Aversion Analysis

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Cahier 2003-06

ALLARD, Marie

BRONSARD, Camille

GOURIÉROUX Christian

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Département de sciences économiques

Université de Montréal

Faculté des arts et des sciences C.P. 6128, succursale Centre-Ville Montréal (Québec) H3C 3J7 Canada http://www.sceco.umontreal.ca SCECO-information@UMontreal.CA Téléphone : (514) 343-6539 Télécopieur : (514) 343-7221

Ce cahier a également été publié par le Centre interuniversitaire de recherche en économie quantitative (CIREQ) sous le numéro 04-2003.

This working paper was also published by the Center for Interuniversity Research in Quantitative Economics (CIREQ), under number 04-2003.

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ABSTRACT

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Keywords : Ulvn dyhuvlrq/ dyhuvlrq wr lpsdwlhqfh/ looltxlglw| dyhuvlrq/ pxowlglphqvlrqdo dyhuvlrqv/ qdqfldo suhplxpv/ Dqwrqhool pdwul{/ dvvhw vxevwlwxdelolw|/ Guë}h0Prgljoldql ghfrpsrvlwlrq/ vxemhfwlyh fhuwdlqw|/ vxuh dqg ulvn| dvvhwv/ lqfrpsohwh pdunhwv1

* Lqvwlwxw g*ìfrqrplh dssoltxìh/ KHF Prqwuìdo/ 6333 fkhplq gh od F÷wh0Vdlqwh0Fdwkhulqh/ Prqwuìdo +Txìehf, K6W 5D:/ pdulh1doodugCkhf1fd/ Gìsduwhphqw gh vflhqfhv ìfrqrpltxhv hw F1U1G1H1/ Xqlyhuvlwì gh Prqwuìdo/ F1S1 945;/ vxffxuvdoh Fhqwuh0ylooh/ Prqwuìdo +Txìehf, Fdqdgd/ K6F 6M:/ dqg Ghsduwphqw ri Hfrqrplfv/ Xqlyhuvlw| ri Wrurqwr dqg FUHVW +Sdulv,/ 48 erxo1 Jdeulho0Sìul/ <5578 Pdodnr Fhgh{/ Iudqfh/ jrxulhurChqvdh1iu1

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1. Introduction

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Prvw ri wkh surriv duh jdwkhuhg lq wkh dsshqglfhv1

2. The model

2.1. Basic concepts

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R

1r%1r ^1r+ ' w1r lq shulrg dqg vwdwh rc r ' c c 7 +515,

Dvvxplqj wkdw wkh frqvxphu*v suhihuhqfhv duh uhsuhvhqwdeoh e| d xwlolw| ixqfwlrq X/ klv rswlpl}dwlrq sureohp lv =

4@

%0cc%17c+XE%0c %11c c %17 +516, vxemhfw wr +514, dqg +515,1

Pdwul{ qrwdwlrq zloo eh xvhg lq wkh vhtxho1

(9)

% ' d%

0c %11c c %17o lv wkh u 0 glphqvlrqdo yhfwru ri frpprglw| txdqwlwlhv zkhuh

u ' u0n [ r u1r > ' 5 9 9 9 9 9 7 R0 f R11 f 111 R17 6 : : : : : 8 lv wkh u E7 n 0 glphqvlrqdo pdwul{ ri frpprglw| sulfhv>

' ' d^0 ^11 ^17o lv wkh E7 n 0 glphqvlrqdo pdwul{ ri dvvhw

sulfhv dqg sd|0rv>

w' dw0c w11c c w17o lv wkh E7 n 0 glphqvlrqdo yhfwru ri lqfrphv1

Zlwk wkh khos ri wklv qrwdwlrq wkh suhylrxv frqvxphu*v sureohp fdq eh zulwwhq = 4@ %c+ XE%

vxemhfw wr _{% n '}_{+ ' w}

+517,

Lq rughu wr glvwlqjxlvk ehwzhhq suhvhqw dqg ixwxuh yduldeohv +d xvxdo glvwlqfwlrq,/ lw lv dovr frqyhqlhqw wr frqvlghu wkh iroorzlqj sduwlwlrq =

% ' %0 %1 c _' R0 f f 1 c ' _' ^0 ' 1 c w ' w0 w1 Wkh lqglylgxdo ghflvlrq sureohp lv wkhq 4@ %0c%1c+ XE%0c %1 vxemhfw wr R 0%0n ^0+ ' w0 1%1 '1+ ' w1 < @ > +518,

Remark 1 : d, Wkh frqvxphu*v sureohp frqvlghuhg khuh hqfrpsdvvhv wkh xvxdo iudphzrun zkhuh wkh xwlolw| ixqfwlrq zrxog eh zulwwhq dv =

XE%0c %11c c %17 ' E%0 n B. ZE%1r ' E%0 n B [ r ZrE%1r c +519,

(10)

zkhuh Zr lv d vxemhfwlyh suredelolw| dqg B d sv|fkrorjlfdo glvfrxqw idfwru1

ixqfwlrqdo irup +519, lqkhulwhg iurp wkh yrq Qhxpdqq0Prujhqvwhuq wudglwlrq kdv ehhq fkdoohqjhg lq rughu wr hudglfdwh lwv sdudgr{lfdo lpsolfdwlrqv ^vhh/ iru lqvwdqfh/ Doodlv +4<86, dqg Hoovehuj +4<94,` zkloh nhhslqj wkh frpsxwdwlrqdo exughq dv orz dv srvvleoh1

e, Hduo| h{dpsohv ri wkh xwlolw| ixqfwlrq zklfk gr qrw vdwlvi| wkh yrq Qhxpdqq0Prujhqvwhuq wudglwlrq fdq eh irxqg lq Pdfklqd +4<;:,1 Pruh uhfhqw rqhv +ru pruh dssursuldwh wr rxu sxusrvhv, duh =

X E% ' [ r Zr E%0c %1r c +51:, X E% ' E%0 n B [ r ZrrE%1r c +51;, X E% ' E%0 n B [ r [ j ) E%1rc %1j ZrZj c +51<, X E% ' E%0 n B + 4? ZMT [ r Zr E%1r , c +5143, X E% ' E%0 n B ; ? = [ r Zr E%1r #₂ [ r Zr % E%1r [ r Zr E%1r &2<_@ > c +5144, X E% ' E%0 n B E%c Z +5145, Wkh vshflfdwlrq +51:,/ zklfk lv qrw qhfhvvdulo| wlph0vhsdudeoh/ zdv xvhg e| Guë}h dqg Prgljoldql +4<:5,1 Wkh vshflfdwlrq +51;,/ zkhuh wkh hohphqwdu| xwlolwlhv r duh vwdwh0ghshqghqw/ zdv iru lqvwdqfh frqvlghuhg e| Frrn dqg Judkdp

+4<::,1 Wkh txdgudwlf irup +51<, zdv d{lrpdwl}hg e| Fkhz/ Hsvwhlq dqg Vhjdo +4<<4,1 Wkh vshflfdwlrq +5143, zdv d{lrpdwl}hg e| Jloerd dqg Vfkphlgohu +4<;<, dqg xvhg lq qdqfh e| Grz dqg Zhuodqj +4<<5,1 Lq wklv irup/ wkh suredelolw| Z lv qrw h{dfwo| nqrzq/ doorzlqj iru dpeljxlw| +Nqljkwldq xqfhuwdlqw|, ^vhh Hsvwhlq +4<<<,/ Hsvwhlq dqg Vfkqhlghu +5334/ 5335,/ Hsvwhlq dqg ]kdqj +5334,/

(11)

dqg Pdfklqd +5334,`1 Wkh vshflfdwlrq +5144, h{whqgv +51;, e| lqwurgxflqj d ulvn fruuhfwlrq wkdw dffrxqwv iru wkh xqfhuwdlqw| ri xwlolw| ohyhov E%1r

Wkh vshflfdwlrq +5145, zdv sursrvhg e| Vdpxhovrq +4<93,/ xvhg e| Pdfklqd +4<<8/ 5333,/ dqg d{lrpdwl}hg e| Pdfklqd dqg Vfkphlgohu +4<<5/ 4<<8,1 Wklv vshflfdwlrq jhqhudol}hv doo wkh rwkhu rqhv h{fhsw +51:,1

2.2. Assumptions

Wkh frqvxphu*v sureohp zloo eh vwxglhg xqghu wkh iroorzlqj dvvxpswlrqv = D14 Wkh xwlolw| ixqfwlrq X lv ghqhg rq wkh frqvxpswlrq vhw [ Ru_{1 Wklv}

vhw lv rshq/ frqyh{/ erxqghg ehorz dqg qrq0hpsw|1 Wkh xwlolw| ixqfwlrq lv wzlfh frqwlqxrxvo| glhuhqwldeoh1 Wkh judglhqw X% lv vwulfwo| srvlwlyh

+vwurqj prqrwrqlflw|,> wkh Khvvldq pdwul{ X%%0 lv vxfk wkdw wkh txdgudwlf irup l_X

%%0l lv vwulfwo| qhjdwlyh zkhqhyhu l 9' f dqg lX_% ' f +vwurqj txdvl0frqfdylw|,1

Dvvxpswlrq D14 frqfhuqv wkh lqglylgxdo*v fkdudfwhulvwlfv dqg lv vwdqgdug lq hfrqrplf wkhru| vlqfh Gheuhx +4<:5, ^vhh/ iru lqvwdqfh/ Edodvnr +4<;;,/ Pdv0Frohoo +4<;8,/ Pdv0Frohoo hw do1 +4<<8,`1 Wkh frqvxpswlrq vhw lv qrw qhfhvvdulo| wkh vwulfwo| srvlwlyh ruwkdqw ri Ru_{/ vlqfh d qhjdwlyh frpsrqhqw ri %}

fdq eh d txdqwlw| ri oderu1 Ixuwkhupruh/ wkh jhqhudo irup ri wkh xwlolw| ixqfwlrq lv jhqhudo hqrxjk wr dgplw/ lq sduwlfxodu/ vwdwh ghshqghqf| ri wkh xwlolw| ri frqvxpswlrq1

D15 Wkh exgjhwdu| frqvwudlqwv duh nqrzq wr wkh frqvxphu1

Lq Dvvxpswlrq D15/ lw lv lpsolflwo| dvvxphg wkdw sulfhv dqg lqfrphv duh h{rjhqrxv gdwd lpsrvhg rq wkh frqvxphu1 Wklv lv qrw vxusulvlqj iru R0c ^0 dqg w0/ exw

dprxqwv wr lpsrvlqj d vshflf +wkrxjk xvxdo, dvvxpswlrq rq R1c ^1 dqg w11 Hy hq

lq wklv fdvh wkh srvvleoh ydoxhv ri wkhvh yduldeohv vkrxog qrw eh vhhq dv ghwhuplqhg e| wkh vwdwh ri qdwxuh +zdu grhv qrw vx!fh wr h{sodlq wkh irupdwlrq ri zdu sulfhv,1 Rq wkh rqh kdqg wkh vsdfh ri wkh srvvleoh sulfhv dqg lqfrphv fdqqrw eh vsdqqhg iurp wkh +vpdoo dqg glvfuhwh, vsdfh ri vwdwhv> rq wkh rwkhu kdqg/ wkh srvvleoh vwdwhv ri wkh zruog duh jlyhq d sulrul/ rqfh dqg iru doo/ zkloh wkh frqvlghuhg sulfhv dqg lqfrphv duh lq sulqflsoh hqgrjhqrxv dw wkh djjuhjdwh ohyho1 Wklv srlqw lv lpsruwdqw ehfdxvh xvxdo qdqfldo dssurdfkhv lghqwli| vwdwhv dqg sulfh hyroxwlrqv ^vhh/ iru lqvwdqfh/ wkh rswlrq sulflqj prgho ri Eodfn dqg Vfkrohv +4<:6,` dqg/ e| grlqj vr/ duh ohg lqwr huuru rq wkh ghjuhh ri pdunhw lqfrpsohwhqhvv1

(12)

