A practical Heckscher-Ohlin model

Transcription

1 QEH Workng Paper Seres QEHWPS70 Workng Paper Number 70 [THIS PAPER HAS BEEN SUPERSEDED BY WORKING PAPER 85 IN THE SAME SERIES, WITH THE EXCEPTION OF PARTS OF SECTIONS 3 AND 4] A practcal Heckscher-Ohln model Adran Wood * adran.wood@qeh.ox.ac.uk Ths paper offers a formalsaton of the nsghts of Heckscher and Ohln that s more consstent wth the evdence than standard models, and smple enough to be used for teachng and polcy analyss, as well as for research. It descrbes both the effects of a country s factor endowments on the commodty composton of ts trade and the effects of trade-related changes n goods prces on factor prces. The model apples to any numbers of goods or factors. Trade costs, ncludng polcy barrers, play a central role n the model, especally by reducng elastctes of demand. May 2009 *Department of Internatonal Development, Queen Elabeth House, Oxford Professor of Internatonal Development, Unversty of Oxford. Wthout help, advce, nstructon and crtcsm over a long perod from many people, I could not have wrtten ths paper. Two specal debts of grattude are to Enrque Alda-Carroll, from whose doctoral thess I learned a lot, and to Alasdar Smth, who provded the hgher-dmensonal formalsaton of Jones n secton 3. I receved valuable comments from Peter Neary and other partcpants n a semnar n Oxford.

2 2. Introducton The compellng ntuton of Heckscher and Ohln that what countres trade depends on ther factor endowments and that the earnngs of factors are affected by trade combned wth the elegance of Samuelson s formalsaton of ths ntuton has gven Heckscher-Ohln theory a promnent place n nternatonal economcs. Emprcally, however, the theory has had a hard tme: from the 950s to the 980s a seres of studes found lttle support for even ts most basc proposton, whch s that countres export goods that use ther abundant factors ntensvely and mport goods that use ther scarce factors ntensvely (Baldwn, 2008, provdes an excellent survey). The low pont was Bowen et al. (987), whch decsvely reected Vanek s (968) factor content reformulaton of Samuelson n hgher dmensons. In the 990s, thngs mproved for Heckscher-Ohln (henceforth H-O). A lvely debate on the causes of rsng wage nequalty n developed countres drew a lot of attenton to H-O theory (e.g. Wood, 994; Leamer, 998). Further emprcal work also yelded a clearer and less dscouragng understandng of ts falure of the Bowen et al. test. In partcular, Trefler (995) and Davs and Wensten (200) showed that the H-O factor content proposton s broadly correct f the assumptons of the Samuelson (HOS) and Vanek (HOV) models are modfed. The key modfcatons are to allow for economywde dfferences n techncal effcency among countres, varaton of factor prces among countres related to ther endowments, and home bas n consumpton. Ths emprcal success, however, has left economcs wthout a practcal H-O model (Deardorff, 2006; Baldwn, 2008). These modfcatons of the assumptons serously undermne the HOS and HOV models. Varaton of factor prces can be allowed for by havng multple cones of dversfcaton, but at a cost n terms of complexty, both theoretcally (Helpman, 984) and emprcally (Schott, 2003). Nor has HOS ever been partcularly useful n hgher dmensons: wth more goods than factors, t mples that the composton of trade s ether ndetermnate or unrealstcally specalsed, and wth more than two factors, smple general relatonshps between endowments and the composton of trade dsappear. HOV solves these hgher-dmensonal problems, but at the cost of abandonng the most practcally relevant contrbuton of H-O, whch s to explan what determnes the pattern of trade n actual goods. The present paper ams to fll ths gap wth an alternatve formalsaton of Heckscher and Ohln that s more consstent wth emprcal evdence, ncludng the fndngs of Trefler and of Davs and Wensten, and that works more consstently and usefully n hgher dmensons. Its key nnovaton s to brng trade costs nto the heart of the H-O model, a suggeston that recurs from Samuelson (953: 6) to Deardorff (2006). Ths paper s thus n the sprt of many other recent studes of the effects of trade costs on the drectons of trade (Anderson and van Wncoop, 2004), economc geography (Henderson et al., 200), the exports of frms (Melt, 2003; Bernard et al., 2007), and the fragmentaton of producton (Markusen and Venables, 2007). The man reason why trade costs matter for H-O theory s that the relatve trade costs of dfferent goods are largely ndependent of ther relatve producton costs. A good that s cheaper to produce n one country than another, for example, s not necessarly cheaper to transport. As a result, the relatve prces that purchasers pay for goods, whch are the sum of producton costs and trade costs, vary across countres and

