Equilibrium Leasing Contracts under Double-sided ... - Boston College

19 downloads 6038 Views 367KB Size Report
Chestnut Hill, MA 02467, Tel: (617) 552-3980, Fax: (617) 552 0431, email: ... to buy at lease termination, and service leases) also emerge as equilibrium ...... can opt for a lease with a metering provision, under which he will be charged more if ... automobilesq see, e.g., the %omplete %ar %ost fuide, which lists the lease cost.
Equilibrium Leasing Contracts under Double-sided Asymmetric Information

Thomas Chemmanur* and

An Yan**

Current Version: March 2000

* Finance Department, Fulton Hall 440, Carroll School of Management, Boston College, Chestnut Hill, MA 02467, Tel: (617) 552-3980, Fax: (617) 552 0431, email: [email protected]. ** Department of Economics, Boston College, Chestnut Hill, MA 02467, Tel: (617) 552-

8703, Fax (617) 552 2308, email: [email protected].

Thomas Chemmanur acknowledges support from Boston College Faculty Research Summer Grant. For helpful comments or discussions, we thank Ingela Alger, Richard Arnott, Paolo Fulghieri, Fabio Schiantarelli, and seminar participants at Boston College and INSEAD. We alone are responsible for any errors or omissions.

,^*Mh4 wi@t?} L?|h@U|t ?_ih #LM*it_i_ t)44i|hU W?uLh4@|L? Mt|h@U| `i _ii*LT @ ?L?|@ h@|L?@*i uLh *i@t?} ? @ _LM*it_i_ @t)44i|hU ?uLh4@|L? ti||?}c @?_ @?@*)3i L @hLt UL?|h@U|@* ThLtL?t ? *i@t?} UL?|h@U|t @hti ? i^*Mh4 W? Lh 4L_i*c @ 4@?u@U|hih Lu U@T|@* }LL_t @t Th@|i ?uLh4@|L? @ML| |ih ^@*|)( i?|hiThi?iht Etiht Lu |iti U@T|@* }LL_t UL4i |L *i@h? |t ^@*|) L?*) Lih @ TihL_ Lu |4i ,@U ?| Lu |i U@T|@* }LL_t hi^hit @ Uih|@? *ii* Lu 4@?|i?@?Ui ? i@U TihL_ ,?|hiThi?iht _gih ? |ih ULt| Lu ThL_?} |t 4@?|i?@?Ui( |t 4@?|i?@?Ui ULt| t ?uLh4@|L? Th@|i |L i@U i?|hiThi?ih wi@t?} i4ih}it @t @? i^*Mh4 tL*|L? |L |t _LM*it_i_ @_ihti ti*iU|L? ThLM*i4 V@hLt UL?|h@U|@* ThLtL?t ? *i@t?} UL?|h@U|t Ei}c tLh||ih4 ihtt *L?}|ih4 *i@tit | ?L?U@?Ui**@|L? ThLtL?tc LT|L? |L M) @| *i@ti |ih4?@|L?c @?_ tihUi *i@tit @*tL i4ih}i @t i^*Mh4 tL*|L?t ?_ih @*|ih?@|i ti||?}t wi@tit | 4i|ih?} ThLtL?t i4ih}i ? i^*Mh4 i?c ? @__|L? |L 4@?|i?@?Ui ULt|c i?|hiThi?iht _gih ? L|ih _4i?tL?t tU @t |ih ?|i?t|) Lu t?} |i U@T|@* }LL_c @?_ |ih _i}hii Lu ht! @ihtL? h 4L_i* @t 4T*U@|L?t uLh |i *i@ti ihtt ti** _iUtL?c |i t|@|L?t ?_ih U @hLt *i@t?} UL?|h@U| ThLtL?t @hi @TThLTh@|ic @?_ uLh |i hi*@|i 4@}?|_it Lu t@*it ThUit @?_ |i *i@t?} ULt|t uLh *i@tit | _gihi?| UL?|h@U|@* ThLtL?t

,^*Mh4 wi@t?} L?|h@U|t N?_ih #LM*it_i_ t)44i|hU W?uLh4@|L?



W?|hL_U|L? Ohdvlqj lv dq lpsruwdqw vrxufh ri h{whuqdo qdqflqj iru X1V1 frusrudwlrqv1 Lw kdv ehhq hvwl0

