Econ674 Economics of Natural Resources and the Environment

Transcription

1 Econ674 Economcs of Natural Resources and the Envronment Sesson 7 Exhaustble Resource Dynamc

2 An Introducton to Exhaustble Resource Prcng 1. The dstncton between nonrenewable and renewable resources can become blurred. 2. Renewable resources can be nonrenewable 3. Nonrenewable resources can, n a sense, be renewed through - the dscovery of new deposts - techncal advances that make t economcally feasble to recover a resource from low-grade materals. 4. We wll follow the conventon of classfyng resources as nonrenewable or renewable dependng on the sgnfcance of ther rates of regeneraton n an economc tme scale. Thus, e.g., ol s nonrenewable and tmber s renewable 5. Are our nonrenewable resources beng depleted rapdly/slowly? 6. What s the optmal use of exhaustble resources?

3 Beyond the use of mneral classfcaton to determne whether a resource s renewable or exhaustble, we also can vew the queston of tme n terms of the earth s energy flows, as shown below.

4 Wth ths perspectve n mnd, what steps are nvolved n analyzng the optmal prcng of exhaustble resources? A standard approach n such an nqury s to 1. derve condtons characterzng socally effcent resource use, 2. then establsh whether a compettve equlbrum realzes these condtons. 3. Examne the results n terms of whether market prcng acheves an optmal soluton, and where market falure arses, devse sutable polcy alternatves consstent wth an gven socal welfare functon. Do theorems of welfare economcs hold n the context of nonrenewable resources? Imperfectons can lead to an neffcent allocaton. Thus, even f a compettve equlbrum s effcent, we must consder the effects of relevant mperfectons and market falures; - Monopoly - Envronmental dsrupton n extractng, convertng, and use - Uncertanty surroundng dscovery and the long-lastng effects of current decsons about ther use.

5 The theory of optmal exhaustble resources contans two key results: A) prce = MC + opportunty cost. Why should opportunty cost be ncluded? Nonrenewable resources are lmted n quantty and are not reproducble. Thus, consumpton today has a future opportunty cost that should be taken nto consderaton. Accountng for the opportunty cost bears crtcally on how to allocate exhaustble resources over tme, and t recalls the varous postons regardng sustanablty already dscussed. One mplcaton s that when opportunty cost s ncluded, there wll be less extracton of a resource today than f t were reproducble. The relatonshp between Prce and Margnal Cost for an exhaustble resource

6 Gven the demand p = p(y), only y* wll be extracted by a planner or resource manager seekng to allocate producton effcently over tme, I.e. there wll be a postve dfference between prce and margnal cost. The dfference between prce and the margnal extracton cost (MEC) s known varously as the user cost, royalty, rent, net prce, or margnal proft. We wll use rent, consstent wth the taxonomy of factor prcng. B. The present value (PV) of the rent for an exhaustble resource must be the same n all perods. Equvalently, undscounted rent must ncrease at the prevalng rate of nterest, or dscount. The net socal beneft from extractng a unt of an exhaustble resource s the dfference between the market prce (or the wllngness of consumers to pay) and the margnal extracton cost). Effcency Condtons under Pure Depleton for a Compettve Industry Followng Hotellng (1931), under pure depleton, f R(t) represents remanng reserves, and q(t) represents producton, then: ( t) = q( t) EQ( 1) R &! The prce of the exhaustble resource, p(t) can be represented as a functon whose form s:! p t = p 0 e t EQ 2 ( ) ( ) ( )

7 Effcency Condtons under Pure Depleton for a Compettve Industry - 1 Under compettve condtons, the rate of extracton at tme t s determned by the followng demand functon: Assumng no extracton costs, ntal reserves wll be exhausted va the followng functon: At t=t, q(t)=0, we then have: Equaton 3-5 determnes p(0), T, and the entre tme-path of extracton. As an example, assume that the demand functon D(A) s lnear: thus: Under the last equaton, wth t=t and q(t)=0, we have Thus: q! T 0 ( t) D( p( t ) EQ( 3) q = q ( T ) = D p( ) ( t) dt = R EQ( 4)! T ( 0 e ) = 0 EQ( 5) ( t) = D( p( t ) = a bp( t) EQ( 6) q! q q! ( t) = a " bp( 0) e t EQ( 7) p "! ( 0) = ae T / b ( ) EQ( 8) " ( t! T ) ( t) = a 1! e

