(M.R)-Systems as a Framework for Modeling Structural Change in a Global Industry

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

(M.R)-SYSIEMS

AS

A FRAMEWORK FOR MODELING SITUCTURAL CHANGE

IN

A GLOBAL INDUSIXY

John Casti

January 1 9 8 5 WP-8 5 - 1

Working Papers a r e interim reports on work of t h e International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily represent those of t h e Institute or of its National Member 3rganizations.

INTERNATIONAL INSTITUTE

FOR

APPL!ED SYSTEMS ANALYSIS 2361 Laxen burg, Austria

The theory of metabolism-repair (M,R)-systems is developed a s a means for mathematically characterizing an industrial firm. a n d a net- work of such (M,R)-systems is proposed as a suitable vehicle for describ- ing an e n t i r e industry composed of several interacting firms.

I t

is shown t h a t virtually all of t h e important features of an industrial process including production, marketing, innovation, growth, decline, emergence of new firms and s o on can be accommodated within the (M,R)- framework.

Theoretical issues associated with time-lags, dynamics, adaptation a n d selection a r e explored from t h e vantage point of (M.R)-systems, a s a r e practical questions involving the application of t h e theoretical ideas t o the world automotive industry. The paper concludes with a discussion of now (M.R)-system can be used a s a means for comparison of entire industries using the mathematical machinery of category theory.

( M . R ) - S Y S E E AS A F'RAWXORK FDR MODELING

SI"RUCTURAL CHANGE

IN

A

GLOBAL

IIJDUSl'KY

John Casti

1. The Problem of Structural Change

With t h e arrival of rapid global communication Facilities via satel- lites and t h e widespread availability of c h e a p worldwide transportation by a i r a n d sea, t h e phenomena of t h e transnational corporations (TNC) h a s e m e r g e d , carrying with i t a total a n d complete re-shaping of t h e s t r a c - t u r e and operation of major industrial activities. Prior to t h e advent of t h e TNC, large industries were considered nationally; now, for example, t h e automotive industry is s c a t t e r e d throughout t h e world with firms engaging in design in one place, production in another a n d marketing a n d sales everywhere. The situation is f u r t h e r compounded a n d con- fused by a m y r i a d of interlocking joint ventures, co-production agree- m e n t s , partial m e r g e r s and so forth. What all of this a m o u n t s t o is a discontinuous shift From one way of doing business to a n o t h e r and from

one industrial paradigm, emphasizing national centralization and domes- tic markets, to a global st,ructure transcending national boundaries and, to a great extent, local governmental control. The problem of industrial structural change is basically how to account for this transition, how t o understand its implications for t h e future evolution and development of a given industry and how to gain some understanding of the way s u c h changes can be directed, or managed, to avoid unnecessary chaos, disorder and economic upheavals during the transition periods from one s t r u c t u r e to another.

To study t h e problem of s t r u c t u r a l change, a suitable conceptual framework is needed within which t h e various firms comprising the industry can play out their roles in interaction with their environment.

Whatever framework is used., it m u s t account for t h e way in which firms execute their design, production and marketing functions, a s well a s incorporate mechanisms whereby t h e firms can expand, merge, or even cease to exist. The conceptual scheme must also allow for t h e m u t u a l interactions of t h e firms of t h e industry, both with each other, a n d with t h e i r external environment. The outside environment includes the sup- pliers of the raw materials a n d resources needed for the firm's activities, t h e consumers of t h e firm's product and the various environmental influences exerted by government regulators acting by the setting of t h e economic climate (through taxes, interest rates, exchange restrictions, etc.), and the business climate (through tariffs, quotas, import restric- tions and the like).

The foregoing requirements for a conceptual framework for t h e s t u d y ol industrial s t r u c t u r a l change have a strongly biological overtone, suggesting t h a t the view of a global industry as a living multicelled organism m a y serve a s a foundational metaphor for t h e framework we seek. The balance of t h i s paper is devoted t o t b e exploration of t h i s idea.

More specifically, t h e notion of an (M.R)-system (metabolism-repair sys- t e m ) is examined a s a candidate for t h e type of theoretical construct needed t o c a p t u r e t h e main features of industrial s t r u c t u r a l change.

Originally, (M,R)-systems were introduced into biology by Rosen [I-21 a s a m e a n s t o study cellular development of organisms by b r e a h n g away from t h e traditional bio-chemical types of analyses, and employing a purely relational analysis emphasizing t h e functional r a t h e r than struc- t u r a l organization of t h e system. This approach leads to t h e s t u d y of c l m s e s of a b s t r a c t biologies and a means for their comparison r a t h e r

t h a n to detailed m a t e r i a l analysis of a single organism. This is exactly t h e type of s c h e m e needed to investigate industrial s t r u c t u r a l change, although a s we go along it will become clear t h a t certain biological aspects of t h e (M,R)-systems will require modifications in t h e industrial context. Nonetheless, t h e (M.R)-framework t h a t follows, does, in our opinion, provide a suitable mathematical skeleton upon which t o build an operational t h e o r y of industrial s t r u c t u r a l change.

