lecture 7 and 8 the efficient and optimal use of natural resources

Lecture 7 and 8

The efficient and optimal use of natural resources

Objective for the two morning lectures are:

• to develop a simple economic model built around a production function in which natural resources are inputs into the production process;

• to identify the conditions that must be satisfied by an economically efficient pattern of natural resource use over time;

• to establish the characteristics of a socially optimal pattern of resource use over time in the special case of a utilitarian social welfare function.

The economy and its production function:

,Q Q K R

, where , , 0.Q AK R A

,

where A, , , >0, + 1, 1 0.

Q A K R

The simple optimal resource depletion model

Production function with two inputs

For example Cobb-Douglas PF

For example CES PF

The natural resource R is essential, if Q = Q(K,R=0) = 0.

Elasticity of substitution, , between capital and resource:

Proportional change in the ratio of capital to the resource in response to a proportionate change in the ratio of the marginal products of capital and the resource, conditional on total output remaining constant.

Q = constant , , .K R

R KK R

d K R d Q QQ Q R Q Q K

K R Q Q

Elasticity of substitution, , ranges between zero and infinity.

K

R0

Q = Q

=

0 < <

= 0

Q = QQ = Q

Figure 14.1 Substitution possibilities and the shapes of production function isoquants (Perman et al.: page 475)

Production function isoquant is the locus of all combinations of inputs which, when used efficiently, yield a constant level of output:

= 0: no input substitution possible, (Leontief PF)

= : perfect substitution possible

0 < < : imperfect substitution possible

The social welfare function and an optimal allocation of natural resources

General social welfare function (SWF): 0 1 2, , ,..., TW W U U U U

Utilitarian SWF: 0 0 1 1 2 2 ,... T TW U U U U

1 2

0 2 ,...1 1 1

TT

U U UW U

0

tt

t

t

W U C e dt

Utility in each period a concave function of consumption level in that period, Ut = U(Ct):

In continuous time:

Objective: what properties for natural resource allocation must be satisfied, that the allocation maximizes a social welfare function.

0

0

t

tS S R d

Two constraints must be satisfied to obtain maximum social welfare.

1. All resource stock is to be extracted and used

Non-renewable resource with a fixed and finite initial stock,

total use of resource over time constraint to the initial stock,

S0 initial stock,

Rt extraction and use of resource at time t.

Resource use constraint:

Constraint states, the stock remaining at time t (St) is equal to the initial stock (S0) less the amount of resource extracted over the time interval from zero to t. The change of the stock of with respect to time is the first derivative of S t:

, t t tS R S dS dt

t t tK Q C

0

0

T t

t T T

T

K K Q C dT

Capital use constraint:

Change in capital:

2. The capital can be either used for consumption or production.

Now, we go to output. Output, Q, is produced through a production function involving two inputs, capital, K, and a non-renewable resource, R: Qt = Q (Kt, Rt). Substituting in the equation for the change in capital stock:

,t t t tK Q K R C

Using the information, we can find the socially optimal intertemporal allocation of the non-renewable resource. The objective is to maximize the economy’s social welfare function subject to the non-renewable resource stock-flow and the capital stock flow constraint.

0

tt

T

t

W U C e dt

Objective function:

Constraints: ,t t t tK Q K R C ,t tS R

Objective: select values for the choice variables Ct and Rt for t = 0, . . ., to maximize social welfare, W.

The rationale of the maximum principle

Optimal - control theory has as its foremost aim the determination of the optimal time path for a control variable, u.

=> once the optimal time-path of the , u*(t), has been found , we can also find the optimal state path, y*(t), that corresponds to it.

What makes a variable a “control”variable? How does it fit into dynamic optimization?

Simple example with a non-renewable resource S with S(0) = S0.

