BOUNDED UNCERTAINTY AND CLIMATE CHANGE ECONOMICS. Christopher Costello, Andrew Solow, Michael Neubert, and Stephen Polasky

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1 BOUNDED UNCERTAINTY AND CLIMATE CHANGE ECONOMICS Christpher Cstell, Andrew Slw, Michael Neubert, and Stephen Plasky Intrductin The central questin in the ecnmic analysis f climate change plicy cncerns the degree t which current cnsumptin shuld be reduced t avid r mitigate future csts f climate change. In answering this questin, tw issues that must be addressed are hw t discunt lsses incurred far in the future and hw t deal with substantial uncertainty abut these lsses. Much f the discussin stimulated by the Stern Review n the Ecnmics f Climate Change (Stern, 2007) revlved arund the first f these issues (Dasgupta, 2007; Nrdhaus, 2007). This paper fcuses n the secnd. Althugh there remains little if any uncertainty abut the reality f anthrpgenic climate change, substantial uncertainty surrunds key aspects f it. Fr example, under ne emissins scenari frm the mst recent assessment by the Intergvernmental Panel n Climate Change (IPCC, 2007), there is a 2/3 prbability that mean glbal surface temperature will increase by C by 2100 with a remaining 1/3 prbability that the increase will fall utside this range. The crrespnding ranges f welfare effects and sciety s willingness t pay t avid them are likely t be very wide and there is a need t balance the risks f under- and ver-reacting. The standard ecnmic calculus fr decisin-making under uncertainty rests n cnsideratin f the expected discunted welfare r utility change. In the face f risk aversin, expected discunted utility can be sensitive t the upper tail f the prbability 1

2 distributin f future temperature change. This pint has been made mst frcefully by Weitzman (2008), wh argued that this calculus essentially falls apart in the case f climate change. Briefly, using a theretical result f Geweke (2001), Weitzman shwed that the cmbinatin f a heavy-tailed prbability distributin fr temperature change and a cmmn mdel f risk aversin implies an infinite expected discunted welfare lss. Althugh Weitzman was inexplicit abut specific plicy implicatins f this result, he criticized analyses such as Nrdhaus (2008) pinting t a mderate respnse. A necessary (but nt sufficient) cnditin fr Weitzman s result is that the upper tail f the distributin f temperature change is unbunded. The purpse f this paper is t explre the effect f placing an upper bund n temperature change. Weitzman argued strenuusly against this kind f truncatin, claiming that the truncatin pint must perfrce be arbitrary and that the results wuld therefre be highly sensitive t this arbitrary chice. As discussed belw, we disagree with the first f these pints and we shw that - whether r nt the chice f truncatin pint is arbitrary - the secnd pint is nt crrect. Mdel We next sketch ut the mdel under which the analysis will prceed. This mdel is highly stylized. Stylized mdels are used in ecnmics, atmspheric science, and ther fields t develp and sharpen qualitative insights. Fr example, Re & Baker (2007) used a stylized mdel f climate feedback t explain the difficulty f reducing 2

3 uncertainty abut temperature sensitivity. The mdel utlined here mirrrs the general set-up f Weitzman and thers. Let T ( dente the temperature increase at time s in the future. We assume that: τ T ( = s s τ 0 s s s < s (1) Under this mdel, temperature increases linearly until time s at which pint temperature levels ff at a final increase f τ. We will assume that s is knwn and fcus n uncertainty in τ. Ading a Bayesian perspective, this uncertainty can be expressed thrugh a prbability density functin f (τ ), abut which mre belw. Let C ( be cnsumptin at time s, nrmalized s that current cnsumptin is 1. In the absence f temperature change, cnsumptin increases at rate g. In this mdel, increasing temperature adversely affects cnsumptin, e.g. thrugh adversely affecting health r by decreasing ecnmic prductivity. We assume that the prprtin f cnsumptin retained is: 2 λ( = exp( β T ( ) (2) s that cnsumptin is given by: 2 C( = exp( g s βt ( ) (3) 3

4 Ecnmists measure the welfare that sciety gains by cnsumptin thrugh a utility functin. Amng ther things, the shape f the utility functin determines the degree f risk aversin held by sciety. A risk averse utility functin implies that uncertainty ver future temperature is cstly and wuld lead us t reduce emissins tday, even if the expected cst f climate change is mdest. Fllwing Weitzman and thers, we ad the cnstant relative risk aversin (CRRA) utility functin: 1 η C U ( C) = (4) 1 η with η 1. When η = 1, utility takes the limiting frm lg C. Under this mdel, the percentage change in utility assciated with a fixed percentage change in cnsumptin is independent f the level f cnsumptin. Finally, the discunt rate is assumed t fllw the s-called Ramsey Rule: r = δ + η g (5) where δ > 0 is the pure rate f time discunting (Dasgupta, 2007). The uncertainty in τ prpagates t utility. Let U ( C( τ ) be certain utility at time s fr fixed τ. The expected discunted utility is given by: 4

5 EU U ( C( τ )exp( r ds = f ( τ ) dτ 0 (6) The inner integral represents discunted utility fr fixed τ and the uter integral averages this ver the distributin f τ. The absent bunds n the uter integral crrespnd t the range f τ ver which f (τ ) is psitive. A central questin cncerns the willingness f sciety t pay t avid the expected lss in utility assciated with climate change. Fllwing Weitzman (2008), ne way t measure this is by the fractin θ f cnsumptin that sciety wuld be willing t freg in perpetuity t avid this lss in utility. Let: 0 V ( θ ) = U ((1 θ ) C( 0) exp( r ds (7) be the certain discunted utility if utility is reduced by a factr θ at each time and climate change is avided (i.e., τ = 0 ). The imal value θ f θ is fund by equating (6) and (7). Sme results The mdel utlined abve includes an ecnmic cmpnent and a climate cmpnent. As a base case n the ecnmic side, we take δ = 0, g = 0.015, and η = 2, which imply a discunt rate f 3%. The specificatin f these parameters was discussed by Dasgupta 5

