CC BY
4Precautionary Policy Rules in an Integrated Climate-Economy Differential Game with Climate Model Uncertainty
Magnus Hennlock
University of Gothenburg,
Department of Economics and Statistics at University of Gothenburg, P. O. Box 640, S-405 30 Gothenburg, Sweden E-mail: Magnus.Hennlock@economics.gu.se
Abstract The paper introduces structural uncertainty in an integrated climate-economy differential game such that the probability distribution of climate sensitivity is unknown. This is generated by perturbing a continuous-time version of the climate model in Nordhaus (1992) and Nordhaus and Yang (1996). Instead of analyzing choices of regional representative consumers, we define social profit from regional production as the payoff to regional policymakers. There are two types of players: Firstly, regional policymakers j = 1, 2, ..., N, who are tied to a region by acting as a sovereign regional social planner who can only enforce regional emissions reduction policies. Secondly, investors i = 1, 2, ..., n, who are not tied to any region who allocate investments between firms (production processes generating emissions) located in all regions j = 1, 2, ..., N. We identify policymakers’ optimal policy responses to firms’ investment responses as well as firms’ optimal investment responses to policymakers’ policy responses in a global subgame perfect Nash equilibrium when policymakers do not cooperate and compare it to when policymakers coordinate national policy rules
Keywords: Differential games, structural uncertainty, feedback Nash equilibrium, closed-loop equilibrium, uncertainty aversion.
1. Introduction
Climate change policy is subject to fundamental uncertainties concerning the underlying scientific information available. Policy makers’ decision to take or not take measures today are based on scientists’ projections, generated by computer climate models and evaluated for different emissions scenarios. A common measure when comparing projections is the change in equilibrium mean atmospheric temperature that results from a doubling of CO2. An increase in atmospheric CO2 changes net radiation at the tropopause (radiative forcing) in the atmosphere, which influences the energy balance of the climate system, and hence, changes the mean atmospheric temperature Tt. When comparing computer climate models, a conclusive component is equilibrium climate sensitivity, which is defined as the ratio between a steady-state change in mean atmospheric temperature AT and a steady-state change in radiative forcing ARt.
AT
ARt
Climate sensitivity depends on several underlying physical feedback processes which are hard to predict. One of the most uncertain is the cloud effect, other are water vapor, albedo and vegetation effect (see e.g. Harvey (2000) and Hansen, Lacis, Rind and Russell, Climate Sensitivity: Analysis of Feedback Mechanisms (1984a)). A recent analysis by Roe and Baker (2007) shows how climate sensitivity and its probability distribution becomes unpredictable due to uncertainties in underlying physical feedback factors when they translate into uncertainty in climate sensitivity. The apparent uncertainty in climate sensitivity as also evident in IPCC, Climate Change 2007: Working Group I Report The Physical Science Basis (2007c) Executive Summary which states ‘The equilibrium climate sensitivity is a measure of the climate system response to sustained radiative forcing. It is not a projection but is defined as the global average surface warming following a doubling of carbon dioxide concentrations. It is likely to be in the range 2°C to 4.5°C with a best estimate of about 3°C, and is very unlikely to be less than 1.5°C. Values substantially higher than 4.5°C cannot be excluded, but agreement of models with observations is not as good for those values.’
Roe and Baker (2007) conclude that for high temperature levels above the IPCC interval of 2.0oC - 4.5oC the probability distribution changes very little to changes in the variance in the underlying physical processes. Hence, their conclusion is that scientific research that reduces uncertainty in the underlying physical processes has little effect in reducing uncertainty in climate sensitivity at high temperature outcomes. For values above 2.0oC - 4.5oC, the upper fat tail of the probability distribution of equilibrium climate sensitivity would remain fat despite progress in understanding the underlying physical processes. Roe and Baker (2007) therefore conclude ‘We do not therefore expect the range presented, in the next IPCC report to be different from that in the 2007 report’ and ‘we are constrained, by the inevitable: the more likely a large warming is for a given forcing (i.e. the greater the positive feedbacks) the greater the uncertainty will be in the magnitude of that warming. ’
What does this message tell policymakers? Clearly, optimal reductions in CO2 emissions would differ whether the decision is based on a climate model predicting an equilibrium temperature of 1.5oC or a model predicting 4.5oC or even higher. Secondly, the fundamental uncertainty concerns not only future outcomes but also future probability distributions of climate sensitivity. True or inferred probability distributions are not available from current data. This deeper type of uncertainty refers to model uncertainty rather than variable uncertainty. When comparing the projections of computer climate models based on current data and knowledge, the probability distributions differ among models as they are based on components resulting from scientists’ ad hoc assumptions and guesswork (Harvey, 2000).
So which climate model should we then base our decisions on when looking for optimal emissions reductions? Nordhaus (1992) used for example a model that predicts an equilibrium temperature of app. 3oC located in the mid range of the IPCC interval. A more precautionary policymaker would perhaps prefer to use a model with 4.5oC equilibrium to be more certain that she is not bad off even in the case a model corresponding to the upper climate sensitivity range becomes true. Our approach in this paper is simply to leave the question unanswered by defining structural uncertainty in the climate model such that policymakers face a set of climate models corresponding to a range of climate sensitivities. These conditions of uncertainty better mimics the scientific uncertainties that IPCC, Climate Change
2007: Working Group I Report The Physical Science Basis (2007c), Roe and Baker (2007) and Allen and Frame (2007) find in the computer climate models.
