natural resource management and intertemporal intergenerational choices
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Natural resource management and intertemporal/intergenerational 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

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natural resource management and intertemporal intergenerational choices

Natural resource management and intertemporal/intergenerational 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 “the tragedy of the commons”

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

So far, the problem of optimal allocation of an environmental good E (i.e., 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 anddistribute 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 dimensionconcerns two different aspects:

How future generations will use/demand this resource

How the resource stock evolves over time

The allocation problem thus becomes to find TODAY the optimal exploitation/extraction 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.

future generations and the problem of discounting
Future generations and the problem of discounting

In principle, the idea of optimality can be maintained: maximization of the Net Social Benefit. Now, however, the “society” is the aggregation of current and next generations and its net benefit is the difference between the flow of benefits B(E)tand 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 generations/periods) is expressed as:

  • Is there a market or a rights’ 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 (intergenerationalcoordination).

This is evident in the case of exhaustible resources:

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 “egoism” of present generations with respect to future generations.

Let’s 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 exhaustible natural resources - 1

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

Optimal intertemporal extraction of exhaustible natural resources - 2

QT - Q1 = Q2 (expresses the extraction of second generation)

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


Optimal intertemporal extraction of exhaustible natural resources - 3

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).




Optimal use of current generation (Q1) under an infinite discount rate (r=∞) expressing the lack of intergenerational coordination









Optimal intertemporal exploitation of renewable natural resources

  • For these resources, the problem of the optimal dynamic exploitation has not only and simply to do with intergenerational coordination (a “fair” 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, i.e. 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 – Xt) depends on the initial level of the stock itself Xt:

Sustainable exploitation of renewable natural 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 Ytis called sustainable equilibrium (or sustainable exploitation) as it allows the resource stock to remain stable over time.
  • Any exploitation level Y>YMAXcan 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 Y<YMAX, it is always possible to find two different stock levels (X1 e X2) making that exploitation, Yt = f(Xt ), sustainable:






  • 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; XMis unstable, too). On the contrary, in X2 we have astable sustainable equilibrium (the stock will spontaneously return to the equilibrium value after a little deviation).

Sustainable exploitation of renewable natural resources - 2

  • 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(X,E), where E indicates the exploitation effort, a synthetic measure of production inputs used for exploitation.
  • It is reasonable to assume that Y/E>0 but also that E/X<0, namely, for a given exploitation level Y the effort must increase as the stock decreases:
  • If c is the unit cost of E, it is easy to see how the more efficient (i.e., 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. Let’s consider two opposite access regimes:

- Free Access

- Exclusive Access





E3 >

E2 >



Sustainability and optimal exploitation

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)>cE. exploitation will stop when Y(E)/E=c. Let’s represent the revenue Y(E)=f(X) and the cost cE in the same diagram:







  • 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 (cE*<cELA). 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 >0). Collectively, however, they are not able to understand that a greater aggregate profit could be obtained with a lower level of exploitation.

Sustainability and optimal exploitation:

free access vs. exclusive access rights - 2

  • 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). He/She decides the level of exploitation Y(E). If he/she is rational, as under free access, he/she aims at maximizing the profit given by Y(E) - cE. Therefore, the optimal solution will be the level for which: Y/E = Yl = c.

The Tragedy

of the Commons

Yl =c




  • Under assignment of exclusive access rights, the consequent exploitation (YP; XP) will be stable and optimal, much better than under LA (XP>XLA, YP>YLAand EP<ELA) : 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.

Sustainability and optimal exploitation:

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.


EP· t*

  • Therefore, the access is not free but it is free the participation to the “market” 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 “convince” free and non-coordinated individuals to stop at YP. The public authority will also obtain an access tax revenue (t*x EP) to be invested on the resource itself or on compensating individuals discouraged by the tax and, thus, that lost benefits.