Heckscher-Ohlin Model

Transcription

1 Chapter 3 Heckscher-Ohlin Model aaaaaaaaa Eli Heckscher (1919), The effect of foreign trade on the distribution of income, Ekonomisk Tidskerift, 21: aaaaaaaaaaaa Bertil Ohlin (1933), Interregional and International Trade, Cambridge University Press. Differences in the endowments of factors of production explain the pattern of international trade. 1. The pattern of trade reflects the relative endowment of productive factors relatively labor-abundant countries tend to export relatively laborintesive commodities. 2. Free trade benefits the relatively abundant factor of production but harms the relatively scarce factor. Assumptions 1. Two goods: Food (F) and Clothing (C): j = C,F 2. Two factors: Labor (L) and Capital (K): i = L,K a ij = units of factor i required to produce a unit of good j. Fixed because of CRS. 3. L = fixed supply of labor K = fixed supply of capital 4. a LC a KC > a LF a KF (1) Clothing is labor-intensive and food is capital-intensive. 1

2 2 CHAPTER 3. HECKSCHER-OHLIN MODEL Full employment conditions:x Food a LC x C +a LF x F = L (2) a KC x C +a KF x F = K (3) x F = L a LF a LC a LF x c (2 ) x F = K a KF a LC a KF x c (3 ) L a LF K a KF K K PPF O L K Clothing a LC a KC An increase in capital supply from K to K expands the production possibility frontier. This expansion is biased towards the production of food. Therefore, an economy with a higher capital-labor ratio produces relatively large quantities of food. The Hat Notation: ˆx dx x = dlogx Let y=f(x). Then the elasticity of y with respect to x is given by: ǫ f ŷ ˆx. This notation will be used a lot in the subsequent proofs. Theorem 1 (Rybcszynski) Let the factor intensity assumption in (1) hold. If the endowment of a factor increases, then the production of the good that uses this factor more intensively increases and the production of the other good decreases.

3 3 Proof: Differentiating the full employment condition (2) we get a LC dx C +a LF dx F = dl a LC x Cˆx C +a LF x Fˆx F = L ˆL a LC x C a LF x F ˆx C + ˆx F = a LC x C +a LF ax F a LC x C +a LF x ˆL F λ LCˆx C +λ LFˆx F = ˆL (4) Similarly, differentating (3) we get λ KCˆx C +λ KFˆx F = ˆK (5) λ ij Fraction of factor i used in the production of good j. (5)-(4) gives ˆK ˆL = (λ KC λ LC )ˆx C +(λ KF λ LF )ˆx F. Notice that λ ic +λ if = 1 for i = L,K Therefore, λ LF = 1 λ LC λ KF = 1 λ KC Notice that ˆK ˆL = (λ KC λ LC )ˆx C +(1 λ KC 1+λ LC )ˆx F = (λ LC λ KC )(ˆx F ˆx C ) a LC x C a KC x C λ LC λ KC = a LC x C +a LF x }{{ F a } KC x C +a KF x }{{ F } L K = 1 LK [a LCa KC x 2 C +a LCa KF x C x K a LC a KF x 2 C a LFa KC x C x F ] = x Cx F LK [a LCa KF a LF a KC ] sign[λ LC λ KC ] = sign[a LC a KF a LF a KC ] Since a LC a KC > a LF a KF a LC a KF a LF a KC > 0, we have λ LC λ KC > 0. Therefore, ˆK > ˆL ˆxF > ˆx C Suppose only capital expands, i.e., ˆK > 0 and ˆL = 0. Then the above implies that the production of food must expand relatively more than that of clothing. Consider (4) again: λ LCˆx C +λ LFˆx F = ˆL (1 λ LF )ˆx C +λ LFˆx F = ˆL λ LF (ˆx F ˆx C ) = ˆL ˆx C

