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
SETTING UP THE MODEL
A simplified “model” of the economy.
First element: the objective of society.
The economy’s (social) utility function at each point in
time: Ut = U(Ct) for all t
Then, the economy’s intertemporal social welfare
function:
W
 t
t 
t 0 U (C t )e dt
(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 stockflow relationship for a non-renewable resource:
 t
S t  S 0   R  d
0
or
S t   R t
(7.10)
where S t = dS/dt.
(7.9)
where
R = environmental resource flow (amount extracted and used
(per period)
S = environmental resource stock
S0 = initial stock (at t = 0)
Finally we specify the economic system:
National income accounting identity:
 t  Q t  C t (7.11)
K
where K = manufactured capital stock
The economy’s production function: Qt = Q (Kt, Rt)
QR = Q/R = marginal product of the resource
QK = Q/K = marginal product of capital
and so
 t  Q( K t , R t )  C t
K
SUMMARY OF THE PROBLEM
Select values for the choice variables Ct and Rt for t = 0,...,
to maximise
t 
W   U (C t ) e t dt
t 0
subject to
S t   R t
and
 t  Q(K t , R t )  C t
K
S(0) = S0, fixed.
Four equations characterise the optimal solution:
U C, t  t
(7.14a)
Pt  t Q R , t
(7.14b)
P t  Pt
(7.14c)

 t   t  Q K , t t
(7.14d)
Hotelling’s Rule
P t   Pt
or
dP
  Pt
dt
where P is the net price (or rent, or royalty) of the resource.
Intuition behind Hotelling’s Rule
Owner of financial resources:
two uses of the capital
Invest capital in a mine
An interest generating
financial asset
Return on mine must also
be r in equilibrium
r = return per period
As mine is not intrinsically productive,
price of mineral must rise at rate r.
Hotelling’s Rule
By integration
dP
  Pt
dt
implies that
Pt  P0e
t
and so …
Pt
Hotelling’s rule: the time path of the resource net price
Pt = P0et
P0
t
Pt
Hotelling’s rule: non-uniqueness of efficient time paths
of the resource net price
Pt = Pbet
Pt = Paet
Pb
Pa
t
The optimal path will be that one which satisfies S  0 as t
Pt
The time paths of the resource net price and stock
St
Remaining
resource stock
Pt
Net
price
P0
t
t
Pt
Hotelling’s rule: with a backstop technology available
Choke
Price
Pt = P0et
P0
t=T
t
The time paths of the resource net price and stock with
a backstop technology.
St
Remaining
resource stock
Pt
Net
price
P0
t=T
t=T
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.
The marginal utility function tells us about willingness to pay (WTP)
and so corresponds to a demand function in a market economy
MUt
MU1 = P1
C1
Ct
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.]
Extending the model to incorporate extraction costs
G = total extraction costs
Gt = G(Rt, St)
(7.19)
The optimisation problem:
Select values for the choice variables Ct and Rt for t
=0,..., to maximise
W
t
t 
t 0 U (C t )e dt
subject to the constraints
S t   R t
and
 t  Q( K t ,R t )  C t  G ( R t ,St )
K
The solution to this problem:
UC  
(7.20a)
P  Q R  G R
(7.20b)
P  P  G S
(7.20c)

    Q K 
(7.20d)
Pt
=
Net
=
price
t Q R
Gross
price
less
t G R
less
Marginal cost