The Optimal Depletion of haustible Resources: A Complete Characterization. Hassan Benchekroun a and Cees Withagen b Abstract We provide the closed form solution to the Dasgupta-Heal-Solow-Stiglitz (DHSS) model. The DHSS model is based on the seminal articles Dasgupta and Heal (Rev. con. Stud.,974), Solow (Rev. con. Stud.,974) and Stiglitz (Rev. con. Stud.,974) and describes an economy with two assets, man-made capital and a nonrenewable resource stock. We eplicitly characterize, for such an economy, the dynamics along the optimal trajectory of all the variables in the model and from all possible initial values of the stocks. We use the analytical solution to prove several properties of the optimal consumption path. In particular, we show that the initial consumption under a utilitarian criterion starts below the maimin rate of consumption if and only the resource is abundant enough and that under a utilitarian criterion, it is not necessarily the present generation that bene ts most from a windfall of resources. JL Classi cation: 2, Q3, C65 Key words: ehaustible resources, Dasgupta-Heal-Solow-Stiglitz economy, eponential integral (a) Department of conomics, CIRQ. McGill University. mail: hassan.benchekroun@mcgill.ca. (b) Department of Spatial conomics, VU University Amsterdam. mail: cwithagen@feweb.vu.nl. ()We thank NWO for nancial support. The rst author also thanks SSHRC and FRQSC for nancial support. This project started while the rst author was a visiting scholar at Tilburg University. We thank Geir Asheim, Saad Benchekroun, Raouf Boucekkine, Lucas Bretschger, Robert Cairns, Michael Hoel, Jack Pezzey, Sjak Smulders, Nico Temme and the participants of SURD28 in Ascona, where an earlier version of this paper was presented, for helpful comments. Usual disclaimers apply.
Introduction We provide a closed form solution to the Dasgupta-Heal-Solow-Stiglitz (from here on DHSS) model. The DHSS model is based on seminal articles by Dasgupta and Heal (974), Solow (974) and Stiglitz (974). It describes an economy with two assets, man-made capital and a nonrenewable resource stock. Together with man-made capital the raw material from the resource is used as an input in the production of a commodity that can be used for consumption and for net investments in man-made capital. In this framework some important questions have been addressed. For instance, in the case where the objective is to maimize the minimum rate of consumption throughout the time horizon, a central question is whether, despite the resource constraint, there eists a sustainable constant positive rate of consumption (Solow, 974). Another stream of the literature adopts a utilitarian objective and studies the optimal consumption and investment paths that maimize a discounted sum of utility from consumption (Stiglitz, 974). In the present paper we give the optimal paths for the DHSS economy under a utilitarian objective. In the literature attention has mainly been given to the case where the production function is Cobb-Douglas and instantaneous utility is logarithmic. ven for these speci cations no closed form solutions of the optimum have been found so far. There has been some progress regarding the characterization of the optimal solution to the DHSS problem, in particular it has been shown without eplicitly nding the solution that consumption can be single peaked (see Pezzey and Withagen, 998, and Hartwick et al., 23). However, in the absence of a closed-form solution it is not possible to address other relevant issues such as understanding the relationship between the instant of time where the peak takes place and the initial stocks of capital and the resource and how this phenomenon depends on the model parameters. Moreover, to actually calculate the optimal path more information is needed on the co-state variables associated with the stocks, which amounts to having a complete solution of the model 2. The closed form solution to the DHSS problem uses the eponential integral function. The eponential integral belongs to a family of special functions which are etensively used in mathematical physics and probability theory. They are particularly helpful to determine solutions to di erential equations encountered in physics (see e.g., Temme, 996, Ch. 5 and Ch. 7). The use of special functions in economic theory is relatively recent. Boucekkine et al. (28a and 28b) show that the solution to a two-sector Lucas-Uzawa model of endogenous growth can be epressed in terms of a speci c type of For high rates of pure time preference consumption always decreases over time and for low rates of time preference consumption monotonically increases during an initial interval of time, reaches a maimum, and eventually decreases. 