The consequences of an endogenous discounting depending on environmental quality

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1 The consequences of an endogenous discounting depending on environmental quality A A K Université de Grenoble 2, Commissariat général du Plan and EUREQua, Université de Paris 1 adayong@univ-paris1fr K S EUREQua, Université de Paris 1 schubert@univ-paris1fr May 2002 Abstract Our intention is to study, in the framework of a very simple optimal growth model, the consequences on the optimal paths followed by consumption and environmental quality of an endogenous discounting Consumption directly comes from the use of environmental services and so is a direct cause of environmental degradation The environmental renewable resource is valued both as a source of consumption and as an amenity For a sustainability concern, we introduce an endogenous discount rate depending on the stock of environmental resource, and compare the optimal growth paths with the ones obtained in the usual case of eogenous and constant discounting We show that endogenous discounting can change qualitatively the nature of the optimal growth path, and, more precisely, that there eists a case in which consumption and environmental quality optimally decrease towards zero under eogenous discounting while they converge towards a positive stationary state with endogenous discounting Keywords: Endogenous discounting, Sustainability, Environment JEL codes: O41,Q0,E6 1 Introduction Optimal growth models usually make the assumption of a constant and strictly positive utility social discount rate This usual practice has been long questioned, but is still widely used, maybe because of the lack of a convincing alternative (Heal (2001) The first doubt goes back to the well known Ramsey and Harrod s criticisms of a strictly positive social discount rate (Ramsey (1928), Harrod (1948)), and has led to the so-called undiscounted utilitarianism But Koopmans (1960) has stressed the drawbacks of this approach: it usually leads to unrealistically high optimal saving rates for the present and near generations, andsotoasacrifice of the present and of the near future instead of the sacrifice of the far future implied by a positive discount rate Nevertheless, the temptation of a zero rate of discount is still very present as far as environment is concerned (see, for a comprehensive view of discounting and the environment, the contributions gathered in the book edited by Portney and Weyant (1999)) The usual approach has then been challenged by authors arguing that the discount rate should neither be constant and positive nor zero, but should be decreasing in the course of Corresponding author Address: EUREQua,UMRn 8594 CNRS, Université Paris I, Maison des Sciences Économiques, bd de l Hôpital, Paris Cede 13, France 1

2 time They claim that empirical evidence support that Both Harvey (1994) and Heal (1998) propose formulas of declining discounting Papers in the spirit of Laibson (see Laibson (1996 and 1997), Cropper and Laibson (1999) among others) use hyperbolic discounting based upon considerations of individual psychology Li and Löfgren (2001) analyse the Brock (1973) growth and environment model with hyperbolic discounting But the choice of a utility discount rate in normative models is an ethical choice, as stressed by Heal (1998 and 2001) or Ayong Le Kama (2001), and it is less than obvious than a central planner should only reflect in this choice the representative consumer s psychology On what basis should then this ethical choice be made? Another strand of literature, disconnected from environmental concerns, studies the question of habit formation, and introduces a utility discount factor depending on past consumption levels (Epstein (1987), Obstfeld (1990)) Obstfeld (1990) gives a formal treatment of the simple optimal growth model with this utility discount factor, and concludes that most of the time this assumption does not qualitatively change the optimal growth paths, in comparison with the case of an eogenous constant discount rate We want here to etend Obstfeld s approach to a model where environment matters We consider two ways by which environmental quality could affect the social intertemporal welfare: the usual direct effect of the current level of environmental quality on instantaneous utility (amenity effect), and a less usual indirect effect of current and past levels of environmental quality on the utility discount factor This discount factor now depends on the path of environmental quality through time Utility is no more time-separable, social tastes are intertemporally dependant Therationalizationoftheethicalchoiceofanendogenousdiscountfactordependingon environmental quality could be a sustainable development motive Society could epress this way a form of intergenerational concern, consisting in deciding to discount the future at a rate all the lower than environmental quality is and has been low That could be a way of implementing sustainability concerns in optimal growth models We then try to elucidate the consequences of such an ethical choice These consequences are very hard to assess in comple growth models with capital accumulation We thus use here a simple framework where consumption directly comes from the use of environmental services, and so directly causes environmental degradation It is in some sense the worst possible case: there eist neither a technical progress enhancing the transformation of environmental services into consumption, nor substitution possibilities between environmental services and man-made capital There is no means of improving environmental quality ecept natural regeneration In such a framework, consumption and environmental quality necessarily evolve together in the long run: both increase, or both decrease, or both converge towards a stationary state We study the evolution of the economy and the environment with an eogenous and constant discount rate (section 2), and then with the endogenous discount factor (section 3) Our objective is to compare the growth paths implied by the two possible choices (eogenous or endogenous discount rate) We show that endogenous discounting can change qualitatively the nature of the optimal growth path, contrary to what Obstfeld (1990) suggests More precisely, there eists a case in which consumption and environmental quality optimally decrease towards zero under eogenous discounting (doomsday) while they converge towards a positive stationary state with endogenous discounting Section 4 gives a comparison of the eogenous and endogenous discounting cases in terms of instantaneous utilities Finally, we try in section 5 to clarify the different concepts of discount rates that are used in the litterature, and to compare them in the endogenous and the eogenous cases 2