8 Effcency Condtons under Pure Depleton for a Compettve Industry - 2 Exhauston of ntal reserves mples: T $ a 1" e # ( t"t ) dt = R EQ 9 Integraton yelds: 0 at ( )! ( )!" T 1 ( 1! e ) a = R EQ( 10) " Pure Depleton under Monopoly Let us now examne the behavor of a monopolst owner of an exhaustble resource. As n the compettve ntal case, we assume zero margnal extracton costs. The monopolst s objectve functon can be expressed as: Subject to: T m 0 #! t ( ( )) ( ) ( ) max " = $ p q t q t e dt EQ 11 ( t) R( ) R EQ( 12) R & =! q 0 = Where p(q(t)) = the nverse of D(p(t)). The monopolst s problem can be formulated as an optmal control problem. The monopolst s current value Hamltonan may be wrtten as: ( q( t ) q( t)! ( t) q( t) EQ( 13) H = p µ The frst-order necessary condtons are: " H " q ( t) = p + q ( t) " p " q () # ( t)! µ ( t) = 0 => MR( t) = S.P( t) EQ( 14)

9 Pure Depleton under Monopoly - 1! H µ& " #µ( t) = "! R H R & = " =! q " µ ( t) ( t) = 0 Equaton 15 mples that.e., the current value shadow prce (CV SP) rses at the rate of nterest. However, by equaton 14, the CV SP s equated to margnal revenue (MR) at each nstant t. Thus the monopolst extracts the resource so that the MR rather than the compettve rent rses at the rate of nterest,.e.: MR If we assume a lnear demand functon, the nverse can be expressed as: and the monopolst s MR functon thus s defned as: MR = a /b " 2q t In terms of our Hamltonan, we have: EQ( 15) ( t) EQ( 16) µ( t)! µ & / = MR ( t) = " EQ( 17) ( t) = a / b! q( t) b EQ( 18) p / ( ) /b EQ( 19) H ( T m ) = 0.e., q(tm )=0 Evaluaton equaton 14 at t = T m our nverse demand functon mples: µ ( T ) = a / b " q( T )/ b! µ ( T ) = a b EQ( 20) m m m /

10 Pure Depleton under Monopoly - 2! ( ) ( ) t m But µ t = µ 0 e and µ ( T m ) = µ ( 0) e Thus from equaton 20, "! whch yelds: µ m ( 0) = ae T / b µ " ( t! m ) ( t) = ae T / b EQ( 21) Equatng equaton 19 wth equaton 21 and solvng for q(t) yelds: q a 2 " ( t! T ) ( t) = 1! e m As before, the condton on total reserves gves: Now we compare the explotaton profles of the compettve and monopolstc ndustres. If Tc denotes the compettve exhauston date, we obtan from equaton 10 and equaton 23: ( ) /# = R /a and T m " 1" e "#T m Snce these condtons are an ncreasng functon of T, t follows that: T c < T m Eq (24) Equaton 22 and 8 show that q c (0)>q m (0): Thus, compettve ndustres explot the resource at a hgher rate ntally and exhaust t more rapdly than a monopolst.! T ( ) EQ( 22)!" m ( 1! e T )/ 2" R EQ( 23) a Tm! a = 2 T c " 1" e "#T ( ) /# = 2R /a

11 An Example of Alternatve Extracton Paths under Competton and Monopoly Let a = 100, b = 10, and R = 1,000 tons. The resultng values of T are: Tc = years and Tm = years. The producton rates q c (t) and q m (t) are shown n the followng fgure: the correspondng prce paths are: P c (t) = a - bq c (t) P m (t) = a - bq m (t). P m (t) starts out hgher than p c (t); they ntersect at t years and s below p c (t) untl t reaches a choke-off prce a/b =10 at T m = Under these condtons, the monopolst appears as the conservatonst s frend (Solow, 1974). Intally, the monopolst restrcts producton and stretches t out over a longer tme horzon.

