WPC2 T9HuYUkOh?W-p YcvhaAӪ%䯌Y,: m2Av&ي7nGo`#x؇nl RyݦKKRi_ℋIO}f$hƇO)#:ߐՙ<~/S/ wũi:yAL"g%MiOpHL ZqB'GV&p*1~geKg졼Q&l)2gnj *88@2g[:/s9=ps}$ǵڌ/VA Jڰw>\9MA`̺,_S8dr\G 0s 0U 0 0t 0  0 0 0 0s 0 0 0 0 0 0] 0B 0s: 0 03 0 0x 07 0  0 0s 0Y 0 0x 0$ 0 0 0UN % 0UWU]<UU 1UUBoU>   D+: 0O 0(O AMw oE @ &o&@*U *U*U&*U.* + 1y+ 1y+ 1y , 1y,,n - 041#!D1Ne2^ g2ws24w22 m2<6 :AutoList4[1]  23  0..<I :AutoList4[2]0..  23  0` ..` <\ :AutoList4[3]0..0` ..`   23  0 .. <o :AutoList4[4]0..0` ..` 0 ..   23  0..< :AutoList4[5]0..0` ..` 0 .. 0..  23  0h..h< :AutoList4[6]0..0` ..` 0 .. 0..0h..h  23  0..< x:AutoList4[7]0..0` ..` 0 .. 0..0h..h0..  23  0..< lb:AutoList4[8]0..0` ..` 0 .. 0..0h..h0..0..  23  0p..p<6 :AutoList3[1]  23  0..<I :AutoList3[2]0..  23  0` ..` <\ :AutoList3[3]0..0` ..`   23  0 .. <o :AutoList3[4]0..0` ..` 0 ..   23  0..< :AutoList3[5]0..0` ..` 0 .. 0..  23  0h..h< :AutoList3[6]0..0` ..` 0 .. 0..0h..h  23  0..< x:AutoList3[7]0..0` ..` 0 .. 0..0h..h0..  23  0..< lb:AutoList3[8]0..0` ..` 0 .. 0..0h..h0..0..  23  0p..p<6 :AutoList2[1]  23  0..<I :AutoList2[2]0..  23  0` ..` <\ :AutoList2[3]0..0` ..`   23  0 .. <o :AutoList2[4]0..0` ..` 0 ..   23  0..< :AutoList2[5]0..0` ..` 0 .. 0..  23  0h..h< :AutoList2[6]0..0` ..` 0 .. 0..0h..h  23  0..< x:AutoList2[7]0..0` ..` 0 .. 0..0h..h0..  23  0..< lb:AutoList2[8]0..0` ..` 0 .. 0..0h..h0..0..  23  0p..p<6 :AutoList1[1]  23  0..<I :AutoList1[2]0..  23  0` ..` <\ :AutoList1[3]0..0` ..`   23  0 .. <o :AutoList1[4]0..0` ..` 0 ..   23  0..< :AutoList1[5]0..0` ..` 0 .. 0..  23  0h..h< :AutoList1[6]0..0` ..` 0 .. 0..0h..h  23  0..< x:AutoList1[7]0..0` ..` 0 .. 0..0h..h0..  23  0..< lb:AutoList1[8]0..0` ..` 0 .. 0..0h..h0..0..  23  0p..p(9 Z6Times New Roman Regular!X<FF:Default ParaXXX-$\ `-XXX-$\ `-\ `<  Y Z$Symbol Regular'4 $359=AEIMQ1\  `*Times New RomanTT\  `&Times New RomanVariable list'dxd Level 1 Level 2 Level 3 Level 4 Level 5('2/$ ?T!   ($,-.+A<< cWPC. . 9513!.  e  .1  &` & MathType "- K  @Times New Roman- 2 2CG+B 2 t} 2  g 2 s 2 ds [Times New Roman- 2 7popppp 2 3 popppp 2 61t> 2 #(popppp@Times New Roman- 2 Z( 2 ) 2 ( 2  exp[ 2 ]) [Times New Roman- 2 *p@Symbol- 2 = 2 -\Symbol- 2 }@Times New Roman- 2 ` 100 2 `100 [Times New Roman- 2 #0p@Symbol- 2 d & "System-0WPC.G .G 9513!.G     .1   &R & MathType @Times New Roman- 2 2CG+B 2 {t} 2 ! g 2 ms 2 Uds [Times New Roman- 2 tfp>>p 2 0 tfp>>p 2 t> 2 dtfp>>p@Times New Roman- 2 ( 2 ) 2 d( 2 E exp[ 2 ]) [Times New Roman- 2  *p@Symbol- 2 = 2 x-\Symbol- 2 = [Times New Roman- 2 Z0p@Symbol- 2 Ed & "System-2'\ `9\ `4 (\ `@ CEKQW]emsAutoList1I.A.1.a.(1)(a)i)a) CEKQW]emsAutoList2I.A.1.a.(1)(a)i)a)  CEKQW]emsAutoList3I.A.1.a.(1)(a)i)a) CEKQW]emsAutoList4I.A.1.a.(1)(a)i)a)3|x \ `s New Roman RegularXXN\  P}XP'4 s New Roman RegularXXN4  pG;Xp'\ `s New Roman RegularXXN\  PkXP9\ `s New Roman RegularXXr\  PnQXP4 s New Roman RegularXXN4  p}G;Xp(\ `s New Roman RegularXXr\  P}QXP( T$  HP LaserJet Series II,,,,,,0b'70'U ?T!   _TheDICE99Excelspreadsheetiscontainedinthefile dice99.xls.I. Worksheets   Thespreadsheetcontainsfourworksheets:  Model ,  SavedOutput ,  Output, and  ESP . d   A. Model     h  @    ContainstheequationsthatmakeuptheDICEmodel.Thisworksheetcontainsfourkindsofrows: ParametersandExogenousVariables, EndogenousVariables, ControlVariables,and KeyShadowPrices.ParameterandExogenousVariablerowscontaineithervalues(asopposedtoformulas)orformulasbasedentirelyonotherParameterandExogenousVariablerows.EndogenousVariablerowscontainformulasbasedonboththeexogenousvariablesandthecontrolvariables.TheControlVariablerowsaredeterminedbytheuser.A policyisachoiceofvaluesfortheserows.TheKeyShadowPricerowsareformulasbasedontheEndogenousVariables.TheKeyShadowPricerowsareincludedinthe Model worksheetbecausethey   indicatewhethertheControlVariablesareoptimal.(SeesectionIII.C.1formoreonthis.)  Important:theusershouldnotenternumbersdirectlyintotheCarbontaxrow( Model , l row99).Thiswilleraseaformula.Instead,whenchoosingvaluesforthecarbontax,theusershouldenterthemintotheUnconstrainedcarbontaxrow( Model ,row110).Thisisexplained H fullyinsectionsII.A.3andII.A.4.b.  B. SavedOutput  L   Containsselectedoutputfromthebasecaseandothercases.(SeesectionIII.Bfordiscussionofcases.)Thisworksheetconsistsentirelyofvalues(Noformulas).Thus,itisnotchangedwhenanewpolicyorotherchangealterstheendogenousvariables.  The SavedOutput worksheetisincludedasanexampleofhowtosaveoutputfroma  modelrun,recordsusefuloutputfromthebasecaseandothercases,andallowsmeasuresofthewelfareimpactsofpoliciestobeconstructedonthe Output worksheet. x    C. Output  @#!   Thisworksheetdisplaysmodeloutput.Consistsofformulaslinkedtothe Model  %l # worksheet.  D. ESP  ( $'     Thisworksheetcalculatestheenvironmentalshadowpriceofcarbon,oneoftheKeyShadowPricesonthemodelworksheet. p-(, II. Modelstructure ..Inthespreadsheet,wefollowtheconventionthatendogenousvariablesareinboldfontwhileexogenousvariablesandparametersareinnormalfont.Whenwerefertoarowonaworksheet,wewillgenerallyrefertoitasVariablename( Worsksheetname ,row#)..Intheformulasbelow, trepresentstime.In1995t=1,in2005t=2,etc...Factorsof10,100,and1000....Sinceeachtimeperiodspansadecadeinthemodel,andsinceflowsandsomeratesofchangearerepresentedonthespreadsheetonaperyearbasis,factorsof10areneededwhenexpressingstocksasfunctionsofflowsorexpressingparametersasfunctionsofannualratesofchange.Forinstance,thegrosschangeinthecapitalstockfromperiodtoperiodis10timesannualinvestment...Wedisplaypercentagesratherthanfractionsonthespreadsheet.Intheformulascreatingthesepercentages,fractionsaremultipliedby100;similarly,informulaslinkedtothepercentagesonthespreadsheetwhichcallforfractionsratherthanpercentages,wedividethepercentagesby100...Theunitsinwhichvariablesonthespreadsheetarecommonlymeasuredareoccasionallyinconsistent.Weusuallyrefertocarbonintensityintermsoftonnesofcarbonper$1000ofGDPbutrefertothecarbontaxintermsof$/tonne,not$1000pertonne.Factorsof10to1000occuroccasionallyinthespreadsheetformulastoallowfortheexpressionofallthevariablesintheirmostcommonlyusedunits...A. ModelWorksheet ....1. ParametersandExogenousVariables ......a. OutputandCapitalAccumulation Row50` ..` 0 .. 0..XXXQ%<  Y Z$Symbol RegularQqXXXc!(9 Z6Times New Roman Regularc10h..h0..Damagecoefficientontemperature....ParameterinClimatedamage( Model ,row74).AlsoentersMarginaleffectoftemperatureonoutput( ESP ,row4)Row6XXXQ%<  Y Z$Symbol RegularQ....0 .. qXXXc!(9 Z6Times New Roman Regularc20..0h..hDamagecoefficientontemperaturesquared..ParameterinClimatedamage( Model ,row74).AlsoentersMarginaleffectoftemperatureonoutput( ESP ,row4).Row70..0` ..` 0 .. b1(t)0..0h..hAbatementcostfunctioncoefficient....ParameterinAbatementcost( Model ,row72).AlsoentersControlrate( Model ,row82)andCarbontax( Model ,row99).AlsoentersformulafortheUnconstrainedcarbontax( Model ,row110)whenthemacroLimCTisrun(insolvingtheemissionslimitscase.SeesectionIII.Bbelow).Row80` ..` 0 .. 0..b1*0h..h0..Initialcostfunctioncoefficient.......DeterminesfirstperiodvalueofAbatementcostfunctioncoefficient.( Model ,row7)Row90..0` ..` 0 .. gb*0..0h..hInitialrateofgrowthinabatementcostcoefficient...ParameteringrowthrateofAbatementcostcoefficient.( Model ,row11)Row10XXXQ%<  Y Z$Symbol RegularQ....dXXXc!(9 Z6Times New Roman Regularcb0..0h..hRateofdeclineingrowthrateofabatementcostcoefficient..ParameteringrowthrateofAbatementcostcoefficient.