SB 350 Study: Economic Model & Methodology · SB 350 Study: Economic Model & Methodology ......

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SB 350 Study: Economic Model & Methodology BEAR Technical Documentation (Early Release Materials to Stakeholders)

March 28, 2016

Berkeley Economic Advising and Research, LLC 1442A Walnut Street, Suite 108 Berkeley, CA 94705 www.bearecon.com

EarlyReleaseMaterials

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SB 350 Study: Economic Model & Methodology BEAR Model Technical Documentation (Early Release Materials to Stakeholders)

Prepared for:

Berkeley Economic Advising and Research

1442A Walnut Street, Suite 108

Berkeley, CA 94705

www.bearecon.com March 28, 2016 © All rights reserved.


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ABSRACT

The following document summarizes the structural characteristics of a dynamicforecastingmodelforthestateofCalifornia,designedtosupportresearchintoclimatechange,policyresponse,andtheireffectsacrossthis largeanddiversestateeconomy.The model integrates detailed treatment of sectoral production, employment, anddemandwithstatewideassessmentofenvironmentalpollutionandenergyuseoverthenexttwodecades.Thismodel iscurrentlyunderdevelopmentandall technicaldetailscoveredinthisoverviewaresubjecttochange.TheBEARmodelwasdevelopedaspartof a larger research exercise sponsored by, among others, the California EnergyCommission, the Energy Foundation, and Next10, whose support is gratefullyacknowledged. I amalso grateful to LarryGoulder for veryhelpful insights.All viewsexpressed here are those of the author and should not be attributed to his affiliatedinstitutions.


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2 TableofContents

1 INTRODUCTION ............................................................................................................... 5

2 PRODUCTION .................................................................................................................. 7

3 INCOME DISTRIBUTION .................................................................................................. 12

4 HOUSEHOLD CONSUMPTION AND SAVINGS ......................................................................... 14

5 OTHER FINAL DEMANDS .................................................................................................. 15

6 GOVERNMENT REVENUES AND SAVING ............................................................................... 16

7 TRADE, DOMESTIC SUPPLY AND DEMAND ........................................................................... 18

8 EQUILIBRIUM CONDITIONS ............................................................................................. 22

9 DETERMINATION OF VINTAGE OUTPUT AND CAPITAL MARKET EQUILIBRIUM ............................ 24

10 MACRO CLOSURE ......................................................................................................... 26

11 DYNAMICS .................................................................................................................. 27

12 EMISSIONS ................................................................................................................. 30

13 DEALING WITH UNCERTAINTY - STOCHASTIC VARIATIONAL ANALYSIS ................................... 34

APPENDEX 1 – THE CES/CET FUNCTIONS ......................................................................... 36

APPENDIX 2 - DATA STRUCTURE FOR THE BEAR MODEL .................................................. 42


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1 Introduction ThispaperprovidesthecompletetechnicalspecificationofacomputablegeneralequilibriummodelfortheCaliforniaeconomy,withdetailedtreatmentofenergyuseandenvironmentalpollution.Suchamodelcanbeusedtosupportabroadspectrumofpolicyanalysis,includingenergypolicyandpolicyresponsestoclimatechange.Thenextsectionprovidesabriefoverviewofthemainfeaturesofthemodel,whichisfollowedbyadetaileddescriptionofeachblockofthemodel.

Production

Allsectorsareassumedtooperateunderconstantreturnstoscaleandcostoptimization.Productiontechnologyismodeledbyanestingofconstant-elasticity-of-substitution(CES)functions.SeeFigure1foraschematicdiagramofthenesting.TheimplementationofthemodelallowsforallpermissiblespecialcasesoftheCESfunction,notablyLeontiefandCobb-Douglas.Ineachperiod,thesupplyofprimaryfactors—capital,land,andlabor—isusuallypredetermined.1Themodelincludesadjustmentrigidities.Animportantfeatureisthedistinctionbetweenoldandnewcapitalgoods.Inaddition,capitalisassumedtobepartiallymobile,reflectingdifferencesinthemarketabilityofcapitalgoodsacrosssectors.2Oncetheoptimalcombinationofinputsisdetermined,sectoraloutputpricesarecalculatedassumingcompetitivesupply(zero-profit)conditionsinallmarkets.

Consumption and Closure Rule

Allincomegeneratedbyeconomicactivityisassumedtobedistributedtoconsumers.Eachrepresentativeconsumerallocatesoptimallyhis/herdisposableincomeamongthedifferentcommoditiesandsaving.Theconsumption/savingdecisioniscompletelystatic:savingistreatedasa“good”anditsamountisdeterminedsimultaneouslywiththedemandfortheothercommodities,thepriceofsavingbeingsetarbitrarilyequaltotheaveragepriceofconsumergoods.3Thegovernmentcollectsincometaxes,indirecttaxesonintermediateinputs,outputsandconsumerexpenditures.Thedefaultclosureofthemodelassumesthatthegovernmentdeficit/savingisexogenouslyspecified.4Theindirecttaxschedulewillshifttoaccommodateanychangesinthebalancebetweengovernmentrevenuesandgovernmentexpenditures.Thecurrentaccountsurplus(deficit)isfixedinnominalterms.Thecounterpartofthisimbalanceisanetoutflow(inflow)ofcapital,whichissubtracted(addedto)thedomesticflowofsaving.Ineachperiod,themodelequatesgrossinvestmenttonet

1 Capital supply is to some extent influenced by the current period’s level of investment. 2 For simplicity, it is assumed that old capital goods supplied in second-hand markets and new capital goods are homogeneous. This formulation makes it possible to introduce downward rigidities in the adjustment of capital without increasing excessively the number of equilibrium prices to be determined by the model (see Fullerton, 1983). 3 The demand system is a version of the Extended Linear Expenditure System (ELES) which was first developed by Lluch (1973). The formulation of the ELES in this model is based on atemporal maximisation — see Howe (1975). In this formulation, the marginal propensity to save out of supernumerary income is constant and independent of the rate of reproduction of capital. 4 In the reference simulation, the real government fiscal balance converges (linearly) towards 0 by the final period of the simulation.


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saving(equaltothesumofsavingbyhouseholds,thenetbudgetpositionofthegovernmentandoutofstatecapitalinflows).Thisparticularclosureruleimpliesthatinvestmentisdrivenbysaving.

Trade

Goodsareassumedtobedifferentiatedbyregionoforigin,includinggoodsfromabroadandfromtherestoftheUnitedStates.Inotherwords,goodsclassifiedinthesamesectoraredifferentaccordingtowhethertheyareproduceddomesticallyorimported.ThisassumptionisfrequentlyknownastheArmingtonassumption.Thedegreeofsubstitutability,aswellastheimportpenetrationsharesareallowedtovaryacrosscommoditiesandacrossagents.ThemodelassumesasingleArmingtonagent.Thisstrongassumptionimpliesthatthepropensitytoimportandthedegreeofsubstitutabilitybetweendomesticandimportedgoodsisuniformacrosseconomicagents.Thisassumptionreducestremendouslythedimensionalityofthemodel.Inmanycasesthisassumptionisimposedbythedata.Asymmetricassumptionismadeontheexportsidewheredomesticproducersareassumedtodifferentiatethedomesticmarketandtheexportmarket.ThisisimplementedusingaConstant-Elasticity-of-Transformation(CET)productionpossibilityfrontier.

Dynamic Features and Calibration

Thecurrentversionoftheprototypehasasimplerecursivedynamicstructureasagentsareassumedtobemyopicandtobasetheirdecisionsonstaticexpectationsaboutpricesandquantities.Dynamicsinthemodeloriginateinthreesources:i)accumulationofproductivecapitalandlaborgrowth;ii)shiftsinproductiontechnology;andiii)theputty/semi-puttyspecificationoftechnology.

(a) Capital accumulation

Intheaggregate,thebasiccapitalaccumulationfunctionequatesthecurrentcapitalstocktothedepreciatedstockinheritedfromthepreviousperiodplusgrossinvestment.However,atthesectorallevel,thespecificaccumulationfunctionsmaydifferbecausethedemandfor(oldandnew)capitalcanbelessthanthedepreciatedstockofoldcapital.Inthiscase,thesectorcontractsovertimebyreleasingoldcapitalgoods.Consequently,ineachperiod,thenewcapitalvintageavailabletoexpandingindustriesisequaltothesumofdisinvestedcapitalincontractingindustriesplustotalsavinggeneratedbytheeconomy,consistentwiththeclosureruleofthemodel.

(b) The putty/semi-putty specification

Thesubstitutionpossibilitiesamongproductionfactorsareassumedtobehigherwiththenewthantheoldcapitalvintages—technologyhasaputty/semi-puttyspecification.Hence,whenashocktorelativepricesoccurs(e.g.theimpositionofanemissionstax),thedemandsforproductionfactorsadjustgraduallytothelong-runoptimumbecausethesubstitutioneffectsaredelayedovertime.Theadjustmentpathdependsonthevaluesoftheshort-runelasticitiesofsubstitutionandthereplacementrateofcapital.Asthelatterdeterminesthepaceatwhichnewvintagesareinstalled,thelargeristhevolumeofnewinvestment,thegreaterthepossibilitytoachievethelong-runtotalamountofsubstitutionamongproductionfactors.


