Methods [for online publication] Economic module The model is a

Methods [for online publication]
Economic module
The model is a stylistic representation of the economy-climate system in the tradition of the
Nordhaus (1993, 2008) and van der Zwaan et al. (2002) Integrated Assessment Models (IAMs).
However, our model introduces a richer specification for the valuation of environmental damages;
taking into account the possibility that preferences for long-term climate goals are not necessarily
consistent across generations. Emissions lead to a build-up of atmospheric CO2, which both tangibly
and intangibly reduces the regulator’s felicity.
Model of the Economy
Per period utility, π‘ˆπ‘‘ , is proportional to population size, 𝐿𝑑 , and depends on consumption , 𝐢𝑑 , as
given by:
(1) π‘ˆπ‘‘ = 𝐿𝑑 ln (𝐢𝑑 /𝐿𝑑 )
Note that 𝐿𝑑 is a stock variable, while 𝐢𝑑 and π‘ˆπ‘‘ are flow variables. Both are measured in terms of
annual flows. Let 𝑁 be the number of years per period. Then 𝑁𝐢𝑑 and π‘π‘ˆπ‘‘ are the per-period flow of
consumption and utility. Future model versions will employ a more flexible utility function with
constant elasticity of marginal utility. The regulator at evaluation period 𝜏 maximizes welfare based
on the discounted stream of future utilities, but also includes social cost associated with the
deterioration of the future state of the climate. The latter element is an important extension of
existing IAMs. The regulator evaluates the economic damages of climate change through climate’s
effect on future output, consumption and utility. But in addition to that, it evaluates intangible
damages such as eco-system losses. These (potential) damages to the global environment are not
discounted over time; the regulator does not differentiate with respect to the time of global
environmental damages, but rather includes the long-term state of the climate directly into its
welfare functional. In the model, we take the atmospheric CO2 content [π΄π‘‘π‘šπ‘‘ , measured in GtCO2] as
the targeted variable that describes the state of the climate, and Ξ› 𝜏 is the ceiling for atmospheric CO2
when evaluated by the regulator at 𝜏:
(2) π‘Šπœ = Ξ₯𝜏 βˆ’ Ξ“(Ξ› 𝜏 )
(3) Ξ₯𝜏 = βˆ‘βˆž
𝑑=𝜏(1 + 𝜌)
(4) Ξ› 𝜏 = max{π΄π‘‘π‘šπ‘‘ ; 𝑑 = 1, … , ∞}
The model can easily be adjusted to take on board cumulative emissions, temperature change or
ocean acidification as the climate target variables. We assume exponential discounting, with 𝜌 the
annual pure rate of time preference, but the formulation is flexible and allows for non-exponential
discounting, such as hyperbolic discounting. Ξ₯𝜏 is the net present value of the utility stream net of
tangible damages in present value terms at 𝜏, Ξ“(Ξ›πœ ) is the welfare cost that the regulator associates
with stabilizing atmospheric CO2 levels at Ξ› 𝜏 . For the numerical calibration, we choose a quadratic
functional form for Ξ“(Ξ› 𝜏 ) Notice that welfare is time-inconsistent. The standard economy-climate model
assumes recursive time-consistent preferences, π‘Šπœ = π‘π‘ˆπ‘‘ + (1 + 𝜌)βˆ’π‘ π‘Šπœ+1 , which ensures that
each subsequent regulator conforms with decisions by previous regulators as long as no new
information arrives. In our case, however, welfare is not recursive and future regulators may
disagree with past regulators as to what is the optimal allocation, and consequently, they may
change policies.
