Transcript Dynamic Efficiency & Hotelling`s Rule

Dynamic Efficiency
& Hotelling’s Rule
[adapted from S. Hackett’s lecture notes]
Dynamic efficiency
Recall static notion of Pareto efficient resource
allocation is that one cannot change how resources
are split to generate larger gains from trade
(without making some one else worse off)
In contrast, dynamic 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.
Dynamic efficiency
The notion of dynamic efficiency is an
intuitive concept.
First, let’s 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)?
Dynamic efficiency
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.
Dynamic efficiency
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 don’t
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 one’s pure rate of time preference.
Dynamic efficiency
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.
Dynamic efficiency
As an aside, why might your present
discounted value of a $10,000 payment 10
years in the future differ from that of
someone else?
Different life circumstances, different investment
opportunities. Other?
Dynamic efficiency
Note: The discount rate (like an interest
rate) reflects the time value of money:
• The rate at which the present value of a payment
shrinks as the time of payment is pushed off further into
the future
•The rate at which the future value of current
interest-earning savings grows over time.
Dynamic efficiency
Since different people have different discount rates,
then at the prevailing market interest rate, some
people are lenders (financial investors), while others
are borrowers.
As with market equilibrium price, the equilibrium
market interest rate reflects a balancing of the
discount rates of those supplying and demanding
loanable funds.
Dynamic efficiency
Finance is an application of economics that
focuses on time value of money. We will limit
ourselves to an elementary application of the
time value of money.
Dynamic efficiency
Suppose that you will receive a single guaranteed
future payment “i” years from the present, and your
discount rate (interest rate) is “r”. Then the present
discounted value (PV) of that future payment (FP) is
given by the following formula:
PVFP = ($ future payment)/(1+r)i
Dynamic efficiency
PV example:
$ future payment is $10,000. “i” = 2 years
from the present. “r” = 10% (0.10). Then:
PVFP =
$10,000/(1+0.1)2
=
$10,000/1.21
=
$8,264.63
Dynamic efficiency
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 $10,000 to be received two
years from now.
Dynamic efficiency
Final point on PV: If you will receive a stream of
payments over time (e.g., social security
payments), then the PV of that stream of
payments is found as follows:
PVFP =
i($ future payment, year i)/(1+r)i
Where i = 0, 1, 2, …, n years.
Dynamic efficiency
Moving on…
Our analysis of dynamic efficiency 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.
Dynamic efficiency
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)
Dynamic efficiency
Simplifying assumptions, continued:
• Marginal cost is constant
• Market demand is “steady state”, meaning that demand in
period 1 is the same as in period 0 (no growing or shrinking
demand)
Dynamic efficiency
Model:
Demand: P = 200 – Q
Supply: P = 10
Discount rate “r” = 10 percent (0.1)
Total resource stock Qtot = 100
Dynamic efficiency
Case 1: Ignore period 1 while in period
0 (“live for today”)
Competitive market equilibrium: 200-Q0 = 10 
Q0 = 190
Problem! Qtot = 100 < 190. Scarcity-constrained
market equilibrium Q0 = 100;
P = 200 – 100 = $100.
Case 1: Consume All in Period 0
250
Price
200
150
100
50
Demand
Supply
0
0
20
40
60
80
100
Quantity
120
140
160
180
200
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/(1+0.1)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
Case 1: Consume All in Period 0
250
Price
200
150
CS = $5000
100
PS = $9000
50
Demand
Supply
0
0
20
40
60
80
100
120
Quantity
PV of total gains from trade = $14,000
140
160
180
200
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
Case 2: Consume Half in Period 0 and Half in Period 1
250
200
Price
CS = $1,250
150
100
PS = $7,000
50
Demand
Supply
0
0
20
40
60
80
100
120
140
160
Quantity
PV of total gains from trade, period 0, = $8,250
180
200
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/(1+0.1)1 = $7,500
Case 2: Consume Half in Period 0 and Half in Period 1
250
200
Price
CS = $1,250
150
100
PS = $7,000
50
Demand
Supply
0
0
20
40
60
80
100
120
140
Quantity
PV of total gains from trade, period 1, = $7500
160
180
200
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.
Methods for solving for the dynamically
efficient allocation of the fixed stock of resource
over time:
Hotelling’s rule: The dynamically efficient
allocation 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.
