EXHAUSTIBLE RESOURCES

NATURAL RESOURCES

ECONOMICS

CHAPTER III

EXHAUSTIBLE

RESOURCES

T O U L O U S E S C H O O L O F E C O N O M I C S

CHAPTER THREE : EXHAUSTIBLE RESOURCES

Natural resources economics

M1-TSE

I N T RODUCT I ON

The main difference between exhaustible resources and renewable resources, like land or water, is that consuming a unit of such a resource today implies to be unable to consume it later. In other words, the management of an exhaustible resource has an explicit temporal dimension. The introduction of time into economic reasoning is a cause of many difficulties. First, the time horizon must be specified (finite time or infinite time), and it may be the case that the time horizon is a decision variable in the problem (when to start and when to stop). Second, a decision takes now the form of a decision rule for any time period depending upon external variables (prices dynamics, speed of the technical progress, changes in tastes and preferences), that is a vector of actions over time, maybe of infinite size (as for infinite time horizon problems).

Such decision rules must satisfy minimal rationality requirements, for example in most cases they should be time consistent. Time consistency implies that if at a given time period t0, the decision maker sets a decision rule to be applied at some future period t1, when getting to the time period t1, the decision maker should not decide to change the rule he (she) has previously determined. Such consistency issues play an important role in commitment analysis and in dynamic games theory.

Intertemporal decision schemes apply typically in more or less uncertain environments and hence, risky decisions considerations have to be explicitly taken into account. Information about future events may also evolve and makes room for anticipation construction and implementation considerations. Last the decision rule space at any time period may be dependant upon history, that is upon past actions or past outcomes of the decision tree. This historical dependence (or inheritance process) may appear as a constraint upon the decision process itself or as a tool to improve the outcome of the decision process (in learning by doing or information gathering processes for example).

We are going to proceed step by step. First, we shall try to describe the general rules for the optimal management of an exhaustible resource. Second we shall introduce market structure considerations, exhaustible resource exploitation being characterized by strong oligopoly positions of the operating firms in this sector. We shall also be interested in substitution between resources issues and in sustainable use of an exhaustible resource.

Part One

OPT I M AL EX PL OI TAT I ON OF AN E XH AUST I BLE R E SOURCE

Let us start with a prime example. A society seeks to determine the best plan of consumption of an exhaustible resource over two time periods: period 0 and period 1. Let

x

₀ and

x

₁ be respectively the resource consumption over the two periods and let us denote by X the size of the stock of the exhaustible resource. The stock constraint would hence be:

x

₀



x

₁



X

We assume that the society tries to maximize a social welfare function defined over the two periods:

U  x

_0,

x

₁



. We shall discuss at some length this problem of social choice criterion later. The Lagrangian of such a social program would be:

L=U  x0,x1X −x0−x1

Denoting by

U

₀ and

U

₁ respectively, the marginal utilities with respect to

x

₀ and

x

₁ , the necessary condition for optimality is simply:

U0x0,x1=U1x0,x1=

The first equality implies that:

U

₀



x

_0,

x

₁



U

1



x

0,

x

1



=1

That is, an optimal exploitation path of an exhaustible resource must equalize the marginal rate of substitution between two time periods to unity. The so-called 'intertemporal marginal rate of substitution' must be equal to unity.

The second equality implies that the marginal utilities of consumption, equal to each other, must also be equal to the marginal opportunity cost (that is the Lagrange multiplier) of the stock constraint. As we shall describe later, the marginal opportunity cost measures also the marginal scarcity rent over the exhaustible resource. We are prima facie in a situation fairly similar to the cases of water and land. This should come at no surprise, since, when working with two periods models, we are simply extending to dated goods (the resource consumption in period 0 and period 1) the usual optimal allocation procedure of the static framework. The marginal opportunity cost is the marginal willingness to pay of the whole society to get one more unit of resource stock.