2. Production and Sales as an Input /Output Process

Our underlying basic hypothesis is t h a t an industry such as the world a u t o industry, is composed of a collection of interacting Lrms receiving inputs from an external environment, processing these inputs

i n t o t h e products of t h e firm, which are in t u r n discharged back into the e n v i r o n m e n t for which t h e firms receive additional r e s o u r c e s (money, usually) t o continue t h e i r activity. For a variety of reasons t h a t will become c l e a r l a t e r , i t is most natural a n d convenient t o regard t h e firms' o u t p u t s a s t h e money they receive from t h e i r products r a t h e r t h a n t h e products, themselves. In short, t h e real mission of a firm is t o make money, n o t products, a n d t h e products a r e t h o u g h t of as only a vehicle t o facilitate t h i s higher-level goal.

For t h e m o m e n t , l e t u s concentrate upon t h e description of a single f i r m

F

a s a n (M,R)-system. We will r e t u r n l a t e r t o t h e case of several firms ( a n industry). Let

R

denote the s e t of environmental inputs received by

F.

In g e n e r a l , the elements of

Q

a r e both physical inputs s u c h a s raw m a t e r i a l s , labor, machinery a n d so forth, a n d t h e external opmutzng inputs s u c h a s t h e economic, political a n d technological con- s t r a i n t s of t h e g e n e r a l environment. The firm a c c e p t s an i n p u t o E

R

and processes i t via some i n t e r n a l production a n d marketing procedures t o produce a m a r k e t a b l e product which is t h e n sold, t h e r e b y generating a n output y c

r,

m e a s u r e d i n monetary units. Here, due to t h e assumption t h a t t h e observed o u t p u t is money, we could take

r ⁼

^R, t h e real n u m b e r s . To m a i n t a i n uniformity of scale between inputs and outputs, we c a n i n t r o d u c e prices for all environmental inputs, thereby converting all i n p u t s i n t o m o n e t a r y equivalents. We shall omit this matnematicslly trivial, b u t possibly economically important s t e p in t h e r e m a i n d e r of t h e paper. Thus a b s t r a c t l y t h e behavior of

F

is r e p r e s e n t e d by a metabolic m a p

(Note: in t h e economics l i t e r a t u r e , it is common t o call f a p r o d u c t i o n f u n c t i o n . To preserve o u r bio!ogical m e t a p h o r , we shall d e p a r t from this convention in t h i s paper). Schematically, we can r e p r e s e n t t h e produc- tion and sales c o m p o n e n t of

F

a s

The foregoing picture is very familiar in t h e m a t h e m a t i c a l s y s t e m theory l i t e r a t u r e , where it is t e r m e d a n e z t e r n a l or i n p u t / o u t p u t description of t h e behavior of F [3-41. If we want t o focus a t t e n t i o n on t h e m a n n e r in which t h e i n p u t s from

R

a r e t r a n s l a t e d i n t o r e v e n u e by m e a n s of specific p r o d u c t s of t h e firm, t h e n we m u s t look a t t h e i n t e r n a l behavior of F. Abstractly, what t h i s m e a n s is t h a t we m u s t "factor" t h e metabolic m a p f t h r o u g h a s t a t e space X, using two m a p s g a n d h s u c h t h a t

In o t h e r words, we m u s t find a s p a c e X a n d m a p s g a n d h s u c h t h a t t h e diagram

commutes.

In our industrial s e t t i n g , t h e space X and t h e m a p s g a n d h have a very interesting i n t e r p r e t a t i o n : X r e p r e s e n t s t h e a c t u a l p r o d u c t s t h a t t h e firm produces (cars, W s , lamps, drugs or whatever), while t h e m a p g specifies how i n p u t s a r e t r a n s f o r m e d into products (a p r o d u c t i o n map).

The m a p h r e p r e s e n t s t h e m a n n e r in which t h e firm t r a n s l a t e s products

into revenue, (i.e., h is a marketing/saLes map).

For a variety of practical a s well as mathematical reasons, it is cus- tomary t o impose t h e additional requirements t h a t t h e m a p g be onto, while t h e .nap h is one-to-one. Such a factorization of f is called canon- i c d , and is essentially unique. These conditions have a very d i r e c t interpretation in t h e business setting: t.he production m a p being o n t o means t h a t any level of production can be achieved if F is supplied with suitable inputs, i.e. t h e r e a r e n o intrinsic limitations on t h e firm's abil- ity to produce products given adequate raw materials a n d o t h e r resources. The r e q u i r e m e n t t h a t t h e marketing/sales m a p be one-to- one just m e a n s t h a t different levels of production g e n e r a t e different amounts of revenue. Or, p u t a n o t h e r way, two distinct levels of produc- tion cannot g e n e r a t e t h e s a m e revenue for F. Procedures for generating such canonical factorizations of a metabolic m a p f a r e well-known in t h e system theory l i t e r a t u r e a n d will n o t be discussed f u r t h e r h e r e (cf. [3- 51).

