Chapter 5: The Economics of Natural Resource Systems

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Chapter 5 The Economics of Natural Resource
Systems

• Steven C. Hackett
• Humboldt State University

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Nonrecyclable, nonrenewable resources

• The theory of dynamically efficient resource
  markets

• Chapter 3 dealt with static market efficiency,
  which is like a photograph at a point in time.

• Dynamic efficiency is more complex
• allows for modeling how future conditions affect
  present supply and demand
• allows for modeling how current conditions
  affect future supply and demand

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The theory of dynamically efficient resource
markets

• Recall that the static notion of efficient
  resource allocation is that one cannot change
  quantity from that of the equilibrium in the
  snapshot time period and generate larger gains
  from trade

In contrast, the dynamic notion of efficient
resource allocation is that one cannot shift
production from one time time period to another
and generate a larger present value of gains from
trade summed across all time periods.
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The theory of dynamically efficient resource
markets

• The notion of dynamic efficiency is, ultimately,
  an intuitive concept. Lets take it one step at a
  time.

First, lets consider the concept of present
(discounted) value.
Would you rather have 10,000 in cash right now
or 10 years from now? Why (or why not)?
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The theory of dynamically efficient resource
markets

• Reasons why most people would rather have 10,000
  today instead of 10 years from now

• If we anticipate inflation (rising prices over
  time), then the purchasing power of 10,000 will
  shrink over time.

• If we take the 10,000 today and invest it in,
  say, government bonds, then we will have more
  than 10,000 in 10 years.

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The theory of dynamically efficient resource
markets

• Reasons why most people would rather have 10,000
  today instead of 10 years from now (continued)

• Pure rate of time preference I want good things
  now and would rather wait for bad things. I dont
  know if I will be alive in 10 years, so why wait?

• Strong current needs (e.g., college expenses,
  health care expenses, basic food and shelter
  needs) heightens ones pure rate of time
  preference.

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The theory of dynamically efficient resource
markets

• Suppose that you have inherited 10,000, which
  will be held in trust for you for 10 years.

• What is the least amount of cash you would
  accept from me RIGHT NOW that would make you
  willing to sign over the inheritance to me?

• Your answer to that question is your present
  (discounted) value of that future 10,000 payment.

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The theory of dynamically efficient resource
markets

• As an aside, why might your present discounted
  value of a 10,000 payment 10 years in the future
  differ from that of someone else?

Different life circumstances, different
investment opportunities. Other?
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The theory of dynamically efficient resource
markets

• Note The discount rate (essentially like an
  interest rate) reflects the time value of money

• The rate at which the present value of a payment
  shrinks as the time of payment is pushed off
  further into the future

• The rate at which the future value of current
  interest-earning savings grows over time.

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The theory of dynamically efficient resource
markets

• Since different people have different discount
  rates, then at the prevailing market interest
  rate, some people are lenders (financial
  investors), while others are borrowers.

As with market equilibrium price, the equilibrium
market interest rate reflects a balancing of the
discount rates of those supplying and demanding
loanable funds.
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The theory of dynamically efficient resource
markets

• Finance is an application of economics that
  focuses on time value of money. We will limit
  ourselves to an elementary application of the
  time value of money.

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The theory of dynamically efficient resource
markets

• Suppose that you will receive a single guaranteed
  future payment i years from the present, and
  your discount rate (interest rate) is r. Then
  the present discounted value (PV) of that future
  payment (FP) is given by the following formula

PVFP ( future payment)/(1r)i
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The theory of dynamically efficient resource
markets

• PV example

future payment is 10,000. i 2 years from
the present. r 10 (0.10). Then PVFP 10,00
0/(10.1)2 10,000/1.21 8,264.63
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The theory of dynamically efficient resource
markets

• Based on the preceding example, the person is
  indifferent between having 8,264.63 right now
  and getting 10,000 two years from now.

