Back to CJL – one quarter year

This code is based on DICE 2007 with 10 year time periods given in the webpage:

This code changes 10 year time periods into 0.25 year time periods with the explicit
Euler finite difference method.
Authors: Yongyang Cai, Hoover Institution
Kenneth L. Judd, Hoover Institution
Thomas S. Lontzek, University of Zurich

SETS T Time periods /1*2400/ ;
scalar deltat length of one time period / 0.25 /;


** Preferences
B_ELASMU Elasticity of marginal utility of consumption / 2.0 /
B_PRSTP Initial rate of social time preference per year / .015 /

** Population and technology
POP0 2005 world population millions /6514 /
GPOP0 Growth rate of population per year /.035 /
POPASYM Asymptotic population / 8600 /
A0 Initial level of total factor productivity /.02722 /
GA0 Initial growth rate for technology per year /.0092 /
DELA Decline rate of technol change per year /.001 /
DK Depreciation rate on capital per year /.100 /
GAMA Capital elasticity in production function /.300 /
Q0 2005 world gross output trill 2005 US dollars /61.1 /
K0 2005 value capital trill 2005 US dollars /137. /

** Emissions
SIG0 CO2-equivalent emissions-GNP ratio 2005 /.13418 /
GSIGMA Initial growth of sigma per year /-.00730 /
DSIG Decline rate of decarbonization per year /.003 /
DSIG2 Quadratic term in decarbonization / .000 /
ELAND0 Carbon emissions from land 2005(GtC per year) / 1.1000 /

** Carbon cycle
MAT2000 Concentration in atmosphere 2005 (GtC) /808.9 /
MU2000 Concentration in upper strata 2005 (GtC) /1255 /
ML2000 Concentration in lower strata 2005 (GtC) /18365 /
b11 Carbon cycle transition matrix /0.9810712 /
b12 Carbon cycle transition matrix /0.0189288 /
b21 Carbon cycle transition matrix /0.0097213 /
b22 Carbon cycle transition matrix /0.9852787 /
b23 Carbon cycle transition matrix /0.005 /
b32 Carbon cycle transition matrix /0.0003119 /
b33 Carbon cycle transition matrix /0.9996881 /

** Climate model
T2XCO2 Equilibrium temp impact of CO2 doubling oC / 3 /
FEX0 Estimate of 2000 forcings of non-CO2 GHG / -.06 /
FEX1 Estimate of 2100 forcings of non-CO2 GHG / 0.30 /
TOCEAN0 2000 lower strat. temp change (C) from 1900 /.0068 /
TATM0 2000 atmospheric temp change (C)from 1900 /.7307 /
C1 Climate-equation coefficient for upper level /.0220 /
C3 Transfer coeffic upper to lower stratum /.300 /
C4 Transfer coeffic for lower level /.0050 /
FCO22X Estimated forcings of equilibrium co2 doubling /3.8 /

** Climate damage parameters calibrated for quadratic at 2.5 C for 2105
A1 Damage intercept / 0.00000 /
A2 Damage quadratic term / 0.0028388 /
A3 Damage exponent / 2.00 /

** Abatement cost
EXPCOST2 Exponent of control cost function /2.8 /
PBACK Cost of backstop 2005 000$ per tC 2005 /1.17 /
BACKRAT Ratio initial to final backstop cost / 2 /
GBACK Initial cost decline backstop pc per year /.005 /
LIMMIU Upper limit on control rate / 1 /

** Participation
PARTFRACT1 Fraction of emissions under control regime 2005 /1 /
PARTFRACT2 Fraction of emissions under control regime 2015 /1 /
PARTFRACT21 Fraction of emissions under control regime 2205 /1 /
DPARTFRACT Decline rate of participation /0 /

** Availability of fossil fuels
FOSSLIM Maximum cumulative extraction fossil fuels / 6000 /

** Scaling and inessential parameters
scale1 Scaling coefficient in the objective function /194 /
scale2 Scaling coefficient in the objective function /381800 / ;

* Definitions for outputs of no economic interest

L(T) Level of population and labor
AL(T) Level of total factor productivity
SIGMA(T) CO2-equivalent-emissions output ratio
R(T) Instantaeous rate of social time preference
RR(T) Average utility social discount rate
GA(T) Growth rate of productivity from 0 to T
FORCOTH(T) Exogenous forcing for other greenhouse gases
GL(T) Growth rate of labor 0 to T
GCOST1 Growth of cost factor
GSIG(T) Cumulative improvement of energy efficiency
ETREE(T) Emissions from deforestation
COST1(t) Adjusted cost for backstop
PARTFRACT(T) Fraction of emissions in control regime
AA1 Variable A1
AA2 Variable A2
AA3 Variable A3
ELASMU Variable elasticity of marginal utility of consumption
PRSTP Variable nitial rate of social time preference per year
LAM Climate model parameter
Gfacpop(T) Growth factor population ;