3. Innovations. Repair and Replication

The s t a n d a r d s y s t e m - t h e o r e t i c framework presented above for describing t h e metabolic behavior of a firm as an input/output m a p f (or, by its equivalent factorization through the production a n d marketing/sales m a p s g a n d h ) would be perfectly adequate for t h e characterization of F if t h e f i r m were operating in a totally s t a b l e environment with n o competition. However, t h e intrusion of real-world considerations i n t o t h e firm's activities results in t h e need for t h e f i r m t o continually engage in changes of i t s product, introduction of new pro-

duction techniques a n d development of a i t e r n a t e marketing strategies if i t i s to r e m a i n a viable enterprise. In biological t e r m s , t h e firm m u s t r e p a i r damages and adapt or else e n t e r a s e n e s c e n t phase ultimately resulting in its extinction. In biological organisms, t h e adaptation a n d r e p a i r is c a r r i e d o u t by genetic p r o g r a m s which re-process the s y s t e m o u t p u t t o renew t h e metabolic behavior of t h e organism. In the c o n t e x t of a n a u t o firm, s u c h a restoration of t h e metabolic activity c a n only r e s u l t from t h e repair of different production and/or marketing pro- c e d u r e s , i.e. renewal of the maps g a n d h . This c a n c o m e about only t h r o u g h technological improvements, b e t t e r managerial procedures a n d / o r incorporation of new knowledge, i.e: through innovation. Our basic question h e r e is: how can the modeling framework introduced ear- l i e r be extended in a natural fashion t o a c c o u n t for t h e firm's need t o

"innovate or die?"

The key t o answering this question is to n o t e t h a t t h e only way t h e f i r m can renew i t s metabolic activity is t o utilize some p a r t of its reve- n u e s to r e g e n e r a t e e i t h e r i t s production processes or i t s marketing approach or both. Thus, the firm's "repair" m e c h a n i s m m u s t ultimately be a m a p t h a t t r a n s f o r m s t h e firm's o u t p u t (revenues) i n t o t h e desired metabolic s t r u c t u r e . If we let H ( R , r ) denote t h e s e t of all possible metabolic processes, a n d if

Q,

denotes t h e r e p a i r map, we have

Note t h a t we explicitly i n d c a t e the dependence of t h e repair m a p upon t h e metabolism j since the objective of the r e p a i r procedure is to repro- d u c e

!

which is an activity of t h e firm and, as such, i s affected by t h e

metabolic activity. We now have t h e f o l l o w i ~ g a b s t r a c t diagram for a firm F a s a m.etabolism-repair (M,R)-system:

f *I

fl +F + ~ ( n , r).

As already noted, f r e p r e s e n t s t h e firm's procedures for operating upon t h e e n v i r o n m e n t t o produce or "metabolize" r e v e n u e s , while

Qf

corresponds t o t h e firm's "genetic" capacity to r e p a i r d i s t u r b a n c e s in metabolism arising from e n v i r o n m e n t a l fluctuations.

To c o m p l e t e t h e m e t a p h o r of t h e f i r m a s a biological organism, we m u s t a d d r e s s t h e issue of how t o r e p a i r t h e r e p a i r e r s . The repair m e c h a n i s m s were i n t r o d u c e d to a c c o u n t for t h e fact t h a t during t h e course of t i m e , t h e f i r m ' s metabolic m a c h i n e r y will e r o d e a n d decay, thereby requiring s o m e s o r t of rejuvenation if t h e f i r m is t o avoid extinc- tion. Precisely t h e s a m e a r g u m e n t applies t o t h e r e p a i r m e c h a n i s m , b u t i t is of n o p a r t i c u l a r h e l p t o i n t r o d u c e r e p a i r e r s for t h e r e p a i r e r s and so forth, ending up in a useless infinite regress. Tbe way o u t of this loop is t o make t h e r e p a i r c o m p o n e n t s self-replicating. In t h i s way, new copies of t h e r e p a i r m e c h a n i s m a r e continually being produced, a n d i t is unnecessary t o a s s u m e t h e r e p a i r functions a r e i m m o r t a l o r t o fall i n t o an infinite r e g r e s s of r e p a i r e r s t o i n s u r e survivability of t h e firm. Let us see how t o i n t r o d u c e t h e idea of replication i n t o t h e foregoing frame- work

Since t h e replication operation involves reproducing t h e g e n e t i c component

af

from t h e metabolic activity of t h e firm, i t follows t h a t t h e replication m a p , call i t

p7,

m u s t be such t h a t

p,

: H

( n . r) + H ( r . ~ ( n . r)).

if it exists a t all. The question is: how can such a m a p

By

be constracted from t h e basic metabolic components

R , r ,

a n d

H ( R .F)

of the firm? To see how this is done, i t is easiest t o consider a somewhat more general situation.

