Kiinori Matyaa, Engel aw in te Global Econoy Engel aw in te Global Econoy: Deand-Indced Pattern of Strctral Cange, Innovation, and Trade* By Kiinori Matyaa** Marc, 8 Abtract: Endogeno deand copoition acro ector de to nonootetic deand, or Engel aw for te brevity, affect i ectoral copoition in eployent and in vale-added, ii variation in innovation rate and in prodctivity cange acro ector, iii interectoral pattern of trade acro contrie; and iv igration of indtrie fro ric to poor contrie. Uing a two-contry odel of directed tecnical cange wit a contin of ector nder nonootetic preference, wic i ric enog to captre all tee effect a well a teir interaction, ti paper offer a nifying perpective on ow econoic growt and globalization affect te pattern of trctral cange, innovation and trade acro contrie and acro ector in te preence of Engel aw. Aong te ain eage i tat globalization aplifie, intead of redcing, te power of endogeno doetic deand copoition difference a a driver of trctral cange. Keyword: Ioelatically nonootetic CES, Iplicit additivity, Diit-Stiglitz-Krgan odel of prodction and trade, Scooler effect, Directed tecnical cange, Hoe aret effect in eployent and pattern of trade, inder effect, Vernon prodct cycle, Ter-of-trade cange, Factor price convergence, eapfrogging, Trade pattern reveral, og-perodlarity, Monotone lieliood ratio, Monotone coparative tatic JE Claification: F6 Ipact of globalization: acroeconoic, F63 Ipact of globalization: econoic developent, O Macroeconoic apect of econoic developent, O9 International linage to developent, O33 Tecnological cange *Ti paper draw and epand te aterial fro Section of Matyaa 5 wit additional analye and new interpretation. A c, it benefit fro te advice and coent received fro any people, not only on an earlier verion of ti paper bt alo on tat paper. Tey inclde alpabetically P. Antra, C.Arolai, S.Atey, G.Barlevy, K.Beren, R.Benabo, F.Bera, A.Cotinot, D.Davi, J.Dingel, M.Doepe, D.Donaldon, P.Fagelba, G.Gancia, C.Gabert, O.Galor, M.Goloov, G.Groan, E.Helpan, J.Kaboi, D.aari, F.ippi, R.ca, EGJ.tter, J.Maren, M.Metieri, Y.Mrata, E.Oberfield, M.Peter, S.Redding, A.Rodrigez-Clare, E.Roi-Hanberg, T.Sapon, R.Staiger, N.Stoey, B.Strlovici, P.Ucev, J.Ventra, X.Vive, J.Vogel, D.Weintein, I.Werning, K-M.Yi, J.Zang, F.Zilibotti, J.Zweieller, a well a worop participant at cronologically Cicago Fed Trade Worop, SE, NBER Ser Intitte Incoe Ditribtion & Macro, Yale, RIETI, Hitotbai conference on trade and FDI 5, Toyo, Keio, CREI, Atonoa, MIT, NES, MSSE-HSE, Colbia, were Matyaa 5 wa preented; and toe at Hitotbai conference on trade and FDI 6, Cicago Fed Developent Worop, UISS, 7 IOSE Conference, NBER Ser Intitte International Trade & Invetent, and NBER Econoic Growt Progra, were an earlier verion of ti paper wa preented. Te al diclaier applie. **Hoepage: ttp:faclty.wca.nortwetern.ed~at; Eail: -atyaa@nortwetern.ed.
Kiinori Matyaa, Engel aw in te Global Econoy. Introdction Wit incoe elaticity difference acro ector, te ependitre are are ore ewed toward iger incoe elatic ector in ricer contrie. A diced in te literatre review, any arged tat c an endogeno deand copoition de to te nonooteticity of deand acro ector, wic we all call Engel aw for te brevity, i an iportant cannel trog wic econoic growt and globalization affect te pattern of trctral cange, innovation, and trade acro contrie and acro ector. For eaple, Engel aw play a central role in acconting for canging ectoral are in eployent and in vale-added. It cold alo affect relative prodctivity growt acro ector, ince te aret ize i a crcial factor in providing incentive for innovation, a pointed ot by Scooler 966 and any oter. It alo affect te interectoral pattern of trade between ric and poor contrie. inder 96 arged tat te difference in te deand copoition acro ric and poor contrie cae te ric poor to develop coparative advantage in te iger lower incoe-elatic ector, wile relying on iporting ore fro te poor ric in te lower iger incoe elatic ector. It cold alo play a crcial role in deterining te igration pattern of indtrie fro ric to poor contrie. Vernon 966, in particlar, propoed te prodct cycle ypotei; indtrie tat prodce incoe-elatic good are firt etablied in ig-incoe contrie, were tey find c of teir deand, and ten igrate to low incoe contrie, a te world econoy grow. Altog oe of tee effect ave been a bect of previo tdie, tey ave been otly treated eparately. Ti cold be ileading, a tee effect are interconnected. For eaple, any tdie in te trctral tranforation literatre a to wat etent te canging pattern of ectoral are in eployent and in vale-added can be acconted for by te deand nonooteticity or by prodctivity growt differential acro ector, nder te often iplicit aption tat prodctivity cange in eac ector i eogeno. Sc an incoe elaticity ver prodctivity growt differential approac i a fale dicotoy in te preence of te Scooler effect, becae te relative prodctivity cange acro ector repond endogenoly to cange in te relative aret ize caed by econoic growt de to te Engel aw. Frterore, te eiting tdie of trctral cange typically e cloed econoy odel, were te doetic pply i necearily eqal to te doetic deand ector-
Kiinori Matyaa, Engel aw in te Global Econoy by-ector. Since te doetic pply copoition doe not need be eqal to te doetic deand copoition in an open econoy, one igt tin, intitively, tat international trade wold ae te doetic deand copoition le iportant a a driver of trctral cange. However, ti i far fro obvio. Firt, international trade wold generate prodctivity gain. Te relting incoe effect wold cae a frter ectoral ift in te ependitre trog Engel aw. Second, in te preence of te inder effect, ric poor contrie wold allocate even ore reorce in iger lower incoe-elatic ector nder trade tan nder atary, wic ean tat te doetic deand copoition wold ave ore tan proportional effect on te doetic pply copoition. Frterore, igration of indtrie fro te ric to te poor contrie wold enable bot contrie to acieve ectoral ift toward ore incoeelatic, if toe indtrie tat igrate fro te ric to te poor are le ore incoe-elatic tan toe operating in te ric poor, a Vernon arged. Ten, prodct cycle old be regarded a an integral part of te interdependent pattern of trctral cange acro te ric and poor contrie. For all tee reaon, globalization cold aplify, intead of redcing, te power of Engel aw