JOURNAL
OF ECONOMIC
THEORY
Unjust
54,
350-371
(1991)
Intergenerational
Allocations
GEIR B. ASHEIM*
The Norwegian
Received
School
N-5035
of Economics and Business
Bergen-Sanduiken.
Norway
March
1.5, 1988; revised
December
Administration,
17, 1990
An intergenerational
allocation
is detined to be unjust if there is a feasible allocation with more total consumption
and less relative inequality.
Unjust allocations
are characterized
in technologies
satisfying
certain
regularity
conditions.
After
ruling
out unjust allocations,
the consequences
of letting
generations
choose
according
to a standard
form of altruistic
preferences
are explored
in to particular
classes of technologies.
A connection
between
excluding
unjust allocations
and
maximizing
the welfare of the worst off generation
is established in these technologies.
Journal CfEronomic
Liturarure
Classification
Number:
D63.
‘1’ 1991 Academic Press, Inc.
I. INTRODUCTION
It has been the purpose of several writers (see, e.g., Page [ 191 for verbal
arguments and Ferejohn and Page [ 141 for an axiomatic analysis) to point
out that the utilitarian criterion with positive discounting may not be an
appropriate criterion for intergenerational justice. A particularly disturbing
outcome occurs in natural resource models where the criterion for any
positive discount rate may force consumption to eventually approach zero
even if unbounded consumption growth is feasible (see Dasgupta and Heal
C8, 91).
Ferejohn and Page [ 14, p. 2741 write:
Our result suggests that the research for a
Instead of searching for the “right”
number.
look to broader
principles
of social choice
equity. Once found, these principles
might
counting procedure
to rule out gross inequities
with a “low” discount
rate.
“fair”
rate of discount
is a vain one.
“the” social rate of discount,
we must
to incorporate
ideas of intertemporal
be used as side conditions
in a disthat can arise with discounting,
even
* This research was initiated
during a visit to Stanford
University,
1985-1986.
version of the paper was presented
at the European
Public Choice Society and
Economic
Association
meetings in 1988. I thank an associate editor and a referee
Bjorn Sandvik.
H. A. A. Verbon, and Bengt-Arne
Wickstrom
for helpful comments.
support
by the Norwegian
Research Council
for Science and the Humanities
is
acknowledged.
350
0022-0531/91
Copyright
All rights
$3.00
:p 199 I by Acadenuc Press. Inc
of reproduction
m any form reserved
An earlier
European
as well as
Financial
gratefully
UNJUST
INTERGENERATIONAL
ALLOCATIONS
351
The present paper follows this program by employing a quasi-ordering
attributed to Sen [23] by Blackorby and Donaldson [4] to exclude allocations of consumption that are not desirable candidates for a social choice.
We call such allocations unjust. Loosely speaking, an allocation is unjust if
there exists another feasible allocation with more total consumption and
less relative inequality. Here we define this quasi-ordering
for infinite
consumption
sequences and demonstrate that in productive technologies
(implying that waiting is productive)
only efficient and nondecreasing
allocations remain after ruling out allocations that are unjust.
Generations are assumed to choose according to a simple recursive form
of nonpaternalistic
altruistic preferences, where the welfare of each generation is an additively separable function of its own utility and the welfare of
the next generation. This corresponds to the traditional utilitarian criterion
with positive discounting. However, generations are here assumed to be
required to subscribe to an overriding ethical principle, deeming unjust
allocations as socially unacceptable.
The implications of combining the exclusion of unjust allocations with
this standard form of altruism is explored in two particular classes of
technologies, viz. the usual one-sector technology as well as a resource
technology in which the unrestricted use of the utilitarian criterion with
discounting leads to undesirable outcomes. Both technologies are shown to
be productive, and consequently, the selected allocations are efficient and
in both cases the optimal allocations
nondecreasing.
Furthermore,
correspond
to outcomes that would arise if generations as an ethical
principle-instead
of excluding unjust allocations-had
maximized the
welfare of the worst off generation.
This approach-that
conceptions of intergenerational justice should be
evaluated by their implications in specific economic environments-is
in
principle supported by Koopmans [ 171, Mishan [IS], and Dasgupta and
Heal [9, pp. 308-31 l] as well as Rawls [20, p. 201. It is argued in this
paper that combining the exclusion of unjust allocations with altruism
yields desirable implications in the two chosen classes of technologies. In
particular, a trade-off exists between present and future consumption
so
that some degree of economic development is allowed without leading to
any gross inequalities. A dilemma posed by Epstein [ 131 (that an economy
has to choose between development and equity; it cannot have both) is
thereby apparently resolved. Moreover, in the two classes of technologies
considered, we obtain allocations in congruence with a view expressed by
Dasgupta and Heal [9, p. 3111, viz. that trading off present consumption
for future consumption
is more appropriate for poorer societies, while
equality considerations should dominate for richer ones.
One may argue that the above mentioned quasi-ordering
is uncontroversial only if each generation is egoistic in the sense that its welfare depends
352
GEIR
B. ASHEIM
solely on its own consumption.
Here, in contrast, each generation is
altruistic: its welfare depends in part on the welfare of the next generation.
However, there is an argument to be made in favor of distinguishing the
conception of justice applied in a society from the forces that are
instrumental in attaining it. Hence, the present paper may be seen to discuss whether altruism as a motivating force is able to implement a weak
conception of justice (by not leading to unjust alloations) in the classes of
technologies considered.
This resembles the distinction, made by Rawls [21], between a political
conception of justice (“the right”) and a religious, philosophical, or moral
doctrine (“the good”), where the right is assumed to set the limits within
which the good may operate. Fitting this distinction to the present analysis,
each generation’s consumption is to be interpreted as an indicator of its
objective well-being. The adopted conception of justice (the right) is concerned with the attainment of an equitable distribution of such well-being
and draws the limit by excluding allocations that are unjust.’ Each generation’s altruistic welfare, on the other hand, is to be interpreted as a
representation of its subjective preferences in which the doctrine that each
generation should care about the welfare of its immediate successors (the
good) has been internalized.*
Still, the above mentioned correspondence between excluding unjust
allocations and maximizing minimal welfare means that this particular conception of justice can be reformulated as a restriction on the distribution of
altruistic welfare, provided attention is confined to the kind of altruism and
technologies considered.
In summary: the original contribution of this paper is to apply a weak
conception of justice as an ethical restriction in problems of intergenerational distribution, and to show that it-combined
with altruism-leads
to
equitable outcomes in two important classes of technologies. The paper is
organized as follows: The quasi-ordering defining unjust allocations is
discussed in Section 2. The consequences of excluding unjust allocations in
productive economies are explored in Section 3. After introducing altruistic
r The result that only efficient and nondecreasing
allocations
are not unjust in productive
technologies
means that this conception
of justice may be looked at as a normative
basis for
the present-day
goal of sustainabiliry
(WCED
1291).
’ Rawls [20, p. 1291 can be interpreted
as supporting
the view that altruism
should not
enter into the conception
of justice: “There is no inconsistency,
then, in supposing
that once
the veil of ignorance
is removed,
the parties lind that they have ties of sentiment
and affection,
and want to advance the interests of others and to see their ends attained.