D16 Lq wkh exgjhwdu| frqvwudlqwv/ / ' dqg w duh vxfk wkdw = d, R0c R11c c R17 duh vwulfwo| srvlwlyh>

e, udqn '1 ' 7>

f, '_{+ 9 f iru dq| +>}

g, [ _ i% m _{% n '}_{+ 'wj 9' > iru dw ohdvw d +1}

Dvvxpswlrq D16 frqfhuqv wkh exgjhw frqvwudlqwv idfhg e| wkh lqglylgxdo1 Xqghu wkhvh frqvwudlqwv d sruwirolr grhv qrw qhfhvvdulo| ehorqj wr wkh srvlwlyh ruwkdqw ri R _{= d qhjdwlyh frpsrqhqw ri + pd| lqglfdwh d edqn eruurzlqj dv zhoo dv d}

vkruw vdoh wudqvdfwlrq1 Dvvxpswlrq D16d lpsolhv/ lq sduwlfxodu/ wkdw wkh Mdfreldq pdwul{ ri wkh 7n exgjhw frqvwudlqwv kdv udqn 7n1 Dvvxpswlrq D16e holplqdwhv uhgxqgdqw dvvhwv > li ' 7/ wkh dvvhw vwuxfwxuh lv frpsohwh > li 7/ wkh dvvhw vwuxfwxuh lv lqfrpsohwh/ dqg wklv lqfrpsohwhqhvv kdv glphqvlrq 71 Dvvhw sulfhv duh duelwudjh0iuhh e| D16f +zlwkrxw wklv dvvxpswlrq wkh frqvxphu*v zhdowk frxog lqfuhdvh lqghqlwho|,1 Dvvxpswlrq D16g lv wkh vxuylydo frqglwlrq1

Iurp d pdwkhpdwlfdo ylhzsrlqw/ qdqfldo dvvhwv fdq eh frqfhlyhg dv d zd| wr uhgxfh wkh qxpehu ri frqvwudlqwv idfhg e| wkh frqvxphu1 Lqghhg/ vlqfh '1 kdv

udqn / lw lv dozd|v srvvleoh wr fkrrvh rxw ri 7n frqvwudlqwv lq rughu wr h{suhvv + lq whupv ri uhyhqxhv dqg h{shqglwxuhv/ dqg wr vxevwlwxwh wkhvh h{suhvvlrqv lq wkh uhvlgxdo frqvwudlqwv = zh wkhq kdyh 7 n lqghshqghqw frqvwudlqwv1 Li ' fc 7 n frqvwudlqwv duh hhfwlyh1 Li ' 7/ wkh ixwxuh frqvwudlqwv ri +518, fdq eh zulwwhq dv + ' d'

1o31d1%1w1o zklfk/ lq wxuq/ fdq eh vxevwlwxwhg lq wkh

suhvhqw frqvwudlqwv > wkxv rqh kdv R

0%0n ^0d'1o311%1 ' w0n ^0d'1o31w1 c

dqg wklv xqltxh hhfwlyh frqvwudlqw lv/ xs wr wkh qrwdwlrq/ wkh frqvwudlqw frqvlghuhg lq wkh vwdqgdug fdvh ri frqvxpswlrq wkhru|1 Li/ pruhryhu/ '

1 ' U7 wkh qdqfldo

dvvhwv uhgxfh wr wkh hohphqwdu| frqwlqjhqw dvvhwv ri Duurz +4<86,1 Vlqfh wkh dvvhw vwuxfwxuh lv frpsohwh/ vwdqgdug uhvxowv dsso|1

Vlqfh wkhuh duh dfwxdoo| 7 n frqvwudlqwv wkh exgjhw vhw lv d olqhdu pdqlirog ri glphqvlrq u 7 n lq Ru_{1 Xqghu Dvvxpswlrq D16/ lw lqwhuvhfwv wkh}

frqvxpswlrq vhw [ 1 Ohw %_{eh d srlqw lq wklv lqwhuvhfwlrq dqg %}_%_{eh dqrwkhu}

(13)

ohvv h{shqvlyh wkdq % _{jlyh d qrq0hpsw| dqg frpsdfw vxevhw ri [ 1 Pd{lpl}lqj}

XE% rq wklv vhw ohdgv wr d vroxwlrq zklfk lv xqltxh iurp vwurqj prqrwrqlflw| dqg vwurqj txdvl0frqfdylw| dvvxpswlrqv1

2.3. First-order conditions and demand systems

Xqghu wkh suhylrxv dvvxpswlrqv/ wkh frqvxphu*v sureohp pd| eh vroyhg zlwk wkh khos ri wkh Odjudqjhdq =

uE%c +( b ' XE% b_d_{% n '}_{+ wo c}

zkhuh b _{' db}

0c b11c c b17o lv d yhfwru ri Odjudqjh pxowlsolhuv1 Wklv yhfwru h{lvwv

dqg lv xqltxh1 Wkh uvw0rughu frqglwlrqv duh qhfhvvdu| dqg vx!flhqw1 Wkh| duh zulwwhq = X% b ' f 'b ' f _{% '}_{+ n w ' f} < @ > c +5146,

zkhq wkh zkroh vwuxfwxuh lv frqvlghuhg>

X0 ' b0R0c X1r' b1rR1rc r ' c c 7 b0^0 ' S rb1r^1r R 0%0n ^0+ ' w0c R1r%1r ^1r+ ' w1rc r ' c c 7 < A @ A > c +5147, zkhq wlph dqg xqfhuwdlqw| duh h{solflwhg1

Wkh frqglwlrqv X% b ' f duh lqwhuqdo wr wkh frqvlghuhg shulrg vwdwh1

Iru lqvwdqfh/ wkh frqglwlrqv X0 ' b0R0 lpso| wkdw pdujlqdo udwhv ri vxevwlwxwlrq

duh htxdo wr uhodwlyh sulfhv zlwklq shulrg 3/ exw gr qrw jlyh ulvh wr dq| olqn dfurvv shulrgv ru vwdwhv1 Wkrvh olqnv duh hvwdeolvkhg e| wkh frqglwlrqv 'b ' f1 Li ' 7 dqg '1 ' U7/ rqh kdv b_b1

0 ' ^0 dqg wkh olqnv duh frpsohwh1 Li 7/

erwk lqwhuwhpsrudo dqg frqwlqjhqw vxevwlwxwlrq dprqj frpprglwlhv duh lpshuihfw1 Li ' f/ qr vxfk vxevwlwxwlrq lv doorzhg/ exw wkh Odjudqjh pxowlsolhuv duh vwloo ghqhg dqg fdq eh xvhg wr vwxg| wkh ghvludelolw| ri wudqvihuv ryhu wlph dqg dfurvv vwdwhv1 Ilqdoo|/ vlqfh D16f lpsolhv wkdw wkh frqvxphu*v zhdowk fdqqrw eh lqfuhdvhg lqghqlwho|/ wkh frqglwlrqv '_{+ ' w}_{% h{suhvv/ xqghu D16/ wkh idfw wkdw/}

iru d xwlolw|0pd{lpl}lqj frqvxphu/ vrph nlqg ri khgjlqj lv d qhfhvvlw|1 Ghhshu lqwhusuhwdwlrqv duh srvvleoh li vrph uhohydqw wrrov duh uvw ghyhorshg1

(14)

Proposition 1 : Ohw xv vhw R ' dR₀c R₁₁c c R₁₇o c ^ ' d^₀c ^₁₁ c c ^₁₇ o Xqghu Dvvxpswlrqv D14/ D15 dqg D16/ wkhuh h{lvw vroxwlrqv ri wkh frqvxphu rswlpl}dwlrq sureohp =

d, % ' %ERc ^cw +frpprglw| ghpdqgv,/ e, + ' +ERc ^cw +dvvhw ghpdqgv,/

f, b ' bERc ^cw +ghvludelolwlhv ri lqfrphv,/ wkdw duh frqwlqxrxvo| glhuhqwldeoh1

Proof = Vhh Dsshqgl{ D1

Dv douhdg| phqwlrqhg/ li wkh dvvhw vwuxfwxuh lv frpsohwh E ' 7 c wkh 7 n exgjhwdu| frqvwudlqwv idfhg e| wkh frqvxphu uhgxfh wr d xqltxh frqvwudlqw1 Dv d uhvxow/ frpprglw| ghpdqgv %c dvvhw ghpdqgv + dqg wkh ghvludelolwlhv ri lqfrphv b zloo ghshqg rq ERc ^c w lq d vshflf zd|1 Lq sduwlfxodu/ wkh ghvludelolwlhv ri ixwxuh lqfrphv fdq eh zulwwhq dv b1E ' b0E '311 ^0c vlqfh '1 lv lqyhuwleoh1

Wkh ghvludelolwlhv ri lqfrph duh qrw lqghshqghqw ri d prqrwrqlf wudqvirupdwlrq ri wkh xwlolw| ixqfwlrq X1 Lq wkh qh{w vhfwlrq/ zh vkdoo wudqvirup wkhp lqwr vxemhfwlyh Duurz sulfhv1

3. Subjective Arrow prices and indirect utility function

Vlqfh ghpdqg ixqfwlrqv duh douhdg| nqrzq wr h{lvw/ dq lqgluhfw xwlolw| ixqfwlrq fdq eh ghqhg e| wkh uhodwlrq =

ERc ^c w X d%ERc ^c wo c +614, zkhuh X d%ERc ^c wo lv wkh pd{lpdo ydoxh ri wkh Odjudqjhdq u ' XE% b_d_{% n}

'_{+ wo1 E| wkh hqyhorsh wkhruhp/ wkh sduwldo ghulydwlyhv ri +614, fdq eh}

frpsxwhg zlwk wkh khos ri wklv Odjudqjhdq1 Iru lqvwdqfh/ rqh kdv = Y Yw bERc ^c w c +615, ru htxlydohqwo| b0 _YwY 0c b1r Y Yw1rc r ' c c 71 Wkh pxowlsolhu b0 lv wkh

(15)

pdujlqdo xwlolw| ri ixwxuh lqfrph lq vwdwh r1 Wkhvh ghvludelolwlhv zloo eh frqyhuwhg lqwr vxemhfwlyh Duurz sulfhv lq vxevhfwlrq 6141 Wkhvh Duurz sulfhv zloo eh xwlol}hg wr uhlqwhusuhw rxu uvw0rughu frqglwlrqv lq vxevhfwlrq 615 dqg wr fkdudfwhul}h wkh gxdo uvw0rughu frqglwlrqv lq vxevhfwlrq 6161

3.1. Arrow prices

D udwlr ri pdujlqdo xwlolwlhv lv d sulfh1 Wkh udwlr b_b1r

0 '

Y*Yw1r

Y*Yw0 lv wkh dprxqw

ri suhvhqw prqh| d frqvxphu lv zloolqj wr sd| iru dq dgglwlrqdo xqlw ri ixwxuh uhyhqxh lq vwdwh r1 Vxfk d +suhvhqw, pdujlqdo zloolqjqhvv wr sd|/ ru vxemhfwlyh Duurz sulfh ri dq hohphqwdu| frqwlqjhqw dvvhw/ zloo eh ghqrwhg >1r1 Ehiruh ehlqj

dfwxdol}hg/ wklv ydoxh +sulfh, lv ghqrwhg >1r vr wkdw rqh kdv >1r ' q>1r zlwk wkh

qrupdol}dwlrq S7_r=1>_1r ' 1 Frqvhtxhqwo|/ q ' S

rb1r

b0 lv d vxemhfwlyh glvfrxqw

idfwru riwhq zulwwhq

n o zkhuh o lv wkh qrplqdo udwh ri lqwhuhvw/ dqg >1r ' b1r

S

rb1r

lv d vxemhfwlyh iruzdug Duurz sulfh1 Dv >1r : f dqg

S

r>1r ' / >1r fdq dovr eh

lqwhusuhwhg dv dq Duurz ulvn0qhxwudo vxemhfwlyh suredelolw|1 Qrwh wkdw >_1r grhv h{lvw zkdwhyhu wkh qxpehu ri dvvhwv + ' f lv dgplvvleoh,1

3.2. Arrow prices and first-order conditions

Zh qrz uhfrqvlghu wkh lqwhusuhwdwlrq ri wkh uvw rughu frqglwlrqv +5147, ri wkh frqvxphu*v sureohp1 Zlwk wkh suhylrxv lqwhusuhwdwlrqv lq plqg/ X0*b0 ' R0 lv

dq htxdolw| ehwzhhq d pdujlqdo zloolqjqhvv wr sd| dqg dq +r!fldo, sulfh1 Vr lv X1r*b1r ' R1r1 Lq hdfk fdvh wkh pdujlqdo zloolqjqhvv wr sd| lv h{suhvvhg lq wkh

vshflf xqlw ri dffrxqw ri wkh fruuhvsrqglqj sulfh1 Ohw xv qrz frqvlghu wkh suhvhqw pdujlqdo zloolqjqhvv wr sd| X1r*b0 h{suhvvhg lq wkh suhvhqw dffrxqw xqlw1 Zh kdyh

X1r*b0 ' Eb1r*b0 R1r ' >_1rR1r Lw fdq eh frpsduhg wr R1r ehfdxvh wkh odwwhu kdv

ehhq frqyhuwhg +yld >1r, lqwr d vxemhfwlyh Duurz0Gheuhx sulfh1 Lq wkh vdph zd|/

wkh frqglwlrq S_rb1r^1r ' b0^0 fdq eh zulwwhq

S

r>1r^1r ' qSr>1r^1r ' ^0/

dqg h{suhvvhv wkdw/ iurp wkh frqvxphu*v ylhzsrlqw/ klv rzq pdujlqdo ydoxdwlrq ri dvvhwv lv htxdo wr wkhlu jlyhq sulfhv1

