The document discusses optimal use of natural resources over time from an economic perspective. It develops a simple model where natural resources are an input in the production process along with capital. Under certain conditions, sustainable development is possible where consumption remains constant over time. The key conditions are: - A high degree of substitutability between capital and natural resources. - A sufficiently high rate of technical progress. - The presence of a permanent "backstop" technology that does not require natural resources.

1. The efficient and optimal use of
natural resources

2. Objectives
• Develop a simple economic model, built around a production function in
which natural resources are inputs into the production process;
• Identify the conditions that must be satisfied by an economically efficient
pattern of natural resource use over time;
• Establish the characteristics of a socially optimal pattern of resource use over
time in the special case of a utilitarian social welfare function.

3. A simple optimal resource depletion model:
the economy and its production function
• The economy produces a single good, Q, which can be either consumed or invested.
• Consumption increases current well-being, while investment increases the capital stock,
permitting greater consumption in the future.
• Output is generated through a production function using as inputs a single ‘composite’
non-renewable resource input, R, and manufactured capital, K:
Q = Q(K, R) (1)

4. Is the natural resource essential?
• Essentialness of a resource could mean several things.
• Here, we interpret this term to mean whether a resource is directly essential for
production (where production and the resource are both conceptualised sources at a high
degree of aggregation, dealing with general classes such as total output and non-
renewable and renewable resources.
• A productive input is defined to be essential if output is zero whenever the quantity of
that input is zero, irrespective of the amounts of other inputs used. That is, R is essential
if Q = Q(K, R = 0) = 0 for any positive value of K….. (2)
• In the case of the CD production function, R and K are both essential, as setting any
input to zero in equation results in Q = 0…. (3)

5. Relevance
• If we wish to answer questions about the long-run properties of economic systems, the
essentialness of non-renewable resources will matter.
• Since, by definition, non-renewable resources exist in finite quantities it is not possible
to use constant and positive amounts of them over infinite horizons.
• However, if a resource is essential, then we know that production can only be
undertaken if some positive amount of the input is used.
• This seems to suggest that production and consumption cannot be sustained indefinitely
if a non-renewable resource is a necessary input to production.
• However, if the rate at which the resource is used were to decline asymptotically to zero,
and so never actually become zero in finite time, then production could be sustained
indefinitely even if the resource were essential.
• Whether output could rise, or at least stay constant over time, or whether it would have
to decline towards zero will depend upon the extent to which other resources can be
substituted for non-renewable resources and upon the behaviour of output as this
substitution takes place.

6. Elasticity of substitution
Expression:
or equivalently:
• where the partial derivative QR = ∂Q/∂R denotes the marginal product of the resource;
QK = ∂Q/∂K denotes the marginal product of capital, and where PR and PK denote the
unit prices of the non-renewable resource and capital, respectively.

7. Q
Q 
 = 0
Q
Q  Q
Q 
0 <  < 
 = 
Figure 1 Substitution possibilities and the shapes of production function
isoquants
K
R
0

8. Substitution possibilities and the shapes of
production function isoquants
• The differing substitution possibilities are reflected in the curvatures of the isoquants.
• In the case where no input substitution is possible (that is, σ = 0), inputs must be
combined in fixed proportions and the isoquants will be L-shaped. (Known as Leontief
functions. They are commonly used in input–output models of the economy.)
• At the other extreme, if substitution is perfect (σ = ), isoquants will be straight lines.
• In general, a production function will exhibit an elasticity of substitution somewhere
between those two extremes (although not all production functions will have a constant
σ for all input combinations). In these cases, isoquants will often be convex to the
origin, exhibiting a greater degree of curvature the lower the elasticity of substitution, σ.
• For a CES production function, we can also relate the elasticity of substitution to the
concept of essentialness. It can be shown that σ = 1/(1 + θ). No input is essential where θ
< 0, and all inputs are essential where θ > 0. Given the relationship between σ and θ, it
can be seen that no input is essential where σ > 1, and all inputs are essential where σ <
1.
• Where σ = 1 (that is, θ = 0), the CES production function collapses to the CD form,
where all inputs are essential.

