Adaptive rolling plans good

Piotr Maćkowiak

*

ADAPTIVE ROLLING PLANS ARE GOOD

0B

Here we prove the goodness property of adaptive rolling plans in a multisector optimal growth model under decreasing returns in deterministic environment. Goodness is achieved as a result of fast convergence (at an asymptotically geometric rate) of the rolling plan to balanced growth path. Further on, while searching for goodness, we give a new proof of strong concavity of an indirect utility function – this result is achieved just with help of some elementary matrix algebra and differential calculus.

1B

Keywords: indirect utility function, good plans, adaptive rolling planning, multisector model.

2B

JEL classification: C61, O41

3B

1. INTRODUCTION

The idea of this paper comes from Kaganovich (1996) where a hypothesis is put forward that adaptive rolling plans are good (in Gale’s sense, see Gale (1967)), if there is uniform strong convexity of technology. Goodness of rolling plans in one sector models was proven in Bala et al. (1991). Fast convergence (at an asymptotically geometric rate) toward turnpike under linear technology (with suitably defined opitmality criterion) is known from Kaganovich (1998). We extend these results to the case where production of goods is described by neoclassical technology.

Rolling planning is a procedure of constructing infinite horizon programs. After finding an optimal process starting from a given initial state and under a fixed and finite horizon length of planning, only the first step of the plan is executed and a new optimal plan is constructed starting from the just achieved state (Goldman 1968). When feasible processes of growth are those in which initial and final state of the economy is the same (changes may occur between initial and final periods), then we deal with adaptive rolling planning procedure. It is known that in one-sector case adaptive rolling plans are efficient and good (Bala et al. 1991)). Kaganovich (1996) proved that

rolling plans converge toward turnpike,F

∗ FF 1

F

which is a necessary but not

∗_{Department of Mathematical Economics, Poznań University of Economics}

 

sufficient condition for goodness. Kaganovich (1998) showed that under linear technology and maximal growth rate as optimality criterion (for constructing adaptive rolling plans) rolling plans approach von Neumann ray at (asymptotically) maximal growth rate that can be achieved among all

balanced growth processes. We prove that rolling plans are goodF

2

F under

neoclassical technology of goods (theorem X5X) and while proving it we use

strong concavity (Vial 1983) of an indirect utility function near turnpike. To this goal, we firstly construct indirect utility function (definition). Our construction differs from a typical one (Venditti 1997) in that we express utility as a function of today’s and tomorrow’s inputs and not as a function of today’s and tommorow’s outputs (stocks of goods). Strong concavity was proven in Venditti (1997) for an economy where there is only one consumption good and all other goods are capital goods. In our case – to be in compliance with Kaganovich’s approach (Kaganovich 1996) – all goods are treated as consumption/production goods at a time, so that Venditti’s approach is not applicable here.F

3 F

We also show that strong concavity of indirect utility function holds (under our assumptions) only if at most one production function is positively homogeneous of degree one and the other are subject to decreasing returns to scale.F

4 F

The next two sections set notation and preliminaries. In section 4 and 5 we included main results. Section 6 is a summary.

4B

2. NOTATION AND CONVENTIONS

Symbol Rn denotes an n-dimensional real linear space, and

R

n₊ is

its non-negative orthant. A point x R∈ n possesses coordinates

x

₁

,

…

,

x

_n. If an element of Rn is named xj, where j is a nonnegative integer, then

) , , ( _{1j nj} j _{x x} x = … . For x,x'∈Rn we write x≥x' if

x

_i

≥

x

'

_i

,

i

=

1

,

…

,

n

;

2 _{So that, the procedure of constructing adaptive rolling plans can be used to build an}

evolutionary mechanism – more on this see in Bala et al. (1991) or Kaganovich (1996).

3_{We tried to prove the strong concavity of indirect utility function when its arguments were}

outputs – but we did not succeed because in that approach we could not determine definiteness of a counterpart of matrix

V

''

(equation 21), which is crucial.

4_{Assumptions on production functions similar to ours were taken in Hirota and Kuga (1971),}

Benhabib and Nishimura (1979a), Benhabib and Nishimura (1979b), Benhabib and Nishimura (1981).

'

x

x> means x≥x' and x≠ x'; x>>x' is equivalent to

n

i

x

x

_i

>

'

_i

,

=

1

,

…

,

. By

int

R

n₊ we denote the interior of

R

n₊, i.e. the set

of positive vectors. If m and n are positive integers, then for a∈R

symbol

a

_m×n denotes a matrix composed of m rows and n columns with

a on each coordinate;

a

_n stands for

a

_n×1. For two matrices

A

,

B

their

(right) Kronecker product is written as A⊗B (see Lancaster

and Tismenetsky (1985), p. 407). The transposition of

A

is denoted by

T

A . Euclid norm of x∈Rn is denoted as x . Writing

n n _B A B A y x, )∈ × , ⊂R , ⊂R

( we mean x∈A,

y

∈

B

. Given two

matrices

A

(m rows, n columns) and

B

(n rows, k columns) and

equation AB=0, we deduce zero on right-hand-side is

0

_m×n (without

writing this explicitly). Analogously: if

x

∈

_R

n

,

x

≥

0

_{, then zero on the}

right-hand-side is

0

_n. The identity matrix of order n is denoted as

I

_n.