(16)

Proposition 2 : Wkh dvvhw sulfhv/ dw gdwh 3/ fdq eh zulwwhq dv d pdwkhpdwlfdo h{shfwdwlrq ri glvfrxqwhg ixwxuh fdvk0 rzv = ^0 ' q S r>1r^1r ' _{n o} S_r>1r^1r zkhuh >_1r ' _Sb1r rb1r

ghqhv d vxemhfwlyh suredelolw| phdvxuh rq wkh vhw ri vwdwhv/ dqg zkhuh q ' n o ' S rb1r b0 lv d vxemhfwlyh glvfrxqw idfwru1 ¥ Remark 2 : Wkh suredelolw| phdvxuh dqg wkh glvfrxqw idfwru duh d sulrul vxemhfwlyh/ exw fdq ehfrph remhfwlyh dffruglqj wr wkh vhw ri h{fkdqjhdeoh dvvhwv1 Iru lqvwdqfh/ li wklv vhw frqwdlqv dq Duurz vhfxulw| dvvrfldwhg zlwk vwdwh r/ lwv sulfh lv q>_1r ' >_1r1 Vlploduo|/ li wkhuh lv d }hur0frxsrq +^1r ' c ;r,/ lwv sulfh dw gdwh

3 lv q ' S_r>1r1 Lq wkh jhqhudo fdvh q>1r ' >1r lv wkh vxemhfwlyh sulfh ri dq

Duurz vhfxulw|/ >1 ' E>11c c >17 d ulvn0qhxwudo vxemhfwlyh suredelolw| phdvxuh

dqg q d vxemhfwlyh ydoxdwlrq ri d }hur0frxsrq ^Vhh Kduulvrq dqg Nuhsv +4<:<,/ Gx!h +4<<5,`1

Ohw xv frqvlghu wkh qr0duelwudjh frqglwlrq1 Wkh Dvvxpswlrq '_{+ ¤ f lv}

htxlydohqw wr wkh h{lvwhqfh ri d E7 n glphqvlrqdo yhfwru k : f vxfk wkdw 'k ' f ^e| Vwlhpnh*v ohppd/ vhh Pdqjdvduldq +4<9<,8_{`1 Wkh uhodwlrq 'k ' f}

lv xvxdoo| olqnhg wr Idundv* ohppd dqg wkxv zulwwhq ^0 ' '1Ek1*k0 Lq wklv

fdvh/ k1*k0 lv d qrq0qhjdwlyh0V0glphqvlrqdo yhfwru +exw qrw qhfhvvdulo| vwulfwo|

srvlwlyh,1 Zkhq wkh dvvhw vwuxfwxuh lv lqfrpsohwh/ vxfk d yhfwru k1*k0 lv qrw

qhfhvvdulo| xqltxh = wkhuh h{lvwv d srvlwlyh frqh ri glphqvlrq 7 +wkh glphqvlrq ri lqfrpsohwhqhvv, ri vxfk yhfwruv ^Urvv +4<:;,/ Euhhghq dqg Olw}hqehujhu +4<:;,/ Yduldq +4<;:,`1 Wkh yhfwru >1 ' b1*b0 lv dq hohphqw ri wkh suhylrxv frqh1 Zkhq

wkh dvvhw vwuxfwxuh lv frpsohwh/ wkh yhfwru k1*k0ehfrphv xqltxh dqg >1lv htxdo wr

wklv xqltxh k1*k01 Dv d uhvxow/ d yhfwru k1*k0 lv riwhq vhhq dv d yhfwru ri vkdgrz

sulfhv dvvrfldwhg zlwk hohphqwdu| frqwlqjhqw dvvhwv dqg frqvlvwhqw zlwk wkh sulfhv ri wudghg dvvhwv1 Wkh uvw0rughu frqglwlrqv 'b ' f dqg wkh srvlwlylw| ri wkh b1r*b0 ' >1r h{suhvv wkh qr0duelwudjh frqglwlrq1 Dv douhdg| vhhq lq Sursrvlwlrq 4/

e| vroylqj klv rswlpl}dwlrq sureohp/ wkh lqglylgxdo fkrrvhv d yhfwru ri vkdgrz sulfhv dprqj wkh pxowlsolflw| ri vkdgrz sulfhv ri hohphqwdu| frqwlqjhqw dvvhwv/ dqg wkh fkrvhq yhfwru ghshqgv rq klv suhihuhqfhv/ klv lqfrph dqg wkh vhw ri jlyhq sulfhv1

(17)

Remark 3 : Vxssrvh wkh frqvxphu*v sureohp lv vroyhg vhtxhqwldoo|1 Diwhu vroylqj iru % dqg b +uvw vwhs,/ wkh vhfrqg vwhs ri wkh rswlpl}dwlrq frqvlvwv ri @%

+ 7ERcw'

_{+c zkhuh 7 lv dq lqgluhfw xwlolw| ixqfwlrq1 Wklv vshflf remhfwly h}

ixqfwlrq lv w|slfdo ri wkh fkdudfwhulvwlf dssurdfk ^Ehfnhu +4<98,/ Odqfdvwhu +4<99, dqg Urvhq +4<:7,`1 Wr hdfk qdqfldo dvvhw lv qdwxudoo| dvvrfldwhg d froxpq ri ' _{dqg vxfk d froxpq +zkrvh frpsrqhqwv duh lqlwldo frvw dqg srvvleoh ixwxuh}

sd|0rv, fdq eh vhhq dv d yhfwru ri fkdudfwhulvwlfv1 '_{+ lv wkhq wkh yhfwru ri}

wrwdo dprxqwv ri fkdudfwhulvwlfv1 Wkhuhiruh/ lw vkrxog qrw eh vxusulvlqj wkdw/ zkloh vroylqj wkh frqvxphu*v sureohp/ rqh rewdlqv qdwxudoo| wkhvh fkdudfwhulvwlf vkdgrz +ru khgrqlf, sulfhv +dv vhhq lq Sursrvlwlrq 5,1

3.3. Arrow prices and the indirect utility function

Wkh Odjudqjhdq u ' XE% b_d_{% n '}_{+ wo fdq dovr eh zulwwhq u ' XE%}

b0E%0R0n +^0 w0

S

rb1rE%1rR1r +^1r w1r1 E| dsso|lqj wkh hqyhorsh

wkhruhp/ zh jhw wkh sduwldo ghulydwlyhv ri ERc ^cw zlwk uhvshfw wr Rc ^ dqg w = Y YR0 ' b0%0c Y YR1r ' b1r%1rc r ' c c 7 c +616, Y Y^0 ' b0+c Y YE^1r ' b1r+c r ' c c 7 c +617, Y Yw0 ' b0c Y Yw1r ' b1rc r ' c c 7 c +618,

zkhuh dq| whup ri wkhvh htxdolwlhv vkrxog eh vhhq dv d ixqfwlrq1

Qrz/ vxssrvh wkh frqvxphu lv idflqj vkrfnv _Rc _^c _w1 Dw uvw rughu/ wkh lpsdfw rq klv xwlolw| ohyho lv = _ ' b0d_w0 %0_R0 +_^0o n [ r b1rd_w1r %1r_R1rn +_^1ro c _ b0 ' d_w0 % 0_R0 +_^0o n [ r >1rd_w1r %1r_R1rn +_^1ro +619,

Lq wkh deryh ghfrpsrvlwlrq/ hdfk eudfnhw uhsuhvhqwv d uhdo yduldwlrq ri lqfrph zlwklq d jlyhq shulrg dqg vwdwh1

(18)

Proposition 3 : Wkh uvw0rughu yduldwlrqv ri xwlolw| +ru vwdqgdug ri olylqj, _ duh olqnhg wr wkh uhdo yduldwlrqv ri lqfrphv e| wkh irupxodv =

_ b0 ' d_w0 % 0_R0 +_^0o n [ r >_1rd_w1r %1r_R1rn +_^1ro ' d_w0 %0_R0 +_^0o n q [ r >_1rd_w1r %1r_R1rn +_^1ro ' d_w0 %0_R0 +_^0o n q.>1d_w1 % 1_R1n +_^1o c

zkhuh .>1 ghqrwhv wkh h{shfwdwlrq zlwk uhvshfw wr wkh suredelolw| glvwulexwlrq

>_1rc r ' c c 7 ¥

Lq rwkhu zrugv/ diwhu djjuhjdwlqj wkh ydulrxv vkrfnv _Rc _^c _w lqwr vsrw yduldwlrqv ri uhdo lqfrphv/ rqh fdq jr rq/ hyhq li wkh dvvhw vwuxfwxuh lv lqfrpsohwh/ dqg djjuhjdwh ryhu wlph dqg dfurvv vwdwhv1 Zkhq grlqj vr/ vxemhfwlyh Duurz sulfhv duh xvhg1 Wkh qdo uhvxow lv wkdw xwlolw| yduldwlrqv duh sursruwlrqdo wr lqwhuwhpsrudo uhdo yduldwlrqv ri zhdowk1 Wklv odvw frqfhsw frqwdlqv d suhvhqw dqg d ixwxuh1 Wkh ixwxuh uhdo zhdowk lv orfdoo| vhhq dv d pdwkhpdwlfdo h{shfwdwlrq zkhuh wkh suredelolwlhv xwlol}hg duh wkh frqvxphu*v ulvn0qhxwudo suredelolwlhv1 Remark 4 : Ohw xv frqvlghu htxdwlrq +619, zkhq _R ' fc _w ' f1 Rqh kdv _ ' b0+d

S

r >1r_^1r _^0o/ zkhuh wkh h{suhvvlrq ehwzhhq eudfnhwv lv d yhfwru

ri vxemhfwlyh h{fhvv uhwxuq fkdqjhv1 Wkh frqvxphu vxppdul}hv lq d vlpsoh 0glphqvlrqdo lqgh{ wkh E7 n qdqfldo sulfh yduldwlrqv1

Proposition 4 : Ur| lghqwlwlhv duh jlyhq e| = %0 ' Y*YR0 Y*Yw0c %1r' Y*YR1r Y*Yw1rc r ' c c 7 c + ' Y*Y^0 Y*Yw0 ' Y*YE^1r Y*Yw1r c r ' c c 7 ¥ Uhpdun 7 h{sodlqv wkh udwkhu shfxoldu orrn ri wkh qdqfldo frpsrqhqwv + ri wkh Ur| lghqwlwlhv1

(19)

4. Premiums and aversions

Duurz sulfhv >_1r ' b1r*b0 kdyh ehhq ghqhg dqg olqnhg wr wkh lqgluhfw xwlolw|

ixqfwlrq lq wkh suhylrxv vhfwlrq1 Lq wklv vhfwlrq/ wkh orfdo vwuxfwxuh ri Duurz sulfhv zloo eh xvhg wr ghqh suhplxpv dqg dyhuvlrq phdvxuhv1 Lq vxevhfwlrq 714/ zh zloo vwxg| wkh orfdo vwuxfwxuh ri Duurz sulfhv +zlwk wkh khos ri dq Dqwrqhool pdwul{, zklfk/ lq wxuq/ zloo eh xvhg wr ghqh d glvxwlolw| suhplxp1 Wkh odwwhu zloo eh vhhq dv d uhvlgxdo suhplxp zklfk phdqv wkdw/ zkloh idflqj d vkrfn/ wkh frqvxphu xvhv wkh qdqfldo pdunhwv wr dgmxvw klpvhoi1 Dq dowhuqdwlyh vhwwlqj zloo eh frqvlghuhg lq vxevhfwlrq 715 = wkh lqglylgxdo vwloo idfhv d vkrfn/ exw lv qr pruh deoh wr dgmxvw qdqfldoo|1 Lq vxfk d frqwh{w/ wkh uhvxowlqj suhplxp dqg ulvn dyhuvlrq zloo eh fdoohg ixqgdphqwdo1 Lq vxevhfwlrq 716/ zh vkrz krz uhvlgxdo dqg ixqgdphqwdo suhplxpv duh olqnhg wrjhwkhu yld wkh Guë}h0Prgljoldql ghfrpsrvlwlrq1 Iru frqyhqlhqfh/ rqo| lqfrph vkrfnv zloo eh frqvlghuhg1 Wklv lv frqvlvwhqw zlwk wkh exon ri wkh olwhudwxuh dqg/ lq sduwlfxodu/ lw lv wkh qdwxudo frqwh{w ri wkh Guë}h0Prgljoldql ghfrpsrvlwlrq1