12 The monopolst s motvaton doesn t derve from concern for future generatons, but smply ncreases the PV by restrctng producton early on. The compettve extracton path s socally optmal, and the monopolstc path s dynamcally neffcent n the sense that the current generaton could more than compensate future generatons for an ncrease n the near term extracton and a reducton n future extracton. To see how ths result s obtaned, assume that the socal welfare from producton q(t) s gven by the area under the nverse demand curve gven by: q( t) U( q( t ) =! p( z) dz 0 so that the socal manager would lke to : max = " If we construct the current value Hamltonan, get the frst-order condtons and solve, we observe that: from equaton 28, and From equaton ( q( t ) EQ( 25) R& =! q( t) ( 0) = R gven EQ( 26) subject to R T U e!# t The the welfare maxmzng and compettve extracton paths are dentcal. dt ( q( t ) = p( q( t ) = p( t) = ( t) U ' µ µ( t ) = p / p( t)! µ & / & =

13 How reasonable s the assumpton that the compettve prce grows at the prevalng rate of nterest? For ths to be so requres that the owner of the exhaustble resource: -estmate the date of depleton and choke-off prce accurately - must use the same dscount rate Ths seems to be an unreasonable assumpton for extractve ndustres n that t requres perfect foresght. It thus gnores: - prce volatlty and other forms of uncertanty - exploraton, dscovery, and technologcal change, whch suggest the possblty of alternatve prce paths. Smooth exponentally ncreasng prce paths have not n fact been observed n extractve ndustres such as mnng. In lght of these consderatons, let us now consder a few modfcatons to the basc Hotellng model: Postve Extracton Costs for a Sngle Exhaustble Resource Suppose the cost of mnng depends only on the rate of extracton,k e. C(t) = C(q(t)) Eq. 31 Assume further that p(t) s exogenous and known n advance. The owner of the resource then would: T " $ t max = # [ p( t) q( t) " C( q( t )] e dt 0 R& = " q( t) ( 0) = R gven R( t)! 0 EQ( 32) subject to R The current value Hamltonan can now be defned as: ( t) q( t)! C( q( t )! ( t) q( t) EQ( 33) H = p µ wth frst-order necessary condtons that mply: ( t) " C' ( q( t )" µ ( t) = 0 EQ( 34) & µ & = ( t) EQ( 35) p!µ Convexty n C(.) mples that the frst-order condtons also are suffcent. We thus assume C (.)>0, and C (.)>0.

14 It now can be shown that: d dt ( p( t) " C' ( q( t ) p( t) " C' ( q( t ) =! EQ( 36) whch mples that the prce net of margnal cost ncreases at the prevalng rate of nterest. The correspondng condton for the monopolst s prce s d dt ( R' () " # C' () " ) R' () " # C' () " =! EQ( 37) Equatons 36 and 37 tell us about dfferences n extractve resource owner problems such as how the tme path of extracton q(t) s determned. Geometrcally, assume a U-shaped cost curve. The resource owner wll never produce at a rate q wth )<q<q*. At q*, MC equals AVC. Unless the extracton shuts down temporarly, we have: q* " q( t) for 0 " t " T EQ( 38)

15 At tme t = T, the extracton operaton stops permanently and q drops to zero. The transversalty condton for a free-termnal tme problem mples: From equaton 33, we obtan: Whle equaton 34 mples that: µ T Hence: H = 0 at t = T C µ T! ( ) = p( T ) C C ' = ( q( T ) so that: q(t) = q* EQ (42) For a known T horzon, the problem s solved. As equaton 40 determnes p(t) and by equaton 35 we have: µ By equaton 34 we have: from whch q(t) s determned for O < t < T. EQ( 39) ( q( T ) EQ( 40) q( T ) ( ) = p( T )! C' ( q( T ) ( q( T ) EQ( 41) q( T ) # ( t" T ) ( t) = e µ ( T ) 0! t! T EQ( 43) C '( q( t ) = p( t)! µ ( t) T s determned from the condton that R(T) = 0 (Assumng that p(t)>c(q )/q* for all t, whch mples that the stock of the resource wll be exhausted eventually. As an example, suppose prce p(t) = p (constant), and assume that C(q) = a + bq 2. Then q * = a / b and equatons mply the condtons: µ C $ ( t! T ) ( T ) = p! C' ( q *) = p! 2 ab µ ( t) = e µ ( T ) 0 " t " T 1 $ ( t! T '( q( t ) = 2bq( t) = p! µ ( t) q( t) = p! e ) ( p! 2 ab) T pt p! 2 ab 1! e # q 0 2b 2b $! $ T ( t) d( t) =! = R 2b [ ]

