Free emission permits and public revenue

Working Paper No 87/00

Free Emission Permits and Public Revenue

by Ottar Mæstad

SNF project no 1090

Environmental Policy towards Internationally Mobile Firms

The project is financed by the Research Council of Norway

SIØS - Centre for International Economics and Shipping

FOUNDATION FOR RESEARCH IN ECONOMICS AND BUSINESS ADMINISTRATION BERGEN,DECEMBER2001

ISSN 0803 - 4028

© Dette eksemplar er fremstilt etter avtale med KOPINOR, Stenergate 1, 0050 Oslo.

Ytterligere eksemplarfremstilling uten avtale og i strid med åndsverkloven er straffbart og kan medføre erstatningsansvar.

SIØS - CENTRE FOR INTERNATIONAL ECONOMICS AND SHIPPING

SIØS – Centre for international economics and shipping – is a joint centre for The Norwegian School of Economics and Business Administration (NHH) and The Foundation for Research in Economics and Business Administration (SNF). The centre is responsible for research and teaching within the fields of international trade and shipping.

International Trade

The centre works with all types of issues related to international trade and shipping, and has particular expertise in the areas of international real economics (trade, factor mobility, economic integration and industrial policy), international macroeconomics and international tax policy. Research at the centre has in general been dominated by projects aiming to provide increased insight into global, structural issues and the effect of regional economic integration. However, the researchers at the centre also participate actively in projects relating to public economics, industrial policy and competition policy.

International Transport

International transport is another central area of research at the centre. Within this field, studies of the competition between different modes of transport in Europe and the possibilities of increasing sea transport with a view to easing the pressure on the land based transport network on the Continent have been central.

Maritime Research

One of the main tasks of the centre is to act as a link between the maritime industry and the research environment at SNF and NHH. A series of projects that are financed by the Norwegian Shipowners Association and aimed directly at shipowning firms and other maritime companies have been conducted at the centre. These projects include studies of Norwegian shipowners' multinational activities, shipbuilding in Northern Europe and the competition in the ferry markets.

Human Resources

The centre’s human resources include researchers at SNF and affiliated professors at NHH as well as leading international economists who are affiliated to the centre through long-term relations. During the last few years the centre has produced five PhDs within international economics and shipping.

Networks

The centre is involved in several major EU projects and collaborates with central research and educational institutions all over Europe. There is particularly close contact with London School of Economics, University of Glasgow, The Graduate Institute of International Studies in Geneva and The Research Institute of Industrial Economics (IUI) in Stockholm. The staff members participate in international research networks, including Centre for Economic Policy Research (CEPR), London and International Association of Maritime Economists (IAME).

Iuhh hplvvlrq shuplwv dqg sxeolf uhyhqxh

Rwwdu Pévwdg

Irxqgdwlrq iru Uhvhdufk lq Hfrqrplfv dqg Exvlqhvv Dgplqlvwudwlrq2VLÛV Khoohyhlhq 63/ Q08378 Ehujhq/ Qruzd|

H0pdlo= Rwwdu1PdhvwdgCvqi1qr

Devwudfw

Lq wkh suhvhqfh ri suhh{lvwlqj glvwruwlrqdu| wd{hv/ lw lv riwhq du0 jxhg wkdw dxfwlrqhg hplvvlrq shuplwv duh suhihudeoh wr qrq0dxfwlrqhg shuplwv/ ehfdxvh wkh iruphu jhqhudwh uhyhqxhv wkdw pd| eh xvhg wr uhgxfh rwkhu wd{hv1 Wklv sdshu vkrzv wkdw zkhq fdslwdo lv lqwhuqdwlrq0 doo| preloh/ lw pd| eh rswlpdo wr xvh d frpelqdwlrq ri qrq0dxfwlrqhg dqg dxfwlrqhg hplvvlrq shuplwv/ iru erwk hqylurqphqwdo dqg vfdo uhd0 vrqv1 E| ohwwlqj wkh qxpehu ri qrq0dxfwlrqhg shuplwv eh d srvlwlyh ixqfwlrq ri wkh dprxqw ri fdslwdo xvhg grphvwlfdoo|/ wkh| zloo dwwudfw fdslwdo wr wkh krph frxqwu|1 Wklv pd| fuhdwh hqylurqphqwdo ehqhwv lq whupv ri uhgxfhg wudqverxqgdu| srooxwlrq dqg pd| ohdg wr lqfuhdvhg sxeolf uhyhqxh ehfdxvh wkh sulfh ri hplvvlrq shuplwv pd| lqfuhdvh1 Lw lv dovr vkrzq wkdw wkh rswlpdo qxpehu ri qrq0dxfwlrqhg shuplwv pd|

lqfuhdvh dv wkh pdujlqdo frvwv ri sxeolf ixqgv lqfuhdvh1 MHO fodvvlfdwlrq= G95/ I54/ T5;1

Ilqdqfldo vxssruw iurp wkh Uhvhdufk Frxqflo ri Qruzd| lv judwhixoo| dfnqrzohgjhg +judqw qr1 45:;4;2:63,1

4

4 Lqwurgxfwlrq

Lq wkh zdnh ri wkh folpdwh qhjrwldwlrqv dqg wkh N|rwr Surwrfro/ zh kdyh zlwqhvvhg dq lqfuhdvlqj lqwhuhvw lq pdq| frxqwulhv iru wudgdeoh hplvvlrq shu0 plwv dv dq lqvwuxphqw iru hqylurqphqwdo srolf| pdnlqj1 Vlqfh wkh qdwlrqdo frpplwphqwv lq wkh N|rwr Surwrfro duh vshflhg lq whupv ri hplvvlrq txr0 wdv/ lw zloo hdvlhu wr hqvxuh frpsoldqfh li hplvvlrq shuplwv duh xvhg udwkhu wkdq hplvvlrq wd{hv1 Pruhryhu/ wkh h{lelolw| phfkdqlvpv exlow lqwr wkh Surwrfro hqvxuh wkdw hplvvlrq txrwdv duh wudgdeoh dprqj qdwlrqv1 E| xvlqj d v|vwhp ri wudgdeoh hplvvlrq shuplwv zlwklq hdfk ri wkh sduwlflsdwlqj frxq0 wulhv/ wkh qdwlrqdo v|vwhpv zloo eh hdv| wr lqwhjudwh zlwk wkh lqwhuqdwlrqdo txrwd pdunhw1

Dorqj zlwk wkh irfxv rq wudgdeoh hplvvlrq shuplwv kdv qdwxudoo| frph d glvfxvvlrq derxw krz wkh shuplwv vkrxog eh doorfdwhg1 D fhqwudo txhvwlrq lv zkhwkhu wkh shuplwv vkrxog eh dxfwlrqhg ru qrw1 Lq dq hfrqrp| zlwk suhh{lvwlqj wd{ glvwruwlrqv/ lw lv riwhq dujxhg wkdw dxfwlrqhg hplvvlrq shuplwv duh suhihudeoh wr qrq0dxfwlrqhg shuplwv/ surylghg wkh uhyhqxhv jhqhudwhg duh xvhg wr fxw lq pdujlqdo wd{ udwhv1 Wklv uhyhqxh0uhf|folqj hhfw kdv ehhq glvfxvvhg e| Rdwhv dqg Vfkzde +4<;;, dqg Srwhued +4<<6,/ dprqj rwkhuv1 Pruh uhfhqwo|/ lw kdv ehhq ghprqvwudwhg wkdw gxh wr lqwhudfwlrq hhfwv zlwk suhh{lvwlqj wd{hv/ hqylurqphqwdo srolflhv wkdw gr qrw h{sorlw wkh ehqhwv ri uhyhqxh0uhf|folqj pd| surgxfh vljqlfdqwo| orzhu zhoiduh ohyhov wkdq srolflhv wkdw gr +Jrxoghuhw do1 +4<<:,/ Sduu| dqg Zlooldpv LLL +4<<<,/ Jrxoghuhw do1 +4<<<,,1

Frqfhuqv derxw uhyhqxh0uhf|folqj lv qrw wkh rqo| uhdvrq wr fduh derxw wkh doorfdwlrq ri hplvvlrq shuplwv/ wkrxjk1 Lq vrph frxqwulhv/ lw kdv ehhq vxjjhvwhg wkdw srooxwlqj upv wkdw duh h{srvhg wr lqwhuqdwlrqdo frpshwlwlrq vkrxog uhfhlyh hplvvlrq shuplwv iuhh ri fkdujh lq rughu wr suhyhqw up for0 vxuh dqg2ru uhorfdwlrq wr rwkhu frxqwulhv1 Vxfk forvxuh2uhorfdwlrq pd| eh d vhulrxv sureohp lq wkh fdvh ri wudqverxqgdu| hqylurqphqwdo sureohpv/ lq sduwlfxodu zkhq vrph frxqwulhv duh xqzloolqj wr frrshudwh lq rughu wr vroyh wkh sureohp1 Wkh folpdwh sureohp lv d fdvh lq srlqw1 Wkh folpdwh sureohp lv d joredo hqylurqphqwdo sureohp1 Exw vlqfh rqo| vrph lqgxvwuldolvhg frxqwulhv duh zloolqj wr frpplw wkhpvhoyhv wr elqglqj hplvvlrq wdujhwv/ wkhuh lv d srv0 vlelolw| wkdw uhgxfhg hplvvlrqv lq wkhvh frxqwulhv zloo ohdg wr kljkhu hplvvlrqv hovhzkhuh1 Wkhuh duh vhyhudo srwhqwldo vrxufhv ri vxfk ohdndjh hhfwv1 Rqh ri wkhp lv wkdw preloh fdslwdo pd| hvfdsh vwulfw uhjxodwlrqv lq wkh krph frxqwu|

dqg pryh wr rwkhu uhjlrqv zkhuh srooxwlqj dfwlylwlhv fdq eh frqwlqxhg1⁴ Lq

4Ohdndjh pd| dovr rffxu wkurxjk wkh surgxfw pdunhwv/ zkhq iruhljq surgxfhuv fds0 wxuh pdunhw vkduhv iurp grphvwlf surgxfhuv gxh wr xqlodwhudo hqylurqphqwdo srolflhv +vhh ehorz,1

5

ghvljqlqj folpdwh srolflhv/ wkh txhvwlrq derxw judqwlqj iuhh hplvvlrq txrwdv wr fhuwdlq lqgxvwulhv wkhuhiruh riwhq sod|v d fhqwudo uroh/ dv zdv wkh fdvh/ h1j1/

lq wkh uhsruwv irup wkh Fdqdgldq Wudgdeoh Shuplwv Zrunlqj Jurxs dqg wkh Qruzhjldq Txrwd Frpplvvlrq1

