ere5: efficient and optimal use of environmental resources a simple optimal depletion model...
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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
= 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