Global and local indeterminacy and optimal environmental public policies in an economy with public abatement activities

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(1)Documento de trabajo E2004/79. Global and local indeterminacy and optimal environmental public policies in an economy with public abatement activities Rafaela Pérez Jesús Ruiz.

(2) Centro de Estudios Andaluces. Documento de Trabajo Serie Economía E2004/79 Global and local indeterminacy and optimal environmental public policies in an economy with public abatement activities1 Rafaela Pérez Jesús Ruiz U. Complutense (Madrid) and Fundación Centro de Estudios Andaluces RESUMEN Estudiamos las propiedades dinámicas de un modelo de crecimiento endógeno con contaminación en el que el gobierno puede controlar la contaminación mediante impuestos distorsionantes sobre las empresas que contaminan y mediante actividades públicas encaminadas a reducir la contaminación. Primero, caracterizamos las condiciones para que se produzca indeterminación de equilibrios cuando el gobierno es benevolente y elige su política fiscal teniendo en cuenta el equilibrio descentralizado. Demostramos que, en este contexto de second-best, puede encontrarse indeterminación global (puede haber dos sendas de crecimiento equilibrado) y local. Por tanto, bajo indeterminación, la política económica óptima no garantiza que la economía convergerá hacia el estado estacionario caracterizado por un nivel de contaminación bajo y tampoco garantiza que, durante la transición, la economía mostrará bajos niveles de emisiones. Segundo, demostramos que la solución del planificador también puede estar indeterminada, en particular, pueden existir dos tipos impositivos medioambientales Pigouvianos. Palabras clave: Indeterminación global y local, Impuestos medioambientales, Contaminación, Actividades de reducción de la contaminación, Impuestos Pigouvianos ABSTRACT We study the dynamic properties of an endogenous growth model with pollution in which the government can control the pollution through distorting taxes on the pollutant firms and through public abatement activities. First, we characterize the conditions for indeterminacy of equilibria when the government is benevolent and chooses its tax policy by taking into account the decentralized competitive equilibrium. Under this second best setup we show that two balanced growth paths can be found (one with a low level of pollution and the other with a high level) both of which can be locally indeterminate. Therefore, under indeterminacy, the optimal public policies do not guarantee that the economy will converge towards the steady state characterized by a low level of pollution and neither guarantee that the economy will display, along the transition, low levels of pollution. Second, we show that the central planner solution might also display indeterminacy; in particular, two Pigouvian taxes can be found. Keywords: Global and local indeterminacy, Environmental taxes, Pollution, Abatement, Pigouvian taxes. JEL Classification: H23, O41, Q28. 1. The authors would like to thank professors A. Novales, S. Smulders and the participants at the 1st. Atlantic Workshop on Energy and Environmental Economics for helpful comments and financial support from Fundación Centro de Estudios Andaluces, Fundación Ramón Areces and from the Spanish Ministry of Education through grant BEC2003-03965. Correspondence: Jesús Ruiz, Departamento de Economía Cuantitativa, Universidad Complutense de Madrid, Campus de Somosaguas, 28223 Madrid, Spain. Fax: (34) 91 394 26 13, e-mail: [email protected]..

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(7) Centro de Estudios Andaluces. Whfkqrorj| fdq dovr eh h{suhvvhg lq shu fdslwd whupv ru shu oderu xqlwv= + ' } 3 k & k c dqg khqfh/ lw lv dvvxphg wkdw sxeolf surgxfwlyh vhuylfhv duh qrw vxemhfw wr frqjhvwlrq1 Wkh uvw rughu frqglwlrq iru fdslwdo dqg oderu duh dv xvxdo= + c & ' E E k + . +7,. o ' E k. 516. +8,. Jryhuqphqw vhfwru. Wkh jryhuqphqw exgjhw lv edodqfhg lq hyhu| shulrg ri wlph= + '}n5 c. +9,. zkhuh + ' }3k &k duh wrwdo wd{ uhyhqxhv dqg } n 5 duh wrwdo jryhuqphqw vshqglqj/ 5 ehlqj wkh h{shqglwxuhv doorfdwhg wr lqfuhdvlqj wkh hqylurqphqwdo txdolw|/ wkdw lv/ wkh rz ri dedwhphqw dfwlylwlhv1 Wkh doorfdwlrq ri wrwdo jryhuqphqw vshqglqj ehwzhhq lqyhvwphqw dqg dedwhphqw lv pdgh e| iroorzlqj wkh uxohv= } ' E }3k &k c 5 ' } 3k &k c. +:, +;,. zkhuh lv wkh vkduh ri sxeolf vshqglqj ghyrwhg wr dedwhphqw dfwlylwlhv1 Zh zloo uhihu wr dv sxeolf vshqglqj frpsrvlwlrq1 Iurp +:, dqg +;,= } ' d & 5 '. 517. E o*k c. +<,. } E . +43,. Srooxwlrq htxdwlrq. Zh dgrsw wkh srooxwlrq ixqfwlrq sursrvhg e| Vpxoghuv dqg Judgxv +4<<9,/# wkdw lv/ srooxwlrq rz ghshqgv srvlwlyho| rq wkh fdslwdo vwrfn xvh dqg qhjdwlyho| rq wkh dedwhphqw f1% Khqfh/ e| xvlqj +<, dfwlylwlhv/ l1h1 ' s Egc ~ ' s E&c 5/$ zkhuh s& : f dqg s5 dqg +43, lq wklv ghqlwlrq= & Q ' 'k 5. & Q . E3. }. l '. & Q 3 . dk E E3k. *k. o. . +44,. Prkwdgl +4<<9, dgrsw d vlplodu ixqfwlrq iru wkh ohyho ri wkh hqylurqphqwdo txdolw|/ zkhuh wkh hqyl0 urqphqwdo txdolw| lv lqyhuvho| uhodwhg wr srooxwlrq rz1 Qrwh wkdw zh dvvxph frqvwdqw srsxodwlrq qrupdol}hg wr rqh1 Khqfh/ N ( n> ] ( }= Ci Zh ghqh i{ ( C{ > iru { ( n>}= # $ %. 9.

(8) Centro de Estudios Andaluces. 518. Ghfhqwudol}hg Frpshwlwlyh Htxloleulxp. Ghqlwlrq 4 D frpshwlwlyh htxloleulxp iru wklv hfrqrp| lv d vhw ri doorfdwlrqv iSc &c ,j |'f dqg d sulfh v|vwhp ic oj |'f vxfk wkdw jlyhq d sulfh v|vwhp dqg d vfdo srolf| i c j|'f = l, iSc &j |'f pd{lpl}hv krxvhkrog*v xwlolw| +4,/ vxemhfw wr +5,/ dqg wdnlqj wkh srooxwlrq ohyho i j|'f dqg i&f j dv jlyhq> ll, i&c ,j |'f vdwlvhv wkh upv surw pd{lpl}dwlrq frqglwlrqv/ dqg lll, doo pdunhwv fohdu/ wkdw lv/ iSc &c }c 5j |'f vdwlvhv wkh djjuhjdwh uhvrxufhv frqvwudlqw +45, ehorz dqg wkh jryhuqphqw exgjhw frqvwudlqw +9, dqg/ lq wkh oderu pdunhw/ , ' = S n & ' E }3k &k . +45,. S ' [E c 4 c S. +46,. [E c ' E k *k d E oE3k*k . +47,. E| xvlqj +6,/ +7, dqg +<,=. zkhuh. Iurp +5,/ +7,/ +8, dqg +<,= & [E c S ' & k &. +48,. Qrwh wkdw/ iurp +46,/ wkh orqj0uxq jurzwk udwh lv srvlwlyh li dqg rqo| li [E c : 4 Khqfh/ zh zloo rqo| frqvlghu edodqfhg jurzwk sdwkv lq zklfk wkh wlph sdwkv iru c jxdudqwhh wkdw wklv frqglwlrq krogv1 Wkh ghfhqwudol}hg htxloleulxp lv joredoo| dqg orfdoo| ghwhuplqdwh1 Lq wkh hqg/ wklv prgho lv txlwh vlplodu wr Eduur*v +4<<3, prgho/ dqg lw lv zhoo nqrzq wkdw Eduur*v prgho lv orfdoo| dqg joredoo| ghwhuplqdwh1. 6. Vhfrqg0ehvw srolf|. Wkh jryhuqphqw fkrrvhv hfrqrplf srolf| i c j wr pd{lpl}h wkh xwlolw| ri wkh uhsuhvhq0 wdwlyh krxvhkrog/ vxemhfw wr wkh frpshwlwlyh htxloleulxp frqglwlrqv1 Wkdw lv= ] " e34| d*? S # *? o _| @%. tSc&cc c. f. r| G S ' S d[E c 4o [E c & ' &S k ' &Q! k E E!k. +49, +4:, ! *k . +4;,. Zh uhihu wr wklv rswlpdo frqwuro sureohp dv wkh Udpvh| sureohp dqg wr wkh dvvrfldwhg doorfdwlrqv dqg srolflhv dv wkh Udpvh| doorfdwlrqv dqg wkh Udpvh| sodq1 Zh dvvxph wkdw :.

