Spaghetti Regionalism
Abstract
This paper examines the welfare implications of multiple free trade agreements in a model of imperfect competition. We show that free trade is the unique Nash equilibrium under the simple rule that any two countries can form a bilateral free trade agreement. Specifically, a country is always better off forming a bilateral trade agreement with every other country, irrespective of previous agreements. This suggests that each new preferential free trade agreement may be a step towards multilateral free trade.
Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 680 September 2000 SPAGHETTI REGIONALISM Caroline L. Freund NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.bog.frb.fed.us.
SPAGHETTI REGIONALISM Caroline L. Freund* Abstract: This paper examines the welfare implications of multiple free trade agreements in a model of imperfect competition. We show that free trade is the unique Nash equilibrium under the simple rule that any two countries can form a bilateral free trade agreement. Specifically, a country is always better off forming a bilateral trade agreement with every other country, irrespective of previous agreements. This suggests that each new preferential free trade agreement may be a step towards multilateral free trade. Keywords: trade, regionalism, and imperfect competition * Economist in the International Finance Division of the Federal Reserve Board. I would like to thank Pravin Krishna and Marcelo Olarreaga for helpful comments. Correspondence: Caroline Freund, International Finance Division, Federal Reserve Board, Washington D.C. 20551, e-mail: caroline.l.freund@frb.gov. The views presented here are solely the responsibility of the author and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or any other person associated with the Federal Reserve System.
4 Lqwurgxfwlrq Pdq| frxqwulhv kdyh vljqhg pxowlsoh iuhh wudgh djuhhphqwv1 Wkh Xqlwhg Vwdwhv kdv suhihuhqwldo wudgh djuhhphqwv zlwk Lvudho/ Fdqdgd/ Ph{lfr/ dqg vhyhudo qdwlrqv lq wkh Fduleehdq1 Ph{lfr kdv iuhh wudgh djuhhphqwv zlwk wkh Xqlwhg Vwdwhv/ Fdqdgd/ Erolyld/ Frvwd Ulfd/ dqg Fkloh1 Fkloh kdv wudgh djuhhphqwv zlwk Ph{lfr/ Yhqh}xhod/ Frorpeld/ Fdqdgd/ dqg wkh Phufrvxu wudgh eorf1 Hxurshdq frxqwulhv kdyh vljqhg rq wr qxphurxv iuhh wudgh djuhhphqwv dv zhoo1 Pruh wkdq 63 wudgh djuhhphqwv dprqj dqg ehwzhhq vhyhudo ri wkh Fhqwudo dqg Hdvwhuq Hxurshdq frxqwulhv dqg phpehuv ri Hxurshdq Xqlrq dqg Hxurshdq Iuhh Wudgh Dvvrfldwlrq zhuh udwl(cid:31)hg ehwzhhq 4<<5 dqg 4<<81 Wkh txhvwlrq zh dgguhvv lq wklv sdshu lv zkhwkhu wklv (cid:19)vsdjkhwwl erzo(cid:20) ri uhjlrqdo djuhhphqwv/ dv Mdjglvk Ekdjzdwl +4<<6, kdv fdoohg lw/ zloo ohdg wr iuhh wudgh1 Wdnlqj wkh wdul(cid:30) ohyho dv (cid:31){hg/ zh vroyh iru wkh rswlpdo qxpehu ri elodwhudo iuhh wudgh djuhhphqwv wkdw d frxqwu| zloo sduwlflsdwh lq1 Zh (cid:31)qg wkdw iuhh wudgh zloo dozd|v eh dfklhyhg xqghu wkh vlpsoh uxoh wkdw dq| elodwhudo wudgh djuhhphqw +EWD, lv shuplwwhg1 Vshfl(cid:31)fdoo|/ lq d wkuhh0frxqwu| prgho zlwk lpshuihfw frpshwlwlrq dqgv|pphwulfwdul(cid:30)v/zhvkrzwkdwdfrxqwu|wkdwlvdphpehurirqhelodwhudodjuhhphqwlvdozd|v ehwwhu r(cid:30) vljqlqj d vhfrqg EWD/ dqg d frxqwu| wkdw lv qrw d phpehu ri wkh lqlwldo EWD lv dozd|v ehwwhu r(cid:30) vljqlqj d vhsdudwh djuhhphqw zlwk hdfk ri wkh lqfxpehqwv1 Khqfh/ wkh vwulfwo| grplqdqw vwudwhj| iru hdfk frxqwu| lv wr kdyh d elodwhudo djuhhphqw zlwk doo rwkhu frxqwulhv/ dqg wkh xqltxh Qdvk htxloleulxp lv rqh zkhuh doo frxqwu| sdluv kdyh iuhh wudgh1 Zh vkrz wkdw wklv uhvxow h{whqgv wr d q0frxqwu| prgho1 Wklv lpsolhv wkdw wkh uhjlrqdo sdwk zloo dozd|v ohdg wr iuhh wudgh/ surylghg frxqwulhv duh shuplwwhg wr vljq pruh wkdq rqh elodwhudo iuhh wudgh djuhhphqw1 Wklv (cid:31)qglqj lv lq frqwudvw wr pxfk ri wkh uhfhqw olwhudwxuh rq zkhwkhu uhjlrqdolvp khos ru klqghu wkh iuhh wudgh surfhvv14 Rqh vwudqg ri olwhudwxuh h{dplqhv wkh h(cid:30)hfw ri d vlqjoh uhjlrqdo 4Vhh Sdqdjdul|d +5333, iru d vxuyh| ri wkh wkhruhwlfdo olwhudwxuh1 5
djuhhphqw rq wkh phpehuv* lqfhqwlyhv iru pxowlodwhudo wdul(cid:30) uhgxfwlrq1 Jurvvpdq dqg Khospdq +4<<8,/ Sdqdjdul|d dqg Ilqgod| +4<<9,/ Ohy| +4<<:,/ dqg Nulvkqd +4<<;, (cid:31)qg wkdw wkh srolwlfdo vxssruw iru pxowlodwhudo iuhh wudgh ghfolqhv vxevhtxhqw wr wkh irupdwlrq ri d suhihuhqwldo wudgh djuhhphqw1 Wkh orjlf lq wkhvh prghov lv wkdw vshfldo lqwhuhvw jurxsv orvh wkh suhihuhqwldo wuhdwphqw lq wkhlu rzqpdunhw dv zhoo dv wkh suhihuhqwldo wuhdwphqw lq wkh rwkhu phpehu*v pdunhw li iuhh wudgh lv dfklhyhg1 Zlwkrxw wkh wudgh eorf/ lqwhuhvw jurxsv kdyh rqo| wkh suhihuhqwldo wuhdwphqw lq wkhlu rzq pdunhw wr orvh1 Wkxv/ li wkh suhihuhqwldo wudgh djuhhphqw lv vxssruwhg e| vshfldo lqwhuhvwv jurxsv/ lw pxvw