Note that it is also equal to the implicit transfer price of the resource from period 0 to period 1, or from period 1 to period 0, which would nullify gains from trade between people living in period 0 and in period 1. The marginal willingness to pay of people living in period 0 for one more unit of resource consumption at that period should be given by the marginal utility of consumption from a social point of view and the same applies to people living in period 1. In other words, optimality implies that the society should be indifferent between consuming one unit of resource at period 0 or at period 1. If this was not the case, it would better to consume more or less today than tomorrow.

The allocation procedure can be easily extended to any fixed horizon problem, that is to any finite given sequence of periods, say of T periods. For each pair of periods, the marginal rate of substitution

should be equal to unity. This would lead to a T-1 system of equations. The total stock constraint would give an extra equation, determining the optimal resource allocation at any time period. Under concavity of preferences, the T-1 non linear system augmented by the linear stock constraint satisfy the sufficient conditions of the Hadamard lemma and hence admit a unique solution.

The following graph illustrates the determination of the optimal resource allocation in the two periods case.

Fig 1: Optimal allocation of an exhaustible resource in the two periods case

Hence, there is no difficulty to extend to any finite fixed horizon problem the procedure of determination of a Pareto optimal exploitation policy for and exhaustible ressource. There remain two problems:

1. Infinite horizon exploitation programs. By extending the resource consumption bundle to an infinite stream of consumption, one may find difficulties to identify an optimal policy. This will involve some additional conditions upon the social welfare criterion and in many cases some conditions at infinity (the so-called transversality conditions) in order to ensure the dynamic consistency of the Pareto criterion.

2. Endogenous time horizon (either finite or infinite). In such cases, the time horizon of the optimization program itself has to be decided in an optimal way. The same may apply for the starting date of the program (when to begin the exploitation of an exhaustible resource ?). Further transversality conditions apply in such a case, giving extra necessary conditions helping to determine the value of the time horizon.

We shall consider these issues step by step. To help understand the problem of endogenous time horizon, let us consider the classic problem of substitution from an exhautible resource to some inexhaustible backstop resource (from oil to some renewable resource like solar energy for example).

x1+x2 ≤ X

U(x1,x2)=U

X

X

x

₂

x

₁

SUBST I T UT I ON FROM AN EX H AUST IBLE RE SO URCE TO A BACKSTOP

Let us condider a finite stock of an exhaustible resource

S

₀ at the beginning of period 0. Let s_t be the level of extraction at period t. We denote by St the remaining amount of the natural

resource stock at the beginning of period t. That is :

S

_t

=

S

₀

−

∑

x=0 t

s

_x

Note that we are back in a cake-eating problem : the available resource at period t is the initial stock reduced by the cumulative consumption over the past periods. Let us first consider a partial optimum version. We denote by V  t , c_t the consumer surplus as a function of the level of resource consumption c_t within the period t and by v t , c_t the marginal consumer surplus, that is:

v t , c

_t

=

d V t , c

t



d c

t

Assume that the surplus function is strictly concave, hence that the marginal surplus function is a strictly decreasing function of the extraction level. Let c be the constant level of the unit cost of extraction, hence equal to the marginal cost.

There exists some renewable substitute which may be used in any amount x_t at a cost k, assumed such that

k c

. Hence the exploitation of the inexhaustible substitute is assumed to be more costly than the exploitation of the exhaustible resource. The total resource consumption is the sum of the exhaustible resource exploitation and of the backstop use, that is: c_t=s_tx_t (the two goods are assumed to be perfectly substituable at the consumption stage). The optimization problem is hence:

Max

∑

t =0 ∞

[

V t , x

_t



s

_t

−

cs

_t

−

kx

_t

]

s.t. S

_t

=

S

₀

−

∑

x=0 t

s

_x

, x

_t

0 , s

_t

0 , S

_t



0

Since

S

t 1

0 implies that

St0

,

limt  ∞St0

appears as a necessary and sufficient

condition to get

S_t0

, t0 . Hence t

he Lagrangian of this program is the following:

L=

∑

t=0 ∞

[

V t , x

_t



s

_t

−

cs

_t

−

kx

_t

][

S

₀

−

∑

t=0 ∞

s

_t

]

