Lipschitz continuous policy functions for strongly concave optimization problems

(1)

E U R O P E A N U N IV E R S IT Y IN S T IT U T E , F L O R E N C E

DEPARTMENT O F ECONOMICS

EUI WORKI NG PAPER

LIPSCHITZ CONTINUOUS

FOR STRONGLY CONCAVE O]

by

Luigi M

ontrucchk

* Dipartimento di Matematica • Politecnico di Torino.

This paper was presented at the Workshop in Mathematical Economics organized by the European University Institute in San Miniato, 8 T 9 September 1986. Financial support from the Scientific Affairs Division o f NATO and the hospitality o f the Cassa di Risparmio di San Miniato

me

gratefully acknowledged.

©

The

Author(s).

European

University

Institute.

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(2)

permission of the author.

(C) Luigi Montrucchio Printed In Italy In December 1986

European University Institute Badla Flesolana - 50016 San Domenico (FI) -

Italy

©

The

Author(s).

European

University

Institute.

version

produced

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(3)

Llpschltz Continuous Policy Functions for Strongly Concave Optimization Problems**

ABSTRACT

We prove that the policy function, obtained by optimizing a discounted Infinite sum of stationary return functions, are Llpschltz continuous when the Instantaneous function Is strongly concave. Moreover, by using the notion of a-concavity, we provide an estimate of the Llpschltz constant which turns out to be a decreasing function of the discount factor.

**

This research was partially supported by a grant from the Italian "Ministero della Pubblica Istruzione". An early version of this paper was discussed at IMSSS (Stanford) in the Summer Seminars 1986.

©

The

Author(s).

European

University

Institute.

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(4)

©

The

Author(s).

European

University

Institute.

version

produced

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(5)

1. Introduction

Several dynamic economic problems can be stated in terms of optimization of a discounted sum of stationary functions subject to stationary constraints. A great deal of research has been directed to finding conditions for the dynamic stability of the optimal solutions of these models. Tipically this requires a rather large discount factor (but see Araujo and Schein-kman(1977) for a notable exception).

It is well known that in these models the optimal paths are generated by a "policy function" x which maps the current state x into the

(5 t

next state x - t

(

x

)

. This policy function depends on the discount

t+I O t

factor 6 .

Recently it has been pointed out that a high discounting of the future utilities may destroy the regular dynamic behavior of the optimal paths and that the system may even reach a chaotic regime Q see Montrucchio (1986),Deneckere and Pelikan(1986).Boldrin and Montrucchio(1986)J . In

. . 2

particular we proved m Boldrin and Montrucchio(1986) that any C dynami

cal system x - 0(x ) can represent-a policy function T. when 5

t+1 t 0

is small enough.

On the other side turnpike results suggest that the policy function

becomes simpler as the discount factor 6 increases.

We propose the Lipschitz constant of the map x as a measure of

o

its degree of "complexity", i.e. the smaller the Lipschitz constant the simpler are the dynamic paths produced by x . A good measure of the' Lipschitz constant would be obtained by computing the norm of the deriva tive of x^ with respect to x . Unfortunately it is not clear wether

x^(x) is differentiable or not,even in the case in which the one-period

©

The

Author(s).

European

University

Institute.

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

Repository.

(6)

return function ic C . To be sure,only continuity of r, has been o

proved to be true under standard conditions.

The main achievement of this paper is to prove that strongly concave return functions produce,under some qualifications, policy functions which are Lipschitz continuous and .furthermore,to give an estimate of the Lipschitz constant which turns out to be a decreasing function of the discount factor.

The paper is organized as follows.

In Section 2 we introduce a general discrete-time model of optimi zation over an infinite horizon.

In Section 3 we characterize the notion of strongly concave functions, by using the a-concavity theory of Rockafellar(1976).

In Section 4 we introduce the transform 0(x,y) — >• T(t) which

associates a real function f(t) po any concave function U(x,y).

The last section contains the central results. We prove that if the return function is strongly concave the value function is strongly concave as well. Moreover one can evaluate its degree of concavity by

means of the T(t) function given in Section 4. Theorem 5.2 gives

the above mentioned estimate of the Lipschitz constant of the policy function x. . Some remarks on the implications of this result are

o

contained in the conclusive section.

©

The

Author(s).

European

University

Institute.

version

produced

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(7)

2. The Model

In this paper we will analyse the dynamic behavior of the solutions

to the problem P(xo ,«) :

00

w

■ ““

2

t=i V(x ,x ) 1 , subject to ( x ^ . x ^ C D , t - 1 , 2 .... and fixed in X . 0

Under the following assumptions :

A.l) X is a compact and convex subset of f?." ;

A . 2) V : X * X — > IR is a continuous and strictly concave function ;

A. 3) D is a closed and convex subset of X x X and pr^(D) - X ;

A . 4) 0 < 5 < 1 is the discount factor.

Under (A.1)-(A.4) problem P(x^,<5) has one and only one optimal

solution (x* ) for any given initial condition xQ in X . Moreover

the "value function" W turns out to be strictly concave on X and 6

to satisfy the Bellman equation :

W^(x) - Max | v ( x , y ) + 6W { (y) ; s.t. (x,y) 6 D j- . (2)

In the theory of Dynamic Programming the optimal sequences (x* )

is generated by the dynamical system :

©

The

Author(s).