In the literature on the theory of decision-making it is common to distinguish between risk and uncertainty. The former refers to a process where actual outcome of variable is unknown but its probability distribution (objective or subjective) is known or can be estimated from samples and corresponds to variable uncertainty. The latter refers to when outcomes as well as probability distributions are unknown and corresponds to model or structural uncertainty. Another approach, taking into account unknown probability distributions or structural uncertainty, is Weitzman (2007) though his analysis is highly abstract by only looking at the effects of uncertainty (unknown probability distribution) with a CRRA utility function in a two-period analysis without specifying the source of uncertainty. The discussion on unknown probability distributions is not new though. Already Knight (1971) suggested that for many choices, the assumption of known probability distributions is too strong. Moreover, Keynes (1921), in his treatise on probability, put forward the question whether we should be indifferent between two scenarios that have equal probabilities, but one of them is based on greater knowledge. Savage (1954) argued that we should, while Ellsberg (1961) showed in an experiment that in reality humans tend not to do so. A person that is facing two uncertain lotteries with the same (subjective) probability to success, but with less information provided in the second lottery, tends to prefer the first lottery where more information is available. Having Ellsberg’s paradox in mind, Gilboa and Schmeidler (1989) formulated a maximin decision criterion, by weakening Savage’s Sure-Thing Principle, to explain the result from the Ellsberg experiment. In plain words, the decision-maker is suggested to maximize expected utility under the belief that the worst case scenario will happen (a maximin decision criterion). This preference is usually referred to as uncertainty aversion (as opposed to risk aversion which is a tendency to avoid uncertain outcomes) which is about avoiding bad since due to the pessimistic climate model is correct. The maximin decision criterion has been applied before in static models by e.g. Chichilnisky (2000) and Bretteville Froyn (2005) with the general result that it leads to an increase in abatement effort. Roseta-Palma and Xepapadeas (2004) apply it in a dynamic model of a water management problem following Hansen and Sargent (2001).
In this paper we start from Roe and Baker (2007) and Allen and Frame (2007), that current data from underlying physical processes are not sufficient to predict climate sensitivity. Firstly, we introduce unknown probability distributions of climate sensitivity in an integrated assessment model based on a continuous-time version of the climate model used in Nordhaus (1992) and Nordhaus and Yang (1996). Secondly, we introduce a type of precautionary preferences among policymakers based on maximin decision criteria by adding a preference parameter for robustness in policymakers’ payoff functions. Nordhaus (1992) and Nordhaus and Yang (1996) as well as Weitzman (2007) use a representative consumer CRRA utility function. However, since we are interested in policymakers’ and firms’ responses to each other rather than the choices of regional representative consumers, we instead define the social profit from physical and natural capital within each region as the payoff to the policymaker. We introduce two types of players, investors i = 1, 2,...,n, investing in firms (production processes generating emissions) located in region j = 1,2,..., N, and who are not physically tied to any specific region, and policy-
makers j = 1, 2,..., N, who are tied to a region by acting as a sovereign regional social planner who can only enforce regional policies while taking the policies of foreign policymakers as well as all investors as given in her optimal choice of regional policy. We then identify an analytically tractable feedback Nash equilibrium in policy strategies for N asymmetric policymakers subject to the climate model with uncertain climate sensitivity. Finding analytically tractable solutions to nonlinear feedback Nash equilibria with asymmetric players is usually difficult. The analytical solution found in this paper opens up for analyzes of different game formulations with asymmetric players in climate models of the type in Nordhaus (1992) and Nordhaus and Yang (1996).
The following sections are organized in the following way. In section 2., the range of climate models and climate change impacts are presented. Section 3. presents players and payoff functions, optimization problems and the optimal noncooperative and cooperative policy rules, which is followed by a summary in 4..
2. The Climate Models
In this section we present the set of climate models which are generated by perturbing a continuous-time version of the climate benchmark model in Nordhaus (1992) and Nordhaus and Yang (1996) describing the relationship between atmospheric concentration rate Mt, radiative forcing Rt, atmospheric temperature Tt and deep ocean temperature Tt. The net radiation balance is Rt = Ft+St where Ft is outgoing radiation and St ingoing radiation. In equilibrium Ft = —St since Rt = 0. Increasing anthropogenic emissions results in radiative forcing ARft which moves the system away from the initial equilibrium towards a new equilibrium as ARft = — ARt and ARt = AFt + ASt. Accordingly, the steady-state mean temperature must change by ATt between the two equilibria.
ATt = A0ARft + Y; fsATt (2)
s
where A0 is the reference climate sensitivity in absence of underlying feedback factors fs such as cloud effect, albedo effect and vegetation effect.1 Denoting f = ^s fs, (2) can be rewritten as
ATt = X0-^—fARft (3)
where 1/(1 — f) is the change in equilibrium temperature that derives from feedback factors also called the gain of the climate system and equals AT/AT0 i.e. the proportion by which the system response has changed due to feedback. In Nordhaus (1992) and Nordhaus and Yang (1996) the discrete-time radiative forcing dynamics Rt is
XMMt/M0)
Rt-----H2)---+ 0t (4)
1 Ao = 1/(4aT3) is found as the balanced radiative forcing at a blackbody planet, by
dT/dF from Stefan-Boltzmann relationship F = — ctT4.
where A1 is a parameter conclusive for equilibrium climate sensitivity and Ot is radiative forcing from non-anthropogenic sources. For analytical tractability of Isaacs-Bellman-Flemming equations, we approximate (4) by the square-root approximation in (6). Compared to (4), this approximation overestimates radiative forcing by at most app 5% in the range from current rate up to a tripling in CO2 rate. However, the advantage is that it makes an analytical solution possible which is necessary for identifying subgame consistent cooperative solutions with time-consistent payment streams in forthcoming analyzes of this game model.
The climate model in (6) - (8) shows the atmospheric concentration rate Mt influence on global mean atmospheric and upper ocean temperature Tt and deep ocean temperature Tt relative to the preindustrial level via the change in radiative forcing Rt (measured in Wm-2) in (6). Emissions Eijt originates from production in j = 1, 2,...,N regions where rqij qijt is abatement, qijt being abatement effort undertaken in firm i in region j and rqij > 0 an firm-specific efficiency parameter. In equation (5), the sum of net emissions flows Eijt — rqijqijt at time t £ [0, <x) from production processes i = 1,2,..., n in regions j = 1,2,..., N accumulates to the global atmospheric concentration CO2 stock Mt. Cj > 0 is the marginal atmospheric retention ratio and Q > 0 the rate of assimilation.
dM
N
EE Cj (Emkt Vmk qmkt) QMt
m= 1 k=1
dt (5)
Ai \J MfJMo + v
R, = —wi—+ ‘
dT — -z— (Rt, — XT — — (Tt, — Tt,) j dt
Ri V r2
(6)
(7)
df=^( — (Tt-Tt))dt (8)
R2 V T2 )
A change in radiative forcing Rt affects the energy balance of the climate system and hence the global mean atmospheric temperature Tt through the relationship (7) via the deep ocean temperature Tt in (8). The physical constants are A as a component in the underlying feedback processes, Ri the thermal capacity of the atmosphere and the upper ocean (i = 1) and the deep ocean (i = 2), respectively. 1/t2 is the transfer rate from the atmosphere and upper ocean layer to the deep ocean layer. The parameters a and v are set to fit (4).
