Nonlinear input-output models and comparative statics

Tilburg University

Nonlinear input-output models and comparative statics

Kaper, B.

Publication date:

1979

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Citation for published version (APA):

Kaper, B. (1979). Nonlinear input-output models and comparative statics. (pp. 1-25). (Ter Discussie FEW).

Faculteit der Economische Wetenschappen.

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No. 79.089

Nonlinear input-output models and Comparative Statics

juli 1979

2

-Introduction 3

1 Statement of the model 5

1.1 Input-output models 5

1.2 N-, M- and P- Matrices 7

1.3 Irreducibility 8

2 Linear input-output models and comparative statics 11

3 Nonlinear input-output models 11~

3.1 Nonlinear analysis 14

3.2 Nonlinear input-output models 19

Appendix 23

3

Introduction.

In this paper we will deal with static nonlinear input-output models. Given a nonnegative final demand vector the problem of the existence and uniqueness of a meaningful solution of a nonlinear input-output model will be examined. In [1] Gale and Nikaido derived a result on monotonicity of the system: if the demand of some commodities increases while others remain constant then under certain conditions the production of none of the

commodities decreases. We will strengthen the conditions in order the pro-duction of all commodities will increase. Furthermore we will extend these results in the following sense: if the production of all coaBnodities in-creases for which the demand dein-creases then all productions increase. Finally we will deduce a result on the increase by the largest proportion for the case just mentioned.

In section 1 we list some definitions and we will deduce some of their basic properties. Preceding the analysis of non linear models we will summarize

in section 2 some results for the linear input-output models analogous to the nonlinear results of section 3.

Notations.

- ~ n ~ . - {1,2,...,n}, n EIN.

- x EIRn; x~ 0" xi ~ 0, iii E ~ n~.

x~ 0 p xi ~ 0, ii i E ~ n~, xi ~ 0 for at least one index i.

x~ O p xi ~ O, ~li E ~ n~.

Similar definitions hold for matrices: A ~ 0(a positive -), A~ 0 (a semi-positive -) and A~ 0(a nonnegative matrix) respectively. - x E IRn; I} (x) -{i ~ i E ~ n~, xi ~ p}

I- (x) -{i ~ i E ~ n~, xi ~ p} I~ (x) -{i ~ i E ~ n~ xi - p}

f ~

x _{- (max {0, xi}, x- - (max {0, - x.}), x - x - x .}

4

- Let G: S C~gn

~~n~

S open and convex, be a differentiable function:

T

G(x) - IG~(x~~...,xn),...,Gn(x~,...,xn)j .

The Jacobian-ma.trix of G at x is denoted by DG(x) - ~d~G~(x)...BnG~(x ~ ,

i

B~Gn(x)...BnGn(x)

- S

Section 1. .STATEMENT OF THE MODEL; PRELIMINARY.

1.1 INPUT-OUTPUT MODELS.

In this paper we study input-output models in which the econor~y is devided into n industries, each producing one comonodity andin which the econo~r is consuming the products of the other industries as well as its own product. The model contains only goods which cease to exist once they are used up in production. There is no production-lag.

Let xi be the total output of industry i, and fij(xj) the amount of output of industry i absorbed by industry j. The net output of each industry, i.e.

n

the excess of xi over jE1 fij(xj), is available for outside use and will meet the final demand. Th~the overall input-output balance of the whole econo~y can be expressed in terms of n equations:

n

x. -_{1 j-1}E f. (x ) f c, i - 1,...,n_{ij j i}

where c. represents the final demand for output i. Let f represent the_{1 n}

vectorvalued

function onlRn with components f., f.(x) -.E

f..(x.).

Then

(1.1) can be written as x - f(x) f c i i- J-1 1J J or simply as F(x) - c _(1.3) where F(x): - x - f(x).

We will call x the _{(gross) production vect~or and c the finaZ}

demand veetor of the model (1.2).

We shall impose the following conditions on fij. i) fij(0) x 0(,F(0) - 0)

ii) f. 1

lj E C(S1), where S1 is an open and convex subset of

6

iii) ajfij(0) ~ 8jfij(x~) ? 0

y xj E S~

(ajfij(x): - afij(x)~axj)

In a linear input-output model we assume fij(xj) - aijxj,

the aij (~ 0) being constant for the input coefficient of commodity i used in industry j. The above system of equations now becomes

n

xi - E ai x f ci, i- 1, ..., n

j-~ j J

In matrix notation (1.5) becomes

or simply

Bx - c

1.2 N-, M- AND P-MATRICES.

Let A be an n-square matrix. A principal mtinor of order p is denoted by A i1...i _{- det a ..} p i1i1 'aili i1...i ; p P . aipil...aipiP

A Zeadting prtincipal minor of order p is denoted

by

- det

-a1

...s1

.