European

University

Institute.

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(8)

x * - t5 ( x*_x> , x * - xQ given in X , (3)

where . x. : X — ^ X is a continuous map (the so-called optimal policy 6

or policy function), which depends continuously on the discount parameter

6 . (2) implies that x is obtained by maximizing V(x,y)+6W p(y) ,

6 6

that is to say :

Max | V(x,y)+6V/j(y) ; s.t. (x,y) e d| = V(x,Tfi (x)

(

t

{ (

x

))

. (4)

It is also well known that the value function turns out to

6

be the unique fixed point of the functional equation U (f) = f ,

6

where

Uj(f)(x) - Max | V(x,y) + 6 f(y) ; s.t. (x,y) e D } (5)

maps the space C^(X;

IK)

into itself. Here C (X;J^) is the space of all continuous functions endowed with the uniform topology.

is in fact a contraction operator : || U^(f)-U^(g) |j ,<d||f - g || ,

for all f,g G C^(X;|R ) . Its unique fixed point is the value function

W6 i U6(W6) - W6 , see (2) .

We recall also that the successive iterates of U ,starting from

(2) (3) 6

the zero function : U p(0) , U. (0) , U. (0) , ... yield the

6 6 6

value functions of the problems with finite horizon . In other words :

V(x^ ^,x^_) ^ ,subject to (6) and Xq fixed in X . D6(T)(0) - Y l V = “ “

tS

the constraints (x ,x ) ^ D

©

The

Author(s).

European

University

Institute.

version

produced

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(9)

As it was mentioned in the introduction,there is no conclusive

evidence about the differentiability of t

. .

However

a

heuristic way

6

to understand the methods we are using is that of looking at the one-di

mensional case ,under the assumption that W „ is twice differentiable.

--- --- g 1

If t is interior, from (4) and the implicit function theorem,

6

we have :

1 Tj /dx ■ - V1 2(x,Tfi(x)) [ v22(x,r5(x)) + 6 1

Therefore we get the estimate :

Id T.(x)/dx I 4 E ( a + S t ) * , where L • Max I V (x,y)| ,

* o o 12

o ■ Min |v (x,y) I and t. ■ Min f W"(x)| , which in turn

22 1 o o

implies :

We shall make (7) rigorous ( see Theorem 5.2 ) by giving an appropriate meaning to L ,

a ,

t. . a and t, will be related to

o o

a measure of "curvature" of V(x,.) and W (.) obtained from the

0 notion of o-concavity.

©

The

Author(s).

European

University

Institute.

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

Repository.

(10)

Although our applications involve finite dimensional spaces, Hilbert spaces are a natural setting for the theory we are developping. Therefore throughout this and the next section the functions f are defined on convex sets X of a Hilbert space H . ii • ii denotes the Hilbert

norm in H . Anagously functions U(x,y) will be defined on a set D

that is a convex subset of H * H^ , where H , H^ are real Hilbert spaces.

3. a-concavity

Definition 3.1 f is said to be a-concave on X , if

2

f(x) + (l/2)a ||x )| is concave on X , or,equivalently,if

f (txl+(l-t)x2> £ t f(x1) + (1-t) f(x2 > + (l/2)at(l-t)|Jx^-

x2 l

l

(8)

holds for any x^

,x^

e X and all t e £ o , lj

Definition 3.2 U(x,y) is said to be a - concave on D if

--- x

2

U(x,y) + (1/2) a || x II is concave on D , o r ,equivalently, if

U(tx1+(l-t)x2,ty1+(l-t)y2)5. t U f c c ^ y p + (l-t)U(x2>y 2) + (l/2)at(l-t)||x1-x2l|2

for all (x1>y1),(x2,y2 > e D and t G [o , lj . (9)

Definition 3.3 We set

(f;X) = Sup a ; f is a-concave on X J

x (U;D) * Sup | a ; U is a^-concave on dJ. (

10)

If f and U are concave : p(f;X) > 0 , p (U;D) £ 0 and x

they are termed the concavity parameter of f and U . The assumptions

©

The

Author(s).

European

University

Institute.

version

produced

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(11)

p(f;X) > 0 and p (U;D) > 0 will denote strong concavity.

We recall that in (10) : Sup { a } • M a x j a J < + 00 in both cases.

In the differentiable case,there are simple criteria to verify

a-concavity. For example t if f is twice differentiable over an open

set containing X , then f is a-concave ( a > 0 ) when

2 2

I w' D f(x) w I £ a H w II for all x e X and w e H .

We need also a notion of directional derivative which replaces

differentiability. If f is a finite concave function and x^ a point

in X , then we can define the directional derivative of f at x^ in the direction h € H , as the limit :

f ’(x0 ; h) - limt^0 + t_1 [ f(xQ + t h) - f(xo )J .

When h is feasible,i.e., ( x^, x^+th) n X

f

/ for some t>0, the above limit f'(x^;h) always exists (finite or infinite). In the

same way U^x.y;!») denotes the partial directional derivative at

(x,y) along h € . In other words :

02<x *y;h ) " lim t ^o+ c 1 £ o ( x .y +th) - U ( x ,y ) J .

Theorem 3.1 Let U : X * T + IR be a finite concave function,

where X c H and Y C are closed and convex subsets of Hilbert

spaces. Assume :

i) U(x,.) is upper-semicontinuous on Y for each x in X ;

ii) U(x,.) is a-concave on Y for each x in X ;

©

The

Author(s).

