مدل آب و هوا تعادل انرژی و تعادل عمومی سیاست های کاهش دهنده مطلوب

In a general equilibrium model of the world economy, we develop a two-dimensional energy balance climate model featuring heat diffusion and anthropogenic forcing driven by global fossil fuel use across the sphere of the Earth. This introduces an endogenous location dependent temperature function, driving spatial characteristics, in terms of location dependent damages resulting from local temperature anomalies into the standard climate-economy framework. We solve the social planner's problem and characterize the competitive equilibrium for two polar cases differentiated by the degree of market integration. We define optimal taxes on fossil fuel use and how they may implement the planning solution. Our results suggest that if the implementation of international transfers across latitudes is not possible then optimal taxes are in general spatially non-homogeneous and may be lower at poorer latitudes. The degree of spatial differentiation of optimal taxes depends on heat transportation. By employing the properties of the spatial model, we show by numerical simulations how the impact of thermal transport across latitudes on welfare can be studied.

The economics of climate change are based on the use of integrated assessment models (IAMs) and focus mainly on the impact of greenhouse gasses (GHGs) emissions on global temperature, as well as the impact from increases of global temperature on economic variables such as output and utility from consumption. IAMs are also used for developing long-term projections regarding climate and economic variables, the cost of climate change, and the formulation of mitigation and climate policies.1 The major IAMs which are structured as optimal growth models with a climate component and which are focusing on cost-benefit analysis and policy simulations (e.g. DICE/RICE, MERGE, FUND) simplify the carbon cycle and the climate system considerably, and provide results regarding temperature at a global level. When however the geographical scale is global in terms of temperature and/or damages due to climate change, important aspects of the problem, which are related both to climate science and economics are obscured. In particular there are natural mechanisms which induce a spatially non-homogeneous distribution of the surface temperature across the globe. The main drivers of these mechanisms are heat transportation that balances incoming and outgoing radiation, and the differences among the local heat absorbing capacity, the local co-albedo, which is relatively lower in ice covered regions and which changes with time as global warming tends to diminish the ice-cups. However, these mechanisms are not accounted for by cost-benefit oriented IAMs, and as a result the existing variations in local temperatures are not modeled and therefore they do not affect local damages and their dynamics.2 This, however, may introduce a serious bias into the result and the policy prescriptions, since it is clear that climate change is going to have impacts with profound regional differentiation across the globe. From the point of view of climate science these IAMs are zero-dimensional models since they do not include spatial aspects such as heat diffusion. This is not however the case for the one- or two-dimensional energy balance climate models (EBCMs) developed by climate scientists which model heat diffusion across latitudes (one-dimensional) or across latitudes and longitudes (two-dimensional) (see e.g. Budyko, 1969, Sellers, 1969 and Sellers, 1976; North, 1975a and North, 1975b; North et al., 1981, North et al., 1983, Kim and North, 1992 and Wu and North, 2007).3 One-dimensional EBCMs predict a concave temperature distribution across latitudes with the maximum temperature at the equator. This non-uniform temperature distribution is important for understanding the so called temperature anomaly which is the difference between the temperature distribution at a given benchmark period and the current period. Data indicate (Hansen et al., 2010) that since 1880 the anomaly has been relatively higher in high latitude zones, relative to zones around the equator, which suggest spatial non-uniformity in the distribution of temperature over time. In previous papers, Brock et al. (forthcoming), Brock et al. (2013a), we have explored the properties of the one-dimensional energy balance climate models, showing how they could be coupled to economic growth models in a tractable manner. The analysis revealed new complexities stemming from climate science which also proved to have potential qualitative effects on optimal mitigation policy. The results that could be derived from these models, were however limited in the geopolitical sense, since the spatial analysis remains constrained to interaction across latitudes as opposed interactions across countries or regions. In the present paper we will take the next natural step following the one-dimensional case, by deriving a two-dimensional model. Here the second dimension will allow for a continuous representation of local temperature anomalies for every latitude and longitude across the surface of the Earth. As with our previous papers, we will be working with a dynamic climate-economy growth model involving both heat diffusion and albedo differentiation across different geographical locations. The solution method is similar to the one-dimensional case but involves finding an alternative orthogonal basis set for the Laplacian of the heat diffusion equation. Due to the added dimensionality this basis set will differ from the Legendre polynomial basis that