Economics of Controlling Climate Change under Uncertainty.

Economics of Controlling Climate Change under Uncertainty. Alexander Golub y Environmental Defense Fund, Washington, DC. Santanu Roy z Southern Methodist University, Dallas, TX. October 18, 2010 Abstract We analyse the economic e ciency of controlling climate change through reduction in greenhouse gas emissions (GHG) in a general stochastic dynamic framework with irreversibility in climate change. Climate change depends on current GHG emissions and an i.i.d. sequence of random shocks. The total and marginal current welfare in each period depends on both current GHG emissions and the current climate state. We outline the economic conditions under which it is optimal to reduce GHG emission when the state of the climate worsens, as well as conditions under which, for any given realization of the random shock, an improvement in current climate state makes it optimal to improve future climate state. Finally, we establish conditions under which it is optimal to improve the state of the climate from a certain current state. These conditions re ect the roles of discounting, uncertainty, the welfare function and the pace of climate change in response to change in GHG emissions. Key Words: Climate Change, Optimal Control, Greenhouse Gases, Uncertainty, Stochastic Dynamic Model. JEL Classi cation: Q 54, Q 01, O 44. Research on this project is supported by a grant from the Environmental Defense Fund. y Senior Research Fellow, Environmental Defense Fund, 1875 Connecticut Avenue NW Suite 600, Washington, DC 20009; E-mail: agolub@edf.org z Department of Economics, Southern Methodist University, Dallas, TX 75275-0496; E-mail: sroy@smu.edu.

1 Introduction Over the last twenty years, a growing literature in economics has focused attention on the control of future climate change through reduction in current greenhouse gas emissions and in particular, on the question of the extent and nature of current abatement that is economically optimal. Much of the analysis is carried out in the framework of integrated assessment models that are designed for calibration and numerical analysis in order to bring out the quantitative e ect of relevant economic and scienti c parameters on optimal emission paths, optimal carbon taxes and other instruments (see Nordhaus, 2008, and Stern, 2006). However, to the best of our knowledge, there is very little by way of general theoretical analysis of the economics of global climate change that can enable us to derive broad qualitative conclusions about the role of various economic fundamentals that determine the intertemporal trade-o s involved. This paper is an attempt to address the need for such general analytical treatment taking into account uncertainty about the sensitivity of climate change, and the well known irreversibilities of the process. We analyze the normative path of optimal climate change through reduction in greenhouse gas emissions (GHG) in a reduced form general stochastic dynamic framework that allows for irreversibility in climate change. The main point of departure of our model from the received models of climate change (such as the DICE model and various extensions of it) is that it makes the current opportunity cost of reduction in greenhouse gases depend on the current state of the climate. More speci cally, the welfare in each period depends on both current GHG emissions and the current climate state. This allows us to incorporate (in a reduced form) the e ects caused by substitutability and complementarity between climate change and the economic bene ts of additional GHG emissions such as consumption of goods and services. Climate change depends on current GHG emissions and an i.i.d. sequence of random shocks. We abstract from issues of economic capital formation. In terms of the existing literature, the theoretical framework in Heal and Kriström (2002) is closest to our model, however our framework has only one state variable that allows us to obtain general qualitative characterization of optimal policy. Our analysis is a carried out in a traditional framework that abstracts from issues of catastrophic change and "fat 1

tailed" uncertainty. 1 Finally, the framework is close to models of optimal stochastic growth with irreversibility (see, for example, Olson, 1989, Dow and Olson, 1992). We establish the dynamic programming foundations of the model including basic results on existence and qualitative nature of optimal policy. Optimal GHG emissions and the optimal climate state next period (for any realization of the random shock) need not be monotonic functions of the current climate state. We outline the economic and environmental conditions under which it is optimal to reduce GHG emission when the state of the climate worsens, as well as conditions under which, for any given realization of the random shock, an improvement in current climate state makes it optimal to improve future climate state. Finally, we establish conditions under which it is optimal to cut greenhouse gases to the extent that the state of the climate improves next period with probability one. These conditions re ect the roles of discounting, uncertainty, the welfare function and the pace of climate change in response to change in GHG emissions. Section 2 outlines the model and a set of basic results that provide the dynamic programming foundations. Section 3 contains our core results on qualitative nature of optimal policy. All proofs are contained in the Appendix. 2 Preliminaries. Consider an in nite horizon, discrete time model of optimal climate change under uncertainty where time is indexed by t = 0; 1; :::::Let s t 2 R + denote the (one-dimensional) state of the climate in period t (for example, the temperature):increase in s t represents adverse climate change. This will be the only state variable. We will abstract from physical capital accumulation in this study. 2 Let g t 2 R + represent the current ow of (GHG) emissions in period t: We view g t as resulting from current consumption, production and abatement activities. Naturally, other things being constant, higher g t creates higher welfare for the current generation. Further, for any given level of current emission g t ; adverse climate change reduces productivity, consumption possibilities and welfare so that a higher level of s t reduces the welfare of current generation. Let g > 0 be an exogenous upper bound on the amount of GHG emission in 1 See, Weitzman (2007). 2 The frequency of investment in physical capital formation and investment in prevention of global climate change as well as the gestation lag for return on such investments are widely di erent. To focus on measures to control GHG for global climate change, the length of each time period has to be su ciently large and may involve many hundreds of periods of investment and return on physical capital formation. 2

