An Introduction to Resource Economics lecture delivered by Aaron Hatcher at the University of Portsmouth, 2008. Shared through the TRUE wiki for Environmental and Resource Economics. Download from http://economicsnetwork.ac.uk/environmental/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 )