2.1. Introducing Climate Model Uncertainty
Roe and Baker (2007) find that for high temperature levels above the IPCC interval of 2.0oC - 4.5oC, the probability distribution of equilibrium climate sensitivity changes very little to changes in the variance in underlying physical processes. Hence, their conclusion is that scientific research that reduces uncertainty in the underlying physical processes has little effect in reducing uncertainty in climate sensitivity. The fatter tail of the probability distribution of climate sensitivity for values above 4.5oC would remain despite progress in understanding the underlying physical processes. Roe and Baker (2007) conclude ‘we are constrained by the
inevitable: the more likely a large warming is for a given forcing (i.e. the greater the positive feedbacks) the greater the uncertainty will be in the magnitude of that warming.’2 If the world warms by 4oC, the conditions in the underlying feedback processes f, such as cloud and water vapor effects, may have changed from current conditions making it impossible to determine when warming stops. Hence, from all climate and physical data we can observe today, we cannot distinguish between a climate sensitivity of 4oC or 6oC. Another consequence is that dramatic changes in the physical processes may not be needed for dramatic changes in the climate sensitivity (Visser et al., 2000).
In this paper, we want to reconstruct the conditions of uncertainty to be closer to real conditions of uncertainty as described in Roe and Baker (2007). We also introduce precautionary preferences based on the Gilboa and Schmeidler (1989) idea, however this turns out to be a problem since their axioms are based on static decision making and not sufficient for dynamic models. Specifically, they do not state how the decision-maker’s beliefs are affected by new information (which could increase or decrease scientific uncertainty as time proceeds). We then follow Hansen, Sargent, Turmuhambetova and Williams, Robustness and Uncertainty Aversion (2001b) and suggest that a rational decision-maker updates her beliefs to new information due to scientific progresses by a rule derived from backward induction.3
However, we start by introducing a new approach to scientific uncertainty in climate modeling by perturbing climate sensitivity in the model (6) - (8), defining the following process
Bt = Bt + I Asds As £ [Amin i Amox] (9)
0
where dB is the increment of the Wiener process Bt on the probability space (H,^,G) with variance a2 > 0 where {Bt : t > 0}. Moreover, {At : t > 0} is a progressively measurable drift distortion, implying that the probability distribution of Bt itself is distorted and the probability measure G is replaced by another unknown probability measure Q on the space (H,^,Q). Hence, (6) is replaced by
a[Xtdt + dB] \JMt/Mo + vdt
---------------1=--------------h utdt (111)
a2
Since both mean and probability distribution of the drift term At are unknown, (9) yields different statistics of climate sensitivity in (10) where the interval [Amin, Amox ] indicates the maximum model specification error, e.g. corresponding to the range of climate sensitivities as the 2o - 4.5o range by IPCC, Climate Change 2007: Working Group I Report The Physical Science Basis (2007c) or even wider ranges that the policymaker is willing to accept. Hence, (9) together with (10) describe a set of
2 Allen and Frame (2007) goes further on this result and conclude that scientific research trying to narrow the uncertainties in the upper end of the tail is perhaps useless, and hence, in the choice of research as well as policy targets one should take this into account.
3 Gilboa and Schmeidler (1989) view uncertainty aversion as a minimization of the set of probability measures while Hansen, Sargent, Turmuhambetova and Williams, Robustness and Uncertainty Aversion (2001b) set a robust control problem and let its perturbations be interpreted as multiple priors in max-min expected utility theory.
climate models with the only restriction that As is bounded by the constraint in (9). Since the mean as well as probability distribution of As is unknown, the problem of the policymakers becomes two-folded: (1) what At should today’s policy decision be based upon and (2) by which rules should At be updated if more scientific knowledge about climate sensitivity is gained as time proceeds?
2.2. Regional Climate Change Impacts
All regions j = 1, 2,..., N face the same change in global mean surface temperature Tt — To while benefit and damage costs may differ significantly across regions j = 1,2,..., N .A broad overview of climate change impacts is given in IPCC, Climate Change 2001: Impacts, Adaptation and Vulnerability. Contribution of Working Group II to the Third Assessment Report of the Intergovernmental Panel on Climate Change (2001b), IPCC, Climate Change 2007: Working Group II Report Impacts, Adaptation and Vulnerability (2007d) as well as Tol, Estimates of the Damage Costs of Climate Change Part I. Benchmark Estimates (2002a) and Tol, Estimates of the Damage Costs of Climate Change Part II. Dynamic Estimates (2002b). The considered impacts are often on natural capitals such as agriculture, forestry, water resources, loss of dry- and wetland (due to sea-level rise) and increased consumption of energy resources (heating and cooling).4 In this model we introduce damages on regional natural capitals xjt that derive from a global mean temperature deviation Tt — T0. Another reason is that the solution to the partial differential equation system is technically simplified by separating benefits and costs to a set of polluting stocks and a set of damaged stocks.
Regional physical capital accumulation in (11) follows the structure of Merton (1975) and Yeung (1995) with a Cobb-Douglas investment function and depreciation rate ¿ij- > 0 where /ijt is the investment by investor i in region j in period t
dkij
11/2k1/2 — & ■ Jijt kijt
dt (11)
dxj =
,1/2
1
1/2
xjt
xjt
dt
(12)
r
j
i = 1, 2,..., n j = 1,2,..., N (13)
The equations of motion of xjt in (12) are specified forms of the relationship used in Hennlock, A Differential Game on the Management of Natural Capital Subject to Emissions from Industry Production (2005a) and Hennlock, An International Marine Pollutant Sink in an Asymmetric Environmental Technology Game (2008c) and consist of a modified natural growth function with intrinsic growth rj > 0 and regional carrying capacity xjt = K?. The loss of xjt due to a deviation in global mean temperature rise Tt — T0, where T0 is the 1990 mean temperature level, is determined by a non-linear endogenous decay rate ^j (Tt — T0)x1/2, suggesting that the damage from a given mean temperature deviation accelerates as the natural capital stock xjt decreases where ^j > 0 is a region-specific damage parameter. (5)
4 Most research has been conducted on the effects of sea level rise e.g. Titus and Narayan (1991).
to (13) define the dynamic system with 2 + M (1 + n) state variables. The introduction of the unknown variable At in (7) implies that the dynamics of the system corresponds to the set [Amin, Amox] of climate models. Hence, climate model uncertainty in (6) - (9) also makes regional impacts uncertain for a given net emissions scenario YWXT{£iji — nijqijt}.
