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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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Page 14

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.

14

Page 15

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

15

Page 16

why it was called "familial altruism" in [Mabrouk 2005]. Hence, despite the

presence of this familial altruism component, Vncan still be considered as an

expression of personal incentives.

Let K be an interior steady-state optimal growth path and suppose that

U0

nl(kn−1,lmaxn(kn−1)) + A0

n(lmaxn(kn−1)) ≤ 0 (8)

and that

U0

nl(kn−1,lminn(kn−1)) + A0

n(lminn(kn−1)) ≥ 0 (9)

Assumptions (8) and (9) make sure that each of Un and An is operative

when gnis to behave according to its personal incentives.

Proposition 22 Under assumptions (8), (9) and assumptions of subsection

1.2, if the interior steady-state optimal growth path K coincide with a sponta-

neous equilibrium then:

u0

h= a0ep

(10)

where epis the maximum growth rate of capital, u0

limu0

utility.

h= limu0

hne−(p1−p)n, u0

l=

lne−(p1−p)n, a0= limA0

ne−(p1−p)nand ep1the maximum growth rate of

The left hand-side of equation (10) is the increase of selfish utility resulting

from an increase of heritage by one unit. The right hand-side is eptimes the

increase of altruistic utility resulting from an increase of bequest by one unit.

To interpret this, observe that the ratio

s =u0

h

a0

(11)

measures selfishness with respect to the heir. Indeed, if one was to choose

between an increase of his own heritage by one unit and the increase of his

heir’s heritage by one unit, one should assess the selfishness ratio s. If s ≥ 1,

the increase of own heritage is better and conversely if s ≤ 1.

Thus, we can write (10) in another way:

s = ep

(12)

what tells that one can be all the more selfish as technical progress, or more pre-

cisely, maximum capital growth rate, is significant. Much more, it is imperative

to be more selfish if technical progress increases.

If s Â ep, agents being too selfish, we can expect accumulation to be insuf-

ficient.

If s ≺ ep, agents are not selfish enough but it is not certain wether the

economy would go over-accumulated, which means that marginal productivity

is lower than the capital depreciation ratio a. As said above (subsection 6.1)

16

Page 17

the source of non-optimality could be that agents damage themselves in trying

to catch up with the level of capital meeting the golden rule of next generation.

The consequence is that everybody is worse off!

To some extent, relation (12) indicates that technical progress compensates

for selfishness.

6.3 Comparison between discounted-sum and Allais-anonymous

criteria

In this subsection, we place ourselves under the assumptions of subsection 1.2

and the assumption (A3).

Another way to present the relation between selfishness and technical change

is to define the altruistic component of utility as next generation utility dis-

counted by an own-generation-preference rate ρ ([Groth 2003, Heidjra-van der Poeg 2002,

Barro 1974]):

An=1

ρ(Un+1+ An+1)

ρVn+1=1(13)

Define as myopic spontaneous equilibrium the state of the economy achieved

when each generation gn maximizes Un+1

(kn+1,kn+2,...) don’t depend on its control variable: kn, and define as rational-

expectations spontaneous equilibrium the state of the economy achieved when

each generation takes fully into account the changes of behavior of all posterior

generations, induced by the variation of its own control variable.

ρ(Un+1+ An+1) considering that

Proposition 23 Let Kmbe an interior steady-state myopic spontaneous equi-

librium. Then, on the path Km:

u0

hn+1+ ρu0

ln= 0(14)

Moreover, the asymptotic selfishness ratio and the asymptotic marginal rate of

substitution between heritage and bequest are both equal to:

ρ

ep1ep

The interpretation of equation (12) is then that the "optimal own-generation-

preference rate" is ep1, the maximum growth rate of utility.

Consider now the discounted-sum criterion:

Φ(G(K)) =

+∞

X

1

Un(kn−1,kn)

ρn

So that the definition domain of Φ contains lp1

ρ Â ep1.

∞, it is necessary to have

17

Page 18

Proposition 24 Let Kdbe an interior steady-state consensual optimum for

the criterion Φ. Then, the path Kdchecks equation (14) and the asymptotic

marginal rate of substitution between heritage and bequest is the same than that

of Km.

Proposition 25 Let Krbe a rational-expectations spontaneous equilibrium with

ρ as own-generation-preference rate. Then equation (14) is checked on the path

Kr.

Proposition 25 shows that, using the same own-generation-preference rate ρ,

equation (14) characterizes also rational-expectations equilibria.

Consequently, since equation (14) characterizes at the same time rational-

expectations equilibria, discounted-sum optima and myopic equilibria, the three

concepts are equivalent (It is not worth being rational!). Thus, since ep1is the

optimal own-generation-preference rate, the condition ρ Â ep1shows that the

optimality defined with a discounted-sum criterion will never coincide

with that defined with an Allais-anonymous criterion.

Economic intuition suggests that the discounted-sum criterion constitutes a

short-run criterion compared to the Allais-anonymous criterion, supporting the

close generations compared to those remote. I have not a general formal proof

for that but, nevertheless, it is easy to check in the discrete-time example of

section 5. Indeed, the discounted-sum optimum Kdchecks

kd

Ln−→ w

n

where w is such that f0(w) = a +

w ≺ w∗(w∗is defined in subsection 5.3). As a result, for n large enough,

Un

nn

subsection 5.4. Moreover, the form of Φ implies, in an obvious way, that Kd

is Pareto-optimal. We can thus deduce that close generations are necessarily

better off in the discounted-sum optimum Kdwhile distant generations are

better off in the Allais-anonymous optimum K∗∗.

