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Factor Endowments1 IntroductionRicardian model insu¢ cient for understanding objectionsto free trade.Cannot explain the e¤ect of trade on distribution of in-come since there is only factor of production.HOS model attempts to …ll this void – introduces a sec-ond factor and usage of factors varies across sectors.Owners of a country’s relatively scarce factor lose fromtrade.New source of comparative advantage in the HOS model:di¤erences in factor endowments (rather than technolo-gies) across countries.2 Simple HOS Model2.1 AssumptionsTwo goods: cloth (C) and wheat (W ).Two countries: Home and Foreign *.Two factors in …xed supply: capital (K) and labor (L).Factors perfectly mobile between sectors but immobileacross countries.Perfect competition in all markets.Countries share a common production technology andidentical tastes.2.2 Production CostsFor any given wage (w) and rent (r), …rms choose thetechnique of production (the mix of the two factors) thatminimizes the cost of producing each unit of output.De…nition 1 A unit isoquant curve represents di¤erentcombinations of capital and labor that yield one unit ofoutput.De…nition 2 The budget line represents combinations ofcapital and labor that cost the same amount at the ex-isting factor prices.De…nition 3 The unit cost function records the mini-mum cost of producing one unit of the good for variousfactor prices.Ci(w; r) = waLi(w; r) + raKi(w; r) (1)The …rst order condition for cost minimization iswdaLi+ rdaKi= 0which implies optimal input mix occurs at tangency be-tween the budget line and isoquant as in Figure A.8daKidaLi= wrThe derivatives of the cost function (1) for each goodwith respect to factor prices give (Shephard’s Lemma)[email protected]@w= aLi+wdaLidw+ rdaKidw|{z }0= aLi(2)@[email protected]= aKi+wdaLidr+ rdaKidr| {z }0= aKi(3)2.3 Isocost CurvesDe…nition 4 An isocost curve plots combinations of wand r for which minimum cost of production is constant.Due to perfect competition in output markets, pro…ts aredriven to zero so the price of a good cannot exceed thecost of producing it in equilibrium (costs can exceed priceif the good is not produced).Thus, an isocost curve represe nts Pi= Ci(w; r) for eachindustry as in Figure A.9.The slope of the isocost curve denotes the capital to la-bor ratio employed in an industry at the prevailing fa ctorprices.dwdr= aKiaLiThe wage-to-rent ratio w=r is indicated by the slopeof a ray from the origin.For any wage-to-rent ratio (such as the ray throughpoint A), cloth is relatively capital intensive i¤ the clothisocost curve is steeper than the wheat isocost curveaKCaLC>aKWaLWsince the slope of each isocost curve re‡ects the capital-to-labor ratio employed in that sector.De…nition 5 If the identity of which good is relativelyintensive in which factor depends on wage-to-rent ratio, then there exists a factor intensity reversal (FIR) atthe point where the capital-to-labor ratios are identical.In Figure 9b, cloth is relatively capital intensive at pointA but relatively labor intensive at point A’.De…nition 6 Let ijdenote the share of revenue thatgoes to factor i in producing good j.By de…nition and then dividing the numerator and de-nominator by raLiLiwaLiwaLi+ raKi= +aKiaLi(4)The revenue shares for a good must sum to oneKi+ Li=raKiPi+waLiPi= 1 (5)and by the de…nition are each positive and strictly lessthan one 0 < ji< 1.Lemma 7 I¤ cloth is relatively capital intensive aKC=aLC>aKW=aLW, then the share of revenue that goes to laboris higher in wheat than cloth LW> LC, and similarlythe share of revenue that goes to capital is higher in cloththan wheat KC> KW.Proof. The labor result follows directly from the de…ni-tion of revenue shares (4)aKCaLC>aKWaLW! +aKWaLW> +aKCaLC! LW> LCand the capital result follows from the property that therevenue shares must sum to one (5)LW> LC! 1 LW< 1 LC! KC> KW2.4 Production EquilibriumThe point of intersection of the two isocost curves repre-sents an equilibrium in which both goods are produced.The intersection then determines the two factor pricesbased on the prices of the two goods. This relationshipwill be important for the Stolper-Samuelson result to fol-low.De…nition 8 The factor-price frontier is the outer fron-tier traced by the two isoc ost curves and denotes thepossible equilibria.At the kink (point A), both goods are produced whereasfor any other point, only one of the two goods is pro-duced since for the other good, price is below marginalcost. Factor intensity reversals allow the possibility ofmore than one kink.De…nition 9 De…ne LiLiLas the fraction of thelabor supply used in sector i and similarly KiKiKasthe fraction of the capital stock used in sector i.Summing the fraction of the labor supply used across thetwo sectors must sum to one as the total labor supplymust be fully employed by the labor constraint LC+LW= L,LC+ LW=LCL+LWL=LL= 1 (6)and similarly for capital KC+KW= 1. By de…nition,the factor shares are each positive and strictly less thanone 0 < ji< 1.Lemma 10 If cloth is relatively capital intensive aKC=aLC>aKW=aLW, then the fraction of the capital stock usedin cloth production is greater than the fraction of the la-bor supply used in cloth production KC> LCandsimilarly the fraction of the labor supply used in wheatproduction is greater than the fraction of the capital stockused in wheat production LW> KW.Proof. Cloth being relatively capital intensive impliesthat the capital-to-labor ratio in the cloth sector mustexceed that in the factor endowme nts, which yields theresult for cloth.aKCaLC>aKWaLW!aKCaLC>KL!KCLC>KL!KCK>LCL! KC> LCUsing that the factor shares must sum to one LW=1 LCand KW= 1 KCfrom (6) gives the resultfor wheatKC> LC! 1 KC< 1 LC! LW> KWLemma 11 An equilibrium where both goods are pro-duced is possible if the capital-to-labor ratios used in thetwo sectors in such an equilibrium span the capital tolabor ratio in the economy as a whole.aKCaLC>KL>aKWaLWProof. Dividing the capital constraint K = KC+ KWby L,KL=KCL+KWLand multiplying each term on the right-hand-side (RHS)byLiLi,KL=LCLCKCL+LWLWKWL!KL=LCLKCLC+LWLKWLWusing the de…nition of Li=LiLKL= LCKCLC+ LWKWLWand …nally substitutingKCLC=aKiaLiKL= LCaKCaLC+ LWaKWaLWhave that the capital-to-labor ratio in the economy asa whole (the ec

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