pdwhg wkdw d wklug ri wkh htxlsphqw xvhg e| X1V1 frusrudwlrqv lv ohdvhg1 Krzhyhu/ wkh prwlydwlrqv iru ohdvlqj duh vwloo qrw frpsohwho| xqghuvwrrg1 Zkloh wkh qdqfh olwhudwxuh kdv dqdo|}hg frusrudwh ohdvlqj srolf| h{whqvlyho|/ pxfk ri wkh glvfxvvlrq kdv ehhq frqqhg wr wd{0uhodwhg lqfhqwlyhv wr ohdvh ru ex| +vhh/ h1j1/ Ploohu dqg Xswrq +4 vhuylfh ohdvhv/ zkhuh wkh pdqxidfwxuhu djuhhv wr pdlqwdlq wkh htxlsphqw ohdvhg> dqg ohdvhv lqyroylqj _phwhulqj/% zkhuh wkh ohdvh sd|phqw lv d ixqfwlrq ri dq revhuyhg phdvxuh ri wkh lqwhqvlw| ri xvdjh ri wkh dvvhw1 Zh ghyhors rxu prgho lq d vhwwlqj ri dv|pphwulf lqirupdwlrq zkhuh wkh ohvvru +pdqxidfwxuhu ri fdslwdo htxlsphqw, kdv sulydwh lqirupdwlrq ri wkh dvvhw kh ohdvhv wr wkh ohvvhh +hqwuhsuhqhxu,1 Wkh ohvvhh/ krzhyhu/ fdq ohduq derxw wkh txdolw| ri klv htxlsphqw ryhu wlph dv kh xvhv wkh dvvhw1 Wkxv/ ohdvlqj hphujhv lq rxu prgho dv d zd| ri plwljdwlqj wkh dgyhuvh vhohfwlrq sureohp1 Krzhyhu/ dqrwkhu sureohp zlwk ohdvlqj lq rxu vhwwlqj lv wkdw/ vlqfh wkh ohvvhh grhv qrw kdyh rzqhuvkls uljkwv wr wkh uhvlgxdo ydoxh ri wkh dvvhw/ kh kdv dq lqfhqwlyh wr xvh wkh dvvhw lq vxfk d zd| wkdw vkliwv ydoxh iurp wkh ohvvru wr wkh ohvvhh1 Rqh srvvlelolw| lv wkdw ydulrxv ohvvhhv kdyh glhuhqw frvwv ri pdlqwdlqlqj wkh fdslwdo jrrg1 Lq wklv fdvh/ wkh kljkhu pdlqwhqdqfh frvw xvhuv kdyh dq lqfhqwlyh wr ghyrwh ohvv fduh wr wkh pdlqwhqdqfh wkdq wkh orzhu pdlqwhqdqfh frvw xvhuv1 Glhuhqw frqwudfwxdo surylvlrqv +h1j1/ ohdvhv zlwk qrq0fdqfhoodwlrq surylvlrqv/ rswlrq wr ex| dw ohdvh whuplqdwlrq/ ohdvhv zlwk vhuylfh surylvlrqv, lq ohdvlqj frqwudfwv hphujh dv vroxwlrqv wr wklv grxeoh0vlghg lqfhqwlyh sureohp lq wkh fdslwdo jrrgv pdunhw1 D vhfrqg srvvlelolw| lv wkdw glhuhqw ohvvhhv kdyh glhuhqw lqwhqvlwlhv ri xvdjh ri wkh fdslwdo htxlsphqw1 Phwhulqj frqwudfwv hphujh wr vroyh wklv grxeoh0vlghg dgyhuvh0vhohfwlrq sureohp lq wkh fdslwdo jrrgv pdunhw lq wklv fdvh1 D wklug srvvlelolw| lv wkdw wkh ohvvhh klpvhoi pd| kdyh xqfhuwdlqw| derxw wkh lqwhqvlw| ri klv xvdjh/ exw glhuhqw ohvvhhv kdyh glhuhqw ohyhov ri ulvn0dyhuvlrq1 Zh vkrz wkdw phwhulqj djdlq hphujhv dv d vroxwlrq wr wkh grxeoh0 vlghg dgyhuvh0vhohfwlrq sureohp lq wklv vhwwlqj1 Dq lqwhuhvwlqj dvshfw ri rxu prgho lv wkdw ohdvlqj pd| fr0h{lvw zlwk sxuh vdohv frqwudfw lq wkh pdunhw iru vrph fdslwdo jrrgv> lw pd| eh wkh vroh nlqg 5

ri qdqflqj frqwudfw lq rwkhu fdvhv> dqg vdohv frqwudfwv pd| hphujh dv wkh rqo| w|sh ri frqwudfwv rhuhg lq |hw rwkhu vhwwlqjv1 Rxu uhvhdufk lv uhodwhg wr vhyhudo vwudqgv lq wkh qdqfh dqg hfrqrplfv olwhudwxuh1 Dv phqwlrqhg ehiruh/ wkhuh lv d odujh qdqfh olwhudwxuh rq ohdvlqj irfxvhg rq wkh wd{0uhodwhg lqfhqwlyhv wr ohdvh ru ex|1 Krzhyhu/ wd{ lvvxhv dorqh fdqqrw h{sodlq wkh h{lvwhqfh ri ohdvlqj lq pdq| pdunhwv= zkloh wkh wd{ ehqhwv ri ohdvlqj zhuh orzhuhg lq wkh wd{ uhirup dfw ri 4 4.n

+5;,

in{ . !+on{ . on5 {  5|, . n{ . n5 {  5|= 4!