3 3 over tme by proportonally less than the relatve prces that producers receve for them (whch n equlbrum equal relatve costs of producton). The elastcty of demand wth respect to producer prces s thus lower than the elastcty wth respect to purchaser prces, to a degree that depends on the se of trade costs. Most other models assume that relatve (varable) trade costs vary n strct proporton to relatve producton costs. However, the dea of ndependent trade costs reducng demand elastctes s not new. It has been studed n ndustral organsaton theory, most famously by Alchan and Allen (964), whose conecture about shppng the good apples out was confrmed emprcally by Hummels and Skba (2004). 2 What s new s to generalse ths dea to all aspects of the composton of trade across goods, not only across dfferent qualtes of goods and to grasp ts mplcaton for any trade theory based on comparatve advantage, whch s that ndependent trade costs make producer-prce elastctes of demand n open economes much lower than the nfnte elastctes assumed n the HOS and HOV (and most other old trade) models. Less-than-nfnte purchaser prce elastctes n trade have been acknowledged snce Armngton (969): even for commodtes such as ol and gran, there are dfferences among suppler countres n the physcal attrbutes of goods and condtons of supply. 3 Purchaser prce elastctes are hgh, typcally between 5 and 0 (Anderson and van Wncoop, 2004: 75-6), whch may be close enough to nfnty for some purposes. However, trade costs are large (estmated at 70% of producton costs on average n developed countres by Anderson and van Wncoop, 2004), and a large share of trade costs s ndependent of producton costs: producer-prce demand elastctes are thus on average probably less than half the se of purchaser-prce elastctes between 2 and 4, whch s a long way from nfnte (Wood, 2008). The techncal mplcatons of fnte elastctes of demand n open economes are well understood from the many models that have used the Armngton assumpton almost all CGE models and some theoretcal H-O models (Robnson and Therfelder, 996; Venables, 2003). They make an open economy behave qualtatvely lke a closed one, n whch factor prces vary wth endowments. The present model s thus able to draw on much earler work, and especally on that of Jones (965). It s related to Alda- Carroll (2003), whch adds trade costs to the Jones model n a bascally smlar though formally dfferent way, and to Romals (2004), n whch factor prces also vary wth endowments n a H-O model because of trade costs. Secton 2 presents the model wth two goods and two factors, and explans the mnor modfcatons that t would requre to standard tools for teachng 2x2 H-O. Secton 3 extends the model to any numbers of goods and factors, n a general and a smplfed form. Secton 4 dstngushes between exports, mports and home sales (ncludng of non-traded goods), whch nvolves more analyss of trade costs. Secton 5 concludes wth a dscusson of ssues that arse n applyng the model emprcally. 2 The conecture was that, snce the cost of transport s the same for all qualtes of a good, whle the cost of producton rses wth qualty, the relatve prce of the better-qualty varetes wll be lower at the pont of sale than at the pont of producton. Hummels and Skba tested ths conecture, usng data on trade n varyng qualtes of the same goods. 3 Another reason s lmted nformaton, as emphassed by Rauch and Trndade (2003): purchasers are less sure that goods from foregn supplers wll be what they want.

4 4 2. Two goods and two factors The low-dmensonal verson of the model n ths secton uses the framework of Jones (965). Its algebra s smlar to that of the closed verson of the Jones model, whch ncludes a goods-demand equaton. Its economc nterpretaton, however, s dfferent, because t refers to an open or tradng economy, n whch prce nelastcty of demand for goods s caused by, and vares wth, trade costs. 2. Producer-prce demand elastcty Consder a sngle country producng ts own varetes of two goods, and, for sale n a sngle world market of whch t has only small shares for both goods. Its relatve sales and outputs of the two goods vary wth the relatve prces t charges to purchasers accordng to a demand functon q q p p (2.) where hats denote proportonal changes, q s sales of good, p s the purchaser prce of good, and β s an average across goods and of the elastctes of substtuton of purchasers among natonal varetes. 4 Because the relatve costs of tradng goods are partly ndependent of the relatve costs of producng them, rather than proportonal to relatve producton costs, as explaned more fully n Wood (2008), relatve purchaser prces vary less than relatve producer prces (those at the factory gate or farm gate). More precsely, p c p c ~ (2.2) where c s the producer prce of good and ~ s an average of ~ ~ t c t (2.3) ~ and the correspondng expresson for good, n whch t s that part of the (varable) cost of tradng good that s ndependent of ts producer prce, and t s the part that s proportonal to ts producer prce. It s assumed that all varable trade costs can be fully decomposed nto these two parts and that there are no fxed trade costs. Equaton (2.2) shows how much ndependent trade costs damp the effects of changes n producer prces on changes n purchaser prces. To understand ths equaton, note ~ s the share of c t n p (snce p c t ~ t ), averaged across goods that and. Wth no change n relatve ~ t s, ths share governs the se of the effect of a change n relatve c s, and thus relatve c t s, on relatve p s, whch s smaller, the 4 As explaned n Wood (2008), ths average s a weghted harmonc mean, n whch the weghts are nversely proportonal to the relatve quanttes of the two goods purchased.

5 5 larger are ndependent trade costs. For example, f ~ were unty, the share would be half, so a 0% change n relatve c s would cause only a 5% change n relatve p s. Substtutng (2.2) nto (2.) yelds q q ~ c c (2.4) whch shows how the country s relatve sales of goods and vary wth ts relatve producer prces. The rato on ts rght-hand sde, n other words, s the producer-prce demand elastcty, whch plays a crucal role n the present model. The se of ths elastcty depends partly on that of the purchaser-prce elastcty, β, and hence vares wth the degree of substtutablty among natonal varetes of the goods concerned, as n many models followng Armngton (969). What s new about the present model s that the se of ths demand elastcty also depends on, and vares wth, trade costs: for goods wth hgher trade costs and countres wth hgher trade barrers, producer-prce demand elastctes are lower than for goods and countres wth lower trade costs. 2.2 Effects of changes n endowments Assumng that nternal competton causes producer prces to be equal to producton costs, expanson of equaton (2.4) shows how changes n relatve factor prces affect the country s relatve sales and outputs of the two goods q q w w ~ (2.5) where the two factors are and, w s the prce of, and θ s the share of factor n c, the producton cost or producer prce of good. The drecton of the effect of a change n factor prces depends on the sgn of (θ θ ): f good s the more factor - ntensve one, a rse n the relatve prce of factor reduces relatve sales of good, to an extent governed by the ses of (θ θ ) and the producer-prce demand elastcty. Relatve factor prces depend nversely on relatve factor endowments, n a way to be explaned shortly but whch can be summarsed by w v v w (2.6) where φ s sgned lke a substtuton elastcty. Substtutng (2.6) nto (2.5) yelds q q v v ~ (2.7) whch shows how a country s relatve sales n the world market (and hence ts outputs of) goods and vary wth ts relatve factor endowments, n the sprt of Heckscher and Ohln. Ths equaton, whch s smlar n form to equaton ( ) n Jones (965), can be nterpreted causally by gong from rght to left: relatve endowments nfluence

