WPC +UQۧKU >gB>dؙF4:{'ũQѳ؆Ls$;ň^YP^=Rv}wֵFlS`]fX/G&+0bp#=ΣcݪV O.H5uHwa;{Y>C;([ lJ1É7`uM?F8pE%bL4aXrRq_1ߠ#DwzǝTؔa3h 2C@y/9B#Q2gJ!f⦮ױRY[B̡cFȅWݥ9/ xE(XA|r_Zt}RxU+ ib n¥FY-_}ZXX[ZNOA]BC>D XiP ePcn6 qZPYȦEU@ % 0 C >O0 2 D+= 0O 0ER AYd=/ld&%d:dG 0N d~ %dc&=,dr ?-7d=9f:1;d<@=U|>d>Ad BhMdYOGaTd U8`dJ-bXwdddfdrg3 hd>hi-Sj@lnnn 04}oow@o4o\pdap;rXXGXX &theta~>~thetaHORZ53VERT35 XXB(XXaB4XXBaXX)B)XXL#aw,,,,}41a(aq,,>61pa,aa)w,,5P0,, 1XXbBx1XXB(XX>Bp0 1XXB,XX6Bp0 2XXB,XX. Bw 1XX B,XX( Bw 0 2XX B,XXB Bq 1XX B,XX BXX B)XX Bdw 1XXTBXXL$aw,,,,~41a(aq,,?60qa,aa)w,,6P0,,!1XXcBx1XXB(XX?Bp0 1XXB,XX7Bp0 2XXB,XX/Bw1XXB,XX)Bw0 2XXB,XXCBq0XXB,XXBXXB)XXBdw1 alignl{~~~(4a)~~~~~INTFROM{w_1^0}TO{w_1^*(q_1,)}~x_1(p_1^0,p_2^0,w_1,w_2^0,q_1,)dw_1~~INTfrom{w_1^0}to{w_1^*(q_0,)}~x_1(p_1^0,p_2^0,w_1,w_2^0,q_0,)dw_1}#~ XXo(XX4XXbXXy)XXsXXP Lw,, ,,1F(rq,,1, V)>\w,,0,,/1XX$X^XXt6,XX6!XXP6(XX6p 01XXX 6,XX 6p 0 2XXP 6,XX 6w 1XXJ 6,XX| 6w" 0 2XXd 6,XX 6q 1XX< 6,XXn 6XX 6)XX8 ,XX w 1XX:dw$U1XXXXC Lw,,,,19(eq,,0,I)1\w,,0,,/1XX$XQXXg6,XX6!XXC6(XX6p 01XXK6,XX}6p02XXC6,XXu6w1XX=6,XXo6w02XXW6,XX6q0XX/6,XXa6XX6)XX+,XXw1XX-dwU1 alignl{~(4b)~~~~~=~~INTfrom{w_1^0}to{w_1^*(q_1,)}~{PARTIALpi(p_1^0,p_2^0,w_1,w_2^0,q_1,)}over{PARTIALw_1}dw_1~~INTfrom{w_1^0}to{w_1^*(q_0,)}~{PARTIALpi(p_1^0,p_2^0,w_1,w_2^0,q_0,)}over{PARTIALw_1}dw_1} XX'(XXi6XXaXX1)XXLp,,'0,,1+Vp,,,,o)1V(Vq,,0+1bV,VV)XXTyO1XX(XX0pO1XX,XX p 0l [2XX ,XX w 0 [1XX ,XX w 0 [2XX ,XX4 q O1XX ,XX XX )XX dpO1XX$XXLp,,0,,1Vp,,M,,8)1V(Vq,,+0+V,MVV)XXyuO1XX(XXp]O1XX,XXpU05[2XX,XXwo0O[1XX,XXw0i[2XX,XXqaO0XX,XXXXI)XXdpSO1XXXX Lp,,0,,1\p,,`,,K/1\(\q,, 11>\,`\\)$X<XXR6,XX6!XX.6(XXp6p1XX 6,XXH 6p 0 2XX 6,XX@ 6w 0 1XX( 6,XXZ 6w 0 2XXB 6,XXt 6q 1XX 6,XXL 6XX 6)XX' ,XX p 1XXpXX Lp,,<0,,'1@\p,,,,/1\(\q,,E10w\,\\)$XuXX6,XX6!XXg6(XX6p 1XXO6,XX6p02XXG6,XXy6w01XXa6,XX6w902XX{6,XX6q0XXS6,XX6XX6)XX`,XXp$1XXdpqU1 alignl{~~~~~~(6a)~~~~~intfrom{p_1^*(q_1,theta)}to{p_1^0}~y_1(p_1,p_2^0,w_1^0,w_2^0,q_1,theta)dp_1~~intfrom{p_1^*(q_0,theta)}to{p_1^0}~y_1(p_1,p_2^0,w_1^0,w_2^0,q_0,theta)dp_1}#~#alignl{~~~~~~~~~~~=~~intfrom{p_1^*(q_1,theta)}to{p_1^0}~{{PARTIALpi(p_1,p_2^0,w_1^0,w_2^0,q_1,theta)}over{PARTIALp_1}}~~intfrom{p_1^*(q_0,theta)}to{p_1^0}~{PARTIALpi(p_1,p_2^0,w_1^0,w_2^0,q_0,theta)}over{PARTIALp_1}~dp_1} XXb(XXb7XX{b)XXL q,,dT1 Zq,,d/0RXXX,XXs!XX,XXqXX]bdqXXbXXL ZcXX bdXX5 b! \~~~~~(7)~~~~INTfrom{q_0}to{q_1}~{PARTIALpi}over{PARTIALq}~dq~~=~~intfromc~dpi  7T!   6.0  Thisassumptionmaybemathematical"overkill"formostsituationsencounteredin  analysisofwellbehavedprofitfunctionswherethisgeneralformofGreen'sTheoremwouldnotbelikelytoberequired.ForadiscussionofseveralversionsofGreen'sTheoremastheconditionsonthepathofintegrationarerelaxedforsituationswherethestructureoftheexternalityeffectmightintroducediscontinuities,seeApostol[1957],pp.283297. XXb(XXb8XXub)XXoLZcXXbdXXOb!XX#bXXLZc,,6/0XXobdXXb!XXbXXw L~ Zc,, /1XX bdXXW b! K~~~~~(8)~~~~~INTfromcdpi~=~INTfrom{c_0}dpi~+~INTfrom{c_1}dpi XXb(XX)b9XXb)XXLZc,,/0XX[b(XXX,XXE!XX,XXw1XXbdw!1XXbX] XX ,XX !XXs ,XX wY 2XX bdw !2XX b)XXw bXXG LN Zc,, /1XXb(XiXX,XX!XX,XXwe1XXbdw!1XXAbXXXY,XX!XX3,XXw2XXqbdw[!2XXb) ~~~~~(9)~~~~~INTfrom{c_0}~({PARTIALpi}over{PARTIALw_1}dw_1~+~{PARTIALpi}over{PARTIALw_2}dw_2)~+~INTfrom{c_1}~({PARTIALpi}over{PARTIALw_1}dw_1~+~{PARTIALpi}over{PARTIALw_2}dw_2) XX(XX10XXaXX=)XXZLw,,NK,,91(q,,0,,N)w,,0,,}1 X*XX@,XX!XX(XX^P0XX: ,XXl w C1XX4 ,XXf w 0 O2XXN ,XX q C0XX& ,XXX XX )XX D,XX Dwy 1XX$ dw1XXXX-ZLxw,,K,,2#(Oq,,0,3)w,,0,,u}2%*XXX,XX9!XX(XXP0XX,XXwO1XX(XX=qC0XX,XXXX)XX,XXwC2XX,XXq[C0XX,XXXXC)XXD,XX{Dw2XXdw2XXXX L w,,y0,,d1g\w,,,,/1\(>\q,,11\,\"\)%*XXX6,XX( 6!XX 6(XX 6P 0XX 6,XX 6wz 1XX 6,XX 6w t 2XX 6(XX& 6q 1XX 6,XX 6XXr6)XX6,XX6qJ1XX6,XX6XX26)XX ,XXj w 1XXdwtU1XXXX Lw,,0,,2\w,,M,,8/2\(\q,,11+\,M\\) X)XX?6,XX6!XX6(XX]6P0XX96,XXk6w01XXS6,XX6w 