∑

t =0 ∞



_t

x

_t



∑

t =0 ∞



_t

s

_t

Note that since the exhaustion period of the non renewable resource is not fixed, we have to introduce an infinite horizon problem to build the optimal solution. Assume that the infinite sums are finite in order to give sense to the expression of the Lagrangian. The first order conditions are:

L

xt

=0 : v t , x

t



s

t

−

k 

t

=0 , L

st

=0 :v t , x

t



s

t

−

c

t

−=

0

together with the complementary slackness conditions:



t

x

t

=0 , 

t

0 , t0



S

₀

−

∑

t =0

∞

s

_t

=0 , 0 , t0

Let us suppose that there exists some period T such that

x

_T

0 , y

_T

0

and hence

_T=_T=0 . This would imply from the optimality conditions that

=

k −c

. In such a case, inserting this expression of



into the optimality conditions, we would obtain for any period t different from T:

v t , x_ts_t=k−_t=k −_t _t=_t

Let us assume, to give an economic sense to the problem, that:

v t ,0k c , ∀ t0

. Under this assumption, at least one resource will be consumed. This will exclude the case x_t=y_t=0 , that is the case



t

0 , 

t

0

. In all other possibilities,



t

=

t

=0

. Thus, optimality requires

that v t , x_ts_t=k , the marginal utility from consumption should be constant over time, and, dividing the condition for two different periods, the marginal rate of substitution between these two periods must be equal to unity.

Now we allow to introduce the idea of cost priority for different resources. Assume to the contrary that there does not exist some simultaneous exploitation period like the period T. This would imply that

≠

k −c

. Assume for example that



k −c

.Since k>c, we get from the optimality conditions for any period such that the non renewable ressource is not exhausted :

k −c−=

_t

−

_t

0  

_t



_t



0

which implies that x_t=0 along an optimal path if S_t0 . Furthermore provided that

v t ,0k c , t0

, the non renewable ressource will be consumed in positive amount:

s

_t

0

. The society will give priority to the exploitation of the least cost resource, that is to the non renewable resource under our cost assumption. The inexhaustible backstop substitute must not be used before the non renewable resource is completely exhausted.

The opposite case k −c may be easily ruled out.. In such a case we would get ≥0 . Thus the non renewable ressource would never be exploited at all which cannot be optimal since it is less costly tha,n the inexhaustible substitute.

There must exist some time period T such that

S

_t

=0 , tT

, that is the non renewable resource has been completely exhausted at the end of the period T, the last period of exploitation of the exhaustible resource. This would lead to:

v t , s

t

=

c , t∈[0, T ]

v T 1, x

T 1

=

k , tT

since T+1 is the first period of exploitation of the substitute. The time profile of the marginal surplus is pictured on the following graph:

Fig2 : Substitution from an exhaustible resource to a backstop

CON TI N U OUS TI ME VE R SI ON

The dynamics of a non renewable resource extraction policy under the presence of a backstop substitute is usually presented in continuous time using the optimal control technique. As an illustration, we know present this classical presentation. But first, we must give the fundamentals of optimal control.

Optimal control

Optimal control theory is involved in the resolution of problems of this kind:

Max

_∫

t0

t1

V  x

_t

, u

_t

, t dt s.t. ˙x

t

=

F  x

t

, u

t

, t , x

t0

=

x

The vector xt is called the vector of state variables at time t. The vector ut is the vector of control variables. The state vector is moving over time through a law of motion depending upon the

level of the state variables and of the control variables:

˙x

t

=

F  x

t

, u

t

, t

. The last condition

k

c+λ

T T+1

v(t,s)

x

_t0

= 

x

is called the initial condition. It states that that at the beginning of the program, the state variables vector must start from some given level.

