The Effects of Aggregation on the Perron Root and its Corresponding Eigenvector

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

THE EFFECTS OF AGGREGATION ON THE PERRON ROOT AND ITS CORRESPONDING EIGENVECTOR

Erik Dietzenbacher

September 1986 WP-86-65

C o Z Z a b o r a t i v e Papers

report work which has not been performed solely at the International Institute for Applied Systems Analysis and which has received only

limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organi- zations supporting the work.

.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS

A-2361 Laxenburg, Austria

FOREWORD

The issue of aggregation in economics in general and in the case of the widely used input-output models is an important problem not yet satisfactory solved. This paper addresses precisely this question and constitutes a contribution to existing literature.

The classical economic problem which is connected with the results discussed in the paper relates to the aggregation of the data of an input-output table into a single sector to determine the rate of profit. This can be done under certain conditions, which are discussed in the paper.

The publication was prepared during the author's participa- tion in the Young Scientists' Summer Program at IIASA.

Alexander B. Kurzhanski Chairman

System and Decision Sciences

Program

The Effects of Aggregation on the Perron Root and _its Corresponding Eigenvector

Erik Dietzenbacher

Econometrics Institute, University of Groningen, P.0 Box 800, 9700 AV Groningen, The Netherlands.

In t h i s p a p e r t h e behaviour of t h e P e r r o n root and i t s corresponding e i g e n v e c t o r i s examined, when t h e underlying matrix is being a g g r e g a t e d . Bounds are p r e s e n t e d f o r t h e P e r r o n root and t h e elements of t h e P e r r o n v e c t o r of t h e r e s u l t a n t matrix. The bounds are mainly e x p r e s s e d in t e r m s of t h e P e r r o n root and v e c t o r of t h e original matrix. A s a n application aggregation in input-output analysis is considered.

1- Introduction.

Eigenvalues and e i g e n v e c t o r s have become useful tools in economic analysis.

Especially t h e P e r r o n root and i t s corresponding eigenvector play a n essential role. In empirical work however are t h e underlying matrices often exposed to errors or obtained a f t e r aggregation. I t t h e r e f o r e seems important to investigate how t h i s a f f e c t s t h e P e r r o n root and i t s v e c t o r . The impacts of small changes in matrixelements are thoroughly discussed in Varga [I9621 and Wilkinson [1965].

When t h e matrix i s changed substantially or i s aggregated, relatively less is known although, of c o u r s e , bounds c a n b e given f o r t h e P e r r o n root and i t s v e c t o r (see e.g. Berman and Plemmons [I9791 and S e n e t a [1981], a l s o f o r bibliographies).

Unfortunately, t h e s e bounds are in g e n e r a l e x p r e s s e d in t e r m s of matrixelements, so t h a t nothing c a n b e concluded a b o u t t h e actual change of t h e P e r r o n vector.

Our primary i n t e r e s t g o e s t o obtaining bounds, f o r t h e new P e r r o n v e c t o r , t h a t are given in t e r m s of t h e old one. Recently E l s n e r , Johnson and Neumann [I9821 exam- ined t h e case where t h e ith r o w of a nonnegative, i r r e d u c i b l e matrix is increased while i t s jth r o w i s d e c r e a s e d . In t h e p r e s e n t p a p e r w e shall consider changes in t h e P e r r o n root and v e c t o r when t h e matrix i s being aggregated.

Aggregation h a s been a topic in economics f o r long. F o r instance any study based on input-output t a b l e s , uses d a t a t h a t are somehow a g g r e g a t e d o v e r pro- ducts and industries o r o v e r regions. Our r e s u l t s may t h e r e f o r e b e applied directly to some of t h e dynamic Leontief-type models (see e.g. Takayama [1985]).

F o r f u r t h e r applications w e may think of Seton's eigenprices (see e.g. Seton [1985]) and Saaty's p r i o r i t y c o n c e p t (see e.g. S a a t y [1980], see Steenge [I9861 f o r links with t h e Leontief framework).

