Non-cooperative and Cooperative Responses to Climate Catastrophes in the Global Economy: A North–South Perspective

Frederick van der Ploeg; Aart de Zeeuw

doi:10.1007/s10640-016-0037-z

Environmental and Resource Economics

November 2016, Volume 65, Issue 3, pp 519–540 | Cite as

Non-cooperative and Cooperative Responses to Climate Catastrophes in the Global Economy: A North–South Perspective

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Frederick van der Ploeg
Aart de Zeeuw

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First Online: 04 June 2016

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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.

Keywords

Global warming Tipping point Precautionary capital Growth Risk avoidance Carbon tax Free riding International cooperation Asymmetries

JEL Classification

D81 H20 O40 Q31 Q38

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Appendix 1: Calibration and Functional Forms

Table 2 summarizes our calibration.

Table 2

Calibration of the two-region model with a tipping point

Variable/parameter
Pure rate of time preference, $\rho $	0.014
Elasticity of substitution, $\sigma $	0.5
Depreciation rate of capital, $\delta $	0.05
Share of capital in value added, $\alpha $	0.3
Share of fossil fuel in value added, $\beta $	0.0623
Total factor productivity, A	8.5044
Eventual climate shock, $\pi $ $_{i}$	0.1, 0.3
Fraction of carbon staying up, $\psi $	0.5
Natural decay of carbon, $\gamma $	0.003
Cost of fossil fuel, d	9 US$/million BTU $=$ 504.3 US$/tC
2010 levels of GDP	42.1 and 20.9 trillion US$
2010 levels of capital, $K_{i0}$	180 and 20 trillion US$
2010 fossil fuel use, $E_{i0}$	5.551 and 2.749 GtC
Initial stock of carbon, $P_{0}$	826 GtC $=$ 388 ppm by vol. $\hbox {CO}_{2}$

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

$$\begin{aligned} Y_i (K_i ,\tau _{i} )=(1-\beta )\left[ {\tilde{A}_i \left( {\frac{\beta }{d+\tau _{i} }} \right) ^{\beta }} \right] ^{\frac{1}{1-\beta }}K_i ^{\frac{\alpha }{1-\beta }}-\delta K_i ,\quad i=1,2. \end{aligned}$$

(26)

The modified golden rules $Y_{iK_i } ({\bar{K}}_i ,{\bar{\tau }}_i )=\rho -{\bar{\theta }}_i$ , $i=1,2,$ yield

$$\begin{aligned} {\bar{K}}_i =\frac{\alpha }{\rho +\delta -{\bar{\theta }}_i }f_i ({\bar{K}}_i ,{\bar{\tau }}_i ),\quad f_i (K_i ,\tau _{i} )\equiv \left[ {\tilde{A}_i \left( {\frac{\beta }{d+\tau _{i} }} \right) ^{\beta }} \right] ^{\frac{1}{1-\beta }}K_i ^{\frac{\alpha }{1-\beta }},\quad i=1,2,\nonumber \\ \end{aligned}$$

(27)

so that the (target) steady-state capital stocks become

$$\begin{aligned} {\bar{K}}_i =\left[ {\tilde{A}_i \left( {\frac{\alpha }{\rho +\delta -{\bar{\theta }}_i }} \right) ^{1-\beta }\left( {\frac{\beta }{d+{\bar{\tau }}_i }} \right) ^{\beta }} \right] ^{\frac{1}{1-\alpha -\beta }},\quad i=1,2. \end{aligned}$$

(28)

Since ${\bar{E}}_i =\beta f_i ({\bar{K}}_i ,{\bar{\tau }}_i ){/}(d+{\bar{\tau }})$, the other (target) steady states are given by