Wkh xvh ri iuhh hplvvlrq shuplwv lq rughu wr uhgxfh ohdndjh hhfwv lv qrw zlwkrxw vxssruw lq wkh wkhruhwlfdo olwhudwxuh1 Lw kdv ehhq vkrzq wkdw vlqfh lqfuhdvhg hplvvlrq ri folpdwh jdvhv lq qrq0frrshudwlqj frxqwulhv lpsrvhv d frvw rq wkh frrshudwlqj rqhv/ lw zloo eh h!flhqw wr lpsohphqw srolflhv wkdw uhgxfh wkh lqfhqwlyhv ri preloh fdslwdo/ hpsor|hg lq srooxwlqj upv/ wr pryh wr rwkhu uhjlrqv1 Krho +4<<9, qgv wkdw d xqlirup fduerq wd{ lq wkh fr0 rshudwlqj frxqwulhv vkrxog eh frpelqhg zlwk dq h{sruw wd{ rq srooxwlqj fdslwdo1 Wkh sureohp zlwk vxfk d srolf| lv wkdw lw lv lpsrvvleoh lq sudfwlfh wr glvfulplqdwh ehwzhhq srooxwlqj dqg qrq0srooxwlqj fdslwdo/ vlqfh lw lv qrw srvvleoh wr wudfh wkh xvh ri wkh fdslwdo rqfh lw kdv pryhg deurdg1 Rqh do0 whuqdwlyh wr vxfk h{sruw wd{hv lv wr vxevlglvh wkh xvh ri srooxwlqj fdslwdo lq wkh frrshudwlqj frxqwulhv +Udxvfkhu +4<<:,/ Pévwdg +5334,,1 Vxfk vxevlglhv vkrxog eh xvhg lq frpelqdwlrq zlwk d xqlirup hplvvlrq wd{ ru d v|vwhp zlwk wudgdeoh hplvvlrq shuplwv1 Rqh zd| ri lpsohphqwlqj wkh vxevlg| zrxog eh wr jlyh upv iuhh hplvvlrq shuplwv sursruwlrqdo wr wkh dprxqw ri fdslwdo hpsor|hg zlwklq wkh frrshudwlqj frxqwulhv +Pévwdg/ 5334,1 Qrwh wkdw wkh qxpehu ri iuhh shuplwv pxvw khuh eh d ixqfwlrq ri d uhdo yduldeoh1 Xqfrqgl0 wlrqdo doorfdwlrq ri iuhh hplvvlrq txrwdv zloo kdyh qr h!flhqf| hhfwv dw doo +fi1 wkh Frdvh Wkhruhp,1 Zh duh wkxv qrw wdonlqj ri judqgidwkhulqj edvhg rq klvwrulf hplvvlrqv/ exw udwkhu ri wdujhwhg xvh ri iuhh hplvvlrq txrwdv lq rughu wr rewdlq h!flhqf| jdlqv1 Wklv lv d nh| glhuhqfh ehwzhhq wkh hqylurqphqwdo hfrqrplfv olwhudwxuh rq wklv wrslf dqg wkh sxeolf qdqfh olwhudwxuh1

D sureohp zlwk suhylrxv vwxglhv ri wklv lvvxh zlwklq wkh hqylurqphqwdo hfrqrplfv olwhudwxuh lv wkdw wkh| ljqruh suhh{lvwlqj glvwruwlrqdu| wd{hv1 Frq0 vhtxhqwo|/ wkh| gr qrw wdnh lqwr dffrxqw wkh srwhqwldo dgydqwdjh ri uhyhqxh froohfwlqj hqylurqphqwdo lqvwuxphqwv ryhu srolflhv wkdw gr qrw froohfw uhy0 hqxh/ vxfk dv iuhh hplvvlrq txrwdv1 Lq wklv sdshu/ zh dvn krz wkh suhvhqfh ri suhh{lvwlqj wd{hv zloo dhfw wkh ghvludelolw| ri wdujhwhg iuhh hplvvlrq shuplwv1 Wkh sdshu wkxv dwwhpswv wr eulgjh wkh jds ehwzhhq wkh sxeolf hfrqrplfv olwhudwxuh ri qrq0dxfwlrqhg hplvvlrq shuplwv/ zkhuh wkh hpskdvlv lv rq wkh orvv ri sxeolf uhyhqxhv/ dqg wkh hqylurqphqwdo hfrqrplfv olwhudwxuh/ zkhuh iuhh hplvvlrq txrwdv pd| sod| d frqvwuxfwlyh uroh dv d phdqv wr uhgxflqj ohdndjh hhfwv1

Wkh sdshu vkrzv wkdw li fdslwdo lv lqwhuqdwlrqdoo| preloh/ lw pd| eh rs0 wlpdo wr xvh erwk qrq0dxfwlrqhg dqg dxfwlrqhg hplvvlrq shuplwv/ hyhq li wkhuh duh suhh{lvwlqj glvwruwlrqdu| wd{hv1 Pruhryhu/ wkh rswlpdo qxpehu ri qrq0dxfwlrqhg shuplwv pd| lqfuhdvh dv wkh pdujlqdo frvwv ri sxeolf ixqgv

6

lqfuhdvh1 Lw lv fuxfldo iru wkh uhvxowv wkdw wkh qxpehu ri qrq0dxfwlrqhg hplv0 vlrq shuplwv lv srvlwlyho| uhodwhg wr wkh dprxqw ri fdslwdo xvhg grphvwlfdoo|1 Qrq0dxfwlrqhg shuplwv zloo wkhq dwwudfw pruh fdslwdo wr wkh krph frxqwu|/

lqfuhdvh wkh ghpdqg iru hplvvlrq shuplwv/ dqg gulyh xs wkh shuplw sulfh/

surylghg wkhuh lv frpsohphqwdulw| ehwzhhq fdslwdo dqg hplvvlrqv lq wkh sur0 gxfwlrq ixqfwlrq1 Li wkh qxpehu ri qrq0dxfwlrqhg shuplwv lv qrw wrr odujh/

wklv pd| ohdg wr lqfuhdvhg sxeolf uhyhqxhv1 Khqfh/ frqvlghudwlrqv derxw orvv lq sxeolf uhyhqxh lv qrw qhfhvvdulo| dq dujxphqw djdlqvw +vhohfwlyh, xvh ri qrq0dxfwlrqhg hplvvlrq shuplwv lq hqylurqphqwdo srolf|1

Wkh sdshu lv rujdqlvhg dv iroorzv1 Vhfwlrq 5 suhvhqwv wkh prgho1 Wkh hfrqrplf frqvhtxhqfhv ri jlylqj iuhh hplvvlrq txrwdv duh glvfxvvhg lq Vhfwlrq 61 Lq Vhfwlrq 7/ wkh rswlpdo qxpehu ri iuhh hplvvlrq txrwdv lv fkdudfwhulvhg xqghu wkh dvvxpswlrq wkdw wkh hplvvlrq olplw dovr lv rswlpdoo| fkrvhq1 D ulfkhu vhw ri srolf| lqvwuxphqwv duh lqwurgxfhg lq Vhfwlrq 8 lq rughu wr vhh zkhwkhu wkh uhvxowv fduu| ryhu wr wkh fdvh ri sulru wd{ srolflhv1 Vhfwlrq 9 frqfoxghv1

5 Prgho

Zh frqvwuxfw d vlpsoh/ sduwldo htxloleulxp prgho zlwk wzr uhjlrqv/ qdphg wkh krph frxqwu| +, dqg wkh iruhljq frxqwu| +s,1 Wkhuh lv rqo| rqh surgxfwlrq vhfwru lq hdfk frxqwu|⁵1 Rxwsxw lq vhfwru ++, lv surgxfhg e| wzr idfwruv 0 fdslwdo +&, dqg d irvvlo ixho +e,1 Whfkqrorj| lv ghqhg e| wkh surgxfwlrq ixqfwlrqv

+ '8E&c e 'c s +4, zlwk wkh surshuwlhv8 :f/8 f/8_&&8_ee8_&e² f/ l1h1/ pdujlqdo surgxfwv duh srvlwlyh dqg ghfuhdvlqj/ dqg wkhuh lv qrq0srvlwlyh uhwxuqv wr vfdoh1

Wkh xvh ri irvvlo ixhov fdxvhv hplvvlrqv ri juhhqkrxvh jdvhv1 Iru vlpsolflw|/

zh qrupdolvh wkh hplvvlrq idfwru wr 41 Khqfh/ wrwdo hplvvlrqv duh ene^s1 Wkh jryhuqphqw ri wkh krph frxqwu| ghqhv dq xsshu olplw .7 rq wkh grphvwlf hplvvlrq ohyho

e .7 +5,

Li wkh hplvvlrq olplw.7lv elqglqj/ hplvvlrq shuplwv zloo kdyh d srvlwlyh pdunhw ydoxh R lq wkh krph frxqwu|1

Wkh jryhuqphqw pd| dovr ghflgh wkdw d fhuwdlq vkduh ri wkh wrwdo hplvvlrq txrwd vkdoo eh doorfdwhg iuhh ri fkdujh1 Li wkhuh duh iuhh hplvvlrq txrwdv/ wkh|

duh doorfdwhg wr grphvwlf surgxfhuv lq sursruwlrq wr wkh dprxqw ri fdslwdo

5Vhh Pévwdg +5333, iru d glvfxvvlrq ri wkh xvh ri iuhh hplvvlrq txrwdv zlwk pruh wkdq rqh surgxfwlrq vhfwru lq hdfk frxqwu|1

7

hpsor|hg lq wkh krph frxqwu|1 Wklv srolf| ghvljq uh hfwv wkh uhvxowv lq Krho +4<<9,/ Udxvfkhu +4<<:, dqg Pévwdg +5334, wkdw ohdndjh wkurxjk fdslwdo pdunhwv vkrxog eh doohyldwhg wkurxjk fdslwdo pdunhw lqwhuyhqwlrqv1 Ohw ^ eh wkh qxpehu ri iuhh hplvvlrq txrwdv shu xqlw ri fdslwdo hpsor|hg lq wkh krph frxqwu|1

Lq jhqhudo/ uhjxodwlrqv ri wkh srooxwlrq ohyho lq wkh krph frxqwu| zloo dhfw erwk surgxfw sulfhv dqg idfwru sulfhv1 Suhylrxv vwxglhv kdyh vkrzq wkdw li surgxfw pdunhw sulfhv duh dhfwhg e| hqylurqphqwdo srolflhv/ ohdndjh pd|

rffxu wkurxjk frpshwlwlrq hhfwv lq surgxfw pdunhwv1 Vlploduo|/ fkdqjhv lq hqhuj| sulfhv lpso| wkdw wkhuh zloo eh srwhqwldo ohdndjh wkurxjk wkh hqhuj|

pdunhw1 Dv vkrzq e| Pévwdg +4<<;/ 5334, dq Krho +4<<9,/ wkh h!flhqw zd|

ri ghdolqj zlwk vxfk ohdndjh hhfwv lv wkurxjk wudgh srolflhv dqg qrw wkurxjk fdslwdo pdunhw lqwhuyhqwlrqv1 Vlqfh rxu irfxv khuh lv rq wkh rswlpdo xvh ri fdslwdo vxevlglhv lq wkh irup ri iuhh hplvvlrq txrwdv wr grphvwlf fdslwdo/

wkhuh lv qr uhdvrq iru eulqjlqj wkhvh rwkhu srwhqwldo vrxufhv ri ohdndjh lqwr wkh prgho1 Wkhuhiruh/ surgxfw sulfhv dqg wkh zruog pdunhw sulfh ri irvvlo ixhov vkdoo eh wuhdwhg dv {hg1 Whfkqlfdoo|/ zh dvvxph wkdw wkh sulfh hodvwlflwlhv ri surgxfw ghpdqg dqg ixho vxsso| duh lqqlwh1

Ilupv wdnh idfwru sulfhv dv jlyhq1 Ohw Eoc eh wkh idfwru sulfhv ri fdslwdo dqg irvvlo ixhov lq frxqwu| / ohw ` ghqrwh wkh +{hg, zruog pdunhw sulfh ri irvvlo ixhov/ dqg ohw-eh wkh lqwhuqdwlrqdo uhqwdo sulfh ri fdslwdo1 Zh wkhq kdyho<sup>s</sup> '-/o '-R^/<sup>s</sup> '`/ dqg '`nR1 Surw pd{lplvlqj ehkdylrxu lpsolhv wkdw pdujlqdo surgxfwv zloo eh htxdo wr idfwru sulfhv=