3 2 The economy with a constant discount rate 21 The model We introduce an optimal growth model with a renewable resource This resource is valued both as a source of consumption and also as a stock of amenities One interpretation, between many others 1, would be forests, which contribute to welfare both as sources of timber and also as stocks, which provide many ecosystem services to society (carbon s sequestration, preservation of biodiversity, etc) For simplification, we will consider that the stock of renewable resource, S, corresponds to an inde of environmental quality which is depleted by consumption C, but regenerates itself at the constant rate m>0 Its dynamics is then described by: St= ms t C t (1) The objective of the central planner is to maimize the present value of life-time utility of the representative consumer over an infinite horizon The representative consumer derives felicity 2 u() not only from consumption, but also from environmental quality The future felicities are discounted at the constant rate ρ > 0 The social planner s program is the following: ma U (C, S) = + 0 e ρt u (C t,s t ) dt Ṡ t = ms t C t st C t 0,S t 0 S 0 given The felicity function u () is continuous, twice differentiable, and possesses the following properties 3 : u C > 0, u S > 0, u CC 0 We also suppose that the felicity function is concave with respect to its two arguments: u CC u SS (u CS ) 2 0 The current-value Hamiltonian associated to the problem (2) writes: H = u (C, S)+λ [ms C], where λ 0 is the costate variable associated to environmental quality The first order necessary conditions then give us: (2) λ = u C λ λ = ρ m u S = ρ m φ (C, S) C u C S, and the transversality condition writes: lim t e ρt λ t S t =0 (3a) (3b) (3c) φ (C, S) Su S Cu C is the ratio of the values of environmental quality and consumption, both evaluated at their marginal felicity φ () then reflects the relative preference for environment of the representative agent Logdifferentiation of equation (3a) leads to the following result: λ λ = η C CC C + η S CS S, 1 For a precise interpretation of the role of this resource, see Heal (1998) and Daily (1977) and for more on ecosystem services, see Heal (2000) 2 We follow the convention of Arrow and Kurz (1970) in referring to the subutility functions as felicities In contrast, the term utility always refers to the planner s intertemporal objective 3 u C and u S are the first partial derivatives of the function u () with respect to its arguments C and S u CC is likewise the second partial derivative, using obvious notation 3