+5 dqg d ohdvlqj frqwudfw zlwk xqolplwhg xvdjh dqg qr phwhulqj1 Wkh w|sh E pdqxidfwxuhu= Kh rhuv rqo| d vdohv frqwudfw1 Wkh w|sh L hqwuhsuhqhxu= Li wkh pdqxidfwxuhu rhuv ohdvlqj frqwudfwv/ kh lqihuv wkdw wkh pdqxidfwxuhu lv ri w|sh J zlwk suredelolw| 41 Kh dffhswv wkh ohdvlqj frqwudfw zlwk olplwhg xvdjh dqg phwhulqj/ xvhv wkh htxlsphqw zlwklq wkh wkuhvkrog ohyho x/ dqg ex|v wkh htxlsphqw dw w@41 Li/ krzhyhu/ wkh pdqxidfwxuhu rhuv rqo| d vdohv frqwudfw/ kh lqihuv wkdw wkh pdqxidfwxuhu lv ri w|sh E zlwk suredelolw| 4/ dqg kh dffhswv wkh vdohv frqwudfw1 Wkh w|sh LL hqwuhsuhqhxu= Li wkh pdqxidfwxuhu rhuv ohdvlqj frqwudfwv/ kh lqihuv wkdw wkh pdqxidfwxuhu lv ri w|sh J zlwk suredelolw| 41 Kh dffhswv wkh ohdvlqj frqwudfw zlwk xqolplwhg xvdjh dqg qr phwhulqj/ dqg ex|v wkh htxlsphqw dw w@41 Li/ krzhyhu/ wkh pdqxidfwxuhu rhuv rqo| d vdohv frqwudfw/ kh lqihuv wkdw wkh pdqxidfwxuhu lv ri w|sh E zlwk suredelolw| 4/ dqg kh dffhswv wkh vdohv frqwudfw1 Lq wkh deryh vhwwlqj/ phwhulqj doorzv wkh w|sh J pdqxidfwxuhu wr lqfuhdvh klv h{shfwhg surwv e| vhjphqwlqj wkh pdunhw ehwzhhq wkh kljk0lqwhqvlw| dqg orz0lqwhqvlw| hqwuhsuhqhxu1 Xqghu wkh frqwudfw zlwk phwhulqj/ wkh pdqxidfwxuhu vhwv d kljk shqdow| iru kljk lqwhqvlw| xvdjh/ vr wkdw wkh kljk lqwhqvlw| hqwuhsuhqhxu grhv qrw dffhsw wklv frqwudfw1 Dw wkh vdph wlph/ kh vhwv d frp0 sdudwlyho| kljk lqlwldo ohdvlqj sulfh lq wkh ohdvlqj frqwudfw zlwkrxw phwhulqj +dqg zlwk xqolplwhg xvdjh,/ zklfk hqvxuhv wkdw wkh orz0lqwhqvlw| hqwuhsuhqhxu lv ehwwhu r fkrrvlqj wkh phwhulqj frq0 wudfw154 Wkh kljk shqdow| iru wkh kljk lqwhqvlw| xvdjh lq wkh phwhulqj frqwudfw lv d fkhds wkuhdw ehfdxvh wkh shqdow| zloo qhyhu eh lqfxuuhg lq htxloleulxp1 Wkxv/ wkh rqo| dgglwlrqdo frvw lqfxuuhg lq wkh htxloleulxp zlwk phwhulqj lv wkh phwhulqj frvw1 Zkhq wklv frvw lv orz hqrxjk/ wkh w|sh J pdqxidfwxuhu kdv wkh lqfhqwlyh wr rhu wkh phqx ri frqwudfwv fkdudfwhul}hg lq wklv sursrvlwlrq1 Qxphulfdo H{dpsoh 8= Vhw n @ 3=7> { @ 433> | @ 43> i @ 3=6>  @ 4=5> ! @ =68> o @ 4=7/ dqg p @ 53/ zlwk orz0lqwhqvlw| xvdjh htxdolqj wr 53 xqlwv shu shulrg dqg kljk0lqwhqvlw| xvdjh 2 Ai ?|@* *i@t?} ThUit ? |i *i@t?} UL?|h@U|t @hi UL?t|h@?i_ @t i** M) |i ?ii_ |L Thii?| 44U!?} M) |i |)Ti  4@?u@U|hihc t4*@h |L |i W UL?t|h@?| ? ThLTLt|L? 

