The document discusses the fundamental concepts of natural resource exploitation, focusing on the difference between renewable and non-renewable resources, and their economic implications. It includes the capital-theoretic approach to natural resources, emphasizing the importance of efficient intertemporal allocation, investment returns, and discount rates. The document also elaborates on Hotelling’s rule, which addresses the optimal extraction rate for non-renewable resources.

1. Natural resource exploitation:
basic concepts
NRE - Lecture 1
Aaron Hatcher
Department of Economics
University of Portsmouth

2. Introduction
I Natural resources are natural assets from which we derive
value (utility)

3. Introduction
I Natural resources are natural assets from which we derive
value (utility)
I Broad de…nition includes amenity value, provision of
“ecosystem services”, etc.

4. Introduction
I Natural resources are natural assets from which we derive
value (utility)
I Broad de…nition includes amenity value, provision of
“ecosystem services”, etc.
I We focus here on natural resources that must be extracted or
harvested

5. Introduction
I Natural resources are natural assets from which we derive
value (utility)
I Broad de…nition includes amenity value, provision of
“ecosystem services”, etc.
I We focus here on natural resources that must be extracted or
harvested
I Distinguish between renewable and non-renewable resources

6. Introduction
I Natural resources are natural assets from which we derive
value (utility)
I Broad de…nition includes amenity value, provision of
“ecosystem services”, etc.
I We focus here on natural resources that must be extracted or
harvested
I Distinguish between renewable and non-renewable resources
I Renewable resources are capable of growth (on some
meaningful timescale), e.g., …sh, (young growth) forests

7. Introduction
I Natural resources are natural assets from which we derive
value (utility)
I Broad de…nition includes amenity value, provision of
“ecosystem services”, etc.
I We focus here on natural resources that must be extracted or
harvested
I Distinguish between renewable and non-renewable resources
I Renewable resources are capable of growth (on some
meaningful timescale), e.g., …sh, (young growth) forests
I Non-renewable resources are incapable of signi…cant growth,
e.g., fossil fuels, ores, diamonds

8. Introduction
I Natural resources are natural assets from which we derive
value (utility)
I Broad de…nition includes amenity value, provision of
“ecosystem services”, etc.
I We focus here on natural resources that must be extracted or
harvested
I Distinguish between renewable and non-renewable resources
I Renewable resources are capable of growth (on some
meaningful timescale), e.g., …sh, (young growth) forests
I Non-renewable resources are incapable of signi…cant growth,
e.g., fossil fuels, ores, diamonds
I In general, e¢ cient and optimal use of natural resources
involves intertemporal allocation

9. A capital-theoretic approach
I Think of natural resources as natural capital

10. A capital-theoretic approach
I Think of natural resources as natural capital
I We expect a capital asset to generate a return at least as
great as that from an alternative (numeraire) investment

11. A capital-theoretic approach
I Think of natural resources as natural capital
I We expect a capital asset to generate a return at least as
great as that from an alternative (numeraire) investment
I Consider the arbitrage equation for an asset
y (t ) = rp (t ) p
˙
where p denotes dp (t ) /dt
˙

12. A capital-theoretic approach
I Think of natural resources as natural capital
I We expect a capital asset to generate a return at least as
great as that from an alternative (numeraire) investment
I Consider the arbitrage equation for an asset
y (t ) = rp (t ) p
˙
where p denotes dp (t ) /dt
˙
I The yield y (t ) should be (at least) equal to the return from
the numeraire asset rp (t ) minus appreciation or plus
depreciation p
˙

13. A capital-theoretic approach
I Think of natural resources as natural capital
I We expect a capital asset to generate a return at least as
great as that from an alternative (numeraire) investment
I Consider the arbitrage equation for an asset
y (t ) = rp (t ) p
˙
where p denotes dp (t ) /dt
˙
I The yield y (t ) should be (at least) equal to the return from
the numeraire asset rp (t ) minus appreciation or plus
depreciation p
˙
I This is sometimes called the short run equation of yield

14. Hotelling’ Rule for a non-renewable resource
s
I A non-renewable resource does not grow and hence does not
produce a yield

15. Hotelling’ Rule for a non-renewable resource
s
I A non-renewable resource does not grow and hence does not
produce a yield
I If y (t ) = 0, we can rearrange the arbitrage equation to get
p
˙
y (t ) = 0 = rp (t ) p
˙ ) =r
p (t )