2XXM6,XX6q0XX%6,XXW6XX6)XX,XXwx2XX#dw U2 alignl{~~~~~(10a)~~~~INTfrom{w_1^0}to{w_1^*(q_0,theta)}~{PARTIALpi(P^0,w_1,w_2^0,q_0,theta)}over{PARTIALw_1}dw_1~+~INTfrom{w_2^0}to{w_2^*(q_0,theta)}~{PARTIALpi(P^0,w_1^*(q_0,theta),w_2,q_0,theta)}over{PARTIALw_2}dw_2}#alignl{~~~~~~~~~~~~~~~~+~INTfrom{w_1^*(q_1,theta)}to{w_1^0}~{PARTIALpi(P^0,w_1,w_2^*(q_1,theta),q_1,theta)}over{PARTIALw_1}dw_1~+~INTfrom{w_2^*(q_1,theta)}to{w_2^0}~{PARTIALpi(P^0,w_1^0,w_2,q_0,theta)}over{PARTIALw_2}dw_2} XX(XX11XX)XX=Lc,,1XX(XXyA1XXdp1XX=XX ye 2XX dp 2XX )XX XXs =Lz c,, 0XXG (XX y 1XX{dpC1XXXXy2XXdpg2XX)XXaXXLYc,,J.1XXa(5X)XXT,XX!XX?,XXp1XX[adp# 1XXa5X XX ,XX$ !XX ,XX ps 2XX adp 2XX a)XXo aXX? LF Yc,, .0XXa(5XaXX,XX!XXw,XXp;1XXadp[ 1XXa5XXX,XX\!XX,XXGp2XXadp 2XX a) =alignl{~~~~~(11)~~~~~INTfrom{c_1}~(y_1~dp_1~+~y_2~dp_2)~~INTfrom{c_0}~(y_1~dp_1~+~y_2~dp_2)}#alignl{~~~~~~~~~~~=~~INTfrom{c_1}~({PARTIALpi}over{PARTIALp_1}dp_1~+~{PARTIALpi}over{PARTIALp_2}dp_2)~~INTfrom{c_0}~({PARTIALpi}over{PARTIALp_1}dp_1~+~{PARTIALpi}over{PARTIALp_2}dp_2)} XX(XX12XX)XX%ZLp,,wK0,,b1{p,,,,}1(<q,,1, ))XXX,XX&!XX(XXpHC1XX,XXp@  O2XX (XX q6 C1XXx ,XX XX )XX` ,XX WX 0XX ,XX q0 C1XXr ,XX XX)XX D,XXm Dp 1XXpdp81XXXXLZLEp,,K0,,2p,,,,}27(cq,,1,G)XXX,XXM!XX(XX p0oO1XX,XXpgC2XX,XXW0XX,XXqyC1XX,XXXXa)XXED,XXDp 2XXdp2XXXX L p,,c0,,N1g\p,,,,/1\((\q,,l10\,\ \))XXX6,XX 6!XX 6(XX 6p4 1XXv 6,XX 6p,  2XX| 6(XX 6q" 0XXd 6,XX 6XX 6)XXL 6,XX~ 6WD0XX6,XX6q0XX^6,XX6XX6)XX ,XXY p 1XX\dp$U1XXXX8 L1p,,0,,u2\p,,,,/2#\(O\q,,10\,\3\)XXX6,XX96!XX6(XX6p{0[1XX6,XX6pS2XX6,XX6W0XX6,XX6qe0XX6,XX6XXM6)XX1,XXp2XXdpmU2 alignl{~~~~~(12)~~~~INTfrom{p_1^*(q_1,theta)}to{p_1^0}~{PARTIALpi(p_1,p_2^*(q_1,theta),W^0,q_1,theta)}over{PARTIALp_1}dp_1~+~INTfrom{p_2^*(q_1,theta)}to{p_2^0}~{PARTIALpi(p_1^0,p_2,W^0,q_1,theta)}over{PARTIALp_2}dp_2}#alignl{~~~~~~~~~~~~~~~~~INTfrom{p_1^*(q_0,theta)}to{p_1^0}~{PARTIALpi(p_1,p_2^*(q_0,theta),W^0,q_0,theta)}over{PARTIALp_1}dp_1~~INTfrom{p_2^*(q_0,theta)}to{p_2^0}~{PARTIALpi(p_1^0,p_2,W^0,q_0,theta)}over{PARTIALp_2}dp_2} XXx(XXx14XXx)XX#xx{C1XXx(XX)xPXXx,XXxWXX{x,XXxqXXx,XXCxXXx)XXQxXX! xxy 71XX x(XX xPXXw x,XX xWXXO x,XX xqXX x,XX x!XX x(XX xPXXO x,XX xWXX'x,XXYxqXXx,XXxXXcx)XXx) Nalignl{~~~~~(14)~~~x_1^theta(P,W,q,theta)~=~x_1(P,W,q,pi(P,W,q,theta))} XX(XX15XX)X/XXE,XXxR1XXuG,XXGqXXXXX,XX)xF1XXG,XXDGqXX1X XX# ,XX x F1XX2 G,XX G!XX RX! XX7 V,XX V!XXC G,XX Gq alignl{~~~~~(15)~~~{partialx_1^theta}over{partialq}~=~{partialx_1}over{partialq}~+~{partialx_1}over{partialpi}~CDOT~{partialpi}over{partialq}} X)XX?,XXxR1XXoG,XXGqXXXX0 ){partialx_1^theta}over{partialq}~=~0 XX'(XX'16XX')RX/UXXE,XX!XXQ,XXqXX'XX_'#XUX1dXXG,XXx1XXb,XXqX1XXG,XXxF1XXVG,XXG! alignl{~~~~~(16)~~~{{partialpi}over{partialq}}~=~``{{partialx_1}over{partialq}}over{{partialx_1}over{partialpi}}} # 7T!   7.0  BockstaelandMcConnell[1993]havedescribedindetailthelimitationsintheNeill J! Larsonapproachtowelfaremeasurement.TheHicksianresponseofconsumerdemandforaprivategoodlinkedtotheenvironmentalresourceisnotobservable.Thesamecommentappliestothecaseofprofitfunctions.Theonlypotentialqualificationarisesifonearguesthatthesourceofthedistributionofquasirents(i.e.,)isobservable  % acrossfirms. 6 7T!   8.0  ThisargumentwasfirstdevelopedforenvironmentalregulationsbyHazillaandKopp ' [1990].Freeman's[1993]overviewofhowthetheoryofbenefitmeasurementshouldbeusedinthemeasurementofthevaluesofenvironmentalresourcesoutlineshowthisperspectivecanbeintegratedintomeasuringthebenefitsandcostsofresourcechangeunderuncertainty.EspinosaandSmith[1995]illustratehowitcanbegeneralizedforthecaseofevaluationsoftradeandenvironmentalpolicychanges.3|x2* `(CG TimesScalableXXx PE37XP( T$  )*+D,-.E/0E127$  R~~~~~(3)~~~~~{PARTIALpi(P,w_1^*,w_2,...,w_m,q)}over{PARTIALq}~~=~~0 XX(XX3XX{)#XXX,XX!XXs(XXPXX/,XXawR1XXW,XXwF2XXQ,XX.XX.XX.XX ,XXK w FmXX6 ,XXh qXX )XXpG,XXGqXX XX 0  7T!   .0  Inarecentoverviewofmodelsforestimatingthevalueofenvironmentalresourcesas  productionfactors,Point[1995]observedthat:"westilllackasufficientunderstandingofthevalueofthese[nonmarketnaturalresources]assetsasproductionfactors.Intheliteraturedevotedtothenaturalassetsvaluationproblem,theproducersidehasreceivedmuchlessattentionthantheconsumerone"(p.23,bracketedtermsadded).  7T!   .0  FreemanandHarrington[1990]describetheconditionsfornecessaryinputsand  outputsasfollows: (#(# 8  0`    Whatisrequiredisthattherebeeithernecessaryinputoranecessaryoutput.Anecessaryoutputisoneforwhichthereissomepositiveminimumpriceatwhichthefirmwillchoosetostopproducingnotonlythatoutputbutallotherproductsaswell;inotherwords,shutdowncompletely....Aninputisdeemedtobenecessaryifthereissomepriceforthisinputatwhichitsderiveddemandfallstozeroandifallofthefirms'outputsfalltozerowhenthisinputissetatzero."(p.900) ` `  0  Thekeyissueinmeasuringthechangeinquasirentsduetoachangeinenvironmentalqualityisthattheprofitfunctionisinvarianttoachangeinthatquality.Onewaytorealizethatoutcomeistoassumealloutputsarezero.Byfocusingonthisconditionasaformofweakcomplementarityinproduction,weseethatitwouldbepossibletofollowBradfordandHildebrandt,adoptingaweakerformofMler's[1974]originalconditionandassumethecontributionoftheenvironmentalresourceisconstantwhenthedemandfortheprivateinputwaszero.However,theyalsoassumedthattheremustexistapricevectorsuchthattheenvironmentalresourcehadzerovalueorintheproductioncasethecontributionwaszero. ){partialx_1^theta}over{partialq}~=~0 X)XX?,XXxR1XXoG,XXGqXXXX0  .   7T!   .0  Theintroductionofproductionexternalitiesintoacompetitivemodeloffirmbehavior h requiresthatweacknowledgetheprospectsforpositivequasirents.Insomeapplications,firmswithinanindustryareaffecteddifferentlybyproductionexternalities.Thecompetitivemodelwouldholdthatunlessallareaffectedequally,theentryandexitoffirmswouldultimatelycausethosewithdifferentialnegativeimpactstoexitfromtheindustry. (#(# 0  Consistentmeasuresofthewelfarelosstofirmsrequiresaframeworkthatrecognizestheselongrunadjustmentsasunlikelytoberelevanttomostapplications.Thus,thePanzarWilligapproacharguingforpositivequasirentsinanindustrypermitssomeentryandexiteffects.Theexternality'simpactisonthequasirentsoffirmsremainingintheindustry. (#(# 0  Thus,theappropriateoutputsupplyandinputdemandfunctionsareintermsofthatconstantquasirentsourcecondition.Thiscanbeindexedbythefactor(inourcase,)   assumedtogiverisetotherents.Bymeasuringthelossesalongwhatareakinto"compensated"inputdemandandoutputsupplyfunctionsweproperlymeasurethereallossesduetotheexternalities,andtakeaccountoftheconcernsraisedbyPanzarandWilligfortheuseofproductmarketanalysesforcompetitivemarketswithpositivequasirents.ArgumentsanalogoustoHausman[1981]orVrtia[1983]canbeusedtorecoverthesefunctionsfromthemanyobservableconstantprofitfunctions.  7T!   .0  Thisassumptionmaybemathematical"overkill"formostsituationsencounteredin  analysisofwellbehavedprofitfunctionswherethisgeneralformofGreen'sTheoremwouldnotbelikelytoberequired.ForadiscussionofseveralversionsofGreen'sTheoremastheconditionsonthepathofintegrationarerelaxedforsituationswherethestructureoftheexternalityeffectmightintroducediscontinuities,seeApostol[1957],pp.283297. " 7T!   .0  BockstaelandMcConnell[1993]havedescribedindetailthelimitationsintheNeill J! Larsonapproachtowelfaremeasurement.TheHicksianresponseofconsumerdemandforaprivategoodlinkedtotheenvironmentalresourceisnotobservable.Thesamecommentappliestothecaseofprofitfunctions.Theonlypotentialqualificationarisesifonearguesthatthesourceofthedistributionofquasirents(i.e.,)isobservable  % acrossfirms. 5 7T!   .0  ThisargumentwasfirstdevelopedforenvironmentalregulationsbyHazillaandKopp ' [1990].Freeman's[1993]overviewofhowthetheoryofbenefitmeasurementshouldbeusedinthemeasurementofthevaluesofenvironmentalresourcesoutlineshowthisperspectivecanbeintegratedintomeasuringthebenefitsandcostsofresourcechangeunderuncertainty.EspinosaandSmith[1995]illustratehowitcanbegeneralizedforthecaseofevaluationsoftradeandenvironmentalpolicychanges. 7T!     2Copyright19952@  WeakComplementarityandQuasiRents     ̀@  JuChinHuangandV.KerrySmith*   @"November2,1995  % $ 7*XXdXXd7   WeakComplementarityandQuasiRents a JuChinHuangandV.KerrySmithAbstract    Thispaperdescribesproductionanalogstotheconditionsusedinconsumertheoryto  P recovermeasuresofwillingnesstopayfornonmarketedenvironmentalresources.TheanalysissuggeststhatbothweakcomplementarityandHicksianneutralityhaveproductionanalogs.Moreover,itindicatesthatpastmeasuresofthewelfarelossesduetopollutionhavefailedtodistinguishconstantprofitandconstantquasirentsourcemeasures,thelateristhetheoreticallyconsistentconcepttobeused.  j  KeyWords: ` Quasirents,Externalities,ProductionLosses,WeakComplementarity q JELClassificationnumber(s): D24,H41,H23   I.  Introduction  h   Estimationoftheeconomicvalueofenvironmentalresourcesfocusesonmeasuresofwhatpeople,asconsumers,arewillingtopayformaintainingorenhancingthem.Improvedhealth,expandedrecreationalopportunities(orquality),andincreasesinamenityservicesareamongthereasonsforthesevalues.Environmentalresourcesalsocontributetoproductionactivitiesbyprovidingcompatiblegrowingconditionsforagriculturalcrops,waterforcoolingsystems,aswellasreceptaclesforresiduals.Totheextentresidualsfromonesetofactivitiesreducetheavailability(orquality)oftheseservicesforotherproductionactivities,thereisaneconomicloss.Asarule,theselosseshavebeendescribedaschangesinthequasirentsattributedtotheenvironmentalresourcesasunpricedfactorinputstoproduction.  WiththeexceptionoftheunpublishedeconometricanalysisofcostfunctionsbyMathTech[1982],effortstomeasurethissourceofbenefits(costs)fromimprovements(deteriorations)inenvironmentalqualityhavebeenprimarilyforagriculture.-  --  -   1      ׀Moreover,  appliedstudieshavegenerallylinkedadamagefunctionforpollutants(oftenderivedfromexperimentaltrials)withanoptimizingmodelofthesupplyprocess-  --  -   2      .WhileFreemanand  @ Harrington[1990]andFreeman[1993]haveoutlinedtheconditionsforconsistentlymeasuringwelfarechangeswithbothsingleoutputandmultipleoutputfirms,theiranalysisfocusesmostofitsattentiononsituationswheretheeffectsoftheenvironmentalresourceontheoutput(s)supplyisknown(orcanbeconstructedinanoptimizationmodel).Therevelationpropertiesoffirms'behaviorhavenotbeenconsideredinwaysthatarecomparabletothesequestionsforconsumers'choicesofmarketedgoodsandservices.  Thispaperdescribesproductionanalogstotheconditionsusedinconsumertheoryto -`'* recovermeasuresofwillingnesstopayfornonmarketedenvironmentalresourcesandoutlinestheirrelevanceforsingleandmultipleoutputproductionprocesses.Morespecificallywedescribefivetypesofweakcomplementarityinproduction:environmentalqualitytoasingleinputoroutput,qualitytomultipleinputs,qualitytomultipleoutputs,andenvironmentalqualitytosetsofinputsandoutputs.Ineachcasewedescribehowestimatesofthechangesinquasirentscanberecoveredininputdemandandoutputsupplyfunctions.Followingthatanalysiswediscusstherelevanceofotherrestrictionsusedinthecontextofrecoveringconsumerwillingnesstopayforestimatinglossestofirmsfromenvironmentalexternalities. II.  ModelingWeakComplementarityinProduction  H   Thereisastrikingcontrastbetweentheattentiongiventorestrictionsonpreferencesaspartofthedevelopmentoftheindirectapproachesformeasuringconsumers'valuesfortheenvironmentandthemodelingoffirms'quasirents.