In t h e n e x t section w e shall discuss non-weighted aggregation, also r e f e r r e d t o as consolidation, where r o w s and columns are simply added. In section 3 w e shall consider t h e consolidation of a n input-output t a b l e which leads to a weighted aggregation of t h e matrix of input-output coefficients. The summary and conclu- sions shall b e p r e s e n t e d in section 4.

2. Non-weighted aggregation.

In this section w e consider a n nxn nonnegative, i r r e d u c i b l e matrix A . With p , y and p w e denote i t s P e r r o n r o o t and t h e r i g h t - and left-hand eigenvector corresponding t o p. To avoid unnecessary notational inconveniences w e p r e s e n t and proof o u r r e s u l t s f o r t h e simplest case where only t h e f i r s t and second s e c t o r are taken Loge t h e r . The a g g r e g a t e d (n-l)x (n-1) nonnegative, i r r e d u c i b l e matrix i s denoted by A , i t s P e r r o n root and v e c t o r s by jj, z and q . Generalizations of t h e theorems are obtained straightforwardly and are p r e s e n t e d without proofs in t h e appendix.

Definitions.

A Y

=

PY P

>

^{0, Y}

>>

^{0, Y'}

=

(Y 2 1 - - * ~ n ) p P A

=

p p ' P

>>

0 , P'

=

b l l ~ z , . . , ~ n >

GAG'

^{with G}

= I1

In ^O'

-, I

where In denotes t h e nxn identity matrix. Throughout t h i s section w e assume Y 1 5 Yz.

Theorem 2.1.

Proof. According t o t h e well known Subinvariance Theorem, t h e left-hand side i s proved when w e find a v e c t o r z

>

0 f o r which Zz

>

p z . Let (A"Z)~ denote t h e ith element of t h e v e c t o r iz, note t h a t in o u r notation i = 2 , . . , n . Take z'

=

( Y ~ + Y Z , Y ~ ~ . . , Y , ) t h e n

For v e c t o r s and m a t r i c e s we adopt the following notation in order t o describe their nonnegatlvlty. Let Z be a n-element v e c t o r , then Z 2 0 means z, 5 0 for each i, Z

>

⁰

means Z 2 0 and Z # 0 , ²

>>

0 means Zi

>

⁰ f o r each i. With 0 we denote the n- element null v e c t o r .

n

t a i l y l

+

a i 2 y 2

+ C

at, y j

=

p y i

=

pxi f o r i

=

3,..,n.

j =3

Thus A z

-

t p x , b u t equality c a n only o c c u r when a i l

=

a t 2

=

⁰ f o r i

=

l , . . , n which would c o n t r a d i c t t h e irreducibility of A . T h e r e f o r e z x

>

^px

.

To prove t h e right-hand s i d e w e t a k e x'

=

( y 2 , y 3 , .

.,

y n ) .

5 PYi

=

PX,

<

^{Y l + Y 2 pxi f o r i}

⁼

3 , . . , n . Y 2

From

Zz <

Y 1 + Y 2

px i t follows t h a t

E <

p Y 1 + Y 2

Y 2 Y 2

W e now p r e s e n t o u r basic theorem which states t h a t t h e r e l a t i v e i n c r e a s e in t h e elements of t h e P e r r o n v e c t o r i s t h e l a r g e s t f o r t h e sector which i s aggre- g a t e d . The proof essentially i s a refinement of t h e one used in E l s n e r , Johnson and Neumann [I9821 f o r p e r t u r b a t i o n s of a single row2).

Theorem 2.2.

Z i P Z 2

-

₅

--

f o r i

=

3,..,n.

Y i

E

^{Y 2}

Proof. Suppose to t h e c o n t r a r y t h a t t h e r e e x i s t s a n index m

>

2 f o r which

2 , U s i n g o u r r e f i n e m e n t , t h e bounds i n t h e i r Theorem 2.1 c a n a l s o be sharpened a c c o r d i n g t o (3).

which is not possible. Note t h a t equality would imply ami

=

0 f o r i

=

^l,..,n which would c o n t r a d i c t irreducibility.