4.1. The local structure of Arrow prices and the disutility premium Zh vwduw iurp Sursrvlwlrq 61 Ohw xv vhw =

> ' b*b0 ' >1 ' dc q>11c c q>17o

Xqghu wklv frqyhqwlrq/ dqg zkhq frpprglw| dqg dvvhw sulfhv duh nhsw frqvwdqw/ wkh uvw0rughu yduldwlrq ri xwlolw| lv olqnhg wr wkh yduldwlrq ri lqfrphv e| wkh uhodwlrq =

_ b0 ' >

_{_w}9 _+714,

Erwk vlghv ri wkh uhodwlrq duh lqyduldqw xqghu d prqrwrqlf wudqvirupdwlrq ri wkh xwlolw| ixqfwlrq1 Wkh h{sdqvlrq ri wkh lqgluhfw xwlolw| ixqfwlrq fdq dovr eh frqvlghuhg dw rughu wzr = { ' _ n ₂_2_{n J}_{n_wn}2 _+vd|, Wkhq e| glhuhqwldwlqj +714,/ rqh kdv = _2 b0 _b0 b0 >_{_w ' _w}0 1_>1 +715, 9 _{Li lqfrph vkrfnv gw zhuh frxsohg zlwk frpprglw| dqg dvvhw sulfh vkrfnv gs dqg gt/}

(20)

Zh vkdoo vwxg| = d, wkh ghfrpsrvlwlrq ri _>1 lqwr vxevwlwxwlrq dqg zhdowk

hhfwv/ e, wkh ghfrpsrvlwlrq ri _2_{/ f, wkh fruuhvsrqglqj irupdwlrq ri d}

glvxwlolw|0suhplxp irupxod/ dqg g, dq h{dpsoh zklfk looxvwudwhv wkh Dqwrqhool pdwul{1

a) The local structure of Arrow prices

Wkh orfdo vwuxfwxuh ri Duurz sulfhv lv vwxglhg e| h{dplqlqj wkh frh!flhqwv ri _>11 Wkhlu pdlq fkdudfwhulvwlfv duh vxppdul}hg lq wkh ohppd ehorz1

Lemma 1 = l, Zh jhw =

_>1 ' 11_w1n _YwY>1 0>

_{_w c}

zkhuh 11 lv dq Dqwrqhool pdwul{ zklfk phdvxuhv wkh hhfwv rq Duurz sulfhv ri d

frpshqvdwhg lqfrph vkrfn = 11' Y>1 Yw 1 ˜ >0_{_w= 0} ' Y>1 Yw 1 Y>1 Yw0> 1 ' d>1 Uro Yb*Yw b0 > 1 Ur ' d>1 Uro Y2_*YwYw b0 > 1 Ur ( ll, 11 lv d v|pphwulf pdwul{ zklfk lv lqghshqghqw ri d prqrwrqlf wudqvirupdwlrq

ri wkh xwlolw| ixqfwlrq/ qhjdwlyh vhpl0ghqlwh/ zlwk udqn 7 +wkh glphqvlrq ri lqfrpsohwhqhvv,/ dqg !ih 11' udqjh '1 ¥

Proof = Vhh Dsshqgl{ E1

Dq Dqwrqhool pdwul{ lv dqdorjrxv wr d Voxwvn| pdwul{ = wkh odwwhu fkdudfwhul}hv yduldwlrqv lq frpprglw| ghpdqgv iroorzlqj frpshqvdwhg sulfh vkrfnv zkloh wkh iruphu fkdudfwhul}hv yduldwlrqv lq wkh +fruuhvsrqglqj, sulfhv iroorzlqj frpshqvdwhg txdqwlw| vkrfnv1 Erwk pdwulfhv fdq eh xvhg wr fkdudfwhul}h vxevwlwxwlrq dqg frpsohphqwdulw| ^vhh/ iru lqvwdqfh/ Vdpxhovrq +4<83, dqg klv vxuyh| +4<:7,`1 E| dqdorj|/ 11 lv fdoohg dq Dqwrqhool pdwul{ vlqfh lw dovr

fkdudfwhul}hv yduldwlrqv lq wkh vxemhfwlyh sulfhv1 Wkh pdwul{ 11c krzhyhu/

phdvxuhv yduldwlrqv lq wkh vxemhfwlyh sulfhv ri wkh lpsolhg +ru yluwxdo, Duurz dvvhwv iroorzlqj d frpshqvdwhg lqfrph vkrfn:_{/ dqg wkh dvvhwv fdq eh wudgdeoh} : _{Uhfdoo wkdw S}0

1{1 @ W1. T01| fdq uhgxfh wr S10{1 @ W1. |= Wkhuhiruh/ lqfrphv fdq eh

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ru qrw1 Wkh ghfrpsrvlwlrq ri _>1 +jlyhq e| Ohppd 4, lqyroyhv vxevwlwxwlrq

dqg zhdowk hhfwv/ dqg wkh Dqwrqhool pdwul{ 11 lv xvhg wr fkdudfwhul}h wkh

vxevwlwxwlrq0frpsohphqwdulw| dprqj +yluwxdo, dvvhwv1 Frqfhuqlqj wkh udqn ri 11/

wkh lqwxlwlrq lv hdvlhu wr jhw li rqh uvw frqvlghuv wkh fdvh zkhuh wkh dvvhw vwuxfwxuh lv frpsohwh1 Lq vxfk d fdvh/ wkh ' 7 wudgdeoh dvvhwv duh Duurz dvvhwv dqg Duurz sulfhv duh h{rjhqrxv ru {hg yduldeohv/ zklfk lpsolhv wkdw kY˜>1

Yw1 l

' f Frqvhtxhqwo|/ wkh udqn ri 11 uhgxfhv wr }hur1 Zkhq wkh dvvhw vwuxfwxuh lv

lqfrpsohwh 7c wkh h{lvwhqfh ri wudgdeoh dvvhwv dprxqwv wr kdylqj {hg frpelqdwlrqv ri wkh 7 Duurz sulfhv >_1r zklfk/ lq wxuq/ uhgxfhv wkh udqn ri 11 wr 7 b) The decomposition of _2 Lemma 2 : Zh jhw = _2_{' b} 0 _w0 111_w1n 2_w 0 1 Y>1 Yw0> _{_w n} Yb0 Yw0 E> _{_w}2 ¥ Proof = vhh Dsshqgl{ F1 Wkh ghfrpsrvlwlrq ri _2_{lqyroyhv d vxevwlwxwlrq hhfw b} 0 _w0 111_w1 ' d_2_o ˜

>0_{_w= 0} dqg d zhdowk hhfw/ wkh odwwhu ehlqj ri ghjuhh wzr1 Lw lv lpsruwdqw wr qrwh wkdw qhlwkhu _2_{qru _}2_*b

0 lv lqyduldqw zlwk uhvshfw wr lqfuhdvlqj

wudqvirupdwlrq ri wkh xwlolw| ixqfwlrq1 Krzhyhu/ wkh uhodwlrq ri Ohppd 5 fdq dovr eh zulwwhq dv = _2 b0 Yb0*Yw0 b0 E> _{_w}2 _{' _w}0 111_w 0 1n 2_w 0 1 Y>1 Yw0> _{_w c} _+716,

zkhuh erwk vlghv vdwlvi| wkh lqyduldqfh surshuw|1 Zh ghgxfh = _ b0 n 2 _2 b0 Yb0*Yw0 b0 E> _{_w}2 ' n _w0 1 Y>1 Yw0 >_{_w n} 2_w 0 111_w1 +717,

(22)

Rxu qh{w sxusrvh lv wr vkrz wkdw vxfk d uhodwlrq ghqhv d suhplxp dqg vwloo d zhoiduh fulwhulrq1

c) The disutility premium

Ohw Ew0_{eh wkh ydoxh ri wkh lqgluhfw xwlolw| ixqfwlrq dw vrph srlqw w}0_{/ dqg}

Ew lwv ydoxh diwhu wkh lqfrph vkrfn _w'ww0 _{+sulfhv duh ghohwhg wr vljqdo}

wkhlu frqvwdqf|,1 Wkhq/ wkhuh h{lvwv d wzlfh frqwlqxrxvo| glhuhqwldeoh ixqfwlrq 4 vxfk wkdw Ew ' w00 4 wc w0_{c w}0 1 c +718,

hyhu|zkhuh rq wkh grpdlq ri 1 Lq wklv uhodwlrq/ dqg 4 duh prqrwrqlf wudqvirupdwlrqv ri hdfk rwkhu1 Wkhuhiruh/ 4 pd| eh vhhq dv d glvxwlolw| ixqfwlrq1 Wkh ydoxh 4 Ewc w0_{lv h{suhvvhg lq suhvhqw prqh|1 Lq wkh odqjxdjh ri zhoiduh}

phdvxuhv/ lw lv dqdorjrxv wr dq htxlydohqw yduldwlrq1 Lwv rssrvlwh/ 4 Ewc w0_{c lv d}

suhplxp1

Ohw xv glhuhqwldwh +718,1 Wklv jlyhv = _ ' b0 w0₀ 4 E c w0₁_4 c _2_{' b} 0 w00 4 E c w01 _2_{4 n} Yb0 Yw0 w00 4 E c w01 E_42

Dw wkh uhihuhqfh srlqw w0_{c 4 Ew}0_{c w}0_{' f Wkh suhylrxv uhodwlrqv ehfrph =}

_4 ' __b 0 c _2_{4 '} _2 b0 Yb0*Yw0 b0 E> _{_w}2 _c zkhuh b0 ' b0Ew00c w01 ' b0Ew0 Wklv |lhogv = _4 ₂_2_{4 '} _ b0 n 2 _2 b0 Yb0*Yw0 b0 E> _{_w}2 c

zklfk lv h{dfwo| wkh ohiw0kdqg vlgh ri +717,1 Ilqdoo|/ wkh Wd|oru h{sdqvlrq ri 4 pd| eh zulwwhq {4 ' _4 n 1

2_

(23)

{4 ' 4 Ewc w0_{4 Ew}0_{c w}0_{' 4 Ewc w}0_{Vr/ zh kdyh ^e| xvlqj +714, dqg} Ohppd 5` = 4 ' __b 0 n 2 _2 b0 Yb0*Yw0 b0 E> _{_w}2 Jn_wn2 ' n _w0 1 Y>1 Yw0 >_{_w n} 2_w 0 111_w1 J n_wn2 _+719,

Wkh dvvhuwlrq frqfhuqlqj uhodwlrq +717,/ wkdw lv/ lw ghqhv d suhplxp dqg lv vwloo d zhoiduh fulwhulrq/ lv suryhg vlqfh 4 lv erwk d glvxwlolw| ixqfwlrq dqg d suhplxp1 Pruhryhu/ uhpdun wkdw 4 lv dovr lqghshqghqw ri d prqrwrqlf wudqvirupdwlrq ri wkh xwlolw| ixqfwlrq/ dv lv wkh ohiw0kdqg vlgh ri +717,1 Iurp qrz rq/ zh vkdoo uhihu wr 4 dv d glvxwlolw| suhplxp1

Proposition 5 : Wkh zhoiduh hhfw ri dq lqfrph vkrfn _w'ww0 _{pd| eh}

phdvxuhg zlwk wkh khos ri d glvxwlolw|0suhplxp ixqfwlrq 4 fkdudfwhul}hg e| wkh uhodwlrq Ew ' dw0

0 4 Ewc w0 c w01o Dw wkh uhihuhqfh srlqw w0c wklv ixqfwlrq

lv vxfk wkdw 4 ' n _w0 1 Y>1 Yw0 >_{_w n} 2_w 0 111_w1 J n_wn2 _c

zkhuh 11 lv dq Dqwrqhool pdwul{1 Orfdoo|/ wkh glvxwlolw| suhplxp lqyroyhv erwk

vxevwlwxwlrq dqg zhdowk hhfwv> dw uvw rughu/ _4 ' >_{_w dqg wkh zhoiduh hhfw}

lv d zhdowk yduldwlrq> dw wkh vhfrqg rughu/ krzhyhu/ 1 2_24 ' 1 2_w 0 111_w1 n _w0

1_YwY˜>1₀>_w dqg wkh zhoiduh hhfw lqyroyhv erwk vxevwlwxwlrq dqg zhdowk hhfwv1

¥ Corollary 1 : Li qdqfldo pdunhwv duh frpsohwh E ' 7 c Duurz sulfhv >1frlqflgh

zlwk pdunhw sulfhv lq wkh vhqvh wkdw >1 ' '311 ^0 G wkhuh duh qr vxevwlwxwlrq hhfwv

11' f/ dqg d zhdowk yduldwlrq kdv qr hhfw rq +lqglylgxdo, Duurz sulfhv _YwY˜>1₀ ' f