For an element

a R

∈

n by

diag

a

we denote diagonal matrix of order

n

n× with

a

₁

,

…

,

a

_n on the diagonal. Symbol ‘:=’ reads as ‘by definition

equals to’.

5B

3. PRELIMINARIES

To achieve our goal we have to give a more detailed description of a

technology set

Z

than was done in Kaganovich (1996). Technology set

Z

is defined as ⎩ ⎨ ⎧ ≤ = ∈ ∃ × ∈ =

∑

= + + + n j j n j n n _{x j n x x} y x Z 1 , , , , 1 , : ) , ( R R R … ⎭ ⎬ ⎫ ≤ ≤ = ∈ ∃

∑

∑

= = + n j n j j j j j j j j n l y f x l l 1 1 ) , ( , 1 , , , 1 , … R (1) where _:R₊n+1 → R₊ j

f is production function of

j

-th good,

) , ,

( _{1j nj}

j _{x x}

x = … represents producible goods inputs and l_j stands for

(i) f_j is continuous on

R

₊n+1, twice continuously differentiable and

strictly increasing over

int

+1

+ n

R

with

(

,

)

0

,

(

,

)

>

0

∂

∂

>

∂

∂

j j j j ij j j j

l

l

x

f

x

l

x

f

,

n

i

=

1 …

,

,

, and strictly concave on interior of its domain and

0 ) , ( j _j >

j x l

f only if _xj >>0,_lj >0_{. Moreover, Hessian of}

j

f is

negatively definite everywhere on

int

+1

+

n

R

.

(ii) There exists

β

>

0

such that if x >

β

, then for

(

x

,

y

)

∈

Z

:

y

≤

x

. (iii)There exists expansible stocks vector

x

∈ R

n₊:

y

>>

x

for some

Z

y

x

,

)

∈

(

.

The construction of set

Z

and assumption (i) guarantee that the set is

closed and convex; free disposal is allowed and

Z

admits weak strict

convexity (external effects) on inputs:

(

x

,

y

)

∈

Z

,

(

x

,'

y

'

)

∈

Z

with x≠ x'

imply that there exists

z

y

y

x

x

y

⎟

∈

Z

⎠

⎞

⎜

⎝

⎛ +

+

>>

,

2

'

:

2

'

. These properties imply that assumptions imposed on production set in Kaganovich (1996) are met, and we can use results obtained therein.

Consumption c is valuated by an instantaneous utility function

+ +

→ R

R

n

U :

which satisfies

(iv) U is continuous, strictly concave and twice continuously

differentiable on

int

R

n₊ with negatively defined Hessian.

(v) Uis strictly increasing on

int

R

n₊:

c

>

c

'

⇒

U

(

c

)

>

U

(

c

'

)

, with

0

)

(

_>

∂

∂

j

c

c

U

.

Let us fix initially available input

x

∈ R

n₊ and a positive integer number

N. A sequence

(

(

x

_t

,

y

_t

,

c

_t

)

)

_tN₌₁

⊂

R

n₊

×

R

n₊

×

R

₊n is called feasible

N-process from x to

b

∈ R

n₊, if

.

,

,

,

,

1

,

,

1

,

,

0

,

)

,

(

0 1

b

x

x

x

N

t

y

c

x

N

t

Z

y

x

N t t t t t

≥

=

=

≤

+

−

=

∈

+

…

…

(2)

Sequence

(

(

x

_t

,

y

_t

,

c

_t

)

)

_t∞₌₁

⊂

R

n₊

×

R

₊n

×

R

₊n is called feasible

∞

-process from x if for all t ≥1 it holds

(

x

_t

,

y

_t+1

)

∈

Z

,

x

_t

+

c

_t

≤

y

_t and

Z

y

x

,

)

∈

(

₁ . An N-feasible process from

x

∈ R

n₊ to

b

∈ R

n₊ is called

N-optimal from x to b if it maximizes

∑

= N t t

c

U

1

)

(

(3)

over the set of all N-feasible processes from x to b. We are interested in properties of adaptive rolling plans defined as follows. F

5 F

Definition 1 Fix

x

∈ R

n₊. A sequence

(

(

x

_t

,

y

_t

,

c

_t

)

)

∞_t=1

⊂

R

₊n

×

R

₊n

×

R

₊n

is called adaptive rolling plan from x if for all

t

=

1

,

2

,

…

,a sequence

))

,

,

(

),

,

,

((

x

_t

y

_t

c

_t

x

_t+1

y

_t+1

c

_t+1

is 2-optimal process from

x

_t−1 to

x

_t−1 where

x

₀

=

x

.

From now on we assume that there exists an adaptive rolling plan for a given initial inputs vector

x

₀.

Definition 2 Triplet

(

x

,

y

,

c

)

∈

R

n₊

×

R

n₊

×

R

n₊ is called turnpike if it is optimal solution of the following problem

.