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(c) Dynamic calibration

Themodeliscalibratedonexogenousgrowthratesofpopulation,laborforce,andGDP.Intheso-calledBusiness-as-Usual(BaU)scenario,thedynamicsarecalibratedineachregionbyimposingtheassumptionofabalancedgrowthpath.Thisimpliesthattheratiobetweenlaborandcapital(inefficiencyunits)isheldconstantovertime.5Whenalternativescenariosaroundthebaselinearesimulated,thetechnicalefficiencyparameterisheldconstant,andthegrowthofcapitalisendogenouslydeterminedbythesaving/investmentrelation.Intheequationswhichfollow,thefollowingindiceswillbeusedextensively.Notethatthetimeindexgenerallybedroppedfromtheequations.

i Production sectors. j is an alias for i.

nf Represents the non-fuel commodities.

e Represents fuel commodities.

l Represents the labor types.

v Represents the capital vintages.

h Represents the households.

g Represents the government expenditure categories.

f Represents the final demand expenditure categories (including g as a subset).

t Time index.

2 Production ProductionisbasedonanestedstructureofConstantElasticityofSubstitution(CES)functions.Eachsectorproducesagrossoutput6,XP,whichgiventheassumptionofconstantreturnstoscaleisundeterminedbytheproducer.Itwillbedeterminedbyequilibriumconditions.TheproducerthereforeminimizescostssubjecttoaproductionfunctionwhichisoftheCEStype.Atthefirstlevel,theproducerchoosesamixofavalueaddedaggregate,VA7,andanintermediatedemandaggregate,ND.8Inmathematicalterms,thisleadstothefollowingformulation:

s.t.

5This involves computing in each period a measure of Harrod-neutral technical progress in the capital-labor bundle as a residual. This is a standard calibration procedure in dynamic CGE modeling — see Ballard et. al. (1985). 6 Gross output is divided into two parts, one part produced with old capital, and the residual amount produced with new capital. 7 The value added bundle also contains demand for energy, see below. 8 Some models of this type assume a top level Leontief, i.e. a substitution elasticity of zero, in which case there is no substitution possibility between intermediate demand and value added. The GAMS implementation of the model can handle all of the special cases of the CES, i.e. Leontief and Cobb-Douglas.


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wherePVAistheaggregatepriceofvalueadded,PN,isthepriceoftheintermediateaggregate,avaandandaretheCESshareparameters,andρistheCESexponent9.TheexponentisrelatedtotheCESelasticity,viathefollowingrelationship:

Notethatinthemodel,theshareparametersincorporatethesubstitutionelasticityusingthefollowingrelationships:

Solvingtheminimizationproblemabove,yieldsEquations(2.1.1)and(2.1.3)inTable2.1.Becauseoftheassumptionofvintagecapital,weareallowingthesubstitutionelasticitiestodifferaccordingtothevintageofthecapital.Dependingontheavailabledata,andduetotheimportanceofenergyintermsofpollution,weseparateenergydemandfromtherestofintermediatedemand,andincorporatethedemandforenergydirectlyinthevalueaddednest.Hence,theequationsbelowarenotspecifiedintermsofavalueaddedbundle,butavalueaddedplusenergybundle.Equation(2.1.1)determinesthevolumeofaggregateintermediatenon-energydemand,byvintage,ND.Equation(2.1.2)determinesthetotaldemandforintermediatenon-energyaggregateinputs(summedovervintages),ND.Equation(2.1.3)determinesthelevelofthecompositebundleofvalueaddeddemandandenergy,KEL.TheCESdualpriceofND,andKEL,PXv,isdefinedbyEquation(2.1.4).Equation(2.1.5)determinestheaggregateunitcost,PX,exclusiveofanoutputsubsidy/tax10.Finally,weallowthepossibilityofanoutputsubsidyortax,generatingawedgebetweentheproducerpriceandtheoutputprice,PP,yieldingEquation(2.1.6).Theproductiontaxismultipliedbyanadjustmentfactorwhichnormallyisfixedatunitvalue.However,itispossibletoendogenizetheaverageleveloftheproductiontaxtoachieveapre-determinedfiscaltarget.

9 The CES is described in greater detail in Appendix 1. 10 The unit cost equation will be affected by production-specific emission taxes. Emission taxes are discussed in section 12.


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Table2.1:TopLevelProductionEquations

(2.1.1)

(2.1.2)

(2.1.3)

(2.1.4)

(2.1.5)

(2.1.6)

ThenextleveloftheCESnestconcernsontheonesideaggregateintermediatedemand,ND,andontheotherside,theKELbundle.ThesplitofNDintointermediatedemandisassumedtofollowtheLeontiefspecification,inotherwordsasubstitutionelasticityof0.(Wealsoassumethatthesharecoefficientsareindependentofthevintage.)Thedemandfornon-fuelintermediategoodsisdeterminedbyEquation(2.2.1).Theintermediatedemandcoefficientsaregivenbyanf,j.Thepriceofaggregateintermediatedemandisgivenbyaddinguptheunitpriceofintermediatedemand.ThisisspecifiedinEquation(2.2.2)inTable2.2.DemandforeachgoodisspecifiedasademandfortheArmingtoncomposite(describedinmoredetailbelow),anaggregationofadomesticgoodandanimportgoodwhichareimperfectsubstitutes.Therefore,whilethereisnosubstitutionofoneintermediategoodforanother,therewillbesubstitutionbetweendomesticdemandandimportdemanddependingontherelativeprices.ThepriceoftheArmingtongoodisgivenbyPA.Atthesamelevel,theKELbundleissplitbetweenlaborandacapital-energybundle,KE.Itisassumedhereaswell,thatthesubstitutionpossibilitiesbetweenlaborandtheKEbundledependonthevintageofthecapital.Theoptimizationproblemissimilartoabove,i.e.costminimizationsubjecttoaCESaggregationfunction.IfAWistheaggregatesectoralwagerate,andPKEisthepriceoftheKEbundle,aggregatelabordemand,ALanddemandfortheKEbundle,aregivenbyEquations(2.2.3)and(2.2.4).αl,iandαke,iaretheCESshareparameters,andσvistheCESelasticityofsubstitution.ThepriceofKELbundle,PKEL,isdeterminedbyEquation(2.2.5),whichistheCESdualprice.


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Table2.2:SecondLevelCESProductionEquations

(2.2.1)

(2.2.2)

(2.2.3)

(2.2.4)

(2.2.5)

Thecombinedlaborbundleissplitintolabordemandbytypeoflabor,eachwithaspecificwagerate,W.11(Thoughlabormarketsareassumedtoclearforeachskillcategory,weallowfordifferentialwageratesacrosssectorsreflectingpotentialdifferentinstitutionalarrangements).Equation(2.3.1)determineslabordemandbyskilltypeineachsector,usingaCESaggregationfunction.Weallowforchangesinlaborefficiencywhichcanbespecifiedbybothskilltypeandbysector.Thedualprice,ortheaveragesectoralwage,AW,isdefinedbyEquation(2.3.2).

Table2.3:LaborDemand

(2.3.1)

(2.3.2)

Thenexttabledescribesthedisaggregationofthecapital-energybundle,KE,intoitsenergyandcapital-landcomponents.Equation(2.4.1)determinesthedemandforaggregateenergy.Equation(2.4.2)determinesthedemandforthecapital-landbundle

11 The current CALIFORNIA SAM has a single aggregate labor account, though both the data processing facility and the model can handle multiple labor accounts. Depending on the level of labor disaggregation, it might be appropriate to have a more detailed nesting structure for labor, rather than a single level nest.


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byvintage,KT,wherePKTisthepriceofthecapital-landbundle.Equation(2.4.3)definesthedualpriceoftheKEbundle.

Table2.4:Capital-LandBundleandEnergyBundleDemand

(2.4.1)

(2.4.2)

(2.4.3)

Thenextlevelinthenestdeterminesthedemandforthecapitalandlandfactors.Equation(2.5.1)defineslanddemandbysectorandvintage,Tv,wherePTisthepriceofland.Similarly,Equation(2.5.2)determinesdemandforcapitalbysectorandvintage,Kv,whereRistherentalrateofcapital.Notethattherentalrateisbothsectorandvintagespecific.Bothequationsincorporatetechnologyshifters(whichwillbeexplainedinthesectionondynamics).12Equations(2.5.4)and(2.5.5)determinerespectivelyaggregatesectorallanddemandandcapitaldemand.

Table2.5:CapitalandLandDemand

(2.5.1)

(2.5.2)

(2.5.3)

(2.5.4)

(2.5.5)

12 The current data for the CALIFORNIA model does not include land as a separate account, therefore the additional capital-land nest, though active, is irrelevant.


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TheenergybundledeterminedbyEquation(2.4.1)isfurtherdisaggregatedbyenergy-type.Thenumberoffueltypeswilldependontheavailabledata.Welettheindexerangeoverthenumberoffueltypes(eventuallythedimensionofecouldevenbe1).Equation(2.6.1)determinesthedemandforthedifferenttypesoffuels.Theλfactorallowsforenergyefficiencyimprovementovertimewhichcanbesectorspecific,aswellasvintagespecific.Equation(2.6.2)determinestheCESdualprice,PEv,oftheenergybundle.