European

University

Institute.

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

Repository.

(12)

iii) | U2(x x ’y : h )-U2(x2’y:h)l ^ L |1 x x~ x2l|||h|| for any feasible

direction h € and x^,x2 € x I

then we have :

a) there exists a unique map 0 : X Y such that

Sup U(x,y) - U(x,0(x)) ;

y S Y

b) 9(x) is Lipschitz continuous on X and

ll9 ^ ) - 8(x2 )|| ^ (L/u) II Xj- x2 || , x i>x2e X

Remark 3.1 In Theorem 3.1 the feasible set D agrees with the

whole domain X x Y . In the case in which D is a strict subset of X x y , we have that if the map 0 such that Sup [u(x,y) ; s.t. (x,y)£

D

1

= U(x,0(x)) is interior , then the conclusion (b) still holds. This is easily verified by looking at the proof of the theorem.

Proof : See Appendix.

We end this section with a few comments.

1) Condition (iii) implies implicitly that U2(x,y;h) < + °° for

any (x,y,h) 6 XxYxB^ with h feasible.

2

2) If 0 is C then one can choose for L a number such that

|| U^2(x,y) || ^ L , for all ( x , y ) e X * Y , whereas a can be

2

taken such that | w ' U22(x,y)w| J a ||w || , -V- w e and (x,y)e

XxY . With these qualifications one obtains a theorem given in

Fleming and Rischel(1975; pag.170). Their assumptions are : U

©

The

Author(s).

European

University

Institute.

version

produced

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(13)

is C and X , Y are finite dimensional spaces. Our extension is not fictious because it is addressed to return functions of the type U(x,y) = V(x,y) + 5 W (y) (see (2)). As it was already

6

2 . . . 2

mentioned,even when V is C it does not imply that W is C

6 to o .

©

The

Author(s).

European

University

Institute.

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

Repository.

(14)

A. The Y - transform

Proposition 4.1 Let U :

from above ( D C X*Y C H*H^ ). Definition 3.2) .then

W(x) - Sup ^ U(x,y) ; s.t.

D - IR be finite and bounded

If U is a -concave on D (see

x

(x , y ) e d} is a-concave on X.

Proof : Take two points x^ , x^ in X and two numbers a , b such that W(Xj) > a and W(x^) > b . By assumptions there exist a

(x^.y') e D and (x^.y") e D such that W(x^)j. U(x^.y') > a

and W(x2 > J U(x2>y") > b . It follows from (9) that :

W(tXj + (l-t)) J U(tx1+(l-t)x2>ty'+(l-t)y") J t i K x ^ y ' ) +(l-t) U(x2>y") + (l/2)at(l-t) ||xl - x2/|2 > t a + (1-t) b + (l/2)at(l-t)||xl - x^ 2 .

As a tend to W(x^) and b to W(x^) respectively, we obtain

W( tx2 + (l-t)x2) j t W(Xl ) + (1-t) W(x2) +(l/2)at(l-t) ||X l - x2l|2

and this completes the proof.

Definition 4.1 Let U : D |R. be a finite concave

function. The ¥ - transform of U is the real function defined on

t

>,

0 :

Y( t ; U.D) - Sup { a 0 ; D(x,y) + ( l / 2 )a||x||2 - (l/2)t/|yl|2

is concave on D } » ( U(x,y) - (l/2)t||y|/2 ; d )

In the sequel the notations ^CtjO) , or even ¥(t) ,

of H'Ct ;U,D) may be met.

instead

©

The

Author(s).

European

University

Institute.

version

produced

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(15)

Proposition 4.2 The Y - transform of any finite concave function

U : D ♦ (R. is a continuous .increasing and concave function

on t > 0 Moreover it is bounded : 0 .$ Y(t) ,< Inf^

Proof : As one can easily verify :

Px ( ♦ c2U2 ) J c x Px (U1) * c2 Px (U2 ) (i d

for c^,c2 J 0 and ,U2 concave on D .

2

If t £ , we can write U(x,y) - (l/2)t||yll

-£u(x , y ) - (1/2) t^lyll2 ^ - (1/2) ( t - t ^ U y l f2 . and .from (1 1),

we have f(t) 5. Y ^ ) . Concavity of 'f(t) follows directly by

(11) . This implies^in turn,that fit) is continuous on (0 ,+ o o ) and that ¥(0) .f lim Y(t) as t -*■ 0+ . On the other hand, it is easily verified that p (.) is "upper semicontinuous". More precisely,

if U is a sequence of concave functions which point-wise converges n

to U , then limsup p (U ) < p (U) . This implies that

n x n ^ x

limsup f(t) < Y(0) ,as t -*• 0+ . Consequently lim Y(t) * Y(0) ,

as t -*• 0+ and therefore Y(t) is continuous at 0 too. The

last part of the statement is obvious.

Remark : ¥(0) - p^dJ >D ) - If U is strongly concave then Y(0)> 0.

Theorem 4.1 Let U(x,y) and f(y) be finite and bounded from

above on X*Y and Y respectively. If f is a-concave on Y ,

then W(x) - S u p ^ U ( x , y ) + f(y) ; s.t. (x,y) e D C XxY } is

Y(a ; U,D)- concave on X .