,

,

,

)

,

(

,

)

(

max

n

c

y

x

Z

y

x

y

c

x

c

U

+

∈

∈

≤

+

R

Under our assumptions turnpike exists and is unique. In what follows we denote the turnpike as

(

x

,

y

,

c

)

and its utility as

U

=

U

(c

)

. We also assume:

(vi) Turnpike consumption

c

is positive, i.e. c >>0.

We shall show that adaptive rolling plans enjoy a goodness property Gale (1967) defined as

Definition 3 Let

x

₀

∈ R

n₊. A feasible

∞

-process from

x

₀,

(

)

∞ =1

)

,

,

(

x

_t

y

_t

c

_{t t} is called good if

∑

= ∞ →

−

>

−∞

N t t N

U

c

U

1

)

)

(

(

inf

lim

. (4)

It is known that for any

∞

-process

lim

sup

of the series on left-hand-side in (4) is always finite and if Alim inEfA is finite then the series converges (Gale

1967). Further, if a process is good then it converges to the turnpike – it is a necessary condition for goodness – and as it has been said this property holds in our setting (by results of Kaganovich 1996). Our goal is to prove that the speed of convergence toward turnpike is high enough to ensure that the condition (4) holds. We need to show that the indirect utility function (to be defined below) is twice continuously differentiable and strictly concave near the turnpike and that its Hessian is negatively definite at the turnpike.

Definition 4 The indirect utility function

_{V :}

R

₊n

×

R

₊n

→

R

_{+ is defined}

asF 6 F ) ' ( max : ) ' , ( ' , ) , ( : , x y U x x V x x Z y x x y y n n − = ∈ ∀ ∈ ≤ ∈ + + R R (5)

Certainly, function V is concave and continuous for

x

,

x

'

near the

turnpike

x

.

6B

4. STRONG CONCAVITY OF THE INDIRECT UTILITY FUNCTION NEAR TURNPIKE

Now we shall use the strength of the definition of technology set

Z

and assumptions. Fix

x

,

x

'

∈ R

n₊. The optimization problem defining function V

,

,

)

,

(

,

'

)

'

(

max

n

y

Z

y

x

y

x

x

y

U

+

∈

∈

≤

−

R

(6)

is – by assumptions (i), (iv) – equivalent to a concave maximization problem

6_{If a set is empty then the maximum value of a function over the set equals}

−

∞

_{, as a usual}

n

j

x

l

x

x

l

x

f

x

x

l

x

f

x

l

x

f

U

n j n j j n j j j j j j n n n

,

,

1

1

0

)

,

(

'

)

'

)

,

(

,

,

'

)

,

(

(

max

1 1 1 1 1 1 1

…

…

=

∈

≤

≥

−

≤

−

−

+ = =

∑

∑

R

(7)

in the following sense: if x1,_…,xn_{, with some choice of}

n

l

l

₁

,

…

,

solves (7), then ( ( 1, ₁), , ( , )) 1 n n n x l f l x f

y= … solves (6) and every

solution of (6) is obtained by some choice of x1,_…,xn _and

n

l

l

₁

,

…

,

solving (7) – in fact this choice is unique (again by assumptions (i), (iv)).

9B

4.1. Non-homogeneous case

We just keep assumption (i) in force.

Lemma 1 There exists a neigbourhood

_W

n n

+ +

×

⊂

R

R

of

( x

x

,

)

such

that function V is a twice continuously differentiable and strongly concave

on W

.

F

7

F

Proof: We divide the proof into three steps

.

F

8

Step 1 Lagrange multipliers

λ

and sectoral inputs xi are twice

continuously differentiable functions of x and x'.

We know that there is a one-to-one relationship between solutions of (6) and (7). We shall show that solution of (7) depends twice continuously

differentiably on

( x

x

,

'

)

in a neighbourhood of

( x

x

,

)

. Let

) ,

( j _j

j

j f x l

y = , for x ,j l_j

j

=

1 …

,

,

n

, solving (7). By assumption (vi),

7_{Symmetric matrix A is called negative-definite (nonpositive-definite) if all its eigenvalues}

are negative (nonpositive). If A is nonpositive-definite and is not negative-definite then we call it negative-semidefinite (Lancaster and Tismenetsky (1985), p. 179). It is known that a twice continuously differentiable concavefunction is strongly concave on W iff its Hessian is negative-definite on W with eigenvalues strictly separated from 0 – proof of this fact and definition of strong concavity (convexity) is contained in Vial (1983).