2 To make progress on the analytical solution, Pezzey and Withagen, 998, and Hartwick et al., 23, have considered the special case where the elasticity of marginal utility is equal to the output elasticity of capital. This is clearly a very narrow class of economies since there no relationship between this two distinct parameters, i.e. the set of economies for which this relationship holds is of measure zero. 2
special functions : the hypergeometric functions 3. In our problem of nding the solution to the canonical model of an economy with resource constraints, it is another type of special function, the eponential integral 4, that turned out to be instrumental in epressing the solution in a closed form. Thus, along with Boucekkine et al. (28a and 28b), this paper is a proof that special functions can play a key role in analyzing dynamic economic problems and characterizing the transition dynamics of all the variables in a dynamic problem and from all possible initial values of the state variables 5. We eploit the eplicit form of the solution to study the behavior of the optimal consumption and investment paths as functions of the parameters of the model. We compare the solution to the DHSS model, under a utilitarian objective, to the solution of the problem where the objective is to maimize the minimum rate of consumption over the whole time horizon. The solution to the latter problem is called the maimin rate of consumption. Sustainability in the DHSS contet requires that the Hartwick rule, i.e. zero genuine savings, holds: the investment rate must equal the etraction rate times the marginal product of the resource. Asheim (994) shows that this condition should hold at all instants of time. Hence, if at some instant of time Hartwick s rule holds it does not mean that the economy is on a sustainable path. The argument used by Asheim rests on the assumption that in a utilitarian optimum the initial rate of consumption is below the maimin rate of consumption if the pure rate of time preference is small enough, and above the maimin rate of consumption if the pure rate of time preference is large. By continuity there is a rate of time preference for which both initial consumption rates coincide, and for which the utilitarian rate of consumption is increasing for an initial period of time. Asheim (994) refers to a graph in Dasgupta and Heal (979) to support this assumption with respect to the rates of time preference. However, Dasgupta and Heal do not provide a formal proof of their claim. Also, no proof is given of continuity. In our case we are able to provide a proof to both claims and show that the ratio of the maimin rate of consumption over the initial rate of consumption is a strictly decreasing continuous function of the rate of discount. Moreover, we show that investments in man-made capital may also be single-peaked. We provide the condition under which overshooting in man-made capital occurs. In particular, this phenomenon arises in relatively natural resource rich economies. Our treatment allows us to study the relationship between the ratio of the maimin rate of consumption over the initial consumption rate under a utilitarian criterion and the stocks of the resource and capital. Using the ratio of the resource stock over the capital stock as an indicator of resource abundance we show that 3 See also Perez-Barahona (2). 4 The eponential integral belongs to a class of special functions called the con uent hypergeometric functions which solve the Kummer di erential equation, a con uent of the Gauss hypergeometric di erential equation. For more details we refer the reader to Temme, 996, Ch. 5 and Ch. 7. 5 Dynamic systems in economics, in particular those involving more than one state variable, have been so far treated rigorously but mostly using qualitative techniques such as phase diagrams accompanied with an analytical study of the behaviour in the neighborhood of a steady state, or using numerical techniques. 3