6 6 relatve factor prces va φ, whch nfluence relatve producer prces va (θ θ ), ~. The effect of relatve endowments whch nfluence relatve outputs va on relatve outputs s smply the product of these three elastctes: ts se depends on ther ses, whle ts drecton depends on the sgn of (θ θ ): a larger endowment of factor ncreases the relatve output of the -ntensve good. To explan the effect of relatve endowments on relatve factor prces, φ, requres a factor market clearng condton v ( ) ( )( v w w q q ) where λ s the share of the endowment of factor used by good, and, (2.8) (2.9) s a weghted average of the elastctes of substtuton between factors and for the two goods, σ and σ. Equaton (2.8) specfes that a rse (say) n the relatve supply of factor must be matched by a rse n relatve demand, whch can be acheved by an ncrease n the -ntensty of the technques used for both goods (the frst term), or by a shft n the sectoral composton of output towards the -ntensve good (the second term), or by some mxture of these two mechansms. The neatly addtve combnaton of the two mechansms n (2.8) s accurate only for small changes, but these two basc mechansms are the relevant ones for changes of any se. Substtutng from equaton (2.5) for q q ) n equaton (2.8) and rearrangng yelds ( w w v v v v ( ) ~ (2.0) whch shows how relatve factor prces vary wth relatve factor endowments (and so provdes a precse defnton of φ ). 5 Equaton (2.0) nvolves both factor market-clearng mechansms. The second term n the denomnator of the bg rato shows how a change n factor prces, by alterng goods prces, shfts the sectoral composton of output n a drecton that lessens the requred change n factor prces. The response of relatve factor prces to a change n relatve endowments thus depends nversely on both the elastcty of substtuton n ~. If natonal producton, σ, and the producer-prce elastcty of demand, varetes were perfect substtutes, so β was nfnte, equaton (2.0) would collapse, because factor prces would be unaffected by endowments (as n HOS). Imperfect substtutablty, though, makes factor prces vary wth endowments, and ths varaton of factor prces s amplfed by ndependent trade costs. Trade costs affect the relatonshp between outputs and endowments n equaton (2.7) n two opposng ways: they reduce the response of relatve sales to relatve costs (and 5 φ s the nverse of Jones s (965: 564) economy-wde elastcty of substtuton between factors.

7 7 hence to relatve factor prces), but they ncrease the response of relatve factor prces to relatve endowments. To assess the net effect, the defnton of φ from (2.0) can be substtuted nto (2.7) and rearranged to yeld q q ~ v v ( ) ( ) (2.) Ths equaton shows that n a 2x2 model, snce (θ θ ) and (λ λ ) must have the same sgn, trade costs make the effect of endowments on outputs smaller, whatever ts drecton (though ths smple result does not generalse to hgher dmensons). If β were nfnte, the bg rato n equaton (2.) would reduce to /(λ λ ), whch s the Rybcynsk relatonshp n Jones (965): changes n relatve endowments would have magnfed effects on relatve outputs. Wth fnte β, the effects are smaller, to a degree dependent on the se of trade costs, and need not be magnfed. An ncreased (say) endowment of factor, by rasng the output of the -ntensve good, lowers the prce of that good and thus also the prce of factor, whch nduces -usng changes n technque that absorb some of the ncreased endowment. 2.3 Changes n foregn prces and trade costs To analyse the effects of changes n foregn goods prces on ths country, contnung to assume that t has only small market shares for both goods, the demand functon n (2.) can be expanded to q q where * * p p p p * p (2.2) s the average purchaser prce charged by foregn supplers of good, and γ s the elastcty of substtuton n consumpton between goods and. The effect of a change n foregn relatve prces on the relatve sales of the country concerned s n the opposte drecton to that of a change n ts own relatve prces. The effect s smaller, too, because a rse n p (for example) not only ncreases ths country s share * of the market for good (to the same degree as a fall n p would) but also offsettngly shrnks the market for good relatve to good, to a degree determned by γ (whch s generally smaller than β ). By substtutng equaton (2.2) nto equaton (2.8), an expresson can be derved that shows how ths country s relatve factor prces are affected by changes n the prces of foregn supplers, holdng ts own endowments constant w w * * p p ( ) (2.3) The change n the relatve sales and outputs of the two goods caused by the change n foregn prces s translated by (λ λ ) nto a change n the relatve demand for the two factors, whch φ translates nto a change n ther relatve prces. As n other H-O models, a rse (say) n the relatve foregn prce of a good rases the relatve prce of the factor n whch that good s ntensve (whch depends on the sgn of λ λ ).

8 8 Equaton (2.3) s smlar n form to (2.7), llustratng the recprocty of H-O models. In both equatons, the sgn and se of the effect depend on the dfference n factor ntensty between the goods, measured n one equaton by (θ θ ) and n the other by (λ λ ). In both equatons, too, the se of the effect s greater, the hgher are the purchaser-prce elastcty, β, and the factor-prce elastcty, φ. There s an asymmetry between equatons (2.7) and (2.3) n the effect of trade costs: n both equatons, trade costs ncrease φ (from equaton 2.0), but n (2.7) trade costs also lower the goods-prce demand elastcty. The net effect of trade costs s thus to reduce the response of outputs to changes n endowments (as shown n 2.), but to ncrease the response of factor prces to changes n foregn prces (though ths smple asymmetry s another result that does not generalse). To analyse more deeply the response of factor prces to foregn prces, the expresson for φ from (2.0) can be substtuted nto (2.3) and rearranged to yeld w w * * p p ( ) ~ (2.4) If β were nfnte and ~ were ero, the rght-hand sde of (2.4) would reduce to * * p p, whch s the Stolper-Samuelson relatonshp n Jones (965): the effect of a change n foregn prces on factor prces would be magnfed. A fnte β tends to reduce ths effect, because of the feedback of a change n factor prces va changes n technque and thus n factor avalabltes, outputs and goods prces (n the frst term of the denomnator of the bg rato n 2.4). Independent trade costs, by contrast, tend to ncrease ths effect, fundamentally because the shock s to purchaser prces, changes n whch, wth ndependent trade costs, requre proportonally bgger changes n producer prces and hence n factor prces. A change n foregn relatve prces may be caused by a change n foregn producton costs, c*, or n trade costs on foregn goods, t*, ncludng polcy barrers such as tarffs (not dstngushng here between ndependent and proportonal trade costs) p * * * * * * p c c T T (2.5) where T t * * * c, and smlarly for good. For example, to show the effects of a change n relatve foregn trade costs, assumng no change n foregn producton costs, * the fnal term p * p * n equaton (2.3) or (2.4) s replaced by T * T. As usual n H-O models, a reducton (say) n the trade costs of foregn supplers of a good would lower the prce of the factor used ntensvely n the producton of that good. To analyse the effects of changes n ths country s own trade costs, equaton (2.2) can be expanded to