Thus, literally, the 8,264.63 is the present
(discounted) value of 10,000 to be received two
years from now.
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The theory of dynamically efficient resource
markets

• Another point on PV If you will receive a stream
  of payments over time (e.g., social security
  payments), then the PV of that stream of payments
  is found as follows

PVFP ?i( future payment, year i)/(1r)i
Where i 0, 1, 2, , n years.
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The theory of dynamically efficient resource
markets

• Final point on PV The preceding formulae are for
  discrete time periods. If you are taking
  integrals, you will want to use the continuous
  discounting formula

PVFP ?FPte-rtdt Where you are integrating over
time t up to some end point T.
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The theory of dynamically efficient resource
markets

• Moving on

Our analysis of dynamically efficient resource
markets will be based on a highly simplified
modeling framework, which will provide an
accessible introduction to the topic, as well as
important insights, without overwhelming you with
complex mathematics.
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The theory of dynamically efficient resource
markets
Simplifying assumptions

• There is a well-functioning competitive market
  for the nonrenewable resource in question (no
  monopolies or cartels)
• Market participants are fully informed of current
  and future demand, marginal production cost,
  market discount rate, available supplies, and
  market price
• We will look at the most basic dynamic case two
  time periods today (period 0) and next year
  (period 1)

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The theory of dynamically efficient resource
markets
Simplifying assumptions, continued

• Marginal production cost is constant (for now we
  assume the law of diminishing marginal returns
  does not hold)
• Market demand is steady state, meaning that
  demand in period 1 is the same as in period 0 (no
  growing or shrinking demand)

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The theory of dynamically efficient resource
markets

• Model
• Demand P 200 Q
• Supply P 10
• Discount rate r 10 percent (0.1)
• Total resource stock Qtot 100

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The theory of dynamically efficient resource
markets

• Case 1 Ignore period 1 while in period 0 (live
  for today)

Competitive market equilibrium 200-Q0 10 ? Q0
190 Problem! Qtot 100 lt 190.
Scarcity-constrained market equilibrium Q0
100 P 200 100 100.
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PV of total gains from trade over periods 0 and 1
Period 0 CS0 (200-100)100/2 5000 PS0
(100-10)100 9000 TS0 14,000 PVTS0
14,000/(10.1)0 14,000 Period 1 Since all of
the resource was consumed in period 0, there are
no gains from trade in period 1. PVTS 14,000
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PV of total gains from trade 14,000
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The theory of dynamically efficient resource
markets

• Case 2 Divide Qtot equally over periods 0 and 1

Period 0 Q0 50, P0 200 50 150. Period 0
gains from trade CS0 (200-150)50/2
1,250 PS0 (150-10)50 7,000 TS0 8,250
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PV of total gains from trade, period 0, 8,250
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Case 2 Divide Qtot equally over periods 0 and 1
Period 1 Q1 50, P1 200 50 150. Period 1
gains from trade CS1 (200-150)50/2
1,250 PS1 (150-10)50 7,000 TS1 8,250 PV
TS1 8,250/(10.1)1 7,500
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PV of total gains from trade, period 1, 7500
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Case 2 Divide Qtot equally over periods 0 and 1
Sum of the PV of total gains from trade over
periods 0 and 1 8,250 7500 15,750
Note that 15,750 in PV of total gains from trade
from dividing the resource equally over periods 0
and 1 EXCEEDS the 14,000 in total gains from
trade when we consumed all of the resource in
period 0. Thus equal division is closer to being
dynamically efficient.
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The theory of dynamically efficient resource
markets

• Methods for solving for the dynamically efficient
  allocation of the fixed stock of resource over
  time

Hotellings rule The dynamic equilibrium (which
we will show is also dynamically efficient)
occurs when the PV of marginal profit (also known
as marginal scarcity rent or marginal Hotelling
rent) for the last unit consumed is equal across
the various time periods.
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Hotellings rule

• (P0-MC)/(1r)0 (P1MC)/(1r)1

Marginal profit, period 0
Marginal profit, period 1
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The Dynamically Efficient Solution
Let demand be given by P a bqi, i 0, 1, ,
n. The integral of demand is total (or gross)
benefits, aqi bqi2/2. Likewise total cost is
cqi (c is constant MC).
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The Dynamically Efficient Solution