The problem is to find a sequence of control variable vectors

{

u

_t0

, ... ,u

_t

, ... , u

_t1

}

which maximizes the integral criterion subject to the state variables law of motion. In order to solve this program, let us introduce the following auxilliary function, called the Hamiltonian function:

Ht=V  xt, ut,t tF  xt, ut, t

where



t is called a vector of costate variables. There exists two sets of optimality conditions:

– The maximum principle. At each time t, the choice of the control variables must maximize the hamiltonian at this time. The idea is that a necessary condition for a choice of a sequence of optimal controls over the whole time range [t_0,t₁] , is that the optimal controls maximize the criterion at each instant of this interval. Hence the maximum principle states that a set of necessary conditions for optimality is:

∂

H

∂

u

_{j ,t}

=

0

∂

V

∂

u

_jt



t

∂

F

∂

u

_jt

=0

for any control variable

{

u

_jt

}

and each time t.

– The costate dynamics conditions. These conditions take the form of a system of differential equations describing the law of motion of the costate variables over time (duality with the state motion).

˙



_it

=−

∂

H

∂

x

_it

=−{

∂

V

∂

x

_it



t

∂

F

∂

x

_it

}

for any state variable {x_it} and each time t.

Let us apply this technique to the cake eating problem. Let us assume that the objective of the society is to maximize the following felicity function:

U =

∫

0 ∞

[

u x

_t



s

_t

−

cs

_t

−

kx

_t

]

e

−t

dt

using the same notation as before.



Is called the social discount rate or the rate of social preference for the present. Under social discounting, the utilities of future consumption are reduced with respect to present utilities. From the point of view of generations living at time t, future utilities are less important tant present ones. This is one of the reason for many criticism of this kind of optimality criterion : it gives more weight to the interest of the present generations with respect to the interest of future generations.

The dynamics of exhaustion of the natural resource, that is the law of motion of the state variable S_t , may be written as :

S

˙

_t

=−

s

_t and the initial condition reads :

S

₀

= 

S

.

● A first phase

[

0,T ]

during which only the exhaustible resource is consumed, the

inexhaustible substitute remaining in the backdoor.

● A second phase



T , ∞

during which only the inexhaustible substitute is used, the non

renewable resource being exhausted at time T.

During the second phase, the optimal level of the substitute consumption is the solution of:

u '  x=k

, let



u k 

be the corresponding present utility level. Hence starting from tT , the criterion is maximized at the level:

_{U T , k =}



_∫

T ∞



u  k  e

−t

dt= 

u  k 



e

−T

The optimization problem may hence be written as:

Max

_∫

0

T

[

u s

_t

−

cs

_t

]

e

−t

_{dt }

_{U T , k  s.t. ˙S}

t

=−

s

t

, S

0

= 

S

Let us apply the optimal control technique. Here st is the control variable and the resource stock

level at time t ,

S

_t is the state variable. The hamiltonian function reads: H =[u s_t−cs_t]e−t

_ts_t The optimality conditions are hence:

– The maximum principle:

∂

H

∂

s

t

=0  u '  s

_t

=

c

_t

e

t

_{, t∈[0,T ]}

– The costate dynamics:

˙



_t

=−

∂

H

∂

S

_t

=

0  

t

≡

Since the state variable does not appear into the expression of the hamiltonian function, we note that the costate variable must be constant over time. Using the maximum principle, we thus deduce from the first equation that:

[

u '  s

t

−

c ]e

−t

=≡

cste

An optimal exploitation policy of the exhaustible ressource must be such that the net marginal surplus in discounted terms must be constant over time. This is an expression of the so-called

« Hotelling Rule » for an optimal policy.