6 (2007) in the cntext f climate change plicy. The IPCC (2007) reprted a range f fractinal lss in cnsumptin f 1-5% fr a warming f 4. Our chice f β = gives a crrespnding value f 3%. Stern (2007) referred t the pssibility f ecnmic lsses f up t 20% f GDP. Fr this mdel, this wuld ccur with a warming f arund 11. On the climate side, we calibrate the base case by referring t the A1F1 scenari f the IPCC (2007). Under this scenari, which is the mst pessimistic f the IPCC, the best estimate is a warming f 4 C between the baseline perid and with a likely range f C. T capture this scenari in a rugh way, we take s t be 100 years in the future and assume that τ has a Cauchy distributin, shifted t have a mde at 4.4 C and truncated n the left at 0 C. The Cauchy distributin which crrespnds t the Student t distributin with 1 degree f freedm - is the cannical heavy tailed distributin. Fr this distributin, the prbability that τ lies between 2.4 C and 6.4 C is apprximately 0.75, which is slightly larger than the value f 0.67 given by IPCC (2007). In the sequel, we will truncate this distributin at relatively high levels f τ. This truncatin has minimal effect n this prbability. As guaranteed by the results f Geweke (2001), EU = when f (τ ) is unbunded n the right, essentially implying that θ = 1. Hwever, the picture changes dramatically when f (τ ) is truncated, even when the truncatin ccurs at very high levels. Fr the base case parameterizatin, this is shwn in Figure 1, where θ is pltted against the truncatin pint τ max fr τ max in the range C. The imal value f θ increases very slwly frm arund at τ 20 t arund 0.01 at τ 50. max = max = 6

7 Althugh ur main interest here is in the sensitivity f θ t τ max and nt in its abslute value, the range f values f θ in Figure 1 is cnsistent with mst mainstream ecnmic analyses, lending a further degree f supprt t the base case calibratin f the mdel. Thus, simply truncating the distributin f temperature increases substantially affects the implied mitigatin t undertake tday. This qualitative result is maintained ver a wide range f parameters. Fr example, Figure 1 shws θ vs. τ max fr a range f risk aversin parameters ( η = {1,2 } ) and temperature affect parameters ( β = {0.0019,.0032} η=2,β=.0019 η=2,β=.0032 η=1,β=.0019 η=1,β= θ τ max 7

8 Discussin The effects f climate change n mankind are uncertain. Previus authrs have argued that this uncertainty essentially paralyzes ur ability t cnduct analysis f the apprpriate respnse. Ading a stylized mdel with ecnmic and climate cmpnents, we have explred the cnsequences f bunding uncertainty ver future temperature increases frm climate change. We find that the imal level f respnse t climate change, as measured by the factr θ, is relatively stable ver a wide range f upper bunds n the uncertainty abut future warming. This finding is imprtant because it implies that there is n need t establish a precise upper bund, and may help refcus debate n the apprpriate level f mitigatin. The questin remains whether it is reasnable t bund uncertainty abut future climate change. We believe that it is. Specifically, temperature sensitivity and hence future warming can be cnstrained by empirical studies f the actual temperature respnse t changes in radiative frcing in bth the mdern and gelgical recrds (REFS). Althugh such studies have nt been aimed directly at establishing an upper bund n temperature sensitivity indeed, sme have been aimed at establishing a lwer bund values admitting a temperature increase in the range f can be ruled ut. Finally, as nted, truncating f (τ ) is sufficient, but nt necessary t avid the result f Weitzman (2008): the result is als avided if the tail f declines at a faster than plynmial rate. We have fcused here n the effect f truncatin fr the technical reasn that the latter result can nly be achieved under the Bayesian mdel f Weitzman (2008) by altering the standard nn-infrmative prir distributin fr τ. While a slavish 8

9 adherence t standard prirs is n virtue, it is n a scientific understanding f climate change and nt the frm f a nn-infrmative (and therefre nn-scientific) prir that we have fcused. References Dasgupta, P Cmmentary: The Stern Review s Ecnmics f Climate Change. Natinal Institute Ecnmic Review. 199: 4-7. Geweke, J A nte n sme limitatins f CRRA utility. Ecnmics Letters. 71: IPCC, Climate Change 2007: The Physical Science Basis. Cntributin f Wrking Grup I t the Furth Assessment Reprt f the Intergvernmental Panel n Climate Change. Cambridge University Press. New Yrk. Nrdhaus, W The stern review n the cnmics f climate change. Jurnal f Ecnmic Literature. 45(3): Re, G. and M. Baker Why is climate sensitivy s unpredictable? Science 318:

10 Stern, N. et al The Ecnmics f Climate Change. Cambridge University Press. New Yrk. Weitzman, M On mdeling and interpreting the ecnmics f catastrphic climate change. The Review f Ecnmics and Statistics. 91(1):

, which yields. where z1. and z2

, which yields. where z1. and z2 The Gaussian r Nrmal PDF, Page 1 The Gaussian r Nrmal Prbability Density Functin Authr: Jhn M Cimbala, Penn State University Latest revisin: 11 September 13 The Gaussian r Nrmal Prbability Density Functin