Article Contents
Article Contents

# Optimal control of a global model of climate change with adaptation and mitigation

• * Corresponding author: Helmut Maurer
• The economy-climate interaction and an appropriate mitigation policy for climate protection have been treated in various types of scientific modeling. Here, we specifically focus on the seminal work by Nordhaus [14, 15] on the economy-climate link. We extend the Nordhaus type model to include optimal policies for mitigation, adaptation and infrastructure investment studying the dynamics of the transition to a low fossil-fuel economy. Formally, the model gives rise to an optimal control problem consisting of a dynamic system with five-dimensional state vector representing stocks of private capital, green capital, public capital, stock of brown energy in the ground, and carbon emissions. The objective function captures preferences over consumption but is also impacted by atmospheric $\mathrm{CO}_2$ and by mitigation and adaptation policies. Given the numerous challenges to climate change policies the control vector is eight-dimensional comprising mitigation, adaptation and infrastructure investment. Our solutions are characterized by turnpike property and the optimal policies that accomplish the objective of keeping the $\mathrm{CO}_2$ levels within bound are characterized by a significant proportion of investment in public capital going to mitigation in the initial periods. When initial levels of $\mathrm{CO}_{2}$ are high, adaptation efforts also start immediately, but during the initial period, they account for a smaller proportion of government's public investment.

Mathematics Subject Classification: Primary: 49N90, 49K15, 49M37; Secondary: 91-08.

 Citation:

• Figure 1.  State and control trajectories for terminal time $t_f = 200$, initial states $K_p(0) = 2.5 , K_g(0) = 0.3, G(0) = 0.8 , M(0) = 3.25 , R(0) = 1, T(0) = 290$, and free terminal state $X(t_f)$. Top row: (left) physical capital $K_p$, green capital $K_g$ and government capital $G$, (middle) CO$_2$ concentration $M$, (right) resource $R$. Middle row: (left) investments $i_p$ and $i_g$ and tax revenue $e_p$, (middle) temperature $T$, (right) extraction rate $u$. Bottom row: (left) consumption $C$ and productivity $Y$, (middle) infrastructure $\nu_1$, (right) adaptation $\nu_2$ and mitigation $\nu_3$.

Figure 2.  Current-value adjoint variables (shadow prices) for terminal time $t_f = 200$, initial states $K_p(0) = 2.5 , K_g(0) = 0.3, G(0) = 0.8 , M(0) = 3.25 , R(0) = 1, T(0) = 290$, and free terminal state $X(t_f)$: (left) adjoint variables $\lambda_{Kp}, \lambda_{Kg}, \lambda_G$, (middle) adjoint variable $\lambda_M$, (right) adjoint variable $\lambda_R$

Figure 3.  State and control trajectories for terminal time $t_f = 300$, initial states $K_p(0) = 2.5 , K_g(0) = 0.3, G(0) = 0.8 , M(0) = 3.25 , R(0) = 1, T(0) = 290$, and free terminal state $X(t_f)$. Top row: (left) physical capital $K_p$, green capital $K_g$ and government capital $G$, (middle) CO$_2$ concentration $M$, (right) resource $R$. Middle row: (left) investments $i_p$ and $i_g$ and tax revenue $e_p$, (middle) temperature $T$, (right) extraction rate $u$. Bottom row: (left) consumption $C$ and productivity $Y$, (middle) infrastructure $\nu_1$, (right) adaptation $\nu_2$ and mitigation $\nu_3$

Figure 4.  State and control trajectories for $t_f = 200$: "small" initial values of capital stocks and high emission stock $K_p(0) = 1 , K_g(0) = 0.02, G(0) = 0.2 , R(0) = 1 , M(0) = 3.25 , T(0) = 290$ and prescribed terminal states $K_p(t_f) = 2.2164 , K_g(t_f) = 0.53731 , G(t_f) = 0.66746$ as approximate stationary values (33). Top row: (left) physical capital $K_p$, green capital $K_g$ and government capital $G$, (middle) CO$_2$ concentration $M$, (right) resource $R$. Middle row: (left) investments $i_p$ and $i_g$ and tax revenue $e_p$, (middle) temperature $T$, (right) extraction rate $u$. Bottom row: (left) consumption $C$ and productivity $Y$, (middle) infrastructure $\nu_1$, (right) adaptation $\nu_2$ and mitigation $\nu_3$

Figure 5.  Current-value adjoint variables for $t_f = 200$: "small" initial values of capital stocks and high emission stock. (left) current value adjoint variables $\lambda_{Kp}, \lambda_{Kg}, \lambda_G$, (middle) current value adjoint variable $\lambda_M$, (right) current value adjoint variable $\lambda_R$