3. Players and Payoffs
There are two types of players, investors i = 1, 2,..., n, investing in firms (production processes) located in region j = 1, 2,..., N, and who are not physically tied to any specific region, and policymakers j = 1,2,..., N, who are tied to a region by acting as a sovereign regional social planner who can only enforce regional policies while taking the policies of foreign policymakers as well as all investors as given in her optimal choice of regional policy. If all n + N players assume that the other players do their best, there is a global Nash equilibrium in policies and investment rates. Identifying a Nash equilibrium as benchmark, several patterns of cooperation are then possible which are further discussed in section 3.4.
3.1. Investor i G [1,n]
Every investor i € [1, n] solves a stochastic optimization problem and allocates total investment Yfe=11ifct in period t between firms (production processes) located in all regions j € [1,N] for the production of a good y that is sold on the world market at unit price. Profit-maximization by each investor i follows Hennlock, A Robust Feedback Nash Equilibrium in a Climate Change Policy Game (2008d), and is obtained by allocating investment (and thereby production activity) between the regions j € [1,N]. The expected payoff of investor i, where e is the expectation operator, is
f°° f r-t \ Cik(qikt)2 \ ~Pit u /u\
maxe y < p(l - uikt)yikt-----------------------\e p' dt (14)
u,fct ^0 £=1 I Eikt J
Investor i seeks the optimal expected cash dividend from each firm i € [1, n] in each period t located in region j € [1,N] by controlling the share Mijt € [0,1] of net profit that is reinvested in regional capital stocks kij. By investing /ijt investor i contributes to total industry output yjt in region j, which is yjt = Ym=1 ^mj k1/j2t. The amount reinvested /ijt in period t is the remainder /ijt = Mijiyiji. Investor i’s discount rate is pi > 0. The last term in (14) is firm i’s abatement cost, which is quadratic in regional abatement effort qijt due to capacity constraints as more local abatement effort qijt is employed. Abatement cost is decreasing in Eijt, suggesting that it requires more expensive techniques as Eijt becomes smaller. cij > 0 is an abatement cost parameter for production i in region j. The total emissions flow from region j is Ejt = Y,m=1 Emjt = Ym=1 ^mjymjt where ^TOj > 0 is a firm-specific emissions parameter and the firm-specific level of q jt set by policymaker j is taken as given by investor i when seeking optimal investment rates u*fct. Every investor i maximizes the payoff function (14) subject to the dynamic system formed by (5) and (7) to (13) and q*jt > Vi € [1,n] and Vj € [1,N]. The investors’ stochastic optimal problems are solved in appendix A.1. The feedback investment rate strategies in terms of parameter values are:
Hj
G [0,1] Vi G [1,n] and Vj G [1,N] (15)
Investor i’s feedback Nash investment rate uj is decreasing in , implying that the share of net profit used for investment is large in the beginning when is low during business start-up. As grows, investor i reduces the share of net profit reinvested in capital stock fcj located in region j. The optimal investment rate rules in (15) are further discussed in Hennlock, A Robust Feedback Nash Equilibrium in a Climate Change Policy Game (2008d).
3.2. Policymaker j G [1,N]
Policymaker j = 1,2,...,N receives a social profit from production (or employment) within region j which is proportional to total production level ^m=i
1/2
fcmjt in region j but also a loss of regional natural capital due to global climate change Tt — T0. In the Nash equilibrium, each policymaker j G [1, N] seeks the optimal emissions reductions nj qj in each firm i located in region j given that the remaining N — 1 policy makers individually seek the optimal nifcVfc = j and that every investor i G [1,n] individually seek optimal investment rate M*jt. The expected payoff of policymaker j G [1, N] is5
n f 2 'I
j UjVmjt + -----£ mjt f e Pltdt + OjR(Q) (16)
m=1 I I
The first term is social benefit of employment that is assumed to be proportional to total regional production ymjt = ^mj- fcmj where the parameter Wj > 0. The second term is the benefit from the regional natural capital in region j € [1, N] where ^j > 0 is a parameter. The last term within the brackets is the abatement cost function. Policy maker j’s discount rate is pj > 0. Following Hansen, Sargent, Turmuhambetova and Williams, Robustness and Uncertainty Aversion (2001b), policymaker j’s payoff function can be written as (16) in a multiplier robust problem where 1/0j > 0 denotes the policy maker’s preference for robustness which together with R(Q), the finite entropy, act as Lagrangian multiplier in (16). The process {As} in (6) is unknown and will change the future probability distribution of having probability measure Q relative to the distribution of having measure G. The Kullback-Leibler distance between Q and G is
R{Q) = j™ eQ(^~y-^ds (17)
As long as R(Q) < in (16) is finite
Ql I l^sl2ds < = 1 (18)
5 Technically, it is straightforward to also let investors be uncertainty averse in this solution.
1
OO
which has the property that Q is locally continuous with respect to G, implying that G and Q cannot be distinguished with finite data, and hence, modeling a situation with a decision-maker that cannot know the future probability distribution when using current data. Following Hansen and Sargent (2001), policymaker j € [1,N] optimal problem of finding optimal emissions reductions for each firm nj qjt, can be written as:
maxmin e /
j JQ
m=1
Wj ymjt + ^j j -
1/2 Cmj<lmjt 0j>H
+
mjt
‘dt (19)
subject to
dfc,.
dt
(20)
dxo-
1/2
jt
rj 11 K.