(9) Centro de Estudios Andaluces. wkh jryhuqphqw kdv vrph frpplwphqw srzhu wr elqg lwvhoi wr lpsohphqw/ dw dq| ixwxuh gdwh/ wkh sodq lw kdv dqqrxqfhg lq wkh uvw shulrg1 Wkh fxuuhqw0ydoxh Kdplowrqhdq lv= M e!4| d*? S # E 2 *? & n k n#2 *? n *?E n *? E n k k [E c 4| 4| ! ! ne > dSE[E c 4o n e >2 &S k zkhuh > dqg >2 duh wkh g|qdplf pxowlsolhuv dvvrfldwhg wr +49, dqg +4:,/ uhvshfwlyho|1 Wkh qhfhvvdu| frqglwlrqv zlwk uhvshfw wr iSc &c > c >2 c c j duh jlyhq e|= YM YS YM Y& YM Y> YM Y>2 YM Y YM Y. . ' > 4> $ > ' n >2 > d[ 4o n 4> c S [ n # E 2 c ' > 2 4>2 $ > 2 ' >2 4 k & ' f $ S ' S d[ 4o c. +53, +54,. [ ' f $ & ' &S c k & # ' f $ f ' 2 n > S [ n >2 [ c k k k & ' f $ f ' #2 n > S [ n >2 [ k E k k Ek & $ f ' #2 n > S Ek [ n >2 [ c k. zkhuh [ ' [E c / [ ' YY[ ' ! k!k [/ [ ' YY[ ' Ek! k. +4<,. +55, +56,. +57,. [ ! 1. E. Sursrvlwlrq 4 Wkh rswlpdo udwlr ri dedwhphqw wr wrwdo jryhuqphqw vshqglqj/ / idoov ehwzhhq f dqg k2 / dqg wkh rswlpdo wd{ udwh/ / idoov ehwzhhq E k dqg 4/ iru doo |1 Surri1 E| xvlqj +56, dqg +57,/ zh rewdlq wkdw k 'k . +58,. Khqfh/ lw lv reylrxv wkdw lv prqrwrqlfdoo| lqfuhdvlqj dqg frqfdyh lq 1 Ehfdxvh dqg pxvw eh ghqhg lq wkh lqwhuydo +3/4,/ dqg jlyhq +58,/ lw lv uhdglo| rewdlqhg wkdw wkh udqjh iru wkh ixqfwlrq +58, lv jlyhq e| d kc dqg wkh grpdlq lv jlyhq e| dfc k2 1 Fruroodu| 5 Wkh yduldwlrq udwh ri rswlpdo lv prqrwrqlfdoo| lqfuhdvlqj lq wkh yduldwlrq udwh ri rswlpdo 1. ;.

(10) Centro de Estudios Andaluces. Surri1 Iurp +58,/ zh rewdlq wkh lqvwdqwhqrxv fkdqjh iru rswlpdo . K. '. k E k . c. +59,. dqg glylglqj wklv h{suhvvlrq e| / zh rewdlq. K. ' . E k E k. +5:,. Jlyhq Sursrvlwlrq +4,/ !EE!!kk : f/ dqg khqfh/ +5:, lv prqrwrqlfdoo| lqfuhdvlqj lq Wkh v|vwhp +4<, 0 +57, frqvlvwv ri vl{ htxdwlrqv dqg vl{ yduldeohv1 Zh uhgxfh lwv glphqvlrqdolw| wr vlpsoli| frpsxwdwlrq1 Wkh qhz yduldeohv yhfwru zloo eh i% c %2 c j/ zkhuh % dqg %2 duh wzr dx{loldu| yduldeohv ghqhg dv= S % c +5;, & +5<, %2 >2 S c vr wkdw erwk % dqg %2 gr qrw jurz lq wkh vwhdg| vwdwh1 Wklv lpsolhv wkdw S dqg & jurz dw wkh vdph udwh/ ghqrwhg e| 1 Glhuhqwldwlqj +5;,/ +5<,/ dqg +56, zlwk uhvshfw wr wlph dqg xvlqj +4<, wr +57,/ zh rewdlq diwhu vrph dojheud= K 4 n % % c % ' [ +63, k %2 ' [ %2 n # E 2 % c k. K. '. Xl c #2. +64,. +65,. zkhuh=& ' [E ' E k *k dE kE n koE!k*k c [ X ' XE '. +66,. kE E n kE k2 [ c E k d k k2 o kE E n k. +67,. # E 2 %2 E k 4#2 k k E k[. +68,. l ' l E%2 c '. Qrz/ wkh vwhdg| vwdwh dqg wkh wudqvlwlrqdo g|qdplf surshuwlhv ri wkh Udpvh| sureohp fdq eh vwxglhg iurp wkh wkuhh0glphqvlrqdo v|vwhp +63,0+65, lq i% c %2 c j1 &. Qrwh wkdw e| xvlqj +58, lq +47,/ zh rewdlq +66,1 <.

(11) Centro de Estudios Andaluces. 614. Vwhdg| vwdwh dqg joredo lqghwhuplqdf|. Wklv vhfwlrq irfxvhv rq wkh vwhdg| vwdwh ru edodqfhg jurzwk sdwk +EJS,1 Dq lqwhulru vwhdg| vwdwh lv d yhfwru +% c %2 c , vdwlvi|lqj wkh htxdwlrqv iru wkh Udpvh| sureohp +lq sduwlfxodu/ wkh htxdwlrqv +63,0+65,,/ vxfk wkdw li lw lv hyhu uhdfkhg/ wkh v|vwhp zloo vwd| / wkh vwhdg| vwdwh lv jlyhq e| dw wkdw srlqw iruhyhu + % ' %2 ' ' f,1 Ghqlqj [ ' [E K. K. % ' 4 n. E k[ c k %. %2 ' # E 2 dqg. n. +69, &. 4k E k[. c. +6:,. lv rewdlqhg iurp wkh lpsolflw htxdwlrq= nk 4# [' 2 # E 2 E n k . +6;,. Wkh uvw vwdjh lv wr qg wkh vroxwlrq iru +6;, lq wkh orqj0uxq wd{ udwh/ 1 Rqfh lw kdv ehhq rewdlqhg/ +69, fdq jlyh wkh frqvxpswlrq wr fdslwdo udwlr lq wkh vwhdg| vwdwh +% , dqg +6:, ghwhuplqhv wkh orqj0uxq ydoxh iru wkh dx{loldu| yduldeoh %2 1 Sursrvlwlrq 6 D qhfhvvdu| frqglwlrq iru wkh frqvxpswlrq orqj0uxq jurzwk udwh wr eh srvlwlyh lv wkdw 2. nk : / wrjhwkhu zlwk : # E 2 c E n k . +6<,. 2. nk E n k . +73,. ru / wrjhwkhu zlwk . # E 2 . Surri1 Iurp +54,/ wkh frqvxpswlrq jurzwk udwh lv srvlwlyh lq htxloleulxp li dqg rqo| li [ : 4 Khqfh/ d qhfhvvdu| frqglwlrq iru srvlwlyh jurzwk lv [ : f Dv orqj dv Sursrvlwlrq +4, lpsolhv wkdw k/ dqg khqfh/ 2 E nnkk : fc lw lv reylrxv iurp +6;, wkdw wkh !. frqglwlrq iru [ : f lv fkdudfwhul}hg e| +6<, ru +73,1. Sursrvlwlrq 7 Wkh vhfrqg ehvw frqvxpswlrq orqj0uxq jurzwk udwh lv srvlwlyh li dqg rqo| li #2 : n k Surri1 Iurp +54, wkh frqvxpswlrq jurzwk udwh lv srvlwlyh lq htxloleulxp li dqg rqo| k l # nk li [ : 4 Iurp +6;,/ [ : 4 li dqg rqo| li #E 2 E nk : Diwhu vrph Q dojheud/ wkh frqglwlrq #2 : n k fdq eh rewdlqhg1 !. !. !. Lq rughu wr vwxg| wkh frqglwlrqv iru joredo lqghwhuplqdf|/ uvw/ zh fkdudfwhul}h wkh vkdsh ri wkh ohiw0kdqg vlgh ri +6;,/ ghqrwhg e| uM7E ' [ ' E k *k dE kE n koE. !. 43. k*k. c.

