Allaisanonymity as an alternative to the discountedsum criterion in the calculus of optimal growth II: Pareto optimality and some economic interpretations
ABSTRACT This paper studies the Paretooptimality of the consensual optimum established in "Allaisanonymity as an alternative to the discountedsum criterion I: consensual optimality" (Mabrouk 2006a). For that, a Paretooptimality criterion is set up by the application of the generalized Karush, Kuhn and Tucker theorem and thanks to the decomposition of the space of geometricallygrowing real sequences. That makes it possible to find sufficient conditions so that a bequestrule path is Paretooptimal. Through an example, it is then shown that the golden rule must be checked to achieve Allaisanonymous optimality. The introduction of an additive altruism makes it possible to highlight the intergenerationalpreference rate compatible with Allaisanonymous optimality. In this approach, it is not any more the optimality which depends on the intergenerationalpreference rate, but the optimal intergenerationalpreference rate which rises from Allaisanonymous optimality.
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Article: Optimization by Vector Spaces
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ABSTRACT: The subjective value given to time, also known as the psychological interest rate, or the subjective price of time, is a core concept of the microeconomic choices. Individual decisions using a unique and constant subjective interest rate will refer to an exponential discounting function. However, many empirical and behavioural studies underline the idea of a nonflat term structure of subjective interest rates with a decreasing slope. Using an empirical test this paper aims at identifying in individual behaviours if agents see their psychological value of time decreasing or not. A sample of 243 individuals was questioned with regard to their time preference attitudes. We show that the subjective interest rates follow a negatively sloped term structure. It can be parameterized using two variables, one specifying the instantaneous time preference, the other characterizing the slope of the term structure. A tradeoff law called “balancing pressure law” is identified between these two parameters. We show that the term structure of psychological rates depends strongly on gender, but appears not linked with life expectancy. In that sense, individual subjective time preference is not exposed to a tempus fugit effect. We also question the cross relation between risk aversion and time preference. On the theoretical ground, they stand as two different and independent dimensions of choices. However, empirically, both time preference attitude and slope seem directly influenced by the risk attitude.Journal of Economic Theory 06/2005; 122(2):254266. · 1.24 Impact Factor
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Allaisanonymity as an alternative to the
discountedsum criterion in the calculus of optimal
growth
II: Pareto optimality and some economic interpretations
Mohamed Mabrouk
Ecole Supérieure de Statistique et d’Analyse de l’Information de Tunisie
(Université 7 November à Carthage), 6 rue des métiers, Charguia II, Tunis,
Tunisia
Faculté des Sciences Economiques et de Gestion (Université Tunis El Manar),
Campus Universitaire, Tunis 1060, Tunisia
Personal address : 7 rue des Lys, El Menzah 5, Tunis 1004, Tunisia
m_b_r_mabrouk@yahoo.fr
tel: 21621141575
version April 4, 2006
A previous version of this paper circulated under the title "Intergenerational
anonymity as an alternative to the discountedsum criterion in the calculus of
optimal growth: II Paretooptimality and some economic interpretations"
1
Page 2
Abstract
This paper studies the Paretooptimality of the consensual optimum estab
lished in "Allaisanonymity as an alternative to the discountedsum criterion I:
consensual optimality" ([Mabrouk 2006a]). For that, a Paretooptimality crite
rion is set up by the application of the generalized Karush, Kuhn and Tucker
theorem and thanks to the decomposition of lp∗
sufficient conditions so that a bequestrule path is Paretooptimal. Through
an example, it is then shown that the golden rule must be checked to achieve
Allaisanonymous optimality.
The introduction of an additive altruism makes it possible to highlight the
intergenerationalpreference rate compatible with Allaisanonymous optimal
ity. In this approach, it is not any more the optimality which depends on the
intergenerationalpreference rate, but the optimal intergenerationalpreference
rate which rises from Allaisanonymous optimality.
∞. That makes it possible to find
JEL classification: D90; C61; D71;D63; O41; O30.