-  -- = -   3      ׀Nonetheless,FreemanandHarrington  [1990]identifyakeyinsightintheanalogieswedevelopbelow.TheynotethattheJust,Hueth,andSchmitz[1982]argumentthatmultiplemarketmeasuresofthewelfarelossesduetopricedistortionscanbeadaptedtoconsiderthecaseofmeasuringchangesinquasirentsthatarisefromchangesinenvironmentalresources.Theyobservethatquasirentscanbemeasuredusingeithertheshiftsinthesupplycurveforthenecessaryoutputfromtheshutdownpricetothecurrentpriceorbetweenthenecessaryinput'sdemandcurvesatthedifferentlevelsoftheresourcefromthechokepricetothecurrentprice.Whilenotdescribedassuch,theseconditionsareproductionanalogstoweakcomplementaritybetweentheenvironmentalresourceandoneoutputoroneinput.-  -- > -   4      ׀Indeed,analyzingthepropermeasures -`'* ofwelfarechangeswithingthecontextofproductioneffectsparallelsthetreatmentinconsumertheory.Becauseofthisparallel,inwhatfollowsweconsidertherelevanceofseveraldifferentformsofweakcomplementarityandNeill's[1988]Hicksianindependence. A.  WeakComplementarityOneInputandMultipleOutputs      Considerajointoutputprofitfunctionwithnoutputsandminputs.̀!=!(p1,...,pn,w1,...,wm,q) @  Ѐ=!(P,W,q), $  ӀwhereP:avectorofoutputprices,pi,I=1,2,...,n p ЀW:avectorofinputprices,wj,j=1,2,...,m b Ѐq:avariablerepresentingthequality T ЀofenvironmentasaquasifixedinputByHotelling'slemma,  QA=zx 0@Xdddddddd@ExddxL@3 (#(#    (#(#whereyiistheithoutputandxjisthejthinput.LetYbethevectorofyisandXbethevector f%! ofxjs.Thesimplestformofweakcomplementarityinvolvescomplementaritybetweenthe J' # environmentalresource,q,andoneinput,sayx1.Itisdefinedbyequation(3): .)"%  ;<RB>zx p@Xdddddddd@E+xBddq zL+@ (#(#  ,&) wherew1*isthechokepriceofx1;i.e.x1(w1*)=0.Equation(3)impliesthat!(Y,0,x2,...,xm,q1) H Є!(Y,0,x2,...,xm,q0)=0.Thatis,ifx1isnotused,achangeinqdoesnotaffectthequasi ,  rents.  Inappreciatingthefullimplicationsofthisresult,itisimportanttoacknowledgethatthequasirentsdonothavetoariseexclusivelyfromq.Wecanassumethatfirmsare h  heterogeneousandearnquasirentsfromothersourcesinacompetitiveequilibrium,followingPanzarandWillig[1978].-  -- A -   5      ׀Whatisatissueisthechangeinquasirentsduetochangesinq. ,  Considerasimpleexamplewithtwoinputsandtwooutputsinvolvedintheproductionprocess(i.e.,n=m=2),then,Pconsistsofp1andp2,andWis[w1,w2].Ifweassumex1isaweak X complementtoenvironmentalqualityq,environmentalquality,q,haspositiveeffectsonboth < outputsandinputs;i.e.,,yi/,q>0and,xj/,q>0.Thechokepriceofx1dependsonthe   levelofq.    Toreflectthissituationwecanaddaparameterto!(P,W,q)andrecognizethatthe  marginalfirm,at VFBz X x 0X @Xdddddddd@E j ddEp h 2,willearnzeroprofits(i.e., VFBz X x 0X @Xdddddddd@E  ddHE"h ).However,those  d with VFBz X x 0X @Xdddddddd@EP$ dd6E$willearnpositivequasirents.Thedistributionofacrossfirmsdeterminesthe $| distributionofgainsandlossesduetochangesinq,butitdoesnotchangethedefinitionofthe ("  appropriatemeasures.Equation(4)definestheareabetweenthederiveddemandsfortheinputweaklycomplementarytoqwithw1*(q,),thechokepricefunction. ,X&$ Ї QA=zx 0@Xdddddddd@EhxddLh@I (#(#   (#(#OrbyHotellingslemmawehave I95zx 0 @ Pdddd@E x9dpq   (#(#   (#(#Thisexpressiondefinesthechangeinquasirentsasin(4c).   Ѐ(4c) ` !(p10,p20,w10,w20,q1,)!(p10,p20,w1*(q1,),w20,q1,) l    `  +!(p10,p20,w1*(q0,),w20,q0,)!(p10,p20,w10,w20,q0,) P Byequation(3),wehaveequation(5): 0 Ѐ(5) ` !(p10,p20,w1*(q1,),w20,q0,)!(p10,p20,w1*(q1,),w20,q1,)=0  Hence,(4a)canbereducedtothechangeofquasirent,!=!(P0,W0,q1,)!(P0,W0,q0,). p Similarlyifoneoutputy1isweaklycomplementaryto(or necessary)q,thenthearea  T betweenthesupplyfunctionsforthatoutputmeasuresthechangeinquasirentduetothechangeinq.Thiscanbeseeninequations(6a)through(6c)withp1*(q,)theshutdown $  price.  d&"  QA=zx 0@Xdddddddd@Ehx3ddkLh@ (#(#     (#(#  (6b)=!(p10,p20,W0,q1,)!(p1*(q1,),p20,W0,q1,)!(p10,p20,W0,q0,)+ `     ` !(p1*(q0,),p20,W0,q0,) D  Byequation(3), l   (6c) ` !