(12) Centro de Estudios Andaluces. dqg wkh vkdsh ri wkh uljkw0kdqg vlgh ri +6;,/ ghqrwhg e| nk 4# 2 -M7E ' # E 2 E n k Sursrvlwlrq 8 Wkh ixqfwlrq uM7E lv prqrwrqlfdoo| ghfuhdvlqj dqg frqfdyh dqg lwv Q!k grpdlq lv jlyhq e| dfc k2 *k E k k o / ; 5 d kc Surri1 Vhh Dsshqgl{1 Sursrvlwlrq 9 Li #E 2 dqg 2 nkk E: / wkhq wkh ixqfwlrq -M7E lv prqrwrqlfdoo| lqfuhdvlqj/ frqfdyh/ dqg lwv grpdlq lv E4c V/ zlwk V : E f Surri1 Vhh Dsshqgl{1 Sursrvlwlrq : Li : #E 2 dqg 2 nkk E: / wkhq wkh ixqfwlrq -M7E lv prqrwrqlfdoo| ghfuhdvlqj dqg frqyh{ dqg lwv grpdlq lv EVc 4/ zlwk V E:f Surri1 Vhh Dsshqgl{1 Ohw dqg eh wkh rswlpdo orqj0uxq ohyhov iru wkh jurzwk udwh dqg wkh vshqglqj frpsrvlwlrq/ uhvshfwlyho|1 Qrz/ wkh frqglwlrqv iru joredo lqghwhuplqdf| fdq eh fkdudfwhul}hg1 Iurp wkh sursr0 vlwlrqv deryh/ irxu w|shv ri vlwxdwlrqv fdq eh rewdlqhg= l, Li -M7E lv prqrwrqlfdoo| lqfuhdvlqj/ frqfdyh/ dqg lwv grpdlq lv E4c V/ zlwk V : f/ wkhq -M7E : f iru $ ! Lq wklv vlwxdwlrq/ vhh jxuh 4d/ wkhuh h{lvwv rqh vlqjoh vroxwlrq iru / zklfk yhulhv wkh qhfhvvdu| frqglwlrq iru srvlwlyh jurzwk udwh1 ll, Li -M7E lv prqrwrqlfdoo| lqfuhdvlqj/ frqfdyh/ dqg lwv grpdlq lv E4c V/ zlwk V f/ wkhq -M7E f iru $ ! Lq wklv fdvh/ wkhuh lv qr vroxwlrq +vhh jxuh 4e,1 lll, Li -M7E lv prqrwrqlfdoo| ghfuhdvlqj dqg frqyh{ dqg lwv grpdlq lv EVc 4/ zlwk V f/ wkhq -M7E f iru $ ! Khqfh/ wkhuh lv rqh vlqjoh vroxwlrq iru +vhh jxuh 5,1 Lq wklv fdvh/ wkh ixoophqw ri wkh qhfhvvdu| frqglwlrq iru srvlwlyh orqj0uxq jurzwk udwh +Sursrvlwlrq 5, xqghu wkh rswlpdo wd{ udwh pxvw eh fkhfnhg/ ehfdxvh lw grhv qrw krog iru $ ! 1 ly, Li -M7E lv prqrwrqlfdoo| ghfuhdvlqj dqg frqyh{ dqg lwv grpdlq lv EVc 4/ zlwk V : f/ wkhq -M7E : fc ; 5 d kc Wkhq wkhuh duh wkuhh srvvlelolwlhv= Iluvw/ uM7E olhv ehorz -M7E c ; 5 d kc > lq wklv fdvh wkhuh lv qr vroxwlrq +vhh jxuh 6d,1 Vhfrqg/ uM7E lv wdqjhqw wr -M7E > lq wklv fdvh/ wkhuh lv rqh vlqjoh vroxwlrq +vhh jxuh 6e,1 Wklug/ uM7E olhv deryh -M7E dv vkrzq lq jxuh 6f> lq wklv fdvh/ wkhuh duh wzr vroxwlrqv1 Wklv odvw vlwxdwlrq jxdudqwhhv wkh h{lvwhqfh ri joredo lqghwhuplqdf|/ wkdw lv/ wkhuh h{lvw wzr orqj0uxq rswlpdo wd{ udwhv dqg 2 vxfk wkdw E k 1 Khqfh/ wkhuh duh wzr Udpvh| doorfdwlrqv fruuhvsrqglqj wr wzr 2 Udpvh| sodq1 Iurp +58, dqg +46,/ wkh rswlpdo E c dqg E2 c 2 fdq eh rewdlqhg/ yhuli|lqj wkdw f 2 k2 dqg : 2 ' '. Qrwh wkdw lq wklv fdvh = yhulhv wkh qhfhvvdu| frqglwlrq iru srvlwlyh jurzwk udwh +Sursrvlwlrq 5,1 44.