ρ

ep1ep− 1. But

¢, where K∗∗is the bequest-rule path defined in

ρ

ep1epÂ epimplies that

¡kd

n−1,kd

¢≺ Un

¡k∗∗

n−1,k∗∗

7 Proofs

7.1Proofs for section 2

Proof of proposition 1: Observe that Hi(K) = 0. Therefore, the question

amounts to find X ∈ lp

According to lemma 22 [Mabrouk 2006a], sp1

X Â 0 ⇔ δHi(K)·X ∈ sp1

for n ∈ [1,i − 1] :

(δHi(K) · X)n= u0

∞such that δHi(K) · X Â 0.

∞++=

◦

lp1

∞+. As a result, δHi(K)·

∞++. If we take, for convenience, x0= 0, we can write

hnxn−1+ u0

lnxn

18

Page 19

and for n ≥ i:

(δHi(K) · X)n= u0

hn)n≥1∈ sp1−p

hn+1xn+ u0

ln+1xn+1

also, we have : (u0

(u0

ln)n≥1.

∞

⇔ liminf |u0

hn|e−(p1−p)nÂ 0, similarly for

If R?≺ ep:

There is ε Â 0 such that

ep

R?+ ε− 1 Â 0

and N integer such that n ≥ N =⇒

u0

−u0

hn

ln

≺ R?+ ε

This gives

ep

u0

hn

−u0

ln

− 1 Â

ep

R?+ ε− 1 Â 0

Take x1≺ 0 and xi≺ 0. For n ∈ [2,i − 1]∪ [i + 1,N] choose xn≺ 0 such

that

u0

hn

−u0

Thus, u0

lnxnÂ 0 for n ∈ [1,i − 1]∪ [i + 1,N] (with x0= 0).

Take xn= xNe(n−N)pfor n ≥ N. Thus, (δHi(K) · X)n−1=

µ

−xNu0

xn≺

ln

xn−1

hnxn−1+ u0

u0

hnxn−1+ u0

lnxn

=u0

hn

xn−1+u0

ln

u0

hn

µ−u0

xn

¶

=

hne(n−1)p

ln

u0

hn

ep− 1

¶

Â 0

and, if we suppose that (u0

hn)n≥1∈ sp1−p

∞

liminf [u0

hnxn−1+ u0

lnxn]e−(n−1)p1

≥−xNep1−pliminf u0

hnen(p−p1)

µ

ep

R?+ ε− 1

¶

Â 0

We have then found X in lp

of (u0

∞such that δHi(K) · X Â 0. If we suppose, instead

, that (u0

∞

hn)n≥1∈ sp1−p

∞

ln)n≥1∈ sp1−p

, we have the same result.

If R?Â ep

There is ε Â 0 such that

1 −

ep

R?− εÂ 0

19

Page 20

and N integer such that n ≥ N =⇒

u0

−u0

hn

ln

Â R?− ε

This gives

1 −

ep

u0

hn

−u0

ln

Â 1 −

ep

R?− εÂ 0

Take x1≺ 0 and for n ∈ [2,i − 1] choose xnsuch that

xn≺

u0

−u0

hn

ln

xn−1

Take xiÂ 0. For n ∈ [i + 1,N] choose xnÂ 0 such that

xn≺

u0

−u0

hn

ln

xn−1

Thus, u0

Take xn= xNe(n−N)pfor n ≥ N. Thus, (δHi(K) · X)n−1=

hnxn−1+ u0

lnxnÂ 0 for n ∈ [1,i − 1]∪ [i + 1,N] (with x0= 0).

u0

hnxn−1+ u0

lnxn

=u0

hn

µ

hne(n−1)p

xn−1+u0

ln

u0

µ

hn

1 −−u0

xn

¶

=xNu0

ln

u0

hn

ep

¶

Â 0

and, if we suppose that (u0

hn)n≥1∈ sp1−p

liminf [u0

∞

hnxn−1+ u0

hlxn]e−(n−1)p1

≥xNep1−pliminf u0

hnen(p−p1)

µ

1 −

er

R?− ε

¶

Â 0

We have δHi(K) · X Â 0. Similarly if (u0

ln)n≥1∈ sp1−p

∞

¥

Proof of proposition 3: Suppose that (A’1) holds. Let K ∈

(L−∪ L+) such that (A3) and (A4) hold.

Tucker theorem, if K is a solution of Pi(K) then there is λ∗∈ lp1∗

all ∆K in lp

∞

δUi(K) · ∆K + hλ∗| δHi(K) · ∆Ki = 0

with λ∗≥ 0.

Now apply lemma 2 of [Mabrouk 2006a]:

◦

D ∩ Dl∩

According to the Karush-Kuhn-

∞such that, for

(15)

λ∗= λ + β

where λ = (λn)n≥1∈ lp1

to δp1

∞.

1and β is such that its restriction to cp1is proportional

20

Page 21

Let ∆K be in cp

hn)n≥1and (u0

0. We see that δUi(K)·∆K is in cp1

ln)n≥1are in lp1−p

0, and since the sequences

(u0

∞

(see proposition 10 of [Mabrouk 2006a]),

lim

n→+∞|u0

|u0

lim

hn∆kn−1+ u0

ln∆kn|e−np1

n→+∞|u0

lim

n→+∞

thus, δHi(K) · ∆K is also in cp1

hβ | δHi(K) · ∆Ki = 0.