(p1*(q0,),p20,W0,q0,)!(p1*(q1,),p20,W0,q1,)=0 0  B.  WeakComplementarityMultipleInputsorMultipleOutputs     BockstaelandKling[1988]generalizedtheconceptofweakcomplementaritytoconsiderthecasewhereanenvironmentalresourcewasweaklycomplementarytoasetofcommodities.Considertheproductionanalog,supposethatx1andx2compriseasetwhichis ", weaklycomplementarytoq.Ajointproductiontechnologyisstillassumed.Twoquestions x$  needtobeaddressed.First,canthesumofareasbetweenthedemandcurvesofx1andx2, \&" respectively,measuretherequiredchangeinquasirent?Second,howcantheareasbecalculated?Thecaseofasetofgoodsissomewhatmorecomplicatedthanthesinglegoodcase.Thechangeofquasirentduetothechangeinenvironmentalqualityisnowgiveninequation(7). -x'* Ї QA=zx 0@Xdddddddd@Ehxdd Lh@5 (#(#%%   (#(#wherecisarectifiableJordancurvefrom(P0,w10,w20,q0,)to(P0,w10,w20,q1,).-  -- B -   6      ׀Theline   integraldefinedinequation(7)ispathindependent.Undertheassumptionofweak   complementarity,thelineintegralfrom(P0,w1*(q0,),w2*(q0,),q0)to h  (P0,w1*(q1,),w2*(q1,),q1)iszero.Equation(7)canbewrittenasthesumoftwolineintegrals L  asinequation(8). "!QA=zx 0@Xdddddddd@Exxdd HLx@t  (#(#  (#(#  c0isarectifiableJordancurvewithinitialpoint(P0,w10,w20,q0,)andterminalpoint  (P0,w1*(q0,),w2*(q0,),q0).c1isalsoarectifiableJordancurvewithinitialpoint  (P0,w1*(q1,),w2*(q1,),q1)andterminalpoint(P0,w10,w20,q1). x   Equation(8)canberewrittenas  \  $#QA=zx 0@Xdddddddd@E"xQddL"@  (#(#&&'  (#(#'  HoldingPandqconstant,theintegrandsofthetwolineintegralsin(9)areexact H(!$ differentialsoftheprofitfunction.Twopathsarechosenforc0andc1as: ,*#&   ,%( Ѐc0:(P0,w10,w20,q0)(P0,w1*(q0,),w20,q0)(P0,w1*(q0,),w2*(q0,),q0) h   c1:(P0,w1*(q1,),w2*(q1,),q1)(P0,w10,w2*(q1,),q1)(P0,w10,w20,q1) L   Followingthesepaths,expression(9)canbewrittenasregularintegrals.    &%QA=zx 0@Xdddddddd@E xddfL @  (#(#     (#(#   ` = !(P0,w1*(q0,),w20,q0,)!(P0,w10,w20,q0,) 4    `  +!(P0,w1*(q0,),w2*(q0,),q0,)!(P0,w1*(q0,),w20,q0,)     `  +!(P0,w10,w2*(q1,),q1,)!(P0,w1*(q1,),w2*(q1,),q1,)     `  +!(P0,w10,w20,q1,)!(P0,w10,w2*(q1,),q1,) l Cancelingtermswehave:  (10b) `  !(P0,w10,w20,q1,)!(P0,w10,w20,q0,) T&"    `  +!(P0,w1*(q0,),w2*(q0,),q0,)!(P0,w1*(q1,),w2*(q1,),q1,) 8(!$   !(P0,w1*(q0,),w2*(q0,),q0,)!(P0,w1*(q1,),w2*(q1,),q1,)iszerobytheassumption +%( ofweakcomplementaritybetweenqandthesetcomprisedbyx1andx2.Expression(10a)then -t'* providesawaytomeasurethechangeofquasirentduetothechangeinenvironmentalquality.Givenweassumethat,x/,q>0,theimprovementofqualitywillincreasethequasi H rentbytheamountcalculatedin(10a),whichisequaltothesumoftheareasbetweenthetwo ,  factordemandcurvesasillustratedinFigure1a.  Considernowthecaseofoutputsy1andy2thatarenotindividuallyweakcomplements   withq,buttogethery1andy2formanoutputsetwhichisweaklycomplementarytoq.This l  casecouldariseifafarmerhasamultipleproductrotationcycle,saycombiningcornandsoybeans.Hewillnotcareabouttheweatherifbothcropsarenotproduced.  Theanalysisofthiscaseissimilartothatoftwoinputs.Thebenefitsofachangeinenvironmentalqualitycanbecalculatedfromthechangesinthesupplyfunctionsfory1andy2. X Thelineintegraldefiningthiscasewithdifferentpathsc1andc0isgiveninequation(11). <  ('VFBz ( 0@Xdddddddd@E uddfL   (#)#(#(#(#(#(( (#)#(#(#(#)#(#(# (#)#(#(#(#)#(#(# (#)#(#(#(#)#(#(# (#(#(#)#(#(#  Asbefore,weassumec1isarectifiableJordancurvewithinitialpoint "8 (p1*(q1,),p2*(q1,),W0,q1)andterminalpoint(p10,p20,W0,q1).c0isalsoarectifiableJordan $  curvewithinitialpoint(p1*(q0,),p2*(q0,),W0,q0)andterminalpoint(p10,p20,W0,q0). h& "   Holding(w1,w2)constant,theintegrandsofthetwolineintegralsin(11)areexact L(!$ differentialsoftheprofitfunction.Hence,thelineintegralsarepathindependent.Aconvenientintegratingpathischosenasfollows: -'* Ѐ(p1*(q),p2*(q),W0,q)U(p10,p2*(q),W0,q)U(p10,p20,W0,q) h Equation(11)canberewrittenasequation(12). ,   *)QA=zx 0@Xdddddddd@E xddL @  (#(#     (#(#   ` =!