But the postulate
of mutual disinterest
in the original position is made to insure that the principles
of justice do
not depend upon strong assumptions.
Recall that the original position is meant to incorporate
widely shared and yet weak conditions.
A conception
of justice should not presuppose,
then,
extensive ties of natural sentiment.”
UNJUSTINTERGENERATIONAL
ALLOCATIONS
353
preferences (Section 4) these results are then applied to a one-sector
technology (Section 5) and a resource technology (Section 6).
2. THE QUASI-ORDERING
Consider a constant population economy where each generation lives for
one period. Let c,~ be a nonnegative scalar denoting the consumption of
generation s. Write .Vc= (c,, c, + , , ... ) and correspondingly for other sequences. Refer to .c as an allocation at time s, let .J, denote a truncated allocation (i.e., ,sc,= (c,, .... c,)), and let .,u, represent the mean consumption of
.sc, (i.e., sP, = Ck=, c,/(t - s + 1)).
The quasi-ordering we introduce in Definition 3 ranks one allocation as
high as another it it compares favorably both w.r.t. total consumption and
w.r.t. relative inequality. Hence, we need a quasi-ordering that ranks allocations according to total consumption as well as one that ranks according
to relative inequality. For total consumption the comparison relies on
von Weizacker’s [28] overtaking criterion.
DEFINITION 1. $ C +c” (,c’ catches up with ,,c” in finite time) iff there is
a 7 such that for all r> 7 ,p:>,p:‘.
For relative inequality
(see below ).
the comparison
relies on weak Lorenz-domination
2. For ,c’, sc” > 0,3 ,c’ E Sc” (,c’ is as egalitarian as sc”) iff
is a i such that for all t> i there is a bistochastic4
1)x (r-s+
1) matrix A, such that .c;/,~:=A,..c:/,~~.
DEFINITION
there
(r-s+
In order to interpret Definition 2, let .E, denote the permutation of ,c,
ordered according to increasing size (i.e., Fi < Zi+, , i = s, .... t - 1, where the
index i does not refer to time). Also, write ,c’ E sc” when the quasi-ordering
E ranks ,c’ strictly above .Vc” (i.e. , .c’ E ,c” iff ,c’ E,c”, but not sc” E .c’)
and correspondingly for other quasi-orderings. We can now give three
equivalent formulations of Definition 2.
LEMMA 1. For .c’ , Sc” > 0, $ E Sc” is equivalent to any of the following
three conditions: There is a i such that for all t > i
’ An allocation $c is nonnegative
( >, 0) if c, > 0 for all t > S, positiue
( $0)
if c, 10 for all
t 2 s, and semi-positive
( > 0) if ,c 2 0, with C,> 0 for some f.
’ A square matrix is bisrochasfic
if all its entries are nonnegative and each of its rows and
columns sums to one.
354
GEIR
(i)
(ii)
formations
B. ASHEIM
(c”i.+ ... +ti)/,p(:>(i’:‘+
... +?J)/,pL;)for
aNje {s, .... t);
,C:/,pi can be obtainedfrom $5: J,Yp:’ by a finite sequence of transof the form
(iii) xi=,
u: R, Y D-a.
,y;+ ’ = xf+e’<xj,
j> i,
.x;+ ’ = xj - e’3 xi,
e’> 0,
.x’+
n ’ = ?y’II,
if
u(cb/,p~)~C~=,
u(cz/,p:‘)
n#i,j;
for
any
concave
function
Moreover, the above equivalence is valid for ,$ E SC if, in addition, there
are infinitely many t such that (i) holds with strict inequality for at least one
j, the sequence of transformations
in (ii) is nonempty, and (iii) holds with
strict inequality for any strictly concave function u.
Proof:
Lemma 2, Dasgupta, Sen, and Starrett [lo].
1
By Lemma 1 (i), Definition 2 entails that for large t, the Lorenz-curve
associated with sc; lies nowhere outside the corresponding curve for .c:‘;
ie. ., sc; weakly Lorenz-dominates
sc:‘. By Lemma 1 (ii), this is equivalent
to ,c;/,,u; being obtainable from ,c:‘/,pr by a finite number of pairwise
transfers (from richer to poorer generations).
We are now able to define the quasi-ordering which will be our main
concern.
3. For ,c’, sc” > 0, .c’ J $ (sc’ is as just as .Yc”) iff .c’ C $
,cJO for all ~20;
.cJO iff .c>O.
DEFINITION
and ,c’E$‘.
Both C and E as well as J are easily seen to be reflexive and transitive;
thus, they are quasi-orderings. Some results are immediate.
2. (i) Form .c’ from $’ by multiplying Sun by a scalar K > 1
. ,c”). Then $_J ,c”.
(ii) Form ,c’ from sc” by transferring e > 0 from generation 5 to
generation t, where c: - c; > e. Then ,c’J #I.
LEMMA
($’ =
K
(iii) Form Sc’ from ,c” by interchanging the consumption
number of generations. Then $ J Sc” and $’ J .c’.
ProojY
(ii)
(iii)
$;=jf.
of a finite
(i) ,c’ C #’ and ,c’ E .$‘.
.c’ C $” and .c’ E,c“ by Lemma 1 (ii).
By Lemma 1 (i) since there is a I such that for all t > i
1
UNJUST
INTERGENERATIONAL
355
ALLOCATIONS
Property (ii) is often called the (strong) principle of transfers and was
originally formulated by Dalton [7]. Birchenhall and Grout [3] pointed
out that for the principle of transfers to be valid in the presence of an
infinite sequence of generations, it is necessary to define the quasi-ordering
by a procedure inspired by von Weizsgcker’s [28] overtaking criterion, as
we have done through Definitions l-3. Property (iii) shows that the quasiordering J is equitable in the sense of Svensson [26] and Epstein [ 133.
Let us further explore the welfare implications of the quasi-ordering J
by introducing two important quasi-orderings of infinite consumption
sequences.
DEFINITION
4 (Lexicographic
maximin).
.c’ L ,Yc” iff there is a 1 such
that for all t 2 2, either there is a Jo (s, .... t) such that c”j> 2,!’ and 2: = F,!
for all s d i <j, or ,e: = $:‘.
DEFINITION
5 (Ulilitarianism).
,sc’ U Sc” iff there is a 2 such that for all
t 3 2, xi=, u(ck) 3 CL=, u(cz), where u : R + -+ R is an increasing and
strictly concave function.
In order to analyze the relation between J and these quasi-orderings,
introduce the notion of a subrelation.
DEFINITION
6. Consider two quasi-orderings, R, and R,. R, is a
subrelation of R, iff (,c’ R, $’ * ,c’ R2 Sc”) and (,c’ & $’ * z~‘&z ,Yc”).
PROPOSITION
quasi-orderings
1. The quasi-ordering J is a subrelation of each of the
L and U (for any increasing and strictly concave function u).