Replace λ∗by λ + β in (15):

ln∆kn|e−np1≤lim

n→+∞|u0

hn∆kn−1|e−np1+

=

hn|e−n(p1−p)|∆kn−1|e−np+

kU0

¯¯¯u0

lne−n(p1−p)¯¯¯e−n(p1−p)|∆kn|e−np

= 0

≤

h

hkp1−p∆kn−1e−np+ kU0

lkp1−p∆kne−npi

0. It implies that hβ | δUi(K) · ∆Ki = 0 and

δUi(K) · ∆K + hλ | δHi(K) · ∆Ki = 0

for all ∆K in cp

0. By development and identification, we obtain:

λi−n

=(−1)nu0

hi...u0

u0

hi−n+1

li−n

u0

hi+1...u0

li−1...u0

for n ∈ [1,i − 1] (16)

λi+n

=(−1)n+1

li...u0

li+n

u0

hi+n+1

for n ∈ [0,+∞[

Since λ is in lp1

(16):

1we haveP+∞

n=1|λn|ep1n≺ +∞. Replace (λj) by its value in

¯¯¯¯¯

+∞

X

n=0

n

Y

j=0

u0

u0

li+jep1

hi+j+1

¯¯¯¯¯≺ +∞(17)

If K is Pareto-optimal, the inequality (17) holds for all i ≥ 1.

Now drop the assumption (A4). We show the same way than in [Mabrouk 2005]

that (17) is still a necessary condition for Pareto-optimality:

Let J = {j/U0

If K is Pareto-optimal, the bequests plan extracted from K and beginning at

the generation gq+1: (kq+1,kq+2,···) is also necessarily Pareto-optimal when

we take (k0,k1,···kq) as fixed parameters.

extracted plan is also in

Dkq∩ (L−∪ L+) and (kq,kq+1) ∈

Q+∞

X

n

nl(kj−1,kj) = 0}. If J is up-bounded, let q = maxJ.

If K is in

◦

D ∩ (L−∪ L+), the

◦

Dq+16.

◦

Since

j=q+1U0

gq+1as first generation. This gives, for all i ≥ 1:

nl(kj−1,kj) 6= 0, it verifies necessarily the condition (17), but with

+∞

n=0

n

Y

j=0

¯¯¯¯¯

u0

u0

lq+i+jep1

hq+i+1+j

¯¯¯¯¯≺ +∞

6Dkqis the set

B = (bq+1,bq+2,···) ∈ lp

∞+/ ∀i ≥ 1 : (bq+i−1,bq+i) ∈ Dq+i

o

21

Page 22

Multiply the above inequality with

q−1

Y

j=0

¯¯¯¯¯

u0

u0

li+jep1

hi+1+j

¯¯¯¯¯

we find again the inequality (17).

If J is not up-bounded, there is episodically a q such that U0

0. Consequently, in the sumP+∞

nl(kq−1,kq) =

p=0

Qp

n=0

¯¯¯

u0

u0

li+nep1

hi+1+n

¯¯¯ there is a finite number of

nonzero terms. Thus, for all i ≥ 1 the sum converges¥

Proof of proposition 4: The first step is to show the stationarity of the

◦

D and i ≥ 1 define the Lagrangian Lifrom D × lp1

R as follows Li(B,µ) = Ui(B) + hµ | Hi(B)i

Qi−1

+∞

X

then we can see that λ ∈ lp1

to B at B = K. Its differential, computed at µ = λ and B = K is δLi(K,λ) =

δUi(K) + hλ | δHi(K)i. Since λ ∈ lp1

Lagrangian. For K ∈

1 to

7, and suppose, if i Â 1, that

j=1u0

If

lj6= 0. The system (16) defines a sequence λ.

n=0

n

Y

j=0

¯¯¯¯¯

u0

u0

li+jep1

hi+j+1

¯¯¯¯¯≺ +∞

1. Like Uiand Hi, Liis differentiable with respect

1, for ∆K ∈ lp

∞we have

δLi(K,λ) · ∆K=δUi(K) | ∆K + λ | hδHi(K) · ∆Ki

δUi(K) | ∆K +

=

lim

n− →+∞(λ1,λ2,··· ,λn) | hδHi(K) · ∆Ki

Replace (λj) by their values in (16), we obtain

δUi(K) | ∆K + (λ1,λ2,··· ,λn) | hδHi(K) · ∆Ki = λnu0

ln+1∆kn+1

which tends to 0 when n tends to infinity.

Thus δLi(K,λ) = 0. In other words, the Lagrangian Li is stationary at

(K,λ).

The second step is to deduce optimality from stationarity.

Let i be such that if i Â 1 we have:Qi−1

result, for all B ∈ D and α ∈ ]0,1[ we have:

j=1u0

lj6= 0. From the system (16), we

deduce that λj≥ 0 for all j ≥ 1. Thus, Liis concave with respect to B. As a

Li((1 − α)K + αB,λ) ≥ (1 − α)Li(K,λ) + αLi(B,λ)

then

Li(K + α(B − K),λ) − Li(B,λ)

α

≥ Li(B,λ) − Li(K,λ)

7This is possible since lp1

1

⊂ lp1∗

∞ and Hi(B) ∈ lp1

∞.

22

Page 23

Since Liis Frechet-differentiable at K, when α tends to 0 we obtain:

δLi(K,λ) | (B − K) ≥ Li(B,λ) − Li(K,λ) (18)

We know that if

+∞

X

n=0

n

Y

j=0

¯¯¯¯¯

u0

u0

li+jep1

hi+j+1

¯¯¯¯¯≺ +∞

we have δLi(K,λ) = 0.