( Model ,row11)Row11....gb(t)0..0h..hGrowthrateofabatementcostcoefficient0..ParameterthatdeterminesAbatementcostfunctioncoefficient.( Model ,row7).Row120` ..` 0 .. b20..0h..hExponentofabatementcostfunction..ParameterinAbatementcost( Model ,row72).AlsoentersControlrate( Model ,row82)andCarbontax( Model ,row99).AlsoentersformulafortheUnconstrainedcarbontax( Model ,row110)whenthemacroLimCTisrun(insolvingtheemissionslimitscase.SeesectiononCasesbelow)Row13XXXQ%<  Y Z$Symbol RegularQ....gXXXc!(9 Z6Times New Roman Regularc0..0h..hCapitalshare..ParameterinOutputgrossofclimatedamageandabatementcosts.( Model ,row71)AlsoentersInterestrate(Model,row103)andOutput,basecasetemperature(Output,row9)Row14XXXQ%<  Y Z$Symbol RegularQ0` ..` 0 .. dXXXc!(9 Z6Times New Roman Regularc0..0h..hRateofdepreciation..ParameterinCapitalstock( Model ,row79).AlsoentersInterestrate( Model ,row103)andOutput,basecasetemperature( Output ,row10).Row150` ..` 0 .. K*0..0h..hInitialcapitalstock..ParameterinCapitalstock( Model ,row79).AlsoentersCapital,basecasetemperature( Output ,row10)...........b. Emissions Row17XXXQ%<  Y Z$Symbol RegularQ0` ..` 0 .. dXXXc!(9 Z6Times New Roman RegularcL0..0h..hRateofdeclineinlandusechangeemissions..ParameterthatdeterminesCarbonemissionsfromlandusechange( Model ,row18)Row180` ..` 0 .. LU(t)0..0h..hCarbonemissionsfromlandusechange..ParameterinTotalemissions( Model ,row81).AlsoentersformulafortheUnconstrainedcarbontax( Model ,row110)whenthemacroLimCTisrun(insolvingtheemissionslimitscase.SeesectionIII.Bbelow)Row190` ..` 0 .. LU*..0h..hInitialcarbonemissionsfromlandusechange..ParameterinEmissionsfromlandusechange( Model ,row18)Row200` ..` XXXQ%<  Y Z$Symbol RegularQ0 .. s(XXXc!(9 Z6Times New Roman RegularctXXXQ%<  Y Z$Symbol RegularQ)XXXc!(9 Z6Times New Roman Regularc0..0h..hIndustrialCO2emissionstooutputratio....ParameterinIndustrialemissions( Model ,row80).AlsoentersControlrate( Model ,row82)andCarbontax( Model ,row99).AlsoentersformulafortheUnconstrainedcarbontax( Model ,row110)whenthemacroLimCTisrun(insolvingtheemissionslimitscase.SeesectionIII.Bbelow)...ThisparameterrepresentstheCO2outputratiointheabsenceofenvironmentalcontrols!inapolicywherethecarbontaxiszero.TheactualCO2outputratioinagivencasewilldependontheControlrate( Model ,row82),whichisafunctionoftheCarbontax( Model ,row99).Row21XXXQ%<  Y Z$Symbol RegularQ0` ..` 0 .. s*XXXc!(9 Z6Times New Roman Regularc0..0h..hInitialindustrialCO2emissionstooutputratio....DeterminesfirstperiodvalueofIndustrialCO2emissionstooutputratio( Model ,row20)Row220` ..` 0 .. gXXXQ%<  Y Z$Symbol RegularQsXXXc!(9 Z6Times New Roman Regularc*0..0h..hInitialgrowthrateofindustrialCO2emissionstooutputratio...ParameterinGrowthrateofindustrialCO2emissionstooutputratio( Model ,row25)Row23XXXQ%<  Y Z$Symbol RegularQ0` ..` 0 .. dsXXXc!(9 Z6Times New Roman Regularc10..0h..hRateofdeclineingrowthrateofsigma..ParameterinGrowthrateofindustrialCO2emissionstooutputratio( Model ,row25)Row24XXXQ%<  Y Z$Symbol RegularQ0` ..` 0 .. ds2XXXc!(9 Z6Times New Roman Regularc0..0h..hAccelerationparameterofgrowthrateofsigma..ParameterinGrowthrateofindustrialCO2emissionstooutputratio(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row25)Row250` ..` 0 .. gXXXQ%<  Y Z$Symbol RegularQsXXXc!(9 Z6Times New Roman Regularc(t)0..0h..hGrowthrateofindustrialCO2emissionstooutputratio0..ParameterthatdeterminesIndustrialCO2emissionstooutputratio(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row20).Row260` ..` 0 .. EL(t)0..0h..hWorldemissionslimit..LimitfortotalCO2emissionsintheemissionslimitcase.(Seedescriptionofcasesbelow.)Themodelusercantryoutdifferentemissionslimitsinthemodelbyenteringdifferentnumbersinthisrow.ThisparameterenterstheUnconstrainedcarbontax(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row111)whenthemacroLimCTisrun.ELhasnoroleinthemodeloutsideoftheemissionslimitcase.Row270` ..` 0 .. CPOT0..0h..hConvergenceparameterforoptimalcarbontax..ParameterdeterminesthestepsizeinthemacroOptCTthatsolvesfortheoptimalcarbontax..CPOTmustbesettoanumberbiggerthan0andnolargerthan1.(SeesectionIII.C.2.bforadviceonchoosingCPOT.)CPOTentersintotheIterationstep.(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row110)......c. Population Row290` ..` 0 .. L(t)0..0h..hPopulation..EntersintoOutputgrossofclimatedamageandabatementcosts(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row71),Percapitaconsumption(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row93),Periodspecificutilityfunction( Model ,row94),GDPpercapita( Output ,row7)andOutput,basecasetemperature( Output ,row9).Row300` ..` 0 .. L*0..0h..hInitialpopulation....DeterminesfirstperiodvalueofPopulation(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row29)Row310` ..` 0 .. gpop*0..0h..hInitialpopulationgrowthrate..ParameterinCumulativepopulationgrowthrate(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row33)Row32XXXQ%<  Y Z$Symbol RegularQ0` ..` 0 .. dXXXc!(9 Z6Times New Roman Regularcpop0..0h..hRateofdeclineinpopulationgrowthrate..ParameterinCumulativepopulationgrowthrate(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row33)Row330` ..` 0 .. CGpop(t)0h..hCumulativepopulationgrowthrate0..ParameterthatdeterminesPopulation(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row29).......d. Productivity Row350` ..` 0 .. TFP(t)0..0h..hTotalfactorproductivity..EntersintoOutputgrossofclimatedamageandabatementcosts(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row71)andOutput,basecasetemperature( Output ,row9)̇Row360` ..` 0 .. TFP*0..0h..hInitialleveloftotalfactorproductivity....DeterminesfirstperiodvalueofTotalfactorproductivity(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row35)Row370` ..` 0 .. gtfp*0..0h..hInitialproductivitygrowthrate..ParameterinProductivitygrowthrate(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row39)Row38XXXQ%<  Y Z$Symbol RegularQ0` ..` 0 .. dXXXc!(9 Z6Times New Roman Regularctfp0..0h..hRateofdeclineinproductivitygrowthrate..ParameterinProductivitygrowthrate(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row39)Row390` ..` 0 .. gtfp(t)0..0h..hProductivitygrowthrate0..ParameterthatdeterminesTotalfactorproductivity(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row35).........e. Concentrations Row410` ..` 0 .. M*0....InitialatmosphericconcentrationofCO2Row420` ..` 0 .. MU*0..0h..hInitialconcentrationofCO2inbiosphere/shallowoceansRow430` ..` 0 .. ML*0..0h..hInitialconcentrationofCO2indeepoceans..Parametersincarboncycleequations(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,rows8486)Rows4451XXXQ%<  Y Z$Symbol RegularQ0` ..` 0 .. f11,f12,0h..hXXXc!(9 Z6Times New Roman RegularcCarboncycletransitioncoefficientsXXXQ%<  Y Z$Symbol RegularQ......f21,f22,f23,......f32,f33XXXc!(9 Z6Times New Roman Regularc..Parametersincarboncycleequations(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,rows8486)XXXQ%<  Y Z$Symbol RegularQfXXXc!(9 Z6Times New Roman Regularcijspecifiesthepercentageofcarboninboxjatthestartofadecadethatmovesintoboxioverthedecade.Row520` ..` 0 .. CL0..0h..hConcentrationslimit..Limitforatmosphericcarbonconcentrationsintheconcentrationslimitcase.