The Heckscher-Ohlin model: Mathematical treatment*

Transcription

1 The Heckscher-Ohlin model: Mathematical treatment* Robert Stehrer Draft and unfinished version Version: April 23, 2013

2 Various approaches Primal approach approach which are partly interrelated

3 Primal approach Production function (i=1,2) Y i = f i (K i, L i ) f i > 0, f i < 0, f (0) =, f ( ) = 0 Rewriting (assuming homogeneity of degree one) Factors marginal productivities Y i = L i f i (K i /L i, 1) = L i f i (k i ) ( L i f i (k i ) ) = L i f i K (k i ) 1 = f i i L (k i ) i ( L i f i (k i ) ) = f i (k i ) + L i f i L (k i ) K i i L 2 i are expressed as functions of the capital labor ratio k i. = f i (k i ) k i f i (k i )

4 Optimal factor utilization requires that price of each factor equals value of its marginal productivity Wage-rental ratio w = p i [f i (k i ) k i f i i )] r = p i f i i ) ω = w r = f i (k i ) f i (k i ) k i The capital-labor ratio is increasing in the wage-rental ratio dω(k i ) = [f i (k i )] 2 f i (k i )f i (k i ) dk i [f i (k i )] 2 1 = f i (k i )fi (k i ) [f i (k i )] 2 > 0

5 Factor intensity curves ω = f i (k i ) f i (k i ) k i are positively sloped as differentiation with respect to ω (and noting that k i (ω)) shows dk i dω 1 = (f i (k i )) 2 dk i dω f i (k i )f i (f i (k i )) 2 = dk i [1 f i (k i )f i (k i ) dω (f i (k i )) 2 1 (f i = (k i )) 2 f i (k i )f i (k i ) > 0 (k i ) dk i dω ] dk i dω Assumption on factor-intensity condition: k 2 (ω) > k 1 (ω) (for all relevant ω s).

6 Factor prices and product prices dp dω = f 2 (k 2) dk 2 dω f 1 (k 1) f 1 (k 1)f 2 (k 2) dk 1 dω [f 1 (k 1)] 2 = = f 2 (k 2) [f 2 (k 2)] 2 f 2 (k 2 )f f 2 (k 1 (k 1) f 1 (k 1)f 2 (k 2) [f 1 (k 1)] 2 2) f 1 (k 1 )f 1 (k 1) [f 1 (k 1)] 2 [f 2 (k 2)] 2 f f 2 (k 2 ) 1 (k 1) f 2 (k 2) [f 1 (k 1)] 2 f 1 (k 1 ) [f 1 (k 1)] 2 [f 2 (k 2)] 2 f f = 2 (k 2 ) 1 (k 1) f 2 [f 1 (k 1)] 2 (k 2) [f 1 (k 1)] 2 f 1 (k 1 ) [f 1 (k 1)] 2 = f 2 (k 2) f 2 (k 2) f 2 (k 2 ) f 1 (k 1) + f 2 (k 2 ) f 1 (k 1) f 1 (k 1 ) f 1 (k 1) p = p > 0 if k 1 (ω) < k 2 (ω) ω + k 1 ω + k 2

7 Autarkic equilibrium (Example) (i = 1, 2) y i = f i (k i )l i with y i = Y i /L, l i = L i /L ω = f i (k i ) f i (k i ) k i p = p 2 = f 1 (k 1) p 1 f 2 (k as p 1 f 1 2) (k 1) = p 2 f 2 (k 2) = r Y = p 1 Y 1 + p 2 Y 2 = Y 1 + py 2 y = y 1 + py 2 with p 1 = 1 (numeraire) y 2 = βy/p which assumes Cobb-Douglas preferences k = k 1 l 1 + k 2 l 2 as k = K L = 1 L (K 1 + K 2 ) = K 1 L 1 L 1 L + K 2 L 2 L 2 L 1 = l 1 + l 2 as L = L 1 + L 2 i.e. 9 independent equations to solve for 9 unknowns y, y i, k i, l i, ω, p