Wkhuhiruh/ wkh glvxwlolw| suhplxp uhgxfhv wr wkh uvw0rughu zhdowk hhfw =

4 ' _4 n Jn_wn2_{' >}_{_w n J}_{n_wn}2 _+71:,

¥ Lq wkh jhqhudo fdvh/ li >_{_w9' f/ wkh uvw0rughu whup +wkdw lv/ wkh zhdowk hhfw}

(24)

h{sdqvlrq ri 4/ hyhq li wkh vkrfn dovr lqyroyhv d uhdoorfdwlrq ryhu wlph dqg dfurvv vwdwhv1 Frqvhtxhqwo|/ wkh lqwhuhvwlqj fdvh wr eh vwxglhg lv zkhq >_{_w' f} Wkh frqglwlrq >_{_w ' f dovr phdqv _4 ' f dqg} _ b0 ' fc wkdw lv/ uvw0rughu frpshqvdwlrq1 Dv >_{_w'} S r>1rd_w0n q_w1ro/ lw dovr phdqv wkdw

wkh pdwkhpdwlfdo h{shfwdwlrq ri zhdowk fkdqjhv ydqlvkhv1 Lq vxfk d fdvh/ wkh vkrfn rqo| lqyroyhv d uhdoorfdwlrq ryhu wlph dqg dfurvv vwdwhv/ wkh fdvh zh vkdoo irfxv rq iurp qrz rq1

Corollary 2 : Li wkhuh lv uvw0rughu frpshqvdwlrq E>_w ' f c wkh glvxwlolw| suhplxp lv htxlydohqw wr = 4 ' ₂_w0 111_w1n J n_wn2 ' ₂_w0 1_>1n J n_wn2_' 2_w 0Yb*Yw 0 b0 _w n J n_wn2 ' 2 _2 b0 n Jn_wn2 _'{ b0 n Jn_wn2 _+71;, ¥ Wkh irupxodv ri Sursrvlwlrq 8 dqg Fruroodu| 5 wdnh lqwr dffrxqw wkh frqvxphu*v dgmxvwphqwv +wr _w, lq frpprglwlhv dqg dvvhwv1 Lq d qdqfldo hfrqrp|/ wkh frqvxphu dgmxvwv klpvhoi e| pdnlqj lqfrph wudqvihuv zklfk/ lq wxuq/ duh pdgh srvvleoh wkurxjk sruwirolr vhohfwlrq1 Idflqj dq V0glphqvlrqdo lqfrph vkrfn/ kh xvhv wkh Q0wudgdeoh dvvhwv wr dmxvw klpvhoi dqg wr vdwlvi| wkh qhz exgjhw frqvwudlqwv1 Wklv lv zk| wkh suhylrxv glvxwlolw| suhplxp lv qhfhvvdulo| d uhvlgxdo suhplxp/ phdqlqj wkdw lw wdnhv lqwr dffrxqw qdqfldo dgmxvwphqw1 Wkh h{whqw ri wklv uhvlgxdo hhfw ghshqgv rq wkh qxpehu dqg fkdudfwhulvwlfv ri qdqfldo dvvhwv/ lq sduwlfxodu/ wkurxjk wkh udqn 7 ri wkh Dqwrqhool pdwul{ 111 Zh vkdoo kdyh

d forvhu orrn dw wklv uhvlgxdo hhfw lq vxevhfwlrq 7161 d) Example

Zh qrz surylgh d vlpsoh h{dpsoh wr looxvwudwh judsklfdoo| wkh hhfw phdvxuhg e| wkh Dqwrqhool pdwul{1 Pruh vshflfdoo|/ zh frqvlghu wkh frqvxphu*v rswlpl}dwlrq sureohp zkhq wkh ixwxuh shulrg frqvlvwv ri rqo| rqh vwdwh E7 ' dqg zkhq qr wudgdeoh dvvhw h{lvwv E ' f Wkh rswlpl}dwlrq sureohp lv

(25)

4@ %0c%11 X E%0c %11 vxemhfw wr R0%0 ' w0 R11%11 ' w11 < A @ A >S

Wkh vroxwlrq lv vwudljkwiruzdug/ jlyhq e| %0 ' w0c %11 ' w11c zkhuh/

iru vlpsolflw|/ zh kdyh dvvxphg R0 ' R11 ' Pruhryhu/ li vroyhg e| phdqv ri

wkh Odjudqjhdq = O E%0c %11c b0c b11 ' X E%0c %11 b0d%0 w0o b11d%11 w11o c

sureohp S4 doorzv wr ghqh wkh Duurz sulfh >11' b_b11₀ ' YF*Y%_YF*Y%11₀ c dqg vlqfh 7 ' c

zh kdyh S_rb1r ' b11 zklfk lpsolhv q ' S b1r b0 ' b11 b0 ' >11 Lq vxfk d fdvh/ wkh Dqwrqhool pdwul{ kdv udqn 7 ' c frqvlvwv ri rqo| rqh frh!flhqw dqg lv jlyhq e| = 11 ' Y>11 Yw11 _w0+˜>11_w11=0 ' Y>11 Yw11 Y>11 Yw0>11 ru/ htxlydohqwo|/ 11 ' Yq Yw11 _w0+q_w11=0 ' _YwYq 11 Yq Yw0q Qrwh wkdw sureohp S4 lv htxlydohqw wr 4@ %0c%11 X E%0c %11 vxemhfw wr %0n q Ew0c w11 %1 ' w0n q Ew0c w11 w11' ¯w c

zkhuh ¯w lv wkh frqvxphu*v suhvhqw zhdowk1

Ohw xv frqvlghu jlyhq ydoxhv ri lqfrphv1 Wkh vroxwlrq ri S4 lv uhsuhvhqwhg e| srlqw D rq Iljxuh 4/ zkhuh q Ew0c w11 ' q/ vd|/ lv wkh vorsh ri wkh wdqjhqw

ri wkh lqglhuhqfh fxuyh X E%0c %11 ' X dqg ¯w lv wkh %00lqwhufhsw ri wkh vdph

wdqjhqw olqh1

Ohw xv wkhq frqvlghu dq lqfrph vkrfn E_w0c _w11 Wkh qhz vroxwlrq/

uhsuhvhqwhg e| srlqw E/ lv %

0 ' w0n_w0c %11' w11n_w11c dqg lqyroyhv d fkdqjh

(26)

q Ew0n _w0c w11c _w11 ' q/ vd|/ dqg ¯wn_ ¯w' Ew0n _w0nqEw11n _w11 c

uhvshfwlyho|1 Wkh zhdowk yduldwlrq _ ¯w kdv wzr frpsrqhqwv = _ ¯w' _w 0n q_~}_w11 1 n Yq Yw0_w0n Yq Yw11_w11 ~} 2 w11 c zkhuh Yq Yw0 ' Yq(w0cw11)

Yw0 Wkh uvw rqh uhihuv wr wkh yduldwlrq zkhq q lv xqfkdqjhg +d sdudooho pryhphqw ri wkh wdqjhqw olqh %0n q%11' ¯w, zkloh wkh vhfrqg rqh lv

gxh wr wkh fkdqjh lq q

Lq rughu wr rewdlq wkh hhfw phdvxuhg e| wkh Dqwrqhool pdwul{/ zh qhhg wr lghqwli| d wklug srlqw/ wkdw lv/ zkdw wkh frqvxphu zrxog fkrrvh li wkh lqfrph vkrfn zhuh frpshqvdwhg1 Ohw %

0c %11

' E%0n _ ˜w0c %11n _ ˜w11 +uhsuhvhqwhg e|

(27)

4, _ ˜w0 vkrxog frpshqvdwh iru _ ˜w1 vr dv wr ohdyh wkh suhvhqw zhdowk ¯wxqfkdqjhg/

wkdw lv _ ˜w0nq Ew0c w11 _ ˜w1 ' f( dqg 5, q ' q Ew0n _ ˜w0c w11n _ ˜w11 vkrxog

eh htxdo wr q _{' q Ew}

0n _w0c w11n _w11 Lw lv hdvlo| fkhfnhg wkdw srlqw F lv

orfdwhg rq wkh wdqjhqw olqh %0n q%11 ' ¯w dqg lv vxfk wkdw _ ˜w lv htxdo wr wkh

vhfrqg frpsrqhqw ri _ ¯w1

Qrwh wkdw lq wklv sduwlfxodu h{dpsoh Yq

Yw0 ' f Wkhuhiruh/ wkh frpshqvdwhg dqg wkh qrq frpshqvdwhg lqfrph vkrfn erwk lqyroyh wkh vdph hhfw rq wkh vorsh q Pruh suhflvho|/ zh jhw 11 ' Yq Yw11 w0+q_w11=0 ' _YwYq 11c dqg wkh hhfw ri wkh

Dqwrqhool pdwul{ fruuhvsrqgv wr wkh fkdqjh ri vorsh iurp srlqw D wr srlqw F1 Wklv h{dpsoh dovr surylghv wkh lqwxlwlrq ri zk|/ lq rxu prgho/ wkh lqgluhfw xwlolw| ixqfwlrq lv qrw txdvl0frqyh{ lq w dv xvxdo/ exw txdvl0frqfdyh lq w1 Lqghhg/ zh kdyh = ER0c R11c w0c w11 ' Ec c w0c w11 ' Ew0c w11 c dqg

frqvhtxhqwo|/ wkh lqgluhfw xwlolw| ixqfwlrq ehkdyhv +urxjko| vshdnlqj, olnh d gluhfw xwlolw| ixqfwlrq1

Lq wkh qh{w vxevhfwlrq/ zh ghqh d ixqgdphqwdo glvxwlolw| suhplxp dv zhoo/ zkhq wkh lqglylgxdo grhv qrw xvh wkh qdqfldo pdunhwv wr dgmxvw klpvhoi1 Lq vxevhfwlrq 716/ zh vhh krz uhvlgxdo dqg ixqgdphqwdo suhplxpv duh wlhg wrjhwkhu yld wkh Guë}h0Prgljoldql ghfrpsrvlwlrq1

4.2. The fundamental premium and the fundamental risk aversion Lq rughu wr ghqh dqg fkdudfwhul}h wkh ixqgdphqwdo glvxwlolw| suhplxp dw w0

+dqg/ xowlpdwho|/ d fruuhvsrqglqj phdvxuh ri ixqgdphqwdo ulvn dyhuvlrq,/ rqh frqvlghuv wkh iroorzlqj vhw0xs1 Wkh frqvxphu vwloo idfhv wkh lqfrph vkrfn _w dw wkh vdph uhihuhqfh srlqw w0_{/ exw lv qrw deoh wr dgmxvw qdqfldoo| ru/ htxlydohqwo|/}

lv hqgrzhg zlwk dq looltxlg sruwirolr1 Wklv vhwwlqj zloo uvw eh xvhg wr vwxg| wkh ixqgdphqwdo glvxwlolw| suhplxp1 Zh wkhq lqwurgxfh d qhz qrupdol}dwlrq +iru >_1r dqg q, lq rughu wr frpsohwho| glvhqwdqjoh wlph dqg xqfhuwdlqw|1 Wklv zloo doorz xv wr ghfrpsrvh wkh ixqgdphqwdo glvxwlolw| suhplxp lqwr dq lpsdwlhqfh dqg d ulvn suhplxp1 Wkh odwwhu ghfrpsrvlwlrq kdv d qdwxudo frxqwhusduw lq whupv ri dyhuvlrq = wkh ixqgdphqwdo mrlqw wlph0ulvn dyhuvlrq zloo eh ghfrpsrvhg lqwr dq dyhuvlrq wr lpsdwlhqfh dqg d ulvn dyhuvlrq1

(28)

a) The fundamental premium

Dv lq wkh fdvh zlwk qdqfldo dgmxvwphqw/ wkh lqfrph vkrfn lv orfdoo| dqdo|}hg durxqg wkh rswlpdo sruwirolr +W _{+wkh vhohfwlrq ri zklfk lv pdgh zlwkrxw dq|}

uhvwulfwlrq,1 Iru lqvwdqfh/ wkh lqfrph vkrfn frxog khuh eh dqdo|}hg e| xvlqj wkh lqgluhfw xwlolw| ixqfwlrq Ew ' 7 Ew '_{7+ zkhuh 7+ ' +}W _{+vhh Vhfwlrq 6/}

Uhpdun 6,1 Frpsduhg zlwk wkh fdvh zlwk qdqfldo dgmxvwphqw/ wkh Duurz sulfhv wdnh wkh vdph lqlwldo +rswlpdo, ydoxhv/ exw wkhlu fkdqjhv duh/ lq sulqflsoh/ glhuhqw1 Ohw _7q/ _3>/ _7>/ dqg 711 ghqrwh/ uhvshfwlyho|/ wkh glvfrxqw idfwru yduldwlrq/ wkh