the initial consumption under a utilitarian criterion starts below the maimin rate of consumption if and only the resource is abundant enough and that under a utilitarian criterion, it is not necessarily the present generation that bene ts most from a windfall of resources. The outline of the paper is as follows. The model is introduced in section 2. Section 3 contains the characterization of the optimum in a series of lemmata and propositions. The mathematical appendi contains the proofs. Section 4 covers some sensitivity analyses and section 5 concludes. 2 The model and preinary results Let K(t) and S(t) denote the stock of man-made capital and the nonrenewable resource at instant of time t; respectively. The variables C(t) and R(t) are rates of consumption and resource etraction at instant of time t and are assumed non-negative. Let be the production elasticity of man made capital ( < < ): The rate of pure time preference is : We assume to be strictly positive. The case of zero discounting is etensively treated in Dasgupta and Heal (979) and is also discussed in Asheim et al. (27). For any variable (t) we adopt the convention _(t) d(t)dt: Consider the following optimal control problem of the DHSS economy, which we refer to as the DHSS problem: subject to Z Ma C e t U (C (t)) dt () _K(t) K(t) R(t) C(t) (2) _S(t) R(t) (3) K () K > and S () S >. (4) with 8 < U (C) : C for 6 ; > ln C for A solution of the problem above is described by a quadruple of paths (C; K; R; S). Let (t) and (t) denote the co-state variables associated with the stock of capital and the natural resource stock, respectively. The current value Hamiltonian is given by H(K; R; C; ; ) U (C (t)) + [K R C] R The maimum principle yields (t) U (C (t)) C(t) (5) ( ) K(t) R(t) (t) (t) (6) 4
_(t) (t) H K (t) K(t) R(t) (t) (7) _(t) (t) H S (t) (8) Any solution that satis es the above system along with the following transversality conditions is a solution to the optimal control problem ()-(4). e t (t) K (t) (9) t! e t (t) S (t) () t! In the sequel we aim at obtaining an eplicit solution to the DHSS problem. The line of attack can be sketched as follows. Using (6), (7) and (8) yields _(t) (t) ( ) ( e t ) (t) () where (), the initial value of ; still to be determined. It is easily veri ed that the solution to this di erential equation reads (t) e t " + ( ) # t where ; the initial value of ; is still to be determined. From (2), (6) and substituting consumption from (5) we have _K(t) K(t) (t) ( )(t) (2) (t) (3) and since and are given functions of time, we solve for K as a function of time, and of the initial and : Net we can determine R(t) from (6) and solve for the resulting resource stock S(t): Finally, we use the transversality conditions (9) and () to solve for the initial values of the co-state variables. Given the strict concavity of the utility and production functions involved, if the DHSS problem has a solution, it is unique. Hence, if we nd a solution satisfying the transversality conditions it is the unique solution to the DHSS problem. Remark : Note that so far we haven t used the resource constraint. The resource constraint will be used below to determine the initial values of the co-state variables as well as the path of the resource stock. It does not a ect the functional form of the paths of consumption, capital accumulation and resource etraction. It turns out that the solution for K from (3), and hence for R and S; can be epressed in terms of a special function, i.e., the eponential integral de ned as a (z) Z 5 e zu u a du