©

The

Author(s).

European

University

Institute.

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

Repository.

(16)

Proof : U(x,y) + f(y) can be written as £ U(x,y)-(l/2)ci|f y ||^J +

£f(y) + (1/2) a II y l . The first addendum is ^(ajlO^-concave,

by definition, whereas the second one is O^-concave . Hence U + f

is surely 'P(ct ;U) -concave on D and the statement follows by

Proposition 4.1 .

To make things more evident let us show two examples of 'F-transforms. The computations are straightforward and,therefore,omitted.

Example 1 : Quadratic case. Let X * Y = fl?n , D » X*Y

and U(x,y) = (1/2) x ’Ax + x*By + (1/2) y ’Cy be concave ( A and C

are negative definite ). The ^-transform of U is

¥(t) * the least eighenvalue of [ B ( C - t 1^) ^B* - A

J

2

Example 2 : The C one-dimensional case. Let D C lR*IR be

2

an open and convex set . Assume U(x,y) to be a concave C function

on D . We have :

l’(t) = Inf [ U ^ ( x , y ) ( U22(x,y)-t)"1- U1 1(x,y) J . (x,y)e D

Here U . . denote the second partial derivatives of U. J-J

©

The

Author(s).

European

University

Institute.

version

produced

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(17)

5. Lipschitzian Policy Functions

In this section we apply the results of Section 3 and 4 to the problem P(x^,6) discussed in Section 2. The notation is the same

of that section. p^CS)

s

p( ^;X) will denote the concavity

parameter of the value function of the truncated problem ( see (6) ). p^CS) - p( W^;X) will denote the concavity parameter of the value function of problem P(x^,5).

Lemma 5.1 Let the return function V(x,y) of P(x q,6) be

strongly concave ( i.e. p^CVjD) > 0 ) and Y(t) ■ ¥(t;V,D) be

the ^-transform of V . We have :

i) The equation ip(61) ■ t has one and only one positive solu

tion t for any 5 fixed in ( 0 , 1 ) . Moreover t_. increases

5 6

continuously as 6 increases ;

ii) the iterative system t - Y(t .) is increasingly

con-n n- 1

vergent to t. for all initial condition t €E [ o , t r) and it is

6 0 6

decreasingly convergent to t for all t €E (t ,+ ® ).

6 « 0 6

o

P roof: i) Let us write

1 At)

- V(5t) . T is again an increasing

concave function for t J. 0. Take a(t) = Y (t) - t . Since a(0) > 0

6

and a(+oo ) - - oo ,it follows that at least a t exists such that

6

It is unique. In fact,suppose not : Y^Ct^)” t^ and

, with t - 0 t and 0 < 0 < 1.

o 2 Z 1 2

But then * s (t ) “ V e V ( l~e) ' 0 9 W * * 1"9* V 0> > *•*■»

t X 0 t + (1-0) ¥ (0) , that implies T(0) .< 0 which contradicts our

1 2 o

©

The

Author(s).

European

University

Institute.

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

Repository.

(18)

continuous dependence on 6 is then easily obtained by using topological degree methods.

ii) Consider now the iterative system t ■ Y (t ,) . For all t €fO,t.)

n 6 n- 1 0 o

has ¥ (t ) € f"o , t ). In fact ,

6 0 * t 0 < t

&

6 0

V

S' S

O * V By

uniqueness it follows ? (t ) < t, .

o 0 6

Moreover V (t ) > t- holds . Indeed,suppose not : ¥,(t ) X t

o 0

6 0

^ ' 1

As ¥ (0) > 0 .there will exist another fixed point t1 of ¥ belonging

o o

to f0 , t^

J.

But this contradicts part (a). Hence the sequence of iterates will be : t < t < t < .... < t and so lim t = t, as

0 1 2

6 n

5

n ♦ oo

One can deal with the case t e (t , + oo ) by a similar method.

0 6

Theorem 5.1 Assume p (V;D) > 0 and t be defined as in

x 6

Lemma 5.1 . We have :

i) P_(«) £ P (V;D) , p(«) >. p (V;D) ;

1 X 00 X

ii) there are two possibilities : either p (6) < t for any T

T 6

and then PT <5) < PT + 1 (fi) and PT (5) ” P— (S) as T -*■ ® ,

or PT (6) J, tfi for some T Q and then

P^(S)

> t{ for all T i T

0 ’

iii) p j d ) J. t6 .

Proof : Let us use the iterative system described in Section 2 (see (5) and (6) ) :

W S,T(x) " { V ( x ’y)

*

5 W5 T - l (y) ’ S ‘t ’ (x,y) 6 D }

©

The

Author(s).

European

University

Institute.

version

produced

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(19)

From Theorem 4.1 we have :

Pt(6) £

'HS

PT - 1 («) ) for T } 2 and p ^ d ) s T(0) = p ^ V ) .

Denote by (t ) the sequence generated by t = Y(S t ) (see

n n n—1

part (ii) of Lenina 5.1). One has p^(d) £ t

,

Y - T .

This implies PT (S) ? ^CO) - p^(V) by Lemma 5.1 .

Since W converges(uniformly) to W • and p (.) is upper

6 ,T o x

semicontinuous, then p (6) « p(W„) % limsup p (6)

i.

lim t = t

” o n n

'

n d

and hence (iii) is true.