16 The last equaton has a unque soluton T>0, so that the problem s solved. Now consder a dscrete-tme verson of the exhaustble resource owner problem. We can state the problem as: T! 1 t = " # q t t= 0 maxv { } subject to R R 0 = 1000 t+ 1 [ 1! q / R ]! R =! q EQ( 44) For δ = 0.10 and T = 10, v 10 = Snce T assumes nteger values, there s no dervatve condton for dscrete-tme problems. To determne whether t s optmal to lengthen or shorten the extracton horzon one needs to: - ncrease or decrease the horzon, - solve the optmal producton schedule (qt), - calculate the present value of net revenues, Vt, and compare the results. t t t t q t Another look at the Compettve Extractve Industry Let an extractve ndustry be comprsed of N-prce-takng owners. Let: -C (q(t)) (I=1,2, N(N)>1) = the cost of extracton for the th frm - R, (I=1,2, N (N)>1) = ntal reserves, T - p(t) = the prce, whch s determned by aggregate producton accordng to: p N & = ' % = 1 # ( t) p$ q ( t)! EQ( 45) "

17 Each frm attempts to maxmze ts profts gven by: T " = % [ p( t)q ( t) # C ( q ( t) )]e #$t dt EQ 46 subject to the reserves constrant: R = "q ( t) R ( 0) = R R ( T) # 0 EQ( 47) From ths we need to determne the prce path, p = p(t): In the statc theory of the frm, we can proceed on the bass of the assumpton of compettve markets alone. In a dynamc settng, we must assume that the frm can correctly predct the entre prce profle p(t) over tme. Ths assumpton s known as ratonal expectatons. The th frm s PV (not CV) Hamltonan can be defned as: H =[p(t)q (t)-c (q (t))]e -δt λ (t)q (t) EQ (48) wth necessary condtons that mply: and: Thus mples: " a constant. The transversalty condton H(T)=0 But, evaluatng EQ(49) at t = T also mples: and thus, 0 p ( )! t =! s the level of producton where: ( )! t ( t) # C' ( q ( t ) = " ( t) e EQ( 49) " = # $H $R t ( ) = 0 EQ( 50) #! T ( T) = [ p( T )# C ( q ( T )/ q ( T )] e EQ( 51) that s, where the AVC of the th resource operaton s mnmzed. #! T [ p( T )# C ( q ( T )] e EQ( 52) " = ' * q ( T ) = q ( q( T ) C ( q ( T ) q ( T ) EQ( 53) C ' = /

18 Substtutng EQ(53) nto EQ(49) mples: C' At ths stage we have 2N+1 unknowns, T 1,,T n, q 1 (t), q N (t), and p(t) [λ 1,,λ N can be determned from EQ(51)] and 2n+1 equatons consstng of EQ(54) wth q 1 (T1)=q * plus EQ(45) and T q t dt = R! 0 The Socal Optmum wth N Frms Wth No externaltes, no stock common pool externaltes, the perfectly compettve ndustry would be socally optmal n the sense of maxmzng ts dscounted socal welfare (DSW). Ths can be verfed more formally, as defned below: The socal welfare functon can be defned as: The correspondng Hamltonan s gven by: ( N N % N ( % "* t H = & U&! q ( )! ( ( ) # t # " C q t e "!) ( t) q( t) EQ( 56) ' ' = 1 $ = 1 $ = 1 Each! ( t ) =!, a constant (snce δh/δr =0) and we now have where: and, " ( ( ) ( ) [ ( ) ( ( )] ( t! T q t = p t! p T! C' q T e ) EQ( 54) thus If all N frms have dentcal reserves and costs, T = T and H(T) = 0 whch mples: ( ) max 0 ( t ) " C ( q ( t ) R& = " q( t) ( 0) = R R ( t)! 0 EQ( 55) subject to R (.. &, U, '- - T = / 0q 0 N = 1 + * N = 1 + ) e * " 1t % dt# $ #! t [ U '( Q( t )# C ( q ( t )] e EQ( 57) " = ' N ( ) q ( ) Q t =! t = t 1! U /! q =! U /! Q "! Q /! q ( Q( t ) = p( Q( t ) p( t) U ' = #! T [ p( T )# C( q( T ) q( T )] e EQ( 58) " = " = /