(SeedescriptionofcasesinsectionIII.B)Themodelusercantryoutdifferentconcentrationslimitsinthemodelbyenteringdifferentnumbersinthiscell.ThisparameterentersMarginaleffectofenvironmentonoutput( ESP ,row4)whenthemacroClimdfunctionisrun.Outsideoftheconcentrationlimitscase,CLplaysnoroleinthemodel.Thedefaultvalueis1192.8GtC,doubletheassumedpreindustrialatmosphericconcentrationofcarbondioxide.......f. Temperature ( ' $(Row540` ..` 0 .. O(t)0..0h..hExogenousforcing....ForcingfromsourcesotherthenCO2emissions.EntersTotalincreaseinradiativeforcingsincepreindustrial(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row89).XXXW(\  `*Times New RomanTTWRow550` ..` 0 .. T*0..0h..hInitialatmospherictemperature..EntersAtmospherictemperature(XXX-&'4 -ModelXXXW(\  `*Times New RomanTTW,row88)Row560` ..` 0 .. TLO*0..0h..hInitialtemperatureofthedeepoceansXXXc!(9 Z6Times New Roman Regularc..XXXW(\  `*Times New RomanTTWEntersLoweroceantemperature(XXX-&'4 -ModelXXXW(\  `*Times New RomanTTW,row90)XXXc!(9 Z6Times New Roman RegularcRow57XXXQ%<  Y Z$Symbol RegularQ0` ..` 0 .. sXXXc!(9 Z6Times New Roman Regularc1󀀀....Speedofadjustmentparameterforatmosphericcarbon..EntersAtmospherictemperature(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row88)Row580` ..` 0 .. CS0..0h..hClimatesensitivity..ParameterinAtmospherictemperature(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row88).Specifiesthelongrunincreaseintemperatureoverpreindustriallevelsduetoadoublingofatmosphericcarbonconcentrationsfrompreindustriallevels.Row59XXXQ%<  Y Z$Symbol RegularQ0` ..` 0 .. sXXXc!(9 Z6Times New Roman Regularc20..0h..hCoefficientofheatlossfromatmospheretooceans..ParameterinAtmospherictemperature(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row88)Row60XXXQ%<  Y Z$Symbol RegularQ0` ..` 0 .. sXXXc!(9 Z6Times New Roman Regularc30..0h..hCoefficientofheatgainbydeepoceans..ParameterinLoweroceantemperature(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row90)Row610` ..` 0 .. TL0..0h..hTemperaturelimit..Limitforatmospherictemperatureinthetemperaturelimitcase.(Seedescriptionofcasesbelow.)Themodelusercantryoutdifferenttemperaturelimitsinthemodelbyenteringdifferentnumbersinthiscell.ThisparameterentersMarginaleffectofenvironmentonoutput( ESP ,row4)whenthemacroTlimdfunctionisrun.Outsideofthetemperaturelimitscase,thisparameterplaysnoroleinthemodel.Thedefaultvalueis2.5degreesCelsius.......g. Welfare Row630` ..` 0 .. XXXQ%<  Y Z$Symbol RegularQԀr(XXXc!(9 Z6Times New Roman RegularctXXXQ%<  Y Z$Symbol RegularQ)XXXc!(9 Z6Times New Roman Regularc0..0h..hSocialrateoftimepreference..0..ParameterinSocialtimepreferencefactor(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row66)Row64XXXQ%<  Y Z$Symbol RegularQ0` ..` 0 .. rXXXc!(9 Z6Times New Roman Regularc*0..0h..hInitialsocialrateoftimepreference..ParameterinSocialrateoftimepreference(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row63)̇Row650` ..` 0 .. gXXXQ%<  Y Z$Symbol RegularQrXXXc!(9 Z6Times New Roman Regularc0..0h..hRateofdeclineofsocialrateoftimepreference..ParameterinSocialrateoftimepreference(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row63)Row660` ..` 0 .. R(t)0..0h..hSocialtimepreferencefactor..ParameterinPeriodspecificutilityterm(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row94)andDiscountfactor(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row96)Row670` ..` 0 .. USM0..0h..hMultiplicativescalingfactorinutilityfunction..ParameterinWelfare(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row95).CalibratedsothatsmallchangesinWelfarefromthebasecaselevelaremeasuredinunitsoffirstperiodconsumption!ie,dW/dC(0)=1inthesolutionofthebasecase.Row680` ..` 0 .. USAD0h..h0..Additivescalingfactorinutilityfunction..ParameterinWelfare(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row95).CalibratedsothatinthebasecaseWelfare equalsPresentvalueofconsumption( Model ,row97).........2. EndogenousVariables ........a. Output Row710` ..` 0 .. Y(t)0..0h..hOutputgrossofclimatedamageandabatementcosts..EntersintoOutputgrossofclimatedamage(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row73),Industrialcarbonemissions(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row80),andIndustrialcarbonintensity( Output ,row27)Row720` ..` 0 .. ACP(t)0h..hAbatementcost(asapercentageofY)0x..x..PercentageofoutputlostduetocostsofabatingCO2emissions.EntersintoOutputgrossofclimatechange(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row73).Row730` ..` 0 .. YGR(t)0h..h0..Outputgrossofclimatedamage..EntersintoOutputnetofclimatedamageandabatementcosts(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row75)andMarginaleffectofenvironmentonoutput( ESP ,row4).Row740` ..` 0 .. D(t)0..0h..hClimatedamage..Percentageofoutputlostduetodamagefromtemperatureincrease.EntersintoOutputnetofclimatedamageandabatementcosts(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row75),Controlrate( Model ,row82),Carbontax( Model ,row99),Climatedamage(Output,row12),andMarginaleffectofenvironmentonoutput( ESP ,row4).̇Row750` ..` 0 .. Q(t)0..0h..hOutputnetofclimatedamageandabatementcosts..Outputavailableforconsumptionandinvestment.EntersintoInvestment(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row77),Consumption(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row92),Interestrate(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row103),Output( Output ,row4),andGDP,differencefromreference( Output ,row5)......b. Capitalaccumulation Row770` ..` 0 .. I(t)0..0h..hInvestment..Grosscapitalaccumulation.EntersintoCapital(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row78)andConsumption( Model ,row92)Row780` ..` 0 .. K(t)0..0h..hCapital..Capitalstock.EntersintoOutputgrossofabatementcostsandclimatedamage(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row71)andinterestrate(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row103)0......c. Emissions Row800` ..` 0 .. E(t)0..0h..hIndustrialCO2emissions..EntersintoTotalCO2emissions(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row81),IndustrialCO2emissions( Output ,row26),andIndustrialcarbonintensity( Output ,row27).Row810` ..` 0 .. ET(t)0..0h..hTotalCO2emissions..EntersintoAtmosphericCO2concentration(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row82)andTotalCO2emissions( Output ,row25).Row82XXXQ%<  Y Z$Symbol RegularQ0` ..` 0 .. mXXXc!(9 Z6Times New Roman Regularc(t)0..0h..hControlrate..Percentdecreaseinemissionsfromtheirzerocarbontaxlevel.EntersintoIndustrialCO2emissions(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row80),Abatementcost(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row72),andControlrate( Output ,row30).TheControlrateisafunctionoftheCarbontax(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row99)...FromII.A.5.1417andII.A.5.20,II.A.2.1..Q(t)=[1/(1+D(t)/100)](1b1(XXXQ%<  Y Z$Symbol RegularQmXXXc!(9 Z6Times New Roman Regularc(t)/100)b2)Y(t)II.A.5.20..W(\  `*Times New RomanTTWE(t)=(1Q%<  Y Z$Symbol RegularQmW(\  `*Times New RomanTTW(t)/100)Q%<  Y Z$Symbol RegularQsW(\  `*Times New RomanTTW(t)Y(t)XXXc!(9 Z6Times New Roman Regularc..II.A.5.20canberewrittenII.A.2.2..XXXQ%<  Y Z$Symbol RegularQmXXXc!(9 Z6Times New Roman Regularc(t)=100[1E(t)/Q%<  Y Z$Symbol RegularQsW(\  `*Times New RomanTTW(t)Y(t)XXXc!(9 Z6Times New Roman Regularc]̇..PluggingII.A.2.2intoII.A.2.1andtakingdQ(t)/dE(t),wegetaformulaforthemarginalproductofemissions:II.A.2.3..dQ(t)/dE(t)={[1/(1+D(t)/100)]b1(t)b2(XXXQ%<  Y Z$Symbol RegularQmXXXc!