with a 2 R and z > (see e.g., Abramowitz and Stegun, 972, and Temme, 996 6 ). A special case of the eponential integral is (z) Z e zu u du A simple change of variable allows to have this useful alternate epression for the eponential integrals The function a (z) is a strictly decreasing function of z and, nally, Z a (z) z a e v v a dv. (4) z (z) e z z : Another property of the eponential integral that will prove useful for our purposes is (see Abramowitz and Stegun,972, p.229, inequality 5..2) 2 e z ln + 2 z < (z) < e z ln + z (5) which implies that (z) and (z). z! z! + 3 Solving the DHSS problem In this section, we provide the steps to determine the solution to the set of conditions given by the maimum principle (5)-(8). In view of (2) it will turn out convenient to de ne ( ) ; (t) + t; (t) (t) ; () ; The following observations will be useful for the rest of the analysis: the case implies, if and only if and the variable (t) is an a ne function of time with (t) + t. (6) 3. Consumption Proposition The optimal consumption path is given by C(t) e Proof: This is straightforward from (2) and (5) t (t) ( ) 6 Both Abramowitz and Stegun (972) and Temme (996) de ne the eponential integral with a integer and allow for z to be comple with Re (z) >. However, the de nition etends naturally to allow a to be real or comple. generalized de nition of the eponential integral that we use. In our analysis the argument, a, is real. It is this 6
3.2 Man-made capital From (3), we have that the stock of man-made capital is given by where K (t) K e R t f(z)dz + Z t g (z) e R t z f(s)ds dz (7) f(t) (t) ( )(t) and g(t) (t) : Since (t) e t and (t) e t (t) from (2) and (6), we have and Z t f(t) z f (z) dz (t) ln (t) (z) We now determine the second term of the right hand side of equation (7). Lemma (8) Z t g (z) e where (t) is de ned in (6). Proof: Appendi A. R t z f(s)ds dz e (t) We can now derive the path of the capital stock. Proposition 2 The optimal path of the stock of capital is K (t) (t) K e (t) (t) (t) (9)! (t) (2) Proof: Substituting (8) and (9) into (7) gives (t) K (t) K e (t) (t) (t) : Factoring (t) ( ) and taking into account that ( ) yields (2) 3.3 traction It follows from (6) and (6) that R(t) (t) K(t) 7
Direct substitution from Proposition 2 yields Proposition 3: The optimal etraction rate is given by R (t) K e (t)! (t) (2) 3.4 The resource stock The optimal path of the stock of the resource is the unique solution to (3) with S () S. Proposition 4. The optimal path of the stock of the resource is given by S (t) S K e e 2 (t)! t with () 2 ( () ()) : Proof: Appendi B. 3.5 Solving for the co-state variables To fully characterize the optimal paths of consumption, the rate of etraction, and the stocks of capital and the resource we still need to determine and. We use the transversality conditions (9) and () to do so. Lemma 2 Given > ; the vector ( ; ) is given by! K e (22) 2 B @ S ( ) C A ( )+ : (23) Proof: Appendi C. 8
De ne A ( )! S K (24) and h () + Lemma 3 For any S and K positive, is the unique solution to h ( ) A. Proof: Appendi D. The determination of the optimal solution to the DHSS problem is now complete. For any given positive values of S and R we solve for using Lemma 3, then derive the initial values of the costate variables and from Lemma 2. We obtain the time paths of all the model s variables from Propositions -4. 4 Sensitivity analysis We eploit the analytical tractability of the solution to the DHSS model to establish some key features of the optimal paths. We focus on the consumption and the investment paths. 4. The optimal consumption path We rst study the conditions under which consumption is increasing for some initial period of time. We highlight the role of the parameters of the model, like the pure rate of time preference and the initial stocks of capital and the natural resource, on the possibility that consumption may rise for an initial period of time. For the ease of eposition only, we focus on the case of a logarithmic utility function:, i.e. U (C) ln C. From the speci cation of consumption in Proposition it follows that the time at which maimum consumption is reached is t + (25) Clearly t > i < which holds i A > A ~ h >. Indeed, we show in Appendi that h is a strictly decreasing function of and therefore, given A and such that h ( ) A we have < i A > ~ A. 9