4 where η CC = Cu CC / u C < 0 is the elasticity of the marginal felicity of consumption with respect to the level of consumption and η CS = Su CS / u C 0 is the elasticity of the same quantity with respect to the level of environmental quality Substituting this relationship into (3b) gives the following relationship: η CC 22 The balanced growth path (BGP) 221 Eistence of a BGP C C + η S CS S = ρ m φ (C, S) C S (4) Given the equation of motion of environmental quality (1), if the growth rate of environmental quality is constant along the optimal path then the ratio C/S is also constant, which indicates that C and S grow at the same rate Let g be this common rate We have C/S = m g Substituting g into (4) leads to the following: ρ (1 + φ (C, S))m g = η CC + η CS φ(c, S) Therefore, the feasibility of the BGP, that is the constancy of g requires: Assumption 1 (i) The elasticities of the marginal felicity of consumption with respect to consumption, η CC, and environmental quality, η CS, are constant and (ii) the relative preference for environment is constant, ie φ (C, S) =φ, (C, S) 4 These restrictions 5 give rise to felicity functions of the Cobb-Douglas form, such as: CS φ 1 1/σ u (C, S) = 1 1/σ, (5) where σ = 1/η CC is the intertemporal elasticity of substitution for consumption We suppose that σ < 1, whichissufficient to ensure the concavity of the felicity function and implies also that the marginal felicity of consumption decreases with the level of environmental quality: u CS < 0 Thus, this negative felicity function (u () 0) is imposed hereafter It is easy to show that, with the specification of the felicity function (5), the growth rate along the BGP is: g = σ m ρ, (6) 1+φ which may be positive or negative, depending on the parameters 6 We can also notice that the optimal growth rate is bounded from above (g σm = lim ρ 0 g) σm therefore reflects the highest growth rate that society can epect; it depends on the regenerative capacities of the resource Besides, as S is growing along the BGP at the rate g and λ at the rate ρ m φ(m g), the transversality condition (3c) is fulfilled if and only if (g m)(1 + φ) < 0 ie g<mgiven that we have assumed σ < 1, the upper bound of g is strictly lower than m: g σm <mthe transversality condition is always satisfied 4 Smulders and Gradus (1996) show that (ii) is a necessary condition for the eistence of a balanced growth path when the stock of environmental resource is a source of felicity 5 Despite the lack of generality that these restrictions imply, they allow us to study the effect of endogenous discounting in a framework where the benchmark the case with constant and eogenous discounting consists in abalancedgrowthpath 6 The growth rate of consumption and environmental quality along the BGP will be positive if the natural regeneration and the preference for environment are sufficiently high, and if the discount rate is low enough 4

5 222 Properties of the BGP Introducing the stationary variable = C S, we easily show, using the equation of motion of environmental quality (1) and equation (4), that the dynamic system characterizing the evolution of the economy and the environment reduces to a single equation in, which writes: ẋ =(1+φ)( ), (7) where >0 is the stationary ratio of consumption to environmental quality along the BGP By (1) and (6), we have: = m g =(1 σ)m + σρ 1+φ (8) Equation (7) is unstable The ratio then takes from the initial time its stationary value, and initial consumption is: C 0 = S 0 = (1 σ)m + σρ S 0 (9) 1+φ 3 The economy with an endogenous discount rate 31 The model We now introduce an economy where the central planner uses a discount factor based on the historical path of environmental quality In all other respects, the economy is the same as in the previous section The intertemporal discounted utility function, with variable discount rate, of a representative consumer is now given by: U (C, S) = + 0 e t u (C t,s t ) dt, (10) where the felicity function u () has the same properties as in the previous model The future felicities are discounted at time t at the cumulated discount rate t 0, whichisassumedto depend on the past and current levels of environmental quality, as described by the following equation: t = t 0 θ(s τ )dτ, (11) where θ(s t ) 0, S t, denotes the discount rate at time t We also require: Assumption 2 (i) θ (S) > 0, and θ (S) < 0 S >0 and (ii) 0 =0 Note that in the case of an endogenous discounting depending on the past and current levels of consumption, there is a considerable disagreement over whether the marginal discount rate should be positive or negative 7 However, our assumption of a discount rate increasing with environmental quality reflects the sustainability motive that underlies in our model endogenous discounting: concerned by intergenerational equity, the society chooses to discount at a rate all the smaller than environmental quality is low, because in this case environmental questions become pressing The idea, that must be confirmed, is that a lower discount rate should in this case help to prevent or to limit further deterioration of environmental quality 7 See Epstein (1987) and Obstfeld (1990) 5

