1. The Economics of Nonrenewable Resources
Jeffrey Krautkraemer (JEL 1998)
Nonrenewable resource scarcity

2. Introduction Natural resources:
Renewable: capacity for regeneration (fish, clean water, forests)
Non–renewable: regeneration only at geological time scale, not human (oil, gas, copper)
Leading to Issue of Sustainability:
Non–declining (per capita) consumption or utility
are non–renewable resource use and sustainable growth incompatible?

3. Introduction (2) Set–up:
Efficient use of non–renewable resources: perfect competition and non–renewable resource exploitation
1a. A simple model (the mine)
1b. A general model (optimal control)
2. Complicating factors

4. 1a. The mine – Lagrangean method
Model of non–renewable resource extraction:
Firm i owns a mine
with initial stock size: Si,0, and
aims to maximize net benefits (profits) over infinite time horizon by choosing the optimal amount extracted (qi,t  0) in every period
Net benefits: V(Si,t, qi,t) = B(Si,t, qi,t) – C(Si,t, qi,t)
Discount factor  (0    1)
Objective function:
Note: constraint implies Si,t – Si,t+1 = qi,

5. The mine (2) Simplest case – perfectly competitive mining industry:
All firms (mine owners) identical in all respects;
Perfect competition; sales price Pt
Same to each individual mine owner
Assume industry D:
Implies max price of
Just sales, no other benefits (or costs) of extraction, no ‘stock effects’
B(Si,t, qi,t) = Pt qi,t; C(Si,t, qi,t) = 0  V(Si,t, qi,t) = Pt qi,t
Objective function for every firm:

6. The mine (3) – first order conditions
Lagrangean:
where  is the Lagrangean multiplier (or the shadow price of the resource)
First order conditions:
 L /  qi,0 = P0 –  = 0
 L /  qi,1 =  P1 –  = 0
 L /  qi,2 = 2 P2 –  = 0 …
 L /  qi,T = T PT –  = 0
So: t Pt –  = 0 for all t = 1, …, T, and hence
P0 = P1 = 2P2 = … = t Pt = … = T PT

7. The mine (4) – arbitrage across time
What if t Pt >  for any t = 1..T, and t Pt   in all other t’s?
Present value of marginal benefits in period t larger than in all other periods
Extract everything in period t, zero in all other periods…
But then that holds for all firms in the industry…
Industry demand:
Price very low in period t, very high in all other periods…
Arbitrage until P0 =  P1 = 2 P2 = … = T PT
Moving resource extraction between periods until, in equilibrium, each individual firm is indifferent when to extract  arbitrage

8. The mine (5) – Hotelling Rule
Interpretation Pt = Pt+1 ?
Note that   1 / (1 + r)  Pt+1 = (1 + r) Pt
Hotelling Price path:
First, Pt+1 = (1 + r) Pt  Pt+1 – Pt = rPt
Continuous time equivalent:
Straightforward integration:

9. The mine - Hotelling’s Rule for extraction
Resource stocks are “natural assets” I.e. the form which assets take in the initial state of the economy (Solow)
Assets must be competitive against all other assets – marginal rate of return on holding asset must be equal to market rate of return (r)
Extraction paths for resource stocks will generate the price path – and so we would expect extraction path to generate the return r on the initial price P
 First point of natural resource economics is that natural resources operate as assets in economy

10. The mine (6) – terminal condition and path
Assume that the price that can be charged for the resource is bounded at
the maximum (or backstop) price for this resource (?)
Unknown variables: C and T…
We know: Q(T) = 0, and also
Insert in
Price path:
Given the price path, we can now find the industry extraction path

11. The Mine – Backstop Price
Resource stock prices rise because they are assets
Resource prices are bounded because there are substitutes
The point at which the initial price of a substitute outcompetes a resource renders it economically obsolete (choke off price)
Extraction path targets this backstop price and works backwards to the present (in recognition the resource’s role as an asset)
Extraction paths and end of use (of a particular resource) are determined by the identification of substitutes to meet the needs identified by the D curve
Role of Exploration
Role of Technological Change

12. The mine (7) – solving paths of extraction (Q)
Combine and
Constraint: ,
To find T, we just need to solve the integral of Q(t) for T

13. The mine (9) – Terminal (depletion) Time
So,
Integration:
And hence

14. The Mine – Terminal Conditions
Cake eating models are about the allocation of finite resources across time (who gets to eat the cake?)
Assume that the definition of the relevant time horizon is that the resource should be economically extinct at the end of it (no sense in leaving leftovers for non-existent future)
Work backwards from absolute depletion toward present recognising that the resource must meet marginal asset returns
The optimal extraction of resources is equivalent to the optimal allocation of their usefulness across the relevant time horizon
Hotelling showed that there were forces within the economy to prevent the first owners/users from eating the entire cake in the first period

15. The Mine - Complete solution:
Depletion time:
Price path:
Industry extraction path:
Consider the example of two different initial stock levels
(same extraction paths but aggregating to different S)

16. Q t P
450
t = t T T

17. The mine (11) Sensitivity analysis:
How does optimal extraction path change if (some of the) parameters change?
Parameters:
First: r increases:
higher rate of price increase 
more extraction in early periods, faster depletion.
Graphically…the curve first shifts in, but then also shifts down (to indicate faster early depletion)
 Greater pressure on resources to perform as assets