Keywords: Intergenerational anonymity; Allaisanonymity; Intergenerational
equity; Optimal growth; Technical change; Timepreference; Discountedsum
criterion; Consensual criterion; Paretooptimality; OG economy.
2
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1 Introduction
1.1Motivation
This paper pursues two goals. First, we seek to set up a criterion of Pareto
optimality applicable to a situation with exogenous technical change, overlap
ping generations, bequests and infinite horizon (sections 2, 3 and 4), with an aim
of judging efficiency of the consensual optimum (with a Allaisanonymous con
sensual criterion) partly characterized in the article "Allaisanonymity as an al
ternative to the discountedsum criterion I: consensual optimality" [Mabrouk 2006a]
and of which this article constitutes the prolongation.
The criterion of Paretooptimality is obtained thanks to the direct applica
tion of the generalized theorem of Karush, Kuhn and Tucker and also thanks
to the decomposition of lp∗
∞(see [Mabrouk 2006a]) which enables to calculate
the adjoint variable of the program defining Paretooptimality. Using a suitable
adaptation of the variables, one realizes that this criterion is in fact nothing but
a particular case of the Paretooptimality criterion of [Cass 1972] or the Pareto
optimality criterion of [BalaskoShell 1980], although the method implemented
here differs by the fact that it has recourse to the tools of the theory of optimiza
tion. It appears indeed that the criterion used here is equivalent to these criteria
in the case of regular bequests plans. On the other hand, it does not constitute
a necessary condition of Paretooptimality in the case of nonregular bequests
plans, whereas it is the case for Cass and BalaskoShell criteria. But the am
bition here is not to establish a complete characterization of Paretooptimality
following the example of [Cass 1972] or [BalaskoShell 1980]. However, although
incomplete, the criterion suggested here does not require a condition of minimal
curvature on the indifference curves and can be thus extended to the case of
unbounded capital without involving differentiability problems for the sequence
of utility functions (section 2).
Section 2 establishes the criterion of Paretooptimality. Section 3 considers
the case where the growth rate of the capital is not the maximum rate, case not
taken into account by section 2, the optimum being then noninterior.
In absence of a general result on the Paretooptimality of a consensual op
timum, whereas there was such a result in the case without technical change
[Mabrouk 2005], section 4 gives some sufficient conditions for a consensual op
timum to be Paretooptimal. That will make it possible in certain cases, as in
the example of section 5, to partly characterize the optimal growth path which
satisfies at the same time consensual optimality and Paretooptimality.
The second goal is to highlight certain properties of the optimal growth
path to draw some economic interpretations from them. Will be successively ap
proached the comparison between goldenrule states (that are shown to coincide
asymptotically with optima) with and without technical change in a discrete
time case (subsection 6.1), the analysis of the stability of the optimal path with
introduction of an additive altruism (subsection 6.2) and finally, the comparison
between Allaisanonymous and discountedsum criteria (subsection 6.3).
All proofs are gathered in section 7 except those relating to the discretetime
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example (section 5) which can be found in appendix A.
The model used in this paper is exactly that of [Mabrouk 2006a]. We start
by recalling its essential features, assumptions and the principal results. To have
more details on the model, it is preferable to consult [Mabrouk 2006a].
1.2 Model, assumptions and results on consensual opti
mality
The economy is constituted by a succession of generations g1,g2,g3..., each gen
eration being made up of only one individual who is at the same time consumer
and producer. At the beginning of its active life, a given generation giinherits a
quantity bi−1of that good. Its only acts during its life are: to consume, produce,
invest in order to increase its future consumption and, at the end of the active
lifetime, to bequeath bi to the descent. In doing so, generation gi achieves a
level of lifeutility Ui(bi−1,bi). Each utility function Uiis defined from Di⊂ R2
to R. Diis strictly included in R2
is concave and of class C2 on Di; U0
derivatives of Uiwith respect to its first and second variable).
