Economics of Controlling Climate Change under Uncertainty.
|
|
- Christian Crawford
- 5 years ago
- Views:
Transcription
1 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; agolub@edf.org z Department of Economics, Southern Methodist University, Dallas, TX ; sroy@smu.edu.
2 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
3 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
4 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
5 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
6 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
7 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
8 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
9 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
10 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
11 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
12 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 w 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
13 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
14 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
15 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
16 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
17 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 )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
18 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 f 2 = 0 f 1 1 = 1: 17
19 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
20 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
21 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
22 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
23 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
24 References [1] Dow, James Jr and Olson, Lars J "Irreversibility and the behavior of aggregate stochastic growth models," Journal of Economic Dynamics and Control 16(2), [2] Heal, G. and Kriström, B Uncertainty and Climate Change. Environmental and Resource Economics 22: Kluwer Academic Publishers. [3] IPCC, Special Report on Emission Scenarios. Climate Change.IPCC, Summary of the report for policy makers. Climate Change. [4] IPCC, Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change [Solomon, S. et al (eds.)]. Cambridge University Press. [5] Nordhaus, William D The Stern Review on the Economics of Climate Change. Journal of Economic Literature [6] Nordhaus, William D A Question of Balance: Weighing the Options on Global Warming Policies. Yale University Press. [7] Olson, Lars "Stochastic growth with irreversible investment," Journal of Economic Theory 47(1), [8] Stern, Nicholas The Economics of Climate Change: The Stern Review. Cambridge, UK: Cambridge University Press. [9] Weitzman, Martin L Role of Uncertainty in the Economics of Catastrophic Climate Change. AEI-Brookings Joint Center Working Paper No
Solow Growth Model. Michael Bar. February 28, Introduction Some facts about modern growth Questions... 4
Solow Growth Model Michael Bar February 28, 208 Contents Introduction 2. Some facts about modern growth........................ 3.2 Questions..................................... 4 2 The Solow Model 5
More informationAdvanced Economic Growth: Lecture 8, Technology Di usion, Trade and Interdependencies: Di usion of Technology
Advanced Economic Growth: Lecture 8, Technology Di usion, Trade and Interdependencies: Di usion of Technology Daron Acemoglu MIT October 3, 2007 Daron Acemoglu (MIT) Advanced Growth Lecture 8 October 3,
More informationDynamic Games with Applications to Climate Change Treaties
Dynamic Games with Applications to Climate Change Treaties Prajit K. Dutta Roy Radner MSRI, Berkeley, May 2009 Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May 2009 1 / 27 Motivation:
More informationEconomic Growth
MIT OpenCourseWare http://ocw.mit.edu 14.452 Economic Growth Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 14.452 Economic Growth: Lecture
More informationSTATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics
STATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics Ph. D. Comprehensive Examination: Macroeconomics Fall, 202 Answer Key to Section 2 Questions Section. (Suggested Time: 45 Minutes) For 3 of
More informationThe Kuhn-Tucker Problem
Natalia Lazzati Mathematics for Economics (Part I) Note 8: Nonlinear Programming - The Kuhn-Tucker Problem Note 8 is based on de la Fuente (2000, Ch. 7) and Simon and Blume (1994, Ch. 18 and 19). The Kuhn-Tucker
More informationCAE Working Paper # Equivalence of Utilitarian Maximal and Weakly Maximal Programs. Kuntal Banerjee and Tapan Mitra.