any period that captures the maximum feasible level of emission through economic activity in any period. We assume: A1. The emission ow in each period t satis es: 0 g t g: The climate state variable evolves over time according to: s t+1 = t+1 f(g t ; s t ) (1) where where f captures our scienti c knowledge about the e ect of emissions on climate change and t+1 is a random shock a ecting the evolution of climate over time. Observe that the second argument of f(g; s) captures the e ect of past emissions (for example, the existing stock of GHG) on current climate change. We make the following assumption on the random shocks: A.2. f t g is a sequence of independent and identically distributed random shocks with common distribution function F whose support is an interval [a; b] R ++. Here are the assumptions imposed on the function f: A.3. f : R 2 +! R + is twice continuously di erentiable on R 2 +, f 1 (g; s) > 0; f 2 (g; s) > 0: A.4. There exists (s) 2 (0; s) such that af(0; s) (s); 8s > 0: (2) In assumption A.4, (s) captures the irreversibility in climate change. Note that we will allow for the possibility that f(0; s) > s so that climate change can occur even if current emissions are zero - for example, for very slowly decaying stocks of GHG, high values of s correspond to large stocks of GHG so that global warming may occur simply because existing concentration of GHG is high. Also, 3

observe that (2) implies that if s t > 0, then s t+1 = t+1 f(g t ; s t ) af(0; s t ) (s t ) > 0; with probability one so that starting from any initial state s 0 > 0; the stochastic process of climate states fs t g generated by (1) always lies in R ++ (independent of the actions chosen). The reduced form welfare function of the generation in each period is given by: w(g; s) where immediate welfare in each period is increasing in its current emission and decreasing in climate state. We assume: A.5. w : [0; g] R ++! R is continuous and strictly concave in (g; s) and twice continuously di erentiable with w 1 > 0; w 2 < 0 on (0; g] R ++ : Further, we assume that some emission of GHG is critical to life: A.6 lim w 1(g; s) = +1; 8s 0: (3) g!0 Given s 0 > 0; the dynamic optimization problem is given by: 1X max E[ t w(g t ; s t )] (4) t=0 subject to s t+1 = t+1 f(g t ; s t ) 0 g t g: An important technical contrast with the standard Ramsey growth model is that here the state variable does not in uence the feasible set of action that can be taken at any point of time. In the growth model, capital determines the level of output available which determines the feasible set of consumption or investment every period. Here, the feasible set from which action g t (the level of current GHG emission) can be chosen is independent of climate state (the negative in uence of climate state on the economy enters directly 4