9 9 p p ~ c c ~ ~ ~ T T ~ t t (2.6) where T t c and smlarly for good. Changes n relatve purchaser prces are a weghted sum of changes n relatve producer prces plus proportonal trade costs (t ) and changes n relatve ndependent trade costs ( ~ t ), where the weghts are ther shares n purchaser prces. Changes n one-plus-proportonal trade cost ratos (T s) have the same effect on purchaser prces as changes n producer prces, and thus the same weght, whch s the average share of c t n p. The effect of changes n relatve ndependent trade costs has a dfferent weght, whch s the average share of ~ t n p. Equaton (2.6) can be used to derve the effect on relatve factor prces of changes n own-trade costs, holdng relatve factor endowments constant, w ~ w ~ ~ ( ) ~ T T ~ t t (2.7) The term n square brackets shows how changes n the country s relatve trade costs affect ts relatve purchaser prces, and the terms before the square bracket show how ths prce change affects relatve sales of the two goods (β ), relatve factor demands (λ λ ) and relatve factor prces (φ ). The sgn s negatve: a fall (say) n own-trade costs on the -ntensve good (trade taxes as well as thngs lke transport costs) rases the relatve prce of factor, unlke a fall n the trade costs of foregn supplers of the -ntensve good, whch as shown above would hurt factor. Allowng for the endogenety of φ, equaton (2.7) can be rewrtten as 6 w w ~ ~ ~ ~ T T t t (2.8) ( ) If β were nfnte and ~ were ero, the rght-hand sde of (2.8) would reduce to * * T T, mplyng magnfcaton. Wth fnte β, the effects of changes n trade costs on factor prces are smaller, because there s some shftng of trade costs onto purchasers. Hgher ndependent trade costs, ~, lessen the effect of changes n 6 Equaton (2.8) dffers from equaton (2.4) n the poston of the ~ term because the fnal square-bracketed term n (2.8) refers to changes n relatve producer prces, whereas the fnal term n (2.4) refers to changes n relatve purchaser prces.

10 0 * proportonal trade costs, * T T (though agan ths result does not generalse), but could ether rase or lower the effect of changes n ndependent trade costs, t ~ t ~. 2.4 Teachng the practcal 2x2 model The man use of the 2x2 verson of H-O s for teachng, so t s worth pausng, before movng to hgher dmensons, to consder how the model descrbed above mght alter the way n whch 2x2 H-O s taught. Snce the basc economc nsghts reman the same, as does much of the model, the same teachng tools could be used, albet wth some sgnfcant modfcatons of emphass and nterpretaton. The model above s of one small country tradng n a world market, whch for some purposes s suffcent, but the teachng of trade clearly has to nvolve more than one country. The hatted changes n varables thus need to be nterpreted as referrng to dfferences across countres as well as to movements over tme n one country. The algebra strctly apples only to small changes and hence also only to small dfferences, but t wll be argued below that the model can be extended wth acceptable accuracy to large changes and dfferences by workng n logs. Extendng the model to many countres requres some addtonal assumpton about technology, of whch the obvous one s that all countres have the same technology up to a scale factor reflectng economy-wde effcency as a result of varaton n, say, the qualty of nsttutons. Ths assumpton allows the mx of factors n each sector to vary wth endowments. It s weaker than the usual HOS one of dentcal technology, and conforms wth the results of Trefler (995) and Davs and Wensten (200). Ths assumpton also underlnes that H-O s a theory of comparatve advantage. The usual H-O theorems all contnue to apply, though n less strong and smple forms than n HOS. Rybcynsk and Stolper-Samuelson were dscussed above, and so was recprocty. Changes n relatve endowments stll act n the usual drecton on relatve outputs, as do changes n relatve foregn prces on relatve factor prces. What s dfferent s the magntudes of these effects. However, the strong and smple theorems of HOS reman useful for explanng the essence of H-O (Leamer, 995). Factor prce equalsaton exsts only n specal cases: relatve factor prces vary among countres wth endowments, and absolute factor prces vary also because of the scale dfferences n technology. What remans s relatve factor prce convergence, whch s what was predcted by Ohln (967: 24-8). If reduced trade costs cause a country s relatve goods prces to move towards the relatve goods prces of other countres, ts relatve factor prces wll also tend to move towards ther relatve factor prces. The H-O theorem that a country s exports use ts abundant factor ntensvely, and vce versa for ts mports can be llustrated n the model by nterpretng the changes n equatons (2.7) or (2.) as referrng to devatons of a country s endowment rato from the world average endowment rato. What endowments alter n the model above s the structure of producton, from whch the effects on exports and mports follow wth the usual addtonal assumpton that consumpton preferences are smlar across countres. In secton 4, exports and mports are modelled separately.