pdnh wkhlu vxssruw iru iuhh wudgh zdqh1 D vhfrqg vwudqg ri olwhudwxuh dqdo|vhv zkhwkhu wkh h{sdqvlrq ri dq h{lvwlqj uhjlrqdo djuhhphqw zloo ohdg wr iuhh wudgh1 Edogzlq +4<<8, dqg \l +4<<9, (cid:31)qg wkdw qrq0phpehuv kdyh pruh lqfhqwlyh wr mrlq d uhjlrqdo djuhhphqw dv lw h{sdqgv1 Wkh lqwxlwlrq lv wkdw dv wkh iuhh wudgh duhd jurzv/ wkh frvw wr d qrq0phpehu ri uhpdlqlqj rxwvlgh wkh djuhhphqw dovr jurzv/ lpso|lqj wkdw d qrq0phpehu zloo kdyh d juhdwhu lqfhqwlyh wr mrlq wkh xqlrq dv lw h{sdqgv1 Exw/ Erqg dqg V|ursrxorv +4<<9, dqg \l +4<<9, vkrz wkdw wkh phpehuvrid xqlrqzloo qrw kdyh dqlqfhqwlyh wrshuplw doo rwkhu frxqwulhv wr mrlq1 Rqfh wkh suhihuhqwldo wudgh duhd lv odujh hqrxjk/ wkh orvv ri suhihuhqwldo wuhdwphqw lq hdfk rwkhuv* pdunhwv rxwzhljkv wkh h!flhqf| jdlq iurp dffhswlqj d qhz phpehu1 Wkh| (cid:31)qg wkdw wkh zhoiduh ri lqfxpehqwv lv pd{lpl}hg zkhq vrph/ exw qrw doo/ frxqwulhv duh phpehuv ri wkh xqlrq1 Wkh prgho lq wklv sdshu vkrzv wkdw doorzlqj iru vhsdudwh elodwhudo djuhhphqwv xqklqjhv wklv lqwxlwlrq ehfdxvh vljqlqj d vhfrqg elodwhudo djuhhphqw grhv qrw d(cid:30)hfw suhihuhqwldo wuhdwphqw lq wkh rwkhu phpehu*v pdunhw1 Vshfl(cid:31)fdoo|/ zkloh d sdlu ri frxqwulhv fdq eh mrlqwo| ehwwhu r(cid:30) zlwk d srvlwlyh h{whuqdo wdul(cid:30)/ hdfk phpehu lv dozd|v lqglylgxdoo| ehwwhu r(cid:30) li lw elodwhudoo| djuhhv wr iuhh wudgh zlwk d wklug sduw|1 Wkxv/ wkh uhjlrqdo rxwfrph lv qrw d Qdvk htxloleulxp ehfdxvh hdfk frxqwu| kdvdqlqfhqwlyhwrvxeplw wrrwkhu elodwhudo iuhh wudgh djuhhphqwv1 Wklvkdvwkh lpsruwdqw srolf| lpsolfdwlrq wkdw/ li iuhh wudgh lv wkh ghvluhg rxwfrph/ qr eorf phpehu vkrxog eh deoh wr 6
suhyhqw dqrwkhu eorf phpehu iurp frpplwwlqj wr iuhh wudgh zlwk d wklug sduw|1 Zkloh wkh uhvxowv vxjjhvw wkdw uhjlrqdolvp pd| eh d yldeoh dowhuqdwlyh wr pxowlodwhudo wdul(cid:30) uhgxfwlrq/ wkh| vkrxog qrw eh lqwhusuhwhg dv lpso|lqj wkdw elodwhudolvp lv ehwwhu wkdw pxowlodwhudo wdul(cid:30) uhgxfwlrq1 Lq sduwlfxodu/ wkh prgho lv vwdwlf1 Lq uhdolw| wkhuh duh frvwv dvvrfldwhg zlwk kdylqj dq dexqgdqfh ri EWDv gxulqj wkh wudqvlwlrq wr iuhh wudgh wkdw zh gr qrw hydoxdwh1 Iru h{dpsoh/ dv Ekdjzdwl dqg Sdqdjdul|d +4<<9, kljkoljkw/ d pxowlwxgh ri djuhhphqwv zloo fdxvh frpsoh{ uxohv ri ruljlq wr (cid:31)qg wkhlu zd| lqwr sudfwlfh1 Lq dgglwlrq/ li wkh lqfxpehqw phpehuv ri d iuhh wudgh djuhhphqw fdq hqirufh frrshudwlrq dprqj wkhpvhoyhv wkhq wkh| pljkw eh deoh wr eorfn wkh iuhh wudgh rxwfrph1 Lq wklv fdvh/ d vpdoo qxpehu ri h{foxvlyh djuhhphqwv frxog eh pdlqwdlqhg1 Ilqdoo|/ uhjlrqdolvp lv vwloo d txhvwlrqdeoh srolf| dv frpsduhg zlwk pxowlodwhudolvp1 Wkhru| vxjjhvwv wkdw lw pd| khos ru klqghu wkh iuhh wudgh surfhvv/ dqg ri frxuvh wkhuh lv wkh gdqjhu ri zhoiduh0uhgxflqj wudgh glyhuvlrq dv d uhvxow ri uhjlrqdolvp1 Vwloo/ li uhjlrqdolvp suhydlov/ dv vhhpv wr eh wkh fdvh suhvhqwo|/ lw lv frpiruwlqj wr nqrz wkdw d (cid:19)vsdjkhwwl erzo(cid:20) ri iuhh wudgh djuhhphqwv pd| yhu| zhoo ohdg wr iuhh wudgh/ dqg wkdw lw lv pruh olnho| wr eulqj derxw iuhh wudgh wkdq d vpdoo qxpehu ri glvwlqfw wudgh eorfv1 Wklv sdshu lv glylghg lqwr irxu vhfwlrqv1 Vhfwlrq 5 ghvfulehv wkh edvlf wkuhh0frxqwu| prgho1 Vhfwlrq 6 jhqhudol}hv wkh prgho wr doorz iru d odujhu qxpehu ri frxqwulhv lq wkh uhvw ri wkh zruog1 Vhfwlrq 7 frqfoxghv1 5 LL1 D Wkuhh0Frxqwu| Prgho Lqwklvvhfwlrq/zhxvhdwkuhh0frxqwu|roljrsrolvwlfprghoriwudghwrh{dplqhwkhrswlpdoqxpehuri elodwhudo djuhhphqwv1 Wklv prgho lv dssursuldwh iru dqdo|}lqj wudgh eorf txhvwlrqv ehfdxvh rswlpdo wdul(cid:30)v duh qrq0}hur dqg ehfdxvh uhjlrqdo djuhhphqwv h{sdqg wkh phpehuv* vkduh ri wkh zruog pdunhw dqg khqfh hqkdqfh sur(cid:31)wv deurdg1 Nulvkqd +4<<;, hpsor|v dq roljrsrolvwlf prgho wr vkrz 7
wkdw wklv sur(cid:31)w h(cid:30)hfw fdq uhgxfh wkh lqfhqwlyh iru pxowlodwhudo iuhh wudgh li wkh jryhuqphqw fduhv uhodwlyho|pruhderxwsurgxfhuv*zhoiduh1 Iuhxqg+5333,vkrzvwkdwhyhqzlwkdzhoiduh0pd{lpl}lqj jryhuqphqw wkhuh h{lvwv d srvlwlyh h{whuqdo wdul(cid:30) vxfk wkdw phpehu frxqwulhv fdq dwwdlq d kljkhu zhoiduh ohyho wkdq wkh| zrxog dw iuhh wudgh1 Wkxv/ lq wklv prgho/ uhjlrqdolvp vhhpv d sulrul olnho| wr klqghu wkh pryh wr iuhh wudgh1 Wkhuh lv rqh jrrg zklfk lv surgxfhg e| d vlqjoh sur(cid:31)w0pd{lpl}lqj (cid:31)up lq hdfk frxqwu| dqg vhjphqwhg pdunhwv ohdg wr wudgh lq wklv jrrg +dv lq Eudqghu dqg Nuxjpdq +4<;6, dqg Eudqghu dqg Vshqfhu +4<;7,,1 Wkhuh duh wkuhh frxqwulhv [> \> dqg ]1 Zh dvvxph wkdw wkh lqyhuvh ghpdqg ixqfwlrq/ S(cid:16)+T,/ lq hdfk frxqwu| lv olqhdu1 Vshfl(cid:31)fdoo|/ S(cid:16)+T,@4(cid:3)dT(cid:16) dqg T(cid:16) @t(cid:16) .t(cid:16) .t(cid:16)> +4, % + 5 zkhuh d lv d frqvwdqw/ T(cid:16) lv wkh wrwdo txdqwlw| frqvxphg lq frxqwu| l/ dqg t(cid:16) lv wkh txdqwlw| (cid:26) surgxfhg e| wkh (cid:31)up lq m