Dynamic Efficiency

1
Dynamic Efficiency Hotellings Ruleadapted
from S. Hacketts lecture notes

2
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.
3
Dynamic efficiency

• The notion of dynamic efficiency is an intuitive
  concept.

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)?
4
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.

5
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 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.

6
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.

7
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?
8
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.

9
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.
10
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.

11
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)/(1r)i
12
Dynamic efficiency

• 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
13
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.
14
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)/(1r)i
Where i 0, 1, 2, , n years.
15
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.
16
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)

17
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)

18
Dynamic efficiency

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

19
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 lt 190.
Scarcity-constrained market equilibrium Q0
100 P 200 100 100.
20
(No Transcript)
21
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
22
PV of total gains from trade 14,000
23
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
24
PV of total gains from trade, period 0, 8,250
25
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
26
PV of total gains from trade, period 1, 7500
27
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.
28

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

Hotellings 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.
29
Hotellings rule

• (P0-MC)/(1r)0 (P1MC)/(1r)1

Marginal profit, period 0
Marginal profit, period 1
30
Hotellings 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

31
Hotellings rule

• Less math-intensive solution method

• If you are not very close to satisfying
  Hotellings 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?

32
Hotellings 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 Hotellings rule. Repeat steps as
needed until you are within a reasonable
approximation of satisfying Hotellings rule.
33
Optional Hotellings 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
34
Optional Hotellings rule

• The dynamically efficient allocation solves the
  following maximization problem

• ?i (aqi bqi2/2 cqi)/(1r)i ?Qtot - ?i qi,
  
• where i 0, 1, 2, , n. If Qtot is constraining,
  then the dynamically efficient solution
  satisfies
• (a bqi c)/(1r)i - ? 0, i 0, 1, , n.
• Qtot - ?i qi 0

35
Optional Hotellings rule

• 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
36
Optional Hotellings rule
(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
37
Optional Hotellings rule
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.
38
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
39
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.
40
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.
41
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.
42
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.
43
Dynamically efficient equilibrium

• Intuition

How will the dynamically efficient allocation of
the fixed resource stock change if the discount
rate r becomes larger? Explain
44
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?
45
Dynamically efficient equilibrium

• Intuition

Under the dynamically efficient solution in our
simplified modeling framework, what is the
trend of price over time? Why?
46
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.

47
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.

48
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?