Duurz sulfhv yduldwlrqv/ wkh iruzdug Duurz sulfhv yduldwlrqv/ dqg wkh Dqwrqhool pdwul{ fruuhvsrqglqj wr wklv fdvh1 Ohw dovr 74 ghqrwh wkh qhz glvxwlolw| suhplxp1 Li zhuh lqwhusuhwhg dv wkh qxpehu ri oltxlg dvvhwv/ wkh dqdo|vlv ri wkh lqfrph vkrfn lq wkh suhvhqw fdvh dprxqwv wr uhzulwlqj Sursrvlwlrq 8 lq wkh sduwlfxodu fdvh zkhuh ydqlvkhv orfdoo|1 Wklv ohdgv wr wkh sursrvlwlrq ehorz =

Proposition 6 : Li wkhuh lv uvw0rughu frpshqvdwlrq E>_w ' f c wkh zhoiduh hhfw ri dq lqfrph vkrfn _w ' w w0 _{zlwkrxw qdqfldo dgmxvwphqw pd| eh}

phdvxuhg zlwk d glvxwlolw| suhplxp 74 dv iroorzv = 74 ' 2_w1711_w1 J n _w n2 ' 2_w 0 1_ 3 >1 J n_wn2 _' 2_w 0Y7b*Yw 0 7b0 _w Jn_wn2 ' 2 _2₇ 7b0 Jn_wn2_' {7 7b0 Jn_wn2 _c _+71<, zkhuh 711 kdv udqn 7

Proof = Wkh udqn ri 711 fdq eh ghgxfhg iurp wkh udqn ri wkh pdwul{ 11

+vhh Ohppd 4, zkhq ' f ¥

Iurp qrz rq/ zh vkdoo uhihu wr 74 dv wkh ixqgdphqwdo glvxwlolw| suhplxp/ phdqlqj wkdw lw grhv qrw wdnh lqwr dffrxqw qdqfldo dgmxvwphqw1 Wkh dlp ri wkh qh{w vxevhfwlrq lv wr ghfrpsrvh wkh ixqgdphqwdo glvxwlolw| suhplxp lqwr dq lpsdwlhqfh dqg d ulvn suhplxp1

(29)

b) Impatience and risk premiums

Ehiruh jrlqj lqwr wkh ghwdlov ri wkh ghfrpsrvlwlrq/ zh uvw jlyh wkh lqwxlwlrq ri krz lw zloo eh grqh1 Ohw xv frqvlghu wkh 70glphqvlrqdo yhfwru ri ixwxuh lqfrphv dqg lqwhusuhw lw dv d sruwirolr frpsrvhg ri 7 frqwlqjhqw Duurz dvvhwv w1rc r ' c c 7 Lq wklv iudphzrun/ >1w1 uhsuhvhqwv wkh suhvhqw sulfh +dw gdwh 3,

ri wklv sruwirolr1 Lwv sulfh dgplwv glhuhqw frpsrqhqwv zklfk uhihu wr wkh frvw ri lqwhuwhpsrudo wudqvihuv/ wkh frvw ri lqvxudqfh/ dqg d frvw iru furvv0hhfwv1 Li zh xvhg wkh qrupdol}hg Duurz sulfhv >1r dqg wkh glvfrxqw idfwru qc wkh fkrlfh ri wkh

qrupdol}dwlrq +wkurxjk wkh fkrlfh ri q, zloo reylrxvo| dhfw wkh ghfrpsrvlwlrq ri wkh sruwirolr sulfh vlqfh >1r' q>1rdqg >1w1 ' q>1w1 Wkh qxpehu ri dgplvvleoh

qrupdol}dwlrqv lv lqqlwh1 Xqwlo qrz/ zh kdyh xvhg wkh fdqrqlfdo qrupdol}dwlrq ri wkh olwhudwxuh1 Lw doorzv wr lqwhusuhw >_1r dv dq Duurz ulvn0qhxwudo vxemhfwlyh suredelolw|1 Wkh glhuhqw uhvxowv ri vxevhfwlrq 714 zhuh hvwdeolvkhg iurp wkh Duurz sulfhv >1r Wkh| duh wkhuhiruh lqghshqghqw ri wkh fkrvhq qrupdol}dwlrq

iru wkh glvfrxqw idfwru q dqg wkh >_1r ' ˜>_1r

q Dv phqwlrqhg deryh/ wklv zloo qrw eh

wkh fdvh iru wkh ghfrpsrvlwlrq ri wkh sruwirolr ydoxh1 Zh zrxog olnh wr uhvwulfw rxu fkrlfh wr d vhw ri qrupdol}dwlrqv wkdw frpsohwho| glvhqwdqjoh wlph dqg xqfhuwdlqw|1 Lq vxfk d fdvh/ wkh sruwirolr sulfh zrxog eh pdgh0xs ri rqo| wzr frpsrqhqwv/ wkdw lv/ d whpsrudo dqg dq lqvxudqfh frpsrqhqw1 Wkh qrupdol}dwlrq xvhg vr idu grhv qrw doorz iru vxfk d vwulfw ghfrpsrvlwlrq1 Ohw xv qrz lqwurgxfh wkh qhz qrupdol}dwlrq1 Ohw xv ghqh q dqg >1 ' ˜ >1 q ' qbb10 vxfk wkdw > 1711>1 ' c +7143,

zkhuh 711' 73111 Ew0 lv d srvlwlyh ghqlwh pdwul{1 Zh ghgxfh

>1 ' b1 b 1711b1 1*2 c +7144, q ' b 1711b1 1*2 b0 +7145,

Iurp wkh qrupdol}dwlrq uhvwulfwlrq +7143, dqg >1 ' q>1c zh ghgxfh wkh

glhuhqwldo uhvwulfwlrqv =

_3>1 ' _7q>1n _7>1q +7146,

>

(30)

Wkh frqglwlrqv deryh fdq eh lqwhusuhwhg lq wkh iroorzlqj zd|1 Wkh Duurz sulfh yduldwlrqv _3>1 fdq eh ghfrpsrvhg dv wkh vxp ri _7q>1 dqg _7>1q Zkhq _w ydulhv/

_7q>1c _7>1qc _ 3

>1 jhqhudwh yhfwru vsdfhv .Uc .UUc .c uhvshfwlyho|1 Wkh| duh vxfk

wkdw

. ' .U_{n .}UU _c

dqg .U _{dqg .}UU _{duh ruwkrjrqdo iru wkh vfdodu surgxfw 7}

11 Lqghhg/

_7q>

1711q_7>1 ' f c

e| uhodwlrq +7147,1 Wklv fdq eh vxppdul}hg lq wkh iroorzlqj ohppd =

Lemma 3 : Decomposition of the space of price variations. Wkh 70glphqvlrqdo yhfwru vsdfh . vsdqqhg e| wkh Duurz sulfhv yduldwlrqv _3>1 lv wkh

gluhfw vxp ri wkh rqh0glphqvlrqdo yhfwru vxevsdfh .U _{vsdqqhg e| wkh yhfwru}

ri qrupdol}hg Duurz sulfhv >1 dqg wkh E7 0glphqvlrqdo yhfwru vxevsdfh .UU

vsdqqhg e| wkh qrupdol}hg Duurz sulfh yduldwlrqv _7>1 Wkh vxevsdfhv .U dqg

.UU _{duh ruwkrjrqdo iru wkh vfdodu surgxfw 7}

11 Wkh surmhfwru A rq .U dorqj .UU

lv ghqrwhg e| =

A ' >1>1711

¥ Zh fdq dovr lqwurgxfh d gxdo ghfrpsrvlwlrq lq wkh vsdfh ri lqfrph vkrfnv1 Lqghhg/ xqghu uvw0rughu frpshqvdwlrq/ 711_w1 ' _ 3 >1 Vlqfh 711 lv lqyhuwleoh/ rqh dovr kdv _w1 ' 73111_ 3 >1 ' 711_ 3 >1 Xvlqj +7146,/ zh jhw = _w1 ' 711_7q>1 711q_7>1 +7148,

Pruhryhu/ wkh ruwkrjrqdo surmhfwru A vdwlvhv = 711A ' A711 Zh ghgxfh

A_{_w} 1 ' A711_7q>1 A711q_7>1 ' 711A _7q>1 711A q_7>1 ' 711_7q>1 c vlqfh A q_7>1 ' f dqg A _7q>1 ' _7q>1 e| wkh ghqlwlrq ri A / dqg wkhuhiruh = EU A_{_w} 1 ' 711q_7>1

(31)

Qrwh dovr wkdw 731

11A ' EA71131c zklfk phdqv wkdw A lv dq ruwkrjrqdo surmhfwru

iru wkh vfdodu surgxfw 731

11 ' 711

Lq uhodwlrq +7148,/ 711 ' 73111 lv d Voxwvn| pdwul{ +ghohwhg ri lwv uvw

olqh dqg froxpq,1 _w1 fdq eh ghfrpsrvhg lqwr _w1 ' _wU1 n _wUU1 zkhuh

_wU

1 ' 711_7q>1 dqg _wUU1 ' 711q_7>1 duh lqwhuwhpsrudo dqg dfurvv vwdwh

uhdoorfdwlrqv/ uhvshfwlyho|1 Wkxv/ 711>1 pd| eh lqwhusuhwhg dv lqwhuwhpsrudo

vxevwlwxwlrq hhfwv dqg 711q dv vxevwlwxwlrq hhfwv dfurvv vwdwhv1

Wkh uhvxowv fdq eh vxppdul}hg lq wkh iroorzlqj ohppd =

Lemma 4 : Decomposition of the space of income reallocations. Wkh 70glphqvlrqdo yhfwru vsdfh ` ' 731

11. vsdqqhg e| wkh lqfrph uhdoorfdwlrqv _w1

lv wkh gluhfw vxp ri wkh rqh0glphqvlrqdo vxevsdfh `U _{' 7}31

11.U ri lqwhuwhpsrudo

uhdoorfdwlrqv dqg wkh E7 0glphqvlrqdo vxevsdfh `UU _{' 7}31

11.UU ri dfurvv vwdwh

uhdoorfdwlrqv1 Wkh vxevsdfhv `U_{c `}UU _{duh ruwkrjrqdo iru wkh vfdodu surgxfw}

711' 71131 Wkh ruwkrjrqdo surmhfwru rq `U dorqj `UU lv A ¥

Wkh lqwhusuhwdwlrqv ri Ohppdv 6 dqg 7 duh hdvlhu wr jhw li/ rqfh djdlq/ rqh lqwhusuhwv w1 dv d sruwirolr ri frqwlqjhqw Duurz dvvhwv dqg _w1 dv lwv uhdoorfdwlrq1

Iluvw/ wkh gxdolw| ehwzhhq wkh wzr yhfwru vsdfhv/ . dqg ` / ehfrphv pruh dssduhqw1 Wkh vsdfh ri Duurz sulfhv yduldwlrqv lv wkh gxdo vsdfh ri wkh vsdfh ri Duurz sruwirolr uhdoorfdwlrqv1 Vhfrqg/ vwduwlqj zlwk dq Duurz sruwirolr w1/

li Duurz sulfhv fkdqjh/ wkh sruwirolr ydoxh ehfrphv >1n _ 3 >1 w11 Li wkhuh lv d

uhdoorfdwlrq ri wkh Duurz sruwirolr/ wkh uhdgmxvwphqw ydoxh lv >1n _ 3 >1 _w11

Xvlqj wkh ruwkrjrqdolw| frqglwlrq +7147,/ lw lv hdvlo| fkhfnhg wkdw wkh uhdgmxvwphqw ydoxh fdq eh zulwwhq dv = >1n _ 3 >1 _w1 ' q n _7q_w 1>1n q_w1_7>1 c zkhuh q n _7q_w

1>1 lv wkh frvw ri lqwhuwhpsrudo wudqvihuv +vdylqj ru fuhglw,

dqg q_w

1_7>1 lv wkh frvw ri lqvxudqfh1 Dv douhdg| phqwlrqhg/ wkh ghfrpsrvlwlrq

lqyroyhv wzr frpsrqhqwv rqo|/ gxh wr wkh vhohfwhg qrupdol}dwlrq1 Wklug/ li rqh frqvlghuv d uhdoorfdwlrq ri wkh Duurz sruwirolr vxfk wkdw _w1 ' _wU1 +dovr qdphg

lqwhuwhpsrudo uhdoorfdwlrqv,/ Ohppdv 6 dqg 7 lpso| wkh iroorzlqj htxlydohqw vwdwhphqwv +vhh Dsshqgl{ G, =

(32)

l, wkh lqvxudqfh sulfh dvvrfldwhg zlwk dq lqwhuwhpsrudo sruwirolr uhdoorfdwlrq lv htxdo wr }hur E_w