Let

X

and Y be arbitrary sets. Then for each z ^t

X .

we can define a m a p

by t h e rule

for all f E

H ( x

,

Y ) .

Thus, we have an embedding of

X

i n t o t h e s e t

H ( H ( x .

Y), Y). Now, msume t h a t t h e m a p 2 has a left inverse z^-I , s o t h a t

Then, we clearly have

for all f E

H ( x ,

Y).

Returning now t o our replication situation, we s e t

and apply t h e foregoing general argument to obtain for each y E

r,

^{a map}

p7 = 9-'

with t h e property t h a t

8,

^:

H ( O

,

0 -+ H ( T . H ( n . 0)

for all

7

possessing a left inverse. In short, the metabolic activity of the firm can be used to reproduce its repair component if the technical con- dition on t h e invertibility of t h e map

7

is satisfied. The economic interpretation of this condition is t h a t

7

is invertible if chfferent innova- tions and R&D activities (i.e. different genetics mechanisms

@I,

#

e) give rise to different production and marketing functions (i.e. different metabolic processes f _{# f 2).} In t h e industrial context we a r e examin- ing, this seems to be a reasonably defensible assumption t h a t will be accepted for t h e remainder of the paper.

Before entering into a more thorough discussion of the implications of the (M,R)-sysem as a paradigm for industrial structuring and opera- tion, it is of interest to consider the actual meaning of the replication process described by t h e map

pT.

We have seen t h a t t h e repair mechan- ism iPf basically provides the prescription by which revenues a r e used to support and renew the production-marketing process f . By t h e s a m e token, t h e replication process

p7

gives t h e instructions by which the genetic process Qlf is duplicated. Thus, since corresponds to innovation/R&D, we can only conclude t h a t

p7

corresponds t o the diffu- sion of innovation/R&D. In short,

p7

is a prescription for growth of the firm by development of new divisions. Alternately, it could represent the start-up procedure of new firms t h a t spin-off from the parent corpora- tion. In either case, the innovation and "know-how" of the parent f i r m is transferred t o a new organization and is then used as part of the meta- bolic operation f to produce revenue from environmental inputs in the usual way.

4. Consequences of the (M.R)-Fkamework for a Single Firm

The minimal s t r u c t u r e i ~ t r o d u c e d thus far to define an (M,R)- system is already sufficient to shed light on a variety of interesticg ques- tions surrounding t h e way a firm can respond to changes in its operating environment, t h e possibility for innovation to occur through environ- mental effect, the circumstances under which environmental changes can be reversed, feedback, and so on. In this section we sketch t h e way in which these issues appear within the (M,R)-framework, and consider the conclusions t h a t can be drawn about firm behavior from this struc- ture.

A. S t a b l e M e t a b o l i c @ e r a t i o m in Changing E n v i r o n m e n t s - imagine the situation in which t h e firm's "usual" input o of raw materiais, labor, etc. is disturbed to a new input Z. The condition for stable operation of the firm is for t h e environment w to be such t h a t

i.e. the metabolic s t r u c t u r e f is stable in the environment o in the sense t h a t the repair mechanism

af

always regenerates f when the environmental input is w . We would say that all o E

R

satisfying ( * ) form a stable environment for t h e firm.

Now suppose t h a t t h e new environment E # o. Then ( * ) will hold only if either

The first case i s trivial in t h e sense that the observed revenues of the firm a r e invariant t o t h e change of environmental inputs. If f (o) # f

(6)

then t h e firm's revenues are not stable with respect to t h e change of environment and we m u s t consider the repair mechanism to see whether or not t h e environmental alterations can be compensated for in t h e sense t h a t

with j ( 7 j )

=

^f (o). i-e. will the genetic mechanism produce a new meta- bolism

7

which will duplicate the revenues of f , but with the i n p u t

o

r a t h e r t h a n w . In this case, t h e entire metabolic activity of t h e firm would be permanently altered. if we had

(7 (3, =

On t h e o t h e r hand, if we had F(G)

=

f (o) or, more generally,

a f ^(f(Z)) = f

^*

t h e n the firm's metabolism would only undergo periodic changes in time.

Finally, we could have t h e situation in which

a f V(Z)) ⁼ f

^{# f ,}

f

and, iterating this process, we see t h a t an environmental change may cause t h e firm to wander about in t h e s e t H(R,

r),

changing i t s production-marketing procedures through a sequence of metabolic processes f ( l ) , f ( 2 ) , f (3),

. . . .

This "hunting process n-ill terrninate if e i t h e r

(i) t h e r e exists an

N

such t h a t

a f ^{(f 'h?} ( G ) ) =

^{f (N)}

o r

(ii) t h e r e exists an N such t h a t

@, ( f ) f , k = 1 . 2 .... N-1 .

In case (i) t h e firm becomes stable in t h e new environment E, while in c a s e (ii) t h e firm undergoes periodic changes in its metabolic s t r u c t u r e . If no such N exists, the firm is unstable and aperiodic. (Note: - This last possibility can occur only if t h e s e t H(R ,

r)

of possible production- marketing procedures is infinite).