4 4 CHAPTER 3. HECKSCHER-OHLIN MODEL Since ˆx F ˆx C > 0, we have ˆL > ˆx C. Similarly, from (5) we get Combining the above two we get λ KC (ˆx F ˆx C ) = ˆx F ˆK ˆx F > x C ˆx F > ˆK ˆx F > ˆK > ˆL > ˆx C If ˆK > 0 and ˆL = 0, then ˆx F > 0 and ˆx C < 0 which is the Rybcszynski result. Relationship between commodity prices and factor prices Constant returns to scale implies that the unit cost of producing a good equals its price in equilibrium. Therefore, a LC w +a KC r = P C (6) a LF w+a KF r = P F (7) r p C a KC p C p C Price of clothing increases wage-rate increases and rental-rate increases p F a KF p F O p F p C w a LF a LC Theorem 2 (Stolper-Samuelson) Let the factor intensity assumption in (1) hold. If p j increases for j = C,F, then the prices of the factor more intensively used in the production of good j increases, while the price of the other factor decreases. Intuition: Suppose p C with p F remaining constant. Then the relative price of clothing goes up. This implies an increase in the production of clothing and a decrease

5 5 in the production of food. 1 unit reduction in the production of food releases a LF relative units of labor. a KF 1 unit expansion of the production of clothing requires a LC relative units of a KC labor. Since a LC > a LF, there is excess (relative) demand for labor. Hence, w a KC a KF r increases. Proof: Differentiating (6) and (7) we get [ ][ ] alc a KC dw = a KF dr a LF [ ] dpc dp F = a LC a LF a LF a KC > 0 because of the factor intensity assumption. Let dp F = 0 and dp C > 0. Then, dw = a LC dp C > 0 dr = a LF dp C < 0 Alternative: Differentiating (6) and (7) we get Where θ LC = etc. θ LC ŵ+θ KCˆr = ˆp C (8) θ LF ŵ+θ KFˆr = ˆp F (9) a LC w share of labor income in the production of food, a LC w+a KC r Since θ Lj +θ Kj = 1 for j = C,F. ˆp C ˆp F = (θ KF θ KC )(ŵ ˆr) ˆp C > p F ŵ > ˆr. Simlar to the proof of Theorem 1, it is to show that ŵ > ˆp C > ˆp F > ˆr Theorem 3 (Factor price equalization) If two countries share a common technology and neither country specializes in one good, then free trade in commodities serves to equalize factor returns even if no international factor mobility is allowed. Proof: Both countries share the same technology implies a ij = a ij for i = L,K and j = C,F Free trade implies p j = p j. If both countries produce both goods in positive quantities, then the same set of equations (6) and (7) determine the equilibrium w and r in both countries which implies that w = w and r = r.

6 6 CHAPTER 3. HECKSCHER-OHLIN MODEL The Pattern of International Trade Let L K > L K, i.e.,homecountryistherelativelylabor-abundantcountry. Then, from Theorem 1 it immediately follows that. Theorem 4 (Heckscher-Ohlin) If the preferences do not differ much between the two countries, then the relatively labor-abundant home country will tend to export clothing and import food. Variable coefficients a ij s are not fixed any more, i.e., there are many different possible combinations of capital and labor to produce 1 unit of a particular good. How to find such combinations? min wa LC +ra KC s.t. f c (L,K) = 1 ā LC = a LC (w,r) ā KC = a KC (w,r) similarly, ā LF = a LF (w,r) ā KF = a KF (w,r) The price equations are: wa LC (w,r)+ra KC (w,r) = p C (6 ) wa LF (w,r) +ra KF (w,r) = p F (7 ) Factor intensity assumptions: a LC (w,r) a KC (w,r) > a LF(w,r) a KF (w,r) w and r Equations (6 ) and (7 ) can be represented as follows:

7 7 r r a LF,a KF a LC ( w, r), a KC ( w, r) O Leontief Paradox w w How can we test Heckscher-Ohlin theorem systematically? Take a country that appears exceptionally well-endowed with some factor, say, capital. Measure the quantities of capital and labor that it uses to produce a unit of each good that it exports and each good that it produces at home in competition with imports(call such goods importables ). Add these product-level inputs to obtain the total amounts of capital and labor in the production of exports (K x and L x ), and that of the importables (K m and L m ), with actual exports and imports serving as weights. The capital-rich nation should then export a bundle K x of goods more capital-intensive than its bundle of importables: > K m. L x L x W.W. Leontief (1953) proposed that after World War II the United States was a standout capital-rich nation suitable for this test. Using the industry-wide data of 1947, he showed that imports of the US were more capital-intensive. Possible explanations for this paradox: US and foreign technologies are not the same. By focusing only on labor and capital, Leontief ignored land. Labor should have been disaggregated by skill (since it would not be surprising to find thath US exports are intensive in skilled labor). The data for 1947 may be unusual, since World War II had just ended. The US was not engaged in free trade, as the HO model assumes. Factor-intensity reversals: Consider again the zero-profit conditions with variable factor intensities: wa LC (w,r)+ra KC (w,r) = p C (6 ) wa LF (w,r) +ra KF (w,r) = p F (7 )

8 8 CHAPTER 3. HECKSCHER-OHLIN MODEL Suppose there is factor-intensity reversals as shown in the following diagram: p p A A (a LF,a KF ) (a LC,a KC ) p B w A B w B (b LF,b KF ) (b LC,b KC ) w At point A, we have a LC > a LF. As the relative wage rate increases, at B, a KC a KF the clothing industry starts using relatively less labor, i.e., b LC b KC < b LF b KF. This phenomenon is called the factor-intensity reversal. Now, let the home country be the relatively labor-abundant economy. Then, it is logical to expect that the home country will have a higher ratio of r to w, i.e., more likely to at point A. Thus, in equilibrium, factor prices will not be equalized. Therefore, we can modify the statement of Theorem 3. Therefore, we can modify the statement of Theorem 3. Theorem 3 If two countries share a common production technology, there are no factor intensity reversals, and neither country specializes in one good, then free trade in commodities serves to equalize factor returns even if no international factor mobility is allowed. More on the Heckscher-Ohlin Model 1. Many goods and many factors Let there be J goods indexed by j=1,2,...,j, and I factors indexed by i=1,...,i. a ij amount of factor i required to produce 1 unit of good j.

9 9 Example I = J = 2 j = C,F i = L,K In general I J. 1.1 Rybczynski Theorem Now there are I full-employment conditions: The above systema can be writeen as a 11 a 12 a 1J a 21 a 22 a 2J... a 11 a 12 a 1J a 11 x 1 +a 12 x a 1J x J = Z 1 (1.1) a 21 x 1 +a 22 x a 2J x J = Z 2 (1.2) =. a I1 x 1 +a I2 x a IJ x J = Z I (1.2) I J Now differentiate (1.1) totally to get: x 1 x 2. x J J 1 z 1 z 2 =. z I I 1 Ax = Z (1) a 11 dx 1 +a 12 dx 2 + +a 1J dx J = dz 1 a 11 x 1ˆx 1 +a 12 x 2ˆx 2 + +a 1J x Jˆx J = z 1 ẑ 1 a 11 x 1 ˆx 1 + a 12x 2 ˆx 2 + a1jx J ˆx J = ẑ 1 z 1 z 1 z 1 Define λ 1j a 1jx j a 1j x j = : fraction of factor 1 z 1 a 11 z 1 + +a ij z 1 + +a 1J z 1 demanded in sector j. Then the full-employment differential condition for factor 1 becomes λ 11ˆx 1 +λ 12ˆx 2 + +λ 1jˆx j + +λ 1Jˆx J = ẑ 1 J λ 1jˆx j = ẑ 1 j=1 Similarly, for factor i we have J λ ijˆx j = ẑ i j=1