Therefore, consumption is initially (i.e., for t < t ) increasing over time, if and only if A > A. ~ For any 2 (; ) we have A S ( ) K > A ~ when S K is large enough or is small enough. More precisely, from lemma 3 and (25) and using the implicit function theorem we have and thus da d d d h ( ) > since h ( ) < and da d < (26) dt d 2 d d < : Also observe that the time where the peak takes place goes to in nity as the rate of pure time preference goes to zero. Indeed,! implies A! ; which implies that in the optimum h ( )! ; so that, according to Appendi D,! : Therefore, for each given instant of time, the di erence between the optimal rate of consumption with zero discounting can be made arbitrarily close to the utilitarian optimum with discounting by choosing the rate of time preference small enough. Note, however, that this convergence does not imply convergence over the entire trajectory. Consequently, by choosing the rate of time preference small enough, we can postpone the moment in time at which consumption decreases below the maimin rate of consumption. We can also determine dt da d da h ( ) > and therefore an increase of S K implies a larger t. Note that the eistence of a phase where consumption is increasing with time depends on the ratio of S and K and not the absolute values of S or K. The peak of consumption can be epressed, after manipulations, as C e S (e ( )) or, in terms of the initial stock of capital, as C e K : ( ) Consider now the rate of consumption that solves the following problem Ma C fmin U (C)g (27) subject to (2)-(4). It can be shown (Solow, 974) that, provided that 2 problem, referred to as the maimin rate of consumption, is < < ; the solution to this ~C (2 ) S K 2 :
The main criticism of the utilitarian criterion is that it discounts future consumption. It is intuitive to think that the solution under such a criterion, which favors present consumption relative to future consumption, would result in larger initial consumption than any consumption rate that would be sustained at all time. We show below that this is not true in its generality. There eists a rate of pure time preference ^ > such that if < ^ the initial rate of consumption is below the maimin rate of consumption. The initial rate of consumption in the utilitarian framework is C C () K e ( ) The analysis of the ratio ~ CC as a function of allows to determine the behaviour of ~ CC as a function of as well as S K in a compact way. The ratio ~ CC can be written as (28) where ~CC (2 ) ( ) ( ) e ( ) ( ) e ( ) : Lemma 4: We have d d < for all > with! ( ) and! + ( ) Proof: See Appendi F: We can now link the ratio ~ CC to and to S K through h ( ) A with A given in (24). Proposition 5a: The ratio ~ CC is a strictly decreasing function of. Moreover there eists ^ > such that ~ CC > i < ^: Proof: From (24) we have since da d < and in Appendi we show that h () < for all >. da d d d h ( ) > (29) Using Lemma 4 along with d d > and the fact that! ( ) and! + ( ) completes the proof Asheim (994) obtains this result using the assumption that the consumption path in a utilitarian optimum has the following properties: (i) consumption is a continuous function of the rate of pure time preference and (ii) there eists ^ > such that if < ^ the initial rate of consumption is below the maimin rate of consumption. These, plausible, properties have been assumed for instance by Asheim (994) to show that if at some instant of time Hartwick s rule holds it does not mean that the economy
is on a sustainable path. Our analysis provides a proof of both properties and shows monotonicity of ~CC with respect to. In the eisting litterature these properties were shown to hold in the very special case where the production function is Cobb-Douglas with constant returns to scale where the production elasticity of capital is assumed equal to the elasticity of marginal utility (see e.g., Pezzey and Withagen (998) and Hartwick et al. (23)). In our analysis we do not rely on this restrictive assumption. Moreover, we can give closed form solutions of all other relevant variables, which will be further eploited. Indeed, our treatment also allows to determine the relationship between the ratio ~ CC and the ratio S K. Proposition 5b: The ratio ~ CC is a strictly decreasing function of S K. Moreover there eists a ratio \ S K > such that ~ CC > i S K > \ S K : Proof: From (24) we have d d (S K ) h ( ) ( ) The proof that h () < for all > is given in Appendi. Using Lemma 4 along with completes the proof! < (3) d d(s K ) < and the fact that S K! ( ) and SK! + ( ) The ratio S K can be considered as an indicator of resource abundance. Proposition 5b states