B. " L & m a ~ c k i m " Changes ⁱⁿ the Repair Process - t h e above &scus- sion of metabolic changes was undertaken subject to t h e tacit assump- tion t h a t t h e r e p a i r m a p

@I

r e m a i n s unchanged. It is of i n t e r e s t t o inquire a s t o whether or not an environmental change w ³ E can gen- e r a t e a "Lamarckian" type of genetic change in

Q ,

through the replica- tion process described in the lasi, section. If s u c h a change were indeed possible, then i t would imply t h a t t h e a c t u a l innovation/R&D process, which r e g e n e r a t e s the metabolic activity f , could be affected by environmental changes alone.

To examine this question, suppose we have the environmental change o

-> o.

Then the replication m a p

p7

associated with t h e input o a n d t h e o u t p u t 7

=

f ( o j is changed t o

4,

^where

^{f =}

^{f ( G ) .} Recalling t h a t

a f t e r applying /3 ,

8-,

respectively to t h e l a s t two relations, we find t n a t 7 7

By(@, ( f

(4)) = $

(@, (f

(a)) ⁼

^{Qf 1}

showing t h a t t h e new replication map @- replicates t h e existing repair 7

component Qf exactly. Thus, an environmental change alone can have no effect upon t h e r e p a i r m a p @,.

Now we a s k w h e t h e r it is possible for a change in t h e metabolic production-marketing procedures t o result in a c h a n g e of t h e firm's

"genotype." Suppose we replace t h e metabolic activity f by some o t h e r production-marketing process b t H (fl ,

r).

By definition,

b(o)

^(@,)

=

@b (h(0)) .

A s s u m i n g c ( w ) is invertible, we apply c-*(w) t o both sides of t h e above relation t o obtain

Thus, t h e i n d u c e d replication m a p reproduces t h e original repair com- ponent of t h e firm u n d e r all conditions. In s h o r t , no Lamarckian changes in t h e metabolic component, e i t h e r in t h e e n v i r o n m e n t fl o r i n t h e metabolic s e t

H(R

,

r),

c a n result in changes in t h e firm's repair mechanism. S u c h "genetic" changes can only c o m e a b o u t through a direct i n t e r v e n t i o n in t h e genetic code itself ( m u t a t i o n ) a n d not via indirect metabolic alterations.

C. F b e d b a c k as an h v i r o n m e n t a l R e g u l a t o r

-

t h e environmental changes discussed t h u s far have been a s s u m e d t o be g e n e r a t e d by actions e x t e r n a l t o t h e firm; however, it m a y often be t h e c a s e t h a t t h e firm's o u t p u t of r e v e n u e is employed a s one of t h e e n v i r o n m e n t a l inputs, i.e. w is a function of 7, ^t;

=

o(y). In this event, t h e firm actually c r e a t e s part of i t s own e n v i r o n m e n t and a s a consequence, c a n pzrtially

regulate its own s t r u c t u r a l a l t e r a t i o n s . An i m p o r t a n t a s p e c t of this gen- e r a l process is t o u n d e r s t a n d a s to what degree adverse e n v i r o n m e n t a l d i s t u r b a n c e s c a n be "neutralized" by suitably c h o s e n feedback policies.

This question is a special case of t h e m o r e g e n e r a l problem of "reachabil- ity", in which we ask a b o u t t h e possibility of a t t a i n i n g any pre-defined metabolic s t r u c t u r e by m e a n s of a sequence of e n v i r o n m e n t a l changes.

This is a topic we shall r e t u r n to l a t e r in connection with discussing t h e dynamical a s p e c t s of t h e firm's morphology.

5. Global Industry as a Network of (M.R)-Systems

So far we have only considered a single firm F a s a n (M,R)-system. If we c o n n e c t several f i r m s t o g e t h e r , with t h e o u t p u t s of s o m e firms serv- ing a s inputs t o o t h e r s a n d t h e repair m e c h a n i s m s of e a c h f i r m requiring t h e o u t p u t from a t l e a s t one f i r m , t h e n we have t h e s t r u c t u r a l basis for c h a r a c t e r i z a t i o n of an e n t i r e industry. Such a network of interdepen- d e n t firms gives r i s e t o a n u m b e r of significant questions involving t h e b i r t h , growth a n d d e a t h of a n industry and of t h e individual firms comprising t h e i n d u s t r y . In this section, we consider how t h e s e issues a r i s e naturally within t h e c o n t e x t of a n (M,R)-network a n d t h e way in which o u r e a r l i e r (M,R)-formalism for a single firm c a n be e x t e n d e d t o form a basis for modeling a n e n t i r e industry.

In order t o fix ideas, consider t h e specific (M,R)-network depicted in Figure 1. The s q u a r e blocks lzbeled F 1 , F2. ^.

. .

, F6 r e p r e s e n t t h e rneta- bolic processes of t h e individual firms, while t h e ovals, denoted

R1,

R2,

. . .