1/2 _ &j(Tt - To)
1/2 J*
Xjt -
Xjt
dt
(21)
dM
N
EE Cj (Emfct nmfc qmfct) i2Mt
m=1 k=1
dt
(22)
1 / u[Ajtdi + dB] y/Mt/Mo + vdt
dT=Ri {---------------^---------------+ °tdi
—XTdt — — (Tt — Tt)dt
T2
(23)
(24)
i = 1, 2,..., n j = 1,2,..., N
(25)
Definition 1. Feedback Nash Equilibrium If there exist N value functions Wj (k, x, M, T, T, t) where
k = (fcn , &12 , . . . , &1N, &21, k22, . . . , k2N, kn1, kn2, . . . , knN)
(26)
and x = (x1, x2,..., xN) that satisfy
Wj- (k, x ,M,T,T,t) = (27)
j jt
m=1
poo
mjt
22 1/2 uj^jt
m=1
E
+
mjt
2
‘dt
oo
2
e
for strategies qj (kj ,t) C R1 and A*t (M, t) C R1 given that
A*t (M, t) = argmin Wj (k, x, L, M, T, T,t) Vj € N and which satisfy the state equations,
dfci.
( j)1/2kj2 - ¿jk
¿ij kijt
dt
(28)
dxo-
1/2
1
'jt \ 1/2 - Го)
jt
1/2
xjt
xjt
dt
(29)
r
j
dM =
N
EE £j (Emfct ^mfc qmfct) ^Mt
m=1fc=1
dt
(30)
1 /" cr[Ajt(áí + dB] Vх Mt/M0 + vdt
,¡T=rA-----------------------------------+ 0,й <31)
—XTdt — — (Tt — Tt)dt
T2
dt
(32)
1, 2,
nj
1,2,
N
(33)
The feedback Nash controls strategies
rjt = {qíj(kij),Л*(kj)} Vi Є [1,n] Vj Є [1, N] (34)
provide a robust feedback Nash equilibrium solution of the game defined in (19) to (25) given (49) to (55) (Basar and Olsder, 1999).
The value functions in definition 3 satisfy the partial differential equation system (35) - (36). Using (49) - (55) and (19) - (25) and definition 2 and 3 yield the Isaacs-
Bellman-Fleming equation (Fleming and Richel, 1975) of policy maker j:
dWi
dt
1/2 _ + Oj^jt \ p-Pjt
m=1
E
mjt
2
N
dkmk L
m=1 k=1
,fct
A dWj
+E
k=i
dxk
+
dM
I 1 ^/Xkt\ t^k(Tt To)
n N
EE Cj (Emkt ^mk qmkt) i?Mt
+
6»T
,m=1 k=1
^ ( a\/Mt /Mo + z/
r2
+ Ot - XT - —(Tt - Tt)
T2
+
dT
+ \^a2Mt
(35)
* = 1, 2,..., n j = 1,2,...,N (36)
The robust feedback Nash controls strategies
■Tj = {qj(kijt),A*(Mt)} V* G [1,n] Vj G [1,N] (37)
are given by maximizing the partial differential equations (35) with respect to (37) for the N policymakers and solving for the robust policy rules.
3.3. Feedback Nash Equilbrium Policy Rules
Since, policymakers’ payoff functions are time autonomous, these policy rules (38) are time consistent and subgame perfect policy strategies or responses to the evolution of firms’ net emissions Eijt. Hence, the policy rules are credible and efficient threats in every subgame starting at t < ro (Dockner et al., 2000), given the policymakers’ preferences for robustness. By announcing the policy rule qijt(Eijt), a policymaker regulates net emissions Eijt — nj qj for each firm i at each instant of time. As expected, the optimal abatement effort is qijt > 0 for all kj > 0 since dW/dMt < 0 Vt.
* f p \ _ I' pjt / o o\
- dMt 2cij Eijte (38)
The policy rules qjt (Eijt) state that abatement effort qijt in firm j is proportional
to firm j’s total emissions levels Eijt and the size of the proportion is determined by a ratio between policymakers’ shadow costs, dW/dMt, of CO2 and the firm’s abatement cost parameter cj. The partials derivatives in (38) are policymakers’ expected
Nash shadow cost of CO2, which are identified in appendix A.2. by differentiating the value functions (84) using (96) and (97). Substituting the Nash shadow costs into (38) yields the robust feedback Nash equilibrium strategies in terms of model parameter values.
bjk_yTr, I - I
* CP \ _ J=1 2 j \R1aV2J £jT)ij
+ Ad - Irfc) ' * - ' ’
The optimal policy rules in (39) are determined by four categories of factors (i) the set of physical parameters in the climate models defined by (5) and (7) to (9), (ii) the sum of climate impacts parameters on regional natural capitals Xj, (iii) the sum of parameters ^¿j from the payoffs of natural capitals and, (iv) the policymaker’s preferences of time pj and robustness 1/0j. Since a single policymaker only takes into account the value of own regional natural capital, j = 0 for all k = j in (39) resulting in that the levels of shadow cost trajectories of CO2 are the share / Yj bjfc of the total level of the globally optimal shadow cost trajectories. Decreased time preference pj or increased preference for robustness 1/0j shifts expected shadow cost trajectory upwards, implying stricter policy rules qj (Eijt), i.e. greater qj for given (Eijt) levels.
Using investor i’s optimal investment rule ujt from (83), which determines the growth of capital kj, gives the policymaker j’s optimal policy response expressed in terms of firms’ investment rate strategies ujt. The net effect is that the policymaker’s best policy response is to reduce emissions reductions to regional firms which re-investment a larger share of profit in regional production.
£?= 1
qijt
1
1+Pj T2
) 2cjjMg^j (pj + ^) 4 u;
0
(40)
ijt
2
a
1
a
j
k = 1, 2,..., n j = 1,2,...,N (41)
The rule A*(Mt) in (42) tells how the policymaker j given her preference for robustness (1/0j) updates climate sensitivity in (6). A policymaker with no preference for robustness (1/0j ^ 0) will not update the climate sensitivity but continue using the benchmark model Amin as in ordinary stochastic optimal control regardless discovery of new climate data.
t
k = 1, 2,...,n j = 1,2,...,N
(44)
The comparative statics of investors’ optimal investment rates in Hennlock, A Robust Feedback Nash Equilibrium in a Climate Change Policy Game (2008d) also convey to this analysis. Regions with high abatement costs attracts investments in dirty capital as the policymaker chooses a lower reductions for given emission flows. High abatement cost regions become pollution havens with high cash-dividends. The introduction of precautionary preference of the policymaker in region j will increase emissions reductions, further lowering firms’ shadow price of capital, which induce a response by investors to increase cash-dividend at the cost of lower reinvestment in the region. Not surprisingly, a region hosting a non-precautionary policymaker (low 1/0j) will attract greater reinvestments in regional firms.