We then deduce from (18) that for all B ∈ D:

Li(B,λ) − Li(K,λ) ≤ 0 (19)

We now show that K solves Pi(K). Suppose there is B in D such that

Ui(B) Â Ui(K) and Hi(B) ≥ 0. We have Hi(K) = 0 so Hi(B) ≥ Hi(K). Since

λ ≥ 0 we have λ | Hi(B) ≥ λ | Hi(K). Finally Ui(B)+λ | Hi(B) Â Ui(K)+λ |

Hi(K). This contradicts (19)¥

7.2 Proofs for section 3

Proof of proposition 9: If we show that if a plan B Pareto-dominates a plan

K, then bn≤ knfor all n ≥ 1, this will prove the proposition. Suppose there

is n ≥ 1 such that bn+1Â kn+1. According to the assumptions on U, it is easy

to show that we then have bnÂ kn. Hence, by induction, b1Â k1and the first

generation g1is better off in K, which contradicts that B Pareto-dominates K¥

7.3Proofs for section4

Proof of proposition 14: Observe that, because of the concavity of Un, we

have

∆kn+1

∆kn

ÂD1Un+1(K)

D2Un+1(K)= ep

∆kn is the decrease of bequest generation gn would have to carry out to

maintain its utility level if g1lowers its bequest by a quantity ∆k1. Suppose

that the sequence (∆kn) is defined for all n ≥ 1. (∆kn+1

to l Â ep, which means that (∆kn) goes out of lp

Thus, for all decrease of first generation’s bequest ∆k1, there is a generation

gnwhich cannot maintain its utility level. We can make the same reasoning for

the decrease of any generation’s bequest. Thus K is Pareto-optimal.¥

∆kn) would then converge

∞. This contradicts D ⊂ lp

∞.

Proof of proposition 16: We see clearly that K is a bequest-rule plan.

Equation (2) shows that for n large enough we have D2Un(kn−1,kn) ≺ 0. When

K is Pareto-optimal starting from a given index, it is also Pareto-optimal start-

ing from n = 1 (see [Mabrouk 2005]). For this reason, we will suppose without

23

Page 24

loss of generality that the inequality D2Un(kn−1,kn) ≺ 0 holds starting from

n = 1. Let π1be the reason of G(Dπ). According to proposition 11 ( section

3), if

+∞

X

then K is Pareto-optimal.

Suppose first that p = p1.For n ≥ 1, we have:

Y

For n ≥ 1 denote

Πn=

n=0

n

Y

j=0

¯¯¯¯¯

u0

l1+jeπ1

u0

h2+j

¯¯¯¯¯≺ +∞

n

j=0

¯¯¯¯¯

u0

l1+jep1

u0

h2+j

¯¯¯¯¯=−u0

n+1

Y

l1ep

u0

h2+n

n+1

Y

j=2

−u0

u0

ljep

hj

j=2

−u0

u0

lj

hj

ep

and take Π0= 1. Observe that the equation (2) implies that the seriesP+∞

Πn

Πn−1

n=0Πn

fulfills Rabee-Duhamel criterion since

=−u0

ln+1

u0

hn+1

ep

Thus, it converges.

But

+∞

X

n=0

n

Y

j=0

¯¯¯¯¯

u0

l1+jep1

u0

h2+j

¯¯¯¯¯=

+∞

X

h= limu0

n=0

−u0

u0

h2+n

l1ep

Πn

Since K is steady state, we can write u0

a result, sinceP+∞

hne−(p1−p)n= limu0

hn. As

n=0Πnconverges, the series

+∞

X

n=0

n

Y

j=0

¯¯¯¯¯

¯¯¯¯¯

u0

l1+jep1

u0

h2+j

¯¯¯¯¯

¯¯¯¯¯

also converges8. Since π1≤ p1the series

+∞

X

n=0

n

Y

j=0

u0

l1+jeπ1

u0

h2+j

converges and K is Pareto-optimal.

If p 6= p1, multiply Unby e(p−p1)n. Again D and G(D) have the same reason.

Apply then the above proof to see that K is still Pareto-optimal.¥

8It is enough to consider the remainder of Cauchy to prove it.

24

Page 25

7.4Proofs for section 6

Proof of proposition 22: Generation gnsolves

max

l

Vn(kn−1,l)

which is characterized by the equation

u0

ln+ a0

n= 0

Since the spontaneous equilibrium and the interior steady-state optimal

growth path coincide, a0= lima0

only to use the bequest rule (1) to see that u0

ne−(p1−p)nexists and is equal to −u0

h= a0ep¥

l. It remains

Proof of proposition 23 ; Generation gnmaximizes

Un

¡km

n−1,km

n

¢+1

ρ

£Un

¡km

n,km

n+1

¢+ An+1

¡km

n+1,km

n+2,...¢¤

with respect to km

n, what gives equation (14). For a steady state, lim

u0

hn+1=

= ρ

u0

hn

u0

ep−p1. Then, since a0

(equation (11)) and s = limsn = ρep−p1. The marginal rate of substitution

between heritage and bequest is rn=

−u0

(14), this gives rn=

u0

ln=

1

ρu0

hn+1, the selfishness ratio is sn=

u0

a0

hn

ln

u0

hn

hn+1

u0

hn

ln=

u0

hn

u0

hn+1

u0

−u0

hn+1

ln. Thanks to equation

u0

hn+1ρ −→ ρep−p1¥

hn

Proof of proposition 24 ; Interior consensual optima for the criterion Φ

check the relation:

δΦ(G(K)) = 0(20)

With the help of proposition 14 of [Mabrouk 2006a] and noting that

0, (20) gives equation (14). The proof for the asymptotic marginal rate of

substitution between heritage and bequest is similar to that of the myopic case¥

∂Φ

∂p1∞=

Proof of proposition 25: Equation (13) implies An+1= ρAn−Un+1. By

induction

An+1

ρn+1=A1

ρ

−

n+1

X

j=2

Uj

ρj

For each value of A1there is a sequence (An)n≥1that fulfills equation (13).