Proof
Case 1. ,c’ and # are both semi-positive. Assume ,Sc’J #‘. Then, for
all t exceeding some i, there is a .c: and a bistochastic matrix A, such that
44 a.4
= SPL;
and
,c:I,/4 = ,cf~/sp: = A,. ,c:‘/,P;.
By Lemma 1 (i) [Lemma 1 (iii)], .c’ L ,Sc” CSc’ U Sc”]. Assume in addition
that ,c’J $‘. Then, either ,p; > ,p: for infinitely many t, or A, is not a permutation5 matrix for infinitely many t. By Lemma 1 (i) [Lemma 1 (iii)],
.c’ L 3cn [.,c’ _v$“I.
Case 2. $’ = 0 so that .c’ JO with ,VcJO only if .c’ > 0. That ,Vc’L 0
[,c’ U 0] for all .c > 0, with .c’ L 0 CSc’ U 0] if ,c’ > 0, follows directly
from Definition 4 [Definition 51. m
’ A permumtion
matrix
is a bistochastic
matrix
with entries
either
0 or 1.
356
GEIR B. ASHEIM
Proposition 1 states that if J ranks two allocations, then L and U (for
any increasing and strictly concave function U) will rank the allocations
accordingly. Proposition 1 therefore implies that J is fairly uncontroversial.
The quasi-ordering J is clearly not complete. In particular, it does not
rank two allocations if the one has less relative inequality, but does not
catch up with the other. In this sense, no trade-off between relative
inequality and total consumption is allowed. Hence, the ordering cannot be
used to select an optimal allocation relative to a feasibility constraint; i.e.,
we cannot find a .c’ such that ,Yc’J,$c for all feasible allocations ,<c.
However, we can determine a set of quasi-optima,
each element of which
has the property that there is no feasible allocation ranked strictly higher
by J. Given a set of feasible allocations, $, and an arbitrary quasiordering, Q, such quasi-optima will be referred to as Q-optima. Formally,
DEFINITION 7. ,c’ E 5 is a Q-optimum
.c E 9 such that ,c Q ,c’.
The notion
allocations.
DEFINITION
of a quasi-optimum
relative
to 9
iff there is no
may be used to exclude undesirable
8. >c E p is unjust relative to 9 iff .c is not a J-optimum
relative to 9.
Proposition
1 implies the following corollary.
COROLLARY.
If .c’ is an L-optimum,
or a U-optimum
increasing and strictly concave u), then .Yc’is a J-optimum.
(for some
This corollary entails that if egoistic generations were to choose the
distribution of consumption in ignorance of their own position (i.e., before
knowing in which sequence they would live, any sequence being equally
probable), then generations being risk-averse with respect to the level of
their own consumption would choose a J-optimal allocation-no
matter
their degree of risk-averseness-provided
the von Neumann-Morgenstern
axioms are satisfied (see Vickrey [27, p. 3291, Harsanyi [ 15, 161).
In the next section, we completely characterize the set of allocations that
are not unjust in a class of technologies satisfying certain regularity conditions.
3. PRODUCTIVE TECHNOLOGIES
A cake-eating technology is defined as follows: Let B”(yS) denote the set
of feasible allocations at time s when the size of the cake at time s is given
by a nonnegative scalar y,. We have that p’(y,) = { ,c 2 0: C,“= s c, d us}.
UNJUST INTERGENERATIONAL
357
ALLOCATIONS
PROPOSITION 2. In a cake-eating technology, there exists a J-optimum
relative to S”( y,) iffy,, = 0.
Proof. (Sufficiency.) Trivial since 9’(O) = {O}.
(Necessity.) We have to show that all feasible allocations are unjust
if y, > 0. First, note that 0 is unjust by Definition 3. If S’(J~,) 3 ,c’> 0,
there exist 5 and t such that c: > c;. Form .scE p”( v,) by c, = (1/2)(c: + c:)
for cr= T, t, and c, = c& otherwise. By Lemma 2 (ii), .c J ,,c’, which shows
that .c’ is unjust. 1
A cake-eating technology is unproductive: The cake cannot be invested
yielding positive net returns. Proposition 2 shows that in such a technology, available consumption cannot be allocated to an infinite number of
generations in a just manner (except for the trivial case where there is no
consumption available for any generation). We therefore turn to a class of
to their productiveness-have
nontrivial
technologies which-due
J-optima.
Let the vector yse R’; denote the n-dimensional state of the economy at
time s; n 3 1. Interpret Y.~as the vector of stocks available at the beginning
or period s. In the example of Section 5, the state is the one-dimensional
output produced in the previous period; in the example of Section 6, the
state is two-dimensional, consisting of output as well as a resource stock.
The set of feasible allocations at time s, denoted E(y,), is determined by
the state at time s y, and a correspondence E: R”+ + iw:. Let a sequence
of correspondences o9 = (&, F,, ...) be referred to as a technology6 if for
any feasible allocation at time s ,c’ EZQy,) there exists a corresponding
sequenceof states ,y’ E IR’Y ic, with y: = Y,~,such that the following holds at
each t>s: (a)(,c’)E%(y;)
and (b)(,c”)E~(y;)J(.~c:~,,
,c”)~fl$y,).
Consider the following conditions.
CONDITION 1 (Costless storage; by inefficiency, costlessaugmentation of
initial consumption). For any ,c’ E9JyX), there is a 6 B 0, strictly positive
iff J’ is inefflcient,’ such that (sc> 0 and CL = F(c, - ck) 6 6 for all t B s)
implies ,c Eps(y,).
CONDITION 2 (Costly acceleration of consumption). Let .c’ EE(y,) be
efficient, with corresponding sequence of states Jy’ $0. Then there is no
,c E 9>(ys) such that 1: = ,,(c, - CL)> 0 for all t 2 s, with strict inequality
for at least one t.
6If 09 is stationary(i.e.,3 = 9 for all I 2 0). then 3 is also referred lo as a technology.
7An allocation.YcEYY(y,)is inejjjcienr relativeto .e(y,) iff thereis an allocation,c’ E9&y,)
such that ,c’ > ,c (i.e..
inefficient.
,c’
>
,C
with
c; > c, for some t > s). An allocation
is efficienr
iff it is not
358
GEIR B. ASHEIM
3 (Existence of an efficient and stationary’ allocation). If
then there is an efficient, positive, and stationary allocation
If y, E y \ q + 9 then ,c EFS(y,) implies that C,“=,Yc, is finite.
CONDITION
Y.,EK+,
,c E E(Y.J
In order to explain Conditions 1 and 2, form ,c from .c’ EPJy,) by
transferring e > 0 from generation f to generation z. First, assume7 < t so
that consumption is stored. If PSsatisfiesCondition 1, then .c ES$(y,) since
CL=, (c, - cb) = --e for t E {z”, .... T - 1} and ckzs (c, - cb) = 0 otherwise.
