4. The multiple use forestry maximum principle. This principle will be derived as in Heaps (1984) by considering perturbations

Transcription

1 4. The muliple use foresry maximum principle This principle will be derived as in Heaps (1984) by considering perurbaions H(; ) of a logging plan H() in A where H(; 0) = H() and H(; ) A is di ereniable in. Le () = H (; 0). As h(; ) = H (; ) a.e., i follows ha () = h (; 0) a.e.. The presen value of a perurbed logging plan is W () = W T ()+W NT () = e rao [p( v(; ))h(; ) C(h(; ))]e r d+ (10) e rao =0 Z w(;) h(; )e r [ E(u)e ru du]d Then leing q() be any absoluely coninuous muliplier de ned for a o W T () = e rao [p( v(; ))h(; ) C(h(; )) (11) +q()(h (; ) h(; ))]e r d Inegraing by pars q()h (; )e r d = lim!1 q()h(; )e r q(a 0 )Ae ra 0 + (rq q)h(; )e r d

2 Imposing he ransersaliy condiion lim!1 q()h(; )e r = 0 and di ereniaing wih respec o hen gives W 0 T () = e rao [( p( v(; ))v (; )h(; ) +(p( v(; )) C 0 (h(; )) q())h (; )) + (rq q)h (; )]e r d and in Heaps (015) (see (1)) his gives W 0 T (0) = e rao [( p( v) v + p(w )e r(w ) + rq q) +(p( v) C 0 (h()) q()) ]e r d For he nonimber values W 0 NT () = e rao =0 Z w(;) fh (; )e r [ E(u)e ru du]+h(; )w E(w(; ) )e rw(;) gd (1) Now w h() = (() (w)) w a.e. can be subsiued in he second erm of

3 W 0 NT (0).1 This erm becomes e ra 0 [ =0 [() we(w )e rw d (w) we(w )e rw ]d] Since () 0 for < a 0, he boom limi of he rs inegral here can be increased o a 0. Making he change of variable = v(s) in he second inegral gives R 1 s=a 0 E(s v(s))(s)e rs ds. Consequenly W 0 NT (0) = e rao [E(w ) we r(w ) E( v)) Z w() +( E(u)e ra du) ]e r d (13) The muliplier q() can hen be chosen for a o o be an absoluely coninuous funcion so ha a.e. (see (17) below) q rq = p(w )e r(w ) p( v) v + E(w ) we r(w ) E( v) (14) 1 From H(w(; ); ) = H(; )+A, di ereniaion wih respec o gives h(w((; ); )w (; )+ H (w((; ); ) = H (; ). Leing! 0, h(w)w = () (w) and muliplying by w gives h()w = (() (w)) w. A furher di ereniaion wih respec o gives h(w)w + h(w)w + h (w)w = 0. I he second di ereniaion is wih respec o, hen h(w) ww + h(w) w + h (w) w h () = 0. For laer use ake w imes he rs of hese equaions minus w imes he second. The resul can be wrien as h()w + h ()w = h(w) ww + h(w)w w. 3

4 and he coe cien of under he inegral sign in W 0 (0) is zero a.e.. This sum of inegrals hen simpli es o W 0 (0) = e rao [p( v) C 0 (h()) q() + Z w() E(u)e ru du)] e r d (15) The curren value Hamilonian for he muliple use foresry maximum principle is hen Z w() J(h) = p( v())h C(h) + h E(u)e ru du qh (16) The opimal h should saisfy W 0 (0) = e rao R 1 a o (@J=@h) e r d 0 for all admissible. In Heaps (1984), i is shown a his sage ha his implies ha he opimal h mus maximize J(h) subjec o h [0; h]. The argumen presened here can be modi ed so ha i also applies for he muliple use foresry age class model. Thus a maximum principle for his model has been obained. Proposiion 3. MUFMP An opimal soluion h for he muliple use foresry age class coninuous model (5 ) maximizes he Hamilonian J(h) subjec o h [0; h] and a o. The oher condiions in ( 5) mus also be sais ed and he muliplier q() (de ned on [a o ; 1)) should be absoluely coninuous and saisfy The argumen may be obained from hp:// 4

