.DICE

Dynamic Integrated Climate Economy model, W. D. Nordhaus Replicated by Tom Fiddaman, MIT System Dynamics Group (tomfid@mit.edu) from: Nordhaus, William D. (1992). The "DICE" Model: Background and Structure of a Dynamic Integrated Climate-Economy Model of the Economics of Global Warming No. Discussion Paper No. 1009). Cowles Foundation for Research in Economics at Yale University. Nordhaus, William D. (1992). An Optimal Transition Path for Controlling Greenhouse Gases. Science, 258(20), 1315-1319. Nordhaus, William D. (1994). Managing the Global Commons. Cambridge, MA: MIT Press. To use the model, you must: - Use a copy of Vensim DSS (or the DSS model reader) in order to read arrays and perform optmization. - Convert the file Nordhaus2.dat (which contains data from Nordhaus' results) to a .vdf dataset on your own platform. Alternatively, you can eliminate the variables that rely on the dataset. Notes: - For optimization runs, be sure to use a final time of at least 2305 to 2505 to prevent finite-horizon effects. Unlike Nordhaus' GAMS implementation, this model does not use transversality coefficients from longer runs. The results from 1965 to 2105 are generally insensitive to time horizons greater than 2305, though. - The efficiency of optimization can be increased by raising the absolute tolerances on the decision variables. - The investment path changes very little from run to run, so for most purposes it is acceptable to use a baseline investment path and optimize only the GHG reduction path (using redonly2.prm). Alternatively, you can use the behavioral investment heuristic and search for optimal values of invest_frac_nonlin and invest_frac_scale; this heuristic very closely mimics the optimal time path. - It's a good idea to use random multi-start optimization to verify your solutions.

.Carbon

Emissions, carbon cycle and related variables.

(001) Atmos_Retention = 0.64

Units: dmnl

Atmospheric Retention Fraction [beta] (dimensionless) Fraction of Greenhouse Gas Emissions which accumulate in the atmosphere. [Cowles, pg. 21]

Uses:

(002) CO2_Emiss = (1-GHG_Reduction_Frac)*CO2_Intensity_of_Output*Output

Units: TonC/year

Greenhouse Gas Emissions [E(t)] (tons carbon equivalent/year) [Cowles, pg. 20]

Causes:

Uses:

(003) CO2_in_Atmos = INTEG(CO2_Net_Emiss - CO2_Storage, 6.77e+011)

Units: TonC

Greenhouse Gases in Atmosphere [M(t)] (tons carbon equivalent) [Cowles, pg. 21]

Causes:

Uses:

(004) CO2_Intens_Dec_Rt_Decline_Rt = CO2_Intens_Decline_Rt*Fact_Prod_Gr_Rt_Dec_Rt

Units: 1/year/year

Causes:

Uses:

(005) CO2_Intens_Decline_Rt = INTEG(- CO2_Intens_Dec_Rt_Decline_Rt, init_co2_intens_decline_rt )

Units: 1/year

Rate of Decline of Greenhouse Gas Intensity of Output [g-sigma] (1/year) Note that Nordhaus decompounds the decadal rate of .1168 to yield an annual rate of .0125. This does not work with time steps smaller than Nordhaus' 10 years, so I have simply divided by 10 to convert the decadal rate to an annual rate. [Managing Global Commons, pg. 21]

Causes:

Uses:

(006) CO2_Intensity_of_Output = INTEG(- Decline_CO2_Intens, 0.000519)

Units: TonC/$

Greenhouse Gas Intensity of Output [sigma(t)] (tons carbon equivalent/$) [Managing Global Commons, pg. 21] Conflicts with value reported on Cowles, pg. 24: .5368*.9875^(TIME-1990)/1000 = .7352/1000

Causes:

Uses:

(007) CO2_Net_Emiss = Atmos_Retention*CO2_Emiss

Units: TonC/year

Net Greenhouse Gas Emissions (tons carbon equivalent/year) Greenhouse gas emissions less short-run uptake from the atmosphere. Where does the portion not retained go in the long run? [Cowles, pg. 21]

Causes:

Uses:

(008) CO2_Rad_Force_Coeff = 4.1

Units: watt/meter/meter

Coefficient of Radiative Forcing from CO2 (W/m^2) Coeff. of additional surface warming from accumulation of CO2. [Cowles, pg. 22]

Uses:

(009) CO2_Rad_Forcing = CO2_Rad_Force_Coeff*LOG(CO2_in_Atmos/Preindustrial_CO2 ,2)

Units: watt/meter/meter

Radiative Forcing from CO2 [F(t)] (W/m^2) Additional surface warming from accumulation of CO2. [Cowles, pg. 22]

Causes:

Uses:

(010) CO2_Storage = (CO2_in_Atmos-Preindustrial_CO2)*Rate_of_CO2_Transfer

Units: TonC/year

Greenhouse Gas removal from the atmosphere and storage by long-term processes. (tons carbon equivalent/year) [Cowles, pg. 21]

Causes:

Uses:

(011) Decline_CO2_Intens = CO2_Intensity_of_Output*CO2_Intens_Decline_Rt

Units: TonC/$/year

Decline of GHG Intensity of Output (tons carbon equivalent/$/year) [Cowles, pg. 20]

Causes:

Uses:

(012) Emiss_Stabilization

Units: dmnl

Fraction of CO2 and CFC Emissions Controlled (dimensionless) Stabilization of Emissions. Estimated from graph in [Science, Fig. 1].

Uses:

(013) Emissions_Scenario = 1

Units: dmnl

Uses:

(014) GHG_Red_Cost_Frac = 1-Red_Cost_Scale*if_then_else(GHG_Reduction_Frac> 0,GHG_Reduction_Frac^Red_Cost_Nonlinearity ,0)

Units: dmnl

Fraction of Output devoted to cost of GHG emissions reductions (dimensionless)

Causes:

Uses:

(015) GHG_Reduction_Frac = Optimal_Red_Switch*Optimal_GHG_Reduction_Frac + (1-Optimal_Red_Switch)*Nord_GHG_Reduction_Frac

Units: dmnl

Fraction of Greenhouse Gas Emissions Abated [mu(t)] May be switched between path from optimization and Nordhaus' path.

Causes:

Uses:

(016) init_co2_intens_decline_rt = 0.01168

Units: 1/year

Uses:

(017) No_Controls = 0

Units: dmnl

Fraction of CO2 and CFC Emissions Controlled (dimensionless) Uncontrolled scenario.

Uses:

(018) Nord_GHG_Reduction_Frac = if_then_else(Emissions_Scenario=1,No_Controls ,if_then_else(Emissions_Scenario =2 ,Optimal_Controls,if_then_else(Emissions_Scenario=3,Emiss_Stabilization,Temp_Stabilization )))

Units: dmnl

Fraction of Greenhouse Gas Emissions Abated [mu(t)] (dimensionless) Selects one of three scenarios. [Cowles, pg. 20]

Causes:

Uses:

(019) Optimal_Controls

Units: dmnl

Fraction of CO2 and CFC Emissions Controlled (dimensionless) Optimal control scenario. [Cowles, table IV-3]

Uses:

(020) Optimal_Red_Switch = 1

Units: dmnl

Switches GHG Reduction Frac between Nordhaus' time path and time path from optimization.

Uses:

(021) Preindustrial_CO2 = 5.9e+011

Units: TonC

Uses:

(022) Rate_of_CO2_Transfer = 0.008333

Units: 1/year

Rate of Storage of Atmospheric Greenhouse Gases [delta-m] (1/year) Inverse yields average residence time of gases (120 years). Note that the validity and stability of this factor is highly questionable. [Cowles, pg. 21]

Uses:

(023) Red_Cost_Nonlinearity = 2.887

Units: dmnl

Nonlinearity of GHG Reduction Cost [b2] (dimensionless) [Cowles, pg. 13 & 24]

Uses:

(024) Red_Cost_Scale = 0.0686

Units: dmnl

Scale of GHG Reduction Cost [b1] (dimensionless) [Cowles, pg. 13 & 24]

Uses:

(025) Temp_Stabilization

Units: dmnl

Fraction of CO2 and CFC Emissions Controlled Stabilization of temperature. Estimated from graph. [Science, Fig. 1].