8_&^s ' -'8_&nR^c +6,

8_e^s ' ` '8_eR +7,

Wkhuh lv d jlyhq wrwdo vwrfn ri fdslwdog1 Fdslwdo lv iuhho| preloh lqwhuqd07 wlrqdoo|1 Wkh uhqwdo sulfh ri fdslwdo lv ghwhuplqhg vr wkdw djjuhjdwh fdslwdo ghpdqg htxdov wkh fdslwdo vwrfn/ l1h1/

&E-R^ n&^sE- ' 7g +8,

6 Wkh lpsdfw ri iuhh hplvvlrq txrwdv

Frqvlghu krz wkh htxloleulxp ri wklv prgho lv dhfwhg e| wkh doorfdwlrq ri iuhh hplvvlrq shuplwv1 Glhuhqwldwlrq ri Htv1 +6,0+8, zlwk uhvshfw wr^ |lhogv

8

wkh iroorzlqj v|vwhp 5 99 99 7

8_&& f f  ^

f 8_&&^s 8_&e^s  f

8_&e f f f 

f 8_&e^s 8_ee^s f f

  f f f

6 :: :: 8

5 99 99 99 7

_&

_&_^^s _e_^^s _-_^

_R_^

_^

6 :: :: :: 8

' 5 99 99 7

Rf ff f

6 :: ::

8 +9,

E| xvlqj Fudphu*v uxoh/ wkh hhfw ri iuhh hplvvlrq txrwdv fdq eh vxppdulvhg dv iroorzv=

_&

_^ ' 

mMmR8_ee^sc +:, _e^s

_^ ' 

mMmR8_&e^sc +;,

_-

_^ ' 

mMmRw^sc +<,

_R

_^ ' 

mMmR8_&e8_ee^sc +43,

zkhuh w '8_&& 8_ee E8_&e ² dqgmMm'w^s 8_&&8_ee^s ^8_&e8_ee^s1

Sursrvlwlrq 4 Li wkhuh duh qrw wrr pdq| iuhh txrwdv dw wkh rxwvhw +t vpdoo, dqg wkhuh lv d elqglqj hplvvlrq frqvwudlqw lq wkh krph frxqwu| +s srvlwlyh,/ d odujhu qxpehu ri iuhh hplvvlrq shuplwv shu xqlw ri fdslwdo zloo ohdg wr fdslwdo lq rz lq wkh krph frxqwu|E_&*_^ :f dqg d ulvh lq wkh lqwhuqdwlrqdo uhqwdo sulfh ri fdslwdo E_-*_^ : f Li wkhuh lv idfwru frpsohphqwdulw| E8_&e : f/

iruhljq hplvvlrqv zloo eh uhgxfhg E_e^s*_^ fc dqg wkh sulfh ri hplvvlrq shuplwv zloo lqfuhdvh E_R*_^ :f

Jlyhq wkdw wkh sulfh ri hplvvlrq txrwdv lv srvlwlyh/ jlylqj iuhh txrwdv lq sursruwlrq wr wkh dprxqw ri fdslwdo hpsor|hg lq wkh krph frxqwu| lv htxlydohqw wr d vxevlg| rq fdslwdo xvh1 Fdslwdo ghpdqg lq wkh krph frxqwu|

lqfuhdvhv/ gulylqj xs wkh lqwhuqdwlrqdo udwh ri uhwxuq dqg ohdglqj wr fdslwdo lq rz wr wkh krph frxqwu|1 Zlwk idfwru frpsohphqwdulw|/ ohvv fdslwdo lq wkh iruhljq frxqwu| uhgxfhv wkh pdujlqdo surgxfw ri irvvlo ixhov lq wkh iru0 hljq frxqwu| dqg wkhuhiruh ohdgv wr orzhu iruhljq hplvvlrqv1 Vlqfh wkhuh lv d elqglqj xsshu olplw rq krph frxqwu| hplvvlrqv/ wrwdo hplvvlrqv duh uhgxfhg dv zhoo1 Dv pruh fdslwdo lv xvhg lq wkh krph frxqwu|/ wkh ghpdqg iru irv0 vlo ixhov zloo lqfuhdvh1 Wklv gulyhv xs wkh sulfh ri hplvvlrq txrwdv/ lqgxflqj vxevwlwxwlrq dzd| iurp irvvlo ixhov wrzdugv fdslwdo1

9

Wkh sxeolf vhfwru*v lqfrph Ciurp vdoh ri hplvvlrq txrwdv lv jlyhq dv

C'RE 7.&^ +44,

Ohw xv lqyhvwljdwh krz d vpdoo lqfuhdvh lq wkh qxpehu ri iuhh hplvvlrq shuplwv dhfw sxeolf uhyhqxh1 Glhuhqwldwlrq ri +44, zlwk uhvshfw wr ^ |lhogv

_C

_^ ' _R

_^

.7&^

R^_&

_^ R& +45,

' 

mMmR8_&e8_ee^s .7&^

R^ 

mMmR8_ee^s R&w^s 8_&&8_ee^s ^8_&e8_ee^s mMm

' R mMm

8_&e8_ee^s.7R^8_ee^s &

w^s 8_&&8_ee^s ' R8_ee^s

mMm

8_&e.7n

8_&& n w^s 8ee^s

& R^

Wr lqfuhdvh wkh qxpehu ri iuhh txrwdv shu xqlw ri fdslwdo uhgxfhv sxeolf lqfrph erwk wkurxjk d gluhfw uhgxfwlrq lq wkh qxpehu ri dxfwlrqhg hplvvlrq shuplwv dqg wkurxjk dq lqfuhdvh lq wkh fdslwdo vwrfn lq wkh krph frxqwu|1 Wkhuhiruh/ lw lv riwhq frqfoxghg wkdw qrq0dxfwlrqhg hplvvlrq shuplwv duh ohvv dwwudfwlyh wkdq dxfwlrqhg shuplwv zkhq rqh sd|v dwwhqwlrq wr wkh suhvhqfh ri suhh{lvwlqj glvwruwlrqdu| wd{hv1 Dv zh mxvw vkrzhg/ krzhyhu/ d odujhu qxpehu ri iuhh txrwdv pd| lqfuhdvh wkh sulfh ri hplvvlrq shuplwv1 Wkh wrwdo hhfwv ri iuhh hplvvlrq txrwdv rq sxeolf lqfrph lv wkxv dpeljxrxv1

Ht1 +45, vkrzv wkdw d qhfhvvdu| frqglwlrq iru kdylqj sxeolf uhyhqxh lq0 fuhdvh zlwk wkh qxpehu ri iuhh hplvvlrq shuplwv lv wkdw fdslwdo dqg hplvvlrqv duh frpsohphqwv +l1h1/ 8_&e : f,1 Wkh lqwxlwlrq lv d iroorzv= Li fdslwdo dqg hplvvlrq duh frpsohphqwv/ d kljkhu grphvwlf fdslwdo vwrfn zloo lqfuhdvh wkh ghpdqg iru hplvvlrq shuplwv dqg wkxv gulyh xs wkh txrwd sulfh1 Khqfh/ e|

dwwudfwlqj pruh fdslwdo wr wkh krph frxqwu|/ d fdslwdo vxevlg| kdv d srwhqwldo srvlwlyh lpsdfw rq sxeolf uhyhqxh wkurxjk wkh txrwd sulfh1 Li wklv hhfw lv vwurqjhu wkdq wkh gluhfw orvv lq uhyhqxh lpsolhg e| wkh vxevlg|/ wrwdo uhyhqxhv zloo lqfuhdvh1

Wklv uhvxow vxjjhvwv wkdw wkhuh pljkw eh fdvhv lq zklfk kljkhu pdujlqdo frvwv ri sxeolf ixqgv zloo uhtxluh d odujhu qxpehu ri iuhh hplvvlrq txrwdv1 Krzhyhu/ lq wkh dqdo|vlv vr idu/ wkh hplvvlrq olplw kdv ehhq wdnhq dv h{rjh0 qrxv1 Lq rughu wr dqvzhu rxu txhvwlrq zh kdyh wr lqyhvwljdwh zkhwkhu wkh rswlpdo qxpehu ri iuhh hplvvlrq txrwdv pd| lqfuhdvh zlwk wkh pdujlqdo frvwv ri sxeolf ixqgv dovr zkhq .7 lv fkrvhq rswlpdoo|1

:

7 Rswlpdo srolflhv

Lq wklv vhfwlrq zh fkdudfwhulvh wkh rswlpdo qxpehu ri iuhh hplvvlrq shuplwv zkhq wkh hplvvlrq olplw lv rswlpdoo| fkrvhq1 Ohw krph frxqwu| zhoiduh eh dq dgglwlyho| vhsdudeoh ixqfwlrq ri lqfrph dqg wkh ohyho ri srooxwlrq/

'EUEc c :f +46,

Lqfrph lv wkh vxp ri surgxfhu vxusoxv +lqfoxglqj wkh ydoxh ri iuhh hplvvlrq txrwdv,/ jryhuqphqw lqfrph/ dqg wkh glhuhqfh ehwzhhq wkh uhwxuq rq fdslwdo lqyhvwhg deurdg ohvv wkh uhqw sdlg wr iruhljq fdslwdo lqyhvwhg lq wkh krph frxqwu|⁶1 Ohwkeh wkh vkduh ri wkh fdslwdo vwrfn rzqhg e| wkh krph frxqwu|1 Wkh qhw iruhljq dvvhwv ri wkh krph frxqwu| duh wkhq kg7 &/ dqg wkh qhw qdqfldo uhwxuq iru wkh krph frxqwu| lv -Ekg7 &1 Zh dvvxph wkdw wkh jryhuqphqw kdv wr qdqfh d exgjhw ri d jlyhq vl}h dqg wkdw sxeolf lqfrph lv qdqfhg e| glvwruwlrqdu| wd{hv1 Ohw b ghqrwh wkh pdujlqdo frvwv ri sxeolf ixqgv1 Khqfh/ rqh groodu ri jryhuqphqw lqfrph lv htxlydohqw wr E nb grooduv ri sulydwh lqfrph1 Vlqfh rxu prgho lv ri wkh sduwldo htxloleulxp w|sh/

zh wuhdw b dv d frqvwdqw1

Wkh folpdwh sureohp lv d joredo hqylurqphqwdo sureohp/ dqg iruhljq hplv0 vlrqv duh wkxv htxdoo| gdpdjlqj dv krph frxqwu| hplvvlrqv1 Zh dvvxph wkdw wkh hplvvlrq olplw lq wkh krph frxqwu| lv elqglqj1 Khqfh/ wkh ohyho ri sro0 oxwlrq lv jlyhq e| ' 7.ne^s1

Wkh zhoiduh ri wkh krph frxqwu| fdq qrz eh zulwwhq⁷

'

8

&c e

E` nRenR^&n-Ekg7 &

nE nbRE 7. &^

.7ne^s Wkh jryhuqphqw fkrrvhv wkh hplvvlrq olplw .7 dqg wkh qxpehu ri iuhh+47, hplvvlrq txrwdv ^ vr dv wr pd{lplvh zhoiduh1 Wkh uvw rughu frqglwlrqv iru dq lqwhulru vroxwlrq duh

E nbR^_&

_^ b

R& _R _^

.7&^ n_-

_^

kg7 &

_e^s _^ ' f

+48,

E nbR^Y&

Y.7 n E nbRnbYR Y.7

.7&^ n Y-

Y.7

kg7 &

 n Ye^s Y.7

' f +49,

6Wkhuh lv qr frqvxphu vxusoxv vlqfh ghpdqg lv dvvxphg wr eh lqqlwho| hodvwlf1

7Lqfrph lv phdvxuhg lq whupv ri lqfrph rq sulydwh kdqgv1

;