in the reduced form welfare function). Also, observe that the model is signi cantly more general than a stochastic stock pollution control model. The dynamic optimization problem above is a convex problem if we assume f to be a convex function. Non-convexity arises if f is assumed to be concave on part of its domain. This contrasts with the standard optimal growth problem where non-convexity arises if the "production function" is convex on part of its domain; the di erence between the models arise because here the state variable whose transition is determined by f is actually a "bad" capital whose accumulation reduces future welfare. Assuming convexity of f captures features such as low e ect of GHG on climate at low levels of emission (a smooth version of the threshold e ect) and sharp e ect on climate change at higher levels of emission and concentration. From a scienti c point of view the e ect of additional GHG emission on climate change might taper o eventually i.e., f may be eventually concave. While this may be important, this " nal" part of climate evolution may not be immediately relevant to current global conditions and decision making. We will, therefore, go ahead and assume that: A.5. f(g; s) is convex on R 2 +: Note that even though f is assumed to be convex, multiplicity of steady states and initial state dependence of optimal paths can potentially arise in this model because immediate welfare depends on the state of the climate; the latter is a far more important source to focus on in the context of this problem than non-convexity in f: Observe that the function w is bounded above on its domain by w(g; 0):It is easy to show that under the assumptions made above, for every initial state s 0 > 0 there exists a solution to the dynamic optimization problem (4). Let V (s) be the value function for the maximization problem in (4), de ned on R ++ : 1X V (s) = maxfe[ t w(g t ; s t )] : subject to (1),s 0 = sg (5) t=0 Standard dynamic programming arguments show that: Lemma 1 V (s) is continuous and strictly decreasing on R ++ and the following functional equation of dynamic programming holds: V (s) = max [w(g; s) + EV [f(g; s)] (6) 0gg 5

Lemma 2 V (s) is strictly concave on R ++ : Standard convex dynamic programming arguments can again be used to show: Lemma 3 9 a unique optimal policy and it is the stationary policy generated by the function g(s) : R ++! [0; g] where g(s) = arg max fw(g; s) + EV (f(g; s))g (7) 0gg which is the unique solution to the maximization problem on the right hand side of (6). The next lemma follows directly from (6) and (3) using standard arguments: Lemma 4 g(s) > 0; 8s 2 R ++ : Note that Lemma 4 implies that the irreversibility constraint (2) is never binding on any optimal path: no matter how terrible the state of the climate, society always prefers to emit a bit. De ne the variable y by f(g; s) = y: Then, if the current climate state is s and an amount g is emitted, then the next period s state is y. In other words, y is the deterministic component of next period s climate state f(g; s). Note that for any s, y is strictly increasing in g:we can think of the dynamic optimization problem as being one where, in each period society chooses y (instead of choosing emission g), i.e., the deterministic component of the the next period s state. Indeed, since f 1 > 0; g can be written as an implicit function of (y; s), g = (y; s): (8) where the identity f((y; s); s) = y (9) holds for all y 2 [f(0; s); f(g; s)]: It is easy to check that (see appendix): 6

Lemma 5 (y; s) : f(y; s) : s > 0; f(0; s) y f(g; s)g! R + is twice continuously di erentiable with Further, (y; s) is concave. on w: We can now de ne welfare as a function of y and s by: 1 = 1 f 1 > 0; 2 = f 2 f 1 < 0 (10) 12 = 1 (f 1 ) 2 [f f 2 11 f 12 ]: (11) f 1 u(y; s) = w((y; s); s) (12) The next lemma follows immediately from the identity (12), Lemma 5 and assumptions Lemma 6 u(y; s) : f(y; s) : s > 0; f(0; s) y f(g; s)g! R + is continuous and concave, twice continuously di erentiable on f(y; s) : s > 0; f(0; s) < y f(g; s)g; u 1 (y; s) = w 1 ((y; s); s) 1 (y; s) > 0 (13) u 2 (y; s) = w 1 ((y; s); s) 2 (y; s) + w 2 ((y; s); s) < 0 (14) lim u 1(y; s) = +1: y!f(0;s) The dynamic optimization problem (4) can then be re-written as follows: Given s 0 > 0; 1X max E[ t u(y t ; s t )] (15) t=0 subject to s t+1 = t+1 y t f(0; s t ) y t f(g; s t ): Recall that g(s) is the unique optimal emission policy function. De ne: y(s) = f(g(s); s): (16) 7

Re-writing the functional equation of dynamic programming (6) and using Lemmas 3 and 4 we have: Lemma 7 8s > 0; f(0; s) < y(s) f(g; s); V (s) = max [u(y; s) + EV (s)] (17) f(0;s)y(s)f(g;s) and y(s) is the unique solution to the maximization problem on the right hand side of (17): We now show that the value function is di erentiable at each s > 0 as long as optimal action is interior. The proof is contained in the appendix. Lemma 8 Suppose that g(s) < g for some s > 0: Then, V is di erentiable at s and V 0 (s) = w 2 (g(s); s) w 1 (g(s); s) f 2(g(s); s) (18) f 1 (g(s); s) = u 2 (y(s); s) (19) Next, we establish the stochastic Euler equation & inequality: Lemma 9 For any s > 0; let Then, es() = f(g(s); s) = y(s): w 1 (g(s); s) E[V 0 (es())f 1 (g(s); s)] (20) = E[fw 1 (g(es()); es()) f 2(g(es()); es()) f 1 (g(es()); es()) w 2 (g(es()); es())gf 1 (g(s); s)] (21) or, equivalently, u 1 (y(s); s) E[ u 2(y(es()); es()) ]: (22) 1 (y(s); s) 8

