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Allais-anonymity as an alternative to the

discounted-sum 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 discounted-sum criterion in the calculus of

optimal growth: II Pareto-optimality and some economic interpretations"

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Abstract

This paper studies the Pareto-optimality of the consensual optimum estab-

lished in "Allais-anonymity as an alternative to the discounted-sum criterion I:

consensual optimality" ([Mabrouk 2006a]). For that, a Pareto-optimality 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 bequest-rule path is Pareto-optimal. Through

an example, it is then shown that the golden rule must be checked to achieve

Allais-anonymous optimality.

The introduction of an additive altruism makes it possible to highlight the

intergenerational-preference rate compatible with Allais-anonymous optimal-

ity. In this approach, it is not any more the optimality which depends on the

intergenerational-preference rate, but the optimal intergenerational-preference

rate which rises from Allais-anonymous optimality.

∞. That makes it possible to find

JEL classification: D90; C61; D71;D63; O41; O30.

Keywords: Intergenerational anonymity; Allais-anonymity; Intergenerational

equity; Optimal growth; Technical change; Time-preference; Discounted-sum

criterion; Consensual criterion; Pareto-optimality; OG economy.

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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 Allais-anonymous con-

sensual criterion) partly characterized in the article "Allais-anonymity as an al-

ternative to the discounted-sum criterion I: consensual optimality" [Mabrouk 2006a]

and of which this article constitutes the prolongation.

The criterion of Pareto-optimality 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 Pareto-optimality. Using a suitable

adaptation of the variables, one realizes that this criterion is in fact nothing but

a particular case of the Pareto-optimality criterion of [Cass 1972] or the Pareto-

optimality criterion of [Balasko-Shell 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 Pareto-optimality in the case of nonregular bequests

plans, whereas it is the case for Cass and Balasko-Shell criteria. But the am-

bition here is not to establish a complete characterization of Pareto-optimality

following the example of [Cass 1972] or [Balasko-Shell 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 Pareto-optimality. 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 non-interior.

In absence of a general result on the Pareto-optimality 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 Pareto-optimal. 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 Pareto-optimality.

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 golden-rule 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 Allais-anonymous and discounted-sum criteria (subsection 6.3).

All proofs are gathered in section 7 except those relating to the discrete-time

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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 life-utility 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≥1|bi|e−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 Frechet-differentiable at K.

Consider a consensual criterion represented by a real valued, Frechet-differentiable

functional Ψ on lp1

∞.

Ψ(G(K)). Suppose also that Ψ is Allais-anonymous 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 non-empty interior; Ui

ihÂ 0 (U0

∞=©B = (b1,b2,...)/bi∈ R andsupi≥1|bi|e−ri≺ +∞ª

ihand U0

ilare respectively the

◦

∞with respect to the norm kBkp= supi≥1|bi|e−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 bequest-rule 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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2 Pareto optimality

2.1 Introduction

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 Frechet-differentiable at K. This implies

that Hiand Uiare also Frechet-differentiable 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 Frechet-differential.

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 Karush-Kuhn-Tucker 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 Pareto-optimal 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 Pareto-optimal 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 Pareto-optimal 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 Pareto-optimal 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.

3 If 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 |kn|e−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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3.2 Necessity:

Proposition 9

that Pareto-dominate 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 Pareto-optimal in Dπ, it is also

Pareto-optimal 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 Pareto-optimal 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

¯¯¯¯¯≺ +∞

3.3 Sufficiency

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 Pareto-optimal 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 Pareto-optimal 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 ∈

4 Is a consensus-optimal plan Pareto-optimal?

An optimal growth path has to be at the same time consensus-optimal and

Pareto-optimal. We have then to select from the set of consensus-optima those

which are Pareto-optimal.

There is not here a general result on the Pareto optimality of consensual

optima or on the Pareto-optimality of bequest-rule plans as in the case without

technical change [Mabrouk 2005].

Indeed, in the case of an Allais-anonymous consensual criterion (which is, I

think, the more interesting case), it is not certain that there exists a consensual

optimum that is Pareto-optimal or a bequest-rule plan that is Pareto-optimal.

Nevertheless, the following propositions should help answer the question of

Pareto-optimality of a bequest-rule plan in some practical cases. They give

sufficient conditions for Pareto-optimality.

Proposition 14 Let K be a bequest-rule 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 Pareto-optimality of K.

If we had only for every ∆k1Â 0 an ε Â 0 such that

∆kn+1

∆kn

Â

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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 Pareto-optimal bequest-rule plan.

5 A discrete-time example

5.1Introduction

Consider the labor-saving 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 labor-saving 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 start-up 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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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.

5.2 Geometricity

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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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 bequest-rule plan.

5.4Pareto-optimality

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 Pareto-optimal bequest-rule 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 discrete-time example, but with neutral technical change

(see [Solow 1956]), and with a Cobb-Douglas 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 bequest-rule 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

6 Some 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 Cobb-Douglas 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

golden-rule level of capital "flees". The source of non-optimality 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.

6.2 Welfare 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 bequest-motives exist and other kinds of altruism can be used

(see [Saez-Marti-Weibull 2005, Lakshmi 2002, Barro 1974]), the proposed wel-

fare analysis is limited to the introduction of a father-son 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 price-taker 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

father-son 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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