This document provides an overview of dynamic efficiency and Hotelling's rule in the context of non-renewable resource extraction. It introduces the concept of maximizing net present value over time by ensuring the net price of the resource increases at the discount rate. A two period example is used to illustrate this, showing extraction quantities that equalize the marginal net benefit between periods. The impact of changing assumptions like the discount rate and demand elasticity are also discussed. Reasons why prices may not follow Hotelling's rule in reality are outlined, such as new discoveries, technological change, and imperfect information.

1. Dynamic Efficiency: Hotelling’s
Rule
Environmental Economics II Spring 2014
Lecture based in part on Harris and Roach 2013 and Field 2008

2. Present Value Calculations
Vo = Vn/(1 + r)n
$1000 next year is worth today:
Vo= $1000/(1.05) = $953
$1000 in seven years is worth today:
Vo= $1000/(1.05)7 = $710

3. Net Present Value
• NPV is the present value of revenues minus
present value of costs.
NPV = R0 + R1/(1+r) + R2/(1+r)2 + R3/(1+r)3 ...
Rn/(1+r)n - C0 - C1/(1+r) - C2/(1+r)2 –
C3/(1+r)3 ... Cn/(1+r)n
• NPV = ∑(Rn/(1+r)n - Cn/(1+r)n )

4. Present Value of Periodic
Payments:
V0= p {1- (1 + r)-n}/r
where p is payment and n is number of
years

5. Number
Of
Payments
Time
Between
Payments
Evaluation
Period
Time of
Value
Formula
Formula
Name
Legend
Future
Future value* of
an amount
Present
V0=Vn/(1 + r)n
Present value of
an amount
Future
One
Vn=Vo(1 + r)n
éæ
êç
êè
ê
ê
ë
Terminating
Vn = p
Terminating
START
ù
n
÷
1 + r ö - 1ú
ø
é
ê
ê
ê
ê
ë
ú
ú
ú
û
r
ç
1 - æ1 + r
è
Present
V0 = p
Future
Present
Present value of
a terminating
annual series
Vn= Infinity
V0 =
-n ù
ö
÷
ú
ø
ú
ú
ú
û
Future value*
terminating
annual series
r
Annual
Perpetual
Series
Present value of
a perpetual
annual series
p
r
Vn = p
éæ
êç
êè
êæ
êç
ëè
Present
V0 = p
é
ê
ê
ê æ
ê ç
ë è
Future
Vn= Infinity
Present
V0 =
Future
Terminating
Periodic
n
ù
t
ú
ú
ú
û
Present value of
a terminating
periodic series
÷
1 + r ö - 1ú
ø
÷
1 + r ö -1
ø
1- 1 + r
æ
ç
è
t
ö
÷
ø
÷
1 + r ö -1
ø
Perpetual
Decision tree from: Klemperer, W. (1996) Forest Resource Economics & Finance. McGraw Hill.
p
t
ö
1 + r ÷ -1
ø
æ
ç
è
V0 = Present value (or initial
value)
Vn = Future value after n years
(including interest)
n = Number of years of
compounding or discounting
p = Amount of fixed payment
each time in a series
(occurring annually or every
t years)
t = Number of years between
periodic occurrences of p
Future value* of
a terminating
periodic series
-n ù
ú
ú
ú
ú
û
r = Annual interest rate/100. (If
payments are fixed in real
terms, r is real; if payments
are fixed in nominal terms, r
is nominal.
Present value of
a perpetual
periodic series
* The future value of any
terminating serires is its
present value formula time
(1 + r)n. Adapted from
Gunter and Haney (1978), by
permission.

6. PV of string of payments using
Excel

7. Present Value Calculations and
Resources
• Renewable Resource Problem: Harvest
such that resource grows at same rate as
money in the bank.
• Non-renewable Resource Problem:
Extract such that price of resource grows
at same rate as money in the bank. –
Hotelling’s Rule

8. Hotelling’s Rule
• Net benefits (CS + PS) over time are
maximized (dynamic efficiency) when net
price increases at the discount rate.
• Managers will extract at this rate.
• Therefore, resource extraction will be
“socially efficient”.
• Let’s test with a simple two-period
example.

9. Dynamic Efficiency Model - Disclaimer
This analysis of dynamic efficiency for nonrenewable resource extraction is based on a
highly simplified modeling framework, in
order to provide an accessible introduction
to the topic, along with important
insights, without complex mathematics.