we used in the one-dimensional models featured in Brock et al. (forthcoming), Brock et al. (2013a), and will now involve an expansion in terms of so called spherical harmonics which are the eigenfunctions of the solution to the two dimensional Laplacian. We believe that this approach which integrates solution methods for two-dimensional spatial climate models, with methods of solving economic models, can help to push the frontier of integrated assessment modeling further by showing how solution methods involving in complex climate models such as Earth system models or general circulation models can be made tractable also when coupled to economic models.4 Since EBCMs most likely are new to most economists we have chosen to focus on the basic general equilibrium properties of the derived model under different assumptions regarding international capital markets and their integration. This approach shows of the basic welfare properties of the model and how the fundamental theorems of welfare economics apply within the context of our energy balance climate economy framework. The approach also makes clear how the optimal carbon tax rates should be chosen in order to implement a social planning problem and should thus constitute an important first step for economists working with these types of models. The economic part of the model is an infinite horizon Ramsey-type model allowing for basic heterogeneity among consumers and firms at each respective geographical location on the surface of the Earth identified by its latitude and longitude coordinate. We have chosen to look at two polar cases here. The first two cases concern economies that are either completely open with respect to trade and transfers or completely closed in the sense of being autarkic. In the first case when international trade and transfers are available capital returns and interest rates will be equal across locations. In this case we show that the optimal carbon tax must also be uniform or equal across locations. In the second case, of autarkic economies, interest rates and tax rates will generally differ across latitudes. In this case if some countries are rich and some poor and international transfers are assumed to be restricted across latitudes, this will in general imply that optimal carbon taxes will be spatially differentiated. This result that in the absence of international transfers a spatially uniform optimal tax rate is in general not possible was first noted by Chichilnisky and Heal (1994). Our result provides new insights into this issue by characterizing the spatial distribution of fossil fuel taxes and linking the degree of spatial differentiation of optimal fossil fuel taxes to the diffusion of heat across latitudes.5 Using these two polar cases we are able to reveal aspects of how heat transport across latitudes matters regarding the prediction of spatial distributions and the corresponding temporal evolutions of temperature, damages and optimal mitigation efforts. Our two-dimensional model further allows us to show how heat diffusion across geographical zones impacts on the size of the spatial differentiation of fossil fuel taxes between poor and wealthy regions. In the numerical section of the paper we finally attempt a tentative calibration exercise and show how the autarkic economy can be solved numerically in order to obtain graphical representations of e.g. damage and temperature distributions. To sum up, the main objective of this paper has been to introduce the economics profession to spatial EBCMs with heat transport as a potentially useful tool for studying the economics of climate change relative to alternative zero dimensional models. By deriving conditions for the spatial distribution for optimal taxes for the two-dimensional coupled climate economic model, we show how the spatial EBCMs can contribute to the current debate regarding how much to mitigate now, whether mitigation policies should be spatially homogeneous or not, and how to derive geographically specific information regarding damages and policy measures.6 This paper is structured as follows. In Section 2 we present the two dimensional energy balance climate model which incorporates human impacts on climate resulting from carbon dioxide accumulation due to the use of fossil fuels, that blocks outgoing radiation. In developing the model we follow North et al. (1983) and use his notation.7 We use solution methods based on spherical harmonics for partial differential equations (PDEs), approximating the solution by a finite set of ordinary differential equations (ODEs). The approach will be used to solve, and numerically approximate location specific temperature and damage functions. Section 3 couples the spatial EBCM with an economic growth model, where a finite stock of fossil fuel is an essential input along with capital and labor. Fossil fuels are extracted by fossil fuel firms which pay taxes on profits and/or taxes per unit of fossil fuel extracted. We solve the model for the social planner and for the competitive equilibrium with taxes. We derive the optimal taxes and their temporal profiles. In Section 4 we use approximate solutions, and simulate the model and in order to derive explicit numerical solutions. Here, we apply an extension related to a paper by Alexeev et al. (2005) which captures a process known as polar amplification which is the empirical observation that for a given amount of global warming, local warming tends to be amplified in the polar regions. This shows how our framework can also quite easily incorporate recent developments in the climate community. The last section concludes.