11 Many of the same dagrams could also stll be used. An mportant one, though not specfcally H-O, llustrates the gans from trade wth a producton possblty fronter, ndfference curves and a world prce rato lne. Two modfcatons are requred. One s to have two world prce rato lnes: the relatve foregn cost of producton, and the ~. relatve purchaser prce derved by rasng ths relatve cost to the power The other modfcaton s to ncrease the curvature of the ndfference curves: the producton possblty fronter refers to the varetes of the two goods produced by the country concerned, for whch the varetes obtaned for consumpton through mports are less-than-perfect substtutes. 7 The nvaluable Lerner dagram can ntally be presented n the usual way and used to llustrate the effects of a shft n country endowments on the nter-sectoral allocaton of factors. It can then be explaned that shfts n the sectoral composton of output usually requre opposte shfts n relatve goods prces and thus n the dstances of the unt value soquants from the orgn (snce the soquants are valued at producer prces, ndependent trade costs amplfy these shfts). Relatve factor prces thus move n the opposte drecton to the endowment rato, and the factor ntensty ratos of both goods move n the same drecton as the endowment rato, whch makes the nter-sectoral reallocaton of the factors smaller than f goods prces were unchanged. 8 Much of trade can be taught wthout smultaneous consderaton of more than one country, and analysng a realstc world of many countres requres smulaton. Twocountry models are also useful, though, ncludng n teachng H-O. A varant of the ntegrated equlbrum dagram of Dxt and Norman (980), wthout factor prce equalsaton but wth only two goods, could be a two-country Lerner dagram. The factor ntensty ratos of the goods dffer between the two countres because of (and n the same drecton) as the dfferences n ther endowment ratos. The endowments of each country, shown by a sngle pont for both countres, are allocated between the goods as n any Lerner dagram, and the country allocatons are summed n another pont to show how world factor supples are dvded between the two goods. Two-country North-South models are also useful (e.g. Wood, 994). In ths context, the easest way to apply the present algebra s to assume that each regon contans many dentcal countres and to focus on representatve ones n each regon. It s then smple to analyse the effects of, say, a change n the South s own relatve trade costs on relatve wages both n the South (equaton 2.7) and n the North (equaton 2.3), snce the change n the South s trade costs also alters foregn prces n the North. The outcome may be modfed by techncal change, ether autonomous or caused by more openness to trade, whose effects on factor prces n a model of ths type are analysed by Jones (965). The relatve effects of sector-based and factor-based change dffer from those n the usual HOS model wth nfntely elastc demand for goods. 7 Wth trade, consumers choose at the margn between the home varety of the export good and foregn varetes of the mported good: the elastcty of substtuton s roughly γ ( /β ), whch gets closer to γ, the more substtutable are home and foregn varetes. There s more curvature n an open than n a closed economy, whch reduces the gan from trade, but not by enough to make t worth drawng. 8 Interpreted lterally, the present model also mples that the endowment rato can never le outsde the range of the factor ntensty ratos. Snce each country produces ts own varety, producton n each of ts sectors can never qute fall qute to ero. It s more nstructve, however, to contnue to nterpret the dagram as mplyng complete specalsaton when the endowment rato s outsde ths range.

12 2 3. Any numbers of goods or factors Ths secton extends the 2x2 model to hgher dmensons. Goods are ndexed by the subscrpt (=,., n) and factors by or k (=,., m). Followng Smth and Wood (2009), a standard hgher-dmensonal Jones-type H-O model contans n equatons lnkng changes n relatve goods prces to changes n relatve factor prces p p m ( k k)( w k w ) k2 (3.) of whch all the terms were defned n the prevous secton, and m factor marketclearng equatons, also n changes, v v (3.2) m n n ( k k ) ( w k w ) ( )( q q ) k2 2 whch are of the same general form as the factor market-clearng equaton n the 2x2 model, but requre a bt more explanaton. The last term dffers only n summng the effects on relatve factor demands of changes n relatve outputs over n pars of goods, rather than one par. The frst term on the rght-hand sde, however, nvolves a redefnton: the σ terms are no longer elastctes of substtuton: nstead k a w k w a k (3.3) s the elastcty wth respect to the prce of factor k of the factor nput coeffcent for good. The partal dervatve n (3.3) can be of ether sgn, but for consstency wth the prevous secton (and dfferently from Smth and Wood, 2009), σ k s defned lke a substtuton elastcty: f a w s negatve, σ k s postve. These elastctes are k weghted by the shares of the endowment of used n good, and summed over all pars of goods and factors. The frst rght-hand sde term of (3.2) thus shows how changes n all relatve factor prces affect the relatve demands for factors and. Equatons (3.) and (3.2) would normally be nterpreted as referrng to an economy facng exogenous goods prces, p. As n the prevous secton, however, ndependent trade costs can be ntroduced nto (3.), whch becomes p p m c c ~ k k k ~ ( )( w w ) k2 (3.4) and another equaton can be added n p p 2 q q (3.5)