The inequality in (3) i s s t r i c t when additionally t h e irreducibility is assumed of the submatrix defined by aij f o r i , j

=

3,..,n. Alternative proofs of (3) c a n b e obtained by using Fiedler and P t d k [1962; Th. 4.21 o r by applying t h e framework of Courtois and Semal [1984]. Both a p p r o a c h e s yield zi/% S z2/y2 f o r i

=

3,..,n from which pzi/yi S pz2/y2 easily follows. Because t h e non-weighted aggregation in this section simply means t h a t t h e f i r s t two r o w s and columns are added, both theorems also hold f o r t h e left-hand P e r r o n v e c t o r s . Thus in (2) and (3) w e may r e p l a c e y i and zi by pi and q i respectively ( f o r i

=

l,..,n).

9. Weighted aggregation.

In t h i s section w e f i r s t consider aggregation in a n input-output framework and p r e s e n t r e s u l t s comparable with theorems 2.1 and 2.2. Secondly, w e show t h a t aggregation of a homogeneous Markov Chain simply implies aggregation of t h e sta- tionary distribution.

The problems which may a r i s e when in a n input-output model s e c t o r s are a g g r e g a t e d , were f i r s t recognized by Leontief [1951]. Hatanaka [I9521 and McManus [I9561 focus on n e c e s s a r y and sufficient conditions f o r t h e aggregation scheme to b e a c c e p t a b l e . S t a r t i n g point is a n input-output t a b l e X , with i t s typical element zij denoting t h e deliveries from sector i to sector j. If, f o r t h e s a k e of simplicity again, we a g g r e g a t e sectors 1 and 2, t h e new table

2

i s obtained as GXG', with G as defined in (1). The matrix of input-output coefficients i s defined as A

=

where 2 denotes t h e diagonal matrix with t h e output v e c t o r z on i t s main diagonal. Hence

=

GXG1(G2G')-I

=

G A R with H'

=

2G'(G2G')-1, or equivalently

" 1 " 2

with w ,

=

^{" 1} and w 2

=

^{2 2} 0 0 I,, -2

2 1 + 2 2 Zl + 1 2

The weights w l and w 2 denote which f r a c t i o n of total output of sectors 1 and 2 Eomes from t h e sectors seperately3). Final demand v e c t o r s are denoted by f and f

=

Gf

.

The input-output equations are given by

The aggregation i s called a c c e p t a b l e if

2 =

^Gz f o r each final demand v e c t o r f .

From (5) and (6) i t follows t h a t under acceptability

2 =

( I

- ^A

^{) - l ~ f}

=

^G ( I- A ) must hold f o r a l l f

,

which implies

ZG =

^GA

.

A r a

') The theorems t o be p r e s e n t e d below a l s o hold f o r t h e more general c a s e i n which i t is o n l y assumed t h a t 0

<

w l , W 2

<

¹ and w l

+

^{W 2}

=

^1.

[I9591 f i r s t took eigenvectors into consideration and showed t h a t under accepta- bility t h e P e r r o n v e c t o r of t h e a g g r e g a t e d matrix i s t h e aggregated P e r r o n v e c t o r of t h e original matrix_. Moreover, t h e P e r r o n root does not change: p y

=

^Ay

implies pGy

=

^GAy

=

^{A G y .} The condition of acceptability i s quite s e v e r e and i t i s unlikely t h a t i t will b e fullfilled in p r a c t i c a l work4). I t also i s not n e c e s s a r y for Gy to b e t h e P e r r o n v e c t o r of A , with p t h e P e r r o n root.

