Abstract
The optimal response to a potential productivity shock which becomes more imminent with global warming is to have carbon taxes to curb the risk of a calamity and to accumulate precautionary capital to facilitate smoothing of consumption. This paper investigates how differences between regions in terms of their vulnerability to climate change and their stage of development affect the cooperative and non-cooperative responses to this aspect of climate change. It is shown that the cooperative response to these stochastic tipping points requires converging carbon taxes for developing and developed regions. The non-cooperative response leads to a bit more precautionary saving and diverging carbon taxes. We illustrate the various outcomes with a simple stylized North–South model of the global economy.
Similar content being viewed by others
Notes
We use \(3\ln (P_t{/}596.4){/}\ln (2)\) for the temperature compared to pre-industrial temperature.
References
Başar T, Olsder G-J (1982) Dynamic noncooperative game theory. Academic Press, New York
Biggs R, Blenckner T, Folke C, Gordon L, Norström A, Nyström M, Peterson G (2012) Regime shifts. In: Hastings A, Gross L (eds) Encyclopedia in theoretical ecology. University of California Press, Berkeley, pp 609–616
Cai Y, Judd KL, Lontzek TS (2012) The social cost of stochastic and irreversible climate change. Working Paper 18704, NBER, Cambridge, MA, USA
Chichilnisky G, Heal G (1994) Who should abate carbon emissions? an international viewpoint. Econ Lett 44(4):443–449
Dockner E, Long N (1993) International pollution control: cooperative versus noncooperative strategies. J Environ Econ Manag 25(1):13–29
Doraszalski U (2003) An R&D race with knowledge accumulation. RAND J Econ 34(1):20–42
Engström G and Gars J (2014) Climate tipping points and optimal fossil fuel use. In: Presented at a conference of the Beyer Institute, Stockholm
Golosov M, Hassler J, Krusell P, Tsyvinski A (2014) Optimal taxes on fossil fuel in general equilibrium. Econometrica 82(1):41–88
Hassler P, Krusell J (2012) Economics and climate change: integrated assessment in a multi-region world. J Eur Econ Assoc 10(5):974–1000
Jaakkola N (2015) Putting OPEC out of business. Research Paper No. 99, OxCarre, Department of Economics, University of Oxford, Oxford
Judd KL (1998) Numerical methods in economics. MIT Press, Cambridge
Lemoine D, Traeger C (2014) Watch your step: optimal policy in a tipping climate. Am Econ J Econ Policy 6(1):137–166
Lenton TM, Ciscar J-C (2013) Integrating tipping points into climate impact assessments. Climatic Chang 117:585–597
Nordhaus W (2008) A question of balance: economic models of climate change. Yale University Press, New Haven
Nordhaus W (2014) Estimates of the social cost of carbon: concepts and results from the DICE-2013R model and alternative approaches. J Assoc Environ Resour Econ 1(1):273–312
Nordhaus W, Boyer J (2000) Warming the world: economic models of global warming. MIT Press, Cambridge
Pindyck R (2013) The climate policy dilemma. Rev Environ Econ Policy 7(2):219–237
Polasky S, de Zeeuw A, Wagener F (2011) Optimal management with potential regime shifts. J Environ Econ Manag 62(2):229–240
Saini V (2012) Endogenous asymmetry in a dynamic procurement auction. RAND J Econ 43(4):726–760
Smulders S, Tsur Y, Zemel A (2014) Uncertain climate policy and the green paradox. In: Moser E, Semmler W, Tragler G, Veliov V (eds) Dynamic optimization in environmental economics. Springer, Berlin
Stern N (2007) The economics of climate change: the Stern review. Cambridge University Press, Cambridge
Tol R (2002) Estimates of the damage costs of climate change. Environ Resour Econ 21:135–160
van der Ploeg F (2014) Abrupt positive feedback and the social cost of carbon. Eur Econ Rev 67:28–41
van der Ploeg F, de Zeeuw A (1992) International pollution control. Environ Resour Econ 2(2):117–139
van der Ploeg F, de Zeeuw AJ (2014) Climate tipping and economic growth: precautionary saving and the social cost of carbon. Discussion Paper No. 9982, CEPR, London
Author information
Authors and Affiliations
Corresponding author
Additional information
We are very grateful for the comments of two anonymous reviewers: they have helped to improve the paper considerably. Van der Ploeg is grateful for support from the ERC Advanced Grant ‘Political Economy of Green Paradoxes’ (FP7-IDEAS-ERC Grant No. 269788) and the BP funded Oxford Centre for the Analysis of Resource Rich Economies. De Zeeuw is grateful for support from the European Commission under the 7th Framework Programme (Socioeconomic Sciences and Humanities—SSH.2013.2.1-1—Grant Agreement No. 613420).