9. Resource substitutability and the consequences
of increasing resource scarcity
• As production continues throughout time, stocks of non-renewable resources must decline.
Continuing depletion of the resource stock will lead to the non-renewable resource price rising
relative to the price of capital. As the relative price of the non-renewable resource rises the resource
to capital ratio will fall, thereby raising the marginal product of the resource and reducing the
marginal product of capital.
• However, the magnitude of this substitution effect will depend on the size of the elasticity of
substitution. Where the elasticity of substitution is high, only small changes in relative input prices
will be necessary to induce a large proportionate change in the quantities of inputs used. ‘Resource
scarcity’ will be of little consequence as the economy is able to replace the scarce resource by the
reproducible substitute.
• Low substitution possibilities mean that as resource depletion pushes up the relative price of the
resource, the magnitude of the induced substitution effect will be small. ‘Resource scarcity’ will have
more serious adverse effects, as the scope for replacement of the scarce resource by the reproducible
substitute is more limited. Where the elasticity of substitution is zero, then no scope exists for such
replacement.

10. The feasibility of sustainable development
• Is sustainable development actually possible?
• To address this question, two things are necessary.
1. A criterion of sustainability is required.
2. We need to describe the material transformation conditions available to society, now and in the future. These
conditions – the economy’s production possibilities – determine what can be obtained from the endowments of
natural and human-made capital over the relevant time horizon.
• We adopt here a conventional sustainability criterion: non-declining per capita
consumption maintained over indefinite time .
• Turning attention to the transformation conditions, it is clear that a large number of
factors enter the picture.
– What is happening to the size of the human population?
– What kinds of resources are available and in what quantities, and what properties do they possess?
– What will happen to the state of technology in the future?
– How will ecosystems be affected by the continuing waste loads being placed upon the environment, and how will ecosystem
changes feed back upon productive potential?

11. Transformation possibilities
• To make progress, simplify and narrow down the scope of the problem, by making an assumption
about the form of an economy’s production function.
• A series of results have become established for several special cases. For the CD and CES functions
we have the following.
CASE A:
• Output is produced under fully competitive conditions through a CD production function with
constant returns to scale and two inputs, a non-renewable resource, R, and manufactured capital, K,
as in the following special case of above equation:
……..(5)
• Then, in the absence of technical progress and with constant population, it is feasible to have
constant consumption across generations if the share of total output going to capital is greater than
the share going to the natural resource (that is, if α > β).
  1
β
α
with
R
K
Q β
α




12. Other cases
Case B:
• Output is produced under fully competitive conditions through a CES production function with
constant returns to scale and two inputs, a non-renewable resource, R, and manufactured capital, K,
as in equation 14.3:
• Then, in the absence of technical progress and with constant population, it is feasible to have
constant consumption across generations if the elasticity of substitution σ = 1/(1 + θ) is greater than
or equal to one.
Case C:
• Output is produced under conditions in which a backstop technology is permanently available. In this
case, the non-renewable natural resource is not essential. Sustainability is feasible, although there
may be limits to the size of the constant consumption level that can be obtained.
    1
β
α
with
βR
αK
A
Q
ε/θ
θ
θ








13. Additional matters
• These results assumed that the rate of technical progress and the rate of population
growth were both zero.
• Results change if one or both of these rates is non-zero.
• The presence of permanent technical progress increases the range of circumstances in
which indefinitely long-lived constant per capita consumption is feasible.
• Constant population growth has the opposite effect.
• However, there are circumstances in which constant per capita consumption can be
maintained even where population is growing provided the rate of technical progress is
sufficiently large and the share of output going to the resource is sufficiently low.
• Similarly, for a CES production function, sustained consumption is possible even where
σ < 1 provided that technology growth is sufficiently high relative to population growth.
• The general conclusion is that sustainability requires either a relatively high degree of
substitutability between capital and the resource, or a sufficiently large continuing rate
of technical progress, or the presence of a permanent backstop technology.