ISSN 1936-5098 CAE Working Paper #09-03 Equivalence of Utilitarian Maximal and Weakly Maximal Programs by Kuntal Banerjee and Tapan Mitra February 2009 Equivalence of Utilitarian Maximal and Weakly Maximal
More informationAdvanced Economic Growth: Lecture 21: Stochastic Dynamic Programming and Applications
Advanced Economic Growth: Lecture 21: Stochastic Dynamic Programming and Applications Daron Acemoglu MIT November 19, 2007 Daron Acemoglu (MIT) Advanced Growth Lecture 21 November 19, 2007 1 / 79 Stochastic
More informationMicroeconomics, Block I Part 1
Microeconomics, Block I Part 1 Piero Gottardi EUI Sept. 26, 2016 Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, 2016 1 / 53 Choice Theory Set of alternatives: X, with generic elements x,
More informationEconomics 202A Lecture Outline #3 (version 1.0)
Economics 202A Lecture Outline #3 (version.0) Maurice Obstfeld Steady State of the Ramsey-Cass-Koopmans Model In the last few lectures we have seen how to set up the Ramsey-Cass- Koopmans Model in discrete
More informationMinimum Wages and Excessive E ort Supply
Minimum Wages and Excessive E ort Supply Matthias Kräkel y Anja Schöttner z Abstract It is well-known that, in static models, minimum wages generate positive worker rents and, consequently, ine ciently
More informationThe Ramsey Model. Alessandra Pelloni. October TEI Lecture. Alessandra Pelloni (TEI Lecture) Economic Growth October / 61
The Ramsey Model Alessandra Pelloni TEI Lecture October 2015 Alessandra Pelloni (TEI Lecture) Economic Growth October 2015 1 / 61 Introduction Introduction Introduction Ramsey-Cass-Koopmans model: di ers
More informationOptimal taxation with monopolistic competition
Optimal taxation with monopolistic competition Leslie J. Reinhorn Economics Department University of Durham 23-26 Old Elvet Durham DH1 3HY United Kingdom phone +44 191 334 6365 fax +44 191 334 6341 reinhorn@hotmail.com
More informationCompetitive Equilibrium and the Welfare Theorems
Competitive Equilibrium and the Welfare Theorems Craig Burnside Duke University September 2010 Craig Burnside (Duke University) Competitive Equilibrium September 2010 1 / 32 Competitive Equilibrium and
More informationProduction Policies for Multi-Product Systems with Deteriorating. Process Condition
Production Policies for Multi-Product Systems with Deteriorating Process Condition Burak Kazaz School of Business, University of Miami, Coral Gables, FL 3324. bkazaz@miami.edu Thomas W. Sloan College of
More informationTime is discrete and indexed by t =0; 1;:::;T,whereT<1. An individual is interested in maximizing an objective function given by. tu(x t ;a t ); (0.
Chapter 0 Discrete Time Dynamic Programming 0.1 The Finite Horizon Case Time is discrete and indexed by t =0; 1;:::;T,whereT
More informationTraining, Search and Wage Dispersion Technical Appendix
Training, Search and Wage Dispersion Technical Appendix Chao Fu University of Wisconsin-Madison October, 200 Abstract This paper combines on-the-job search and human capital theory to study the coexistence
More informationDepartment of Economics Working Paper Series
Department of Economics Working Paper Series On the Existence and Characterization of Markovian Equilibrium in Models with Simple Non-Paternalistic Altruism Olivier F. Morand University of Connecticut
More informationConcave Consumption Function and Precautionary Wealth Accumulation. Richard M. H. Suen University of Connecticut. Working Paper November 2011
Concave Consumption Function and Precautionary Wealth Accumulation Richard M. H. Suen University of Connecticut Working Paper 2011-23 November 2011 Concave Consumption Function and Precautionary Wealth
More informationNon-convex Aggregate Technology and Optimal Economic Growth
Non-convex Aggregate Technology and Optimal Economic Growth N. M. Hung y, C. Le Van z, P. Michel x September 26, 2007 Abstract This paper examines a model of optimal growth where the aggregation of two
More informationInternet Appendix for The Labor Market for Directors and Externalities in Corporate Governance
Internet Appendix for The Labor Market for Directors and Externalities in Corporate Governance DORON LEVIT and NADYA MALENKO The Internet Appendix has three sections. Section I contains supplemental materials
More informationTheoretical and Computational Appendix to: Risk Sharing: Private Insurance Markets or Redistributive Taxes?
