Exploration of non renewable resources a dynamic approach

Abstract

Arrow studies a model of exploration of non renewable natural resources, in continuous time (see |4|), where, the quantity of resources, to be found is a random variable.

In our model there is a T planning period to go.

If society consumes, in oen period, an amount c of non renewable resource, it receives a utility u(c). New resources can be found through exploration, α reflecting the effort applied to get them, spending p per unit of effort.

The number of units found when effort α is employed is given by a random variable w_α, having a Poisson distribution with parameter λα.

Society is then faced with the problem of maximizing the descounted total utility:

$$V^T \left( y \right) = Max\left[ {u\left( \alpha \right) - p\alpha + \delta {\rm E}V^{T - 1} \left( {y - 1c + W_\alpha } \right)} \right]$$

Subject to: 0≤c≤y, α≥0 where y is the stock of resources available at the begining of the initial period and δ is the discount factor.

The new fact in the above problem, in relation with similar ones in literature, is that the probability distribution is defined by choice variable.

Concavity plays a major role in thes kind of model, it can be proved that, under some conditions [related to decreasing risk aversion for u′(c) and u″(c)], V(y) is a concave function.

The main result is that there exists y such that for y>y, α=0 (there is no exploration).