13 3 whch makes the country s relatve sales and thus outputs of goods and depend on the relatve purchaser prces of ts varetes of these and all other goods. Combnng equatons (3.4) and (3.5) yelds n equatons of the form q q n m k k k ~ ( )( w w ) 2 k2 (3.6) whch n conuncton wth the m factor market-clearng equatons (3.2) descrbe how changes n all relatve factor prces and relatve outputs depend on changes n the exogenous relatve factor endowments. Changes n the relatve purchaser prces of all goods are determned by equatons (3.4). Expanson of (3.5) along the lnes of (2.2) and (2.6) would allow analyss of the effects on factor prces and outputs of changes n foregn goods prces and trade costs. As n ts 2x2 verson, the model depends on the β s beng fnte: f dfferent natonal varetes of goods were perfect substtutes, t would collapse back to the standard system of equatons (3.) and (3.2). Ths model has the basc H-O feature of showng how the composton of a country s output s affected by the composton of ts factor endowments, but has three notable advantages over most hgher-dmensonal H-O models. It apples to (and behaves n the same way wth) any numbers of goods and factors: these numbers do not have to be even, and nor does the model become ndetermnate f, as n realty, the number of goods exceeds the number of factors. A second advantage s that relatve factor prces are always nfluenced by relatve endowments, as they appear to be n realty but are not n the usual one-cone HOS model. A thrd advantage s that ths model shows how outcomes are affected by trade costs. 3. Smplfcaton of the model In these three respects, ths general hgher-dmensonal verson of the present model can clam to be practcal. What s not practcal about t, however, lke other hgherdmensonal H-O models, s that t s complcated and that changes n endowments could n prncple affect outputs and factor prces n almost any way. To get a more tractable model and clearer results, as Blss (2007: 28) emphasses, t s necessary to make smplfyng assumptons. The rest of ths secton wll therefore formulate and analyse a smpler verson of the present hgher-dmensonal model. A frst smplfcaton s to omt the effects on a country s relatve sales of any par of (ts varetes of) goods of the prces of (ts varetes of) other goods: t seems unlkely that choces between (say) French cars and French shrts depend much on the prce of French shoes n partcular. Each of the n equatons (3.6) thus becomes q q m k k k ~ ( )( w w ) k2 (3.7) whch can be substtuted nto equatons (3.2) to yeld

14 4 v v m n n m ( k k ) ( w k w ) ( ) ( k k)( w k w ) k2 2 ~ k2 (3.8) A much more radcal smplfcaton s then to convert each of the equatons (3.8) nto a relatonshp between the relatve prces and endowments of a sngle par of factors, and. Algebracally, ths nvolves suppressng the two summatons over k: n the frst rght-hand sde term, t omts the effects on the choce of technque as between factors and of the prces of all factors other than and ; n the second term, t omts the effects on the relatve prces, and hence the relatve outputs, of goods and of the relatve prces of all factors other than and. Wth these smplfcatons, (3.8) can be rearranged to yeld m elastctes, φ, that show how each par of relatve factor prces depends, nversely, on the correspondng par of factor endowments w v w v n ( ) n 2 ( ) ~ ( ) (3.9) The elastcty φ was used above n the 2x2 verson, and equaton (3.9) s smlar to (2.0), except that t nvolves summatons over all pars of goods (underlnng that φ s an economy-wde relatonshp). 9 The k subscrpt s dropped, snce only two factors are nvolved n each equaton, but the trple subscrpt on σ s retaned as a remnder of one of the smplfcatons made n dervng ths expresson. As wth all smplfcaton of models, the beneft of these smplfcatons of (3.8) needs to be weghed aganst ther cost. The beneft, a much clearer hgher-dmensonal H-O model, wll emerge n the next sub-secton. The cost, a reducton n the accuracy of the model, wll then be explaned and dscussed. 3.2 Hgher-dmensonal H-O relatonshps Usng (3.9), equaton (3.7) can be extended nto q q m v v ~ 2 (3.0) a set of n equatons showng how the relatve outputs of all pars of goods depend on the relatve endowments of all pars of factors. These equatons are smlar to (2.7) n the 2x2 verson of the model, except that there are more of them and that each of them nvolves many pars of factors rather than only one par. Ther meanng can be brought out by wrtng the equaton for (say) good 2 as a seres of m terms, each of whch s a change n one factor endowment rato multpled by a coeffcent 9 The expresson beng summed n the second term n the denomnator of (3.9) may not be postve for all pars of goods, snce n hgher dmensons the λ and θ dfference terms may not have the same sgn, but ts sum over all pars of goods, and hence (3.9) as a whole, seems certan to be postve.

15 5 q 2 v v... v v 2 2 q ~ ~ m2 m m m 2 2 (3.) The negatve sgns on β 2 and the φ s cancel, so the sgn of each coeffcent depends on that of ts (θ 2 θ ) term: f good 2 uses factor more ntensvely than good, the coeffcent s postve, and f less ntensvely, s negatve. The (θ 2 θ ) term affects also the se of the coeffcent: the bgger the dfference between goods and 2 n the ntensty of ther use of factor, the larger s the coeffcent (and f the two goods were of equal factor ntensty, the term would vansh). The se of each coeffcent depends also on the se of the relevant φ, whch can vary among pars of factors, dependng on the parameters of equaton (3.9). All the coeffcents n (3.) depend n the same way on 2 ~ 2: the lower s the average degree of substtutablty among varetes of goods and 2, and the hgher are ndependent trade costs on these two goods, the smaller are the effects of all pars of factor endowments on the relatve outputs of goods and 2. Independent trade costs pull n the other drecton, too, by makng the φ s larger amplfyng the effects of endowments on factor prces. Ths amplfcaton depends not on ~ 2, however, but (from 3.9) on average ~ s across all pars of goods. The net effect of a hgher ~ 2, holdng other ~ s constant, s to reduce the response of the relatve outputs of goods and 2 to endowments (as n the 2x2 verson). Rasng other ~ s, by contrast, makes the relatve outputs of goods and 2 more responsve to endowments. The assumptons used n dervng equaton (3.9) also facltate analyss of the effects of changes n foregn prces and own trade costs on relatve factor prces, as n the 2x2 model. Consderng frst changes n foregn purchaser prces, whch could be caused by changes n trade costs on (ncludng polcy barrers to) foregn goods, the demand equatons (3.7) become q q * * p p m ~ ( k k)( wk w ) k2 (3.2) whch can be substtuted nto the factor market-clearng equaton and smplfed as before, holdng endowments constant, to yeld, after rearrangement, a set of m equatons whch descrbe how each par of factor prces s affected by changes n the relatve foregn purchaser prces of all pars of goods w w * * p p n ( ) 2 (3.3) Equatons (3.3) can agan be explaned by wrtng out one of them (for factors and 2, say) as a seres of n terms, each of whch s the change n one foregn prce rato multpled by a coeffcent, as n the correspondng 2x2 equaton (2.3)