10 10 CHAPTER 3. HECKSCHER-OHLIN MODEL We will have I such equations which can be written as follows: λ 11 λ 12 λ 1J ˆx 1 ẑ 1 λ 21 λ 22 λ 2J ˆx 1 ẑ 1 = λ I1 λ I2 λ IJ ˆx J ẑ J I J J 1 I 1 Lˆx = ẑ (2) Go back to equation (1), and let I=J=2. Then (1) becomes Let det (A)= a 11 a 22 > a 21 a 12. > 0 a 11 a 22 > a 21 a 12 a 11 x 1 +a 12 x 2 = z 1 a 21 x 1 +a 22 x 2 = z 2 [ ][ ] [ ] a11 a 12 x1 z1 = a 21 a 22 x 2 z 2 a 11 a 21 > a 12 a 22 Good 1 uses factor 1 more intensively. So, we have: Good 1 clothing Factor 1 Labor Good 2 Food Factor 2 Capital a 11 = a LC a 12 = a LF a 21 = a KC a 22 = a KF How to represent equation (1) graphically? Z 2 (Z 1,Z 2 ) x 2 (a 12,a 22 ) (a 12,a 22 ) x 2 (a 11,a 21 ) (a 11,a 21 ) O Z 1 Cone of diversification cone (A) Output Determination

11 11 x 1 = a 22z 1 a 1 2z 2 x 2 = a 11z 2 a 2 1z 1 a 21 a 11 < z 1 z 2 < a 22 a 12 > 0 a 22 z 1 > a 12 z 2 z 1 z 2 > a 12 a 22 z 2 z 1 > a 22 a 12 > 0 a 11 z 2 > a 21 z 1 z 2 z 1 > a 21 a 11 So, a necessary condition for both outputs to be produced at strictly positive quantities is that z cone(a). If z / cone (A), then at least one good must not be produced. Now suppose z 1 increases to z 1 with z 2 remaining constant. Then there is a unique combination of (x 1,x 2 ) that can be achieved by the new input vector (z 1,z 2 ) since the factor intensity assumption must hold. Graphically, Z 2 x 2 (a 12,a 22 ) x 2 (a 12,a 22 ) x 1 (a 11,a 21 ) (Z 1,Z 2 ) x 1 (a 11,a 21 ) (Z 1,Z 2) O Rybczynski Theorem Z 1 Equation (2) is a generalization of the 2 2 Rybczinski Theorem Stolper-Samuelson Theorem Next, consider the zero-profit conditions in J sectors. Start with sector j whose zero-profit condition is given by: a 1j w 1 +a 2j w 2 + +a Ij w I = p j, j = 1.,J. The above system can be writeen as: A w = p (3) Differentiating the j-th equation of the above system (3), we get

12 12 CHAPTER 3. HECKSCHER-OHLIN MODEL a 1jw 1 p 1 a 1j dw 1 + +a Ij dw I = dp j ŵ a Ijw I ŵ I = ˆp j p j I θ ij ŵ i = ˆp j (4) Where θ ij a ijw i a ij w i = p I j i=1 a is the share of factor i in the total unit cost ijw i of producing good j. Let I=J=2. Then (3) becomes [ ][ ] [ ] a11 a 21 w1 p1 = a 12 a 22 w 2 p 2 det(a)= a 11 a 22 a 12 a 21 > 0 because of the factor intensity assumption. w 1 = a 22p 1 a 21 p 2 w 2 = a 11p 2 a 12 p 1 a 12 a 11 < p 2 p 1 < a 22 a 21 i=1 > 0 a 22 p 1 > a 21 p 2 p 2 p 1 < a 22 a 21 > 0 a 11 p 2 > a 12 p 1 p 2 p 1 > a 12 a 11 So, a necessary condition for the prices of both factors to be strictly positive is that p (A ). p 2 (p 1,p 2 ) w 2 (a 21,a 22 ) (a 21,a 22 ) w 1 (a 11,a 12 ) (a 11,a 12 ) O Factor price determination p 1 Now differentiating condition (3) we get the general version of the Stolper- Samuelson Theorem which is as follows. Tŵ = ˆp (4)