Part (ii) is easily proved by Lemma 5.1. In fact, suppose p (d) e

n (0 ,t.) for all n. Then p , (d) i Y.(p (d)) > p (d) and thus

d n+1 d n n

P, < P,(d) < p,(d) < ....< t . This implies that p (d) -*■ t_.

1 2 3 o n d

The other case is p^, > t^. But the set (t^, + D O ) is invariant

To

under by Lemma 5.1 and p^(d) J. t^ ,

T >,

, follows.

Theorem 5.2 Assume :

i) I

-

V^(x2>y;h) I < L H x ^ - x21| || hl| , for any

X l ’X2,y e X and any feasible b e ^ ’

ii) either D « X*X or (x,^(x))is interior to D for any x e X ;

iii) Px (V;D) > 0 ;

iv) p (V ( x , . ) ) j . a j . 0 for any x e X .

©

The

Author(s).

European

University

Institute.

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

Repository.

(20)

V X

1 } - W

1

L(a + « t^)

-1

X1 " X2l

( 1 2 )

for any x ,x € X and where t is defined as in Lemma 5.1.

1 2 o

Proof : Take U(x,y) = V(x,y) + 6 W„(y) . Suppose at the moment

--- o

that D = X*X . Because Max U(x,y) =» U(x,t p(x)) (see (2)) , we

y

6

are in position to apply Theorem 3.1 . From assumption (iii) and

by Theoerem 5.1 we have p( H ) J ! t . Hence

0 6

p( U(x,.)) > a + 5 t . Let us suppose W*(y;h) < +

oo

. Then

^

6 6

U*(x,y;h) * V'(x,y;h) + 6 W*(y;h) and thus we have :

2 2 6

I

U’ ^ . y p h ) “ ° 2 (x2 , y ; h ) l 4 L 11*2 " x2ff W

hll

•

All the assumptions of theorem 3.1 are fullfilled and we can conclude the given estimate for T .

6

Remark The assumption W*(y;h) < + do is not essential. In fact, 6

replace the initial problem with Max £ü(x,y) ;s.t. y € XC } ,

where X G c X is an £ - contraction of X ,i.e., X G - fG (X) and

£ £

f : X ■+■ X is defined as f (x) * (l-e)x + e x q , x q is a point

in the relative interior of X and e is a small positive number. Consequently we have :

Max { U(x,y) ; s.t. y e XE } - U( x , t E (x)) and tE (x) satisfyes

|| T (x ^ ) - tE(x2 ) II £ L ( a + t fi) x^ - x 2 II b ecau se W ^(y;h )< OO

^or all x € rei int X .

On the other hand t

E(

x

)

-*• x^(x) as t ♦ 0 , by the Maximum Principle, and therefore we get again the desired estimate.

By Remark 3.1 we get the same estimate in the case in which D is

©

The

Author(s).

European

University

Institute.

version

produced

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

Repository.

(21)

It is not difficult to understand the use of Theorem 5.2 to obtain stability conditions. In fact , L(a+6t ) < i sufficies ,by

6

Banach fixed point theorem,to assure that all the trajectories of P(x ,6) converge to a unique stationary trajectory "x (i.e. to

o

the unique fixed point of t. ). o

We conclude this paper by giving an example of this fact.

E x a mple. Take the problem P(x^,5) with the return function :

2

V :

C°

. *

Co

, lj -*■ IR given by V(x,y) - - 4 x y +

4 xy -(1/2) y2 - (R/2) x2 , R > 16 and D - X x X .

It was shown in Montrucchio(1986) that V is concave on X*X

and that the policy t is chaotic for small 6 . Let us find out

6

which value of 5 eliminates such a behavior. According to example 2

of Section 4 , the f-transform of V is ¥(t) = R - 16(1 - t) * and

t{ - [ S R - 1 + (1 + 62R2 - 64 6 * 2 SR )* ] / 26 .

In addition we have L » 4 and a = 1” . By a straightforward computation we get : L(a+ 6t ) ^ < 1 iff 6 > 3 (R - 4) ^

6

which is the desired "turnpike" condition . Notice that 3(R-4) 1/4 for all R > 16 .

©

The

Author(s).

European

University

Institute.

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

Repository.

(22)

6. Concluding Remarks

As the example of the last section shows, the estimate (12) pro vides a powerful tool to compute stability conditions for the discounted Ramsey models. Note the case

t/a

< 1 . Because L/a J L/(a+6t ) ,

o we can deduce global stability of the model ,indipendently of the the discount factor.

A partial answer to a conjecture formulated by some authors about the role played by the discount factor is another issue of Theorem 5.2. It seems to be resonably true that the complexity of the optimal paths

decreases as the rate of impatience decreases. This fact can be formu

lated more precisely by using the topological entropy h(f) as an

indicator of the dynamical complexity of a map f . From (12) it is

not difficult to give the estimate

h( t ) < dim (X) log ( L/(a + 6 t ) ) , where

<5 0

log (u) = Max ( 0 , log u ) .

A last comment to be made concerns with the differentiability

of the policy function. By the classical Rademacher*s theorem , it

follows that the policy functions are almost everywhere differentiable

on X .

©

The

Author(s).