6 Following Obstfeld (1990), we consider t as a second state variable that accounts for accumulated impatience 8 Differentiating (11) with respect to time yields: t = θ(s t ) (12) The central planner s program is therefore to maimize (10) subjects to (1) and (12) together with the initial conditions (S 0 and 0 given) and non-negativity constraints (C t 0,S t 0) As in Palivos, Wang and Zhang (1997), we further impose the following assumption: Assumption 3 u (C, S)/θ(S) Qe β (S) C, S > 0, where <Q<0 and β < 1 This assumption ensures that the intertemporal utility function U () in (10), because u () is taken to be negative, is bounded from below and hence the optimization problem is well-defined 9 See Palivos, Wang and Zhang (1997) for a proof The constant-value Hamiltonian associated to the central planner s problem writes: H C, S,, λ, µ = u (C, S) e + λ [ms C] µθ(s), where λ 0 is the costate variable associated to environmental quality, and µ is the costate variable associated to the stock of accumulated impatience (Obstfeld (1990)) Given assumption 1 and applying the Pontryagin s maimum principle, the first-order necessary optimality conditions are: u C = λ λ λ = θ(s) m φc S + µ λ θ (S) µ u(c, S) = θ(s) µ µ (13a) (13b) (13c) where λ = λe and µ = µe (thus λ θ(s)λ = λ e ), together with (1) and (12) and the transversality condition 10 : lim H (t) =0 (14d) t First, note that, as Obstfeld (1990) pointed out, when µ converges to a definite long term value, as it will be the case below (given assumption 3), the third equation (13c) of this system can be integrated into: µ t = u(c s,s s )e s t θ(sτ )dτ ds = e t e s u(c s,s s )ds (14) t which means that µ t corresponds to the discounted present value of the future flow of felicities from the standpoint of time t Given that we have assumed a negative felicity function, we also have µ t 0 Net, we reduce the system (13a)-(14d) into a more tractable one Following Palivos et al (1997), recall that, along the optimal path, dh/ dt = H/ t Since the central planner s 8 Due to the non constant discount rate, Pontryagin s maimum principle cannot be applied directly We need this state variable to solve the problem within the standard optimal control approach 9 For the optimization problem to be completely well-defined, that is for the problem to satisfy the entire conditions for monotonicity and concavity in Epstein (1987), we suppose also that the current value Hamiltonian (specified below) is jointly strictly concave in C and S 10 On the transversality conditions in infinite horizon problems, see Michel (1982) t 6

7 program considered here is autonomous, H/ t =0, thus the Hamiltonian is independent of time along the optimal path This and the transversality condition (14d) imply that: H (t) =0 t along the optimal path We therefore can deduce: The system (13a)-(13b) then writes: 32 The optimal paths µ = 1 [u (C, S)+λ (ms C)] (15) θ(s) λ = u C (16a) λ λ = θ(s) m φc S + θ (S) u (C, S) (C ms)u C (16b) θ(s) u C Given the felicity function in (5) and assumption 1, the differentiation of the first-order condition (16a) with respect to time gives: λ λ = 1 C σ C + φ(1 1 σ )Ṡ S, and so, substituting the dynamics of the environmental quality (1) and the first-order condition (16b) into this equation, we show that the dynamics of consumption along the optimal path is: C C = φc S 1 C σθ(s)+(σ φ(1 σ)) m + σε(s) 1 σ S m, (17) where ε(s) = Sθ (S) θ(s) is the elasticity of utility discount rate θ () with respect to environmental quality ε(s) is positive under assumption 2 Now, if we reintroduce the stationary variable = C/S, the dynamic system characterizing the evolution of this endogenous discounted economy is then given by the motion of the ratio of consumption to environmental quality along the optimal path: = 1+φ + σ 1 σ ε(s) [ (1 σ)m] σθ(s), (18) together with the one of environmental quality (1) 321 The asymptotically balanced growth path Eistence of an asymptotically BGP Following Palivos et al (1997), let us first try to find an asymptotically balanced growth solution of the system (1), (18) Palivos et al (1997) show that necessary and sufficient conditions for the eistence of an asymptotically BGP solution of this type of problems is an asymptotically constant discount rate and an asymptotically constant elasticity of marginal felicity The second condition is by assumption fulfilled here Let us then assume that the utility discount rate is bounded from above: Assumption 4 (i) lim S θ(s) =θ is finite, and (ii) lim S 0 θ(s) =θ 0 7