Uses:

.Climate

(026) A_UO_Heat_Cap = 44.248

Units: watt*year/DegreesC/(meter*meter)

Atmosphere & Upper Ocean Heat Capacity per Unit Area [1/R1] (W-yr/m^2/degrees C) Note: equals 1/0.0226 [Managing the Global Commons, pg. 21]

Uses:

(027) Atmos_UOcean_Temp = INTEG(Chg_A_UO_Temp, 0.2)

Units: DegreesC

Temperature of the Atmosphere and Upper Ocean [T] (degrees C) [Cowles, pg. 24]

Causes:

Uses:

(028) Chg_A_UO_Temp = (Radiative_Forcing-Feedback_Heating-Heat_Transfer)/A_UO_Heat_Cap

Units: DegreesC/year

Change in the Atmosphere & Upper Ocean Temperature (degrees C/yr) [Cowles, pg. 27]

Causes:

Uses:

(029) Chg_DO_Temp = Heat_Transfer/DO_Heat_Cap

Units: DegreesC/year

Change in the Deep Ocean Temperature (degrees C/yr) [Cowles, pg. 30]

Causes:

Uses:

(030) Climate_Damage_Frac = 1/(1+Climate_Damage_Scale*(Atmos_UOcean_Temp/Reference_Temperature)^Climate_Damage_Nonlinearity )

Units: dmnl

Fraction of Output lost to combating Climate Change (1/Degrees C^2)

Causes:

Uses:

(031) Climate_Damage_Nonlinearity = 2

Units: dmnl

Nonlinearity of Climate Damage Cost Fraction [Theta2] (dimensionless) [Cowles, pg. 13 & 24]

Uses:

(032) Climate_Damage_Scale = 0.013

Units: dmnl

Climate Damage Fraction at Reference Temperature [part of Nordhaus' variable Theta1] (dimensionless) [Managing Global Commons, pg. 18 and 21]

Uses:

(033) Climate_Feedback_Param = 1.41

Units: watt/meter/meter/DegreesC

Climate Feedback Parameter [lambda] (W-m^2/degree C) The crucial climate sensitivity parameter - determines feedback warming from temperature increase. The Schneider-Thompson 2-stock model uses 1.33 [Cowles, Table III-B1]. [Managing Global Commons, pg. 21]

Uses:

(034) Deep_Ocean_Temp = INTEG(Chg_DO_Temp, 0.1)

Units: DegreesC

Temperature of the Deep Ocean [T*] (degrees C) [Cowles, pg. 24]

Causes:

Uses:

(035) DO_Heat_Cap = Heat_Capacity_Ratio*Heat_Trans_Coeff

Units: watt*year/DegreesC/meter/meter

Deep Ocean Heat Capacity per Unit Area [R2] (W-yr/m^2/degrees C) Note: Managing Global Commons uses .44*Heat_Trans_Coeff = 220; Cowles report uses 223.7 (page 30). [Managing Global Commons, pg. 21]

Causes:

Uses:

(036) Feedback_Heating = Atmos_UOcean_Temp*Climate_Feedback_Param

Units: watt/meter/meter

Feedback Heating (W/m^2) Additional heating of the atmosphere/upper ocean system from feedback effects of warming. [Cowles, pg. 27]

Causes:

Uses:

(037) Heat_Capacity_Ratio = 0.44

Units: watt/(meter*meter*DegreesC)

Ratio of Thermal Capacity of Deep Ocean to Heat Transfer Time Constant [R2/Tau12] [Managing Global Commons, pg. 21]

Uses:

(038) Heat_Trans_Coeff = 500

Units: year

Heat Transfer Coefficient [tau12] (years) Coefficient of heat transfer between the atmosphere & upper ocean and the deep ocean. May be interpreted as a mixing time constant. Schneider & Thompson use a slightly higher estimate of 550. [Cowles, pg. 31]

Uses:

(039) Heat_Transfer = Temp_Diff*DO_Heat_Cap/Heat_Trans_Coeff

Units: watt/meter/meter

Heat Transfer from the Atmosphere & Upper Ocean to the Deep Ocean

Causes:

Uses:

(040) Other_GHG_Rad_Forcing

Units: watt/meter/meter

Radiative Forcing from Other GHGs (W/m^2) Additional surface warming from accumulation of other GHGs (NOx and Methane). [Table 4.9B, Managing Global Commons, pg. 73]