8 _{The first one is rather standard as regards its idea, see Benhabib and Nishimura (1979a),}

j j j

j x l x

f ( , )> ,

j

=

1 …

,

,

n

. Obviously, by assumptions (i) and (v), for

any solution x1,_…,xn_,

n

l

l

₁

,

…

,

of (7) (under any given positive

(

x

,

x

'

)

),

it holds that

∑

= = n j j x x 1

and _xj >>0 _near

₍

_x

_,

_x

₎

_{, since solution of (7)} depends continuously on

(

x

,

x

'

)

(by Berge’s maximum theorem, Lucas and Stokey (1989), p. 62). Therefore Lagrange function for

( x

x

,

'

)

near

( x

x

,

)

can be written as = + , , ') , , , , , , , , (_x1 _l1 _{x l 1 1 x x} L _… n n

_λ

_…

_λ

_n

_λ

_n ), 1 ( ) ( ) ' ) , ( , , ' ) , ( ( 1 1 1 1 1 1 1 1

∑

∑

∑

= = = + − + − + − − = n j j n i n n j ij i i n n n n x l x x x l f x l x f U …

λ

λ

(8)

where

λ

_i

,

i

=

1

,

…

,

n

, denotes Lagrange multiplier. Necessary and

sufficient conditions for optimality of a feasible solution x1,_…,xn_,

n

l

l

₁

,

…

,

of (7) at

(

x

,

x

)

read as (Takayama 1985, p. 91)

,

0

)

,

(

)

(

,

0

)

,

(

)

(

1

=

−

∂

∂

∂

−

∂

=

−

∂

∂

∂

−

∂

+ n j j j j j i ij j j j j

l

l

x

f

c

x

F

U

x

l

x

f

c

x

F

U

λ

λ

(9)

for all

i

,

j

=

1

,

…

,

n

some positive optimal multipliers

λ

_i , where ) , , , , ( 1 ₁ n n _l x l x F F = … and )) , ( , ), , ( ( ) , , , , ( 1 ₁ 1 1 1 n n n n n _{l f x l f x l} x l x F … = …

Conditions (9) can be written in matrix notation as

0

1

'

'

F

−

_1×n

⊗

λ

=

U

(10)

where

U

'

is the first derivative of U evaluated at

x l x l x F_{( , , ,} n_{, n )−} 1 1 _{… , ' '( , , , , )} 1 1 n n _l x l x F F = … and

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

∂

∂

∂

∂

∂

∂

=

n n n n n

f

f

f

l

x

l

x

F

x

x

x

x

x

x

)

(

0

0

0

0

0

)

(

0

0

0

0

)

(

)

,

,

,

,

(

'

2 2 2 1 1 1 1 1

…

…

…

…

…

…

…

…

…

where ), , ( j _j j _{= x l} x

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

∂

∂

∂

∂

∂

∂

=

∂

∂

=

j j j j nj j j j j j j j j j j j j

l

l

x

f

x

l

x

f

x

l

x

f

f

f

(

)

:

(

)

(

,

)

(

,

)

(

,

)

1 '

_…

x

x

x

and

]

[

_{1 +1}

=

λ

λ

_n

λ

_n

λ

…

.

Define function

:

int

( 1)

int

1

int

int

( 1) +1

+ + + + + + + + +n

×

n

×

n

×

n

→

nn

×

n n

G

R

R

R

R

R

R

as

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−

⊗

−

−

=

∑

∑

= = × n j j n j j n n n

_x

_x

_U

_F

_x

_F

_x

_x

_l

G

1 1 1 1 1

_,

_,

_,

_,

_,

_,

_,

_'

₎

_('

₍

₎

_)'

_'

₍

₎

₁

_,

_,

₁

(

λ

λ

λ

λ

x

x

x

x

x

…

…

(11) By assumption of optimality of

(

,

,

,

,

)

1 1 1

_…

n n n

_l

x

l

x

x x

x

=

at

( x

x

,

)

it holds that

0

)

,

,

,

(

x

x

=

G

x

λ

and since G is of class C1 in a neighbourhood of

(

x

,

λ

,

x

,

x

)

, then – by the implicit function theorem – if we knew that

) , ( ) , , , (

λ

λ

x x ∂ ∂G x x were

invertible then we could express

x

and

λ

as continuously differentiable

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

=

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

−

⊗

⊗

−

⊗

+

=

∂

∂

+ × + + × + + × + × + ) 1 ( ) 1 ( ) 1 ( ) 1 ( 1 1 1 1 1

0

:

0

1

1

''

)

)'

diag

((

'

''

'

)

,

(

)

,

,

,

(

n n T n n n n n n n T

_B

B

A

I

I

F

I

U

F

U

F

x

x

G

λ

λ

x

x

(12),

where

U

''

=

U

''

(

F

(

x

)

−

x

)

is Hessian of U evaluated at

F

(x

)

−

x

,

)

(

''

''

F

x

F

=

and

(

( ), , ( )

)

diag ) ( '' '' 1 '' 1 n n f f F x = x … x

is a block-diagonal matrix of rank

(

n

+

1

)

2 with Hessians

) ( , ), ( 1 '' '' 1 n n f

f x … x on the diagonal. We shall show first that

A

(defined in

(12)) is a negative definite matrix. Certainly U '' is a negative definite and so is

F

''

. Further ((diagU')⊗I_n+1)F '' is negative definite by assumption (i), so that A is negative definite. It is easily seen that for any

0

_≠

_x

_∈

_R

n(n+1)

which satisfies

(

−

1

_1×n

⊗

I

_n+1

)

x

=

0

, it holds that

(

,

₂

,

,

)

0

2

<

∂

∂

x

x

x

x

T

L

λ

x

x

_.