There is one consumption (or final) good, for which we normalize the price to one. All prices are
measured relative to the price of the final good. Final output, π‘Œπ‘‘ can be used for consumption, 𝐢𝑑 , or
net investment:
(5) 𝑁𝐢𝑑 + 𝐾𝑑+1 βˆ’ (1 βˆ’ 𝛿)𝑁 𝐾𝑑 = π‘π‘Œπ‘‘
𝐾𝑑 , denotes capital, and 𝛿 is the annual depreciation rate for capital so that (1 βˆ’ 𝛿)𝑁 is the remaining
stock after N years. For future reference, we use 𝛿𝑁 = 1 βˆ’ (1 βˆ’ 𝛿)𝑁 as the per-period depreciation of
We define gross output, 𝑋𝑑 , while net final output π‘Œπ‘‘ is defined as gross output minus climate
damages and costs of emission reduction measures. Gross output uses capital and labor as
production factors. For a given labor size, 𝐿𝑑 , and labour productivity 𝐴𝑑 , gross output 𝑋𝑑 is given by:
(6) 𝑁𝑋𝑑 = 𝐾𝑑𝛼 (𝐴𝑑 𝑁𝐿𝑑 )1βˆ’π›Ό
where 𝛼 is the output elasticity of capital. Note that labour supply Lt is also multiplied by the number
of years per period, but the capital stock is not.
Net output, π‘Œπ‘‘ , and emissions, 𝑍𝑑 , depend on gross output, climate damages dependent on the
temperature, and the abatement rate πœ‡π‘‘ . We assume that abatement costs are quadratic in the
current abatement rate.
(7) π‘Œπ‘‘ = Ξ©(π‘‡π‘’π‘šπ‘π‘‘ )(1 βˆ’ 2 πœƒπ‘‘ πœ‡π‘‘ 2 )𝑋𝑑
(8) Ω𝑑 = 𝑒π‘₯𝑝(βˆ’Ξ”π‘Œ π‘‡π‘’π‘šπ‘π‘‘ 2 )
(9) 𝑍𝑑 = (1 βˆ’ πœ‡π‘‘ )πœŽπ‘‘ 𝑋𝑑
π‘‡π‘’π‘šπ‘π‘‘ is the temperature anomaly relative to the pre-industrial state, and parameter Ξ”π‘Œ describes
the intensity of output loss due to temperature change, such that Ω𝑑 = 1 in pre-industrial state. The
parameter πœŽπ‘‘ [tCO2/euro] is the emission intensity per unit of gross output if no abatement effort is
undertaken. Therefore, πœƒπ‘‘ /πœŽπ‘‘ gives the marginal costs of emissions reductions when abatement is
100%, that is, the carbon price at which emissions become zero. The value of πœƒπ‘‘ is the relative
income loss associated with a zero-emission policy, where the parameter πœƒπ‘‘ describes the extra costs
relative to output.
Labor and technology, both supplied inelastically, are assumed to follow a logistic growth path:
𝐿𝑑+1 = 𝐿𝑑 + 𝛾𝐿 𝐿𝑑 (1 βˆ’ 𝐿𝑑 /πΏπ‘šπ‘Žπ‘₯ )
𝐴𝑑+1 = 𝐴𝑑 + 𝛾𝐴 𝐴𝑑 (1 βˆ’ 𝐴𝑑 /π΄π‘šπ‘Žπ‘₯ )
where 𝛾𝐿 and 𝛾𝐴 are labor and technology growth rates at low levels, while πΏπ‘šπ‘Žπ‘₯ and π΄π‘šπ‘Žπ‘₯ are the
long-term labor and technology levels to which the variables converge.
Model of Climate Change
The carbon cycle of this model presents a reduced form for the atmospheric CO2 taken up by land
biomass, and separately for CO2 taken up by upper-layer and the deep ocean as depicted in Fig. 7.3
of the IPCC-AR4-WG1 report.1 The reduced form model has three reservoirs: the atmosphere plus
upper-layer of the ocean (label 0), biomass (label 1), and the deep ocean (label 2). Because of the
dynamic complexity of a 3-reservoir model, we do not write the variables and parameters as
describing annual processes, but directly calibrate the model dependent on the period length and
use variables and parameters that are consistent with the specific model period length.
Atmosphere and the upper-layer ocean are in equilibrium at a short time scale (about 1
year), so that for a decadal model we can consider atmospheric and ocean upper-layer CO2 as wellmixed, described by one variable, 𝑆0,𝑑 . Atmospheric CO2 can, therefore, be computed as a fixed share
of the CO2 stock in this first box:
π΄π‘‘π‘šπ‘‘ =
1+𝐻0 0,𝑑
𝐻0 is the relative absorptive capacity of CO2 of the ocean upper layer vis-a-vis the atmosphere.
