Intergenerational ethics in a
resource context
Lectures in resource economics
Spring 2004, Part 2
G.B. Asheim, nat.res. 2, updated 28.01.2004
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A two-consumer economy
C A : Well - being of consumer A
U
U (C )
B
C : Well - being of consumer B
U (C ) : Utility derived from C
U A = U (C A ) : Utility of consumer A
C
U B = U (C B ) : Utility of consumer B
A utilitarian social welfare function: W = U A + U B
A Rawlsian social welfare function: W = min{U A ,U B }
Maximin: Maximize the well-being of the worst-off.
Leximin: Maximize the well-being of the worst-off; if tie between
two distributions, maximize the well-being of the second worst-off.
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G.B. Asheim, nat.res. 2, updated 28.01.2004
UB
Illustration
Utilitarian optimum
Rawlsian optimum
Properties:
1. Strongly Pareto
efficient
2. Equitable in the
sense that a
permutation
would lead to the
same welfare level
UA
G.B. Asheim, nat.res. 2, updated 28.01.2004
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Intergenerational distribution
Ct : Well - being of generation t
An infinite number of generations: 0, 1, K , t , K
U t = U (Ct ) : Utility of generation t
A social welfare function W = W (U 0,U1 , K, U t , K)
represents social preferences over intergen. distributions.
What properties should and can such preferences satisfy?
„
„
„
„
„
Efficiency (Strong Pareto)
Equity (invariance for finite permutations)
Completeness and continuity
Dynamic consistency
Unit comparability
G.B. Asheim, nat.res. 2, updated 28.01.2004
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Properties for social preferences
Equity
Efficiency
Continuity
Completeness
UC
DC
G.B. Asheim, nat.res. 2, updated 28.01.2004
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Maximin satisfies …
W = inf U t
t ≥0
Rawls (1971)
Solow (1974)
Equity
Efficiency
Continuity
Completeness
UC
DC
G.B. Asheim, nat.res. 2, updated 28.01.2004
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2
Leximin satisfies …
Equity
Efficiency
Continuity
Completeness
UC
DC
G.B. Asheim, nat.res. 2, updated 28.01.2004
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No social preferences satisfy …
Equity
Efficiency
Continuity
Completeness
UC
DC
G.B. Asheim, nat.res. 2, updated 28.01.2004
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Only the infinite future matters if …
W = lim U t
t →∞
Equity
Efficiency
Continuity
Completeness
UC
DC
G.B. Asheim, nat.res. 2, updated 28.01.2004
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3
Undiscounted utilitarianism satisfy …
T
max ∑ U t when T → ∞
t =0
Ramsey (1928)
von Weizsäcker (1965)
Equity
Efficiency
Continuity
Completeness
UC
DC
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G.B. Asheim, nat.res. 2, updated 28.01.2004
Discounted utilitarianism satisfy …
∞
∞
1
Ut
t
t = 0 (1 + ρ )
Koopmans
Equity
(1960)
W =∑
Efficiency
W = ∫ U t e − ρt dt
0
Continuity
Completeness
UC
DC
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G.B. Asheim, nat.res. 2, updated 28.01.2004
Two models of optimal growth
One sector model: ∞
(a)
max ∫ U (Ct )e − ρt dt subject to
0
Dasgupta-HealSolow model: ∞
(b)
K& t = Q( K t ) − Ct
K& t = Q( K t , Rt ) − Ct
− ρt
max ∫ U (Ct )e dt subject to
0
Dasgupta & Heal (1974)
S&t = − Rt
∞
S = ∫ Rt dt
0
Assumptions on the technology: Q( K t , Rt )
Consequences of discounted utilitarianism.
G.B. Asheim, nat.res. 2, updated 28.01.2004
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rate of conDiscount rate vs. interest rate sumption
growth
Q′(K )
∂Q( K , R )
∂K
interest rate = ρ −
U ′′(C )C C&
⋅
U ′(C ) C
elasticity of
marginal utility
The consumption interest rate is … U
U (C )
… greater than the utility discount
rate if positive consumption growth
… smaller than the utility discount
rate if negative consumption growth
C
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G.B. Asheim, nat.res. 2, updated 28.01.2004
Consequences of maximin in the Dasgupta-HealSolow model (Solow, 1974)
K& t = Q( K t , Rt ) − Ct
(b) max inf U (Ct )
t ≥0
subject to
S&t = − Rt
∞
Leads to constant utility.
S = ∫ Rt dt
0
Hartwick’s rule:
(Hartwick, 1977; Dixit, Hammond & Hoel, 1980)
∂Q( K t , Rt )
K& t =
Rt
∂Rt
… holds at all times: Accumulation of manmade
capital equals the value of resource extraction
G.B. Asheim, nat.res. 2, updated 28.01.2004
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