^, R6, r e p r e s e n t t h e respective firms' r e p a i r m e c h a n i s m s . The requirements t h a t we impose for any s u c h network a r e modest:

External Inputs

External Outputs

FIGURE 1. A Typical (M,R)-Ne twork

i) each firm m u s t receive a t l e a s t one input, either from t h e exter- nal world or from the output of a n o t h e r firm;

ii) each firm produces a t l e a s t one output;

iii) e a c h repair mechanism receives the output of a t least one firm in the network.

We have s t a t e d t h e r e q u i r e m e n t s for an (M,R)-network in q u i t e g e n e r a l t e r m s . In a typical industry, s u c h as the automotive i n d u s t r y , i t is likely t h a t e a c h firm will receive its i n p u t from t h e outside world a n d discharge a t l e a s t p a r t of i t s o u t p u t of revenues back t o t h e e x t e r n a l world in t h e f o r m of p a y m e n t for goods a n d services a n d r e t u r n s t o shareholders.

Also, t h e repair m e c h a n i s m s will, for the m o s t p a r t , receive only t h e out- p u t f r o m t h e i r corresponding f i r m since the c a s e of one firm devoting i t s r e s o u r c e s toward supporting a n o t h e r ( a s with t h e firm F4 of Figure 2 ) s e e m s r a t h e r improbable u n d e r most c i r c u m s t a n c e s , although c e r t a i n l y n o t impossible.

The f i r s t g e n e r a l issue t o consider for an (M,R)-network is t h e depen- d e n c y s t r u c t u r e . We a r e c o n c e r n e d with t h e question of how t h e removal of a given f i r m from t h e network affects t h e e x i s t e n c e a n d operation of o t h e r firms in t h e industry. For instance, referring t o Figure 1 we s e e t h a t t h e failure of F5 r e s u l t s in t h e failure of firm F6, a s well, s i n c e F6 receives i t s only i n p u t from F5. F u r t h e r m o r e , t h e failure of F5 m a y influence t h e operation of F3, a s F3 receives p a r t of i t s i n p u t f r o m F5.

Thus, in this c a s e we would consider firms F3 a n d F6 t o c o m p r i s e t h e d e p e n d e n c y s e t of f i r m F5. Any firm whose failure affects e i t h e r t h e e x i s t e n c e o r operation of all f i r m s in the industry will be c a l l e d a c e n t r a l firm, i.e. t h e dependency s e t of a c e n t r a l f i r m is t h e e n t i r e industry.

Since we h o w t h a t in .the absence of t h e r e p a i r m e c h a n i s m a n y firm will go o u t of e x i s t e n c e a f t e r s o m e finite lifetime, i t is c l e a r t h a t o t h e r f i r m s in t h e d e p e n d e n c y s e t of a given firm will also go o u t of e x i s t e n c e when t h a t firm does. However, with t h e repair m e c h a n i s m in operation,

it is quite possible t h a t a given firm could "come back t o life" even after its initial demise. For example, firm F6 in Figure 1 may c e a s e metabolic operation a n d be removed from t h e network; however, t h e repair mechanism

R6

receives i t s necessary inputs from firms F4 a n d F5 indi- cating t h a t whatever "shock" caused t h e extinction of F6, t h e firm will be re-inserted into t h e network after some c h a r a c t e r i s t i c delay time depending upon t h e r e p a i r mechanism Re. In o t h e r words, copies of F6 will continue t o be m a n u f a c t u r e d even after t h e removal of F6 from t h e network Firms like F6 will be called r e - e s t a b l . i s h a b l e , while all other firms a r e t e r m e d n o n - r e e s t a b l i s h a b l e (e.g. F 1 , F5,

. .

. ). There is a n important relationship between t h e notion of re-establishability and the concept of a c e n t r a l component expressed by t h e following r e s u l t .

R e o r e m 1 . f i e r y (M,R)-network m u s t c o n t a i n a t l e a s t o n e n o n - r e e s t a b l i s h a b l e f w n .

CoroLlary.

f l

an (M,R)-network c o n t a i n s o n l y o n e non- r e e s t a b l i s h a b l e f i r m , t h e n t h a t f i r m is a c e n t r a l c o m p o n e n t . The proofs of these results c a n be found in the papers cited in t h e references.

The significance of t h i s result is twofold:

i) every industry m u s t contain at least one Firm whose metabolic failure c a n n o t be r e p a i r e d This conclusion follows only from t h e con- nective s t r u c t u r e of t h e (M,R)-network and is completely independent of t h e specific industry, the procedures of t h e firms, t h e i r products or marketing strategies. I t is solely a consequence of t h e meaning of the metabolism-repair functions and the replication process.

ii) in o r d e r to be "resilient" to unforeseen disturbances, one would desire an i n d u s t r y to consist of a large n u m b e r of re-establishabie firms.