For r ≥ 0, denote lr
the Banach space normed by kBkr= supi≥1bie−ri.
The set D = {K = (k1,k2,···) / for all i ≥ 1 : (ki−1,ki) ∈ Di} is assumed
to be strictly geometric of reason p ≥ 0 (see [Mabrouk 2006a] section 5 for the
definition of strict geometricity) and G(D) strictly geometric of reason p1≥ 0,
G being the mapping that associates to K ∈ D, G(K) = (Ui(ki−1,ki))i≥1.
Denote
D the interior of D in lp
It has been proved in [Mabrouk 2006a], section 6, that if G is linear at infinity
◦
D for the reasons (p,p1) then G is Frechetdifferentiable at K.
Consider a consensual criterion represented by a real valued, Frechetdifferentiable
functional Ψ on lp1
∞.
Ψ(G(K)). Suppose also that Ψ is Allaisanonymous and sensitive to long run
interest (definitions in [Mabrouk 2006a], section 7).
It has been proved in [Mabrouk 2006a], section 7, theorem 18, that if a
◦
D is a consensual optimum for the criterion Ψ then
+
+, closed and with a nonempty interior; Ui
ihÂ 0 (U0
∞=©B = (b1,b2,...)/bi∈ R andsupi≥1bie−ri≺ +∞ª
ihand U0
ilare respectively the
◦
∞with respect to the norm kBkp= supi≥1bie−pi.
at K ∈
The consensual value of a state K ∈ D is given by
steady state1K in
u0
he−p+ u0
l= 0(1)
where u0
Equation (1) is the bequestrule and characterizes consensual optimality.
h= limU0
nh(kn−1,kn)e−(p1−p)nand u0
l= limU0
nl(kn−1,kn)e−(p1−p)n.
1definition 12 in [Mabrouk 2006a]
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2Pareto optimality
2.1Introduction
The criterion Ψ not being strictly increasing, it is not sure that any solution of
the first order condition (1) is Pareto optimal. That’s why an efficiency criterion
is needed. It is the objective of the present section.
Define Dl = {B ∈ D/ for all i ≥ 1, U0
quests plan which is not in Dlis not of interest since it cannot be Pareto optimal
. We will henceforth look for solutions in Dl.
Suppose2
◦
D ∩ Dl6= ∅
Let
K ∈
Suppose
(Ui)i≥1linear at infinity at K
We will first consider the case where
il(bi−1,bi) ≤ 0}. Observe that a be
(A1)
◦
D ∩ Dl
(A2)
(A3)
U0
nl(kn−1,kn) ≺ 0 for all n ≥ 1(A4)
The latter assumption will be used to set regularity and then dropped.
Let B ∈ D. For i ≥ 1, let Tibe the transformation which suppresses the ith
component of an element of lp
∞, replaces it by the next one and shifts all the
following components backward. Let eibe the sequence of lp
are all 0 except the ithequal to 1.
Denote Hi(B) = Ti(G(B) − G(K)).
Under the above assumptions, G is Frechetdifferentiable at K. This implies
that Hiand Uiare also Frechetdifferentiable at K and we have
∞which components
δHi(K) = Ti(δG(K))
and
δUi(K) = ei δG(K)
where δ preceding a transformation means its Frechetdifferential.
The program Pi(K) which gives Pareto optimality, can be written
max
B∈Dei G(B)
subject to:Hi(B) ≥ 0
2◦
D∩Dl= ∅ would mean that in the interior of D, U0
be interior to D. For example if U0
of interest between a generation and the following generations. The optimum would consist
in always bequeathing the maximum.
il≥ 0. Thus, the optimum would not
ilis everywhere positive, there would not be a real conflict
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2.2Regularity of K for the inequality Hi(B) ≥ 0
To apply the KarushKuhnTucker theorem to Pi(K), we have to make sure that
K is a regular point of the inequality Hi(B) ≥ 0. This means that Hi(K) ≥ 0
and that there is X ∈ lp
all components are strictly positive and that the sequence is strictly of reason
p1).