Therefore matrix (12) is non-singular. Further,

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

=

∂

∂

× × × × + n n n n n T n n n

I

U

F

x

x

x

x

G

1 1 ) 1 (

0

0

0

''

'

0

)

'

,

(

)

,

,

,

(

x

λ

(13) By the implicit function theorem (Nikaido 1968, p. 85) there exists a

neighbourhood Wof

( x

x

,

)

and continuously differentiable function

n n n

W

g

+ ₊ +

×

→

int

R

int

R

:

( 1) _{such that}

W

x

x

∈

∀

(

,

'

)

:

G

(

g

(

x

,

x

'

),

x

,

x

'

)

=

0

and if

g

(

x

,

x

'

)

≠

(

x

,

λ

)

∈

W

then

0

)

'

,

,

,

(

x

x

≠

G

x

λ

and it follows that

x

does not solve (7) at

( x

x

,

'

)

. Since

( 1) ( , ') int n n int n g x x ∈ R₊ + × R it holds ₊

∀

(

x

,

x

'

)

∈

W

:

)

'

))

'

,

(

(

(

)

'

,

(

x

x

U

F

x

x

x

V

=

x

−

, (14)

)

'

,

(

)

,

,

,

(

)

,

(

)

,

,

,

(

)

'

,

(

)

,

(

1

x

x

x

x

G

x

x

G

x

x

x

x

g

∂

∂

⎥

⎦

⎤

⎢

⎣

⎡

∂

∂

−

=

∂

∂

−

λ

λ

λ

x

x

x

, (15) where

g

(

x

,

x

'

)

=

(

x

(

x

,

x

'

),

λ

(

x

,

x

'

))

.

By the envelope theorem (see Takayama 1985, p. 138) we get from (8)

W

x

x

∈

∀

(

,

'

)

(

(

,

'

),

,

(

,

'

),

'

(

(

(

,

'

))

'

)

)

)

'

,

(

)

'

,

(

1

x

x

x

x

U

F

x

x

x

x

x

x

x

V

n

−

−

=

∂

∂

x

λ

λ

…

(12)

and since

λ

(

⋅ x

,

⋅

),

(

⋅

,

⋅

)

are continuously differentiable on W then V is twice continuously differentiable in W.

Step 2 Hessian of the indirect utility function

All we need now is to show that Hessian

V

''

(

x

,

x

)

is negative definite. It holds (arguments omitted in the last line)

= ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ∂ − ∂ − ∂ − ∂ − ∂ ∂ ∂ ∂ = = ' ) ' )) , ( ( ( ' ) ' )) , ( ( ( ' ' ) , )( , , ( ) , )( , , ( ) , ( '' :' ' 1 1 x x x x F U x x x x F U x x x x x x x x V V n n x x

λ

λ

λ

λ

… …

=

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

+

∂

∂

−

∂

∂

−

∂

∂

∂

∂

=

''

'

)

,

(

'

''

)

,

(

'

''

'

)

,

)(

,

,

(

)

,

)(

,

,

(

_{1 1}

U

x

x

x

F

U

x

x

x

F

U

x

x

x

x

x

x

_n n

x

x

λ

λ

λ

λ

…

…

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = × × × × × × + × '' 0 0 0 ' ' 0 0 ' '' 0 0 1 1 ) 1 ( U x x x x F U I n n n n n n n n n n n n n n

λ

λ

x x (17) By (14), (15) and (17)

⎥

⎦

⎤

⎢

⎣

⎡

+

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

∂

∂

⎥

⎦

⎤

⎢

⎣

⎡

−

=

× × × × × × + × + × × × + ×

''

0

0

0

0

0

0

''

'

0

)

,

(

)

,

,

,

(

0

0

'

''

0

0

''

1 1 ) 1 ( ) 1 ( 1 1 ) 1 (

U

I

U

F

x

x

G

F

U

I

V

n n n n n n n n n n n T n n n n n n n n n n n

λ

λ

x

x

(18) By (12) we can write

⎥

⎦

⎤

⎢

⎣

⎡

=

⎥

⎦

⎤

⎢

⎣

⎡

∂

∂

−

E

D

D

C

x

x

G

T 1

)

,

(

)

,

,

,

(

λ

λ

x

x

(19)

whereF 9 F , : , : , ) ( : 1 1 1 1 1 1 − − − − − − + = − = − = A BEB A A C BE A D B A B E T T (20) and by (18) we get

[

]

_[

_]

⎥

⎦

⎤

⎢

⎣

⎡

+

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎣

⎡

−

⎥

⎦

⎤

⎢

⎣

⎡

⎥

⎦

⎤

⎢

⎣

⎡

−

=

× × × × × × ×

''

0

0

0

''

'

'

''

''

'

0

0

'

''

0

0

''

1 1 1 1

U

U

F

C

F

U

U

F

D

I

I

D

F

U

I

E

I

V

n n n n n n T T T n n n n n n n n (21)

Step 3 Negative definiteness of Hessian

Since V is concave, then V '' is at least nonpositive definite. To show that ''