8 Existence of globally stable static equilibrium Define excess demand function E = βy p y 2 Inserting for y, p, y 2 yields equation in ω (for given k = K/L) [ ] (ω + E(ω) = βf 2 (k 2)[ω + k] f 2 (k k2 )(k 1 k) 2) k 1 k 2 At ω min only good 1 is produced, i.e. l 1 = 1, l 2 = 0 and k 1 = k and E(ω min ) = βf 2 (ωmin + k) > 0 At ω max only good 2 is produced, i.e. l 1 = 0, l 2 = 1 and k 2 = k and E(ω) is continuous and Therefore ω exists where E(ω ) = 0 E(ω max ) = (1 β)f 2 (ωmax + k) < 0 de(ω) dω < 0

9 Existence: Brouwers fix-point theorem Set of ω s, Ω, is non-empty, convex and compact E(ω) is a continuous mapping onto itself Satisfies conditions of Browers fix-point theorem, therefore at least one ω exists for which E(ω ) = 0 Local stability:... Global stability: Lyapunov function Walrasian tatonnement process: ω = be(ω) Walrasian auctioneer decreases (increases) ω if E(ω) > 0 (E(ω) < 0) Distance between arbitrary ω and ω is measured by Lyapunov function M = (ω ω ) 2 Differentiation with respect to time Ṁ = 2(ω ω ) ω = 2b(ω ω )E(ω) If ω > ω then E(ω) < 0, therefore Ṁ < 0 If ω < ω then E(ω) > 0, therefore Ṁ < 0

10 International equilibrium (Example); i=1,2; c=a,b yi c = f i (ki c )lc i + zi c yi c = Yi c /L c, li c ω c = f i (ki c) f i (kc i ) kc i p = p 2 = f 1 (kc 1 ) p 1 f 2 (kc 2 ) y c = y1 c + py 2 c y2 c = βy c /p k c = k c 1 lc 1 + kc 2 lc 2 1 = l c 1 + lc 2 v c = L c /(L A + L B ) z c 1 + pzc 2 = 0 BoP-equation = L c i /Lc v A z i + v B z B i = 0 Quantity of exports equals quantity of imports 24 equations and 23 unknowns; but last four equations are not independent; therefore 23 equations in 23 unknowns; proof of existence and stability of equilibrium similar to above (but using z1 c as a function of ω)

11 Proofs existence, uniqueness and (global) stability However, economic analysis requires More specific assumptions on production and consumption (e.g. elasticities) Simulation studies (e.g. with GAMS)

12 The dual approach Unit cost function c i (w, r) = min {wa Li + ra Ki f i (L i, K i ) 1} L i,k i 0 with properties c i w = a L,i (w, r), c i r = a K,i (w, r), a Li w Perfect competition Full employment of factors 0, a Ki r p i = c i = wa Li + ra Ki 0, a Li r 0, a Ki w 0 L = a L1 Y 1 + a L2 Y 2 K = a K1 Y 1 + a K2 Y 2 The overall capital-labor ratio is a weighted sum of sector-specific capital labor ratios: K = K 1 + K 2 K L = K 1 L 1 L 1 L + K 2 L 2 L 2 L k = k 1l 1 + k 2 l 2

13 (Total) differentiation of price equation and rearranging yields dp i = p i w dw + p i r dr + p i a Li da Li + p i a Ki da Ki dp i 1 p i = p i w dw w w 1 p i + p i r dr r r 1 p i + p i a Li da Li a Li a Li 1 p i +... w r 1 1 ˆp i = ŵa Li + ˆra Ii + aˆ Li wa Li + aˆ Ki ra Ki p i p i p i p i ˆp i = ŵθ Li + ˆrθ Ki + [θ Li aˆ Li + θ Ki aˆ Ki ] }{{} =0 where dx i /x i = ˆx i and a Li w/p i = a Li w/c i = θ Li, etc. are cost shares. Further, for fixed factor costs, the envelope theorem states dc i = wda Li + rda Ki = 0 dc i = wa Li + ra Ki c i c i c i da Li a Li dc i = θ Li aˆ Li + θ Ki aˆ Ki = 0 c i da Ki a Ki = 0