Hence, Condition 1 entails that consumption can be postponed without
cost. Next, assume t <? so that consumption is accerelated. If E satisfies
Condition 2 and ,c’ is an efficient allocation with corresponding sequence
of states 5yP 0, then ,c 4 FS(yS) since x: = ,s(c, - rk) = e for t E (z, .... T- 11
and CL = s (c, - c&) = 0 otherwise. Hence, Condition 2 entails that the
transfer of consumption from a later to an earlier generation is costly in the
sensethat the later generation has to give up more than the earlier one
receives.
A technology o9 satisfying Conditions l-3 for any s > 0 will be referred
to as a productive technology. A cake-eating technology does not satisfy
Conditions 2 and 3. For illustrative purposes, consider technologies
satisfying only two out of the three conditions above. In order to facilitate
these examples, introduce the concept of a linear technology o9-i. defined
by an exogeneously given positive price sequence op. We have that
F:(yS)= ($20:
Cc=, pack ~:,~y,.), where y, is a nonnegative scalar
and where .y’ is defined recursively by y: =y, and p, ,: =P, ._,(y: ~~,- c: ~ ,)
for t > S. This technology is productive iff p, >p, + , for all t 2 0 and
C:=oPb<~~
EXAMPLE 1. A linear technology o9’ with P, >P,+ , for all t 2 0 and
CF’, pb diverging, satisfies only Conditions 1 and 2.
EXAMPLE 2. A linear technology oFz with p,, =pl, p, >p,+ 1 for all
t 3 1, and C,“=, pg < co, satisfies only Conditions 1 and 3.
EXAMPLE 3. Let ,P3 be a linear technology with p, = a .p, + , for all
t 3 1; a > 1. Let Yi(y,)
denote the following transformation set (illustrated
in Fig. 1):
(k, y):O<y<a.kforkE
[ 0j y,n> u [f,v,]and
}.
aAn allocation,c is
s/a/ionar~v
iff c, = c, _ , for all I > s.
UNJUST
INTERGENERATIONAL
359
ALLOCATIONS
k
FIGURE
Define Fi(y,)
1
by
~~(yo)={,c~O:~(~o,y,)s.t.c,=y,-~o,(~,,I’,)~~~(y,)
and Ic E S:(.V,)}.
The resulting technology OF3 satisfies Conditions 2 and 3, but not Condition 1 since, given an inefficient allocation with cO= y, - y,/cr, one need not
be able to augment the consumption of generation 0 without reducing the
consumption of a later generation.
The following three propositions fully characterize the set of J-optima in
productive technologies.
PROPOSITION3.
is productive,
9 An allocation
Zj” ,c’ E Ts(y,) is a J-optimum
then .c’ is nondecreasing.g
,c is nondecreasing
iff c, < c, + I for all / > s.
relative
to z.(ys),
and ,+F
360
GEIR B. ASHEIM
ProoJ: Suppose c; > c; + , for some t. By Condition 1, there is a
.c EK(y,) formed by $’ by c, = (1/2)(c; + c: + , ) for 0 = t, t + 1, and c, = CL
otherwise. By Lemma 2(ii), .cJ .c’, which contradicts that .c’ is a
J-optimum. a
By the proof of Proposition 3, if ,c’ were not nondecreasing, a carefully
chosen postponement of consumption would decrease relative inequality
without reducing total consumption, thereby producing a J-improvement.
Hence, by Condition 1, in a productive technology a J-optimal allocation
is nondecreasing. This result also implies Proposition 4, stating that in a
productive technology with a positive initial state a J-optimal allocation is
efficient. Becauseif it were not, by Condition 1, the consumption of the first
least favored generation(s) could be augmented without cost, yielding less
relative inequality and more total consumption.
PROPOSITION 4. Let y.,~lX”++. If ,c’ l z.(y,) is a J-optimum relative to
R(Y,~), and ,,S is productive, then ,c’ is efficient.
ProoJ Suppose .c’ is a J-optimum that is not efficient. By Proposition 3, .c’ is nondecreasing. By Condition 3 and Lemma 2 (i), ,c’
is not stationary. Hence, ,c’ has a finite set (s, .., t> of least favored
generations: cl = c,:., i = . . = c:- 1= c,, < c,I + , . By Condition 1, for any
EE (0, 6/( t -s + 1)), given some 6 > 0, there is a .c Ee(y,) formed from .c’
by c, = cb + E for 0 E {s, .... t } and c, = CL otherwise. By Lemma 1 (ii),
,c_J,c’ (for any a>0 satisfying c:+E<c:+,)
since for s>t, sp,>,&,
and
Jc,/,pL, can be obtained from ,c:/,,& by a finite number of pairwise
transfers. This contradicts that .c’ is a J-optimum. 1
Conversely, if .Yc is efficient and nondecreasing, reducing relative
inequality requires that consumption be transferred from a later to an
earlier generation. However, by Condition 2, in a productive technology
this can only be achieved by reducing total consumption. Hence, as stated
in Proposition 5, in a productive technology with a positive initial state
there exists no feasible J-improvement of an efficient and nondecreasing
allocation.
is efficient and nonPROPOSITION
5. Let yS~ Iw; +. If $ Ee(y,)
decreasing, and &F is productive, then .c’ is a J-optimum relative to 9(y,).
Proof: By Condition 3, there is a f and an E> 0 such that c: 3 E for all
r > ?, and thus, yb E rW: + for all 0 > s. Hence, Condition 2 applies. Suppose
,c E9$(y,) satisfies .cJ .c’. Form ,E, and .I?; for any t 2 s. Then
C’,=, c,,aCI=, ?, for all Jo (s, .... t}, and ,c: = ,S: since .c’ is nondecreasing. It therefore follows from Definition 3 and Lemma 1 (i) that
UNJUST
INTERGENERATIONAL
ALLOCATIONS
361
there is a i such that t > i implies CL=,(c, - cb) b 0 for all je {s, .... t $. By
Condition 2, ,,c E e.(ys) implies .c = c’, which contradicts ,VcJ ,c’ 1
We have proven that if the technology is productive and the initial state
Y,~is positive, then the set of J-optima is identical to the set of efficient and
nondecreasing allocations. Note that Condition 3 implies that this set is
nonempty; it contains at least an efficient and stationary
allocation.
Example 1 above does not satisfy Condition 3. A proof similar to the one
of Proposition 2 shows that there exists no J-optimum relatve to YA(yo) if
y. > 0. In the case of Example 2 (which does not satisfy Condition 2) a
J-optimum
relative to F$yo)
requires that c, = c,; i.e., not all efficient
and nondecreasing allocations are J-optima. In the case of Example 3
(which does not satisfy Condition 1) there are inefficient J-optima. In
particular,
0c E Si( y,) with co = y. - .rO/sr and c, = c E ([a - 1] (~,/a) .
[l - ((a-a))la)“],
[a- l] . (y,,/c()) for all t 3 1 is inefficient, but still a
J-optimum,
since a reduction in relative inequality, involving increased
consumption at time 0, can be achieved only by reducing total consumption.
