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- 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 )

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