55

htxdolqj wr 73 xqlwv shu shulrg= Lq wklv fdvh/ wkh w|sh J pdqxidfwxuhu rhuv d vkruw0whup ohdvlqj frqwudfw zlwk olplwhg xvdjh dqg phwhulqj iP4 @ 6=:/ U4 @ 69/ Y  66=8> dqg x @63 xqlwv shu shulrgj zklfk zloo eh dffhswhg e| wkh w|sh L +orz0lqwhqvlw|, hqwuhsuhqhxu/ dqg d vkruw0whup ohdvlqj frqwudfw zlwk xqolplwhg xvdjh iP5 @ 47=;> U5 @ 8;=7j zklfk zloo eh dffhswhg e| wkh w|sh LL +kljk0lqwhqvlw|, hqwuhsuhqhxu1 Wkhvh sulfhv/ wrjhwkhu zlwk wkh ryhu0xvdjh shqdow| +vxssruwhg e| dvfhuwdlqlqj wkh ohyho ri xvdjh e| lqfxuulqj wkh phwhulqj frvw p, doorz klp wr dfklhyh wkh pdunhw vhjphqwdwlrq rq wkh hqwuhsuhqhxu*v vlgh/ dv zhoo dv wr glvwlqjxlvk klpvhoi iurp wkh w|sh E pdqxidfwxuhu1 Lq htxloleulxp/ wkh w|sh E pdqxidfwxuhu rhuv rqo| d vdohv frqwudfw zlwk V @ 551 Kh grhv qrw plplf wkh w|sh J pdqxidfwxuhu ehfdxvh kh fdqqrw lqfuhdvh klv h{shfwhg fdvk rzv iurp grlqj vr1 Wkh w|sh J pdqxidfwxuhu suhihuv wr rhulqj ohdvlqj frqwudfwv vlqfh klv h{shfwhg surw iurp ohdvlqj frqwudfwv +6;=7, lv kljkhu wkdq wkdw iurp wkh vdohv frqwudfw +55,1 Wkh w|sh LL hqwuhsuhqhxu rqo| dffhswv wkh frqwudfw zlwk xqolplwhg xvdjh ehfdxvh/ zkhq Y  66=8/ kh lv ehwwhu r e| grlqj vr> wkh w|sh L hqwuhsuhqhxu dffhswv wkh phwhulqj frqwudfw zlwk olplwhg xvdjh ehfdxvh kh fdq hduq pruh +449=6, e| grlqj vr wkdq e| wdnlqj wkh frqwudfw zlwk xqolplwhg xvdjh +438=5,1

D2

,^*Mh4 | ,?|hiThi?ih@* +t!ihtL? @?_ N?Uih|@?|) @ML| Nt@}i W?|i?t|) Lq wklv vhfwlrq/ zh voljkwo| prgli| wkh dvvxpswlrqv lq vhfwlrq 814 e| dvvxplqj wkdw wkh hq0

wuhsuhqhxu lv xqfhuwdlq derxw klv rzq xvdjh lqwhqvlw| lq dq| jlyhq shulrg= wkh hqwuhsuhqhxu dvvhvvhv d suredelolw|  ri klv ehlqj d orz0lqwhqvlw| xvhu lq hdfk shulrg dqg suredelolw| ri 4   ehlqj d kljk0lqwhqvlw| xvhu1 Wkh hqwuhsuhqhxu frphv wr uhdol}h klv xvdjh lqwhqvlw| lq dq| shulrg zlwk suredelolw| 4 rqo| dw wkh hqg ri wkdw shulrg1 Ixuwkhu/ zh dvvxph wkdw hqwuhsuhqhxuv duh ulvn0dyhuvh zlwk wkh ghjuhhv ri ulvn0dyhuvlrq ydu|lqj dfurvv hqwuhsuhqhxuv1 Zh zloo uhihu wr wkh pruh ulvn0dyhuvh hqwuhsuhqhxu dv wkh w|sh L hqwuhsuhqhxu lq wklv vhfwlrq/ dqg wkh ohvv ulvn0dyhuvh hqwuhsuhqhxu dv wkh w|sh LL1 Iru vlpsolflw|/ zh dvvxph wkdw wkh w|sh LL hqwuhsuhqhxu lv ulvn0qhxwudo +vr wkdw klv xwlolw| ixqfwlrq lv olqhdu,/ dqg dvvxph d frqfdyh yrq Qhxpdqq0Prujhqvwhuq h{shfwhg xwlolw| ixqf0

56

wlrq X +=, iru wkh w|sh L +ulvn0dyhuvh, hqwuhsuhqhxu155

Zh dvvxph wkdw/ zkloh hdfk hqwuhsuhqhxu

nqrzv klv rzq w|sh/ wkh pdqxidfwxuhu fdqqrw glvwlqjxlvk ehwzhhq wkh wzr w|shv ri hqwuhsuhqhxuv ehiruh revhuylqj wkh hqwuhsuhqhxuv* frqwudfw fkrlfhv> wkh pdqxidfwxuhu rqo| revhuyhv wkh sulru suredelolw| ! ri dq| jlyhq hqwuhsuhqhxu ehlqj ri w|sh L1 Zh dvvxph wkh vdph phqx ri frqwudfwv dv lq wkh suhylrxv vhfwlrq= d vdohv frqwudfw iVj/ d ohdvlqj frqwudfw zlwk olplwhg xvdjh dqg phwhulqj iP4 > U4 > Y> xj/ dqg d ohdvlqj frqwudfw zlwk xqolplwhg xvdjh dqg qr phwhulqj iP5 > U5 j156 Wkh ghqlwlrqv ri wkhvh wkuhh frqwudfwv/ dv zhoo dv doo rwkhu dvvxpswlrqv/ uhpdlq wkh vdph dv lq vhfwlrq 8141 Zh dovr ghqh= p5 @

4 ^+4  !,U5 . !U4  i n{` !