$$\begin{aligned} {\bar{P}}= & {} \sum _{j=1}^2 {\left[ {\frac{\psi }{\gamma }\left( {\frac{\beta }{d+{\bar{\tau }}_{j} }} \right) \left( {\frac{\rho +\delta -{\bar{\theta }}_{j} }{\alpha }} \right) {\bar{K}}_{j} } \right] } ,{\bar{\theta }}_i =H\left( {\bar{P}}\right) \left[ {\frac{{\bar{C}}_i^{1/\sigma } }{C_{i}^{A} \left( {\bar{K}}_i\right) ^{1/\sigma }}-1} \right] , \nonumber \\ {\bar{C}}_i= & {} \left[ {\left( {1-\beta +\frac{\beta {\bar{\tau }}_i }{d+{\bar{\tau }}_i }} \right) \left( {\frac{\rho +\delta -{\bar{\theta }}_i }{\alpha }} \right) -\delta } \right] {\bar{K}}_i , \nonumber \\ {\bar{\tau }}_i= & {} \frac{\psi H^{\prime }\left( {\bar{P}}\right) {\bar{C}}_i ^{1/\sigma }{\mathop {\sum }\nolimits _{j=1}^{2}} {\left[ {\frac{{\bar{C}}_{j} ^{1-1{/}\sigma }}{1-1{/}\sigma }-\rho V_j^A \left( {\bar{K}}_{j}\right) } \right] } }{\left[ {\rho +\gamma +H\left( {\bar{P}}\right) } \right] \left[ {\rho +H\left( {\bar{P}}\right) } \right] },\quad i=1,2, \end{aligned}$$

(29)

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

$$\begin{aligned} {\bar{K}}_{i}^{A}= & {} \left[ {(1-\pi _{i} )A\left( {\frac{\alpha }{\rho +\delta }} \right) ^{1-\beta }\left( {\frac{\beta }{d}} \right) ^{\beta }} \right] ^{\frac{1}{1-\alpha -\beta }}, \nonumber \\ {\bar{C}}_{i}^{A}= & {} \left[ {\left( {1-\beta } \right) \left( {\frac{\rho +\delta }{\alpha }} \right) -\delta } \right] {\bar{K}}_{i}^{A} ,\quad i=1,2. \end{aligned}$$

(30)

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:

$$\begin{aligned} C_{iK_i }^A \left( {\bar{K}}_{i}^{A}\right) =\mathop {\lim }\limits _{K_i \rightarrow {\bar{K}}_{i}^{A} } \frac{\sigma \left[ {Y_{iK_i } (K_i )-\rho } \right] C_{iK_i }^A (K_i )+\sigma Y_{iK_i K_i } (K_i)C_{i}^{A} (K_i )}{Y_{iK_i } (K_i )-C_{iK_i }^A (K_i )},\quad i=1,2.\nonumber \\ \end{aligned}$$

(31)

This yields the simple quadratic equation

$$\begin{aligned} \left( {C_{iK}^A \left( {\bar{K}}_{i}^{A}\right) } \right) ^{2}-\rho C_{iK}^A \left( {\bar{K}}_{i}^{A}\right) +\sigma Y_{iK_i K_i } \left( {\bar{K}}_{i}^{A} \right) {\bar{C}}_{i}^{A} =0,\quad i=1,2. \end{aligned}$$

(32)

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:

$$\begin{aligned} C_{i}^{A} (K_i )\cong {\bar{C}}_{i}^{A} \left( {\frac{K_i }{{\bar{K}}_{i}^{A} }} \right) ^{\phi _{i}},\quad \phi _{i} \equiv \frac{C_{iK}^A \left( {\bar{K}}_{i}^{A}\right) {\bar{K}}_{i}^{A} }{{\bar{C}}_{i}^{A} }>0,\quad i=1,2. \end{aligned}$$

(33)

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:

$$\begin{aligned} V_{i}^{A} (K_i )=\frac{U\left( {C_{i}^{A} (K_i )} \right) +U^{\prime }\left( {C_{i}^{A} (K_i )} \right) \left[ {Y_i (K_i )-C_{i}^{A} (K_i )} \right] }{\rho },\quad i=1,2. \end{aligned}$$

(34)

With $g_i (K_i )\equiv f_i (K_i ,0)$ we can calculate the after-tipping values as

$$\begin{aligned} V_{i}^{A} (K_i )=\frac{C_{i}^{A} (K_i )^{1-1{/}\sigma }}{\rho (\sigma -1)}+\frac{C_{i}^{A} (K_i )^{-1{/}\sigma }\left[ {(1-\beta )g_i (K_i )-\delta K_i } \right] }{\rho },\quad i=1,2, \end{aligned}$$

(35)

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

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):

$$\begin{aligned} \dot{K}_i= & {} Y_i +\tau _{i} E_i -C_i ,\quad K_i (0)=K_{i0} ,\quad i=1,2, \end{aligned}$$

(36a)

$$\begin{aligned} \dot{P}= & {} \psi (E_1 +E_2 )-\gamma P,\quad P(0)=P_0, \end{aligned}$$