Let us sketch an economic interpretation of this rule. Consider an optimal exploitation path

{

s

_t

∗

, t∈[0, T ]}

. Let us suppose that along some finite time interval



0

=[

t

0,

t

0



dt ]

, the

society decides to increase the resource consumption by an amount ds at each time during this interval. The net increase in net discounted utility with respect to the original optimal plan over 0 would be



U

₀

=

∫

t0

t0dt

[

u s

_

∗

ds−u s

_

−

c ds ]e

−

_{d }

For dt sufficiently small, we get the following approximation:

U₀≈[u s_t0ds −u  s_t0−cds ]e−t0dt

And for ds sufficiently small:

u ' s

t0

≈

u  s

_t0



ds−u s

_t0



ds

Hence:

U₀≈[u '  s_t0−c ]e−t0ds dt

Over some further time interval



1

=[

t

1,

t

1



dt ]

, t1t0dt , the society decides to get back

to the original optimal path. In order to restore the remaining resource stock level to its original level, this implies to cut down the exploitation rate by an amount ds during this second time interval. Such a decrease in the exploitation rate will induce a decrease in utility over ₁ . Following the same procedure as before, this decrease in utility can be approximated by:



U

₁

≈−[

u ' s

_t1

−

c]e

−t1

dsdt

The balance sheet of such an arbitrage in utility terms would hence be equal to:

U = U0U1=[u '  st0−ce

−t0_−u ' s

t1−c e

−t1_]ds dt

For the original exploitation path to be optimal, the society should not gain any utility by doing a perturbation of the original path, here consuming more resource over the first time interval



0 at the

expense of a lower resource consumption over the second time interval 1 . This thus implies that



U =0

. That is:

u '  st0−c e

−t0_=u ' s

t1−c e

−t1

This is exactly the expression of the optimality condition, we called the Hotelling rule. Under the Hotelling rule, the society is made perfectly indifferent between consuming more resource today at the expense of a lower consumption tomorrow.

We can now compute the dynamics of the exploitation rate over time. To accomplish that, take a logarithmic version of the first order condition:

log u '  s

t

−

c−t=log

Since the derivative of the logarithm of any function V is given by dV/V, we get by differentiating the previous condition with respect to time:

d

dt

[

u '  s

t

−

c ]

u '  s

_t

−

c

−=0

That is:

u ' '  s

_t



d s

t

dt

=[

u ' s

t

−

c]

Since the utility function has been assumed to be a strictly concave function (that is

u ' '  s0

), this implies that ds_t/dt0 . Along an optimal path, the rate of exploitation of the natural resource should decrease over time.

Such a decrease is the consequence of two facts. The first is the discounting of utility which gives more weight to present utilities than to future utilities. Hence the society prefers to consume more today than tomorrow. The present consumption level should thus be higher than future consumption levels, hence the exploitation rate should diminish over time. But the impatience of society is a necessary, but not a sufficient, condition for this decreasing pattern of consumption. As seen from our previous calculation, the utility function must also be concave.

From the Hotelling rule, we know that the net marginal utility should be constant in discounted terms along an optimal trajectory. This implies that in present value terms, the net marginal utility should increase in order to compensate for the depriving effect of discounting upon the future social value of the use of the natural resource. Hence the main consequence of discounting along an optimal trajectory is to increase the net marginal utility in present terms through time. In order for such an increase to induce a reducing pattern of the resource consumption over time, the present marginal utility should be a decreasing function of the consumption level. Under this concavity assumption, an increase in net marginal utility will entail a decrease of the consumption level, hence a decreasing exploitation pattern over time.

As we have seen in the introduction, the actual consumption patterns of the main exhaustible resources (coal, oil, iron, copper,...) show a sharp increase over the last two centuries, contradicting this prediction of the model. This is an indication that other economic forces than preferences and discounting affect the dynamics of the exhaustible resources use. We shall come to that later.

Let us proceed to the description of the optimal path in this model. Along the first phase [0,T ] , where only the exhaustible resource is consumed by the society, the net present marginal utility increase at the social discount rate. At the exhaustion time T, the present marginal utility should equalize at the constant level k, a level corresponding to the optimal level of the inexhaustible substitute use, that is, at time T:

u ' s

_T

=

c e

T

=

k

Note that this implies a downward jump of the extraction level st from a strictly positive level

s_T0 at T, to 0 for tT . Using the previous relation, we can express the exhaustion time T as a function of λ , a function we denote by

T 

.