16 6 w * * * * p p... ( ) p p 2 w 2( 22 2 ) n n n n n (3.4) The proxmate effect of a change n a relatve foregn prce s to alter the relatve sales and outputs of ths country s varetes of the two goods, to an extent determned by the se of the purchaser-prce elastcty, (β γ ). Ths change n relatve outputs alters the relatve demand for factors and 2, n a drecton and to an extent governed by the dfference between the shares of factor 2 used by the two goods, (λ 2 λ 2 ), whose sgn determnes that of the coeffcent. The se of each coeffcent depends also on that of φ 2, common to all the coeffcents. Trade costs act through φ 2 : the hgher the average level of (ndependent) trade costs across all goods n the country concerned, the larger s φ 2 and hence the more do the relatve prces of factors and 2 respond to a change of gven se n relatve demand for these two factors. Equaton (3.3) thus mples that hgher trade costs ncrease the effect of changes n foregn prces on factor prces. Ths mplcaton should be treated cautously, though, and s modfed below, because equaton (3.3), lke all the others so far, assumes that the country has only small market shares for all ts goods The recprocty of H-O models can be seen from the smlarty of form between the quantty equatons (3.0) and the prce equatons (3.3). Each of these equatons s a seres of terms nvolvng a coeffcent whch ncludes a prce elastcty common to all the terms (n the quantty equatons a goods-prce elastcty and n the prce equatons a factor-prce elastcty). Each coeffcent also ncludes a measure of dfference n factor use between goods (based on θs n the quantty equaton and on λs n the prce equaton) and another prce elastcty whch vares among the terms (n the quantty equatons a factor-prce elastcty and n the prce equatons a goods-prce elastcty). To analyse the effects of changes n relatve own-trade costs, equaton (2.6) can be adapted and used to yeld w w n 2 ( ) ~ ~ T T t ~ t ~ ~ (3.5) whose form s smlar to that of the foregn-prce equaton (3.3), apart from ts fnal terms, but wth the opposte sgn. A rse n the foregn prce of a good, caused say by an ncreased tarff, rases the prce of the factor that the good uses ntensvely, whle a rse n own-trade costs on that good lowers the prce of that factor. 3.3 Cost of the smplfcatons Equatons (3.9), (3.0), (3.3) and (3.5) are basc H-O relatonshps: for any numbers of goods and factors, they show how the composton of a country s output and ts relatve factor prces are nfluenced by the composton of ts endowments, trade costs and foregn prces. They also make these relatonshps far clearer than n the general hgher-dmensonal verson of the model n equatons (3.2) and (3.6). Ths clarty was acheved, however, by radcal smplfcaton of the factor market-clearng equatons (3.8), omttng (a) the effects on the choce of technque as between factors and of

17 7 the prces of all factors other than and, and (b) the effects on the relatve prces of goods and of the relatve prces of factors other than and (though these effects reman n other parts of the model, such as equatons 3.0). The smplfed verson of the model s thus less accurate than the general verson. The naccuracy can be explaned by consderng, say, the frst term n equaton (3.) 2 q q ~ v v (3.6) n the case of an ncrease n the endowment of factor 2. The expresson for φ 2 n (3.9), and more specfcally the second term n ts denomnator, s naccurate because a fall n the relatve prce of factor 2 as a result of ts ncreased supply would also alter the relatve prces of goods other than and 2, so the fall n the relatve prce of factor 2 could be larger or smaller than (3.9) mples. Moreover, ncreasng the endowment of factor 2 would lower the prces of other factors for whch t was a substtute (and rase the prces of factors for whch t was a complement), whch would affect the relatve producton costs of goods and 2, and thus ther relatve purchaser prces and relatve outputs. Equaton (3.6) should thus be expanded to somethng lke m 2 q q ~ v2 v... 2 (3.7) n whch the added summaton could be of ether sgn, snce the φ 2 s could be of ether sgn (though most would probably be postve, snce substtutablty s more common than complementarty) and so could the (θ 2 θ )s. For these reasons, the effect of a change n the relatve endowments of any gven par of factors on the relatve outputs of any gven par of goods s not exactly as specfed n equatons (3.9) and (3.0): t could be ether larger or smaller. Much the same s true of the relatonshps n equatons (3.3) between changes n relatve foregn prces and n relatve factor prces, because of the naccuracy of the φ terms. Shfts n the relatve demand for factors and 2 caused by changes n relatve foregn prces could have larger or smaller effects on the prces of factors and 2 than equatons (3.3) and (3.9) specfy. Ths s because changes n the prces of these two factors affect the prces of other factors for whch they are substtutes or complements, whch has ndrect effects on the relatve demands for, and prces of, these two. The costs of these naccuraces, relatve to the benefts of the smplfed verson of the hgher-dmensonal model, depend on the purpose for whch the model s beng used. The naccuraces of the smplfed equatons, and the more accurate general verson of the model n equatons (3.2) and (3.6), should not be forgotten, and for some uses, the general verson s preferable. But for most practcal purposes of analyss, teachng, estmaton and applcaton to actual country cases the smplfed verson s lkely to be more useful, provded that readers or students are warned of ts lmtatons and that econometrc results are nterpreted wth due cauton.