If, further, g(es) < g for all es 2 [af(g(s); s); bf(g(s); s)], then w 1 (g(s); s) = E[V 0 (es())f 1 (g(s); s)] (23) = E[fw 1 (g(es()); es()) f 2(g(es()); es()) f 1 (g(es()); es()) w 2 (g(es()); es())gf 1 (g(s); s)] (24) or, equivalently, u 1 (y(s); s) = E[ u 2(y(es()); es()) ]: (25) 1 (y(s); s) The proof the lemma is contained in the appendix. The following result follows immediately: Corollary 10 Consider the stochastic process fg t ; s t ; y t g generated from initial state s 0 by the optimal policy. Then, conditional on the information set at each t 0; the following holds almost surely: w 1 (g t ; s t ) E t [fw 1 (g t+1 ; s t+1 ) f 2(g t+1 ; s t+1 ) f 1 (g t+1 ; s t+1 ) w 2 (g t+1 ; s t+1 )g t+1 f 1 (g t ; s t )] (26) = E t [ u 2(y t+1 ; s t+1 ) 1 (y t ; s t ) t+1 ]: (27) In addition, if g t < g almost surely 8t 0; then w 1 (g t ; s t ) = E t [fw 1 (g t+1 ; s t+1 ) f 2(g t+1 ; s t+1 ) f 1 (g t+1 ; s t+1 ) w 2 (g t+1 ; s t+1 )g t+1 f 1 (g t ; s t )] (28) = E t [ u 2(y t+1 ; s t+1 ) 1 (y t ; s t ) t+1 ] (29) (In the above expression, E t is the conditional expectation with respect to the information set at the beginning of period t:) 3 Properties of the Optimal Policy We begin with a result that establishes a condition under which optimal emission reduction is strictly positive. The proof is contained in the appendix. 9

Proposition 11 For any s > 0;let u 0 (s) > 0 be de ned by u 0 (s) = inf [w 1 (g; es) f 2(g; es) 0gg f 1 (g; es) af(g;s)esbf(g;s) w 2 (g; es)] If w 1 (g; s) < u 0 (s)e()f 1 (g; s)] (30) then, g(s) < s and optimal emission reduction is strictly positive. In the above proposition, starting from a situation where no emission reduction is being undertaken at all, u 0 (s) is a uniform lower bound on the marginal welfare gain of a small improvement in next period s climate state. E()f 1 (g; s) gives us the "average" extent of climate change that occurs next period from having some emission reduction today. On the other hand, w 1 (g; s) is the current welfare sacri ce involved in moving from doing nothing to having a small emission reduction. Condition (30) ensures that in a situation where no emission reduction is being undertaken, the current marginal welfare cost of emission reduction is below the future marginal value gain occurring through improvement in climate. It is di cult to sign, in general, the e ect of an increase in s on u 0 (s): However, it is easy to see that if, as we assume in the next section, w 12 (g; s) < 0; f 12 (g; s) 0; then an increase in s (or, worsening of current climate state) reduces w 1 (g; s), the marginal utility from current emission generation or consumption, and increases f 1 (g; s); the marginal e ect of current emission on climate change. In that sense, condition (30) is more likely to hold as the current climate state worsens. Indeed, under if w 12 (g; s) < 0 and f 12 (g; s) 0; then it is easy to check that condition (30) holds if: w 1 (g; s) w 1 (g; bf(g; s)) < [f 2(0; af(g; s)) f 1 (g; bf(g; s)) E()f 1(g; s)]; which is easy to verify from the fundamentals of the model. We will now impose a set of assumptions that ensure certain monotonicity properties of the optimal policy. The following assumption ensures that it is optimal for society to reduce emission as the climate worsens: A.6. w 12 (g; s) < 0; f 12 (g; s) 0 on the set f(g; s) : s > 0; 0 < g gg: 10