(9 Z6Times New Roman Regularc(t)/100)(b21)}/XXXQ%<  Y Z$Symbol RegularQsXXXc!(9 Z6Times New Roman Regularc(t)..EconomicagentsproduceIndustrialcarbondioxideemissions( Model ,row80)untilthemarginalproductequalsthemarginalcost.Inourmodel,emissionshavezeroeconomiccost;thus,themarginalcostofanemissionistheCarbontax( Model ,row99).SettingthemarginalproductofemissionsequaltotheCarbontax,wecansolvefortheControlrateinducedbyagivenCarbontax.......d. Carboncycle Row840` ..` 0 .. M(t)0..0h..hAtmosphericcarbonconcentration..EntersintoBiosphere/upperoceancarbondioxideconcentration( Model ,row85),Totalincreaseinradiativeforcingsincepreindustrial(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row89)anddF(t)/dE(i)( ESP ,rows78112).WhenthemacroClimdfunctionhasbeenrun,thisvariableentersintoMarginaleffectoftheenvironmentonoutput( ESP ,row4)...0` ..` Row850` ..` 0 .. MU(t)0h..h0..Biosphere/upperoceancarbondioxideconcentration..EntersintoAtmosphericCO2concentration(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row84)andDeepoceanCO2concentration(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,86)Row860` ..` 0 .. ML(t)0..0h..hDeepoceanCO2concentration0..EntersintoBiosphere/upperoceanCO2concentration(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,85)......e. Temperature Row880` ..` 0 .. T(t)0..0h..hAtmospherictemperature..ThisvariableservesasacomprehensivemeasureofclimatechangeintheDICEmodel.AllharmfulenvironmentaleffectsofgreenhousegasesoccurthroughtheeffectofthisvariableonGDP.EntersClimatedamage(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row72),Loweroceantemperature(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row90),Atmospherictemperature( Output ,row13),Globalmeantemperature,differencefrombase( Output ,row15),andMarginaleffectofenvironmentonoutput( ESP ,row4).....Row890` ..` 0 .. F(t)0..0h..hIncreaseinradiativeforcingsincepreindustrialtimes..EntersAtmospherictemperature(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row88)...The596.4inII.A.5.28isourestimateofpreindustrialconcentrationsofcarbondioxide.Adoublingofcarbondioxiderelativetoitspreindustriallevelwillraiseradiativeforcingby4.1W/m2overitspreindustriallevel,accordingtothisequation.0..Row900` ..` 0 .. TLO(t)0h..h0..Loweroceantemperature..EntersAtmospherictemperature(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row90)......f. Welfare Row920` ..` 0 .. C(t)0..0h..hConsumption..EntersintoConsumptionpercapita(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row93)Row930` ..` 0 .. CPC(t)0h..h0..Consumptionpercapita..EntersintoPeriodspecificutilityfunction(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row95),Discountfactor(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row96),andConsumption( Output ,row17).Row940` ..` 0 .. U(t)0..0h..hPeriodspecificutilityterm..Individualperiodtermsinthetimeseparablesocialwelfarefunction.EntersWelfare(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row95).Row950` ..` 0 .. W0..0h..hWelfare..Thiscelldisplaysthevalueofthesocialwelfarefunction,themeasureoftheoverallwelfareofsociety.Row960` ..` 0 .. DF(t)0..0h..hDiscountfactor..Ratioofmarginalsocialwelfareofconsumptionintimettothemarginalsocialwelfareofconsumptioninthefirstperiod,orthevalueofaunitoffutureconsumptionintermsofunitsofconsumptiontoday.EntersintoPresentvalueofconsumption( Model ,row97)andDiscountrate( Model ,row104).(II.A.2.4)0` ..` DF(t)=R(t)L(t)C(0)/[C(t)L(0)]..ThisisderivedbytakingthederivativeofII.A.5.32withrespecttoC(t).Row970` ..` 0 .. PVC0..0h..hPresentvalueofconsumption..AnalternativewelfaremeasuretoWelfare,thisisthesumacrossperiodsofconsumption,discountedbythediscountfactor.̇(II.A.2.5)0` ..` PVC=XXXQ%<  Y Z$Symbol RegularQSXXXc!(9 Z6Times New Roman RegularcԀ10DF(t)C(t)......t....Note:Ifoneiscomparingthepresentvalueofconsumptionacrosspolicies,oneshouldusethesamediscountfactorstocalculatebothdiscountedsums.....3. ControlVariables Row99XXXQ%<  Y Z$Symbol RegularQ0` ..` 0 .. tXXXc!(9 Z6Times New Roman Regularc0..0h..hCarbontax..Thetaxthatmustbepaidforeachmetrictonofindustrialemissions.Thisvariableisthepolicymakersmeansofcontrollingemissions.Byincreasingthetax,thepolicymakermakesemissionsmorecostly,thusdecreasingtheiruse.ThisvariableenterstheformulafortheControlrate(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row82),Iterationstep(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row109),andCarbontax( Output ,row32)...ThecontrolrateformulaallowstheControlrate( Model ,row82)tobeover100%forhighenoughXXXQ%<  Y Z$Symbol RegularQtXXXc!(9 Z6Times New Roman Regularc.ThiswouldimplynegativeIndustrialCO2emissions.TopreventtheCarbontaxfromenteringthisrange,wehavemadethecarbontaxrowaformulathatcapstheCarbontaxatthevaluethatsetstheControlrateto100.Modelusersshouldnotentervaluesintothecarbontaxrowdirectly;thiswillerasetheformula.Instead,valuesfortheCarbontaxshouldbeenteredintotheUnconstrainedcarbontax( Model ,row110).Thevaluethatappearsinthecarbontaxrowwillthenbetheminimumoftheuserenteredtaxrateandthezeroindustrialemissionstaxrate:....XXXQ%<  Y Z$Symbol RegularQ(II.A.3.1)0` ..` tXXXc!(9 Z6Times New Roman Regularc(t)=min{XXXQ%<  Y Z$Symbol RegularQtXXXc!(9 Z6Times New Roman RegularcUC(t)XXXQ%<  Y Z$Symbol RegularQ,1000b1(XXXc!(9 Z6Times New Roman RegularctXXXQ%<  Y Z$Symbol RegularQ)b2/[(1+XXXc!(9 Z6Times New Roman RegularcD(t)XXXQ%<  Y Z$Symbol RegularQ/100)s(XXXc!(9 Z6Times New Roman RegularctXXXQ%<  Y Z$Symbol RegularQ)]}XXXc!(9 Z6Times New Roman Regularc..Thezeroindustrialemissionscarbontax,orthesecondtermontherighthandsideof(II.A.3.1)isderivedbysettingXXXQ%<  Y Z$Symbol RegularQmXXXc!(9 Z6Times New Roman Regularc(t)=100inII.A.5.22andsolvingforXXXQ%<  Y Z$Symbol RegularQtXXXc!(9 Z6Times New Roman Regularc(t).Row1000` ..` 0 .. s(t)0..0h..hSavingsrate..TheSavingsratedetermineswhichfractionofoutputisdevotedtoconsumptionandwhichtoinvestment.ItenterstheformulaforInvestment(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row77)andCapital,basecasetemperature( Output ,row10).TheuserentershischoiceforthesavingsratesdirectlyintotheSavingsraterow.....4. Keyshadowprices ......a. Capital Row1030` ..` 0 .. i(t)0..0h..hInterestrate..Rateofreturnoninvestment,expressedasanannualrate.EntersintotheDifferencebetweentheinterestrateanddiscountrate(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row105)...Anadditionalunitofinvestmentinthecurrentperiodhastwobenefitsinthenextperiod.Itincreasesoutputandreducestheinvestmentnecessarytoattainagiventargetforthecapitalstock.Thus,theinterestratethatappliestoinvestmentinthecurrentperiodcanbederivedbyaddingthemarginalproductofcapitalinthenextperiodtothefractionofinvestmentthatremainsafteronedecadesdepreciation,andthenconvertingthesumintoanannualrate:(II.A.4.1)..[1+i(t)/100]10=10XXXQ%<  Y Z$Symbol RegularQgXXXc!(9 Z6Times New Roman RegularcQ(t+1)/K(t+1)+(1XXXQ%<  Y Z$Symbol RegularQdXXXc!(9 Z6Times New Roman Regularc/100)10..XXXS)\  `&Times New RomanS..Thereisnofactorof10onthesecondterminthefirstequationabove,becauseanincreaseof10inendofperiodcapitalstockreducestheperyearinvestmentnecessarytoattainagiventargetbyonly1...Rearranging(II.A.4.1)givesusthespreadsheetformulafortheinterestrate:XXXc!(9 Z6Times New Roman Regularc(II.A.4.2)0` ..