On the o t h e r h a n d , t h e above resu!ts show t h a t if only a small n u m b e r of f i r m s a r e non-reestablishable, t h e n t h e r e is a high likelihood t h a t one of t h e m will be a c e n t r a l c o m p o n e n t whose failure will destroy the e n t i r e industry. Thus, a n i n d u s t r y with a large n u m b e r of re-establishable f i r m s will be able to survive many types of shocks a n d surprises, b u t t h e r e will be c e r t a i n t y p e s of disturbances t h a t will effectively cripple t h e whole i n d u s t r y . Consequently, it m a y be b e t t e r t o have a n industry with a relatively large n u m b e r of non-reestablishable firms if i t is desir- able t o p r o t e c t t h e i n d u s t r y from complete breakdown.

6. Time- Lags and Dynamics

Up to t h i s point, i t h a s been assumed t h a t t h e metabolic a n d r e p a i r functions of t h e firm t a k e place instantaneously, i.e. i n p u t s a r e transformed i n t o r e v e n u e s immediately a n d t h e r e is n o delay in e i t h e r repairing t h e metabolic p r o c e s s itself o r in t h e replication of t h e repair mechanism. Needless t o s a y , t h e s e assumptions a r e p u r e fiction; pro- duction of r e v e n u e a n d repair/replication t a k e s t i m e a n d t h e delays involved often spell t h e difference between s u c c e s s o r failure for 3. firm.

While t h e r e is no s p a c e h e r e t o e n t e r i n t o a detailed discussion of t h e m a t t e r , l e t u s simplify t h e situation by assuming only two types of delays. The first we t e r m t h e production d e l a y , correspondmg to t h e t i m e required t o t r a n s f o r m a given input of m a t e r i a l s , manpower a n d knowledge i n t o a n observable a m o u n t of revenue or t h e t i m e r e q u i r e d for a r e p a i r function t o r e s t o r e a metabolic operation. The second type of

delay we shall call t h e t r a n s p o ~ t d e l a y . It corresponds t o t b e time needed t o t r a n s p o r t t h e o u t p u t of a firm to where i t can be utilized as t h e i n p u t t o e i t h e r a n o t h e r firm or a repair mechanism (or t o the exter- nal world).

R g u r e 2. A Single Firm

As an illustration of how time-lags can influence t h e behavior of an (M,R)-model of a firm, consider t h e case of the single firm F depicted in Figure 2. The firm is clearly non-reestablishable in the sense discussed earlier. If t h e combined delay time of t h e production delay of

R

and the t r a n s p o r t delay from F to

R

is T units, ar,d if a t time t

=

0 F produces a n o u t p u t a n d is then removed from t h e network, T u n i t s l a t e r R will pro- d u c e a copy of

F

and

F

will be built back into t h e network even though F is graph-theoretically non-reestablishable. However, if

F

is not just removed from t h e industrial network, b u t is suppressed for an zmount of t i m e

t

s T, t h e n irreversible damage will have occurred a n d F will be removed from the network forever. The interplay between the various

time-lags involved when several firms are coupled together i n t o an industry is a delicate m a t t e r and will be taken u p in a l a t e r paper.

Closely r e l a t e d t o t h e time-lag problem is t h e m a t t e r of s y s t e m dynamics. There a r e several issues surrounding this topic, n o t all of t h e m mutually consistent. For simplicity, l e t us consider h e r e only t h e c a s e of a single firm F modeled a s an (M,R)-system. Abstractly, t h e diagram for

F

is

a n d ask in what m a n n e r

F

can be regarded a s a dynarnical system. If it were not f o r t h e r e p a i r a n d replication m a p s

af

a n d #?, t h i s would be a straightforward question addressable via normal system-theoretic reali- zation t h e o r y procedures, i.e. we would have t h e problem of constructing a canonical i n t e r n a l model of t h e firm

whose i n p u t / o u t p u t behavior duplicates t h a t of t h e given metabolic m a p f

.

Techniques for handling this question a r e readily available in t h e m a t h e m a t i c a l s y s t e m t h e o r y l i t e r a t u r e [3-51.

Let u s ignore for t h e m o m e n t t h e factorization of t h e firm's m e t a - bolism f t h r o u g h t h e production-marketing m a p s g and h , a n d consider t h e (M,R)-system

where j E H ( n ,

r),

^{(pf E} H ( r ,H(R ^,

i ' ) ) .

We wish t o show how t h i s a b s t r a c t

model of a firm c a n be considered as a sequential machine, i.e. as a discrete-time dynamical input/output system.

Let us recall t h a t a sequential machine M is a composite M

=

( A ,B ,S ,6 ,A), where A , B

.

and S a r e s e t s (possibly infinite), while 6 : A x S j S, A : S x A +B a r e maps. We i n t e r p r e t A a s the input alphabet of

M ,

_B as t h e output set, S as the set of s t a t e s , with 6 and A the s t a t e - transition and o u t p u t maps of t h e machine, respectively. At each discrete instant of t i m e

t =

0 ,1 , 2 ,

.

.

.

