We begin our study of international trade with the classic Ricardian model,

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1 Copyrght, Prnceton Unversty Press. No prt of ths book my be Prelmnres: Two-Sector Models We begn our study of nterntonl trde wth the clssc Rcrdn model, whch hs two goods nd one fctor (lbor. The Rcrdn model ntroduces us to the de tht technologcl dfferences cross countres mtter. In comprson, the Heckscher-Ohln model dspenses wth the noton of technologcl dfferences nd nsted shows how fctor endowments form the bss for trde. Whle ths my be fne n theory, the model performs very poorly n prctce: s we show n the next chpter, the Heckscher-Ohln model s hopelessly ndequte s n explnton for hstorcl or modern trde ptterns unless we llow for technologcl dfferences cross countres. For ths reson, the Rcrdn model s s relevnt tody s t hs lwys been. Our tretment of t n ths chpter s smple revew of undergrdute mterl, but we wll present more sophstcted verson of the Rcrdn model (wth contnuum of goods n chpter 3. After revewng the Rcrdn model, we turn to the two-good, two-fctor model tht occupes most of ths chpter nd forms the bss of the Heckscher- Ohln model. We shll suppose tht the two goods re trded on nterntonl mrkets, but do not llow for ny movements of fctors cross borders. Ths reflects the fct tht the movement of lbor nd cptl cross countres s often subject to controls t the border nd s generlly much less free thn the movement of goods. Our gol n the next chpter wll be to determne the pttern of nterntonl trde between countres. In ths chpter, we smplfy thngs by focusng prmrly on one country, tretng world prces s gven, nd exmne the propertes of ths twoby-two model. The student who understnds ll the propertes of ths model hs lredy come long wy n hs or her study of nterntonl trde. RICARDIAN MODE Indexng goods by the subscrpt, let denote the lbor needed per unt of producton of ech good t home, whle s the lbor need per unt of producton n the foregn country, =,. The totl lbor force t home s nd brod s. bor s perfectly moble between the ndustres n ech country, but mmoble cross countres. Ths mens tht both goods re produced n the home country only f the wges erned n the two ndustres re the sme. Snce the mrgnl product of lbor n ech ndustry s /, nd workers re pd the vlue of ther mrgnl

2 Copyrght, Prnceton Unversty Press. No prt of ths book my be Chpter y y * p * * B* A C p C* A* p B p * y * * y* ( Home Country (b Foregn Country Fgure. products, wges re equlzed cross ndustres f nd only f p/ = p/, where p s the prce n ech ndustry. ettng p= p/ p denote the reltve prce of good (usng good s the numerre, ths condton s p= /. These results re llustrted n fgure.( nd (b, where we grph the producton possblty fronters (PPFs for the home nd foregn countres. Wth ll lbor devoted to good t home, t cn produce / unts, =,, so ths estblshes the ntercepts of the PPF, nd smlrly for the foregn country. The slope of the PPF n ech country (gnorng the negtve sgn s then / nd /. Under utrky (.e., no nterntonl trde, the equlbrum reltve prces p nd p must equl these slopes n order to hve both goods produced n both countres, s rgued bove. Thus, the utrky equlbrum t home nd brod mght occur t ponts A nd A. Suppose tht the home country hs comprtve dvntge n producng good, menng tht / < /. Ths mples tht the home utrky reltve prce of good s lower thn tht brod. Now lettng the two countres engge n nterntonl trde, wht s the equlbrum prce p t whch world demnd equls world supply? To nswer ths, t s helpful to grph the world reltve supply nd demnd curves, s llustrted n fgure.. For the reltve prce stsfyng p p = / nd p p = / both countres re fully speclzed n good (snce wges erned n tht sector re hgher, so the world reltve supply of good s zero. For p p p, the home country s fully speclzed n good wheres the foregn country s stll speclzed n good, so tht the world reltve supply s ( / /( /, s lbeled n fgure.. Fnlly, for p p nd p p, both countres re speclzed n good. So we see tht the world reltve supply curve hs str-step shpe, whch reflects the lnerty of the PPFs. To obtn world reltve demnd, let us mke the smplfyng ssumpton tht tstes re dentcl nd homothetc cross the countres. Then demnd wll be ndependent of the dstrbuton of ncome cross the countres. Demnd beng homothetc mens tht reltve demnd d /d n ether country s downwrd-slopng functon of the reltve prce p, s llustrted n fgure.. In the cse we hve shown, reltve demnd ntersects reltve supply t the world prce p tht les between p nd p, but ths does

3 Copyrght, Prnceton Unversty Press. No prt of ths book my be p Prelmnres: Two-Sector Models 3 p * Reltve Supply p p Reltve Demnd ( (* * (y + y * (y + y* Fgure. not need to occur: nsted, we cn hve reltve demnd ntersect one of the flt segments of reltve supply, so tht the equlbrum prce wth trde equls the utrky prce n one country. Focusng on the cse where p p p, we cn go bck to the PPF of ech country nd grph the producton nd consumpton ponts wth free trde. Snce p p, the home country s fully speclzed n good t pont B, s llustrted n fgure.(, nd then trdes t the reltve prce p to obtn consumpton t pont C. Conversely, snce p p, the foregn country s fully speclzed n the producton of good t pont B n fgure.(b, nd then trdes t the reltve prce p to obtn consumpton t pont C. Clerly, both countres re better off under free trde thn they were n utrky: trde hs llowed them to obtn consumpton pont tht s bove the PPF. Notce tht the home country exports good, whch s n keepng wth ts comprtve dvntge n the producton of tht good, / /. Thus, trde ptterns re determned by comprtve dvntge, whch s deep nsght from the Rcrdn model. Ths occurs even f one country hs n bsolute dsdvntge n both goods, such s nd, so tht more lbor s needed per unt of producton of ether good t home thn brod. The reson tht t s stll possble for the home country to export s tht ts wges wll djust to reflect ts productvtes: under free trde, ts wges re lower thn those brod. Thus, whle trde ptterns n the Rcrdn model re determned by comprtve dvntge, the level of wges cross countres s determned by bsolute dvntge. Ths occurs f one country s very lrge. Use fgures. nd. to show tht f the home country s very lrge, then p= p nd the home country does not gn from trde. The home country exports good, so wges erned wth free trde re w= p/. Conversely, the foregn country exports good (the numerre, nd so wges erned there re w = / p /, where the nequlty follow snce p / n the equlbrum beng consdered. Then usng, we obtn w= p/ < p / w.