Denote henceforth u0
Define R?and R?as follows
∞such that Hi(K)+δHi(K)·X Â 0 (which means that
hn= U0
nh(kn−1,kn) and u0
ln= U0
nl(kn−1,kn).
R?= limsupu0
hn
−u0
ln
, R?= liminf
u0
−u0
hn
ln
According to proposition 10 of [Mabrouk 2006a], the sequences (u0
ln)n≥1are in lp1−p
sp1−p
∞
hn)n≥1and
ln)n≥1are in
(u0
∞
. We need to assume that either (u0
to set regularity.
hn)n≥1or (u0
Proposition 1
either (u0
Under assumptions (A1, A2, A3 and A4), K is regular if
hn)n≥1or (u0
ln)n≥1is in sp1−p
∞
and if either R?≺ epor R?Â ep.
Remark 2 If p1 was not the strict reason of G(D), there would be p0
such that (u0
(u0
∞
tion of strict geometricity of G(D) is crucial for the necessity of the Pareto
optimality criterion given by proposition 3. The strict geometricity of D, as for
it, is crucial for both necessity and sufficiency as far as we need definition sets
with non empty interiors to use optimization theorems.
1≺ p1
hn)n≥1or
hn)n≥1and (u0
. Thus, we would not have regular points. Hence, the assump
ln)n≥1are in lp0
1−p
∞
which is contrary to (u0
ln)n≥1∈ sp1−p
Denote
L+=
(
K ∈ D/limsup
either (u0
u0
−u0
ln)n≥1is in sp1−p
hn
ln≺ epand
hn)n≥1or (u0
∞
)
and
L−=
(
K ∈ D/liminf
either (u0
u0
−u0
hn
lnÂ epand
ln)n≥1is in sp1−p
hn)n≥1or (u0
∞
)
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2.3Necessity
Suppose
◦
D ∩ Dl∩ (L−∪ L+) 6= ∅
(A’1)
Proposition 3 Under the assumptions (A’1), K ∈
if K is a Paretooptimal bequests plan then, for all i ≥ 1, we have:
+∞
X
◦
D ∩ (L−∪ L+) and (A3),
n=0
n
Y
j=0
¯¯¯¯¯
u0
u0
li+jep1
hi+j+1
¯¯¯¯¯≺ +∞
2.4Sufficiency
Proposition 4 Under assumptions (A1) , (A2) and (A3), let i such that if
i Â 1 we have:Qi−1
If
+∞
X
then K is solution of Pi(K).
j=1u0
lj6= 0.
n=0
n
Y
j=0
¯¯¯¯¯
u0
u0
li+jep1
hi+j+1
¯¯¯¯¯≺ +∞
If K is such that for all i ≥ 1 we have D2Ui(ki−1,ki) ≺ 0, then we have:
⎛
n=0
j=0
u0
h2+j
n=0
⎝
+∞
X
If K is a solution of Pi(K) for all i ≥ 1, then K is a Paretooptimal bequests
plan. Thus, we can state:
n
Y
¯¯¯¯¯
u0
l1+jep1
¯¯¯¯¯≺ +∞
⎞
⎠=⇒
⎛
⎝
+∞
X
n
Y
j=0
¯¯¯¯¯
u0
u0
li+jep1
hi+j+1
¯¯¯¯¯≺ +∞
⎞
⎠for all i ≥ 1
Proposition 5 Under the assumptions (A1) , (A2), (A3) and (A4), if
¯¯¯¯¯
then K is a Paretooptimal bequests plan.
+∞
X
n=0
n
Y
j=0
u0
l1+jep1
u0
h2+j
¯¯¯¯¯≺ +∞
Remark 6 (a)The condition K ∈
bility at K. If, besides, we are sure that G is differentiable at K, we don’t need
anymore this interiority condition. (b)In propositions 4 and 5, one could omit
the assumption of strict geometricity of G(D).