(p10,p2*(q1,),W0,q1,)!(p1*(q1,),p2*(q1,),W0,q1,) P    ` +!(p10,p20,W0,q1,)!(p10,p2*(q1,),W0,q1,) 4    ` !(p10,p2*(q0,),W0,q0,)+!(p1*(q0,),p2*(q0,),W0,q0,)     ` !(p10,p20,W0,q0,)+!(p10,p2*(q0,),W0,q0,)  Cancelingliketerms,wehave(13).  L   (13) ` !(p10,p20,W0,q1,)!(p10,p20,W0,q0,)+!(p1*(q0,),p2*(q0,),W0,q0,) x$     ` !(p1*(q1,),p2*(q1,),W0,q1,) \&"   Thedefinitionofjointweakcomplementaritybetweeny1,y2andqimpliesthatthelast *#& twotermsin(13)cancel.Thus,theareasbetweenthetwosupplycurvescanmeasurethe ,%( desiredchangeinquasirent,asinFigure1b. -|'*   WeakcomplementaritycanservetoresolvethedifficultiesidentifiedbyFreemanandHarringtoninmeasuringchangesinquasirentsduetochangesinenvironmentalquality.Moreover,wecanconsidercomplementaritywithanindividualinputoroutput,multipleinputsormultipleoutputs,orsetscomprisinginputsandoutputs.Thebasicanalysisisthesame.Ofcourse,thesamequalificationstotheresultsalsohold.Thatis,theassumptionofweakcomplementarityisapriorrestrictionthatmustbetreatedasamaintainedhypothesis.Estimatesofthechangeinquasirentarethereforeconditionaltothatassumption.Thereis,however,animportantdistinctionbetweentheseapplicationsandtheuseofrestrictionsforconsumerbehavior.Theexistingliteraturehasdemonstratedthatitispossibletousefieldexperimentstobetterunderstandthenatureoftheproductiontechnologiesand,therefore,theprospect,inprinciple,fortestingspecificformsofweakcomplementarityrestrictionsforspecifictechnologies. III.  DoOtherRestrictionsHaveRelevanceforMeasuringChangesinQuasiRents?  `   Theshortanswerisyes,becausethestructureofourmodelsofthedecisionprocessiscompletelycomparable.ConsidertwoexamplesperfectsubstitutionandHicksneutralityinproduction(Neill[1988]andLarson[1992,1993)].Thefirstfollowsfromtheanalysisofwhatcanbelearnedfromavertingbehavior(seeMler[1985]andSmith[1991]).Ifoneinputprovidesaperfectsubstitutefortheservicesprovidedtothefirmbyanenvironmentalresourcethenthechangeinexpendituresonthatinputwithachangeinqmeasuresthedesiredchange *#& inquasirents.  Neillhasdemonstratedpriorknowledge(ormaintainedassumption)thatone -d'* commodity'sHicksiandemandfunctionisindependentofchangesintheenvironmentalresourcecanbeusedtobound,orwithadditionalstructure,toestimatethemarginalvalueoftheresource.Adaptingthisapproachtotheproductioncontextimpliesrestrictingeitherthederiveddemandorsupplyfunctions.Considertheconstantquasirentinputdemandforx1as   inequation(14): ,+QA=zx 0@Xdddddddd@Ex3ddkL@ (#(#  (#(#AsinthecaseofthistypeofderivationoftheSlutskyequation(Cook[1972]),wehaveanexpressiondescribinghowtheconstantquasirentdemand(x1)canbelinkedtotheobservable L constantprofitdemand(x1)asin(15). 0  .-QA=zx 0@Xdddddddd@ExBdd@ zL@ (#(#++  (#(#Hicksianneutralityintheproductioncontextimplies) 0/VFBz X x 0X @Xdddddddd@E x<R" Z)*) ?@WGCz X x pX @Xdddddddd@E x<R" Z)*,* ?@ZJFz \ xG pX @Xdddddddd@E_xG<#RZ*,, ?@ZJFz \ x1 pX @Xdddddddd@E1dzRp,sowecansolveequation(15) l forthedesiredchangeinquasirentswithachangeinq. 4#  21QA=zx 0@Xdddddddd@E%xVddL%@ (#(#    (#(# ,0&&   Asimilarargumentcanbedevelopedforoutputsupplyfunctionsusingtheenvelopeargument(i.e.,yj(P,W,q,)=yj(P,W,q,!(P,W,q,))),andthesameargumentsfollow.With H specificassumptionsaboutthedemand(orsupply)functions(comparabletotheargumentinLarson[1992,1993])theincrementtoquasirentscanbemeasuredfromthedemandorsupplyfunctions.  Ofcourse,asinthecaseofLarson'sanalysis,theplausibilityofthisstrategyformeasurementreliesonthemaintainedassumptionofneutrality.- 3 -- C -   7      ׀Nonetheless,itmightbe D  arguedthattotheextentwecanidentifythesourcesofquasirents(i.e.,thefactorsgivingrisetoabovenormalreturns,representedbyinourprofit,inputdemandandoutputsupply l functions),wemayhavebetterprospectsforverifyingthesemaintainedhypothesesthaninthecaseofconsumerapplications.  