Uses:

(041) Radiative_Forcing = CO2_Rad_Forcing+Other_GHG_Rad_Forcing

Units: watt/meter/meter

Radiative Forcing from All GHGs (W/m^2) Additional surface warming from accumulation of CO2 & CFCs. [Cowles, Sec. III.F]

Causes:

Uses:

(042) Reference_Temperature = 3

Units: DegreesC

Reference Temperature for Calculation of Climate Damages [part of Nordhaus' variable theta1] [Managing Global Commons, pg. 18 and 21]

Uses:

(043) Temp_Diff = Atmos_UOcean_Temp-Deep_Ocean_Temp

Units: DegreesC

Temperature Difference between Upper and Deep Ocean (degrees C)

Causes:

Uses:

.Control

(044) FINAL_TIME = 2105

Units: year

(045) INITIAL_TIME = 1965

Units: year

Uses: (000)Time - Internally defined simulation time.

(046) SAVEPER = 5

Units: year

(047) TIME_STEP = 5

Units: year

.Data

(048) IPCC_CO2_CFC_Rad_Force

Units: watt/meter/meter

IPCC Scenario for Radiative Forcing from CO2 and CFCs (W/m^2) As interpolated by Nordhaus. [Cowles, Table III.E-5]

(049) Nord_CO2_in_Atm

Units: GTonC

Nordhaus' CO2 & CFC Concentrations (Gt Carbon Equivalent) Uncontrolled scenario [Cowles, Table IV-4].

(050) Nord_CO2_Intensity

Units: GTonC/$

(051) Nord_Emiss

Units: GTonC/year

Nordhaus' CO2 & CFC Emissions (Gt Carbon Equivalent) Uncontrolled scenario [Cowles, Table IV-4].

(052) Nord_Output

Units: $/year

Nordhaus' Output ($/year) [Cowles, Table IV-1]

(053) Nord_Temp

Units: DegreesC

Nordhaus' Atmospher & Upper Ocean Temperature Difference (degrees C) Uncontrolled scenario [Cowles, Table IV-5].

.Econ

(054) Behav_Invest_Frac = Invest_Frac_Scale*(Marg_Return_Capital/Norm_Return_Capital )^Invest_Frac_Nonlin

Units: dmnl

A simple behavioral heuristic for investment; closely replicates results of the optimal time path.

Causes:

Uses:

(055) Capital = INTEG(Investment - Depreciation, 1.6e+013)

Units: $

Capital ($) Capital stock in 1989 dollars. [Managing Global Commons, pg. 21]

Causes:

Uses:

(056) Capital_Elast_Output = 0.25

Units: dmnl

Capital Elasticity of Output [alpha] (dimensionless) Derived from share of capital in national income. [Cowles, pg. 17]

Uses:

(057) Consumption = Output-Investment

Units: $/year

Consumption ($/year) Output less investment (savings).

Causes:

Uses:

(058) Depreciation = Capital*Depreciation_Rate

Units: $/year

Depreciation ($/year)

Causes:

Uses:

(059) Depreciation_Rate = 0.065

Units: 1/year

Depreciation Rate [delta-k] (1/year) Note that Nordhaus assumes a 10-year capital life, then chooses a value of 0.065 to correct for the lack of compounding in the 10-year time step he uses. This is simply wrong, as the capital stock has an inflow as well as an outflow, and it is the net rate (investment-depreciation) that must be compounded. Also, using a value of 0.065 results in an average residence time of units in the capital stock of 15 years, even with the 10-year time step. I have preserved the value 0.065 for replication; a 15-year capital life is perfectly reasonable anyway. [Managing Global Commons, pg. 21]

Uses:

(060) Fact_Prod_Gr_Rt_Dec_Rt = 0.011

Units: 1/year

Rate of Decline of Factor Productivity Growth Rate [delta-A] (1/year/year) Factor productivity growth rate declines 11% per decade. [Cowles, pg. 18]

Uses:

(061) Fact_Prod_Gr_Rt_Decline_Rt = Fact_Prod_Growth_Rt*Fact_Prod_Gr_Rt_Dec_Rt

Units: 1/year/year

Decline of Factor Productivity Growth Rate (1/year/year)

