1. ERE5: Efficient and optimal use of environmental resources
A simple optimal depletion model
Resource substitutability
Static and dynamic efficiency
Hotelling‘s rule
Optimality
An example
Extraction costs
Renewable resources
Complications

2. Last week Efficiency and optimality Market failure & public policy
Static efficiency
Optimality
Dynamic efficiency and optimality
Market efficiency
Market failure & public policy
Externalities
Public policy

3. Exhaustion, Production & Welfare
One can construct production functions with various degrees of essentialness, but the question is of course empirical
The question whether it matters that a resource gets exhausted depends on its substitutability; or, if you can‘t do without, don‘t lose it
So far, we have not run out of anything essential, but that does not mean we won‘t
Human ingenuity is the ultimate resource, but also works to create problems

4. K
 = 0
0 <  < 
 =  R
Substitution possibilities and the shapes of production function isoquants

5. Substitutability and Scarcity
Feasibility of sustainable development depends on
Substitutability
Technical progress
Backstop technology
The magnitude of substitution possibilities
Economists: relatively high
Natural scientists and ecologists: limited
However, it matters what services we look at

6. Optimal Resource Extraction - Discrete Time
Social welfare function:
Extraction:
Investment:
Production function:

7. Optimal Resource Extraction – Discrete Time (2)

8. Optimal Resource Extraction – Continuous Time
Social welfare function:
Extraction:
Investment:
Production function:

9. The Maximum Principle Objective function: Subject to:
J depends on control variables (u), state variables (x), and time
State variables describe the economy at any time; the equation of motion governs its evolution over time
Control variables are time-dependent policy instruments
To obtain the solution we construct a current value Hamiltonian

10. The Hamiltonian
The Hamiltonian only contains the current state and controls; current optimality is a necessary condition for intertemporal optimality
The co-state variables (l) secure intertemporal optimality; they are like Lagrange multipliers, indeed measure the shadow price
FOC:

11. Optimal Resource Extraction - Continuous Time (2)
Social welfare function:
Equations of motion:
Hamiltonian:
Necessary conditions:

12. Static Efficiency
Marginal utility of consumption equals the shadow price of capital
Marginal product of the natural resource equals the shadow price of the resource stock

13. Dynamic Efficiency
The growth rate of the shadow price of the resource equals the discount rate
The return to capital equals the discount rate

14. Hotelling‘s Rule Dynamic efficiency required:
The growth rate of the shadow price equals the discount rate
An alternative interpretation:
The discounted price is constant along an efficient resource extraction path
Thus, environmental resources are like other assets

15. Growth Rate of Consumption
The growth rate of consumption along the optimal time path:
Since η>0, consumption grows if the marginal product of capital exceeds the discount rate
The intuition:

16. Hotelling‘s rule and Optimality
PtB = P0Bet
PtA = P0Aet
P0B
P0A t

17. Extraction Costs Social welfare function: Constraints:
Production function:
Extraction costs:

18. Extraction Costs and Resource Stock
Gt (for given value of Rt = )
(iii)
(ii)
(i) S0
Remaining resource stock, St

19. Extraction Costs –2
Hamiltonian:
Necessary conditions:

20. Resource Price Net price = Gross price – marginal extraction cost
Gross price = Marginal contribution to output, income (measured in utils)
Net price = Marginal value of the resource in situ
Net price = Rent = Royalty

21. Hotelling‘s Rule -2
The growth rate of the shadow price of the resource is lower if extraction costs rise with falling resource stocks
The discount rate equals the rate of return of holding the resource, which equals its price appreciation plus the foregone increase in extraction costs

22. Graphical solution Net price Pt PT =K P0 Pt 45° T R R0 Time t Rt
Demand P0 Pt
45° T R R0
Time t Rt
Area =
= total resource stock T
Time t

23. Renewable Resources Social welfare function: Constraints:
Production function:

24. Renewable Resources -2
Hamiltonian:
Necessary conditions:

25. Hotelling‘s Rule -3
The growth rate of the shadow price of the resource is lower for renewable resources
The discount rate equals the rate of return of holding the resource, which equals its price appreciation plus the increase in the resource growth

26. Complications The total stock is not known with certainty
New discoveries increase the known stock
There is a distinction between physical quantity and economically viable stock size
Technical progress and R&D
Heterogeneous quality
Extraction costs differ
Availability of backstop-technology