We have to choose the value of A1that fits best our economic question. That

is why it is necessary to choose A1 such that

A1

ρ−

+∞

X

j=2

Uj

ρj = 0. Otherwise,

the sequence (An)n≥1would grow at the rate ρ which is larger than ep1, the

maximum growth rate of (Un)n≥1. This would imply that the selfish part of the

25

Page 26

utilityUn

Vntends to zero, which contradicts the idea of own-generation-preference.

+∞

X

We then have A1=

j=2

Uj

ρj−1 and, by induction

Vn= Un+ An=

+∞

X

j=0

Un+j

ρj

Denote Φn(G(K)) =

+∞

X

j=0

Un+j(K)

ρj

. Generation gnwill choose the capital to

bequeath to gn+1anticipating the behavior of gn+1, gn+2...Denote Kn+1(h) =

¡kn

¡h,Kn+1(h)¢

by Gn

gnsolves

max

h

The first order condition gives (rigorously, we should prove that kn

the expectation functions consistent with rational-expectations equilibria, are

derivable):

µ

n+1(h),kn

tions, to a bequest kn= h by gn. Denote¡h,kn

and (Un(kn−1,h),Un+1(h,kn+1),Un+2(kn+1,kn+2)...)

¡kn−1,h,Kn+1(h)¢.

Φn

n+2(h),kn

n+3(h)...¢

the awaited response of the posterior genera-

n+1(h),kn

n+2(h),kn

n+3(h)...¢by

¡Gn

¡kn−1,h,Kn+1(h)¢¢

n+j(h),

u0

ln+1

ρu0

hn+1+k0n

n+1

ρ

u0

ln+1+1

ρu0

hn+2

¶

+k0n

n+2

ρ2

µ

u0

ln+2+1

ρu0

hn+3

¶

+ ... = 0

(21)

Inheriting h,

gn+1bequeaths h1on the basis of its anticipations kn+1

Rational expectations imply kn

kn+1

n+2n+3

k0n+1

n+1...and (21) gives

⎡

⎣

But, since gn+1solves

n+2(h1),kn+1

n+3(h1)....

n+2(h) =

n+2k0n

n+1(h) = h1. We can then write kn

¡kn

³

ln+2+1

hn+3

+

¡kn

n+1(h)¢, kn

n+3(h) = kn+1

n+1(h)¢...Thus, k0n

n+2= k0n+1

n+1, k0n

n+3=

n+3k0n

u0

ln+1

ρu0

hn+1+k0n

n+1

ρ

⎢

u0

ln+1+1

´

ρu0

k0n+1

n+3

hn+2

´

u0

+

k0n+1

n+2

ρ

∙³

u0

ρu0

ρ

³

ln+3+1

ρu0

hn+4

´

...

¸

⎤

⎦= 0

⎥

(22)

max

h1

Φn+1

¡Gn+1

¡kn,h1,Kn+2(h1)¢¢

we can replace n by n + 1 in (21):

µ

u0

ln+1+1

ρu0

hn+2

¶

+k0n+1

n+2

ρ

"µ

u0

ln+2+1

ρu0

hn+3

¶

+k0n+1

n+3

ρ

µ

u0

ln+3+1

ρu0

hn+4

¶

(23)

...

#

= 0

26

Page 27

(22) and (23) imply

u0

ln+1

ρu0

hn+1= 0

¥

AProofs for the discrete-time example

Proof of proposition 17:

Ui(h,l)=u

µ

∙

Li−1f(

h

Li−1) − ah − (l − h)

h

Li−1) − a

¶

=uLi−1

µ

f(

h

Li−1−

µ

l

Li−1−

h

Li−1

¶¶¸

We see that

Ui(h,l) = (ULi−1)1(

h

Li−1,

l

Li−1)

where (ULi−1)1is the function obtained by replacing u by uLi−1in the equation

(3) defining U1.

This implies that

(h,l) ∈ Di⇐⇒ (

h

Li−1,

l

Li−1) ∈ D1

(24)

therefore

Di= Li−1(D1)

where Li−1(.) denotes the homothety with center O and factor Li−1.

Denote ∂Di the upper frontier of Di.

lmaxi(h) = sup{l/(h,l) ∈ Di}.

The concavity of f implies that Diis convex and ∂Diconcave. Since ∂Di=

Li−1(∂D1), ∂Diand ∂D1have the same asymptotic directions. This asymptotic

direction is limf(h)+(1−a)h

h

= 1 + limy→+∞D2F(1,y) − a ≺ 1.

Denote ∆Lthe straight line {l = L.h}.

L Â 1 , lmaxi(0) Â 0, ∂Diconcave and its asymptotic direction is strictly

smaller than 1 implies ∆L cuts ∂Dione time for each i ≥ 1. Let Mi be the

intersection point between ∆Land ∂Diand (wi−1,xi−1) the coordinates of Mi.

We have

∂Di = {(h,l)/l = lmaxi(h)} where

L(∆L) = ∆Land L(∂Di) = ∂Di+1

then

L(Mi) = L(∆L∩ ∂Di) = L(∆L) ∩ L(∂Di) = ∆L ∩ ∂Di+1= Mi+1

which implies

wi= Lwi−1and xi= Lxi−1

27

Page 28

But Mi∈ ∆L=⇒ xi−1= Lwi−1hence xi−1= wiand the coordinates of Mi

are (wi−1,wi) = Li−1w0.(1,L).