Table2.6:DecompositionoftheEnergyBundle

(2.6.1)

(2.6.2)

3 Income Distribution Productiongeneratesincome,bothwageandnon-wage,whichisdistributedinsomeformtothreemaininstitutions:households,government,andfinancialinstitutions(bothdomesticandoutofstate).Equation(3.1.1)determinesgrossoperatingsurplus,KY.Itisthesumacrossallvintagesandallsectorsofcapitalremuneration.Equation(3.1.2)definescompanyincome,CY,itisequaltoashareofgrossoperatingsurplus(therestbeingdistributedtohouseholds).Equation(3.1.3)determinescorporatetaxes,Taxc.Thebasetaxrateisgivenbytheparameterκc.However,corporatetaxescanbemadeendogenous(inordertomeetafiscaltarget,forexample),inwhichcasetheadjustmentparameter,δc,becomesendogenous.Equation(3.1.4)definesretainedearnings,i.e.corporatesaving.Corporatesavingisequaltoaresidualshareofafter-taxcompanyincome.13Theremainingamountofnetcompanyincomeisdistributedtohouseholds.

13 In the reference simulation, both the private corporate saving rate and the household saving rate are adjusted (upwards), under the assumption that domestic saving, as a share of GDP, will increase in the future. The adjustments are based on rules of thumb, but could be made explicit in the model.


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Table3.1:CorporateEarningsEquations

(3.1.1)

(3.1.2)

(3.1.3)

(3.1.4)

Householdincomederivesfromtwomainsources,capitalandlaborincome.Additionally,householdsreceivetransfersfromthegovernment.Equation(3.2.1)definestotallaborincome,YLastheproductoftotallabordemandandthewagerate.Laborincomeisdistributedtothehouseholds.Equation(3.2.2)definestotalhouseholdincome,YH.Itisthesumoflaborincome,distributedcapitalincomeandnetcompanyincome,incomefromland,andtransfersfromthegovernment,TRgh.Capital,company,andlandincomearedistributedtohouseholdsusingfixedshares.TheadjustmentfactorδHTrongovernmenttransferscanbeusedasafiscalinstrumentinordertoachieveaspecifiedtarget,similartotheadjustmentfactorsonothertaxesinthemodel.Householddirecttax,Taxh,isgivenbyEquation(3.2.3),whereκhisthetaxrate.TheadjustmentfactorδHTxcanbeendogenousifthegovernmentsaving/deficitisexogenous.Inthiscase,thehouseholdtaxschedulesshiftsinorouttoachievethenetgovernmentbalance.Otherwise,thehouseholdtaxscheduleisexogenous,andthefactorstaysatitsinitialvalueof1.FinallyEquation(3.2.4)defineshouseholddisposableincome,YD.Disposableincomeisequaltototalhouseholdincomelesstaxes.

Table3.2:HouseholdIncomeEquations

(3.2.1)

(3.2.2)

(3.2.3)

(3.2.4)


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4 Household Consumption and Savings Householddisposableincomeisallocatedtogoods,services,labor,andsavingsusingtheExtendedLinearExpenditureSystem(ELES)specification.14Theconsumerproblemcanbesetupasfollows:

whereUistheutilityfunction,Ciisconsumptionbycommodity,Sishouseholdsaving,PCisthevectorofconsumerprices,andYDisdisposableincome.µandθareparameterswhichwillbegivenaninterpretationbelow.Scanbethoughtofasdemandforafuturebundleofconsumergoods.Forreasonsofsimplification,itisassumedthatthesavingbundleisevaluatedusingtheconsumerpriceindex,cpi.Lluchprovidesamoredetailedtheoreticalanalysisofhowsavingsenterstheutilitymaximizationproblem.Solvingtheaboveoptimizationproblemleadstothefollowingdemandfunctions:

Consumptionisthesumoftwoparts,θ,whichisoftencalledthesubsistenceminimaorfloorconsumption,andafractionofY*,whichisoftencalledthesupernumeraryincome.Y*isequaltodisposableincomelesstotalexpendituresonthesubsistenceminima.Table4.1presentstheequationsoftheconsumerdemandsystem.Equation(4.1.1)definestheconsumerpricevector(forgoodsandservices),PC,itistheArmingtonpriceincorporatinghouseholdspecificindirecttaxesandsubsidies.Equation(4.1.2)definessupernumeraryincome,thatis,disposableincomelesstotalexpendituresonthesubsistenceminima.(Thesubsistenceminimaareadjustedeachtimeperiodbythegrowthrateinpopulation).ConsumerdemandforgoodsandservicesisgivenbyEquation(4.1.3).15HouseholdsavingsisdeterminedasaresidualandisgiveninEquation(4.1.4).AggregatehouseholdsavingisdeterminedbyEquation(4.1.5).Equation(4.1.6)definestheconsumerpriceindex.

14 For references, see Lluch (1973) or Deaton and Muellbauer (1980). 15 As noted earlier, the m parameters are adjusted in the reference simulation in order to increase the level of domestic saving.


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Table4.1:HouseholdConsumptionandSavingsEquations

(4.1.1)

(4.1.2)

(4.1.3)

(4.1.4)

(4.1.5)

(4.1.6)

5 Other Final Demands Allotherfinaldemandaccounts(exceptstockchanges)areintegratedintoasingledemandmatrixcomponent.Inthemostgeneralversionofthemodel,thefinaldemandcomponentsaregovernmentcurrentexpenditures,governmentcapitalexpenditures,privatecapitalexpenditures,tradeandtransportmarginsfordomesticsales,andtradeandtransportmarginsforimports.Allthefinaldemandvectorsareassumedtohavefixedexpenditureshares.Theclosureofthefinaldemandaccountswillbediscussedbelow.Equation(5.1.1)determinesthecompositionoffinaldemandforeachofthefinaldemandcomponents.Demandforgoodsisdeterminedasconstantsharesofthevolumeoftotalfinaldemand,TFD.Theindexfcoversgovernmentcurrentandcapitalexpenditures,privatecapitalexpenditure,inventorychange,andbothdomesticandimporttrademarginexpenditures.Equation(5.1.2)determinesthevalueoffinaldemandexpenditures,TFDV.Equation(5.1.3)determinesthepriceoffinaldemandexpendituresinclusiveoftaxesandsubsidies,PFD.Equation(5.1.4)determinestheaggregatefinaldemandpricedeflatorforeachtypeoffinaldemandaccount,PTFD.

Table5.1:FinalDemandExpenditureEquations

(5.1.1)

(5.1.2)

(5.1.3)

(5.1.4)


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Governmentcurrentexpendituresincludeexpendituresongoodsandservices.Governmentaggregateexpendituresongoodsandservicesarefixedinrealterms.Totalnominalgovernmentexpenditures,GExp,isdeterminedbyEquation(5.2.1)inTable5.2.Thereareseveralexogenouselementswhichenterthisequationincludingtransferstohouseholds,TRgh.Notethepotentialadjustmentfactorattachedtothehouseholdtransfervariable.Alsonotethatalldomestictransfersaretypicallyheldfixedandaremultipliedbyapriceindexinordertoinsurethehomogeneityofthemodel.Equation(5.2.2)definesthegovernmentexpendituredeflator,PG.Finally,Equation(5.2.3)issimplyanidentitywhichequatesaggregaterealgovernmentexpenditurestothevariableTFD(fortheaccountsindexedbyg).

Table5.2:CurrentGovernmentExpenditureEquations

(5.2.1)

(5.2.2)

(5.2.3)

6 Government Revenues and Saving Governmentderivesmostofitsrevenuesfromdirectcorporateandhouseholdtaxes,andindirecttaxes.Subsidiesarealsoprovidedwhichenterasnegativerevenues.Equations(6.1.1)-(6.1.4)inTable6.1listallthedifferentindirecttaxespaidbyproductionactivities,householdconsumption,finaldemandexpenditures,andexports,respectively,PITx,SITx,HITx,FDITx,andEITx.Equation(6.1.5)describesthesumofallindirecttaxes.


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Table6.1:IndirectTaxEquations

(6.1.1)

(6.1.2)

(6.1.3)

(6.1.4)

(6.1.5)

Equations(6.2.1)-(6.2.3)inTable6.2definethelevelofsubsidiesforhouseholdconsumption,otherfinaldemandexpenditures,andexports,respectively,HSubs,FDSubs,andESubs.TotalsubsidiesisgivenbyEquation(6.2.4).

Table6.2:SubsidyEquations

(6.2.1)

(6.2.2)

(6.2.3)

(6.2.4)

Table6.3definesfiscalclosureforthegovernment.Equation(6.3.1)describestotalincomefromimporttariffs,whereWPMareworldprices,τmaretariffs,andXMrrepresentsimportvolumes.Alltherelevantimportvariablesaredoublyindexedsincetheyrepresentvariablesbysectorandregionoforigin.Theexchangerateisusedtoconvertworldprices(e.g.indollars)intolocalcurrency.ThereisanadditionaladjustmentfactorδTarwhichallowstheaggregatetariffratetovaryendogenously.Equation(6.3.2)identifiesmiscellaneousgovernmentrevenuesources,theseareallrevenueslesshouseholddirecttaxes.Equation(6.3.3)providestotalcurrentgovernmentnominalrevenues,GRev.Equation(6.3.4)and(6.3.5)definerespectivelythenominalandreallevelofgovernmentsaving.Twogovernmentclosurerulesareimplemented.Underthedefaultrule,governmentsavingisheldfixed(typicallyatitsbasevalue),andoneofthetaxes(orgovernmenttransferstohouseholds)isallowedtoadjust(uniformly)toachievethegovernmentfiscaltarget.Underthesecondclosurerule,alltaxlevelsandtransfersarefixed,andrealgovernmentsavingisendogenous.


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Thislatterrulecanhavesignificantconsequencesonthelevelofinvestmentsinceinvestmentissavingsdriven.