In this paper we have developed a model of climate change consisting of a two-dimensional energy balance climate model which we coupled to a model of economic growth. We believe that modeling heat transport in the coupled model is the main contribution of our paper since it allows, for the first time, as far as we know, the derivation of latitude dependent temperature and dam- age functions, as well as optimal mitigation policies, in the form of optimal carbon taxes, which are all determined endogenously through the interaction of climate spatiotemporal dynamics with optimizing forward looking economic agents. We derive Pareto optimal solutions for a social planner who seeks to implement optimal allocations with taxes on fossil fuels and we show the links between welfare weights and international transfers across locations and the spatial structure of optimal taxes. Our results suggest when per capita consumption across latitudes can be adjusted through costless transfers for any set of non-negative welfare weights, so that marginal valuations across latitudes are equated, or alternatively that transfers are zero due to Negishi weighting, then optimal carbon taxes will be spatially homogeneous. On the other hand when marginal valuations across latitudes are not equated, due to institutional/political constraints, optimal carbon taxes will instead be spatially differentiated. We show that if economies are autarkic and the planner is not constrained by Negishi weighting, then taxes on fossil fuels could be lower in relatively poor geographical areas. The degree of geographical tax differentiation will depend on the heat transport across latitudes. Without appropriate implementation of international transfers, and without Negishi weights that keep the existing international distribution invariant, carbon taxes will be latitude specific and their sizes will depend on the heat transfer across locations. We also show how to derive numerical results for the optimization problem of the unconstrained planning problem when economies are autarkic. These results show how the optimization model generates surface plots of temperature and damages across latitudes and over time and how these depend on the diffusion of temperature. The climate module of our model is sometimes referred to as a surface EBCM where the impact of oceans is reflected in the carbon decay parameter m, with no further modeling of the deep ocean component is undertaken. Further extensions of our simple one-dimensional model to richer climate models (e.g. Kim and North, 1992 and Wu and North, 2007) with a ocean and with simple atmospheric layers added and where tipping phenomena are possible may help to understand results like those of Challenor et al. (2006), which found higher probabilities of extreme climate change than expected. Challenor et al. (2006) suggest several reasons for their findings including “The most probable reason for this is the simplicity of the climate model, but the possibility exists that we might be at greater risk than we believed”. We emphasize that we are still doing what economists call a “finger exercise” in this paper where one deliberately posits an over simplified “cartoon” model in order to illustrate forces that shape, for example, an object of interest like, a socially optimal fossil fuel tax structure over time and space that might be somewhat robust to introduction of more realism into the toy model. For example, we believe that the interaction of spatial heat transport phenomena and difficulties in implementing income transfers (or their equivalent, e.g. allocations of tradable carbon permits) will play an important role in determining the shape of the socially optimal tax schedule over different parts of space in more complex and more realistic models. Our simple model is useful in making this type of point under institutions where income transfers are possible and where they are politically infeasible, i.e. essentially impossible. The two-dimensional model allows the exploration of issues which cannot be fully analyzed in conventional zero-dimensional models. In particular two-dimensional models with spatially dependent co-albedo allow the introduction of latitude dependent damage reservoirs like endogenous ice-lines and permafrost. Since reservoir damages are expected to arrive relatively early and diminish in the distant future, because the reservoir will be exhausted, the temporal profile of the policy ramp could be declining or U-shaped.57 A U-shaped policy ramp could be explained by the fact that high initial damages due to the damage reservoirs will start declining as the reservoir is exhausted, giving rice to a declining policy ramp, damages from the increase of the overall temperature will dominate causing the policy ramp