Theoretical and Computational Appendix to: Risk Sharing: Private Insurance Markets or Redistributive Taxes? Dirk Krueger Department of Economics Stanford University Landau Economics Building 579 Serra
More informationIn the Ramsey model we maximized the utility U = u[c(t)]e nt e t dt. Now
PERMANENT INCOME AND OPTIMAL CONSUMPTION On the previous notes we saw how permanent income hypothesis can solve the Consumption Puzzle. Now we use this hypothesis, together with assumption of rational
More informationMacroeconomics IV Problem Set I
14.454 - Macroeconomics IV Problem Set I 04/02/2011 Due: Monday 4/11/2011 1 Question 1 - Kocherlakota (2000) Take an economy with a representative, in nitely-lived consumer. The consumer owns a technology
More informationSolutions to Problem Set 4 Macro II (14.452)
Solutions to Problem Set 4 Macro II (14.452) Francisco A. Gallego 05/11 1 Money as a Factor of Production (Dornbusch and Frenkel, 1973) The shortcut used by Dornbusch and Frenkel to introduce money in
More informationThe MBR social welfare criterion meets Rawls view of intergenerational equity
The MBR social welfare criterion meets Rawls view of intergenerational equity Charles Figuières, Ngo Van Long y and Mabel Tidball z February 3, 200 Abstract In this paper we further explore the properties
More informationVolume 29, Issue 4. Stability under learning: the neo-classical growth problem
Volume 29, Issue 4 Stability under learning: the neo-classical growth problem Orlando Gomes ISCAL - IPL; Economics Research Center [UNIDE/ISCTE - ERC] Abstract A local stability condition for the standard
More informationSession 4: Money. Jean Imbs. November 2010
Session 4: Jean November 2010 I So far, focused on real economy. Real quantities consumed, produced, invested. No money, no nominal in uences. I Now, introduce nominal dimension in the economy. First and
More informationCapital Structure and Investment Dynamics with Fire Sales
Capital Structure and Investment Dynamics with Fire Sales Douglas Gale Piero Gottardi NYU April 23, 2013 Douglas Gale, Piero Gottardi (NYU) Capital Structure April 23, 2013 1 / 55 Introduction Corporate
More informationECON 582: Dynamic Programming (Chapter 6, Acemoglu) Instructor: Dmytro Hryshko
ECON 582: Dynamic Programming (Chapter 6, Acemoglu) Instructor: Dmytro Hryshko Indirect Utility Recall: static consumer theory; J goods, p j is the price of good j (j = 1; : : : ; J), c j is consumption
More informationLecture 2 The Centralized Economy: Basic features
Lecture 2 The Centralized Economy: Basic features Leopold von Thadden University of Mainz and ECB (on leave) Advanced Macroeconomics, Winter Term 2013 1 / 41 I Motivation This Lecture introduces the basic
More information4- Current Method of Explaining Business Cycles: DSGE Models. Basic Economic Models
4- Current Method of Explaining Business Cycles: DSGE Models Basic Economic Models In Economics, we use theoretical models to explain the economic processes in the real world. These models de ne a relation
More informationUNIVERSITY OF NOTTINGHAM. Discussion Papers in Economics WEAK LINKS, GOOD SHOTS AND OTHER PUBLIC GOOD GAMES: BUILDING ON BBV
UNIVERSITY OF NOTTINGHAM Discussion Papers in Economics Discussion Paper No. 06/09 WEAK LINKS, GOOD SHOTS AND OTHER PUBLIC GOOD GAMES: BUILDING ON BBV by Richard Cornes & Roger Hartley August 12 th 2006
More informationFluctuations. Shocks, Uncertainty, and the Consumption/Saving Choice
Fluctuations. Shocks, Uncertainty, and the Consumption/Saving Choice Olivier Blanchard April 2002 14.452. Spring 2002. Topic 2. 14.452. Spring, 2002 2 Want to start with a model with two ingredients: ²
More informationThe Quest for Status and Endogenous Labor Supply: the Relative Wealth Framework
The Quest for Status and Endogenous Labor Supply: the Relative Wealth Framework Walter H. FISHER Franz X. HOF y November 2005 Abstract This paper introduces the quest for status into the Ramsey model with
More informationLabor Economics, Lecture 11: Partial Equilibrium Sequential Search
Labor Economics, 14.661. Lecture 11: Partial Equilibrium Sequential Search Daron Acemoglu MIT December 6, 2011. Daron Acemoglu (MIT) Sequential Search December 6, 2011. 1 / 43 Introduction Introduction
More informationNotes on the Thomas and Worrall paper Econ 8801
Notes on the Thomas and Worrall paper Econ 880 Larry E. Jones Introduction The basic reference for these notes is: Thomas, J. and T. Worrall (990): Income Fluctuation and Asymmetric Information: An Example
More informationLecture Notes 8
14.451 Lecture Notes 8 Guido Lorenzoni Fall 29 1 Stochastic dynamic programming: an example We no turn to analyze problems ith uncertainty, in discrete time. We begin ith an example that illustrates the
More informationSolving Extensive Form Games
Chapter 8 Solving Extensive Form Games 8.1 The Extensive Form of a Game The extensive form of a game contains the following information: (1) the set of players (2) the order of moves (that is, who moves