Table6.3:GovernmentRevenuesandClosureEquations

(6.3.1)

(6.3.2)

(6.3.3)

(6.3.4)

(6.3.5)

7 Trade, Domestic Supply and Demand SimilartomanytradeCGEmodels,wehaveassumedthatimportedgoodsarenotperfectsubstitutesforgoodsproduceddomestically.16Thedegreeofsubstitutionwilldependonthelevelofdisaggregationofthecommodities.Forexample,wheatismoresubstitutableasacommoditythangrains,whichinturnaremoresubstitutablethanacommoditycalledprimaryagriculturalproducts.TheArmingtonassumptionreflectstwostylizedfacts.Tradedatashowstheexistencetwo-waytradewhichisconsistentwiththeArmingtonassumption.Aswell,andrelated,theArmingtonassumptionleadstoamodelwhereperfectspecialization,whichisrarelyobserved,isavoided.Inthisversionofthemodel,wehaveadaptedtheCESfunctionalspecificationfortheArmingtonassumption.Thishassomeundesirablepropertieswhichhavebeenexploredinmoredetailelsewhere17,butalternativeformulationshaveproventobedeficientaswell.TheadoptionoftheConstantElasticityofTransformation(CET)specificationforexportsalleviatestosomeextentthedeficienciesoftheArmingtonCESspecification.WealsoassumethatthereisonlyonedomesticArmingtonagent,thisissometimesknownasborder-levelArmingtonspecification.Itisparsimoniousinbothdatarequirementsandcomputationalresources.Toallowfortheexistenceofmultipletradingpartners,themodeladoptsatwo-levelCESnestingtorepresenttheArmingtonspecification(seeFigure2).18Atthetoplevel,agentschooseanoptimalcombinationofthedomesticgoodandanimportaggregatewhichisdeterminedbyasetofrelativepricesandthedegreeofsubstitutability.LetXArepresentaggregatedemandforanArmingtoncomposite,withtheassociatedArmingtonpriceofPA.EachagentthenminimizesthecostofobtainingtheArmingtoncomposite,subjecttoanaggregationfunction.Thiscanbeformulatedby:

16 This is known as the Armington assumption — see Armington (1969). 17 See for example Robinson et. al. (1992). 18 The current CALIFORNIA SAM has a single rest of the world account, i.e. an aggregate trading partner. The dual nesting is therefore redundant. However, both the data processing facility and the model retain the multiple trading partner specification in order to maintain flexibility for future data developments.


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whereXDisdemandforthedomesticgood,PDisthepriceofobtainingthedomesticgood,XMisdemandfortheaggregateimportedgood,PMistheaggregateimportprice,aaretheCESshareparameters,andρistheCESexponent.ρisrelatedtotheCESsubstitutionelasticityviathefollowingrelation:

Atthesecondlevelofthenest,agentschoosetheoptimalchoiceofimportsacrossregions,againasafunctionoftherelativeimportpricesandthedegreeofsubstitutionacrossregions.Notethattheimportpricesareregionspecific,asarethetariffrates.ThesecondlevelnestalsousesaCESaggregationfunction.TheCESformulationimpliesthatthesubstitutionbetweenanytwopairsofimportingpartnersisidentical.Table7.1liststhesolutionoftheoptimizationproblemdescribedabove.Equation(7.1.1)determinesdomesticdemandfortheArmingtonaggregateacrossallagentsoftheeconomy,XA.Equations(7.1.2)and(7.1.3)determinerespectively,theoptimaldemandforthedomesticcomponentoftheArmingtonaggregate,XD,andaggregateimportdemand,XM.Equation(7.1.4)definesthepriceoftheArmingtonbundle,PA,whichistheCESdualprice.

Table7.1:Top-levelArmingtonEquations

(7.1.1)

(7.1.2)

(7.1.3)

(7.1.4)

TheequationsinTable7.2describethedecompositionoftheaggregateimportbundle,XMintoitscomponents,i.e.importsbyregionoforigin.Eachdemandcomponentwillbeafunctionofthepriceoftheexportingpartner,aswellaspartner-specifictariffrates.Equation(7.2.1)determinesimportvolumebysectorandregionoforigin,XMr,wherePMristhepartnerspecificimportprice,indomesticcurrencyandinclusiveoftariffs.Equation(7.2.2)definesthepriceoftheaggregateimportbundle,PM,whichistheCESdualprice.Finally,Equation(7.2.3)definesthedomesticimportprice,PMr,whichisequaltotheimportpriceofthetradingpartner,convertedintolocalcurrency,andinclusiveofthepartner-specifictariffrate.


-- 20

Table7.2:Second-levelArmingtonEquations

(7.2.1)

(7.2.2)

(7.2.3)

Treatmentofdomesticproductionissymmetrictothetreatmentofdomesticdemand.Domesticproducersareassumedtoperceivethedomesticmarketasdifferentfromtheexportmarket.Thereasonissimilar:ahighlevelofaggregation.Further,exportmarketsmightbemoredifficulttopenetrate,forcingperhapsdifferentqualitystandardsthanthoseapplicableforthedomesticmarket.Thisformulationassumesaproductionpossibilitiesfrontierwhereeachproducermaximizessales,subjecttobeingonthefrontier,andinfluencedbyrelativeprices.Theoptimizationproblemisformulatedsomewhatdifferentlysincetheobjectofthelocalproduceristomaximizesales,nottominimizecosts.Wethereforehave:

whereXDisaggregatedomesticsalesofdomesticproduction,ESisoutofstatesalesofdomesticproduction(exports),withaproducerexportpriceofPE,XPisaggregatedomesticproductionwithaproducerpriceofPP,γaretheCETshareparameters,andlistheCETexponent.TheCETexponentisrelatedtotheCETsubstitutionelasticity,Λviathefollowingrelation:

AnalogoustotheArmingtonspecification,producersupplydecisionsareassumedtobeundertakenittwosteps(seeFigure3).First,producerschoosetheoptimalcombinationofdomesticsupplyandaggregateexportsupply.Then,anadditionalstepwhichoptimizesexportsupplyacrosstradingpartners.Thetop-levelproducersupplydecisions,inreducedform,aregivenbyEquations(7.3.1)and(7.3.2),wheretheshareparametersareαtandtheCETsubstitutionelasticityisσt.19Equation(7.3.3)istheCETdualpricefunction,whichdeterminessectoraldomesticoutput.IftheCETelasticityisinfinite,producersperceivenodifferentiationacrossmarkets,inwhichcasebothdomesticandexportgoodsaresoldattheuniformproducerprice,PP,andoutputissimplythesumofdomesticsupplyandexportsupply.(Theformulasreflectanadjustmentforstockbuilding.Stockbuildingisassumedtooccurusingonly19 Note the difference between the Armington CES and the CET. First, the relation between the exponent and the substitution elasticity is different. Second, the ratio of the prices and the share parameter in the reduced forms are inverted. This is logical since the goal of the producer is to maximize revenues. For example, an increase in the price of exports, relative to the composite aggregate price, will lead to an increase in export supply.


-- 21

domesticallyproducedgoods,whicharepricedattheaggregateproducerprice,PP.Sectoralstockbuildingismodeledasafixedshareofavolumeofstockbuilding,StB.Thisformulationimpliesthatstockbuildingissimplysubtracted(added)from(to)totalcurrentoutput,XP.)

Table7.3:Top-levelCETEquations

(7.3.1)

(7.3.2)

(7.3.3)

Thesecond-levelCETnestdeterminestheoptimalsupplyofexportstoindividualtradingpartners,ESr.Equation(7.4.1)definesexportsupplybyregionofdestination.Equation(7.4.2)determinestheaggregateexportprice,PE.

Table7.4:Second-levelCETEquations

(7.4.1)

(7.4.2)

Table7.5presentstheequationswhichdetermineexportdemandbytheregionaltradingpartners,andtheexportmarketequilibriumcondition.Equation(7.5.1)definesexportdemandbytradingpartner,ED.Iftheexportingcountryhassomemarketpower,itwillfaceadownwardslopingdemandcurve.Thisisimplementedusingaconstantelasticityfunction,withtheelasticitygivenbyσe.Exportdemandwillalsobeinfluencedbythepriceofcompetingexports.ThisisreflectedinthevariableWPINDEX,whichisexogenoussinceitisassumedthedomesticeconomydoesnotinfluenceexportpricesof


-- 22

itstradingpartners.(ChangesintheWPINDEXcouldshowtheimpactsofexogenouschangesintheterms-of-trade).Underthesmall-countryassumptiontheexportdemandelasticityisinfinity,andtheexportingcountryfacesaflatdemandcurve,i.e.theexportpriceisfixed(indollarterms).Equation(7.5.2)convertsthedomesticexportproducerpriceintothedomesticexportpriceinclusiveoftaxesandsubsidies(however,itisstillinlocalcurrency).Equation(7.5.4)definestheexportmarketequilibrium,i.e.theequalitybetweendomesticexportsupplyandoutofstatedemand.