Now let K ∈ D. Suppose there is i ≥ 1 such that ki−1≤ wi−1.

Since Mi∈ ∂Di, with an heritage wi−1, generation gicould not bequeath

more than wi. It follows that with an heritage ki−1smaller than wi−1, generation

gicould not bequeath more than wi. So, ki≤ wi.

Consequently, for all j ≥ i we have kj≤ wj= Ljw0. Hence supkje−jLogL≺

+∞ and K ∈ lLogL

Suppose now that for all i ≥ 1 we have ki−1 Â wi−1. Since the line ∆L

comes out of Diat the point Mi= (wi−1,wi), the point (ki−1,Lki−1), which

belongs to ∆L, is out of Di. Thus giwith an heritage ki−1cannot bequeath as

much as Lki−1. Then, ki≺ Lki−1. This implies ki≺ Lik0for all i ≥ 1 and we

have also K ∈ lLogL

This proves that D ⊂ lLogL

We show now that the interior of D in lLogL

that LogL = inf {α/D ⊂ lα

{k0} × [k1− ξ,k1+ ξ] ⊂ D1and [k1− ξ,k1+ ξ] × [Lk1− Lξ,Lk1+ Lξ] ⊂ D2

Denote K =¡k1,Lk1,L2k1,...¢.

for all i ≥ 0, Li−1[k1− ξ,k1+ ξ] ⊂ Diand

×+∞

K is interior to D and

D is not empty¥

∞

.

∞

.

∞

.

∞

is not empty, which also implies

∞}. Take ξ Â 0 and k1such that

According to (24),

i=1Li−1[k1− ξ,k1+ ξ] = S (K,ξ). This shows that S (K,ξ) ⊂ D. Thus,

◦

Proof of proposition 18: As seen in the proof of proposition 17, Ui(h,l) =

(ULi−1)1(

Li−1,

Moreover, u concave =⇒ u(λx+(1−λ)y) ≥ λu(x)+(1−λ)u(y) for all x, y

in [0,+∞[ and λ in [0,1]. Take y = 0 and λ =1

c =x

L. Thus, L u(c) ≥ u(Lc) for all c in [0,+∞[.

For i ≥ 2, this implies

h

Li−1,

LUi−2(h

hl

Li−1).

L, then u(x

L) ≥

1

Lu(x). Denote

Ui(h,l)=(ULi−1)1(

l

Li−1) ≤ L(ULi−2)1(

h

Li−1,

l

Li−1)

=

L,lL)

We then easily prove that G(D) ⊂ lLogL

∞

¥

Proof of proposition19:

limy→+∞f0(y) ≺ a and f0(w∗) = a+L−1 imply that f (w∗) Â (a + L − 1)w∗.

Moreover if we denote x the solution of f (x) = (a + L − 1)x, we have w∗≺ x.

f (w∗) Â (a + L − 1)w∗implies that there is ξ Â 0 such that for all (y,z)

in [w∗− ξ,w∗+ ξ]2, f (x)−ax−(Ly − x) Â 0. Hence, if (kj−1,kj) is such that

³kj−1

£Lj−1(w∗− ξ),Lj−1(w∗+ ξ)¤×£Lj(w∗− ξ),Lj(w∗+ ξ)¤⊂ Dj

28

Observe that f strictly concave, f (0) ≥ 0,

Lj−1,kj

Lj

´

∈ [w∗− ξ,w∗+ ξ]2, (kj−1,kj) ∈ Dj. Thus

(25)

Page 29

Denote K the maximum bequests plan (obtained with zero consumption).

kj

Lj → x. Thus, the inequality w∗≺ x shows that asymp-

totically, the plan

is below K. Starting from k0, one can then reach

the plan¡Ljw∗¢by bequeathing a little less than the maximum bequest to be

reaches the plan¡Ljw∗¢and that the path is (k∗

We easily prove that

¡Ljw∗¢

interior to the definition domain. Suppose that generation gi+1is the one that

1,k∗

2,...,k∗

i). Then, with (25),

we deduce that the plan¡k∗

Proof of proposition 20: According to theorem 18 of [Mabrouk 2006a], a

◦

D is a consensual optimum if and only if

1,k∗

2,...,k∗

i,Li+1w∗,Li+2w∗,...¢is interior to D¥

steady state K in

u0

L+ u0

h

l= 0

Thus, if an interior steady state K checks

limu0

hn

−u0

ln

= L

it is a consensual optimum.

But

u0

−u0

hn

ln

= f0(kn−1

Ln−1) − a + 1

and (7) =⇒ limf0(kn−1

Then

Ln−1) = a + L − 1.

u0

−u0

hn

ln

−→ L

Consequently, the interiority of K being warranted by (6), K is an interior

steady state consensual optimum. ¥

Proof of

and K∗∗is a consensual optimum.

We have

proposition 21: Since limf0−1(xn) = w∗, (7) is also checked

−u0

u0

ln

hn

L=

L

´

f0³k∗∗

1 −

n−1

Ln−1

ξ

n + 1+ rn+1

− a + 1

=

We can then apply proposition 16 and conclude that K∗∗is also Pareto-

optimal¥

The following propositions set conditions (4) and (5) respectively for the case

u(c) = αc +1 −

and 29).

1

c+1(propositions 26 and 27) and u(c) = c1−θ(propositions 28

29

Page 30

Proposition 26 In the case u(c) = αc + 1 −

with p1= LogL.