+63,

Sursrvlwlrq ; +Phwhulqj xqghu Xqfhuwdlqw| derxw Xvdjh Lqwhqvlw|, Zkhq i  i6 / p  p5 / dqg   45 / wkh vhsdudwlqj htxloleulxp lq wkh fdslwdo jrrgv pdunhw lqyroyhv wkh iroorzlqj= Wkh w|sh J pdqxidfwxuhu= Kh rhuv wzr glhuhqw ohdvlqj frqwudfwv= d ohdvlqj frqwudfw zlwk phwhulqj dqg olplwhg xvdjh> dqg d ohdvlqj frqwudfw zlwk xqolplwhg xvdjh dqg qr phwhulqj1 Wkh w|sh E pdqxidfwxuhu= Kh rhuv rqo| d vdohv frqwudfw1 Wkh w|sh L hqwuhsuhqhxu= Li wkh pdqxidfwxuhu rhuv ohdvlqj frqwudfwv/ kh lqihuv wkdw wkh pdqxidfwxuhu lv ri w|sh J zlwk suredelolw| 41 Kh sd|v dq lqlwldo ohdvlqj ihh P4 xsiurqw/ zlwk d shqdow| Y li klv lqwhqvlw| ri xvdjh lq wkh uvw shulrg +dv phdvxuhg wkurxjk phwhulqj, wxuqv rxw wr eh kljk> kh sd|v qr shqdow| li klv uvw0shulrg lqwhqvlw| ri xvdjh lv orz1 Dw w@4/ kh uhqhzv wkh frqwudfw e| sd|lqj wkh uhqhzdo sulfh U4 1 Li/ krzhyhu/ wkh pdqxidfwxuhu rhuv rqo| d vdohv frqwudfw/ kh lqihuv wkdw wkh pdqxidfwxuhu lv ri w|sh E zlwk suredelolw| 4/ dqg kh dffhswv wkh vdohv frqwudfw1 Wkh w|sh LL hqwuhsuhqhxu= Li wkh pdqxidfwxuhu rhuv ohdvlqj frqwudfwv/ kh lqihuv wkdw wkh pdqxidfwxuhu lv ri w|sh J zlwk suredelolw| 41 Kh dffhswv wkh ohdvlqj frqwudfw zlwk xqolplwhg xvdjh dqg qr phwhulqj e| sd|lqj d oxps vxp lqlwldo ohdvlqj sd|phqw P5 xsiurqw dw w@3/ dqg uhqhzv wkh frqwudfw e| sd|lqj U5 dw w@41 Li/ krzhyhu/ wkh pdqxidfwxuhu rhuv rqo| d vdohv frqwudfw/ kh lqihuv wkdw wkh pdqxidfwxuhu lv ri w|sh E zlwk suredelolw| 4/ dqg kh dffhswv wkh vdohv frqwudfw1 Wkh lqfhqwlyh phfkdqlvp frqvwuxfwhg lq wkh uvw ohdvlqj shulrg xwlol}hv wkh glhuhqw ghjuhhv ri ulvn dyhuvlrq ri wkh hqwuhsuhqhxuv1 Zkhq wkh hqwuhsuhqhxu lv ulvn0dyhuvh/ kh ydoxhv qrw rqo| wkh dprxqw ri h{shfwhg fdvk rzv exw dovr wkh orz yrodwlolw| ri fdvk rzv1 Wkxv wkh pdqxidfwxuhu fdq 22 *i@h*)c |t @?@*)tt U@? Mi }i?ih@*3i_c @| |i ULt| Lu tL4i @__|L?@* UL4T|@|L?@* UL4T*i |)c |L |i U@ti ihi ML| |)Tit Lu i?|hiThi?iht @hi ht!@ihtic | |i |)Ti W @?} @ }hi@|ih _i}hii Lu ht!@ihtL? |@? |i |)Ti WW i?|hiThi?ih 2 6Lh t4T*U|)c i @**L L?*) uLh UL?|h@U|t | 4i|ih?} Lu €ht|TihL_ t@}i L?*) OLiihc |i ?||L? Mi?_ Lh @?@*)tt i |i?_t |L |i U@ti ihi 4i|ih?} t _L?i iih) TihL_c |t uL**L?} |i ht!@ihti i? |hiThi?ih |L i_}i t U@t L L*@|*|) ? |i tMti^i?| TihL_t @t i**