European

University

Institute.

version

produced

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(23)

Appendix

Lemma A.l Let f be a real function defined on X . f

2

a-concave on X iff f(y)-f(x) ^ f'(x;y-x) - (l/2)a ||y-x || for every x,y e X.

P roof: (13) follows trivially by concavity of f (x) + (l/2)a || x if” •

Lemma A.2 If f is a-concave on X .then

2 ally

“ x II £ f'(y;x-y) + f'(x;y-x)

holds for every x.y £ X.

P r o o f : Exchanging x and y in (13) ,we have f(x) - f (y) < f ' (y;x-y) -(l/2)a II y - x || 2

Summing up (13) and (15),we get (14).

Lemma A. 3 Let f : X IR be an upper semicontinuous and

a-concave function on a closed and convex set X C H , a > 0 . f has a unique point

x E*

X that maximizes f on X a n d ,furthermore :

a || x - x*||2 .< f'(x; x*- x ) , ¥ x

e

X . (16)

P roof: The sections x X ; f(x) > X } are closed because

f is upper semicontinuous. We prove now that are relatively

compact for the weak topology. In fact, f is superdifferentiable in some

point xQ fe X (actually it is superdifferentiable over a dense subset of X ). Let no w u £ 3f(xQ)

? JX

be .where 3f(x^) is the set of

the superdifferential of f at x^. Let A = { x d X ; f(x)if(x^)y.

2

(14) (15) (13)

©

The

Author(s).

European

University

Institute.

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

Repository.

(24)

Thus (1/2) a||x-Xgl! £ < u , x - x ^ > < II u II l|x-x^f| .that implies

|| x — x || 4 (2/a)ll u II , V x fe A .

This sufficies to infer the existence of a maximal point x * in X (see

for example Ekeland and Teman (19 76 ). The maximal point is then

u-nique by the strong concavity of f.

Put now y - x * in (14). We have a || i£x II 4 f ' (x*;x-x*)+f' (x;x*-x) . On the other hand f'(x*;x-x*) 4 0 for any x in X .because x* maximizes f and thus we get (16).

2

Proof of Theorem 3.1 : (a) follows immediately from Lemma A.3 .

Let now x ^ , x ^ € X be . Take (16) with f - U(x^,.) . We have :

o || y - 9 (x^) ll 4 U ^ ( x1>y;9(x1)-y) for all y t Y . (17)

Putting y - 9(x^) in (17) .one obtain

“ II 9(x2) - 9 (x^) H2 ^ 0-(x1,9(x2); 9(x2)-9(x2)).

On the other hand D 2^X 2’ 0 ^*2^ ’ 8(x )-9(x2).) 4 0 holds because 9(x2) maximizes U ( x 2 > .) . We can write

a || 9 (x2)-9(x1) II 24 U ^ ( x l>9(x2);9(xl)-9(x2))- U* (x2>9(x^ ; 9 ( x ^ - 9 ( x ^ ) (18)

From assumption (iii),the right side of (18) is smaller that

L AX1 " *2* 9(x2) *' ‘ HenC* 5

o || 9 ( x ^ - 9 (x2)||2 4 L l| Xj- x2 l| II9 (x2) - 9(x<)|| ,i.e.,

|| 9 (x^) - 0(x2)|| ^ (L/a) || - x^ || and this completes the

proof of (b).

©

The

Author(s).

European

University

Institute.

version

produced

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

Repository.

(25)

References

Araujo.Aloisio and Jose A. Scheinkman,1977,Smoothness,comparative dynamics, and the turnpike property.Ecopometrica 1*5,601-620.

Deneckere,Raymond and Steve Pelikan,1986,Competitive caos,in:Jean M. Grand- mont.ed..Nonlinear Economic Dynamics(Academic Press,New York).

Boldrin.Michele and Luigi Montrucchio,1986,On the indeterminacy of capital accumulation paths,in:Jean M.Grandmont,ed..Nonlinear Economic Dynamics

(Academic Press,New York).

Ekeland.Ivar and Roger Teman,19T6,Convex Analysis and Variational Problems (North-Holland,New York).

Feming.Wendell H. and Raymond W.Rischell,1975,Deterministic and Stochastic Optimal Control(Spriger-Verlag,New York).

Montrucchio,Luigi,1986,Optimal Decisions over time and strange attractors: an analysis by the Bellman Principle,Mathematical Modelling,forthcoming. Rockafellar.Tyrrell R . ,1976,Saddle points of hamiltonian systems in

convex Lagrange problems having a nonzero discounting rate,Journal of Economic Theory 12,71-113.

©

The

Author(s).

European

University

Institute.

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(26)

©

The

Author(s).

European

University

Institute.

version

produced

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(27)

WORKING PAPERS ECONOMICS DEPARTMENT

85/155: François DUCHENE 85/156: Domenico Mario NUTI

85/157: Christophe DEISSENBERG

85/161: Domenico Mario NUTI

Beyond the First C.A.P.