The build-up of CO2 in the atmosphere and upper-layer ocean (i.e., the first box) is proportional
to emissions, but decays as the stock of CO2 dissolves into the deep ocean and is absorbed by land
and biomass:
𝑆0,𝑑+1 = 𝑆0,𝑑 + π‘Ž0 (𝑍𝑑 + 𝐸𝑑 ) βˆ’ 𝛿1 (𝐻1 𝑆0,𝑑 βˆ’ 𝑆1,𝑑 ) βˆ’ 𝛿2 (𝐻2 (𝑆0,𝑑 ) βˆ’ 𝑆2,𝑑 ))
𝐸𝑑 , are projections for emissions from cement and land use change as assessed from Nordhaus
(2008), which follow an exogenous path. As our period covers ten years, part of the emissions enter
the atmosphere, but also part enters the biomass and deep ocean box within the period: π‘Ž0 is the
emissions’ uptake by the atmosphere and upper-layer ocean together within the period. The
remainder of the emissions is taken up by land biomass (π‘Ž1 ) and deep ocean (π‘Ž2 ) within our time
span of a decade.
π‘Ž0 = 1 βˆ’ π‘Ž1 βˆ’ π‘Ž2
The equilibrium excess (relative to pre-industrial) CO2 stored as land biomass is assumed
proportional to the perturbation in atmosphere and upper-layer CO2, and the within-period CO2
uptake as land biomass is proportional to the gap between equilibrium and current stocks.
𝑆1,𝑑+1 = 𝑆1,𝑑 + π‘Ž1 (𝑍𝑑 + 𝐸𝑑 ) + 𝛿1 (𝐻1 𝑆0,𝑑 βˆ’ 𝑆1,𝑑 ))
𝐻1 is the storage capacity of biomass relative to the atmosphere plus upper-layer ocean together,
and 𝛿1 is the speed at which the biosphere moves to its equilibrium.
The deep ocean equilibrium storage of CO2 is the largest reservoir, and its dynamics determine
the long-run climate outcome of the model. It is assumed non-linear with respect to atmospheric
CO2, consistent with Figure 6.3 from the IPCC special report on CCS, chapter 6 (Caldeira and Akai
2005). We model this non-linear uptake as:
𝑆2,𝑑+1 = 𝑆2,𝑑 + π‘Ž2 (𝑍𝑑 + 𝐸𝑑 ) + 𝛿2 (𝐻2 (𝑆0,𝑑 ) βˆ’ 𝑆2,𝑑 ))
𝐻2 (𝑆0,𝑑 ) = 𝐻2π‘š (1 βˆ’ 𝑒 βˆ’π‘†0,𝑑 𝐻2 /𝐻2 )
where, 𝐻2π‘š is the maximum CO2 absorption by the deep ocean [TtCO2], 𝐻20 is the relative absorptive
capacity of the deep ocean at low CO2 levels, and 𝛿2 is the speed at which the deep ocean moves to
The atmospheric CO2 drives atmospheric forcing, which drives the enhanced greenhouse effect. We
specify temperature change as following a simple one-box adjustment to equilibrium temperature
π‘‡π‘’π‘šπ‘π‘‘+1 = π‘‡π‘’π‘šπ‘π‘‘ + πœ€π· (πΆπ‘™π‘–π‘šπ‘†π‘’π‘›π‘  (
⁄𝑆 )
) βˆ’ π‘‡π‘’π‘šπ‘π‘‘ )
where πœ€π· is the adjustment speed for the surface temperature. Temperature change is log-linear in
atmospheric CO2, stressing the convexity of the temperature-atmospheric CO2 relation.
As carbon dioxide does not escape the reservoirs within the time scale of our model, we can
determine the committed long-term climate change cumulated by past emissions through a steady
state analysis. We use the steady state conditions for carbon diffusion and temperature and denote
the stationary long-run outcomes by a tilde on top of the variables.