More informationDynamic Optimization with a Nonsmooth, Nonconvex Technology: The Case of a Linear Objective Function
Dynamic Optimization with a Nonsmooth, Nonconvex Technology: The Case of a Linear Objective Function Takashi Kamihigashi* RIEB Kobe University tkamihig@rieb.kobe-u.ac.jp Santanu Roy Department of Economics
More informationOnline Appendix for Investment Hangover and the Great Recession
ONLINE APPENDIX INVESTMENT HANGOVER A1 Online Appendix for Investment Hangover and the Great Recession By MATTHEW ROGNLIE, ANDREI SHLEIFER, AND ALP SIMSEK APPENDIX A: CALIBRATION This appendix describes
More informationFEDERAL RESERVE BANK of ATLANTA
FEDERAL RESERVE BANK of ATLANTA On the Solution of the Growth Model with Investment-Specific Technological Change Jesús Fernández-Villaverde and Juan Francisco Rubio-Ramírez Working Paper 2004-39 December
More informationSimple Consumption / Savings Problems (based on Ljungqvist & Sargent, Ch 16, 17) Jonathan Heathcote. updated, March The household s problem X
Simple Consumption / Savings Problems (based on Ljungqvist & Sargent, Ch 16, 17) subject to for all t Jonathan Heathcote updated, March 2006 1. The household s problem max E β t u (c t ) t=0 c t + a t+1
More informationRBC Model with Indivisible Labor. Advanced Macroeconomic Theory
RBC Model with Indivisible Labor Advanced Macroeconomic Theory 1 Last Class What are business cycles? Using HP- lter to decompose data into trend and cyclical components Business cycle facts Standard RBC
More informationAlmost sure convergence to zero in stochastic growth models
Forthcoming in Economic Theory Almost sure convergence to zero in stochastic growth models Takashi Kamihigashi RIEB, Kobe University, Rokkodai, Nada, Kobe 657-8501, Japan (email: tkamihig@rieb.kobe-u.ac.jp)
More informationEndogenous timing in a mixed duopoly
Endogenous timing in a mixed duopoly Rabah Amir Department of Economics, University of Arizona Giuseppe De Feo y CORE, Université Catholique de Louvain February 2007 Abstract This paper addresses the issue
More informationStochastic Dynamic Programming. Jesus Fernandez-Villaverde University of Pennsylvania
Stochastic Dynamic Programming Jesus Fernande-Villaverde University of Pennsylvania 1 Introducing Uncertainty in Dynamic Programming Stochastic dynamic programming presents a very exible framework to handle
More informationThe Solow Model in Discrete Time Allan Drazen October 2017
The Solow Model in Discrete Time Allan Drazen October 2017 1 Technology There is a single good Y t produced by means of two factors of production capital K t in place at the beginning of the period and
More informationChapter 6. Maximum Likelihood Analysis of Dynamic Stochastic General Equilibrium (DSGE) Models
Chapter 6. Maximum Likelihood Analysis of Dynamic Stochastic General Equilibrium (DSGE) Models Fall 22 Contents Introduction 2. An illustrative example........................... 2.2 Discussion...................................
More informationEconomics Bulletin, 2012, Vol. 32 No. 1 pp Introduction. 2. The preliminaries
1. Introduction In this paper we reconsider the problem of axiomatizing scoring rules. Early results on this problem are due to Smith (1973) and Young (1975). They characterized social welfare and social
More informationMacroeconomics II Dynamic macroeconomics Class 1: Introduction and rst models
Macroeconomics II Dynamic macroeconomics Class 1: Introduction and rst models Prof. George McCandless UCEMA Spring 2008 1 Class 1: introduction and rst models What we will do today 1. Organization of course
More informationproblem. max Both k (0) and h (0) are given at time 0. (a) Write down the Hamilton-Jacobi-Bellman (HJB) Equation in the dynamic programming
1. Endogenous Growth with Human Capital Consider the following endogenous growth model with both physical capital (k (t)) and human capital (h (t)) in continuous time. The representative household solves
More informationIntroduction: structural econometrics. Jean-Marc Robin
Introduction: structural econometrics Jean-Marc Robin Abstract 1. Descriptive vs structural models 2. Correlation is not causality a. Simultaneity b. Heterogeneity c. Selectivity Descriptive models Consider
More informationLecture Notes - Dynamic Moral Hazard
Lecture Notes - Dynamic Moral Hazard Simon Board and Moritz Meyer-ter-Vehn October 27, 2011 1 Marginal Cost of Providing Utility is Martingale (Rogerson 85) 1.1 Setup Two periods, no discounting Actions
More informationLECTURE 12 UNIT ROOT, WEAK CONVERGENCE, FUNCTIONAL CLT
MARCH 29, 26 LECTURE 2 UNIT ROOT, WEAK CONVERGENCE, FUNCTIONAL CLT (Davidson (2), Chapter 4; Phillips Lectures on Unit Roots, Cointegration and Nonstationarity; White (999), Chapter 7) Unit root processes
More informationSubjective Recursive Expected Utility?