8 The first part of this assumption seams reasonable: it means that when environmental quality becomes very high the discount rate remains bounded θ can be interpreted as the utility discount rate that the central planner would choose if economic activity did not harm environmental quality, which could remain always high The second part of this assumption just introduces the notation used for the lower bound of the utility discount rate, eventually equal to zero With this assumption, we obtain the following Proposition 1 :Underassumptions1-4,ifθ <m(1 + φ) (ie if the upper bound of the utility discount rate is low enough vis-à-vis the preference for environment and the natural regeneration rate), there eists an asymptotically balanced growth path characterized by: g = σ m 1+φ θ =(1 σ)m + σθ 1+φ (19) Proof Let us suppose that there eists a constant and positive growth rate g such that Ṡ lim t S = lim t Ċ C = g Because g>0, wehavelim t S =+ and assumption 4 can be used Moreover, Palivos et al (1997) show (Lemma 1) that under assumption 4 we have lim S Sθ (S) =0, ie lim S ε(s) =0 The long term of the system (1), (18) is then given by: g = m 0=(1+φ)( (1 σ)m) σθ, from which we deduce (19) By construction, this solution is valid if and only if g>0, ie θ <m(1 + φ), ie if and only if the maimal utility discount rate is sufficiently low vis-à-vis the preference for environment and the natural regeneration rate We may notice that the asymptotic growth rate g is eactly the same than the balanced growth rate g of the problem with eogenous discounting, provided that θ = ρ (see equation (6)) Then if the upper bound of the utility discount rate is eactly the rate that would be choosen by the central planner in case of eogenous discounting, the endogenous discounting economy follows in the long run the same path that the eogenous one, provided that this rate is low enough Convergence towards the asymptotically BGP The system (1), (18) is a dynamic system involving two variables: the environmental quality S, which grows asymptotically at a constant rate, and the ratio of consumption to environmental quality which is stationary in the long run Moreover, the second equation makes the growth rate of depend on the level of S, and so S cannot be easily eliminated to obtain a system involving two variables stationary in the long run Nevertheless, we obtain the following results Proposition 2 : Under assumption 1-4, if the upper bound of the utility discount rate is low enough (ie if θ <m(1 + φ)), then along the transition towards the asymptotically BGP: (i) is lower than its long run value, ie t < t; (ii) the growth rate of environmental quality is higher than its long run value, ie Ṡ t /S t > g t Proof It is easy to show that equation (18) can be written as: = 1+φ + σ 1 σ ε(s) ( )+σ θ θ(s) σ 2 θε(s) + (1 σ)(1+φ) 8

9 This equation shows that if, then ẋ/ > 0: diverges, which is impossible We therefore deduce that < t (this shows the first part of the proposition) We then have Ṡ/S = m = g + >g Propositions 1 and 2 indicate that endogenous discounting associated with an upper bound of the utility discount rate relatively low leads to an asymptotic growth path identical to the growth path of the eogenous discounting case (provided that the upper bound in the endogenous case is equal to the eogenous rate), but that in the short run environmental quality grows faster at the epense of a lower consumption Society is less impatient to consume, as long as environmental quality is not high enough See figure 1 m = eogenous disc endogenous disc ẋ =0 Ṡ =0 S Figure 1: The (asymptotically) balanced growth paths Case θ = ρ <m(1 + φ) ie <m 322 The stationary state Eistence of a stationary state Let us now suppose that the condition on parameters for a positive g is not fulfilled, ie θ m(1 + φ) (the upper bound of the utility discount rate is relatively high) In this case, an asymptotically balanced growth path does not eist, and we then look at the eistence of a stationary state of the dynamic system (1), (18) A stationary solution (,S ) of this dynamic system is characterized by ẋ=ṡ =0 Thus, equation (1) implies = m Substituting this into (18), we easily show that there will eist stationary solutions if values S eist such that: θ(s) =m 1+φ + σ 1 σ ε(s) (20) Assumption 5 Sθ (S) θ (S) 1 ε(s), S This assumption demands that the elasticity of marginal discounting, θ (S), is sensitive enough with respect to changes in the level of environmental quality Proposition 3 : Under assumptions 1-5, for high enough values of the upper bound of the utility discount rate (ie θ m(1 + φ)), there eists a unique stationary equilibrium (,S ) 9