(13) Centro de Estudios Andaluces. Rq wkh rwkhu kdqg/ li Q : wkh srooxwlrq jurzwk udwh zloo eh srvlwlyh dqg iru Q zloo eh odujhu wkdq iru 1 Ixuwkhupruh/ wkh vkduh ri sxeolf vshqglqj ghyrwhg wr dedwhphqw lv orzhu wkh orzhu wkh wd{ udwh/ dqg khqfh/ wkh srooxwlrq ohyho zloo dovr eh odujhu iru Q fhwhulv sdulexv1 Li Q / wkh srooxwlrq jurzwk udwh lv qhjdwlyh/ dqg khqfh/ srooxwlrq zloo idoo dw d idvwhu udwh wkh orzhu 1 Krzhyhu/ wkh dedwhphqw dfwlylwlhv duh orzhu iru Q vr wkdw wkh rqo| gxulqj wkh uvw shulrgv1 srooxwlrq sdwk iru Q olhv deryh wkh srooxwlrq sdwk iru Wkh hfrqrplf lpsolfdwlrq ehklqg joredo lqghwhuplqdf| lv wkdw wzr hfrqrplhv zlwk wkh vdph ixqgdphqwdov frxog fkrrvh glhuhqw rswlpdo srolflhv +wd{hv dqg vshqglqj frpsrvl0 wlrq, wkdw zrxog |lhog glhuhqw rswlpdo sdwkv ri srooxwlrq1 Wklv uhvxow lv deoh wr h{sodlq wkh odfn ri frqyhujhqfh lq wkh srooxwlrq ohyhov ri frxqwulhv zlwk vlplodu ixqgdphqwdov1 Lw lv hdv| wr surylgh vhyhudo qxphulfdo h{dpsohv ri d xqltxh lqwhulru EJS ru wzr lqwhulru EJSv1 Wkh uvw h{dpsoh fruuhvsrqgv wr d xqltxh lqwhulru EJS wkdw fdq eh rewdlqhg iru wkh iroorzlqj vhw ri vwuxfwxudo sdudphwhuv= k ' f./ ' / 4 ' *L}EfbD/ # ' e/ Q ' fH/ ' f Wkh rswlpdo wd{ udwh fruuhvsrqglqj wr wklv iudphzrun lv ' fHH/ wkh rswlpdo vshqglqj frpsrvlwlrq lv ' feS dqg wkh rswlpdo jurzwk udwh lv ' fffb1 Lw lv hdv| wr fkhfn wkdw zlwk wkhvh sdudphwhu ydoxhv wkh hfrqrp| lv sodfhg lq wkh vlwxdwlrq glvsod|hg lq jxuh 4d1 Wkh vhfrqg h{dpsoh |lhogv d xqltxh lqwhulru EJS fruuhvsrqglqj wr wkh vlwxdwlrq lq jxuh 6e1 Wkh vhw ri vwuxfwxudo sdudphwhuv lv jlyhq e|= k ' f./ ' / 4 ' *L}EfbD/ # ' f.2D/ Q ' fe/ ' f2 Wkh rswlpdo wd{ udwh lv ' f.SH/ wkh rswlpdo vshqglqj frpsrvlwlrq lv ' fe2. dqg wkh rswlpdo jurzwk udwh lv ' ffS Wkh wklug h{dpsoh ohdgv wr wzr lqwhulru EJS olnh wkh vlwxdwlrq glvsod|hg lq jxuh 6f1 Wkh vhw ri sdudphwhuv lv= k ' f./ ' f./ 4 ' *L}EfbD/ # ' / Q ' f./ ' fS Iru wkh uvw EJS/ wkh rswlpdo wd{ udwh lv Q ' fDH/ wkh rswlpdo vshqglqj frpsrvlwlrq lv Q ' f e dqg wkh rswlpdo jurzwk udwh lv Q ' ffS. Wkh vhfrqg EJS lv ghqhg e| ' f.H/ ' fe / ' ffeD. 615. Vrxufh ri lqghwhuplqdf|. Lq wklv prgho/ wkh hqylurqphqwdo h{whuqdolw| lq wkh xwlolw| ixqfwlrq lv nh| wr rewdlq wkh uhvxow ri joredo lqghwhuplqdf|1 Zh vkrz wkdw lqghwhuplqdf| lv qrw suhvhqw lq wkh fdvh ri # ' f/ wkdw lv/ srooxwlrq grhv qrw dhfw wkh djhqw*v zhoiduh= li # ' f/ htxdwlrq +56, ehfrphv & [ ' f +74, >Q S n > k Wkh whup lq eudfnhwv lv qhyhu }hur/: vr wkh h{suhvvlrq deryh rqo| krogv li [ ' f Wklv frqglwlrq lv rqo| jxdudqwhhg iru ' k1 Wkh vlpsolfdwlrq dfklhyhg e| lpsrvlqj # ' fc ohdgv wr Eduur +4<<3, prgho/ dqg ehfdxvh ri wkdw/ wkh rswlpdo wd{ udwh lv wkh vdph rewdlqhg e| klp1 Lq wklv vlpsolhg hfrqrp|/ wkh rswlpdo shufhqwdjh ri sxeolf vshqglqj ghyrwhg wr dedwhphqw/ / lv }hur/ vlqfh wkhuh lv qr lqfhqwlyh wr uhgxfh wkh srooxwlrq ohyho1. Glhuhqwldwlrq ri wkh whup lq eudfnhwv zlwk uhvshfw wr wlph |lhogv dq h{suhvvlrq wkdw lv qrw frpsdwleoh zlwk wkh rwkhu uvw rughu frqglwlrqv +4<,0+55,1 :. 45.

(14) Centro de Estudios Andaluces. Rq wkh frqwudu|/ wkh rwkhu w|sh ri h{whuqdolw| lq rxu ehqfkpdun hfrqrp| lv qrw fuxfldo wr dfklhyh wkh lqghwhuplqdf| uhvxow1 Li wkh sxeolf vhuylfhv gr qrw surylgh dq| surgxfwlrq h{whuqdolwlhv +k ' ,/ vwloo lv srvvleoh wr rewdlq d joredo lqghwhuplqdf| uhvxow1 Li k ' / wkh rswlpdo lv htxdo wr 4/ wkdw lv/ wkhuh lv qr lqfhqwlyhv wr doorfdwh sxeolf vshqglqj wr sxeolf vhuylfhv lq surgxfwlrq1 Lq wklv fdvh/ wkh htxdwlrq +56, ehfrphv & # +75, ' >Q S n > E k Glhuhqwldwlqj zlwk uhvshfw wr wlph wkh h{suhvvlrq +75, dqg xvlqj +4<,0+55,/ zh rewdlq E 2 # 4# ' f +76, n E n #EQ n E E Iurp +76,/ lq wkh vwhdg| vwdwh + ' f, wkh rswlpdo wd{ udwh + , pxvw yhuli| d #EQ o '. 4# / ; 5 Efc E . +77,. Lw lv uhdglo| rewdlqhg wkdw wkh uljkw0kdqg vlgh ri htxdwlrq +77, lv srvlwlyh/ lqfuhdvlqj/ c 4 Rq wkh rwkhu kdqg/ wkh ohiw0kdqg vlgh frqyh{/ zlwk udqjh +3/4, dqg grpdlq 4# lv d ud| zlwk vorsh #EQ Li #EQ c wkhuh lv qr vroxwlrq iru 5 Efc yhuli|lqj +77,1 Exw li : #EQ c wkhq zh fdq qg qr vroxwlrq/ rqh ru wzr vroxwlrqv iru Wklv odvw fdvh fruuhvsrqgv wr d joredo lqghwhuplqdf| vlwxdwlrq1. 616. Wudqvlwlrqdo g|qdplfv dqg orfdo lqghwhuplqdf|. Olqhdul}lqj wkh g|qdplf v|vwhp +63,0+65, durxqg wkh wkh iroorzlqj olqhdu v|vwhp= 5 6 5 % % f % [ 9 : 7 7 %2 8 ' # E 2 [ k %2 [ f s K. K. EJS lq +69,0+6;,/ doorzv wr rewdlq 6 65 % % k k 8 7 %2 %2 8 4. [ kE kE E nkE k k k zkhuh= [ ' Y [E Y ' kE E nk [c dqg s ' k# dE kE k k kE E nko Ohw K eh wkh Mdfreldq pdwul{ lq +78, dqg wkh fkdudfwhulvwlf htxdwlrq ri K lv= ! !. !. !. !. ! !. !. ! !. ! !. !. !. !. |oEK 2 n lEK _i|EK ' f c zkhuh= |oEK ' 24 c 6 5 2 E E k [ 8 l c _i|EK ' 4% [ 7 n k 2 k 2 E k[ [ l k %2 [ s n % [ n 4[ n % 4 lEK ' k 46. +78,.

