14.581 International Trade — Lecture 9: Factor Proportion Theory (II) — 14.581 Week 5 Spring 2013 14.581 (Week 5) Factor Proportion Theory (II) Spring 2013 1 / 24
Today’s Plan Two-by-two-by-two Heckscher-Ohlin model 1 Integrated equilibrium 1 Heckscher-Ohlin Theorem 2 High-dimensional issues 2 Classical theorems revisited 1 Heckscher-Ohlin-Vanek Theorem 2 Quantitative Issues 3 14.581 (Week 5) Factor Proportion Theory (II) Spring 2013 2 / 24
Two-by-two-by-two Heckscher-Ohlin model Basic environment Results derived in previous lecture hold for small open economies relative good prices were taken as exogenously given We now turn world economy with two countries, North and South We maintain the two-by-two HO assumptions: there are two goods, g = 1,2, and two factors, k and l identical technology around the world, y g = f g ( k g , l g ) identical homothetic preferences around the world, d c g = α g ( p ) I c Question What is the pattern of trade in this environment? 14.581 (Week 5) Factor Proportion Theory (II) Spring 2013 3 / 24
Two-by-two-by-two Heckscher-Ohlin model Strategy Start from Integrated Equilibrium � competitive equilibrium that would prevail if both goods and factors were freely traded Consider Free Trade Equilibrium � competitive equilibrium that prevails if goods are freely traded, but factors are not Ask: Can free trade equilibrium reproduce integrated equilibrium? If factor prices are equalized through trade , the answer is yes In this situation, one can then use homotheticity to go from di¤erences in factor endowments to pattern of trade 14.581 (Week 5) Factor Proportion Theory (II) Spring 2013 4 / 24
Two-by-two-by-two Heckscher-Ohlin model Integrated equilibrium Integrated equilibrium corresponds to ( p , ω , y ) such that: p = A 0 ( ω ) ω ( ZP ) : (1) � � ω 0 v ( GM ) : y = α ( p ) (2) ( FM ) v = A ( ω ) y : (3) where: � � p � ( p 1 , p 2 ) , ω � ( w , r ) , A ( ω ) � a fg ( ω ) , y � ( y 1 , y 2 ) , v � ( l , k ) , α ( p ) � [ α 1 ( p ) , α 2 ( p )] A ( ω ) derives from cost-minimization α ( p ) derives from utility-maximization 14.581 (Week 5) Factor Proportion Theory (II) Spring 2013 5 / 24
Two-by-two-by-two Heckscher-Ohlin model Free trade equilibrium Free trade equilibrium corresponds to ( p t , ω n , ω s , y n , y s ) such that: p t � A 0 ( ω c ) ω c for c = n , s ( ZP ) : (4) � p t � � ω n 0 v n + ω s 0 v s � y n + y s = α ( GM ) : (5) v c = A ( ω c ) y c for c = n , s ( FM ) : (6) where ( 4 ) holds with equality if good is produced in country c De…nition Free trade equilibrium replicates integrated equilibrium if 9 ( y n , y s ) � 0 such that ( p , ω , ω , y n , y s ) satisfy conditions ( 4 ) - ( 6 ) 14.581 (Week 5) Factor Proportion Theory (II) Spring 2013 6 / 24
Two-by-two-by-two Heckscher-Ohlin model Factor Price Equalization (FPE) Set De…nition ( v n , v s ) are in the FPE set if 9 ( y n , y s ) � 0 such that condition ( 6 ) holds for ω n = ω s = ω . Lemma If ( v n , v s ) is in the FPE set, then free trade equilibrium replicates integrated equilibrium Proof: By de…nition of the FPE set, 9 ( y n , y s ) � 0 such that v c = A ( ω ) y c So Condition ( 6 ) holds. Since v = v n + v s , this implies v = A ( ω ) ( y n + y s ) Combining this expression with condition ( 3 ) , we obtain y n + y s = y . Since ω n 0 v n + ω s 0 v s = ω 0 v , Condition ( 5 ) holds as well. Finally, Condition ( 1 ) directly implies ( 4 ) holds. 14.581 (Week 5) Factor Proportion Theory (II) Spring 2013 7 / 24
Two-by-two-by-two Heckscher-Ohlin model Integrated equilibrium: graphical analysis Factor market clearing in the integrated equilibrium: v y 2 a 2 ( ω ) k a 2 ( ω ) y 1 a 1 ( ω ) a 1 ( ω ) O l 14.581 (Week 5) Factor Proportion Theory (II) Spring 2013 8 / 24
Two-by-two-by-two Heckscher-Ohlin model The “Parallelogram” FPE set � ( v n , v s ) inside the parallelogram l s O s k s v y 2 a 2 ( ω ) v s k n v n a 2 ( ω ) y 1 a 1 ( ω ) a 1 ( ω ) O n l n When v n and v s are inside the parallelogram, we say that they belong to the same diversi…cation cone This is a very di¤erent way of approaching FPE than FPE Theorem Here, we have shown that there can be FPE i¤ factor endowments are not too dissimilar, whether or not there are no FIR Instead of taking prices as given—whether or not they are consistent with integrated equilibrium—we take factor endowments as primitives 14.581 (Week 5) Factor Proportion Theory (II) Spring 2013 9 / 24