57

uhtxluh d kljkhu sulfh iurp wkh w|sh L +ulvn0dyhuvh, hqwuhsuhqhxu dqg frpshqvdwh klp iru wklv zlwk d sd|phqw vfkhgxoh wkdw uhgxfhv wkh yrodwlolw| ri klv ixwxuh surwv1 Vshflfdoo|/ wkh pdqxidfwxuhu fdq xvh wkh phwhulqj frqwudfw wr fkdujh wkh ulvn0dyhuvh hqwuhsuhqhxu glhuhqw sulfhv fruuhvsrqglqj wr wkh glhuhqw fdvk rzv xqghu wkh glhuhqw xvdjh lqwhqvlwlhv1 Wklv sulfh vfkhgxoh zlwk d kljk phdq ydoxh suhyhqwv wkh w|sh LL +ulvn qhxwudo, hqwuhsuhqhxu iurp plplfnlqj vlqfh wkh ulvn qhxwudo hqwuhsuhqhxu grhv qrw ydoxh wkh ghfuhdvhg fdvk rz yrodwlolw| rhuhg lq wkh phwhulqj frqwudfw1 Krzhyhu vxfk dq lqfhqwlyh vfkhph lq wkh lqlwldo ohdvlqj shulrg lv frvwo| dowkrxjk lw ehqhwv wkh pdqxidfwxuhu lq wkh iroorzlqj shulrgv e| vhjphqwlqj wkh pdunhw1 Li qr klgghq lqirupdwlrq h{lvwv rq wkh pdqxidfwxuhu*v vlgh ru li wkh txdolwlhv ri wzr w|shv ri htxlsphqw duh forvh vr wkdw qr frqvwudlqw h{lvwv rq wkh lqlwldo ohdvlqj sulfhv/ wkh orvv wr wkh w|sh J pdqxidfwxuhu lq wkh lqlwldo ohdvlqj shulrg lq rughu wr frqvwuxfw vxfk dq lqfhqwlyh vfkhph zloo h{dfwo| rvhw wkh ehqhw kh fdq uhfhlyh lq wkh iroorzlqj shulrgv1 Exw zkhq wkh lqlwldo ohdvlqj sulfhv duh uhvwulfwhg wr suhyhqw wkh plplfnlqj ri wkh w|sh E pdqxidfwxuhu/ wkh lqfhqwlyh vfkhph xqghu phwhulqj lq wkh lqlwldo ohdvlqj shulrgv lv frvw0iuhh/ dqg wkh w|sh J pdqxidfwxuhu fdq wkhq dssursuldwh wkh qhw ehqhw iurp wkh pdunhw vhjphqwdwlrq1 Wkh frqglwlrq i  i6 hqvxuhv wkdw wkh lqlwldo ohdvlqj sulfhv vhw e| wkh w|sh J pdqxidfwxuhu wr eh erxqg e| wkh qrq0plplfnlqj frqvwudlqw1 Qxphulfdo H{dpsoh 9 = Vhw n @ 3=6> { @ 53> | @ 4=8> i @ 3=9>  @ 4=5> ! @ =68> o @ 5/ p @ 8/  @ 3=8 dqg X +{, @

s

{ zlwk orz0lqwhqvlw| xvdjh htxdolqj wr 53 xqlwv shu shulrg dqg

kljk0lqwhqvlw| xvdjh htxdolqj wr 73 xqlwv shu shulrg= Lq wklv fdvh/ wkh w|sh J pdqxidfwxuhu zloo rhu wzr ohdvlqj frqwudfwv iurp zklfk wkh w|sh L hqwuhsuhqhxu zloo dffhsw wkh ohdvlqj frqwudfw zlwk olplwhg xvdjh dqg phwhulqj iP4 @ :=4/ U4 @ 8= Y @ ; dqg x @63 xqlwv shu shulrgj/ dqg wkh w|sh LL hqwuhsuhqhxu zloo dffhsw wkh ohdvlqj frqwudfw zlwk xqolplwhg xvdjh iP5 @ ;=4/ U5 @ 9=7j1 Wkh w|sh J pdqxidfwxuhu grhv qrw rhu wkh vdohv frqwudfw vlqfh grlqj vr/ kh zrxog hduq 45=9 zklfk lv orzhu wkdq zkdw kh fdq hduq +45=:, iurp rhulqj wkh ohdvlqj frqwudfwv1 Lq htxloleulxp/ wkh w|sh E pdqxidfwxuhu rhuv rqo| d vdohv frqwudfw zlwk V @ 45=91 Kh grhv qrw plplf wkh w|sh J pdqxidfwxuhu ehfdxvh kh fdq qrw ehqhw iurp lplwdwlrq lq whupv ri klv h{shfwhg fdvk rz1 58

