sustainability

chapter 5

what else besides efficiency?• 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

two-period model

• 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

two-period model

• 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

what happens if supply < 30?• 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?

Period 1 Period 2

6

15 units in period 1 / 5 units period 2

Period 1 Period 2

4

10 units in period 1 / 10 units period 2

4

check 10 and 10

• 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.

how to solve for efficient allocation?

highest PVNB (compare to others

we tried)

Total PVNB is area under NB curves.Try to allocate another way, and total area will decline.

Period 1 Period 2

4.0952

10.238 units in period 1 / 9.762 units period 2

3.9048

10.238 9.762

check 10.238 and 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

marginal user cost

• 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

marginal user cost

• 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

efficient price

• no scarcity:P = marginal extraction cost

• with scarcity:P = marginal extraction cost +

MUC

solve for efficient prices under scarcity• plug efficient quantities into

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

MUC is the difference btw price and MC (extraction cost)

$3.90-$2.00 = $1.90 $4.09-$2.00 = $2.09

(1.90)*(1+r)

discount rate

• affects size of the MUC and the allocation between periods

• because of discounting, 1st period gets more than 2nd period

• higher discount rates skew resource extraction towards the present because gives future less weight

defining intertemporal fairness• 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?

sustainability criterion • 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 1st 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

Period 1 Period 2

4

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

4

Period 1 Period 2

4.0952

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

3.9048

10.238 9.762

sharing the efficient allocation better than the equal allocation• 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

applying the sustainability criterion• 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

example

• 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

applied to the environment• 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

substitutability

• can physical capital readily substitute for natural capital?

• other definitions of sustainability?

strong sustainability

• maintain value of stock of natural capital

environmental sustainability• certain physical flows of certain

individual resources should be maintained

• not sufficient to maintain the value of an aggregate

sustainability vs. efficiency• not all efficient allocations are

sustainable• not all sustainable allocations

are efficient

• choose the sustainable allocation that satisfies dynamic efficiency!