4 Copyrght, Prnceton Unversty Press. No prt of ths book my be 4 Chpter TWO-GOOD, TWO-FACTOR MODE Whle the Rcrdn model focuses on technology, the Heckscher-Ohln model, whch we study n the next chpter, focuses on fctors of producton. So we now ssume tht there re two fctor nputs lbor nd cptl. Restrctng our ttenton to sngle country, we wll suppose tht t produces two goods wth the producton functons y= f(,, =,, where y s the output produced usng lbor nd cptl. These producton functons re ssumed to be ncresng, concve, nd homogeneous of degree one n the nputs (,. 3 The lst ssumpton mens tht there re constnt returns to scle n the producton of ech good. Ths wll be mntned ssumpton for the next severl chpters, but we should be pont out tht t s rther restrctve. It hs long been thought tht ncresng returns to scle mght be n mportnt reson to hve trde between countres: f frm wth ncresng returns s ble to sell n foregn mrket, ths expnson of output wll brng reducton n ts verge costs of producton, whch s n ndcton of greter effcency. Indeed, ths ws prncpl reson why Cnd entered nto free-trde greement wth the Unted Sttes n 989: to gve ts frms free ccess to the lrge Amercn mrket. We wll return to these nterestng ssues n chpter 5, but for now, gnore ncresng returns to scle. We wll ssume tht lbor nd cptl re fully moble between the two ndustres, so we re tkng long run pont of vew. Of course, the mount of fctors employed n ech ndustry s constrned by the endowments found n the economy. These resource constrnts re stted s #, #, (. where the endowments nd re fxed. Mxmzng the mount of good, y= f(,, subject to gven mount of good, y= f(,, nd the resource constrnts n (. gve us y= h( y,,. The grph of y s functon of y s shown s the PPF n fgure.3. As drwn, y s concve functon of y, hy (,, / y 0. Ths fmlr result follows from the fct tht the producton functons f(, re ssumed to be concve. Another wy to express ths s to consder ll ponts S= ( y, y tht re fesble to produce gven the resource constrnts n (.. Ths producton possbltes set S s convex, menng tht f y = ( y, y nd y b = ( y b, y b b re both elements of S, then ny pont between them my ( m y s lso n S, for 0 # m #. 4 The producton possbltes fronter summrzes the technology of the economy, but n order to determne where the economy produces on the PPF we need to dd some ssumptons bout the mrket structure. We wll ssume perfect competton n the product mrkets nd fctor mrkets. Furthermore, we wll suppose tht product prces re gven exogenously: we cn thnk of these prces s estblshed on world mrkets, nd outsde the control of the smll country beng consdered. 3 Students not fmlr wth these terms re referred to problems. nd.. 4 See problems. nd.3 to prove the convexty of the producton possbltes set, nd to estblsh ts slope.

5 Copyrght, Prnceton Unversty Press. No prt of ths book my be Prelmnres: Two-Sector Models 5 y PP Fronter y = (y, y λy + ( λy b PP Set b b y b = (y,y y Fgure.3 GDP FUNCTION Wth the ssumpton of perfect competton, the mounts produced n ech ndustry wll mxmze gross domestc product (GDP for the economy: ths s Adm Smth s nvsble hnd n cton. Tht s, the ndustry outputs of the compettve economy wll be chosen to mxmze GDP: G( p, p,, = mx p y p y s.t. y = h( y,,. (. y, y To solve ths problem, we cn substtute the constrnt nto the objectve functon nd wrte t s choosng y to mxmze py ph ( y,,. The frst-order condton for ths problem s p p ( h/ y = 0, or, p h y p = = =. (.3 p y y Thus, the economy wll produce where the reltve prce of good, p= p/ p, s equl to the slope of the producton possbltes fronter. 5 Ths s llustrted by the pont A n fgure.4, where the lne tngent through pont A hs the slope of (negtve p. An ncrese n ths prce wll rse the slope of ths lne, ledng to new tngency t pont B. As llustrted, then, the economy wll produce more of good nd less of good. The GDP functon ntroduced n (. hs mny convenent propertes, nd we wll mke use of t throughout ths book. To show just one property, suppose tht we dfferentte the GDP functon wth respect to the prce of good, obtnng 5 Notce tht the slope of the prce lne tngent to the PPF (n bsolute vlue equls the reltve prce of the good on the horzontl xs, or good n fgure.4.

6 Copyrght, Prnceton Unversty Press. No prt of ths book my be 6 Chpter y A B p y Fgure.4 G y y = y p p. p d p n (.4 p It turns out tht the terms n prentheses on the rght of (.4 sum to zero, so tht G/ p= y. In other words, the dervtve of the GDP functon wth respect to prces equls the outputs of the economy. The fct tht the terms n prentheses sum to zero s n pplcton of the envelope theorem, whch sttes tht when we dfferentte functon tht hs been mxmzed (such s GDP wth respect to n exogenous vrble (such s p, then we cn gnore the chnges n the endogenous vrbles (y nd y n ths dervtve. To prove tht these terms sum to zero, totlly dfferentte y= h( y,, wth respect to y nd y nd use (.3 to obtn pdy= pdy, or pdy pdy= 0. Ths equlty must hold for ny smll movement n y nd y round the PPF, nd n prtculr, for the smll movement n outputs nduced by the chnge n p. In other words, p( y/ p p( y/ p = 0, so the terms n prentheses on the rght of (.4 vnsh nd t follows tht G/ p = y. 6 EQUIIBRIUM CONDITIONS We now wnt to stte succnctly the equlbrum condtons to determne fctor prces nd outputs. It wll be convenent to work wth the unt-cost functons tht re dul to the producton functons f(,. These re defned by c ( w, r = mn { w r f(, $ }. (.5, $ 0 6 Other convenent propertes of the GDP functon re explored n problem.4.