◦
D is needed only to ensure G’s differentia
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From proposition 3 and proposition 5, we deduce the following theorem:
Theorem 7
(A4), K is a Paretooptimal bequests plan if and only if
Under the assumptions (A’1), K ∈
◦
D ∩ (L−∪ L+), (A3), and
+∞
X
n=0
n
Y
j=0
¯¯¯¯¯
u0
l1+jep1
u0
h2+j
¯¯¯¯¯≺ +∞
This condition implies that, for ”most” generations we have:
−U0
nl(kn−1,kn)ep1≺ U0
n+1h(kn,kn+1)
which means that if generation gndecreases its bequest by one unit, it
wins ep1times less than what is lost by generation gn+1. That suggests
that the agents can be all the more selfish as p1is large, because the reduction
of heritage by a generation, without damage for all the line, is all the more high
as p1is large. This idea will be specified in section 6.
3If K has not the maximum growth rate
3.1Introduction
We supposed above that K ∈
and kngrows at the maximum rate ep. We then need another method to test
Pareto optimality for a bequests plan K which doesn’t grow at the maximum
rate.
◦
D. But since
◦
D ⊂ sp
∞++, liminf kne−pnÂ 0
Definition 8 Let π ∈ [0,p] , Dπ= D∩lπ
◦
Dπ, define linearity at infinity in lπ
after having replaced p and p1respectively by π and π1.
∞and π1=reason of G(Dπ). For K ∈
∞exactly as in definition 8 of [Mabrouk 2006a]
In this section, the condition of linearity at infinity refers to linearity at
infinity in lπ
∞for the reasons (π,π1).
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3.2Necessity:
Proposition 9
that Paretodominate it are also in lπ
If a bequests plan is in Dπwhere π ∈ [0,p[, all bequests plans
∞.
Consequently, if a bequests plan K in Dπis Paretooptimal in Dπ, it is also
Paretooptimal in D.
◦
Dπ 6= ∅ (using the appropriate norm k.kπ) and G(Dπ) strictly
geometric of reason π1. Let K ∈
K ∈ sπ
(
either (u0
Suppose
◦
Dπ. Since
◦
Dπ ⊂ sπ
∞++this implies that
∞++. Observe that the growth rate of K is now eπ≺ ep. Denote
Lπ+=
K ∈ D/limsup
hn)n≥1or (u0
u0
−u0
ln)n≥1is in sπ1−π
hn
ln≺ eπand
∞
)
and
Lπ−=
(
K ∈ D/liminf
either (u0
u0
−u0
hn
lnÂ eπand
ln)n≥1is in sπ1−π
hn)n≥1or (u0
∞
)
Now change p by π in the proposition 3,
It then gives:
Proposition 10
(A3)3, if K is a Paretooptimal bequests plan in D then for all i ≥ 1 we have:
+∞
X
Under the assumptions (A’1), K ∈
◦
Dπ∩ (Lπ−∪ Lπ+) and
n=0
n
Y
j=0
¯¯¯¯¯
u0
u0
li+jeπ1
hi+j+1
¯¯¯¯¯≺ +∞
3.3Sufficiency
Let π ∈ [0,p] and K ∈
π1the reason of G(Dπ). As we have done above, change p by π and p1by π1
in proposition 5, theorem 7 and in assumptions (A’1, A3, and A4).
It then gives:
◦
Dπsuch that G is linear at infinity at K in lπ
∞. Denote
Proposition 11 If
+∞
X
n=0
n
Y
j=0
¯¯¯¯¯
u0
l1+jeπ1
u0
h2+j
¯¯¯¯¯≺ +∞
then K is a Paretooptimal bequests plan.
3For (A’1) and (A3), use (π,π1) instead of (p,p1).
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Remark 12
ometricity of G(Dπ).