14. Sustainability and the Hartwick rule
• John Hartwick (1977, 1978): identified two sets of conditions which were sufficient to
achieve non-declining consumption through time:
1. a particular savings rule, known as the Hartwick rule, which states that the rents derived from
an efficient extraction programme for the non-renewable resource are invested entirely in
reproducible (physical and human) capital;
2. conditions pertaining to the economy’s production technology, described on a previous slide.
• We discuss the implications of the Hartwick rule further in Chapter 19. But three
comments about it are worth making at this point.
1. The Hartwick rule is essentially an ex post description of a sustainable path. Hence if an economy
were not already on a sustainable path, then adopting the Hartwick rule is not sufficient for
sustainability from that point forwards.
2. Even were the economy already on a sustainable path, the Hartwick rule requires that the rents be
generated from an efficient resource extraction programme in a competitive economy.
3. Even if the Hartwick rule is pursued subject to this qualification, the savings rule itself does not
guarantee sustainability. Technology conditions may rule out the existence of a feasible path.

15. The social welfare function and an optimal
allocation of natural resources: model
Social welfare function (SWF)
In general form: (6)
• where Ut, t = 0,. . ., T, is the aggregate utility in period t.
Assume that the SWF is utilitarian in form:
(7)
• This defines social welfare as a weighted sum of the utilities of the relevant aggregate of
individual persons living at a sequence of points in time.
• We assume that utility in each period is a concave function of the level of consumption
in that period, so that Ut = U(Ct) for all t, with UC > 0 and UCC < 0.
 
T
2
1
0
U
,
...
,
U
,
U
,
U
W
W 
     T
...














1
U
1
U
1
U
1
U
U
W T
2
2
2
1
1
0

16. SWF in continuous time form with infinite
horizon
• For convenience, we switch from discrete-time to continuous-time notation,
and assume that the relevant time horizon is infinite.
• This leads to the following special case of utilitarian SWF:
•
.......(8)
  dt
e
C
U
W ρt
t
0
t
t







17. Constraints
Two constraints that must be satisfied by any optimal solution.
Constraint 1: Resource stock-flow constraint
• The resource stock is to be extracted and used by the end of the time horizon.
• Given this, together with the fact that we are considering a non-renewable resource for which there is
a fixed and finite initial stock, the total use of the resource over time is constrained to be equal to the
fixed initial stock.
• Denoting the initial stock (at t = 0) as S0 and the rate of extraction and use of the resource at time t as
Rt, we can write this constraint as
…(9 and 10)
t
R








dt
dS
S
implies
which
dτ
R
S
S
t
t
t
τ
0
τ
τ
0
t


18. Second constraint: accounting identity relating consumption,
output and the change in the economy’s stock of capital
• Output is shared between consumption goods and capital goods, and so that part of the economy’s
output which is not consumed results in a capital stock change. Writing this identity in continuous-
time form we have:
….(11)
• Given that output is produced through a production function involving two inputs, capital and a non-
renewable resource, Qt = Q(Kt, Rt), we can write the constraint as:
• …(12)
t
t
t
C
Q
K 


  t
t
t
C
K
Q
K 
 t
R
,


19. The optimisation problem

20. First order conditions
QK (= ∂Q/∂K) and QR (= ∂Q/∂R) are the partial derivatives of output with respect to capital and the non-renewable
resource. (i.e. the marginal products of capital and the resource).
Time subscripts are attached to these marginal products to make explicit the fact that their values will vary over time in
the optimal solution.
The terms Pt and ωt are the shadow prices of the two productive inputs, the natural resource and capital. These two
variables carry time subscripts because the shadow prices will vary over time. The solution values of Pt and ωt, for t = 0,
1, . . ., , define optimal time paths for the prices of the natural resource and capital.
The quantity being maximised in equation 8 is a sum of (discounted) units of utility. Hence the shadow prices are
measured in utility, not consumption (or money income), units.

21. First order conditions
Equation 14.14a: In each period, the marginal utility of consumption
UC,t must be equal to the shadow price of capital ωt . An efficient
outcome will be one in which the marginal net benefit of using one
unit of output for consumption is equal to its marginal net benefit
when it is added to the capital stock.

22. First order conditions
Equation 14.14b: The value of the marginal product of the natural
resource must be equal to the marginal value (or shadow price) of the
natural resource stock, Pt. The value of the marginal product of the
resource is the marginal product in units of output (QR,t) multiplied by
the value of one unit of output(, ωt).