Causes:

Uses:

(062) Fact_Prod_Growth_Rt = INTEG(- Fact_Prod_Gr_Rt_Decline_Rt, 0.015)

Units: 1/year

Growth Rate of Factor Productivity [gA(t)] (1/year) Growth rate declines over time. Value reported in [Cowles, pg. 17]: .0152 for period 1965-1987, matches statement in [Science, pg. 1317] that average was 1.3% from 1965-1989, with an 11%/decade rate of decline. Note that Nordhaus decompounds the decadal rate of .150 to yield an annual rate of .0141; I have simply divided by 10 to convert the decadal rate to an annual rate. [Managing Global Commons, pg. 21] [Managing the Global Commons, pg. 21]

Causes:

Uses:

(063) Fact_Prod_Incr_Rt = Factor_Productivity*Fact_Prod_Growth_Rt

Units: 1/year

Change in Factor Productivity (1/year)

Causes:

Uses:

(064) Factor_Productivity = INTEG(Fact_Prod_Incr_Rt, 1)

Units: dmnl

Total Factor Productivity [A(t)] (dimensionless) May be interpreted as level of technology. [Cowles pg. 17]

Causes:

Uses:

(065) Invest_Frac_Nonlin = 1

Units: dmnl

Uses:

(066) Invest_Frac_Scale = 0.2

Units: dmnl

Uses:

(067) Investment = Output*Investment_Frac

Units: $/year

Gross Investment ($/year)

Causes:

Uses:

(068) Investment_Frac = if_then_else(Optimal_Invest_Switch=1,Optimal_Invest_Frac , if_then_else(Optimal_Invest_Switch=2,Behav_Invest_Frac,Nord_Investment_Frac ))

Units: dmnl

Fraction of Output Invested May be switched between path derived from optimization and Nordhaus' path

Causes:

Uses:

(069) Net_CC_Impact = GHG_Red_Cost_Frac*Climate_Damage_Frac

Units: dmnl

Net Climate Change Impact [Omega(t)] (dimensionless) The fraction of output lost to GHG emissions reduction and climate change damage costs. [Cowles, pg. 13]

Causes:

Uses:

(070) Nord_Investment_Frac

Units: dmnl

Fraction of Output allocated to Investment (dimensionless) Time path derived from results of optimization reported in [Cowles, Table IV-2, Optimal]. Intermediate points interpolated linearly. Points after 2075 estimated from [Cowles, Fig. IV-5].

Uses:

(071) Norm_Return_Capital = 0.08

Units: 1/year

Uses:

(072) Optimal_Invest_Switch = 1

Units: dmnl

Switches Investment Frac between Nordhaus' time path and time path from optimization.

Uses:

(073) Output = Reference_Output*Net_CC_Impact

Units: $/year

Output [Q(t)] ($/year) Cobb-Douglas capital-labor formulation. [Cowles, pgs. 17 & 24]

Causes:

Uses:

(074) Output_in_1965 = 8.519e+012

Units: $/year

Output in 1965 ($/yr) [Managing Global Commons, pg. 21]

Uses:

(075) Reference_Output = Output_in_1965*Factor_Productivity*(Capital/INIT(Capital ))^Capital_Elast_Output *(Population/INIT(Population))^(1-Capital_Elast_Output)

Units: $/year

Reference Output before effects of climate damage and emissions abatement are considered

Causes:

Uses:

.Indices

(076) Capital_Labor_Ratio = Capital/Population

Units: $/person

Ratio of Capital Inputs to Labor Inputs ($/person)

Causes:

(077) Capital_Output_Ratio = Capital/Output

Units: $/($/year)

Capital per Unit Output ($ per $/year)

Causes:

(078) Climate_Damages = Reference_Output*(1-Climate_Damage_Frac)

Units: $/year

Flow of damages from climate change.