Note that w 12 (g; s) < 0 implies that adverse climate change reduces the marginal welfare from current emission - this may re ect direct e ect of climate on marginal utility of consumption caused by emission or indirect e ect on productivity. f 12 (g; s) 0 implies that worsening of the existing climate state (for example, re ecting a higher level of existing concentration of GHG) intensi es the e ect of current emission on climate change. We now show that under the assumptions made so far (particularly A.6), optimal emission declines as the state of the climate worsens. The proof of this result is contained in the appendix. Proposition 12 g(s) is decreasing in s: It is however important to note that if assumption A.6 does not hold i.e., w 12 > 0 or f 12 < 0; it may be optimal to emit more as climate worsens. This may be because adverse climate change increases the marginal welfare from current emission for example, emission related to measures to protect against the adverse e ects of climate change and more generally, when current emission is generated by activities that substitute or compensate for adverse climate change. It may also be because adverse climate change reduces the marginal (stock) e ect of further new emission on future climate change. The next assumption ensures that for each xed realization of the random shock, the state of the climate next period is increasing in the current state. In other words, it is optimal for society to reduce emission as the climate worsens: A.7. u(y; s) =w((y; s); s) is strictly supermodular in (y; s) on f(y; s) : s > 0; f(0; s) < y f(g; s)g: Note that under our assumptions so far, u(y; s) is twice di erentiable on the set fs > 0; f(0; s) < y f(g; s)g and therefore, A.7 is equivalent to the requirement that u 12 (y; s) > 0; 8(y; s) 2 f(y; s) : s > 0; f(0; s) < y f(g; s)g: (31) where u 12 = w 1 12 + w 11 1 2 + w 12 1 f 2 = w 1 [f 12 f 1 f 2 f 11 ] w 11 (f 1 ) 2 + w 1 12 f 1 11

Note that strict concavity of w implies that w 11 < 0; and so A.7 would be satis ed if 12 = [f 12 f 1 f 2 f 11 ] 0 and that w 12 (assumed to be negative in assumption A.6) is relatively small in absolute value. The next proposition shows that the state of the climate next period is increasing in the current state. The proof of this proposition is contained in the appendix. Proposition 13 y(s) = f(g(s); s)) is strictly increasing in s so that for each realization of t+1 ;the state s t+1 of the climate next period under the optimal policy determined by the transition function: s t+1 (; s t ) = f(g(s t ); s t ) = y(s t ): is a continuous and strictly increasing function of s t : Thus, the Markov process fs t g is a stationary process generated by iterations of the i.i.d. map t+1 y(s) where y(s) is (strict) monotone increasing. In this case, starting from a better current state of the climate, the distribution of next period s climate state is abetter (in a rst order stochastic sense). Finally, we establish a condition under which improvement in climate is optimal from a certain current state. The proof of this proposition is contained in the appendix. Proposition 14 Suppose that bf(0; s) < s bf(g; s) and, further let bg(s) be de ned by: bf(bg(s); s) = s: If [w 1 (g; a b s)f 2(g; a b s) f 1 (g; a b s) w 2 (g; a b s)][ 1 w 1 (bg(s); s) ]E()f 1(bg(s); s) 1: (32) then, it is optimal to improve the climate almost surely from state s i.e., if s t = s, then s t+1 < s t with probability one. The inequality in (32) provides a characterization of environmental and economic conditions under which, given a certain current state s, it is optimal to improve the state of the climate through current action (control of emission). To interpret this condition, rst consider the de nition of bg(s) : bf(bg(s); s) = s: 12

bg(s) is the level of current GHG emission that would allow society to sustain the current climate state under the worst environmental condition (a ecting the relationship between GHG emission and climate state). In other words, bg(s) is the highest level of current emission consistent with almost sure (probability one) sustainability of current climate state. Now, consider the marginal e ect of an increase in emission when the current emission is exactly at the "sustainable" level bg(s):there are two e ects. First, the current welfare increases because of greater emission control and this welfare e ect is captured by: w 1 (bg(s); s): (33) Second, depending on the realization of the random shock, the next period s climate state bs() is given by: bs() = f(bg(s); s): Therefore, an increase in emission when the current emission is at bg(s) implies that the next period s climate state worsens at the margin by an amount : f 1 (bg(s); s):the welfare consequence of this is given by: V 0 (bs())f 1 (bg(s); s) where V 0 captures the marginal loss of intertemporal future welfare due to (or the shadow price of) this worsening of the climate state. As we have seen, a dynamic envelope condition ensures that from any state bs V 0 (bs) = w 1 (g; bs) f 2(g; bs) f 1 (g; bs) + w 2(g; bs) where g is the optimal emission from state bs: The e ect of climate worsening is to reduce welfare by a direct e ect (w 2 (g; bs)) and by an indirect e ect - through increase in future optimal emission control because of worsening of climate state (captured by w 1 (g; bs) f 2(g;bs) f 1 (g;bs) ):Of course, in a general framework, we do not know (ex ante) what the optimal emission next period (g) is going to be from each possible state bs(): But we know that g is bounded above by g: We also know that bs() is bounded below by bs() = f(bg(s); s) af(bg(s); s) = a b bf(bg(s); s) = a b s: 13