Lemma

3.1.

pGy

=

^ZGy if t h e r e e x i s t s a v e c t o r t

>>

⁰ such t h a t y

=

^SGft

Proof. pGy

=

GAY

=

^GAZGft

=

lic%cA~y

=

This l e m m a implies t h a t when w e start f r o m t h e matrix A instead of f r o m t h e input- output t a b l e X, w e c a n always find weights w and w 2 t h a t provide a n aggregation which r e s u l t s in Gy being t h e new P e r r o n v e c t o r . The condition y

=

S G ' ~ states t h a t y l / x l

=

y 2 / x 2 , and t h u s t h e weights become w 1

=

y l / ( y l

+

^{y 2 )} and w

=

y 2 / ( y

+

^{y 2 ) .} In p r a c t i c a l work, s t a r t i n g f r o m

X I

i t i s unlikely t h a t t h i s con- dition i s m e t although i t i s weaker t h a n acceptability. In g e n e r a l t h e P e r r o n root will change, bounds f o r which are given by t h e following theorem.

Theorem 3.2.

Proof. W e only show t h e left-hand side and again use t h e Subinvariance Theorem.

W e c o n s t r u c t a positive v e c t o r z'

=

( Z ~ , Z ~ , . . , Z , ) f o r which

F i r s t t a k e zi

=

y i f o r i

=

3,..,n t h e n

Now taking z 2

=

max

1:; ^-

^I

^- zz 1

^gives

"

S e e Theil 119571 and Ara 119591 f o r conditions under which a m a t r i x can acceptably be eggregated.

which completes t h e proof.

For a n input-output matrix t h e weights are defined as output f r a c t i o n s and (7) c a n be r e s t a t e d as

N o t e t h a t when zl/yl

=

z2/y2 w e find

F =

p and z 2

=

y l / w l

=

y 2 / w 2

=

y1

+

y 2 which conforms with lemma 3.1.

The non-weighted aggregation simply meant addition of t h e f i r s t and second r o w and t h e n summing t h e f i r s t t w o columns. T h e r e f o r e w e could in o u r theorems r e p l a c e right- by left-hand P e r r o n v e c t o r s . Weighted aggregation implies addition of t h e f i r s t t w o r o w s and t h e n taking t h e weighted sum of t h e f i r s t and second column. W e t h u s need d i f f e r e n t e x p r e s s i o n s f o r t h e left-hand P e r r o n v e c t o r . W e f i r s t p r e s e n t t h e equivalent of (7), w e assume throughout t h i s section t h a t pl

r

p2.

Theorem 3.3.

Proof. To p r o v e t h e left-hand side t a k e q'

=

(p I,P 3,. ..P, t h e n

P 2

Thus

q'Z

^{2 p(w}

+

-w 2)qr. F o r t h e right-hand side t a k e q'

=

(p 2 , p 3 , .

.

, p n ).

P l

W e now p r e s e n t t h e bounds f o r t h e elements of t h e P e r r o n v e c t o r , theorem 3.4 for t h e right-hand and 3.5 f o r t h e left-hand v e c t o r . In both theorems w e have to distinguish between a n increasing, decreasing or constant P e r r o n root.

Thearem3.4. F o r i = 3

,..,

ⁿ a n d j

=

^1,2

Proof. W e f i r s t p r o v e (9). Suppose to t h e c o n t r a r y t h a t t h e r e e x i s t s a n index m

>

2 s u c h t h a t

-

zm

₌

_max-

_>

_{max ,then}

Ym i Yi

which i s n o t possible. ( 1 0 ) i s proved analogously. Finally w e p r o v e only t h e right- hand s i d e of ( 1 1 ) . Suppose t o t h e c o n t r a r y t h a t t h e r e e x i s t s a n index m

>

2 such t h a t

-

=

max-

>

max ym

and suppose f u r t h e r m o r e t h a t t h e s e c t o r s 3 , .

.

^,n are r e - o r d e r e d such t h a t

zi zn

m = n , - = - f o r i

=

k , . . , n

, -

Z j <% f o r j = 3

,...

k - 1 with k = 3

,..,

n.