5 a.e. he adjoin equaion (14). The ransversaliy condiion lim!1 qh()e r = 0 should also hold. Also useful are he inegral forms of he adjoin equaion which are q() = Z w() [ p(s v(s)) v + E(s v)]e r(s ) ds (17) = Z [ p(w(s) s) + E(w s) w]e r(w(s) ) ds These formulas show ha q() is well de ned for any opimal logging policy and a 0. I should also be noed ha he MUFMP is su cien as saed nex in Proposiion 4. The proof is a sraigh forward exension of he proof of he similar proposiion in Heaps (1984). 3 Proposiion 4. The MUFMP is su cien o describe he opimal soluion o he exended foresry age class coninuous model. Moreover, an opimal logging policy is unique. 3 The proof may be obained from hp:// pr.pdf. 5

6 Proof: From Heaps (1984, (34)) i follows ha W 00 T (0) = e rao [( p( v) r p( v))h(v)v C 00 (h()) ]e r d Fo he nonimber value funcion, di ereniaing WNT 0 () gives W 00 NT () = e rao f[h ()w +h()w rh()w ]E(w )+h()w E(w =0 )ge rw d Now d d [h(w)w E(w )e rw ] = f[ h(w) ww + h(w) w w rh(w) ww ]E(w ) +h(w)( w 1)w E(w )ge rw g I follows ha W 00 NT () = e rao f d d [h(w)w E(w =0 )e rw ] + h(w)w E(w )e rw gd = e rao h(w)w E(w =0 )e rw d 6

7 when only permuaions for which w(0; ) = w(0) and hus w (0) = 0 are used. Now make he change of variable = v(s) so s = w(). WNT 00 () = e rao h(s)w (v(s)) E(s s=a 0 v(s)) v(s)e rs ds Furher h(s) = h(v) v(s) and from w(v(s; ); ) = s i follows ha w(v(s))v + w (v) = 0 and w (v) v(s) = v. Thus WNT 00 () = e rao h(v)v (s) E(s s=a 0 v(s))e rs ds In sum and leing! 0 W 00 NT (0) = e rao [( p( v) r p( v) + E(s v(s))h(v)v C 0 (h()) ]e r d This is negaive for an opimal harvesing policy since v a 8 and by condiion (f) p(a) r p(a) + E(a) < 0 8a a. Now le H 1 be he opimum soluion of he foresry age class model and H some oher soluion of he exended foresry maximum principle. Then for 0 1, H(; ) = H 1 () + (1 )H () is an admissible cumulaive harvesing plan because he se of admissible harvesing plans is convex (see below). Leing 7

8 W () be he ne presen value of his logging policy, W () mus have a relaive minimum on [0; 1] for some < 1. Now = H = H 1 H, = h 1 h and H = 0 = h. The calculaion of he second variaion as in (4.3) hen applies and i was esablished ha W 00 () < 0. This rules ou an inerior so = 0 mus be he relaive minimum. The derivaion of he exended foresry maximum principle conains he following formula (under (4)). W 0 (0) = 0 (@J=@h)(h 1 h )e r d is evaluaed a (h ; v ). Since h maximizes J subjec o 0 h h, i follows 1 h ) 0 and W 0 (0) 0. This, ogeher wih W 00 (0) < 0 rules ou 0 as a relaive minimum. In summaion, he exended foresry maximum principle has a unique soluion and his mus be he opimum for he foresry age class model. Concerning he convexiy of A, suppose h i A for i = 0; 1 and [0; 1]. Le h = (1 )h 0 + h 1 and use similar noaion for H and w. I is obvious ha h sais es all he condiions of (10A) excep possibly (d) ea w () for 0. Noe ha H (w ()) = H () + A = (1 )(H 0 () + A) + (H 1 () + A) = (1 )H 0 (w 0 ()) + H 1 (w 1 ()). Le w() = minfw 0 (); w 1 ()g. Then H (w ()) 8

9 H (w()) which implies w () w() and also w () minfw 0 () ; w 1 () g ea. 9