13 13 where T=[θ ij ] J I and θ ij a ijw i p j. p 2 w 2 (a 21,a 22 ) (p 1,p 2 ) (p 1,p 2) w 2 (a 21,a 22 ) w 1 (a 11,a 12 ) w 1 (a 11,a 12 ) O p 1 Stolper-Samuelson Theorem 1.3 Determination of equilibrium outputs and factor prices The equilibirium is determined by the following two systems of equations: Ax = z (1) A w = p (2) There are a total of I+J equations in I+J unkowns: x 1,,x J and w 1,,w I Same number of goods and factors Consider the case where I=J. Then, system (2) consists of J equations in J unkowns. The question is if we can obtain a solution for the factor prices vector w, and if the solution is unique. At this point we have to know what does it mean by factor intensity reversal (FIR). We have seen that if there is FIR, then the iso-cost curves of the two industries intersect twice, then there exist a price vector p at which the iso-cost are tangent to each other. In terms of the general model this means that two columns of the matrix A are linearly independent, i.e., A is singular. Hence, we need to impose stronger conditions on the matrix A that can guarantee a unique solution to the system (2), which are a 11 > 0, a 11 a 21 a 12 a 22 > 0,,.. a 11 a 1J a J > 0 a ajj

14 14 CHAPTER 3. HECKSCHER-OHLIN MODEL Provided the matrix A is non-singular, the generalized version of Rybczynski and Stolper-Samuelson Theorems hold. Rybczynski Theorem For an increase in the endowment of each factor, there must be a good whose output rises and another good whose output falls. Sotlper-Samuelson Theorem For an increase in the price of each good, there exists some factor that gains in real terms and another that loses More factors than goods Consider now I > J. Then the generalized version of the Stolper-Samuelson Theorem continues to hold, but that of the Rybczyski Theorem may not. To see this consider J=2 and I=3. Then system (3) becomes a 11 w 1 +a 21 w 2 +a 31 w 3 = p 1 (3.1) a 12 w 1 +a 22 w 2 +a 23 w 3 = p 2 (3.2) Differentiating the above we get θ 11 ŵ 1 +θ 21 ŵ 2 +θ 31 ŵ 3 = ˆp 1 (4.1) θ 12 ŵ 1 +θ 22 ŵ 2 +θ 32 ŵ 3 = ˆp 2 (4.2) The above system has two equations in three unknowns, yet we can derive the Stolper-Samuelson result. Let ˆp 2 = 0. Then from (5.2) it follows that [ θ12 ŵ 3 = ŵ 1 + θ ] 22 ŵ 2 (6) θ 32 θ 32 Substituting the above in (5.1) we get: ( θ 11 θ ) ( ) 31θ 12 ŵ 1 + θ 21 θ 31 θ22 ŵ 2 (4.1 ) θ 32 θ 32 }{{}}{{} θ 1 θ 2 Supposing θ 1,θ 2, ˆp > 0, we may conclude that at least one of the two factor price changes, i.e., ŵ 1 and ŵ 2, must be positive. Suppose both ŵ 1 > 0 and ŵ 2 > 0. Thenfrom(6) itfollowsthat ŵ 3 < 0, which conformstothe Stolper-Samuelson result. Next consider the full-employment conditions (1) a 11 x 1 +a 12 x 2 = z 1 (1.1) a 21 x 1 +a 22 x 2 = z 2 (1.2) a 31 x 1 +a 32 x 2 = z 3 (1.3)

15 15 Differentiating the above system we get: λ 11ˆx 1 +λ 12ˆx = ẑ 2 (2.1) λ 21ˆx 1 +λ 22ˆx = ẑ 2 (2.2) λ 31ˆx 1 +λ 32ˆx = ẑ 2 (2.3) The above system is clearly over-determined, and a Rybczynski-like conclusion is difficult to draw.