23. Static and dynamic efficiency conditions
The static efficiency conditions
• As with any asset, static efficiency requires that, in each use to which a resource is put, the marginal
value of the services from it should be equal to the marginal value of that resource stock in situ. This
ensures that the marginal net benefit (or marginal value) to society of the resource should be the same
in all its possible uses.
• This is what equations 14(a) and 14(b) imply.
• Equation 14a: In each period, the marginal utility of consumption UC,t must be equal to the shadow
price of capital ωt . An efficient outcome will be one in which the marginal net benefit of using one
unit of output for consumption is equal to its marginal net benefit when it is added to the capital
stock.
• Equation 14b: The value of the marginal product of the natural resource must be equal to the
marginal value (or shadow price) of the natural resource stock, Pt. The value of the marginal product
of the resource is the marginal product in units of output (QR,t) multiplied by the value of one unit of
output(, ωt).

24. The dynamic efficiency conditions
• Dynamic efficiency requires that each asset or resource earns the same rate
of return, and that this rate of return is the same at all points in time, being
equal to the social rate of discount.
• Equations 14.14c and 14.14d ensure that dynamic efficiency is satisfied.
• Equation 14.14c: Dividing each side by P we obtain which states that the
growth rate of the shadow price of the natural resource (that is, its own rate of
return) should equal the social utility discount rate.
• Equation 14.14d: Dividing both sides of 14.14d by ω, we obtain an expression
which states that the return to physical capital (its capital appreciation plus its
marginal productivity) must equal the social discount rate.

25. First order conditions
Equation 14.14c: Dividing each side by P the expression states that
the growth rate of the shadow price of the natural resource (that is, its
own rate of return) should equal the social utility discount rate.

26. First order conditions
Equation 14.14d: Dividing both sides of 14.14d by ω, we obtain an
expression
which states that the return to physical capital (its capital appreciation
plus its marginal productivity) must equal the social discount rate.





t
,
K
t
t
Q


27. Hotelling’s rule: two interpretations
• Equation 14.14c is known as Hotelling’s rule for the extraction of non-renewable
resources. It is often expressed in the form
(15)
• The Hotelling rule is an intertemporal efficiency condition which must be satisfied by
any efficient process of resource extraction.
• A few lines of algebra ( given in the text) yields the following expression:
• which give a second interpretation: the discounted price of the natural resource is
constant along an efficient resource extraction path.


t
t
P
P

0
P
e
P
P t
t
*
t

 


28. Hotelling’s rule: an implication
• Note the effect of changes in the social discount rate on the
optimal path of resource price.
• The higher is ρ, the faster should be the rate of growth of
the natural resource price.
Look at Appendix 14.2

29. Intuition
• The social discount rate, ρ, reflects impatience for future consumption
• QK (the marginal product of capital) is the pay-off to delayed consumption.
The relations imply that along an optimal path:
1. consumption is increasing when ‘pay-off’ is greater than ‘impatience’;
2. consumption is constant when ‘pay-off’ is equal to ‘impatience’;
3. consumption is decreasing when ‘pay-off’ is less than ‘impatience’.
• Therefore, consumption is growing over time along an optimal path if the marginal product of capital
(QK) exceeds the social discount rate (ρ); consumption is constant if QK = ρ; and consumption
growth is negative if the marginal product of capital is less than the social discount rate. This makes
sense, given that:
1. when ‘pay-off’ is greater than ‘impatience’, the economy will be accumulating K and hence growing;
2. when ‘pay-off’ and ‘impatience’ are equal, K will be constant;
3. when ‘pay-off’ is less than ‘impatience’, the economy will be running down K.