Appendices
Appendix 1: Calibration and Functional Forms
Table 2 summarizes our calibration.
The utility functions \(U(C_i )=C_i ^{1-1{/}\sigma }{/}(1-1{/}\sigma )\), \(i=1,2\), have a constant elasticity of intertemporal substitution of \(\sigma = 0.5\) (implying a constant coefficient of intergenerational inequality aversion of 2) and a pure rate of time preference of \(\rho \) = 0.014. The Cobb–Douglas production functions \(F(K_i ,E_i )=K_i ^{\alpha }E_i ^{\beta }\), \(i=1,2\), have a capital share of \(\alpha = 0.3\) and an energy share of \(\beta \) = 0.0623. The depreciation rate is set to \(\delta = 0.05\).
We calibrate to the business-as-usual (i.e., negligible carbon taxes and no precautionary capital accumulation) outcome for the world economy for the year 2010. Data sources are the BP Statistical Review and the World Bank Development Indicators.
The initial 2010 capital stocks are set to \(K_{1}(0) = 180\) and \(K_{2}(0) = 20\) trillion US dollars, and the 2010 level of world GDP is 63 trillion US dollars. We measure fossil fuel in GtC, so the emission-input ratio equals one. We use a market price for fossil fuel of \(d = 504.3\) US$/t of carbon (or 9 US$/million BTU). Global fossil fuel use in 2010 is 8.3 GtC (or 468.3 million GBTU). Using \(\frac{E_1 (0)}{E_2 (0)}=\left( {\frac{K_1 (0)}{K_2 (0)}} \right) ^{\frac{\alpha }{1-\beta }}=9^{\frac{0.3}{1-0.0623}}=2.02\), we get \(E_{1}(0) = 5.551\) and \(E_{2}(0) = 2.749\) GtC in 2010. We thus have 42.1 and 20.9 trillion US dollars for GDP in the developed and developing part of the global economy. The level of total factor productivity that matches these levels of output and inputs in both regions is \(A = 8.5044\).
The initial capital stocks of 180 and 20 trillion US dollars are below the steady-state levels of 211 and 104 trillion US dollars to reflect that the developed region is still catching up. The fraction of carbon staying in the atmosphere is set equal to \(\psi = 0.5\) and the rate of decay of atmospheric carbon is set equal to \(\gamma = 0.003\).
Appendix 2: Before- and After-Catastrophe Steady States
Our specification with CES utility and Cobb–Douglas production functions yields tractable forms for the crucial variables in the analysis. Combining (3), (6), (13) and the equivalent expression for the non-cooperative case, it follows that optimal fossil-fuel use is generally given by \(E_i =\beta \tilde{A}_i F{/}(d+\tau _{i} )\), so that output net of fossil fuel costs and capital depreciation becomes
The modified golden rules \(Y_{iK_i } ({\bar{K}}_i ,{\bar{\tau }}_i )=\rho -{\bar{\theta }}_i\) , \(i=1,2,\) yield
so that the (target) steady-state capital stocks become
Since \({\bar{E}}_i =\beta f_i ({\bar{K}}_i ,{\bar{\tau }}_i ){/}(d+{\bar{\tau }})\), the other (target) steady states are given by
where in the non-cooperative case the summation in the expression for the carbon tax (the last steady state) reduces to only the \(j = i\) term.