and te endogeno doetic deand copoition difference acro contrie a a driver of trctral cange in te global econoy. Te goal of ti paper i to offer a nifying perpective on ow econoic growt and globalization affect te interdependent pattern of trctral cange, innovation, and trade acro contrie and acro ector in te preence of Engel aw. Wit ti goal in ind, it develop a two-contry tatic odel of directed tecnological cange wit a contin of ector nder nonootetic preference, wic i ric enog to captre all tee effect of Engel aw a well a teir interaction. At te ae tie, it deliberately abtract fro all oter factor to iolate te effect of Engel aw. Here i a roadap of te paper. Section introdce ti odel and derive it eqilibri condition. Te odel a a ingle nontradeable factor of prodction, labor, and two contrie, wic differ in te poplation ize, N, and labor prodctivity,, wic alo ean tat tey differ in ize, eared in te total effective labor pply, = N. Tere i a contin of nontradeable conption good ector, indeed by I, were I i an open real interval, and preference over tee good are ioelatically nonootetic wit contant elaticity of btittion CES. A eplained in ore detail in Appendi A, ti cla of preference, wic 3
Kiinori Matyaa, Engel aw in te Global Econoy atifie bot direct and indirect iplicit additivity, a everal featre tat ae it niqely well-ited for or prpoe. Firt, it allow for any nber of ector wit ector-pecific incoe elatic paraeter,, wile eeping te contant elaticity of btittion acro ector,, a a eparate paraeter. Ti ae it poible to control for te incoe elaticity difference witot affecting te price elaticity, wic elp to iolate te role of incoe elaticity difference. Second, te CES paraeter,, can be eiter greater tan one te cae of gro btitte or le tan one te cae of gro copleent. Tird, being a CES, it retain c of te tractability of te tandard CES, in pite of te nonooteticity. Frterore, wit teir incoe elaticity paraeter being te only fndaental eterogeneity acro ector, te ector can be indeed c tat ector-pecific incoe elaticity i increaing in I. Ti iplie tat te weigt attaced on eac good in or nonootetic CES tility fnction i log-perodlar in I and in te per capita real ependitre and incoe. Ti in trn iplie tat, olding price given, a iger per capita real incoe ift te denity of te ependitre are toward iger-indeed good in te ene of te onotone lieliood ratio MR, wic alo iplie tat it clative ditribtion fnction ift to te rigt in te ene of te firt-order tocatic doinance FSD. 3 On te prodction and trade ide, we deliberately e te tandard onopolitic copetition odel of trade de to Diit- Stiglitz 977: Section I and Krgan 98, in order to iolate te role of Engel aw. Eac nontradeable conption good i prodced by a copetitive ector, wic aeble tradable differentiated interediate inpt, ing te CES aggregator. Eac differentiated interediate In te original forlation of ti cla of preference by Hanoc 975; ec..5ii and ec.3.5ii, a well a in it recent application by Coin, aari, and Metieri 5, te et of conption good i aed to be finite. Here, we conider te cae were te et of conption good, I, i an open real interval tat i, it i a totally ordered et wit a contin of eleent, a it facilitate te caracterization of te eqilibri and coparative tatic, a in Dornbc, Ficer, and Saelon 98 and Cotinot and Vogel, 5. Te ability to deal wit te cae of gro copleentarity acro ector, η <, i eential in or contet, becae te epirical etiate of Engel crve are in te range of η.6.8; ee Herrendorf, Rogeron, and Valentinyi 4 and Coin, aari, and Metieri 5. In contrat, nonootetic preference tat rely on oe notion of vertical or qality differentiation acro good witin a ector, ed by Fla and Helpan 987, Stoey 99, or Fagelba, Groan, and Helpan, necearily iply tat good are gro btitte, and ence not well-ited for tdying Engel aw, i.e., nonooteticity of deand acro ector tat prodce copleentary good. At te ae tie, te ability to deal alo wit te cae of gro btitte, η >, igt ae or cla of preference potentially efl a a redced for way of captring te nonooteticity of deand acro good wit different qality level witin a ector. 3 See Atey and Vive 999; C..7 for log-perodlarity and onotone coparative tatic. Cotinot 9 and Cotinot and Vogel, 5 are aong te firt to apply te in international trade. 4
Kiinori Matyaa, Engel aw in te Global Econoy inpt i pplied by a onopolitically copetitive fir, ing labor for bot prodction and entry or innovation. Tee differentiated inpt are tradable, bect to te iceberg trade cot, a in Krgan 98. One ey featre of ti etp i tat prodctivity of eac conption good ector in eac contry i endogenoly deterined, a it depend on te availability of differentiated inpt, wic cange trog entryeit of onopolitically copetitive fir a well a trog trade. 4 Section 3 loo at te cloed econoy eqilibri. An increae in or N iprove welfare, wic can be eared by te per capita real incoe. Ti ift te relative aret ize toward te iger-indeed ector de to Engel aw, caing a proportional cange in te eployent are acro ector. Ti cange in te relative aret ize alo lead to oe entrie of inpt prodcer to te iger-indeed ector a well a to oe eit fro te lowerindeed ector. Te relting cange in te relative prodctivity acro ector te Scooler effect ae te iger-indeed good relatively ceaper, wic oderate aplifie te ectoral ift wen te conption good are gro copleent gro btitte. Section 4 caracterize te cro-contry variation in a trade eqilibri. Te wage rate per efficiency nit i lower in te contry aller in ize at any poitive trade cot, becae te aller econoy a diadvantage of aving te aller doetic aret. In contrat, te contry raning i.e., wic contry i ricer eared by te per capita real incoe i endogenoly deterined. Wen te trade cot i fficiently ig, te contry wit iger labor prodctivity cold be poorer if it i fficiently aller in ize, becae of it diadvantage of being aller. Bt it i ricer wen te trade cot i fficiently low, wic ae ti diadvantage fficiently all. At any given eqilibri, te doetic deand copoition i ore le ewed toward te iger-indeed ector in te contry tat i ricer poorer at tat eqilibri, de to Engel aw. Wit a poitive trade cot, ti cro-contry difference in 4 Tere i an alternative interpretation of or odel, a ed in Matyaa 5, Section. Coner ave ioelatically nonootetic CES preference over a contin of conption categorie. And te tility of coning eac category i given by a Diit-Stiglitz CES aggregator of a variety of tradeable differentiated coner prodct. And eac tradeable differentiated coner prodct i pplied by a onopolitic copetitive fir and bect to te iceberg trade cot. Here, by following Etier 98 and Roer 99, we intead interpret te tradeable differentiated prodct a interediate inpt to te prodction of nontradeable conption good for two reaon. Firt, it allow to tal abot endogeno ectoral prodctivity, wic ae it eaier to dic it iplication on te Scooler effect and trctral cange. Second, c of global trade in anfactring conit of interediate inpt, not conption good; ee, for eaple, Antra 5. 5