30. 0
0.5
1
1.5
2
2.5
3
3.5
0 100 200 300 400 500 600 700 800 900 1000
C
U
=
log(C+1)
Figure 2 The logarithmic utility function

31. P0
A
P0
B
Pt
Pt
A = P0
Aet
Pt
B = P0
Bet
Figure 3 Two price paths, each satisfying Hotelling’s rule
time, t

32. Extending the model to incorporate extraction costs
Modelling extraction costs
• Denote total extraction costs as ; the amount of the resource being extracted as R; and remaining
resource stock as S.
• We would expect that  will be an increasing function of R. Also, in many circumstances, costs will
depend on the size of the remaining stock of the resource, typically rising as the stock becomes more
depleted.
• Letting St denote the size of the resource stock at time t (the amount remaining after all previous
extraction) we can write extraction costs as
(16)
)
S
,
Γ(R
Γ t
t
t


33. S0
(iii)
t
(for given value of
Rt = )
R
(i)
0
(ii)
Remaining resource stock, St
Figure 4 Three possible examples of the relationship between extraction
costs and remaining stock for a fixed level of resource extraction, R

34. The optimal solution to the resource depletion
model incorporating extraction costs

38. Conclusion
• The presence of costs related to the level of resource extraction raises the gross
price of the resource above its net price but has no effect on the growth rate of
the resource net price.
• In contrast, a resource stock size effect on extraction costs will slow down the
rate of growth of the resource net price.
• In most circumstances, this implies that the resource net price has to be higher
initially (but lower ultimately) than it would have been in the absence of this
stock effect. As a result of higher initial prices, the rate of extraction will be
slowed down in the early part of the time horizon, and a greater quantity of the
resource stock will be conserved (to be extracted later).

39. Generalisation to renewable resources
• Only brief comment here as lengthy analysis of the allocation of
renewable resources covered in Chapters 17 and 18.
• Assume that the amount of natural growth of non-renewable resources,
Gt, is some function of the current stock level, so that Gt = G(St).
• Then write the relationship between the change in the resource stock
and the rate of extraction (or harvesting) of the resource as
(21)
  t
t
t
R
S
G
S 



40. Generalisation to renewable resources (2)
• The efficiency conditions required by an optimal allocation of resources are now
different from the case of non-renewable resources.
• However, a modified version of the Hotelling rule for rate of change of the net price of
the resource still applies, given by
…..(22)
• Inspection of this modified Hotelling rule for renewable resources demonstrates that the
rate at which the net price should change over time depends upon GS, the rate of change
of resource growth with respect to changes in the stock of the resource.
 
S
PG
ρP
P 



41. Steady-state harvesting
• A steady-state harvesting of a renewable resource exists when all stocks and flows are
constant over time.
• In particular, a steady-state harvest will be one in which the harvest level is fixed over
time and is equal to the natural amount of growth of the resource stock.
• Additions to and subtractions from the resource stock are thus equal, and the stock
remains at a constant level over time.
• If demand for the resource is constant over time, the resource net price will remain
constant in a steady state, as the quantity harvested each period is constant.
• In a steady state, the Hotelling rule simplifies to
ρP = PGS (23)
and so
ρ = GS (24)

42. S
G
G*
0

S*
Figure 5 The relationship between the resource
stock size, S, and the growth of the resource, G
MSY = GMAX
Ŝ

43. Complications & extensions
• The model set up assumes that there is a single, known, finite stock of a non-renewable
resource. Furthermore, the whole stock was assumed to have been homogeneous in
quality. In practice, both of these assumptions are often false.
The following situations are likely:
1. The total stock is not known with certainty.
2. New discoveries increase the known stock of the resource.
3. A distinction needs to be drawn between the physical quantity of the stock and the
economically viable stock size.
4. Research and development, and technical progress, take place, which can change
extraction costs, the size of the known resource stock, the magnitude of economically
viable resource deposits, and estimates of the damages arising from natural resource use.
5. Even when we focus on a particular kind of non-renewable resource, the stock is likely
to be heterogeneous. Different parts of the total stock are likely to be uneven in quality,
or to be located in such a way that extraction costs differ for different portions of the
stock.

44. Backstop technology
• By treating all non-renewable resources as one composite good, our analysis in this
chapter had no need to consider substitutes for the resource in question (except, of
course, substitutes in the form of capital and labour).
• But once our analysis enters the more complex world in which there are a variety of
different non-renewable resources which are substitutable for one another to some
degree, analysis inevitably becomes more complicated.
• One particular issue of great potential importance is the presence of backstop
technologies (see Chapter 15).
• The existence of a backstop technology will set an upper limit on the level to which the
price of a resource can go. If the cost of the ‘original’ resource were to exceed the
backstop cost, users would switch to the backstop.