19 * * ( ) and: q T = q = q If the N frms are not dentcal, t can be shown that: #! T [ p( T )# C( q ( T ) q( T )] e EQ( 59) " = / The same system of equatons s obtaned as n the compettve model under the ratonal expectatons assumpton. Both lead to the same soluton. Scarcty Defned n Economc Terms Economcs does not consder scarcty as a physcal concept but as a value concept. The rent for a non-renewable resource s gven by the co-state varable: µ Ths reflects the dfference between the prce and the margnal extracton cost at nstant t. In a compettve market, ths s the dfference between what socety would be wllng to pay for an addtonal unt of R(t) and the cost ncurred n ts extracton. If ths dfference: - s postve and large, then the resource s scarce. - ncreases over tme ( µ& > 0).e., the resource s becomng more scarce - decreases over tme ( µ& < 0).e., the resource s becomng less scarce. ( t ) = p( t) " MC( t)! t EQ( 60)

20 Exploraton Exploraton and dscovery ncrease reserves, whch may lower extracton costs (e.g., C(q,R 2 )<c(q, R 1 ); R 1 <R 2 ). Thus, there are economc ncentves to add to known reserves. Followng Pndyck (1978), we state the queston of exploraton as: and: ( w( t) X ( t ) q( t) EQ( 61) R & = f,! X & = f, ( w( t) X ( t ) EQ( 62) where f(.) s a dscovery rate as a functon of: -w(t), the exploratory effort, and -X(t), the level of cumulatve dscoveres. Assumng that q(t) s a lnear cost functon, the total extracton cost per unt of tme can be defned as: C 1 ( R( t) )q( t) EQ(63) C1(.) s a unt extracton cost functon that s dependent on remanng reserves. Exploraton costs are assumed to be a convex functon gven as C 2 (w(t)) Thus, net revenue at nstant t, gven p(t), s the per unt prce of the exhaustble resource: t q t! C R t q t C w t EQ 64 ( ) ( ) ( ( ) ( ) ( ( ) ( ) p 1! 2 Now assume a compettve extractve ndustry: a large and dentcal number of frms. The ndvdual frm takes p(t) as exogenous, and under ratonal expectatons, attempts to maxmze the followng functon: T max # ( p( t)q( t) " C 1 ( R( t) )q( t) " C 2 ( w( t) )) e "$t dt EQ(65) 0 subject to R = f w( t), X( t) ( ) X = f w t R( 0),X 0 ( ) " q t ( ( ),X ( t) ) ( ) gven