Political and Economic Fluctuations in the Socialist System

On the Determination of Macroeconomic Policies with Robust Outcome

A Critique of Orwell’s Oligarchic Collectivism as an Economic System

85/162: Will BARTLETT Optimal Employment and Investment

Policies in Self-Financed Producer Cooperatives

85/169: Jean JASKOLD GABSZEWICZ Asymmetric International Trade

Paolo GARELLA

85/170: Jean JASKOLD GABSZEWICZ Paolo GARELLA

85/173: Berc RUSTEM

Kumaraswamy VELUPILLAI

85/178: Dwight M. JAFFEE

85/179: Gerd WEINRICH

85/180: Domenico Mario NUTI

85/181: Will BARTLETT

Subjective Price Search and Price Competition

On Rationalizing Expectations

Term Structure Intermediation by Depository Institutions

Price and Wage Dynamics in a Simple Macroeconomic Model with Stochastic Rationing

Economic Planning in Market Economies: Scope, Instruments, Institutions Enterprise Investment and Public Consumption in a Self-Managed Economy

85/186: Will BARTLETT Gerd WEINRICH

Instability and Indexation in a Labour- Managed Economy - A General Equilibrium Quantity Rationing Approach

85/187: Jesper JESPERSEN Some Reflexions on the Longer Term Con

sequences of a Mounting Public Debt

85/188: Jean JASKOLD GABSZEWICZ Scattered Sellers and Ill-Informed Buyers

Paolo GARELLA A Model of Price Dispersion

©

The

Author(s).

European

University

Institute.

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(28)

85/195: 85/196: 85/198: 85/200: 85/201: 86/206:

86

/

212

: 86/214: 86/218: 8 6 /2 2 2 : 86/223: 86/224: 86/225: 86/227: 86/228: 86/229: Pierre DEHEZ Jean-Paul FITOUSSI Werner HILDENBRAND Will BARTLETT Milica UVALIC Domenico Mario NUTI

Ernesto SCREPANTI

Volker DEVILLE

Emil CLAASSEN Melvyn KRAUSS Alberto CHILOSI

Wage Indexation and Macroeconomic Fluctuations

A Problem in Demand Aggregation: Per Capita Demand as a Function of Per Capita Expenditure

Bibliography on Labour-Managed Firms and Employee Participation

Hidden and Repressed Inflation in Soviet- Type Economies: Definitions, Measurements and Stabilisation

A Model

of

the Political-Economic Cycle in Centrally Planned Economies

Bibliography on The European Monetary System and the European Currency Unit. Budget Deficits and the Exchange Rate

The Right to Employment Principle and Self-Managed Market Socialism: A Historical Account and an Analytical Appraisal of some Old Ideas

Emil CLAASSEN

Edmund S. PHELPS

Giuliano FERRARI BRAVO

Jean-Michel GRANDMONT Donald A.R. GEORGE

Domenico Mario NUTI

Marcello DE CECCO

The Optimum Monetary Constitution: Monetary Integration and Monetary Stability

Economic Equilibrium and Other Economic Concepts: A "New Palgrave" Quartet

Economic Diplomacy. The Keynes-Cuno

Affair

Stabilizing Competitive Business Cycles Wage-earners' Investment Funds: theory, simulation and policy

Michal Kalecki's Contributions to the Theory and Practice of Socialist Planning Codetermination, Profit-Sharing and Full Employment

Currency, Coinage and the Gold Standard

©

The

Author(s).

European

University

Institute.

version

produced

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(29)

86/230: Rosemarie FEITHEN D e terminants of Labour Migration in an Enlarged European Community

86/232: Saul ESTRIN Derek C. JONES

/

Are There Life Cycles in Labor-Managed Firms? Evidence for France

86/23,6: Will BARTLETT Milica UVALIC

Labour Managed Firms, Employee Participa tion and Profit Sharing - Theoretical Perspectives and European Experience.

86/240: Domenico Mario NUTI Information, Expectations and Economic

Planning

86/241: Donald D. HESTER Time, Jurisdiction and Sovereign Risk

86/242: Marcello DE CECCO Financial Innovations and Monetary Theory

86/243: Pierre DEHEZ Jacques DREZE

Competitive Equilibria with Increasing Returns

86/244: Jacques PECK Karl SHELL

Market Uncertainty: Correlated Equilibrium and Sunspot Equilibrium in Market Games

86/245: Domenico Mario NUTI Profit-Sharing and Employment: Claims and

Overclaims

86/246: Karl Attila SOOS Informal Pressures, Mobilization, and

Campaigns in the Management of Centrally Planned Economies

86/247: Tamas BAUER Reforming or Perfectioning the Economic

Mechanism in Eastern Europe

86/257: Luigi MONTRUCCHIO Lipschitz Continuous Policy Functions for

Strongly Concave Optimization Problems

Spare copies of these working papers and/or a complete list of all working papers that have appeared in the Economics Department series can be obtained from the Secretariat of the Economics Department.

©

The

Author(s).

European

University

Institute.

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(30)

©

The

Author(s).

European

University

Institute.

version

produced

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(31)

F.UT Working Papers are published and distributed by the European University Institute, Florence.

A complete list and copies oT Working Papers can be obtained free of charge — depending on the availability of stocks — from!

The Publications Officer European University Institute

Badia Fiesolana

J-50016 San Domenico di Fiesole (FI)

©

The

Author(s).

European

University

Institute.

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(32)

To The Publications OTTicer European University Institute lindi a Fiesolann

1-50016 Han Domenico di Fiesole (FI) I tal y

F r om N a m e ... ...

Addr e s s ...

Please send me:

[ZJ

a complete list of Kill Working Papers

I 1 the following FUI Working raper(s):

No. : ... Author, title:... Da te : Signature :

©

The

Author(s).