Yi Y n j Y n

Then f o r i

=

k , . . , n

S t r i c t inequality must hold f o r at l e a s t one i because equality f o r i

=

k , . . , n would imply ail

=

ai

=

ai

=

⁰ with j

=

3 , .

.

, k -1, which c o n t r a d i c t s with t h e irreducibil- ity of A .

Note t h a t when w l / y l

=

w 2 / y 2 t h e left- as well as t h e right-hand s i d e of ( 1 1 ) equal z 2 / ( y 1

+

y 2 ) . From lemma 3.1 i t follows t h a t z 2

=

y l

+

y 2 a n d z i

=

y i .

Theorem 3.5. F o r i

=

3 , .

.

, n

Proof. Analogous t o t h e proof of Theorem 3 . 4 .

In t h e Appendix theorems 3.2

-

3.5 are p r e s e n t e d f o r t h r e e generalized types of aggregation.

A s a f u r t h e r application w e n e x t consider t h e aggregation of a homogeneous Markov Chain (MC). W e show t h a t this r e s u l t s in a non-homogeneous MC and t h e r e - f o r e eigenvectors no longer play a r o l e . The s t a t i o n a r y distribution of t h e aggre- gated MC however equals t h e a g g r e g a t e d s t a t i o n a r y distribution of t h e original MC.

Let t h e f i n i t e homogeneous MC b e d e s c r i b e d b y rr(t)'

=

r r ( ~ ) ' ~ ~ , w h e r e rr(t ) d e n o t e s t h e p r o b a b i l i t y d i s t r i b u t i o n at time t , ~ ( 0 ) t h e initial d i s t r i b u t i o n and P t h e t r a n s i t i o n matrix. Let t h e s t a t i o n a r y d i s t r i b u t i o n b e d e n o t e d by v , with v ' 1

=

1 w h e r e 1'

=

( 1 , . . , 1 ) . When P i s primitive p t -, 1v' elementwise f o r t -,

-.

Consequently rr(t)' -, rr(0)'1v1

=

v' elementwise f o r t -, w. By using conditional p r o b a b i l i t i e s i t i s e a s i l y s e e n t h a t , wken t h e f i r s t t w o setes are a g g r e g a t e d , t h e n_ew MC becomes % ( t + 1 ) ' = % ( t ) ' P ( t + l ) . H e r e P ( t + 1 ) i s defined as P ( t + I )

=

H(t)PG1 with G as defined in ( 1 ) a n d H ( t ) as defined in ( 4 ) with t h e fol-

lowing weights.

~ ( t

w 1

=

a n d w z

=

+

I z

⁺

l2

where rr(t ) * d e n o t e s t h e ith element of t h e p r o b a b i l i t y v e c t o r rr(t ). The weight w now i s t h e p r o b a b i l i t y to be-in state 1 at time t , given t h a t o n e i s in state 1 or 2 . Thus, t h e t r a n s i t i o n m a t r i x P i s n o l o n g e r independent from t. If %(t )

=

Grr(t ) t h e n

%(t +I)'

=

n(t)'GIH(t)PG'

=

rr(t)'PG1

=

rr(t + l ) ' G 1 . T h e r e f o r e , when t h e distribution of t h e a g g r e g a t e d MC e q u a l s t h e a g g r e g a t e d d i s t r i b u t i o n of t h e o r i g i n a l MC at time t , i t also d o e s at time t + l . This obviously i s t h e case f o r t h e initial p r o b a b i l i t y v e c t o r , %(O)

=

Grr(0). W e t h e n o b t a i n f o r t -, : %(t )'

=

rr(t)'G1 -, vlG'

= c'.

^This

a l s o is t h e s t a t i o n a r y d i s t r i b u t i o n as follows from

5' =

C'P, w h e r e i s t h e a g g r e - gation of P using weights w

=

v l / ( v l

+

v z ) a n d w z

=

v z / ( v l

+

v 2 ) .

