ERE5: Efficient and optimal use of environmental

resources• A simple optimal depletion model

– Resource substitutability– Static and dynamic efficiency– Hotelling‘s rule– Optimality– An example

• Extraction costs• Renewable resources• Complications

Last week

• Efficiency and optimality– Static efficiency– Optimality– Dynamic efficiency and optimality– Market efficiency

• Market failure & public policy– Externalities– Public policy

Exhaustion, Production & Welfare

• One can construct production functions with various degrees of essentialness, but the question is of course empirical

• The question whether it matters that a resource gets exhausted depends on its substitutability; or, if you can‘t do without, don‘t lose it

• So far, we have not run out of anything essential, but that does not mean we won‘t

• Human ingenuity is the ultimate resource, but also works to create problems

QQ

= 0

QQ QQ

0 < <

=

K

R0

Substitution possibilities and the shapes of production function isoquants

Substitutability and Scarcity

• Feasibility of sustainable development depends on – Substitutability– Technical progress– Backstop technology

• The magnitude of substitution possibilities– Economists: relatively high– Natural scientists and ecologists: limited

• However, it matters what services we look at

Optimal Resource Extraction - Discrete Time

, 0

( )max

(1 )t t

ttC R t

U CW

1

1 1 00

t

t t tS S R S R

1 1 1t t t tK K Q C

( , )t t tQ Q K R

Extraction:

Social welfare function:

Investment:

Production function:

1t tS R

1 1t t tK Q C

Optimal Resource Extraction – Discrete Time (2)

0

1 1 1 1 10 0

( )(1 )

tt

t

t t t tt t t t tt t

U CL

S S R K K Q C

0 1

0 1

0; 0;...; 0;...

0; 0;...; 0;...

t

t

L L LC C C

L L LR R R

0 1

0 1

0; 0;...; 0;...

0; 0;...; 0;...

t

t

L L L

L L L

Optimal Resource Extraction – Continuous Time

,0

max ( ) dt t

tt

C Rt

W U C e t

t tS R

t t tK Q C

( , )t t tQ Q K R

Social welfare function:

Extraction:

Investment:

Production function:

00

dt

tK K Q C

00

dt

tS S R

The Maximum Principle

• J depends on control variables (u), state variables (x), and time

• State variables describe the economy at any time; the equation of motion governs its evolution over time

• Control variables are time-dependent policy instruments

• To obtain the solution we construct a current value Hamiltonian

0

max ( , , ) dt

uJ L x u t e t

0 0: ( , , ); ( )x

x f x u t x t xt

Subject to:

Objective function:

The Hamiltonian

• The Hamiltonian only contains the current state and controls; current optimality is a necessary condition for intertemporal optimality

• The co-state variables () secure intertemporal optimality; they are like Lagrange multipliers, indeed measure the shadow price

( , , , ) ( , , ) ( , , )H x u t L x u t f x u t

0 and

H Hu x

FOC:

Optimal Resource Extraction - Continuous Time (2)

( ) ( ) ( ( , ) )t t t t t t t tH U C P R Q K R C

0t

ttC

t

HU

C

0t

tt t R

t

HP Q

R

tt t t

t

HP P P

S

t

tt t t t K

t

HQ

K

Social welfare function: ,

0

max ( ) dt t

tt

C Rt

W U C e t

Equations of motion: and t t t t tS R K Q C

Hamiltonian:

Necessary conditions:

Static Efficiency

• Marginal utility of consumption equals the shadow price of capital

• Marginal product of the natural resource equals the shadow price of the resource stock

0t t

tt tC C

t

HU U

C

0t t

tt t t tR R

t

HP Q P Q

R

Dynamic Efficiency

• The growth rate of the shadow price of the resource equals the discount rate

• The return to capital equals the discount rate

tt t t t

t

H PP P P P

S P

t t

tt t t t K K

t

HQ Q

K

Hotelling‘s Rule

• Dynamic efficiency required:

The growth rate of the shadow price equals the discount rate

• An alternative interpretation:

The discounted price is constant along an efficient resource extraction path

• Thus, environmental resources are like other assets

PP

0 *t t

t t t t tP P P Pe Pe P

Growth Rate of Consumption

• The growth rate of consumption along the optimal time path:

• Since η>0, consumption grows if the marginal product of capital exceeds the discount rate

• The intuition:

tK

t

QCC

0 ; 0 ; 0K K K

C C CQ Q Q

C C C

P0A

P0B

Pt

PtA = P0

Aet

PtB = P0

Bet

t

Hotelling‘s rule and Optimality

Extraction Costs

,0

max ( ) dt t

tt

C Rt

W U C e t

t tS R

( , ) ( , )t t t t t tK Q K R C G R S

( , )t t tQ Q K R

( , )t t tG G R S

Social welfare function:

Constraints:

Production function:

Extraction costs:

S0

(iii)

Gt

(for given value of Rt = )R

(i)

0

(ii)

Remaining resource stock, St

Extraction Costs and Resource Stock

Extraction Costs –2 Hamiltonian:

Necessary conditions:

( ) ( ) , ,t t t t t t t t t tH U C P R Q K R C G R S

0t

ttC

t

HU

C

0t t

tt t tR R

t

HP Q G

R

t

tt t t t S

t

HP P P G

S

tt t t KQ

Resource Price

• Net price = Gross price – marginal extraction cost

• Gross price = Marginal contribution to output, income (measured in utils)

• Net price = Marginal value of the resource in situ

• Net price = Rent = Royalty

0t t

t t

tt t tR R

t

t t tR R

HP Q G

R

P Q G

Hotelling‘s Rule -2

• The growth rate of the shadow price of the resource is lower if extraction costs rise with falling resource stocks

• The discount rate equals the rate of return of holding the resource, which equals its price appreciation plus the foregone increase in extraction costs

tt t t SP P G

tt St

t t

GPP P

Net price Pt

Time t

Time t

P0

Pt

T

T

R R0

Area =

= total resource stock

S

Rt

Demand

PT =K

45°

Graphicalsolution

Renewable Resources

,0

max ( ) dt t

tt

C Rt

W U C e t

( )t t tS S R

t t tK Q C

( , )t t tQ Q K R

Social welfare function:

Production function:

Constraints:

Renewable Resources -2Hamiltonian:

Necessary conditions:

( ) ( ( ) ) ,t t t t t t t t tH U C P S R Q K R C

0t

ttC

t

HU

C

0t

tt t R

t

HP Q

R

t

tt t t S

t

HP P P P

S

t

tt t t t K

t

HQ

K

Hotelling‘s Rule -3

• The growth rate of the shadow price of the resource is lower for renewable resources

• The discount rate equals the rate of return of holding the resource, which equals its price appreciation plus the increase in the resource growth

tt t t SP P P

t

tS

t

PP

Complications

• The total stock is not known with certainty• New discoveries increase the known stock• There is a distinction between physical

quantity and economically viable stock size

• Technical progress and R&D• Heterogeneous quality • Extraction costs differ• Availability of backstop-technology