Taking this upper bound on emission tomorrow and the lower bound on the climate state tomorrow, we can derive a lower bound on the (absolute value of) welfare decline resulting from a worsening of future climate state and this is given by: [w 1 (g; a b s)f 2(g; a b s) f 1 (g; a b s) w 2 (g; a b s)]: Thus, the net future welfare cost of a marginal worsening of the climate when the current state is s and the current emission control is exactly at the sustainable level bg(s) is at least as large as: [w 1 (g; a b s)f 2(g; a b s) f 1 (g; a b s) w 2 (g; a b s)]f 1(bg(s); s) and its expected present value (in terms of current period) : [w 1 (g; a b s)f 2(g; a b s) f 1 (g; a b s) w 2 (g; a b s)]e()f 1(bg(s); s): (34) Therefore, it is optimal to refrain from further worsening of the climate when current emission is exactly at the sustainable level as long as the marginal bene t given by (33) is below the marginal future welfare cost given by (34). This is exactly what the inequality above requires. Note that the condition (32) for climate improvement from current state s is veri able on the basis of the exogenous elements of the model - it does not involve any endogenous variable (other than current state s). There are certain implications of the condition (32) that can be readily observed. The following make the condition for climate improvement more likely to hold (i) Milder discounting (" in ) (as would be expected) (ii) An increase E()f 1 (bg(s); s) - the "expected" rate at which climate state worsens with an increase in current emission when this rate is measured at the sustainable level (iii) An increase in of climate state w 2, the absolute (direct) marginal welfare decline from worsening (iv) An increase in f 2 f 1 : the marginal rate of substitution between current emission and current climate state in the determination of the future climate state (higher value of this marginal rate implies that a marginal worsening of future climate state will require more severe control on future emission in order to retain a certain climate state). 14

Note that w 1, the marginal welfare gain from higher emission, enters both the current bene t from higher emission as well as the future welfare loss from higher emission control necessitated by worsening of the future state of climate. The most important qualitative conclusion from the above analysis is the important role played by the welfare function - in addition to the physical laws of climate change and the discount factor - in determining the dynamic e ciency of climate improvement. 15

APPENDIX. Proof of Lemma 2. Let! = (! t ) 1 t=1 be an element of the probability space where! tis a vector of realization of the random variables observable at the end of period (t 1):Consider optimal processes fgt(!); i s i t(!)g from initial states s i 0 2 R ++; i = 1; 2; where s 1 0 6= s2 0 : Let s = s 1 0 + (1 )s 2 0 for any given 2 (0; 1): We will show that V (s) > V (s 1 0) + (1 )V (s 2 0): To see, consider the process fg t (!); s t (!)g de ned by g t (!) = g 1 t (!) + (1 )g 2 t (!); s 0 (!) = s and s t+1 (!) = t+1 (!)f(g t (!); s t (!)); t 0: Then fg t (!); s t (!)g is feasible from initial state s: Further, by induction, if s t (!) s 1 t (!) + (1 )s 2 t (!); then using convexity of f, s t+1 (!) = t+1 (!)f(g t (!); s t (!)) t+1 (!)f(g 1 t (!); s 1 t (!)) + (1 ) t+1 (!)f(g 2 t (!); s 2 t (!)) = s 1 t+1(!) + (1 )s 2 t+1(!): 16