European

University

Institute.

version

produced

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(33)

PUBLICATIONS OF THE EUROPEAN UNIVERSITY INSTITUTE SEPTEMBER 1986

The Ambiguity of the American Reference in the French and Italian Intellectual Renewal of the Late 1950's

86/217: Michela NACC1 Un'Immagine della modernità:

L'America in Francia negli Anni Trenta

86/218: Emi1-Maria CLAASSEN The Optimum Monetary Constitution:

Monetary Integration and Monetary Stability *

86/219:Stuart WOOLF The Domestic Economy of the Poor of

Florence in the Early Nineteenth Century

86/220:Raul MERZARIO Il Capitalismo nelle Montagne

L'evoluzione delle strutture famigliari nel comasco durante la prima fase di industrializzazione (1746-1811)

86/221 .-Alain DROUARD Relations et Reactions des Sciences

Sociales "Françaises" Face Aux Sciences Sociales "Américaines"

86/222:Edmund PHELPS Economie Equilibrium and Other

Economic Concepts: A "New Paigrave" Quartet

86/223:Giuliano FERRARI BRAVO Economic Diplomacy: The Keynes-Cuno

Affair

86/224:Jean-Michel GRANDMONT Stabilising Competitive Business

Cycles

86/225:Donald GEORGE Wage-Earners' Investment Funds:

Theory, Simulation' and "Policy

86/226:Jean-Pierre CAVAILLE Le Politique Révoqué

Notes sur le statut du politique dans la philosophie de Descartes

86/227:Domenico Mario NUTI Michal Kalecki's Contributions to the

Theory and Practice of Socialist Planning

©

The

Author(s).

European

University

Institute.

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

Repository.

(34)

86/229: Marcello DE CECCO Currency, Coinage and the Gold Standard

86/230: Rosemarie FLEITHEN Determinants of Labour Migration in an

Enlarged European Community

86/231: Gisela BOCK Scholars'Wives, Textile Workers and

Female Scholars’ Work: Historical

Perspectives on Working Women's Lives 86/232: Saul ESTRIN and

Derek C. JONES

Are there life cycles in labor-managed firms? Evidence for France

i

86/233: Andreas FABRITIUS Parent and Subsidiary Corporations

under tl.S. Law - A Functional Analysis of Disregard Criteria

86/234: Niklas LUHMANN Closure and Openness: On Reality in

the World of Law

86/235: Alain SUPXOT Delegalisation and Normalisation

86/236: Will BARTLETT/ Milika UVALIC

Labour managed firms

Employee participation and profit- sharing - Theoretical Prospectives and European Experience

86/237: Renato G IANNETTI The Debate on Nationalization o f the

Electrical Industry in Italy after the Second World

War

(1945-47)

86/238: Daniel ROCHE Paris capitale des pauvres: quelques

réflexions sur le paupérisme parisien entre XVII et XVIII siècles

86/239: Alain COLLOMP Les draps de laine, leur fabrication

et leur transport en Haute-Provence; XVII - XIX siècle: #

univers familiaux, de l'ère pré industrielle à la

protoindustrialisation

86/240: Domenico Mario NOTI Information, Expectations and Economie

Plannirtg

86/241: Donald D. HESTER Time, Jurisdiction and Sovereign Risk

86/242: Marcello DE CECCO Financial Innovations and Monetary

Theory

©

The

Author(s).

European

University

Institute.

version

produced

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

(35)

PUBLICATIONS OF TME EUROPEAN UNIVERSITY INSTITUTE DECEMBER 1986

86/243: Pierre DEHEZ and Jacques DREZE

Competitive Equilibria With Increasing Returns

86/244: James PECK and Karl SHELL

Market Uncertainty: Correlated Equilibrium and Sunspot Equilibrium in Market Games

86/245: Domenico Mario NUTI Profit-Sharing and Employment: Claims

and Overclaims

86/246: Karoly Attila SOOS Informal Pressures, Mobilization and

Campaigns in the Management of Centrally Planned Economies

86/247: Tamas BAUER Reforming or Perfectioning the

Economic Mechanism in Eastern Europe

86/248: Francese MORATA Autonomie Régionale et Integration

Européenne :

la participation des Régions espagnoles aux décisions commuhautaires

86/249: Giorgio VECCHIO Movimenti Pacifisti ed

Antiamericanismo in Italia (1948-1953)

86/250: Antonio VARSORI Italian Diplomacy and Contrasting

Perceptions of American Policy After World War II (1947-1950)

86/251: Vibeke SORENSEN Danish Economic Policy and the

European Cooperation on Trade and Currencies, 1948-1950

86/252: Jan van der HARST The Netherlands an the European

Defence Community

86/253: Frances LYNCH The Economic Effects ofr tKe Korean War

in France, 1950-1952 86/254: Richard T. GRIFFITHS

Alan S. MILWARD

The European Agricultural Community, 1948-1954

86/255: Helge PHARO The Third Force, Atlanticism and

Norwegian Attitudes Towards European Integration

©

The

Author(s).

European

University

Institute.

by

the

EUI

Library

in

2020.

Available

Open

Access

on

Cadmus,

European

University

Institute

Research

Repository.

(36)

86/257: Luigi MONTnUCCHIO Lipschitz Continuous Policy Functions for Strongly Concave Optimization Problems