21 The problem has: - two state varables, R, X, and - two control varables q, w=>, and s more dffcult than earler problems The current-value Hamltonan s: H = p( t)q( t) " C 1 ( R)q( t) " C 2 w The frst-order condtons nclude: "H "q t "H "w t ( ) + µ 1 [ f (#) " q t fx and fw are partals of f() wth respect to X(t) and w(t). Takng the tme dervatve of 67 mples µ & 1 ( t ) = p& ( t) " C' ()R& 1! Substtutng ths expresson nto EQ(69) yelds: Equaton 68 may be solved for µ 2 (t) to get: µ Takng the tme derve of ths expresson yelds: µ 2 = C'' 2 (") Substtutng the expresson for µ 2 (t) nto 70 and notng from 68 that µ 1 ( t ) + µ 2( t ) = C' 2! / We obtan: µ 2 = " C' 2 # ( )] + µ 2 f # ( ) EQ( 66) ( ) = p ( t ) # C 1 ( $ ) # µ 1 ( t) = 0 EQ(67) ( ) = #C' $ 2 ( ) + ( µ 1 ( t) + µ 2 ( t) ) f w = 0 EQ(68) µ 1 "#µ 1 = " $H $R t µ 2 "#µ 2 = " $H $X t ( ) = " p( t) # C 1 ( $ ) p t ( ) = C' % 1 ( )q( t) EQ(69) ( ) = " ( µ 1( t) + µ 2 ( t) ) f x EQ(70) ( ) + C' 1 $ ( t) C' ()! f " p( t) + C ()! 2 = 2 / w 1 ( ) f ( $ ) EQ( 71) w / f w # ( f w,w w + f w,x X )C' 2 f #2 w # p + C' 1 R EQ 72 ( ) () f w ( ) [ µ & =!µ " ( µ ( ) + ( t ) ] t µ 2 ( ( ) / f w $ p( t) + C 1 (#)) $ C' 2 (#) f x / f w EQ( 73) f x

22 Equatng 72 and 73 and solvng for the rate of change n w yelds: w = % & ' f w,x X C' 2 f + p " C' 2 1 w By substtutng 3.70 for the rate of change n p and smplfyng: %[( f w w,x / f w ) f (") + # $ f x ]C' 2 +C' 1 (")q( t) f ( w = ' * &' ( C'' 2 (") $ f w,w C' 2 / f w ) )* Equatons 61, 62, 71, and 74 are a four-equaton dynamcal system for the fve unknown functons R(t), X(t), w(t), p(t), and q(t). The latter functon may be elmnated by the demand equaton: q(t) = D(p(t)) ( ) R + # C' 2 $ f w "#p( t) + #C 1 ( $ ) " C' 2 ( $ ) f x The termnal condtons for the rates of change n p and w depend on C' 2 / f w as t => T. Ths expresson defnes the rato of the margnal cost of exploraton to the margnal product of exploratory effort, and s referred to as the margnal dscovery cost. If C2 (O)/fw(0,X)=0, then w(t+ q(t) = 0 smultaneously at t = T. It wll also be the case that : µ 2 ( T ) = 0 and µ 1 ( T ) = p( T )! C1( R( T ) = 0 Ths means that no addtonal proft can be obtaned from further extracton. If C' 2 ( 0) / f w( 0, X ) =! > 0 then exploratory effort wll become 0 before extracton, I.e., there wll exst an nterval T1<t<T where w(t) = 0, but q(t) >0. At t = T, µ ( T 1) 0 and wth w(t) = 0, µ& 0 Then for all t n T, 2 = p 2 = ( t) () " = µ =! # C 1 1 Ths mples µ& 1 = 0 and that % C' 1 () $ q( t) /" #! at t approaches T. thus for all t n T, both p(t) - C 1 () and c 1 (.)q(t) reman constant, mplyng p(t), C 1 (), and c 1 () rse as q(t) falls. f w ( + )- *, f w 2 f w C'' 2 $ ( ) " f w,w C' 2. 0 EQ 74 / ( ) EQ( 75)

23 The last unt of reserves should be dscovered when ts MC of dscovery equals the sum of the net revenue obtaned upon extracton, and sale, and the value of cost savngs after dscovery, but before extracton. Ths sum s the PV of net revenue of the margnal dscovery. The system R, X, p, and w leads to several dynamc possbltes. In graphcal terms, we may portray the dynamc effects n terms of the followng graphs shown below:

24 Usng Dscrete-Tme Models to Solve for the Optmal Rate of Extracton of an Exhaustble Resource

25 Usng Dscrete-Tme Models to Solve for the Optmal Rate of Extracton of an Exhaustble Resource - 1

26 Usng Dscrete-Tme Models to Solve for the Optmal Rate of Extracton of an Exhaustble Resource - 2

27 Usng Dscrete-Tme Models to Solve for the Optmal Rate of Extracton of an Exhaustble Resource - 3

28 Usng Dscrete-Tme Models to Solve for the Optmal Rate of Extracton of an Exhaustble Resource - 4