1

p

ap1...apP"

- The n-square matrix A will be called a P-matr2x

if the leading

principal minors of arbitrary order p, p E ~n~, are strictly positive.

- The n-square matrix A will be called an N-matrix if a._{i ~ j,} ~ 0, ij

-The n-square matrix A will be called an M-matrix if A is an N- and a P-matrix.

1.2.1 Lemma.

Let A be an n-square N matrix. A is a P-matrix ifl' A-1 ~ 0. (By some authors the condition of lemma 1.2.1 is used for the definition of an M matrix).

1.2.2 Lemma.

Let A be an M-matrix, B an N-matrix such that A ~ B. Then B is

an M-matrix.

1.3 IRREDUCIBILITY.

- An n-square matrix A is called reducibeZ if there exists an

index set I C ~n~ such that

-~ (~)

akj - 0 ii (k, j) E I x I.

- An n-square matrix A is called irreducibeZ if for each index

~

set I C ~n~ there exists a pair of indices ( k,j) E I x I with a ~ ~ 0 .

1.3.1 Consequence

Let A be a nonnegative,

irreducibel n-square matrix.

For each x and ~ E IRn with x ~~ there exists at least one index k E I~(x - y) for which

xk - (Ax)k ~ yk - (A~r)k.

Proof.

As A is irreducibel there exists a pair of indices (k,j) E ID x

~Ip(x - ~) such that akj ~ 0. Consider the kth equation of x-~:

n

~ - (Ax)k - ~ - 1~1 ~lxl - ~ - lEI

~lXl - l~I

~lxl

-0

-

0

e ~ - lEI ~l - 1~I ~xl - ~j xJ 0 0

l~j

~yk- (~)k.

In terms of a static linear input-output model with monotonic increasing productions irreducibility of the technology matrix A means that of all industries with non-increasing production there is at least one

-1

industry with decreasing demand.

It is immediately clear that a posítive matrix

is irreducibel.

1.3.2 Lemma.

Let A be an irreducibel P-matrix. Then Á 1~ 0(or equivalently Ax ~ A~ ~

x ~~) c. f. Varga [].

- p

- Let G: S C~n ~~n, _{S-open and convex, be a differentiable function.} The function G is called irreducibel in x E S if for each pair of vectors (x, ~) E S X S with x ~~ there exists a k E ID(x~) such that Gk(x) ~ Gk(~). In the sequel the concept of illeducibility will be used in connection with inverse-isotor{y (c.f. lemma 1.3.2).

1.3.3 Definition. (c.f. Ortega ( 3] ).

A function G: S CIRn ~IRn is inverse (strict) isotone on S if

G(x) ~ G(~) ~ x ~ y(x ~~) for each pair x, ~ E S x S p

1.3.~ Consequence.

If G is irreducibel and inverse isotone then G is inverse strict isotone. Proof. G(x) ~ G(~) ~ x ~~. Suppose IG(x -~) ~~.

Then by the irreducibility of G there exists an index k E I~ such that Gk(x) ~ Gk(~).which contradicts our assumption. p

Another way of introducing the concept of irreducibility of nonlinear n-dimensional functions is by means of their Jacobian-matrix.

1.3.5 Lemma

Let G: S C~n -,~n, S open and convex, be a continuously differentiable

function on S whose Jacobian matrix DG is an irreducibel M-matrix at each

x E S. Then G is inverse strict isotone.

Proof .

Let x, ~ E S with G(x) ~ G(~). We will apply a meanvalue theorem on G based on the fundamental theorem of integration theory

As DG is an irreducible M-matrix the matrix

Rx~~ :- ~I1 DG(x f t(~ - x)) dt

will have a positive inverse (leimna 1.3.2). Hence x ~~

O

Section 2. LINEAR INPUT-OUTPUT MODELS AND COMPARATIVE STATICS.

In general for linear and non-linear models we are concerned with the following questions.

1. given a nonnegative demand vector, does the model admit a(unique)

non-negative solution.

2. what happens with the (gross) production vector if the demand vector changes.

In the case of a linear input-output model globally we have three sets of necessary and sufficient _{conditions for the existence of a meaningful} solution: the generalised Hawkins-Simon conditions (formulated in terms of determinants), the diagonal-dominance condition and the Perron-Frobenius conditions (stated in terms of the characteristic roots). In view of the results for nonlinear input-output models we state without proof the generalized Hawkins-Simon theorem. (c.f. Nikaido [2]).

2.1 Theorem.

Consider the linear input-output model with technology matrix A and final demand vector c,

x- Ax f c or equivalent Bx - c (2,1)

(B is an N-matrix). Then the following conditions are equivalent i) _{for some vector c~ 0(2.1) has a solution x~ 0}

ii) for ar~y _{vector c~ 0(2.1) has a solution x~ 0}

iii) B is an M-matrix

--

p

The necessary and sufficient character of the conditions establishes the equivalence of the Hawkins-Simon, Perron and diagonal dominance conditions. Moreover, if the matrix A, (A - I- B) is irreducible then given c~ 0

(2.1) has a solution x~ 0.

