The permafrost carbon feedback in DICE-2013R modeling and empirical results

Abstract

Climate feedback mechanisms that have the potential to intensify global warming have been omitted almost completely in the integrated assessment of climate change and the economy so far. In the present paper, we incorporate the permafrost carbon feedback (PCF) into the well-known integrated assessment model DICE-2013R. We calibrate the parameters for our extended version of DICE-2013R and compute the optimal emission mitigation rates that maximize welfare. Our results indicate that accounting for the PCF leads to an increase in mitigation. Finally, we quantify the economic losses resulting from a climate policy which ignores the impacts of the PCF.

Appendix: Short description of the DICE-2013R model

The utility in DICE is expressed as a standard constant-relative-risk-aversion utility function for neoclassical growth models:

$$U[c(t),L(t)] = L(t)\left[ {\frac{{c(t)^{1 - \alpha } }}{1 - \alpha }} \right]$$

(1)

with t indicating the specific period (one period accounts for 5 years), c(t) is per capita consumption, L(t) is the population and α is the elasticity of marginal utility of consumption. The objective is to maximize the welfare function W. The latter consists of the discounted utility summed over a finite time horizon:

$$W[c(t),\;L(t)] = \sum\limits_{t = 1}^{T} {\frac{1}{{(1 + \rho )^{t - 1} }}} U[c(t),L(t)].$$

(2)

The parameter ρ is the pure rate of social time preference such that ${1\mathord{\left/ {\vphantom {1 {(1 + \rho )^{t - 1} }}} \right. \kern-0pt} {(1 + \rho )^{t - 1} }}$ is the discount factor. The production function is of Cobb–Douglas type:

$$Y(t) = A(t)K(t)^{\gamma } L(t)^{1 - \gamma } .$$

(3)

A(t) is the total factor productivity, K(t) is the capital stock, L(t) is not only the population but also the labor input and γ is the elasticity of output with respect to capital. The link to the climate module is formed via greenhouse gas emissions which are caused by production due to an exogenous emission coefficient [see also Eq. (7)]. These emissions accumulate in the atmosphere. The carbon dioxide concentrations in the atmosphere and in the oceans are interrelated, since oceans are considered a huge sink for emissions (Nordhaus 2008 p. 43). As described below by Eqs. (9)–(14), the accumulated emissions lead to a higher atmospheric greenhouse gas concentration which causes the radiative forcing to increase and ultimately cause the surface temperature to increase. The impact of this temperature increase is given by the following damage function Ω(t) which indicates the share of output that is lost due to climate damages:

$$\varOmega (t) = \sigma_{1} \times \Delta T_{AT} (t) + \sigma_{2} \times \Delta T_{AT} (t)^{2} .$$

(4)

ΔT _AT(t) is the increase of the atmospheric global mean temperature compared to the pre-industrial level, and σ ₁ as well as σ ₂ are parameters that determine the shape of the damage function. To avoid damages, emissions can be reduced by mitigation activities. The accompanying costs Λ(t) are expressed as the share of output that is lost due to mitigation activities. The cost function describing Λ(t) is given by:

$$\varLambda (t) = \psi_{1} \cdot \mu (t)^{{\psi_{2} }}$$

(5)

with μ(t) indicating the share of avoided emissions, and ψ ₁ as well as ψ ₂ are parameters that determine the shape of the mitigation cost function.

To sum up, Ω(t) indicates the share of output lost due to climate damages, whereas Λ(t) indicates the share of output lost due to mitigation activities. Consequently, weighting the gross output Y(t) by the multipliers [1 − Ω(t)] and [1 − Λ(t)] yields the remaining net output:

$$Y_{net} (t) = [1 - \varOmega (t)][1 - \varLambda (t)]A(t)K(t)^{\gamma } L(t)^{1 - \gamma } .$$

(6)

Equation (6) highlights the typical trade-off in climate policy: More emission mitigation leads to higher mitigation costs Λ(t) resulting in a decreasing net output. However, at the same time, more emission mitigation leads to lower damages Ω(t) resulting in an increasing net output.