Figure 6.  State and control trajectories for $t_f = 200$: "small" initial values of capital stocks and low emission stock $K_p(0) = 1 , K_g(0) = 0.02, G(0) = 0.2 , R(0) = 1 , M(0) = 2.6 , T(0) = 290$ and prescribed terminal stationary values $K_p(t_f) = 2.2164 , K_g(t_f) = 0.53731 , G(t_f) = 0.66746$ in (33). Top row: (left) physical capital $K_p$, green capital $K_g$ and government capital $G$, (middle) CO$_2$ concentration $M$, (right) resource $R$. Middle row: (left) investments $i_p$ and $i_g$ and tax revenue $e_P$, (middle) temperature $T$, (right) extraction rate $u$. Bottom row: (left) consumption $C$ and productivity $Y$, (middle) infrastructure $\nu_1$, (right) adaptation $\nu_2$ and mitigation $\nu_3$

Figure 7.  State and control trajectories for large terminal time $t_f = 300$: "small" initial value of capital stocks and low emission stock $K_p(0) = 1 , K_g(0) = 0.02, G(0) = 0.2 , R(0) = 1 , M(0) = 2.6 , T(0) = 290$ and prescribed terminal stationary values $K_p(t_f) = 2.2164 , K_g(t_f) = 0.53731 , G(t_f) = 0.66746$ in (33). Top row: (left) physical capital $K_p$, green capital $K_g$ and government capital $G$, (middle) CO$_2$ concentration $M$, (right) resource $R$. Middle row: (left) investments $i_p$ and $i_g$ and tax revenue $e_p$, (middle) temperature $T$, (right) extraction rate $u$. Bottom row: (left) consumption $C$ and productivity $Y$, (middle) infrastructure $\nu_1$, (right) adaptation $\nu_2$ and mitigation $\nu_3$

Figure 8.  State and control trajectories for $t_f = 200$: "large" initial values of capital stocks and high emission stock $K_p(0) = 3 , K_g(0) = 0.5, G(0) = 1.0, M(0) = 3.25, R(0) = 1$ and terminal constraints $K_p(t_f) = 2.2164 , K_g(t_f) = 0.53731 , G(t_f) = 0.66746$. Top row: (left) physical capital $K_p$, green capital $K_g$ and government capital $G$, (middle) CO$_2$ concentration $M$, (right) resource $R$. Middle row: (left) investments $i_p$ and $i_g$ and tax revenue $e_P$, (middle) temperature $T$, (right) extraction rate $u$. Bottom row: (left) consumption $C$ and productivity $Y$, (middle) infrastructure $\nu_1$, (right) adaptation $\nu_2$ and mitigation $\nu_3$

Figure 9.  Current-value adjoint variables for "large" initial values and high emission stock. (left) current value adjoint variables $\lambda_{Kp}, \lambda_{Kg}, \lambda_G$, (middle) current value adjoint variable $\lambda_M$, (right) current value adjoint variable $\lambda_R$