In productive linear technologies, the exclusion of unjust allocations is
related to Epstein’s
[ 121 Efficiency and Equity axioms: .?c’E 9f(y,)
satisfies Efficiency iff c,“=, p,c& = p,~, and Equity iff for all T, f > s,
p, >p, 3 c: d c:. Clearly, when @’ is productive, ,c’ satisfies Efficiency
and Equity iff .c’ is efficient and nondecreasing; i.e., iff ,c’ is J-optimal.
However, when o*’ is not productive, the Efficiency and Equity axioms
are not sufficient for J-optimality
because they admit .c with c, > c, for T,
t satisfying p, =P,. Such ,c are not J-optimal relative to 9f.l’
For y., 9 0 in a productive technology, the continuation at time r ( >s)
of a J-optimal allocation 5c’ E q(y,),
,c’, is J-optimal relative to *(y\),
since y: $0 and ,c’ is efficient and nondecreasing. However, by choosing an
alternative J-optimal allocation , c”E$(y:)
at time t, generation t may
destroy the J-optimality
relative to Ys(y,) of the resulting allocation
(*C , , ,c”) E ) E 9Jyz). This will occur when c:-, > c;‘. We will return to
this problem of time-inconsistency.
Having characterized the set of J-optima in productive technologies as
the set of efficient and nondecreasing allocations, one question remains:
Does the imposition of J-optimality lead to the selection of more desirable
allocations? Such an evaluation of J-optimality
as an ethical principle is
provided by the next three sections where the consequences of combining
J-optimality
with a simple recursive form of altruistic preferences are
explored in two classes of productive technologies.
I” In particular, in
and c,<c,+,
for all
the case of ,#’ all efficient allocations “c E 5(y,)
with
t > 2 satisfy Efficiency and Equity; i.e.. we need not have
cu < c2, Cl < c2,
cg = c,,
362
GEIR
4.
ALTRUISTIC
B. ASHEIM
PREFERENCES
Assume in the sequel that the subjective preferences of any generation s
are altruistic
and can be represented
by an ancestor-insensitive
and
stationary altruistic welfare function w: rWT r* R, defined by
W(,C)E
f b”--” .u(c,)=~(c,)+b.~‘(,+,c),
c7= s
where U: lh?, -+ LT.+ is a stationary
one-period
O<b<l,
utility function
24 is continuous,
strictly
increasing,
(u.1)
continuously differentiable at any c > 0.
(~2)
du/dc+
and
satisfying
concave;
it
is
cc as cl0
and where b is the utility discount factor. Note that the welfare function has
both a paternalistic and a nonpaternalistic representation, the latter being
of a simple recursive form. (See, e.g., Ray [22].)
If each generation maximizes M’constrained only by the technology, the
result corresponds to the traditional utilitarian criterion with positive
discounting. As mentioned in the introduction, this criterion prescribes
ethically questionable intergenerational allocations, especially in resource
technologies. This provides a motivation for requiring generations to
subscribe to J-optimality as an overriding ethical principle, deeming unjust
allocations socially unacceptable.”
Hence, if generation s inherits yi in the technology #, it seeksto
(W)
maximize w(,c) over the class of J-optimal .$ E E(yl).
Let Oc’ denote the choice of generation 0 with “y’ as the corresponding
sequence of states. The allocation ,,c’ is said to be time-consistent relative
to (W), if, for all s 3 0, ,c’ solves (W). If the present generation cannot
dictate the future to follow its chosen allocation, time-consistency is
essential. Otherwise, a game-theoretic approach is called for.
In the two classesof productive technologies we consider in Sections 5
and 6, productivity of waiting (defined as the marginal rate of technical
transformation between consumption in one period and the next) is nonincreasing along any efficient and nondecreasing allocation. This ensures
time-consistency relative to (W).”
‘I Harsanyi
[ 16, p. 3151 uses the term ethical preferences
(here choosing
J-optimal
allocations)
as opposed to subjecrioe preferences
(here represented
by the function
w).
I2 In a productive
technology,
J-optimality
constrains
generation
0 to choose ,,c’ E .F(y,,)
with c:> c;-, for any s>O. However.
generation
s does not face this constraint.
It can be
shown that generation
s will not choose ,c”e.F(y:)
with c; -CC:-,
if the productivity
of
waiting at time s is nonincreasing.
See Asheim [ 1, proof of Lemma 41 for a formal demonstration of this point.
UNJUSTINTERGENERATIONAL
363
ALLOCATIONS
Sen [24, p. 15591 denotes by welfarism “[tlhe general approach of
making no use of any information about the social states other than that
of personal welfares generated in them....“13 Note that respecting the
subjective preferences of each generation, only within the restricted class of
J-optimal allocations, is not in accordance with welfarism: The ethical
principle of imposing J-optimality is based directly on the intergenerational
allocations of consumption and works by excluding allocations that are
unjust. Hence, it does not reflect the subjective preferences of the generations involved. As an alternative, consider the case where each generation
is required to subscribe to maximin as an ethical principle; i.e., given some
bequest -vi., generation s seeks to
(Z7) maximize 7&c) over all >c E E(yi),
with ?c:R~-+R+
defined by rc(.c) E inf, a s w(,c). Here welfarism holds
since the ethical principle depends solely and in an invariant way on the
subjective preferences as represented by w. A chosen allocation ,,c’ (with
corresponding sequence of states Oy’) is said to be time-consistent relative
to (IQ, if, for all s 3 0, ,c’ solves (II).
Note that maximin combined with altruism allows for consumption
growth (see Calvo [S], Asheim Cl]). It turns out that in the productive
technologies considered in the two subsequent sections J-optimality
combined with altruism (i.e., (W)) gives rise to the same allocations as maximin
combined with altruism (i.e., (Z7)) provided that time-consistency relative
to (ZI) is imposed.
5. A ONE-SECTOR
TECHNOLOGY
First, consider a technology where total output y is split between
consumption c and capital k, the latter producing the total output available
in the next period. Assume that the stationary production function
g: R, c, R, satisfies
(g.1) g is continuous, strictly increasing, and strictly concave; it is
continuously differentiable at any k > 0.
(g)
g(O)=O;dg/dk+ooaskJO,dg/dkJOask-rco.
A total output function f is defined by f(k) -g(k) + k. Production
possibilities are described by a stationary transformation
set Yp of
input--output pairs:
P={(k,y):O<y<f(k);k>O}.
I3 It should
be noted
that Sen [24, pp. 1559-15621
presents
arguments
against
welfarism.
364
GEIR B. ASHEIM
The stationary technology Rfl’, describing the set of feasible allocations,
defined by
9p(y,Y)=
(,c>O:
3(,y, .sk), V’t>s, c,=y,-k,and
is
(k,, yr+,)~YQ}.
The pair (,y, ,k) is called the associated program
LEMMA
3. The technology FV satisfies Conditions 1-3; i.e., it is produc-
tive.
Proof
See the Appendix.
1
Consider maximizing w(~c) over the class of J-optimal ,,c E Fp( y,). With
y, = 0, this problem is trivial since Fp(0) = (0). Therefore turn to the case
with y,> 0. Define k, by b .df(k,,)/dk s 1, ym by y, =f(k,),
and c,, by
= YCC-k,.
