Natural resource management and intertemporal/intergenerational choices

1

Facoltagrave di Economia ldquoG FuagraverdquoFacoltagrave di Economia ldquoG FuagraverdquoUniversitagrave Politecnica delle MarcheUniversitagrave Politecnica delle MarcheFacoltagrave di Economia ldquoG FuagraverdquoFacoltagrave di Economia ldquoG FuagraverdquoUniversitagrave Politecnica delle MarcheUniversitagrave Politecnica delle Marche

Natural resource Natural resource management and management and intertemporalintergeneraintertemporalintergenerational choicestional choices

The problem of next generations Non-renewable resources the problem of the

discounted value Renewable resources sustainable exploitation Sustainable and optimal exploitation (extraction)

rate Access regime and ldquothe tragedy of the

commonsrdquo

2

Optimal intertemporal (intergenerational) Optimal intertemporal (intergenerational) use of environmental-natural resourcesuse of environmental-natural resources

So far the problem of optimal allocation of an environmental good E (ie of pursuing the maximum net social benefit) has been worked out by comparing CURRENT costs and benefits associated to the use of this good

This static representation however does not fit the actual concerns related to the use of many natural resources where costs and benefits differently occur and distribute over time

We need to make explicit that these resources behave as stock that can be used either in the current period or in next periods Therefore choices about resource use have an inherent dynamic (intertemporal) dimension Such dimension concerns two different aspects How future generations will usedemand this resource How the resource stock evolves over time

The allocation problem thus becomes to find TODAY the optimal exploitationextraction rate Such optimal resource exploitation substantially differs for the two different kind of natural resources NON-RENEWABLE (EXHAUSTIBLE) RESOURCES fossil energy mineral

resources etc RENEWABLE RESOURCES forestry resources fishery resources water

resources etc

3

Future generations and the Future generations and the problem of discountingproblem of discounting

In principle the idea of optimality can be maintained maximization of the Net Social Benefit Now however the ldquosocietyrdquo is the aggregation of current and next generations and its net benefit is the difference between the flow of benefits B(E)t and of costs C(E)t over time

Algebraic summation of such benefits and costs however incur the problem of comparing monetary values over different periods of time This problem is tackled by comparing the current value of benefits and costs therefore by discounting all values at the discount rate r

Therefore the maximization of the current (discounted) intertemporal net social benefit (SB0) (thus achieving the optimal allocation of E across generationsperiods) is expressed as

000

1

)(

1

)(

ttt

ttt

E r

EC

r

EBSBMAX

t

Is there a market or a rightsrsquo negotiation mechanism allowing this optimal solution to be achieved Here market-negotiation can not be afforded for the simple reason that some generations can not express their preference now

therefore transactions can not be carried out Present generation therefore may tend to impose a too-high (egoistic) value of r As a consequence appropriate policy measures

should aim at restoring this hypothetical optimal transaction among generations by fixing an appropriate value for r (intergenerational coordination)

This is evident in the case of exhaustible resources

4

Under this circumstance the resource is available in an absolutely scarce quantity the stock QT Therefore the problem in resource management is to decide how much of QT has to be extracted (used) by the present generation and how much has to be left to the next generations Any generation will obtain a net benefit B(Q)t from resource extraction Due to absolute scarcity quantity used by time t generation is definitively missed for time (t+n) generations Therefore B(Q) t+n becomes an opportunity cost associated to B(Q)t in other words it is the option value of the resource itself Without an intergenerational coordination in any period t there will be tendency to over-utilize the resource to the level Q for which Bm(Q)t = 0 At such exploitation rate however there will correspond an opportunity cost for the following periods whose discounted value is B(Q)t+1(1+r) A generation that is not aware of this implicit cost implicitly assumes a very high discount rate that makes this opportunity cost negligible Therefore the discount rate in such context is somehow a measure of the degree of ldquoegoismrdquo of present generations with respect to future generations

Letrsquos consider this problem of intergenerational coordination in an oversimplified situation (model) one good (E) and only two generations (t = 1 2)

Optimal intertemporal extraction of Optimal intertemporal extraction of exhaustible natural resources - 1exhaustible natural resources - 1