iru pdunhw l1 Frpshwlwlrq lv Frxuqrw1 Sur(cid:31)wv wr wkh (cid:31)up iurp frxqwu| [> lq frxqwu| l/ (cid:28)(cid:16)> duh (cid:28)(cid:16) @{(cid:16)S(cid:16)+T(cid:16),(cid:3)ft(cid:16) (cid:3)w(cid:16)t(cid:16)> +5, % % % zkhuh> f lv d frqvwdqw pdujlqdo frvw ri surgxfwlrq/ dqg w(cid:16) lv wkh wdul(cid:30) wkdw wkh (cid:31)up iurp frxqwu| (cid:26) m idfhv lq pdunhw l +w(cid:16) @ 3 dqg w(cid:26) (cid:24) 3,1 Vroylqj iru wkh sur(cid:31)w pd{lpl}lqj txdqwlw| ri d (cid:31)up iurp (cid:16) (cid:16) frxqwu| [ lq pdunhw l/ zh kdyh 4(cid:3)f.w(cid:16) (cid:3)6w(cid:16) t(cid:16) @ 3% %= +6, % 7d D vx!flhqw frqglwlrq iru hdfk (cid:31)up wr vhoo lq hdfk pdunhw lv wkdw w(cid:16) ? E(cid:129)3S(cid:28)= (cid:26) (cid:12) Wkh jryhuqphqw*v zhoiduh ixqfwlrq/ Z/ lq frxqwu| [/ lv wkh vxp ri frqvxphu vxusoxv/ surgxfhu vxusoxv/ dqg wdul(cid:30) uhyhqxh1 [ Z @X+T,(cid:3)S%T. +{(cid:16)S(cid:16)(cid:3)f{(cid:16) (cid:3)w(cid:16)t(cid:16),.w%t%.w%t%= +7, % % + + 5 5 (cid:16)’%c+c5 8
Lqhtxdwlrq+7,/ X+T, lvwkh frqvxphu xwlolw|iurpwkdwtxdqwlw|1 Soxjjlqjlqwkhsur(cid:31)wpd{lpl}lqj txdqwlwlhv iurp htxdwlrq +6,/ zh kdyh # $ S 2 (cid:3) (cid:4) 4(cid:3)f. w(cid:16) (cid:3)6w(cid:16) 6+4(cid:3)f,(cid:3)w%(cid:3)w% 2 [ (cid:26) % Z @ + 5 . (cid:26)’+c5 +8, 65d 49d (cid:16)’%c+c5 4(cid:3)f.w%(cid:3)6w% 4(cid:3)f.w%(cid:3)6w% . w + 5 +,.w + + 5,> + 7d 5 7d zkhuh wkh (cid:31)uvw whup lv frqvxphu vxusoxv/ wkh vhfrqg whup lv surgxfhu vxusoxv/ dqg wkh odvw wzr whupv duh wdul(cid:30) uhyhqxh1 Liwkhuh duh qriuhh wudgh djuhhphqwvwkhqwkh jryhuqphqw pd{lpl}hv zhoiduh/ htxdwlrq+8,/ ryhu w% dqg w%> wdnlqj rwkhu frxqwulhv* wdul(cid:30)v dv jlyhq1 Zlwk v|pphwulf frxqwulhv/ wkh wdul(cid:30) rq \ zloo + 5 eh htxlydohqw wr wkh wdul(cid:30) rq ]/ uhvxowlqj lq dq rswlpdo wdul(cid:30) ri 6+4(cid:3)f, wW @ = 43 Qrwhwkdw/lqwklvprgho/wkhrswlpdowdul(cid:30)lqrqhfrxqwu|lvqrwdixqfwlrqriwkhwdul(cid:30)vlpsohphqwhg deurdg1 Wklvlpsolhvwkdwuhjdugohvvrizkdwsrolf|rwkhufrxqwulhviroorz/dvlqjohfrxqwu|lvdozd|v ehwwhu r(cid:30) zlwk d srvlwlyh wdul(cid:30)1 Krzhyhu/ li hdfk frxqwu| lqvwdoov lwv rswlpdo wdul(cid:30) wkhq wkh| duh doo zruvh r(cid:30) wkdq wkh| zrxog eh dw iuhh wudgh1 Hdfk frxqwu| lv pdgh ehwwhu r(cid:30) li vrph uhflsurfdo elodwhudo ru pxowlodwhudo wdul(cid:30) uhgxfwlrq lv dfklhydeoh1 514 Elodwhudo Djuhhphqwv Lq wklv vhfwlrq/ zh h{dplqh wkh lqfhqwlyh iru d frxqwu| wr mrlq d elodwhudo iuhh wudgh djuhhphqw/ frqvlghulqj doo srvvleoh shupxwdwlrqv ri elodwhudo djuhhphqwv wkdw frxog h{lvw lq d wkuhh0frxqwu| prgho1 Zlwk wkuhh frxqwulhv/ lw lv rqo| vhqvleoh irfxv rq elodwhudo djuhhphqwv> krzhyhu/ rxu uhdvrqv irufrqvlghulqjrqo|EWDvduhvrphzkdwpruhjhqhudo1 Dfrxqwu|fdqdozd|vdjuhhwrdqdgglwlrqdo elodwhudo wudgh djuhhphqw/ qr pdwwhu krz pdq| djuhhphqwv lw lv d phpehu ri ru zkhwkhu lw lv d 9
phpehuridodujhwudgheorf1 Hydoxdwlqjpxowlsohelodwhudodjuhhphqwv/ wkhuhiruh/ h(cid:30)hfwlyho|doorzv iru doo srvvleoh shupxwdwlrqv ri wudgh djuhhphqwv/ dqg lw lv wkh prvw vwudljkw0iruzdug phwkrg wr ghwhuplqh wkh rxwfrph zkhq pxowlsoh ryhuodsslqj iuhh wudgh djuhhphqwv h{lvw1 Iru vlpsolflw|/ zh dvvxph wkh jdph lv d vlpxowdqhrxv pryh jdph/ hdfk frxqwu| ghflghv krz pdq| elodwhudo djuhhphqwv wr mrlq/ wkhq doo EWDv/ zkhuh erwk phpehuv zdqw dq djuhhphqw/ duh iruphg1 Wkh rxwfrph grhv qrw ghshqg rq vlpxowdqhrxv pryhv/ lw rqo| uhtxluhv wkdw wkhuh lv qr frrshudwlrqdprqjphpehuvri rqhdjuhhphqw/ vxfkwkdw wkh| fdqsuhyhqw hdfkrwkhu iurpiruplqj dq dgglwlrqdo iuhh wudgh djuhhphqw15 Zlwkrxw orvv ri jhqhudolw|/ zh dvvxph wkdw wdul(cid:30)v duh srvlwlyh exw qrw deryh wkh surklelwlyh ohyho +3 ? w (cid:23) E(cid:129)3S(cid:28),= Zh dvvxph wkh wdul(cid:30) lv juhdwhu wkdq }hur ehfdxvh li w htxdov }hur d elodwhudo (cid:12) djuhhphqw kdv qr h(cid:30)hfw dqg wkh sureohp lv qrw lqwhuhvwlqj1 Zh dvvxph wkh wdul(cid:30) ohyho lv ohvv wkdq ru htxdo wr wkh surklelwlyh ohyho lq rughu wr suhyhqw qhjdwlyh txdqwlwlhv iurp hqwhulqj wkh zhoiduh ixqfwlrq1 Li wkh wdul(cid:30) lv deryh wkh surklelwlyh ohyho/ zhoiduh lv h{dfwo| htxdo wr lwv ydoxh dw wkh surklelwlyh ohyho(cid:24)wkhuh lv qr pdujlqdo h(cid:30)hfw rq zhoiduh ri d wdul(cid:30) lqfuhdvh eh|rqg wkh surklelwlyh ohyho1 Zh ixuwkhudvvxph wkdwdoosrvlwlyhwdul(cid:30)vduhv|pphwulf+l1h1 lidfrxqwu|kdvdsrvlwlyhwdul(cid:30) rq dqrwkhu frxqwu| lw lv w, dqgwkdw h{whuqdo wdul(cid:30)v duh sduw ridqh{rjhqrxv pxowlodwhudo djuhhphqw dqgwkhuhiruhgrqrwfkdqjhdvwkhuhvxowriwkhirupdwlrqridelodwhudodjuhhphqw16 Wklvdvvxpswlrq 5\l +4<<9, h{dplqhv wkh htxloleulxp vl}h ri d iuhh wudgh duhd li phpehuv dfw dv d frdolwlrq dqg fdq suhyhqw qrq0phpehuv iurp mrlqlqj wkh wudgh eorf1 6Hwklhu +4<<;, dqg Iuhxqg +5333, dovr wuhdw uhjlrqdolvp dv hqgrjhqrxv zlwk uhvshfw wr wkh wdul(cid:30) ohyho1 Wkh| frqvlghu