BackgroundThe ModelWelfare, Version 1The Climate EconomyThe CodeThe QuestionsA model on numerical DP and stylized integrated assessment1 BackgroundThe problem replicates a stylized version of William Nordhaus’s (2008) DICE modelas a recursive dynamic programming model. DICE is an open source integrated as-sessment model. Basically, it is a Ramsey growth model that is en r i ched by emissions,pollution stock, and temperature related damages to world GDP. Some of the keyexogenous parameters in the original model are time dependent. We omit the time ofthese parameters. In contrast to DICE, t h i s problem allows you to model uncertaintyin a coher ent way. Uncertainty is persistent and affects decisions in every period.Uncertainty enters as an iid shock on climate sensitivity, which characterizes thetemperature response to the radiative forcing caused by a doubling in atmosphericCO2concentrations with respect to the preindustrial level.1The climate sensiti vi typarameter is one of the key unknowns in modeling global warming because of sev-eral feedback processes are involved when translating CO2 increase into t em per at u r echange. We have an annual time step and face an infinite planning horizon. Themodel sol ves for the optimal expected trajectories of the climate economy controllingfor emissions and investment.2 The Model2.1 Welfare, Version 1We use the intertemporall y addit i ve expected utility standard model. It evaluatesscenarios by aggregating instantaneous welfare linearly over time and aggregatinglinearly over risk states by taking expected values. Thus, th e social plann e r maximizesU =XtβtLtuCtLt=XtβtLtCtLtρρ= L1−ρtXtβtCρtρwhere Ltis th e populatio n size and β i s the discount factor stemming from theadditional assumption of stationary preferences. The assum p t i on of a power utilityfunction retur n s to us a constant elasticity of intertemporal substitution. We candrop Ltin the welfare eq u at i on because it comes down to a multiplicative constantof welfare having no real effects. Antici p at i n g that the clima t e-e con o my equationsintr odu ced in section 2.2 will depend on the two state variables capital Ktand CO2pollution stock Mtand will be controlled by t h e consum p t i on - i nvestment decisio n andthe abatement rate µtwe can wr it e the according dynamic p r o gr am m i n g equation asV (Kt, Mt) = maxCt,µtCρtρ+ β V (Kt+1, Mt+1) ,1Obviously the iid nature is not quite realistic bu t saves us additional state variables.1A model on numerical DP and stylized integrated assessmentwith control variables consumption ctand emission control rate µt. Because wewill assume a constant population, we can substitute per capita consumption ctbyaggregate global consumption Ct.22.2 The Climate EconomyThe decision maker max i m iz es h i s value functions under the constraints of th e fol -lowing stylized m od el of a climate enr i ched economy. The model is largely a (very)reduced form of Nordh aus (200 8) DICE-2007 model. All parameters are characterizedand qu a ntified in table 2.2 on page 4. The economy accumulates capital according toKt+1= (1 − δK) Kt+ Yt− Ct,where δKdenotes the depreciati on rate, Ytdenotes net production (net of abatementcosts and climate dam age) , and Ctdenotes aggregate global consumption of producedcommodities. Instead of trying to model the full carbon cycle, which would be verycostly i n terms of stock variables, we assume an exponential decay of CO2in theatmosphere at rate δMMt+1= Mt(1 − δM) + Et.The variable Etcharact er i zes overall yearly CO2emissions. We use values for Mtand Etcharact er i zing CO2only. However, at the given level of abstraction a rescaledversion of Mtcould be thought of as representing greenhouse gases in CO2equiva-lents more generally. Emission are composed of industri al emission (first term) andemissions from land use change an forestr y B1(which a re assumed to be constant)Et= σ (1 − µt) (AL)1−κKκt+ BThe constant σ specifies the em is si on s to GDP ratio and the control variable µtisthe abatement rate. The constant AL r ep r ese nts effective labor (technology level andlabor). The product (AL)1−κKκtis th e global gross product (gross of abatementcosts and climate damage) . The net pro d u ct is obtained from the gross product asfollowsYt=1 − Λ(µt)1 + D(Tt)(AL)1−κKκt=1 − a1µa2t1 + b1Tt+ b2Tb3t(AL)1−κKκtwhereΛ(µt) = Ψµa2t2The change res ul ts in an affine transformation of th e value function that leaves the decisionproblem unchanged.2A model on numerical DP and stylized integrated assessmentcharact er i zes abatement costs as percent of GDP depending on the emission controlrate µtandD(Tt) = b1Tt+ b2Tb3tcharact er i zes the c li m at e damage as percent of GDP de pen d i n g on t he t emp e ra t u r edifference Ttof current with respect to preindustrial temperatu r es.In the m od el , temperature is an immediate response to the radiative forcing causedby th e stock of CO2in the atmosphereTt= stlnMtMpreindln 2+EFη. (1)where stdenotes climate sensitivity, i.e. the temperature response to a doubling ofCO2in the atmosphere with respect to preindu s t r ia l concentrations. The second termin equation (1) represents external forcing that is caused by other greenhouse gas es.The model uses a climat e sensitivity of st= s ≈ 3. Finally the syst em is bound bythe followi n g constraints on th e co ntrol var i ab l es consumption0 ≤ Ct≤ [1 − Λ(µt)]Y∗tand abatement0 ≤ µt≤ 1 .The constraint on consumption uses gross output less climate damages Y∗t=(AL)1−κKκt1+D(Tt).Rewriting the constraints in terms of abatement expenditure Λtrather than the abate-ment ra te µt= (ΛtΨ)1a2makes th e ‘r i ght hand side’ constraints linear in the controlsCt+ ΛtY∗t≤ Y∗tΛt≤ Ψ .As linear constraints are preferr ed by num er i cal solvers, we actually use Λtratherthan µtas the control variable in the num e ri ca l implementation of this model.3 The CodeThe model is an implementation of t h e abstract procedures we discussed in the classwhen introducing som e of the mechanics of solving numerical problems. For eval-uating the Chebyshev polynomials we use t h e package by ?. The same is true forcalculating the optimal nodes wh er e to approximate the value function and for fittingthe new coefficients. You can downl oad the package from33Of course you are happily invited to add the Matlab code for the according procedures to