` i(t)=100{[10XXXQ%<  Y Z$Symbol RegularQgXXXc!(9 Z6Times New Roman RegularcQ(t+1)/K(t+1)+(1XXXQ%<  Y Z$Symbol RegularQdXXXc!(9 Z6Times New Roman Regularc/100)10]1/101}XXXS)\  `&Times New RomanS    5+ 4 <DL!5XXXc!(9 Z6Times New Roman RegularcRow10404..0 ..P DR(t)0 .. 0<..Discountrate0..0..Ratioofmarginalsocialwelfareofconsumptionincurrentperiodtothatinnextperiod,expressedasanannualrate.EntersintotheDifferencebetweentheinterestrateanddiscountrate(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row105).(II.A.4.3)..[1+DR(t)/100]10=[dW/dC(t)]/[dW/dC(t+1)]=DF(t)/DF(t+1)(II.A.4.4)04..DR(t)=100([DF(t)/DF(t+1)]1/10󀄀1)% * Row10504..0 ..P DiffC(t)0<..0..XDifferencebetweeninterestrateanddiscountrate....ServesasacriteriontojudgewhethertheSavingsrateisoptimal.OptimalsavingsrateswillsetDiffCtozero.(II.A.4.5)04..DiffC(t)=100[i(t)DR(t)]/DR(t)......b. Emissions Row10704..0 ..P ESP(t)0<..0..XEnvironmentalshadowpriceofcarbon0..Societalwillingnesstopaytoavoidthemarginalenvironmentalconsequencesofcarbondioxideemissions.ThecalculationofESPisdoneonthe ESP worsksheet.EntersintoDifferencebetweencarbontaxandESP(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row108),Iterationstep(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row109),andEnvironmentalshadowpriceofcarbon( Output ,row33)Row10804..0 ..P DiffE(t)0<..0..XDifferencebetweencarbontaxandESP0..ServesasacriteriontojudgewhethertheCarbontaxisoptimal.TheoptimalCarbontaxsetsDiffEtozero!themarginalproductofemissionsshouldequaltheirmarginalenvironmentalcost.̇(II.A.4.6)04..DiffE(t)=100[ESP(t)XXXQ%<  Y Z$Symbol RegularQtXXXc!(9 Z6Times New Roman Regularc(t)]/ESP(t)Row10904..0 ..P IS(t)0 .. 0<..Iterationstep0..ApointmidwaybetweentheCarbontaxandESP.UsedinthemacroOptCTthatfindstheoptimalCarbontax.IS(t)isthenextvaluethatthealgorithmwilltryintheUnconstrainedcarbontaxrow( Model ,row110).(II.A.4.7)..IS(t)=XXXQ%<  Y Z$Symbol RegularQtXXXc!(9 Z6Times New Roman Regularc(t)+CPOT(ESP(t)XXXQ%<  Y Z$Symbol RegularQtXXXc!(9 Z6Times New Roman Regularc(t))Row110XXXQ%<  Y Z$Symbol RegularQ04..0 ..P tXXXc!(9 Z6Times New Roman Regularcuc(t)0 .. 0<..Unconstrainedcarbontax@..0..InthisrowtheuserspecifieshischoicefortheCarbontax.Theusercanplaceashighanumberashewouldlikeinthisrow.Theformulainthecarbontaxrow(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row99)preventstheCarbontaxthatentersthecontrolrateformula(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row82)fromexceeding100%.....5. Listofequations ..Thislistcontainsallformulasonthe Model worksheetuptorow95.ThesecomprisetheequationsoftheDICE99model.(II.A.5.1)..b1(t)=b1(t1)/[1+gb(t)/100]Row7....b1(1)=b1*(II.A.5.2)04..gb(t)=gb*exp(XXXQ%<  Y Z$Symbol RegularQ10(dXXXc!(9 Z6Times New Roman Regularcb/100)t)Row11(II.A.5.3)04..LU(t)=LU*(1[XXXQ%<  Y Z$Symbol RegularQdXXXc!(9 Z6Times New Roman RegularcL/100])t....Row18(II.A.5.4)04..XXXQ%<  Y Z$Symbol RegularQsXXXc!(9 Z6Times New Roman Regularc(t)=XXXQ%<  Y Z$Symbol RegularQsXXXc!(9 Z6Times New Roman Regularc(t1)/[1+gXXXQ%<  Y Z$Symbol RegularQsXXXc!(9 Z6Times New Roman Regularc(t)/100]Row20..XXXQ%<  Y Z$Symbol RegularQ..sXXXc!(9 Z6Times New Roman Regularc(1)=XXXQ%<  Y Z$Symbol RegularQsXXXc!(9 Z6Times New Roman Regularc*(II.A.5.5)04..gXXXQ%<  Y Z$Symbol RegularQsXXXc!(9 Z6Times New Roman Regularc(t)=gXXXQ%<  Y Z$Symbol RegularQsXXXc!(9 Z6Times New Roman Regularc*exp(XXXQ%<  Y Z$Symbol RegularQ(dsXXXc!(9 Z6Times New Roman Regularc/100)tXXXQ%<  Y Z$Symbol RegularQds2XXXc!(9 Z6Times New Roman Regularct)Row25(II.A.5.6)..L(t)=L*exp[CGpop(t)]Row29(II.A.5.7)04..Row33.. )  )  ) [/1J:6z \  `. @@@[      ....(II.A.5.8)04..TFP(t)=TFP(t1)/[1+gtfp(t)/100]Row35....TFP(1)=TFP*(II.A.5.9)04..gtfp(t)=gtfp*exp(XXXQ%<  Y Z$Symbol RegularQ(dXXXc!(9 Z6Times New Roman Regularctfp/100)t)Row37......(II.A.5.9)Row3904.. )  )  ) [/3J:6z \,  `.G @@@[      XXXW(\  `*Times New RomanTTW(II.A.5.10)0 ..P O(t)=0.1965+0.13465t0..0D..t<12Row54..=1.150 .. 0<..0..X0..t>11XXXc!(9 Z6Times New Roman Regularc......................c!(9 Z6Times New Roman Regularc(II.A.5.11)Q%<  Y Z$Symbol RegularQԀ0 ..P rW(\  `*Times New RomanTTW(t)=Q%<  Y Z$Symbol RegularQrW(\  `*Times New RomanTTW*exp(10(gQ%<  Y Z$Symbol RegularQr/100)W(\  `*Times New RomanTTWt)XXXc!(9 Z6Times New Roman RegularcRow63......(II.A.5.12)..W(\  `*Times New RomanTTWR(t)=R(t1)/[1+#(t1)]10XXXc!(9 Z6Times New Roman RegularcRow66......R(1)=1(II.A.5.13)..Y(t)=TFP(t)(K(t)XXXQ%<  Y Z$Symbol RegularQgXXXc!(9 Z6Times New Roman Regularc)(L(t)1XXXQ%<  Y Z$Symbol RegularQgXXXc!(9 Z6Times New Roman Regularc)Row71(II.A.5.14)..ACP(t)=100b1(XXXQ%<  Y Z$Symbol RegularQmXXXc!(9 Z6Times New Roman Regularc(t)/100)b2Row72(II.A.5.15)0 ..P YGR(t)=[1(ACP(t)/100)]Y(t)Row73(II.A.5.16)..D(t)=100*(XXXQ%<  Y Z$Symbol RegularQqXXXc!(9 Z6Times New Roman Regularc1T(t)+XXXQ%<  Y Z$Symbol RegularQqXXXc!(9 Z6Times New Roman Regularc2T(t)2)Row74(II.A.5.17)0 ..P Q(t)=YGR(t)/(1+D(t)/100)Row75(II.A.5.18)0 ..P I(t)=(s(t)/100)Q(t)Row77̇(II.A.5.19)..K(t)=10I(t)+K(t1)(1XXXQ%<  Y Z$Symbol RegularQdXXXc!(9 Z6Times New Roman Regularc/100)10Row78..K(1)=K*W(\  `*Times New RomanTTW(II.A.5.20)0 ..P E(t)=(1Q%<  Y Z$Symbol RegularQmW(\  `*Times New RomanTTW(t)/100)Q%<  Y Z$Symbol RegularQsW(\  `*Times New RomanTTW(t)Y(t)XXXc!(9 Z6Times New Roman RegularcRow80(II.A.5.21)0 ..P ET(t)=E(t)+LU(t)Row81XXXQ%<  Y Z$Symbol RegularQ(II.A.5.22)0 ..P mXXXc!(9 Z6Times New Roman Regularc(t)=100[(XXXQ%<  Y Z$Symbol RegularQt(XXXc!(9 Z6Times New Roman RegularctXXXQ%<  Y Z$Symbol RegularQ)/1000)s(XXXc!(9 Z6Times New Roman RegularctXXXQ%<  Y Z$Symbol RegularQ)(1+XXXc!(9 Z6Times New Roman RegularcD(t)/100)/(b1(t)b2)]1/(b21)Row82(II.A.5.23)0 ..P M(t)=10ET(t1)+(XXXQ%<  Y Z$Symbol RegularQfXXXc!(9 Z6Times New Roman Regularc11/100)M(t1)+(XXXQ%<  Y Z$Symbol RegularQfXXXc!(9 Z6Times New Roman Regularc12/100)MU(t1)Row84..M(1)=M*(II.A.5.24)0 ..P MU(t)=(XXXQ%<  Y Z$Symbol RegularQfXXXc!(9 Z6Times New Roman Regularc21/100)M(t1)+(XXXQ%<  Y Z$Symbol RegularQfXXXc!(9 Z6Times New Roman Regularc22/100)MU(t1)+(XXXQ%<  Y Z$Symbol RegularQfXXXc!(9 Z6Times New Roman Regularc23/100)ML(t1)Row85..MU(1)=MU*(II.A.5.26)0 ..P ML(t)=(XXXQ%<  Y Z$Symbol RegularQfXXXc!(9 Z6Times New Roman Regularc32/100)MU(t1)+(XXXQ%<  Y Z$Symbol RegularQfXXXc!(9 Z6Times New Roman Regularc33/100)ML(t1)Row86..ML(1)=ML*(II.A.5.27)0 ..P T(t)=T(t1)+XXXQ%<  Y Z$Symbol RegularQsXXXc!(9 Z6Times New Roman Regularc1{F(t1)[4.1/CS]T(t1)XXXQ%<  Y Z$Symbol RegularQsXXXc!(9 Z6Times New Roman Regularc2[T(t1)TLO(t1)]}Row88..T(1)=T*(II.A.5.28)0 ..P F(t)=4.1[ln(M(t)/596.4)/ln(2)]Row89(II.A.5.29)0 ..P TLO(t)=TLO(t1)+XXXQ%<  Y Z$Symbol RegularQsXXXc!(9 Z6Times New Roman Regularc3[T(t1)TLO(t1)]Row9004..TLO(1)=TLO*(II.A.5.30)0 ..P C(t)=Q(t)I(t)Row92(II.A.5.31)0 ..P CPC(t)=1000C(t)/L(t)Row93(II.A.5.32)0 ..P U(t)=R(t)L(t)ln(CPC(t)/1000)Row94(II.A.5.33)..W=USAD+XXXQ%<  Y Z$Symbol RegularQSXXXc!(9 Z6Times New Roman RegularcԀU(t)/USMRow95....t..B. SavedOutput Thisworksheetwascreatedbysolvingsomeofthecasesbelow(III.B)andcopyingthevaluesfromsomeoftherowsinthe Output worksheet.Therearenoformulasonthispage;therowsconsistentirelyofvalues.Thevariabledefinitionsarethesameasthecorrespondingvariablesonthe Output worksheet.Someoftherowsonthispageareusedtoconstructvariablesonthe Output worksheetthatcomparetheuserschoiceofpolicytothebasecase.TheserowsareRow504..0 ..P 0 .. QB(t)0<..0..XBasecaseoutput0..EntersGDP,differencefromreference( Output ,row5)andGDP,differencefromreference,percent( Output ,row6)Row904..0 ..P 0 .. DB(t)0<..0..XBasecaseclimatedamage0..EntersOutput,basecasetemperature( Output ,row9),Globalmeantemperature,differencefrombase( Output ,row15)Row1004..0 ..P TB(t)0 .. 