4. Summary and conclusions.

In t h i s p a p e r w e h a v e d e r i v e d bounds f o r t h e P e r r o n root a n d f o r t h e ele- ments of i t s c o r r e s p o n d i n g e i g e n v e c t o r when t h e underlying matrix i s being a g g r e - g a t e d . W e h a v e distinguished two t y p e s of aggregation. F i r s t t h e case where a g g r e g a t i o n simply meant adding rows a n d columns a n d secondly t h e case where a g g r e g a t i o n w a s a p p l i e d within a n input-output framework which led to t h e u s e of weighted sums. The bounds are mainly e x p r e s s e d in t e r m s of t h e o r i g i n a l P e r r o n root a n d v e c t o r . A s s u c h t h e s e bounds p r o v i d e i m p o r t a n t information o n t h e b e h a v i o u r of t h e P e r r o n root a n d v e c t o r u n d e r a g g r e g a t i o n .

5. References.

A r a , K.: 'The Aggregation P r o b l e m in Input-Output Analysis," Econometrica, 27 ( 1 9 5 9 ) , 257-262.

Berman, A. a n d R.J. Plemmons: Nonnegative Matrices i n the Mathematical Sci- ences. N e w York, Academic P r e s s , 1979.

Courtois, P.J. a n d P. Semal: "Ekror Bounds f o r t h e Analysis b y Decomposition of Non-Negative Matrices," pp. 209-224 in Mathematical Computer Perfirmance a n d R e l i a b i l i t y e d . b y G . Iazeolla, P.J. Courtois a n d A. Hordijk. Amsterdam, North-Holland, 1984.

E l s n e r , L., C.R. Johnson a n d M. Neumann: "On t h e E f f e c t of t h e P e r t u r b a t i o n of a Nonnegative Matrix on i t s P e r r o n Eigenvector,

"

Czechoslovak Mathematical J o u r n a l , 32 ( 1 9 8 2 ) , 99-109.

Fiedler, M . and V. Ptdk: "On Matrices with Non-positive Off-diagonal elements and Positive Principal Minors," Czechoslovak Mathematical J o u r n a l , 1 2 (1962), 382-400.

Hatanaka,

M.:

"Note on Consolidation within a Leontief System," Econometrica, 20 (1952), 301-303.

Leontief, W.W.: The S t r u c t u r e of t h e A m e r i c a n Economy, 2929-2939, 2 n d e d . New York, Oxford University P r e s s , 1951.

McManus, M. : "On Hatanaka's Note on Consolidation,

"

Econometrica, 24 (1956), 482-487.

Saaty, T.L.: The A n a l y t i c H i e r a r c h y Process. New York, McGraw-Hill, 1980.

S e n e t a , E.: Non-negative Matrices a n d Markov C h a i n s , 2 n d e d . New York, S p r i n g e r Verlag, 1981.

Seton, F.: Cost, Use a n d Value: The E v a l u a t i o n of Performance, S t r u c t u r e a n d P r i c e s a c r o s s Time, *ace a n d Economic S y s t e m s . Oxford, Oxford University P r e s s , 1985.

Steenge, A.E.: "Saaty's Consistency Analysis: An Application to Problems in S t a t i c and Dynamic Input-Output Models," Socio-Econ. P l a n n . Sci., 20 (1986), 173- 180.

Takayama, A.: Mathematical Economics, 2 n d e d . Cambridge, Cambridge University P r e s s , 1985.

Theil, H.: ' Z i n e a r Aggregation in Input-Output Analysis," Econometrica, 2 5 (1957), 111-122.

Varga, R.S.: M a t r i x I t e r a t i v e A n a l y s i s , Englewood Cliffs, N.J., Prentice-Hall, 1962.

Wilkinson, J.H.: The Algebraic E i g e n v a l u e Problem, London, Oxford University P r e s s , 1965.

6 . Appendix.