Kiinori Matyaa, Engel aw in te Global Econoy te doetic aret copoition cae relatively ore inpt prodcer to operate in te igerlower-indeed ector in te ricerpoorer contry. A a relt, te ricer contry a relatively iger prodctivity in iger incoe-elatic ector te Scooler effect and allocate diproportionately ore labor in toe ector te Hoe Maret Effect in eployent. Ti diproportionate effect on te cro-contry difference in te doetic deand copoition alo ow p in te inter-ectoral pattern of intra-ectoral trade. Altog tere are two-way flow of differentiated inpt witin eac ector, tere i a niqe ctoff ector, c I, c tat te ricer contry rn a trade rpl in te ector above te ctoff and te poorer rn a trade rpl in te ector below te ct-off. T, te ricer poorer becoe a net eporter in te iger lower incoe-elatic ector, becae it doetic deand copoition i ore ewed toward iger lower incoe-elatic te inder effect. 5 Section 5 condct oe coparative tatic of te trade eqilibri. Section 5. loo at labor prodctivity growt tat i nifor acro contrie. Sc a cange doe not affect te ter-of-trade te relative wage nor te raning of te two contrie. It ift bot te ependitre and eployent are toward iger-indeed ector in bot contrie. It alo ift te ctoff-ector, c I. T, te ricer contry witce fro a net eporter to a net iporter in oe iddle ector, generating oeting ain to Vernon prodct cycle. Te intition beind ti relt i eay to grap. A bot contrie becoe ricer and ift teir ependitre toward iger-indeed ector, te weigt of te iger indeed ector, in wic te ricer contry rn a rpl, becoe iger. In order to eep te overall trade accont between te two contrie in balance, te ectoral trade accont of te ricer contry t deteriorate in eac ector. Ti i wy it ectoral trade balance witc fro being poitive to negative in oe iddle ector. Frterore, igrating ector in te iddle range fro te 5 Note tat it i not te relative contry ize bt te relative per capita real incoe tat deterine te direction of te pattern of interectoral trade in ti odel. Te relative contry ize doe atter bt only indirectly trog it effect on te relative per capita real incoe. For eaple, iagine tat one contry, ay Switzerland, i c aller bt it coner enoy iger per capita real incoe tan toe living in te contry tat i c larger, ay Cina. Or odel predict tat Switzerland i a net-eporter in ig incoe elatic ector, even tog te Cinee doetic aret igt be larger tan te Swi doetic aret in all ector, inclding ig incoe elatic ector. Ti i becae te inder effect or Hoe Maret effect in ti odel i de to te difference in te doetic deand copoition, a in Krgan 98, and not in te abolte ize of te doetic deand in eac ector, a in Helpan and Krgan 985, C..4. 6
Kiinori Matyaa, Engel aw in te Global Econoy ricer to te poor contrie cae te ectoral are in eployent to ift toward igerindeed ector in bot contrie. How welfare gain fro c a cange are ditribted acro te two contrie depend on te elaticity of btittion acro ector. By increaing te relative aret ize of ig-indeed ector and ence by redcing te relative price of toe ector in wic te ricer contry a coparative advantage trog te Scooler effect, a nifor labor prodctivity growt cae te welfare gap between te two contrie, a eared by te ratio of te per capita real incoe, to narrow widen, wen te conption good prodced in different ector are gro copleent btitte. Te effect of globalization, captred by a redction in te iceberg trade cot nifor acro ector, are iilar to nifor labor prodctivity growt, ecept tere are additional ter of trade effect wen te two contrie differ in ize, eiter eared in te total effective labor pply or in GDP. Wen te two contrie are eqal in ize Section 5., te wage rate are alway eqalized acro te contrie and ence te ter of trade are not affected by a redction in te trade cot. Ti ean tat te contry wit iger labor prodctivity a iger per capita real incoe in ti cae. And witot caing any ter of trade cange, te effect of globalization are ioorpic to toe of nifor labor prodctivity growt. Te intition i, again, eay to grap. A lower trade cot allow bot contrie to ave better acce to te differentiated inpt prodced abroad, wic generate prodctivity gain ioorpic to labor prodctivity growt. Ti incoe effect of prodctivity gain fro trade cae bot contrie to ift teir ependitre toward iger-indeed ector, and te ricer te poorer to witc fro a net eporter iporter to a net iporter eporter in oe iddle ector, generating prodct cycle, depite tat te decline in te trade cot i nifor acro ector. Again, a globalization cae te welfare gap between te two contrie to narrow widen wen te conption good prodced in different ector are gro copleent btitte. Wen te two contrie are neqal in ize Section 5.3, te factor price i lower in te aller contry, reflecting it diadvantage of being aller in ti world of aggregate increaing retrn de to te variety effect. Globalization redce bt never eliinate ti diadvantage, and cae te factor price to converge bt never copletely eqalize and ence te ter of trade to cange in favor of te aller contry. Ti generate oe additional effect. If labor 7