5

If we wish to define the optimal allocation of the given stock QT between the two generations the problem to be solved is

r

QQBQBMAX T

Q 1

)1()1(

1

It can be easily found that the optimal solution is that level of current (generation 1) use Q1 such that

r

QBmQBm

1

)2()1(

The intuitive explanation is that any further unit of current exploitation would generate an additional benefit for generation 1 that is lower than the discounted value of the benefit subtracted (opportunity cost) to generation 2

We can better appreciate this result graphically


QT - Q1 = Q2 (expresses the

extraction of second generation)

6

Q1

Q2Q

0

Bm(Q)1

Bm(Q)2

Bm(Q)2(1+r)

Q1S

Q2S

Optimal intergenerational allocation of stock QT under a non-null discount rate The higher is r

the larger is the use of current generation (Q1) the lower the

amount left to generation 2 (Q2)

Optimal intergenerational allocation of stock QT only when the discount rate is null (r = 0 no

intertemporal preference)

QT

Optimal use of current generation (Q1) under an

infinite discount rate (r=infin) expressing the

lack of intergenerational coordination


7

Optimal intertemporal exploitation of Optimal intertemporal exploitation of renewable natural resourcesrenewable natural resources For these resources the problem of the optimal dynamic exploitation has not only and simply to do with intergenerational coordination (a ldquofairrdquo distribution across generations) Before dealing with optimality in fact the issue is to pursue sustainability in the use of the resource Its available quantity X in fact is not absolutely scarce (a stock QT) as it depends on a natural accumulation process usually based on biological processes ie on a growth function

In the (classical) case of biological population this growth typically follows a logistic function According to this function the the resource (for instance a forest) stops growing at a given maximum level of the stock X representing its dynamic biological equilibrium

This function also implies that at any time t the resource stock growth (Xt+1=Xt+1 ndash Xt) depends on the initial level of the stock itself Xt

Xt+1

Xt

tempo

Xt

time (t)

8

Sustainable exploitation of renewable Sustainable exploitation of renewable natural resources - 1natural resources - 1

If the resource growth is Xt+1 = f(Xt) it will be evidently possible to use (extract or exploit) in the unit of time such level (quantity) of the resource itself Yt = Xt+1 = f(Xt) without affecting the initial available stock for any following period that is maintaining the initial stock constant at Xt This level of use Yt is called sustainable equilibrium (or sustainable exploitation) as it allows the resource stock to remain stable over time

Any exploitation level YgtYMAX can never be sustainable regardless the initial stock as this is never able to regenerate the same amount of the resource On the contrary for any YltYMAX it is always possible to find two different stock levels (X1 e X2) making that exploitation Yt = f(Xt ) sustainable

Xt+1

Xt

Yt

X1 X2XM

YMAX

These two sustainable equilibria however are not equivalent The equilibrium corresponding to the smaller steady stock (X1) is an unstable sustainable equilibrium even a little movement of the stock from X1 will cause a permanent departure from the equilibrium (YMAX XM is unstable too) On the contrary in X2 we have a stable sustainable equilibrium (the stock will spontaneously return to the equilibrium value after a little deviation)

9

Beside stability is (Yt X2) also more economic efficient compared to (Yt X1) To deal with economic efficiency in this context we have to introduce the cost associated to resource exploitation The exploitation level Y evidently has a cost according to this sort of production function Y=g(XE) where E indicates the exploitation effort a synthetic measure of production inputs used for exploitation It is reasonable to assume that YEgt0 but also that EXlt0 namely for a given exploitation level Y the effort must increase as the stock decreases Xt+1

XtX3 X1X2

E E1E2 gtE3 gt

If c is the unit cost of E it is easy to see how the more efficient (ie lower cost) solutions correspond to the higher stock levels (the stable ones)

It is also interesting to notice that this result has a lot to do with the access regime for the resource Letrsquos consider two opposite access regimes - Free Access- Exclusive Access


10

Sustainability and optimal exploitationSustainability and optimal exploitationfree access vs exclusive access rights - 1 free access vs exclusive access rights - 1