Finally, the net output is divided into consumption and investment: $Y_{net} (t) = C(t) + I(t)$. This creates the typical trade-off in neoclassical growth models. Output is either consumed directly or invested in physical capital to increase the consumption possibilities in the future.

Emissions are caused by production depending on an exogenous emission coefficient τ(t) which declines over time to simulate carbon-saving technological change. Accounting for abatement activities, the remaining emissions from production are given by:

$$E_{ind} (t) = \tau (t)[1 - \mu (t)]A(t)K(t)^{\gamma } L(t)^{1 - \gamma }$$

(7)

with μ(t) indicating the mitigation rate, i.e., the share of emissions avoided. Besides emissions from production the model also accounts for exogenously given emissions from land use changes (e.g., deforestation) which are denoted by $E_{\text{def}} (t)$. Hence, the complete emissions are given by:

$$E(t) = E_{\text{ind}} (t) + E_{\text{def}} (t).$$

(8)

These emissions accumulate in the atmosphere and cause the atmospheric carbon concentration to increase. The latter is interrelated with the concentrations in different layers of the oceans. The concentrations in the atmosphere M _AT(t), in the upper ocean M _UO(t) and in the lower ocean M _LO(t) and their interrelationship are shown in Eqs. (9)–(11):

$$M_{AT} (t) = E(t) + \left[ {1 - \varphi_{12} } \right] \times M_{AT} (t - 1) + \varphi_{21} \times M_{UO} (t - 1),$$

(9)

$$M_{UO} (t) = \varphi_{12} \times M_{AT} (t - 1) + \varphi_{22} \times M_{UO} (t - 1) + \varphi_{32} \times M_{LO} (t - 1) ,$$

(10)

$$M_{LO} (t) = \varphi_{23} \times M_{UO} (t - 1) + \varphi_{33} \times M_{LO} (t - 1).$$

(11)

The atmospheric concentration M _AT(t) is composed of the current emissions, the share of the concentration remaining from the previous period plus the share of the upper oceanic concentration from the previous period that diffuses into the atmosphere. The upper oceanic concentration M _UO(t) consists of the remaining share from the previous period plus the absorptions from the atmosphere and the lower oceans. The concentration in the lower oceans M _LO(t) is the remaining share of the previous period plus the absorption from the upper oceans. The parameters φ _ij control these relationships between different reservoirs and periods.

In the next step, the atmospheric carbon concentration M _AT(t) increases the radiative forcing from the sun F(t) represented by:

$$F(t) = \eta \times \frac{{M_{\text{AT}} (t)}}{{M_{\text{AT}} ({\text{preind}})}} + F_{\text{exog}} (t)$$

(12)

with η being a parameter that controls the impact of increasing greenhouse gas concentrations and M _AT(preind) indicating the pre-industrial level of these concentrations. The term F _exog(t) covers the additional radiative forcing caused by other greenhouse gases or aerosols that are exogenous in the model.

Finally, the increasing radiative forcing results in an increase of temperatures in the atmosphere ΔT _AT(t) as well as in the oceans ΔT _O(t) as given by Eqs. (13) and (14):

$$\Delta T_{\text{AT}} (t) = \Delta T_{\text{AT}} (t - 1) + \omega_{1} \left[ {F(t) - \omega_{2} \Delta T_{\text{AT}} (t - 1) - \omega_{3} \left( {\Delta T_{\text{AT}} (t - 1) - \Delta T_{O} (t - 1)} \right)} \right] ,$$

(13)

$$\Delta T_{\text{O}} (t) = \Delta T_{\text{O}} (t - 1) + \omega_{4} \left[ {\Delta T_{\text{AT}} (t - 1) - \Delta T_{\text{O}} (t - 1)} \right] .$$

(14)

The change in atmospheric temperatures according to (13) results from the change of the previous period, as well as from the current radiative forcing that is corrected for the previous period and the interference between atmosphere and oceans. Analogously, the temperature change in the oceans according to (14) is computed from the change of the previous period that is corrected for the interference between the oceans and the atmosphere. These relationships between radiative forcing and the temperature in different carbon reservoirs or different periods, respectively, are controlled by the parameters ω _i.