Causes:

(079) CO2_And_CFC_Intens_Capital = CO2_Emiss/Capital

Units: TonC/year/$

CO2 and CFC Emissions per Unit of Capital (tons carbon equiv/year/$)

Causes:

(080) Labor_Output_Ratio = Population/Output

Units: person/($/year)

Ratio of Labor to Output (persons/$)

Causes:

(081) Marg_Prod_Capital = Capital_Elast_Output*Output/Capital

Units: 1/year

Marginal Productivity of Capital

Causes:

Uses:

(082) Marg_Prod_Carbon = Reference_Output/Reference_CO2_Emissions*Red_Cost_Scale *Red_Cost_Nonlinearity *if_then_else(GHG_Reduction_Frac&gt0,(GHG_Reduction_Frac)^(Red_Cost_Nonlinearity -1),0)

Units: $/TonC

Marginal Productivity of CO2 Emissions

Causes:

(083) Marg_Return_Capital = Marg_Prod_Capital-Depreciation_Rate

Units: 1/year

Marginal Return to Capital Equals the marginal product of capital less depreciation.

Causes:

Uses:

(084) Net_Investment = Investment-Depreciation

Units: $/year

Net Investment Investment less depreciation

Causes:

Uses:

(085) Net_Savings_Rate = Net_Investment/Output

Units: dmnl

Net Savings Rate Equal to the ratio of net investment to output.

Causes:

(086) Reduction_Costs = (1-GHG_Red_Cost_Frac)*Reference_Output

Units: $/year

Flow of greenhouse gas abatement costs.

Causes:

(087) Reference_CO2_Emissions = Reference_Output*CO2_Intensity_of_Output

Units: TonC/year

Reference CO2 Emissions Emissions at normal CO2 intensity, with no abatement.

Causes:

Uses:

.Optimization

Structures for allowing optimization of decisions as an arbitrary time path.

(088) GHG_Red_Fracs[T] = INTEG(Zero,Init_GHG_Red_Fracs[T])

Units: dmnl

GHG Reduction Fractions at policy time T

Causes:

Uses:

(089) Init_GHG_Red_Fracs[T] = 0,0,0,0,0,0,0,0,0,0

Units: dmnl

GHG Reduction Fractions at policy time T

Uses:

(090) Init_Invest_Fracs[T] = 0.17,0.17,0.17,0.17,0.17,0.18,0.19,0.2,0.21,0.22

Units: dmnl

Investment Fractions at policy time T

Uses:

(091) Init_Policy_Times[T] = 2305,2205,2105,2050,2025,2005,2000,1995,1985,1965

Units: year

Year of implementation of Tth policy

Uses:

(092) Interpolation_Frac = max(0,zidz(Time-Policy_Times[T10],Policy_Times[T9 ]-Policy_Times[T10]))

Units: dmnl

Fraction of interval between policy times elapsed. (000)Time - Internally defined simulation time.

Causes:

Uses:

(093) Invest_Fracs[T] = INTEG(Zero,Init_Invest_Fracs[T])

Units: dmnl

Investment Fractions at policy time T

Causes:

Uses:

(094) Optimal_GHG_Reduction_Frac = GHG_Red_Fracs[T10] + (GHG_Red_Fracs[T9]- GHG_Red_Fracs[T10])*Interpolation_Frac

Units: dmnl

GHG Reduction Fraction derived from optimization.

Causes:

Uses:

(095) Optimal_Invest_Frac = Invest_Fracs[T10] + (Invest_Fracs[T9]-Invest_Fracs [T10])*Interpolation_Frac

Units: dmnl

Investment Fraction derived from optimization.

Causes:

Uses:

(096) Policy_Times[T] = INTEG(0,Init_Policy_Times[T])

Units: year

Year of implementation of Tth policy

Causes:

Uses:

(097) Shift_Invest = SHIFT_IF_TRUE(Invest_Fracs[T1],shift_switch=1,T10,0,Invest_Fracs [T1])

Units: dmnl

Shifts investment stack values. (000)T10 -

Causes:

(098) Shift_Red = SHIFT_IF_TRUE(GHG_Red_Fracs[T1],shift_switch=1,T10,0,GHG_Red_Fracs [T1])

Units: dmnl

Shifts reduction stack values. (000)T10 -

Causes:

(099) shift_switch = if_then_else(Time > Policy_Times[T9],1,0)

Units: dmnl

(000)Time - Internally defined simulation time.