Table7.5:ExportDemandandMarketEquilibrium

(7.5.1)

(7.5.2)

(7.5.3)

8 Equilibrium Conditions Thefirstfactormarketequilibriumconditionconcernslabor.Labordemand,byskilltypeisgeneratedbyproductiondecisions.Intermsofsupply,themodelimplementsasimplelaborsupplycurve,wherelaborsupplyisafunctionoftherealwage.Equation(8.1.1)definesthelaborsupplycurve.Ifthesupplyelasticityislessthaninfinity,laborsupplyisafunctionoftheequilibriumrealwagerate.Intheextremecasewheretheelasticityiszero,laborisfullyemployedandfixed.Iftheelasticityisinfinite,therealwageisfixedandthereisnoconstraintonlaborsupply.Thismaybeanappropriateassumptionincaseswherethelevelofunemploymentisrelativelyhigh.Equation(8.1.2)determinesequilibriumonthelabormarket.Ifthelaborsupplycurveisnotflat,itdeterminestheequilibriumwagerate.Ifthelaborsupplycurveisflat,itsetslaborsupplyidenticallyequaltoaggregatelabordemand.Laborbyskilltypeisassumedtobeperfectlymobileacrosssectors,thereforeEquation(8.1.2)determinestheuniformwagebyskilltype.Becausethemodelallowsforwagestovaryacrosssectors,theuniformwageisactuallytheaggregatewagewhichvariesuniformlyacrosssectorsforeachskilltype,andtherelativewagesacrosssectorsareheldfixedattheirbaselevels.


-- 23

Table8.1:EquilibriumConditionsfortheLaborMarket

(8.1.1)

(8.1.2)

Landmarketdemand,similartodemandforlaborandcapital,isgeneratedbytheproductionsector.LandsupplyismodeledusingtheCETspecification.Iftheelasticityisinfinite,landisperfectlymobileacrosssectors.Iftheelasticityiszero,landisfixedandsector-specific.Betweenthesetwoextremevalues,landispartiallymobileandsectoralsupplywillreflecttherelativerate-of-returnoflandacrosssectors.Equations(8.2.1)-(8.2.3)reflecteithersituation(finiteorinfinite).InthecaseofafiniteCETelasticity,Equation(8.2.1)determinestheaggregatepriceofland,PLand,whichistheCETdualprice.TLandisaggregatelandsupplywhichisexogenous.Equation(8.2.2)determinessectoralsupplyofland,Ts,andEquation(8.2.3)istheequilibriumconditionwhichdeterminesthesector-specificlandprice,PT.Inthecaseofinfiniteelasticity,Equation(8.2.1)determinestheaggregate(uniform)priceoflandthroughanequilibriumconditionwhichequatestotallandsupply,TLand,toaggregatelanddemand.Equation(8.2.2)triviallysetsthesectorallandpriceequaltotheeconomy-widelandprice,andEquation(8.2.3)equatessectoralsupplytosectoraldemand.

Table8.2:LandSupplyandMarketEquilibrium

(8.2.1)

(8.2.2)

(8.2.3)


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9 Determination of Vintage Output and Capital Market Equilibrium

Themodelissetuptorunineithercomparativestaticmodeorrecursivedynamicmode.Capitalmarketequilibriumisdifferentinthetwocases,andeachwillbedescribedseparately.ComparativeStaticCapitalMarketEquilibriumIncomparativestaticmode,thereisnodistinctionmadebetweenoldandnewcapital.Eachsectordeterminesdemandforasingleaggregatecapitalgood.Onthesupplyside,themodelimplementsaCETsupplyallocationfunction(similartolandabove).Thereisasingle“capitalist”whoownsallthecapitalintheeconomy,andsuppliesittothedifferentsectorsbasedoneachsector’srateofreturn.Capitalmobilityacrosssectorsisdeterminedbythe“capitalist’s”CETsubstitutionelasticity.Thesubstitutionelasticityisallowedtovaryfrom0toinfinity.Iftheelasticityis0,thereisnocapitalmobility.Thisisanadequatedescriptionofashorttermscenario.Inthepolarcase,thesubstitutionelasticityisinfiniteandthereisperfectcapitalmobility.Anintermediatevaluewouldallowforpartialcapitalmobility.TheequationsinTable9.1determinetheequilibriumconditionsforthecapitalmarketincomparativestaticmode.Equation(9.1.1)determinestheaggregaterentalrate.Ifthereispartialcapitalmobility,theaggregaterentalrateistheCETdualpriceofthesectorspecificratesofreturn.Ifthereisperfectcapitalmobility,theaggregaterentalrateisdeterminedbyanequilibriumconditionwhichequatesaggregatecapitaldemandtototalcapitalsupply.Equation(9.1.2)determineseithersectoralcapitalsupply,orthesectoralrentalrate.Ifcapitalispartiallymobile,sectoralcapitalsupplyisdeterminedbytheCETfirstordercondition,i.e.sectoralcapitalsupplyisafunctionofeachsectorsrelativerateofreturn.Ifcapitalisperfectlymobile,theequivalentconditionidenticallysetsthesectoralrateofreturntotheeconomy-widerateofreturn.Finally,Equation(9.1.3)determinesthesectoralrateofreturninthecaseofpartialcapitalmobility.Underperfectcapitalmobility,ittriviallyequatescapitalsupplytocapitaldemand.

Table9.1:CapitalMarketEquilibriuminComparativeStatic

(9.1.1)

(9.1.2)

(9.1.3)

Recursive-DynamicCapitalMarketEquilibrium


-- 25

Sectoraloutputisessentiallydeterminedbyaggregatedemandfordomesticoutput,seeEquation(7.3.3).(Inthesimplestcase,withnomarketdifferentiation,outputisequaltothesumofdomesticdemandfordomesticoutput,plusexportdemand,i.e.XP=XD+ED.)Theproducerdecidestheoptimalwaytodivideproductionoftotaloutputacrossvintages.Atfirst,theproducerwilluseallthecapitalinstalledatthebeginningofthecapital,thisisthedepreciatedinstalledcapitalfromthepreviousperiod.Ifdemandexceedswhatcanbeproducedwiththeoldcapital,theproducerwilldemandnewcapital.Ifdemandislowerthantheoutputwhichcanbeproducedwiththeoldcapital,theproducerwilldisinvestsomeoftheinstalledcapital.Equation(9.2.1)providesthecapital/outputratioforoldcapital,χ(notethatKvd,Oldreflectstheoptimalcapitaldemandforoldcapitalbytheproducer).Oncethecapital/outputratioisdetermined,itiseasytodeterminetheoptimaloutputusingoldcapital.Equation(9.2.2)determinesthisquantity,XPvOld,whereanupperboundisgivenbytotaloutput.Iftheproducerownstoomucholdcapital,i.e.thedesiredoutputexceedstotaldemand,theproducerwilldisinvestthedifferencebetweentheinitialcapitalstockandthecapitalstockwhichwillproducethedesireddemand.Equation(9.2.3)determinesoutputproducedwithnewcapitalasaresidual.

Table9.2:DeterminationofVintageOutput

(9.2.1)

(9.2.2)

(9.2.3)

Ifasectorisindecline,i.e.ithastoomuchinstalledcapitalgivenitsdemand,itwilldisinvest.Thecapitalsupplycurveisasimpleconstantelasticityfunctionoftherelativerentalrates.Thehighertherentalrateonoldcapital,thehigherthesupplyofoldcapital.Theformulawhichisusedis:

whereηkisthedisinvestmentelasticity.Anotherwaytothinkofthisistosubtractthetwocapitalnumbers,i.e.

Thisrepresentsthesupplyofdisinvestedcapital,whichincreasesastherelativerentalrateofoldcapitaldecreases.Atthelimit,whentherentalratesareequalized,thereisnodisinvestedcapital.Atequilibrium,demandforoldcapital(ineachdecliningsector),mustequalsupplyofoldcapital.Wecanthereforeinvertthefirstequationtodeterminetherentalrateonoldcapital,assumingthesectorisindeclineandsupplyequalsdemand.Equation(9.3.1)determinestherelativerentalrateonoldcapitalforsectorsindecline,i.e.itistheratiooftheoldrentalratetothenewrentalrate.Itisboundedabove


-- 26

by1,becausetherentalrateonoldcapitalindecliningsectorsisnotallowedtoexceedtherentalrateonnewcapital.Equation(9.3.2)determinestherentalrateonmobilecapital.Mobilecapitalisthesumofnewcapital,disinvestedcapital,andinstalledcapitalinexpandingsectors.Itisnotnecessarytosubtractimmobilecapitalfromeachsideofthecapitalequilibriumcondition,i.e.therentalrateonmobilecapitalcanbedeterminedfromtheaggregatecapitalequilibriumcondition.Equation(9.3.3)isanidentitysettingtherentalrateonnewcapitalequaltotherentalrateonmobilecapital.Equation(9.3.4)determinestherentalrateofoldcapital.Ifasectorisdisinvesting,therentalrateonoldcapitalisessentiallydeterminedbyEquation(9.3.1).Ifasectorisexpanding,thanRRisequalto1,andthereforetherentalrateonoldcapitalinexpandingsectorswillbeequaltotherentalrateofnewcapital.

Table9.3:CapitalMarketEquilibrium

(9.3.1)

(9.3.2)

(9.3.3)

(9.3.4)

10 Macro Closure Governmentclosurewasdiscussedabove,currentgovernmentsavingsaredeterminedeitherendogenouslywithfixedtaxrates,orexogenously,withoneofthetaxadjustmentfactorsendogenous.Equation(10.1.1)istheubiquitoussavingsequalsinvestmentequation.InEquation(10.1.1),TFDVzpisthevalueofprivateinvestmentexpenditures,whosevaluemustequaltotalresourcesallocatedtotheprivateinvestmentsector:retainedcorporateearning, ,totalhouseholdsavings,Sh,governmentsavings,Sg,thesumacrossregionsofoutofstatecapitalflows,Sf,andnetofstockbuildingexpenditures.Thelastclosureruleconcernsthebalanceofpayments.First,wemakethesmallcountryassumptionforimports,i.e.localconsumptionofimportswillnotaffecttheborderpriceofimports,WPM.Equation(10.1.2)istheoverallbalanceofpaymentsequation.Thevalueofimports,atworld(border)prices,mustequalthevalueofexports,atborderprices(i.e.inclusiveofexporttaxesandsubsidies)plusnettransfersandfactorpayments,andplusnetcapitalinflows.ThebalanceofpaymentsconstraintisdroppedfromthemodelduetoWalras’LawThefinalequationsofthemodel,Equations(10.1.3)-(10.1.5)areusedtocalculatethedomesticpriceindexwhichisusedtoinflaterealdomestictransfers.NotethatrealGDPismeasuredinefficiencyunits.Thenuméraireofthemodelistheexchangerate.