T =

1



log 

k −c

Note that this function is well defined because we have assumed that the inexhaustible substitute is more costly than the non renewable resource , that is by assumption

k c

. As a function of λ,

T 

verifies:

d T 

d 

0

, lim

 0

T =∞ , lim

 k −c

T =0

The only thing which remains to be determined is the value of



. To compute this value, one has to use the stock condition, that is, in the wording of optimal control theory, the initial condition

S

₀

= 

S

. This implies to express the current level of consumption as a function of



. Going back to the optimality condition, we can implicitely define such a function:

s

_t

=

u '

−1



c e

t

≡

s t , 

As a function of



:

d s t ,

d 

=

e

t

u ' '  s

t



0

Let us denote by

X 

the cumulated amont of resource consumed over the interval [0,T ]

along the optimal trajectory:

X =

_∫

0

T 

s t ,  dt

Since we have shown that

T 

and

s t ,

are both decreasing functions of



, we conclude that

X 

is also a strictly decreasing function of



. Moreover:

lim

_{ 0}

X =∞ , lim

_k−c

X =0

This implies that the stock condition X = S determines a unique value of



.

This unique value is the marginal opportunity cost of the stock constraint. That is,



is the social marginal willingness to pay for an extra unit availability of the natural resource. The marginal opportunity cost



is the marginal scarcity rent over the limited amount of the exhaustible resource stock. It is also called the Hotelling rent of the resource. The Hotelling rent must be constant over time in discounted value along an optimal exploitation trajectory. The Hotelling rent in present terms should increase at the social discount rate as an expression of the Hotelling rule.

The following graphs illustrate the dynamics of the optimal trajectories of values and quantities in our model.

3. : (

Fig a The dynamics of marginal utilities or of the resource « value )»

3. :

Fig b Quantities dynamics

U'(s)

k

c+λe

ρt

0

T

t

c

S

s

_t

T

t

Summary

● An exhaustible resource requires an explicit intertemporal framework of study since present

consumption decisions have irreversible effects upon the future availability of the resource

● The general rule of optimality is to equalize for any pair of dates the marginal utility of

consumption of the resource. This implies that the intertemporal marginal rate of substitution should be equal to unity.

● When there exists some inexhaustible substitute to the resource but more costly (a backstop

good), priority should be given to the least cost resource, that is to the exhaustible resource in that

case.

● In such a case, an optimal consumption plan will be split into two successive phases : a first

phase of finite duration, where only the non renewable resource is consumed up to exhaustion, followed by a phase of infinite duration during which only the inexhaustible backstop substitute is used.

● If the felicity function (the intertemporal social choice criterion) has an additive discounted

form for some strictly positive social discount rate, then the optimality rule implies that the discounted net marginal utilities should be equal at each time along an optimal exploitation trajectory of the exhaustible resource.

● This rule is called the Hotelling rule for optimal plans. The rule implies that the net marginal

utilities in present terms should increase at the social discount rate.

● At each time, the present marginal utility should be equal to the marginal cost augmented by a

term increasing over time at the social discount rate, this term being the marginal opportunity

cost of the limited total resource availability constraint.

● This marginal opportunity cost measures the social willingness to pay for an extra amount of

the natural resource stock. It is the marginal scarcity rent of the exhaustible resource, also called the Hotelling rent.

● With positive time discounting and under strict concavity of the present utility function, the

A G E NE R AL E QUI LI BRI U M EX AM PLE : CON SUMPT I ON OR LEI SUR E ?

Since the end of the nineties, the theme of « zero growth », or even « negative growth », became popular in the french ecological parties. The general ideas are the following. First, there exists a structural incompatibility between the exhaustion process of the non renewable resources (like oil) and the evergrowing development of consumption. Hence, sustainability can only be achieved through a stagnation of the consumption level and better by a constant decrease of the consumption level, down to a level that would be compatible with a pure renewable ressource economy. But such a consumption decrease would imply a diminishing welfare level of the population, a rather politically hard to be accepted perspective.