Economic Theory manuscript No. (will be inserted by the editor) Subjective Recursive Expected Utility? Peter Klibano 1 and Emre Ozdenoren 2 1 MEDS Department, Kellogg School of Management, Northwestern
More informationRice University. Fall Semester Final Examination ECON501 Advanced Microeconomic Theory. Writing Period: Three Hours
Rice University Fall Semester Final Examination 007 ECON50 Advanced Microeconomic Theory Writing Period: Three Hours Permitted Materials: English/Foreign Language Dictionaries and non-programmable calculators
More informationLecture 3: Dynamics of small open economies
Lecture 3: Dynamics of small open economies Open economy macroeconomics, Fall 2006 Ida Wolden Bache September 5, 2006 Dynamics of small open economies Required readings: OR chapter 2. 2.3 Supplementary
More informationComprehensive Exam. Macro Spring 2014 Retake. August 22, 2014
Comprehensive Exam Macro Spring 2014 Retake August 22, 2014 You have a total of 180 minutes to complete the exam. If a question seems ambiguous, state why, sharpen it up and answer the sharpened-up question.
More information"A Theory of Financing Constraints and Firm Dynamics"
1/21 "A Theory of Financing Constraints and Firm Dynamics" G.L. Clementi and H.A. Hopenhayn (QJE, 2006) Cesar E. Tamayo Econ612- Economics - Rutgers April 30, 2012 2/21 Program I Summary I Physical environment
More informationDynamic Macroeconomic Theory Notes. David L. Kelly. Department of Economics University of Miami Box Coral Gables, FL
Dynamic Macroeconomic Theory Notes David L. Kelly Department of Economics University of Miami Box 248126 Coral Gables, FL 33134 dkelly@miami.edu Current Version: Fall 2013/Spring 2013 I Introduction A
More informationECON0702: Mathematical Methods in Economics
ECON0702: Mathematical Methods in Economics Yulei Luo SEF of HKU January 14, 2009 Luo, Y. (SEF of HKU) MME January 14, 2009 1 / 44 Comparative Statics and The Concept of Derivative Comparative Statics
More informationOn Stollery s Maximin Model of Global Warming
On Stollery s Maximin Model of Global Warming A. D. d Autume, J.M. Hartwick and K. Schubert May, 2008 Abstract We characterize a solution to Stollery s analysis of maximin with an exhaustible resource
More informationAn adaptation of Pissarides (1990) by using random job destruction rate
MPRA Munich Personal RePEc Archive An adaptation of Pissarides (990) by using random job destruction rate Huiming Wang December 2009 Online at http://mpra.ub.uni-muenchen.de/203/ MPRA Paper No. 203, posted
More informationKyoto Project Mechanisms and Technology. Di usion
Kyoto Project Mechanisms and Technology Di usion Matthieu Glachant, Yann Ménière y February 12, 2007 Abstract This paper deals with the di usion of GHG mitigation technologies in developing countries.