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Table10.1:ClosureEquations

(10.1.1)

(10.1.2)

(10.1.3)

(10.1.4)

(10.1.5)

11 Dynamics Pre-DeterminedVariablesThefirsttablepresentsthevariableswhicharepre-determined,i.e.theydonotdependonanycontemporaneousendogenousvariables.Equation(11.1.1)determinesthelaborsupplyshiftfactorwhichisequaltothepreviousperiod’slaborsupplyshiftfactormultipliedbyanexogenouslyspecifiedlaborsupplygrowthrate.(Alldynamicequationsreflectthefactthatthetimestepsmaynotbeofequalsize.Thegrowthratesarealwaysgivenaspercentperannumincreases.)Equation(11.1.2)providesasimilarequationforpopulation.Thepopulationandlaborgrowthratesareallowedtodiffer.Government(real)expendituresandthetransfersbetweengovernmentandhouseholdsgrowattherateofgrowthofGDP.Thislattergrowthrateisexogenouslyspecified(fortheBaUscenario).Equations(11.1.3)-(11.1.4)providetherelevantformulas.Userscaninputthereownexogenousassumptionsaboutthesevariables.Equation(11.1.5)determinestheamountofinstalledcapitalatthebeginningoftheperiod.Ifasectorisexpanding,thiswillequaltheamountofoldcapitalinthesectorattheendoftheperiod.Ifasectorisdeclining,theamountofoldcapitalattheendoftheperiodwillbelessthantheinitialinstalledcapital.Thedepreciationrateisexogenous.


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Table11.1:Pre-DeterminedVariables

(11.1.1)

(11.1.2)

(11.1.3)

(11.1.4)

(11.1.5)

CapitalStockThemotionequationfortheaggregatecapitalstockisgivenbythefollowing1-stepformula:

whereKistheaggregatecapitalstock,δistheannualrateofdepreciation,It-1isthelevelofrealinvestmentinthepreviousperiod.Usingmathematicalinduction,wecandeducethemulti-periodtransitionequation:

Ifthestepsizeifgreaterthan1,themodeldoesnotcalculatetheintermediatevaluesforthepathofrealinvestment.Theinvestmentpathisestimatedusingasimplelineargrowthmodel,i.e.

where

Notethattheformulafortheinvestmentgrowthdependsonthecontemporaneouslevelofrealinvestment.Thisexplainswhythecurrentcapitalstockisnotpre-determined.Ifrealinvestmentincreases(e.g.becauseoutofstatetransfersincrease),thiswillhavesomeeffectonthecurrentcapitalstockviaitsinfluenceontheestimatedgrowthrateofrealinvestment.Insertingtheformulafortheestimatedrealinvestmentstreaminthecapitalstockequation,wederive:

Alittlebitofalgebra,yieldsEquation(11.2.1)fortheaggregatecapitalstock.Equation(11.2.2)definestheannualizedgrowthrateofrealinvestmentwhichisusedtocalculatetheaggregatecapitalstock.Equation(11.2.3)determinesthelevelofnormalizedcapital.Therearetwoindicesofcapitalstock.Thefirstindexisthenormalizedlevelofcapitalstock.Thisindexiscallednormalizedbecauseitisthelevelofcapitalstockineachsectorwhichyieldsarentalrateof1.Thesecondindexisthe


-- 29

actuallevelofthecapitalstock,giveninbaseyearprices.Thelattervariableisonlyusedintwoequations.Itisusedtodeterminethedepreciationallowance,anditisusedtoupdatethelevelofthecapitalstockinEquation(11.2.1)(becauseitisinthesameunitsasthelevelofrealinvestment).20

Table11.2:AggregateCapitalStock

(11.2.1)

(11.2.2)

(11.2.3)

ProductivityProductivityentersthevalueaddedbundle—labor,land,andcapital—asseparateefficiencyparametersforthethreefactors,differentiatedbysectorandbyvintage.Inthecurrentversionofthemodel,andforlackofbetterinformation,thelaborefficiencyfactor(andtheenergyefficiencyfactor)areexogenous.Indefiningthereferencesimulation,thegrowthpathofrealGDPispre-specified,andasingleeconomy-wideefficiencyfactorforlandandcapitalisdeterminedendogenously.Insubsequentsimulations,i.e.withdynamicpolicyshocks,thecapitalandlandefficiencyfactorsareexogenous,andthegrowthrateofrealGDPisendogenous.Equation(11.3.1)definesthegrowthrateofrealGDP.Indefiningthereferencesimulation,bothlaggedrealGDPandthegrowthrateγyareexogenous,thereforetheequationisusedtodeterminethecommonefficiencyfactorforlandandcapital.Insubsequentsimulations,Equation(11.3.1)determinesγy,i.e.thegrowthrateofrealGDP.Equations(11.3.2)and(11.3.3)determinerespectivelytheefficiencyfactorsforcapitalandland.Botharesettotheeconomy-wideefficiencyparameterdeterminedbyEquation(11.3.1),however,themodelallowsforapartitionofsectors,wheretheindexi'indexesasubsetofallthesectors.Itisassumedthatthesectorsnotindexedbyi'havenoefficiencyimprovementinland-capital.Equation(11.3.4)determinesthecommoncapital-laborefficiencygrowthfactor,whichisstoredinafileforsubsequentsimulations.Therearealternativemethodsforspecifyingandimplementingthereferencescenario.20 The following numerical example may clarify issue. Assume the value of the capital stock is 100. Assume, as well, that capital remuneration is 10. Capital remuneration is simply rK where r is the rental rate and K the demand for capital. In this example, rK is equal to 10, which implies a rental rate of 0.1. The model assumes a normalisation rule such that the rental rate is 1, and normalizes the capital data to be consistent with the normalisation rule. In other words, the normalized capital demand is 10, and it is really an index of capital volume. The non-normalized level of capital is used only in the accumulation function and in determining the value of the depreciation allowance. All other capital stock equations use the normalized value of capital.


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Table11.3:Capital-LandEfficiency

(11.3.1)

(11.3.2)

(11.3.3)

(11.3.4)

VintageRe-CalibrationAtthebeginningofeachnewperiod,theparametersoftheproductionstructureneedtobemodifiedtoreflectthechangingcompositionofcapital.Asanewperiodbegins,whatwasnewcapitalgetsaddedtooldcapital,i.e.thenewOldcapitalhasadifferentcompositionfromthepreviousOldcapital.Asimpleruleisusedtore-calibratetheproductionstructure:theparametersarecalibratedsuchthattheycanre-producethepreviousperiod’soutputusingtheaggregatecapitalofthepreviousperiod,butwiththeOldelasticities.(TheparametersoftheNewproductionstructurearenotmodified.)Therelevantformulasarenotre-producedherebutcanbefoundintheGAMScode.

12 Emissions EmissionsdatafortheBEARmodelhavebeenassembledfromavarietyofstate,Federal,andindependentsources.Inthecurrentversionofthemodel,thirteentypesofindividualandcompositeemissionsarerepresented(Table12.1).TheprimarydatarelevanttoAB32comefromtheCalifornia'sownemissionsinventory,catalogedformvarioussourcesbytheCaliforniaEnergyCommission.


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Table12.1:EmissionTypes

AirPollutants 1. Suspendedparticulates PART 2. Sulfurdioxide(SO2) SO2 3. Nitrogendioxide(NO2) NO2 4. VolatileorganiccompoundsVOC 5. Carbonmonoxide(CO) CO 6. Toxicairindex TOXAIR 7. Biologicalairindex BIOAIR WaterPollutants 8. Biochemicaloxygendemand BOD 9. Totalsuspendedsolids TSS 10. Toxicwaterindex TOXWAT 11. Biologicalwaterindex BIOWAT LandPollutants 12. Toxiclandindex TOXSOL 13. Biologicallandindex BIOSOLTomodeltheemissiongenerationprocess,weincorporateemissionsproductionintosectoralproductionfunctions.Inmuchworkofthiskind,emissiondatahavebeendirectlyassociatedwiththevolumeofoutput.Thishasseveralconsequences.First,theonlywaytoreduceemissions,withagiventechnology,istoreduceoutput.Thiscanleadtounpleasantmessagesforpolicymakers.Asecondconsequenceisthatitignoresimportantsourcesofpollutionoutsidetheproductionsideoftheeconomy,namelyhouseholdconsumption.Inanattempttoamelioratetheseshortcomings,thepollutiondataoftheCaliforniahasbeenregressedonasmallsubsetofinputsofthestate'scorrespondinginput-outputtable.Usingthiseconometricapproach,wehaveshownelsewherethatthelevelofemissionscanbeexplainedbyaverysmallsubsetofinputs(primarilyfuelsandchemicals).21Incorporatingtheseinputsintovalueaddedthenallowsproductioninthemodeltosubstituteawayfrompollutinginputs.Weassumethathouseholdemissionsareaffectedbytherelativepricesofpollution-intensivegoodsandservicesinconsumption.Equation(12.2.1)definesthetotallevelofemissionsforeachpollutantp.Thebulkofthepollutionisassignedtofinaldemandforgoodsandservices,representedbythebracketedsecondtermintheexpression.ThefirstterminEquation(12.2.1)representswhatwecallprocesspollution.Itistheresidualamountofpollutioninproductionwhichisnotexplainedbytheconsumptionofinputs.Intheestimationprocedure,aprocessdummyprovedtobesignificantincertainsectors.Ifanemissionfee(ortax)isexogenous,theyarespecifiedinphysicalunits,i.e.dollarsperton.Equation(12.2.2)convertsthisintoanominalamount.