C
It is well known that for any y,>O, the modified Ramsey
maximizing
w(~c) over all
p;ogram (associated with the allocation
,yzE 9”( yo)) converges to (y,=, k, ).
PROPOSITION 6. For any y, > 0, there is an allocation ,,c’ E Fslc(yO) maximizing w(~c) over the class of J-optimal +T EFp( yO). Moreover, Oc’ is unique
and time-consostent relative to (W). For y0 E (0, y,*), c: < c:, 1for all t b 0,
with c:t c, as t+oO. For y,~y%, c; = c0= y0 - k, for all t b 0 (where k,
is defined bv y, zf (k,)).
Prooj
Case 1. y0 E (0, y,,). It is well known (see Beals and Koopmans [2])
that there is an allocation 0c E Fp(( yO) with associated program (Oy’, ,k’)
such that for all s L 0, .c’ uniquely maximizes w(,c) over all ,c E Rp( yi).
Furthermore, c; < c; + i and 0 < vj < y, for all t 2 0. By Proposition 5 and
Lemma 3, ,c’ is J-optimal for all s 2 0, since ,c’ is efficient and nondecreasing. Hence, for all s > 0, .c’ uniquely maximizes w(,c) over the class
of J-optimal .c E YV( y:).
Define the stationary allocation ,,c’E~@( y,,) by
Case 2. yO>y,.
c: = c0 (Vt 3 0) with associated program (Oy’, ,k’) given by (y;, k:) =
(yO, k,) (Vt 2 0). Write
83
so that
l
1,~ [(l -/?)/(l
pI = I., . du(c,)/dc
(Vt2.Y)
df &JW’
-b)]
./Y”,
and
UNJUST
INTERGENERATIONAL
5 L,=l/(l
CT=5
-b)=
ALLOCATIONS
2 b”-”
o=s
V’tas,
P ,+,Y:-p,k:~PI+,y-p,k,
365
(1)
V[k, .v]~cT’
(2)
with strict inequality if [k, V] # [k:, y:+ ,I. Hence, for any s >, 0 and for
any ,sc”E FP( ,:) with associated program (.y”, ,k”),
i: A,. [u(C) - 4cb)l
v=s
I
d 1 p;
cc::-c&l
0= 5
by (u.1) since ck = c,, (Vo 3 S)
= i p;[(y,“-k::)-(y&-k;)]
0= 5
~p,.(k:-k:‘)-p;o,:-y:()
by (2)
with strict inequality if (,y”. .k”) # (,y’, ,k’). Since p, JO as t -+ 00, k; = k,
and k:’ Z 0 (Vt 3 0), and y,r = JJ; = y,, it follows that .c’ is efficient:
lim sup i: A,. [u(c:) - u(c;)] d 0
r-r
rJ=s
(3)
with strict inequality if (,y”, ,k”) # (,y’, .k’).
As for Case 1, Sc’ is J-optimal. It remains to be shown that .c’ uniquely
maximizes w( ,c) over the class of J-optimal ,c E 9@( y:). By Proposition 3,
it suffices to show that .c’ uniquely maximizes w(,c) over the class of nondecreasing .c E g”(y,i). If $ E 9”( y:) is nondecreasing with associated
program (Sy”, .Tk”), then
limsup i b”-‘.[u(ci)--u(cb)]
I--r’?2 o=s
= lim sup i [b”~“.u(c~)-l;u(cb)]
,-3c g=s
by (l)sincecb=c,
(Vo2.r)
<lim
sup i [b”~“.~(c%)--~.u(c~)]
r--r= o=s
by (3)
( < if Ly”. ,k”) # (A,
.&‘)I
366
GEIR
= lim sup i (b”-’
I + a’ D= ,y
B. ASHEIM
- i,).
[u(cz)
-u(C)]
by (1) for any constant C
GO
if C is chosen such that c:-,
< C < c:.
and where r satisfies b’-” - A,2 0 for t < r
and b’-“-II,<Ofor
t>r.i4
1
The proof of Proposition
6 shows that ,c’, denoting the allocation
in Y*I”(yO) that uniquely maximizes w(~c) over the class of J-optimal
allocations in Tp(yO), follows a modified Ramsey program if this is nondecreasing; otherwise ,,c’ is efficient and stationary. It it in this latter case
that the process of disallowing unjust allocations affects the selection made
by the utilitarian criterion with positive discounting:
Along a modified
Ramsey program when y0 > yor,, the current generation enjoys a binge at
the expense of all future generations. Compared to the efficient stationary
allocation, this leads to more relative inequality and less total consumption.
It is of interest to note that substituting maximin combined with altruism
(i.e., (Z7)) for J-optimality combined with altruism (i.e., (W)) is of no consequence in this particular technology: For any y, > 0, Oc’ of Proposition 6,
in addition to maximizing w(*c) over the class of J-optimal allocations in
Fp( yO), also uniquely maximizes IC(~C) over all allocations in Fp(?lO) (see
Calvo [S] ). In the next section we show that this result carries over (subject to the imposition of time-consistency
relative to (ZZ)) to a resource
technology in which the altruistic preferences of Section 4 implement ethically questionable intergenerational
allocations when not combined with an
ethical principle.
6. A. RESOURCE TECHNOLOGY
Following Dasgupta and Mitra [ 111, consider a technology where
capital k, resource extraction r, and labor z produce the total output y
available in the next period. Assume that the stationary production
function G: LQ: -+ Iw, satisfies
(G.l)
G(k, r, 2) is continuous, nondecreasing, concave, and
homogeneous of degree one; it is continuously twice differentiable at any
(k r, z)%O.
(G.2)
WA r, z) = 0 = G(k, 0, z); (G,, G,, G,) B 0 for (k, r, 2) $ 0.15
I4 Such a 5 exists by (1) since )‘0 a~,
implies that k, 2 k I and fi > b.
I5 Write
Gk = aG/ak, Fk, = a2F/2;k&,
and so forth.
UNJUST
INTERGENERATIONAL
367
ALLOCATIONS
The labor force is assumed to be stationary ( = 1). Given z = 1, let the ratio
of the resource share in output to the labor share in output be bounded
away from zero:
(G.3) Given any (E, r) $0, there is rj >O such that for all (k, r)
satisfying k&z, O<r<F,
we have [rG,(k, r, I)]/G,(k,
r, l)>Q.
A total output function F, defined by F(k, r) z G(k, Y, 1) + k, satisfies
(F)
F(k, Y) is strictly concave; F,, >, 0.
Let rrr denote the resource stock. Then production possibilities are
described by a stationary transformation set F/I” of input-output
pairs:
~~=~[C(k,m),(y,m+,)l:Ofy~F(k,v));
O<r=m-m,,;
(k,m+,)kO}.
The stationary technology 9”, describing the set of feasible allocations,
defined by
is
The triple (,y, sm, ,Yk) is called the associated program. (G.l)-(G.3) are not
sufhcient to ensure the existence of a stationary allocation with positive
consumption (Solow [ZS], Cass and Mitra [6]). Therefore assume
(A) ,c’ E 9”(y,, m,) with cj = c: > 0 (Vr >s) exists if (I’,, m,) >>0.