7 Copyrght, Prnceton Unversty Press. No prt of ths book my be Prelmnres: Two-Sector Models 7 In words, c ( w, r s the mnmum cost to produce one unt of output. Becuse of our ssumpton of constnt returns to scle, these unt-costs re equl to both mrgnl costs nd verge costs. It s esly demonstrted tht the unt-cost functons c ( w, r re non-decresng nd concve n (w, r. We wll wrte the soluton to the mnmzton n (.5 s c( w, r = w r, where s optml choce for, nd s optml choce for. It should be stressed tht these optml choces for lbor nd cptl depend on the fctor prces, so tht they should be wrtten n full s (w, r nd (w, r. However, we wll usully not mke these rguments explct. Dfferenttng the unt-cost functon wth respect to the wge, we obtn c = w r. w c m (.6 w w As we found wth dfferenttng the GDP functon, t turns out tht the terms n prentheses on the rght of (.6 sum to zero, whch s gn n pplcton of the envelope theorem. It follows tht the dervtve of the unt-costs wth respect to the wge equls the lbor needed for one unt of producton, c/ w=. Smlrly, c/ r=. To prove ths result, notce tht the constrnt n the cost-mnmzton problem cn be wrtten s the soqunt f(, =. Totlly dfferentte ths to obtn fd f d = 0, where f / f/ nd f / f/. Ths equlty must hold for ny smll movement of lbor d nd cptl d round the soqunt, nd n prtculr, for the chnge n lbor nd cptl nduced by chnge n wges. Therefore, f( / w f ( / w = 0. Now multply ths through by the product prce p, notng tht pf = w nd pf r = from the proft-mxmzton condtons for compettve frm. Then we see tht the terms n prentheses on the rght of (.6 sum to zero. The frst set of equlbrum condtons for the two-by-two economy s tht profts equl zero. Ths follows from free entry under perfect competton. The zero-proft condtons re stted s p = c ( wr,, p = c ( wr,. (.7 The second set of equlbrum condtons s full employment of both resources. These re the sme s the resource constrnts (., except tht now we express them s equltes. In ddton, we wll rewrte the lbor nd cptl used n ech ndustry n terms of the dervtves of the unt-cost functon. Snce c/ w= s the lbor used for one unt of producton, t follows tht the totl lbor used n = y, nd smlrly the totl cptl used s = y. Substtutng these nto (., the full-employment condtons for the economy re wrtten s y y =, : : y y =. ; ; (.8 Notce tht (.7 nd (.8 together re four equtons n four unknowns, nmely, ( wr, nd ( y, y. The prmeters of these equtons, p, p,, nd, re gven exogenously. Becuse the unt-cost functons re nonlner, however, t s not enough to just count equtons nd unknowns: we need to study these equtons n detl to

8 Copyrght, Prnceton Unversty Press. No prt of ths book my be 8 Chpter understnd whether the solutons re unque nd strctly postve, or not. Our tsk for the rest of ths chpter wll be to understnd the propertes of these equtons nd ther solutons. To gude us n ths nvestgton, there re three key questons tht we cn sk: ( wht s the soluton for fctor prces? ( f prces chnge, how do fctor prces chnge? (3 f endowments chnge, how do outputs chnge? Ech of these questons s tken up n the sectons tht follow. The methods we shll use follow the dul pproch of Woodlnd (977, 98, Muss (979, nd Dxt nd Normn (980. DETERMINATION OF FACTOR PRICES Notce tht our four-equton system bove cn be decomposed nto the zero-proft condtons s two equtons n two unknowns the wge nd rentl nd then the full-empoyment condtons, whch nvolve both the fctor prces (whch ffect nd nd the outputs. It would be especlly convenent f we could unquely solve for the fctor prces from the zero-proft condtons, nd then just substtute these nto the full-employment condtons. Ths wll be possble when the hypotheses of the followng lemm, re stsfed. EMMA (FACTOR PRICE INSENSITIVITY So long s both goods re produced, nd fctor ntensty reversls (FIRs do not occur, then ech prce vector ( p, p corresponds to unque fctor prces ( wr., Ths s remrkble result, becuse t sys tht the fctor endowments (, do not mtter for the determnton of ( wr., We cn contrst ths result wth one-sector economy, wth producton of y= f(,, wges of w= pf, nd dmnshng mrgnl product f 0. In ths cse, ny ncrese n the lbor endowments would certnly reduce wges, so tht countres wth hgher lbor/cptl endowments (/ would hve lower wges. Ths s the result we normlly expect. In contrst, the bove lemm sys tht n two-by-two economy, wth fxed product prce p, t s possble for the lbor force or cptl stock to grow wthout ffectng ther fctor prces! Thus, emer (995 refers to ths result s fctor prce nsenstvty. Our gol n ths secton s to prove the result nd lso develop the ntuton for why t holds. Two condtons must hold to obtn ths result: frst, tht both goods re produced; nd second, tht fctor ntensty reversls (FIRs do not occur. To understnd FIRs, consder fgures.5 nd.6. In the frst cse, presented n fgure.5, we hve grphed the two zero-proft condtons, nd the unt-cost lnes ntersect only once, t pont A. Ths llusttes the lemm: gven ( p, p, there s unque soluton for ( wr., But nother cse s llustrted n fgure.6, where the unt-cost lnes nteresect twce, t ponts A nd B. Then there re two possble solutons for ( wr,, nd the result stted n the lemm no longer holds. The cse where the unt-cost lnes ntersect more thn once corresponds to fctor ntensty reversls. To see where ths nme comes from, let us compute the lbor nd cptl requrements n the two ndustres. We hve lredy shown tht nd re the dervtves of the unt-cost functon wth respect to fctor prces, so t follows tht the vectors (, re the grdent vectors to the so-cost curves for the two ndustres

9 Copyrght, Prnceton Unversty Press. No prt of ths book my be r Prelmnres: Two-Sector Models 9 (, (, A p = c (w,r p = c (w,r w Fgure.5 n fgure.5. Recll from clculus tht grdent vectors pont n the drecton of the mxmum ncrese of the functon n queston. Ths mens tht they re orthogonl to ther respectve so-cost curves, s shown by (, nd (, t pont A. Ech of these vectors hs slope ( /, or the cptl-lbor rto. It s cler from fgure.5 tht (, hs smller slope thn (,, whch mens tht ndustry s cptl ntensve, or equvlently, ndustry s lbor ntensve. 7 In fgure.6, however, the stuton s more complcted. Now there re two sets of grdent vectors, whch we lbel by (, nd (, t pont A nd by (b, b nd (b, b t pont B. A close nspecton of the fgure wll revel tht ndustry s lbor ntensve ( / / t pont A, but s cptl ntensve ( b / b b/ b t pont B. Ths llustrtes fctor ntensty reversl, whereby the comprson of fctor ntenstes chnges t dfferent fctor prces. Whle FIRs mght seem lke theoretcl curosum, they re ctully qute relstc. Consder the footwer ndustry, for exmple. Whle much of the footwer n the world s produced n developng ntons, the Unted Sttes retns smll number of plnts. For snekers, New Blnce hs plnt n Norrdgewock, Mne, where employers ern bout $4 per hour. 8 Some operte computerzed equpment wth up to twenty sewng mchne heds runnng t once, whle others operte utomted sttchers guded by cmers, whch llow one person to do the work of sx. Ths s fr cry from the plnts n As tht produce shoes for Nke, Reebok, nd other U.S. producers, usng centuryold technology nd pyng less thn $ per hour. The technology used to mke snekers n As s lke tht of ndustry t pont A n fgure.5, usng lbor-ntensve 7 Alterntvely, we cn totlly dfferentte the zero-proft condtons, holdng prces fxed, to obtn 0 = dw + dr. It follows tht the slope of the so-cost curve equls dr/dw = / = /. Thus, the slope of ech so-cost curve equls the reltve demnd for the fctor on the horzontl xs, wheres the slope of the grdent vector (whch s orthogonl to the so-cost curve equls the reltve demnd for the fctor on the vertcl xs. 8 The mterl tht follows s drwn from Aron Bernsten, ow-sklled Jobs: Do They Hve to Move? Busness Week, Februry 6, 00, pp