21 See Dessus et. al. (1994).


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TheequationsinTable12.3describehowthemodelismodifiestoaccountforpollutionfee/taxincidence.Theunderlyingfee/taxcanbegeneratedinoneoftwoways,eitherbespecifiedexogenously(inwhichcaseitismultipliedbyapriceindextopreservethehomogeneityofthemodel),generatedendogenouslybespecifyingaconstraintonthelevelofemission.Thelattercase,forexample,correspondstoaCapandTradeprogram,andEquation(12.2.1)isusedtodefinethepollutionconstraint,whilethefee/tax(e.g.permitprice)generatedbytheconstraintistheshadowpriceofEquation(12.2.1),andEquation(12.2.2)isnotactive.

Table12.2:EmissionLevels

(12.2.1)

(12.2.2)

Foraccountingpurposes,thetaxcanbeimplementedasanexcisetax,i.e.itisimplementedasataxperunitofemissionincurrencyunits,i.e.$xpertonofemission.Itisconvertedtoapricewedgeontheconsumptionofthecommodity(asopposedtoataxontheemission),usingthecommodityspecificemissioncoefficient.Forexample,inEquation(2.1.5'),thetaxaddsanadditionalpricewedgebetweentheunitcostofproductionexclusiveofthepollutiontax,andthefinalunitcostofproduction.Letproductionequal100(milliondollarsforexample),andlettheamountofpollutionbeequalto1tonofemissionper10milliondollarsofoutput.Thenthetotalemissioninthiscaseis10tons.Ifthefee/taxisequalto$25pertonofemission,thetotaltaxbillforthissectoris$250.Intheformulabelow,νisequalto0.1(tonspermilliondollarsofoutput),XPisequalto100(milliondollars),andtpisequalto$25.TheconsumptionbasedpollutiontaxisaddedtotheArmingtonprice,seeEquation(7.1.4').However,theArmingtondecompositionoccursusingbasicprices,therefore,thetaxesareremovedfromtheArmingtonpriceinthedecompositionformulae,seeEquations(7.1.2')and(7.1.3').Equation(6.3.3')determinesthemodificationtothegovernmentrevenueequation.


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Table12.3:EmissionPriceWedges

(2.1.5')

(7.1.4')

(7.1.2')

(7.1.3')

(6.3.3')


-- 34

13 Dealing with Uncertainty - Stochastic Variational Analysis Economicpolicyissubjecttoabroadrangeofsystemicuncertainties,duringdevelopment,implementation,andbeyond.Inotherhumanendeavors,uncertaintyisalsopervasive,butinsomefieldsithasbeeneffectivelymanagedwithstatisticalmethods.Inengineeringgenerallyandtheelectricpowersectorinparticular,reliabilityanalysisiscriticaltohedgeagainstrisksofuncertainspecification,design,materials,andoperatingconditions.Indeed,thereisalargeliteratureonmostcomponentsofmodernenergysystems,includinggenerationtechnologies,transmissionsystems,etc.22Likewise,financialmarketsmanageextensiveriskpatternswithstatisticalmethods,includingthesameMonteCarlomethodspopularizedbyengineers.Simpler“stresstest”modelsofstochasticnetpresentvalueinformmostlargeprojectinvestments,butthespiritoftheseapproachesisthesame.Ineconomicforecasting,thereisalsoalongMonteCarlotraditionof“sensitivityanalysis”,mainlyintendedtoovercomeuncertaintyinestimatesofbehavioralparameters.23Whathasbeenlargelymissingisanefficientmethodologyforwhatmightbeterms“policyreliabilityanalysis,”atractableempiricalframeworkthatcanquantifythepotentialcostsofuncertaintyfacingeconomicdecisionmakers.Itissomewhatsurprisingthatmostforecastersstillreportpointestimatesforeventsinthedistantfuture,usingscenarioanalysistocompareseeminglydeterministicdifferencesinoutcomesofqualitativelydifferentpoliciesorstatesofnature.Inreality,itisonlypossibletoanticipateanintervalofoutcomesfromanyaction,hopefullywithacorrespondingdegreeofconfidence.Thisapproachmightbemoreresponsibilityforthosewhoforecast,butitoffersanimportantdegreeofrobustnessagainstveryrealrisksfacedbythosewhoenactandimplementpolicies.Untilnow,MonteCarlomethodswouldhavebeenthetoolofchoiceforthiskindofpolicyresearch.Unfortunately,thestatisticalpropertiesofthis(randomizeddrawingapproach)havemanylimitations,includingresourcerequirementsandinstabilityinsomeapplications.Fortunately,anewgenerationofnumericalintegrationmethodsfromphysicsandappliedmathematicspromisestogreatlyimproveboththeefficiencyandaccuracyofstochasticmethods,andweapplythisinthepresentreport.GaussianQuadratureAsanalternativetoMonteCarlomethods,anumericalmethodcalledGaussianQuadraturecanapproximatethedistributionfunctionsneededtodomeanandvarianceanalysisagainstparametricuncertainty.Thebasicgoalistoapproximatemeanandvariance,definedbyintegralsofthedistributionofCGEmodelparametersorexogenousvariablevaluesa.Generally,theGaussianquadratureapproachyieldsnodesxiandweightsωithatcanapproximateacorrespondingintegral

22 See e.g. Mazumdar and co-authors, Snyder and Stremel (1990), Scully et al (1992), and others cited below. 23 See, e.g. Thissen (1998) for a survey, as well as Abler et al (1999), Belgodere et al (2011). In energy modeling, see also Borenstein and co-authors.


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Inthepresentcase,weseeknodesaiandweightsgitoapproximatemeansandvariancesofvariablesforecastbytheCGEmodel,i.e.

Basedonthisperspective,Hermelingetal(2013)developaversionofGaussquadraturebasedonorthogonalpolynomials.ThisextendsatraditionalGaussquadraturealgorithmpreviouslyusedineconomics(cf.Arndt1996andDeVuystandPreckel1997),usingfunctionalformsbetteradaptedtounderlyingstandardprobabilitydistributionsandthusrealizingsubstantialgainsincomputationalefficiency/accuracy.WesummarizetheirapproachhereasitwillbeappliedtotheBEARmodel.Defineorthogonalitybytheintegralscalarproduct

Astandardresultoffunctionalanalysis(cf.Rudin1976,ch.7,p.159)guaranteesthatanycontinuous(e.g.density)fuction,canbeapproximateywitharbitraryaccuracybyorthogonalpolynomialswhoserootsdefineafinitesetofapproximationofapproximationpoints.Thebest-knownexamplesoforthogonalpolynomialssuitedtomodelingprobabilitydensitiesareLegendre,Tchebychev,LaguerreandHermite,summarizedbelow:


-- 36

Inpractice,then,foragivendensityfunctiong(a),wecalculatethezerosa1,...,akofthecorrespondingorthogonalpolynomialofdegreek.Calculatingtheweightsg1,...,gkfromasuitablesystemoflinearequations,weobtainanintegrationformulathatintegratespolynomialsuptodegree2k-1exactly,usingweightsfromthedensityfunctiong(a).Forhigherdimensionalvariationalanalysis(n>1),jointdensityintegrationcanbeapproximatedbyusinganalogouspolynomialsandproductrules,combiningone-dimensionalnodesandweights

TheadditivityofnodesmakesthisapproachdramaticallymoreefficientthanMonteCarlomutlivariateanalysis.Withtheformer,functionevaluationsincreasemultiplicatively,whilewithMCtheyincreaseexponentially.

4 APPENDEX1–TheCES/CETFunctionsBecauseofthefrequentuseoftheconstantelasticityofsubstitution(CES)function,thisappendixwilldevelopsomeofthepropertiesoftheCES,includingsomeofitsspecialcases.TheCESfunctioncanbeformulatedasacostminimizationproblem,subjecttoatechnologyconstraint:

whereVistheaggregatevolume(ofproduction,forexample),Xaretheindividualcomponents(“inputs”)oftheproductionfunction,Parethecorrespondingprices,andaandλaretechnologicalparameters.aaremostoftencalledtheshareparameters.λaretechnologyshifters.TheparameterρistheCESexponent,whichisrelatedtotheCESelasticityofsubstitution,whichwillbedefinedbelow.Abitofalgebraproducesthefollowingderiveddemandfortheinputs,assumingVandthepricesarefixed:

(1)

wherewedefinethefollowingrelationships:


-- 37

and

(2)

PiscalledtheCESdualprice,itistheaggregatepriceoftheCEScomponents.Theparameterσ,iscalledthesubstitutionelasticity.ThistermcomesfromthefollowingrelationshipwhichiseasytoderivefromEquation(1):

Inotherwords,theelasticityofsubstitutionbetweentwoinputs,withrespecttotheirrelativeprices,isconstant.(Note,weareassumingthatthesubstitutionelasticityisapositivenumber).Forexample,ifthepriceofinputiincreasesby10percentwithrespecttoinputj,theratioofinputitoinputjwilldecreaseby(around)σtimes10percent.TheLeontiefandCobb-DouglasfunctionsarespecialcasesoftheCESfunction.InthecaseoftheLeontieffunction,thesubstitutionelasticityiszero,inotherwords,thereisnosubstitutionbetweeninputs,nomatterwhattheinputpricesare.Equations(1)and(2)become:

(1')

(2')

Theaggregatepriceistheweightedsumoftheinputprices.TheCobb-Douglasfunctionisforthespecialcasewhenσisequaltoone.ItshouldbeclearfromEquation(2)thatthiscaseneedsspecialhandling.ThefollowingequationsprovidetherelevantequationsfortheCobb-Douglas:

(1'')

(2'')

wheretheproductionfunctionisgivenby:


-- 38

and

Note that in Equation (1'') the value share is constant, and does not depend directly on technology change.