(36b)

$$\begin{aligned} \dot{C}_i= & {} \sigma \left( {Y_{iK_i } -\rho +H(P)\left[ {\frac{C_i^{1{/}\sigma } }{C_{i}^{A} (K_i )^{1{/}\sigma }}-1} \right] } \right) C_i ,\quad i=1,2, \end{aligned}$$

(36c)

$$\begin{aligned} \dot{\tau }_i= & {} \left[ {Y_{iK_i } +\gamma +H(P)\frac{C_i^{1{/}\sigma } }{C_{i}^{A} (K_i )^{1{/}\sigma }}} \right] \tau _i -\psi H^{\prime }(P)C_i^{1{/}\sigma } \sum _{j=1}^2 {\left[ {V_{j} \!-\!V_j^A (K_{j} )} \right] } ,\quad i\!=\!1,2,\nonumber \\ \end{aligned}$$

(36d)

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

$$\begin{aligned}&\sum _{i=1}^2 {V_i (K_i ,P)}\\&\quad =\frac{{\mathop {\sum }\limits _{i=1}^{2}} {\left\{ {\frac{C_i^{1-1{/}\sigma } }{\sigma \!-\!1}\!+\!C_i^{-1{/}\sigma } (Y_i \!+\!\tau _{i} E_i )} \right. } \left. {\!+H(P)V_{i}^{A} (K_i )} \right\} \!+\!(V_{1P} \!+\!V_{2P} )\left( {\psi {\mathop {\sum }\nolimits _{j=1}^{2}} {E_{j} } -\gamma P} \right) }{\rho \!+\!H(P)} \end{aligned}$$

or using $V_{1P} +V_{2P} =-C_1^{-1{/}\sigma } \tau _1{/}\psi =-C_2^{-1{/}\sigma } \tau _2{/}\psi ,$ we get

$$\begin{aligned} \sum _{i=1}^2 {V_i (K_i ,P)} =\frac{{\mathop {\sum }\nolimits _{i=1}^{2}} {\left\{ {\frac{C_i^{1-1/\sigma } }{\sigma -1}+C_i^{-1/\sigma } \left( {Y_i +\frac{1}{2}\tau _{i} \frac{\gamma P}{\psi }} \right) +H(P)V_{i}^{A} (K_i )} \right\} } }{\rho +H(P)}, \end{aligned}$$

(37)

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

$$\begin{aligned}&V_i =\frac{C_i^{1-1/\sigma } }{\left[ {\rho +H(P)} \right] (\sigma -1)}+\frac{C_i^{-1/\sigma } }{\rho +H(P)}\left( {Y_i -\tau _{i} E_{j}+\tau _{i} \frac{\gamma P}{\psi }} \right) \nonumber \\&\quad +\frac{H(P)V_{i}^{A} (K_i )}{\rho +H(P)}, \quad i=1,2,j\ne i, \end{aligned}$$

(38)

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):

$$\begin{aligned} Y=(1-\beta )f-\delta K,\quad E=\frac{\beta }{d+\tau }f,\quad f_K =\frac{\alpha }{(1-\beta )K}f\quad \hbox { and }\quad f_\tau =\frac{-E}{1-\beta }. \end{aligned}$$

(39)

It follows that

$$\begin{aligned} Y_K= & {} \frac{\alpha f}{K}-\delta ,\quad Y_{\textit{KK}} =\frac{-\alpha (1-\alpha -\beta )f}{(1-\beta )K^{2}},\quad Y_\tau =-E,\quad Y_{K\tau } =\frac{\alpha f_\tau }{K}, \nonumber \\ E_K= & {} \frac{\beta }{d+\tau }f_K\quad \hbox { and }\quad E_\tau =\frac{-E}{(d+\tau )(1-\beta )}. \end{aligned}$$

(40)