10 characterized by: = m θ(s )=m 1+φ + σ 1 σ ε(s ) if and only if m 1+φ + σ 1 σ ε(0) > θ ie if and only if θ is low enough 11 Proof Let us consider the function f (S) =m 1+φ + σ 1 σ ε(s) > 0 which corresponds to the RHS of (20) A solution S of (20) will be such that θ(s )=f(s ) We know, by assumptions 2, 4 and 6, that θ(s) is strictly increasing from its lower bound θ (eventually equal to zero) to its upper bound θ Now, it easy to show that the first derivative of f () with respect to S is f (S) =m σ 1 σ ε(s) S 1 ε(s)+ Sθ (S) θ (S) f (S) is positive and decreases monotically for S =0to + from f(0) = m (21) Thus, under assumptions 2 and 5 above, f (S) < 0, and 1+φ + σ 1 σ ε(0) to m(1 + φ) There eists therefore a unique point S so that θ(s )=f (S ) provided that f(0) > θ Convergence towards the stationary state Proposition 4 : Under assumptions 1-5, the unique stationary equilibrium (,S ) is a saddlepoint Proof We can show that the Jacobian matri of the dynamic system is: 0 S J = σm σ 1 σ mε (S ) θ (S ) θ(s ) We then have the following determinant: σ det J = σms 1 σ mε (S ) θ (S ), with ε (S )= ε(s ) S 1 ε(s )+ S θ (S ) θ (S ) 0 by assumptions 2 and 5 So det J<0 Furthermore,wehavetrJ = θ(s ) > 0 Propositions 3 and 4 state that endogenous discounting associated with a high upper bound of the utility discount rate and a small lower bound of this rate leads to the convergence of the economy towards a locally stable stationary state (assumption 5 is also needed), while in the eogenous discounting case with a discount rate equal to the upper bound the economy follows a path where consumption and environmental quality continuously decrease towards zero This result means that endogenous discounting with a utility discount rate becoming sufficienty high when environmental quality is high and sufficiently low when environment is very depleted prevents any solution of collapse of environmental quality, as it would be the case with eogenous discounting See figure 2, where the ẋ =0curve is growing under assumption 5, and admits as an asymptot On figure 2 is depicted the case S 0 >S Then, in the eogenous discounting case, S decreases towards 0 while remains constant at, while in the endogenous dicounting case S and decrease, respectively towards S and Notice that when S 0 <S, the growth path does not change in the eogenous discounting case, while and S now grow towards their stationary values in the endogenous discounting case 11 If θ =0, this condition is always fulfilled 10

11 = m eogenous discounting endogenous discounting ẋ =0 Ṡ =0 S S Figure 2: The optimal paths Case θ = ρ m(1 + φ) ie m and θ <f(0) ie <m 323 The asymptotical depletion of environmental quality Eistence We now consider the last feasible case This case involves a high upper bound of the utility discount rate (ie θ m(1 + φ)), as in the previous case, but also a high lower bound, such that f(0) < θ and a stationary state no longer eists Proposition 5 : Under assumptions 1 5, if θ >m 1+φ + 1 σ σ ε(0) (ie if the lower bound of the utility discount rate is high enough vis-à-vis the preference for environment and the natural regeneration rate), an symptotical depletion of environmental quality occurs, with: g = σ m < 0 =(1 σ)m + θ 1+φ+ σ 1 σ ε(0) σθ (22) 1+φ+ σ ε(0) 1 σ Proof When environmental quality decreases towards 0, the long term of the system (1), (18) is given by: g = m 0=(1+φ + 1 σ σ ε(0))( (1 σ)m) σθ, from which we deduce (22) By construction, this solution is valid if and only if g < 0, ie θ >m 1+φ + 1 σ σ ε(0) = f(0) Convergence Proposition 6 : Under assumptions 1 5, if θ >m 1+φ + 1 σ, σ ε(0) then along the transition towards the asymptotical depletion of environmental quality: (i) is higher than its long run value, ie t > t; (ii) the growth rate of environmental quality is lower than its long run value, ie Ṡ t /S t <g t Proof It is easy to show that equation (18) can be written as: = 1+φ + σ 1 σ ε(s) σ 2 θ (ε(s) ε(0)) ( )+σ (θ θ(s)) + (1 σ) 1+φ + 1 σ σ ε(0) 11