18 8 4. Exports, home sales and mports All the analyss so far has been of one country producng and sellng (ts varetes of) goods n a sngle world market (ncludng ts domestc market) of whch t has small shares for all goods. Ths secton dstngushes among dfferent sorts of markets, n whch outcomes dffer manly because trade costs dffer (assumng away dfferences n tastes, as n most H-O models). Ths extenson of the model s needed for explct analyss of the two basc elements of nternatonal trade, exports and mports. Some refnements of notaton are requred. Countres of orgn (or suppler countres) are ndexed by a superscrpt (=,, Z), and markets (or countres of destnaton) by a second superscrpt, ž (=,, Ž). For example, country s sales of good n market ž are q, and ts purchaser prce n that market s p, but ts producer prce, common to all ts markets, s ust c. The superscrpt * can be substtuted for ether or ž to refer to the rest of the world as a whole: for example, the country s exports are q *. Trade costs are splt between nternal costs, ncurred by all supplers to a market, wth superscrpt D, and nternatonal costs, ncurred only by foregn supplers, wth superscrpt F. For example, t s the varable nternatonal trade cost of exports of a F* unt of good from country, averaged over all ts export markets. 4. Producer-prce demand elastctes What vares among markets s the producer-prce elastcty of demand, whch n the ~. The general form of ths elastcty s two prevous sectons was ~ (4.) where ~ s the purchaser-prce elastcty, whch so far has been β, and p p D F c c ~ ~ s the prce-rato elastcty, whch so far has been ~ (4.2). Both the ngredents of the producer-prce elastcty n equaton (4.) have and ž superscrpts, because both of them vary wth trade costs. The reason for varaton of the prce-rato elastcty s D F obvous, namely that the rato of (ndependent) trade costs to other costs, ~ ~, depends on characterstcs of both the suppler and the market. The reason for varaton of the purchaser-prce elastcty requres more explanaton. The response of relatve sales of a country s varetes of two goods to changes n ts relatve purchaser prces depends n prncple not only on the degree of substtutablty among dfferent varetes of each good, measured by β, but also on the degree of substtutablty between the two goods, measured by γ. The relatve mpact of these two sorts of substtutablty depends on the country s shares of the market for the two goods: f ts shares are small, ~ s determned almost entrely by β, but wth larger

19 9 shares, ~ depends also on γ, snce the prces charged by the country n queston for ts varetes then have a sgnfcant effect on the average prces of the goods. More precsely, as n Wood (2008), followng Sato (967), wth CES preferences s a weghted harmonc mean of β, β and γ, n whch the weghts nclude country s shares of market ž for the two goods. A more tractable approxmaton for purposes of exposton s the weghted arthmetc mean ~ s s s (4.3) where β s an average of β and β, and s an average of country s shares of market ž for goods and. If s 0, ~, as before, whle f s, as n a closed economy or for two non-traded goods, ~. Assumng preferences to be dentcal and homothetc avods the need for ž superscrpts on β and γ. Trade costs therefore nfluence purchaser-prce elastctes by affectng market shares. The relatonshp s analysed n Wood (2008), but ts essence can be conveyed by s ~ s F* F t t s 0 s (4.4) Ths equaton says that country s average share of market ž for goods and rses wth the nternatonal trade costs ncurred by the rest of the world n supplyng market ž, relatve to those that country ncurs. In ts home market, country has a trade cost F advantage, snce t 0, whch tends to rase ts market share, especally f natural or polcy barrers to mports are hgh. In export markets, country wll always be at a dsadvantage relatve to local supplers, and may be at an advantage or a dsadvantage compared to the average of other foregn supplers. Note that the trade cost terms n (4.4) do not bear tldes: ndependent and proportonal costs both matter for purchaser-prce elastctes, though only ndependent ones matter for prce-rato elastctes. Note also that country s average share of any market for any par of goods depends mportantly on other thngs, too, ncludng ts producton costs and ts se: larger countres tend to have larger shares of all markets. That trade costs put home supplers at an advantage n home markets helps to explan why emprcal studes consstently fnd home bas n patterns of consumpton (Trefler, 995; Davs and Wensten, 200). In the present model, home bas arses also from the varaton of relatve factor prces among countres wth ther endowments: more s consumed of goods whch a country can produce more cheaply (Romals, 2004: 70). Pullng the precedng dscusson together, the producer-prce elastcty of demand for a country s exports s usually smlar n form to that of earler sectons * ~ D* ~ F* (4.5)

20 20 because export market shares are usually small. The only change s the expanson of the denomnator to separate the trade costs that country ncurs n sendng goods to ts export markets from the trade costs that t ncurs wthn those markets (whch are usually smlar for all supplers). Equaton (4.5) refers to total exports, but could be modfed to refer to any ndvdual export market by substtutng ž ( ) for *. The dstrbuton of a country s exports across ndvdual markets depends on the relatve ses and trade costs of the markets, n ways that are analysed n gravty models. There s more of a dfference from earler sectons n the producer-prce elastcty of demand for home market sales, whch s approxmately s F* t ~ s D 0 (4.6) The purchaser-prce elastcty n the numerator s smaller than for exports, to an extent that depends on how much smaller γ s than β and on the degree of protecton of the home market for these goods by the nternatonal trade costs of foregn supplers. By contrast, the prce-rato elastcty for home sales s usually hgher than for exports, because the denomnator of (4.6) ncludes only nternal trade costs. The producerprce elastcty for home market sales could thus n prncple be ether lower or hgher than for exports. In the lmtng case of a closed economy or a par of non-traded goods, the home market producer-prce elastcty s D ~ (4.7) whch s lower than for home market sales of otherwse smlar traded goods (and s usually, though not necessarly, lower than for exports of such goods). The producer-prce elastcty of demand for a country s total outputs of goods and, * whch must le between and, s thus s ~ (4.8) where s and ~ are averages across all the country s markets, abroad and at home, weghted by ts sales n each of them. The share n each market s affected by trade costs, as n equaton (4.4), whle the weght depends also on the se of that market. 0 Hgher ndependent trade costs tend to lower by rasng ~. Hgher nternatonal trade costs, both ndependent and proportonal, tend to rase s, but also to reduce by rasng the share of output sold n the home market, where the country s producers ncur only nternal trade costs, so ther net effect on could be n ether drecton. ~ 0 2 Algebracally, s s s, where μ ž s the overall se of market ž.