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

$$\begin{aligned} B=\left( {{\begin{array}{ccccccc} {Y_{1K_1 } +\tau _1 E_{1K_1 } }&{}\quad 0&{}\quad 0&{}\quad {-1}&{}\quad 0&{}\quad {\tau _1 E_{1\tau _1 } }&{}\quad 0 \\ 0&{}\quad {Y_{2K_2 } +\tau _2 E_{2K_2 } }&{}\quad 0&{}\quad 0&{}\quad {-1}&{}\quad 0&{}\quad {\tau _2 E_{2\tau _2 } } \\ {\psi E_{1K_1 } }&{}\quad {\psi E_{2K_2 } }&{}\quad {-\gamma }&{}\quad 0&{}\quad 0&{}\quad {\psi E_{1\tau _1 } }&{}\quad {\psi E_{2\tau _2 } } \\ {\xi _{1,1} }&{}\quad 0&{}\quad {\xi _{2,1} }&{}\quad {\xi _{3,1} }&{}\quad 0&{}\quad {\sigma Y_{1K_1 \tau _1 } C_1 }&{}\quad 0 \\ 0&{}\quad {\xi _{1,2} }&{}\quad {\xi _{2,2} }&{}\quad 0&{}\quad {\xi _{3,2} }&{}\quad 0&{}\quad {\sigma Y_{2K_2 \tau _2 } C_2 } \\ {\xi _{4,1} }&{}\quad {\xi _{5,1} }&{}\quad {\xi _{6,1} }&{}\quad {\xi _{7,1} }&{}\quad 0&{}\quad {\xi _{8,1} }&{}\quad {\xi _{9,1} } \\ {\xi _{5,2} }&{}\quad {\xi _{4,2} }&{}\quad {\xi _{6,2} }&{}\quad 0&{}\quad {\xi _{7,2} }&{}\quad {\xi _{9,2} }&{}\quad {\xi _{8,2} } \\ \end{array} }} \right) ,\qquad \end{aligned}$$

(41)

where $\xi _{1,i} \equiv \sigma \left[ {Y_{iK_i K_i } -H(P)\frac{C_i^{1/\sigma } C_{iK_i }^A (K_i )}{\sigma C_{i}^{A} (K_i )^{1+1/\sigma }}} \right] C_i ,\hbox { }\xi _{2,i} \equiv \sigma H^{\prime }(P)\left[ {\frac{C_i^{1/\sigma } }{C_{i}^{A} (K_i )^{1/\sigma }}-1} \right] C_i ,$

$$\begin{aligned} \xi _{3,i}\equiv & {} H(P)\left[ {\frac{C_i^{1/\sigma } }{C_{i}^{A} (K_i )^{1/\sigma }}} \right] ,\\ \xi _{4,i}\equiv & {} \left[ {Y_{iK_i K_i }-H(P)\frac{C_i^{1/\sigma } C_{iK_i }^A (K_i )}{\sigma C_{i}^{A} (K_i )^{1+1/\sigma }}} \right] \tau _{i}-\psi H^{\prime }(P)\left[ {1-\frac{C_i^{1/\sigma } }{C_{i}^{A} (K_i )^{1/\sigma }}} \right] ,\\ \xi _{5,i}\equiv & {} -\psi H^{\prime }(P)C_i^{1/\sigma } [C_j^{-1/\sigma } -C_j^A (K_{j} )^{-1/\sigma }],\quad j\ne i, \quad i=1,2, \end{aligned}$$

which in the non-cooperative case becomes

$$\begin{aligned} \xi _{5,i}\equiv & {} 0,\\ \xi _{6,i}\equiv & {} H^{\prime }(P)\frac{C_i^{1/\sigma } }{C_{i}^{A} (K_i )^{1/\sigma }}\tau _1 +H^{\prime }(P)\tau _{i}\\ \xi _{7,i}\equiv & {} H(P)\frac{C_i^{-1+1/\sigma } }{\sigma C_{i}^{A} (K_i )^{1/\sigma }}\tau _{i} -\psi H^{\prime }(P)\frac{C_i^{-1+1/\sigma } }{\sigma }\mathop {\sum }\nolimits _{j=1}^{2} {\left[ {V_{j} -V_j^A (K_{j} )} \right] }, \end{aligned}$$

where in the non-cooperative case the summation reduces to only the $j = i$ term,

$$\begin{aligned} \xi _{8,i} \equiv \rho +\gamma +H(P)\frac{C_i^{1/\sigma } }{C_{i}^{A} (K_i )^{1/\sigma }}+Y_{iK_i \tau _{i} } \tau _{i} -\frac{\psi H^{\prime }(P)}{\rho +H(P)}\left( {-E_i +\frac{\gamma P}{2\psi }} \right) , \end{aligned}$$

which in the non-cooperative case becomes $\xi _{8,i} \equiv \rho +\gamma +H(P)\frac{C_i^{1/\sigma } }{C_{i}^{A} (K_i )^{1/\sigma }}+Y_{iK_i \tau _{i} } \tau _{i} ,$

$$\begin{aligned} \xi _{9,i} \equiv \frac{-\psi H^{\prime }(P)C_i^{1/\sigma } C_j^{-1/\sigma } }{\rho +H(P)}\left( {-E_{j} +\frac{\gamma P}{2\psi }} \right) ,\quad j\ne i, \end{aligned}$$

which in the non-cooperative case becomes $\xi _{9,i} \equiv \frac{\psi H^{\prime }(P)\tau _{i} E_{j\tau _{j} } }{\rho +H(P)},\,j\ne i.$