As in proposition 5, one could omit the assumption of strict ge
Theorem 13 Let K ∈
if and only if
◦
Dπ∩(Lπ−∪Lπ+). K is a Paretooptimal bequests plan
¯¯¯¯¯
Remark: For a bequests plan K which has not a strict reason, we cannot
apply the propositions and theorem of this section since there is not π such that
◦
Dπ.
+∞
X
n=0
n
Y
j=0
u0
l1+jeπ1
u0
h2+j
¯¯¯¯¯≺ +∞
K ∈
4Is a consensusoptimal plan Paretooptimal?
An optimal growth path has to be at the same time consensusoptimal and
Paretooptimal. We have then to select from the set of consensusoptima those
which are Paretooptimal.
There is not here a general result on the Pareto optimality of consensual
optima or on the Paretooptimality of bequestrule plans as in the case without
technical change [Mabrouk 2005].
Indeed, in the case of an Allaisanonymous consensual criterion (which is, I
think, the more interesting case), it is not certain that there exists a consensual
optimum that is Paretooptimal or a bequestrule plan that is Paretooptimal.
Nevertheless, the following propositions should help answer the question of
Paretooptimality of a bequestrule plan in some practical cases. They give
sufficient conditions for Paretooptimality.
Proposition 14 Let K be a bequestrule plan. Take ∆k1Â 0. For n ≥ 1define
the sequence (∆kn) as follows:
Un+1(kn− ∆kn,kn+1− ∆kn+1) = Un+1(kn,kn+1)
If for all ∆k1 Â 0 the sequence (∆kn+1
optimal.
∆kn) is increasing, then K is Pareto
Remark 15
ep+ ε , this would be anyway a proof of the Paretooptimality of K.
If we had only for every ∆k1Â 0 an ε Â 0 such that
∆kn+1
∆kn
Â
10
Page 11
Proposition 16 Let α be a real, m a positive integer and (rn) a sequence of
reals such that α Â 1, limrn= 0 and
and a steady state K in
Dπ such that G is linear at infinity at K in lπ
such that:
−U0
U0
X
rn≺ +∞. If there is a real π in [0,p]
◦
∞and
nl(K)
nh(K)ep= 1 −
α
n + 1+ rn+1for all n ≥ m (2)
then K is a Paretooptimal bequestrule plan.
5 A discretetime example
5.1Introduction
Consider the laborsaving technical change case with a unique period. The
agent gets born in the beginning of the period, immediately inherits a capital
h and begins to produce with this capital. At the end of the period, the agent
consumes c, the capital is depreciated of ah, the agent bequeaths l and dies.
Consumption of generation giis c = F(Li−1,h) − ah − (l − h) where F is
the production function and L is the exogenous laborsaving technical change
factor (L Â 1).
The satisfaction level achieved by the generation giis then
Ui(h,l) = u(c) = u¡F(Li−1,h) − ah − (l − h)¢
with the constraint c ∈£0,F(Li−1,h)¤.
duction and utility functions, u and F are strictly concave, increasing, continu
ous and twice derivable on their definition domains. Then, we check easily that
all needed assumptions on (Ui)i≥1are fulfilled.
Suppose F homogenous (F(λX,λY ) = λF(X,Y ) ), F(1,0) ≥ 0, u(0) ≥ 0
and limy→0D2F(1,y) Â a + L − 1.
There is not much lost of generality in the two latter assumptions since we
don’t change the problem when we add a constant to u and if limy→0D2F(1,y) ≤
a + L − 1, we will see further that the productivity of capital would be so low
that it will not be interesting any more to accumulate.
Also without loss of generality, suppose, to simplify, that the startup capital
k0is strictly positive.
Suppose that a ∈ ]0,1[. It means that capital does depreciate, but it can
never disappear completely from the only fact of its depreciation.
Lastly, suppose limy→+∞D2F(1,y) ≺ a . As we shall see, this guarantees
the geometricity of bequests. It means that for the first generation, from a
given level of accumulation, marginal productivity falls under the rate of capital
depreciation. Consequently, at this level, it would not be worth accumulating
any more.