CalibrationTypically,thebasedatasetalongwithagivensubstitutionelasticityareusedtocalibratetheCESshareparameters.Equation(1)canbeinvertedtoyield:

assumingthetechnologyshiftershaveunitvalueinthebaseyear.Moreover,thebaseyearpricesareoftennormalizedto1,simplifyingtheaboveexpressiontoatruevalueshare.Let’staketheArmingtonassumptionforexample.Assumeaggregateimportsare20,domesticdemandfordomesticproductionis80,andpricesarenormalizedto1.TheArmingtonaggregatevolumeis100,andtherespectiveshareparametersare0.2and0.8.(Notethatthemodelalwaysusestheshareparametersrepresentedbyα,nottheshareparametersrepresentedbya.Thissavesoncomputetimesincetheaparametersneverappearexplicitlyinanyequation,whereasaraisedtothepowerofthesubstitutionelasticity,i.e.α,occursfrequently.)Withlessdetail,thefollowingdescribestherelevantformulasfortheCETfunctionwhichissimilartotheCESspecification.

whereVistheaggregatevolume(e.g.aggregatesupply),Xaretherelevantcomponents(sector-specificsupply),Parethecorrespondingprices,garetheCETshareparameters,andλistheCETexponent.TheCETexponentisrelatedtotheCETsubstitutionelasticity,Λviathefollowingrelation:

Solutionofthismaximizationproblemleadstothefollowingfirstorderconditions

wheretheγparametersarerelatedtotheprimalshareparameters,g,bythefollowingformula:


-- 39

5 FiguresFigure1:ProductionNesting

Notes: 1. EachnestrepresentsadifferentCESbundle.ThefirstargumentintheCESfunctionrepresentsthesubstitutionof

elasticity.Theelasticitymaytakethevaluezero.Becauseoftheputty/semi-puttyspecification,thenestingisreplicatedforeachtypeofcapital,i.e.oldandnew.Thevaluesofthesubstitutionelasticitywillgenerallydifferdependingonthecapitalvintage,withtypicallylowerelasticitiesforoldcapital.ThesecondargumentintheCESfunctionisanefficiencyfactor.InthecaseoftheKEbundle,itisonlyappliedonthedemandforcapital.Inthecaseofthedecompositionoflaborandenergy,itisappliedtoallcomponents.

2. Intermediatedemand,bothenergyandnon-energy,isfurtherdecomposedbyregionoforiginaccordingtotheArmingtonspecification.However,theArmingtonfunctionisspecifiedattheborderandisnotindustryspecific.

3. Thedecompositionoftheintermediatedemandbundle,thelaborbundle,andtheenergybundlewillbespecifictothelevelofaggregationofthemodel.ThediagramrepresentsonlyschematicallythedecompositionandisnotmeanttoimplythattherearethreecomponentsintheCESaggregation.


-- 40

Figure2:ArmingtonNesting

Note(s): 1. ThebaseSAMincludesasingletradingpartnerwithCALIFORNIA,thoughthespecificationofimportdemandusesthe

multiplenestingapproachinordertoprovideflexibilityforthefutureastradedataisdevelopedfurther.ImportdemandismodeledasanestedCESstructure.Agentsfirstchoosetheoptimallevelofdemandfortheso-calledArmingtongood(XA).Inasecondstage,agentsdecomposetheArmingtonaggregategoodintodemandforthedomesticallyproducedcommodity(XD),andanaggregateimportbundle(XM).Atthethirdandfinalstage,agentschoosetheoptimalquantitiesofimportsfromeachtradingpartner.Importpricesandtariffsarespecifictoeachofthetradingpartners.


-- 41

Figure3:OutputSupply(CET)Nesting

Note(s): 1. ThemarketfordomesticoutputismodeledasanestedCETstructure(similartothenoteabove,thecurrentversionof

theCALIFORNIAdataonlyconcernsasingletradingpartner).Producersfirstchoosetheoptimallevelofoutput(XP)24.Inasecondstage,producerschoosetheoptimalmixofgoodssuppliedtothedomesticmarket(XD),andanaggregateexportsupply(ES).Atthethirdandfinalstage,producerschoosetheoptimalmixofexportstoeachoftheindividualtradingpartners.Theexportpriceofeachtradingpartnerisregion-specific.Underthesmall-countryassumption,theexportpriceisfixed(inoutofstatecurrencyterms),otherwize,eachtradingpartnerhasadownwardslopingdemandcurve,andtheexportpriceisdeterminedendogenouslythroughanequilibriumcondition.

24 Note that in a perfectly competitive framework, output is determined by equilibrium conditions, and is not a producer decision.

6 Appendix2-DataStructurefortheBEARModelTheBEARmodeliscalibratedtoa2010SocialAccountingMatrixfortheCaliforniaeconomy.Thistableisbasedonanextensivesynthesisofdatafromdiverseofficialandindependentsources.WhiletheCalifornia2010SAMisfullydocumentedelsewhere,wesummarizethemaindimensionsofthedatainthisappendixastheypertaintotheBEARmodel'sstructure.ThedimensionalityoftheoverallSAMisdescribedinTableA2.1,whilecomponentstructuresusedintheBEARmodelarediscussedindividually.TableA2.1:InstitutionalStructureofthe2010CaliforniaSAM

124 production activities

124 commodities (includes trade and transport margins)

4 factors of production

2 labor categories (skilled and unskilled)

Capital

Land

7 Household types, defined by income tax bracket

Enterprises

Federal Government (7 fiscal accounts)

State Government (27 fiscal accounts)

Local Government (11 fiscal accounts)

Consolidated capital account

External Trade Accounts

Rest of United States

Rest of the World

Early Release Materials

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TableA2.2:SectorAggregationLabel Description A01Agric Agriculture A02Cattle Cattle and Feedlots A03Dairy Dairy Cattle and Milk Production A04Forest Forestry, Fishery, Mining, Quarrying A05OilGas Oil and Gas Extraction A06OthPrim Other Primary Products A07DistElec Generation and Distribution of Electricity A08DistGas Natural Gas Distribution A09DistOth Water, Sewage, Steam A10ConRes Residential Construction A11ConNRes Non-Residential Construction A12Constr Construction A13FoodPrc Food Processing A14TxtAprl Textiles and Apparel A15WoodPlp Wood, Pulp, and Paper A16PapPrnt Printing and Publishing A17OilRef Oil Refining A18Chemicl Chemicals A19Pharma Pharmaceutical Manufacturing A20Cement Cement A21Metal Metal Manufacture and Fabrication A22Aluminm Aluminum A23Machnry General Machinery A24AirCon Air Conditioning and Refrigeration A25SemiCon Semi-conductor and Other Computer Manufacturing A26ElecApp Electrical Appliances A27Autos Automobiles and Light Trucks A28OthVeh Vehicle Manufacturing A29AeroMfg Airplane and Aerospace Manufacturing A30OthInd Other Industry A31WhlTrad Wholesale Trade A32RetVeh Retail Vehicle Sales and Service A33AirTrns Air Transport Services A34GndTrns Ground Transport Services A35WatTrns Water Transport Services A36TrkTrns Truck Transport Services A37PubTrns Public Transport Services A38RetAppl Retail Electronics A39RetGen Retail General Merchandise A40InfCom Information and Communication Services A41FinServ Financial Services A42OthProf Other Professional Services A43BusServ Business Services A44WstServ Waste Services A45LandFill Landfill Services A46Educatn Educational Services A47Medicin Medical Services


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A48Recratn Recreation Services A49HotRest Hotel and Restaurant Services A50OthPrSv Other Private Services Inadditiontodetailedinformationonsectoralstructureofproduction,demand,andtrade,wedetailincomeandexpenditureaccountsforavarietyofhouseholdstobettercapturedistributionalimpactsofpolicy.UsingdatasourcesthatcombineCaliforniaDepartmentofFinanceinformationwithhouseholdsurveyandinput-outputdata,wetrackhouseholdsinseventaxbrackets.

Table A2.2: California Households and Population by Income Tax Bracket

(California Department of Finance: 2006, millions of people)

Households Cumulative Population Cumulative Percent

1 <$12k 1.220 1.340 3.575 3.926 9.7522 $12-28k 2.360 3.580 6.915 10.489 18.863 $28-40k 1.650 5.230 4.835 15.324 13.194 $40-60k 2.110 7.340 6.182 21.506 16.875 $60-80k 1.650 8.990 4.835 26.341 13.196 $80-200k 3.140 12.130 9.200 35.541 25.107 $200k+ 0.380 12.510 1.113 36.654 3.04

Total 12.510 36.654 100


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