Sustainable Use of Renewable Resources
Geoffrey Heal1998, Springer eBooks
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https://www.academia.edu/72306516/ ;
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Munich Personal RePEc Archive
Sustainable use of renewable resources,
Chapter 2.1
Graciela Chichilnisky and Andrea Beltratti and Geoffrey Heal
1998
Online at http://mpra.ub.uni-muenchen.de/8815/
MPRA Paper No. 8815, posted 22. May 2008 05:46 UTC
Sust ai nabi l i t y : Dynami cs and Uncer t ai nt y
Gr aci el a Chi chi l ni sy, Geof f r ey Heal , and Al l essandr o
Ver cel l i
Edi t or s
Kl uwer Academi c Publ i sher s, 1998
ANDREA BELTRATTI , GRACI ELA CHI CHI LNI SKY ANDGEOFFREY HEAL
2 . 1 . Sust ai nabl e Use of Renewabl e Resour ces
1. I nt r oduct i on
We consi der her e opt i mal use pat t er ns f or r enewabl e r esour ces . Many i mpor t ant r esour ces ar e i n t hi s cat egor y : obvi ous_, ones ar e f i sher i es and f or est s .
Soi l s, cl ean wat er , l andscapes, and t he capaci t i es of ecosyst ems t o assi mi l at e and degr ade wast es ar e ot her l ess obvi ous exampl es . ' Al l of t hese have
t he capaci t y t o r enew t hemsel ves, but i n addi t i on al l can be over used t o t he
poi nt wher e t hey ar e i r r ever si bl y damaged. Pi cki ng a t i me- pat h f or t he use of
such r esour ces i s cl ear l y i mpor t ant : i ndeed, i t seems t o l i e at t he hear t of any
concept of sust ai nabl e economi c management .
We addr ess t he pr obl em of opt i mal use of r enewabl e r esour ces under a
var i et y of assumpt i ons bot h about t he nat ur e of t he economy i n whi ch t hese
r esour ces ar e embedded and about t he obj ect i ve of t hat economy. I n t hi s second r espect , we ar e par t i cul ar l y i nt er est ed i n i nvest i gat i ng t he consequences
of a def i ni t i on of sust ai nabi l i t y as a f or mof i nt er t empor al opt i mal i t y r ecent l y i nt r oduced by Chi chi l ni sky [ 7] , and compar i ng t hese consequences wi t h
t hose ar i si ng f r om ear l i er def i ni t i ons of i nt er t empor al opt i mal i t y. I n t er ms
of t he st r uct ur e of t he economy consi der ed, we r evi ew t he pr obl emi ni t i al l y
i n t he cont ext of a model wher e a r enewabl e r esour ce i s t he onl y good i n
t he economy, and t hen subsequent l y we ext end t he anal ysi s t o i ncl ude t he
accumul at i on of capi t al and t he exi st ence of a pr oduct i ve sect or t o whi ch t he
r esour ce i s an i nput .
Al t hough we f ocus her e on t he t echni cal economi c i ssues of def i ni ng and
char act er i zi ng pat hs whi ch ar e opt i mal , i n var i ous senses, i n t he pr esence of
r enewabl e r esour ces, one shoul d not l oose si ght of t he ver y r eal mot i vat i on
under l yi ng t hese exer ci ses : many of t he ear t h' s most i mpor t ant bi ol ogi cal and
ecol ogi cal r esour ces ar e r enewabl e, so t hat i n t hei r management we conf r ont
' Thi s paper dr aws heavi l y on ear l i er r esear ch by one or mor e of t he t hr ee aut hor s, namel y
Bel t r at t i , Chi chi l ni sky andHeal [ 1- 3] , Chi chi l ni sky [ 7, 81, and i n par t i cul ar many of t he r esul t s
her e wer e pr esent ed i n Heal [ 18] .
49
G. Chi chi l ni sky et al . ( eds . ) . Sust ai nabi l i t y : Dynami cs and Uncer t ai nt y, 49- 76.
© 1998 Kl uwer Academi c Publ i sher s . Pr i nt ed i n t he Net her l ands .
50
A. Bel t r at t i et al .
t he f undament al choi ce whi ch under l i es t hi s paper , namel y t hei r ext i nct i on,
or t hei r pr eser vat i on as vi abl e speci es . I n t hi s cont ext t he r ecent di scussi on of
sust ai nabi l i t y or sust ai nabl e management of t he ear t h' s r esour ces i s cl osel y
r el at ed t o t he i ssues of concer n t o us . ( For a mor e compr ehensi ve di scussi on
of i ssues r el at i ng t o sust ai nabi l i t y and i t s i nt er pr et at i on i n economi c t er ms,
see [ 18] . For a r evi ew of t he basi c t heor y of opt i mal i nt er t empor al use of
r esour ces, see [ 10, 11, 15] . )
We assume, as i n [ 19] and i n ear l i er wor k by some or al l of us [ 1- 3] t hat
t he r enewabl e r esour ce i s val ued not onl y as a sour ce of consumpt i on but al so
as a sour ce of ut i l i t y i n i t s own r i ght : t hi s means t hat t he exi st i ng st ock of
t he r esour ce i s an ar gument of t he ut i l i t y f unct i on . The i nst ant aneous ut i l i t y
f unct i on i s t her ef or e u ( c, s) , wher e c i s consumpt i on and s t he r emai ni ng
st ock of t he r esour ce . Thi s i s cl ear l y t he case f or f or est s, whi ch can be used
t o gener at e a f l ow of consumpt i on vi a t i mber , and whose st ock i s a sour ce of
pl easur e . Si mi l ar l y, i t i s t r ue f or f i sher i es, f or l andscapes, and pr obabl y f or
many mor e r esour ces . I ndeed, i n so f ar as we ar e deal i ng wi t h a l i vi ng ent i t y,
t her e i s a mor al ar gument , whi ch we wi l l not eval uat e her e, t hat we shoul d
val ue t he st ock t o at t r i but e i mpor t ance t o i t s exi st ence i n i t s own r i ght and not
j ust i nst r ument al l y as a sour ce of consumpt i on .
The Hami l t oni an i n t hi s case
H= u ( ct ,
st ) e
-i t
Maxi mi zat i on wi t h r espect t c
mar gi nal ut i l i t y of consumpt i
l evel s :
uC ( ct ,
SO = Xt
and t he r at e of change of t he
( , Xt e- i t ) =
[u
To si mpl i f y mat t er s we shal
and s : u ( c, s) = ul ( c) + u2
di f f er ent i abl e . I n t hi s case a
ui ( ct ) =
st = r ( st )
~t - JAt = - u2 ( St
I n st udyi ng t hese equat i ons,
t hen exami ne t he dynami cs c
2. The Ut i l i t ar i an Case wi t hout Pr oduct i on
We begi n by consi der i ng t he si mpl est case, t hat of a convent i onal ut i l i t ar i an
obj ect i ve wi t h no pr oduct i on : t he r esour ce i s t he onl y good i n t he economy. For
t hi s f r amewor k we char act er i ze t he ut i l i t ar i an opt i mum, and t hen ext end t hese
r esul t s t o ot her f r amewor ks . The maxi mand i s t he di scount ed i nt egr al of ut i l i t i es f r omconsumpt i on and f r omt he exi st ence of a st ock,
f o u ( c, s) e- bt dt ,
wher e S > 0 i s a di scount r at e . As t he r esour ce i s r enewabl e, i t s dynami cs ar e
descr i bed by
ht = r ( st ) - ct .
Her e r i s t he gr owt h r at e of t he r esour ce, assumed t o depend onl y on i t s
cur r ent st ock . Mor e compl ex model s ar e of cour se possi bl e, i n whi ch sever al
such syst ems i nt er act : a wel l - known exampl e i s t he pr edat or - pr ey syst em. I n
gener al , r i s a concave f unct i on whi ch at t ai ns a maxi mumat a f i ni t e val ue of
s, and decl i nes t her eaf t er . Thi s f or mul at i on has a l ong and cl assi cal hi st or y,
whi ch i s r evi ewed i n [ 11] . I n t he f i el d of popul at i on bi ol ogy, r ( st ) i s of t en
t aken t o be quadr at i c, i n whi ch case an unexpl oi t ed popul at i on ( i . e . , ct = 0 dt )
gr ows l ogi st i cal l y. Her e we assume t hat r ( 0) = 0, t hat t her e exi st s a posi t i ve
st ock l evel s at whi ch r ( s) = 0 Vs >_ s, and t hat r ( s) i s st r i ct l y concave and
t wi ce cont i nuousl y di f f er ent i abl e f or s E ( 0, s) . The over al l pr obl emcan now
be speci f i ed as
max
f 00 u ( c, s) e - at dt s . t . ht = r ( s t ) - ct , so gi ven.
0
2. 1 . St at i onar y Sol ut i ons
At a st at i onar y sol ut i on, by
addi t i on, t he shadow pr i ce i s
Sui ( ct ) = u2 ( st ) +
Hence:
PROPOSI TI ON 1 . A st at i onc
( 2) sat i sf i es
r ( st ) = ct
- J - r ' ( st )
u~( st )
u; ( t =t )
The f i r st equat i on i n ( 3) j us
t he cur ve on whi ch consun
t hi s i s obvi ousl y a pr er equi s
r el at i onshi p bet ween t he sl o:
t he sl ope of t he r enewal f i u
cur ve cut s t he r enewal f unct i
Fi gur e 1 . Thi s i s j ust t he r est
equal t he di scount r at e i f r '
[ 17, 18] .
Sust ai nabl e Use of Renewabl e Resour ces
Japer , namel y t hei r ext i nct i on,
ont ext t he r ecent di scussi on of
i e ear t h' s r esour ces i s cl osel y
or e compr ehensi ve di scussi on
r pr et at i on i n economi c t er ms,
opt i mal i nt er t empor al use of
~y some or al l of us [ 1- 3] t hat
our ce of consumpt i on but al so
ans t hat t he exi st i ng st ock of
i on. The i nst ant aneous ut i l i t y
zmpt i on and s t he r emai ni ng
or f or est s, whi ch can be used
nd whose st ock i s a sour ce of
l andscapes, and pr obabl y f or
-e deal i ng wi t h a l i vi ng ent i t y,
- val uat e her e, t hat we shoul d
>t ence i n i t s own r i ght and not
51
The Hami l t oni an i n t hi s case i s
[r
.
ct ]
H= u ( ct , st ) e- at + At e- 6t
( St )
Maxi mi zat i on wi t h r espect t o consumpt i on gi ves as usual t he equal i t y of t he
mar gi nal ut i l i t y of consumpt i on t o t he shadow pr i ce f or posi t i ve consumpt i on
l evel s :
uc ( ct , SO = At
and t he r at e of change of t he shadow pr i ce i s det er mi ned by
( , \ t e- 5t )
- bt
=[ us ( ct , st ) e- bt + at e
r
( st ) ] .
To si mpl i f y mat t er s we shal l t ake t he ut i l i t y f unct i on t o be separ abl e i n c
and s : u ( c, s) = ui ( c) + u2 ( s) , each t aken t o be st r i ct l y concave and t wi ce
di f f er ent i abl e . I n t hi s case a sol ut i on t o t he pr obl em ( 1) i s char act er i zed by
ui ( ct ) = Xt
st = r ( st ) - ct
at - b,\ t = - u2 ( st ) - At r ' ( St )
( 2)
I n st udyi ng t hese equat i ons, we f i r st anal yze t hei r st at i onar y sol ut i on, and
t hen exami ne t he dynami cs of t hi s syst em away f r omt he st at i onar y sol ut i on.
t of a convent i onal ut i l i t ar i an
) nl y good i n t he economy. For
t i mum, and t hen ext end t hese
i e di scount ed i nt egr al of ut i l ) f a st ock, J u ( c, s) e- 6t dt ,
r enewabl e, i t s dynami cs ar e
2. 1 . St at i onar y Sol ut i ons
At a st at i onar y sol ut i on, by def i ni t i on s i s const ant so t hat r ( St ) = ct : i n
addi t i on, t he shadow pr i ce i s const ant so t hat
o`
zmed t o depend onl y on i t s
se possi bl e, i n whi ch sever al
t he pr edat or - pr ey syst em. I n
naxi mumat a f i ni t e val ue of
a l ong and cl assi cal hi st or y,
at i on bi ol ogy, r ( s t ) i s of t en
: d popul at i on ( i . e. , ct = 0 b' t )
0, t hat t her e exi st s a posi t i ve
r ( s) i s st r i ct l y concave and
' he over al l pr obl em can now
ct , so gi ven.
8ui ( ct ) = u2 ( St ) + ui ( ct )
r
( st )
Hence:
PROPOSI TI ON 1 . A st at i onar y sol ut i on t o t he ut i l i t ar i an opt i mal usepat t er n
( 2) sat i sf i es
r
= ct
( st )
.
( 3)
u'
=b- r ' ( St ) ~
u, ( at )
The f i r st equat i on i n ( 3) j ust t el l s us t hat a st at i onar y sol ut i on must l i e on
t he cur ve on whi ch consumpt i on of t he r esour ce equal s i t s r enewal r at e :
t hi s i s obvi ousl y a pr er equi si t e f or a st at i onar y st ock. The second gi ves us a
r el at i onshi p bet ween t he sl ope of an i ndi f f er ence cur ve i n t he c- s pl ane and
t he sl ope of t he r enewal f unct i on at a st at i onar y sol ut i on : t he i ndi f f er ence
cur ve cut s t he r enewal f unct i on f r om above. Such a conf i gur at i on i s shown i n
Fi gur e 1 . Thi s i s j ust t he r esul t t hat t he sl ope of an i ndi f f er ence cur ve shoul d
equal t he di scount r at e i f r ' ( s) = 0 Vs, i . e . , i f t he r esour ce i s non- r enewabl e
[ 17, 18] .
52
A. Bel t r at t i et al .
S
3 . on t he cur ve r ( s) = c, s i s
4. f r om( 2) , t he r at e of chang.
ui ( c) 6 = ui ( c) [ b - r '
The f i r st t er mher e i s nega
negat i ve and l ar ge f or sma
c i s r i si ng f or smal l s and ,
when t he r at e of change o
of posi t i ve sl ope cont ai ni n
5 . by l i near i zi ng t he syst em
ui ( c)
= ui ( c) [ E - s
ht = r ( st )
ar ound t he st at i onar y sol ut
poi nt . The det er mi nant of
Fi gur e 1. Dynami cs of t he ut i l i t ar i an sol ut i on.
Ther e i s a st r ai ght f or war d i nt ui t i ve i nt er pr et at i on t o t he second equat i on
i n ( 3) . Consi der r educi ng consumpt i on by an amount Ac and i ncr easi ng t he
st ock by t he same amount . The wel f ar e l oss i s Ocui : t her e i s a gai n f r om
i ncr easi ng t he st ock of Acu' , whi ch cont i nues f or ever , so t hat we have t o
comput e i t s pr esent val ue. But we al so have t o r ecogni ze t hat t he i ncr ement
t o t he st ock wi l l gr ow at t he r at e r ' : hence t he gai n f r omt he i ncr ease i n st ock
i s t he pr esent val ue of an i ncr ement whi ch compounds at r at e r ' . Hence t he
t ot al gai n i s
Ac
Jo
00 u2er ' t e- at dt
= u20c/ ( r ' - 8) .
When gai ns and l osses j ust bal ance out , we have
ui +u2/ ( r ' - S) =0
whi ch i s j ust t he second equat i on of ( 3) . So ( 3) i s a ver y nat ur al and i nt ui t i ve
char act er i zat i on of opt i mal i t y.
r ( s) { b - r ( s) }
-
ui
whi ch i s negat i ve f or an)
sust ai nabl e yi el d.
Hence t he dynami cs of pat h
mal i t y ar e as shown i n f i gur e I
PROPOSI TI ON2. 2 For smal l ,
t he der i vat i ves r ' , r " or ui , al ,
t end t o t he st at i onar y sol ut i on
or der condi t i ons ( 2) , andf ol l oi
Fi gur e 1 l eadi ng t o t he st at i ona
so, t her e i s a cor r espondi ng v.
of t he st abl e br anches l eadi ng
st at i onar y sol ut i on depends or ,
of t he st at i onar y st ock as t hi s
t ends t o a poi nt sat i sf yi ng u2/
an i ndi f er ence cur ve of u ( c, s
gr aph of t he r enewal f unct i on .
2. 2 . Dynami c Behavi or
What ar e t he dynami cs of t hi s syst em out si de of a st at i onar y sol ut i on? These
ar e al so shown i n Fi gur e 1 . They ar e der i ved by not i ng t he f ol l owi ng f act s :
1 . beneat h t he cur ve r ( s) = c, s i s r i si ng as consumpt i on i s l ess t han t he
gr owt h of t he r esour ce .
2. above t he cur ve r ( s) = c, s i s f al l i ng as consumpt i on i s gr eat er t han t he
gr owt h of t he r esour ce .
Thi s r esul t char act er i zes opt i n
t he exi st ence of such pat hs . T
l i shes t hat an opt i mal pat h exi
char act er i zed i n t hi s paper .
Not e t hat i f t he i ni t i al r esow
consumpt i on, st ock and ut i l i t y
t hat because t he r esour ce i s r e
Sust ai nabl e Use of Renewabl e Resour ces
53
3. on t he cur ve r ( s) = c, s i s const ant .
4. f r om( 2) , t he r at e of change of c i s gi ven by
ui ( c) c = ui ( c) [ ~ Ut i l i t ar i an st at i onar y
sol ut i on
r (s)] -
u2 ( s) .
The f i r st t er m her e i s negat i ve f or smal l s and vi ce ver sa: t he second i s
negat i ve and l ar ge f or smal l s and negat i ve and smal l f or l ar ge s . Hence
c i s r i si ng f or smal l s and vi ce ver sa: i t s r at e of change i s zer o pr eci sel y
when t he r at e of change of t he shadowpr i ce i s zer o, whi ch i s on a l i ne
of posi t i ve sl ope cont ai ni ng t he st at i onar y sol ut i on.
5 . by l i near i zi ng t he syst em
ui ( c) c = ui ( c) [ b - r ' ( s) ] - u2 ( s)
ht = r ( st ) - ct
r ent al st ock s
r i an sol ut i on .
t at i on t o t he second equat i on
t mount Ac and i ncr easi ng t he
. s Ocui : t her e i s a gai n f r om
s f or ever , so t hat we have t o
r ecogni ze t hat t he i ncr ement
; si n f r omt he i ncr ease i n st ock
upounds at r at e r ' . Hence t he
re
i s a ver y nat ur al and i nt ui t i ve
a st at i onar y sol ut i on? These
y not i ng t he f ol l owi ng f act s :
consumpt i on i s l ess t han t he
) nsumpt i on i s gr eat er t han t he
ar ound t he st at i onar y sol ut i on, one can show t hat t hi s sol ut i on i s a saddl e
poi nt . The det er mi nant of t he mat r i x of t he l i near i zed syst em i s
r ( s) t a -
r ( s) } -
u,
{ ui r "
+ u2}
I
whi ch i s negat i ve f or any st at i onar y st ock i n excess of t he maxi mum
sust ai nabl e yi el d.
Hence t he dynami cs of pat hs sat i sf yi ng t he necessar y condi t i ons f or opt i mal i t y ar e as shown i n f i gur e 1, and we can est abl i sh t he f ol l owi ng r esul t :
PROPOSI TI ON 2. 2 For smal l val ues of t he di scount r at e E or l ar ge val ues of
t he der i vat i ves r ' , r " or ui , al l opt i mal pat hs f or t he ut i l i t ar i an pr obl em ( 1)
t end t o t he st at i onar y sol ut i on ( 3) . They do so al ong apat h sat i sf yi ng t hef i r st
or der condi t i ons ( 2) , andf ol l ow one of t he t wo br anches of t he st abl e pat h i n
Fi gur e I l eadi ng t o t he st at i onar y sol ut i on. Gi ven any i ni t i al val ue of t he st ock
so, t her e i s a cor r espondi ng val ue of co whi ch wi l l pl ace t he syst em on one
of t he st abl e br anches l eadi ng t o t he st at i onar y sol ut i on . The posi t i on of t he
st at i onar y sol ut i on depends on t he di scount r at e, and moves t o hi gher val ues
of t he st at i onar y st ock as t hi s decr eases. As b - + 0, t he st at i onar y sol ut i on
t ends t o a poi nt sat i sf yi ng u2/ ui = r ' , whi ch means i n geomet r i c t er ms t hat
an i ndi f f er ence cur ve of u ( c, s) i s t angent t o t he cur ve c = r ( s) gi ven by t he
gr aph of t he r enewal f unct i on .
Thi s r esul t char act er i zes opt i mal pat hs f or t he pr obl em( 1) . I t does not pr ove
t he exi st ence of such pat hs . The Appendi x gi ves an ar gument whi ch est abl i shes t hat an opt i mal pat h exi st s f or al l of t he pr obl ems whose sol ut i ons ar e
char act er i zed i n t hi s paper .
Not e t hat i f t he i ni t i al r esour ce st ock i s l ow, t he opt i mal pol i cy r equi r es t hat
consumpt i on, st ock and ut i l i t y al l r i se monot oni cal l y over t i me . The poi nt i s
t hat because t he r esour ce i s r enewabl e, bot h st ocks and f l ows can be bui l t up
54
A. Bel t r at t i et al .
over t i me pr ovi ded t hat consumpt i on i s l ess t han t he r at e of r egener at i on, i . e. ,
t he syst em i s i nsi de t he cur ve gi ven by t he gr aph of t he r enewal f unct i on r ( s) .
I n pr act i ce, unf or t unat el y, many r enewabl e r esour ces ar e bei ng consumed at
a r at e gr eat l y i n excess of t hei r r at es of r egener at i on : i n t er ms of Fi gur e 1, t he
cur r ent consumpt i on r at e ct i s much gr eat er t han r ( st ) . So t aki ng advant age of
t he r egener at i on possi bi l i t i es of t hese r esour ces woul d i n many cases r equi r e
shar p l i mi t at i on of cur r ent consumpt i on. Fi sher i es ar e a wi del y- publ i ci zed
exampl e: anot her i s t r opi cal har dwoods and t r opi cal f or est s i n gener al . Soi l i s
a mor e subt l e exampl e: t her e ar e pr ocesses whi ch r enew soi l , so t hat even i f i t
suf f er s a cer t ai n amount of er osi on or of depl et i on of i t s val uabl e component s,
i t can be r epl aced. But t ypi cal l y human use of soi l s i s depl et i ng t hemat r at es
f ar i n excess of t hei r r epl eni shment r at es .
Pr oposi t i on 2 gi ves condi t i ons necessar y f or a pat h t o be opt i mal f r om
pr obl em ( 1) . Gi ven t he concavi t y of u ( c, s) and of r ( s) , one can i nvoke
st andar d ar gument s t o show t hat t hese condi t i ons ar e al so suf f i ci ent ( see, f or
exampl e, [ 22] ) .
3. Renewabl e Resour ces and t he Gr een Gol den Rul e
We can use t he r enewabl e f r amewor k t o ask t he quest i on : what conf i gur at i on
of t he economy gi ves t he maxi mumsust ai nabl e ut i l i t y l evel ? 3 Ther e i s a
si mpl e answer .
Fi r st , not e t hat a sust ai nabl e ut i l i t y l evel must be associ at ed wi t h a sust ai nabl e conf i gur at i on of t he economy, i . e . , wi t h sust ai nabl e val ues of consumpt i on and of t he st ock. But t hese ar e pr eci sel y t he val ues t hat sat i sf y t he
equat i on
ct = r ( st )
f or t hese ar e t he val ues whi ch ar e f easi bl e and at whi ch t he st ock and t he
consumpt i on l evel s ar e const ant . Hence i n Fi gur e 1, we ar e l ooki ng f or val ues
whi ch l i e on t he cur ve ct = r ( s t ) . Of t hese val ues, we need t he one whi ch
l i es on t he hi ghest i ndi f f er ence cur ve of t he ut i l i t y f unct i on u ( c, s) : t hi s poi nt
of t angency i s shown i n t he f i gur e. At t hi s poi nt , t he sl ope of an i ndi f f er ence
cur ve equal s t hat of t he r enewal f unct i on, so t hat t he mar gi nal r at e of subst i t ut i on bet ween st ock and f l ow equal s t he mar gi nal r at e of t r ansf or mat i on
al ong t he cur ve r ( s) . Hence:
PROPOSI TI ON 3. 4 The maxi mumsust ai nabl e ut i l i t y l evel ( t he gr een gol den
r ul e) sat i sf i es
U, ( 80
ui ( ct )
Recal l f r om ( 3) t hat as t he di scount r at e goes t o zer o, t he st at i onar y sol ut i on t o t he ut i l i t ar i an case t ends t o such a poi nt . Not e al so t hat any pat h
whi ch appr oaches t he t ange
t i on f unct i on, i s opt i mal acc(
or l ong- r un ut i l i t y. I n ot her m
t he l i mi t i ng behavi or of t he
appr oached. Thi s cl ear l y i s
t he gr een gol den r ul e, some
woul d l i ke t o knowwhi ch of
a best . I t t r anspi r es t hat i n g
3. 1 . Ecol ogi cal St abi l i t y
An i nt er est i ng f act i s t hat
di scount r at es t he ut i l i t ar i an
i n excess of t hat gi vi ng t he
t he st ock at whi ch t he max
onl y r esour ce st ocks i n exc
ar e st abl e under t he nat ur a
ar e ecol ogi cal l y st abl e . To
t he r esour ce dynami cs i s j u
s =r ( s ) - d.
For d < max, r ( s) , t her e a
t o t hi s equat i on, as shown
Cl ear l y f or s > s2, s < 0.
as shown i n Fi gur e 2. Onl y
yi el d i s st abl e under t he na
r at es, and ut i l i t ar i an opt i n
an ar gument of t he ut i l i t ;
maxi mumsust ai nabl e yi el
4. The Rawl si an Sol ut i c
Consi der t he i ni t i al st ock
t hi s i s t o f ol l ow t he pat h t
consumpt i on, st ock and u
l east wel l of f , i s t he f i r st
pr esent model , wi t h i ni t i
set t i ng c = r ( s 1) f or eve
hi ghest ut i l i t y l evel f or t h
bei ng no l ower . Thi s r er r
associ at ed wi t h t he gr een
en r ul e i s a Rawl si an opt i
Sust ai nabl e Use of Renewabl eResour ces
t he r at e of r egener at i on, i . e . ,
) f t he r enewal f unct i on r ( s) .
i r ces ar e bei ng consumed at
i on : i n t er ms of Fi gur e 1, t he
( st ) . So t aki ng advant age of
woul d i n many cases r equi r e
i es ar e a wi del y- publ i ci zed
cal f or est s i n gener al . Soi l i s
i r enew soi l , so t hat even i f i t
i o f i t s val uabl e component s,
i l s i s depl et i ng t hemat r at es
a pat h t o be opt i mal f r om
i d of r ( s) , one can i nvoke
s ar e al so suf f i ci ent ( see, f or
I
en Rul e
l uest i on : what conf i gur at i on
l e ut i l i t y l evel ? 3 Ther e i s a
st be associ at ed wi t h a suss sust ai nabl e val ues of conl y t he val ues t hat sat i sf y t he
at whi ch t he st ock and t he
; 1, we ar e l ooki ng f or val ues
aes, we need t he one whi ch
y f unct i on u ( c, s) : t hi s poi nt
t he sl ope of an i ndi f f er ence
at t he mar gi nal r at e of subgi nal r at e of t r ansf or mat i on
t i l i t y l evel ( t he gr een gol den
o zer o, t he st at i onar y sol unt . Not e al so t hat any pat h
55
whi ch appr oaches t he t angency of an i ndi f f er ence cur ve wi t h t he r epr oduct i on f unct i on, i s opt i mal accor di ng t o t he cr i t er i on of maxi mi zi ng sust ai nabl e
or l ong- r un ut i l i t y. I n ot her wor ds, t hi s cr i t er i on of opt i mal i t y onl y det er mi nes
t he l i mi t i ng behavi or of t he economy : i t does not det er mi ne how t he l i mi t i s
appr oached. Thi s cl ear l y i s a weakness : of t he many pat hs whi ch appr oach
t he gr een gol den r ul e, some wi l l accumul at e f ar mor e ut i l i t y t han ot her s . One
woul d l i ke t o know whi ch of t hese i s t he best , or i ndeed whet her t her e i s such
a best . I t t r anspi r es t hat i n gener al t her e i s not . We r et ur n t o t hi s l at er.
3 . 1 . Ecol ogi cal St abi l i t y
An i nt er est i ng f act i s t hat t he gr een gol den r ul e, and al so f or l ow enough
di scount r at es t he ut i l i t ar i an sol ut i on, r equi r e st ocks of t he r esour ce whi ch ar e
i n excess of t hat gi vi ng t he maxi mum sust ai nabl e yi el d, whi ch i s of cour se
t he st ock at whi ch t he maxi mumof r ( s) occur s . Thi s i s i mpor t ant because
onl y r esour ce st ocks i n excess of t hat gi vi ng t he maxi mumsust ai nabl e yi el d
ar e st abl e under t he nat ur al popul at i on dynami cs of t he r esour ce [ 21] : t hey
ar e ecol ogi cal l y st abl e. To see t hi s, consi der a f i xed depl et i on r at e d, so t hat
t he r esour ce dynami cs i s j ust
s =r ( s ) - d .
For d < max, r ( s) , t her e ar e t wo val ues of s whi ch gi ve st at i onar y sol ut i ons
t o t hi s equat i on, as shown i n Fi gur e 2. Cal l t he smal l er s 1 and t he l ar ger s2 .
Cl ear l y f or s > 82, s < 0, f or sl < s < s2, s > 0, and f or s < s I , s < 0,
as shown i n Fi gur e 2. Onl y t he st ock t o t he r i ght of t he maxi mumsust ai nabl e
yi el d i s st abl e under t he nat ur al popul at i on adj ust ment pr ocess : hi gh di scount
r at es, and ut i l i t ar i an opt i mal pol i ci es when t he st ock of t he r esour ce i s not
an ar gument of t he ut i l i t y f unct i on, wi l l gi ve st at i onar y st ocks bel ow t he
maxi mumsust ai nabl e yi el d.
4. The Rawl si an Sol ut i on
Consi der t he i ni t i al st ock l evel sl i n Fi gur e 1 : t he ut i l i t ar i an opt i mum f r om
t hi s i s t o f ol l ow t he pat h t hat l eads t o t he saddl e poi nt . I n t hi s case, as not ed,
consumpt i on, st ock and ut i l i t y ar e al l i ncr easi ng . So t he gener at i on whi ch i s
l east wel l of f , i s t he f i r st gener at i on . What i s t he Rawl si an sol ut i on i n t he
pr esent model , wi t h i ni t i al st ock sl ? I t i s easy t o ver i f y t hat t hi s i nvol ves
set t i ng c = r ( s 1) f or ever : t hi s gi ves a const ant ut i l i t y l evel , and gi ves t he
hi ghest ut i l i t y l evel f or t he f i r st gener at i on compat i bl e wi t h subsequent l evel s
bei ng no l ower. Thi s r emai ns t r ue f or any i ni t i al st ock no gr eat er t han t hat
associ at ed wi t h t he gr een gol den r ul e : f or l ar ger i ni t i al st ocks, t he gr een gol den r ul e i s a Rawl si an opt i mum. For mal l y,
56
A. Bel t r at t i et al .
max a
f
0
u ( ct , st )
s . t . At = I
wher e f ( t ) i s a f i ni t e count ab'
The change i n opt i mal pol i
opt i mal i t y i s qui t e dr amat i c . N
f ( t ) gi ven by an exponent i al
sol ut i on t o t he over al l opt i mi z
t akes a di f f er ent , non- expont
r at e whi ch t ends asympt ot i cE
i n an unexpect ed way wi t h r e
t he f ut ur e : t her e i s empi r i cal
choi ces act as i f t hey have n
t i me . For mal l y:
PROPOSI TI ON 5. 6 The pr ob
pat t er n of use of a r enewabl e
a const ant di scount r at e.
Pr oof . Consi der f i r st t he p
Fi gur e 2.
The dynami cs of t he r enewabl e r esour ce under a const ant depl et i on r at e .
00
max
v. ( ct , st ) e0
PROPOSI TI ON 4. For an i ni t i al r esour ce st ock sl l ess t han or equal t o t hat
associ at ed wi t h t he gr een gol den r ul e, t he Rawl si an opt i mumi nvol ves set t i ng
c = r ( s ) f or ever . For s 1 gr eat er t han t he gr een gol den r ul e st ock, t he gr een
gol den r ul e i s a Rawl si an opt i mum.
5.
Chi chi l ni sky' s Cr i t er i on
Next , we ask howt he Chi chi l ni sky cr i t er i on ( 7, 8) al t er s mat t er s when appl i ed
t o an anal ysi s of t he opt i mal management of r enewabl e r esour ces . Recal l t hat
Chi chi l ni sky' s cr i t er i on r anks pat hs accor di ng t o t he sumof t wo t er ms, one
an i nt egr al of ut i l i t i es agai nst a f i ni t e count abl y addi t i ve measur e and one
a pur el y f i ni t el y addi t i ve measur e def i ned on t he ut i l i t y st r eam of t he pat h .
The f or mer i s j ust a gener al i zat i on of t he di scount ed i nt egr al of ut i l i t i es
( gener al i zed i n t he sense t hat t he f i ni t e count abl y addi t i ve measur e need
not be an exponent i al di scount f act or ) . The l at t er t er m can be i nt er pr et ed as a
sust ai nabl e ut i l i t y l evel : Chi chi l ni sky shows t hat any r anki ng of i nt er t emp . or al
pat hs whi ch sat i sf i es cer t ai n basi c axi oms must be r epr esent abl e i n t hi s way .
The pr obl emnow i s t o pi ck pat hs of consumpt i on and r esour ce accumul at i on
over t i me t o :
The dynami cs of t he sol ut i o:
ur e 3 .
I t di f f er s f r om t he pr obl e
i n l i mi t i ng ut i l i t y i n t he ma
Fi gur e 3. Pi ck an i ni t i al va
saddl e- poi nt , and f ol l ow t he
condi t i ons gi ven above:
ui ( c) b = ui ( c) ( 6
At = r ( St ) Denot e by vo t he 2- vect or c
{ ' Et , st } ( vo) . Fol l ow t hi s pat ]
t he gr een gol den r ul e, i . e . ,
9t , = s* , and t hen at t = t '
t o t he gr een gol den r ul e, i . (
because ct < r ( s t ) al ong
For mal l y, t hi s pat h i s ( c t , s
st , = s* , and ct = r ( s* ) , st
Any such pat h wi l l sat i sf 3
up t o t i me t ' and wi l l l ea(
t her ef or e at t ai n a maxi mum
However , t he ut i l i t y i nt egr a
Sust ai nabl e Use of Renewabl e Resour ces
57
00
max a ~
u ( ct , st ) f ( t ) dt + ( I - a) sl i m u ( ct , st )
( 4)
s . t . ht = r ( st ) - ct , so gi ven.
Ut i l i t ar i an st at i onar y
i ol ut i on
ar mgown r uk
ader a const ant depl et i on r at e.
sl l ess t han or equal t o t hat
i an opt i mumi nvol ves set t i ng
gol den r ul e st ock, t he gr een
al t er s mat t er s when appl i ed
; wabl e r esour ces . Recal l t hat
o t he sumof t wo t er ms, one
y addi t i ve measur e and one
. e ut i l i t y st r eam of t he pat h.
; ount ed i nt egr al of ut i l i t i es
abl y addi t i ve measur e need
t er m can be i nt er pr et ed as a
any r anki ng of i nt er t empor al
) e r epr esent abl e i n t hi s way.
i and r esour ce accumul at i on
wher e f ( t ) i s a f i ni t e count abl y addi t i ve measur e .
The change i n opt i mal pol i cy r esul t i ng f r omt he change i n t he cr i t er i on of
opt i mal i t y i s qui t e dr amat i c . Wi t h t he Chi chi l ni sky cr i t er i on and t he measur e
f ( t ) gi ven by an exponent i al di scount f act or , i . e . , f ( t ) = e- bt , t her e i s no
sol ut i on t o t he over al l opt i mi zat i on pr obl ems Ther e i s a sol ut i on onl y i f f ( t )
t akes a di f f er ent , non- exponent i al f or m, i mpl yi ng a non- const ant di scount
r at e whi ch t ends asympt ot i cal l y t o zer o . Chi chi l ni sky' s cr i t er i on t hus l i nks
i n an unexpect ed way wi t h r ecent di scussi ons of i ndi vi dual at t i t udes t owar ds
t he f ut ur e : t her e i s empi r i cal evi dence t hat i ndi vi dual s maki ng i nt er t empor al
choi ces act as i f t hey have non- const ant di scount r at es whi ch decl i ne over
t i me . For mal l y :
PROPOSI TI ON 5 . 6 Thepr obl em( 4) has no sol ut i on, i . e. , t her e i s no opt i mal
pat t er n of use of a r enewabl e r esour ce usi ng t he Chi chi l ni sky cr i t er i on wi t h
a const ant di scount r at e.
Pr oof . Consi der f i r st t he pr obl em
max
f 0" u ( ct , s t ) e- bt dt s . t . st = r ( s t ) - ct , s o gi ven .
0
The dynami cs of t he sol ut i on i s shown i n Fi gur e 1, r epr oduced her e as Fi gur e 3 .
I t di f f er s f r om t he pr obl em under consi der at i on by t he l ack of t he t er m
i n l i mi t i ng ut i l i t y i n t he maxi mand. Suppose t hat t he i ni t i al st ock i s so i n
Fi gur e 3 . Pi ck an i ni t i al val ue of c, say co, bel ow t he pat h l eadi ng t o t he
saddl e- poi nt , and f ol l ow t he pat h f r om co sat i sf yi ng t he ut i l i t ar i an necessar y
condi t i ons gi ven above:
u" ( c) c = ui ( c) [ b - r ' ( s) ] - u' ( s) ,
St = r ( st ) - ct Denot e by vo t he 2- vect or of i ni t i al condi t i ons : v0 = ( co, so) . Cal l t hi s pat h
{ ct , st } ( vo) . Fol l ow t hi s pat h unt i l i t l eads t o t he r esour ce st ock cor r espondi ng
t he gr een gol den r ul e, i . e. , unt i l t he t ' such t hat on t he pat h { ct , st } ( vo) ,
s t , = s* , and t hen at t = t ' i ncr ease consumpt i on t o t he l evel cor r espondi ng
t o t he gr een gol den r ul e, i . e . , set ct = r ( s* ) f or al l t > t ' . Thi s i s f easi bl e
because ct < r ( st ) al ong such a pat h. Such a pat h i s shown i n Fi gur e 3.
For mal l y, t hi s pat h i s ( ct , st ) = { at , st } ( vo) dt <_ t ' wher e t ' i s def i ned by
st , =s* , and ct =r ( s* ) , st =s* ` dt >t ' .
Any such pat h wi l l sat i sf y t he necessar y condi t i ons f or ut i l i t ar i an opt i mal i t y
up t o t i me t ' and wi l l l ead t o t he gr een gol den r ul e i n f i ni t e t i me. I t wi l l
t her ef or e at t ai n a maxi mumof t he t er m l i mt , u ( ct , st ) over f easi bl e pat hs .
However , t he ut i l i t y i nt egr al whi ch const i t ut es t he f i r st par t of t he maxi mand
mu~i
58
A. Bel t r at t i et al .
Hence t her e i s no sol ut i on t o
I nt ui t i vel y, t he non- exi st ence
t o post pone f ur t her i nt o t he f
cost i n t er ms of l i mi t i ng ut i l i t
of ut i l i t i es . Thi s i s possi bl e be
i s no equi val ent phenomenon
5. 1 . Decl i ni ng Di scount Rat e
Wi t h t he Chi chi l ni sky cr i t er i c
f 0 00 ' u ( ct , st ) e - qt ,r
Fi gur e 3. A sequence of consumpt i on pat hs wi t h i ni t i al st ock so and i ni t i al consumpt i on
l evel bel ow t hat l eadi ng t o t he ut i l i t ar i an st at i onar y sol ut i on and conver gi ng t o i t . Once t he
st ock r eaches s* consumpt i on i s set equal t o r ( s* ) . The l i mi t i s a pat h whi ch appr oaches t he
ut i l i t ar i an st at i onar y sol ut i on and not t he gr een gol den r ul e .
t her e i s no sol ut i on t o t he pi
r esour ce . I n f act as not ed t he d
f unct i on of t i me . The cr i t er i on
i s st i l l consi st ent wi t h Chi chi l
sol vi ng t he r enewabl e r esour c
t hat we have not ed bef or e, n, t he di scount r at e goes t o zer o,
r ul e . We shal l , t her ef or e, con
a
can be i mpr oved by pi cki ng a sl i ght l y hi gher i ni t i al val ue co f or consumpt i on,
agai n f ol l owi ng t he f i r st or der condi t i ons f or opt i mal i t y and r eachi ng t he gr een
gol den r ul e sl i ght l y l at er t han t ' . Thi s does not det r act f r om t he second t er m i n
t he maxi mand. By t hi s pr ocess i t wi l l be possi bl e t o i ncr ease t he i nt egr al t er m
i n t he maxi mand wi t hout r educi ng t he l i mi t i ng t er m and t hus t o appr oxi mat e
t he i ndependent maxi mi zat i on of bot h t er ms i n t he maxi mand : t he di scount ed
ut i l i t ar i an t er m, by st ayi ng l ong enough cl ose t o t he st abl e mani f ol d l eadi ng
t o t he ut i l i t ar i an st at i onar y sol ut i on, and t he l i mi t ( pur el y f i ni t el y addi t i ve)
t er m by movi ng t o t he gr een gol den r ul e ver y f ar i nt o t he f ut ur e .
Al t hough i t i s possi bl e t o appr oxi mat e t he maxi mi zat i on of bot h t er ms i n
t he maxi mand i ndependent l y by post poni ng f ur t her and f ur t her t he j ump t o
t he gr een gol den r ul e, t her e i s no f easi bl e pat h t hat act ual l y achi eves t hi s
maxi mum. The supr emum of t he val ues of t he maxi mand over f easi bl e pat hs
i s appr oxi mat ed ar bi t r ar i l y cl osel y by pat hs whi ch r each t he gr een gol den
r ul e at l at er and l at er dat es, but t he l i mi t of t hese pat hs never r eaches t he
gr een gol den r ul e and so does not achi eve t he supr emum. Mor e f or mal l y,
consi der t he l i mi t of pat hs ( ct , st ) _ { ct ,
st } ( vo) b' t _< t ' wher e t ' i s def i ned
by s t , = s* , and ct = r ( s* ) , s t = s* bt > t ' as co appr oaches t he st abl e
mani f ol d of t he ut i l i t ar i an opt i mal sol ut i on. On t hi s l i mi t i ng pat h s t < s* Vt .
f
00
0
u ( ct , SO A( t )
wher e A( t ) i s t he di scount f a
r at e q ( t ) at t i me t i s t he pr op
and we assume t hat t he di sco
0.
t ~~ q ( t ) =
So t he over al l pr obl em i s noN
max a
f
00
0
u ( ct , st )
s . t . ht =
wher e t he di scount f act or A
r at e goes t o zer o i n t he l i mi t
sol ut i on: i n f act , i t i s t he sol u
t he f i r st t er m i n t he above ma:
t he ut i l i t y f unct i on t o be sepa
For mal l y,
Sust ai nabl e Use of Renewabl e Resour ces
Hence t her e i s no sol ut i on t o ( 4) .
i vn s ut i ons r y
I*
i m
st ock so and i ni t i al consumpt i on
i n and conver gi ng t o i t . Once t he
ni t i s a pat h whi ch appr oaches t he
d val ue co f or consumpt i on,
Lal i t y and r eachi ng t he gr een
r act f r omt he second t er m i n
t o i ncr ease t he i nt egr al t er m
r mand t hus t o appr oxi mat e
maxi mand : t he di scount ed
t he st abl e mani f ol d l eadi ng
i t ( pur el y f i ni t el y addi t i ve)
i nt o t he f ut ur e .
dmi zat i on of bot h t er ms i n
i er and f ur t her t he j ump t o
t hat act ual l y achi eves t hi s
xi mand over f easi bl e pat hs
ch r each t he gr een gol den
se pat hs never r eaches t he
; upr emum. Mor e f or mal l y,
b' t < t ' wher e t ' i s def i ned
us co appr oaches t he st abl e
i s l i mi t i ng pat h st < s* Vt .
59
o
I nt ui t i vel y, t he non- exi st ence pr obl em ar i ses her e because i t i s al ways possi bl e
t o post pone f ur t her i nt o t he f ut ur e movi ng t o t he gr een gol den r ul e, wi t h no
cost i n t er ms of l i mi t i ng ut i l i t y val ues but wi t h a gai n i n t er ms of t he i nt egr al
of ut i l i t i es . Thi s i s possi bl e because of t he r enewabi l i t y of t he r esour ce . Ther e
i s no equi val ent phenomenon f or an exhaust i bl e r esour ce [ 18] .
5 . 1 . Decl i ni ng Di scount Rat es
Wi t h t he Chi chi l ni sky cr i t er i on f or mul at ed as
a
f 00 u ( ct , st ) e- at dt + ( 1 - a) t l i m u ( ct , st ) ,
0
t her e i s no sol ut i on t o t he pr obl em of opt i mal management of a r enewabl e
r esour ce . I n f act as not edt he di scount f act or does not have t o be an exponent i al
f unct i on of t i me. The cr i t er i on can be st at ed sl i ght l y di f f er ent l y, i n a way whi ch
i s st i l l consi st ent wi t h Chi chi l ni sky' s axi oms and whi ch i s al so consi st ent wi t h
sol vi ng t he r enewabl e r esour ce pr obl em. Thi s r ef or mul at i on bui l ds on a poi nt
t hat we have not ed bef or e, namel y t hat f or t he di scount ed ut i l i t ar i an case, as
t he di scount r at e goes t o zer o, t he st at i onar y sol ut i on goes t o t he gr een gol den
r ul e . We shal l , t her ef or e, consi der a modi f i ed obj ect i ve f unct i on
a
00
f o u ( ct , st ) A( t ) dt + ( 1 - a) t l i m u ( ct , st ) ,
wher e A( t ) i s t he di scount f act or at t i me t , f o A ( t ) dt i s f i ni t e, t he di scount
r at e q ( t ) at t i me t i s t he pr opor t i onal r at e of change of t he di scount f act or :
(t)
0 (t)
q (t) _ -
and we assume t hat t he di scount r at e goes t o zer o wi t h t i n t he l i mi t :
o q (t)
0.
=
ta
i
So t he over al l pr obl emi s now
max a
( 5)
f 00 u ( ct , s t ) A( t ) dt + ( 1 - a) t l i i o u ( et , st )
0
s. t . st = r ( st ) - ct , so gi ven,
wher e t he di scount f act or A( t ) sat i sf i es t he condi t i on ( 5) t hat t he di scount
r at e goes t o zer o i n t he l i mi t . We wi l l show t hat f or t hi s pr obl em, t her e i s a
sol ut i on : ? i n f act , i t i s t he sol ut i on t o t he ut i l i t ar i an pr obl emof maxi mi zi ng j ust
t he f ast t er m i n t he above maxi mand, f o u ( ct , st ) A( t ) dt . As bef or e we t ake
t he ut i l i t y f unct i on t o be separ abl e i n i t s ar gument s : u ( c, s) = ul ( c) + u2 ( s) .
For mal l y,
60
A. Bel t r at t i et al .
PROPOSI TI ON 6 . 8 Consi der t hepr obl em
max a
f 00 { u1 ( c) + u2 ( s) } A ( t ) dt
i s t he same as t hat of t he a
+ ( 1 - a) t l i y m { ul ( c) + u2 ( s) } ,
0 < a < 1, s . t. ht = r ( st ) - ct , so gi ven,
wher e q( t ) = - ( 0( t ) / 0( t ) ) andl i mt , q ( t ) = 0. Asol ut i on t ot hi spr obl em
i s i dent i cal t o t he sol ut i on of " max f ' { u1( c) + u2 ( s) } 0 ( t ) dt subj ect t o t he
same const r ai nt " . I n wor ds, t he condi t i ons char act er i zi ng a sol ut i on t o t he
ut i l i t ar i an pr obl em wi t h t he var i abl e di scount r at e whi ch goes t o zer o al so
char act er i ze a sol ut i on t o t he over al l pr obl em.
Pr oof. Consi der f i r st t he pr obl em max a f ° ° { ul ( c) + u2 ( s) JA ( t ) dt s . t .
ht = r ( st ) - ct , so gi ven. We shal l show t hat any sol ut i on t o t hi s pr obl em
appr oaches and at t ai ns t he gr een gol den r ul e asympt ot i cal l y, whi ch i s t he
conf i gur at i on of t he economy whi ch gi ves t he maxi mumof t he t er m ( 1 a) l i mt4
-0
. u ( c t , s t ) . Hence t hi s sol ut i on sol ves t he over al l pr obl em. The
Hami l t oni an f or t he i nt egr al pr obl emi s now
~t = - u2 ( St ) ht = r ( st .
whi ch di f f er s onl y i n t hat t
zer o . 9 The pai r of Equat i o:
st abi l i t y pr oper t i es of or i l
associ at ed l i mi t i ng aut ono
by t he st andar d t echni que:
and ct = r ( s t ) , so t hat
u2
u1
= -r'
and
whi ch i s j ust t he def i ni t i on
ment s used above we can
of t he syst em( 8) , as sho) N
H= { u1( c) + u2 ( s) } A ( t ) + At A( t ) [ r ( st ) - Ct ]
00
" maxi mi ze
and maxi mi zat i on wi t h r espect t o consumpt i on gi ves as bef or e
i s f or any gi ven i ni t i al st e
t hat ( co, so) i s on t he st y
appr oaches t he Gr een Gc
t o t he maxi mum possi bl ,
t her ef or e l eads t o a sol ut i
The r at e of change of t he shadowpr i ce At i s det er mi ned by
_ - [ u2 ( st ) 0 ( t ) + At A ( t )
r (st )] .
The r at e of change of t he shadow pr i ce i s, t her ef or e,
a t A( t )
+ At , & ( t ) = - u2 ( st ) A( t ) - At A ( t ) r ' ( st ) .
( 6)
As , & ( t ) depends on t i me, t hi s' equat i on i s not aut onomous, i . e . , t i me appear s
expl i ci t l y as a var i abl e. For such an equat i on, we cannot use t he phase por t r ai t s
and associ at ed l i near i zat i on t echni ques used bef or e, because t he r at es of
change of c and s depend not onl y on t he poi nt i n t he c- s pl ane but al so on
t he dat e . Rear r angi ng and not i ng t hat A( t ) / A ( t ) = q ( t ) , we have
at
0
subj ect t o At
u 1 ( ct ) = At .
d ( at o( t ) )
f {
+ At q ( t ) = - u2 ( st ) - ui ( ct ) r ' ( st )
But i n t he l i mi t q = 0, so i n t he l i mi t t hi s equat i on i s aut onomous : t hi s equat i on
and t he st ock gr owt h equat i on f or mwhat has r ecent l y been cal l ed i n dynami cal syst ems t heor y an asympt ot i cal l y aut onomous syst em [ 4] . Accor di ng t o
pr oposi t i on 1 . 2 of [ 4] , t he asympt ot i c phase por t r ai t of t hi s non- aut onomous
syst em
~ t + At q ( t ) ' = - u2 ( st ) - ui ( ct ) r ' ( st )
ht = r ( st ) - ct
Fi gur e 4 shows t he behal
can see what dr i ves t hi s 1
const ant di scount r at e ar
of t he pat h t hat maxi mi ze
pat h t hat maxi mi zes t he
t o zer o i n t he l i mi t , t hat
r esol ved onl y i n t hi s case
PROPOSI TI ON 7. 1o Cot
00
max a (
J0
{ u1( c)
0<~
wher e q( t ) = - ( 0( t ) / L
= 0. I n t hi s case, t he sol e
act er i ze t he sol ut i on t o "
const r ai nt " .
-
Sust ai nabl e Use of Renewabl e Resour ces
61
i s t he same as t hat of t he aut onomous syst em
t l +00
i MJU1
ct , so gi ven,
a)
( c)
+ u2
( S) } ,
0. 14 sol ut i on t o t hi spr obl em
12 ( s) } A( t ) dt subj ect t o t he
- act er i zi ng a sol ut i on t o t he
2t e whi ch goes t o zer o al so
+ u2 ( s ) } 0 ( t ) dt s . t .
ny sol ut i on t o t hi s pr obl em
sympt ot i cal l y, whi ch i s t he
naxi mum of t he t er m ( 1 ; t he over al l pr obl em. The
. 2c1 ( c)
( st ) - ct ]
at =
-
ua ( st )
- ui ( ct ) r ' ( st )
=
r ( St ) - Ct
St
whi ch di f f er s onl y i n t hat t he non- aut onomous t er mq( t ) has been set equal t o
zer o . 9 The pai r of Equat i ons ( 8) i s an aut onomous syst em and t he asympt ot i c
st abi l i t y pr oper t i es of or i gi nal syst em ( 7) wi l l be t he same as t hose of t he
associ at ed l i mi t i ng aut onomous syst em( 8) . Thi s l at t er syst emcanbe anal yzed
by t he st andar d t echni ques used bef or e. At a st at i onar y sol ut i on of ( 8) , . f i t = 0
and ct = r ( st ) , so t hat
and
U2
ul
ct = r ( st )
whi ch i s j ust t he def i ni t i on of t he gr een gol den r ul e . Fur t her mor e, by t he ar gument s used above we can est abl i sh t hat t he gr een gol den r ul e i s a saddl epoi nt
of t he syst em ( 8) , as shown i n Fi gur e 3. So t he opt i mal pat h f or t he pr obl em
00
" maxi mi ze
i ves as bef or e
( 8)
J0
{ ul ( c) + u2 ( s ) } 0 ( t )
dt
subj ect t o st = r ( st ) - ct , so gi ven"
i s f or any gi ven i ni t i al st ock so t o sel ect an i ni t i al consumpt i on l evel co such
t hat ( co, so) i s on t he st abl e pat h of t he saddl e poi nt conf i gur at i on whi ch
appr oaches t he Gr een Gol den Rul e asympt ot i cal l y. But t hi s pat h al so l eads
t o t he maxi mum possi bl e val ue of t he t er m l i mt , , , . { u l ( c) + u2 ( s) } , and
o
t her ef or e l eads t o a sol ut i on t o t he over al l maxi mi zat i on pr obl em.
r mi ned by
r ' ( s t )] .
) r e,
( t ) r ' ( st ) .
( 6)
: onomous, i . e. , t i me appear s
annot use t he phase por t r ai t s
Mor e, because t he r at es of
i n t he c- s pl ane but al so on
i = q ( t ) , we have
i s aut onomous : t hi s equat i on
ent l y been cal l ed i n dynamas syst em [ 4] . Accor di ng t o
r ai t of t hi s non- aut onomous
Fi gur e 4 shows t he behavi or of an opt i mal pat h i n t hi s case . I nt ui t i vel y, one
can see what dr i ves t hi s r esul t . The non- exi st ence of an opt i mal pat h wi t h a
const ant di scount r at e ar ose f r om a conf l i ct bet ween t he l ong- r un behavi or
of t he pat h t hat maxi mi zes t he i nt egr al of di scount ed ut i l i t i es, and t hat of t he
pat h t hat maxi mi zes t he l ong- r un ut i l i t y l evel . When t he di scount r at e goes
t o zer o i n t he l i mi t , t hat conf l i ct i s r esol ved. I n f act , one can show t hat i t i s
r esol ved onl y i n t hi s case, as st at ed by t he f ol l owi ng pr oposi t i on .
PROPOSI TI ON 7. 10 Consi der t he pr obl em
max a
J0
00{ u l ( c) + u2 ( s ) } 0 ( t )
{ u l ( c) + u2
t _+00
liM
0 < a < 1, s. t . §t = r ( st ) - ct , so gi ven,
dt + ( 1 - a)
( s) } ,
wher e q ( t ) = - ( , &( t ) / A ( t ) ) . Thi spr obl emhas a sol ut i on onl y i f l i mt , " ' , q ( t )
= 0. I n t hi s case, t he sol ut i on i s char act er i zed by t he condi t i ons whi ch char act er i ze t he sol ut i on t o " max f o { ui ( c) + u2 ( s) } 0 ( t ) dt subj ect t o t he same
const r ai nt " .
62
A. Bel t r at t i et al .
and
5. 3 . Empi r i cal Evi dence on D
Fi gur e 4. Asympt ot i c dynami cs of t he ut i l i t ar i an sol ut i on f or t he case i n whi ch t he di scount
r at e f al l s t o zer o .
Pr oof. The " i f ' par t of t hi s was pr oven i n t he pr evi ous pr oposi t i on, Pr oposi t i on 6. The " onl y i f ' par t can be pr oven by an ext ensi on of t he ar gument s
i n Pr oposi t i on 5, whi ch est abl i shed t he non- exi st ence of sol ut i ons i n t he case
of a const ant di scount r at e . To appl y t he ar gument s t her e, assume cont r ar y t o
t he pr oposi t i on t hat l i mi nf t _. ~, , . q ( t ) = q > 0, and t hen appl y t he ar gument s
of Pr oposi t i on 5 .
p
Exi st ence of a sol ut i on t o t hi s pr obl em i s est abl i shed i n t he Appendi x .
5 . 2 . Exampl es
To compl et e t hi s di scussi on, we r evi ew some exampl es of di scount f act or s
whi ch sat i sf y t he condi t i on t hat t he l i mi t i ng di scount r at e goes t o zer o . The
most obvi ous i s
t t
O( t ) = e- a ( ) , wi t h
t l i , 1 b ( t ) = 0.
Anot her exampl es 1 i s
A( t ) = t - a,
a > 1.
Taki ng t he st ar t i ng dat e t o be t = 1, 12 we have
00
1
t ` dt =
a1
f,
Pr oposi t i on 7 has subst ant i al
mal i t y wi t h a cr i t er i on sensi t i
wi t h non- r enewabl e r esour ce :
behavi or of t he di scount r at e :
ut i l i t i es symmet r i cal l y i n t he e
sense, t he t r eat ment of pr esen
si st ent wi t h t he pr esence of t r
posi t i ve wei ght on t he ver y l os
Ther e i s a gr owi ng body of
l i ke t hi s i n eval uat i ng t he f uh:
mor e compr ehensi ve di scussi o
whi ch peopl e appl y t o f ut ur e
f ut ur i t y of t he pr oj ect . Over r e
t hey use di scount r at es whi ch ;
r egi on of 15%or mor e . For pi
di scount r at es ar e cl oser t o st
ext ends t he i mpl i ed di scount r
year s and down t o of t he or der ,
f r amewor k f or i nt er t empor al of
f ut ur e gener at es an i mpl i cat i on
per sonal behavi or t hat hi t her t o
Thi s empi r i cal l y- i dent i f i ed l ;
sci ences whi ch f i nd t hat humar
l i near , and ar e i nver sel y pr opor l
i s an exampl e of t he Weber - Fec
t hat human r esponse t o a chang
pr e- exi st i ng st i mul us . I n symbc
dr _ _K
_
or r = K
ds
s
wher e r i s a r esponse, s a st i m
t o appl y t o human r esponses t o
We not ed t hat t he empi r i cal r e ;
somet hi ng si mi l ar i s happeni ng
of an event : a gi ven change i s
l eads t o a smal l er r esponse i n t o
t he event al r eady i s i n t he f ut ur e
appl i ed t o r esponses t o di st ance
Sust ai nabl e Use of Renewabl e Resour ces
63
and
0=t
-+ 0
as
t - + oo .
5. 3 . Empi r i cal Evi dence on Decl i ni ng Di scount Rat es
I f i mal pat hs
a f or t he case i n whi ch t he di scount
pr evi ous pr oposi t i on, Pr opon ext ensi on of t he ar gument s
, t ence of sol ut i ons i n t he case
: nt s t her e, assume cont r ar y t o
i nd t hen appl y t he ar gument s
0
i shed i n t he Appendi x .
I - xampl es of di scount f act or s
; count r at e goes t o zer o . The
Pr oposi t i on 7 has subst ant i al i mpl i cat i ons . I t says t hat when we seek opt i mal i t y wi t h a cr i t er i on sensi t i ve t o t he pr esent and t he l ong- r un f ut ur e, t hen
wi t h non- r enewabl e r esour ces exi st ence of a sol ut i on i s t i ed t o t he l i mi t i ng
behavi or of t he di scount r at e: i n t he l i mi t , we have t o t r eat pr esent and f ut ur e
ut i l i t i es symmet r i cal l y i n t he eval uat i on of t he i nt egr al of ut i l i t i es . I n a cer t ai n
sense, t he t r eat ment of pr esent and f ut ur e i n t he i nt egr al has t o be made consi st ent wi t h t he pr esence of t he t er ml i mt , , , , , { u l ( c) + u2 ( s) } whi ch pl aces
posi t i ve wei ght on t he ver y l ong r un.
Ther e i s a gr owi ng body of empi r i cal evi dence t hat peopl e act ual l y behave
l i ke t hi s i n eval uat i ng t he f ut ur e ( see, f or exampl e, [ 20] ; see al so [ 18] f or a
mor e compr ehensi ve di scussi on) . The evi dence suggest s t hat t he di scount r at e
whi ch peopl e appl y t o f ut ur e pr oj ect s depends upon, and decl i nes wi t h, t he
f ut ur i t y of t he pr oj ect . Over r el at i vel y shor t per i ods up t o per haps f i ve year s,
t hey use di scount r at es whi ch ar e hi gher even t han commer ci al r at es - i n t he
r egi on of 15%or mor e . For pr oj ect s ext endi ng about t en year s, t he i mpl i ed
di scount r at es ar e cl oser t o st andar d r at es - per haps 10%. As t he hor i zon
ext ends t he i mpl i ed di scount r at es dr ops, t o i n t he r egi on of 5%f or 30 t o 50
year s and down t o of t he or der of 2%f or 100 year s . I t i s of gr eat i nt er est t hat a
f r amewor k f or i nt er t empor al opt i mi zat i on t hat i s sensi t i ve t o bot h pr esent and
f ut ur e gener at es an i mpl i cat i on f or di scount i ng t hat may r at i onal i ze a f or mof
per sonal behavi or t hat hi t her t o has been f ound i r r at i onal .
Thi s empi r i cal l y- i dent i f i ed behavi or i s consi st ent wi t h r esul t s f r omnat ur al
sci ences whi ch f i nd t hat human r esponses t o a change i n a st i mul us ar e nonl i near , and ar e i nver sel y pr opor t i onal t o t he exi st i ng l evel of t he st i mul us . Thi s
i s an exampl e of t he Weber - Fechner l aw, whi ch i s f or mal i zed i n t he st at ement
t hat human r esponse t o a change i n a st i mul us i s i nver sel y pr opor t i onal t o t he
pr e- exi st i ng st i mul us . I n symbol s,
_dr _ _K
or r = Kl og s,
ds
s
wher e r i s a r esponse, s a st i mul us and K a const ant . Thi s has been f ound
t o appl y t o human r esponses t o t he i nt ensi t y of bot h l i ght and sound si gnal s .
We not ed t hat t he empi r i cal r esul t s on di scount i ng ci t ed above suggest t hat
somet hi ng si mi l ar i s happeni ng i n human r esponses t o changes i n t he f ut ur i t y
of an event : a gi ven change i n f ut ur i t y ( e . g. , post ponement by one year )
l eads t o a smal l er r esponse i n t er ms of t he decr ease i n wei ght i ng, t he f ur t her
t he event al r eady i s i n t he f ut ur e . I n t hi s case, t he Weber - Fechner l awcan be
appl i ed t o r esponses t o di st ance i n t i me, as wel l as t o sound and l i ght i nt ensi t y,
64
A. Bel t r at t i et al ,
wi t h t he r esul t t hat t he di scount r at e i s i nver sel y pr opor t i onal t o di st ance i nt o
t he f ut ur e . Recal l i ng t hat t he di scount f act or i s A( t ) and t he di scount r at e
q ( t ) = - A( t ) / A ( t ) , we can f or mal i ze t hi s as
1 dO
q (t) =
A dt
K
t
or
A( t ) =e K
l og
t
=tK
f or K a posi t i ve const ant . Such a di scount f act or can meet al l of t he condi t i ons
we r equi r ed above: t he di scount r at e q goes t o zer o i n t he l i mi t , t he di scount
f act or A( t ) goes t o zer o and t he i nt egr al f l' A( t ) dt = f l' eK l og t dt =
f l , t K dt conver ges f or Kposi t i ve, as i t al ways i s . I n f act , t hi s i nt er pr et at i on
gi ves r i se t o t he second exampl e of a non- const ant di scount r at e consi der ed
i n t he pr evi ous sect i on. A di scount f act or A( t ) = eK 109' has an i nt er est i ng
i nt er pr et at i on : t he r epl acement of t by l og t i mpl i es t hat we ar e measur i ng
t i me di f f er ent l y, i . e . by equal pr opor t i onal i ncr ement s r at her t han by equal
absol ut e i ncr ement s .
5. 4 . Ti me Consi st ency
An i ssue whi ch i s r ai sed by t he pr evi ous pr oposi t i ons i s t hat of t i me consi st ency . Consi der a sol ut i on t o an i nt er t empor al opt i mi zat i on pr obl emwhi ch i s
comput ed t oday and i s t o be car r i ed out over some f ut ur e per i od of t i me st ar t i ng t oday. Suppose t hat t he agent f or mul at i ng i t - an i ndi vi dual or a soci et y may at a f ut ur e dat e r ecomput e an opt i mal pl an, usi ng t he same obj ect i ve and
t he same const r ai nt s as i ni t i al l y but wi t h i ni t i al condi t i ons and st ar t i ng dat e
cor r espondi ng t o t hose obt ai ni ng when t he r ecomput at i on i s done . Then we
say t hat t he i ni t i al sol ut i on i s t i me consi st ent i f t hi s l eads t he agent t o cont i nue
wi t h t he i mpl ement at i on of t he i ni t i al sol ut i on. Anot her way of sayi ng t hi s i s
t hat a pl an i s t i me consi st ent i f t he passage of t i me al one gi ves no r eason t o
change i t . The i mpor t ant poi nt i s t hat t he sol ut i on t o t he pr obl emof opt i mal
management of t he r enewabl e r esour ce wi t h a t i me- var yi ng di scount r at e,
st at ed i n Pr oposi t i on 7, i s not t i me- consi st ent . A f or mal def i ni t i on of t i me
consi st ency i s : 13
DEFI NI TI ON 8 . Let ( ct , st
max a
f o 00 { ul
) t =o, 00
( c) + u2
be t he sol ut i on t o t he pr obl em
( s) } A ( t )
dt + ( 1 - a) t hi n { ul ( c) + u2
t - +00
( s) } ,
0 < a < 1, s . t . At = r ( st ) - ct , so gi ven .
Let ( ct , 9' t ) t =T, Oo be t he sol ut i on t o t he pr obl emof opt i mi zi ng f r omTon, gi ven
t hat t he pat h ( ct , s t ) t =o, o0 has been f ol l owed up t o dat e T. , i . e . , ( Ft , st ) t =T, m
sol ves
max a ,
f ' { ul
1'
( c) + u2
( t - T) dt + ( 1 - a) l i m { ul ( c) +
t - r oo
0 < a < 1, s . t . At = r ( st ) - ct , sT gi ven.
( s) } 0
u2 ( s) } ,
Then t he or i gi nal pr obl em so
( Et , St ) t =T, . = ( Ct , st ) t =T, oc
per i od [ T, oo] i s al so a sol ut i c
st ock sT, f or any T.
I t i s shown i n [ 14] t hat t he sc
i n gener al t i me consi st ent onl
l owi ng r esul t i s an i l l ust r at i on
PROPOSI TI ON 9. 14 The sol u,
r enewabl e r esour ce wi t h a di s
t i me consi st ent , i . e . , t he sol ut i
max a
0<a<
f
00
{ ul
( c) + u2
0
l,
s . t.
At = r ( st )
i s not t i me consi st ent .
Pr oof . Consi der t he f i r st o
whi ch ar e gi ven i n ( 7) and r ef
u1 ( ct ) ct + ui ( ct ) q i
At
Let ( ct , st ) t =o, . be a sol ut i on
consumpt i on on t hi s at a dat e 7
be a sol ut i on t o t he pr obl em
condi t i ons at T gi ven by ( cr
st ar t i ng dat e T, t he val ue of
val ue of t he di scount f act or . H
A( T - T) , whi l e i t i s A( T) ,
t he t wo pat hs wi l l have di f f er t
i n excess of T. Thi s est abl i shc
T > 0, t hen t he i ni t i al pl an wi
These ar e i nt er est i ng and sw
opt i mal pat h whi ch bal ances
Chi chi l ni sky' s axi oms, we ha
t ent . Of cour se, t he empi r i ca:
behavi or must al so be i ncons
i ng what i ndi vi dual s appar en
al ways r egar ded t i me consi st e
r al choi ce. Mor e r ecent l y, t hi s
and psychol ogi st s have not ed
or hi s l i f e can r easonabl y be
per spect i ves on l i f e and di f f er
wi t h i nconsi st ent choi ces cl ea
Sust ai nabl e Use of Renewabl e Resour ces
pr opor t i onal t o di st ance i nt o
A( t ) and t he di scount r at e
) gt
=t
K
can meet al l of t he condi t i ons
er o i n t he l i mi t , t he di scount
A( t ) dt = f l oo eK t o g t dt =
i s . I n f act , t hi s i nt er pr et at i on
r ant di scount r at e consi der ed
= eK i og t has an i nt er est i ng
pl i es t hat we ar e measur i ng
ement s r at her t han by equal
; i t i ons i s t hat of t i me consi sAi mi zat i on pr obl emwhi ch i s
i e f ut ur e per i od of t i me st ar t - an i ndi vi dual or a soci et y usi ng t he same obj ect i ve and
condi t i ons and st ar t i ng dat e
i mput at i on i s done . Then we
i s l eads t he agent t o cont i nue
mot her way of sayi ng t hi s i s
me al one gi ves no r eason t o
mt o t he pr obl emof opt i mal
t i me- var yi ng di scount r at e,
A f or mal def i ni t i on of t i me
n t o t he pr obl em
u2 ( S) J,
- a) l i m 1ul ( C) +
. ct , so gi ven.
' opt i mi zi ng f r omTon, gi ven
t o dat e T. , i . e. , ( Et , st ) t =T, oo
- a) t l un{ u1 ( c) + u2 ( s) } ,
t , sT gi ven.
65
Then t he or i gi nal pr obl emsol ved at t = 0 i s t i me consi st ent i f and onl y i f
( Ft , St ) t =T, oo = ( ci st ) t =T, , , i . e. , i f t he or i gi nal sol ut i on r est r i ct ed t o t he
per i od [ T, oc] i s al so a sol ut i on t o t he pr obl em wi t h i ni t i al t i me T and i ni t i al
st ock sT, f or any T.
I t i s shown i n [ 14] t hat t he sol ut i ons t o dynami c opt i mi zat i on pr obl ems ar e
i n gener al t i me consi st ent onl y i f t he di scount f act or i s exponent i al . The f ol l owi ng r esul t i s an i l l ust r at i on of t hi s f act .
PROPOSI TI ON 9 . 14 The sol ut i on t o t hepr obl emof opt i mal management of a
r enewabl e r esour ce wi t h a di scount r at ef al l i ng asympt ot i cal l y t o zer o i s not
t i me consi st ent , i . e. , t he sol ut i on t o
max a ,
f 00 { ul ( c) + u2 ( s) } 0 ( t ) dt + ( 1 -
a) t hiy n { ul ( c) + uZ ( s) } ,
0 < a < 1, s . t . st = r ( st ) - ct , so gi ven, q ( t ) _ -
, l i mt +00 q ( t ) = 0.
i s not t i me consi st ent .
Pr oof . Consi der t he f i r st or der condi t i on f or a sol ut i on t o t hi s pr obl em,
whi ch ar e gi ven i n ( 7) and r epeat ed her e wi t h t he subst i t ut i on A = ui :
ui ( ct ) 6t + ui ( ct ) q ( t ) = - ui ( st ) - ui ( ct ) r ' ( st )
.
( 9)
st = r ( st ) - ct
Let ( ct , st ) t =o 0o be a sol ut i on comput ed at dat e t = 0. The r at e of change of
consumpt i on on t hi s at a dat e T > 0 wi l l be gi ven by ( 9) . Nowl et ( Et , st ) t =T, o0
be a sol ut i on t o t he pr obl em wi t h st ar t i ng dat e T, 0 < T < T, and i ni t i al
condi t i ons at T gi ven by ( cr , sT) . When t he pr obl em i s sol ved agai n wi t h
st ar t i ng dat e T, t he val ue of A( t ) at cal endar t i me T i s A( 0) , t he i ni t i al
val ue of t he di scount f act or . Hence on t hi s pat h t he val ue of A( t ) at dat e T i s
A( T - T) , whi l e i t i s A( T) on t he i ni t i al pat h . Hence q ( T) wi l l di f f er , and
t he t wo pat hs wi l l have di f f er ent r at es of change of consumpt i on f or al l dat es
i n excess of T. Thi s est abl i shes t hat i f t he opt i mumi s r ecomput ed at any dat e
T > 0, t hen t he i ni t i al pl an wi l l no l onger be f ol l owed.
These ar e i nt er est i ng and sur pr i si ng r esul t s : t o ensur e t he exi st ence of an
opt i mal pat h whi ch bal ances pr esent and f ut ur e " cor r ect l y" accor di ng t o
Chi chi l ni sky' s axi oms, we have t o accept pat hs whi ch ar e not t i me consi st ent . Of cour se, t he empi r i cal evi dence ci t ed above i mpl i es t hat i ndi vi dual
behavi or must al so be i nconsi st ent , so soci et y i n t hi s case i s onl y r epl i cat i ng what i ndi vi dual s appar ent l y do. Tr adi t i onal l y, wel f ar e economi st s have
al ways r egar ded t i me consi st ency as a ver y desi r abl e pr oper t y of i nt er t empor al choi ce . Mor e r ecent l y, t hi s pr esumpt i on has been quest i oned : phi l osopher s
and psychol ogi st s have not ed t hat t he same per son at di f f er ent st ages of her
or hi s l i f e can r easonabl y be t hought of as di f f er ent peopl e wi t h di f f er ent
per spect i ves on l i f e and di f f er ent exper i ences . 15 The i mpl i cat i ons of wor ki ng
wi t h i nconsi st ent choi ces cl ear l y need f ur t her r esear ch .
66
6.
A. Bel t r at t i et al .
7. 1 . St at i onar y Sol ut i ons
Capi t al and Renewabl e Resour ces
Nowwe consi der t he most chal l engi ng, and per haps most r eal i st i c and r ewar di ng, of al l cases: an economy i n whi ch a r esour ce whi ch i s r enewabl e and so
has i t s own dynami cs can be used t oget her wi t h pr oduced capi t al goods as an
i nput t o t he pr oduct i on of an out put . The out put i n t ur n can as usual i n gr owt h
model s be r ei nvest ed i n capi t al f or mat i on or consumed. The st ock of t he
r esour ce i s al so a sour ce of ut i l i t y t o t he popul at i on . So capi t al accumul at i on
occur s accor di ng t o
k =F( k , v ) - c
and t he r esour ce st ock evol ves accor di ng t o
wher e k i s t he cur r ent capi t al st ock, Q t he r at e of use of t he r esour ce i n
pr oduct i on, and F( k, v) t he pr oduct i on f unct i on . As bef or e r ( s) i s a gr owt h
f unct i on f or t he r enewabl e r esour ce, i ndi cat i ng t he r at e of gr owt h of t hi s when
t he st ock i s s .
As bef or e, we shal l consi der t he opt i mum accor di ng t o t he ut i l i t ar i an cr i t er i on, t hen char act er i ze t he gr een gol den r ul e, and f i nal l y dr awon t he r esul t s
of t hese t wo cases t o char act er i ze opt i mal i t y accor di ng t o Chi chi l ni sky' s
cr i t er i on.
A l i t t l e al gebr a shows t h
under l yi ng di f f er ent i al ec
sol ut i on :
S=Fk ( 1
c =F( k
us ( c+ s ) =
(I
Fa
Uc ( c, s)
Thi s syst em of f our equal
t he var i abl es k, s, v and c .
I t i s i mpor t ant t o under
model , and i n par t i cul ar t h
st ock s acr oss al t er nat i ve ,
Fi r st , consi der t hi s r el al
t he capi t al st ock k : i n t hi s
c = F( k,
and so we have
C9C
Q( s) ) ,
- F,
198 k f i xed
7. The Ut i l i t ar i an Opt i mum
The ut i l i t ar i an opt i mumi n t hi s f r amewor k i s t he sol ut i on t o
max
f ~u
( Ct ) St )
e - dt dt subj ect t o
.
( 10)
k =F( k , a) - c ands =r ( s ) - a
We pr oceed i n t he by- now st andar d manner , const r uct i ng t he Hami l t oni an
H = u ( c, s) e- 6t + Ae - 6t { F ( k,
Q) -
c} + me - 6t { r ( s) - o}
and der i vi ng t he f ol l owi ng condi t i ons whi ch ar e necessar y f or a sol ut i on t o
( 10) :
uC
AF,
= ~,
( 11)
= ~~
( 12)
- Sa = - AFk,
( 13)
- 8I - L = - ' us - hr s 1
( 14)
wher e r s i s t he der i vat i ve of r wi t h r espect t o t he st ock s .
As FQ i s al ways posi t i ve,
and t hen swi t ches t o neg;
bet ween c and s f or f i xe,
acr oss st at i onar y st at es f
gr owt h f unct i on r ( s) and
I n gener al , however , k
on v vi a Equat i on . ( 15) . T
as an i mpl i ci t f unct i on, w
acr oss st at i onar y st at es :
do _
Fk
_
_
rs ( - b
ds Fkk
whi ch, mai nt ai ni ng t he as
agai n i nher i t s t he shape of
1090S) k f i xed I and t hat t l
zer o . The var i ous cur ves r e
i n Fi gur e 5: f or Fk, > 0 t l
r i ses and f al l s mor e shar p.
f r ombel ow whi l e i ncr easi
st at i onar y sol ut i on wi t h a
Sust ai nabl e Use of Renewabl e Resour ces
67
7. 1 . St at i onar y Sol ut i ons
s most r eal i st i c andr ewar dwhi ch i s r enewabl e and so
oduced capi t al goods as an
t ur n can as usual i n gr owt h
asumed. The st ock of t he
n. So capi t al accumul at i on
of use of t he r esour ce i n
As bef or e r ( s) i s a gr owt h
r at e of gr owt h of t hi s when
i r di ng t o t he ut i l i t ar i an cr i i f i nal l y dr aw on t he r esul t s
; cor di ng t o Chi chi l ni sky' s
A l i t t l e al gebr a shows t hat t he syst em ( 11) t o ( 14) , t oget her wi t h t he t wo
under l yi ng di f f er ent i al equat i ons i n ( 10) , admi t s t he f ol l owi ng st at i onar y
sol ut i on :
b = F' k ( k, ~)
( 15)
= r ( s) ,
( 16)
c = F ( k, v) ,
( 17)
u s ( c' s)
= FQ ( k, o) ( 5 - r . , )
u, ( c, s)
( 18)
Q
Thi s syst em of f our equat i ons suf f i ces t o det er mi ne t he st at i onar y val ues of
t he var i abl es k, s, o, and c .
I t i s i mpor t ant t o under st and f ul l y t he st r uct ur e of st at i onar y st at es i n t hi s
model , and i n par t i cul ar t he t r ade- of f bet ween consumpt i onc and t he r esour ce
st ock s acr oss al t er nat i ve st at i onar y st at es .
Fi r st , consi der t hi s r el at i onshi p acr oss st at i onar y st at es f or a gi ven val ue of
t he capi t al st ock k : i n t hi s case we can wr i t e
c = F( k, v ( s) ) ,
and so we have
ac
Fr
Qr s. .
( 19)
( 7S k f i xed -
ol ut i on t o
( 10)
r uct i ng t he Hami l t oni an
+
Fee
- 8t
f r ( s)
_
Q}
necessar y f or a sol ut i on t o
, t ock s .
As FQ i s al ways posi t i ve, t hi s has t he si gn of r s , whi ch i s i ni t i al l y posi t i ve
and t hen swi t ches t o negat i ve : hence we have a si ngl e- peaked r el at i onshi p
bet ween c and s f or f i xed k acr oss st at i onar y st at es . The c- s r el at i onshi p
acr oss st at i onar y st at es f or a f i xed val ue of k r epl i cat es t he shape of t he
gr owt h f unct i on r ( s) and so has a maxi mumf or t he same val ue of s .
I n gener al , however , k i s not f i xed acr oss st at i onar y st at es, but depends
on Q vi a Equat i on ( 15) . Taki ng account of t hi s dependence and t r eat i ng ( 15)
as an i mpl i ci t f unct i on, we obt ai n t he t ot al der i vat i ve of c wi t h r espect t o s
acr oss st at i onar y st at es :
( 11)
( 12)
( 13)
( 14)
+ FQ) ,
( 20)
Fkkk
whi ch, mai nt ai ni ng t he assumpt i on t hat Fk, > 0, al so has t he si gn of r , s and
agai n i nher i t s t he shape of r ( s) . Not e t hat f or a gi ven val ue of s : I ( dcl ds) ( >
1 * 149s) k &ed+ and t hat t he t wo ar e equal onl y i f t he cr oss der i vat i ve Fk, i s
zer o . The var i ous cur ves r el at i ng c and s acr oss st at i onar y sol ut i ons ar e shown
i n Fi gur e 5 : f or Fk Q > 0 t he cur ve cor r espondi ng t o k f ul l y adj ust ed t o s bot h
r i ses and f al l s mor e shar pl y t han t he ot her s, and cr osses each of t hese t wi ce,
f r ombel owwhi l e i ncr easi ng and f r om above whi l e decr easi ng, as shown. A
st at i onar y sol ut i on wi t h a capi t al st ock k must l i e on t he i nt er sect i on of t he
TS =
r
s
(-b
68
A. Bel t r at t i et al .
Ful l y- adj ust ed r el at i onshi p
bet ween c and s acr oss
st at i onar y st at es .
7. 2 . Dynami cs of t he Ut i l i t
The f our di f f er ent i al equat
k =F( k , a) - c ,
s =r ( s ) - a( pt ;
~_b11 =- us .
The mat r i x of t he l i near i ze
6 - Fo ) , Fok
Fov
-
AFkk + Fk, a
0
Fi gur e S. Aut i l i t ar i an st at i onar y sol ut i on occur s wher e t he f ul l y- adj ust ed c- s r el at i onshi p
cr osses t he same r el at i onshi p f or t he f i xed val ue of k cor r espondi ng t o t he st at i onar y sol ut i on .
cur ve cor r espondi ng t o a capi t al st ock f i xed at k wi t h t he cur ve r epr esent i ng
t he f ul l y- adj ust ed r el at i onshi p . At t hi s poi nt , c, s and k ar e al l f ul l y adj ust ed
t o each ot her . ( I n t he case of Fk, = 0 t he cur ves r el at i ng c and s f or k f i xed
and f ul l y adj ust ed ar e i dent i cal , so t hat i n t he case of a separ abl e pr oduct i on
f unct i on t he dynami cs ar e si mpl er , al t hough qual i t at i vel y si mi l ar. )
The st at i onar y f i r st or der condi t i on ( 18) r el at es most cl osel y t o t he cur ve
connect i ng c and s f or a f i xed val ue of k ( t he onl y r el evant cur ve f or Fk a = 0) ,
and woul d i ndi cat e a t angency bet ween t hi s cur ve and an i ndi f f er ence cur ve
i f t he di scount r at e 5 wer e equal t o zer o . For posi t i ve J, t he case we ar e
consi der i ng now, t he st at i onar y sol ut i on l i es at t he poi nt wher e t he c- s cur ve
f or t he f i xed val ue of k associ at ed wi t h t he st at i onar y sol ut i on cr osses t he c- s
cur ve al ong whi ch k var i es wi t h s . At t hi s poi nt , an i ndi f f er ence cur ve cr osses
t he f i xed- k c- s cur ve f r omabove: t hi s i s shown i n Fi gur e 5 . Not e t hat as we
var y t he di scount r at e b, t he capi t al st ock associ at ed wi t h a st at i onar y sol ut i on
wi l l al t er vi a Equat i on ( 15) , so t hat i n par t i cul ar l ower i ng t he di scount r at e
wi l l l ead t o a st at i onar y sol ut i on on a f i xed- k c- s cur ve cor r espondi ng t o a
l ar ger val ue of k and t her ef or e out si de t he cur ve cor r espondi ng t o t he i ni t i al
l ower di scount r at e .
To est abl i sh cl ear gene
mat r i x, we have t o make si
mar gi nal pr oduct i vi t y of t h
t hen t he ei genval ues of t ht
u2CJ Z + 4u cAFkk . Ther
st at i onar y sol ut i on . I n t hi s
saddl e poi nt .
PROPOSI TI ON 10 . 16 A .
sol ut i on t o be l ocal l y a sac
pr oduct i vi t y of t he r esour c
Ther e ar e ot her cases i n
poi nt , i nvol vi ng addi t i ve s
sol ut i on t o t he ut i l i t ar i an f
8.
The Gr een Gol den R
Acr oss st at i onar y st at es, t h
st ock sat i sf i es t he equat i on
c = F ( k, r ( s) ) ,
so t hat at t he gr een gol de
l evel wi t h r espect t o t he i n
max u ( F ( k, r ( s) ;
,
Maxi mi zat i on wi t h r espec
Sust ai nabl e Use of Renewabl e Resour ces
69
7. 2 . Dynami cs of t he Ut i l i t ar i an Sol ut i on
The f our di f f er ent i al equat i ons gover ni ng a ut i l i t ar i an sol ut i on ar e
k = F ( k, v) - c ( st , At )
s = r ( s) - a ( pt , At , kt )
a- SA=- AFk
~_J M= - us - pr s
r i an st at i onar y sol ut i on
i e f ul l y- adj ust ed c- s r el at i onshi p
i ondi ng t o t he st at i onar y sol ut i on.
wi t h t he cur ve r epr esent i ng
and k ar e al l f ul l y adj ust ed
r el at i ng c and s f or k f i xed
e of a separ abl e pr oduct i on
i t at i vel y si mi l ar . )
s most cl osel y t o t he cur ve
- el evant cur ve f or Fk, = 0) ,
e and an i ndi f f er ence cur ve
) osi t i ve b, t he case we ar e
e poi nt wher e t he c- s cur ve
ar y sol ut i on cr osses t he c- s
a i ndi f f er ence cur ve cr osses
n Fi gur e 5. Not e t hat as we
- d wi t h a st at i onar y sol ut i on
l ower i ng t he di scount r at e
s cur ve cor r espondi ng t o a
cor r espondi ng t o t he i ni t i al
The mat r i x of t he l i near i zed syst em i s
b - F° A
UCC
rs
Faa
-
1
_ Fa Fo _
AFaa
ucc
Fa
AFaa
t ea
ss
Foa
0
AFkk + Fkva
6
Fa
AF, ,
_
1
AF
aa
+
Faa
(~ - r ,
ucc
To est abl i sh cl ear gener al r esul t s on t he si gns of t he ei genval ues of t hi s
mat r i x, we have t o make si mpl i f yi ng assumpt i ons . I f FQQ i s l ar ge, so t hat t he
mar gi nal pr oduct i vi t y of t he r esour ce dr ops r api dl y as mor e of i t i s empl oyed,
t hen t he ei genval ues of t he above mat r i x ar e : r s , b - r s , 1/ ( 2uo, ) ( 2u, 6 f
u. ' J2 + 4u cc AFkk Ther e ar e t wo negat i ve r oot s i n t hi s case, as r s < 0 at a
.
st at i onar y sol ut i on. I n t hi s case t he ut i l i t ar i an st at i onar y sol ut i on i s l ocal l y a
saddl e poi nt .
0
USCUc s
ucc
-
u ss
_
l -~r
s
- U
PROPOSI TI ON 10. 16 A suf f i ci ent condi t i on f or t he ut i l i t ar i an st at i onar y
sol ut i on t o be l ocal l y a saddl e poi nt i s t hat Faa i s l ar ge, so t hat t he mar gi nal
pr oduct i vi t y of t he r esour ce di mi ni shes r api dl y i n pr oduct i on.
Ther e ar e ot her cases i n whi ch t he st at i onar y sol ut i on i s l ocal l y a saddl e
poi nt , i nvol vi ng addi t i ve separ abi l i t y of t he ut i l i t y f unct i on. 17 Exi st ence of a
sol ut i on t o t he ut i l i t ar i an pr obl emi s est abl i shed i n t he Appendi x.
8.
The Gr een Gol den Rul e wi t h Pr oduct i on and Renewabl e Resour ces
Acr oss st at i onar y st at es, t he r el at i onshi p bet ween consumpt i on and t he r esour ce
st ock sat i sf i es t he equat i on
c = F( k, r ( s) ) ,
so t hat at t he gr een gol den r ul e we seek t o maxi mi ze t he sust ai nabl e ut i l i t y
l evel wi t h r espect t o t he i nput s of capi t al k and t he r esour ce st ock s :
max u ( F ( k, r ( s) ) , s) .
s, k
Maxi mi zat i on wi t h r espect t o t he r esour ce st ock gi ves
( 21)
70
A. Bel t r at t i et al .
whi ch i s pr eci sel y t he condi t i on ( 18) char act er i zi ng t he st at i onar y sol ut i on t o
t he ut i l i t ar i an condi t i ons f or t he case i n whi ch t he di scount r at e 8 i s equal t o
zer o . So, as bef or e, t he ut i l i t ar i an sol ut i on wi t h a zer o di scount r at e meet s t he
f i r st or der condi t i ons f or maxi mi zat i on of sust ai nabl e ut i l i t y wi t h r espect t o
t he r esour ce st ock. Of cour se, i n gener al t he ut i l i t ar i an pr obl emmay have no
sol ut i on when t he di scount r at e i s zer o . Not e t hat t he condi t i on ( 21) i s qui t e
i nt ui t i ve and i n keepi ng wi t h ear l i er r esul t s. I t r equi r es t hat an i ndi f f er ence
cur ve be t angent t o t he cur ve r el at i ng c t o s acr oss st at i onar y st at es f or k
f i xed at t he l evel k def i ned bel ow: i n ot her wor ds, i t agai n r equi r es equal i t y of
mar gi nal r at es of t r ansf or mat i on and subst i t ut i on bet ween st ocks and f l ows .
The capi t al st ock k i n t he maxi mand her e i s i ndependent of s . Howi s t he
capi t al st ock chosen? I n a ut i l i t ar i an sol ut i on t he di scount r at e pl ays a r ol e i n
t hi s t hr ough t he equal i t y of t he mar gi nal pr oduct of capi t al wi t h t he di scount
r at e ( 15) : at t he gr een gol den r ul e t her e i s no equi val ent r el at i onshi p .
We cl ose t he syst em i n t he pr esent case by supposi ng t hat t he pr oduct i on
t echnol ogy ul t i mat el y di spl ays sat i at i on wi t h r espect t o t he capi t al i nput al one :
f or each l evel of t he r esour ce i nput o t her e i s a l evel of capi t al st ock at whi ch
t he mar gi nal pr oduct of capi t al i s zer o . Pr eci sel y,
k ( c) =mi nk :
( k, o) = 0.
( 22)
ak
We assume t hat T ( o, ) exi st s f or al l o >_ 0, i s f i ni t e, cont i nuous and nondecr easi ng i n c . Essent i al l y assumpt i on ( 22) says t hat t her e i s a l i mi t t o t he
ext ent t o whi ch capi t al can be subst i t ut ed f or r esour ces : as we appl y mor e
and mor e capi t al t o a f i xed i nput of r esour ces out put r eaches a maxi mum
above whi ch i t cannot be i ncr eased f or t hat l evel of r esour ce i nput . I n t he
case i n whi ch t he r esour ce i s an ener gy sour ce, t hi s assumpt i on was shown by
Ber r y et al . [ 5] t o be i mpl i ed by t he second l aw of t her modynami cs : t hi s i ssue
i s al so di scussed by Dasgupt a and Heal [ 11] . I n gener al , t hi s seems a ver y
mi l d and r easonabl e assumpt i on. Gi ven t hi s assumpt i on, t he maxi mi zat i on of
st at i onar y ut i l i t y wi t h r espect t o t he capi t al st ock at a gi ven r esour ce i nput ,
m
kax u ( F ( k, r ( s) ) , s)
r equi r es t hat we_pi ck t he capi t al st ock at whi ch sat i at i on occur s at t hi s r esour ce
i nput , i . e . , k = k ( r ( s) ) . Not e t hat
ak ( r ( s) )
ak ( c)
_ as
ao
so t hat k i s i ncr easi ng and t hen decr easi ng i n s acr oss st at i onar y st at es : t he
der i vat i ve has t he si gn of r s . I n t hi s case, t he gr een gol den r ul e i s t he sol ut i on
t o t he f ol l owi ng pr obl em:
t he capi t al st ock maxi mi z
r el at i onshi p bet ween consi
st at es when at each r esour
k ( r ( s) ) has t he f ol l owi ng
dc
ds
= F~
wher e at each val ue of t he r esour ce st ock s t he i nput and t he r esour ce and
t he capi t al st ock ar e adj ust ed so t hat t he r esour ce st ock i s st at i onar y and
rs +.
and by t he def i ni t i on of k t f
t he cur ve r el at i ng c and s v
t he sl ope when t he capi t al s
of t he cur ves f or f i xed val u
The t ot al der i vat i ve of
r esour ce i s now
du _
_
ds -
ucFk
_
8k
ao
rs
By assumpt i on ( 22) and t l
t hi s t o zer o f or a maxi mun
si on ( 21) . The gr een gol de
i ndi f f er ence cur ve and t he
st at i onar y st at es f or f i xed c
PROPOSI TI ON 11 . 18 I n a)
abl e r esour ces, under t he
f unct i on wi t h r espect t o cal
condi t i on us / u c = - Fr ,
cur ve and t he out er envek
capi t al st ock of k ( _o ( s* ) ) ,
r esour ce st ock and k denot e
of capi t al f i r st becomes zen
What i f pr oduct i on does not
I n t hi s case t her e i s no may
a gi ven r esour ce f l ow and s
sat i at i on of pr ef er ences wi t
not wel l def i ned. l 9
9.
Opt i mal i t y f or t he Chi
Nowwe seek t o sol ve t he p
max u ( F ( k ( r ( s) ) , r ( s) ) , s) ,
s
ac
max a
f
00
u ( ct ,
0
s. t . k = F( kt , at )
Sust ai nabl e Use of Renewabl e Resour ces
: i ng t he st at i onar y sol ut i on t o
he di scount r at e b i s equal t o
3 zer o di scount r at e meet s t he
f i nabl e ut i l i t y wi t h r espect t o
l i t ar i an pr obl emmay have no
pat t he condi t i on ( 21) i s qui t e
r equi r es t hat an i ndi f f er ence
ar oss st at i onar y st at es f or k
i , i t agai n r equi r es equal i t y of
n bet ween st ocks and f l ows .
i ndependent of s . How i s t he
di scount r at e pl ays a r ol e i n
A of capi t al wi t h t he di scount
ui val ent r el at i onshi p .
upposi ng t hat t he pr oduct i on
pect t o t he capi t al i nput al one :
evel of capi t al st ock at whi ch
Y,
( 22)
71
t he capi t al st ock maxi mi zes out put f or t hat st at i onar y r esour ce i nput . The
r el at i onshi p bet ween consumpt i on and t he r esour ce st ock acr oss st at i onar y
st at es when at each r esour ce st ock t he capi t al st ock i s adj ust ed t o t he l evel
k ( r ( s) ) has t he f ol l owi ng sl ope:
dc
ak
ds
Fk 8o r s
+ Far .,
and by t he def i ni t i on of k t he f i r st t er mon t he r i ght i s zer o, so t hat t he sl ope of
t he cur ve r el at i ng c and s when t he capi t al st ock i s gi ven by Jc i s t he same as
t he sl ope when t he capi t al st ock i s f i xed. Thi s cur ve i s t hus t he out er envel ope
of t he cur ves f or f i xed val ues of t he capi t al st ock.
The t ot al der i vat i ve of t he ut i l i t y l evel wi t h r espect t o t he st ock of t he
r esour ce i s now
du
o9k
rs
us
u, Fk
ds + ucFar s +
.
By assumpt i on ( 22) and t he def i ni t i on of k, Fk = 0 her e : hence equat i ng
t hi s t o zer o f or a maxi mum sust ai nabl e ut i l i t y l evel gi ves t he ear l i er expr essi on ( 21) . The gr een gol den r ul e i s char act er i zed by a t angency bet ween an
i ndi f f er ence cur ve and t he out er envel ope of al l cur ves r el at i ng c t o s acr oss
st at i onar y st at es f or f i xed capi t al st ocks .
f i ni t e, cont i nuous and nonys t hat t her e i s a l i mi t t o t he
r esour ces : as we appl y mor e
out put r eaches a maxi mum
vel of r esour ce i nput . I n t he
i t s assumpt i on was shown by
i f t her modynami cs : t hi s i ssue
n gener al , t hi s seems a ver y
nnpt i on, t he maxi mi zat i on of
k at a gi ven r esour ce i nput ,
PROPOSI TI ON 11 . 18 I n an economy wi t h capi t al accumul at i on and r enewabl e r esour ces, under t he assumpt i on ( 22) of sat i at i on of t he pr oduct i on
f unct i on wi t h r espect t o capi t al , t he gr een gol den r ul e sat i sf i es t hef i r st or der
condi t i on us / u, = - Far ., whi ch def i nes a t angency bet ween an i ndi f f er ence
cur ve and t he out er envel ope of c- s cur ves f or f i xed val ues of k. I t has a
capi t al st ock of k ( o, ( s* ) ) , wher e s* i s t he gr een gol den r ul e val ue of t he
r esour ce st ock and k denot es t he capi t al st ock at whi ch t he mar gi nal pr oduct
of capi t al f i r st becomes zer of or a r esour ce i nput of v ( s* ) .
. t i at i on occur s at t hi s r esour ce
What i f pr oduct i on does not di spl ay sat i at i on wi t h r espect t o t he capi t al st ock?
I n t hi s case t her e i s no maxi mumt o t he out put whi ch can be obt ai ned f r om
a gi ven r esour ce f l ow and so f r om a gi ven r esour ce st ock. Unl ess we assume
sat i at i on of pr ef er ences wi t h r espect t o consumpt i on, t he gr een gol den r ul e i s
not wel l def i ned. 19
acr oss st at i onar y st at es : t he
en gol den r ul e i s t he sol ut i on
9. Opt i mal i t y f or t he Chi chi l ni sky Cr i t er i on
i nput and t he r esour ce and
ar ce st ock i s st at i onar y and
Nowwe seek t o sol ve t he pr obl em
max a
f 00 u ( ct , st ) e - at dt + ( I - a)
s . t . k = F( k t , at ) - ct
&i t
st )
l i omo u ( ct ,
= r ( s) - o' t , st > 0 ` dt .
( 23)
72
A. Bel t r at t i et al .
I n t he case of sat i at i on of t he pr oduct i on pr ocess wi t h r espect t o capi t al , as capt ur ed by assumpt i on ( 22) , t he si t uat i on r esembl es t hat wi t h t he
Chi chi l ni sky cr i t er i on wi t h r enewabl e r esour ces above: t her e i s no sol ut i on
unl ess t he di scount r at e decl i nes t o zer o . For mal l y :
. 2° Assume t hat condi t i on ( 22) i s sat i sf i ed. Then pr obl em
PROPOSI TI ON 12
( 23) has no sol ut i on .
Pr oof. The st r uct ur e of t he pr oof i s t he same as t hat used above. The i nt egr al
t er m i s maxi mi zed by t he ut i l i t ar i an sol ut i on, whi ch r equi r es an asympt ot i c
appr oach t o t he ut i l i t ar i an st at i onar y st at e. The l i mi t t er mi s maxi mi zedon any
pat h whi ch asympt ot es t o t he gr een gol den r ul e . Gi ven any f r act i on , 0 E [ 0, 1]
we can f i nd a pat h whi ch at t ai ns t he f r act i on, Q of t he payof f t o t he ut i l i t ar i an
opt i mumand t hen appr oaches t he gr een gol den r ul e . Thi s i s t r ue f or any val ue
of Q < 1, but not t r ue f or B = 1 . Hence any pat h can be domi nat ed by anot her
o
cor r espondi ng t o a hi gher val ue of Q.
We nowconsi der i nst ead opt i mi zat i on wi t h r espect t o Chi chi l ni sky' s cr i t er i on
wi t h a di scount r at e whi ch decl i nes t o zer o over t i me :
10 .
Concl usi ons
A r evi ew of opt i mal pat t er r
i nt er est i ng concl usi ons . The
i t gi ves t he hi ghest sust ai nal
i t i ve di scount r at e wi l l accu
associ at ed wi t h t he gr een go
t he di scount r at e used i n t he i
a zer o di scount r at e, t her e i s
Chi chi l ni sky' s cr i t er i on i n s4
concept s of opt i mal i t y : a sol
i nt egr al t er m of Chi chi l ni sk;
whi ch case maxi mi zat i on of
ut i l i t i es - l eads one t o t hezyxw
¬
wi t h t he i ncl usi on of pr oduc
not qual i t at i vel y di f f er ent . I r
pl e di spl ay decl i ni ng di scow
behavi or i s qui t e consi st ent
human choi ce and summar b
PROPOSI TI ON 13 . 21 Consi der t hepr obl em
max a
f
00
ul ( c, s) A( t ) dt + ( 1 - a) t l i m ul ( c, s) , 0 < a < 1,
0
11 . Appendi x
400
s . t . k = F( kt , ut ) - ct &ht = r ( st ) - ct , so gi ven.
wher e q ( t ) = - ( 0( t ) / A( t ) ) and l i mt , c) q ( t ) = 0. Any sol ut i on t o t hi s
pr obl em i s al so a sol ut i on t o t he pr obl emof maxi mi zi ng f o ul ( c, s) A( t ) dt
subj ect t o t he same const r ai nt . I n wor ds, sol vi ng t he ut i l i t ar i an pr obl emwi t h
t he var i abl e di scount r at e whi ch goes t o zer o sol ves t he over al l pr obl em.
Pr oof. The pr oof i s a st r ai ght f or war d adapt at i on of t he pr oof of Pr oposi t i on 6, and i s omi t t ed .
O
As bef or e, t he exi st ence of a sol ut i on i s est abl i shed i n t he Appendi x .
What does t he Chi chi l ni sky- opt i mal pat h l ook l i ke i n t hi s case? I t i s si mi l ar
i n gener al t er ms t o t he set of pat hs shown i n Fi gur e 4, except t hat t he gr aph
of t he gr owt h f unct i on r ( s) i s r epl aced by t he out er envel ope of t he cur ves
r el at i ng c and s f or f i xed val ues of k . The opt i mal pat h moves t owar ds t he
gr een gol den r ul e, whi ch i s a poi nt of t angency bet ween an i ndi f f er ence cur ve
and t he out er envel ope of t he cur ves r el at i ng c and s f or f i xed val ues of k . Thi s
poi nt i s t he l i mi t of ut i l i t ar i an st at i onar y sol ut i ons as t he associ at ed di scount
r at e goes t o zer o .
I n t hi s appendi x we est abl i s :
t i ons t o t he var i ous i nt er t emy
si t i ons 2, 6, 7, 10 and 13 of
devel oped i ni t i al l y by Chi ch
enwal d [ 9] . Thi s i s a ver y di
of f easi bl e sol ut i ons t o t he cc
f unct i on i s a cont i nuous f un
t i nuous f unct i on on a compa
i s t o f i nd a t opol ogy i n whi cl
sonabl e assumpt i ons about t
as i nt r oduced i n Chi chi l ni sk
We consi der t he ut i l i t ar i a
t hi s i s t he most compl ex of
t he paper ar e speci al cases c
i mpl i es t he exi st ence of soi l
pr obl em i s :
max
00 u ( ct , st ) e
0
k =F( k , v ) - c ar
We make t he f ol l owi ng assu
ocess wi t h r espect t o capi on r esembl es t hat wi t h t he
above : t her e i s no sol ut i on
l y:
?) i s sat i sf i ed. Then pr obl em
t hat used above. The i nt egr al
hi ch r equi r es an asympt ot i c
ni t t er mi s maxi mi zed on any
3i ven any f r act i on , 3 E [ 0, 1]
f t he payof f t o t he ut i l i t ar i an
Ae . Thi s i s t r ue f or any val ue
San be domi nat ed by anot her
ct t o Chi chi l ni sky' s cr i t er i on
t i me :
Sust ai nabl e Use of Renewabl e Resour ces
10 .
73
Concl usi ons
A r evi ew of opt i mal pat t er ns of use of r enewabl e r esour ces has suggest ed
i nt er est i ng concl usi ons . The gr een gol den r ul e i s an at t r act i ve conf i gur at i on:
i t gi ves t he hi ghest sust ai nabl e ut i l i t y l evel . Ut i l i t ar i an sol ut i ons wi t h a posi t i ve di scount r at e wi l l accumul at e a smal l er st ock of t he r esour ce t han t hat
associ at ed wi t h t he gr een gol den r ul e, al t hough t he di f f er ence goes t o zer o as
t he di scount r at e used i n t he ut i l i t ar i an f or mul at i on get s smal l er. Of cour se, f or
a zer o di scount r at e, t her e i s t ypi cal l y no ut i l i t ar i an opt i mum. I nvest i gat i on of
Chi chi l ni sky' s cr i t er i on i n some measur e br i dges t he gap bet ween t hese t wo
concept s of opt i mal i t y : a sol ut i on exi st s i f and onl y i f t he di scount r at e i n t he
i nt egr al t er m of Chi chi l ni sky' s maxi mand decl i nes asympt ot i cal l y t o zer o, i n
whi ch case maxi mi zat i on of t he i nt egr al t er m al one - t he sumof di scount ed
ut i l i t i es - l eads one t o t he gr een gol den r ul e . Thi s r esul t r emai ns t r ue even
wi t h t he i ncl usi on of pr oduct i on : mat t er s ar e mor e compl ex i n t hat case, but
not qual i t at i vel y di f f er ent . I nt er est i ngl y, t her e i s empi r i cal evi dence t hat peopl e di spl ay decl i ni ng di scount r at es i n t hei r behavi or t owar ds t he f ut ur e . Such
behavi or i s qui t e consi st ent wi t h behavi or pat t er ns f ound i n ot her aspect s of
human choi ce and summar i zed as t he Weber - Fechner l aw.
l i m ul ( c, S) , 0 < a < 1,
11 .
( st ) - ct , so gi ven.
I n t hi s appendi x we est abl i sh condi t i ons suf f i ci ent f or t he exi st ence of sol ut i ons t o t he var i ous i nt er t empor al opt i mi zat i on pr obl ems consi der ed i n Pr oposi t i ons 2, 6, 7, 10 and 13 of t he t ext . We use an appr oach and a set of r esul t s
devel oped i ni t i al l y by Chi chi l ni sky [ 6] and appl i ed by Chi chi l ni sky and Gr uenwal d [ 9] . Thi s i s a ver y di r ect and i nt ui t i ve appr oach : we show t hat t he set
of f easi bl e sol ut i ons t o t he const r ai nt s i s a compact set , and t hat t he obj ect i ve
f unct i on i s a cont i nuous f unct i on, and i nvoke t he st andar d r esul t t hat a cont i nuous f unct i on on a compact set at t ai ns a maxi mum. The del i cat e st ep her e
i s t o f i nd a t opol ogy i n whi ch we have compact ness and cont i nui t y under r easonabl e assumpt i ons about t he pr obl em: f or t hi s we use wei ght ed Lr spaces,
as i nt r oduced i n Chi chi l ni sky [ 6] .
We consi der t he ut i l i t ar i an opt i mal i t y pr obl em anal yzed i n Sect i on 7, as
t hi s i s t he most compl ex of t he pr obl em i n t he paper. Ear l i er pr obl ems i n
t he paper ar e speci al cases of t hi s, so t hat t he exi st ence of a sol ut i on t o t hi s
i mpl i es t he exi st ence of sol ut i ons t o t he ear l i er pr obl ems . The opt i mi zat i on
pr obl emi s :
= 0. Any sol ut i on t o t hi s
ci mi zi ng f ool , ul ( c, s) A( t ) dt
t he ut i l i t ar i an pr obl emwi t h
I ves t he over al l pr obl em.
l i on of t he pr oof of Pr oposi 0
hed i n t he Appendi x .
l i ke i n t hi s case? I t i s si mi l ar
gur e 4, except t hat t he gr aph
out er envel ope of t he cur ves
mal pat h moves t owar ds t he
Bet ween an i ndi f f er ence cur ve
Ld s f or f i xed val ues of k. Thi s
i ns as t he associ at ed di scount
Appendi x
max
fo
00
u ( ct , st ) e- at dt subj ect t o
k =F( k , v ) - c ands =r ( s ) - v
We make t he f ol l owi ng assumpt i ons :
.
( 24)
74
A. Bel t r at t i et al .
1 . u ( c, s) i s concave, i ncr easi ng and di f f er ent i abl e . I t sat i sf i es t he
Car at heodor y condi t i on, namel y i t i s cont i nuous wi t h r espect t o c and
s f or al most al l t and measur abl e wi t h r espect t o t f or al l val ues of c and
i nequal i t y i s sat i sf i ed f or p
opt i mum.
s.
2 . r ( 0) = 0, 3s > 0 s . t . r ( s) = 0 Vs >_ s, max s r ( s) _< b1 < oo, and r ( s)
i s concave f or s E [ 0, s] .
3 . For any a3b2 ( Q) < oo s . t . F( k, er ) < b2 ( Q) .
W
e have nowpr oven t he ex
t he opt i mi zat i on pr obl ems
t he si mpl er pr obl ems can
4. 3b3 < oo s . t . Js) < b3 .
< b4 .
5. 3b4< oo s . t .
I kI
The f i r st t wo condi t i ons ar e convent i onal . The t hi r d i mpl i es t hat bounded
r esour ce avai l abi l i t y i mpl i es bounded out put : i t i s a f or m of t he assumpt i on
made by Dasgupt a and Heal [ 10] t hat t he r esour ce i s essent i al t o pr oduct i on.
I t i s a r est at ement of assumpt i on ( 22) i n t he t ext . The f i nal t wo assumpt i ons
i mpl y t hat i t i s not possi bl e f or ei t her t he r esour ce st ock or t he
t o change i nf i ni t el y r api dl y. These seemt o be ver y r easonabl e
However , we shal l i n t he end not r equi r e t hem: we shal l pr ove
of an opt i mal pat h under t hese assumpt i ons, and t hen not e t hat
capi t al st ock
assumpt i ons .
t he exi st ence
a pat h whi ch
i s opt i mal wi t hout t hese assumpt i ons i s st i l l f easi bl e and opt i mal wi t h t hem.
PROPOSI TI ON 14 . Under assumpt i ons ( 1) t o ( S) above, t he ut i l i t ar i an opt i -
mi zat i on pr obl em( 24) has a sol ut i on .
Pr oof. Under t he above assumpt i ons, t he set of f easi bl e t i me pat hs of t he
r esour ce st ock s and consumpt i on c ar e uni f or ml y bounded above . ( Not e t hat
s i s bounded by ( 2) , and c by ( 3) and ( 5) . ) They ar e non- negat i ve and so
bounded bel ow. Hence t he pat hs of s and c ar e i nt egr abl e agai nst some f i ni t e
measur e and so ar e el ement s of a wei ght ed L1 space . Denot e by P t he set of
f easi bl e pat hs st and ct , 0 _< t <_ oo : as a subset of L1, P i s cl osed and nor m
bounded, so t hat by t he Banach- Al aogl u t heor em i t i s weak- * compact . By
Lebesgue' s bounded conver gence t heor em, i t i s al so compact i n t he nor mof
Ll .
The obj ect i ve U = f o u ( ct , SOe- bt dt maps Pt o t he r eal l i ne f i t . To compl et e t he pr oof we need t o showt hat U i s cont i nuous i n t he nor mof L1 . Thi s
f ol l ows i mmedi at el y f r om t he char act er i zat i on of LP cont i nui t y gi ven i n [ 6] :
LEMMA 15 ( Chi chi l ni sky) . Let W= f i t u ( c t , t ) dv ( t ) f or a f i ni t e measur e
v ( t ) , wi t h u ( ct , t ) sat i sf yi ng t he Car at heodor y condi t i on. Then W
def i nes a
nor m- cont i nuous f unct i on f r om LP t o Rf or some coor di nat e syst em of Lp i f
and onl y i f I u ( ct , t ) ~ < a ( t ) + b I ct JP, wher e a ( t ) > 0, f i t a ( t ) dv ( t ) < oo
and b > 0.
I n t he case of our obj ect i ve t he r ol e of u ( ct , t ) i s pl ayed by u ( ct , s t ) e- Jt . An
ext ensi on of Chi chi l ni sky' s l emma t o f unct i ons u def i ned on R2 i s st r ai ght f or war d. As u i s def i ned onl y on R2 , concavi t y i mpl i es t hat Chi chi l ni sky' s
( 4) and ( 5) above, whi ch b
t he r esour ce and capi t al st o,
of t he paper . However , not e
sol ut i ons t o t he pr obl ems
do i n f act have bounded r ,
of capi t al . Hence f or suf f i c
t he r at es of change of st ock
pr obl ems . I t f ol l ows t hat w
f or t he unbounded opt i mi z,
Not es
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11 .
12.
13 .
14.
15 .
16 .
17 .
18 .
19 .
20 .
21 .
See [ 12] f or a det ai l ed l i st i ng
Thi s pr oposi t i on, whi ch was
El sewher e we have cal l ed t hi
Thi s r esul t , and t he associ at e
and [ 3] .
We ar e gr at ef ul t o Kenn J ud
Thi s r esul t was i nt r oduced i n
We ar e gr at ef ul t o Har l Ryde .
it.
Thi s r esul t was f i r st pr oven f t
Thi s equal i t y i s not al ways
aut onomous syst emt o t he au
Thi s r esul t was f i r st pr oven i t
Due t o Har l Ryder .
Thi s di scount f act or i s i nf i ni t ,
Fur t her di scussi ons of t i me a
Thi s r esul t was f i r st pr oven i t
For a f ur t her di scussi on, see [
Thi s r esul t was f i r st pr oven i t
See [ 18] f or det ai l s .
Thi s r esul t was f i r st pr oven i t
See [ 18] f or det ai l s .
Thi s r esul t was f i r st pr oven i r
Thi s r esul t was f i r st pr oven i n
Sust ai nabl e Use of Renewabl e Resour ces
; r ent i abl e . I t sat i sf i es t he
uuous wi t h r espect t o c and
ct t o t f or al l val ues of c and
is
r ( s) < b1 < oo, and r ( s)
e t hi r d i mpl i es t hat bounded
i s a f or m of t he assumpt i on
ce i s essent i al t o pr oduct i on .
t . The f i nal t wo assumpt i ons
-ce st ock or t he capi t al st ock
er y r easonabl e assumpt i ons .
we shal l pr ove t he exi st ence
i t hen not e t hat a pat h whi ch
; i bl e and opt i mal wi t h t hem.
i ) above, t he ut i l i t ar i an opt i of f easi bl e t i me pat hs of t he
[ y bounded above. ( Not e t hat
i ey ar e non- negat i ve and so
at egr abl e agai nst some f i ni t e
pace . Denot e by Pt he set of
of L1, Pi s cl osed and nor m
; m i t i s weak- * compact . By
al so compact i n t he nor mof
Pt o t he r eal l i ne R. To comuous i n t he nor mof L 1 . Thi s
L P cont i nui t y gi ven i n [ 6] :
t ) dv ( t ) f or af i ni t e measur e
condi t i on. Then W
def i nes a
t o coor di nat e syst em of Lp i f
( t ) > 0, f gt a ( t ) dv ( t ) < oc
pl ayed by u ( ct , st ) e- bt . An
u def i ned on R2 i s st r ai ght , i mpl i es t hat Chi chi l ni sky' s
75
i nequal i t y i s sat i sf i ed f or p = 1 . Thi s compl et es t he pr oof of exi st ence of an
opt i mum.
El
We have nowpr oven t he exi st ence of an opt i mal pat h f or t he most compl ex of
t he opt i mi zat i on pr obl ems di scussed i n t he paper : exi st ence of an opt i mumf or
t he si mpl er pr obl ems can be deduced f r omt hi s . Our pr oof used assumpt i ons
( 4) and ( 5) above, whi ch bound r espect i vel y s and k, t he r at es of change of
t he r esour ce and capi t al st ocks . These assumpt i ons wer e not made i n t he body
of t he paper. However , not e f r omt he char act er i zat i on r esul t s i n t he paper t hat
sol ut i ons t o t he pr obl ems wi t hout bounds on t he r at es of change of st ocks
do i n f act have bounded r at es of change of t he st ocks of t he r esour ce and
of capi t al . Hence f or suf f i ci ent l y l ar ge bounds, t he i mposi t i on of bounds on
t he r at es of change of st ocks cannot change t he sol ut i ons t o t he opt i mi zat i on
pr obl ems . I t f ol l ows t hat we have al so est abl i shed t he exi st ence of sol ut i ons
f or t he unbounded opt i mi zat i on pr obl ems .
Not es
1.
2.
3.
4.
5.
6.
7.
8.
9.
10 .
11 .
12 .
13 .
14 .
15 .
16 .
17 .
18 .
19 .
20 .
21 .
See [ 12] f or a det ai l ed l i st i ng of many mor e exampl es .
Thi s pr oposi t i on, whi ch was f i r st pr oved i n [ 18] , i s a st r engt heni ng of r esul t s i n [ 3] .
El sewher e we have cal l ed t hi s t he gr een gol den r ul e [ 3] .
Thi s r esul t , and t he associ at ed concept of t he gr een gol den r ul e, wer e i nt r oduced i n [ 1]
and [ 3] .
We ar e gr at ef ul t o Kenn Judd f or t hi s obser vat i on .
Thi s r esul t was i nt r oduced i n [ 18] .
We ar e gr at ef ul t o Har l Ryder f or suggest i ng t hi s r esul t and out l i ni ng t he i nt ui t i on behi nd
it.
Thi s r esul t was f i r st pr oven i n [ 18] .
Thi s equal i t y i s not al ways t r ue : i t r equi r es l ocal l y uni f or m conver gence of t he nonaut onomous syst em t o t he aut onomous syst em. For det ai l s, see [ 4] .
Thi s r esul t was f i r st pr oven i n [ 18] .
Due t o Har l Ryder .
Thi s di scount f act or i s i nf i ni t e when t = 0: hence t he need t o st ar t f r om t = 1 .
Fur t her di scussi ons of t i me consi st ency can be f ound i n [ 14] .
Thi s r esul t was f i r st pr oven i n [ 18] .
For a f ur t her di scussi on, see [ 13] and r ef er ences t her ei n .
Thi s r esul t was f i r st pr oven i n [ 18] .
See [ 18] f or det ai l s.
Thi s r esul t was f i r st pr oven i n [ 18] .
See [ 18] f or det ai l s.
Thi s r esul t was f i r st pr oven i n [ 18] .
Thi s r esul t was f i r st pr oven i n [ 18] .
76
A. Bel t r at t i et al .
Ref er ences
Bel t r at t i , A., G. Chi chi l ni sky and G. M
. Heal . " Sust ai nabl e gr owt h and t he gr een gol den
r ul e" , i nAppr oaches t o Sust ai nabl eEconomi cDevel opment , I an Gol di n and Al an Wi nt er s
( eds . ) , Par i s, Cambr i dge Uni ver si t y Pr ess f or t he OECD, 1993, pp . 147- 172 .
2. Bel t r at t i , A. , G. Chi chi l ni sky and G. M.Heal . " The Envi r onment and t he Long Run: A
Compar i son of Di f f er ent Cr i t er i a" , Ri cer che Economi che 48, 1994, 319- 340.
3 . Bel t r at t i , A., G. Chi chi l ni sky and G. M
. Heal . " The Gr een Gol den Rul e" , Economi cs
Let t er s 49, 1995, 175- 179.
4. Bena-i m, M
. and M
. W
. Hi r sch . " Asympt ot i c Pseudot r aj ect or i es, Chai n Recur r ent Fl ows
and St ochast i c Appr oxi mat i ons" , Wor ki ngPaper , Depar t ment of Mat hemat i cs, Uni ver si t y
of Cal i f or ni a at Ber kel ey, 1994.
5. Ber r y, S. , G. M
. Heal and P. Sal omon. " On t he Rel at i on bet ween Economi c and Ther modynami c Concept s of Ef f i ci ency i n Resour ce Use" , Resour ces and Ener gy 1, 1978,
125- 137 . ( al so r epr i nt ed i n [ 16] ) .
6. Chi chi l ni sky, G. " Nonl i near Funct i onal Anal ysi s and Opt i mal Economi c Gr owt h" , Jour nal of Opt i mi zat i on Theor y and Appl i cat i ons 61( 2) , 1977, 504- 520 .
7. Chi chi l ni sky, G. " What I s Sust ai nabl e Devel opment ?" , Wor ki ng Paper , St anf or d I nst i t ut e
f or Theor et i cal Economi cs, 1993 .
8. Chi chi l ni sky, G. " Sust ai nabl e Devel opment : An Axi omat i c Appr oach" , Soci al Choi ce
and Wel f ar e 13( 2) , 1996, 219- 248.
9. Chi chi l ni sky, G. and P. F. Gr uenwal d. " Exi st ence of an Opt i mal Gr owt h Pat h wi t h
Endogenous Techni cal Change" , Economi cs Let t er s 48, 1995, 433- 439.
10 . Dasgupt a, P. S. and G. M
. Heal . " The Opt i mal Depl et i on of Exhaust i bl e Resour ces" ,
Revi ew of Economi c St udi es, Speci al I ssue on Exhaust i bl e Resour ces, 1974, 3- 28 .
11 . Dasgupt a, P. S. and G. M.Heal . Economi c Theor y and Exhaust i bl eResour ces, Cambr i dge
Uni ver si t y Pr ess, 1979 .
12 . Dai l y, G. Nat ur e' s Ser vi ces, Soci et al Dependence on Nat ur al Ecosyst ems, I sl and Pr ess,
Washi ngt on DC, 1997.
13 . Har vey, C. " The Reasonabl eness of Non- Const ant Di scount i ng" , Jour nal of Publ i c Economi cs 53, 1994, 31- 51 .
14 . Heal , G. M.The Theor y of Economi c Pl anni ng, Advanced Text s i n Economi cs, Amst er dam, Nor t h- Hol l and, 1973 .
15 . Heal , G. M
. " The Opt i mal Use of Exhaust i bl e Resour ces" Handbook of Nat ur al Resour ce
and Ener gy Economi cs, Vol . I I I , Al an Kneese and James Sweeney ( eds. ) , Amst er dam,
. NewYor k and Oxf or d, Nor t h- Hol l and, 1993, pp. 855- 880 .
16. Heal , G. M
. The Economi cs of Exhaust i bl e Resour ces, I nt er nat i onal Li br ar y of Cr i t i cal
Wr i t i ngs i n Economi cs, Edwar d El gar , 1993 .
17. Heal , G. M
. " I nt er pr et i ng Sust ai nabi l i t y" , i n Soci al Sci ences and t he Envi r onment , L.
Quesnel ( ed . ) , Uni ver si t y of Ot t awa Pr ess, 1995, pp. 119- 143 .
18 . Heal , G. M
. Lect ur es on Sust ai nabi l i t y, Li ef Johansen Lect ur es, Uni ver si t y of Osl o, 1995.
Ci r cul at ed as a wor ki ng paper of t he Depar t ment of Economi cs, Uni ver si t y of Osl o, and
f or t hcomi ng as Val ui ng t he Fut ur e: Economi c Theor y and Sust ai nabi l i t y, Col umbi a
Uni ver si t y Pr ess .
19 . Kr aut kr amer , J . A. " Opt i mal Gr owt h, Resour ce Ameni t i es and t he Pr eser vat i on of Nat ur al
Envi r onment s" , Revi ew of Economi c St udi es 52, 1985, 153- 170 ( al so r epr i nt ed i n [ 16] ) .
20 . Lowenst ei n, G. and R. Thal er . " I nt er t empor al Choi ce" , Jour nal of Economi c Per spect i ves
3, 1989, 181- 193 .
21 . Roughgar den, J . and F. Smi t h . " Why Fi sher i es Col l apse and What t o Do about I t " , i n
Pr oceedi ngs of t he Nat i onal Academy of Sci ences, f or t hcomi ng .
22 . Sei er st ad, A. and K. Sydsxt er . Opt i mal Cont r ol Theor y wi t h Economi c Appl i cat i ons,
Advanced Text s i n Economi cs, Amst er dam, Nor t h- Hol l and, 1987 .
RALPH ABRAHAM, GRAC
1.
2 . 2 . Nor t h- Sout h Tr y
of t he Envi r onm
1.
I nt r oduct i on
Thi s paper devel ops a dyr
envi r onment pl ays an i mpc
Nor t h- Sout h model f or t he
of t he wor l d economy . The
W
e i nt r oduce dynami cs i n
endogenous accumul at i on c
duce her e a var i abl e whi cl
envi r onment al asset whi ch
coul d r epr esent , f or exampl
i s ext r act ed t o be used as a
pr oper t y r i ght s on wat er v
goods f or expor t .
The paper expl ai ns mat h
of a t wo- r egi on wor l d . Th
pr oduct i on . Capi t al i s one
t i me as a f unct i on of pr of i t ,
t he envi r onment t he dynam
ar e t he pr oper t y r i ght s, t he
The model s whi ch r esul t
Neumann i n 1932 and ext er
W
e est abl i sh, i n a sequenc
coupl ed l ogi st i c maps st udi o
i dea i s t o al t er [ I ] t o al l ow c
t he appr oach t o equi l i br i um
our model , whi ch ar e not f
evol ut i on of capi t al st ock t h
G. Chi chi l ni sky et al ( eds) , Sust ai nat
© 1998 Kl uwer Academi c Publ i sher s
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