V is negative definite we need to prove that it is non-singular. Suppose that there exists _(x,x')∈Rn_×Rn _{such that}

[

]

[

]

. 0 ' '' ' '' ' ' '' 0 ' '' , 0 ' '' ' 0 0 0 1 1 1 1 = + − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − × × × × x U x U F C F U x I D F U x U F D I x I E I T n n T T n n n n n n

This system of equations is equivalent (by (20)) to , 0 ' '' ' 0 1 1 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − × a x U F A EB x E T T n ₍₂₂₎

,

0

'

''

'

''

'

'

''

'

''

'

'

''

0

'

''

1

₋

1

₋

1 1

₊

₌

⎥

⎦

⎤

⎢

⎣

⎡

−

_U

_F

_A

−

_BE

x

_U

_F

_A

−

_F

T

_U

_x

_U

_F

_A

−

_BEB

T

_A

−

_F

T

_U

_x

_U

_x

(23) 9_{See (12)}

where a is some real number. Substituting

_⎥

⎦

⎤

⎢

⎣

⎡

0

x

E

into (23) we get , 0 ' '' ' '' ' ' '' 0 ' '' 1 1 ₋ 1 _{+ =} ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ _{× −} − _{U F A F U x U x} a B A F U n T

which is equivalent to (after left-multiplying by

F'

T)F 10 F

,

0

0

'

''

'

1 1 1

⎥

⎦

⎤

⎢

⎣

⎡

−

=

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎥

⎦

⎤

⎢

⎣

⎡

−

× × −

a

B

a

B

x

U

F

RA

T n n ₍₂₄₎ where

''

F

I)

'

U

R

:

=

(

diag(

)

⊗

. (25) By invertibility of

R

,

A

and since _AR−1 _=I+F'T_{U ''F'R}−1 _{we get}

from (24) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = − × a B R F U F x U F_'T _{'' ' '}T _{'' '} 1 0n1 _, which is equivalent to ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = − × a B R F x_{' '} 1 0n 1 _.

Putting x' into (22), observing that

n T

T

T_{A F U F R B EB R B I}

EB −1 _{' '' '} −1 = −1 − _{and due to invertibility of}

_E

we get , 0 0 1 1 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ _{− ×} a B R B x _{T n} (26) But by definition of

R

(see (25))

R

−1 is a block-diagonal matrix with negative definite matrices

1 ''

₍

₎

)

)

(

(

−

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

∂

−

∂

j j j

x

f

c

x

F

U

x

on the diagonal. Moreover

B

=

−

1

_n×1

⊗

I

_n+1 and (26) imply

10

,

0

)

(

)

)

(

(

0

1 1 1 ''

⎥

⎦

⎤

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

∂

−

∂

=

⎥

⎦

⎤

⎢

⎣

⎡

_× = −

∑

f

x

_a

c

x

F

U

x

n _n j j j j

x

which is possible only if

a

= x

0

,

=

0

. From this we get x'=0 (by the

equation above to (26)), so that we have shown that equality

0

'

''

_⎥

=

⎦

⎤

⎢

⎣

⎡

x

x

V

(see (21)) is possible only if x= x'=0, therefore V '' is negative definite and w.l.o.g. we can assume that

V

''

(

x

,

x

'

)

is negative definite on W, which ends the proof.

Remark 1 It should be noted that the above proof „works” for all points

n n

x

x

,

'

)

∈

int

R

₊

×

int

R

₊

(

for which optimal consumption (see (7)) is

positive. This observation allows us to broaden the class of ‘base’ models for which the indirect utility function is strongly concave: models of type (7) generate strongly concave indirect utility functions if assumptions (i)-(vi) are

met and for every

(

x

,

x

'

)

∈

int

R

n₊

×

int

R

n₊ it holds that optimal

consumption level is positive.F

11 F

10B

4.2. Homogeneous production functions

Let us put aside the assumption of strict concavity and negative definiteness of Hessians of production functions. Suppose that for at least two j’s (w.l.o.g.

j

=

1

,

2

,

…

) it holds

(vii) f_j is continuous on

R

₊n+1, positively homogeneous of degree 1 and twice continuously differentiable and strictly increasing over

int

+1

+ n

R

with

0

)

,

(

,

0

)

,

(

>

∂

∂

>

∂

∂

j j j j ij j j j

l

l

x

f

x

l

x

f

,

i

=

1 …

,

,

n

and concave on interior of its domain and ( j, _j)>0

j x l

f only if _xj >>0,_lj >0_{. Moreover, rank of}

negatively semidefinite Hessian of f_j is n everywhere on

int

+1 +

n

R

.