that the initial consumption under a utilitarian criterion starts below the maimin rate of consumption if and only the resource is abundant enough. This is not a priori intuitive. The utilitarian criterion is generally considered as biased towards present generations and therefore under such a criterion it may be intuitive to epect that abundance of resources will be heavily eploited by present generations at the detriment of future generations. We have shown that a more abundant resource increases both the maimin rate of consumption and the initial consumption rate under a utilitarian criterion. However, the latter increase is smaller than the former. Thus, under a utilitarian criterion, it is not necessarily the present generation that bene ts most from a windfall of resources. 4.2 The optimal investment path The optimal investment path can be obtained by direct substitution of C (t), K (t) and R (t) _K (t) (t) e K (3) (t) (t) e t (t) 2
I,C,C.25 Investmentand Consumption, S.5 Investmen Consumption.2.5 Cmaimin..5.5 2 4 6 8 t Figure : The investment path can also initially increase over time and decline after reaching a peak and eventually become negative. This was shown earlier by Asheim (994). In particular he proves that if the rate of time preference is such that C ~ C this will be the behavior of the investment path. Here we provide a numerical eample that illustrates the investment and consumption patterns as a function of resource abundance. More precisely we set ; K ; :6 and :3 and we plot the optimal investment and consumption paths under a utilitarian objective and the maimin consumption rate for di erent values of the stock of the resource. We rst set S :5, we have that C < ~ C and that investment is initially increasing over time and declines after reaching a peak and eventually becomes negative, see Figure. When we set S :, we have C > ~ C and investment is always decreasing over time and eventually becomes negative, see Figure 2. There eists a threshold stock of the resource ~ S for which C ~ C. For our numerical eample the approimate value of ~ S is :825. In Figure 3 we plot the case where S :825. Note that investment becomes negative before the moment beyond which the optimal consumption path under the utilitarian objective falls below ~ C forever. numerical simulations we have conducted. It can be shown that, when 2 ; investment at time zero is equal to _K () 2 2e ( ) e ( ) S. We obtained this qualitative result for all From (48) (see Appendi F) we know the denominator is negative. Therefore, we have positive investment 3
.8.6 I,C,C Investmentand Consumption, S. Investment Consumption Cmaimin.4.2.2 2 4 6 8 t Figure 2: I,C,C. Investment and Consumption, S.825 Investment Consumption Cmaimin.5 2 4 6 8 t Figure 3: 4
at time zero i e ( ) > 2. Moreover, also from (48) (see Appendi F), we have d (e ( )) d + e ( ) < : Therefore, there eists ( :289) such that overshooting in man made capital occurs (i.e., _K () > ) i <, i.e. i or K small enough or S large enough. Thus, it is relatively resource rich countries that are more likely to overshoot in man made capital. 4.3 The case of decreasing returns to scale We nally consider the case of decreasing returns to scale, where the production function is given by Y (t) K(t) R(t) with + and ; (32) Trivially this function can be rewritten as Y (t) K(t) R(t) K(t) ~ R(t) (33) where ~ R (:) is such that ~ R(t) R(t) for all t. The resource constraint then reads _S(t) ~R (t) (34) As was noted in Remark in Section 2, the functional forms of consumption, capital accumulation and resource etraction, are not a ected by the resource constraint hence the optimal consumption as well man-made capital along the optimal path still have the same form as in Proposition and 2 7 8. We can use this observation to derive an interesting result with respect to the initial level of consumption. It follows from the necessary conditions (5) -(8) that _C()C() > if and only if < ( ~ R()K ) The right hand side of the condition depends on the endogenous variable ~ R; but, clearly, the condition is satis ed for a large enough initial resource stock. Hence, we may conclude that for a large enough initial resource/capital ratio or for a rate of time preference small enough initial consumption will be increasing. Moreover, for a high rate of time preference consumption will decrease monotonically, and it will start above the maimin level. 7 Proposition 3 can be invoked to obtain the functional form of R ~ (t). The optimal etraction rate R (t) is thus given by ~R (t). 8 While the functional forms of the optimal paths in Proposition -2 carry over to the case of a production function that ehibits decreasing returns to scale the initial values of the co-state variables are obviously di erent when the case where 6. from 5