Throughout this p a p e r w e have only examined t h e simplest case of aggregation where t h e f i r s t two sectors w e r e t a k e n t o g e t h e r . Consequently m o s t notational inconveniences could b e avoided. Our r e s u l t s however c a n b e adapted to o t h e r f o r m s of aggregation without undue efforts. Below w e p r e s e n t t h e equivalences of theorems 3.2

-

3.5 for t h r e e t y p e s of aggregation. The proofs are omitted as t h e y are basically t h e s a m e as t h e p r o o f s p r e s e n t e d for t h e original theorems.

Type i: aggregation of k sectors into a single new s e c t o r , n sectors remain unchanged. W e s h a l l use t h e following indexation of t h e sectors.

old : 1 , 2

,....,

n , n +I,.., n +k new: 1 , 2

,....,

n , n+1

Type ii : aggregation of k , sectors into s new s e c t o r s , n s e c t o r s remain t h e s a m e .

old : 1,2

,....,

n ,n +I,.., n +kl

,....,

n +kT-l+l ,.., n +k

,,....,

n +kSw1+1 ,.., n +k,

- I -

new:1,2

,....,

n , n + 1

,....,

n +r ,...., n +s Type iii : a l l k , sctors are a g g r e g a t e d i n t o s new sectors.

new: 1

,....,

r

,....,

s

with j

=

n +1,..,n +k

with r

=

1

,..,

s a n d j

=

n +kT -1+1 ,.., n + k T , where k o

=

0 ii. p m i n I

iii. as ii with j

=

kT-l+l

,..,

kT Theorem 3.3.

FyjJrnp[$1

with j

=

n +l,..,n +k

+ B ' P ~ ~ x

^I

with r

=

1

,..,

s a n d j

=

n +kT -l +l,.., n +kT

,

w h e r e k o

=

0 ii. p min

T

iii. as ii with j

=

kT-l+l

,..,

^{k ,}

Theorem 3.4. W e only p r e s e n t t h e e x p r e s s i o n s f o r t h e c a s e where

>

p .

C

^{W J P ~}

-

m a x ( P j l

f

5

E

s p max

T

C

^wjJ)j

L

m i n ( P j ) f

with i

=

1

,..,

n and j

=

n+1,.., n +k ii.

-

P

Y i P

with i

=

1

,..,

n , ^r

=

1

,..,

s and j

=

n +kr-l+l

,..,

n +k,, where k o

=

⁰

P P

iii.

:

_P min _f [zi

m p [$]I ^{s -} _{c v j} ^s

^y _P ^max _i ^[zi

^{m y} [$]I

j

with i

=

l , . . , s and j

=

ki-l+l,..,ki

Theorem _3.5. If

>

^{p then}

with i = l , . . , n and j = n + l , . . , n + k

with i

=

1

,..,

n', r

=

1

,..,

s and j

=

n +k, -1+1

,..,

n +k,, where k o

=

0

Q i P

ii.

- s -

^max

P i

ii

^r

:

^m;n@,)

^{+ I}

with i

=

1

,..,

s and j

=

ki -1+1

,..,

ki.

S P max p i

If in this l a s t theorem w e r e p l a c e qi and pi by zi and yi respectively, f u r t h e r - more s e t w j

=

1 f o r all j , then w e obtain t h e generalizations of theorem 2.2 f o r non-weighted aggregation.

Q i

min @ j )

* j

Note t h a t t h e bounds become weaker when aggregation r e s u l t s in more than one new s e c t o r , Consider f o r instance t h e generalization of theorem 3.5. For aggregation of type i. we may choose q , +l as t h e numdraire a f t e r which bounds f o r all o t h e r elements of t h e P e r r o n v e c t o r are given. In c a s e of type ii. aggrega- tion w e may s e t q,

+.

^at unity f o r e a c h r

=

l,.. ,s and thus obtain s s e t s of bounds.

When s i s small t h i s may still provide useful information, although t h e sets of bounds c a n not be compared with e a c h o t h e r as long as q,+, i s unknown. The expressions f o r aggregation of type iii. a r e given f o r t h e s a k e of completeness, t h e i r p r a c t i c a l use however i s little.