wkh lqfhqwlyhv wr mrlq d vlqjoh uhjlrqdo djuhhphqw/ dv d ixqfwlrq ri wkh wdul(cid:30) ohyho1 Hwklhu +4<<;, (cid:31)qgv wkdw d vpdoo frxqwu| fdq pruh hdvlo| dfklhyh uhirup diwhu olqnlqj xs zlwk d odujh frxqwu| ehfdxvh wkh vpdoo frxqwu| zloo kdyh d pdujlqdo dgydqwdjh ryhu rwkhu vpdoo frxqwulhv lq dwwudfwlqj iruhljq gluhfw lqyhvwphqw1 Zkhq wdul(cid:30)v duh orz/ wkh ydoxh ri iruhljq lqyhvwphqw lv juhdwhu/ pdnlqj vpdoo frxqwulhv pruh hdjhu wr vljq iuhh wudgh djuhhphqwv1 Lq d prgho zlwk lpshuihfw frpshwlrq/ Iuhxqg +5333, vkrzv wkdw dw d orz wdul(cid:30) ohyho/ erwk suhihuhqwldo dqg pxowlodwhudo wdul(cid:30) uhgxfwlrq kdyh d vlplodu sur0frpshwlwlyh h(cid:30)hfw/ exw suhihuhqwldo djuhhphqwv duh uhodwlyho| ohvv frvwo| ehfdxvh :
lv mxvwl(cid:31)hg e| wdul(cid:30) ghwhuplqdwlrq zlwklq wkh Zruog Wudgh Rujdql}dwlrq +ZWR,1 Phpehuv ri wkh ZWRidfhfrqvwudlqwvwkdwuxohrxwsuhihuhqwldowdul(cid:30)uhgxfwlrqv/dsduwiurpfrpsohwholehudol}dwlrq1 Lq dgglwlrq/ wkh irupdwlrq ri d qhz uhjlrqdo djuhhphqw lv frqglwlrqdo rq h{whuqdo wdul(cid:30)v uhpdlqlqj xqfkdqjhg dw wkh prvw idyruhg qdwlrq ohyho/ vxevhtxhqw wr wkh irupdwlrq ri d suhihuhqwldo wudgh djuhhphqw1 Wr hqgrjhql}h wkh qxpehu ri elodwhudo djuhhphqwv/ zh hydoxdwh wkh zhoiduh ri d uhsuhvhqwdwlyh frxqwu|/ vd| [/ zlwk doo srvvleoh elodwhudo djuhhphqwv lq wkh zruog1 Zlwk wkuhh frxqwulhv/ wkhuh duh wkuhh srvvleoh elodwhudo djuhhphqwv/ iru hdfk zh dvvljq dq lqglfdwru yduldeoh L > zklfk lv rqh li l (cid:16)(cid:26) dqg m kdyh d elodwhudo djuhhphqw dqg }hur rwkhuzlvh1 Iurp htxdwlrq +8, [*v zhoiduh lv +6+4(cid:3)f,(cid:3)+5(cid:3)L (cid:3)L ,w,2 +4(cid:3)f.+5(cid:3)L (cid:3)L ,w,2 +4(cid:3)f.+4(cid:3)L ,w(cid:3)6+4(cid:3)L ,w,2 Z @ %+ %5 . %+ %5 . +5 %+ 65d 49d 49d +4(cid:3)f.+4(cid:3)L ,w(cid:3)6+4(cid:3)L ,w,2 4(cid:3)f(cid:3)+5.L .L ,w . +5 %5 .+5(cid:3)L (cid:3)L ,w+ %+ %5 ,> +9, 49d %+ %5 7d zkhuh wkh (cid:31)uvw whup rq wkh uljkw0kdqg vlgh lv frqvxphu vxusoxv/ wkh vhfrqg whup lv [*v sur(cid:31)wv lq wkh krph pdunhw/ wkh wklug whup lv [*v sur(cid:31)wv lq \*v pdunhw/ wkh irxuwk whup lv [*v sur(cid:31)wv lq ]*v pdunhw/ dqg wkh odvw whup lv wdul(cid:30) uhyhqxh1 Vljqlqj d EWD zlwk \/ iru h{dpsoh/ d(cid:30)hfwv doo ri wkh whupv h{fhsw sur(cid:31)wv lq frxqwu| ]1 Dv d uhvxow ri wkh djuhhphqw/ wkh (cid:31)up iurp \ h{sruwv pruh wr [/ zklfk pdnhv frqvxphu vxusoxv +wkh (cid:31)uvw whup, lqfuhdvh dqg grphvwlf sur(cid:31)wv +wkh vhfrqg whup, idoo1 Sur(cid:31)wv lq \*v pdunhw +wkh wklug whup, lqfuhdvh ehfdxvh wkh wdul(cid:30) idflqj [ lq \*v pdunhw lv uhpryhg1 Sur(cid:31)wv lq ]*v pdunhw +wkh irxuwk whup, gr qrw fkdqjh vlqfh ]*v wdul(cid:30) suhihuhqfhv rq [ dqg \ grhv qrw fkdqjh dv d uhvxow ri wkh elodwhudo djuhhphqw ehwzhhq [ dqg \1 Wdul(cid:30) uhyhqxh ghfolqhv ehfdxvh \ kdv gxw| iuhh dffhvv1 Iurp htxdwlrq +9, zh fdq fdofxodwh [*v zhoiduh iurp kdylqj EWDv zlwk \ dqg ] xqghu doo srvvleoh shupxwdwlrqv ri rwkhu djuhhphqwv1 Vshfl(cid:31)fdoo|/ zlwk uhvshfw wr frxqwu| [/ wkhuh duh 9 srvvleoh rxwfrphv1 +l, Wkhuh duh qr EWDv1 +ll, Wkhuh lv rqo| rqh EWD dqg [ lv d phpehu1 +lll, sur(cid:31)wv dqg wdul(cid:30) uhyhqxh idoo e| ohvv1 Wklv dovr pdnhv suhihuhqwldo wudgh djuhhphqwv pruh olnho| zkhq wdul(cid:30)v duh orz1 ;
Wkhuh lv rqo| rqh EWD dqg [ lv qrw d phpehu1 +ly, Wkhuh duh wzr EWDv dqg [ lv d phpehu ri erwk1 +ly, Wkhuh duh wzr EWDv dqg [ lv d phpehu ri rqh1 Dqg +yl,/ doo sdluv ri frxqwulhv kdyh d EWD(cid:24)zklfk lv h(cid:30)hfwlyho| iuhh wudgh1 Zh qrz h{dplqh wkh zhoiduh ri [ xqghu hdfk ri wkhvh vfhqdulrv/ zklfk e| v|pphwu| lv wkh vdph dv wkh zhoiduh ri hdfk rwkhu frxqwu| lq wkdw vfhqdulr1 Wkh froxpqv lq Wdeoh 4 vkrz [*v sd|r(cid:30) iurp qrw mrlqlqj d EWD/ iurp mrlqlqj rqh EWD/ dqg iurp kdylqj elodwhudo djuhhphqwv zlwk erwk [ dqg \/ dv frpsduhg zlwk wkh fdvh zkhuh wkhuh duh qr djuhhphqwv lq wkh zruog +l1h1 zkhuh L @ 3> iru doo l dqg m,1 Wkh (cid:31)uvw urz vkrzv [*v sd|r(cid:30)/ (cid:16)(cid:26) dvvxplqj wkhuh lv qr elodwhudo djuhhphqw ehwzhhq \ dqg ] +L @ 3,1 Lq wklv fdvh/ [ lv ehwwhu r(cid:30) +5 vljqlqj d elodwhudo djuhhphqw zlwk erwk \ dqg ]1 Wkh vhfrqg urz vkrzv [*v sd|r(cid:30) zkhq \ dqg ] kdyh d elodwhudo djuhhphqw +L @ 4,1 Djdlq/ frxqwu| [ lv ehwwhu r(cid:30) kdylqj dq djuhhphqw zlwk +5 erwk \ dqg ]1 Wkxv/ wkh vwulfwo| grplqdqw vwudwhj| iru frxqwu| [ lv wr kdyh d elodwhudo djuhhphqw zlwk erwk \ dqg ]1 Vlqfh wkh jdph lv v|pphwulf/ hdfk frxqwu|*v ehvw vwudwhj| lv wr djuhh wr wzr elodwhudo djuhhphqwv dqg h(cid:30)hfwlyho| wkhuh zloo eh pxowlodwhudo iuhh wudgh1 Qrwh wkdw wklv lv qrw dozd|v wkh zhoiduh pd{lpl}lqj vwudwhj|1 Dw orz wdul(cid:30) ohyhov/ li wkhuh lv rqo| rqh elodwhudo djuhhphqw/ wkh wzr phpehuv duh ehwwhu r(cid:30) wkdq wkh| zrxog eh dw iuhh wudgh1 Vshfl(cid:31)fdoo|/ wkh ydoxh ri