10. Simplifying Assumptions
1. Marginal extraction cost is constant
2. Demand is constant
3. There is a competitive market with no market
irregularities such as cartels
4. There is perfect information. i.e. Market participants
are fully informed of current and future
demand, marginal extraction costs, the discount
rate, available stocks, and market price
5. No externalities!
We will look at the most basic case with just two time
periods: today (period 1) and next year (period 2)

11. Dynamic Efficiency Example: Model
Demand curve: P = $300 – 0.25Q
Supply curve: P = $20
Total stock = 1,000 barrels
r = .05

12. Dynamic Efficiency Example
Demand curve: P = $300 – 0.25Q
Supply curve: P = $20
Total stock = 1,000 barrels
r = .05
Sell all the first year: Q = 1,000
•
•
•
•
P = $300 – 0.25(1,000) = $50
Net benefit = CS + PS
CS = (300 – 50) x 1000/2 = $125,000
PS = (50 – 20) x 1000 = $30,000
• Net benefit = $125k + $30K = $155k

13. P
Net Benefits = CS + PS
300
CS
Demand curve
50
20
PS
Supply
1000
Stock
1200
Q

14. P
Marginal Net Benefits
280
Marginal Net Benefits
Total Net Benefits
1000
Stock
1120
Q

15. Sell half each year. Q1 = Q2 = 500
Demand curve: P = $300 – 0.25Q
Supply curve: P = $20
Total stock = 1,000 barrels
r = .05
• The first year:
•
•
•
•
P = $300 – 0.25(500) = $175
CS = (300 – 175) x 500/2 = 31,250
PS = (175 – 20) x 500 = 77,500
CS + PS = $108,750
• Year 2: $108,750/(1.05) =$103,571
• Total = 108,750 + 103,571 = $212,321

16. Dynamic Efficiency: (P2– MC)/(P1– MC) = 1+ r
Demand curve (MWTP): P = $300 – 0.25Q
Supply curve (MC): P = $20
Total stock = 1,000 barrels
r = .05
• Maximize net benefits by selling quantities
such that marginal net benefits (MWTP –
MC) increases at the rate of interest: 5%
(P1 - 20) = (P2 – 20)/1.05
$300 – 0.25Q1 – $20 = ($300 – 0.25Q2– $20 )/1.05

17. Dynamic Efficiency Example Cont.
$280 – 0.25Q1 = ($280 – 0.25Q2)/1.05
1.05 ($280 – 0.25Q1)= ($280 – 0.25Q2)
294 - 0.2625Q1 = 280 – 0.25Q2
Q2 = 1000 – Q1
294 - 0.2625Q1 = 280 – 0.25(1000 – Q1)
294 - 0.2625Q1 = 280 – 250 + 0.25Q1
264 = 0.5125Q1
Q1 = 515.1
Q2 = 1000 – Q1 = 484.9

18. Q1 = 515.1
Q2=484.9
• Net Price (price – cost) should increase by
5%
• Find price for each period by filling in the
demand equation P = 300 - .25Q
• P1 = $300 – 0.25(515.1) = $171.225
• P2 = $300 – 0.25(484.9) = $178.775
• (178.8 – 20)/(171.2 – 20) = 1.05 TA DA!

19. Q1 = 515.1 P1 =$171.2
Q2=484.9 P2 =$178.8
• Calculate present value of net benefits.
CS period 1= (300 – 171.2) x 515.1 / 2 = 33,172
PS period 1 = (171.2 – 20) x 515.1 = 77,883
CS period 2= (300 – 178.8) x 484.9/2 = 29,385
PS period 2 = (178.8 – 20) x 484.9 = 77,002
• 33,172 + 77,883 + (29,385 + 77.002)/1.05
=$212,376

20. P
280
Graphic Dynamic Efficiency
Demand curve: P = $300 – 0.25Q
Supply curve: P = $20
Total stock = 1,000 barrels
r = .05
P/1.05
267 = (280/1.05)
Q2
500
500
515
484
1000
Q1

21. Dynamically Efficient Equilibrium and
Discount Rate
How will the dynamically efficient allocation of
the fixed resource stock change if the discount
rate (r) becomes larger?
Intuition?

22. P
Graphic Dynamic Efficiency r= 10%
280
267 = (280/1.05)
255 = (280/1.10)
Q2
515
484
1000
More extraction in period 1
Q1

23. Dynamically Efficient Equilibrium Intuition:
why this model is amazing
If the net price increases at the interest
rate, then the present value of marginal profit
is equal across time periods (Hotelling’s rule).
Resource managers have no incentive to
change this production path over time, ceteris
paribus. This solution also generates the
largest PV of total net benefits (CS + PS) over
time.

24. Dynamically Efficient
Equilibrium, Profits, and Size of Stock
• When a resource is abundant, then consumption today does
not involve an opportunity cost of lost profit in the future, since
there is plenty available for both today and the future. In a
perfectly competitive market, P = MC and marginal profit
would be zero.
• As the resource becomes increasingly scarce, consumption
today involves an increasingly high opportunity cost of
foregone profit in the future. As resources become
increasingly scarce, P increases relative to MC and profits
grow.

25. Hotelling Rent or Scarcity Rent
• The profit due to resource scarcity in competitive
markets.
• Economic profit that can persist in certain natural
resource cases due to the fixed stock of the
resource.
• Due to fixed stock, consumption of a resource
unit today has an opportunity cost equal to the
present value of the marginal profit from selling
the resource in the future.