𝑆̃1,𝑑 = 𝐻1 𝑆̃0,𝑑
𝑆̃2,𝑑 = 𝐻2 (𝑆̃0,𝑑 )
𝑆0,𝑑 + 𝑆1,𝑑 + 𝑆̃2,𝑑 = 𝑆0,𝑑 + 𝑆1,𝑑 + 𝑆2,𝑑
Μƒ 𝑑 = πΆπ‘™π‘–π‘šπ‘†π‘’π‘›π‘  ln(1+π΄π‘‘π‘šπ‘‘β„π‘†0,0 )
The committed equilibrium
A regulator in a committed equilibrium sets the emission and consumption pathways for the entire
future, and all future regulators comply. Welfare optimization, specifically the first-order condition
for consumption leads to a Ramsey rule for inter-temporal exchange: the marginal rate of
substitution for consumption between periods should equal the discounted marginal return on
capital. Together with the first-order condition for capital investments, we can identify the optimal
consumption-investment decision for the capital-consumption (𝐢𝑑 , 𝐾𝑑 ) choice:
(1 + π‘Ÿπ‘‘+1 ) = (1 + 𝜌)
𝐿𝑑 𝐢𝑑+1
𝐿𝑑+1 𝐢𝑑
(π‘Ÿπ‘‘ + 𝛿𝑁 )𝐾𝑑 = π›Όπ‘žπ‘‘ 𝑋𝑑
π‘Ÿπ‘‘ is the return in period 𝑑 on investments from the previous period, and π‘žπ‘‘ is the shadow price of
gross output in period 𝑑.2 Equation (24) says that the return on capital should equal its marginal
productive value for every period 𝑑.
Let πœ“π‘‘ be the marginal social costs of emissions, the first-order condition for abatement effort
requires optimal climate costs for emissions reduction to equal the marginal productivity of
πœ“π‘‘ =
πœƒπ‘‘ πœ‡π‘‘
The marginal costs of emission reductions, in terms of foregone output, are proportional to the
abatement level.
The regulator seeks a cost effective path for emissions to balance the consumption loss with the
direct welfare loss of the climate target Ξ› 𝜏 . Denote by πœ†πœ,𝑑 the marginal costs in terms of the final
good as consumed in period 𝑑, on the production side, of meeting the target as evaluated by
regulator 𝜏, and by 𝑝𝑑 the marginal utility of consumption:
πœ†πœ,𝑑 =
𝑝𝑑 =
The shadow price of gross output Xt and the final consumption good Yt are not equal because of the
correction for climate damages and abatement costs in equation (7).
then, an optimal target, from the perspective of the regulator 𝜏, must satisfy the first order condition
for all 𝑑 β‰₯ 𝜏:
Ξ“ β€² (Ξ› 𝜏 ) = (1 + 𝜌)βˆ’π‘(π‘‘βˆ’πœ) 𝑝𝑑 πœ†πœ,𝑑
Equation (28) presents the time-consistency problem. Given a path as chosen by regulator 𝜏, with
target Ξ› 𝜏 , if the next regulator 𝜏 + 1 follows the same path, so that Ξ“ β€² (Ξ› 𝜏+1 ) = Ξ“ β€² (Ξ› 𝜏 ) and πœ†πœ+1,𝑑 =
πœ†πœ,𝑑, its first order conditions will not be satisfied. The next regulator will perceive the costs of the
target (RHS of the equation) as higher compared to the previous regulator, while it perceives the
benefits equally valuable (LHS of equation). The result is a higher stabilization threshold in each
next period.
Model calibration
The economy and climate modules are calibrated separately. The economy module is calibrated to
reproduce the output-capital-consumption allocation during the initial period. We derive a set of
equations that facilitate the calibration and run the economic calibration as part of the GAMS model
code. In contrast, the climate module is calibrated outside of the GAMS model code. Its calibration is
set up to reproduce atmospheric CO2 readings over the period 1960-2010.3 A period in our model is
10 years (a decade), with the initial period running from 1996-2005. Thus, annual parameters and
rates are adjusted to reflect decadal period length, while the model is initiated at the period 2000
(i.e., the base year).