ehlqj d phpehu ri wkh vlqjoh iuhh wudgh djuhhphqw lv juhdwhu wkdq wkh ydoxh ri pxowlodwhudo iuhh wudgh/ +wkh ydoxh lq wkh (cid:31)uvw urz dqg vhfrqg froxpq ri Wdeoh 4 lv juhdwhu wkdq wkh ydoxh lq wkh erwwrp uljkw, zkhq wkh wdul(cid:30) ohyho lv ohvv wkdq 2E(cid:129)3S(cid:28)17 Wklv uhvxow/ wkdw (cid:129)(cid:129) wzr frxqwulhv lq d uhjlrqdo djuhhphqw fdq eh ehwwhu r(cid:30) wkdq dw iuhh wudgh dqg wkxv pljkw qrw kdyh dq lqfhqwlyh wr olehudol}h pxowlodwhudoo|/ lv wkh uhvxow wkdw rwkhuv kdyh irfxvhg rq1 Exw/ surylghg dq|elodwhudodjuhhphqw lvshuplwwhg/ pdlqwdlqlqjrqh uhjlrqdodjuhhphqw lvqrwdQdvkhtxloleulxp ehfdxvhhdfkphpehu lvdozd|vehwwhur(cid:30)kdylqjdqdjuhhphqw zlwkwkh wklugsduw|qrpdwwhu zkdw 7Nhqqdq dqg Ulh}pdq +4<<3, dovr vkrz wkdw phpehuv ri d odujh fxvwrpv xqlrq fdq eh ehwwhu r(cid:30) zlwk wkh xqlrq wkdq dw iuhh wudgh1 <
wkh rwkhu phpehu grhv/ dqg wkh wklug sduw| zloo eh zloolqj wr vljq erwk djuhhphqwv1 Wdeoh4= Ihdvleoh Vwudwhjlhv Qr djuhhphqw [ vljqv 4 djuhhphqw [ vljqv 5 djuhhphqwv \ dqg ] kdyh qr djuhhphqw 3 wSE(cid:129)3S(cid:28)3.| w(cid:129)2E(cid:129)3S(cid:28)nH| (cid:12)2@ (cid:12)2@ \ dqg ] kdyh d elodwhudo djuhhphqw (cid:3)wHE(cid:129)3S(cid:28)32f| (cid:3)w2E(cid:129)3S(cid:28)3| weE(cid:129)3S(cid:28)ne| (cid:12)2@ (cid:12)2@ (cid:12)2@ Zlwk v|pphwulf h{whuqdo wdul(cid:30)v dqg wkuhh frxqwulhv/ li dq| elodwhudo djuhhphqw lv hqirufhdeoh/ iuhh wudgh lv wkh xqltxh Qdvk rxwfrph1 Wkh orjlf lv vlpsoh dqg lv olnho| wr fduu| ryhu wr rwkhu prghov1 Wkh dgydqwdjh ri delodwhudo djuhhphqw lvwkdw lw jlyhvhdfkphpehu suhihuhqwldo wuhdwphqw lq wkh rwkhu phpehu*v pdunhw1 Wklv phdqv wkdw sur(cid:31)wv deurdg lqfuhdvh dqg vxsso| lq wkh grphvwlf pdunhw h{sdqgv vr frqvxphu vxusoxv jrhv xs1 Wkh glvdgydqwdjh lv wkdw grphvwlf sur(cid:31)wv dqg wdul(cid:30) uhyhqxh idooehfdxvh wkh rwkhu phpehu kdvgxw|0iuhh dffhvvdw krph1 Zh kdyh vkrzqwkdw wkhh(cid:30)hfw rq sur(cid:31)wv deurdg dqgrq frqvxphu vxusoxv dozd|vrxwzhljk wkh h(cid:30)hfw rqgrphvwlf sur(cid:31)wv dqgwdul(cid:30) uhyhqxh1 Wklv lpsolhv wkdw dq| wzr frxqwulhv dozd|v kdyh wkh lqfhqwlyh wr vljq d elodwhudo iuhh wudgh djuhhphqw1 Wkxv/ li dq| elodwhudo djuhhphqw lv shuplwwhg/ pdlqwdlqlqj d vlqjoh elodwhudo djuhhphqw lv qrw d Qdvk htxloleulxp vlqfh hdfk phpehu kdv wkh lqfhqwlyh wr mrlq d vhfrqg rqh1 Wklv uhvxow gl(cid:30)huv iurp rwkhu prghov ehfdxvh wkh| frqvlghu wkh lqfhqwlyh ri wzr frxqwulhv lq d vlqjoh uhjlrqdo djuhhphqw wr jr wr iuhh wudgh yld pxowlodwhudo wdul(cid:30) uhgxfwlrq ru wkurxjk h{sdqvlrq ri wkh uhjlrqdo djuhhphqw1 Vlqfh erwk ri wkhvh d(cid:30)hfw hdfk frxqwu|*v sur(cid:31)wv lq doo pdunhwv/ wkh lqfxpehqw phpehuv ri d elodwhudo iuhh wudgh djuhhphqw zloo qrw kdyh dq lqfhqwlyh wr dfklhyh iuhh wudgh li sur(cid:31)wv lq wkh frpprq pdunhw duh uhgxfhg vx!flhqwo|1 Lq frqwudvw/ hqwhulqj d qhz elodwhudo djuhhphqw d(cid:30)hfwv pdunhw dffhvv rqo| lq grphvwlf pdunhw dqg lq wkh qhz pdunhw/ vr wkh h(cid:30)hfw ri wkh djuhhphqw rq sur(cid:31)wv lv vpdoohu wkdq lw lv iru wdul(cid:30) uhgxfwlrq ru EWD h{sdqvlrq1 Lq wkh qh{w vhfwlrq zh vkrz wkdw wklv uhvxow lv urexvw wr dq q0frxqwu| prgho1 43
6 Pruh wkdq wkuhh frxqwulhv Dv wkh qxpehu ri frxqwulhv lq wkh uhvw ri wkh zruog lqfuhdvhv/ wkh ehqh(cid:31)fldo h(cid:30)hfwv ri mrlqlqj d elodwhudo djuhhphqw fkdqjhv1 Lq wklv vhfwlrq/ zh vkrz wkdw hyhq li wkh qxpehu ri frxqwulhv lv odujh/ wkhuh lv dozd|v dq lqfhqwlyh iru hdfk frxqwu| wr irup d elodwhudo djuhhphqw zlwk doo rwkhu frxqwulhv1 Dv qrwhg deryh/ zkhq d frxqwu| vljqv d EWD lw jdlqv frqvxphu vxusoxv dw krph dqg sur(cid:31)wv lq wkh phpehu*v pdunhw/ dqg lw orvhv wdul(cid:30) uhyhqxh dqg grphvwlf sur(cid:31)wv ehfdxvh wkh iruhljq (cid:31)up kdv wdul(cid:30)0iuhh pdunhw dffhvv1 Zh vkrz wkdw wkh iruphu wzr h(cid:30)hfwv dozd|v grplqdwh wkh odwwhu wzr/ surylghg hdfk frxqwu| vljqv rq wr qxphurxv elodwhudo djuhhphqwv1 Lq wklv vhfwlrq/ zh hydoxdwh wkh zhoiduh ri d uhsuhvhqwdwlyh frxqwu| m1 Dvvxph wkhuh duh q frxqwulhv/ y elodwhudo djuhhphqwv lq wkh krph frxqwu|/ wkh wdul(cid:30) ohyho lv w> dqg hdfk iruhljq frxqwu| l kdv z djuhhphqwv1 Xvlqj wkh sulfh dqg sur(cid:31)w ixqfwlrq gh(cid:31)qhg lq htxdwlrqv +4, dqg +5,/ dqg wrwdo (cid:16) S txdqwlw| T @ t(cid:16)> zh fdq vroyh iru wkh txdqwlw| wkdw wkh (cid:31)up iurp frxqwu| m h{sruwv wr iruhljq (cid:26) frxqwu| l1 Dvvxplqj l dqg m gr qrw kdyh dq djuhhphqw/ m h{sruwv +4(cid:3)f.+q(cid:3)z(cid:3)5,w(cid:3)qw, t(cid:16) @ > (cid:26) +q.4,d zkhuh z lv wkh qxpehu ri iruhljq frxqwulhv wkdw kdyh d EWD zlwk l1 Wkh txdqwlw| wkdw d (cid:31)up iurp m surgxfhv iru wkh grphvwlf pdunhw lv +4(cid:3)f.+q(cid:3)4(cid:3)y,w,> t(cid:26) @ > (cid:26) +q.4,d zkhuh y lv wkh qxpehu ri elodwhudo djuhhphqwv frqwdlqlqj m1 Wkh txdqwlw| wkdw wkh (cid:31)up h{sruwv wr d frxqwu| wkdw lw kdv d iuhh wudgh djuhhphqw zlwk lv wkh vdph dv t(cid:26)/ h{fhsw wkdw y lv uhsodfhg zlwk (cid:26) z1 Xvlqj wkh h{sruw txdqwlwlhv zh fdq vroyh iru zhoiduh1 Zhoiduh lq frxqwu| m lv +q+4(cid:3)f,(cid:3)+q(cid:3)4(cid:3)y,w,2 +4(cid:3)f.+q(cid:3)4(cid:3)y,w,2 [(cid:15) +4(cid:3)f.+q(cid:3)4(cid:3)z ,w,2 Z+y, @ . . (cid:16) +:, 5d+q.4,2 +q.4,2d +q.4,2d (cid:16)’(cid:129) ?[3(cid:129) +4(cid:3)f(cid:3)+z .5,w,2 +4(cid:3)f(cid:3)+y.5,w, . (cid:16) .+q(cid:3)4(cid:3)y,w = +q.4,2d +q.4,d (cid:16)’(cid:15)n(cid:129) 44
Wkh (cid:31)uvw whup lv frqvxphu vxusoxv/ wkh vhfrqg whup lv grphvwlf sur(cid:31)wv/ wkh wklug whup lv sur(cid:31)wv lq pdunhwv zkhuh wkh frxqwu| kdv d elodwhudo djuhhphqw/ wkh irxuwk whup lv sur(cid:31)wv lq pdunhwv zkhuh wkh grphvwlf frxqwu| grhv qrw kdyh d elodwhudo djuhhphqw/ dqg wkh (cid:31)qdo whup lv wdul(cid:30) uhyhqxh1 Wkh qxpehu ri djuhhphqwv wkdw frqwdlq m/ y> d(cid:30)hfwv frqvxphu vxusoxv/ grphvwlf sur(cid:31)wv/ dqg wdul(cid:30) uhyhqxh1 Dv y lqfuhdvhv/ frqvxphu vxusoxv h{sdqgv ehfdxvh d juhdwhu qxpehu ri frxqwulhv kdyh gxw|0iuhh dffhvv vr frpshwlwlrq lv pruh lqwhqvh1 Wkh frpshwlwlrq h(cid:30)hfw dovr phdqv wkdw grphvwlf sur(cid:31)wv idoo1 Wdul(cid:30) uhyhqxh idoov ehfdxvh pruh frxqwulhv kdyh gxw|0iuhh dffhvv dqg ehfdxvh frxqwulhv wkdw vwloo idfh d wdul(cid:30) h{sruw ohvv1 Wkh qxpehu ri djuhhphqwv lq hdfkiruhljq frxqwu|/ z > d(cid:30)hfwv m*v (cid:16) sur(cid:31)wv lq wkdw frxqwu|1 Dv z lqfuhdvhv/ m*v sur(cid:31)wv deurdg idoo ehfdxvh d juhdwhu qxpehu ri rwkhu (cid:16) frxqwulhv kdyh wdul(cid:30)0iuhh dffhvv wr iruhljq pdunhwv1 Wkhuh duh vrph uhvwulfwlrqv rq y dqg z 1 Iru h{dpsoh/ y fdqqrw htxdo rqh li hdfk z lv }hur1 (cid:16) (cid:16) Wkdw lv/ li m kdv d elodwhudo djuhhphqw zlwk rqh frxqwu|/ wkhq dw ohdvw rqh rwkhu frxqwu| pxvw eh d phpehu ri d elodwhudo djuhhphqw1 Lq htxloleulxp wklv lv qrw d sureohp/ vlqfh e| v|pphwu| hdfk frxqwu| zloo fkrrvh wr kdyh wkh vdph qxpehu ri djuhhphqwv/ z @ z @ y1 Khqfh/ zlwk uhvshfw wr (cid:16) frxqwu| m/ zh fdq vlpsoli| wkh htxdwlrq deryh dv iroorzv1 +q+4(cid:3)f,(cid:3)+q(cid:3)4(cid:3)y,w,2 +4(cid:3)f.+q(cid:3)4(cid:3)y,w,2 +4(cid:3)f.+q(cid:3)4(cid:3)z,w,2 Z+y, @ . .y +;, 5d+q.4,2 +q.4,2d +q.4,2d +4(cid:3)f(cid:3)+z.5,w,2 +4(cid:3)f(cid:3)+y.5,w, .+q(cid:3)4(cid:3)y, .+q(cid:3)4(cid:3)y,w = +q.4,2d +q.4,d Wkh frxqwu|*v sureohp lv wr pd{lpl}h zhoiduh ryhu wkh qxpehu ri djuhhphqwv/ vxemhfw wr wkh frqvwudlqw wkdw wkh qxpehu ri djuhhphqwv lv ehwzhhq 3 dqg q (cid:3) 4> dqg wdnlqj wkh qxpehu ri djuhhphqwv wkdw rwkhu frxqwulhv kdyh/ z/ dv jlyhq/ zkhuh z lv dovr ehwzhhq 3 dqg q(cid:3)41 Wr vroyh wkh sureohp/ zh (cid:31)uvw h{dplqh krz zhoiduh fkdqjhv zlwk uhvshfw wr d fkdqjh lq wkh qxpehu ri djuhhphqwv/ y1 Wkh sduwldo ghulydwlyh ri Z zlwk uhvshfw wr y lv CZ +5q(cid:3)4,+4(cid:3)f,.w+5qy(cid:3)5z(cid:3)5qz.8y.6(cid:3)6q, @w = +<, Cy d+q.4,2 45
Wkh vhfrqg ghulydwlyh lv = C2Z w+5q.8, @w A3= Cy2 d+q.4,2 Wkh zhoiduh ixqfwlrq lv frqwlqxrxv dqg frqyh{ lq y> zklfk phdqv wkdw wkh fulwlfdo srlqw lv d plql0 pxp/ vrwkhpd{lpxppxvwrffxudwrqhriwkhhqgsrlqwv1 Dvderyhdqgzlwkrxworvvrijhqhudolw|/ zh dvvxph w lv juhdwhu wkdq }hur dqg ohvv wkdq ru htxdo wr wkh surklelwlyh ohyho +3(cid:23)w(cid:23) (cid:129)3S ,1 Dw (cid:25)n2 wkh pd{lpxp ydoxh ri y> y @q(cid:3)4/ wkh zhoiduh ixqfwlrq lv lqfuhdvlqj lq y1 CZ +5q(cid:3)4,+4(cid:3)f,.5w+q2(cid:3)z(cid:3)qz(cid:3)4, +y @q(cid:3)4, @w A3> Cy d+q.4,2 iru doo ihdvleoh w dqg z1 Zkhq y @3 wkh ghulydwlyh fdq eh srvlwlyh ru qhjdwlyh/ CZ +5q(cid:3)4,+4(cid:3)f,.w+(cid:3)5z(cid:3)5qz.6(cid:3)6q, +y @3, @w :3> Cy d+q.4,2 ghshqglqj rq w dqg z1 Li w dqg z duh odujh/ wkh zhoiduh ixqfwlrq lv ghfuhdvlqj lq y dw w @ 31 Wklv phdqv wkdw ghshqglqj rq wkh qxpehu ri djuhhphqwv rwkhu frxqwulhv kdyh/ z> dqg wkh wdul(cid:30) ohyho/ w/ wkhplqlpxpydoxh riwkhzhoiduhixqfwlrqfdqolh zlwklqwkhuhvwulfwhgudqjhiruy ru wrwkhohiw riy +y ?3,1 Wr (cid:31)qg wkh pd{lpxp/ zh hydoxdwh wkh zhoiduh ixqfwlrq dw hdfk hqgsrlqw1 Wkh vroxwlrq lv wkh fruqhu vroxwlrq/ y @q(cid:3)4 +vhh dsshqgl{ iru surri,1 Lq htxloleulxp/ doo frxqwulhv kdyh wkh vdph qxpehu ri djuhhphqwv dqg y @ z @ q(cid:3)41 Khqfh hdfk frxqwu| kdv elodwhudo djuhhphqwv zlwk doo rwkhu frxqwulhv1 Iljxuhv 4 dqg 5 looxvwudwh krz wkh qxpehu ri elodwhudo djuhhphqwv dqg krz wkh wdul(cid:30) ohyho d(cid:30)hfw wkh zhoiduh ohyho lq rqh frxqwu|1 Iljxuh 4 vkrzv wkh zhoiduh vxuidfh ri frxqwu| m> dv wkh qxpehu ri iruhljq elodwhudo djuhhphqwv +z, dqg grphvwlf elodwhudo djuhhphqwv +y, fkdqjh +dvvxplqj d@4/ f @ 3=4> w @ 3=34 dqg q @ 433,= Iru dq| jlyhq z/ frxqwu| m lv dozd|v ehwwhu r(cid:30) kdylqj djuhhphqwv zlwk doo iruhljq frxqwulhv1 Zhoiduh lv juhdwhvw zkhq m kdv djuhhphqwv zlwk doo rwkhu frxqwulhv dqg doo rwkhu frxqwulhv kdyh qr djuhhphqwv zlwk hdfk rwkhu +z @3,/ wkrxjk wklv lv qrw dq htxloleulxp1 46
Iljxuh 5vkrzvwkhzhoiduhvxuidfhiru frxqwu| m lqhtxloleulxp/ ryhu wdqgy +zkhuhdoofrxqwulhv kdyh wkh vdph qxpehu ri djuhhphqwv/ y @ z> dqg djdlq dvvxplqj d @ 4> f @ 3=4> dqg q @ 433,1 Iru dq| jlyhq wdul(cid:30) ohyho/ w/ dv wkh qxpehu ri elodwhudo djuhhphqwv lqfuhdvhv zhoiduh h{sdqgv1 Wkxv luuhvshfwlyh ri wkh wdul(cid:30) ohyho/ d frxqwu| lv dozd|v ehwwhu r(cid:30) vljqlqj djuhhphqwv zlwk doo rwkhu frxqwulhv1 Dqrwkhu srlqw/ zklfk lv hylghqw iurp Iljxuh 5/ lv wkdw dv wkh qxpehu ri uhjlrqdo djuhhphqwv pxowlsolhv/ hdfk frxqwu|*v zhoiduh lpsuryhv1 Lq idfw/ wkh h(cid:30)hfw ri h{sdqglqj uhjlrqdolvp rq zhoiduh lv yhu| vlplodu wr wkh h(cid:30)hfw ri d ghfolqlqj pxowlodwhudo wdul(cid:30)1 Surylghg frxqwulhv dfw v|pphwulfdoo|/ wklv zloo dozd|v eh wuxh1 Wr vhh wklv/ qrwh wkdw lq htxloleulxp/ zkhq doo frxqwulhv kdyh wkh vdph qxpehu ri djuhhphqwv +y @ z,/ htxdwlrq </ zklfk vkrzv krz zhoiduh fkdqjhv zlwk uhvshfw wr d fkdqjh lq y/ ehfrphv CZ +5q(cid:3)4,+4(cid:3)f,.6w+y.4(cid:3)q, @w = Cy d+q.4,2 Dvvxplqjwkhwdul(cid:30)lvohvvwkdqruhtxdowrwkhsurklelwlyhohyho/w (cid:23) (cid:129)3S>zhoiduhlvdozd|vlqfuhdvlqj (cid:15)n2 lq wkh qxpehu ri uhjlrqdo djuhhphqwv +Y‘ A3,1 Y(cid:15) Lq wklv vhfwlrq/ zh kdyh vkrzq wkdw hdfk frxqwu|*v vwulfwo| grplqdqw vwudwhj| lv wr kdyh d elodwhudo iuhh wudgh djuhhphqw zlwk doo rwkhu frxqwulhv1 Wklv lpsolhv wkdw pxowlodwhudo iuhh wudgh zloo h(cid:30)hfwlyho| eh dfklhyhg wkurxjk elodwhudo djuhhphqwv1 Lq dgglwlrq/ zh kdyh vkrzq wkdw zhoiduh lv prqrwrqlfdoo| lqfuhdvlqj lq wkh qxpehu ri elodwhudo djuhhphqwv wkdw hdfk frxqwu| lv d phpehu ri/ dvvxplqj doo frxqwulhv dfw v|pphwulfdoo|1 7 Frqfoxvlrq Wklv sdshu kdv vkrzq wkdw li wkh qxpehu ri elodwhudo djuhhphqwv lv ghwhuplqhg hqgrjhqrxvo|/ iuhh wudgh lv wkh xqltxh Qdvk htxloleulxp1 Li frxqwulhv fkrrvh wkh qxpehu ri djuhhphqwv wr mrlq/ wkh 47
vwulfwo| grplqdqw vwudwhj| lv wr kdyh d elodwhudo djuhhphqw zlwk hdfk rwkhu frxqwu|1 Wklv lpsolhv wkdw lq htxloleulxp hdfk frxqwu| kdv d iuhh wudgh djuhhphqw zlwk hyhu| rwkhu frxqwu|1 Rqh lqwhusuhwdwlrq ri wklv uhvxow lv wkdw doorzlqj iru pxowlsoh elodwhudo djuhhphqwv h(cid:30)hfwlyho| xqudyhov wkh rswlpdo0wdul(cid:30) htxloleulxp1 Lq wkh prgho zlwkrxw elodwhudo djuhhphqwv/ hdfk frxqwu| lqglylgxdoo| lv ehwwhu r(cid:30) pdlqwdlqlqj d srvlwlyh wdul(cid:30)/ exw lq wkh uhvxowlqj htxloleulxp doo frxqwulhv duhzruvhr(cid:30)wkdqwkh|zrxogehlqiuhhwudgh1 Lqfrqwudvw/zlwkelodwhudodjuhhphqwv/hdfklqglylgxdo frxqwu|lvehwwhur(cid:30)kdylqjdelodwhudodjuhhphqwzlwkhdfkrwkhufrxqwu|/luuhvshfwlyhrizkdwsrolf| rwkhu frxqwulhv iroorz1 Uhjlrqdolvp lv lqihfwlrxv/ vlpso| dffrxqwlqj iru elodwhudo djuhhphqwv doorzv frxqwulhv wr mxps iurp dq htxloleulxp zlwk d srvlwlyh wdul(cid:30) ohyho wr iuhh wudgh1 Zkloh wklv sdshu kdv hpsor|hg d vlpsoh roljrsrolvwlf prgho/ wkh uhvxowv duh olnho| wr fduu| ryhu wr rwkhu prghov dv zhoo1 Iru h{dpsoh/ lq d prgho zkhuh whupv0ri0wudgh h(cid:30)hfwv jhqhudwh srvlwlyh wdul(cid:30)v/ hqjdjlqj lq qxphurxv elodwhudo djuhhphqwv lv dovr olnho| wr eh zhoiduh lpsurylqj1 Wkh orjlf lv wkdw li d frxqwu| mrlqv d qxpehu ri EWDv/ ghpdqg iru wkh krph jrrg lqfuhdvhv e| uhodwlyho| pruh wkdq wkh ghpdqg iru iruhljq jrrgv/ lpso|lqj wkdw wkh uhodwlyh sulfh ri wkh grphvwlf jrrg pxvw lqfuhdvh1 Ghpdqg iru wkh krph jrrg lqfuhdvhv e| pruh ehfdxvh wdul(cid:30)v duh holplqdwhg lq doo iruhljq pdunhwv dv d uhvxow ri lwv EWDv/ krzhyhu wkh pxowlsoh djuhhphqwv kdyh olwwoh h(cid:30)hfw rq zruog ghpdqg iru wkh iruhljq jrrgv vlqfh elodwhudo djuhhphqwv rqo| holplqdwh wdul(cid:30)v lq wkh krph pdunhw1 Khqfh/ doo frxqwu| sdluv duh olnho| wr kdyh iuhh wudgh djuhhphqwv lq htxloleulxp1 48
Uhihuhqfhv ^4‘ Edogzlq/ U1/ 4<<81 (cid:19)D Grplqr Wkhru| ri Uhjlrqdolvp1(cid:20) Lq= Edogzlq/ U1/ Kddsdudqwd S1/ NldqghuM1+Hgv1,/ H{sdqglqj Phpehuvkls ri wkh Hxurshdq Xqlrq1Fdpeulgjh Xqlyhuvlw|Suhvv/ Fdpeulgjh1 ^5‘ Ekdjzdwl/ M1/ 4<<61 (cid:19)Uhjlrqdolvp dqg Pxowlodwhudolvp= Dq Ryhuylhz1(cid:20) Lq= Sdqdjdul|d D1/ Gh Phor M1 +Hgv1,/ Qhz Glphqvlrqv lq Uhjlrqdo Lqwhjudwlrq1 R{irug Xqlyhuvlw| Suhvv/ R{irug1 ^6‘ Ekdjzdwl/ M1/ dqg D1 Sdqdjdul|d/ 4<<91 (cid:19)Suhihuhqwldo Wudglqj Duhdv dqg Pxowlodwhudolvp= Vwudqjhuv/ Iulhqgv/ ru IrhvB(cid:20) Lq= Ekdjzdwl M1/ Sdqdjdul|d D1 +Hgv1,/ Wkh Hfrqrplfv ri Suhihuhqwldo Wudgh Djuhhphqwv1 DHL Suhvv/ Zdvklqjwrq GF1 ^7‘ Erqg/ H1 dqg F1 V|ursrxorv/ 4<<9 (cid:20)Wkh Vl}h ri Wudglqj Eorfv/ Pdunhw Srzhu dqg Zruog Zhoiduh H(cid:30)hfwv/(cid:20) Mrxuqdo ri Lqwhuqdwlrqdo Hfrqrplfv 73=607/ ss1 744076:1 ^8‘ Eudqghu M1/ NuxjpdqS1/ 4<;61 (cid:19)D cUhflsurfdo Gxpslqj* Prgho riLqwhuqdwlrqdo Wudgh1(cid:20)Mrxu0 qdo ri Lqwhuqdwlrqdo Hfrqrplfv 48/ 64606561 ^9‘ Eudqghu M1/ Vshqfhu E1/ 4<;71 (cid:19)Wdul(cid:30) Surwhfwlrq dqg Lpshuihfw Frpshwlwlrq1(cid:20) Lq= K1 Nlhu}nrzvnl +Hg1,/ Prqrsrolvwlf Frpshwlwlrq dqg Lqwhuqdwlrqdo Wudgh1 R{irug Xqlyhuvlw| Suhvv/ R{irug1 ^:‘ Hwklhu/ Z1/ 4<<;1 (cid:19)Uhjlrqdolvp lq d Pxowlodwhudo Zruog1(cid:20) Mrxuqdo ri Srolwlfdo Hfrqrp| 439/ 4547045781 ^;‘ Iuhxqg/ F153331(cid:19)PxowlodwhudolvpdqgwkhHqgrjhqrxvIrupdwlrqriSuhihuhqwldoWudghDjuhh0 phqwv1(cid:20) iruwkfrplqj Mrxuqdo ri Lqwhuqdwlrqdo Hfrqrplfv1 ^<‘ Jurvvpdq/ J1/ Khospdq H1/ 4<<81 (cid:19)Wkh Srolwlfv ri Iuhh Wudgh Djuhhphqwv1(cid:20) Dphulfdq Hfr0 qrplf Uhylhz ;8/ 99:09<31 49