Causes:

Uses:

(100) Shift_Times = SHIFT_IF_TRUE(Policy_Times[T1],shift_switch=1,T10,0,Policy_Times [T1])

Units: dmnl

Shifts time stack values. (000)T10 -

Causes:

(101) T : (T1-T10) Subscript for policy optimization arrays

(102) Zero = 0

Units: 1/year

Dummy variable to provide a 0 with units 1/year.

Uses:

.Population

(103) Consumption_per_Cap = Consumption/Population

Units: $/person/year

Consumption per Capita ($/person/year)

Causes:

Uses:

(104) Decline_Pop_Gr_Rt = Pop_Growth_Rate*Pop_Gr_Rt_Decline_Rt

Units: 1/year/year

Decline of Population Growth Rate (1/year/year)

Causes:

Uses:

(105) Net_Pop_Incr = Population*Pop_Growth_Rate

Units: person/year

Net Population Increase (persons/year)

Causes:

Uses:

(106) Pop_Gr_Rt_Decline_Rt = 0.0195

Units: 1/year

Rate of Decline of Population Growth Rate [delta-pop] (1/year) 19.5 % per decade. [Cowles, pg. 16] Real data looks closer to 10 % per decade before 1990. Note that Nordhaus decompounds the decadal rate of .195 to yield an annual rate of .02; I have simply divided by 10 to convert the decadal rate to an annual rate. [Managing Global Commons, pg. 21]

Uses:

(107) Pop_Growth_Rate = INTEG(- Decline_Pop_Gr_Rt, 0.0224)

Units: 1/year

Population Growth Rate [gpop(t)] (1/year) Note that Nordhaus decompounds the decadal rate of .224 to yield an annual rate of .0203; I have simply divided by 10 to convert the decadal rate to an annual rate. [Managing Global Commons, pg. 21]

Causes:

Uses:

(108) Population = INTEG(Net_Pop_Incr, 3.369e+009)

Units: person

Population [L(t)] (persons) [Cowles, pg. 16]

Causes:

Uses:

.Utility

(109) Base_Year = 1989

Units: year

Base Year for Discounting (year) Model is denominated in 1989 dollars, and discounting is performed relative to 1989.

Uses:

(110) Cum_Disc_Utility = INTEG(Discounted_Utility, 0)

Units: utiles

Cumulative Discounted Utility (log$) This is Nordhaus' objective function. The results in [Science, Table 1] apparently accumulate only the period from 1990-2045. [Cowles, pg. 15]

Causes:

(111) Discount_Factor = EXP(-Rate_of_Time_Pref*(Time-Base_Year))

Units: dmnl

(000)Time - Internally defined simulation time.

Causes:

Uses:

(112) Discounted_Utility = Utility*Discount_Factor

Units: utiles/year

Discounted Current Utility (log$/year) Current Utility discounted to 1989.

Causes:

Uses:

(113) Rate_of_Inequal_Aversion = 1

Units: dmnl

Rate of Inequality Aversion [alpha] (dimensionless) Measure of marginal utility or social valuation of different levels of consumption. [Cowles, pg. 16]

Uses:

(114) Rate_of_Time_Pref = 0.03

Units: 1/year

Pure Rate of Social Time Preference [rho] (1/year) The social discount rate. [Cowles, pg. 15]

Uses:

(115) Ref_Cons_per_Cap = 1000

Units: $/person/year

Reference Consumption per Capita

Uses:

(116) Utility = Utility_Coeff*Population*if_then_else(Rate_of_Inequal_Aversion =1,LN(Consumption_per_Cap/Ref_Cons_per_Cap ), ((Consumption_per_Cap/Ref_Cons_per_Cap)^(1-Rate_of_Inequal_Aversion)-1)/(1- Rate_of_Inequal_Aversion ))

Units: utiles/year

Current Utility [U(t)] (utiles/year) Reduces to Logarithmic or Bernoullian utility function: Population*(Log(Consumption_per_Cap)) when the Rate of Inequality Aversion -&gt 1 Note that doubling your population with half the consumption per capita is an improvement with this formula. [Cowles, pg. 16]

Causes:

Uses:

(117) Utility_Coeff = 1

Units: utiles/person/year

Reference Rate of Utility Generation (utiles/person/year)

Uses: