2HotellingandParkEEP1

7/18/2019 2HotellingandParkEEP1

http://slidepdf.com/reader/full/2hotellingandparkeep1 1/29

Natural Resource Theory

Copyright, 1998 by Peter Berck



Introduction

Natural Resource Theory is theeconomic theory o e!haustible andrene"able resources# Theseresources last or more than oneperiod o time and so unction as atype o capital# They are also used

or ood, $ber and energy and sounction as ordinary goods



%!haustible Resource

• &il, Coal, etc#

• &ld gro"th trees

•

These pro'ide 'alue by being usedup#



(otelling)s *odel+ %-uations

• 1# Capital *arket %-uilibrium

• .# /easibility

•

# /lo" *arket %-uilibria#



The Capital *arket

• 0ssets %arn a return through

• i'idends

• i'idends paid out rom earnings o $rms#

• Car actory adds 'alue to steel, labor etc#

• Capital 2ains

• 3tock can also ha'e capital gain+ its price

goes up#• 0ll %!haustible Nat Resource returns must

come rom price change



Price 2oes 4p a Rate oInterest

• (otelling)s Rule

• Price o resource is P5t6

•

price at time 1 is P516# 5e#g# 7:thbd t or red"ood#6

Put P(1) $ inbank

Buy one unitof Resource

Period 1 ha'e 7P516 ha'e 7P516"orth oresource

Period . ha'e 751;r6P516

ha'e 7P5.6"orth o

resource



so<

• =hen is it a good idea to by theresource>

• "hen p5.6 ?@ 51;r6 p516

• I ?, then e'eryone "ould "ant theresource and nobody "ould "ant anythingelse in their portolio#

•

I A, then nobody "ould "ant the resource• 3o p5.6 @ 51;r6 p516

• 0nd in general P5t;16 @ P5t6 51;r6#



Real &ld 2ro"th Red"oodPrice

53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 830

50

100

150

200

250

300

350

400

450

year

$ / M

B F



4se no more than there is

• 3econd, the sum o the stumpage

cut, -5t6, o'er time e-uals theoriginal stock o stumpage,

X = Q0

+ Q1

+ . . . + QT

.



emand and 3upply

• 5p, h6 is the demand "hen price is pand some shit 'ariable 5housingstarts i this is oldgro"th stumpage6

is h#

• 56 Q(t) = D(p(t) , h).

•



3ol'ing the *odel

• p = p0(1+r)t, where p0 is initial price

• Q(t) = D(p(t) , h).

•

SO• Q(t) = D(p0(1+r)t, h).

0

,

( (1 ) , )t

t o T

X D p r h=

= +∑



et &'er

• =hy is it e-uality>

• =hy not ?

•

=hy not A• Not so ar etched# 3uppose global

"arming bites and "e gi'e up coalmining# =hat "ill coal then be "orth>

0

,

( (1 ) , )t

t o T

X D p r h=

= +∑



The solution

0

,

( (1 ) , )t

t o T

X D p r h=

= +∑

3o all "e need to do is to $nd p56

and possibly T i its not in$nity

Then "e "ill kno" p and D or e'ery ti



Backstop

• inear demand cur'e 5and any other onethat hits the a!is6 has a choke price#

• Choke price is the price that chokes oE

demand or the resource#• 0t choke price some other technology is

used to meet demand 5e#g# coal instead ooil#6

• P 51;r6 T@ Choke

• I you kno" p, and choke, you kno" T,e!haustion time#



(otelling in FGDuadrants

t

p

-

-

demand

p

P51;r6t

FH

line

Note+ choke price, T#



(otelling in FGDuadrants

t

p

-

-

demand

p

P ert

FH

line



%-uilibrium+ 2reen is 56

t

p

-

-

demand

p

P ert

FH

line

0rea under cur'e is sum o all D)s#



Too o" a

P

t

p

-

-

demand

p

P ert

FH

line

p

0 choice o Po belo"

the e-uilibrium 'alue

leads to more - in eac

period, "hich is more

than 56 by the red#



Increase in r

t

p

-

-

p

P ert

FH

line

p

Red Bounded area must

e-ual green area

Initial price lo"er T sooner

T"o PriceGtime paths

must cross



Taking o the Red"ood Park

• In 19J8 and again in 198 the 43took a total o #1 billion bd t ostanding timber rom pri'ate

companies to orm the Red"oodNational Park

• The amount by "hich the price o

Red"ood "ent up as a result o thetake is called enhancement



%nhancement+ o"ering56

t

p

-

-

Red price path is

result o red 560rro" sho"s siKe0rro" sho"s siKe

o enhancemento enhancement

p

P ert

FH

line

p



Recall

( ) 0

,

( (1 ) , ) j

j t T

X t D p r h=

= +∑

ets call the solution to this P5!5t66, the price

=hen stock is ! at time t#

P5!56 6@ p

P5!5t66 @ p 51;r6t



/olded iagram *odel

p(x(t)) = p p(x(t)) = p00(1+r)(1+r)tt

• price as function of stock is same as price as function of time price as function of stock is same as price as function of time

• price in year t + 1 is just p(x(t) – (t)) !"ic" is a#so p0(1+r)(t+1)

• p(x(t) – ) = p0(1+r)(t+n)

• price after n years of cuttin$ eua#s t"e price at

• time t ,(p0(1+r)

t

) times t"e interest factor for n years (1+r)

n

.• %"oose n so t"at t"e &ark takin$ eua#s

( )

n

t

q t ∑

( )n

t

q t ∑



%nhancement+ Lears*ethod

t

p

-

-

56 is again red area#

0rro" sho"s number o 0rro" sho"s number o years need to "ait toyears need to "ait to

$nd e-ui'alent$nd e-ui'alent

p

P ert

FH

line

p



Malue o %nhancement

• The 198 Park taking "as 1#F billionboard eet, "hich is the e-ui'alent o.#.J years o cutting#

• price 198, "as 711 per *B/#

• real interest rateO percent

•

.#.J years at percent real per yearor 1 percent o price



Conclusion

• 2o')t paid 7J89 million or secondtake

• enhancement "as 7H8 million

• Thereore the 43 paid nearly t"iceor the park



Continuous time e!ample

• Interest is e!p5Grt6 rather than 51;r6Gt

• Integral replaces summation

•

emand unction has no choke price5makes it easier6



%!ample+ D @ pGα

( ) rt

0

0

0 Q(p e ) . x dt

∞

= ∫

( ) ∫ ∞

=0

rt''

0 .e p0 dt x α α

( )r r

xα α

α α

α

'

0

0

rt''

0

pe p0 =

−=

∞



%!ample Concluded+Reduced /orm

( )r r

xα α

α α

α

'

0

0

rt''

0

pe p0 =

−=

∞

( )

α

α

1

00

1 p

=

xr

( ))0(#n()#n()#n(1)#n(p0 xr ++−= α α

2HotellingandParkEEP1

Documents

Transcript of 2HotellingandParkEEP1