12 This equation shows that if, then ẋ/ < 0: diverges, which is impossible We therefore deduce that > t (this shows the first part of the proposition) We then have Ṡ/S = m = g + <g Propositions 5 and 6 indicate that endogenous discounting associated with an upper bound of the utility discount rate relatively high and a lower bound also relatively high leads to an optimal depletion of environmental quality, as in the eogenous discounting case (provided that the upper bound in the endogenous case is equal to the eogenous rate), and that in the short run environmental quality decreases faster, which allows to consume more See figure 3 eogenous discounting = ẋ =0 endogenous discounting m Ṡ =0 Figure 3: The (asymptotically) balanced growth paths Case θ = ρ m(1 + φ) ie m and θ f(0) ie m S 4 A comparison of the two cases We can now try to compare the optimal paths involved by the choice of an eogenous or an endogenous discounting, in the specific framework we have used in this paper To make this comparison possible, that is to say to have a common reference for the two cases, we have made the assumption that the utility discount rate that would be choosen by the central planner in the constant and eogenous case is also the upper bound of the variable endogenous rate We then see that in some cases the asymptotic behaviour of the endogenous discounting economy is the same that the one of the eogenous one, but that there eists a situation in which it is fundamentally different Namely, endogenous discounting associated with a high upper bound of the utility discount rate and a small lower bound of this rate leads to the convergence of the economy towards a stationary state, while in the eogenous discounting case the economy follows a path where consumption and environmental quality continuously decrease towards zero: economy and environment stabilize in the first case while they optimally collapse in the second one This situation occurs when the upper bound of the utility discount rate is high enough and the lower one small enough, this latter characteristic of the endogenous discount function being the important one: the discount rate must become very small when environmental quality is very low for this situation to happen Figure 4 shows the evolution of felicity over time In the eogenous discounting case, for an initial value of the stock of environment S 0, felicity is at the beginning of the trajectory equal to u(s 0,S 0 ) and then decreases at a rate equal to (1 + φ) 1 σ σ g as S decreases at a rate equal to g < 0 In the endogenous discounting case, felicity is at the beginning of the trajectory equal to u( 0 S 0,S 0 ) andthenconvergestou(ms,s ) Figure 4 shows the case where 12

13 S 0 >S : environmental quality is better in the short term that it will be in the stationary state Then 0 <m< (see figure 2) and u( 0 S 0,S 0 ) <u(ms,s ) Felicity is non-decreasing over time, and the growth path is sustainable in this sense, contrary to the optimal path of the eogenous discounting case Endogenous discounting allows, under mild assumptions, to obtain a sustainable growth u(c t,s t ) u( S 0,S 0 ) u(ms,s ) t u( 0 S 0,S 0 ) endogenous discounting eogenous discounting Figure4:Evolutionoffelicityovertimeinthecasesofeogenousandendogenousdiscounting Case θ = ρ m(1 + φ), θ <f(0) and S 0 <S 5 Discount rates: a clarification of the concepts We now try to clarify the different concepts of discount rates that are used in the litterature, and to compare them in the eogenous and the endogenous cases 12 We can find for eample in Dasgupta (2000) the following definition: Let a numeraire be chosen for the accounting price system At any given date, the social rate of discount is the percentage rate at which the accounting price of the numeraire declines Formally, let R t be the quantity of the numeraire to be delivered at time t along the optimal programme Assuming continuous time and a differentiable accounting price, R 1 dr t t dt is the social rate of discount at t We must specify what eactly is the accounting price of the numeraire It is defined as the value of an increment of the numeraire at some date t in terms of its contribution to the objective function, along the optimal path (see for eample Heal (2001)) Let us first take felicity as the numeraire Then the quantity of numeraire you have to pay today for a unit of numeraire at time t is simply R t = e ρt in the eogenous case, and R t = e t in the endogenous one We have called in what precedes this quantity the utility discount factor We have also called the corresponding discount rate the utility discount rate Wecouldhave called it, following several authors, the pure rate of time preference Itisequaltoρ in the eogenous case, and t = θ(s t ) in the endogenous one It is a measure of the impatience of society This impatience is constant in the eogenous case, variable in the endogenous one, where it is low when environmental quality is low, and vice versa The ethical choice of society lies in this characterization of its impatience 12 This presentation is slightly different from the one of Epstein (1987) 13