More informationOn the Possibility of Extinction in a Class of Markov Processes in Economics
On the Possibility of Extinction in a Class of Markov Processes in Economics Tapan Mitra and Santanu Roy February, 2005. Abstract We analyze the possibility of eventual extinction of a replenishable economic
More informationa = (a 1; :::a i )
1 Pro t maximization Behavioral assumption: an optimal set of actions is characterized by the conditions: max R(a 1 ; a ; :::a n ) C(a 1 ; a ; :::a n ) a = (a 1; :::a n) @R(a ) @a i = @C(a ) @a i The rm
More informationLecture 1: The Classical Optimal Growth Model
Lecture 1: The Classical Optimal Growth Model This lecture introduces the classical optimal economic growth problem. Solving the problem will require a dynamic optimisation technique: a simple calculus
More informationNonlinear Programming (NLP)
Natalia Lazzati Mathematics for Economics (Part I) Note 6: Nonlinear Programming - Unconstrained Optimization Note 6 is based on de la Fuente (2000, Ch. 7), Madden (1986, Ch. 3 and 5) and Simon and Blume
More informationLecture 6: Contraction mapping, inverse and implicit function theorems
Lecture 6: Contraction mapping, inverse and implicit function theorems 1 The contraction mapping theorem De nition 11 Let X be a metric space, with metric d If f : X! X and if there is a number 2 (0; 1)
More informationAdvanced Economic Growth: Lecture 3, Review of Endogenous Growth: Schumpeterian Models
Advanced Economic Growth: Lecture 3, Review of Endogenous Growth: Schumpeterian Models Daron Acemoglu MIT September 12, 2007 Daron Acemoglu (MIT) Advanced Growth Lecture 3 September 12, 2007 1 / 40 Introduction
More informationEquilibria in Second Price Auctions with Participation Costs
Equilibria in Second Price Auctions with Participation Costs Guofu Tan and Okan Yilankaya January 2005 Abstract We investigate equilibria of sealed-bid second price auctions with bidder participation costs
More informationRevisiting independence and stochastic dominance for compound lotteries
Revisiting independence and stochastic dominance for compound lotteries Alexander Zimper Working Paper Number 97 Department of Economics and Econometrics, University of Johannesburg Revisiting independence
More information1 The Basic RBC Model
IHS 2016, Macroeconomics III Michael Reiter Ch. 1: Notes on RBC Model 1 1 The Basic RBC Model 1.1 Description of Model Variables y z k L c I w r output level of technology (exogenous) capital at end of
More informationAdvanced Microeconomics Fall Lecture Note 1 Choice-Based Approach: Price e ects, Wealth e ects and the WARP
Prof. Olivier Bochet Room A.34 Phone 3 63 476 E-mail olivier.bochet@vwi.unibe.ch Webpage http//sta.vwi.unibe.ch/bochet Advanced Microeconomics Fall 2 Lecture Note Choice-Based Approach Price e ects, Wealth
More informationUniversity of Warwick, EC9A0 Maths for Economists Lecture Notes 10: Dynamic Programming
University of Warwick, EC9A0 Maths for Economists 1 of 63 University of Warwick, EC9A0 Maths for Economists Lecture Notes 10: Dynamic Programming Peter J. Hammond Autumn 2013, revised 2014 University of
More informationOn Sustainability and Social Welfare
On Sustainability and Social Welfare Marc Fleurbaey y June 2013 Abstract This paper proposes to de ne sustainability in terms of making it possible for future generations to sustain whatever what one wants
More informationSome Notes on Adverse Selection
Some Notes on Adverse Selection John Morgan Haas School of Business and Department of Economics University of California, Berkeley Overview This set of lecture notes covers a general model of adverse selection
More informationSimultaneous Choice Models: The Sandwich Approach to Nonparametric Analysis
Simultaneous Choice Models: The Sandwich Approach to Nonparametric Analysis Natalia Lazzati y November 09, 2013 Abstract We study collective choice models from a revealed preference approach given limited
More informationEstimation of Dynamic Nonlinear Random E ects Models with Unbalanced Panels.