(3)
Suppose that, on top of meeting standard assumptions of respectively pro
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Page 12
Denote f(k) = F(1,k). Then
Ui(h,l) = u
µ
Li−1f(
h
Li−1) − ah − (l − h)
¶
and the definition domain of Uiis
Di=
½
(h,l)/Li−1f(
h
Li−1) − ah − (l − h) ≥ 0
¾
For subsections 5.3 and 5.4, we will adopt the following assumptions:
G(D) is strictly geometric of reason p1
(4)
G is linear at infinity on
◦
D for the reasons (LogL,p1) (5)
where Log denotes the Napierian logarithm.
The proofs of this section are in appendix A.
5.2Geometricity
Proposition 17
strictly geometric of reason L.
Under the assumptions of subsection 5.1 on u and F, D is
Proposition 18 Under the assumptions of subsection5.1 on u and F, G(D) ⊂
lLogL
∞
.
The above proposition indicates that if G(D) admits a strict reason, it is
lower than LogL. But it doesn’t give the exact reason of G(D). For this, it is
necessary to specify u, what I do in subsection 5.5.
5.3Consensual optimality
Denote
w∗= f0−1(a + L − 1)
Proposition 19 Under assumptions of subsection 5.1, there is i ≥ 1 and
(k∗
such that the plan K∗=¡k∗
1,k∗
i,Li+1w∗,Li+2w∗,...¢is interior to D.
2,...,k∗
i)
1,k∗
2,...,k∗
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Page 13
Let χ Â 0 such that S (K∗,χ) ⊂ D. Let K be a plan such that
K ∈
◦
S (K∗,χ) (6)
and that
limkn
Ln= w∗
(7)
then:
Proposition 20 For a consensual criterion meeting the assumptions of subsec
tion 1.2 and under the assumptions of subsection 5.1 on u and F and assump
tions (4), (5), (6) and (7), K is an interior bequestrule plan.
5.4Paretooptimality
Let ξ be a real and (rn) a sequence of reals such that ξ Â 1, limrn= 0 and
X
xn= a +
³
We have limxn = a + L − 1. Therefore, we can define a bequests plan
K∗∗such that (6) is checked and that, from a given index, we have:
f0−1(xn). K∗∗is built of kind to meet the assumptions of propositions 20 and
16.
rn≺ +∞. Denote
1
1 −
ξ
n+1− rn+1
´L − 1
k∗∗
n−1
Ln−1 =
Proposition 21 Under the same assumptions that proposition 20, K∗∗is an
interior Paretooptimal bequestrule path.
5.5Checking of assumptions (4) and (5) for two particular
utility functions
Assumptions (4) and (5) doesn’t necessarily hold for all functions u and f. For
example, (4) doesn’t hold for u(c) = Logc.4
I have studied the case of an hyperbolic function of utility : u(c) = αc +
1 −
except for assumptions (4) and (5), it is easy to see that all the other desired
1
c+1with α Â 0 and the case u(c) = c1−θwith θ in ]0,1[. In the first case,
4It is probable that these questions are primarily of a mathematical nature. With bet
ter mathematics, it should be possible, I believe, to extend the results to these cases.
For example, for the case u(c) = Logc, we could plunge G(D) in the space lp1
n
We have also to change the criterion Ψ.
∞(n) =
bi
ep1ii.
B = (b1,b2,...)/ bi∈ R and supi≥1
bi
ep1ii≺ +∞
o
and use the norm kBk = supi≥1
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assumptions hold. But in the case u(c) = c1−θ, Uiis not differentiable at 0.
However, one can check that for a bequests plan K which is "candidate" to be
an optimal growth path, it is possible to find a strictly positive real ε such that
inf
i≥1
B∈S(K,ε)
ciÂ 0
Thus, assumptions of subsection 1.2 are fulfilled on S (K,ε), which is enough
for the validity of the results.