10 0 Chpter Copyrght, Prnceton Unversty Press. No prt of ths book my be r (, r A A (, (b,b (b,b r B B p = c (w,r p = c (w,r w A w B w Fgure.6 technology nd pyng low wges w A, whle ndustry n the Unted Sttes s t pont B, pyng hgher wges w B nd usng cptl-ntensve technology. As suggested by ths dscusson, when there re two possble solutons for the fctor prces such s ponts A nd B n fgure.6, then some countres cn be t one equlbrum nd others countres t the other. How do we know whch country s where? Ths s queston tht we wll nswer t the end of the chpter, where we wll rgue tht lbor-bundnt country wll lkely be t equlbrum A of fgure.6, wth low wge nd hgh rentl on cptl, wheres cptl-bundnt country wll be t equlbrum B, wth hgh wge nd low rentl. Generlly, to determne the fctor prces n ech country we wll need to exmne ts full-employment condtons n ddton to the zero-proft condtons. et us conclude ths secton by returnng to the smple cse of no FIR, n whch the lemm stted bove pples. Wht re the mplctons of ths result for the determnton of fctor prces under free trde? To nswer ths queston, let us sketch out some of the ssumptons of the Heckscher-Ohln model, whch we wll study n more detl n the next chpter. We ssume tht there re two countres, wth dentcl technologes but dfferent fctor endowments. We contnue to ssume tht lbor nd cptl re the two fctors of producton, so tht under free trde the equlbrum condtons (.7 nd (.8 pply n ech country wth the sme product prces ( p, p. We cn drw fgure.5 for ech country, nd n the bsence of FIR, ths unquely determnes the fctor prces n ech countres. In other words, the wge nd rentl determned by fgure.5 re dentcl cross the two countres. We hve therefore proved the fctor prce equlzton (FPE theorem, whch s stted s follows. FACTOR PRICE EQUAIZATION THEOREM (SAMUESON 949 Suppose tht two countres re engged n free trde, hvng dentcl technologes but dfferent fctor endowments. If both countres produce both goods nd FIRs do not occur, then the fctor prces ( wr, re equlzed cross the countres.

11 Copyrght, Prnceton Unversty Press. No prt of ths book my be Prelmnres: Two-Sector Models The FPE theorem s remrkble result becuse t sys tht trde n goods hs the blty to equlze fctor prces: n ths sense, trde n goods s perfect substtute for trde n fctors. We cn gn contrst ths result wth tht obtned from one-sector economy n both countres. In tht cse, equlzton of the product prce through trde would certnly not equlze fctor prces: the lbor-bundnt country would be pyng lower wge. Why does ths outcome not occur when there re two sectors? The nswer s tht the lbor-bundnt country cn produce more of, nd export, the lbor-ntensve good. In tht wy t cn fully employ ts lbor whle stll pyng the sme wges s cptl-bundnt country. In the two-by-two model, the opportunty to dsproportontely produce more of one good thn the other, whle exportng the mounts not consumed t home, s wht llows fctor prce equlzton to occur. Ths ntuton wll become even clerer s we contnue to study the Heckscher-Ohln model n the next chpter. CHANGE IN PRODUCT PRICES et us move on now to the second of our key questons of the two-by-two model: f the product prces chnge, how wll the fctor prces chnge? To nswer ths, we perform comprtve sttcs on the zero-proft condtons (.7. Totlly dfferenttng these condtons, we obtn dp w dw r dr dp = dw dr & =, =,. (.9 p c w c r The second equton s obtned by multplyng nd dvdng lke terms, nd notng tht p= c( wr,. The dvntge of ths pproch s tht t llows us to express the vrbles n terms of percentge chnges, such s dln w= dw/ w, s well s cost-shres. Specfclly, let = w/ c denote the cost-shre of lbor n ndustry, whle = r / c denotes the cost-shre of cptl. The fct tht costs equl c = w r ensures tht the shres sum to unty, =. In ddton, denote the percentge chnges by dw/ w= wt nd dr/ r=t. r Then (.9 cn be re-wrtten s tp = wt r t, =,. (.9l Expressng the equtons usng these cost-shres nd percentge chnges follows Jones (965 nd s referred to s the Jones lgebr. Ths system of equtons cn be wrtten n mtrx form nd solved s tp wt wt t p d t n= d nc m& c m= d nd, p rt rt t n (.0 p where denotes the determnnt of the two-by-two mtrx on the left. Ths determnnt cn be expressed s = = ( ( (. = = where we hve repetedly mde use of the fct tht =.

12 Chpter Copyrght, Prnceton Unversty Press. No prt of ths book my be In order to fx des, let us ssume henceforth tht ndustry s lbor ntensve. Ths mples tht ts lbor cost-shre n ndustry exceeds tht n ndustry, 0, so tht 0 n (.. 9 Furthermore, suppose tht the reltve prce of good ncreses, so tht tp= tp t p 0. Then we cn solve for the chnge n fctor prces from (.0 nd (. s wt = snce tp t p 0, nd, rt = tp t p ( t p ( tp tp = t p, ( tp tp ( tp ( tp tp = t p, ( (. (.b snce tp t p 0. From the result n (., we see tht the wge ncreses by more thn the prce of good, wt tp t p. Ths mens tht workers cn fford to buy more of good (w/p hs gone up, s well s more of good (w/p hs gone up. When lbor cn buy more of both goods n ths fshon, we sy tht the rel wge hs ncresed. ookng t the rentl on cptl n (.b, we see tht the rentl r chnges by less thn the prce of good. It follows tht cptl-owner cn fford less of good (r/p hs gone down, nd lso less of good (r/p hs gone down. Thus the rel return to cptl hs fllen. We cn summrze these results wth the followng theorem. STOPER-SAMUESON (94 THEOREM An ncrese n the reltve prce of good wll ncrese the rel return to the fctor used ntensvely n tht good, nd reduce the rel return to the other fctor. To develop the ntuton for ths result, let us go bck to the dfferentted zeroproft condtons n (.9l. Snce the cost-shres dd up to unty n ech ndustry, we see from equton (.9l tht t p s weghted verge of the fctor prce chnges wt nd rt. Ths mples tht t p necessrly les n between wt nd rt. Puttng these together wth our ssumpton tht tp t p 0, t s therefore cler tht wt tp tp r. t (.3 Jones (965 hs clled ths set of nequltes the mgnfcton effect : they show tht ny chnge n the product prce hs mgnfed effect on the fctor prces. Ths s n extremely mportnt result. Whether we thnk of the product prce chnge s due to export opportuntes for country (the export prce goes up, or due to lowerng mport trffs (so the mport prce goes down, the mgnfcton effect sys tht there wll be both gners nd losers due to ths chnge. Even though we wll rgue n chpter 6 tht there re gns from trde n some overll sense, t s stll the cse tht trde opportuntes hve strong dstrbutonl consequences, mkng some people worse off nd some better off! 9 As n exercse, show tht / / +. Ths s done by multpyng the numertor nd denomntor on both sdes of the frst nequlty by lke terms, so s to convert t nto cost-shres.