(15) Centro de Estudios Andaluces. Zh zloo xvh wkh Urxwk*v Wkhruhp lq rxu dqdo|vlv +vhh Jdqwpdfkhu +4<93/ Fkdswhu [Y,,1 Lq rxu sduwlfxodu fdvh wklv wkhruhp vd|v wkdw wkh qxpehu ri urrwv ri K zlwk srvlwlyh uhdo sduwv lv htxdo wr wkh qxpehu ri yduldwlrqv ri vljq lq wkh vhtxhqfh= c |oEKc lEK n _i|EK |oEK c _i|EK Iurp qrz/ wklv vhtxhqfh zloo eh fdoohg wkh Urxwk*v Vhtxhqfh1 Dowkrxjk lw lv qrw srvvleoh wr nqrz iurp dq dqdolwlfdo srlqw ri ylhz wkh vljq iru wkh wklug whup lq wkh vhtxhqfh/ zh fdq fodlp wkh iroorzlqj vx!flhqw frqglwlrq iru orfdo lqghwhuplqdf|1. !k [ : f wkhq wkhuh h{lvw orfdo lqghwhuplqdf|1 Sursrvlwlrq ; Li n EQ!E!E!!k[ ![ o d !k [ : f wkhq _i|EK f1 Khqfh/ jlyhq wkh Urxwk*v Vhtxhqfh/ Surri1 Li n EQ !E!E!!k[ ![ o d wkhuh duh rqo| wzr yduldwlrqv ri vljq/ zkdwhyhu wkh vljq ri lEKn _i|EK |oEK eh1 Frqvhtxhqwo|/ wkhuh duh wzr urrwv zlwk srvlwlyh uhdo sduw dqg rqh urrw zlwk qhjdwlyh uhdo sduw1 Vlqfh wkh wkuhh yduldeohv lq wkh v|vwhp +78, duh mxps/ wkhuh h{lvw orfdo ghwhuplqdf| rqo| li wkh !k [ : fc wkuhh urrwv kdyh srvlwlyh uhdo sduw1 Khqfh/ xqghu wkh frqglwlrq n EQ!E!E!!k[ ![ o d wkh EJS lv orfdoo| lqghwhuplqdwh1 !k [ Qrwh wkdw lqghwhuplqdf| frxog dovr dulvh li n EQ!E!E!!k[ f dqg lEK n ![ o d _i|EK |oEK : f Lq wklv fdvh/ wkhuh duh wzr urrwv zlwk qhjdwlyh uhdo sduw ehfdxvh rqo| rqh yduldwlrq ri vljq zrxog h{lvw lq wkh Urxwk*v Vhtxhqfh1 Wkh lpsolfdwlrq ri orfdo lqghwhuplqdf| lv wkdw wkhuh zloo eh pxowlsoh sdwkv frqyhujlqj wr d jlyhq vwhdg| vwdwh1 Khqfh/ lqghwhuplqdf| jxdudqwhhv h{lvwhqfh ri d frqwlqxxp ri vxqvsrw vwdwlrqdu| htxloleuld/ l1h1/ vwrfkdvwlf udwlrqdo h{shfwdwlrqv htxloleuld ghwhuplqhg e| shuwxuedwlrqv xquhodwhg wr wkh xqfhuwdlqw| lq hfrqrplf ixqgdphqwdov1 Wkh lqwhuhvw ri vxqvsrw htxloleuld lv wkdw wkh| surylgh d wkhruhwlfdo mxvwlfdwlrq iru *dqlpdo vslulwv* xqghuo|lqj hfrqrplf lqvwdelolw|1 Lq rxu prgho/ lqghwhuplqdf| lv deoh wr h{sodlq zk| wzr frxqwulhv zlwk vlplodu ixqgdphqwdov +suhihuhqfhv/ whfkqrorj| dqg lqlwldo fdslwdo vwrfn, pljkw glvsod| glhuhqw srooxwlrq sdwkv hyhq li vxfk hfrqrplhv fkrrvh wkh vdph sxeolf srolflhv1 Lq wklv vhqvh/ wkh uhvxow ri orfdo lqghwhuplqdf| lv deoh wr h{sodlq wkh odfn ri hpslulfdo uhjxodulw| lq wkh srooxwlrq h{shulhqfhv ri glhuhqw hfrqrplhv> iru h{dpsoh/ wkh gl!fxow| wr qg d uhjxodulw| zlwk uhvshfw wr wkh Nx}qhwv fxuyh/ wkh uhvxowv ehlqj ghshqghqw rq wkh vdpsoh ri frxqwulhv dqg |hduv vwxglhg1 Wkdw lv/ wkh sdvw srooxwlrq h{shulhqfh ri d ghyhorshg frxqwu| grhv qrw jxdudqwhh wkdw d ghyhorslqj frxqwu| zloo iroorz wkh vdph sdwk dorqj wkh wlph dowkrxjk wkh lqlwldo frqglwlrqv dqg wkh hqylurqphqwdo srolflhv duh vlplodu1 Wkh qxphulfdo h{dpsohv vkrzq lq wkh suhfhglqj vxevhfwlrq duh dovr xvhixo wr looxvwudwh glhuhqw vlwxdwlrqv ri orfdo lqghwhuplqdf|1 Wkh uvw h{dpsoh/ ghqhg e| wkh iroorzlqj sdudphwhu vhw/ k ' f./ ' / 4 ' *L}EfbD/ # ' e/ ' fH/ 2 ' f/ glvsod|v orfdo dqg joredo ghwhuplqdf|1 Wkdw lv/ wkhuh h{lvwv rqh vlqjoh frqyhujlqj sdwk wrzdugv wkh vlqjoh vwhdg| vwdwh1 Wkh vhfrqg h{dpsoh +k ' f./ ' / 4 ' *L}EfbD/ # ' f.2D/ ' fe/ 2 ' f2, dovr glvsod|v orfdo dqg joredo ghwhuplqdf|1 47.

(16) Centro de Estudios Andaluces. Wkh wzr EJSv ri wkh wklug h{dpsoh +k ' f./ ' f./ 4 ' *L}EfbD/ # ' / ' f./ 2 ' fS, duh orfdoo| lqghwhuplqdwh1 Wkdw lv/ wkhuh h{lvw d frqwlqxxp ri htxloleuld frqyhujlqj wr hdfk EJS1 Lw lv dovr srvvleoh wr qg h{dpsohv ri joredo lqghwhuplqdf| zlwk rqh orfdoo| ghwhuplqdwh EJS dqg rqh orfdoo| lqghwhuplqdwh EJS1 Wkh iroorzlqj vhw ri sdudphwhuv k ' f./ ' / 4 ' *L}EfbD/ # ' / ' fe/ 2 ' f2/ |lhogv wzr EJSv fkdudfwhul}hg e| ' f.S/ ' fe2/ ' ffS. dqg 2 ' f.H/ 2 ' fe / 2 ' ffDb Wkh uvw EJS lv orfdoo| ghwhuplqdwh dqg wkh vhfrqg orfdoo| lqghwhuplqdwh1. 7. Sodqqhu vroxwlrq. Zh qrz ghulyh wkh fhqwudo sodqqhu vroxwlrq dv d uvw0ehvw uhihuhqfh srlqw1 E| frpsdulqj wkh uvw0rughu frqglwlrqv ri wkh ghfhqwudol}hg vroxwlrq zlwk wkh fruuhvsrqglqj uvw0rughu frqglwlrqv ri wkh sodqqhu vroxwlrq/ zh duh deoh wr ghwhuplqh wkh uvw0ehvw wd{ udwhv dqg vshqglqj frpsrvlwlrq +sljrxyldq wd{ udwh dqg vshqglqj frpsrvlwlrq,1 Vxfk sljrxyldq wd{ udwh dqg vshqglqj frpsrvlwlrq ohw wkh hfrqrp| uhdfk wkh h!flhqw sdwk dv d frpshwlwlyh htxloleulxp/ wkdw lv/ wkh| lqwhuqdol}h wkh h{whuqdolwlhv1 Zh zloo vkrz ehorz wkdw wkhuh duh vwuxfwxudo sdudphwhu vhwv iru zklfk lv qrw srvvleoh wr qg d wd{ udwh vr wkdw wkh frpshwlwlyh htxloleulxp uhsolfdwhv wkh h!flhqw sdwk/ l1h1/ wkh sljrxyldq wd{ udwh grhv qrw h{lvw1 Wkh ehqhyrohqw fhqwudo sodqqhu fkrrvhv frqvxpswlrq/ sxeolf dedwhphqw dfwlylwlhv dqg sxeolf surgxfwlrq vhuylfhv lq rughu wr pd{lpl}h wkh olihwlph xwlolw| ri wkh uhsuhvhqwd0 wlyh krxvhkrog vxemhfw wr wkh rz uhvrxufh frqvwudlqw ri wkh hfrqrp| dqg wkh srooxwlrq ixqfwlrq= ] " e34| d*? S # *? o _| @% tSc&cc}c5 B. vxemhfw wr S n & n } n 5 ' }3k &k & Q ' 5 &f jlyhq1 Wkh fxuuhqw0ydoxh Kdplowrqldq iru wklv rswlpl}dwlrq sureohp lv M ' *? S # *? & n #2 *? 5 n D }3k &k S } 5 c. 48.

