THE EFFICIENT AND OPTIMAL USE OF

ENVIRONMENTAL RESOURCES

SDP ENRE Session 1 Part 2

NON-RENEWABLE RESOURCES

REQUIRED READING:

Perman et al (2nd ed): Chapters 7, 8.

Perman et al (3rd ed): Chapters 14, 15.

Equation numbering

• Please note that all equation numbers refer to Perman et al 2nd edition.

• If you are using 3rd edition, simply replace 7 by 14. So, for example, equation 7.8 becomes equation 14.8

S E T T I N G U P T H E M O D E L

A s i m p l i f i e d “ m o d e l ” o f t h e e c o n o m y .

F i r s t e l e m e n t : t h e o b j e c t i v e o f s o c i e t y .

T h e e c o n o m y ’ s ( s o c i a l ) u t i l i t y f u n c t i o n a t e a c h p o i n t i n

t i m e : U t = U ( C t ) f o r a l l t

T h e n , t h e e c o n o m y ’ s i n t e r t e m p o r a l s o c i a l w e l f a r e

f u n c t i o n :

dte)C(UWtt

0t t

( 7 . 8 )

where

C = consumption

U = aggregate utility flow

= social utility discount rate

Variables are indexed by the time subscript t, where t = 0,...,

T, with t = 0 being the initial period and t = T (where T may

be infinity) being the final period.

Next we specify the environmental resource stock-

flow relationship for a non-renewable resource:

dRSS

t

00t (7.9)

or

tt RS (7.10)

where tS= dS/dt.

where

R = environmental resource flow (amount extracted and used

(per period)

S = environmental resource stock

S0 = initial stock (at t = 0)

F i n a l l y w e s p e c i f y t h e e c o n o m i c s y s t e m :

N a t i o n a l i n c o m e a c c o u n t i n g i d e n t i t y :

ttt CQK ( 7 . 1 1 ) w h e r e K = m a n u f a c t u r e d c a p i t a l s t o c k

T h e e c o n o m y ’ s p r o d u c t i o n f u n c t i o n : Q t = Q ( K t , R t )

Q R = Q / R = m a r g i n a l p r o d u c t o f t h e r e s o u r c e

Q K = Q / K = m a r g i n a l p r o d u c t o f c a p i t a l

a n d s o

tttt C)R,K(QK

S U M M A R Y O F T H E P R O B L E M

S e l e c t v a l u e s f o r t h e c h o i c e v a r i a b l e s C t a n d R t f o r t = 0 , . . . ,

t o m a x i m i s e

dte)C(UW tt

0tt

s u b j e c t t o

tt RS

a n d

tttt C)R,K(QK

S ( 0 ) = S 0 , f i x e d .

F o u r e q u a t i o n s c h a r a c t e r i s e t h e o p t i m a l s o l u t i o n :

tt,CU ( 7 . 1 4 a )

t,Rtt QP ( 7 . 1 4 b )

tt PP ( 7 . 1 4 c )

tt,Ktt Q ( 7 . 1 4 d )

Hotelling’s Rule

or

where P is the net price (or rent, or royalty) of the resource.

tt PP

tPdtdP

Intuition behind Hotelling’s Rule

Owner of financial resources: two uses of the capital

An interest generating financial asset

r = return per period

Invest capital in a mine

Return on mine must also be r in equilibrium

As mine is not intrinsically productive, price of mineral must rise at rate r.

Hotelling’s Rule

By integration

implies that

and so …

tPdtdP

tePP 0t

Pt

t

Pt = P0et

P0

Hotelling’s rule: the time path of the resource net price

Pt

t

Pt = Paet

Pa

Hotelling’s rule: non-uniqueness of efficient time paths of the resource net price

Pt = Pbet

Pb

The optimal path will be that one which satisfies S 0 as t

Pt

t

Pt

P0

The time paths of the resource net price and stock

St

t

Net price

Remaining resource stock

Pt

t

Pt = P0et

P0

Hotelling’s rule: with a backstop technology available

Choke Price

t=T

t=T

Pt

P0

The time paths of the resource net price and stock with a backstop technology.

St

t=T

Net price

Remaining resource stock

DOES A MARKET ECONOMY YIELD AN

OPTIMAL AND EFFICIENT

ALLOCATION OF RESOURCES?

Answer: Yes (both efficient and optimal) under

certain circumstances.

The efficiency of the market mechanism: intuition:

In a competitive market economy, profit maximisation

requires that firms take proper account of their revenue

and cost functions.

The utility function provides appropriate information

about the market demand curve, and so provides

appropriate information about firms’ revenues.

Any costs of production and extraction will be taken

into account by businesses.

MUt

CtC1

MU1 = P1

The marginal utility function tells us about willingness to pay (WTP)and so corresponds to a demand function in a market economy

A market economy will probably not deliver an efficient and optimal allocation of non-renewable resources because:

•Monopoly: depletion too slow.

•Social costs of resource depletion not considered. (e.g. pollution externalities).

•Private (market) interest rate above the social discount rate.

There may be other forms of market failure:

Presence of public goods

Absence of well-defined and enforceable property rights

Incomplete information

And some of the agents’ functions may depend upon environmental

quality:

U depends on environmental quality

Q depends on environmental quality

Two asides:

(1) Optimality and distribution of initial endowments: we

have made social utility depend only on the aggregate total

of consumption, and not on its distribution. But market

demand curves WILL depend on distribution of

endowments, and may not correspond to an economy’s

(social) utility function if that latter function is defined

differently.

Second …

Sustainability: optimal outcomes may not be sustainable. To

achieve sustainability, may need to impose an additional

constraint on the above optimisation exercises, of the form:

Ut Ut-s for all s > 0 or Ct Ct-s for all s > 0

[Of course, sustainability may not even be feasible.]

E x t e n d i n g t h e m o d e l t o i n c o r p o r a t e e x t r a c t i o n c o s t s

G = t o t a l e x t r a c t i o n c o s t s

G t = G ( R t , S t ) ( 7 . 1 9 )

T h e o p t i m i s a t i o n p r o b l e m :

S e l e c t v a l u e s f o r t h e c h o i c e v a r i a b l e s C t a n d R t f o r t

= 0 , . . . , t o m a x i m i s e

dte)C(UWtt

0t t

s u b j e c t t o t h e c o n s t r a i n t s

tt RS

a n d

)S,R(GC)R,K(QK tttttt

T h e s o l u t i o n t o t h i s p r o b l e m :

CU ( 7 . 2 0 a )

RR GQP ( 7 . 2 0 b )

SGPP ( 7 . 2 0 c )

KQ ( 7 . 2 0 d )

tP

= Rt Q

l e s s Rt G

N e t

p r i c e

= G r o s s

p r i c e

l e s s M a r g i n a l c o s t