16. Hotelling’ Rule for a non-renewable resource
s
I A non-renewable resource does not grow and hence does not
produce a yield
I If y (t ) = 0, we can rearrange the arbitrage equation to get
p
˙
y (t ) = 0 = rp (t ) p
˙ ) =r
p (t )
I This is Hotelling’ Rule (1931) for the e¢ cient extraction of
s
a non-renewable resource

17. Hotelling’ Rule for a non-renewable resource
s
I A non-renewable resource does not grow and hence does not
produce a yield
I If y (t ) = 0, we can rearrange the arbitrage equation to get
p
˙
y (t ) = 0 = rp (t ) p
˙ ) =r
p (t )
I This is Hotelling’ Rule (1931) for the e¢ cient extraction of
s
a non-renewable resource
I The value (price) of the resource must increase at a rate equal
to the rate of return on the numeraire asset (interest rate)

18. A yield equation for a renewable resource
I A renewable resource can produce a yield through growth

19. A yield equation for a renewable resource
I A renewable resource can produce a yield through growth
I Suppose p = 0, then from the arbitrage equation we can …nd
˙
y (t )
y (t ) = rp (t ) ) =r
p (t )

20. A yield equation for a renewable resource
I A renewable resource can produce a yield through growth
I Suppose p = 0, then from the arbitrage equation we can …nd
˙
y (t )
y (t ) = rp (t ) ) =r
p (t )
I Here, we want the yield to provide an internal rate of return
at least as great as the interest rate r

21. A yield equation for a renewable resource
I A renewable resource can produce a yield through growth
I Suppose p = 0, then from the arbitrage equation we can …nd
˙
y (t )
y (t ) = rp (t ) ) =r
p (t )
I Here, we want the yield to provide an internal rate of return
at least as great as the interest rate r
I In e¤ect, we require that the growth rate of the resource
equals the interest rate

22. Discounting
I In general, individuals have positive time preferences over
consumption (money)

23. Discounting
I In general, individuals have positive time preferences over
consumption (money)
I This gives the social discount rate or “pure” social rate of
time preference r

24. Discounting
I In general, individuals have positive time preferences over
consumption (money)
I This gives the social discount rate or “pure” social rate of
time preference r
I High discount rates heavily discount future bene…ts and costs

25. Discounting
I In general, individuals have positive time preferences over
consumption (money)
I This gives the social discount rate or “pure” social rate of
time preference r
I High discount rates heavily discount future bene…ts and costs
I The discount rate and the interest rate measure essentially
the same thing

26. Discounting
I In general, individuals have positive time preferences over
consumption (money)
I This gives the social discount rate or “pure” social rate of
time preference r
I High discount rates heavily discount future bene…ts and costs
I The discount rate and the interest rate measure essentially
the same thing
I Hence, the discount rate re‡ects the opportunity cost of
investment (saving)

27. Discounting
I In general, individuals have positive time preferences over
consumption (money)
I This gives the social discount rate or “pure” social rate of
time preference r
I High discount rates heavily discount future bene…ts and costs
I The discount rate and the interest rate measure essentially
the same thing
I Hence, the discount rate re‡ects the opportunity cost of
investment (saving)
I Market interest rates also re‡ect risk, in‡ation, taxation, etc.

28. Discounting and present value
I From Hotelling’ Rule
s
p
˙
=r ) p = rp (t )
˙
p (t )

29. Discounting and present value
I From Hotelling’ Rule
s
p
˙
=r ) p = rp (t )
˙
p (t )
I Then it follows that
p (t ) = p (0) e rt , p (0) = p (t ) e rt

30. Discounting and present value
I From Hotelling’ Rule
s
p
˙
=r ) p = rp (t )
˙
p (t )
I Then it follows that
p (t ) = p (0) e rt , p (0) = p (t ) e rt
I Here, p (0) is the present value of p (t ) at t = 0

31. Discounting and present value
I From Hotelling’ Rule
s
p
˙
=r ) p = rp (t )
˙
p (t )
I Then it follows that
p (t ) = p (0) e rt , p (0) = p (t ) e rt
I Here, p (0) is the present value of p (t ) at t = 0
I Thus, Hotelling’ Rule implies that the discounted resource
s
price is constant along an e¢ cient extraction path