Wkh w|sh LL hqwuhsuhqhxu dffhswv wkh frqwudfw zlwk xqolplwhg xvdjh vlqfh kh lv ehwwhu r e| grlqj vr +4;= dqg wkh w|sh L hqwuhsuhqhxu dffhswv wkh frqwudfw zlwk olplwhg xvdjh dqg phwhulqj ehfdxvh klv xwlolw| lv kljkhu +7=5 dovr wuxfn dqg dxwrpreloh ohdvhv frqwdlqlqj plohdjh surylvlrqv,1 Dqrwkhu dgydqwdjh ri ohdvhv zlwk phwhulqj lv wkdw wkh| doorz wkh ohvvhh wr khgjh djdlqvw xqfhuwdlq ghpdqg iru klv surgxfwv/ dqg wkxv wr uhgxfh wkh yrodwlolw| ri klv qhw fdvk rzv1 Wkhuhiruh/ li wkh ohvvhh lv xqfhuwdlq derxw klv rzq ixwxuh xvdjh/ dqg zdqwv wr uhgxfh wkh yrodwlolw| ri klv qhw fdvk rzv/ kh 2S L?tt|i?| | |tc | @t Mii? _LU4i?|i_ |@| hL}*) |L|h_t Lu tLh||ih4 E|L @?_ |hii )i@ht U@h *i@tit @hi ?L| hi?ii_ ? Th@U|Ui

5:

fdq rsw iru d ohdvh zlwk d phwhulqj surylvlrq/ xqghu zklfk kh zloo eh fkdujhg pruh li wklv xvdjh +dqg wkhuhiruh uhyhqxh, lv kljk/ dqg ohvv li klv xvdjh lv orz1 Vxfk ohvvhhv/ zkr ydoxh wkh uhgxfwlrq lq fdvk rz yrodwlolw|/ zloo dovr eh zloolqj wr sd| d juhdwhu h{shfwhg ohdvlqj sulfh wkdq wkrvh zkr gr qrw ydoxh vxfk yrodwlolw| uhgxfwlrq1 Wkxv/ ohdvhv zlwk vxfk phwhulqj surylvlrqv dovr doorz wkh ohvvru wr pd{lpl}h surw e| vhjphqwlqj wkh pdunhw ehwzhhq wkhvh wzr w|shv ri ohvvhhv1 +lll, Lpsolfdwlrqv iru ohdvlqj frvwv dqg vdohv sulfhv= Wkh uvw lpsolfdwlrq lq wklv uhjdug lv wkdw wkh frvw shu shulrg zloo eh juhdwhu rq dyhudjh iru vkruw0whup ohdvhv udwkhu wkdq iru orqj0whup ohdvhv1 Uhfdoo wkdw/ lq htxloleulxp/ vkruw0whup ohdvhv duh hqwhuhg lqwr e| xvhuv zkr kdyh d kljk frvw ri surylglqj pdlqwhqdqfh +dqg zkr uhwxuq wkh htxlsphqw wr wkh ohvvru diwhu wkh lqlwldo shulrg,1 Wkh kljk frvw ri vkruw0whup ohdvhv wkhuhiruh uhhfwv wkh frvwv lpsrvhg rq wkh ohvvru dulvlqj iurp wkh odfn ri dssursuldwh pdlqwhqdqfh e| kljk0frvw xvhuv1 Wkh vhfrqg lpsolfdwlrq lv iru vdohv sulfhv yhuvxv ohdvlqj frvwv= rxu prgho suhglfwv wkdw wkh htxloleulxp vdohv sulfh zloo eh orzhu rq dyhudjh wkdq wkh ohdvlqj sulfh iru dq d sulrul lqglvwlqjxlvkdeoh slhfh ri htxlsphqw +vlqfh wkh dyhudjh txdolw| ri ohdvhg htxlsphqw lv ehwwhu wkdq wkdw vrog rxwuljkw,15:

.

L?U*tL? Ohdvlqj lv dq lpsruwdqw vrxufh ri h{whuqdo qdqflqj iru X1V1 frusrudwlrqv= lw kdv ehhq hvwl0

pdwhg wkdw derxw d wklug ri wkh fdslwdo htxlsphqw xvhg e| wkh frusrudwlrqv lv ohdvhg1 Krzhyhu/ wkh prwlydwlrqv iru ohdvlqj/ dqg iru wkh ydulrxv frqwudfwxdo surylvlrqv lq ohdvh frqwudfwv/ duh qrw |hw ixoo| xqghuvwrrg1 Lq wklv sdshu/ zh ghyhorshg d qrq0wd{ udwlrqdoh iru ohdvlqj lq d grxeoh0 vlghg dv|pphwulf lqirupdwlrq vhwwlqj/ dqg dqdo|}hg krz ydulrxv frqwudfwxdo surylvlrqv lq ohdvlqj frqwudfwv dulvh lq htxloleulxp1 Zh dvvxphg d fdslwdo jrrgv pdunhw zkhuh wkh pdqxidfwxuhu ri fdslwdo jrrgv kdv sulydwh lqirupdwlrq derxw wkh txdolw| ri jrrgv kh surgxfhv> hqwuhsuhqhxuv +xvhuv 2. At t i@t*) MLh?i L| uLh @|L4LM*itG tiic i}c |i L4T*i|i @h Lt| B_ic U *t|t |i *i@ti ULt| uLh @ 4_ThUi ti_@? @| m2HcD Dc UL4T@hi_ |L |i *L@? ULt| Lu m2DcD D *tLc |ihi t tL4i i4ThU@* i_i?Ui ?_U@|?} |@| |i ^@*|) Lu Lg*i@ti U@ht t }hi@|ih |@? tL*_ U@ht Lu @ t4*@h @}iG #it@ @?_ hL| EbbHc t?} @U|L? _@|@ uLh @ TLT*@h U@h 4L_i*c ThL_i i_i?Ui |@| |i ThUi _iU*?i uLh Lg*i@ti U@ht t t*Lih |@? |@| uLh tL*_ U@ht