14 Differentiation of full employment equations yields with λ Li = L i /L and λ Ki = K i /K. Note that λ L1 Ŷ 1 + λ L2 Ŷ 2 = ˆL [λ L1 aˆ L1 + λ L2 aˆ L2 ] λ K1 Ŷ 1 + λ K2 Ŷ 2 = ˆK [λ K1 a K1 ˆ + λ K2 a K2 ˆ ] θ Li + θ Ki = 1 for i = 1, 2 λ j1 + λ j2 = 1 for j = K, L

15 Define elasticity of substitution of sector i as σ i = Inserting into θ Li aˆ Li + θ Ki aˆ Ki = 0 yields aˆ Ki aˆ Li ŵ ˆr θ Li aˆ Li = θ Ki aˆ Ki θ Li aˆ Li = θ Ki [σ i (ŵ ˆr) + aˆ Li ] θ Li aˆ Li + θ Ki aˆ Li = θ Ki σ i (ŵ ˆr) aˆ Li (θ Li + θ Ki ) }{{} = θ Ki σ i (ŵ ˆr) =1 aˆ Li = θ Ki σ i (ŵ ˆr) and analogously aˆ Ki = θ Li σ i (ŵ ˆr)

16 Inserting λ L1 Ŷ 1 + λ L2 Ŷ 2 = ˆL ( [λ L1 θk1 σ 1 (ŵ ˆr) ) ( + λ L2 θk2 σ 2 (ŵ ˆr) ) ] λ K1 Ŷ 1 + λ K2 Ŷ 2 = ˆK ( [λ K1 θl1 σ 1 (ŵ ˆr) ) ( + λ K2 θli σ i (ŵ ˆr) ) ] and rearranging λ L1 Ŷ 1 + λ L2 Ŷ 2 = ˆL ( ) ( ) (ŵ ˆr)[λ L1 θk1 σ 1 + λl2 θk2 σ 2 ] λ K1 Ŷ 1 + λ K2 Ŷ 2 = ˆK ( ) ( ) (ŵ ˆr)[λ K1 θl1 σ 1 + λk2 θl2 σ 2 ] gives λ L1 Ŷ 1 + λ L2 Ŷ 2 = ˆL + δ L (ŵ ˆr) λ K1 Ŷ 1 + λ K2 Ŷ 2 = ˆK δ K (ŵ ˆr)

17 Summarizing these exercises leads to the equations of change: ˆp 1 = ŵθ L1 + ˆrθ K1 ˆp 2 = ŵθ L2 + ˆrθ K2 λ L1 Ŷ 1 + λ L2 Ŷ 2 = ˆL + δ L (ŵ ˆr) λ K1 Ŷ 1 + λ K2 Ŷ 2 = ˆK δ K (ŵ ˆr)

18 Define ( ) θl1 θ θ = K1 θ L2 θ K2 ( ) λl1 λ and λ = L2 λ K2 λ K2 Determinants of these matrices are θ = θ L1 θ L2 = θ K2 θ K1 λ = λ L1 λ K1 = λ K2 λ L2 as e.g. θ = θ L1 θ K2 θ L2 θ K1 = θ L1 (1 θ L2 ) θ L2 (1 θ L1 ) = θ L1 θ L2 Using definitions and zero-profit conditions these can be rewritten as θ = wl 1 wl 1 + rk 1 wl 2 wl 2 + rk 2 = λ = L 1 L K 1 K = L 1L 2 (k 2 k 1 ) LK ω ω + k 1 ω ω + k 2 = i.e. θ > 0 and λ > 0 if and only if sector 1 is labor intensive. ω(k 2 k 1 ) (ω + k 1 )(ω + k 2 )

19 With this framework one can establish the important theorems: Stolper-Samuelson theorem Rybczynski-theorem Factor price equalisation theorem

20 Stolper-Samuelson theorem Solving yields ( ) (ŵ ) θl1 θ K1 θ L2 θ K2 ˆr (ŵ ) ˆr = = = ) (ˆp1 ˆp 2 ( ) θl1 θ (ˆp 1 ) K1 1 θ L2 θ K2 ˆp 2 ( ) ) 1 θk2 θ K1 (ˆp1 θ θ L2 θ L1 ˆp 2 ŵ = θ K2ˆp 1 θ K1ˆp 2 θ ˆr = θ L1ˆp 1 θ L2ˆp 2 θ = θ K2ˆp 1 θ K1ˆp 2 θ K2 θ K1 = θ L1ˆp 1 θ L2ˆp 2 θ L1 θ L2