Hotelling’s rule
(P0-MC)/(1+r)0 = (P1–MC)/(1+r)1
Marginal profit,
period 0
Marginal profit,
period 1
Hotelling’s rule
Less math-intensive solution method:
1. Select an initial way to divide the resource stock
over time (hint: usually more in period 0, less in
period 1, due to time preference)
2. Derive prices in both periods using these
quantities
3. Calculate PV of marginal profit in both periods
Hotelling’s rule
Less math-intensive solution method:
4. If you are not very close to satisfying
Hotelling’s rule, then change the way you
allocated the resource stock. Increase Q in the
time period that had the larger PV of marginal
profit, and decrease Q in the other time period.
Note: Profit maximizing firms will automatically
have this incentive to redistribute production.
Why?
Hotelling’s rule
Less math-intensive solution method:
5. Re-derive prices in both periods using these
new quantities
6. Re-calculate PV of marginal profit in both
periods
7. See if you are closer to satisfying Hotelling’s
rule. Repeat steps as needed until you are within
a reasonable approximation of satisfying
Hotelling’s rule.
Optional Hotelling’s rule
More math-intensive solution method
(optional):
In the “simple” two-period case considered here,
let demand be given by P = a –bqi. The integral of
demand is total benefits, aqi – bqi2/2. Likewise total
cost is cqi (c is constant MC). If the available
resource stock is Qtot, then the dynamically
efficient allocation of a resource over “n” years is
the solution to the following maximization
problem:
Optional Hotelling’s rule
The dynamically efficient allocation solves the
following maximization problem:
i (aqi – bqi2/2 – cqi)/(1+r)i + [Qtot - i qi],
where i = 0, 1, 2, …, n. If Qtot is constraining, then
the dynamically efficient solution satisfies:
•(a – bqi – c)/(1+r)i -  = 0, i = 0, 1, …, n.
•[Qtot - i qi] = 0
Optional Hotelling’s rule
Now let’s 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)/(1+0.1)0 = 
(200 – q1 – 10)/(1+0.1)1 = 
100 = q0 + q1
Optional Hotelling’s rule
(200 – q0 – 10)/(1+0.1)0 = (200 – q1 – 10)/(1+0.1)1.
Since q1 = 100 - q0, substitute (100 - q0) for q1 and
simplify:
190 - q0 = (190 - (100 - q0))/(1.1) 
-q0(1+0.9091) = 0.9091*90 – 190 
q0 = 108.182/1.9091 = 56.667 
q1 = 100 – 56.667 = 43.333
Optional Hotelling’s rule
Test:
P0 = 200 – 56.667 = 143.333
(P0 – MC)/(1+0.1)0 = $133.33
P1 = 200 – 43.333 = 156.667
(P1 – MC)/(1+0.1)1 = $133.33
Therefore, Hotelling’s rule is satisfied.
Dynamically Efficient Market Allocation
Period 0 gains from trade:
CS = (200 - 143.333)*56.667/2 = $1,605.55
PS = (143.333-10)*56.667 = $7,555.56
PV(TS) = $9,161.11
Dynamically Efficient Market Allocation
Period 1 gains from trade:
CS = (200-156.667)*43.333/2 = $938.87
PS = (156.667-10)*43.333 = $6,355.48
PV(TS) = $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.
Dynamically efficient equilibrium
Intuition
If the PV of marginal profit is equal across time
periods (Hotelling’s 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.
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.
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.
Dynamically efficient equilibrium
Intuition
How will the dynamically efficient allocation of the
fixed resource stock change if the discount rate “r”
becomes larger? Explain…
Dynamically efficient equilibrium
Intuition
Suppose that the discount rate remains the same,
but the resource stock increases or decreases. How
will this affect the dynamically efficient allocation
of the resource stock?
Dynamically efficient equilibrium
Intuition
Under the dynamically efficient solution in our
“simplified” modeling framework, what is the
trend of price over time? Why?
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.
Dynamically efficient equilibrium
Further Study
In a graduate natural resources economics class you could
evaluate dynamically efficient resource allocation for these
more complex and real-world cases:
•more than 2 time periods
•varying and/or uncertain demand
•increasing and/or uncertain marginal cost of production,
and
•"backstop" technologies allowing for substitutes.
Practice Problem – Dynamic Efficiency
Demand: P = 200 – Q
Supply: P = 10
Discount rate “r” = 20 percent (0.2)
Total resource stock Qtot = 100
1. Solve for the dynamically efficient allocation (within $1 of
marginal profit)
2. How does this increase in the discount rate affect the
dynamically efficient allocation?
3. Now suppose that “r” = 0.1 but Qtot = 60. Solve for the
dynamically efficient allocation (within $1 of marginal profit). How
does a reduction in resource stock affect the dynamically efficient
allocation?