Thisargument,aswellastheoneweofferedinsupportofweakcomplementarityinproductionisaconjecture.Untilthereisthedetailedplantlevelanalysisofactivitiesaffectedbypollutiontheseenhancedtheoreticalrestrictionswillnotmovethiscomponentofafullbenefitcostanalysisanyclosertounderstandinghowtoproperlymeasurethelossesinquasirentsduetopollutionrelatedexternalities. IV.  Implications  P&"   Consistentmeasurementofthewelfarelossesrequiresanintegratedgeneralequilibriumframeworkthatreflectsgeneralequilibriumeffectsassociatedwithchangesinenvironmentalquality.Theanalysisshouldtreataproposedactionasoneimplyingchangesinallproductpricesandincomes(i.e.,valuesoffixedendowments)inadditiontotheexogenouschangein -h'* theenvironmentalresource.- 4 -- D -   8      ׀Untilthemodelingofeconomicactivitiesallowsthislevelof h integrationitwillbeimportanttoconsistentlymeasuretheseparatecomponentsofeconomicsurplus.  Therehasbeenlimitedattentiongiventothedefinitionandconsistentmeasurementofchangesinquasirentsduetochangesinenvironmentalresources.Oncetheseeffectsareconsideredinaframeworkthatacknowledgestheprospectsfordifferentimpacts(e.g.duetolocation)ondifferentfirmswithinthesameindustry,consistentmeasurementoflosses(orgains)requiresconsiderationofthesourceofdifferentialquasirentsacrossfirms.ThePanzarWilligmodelprovidestherequiredstructure.Moreover,aswenoted(footnote#5),ithasfurtherimplications.IfthePanzarWilligframeworkisused,thenitisalsoimportanttoacknowledgethatvirtuallyallpastappliedstudieshaveincorrectlymeasuredtheproductionsidelossesduetoexternalities.Theyhaveusedconstantprofitandnotwhatwehavetermedtheconstantquasirentsourcemeasures.ThisdisparitybetweenwhathasbeenmeasuredincomparisontowhatisrequiredforawelfareconsistentconceptparallelstheearlyworkmeasuringMarshallianconsumerlossespriortothewidespreadadoptionofHausman[1981]orVrtias[1983]methodsformeasurementoftheHicksianmeasuresforconsumergainsandlosses.Dualityimpliesanalogousrelationshipsforimpactdemandandoutputsupplyfunctions.  Thus,usingthewelldevelopedtheoreticalconditionsderivedforconsumerwelfaremeasurementwehavearguedthattherearebothopportunitiesandchallengesincorrectlyapplyingtheconceptsofweakcomplementarityandHicksianneutralitytodescriptionoffirm'sresponsestochangesinenvironmentalquality. -`'*   Theserestrictionsdopermitmeasurementofthechangesinquasirentsfromindividualinputdemandandoutputsupplyfunctionsorfromsetsofthesefunctionswhenthelinkageisbetweentheenvironmentalresourceandthesetofinputsoroutputs.Themeasuresderivedareconditionaltothesemaintainedassumptions.Nonetheless,becausetheremaybemoreabilitytoobservetheresponseofproductiontechnologiestoenvironmentalresourcesortoidentifyfirm(orplant)characteristicsgivingrisetoquasirents,inprinciplethereisgreatprospectforverifyingtheserestrictions.Equallyimportant,byincludingeconomistsinterestedinmeasuringtheselossesintheplanningstagesofexperimentalstudiesofdamagefunctions,thereistheprospectofidentifyingresponserelationshipsthatfitthepropertiesweassociatewithweakcomplementarityinproduction.Thesetypesofexperimentalstudieswouldimprovetheabilitytolinkdamagefunctionstoeconomicmodelsofproductionrelatedenvironmentallosses.Theymightalsoservetoextendtherelevanceoftheseexperimentaltrials,byidentifyingthetypesofrestrictionsthatcouldbeimposedinareaswhereexperimentaldamagefunctionsarenotavailable.  ` @& ENDNOTES p  h  Ӏ*AssistantProfessor,DepartmentofEconomics,EastCarolinaUniversity,andArts H 0  andSciencesProfessor,DepartmentofEconomics,DukeUniversityandResourcesfortheFutureUniversityFellow,respectively.PartialsupportforSmith'sresearchwasprovidedbytheUNCSeaGrantprogramunder._________./Grant#R/MRD/0/0/01251. 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