1_7>1 ' f (

ll, wkh lpsolhg prglfdwlrq ri Duurz sulfhv uhgxfhg wr d uhdgmxvwphqw ri wkh glvfrxqw idfwru

_3>1 ' _7q>1

(

lll, wkh lqwhuwhpsrudo sruwirolr uhdoorfdwlrq lpsolhv qr fkdqjh lq wkh iruzdug Duurz sulfhv E_7>1 ' f

Ilqdoo|/ zh fdq lqwhusuhw wkh vsdfh `U _{+uhvshfwlyho|/ wkh gluhfwlrq 7} 11>1,

l, d Yrq Qhxpdqq Prujhqvwhuq +YQP, xwlolw| ixqfwlrq E%0 n BP

rZr E%r

^vhh +519,`/

ll, zklfk lv vwulfwo| frqfdyh/ dqg

lll, dq dgmxvwphqw lq d qhljkerukrrg ri d vwdqgdug fhuwdlqw| srlqw = w1 n '17+

sursruwlrqdo wr e ' Ec c ;

Lq wkh Duurz0Sudww iudphzrun/ 711>1 ' _e0e_>

1 +wkh 70glphqvlrqdo yhfwru zkrvh frpsrqhqwv duh doo htxdo wr 1

e0_>

1,1 Pruhryhu/ wkh surmhfwru A

_{rq `}U _{dorqj `}UU _lv

htxdo wr A _{' 7}

11>1>1 ' eZ zkhuh Z lv wkh 70glphqvlrqdo yhfwru ri suredelolwlhv

lqyroyhg lq wkh YQP xwlolw| ixqfwlrq +vhh Dsshqgl{ G,1 A_{_w} 1 ' P rZr_w1r e phdvxuhv wkh h{shfwhg lqfrph vkrfn zkhuhdv dU A_{o _w} 1 surylghv wkh ghphdqhg

vkrfn/ wkdw lv wkh ulvn| frpsrqhqwv ri wkh vkrfn1 Wkxv/ lq wkh jhqhudo iudphzrun/ 7

11>1 lv wkh qdwxudo h{whqvlrq ri wkh vwdqgdug fhuwdlqw| gluhfwlrq e/ zkhuhdv

A_{_w}

1 ' _wU1 lv d vxuh uhdoorfdwlrq dqg dU Ao _w1 ' _wUU1 wkh ulvn| uhdoorfdwlrq1 ; _{Lq idfw/ wkh Duurz0Sudww iudphzrun dvvxphv pruh vshflf dvvxpswlrqv1 Wkh dgglwlrqdo}

(33)

Proposition 7 : Lq wkh Duurz0Sudww iudphzrun/ wkh vsdfh `U lv vsdqqhg e| wkh yhfwru e zlwk xqlwdu| frpsrqhqwv Dq lqfrph uhdoorfdwlrq _w1lv ghfrpsrvhg

lqwr lwv h{shfwhg ydoxh dqg lwv ghphdqhg ydoxh1 Lq wkh jhqhudo fdvh/ wkh vsdfh `U

zloo eh fdoohg wkh vxuh uhdoorfdwlrq vsdfh1 ¥ Lq wkh olwhudwxuh/ e lv frqvlghuhg dv wkh vwdqgdug fhuwdlqw| gluhfwlrq1 Lw lv qrw lqyduldqw xqghu d fkdqjh ri wkh ydoxh ri wkh qxphudluh lq wkh glhuhqw vwdwhv dqg/ wkhuhiruh/ lw kdv qr lqwhusuhwdwlrq lq uhdo whupv1 Dw wkh rssrvlwh/ lwv qdwxudo h{whqvlrq 711>1 lv orfdoo| lqyduldqw1 Vrph frqixvlrq frxog eh srvvleoh rzlqj wr

wkh sursruwlrqdolw| ehwzhhq 711>1 dqg e lq wkh Duurz0Sudww iudphzrun1

Dq dowhuqdwlyh udwlrqdoh fdq eh jlyhq iru lqwhusuhwlqj `U _{dv wkh vxemhfwlyh}

fhuwdlqw| ru vxuh uhdoorfdwlrq vsdfh1 Ohw xv frqvlghu wkh uhdoorfdwlrq _wU 1 '

A_{_w}

1 ' 711_7q>1 Lw lv hdvlo| fkhfnhg wkdw _wU1 lv wkh rswlpdo _w1 dq

lqglylgxdo zrxog fkrrvh wr plqlpl}h wkh glvxwlolw| suhplxp +orvv, jlyhq e| 74 ' 1

2_w1711_w1c zkhq >1_w1 ' _w_q0 zlwk _w0 ehlqj h{rjhqrxv1< Dv d uhvxow/

wkh uhdoorfdwlrq _wU

1 +uhvshfwlyho|/ wkh gluhfwlrq 711>1, dsshduv dv wkh ehvw ru

suhihuuhg uhdoorfdwlrq +uhvshfwlyho|/ ehvw ru suhihuuhg gluhfwlrq, lq wkh xqfhuwdlq zruog frqvlghuhg143 _{Ilqdoo|/ qrwh wkdw 7}

11>1 kdv douhdg| ehhq lqwhusuhwhg

dv d yhfwru ri lqwhuwhpsrudo vxevwlwxwlrq hhfwv1 Erwk lqwhusuhwdwlrqv/ wkdw lv vxemhfwlyh fhuwdlqw| gluhfwlrq dqg lqwhuwhpsrudo vxevwlwxwlrq hhfwv/ duh wkxv htxlydohqw1 Wkh ghfrpsrvlwlrq ri wkh lqfrph uhdoorfdwlrqv vsdfh dqg wkh ruwkrjrqdolw| frqglwlrq _wU 1 ₇ 11_wUU1 ' f zloo qrz eh xvhg wr rewdlq d ghfrpsrvlwlrq ri wkh

ixqgdphqwdo glvxwlolw| suhplxp1

< _{Wkh fruuhvsrqglqj dx{loldu| sureohp fdq eh zulwwhq dv =}

plq O @ 4 5gw01D11gw1. 0 1gw1.gw0 > zkhuh lv d Odjudqjh pxowlsolhu1

43 _{Wklv lqwhusuhwdwlrq lv dqdorjrxv wr wkdw ri wkh ohdvw ulvn| sruwirolr lq wkh vwdqgdug}

(34)

Proposition 8 : Decomposition of the fundamental disutility premium. Xqghu wkh dvvxpswlrqv ri Sursrvlwlrq 9/ wkh ixqgdphqwdo glvxwlolw| suhplxp 74 ghfrpsrvhv lqwr dq lpsdwlhqfh dqg d ulvn suhplxp1 Pruh irupdoo|/

74 ' 74U_{n 74}UU _{n J}_{n _w n}2 _c zkhuh 74U _'1 2 _wU 1 ₇ 11_wU1 zklfk ghshqgv rq wkh vxuh uhdoorfdwlrq _wU1 lv dq lpsdwlhqfh suhplxp/ dqg 74UU _'1 2 _wUU 1 ₇ 11_wUU1 zklfk ghshqgv rq wkh ulvn| uhdoorfdwlrq _wUU 1 lv d ulvn suhplxp1 ¥

c) Impatience and risk aversions

Wkh ixqgdphqwdo glvxwlolw| suhplxp ohdgv qdwxudoo| wr d phdvxuh ri dyhuvlrq zklfk/ iru reylrxv uhdvrqv/ zloo eh fdoohg wkh ixqgdphqwdo mrlqw wlph0ulvn dyhuvlrq1

Wkh ghfrpsrvlwlrq ri wkh ixqgdphqwdo glvxwlolw| suhplxp ohdgv wr d vlplodu ghfrpsrvlwlrq ri wkh ixqgdphqwdo mrlqw wlph0ulvn dyhuvlrq lqwr dq dyhuvlrq wr lpsdwlhqfh dqg d ulvn dyhuvlrq/ dqg wklv lv zkdw zh lqwhqg wr vkrz1 Ehiruh grlqj vr/ lw vkdoo eh khosixo wr uhfrqvlghu wkh ulvn suhplxp ri Sursrvlwlrq ; dqg lwv irupxodwlrq lq wkh Duurz0Sudww iudphzrun1

Lq rxu prgho/ wkh Duurz0Sudww iudphzrun ohdgv wr = l, 7 Ew '_{7+ ' 7 Ew} 0 ^07+c w1n '17+ ' Ew0 ^07+ n BP_rZr Ew1rn ^1r7+ zklfk lpsolhv b0 ' _YwY7 0 Ew0 ^ 07+ ' Y Y%0 Ew0 ^ 07+ ' 0 c vd|/ Y7b Yw0 ' Y2₇ Yw2 0 Ew0 ^07+ ' Y2 Y%2 0 Ew0 ^07+ ' 0 c vd|/ b1r ' _YwY7 1r Ew1rn ^ 1r7+ ' BZr_Y%Y 1rEw1rn ^ 1r7+ c r ' c c 7 c Y7b Yw ' 5 9 9 9 9 9 9 7 Y2 Y%2 0 Ew0 ^ 07+ f 111 f BZr_Y%Y22 1r Ew1rn ^ 1r7+ 6 : : : : : : 8 (

(35)

ll, doo ghulydwlyhv ri 7 zlwk uhvshfw wr ixwxuh lqfrphv duh wdnhq dw d fhuwdlqw| srlqw = w1n '17+ sursruwlrqdo wr e ' Ec c c vd| E@c c @ Wklv lpsolhv

Y¯b1r Yw1r ' Y2_¯ Yw2 1r E@ ' BZr Y2 Y%2 1r E@ ' BZr _{E@ c r ' c c 7c vr wkdw rqh fdq zulwh} Y7b*Yw b0 ' 5 9 7 00 0 0 0 f f B00₍_@) 0 0 Z 6 : 8 c

zkhuh Z lv wkh gldjrqdo pdwul{ gldj Z

Proposition 9 = Lq wkh Duurz0Sudww iudphzrun/ wkh ulvn suhplxp uhgxfhv wr = 74UU _' 2_w1dU Ao 711 dU A_{o _w} 1 ' ₂B _E@ 0 _w 1dU Zeo Z dU eZo _w1 ' 2B _E@ _E@ _E@ 0 j2 _w1 c +7149, zkhuh j2 _w1 ' S rZrE_w1r .Z_w12 lv wkh yduldqfh ri wkh lqfrph vkrfn _w1 +vhh Dsshqgl{ H, ¥

Wkh uljkw0kdqg vlgh ri +7149, frlqflghv xs wr wkh vfdodu _{E@ *} 0

zlwk wkh Duurz0Sudww ulvn suhplxp1 Wklv lv edvlfdoo| zk| wkh pdwul{ dU Ao 711

dU A_{o zloo dsshdu dv d jhqhudol}hg phdvxuh ri ulvn dyhuvlrq1 Wkh}

vfdodu _{E@ *} 0 lv dovr d glvfrxqw idfwru1 Wkh ruwkrjrqdolw| frqglwlrq _wU 1 ₇ 11_wUU1 ' f dovr |lhogv wkh qh{w sursrvlwlrq1

Proposition 10 : Decomposition of the fundamental joint time-risk aversion. Xqghu wkh dvvxpswlrqv ri Sursrvlwlrq 9/ wkh ixqgdphqwdo mrlqw wlph0ulvn dyhuvlrq ghfrpsrvhv lqwr dq dyhuvlrq wr lpsdwlhqfh dqg d ulvn dyhuvlrq1 Pruh irupdoo|/ 711 ' A 711 A_{n dU Ao}₇ 11 dU A_{o c} zkhuh 711

fdq eh lqwhusuhwhg dv d mrlqw wlph0ulvn dyhuvlrq/ A 711

A _{dv dq}

dyhuvlrq wr lpsdwlhqfh dqg dU A o 711

dU A_{o dv d ulvn dyhuvlrq1 Pruhryhu/}

(36)

Ehvlghv wkhlu lqyduldqfh/ wkh pdlq ihdwxuh ri wkh jhqhudol}hg dyhuvlrq phdvxuhv lv wkhlu pxowlglphqvlrqdolw|1 Zkloh wkh lqyduldqfh vkrxog eh fohdu iurp zkdw kdv ehhq vdlg xs wr qrz/ wkh pxowlglphqvlrqdolw| uhtxluhv vrph h{sodqdwlrq1