21 Assume ˆp 1 > ˆp 2 and ˆp 1 = ˆp 2 + ɛ, then As θ > 0 by assumption we get (Magnification effect; Jones 1965) ŵ = ˆp 1 + θ K1 θ ɛ ˆr = ˆp 2 θ L2 θ ɛ ŵ > ˆp 1 > ˆp 2 > ˆr Similarly, for ˆp 2 > ˆp 1 we get ˆr > ˆp 1 > ˆp 2 > ŵ

22 Special case that only one commodity price is increasing: Stolper-Samuelson theorem: A rise in the price of a commodity will increase the real reward of the factor used intensively in the sector and decrease the real reward of the other factor.

23 Rybczynski theorem Solving (where we assume constant commodity prices, i.e. ˆp i = 0) ( ) ) λl1 λ L2 (Ŷ1 λ K1 λ K2 Ŷ 2 ) (Ŷ1 Ŷ 2 = = = ( ) ˆL ˆK ( ) λl1 λ ( 1 ) L2 ˆL λ K1 λ K2 ˆK ( ) ( ) 1 λk2 λ L2 ˆL λ λ K1 λ L1 ˆK yields Ŷ 1 = λ K2ˆL λ L2 ˆK λ Ŷ 2 = λ L1 ˆK λ K1ˆL λ

24 Similar to above one can show that Ŷ 1 > ˆL > ˆK > Ŷ2 Ŷ 2 > ˆK > ˆL > Ŷ 1 if ˆL > ˆK if ˆL < ˆK (Magnification effect, Jones 1964) The special case that only one factor endowment is increasing: Rybczynski theorem: An increase in a factor endowment will increase the output of the industry using it intensively, and decrease the output of the other industry.

25 Cone of diversification K Cone of diversification (L,K) K Rybczynski theorem (L,K) (L',K) Y 2(a L2,a K2) Y 2(a L2,a K2) (a L2,a K2) Y' 2(a L2,a K2) (a L2,a K2) Y' 1(a L1,a K1) Y 1(a L1,a K1) Y 1(a L1,a K1) (a L1,a K1) (a L1,a K1) L L

26 Factor price equalization theorem (Samuelson, 1949) 4 equations with 2 zero-profit conditions and full employment conditions p i = wa Li + ra Ki a L1 Y 1 + a L2 Y 2 = L a K1 Y 1 + a K2 Y 2 = K One can solve for factor prices from zero-profit conditions and then substitute into full-employment conditions if the following is satisfied:

27 Factor price insensitivity: So long as both goods are produced, and factor intensity reversals do not occur, then each price vector (p 1, p 2 ) corresponds to unique factor prices (w, r). Factor endowments do not matter for determination of (w, r) (if commodity prices are fixed) Growth of capital stock or labor would not affect factor prices. Two conditions must hold: Both goods are produced Factor intensity reversals do not occur

28 If this holds then the following holds Suppose that two countries are engaged in free trade, having identical technologies but different factor endowments. If both countries produce both goods and factor intensity reversals do not occur, then the factor prices (w, r) are equalized across countries. i.e. trade in goods is a perfect substitute for trade in factors.

29 Jones (1965) started with a description of technology (relating to activity analysis ) using ( ) al1 a A = L2 a K1 a K2 where coefficients a ij are technologically given (either fixed or dependent on input prices). From this also the equations of change can be derived.

30 Discussion Factor price equalization revisited Factors move until factor prices are equalized: Integrated world equilibrium World supply of K O* O L 2 L 1 World supply of labor L

31 Factor price equalization set World supply of K O* B' O L 2 L 1 World supply of labor L For any allocation of labor and capital within parallelogram (FPE set) both countries remain diversified and the same equilibrium prices as in integrated equilibrium occur. Factor prices remain equalized across countries for allocations of labor and capital within parallelogram. For endowments outside FPE (B ) at least one country would have to fully specialize; FPE would not longer hold.