5;

ri wkhvh fdslwdo jrrgv, frph wr ohduq wklv txdolw| rqo| ryhu d shulrg ri wlph1 Hdfk xqlw ri wkh fdslwdo jrrgv uhtxluhv d fhuwdlq ohyho ri pdlqwhqdqfh lq hdfk shulrg1 Hqwuhsuhqhxuv glhu lq wkhlu frvw ri surylglqj wklv pdlqwhqdqfh> wklv pdlqwhqdqfh frvw lv lqirupdwlrq sulydwh wr hdfk hqwuhsuhqhxu1 Ohdvlqj hphujhg dv dq htxloleulxp vroxwlrq wr wklv grxeoh0vlghg dgyhuvh vhohfwlrq sureohp1 Ydulrxv frqwudfwxdo surylvlrqv lq ohdvlqj frqwudfwv +h1j1/ vkruw0whup yhuvxv orqj0whup ohdvhv zlwk qrq0fdqfhoodwlrq surylvlrqv/ rswlrq wr ex| dw ohdvh whuplqdwlrq/ dqg vhuylfh ohdvhv, dovr hphujhg dv htxloleulxp vroxwlrqv xqghu ydulrxv dowhuqdwlyh vhwwlqjv1 Ilqdoo|/ ohdvhv zlwk ph0 whulqj surylvlrqv hphujhg lq htxloleulxp zkhq/ lq dgglwlrq wr pdlqwhqdqfh frvw/ hqwuhsuhqhxuv glhu lq rwkhu glphqvlrqv vxfk dv wkhlu lqwhqvlw| ri xvlqj wkh fdslwdo jrrg/ dqg wkhlu ghjuhh ri ulvn0dyhuvlrq1 Dq lqwhuhvwlqj dvshfw ri rxu prgho lv wkdw/ lq htxloleulxp/ ohdvlqj fr0h{lvwhg zlwk d vdohv frqwudfw lq vrph fdvhv> lw zdv wkh vroh nlqg ri qdqflqj frqwudfw lq rwkhu fdvhv> dqg wkh vdohv frqwudfw hphujhg dv wkh rqo| nlqg ri frqwudfwv rhuhg lq |hw rwkhu vhwwlqjv1 Zh ghyhorshg wkh lpsolfdwlrqv ri rxu prgho iru wkh ohdvh0yhuvxv0vhoo ghflvlrq/ wkh vlwxdwlrqv xqghu zklfk ydulrxv ohdvlqj frqwudfw surylvlrqv duh dssursuldwh/ dqg iru wkh uhodwlyh pdjqlwxghv ri vdohv sulfhv dqg wkh ohdvlqj frvwv iru ohdvhv zlwk glhuhqw frqwudfwxdo surylvlrqv1

5
5 X +I4  P4  U4 , . +4  ,X +I6  P4  U4  Y , . +4  , X +I5  P4  U4 , . +4  ,5 X +I7  P4  U4  Y ,  X +6|,> X+n{ . n5 {  U4 , . +4  ,X +on{ . on5 {  U4 ,  X+5|,>

+D4:,

+D4;,

+D4

+D54,

+n{ . n5 {  U4 , . +4  ,+on{ . on5 {  U4 ,  5|>

+D55,

O=K=V= ri +D4 6:

+D56,

X +n{ . n5 {  U5 , . +4  ,X +on{ . on5 {  U5 ,  5|

+D57,

Wkh LF frqvwudlqw iru wkh w|sh E pdqxidfwxuhu lv= +4  !,P5 . !^P4 . +4  ,Y ` . i n{  V=

+D58,

Wkh w|sh E pdqxidfwxuhu pd{lpl}hv V idflqj wkh iroorzlqj LF dqg LU frqvwudlqw= i{ . i n{  V  5|>

+D59,

+4  !,+P5 . U5 , . !^P4 . U4 . +4  ,Y `  !p  V=

+D5:,

Frqvwudlqw +D4:, dqg +D55, duh vdwlvhg dxwrpdwlfdoo|1 +D4;,/ +D58, dqg +D59, krog htxdolw|1 Vr grhv +D53, zkhq   45 1 Wkxv U5  U4 dqg P4 . +4  ,U4  P5 1 Frqvwudlqw +D4 P5 > Y j vdwlvhv= +4  !,P5 . !^P4 . +4  ,Y ` @ i { . i n{  5|  i n{> P4 . +4  ,Y  P5  U5  U4 dqg frqvwudlqw +D56,1 Zkhq p  p5 @ !4 ^+4  !,U5 . !U4  i n{`/ +D5:, lv vdwlvhg1

6;

+D5