• The dynamic equilibrium is found by solving the
  following maximization problem

• Max Z ?i (aqi bqi2/2 cqi)/(1r)i ?Qtot
  - ?i qi,
• where i 0, 1, 2, , n. If Qtot is constraining,
  and with the profit function being strictly
  concave, the dynamically efficient solution
  satisfies
• (a bqi c)/(1r)i - ? 0, i 0, 1, , n.
• Qtot - ?i qi 0

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Solving for Dynamically Efficient Quantities in
2-Period Case

• Algebraic solution in the 2-period case is found
  by applying Hotellings rule

(a bq0 c)(1r) (a bq1 c). Substitute
(Qtot q0) for q1 above. Solve algebraically q0
bQtot r(a-c)/b(2r) q1 Qtot - q0
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The Dynamically Efficient Solution

• Instead of the preceding discrete approach, one
  can instead maximize the integral of net benefits
  over the relevant time horizon. One would also
  want to use the continuous discounting formula
  e-rt instead of the discrete-period formula
  1/(1r)t.

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The Dynamically Efficient Solution The r
Percent Rule of Extraction

• If we compare any two sequential periods i and
  (i1) we can derive the r percent rule of
  extraction

(a bqi c)1/(1r)i (a bqi1
c)1/(1r)i1 Since we were given that Pi a
bqi, the above equation can be expressed as
Hotellings rule PV(Pi MC) PV(Pi1 MC)
Note 1/(1r)i cancels on both sides,
yielding (a bqi c)(1r) (a bqi1 c)
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The Dynamically Efficient Solution The r
Percent Rule of Extraction

• If we compare any two sequential periods i and
  (i1) we can derive the r percent rule of
  extraction

(a bqi c)(1r) (a bqi1 c). Solving for
r ? r (a bqi1 c) - (a bqi c)/(a
bqi c) In words The discount rate r equals
the rate of growth in (undiscounted) marginal
profit over time.
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The Dynamically Efficient Solution The r
Percent Rule of Extraction

• r percent rule of extraction

r (a bqi1 c) - (a bqi c)/(a bqi
c) Simplifies to r b(qi qi1)/(a bqi
c) The equation in green above says that the
larger is the discount rate r, the faster that
the quantity allocated to future periods
declines. Thus we have a monotonically decreasing
extraction path (q0, q1, , qn) for r gt 0.
Why?
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r Percent Rule of Extraction
On the dynamically efficient path, quantity
shrinks in each successive period
On the dynamically efficient path, price rises in
each successive period

• r percent rule of extraction

Period t
Period t1
p
p
(a bqi c)
(a bqi1 c)

Pt1
pt

MC
MC
D
D
qt1
qt
qt
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r Percent Rule of Extraction

• r percent rule of extraction

Question How would this price path change if the
discount rate was equal to 0? What would this
mean?
P
Question How would this price path change if
discount rate r increased?
Dynamically Efficient price path
0
Time
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r Percent Rule of Extraction

• r percent rule of extraction

Note that the r percent rule of extraction is
derived from profit maximization by firms in a
competitive resource market, and thus arises from
Smiths Invisible Hand in a well-functioning
competitive market.
Suppose that the federal government pursued a
cheap oil policy of intervening in the market
to subsidize current consumption of oil. How
would that distort the price path from what would
be dynamically efficient?
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r Percent Rule of Extraction

• r percent rule of extraction

Note Any gap between price and marginal cost in
a market is called rent in economics. As we will
discuss in more detail later, the rent that
derives from resource scarcity is variously
called user cost, royalty, dynamic rent, scarcity
rent, or Hotelling rent.
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Dynamically Efficient Solution Simplified 2
Period Case

• Now lets apply the parameters from our problem
  (a 200, b 1, c 10, r 0.1, 2 periods). The
  dynamically efficient solution satisfies