,t tS R

R(t) denotes the rate of extraction at time t. R(T) is control variable, because it posses the following two properties:

1. Subject to our discretionary choice.

2. Our choice impinges upon the variable S(t) which indicates the state of the resource at every point in time.


The simplest problem of optimal control:

0

Maximize , ,

subject to , ,

0 free (A, T given)

and for all 0,

T

V F t y u dt

y f t y u

y A y T

u t t T

A

bounded control setA :


The Maximum Principle

Variables and equations:

• control variables, u: variable that can be influenced by the DM, like extraction of resource

• state variables, y: indicate the state of a variable, like stock of resource, S, over time

• equations of motion, dy/dt: provide the mechanism whereby our choice of the control variable can be translated into a specific of movement of the state variable

• co-state variables: measure the shadow price of an associated state variable and can take different values at different point in times.

• Hamiltonian function: the vehicle through which the costate variable gains entry into the optimal control problem. Denoted by H, the Hamiltonian is defined as:

, , , , , , ,H t y u F t y u t f t y u

uMax , , , for all 0,

[equation of motion for y]

[equation of motion for ]

T 0 [transversality condition]

H t y u t T

Hy

H

y

The maximum principle conditions are:


The first line means that the Hamiltonian is to be maximized with respect to u alone as the choice variable. A different way of writing this condition is:

*, , , , , , for all 0,H t y u H t y u t T

0

0

Maximize , ,

subject to , ,

0 (given)

T

V F t y u dt

y f t y u

y y

Explaining the rationale of the maximum principle.

Assuming variable u is unconstrained, u* is an interior solution.

Initial point fixed, terminal point is allowed to vary.

The problem is then:

First step: include equation of motion in objective function. Note, that we get:

0

, , 0T

t f t y u y dt

The objective function can be augmented without changing the solution:

0

0

, ,

, , , ,

T

T

V t f t y u y dt

F t y u t f t y u y dt

V

will have the same value as V, as long as the equation of motion is adhered to at all times.

Substituting the Hamiltonian function in :

It is important to distinguish between the second term in the Hamiltonian, (t)f(t,y,u) on the one hand and the Lagrange multiplier expression on the other hand. The later explicitly includes dy/dt, whereas the former does not. Integrating the last term of by parts, we get:

0

0 0

, , ,

, , ,

T

T T

H t y u t y dt

H t y u dt t ydt

V

0

0 0

0T T

Tt ydt T Y Y y t dt

, , , , , , ,H t y u F t y u t f t y u Remember:

Substituting in the expression:

The new objective function :

0

0 0

0T T

Tt ydt T Y Y y t dt

2 3

1

0

0

terminal initial

, , , 0T

TH t y u y t dt T Y Y

V

We have completed the first step and included the equation of motion in the objective function.

Second step: the Lagrangian multiplier differs from the other variables y and u. The choice of the path (t) will have no effect on as long as the equation of motion is adhered to, that is as long as:

for all 0,H

y t T

This a necessary condition for the maximization of , as otherwise we still would have to worry about .

Third step: We now turn to the u(t) path and its effect on the y(t) path. If we have a known u*(t) path, and if we perturb the u*(t) path with a perturbing curve p(t), we can generate “neighboring” control paths for each value of :

*u t u t p t

*y t y t q t

*T T T *T T Ty y y

implying and TT

dT dyT y

d d

According to the equation of motion, there will then occur for each a corresponding perturbation in the y*(t) path. The neighboring y paths can be written as:

We also have, if T and yT are variable:

We can know express in terms of ,so that we can apply the first order condition d/d=0. If we do that we have identified u* and y*. The new version of is:

* * *

0

0

, ,

0

T

T

H t y q t u p t y q t dt

T Y Y

V

We now have completed step three.

