have :

K＝w1(S12S13−S12S23)＝(w1w2/w3)(S11S22−S12S21)＞0.

Q.E.D Thus unlike constant returns to scale technology (T＝0), we can not determine the sign ofΔ. To this end, letting Z＝X/Y, we may compute from (7) :

The Two Commodities and Three Factors

Model with Increasing Returns to Scale Technology（Ide） −１７１−

（ ７ ）

dZ dp K

'˜(1T)pZ1

w₁w₂w₃Y (16)

The supply price curve is upward sloping ifΔ＜0, and downward sloping ifΔ

＞0.

3. COMPARATIVE STATICS

In this section, we obtain the expressions for ∂wi/∂Vh, ∂wi/∂pj, ∂X/∂Vi, and∂Y/∂Vi, i, h＝1, 2, 3, j＝X, Y from (7). Firstly, we may compute∂wi/∂

Vhby using Cramer’s rule as :

ww₁ wV₁

1

'ª(1T)D₁²pR_X(a_2YG₃a_3YG₂)

¬ º

¼ (17a)

ww₁ wV₂

ww₂ wV₁

1

'

>

@

wV₃ ww₃

wV₁ 1

'

>

@

ww₂ wV₂

1

'ª(1T)D₂²pR_X(a_1YP3a_3YP1)

¬ º

¼ (17d)

ww₂ wV₃

ww₃ wV₂

1

'

>

@

ww₃ wV₃

1

'ª(1T)D₃²pR_X(a_1YS2a_2YS1)

¬ º

¼ (17f)

whereδi≡a1YS3i−a2YS2i,μi≡a1YS3i−a3YS1i, andπi≡a1YS2i−a2YS1i, i＝1, 2, 3.

Note that if T＝0(RX＝0), then∂wi/∂Vh’s are independent of Sih, however, this

−１７２−

（ ８ ）

ww₁ wV₁

ww₁ wV₂

ww₁ wV₃ ww₂

wV₁ ww₂

wV₂ ww₂

wV₃ ww₃

wV₁ ww₃ wV₂

ww₃ wV₃ ª

¬

««

««

««

«

º

¼

»»

»»

»»

»

ª

¬

««

«

º

¼

»»

»

!

is not a case for T≠0. Due to (13), we may observe :

δ1＞0,δ2＞0,δ3＞0,μ1＞0,μ2＞0 (18a)

μ3＜0,π1＞0,π2＞0,π3＞0 (18b)

Ruffin (1981) calls factors i and h are friends if∂wi/∂Vh(∂wh/∂Vi)＞0, and enemies if∂wi/∂Vh＜0. For T＝0, we have the following :

(19)

Thus, for T＝0, the two extreme factors are always enemies, and the middle factor is the friend of either extreme factors. However, in view of (17), the sign of∂wi/∂Vh’s are indetermined in general without additional condition when T≠

0.

We may now compute the expressions of the matrix [∂wi/∂pj] known as the Stolper-Samuelson matrix, and the matrix [∂X/∂Vi, ∂Y/∂Vi] known as the Rybczynski matrix. They are obtained as :

ww₁

wp_x (1T)˜wX

wV₁ (1T)

' ˜(a_2YE₃a_3YE₂) (20a)

ww₂

wp_x (1T)˜ wX wV₂

(1T)

' ˜(a_1YE3a_3YE1) (20b)

ww₃

wp_x (1T)˜ wX

wV₃ (1T)

' ˜(a_1YE₂a_2YE₁) (20c)

The Two Commodities and Three Factors

Model with Increasing Returns to Scale Technology（Ide） −１７３−

（ ９ ）

Z a p * p_S(Z)1 ª

¬« º

¼» { <(Z), a!0, Z& dZ dt

Z p_S˜dp_S

dZ '

K₁˜

>

@

pK w₁w₂w₃X ww₁

wp_y wY wV₁

1

'˜ (ac_2XE₃ ca_3XE₂)p_xR_X w₂w₃˜K ª

¬« º

¼» (20d)

ww₂ wp_y

wY wV₂

1

'˜ (ac_3XE1 ca_1XE3)p_xR_X w₁w₃˜K ª

¬« º

¼» (20e)

ww₃ wp_y

wY wV₃

1

'˜ (a_1Xc E2 ca_2XE1)p_xR_X w₁w₂˜K ª

¬« º

¼» (20f)

As it can be seen from (20) that Samuelson’s reciprocity theorem holds almost for T≠0. For T＝0, from (14) we have the following :

ww₁ wp_x

wX

wV₁!0, ww₂ wp_x

wX

wV₂0, ww₃ wp_x

wX wV₃

!

0 (21a)

ww₁ wp_y

wY

wV₁0, ww₂ wp_y

wY

wV₂!0, ww₃ wp_y

wY wV₃

!

0 (21b)

For T≠0, the sign of ∂wi/∂pj, ∂X/∂Vi and ∂Y/∂Vi depend on the sign ofΔ.

To close the model, we assume the country is a small open economy. Define the Marshallian adjustment process by :

Z (22)

where p^＊ is the world price and pS(Z) is the supply price defined by (16).

Also, define the elasticity of supplyεby :

(23)

−１７４−

（ １０ ）

Assuming an interior solution Z^＊ to (22), we may obtain the following result for a small open economy : An equilibrium is Marshallian stable if and only if Ψ’(Z^＊)＜0 if and only ifε＞0.

In view of (23), we may conclude thatε＞0 if and only if Δ＜0. We can now determine the sign of ∂wi/∂Vhin (17). Assuming∂wi/∂Vi＜0, we may obtain from (18) the same sign pattern as given in (19) under the Marshallian stable equilibrium, in which we use (∂wi/∂V1)a1Y＋(∂wi/∂V2)a2Y＋(∂wi/∂V3) a3Y＝0, i＝1, 2, 3. Hence, we have the following ;

THEOREM 1

Under the Marshallian stable equilibrium, the two extreme factors are always enemies, and the middle factor is the friend of either factors.