The free access regime (regime LA) means that there is no cost associated to the access Still a cost must be borne for extracting the resource (c for any unit of effort E) The exploitation under free access will thus continue (increase) until revenues are greater than exploitation costs that is until Y(E)gtcE exploitation will stop when Y(E)E=c Letrsquos represent the revenue Y(E)=f(X) and the cost cE in the same diagram

XLA

ELA

Under free access the consequent sustainable exploitation (YLA XLA) is unstable and above all is clearly inefficient as the same exploitation (YLA)

can be obtained with a stable stock (X) and a much lower cost (cEltcELA) Nonetheless free access determines over-exploitation not because agents are irrational but only because they are not coordinated Individually they continue to have access to and to extract the resource until revenues are larger than costs (therefore profit gt0) Collectively however they are not able to understand that a greater aggregate profit could be obtained with a lower level of exploitation

E

Xt

Y(E)

cE

X

YLA

E

11

To make the exploitation stable and efficient (optimal) it is thus necessary to introduce forms of coordination The easiest way is to assign an exclusive property right on the resource to a single individual (regime P) HeShe decides the level of exploitation Y(E) If heshe is rational as under free access heshe aims at maximizing the profit given by Y(E) - cE Therefore the optimal solution will be the level for which YE = Yl = c

Under assignment of exclusive access rights the consequent exploitation (YP XP) will be stable and optimal much better than under LA (XPgtXLA YPgtYLA

and EPltELA) The Tragedy of the Commons Once more for a rival but non-

excludable resource (a common good) forms of regulation (or privatization) are apparently needed to avoid the undesired consequences of freedom and to achieve the positive effects of coordination among individuals

Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

The Tragedy

of the Commons


12

Sustainability and optimal exploitationSustainability and optimal exploitationthe access tax the access tax

Assigning (or privatizing) access rights however does not necessarily conflict with freedom of access The same result obtainable under exclusive rights can be achieved by allowing access to the resource upon the payment of an access price (or tax) t Therefore a viable compromise between freedom and coordination is to assign exclusive access rights to a public authority that then sells these rights at price t for unit of E to individuals willing to have access to the resource Therefore the access is not free but it is free the participation to the ldquomarketrdquo of access rights Therefore any individual allowed to extract the resource has to bear a unit cost (c+t) and exploitation will continue until Y(E) = (c+t) As the optimality condition is Yl(E) = c the optimal access tax should be fixed at t = YP(E)EP - Yl(E) This optimal access tax will ldquoconvincerdquo free and non-coordinated individuals to stop at YP The public authority will also obtain an access tax revenue (tx EP) to be invested on the resource itself or on compensating individuals discouraged by the tax and thus that lost benefits

Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

(c+t)E

EP t

2

Optimal intertemporal (intergenerational) Optimal intertemporal (intergenerational) use of environmental-natural resourcesuse of environmental-natural resources

So far the problem of optimal allocation of an environmental good E (ie of pursuing the maximum net social benefit) has been worked out by comparing CURRENT costs and benefits associated to the use of this good

This static representation however does not fit the actual concerns related to the use of many natural resources where costs and benefits differently occur and distribute over time

We need to make explicit that these resources behave as stock that can be used either in the current period or in next periods Therefore choices about resource use have an inherent dynamic (intertemporal) dimension Such dimension concerns two different aspects How future generations will usedemand this resource How the resource stock evolves over time

The allocation problem thus becomes to find TODAY the optimal exploitationextraction rate Such optimal resource exploitation substantially differs for the two different kind of natural resources NON-RENEWABLE (EXHAUSTIBLE) RESOURCES fossil energy mineral

resources etc RENEWABLE RESOURCES forestry resources fishery resources water

resources etc

3





000

1

)(

1

)(

ttt

ttt

E r

EC

r

EBSBMAX

t





4




5


r

QQBQBMAX T

Q 1

)1()1(

1


r

QBmQBm

1

)2()1(






6

Q1

Q2Q

0

Bm(Q)1

Bm(Q)2

Bm(Q)2(1+r)

Q1S

Q2S






QT





7




Xt+1

Xt

tempo

Xt

time (t)