11 _{Our approach eliminates inputs x, x' and consumption c=y−x' with 0 entries –}

We shall proceed keeping in mind that – by Euler’s theorem (Lancaster 1968, p. 335-336) – if

f

_j satisfies assumption (X4.2X) then

j j j T j j j j j y y f y f f x x x x x λ λ ∈ = ∃ ⇔ = = : R 0 ) ( ) ( ) ( '' '

where x =( , )∈intR₊n ×intR₊

j j

j _{x l .}

We shall show that if functions f₁ and

f

₂ fulfill (X4.2X) (and the other

ones satisfy (i) or (X4.2X)) then Hessian of the indirect utility function V is

nonpositive definite. Since the way of construction of V is as before, then to

show that V is twice continuously differentiable (near

( x

x

,

)

) it is

sufficient to show that matrix

A

(see (12)) is negative definite. Certainly,

A

is nonpositive definite. It is negative semidefinite if singular. Suppose that

for some 0 ( 1, , ) ( +1):

+

∈ =

≠_{x x … x}n _Rn n _{Ax=0. It implies x}T_Ax=0,

which is possible only if xTF'TU ''F'x=0 _{and x}T_F'T _{=0. Since}

) (

'' j j

f x ,

j

=

3 …

,

,

n

, are negatively defined and by construction of

F

''

,

then

_x

3

₌

_…

₌

_x

n

₌

0

_{. By Euler’s theorem and assumption (}

X

4.2X) on rank of

Hessian f_j'' there exist scalars

α

_j such that xj =

α

_jxj, j=1, 2 (this observation comes from Hirota and Kuga (1971)). Since xTF'TU ''F'x=0 only if F'x=0, then using Euler’s theorem again brings

)

(

)

(

0

j j j j j j

f

f

x

x

x

x

α

=

∂

∂

=

,

which is possible only if

α

_j

=

0

, since ( j)>0

j

f x by assumption. This

implies x=0 – contradiction, so that A is non-singular, and therefore

negatively definite. We can use (21) to express V ''. Hessian V '' is

negatively definite if solution x,x' of (22), (23) (or equivalently (22), (24)) is trivial (if it exists for a given value of a). We know that turnpike labour inputs are positive:

l

j

>

0

,

j

=

1 …

,

,

n

. Take any

α

₁,

α

₂

non-vanishing simultaneously, such that

α

₁

l

1

+

α

₂

l

2

=

0

. Let us

define

_x

=

(

_x

1

,

_…

,

_x

n

)

∈

_R

₊n(n+1) _as j _j j _{j n} j j _{= x , =1,2, x =0, =3,…,} x

α

. Put a=0, x'=( f (x ), f (x2),0, ,0)∈R₊n 2 2 1 1 1

α

…

α

, and

n x R∈ 2 2 1 1

0

⎥

_⎦

=

α

x

+

α

x

⎤

⎢

⎣

⎡x

. Substituting the values into system (22), (24) and

observing that x_{'=[ 0 0]}F_{' ,} F_'TU _''F_'= A₋R

2

1

α

… x

α

and E

is non-singular we see that

x

, x

'

solve the system for

α

=0 and x'≠0. This means that V '' is singular and therefore negative semidefinite.

Suppose now that only f₁ satisfies (X4.2X) and

f

2

,

…

,

f

n satisfy (i). We

shall show first that system (22), (24) has solution if a=0. Let

) 1 ( 1_{,0, ,0)} ( ∈ ₊ + = _{x R}n n x … . Left-multiplying (24) by AxT E A we get

,

0

0

'

''

'

0

1 1 1

_a

_l

₁

a

B

a

B

x

U

F

RA

T n T n T

₌

⎥

⎦

⎤

⎢

⎣

⎡

−

=

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎥

⎦

⎤

⎢

⎣

⎡

−

=

_x

− ×

_x

×

which is possible only if a=0. So that (22), (24) become 1 1 ' '' ' 0, 0 ' '' ' 0, T T x BA F U x RA F U x − − ⎡ ⎤ −_{⎢ ⎥}− = ⎣ ⎦ = (27)

which – by Euler’s theorem – implies 1 ' '' ' ( 1,0 , ,0 )

n n T_{U x}

F

A− =

α

x …

for some

α

∈Rand

.

'

''

'

0

1 1

₌

₋

_α

_x

−

=

⎥

⎦

⎤

⎢

⎣

⎡

x

_B

T

_A

−

_F

T

_U

_x

Since l1 >0 it can hold only if x=0,

α

=0, but this shows that system (22), (24) has only a trivial solution, under

α

=0 (if a ≠ 0 then it is

an inconsistent system). We have shown that

0

'

''

_⎥

=

⎦

⎤

⎢

⎣

⎡

x

x

V

if x= x'=0, so

that Hessian V '' is negatively definite. Now we can stateF 12

F

Theorem 1 _{Suppose assumptions (ii)-(vi) hold and production function fj}

satisfies assumption (i) or (X4.2X),

j

=

1 …

,

,

n

. The indirect utility function

12 _{Similar results, but for social production frontier only (not for utility), were}

V (see (6)) is strongly concave in a neighbourhood of turnpike if and only if

the number of production functions satisfying (X4.2X) is less than 2.

7B

5. ROLLING PLANS ARE GOOD

From Kaganovich (1996) we know that for every rolling plan

(

)

∞ =1

)

,

,

(

x

_t

y

_t

c

_{t t} it holdsF 13 F ) , , ( ) , , ( lim x_t y_t c_t x y c t→∞ = (28)

To prove that a rolling-plan is good we shall show that it converges toward turnpike fast in a neighbourhood of turnpike. The main result of the paper is

Theorem 2 Fix _x n

+

∈ R . Suppose that

(

(

x

_t

,

y

_t

,

c

_t

)

)

∞_t=1 is an adaptive rolling plan from x. The sequence is a good process.