0<..Basecaseatmospherictemperature..EntersGlobalmeantemperature,differencefrombase( Output ,row15)Row1404..0 ..P DFB(t)0 .. 0<..Basecasediscountfactor..EntersBasecasediscountfactor( Output ,row18)Row1504..0 ..P PVCB0 .. 0<..Basecasepresentvalueofconsumption0..EntersTotalabatementcostofpolicy( Output ,row21)andTotaleconomicimpactofpolicy( Output ,row23)Row1704..0 ..P ETB(t)0 .. 0<..Basecasetotalcarbondioxideemissions..EntersCumulativecarbondioxideemissions,differencefromreference( Output ,row29)....C. Output ....1. Output Row404..0 ..P 0 .. Q(t)0<..0..XOutput0..ThisequalsOutputnetofclimatedamageandabatementcostsontheXXX-&'4 -ModelXXXc!(9 Z6Times New Roman RegularcԀworksheet.(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row75).EntersGDPpercapita( Output ,row7)..Row504..0 ..P 0 .. QDR(t)0..XGDP,differencefromreference0..DifferencebetweenGDPunderuserschoiceofpolicyandGDPinbasecase.EntersGDP,differencefromreference,percent( Output ,row6)(II.C.1.1)..QDR(t)=Q(t)QB(t)Row604..0 ..P 0 .. QDRP(t)0..X0..GDP,differencefromreference,percent..DifferencebetweenGDPandbasecaseGDP,expressedaspercentofbasecaseGDP(II.C.1.2)..QDRP(t)=100[QDR(t)]/QB(t)Row704..0 ..P 0 .. QPC(t)0..X0..GDPpercapita..EntersGDPpercapita,ratioto1995( Output ,row8)(II.C.1.3)..QPC(t)=1000Q(t)/L(t)Row804..0 ..P 0 .. QPCR95(t)0..XGDPpercapita,ratioto1995..RatioofGDPpercapitatoits1995level(II.C.1.4)..QPCR95(t)=QPC(t)/QPC(1995)Row904..0 ..P 0 .. QBT(t)0..X0..Output,basecasetemperature0..Acounterfactual!whatGDPwouldhavebeenunderthecurrentpolicybutwiththesametimepathoftemperatureasinthebasecase.EntersConsumption,basecasetemperature( Output ,row20).(II.C.1.5)..QBT(t)=[1/(1+DB(t)/100)](1ACP(t)/100)A(t)KBT(t)XXXQ%<  Y Z$Symbol RegularQgXXXc!(9 Z6Times New Roman RegularcL(t)1XXXQ%<  Y Z$Symbol RegularQgXXXc!(9 Z6Times New Roman RegularcRow1004..0 ..P KBT(t)0<..0..XCapital,basecasetemperature0..WhatCapitalstockwouldhavebeenundercurrentpolicybutwiththesametimepathoftemperatureasthebasecase.EntersOutput,basecasetemperature( Output ,row9)(II.C.1.6)..KBT(t)=KBT(t1)(1XXXQ%<  Y Z$Symbol RegularQdXXXc!(9 Z6Times New Roman Regularc/100)10+10(s(t1)/100)YBT(t1)....2. Environment Row1204..0 ..P D(t)0 .. 0<..Climatedamage..EqualsClimatedamagefromXXX-&'4 -ModelXXXc!(9 Z6Times New Roman RegularcԀworksheet.(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row74)0..Row1304..0 ..P T(t)0 .. 0<..Atmospherictemperature0....EqualsAtmospherictemperatureinXXX-&'4 -ModelXXXc!(9 Z6Times New Roman RegularcԀworksheet.(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row88)Row1404..0 ..P M(t)0 .. 0<..Atmosphericcarbondioxideconcentration..EqualsAtmosphericcarbondioxideconcentrationonXXX-&'4 -ModelXXXc!(9 Z6Times New Roman RegularcԀworksheet(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row84)Row1504..0 ..P TDR(t)0<..0..XGlobalmeantemperature,differencefromreference(II.C.2.1)..TDR(t)=T(t)TB(t)....3. Welfare Row1704..0 ..P C(t)0 .. 0<..Consumption0..EqualsConsumptioninXXX-&'4 -ModelXXXc!(9 Z6Times New Roman RegularcԀworksheet(XXX-&'4 -ModelXXXc!(9 Z6Times New Roman Regularc,row92)EntersintoTotaleconomicimpactofpolicy( Output ,row23)Row1804..0 ..P DFB(t)0 .. 0<..Basecasediscountfactor0..EqualsBasecasediscountfactor( SavedOutput, row13).EntersintoTotalabatementcostofpolicy( Output ,row21)andPresentvalueofconsumption,basecasediscountfactors(XXX-&'4 -OutputXXX-4'\ `-,row19).Thisrowconsistsofvalues,notformulas.ItwascreatedbysolvingthebasecaseandpastingthevaluesoftheDiscountfactor(Model,row96)..Inmakingwelfarecomparisonsbetween2differentpolicies,thesamerelativepricesshouldbeusedtodiscountthefutureconsumptionstreamsthatresultfrombothpolicies.Thus,inconstructingthecomparisonmeasuresTotalabatementcostofpolicy( Output ,Row21)andTotaleconomicimpactofpolicy( Output ,row23),weusethebasecaserelativepricestodiscountbothbasecaseconsumptionandconsumptionundercurrentpolicy.Row1904..0 ..P PVCBDF0<..Presentvalueofconsumption,basecasediscountfactors0..ThisisidenticaltoPresentvalueofconsumption(XXX-&'4 -ModelXXX-4'\ `-Ԁrow97)exceptthatthediscountfactorsusedaretheBasecasediscountfactors( Output ,row18).EntersTotaleconomicimpactofpolicy( Output ,row23)(II.C.3.1)..PVCBDF=XXX-59\ `-SXXX-4'\ `-Ԁ10DFB(t)C(t)....tRow2004..0 ..P CBT(t)0<..0..XConsumption,basecasetemperature0..Consumption,assumingcurrentpolicyandthetimepathoftemperatureinthebasecase.Calculatingthisvariableallowsustoseparatethecostsofcarbonabatementfromtheenvironmentalbenefitsofapolicy.EntersTotalabatementcostofpolicy( Output ,row21)(II.C.3.2)..CBT(t)=(1s(t)/100)QBT(t)Row2104..0 ..P TAC0 .. 0<..Totalabatementcostofpolicy0..Thetotalimpactofapolicycanbeconceptuallybrokendownintothecostsofabatingcarbonemissionsandtheenvironmentaleffectsoftheabatement.TACprovidesameasureofthefirstcomponent.ItisthedifferencebetweenBasecasepresentvalueofconsumption( SavedOutput ,row15)andthepresentvalueofconsumptionassumingthecurrentpolicyhasnoeffectonthetimepathoftemperature.EntersEnvironmentalbenefitofpolicy( Output ,row22)(II.C.3.3)..TAC=1000[PVCB󀄀XXX-59\ `-SXXX-4'\ `-Ԁ10DFB(t)CBT(t)]........tRow2204..0 ..P EB0 .. 0<..Environmentalbenefitofpolicy0..Definedasaresidual:theTotaleconomicimpactofapolicyequalstheEnvironmentalbenefitofthepolicyminustheTotalabatementcost:(II.C.3.4)04..EB=TEI+TACRow2304..0 ..P TEI0 .. 0<..Totaleconomicimpactofpolicy..Differencefromthebasecaseinthepresentvalueofconsumption.Thisisonepossiblemeasureoftheoveralleffectofapolicyonwelfare.Analternativewaytomeasurethewelfareeffectswouldbetocalculatethecompensatingvariationorequivalentvariationintermsoffirstperiodconsumption,usingthevalueofthesocialwelfarefunction(XXX-&'4 -ModelXXX-4'\ `-,row95).Notethatsinceallofthenegativeeffectsofclimatechangearesummarizedbytheireffectsonconsumptioninthemodel,Totaleconomicimpactofpolicyincludestheenvironmentaleffect.(II.C.3.5)..TEI=PVBDFPVCB....4. Emissions Row2504..0 ..P ET(t)0 .. 0<..Totalcarbondioxideemissions..EqualsTotalcarbondioxideemissionsontheXXX-&'4 -ModelXXX-4'\ `-Ԁworksheet(XXX-&'4 -ModelXXX-4'\ `-,row81)Row2604..0 ..P E(t)0 .. 0<..Industrialcarbondioxideemissions..EqualsIndustrialcarbondioxideemissionsontheXXX-&'4 -ModelXXX-4'\ `-Ԁworksheet(XXX-&'4 -ModelXXX-4'\ `-,row80)Row2704..0 ..P ICI(t)0 .. 0<..Industrialcarbonintensity..0..RatioofindustrialemissionstoGDP.EntersIndustrialcarbonintensitygrowthrate( Output ,row28)(II.C.4.1)..ICI(t)=E(t)/Y(t)Row2804..0 ..P gICI(t)....Industrialcarbonintensitygrowthrate(II.C.4.2)..gICI(t)=100([ICI(t+1)/ICI(t)]1/10󀄀1)Row2904..0 ..P CumEDB(t)..Cumulativecarbondioxideemissions,differencefromthe..........reference(II.C.4.3)..CumEDB(t)=CumEDB(t1)+10[ET(t)ETB(t)]....CumEDB(1)=0Row3004..0 ..P XXX-59\ `-mXXX-$\ `-(t)....Controlrate..EqualsControlrateonXXX-64 -ModelXXX-$\ `-Ԁworksheet(XXX-64 -ModelXXX-$\ `-,row82)....5. Prices Row32XXX-7(\ `-04..0 ..P t(XXX-$\ `-tXXX-7(\ `-)XXX-$\ `-....Carbontax..EqualsCarbontaxonXXX-64 -ModelXXX-$\ `-Ԁworksheet(XXX-64 -ModelXXX-$\ `-,row99)Row3304..0 ..P ESP(t)0 .. 