Two-by-two-by-two Heckscher-Ohlin model Heckscher-Ohlin Theorem: graphical analysis Suppose that ( v n , v s ) is in the FPE set HO Theorem In the free trade equilibrium, each country will export the good that uses its abundant factor intensively l s O s k s v v s k n v n C Slope = w/r O n l n Outside the FPE set, additional technological and demand considerations matter (e.g. FIR or no FIR) 14.581 (Week 5) Factor Proportion Theory (II) Spring 2013 10 / 24
Two-by-two-by-two Heckscher-Ohlin model Heckscher-Ohlin Theorem: alternative proof HO Theorem can also be derived using Rybczynski e¤ect: Rybczynski theorem ) y n 2 / y n 1 > y s 2 / y s 1 for any p 1 Homotheticity ) c n 2 / c n 1 = c s 2 / c s 1 for any p 2 This implies p n 2 / p n 1 < p s 2 / p s 1 under autarky 3 Law of comparative advantage ) HO Theorem 4 14.581 (Week 5) Factor Proportion Theory (II) Spring 2013 11 / 24
Two-by-two-by-two Heckscher-Ohlin model Trade and inequality Predictions of HO and SS Theorems are often combined: HO Theorem ) p n 2 / p n 1 < p 2 / p 1 < p s 2 / p s 1 SS Theorem ) Moving from autarky to free trade, real return of abundant factor increases, whereas real return of scarce factor decreases If North is skill-abundant relative to South, inequality increases in the North and decreases in the South So why may we observe a rise in inequality in the South in practice? Southern countries are not moving from autarky to free trade Technology is not identical around the world Preferences are not homothetic and identical around the world There are more than two goods and two countries in the world 14.581 (Week 5) Factor Proportion Theory (II) Spring 2013 12 / 24
Two-by-two-by-two Heckscher-Ohlin model Trade volumes Let us de…ne trade volumes as the sum of exports plus imports Inside FPE set, iso-volume lines are parallel to diagonal (HKa p.23) the further away from the diagonal, the larger the trade volumes factor abundance rather than country size determines trade volume volumes l s O s k s y 2 a 2 ( ω ) k n a 2 ( ω ) y 1 a 1 ( ω ) a 1 ( ω ) O n l n 4 . pdf If country size a¤ects trade volumes in practice, what should we infer? 14.581 (Week 5) Factor Proportion Theory (II) Spring 2013 13 / 24
High-Dimensional Predictions FPE (I): More factors than goods Suppose now that there are F factors and G goods By de…nition, ( v n , v s ) is in the FPE set if 9 ( y n , y s ) � 0 s.t. v c = A ( ω ) y c for c = n , s If F = G (“even case”), the situation is qualitatively similar If F > G , the FPE set will be “measure zero”: f v j v = A ( ω ) y c for y c � 0 g is a G -dimensional cone in F -dimensional space Example: “Macro” model with 1 good and 2 factors l s O s k s v s k n v n a( ω ) O n l n 14.581 (Week 5) Factor Proportion Theory (II) Spring 2013 14 / 24
High-Dimensional Predictions FPE (II): More goods than factors If F < G , there will be indeterminacies in production, ( y n , y s ) , and so, trade patterns, but FPE set will still have positive measure Example: 3 goods and 2 factors l s O s y 1 a 1 ( ω ) y 2 a 2 ( ω ) k s v v s y 3 a 3 ( ω ) k n v n a 3 ( ω ) a 2 ( ω ) a 1 ( ω ) O n l n By the way, are there more goods than factors in the world? 14.581 (Week 5) Factor Proportion Theory (II) Spring 2013 15 / 24
High-Dimensional Predictions Stolper-Samuelson-type results (I): “Friends and Enemies” SS Theorem was derived by di¤erentiating zero-pro…t condition With an arbitrary number of goods and factors, we still have b p g = ∑ f θ fg b w f (7) where w f is the price of factor f and θ fg � w f a fg ( ω ) / c g ( ω ) Now suppose that b p g 0 > 0, whereas b p g = 0 for all g 6 = g 0 Equation ( 7 ) immediately implies the existence of f 1 and f 2 s.t. w f 1 b � b p g 0 > b p g = 0 for all g 6 = g 0 , b < b p g = 0 < b p g 0 for all g 6 = g 0 . w f 2 So every good is “friend” to some factor and “enemy” to some other (Jones and Scheinkman 1977) 14.581 (Week 5) Factor Proportion Theory (II) Spring 2013 16 / 24
High-Dimensional Predictions Stolper-Samuelson-type results (II): Correlations Ethier (1984) also provides the following variation of SS Theorem If good prices change from p to p 0 , then the associated change in factor prices, ω 0 � ω , must satisfy � � � � > 0, for some ω 0 between ω and ω 0 ω 0 � ω p 0 � p A ( ω 0 ) Proof: De…ne f ( ω ) = ω A ( ω ) ( p 0 � p ) . Mean value theorem implies � ω 0 � = ω A ( ω ) � � + � � [ A ( ω 0 ) + ω 0 dA ( ω 0 )] � � p 0 � p ω 0 � ω p 0 � p f for some ω 0 between ω and ω 0 . Cost-minimization at ω 0 requires ω 0 dA ( ω 0 ) = 0 14.581 (Week 5) Factor Proportion Theory (II) Spring 2013 17 / 24
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