5 Conclusion We have given a closed form solution to the seminal model of an economy with man-made capital and ehaustible resources, based on Dasgupta and Heal (974), Solow (974) and Stiglitz (974). For this economy with two assets, we give a closed form representation of the dynamics of all the variables in the model and from all possible initial values of the state variables. We establish several features that the solution may ehibit. In particular, we determine the condition under which the consumption is initially increasing with time and the condition under which initial investment is positive. We have shown that the initial consumption under a utilitarian criterion starts below the maimin rate of consumption if and only the resource is abundant enough and that under a utilitarian criterion, it is not necessarily the present generation that bene ts most from a windfall of resources. 6
Appendi A: Proof of Lemma From (8) we have Hence; since g (t) Z t (t) g (z) e e R t z f(s)ds (t) (z) and (t) e t (t) by de nition, it holds that R t z f(s)ds dz From (5) we have (t) (t) (z) ; z Z t g (z) e Consider the following change of variable Then dw du: Hence Z t (t) R t z f(s)ds dz (t) e z (z) Z t e z (z) and d(z) dz: Hence Z (t) e e (t) e (t) w u u (t) dz (z) (z) ) Z (t) e Z (t) e dz (z) d(z) u u du u u du Z (t) e u u du Z (t) e w w + Z (t) + Z dw e w w dw e w w dw Z (t) e w w dw! We also have Let! w (t) : Then d! (t) dw and Z e w w dw (35) Z (t) e w w dw Z e (t) + Z (t) (t) (t)!! (t) d! + (t) e (t)!! d! 7
Hence So Z t g (z) e Z (t) e w w dw R t z f(s)ds dz e (t) (t) + (t) (t) (t) Appendi B: Proof of Proposition 4 The resource etraction path is given by (2): R (t) K e (t)! (t) (36) S (t) S () Z t R(s)ds (37) Z t K! e dz + e Z t (z) (z) dz K! e t e Z t (z) (z) dz To complete the determination of the path of the stock of resource we need to determine R t (z) (z) dz. We have (t) + t Consider the following change of variable: (z) + z! + t! (z): Then z 8
and d dz and therefore Z t (z) (z) dz Z (t) () d 2 Z (t) 2 Z () d (t) 2 (t) () d Z () d! (38) where where Z B (:) is the Gamma function and a a () d (B) + (2 a) (39) (B) B 2 a ( a (B) a (B)) To show (39), we rst use (4) and then integrate by parts. We have Z B a a () d Z B F () d BF (B) Z B F () d where This gives or Z B Z B F () Z e t t a dt: a a () d B 2 a a (B) + a a () d B 2 a a (B) + Z Z B a e d a e d Z B a e d: Using (4), the last part of the right-hand side can be substituted by B 2 a a (B) and we have where Z B a a () d B 2 a a (B) + (2 a) B 2 a a (B) is the Gamma function. Thus, using (38), (37) and noting that completes the proof simpli es into Appendi C: Proof of Lemma 2 9
t! e t (t) K (t) t! + t (t) K t! (t) K e e (t)! (t) (t)! (t) (4) The series epansion of the eponential integral when z tends to in nity (see Abramowitz and Stegun (972), 5..5) reads (z) e z z 2 + O z!! 3 z (4) Therefore t! (t)2 (t) t! (t) 2 e (t) (t) + O!! 2 (t) For any 2 R we have t! (t) 2 e (t)! 2 e which implies (t) t! e (t) (t) The transversality condition gives (t) t! e (t) (t) t! e (t) (t) 2 (t) K t!! e This is satis ed if K This is true for all >. Solving for gives (22). e (42) Net we prove (23). We start again from Proposition 4 and take the transversality condition for K into account. Recalling that () 2 ( () ()) we nd S () e (t) 2 t! 2
Using the epansion (4) above gives t! ()! 2 ( () ()) So, the transversality condition becomes S which, after substitution of and noting that B @ e () 2 2 (( )+) S ; gives ( ) C A (43) Appendi D: Proof of Lemma 3 The proof is divided in two steps: (i) proof that must be solution to h () A and (ii) proof that there eists a solution to h () A. and with (i) We have Therefore Substituting ( ) into (45) gives Using the following relationship yields S e 2 K e S K S K ( ) 2 and thus 2 ( )! 2 S K (44) (45) : + 2