(17) Centro de Estudios Andaluces. zkhuh D lv wkh fr0vwdwh yduldeoh1 Wkh uvw0rughu frqglwlrqv duh jlyhq e| YM YS YM Y& YM Y}. ' f$. ' Dc S. +79,. D ' # n k} 3k &k3 4c D &D 3k k ' f $ D E k} & ' fc *k & c ' $ } E k YM ' f $ D ' #2 Y5 5 ' 4D n D $ . +7:,. +7;, +7<,. Wkh frqvxpswlrq wr sxeolf dedwhphqw dfwlylwlhv udwlr lv rewdlqhg iurp +79, dqg +7<, S c ' 5 #2. +83,. zklfk lv d frqvwdqw iru doo |/ dqg wkh fdslwdo wr sxeolf surgxfwlyh vhuylfhv udwlr lv dovr frqvwdqw +7;,1 Diwhu holplqdwlqj wkh fr0vwdwh yduldeoh e| xvlqj htxdwlrqv +79,/ +7:, dqg +7;,/ wkh vwdqgdug Hxohu frqglwlrq iru wkh fhqwudo sodqqhu vroxwlrq fdq eh ghulyhg= S S ' # n *k kE kE3k*k 4 S &. +84,. Xvlqj wkh uhvrxufhv frqvwudlqw/ dqg wkh htxdwlrqv +7;, dqg +83, zh fdq rewdlq & S ' *k E kE3k*k E n #2 dE ko*k & &. +85,. Sursrvlwlrq < Wklv hfrqrp| odfnv wudqvlwlrqdo g|qdplfv1 Surri1 Ohw % eh wkh frqvxpswlrq wr fdslwdo udwlr &S Wkh yduldwlrq udwh ri % lv % S & ' % S & Iurp +84, dqg +85, zh rewdlq % ' d # E 2 o %2 4%. +86,. Dorqj d edodqfhg jurzwk sdwk frqvxpswlrq dqg fdslwdo zloo jurz dw wkh vdph frqvwdqw udwh dqg % zloo eh frqvwdqw1 Khqfh/ iurp +86,/ wkh vwhdg| vwdwh iru % lv %W '. 4 d # E 2 o. Qrwh wkdw %W : f li dqg rqo| li : # E 2 49. +87,.

(18) Centro de Estudios Andaluces. Wkh vroxwlrq iru glhuhqwldo htxdwlrq +86, lv %E| '. k. 4. d # E 2 o d # E 2 o . 4. %Ef. l. e4|. . +88,. Iurp +88,/ rqh fdq rewdlq wkdw rqo| li % ' %W c ;|c wkhq ,6 %E| ' %W Li %Ef 9' %W c |<". wkhq ,6 %E| ' fc wkdw lv/ wkh hfrqrp| ydqlvkhv zlwk wkh wlph1 Khqfh/ wkh frqvxpswlrq |<" wr fdslwdo udwlr lv dozd|v sodfhg rq wkh EJS/ dqg vr lw lv wkh fdslwdo wr sxeolf surgxfwlyh vhuylfhv udwlr +vhh +7;,, dqg wkh frqvxpswlrq wr sxeolf dedwhphqw dfwlylwlhv udwlr +vhh +83,,1 Wkhuhiruh/ wkh frppdqg hfrqrp| lv joredoo| dqg orfdoo| ghwhuplqdwh vlqfh jlyhq &Ef/ zh rewdlq SEf ' %W &Efc 5Ef ' #2 %W &Efc }Ef ' dE ko*k &Efc ; dqg ^E| ' e | ^Efc ^ ' &c Sc 5c }c zkhuh lv wkh uvw0ehvw jurzwk udwh/ zklfk fdq eh rewdlqhg iurp +84, dqg +87,= ' #. 4 n *k kE kE3k*k 4 d # E 2 o. +89,. Ixuwkhupruh/ wkh vhfrqg0ehvw hfrqrp| fdq eh ghqhg iru # E 2 ru #E 2 : c zklovw wkh frppdqg hfrqrp| lv rqo| ghqhg iru # E 2 Qrz zh dgguhvv wkh ghvljq ri wkh hqylurqphqwdo srolf| wkdw doorzv wkh hfrqrp| uhdfk wkh h!flhqw sdwk dv d pdunhw htxloleulxp1 Wkxv/ zh loxvwudwh krz sljrxyldq wd{ dqg vshqglqj frpsrvlwlrq/ li wkh| h{lvw/ fdq eh xvhg wr surprwh h!flhqf|1 Krzhyhu/ xqghu fhuwdlq sdudphwhu vhwv/ qglqj wkh uvw0ehvw wd{ udwh lv qrw srvvleoh1 Lq wkhvh fdvhv/ wkh lpsohphqwdwlrq ri wkh Udpvh|0sodq fdq eh mxvwldeoh1 ' 5} c zkhuh Wkhuhiruh/ zh frpsxwh wkh sdlu + c , vxfk wkdw ' dqg 5} . . lv jlyhq e| +46,/ xvlqj +87,/ +83,/ dqg +7;,/ wkh uvw0ehvw 5 wr } udwlr lv jlyhq e| #2 4 5 ' c *k } dE k o d # E 2 o dqg 5} lv rewdlqhg iurp +43,1 ' 5} c wkh sljrxyldq vshqglqj frpsrvlwlrq fdq eh re0 Iurp wkh frqglwlrq 5} wdlqhg= '. . n. Q*k d3# E dE3ko. # 4. Q. !. o. . +8:,. Zkdwhyhu wkh vwuxfwxudo sdudphwhu vhw/ qglqj d sljrxyldq vshqglqj frpsrvlwlrq lv dozd|v srvvleoh/ vlqfh dozd|v idoov ehwzhhq }hur dqg rqh1. Qrwh wkdw li wudqvlwlrqdo g|qdplf zrxog h{lvw/ wkhuh zrxog eh d frqwlqxxp ri frqyhujlqj sdwkv wrzdugv wkh EJS/ wkdw lv/ iru hdfk {+3, +ru f+3,/ vlqfh n+3, lv jlyhq, wkhuh h{lvwv d htxloleulxp sdwk1 Vxfk frqwlqxxp ri htxloleuld frxog eh lqgh{hg e| {+3,1 Lq wklv fdvh/ wkh prgho zrxog glvsod| orfdo lqghwhuplqdf|1 ;. 4:.