---If we compare solutions of linear input-output models that belong to

2.2 Theorem.

Consider the linear input-output model with technology matrix A and final demand vectors c and d,

(I - A)x - c

, (I - A)~ - d .

Suppose that A is an irreducible M-matrix. If i) c ~ d then x ~~.

ii) d- c f j~, where ~ is the kth unit vector ofIRn then

x.

1 ~- y i E ~n~ yi - Yk

a

Economically i) asserts that if the demand of some commodities increases where all others remain constant then~the productions-of a11 commodities increase. If the demand of only one good increases then the increase of its production will relatively be the largest.

Recently, Sierksma [4] considered linear input-output models in which the production increases of all commodities for which the final demand decreases. He proved in that case that the production of all commodities should increase if the production matrix A is irreducible.

2.3 Theorem

Consider the linear input-output model with final demand vectors c and d,

(I-A)x - c,

(I-A)~ - d,

satisfying the condition [ j E I'- (d - c) ~ xj ~ yj] .

If the production matrix A is an irreducible M-matrix then i) x~~

and moreover

ii) _{Yi - xi ~ max {0, sup (Yj - xj)}}

13

Section 3. NONLINEAR INPUT-OUTPUT MODELS.

In the first part of this section we develop some preliminary analysis on non-linear n-dimensional functions. In the second part these results will be applied to non linear input-output models. Although some of the results of this second part could be formulated for general non-linear n-dimensional functions we will not do so. The problem formulation is intuitively clear for input~output systems and would be rather artifically in tYie case of n-dimensional functions.

3.1 NONLINEAR ANALYSIS.

Let G: S C~n -,~n be a local homeomorphism at each point x E S. In this section we will be concerned with the existence an3 uniqueness of nonnegative solutions of the function system G(x) - c, where c~ 0. The continuation property will play a fundamental role.

3. 1. 1 Definition .( c. f. Ortega [ 3] )

The mapping G: S C~n ~~n has the continuation property for a given continuous function q: [0,1] CIR1 -~IRn if the existence of a continuous function p: [ O,a) -~ S, a E (0,1] stich that G(p(t )) - q(t ) for all

t E[O,a) implies that lim p(t) p(a) exists with p(a) E S and G(p(a)) -tTa

9(a). ~

3.1.3 Theorem (global existence-)

Let G: S C IRn -i 1Rn, S open and S~ IR}, be a local homeomorphism in each point of S, G(0) - 0 and let D-{x E 7R} I G(x) ~ 0}.

If G is norm-coercive on D, i.e, for any Y~ 0 there exists a c~ 0 such that IIG(x)p ~ y yx E D with nxh ~ c, and if G is inverse isotone on ]R} then G ( D ) - IR} .

Proof .

We will show first that G on D has the continuity property for linear functions q(t) -(1-t)c0 f t-c1, t E[4,1] where c~, c1 EIR}, c0 ~ c1.

t E[O,a]}. Then there exists a c~ 0 such that UG(x)p ~ Y for all x E D with ~~ xU ~ c. The set Bc -{x E D I U xU ~ c} is closed and hence it follows from the compactness of Bc that there exists a sequence {tk} C[O,a)

such that lim tk - a and lim p(tk) - x E D.

k~ k-~ ~

From the continuity of G we have G(x) - q(a). Let U and V be open neighbor-hoods of x and q(a) respectively such that the restriction GU of G to U is a homeomorphism from U onto V. Then there exists a t1 ~ a such that p(tk) E U for tk E(t1,a) and G(p(t)) E V for t1 ~ t ~ a. The function

p(t) - Gu1 (G(p(t))), t E(t1,a) satisfies p(tk) - p(tk) for all tk E (t1,a).

On arguments similar to uniqueness we conclude that p(t) - p(t) for all

t E(t1,a). By the continuity of GU 1 we have lim p(t) lim p(t) -tTe, t?a

GU1(G(p(t))) - x. Hence G has the continuity property for linear functions q.of the above type.

Let us now take ancG E G(D) for which there exists an xG EIR} such that G(x~) - c~. We will next show that there exists a continuous function p:[0,1] -~ D as expressed in the above section of the proof with p(0) - x~ and for which holds G(p(t ))- q(t ) for all t E[ 0,1] .

