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 intangiblyreduces the regulator’s felicity.
Model of the Economy
Per period utility,, is proportional to population size, , and depends on consumption , as given by:
(1)
Note that is a stock variable, while and are flow variables. Both are measured in terms of annual flows. Let be the number of years per period. Then and 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 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 [ measured in GtCO2] as the targeted variable that describes the state of the climate, and is the ceiling for atmospheric CO2 when evaluated by the regulator at :
(2)
(3)
(4)
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 the annual pure rate of time preference, but the formulation is flexible and allows for non-exponential discounting, such as hyperbolic discounting. is the net present value of the utility stream net of tangible damages in present value terms at , is the welfare cost that the regulator associates with stabilizing atmospheric CO2 levels at . For the numerical calibration, we choose a quadratic functional form for Notice that welfare is time-inconsistent. The standard economy-climate model assumes recursive time-consistent preferences, , 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, can be used for consumption, , or net investment:
(5)
, denotes capital, and is the annual depreciation rate for capital so that is the remaining stock after N years. For future reference, we use as the per-period depreciation of capital.
We define gross output, , while net final output 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, , and labour productivity , gross output is given by:
(6)
where 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,, and emissions, , depend on gross output, climate damages dependent on the temperature, and the abatement rate . We assume that abatement costs are quadratic in the current abatement rate.
(7)
(8)
(9)
is the temperature anomaly relative to the pre-industrial state, and parameter describes the intensity of output loss due to temperature change, such that in pre-industrial state. The parameter [tCO2/euro] is the emission intensity per unit of gross output if no abatement effort is undertaken. Therefore, gives the marginal costs of emissions reductions when abatement is 100%, that is, the carbon price at which emissions become zero. The value of is the relative income loss associated with a zero-emission policy, wherethe parameter describes the extra costs relative to output.
Labor and technology, both supplied inelastically, are assumed to follow a logistic growth path:
(10)
(11)
where and are labor and technology growth rates at low levels, while and 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 well-mixed, described by one variable, . Atmospheric CO2 can, therefore, be computed as a fixed share of the CO2 stock in this first box:
(12)
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:
(13)
, are projections for emissions from cement and land use change as assessed fromNordhaus (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: 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 () and deep ocean () within our time span of a decade.
(14)
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.
(15)
is the storage capacity of biomass relative to the atmosphere plus upper-layer ocean together, and 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-linearuptake as:
(16)
(17)
where, is the maximum CO2 absorption by the deep ocean [TtCO2], is the relative absorptive capacity of the deep ocean at low CO2 levels, and is the speed at which the deep ocean moves to equilibrium.
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 levels:
(18)
where 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.
(19)
(20)
(21)
(22)
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 choice:
(23)
(24)
is the return in period on investments from the previous period, and is the shadow price of gross output in period .[2]Equation (24) says that the return on capital should equal its marginal productive value for every period .
Let 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 emissions:
(25)
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 . Denote by the marginal costs in terms of the final good as consumed in period , on the production side, of meeting the target as evaluated by regulator , and by the marginal utility of consumption:
(26)
(27)
then, an optimal target, from the perspective of the regulator , must satisfy the first order condition for all
(28)
Equation (28) presents the time-consistency problem. Given a path as chosen by regulator with target , if the next regulator follows the same path, so that and , 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, , , , so that . We assume the equilibrium is close to a balanced growth, and obtain the following system of three equations, to calibrate three parameters :
(29)
(30)]
(31)
, , and 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 and .
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, of 29.4% [/decade].
We set the gross economic growth at 41% [/decade], using Equation (29), we then calibrate for periodical labor productivity growth at 25% [/decade]. Labor productivity is assumed to grow by factor 10 in the long run. We use (29) to derive as 27% [/decade]
Assuming a real return on capital of 63% [/decade], equation (30) calibrates the social rate of time preference , obtained as 28% [/decade]. Finally, using Equation (31) we calibrate for the capital-labor elasticity of substitution, , 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 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 , of 1.1% per year. Similarly 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, setting.
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 (), and two parameters that describe the speed of the processes towards equilibrium ( and ). 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 ‘observations’, so that in total we have 6 degrees of freedom to match 9 ‘observations’.
From Fig. 7.3 of the IPCC-AR4-WG1 report, we calculate as the ratio of the estimated anthropogenic cumulated CO2 in the upper layer ocean versus the atmosphere:
(32)
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. 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 , 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, and . 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 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.
Figure 1: Mauna Loa versus carbon cycle model
Parameter values
Table 1: Derived and observed economic parameters for calibrating the economic module
Parameter / Explanation / ValueObserved
First periodcentralyear / 2000
/ Annualized first period population (billion) / 6.1
/ Max. population (billion) / 11
/ Annualized first period capital stock (billion Euro) / 150
/ Annualized first period GWP (trillion Euro) / 42
/ Annualized first period carbon emissions (GtCO2) / 24
/ First period labor productivity (Euro/labor unit) / 1.504
/ Max. labor productivity (Euro/labor unit) / 15.04
/ Intensity of output loss due to temperature change / 0
/ Maximum relative carbon cost for zero-emission policy x2 / 0.2
/ First period cement and land use emissions (GtCO2) / 10
/ Population growth rate / (1.24%/yr)
/ Output growth rate / (3.5%/yr)
/ Rate of capital depreciation / (7%/yr)
/ Real interest rate / (5%/yr)
/ Technical improvements in CO2 intensity per unit of output / (1.1%/yr)
Derived
/ Output elasticity of capital / 0.409
/ Emission rate (tCO2/Euro) / 0.571
/ Social rate of time preference / (2.5%/yr)
/ Maximum population growth rate / (2.13%/yr)
/ Maximum rate of labor productivity growth rate / (2.72%/yr)
Table2: Climate change calibration parameters