An uncertainty averse (high 1/0j) policymaker j faces a greater expected Nash shadow cost of CO2 compared to a policymaker k = j with (1/0k < 1/0j) which induces greater levels of qj for given (Ej) levels. For example, a one percentage increase in 1/0j results in one percentage shift upwards in expected shadow cost of CO2 trajectory for given CO2 levels which in turn alters the policy rule q jt (Ej) by one percentage greater abatement qjt for given emissions levels (Ejt). As 0j ^ 0, policymaker j’s expected Nash shadow cost of Mt increases toward infinity and the policy maker bases policy on the Amox climate model. A low 0j also makes expected Nash shadow cost of M highly sensitive (quadratic) to the variance a of underlying climate sensitivity factors and thermal capacity of atmosphere and upper ocean R1 in the climate model, resulting in significant increases in feedback Nash emissions reduction strategies qjt V G [1, n]. M
3.4. Cooperative Policy Rules
The analysis of cooperative solutions in this game are left to forthcoming studies. However, there are several interesting cooperative structures possible in the game: (i) cooperation between policymakers across regional borders, (ii) cooperation between the policymaker and the regional investors within each region, (iii) cooperation between investors within each region, (iv) cooperation between investors across regions while policymakers choose policies individually, (v) bilateral coalitions where policymakers cooperate across regions in one coalition while investors cooperate across regions in another coalition, (vi) global cooperation between all policymakers and all investors on policies and allocation of investments.
Due lack of space we only identify coordinated regional policy rules in case (i) when policymakers’ form a ‘grand coalition’ and investors play individual Nash investment rate strategies, assuming that the payment streams fulfill individual rationality and some time consistent burden-sharing (Yeung and Petrosjan, Subgame Consistent Cooperative Solutions in Stochastic Differential Games (2004a) and Yeung and Petrosjan, Cooperative Stochastic Differential Games (2006b)). The cooperative policy rules qjt (Ejt) are obtained by maximizing the sum of policymakers’ payoffs given the dynamics in (5), the climate model in (6) - (9) and that each investor i individually solves for optimal investment rate trajectories ujt. The expected joint payoff to all policymakers j G [1,N] is
max min e
qjt Xjt
^j ymjt + ^
0 0 — 1 m=1 I
j—1 m—1
+ ~y > e-ptdt (45)
The preferences of time p and robustness 1/0 are assumed to be weighted averages of policymakers’ individual preferences i.e. p = Yj=i ojPj and 0 = Yj=i oj0j with £j=i = 1, assuming that not only instantaneous payoffs but also preferences of time and robustness are reflected by policymakers’ bargaining power in negotiations. From appendix A.3, the grand coalition’s optimal policy rules are
£?=i
3 =1 2 3 \RlCt^2
P + R2 (1------------------—'
^ #1T2 ' 1+ PT2-
2cjj Mo0(p +
Eijt ept > 0
(46)
AO
2^j=1 2 w3
0
(47)
2
a
j = 1, 2,...,N (48)
The optimal abatement levels and updating of climate sensitivity will have the same structural form as (38) and (42) but global optimal shadow cost of atmospheric concentration rate dW/dMt and global mean temperature dW/dTt will be different as they carry the total marginal cost that each firm incurs on all regional natural capital stocks via the climate model dynamics. By symmetry with (43), A(Mt) tells how the global social planner, given 0, would update equilibrium climate sensitivity in (6). Substituting the optimal polcy rules in (42) and (43) and (46) and (47) in the dynamic system (5) and (13) and solving gives the dynamics that corresponds to the Nash equilbrium and the cooperative solution respectively and is a straightforward operation left for the reader.
4. Concluding Comments
In the article ‘Why is Climate Sensitivity So Unpredictable?’ Roe and Baker (2007) analyze the effects of uncertainty in underlying physical feedback processes on mean outcome and shape of probability distribution of climate uncertainty. They explain why the shape of probability distribution of climate sensitivity gets a thick high-temperature tail and that it is not likely that progress in understanding underlying physical processes will narrow the tail. Having this in mind, we introduced a new approach of uncertainty to integrated assessment modeling by letting policymakers face a set of climate models with different equilibrium climate sensitivity, and without knowing which of them is correct or which probability distribution of equilibrium climate sensitivity is correct. The climate models were generated by perturbing a continuous-time version of the climate model in Nordhaus (1992) and Nordhaus and Yang (1996) making it statistically impossible for policymakers to infer correct future probability distributions about climate sensitivity by using current data. These conditions of uncertainty better describe the real conditions that policymakers today actually are facing as shown in Roe and Baker (2007), Allen and Frame (2007) and IPCC, Climate Change 2007: Working Group I Report The Physical Science Basis (2007c). Our policymakers update their policy rules in the light of new climate data according to optimal rules which will change the predictions of equilibrium climate sensitivity and alter the climate model continuously
as the game evolves. This is accordance with IPCC, Climate Change 1995: Economic and Social Dimensions of Climate Change. Contribution of Working Group III to the Second Assessment of the Intergovernmental Panel on Climate Change (1995a) which stated ‘The challenge is not to find the best policy today for the next 100 years, but to select a prudent strategy and to adjust it over time in the light of new information’. On the other hand, policymakers using ordinary stochastic optimal control and the expected utility criterion, would stick to the original climate model used as base for policy decisions, despite discovery of new climate data that would reject the model.