Numerical Spectral Decomposition Algorithm

We can write the log-linear approximation around the steady state as follows:

$$\begin{aligned}&\dot{x}=Ax=\left( {{\begin{array}{cc} {A_{pp} }&{}\quad {A_{pn} } \\ {A_{np} }&{}\quad {A_{nn} } \\ \end{array} }} \right) \left( {{\begin{array}{l} {x_p } \\ {x_n } \\ \end{array} }} \right) \quad \hbox { with}\,x_p \equiv \left( {{\begin{array}{c} {{\begin{array}{c} {\ln \left( K_1 /{\bar{K}}_1\right) } \\ {\ln \left( K_2 /{\bar{K}}_2\right) } \\ \end{array} }} \\ {\ln \left( P/{\bar{P}}\right) } \\ \end{array} }} \right) \quad \hbox { and }\quad \nonumber \\&\quad x_n \equiv \left( {{\begin{array}{c} {{\begin{array}{l} {\ln \left( C/{\bar{C}}_1\right) } \\ {\ln \left( C_1 /{\bar{C}}_2\right) } \\ \end{array} }} \\ {{\begin{array}{l} {\ln \left( \tau _1 /{\bar{\tau }}_1\right) } \\ {\ln \left( \tau _2 /{\bar{\tau }}_2\right) } \\ \end{array} }} \\ \end{array} }} \right) , \end{aligned}$$

(42)

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):

$$\begin{aligned} \left[ {A_{ij} } \right] =\left[ {B_{ij} } \right] {\bar{x}}_{j} /{\bar{x}}_i ,\quad i=1,\ldots ,3,\quad j=1,\ldots ,4. \end{aligned}$$

(43)

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

$$\begin{aligned} \dot{y}=\Lambda y\quad \hbox {for}\,y=Nx, \end{aligned}$$

(44)

which has the stable solution

$$\begin{aligned} y_{p,i} (t)=e^{\lambda _{p,i} t}y_{p,i} (0),\hbox { }i=1,\ldots ,3\quad \hbox {and}\quad y_{n,j} (t)=0,\quad j=1,\ldots ,4. \end{aligned}$$

(45)

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).

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Frederick van der Ploeg
- 1
- 2
- 3
Email author
Aart de Zeeuw
- 4
- 5

1.Department of Economics, OXCARREUniversity of OxfordOxfordUK
2.VU University AmsterdamAmsterdamThe Netherlands
3.Laboratory Economic Performance and the EnvironmentSt. Petersburg State UniversitySt. PetersburgRussia
4.CentER and TSCTilburg UniversityTilburgThe Netherlands
5.Beijer Institute of Ecological EconomicsStockholmSweden

Non-cooperative and Cooperative Responses to Climate Catastrophes in the Global Economy: A North–South Perspective

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Variable/parameter
Pure rate of time preference, \(\rho \)	0.014
Elasticity of substitution, \(\sigma \)	0.5
Depreciation rate of capital, \(\delta \)	0.05
Share of capital in value added, \(\alpha \)	0.3
Share of fossil fuel in value added, \(\beta \)	0.0623
Total factor productivity, A	8.5044
Eventual climate shock, \(\pi \) \(_{i}\)	0.1, 0.3
Fraction of carbon staying up, \(\psi \)	0.5
Natural decay of carbon, \(\gamma \)	0.003
Cost of fossil fuel, d	9 US$/million BTU \(=\) 504.3 US$/tC
2010 levels of GDP	42.1 and 20.9 trillion US$
2010 levels of capital, \(K_{i0}\)	180 and 20 trillion US$
2010 fossil fuel use, \(E_{i0}\)	5.551 and 2.749 GtC
Initial stock of carbon, \(P_{0}\)	826 GtC \(=\) 388 ppm by vol. \(\hbox {CO}_{2}\)