Cass and Mitra [6] give a necessary and sufficient condition for (A) to
hold.
LEMMA
4.
The technology 9” satisfies Conditions
1-3; i.e., it is produc-
tive.
Proof:
See the Appendix.
f
Consider maximizing w(~c) over the class of J-optimal 0c E Y”( y,, mo).
With y0 = 0, this problem is trivial since 9”(0, m,) = (0). With m, = 0, we
are faced with a cake-eating problem for which no J-optimum
exists if
yO>O (see Proposition 2). Therefore turn to the case with (y,, mo) $0.
hOPOSITION
7.
For
any
(yO, m,) + 0, there
is an allocation
maximizing
over the class
of J-optimal
6’ E W3h,
md
4&l
,C E 9”(y0,
mo). Moreover, get is unique and time-consistent relative to (W).
The allocation ,c’ is nondecreasing with an eventual stationar)) phase. For
368
GEIR
some (y,,, mo), this eventual
increasing consumption.
phase
B. ASHEIM
is preceded
by an initial
phase
with
Proof
By Asheim [ 1, Proposition 2 and Lemma 4(a)], there is an
efficient and nondecreasing allocation Oc’EF”(y,,, m,) with associated
program (Oy’, ,m’, ,k’) such that for all s 3 0, ,,c’ uniquely maximizes w(,c)
over the class of nondecreasing .YcE.FV( y,;, m,:). By Proposition 5 and
Lemma 4, .c’ is J-optimal for all s > 0. By Proposition 3, it suffices to consider the class of nondecreasing .c EF(y,i, ml). The two phase structure of
Oc’ is shown in the proof of Lemma 4 of Asheim [ 1J. 1
The allocation Oc’ of Proposition 7 has the following desirable distributional properties (discussedin Asheim [ 1, p. 4743): It allows for consumption (and welfare) growth in an economy that is highly productive due to
a small capital stock and a large resource stock, while the eventual
stationarity protects distant generations from the grave consequences of
utility discounting when the productivity of waiting is low and diminishing.
Not restricting the maximization of w(~c) to J-optimal allocations would
have forced consumption to eventually approach zero. Compared to a
stationary allocation with positive consumption, this leads to an allocation
with more relative inequality and less total consumption. Hence, such an
allocation is unjust.
Since the restriction to J-optimal allocations is not in accordance with
welfarism, Oc’ of Proposition 7 need not be Pareto-efficient in the senseof
there being no alternative allocation in 9”(y0, m,,) increasing the welfare
u~(,~c)of some generation s without decreasing the welfare of any other
generation. It can be shown (see Asheim [ 1, p. 4751) that, for a class of
initial states, Oc’ of Proposition 7 is in fact Pareto-inefficient, being Paretodominated by an allocation which is not nondecreasing.
This provides a motivation for considering maximin combined with
altruism (i.e., (n)) as an alternative to J-optimality combined with altruism
(i.e., (W)). Denote by Oc” an allocation in 9”(y,, m,) maximizing TC(,$)
over all allocations in 9’(y,, nt,). It turns out that Oc” is not nondecreasing and, hence, not time-consistent relative to (Z7) for the sameclass
of initial states as the one for which Oc’ of Proposition 7 is Paretoinefficient [l, Theorem 11. Moreover, if generation 0 maximizes rc(,$) over
the classof allocations in F”( y,, mo) that future generations subscribing to
(Z7) actually will carry out, then Oc’ of Proposition 7 is its unique optimal
choice Cl, Theorem 2].16 In this sense, (W) and (n) result in the same
allocations, also in this technology.
I6 In Asheim [ 11. I assume that an allocation
that is time-consistent
followed
as soon as one exists. Then &’ of Proposition
7 is the
equilibrium
allocation
in an intergenerational
game.
relative to (f7) will be
unique subgame-perfect
UNJUSTINTERGENERATIONAL
7. CONCLUDING
369
ALLOCATIONS
REMARKS
In this paper, we have demonstrated that combining the exclusion of
unjust intergenerational allocations of consumption (J-optimality) with a
simple recursive form of altruistic preferences yields desirable outcomes in
two important classesof technologies. In particular, while this form of
altruism alone may cause gross inequities, and maximin combined with
egoistic preferences may perpetuate poverty,17 the combination of
J-optimality with altruism allows for development when the economy is
highly productive, ensuring equality otherwise. It is noteworthy that this is
achieved also in the resource technology considered, since this technology
posesan awkward problem of intergenerational distribution when standard
criteria are used.
We have also shown, for the two classes of technologies, that
J-optimality combined with altruism (i.e., (W)) give rise to the same
allocations as maximin combined with altruism (i.e., (n)) given that timeconsistency relative to (Z7) is imposed. It is natural to ask whether this
result carries over to more general classesof technologies. That (W) and
(n) are not equivalent is most easily demonstrated by considering a cakeeating technology: For a positive initial stock, no allocation is J-optimal
while all allocations are maximin.‘” It is my conjecture, though, that (W)
and (Z7) will lead to the same allocations in a more general class of
productive technologies (which includes the one-sector technology and the
resource technology as special cases)having the property that the productivity of waiting is nonincreasing along any efficient and nondecreasing
allocation.
APPENDIX
Proof of Lemma 3.
Condition 1. (a) Consider .c‘ E9”(yS) associated with (,y’, ,k’). Let
,c 2 0 satisfy Ci =s (c, - cb) d 0 for all t 3 s. Construct (,y, .k):
k,=k;-
i
(c,-c&)[2k;>O]
and
.v, = c, + k,
(Vt 3 s).
Hence, ,$EY~‘(~~)
since for all t3s,
O<v,+,-k,=y;+,--k;g
f(k:)-k:bf(k,)-k,.
(b) If .c’ EFfl(y,) is inefficient, we proceed to show that there is a
hi” E YP(yS)
such that CT= c,: + 6 (6 > 0), ci = cb otherwise, and then
repeat part (a), considering $‘.
” See Solow [25]
” For any feasible
for this criticism.
allocation
gz. we have n(,c)
= u(O)/( 1 -/I).
Hence, they are all maximin.
370
GEIR B. ASHEIM
Condition 2. Let ,$c’Ey”l”(y,) be efficient with (,y’, ,k’) + 0. Suppose
,c EF”l”(y,) associated with (,y, .yk) satisfies CL =,s (c, - c&) 2 0 for all t 2 s
with strict inequality for at least one t. W.1.o.g. we may assume c, > c:.
Hence, k,: - k, = c, - c: > 0 since ~1,:.
=.v,~. The efficiency of ,c’ implies
y:+, =f(k\) for all t 3 s. Thus,
since g(k) =f(k) - k is strictly increasing and concave. Hence, there is a
$” EFfl(cfs) such that c,: = cr, c:‘+ 1= c,~+, + (c, - c:) df(k;)/dk, c; = c,
otherwise. By Condition 1, there is a 3c”’ EsV(y,) such that c,:.”= c:,
cy+,=c:+,
+ (c,~- c.:). [df(kj)/dk - 11, cy = cb otherwise, since c,: = cs’
for t>s.