Thus, 1X V (s) E[ t w(g t (!); s t (!))] t=0 1X E[ t w(g t (!); s 1 t (!) + (1 )s 2 t (!)) t=0 1X = E[ t w(gt 1 (!) + (1 )gt 2 (!); s 1 t (!) + (1 )s 2 t (!))] t=0 1X > E[ t w(gt 1 (!); s 1 t (!))] + (1 t=0 t=0 1X )E[ t w(gt 2 (!); s 2 t (!))] = V (s 1 0) + (1 )V (s 2 0): This concludes the proof of Lemma 2. Proof of Lemma 5 Only the concavity of (y; s) requires a proof (the rest follow from the implicit function theorem and the fact that f 1 > 0; f 11 > 0) 3. Suppose, to the contrary, that is not concave. Then, there exists s 1 ; s 2 2 R ++ ; y 1 2 [f(0; s 1 ); f(g; s 1 )]; y 2 2 [f(0; s 1 ); f(g; s 1 )]; 2 [0; 1] such that (y 1 + (1 )y 2 ; s 1 + (1 )s 2 ) < (y 1 ; s 1 ) + (1 )(y 2 ; s 2 ): (35) From (9), y 1 + (1 )y 2 = f((y 1 + (1 )y 2 ; s 1 + (1 )s 2 ); s 1 + (1 )s 2 ) < f((y 1 ; s 1 ) + (1 )(y 2 ; s 2 ); s 1 + (1 )s 2 ); using (35) and f 1 > 0, f((y 1 ; s 1 ); s 1 ) + (1 )f((y 2 ; s 2 ); s 2 ); using A.5 = y 1 + (1 )y 2 ; using (9) again 3 Implicit di erentiation of (9) with respect to y; s yields: f 1 2 + f 2 = 0 f 1 1 = 1: 17

which yields a contradiction. Proof of Lemma 8 From Lemma 4, we know that g(s) > 0; 8s > 0: Thus, 0 < g(s) < g: As V is strictly concave on R ++ ; the one-sided derivatives V+; 0 V 0 are well de ned and V 0 +(s) V 0 (s); 8s > 0: (36) As 0 < g(s) < g; f(0; s) < y(s) < f(g; s): Choose any > 0 su ciently small. Then, f(0; s + ) < y(s) < f(g; s + ) and Note that f(0; s ) < y(s) < f(g; s ): V (s) = w((y(s); s); s) + EV (y(s)) V (s + ) = max [w(g; s + ) + EV (f(g; s + ))] 0gg = max [w((y; s + ); s + ) + EV (y)] f(0;s+)yf(g;s+) w((y(s); s + ); s + ) + EV (y(s))] and V (s ) = max [w(g; s ) + EV (f(g; s ))] 0gg = max [w((y; s ); s ) + EV (y)] f(0;s )yf(g;s ) w((y(s); s ); s ) + EV (y(s))] so that: while V (s + ) V (s) w((y(s); s + ); s + ) w((y(s); s); s) V (s) V (s ) w((y(s); s); s) w((y(s); s ); s ) 18

and dividing by both sides of the two above inequalities and taking limit as! 0 we obtain: and using (36) we have V 0 +(s) w 1 ((y(s); s); s) 2 (y(s); s) + w 2 ((y(s); s); s) V 0 V 0 +(s) = V 0 (s) = w 1 ((y(s); s); s) 2 (y(s); s) + w 2 ((y(s); s); s) so that V is di erentiable at s and The proof is complete. Proof of Lemma 9 V 0 (s) = w 1 ((y(s); s); s) 2 (y(s); s) + w 2 ((y(s); s); s) = w 1 (g(s); s) f 2(g(s); s) f 1 (g(s); s) + w 2(g(s); s); using (??) = u 2 (y(s); s); using (14) and (??). Follows from (7), the rst order necessary conditions associated with the maximization problem on the right hand side of (7), the fact that es() > 0 almost surely and Lemma 8. Proof of Proposition 11. Suppose g(s) = g: Then next period s climate state is es() = f(g; s):then, using (21), w 1 (g; s) E[fw 1 (g(es()); es()) f 2(g(es()); es()) f 1 (g(es()); es()) w 2 (g(es()); es())gf 1 (g; s)] E[u 0 f 1 (g; s)] which violates (30). Proof of Proposition 12. Let h(g; s) = w(g; s) + EV [f(g; s)] Under assumption A.6, w(g; s) is submodular and f(g; s) is supermodular on f(g; s) : s > 0; 0 < g gg. Since V is decreasing and strictly concave, V [f(g; s)] is submodular in (g; s) 19