Kiinori Matyaa, Engel aw in te Global Econoy prodctivity i lower in te aller contry--wic inclde te cae were te two contrie ave te eqal poplation ize--, ti contry a lower per capita real incoe regardle of te trade cot. However, if labor prodctivity i iger in te aller contry, globalization cae a leapfrogging de to a ter-of-trade cange and a factor price convergence. 6 At a ig trade cot, te aller contry wit iger labor prodctivity igt ave a lower per capita real incoe tan te larger contry wit lower labor prodctivity, becae of teir diadvantage of aving te aller doetic aret. Globalization redce ti diadvantage of te aller contry enog o tat it per capita real incoe becoe iger. In or etp, ti lead to a reveral of te pattern of trade. Te aller contry wit iger labor prodctivity can be a net eporter in te lower incoe-elatic ector at a iger trade cot, and a net eporter in te iger incoe-elatic ector at a lower trade cot. Section 6 dice etenively te relation to te eiting tdie. Section 7 offer conclding rear, inclding oe ggeted direction for ftre reearc. Two Appendice follow. Appendi A eplain wy or cla of nonootetic preference i niqely wellited for or prpoe. Appendi B offer two lea on log-perodlarity and onotone coparative tatic ed trogot te analyi.. Te Model Iagine te world econoy tat conit of two contrie, indeed by or = or. Generally, i ed to indicate te location of prodction, and tat of conption. Tere i a ingle nontradeable factor of prodction, wic all be called labor. Contry i poplated by N oogeno agent, eac of wo pplie wage rate, nit of effective labor inelatically at te w. T, te per capita noinal 7 ependitre and incoe in i te total effective labor pply in i N. Te poplation ize, E w N, and it effective and 6 Ti poibility of leapfrogging de to te ter-of-trade effect of globalization i not de to te nonooteticity, a own by Matyaa 5, ection 3 in an alternative odel, wic differ fro te preent odel in tat te doetic deand copoition difference acro te two contrie i de to te eogeno difference in tate. 7 We call E te per capita noinal incoe to ditingi it fro te per capita real incoe introdced later a a eare of te welfare level, U E P, were P i te eact price inde in. It i not noinal in te ene of being eared in oe crrent nit. Ti i not a onetary odel, and tere i no crrency in te odel. 8
Kiinori Matyaa, Engel aw in te Global Econoy labor pply per agent acro te two contrie., or labor prodctivity, are te only poible orce of eterogeneity. Nonootetic Preference and Ependitre Sare: Tere i a contin of ector, indeed by conption good, alo indeed by eac agent, U ~ = UC I I, were,, are given iplicitly by ~ U ;, and, C d I wit and globally qai-concave in, wic enre tat I C, I, eac prodcing a nontradeable I Ri an open interval. Te preference of U ~ = UC I, i globally onotone increaing I. Witot frter lo of generality, let I, and noralize c tat d. 8 Te tility fnction i directly iplicitly additive, wit contant elaticity of btittion CES. 9 In addition, te weigt of eac ector, ~ ~, U U, te coefficient on te ter C, i ioelatic in U ~ i.e., it i a power fnction of U ~. If for all I, becoe te tandard ootetic CES: ~ U C d, I 8 To ee wy ti i witot lo of generality, ppoe d c c c, ˆ U I. Since ~ U > i an order-preerving onotone tranforation of c ˆ, wic atifie ˆ d I ˆ U ~, ˆ U = ˆ ~ U c = U iplie >. Ten, for. Hence, ti preerve ordinal propertie of te preference. We e ti noralization becae it a a nice cardinal property. A U ~ own below tat, nder ti noralization, te aiized vale of nder te bdget contraint i eqal to te per capita real incoe, and ε can be interpreted a te incoe elaticity of ector-. 9 Appendi A eplain different notion of additivity eplicit v. iplicit, direct v. indirect, wic are iportant for ndertanding wy or togt eperient neceitate te e of ti particlar cla of preference. 9
Kiinori Matyaa, Engel aw in te Global Econoy wic i directly eplicitly additive. By letting dependent on, ti cla of tility fnction, no longer directly eplicitly additive bt till directly iplicitly additive, allow te incoe elaticity to differ acro ector, wile eeping te price elaticity, η, contant and te ae acro ector. In wat follow, we ae tat te ector can be ordered c tat i trictly increaing in I. Ten,, U ~ ~ U i log-perodlar in and ~ By applying ea ee Appendi B for ˆ ; U g = U ~. ~ U, ti iplie tat, a U ~ goe p, te agent care ore abot te iger-indeed good in te ene tat te denity fnction of te weigt attaced to different ector atifie te onotone lieliood ratio MR property and ence tat it clative ditribtion fnction atifie te firt-order tocatic doinance FSD property. contraint: Eac agent in cooe C, I, to aiize U ~, bect to and te bdget P C d E w, I taing P, I, te nontradeable conption good price in, given. 3 Te oltion can be epreed in ter of te ependitre are on good, P C E U P E I t U P t U P t :, wit d dt I were U i te aiized vale of U ~ and given by te indirect tility fnction of : d 4 I P E U, wic i indirectly iplicitly additive. Recall te paraeter retriction,, wic enre te global onotonicity of te tility fnction,. Ti retriction iplie tat Note tat, wit, doe not atify wat i coonly called indirect additivity, wic old be called ore preciely indirect eplicit additivity, becae it indirect tility fnction 4 i not eplicitly additive.
Kiinori Matyaa, Engel aw in te Global Econoy HS of 4 i trictly increaing in U, and ence given. Frterore, eq. 4 can be rewritten a: were E U, were U P d I P P, U i trictly increaing in E, olding price P i te eact price inde of te conption good for te agent in becae logdifferentiating P yield U P t U P logp P P P P d = logp I P P I t t dt I and ltiplying bot te nerator and te denoinator by U lead to were follow,. P P P P I d i given by 3. Hence, U i te real ependitre and incoe per capita. In wat U all be called intercangeably te welfare and te per capita real incoe in contry Notice tat te nerator of eq.3, U P U. Hence, by applying ea for g ˆ, U U P te price contant, te agent wit a iger per capita real incoe, d, i log-perodlar in and, eq.3 ow tat, olding U allocate larger are of teir ependitre toward iger-indeed ector in te ene tat te denity fnction of te ependitre are acro ector atifie te MR property and tat it clative ditribtion Needle to ay, coparing te per capita real incoe acro different contrie or different period poe an epirical callenge, becae te ependitre are of eac good,, te weigt ed to calclate te aggregate price inde, cange wen te relative price and te total ependitre cange dicretely. However, c an epirical callenge i not niqe to or preference. Even in te tandard ootetic CES, a dicrete cange in te relative price ae it ipoible to calclate te eact price inde becae te ependitre are cange nle it i Cobb-Dogla, and need to be approiated by apeyre, Paace, Diviia, or oter indice. Te only difference i tat, in te tandard practice, te epirical callenge aociated wit an endogeno cange in te weigt caed by a cange in te total ependitre i ignored by aing tat te preference are ootetic. I tan Erzo GJ tter for i note on ti point, tter 7.
Kiinori Matyaa, Engel aw in te Global Econoy fnction atifie te FSD property. Note alo tat, fro eq. 3, one cold epre te relative ependitre are of any two ector a: log = log ' log U + log P P. ' ' and te relative deand of any two ector a: ' log C C = log ' log U log P P. ' ' Ti ow not only tat te relative deand for a iger-indeed ector a iger incoe elaticity. It alo ow tat te lope of te Engel crve, log C C log U = ', i independent of te per capita real incoe, U. Frterore, tee incoe elaticity paraeter are not lined to te price elaticity, nlie in oter for of nonootetic preference. 3. Prodction and Trade: We eep te ret of te odel deliberately tandard, ing te Diit-Stiglitz-Krgan onopolitic copetitive odel of prodction and trade to iolate te role of Engel aw... Copetitive Nontradeable Conption Good Sector: ' ' Coin, aari, and Metieri 5 review te epirical evidence tat incoe elaticity difference acro ector are table over a wide range of per capita incoe level. Ti i contrary to te Stone-Geary preference, wic iplie tat incoe elaticity difference acro ector decline wit per capita incoe. Ti ae Stone- Geary nited for odelling Nort-Sot trade, a well a long rn developing procee, a pointed ot by Bera and Kaboi 9. See Matyaa 6 for a ore etenive dicion on te retrictive natre of Stone-Geary and oter directly eplicitly additive nonootetic preference. 3 For eaple, by ing te Contant Ratio of Incoe Elaticity CRIE preference, wic are directly eplicitly additive, bot Fieler, eq.9 and Caron, Fally, and Maren 4, eq. derive te relative ependitre crve, wic can be rewritten a: log = cont. ' log + log P ' log P, ' ' and ence it relative deand crve a: log C C = cont. ' log log P ' log P were ' ', te agrange ltiplier aociated wit te bdget contraint, i inverely related to E at any given price. Notice tat and ' appear alo in te coefficient of te log of price. Under CRIE in fact, alo nder Stone-Geary, or any oter directly eplicitly additive preference, te ratio of incoe elaticity and price elaticity i contant acro ector te o-called Pigo aw, and ence it i infeaible to dientangle te effect of incoe elaticity difference and toe of price elaticity difference. Frterore, Deaton 974 and any oter wo ave etiated c log-linear conption deand yte ave reected te Pigo aw, bt ave not been able to reect a coon price elaticity of btittion, iplied by or preference. Coin, aari, and Metieri 5 review te epirical evidence in pport of te log-linear Engel crve iplied by or iplicitly additive ioelatically nonootetic CES and againt toe iplied by direct eplicit additivity.
Kiinori Matyaa, Engel aw in te Global Econoy indeed by Eac nontradeable conption good, I, i prodced in a copetitive ector, alo I, by aebling a contin of tradable differentiated inpt, indeed by, wit te yetric CES aggregator, 5 Y q were d ; I ; σ > Ma{, η}, Y i te otpt; q te qantity of inpt variety ed in ector I ; et of tradeable differentiated inpt available for I, and, were te = or are te dioint et of differentiated inpt prodced in contry in eqilibri. Te retriction on σ, te elaticity of btittion between inpt witin eac ector, iplie not only σ > bt alo σ > η, o tat differentiated inpt are cloer btitte witin eac ector tan acro ector. 4 Given p, te nit price of inpt variety v in, eac copetitive conption good ector cooe te inpt cobination to iniize it cot, wic yield te nit cot and ence te nit price of te conption good, 6 P p d I in : wic i te CES price aggregator, te dal of 5, and te qantity of inpt variety in : v ed 7 p q Y P... Iceberg Cot and Deand for Differentiated Inpt: Te nit price of eac inpt variety, p,, depend on, becae of te iceberg trade cot; To deliver one nit of in contry, nit need to be ipped 4 For te epirically ore relevant cae of gro copleent, η <, ti ipoe no additional retriction. For te cae of gro btitte, it i neceary to ae σ > η >. If η > σ >, two differentiated inpt ed in te ae ector wold becoe Hic-Allen copleent and te entry of two onopolitic copetitive fir into te ae ector wold becoe trategic copleent, leading to ltiple eqilibria for te ae reaon diced at lengt in Matyaa 995. 3
Kiinori Matyaa, Engel aw in te Global Econoy fro. T, wit te nit factory price, p,, p = p. Ten, p fro 3 and 7, and ing Y N C, te deand for by contry i: q N E U P p N E U P p N E U P = were p,. T, te aggregate deand for can be epreed a: 8 D = A p, were 9 A b ; E b U N P U P N P were A ay be interpreted a te aggregate deand ift paraeter for a variety prodced in ector- in contry ; b a te aggregate deand ift paraeter for ector- in contry ; and i te weigt attaced to te aggregate pending by contry of varietie prodced in contry. Eq. 8- ow tat te deand crve for eac variety a a contant price elaticity wit it deand ift paraeter, A, wic depend on te trade cot in a anner failiar in te Diit-Stiglitz-Krgan onopolitic copetition odel of trade. Wat i new i tat U a differential ipact on te deand ift paraeter acro ector de to te nonooteticity of preference. For te reainder of ti paper, we follow Krgan 98 and any oter by aing tat and, o tat and <. T, [, eare ow c eac contry pend on an iported variety relative to wat it wold pend in te abence of te trade cot; it i inverely related to, wit for and for...3 Prodction and Pricing by Monopolitically Copetitive Fir: 4
Kiinori Matyaa, Engel aw in te Global Econoy Eac differentiated inpt variety i prodced by a onopolitically copetitive fir. Prodcing one nit of eac differentiated inpt in ector- reqire arginal cot i eqal to w for nit of labor, o tat te. Since Eq. 8 ow tat te price elaticity of deand for eac inpt i contant,, all te inpt varietie are priced wit te ae ar-p rate: p w p for all and ence tey are prodced by: 3 y A p. By inerting into 6,, 4 P p d = p were V i te ebege eare of active fir in ector- of contry. d = V p, te eqilibri eare of varietie prodced and of..4. Free Entry Condition and Sectoral Sare in Eployent: Ti eqilibri eare, V, i deterined by te free entry condition. To enter ector, all onopolitically copetitive fir need to pay te etp cot per variety,, in labor, and tey ave incentive to do o, a long a te profit i non-negative. T, in eqilibri, eiter a poitive eare of fir and varietie enter, in wic cae tey all brea even V p y w y, or no fir and varietie enter, becae tey wold earn negative profit if tey were to enter p y w y V. Uing eq. and 3, ti free entry condition can be written a te copleentarity lacne condition: V ; y A p. Ti iplie tat eac active fir in ector- baed in contry ire y = nit of labor, o tat labor deand by ector- in contry i V and it are in eployent i 5
Kiinori Matyaa, Engel aw in te Global Econoy f V rewritten a: 5 wit f I. T, te above copleentarity lacne condition can be frter V 6 f d, ; y A p wic i noting bt te labor aret clearing condition,, d N. I In wat follow, we e te following noralization to iply te notation. Firt, cooe te nit of eac differentiated inpt in ector- c tat. Ti iplifie to p p w for all and all I. Second, cooe te nit of variety in ector- c tat. Tee two noralization togeter iplifie 5 to: 5 f V ; y A w for all I and = and. In oter word, witot lo of generality, we cooe te nit of eareent c tat eac active fir prodce by y, ire labor by y and ell it otpt at p w, to brea even in eqilibri, and te labor deand by ector- of contry i.3 Eqilibri Condition: V. We are now ready to conolidate all te eqilibri condition. Firt, ing, and 5, eq. 4 becoe 4 P f w f w By introdcing ; w w P f w f w, I,, te relative wage or te factoral ter of trade and alo te relative price of inpt varietie prodced in te two contrie in te ae ector, 4 can be frter iplified to: 7 w P f f ; w P f f, I, were w P i te TFP of ector- in contry. 6
Kiinori Matyaa, Engel aw in te Global Econoy Second, fro 9,, 5, te copleentary lacne condition for free entry in ector- in eac contry can be written a 8 f ; b b w & f ; b b w Ti can be frter rewritten a. 9 f ; d d ; & f ; d d, I. by introdcing d b w, wic i te doetic aret are in te revene of a differentiated inpt prodcer baed in. 5 Ti variable, d b w, can be epreed in two different way fro 3 and. Firt, by eliinating d b w Alternatively, by eliinating d b w = N U P fro 3 and,, = and. U fro 3 and and ing w P. Uing 7, ti can be frter rewritten to: d b w = ; f f E N d w w N w, b = f f Finally, te ependitre are or te aret ize ditribtion, a well a te eployent are acro ector t add p to one in eac contry. I d for = and. I 3 f d for = and. Note tat eq. 9,, and ipoe te condition on i fnction of I, and tat eq. and 3 ipoe for additional condition, bt one of te i redndant de to te Walra aw. Tee condition, eq. 9-3, altogeter deterine i endogeno fnction of I ; d te doetic aret are in te revene of an inpt prodcer in eac ector in eac 5 On te oter and, te eport aret are in te revene of an inpt prodcer in i w w d,, wic i eqal to te doetic aret are in te revene of an inpt prodcer in, w w d, ltiplied by, de to te relative price between tee two prodcer, and ltiplied by de to te trade cot. 7
Kiinori Matyaa, Engel aw in te Global Econoy contry, te aret ize ditribtion in eac contry, and eac contry, a well a tree endogeno variable, incoe in eac contry and te ter-of-trade. f te eployent are in U te welfare or te per capita real Before proceeding to olve for te eqilibri, it i wort pointing ot one notable or perap nal featre of ti et of te eqilibri condition; it contain U, = and. Norally, wen we analyze a general eqilibri odel, we firt olve for te eqilibri allocation and price by condcting a poitive analyi. Ten, we plg toe eqilibri allocation into te tility fnction to obtain te welfare level by condcting a norative analyi. Here, de to te iplicit natre of te tility fnction, te coner deand depend on te welfare level, wic in trn affect te eqilibri allocation, wic in trn affect te welfare level. Terefore, it i ore efficient to olve for te eqilibri allocation and price and for te welfare level togeter, witot te eparation of te poitive and norative analye. Indeed, wen olving for te eqilibri below, variable tat will be pinned down. U, = and are aong te firt endogeno 3 Pattern of Strctral Cange in a Cloed Econoy Eqilibri Firt, let conider te cae of ρ =, were eac contry t prodce all differentiated inpt ed in every ector. T, for all 8, 4 d =. Inerting ti to 9 and yield f N U. Sbcript i added ere to indicate tat I and for = and, f >, and ence, fro U i te eqilibri per capita real incoe acieved wen ρ =. Note alo tat eq. 4 ow tat te eployent and vale-added i ditribted proportionately wit te aret ize acro ector in a cloed econoy. By integrating 4 acro all te ector and ing or 3, we can pin down N U I d, wic can be written ore copactly a U a 8
Kiinori Matyaa, Engel aw in te Global Econoy 9 5 U wit N, were i defined iplicitly by 6 I d. ea -i in Appendi B ow tat, defined in eq.6, i a trictly increaing fnction. T, te welfare, or te per capita real incoe, in te cloed econoy increae wit N. Again, bcript i ed to indicate ρ =. Eq.5 ow tat U increae not only in labor prodctivity,, bt alo in N. Ti i de to te aggregate increaing retrn to cale in te preence of love for variety and te fied cot, a failiar featre of te Diit-Stiglitz onopolitic copetition odel. Ti can be alo een, fro eq.7 and ρ =, f P w, o tat te ectoral TFP i increaing in te total eployent in tat ector. Notice alo tat te condition for U < U can be epreed a, wic occr even if wen. In oter word, te contry wit iger labor prodctivity ay ave a lower per capita real incoe wen it i fficiently aller. Ti i becae toe living in a all contry a diadvantage in te preence of aggregate increaing retrn. 6 Net, plgging 5 and 6 into 4 yield te eqilibri denity fnction of eployent and aret ize acro ector a follow: 7 f I t t dt. 6 Ti relt doe not contradict wat we noted earlier, i.e., eq.4 ow tat te agent tility i increaing in E and ence in, olding te price given. Wen coparing te two contrie in eqilibri, te price differ acro te two contrie becae te eare of varietie ed in eac ector in eac contry i endogenoly deterined by te free entry condition.
Kiinori Matyaa, Engel aw in te Global Econoy Te nerator of 7 i log-perodlar in and. T, by applying ea for, ˆ g, eq.7 ow tat, for U U, contry, woe per capita real incoe i iger tan contry, pend relatively ore on iger-indeed good in te ene tat i trictly decreaing in tat i, te denity fnction of eqilibri aret ize ditribtion acro ector atifie te MR property a well a in te ene tat te clative ditribtion fnction for contry firt-order tocatically doinate FSD te clative ditribtion fnction for contry. Notice te difference between te two epreion of, eq.3 and eq.7, in particlar ow it depend on te welfare or per capita real incoe. Eq.3 iplie tat, olding te price given, te relative aret ize of two ector, ', repond to an increae in U a U log log ' = '. In contrat, eq.7 ow tat, in eqilibri, te relative aret ize of two ector repond a log log ' d d = '. Ti i de to te Scooler effect. A cange in te relative aret ize cae oe entrie into iger-indeed ector and oe eit fro lower-indeed ector, wic lead to a iger lower prodctivity in iger-lower-indeed ector, wic redce te relative price of iger-indeed good. Forally, by etting ρ = in eq.7, f P w, o tat ' ' ' f f P P. Ti cange in te relative price oderate aplifie te ift in ependitre are if different ector prodce gro copleent gro btitte. Indeed, fro eq.3 and ing te above epreion, te total effect can be calclated a log log ' d d = log log ' + d P P d ' ' log log log log ' d d = log log ' + log log ' d d,
Kiinori Matyaa, Engel aw in te Global Econoy fro wic d log = d log ' log log ', wic i eactly wat we obtained fro eq.7. Te oderation de to te Scooler nder gro copleent i captred by < for η <, and te aplification de to te Scooler effect nder gro btitte i captred by > for η >. In te literatre of trctral tranforation, it i coon to treat te incoe elaticity difference acro ector and te prodctivity growt difference acro ector a two eparate eogeno cae of trctral cange. Te above relt gget tat, in te preence of te Scooler effect, c a dicotoy can be ileading, a oe prodctivity growt difference ay be indced by te incoe elaticity difference. in Te above aplification or oderation effect alo affect te welfare ipact of a cange. Fro ea -ii own in Appendi B, log d log increae in d = ' i increaing decreaing in if η ><. In word, welfare gain fro a percentage i iger lower at a iger if η ><. Ti iplie, aong oter ting, tat a nifor labor prodctivity growt,, redce agnifie te welfare per capita real incoe gap between te two contrie, U U, if different ector prodce gro copleent gro btitte. Before proceeding, it i wort pointing ot tat, wen ρ =, te eqilibri condition, eq. 9,,, and 3, do not depend on ω. In oter word, ω i indeterinate; tere i noting to pin down te relative wage of te two contrie tat are iolated fro eac oter. Ti i no longer te cae, wen ρ >. Indeed, a te firt tep to olve for a trade eqilibri, we need to deterine te relative wage or te ter-of-trade between te two contrie. 4 Trade Eqilibri: Cro-Contry Variation Ti ection foce on ow te two contrie differ in te trade eqilibri for a given et of te paraeter vale. Te net ection will deal wit coparative tatic. 4. Factoral Ter of Trade
Kiinori Matyaa, Engel aw in te Global Econoy In wat follow, let foc on te cae were f and f for all I iplifie eq. 9 to d d and d d, fro wic 8 d and d for all I. Inerting 8 into yield. Ti 9 f f and f f Integrating tee epreion acro all ector and ing and 3, for I. and. Tee two epreion are eqivalent. Indeed, eiter of te can be rewritten a: 3 were ; = ; i trictly increaing in, ;, and ; li, and atifie ; li. Figre illtrate eq.3, wic deterine te factor ter of trade w w a a fnction of te relative labor pply,,. It ow tat w w i trictly increaing in, for a given and w w if and only if. T, te factor price i iger in te larger econoy, wic reflect te aggregate increaing retrn to cale pointed ot earlier. 7 It alo ow te lower and pper bond for te ter of trade,,. Te arrow indicate te effect of an increae in ρ. A own, it flatten te grap, tereby caing a factor price convergence. Ti i becae globalization, captred by a redction in τ and ence an increae in ρ, redce te aller contry diadvantage., 7 Note tat eq.3 iplie w w ;, wic i trictly increaing in ence alo in and w w < if and only if < ence alo if and only if <. T, te larger econoy i larger regardle of weter it i eared in te total labor pply or in te aggregate GDP.
Kiinori Matyaa, Engel aw in te Global Econoy It i alo wort pointing tat, becae i trictly increaing in wit te range, and = for = for any ρ,, eq.8 iplie: i d d i trictly increaing decreaing in : tat i, te doetic aret accont ore for te revene of te inpt prodcer baed in te larger contry; ii d and d a ; d and d a : tat i, te doetic eport aret accont for ot of te revene of toe operating in a very large all contry; and iii d = d = +ρ > ½: tat i, wen te two contrie are eqal in ize, te doetic aret accont ore tan a alf of teir revene in te preence of te trade cot. 4. Per Capita Real Incoe and Maret Size Ditribtion Net, cobining 8 and yield N 3 U, U N. Here, te bcript ρ i added to indicate tat U, te eqilibri per capita real incoe acieved in eac contry nder trade, depend on ρ. By integrating 3 acro all te ector and ing, we obtain 3 U, wit U, wit N ; N, were i te ae increaing fnction defined iplicitly by 6. Note tat te welfare effect of globalization on eac contry are arized by a ingle inde,. Note alo tat te lower and pper bond on te ter of trade etablied earlier,,, wic can be 3
Kiinori Matyaa, Engel aw in te Global Econoy 4 een in Figre, enre gain fro trade for bot contrie; iplie, ence U > U and iplie, ence U > U. Plgging 3 bac into 3 and ing te definition of, given by 6, yield te eqilibri denity fnction of te aret ize ditribtion acro ector in eac contry a follow. 33 I t t dt for = and. Note tat i log-perodlar in and. Hence, by applying ea for, ˆ g, it follow fro eq. 33 tat, for U < U, contry, woe per capita real incoe i iger tan toe in contry, pend relatively ore on iger-indeed in te ene tat i trictly decreaing in tat i, te denity fnction of te eqilibri aret ize ditribtion acro ector atifie te MR property a well a in te ene tat te clative ditribtion fnction for contry firt-order tocatically doinate FSD te clative ditribtion fnction for contry. In ort, te doetic deand copoition i ore ewed toward te iger-indeed in te contry wit iger per capita real incoe. Te MR property can alo be een by taing te ratio fro 33 to obtain 34. Clearly, ti i trictly decreaing in if U < U and trictly increaing in if U > U. 4.3 Hoe Maret Effect in Eployent and in Pattern of Trade Unlie in te cloed econoy cae, te eployent ditribtion in eac contry i no longer proportional to te aret ize ditribtion in tat contry. By olving 9 for f and