Since (c,-CC:).
and CL=, (c, - ck) > 0 implies CL=., (ci -ci)<O
[df(kj)/dk - 1] > 0, ,Y~“’E 9”(~,~) contradicts the efficiency of .c’.
Condition 3. For any y,>O, it is easy to verify that the stationary
allocation .,c’ER~(JJ.,), satisfying for all t as, c; = c,=y, - k, (with k,s
defined by JJ,~
-f(k,)),
is efficient (see proof of Proposition 6). If y,=O,
then F”l”(y,,) = (0). 1
Proof of Lemma 4.
.yc’e F’(ys, m,)
associated
with
Condition 1. (a)
Consider
(,Vy’, .m’, ,k’). Define f, by f,(k) E F(k, mj - rn: + ,). We can now repeat part
l(a) of the proof of Lemma 3 sincef,(k) - k is nondecreasing.
(b) If .c’ EYV(~~,s,m,) is inefficient, we can correspondingly repeat
part I(b) of the proof of Lemma 3.
Condition 2. Let .$~9”(y,,
m,) be efficient with (,y’, ,Ym’,.k’) $0.
We need to show that m: >m:+,
for all t>s.
Suppose rn:-, =m;
for some t. Since efficiency implies rn: JO as 5 --, co, we may w.l.0.g.
assume that rn: > rn: + 1. Consider .c associated with (,y, .Ym,.k) : m, =
rn:-, -m,), k, satisfiesF(k,, m, -m;+,) =
(l/2)(&,
+ml+,h _y, =F(ki-,,
A+ 1; (Y,, m,, k,) = (J$, rn;, kk) for all g # t. Hence, ,c’ is inefficient since
.c Efl’(v,,
m,), c, =v, - k > c: by (Gl) and (G2), and c, = CLfor all r~# t.
Thus, rn: > rn: + 1 for all t 3 s, and f, defined by f,(k) = F(k, m: -m;+ ,) is
concave and f,(k) -k strictly increasing. Now, repeat part 2 of the proof of
Lemma 3.
Condition 3. For (y,, m,) $0, see Theorem 1 of Dasgupta and Mitra
[11].If~,=Oorm,=O,then.c~~~(~,,m,)impliesthat~,“~,~c,~~~,.
1
REFERENCES
1. G. B. ASHEIIM, Rawlsian
integenerational
justice
resource technology.
Rear. Econ. Stud. 55 (1988),
as a Markov-perfect
469484.
equilibrium
in a
UNJUST
INTERGENERATIONAL
ALLOCATIONS
371
2. R. BEALS AND T. C. KOOPMANS. Maximizing
stationary
utility in a constant technology.
SIAM .I. A&. Math. 17 (1969), 100-1015.
3. C. R. BIRCHENHALL
AND P. GROUT. On equal plans with an infinite horizon,
J. Econ.
Theory 21 (1979), 249-264.
4. C. BLACKORBY AND D. DONALDSON,
Utility
vs. equity: Some plausible
quasi-orderings,
J. Public. Econ. 7 (1977), 365-381.
5. G. CALVO, Some notes on time inconsistency
and Rawls’ maximin
criterion.
Rel;. Econ.
Stud. 45 (1978), 97-102.
6. D. CAB AND T. MITRA. “Persistence
of Economic
Growth
Despite Exhaustion
of Natural
Resources,”
Working
Paper 79-27, University
of Pennesylvania,
Center
for Analytic
Research in Economics
and the Social Sciences, 1979.
7. H. DALTON, The measurement
of the inequality
of incomes, Econ. J. 30 (1920). 348-361.
8. P. DASGUPTA ANU G. HEAL, The optimal depletion
of exhaustible
resources, Rev. Econ.
Stud. (Symposium)
(1974), 3-28.
9. P. DASGUPTA AND G. HEAL, “Economic
Theory
and Exhaustible
Resurces,”
Cambridge
Univ. Press, Cambridge,
1979.
IO. P. DASGUPTA, A. SEN, AND D. STARRETT. Notes on the measurement
of inequality,
J. Econ. Theory 6 (1973). 180-187.
11. S. DASGUP~A
AND T. MITRA,
Intergenerational
equity
and efficient
allocations
of
exhaustible
resources, Int. Econ. Reo. 24 (1983), 133-153.
12. L. EPSTEIN, Intergenerational
consumption
rules: An axiomatization
of utilitarianism
and
egalitarianism,
J. Econ. Theory 38 (1986), 280-297.
13. L. EPSTEIN. Intergenerational
preference orderings,
Sot. Choice Welfare 3 (1986). 151-160.
14. J. FEREJOHN AND T. PAGE. On the foundation
of intertemporal
choice, Amer. J. Agr. Econ.
60 (1978), 269-275.
15. J. C. HARSANYI, Cardinal
utility in welfare economics
and in the theory of risk-taking,
J. Polit. Econ. 61 (1953), 434-435.
16. J. C. HARSANYI, Cardinal
welfare, individualistic
ethics, and interpersonal
comparisons
of
utility. J. Polit. Bon. 63 (1955). 309-321.
17. T. C. KOOPMANS, Intertemporal
distribution
and optima1 aggregate economic growth,
in
“Ten Economic
Studies in the Tradition
of Irving Fisher.”
Wiley. New York, 1967.
18. E. MISHAN,
Economic
criteria
for intergenerational
comparisons,
Futures
9 (1977),
383-403.
19. T. PAGE, “Conservation
and Economic
Efficiency,”
Johns Hopkins
Press, Baltimore,
1977.
20. J. RAWLS, “A Theory of Justice,” Oxford
Univ. Press, Oxford,
1971.
21. J. RAWLS. The priority
of right and ideas of the good, Phi/ox Public AJJairs 17 (1988),
251-276.
22. D. RAY, Nonpaternalistic
intergenerational
altruism, J. Econ. Theory 41 (1987). 112-132.
23. A. SEN. “On Economic
Inequality.”
Oxford
Univ. Press (Clarendon),
London/New
York,
1973.
24. A. SEN, On weights and measures: Informational
constraints
in social welfare analysis,
Econometrica
45 (1977), 1539-1572.
25. R. SOLOW, Intergenerational
equity and exhaustible
resources,
Rec. Econ. Stud. (Symposium) (1974). 29-45.
26. L. G. SVENSSON, Equity among generations,
Econometrica
48 (1980). 1251-1256.
27. W. VICKREY, Measuring
marginal
utility by reactions
to risk, Econometrica
13 (1945).
319-333.
28. C. C. VON WEIZS~~CKER, Existence of optimal programmes
of accumulation
for an infinite
time horizon,
Reti. Econ. Stud. 32 (1965). 85-104.
29. WCED
(WORLD
COMMISSION
ON ENVIRONMENT
AND DEVELOPMENT),
“Our
Common
Future,”
Oxford
Uni. Press. Oxford,
1987.