Next we may determine the sign pattern of the Stolper-Samuelson matrix and the Rybczynski matrix. Recalling (14) withΔ＜0, we may obtain the same sign pattern as given in (21). Thus, we have the following result :

THEOREM 2

Under the Marshallian stable equilibrium, we have the following ; Assuming constant commodity prices, an increase in the endowment of one extreme factor increases the output of the industry which is intensive in the use of that factor and lowers the output of the other industry. The effect of an increase in the endowment of the middle factor upon the output of either of the two industries is indeterminate.

Assuming constant factor endowments, an increase in the price of one commodity increases the price of the extreme factor which relatively

The Two Commodities and Three Factors

Model with Increasing Returns to Scale Technology（Ide） −１７５−

（ １１ ）

' ˜dZ

Z T˜(E K₁˜ w_i i 1

3

¦ V_i)˜dJ

J K₁˜

>

@

S

J p_S˜dp_S

dJ

T˜(E K₁˜¦_{i 1}³ w_iV_i) K₁˜

>

@

intensively used in that industry and lowers the price of the other extreme factor, while its effect on the price of the middle factor is indeterminate.

Note that for a specific factor model (a1Y＝a2X＝0),∂w3/∂px＝(1−T)∂X/∂V3

＞0, however, the sign of ∂w3/∂py＝∂Y/∂V3 is still indeterminate. For the useful and interesting interpretation of Theorem 2, see Takayama (1982), p.20.

In conclusion, it is shown that the same results hold as in the case of constant returns to scale technology under the Marshallian stable equilibrium.

Lastly we show the result of the shift in pS when the endowments of factors increase proportionally byγ. From (7), we may compute :

(24)

Letting dZ＝0 in (24), we may obtain :

(25)

Thus proportionate increase of the endowments shifts the supply curve downward. Note that the shift in pS does not depend on theΔ.

4. COMPARATIVE ADVANTAGE

Let Vi＊

denotes the endowments of Viin a foreign country. In order to avoid a Heckscher-Ohlin bias, assume Vi＊＝θVi, θ＞1, that is, the endowments of a foreign country is larger than that of a home country by the same proportion.

We introduce the homothetic demand function by ; pD＝h(Z), h’(Z)＜0. Define

−１７６−

（ １２ ）

T p˜dp

dT

T˜(E K₁˜¦_{i 1}³ w_iV_i)

K₂˜(V_D H) , K₂ K₁

V_D˜

>

@

Z b p_D(Z) p_S(Z)1 ª

¬« º

¼» { )(Z), b!0

the elasticity of demand by ;σD≡−(Z/pD)(dpD/dZ)＝−h’(Z)Z/h＞0. From (23) and (24), we may compute :

(26)

As it can be seen from (26), the sign of dp/dθdepends upon the sign of (σD

＋ε). We now introduce the Marshallian adjustment process by :

(27)

Assuming an interior solution Z^＊, we obtain the following ; An equilibrium is Marshallian stable if and only if φ(Z^＊)＜0 if and only ifσD＋ε＞0. Hence, we may conclude :

THEOREM 3

Under the Marshallian stable equilibrium, a larger country will have a comparative advantage over a good which an increasing returns to scale prevails.

Note that to obtain Theorem 3 we might need an assumption of a unique equilibrium due to the possibility of multi-equilibrium. Theorem 3 may provide an another interpretation on the U.S. pattern of trade, which is known as the Leontief paradox, i.e., the U.S. imports capital intensive commodities. To this end, call factors 1, 2, 3, respectively, “skilled labor,” “capital,” and “unskilled labor.” Assume that a commodity X is skilled labor intensive (relative to unskilled labor), and a commodity Y is capital intensive (relative to unskilled labor). Also,

The Two Commodities and Three Factors

Model with Increasing Returns to Scale Technology（Ide） −１７７−

（ １３ ）

assume that the extreme factors, skilled labor and capital, are complements.

Finally, assume that a larger country is U.S.. Then, from Theorem 3, we may conclude that U.S. exports a skilled labor intensive commodity X, and imports a capital intensive commodity Y. Thus, Leontief paradox is no longer a paradox.

For an interpretation of the Leontief paradox under constant returns to scale technology, see Takayama (1982), p.23.

5. CONCLUSION

In section 3, we have shown that the sign patterns of the Stolper-Samuelson matrix and the Rybczynski matrix are the same as the ones obtained under constant returns to scale technology at the Marshallian stable equilibrium. In section 4, we have shown that the existence of increasing returns to scale technology can explaine the Leontief paradox. That is, the U.S. exports a skilled labor intensive commodity, and imports a capital intensive commodity. In this paper, we assumed increasing returns to scale technology for only one sector.

However, the results obtained here can be easily extended to more general form in the 2×3 model.

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Haberler, G. (1936), The Theory of International Trade, London, William Hodge (translated from German, Internationale Handel, Berlin, 1933).

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（ １４ ）

Harrod, R. F. (1957), International Economics, London, Nisbet, 4th ed.

Ide, T. and Takayama, A. (1991), “Variable Returns to Scale, Paradoxes, and Global Correspondences in the Theory of International Trade,” in Trade, Policy, and International Adjustments, ed. by A. Takayama, M. Ohyama, and H. Ohta, San Diego, Calf. Academic Press, 108‐154.

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The Two Commodities and Three Factors

Model with Increasing Returns to Scale Technology（Ide） −１７９−

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