In order to circumvent such a political difficulty, some thinkers of the ecological party tried to promote an arbitrage from consumption to leisure. A society which consumes less is also a society which can work less. The loss of welfare from consumption could hence be compensated by an increase of welfare from leisure time. Hence the welfare that would be lost by the individual as a consumer would be gained over by him as an hedonic free time enjoyment. This theme became very popular in the 35 hours vein in France at the turn of the XXIth century. As we are going to show, matters are in fact a bit more complicated than that. It will be appear very dubtious that the substitution from consumption to leisure could help in any way to sustain a long run pattern of use of non renewable resources, without suffering from adverse welfare negative growth effects. Of course, the issue of determining if we should work more or less in our modern societies will remain unsettled (and perhaps impossible to decide in any reasonable general way). This kind of problem provides a motivation for a simple general equilibrium exercise we shall study now.

Let us consider a society which derives welfare at each time from the consumption of natural resources in amount

c

_t at each time and from available leisure

l

_t . Let

U c

_t

, l

_t



be the corresponding instantaneous utility function. For the sake of simplicity, assume that the utility function has the following form :

U c_t, l_t≡u c_tvl_t

The results would not be altered qualitatively under more general forms of the utility function. The utility from consumption function

u c

_t



is assumed to be strictly increasing and concave.

The society disposes of two kinds of resources. The first one is a renewable resource available in unlimited amount. The second one is an exhaustible resource. We assume that the two resources are perfectly substituable at the consumption level. At each time period, the society gets by definition one unit of time. Let the population be constant and normalized to unity. Hence at each time, the society has to split one time unit between leisure and work. Let



be the number of labor units needed to get one unit of the renewable resource for consumption. Let



be the number of labor units needed to get one unit of the non renewable resource.

Suppose first that only the renewable ressource is available to the society. What would be the optimal amount of consumption of the renewable resource and the corresponding level of labor ? Let x_t be the resource level of consumption. In the one resource case,

c

_t

=

x

_t and the amount of labor needed is equal to xt . Hence the leisure level is defined by the following transformation line :

The following graph illustrates the determination of the optimal consumption-leisure bundle:

4 : .

Fig Consumption-leisure in the renewable case

In this case, the optimal leisure-consumption bundle (x*=c*,l*) is simply determined by the tangency point of the highest possible indifference curve with the transformation line. The corresponding welfare level U* = u(c*) +vl* may be sustained forever and hence be equal to the sustainable welfare level of the society.

Now let us introduce the exhaustible resource. In order to give an economic sense to the problem, assume that



, that is the inexhaustible resource is more costly to use than the non renewable resource in labor terms. Note that in the opposite case, the society would prefer to consume forever the renewable resource without ever exploiting the exhaustible resource. Let y_t be the instantaneous level of consumption of the non renewable resource and

Y

t the level of the remaining resource stock at

time t.

Let us assume that the society wants to maximize a discounted sum of instanteneous utilities from consumption and leisure at some constant strictly positive discount rate ρ. The optimization problem would hence be:

Max

∫

0 ∞

[

u  x

_t



y

_t



v 1− x

_t

−

y

_t

]

e

−t

_{s.t. ˙}

_Y

t

=−

y

t

, x

t

0 , y

t



0 , Y

0

= 

Y

where

_Y



_{is the initial level of the non renewable resource stock.}

This kind of optimization problem incorporates positivity constraints and hence cannot directly be solved by our previous optimal control results. But there exists a direct extension of the basic results relying upon the introduction of a Lagrangian function defined as the sum of the Hamiltonian function and a weighted sum of the constraints, the weights being the Lagrange multipliers associated to the constraints.