Estimation of Dynamic Nonlinear Random E ects Models with Unbalanced Panels. Pedro Albarran y Raquel Carrasco z Jesus M. Carro x June 2014 Preliminary and Incomplete Abstract This paper presents and evaluates
More informationOptimal Target Criteria for Stabilization Policy
Optimal Target Criteria for Stabilization Policy Marc P. Giannoni Columbia University y Michael Woodford Columbia University z February 7, Abstract This paper considers a general class of nonlinear rational-expectations
More informationLong-tem policy-making, Lecture 5
Long-tem policy-making, Lecture 5 July 2008 Ivar Ekeland and Ali Lazrak PIMS Summer School on Perceiving, Measuring and Managing Risk July 7, 2008 var Ekeland and Ali Lazrak (PIMS Summer School Long-tem
More informationOnline Appendix for Precautionary Saving of Chinese and US Households
Online Appendix for Precautionary Saving of Chinese and US Households Horag Choi a Steven Lugauer b Nelson C. Mark c May 06 Abstract This online appendix presents the analytical derivations and estimation
More informationAsymmetric Information and Bank Runs
Asymmetric Information and Bank uns Chao Gu Cornell University Draft, March, 2006 Abstract This paper extends Peck and Shell s (2003) bank run model to the environment in which the sunspot coordination
More informationAppendix (For Online Publication) Community Development by Public Wealth Accumulation
March 219 Appendix (For Online Publication) to Community Development by Public Wealth Accumulation Levon Barseghyan Department of Economics Cornell University Ithaca NY 14853 lb247@cornell.edu Stephen
More informationUNIVERSITY OF VIENNA
WORKING PAPERS Cycles and chaos in the one-sector growth model with elastic labor supply Gerhard Sorger May 2015 Working Paper No: 1505 DEPARTMENT OF ECONOMICS UNIVERSITY OF VIENNA All our working papers
More informationEnvironmental R&D with Permits Trading
Environmental R&D with Permits Trading Gamal Atallah Department of Economics, University of Ottawa Jianqiao Liu Environment Canada April 2012 Corresponding author: Gamal Atallah, Associate Professor, Department
More informationSome Notes on Costless Signaling Games
Some Notes on Costless Signaling Games John Morgan University of California at Berkeley Preliminaries Our running example is that of a decision maker (DM) consulting a knowledgeable expert for advice about
More informationEconomic Growth: Lectures 10 and 11, Endogenous Technological Change
14.452 Economic Growth: Lectures 10 and 11, Endogenous Technological Change Daron Acemoglu MIT December 1 and 6, 2011. Daron Acemoglu (MIT) Economic Growth Lectures 10 end 11 December 1 and 6, 2011. 1
More information1. Money in the utility function (start)
Monetary Economics: Macro Aspects, 1/3 2012 Henrik Jensen Department of Economics University of Copenhagen 1. Money in the utility function (start) a. The basic money-in-the-utility function model b. Optimal
More informationDynamic Sanitary and Phytosanitary Trade Policy 1
Dynamic Sanitary and Phytosanitary Trade Policy 1 Lars J. Olson, University of Maryland, College Park, MD. Santanu Roy, Southern Methodist University, Dallas, TX. September 19, 2008 1 This research was
More informationResearch Division Federal Reserve Bank of St. Louis Working Paper Series
Research Division Federal Reserve Bank of St. Louis Working Paper Series Imperfect Competition and Sunspots Pengfei Wang and Yi Wen Working Paper 2006-05A http://research.stlouisfed.org/wp/2006/2006-05.pdf
More informationIntroduction to Continuous-Time Dynamic Optimization: Optimal Control Theory
Econ 85/Chatterjee Introduction to Continuous-ime Dynamic Optimization: Optimal Control heory 1 States and Controls he concept of a state in mathematical modeling typically refers to a specification of
More informationNeoclassical Growth Model / Cake Eating Problem
Dynamic Optimization Institute for Advanced Studies Vienna, Austria by Gabriel S. Lee February 1-4, 2008 An Overview and Introduction to Dynamic Programming using the Neoclassical Growth Model and Cake
More informationMean-Variance Utility
Mean-Variance Utility Yutaka Nakamura University of Tsukuba Graduate School of Systems and Information Engineering Division of Social Systems and Management -- Tennnoudai, Tsukuba, Ibaraki 305-8573, Japan
More informationAdvanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2
Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 Prof. Dr. Oliver Gürtler Winter Term 2012/2013 1 Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2 JJ N J 1. Introduction
More informationLayo Costs and E ciency with Asymmetric Information
Layo Costs and E ciency with Asymmetric Information Alain Delacroix (UQAM) and Etienne Wasmer (Sciences-Po) September 4, 2009 Abstract Wage determination under asymmetric information generates ine ciencies
More informationOn the Optimality of Financial Repression. V.V. Chari
On the Optimality of Financial Repression V.V. Chari University of Minnesota and Federal Reserve Bank of Minneapolis Alessandro Dovis Penn State Patrick J. Kehoe University of Minnesota and University
More informationAddendum to: International Trade, Technology, and the Skill Premium
Addendum to: International Trade, Technology, and the Skill remium Ariel Burstein UCLA and NBER Jonathan Vogel Columbia and NBER April 22 Abstract In this Addendum we set up a perfectly competitive version
More informationLearning and Risk Aversion
Learning and Risk Aversion Carlos Oyarzun Texas A&M University Rajiv Sarin Texas A&M University January 2007 Abstract Learning describes how behavior changes in response to experience. We consider how
More information