Dipartimento di Metodi Quantitativi per le Scienze Economiche ed Aziendali
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Rapporto n. 197
Production and distribution in a network:
the system structure predominates the individual
proficiency
Daniele Felletti
system stru ture predominates the individual
pro ien y
DanieleFelletti
Keywords: networks,systemdynami s, sto hasti matrix,irredu ible matrix, Perron{Frobenius theorem, orporate value.
Abstra t
A system of produ ing rms is onsidered here. The rms own ea h other through xed quotas of sto ks, sothey periodi ally share thein omesandthelosses(i.e.,they losethebooks).
Anetworkmodelisintrodu edtodes ribethesystem. Thevalues ofthermsevolvedynami allya ordingtothenan ial ows(whi h dependonthetopologyofthenetwork)andtothedistributionofthe individualprodu tivities,nevertheless,onthelongrun,onlythesystem stru ture matters. In the limit ase in whi h the shares matrix is irredu ible(itrepresentsastrongly onne tedgraph),thevaluesofthe rmsaredeterminedratiosofthetotalprodu tion,whi hevolveslikea Brownianmotion. Sothevaluestendtobeperfe tly orrelated. These ratiosaredetermined onlybytheshares matrix,whilethe individual pro ien iesae tonlythetotalprodu tionofthesystem. Whenthe sharesmatrixisredu ible (the graphis onne ted,but notstrongly), somermsin reasetheirvaluemu hmorethantheotherones.
1 Introdu tion
The value of a ompany is learly related to the revenues oming from its produ tivity. However, a ompany an ownsome shares theother ones. In this ase its value is ae ted notonly by its own produ tivity, butalso by the value of the other ones. Several works have investigated the ee t of mergers and a quisitions on the sto k pri e (e.g., [9 ℄, [6 ℄). They a tually bindea h other ina network ofrelationshipsso thatit isoften noteasy to understandwho ontrolswho. Thenetworkstru tureof ompaniesisafa t, besides itseems tobes ale{free([3 ℄). Itintrodu esafeedba kee t onthe dynami sand isa ountable forthesystemi risks([2 ℄and ([4 ℄).
a ounts for their individual pro ien y), share their values a ording to xed quotas. The result is a N{dimensional, dis retetime, sto hasti pro- ess. However, a ording to the topologyof the network, thevalueson the long run are weakly related to the individualpro ien y. If the stru ture is strongly onne ted (it is a dire ted, weighted graph), they a tually are determined only by the topology and by the total value of the system. If therearesome onne ted omponents,ea honeevolvesautonomously,while whenit is onne ted, butnotstrongly,one setof rmshasnoshares ofthe remaining rms. In this ase the rms of the latter group in rease their valuemu h more qui klythanthose of theformerset.
2 The dynami s of the value
Let x t
be theve tor of the valuesof the rms (their ash) at time t. x t
is thevalue ve tor. At timet+1,every rm loses thebooks,distributingits ash among the owners a ording to the shares matrix P. Then i-th rm re eivesafra tionP
ij
ofj-thrm value x j;t
,besidesits ashis in reasedby the revenue oming from the individual produ tion "
i;t+1
relatively to the period [t;t+1℄. The random ve tor "
t
is theoutput ve tor. Clearly, ifx j;t isnegative,theowners mustpay,a ordingto thesameratios,to avoidthe bankrupt y.
1
The followingre ursiveidentityholds
x i;t+1 = N X j=1 P ij x i;t +" j;t+1 () x t+1 =Px t +" t+1 (1) By ba kward re urren e: x t =P t x 0 + t X j=1 P t j " j (2)
3 Gaussian produ tions
Most of the results do not depend on the probability distribution of the output ve tor, however if "
t
are Gaussian (and x 0
is xed or Gaussian), the values are Gaussian for every t. Then, to onsider Gaussian output ve tors improves the on ision of the omments. Besides, thanks to the lawoflargenumbersandthe entrallimittheorem,mostoftheresultshold independentlyontheprobabilitydistributionoftheoutputve torsin ethey on ernthedynami sof thesystemon thelong run.
1
be orrelated, besides the probability distribution of the output ve tor is assumed onstant: " t N(; ) () E [" i;t ℄ = i ; ov[" i;s ;" j;t ℄ = ij Æ st whereÆ st
istheKrone kerdelta,while isthe ovarian ematrixof simulta-neousprodu tions. Finallytheinitialvalue x
0
isassumed to bea onstant (i.e.,nota randomve tor).
The followingresultshold
E[x t ℄=P t x 0 + t 1 P j=0 P j ! C= t 1 P j=0 P j P j T (3) whereC ij = ov[x i;t ;x j;t ℄ .
Equations(1)and (2) learlyshow thesto hasti dynami sof thevalue ve tor. The hypothesis of Gaussian output ve torssimplies thereasoning sin e,beingthevalueve torGaussianforeveryt,thevalueve toristotally identiedbytheve tor E[x
t
℄andthe matrixC. 2
4 The dynami s of the value in the long run
Theshares matrixisne essarily a sto hasti matrixsin e
P ij 2[0;1℄ ^ u T P=u (4) whereu T
=(1;1;1;:::;1). Therst onditionisobvious,whilethese ond onditionstatesthat the olumnssumto one. Thisisne essarily truesin e i-th olumn onsistsofthequotasofownershipofi-thrm,whi hmustsum to one.
Consequently1is aneigenvalue forPand u isa lefteigenve tor. Perron{Frobenius theorem and Wielandt's theorems ([7 ℄) are useful to understandthesystemdynami swhenthesharesmatrixPisirredu ibleand when it isprimitive. Even though itis not usedin thiswork, to gure out theJordan anoni alform ofP([1 ℄)mayhelpto understandthedynami s. SeetheAppendixforthedetails.
ThetheorystatesthatP annothave( omplex) eigenvalueswithnorm greaterthan 1,thatis its spe tralradius. Besides, when u
T
6=0,there is noneedtosolvethesumsto knowtheasymptoti behaviorofx
t
forlarget. 2
WhenPisprimitiveallthe(eventually omplex)eigenvalueshavenormless than 1. The ve tor u alone generates the left eigenspa e of 1. Let v be a righteigenve torforA(therighteigenspa eis learlyone{dimensionaltoo, soall therighteigenve tors of1 aremultiplesof ea hother).
Equation (6)states that
lim t!+1 P t = 1 u T v vu T =v~u T where ~ v= v u T v
istheonlyrighteigenve tor forPwhoseentriessumtoone(i.e.,thePerron ve tor of P). The rate of onvergen e dependson the se ond eigenvalueof P([5 ℄). Then Theorem(7)intheAppendiximplies
t P j=1 P t j =tv~u T +o(t) t P n=1 P t n P t n T =t 2 ~ vu T u~v T +o t 2 =t 2 u T u ~ v~v T +o t 2
Consequently,inthelongrun(t!+1),ifu T
6=0,thevaluesoftherms are x t t u T ~ v i.e., E[x t ℄=t u T ~ v+o(t) C=t 2 vv T +o t 2 where =u T u Let 3 G t =u T t X j=1 " j
thetotal produ tion. Sin e
E[G t ℄=<G>= u T t ; Var[ G t ℄= t 2 3 To onsider ~ G t =u T x 0 + t
P
j=1 " j!
doesnot hangetheresults,ex eptfor
x t G t ~ v
That is: the value of i-th rm tends to be a xed quota v i of the total produ tionG t ,beingv i
determinedonlybythetopologyofthenetworkP. Itimpliesthatthe orrelationstendto 1:
4 orr[x i;t ;x j;t ℄= t 2 v i v j +o t 2 q t 2 v 2 i +o( t 2 ) q t 2 v 2 j +o( t 2 ) 1
5 Irredu ible shares matrix
When Pis irredu ible,thePerron ve tor existssin ethe eigenspa eof 1 is one{dimensional, but there are h unitary eigenvalues. However the eigen-valuesareintheform
k =e 2 k h i
withk =0;:::;h, whereh is theindex of imprimitivity. Thisis thereason why a non{negative, irredu ible but not primitive matrix is said periodi . Besidesevery eigenvalueis simple(i.e.,its eigenspa eisone{dimensional).
P t
does not onverge in this ase, however it is similar to a matrix in thefollowingform:
~ P= 0 B B B B B B 1 0 0 ::: 0 0 0 e i 0 ::: 0 0 0 0 e 2i ::: 0 0 ::: ::: ::: ::: ::: ::: 0 0 0 ::: e (h 1)i 0 0 0 0 ::: 0 A 1 C C C C C C A where = 2 h
and A is a matrix with spe tral radius less than 1. That is P=M
~ PM
1
for some non{singular ( omplex) matrix M. As a onse-quen eP j =M ~ P j M 1 . ~
Pisa blo k{diagonal matrix,then
~ P j = 0 B B B B B B 1 0 0 ::: 0 0 0 e ji 0 ::: 0 0 0 0 e 2ji ::: 0 0 ::: ::: ::: ::: ::: ::: 0 0 0 ::: e (h 1)ji 0 0 0 0 ::: 0 A j 1 C C C C C C A 4
with A ! 0 as t ! +1. ~
P an not onverge (and so does P) be ause thediagonalelements
~ P 22 , ~ P 33 ,:::, ~ P hh
keepon y ling. However boththe matri eshave a Cesarolimit,whi h,forP, is:
lim t!+1 1 t t 1 X j=1 P j =v~u T (5)
It is easy to understand, onsidering the Cesaro limitof ~
P and reminding that the rst olumn of M must be a multiple of ~v while the rst row of M
1
mustbeamultipleofu T
bya oeÆ ientwhi histhere ipro alofthe former.
Every sto hasti matrix hasa niteCesaro limit,but it has a dierent formwhen thematrixis redu ible.
Equation (5)implies t 1 X j=0 P j =tv~u T +o(t)
forlarget. Sothat, again, 5 E(x t )=t u T ~ v+o(t)
Withsome more al ulations, one obtains
C=t 2 vv T +o t 2
Thus the values of the rms tend to hange oherently even though the sharesmatrixis notprimitive. The irredu ibilityis suÆ ient.
P is irredu ible if and onlyif the asso iated dire ted graph is strongly onne ted ([7 ℄ or [8 ℄). This means that for every ordered pair of rms f
1 , f 2 , a sequen e s 1 , ::: , s n
of rms an be found su h that f 1
owns some shares of s
1
, whi h ownssome shares of s 2
and soon, untils n
whi h owns some sharesof f
2
. Theoppositemust betrue too(in generalbya dierent sequen e).
Clearly,ifthesharesmatrixisnot onne ted,every onne ted omponent evolvesautonomously.
6 Redu ible shares matrix
Only one ase remains: when the sharesmatrix is onne ted butredu ible (i.e.,not strongly onne ted). Inthis ase P stillhasa niteCesaro limit,
5 Sin e
~ P
j
k k,withk=2;:::;h, y le,thesumonjremainsbounded. A j
form P= T 11 0 T 21 T 22 withT 11
lower{triangularblo kmatrix, and
T 22 = 0 B A 1 . . . A m 1 C A where A k
are irredu ible. This means that the rms an be qualitatively grouped into "owners" (the lastlistedone, when thesharesmatrix hasthe above{mentionedform)and"ownedrms"(eventhoughtheysimplydonot haveshares ofthe se ondgroup ofrms). Then, theCesarolimit is
lim t!+1 1 t t 1 X j=1 P j = 0 0 ET 21 (I T 11 ) 1 E where E= 0 B ~ v 1 u T . . . ~ v m u T 1 C A ~ v k
isthePerron ve tor of thematrixA k . Furthermore lim t!+1 P t = 0 0 ET 21 ( I T 11 ) 1 E ifand onlyifA k
areall primitive. Otherwise thelimitdoesnotexist. Thismeansthatthevaluetendto on entrate ontheowners(whi hare "more onne ted"),whiletheownedrmstendto(relatively)de reasetheir value. A tually themean value ofthe owners isof order t, whilethe oneof theowned rmsis justo(t) (so itdoesnotne essarily tendto zero).
7 Con lusions
omesstrongly onne ted, theprodu tivitiessimply ontribute to thetotal value of the system, whi h tends asymptoti ally to be shared among the rmsa ording toquotasthatdependonlyonthenetworktopology. When the network (whi h is a weighted, dire ted graph) is onne ted, but not strongly onne ted, the rms an be grouped into two lasses. The "less onne ted"ones arepenalizedwith respe t to the"more onne ted"ones.
The model an a tually be applied to a wider range of systems, sin e it des ribes the produ tion and the distribution on a network and shows thatstronger the onne tion,weaker thedependen e of theindividual per-forman e on the personalpro ien y. Besides it shows that networks tend to get more stiening and to trap their dynami s as they in rease their onne tion.
Appendix
A Irredu ible and primitive matri es
Hereare itedsome usefuldenitionsand properties:
Amatrix Ais redu ible([8 ℄)ifthere is apermutationB su h that
BAB 1 = K 11 K 12 0 K 22 where K 11 and K 22
are square matri es. Otherwise it is irredu ible. With the model and the onsequent formalism used in this work, a matrixisredu ibleif BAB 1 = K 11 0 K 21 K 22
A non{negative matrixA is primitiveif there is k >0 su h that A k ispositive.
Aprimitivematrix mustbeirredu ible.
Anirredu iblematrixisprimitiveifandonlyifthereisonlyone eigen-value on itsspe tral ir le.
forA.
Anon{negative,irredu iblematrixisimprimitiveifthere areh eigen-valueson its spe tral ir le. his theindexof imprimitivity.
if (x) = x n + 1 x n k 1 + 2 x n k 2 + 3 x n k 3 ++ s x n ks is the hara teristi polynomial of an imprimitive matrix A in whi h only the non{zero terms are listed, then the index of imprimitivity is the greatest ommondivisorof k
1 ,k 2 ,:::,k s .
B Perron-Frobenius theorem
IfA isnon{negativeand irredu iblethen
thespe tralradius (A)is an eigenvalue forA;
theeigenspa eof thespe tralradiusis one{dimensional;
there isa positiveeigenve tor of thespe tral radiusforA;
BesidestheCollatz{Wielandtformula holdsforall non{negative matri es:
(A)= max x0 x6=0 0 B B B B min 1jn x j 6=0 [Ax℄ j x j 1 C C C C A C Wielandt's theorems C.1
IfjBjAandAisirredu iblethen(B) (A). Inthe ase(B)=(A) (i.e.,
B
=(A)e i
forsome), then
B=e i DAD 1 withD ij = ( e i i j=i 0 j6=i C.2
If A is non{negative and irredu ible and has h eigenvalues 1
, 2
, :::, h onits spe tral ir le then
D.1 Equation 3 ov[ x i;t ;x j;t ℄= t X n=1 t X m=1 X a X b P t n ia P t m jb ov[ " a;n ;" b;m ℄= = t X n=1 t X m=1 X a X b P t n ia P t m jb Æ mn ab = = t X n=1 X a X b P t n ia P t n jb ab = t X n=1 h P t n P t n T i ij = = " t X n=1 P t n P t n T # ij D.2 Theorem If M n
is a onverging sequen e of matri es and M = lim n!1 M n , then as n!1 n X j=1 M j =nM+o(n)nM (7) Proof: Sin eM n
=M+o(1), forevery ">0there is n " 2N su h that max a;b=1;:::;N j [M n M℄ ab j<" 8n>n " Then 1 n n X j=1℄ M j nM = 1 n n X j=1℄ (M j M) = = 1 n n" 1 X j=1℄ (M j M)+ n X j=n" ( M j M) 1 n n" 1 X j=1 (M j M) + 1 n n X j=n" jM j Mj A n + 1 n n X j=n " N 2 "= A n + (n n " +1)N 2 " n A n +B" n!1 ! 0
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