or Recall that ( )! S K A ( ) +! S K and then solves h () + h ( ) A This completes (i). (ii) We now argue that h ( ) and! + h ( )! which given the continuity of h over (; ) proves, the eistence of a sulution. 5..4) We start by rewriting h () using the recurrence relationship (see Abramowitz and Stegun (972), which gives () () (z) e z z (z) 2 2 () () e ( ) () () and thus h () () () () () e ( ) () () or h () () e () ( ) From Abramowitz and Stegun (972), 5..5, we have that (z) is asymptotically equal (and we use the symbol ) to (z) e z z ( + ) ( + ) ( + 2) + z z 2 z 3 ::: (46) 22
and therefore z e z (z) z This combined with 2 (; ) gives ( + ) ( + ) ( + 2) + z z 2 z 3 ::: (47) z e z (z) and thus h (). z!! We now turn to the case where! +. Let n denote the smallest integer strictly smaller than. It is straightforward to show that () is a strictly decreasing function of. Hence, since n, we have We have for all 2 (; ] h () () () n ()! + () Using L Hopital s rule we have, for > and similarly for n ()! + () n ()! + () () n () () n () e! + n () () :::! + n () n ()! + () n +() () :::! + +()! + n () We now show that! + e n () e! +! + Note that n 2 (; ]. We distinguish between n 2 (; ) and n : n () e n () : Suppose that n 2 (; ). The asymptotic behavior of n (z) when z! is given by n (z) z n ( ( n )) which is found by using the following relationship X n ( ) n z n n! ( ( n ) + n) where a (z) z a ( a) z a ( a; z) is the Gamma function and is called the incomplete Gamma function, and by the asymptotic behavior in the neighborhoud of zero (see Temme, 996, p. 279). Therefore, and since n 2 (; ) e n () e n ( ( n )) e! + e () 23 X n ( ) n n n! ( ( n ) + n)
or and thus using gives thus since For the case where we have and thus To sum-up we have,! + e () h () :! + n (i.e., integer) we have! + e () 2 e ln + 2 < ( ) < e ln + 2 ln + 2 < e ( ) < ln + < ln + e ( ) < 2 ln + 2! + ln +! + e () h () :! + h ( ) and! + h ( )! where h (:) is a continuous function of over (; ) :Therefore there eists at least one solution > to h ( ) A; for any A > : Taking into account that the solution of the optimal control problem under consideration is unique, we thereby also establish uniqueness of Appendi : Proof that h () < for all >. We compute the derivative of the function h and nd h () (e ) 2 ( ) (e ( ) + e ) () + ( ())) 2 ( ()) 2 + ( ) The sign of the denominator is positive. Therefore the sign of h () is the same as the sign of the numerator, denoted by N(). Using the property of the eponential integral that 2 e ln + 2 < () < e ln + 24
it holds for all 2 (; ) that N () < e 2 Z () where Z () ( ) ( ( ) + ) 2 ln + 2! + ln + 2. For the derivative of Z () we get Z () 2 ( ) ln( + 2 ( ) + ) + ( + 2) + ln( + 2 ) 2 + ln( + ) The second derivative is Z () 2 2 3 + 2 2 2 + 4 + 4 2 + 3 ln ( + ) + ( + ) 2 ( + 2) 2 2 Using we have + < ln + < 2 3 + 2 2 2 + 4 + 4 2 + 3 ln( ( + )) + > 2 3 + 2 2 2 + 4 + 4 2 + 3 + + and thus Z () < ( 2) which after simpli cations becomes 2 3 + 2 2 2 + 4 + 4 2 + 3 + ( + ) 2 ( + 2) 2 2 Z () < ( 2) (3 + ) + (3 ) 2 + 3 ( ) + ( + ) 3 ( + 2) 2 2 < So Z () < for all 2 [; ] and all 2 (; ) and therefore + Z () 2 (Z () ; Z ()) with! Z () and! + Z () 25
that is Z () > for all 2 (; ) and therefore with Z () 2 (Z () ; Z ()) Z () and Z ()!! + Since Z () < all 2 (; ) we have N () < and h () < all 2 (; ) Appendi F: Proof of Lemma 4 We show that (i) d(e ( )) d > with < e ( ) < for all > and (ii) d(e ( )) d < with < e ( ) for all >. (i) We have d(e ( )) d +( + ) e ( ). Using the fact that (see Abramowitz and Stegun (972) p. 229 Inequality 5..9) we have z + < ez (z) < z gives d(e ( )) d > and therefore e ( ) (; ) with + < e ( ) for z > (48) is a strictly decreasing function of over the domain using gives thus since we have and! + e () 2 e ln + 2 < ( ) < e ln + 2 ln + 2 < e ( ) < ln + < ln + e ( ) < 2 ln + 2!! + ln +! + e () @ 2 A z ln + 2 z! + ln ( + z) 26
therefore e () is a strictly decreasing function from (; ) into (; ). (ii) d(e ( )) d + e ( ) from e ( ) < we have d(e ( )) d < for all > with 2 ln + 2 < e ( ) < ln + and therefore and! e () +! e () 27
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