Economy calibration
With no climate policy, the reduced model satisfies, 𝑍𝑑 = πœŽπ‘‘ 𝑋𝑑 , πœ“π‘‘ = 0, π‘Œπ‘‘ = 𝑋𝑑 , so that π‘ž = 1. We
assume the equilibrium is close to a balanced growth, and obtain the following system of three
equations, to calibrate three parameters (𝑔𝐴 , 𝜌, 𝛼):
1 + π‘”π‘Œ = (1 + 𝑔𝐴 )(1 + 𝑔𝐿 )
1 + π‘Ÿ = (1 + 𝜌)[1 + 𝑔𝐴 ]
𝐾𝑑 /π‘Œπ‘‘ =
𝑔𝐿 , π‘”π‘Œ , and 𝑔𝐴 are growth rates for labor, output, and labor productivity, respectively. For low labor
and technology levels, the economy is almost on a balanced growth such that 𝑔𝐿 = 𝛾𝐿 and 𝑔𝐴 = 𝛾𝐴 .
Table 1 in the Appendix summarizes all relevent (observed and calibrated) parameters for the
economic module.
The model is calibrated to the year 2000. The simulations start typically in the second or third
period, labeled 2010 and 2020, respectively. The first-period global population size is set to 6.1
[billion], with initial population growth rate of 13% [/decade], and assuming long-term global
population to stabilize at 11 [billion] based on World Bank forecasts. By Equation (10), we obtain a
maximum population growth rate, 𝑔𝐿 (= 𝛾𝐿 ) of 29.4% [/decade].
We set the gross economic growth at 41% [/decade], using Equation (29), we then calibrate for
periodical labor productivity growth 𝑔𝐴 at 25% [/decade]. Labor productivity is assumed to grow by
factor 10 in the long run. We use (29) to derive 𝛾𝐴 as 27% [/decade]
Assuming a real return on capital of 63% [/decade], equation (30) calibrates the social rate of
time preference 𝜌, obtained as 28% [/decade]. Finally, using Equation (31) we calibrate for the
The GAMS model code and Excel file that calibrates the climate module are available on request.
capital-labor elasticity of substitution, 𝛼, which we obtain as 0.409 based on initial annualized
capital stock of 150 trillion Euro and initial annualized GWP of 42 [trillion Euro].
The emission intensity πœŽπ‘‘ is calibrated based on annualized initial period emissions of 24
GtCO2 and annualized GWP of 42 trillion Euro, giving an initial emissions-output ratio of 0.625
kgCO2 per Euro output. Historically, emission intensities have been declining due to more efficient
usage of energy production, and the growing contribution of the service sector relative to
manufacturing in GWP. These decline rates vary widely by country, with improvements generally
observed for developed countries, and increasing intensities for developing countries. There is not
much hard evidence to forecast for intensity improvements. Based on past trends and since our
model does not differentiate between countries and sectors, we assume an average improvement in
πœŽπ‘‘ , of 1.1% per year. Similarly πœƒπ‘‘ declines at the same rate from an initial value of 0.2, meaning that
the marginal costs for emission reductions can reach a long-run maximum (after transition costs) of
320 €/tCO2. For the simulations used in the manuscript, we abstracted from direct climate damages,
settingΞ”π‘Œ = 0.
Climate Change calibration
The carbon cycle module has four parameters that describe the capacity of the ocean upper layer,
the land biomass and the deep ocean to take up anthropogenic CO2 (𝐻0 , 𝐻1 , 𝐻20 ; 𝐻2π‘š ), and two
parameters that describe the speed of the processes towards equilibrium (𝛿1 and 𝛿2 ). We calibrate
these parameters such that our model reproduces the Mauna Loa concentration data for 1959-2008
in combination with CDIAC emission data.4
The CO2 concentration curve shown in Figure 1 requires 3 parameters (degrees of freedom) out
of the 6 to reproduce its average level, its average slope, and its curvature. Furthermore, we
reproduce the estimated stocks in the 4 boxes (165, 18, 101, and 100, all in GtC, for the atmosphere,
ocean upper layer, land biomass (excluding changed related to land use) and deep ocean) of Fig 7.3
from AR4-WG1, and the net flows between biomass and the atmosphere and the deep ocean and
ocean upper layer (2.6 and 1.6 GtC/yr), from the same figure. These add 6 β€˜observations’, so that in
total we have 6 degrees of freedom to match 9 β€˜observations’.