0<..Environmentalshadowpriceofcarbon..EqualsEnvironmentalshadowpriceofcarbonfromXXX-64 -ModelXXX-$\ `-Ԁworksheet(XXX-64 -ModelXXX-$\ `-,row33)Row3404..0 ..P i(t)0 .. 0<..Interestrate..EqualsInterestrateonXXX-64 -ModelXXX-$\ `-Ԁworksheet(XXX-64 -ModelXXX-$\ `-,row103)..D. ESP ..Thecalculationoftheenvironmentalshadowpriceisatechnicalmatterwhosedetailsneednotconcernmostmodelusers.ThedefaultformulaforESPisderivedfromtheequationsonthe Model worksheet,soisconsistentwiththerestofthemodel.Inthetemperatureandconcentrationlimitscases,weusedifferentformulastocalculatetheESP.....1. ESPcalculation Row304..0 ..P 0 .. ESP(t)0<..0..XEnvironmentalshadowpriceofcarbon..Societalwillingnesstopaytoavoidthemarginalenvironmentalconsequencesofcarbondioxideemissions.EntersEnvironmentalshadowpriceofcarbon(XXX-64 -ModelXXX-$\ `-,row107).ESPconsistsofthediscountedsumoverallfutureperiodsoftheMarginaleffectoftheenvironmentonoutput(ESP,row4)timesthemarginaleffectofacurrentperiodemissionontheenvironment.........NOP(II.D.1.1)..ESP(t)=1000XXX-7(\ `-SXXX-$\ `-Ԁ[DF(i)/DF(t)]MCD(i)[dT(i)/dE(t)]........i=t+10..whereNOPisthenumberofperiodsinthemodel.Whenthemacroclimdfunctionisrun,ESPtakestheform........NOP..(II.D.1.2)..ESP(t)=1000XXX-7(\ `-SXXX-$\ `-Ԁ[DF(i)/DF(t)]MCD(i)[dM(i)/dE(t)]........i=t+1..Row404..0 ..P 0 .. MCD(t)0..XMarginaleffectofenvironmentonoutput0..EntersEnvironmentalshadowpriceofcarbon( ESP ,row3)ThiscanbederivedfromII.A.5.16andII.A.5.17......(II.D.1.3)..MCD(t)=YGR(t)[XXX-7(\ `-qXXX-$\ `-1+2XXX-7(\ `-qXXX-$\ `-2T(t)]/(1+D(t)/100)2..WhenthemacroTlimdfunctionisrun,theequationbecomes(II.D.1.4)..MCD(t)=YGR(t)[CDCexp(T(t)EDC/TL)][EDC/TL]/[1+CDCexp(T(t)EDC/TL)]2..ThisisderivedfromII.A.5.17andIII.B.1.2.WhenthemacroClimdfunctionisrun,theequationbecomes% 8 (II.D.1.5)..MCD(t)=YGR(t)[CDCexp(M(t)EDC/CL)][EDC/CL]/[1+CDCexp(M(t)EDC/CL)]2..ThisisderivedfromII.A.5.17andIII.B.1.3.Rows6400 ..P 0 .. dT(i)/dE(t)0..XMarginaleffectontemperatureofemissionsattimet0..EntersEnvironmentalshadowpriceofcarbon( ESP ,row3)andMarginaleffectontemperatureofdeepoceansofemissionsattimet( ESP ,rows4276).FromII.A.5.27:(II.D.1.6)04..dT(i)/dE(t)=(14.1XXX-7(\ `-sXXX-$\ `-1/CSXXX-7(\ `-sXXX-$\ `-1XXX-7(\ `-sXXX-$\ `-2)[dT(i1)/dE(t)]+XXX-7(\ `-sXXX-$\ `-1[dF(i1)/dE(t)]........+XXX-7(\ `-sXXX-$\ `-1XXX-7(\ `-sXXX-$\ `-2[dTLO(i1)/dE(t)]....dT(i)/dE(t)=0,i=10..Themarginalemissioninperiodthasnoeffectonclimaticvariablesinperiodt.Inperiodt+1itaffectsatmosphericconcentrationsandforcing,butnottemperature,sinceforcingaffectstemperaturewithalag.Onlyinperiodt+2willitaffectatmospherictemperature,throughitseffectonforcingint+1.Thus,(II.D.1.7)..dT(i)/dE(t)=0,it+1Rows1501840 ..P 0 .. dMU(t)/dE(i)0..X0..Marginaleffectonbioshpere/upperoceancarbondioxideconcentrationofemissionsattimet0..EntersMarginaleffectonatmosphericcarbondioxideconcentrationofemissionsattimet( ESP ,rows114148)andMarginaleffectondeepoceancarbondioxideconcentrationofemissionsattimet( ESP ,rows186220).FromII.A.5.24,(II.D.1.13)..dMU(i)/dE(t)=0,it+1Rows1862200 ..P 0 .. dML(t)/dE(i)0..X0..Marginaleffectondeepoceancarbondioxideconcentrationofemissionsattimet0..EntersMarginaleffectonbiosphere/upperoceancarbondioxideconcentrationofemissionsattimet( ESP ,rows150184)FromII.A.5.26,(II.D.1.14)0 ..P dML(i)/dE(t)=0,it+2....2. Steepdamagecurveparameters ....Intheconcentrationlimitcase,wecalculateMCDasifD(t)were(II.D.2.1)..DSDC(t)=100CDCexp(M(t)EDC/CL)0..Inthetemperaturelimitscase,replaceM(t)withT(t)andCLwithTLinII.D.2.1.DSDC(t)willbepracticallyzerountilMorTgetsnearCLTLandthenwillincreaserapidly.TheusercancontrolDSDCandthusMCDintheconcentrationandtemperaturelimitcasesviatheparametersCDCandEDC.TheusercontrolsCDCandEDCviatheparametersLRP,URP,PLRP,andPURP,whichhavemoreintuitiveinterpretations.ChangingtheseparameterscanchangethesteepnessoftherapidincreaseinDSDCandtherangeoverwhichitoccurs...Row22304..0 ..P LRP0 .. 0<..Lowerreferencepoint0..PointatwhichDSDCbecomesPLRP.EntersintoExponentonsteepdamagecurve( ESP ,row227)andCoefficientonsteepdamagecurve( ESP ,row228)Row22404..0 ..P URP0 .. 0<..Upperreferencepoint0..PointatwhichDSDCbecomesPURP.EntersintoExponentonsteepdamagecurve( ESP ,row227)andCoefficientonsteepdamagecurve( ESP ,row228)Row22504..0 ..P PLRP0 .. 0<..Percentageofoutputlostatlowerreferencepoint..EntersintoExponentonsteepdamagecurve( ESP ,row227)andCoefficientonsteepdamagecurve( ESP ,row228)Row22604..0 ..P PURP0 .. 0<..Percentageofoutputlostatupperreferencepoint..EntersintoExponentonsteepdamagecurve( ESP ,row227)andCoefficientonsteepdamagecurve( ESP ,row228)..LRP,URP,PLRP,andPURParebestexplainedbyexample,usingtheconcentrationscase.Fortheequivalentconditionsinthetemperaturelimitcase,substituteT(t)forM(t)andTLforCLinII.D.2.2andII.D.2.3:(II.D.2.2)..WhenM=(1LRP/100)CLDSDC=PLRP(II.D.2.3)..WhenM=(1+URP/100)CLDSDC=PURP0..Row22704..0 ..P EDC0 .. 0<..Exponentonsteepdamagecurve..EntersMarginaleffectofenvironmentonoutput( ESP ,row4)wheneitherClimdfunctionorTlimdfunctionarerun.Row22804..0 ..P CDC0 .. 0<..Coefficientonsteepdamagecurve..EntersMarginaleffectofenvironmentonoutput( ESP ,row4)wheneitherClimdfunctionorTlimdfunctionarerun.0..WederiveEDCandCDCbysettingupthefollowingsystemofequations,whichweobtainfrom(II.D.2.1),(II.D.2.2),and(II.D.2.3):(II.D.2.4)..PLRP=100CDCexp((1LRP/100)EDC)(II.D.2.5)..PURP=100CDCexp((1+URP/100)EDC)..Whichgivesus..(II.D.2.6)..EDC=[ln(PLRP/100)ln(CDC)]/[1LRP/100](II.D.2.7)..CDC=exp{[(1+URP/100)ln(PLRP/100)(1LRP/100)ln(PURP/100)]/........[LRP/100+URP/100]}̇III. SolvingtheXXX-64 -Model XXX-$\ `-..A. UserControl ....Theusercantryoutdifferentpoliciesbyvaryingthevaluesinthecontrolvariablerows.Theusercanalsoalterthemodelbychangingtheequationsoralteringtheexogenousvariablesandparameters.Thelattertypeofchange,however,shouldnotbeviewedastheimplementationofapolicy;itisonlyjustifiableiftheuserthinksadifferentviewofnature,oftheeconomicandenvironmentalconstraintsfacingsociety,isinorder.Onepossibleexceptionisthealteringofthedamagecoefficients( Model ,rows5and6)inthegeoengineeringcase(sectionIII.B)...OneapproachtosettingpolicyistodirectlyenterthedesiredvaluesfortheCarbontaxandtheSavingsrateintothecontrolvariablerows.Anotherapproachistospecifyconditionsforapolicytomeetandthentofindthevaluesofthecontrolvariablesthatmeettheseconditions.Asetofsuchconditionsiscalleda case.Thevaluesofthecontrolvariablesthatsatisfytheconditionsofacasearesaidtosolvethatcase,andthevaluesoftheendogenousvariablesthatresultfromthesolutionaretheoutputofthatcase.Someofthemostinterestingcasesarelistedbelow...B. Cases ....1. Base ....InthebasecasetheCarbontaxissettozeroandtheSavingsrateisoptimized.Thisrepresentsthemodelsprojectionofwhatwillhappentooutput,emissions,andclimateifnofurthergovernmentactionistakentoreduceclimatechange.Itcanbethoughtofastheunregulatedmarketoutcome:whathappensifthemarketislefttoitsownforceswithnoattempttointernalizetheclimateexternality.Thedefaultvaluesofthecontrolvariablesandthusoftheendogenousvariablesaretheirbasecasevalues.....2. Optimal ......BoththeSavingsrateandtheCarbontaxareoptimized.Thatis,theCarbontaxineachperiodissetatthelevelthatstrikestheoptimalbalancebetweenthecostsandbenefitsofcarbonabatement.....3. Emissionslimits ....TheCarbontaxissetineachperiodsothatTotalcarbondioxideemissions( Model ,row81)donotexceedtheWorldemissionslimit( Model ,row26)inthatperiod.TheSavingsrateisagainoptimized.....4. Geoengineering ....Inthegeoengineeringcase,thedamagecoefficientsXXX-7(\ `-qXXX-$\ `-1andXXX-7(\ `-qXXX-$\ `-2(XXX-64 -ModelXXX-$\ `-,rows5and6)aresettozeroandtheSavingsrateisoptimized.TheCarbontaxissettozero.....Thegeoengineeringcase,inwhichtheclimateproblemisremovedwiththewaveofahand,allowsustoputadollarvalueonthedamagesfromclimatechange.Itcanbethoughtofastheresultoftheapplicationofacostlessgeoengineeringtechnologytosolvetheclimatechangeproblem.RealisticallycostedgeoengineeringsolutionscanbeevaluatedbysubtractingtheircostsfromOutputnetofabatementcostsandclimatedamage(Model,row75)ineachperiod...........................5. Temperaturelimits ....Atmospherictemperature( Model ,row88)isconstrainednottoexceedtheTemperaturelimit( Model ,row61).ThetimepathoftheCarbontaxisoptimizedsubjecttothisconstraint,andtheSavingsrateisoptimizedaswell.....6. Concentrationslimits ....Atmosphericconcentrationofcarbondioxide( Model ,row84)isconstrainednottoexceedtheConcentrationslimit(Model,row52).ThetimepathoftheCarbontaxisoptimizedsubjecttothisconstraint,andtheSavingsrateisoptimizedaswell...C. Solvingcases ....1. Mathematicalconditions ..Casesaresolvedbytranslatingtheirconditionsintomathematicalconditionsandthenfindingthevaluesofthecontrolvariablesthatcausethemathematicalconditionstobemet...Table1displaysthemathematicalconditionsforsolutionforthecasesinpartB,alongwiththecontrolvariablewhichcanbealteredinordertosatisfyeachcondition.Notethatwhenthe Table1.ConditionsforSolutioninDICE99  %%CaseConditionforsolutionMathematicalconditionControlvariable%%1. Base OptimalsavingsbehaviorDiffC=0(XXX-64 -ModelXXX-$\ `-,row105)Savingsrate%%CarbonexternalityignoredXXX-7(\ `-tXXX-$\ `-Ԁ=0(XXX-64 -ModelXXX-$\ `-,row99)Carbontax%%2. Optimal OptimalsavingsbehaviorDiffC=0(XXX-64 -ModelXXX-$\ `-,row105)%%ҴSavingsrateOptimalcarbonabatement%%DiffE=0(XXX-64 -ModelXXX-$\ `-,row108)Carbontax3. Emissionslimit Optimalsavingsbehavior%%DiffC=0(XXX-64 -ModelXXX-$\ `-,row105)SavingsrateTotalemissionsequalworldemissionslimit%%ET(t)=EL(t)(XXX-64 -ModelXXX-$\ `-,rows81and26)Carbontax4. Geoengineering Optimalsavingsbehavior%%DiffC=0(XXX-64 -ModelXXX-$\ `-,row105)SavingsrateTemperaturedamageeliminated%%XXX-7(\ `-qXXX-$\ `-1=0,XXX-7(\ `-qXXX-$\ `-2=0(XXX-64 -ModelXXX-$\ `-,rows5and6)5. Temperaturelimit Optimalsavingsbehavior%%DiffC=0(XXX-64 -ModelXXX-$\ `-,row105)SavingsrateOptimalcarbonabatementsubjecttotemperaturelimit%%DiffE=0,withdamagesasteepfunctionoftemperatureCarbontax6. Concentrationlimit Optimalsavingsbehavior%%DiffC=0(XXX-64 -ModelXXX-$\ `-,row105)SavingsrateOptimalcarbonabatementsubjecttoconcentrationlimit%%DiffE=0,withdamagesasteepfunctionofconcentrationCarbontaxɴ  carbontaxisspecifiedasthecontrolvariableinTable1,theusersearchesbyalteringtheUnconstrainedcarbontax(XXX-64 -ModelXXX-$\ `-,row110).....2. Notesonindividualconditions ......a. Optimalsavingsrates ..TheeasiestwaytofindtheoptimalsavingsratesistousetheGoalSeekcommand.FirstusetheGoalSeekcommandtosetDiffC(1)equaltozerobyvaryings(1),thendosoforperiod2,etc.Inthelastperiod,theSavingsrateshouldbesetsothatInvestmentisequalto2%oftheCapitalstock.ThisisdonebyusingtheGoalSeekcommandtosettheTerminalconditionforK(XXX-64 -ModelXXX-$\ `-,row77,columnAK)equaltozerobyvaryings(NOP).(NOP=numberofperiods.)..ThesequenceofcommandsdescribedintheaboveparagraphcanbecarriedoutbyexecutingthemacroFOSR...SincetheDiscountrate(Model,row104)forperiodtdependsontheSavingsrateinperiodt+1,theusermayneedtorepeatFOSRoneormoretimesinordertosetDiffC=0forallt.......b. Optimalcarbontax ..TheGoalSeekcommandcouldalsobeusedtofindtheCarbontaxthatsetsDiffE=0.AneasierwaytofindtheoptimalcarbontaxistotryvaluesintheUnconstrainedcarbontaxrowthataresomewherebetweentheCarbontaxandtheEnvironmentalshadowprice.RepeatingthiswillgenerallycauseDiffEtobecomesmallerandsmaller.Thatis,simplycopythevaluesoftheIterationstep(XXX-64 -ModelXXX-$\ `-,row109),whichisXXX-7(\ `-tXXX-$\ `-Ԁ+CPOT(ESPXXX-7(\ `-tXXX-$\ `-),intotheUnconstrainedcarbontaxrow(XXX-64 -ModelXXX-$\ `-,row110)andrepeatuntiltheCarbontaxissatisfactorilyclosetotheEnvironmentalshadowprice...ThemacroOptCTcarriesouttheoperationdescribedintheaboveparagraph100times...IncreasingCPOT( Model ,row27)willincreasetheiterationstepsize;thesteptakenineachiterationtowardESPwillbelarger.Thiswillreducethenumberofiterationsneededtoachieveconvergence,assumingthealgorithmremainsstable.ThedrawbackofincreasingCPOTisthatitincreasesthelikelihoodthatthealgorithmbecomesunstable.Intheoptimalcase,whereESPiscalculatedfromthedefaultdamagefunction,thealgorithmisgenerallystableevenifCPOTis1...Thealgorithmtofindtheoptimalcarbontaxisthesameintheoptimal,temperaturelimit,andconcentrationlimitcases.WhatdiffersacrossthesecasesisthecalculationoftheEnvironmentalshadowprice...Inordertouseouroptimalcarbontaxalgorithmtofindoptimalcarbonabatementsubjecttotemperatureorconcentrationlimits,wetransformthelimitsintodamagefunctions.Optimizingcarbonabatementsubjecttotheselimitsisthesameasoptimizingsubjecttoadamagefunctionthatiszerobeforethelimitisreachedandunacceptablyhighuponreachingthelimit.Sinceouralgorithmforfindingtheoptimalcarbontaxesrequiresacontinuousdamagefunction,weusedamagefunctionsthatareverysteepnearthelimitratherthancompletelyverticalatthelimit;thecarbontaxesthatwegetoutofouralgorithmwithsuchfunctionswillbeveryclosetotheoptimalones,althoughnotexactlythesame...Intheoptimalcase,wecalculatetheESPassuming(III.B.1.1)..D(t)=100(XXX-7(\ `-qXXX-$\ `-1T(t)+XXX-7(\ `-qXXX-$\ `-2T(t)2)Inthetemperaturelimitscase,wecalculateESPassuming(III.B.1.2)..D(t)=DSTD(t)=100CDCexp(T(t)EDC/TL)Andintheconcentrationlimitscase,wecalculateESPassuming(III.B.1.3)..D(t)=DSTD(t)=100CDCexp(M(t)EDC/CL)(D(t)isClimatedamage(Model,row74).)..BychoiceoftheparametersCDCandEDC,wecanmakeIII.B.1.2andIII.B.1.3asclosetoastepfunctionaswewantatthetemperatureorconcentrationlimit...Notethatinthetemperatureandconcentrationlimitcases,thesealternativeassumptionsaffectonlytheESPworksheet,notD(t)ontheXXX-64 -ModelXXX-$\ `-Ԁworksheet.Theactualdamagesdonotchangeinthesecases,onlytheincentivesgiventothemarketviathesolutionCarbontax...ToswitchfromthedefaultESPtothetemperature(concentration)limitscaseESP,usethemacroTlimdfunction(Climdfunction).Toswitchback,useNormaldfunction...Becauseofthesedamagefunctionsaresosteep,thealgorithmtofindtheoptimalcarbontaxwillgenerallybeunstableunlessCPOT( Model ,row27)issmall.Forexample,ifAtmospherictemperatureinalateperiodisnearthelimit,theESPforanearlyperiodwillbeveryhigh.IfCPOTisone,thealgorithmwilltryahighvalueoftheCarbontaxintheearlyperiods,whichwouldcauseemissionstobereducedintheearlyperiodsandlateperiodtemperaturetobedrivenawayfromthelimit.Sincethefunctionissosteep,thismightwelldrivetheESPdowntozero...ThedefaultvalueofCPOTis0.0025.....c. Emissionsequaltoworldlimit FromII.A.5.20andII.A.5.21,wecansolveforthecontrolrateXXX-7(\ `-mXXX-$\ `-ELneededtosatisfythesecondconditionforasolutionincase3,Table1:(III.B.1.4)..XXX-7(\ `-mXXX-$\ `-EL(t)=1[WEL(t)LU(t)]/[XXX-7(\ `-sXXX-$\ `-(t)Q(t)]..Intheemissionslimitcase,thelimitisacap,notatargettobemet;ifunregulatedemissionswouldbebelowthelimitinacertainperiod,thecontrolratewillsimplybezerointhesolution,notnegative.Thus,(III.B.1.5)..XXX-7(\ `-mXXX-$\ `-EL(t)=0,XXX-7(\ `-sXXX-$\ `-(t)Q(t)