* Unimportant definitions to reset runs
TLATE(T) = YES$(ORD(T) GE 21);
AA1 = A1;
AA2 = A2;
AA3 = A3;

b12 = deltat*b12;
b23 = deltat*b23;
b11 = 1 – b12;
b21 = 587.473*B12/1143.894;
b22 = 1 – b21 – b23;
b32 = 1143.894*b23/18340;
b33 = 1 – b32 ;

C1 = C1*deltat;
C4 = C4*deltat;

GPOP0 = GPOP0*deltat;
GA0 = GA0*deltat;
DELA = DELA*deltat;
DSIG = DSIG*deltat;
ELAND0 = ELAND0*deltat;
GBACK = GBACK*deltat;

* Important parameters for the model
Gfacpop(T) = (exp(gpop0*(ORD(T)-1))-1)/exp(gpop0*(ORD(T)-1));
L(T)=POP0* (1- Gfacpop(T))+Gfacpop(T)*popasym;
al(“1”) = a0;
LOOP(T, al(T+1)=al(T)/((1-ga(T))););
cost1(T) = (PBACK*SIGMA(T)/EXPCOST2)* ( (BACKRAT-1+ EXP (-gback* (ORD(T)-1) ) )/BACKRAT);

ETREE(T) = ELAND0*(1-0.01*deltat)**(ord(T)-1);
FORCOTH(T)= FEX0+ 0.01*deltat*(FEX1-FEX0)*(ORD(T)-1)$((ORD(T)-1)*deltat<=100)+ 0.36$((ORD(T)-1)*deltat>100);