After tipping where \({\bar{\theta }}_i ={\bar{\tau }}_i =0\), \(i=1,2\), and \(\tilde{A}_i =(1-\pi _{i} )A\), \(i=1,2\), this reduces to
The figures for the before- and after-catastrophe steady states corresponding to the calibration of “Appendix 1” are reported in Table 1.
Appendix 3: Approximation to the After-Catastrophe Stable Manifold
The solution trajectories for the after-catastrophe outcomes are well approximated by a log-linear approximation of the stable manifold relating aggregate consumption to only the aggregate capital stock in each country. The reason for not having to relate the manifold to the atmospheric carbon stock is that, with our specification of one-off catastrophic damages, changes in the degree of global warming do not affect the after-catastrophe value functions and consumption manifolds. Furthermore, since the North and the South are isolated after the catastrophe, the cooperative and non-cooperative outcomes coincide. It is relatively easy to calculate the optimal consumption functions \(C_{i}^{A} (K_i)\) as the log-linear approximations to the stable manifolds of the after-catastrophe system (9) because we can use l’Hôpital’s rule to determine the slopes \(\dot{C}_i{/}\dot{K}_i \) of the stable manifolds in these steady states:
This yields the simple quadratic equation
The positive solutions to (32) exceed the value \(\rho > 0\) and are the slopes of the stable manifolds in the steady states. Using logarithmic differentiation of the state equations, we thus obtain the following expressions for the approximations of the after-catastrophe stable manifolds:
For the calibration discussed in “Appendix 1”, the log-linear approximations \(C_{i}^{A} (K_i )\) to the stable manifolds of the systems (9) are given by \(C_1^A (K_1 )\cong 3.001K_1 ^{0.4303}\) and \(C_2^A (K_2)\cong 2.398K_2 ^{0.4303}\). One can demonstrate that the speed of convergence of the after-catastrophe Ramsey growth systems increase with the rate of discount \(\rho \) and with the elasticity of intertemporal substitution \(\sigma \), but decrease with the share of capital in value added.
Equation (7) gives the after-tipping value functions:
With \(g_i (K_i )\equiv f_i (K_i ,0)\) we can calculate the after-tipping values as
and from the first-order conditions we have that \(V_{iK_i }^A (K_i )=C_{i}^{A} (K_i )^{-1{/}\sigma }\).
Appendix 4: Approximation of the Before-Catastrophe Stable Manifolds
1.1 The State-Space System for the Specific Functional Forms
Before tipping we have a higher-dimensional dynamic system: \((2^\prime )\), \((4^\prime )\), (15) and (16) in the cooperative case and \((2^{\prime \prime })\), \((4^{\prime \prime })\), (22) and (23) in the non-cooperative case. These systems can for our specification with CES utility and Cobb–Douglas production functions be given by (omitting some dependencies to save space):
where in the non-cooperative case the summation in (36d) reduces to only the \(j = i\) term. The steady state of the system (36) is given by (27) and (29). Note that the functional forms for \(Y_{i}\), \(f_{i}\) and \(E_{i}\) are the same for \(i = 1\) and \(i = 2\) because total factor productivity A before tipping is the same in both regions [see Eqs. (26), (27)].
From (11) to (13), and using the CES utility function, we can rewrite the before-catastrophe value functions in the cooperative case as
or using \(V_{1P} +V_{2P} =-C_1^{-1{/}\sigma } \tau _1{/}\psi =-C_2^{-1{/}\sigma } \tau _2{/}\psi ,\) we get
and from the first-order conditions we have that \(V_{iK_i } =C_i ^{-1{/}\sigma },\mathop {\sum }\nolimits _{j=1}^2 {V_{jP} } =-\tau _{i} C_i^{-1{/}\sigma } {/}\psi \).
From (19) and (20) and using the CES utility function, we can write the before-catastrophe value functions in the non-cooperative case as
and from the first-order conditions we have \(V_{iK_i } =C_i ^{-1{/}\sigma },V_{iP} =-\tau _{i} C_i^{-1{/}\sigma }{/}\psi \). Note that \(V_{iC_i}\) boils down to zero in steady state.