32. Discounting and present value
I From Hotelling’ Rule
s
p
˙
=r ) p = rp (t )
˙
p (t )
I Then it follows that
p (t ) = p (0) e rt , p (0) = p (t ) e rt
I Here, p (0) is the present value of p (t ) at t = 0
I Thus, Hotelling’ Rule implies that the discounted resource
s
price is constant along an e¢ cient extraction path
I In discrete time notation...
t
1
p0 = pt , t = 1, 2, ...T
1+δ

33. Discounting and present value
I From Hotelling’ Rule
s
p
˙
=r ) p = rp (t )
˙
p (t )
I Then it follows that
p (t ) = p (0) e rt , p (0) = p (t ) e rt
I Here, p (0) is the present value of p (t ) at t = 0
I Thus, Hotelling’ Rule implies that the discounted resource
s
price is constant along an e¢ cient extraction path
I In discrete time notation...
t
1
p0 = pt , t = 1, 2, ...T
1+δ
I Remember that
r 1
e = , r = ln (1 + δ)
1+δ

34. Discounting and present value contd.
I The present value of a stream of payments or pro…ts v (t ) is
given by
Z T
rt
v (t ) e dt
0

35. Discounting and present value contd.
I The present value of a stream of payments or pro…ts v (t ) is
given by
Z T
rt
v (t ) e dt
0
I Or in discrete time notation
T t
1
∑ 1+δ
vt
t =0
2 T
1 1 1
= v0 + v1 + v2 + ... + vT
1+δ 1+δ 1+δ

36. A simple two-period resource allocation problem
I The owner of a non-renewable resource x0 seeks to maximise
2
1 1
v1 (q1 ) + v2 (q2 )
1+δ 1+δ
subject to the constraint
q1 + q2 = x0

37. A simple two-period resource allocation problem
I The owner of a non-renewable resource x0 seeks to maximise
2
1 1
v1 (q1 ) + v2 (q2 )
1+δ 1+δ
subject to the constraint
q1 + q2 = x0
I The Lagrangian function for this problem is
2
1 1
L v1 (q1 ) + v2 (q2 ) + λ [x0 q1 q2 ]
1+δ 1+δ

38. A simple two-period resource allocation problem
I The owner of a non-renewable resource x0 seeks to maximise
2
1 1
v1 (q1 ) + v2 (q2 )
1+δ 1+δ
subject to the constraint
q1 + q2 = x0
I The Lagrangian function for this problem is
2
1 1
L v1 (q1 ) + v2 (q2 ) + λ [x0 q1 q2 ]
1+δ 1+δ
I The two …rst order (necessary) conditions are
2
1 1
v 0 (q ) λ = 0, 0
v2 (q2 ) λ=0
1+δ 1 1 1+δ

39. A simple two-period resource allocation problem contd.
I Solving the FOCs for the Lagrange multiplier λ we get
0
v2 (q2 ) 0 0
v2 (q2 ) v1 (q1 )
0 = 1+δ , 0 (q ) =δ
v1 (q1 ) v1 1
which is Hotelling’ Rule (in discrete time notation)
s

40. A simple two-period resource allocation problem contd.
I Solving the FOCs for the Lagrange multiplier λ we get
0
v2 (q2 ) 0 0
v2 (q2 ) v1 (q1 )
0 = 1+δ , 0 (q ) =δ
v1 (q1 ) v1 1
which is Hotelling’ Rule (in discrete time notation)
s
I If vt (qt ) pt qt (zero extraction costs) then vt0 (qt ) = pt and
we have
p2 p2 p1
= 1+δ , =δ
p1 p1

41. A simple two-period resource allocation problem contd.
I Solving the FOCs for the Lagrange multiplier λ we get
0
v2 (q2 ) 0 0
v2 (q2 ) v1 (q1 )
0 = 1+δ , 0 (q ) =δ
v1 (q1 ) v1 1
which is Hotelling’ Rule (in discrete time notation)
s
I If vt (qt ) pt qt (zero extraction costs) then vt0 (qt ) = pt and
we have
p2 p2 p1
= 1+δ , =δ
p1 p1
I In continuous time terms this is equivalent to
p
˙
=r
p (t )

42. A simple two-period resource allocation problem contd.
I Instead, we could attach a multiplier to a stock constraint at
each point in time
2
1 1 1
L v1 (q1 ) + v2 (q2 ) + λ1 [x0 x1 ]
1+δ 1+δ 1+δ
2 3
1 1
+ λ 2 [ x1 q1 x2 ] + λ3 [x2 q2 ]
1+δ 1+δ