Lq Ht1 +48,/ wkh qhw frvwv ri jlylqj iuhh hplvvlrq txrwdv wr fdslwdo duh glylghg lqwr wkuhh frpsrqhqwv> +4, wkh frvwv ri lqh!flhqw xvh ri fdslwdo/ +5, wkh frvwv gxh wr dq| orvvhv lq sxeolf uhyhqxh/ dqg +6, jdlqv ru orvvhv wkurxjk fkdqjhv lq wkh whupv ri wudgh lq wkh lqwhuqdwlrqdo fdslwdo pdunhw1 Lq rswlpxp/

wkhvh frvwv vkrxog eh edodqfhg djdlqvw wkh qhw ehqhwv ri uhgxfhg srooxwlrq iurp deurdg1 Ht1 +49,/ vkrzlqj wkh pdujlqdo frvw dqg wkh pdujlqdo ehqhwv ri fkdqjlqj wkh grphvwlf hplvvlrq olplw/ kdv d vlplodu lqwhusuhwdwlrq1

Ht1 +48, fdq eh ixuwkhu vlpsolhg e| xwlolvlqj wkh frpsdudwlyh vwdwlfv ri Htv1 +:,0+43,1 Zh rewdlq

^ ' 

RE nb

%

8_&e^s 8ee^s w^s

8ee^s

kg7 &

nb

8_&e.7n w^s

8ee^s n8_&&

&

&

Lq rughu wr vlpsoli| Ht1 +49,/ zh qhhg wr qg wkh htxloleulxp hhfwv ri+4:, fkdqjhv lq wkh hplvvlrq olplw1 Glhuhqwldwlrq ri wkh frpsohwh prgho +Htv1 +6,0+8,, zlwk uhvshfw wr .7 |lhogv

5 99 99 7

8_&& f f  ^

f 8_&&^s 8_&e^s  f

8_&e f f f 

f 8_&e^s 8_ee^s f f

  f f f

6 :: :: 8

5 99 99 97

_&

_ 7. _&^s _ 7. _e^s _ 7. _-_ 7_R. _ 7.

6 :: :: :8'

5 99 99 7

8_&e f8_ee ff

6 :: ::

8 +4;,

E| xvlqj Fudphu*v uxoh/ zh rewdlq _&

_.7 '  mMm8_ee^s

8_&e n8_ee^

c +4<,

_e^s

_.7 '  mMm8_&e^s

8_&e n8_ee^

c +53,

_-

_.7 '  mMmw^s

8_&e n8_ee^

c +54,

_R

_.7 '  mMm

8_ee^swn8_eew^s

+55,

Zh qrz lqvhuw wkhvh h{suhvvlrqv lqwr Ht1 +49, dqg vroyh Htv1 +49, dqg +4:, vlpxowdqhrxvo| zlwk uhvshfw wr R dqg ^1 Wkh rswlpdo srolflhv fdq eh fkdudfwhulvhg dv iroorzv

R^W ' 

>E nb

c +56,

<

^^W '>

# 8_&e^s

8ee^s

w^s 8ee^s

kg7 &$

n E>

3

C8_&e.7n

w^s

8ee^s n8_&&

&

8_ee.7n8_&e&

4 Dc +57, zkhuh >*EbE8_ee.7n8_&e&

Ht1 +57, vkrzv wkdw wkh rswlpdo qxpehu ri iuhh hplvvlrq txrwdv shu xqlw ri fdslwdo fdq eh h{suhvvhg dv d zhljkwhg vxp ri wzr whupv1 Li wkh pdujlqdo frvwv ri sxeolf ixqgv duh }hur +b ' f,/ > lv htxdo wr rqh1 Wkh odwwhu whup wkhq glvdsshduv1 Zh ghqrwh wklv whup wkh vfdo dujxphqw iru wkh xvh ri iuhh hplvvlrq txrwdv1

Wkh uvw whup kdv wzr frpsrqhqwv1 Wkh uvw rqh lv uhodwhg wr wkh uroh ri iuhh hplvvlrq txrwdv dv d phdqv wr uhgxflqj wkh ohyho ri srooxwlrq jhqhudwhg lq wkh iruhljq frxqwu|1 Lq rughu wr vhh wklv pruh fohduo|/ wrwdoo| glhuhqwldwh wkh uvw rughu frqglwlrq iru rswlpdo xvh rie^s +vhh Ht1 +7,,1 Diwhu uhduudqjlqj whupv/ zh rewdlq

_e^s

_&^s '8_&e^s

8ee^s +58,

Zkhq iuhh hplvvlrq txrwdv duh xvhg/ ohvv fdslwdo zloo eh doorfdwhg lq wkh iru0 hljq frxqwu|1 Wkh h{suhvvlrq 8_&e^s*8_ee^s vkrzv zklfk hhfw vxfk uhdoorfdwlrq ri fdslwdo kdv rq wkh iruhljq hplvvlrq ohyho1 Ht1 +2e, vkrzv wkdw wklv whup glvdsshduv li wkh pdujlqdo frvw ri srooxwlrq lv }hur + ' f,/ uh hfwlqj wkh idfw wkdw wklv whup kdv wr gr zlwk wkh hqylurqphqwdo uhdvrq iru xvlqj iuhh hplvvlrq shuplwv1

Wkh vhfrqg frpsrqhqw lv uhodwhg wr wkh whupv ri wudgh hhfw ri iuhh hplv0 vlrq shuplwv1 Vlqfh fdslwdo vxevlglhv lq wkh krph frxqwu| zloo lqfuhdvh fdslwdo ghpdqg dqg wkxv whqg wr gulyh xs wkh lqwhuqdwlrqdo udwh ri lqwhuhvw/ wkhuh zloo eh d whupv ri wudgh jdlq +orvv, iru fdslwdo h{sruwlqj +lpsruwlqj, frxqwulhv1 Wklv frpsrqhqw zloo glvdsshdu li wkhuh duh frqvwdqw uhwxuqv wr vfdoh lq wkh iruhljq frxqwu| +w^s ' f,/ ehfdxvh wkh lqwhuqdwlrqdo uhqwdo udwh ri fdslwdo zloo wkhq eh {hg1

Lq rughu wr vhh krz wkh rswlpdo qxpehu ri iuhh hplvvlrq txrwdv ydulhv zlwk wkh pdujlqdo frvwv ri sxeolf ixqgv/ zh qhhg wr glhuhqwldwh Ht1 +57, zlwk uhvshfw wr b Wkdw lv d frpsoh{ wdvn dqg lv olnho| wr |lhog xqdpeljxrxv uhvxowv/ vlqfh d fkdqjh lq b grhv qrw rqo| dhfw wkh rswlpdo hplvvlrq olplw .7 dqg wkh htxloleulxp fdslwdo doorfdwlrq &/ exw dovr doo 8_& Zh surfhhg e|

glvfxvvlqj wkh whupv lq Ht1 +57, rqh dw d wlph1 Zh vwduw e| vlqjolqj rxw wkh vfdo frpsrqhqw/ e| dvvxplqj wkdw wkhuh duh qr hqylurqphqwdo sureohpv + ' f, dqg wkdw wkhuh duh frqvwdqw uhwxuqv wr vfdoh lq wkh iruhljq frxqwu|

+w^s ' f,1 Wkh odwwhu dvvxpswlrq lv whfkqlfdoo| htxlydohqw wr wkh vpdoo rshq 43

hfrqrp| dvvxpswlrq1 Wkhuhdiwhu/ zh vwxg| wkh hqylurqphqwdo frpsrqhqw/

e| dvvxplqj wkdw wkh pdujlqdo frvw ri sxeolf ixqgv lv }hur +b ' f,/ zkloh uhwdlqlqj wkh vpdoo rshq hfrqrp| iudphzrun1 Ilqdoo|/ zh frqvlghu d odujh hfrqrp| prgho zlwk hqgrjhqrxv uhqwdo udwh ri fdslwdo1

714 Vpdoo rshq hfrqrp| 0qr hqylurqphqwdo sureohpv

Frqvlghu uvw wkh fdvh zkhq wkh krph frxqwu| lv d vpdoo rqh/ xqdeoh wr lq xhqfh wkh lqwhuqdwlrqdo uhqwdo udwh ri fdslwdo1 Wklv fdvh fdq eh dqdo|vhg e| uhsodflqj wkh htxloleulxp frqglwlrq +8, zlwk d jlyhq lqwhuqdwlrqdo lqwhuhvw udwh-^WDowhuqdwlyho|/ zh fdq dvvxph wkdw wkh iruhljq surgxfwlrq whfkqrorj|

h{klelwv frqvwdqw uhwxuqv wr vfdoh1 Zh iroorz wkh odwwhu dssurdfk vlqfh zh wkhq gr qrw qhhg wr uhgr wkh prgho> lw vx!fhv wr uhsodfh w^s e| }hur1 Zh dovr gr dzd| zlwk wkh hqylurqphqwdo dujxphqwv iru iuhh hplvvlrq txrwdv lq rughu wr lvrodwh wkh dujxphqwv uhodwhg wr sxeolf qdqfh1 E| vhwwlqj ' f/

wkh rswlpdo srolf| fdq eh zulwwhq dv

R ' b

 nb

8_ee.7n8_&e&

c +59,

^ ' 8_&e.7n&8_&&

8_ee.7n8_&e& +5:,

Sursrvlwlrq 5 Lq d vpdoo rshq hfrqrp| zlwkrxw hqylurqphqwdo sureohpv/

wkh rswlpdo qxpehu ri iuhh hplvvlrq shuplwv shu xqlw ri fdslwdo pd| eh hlwkhu qhjdwlyh ru srvlwlyh dqg pd| lqfuhdvh zlwk wkh pdujlqdo frvwv ri sxeolf ixqgv1 Surri1 Dv fdq eh vhhq lpphgldwho| iurp Ht1 +5:,/^zloo dozd|v eh qhjd0 wlyh zkhq8_&e f1 Lq rughu wr suryh wkdw^pd| eh srvlwlyh dqg lqfuhdvlqj lq b/ zh qhhg wr wxuq wr qxphulfdo vlpxodwlrqv1 Ohw wkh surgxfwlrq whfkqrorj|

wdnh wkh iroorzlqj irup

8E&c e 'q_&q₂E&²qE&n_e₂Ee²nB&ec 'c sc +5;, dqg ohw wkh sdudphwhu ydoxhv eh dv iroorzv

q_ q₂ q _{ 2} B

' 81<8 13348 14 h0: 833 43 157 's 716:8 13348958 3 833 43 158

Ohw g ' ffff dqg ` ' Df Wkh rswlpdo srolflhv dqg wkh htxloleulxp ydoxhv ri vhohfwhg yduldeohv duh vkrzq lq wkh wdeoh ehorz

44

b ' f b'D b'

^ q1d1 133:9; 133:;:

R 3 841< ;717

& 8333 7777 7448

. ;518 :615 9:1:

e^s ;8 <41< <914

- 43 43 43

Wklv suryhv wkdw ^ pd| eh srvlwlyh dqg lqfuhdvlqj lq b

Wkh prgho zlwkrxw hqylurqphqwdo sureohpv dqg zlwkrxw whupv ri wudgh hhfwv lq wkh fdslwdo pdunhw lv lqghhg d sxuh prgho ri rswlpdo wd{dwlrq zlwk lqwhuqdwlrqdoo| preloh fdslwdo1 Rxu uhvxowv fdq eh lqwhusuhwhg wr vd| wkdw wkh rswlpdo wd{ udwh rq fdslwdo pd| eh hlwkhu srvlwlyh ru qhjdwlyh +l1h1 d fdslwdo vxevlg|,/ dqg wkdw wkh rswlpdo fdslwdo vxevlg| pd| lqfuhdvh zlwk wkh pdu0 jlqdo frvwv ri sxeolf ixqgv1 Wkh uhdvrq lv wkdw fdslwdo vxevlglhv pd| dwwudfw fdslwdo iurp deurdg dqg wkxv lqfuhdvh wkh wd{ edvh dqg wkh jryhuqphqw uhy0 hqxh1 Ri frxuvh/ dv orqj dv fdslwdo lv vxevlglvhg/ wkh gluhfw lpsdfw ri d odujhu grphvwlf fdslwdo vwrfn zloo eh wr uhgxfh jryhuqphqw lqfrph1 Exw lw pd| kdyh dq lqgluhfw srvlwlyh lpsdfw rq uhyhqxhv wkurxjk lqfuhdvhg ghpdqg iru rwkhu surgxfwlrq idfwruv wkdw duh vxemhfw wr srvlwlyh wd{ udwhv1 Lq rughu iru wklv wr kdsshq/ wkh surgxfwlrq idfwruv qhhg wr eh frpsohphqwv/ lq wkh vhqvh wkdw wkh pdujlqdo surgxfw ri rqh idfwru lqfuhdvhv zlwk wkh xvh ri wkh rwkhu idfwru +8_&e : f,1 Li idfwruv duh qrq0frpsohphqwv +8_&e f,/ wkhvh lqgluhfw hhfwv duh qrq0srvlwlyh/ dqg d srvlwlyh wd{ udwh rq fdslwdo lv wkhq rswlpdo1

Lq rxu h{dpsoh/ lw lv rswlpdo wr uhvwulfw wkh xvh ri irvvlo ixhov zkhqhyhu b : f1 Wklv lv htxlydohqw wr lpsrvlqj d srvlwlyh wd{ udwh rq wklv idfwru1 Lq dgglwlrq/ d fdslwdo vxevlg| lv lpsohphqwhg lq rughu wr uhgxfh wkh fdslwdo ljkw wr wkh iruhljq frxqwu|1 Vlqfh fdslwdo dqg irvvlo ixhov duh frpsohphqwv/

d odujhu fdslwdo vwrfn lq wkh krph frxqwu| lqfuhdvhv wkh ghpdqg iru irvvlo ixhov1 Lq idfw/ wkh frpsohphqwdulw| lv vr vwurqj wkdw kljkhu lqfrph rq txrwd vdoh pruh wkdq rxwzhljkv wkh orvv ri uhyhqxh lpsolhg e| vrph txrwdv ehlqj jlyhq dzd| iuhh ri fkdujh1

Rxu dqdo|vlv lv uhodwhg wr Ud}lq dqg Vdgnd +4<<5,/ zkr vkrz wkdw li iruhljq lqfrph fdqqrw eh wd{hg/ lw lv rswlpdo wr lqwurgxfh fdslwdo frqwurov lq rughu wr hqkdqfh wkh grphvwlf wd{ edvh1 E| frpelqlqj vxfk frqwurov +h1j1/

dq h{sruw wd{ rq fdslwdo, zlwk d srvlwlyh wd{ udwh rq fdslwdo lq wkh krph frxqwu|/ wkh uvw ehvw pd| eh dwwdlqhg1 Lq rxu prgho/ wkhuh lv qr h{sruw wd{

rq fdslwdo1 Zh uxohg rxw vxfk lqvwuxphqwv e| dvvxpswlrq gxh wr rxu irfxv rq hqylurqphqwdo ohdndjh wkurxjk fdslwdo pdunhwv/ zklfk lv dvvrfldwhg qrw zlwk fdslwdo pryhphqwv lq jhqhudo exw zlwk fdslwdo pryhphqwv lqwr sduwlfxodu srooxwlqj lqgxvwulhv lq wkh iruhljq frxqwu|1 Qrwh/ krzhyhu/ wkdw dq h{sruw

45

wd{ fdq eh ghfrpsrvhg lqwr d vxevlg| rq grphvwlf xvh dqg d wd{ rq grphvwlf vxsso|1 Wkxv/ lw lv qrw fohdu/ zlwklq wkh iudphzrun xvhg e| Ud}lq dqg Vdgnd/

zkhwkhu ru qrw wkh rswlpdo qhw wd{ rq grphvwlf fdslwdo xvh lv srvlwlyh ru qhjdwlyh1 Lq rxu prgho/ zh vkrz wkdw wkh qhw wd{ pd| lqghhg e| qhjdwlyh/

lpso|lqj wkdw grphvwlf fdslwdo xvh vkrxog eh vxevlglvhg1 Wkh suhfrqglwlrq lv/ krzhyhu/ wkdw d fdslwdo vxevlg| sod|v d vlplodu uroh dv fdslwdo frqwurov lq wkh prgho ri Ud}lq dqg Vdgnd +4<<5,/ l1h1/ wkdw lw lqfuhdvhv wkh grphvwlf wd{

edvh +e| lqfuhdvlqj wkh ghpdqg iru frpsohphqwdu| surgxfwlrq idfwruv wkdw duh vxemhfw wr srvlwlyh wd{ udwhv,1

715 Vpdoo rshq hfrqrp| 0zlwk hqylurqphqwdo sure0 ohpv

Zh qrz lqwurgxfh wudqverxqgdu| hqylurqphqwdo sureohpv lqwr wkh prgho1 Zh uhwdlq wkh vpdoo rshq hfrqrp| iudphzrun/ lpso|lqj wkdw wkh lqwhuqdwlrqdo uhqwdo udwh ri fdslwdo lv wdnhq dv jlyhq1 Wkh rswlpdo srolflhv duh qrz jlyhq dv

R' 

>E nb

' b

 nbE8_ee.7n8_&e& n 

 nb

c +5<,

^'>

# 8_&e^s

8ee^s

$

n E>

8_&e.7n&8_&&

8_ee.7n8_&e&

c +63,

Iluvw/ zh qrwlfh wkdw iru dq| +.c &7 ,/ wkh txrwd sulfh R zloo eh kljkhu wkdq R +vhh Ht1 +59,,1 Wklv vkrzv wkdw hqylurqphqwdo sureohpv dozd|v pdnh lw ghvludeoh wr lqfuhdvh wkh sulfh rq srooxwlqj idfwruv/ hyhq wkrxjk fdslwdo lv lqwhuqdwlrqdoo| preloh dqg srooxwlrq lv wudqverxqgdu|1 Wkh uhdvrq lv wkdw ohdndjh lv pruh h!flhqwo| ghdow zlwk e| lvvxlqj iuhh hplvvlrq txrwdv wr grphvwlf upv1

Wkh rswlpdo qxpehu ri iuhh hplvvlrq txrwdv shu xqlw ri fdslwdo lv qrz d zhljkwhg vxp ri wkh wudqverxqgdu| srooxwlrq frpsrqhqw dqg wkh vfdo frpsrqhqw glvfxvvhg deryh1 Iru dq| +.c &7 , wkh rswlpdo qxpehu ri iuhh hplvvlrq shuplwv pd| qrz eh kljkhu ru orzhu wkdq lq wkh suhfhhglqj vhfwlrq1 Wkh fulwlfdo idfwru lv zkhwkhu ru qrw wkh wudqverxqgdu| srooxwlrq whupv duh juhdwhu ru vpdoohu wkdq wkh vfdo whup1

Zh duh qrz deoh wr vwuhqjwkhq wkh uhvxow rewdlqhg deryh1

Sursrvlwlrq 6 Lq d vpdoo rshq hfrqrp|/ wkh rswlpdo qxpehu ri iuhh hplv0 vlrq shuplwv shu xqlw ri fdslwdo pd| eh hlwkhu qhjdwlyh ru srvlwlyh dqg pd|

lqfuhdvh zlwk wkh pdujlqdo frvwv ri sxeolf ixqgv1 46

Surri1 Dv fdq eh vhhq lpphgldwho| iurp Ht1 +6 3,/ ^zloo eh qhjdwlyh zkhq 8_&e f dqg > lv vx!flhqwo| vpdoo1 Lq rughu wr suryh wkdw ^pd| eh lqfuhdvlqj lq b/ zh uvw suryh wkdw wkh vfdo whup dqg wkh wudqverxqgdu|

hqylurqphqwdo whup pd| eh lqghshqghqw frqvwdqwv1 Dv qrwhg hduolhu/ wkh wudqverxqgdu| hqylurqphqwdo whup lv dozd|v d frqvwdqw zkhq wkhuh duh frq0 vwdqw uhwxuqv lq wkh iruhljq frxqwu| dqg iruhljq idfwru sulfhv duh jlyhq1 Lq rughu wr suryh wkdw wkh vfdo frpsrqhqw pd| dovr eh d frqvwdqw/ ohw wkh surgxfwlrq ixqfwlrq lq wkh krph frxqwu| wdnh rq d txdgudwlf irup>

+'q_&q₂E&²n_e₂Ee²nB&e +64, Zh fdq zulwh pdujlqdo surgxfwv dv iroorzv>

8_& ' q_n8_&&&n8_&e.c7 +65,

8_e ' _n8_ee.7n8_&e& +66,

E| xwlolvlqj wkh uvw rughu frqglwlrq iru surw pd{lpl}dwlrq/ Htv1 +6,0+7,/

wrjhwkhu zlwk Htv1 +65, dqg +66,/ zh rewdlq wkh iroorzlqj lpsolflw h{suhvvlrqv iru wkh rswlpdo srolf|

R' 

 nb

b

 nbE` nR_c +67,

^'>ER

# 8_&e^s

8ee^s

$

n E>ER

-^WR^q_

` nR_

c +68,

zkhuh >ER ' *EbE` nR_ E| vroylqj iru ^ dqgR/ zh rewdlq

R ' 

 n 2b

nbE_`

c +69,

^ ' >

# 8_&e^s

8ee^s

$

n E>

-^Wq_{ }`

c +6:,

zkhuh > ' *E nbE_ ` Vlqfh -^W lv {hg dv orqj dv wkhuh duh frqvwdqw uhwxuqv lq wkh iruhljq frxqwu|/ wklv vkrzv wkdw wkh vfdo whup lv d frqvwdqw lq wklv fdvh1 Lq rughu wr vkrz wkdw ^pd| eh lqfuhdvlqj lq bc qrwh wkdw li dqg duh frqvwdqwv/ > lv prqrwrqlfdoo| lqfuhdvlqj lq b Khqfh/ ^ zloo eh lqfuhdvlqj lq b dv orqj dv wkh ydoxh ri wkh vfdo whup lv odujhu wkdq wkh ydoxh ri wkh wudqverxqgdu| hqylurqphqwdo whup1

Wklv ghprqvwudwhv wkdw zkhq zh lqwurgxfh hqylurqphqwdo sureohpv lqwr wkh prgho/ wkh rswlpdo qxpehu ri iuhh hplvvlrq txrwdv shu xqlw ri fdslwdo pd|

47

lqfuhdvh iru rwkhu uhdvrqv wkdq lq wkh prgho zlwk sxuh vfdo frqvlghudwlrqv1 Lq wkh odwwhu fdvh/ zh vkrzhg wkdw ^pd| lqfuhdvh zlwkb vlpso| ehfdxvh wkh vfdo whup pd| lqfuhdvh zlwk b Wkdw pd| ri frxuvh vwloo eh wkh fdvh zkhq hqylurqphqwdo sureohpv duh lqwurgxfhg1 Exw lq dgglwlrq frph wzr rwkhu idfwruv= +4, gxh wr kljkhu zhljkw rq wkh vfdo whup dv b lqfuhdvhv/ ^ pd|

lqfuhdvh li wkh ydoxh ri wkh vfdo whup lv odujhu wkdq wkh hqylurqphqwdo whup/

dqg +5, wkh hqylurqphqwdo whup lq lwvhoi pd| lqfuhdvh zlwk b Lw lv wkh uvw ri wkhvh idfwruv wkdw lv xwlolvhg lq wkh surri deryh1 Wkh odwwhu rqh zloo eh glvfxvvhg pruh ixoo| lq Vhfwlrq 81