(200 q0 10)/(10.1)0 ? (200 q1
10)/(10.1)1 ? 100 q0 q1
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Dynamically Efficient Solution 2 Period Case
(200 q0 10)/(10.1)0 (200 q1
10)/(10.1)1. Since q1 100 - q0, substitute
(100 - q0) for q1 and simplify 190 - q0 (190 -
(100 - q0))/(1.1) ? -q0(10.9091) 0.909190
190 ? q0 108.182/1.9091 56.667 ? q1 100
56.667 43.333
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Dynamically Efficient Solution 2 Period Case
Test P0 200 56.667 143.333 (P0
MC)/(10.1)0 133.33 P1 200 43.333
156.667 (P1 MC)/(10.1)1 133.33 Therefore,
Hotellings rule is satisfied.
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Dynamically Efficient Solution 2 Period Case
Period 0 gains from trade CS0 (200 -
143.333)56.667/2 1,605.55 PS0
(143.333-10)56.667 7,555.56 PVTS0 9,161.11
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Dynamically Efficient Solution 2 Period Case
Period 1 gains from trade CS1
(200-156.667)43.333/2 938.87 PS1
(156.667-10)43.333 6,355.48 PVTS1
7,294.35/1.1 6,631.23
Sum of PV of total gains from trade, periods 0
and 1 9,161.11 6,631.23 15,792.34. This
is 42.34 larger than a 50/50 split in Case 2.
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Dynamically efficient equilibrium

• Intuition

If the PV of marginal profit is equal across time
periods (Hotellings rule), then firms have no
incentive to re-arrange production over time.
This solution also generates the largest PV of
total gains from trade over time.
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Dynamically efficient equilibrium

• Intuition

When a resource is abundant then consumption
today does not involve an opportunity cost of
foregone marginal profit in the future, since
there is plenty available for both today and the
future. Thus, when resources traded in a
competitive market are abundant, P MC and thus
marginal profit is zero. As the resource becomes
increasingly scarce, however, consumption today
involves an increasingly high opportunity cost of
foregone marginal profit in the future. Thus as
resources become increasingly scarce relative to
demand, marginal profit (P-MC) grows.
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Dynamically efficient equilibrium

• Intuition

The profit created by resource scarcity in
competitive markets is called Hotelling rent
(also known as resource rent or by the Ricardian
term scarcity rent). Hotelling rent is economic
profit that can be earned and can persist in
certain natural resource cases due to the fixed
supply of the resource. Due to fixed supply,
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.
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Dynamically efficient equilibrium

• Intuition

How will the dynamically efficient allocation of
the fixed resource stock change if the discount
rate r becomes larger (and all else remains the
same)? Why?
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Dynamically efficient equilibrium

• Intuition

Suppose that the discount rate, demand, and
marginal cost remain the same, but the resource
stock increases. How will this change affect the
marginal profit in any given period?
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Dynamically efficient equilibrium

• Intuition

Given the r percent rule of extraction, if
demand and marginal cost are stationary over
time, what is the resource price path over time?
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Dynamically efficient equilibrium

• Intuition

• Real world Natural resource commodity prices may
  rise or fall over time because
• Marginal production cost might decrease
  (technology improves) or increase (exploit
  cheapest sources first).
• Demand may grow over time unless a new
  technology displaces this demand (e.g., coal
  replaced firewood, natural gas replaced coal,
  alt. energy replaces natural gas?),
• Future demand and marginal cost cannot be known
  with certainty.

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Practice Exercise

• Demand P a bQ (let a 200 and b
  1)
• Supply P c (let c 10)
• Total resource stock Qtot 100 2 periods (0 and
  1)
• From the exact solution q0 (bQtot r(a
  c))/b(2r), and q1 Qtot - q0
• 1. Solve for the dynamically efficient resource
  allocation for r 0, 0.05, 0.10, 0.2, and 0.5.
  Build a table that with columns showing r,
  q0, q1, PV of marginal profit period 0 and
  PV of marginal profit period 1. There will be
  five rows, one for each r value above. Provide
  a brief interpretation of the impacts of rising
  discount rates on the dynamically optimal price
  and quantity path over time.
• 2. Now suppose that Qtot 70. Build a second
  table like the one in question 1 above. Provide a
  brief interpretation of the impact of a smaller
  resource stock on the optimal price and quantity
  path.