18. P Q
450 T t
t = t T t

19. The mine (13) Discovery of additional reserves S:
r unchanged, Q(t) should be increased in all t to prevent being hit while there are still reserves left
Price jumps down, price path becomes flatter (because 100r% of a smaller sum of money is smaller in absolute terms than 100r% of a larger sum)
Depletion postponed (higher T);
As we saw above

20. Q t P
450
t = t T T

22. The Mine - Hotelling Diagram
A Hotelling diagram charts out the extraction paths and price paths for a given D, given S, and given P bar
The diagram illustrates how the market will itself allocate resources across time, given their function as assets
The primary force that causes the path to deviate from this:
“technological change”
- meeting needs represented by this D differently
- exploration for additional stocks
- identification of new substitutes
 Technological change is the means by which an economy alters the role and extraction paths of different resources (“peak oil”, “peak whale oil”)

23. The mine (15)
Demand Shift (Lower ): D function shifts out (flatter, intercept unchanged):
along original price path, for any price the quantity demanded of resource is higher  reserves depleted before is hit
Current price should be increased to reduce current extraction, but depletion still occurs sooner

24. The mine (16) Lower backstop price ( ):
- in this specification: no effect here because of particular specification of the industry demand function (creates linear extraction path)
In general (with nonlinear extraction path): initial price would be reduced to stimulate current consumption, resource depletion occurs sooner.

25. Moving toward Continuous Time NR Econ
The same points are made as before
These points are a bit more elegant when using dynamic optimisation techniques (used for growth theory in general)

26. The mine (17) More general case – away from simple Hotelling model
What if there are positive extraction costs (C(q,S))?
Costs vary with amount extracted (Cq > 0, Cqq > 0)
Extraction costs are stock–dependent: smaller stock, need to drill deeper (CS < 0, CSS > 0)
And what if there are stock–dependent benefits too (B(q,S), BS > 0, BSS < 0)?
Scenic value: larger stock, smaller cumulative extraction, landscape less devastated. Or greenhouse effect; larger stock, less cumulative extraction and emission of carbon…
So, define V(q,S) = B(q,S) – C(q,S)
We need to do optimal control…

27. 1b. Optimal control
Optimal control: solving current value Hamiltonian…
If confronted with
Note that constraint implies
Solution: write down current–value Hamiltonian
Necessary conditions are:

28. Optimal control (2) – simple hotelling rule again
Let’s check: correct for simple case?
Note V(q,S) = Pq, and 
Claim: solution is
Combine:

29. Optimal control (3) – basics again
Road map in Conrad and Clark:
Problem we need to solve:
We end up with current value Hamiltonian
And we show analogy with just straightforward Lagrangean approach (discrete time)
For that, we need the present value Hamiltonian

30. Optimal control (4) So, if confronted with
Write up current value Hamiltonian:
Necessary conditions for optimum:
So what is interpretation?

31. Optimal control (5) Suppose V(q,S) = Pq – c(S) Foc’s … Interpretation:
Net marginal benefits (Vq) of extracting today need to be equal to the shadow value (= user cost of current extraction)
Net benefits of extracting an extra unit today (Vq) should be equal to the net benefits of leaving it in the ground (Hotelling)

32. Optimal control (6)
So much for optimal control, back to the questions…
What happens to price path if extraction costs vary with amount extracted (Cq > 0, Cqq  0)?
Suppose C = (t) q(t), with d(t)/dt  0
Hamiltonian:

33. Optimal control (7) So, If In the limit, rate of price increase is r
Price may fall in early periods, but in the limit increases at rate r 
Price path may be U–shaped (if technical progress in extraction technology is sufficiently large)

34. Optimal control (8) And what if benefits or costs stock–dependent?
BS > 0, BSS  0, or
CS < 0, CSS  0
 VS = BS – CS > 0
Suppose BS = 0. Hamiltonian:
Slower extraction, and maybe even depletion not economically feasible…

35. 2. Complicating factors World more complex than captured here…
Endogenous exploration
Extraction capital intensive
Heterogeneity in resource quality
Market imperfections

36. Complicating factors (2)
Endogenous exploration
Finding new reserves not accidental... Is result of deliberate action
More effort if current reserves are closer to depletion
Consequence:
Small reserves, high extraction costs 
more exploration effort  new reserves discovered 
extraction costs fall  price jumps down 
more extraction  price increases again…
Price path U–shaped

37. Complicating factors (3)
Extraction is capital–intensive
C = C (q,S,K)
Higher interest rate, always faster depletion?
Higher interest rate -> less investments in extractive capital
And backstop technology maybe also more expensive if interest rate is higher… even slower extraction
 Capital costs complicate implications of higher r

38. Complicating factors (4)
Heterogeneity across reserves:
Differences in quality (higher and lower grade ores)
Differences in extraction costs (higher and lower).
Optimal extraction: most profitable option exploited first (discounting of future costs –extraction costs, or lower quality ores)
Captured in making costs stock dependent

39. Meaning of Hamiltonian (2)
Hotelling valuation approach:
Can we deduce anything about the value (W) of mining?
Suppose B = P(t)q(t), C = (t) q(t) 
Current value Hamiltonian:

40. Hamiltonian (3) Current value Hamiltonian: Foc’s
That makes life very easy…:

41. Hamiltonian (4) Hotelling valuation approach:
W(0) is firm’s current value (say at stock exchange) and is product of current price, current technical change and reserves