8




Xt+1

Xt

Yt

X1 X2XM

YMAX


9


XtX3 X1X2

E E1E2 gtE3 gt




10



XLA

ELA



E

Xt

Y(E)

cE

X

YLA

E

11





Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

The Tragedy

of the Commons


12



Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

(c+t)E

EP t

3





000

1

)(

1

)(

ttt

ttt

E r

EC

r

EBSBMAX

t





4




5


r

QQBQBMAX T

Q 1

)1()1(

1


r

QBmQBm

1

)2()1(






6

Q1

Q2Q

0

Bm(Q)1

Bm(Q)2

Bm(Q)2(1+r)

Q1S

Q2S






QT





7




Xt+1

Xt

tempo

Xt

time (t)

8




Xt+1

Xt

Yt

X1 X2XM

YMAX


9


XtX3 X1X2

E E1E2 gtE3 gt




10



XLA

ELA



E

Xt

Y(E)

cE

X

YLA

E

11





Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

The Tragedy

of the Commons


12



Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

(c+t)E

EP t

4




5


r

QQBQBMAX T

Q 1

)1()1(

1


r

QBmQBm

1

)2()1(






6

Q1

Q2Q

0

Bm(Q)1

Bm(Q)2

Bm(Q)2(1+r)

Q1S

Q2S






QT





7




Xt+1

Xt

tempo

Xt

time (t)

8




Xt+1

Xt

Yt

X1 X2XM

YMAX


9


XtX3 X1X2

E E1E2 gtE3 gt




10



XLA

ELA



E

Xt

Y(E)

cE

X

YLA

E

11





Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

The Tragedy

of the Commons


12



Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

(c+t)E

EP t

5


r

QQBQBMAX T

Q 1

)1()1(

1


r

QBmQBm

1

)2()1(






6

Q1

Q2Q

0

Bm(Q)1

Bm(Q)2

Bm(Q)2(1+r)

Q1S

Q2S






QT





7




Xt+1

Xt

tempo

Xt

time (t)

8




Xt+1

Xt

Yt

X1 X2XM

YMAX


9


XtX3 X1X2

E E1E2 gtE3 gt




10



XLA

ELA



E

Xt

Y(E)

cE

X

YLA

E

11





Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

The Tragedy

of the Commons


12



Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

(c+t)E

EP t

6

Q1

Q2Q

0

Bm(Q)1

Bm(Q)2

Bm(Q)2(1+r)

Q1S

Q2S






QT





7




Xt+1

Xt

tempo

Xt

time (t)

8




Xt+1

Xt

Yt

X1 X2XM

YMAX


9


XtX3 X1X2

E E1E2 gtE3 gt




10



XLA

ELA



E

Xt

Y(E)

cE

X

YLA

E

11





Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

The Tragedy

of the Commons


12



Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

(c+t)E

EP t

7




Xt+1

Xt

tempo

Xt

time (t)

8




Xt+1

Xt

Yt

X1 X2XM

YMAX


9


XtX3 X1X2

E E1E2 gtE3 gt




10



XLA

ELA



E

Xt

Y(E)

cE

X

YLA

E

11





Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

The Tragedy

of the Commons


12



Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

(c+t)E

EP t

8




Xt+1

Xt

Yt

X1 X2XM

YMAX


9


XtX3 X1X2

E E1E2 gtE3 gt




10



XLA

ELA



E

Xt

Y(E)

cE

X

YLA

E

11





Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

The Tragedy

of the Commons


12



Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

(c+t)E

EP t

9


XtX3 X1X2

E E1E2 gtE3 gt




10



XLA

ELA



E

Xt

Y(E)

cE

X

YLA

E

11





Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

The Tragedy

of the Commons


12



Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

(c+t)E

EP t

10



XLA

ELA



E

Xt

Y(E)

cE

X

YLA

E

11





Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

The Tragedy

of the Commons


12



Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

(c+t)E

EP t

11





Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

The Tragedy

of the Commons


12



Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

(c+t)E

EP t

12



Yl =c

XLA

ELAE

Xt

Y(E)

cE

X

YLA

E

XP

YP

EP

(c+t)E

EP t

Natural resource management and intertemporal/intergenerational choices

Documents

Transcript of Natural resource management and intertemporal/intergenerational choices