14 But felicity is not generally chosen as numeraire It is consumption that usually plays this part We then define the consumption discount rate Ψ C t astherateatwhichtheaccounting price of consumption declines This price is simply the present value of the marginal utility of consumption at t, ie R t = e ρt u C (C t,s t ) in the eogenous case, and R t = e t u C (C t,s t ) in the endogenous one We then have: Ct Ṡt ρ η CC C t η CS S t Ψ C t = 1 R t dr t dt = θ(s t ) η CC Ct C t η CS Ṡt S t in the eogenous case in the endogenous case, (23) with η CC = Cu CC / u C and η CS = Su CS / u C the elasticities of the marginal utility of consumption previously defined The consumption discount rate then depends on the pure rate of time preference and on the growth rates of consumption and environmental quality As the elasticity η CC is negative, a positive growth rate of consumption makes the consumption discount rate higher than the pure rate of time preference It is the so-called wealth effect Besides this wealth effect appears an effect of the growth rate of environmental quality For intertemporal cost-benefit analysis the consumption discount rate is not sufficient We must also describe the evolution of the system of relative prices, which reduces here to the accounting price of environmental quality relative to consumption This relative price is the ratio of the value of an increment of environmental quality at some date t in terms of its contribution to the objective function to the value of an increment of consumption in terms of its contribution to the objective function We have just seen that for consumption this value is simply e ρt u C (C t,s t ) in the eogenous case, and e t u C (C t,s t ) in the endogenous one As far as environmental quality is concerned, this value is equivalently e ρt u S (C t,s t ) in the eogenous case, and the relative price is then simply p t = u S(C t,s t) u C (C t,s t) Things are less simple in the endogenous case Then the preference structure, as specified in (10), is recursive in the sense that is allowed to depend on the past and current levels of environmental quality, as described by equation (11) Thus, a change in the present level of environmental quality will not only have an effect on the current level of felicity, but also on the entire future felicity stream The contribution to the objective function must be measured by the increase in the lifetime utility function at time t caused by an increment of environmental quality at and near its starting point (see Obstfeld (1990)) This is a generalization of the notion of marginal utility at time t, correspondingtothevolterra derivative 13 Using the Volterra derivative, it can be shown 14 that the marginal utility of environmental quality at time t is e t u S (C t,s t ) µ t θ (S t ), with µ definedin(14) Giventhepropertiesofthefelicity function (u () 0, thus µ 0 and u C > 0, u S > 0) and assumption 2, this marginal utility is positive The relative price is then p t = u S(C t,s t) µ t θ (S t) u c(c t,s t) Everything being equal, it is higher than in the eogenous case For the specifications used here, we can sum up the previous results along the optimal paths: eogenous case endogenous case pure rate of time preference ρ θ(s t ) consumption discount rate Ψ C t ρ + 1+φ(1 σ) σ g θ(s t )+ σ 1 ẋt t + 1+φ(1 σ) Ṡ t σ S t relative price of environment p t φ φ t ε(s t ) m 1 1 σ t In the eogenous case, the pure rate of time preference, the consumption discount rate and the relative price of environment are constant along the optimal path, while neither of them are constant in the endogenous case, ecept of course in the long run, where they are equal to the 13 For more on the application of this concept see Ryder and Heal (1973), Epstein (1987) and Obstfeld (1990) 14 For the details of calculus see Pittel (2000) 14

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