In the gap of knowledge concerning climate sensitivity, we introduced precautionary preference among policymakers. Experiments like Ellsberg’s, show that Savage’s Sure-Thing Principle is badly supported in real decisions. People rather tend to choose the alternative where there is more certain information about outcome. Following the technique by Hansen and Sargent (2001), we introduced a preference for robustness 1/0 among policymakers, where robustness means a preference to choose policy trajectories that are optimal in case a climate model with high equilibrium climate sensitivity should be found to be correct in the future. A policymaker with a preference for precaution faces a greater expected shadow cost of atmospheric CO2 which results in stricter policy rules q*(Et) in the game. The stronger the precautionary preference, the policymaker’s expected shadow cost of CO2 trajectory shifts upwards, the stricter are policy rules q*(Et), i.e. greater abatement for given emissions. This lowers investors’ expected shadow price of physical capital and hence, investors’ responses are to increase current cash dividends at the cost of lower reinvestments in the region.
A low preference of time (p) and/or a high preference for precaution (1/0) among policymakers shifts expected shadow cost of atmospheric CO2 upward moving the non-cooperative Nash trajectories of emissions reduction closer toward cooperative trajectories and they may even coincide at finite upper capacity bounds of abatement effort. Hence, a revalue of time preferences and precautionary among policymakers in general toward lower time preferences and higher precautionaries will not only result in stricter policy rules but also support efforts to reach cooperative outcomes.
Evident forthcoming studies would be simulations using empirical data in this game model with asymmetric profit and cost functions and varying precautionary preferences across policymakers. The major contribution of this paper though, should rather lie in the opportunities to analyze cooperative structures given climate model uncertainty. The structural uncertainty in this model was embedded in the integrated assessment differential game in Hennlock, Optimal Policy Rules in an Integrated Climate-Economy Differential Game (2007b), using a continuoustime version of the climate model used in Nordhaus (1992) and Nordhaus and Yang (1996) such that analytical closed-loop solutions can be defined analytically. The advantage of analytical solutions, compared to numerical simulations, which in general have poorer reliability, is not only that they allow for deeper understandings in for example sensitivity analyzes, but they also make analyzes of cooperative structures possible when it comes to the study of fulfilling conditions for individual rationality and subgame consistent payment streams, which are conclusive for the stability of long-term cooperative solutions.
5. Appendix
5.1. Appendix A.1 Investor i £ [1,n] Optimization Problem
Every investor i maximizes payoff function (14) subject to the dynamic system (5) to (13) and > Vi <G [1, n] and Vj <G [1,N] in definition 1. Investor i’s stochastic optimal problem is:
subject to
max
uiht J0
i” i , 1 s ^kiVikt)2 \ -Piti,
e I (1 — Uikt)yikt--------^------- ( e dt
Jo k=i ^ >
11/2 k1/2 — A -fr- •*
dt
(49)
(50)
dx.
1
i/2'
if,-
1/2 (Tt — To)
j —
'jt
dt
(51)
dM
n N
EE Cj (Emkt qmkt) i2Mt
m=1 7=1
dt
(52)
dT = f + dBWMt/M0 + I/di + 0tdi _ _ Ä2 № _ ft)di | (53)
Ri V 2 T2
(54)
i = 1, 2,..., n j = 1,2,..., N Definition 2. If there exist n value functions Vi(k, x, M, T, T,t) where
k = (fcii , fci2, . . . , fciW, &21, &22, . . . , k2N, fcni , kn2, . . . , )
and x = (xi, X2,..., x j) that satisfy
Vi(fc, x ,M,T,T,t) = J o k=1 ikt
a. r¿{tt-%*)»»
Jo r=T I Eifct J
7=1
(55)
(56)
(57)
for strategies u|jt(fcj, t) C R1 Vi € [1, n] and Vj € [1,N] which satisfy the state equations,
dfc,'
r1/2k1/2 _ A..k..t Jijt kjt kjt
dt
(58)
r
.
dxj =
1
i/2'
xjt
K.
1/2 &j(Tt - To)
1/2
xjt
xjt
dt
(59)
dM =
n N
EE (Emk ^mfc qmkt) ^Mt
m=1k=1
dt
(60)
dT = 4- f + dBWMt/M0 + I/di + Gtdi _ _ ^2 (Tt _ ft)di j (61)
R1 V 2 T2
(62)
i = 1, 2,..., n j = 1,2,..., N The feedback Nash controls strategies
(63)
rj = {ujt(kijt)} Vi G [1,n] Vj G [1,N]
(64)
provide a feedback Nash equilibrium solution of the game defined in (49) to (55) given (19) to (25) (Basar and Olsder, 1999).
The value functions in definition 2 satisfy the partial differential equation system (65) - (66). Using (49) - (55) and definition 1 and 2 yield the dynamic programming problem (Fleming and Richel, 1975) of investor i
dVi v'1, i/1 \ cik{likt)2 1 -p t
—= max > < (1 — Uikt)Vikt--------f p
dt lktWlkt Eikt J
k=1
n N
+ EE
<9fc
+E —
dxk k=1 k
m=1 k=1
■\f&kt \
mk
11/2 k1/2 X k Jmfct kmkt — °mk kmkt
ft(Tt - To)
%kt
+
dM
N
EE Cj (Emkt ^mkqmkt) ^Mt
+
<9T
m=1 k=1
1 / a Xjt \J Mt / Mo + i/
#i V a\/2
R2
+ Ot - AT - — (Tt - Tt)
T2
1 d2 Vi
V2Mt
+-
6»f
2 dT2
— (Tt-ft) R2 V T2
(65)
r
j
i = 1, 2,
, n j = 1,2,..., N
(66)
The feedback Nash controls strategies
rjt = K-(kjt)} V* G [1,n] and Vj G [1,N] (67)
are given by maximizing the partial differential equations (65) with respect to (67) for n players and solving for the feedback Nash control variables.
2 1/2
U*ijt=(\w~) ~^~e2P't V*e[1>n] and Vi e [1, A^] (68)
The partials derivatives in (68) are investor i’s expected feedback Nash shadow price of fcj-. In order to identify shadow price paths, the n value functions Vi(k, x, M, T, T, t) that satisfy definition 2 and the partial differential equation system formed by (65) - (66) must be identified.
Proposition 1.
Vi(k, x, M, T, T, t) = (69)
The value functions Vi G [1,n] satisfy definition 2 and the partial differential equation system formed, by system (65) - (66).