to become increasing. Some of these aspects were explored in Brock et al. (forthcoming) in the context of a one-dimensional EBCM. However, we believe that some of the most important areas for future research are the extensions of our framework to precipitation fields as well as temperature fields and to robust control. Our two dimensional energy balance model has been deliberately constructed to allow easy integration with the combined two dimensional energy balance moisture balance model of Fanning and Weaver (1996). This is so because our series expansion methods to approximate solutions to the model apply also to Fanning and Weaver (1996) model. One of the main gaps in received work by economists in the climate area is neglect of the precipitation and evaporation part of climate dynamics as well as ignoring transport phenomena in the dynamics of the temperature field across the globe. Many important sources of damages are more related to precipitation and evaporation than to temperature (e.g. Elliott and Fullerton, 2013 damages to agriculture). Our two dimensional framework allows a direct extension to robust control which, in turn leads to identification of potential hot spots at various locations on the globe. We have already worked out some of the appropriate conceptualization of different kinds of hot spots and we have worked out quite a bit of the analytics ( Brock et al., 2012 and Brock et al., 2013b). It is essential that our approach be extended to incorporate model uncertainty and, hence, some form of robust control to take into account deep uncertainty in the dynamics of damages, climate dynamics, and especially economic dynamics where long term prediction is difficult due to the difficulties in predicting long term socio-economic response to climate change. So this is an extension we will undertake in the very near future. Incorporation of robust control into discussions of geoengineering ( Manousi and Xepapadeas, 2013) is necessary in order to take into account deep uncertainties about potential effects on temperature and precipitation fields. Climate scientists have expressed serious concerns in this direction ( Bala, 2012, Bala et al., 2008, Bala et al., 2011, Tuana et al., 2012 and Goes et al., 2011). We share their concerns. Our framework seems like a useful platform for a robust control analysis of geoengineering, especially if our energy balance model is extended to an energy balance-moisture balance model like that of Fanning and Weaver (1996). We also need to extend our work to deal with issues raised by potential threshold effects and other concerns raised by climate scientists that must be addressed in formulation of robust climate policies ( Keller et al., 2008 and Hall et al., 2012). Another important direction for future research is extending our work to include more modeling of basic economic phenomena that we have left out in order to focus more on spatial climate dynamics. Desmet and Rossi-Hansberg (2012) is a recent paper that has costly trade, diffusion of technology, modeling of innovations, migration of labor, that we lack. Although they work on a hemisphere with latitude belts, they do not have heat transport climate dynamics like we do. They have a reduced form carbon cycle that includes the upper and lower ocean which we do not have. However we can add explicit coupling of spatial models of upper ocean and lower ocean to the dynamics of the atmospheric temperature field as in Rose and Marshall (2009) and our analytical methods will still apply. Desmet and Rossi-Hansberg (2012) are not able to address the potential impact of human activities on spatial phenomena such as amplification of the dynamics of the temperature field at high latitudes ( Alexeev et al., 2005 and Alexeev and Jackson, 2012) much less the potential impacts on spatial dynamics of precipitation and evaporation fields in models like ( Fanning and Weaver, 1996 and Weaver et al., 2001). But their treatment of the economic side is more thorough than ours. Our solution methods for the climate module using spherical harmonics and our closed form solutions for the economic module (under the same set of sufficient conditions for existence of a closed form solution) carry right over to Fanning and Weaver (1996) energy balance moisture balance model as well. Explicit modeling of the spatial dynamics of components of climate as we are doing combined with the explicit modeling of important economic components as in Desmet and Rossi-Hansberg (2012), more sectors as in Engström (2012), Hassler et al. (2012), is urgently needed. This kind of work will allow us to do a better job of accounting for implementation issues involving carbon pricing, for example, the very important issue of potential carbon leakage (Elliott and Fullerton, 2013).