Let U and V be open neighborhoods of x0 and G(x~) respectively such that GU: - GIU be a homeomorphism of U onto V. Then there exists a t1 E(0,1]

such that q(t) E V for t E[O,t1) and hence we can define a continuous function p:[O,t1) -~ U defined by p(t) - GU1(q(t)), t E[O,t1). As G is inverse isotone onlR} p maps into D, p:[O,t1) -~ U n D. By the continuation property p(t1) - lim p(t) exists and G(p(t1)) - q(t1). If t1 ~ 1 then we

t?t1

can repeat the process and continue p successively to points t2 ~ t3 ~.. ~ 1. Now let i ~ 1 be the maximum value of t for which p may be continued in this way, ~ aup ti. Then, either 1; tN for some N, in which case G(p(t))

-q(t) for all t E[ 0,~] , or else G(p(t)) - -q(t) for all t E[ 0,~) so that, by the continuation property again G(p(~)) - q(~). Since p(~) E S, p(0) ~

n

-p(~), p(~) EIR} a,nd S open we may therefore apply the same process to con-tinue p beyond ~. But this would contradict the maximality of ~ and hence ~ must equal 1.

16

-In [1] Gale and Nikaido proved the univalence of a differentiable function G:S C~gn -,~n~ S open and convex, assuming that the Jacobian-matrix DG is a P-matrix at each point x of S. If moreover the Jacobian matrix DG of G is an M-matrix then the function G is inverse isotone, i.e. G(x) ~ G(~)

implies x ~ ~r.

Conversely, if the inverse function G 1 is differentiable and G is inverse isotone then the Jacobian matrix is a P-matrix.

The results of Gale and Nikaido mstter to the development of some of our results. Because of an indistinctness in the region of validity of the theorems of Gale and Nikaido we have given a reformulation of the theorems in appendix A.

Combining the assumptions and results of theorem A.2, A.3 with theorem 3.1.3 we get

3.1.4 Theorem

Let G: S C g~n ~~n~ S open and convex, S~ IR}, G( 0) - 0, be a continuous

differentiable function.

If G is norm-coercive on D, i.e. for any Y~ 0 there exists a c~ 0 such that IIG(x)II ~ y for all x E D with ~xll ~ c, and the Jacobian matrix DG(x) is an M-matrix onlR} then for each vector c EIR} there exists one and only one vector x E1R} such that G(x) - c.

-Proof .

As the Jacobian matrix DG(x) is an M-matrix the function G satisfies the conditions of theorem A.2 as well as of theorem A.3. Hence G is a univalent function on S and G is inverse isotone. Then the conditions of theorem 3.1.3

on norm-coerciveness and inverse isotony are also met which completes the

theorem. ~

As an immediate consequence of theorem 3.1.~ we have

3.1.5 Consequence.

Under the conditions of theorem 3.1.1~ for each vector c? 0 there exists exactly one vector x? 0 such that G(x) - c. 0

3.1.6 Theorem (inverse strict isotone-)

Let G: S C~gn ~~n~ S open and convex, be a continuous differentiable function. If the Jacobian -matrix DG is an M-matrix on S and DG(x) is irreducibel at x E S then G is strict inverse-isotone at x E S, i.e. G(x) ~ G(~) implies x ~~ for any ~E S.

Proof.

Consider the continuous function q:[ 0,1] -i IRn~ q(t )-( 1-t ) G(x) f tG(~) for which there exists a function p:[0,1] ~ S such that G(p(t)) q(t), then p(0) -- x and p(1) -- ~r.

By assumption the Jacobian matrix DG of G at x is an irreducibel matrix. Continuity implies that there exist open neighborhoods U and V of x and G(x) respectively such that GU: - GU is a homeomorfism of U onto V that is strict

inverse isotone.

Hence there exists a t1 E(0,1] such that p(0) ~ p(t) for each t with 0 ~ t ~ t1. If t1 ~ 1 then from theorem A.3 we already know that G is an

inverse - isotone funetion hence p(t1) ~ p(t) ~ p(1). Finally we may conclude

that p(0) ~ p(1). O

3.1.7 Theorem.

Let G: S C~gn ~~n~ S open and convex, S~IR}, G(0) - 0, be a continuous differentiable function and let D-{x EIR}IG(x) ~ 0}.

If G is norm-coercive on D and if the Jacobian -matrix DG is an M-matrix onIR} and is irreducibel at 0 then for each vector c EIR} there exists one and only one vector x~ 0 such that G(x) - c.

Proof.

From theorem 3.1.1~ we know that there exists exactly one vector x? 0 such that G(x) - c. The irreducibility of DG at 0 implies that G is strict inverse

isotone at 0, i.e. 0 ~ x ~

3.1.8 Theorem (global univalence -~

Let G: S C1RH -~IRn, S open and convex, be a continuous differentiable function. If the symmetric part of the Jacobian matrix is a P-matrix on S t hen G is

globally univalent on S.

Proof.

Let x and ~ be two different points of S.

Define functions X:[ 0,1] -~IRn, ~ t) -(1-t)~ftx. and c~:[ 0,1] -~]Rn, cp:[ 0,1] -~ ~~ ~P(t) - G(x(t)) - G(~).x-~.

Convexity of S implies X(t) E S; ~(0) - 0. If we can prove cp(1) ~ 0 then G(x) ~ G(~r) .