L=H 

_t

x

_t



_t

y

_t

=[

u  x

_t



y

_t



v 1− x

_t

−

y

_t

]

e

−t

−

_t

y

_t



_t

x

_t



_t

y

_t

Note that the Lagrange multipliers are now time functions. The general results of optimal control theory still apply, that is the maximum principle and the dynamical conditions over the costate variables, but now to the Lagrangian instead of the Hamiltonian. Hence :

l

1

1- ηc

u(c)+vl=U

c

x*

l*

˙



_t

=−

d L

d Y

_t

=0 

t

=≡

cste

The maximum principle applied to the control variable

x

_t

, y

_t implies that:

d L

d x

_t

=0 [u '  c

t

−

v  ]e

−t



_t

=

0

dL

d y

_t

=0 [u ' c

t

−

v ]e

−t



_t

=

Moreover the complementary slackness conditions apply that is:

txt=0 , t0 ; tyt0 , t0

Assume that

x

_t

0 , y

_t

0

over some non degenerate interval T. This would imply that

t=t=0 and hence that : u ' ct=v  and hence that:

v − e

−t

=

This is possible only at one instant, let T be that instant. Furthermore substracting the first relation to the second one, we get:

v − e−t  _t−_t==v − e−T Hence:

v −[e

−t

−

e

−T

]=

_t

−

_t We conclude that if tT :

e

−t



e

−T



_t



_t

0  x

_t

=0 , y

_t

0

Conversely if tT one gets xt0 , yt=0 . Hence T appears at the exhaustion time of the

non renewable resource. The optimal trajectory is, as before, composed of two successive phases : a first phase

[

0,T ]

where only the renewable resource is consumed, followed by an infinite duration phase



T , ∞

, where only the renewable resource is consumed, the non renewable resource being exhausted. Along this second phase, we are back in the situation with only one renewable resource. Hence x*=c* is the solution for this phase.

During the first phase, since only the exhaustible resource is consumed, we get c_t=y_t . The



c

_t

,l

_t



pair must lie along the transformation line for the exhaustible ressource, that is lt=1−ct . We get from the optimality condition:

u ' c

_t

=

v  e

t

which defines implicitely a function ct=c t ,  where

d c t , /d 0

under the strict

T ≡T =

1



log

v −





a decreasing function of



. Let

C 

be the total amount of resource consumed over the time interval

[

0,T ]

.

C =

∫

0

T 

c t ,  dt

This a decreasing function of



. Hence the equation

C = Y

admits only one solution, which is the optimal value of



. The following picture illustrates the corresponding dynamics of consumption and leisure:

Fig 5 : Leisure-consumption dynamics

We can now proceed to a discussion of our introducing concerns. The optimal path appears to be a sequence of three moments, as shown on the graph.

.

1 _{During the first phase, only the exhaustible resource is consumed. Because of time}

discounting and the concavity of the utility function, the natural resource exploitation should decrease, hence the consumption should decrease. The consumption-leisure pair must lie upon the transformation line for the exhaustible resource. Alongside with the consumption decrease, leisure will increase, illustrating the substitution between consumption and leisure (first black arrow on the picture). Note that during this move the consumption-leisure trajectory cross indifference curves of decreasing utility levels. Hence welfare decreases along the first phase, both in present and in discounted terms. In other words, the substitution from consumption to leisure may alleviates the overall decrease of welfare but not prevent it. Despite leisure increase, the society experiences negative growth of consumption combined with a decrease in welfare.

l

l=1-μc

l=1-ηc

c

0

x*

l*

1

2

3

.

2 At time T, the non renewable resource is exhausted. The exploitation of the resource is stopped at the level y*=c* before jumpiçng down to zero. The consumption level is a continuous time function at time T, which means that the exploitation of the inexhaustible substitute swaps with the exhaustible resource use, jumping from 0 up to x*=c* at time T . Note however that the leisure level should jump downwards, since the inexhaustible substitute requires more labor (see the second black arrow). Hence the utility level also jumps downwards at time T. But now this is because labor has to increase, that is leisure has to decrease, in order to sustain the same consumption level with a more costly resource.

.

3 _{After time T, consumption and leisure remain constant forever. Note that the corresponding}

utility level is the lower bound of the utility levels of the optimal trajectory. That is, over an infinite duration, the society will remain in the poorest situation in terms of welfare.