32 Factor intensity reversals Multiple solutions to zero-profit conditions In combination with full employment assumption one can show Labor abundant country has low wage and high rental rate Capital abundant country has high wage and low rental rate i.e. factor prices depend on factor endowments

33 K Cone of diversification (L,K) K Factor intensity reversals Capital abundant country (a L2,a K2) Y 2(a L2,a K2) Y* 1(a* L1,a* K1) Y* 2(a* L2,a* K2) Labor abundant country Y 2(a L2,a K2) Y 1(a L1,a K1) (a L1,a K1) Y 1(a L1,a K1) L L Empirical evidence: Leamer (1987); Debaere and Demiroglu (2003); Harrigan and Zakrajsek (2000); Xu (2000); Schott (2003).

34 approach (or revenue function) Analogous to restricted profit function in microeconomics Can easily be extended to higher dimensional frameworks Aggregate supplies of goods and market factor prices can be obtained by differentiating the (rather than solving system of equations) We assume fixed commodity prices.

35 Maximize GDP G(p 1, p 2, L, K) = max Y 1,Y 2 p 1 Y 1 + p 2 Y 2 s.t. Y 2 = h(y 1, L, K) Inserting constraint into objective function and maximizing yields G = p 1 Y 1 + p 2 h(y 1, L, K) dg h = p 1 + p 2 = 0 dy 1 Y 1 p 1 = h = Y 2 p 2 Y 1 Y 1

36 Properties of Derivative of the with respect to prices equals outputs of the economy: ( ) G Y 1 Y 2 = Y i + p 1 + p 2 p i p i p i }{{} =0 Derivative of the with respect to endowments: G L = w and G K = r Discussed in more detail for empirical applications.

37 Appendix: Envelope theorem When differentiating a function that has been maximized with respect to an exogenous variable, then one can ignore the changes in endogenous variables in this derivative. Let f (x, a) where a is exogenous; x is chosen to maximize the function. Under sufficiently regular conditions we can write x(a), i.e. the optimal choice of x given a. Define the (optimal) value function as M(a) = f ( x(a), a ). How does the optimal value M changes when a changes? or dm(a) da = f ( x(a), a ) x x=x }{{} =0 = f ( x(a), a ) a x(a) a + f ( x(a), a ) a

38 dm(a) da = f ( x(a), a ). a x=x(a) The total derivative of the value function with respect to a equals the partial derivative when the derivative is evaluated at the optimal choice. A change in a has two effects: (1) Direct effect on f (2) An (indirect) effect on x which in turn affects f. But if x is chosen optimally, an (infinitesimal) small change of x has a zero effect on f, thus the indirect effect drops out.

39 Example 1: Profit function Primal approach π(p, w) = p f ( L(p, w) ) wl(p, w) Assume that p is given. Differentiating with respect to w gives π = p f ( L(p, w) ) L w L w w L L(p, w) w [ = p f ( L(p, w) ) ] w L }{{} =0 as f (L(p,w)) L = w L L(p, w) p w = L(p, w). (1) Increasing the price of an input w, profits must fall; thus the negative sign. (2) Direct effect: A change in the input price w, reduces profit (at the same amount of input). (3) Indirect effect: A (infinitesimal) increase in input price induces the firm to reduce input; but as we are already at the optimal level this does not change profits. Thus only the direct effect remains.

40 Example 2: Primal approach ( ) G y 1 y 2 = y i + p 1 + p 2 p i p i p i }{{} =0 Totally differentiating y 2 = h(y 1, L, K) with respect to y 1 yields dy 2 = h y 1 dy 1 = p 1 p 2 dy 1 p 2 dy 2 = p 1 dy 1 which holds for any (infinitesimal) small movement in y 1 and y 2 ; in particular, it holds for small changes in y i induced by price changes, or p 1 dy 1 + p 2 dy 2 = 0 rewritten as p 1 y 1 p i + p 2 y 2 p i = 0 (1) Increase in price of an output p i, increases GDP. (2) Direct effect: A change in the output price p i, increases GDP (at the same amount of output). (3) Indirect effect: A (infinitesimal) increase in output price induces the economy to increase output; but as we are already at the optimal level this would not change GDP. Thus only the direct effect remains.