13 Copyrght, Prnceton Unversty Press. No prt of ths book my be r Prelmnres: Two-Sector Models 3 A* r 0 r A B p = c (w,r p ʹ = c (w,r p = c (w,r w 0 w* w w Fgure.7 We conclude ths secton by llustrtng the Stolper-Smuelson theorem n fgure.7. We begn wth n ntl fctor prce equlbrum gven by pont A, where ndustry s lbor ntensve. An ncrese n the prce of tht ndustry wll shft out the so-cost curve, nd s llustrted, move the equlbrum to pont B. It s cler tht the wge hs gone up, from w 0 to w, nd the rentl hs declned, from r 0 to r. Cn we be sure tht the wge hs ncresed n percentge terms by more thn the reltve prce of good? The nswer s yes, s cn be seen by drwng ry from the orgn through the pont A. Becuse the unt-cost functons re homogeneous of degree one n fctor prces, movng long ths ry ncreses p nd ( wr, n the sme proporton. Thus, t the pont A, the ncrese n the wge exctly mtched the percentge chnge n the prce p. But t s cler tht the equlbrum wge ncreses by more, w w, so the percentge ncrese n the wge exceeds tht of the product prce, whch s the Stolper-Smuelson result. CHANGES IN ENDOWMENTS We turn now to the thrd key queston: f endowments chnge, how do the ndustry outputs chnge? To nswer ths, we hold the product prces fxed nd totlly dfferentte the full-employment condtons (.8 to obtn dy dy = d, dy dy = d. (.4 Notce tht the j coeffcents do not chnge, despte the fct tht they re functons of the fctor prces ( wr., These coeffcents re fxed becuse p nd p do not chnge, so from our erler lemm, the fctor prces re lso fxed. By rewrtng the equtons n (.4 usng the Jones lgebr, we obtn y y dy y dy = y y dy y dy = y y d d y y & m t m t = t m yt m yt = t. (.4l

14 4 Chpter Copyrght, Prnceton Unversty Press. No prt of ths book my be To move from the frst set of equtons to the second, we denote the percentge chnges dy/ y= t y, nd lkewse for ll the other vrbles. In ddton, we defne m / ( y / = ( /, whch mesures the frcton of the lbor force employed n ndustry, where m m=. We defne m nlogously s the frcton of the cptl stock employed n ndustry. Ths system of equtons s wrtten n mtrx form nd solved s m m yt t yt m m t = G &, m m d n t = y d n = t d n = G ty m m m d n t (.5 where m denotes the determnnt of the two-by-two mtrx on the left, whch s smplfed s m = m m m m = m ( m ( m m = m m = m m, (.6 where we hve repetedly mde use of the fct tht m m = nd m m =. Recll tht we ssumed ndustry to be lbor ntensve. Ths mples tht the shre of the lbor force employed n ndustry exceeds the shre of the cptl stock used there, m m 0, so tht m 0 n (.6. 0 Suppose further tht the endowments of lbor s ncresng, whle the endowments of cptl remns fxed such tht t 0, nd t = 0. Then we cn solve for the chnge n outputs from (.5 (.6 s m m ty = t t 0 nd t y 0. ( m m = t (.7 m From (.7, we see tht the output of the lbor-ntensve ndustry expnds, wheres the output of ndustry contrcts. We hve therefore estblshed the Rybczynsk theorem. RYBCZYNSI (955 THEOREM An ncrese n fctor endowment wll ncrese the output of the ndustry usng t ntensvely, nd decrese the output of the other ndustry. To develop the ntuton for ths result, let us wrte the full-employment condtons n vector notton s: c my y. c m = c m (.8l We hve lredy llustrted the grdent vectors (, to the so-cost curves n fgure.5 (wth not FIR. Now let us tke these vectors nd regrph them, n fgure 0 As n exercse, show tht / / / + m m nd m m.

15 Copyrght, Prnceton Unversty Press. No prt of ths book my be Prelmnres: Two-Sector Models 5 (, y (, (, y (, (, Cone A Fgure.8.8. Multplyng ech of these by the output of ther respectve ndustres, we obtn the totl lbor nd cptl demnds y (, nd y (,. Summng these s n (.8l we obtn the lbor nd cptl endowments (,. But ths exercse cn lso be performed n reverse: for ny endowment vector (,, there wll be unque vlue for the outputs (y, y such tht when (, nd (, re multpled by these mounts, they wll sum to the endowments. How cn we be sure tht the outputs obtned from (.8l re postve? It s cler from fgure.8 tht the outputs n both ndustres wll be postve f nd only f the endowment vector (, les n between the fctor requrement vectors (, nd (,. For ths reson, the spce spnned by these two vectors s clled cone of dversfcton, whch we lbel by cone A n fgure.8. In contrst, f the endowment vector (, les outsde of ths cone, then t s mpossble to dd together ny postve multples of the vectors (, nd (, nd rrve t the endowment vector. So f (, les outsde of the cone of dversfcton, then t must be tht only one good s produced. At the end of the chpter, we wll show how to determne whch good t s. For now, we should just recognze tht when only one good s produced, the fctor prces re determned by the mrgnl products of lbor nd cptl s n the onesector model, nd wll certnly depend on the fctor endowments. Now suppose tht the lbor endowment ncreses to l, wth no chnge n the cptl endowment, s shown n fgure.9. Strtng from the endowments (l,, the only wy to dd up multples of (, nd (, nd obtn the endowments s to reduce the output of ndustry to y l, nd ncrese the output of ndustry to y l. Ths mens tht not only does ndustry bsorb the entre mount of the extr lbor See problem.5.

16 6 Chpter Copyrght, Prnceton Unversty Press. No prt of ths book my be (, (ʹ, y (, y ʹ (, (, y ʹ (, y (, (, Fgure.9 endowment, but t lso bsorbs further lbor nd cptl from ndustry so tht ts ultmte lbor/cptl rto s unchnged from before. The lbor/cptl rto n ndustry s lso unchnged, nd ths s wht permts both ndustres to py exctly the sme fctor prces s they dd before the chnge n endowments. There re mny exmples of the Rybczynsk theorem n prctce, but perhps the most commonly cted s wht s clled the Dutch Dsese. Ths refers to the dscovery of ol off the cost of the Netherlnds, whch led to n ncrese n ndustres mkng use of ths resource. (Shell Ol, one of the world s lrgest producers of petroleum products, s Dutch compny. At the sme tme, however, other trdtonl export ndustres of the Netherlnds contrcted. Ths occurred becuse resources were ttrcted wy from these ndustres nd nto those tht were ntensve n ol, s the Rybczynsk theorem would predct. We hve now nswered the three questons rsed erler n the chpter: how re fctor prces determned; how do chnges n product prces ffect fctor prces; nd how do chnges n endowments ffect outputs? But n nswerng ll of these, we hve reled on the ssumptons tht both goods re produced, nd lso tht fctor ntensty reversls do not occur, s ws stted explctly n the FPE theorem. In the remnder of ths chpter we need to nvestgte both of these ssumptons, to understnd ether when they wll hold or the consequences of ther not holdng. We begn by trcng through the chnges n the outputs nduced by chnges n endowments, long the equlbrum of the producton possblty fronter. As the lbor endowment grows n fgure.9, the PPF wll shft out. Ths s shown n fgure.0, where the outputs wll shft from pont A to pont Al wth n ncrese of good nd reducton of good, t the unchnged prce p. As the endowment of lbor rses, we See, for exmple, Corden nd Nery (98 nd Jones, Nery, nd Rune (987.

17 Copyrght, Prnceton Unversty Press. No prt of ths book my be Prelmnres: Two-Sector Models 7 y Rybczynsk ne for Rybczynsk ne for A Aʹ Slope = p y Fgure.0 cn jon up ll ponts such s A nd Al where the slopes of the PPFs re equl. These form downwrd-slopng lne, whch we wll cll the Rybczynsk lne for chnges n lbor (D. The Rybczynsk lne for D ndctes how outputs chnge s lbor endowment expnds. Of course, there s lso Rybczynsk lne for D, whch ndctes how the outputs chnge s the cptl endowment grows: ths would led to n ncrese n the output of good, nd reducton n the output of good. As drwn, both of the Rybczynsk lnes re llustrted s strght lnes: cn we be sure tht ths s the cse? The nswer s yes: the fct tht the product prces re fxed long Rybczynsk lne, mplyng tht fctor prces re lso fxed, ensures tht these re strght lnes. To see ths, we cn esly clculte ther slopes by dfferenttng the full-employment condtons (.8. To compute the slope of the Rybczynsk lne for D, t s convenent to work wth the full-employment condton for cptl, snce tht endowment does not chnge. Totl dfferenttng (.8 for cptl gves dy y y = & dy dy = 0 & =. (.8 dy Thus, the slope of the Rybczynsk lne for D s the negtve of the rto of cptl/ output n the two ndustres, whch s constnt for fxed prces. Ths proves tht the Rybczynsk lnes re ndeed strght. If we contnue to ncrese the lbor endowment, outputs wll move downwrd on the Rybczynsk lne for D n fgure.0, untl ths lne hts the y xs. At ths pont the economy s fully speclzed n good. In terms of fgure.9, the vector of endowments (, s concdent wth the vector of fctor requrements (, n ndustry. For further ncreses n the lbor endowment, the Rybczynsk lne for D then moves rght long the y xs n fgure.0, ndctng tht the economy remns speclzed n good. Ths corresponds to the vector of endowments (,

18 8 Chpter Copyrght, Prnceton Unversty Press. No prt of ths book my be lyng outsde nd below the cone of dversfcton n fgure.9. Wth the economy fully speclzed n good, fctor prces re determned by the mrgnl products of lbor nd cptl n tht good, nd the erler fctor prce nsenstvty lemm no longer pples. FACTOR PRICE EQUAIZATION REVISITED Our fndng tht the economy produces both goods whenever the fctor endowments remn nsde the cone of dversfcton llows us to nvestgte the FPE theorem more crefully. et us contnue to ssume tht there re no FIRs, but now rther thn ssumng tht both goods re produced n both countres, we wll nsted derve ths s n outcome from the fctor endowments n ech country. To do so, we engge n thought experment posed by Smuelson (949 nd further developed by Dxt nd Normn (980. Intlly, suppose tht lbor nd cptl re free to move between the two countres untl ther fctor prces re equlzed. Then ll tht mtters for fctor prces re the world endowments of lbor nd cptl, nd these re shown s the length of the horzontl nd vertcl xs n fgure.. The mounts of lbor nd cptl choosng to resde t home re mesured reltve to the orgn 0, whle the mounts choosng to resde n the foregn country re mesured reltve to the orgn 0 ; suppose tht ths llocton s t pont B. Gven the world endowments, we estblsh equlbrum prces for goods nd fctors n ths ntegrted world equlbrum. The fctor prces determne the demnd for lbor nd cptl n ech ndustry (ssumng no FIR, nd usng B* * O* A * FPE Set B B* B B Bʹ A O Fgure.

19 Copyrght, Prnceton Unversty Press. No prt of ths book my be Prelmnres: Two-Sector Models 9 these, we cn construct the dversfcton cone (snce fctor prces re the sme cross countres, then the dversfcton cone s lso the sme. et us plot the dversfcton cone reltve to the home orgn 0, nd gn reltve to the foregn orgn 0. These cones form the prllelogrm 0A 0 A. For lter purposes, t s useful to dentfy precsely the ponts A nd A on the vertces of ths prllelogrm. The vectors 0A nd 0 A re proportonl to (,, the mount of lbor nd cptl used to produce one unt of good n ech country. Multplyng (, by world demnd for good, D w, we then obtn the totl lbor w nd cptl used to produce tht good, so tht A= (, D. Summng these gves the totl lbor nd cptl used n world demnd, whch equls the lbor nd cptl used n world producton, or world endowments. Now we sk whether we cn cheve exctly the sme world producton nd equlbrum prces s n ths ntegrted world equlbrum, but wthout lbor nd cptl moblty. Suppose there s some llocton of lbor nd cptl endowments cross the countres, such s pont B. Then cn we produce the sme mount of ech good s n the ntegrted world equlbrum? The nswer s clerly yes: wth lbor nd cptl n ech country t pont B, we could devote 0B of resources to good nd 0B to good t home, whle devotng 0 B to good nd 0 B towrd good brod. Ths wll ensure tht the sme mount of lbor nd cptl worldwde s devoted to ech good s n the ntegrted world equlbrum, so tht producton nd equlbrum prces must be the sme s before. Thus, we hve cheved the sme equlbrum but wthout fctor moblty. It wll become cler n the next chpter tht there s stll trde n goods gong on to stsfy the demnds n ech country. More generlly, for ny llocton of lbor nd cptl wthn the prllelogrm 0A 0 A both countres remn dversfed (producng both goods, nd we cn cheve the sme equlbrum prces s n the ntegrted world economy. It follows tht fctor prces remn equlzed cross countres for lloctons of lbor nd cptl wthn the prllelogrm 0A 0 A, whch s referred to s the fctor prce equlzton (FPE set. The FPE set llustrtes the rnge of lbor nd cptl endowments between countres over whch both goods re produced n both countres, so tht fctor prce equlzton s obtned. In contrst, for endowments outsde of the FPE set such s pont Bl, then t lest one country would hve to be fully speclzed n one good nd FPE no longer holds. FACTOR INTENSITY REVERSAS We conclude ths chpter by returnng to queston rsed erler: when there re fctor ntensty reversls gvng multple solutons to the zero-proft condtons, how do we know whch soluton wll prevl n ech country? To nswer ths, t s necessry to combne the zero-proft wth the full-employment condtons, s follows. Consder the cse n fgure.6, where the zero-proft condtons llows for two solutons to the fctor prces. Ech of these determne the lbor nd cptl demnds shown orthogonl to the so-cost curves, lbeled s (, nd (,, nd (b, b nd (b, b. We hve redrwn these n fgure., fter multplyng ech of them by the outputs of ther respectve ndustres. These vectors crete two cones of dversfcton, lbeled s cones A nd B. Intlly, suppose tht the fctor endowments for ech country le wthn one cone or the other (then we wll consder the cse where the endowments re outsde both cones.

20 0 Chpter Copyrght, Prnceton Unversty Press. No prt of ths book my be ( B, B y (b,b ( A, A y (b,b y (, Cone B y (, Cone A Fgure. Now we cn nswer the queston of whch fctor prces wll pply n ech country: lbor-bundnt economy, wth hgh rto of lbor/cptl endowments, such s A (, A n cone A of fgure., wll hve fctor prces gven by (w A, r A n fgure.6, wth low wges; wheres cptl-bundnt economy wth hgh rto of cptl/lbor B endowments such s shown by (, B n cone B of fgure., wll hve fctor prces gven by (w B, r B n fgure.6, wth hgh wges. Thus, fctor prces depend on the endowments of the economy. A lbor-bundnt country such s Chn wll py low wges nd hgh rentl (s n cone A, whle cptl-bundnt country such s the Unted Sttes wll hve hgh wges nd low rentl (s n cone B. Notce tht we hve now rentroduced lnk between fctor endowments nd fctor prces, s we rgued erler n the one-sector model: when there re FIR n the two-by-two model, fctor prces vry systemtclly wth endowments cross the cones of dversfcton, even though fctor prces re ndependent of endowments wthn ech cone. Wht f the endowment vector of country does not le n ether cone? Then the country wll be fully speclzed n one good or the other. Generlly, we cn determne whch good t s by trcng through how the outputs chnge s we move through the cones of dversfcton, nd t turns out tht outputs depend non-monotonclly on the fctor endowments. 3 For exmple, textles n South ore or Twn expnded durng the 960s nd 970s, but contrcted lter s cptl contnued to grow. Despte the complexty nvolved, mny trde economsts feel tht countres do n fct produce n dfferent cones of dversfcton, nd tkng ths possblty nto ccount s topc of reserch. 4 3 See problem.5. 4 Emprcl evdence on whether developed countres ft nto the sme cone s presented by Debere nd Demroğlu (003, nd the presence of multple cones s explored by emer (987, Hrrgn nd Zkrjšek (000, nd Schott (003.

21 CONCUSIONS Copyrght, Prnceton Unversty Press. No prt of ths book my be Prelmnres: Two-Sector Models In ths chpter we hve revewed severl two-sector models: the Rcrdn model, wth just one fctor, nd the two-by-two model, wth two fctors, both of whch re fully moble between ndustres. There re other two-sector models, of course: f we dd thrd fctor, tretng cptl s specfc to ech sector but lbor s moble, then we obtn the Rcrdo-Vner or specfc-fctors model, s wll be dscussed n chpter 3. We wll hve n opportunty to mke use of the two-by-two model throughout ths book, nd thorough understndng of ts propertes both the equtons nd the dgrms lbor-bundnt economy s essentl for ll the mterl tht follows. One specl feture of ths chpter s the dul determnton of fctor prces, usng the unt-cost functon n the two ndustres. Ths follows the dul pproch of Woodlnd (977, 98, Muss (979, nd Dxt nd Normn (980. Smuelson (949 uses qute dfferent dgrmtc pproch to prove the FPE theorem. Another method tht s qute commonly used s the so-clled erner (95 dgrm, whch reles on the producton rther thn cost functons. 5 We wll not use the erner dgrm n ths book, but t wll be useful to understnd some rtcles, for exmple, Fndly nd Grubert (959 nd Derdorff (979, so we nclude dscusson of t n the ppendx to ths chpter. Ths s the only chpter where we do not present ny ccompnyng emprcl evdence. The reder should not nfer from ths tht the two-by-two model s unrelstc: whle t s usully necessry to dd more goods or fctors to ths model before confrontng t wth dt, the reltonshps between prces, outputs, nd endowments tht we hve dentfed n ths chpter wll crry over n some form to more generl settngs. Evdence on the pttern of trde s presented n the next chpter, where we extend the two-by-two model by ddng nother country, nd then mny countres, trdng wth ech other. We lso llow for mny goods nd fctors, but for the most prt restrct ttenton to stutons where fctor prce equlzton holds. In chpter 3, we exmne the cse of mny goods nd fctors n greter detl, to determne whether the Stolper-Smuelson nd Rybczynsk theorems generlze nd lso how to estmte these effects. In chpter 4, evdence on the reltonshp between product prces nd wges s exmned n detl, usng model tht llows for trde n ntermedte nputs. The reder s lredy well prepred for the chpters tht follow, bsed on the tools nd ntuton we hve developed from the two-by-two model. Before movng on, you re encourged to complete the problems t the end of ths chpter. APPENDIX: THE ERNER DIAGRAM AND FACTOR PRICES The erner (95 dgrm for the two-by-two model cn be explned s follows: Wth perfect competton nd constnt returns to scle, we hve tht revenue = costs n both ndustres. So let us choose specl soqunt n ech ndustry such tht revenue =. In ech ndustry, we therefore choose the soqunt py =, or Y = f(, = / p & w r =. 5 Ths dgrm ws used n semnr presented by Abb erner t the ondon School of Economcs n 933, but not publshed untl 95. The hstory of ths dgrm s descrbed t the Orgns of Terms n Interntonl Economcs, mntned by Aln Derdorff t /glossry/org.html. See lso Smuelson (949, 8 n..

22 Chpter Copyrght, Prnceton Unversty Press. No prt of ths book my be r (, (, f (, = p Cone of Dversfcton w + r = f (, = p w Fgure.3 Therefore, from cost mnmzton, the /p soqunt n ech ndustry wll be tngent to the lne w r=. Ths s the sme lne for both ndustres, s shown n fgure.3. Drwng the rys from the orgn through the ponts of tngency, we obtn the cone of dversfcton, s lbeled n fgure.3. Furthermore, we cn determne the fctor prces by computng where w r= ntersects the two xs: = 0& = / r, nd = 0& = / w. Therefore, gven the prces p, we determne the two soqunts n fgure.3, nd drwng the (unque lne tngent to both of these, we determne the fctor prces s the ntercepts of ths lne. Notce tht these equlbrum fctor prces do not depend on the fctor endowments, provded tht the endowment vector les wthn the cone of dversfcton (so tht both goods re produced. We hve thus obtned n lterntve proof of the fctor prce nsenstvty lemm, usng prml rther thn dul pproch. Furthermore, wth two countres hvng the sme prces (through free trde nd technologes, then fgure.3 holds n both of them. Therefore, ther fctor prces wll be equlzed. erner (95 lso showed how fgure.3 cn be extended to the cse of fctor ntensty reversls, n whch cse the soqunts ntersect twce. In tht cse there wll be two lnes w r= tht re tngent to both soqunts, nd there re two cones of dversfcton. Ths s shown n fgure.4. To determne whch fctor prces pply n prtculr country, we plot ts endowments vector nd note whch cone of dversfcton t les n: the fctor prces n ths country re those pplyng to tht cone. For exmple, the endowments ( A, A wll hve the fctor prces (w A, r A, nd the endowments ( B, B wll hve the fctor prces (w B, r B. Notce tht the lbor-bundnt country wth endowments ( A, A hs the low wge nd hgh rentl, wheres the cptl-bundnt country wth endowments ( B, B hs the hgh wge nd low rentl. How lkely s t tht the soqunts of ndustres nd ntersect twce, s n fgure.4? erner (95, correctly suggested tht t depends on the elstcty of substtuton between lbor nd cptl n ech ndustry. For smplcty, suppose tht ech

23 Copyrght, Prnceton Unversty Press. No prt of ths book my be Prelmnres: Two-Sector Models 3 r B ( B, B ( A, A r A f (, = p f (, = p w B w A Fgure.4 ndustry hs constnt elstcty of substtuton producton functon. If the elstctes re the sme cross ndustres, then t s mpossble for the soqunts to ntersect twce. If the elstctes of substtuton dffer cross ndustres, however, nd we choose prces p, =,, such tht the /p soqunts ntersect t lest once, then t s gurnteed tht they ntersect twce. Under exctly the sme condtons, the so-cost lnes n fgure.6 ntersect twce. Thus, the occurrence of FIR s very lkely once we llow elstctes of substtuton to dffer cross ndustres. Mnhs (96 confrmed tht ths ws the cse emprclly, nd dscussed the mplctons of FIR for fctor prces nd trde ptterns. Ths lne of emprcl reserch ws dropped therefter, perhps becuse FIR seemed too complex to del wth, nd hs been pcked up gn more recently (see note 4 of ths chpter. PROBEMS. Rewrte the producton functon y= f(, s y= f( v, nd smlrly, b b y= f( v. Concvty mens tht gven two ponts y= f( v nd y= f( v, nd 0# m #, then f ( v ( v b y ( y b m m $ m m. Smlrly for the producton functon y= f( v. Consder two ponts y = ( y, y nd y b = ( y b, y b, both of whch cn be produced whle stsfyng the full- employment condtons v v # V nd v b v b # V, where V represents the endowments. Consder producton pont mdwy between these, my ( m y. Then use b the concvty of the producton functons to show tht ths pont cn lso be produced whle stsfyng the full-employment condtons. Ths proves tht the producton possbltes set s convex. (Hnt: Rther thn showng b tht my ( m y cn be produced whle stsfyng the full-employment

24 4 Chpter Copyrght, Prnceton Unversty Press. No prt of ths book my be b condtons, consder nsted lloctng mv ( m v of the resources to ndustry, nd mv ( m v of the resources to ndustry. b. Any functon y= f( v s homogeneous of degree f for ll m 0, f( m v = m fv (. Consder the producton functon y= f(,, whch we ssume s homogeneous of degree one, so tht f( m, m = mf(,. Now dfferentte ths expresson wth respect to, nd nswer the followng: Is the mrgnl product f (, homogeneous, nd of wht degree? Use the expresson you hve obtned to show tht f( /, = f(,..3 Consder the problem of mxmzng y= f(,, subject to the full-employment condtons # nd #, nd the constrnt y = f(,. Set ths up s grngn, nd obtn the frst-order condtons. Then use the grngn to solve for dy /dy, whch s the slope of the producton possbltes fronter. How s ths slope relted to the mrgnl product of lbor nd cptl?.4 Consder the problem of mxmzng p f(, p f(,, subject to the full-employment constrnts # nd #. Cll the result the GDP functon G( p,,, where p= ( p, p s the prce vector. Then nswer the followng: ( Wht s G/ p? (Hnt: we solved for ths n the chpter. (b Gve n economc nterpretton to G/ nd G/. (c Gve n economc nterpretton to G/ p= G/ p, nd G/ p = G/ p..5 Trce through chnges n outputs when there re fctor ntensty reversls. Tht s, construct grph wth the cptl endowment on the horzontl xs, nd the output of goods nd on the vertcl xs. Strtng t pont of dversfcton (where both goods re produced n cone A of fgure., drw the chnges n output of goods nd s the cptl endowment grows outsde of cone A, nto cone B, nd beyond ths.