Fourth step: We now apply the condition d/d=0. Considering for the total derivative of a definite integral :

0

T

t T

H H dTqt pt q t dt H y

y u d

V

, , , ,b x b x

a a

dK FK x F t x dt t x dt F b x x b x

dx x

From:

* * *

0

0

, ,

0

T

T

H t y q t u p t y q t dt

T Y Y

V

We get for the integral:

For the derivative of the second term, applying the product rule, we get from:

TT T T

d Tdy dTT y T y y T T

d dT d

Tt T

dTy T y T

d

* * TT

dT dyT y

d d

The following:

The term (0)y0 drops out in differentiation. We can also substitute in the result for the integral using the result above:

0

0T

Tt T

d H Hqt pt dt H T T y

d y u

V

We get as the result of d/d:

0

d

d

T

t T

T T T

H H dTqt pt q t dt H

y u d

T y T T y y T T

V

After rearrangement we get for the first-order condition d/d =0:

The three components of the derivative relate to arbitrary things: The integral contains arbitrary perturbing curves p(t) and q(t), the other two involve T and yT. Each of them must be zero to fulfill dV/d =0.

0

0T

Tt T

d H Hqt pt dt H T T y

d y u

V

and 0H H

y u

0T

By setting the integral to zero we get:

The first results gives us the equation of motion for the costate variable .

The second represents a weaker form of the “MaxH”. Weaker in the sense that it is predicted on the assumption that H is differentiable with respect to u and there is an interior solution.

Under a fixed T is the term T zero. Under a free yT, (T) has to be zero (Transversality

Condition):

0

, , , :T

H t y u t y dt V

H

y

0T

To summarize, we have the following optimality condition for

for all 0,H

y t T

0

H

u

We can now go back to our problem:

0

tt

T

t

W U C e dt

Objective function:

Constraints: ,t t t tK Q K R C ,t tS R

Objective: select values for the choice variables Ct and Rt for t = 0, . . ., to maximize social welfare, W.

The current value Hamiltonian for this problem is:

, : costate variables

tC t t t t t t

t t

H U C P R Q C

P

After substituting from the PF:

Necessary conditions for a maximum with control variables C and R:

,tC t t t t t t tH U C P R Q K R C

,

1

, 2

0

0

,

,

t

t

t

t

t

CC t

t

Ct t R t

t

C tt t t t t

t

C tt t t t K t t t

t

HU

C

HP Q

R

HP P P P P e

S

HQ e

K

Economic interpretation of the results:

,

1.

2. tC t

t t R t

U

P Q

1. In each period the marginal utility of consumption has to be equal to the shadow price of capital. A marginal unit of output can be used for consumption or be added to the capital stock. An efficient outcome will be where the marginal benefits of using one unit of output for consumption equal those for adding to the capital stock in utility units.

2. The value of the marginal product of the resource must be equal to the marginal value (or shadow price) of the natural resource stock. The shadow price is P t in utility units.

Economic interpretation of the results:

, ,

3. or

4. - or

t

t tt

tt t K t t K t

t

PP P

P

Q Q

3. The result state, that the growth rate of the shadow price of the natural resource that is its own rate of return, should equal the social utility discount rate.

4. The return to physical capital, its capital appreciation plus its marginal productivity, must equal the social discount rate.

3. or t

t tt

PP P

P

The result is also known as Hotelling’s rule, which has several interpretations. It is an intertemporal efficiency condition, which must be satisfied by any efficient process of resource extraction.

We get a second interpretation of the rule by integrating 3.

Hotelling’ s Rule

0 0 or t tt tP P e P Pe

The discounted price of the natural resource is constant along an efficient resource extraction path.

Pt

P0B

P0A

t

PtA = P0

A et

PtB = P0

B et

Figure 14.3 Two price paths, each satisfying Hotelling’s rule (Perman et al.: page 486)

Hotelling’s rule is necessary but not sufficient for an optimal price path of the resource

An expression for the growth rate of consumption along the optimal time path can be obtained by combining solution one and two.