(19) Centro de Estudios Andaluces. Xvlqj +8:, lq wkh frqglwlrq ' c wkh sljrxyldq wd{ udwh fdq eh rewdlqhg iurp n 4 Ek!*k ' +8;, *k E k *k ! k E Wkh -M7 ri +8;, lv olqhdu dqg ghfuhdvlqj lq 5 Efc c zlwk grpdlq Efc Rq wkh rwkhu kdqg/ wkh uM7 ri +8;, lv prqrwrqlfdoo| ghfuhdvlqj dqg frqyh{ lq 5 Efc c zlwk grpdlq kQ*k E!n4E!k*k c 4 Dv d frqvhtxhqfh/ wkhuh fdq eh wzr vroxwlrqv/ rqh vroxwlrq ru qr vroxwlrq iru Khqfh/ dowkrxjk wkh uvw0ehvw doorfdwlrqv duh joredoo| ghwhuplqdwh/ wkh uvw0ehvw wd{ srolf| fdq eh lqghwhuplqdwh1 Wkh h{dpsohv xvhg lq wkh suhylrxv vhfwlrqv duh xvhixo wr vwxg| wkh sljrxyldq wd{ udwh dqg vshqglqj frpsrvlwlrq1 Krzhyhu/ rqo| wkrvh h{dpsohv yhuli|lqj : # E 2 duh ydolg lq wkh uvw0ehvw iudphzrun1 Wkh sdudphwhu vhw jlyhq e| k ' f./ ' f./ 4 ' *L}EfbD/ # ' / ' f./ 2 ' fS |lhogv wkh iroorzlqj vhfrqg0ehvw< dqg uvw0ehvw vroxwlrqv= vhfrqg0ehvw uvw0ehvw eDI 49( S.fI e I 57( eI .HI qr vroxwlrq DHI Qrwh wkdw wkh hqylurqphqwdo txdolw| lv zruvh xqghu wkh sodqqhu vroxwlrq ehfdxvh l, srooxwlrq lv jurzlqj dw d odujhu udwh/ dqg ll, wkh vkduh ri sxeolf vshqglqj ghyrwhg wr dedwhphqw lv orzhu1 Lq wklv fdvh/ qglqj d sljrxyldq wd{ iru zklfk wkh hfrqrp| uhdfk wkh h!flhqw sdwk dv d frpshwlwlyh pdunhw lv qrw srvvleoh1 Wkh sdudphwhu vhw jlyhq e| k ' f./ ' / 4 ' *L}EfbD/ # ' / ' fe/ 2 ' f2 |lhogv wkh iroorzlqj vhfrqg0ehvw dqg uvw0ehvw vroxwlrqv= vhfrqg0ehvw uvw0ehvw DbI 4919( S.I e I 69( e2I .HI DI .SI DfI Lq wklv fdvh wkh hqylurqphqwdo txdolw| lv dovr zruvh xqghu wkh sodqqhu vroxwlrq iru wkh vdph uhdvrqv flwhg deryh1 Qrwh wkdw iru wklv vwuxfwxudo sdudphwhu vhw wkhuh h{lvw wzr sljrxyldq wd{hv1. 8. Frqfoxvlrq. Lq d vhfrqg0ehvw iudphzrun/ zh vwxg| wkh g|qdplf surshuwlhv ri d jhqhudo htxloleulxp prgho ri hqgrjhqrxv jurzwk zlwk hqylurqphqwdo h{whuqdolwlhv1 Lq sduwlfxodu/ zh lqyhv0 <. Qrwh wkdw wkhuh duh wzr EJS iru wkh Udpvh| sureohp iru wklv sdudphwhu vhw1 4;.

(20) Centro de Estudios Andaluces. wljdwh wkh frqglwlrqv iru joredo dqg orfdo lqghwhuplqdf|/ dqg wkhlu lpsolfdwlrqv iru wkh jryhuqphqwdo frqwuro ri srooxwlrq1 Wkh prghoohg hfrqrp| lv fkdudfwhul}hg e|= d, zh dvvxph wkdw wkh fdslwdo vwrfn xvhg lq surgxfwlrq fdxvhv d qhjdwlyh hqylurqphqwdo h{whuqdolw| dv d vlgh surgxfw dqg wklv h{whu0 qdolw| dhfwv lqglylgxdov* xwlolw|> djjuhjdwh srooxwlrq lv d sxeolf edg zklfk fdq eh uhgxfhg e| phdqv ri sxeolf dedwhphqw dfwlylwlhv> e, wkh jryhuqphqw h{shqglwxuhv duh ghyrwhg wr sxeolf dedwhphqw dfwlylwlhv dqg wr sxeolf surgxfwlyh vhuylfhv dqg wkhvh h{shqglwxuhv duh qdqfhg e| d srooxwlrq wd{ wkdw wkh upv pxvw sd|> f, wkh jryhuqphqw lv ehqhyrohqw dqg fkrrvhv lwv wd{ srolf| dqg frpsrvlwlrq ri sxeolf vshqglqj e| wdnlqj lqwr dffrxqw wkh ghfhqwudol}hg frpshwlwlyh htxloleulxp1 Zh vkrz wkdw zkhq zh lqfoxgh srooxwlrq hqwhulqj lq wkh xwlolw| ixqfwlrq wrjhwkhu zlwk wkh dedwhphqw sxeolf dfwlylwlhv/ wkh lqwurgxfwlrq ri hqgrjhqrxvo| fkrvhq hfrqrplf srolf| fdq jhqhudwh joredo dqg orfdo lqghwhuplqdf|/ wkdw lv/ wkhuh fdq eh wzr vwhdg| vwdwhv +joredo lqghwhuplqdf|, erwk ri zklfk fdq eh orfdoo| lqghwhuplqdwh +wkdw lv/ wkhuh h{lvw pxowlsoh frqyhujlqj sdwkv wr hdfk vwdwh,1 Rq wkh frqwudu|/ wkh rwkhu w|sh ri h{whuqdolw| lq rxu hfrqrp| +sxeolf surgxfwlrq vhuylfhv,/ lv qrw fuxfldo wr dfklhyh wkh lqghwhuplqdf| uhvxow/ wkdw lv/ li wkh sxeolf vhuylfhv gr qrw surylgh dq| surgxfwlrq h{whuqdolw|/ vwloo lv srvvleoh wr rewdlq d joredo lqghwhuplqdf| uhvxow1 Joredo dqg orfdo lqghwhuplqdf| duh deoh wr h{sodlq zk| wzr frxqwulhv zlwk vlplodu ixqgdphqwdov +suhihuhqfhv/ whfkqrorj| dqg lqlwldo fdslwdo vwrfn, pljkw glvsod| glhuhqw vkruw dqg orqj uxq srooxwlrq sdwkv uhjdugohvv ri wkh hfrqrplf ixqgdphqwdov1 Ilqdoo|/ zh vwxg| wkh sodqqhu vroxwlrq lq rughu wr ghulyh wkh uvw0ehvw wd{ udwh e| frpsdulqj wkh uvw rughu frqglwlrqv ri wkh frpshwlwlyh htxloleulxp zlwk wkh uvw rughu frqglwlrqv ri wkh vrfldo sodqqhu vroxwlrq1 Zh qg wkdw wkhuh fdq eh wzr/ rqh ru qr sljrxyldq wd{ udwh1 Zkhq wkhuh h{lvw wzr ru rqh sljrxyldq wd{ udwh/ zh vkrz wkdw wkh sljrxyldq wd{ lv orzhu wkdq wkh vhfrqg0ehvw wd{1 Ixuwkhupruh/ xqghu wkh sodqqhu vroxwlrq wkh hqylurqphqwdo txdolw| lv zruvh vlqfh srooxwlrq lv jurzlqj dw d odujhu udwh/ dqg wkh vkduh ri sxeolf vshqglqj ghyrwhg wr dedwhphqw lv orzhu1. Uhihuhqfhv ^4` Dl|djdul/ V1U1+4<<8,/ Wkh hfrqrphwulfv ri lqghwhuplqdf|= dq dssolhg vwxg|/ d frp0 phqw/ Fduqhjlh0Urfkhvwhu Frqi1 Vhu1 Sxeolf Srolf|/ 76/ ss1 5:605;51 ^5` Eduur/ U1 +4<<3,/ Jryhuqphqw vshqglqj lq d vlpsoh prgho ri hfrqrplf jurzwk/ Mrxuqdo ri Srolwlfdo Hfrqrp|/ <;/ ss1 V4360V4581 ^6` Ehqkdele/ M1 dqg U1H1D1 Iduphu +4<<7,/ Lqghwhuplqdf| dqg lqfuhdvlqj uhwxuqv/ Mrxuqdo ri Hfrqrplf Wkhru|/ 96/ ss1 4<0741 ^7` Ehqkdele/ M1 dqg U1H1D1 Iduphu +4<<9,/ Lqghwhuplqdf| dqg vhfwru vshflf h{whu0 qdolwlhv/ Mrxuqdo ri Prqhwdu| Hfrqrplfv 6:/ ss1 75407761 ^8` Ehqkdele/ M1 dqg Qlvklpxud/ N1 +4<<;,/ Lqghwhuplqdf| dqg vxqvsrwv zlwk frqvwdqw uhwxuqv/ Mrxuqdo ri Hfrqrplf Wkhru|/ ;4/ ss1 8;0<91 4<.