Let us differentiate cp with respect to t:

(c.f.

n

(t) - E

a, Gi(x(t) (xi-yi) (xj-yj) ? K

i,j-1

~

Berger _{[ 7] .) Hence cp'(t~ ~ C for t E (0,1] .}

Application of the meanvalue theorem on ~~ gives

w(1) - w(o) -~'(o), o ~ o ~ 1

.2 NON-LINEAR INPUT-OUTPUT MODELS.

In this section we will apply some of the foregoing results on non-linear input-output models, described in the introductory section 1, c.f. formulae

(1.1) -(1.3). We assume that the conditions (1.4) are satisfied throughout

this section.

3.2.1 Theorem.

Consider the nonlinear input-output model x - f(x) - c

F(x) - c

where f satisfies the conditions or equivalently

i) fij(C) - 0, ii) fij E C1(S1)~S1 ~~t~

(3.2.1a)

(3.2.1b)

iii) 8j fij(0) ~ 8j fij(xj) ? 0 (3.2.1c)

If Df(0) is a P-matrix then for each final demand vector c(EIR~.) there exists one and only one production vector x(EIR}) such that x- f(x) - c.

Yroof.

From the third condition of (3.2.1c) it follows that Df(0) ~ Df(x) ~ 0 and hence I- Df(0) ~ I- Df(x) ~ I, or equivalently DF(0) ~ DF(x) ~ I. Then the off-diagonal elements of DF(x) are negative implying that DF(x) is an N-matrix.

By assumption Df(0) is a P matrix.

Then DF(x), majorizing DF(0), is an M-matrix for each x E S(c.f. lemma 1.2.2) where S is the Cartesian product of n sets S1.

We will show next that F satisfies the norm-coerciveness condition of the global existence theorem 3.1.k. We will apply a meanvalue theorem on F:

F(x) - F(~) - Ry~x.x - ~ where

R_~,x'- ₀'I1 DFE~ f t(x - ~))dt.

20

-IIFxq2-aR.xp2-xTRTR

x~ CNxg2

- X - - X ~ X - -

-The last inequality follows from

the symmetry of RX. Rx and from the

con-dition that R is a P matrix. x

Hence, F satisfies the conditions of theorem 3.1.~ from which follows this

theorem.

O

3.2.2 Theorem.

Consider the nonlinear input-output system (3.2.1) for which Df(0) is a P-matrix. For each pair of demand vectors c and d with c ~ d there exists a unique pair of production vectors x and ~ respectively for which x ~~r. If moreover Df(x) is irreducibel at x then x ~~.

Proof.

Existence and uniqueness of production vectors follows from theorem 3.2.1. Inverse strict isotony of f is a direct consequence of theorem 3.1.6. ~

Next we will derive a theorem on comparative statics that includes the in-verse isotor~y of theorem 3.2.2. It generalises the results obtained by Sierksma [] for linear systems to nonlinear input-output models.

3.2.3 Theorem (comparative statics).

Consider the nonlinear input-output system (3.2.1) where DF(0) is a P-matrix. Then, for each pair of production vectors x,~ E S for which

[ ~ E I'-(F(~r) - F(x) )~ y~ - x~ ~ 0]

we have x ~ ~.

Proof.

Let x and ~ E S, where S is the Cartesian product of n sets S1, for which (3.2.1) holds.

If r-(F(~) - F(x))- ~6 then F(x) ~ F(~). According to lemma A.3 we get x ~~ which proves the theorem.

Let us suppose that I'-(F(~) - F(x)) ~~ and assume that r- ~`) consists of in elements (indices). Determine a point z with z~ - x~,

j~ r- and zj - yj, _{j E r-, Then xj ~ zj, j E r-, From the third} condition of (3.2.1c) it follows that DF(x) is an N-matrix on S. We apply

a meanvalue theorem on the components Fj, j ~ r- of F:

Fj(z) - Fj(x) f

n

i

i-1

E

iE r

aiFj(T(j)) (zi - xi) a1FJ(T(J)) _{(zi - xi)} where T(J) is a point in between z and x.

Hence

Fj(z) ~ Fj(x) ~ Fj(~)

,

_{j ~ r-} (3.2.2)

We define three projectivities by means of their matrixrepresentations: E is the matrix obtained from I by replacing the row vectors

ejT, j~ r-, by the nul vector Ó; E2: - In - E1 and P1 -(eT). E r. -J J

Consider the linear (n-m)-dimensional variety L(P;0), where p:- E1~ ánd D:- E2 (ff2n ), and the mapping P: S C]Rn -~ L, P- p f E~. Then ( P~ ) j - yj for all j E r- _{and (P~)j -~j for all j~ r-.} The~composite function P1oF restricted to the variety L defines a function F: S c~n-m ~~n-m~ S:- P1S ~ such that

Fi(~) - F.(n) for some j~ r-, i - 1,...,n - m_J -where

~-P1n~~EL.

From (3.2.2) there follows for the function F

F(~) ~ F(~)

where ~- P1z and n- P1~. F is again a differentiable function on S- P1S, S open and convex whose Jacobi matrix is an M-matrix.

According to theorem A.3 we get ~ ~ r~ where ~j - zj and nj - yj, j~ r-,

Hence zj ~ yj, j~ r-, For j~ r- we supposed xj - zj so that xj ~ yj, j~ r-. Now for all j E ~n~ we have xj ~ yj, hence x ~~. O

3.2.1~ Theorem.

Consider the nonlinear input-output system (3.2.1) where DF(0) is a P-matrix. If for each pair of vectors (x,~) E S x S with F(x) ~ F(~) we have

i) DF(x) is irreducibel

ii )[ j E I'-(F(~) - F(x) )~ yj - xj ~ 0] then x ~ ~.

Proof .

If I'-(F(~) - F(x)) - ~ then F(x) ~ F(~) and according to theorem 3.2.2 there holds x ~ ~.

Let us suppose that I'-(F(~) - F(x) ~~d. Together with DF(x) all submatrices

of DG(x) are irreducibel, especially the Jacobian DG(x).

Appendix

The following lemma is essential in the proof of the univalence theorem of Gale and Nikaido.

A.1. Lemma.

Let G: S C~gn ~~n~ S open and convex, be a differentiable function.

If the Jacobian matrix DG(x) is a P-m,atrix on S then for each a E S we have

X:- {x E S I G(x) ~ G(a), x~ a} -{a},

Proof.

The proof breaks up into two parts. In the first part we will show the theorem for any compact subset SD C S which contains the vector a:

XS:- {x E SG I G(x) ~ G(a), x~ a} -{a}

(A.1)

In that case we could also formulate Xp as follows:

XG - G-1(~ ~ G(a)) n {x E SS, x~ a}.

The first set of the intersecting sets is closed as original of a closed set under a continuous function whereas the second set is compact, hence X~ is compact. The proof of (A.1) now develops along the same lines as that gives in [1 ] and will therefor not be repeated here.

In the second part of the proof of lemma A.1 we assume that there exists a vector b E S, b~ a such that b E X. As S is a convex set there should exist a compact set SD C S, a, b E S~, This contradicts the first part of

the proof. p

A.2 Theorem (univalence-).

Let G: S CIR --~1Rn, S open and convex, be a differentiable function.

A.3 Theorem (inverse isotone).

Let G: S C~gn ~~n~ S open and convex, be a differentiable function.

If the Jacobian matrix DG(x) is an M matrix on S then G is inverse isotone,

References:

[1] Gale, D. and H. Nikaido: The Jacobian matrix and Global univalence of Mappings. Math. Annalen, ~9 (1965), 81-93.

[2] Nikaido, H. : Introduction to sets and mappings in modern economics; 1970, North Holland publishing company, Amsterdam.

[g] Ortega, J.M. and W.C. Rheinboldt: Iterative solution of nonlinear

equations in several variables; 1970, Academic Press, N.Y.

[~t] Sierksma, G.

: Nonnegative Matrices; The open Leontief model;

ws-7701, 1977.

[51 Varga, R.S. : Matrix Iterative Analysis; 1962, Prentice Hall, Inc., New Yersey.

[6] Woods, J.E. : Mathematical Economics; 1978, Longman, London.

z6

In de Reeks ter Discussie ziin verschenen:

1.H.H. ii~;geiaar 2.J.P.C.Kleijnen 3.J.J. Kriens 4. L. R. J. Westermarin S.W. van Hulst J.Th. ~~an Lieshout 6.M.H.C.Paardekooper 7.J.P.C. Kleijnen B.J. Kriens 9.L.R..T. Westermann 10.B.C.J. van Velthoven 11.J.P.C. Kleijnen 12.F.J. Vandamme 13.A. van Schaik 14.J.vanLieshout J.Ritzen J.Roemen 15.J.P.C.Kleijnen 16.J.P.C. Kleijnen 17.J.P.C. Kleijnen 18. F. J . Vandaffine 19.J.P.C. Kleijnen 20.H.H. Tigelaar 21.J.P.C. Kleijnen 22.W.Derks 23.B. Diederen Th. Reijs W. Derks 24.J.P.C. Kleijnen

2~.B. van Velthoven

Spectraalanalyse en stochastische lineaire differentievergelijkingen. De rol van simulatie in de algeme-ne econometrie.

A stratification procedure for

typical auditing problems.

On bounds for Eigenvalues

Investment~financial planning with endogenous lifetimes:

a heuristic approach to mixed integer progremming.

Distribution of errors among input and output variables.

Design and analysis of simulation Practical statistical techniques. Accountantscontrole met behulp

van steekproeven.

A note on the regula falsi

Analoge simulatie van ekonomische

modellen.

Het ekonomisch nut van nauwkeurige informatie: simulatie van onder-nemingsbeslissingen en informatie. Theory change, incompatibility and non-deductibility.

De arbeidswaardeleer onderbouwd?

Input-ouputanalyse en gelaagde

planning.

Robustness of multiple rsnking

procedures: a Monte Carlo

ex-periment illustrating design

and analysis techniques.

Computers and operationa

research: a survey.

Statistical problems in the

simulation of computer systems.

Towards a more natural deontic

logic.

Design and analysis of simulation: practical, statistical techniques. Identifiability in models with lagged variables.

Quantile estimation in regenerative simulation: a case study.

Inleiding tot econometrische

mo-dellen van landen van de E.E.G.

Econometrisch model van België.

Principles of Economics for com-puters.

2~

2b.F. Cole _{1~'orecrssting by crxDuuentia,l}

september '"jb r;ruoot.l,inl;, Ltr~~ _{Ifux rirrci Jenkins}

proceciurt uncl :~pe~ct.rul euraly-s1S. A SimultttiUn ~t11dy.

~'(.ti,. tleuts _{;;c~mc: re1'urmulati~ns r3nci extension:; juli "j~} irr the univaririte Box-Jenkins

time series analysis. ~~ 2t3.W. Derks _{Vier econometrische modellen.}

29.J. Frijns _{Estimation methods for multi- oktober '76} variate dynamic models.

30.P. Meulendijks _{Keynesiaanse theorieën van oktober '76} handelsliberalisatie.

31.W. Derks _{Structuuranalyse van econometrische september '76} modellen met behulp van

Grafentheo-rie. Deel I: inleiding in de Grafentheorie.

32.W. Derks _{Structuuranalyse van econometrische oktober '76} modellen met behulp van

Grafentheo-rie. Deel II: Formule van Mason.

33. A. van Schaik _{Een direct verband tussen economische} veroudering en

bezettingsgraadver-liezen. _{september '7E~}

3~. W. Derks Structuuranalyse van Econometrische Modellen met behulp van Grafentheorie. Deel IIT.De graaf van dynamische

modellen met één vertraging. oktober '76 35. W. Derks Structuuranalyse van Econometrische

Modellen met behulp van Grafentheorie. Deel IV. Formulé van Mason en

dyna-mische modellen met éé.n vertraging. oktober '76 36. J. Roemen _{De ontwikkeling van de}

omvangsverde-ling in de levensmiddelenindustrie

in de D.D.R. oktobt,r ''((~

37. W. Derks Structuuranalyse van Econometrische modellen met behirlp van

grafentheo-rie.

Deel V. De graaf' van dynamische

mo-dellen met meerdere vertragingen. uktobPr '"(f, 3t~. A. vt~n Schaik r;en direkt verband tussen

economi-sche verouderirrg en bezettings-graadverliezen.

Deel II: gevoeligheidsanalyse. december '7E;

3~~. W. Derks Structuuranalyse van Econometri-sche modellen met behulp van Grafentheorie.

Deel VI. Model I van Klein,

sta-tisch. december '7f,

k0. J. Kleijnen _{Information Economics: Inleiding}

en kritiek november ''(~,

1~1. M. v.d. Tillaart. _{De spectrale representatie var,} mutivariate zwak-stationaire stochastische processen met

dis-crete tijdparameter. n~,v~~rnt,..r- ' Ir.

1~2. W. Groenendaal _{Een econometrisch model van}

Th. Dunnewijk Engeland _december

'(r-~3. R. Heuts _{Capital market models for}

4~. J. Kleijnen en P. Rens 45. J. Kleijnen en P. Rens

46. A. Willemstein

47. W. Derks ~i8. L. Westermann 1~9. W. Derks 50. W.v. Groeneiidaal en Th. Dunnewijk 51. J. Kleijnen en P. Rens 52. J.J.A. Moors 53. ~t.M.J. Fieuts 54. B.B. v.d. Genugten 55. P.A. Verheyen 56. W.v.den Bogaard en J.Kleijnen 57. W. Derks

58. R. Heuts

59. A.P. Willemstein 60. Th. Dunnewijk W. van Groenendaal 61. A. Plaisier

A. Hempenius

28

A critical analysis oi IBM's inventory

package ímpact. december '76 Computerized inver.tory management:

A critical analysis of IBM's impact system.

Evaluatie en foutenanalyse van eco-nometrische modellen.

Deel I.

december '76

Een identificatie met:iode voor een li-neair discreet systeem met stor.ingen

op input, outpst en structuur. januari '77 Structuuranalyse van econometrische

modellen met behulp van grafentheorie.

Deel VII. Model I van Klein, dynamisch.februari '77 On systems of linear inequalities

overlRn. Februari '77

Structuuranalyse van econometrische modellen met behulp van Grafentheorie. Deel VIII.

Klein-Goldberger model.

Een econometrisch model van het Verenigd Koninkrijk

A critical analysis of IBM's inventory package "IMPACT"

Estimation in truncated parameter-spaces

Dynamic transfer function-noise modelling (Some thecretical con-siderations}

Limit theorems for LS-estimators in linear regression models with independent errors.

Economische interpretatie in model-len betreffende levensduur van kapitaalgoederen

Minimizing wasting times using priority classes

februari '77

februari '77

februari '77

maart

'77

december '76

mei

juni

juni Structuuranalyse van Econometrische

Modellen met behulp van Grafen-theorie. Deel IX. b'fodel van

lan-den van de E.E.G. juni '77 Capital market models for pcrtfolio

selection (a revised version) j~ir~i '77 Evaluatie en foutezanalyse van

eco-nometrische modellen. Deel II. Het Modeï I van L.R. Klein.

An econometric Model of the Federal

aug.

Republic of Germany 1953-1~73

aug.

'77

Slagen of zakken. Eer. intern rapport

over de studieresultaten

2g

62. A. Hempenius Over een maat voor de juistheid

van voorspellingen aug. '77

63. R.M.J. Heuts

61~. R.M.J. Heuts

Some reformulations and extensions in the univariate Box-Jenkins time

series analysis approach

(a revised version)

sept . ' 77

Applications of univariate time series modelling of U.S, monetary

and business indicator data sept. '77

65. A. Hempenius en J. Frijns Soorten van

prijsheteroskedasti-citeit in marktvraagfunkties. okt. '77

66. Fi.H. Tígelaar ldentifiability in Multiple

Time Series okt. '77

67. J. van Lieshout en P. Verheyen

Levensduur in een jaargangen- nov. '77 model

68. Pieter J.F.G. Meulendijks "De macro-economische betekenis van geinduceerde technische ontwikke-ling; een meer-sektoren model met

jaargangentheorie" nov. '77 69. R.M.J. Heuts en P.J. Rens A Monte Carlo study to obtain the

percentage points of some goodness of fit tests in testing normality, when observations satisfy a

certain low order ARMA-scheme dec. '77 70. J.C.P. Kleijnen Generalizing Simulation Results

through Metamodels dec. '77 71. Th. van de Klundert

72. B.B. van der Genugten

73. A.P. Willemstein

Winstms.~cimalisatie in het Jaar-gangenmodel met vaste technische coëfficiënten; een

int-enta-risatie van de problematiek dec. '77

A central limit theorem with applications in regression

analysis

_{jan. '78}

Evaluatie en foutenanalyse van econometrische modellen. DEEL III Stochastische fluctuaties op de parameters en heteroscedasticiteit

in een lineair model jan. '78

74. Pieter J.F.G.Meulendijks Some reflections on macro-econo-mic planning, policy and de-velopment in the Netherlands:

1918-1978 75. J.H.J. Roemen

apr. '78

Qvereenkomsten en verschillen in uitgangspunten tussen

76. J.D. Sylwestrowicz

77. Pieter J.F.G. Meulendijks

78. B.B. van der Genugten

79. J.M.G. Frijns 80. H.H. Tigelaar 81. Jack P.C. Kleijnen 82. J.M.G. Frijns 83. M.H.C. Paardekooper 8~. J.H.J. Roemen

85. J.H.J. Roemen

86. G.J. de Nooij Th. M.M. Verhallen

87. J. Kriens

88. J.D. Sylwestrowicz

3v

Applications of Hybrid Computers

in Econometrics - Part I. juli '78 On a disequilibrium analysis of

the labour market. Review of and comments upon R.S.G. Lenderink and J.C. Siebrand, A disequilibrium analysis of the labour market,

Rotterdam University Press, '76, aug. '78

Asymptotic normality of 2SLS-estimators in simultaneous

equa-tion systems.

aug. '78

Imperfect competition of the

labour market(s) and the adjustment

of factor inputs. okt. '78 The identifiability problem in

dynamic simultaneous equations with

moving average errors.

nov. 178

Bayesian information economics:

an evaluation.

dec. '78.

Instrumental Variable Estimators maart '79 Applied on Pooled Time Series

Cross Section Models.

Inverse perturbations for approxi- maart '79 mations of least squares solutions.

Kostenverbijzondering op basis van een input-output-model voor een situatie met vaste en variabele

kosten .

maart '79

Een schattingsprocedure op basis van macro data voor de (stationaire) overgangswaarschijnlijkheden in een Markovmodel ter beschrijving van de ontwikkeling van de omvangsverdeling

van ondernemingen. april '7y

Marketing mix sensitivity, april '79 Statistical Sampling in Auditing april '79 Applications of hybrid computers

in econometrics, Part II. An implementation of a medium size