26. User Costs
• Value of the resource in its natural state, such as oil in the
ground.
• Equal to the opportunity costs associated with using the
resource now such that it will not be available in the future.
• The marginal user costs (MUC) are the opportunity cost
associated with using one more unit today instead of saving it
for the future.
• In theory, the user cost should be the cost an extractor pays
to the owner of the resource: royalty or rent.
• Royalty or rent is money derived, not from having special
skills or timing or insights, but simply fromowning or having
access to a resource.

27. Oil Extraction and Externality Costs and User Costs
MEX = marginal extraction cost
MEC = marginal external costs
MUC= marginal user costs
p=private e=external s=social
P
MEX + MEC + MUC (= MTC in book)
MEX + MEC
MEX (i.e. marginal private cost)
Demand
Qs Qe Qp
Q oil

28. Oil Extraction and Externality Costs and User Costs
MEX = marginal extraction cost
MEC = marginal external costs
MUC= marginal user costs
p=private e=external s=social
P
MEX + MEC + MUC (= MTC in book)
Dead weight loss due to EC and UC
MEX + MEC
Dead weight loss due to external costs
MEX (i.e. marginal private cost)
Demand
Qs Qe Qp
Q oil

29. Finding the optimal extraction using Excel

30. Price-cost ratio is where PV of net
benefits is maximized
Q1

31. Another example

32. Comparison
• Both stock of 1000 and r = .05 and
• Supply: P = 20
Example 1
– Demand P = 300 - .25Q
– Q1 = 515, P1 = $171
Example 2
– Demand P= 100 - .01Q
– Q1 = 685, P1 = $93
Why are the extraction rates so different?

33. Calculate the demand elasticity
• ΔQ/ΔP x P/Q
• Example 1: 171/515 x 1/.25 = 1.33
• Example 2: 93/685 x 1/.01 = 13.58

34. Effect of Elasticity on Quantity
Extracted – Less Elastic Demand
P
Q

35. Effect of Elasticity on Quantity
Extracted – Very Elastic Demand
P
Q

36. Graphic Dynamic Efficiency:
P
Demand P = 100 - .01Q
Supply: P = 20
Stock = 1000
P/1.05
100
95.3 = (100/1.05)
Q2
680
320
1000
Q1

37. Impact of Elasticity on
Extraction
• More elastic the demand, more resources
extracted in the present.
Intuition?

38. Why might prices not follow
Hotelling’s Rule?
1. Stock increases due to new discoveries
increased extraction now
2. Technological change
–
–
–
–
More new discoveries
MC of extraction decreasing in future  less
extraction now
MC decreases in all periods  slightly more
extraction now
New substitutes  more extraction now
3. Demand increase over time
– Unpredicted  no change in extraction rate
– Predicted  Increased extraction now

39. Why might prices not follow
Hotelling’s Rule?
4. Awareness of scarcity increases  extract
less now
5. r increases  extract more now
6. Government regulation coming  extract
more now (like MC increasing in future)
7. Market irregularities  extraction and price
irregularities
8. Expectations:
Increase scarcity  decrease extraction now
Government regulation  increase extraction now

40. Why might prices not follow
Hotelling’s Rule?
9. Backstop technology  extract more now
10. Imperfect information  erratic pricing
and extraction

41. P
100
Graphic Dynamic Efficiency:
Demand curve: P = $300 – 0.25Q
Supply curve: P = $20
Total stock = 1,500 barrels
r = .05
95.2
Q2
515
484
Q1 1500
More extraction in period 1 (and 2)

42. P
Graphic Dynamic Efficiency: MC in period
2 falls: Net benefits increases in period
2.
276 = (280 – 10)/1.05
280
267 = (280/1.05)
Q2
Less extraction in period 1
515
484
1000
Q1

43. P
Graphic Dynamic Efficiency: MC in both periods
fall: Net benefits increases both periods.
290
276 = (290 – 10)/1.05
280
267 = (280/1.05)
Q2
Less extraction in period 1
515
484
1000
Q1

44. P
Graphic Dynamic Efficiency: Demand is
predicted to increase in period 2
280
267 = (280/1.05)
Q2
Less extraction in period 1
515
484
1000
Q1

45. Second Example
Demand: P = 100 - .01Q

46. MC in period 2 falls. Less extraction now.

47. MC in both periods fall. Slightly more
extraction now.

48. Demand in period 2 increase - predicted. Less
extraction now.

49. rincreases – More extraction now

50. In-Class Exercises

51. For your math pleasure: Hotelling’srule for
multiple periods (totally optional!)
Demand: Pi = a – bqi
Supply is fixed: P = c
(aqi – bqi2/2 – cqi)/(1+r)i + [Qtot i
iqi],
where i = 0, 1, 2, …, n.
If Qtot is constraining, then the dynamically
efficient solution satisfies:
(a – bqi – c)/(1+r)i [Qtot -
iqi]
=0
= 0, i = 0, 1, …, n.