1

c+1, assumption (4) is checked

Proof : We have

Ui(bi−1,bi) = u

Let ci= Li−1³

Li−1

also in lLogL

∞

c =1

L[f(k) − ak − (Lk − k)]. Since

we can choose k such that [f(k) − ak − (Lk − k)] Â 0. Then

ci

=

Lic

u(ci)

Li

−→

and the sequence G((kLn)) = (u(ci)) is convergent and strictly of reason LogL¥

∙

Li−1

µ

f(bi−1

Li−1) − abi−1

³

We see easily that Ui(bi−1,bi) = u(ci) is

Then G(D) ⊂ lLogL

◦

D 6= ∅,

Li−1−

µ

´´

bi

Li−1−bi−1

Li−1

¯¯¯bi−1

¶¶¸

¯¯¯ ≤ kBkand

f(bi−1

Li−1) − abi−1

Liis bounded.

Li−1−

bi

Li−1−

bi−1

Li−1

. Since

Li−1

¯¯

bi

¯¯≤ LkBk,

ci

.

∞

. Let (kLn) be a sequence in D. Let

£Li−1(f(k) − ak − (Lk − k))¤

αc +1

Li−

Li(cLi+ 1)

αc when i −→ +∞

=

=

1

Proposition 27 In the case u(c) = αc + 1 −

Proof: Let’s start with two preliminary remarks. First, since f0is positive

and decreasing, f0(x) −→ f0

formula, for x,y ≥ h0, there is z in [x,y] such that

f00(z) =f0(x) − f0(y)

1

c+1, assumption (5) holds.

∞≥ 0 when x −→ +∞ . According to Taylor’s

x − y

Make x −→ +∞. It comes that

lim

z− →+∞f00(z) = 0

Secondly, since maximum consumption is the production Li−1f¡

strictly positive for all B ∈ D and n ≥ 1.

In addition, u0(c) = α +

h

Li−1

¢, min-

imum bequest is l = (1 − a)h. Hence, with a start-up capital k0 Â 0, bn is

1

(1+c)2 , u00(c) =

−2

(1+c)3 and:

U00

ih2(bi−1,bi)=u00(ci)

∙

f0

∙

µ

f0

h

Li−1

µ

¶

− a + 1

¶

¸2

+ u0(ci)f00¡

¸

h

Li−1

Li−1

¢

(26)

U00

ihl(bi−1,bi)

U00

il2(bi−1,bi)

=−u00(ci)

u00(ci)

h

Li−1

− a + 1

=

30

Page 31

K∗∗∈

◦

D =⇒there is β0Â 0 such that S¡K∗∗,β0¢⊂

([Mabrouk 2006a], proposition 5), K∗∗∈

0. Take β00=

inf¡β0,β00¢. Then for all B in S (K∗∗,β), B ∈

Ln

Ln

an upper bound.

Because of the concavity of f, f0(y) Â a + L − 1 implies that f (w∗) −

(a + L − 1)w∗Â 0. Since f is continuous, there is γ Â 0 such that

inf

x∈[w∗−γ,w∗+γ]{f (x) − (a + L − 1)x} Â 3Lγ

and since limk∗∗

∙k∗∗

For B ∈ S¡K∗∗,γ

Ln−1,bn

Ln

◦

D. Since

◦

D ⊂ sp

∞++

n

Ln Â

◦

D =⇒ liminfk∗∗

n

Ln Â 0. Then infk∗∗

Ln ≥

D and infbn

1

2infk∗∗

n

Ln. For all B in S¡K∗∗,β00¢, infbn

¢¯¯and¯¯f00¡bn

β00

2. Let β =

◦

Ln ≥β

2Â 0. We then

see that

¯¯f0¡bn

¢¯¯are bounded when B ∈ S (K∗∗,β). Let M0be

n

Ln = w∗there is N such that

n ≥ N =⇒

n

Ln−γ

2,k∗∗

n

Ln−γ

2

¸

⊂ [w∗− γ,w∗+ γ]

2

¢and n ≥ N + 1, we then have

∈ [w∗− γ,w∗+ γ]2

µbn−1

µbn−1

¶

thus

1

L

∙

f

Ln−1

¶

− (a + L − 1)bn−1

Ln−1

¸

Â 3γ (27)

and

−bn

Ln+bn−1

Ln−1≥ −2γ (28)

(27) + (28)=⇒

1

L

∙

f

µbn−1

Ln. We then have, for all B ∈ S¡K∗∗,γ

cn

LnÂ γ Â 0

¢. With the help of equations (26), we obtain for all B ∈

2(M0+ a + 1)2

(1 + γLn)3

2(M0+ a + 1)

(1 + γLn)3

2

(1 + γLn)3

Ln−1

¶

− (a − 1)bn−1

Ln−1−

bn

Ln−1

¸

Â γ

The left hand-side iscn

2

¢and n ≥ N+1:

Let ε = inf¡β,γ

2

S (K∗∗,ε) and n ≥ N + 1:

|U00

nh2(bn−1,bn)|≤ +(1 + α)M0

Ln−1

,

|U00

nhl(bn−1,bn)|≤ and

|U00

nl2(bn−1,bn)|≤

31

Page 32

These inequalities show that we can find 3 reals M1, M2and M3such that

Ln−2|U00

2Ln−1|U00

Ln|U00

nh2(bn−1,bn)|

nhl(bn−1,bn)|

nl2(bn−1,bn)|

≤

≤

≤

M1,

M2and

M3

Thus

sup

n≥N+1

M1+ M2+ M3

∙

L2(n−1)¯¯U00

nh2(bn−1,bn)¯¯+ 2L2n−1|U00

nlh(bn−1,bn)|+

L2n¯¯U00

nl2(bn−1,bn)¯¯

¸

L−n

≤

For n in [1,N], we can use the inequalities : |u00(c)| ≤ 2 and |u0(c)| ≤ 1+α

for all c ≥ 0. In this way, we find easily a real M00such that, for n in [1,N]:

∙

For B ∈ S (K∗∗,ε) and X ∈ lLogL

kΘ(B,X)kLogL=

∙¯¯¯¯

≤

n≥1

≤

Thus, linearity at infinity holds¥

L2(n−1)¯¯U00

nh2(bn−1,bn)¯¯+ 2L2n−1|U00

nlh(bn−1,bn)|+

L2n¯¯U00

nl2(bn−1,bn)¯¯

∞

¸

L−n≤ M00

such that kXk ≤ 1, we can write

sup

n≥1

x2

n−1U00

L2(n−1)¯¯U00

nh2(bn−1,bn) + 2xn−1xnU00

x2

nl2(bn−1,bn)

nh2(bn−1,bn)¯¯+ 2L2n−1|U00

sup(M00,M1+ M2+ M3) = M

nlh(bn−1,bn)+

nU00

¯¯¯¯L−n

¸

L−n

sup

∙

nlh(bn−1,bn)|+

L2n¯¯U00

nl2(bn−1,bn)¯¯

¸

Proposition 28 In the case u(c) = c1−θ, assumption (4) is checked with p1=

LogL(1−θ).

Proof: We have seen above that the sequence (cn)n≥1is in lLogL

(cn)n≥1

reason LogL, u(cn)n≥1

p1= LogL(1−θ)¥

∞

. Thus,

u

³

´

is in lLogL(1−θ)

∞

³

and G(D) ⊂ lLogL(1−θ)

is strictly of reason LogL(1−θ). Hence, (4) holds with

∞

. When (cn)n≥1is strictly of

´

Proposition 29 In the case u(c) = c1−θ, assumption (5) holds.

Proof: u0(c) = (1 − θ)c−θ, u00(c) =

1 −

integer such that:

−θ(1−θ)

c1+θ

. As in the case u(c) = αc +

1

c+1, we build a sphere S (K∗∗,ε) such that there is β Â 0, γ Â 0 and N

ε Â 0, S (K∗∗,ε) ⊂

◦

D

32

Page 33

and

inf

B∈S(K∗∗,ε), n≥1

bn

LnÂβ

2Â 0(29)

and such that for all B ∈ S (K∗∗,ε) and n ≥ N + 1:

cn

LnÂ γ Â 0 (30)

Moreover, S (K∗∗,ε) ⊂

cnÂ 0. But

◦

D =⇒for all B ∈ S (K∗∗,ε) and n ≥ 1 we have

cn=

∙

f

µbn−1

Ln−1

¶

− (a − 1)bn−1

Ln−1− Lbn

Ln

¸

Ln−1

Thus

mn=min

(x,y)∈

∙

k∗∗

n−1

Ln−1−ε,

k∗∗

n−1

Ln−1+ε

¸

×

hk∗∗

n

Ln−ε,k∗∗

n

Ln+ε

i[f (x) − (a − 1)x − Ly]Ln−1Â 0

Denote

m = min

1≤n≤Nmn

m is strictly positive and, for all B ∈ S (K∗∗,ε) and 1 ≤ n ≤ N we have

cn≥ m

Thanks to inequalities (29), (31) and to equations (26), we see that we can

find M00such that, for all B ∈ S (K∗∗,ε) and X ∈ lLogL

∙¯¯¯¯

Thanks to (29) and (30), we have also, for all B ∈ S (K∗∗,ε) and n ≥ N +1:

θ(1 − θ)

(31)

∞

such that kXk ≤ 1:

¯¯¯¯L−n(1−θ)

sup

1≤n≤N

x2

n−1U00

nh2(bn−1,bn) + 2xn−1xnU00

x2

nl2(bn−1,bn)

nlh(bn−1,bn)+

nU00

¸

≤ M00

|U00

nh2(bn−1,bn)|≤

γθLn(1+θ)(M0+ a + 1)2+(1 − θ)M0

θ(1 − θ)

γθLn(1+θ)(M0+ a + 1) and

θ(1 − θ)

γθLn(1+θ)

γθLnθLn−1,

|U00

nhl(bn−1,bn)|≤

|U00

nl2(bn−1,bn)|≤

We can write this in another way:

|U00

nh2(bn−1,bn)|≤

M1

L(1+θ)n,

M2

L(1+θ)nand

M3

L(1+θ)n

2|U00

nhl(bn−1,bn)|≤

|U00

nl2(bn−1,bn)|≤

33

Page 34

Thus, for all B ∈ S (K∗∗,ε) and X ∈ lLogL

∙¯¯¯¯

As a result, for all B ∈ S (K∗∗,ε) and X ∈ lLogL

kΘ(B,X)kLogL=

∙¯¯¯¯

and linearity at infinity holds¥

∞

such that kXk ≤ 1:

sup

n≥N+1

M1+ M2+ M3

x2

n−1U00

nh2(bn−1,bn) + 2xn−1xnU00

x2

nl2(bn−1,bn)

nlh(bn−1,bn)+

nU00

¯¯¯¯L−n(1−θ)

¸

≤

∞

such that kXk ≤ 1,

sup

n≥1

sup(M00,M1+ M2+ M3) = M

x2

n−1U00

nh2(bn−1,bn) + 2xn−1xnU00

x2

nl2(bn−1,bn)

nlh(bn−1,bn)+

nU00

¯¯¯¯L−n(1−θ)

¸

≤

34

Page 35

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