Natural resource exploitation: basic concepts
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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.
![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](https://image.slidesharecdn.com/nre-slides-1-100430053841-phpapp01/85/Natural-resource-exploitation-basic-concepts-37-320.jpg)
![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](https://image.slidesharecdn.com/nre-slides-1-100430053841-phpapp01/85/Natural-resource-exploitation-basic-concepts-38-320.jpg)
![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](https://image.slidesharecdn.com/nre-slides-1-100430053841-phpapp01/85/Natural-resource-exploitation-basic-concepts-42-320.jpg)
![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](https://image.slidesharecdn.com/nre-slides-1-100430053841-phpapp01/85/Natural-resource-exploitation-basic-concepts-43-320.jpg)
![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](https://image.slidesharecdn.com/nre-slides-1-100430053841-phpapp01/85/Natural-resource-exploitation-basic-concepts-48-320.jpg)
![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](https://image.slidesharecdn.com/nre-slides-1-100430053841-phpapp01/85/Natural-resource-exploitation-basic-concepts-49-320.jpg)
Natural resource exploitation: basic concepts
- 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 )