Proof. The values of the undetermined coefficients (auj, bj, d, e^,/,gj) for all investors i G [1, n] and regions j G [1, N] are determined by substituting (83) into the partial differential equations (65) for all i G n forms the n indirect Isaacs-Bellman-Fleming equations of investors i = 1, 2,..., n. The coefficients of the indirect values functions in proposition 1 are then determined by the block recursive equation system
PiOiii = Phi - ^ (70)
4Cj 2
+ d% = 0
2Cij
Pi^imj — 2 (^1)
i j & dj £j nmj V^mj ^mj w , ■
2Cmj
. = (72)
Pidi = —— ( —I - diü (73)
^ M0ej\Ria^] * V '
A bij ÿj A R2 , 1
PA — - 2^ - et - et + /j (74)
z—, 2 Ri R1T2 T2
j=i
Pifi = ei-^-- fi- (75)
R1T2 T2
& (p + - L-^) )
¿iij
Pi + ^ij /2
aimj = di£k<pmj(f>mj [ 1 + k^m3 I Vm^î (77)
mj
2
ei / CT \ 1
2^j=i 2 w3
Pi + lt + Th^i1 ~ T+^)
R2
(76)
bij = 0 (78)
(80)
/* =e* B , , M (81)
RiT2(pi + ^)
i = 1, 2,...,n j = 1, 2, ...,N m G [1,n] (82)
Since bij = 0, then aimj = 0, di = 0 and gi = 0, which further simplify (76). The
coefficients gi in proposition 1 are uniquely determined by the coefficients in (76) -
(82) and (92) - (99).
Substituting the feedback Nash shadow prices and costs into (68) yields the feedback strategies in terms of parameter values:
e
uijt = (tt) ------“172 G t0’1] V* G t1’™] and V-7 G t1’^] (83)
V2^/ kijt
5.2. Appendix A.2 Policymaker j G [1,N] Optimization Problem
Every policymaker j maximizes payoff function (19) subject to the dynamic system (20) to (25) and u> Vi G [1, n] and Vj G [1, N].
Proposition 2.
W(k, x, M, T, T, t) = (84)
✓ n N N \
EE ajrrik k^p k + E bjkXfo dj Al -\- GjT -\- fjT -\- Qj J 6 ^
^m=1 k=1 k 1 /
k=1
The value functions Vj G [1, N] satisfy definition 1 and the partial differential equation system formed, by system (35) - (36).
Proof. Substituting (39) and (43) into the partial differential equations (35) for all j G [1, N] forms the N indirect Isaacs-Bellman-Fleming equations of policy makers j = 1,2,..., N. The coefficients of the indirect values functions in proposition 2 are then
pj ajij = Wj ^ij
(dj r/jj )2 (fijj (f>jj ajjj Sj.
4ci
dj ^j (fiij <f>ij
, f, djÇjrjij(pij<f>ij 'UjÇj'hj 9
2Cij
(85)
ajik^ik , ^ i f-. dkCknik A 7 / •
Pjttjik = ^ djÇkipikÇik I 1 + J / 3
pj bjj = ^j
(86)
(87)
(88)
Pj dj
Mo^j \R1a
— dj
(89)
N
p^' = -E^ A
j=1
^2 „ 1
— 60 —— — Co —-----------------H- Tî —
3Ri 3Rml3T2
(90)
pj fj =
(91)
pj + ^ij /2
(djîjVij)2V>ij
Ac ■ ■
ij
+ dj Cj Vij
(92)
djCk^ik^ik A dkCk^ik \ wi / •
2
e
CT
j
e
a
bjj = j r, (94)
Pi "T" 2K3
bjk = 0 k = j (95)
<h = Tàr[-^-K) (96)
Modj \R±a\/2J Pj +
(97)
¿^7 = 1 2 “j
+ fl7 + 7Ï^(1 “
R 1 ' R 1 T2 1 + pj T2 '
2
f. = e_________________________________________
(98)
i = 1, 2,...,n j = 1, 2,...,N m G [1,n] (99)
The undetermined coefficients in appendices A.2 and A.3 are uniquely defined, and hence, this corresponding feedback Nash equilibrium is unique. The coefficients gj in proposition 2 are uniquely determined by the coefficients in (76) - (82) and (92) - (99).
5.3. Appendix A.3 Global Optimal Policy Responses
The global planner c maximizes the sum of payoff functions (45) subject to the dynamic system (20) to (25) and uj > Vi G [1,n] and Vj G [1,N]. The value function Wc(k, x, M,T, T, t) is:
Proposition 3.
Wc(k, x, M, T, T, t) = (100)
. n N N .
EE acmkkçmk + E bckxk + dM + eT + /T g je pt
'm=1 k=1 k=1 '
The value function satisfy the partial differential equation system formed, by system (35) - (36) but replacing the payoff function in (45).
Proof. Substituting (46) - (47) into the partial differential equations (35) and replacing payoff functions to (45) forms the indirect Isaacs-Bellman-Fleming equations of the global planner. The coefficients of the indirect value function in proposition 3 are then
pacij ---- ^j ^¿j +
(dcCj nij ) ^¿j j ajij ^¿j
2c
¿j
(101)
2
(102)
pbcj = (103)
N bij Ro 1
j=1
Ro 1
p/c = ec----------/c— (106)
Rl T2 T2
</>¿7 (dc^jriij)2(pij
----------;— I Cu-i ----------
P+^ij/2 \ 2cjj
dcCfc^¿fc^¿fc dcCfcnik \ w, , • /moA
= TTW2-i1 + — J VMj (108)
(109)
P + 2 if.
dc = ^('_f_y^_ (110)
Mq0 \Ria\/2 / p +
_ ¿^¿=1 2 “j
e° _ n I _R2_n__________^
^ RlT2' 1 + PT2 '
(111)
/c = ec f2 t (112)
Rir2(p+ ^)
i = 1, 2,..., n j = 1, 2,...,N m G [1,n] (113)
The undetermined coefficients in appendices A.1 and A.3 are uniquely defined, and hence, this corresponding feedback Nash equilibrium is unique. The coefficient gc in proposition 7 is uniquely determined by the coefficients in (76) - (82) and (92) - (99).
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