(21) Centro de Estudios Andaluces. ^9` Ehqkdele/ M1 dqg Shuol/ U1 +4<<7,/ Xqltxhqhvv dqg lqghwhuplqdf|= wudqvlwlrqdo g|0 qdplfv zlwk pxowlsoh htxloleuld/ Mrxuqdo ri Hfrqrplf Wkhru|/ 96/ ss1 44604751 ^:` Ehqqhww/ U1O1 dqg U1H1D1 Iduphu +5333,/ Lqghwhuplqdf| zlwk qrq0vhsdudeoh xwlolw|/ Mrxuqdo ri Hfrqrplf Wkhru| <6/ 44;04761 ^;` Eryhqehuj/ D1 O1 dqg U1 D1 gh Prrlm +4<<:,/ Hqylurqphqwdo wd{ uhirup dqg hq0 grjhqrxv jurzwk1 Mrxuqdo ri Sxeolf Hfrqrplfv 96/ ss1 53:056:1 ^<` gh Eux|q/ V1P1 dqg U1M1 Khlqw} +4<<<,/ Wkh hqylurqphqwdo Nx}qhwv fxuyh k|srwk0 hvlv lq= M1 ydq ghq Ehujk/ Kdqgerrn ri Hqylurqphqwdo dqg Uhvrxufh Hfrqrplfv/ Fkhowhqkdp= Hgjdu Hojdu1 ^43` Iduphu/ U1H1D1 dqg M1W1 Jxr +4<<7,/ Uhdo exvlqhvv f|fohv dqg wkh dqlpdo vslulwv k|srwkhvlv/ Mrxuqdo ri HfrqrplfWkhru|/ 96/ ss1 750:61 ^44` Ihuqäqgh}/ H1/ D1 Qrydohv dqg M1 Uxl} +5336,/ Lqghwhuplqdf| xqghu qrq0vhsdudelolw| ri sxeolf frqvxpswlrq dqg ohlvxuh lq wkh xwlolw| ixqfwlrq/ Hfrqrplf Prghoolqj/ 54/ ss1 73<075;1 ^45` Jdqwpdfkhu/ I1U1 +4<93,/ Wkh Wkhru| ri Pdwulfhv/ Qhz \run/ Fkhovhd/ yro1 LL1 ^46` Oljwkduw/ M1 H1 dqg I1 ydq ghu Sorhj +4<<7,/ Srooxwlrq/ wkh frvw ri sxeolf ixqgv dqg hqgrjhqrxv jurzwk/ Hfrqrplfv Ohwwhuv/ 79/ ss1 68406941 ^47` Shuol/ U1/ +4<<;, Lqghwhuplqdf|/ krph surgxfwlrq dqg wkh exvlqhvv f|foh/ Mrxuqdo ri Prqhwdu| Hfrqrplfv/ 74/ ss1 43804581 ^48` Prkwdgl/ K1 +4<<9,/ Hqylurqphqw/ jurzwk/ dqg rswlpdo srolf| ghvljq/ Mrxuqdo ri Sxeolf Hfrqrplfv/ 96/ ss1 44<04731 ^49` Sdun/ K1/ dqg D1 Sklolssrsrxorv +5337,/ Lqghwhuplqf| dqg vfdo srolflhv lq d jurzlqj hfrqrp|/ Mrxuqdo ri Hfrqrplf G|qdplfv dqg Frqwuro/ 5;/ ss1 97809931 ^4:` Vpxoghuv/ V1 dqg U1 Judgxv +4<<9,/ Srooxwlrq dedwhphqw dqg orqj0whup jurzwk/ Hxurshdq Mrxuqdo ri Srolwlfdo Hfrqrp|/ 45/ ss1 83808651 ^4;` Zhghu/ P1 +4<<;, Ilfnoh frqvxphuv/ gxudeoh jrrgv dqg exvlqhvv f|fohv/ Mrxuqdo ri Hfrqrplf Wkhru|/ ;4/ ss1 6:08:1 ^4<` Zhq/ \1/ +4<<;,/ Fdsdflw| xwlol}dwlrq xqghu lqfuhdvlqj uhwxuqv wr vfdoh/ Mrxuqdo ri Hfrqrplf Wkhru|/ ;4/ ss1 :0691 ^53` [lh/ G1 +4<<7,/ Glyhujhqfh lq hfrqrplf shuirupdqfh= wudqvlwlrqdo g|qdplfv zlwk pxowlsoh htxloleuld/ Mrxuqdo ri Hfrqrplf Wkhru|/ 96/ ss1 <:04451. 53.

(22) Centro de Estudios Andaluces. 9. Dsshqgl{. 914. Surri ri sursrvlwlrq 8. Jlyhq uM7E ' [ ' E k Q*k dE kE n koE!k*k c uvw zh vkrz wkdw lw lv ghfuhdvlqj iru 5 d kc 1 Vlqfh YuM7E c ' *k E kE!k*k E n kE!2k*k k k2 Y wkhq YuM7E Y. f c iru. : k k2 . Vlqfh : k : k k2 c wkhq uM7E lv ghfuhdvlqj iru wkh udqjh ri Vhfrqg/ zh vkrz wkdw uM7E lv frqfdyh lq 1 Y 2 uM7E [ ' k k2 E2 n kk E E n kk 2 2 Y dE E n kko. fc. vlqfh : k k2 c olnh zh vkrzq suhylrxvo|1 Ixuwkhupruh= ,6 n [ ' k2 *k dE koE3k*k : f( ,6! [ ' f Wkhuhiruh/ wkh < <E3k ixqfwlrq lv srvlwlyh ; 5 d kc 1. 915. Surri ri Sursrvlwlrq 9 dqg :. Ghqh { ' # E 2 dqg ' 2 nkk 1 Zkhq : E k/ -M7E ' 4# nk 2 nk3 lv frqwlqxrxv/ dqg ghfuhdvlqj +lqfuhdvlqj, lq li { : E f= { 4# Y-M7E c ' Y { dE n k o2 Y 2 -M7E 24# ' c 2 { dE n k o Y dqg khqfh= l, Li { f dqg f= Y-M7E Y 2 -M7E f( : f( Y Y 2 *4 n -M7E ' 4( *4! -M7E ' V : f c. < 3k E. <. . 54. .

(23) Centro de Estudios Andaluces. wkhuhiruh/ -M7E lv lqfuhdvlqj dqg frqfdyh/ zlwk grpdlq E4c V ll, Li { f dqg : f= Y 2 -M7E Y-M7E f( : f( Y Y 2 *4 n -M7E ' 4( *4! -M7E ' V. < 3k E. <. . . f c. wkhuhiruh/ -M7E lv lqfuhdvlqj dqg frqfdyh/ zlwk grpdlq E4c V lll, Li { : f dqg f= Y-M7E Y 2 -M7E : f( f( Y Y 2 *4 n -M7E ' 4( *4! -M7E ' V. < 3k E. <. . . f c. wkhuhiruh/ -M7E lv ghfuhdvlqj dqg frqyh{/ zlwk grpdlq EVc 4 ly, Li { : f dqg : f= Y-M7E Y 2 -M7E : f( f( Y Y 2 *4 n -M7E ' 4( *4! -M7E ' V : f c. < 3k E. <. . . wkhuhiruh/ -M7E lv ghfuhdvlqj dqg frqyh{/ zlwk grpdlq EVc 4. 55.

(24) Centro de Estudios Andaluces. LHS (τ ). 1−α. RHS (τ ). τ. 1. τ. Figure 1a. LHS (τ ). 1−α. 1 RHS (τ ) Figure 1b. τ.

(25) Centro de Estudios Andaluces. LHS (τˆ). RHS (τˆ). 1−α. τˆ. 1. Figure 2. τ.

(26) Centro de Estudios Andaluces. RHS (τ ). RHS (τ ). LHS (τ ). 1−α. Figure 3a. LHS (τ ). LHS (τ ). RHS (τ ). 1. τ. 1−α. τ Figure 3b. 1. τ. 1−α. τ 1. τ 2. Figure 3c. 1. τ.

(27)