In appendix A, I give four propositions that prove that conditions (4) and
(5) are checked, respectively for the case u(c) = αc + 1 −
and 27) and u(c) = c1−θ(propositions 28 and 29).
1
c+1(propositions 26
5.6The neutral case
Consider now the same discretetime example, but with neutral technical change
(see [Solow 1956]), and with a CobbDouglas production function5. The utility
of generation giis in this case : Ui(h,l) = u¡Ni−1fhη− ah − (l − h)¢, where η
exogenous neutral technical change factor (N Â 1).
Although this case is not presented in detail here (because of the length of
the calculus), it is interesting to quote the following results:
is the share of the income of the capital in the total production and N is the
• As in the LS case, asymptotically, the bequestrule path doesn’t depend
on the utility function, as long as the needed assumptions hold.
• Capital and production grow at the rate N
change, whereas growth rate is the same for capital, production and tech
nical change in the LS case.
1
1−η , faster than technical
6Some implications of the bequest rule
6.1Golden rule
The following analysis is based on the example of section 5.
The equation (7) indicates that the marginal productivity of capital mpc =
D2F (Ln,kn) = Lnf0¡kn
technical change ([Mabrouk 2005]). This means that the optimal level of capital
with technical change is always lower than the one without technical change,
Ln
¢tends to a + L − 1. The equation mpc∗= a + L − 1
replaces the golden rule mpc0 = a characterizing the optimal path without
5Contrary to the LS case, marginal productivity has to tend to 0 for infinite capital,
which is the case for the CobbDouglas function. Otherwise, economy would grow faster than
geometrically.
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more precisely, the level of capital which would be optimal if technical progress
had suddenly stopped.
Why isn’t it optimal to reach the level that guarantees mpc0= a, despite
that, at the level mpc∗= a+L−1, an increase of capital by one unit implies a
net increase of production?
The answer is that it would be too expensive for a father to bequeath a
capital meeting the golden rule of his son. Indeed, with technical change, the
goldenrule level of capital "flees". The source of nonoptimality in trying to
catch up with it, is that the effort made by a generation to enhance the satisfac
tion of its heir, hurts its own satisfaction more than what this generation gains
from a similar effort by its predecessor.
In the neutral case, like in the LS case, the optimal marginal productivity
of capital (mpc∗) is equal to a + N − 1, where N denotes the rate of neutral
technical change.
In addition, asymptotically, the optimal growth path doesn’t depend on the
choice of utility function, as long as the needed assumptions hold.
6.2Welfare analysis
I try here to assess to what extent a behavior led by personal incentives is
coherent with the optimal growth path. The observation that bequest consti
tutes ultimately the current shape of transmission of capital from generation
to generation, led me to give to bequests a crucial role in the model.
though other bequestmotives exist and other kinds of altruism can be used
(see [SaezMartiWeibull 2005, Lakshmi 2002, Barro 1974]), the proposed wel
fare analysis is limited to the introduction of a fatherson altruism, additively
separable from selfish utility and supposed to justify bequests.
Let’s call "spontaneous equilibrium" the state of the economy achieved when
agents behave according to their personal incentives. I do not use the name
"competitive equilibrium" because it refers to the general equilibrium approach
and to the assumptions which are attached to it, in particular the assumption of
competition and pricetaker behavior. The latter assumption is not adapted to
a model which disregards the space dimension of the economy and where each
agent is alone and cannot thus consider that the prices are imposed to him.
The question then amounts to find out what must be the intensity of the
fatherson altruism so that the spontaneous equilibrium it achieves coincides
with the optimal growth path.
As in [Mabrouk 2005], personal incentives are modeled by a utility function
of the form:
Vn(h,l) = Un(h,l) + An(l)
Al
where Unis the selfish component of generation gn’s utility and Anis the al
truistic component. Anrepresents the feelings of generation gnfor its heir and
is supposed to be increasing with respect to l, C1and strictly concave. Conse
quently, the altruism expressed by Anis limited to the agent’s family. That’s
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