Zh qrwlfh wkdw R pd| dovr lqfuhdvh zlwk bc lpso|lqj wkdw wkh rswlpdo vxevlg|R^ lv dq lqfuhdvlqj ixqfwlrq ri wkh pdujlqdo frvwv ri sxeolf ixqgv1 Lq rxu h{dpsoh/ zh kdyh wkdw

_R

_b ' 

E n 2b²

2

n E_`

+6;,

Surylghg wkdw _ ` : fc wklv h{suhvvlrq lv srvlwlyh iru orz ohyhov ri Wkh hvvhqwldo srlqw lv wkdw li d fdslwdo vxevlg| lv fdoohg iru iurp d vfdo srlqw ri ylhz/ d kljkhu pdujlqdo frvw ri sxeolf ixqgv zloo ohdg wr dq lqfuhdvh lq wkh rswlpdo txrwd sulfh1 Wkh lqwurgxfwlrq ri wudqverxqgdu| hqylurqphqwdo sureohpv lqwr wkh prgho zloo zrun lq wkh rssrvlwh gluhfwlrq lq rxu h{dpsoh/

exw iru orz ohyhov ri wklv hhfw lv qrw vwurqj hqrxjk wr rxwzhljk wkh vfdo dujxphqw iru lqfuhdvlqj wkh ohyho ri R

716 Odujh hfrqrp| 0qr hqylurqphqwdo sureohpv

Zh qrz zdqw wr lqyhvwljdwh krz whupv ri wudgh hhfwv lq wkh lqwhuqdwlrqdo fdslwdo pdunhw pd| dhfw wkh rswlpdo qxpehu ri iuhh hplvvlrq txrwdv1 Lq rughu wr irfxv rq rqh hhfw dw d wlph/ zh qrz ljqruh wkh hqylurqphqwdo dujxphqw iru wkh xvh ri iuhh hplvvlrq txrwdv/ e| dvvxplqj wkdw lv }hur1 Wkh rswlpdo srolf| lv wkhq ghqhg e| wkh iroorzlqj frqglwlrqv

R' b

 nbE8_ee.7n8_&e&c +6<,

^ '  b

w^s 8ee^s

kg7 &

8_ee.7n8_&e& 8_&e.7n

w^s

8ee^s n8_&&

&

8_ee.7n8_&e& +73, Wkh irupxod ghvfulelqj wkh rswlpdo txrwd sulfh lv wkh vdph dv lq wkh vpdoo hfrqrp| prgho zlwkrxw hqylurqphqwdo sureohpv1 Wkxv/ wkh odujh hfrqrp|

48

dvvxpswlrq grhv qrw dhfw wkh rswlpdo wd{dwlrq ri wkh srooxwlqj idfwru1 Wkh odujh hfrqrp| dvvxpswlrq grhv/ krzhyhu/ dhfw wkh rswlpdo qxpehu ri iuhh hplvvlrq shuplwv1 Zh uhfrjqlvh wkh odvw whup lq Ht1 +73, dv wkh vfdo whup/

zklfk qrz kdv d vrphzkdw glhuhqw vwuxfwxuh wkdq lq wkh vpdoo hfrqrp|

prghov1 Lq rughu wr xqghuvwdqg wkh vljqlfdqfh ri wkh dgglwlrqdo idfwru lq wkh qrplqdwru/ qrwlfh wkdw wkh vdph whup dsshduv lq wkh h{suhvvlrq iru_R*_^

lq Ht1 +43,>

_R

_^ ' R8_&e

w^s

8ee^s n8_&&

^8_&e +74,

Wkh lqwxlwlrq lv wkdw zkhq wkhuh duh ghfuhdvlqj uhwxuqv lq wkh iruhljq frxqwu|/

lpso|lqj wkdw w^s :fc iuhh hplvvlrq txrwdv zloo eh ohvv srzhuixo lq whupv ri udlvlqj wkh sulfh ri hplvvlrq shuplwv lq wkh krph frxqwu|1 Lw zloo wdnh d odujhu fdslwdo vxevlg| wr dwwudfw d jlyhq dprxqw ri iruhljq fdslwdo1 Wkh uhvxow lv wkdw wkh vfdo whup zloo eh vpdoohu lq d odujh frxqwu| prgho wkdq lq d vpdoo frxqwu| prgho/ wkxv fdoolqj iru ihzhu iuhh hplvvlrq shuplwv shu xqlw ri fdslwdo1 Wkh odujh hfrqrp| dvvxpswlrq dovr lpsolhv wkdw whupv ri wudgh hhfwv lq wkh fdslwdo pdunhw zloo sod| d uroh iru rswlpdo srolflhv1 Whup ri wudgh frqvlghudwlrqv duh uh hfwhg lq wkh uvw whup ri Ht1 +73,1 Dv orqj dv wkh txrwd sulfh lv srvlwlyh +l1h1 8_ee.7 n8_&e& fc wklv whup zloo eh srvlwlyh +qhjdwlyh, zkhq wkh krph frxqwu| lv d fdslwdo h{sruwhu +lpsruwhu,1 Wklv lv d vwdqgdug uhvxow> lw lv ghvludeoh iru d fdslwdo h{sruwlqj frxqwu| +kg : &7 , wr pdnh hruwv wr lqfuhdvh wkh uhqwdo sulfh ri fdslwdo1 Lq rxu prgho/ wklv lv dfklhyhg e| vxevlglvlqj fdslwdo xvh lq wkh krph frxqwu| +l1h1 R^ : f Frqyhuvho|/ d fdslwdo lpsruwlqj frxqwu| +kg &7 , zloo wd{ grphvwlf fdslwdo lq rughu wr uhgxfh fdslwdo ghpdqg dqg khqfh uhgxfh wkh uhqwdo sulfh ri fdslwdo1 Krz duh rswlpdo srolflhv lq wklv prgho dhfwhg e| fkdqjhv lq wkh pdujlqdo frvwv ri sxeolf ixqgvB Ht1 +73, vxjjhvwv wkdw zkhq b dssurdfkhv }hur/ wkh whupv ri wudgh whup pd| srwhqwldoo| ehfrph yhu| odujh/ hlwkhu srvlwlyh ru qhjdwlyh1 Wklv pxvw eh xqghuvwrrg lq frqqhfwlrq zlwk wkh idfw wkdw wkh rswlpdo txrwd sulfh R lv yhu| vpdoo xqghu wkhvh flufxpvwdqfhv1 Lq rughu wr eh deoh wr vxevlglvh +ru wd{, wkh xvh ri fdslwdo dw d jlyhq udwh/ wkh rswlpdo qxpehu ri iuhh hplvvlrq shuplwv shu xqlw ri fdslwdo pljkw wkhq ehfrph yhu|

odujh +ru vwurqjo| qhjdwlyh,1 Frqvlghu wkh fdvh zkhq k lv forvh wr }hur/

lpso|lqj wkdw whupv ri wudgh frqvlghudwlrqv fdoo iru d fdslwdo wd{ +^ f1 Dvvxph dovr wkdw blv forvh wr }hur1 Lq wklv fdvh/^^W zloo eh vwurqjo| qhjdwlyh1 Li zh qrz ohw b lqfuhdvh/ wkh rswlpdo qxpehu ri iuhh hplvvlrq txrwdv zloo dssurdfk wkh ohyho ri ^ wkdw lv rswlpdo iurp d sxuh vfdo shuvshfwlyh1 Lw lv txlwh reylrxv lq wklv fdvh wkdw wkh rswlpdo^ pd| lqfuhdvh zlwk wkh pdujlqdo frvwv ri sxeolf ixqgv1 Wkh rssrvlwh zloo eh wkh fdvh zkhq klv forvh wr rqh> d yhu| kljk qxpehu ri iuhh hplvvlrq shuplwv zloo wkhq eh fdoohg iru zkhqbjrhv

49

Iljxuh 4= Rswlpdo qxpehu ri iuhh hplvvlrq shuplwv shu xqlw ri fdslwdo dv d ixqfwlrq ri wkh rzqhuvkls ri fdslwdo +D, dqg wkh pdujlqdo frvwv ri sxeolf ixqgv +O,1

wr }hur1 Li b lqfuhdvhv/ wkh rswlpdo qxpehu ri iuhh hplvvlrq txrwdv pd| zhoo ghfuhdvh lq rughu wr uh hfw wkdw vfdo frqvlghudwlrqv fdoo iru d pruh prghudwh ohyho ri iuhh hplvvlrq txrwdv1

Sursrvlwlrq 7 Lq d odujh frxqwu| prgho zlwk qr hqylurqphqwdo sureohpv/

wkh rswlpdo qxpehu ri iuhh hplvvlrq txrwdv shu xqlw ri fdslwdo pd| lqfuhdvh ru ghfuhdvh zlwk wkh pdujlqdo frvwv ri sxeolf ixqgv1

Surri1 Wkh sursrvlwlrq zloo eh irupdoo| suryhg e| d qxphulfdo h{dpsoh1 Ohw doo sdudphwhuv wdnh rq wkh vdph ydoxhv dv lq wkh suhylrxv qxphulfdo h{dpsoh/ zlwk wzr h{fhswlrqv> ohw wkhuh eh d txdgudwlf whfkqrorj| lq erwk frxqwulhv +l1h1 q ' fc dqg ohw wkh sdudphwhuv ri wkh iruhljq surgxfwlrq ixqfwlrq eh lghqwlfdo wr wkh sdudphwhuv lq wkh krph frxqwu|1 Wkh rswlpdo ^c dv d ixqfwlrq ri k dqgb/ lv vkrzq lq Iljxuh 41

Wkh jxuh fohduo| vkrzv wkdw whupv ri wudgh hhfwv pd| pdnh wkh rswlpdo qxpehu ri iuhh hplvvlrq txrwdv lqfuhdvh ru ghfuhdvh zlwk wkh pdujlqdo frvwv ri sxeolf ixqgv/ ghshqglqj rq zkhwkhu wkh krph frxqwu| lv d qhw fdslwdo h{sruwhu ru qhw fdslwdo lpsruwhu1

4:

Iljxuh 5= Rswlpdo fdslwdo vxevlg| dv d ixqfwlrq ri wkh rzqhuvkls ri fdslwdo +D, dqg wkh pdujlqdo frvwv ri sxeolf ixqgv +O,

Lw lv lqwhuhvwlqj wr qrwh wkdw lq rxu h{dpsoh/ hyhq wkrxjk^ghfuhdvhv zlwk b dw kljk ydoxhv rikcwkh rswlpdo fdslwdo vxevlg| +R^, pd| vwloo eh lqfuhdvlqj lq b1 Iljxuh 5 ghslfwv wkh rswlpdoR^ dv d ixqfwlrq rib dqgkWklv uhvxow lv krzhyhu qrw urexvw wr fkdqjhv lq wkh sdudphwhu ydoxhv1

