Notes on the Heckscher-Ohlin Theorem

Transcription

1 Notes on the Hecscher-Ohlin Theorem John S Chipman University of Minnesota Fall 00 We assume that there are two countries each producing two commodities with two factors We denote, for i, j, =, : x j = country s consumption of commodity j; y j = country s production of commodity j; l i p j = country s endowment of factor i; = the price of commodity j on country s marets; w i = the rental of factor i in country We assume that l /l l /l (ie, the countries differ in their relative factor endowments), and choose the suffixes i, such that () l /l >l /l We assume: Material balance This states that the world consumption of each commodity is equal to the world production of this commodity, or equivalently, that one country s export is the other country s import: () x j + x j = y j + y j, or x j y j = y j x j Free trade There are no tariffs, transport costs, or other impediments to trade Thus, p j = p j = p j for j =,, ie, the prices of the two commodities are equal in the two countries

2 3 Positive production of both commodities in each country It follows from this that with competitive marets, prices of the two commodities are equal to their minimum unit costs 4 Identical technologies as between countries, characterized by identical concave, strictly quasi-concave, differentiable, and homogeneous-of-degree- production functions (3) y j = f j (v j,v j ) (j, =, ) and thus identical concave, strictly quasi-concave, differentiable, and homogeneous-of-degree- minimum-unit-cost functions (4) p j = g j (w,w ) (j, =, ), where the prices are equal to the commodities minimum units costs, from assumption 3 5 Nonreversal of factor intensities Denoting by the homogeneous-ofdegree-zero functions (5) b ij (w,w )= g j(w,w )/ w i (i, j, =, ) the amount of factor i used to produce one unit of commodity j in country, as a function of the factor rentals, then for a suitable labelling of the commodities we have B(w (6) ) b (w ) b (w ) b (w ) b (w ) = b (w )b (w ) [ b (w ) b (w ) b (w ] ) > 0 b (w ) for all w =(w,w ) Since the matrix B(w ) of (6) is (by (5)) the Jacobian of the transformation g,w ) = p (7) g (w,w ) = p, it follows that the solution of (7) is unique, ie, wi = w i = w i (factor rentals are equalized between the countries) The Rybczynsi functions yj = Π(p,p,l,l ) (8) =ŷ j (p,p,l p,l ) (j, =, ) j This assumptions is stronger than needed to establish the Hecscher-Ohlin theorem If one or both countries specialize in the production of one commodity, the result can be proved by a separate argument This assumption is also stronger than needed to establish the Hecscher-Ohlin theorem It is enough to assume that both countries have their factor endowments in the same diversification cone This automatically assures assumption 3 as well

3 (where Π is the domestic-product function the same for both countries) are then the solutions of the resource-allocation equations (9) b (ŵ(p))y + b (ŵ(p))y = l b (ŵ(p))y + b (ŵ(p))y = l, ie, the linear functions (0) y = ŷ (p,p,l )=b (p)l + b = ŷ (p,p,l,l )=b (p)l + b (p)l, where the b ij (p) are the elements of the inverse matrix [B(ŵ(p)] and ŵ(p) denoted the inverse of the cost mapping g(w) = p Thus the countries Rybczynsi functions are linear and single-valued, having the same form in the two countries, their values differing only according to the countries different factor endowments 6 Identical homothetic preferences These are characterized by demand functions with the property () x j = h j(p,p,y )=Y h j (p,p, ) (where Y is country s national income) so that () x j + x j =(Y + Y )h j (p,p, ) = h j (p,p,y + Y ) 7 Balanced trade For country this may be stated as either (3) p x + p x = p y + p y (the value of expenditure equals the value of output, ie, there is no borrowing or lending by country ), or (4) p (x y )+p (x y )=0 (the value of country s imports (resp exports) is equal to the value of its exports (resp imports) Note that if (4) holds for =, then by assumption (material balance) it automatically holds for = Hecscher-Ohlin Theorem Under the above assumptions, in competitive world equilibrium country will export commodity to and import commodity from country Proof The logic of the proof goes as follows First we will show that from assumptions and 6 we must have (5) x x = x x 3

4 Next we will show that from () and assumptions 3, 4, and 5, we must have (6) y > Finally we shall show that from (), (5), (6), and assumptions and 7, we must have (7) x <y, hence x < To establish (5) we need simply observe from () that, owing to assumption, x x = h (p,p, ), independently of h (p,p, ) To establish (6) we note first that since the domestic-product function Π(p,p,l,l ) is homogeneous of degree in (l,l ), so are the Rybczynsi functions ŷ j = Π/ p j, hence (again using assumption ) y = ŷ(p,p,l /l, ) ŷ (p,p,l /l, ) Thus, differentiating with respect to the third argument l /l we obtain (ŷ /ŷ ) (l/l ) = ŷ ŷ / l ŷ ŷ / l (8) (ŷ ) We now prove that this expression is positive, as follows As observed in assumption 5, formula (9), the Rybczynsi functions are the solution of the linear system [ ][ ] [ ] b b y l (9) = b b y l, where the b ij (ŵ(p,p,l,l )) are (within the diversification cone) independent of (l,l ) (Here, ŵ i denotes Π/ l i, which coincides with g (p) within the diversification cone) Denoting this solution as above by [ ] [ y b b = ][ ] l (0) y b b l, where [ b b ] [ ] b b = B b b (), b b and B > 0 from (6), we have from (0) and () () ŷ l = b B > 0 and ŷ l 4 = b B < 0,

5 so that (8) yields (ŷ /ŷ ) (l/l ) > 0 Thus, given (), (6) holds Finally, by way of contradiction suppose that (7) does not hold, ie, suppose that (3) x y Then from (4) we have also (4) From (3) and (4) it follows that x y (5) x x y x y Now from the material-balance condition () it follows that the inequalities (3) and (4) are respectively equivalent to the inequalities (6) x y and x y From (6) it then follows that (7) y x y x x Putting together (5), (6), and (7) we obtain (8) x x y y > y x x But this contradicts assumption (5); thus the supposition (3) has led to a contradiction, and therefore (7) holds 5