43. A simple two-period resource allocation problem contd.
I Instead, we could attach a multiplier to a stock constraint at
each point in time
2
1 1 1
L v1 (q1 ) + v2 (q2 ) + λ1 [x0 x1 ]
1+δ 1+δ 1+δ
2 3
1 1
+ λ 2 [ x1 q1 x2 ] + λ3 [x2 q2 ]
1+δ 1+δ
I The FOCs for q1 and q2 are now
2
1 1
v 0 (q ) λ2 = 0
1+δ 1 1 1+δ
2 3
1 0 1
v2 (q2 ) λ3 = 0
1+δ 1+δ

44. A simple two-period resource allocation problem contd.
I If the Lagrangian is maximised by q1 , it should also be
maximised by x2 , so that we can add another FOC
2 3
1 1
λ2 + λ3 = 0
1+δ 1+δ

45. A simple two-period resource allocation problem contd.
I If the Lagrangian is maximised by q1 , it should also be
maximised by x2 , so that we can add another FOC
2 3
1 1
λ2 + λ3 = 0
1+δ 1+δ
I This condition implies
1
λ2 = λ3
1+δ

46. A simple two-period resource allocation problem contd.
I If the Lagrangian is maximised by q1 , it should also be
maximised by x2 , so that we can add another FOC
2 3
1 1
λ2 + λ3 = 0
1+δ 1+δ
I This condition implies
1
λ2 = λ3
1+δ
I Hence, the discounted shadow price is also constant across
time

47. A simple two-period resource allocation problem contd.
I If the Lagrangian is maximised by q1 , it should also be
maximised by x2 , so that we can add another FOC
2 3
1 1
λ2 + λ3 = 0
1+δ 1+δ
I This condition implies
1
λ2 = λ3
1+δ
I Hence, the discounted shadow price is also constant across
time
I Substituting for λt , we again get
0 1
v1 (q1 ) = v 0 (q )
1+δ 2 2

48. A simple renewable resource problem
I We can set the problem in terms of a renewable resource by
incorporating a growth function gt (xt ) into each of the stock
constraints
t
1
λt [xt 1 + gt 1 (xt 1) qt 1 xt ]
1+δ

49. A simple renewable resource problem
I We can set the problem in terms of a renewable resource by
incorporating a growth function gt (xt ) into each of the stock
constraints
t
1
λt [xt 1 + gt 1 (xt 1) qt 1 xt ]
1+δ
I Solving the Lagrangian as before, we get
2 2 3
1 0 1 1 0 1
v1 (q1 ) = λ2 , v2 (q2 ) = λ3
1+δ 1+δ 1+δ 1+δ
and
2 3
1 1 0
λ2 = λ3 1 + g2 (x2 )
1+δ 1+δ

50. A simple renewable resource problem contd.
I Solving for λt , we now …nd the intertemporal rule as
0
v2 (q2 ) 1+δ
0 (q ) = 1 + g 0 (x )
v1 1 2 2

51. A simple renewable resource problem contd.
I Solving for λt , we now …nd the intertemporal rule as
0
v2 (q2 ) 1+δ
0 (q ) = 1 + g 0 (x )
v1 1 2 2
I In continuous time, this is equivalent to
dv 0 (q ) /dt
=r g 0 (x )
v 0 (q )

52. A simple renewable resource problem contd.
I Solving for λt , we now …nd the intertemporal rule as
0
v2 (q2 ) 1+δ
0 (q ) = 1 + g 0 (x )
v1 1 2 2
I In continuous time, this is equivalent to
dv 0 (q ) /dt
=r g 0 (x )
v 0 (q )
I Or, if v 0 (q ) = p,
p
˙
=r g 0 (x )
p

53. A simple renewable resource problem contd.
I Solving for λt , we now …nd the intertemporal rule as
0
v2 (q2 ) 1+δ
0 (q ) = 1 + g 0 (x )
v1 1 2 2
I In continuous time, this is equivalent to
dv 0 (q ) /dt
=r g 0 (x )
v 0 (q )
I Or, if v 0 (q ) = p,
p
˙
=r g 0 (x )
p
I If p = 0, we get the yield equation
˙
p g 0 (x ) y (t )
=r
p p (t )