From Fig. 7.3 of the IPCC-AR4-WG1 report, we calculate 𝐻0 as the ratio of the estimated
anthropogenic cumulated CO2 in the upper layer ocean versus the atmosphere:
𝐻0 = 18⁄
(165 + 18) β‰ˆ 9.8%
The calibration of the deep ocean equilibrium storage of CO2 is based on Figure 6.3 from the IPCC
special report on CCS, chapter 6 (Caldeira and Akai 2005). Atmospheric CO2 dissolves in water
forming H2CO3, which equilibrates with H+ and HCO3–. In turn, this equilibrates with 2H+ and CO32–.
The CO32– ions are provided by the solution of CaCO3, but as more of the CaCO3 is dissolved as a
consequence of the CO2 uptake, the ocean uptake of CO2 becomes less responsive. For low
anthropogenic emissions, the oceans take up about 4 times the atmospheric excess CO2; this defines
the parameter 𝐻20. For atmospheric CO2 concentrations of 1000 ppmv, the ocean uptake has
decreased to less than factor two. We use this to calibrate the value for 𝐢2π‘š , the maximum CO2
absorption by the deep ocean.
We are left with three parameters, the relative absorption capacity of biomass, and speed at
which biomass and the deep ocean move to equilibrium, 𝛿1 and 𝛿2 . These 3 parameters are used to
calibrate the net flows between biomass and the atmosphere and the deep ocean and ocean upper
layer (2.6 and 1.6 GtC/yr in 1990), and the Mauna Loa concentration data for 1959-2008 in
See Includes emission estimates for land use
change, cement production, and fossil fuel use, from 1751 to 2008.
combination with CDIAC emission data. The reduced model turns out to reproduce the data very
well as shown in Figure 1. It only has some difficulties to fully reproduce the increase in atmospheric
concentrations in the 21st century. All observed and calibrated parameters for the climate module
are summarized in Table 2.
Observations Mauna Loa
Calibrated Model
Figure 1: Mauna Loa versus carbon cycle model
Parameter values
Table 1: Derived and observed economic parameters for calibrating the economic module
First period central year
Annualized first period population (billion)
Max. population (billion)
Annualized first period capital stock (billion
Annualized first period GWP (trillion Euro)
Annualized first period carbon emissions
First period labor productivity (Euro/labor
Max. labor productivity (Euro/labor unit)
Intensity of output loss due to temperature
Maximum relative carbon cost for zeroemission policy x2
First period cement and land use emissions
Population growth rate
Output growth rate
Rate of capital depreciation
Real interest rate
π‘”πœŽ = π‘”πœ—
Technical improvements in CO2 intensity per
unit of output
Output elasticity of capital
Emission rate (tCO2/Euro)
Social rate of time preference
Maximum population growth rate
Maximum rate of labor productivity growth rate
Table 2: Climate change calibration parameters
π‘Ž0 ; π‘Ž1 ; π‘Ž2
Relative ocean upper layer capacity
Relative land biomass capacity (rel to atm+ocean ul)
Short term relative deep ocean capacity
Long term maximum deep ocean capacity
Relative uptake of periodical emissions by boxes
Adjustment per decade of land biomass
Adjustment per decade of deep ocean
Adjustment speed for surface temperature
Equilibrium temperature per atmospheric CO2 doubling
Preindustrial atmospheric CO2
Atmosphere + ocean upper layer stock in 1996
Land biomass stock in 1996
deep ocean stock in 1996
Surface temperature increase in 1996
13.212 TtCO2
0.281 /decade
0.032 /decade
0.182 /decade
2.129 TtCO2
0.750 TtCO2
0.408 TtCO2
0.396 TtCO2
0.77 K
Caldeira K. and M. Akai (eds) 2005, Ocean storage, Ch 6 in IPCC special report on carbon dioxide
capture and storage, edited by Metz B., O. Davidson, H. de Coninck, M. Loos, and L. Meyer,
Cambridge University Press.
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