Kiinori Matyaa, Engel aw in te Global Econoy 5 f and ing 3, we obtain te eqilibri denity fnction of eployent acro ector in eac contry a follow: 35 f > ; f >, wic reqire. Frterore, te ratio of te two, 36 f f i increaing in and atifie f f, f f, or f f. Figre illtrate eq.34 and eq.36 for te cae of U < U. In ti cae, i trictly decreaing in. Ti cae f f to be alo trictly decreaing in. Frterore, tere i a niqe ctoff ector, c I, c tat f f old below te ctoff and f f above te ctoff. And te grap of f f i teeper tan te grap of. T, diproportionately larger fraction of labor are eployed in lower iger incoe elatic ector in te contry wit lower iger per capita real incoe, preciely becae it doetic deand copoition i ore ewed toward aret in lower iger incoe elatic ector. Ti i in trong contrat to te cloed econoy cae, were labor i allocated acro ector proportionately to te aret ize ditribtion acro ector, o tat f f. In oter word, international trade agnifie te power of te doetic deand copoition in dictating te allocation of reorce acro ector. Ti relt igt coe a a rprie to toe wo addre te qetion of trctral cange witin a cloed econoy etting. In a cloed econoy, te doetic pply i necearily eqal to te doetic deand in eac ector, and ence a cange in te doetic deand copoition acro ector wold cae a proportional cange in te copoition of prodction and ence te ectoral allocation of reorce. Many people ee to believe tat, in an open econoy, te doetic deand copoition becoe le iportant, becae te
Kiinori Matyaa, Engel aw in te Global Econoy doetic pply need not be eqal to te doetic deand in eac ector. Ti logic i fale. Tat te doetic pply i no longer eqal to te doetic pply in eac ector ean tat te ipact of te doetic deand copoition i no longer proportional; intead, it cold be ore tan proportional. Indeed, a long a te trade cot i not zero, te difference in te doetic deand copoition acro contrie give different incentive for entry of fir or ore generally innovation acro ector in different contrie in te preence of te Scooler effect. Trog c differential Scooler effect acro contrie, te ricer poorer contry develop coparative advantage in iger lower incoe-elatic ector, wic i te inder effect. And a lower trade cot cae te ricer poorer contry to allocate even ore reorce toward iger lower incoe-elatic ector by iporting even ore fro te poorer ricer contry in lower iger incoe-elatic ector. Hence, globalization agnifie, intead of weaening, te power of te doetic deand copoition difference in dictating te pattern of trctral cange. 4.4 Te inder Effect, or te Hoe Maret Effect in Inter-Sectoral Pattern of Intra- Sectoral Trade A te above paragrap gget, te diproportional effect of te aret ize ditribtion on te eployent ditribtion nder trade anifet itelf in te inter-ectoral pattern of intraectoral trade. Indeed, tey are te two ide of te ae coin. A indicated in Figre, te contry wit iger lower per capita real incoe becoe a net eporter iporter above te ctoff and a net iporter eporter above te ctoff. To ee ti, recall tat contry pend b p b p b w per variety prodced in ector- of contry. Wit te eare of varietie prodced in ti ector, ector- i V given by NX b V, te total gro eport vale fro to in w f b w. T, te net eport vale fro to in ector- i = f b w f b w. Uing 8, 3 and 35, ti can NX be frter rewritten a: 37 NX w NX w =. 6
Kiinori Matyaa, Engel aw in te Global Econoy T, U NX > for < c and NX NX < for > c wen U < NX. Ti i noting bt te inder effect. It ay alo be viewed a a variant of te Hoe Maret Effect of Krgan 98. Te ey difference i tat te cro-contry difference in te aret ize ditribtion acro ector i de to nonootetic preference in ti odel, not de to te eogeno cro-contry variation in tate a aed in Krgan 98. It i alo wort epaizing tat contry becoe a net eporter in ector were old, wic are not necearily ector were w w old. Wat deterine te direction of net ectoral trade flow in a general eqilibri odel of te oe aret effect i not te cro-contry difference in te aret ize in eac ector. Wat atter i te cro-contry difference in te deand copoition, i.e., in te cro-contry difference in te aret ize ditribtion acro ector. 8 4.5 Raning te Contrie: Trade-off between abor Prodctivity and Contry Size Or reaining ta i to ran te two contrie in ter of te per capita real incoe. Ti i iple wen te two contrie are in eqal ize,. In ti cae, ω = o tat N w w, and ence, = =. T, te contry wit iger labor prodctivity a iger per capita incoe and iger per capita real incoe. Generally, te condition nder wic Contry a lower per capita real incoe tan Contry, U < U or can be written a: 8 Te Hoe Maret Effect i often decribed iply a relatively large doetic deand give copetitive advantage to eporting fir. To ti, we ave eard oe IO people ay oeting to te effect tat te are of te doetic ale t be trivial for any fir baed in all econoie lie Denar or Switzerland. Te relt ere old eplain wy c a critici i nwarranted. Even if te Swi doetic aret igt be aller tan te Cinee doetic aret in every ector in abolte ter, oe ector old accont for larger are in te Swi ependitre tan in te Cinee ependitre, a long a te two contrie differ in te deand copoition. And tat i wat deterine te pattern of coparative advantage in a general eqilibri odel of te Hoe Maret Effect. Matyaa 5, ection 3 deontrated ti by etending te Krgan odel wit a contin of ector and wit two contrie of neqal ize. In te contet of te preent odel, even if Switzerland ay be c aller tan Cina, and coneqently te doetic aret accont for a tiny are of te revene for te Swi fir operating in any ector recall te relt own earlier tat d a, Switzerland old becoe a net-eporter in ig-incoe elatic ector and Cina a net-iporter in low-incoe elatic ector, a long a Switzerland a iger per capita real incoe tan Cina. 7
Kiinori Matyaa, Engel aw in te Global Econoy 8 N N, wic can be frter rewritten a: 38 ; ; ; ~. To ndertand ti condition, it wold be efl to copare it wit te condition nder wic Contry i poorer nder atary, U < U, wic can be written a:, and te condition nder wic Contry a lower per capita noinal incoe, E w w E, ; ; ;. Figre 3 illtrate tee condition. Te blac crve depict te grap of ; ~ on wic U = U old. It i downward-loping, and U < U old below and to te left of ti crve, and U > U old above and to te rigt of ti crve. Te red crve depict te grap of, on wic U = U old. It i alo downward-loping and U < U old below and to te left of ti crve, and U > U old above and to te rigt of ti crve. Te ble crve depict te grap of ;, on wic w w old. It i alo downward-loping and w w old below and to te left of ti crve, and w w old above and to te rigt of ti crve. It i alo eay to verify tat ; = ; ~ = and