Wr vxp xs/ wklv vhfwlrq kdv xqfryhuhg vhyhudo uhdvrqv zk| wkh rswlpdo qxpehu ri iuhh hplvvlrq shuplwv shu xqlw ri fdslwdo pd| lqfuhdvh zlwk wkh pdujlqdo frvwv ri sxeolf ixqgv1 Iluvw/ wkhuh pd| eh sxuh vfdo dujxphqwv iru vxevlglvlqj fdslwdo lq d zruog zlwk lqwhuqdwlrqdo fdslwdo prelolw| dqg idfwru frpsohphqwdulw|1 Li d fdslwdo vxevlg| lv fdoohg iru dw wkh rxwvhw/ d kljkhu pdujlqdo frvwv ri sxeolf ixqgv zloo w|slfdoo| lqfuhdvh wkh rswlpdo udwh ri vxevlg|1 Iru fhuwdlq whfkqrorjlhv/ dq lqfuhdvh lq ^ zloo wkhq eh fdoohg iru1 Vhfrqgo|/ zh kdyh vkrzq wkdw li hqylurqphqwdo frqvlghudwlrqv fdoo iru ihzhu iuhh hplvvlrq shuplwv wkdq gr wkh sxuh vfdo frqvlghudwlrqv/ dq lqfuhdvh lq wkh pdujlqdo frvwv ri sxeolf ixqgv pd| lqfuhdvh wkh rswlpdo^vlpso| ehfdxvh d kljkhu ydoxh ri b sxwv juhdwhu hpskdvlv rq wkh vfdo dujxphqw1 Wklugo|/

zh kdyh vkrzq wkdw iru d odujh hfrqrp|/ whupv ri wudgh hhfwv zloo fdoo iru yhu| orz ydoxhv ri ^ zkhq wkh krph frxqwu| lv d odujh fdslwdo lpsruwhu +k orz, dqg wkh pdujlqdo frvwv ri sxeolf ixqgv duh orz +b orz,1 Dq lqfuhdvh lq b zloo wkhq lqgxfh d kljkhu ohyho ri ^ surylghg wkh vfdo dujxphqw fdoov iru d kljkhu ydoxh ri ^ wkdq wkh whupv ri wudgh dujxphqw lq lwvhoi1 Ilqdoo|/ lq doo

4;

wkh fdvhv frqvlghuhg/ zh kdyh ghprqvwudwhg wkdw qrw rqo| pd| wkh rswlpdo qxpehu ri iuhh txrwdv shu xqlw ri fdslwdo +^, lqfuhdvh zlwkb(wkh vdph pd|

krog iru wkh rswlpdo udwh ri fdslwdo vxevlg| +R^dv zhoo1

D zhdnqhvv zlwk wkh dqdo|vlv vr idu lv wkdw zh kdyh qrw glvfxvvhg wkh srvvlelolw| wkdw wkhuh pd| eh wd{ lqvwuxphqwv lq sodfh dw wkh rxwvhw zklfk ghdo zlwk wkh sxuh vfdo lvvxhv dqg wkh whupv ri wudgh dujxphqwv1 Rxu prgho lv prwlydwhg iurp wkh hqylurqphqwdo vlgh> zh dvn krz dq lqwhuqdwlrqdo hqylurqphqwdo sureohp vkrxog eh ghdow zlwk xqlodwhudoo| e| rqh frxqwu| +ru d jurxs ri frxqwulhv,/ dqg zh xvh d sduwldo htxloleulxp dssurdfk lq rughu wr irfxv rq krz hplvvlrq shuplwv vkrxog eh doorfdwhg wr srooxwlqj vhfwruv zkhuh fdslwdo lv lqwhuqdwlrqdoo| preloh1 Lq rxu prgho/ wkh rswlpdo doorfdwlrq ri iuhh hplvvlrq shuplwv uh hfwv frqvlghudwlrqv +vxfk dv wkh vfdo dujxphqw dqg wkh whupv ri wudgh dujxphqw, zklfk duh uhohydqw qrw vroho| iru wkh srooxwlqj vhfwruv exw iru wkh hfrqrp| dv d zkroh1 Lw pd| eh wr ryhuordg wkh prgho wr ohw wkhvh frqvlghudwlrqv hqwhu lqwr wkh ghflvlrqv derxw wkh rswlpdo doorfdwlrq ri hplvvlrq shuplwv1 Lq wkh qh{w vhfwlrq/ zh wkhuhiruh dvvxph wkdw wkhuh duh rwkhu wd{ srolflhv lq sodfh dw wkh rxwvhw wkdw ghdo zlwk doo lvvxhv wkdw duh qrw uhodwhg wr wkh hqylurqphqwdo glphqvlrq1

8 Sulru wd{ srolflhv

Frqvlghu d prgho zkhuh rswlpdo vfdo srolflhv duh lq sodfh ehiruh hqylurq0 phqwdo sureohpv dqg hqylurqphqwdo srolflhv duh lqwurgxfhg1 Dvvxph wkdw wkhuh lv d vfdo wd{ |& rq fdslwdo xvh dqg d vfdo wd{ |e rq wkh xvh ri irvvlo ixhov1 Lw fdq eh vkrzq wkdw wkh rswlpdo vfdo wd{ udwhv duh jlyhq dv

|^W_e ' b

 nb

8_&e&n8_eee

c +75,

|^W_& ' 

 nb w^s 8ee^s

kg&

b

 nb

8_&een

8_&& n w^s 8ee^s

&

+76, Wkh ghulydwlrq ri wkhvh h{suhvvlrqv lv qrw vkrzq khuh/ vlqfh wkh| iroorz txlwh vwudljkwiruzdugo| iurp Htv1 +56, dqg +57,1 Mxvw uhdolvh wkdw d wd{ rq wkh xvh ri irvvlo ixhov lv htxlydohqw wr d srvlwlyh sulfh rq hplvvlrq shuplwv dqg wkdw d wd{ rq wkh xvh ri fdslwdo lv htxlydohqw wr d qhjdwlyh vxevlg| +R^,1 Wkhq/ e| vhwwlqj wkh pdujlqdo frvw ri srooxwlrq htxdo wr }hur + ' f,/ wkh wd{ udwhv lq Htv1 +75, dqg +76, duh uhdglo| rewdlqhg1

Zh qrz lqwurgxfh wudqverxqgdu| hqylurqphqwdo sureohpv lqwr wkh prgho dqg vroyh iru wkh rswlpdo txrwd sulfh dqg wkh rswlpdo qxpehu ri iuhh hplv0 vlrq txrwdv xqghu wkhvh flufxpvwdqfhv1 Qrwh wkdw d srvlwlyh txrwd sulfh lq wklv fdvh zloo uhsuhvhqw dq h{wud wd{ rq hplvvlrq eh|rqg dq| vfdo wd{hv1

4<

Wkurxjkrxw wkh dqdo|vlv zh dvvxph wkdw wkh vfdo wd{hv|_edqg|_& duh dozd|v vhw dw wkhlu rswlpdo ohyhov1

Lq wklv fdvh/ wkh rswlpdo hqylurqphqwdo srolflhv duh dv iroorzv

R^W ' 

 nb

c +77,

^^W ' 8_&e^s

8ee^s +78,

Ht1 +77, iroorzv gluhfwo| iurp Htv1 +56, dqg +75,1 Ht1 +78, fdq eh rewdlqhg e| ghgxfwlqj wkh fdslwdo wd{|^W_&iurp wkh rswlpdo fdslwdo wd{ +R^W^^W, jlyhq e| Htv1 +56, dqg +57, dqg wkhq glylglqj e| wkh rswlpdo txrwd sulfh lq Ht1 +77,1

Krz zloo wkh rswlpdo qxpehu ri iuhh hplvvlrq txrwdv ydu| zlwk wkh pdu0 jlqdo frvwv ri sxeolf ixqgv lq wklv prghoB Dv fdq eh vhhq iurp Ht1 +78,/ ^^W lv d frqvwdqw dv orqj dv 8^s duh frqvwdqwv/ dv zloo eh wkh fdvh zlwk d txdgudwlf surgxfwlrq whfkqrorj| lq wkh iruhljq frxqwu|1 Lw wxuqv rxw wkdw plqru shuwxu0 edwlrqv ri wkh txdgudwlf surgxfwlrq ixqfwlrq zloo pdnh^^Wlqfuhdvh ru ghfuhdvh zlwk wkh pdujlqdo frvwv ri sxeolf ixqgv1 W|slfdoo|/ R^W zloo qrz ghfolqh zlwk bc exw iru fhuwdlq whfkqrorjlhv wkh ghfolqh lq R^W zloo eh vpdoohu wkdq wkh lq0 fuhdvh lq ^^Wc lpso|lqj wkdw wkh rswlpdo fdslwdo vxevlg| R^W^^W lqfuhdvhv zlwk wkh pdujlqdo frvwv ri sxeolf ixqgv1

Sursrvlwlrq 8 Zkhq wkhuh duh vhsdudwh wd{ lqvwuxphqwv wkdw ghdo zlwk vfdo frqvlghudwlrq dqg whupv ri wudgh hhfwv/ wkh rswlpdo qxpehu ri iuhh hplvvlrq shuplwv shu xqlw ri fdslwdo +^, pd| lqfuhdvh zlwk wkh pdujlqdo frvwv ri sxeolf ixqgv +b1 Wkh rswlpdo fdslwdo vxevlg| +R^, dqg wkh rswlpdo wrwdo qxpehu ri iuhh hplvvlrq txrwdv +&^ pd| lqfuhdvh dv zhoo1

Surri1 Wkh uhvxow zloo eh suryhg e| d qxphulfdo h{dpsoh1 Dvvxph wkh iroorzlqj surgxfwlrq whfkqrorj|/ lghqwlfdo iru erwk frxqwulhv>

8E&c e ' 2D&ffDE&²n 2DffeHDfEe²n f&Ee²c 'c s +79, Ohw g ' ffffc ` ' Dfc k 'Dc dqg ohw ' ff dqg '  Wkh rswlpdo srolflhv dqg wkh htxloleulxp ydoxhv ri vhohfwhg yduldeohv duh vkrzq lq wkh wdeoh ehorz

53

b ' f b'

^ 133433 13344<

R 433 <31<4

|& 13339: 4176

|e 3 085179

& 8333167:37 .7 6169 614;

e^s 6183 61;5

- 4415 4319

Wkh lqfuhdvh lq b iurpfwr fdxvhv dbshu fhqw lqfuhdvh lq ^^W/ zklfk lv d odujhu lqfuhdvh wkdq wkh fruuhvsrqglqj uhgxfwlrq lq R^W dqg&^W

Zkdw h{sodlqv wkhvh uhvxowvB Zkhq wkh pdujlqdo frvwv ri sxeolf ixqgv lqfuhdvh/ vfdo wd{hv zloo eh dgmxvwhg lq rughu wr jhqhudwh pruh sxeolf uhy0 hqxh1 Lq rxu h{dpsoh/ wkh rswlpdo wd{ vwuxfwxuh hqwdlov d srvlwlyh fdslwdo wd{

+|&:fdqg d vxevlg| rq wkh xvh ri irvvlo ixhov +|e f⁸ Wkh sxusrvh ri wklv vxevlg| lv wr hqkdqfh wkh wd{ edvh> d vxevlg| rq irvvlo ixhov zloo lqfuhdvh wkh ghpdqg iru irvvlo ixhov/ dqg pruh iruhljq fdslwdo zloo wkhq eh dwwudfwhg wr wkh krph frxqwu| gxh wrw kh frpsohphqwdulw| ehwzhhq fdslwdo dqg irvvlo ixhov1 Lq rxu h{dpsoh/ wkh orz ohyho ri.7 lpsolhv wkdw vxfk d vxevlg| lv d uhodwlyho|

lqh{shqvlyh phdvxuh iru dwwudfwlqj iruhljq fdslwdo +vhh Ht1 +75,,1

Zkhq lw frphv wr wkh lvvxh ri wudqverxqgdu| srooxwlrq/ lw lv lqwhuhvwlqj wr qrwh wkdw wkh pdujlqdo frvwv ri sxeolf ixqgv gr qrw kdyh d gluhfw lpsdfw rq wkh rswlpdo qxpehu ri iuhh hplvvlrq shuplwv shu xqlw ri fdslwdo1 Wkh ohyho ri

^ lv ghwhuplqhg e| wkh hhfwlyhqhvv ri fdslwdo uhorfdwlrqv lq whupv ri uhgxflqj iruhljq srooxwlrq +l1h1/ _e^s*_&^s1 Lq rxu h{dpsoh/ zh kdyh

