chapter 5

• fairness or justice should accompany efficiency concern
• this chapter considers one particular concern: the treatment of future generations
• how to allocate a depletable resource over time?
• using a numerical example, trace out temporal allocation and show how it is affected by changes in the discount rate

• dynamic efficiency: balances present and future uses of a depletable resource by maximizing PVNB
• simplest of models: 2 periods
• assume:
– fixed supply of depletable resource
– demand is constant
– MWTP: P = 8-0.4Q
– MC = $2 (constant)

Period 1 Period 2

• notice if total supply 30 or greater
– efficient allocation would just be to produce 15 in each period, regardless of discount rate
– supply is sufficient to cover demand in both periods

• suppose only have 20 units
• try 15 units period 1 / 5 units period 2
• what is PVNB?
• period 1 = ½ *6*15 = $45
• period 2 = $45 – ½ *4*10 = $25
– (then discount)
– assume r=10%
– $25/(1+r) = $22.73
• total PVNB = $67.73
• is this the best?

15 units in period 1 / 5 units period 2
Period 1
6
Period 2

10 units in period 1 / 10 units period 2
Period 1 Period 2
4 4

• what is PVNB?
• period 1 = $45 – ½ *2*5 = $40
• period 2 = $45 – ½ *2*5 = $40
– (then discount)
– assume r=10%
– $40/(1+r) = $36.36
• total PVNB = $76.36
• better, but is it the best? how do we know?

how to find the best (optimal) allocation?
• dynamically efficient allocation rule = present value of marginal net benefit from the last unit in period 1 equals the PVNB in period 2.

(MB = P-MC)
Period 1: left to right. Why intersects at $6 and Q = 15?
Period 2: right to left (and discounted, 6 / 1.10 = 5.45)
Draw it this way so that any allocation between periods equals 20

Efficient allocation (10.238, 9.762) is where two curves intersect.

highest PVNB (compare to others we tried)
Total PVNB is area under NB curves.
Try to allocate another way, and total area will decline.

10.238 units in period 1 / 9.762 units period 2
Period 1 Period 2
4.0952
3.9048
10.238
9.762

• what is PVNB?
• pd. 1 = $45 – ½ * $1.9048 *4.762 = $40.4646
• pd. 2 = $45 – ½ *$2.0952*5.238 = $39.5127
– (then discount)
– assume r=10%
– $39.5127/(1+r) = $35.921
– total PVNB = $76.39
• highest PVNB

• when resources are scarce, greater current use diminishes future opportunities
• MUC = present value of those forgone opportunities at the margin
• uses of resources that would have been appropriate in the absence of scarcity may no longer be appropriate once scarcity is present

• with 30 or more units, no MUC
• with only 20 units, what is MUC?
• PV of MUC? Height btw quantity axis and intersection (approx. $1.90)
• if 15 each period, would not be any intersection. So height represents opportunity cost / MUC due to scarcity

• no scarcity:
P = marginal extraction cost
• with scarcity:
P = marginal extraction cost + MUC

• plug efficient quantities into price equation, P = 8– 0.4Q
• P1 = $3.90
• P2 = $4.09
• supply and demand given by…

$3.90-$2.00 = $1.90
$4.09-$2.00 = $2.09
(1.90)*(1+r)

• affects size of the MUC and the allocation between periods
• because of discounting, 1 st period gets more than 2 nd period
• higher discount rates skew resource extraction towards the present because gives future less weight

• what do we “owe” the future?
• one starting point: A Theory of Justice, by
John Rawls
– “veil of ignorance,” born not knowing eventual position (generation) in society – make up rules they would later have to live by
– what kind of rule might come out of this?

• at a minimum, future generations should be left no worse off than current generations
• are efficient allocations fair?
– in the 2 period model, more is allocated to current than future
– but 1 st period can save net benefits for the future
– if an allocation is dynamically efficient, always possible to make next generation at least as well off as any other allocation scheme

is being “fair” optimal?
10 units in period 1 / 10 units period 2:
$40 period 1 / $40 period 2 (now current value!)
Period 1 Period 2
4 4

efficient allocation
10.238 units in period 1 / 9.762 units period 2:
$40.46 period 1 / $39.51 period 2 (now current value!)
Period 1 Period 2
4.0952
3.9048
10.238
9.762

• suppose first generation spends $40 and saves $0.46 for future
• at 10% interest, this would grow to $0.513
• $39.51 + $0.51 > $40
• future better off accepting the dynamically efficient allocation than the equal allocation

• how to make this operational?
• Hartwick rule (“weak sustainability”)
– constant consumption can be maintained perpetually if all scarcity rent / profit invested in capital
– value of total capital stock will not decline

• receive inheritance of $10,000
• put it in bank w/ 10% interest
• if you spend exactly $1,000 per year, always be $10,000 and the income will last forever
• spending $1,000 (from interest) is sustainable

• current generation has been given an endowment (total capital)
– total capital = natural + physical
– sustainable use implies we keep the principle
(endowment) and live off of the flow of services provided
– ensure that the value of the total capital stock is maintained

• can physical capital readily substitute for natural capital?
• other definitions of sustainability?

• maintain value of stock of natural capital

• certain physical flows of certain individual resources should be maintained
• not sufficient to maintain the value of an aggregate

• not all efficient allocations are sustainable
• not all sustainable allocations are efficient
• choose the sustainable allocation that satisfies dynamic efficiency!