Ohw xv frqvlghu wkh ulvn suhplxp 74UU _' 2_w 0 1dU A o 711 dU A_{o _w} 1 Lq wklv uhodwlrq/ dU Ao 711dU Ao _w1 ' q_7>1c vlqfh 711dU Ao _w1 ' q_7>1 dqg 711dU Ao ' dU Ao 711dU Ao e| wkh ruwkrjrqdolw| frqglwlrq Zh ghgxfh 74UU _' 2_w 0 1q_7>1 ' 2_w 0 1dU A o 711 dU A_{o _w} 1 c 74UU _' 2q [ r _w1r_7>1r ' ₂ [ r [ j kUU rj _w1r_w1j c +714:, zkhuh kUU rj ' dU A o 711 dU A_{o c r/ j ' c c 7 Zh vkdoo frph edfn wr} wkh lqwhusuhwdwlrq ri wkh frh!flhqwv kUU rj lq d prphqw1 Ohw xv vhw 74UU r ' ₂q_w1r_7>1r c r ' c c 7 c +714;, zkhuh 74UU r lv wkh ulvn suhplxp vshflf wr vwdwh r/ phdqlqj wkdw lw lv dvvrfldwhg zlwk wkh vshflf vkrfn _w1rc d fkdqjh lq wkh txdqwlw| ri wkh r0hohphqwdu| Duurz dvvhw1 Lq idfw/ zkloh q_w0

1_7>1 lv wkh frvw ri lqvxudqfh lq wkh uhdgmxvwphqw ydoxh

ri d uhdoorfdwlrq ri wkh Duurz sruwirolr ^vhh vxevhfwlrq 715e,`/ q_w1r_7>1r fdq eh

vhhq dv wkh frvw ri lqvxudqfh lq wkh uhdgmxvwphqw ydoxh ri d vshflf frpsrqhqw +dvvhw, ri wklv sruwirolr1 Pruhryhu/ li zh zulwh q_7>_1r ' 2¯4UU_r

_w1rc _7>1r lv wkh udwh ri wklv vshflf suhplxp1 Ohw xv qrz frph wr wkh lqwhusuhwdwlrq ri wkh frh!flhqwv kUU rj Zh uvw qrwh iurp uhodwlrq +714:, wkdw q_7>1r ' [ j kUU rj _w1j c r ' c c 7 +714<,

Qrz/ ohw xv vxssrvh _w1j ' f iru j 9' r dqg vhw 74UUrr ' 12kUUrrE_w1r

2_{Wklv |lhogv} 274UU rr E_w1r2 ' k UU rr c

(37)

zkhuhkUU rr

lv d gluhfw hohphqwdu| frh!flhqw ri ulvn dyhuvlrq/ zklfk/ dsduw iurp ehlqj hohphqwdu|/ lv yhu| vlplodu wr zkdw zh jhw lq wkh xvxdo fdvh +wkh jhqhudo frh!flhqw ri devroxwh ulvn dyhuvlrq lv wzlfh wkh ulvn suhplxp shu xqlw ri yduldqfh,1 E| +714<, dqg +714;, wkh vshflf ulvn suhplxp fdq dovr eh zulwwhq

74UU r ' ₂_w1r [ j kUU rj _w1j ' ₂ [ j 74UU rj c +7153,

zkrvh vxppdwlrq ryhu rc r ' c c 7c jlyhv wkh ulvn suhplxp 74UU

Zkhuhdv kUU

rr lv d gluhfw hohphqwdu| frh!flhqw/ kUUrjEj 9' r lv d furvv

hohphqwdu| frh!flhqw1 Wkhvh furvv hohphqwdu| dyhuvlrqv uhsuhvhqw dyhuvlrqv wr furvv hohphqwdu| ulvnv wkdw vxp xs zlwk wkh gluhfw hohphqwdu| ulvn wr jlyh wkh vshflf ulvn1 Dv fdq eh vhhq iurp +7153,/ wkh vdph udwlrqdoh dssolhv wr hohphqwdu| dqg vshflf suhplxpv1

Lw lv qrwhzruwk| wkdw/ lq wkh Duurz0Sudww fdvh/ wkh jhqhudo +djjuhjdwhg, dyhuvlrq phdvxuh lv/ lq idfw/ wkh vxppdwlrq ri vshflf dyhuvlrq phdvxuhv1 Lqghhg/ lq wklv fdvh/ rqh kdv = dU A o 711 dU A_{o ' dU Ze}_o_B ZE@ 0 dU eZ_{o c}

dqg wkh vwdqgdug Duurz0Sudww frh!flhqw lv wkh wudfh ri wkh pdwul{ ri vshflf ulvn dyhuvlrqv kB Z00_(@) 0_(@) 0_(@) 0 0 l

Ohw xv qrz wxuq wr wkh lpsdwlhqfh dyhuvlrq dqg frqvlghu wkh lpsdwlhqfh suhplxp 74U _' 2_w 0 1A 711 A_{_w} 1 Lq wklv uhodwlrq/ A 711A_w1 ' _7q>1 vlqfh 711A_w1 ' _7q>1 dqg 711A 0 ' A 711A e| ghqlwlrq ri wkh ruwkrjrqdo surmhfwru A0 Zh ghgxfh 74U _' 2_w 0 1_7q>1 ' 2_w 0 1A 711 A_{_w} 1 c 74U _' 2 [ r _w1r_7q>1r ' ₂ [ r [ j kU rj _w1r_w1j c +7154, zkhuh kU rj ' A 711 A0 c rc j ' c c 7 Wkh lqwhusuhwdwlrq ri wkhvh frh!flhqwv fdq eh rewdlqhg lq d yhu| vlplodu zd| wr wkdw xvhg wr lqwhusuhwkUU

rj

Zkloh kU

rr lv d gluhfw hohphqwdu| frh!flhqw ri lpsdwlhqfh dyhuvlrq/ kUrjEj 9' r

(38)

4.3. The residual risk aversion and the liquidity premium

Wkh ixqgdphqwdo glvxwlolw| suhplxp 74 ri Sursrvlwlrq 9 dqg wkh uhvlgxdo glvxwlolw| suhplxp 4 ri Fruroodu| 5 duh wlhg wrjhwkhu lq wklv vxevhfwlrq1 Wkhlu glhuhqfh 744 zloo dsshdu dv d oltxlglw| suhplxp1 Dffruglqjo|/ wkh ixqgdphqwdo Dqwrqhool pdwul{ 711 ri Sursrvlwlrq 9 dqg wkh uhvlgxdo Dqwrqhool pdwul{ 11 ri Fruroodu| 5

+uhvshfwlyho|/ wkh ixqgdphqwdo dqg wkh uhvlgxdo mrlqw wlph0ulvn dyhuvlrq phdvxuhv, duh dovr olqnhg wrjhwkhu lq wklv vxevhfwlrq1 Wkhlu glhuhqfh 711 11 zloo dsshdu

dv d phdvxuh ri dyhuvlrq wr looltxlglw|1 Wkh pdlq wrro wr vkrz wkhvh olqnv lv wkh Guë}h0Prgljoldql ghfrpsrvlwlrq dqg lwv lqyduldqw jhqhudol}dwlrq1 Wkhuhiruh/ zh vkdoo uvw uhfdoo wkh Guë}h0Prgljoldql ghfrpsrvlwlrq dqg jlyh lwv uhirupxodwlrq lq rxu vhwwlqj1 Wkh odwwhu zloo eh xvhg iru wkh frpsxwdwlrq ri wkh oltxlglw| suhplxp 74 4 D jhqhudol}dwlrq ri wkh Guë}h0Prgljoldql ghfrpsrvlwlrq wrjhwkhu zlwk wkh dvvxpswlrq wkdw wkhlu h{lvwv dq wudgdeoh ulvn0iuhh dvvhw zloo wkhq eh xvhg wr ghfrpsrvh wkh ixqgdphqwdo mrlqw wlph0ulvn dyhuvlrq lqwr dq dyhuvlrq wr lpsdwlhqfh/ d uhvlgxdo dyhuvlrq dqg d qdqfldo dyhuvlrq1

a) Liquidity premium and aversion to illiquidity

Wkh uhvxowv ri vxevhfwlrqv 715 dqg 716 duh qdwxudoo| olqnhg wrjhwkhu e| wkh lqgluhfw xwlolw| ixqfwlrq 7 +vhh Vhfwlrq 6/ Uhpdun 6,1 Diwhu rswlpl}lqj 7 Ew '₊

zlwk uhvshfw wr +c zh kdyh =

Ew ' 7 Ew '_{+ Ew} _+7155,

E| glhuhqwldwlqj wklv uhodwlrq zlwk uhvshfw wr w/ zh kdy h = b ' 7b _YwY+'7b ' 7b +vlqfh '7b ' f, / zkhuh b Ew ' Y

YwEw dqg 7b Ew ' YwY¯ Ew '+ Ew duh wkh ghvludelolwlhv

ri lqfrphv zkhq wkh lqfrph yduldwlrq lv dqdo|}hg zlwk dqg zlwkrxw qdqfldo dgmxvwphqwv uhvshfwlyho|1 D vhfrqg glhuhqwldwlrq jlyhv =

Yb Yw ' Y7b Yw U ' Y+ Yw 1 +7156, Vlqfh 'Yb Yw0 ' fc zh jhw e| suhpxowlso|lqj +7156, e| Y+ 0 Yw' = Y+ Yw' Y7b Yw ' Y+ Yw '_YwY7b' Y+ Yw ' Y7b Yw' _+ Yw / +7157,

(39)

gxh wr wkh v|pphwu| ri wkh vhfrqg whup1 Vxevwlwxwlqj lqwr wkh uhodwlrq +7156, dqg uhduudqjlqj whupv jlyh wkh Guë}h0Prgljoldql ghfrpsrvlwlrq1

Lemma 5 : Drèze-Modigliani decomposition. Wkh yduldwlrqv ri wkh ghvludelolwlhv ri lqfrphv zkhq qdqfldo dgmxvwphqwv duh wdnhq lqwr dffrxqw

Yb*Yw0

b0 duh wkh vxp ri l, wkh yduldwlrqv ri wkh ghvludelolwlhv ri lqfrphv zkhq qdqfldo dgmxvwphqwv duh qrw wdnhq lqwr dffrxqw Y¯b*Yw0

b0 dqg ll, d fruuhfwlyh whup Y+0 Yw k 'Y¯b*Yw0 b0 ' l Y+ Yw0 = Yb*Yw_b 0 ' Y7b*Yw b0 n Y+ Yw 'Y7b*Yw_b 0 ' Y+ Yw c

zkhuh wkh fruuhfwlyh whup lv lqyduldqw xqghu d prqrwrqlf wudqvirupdwlrq ri wkh xwlolw| ixqfwlrq1

Wkh ghfrpsrvlwlrq ri Ohppd 8 lv yhu| vlplodu wr wkh irupxod ghulyhg e| Guë}h dqg Prgljoldql1 Wkh odwwhu/ krzhyhu/ zdv rewdlqhg lq d udwkhu glhuhqw frqwh{w1 Wkhlu irupxod lv qrw pxowlglphqvlrqdo dqg uhihuv wr d vshflf vwdwh ri wkh zruog1 Lw lv xvhg iru frpsdulqj d vlwxdwlrq zkhuh wkhuh h{lvwv d qrplqdo vxuh dvvhw +ulvn0iuhh dvvhw, zlwk d vlwxdwlrq zkhuh wkhuh h{lvw wudgdeoh dvvhwv1 Ilqdoo|/ wkhlu fruuhfwlyh whup phdvxuhv wzlfh wkh h{shfwhg ydoxh ri shuihfw lqirupdwlrq shu xqlw ri yduldqfh +111, iru lqqlwhvlpdo ulvnv ^vhh Guë}h dqg Prgljoldql +4<:5,/ s1 647`1 Dv phqwlrqhg deryh/ Ohppd 8 zloo qrz eh xvhg wr olqn wrjhwkhu wkh ixqgdphqwdo dqg wkh uhvlgxdo glvxwlolw| suhplxpv1

E| suhpxowlso|lqj dqg srvwpxowlso|lqj wkh Guë}h0Prgljoldql htxdwlrq e| _w

dqg _w/ dqg e| xvlqj Fruroodu| 5/ Sursrvlwlrq 9 dqg wkh idfw wkdw b0 ' 7b0c zh jhw

wkh iroorzlqj sursrvlwlrq1 Proposition 11 : Li wkh lqfrph vkrfn lv frpshqvdwhg E>_w ' f c l, wkh oltxlglw| suhplxp lv = 74 4 ' 2 _2₇ b0 _2 b0 ' {7 b0 { b0 ' 2_w12g22312_w1 f c zkhuh 2 ' _YwY+0 1 Y+ Yw0> 1 lv d pdwul{ ri frpshqvdwhg lqfrph hhfwv dqg g22 ' k 'Y¯b*Yw_b₀ 0'l31 _' _' 1711'1 ₃₁

lv d v|pphwulf qhjdwlyh ghqlwh pdwul{ zlwk udqn zklfk fkdudfwhul}hv vxevwlwxwlrq dqg frpsohphqwdulw| dprqj dvvhwv>