^43‘ Nhqqdq/M1dqgU1Ulh}pdq/4<<31(cid:19)RswlpdoWdul(cid:30)HtxloleuldzlwkFxvwrpvXqlrqv(cid:20)Fdqdgldq Mrxuqdo ri Hfrqrplfv 56=4/ :30;6 ^44‘ Nulvkqd/ S1/ 4<<;1 (cid:19)Uhjlrqdolvp dqg Pxowlodwhudolvp= D Srolwlfdo Hfrqrp| Dssurdfk1(cid:20) Txdu0 whuo| Mrxuqdo ri Hfrqrplfv 446/ 55:05841 ^45‘ Ohy|/ S1/ 4<<:1 (cid:19)D Srolwlfdo0Hfrqrplf Dqdo|vlv ri Iuhh0Wudgh Djuhhphqwv1(cid:20) Dphulfdq Hfr0 qrplf Uhylhz ;:/ 83904<1 ^46‘ Sdqdjdul|d/ D1 5333/ (cid:19)Suhihuhqwldo Wudgh Olehudol}dwlrq= Wkh Wudglwlrqdo Wkhru| dqg Qhz Ghyhorsphqwv1(cid:20) Mrxuqdo ri Hfrqrplf Olwhudwxuh Yro1 6;/ ss15;:06641 ^47‘ Sdqdjdul|d/ D1 dqg U1 Ilqgod| 4<<9/ (cid:19)D Srolwlfdo Hfrqrp| Dqdo|vlv ri Iuhh Wudgh Duhdv dqg Fxvwrpv Xqlrqv1(cid:20) Ihhqvwud/ U1 Jurvvpdq/ J1/ Luzlq/ G1/ Hgv1 Wkh Srolwlfdo Hfrqrp| ri Wudgh Srolf|= Sdshuv lq krqru ri Mdjglvk Ekdjzdwl1 Fdpeulgjh dqg Orqgrq= PLW Suhvv/ sdjhv 5980;:1 ^48‘ \l/ V1/ 4<<91 (cid:19)Hqgrjhqrxv Irupdwlrq ri Fxvwrpv Xqlrqv xqghu Lpshuihfw Frpshwlwlrq= Rshq Uhjlrqdolvp lv Jrrg1(cid:20) Mrxuqdo ri Lqwhuqdwlrqdo Hfrqrplfv 74/ 48604::1 4:
8 Dsshqgl{ Wklv surri e| frqwudglfwlrq vkrzv wkdw wkh ydoxh ri zhoiduh dw y @ q(cid:3)4 lv juhdwhu wkdq dw y @ 31 Zhoiduh dw y @3 lv +q+4(cid:3)f,(cid:3)+q(cid:3)4,w,2 +4(cid:3)f.+q(cid:3)4,w,2 Z%+y @ 3,@ . . +D4, K 5d+q.4,2 +q.4,2d +4(cid:3)f(cid:3)+z.5,w,2 +4(cid:3)f(cid:3)5w, +q(cid:3)4, .+q(cid:3)4,w = +q.4,2d +q.4,d Zhoiduh dw y @q(cid:3)4 lv +q+4(cid:3)f,,2 +4(cid:3)f,2 +4(cid:3)f.+q(cid:3)4(cid:3)z,w,2 Z%+y @q(cid:3)4,@ . .+q(cid:3)4, = K 5d+q.4,2 +q.4,2d +q.4,2d Wkh gl(cid:30)huhqfh ehwzhhq wkh wzr lv (cid:3) (cid:4) 4 +7q2(cid:3)9q.5,+4(cid:3)f,.w+q(cid:3)4, 5q2.4(cid:3)6q(cid:3)7qz(cid:3)7z Z%+y @q(cid:3)4,(cid:3)Z%+y @3, @ w > K K 5 d+q.4,2 +D5, zklfk lv ohvv wkdq }hur li wkh ydoxh dw y @ 3 lv juhdwhu wkdq dw y @ q(cid:3)41 Wkh (cid:31)uvw whup lq wkh qxphudwru lv srvlwlyh/ vr lw lv qhjdwlyh rqo| li wkh vhfrqg whup lv qhjdwlyh dqg juhdwhu lq devroxwh ydoxh wkdq wkh (cid:31)uvw whup1 Wkh vhfrqg whup lv ghfuhdvlqj lq w1 Wkh pd{lpxp ydoxh ri w lv w@ (cid:129)3S= (cid:25)n2 Dvvxplqj wkh vhfrqg whup lq wkh qxphudwru lv ohvv wkdq }hur/ dqg soxjjlqj lq wkh pd{lpxp ydoxh ri w/ wkh iroorzlqj frqglwlrq pxvw krog li y @3 lv wkh pd{lpxp= 4(cid:3)f (cid:3) (cid:4) 5q2.8q(cid:3)9z(cid:3)6 Q (cid:24)+7q2(cid:3)9q.5,+4(cid:3)f,. +q(cid:3)4, 5q2.4(cid:3)6q(cid:3)7qz(cid:3)7z @+4(cid:3)f,+q(cid:3)4, > z.5 z.5 +D6, zkhuh Q lv wkh qxphudwru ri wkh h{suhvvlrqrq wkh uljkw0kdqg vlgh ri htxdwlrq D41 LiQ lv ohvv wkdq }hur wkhq wkh h{suhvvlrq rq wkh uljkw vlgh ri htxdwlrq D6 pxvw dovr eh ohvv wkdq }hur1 Wklv lpsolhv wkdw z A E?n(cid:12)(cid:28)E2?3(cid:129)(cid:28)> exw z pxvw eh ohvv wkdq q(cid:3)4> d frxqwu| fdq qrw kdyh djuhhphqwv zlwk pruh S frxqwulhv wkdq h{lvw1 Khqfh/ y @q(cid:3)4 lv wkh pd{lpxp1 4;
0.43 0.425 0.42 0.415 0.41 0.405 0.4 50 00 20 40 v 60 80 100 w 100 = Iljxuh 4= Grphvwlf dqg Iruhljq Elodwhudo Djuhhphqwv 0.4 0.39 0.38 0.37 0.36 0.35 0.4 0.34 0.2 0.33 t 00 20 40 v 60 80 100 Iljxuh 5= Elodwhudo Djuhhphqwv/ Wdul(cid:30)v/ dqg Zhoiduh 4<
Cite this document
Caroline Freund (2000). Spaghetti Regionalism (IFDP 2000-680). Board of Governors of the Federal Reserve System, International Finance Discussion Papers. https://whenthefedspeaks.com/doc/ifdp_2000-680
@techreport{wtfs_ifdp_2000_680,
author = {Caroline Freund},
title = {Spaghetti Regionalism},
type = {International Finance Discussion Papers},
number = {2000-680},
institution = {Board of Governors of the Federal Reserve System},
year = {2000},
url = {https://whenthefedspeaks.com/doc/ifdp_2000-680},
abstract = {This paper examines the welfare implications of multiple free trade agreements in a model of imperfect competition. We show that free trade is the unique Nash equilibrium under the simple rule that any two countries can form a bilateral free trade agreement. Specifically, a country is always better off forming a bilateral trade agreement with every other country, irrespective of previous agreements. This suggests that each new preferential free trade agreement may be a step towards multilateral free trade.},
}