To get the Jacobian of the cooperative and non-cooperative system (36) with (37) and (38), respectively, we need some intermediate results (omitting dependencies):
It follows that
1.2 Linearization of the State-Space System
The cooperative and non-cooperative systems (36) for the before-catastrophe outcomes are boundary-value problems with initial conditions on the states \((K_1 ,K_2 ,P)\), transversality conditions on the co-states \((C_1 ,C_2 ,\tau _1 ,\tau _2 )\) and a saddle-point stable steady state. We present the Jacobian matrix of the linearized cooperative system, with the changes that occur in the non-cooperative case. This follows from Eqs. (37) to (40) and is given by
where \(\xi _{1,i} \equiv \sigma \left[ {Y_{iK_i K_i } -H(P)\frac{C_i^{1/\sigma } C_{iK_i }^A (K_i )}{\sigma C_{i}^{A} (K_i )^{1+1/\sigma }}} \right] C_i ,\hbox { }\xi _{2,i} \equiv \sigma H^{\prime }(P)\left[ {\frac{C_i^{1/\sigma } }{C_{i}^{A} (K_i )^{1/\sigma }}-1} \right] C_i ,\)
which in the non-cooperative case becomes
where in the non-cooperative case the summation reduces to only the \(j = i\) term,
which in the non-cooperative case becomes \(\xi _{8,i} \equiv \rho +\gamma +H(P)\frac{C_i^{1/\sigma } }{C_{i}^{A} (K_i )^{1/\sigma }}+Y_{iK_i \tau _{i} } \tau _{i} ,\)
which in the non-cooperative case becomes \(\xi _{9,i} \equiv \frac{\psi H^{\prime }(P)\tau _{i} E_{j\tau _{j} } }{\rho +H(P)},\,j\ne i.\)
1.3 Numerical Spectral Decomposition Algorithm
We can write the log-linear approximation around the steady state as follows:
where \(x_{p}\) denotes the vector of predetermined variables with initial conditions and \(x_{n}\) denotes the vector of non-predetermined variables. The matrix A is the state transition matrix of the log-linearized system [taking due care of the additional terms in (15) and (22) for the cooperative and non-cooperative case to allow for the precautionary returns] and follows from the Jacobian of the linearized system B in (41):
The saddle-point property requires that the matrix A has three eigenvalues with negative real parts corresponding to the predetermined state variables and four eigenvalues with positive real parts corresponding to the non-predetermined state variables. Spectral decomposition gives \(A=M\Lambda M^{-1}=N^{-1}\Lambda N,\) where the diagonal matrix \(\Lambda =\left( {{\begin{array}{cc} {\Lambda _p }&{}\quad 0 \\ 0&{}\quad {\Lambda _n } \\ \end{array} }} \right) \) has the eigenvalues of A on its diagonal. The eigenvalues associated with the predetermined variables are collected in the diagonal sub-matrix \(\Lambda _p\) and the others are collected in the diagonal sub-matrix \(\Lambda _n\). Diagonalization of (42) gives
which has the stable solution
The solution to (42) is \(x(t)=My(t).\) The stable manifold is \(x_n (t)=M_{np} M_{pp} ^{-1}x_p (t).\) With this log-linear approximation of the stable manifolds, we can readily calculate the solution trajectories for the log-linear deviations from the steady state and thus the trajectories for the state variables themselves.
Our spectral decomposition algorithm of the log-linearized state-space system yields answers that make sense for theoretically oriented continuous-time problems. The more frontier numerical infinite-horizon optimization methods for discrete-time problems use efficient discretization methods with ad hoc terminal values, and maximize welfare directly rather than solving the first-order conditions (e.g., Cai et al. 2012).
Rights and permissions
About this article
Cite this article
van der Ploeg, F., de Zeeuw, A. Non-cooperative and Cooperative Responses to Climate Catastrophes in the Global Economy: A North–South Perspective. Environ Resource Econ 65, 519–540 (2016). https://doi.org/10.1007/s10640-016-0037-z
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10640-016-0037-z
Keywords
- Global warming
- Tipping point
- Precautionary capital
- Growth
- Risk avoidance
- Carbon tax
- Free riding
- International cooperation
- Asymmetries