Table 1.  Parameter values

 Parameter Value Definition $\rho$ 0.03 Pure discount rate $n$ 0.015 Population Growth Rate $\eta$ 0.1 Elasticity of transfers and public spending in utility $\epsilon$ $1.1$ Elasticity of $\mathrm{CO}_2$-eq concentration in (dis)utility $\omega$ 0.05 Elasticity of public capital used for adaptation in utility $\sigma$ 2 Intertemporal elasticity of instantaneous utility $A$ $\in [1, 10]$ Total factor productivity $A_{g}$ $\in [\, 1\, , \, 5\, ]$ Efficiency index of green capital ${A_{u}}$ $\in [100, 400]$ Efficiency index of the non-renewable resource $\alpha$ 0.1 Output elasticity of inputs, $(A_g K_g + A_u u)^{\alpha}$ $\beta$ 0.5 Output elasticity of public infrastructure, $(\nu_1 G)^{\beta}$ $\psi$ 1 Scaling factor in marginal cost of resource extraction $\tau$ 2 Exponential factor in marginal cost of resource extraction $\delta_{p}$ 0.1 Depreciation rate of physical capital $\delta_{g}$ 0.05 Depreciation rate of private capital $\delta_{G}$ 0.05 Depreciation rate of public capital $\chi_p$ $\frac{1}{(\delta_p+n) \Omega_p }$ $\chi_g$ $\frac{1}{(\delta_g+n) \Omega_g }$ $\Omega_p$ $\in [5, 15]$ q-elasticity of investment spending on private capital $\Omega_g$ $\in [5, 15]$ q-elasticity of investment spending on public capital $\alpha_{1}$ 0.2 Proportion of tax revenue allocated to new public capital $\alpha_{2}$ 0.5 Proportion of tax revenue allocated to transfers and public consumption $\bar r$ 0.07 World interest rate (paid on public debt) $\widetilde{M}$ 2.5 equilibrium concentration of $\mathrm{CO}_2$ $\kappa$ 1.2 Atmospheric concentration stabilization ratio (relative to $\widetilde M$) $\bar{M}$ 4.5 value in disutility term in welfare (11) $\gamma$ 0.9 Fraction of greenhouse gas emissions not absorbed by the ocean $\mu$ 0.01 Decay rate of greenhouse gases in atmosphere $\theta$ 0.01 Effectiveness of mitigation measures ${\phi}$ $\in {[\, 0.2, \, 1\, ]}$ exponent in mitigation term $\, (\nu_3\, g)^{\phi}$
•  [1] J. T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, 2$^{nd}$ edition, Advances in Design and Control, Philadelphia, 2010. doi: 10.1137/1.9780898718577. [2] T. Bonen, P. Loungani, W. Semmler and S. Koch, Investing to Mitigate and Adapt to Climate Change: A Framework Model, IMF working paper WP no 16/164, International Monetary Fund, Washington, 2016. [3] C. Büskens and H. Maurer, SQP–methods for solving optimal control problems with control and state constraints: Adjoint variables, sensitivity analysis and real–time control, J. Comput. Appl. Math., 120 (2000), 85-108.  doi: 10.1016/S0377-0427(00)00305-8. [4] T. Faulwasser and L. Grüne, Turnpike properties in optimal control: An overview of discrete-time and continuous-time results, Handbook of Numerical Analysis, 23 (2022), 367-400. [5] F. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks-Cole Publishing Company, 1993. [6] L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, J. Ind. Manag. Optim., 10 (2014), 413-441.  doi: 10.3934/jimo.2014.10.413. [7] A. Greiner, L. Grüne and W. Semmler, Growth and climate change: Threshold and multiple equilibria, In Dynamic Systems, Economic Growth, and the Environment, (eds. J. Crespo Cuaresma, T. Palokangas and A. Tarasyev), Springer, Heidelberg and New York, (2010), 63–78. [8] L. Grüne, M. A. Müller, C. M. Kellet and S. R. Weller, Strict dissipativity for discrete discounted optimal control problems, Math. Control Relat. Fields, 11 (2021), 771-796.  doi: 10.3934/mcrf.2020046. [9] R. F. Hartl, S. P. Sethi and R. G. Vickson, A survey of the maximum principles for optimal control problems with state constraints, SIAM Rev., 37 (1995), 181-218.  doi: 10.1137/1037043. [10] M. Hestenes, Calculus of Variations and Optimal Control Theory, John Wiley & Sons, Inc., New York-London-Sydney 1966. [11] Global Warming of 1.5 ℃, Intergovernmental Panel of Climate Change, 2018. [12] H. Maurer, J. J. Preuß and W. Semmler, Policy scenarios in a model of optimal economic growth and climate Change, Chapter 5 in The Oxford Handbook of the Macroeconomics of Global Warming, (eds. L. Bernard and W. Semmler), Oxford University Press, 2015. [13] H. Maurer and W. Semmler, Expediting the transition from non-renewable to renewable energy via optimal control, Discrete Contin. Dyn. Syst., 35 (2015), 4503-4525.  doi: 10.3934/dcds.2015.35.4503. [14] W. Nordhaus, The Question of Balance, New Haven, Yale University Press, New Haven, 2008. [15] W. Nordhaus, Revisiting the social cost of carbon, PNAS, 114 (2017), 1518-1523.  doi: 10.1073/pnas.1609244114. [16] W. Nordhaus and J. Boyer, Warming the World. Economic Models of Global Warming, Cambridge: MIT-Press, Cambridge, 2000. [17] S. Orlov, E. Rovenskaya, W. Semmler and J. Puaschunder, Green bonds, transition to a low-carbon economy, and intergenerational fairness: Evidence from an extended DICE model, IIASA Working Paper, WP-18-001, (2018). doi: 10.2139/ssrn.3086483. [18] L. S. Pontryagin, V. G. Boltyanski, R. V. Gramkrelidze and E. F. Miscenko, The Mathematical Theory of Optimal Processes, Translated by D. E. Brown A Pergamon Press Book The Macmillan Company, New York 1964. [19] , W. Roedel, Private communication. [20] W. Roedel and T. Wagner, Physik Unserer Umwelt: Die Atmosphäre, Springer, Berlin, Heidelberg, 2011. [21] A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.

Figures(9)

Tables(1)