for each : It follows that h(g; s) is submodular 4 i.e., for any (g 1 ; s 1 ); (g 2 ; s 2 ) 2 f(g; s) : s > 0; 0 < g gg; g 1 g 2 ; s 1 s 2 ; the following holds: h(g 1 ; s 1 ) + h(g 2 ; s 2 ) h(g 1 ; s 2 ) + h(g 2 ; s 1 ): Choose any s 1 ; s 2 2 R ++ such that s 1 < s 2 :Let g i = g(s i ); i = 1; 2: We show that g 1 g 2 : Suppose, to the contrary, that g 1 < g 2 : Note that g i > 0. Then, using the submodularity of the function h(g; s) on f(g; s) : s > 0; 0 < g gg we have: h(g 1 ; s 1 ) + h(g 2 ; s 2 ) h(g 1 ; s 2 ) + h(g 2 ; s 1 ): As g 1 and g 2 are unique solutions to the maximization problem on the right hand side of (6) at s = s 1 and s = s 2, respectively: g i uniquely maximizes h(g; s i ) with respect to g 2 [0; g] so that and Thus, h(g 1 ; s 1 ) > h(g 2 ; s 1 ) h(g 1 ; s 2 ) > h(g 2 ; s 2 ): h(g 1 ; s 1 ) + h(g 2 ; s 2 ) > h(g 1 ; s 2 ) + h(g 2 ; s 1 ); a contradiction. Proof of Proposition 13. H(y; s) be de ned by: H(y; s) = [u(y; s) + EV (y)] 4 Though this can be shown even without di erentiability, suppose that that V is twice di erentiable: h 12 = w 12 + E[V 00 (:) 2 f 1f 2 + V 0 (:)f 12] and the fact that strict concavity of V implies that V 00 < 0; V decreasing implies V 0 0: 20

The maximization problem on the right hand side of (17) can be written as: max H(y; s) y2 (s) where (s) = [f(0; s); f(g; s)]: Observe that since EV (y) is independent of at each s > 0, under A.7, H(y; s) is strictly supermodular in (y; s) on the set f(y; s) : s > 0; f(0; s) < y f(g; s)g: Choose s 1 ; s 2 2 R ++ such that s 1 < s 2 :Let y i = y(s i ) = f(g i ; s i ); i = 1; 2: We show that y 1 < y 2 : Suppose, to the contrary, that y 1 y 2 : Since g(s i ) > 0; y i > f(0; s i ):Using the strict supermodularity of the function H(y; s) we have: H(y 1 ; s 1 ) + H(y 2 ; s 2 ) < H(y 1 ; s 2 ) + H(y 2 ; s 1 ): (37) Further, y 1 y 2 ; f(0; s i ) < y i f(g; s i ); i = 1; 2 and the assumption that f(y; s) is strictly increasing in s implies f(0; s 1 ) < f(0; s 2 ) < y 2 y 1 f(g; s 1 ) < f(g; s 2 ) so that y 1 2 (s 2 ); y 2 2 (s 1 ): Using Lemma 7 and the fact that y i (s i ); i = 1; 2; y 1 > y 2 ; we have: = y(s i ); is the unique maximizer of H(:; s i ) on H(y 1 ; s 1 ) H(y 2 ; s 1 ) H(y 1 ; s 1 ) H(y 2 ; s 1 ) that together contradicts (37). Proof of Proposition 14. Note that for any s > 0; V 0 (s) = w 2 (g(s); s) w 1 (g(s); s) f 2(g(s); s) f 1 (g(s); s) = U 2 (y(s); s) 21

where U(y; s) = w((y; s); s): Note that U 2 = H 2 and under our assumption that H 12 > 0; U 12 > 0;so that V 0 (s) = U 2 (y(s); s) U 2 (f(g; s); s) = w 2 (g; s) w 1 (g; s) f 2(g; s) f 1 (g; s) = U 2 (s); say. Suppose that es(b) = bf(g(s); s) = by(s) s: Then, bf(0; s) s bf(g(s); s) bf(g; s) so that g(s) bg(s): w 1 (bg(s); s) = w 1 (g(s); s) = E[V 0 (es())f 1 (g(s); s)] = E[V 0 (y(s))f 1 (g(s); s)] = E[V 0 ( b by(s))f 1(g(s); s)] E[V 0 ( b s)f 1(g(s); s)] V 0 ( a b s)e()f 1(g(s); s) U 2 ( a b s)e()f 1(bg(s); s) which implies: which violates (32). U 2 ( a b [ s) w 1 (bg(s); s) ]E()f 1(bg(s); s) 1 22

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