1. Differentiate solution one according to time

The Growth Rate of Consumption

Substitute the result in solution four:

1. ; t

tC t

UCU U C C

t t

,4. - t t K t tQ


Substitute the result in solution four: ,4. - t t K t tQ

- -

--

1-

K K

KK

K

U C C Q CU Q

CU C U C QCU C U C Q

C C

CQ

U C CC

U C


Consider the elasticity of marginal utility with respect to consumption to the result for the growth rate of consumption:

and we get

1- K

K

U C U C U C C

C C U C

C QQ

C


As we consider a normal behaved utility function, will be positive it follows:

0 C increases when pay-off greater than impatience

0 C is constant when pay-off equals impatience

0 C is decreasing when pay-off less than impatience

K

K

K

CQ

C

CQ

C

CQ

C

Another economic interpretation of the Maximum principle

Consider:

Resource owner who wants to maximize profit over time interval [0,T]

K: capital stock, state variable

u: decision that resource owner has to make at each moment in time, control variable

K0: initial stock of capital

(t,K,u): at any moment of time, the profit of the owner depends on the amount of capital K she currently holds and the policy u it currently selects.

the policy selection has an impact on the rate of capital K change over time

, , :K f t K u

0

Maximize , ,T

t K u dt

0 0

subject to , ,

and 0 free ( , T given)

and for all 0,

K f t K u

K K K T K

u t t T

A

Optimal control problem:

,H

K

0,T , for all 0,H

K t T

0,

H

u

, , , , , , ,H t K u F t K u t f t K u Hamiltonian:

Optimality conditions:

* * * * * * * * *0

0

, , , 0T

H t K u K t dt T K T K

Control variable u and state variable K have an economic interpretation.

=> Economic interpretation of costate variable?

We know:

Partial differentiation of * with respect to given initial and optimal terminal:

* *

* **

0

0 and TK K T

Interpretation:

* *

* **

0

0 and TK K T

*(0): shadow price of a unit of initial capital K0.

*(T): shadow price of a unit of terminal capital stock K*(T).

Question: Why is negative? *

*K T

Answer: If we wished to preserve one more unit (use up one unit less) of capital stock at the end of the planning period, then we would have to sacrifice our total profit by the amount *(T).

=> In general, then, *(t) for any t is the shadow price of capital at that particular point of time.

Question: Why is negative?

*

*K T

* * * * * * * * *0

0

, , , 0T

H t K u K t dt T K T K

Going back:

current profit based on current policy

, , , ,H t K u t f t K u

Hamiltonian:

shadow price of one unit of capital

rate of change of physical capital corresponding to

policy u

current-profit effect future-profit effect

Optimal policy u requires:

0H f

tu u u

ft

u u

marginal future-profit effectmarginal current-profit effect =

Equations of Motions

a) , ,

b) or

K f t K u

H ft

K K Kf

tK K

marginal contribution of capital to current

profit

rate of depreciation of shadow price

marginal contribution of capital to increase in

future profits

Transversality Conditions

T free T 0K

shadow price of capital should be driven to zero at terminal time T.

=> only profits made in within the period [0,T] matter!

0

Maximize , ,T

tV G t y u e dt

0 0

subject to , ,

and 0 free ( , T given)

and for all 0,

y f t y u

y y y T y

u t t T

A

Optimal control problem:

Current Value Hamiltonian

, , , , tF t y u G t y u e

Present Value Hamiltonian

, , , ,tH G t y u e f t y u

, , , ,

, , , , , ,

t t

t t t

He G t y u e f t y u

He G t y u mf t y u m e me

, , , ,tcH He G t y u mf t y u




equation of motion for state variable y: cHy

m

equation of motion for co-state variable : H

y

equation of motion for :H

y

1.

t

t tme

me met

2. tcHHe

y y

t t tc

c

Hme me e

y

Hm m

y

equation of motion for m:



transversality conditions:

0 0

0

t

t T

T

T me

m T e

0 0

0

tct T t T

tc t T

H T H e

H e

, and ,G G y u f f y u

Autonomous Problem

0

Maximize ,T

tV G y u e dt subject to ,

and boundary conditions

y f y u

, and ,G G y u f f y u

Autonomous Problem

*

, ,

c c c c c

c

H G y u mf y u

H H H H Hy m m y y

t y m y m

Hmy m m

y

Hc* is not constant over time.

lecture 7 and 8 the efficient and optimal use of natural resources

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