partfract(t) = partfract21;
PARTFRACT(T)$(ord(T)<500) = Partfract21 + (PARTFRACT2-Partfract21)*exp(-DPARTFRACT*(ORD(T)-2)); partfract(“1”)= PARTFRACT1; VARIABLES MIU(T) Emission control rate GHGs FORC(T) Radiative forcing in watts per m2 TATM(T) Temperature of atmosphere in degrees C TOCEAN(T) Temperatureof lower oceans degrees C MAT(T) Carbon concentration in atmosphere GtC MATAV(T) Average concentrations MU(T) Carbon concentration in shallow oceans Gtc ML(T) Carbon concentration in lower oceans GtC E(T) CO2-equivalent emissions GtC C(T) Consumption trillions US dollars K(T) Capital stock trillions US dollars CPC(T) Per capita consumption thousands US dollars PCY(t) Per capita income thousands US dollars I(T) Investment trillions US dollars S(T) Gross savings rate as fraction of gross world product RI(T) Real interest rate per annum Y(T) Gross world product net of abatement and damages YGROSS(T) Gross world product GROSS of abatement and damages YNET(T) Output net of damages equation DAMAGES(T) Damages ABATECOST(T) Cost of emissions reductions CCA(T) Cumulative industrial carbon emissions GTC PERIODU(t) One period utility function UTILITY; POSITIVE VARIABLES MIU, TATM, TOCE, E, MAT, MATAV, MU, ML, Y, YGROSS, C, K, I, CCA ; EQUATIONS CCTFIRST(T) First period cumulative carbon CCACCA(T) Cumulative carbon emissions UTIL Objective function YY(T) Output net equation YNETEQ(T) Output net of damages equation YGROSSEQ(T) Output gross equation DAMEQ(T) Damage equation ABATEEQ(T) Cost of emissions reductions equation CC(T) Consumption equation KK(T) Capital balance equation KK0(T) Initial condition for capital KC(T) Terminal condition for capital CPCE(t) Per capita consumption definition PCYE(T) Per capita income definition EE(T) Emissions equation SEQ(T) Savings rate equation RIEQ(T) Interest rate equation FORCE(T) Radiative forcing equation MMAT0(T) Starting atmospheric concentration MMAT(T) Atmospheric concentration equation MMATAVEQ(t) Average concentrations equation MMU0(T) Initial shallow ocean concentration MMU(T) Shallow ocean concentration MML0(T) Initial lower ocean concentration MML(T) Lower ocean concentration TATMEQ(T) Temperature-climate equation for atmosphere TATM0EQ(T) Initial condition for atmospheric temperature TOCEANEQ(T) Temperature-climate equation for lower oceans TOCEAN0EQ(T) Initial condition for lower ocean temperature PERIODUEQ(t) Instantaneous utility function equation ; ** Equations of the model CCTFIRST(TFIRST).. CCA(TFIRST)=E=0; CCACCA(T+1).. CCA(T+1)=E=CCA(T)+ E(T); KK(T).. K(T+1) =L= (1-DK)**deltat *K(T)+deltat*I(T); KK0(TFIRST).. K(TFIRST) =E= K0; KC(TLAST).. .02*K(TLAST) =L= I(TLAST); EE(T).. E(T)=E=deltat*SIGMA(T)*(1-MIU(T))*AL(T)*L(T)**(1-GAMA)*K(T)**GAMA + ETREE(T); FORCE(T).. FORC(T) =E= FCO22X*((log((Matav(T)+.000001)/596.4)/log(2)))+FORCOTH(T); MMAT0(TFIRST).. MAT(TFIRST) =E= MAT2000; MMU0(TFIRST).. MU(TFIRST) =E= MU2000; MML0(TFIRST).. ML(TFIRST) =E= ML2000; MMAT(T+1).. MAT(T+1) =E= MAT(T)*b11+MU(T)*b21 + E(T); MMATAVEQ(t).. MATAV(T) =e= MAT(T) ; MML(T+1).. ML(T+1) =E= ML(T)*b33+b23*MU(T); MMU(T+1).. MU(T+1) =E= MAT(T)*b12+MU(T)*b22+ML(T)*b32; TATM0EQ(TFIRST).. TATM(TFIRST) =E= TATM0; TATMEQ(T+1).. TATM(T+1) =E= TATM(t)+C1*(FORC(t)-LAM*TATM(t)-C3*(TATM(t)-TOCEAN(t))); TOCEAN0EQ(TFIRST).. TOCEAN(TFIRST) =E= TOCEAN0; TOCEANEQ(T+1).. TOCEAN(T+1) =E= TOCEAN(T)+C4*(TATM(T)-TOCEAN(T)); YGROSSEQ(T).. YGROSS(T) =e= AL(T)*L(T)**(1-GAMA)*K(T)**GAMA; DAMEQ(T).. DAMAGES(t) =E= YGROSS(T)- YGROSS(T)/(1+aa1*TATM(T)+ aa2*TATM(T)**aa3); YNETEQ(T).. YNET(T) =E= YGROSS(T)/(1+aa1*TATM(T)+ aa2*TATM(T)**aa3); ABATEEQ(T).. ABATECOST(T) =E= (PARTFRACT(T)**(1-expcost2))*YGROSS(T)*(cost1(t)*(MIU(T)**EXPcost2)); YY(T).. Y(T) =E= YGROSS(T)*((1-(PARTFRACT(T)**(1-expcost2))*cost1(t)*(MIU(T)**EXPcost2)))/(1+aa1*TATM(T)+ aa2*TATM(T)**aa3); SEQ(T).. S(T) =E= I(T)/(.001+Y(T)); RIEQ(T).. RI(T) =E= GAMA*Y(T)/K(T)- DK ; CC(T).. C(T) =E= Y(T)-I(T); CPCE(T).. CPC(T) =E= C(T)*1000/L(T); PCYE(T).. PCY(T) =E= Y(T)*1000/L(T); PERIODUEQ(T).. PERIODU(T) =E= ((C(T)/L(T))**(1-ELASMU)-1)/(1-ELASMU); UTIL.. UTILITY =E= SUM(T, RR(T)*L(T)*deltat*(PERIODU(T))/scale1)+ scale2 ; ** Upper and Lower Bounds: General conditions for stability K.lo(T) = 100; MAT.lo(T) = 10; MU.lo(t) = 100; ML.lo(t) = 1000; C.lo(T) = 20; TOCEAN.up(T) = 20; TOCEAN.lo(T) = -1; TATM.up(t) = 20; miu.up(t) = LIMMIU; partfract(“1”)= 0.25372; * First period predetermined by Kyoto Protocol miu.fx(“1”) = 0.005; ** Fix savings assumption for standardization if needed s.fx(t)=.22; ** Cumulative limits on carbon use at 6000 GtC CCA.up(T) = FOSSLIM; ** Solution options option iterlim = 99900; option reslim = 99999; option solprint = on; option limrow = 0; option limcol = 0; model CO2 /all/; * Optimal run * Solution for optimal run solve CO2 maximizing UTILITY using nlp ; solve CO2 maximizing UTILITY using nlp ; solve CO2 maximizing UTILITY using nlp ; solve CO2 maximizing UTILITY using nlp ; solve CO2 maximizing UTILITY using nlp ; solve CO2 maximizing UTILITY using nlp ; Parameters opt_tax(t) opt_mcemis(t) ; opt_tax(t)=-1*ee.m(t)*1000/(kk.m(t)+.00000000001) ; opt_mcemis(t)= expcost2*cost1(t)*miu.l(t)**(expcost2-1)/sigma(t)*1000; display opt_tax, opt_mcemis; file CJL_QuarterYear_CarbonTax; put CJL_QuarterYear_CarbonTax; CJL_QuarterYear_CarbonTax.nw = 12; = 2; = 1e-15; loop(t, put; put opt_tax(t):14:6; put opt_mcemis(t):14:6; put /; );