Proof: By (28) we have lim _t

t→∞x = . There exists a neighbourhood 'x W

of x such that W W'× '⊂W where W satisfies the thesis of lemma X4.1X.

Since x is strictly positive, then for sufficiently large t’s _xt+1 solves uniquely

{

}

' '

max ( , ')_t ( ', )_t

x W∈ V x x +V x x (29)

To prove the thesis it suffices (by concavity of V ) to show that mapping

{

}

arg max ( , ') ( ', ) : '

x V x x +V x x x W∈ is contractive at x .F

14

F Since

restriction V is a function of _{W C -class over W it must hold for large t :} 2

1 ' t x₊ ∈W and ( , 1) ( 1, ) ₀ ' t t t t V x x V x x x x + + ∂ ∂ + =

∂ ∂ . Let us define a function

: ' ' n S W W× → R S as follows: ( , ') ( ', ) ( , ') ' ' ( , ') ' V x x V x x x x W W S x x x x ∂ ∂ ∀ ∈ × = + ∂ ∂

13_{After some mild modification of proof of theorem 1, p. 181, in Kaganovich (1996).}

14 _{For a neighbourhood W of x we call mapping :h W'→W' contractive at x if}

[0,1) _{x W}: _{h x}q( ) _{h x}q( ) _{x x}

α α

∃ ∈ ∀ ∈ − ≤ − , where q is a fixed positive integer number and q( ) : ( )

q times

Then 2 2 2 2 2 2 ( , ') ( ', ) ( , ') ' ' ( , ') ( ', ) ( , ') ' ' T V x x V x x S x x x x x x x V x x V x x S x x x x x ∂ ∂ ∂ = + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ _{= +} ∂ ∂ ∂

For every ( , ')x x ∈ ×W W' '' it holds that ( , ') '

S x x x

∂

∂ is an invertible matrix

and therefore, since ( , ) 0S x x = , then there existsF 15

F a function :h W'→W'

such that ∀ ∈x W S x h x: ( , ( )) 0= and h is continuously differentiable on ' W . Moreover 1 ( , ) ( , ) '( ) ' S x x S x x h x x x − ∂ ∂ ⎡ ⎤ = − ⎢_{⎣ ∂} ⎥_{⎦ ∂} . Denote 2 2 2 21 11 2 22 2 ( , ) ( , ) ( , ) , , ' ' V x x V x x V x x V V V x x x x ∂ ∂ ∂ = = = ∂ ∂ ∂ ∂ . We shall show that 1 11 22 21 21 [_{V V} ] [− _VT _V ]

− + + possesses no eigenvalue with modulus greater or

equal to one – this will finish the proof, since then h is contractive at x . By

symmetry of V₁₁+V₂₂ and _V21T _{+V21 the eigenvalues of interest are real.}

Suppose that for some 0_{≠ ∈ R and x} n _{λ∈ R it holds} x x V V V V ₊ T _{+ =}

_λ

−_{[ ]}−_{[ ]} 21 21 1 22 11 , which is equivalent to

x

V

V

x

V

V

T

_]

_[

_]

[

₂₁

+

₂₁

=

₁₁

+

₂₂

−

λ

.

Left-multiplying last equality by _{x , and using negative definiteness of} T

11 22 V +V we get x V V x x V V x T T T ] [ ] [ 22 11 21 21 + + − =

λ

(30) By assumption, 11 21 21 22 '' ''( , ) T V V V V x x V V ⎡ ⎤ = _{= ⎢ ⎥}

⎣ ⎦ is negatively definite so that

we have

11 21 11 22 21 21 21 22 0 [ ] [ ] . T T T V V x T T T x x x V V x x V V x x V V ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ >_⎣ − _{⎦⎢ ⎥ ⎢ ⎥}= + − + − ⎣ ⎦ ⎣ ⎦ and 11 21 11 22 21 21 21 22 0 [ ] [ ] . T T T V V x T T T x x x V V x x V V x x V V ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ >_{⎣ ⎦⎢ ⎥ ⎢ ⎥}= + + + ⎣ ⎦ ⎣ ⎦ Therefore 11 22 21 21 11 22 [ ] [ ] [ ] T T T T x V +V x<x V +V x< −x V +V x and 21 21 11 22 [ ] 1 1 [ ] T T T x V V x x V V x + − < < + ,

which shows that λ < (see 30). This ends the proof. 1

8B

6. SUMMARY

In this paper we have shown that adaptive rolling plans are good under the assumption of neoclassical technology. We have also shown (by use of rather elementary tools) strong concavity of indirect utility function. As we mentioned, in Bala et al. (1991) it was proven that in one-sector case adaptive rolling plans are good and efficient. ”Efficiency puzzle” of adaptive rolling plans in multiproduct economy seems to have been unsolved, so far.

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