^^W '8_&e^s

8ee^s ' _e^s

_&^s ' 2e^s

.ff n2&^s +7:, Dq lqfuhdvh lq wkh pdujlqdo frvwv ri sxeolf ixqgv zloo lqfuhdvh wkh iruhljq fdslwdo vwrfn/ gxh wr kljkhu fdslwdo wd{hv lq wkh krph frxqwu|1 Wklv zloo dovr ohdg wr d kljkhu ohyho ri iruhljq hplvvlrqv1 Jlyhq rxu whfkqrorjlfdo dvvxpswlrq/ erwk wkhvh hhfwv zloo lqfuhdvh wkh ydoxh ri _e^s*_&^s Lq rwkhu zrugv> wkh hhfwlyhqhvv ri iuhh hplvvlrq txrwdv zloo eh hqkdqfhg zkhq wkh pdujlqdo frvwv ri sxeolf ixqgv lqfuhdvhv1 Khqfh/ ^^W lqfuhdvhv1

Rq wkh rwkhu kdqg/ dq lqfuhdvh lq wkh pdujlqdo frvwv ri sxeolf ixqgv pdnhv lw pruh frvwo| wr vxevlglvh fdslwdo lq rughu wr uhgxfh wkh ohdndjh hhfw1 Wklv

8Qrwh wkdwwn dovr wdnhv fduh ri wkh whupv ri wudgh dujxphqw1 Vlqfh wkh krph frxqwu|

lv d fdslwdo h{sruwhu zkhq n^kA8333> wkh whupv ri wudgh hhfw lpsolhv d voljkwo| srvlwlyh wn zkhq@ 3dqg d qhjdwlyh wn zkhq@=4=

54

lv uh hfwhg lq rxu h{dpsoh e| d uhgxfwlrq lq R^Wc zklfk fhwhulv sdulexv zloo uhgxfh wkh ohyho ri wkh fdslwdo vxevlg|1 Ri frxuvh/ wkh sulfh ri wkrvh txrwdv wkdw duh dxfwlrqhg zloo wkhq dovr ghfolqh/ dqg rqh pljkw wkhuhiruh dvn zk|

vxfk d srolf| lv ghvludeoh zkhq wkh pdujlqdo frvwv ri sxeolf ixqgv lqfuhdvhv1 Lq rughu wr xqghuvwdqg wklv uhvxow/ zh vkrxog ehdu lq plqg wkdw vlqfh wkhuh duh rwkhu wd{ lqvwuxphqwv lq sodfh wkdw ghdo zlwk wkh sxeolf uhyhqxh lvvxh lq d frvw h!flhqw zd|/ d srvlwlyh ydoxh riRzloo dozd|v lpso| dq h!flhqf| orvv +zkhq zh ljqruh wkh hqylurqphqwdo ehqhwv,1 Zkhq wkh pdujlqdo frvwv ri sxeolf ixqgv lqfuhdvh/ juhdwhu hpskdvlv lv sxw rq sxeolf uhyhqxh frqvlghudwlrqv uhodwlyh wr wkh wudqverxqgdu| hqylurqphqwdo dujxphqw1 Wklv fdoov iru d pryhphqw wrzdugv wkh vroxwlrq wkdw zrxog eh rswlpdo lq wkh devhqfh ri hqylurqphqwdo sureohpv1 Khqfh/ lw lv rswlpdo wr uhgxfh R zkhq b lqfuhdvhv1

Lq rxu h{dpsoh/ wkh lqfuhdvh lq^^Wlv odujhu wkdq wkh uhgxfwlrq lqR^W/ lpso|0 lqj wkdw wkh rswlpdo fdslwdo vxevlg| lqfuhdvhv dv wkh pdujlqdo frvwv ri sxeolf ixqgv lqfuhdvh1 Krzhyhu/ wklv uhvxow fohduo| ghshqgv rq sdudphwhu ydoxhv ri wkh prgho dqg frxog hdvlo| eh uhyhuvhg iru rwkhu whfkqrorjlfdo dvvxpswlrqv1 Wkh lqfuhdvh lq pdujlqdo frvwv ri sxeolf ixqgv ohdgv wr dq dgglwlrqdo dujxphqw iru nhhslqj wkh fdslwdo vwrfn lq wkh krph frxqwu| dw d kljk ohyho/

ehfdxvh wklv zloo lqfuhdvh wkh wd{ edvh1 Wklv dujxphqw lv kdqgohg lq rxu prgho e| wkh vxevlg| rq wkh xvh ri irvvlo ixhov lq wkh krph frxqwu| dqg lpsolhv wkdw wkh ghfolqh lq wkh fdslwdo vwrfn zloo qrw eh dv odujh dv lw rwkhuzlvh zrxog eh1 Wkxv/ zlwk d uhodwlyho| vpdoo ghfolqh lq wkh fdslwdo vwrfn/ dq lqfuhdvh lq wkh pdujlqdo frvwv ri sxeolf ixqgv lv pruh olnho| wr lqfuhdvh wkh wrwdo qxpehu ri iuhh hplvvlrq shuplwv &^1

9 Ilqdo uhpdunv

Wklv sdshu vkhgv oljkw rq krz wkh suhvhqfh ri d sxeolf vhfwru wkdw lpsrvhv glvwruwlrqdu| wd{hv dhfwv wkh ghvludelolw| ri xvlqj iuhh hplvvlrq txrwdv dv d phdqv wr uhgxflqj ohdndjh hhfwv zkhq wkhuh duh wudqverxqgdu| hqylurq0 phqwdo sureohpv1

Zh qg wkdw zkhq wkh doorfdwlrq ri iuhh hplvvlrq txrwdv lv lpsohphqwhg lq d zd| wkdw uhgxfhv wkh ohdndjh hhfw +dqg qrw e| judqgidwkhulqj ri hplvvlrq txrwdv,/ wkhuh lv dq dpeljxrxv uhodwlrqvkls ehwzhhq wkh pdujlqdo frvwv ri sxeolf ixqgv dqg wkh rswlpdo qxpehu ri iuhh hplvvlrq txrwdv1 Lq rxu prgho/

zkhuh ohdndjh rffxuv wkurxjk lqwhuqdwlrqdo fdslwdo pryhphqwv/ h!flhqf| uh0 txluhv wkdw iuhh hplvvlrq txrwdv duh doorfdwhg rq wkh edvlv ri wkh ohyho ri fdslwdo xvh lq wkh krph frxqwu|1 Lw lv vkrzq wkdw wkh rswlpdo qxpehu ri iuhh hplvvlrq txrwdv shu xqlw ri fdslwdo pd| lqfuhdvh zlwk wkh pdujlqdo frvwv ri sxeolf ixqgv iru vhyhudo uhdvrqv> +4, ehfdxvh kljkhu fdslwdo vxevlglhv pd|

55

eh fdoohg iru lq rughu wr dwwudfw pruh fdslwdo dqg khqfh lqfuhdvh wkh xvh ri frpsohphqwdu| idfwruv wkdw duh vxemhfw wr d srvlwlyh wd{ udwh/ +5, ehfdxvh vfdo dujxphqwv ri wklv nlqg zloo ehfrph pruh lpsruwdqw uhodwlyh wr hqyl0 urqphqwdo dujxphqwv/ +6, ehfdxvh vfdo dujxphqwv ri wklv nlqg zloo ehfrph pruh lpsruwdqw uhodwlyh wr whupv ri wudgh dujxphqwv/ dqg +7, ehfdxvh wkh hhfwlyhqhvv ri iuhh hplvvlrq txrdwv dv dq lqvwuxphqw ri uhgxflqj ohdndjh iurp deurdg pd| lqfuhdvh lq uhvsrqvh wr fkdqjhv lq wkh idfwru xvdjh lq wkh iruhljq frxqwu|1

Krzhyhu/ rqo| wkh odwwhu ri wkhvh phfkdqlvpv vxuylyh li wkhuh duh vhsdudwh wd{ lqvwuxphqwv lq sodfh wkdw ghdo zlwk vfdo lvvxhv dqg whupv ri wudgh hhfwv1 Lq vxp/ wkh sdshu vkrzv wkdw sxeolf uhyhqxh frqvlghudwlrqv gr qrw qhf0 hvvdulo| uhgxfh wkh dwwudfwlyhqhvv ri xvlqj iuhh hplvvlrq txrwdv dv d phdqv ri ghdolqj zlwk hqylurqphqwdo ohdndjh hhfwv1

Uhihuhqfhv

^4` Jrxoghu/ O1 K1/ L1 Z1 K1 Sduu|/ dqg G1 Exuwudz +4<<:, Uhyhqxh0udlvlqj yhuvxv rwkhu dssurdfkhv wr hqylurqphqwdo surwhfwlrq= Wkh fulwlfdo vljqli0 lfdqfh ri suhh{lvwlqj wd{ glvwruwlrqv/ UDQG Mrxuqdo ri Hfrqrplfv 5;/

:3;0:641

^5` Jrxoghu/ O1 K1/ L1 Z1 K1 Sduu|/ U1 F1 Zlooldpv LLL/ dqg G1 Exuwudz +4<<<, Wkh Frvw0Hhfwlyhqhvv ri Dowhuqdwlyh Lqvwuxphqwv iru Hqylurq0 phqwdo Surwhfwlrq lq d Vhfrqg0Ehvw Vhwwlqj/ Mrxuqdo ri Sxeolf Hfr0 qrplfv :5/ 65<06931

^6` Krho/ P1 +4<<9, Vkrxog d Fduerq Wd{ eh Glhuhqwldwhg dfurvv VhfwruvB Mrxuqdo ri Sxeolf Hfrqrplfv 8</ 4:0651

^7` Pévwdg/ R1 +4<<;, Rq wkh h!flhqf| ri juhhq wudgh srolf|/Hqylurqphq0 wdo dqg Uhvrxufh Hfrqrplfv 44/ 404;1

^8` Pévwdg/ R1 +5334, H!flhqw Folpdwh Srolf| zlwk Lqwhuqdwlrqdoo| Preloh Ilupv/Hqylurqphqwdo dqg Uhvrxufh Hfrqrplfv 4</ 59:05;71

^9` Rdwhv/ Z1 H1 dqg U1 P1 Vfkzde +4<;;, Hfrqrplf Frpshwlwlrq Dprqj Mxulvglfwlrqv= H!flhqf| Hqkdqflqj ru Glvwruwlrq LqgxflqjB Mrxuqdo ri Sxeolf Hfrqrplfv 68/ 66606871

^:` Sduu|/ L1 Z1 K1 dqg U1 F1 Zlooldpv LLL +4<<<, D vhfrqg0ehvw hydoxdwlrq ri hljkw srolf| lqvwuxphqwv wr uhgxfh fduerq hplvvlrqv/ Uhvrxufh dqg Hqhuj| Hfrqrplfv 54/67:06:61

56

^;` Srwhued/ M1 P1 +4<<6, Joredo Zduplqj Srolf|= D Sxeolf Ilqdqfh Shu0 vshfwlyh/Mrxuqdo ri Hfrqrplf Shuvshfwlyhv :/ 7:0961

^<` Udxvfkhu/ P1 +4<<:, Lqwhuqdwlrqdo Wudgh/ Idfwru Pryhphqwv dqg wkh Hqylurqphqw1R{irug= Foduhqgrq Suhvv1

^43` Ud}lq/ D1 dqg H1 Vdgnd +4<<4, H!flhqw lqyhvwphqw lqfhqwlyhv lq wkh suhvhqfh ri fdslwdo ljkw/ Mrxuqdo ri Lqwhuqdwlrqdo Hfrqrplfv 64/ 4:40 4;41

57