,

Y

receives an input symbol from A, e m i t s an o u t p u t i n B and t h e state is changed according to t h e rule 6, a n d t h e process continues from the time

t +

1. F u r t h e r details on t h e properties of sequential machines can be found, for example, in [4,

61.

In o r d e r t o c h a r a c t e r i z e the firm F as a sequential machine, we make t h e identifications

Thus, in general any firm can be regarded a s a sequential machine in which t h e s e t of "states" of the machine correspond to t h e s e t of possible

"phenotypes" of t h e f i r m , while t h e input a n d o u t p u t s e t s of t h e machine a r e t h e inputs and revenues of the firm, respectively.

Putting t h e above ideas together, we arrive a t t h e following s c h e m e for characterizing t h e dynamics of t h e firm:

i) regarding t h e f i r m - a s a sequential machine formed from t h e ele- m e n t s

F = (R

,

r , ^{H ( R}

^,

r).

^{f ),} we compute t h e metabolic process f

a t t h e time

t =

0 ;

ii) using f ,

R

,

I?,

we employ realization theory to form a canonical model for t h e s t a t e s p a c e X and t h e production/marketing maps g and h ;

iii) Let

t -> t +

1 a n d use t h e sequential machine t o calculate the new metabolism f . If i t is t h e s a m e as a t t h e previous time-step, then continue t o use t h e e a r l i e r X , g a n d h ; if f changes, calculate a new canonical produrtion/marketing model and continue t h e process with t h e new model until t h e next time period.

It is of i n t e r e s t t o note t h a t in t h e above s c h e m e , a change of meta- bolism implies t h a t t h e production process, t h e marketing procedure and/or t h e a c t u a l product has been c h a n g e d This c a n c o m e about only if t h e repair m a p fails to reproduce f . 'we have already seen t h a t this m a y c o m e a b o u t only by means of environmental c h a n g e s , in general, unless t h e replication m a p fails t o exist. But t h i s l a s t situation depends entirely upon t h e size of the s e t H(R , H ( R ,

I')),

t h e space of all possible repair maps. If i t is e i t h e r too large or too small, t h e n n o replication is possible. I t would take us too far afield t o e n t e r i n t o t h e details of this a r g u m e n t h e r e , b u t t h e implications a r e t h a t it is only in a highly res- tricted class of c a t e g o r i e s t h a t replicating (M,R)-systems exist, a n d it is within t h i s class t h a t we m u s t s e a r c h for viable models of industrial growth and decline.

7. Attainable Production-Marketing Processes

The a r g u m e n t s given earlier show t h a t t h e metabolic component of a firm c a n be altered by changes in i t s environmental inputs, while s u c h changes leave t h e firm's r e p a i r m e c h a n i s m unaffected. By t u r n i n g t h e s e a r g u m e n t s around, we c a n investigate t h e degree to which environmen- tal changes c a n be used in o r d e r t o bring t h e firm's production a n d marketing processes to some p r e - - s i g n e d s t a t e . An important special c a s e of this "reachability" question is t o a s k if a metabolic s t r u c t u r e r e a c h e d by some sequence of environmental changes can be r e v e r s e d by a n o t h e r appropriate sequence of changes in t h e environment. Questions of t n e above type strike t o t h e h e a r t of many important industrial issues having t o do with t h e way in which changes in materials, m e n , a n d m a c h i n e s c a n be employed t o affect t h e overall productive capabilities ~f t h e firm.

In t e r m s of machines, t h e reachability problem can be s t a t e d as:

given a m a c h i n e in a specified initial s t a t e , does t h e r e exist a s e q u e n c e of i n p u t s t h a t will bring t h e m a c h i n e t o some preassigned s t a t e ( p e r h a p s also a t a preassigned time)? In general, t h e answer t o this question is no. Machines having t h i s "complete reachability" property a r e called strongly c o n n e c t e d , a n d we c a n a s k whether or not machines t h a t correspond t o (M,R)-systems a r e strongly connected.

Generally speaking, m a c h i n e s corresponding t o (M,R)-systems m a y fail t o be strongly connected; hence, t h e r e m a y exist abstract "firms"

t h a t m a y be unreachable from a n y initial configuration by m y s e q u e n c e of environmental alterations. In t h e usual theory of sequential m a c h i n e s , this difficulty c a n be formally by-passed by enlarging t h e s e t

of inputs a n d by appropriately extending t h e maps

6

a n d A. For

(M,R)-

s y s t e m s this is a m u c h m o r e s u b t l e bllsiness for t h e following reasons:

( a ) t h e o u t p u t s e t

R

a n d t h e s t a t e s e t S

=

H ( R ,

r)

a r e r e l a t e d in t h e (M,R)-systems a n d we c a n n o t e n l a r g e

Q

without also enlarging S;

(b) by extending t h e maps Qf a n d f in t h e (M,R)-systems, we move t h e mappings from t h e s e t s H ( R ,

I')

a n d H ( I ' , H ( f l , r ) ) t o new s e t s H(R' ,

r), H ( I ' ,

H ( R ' ,

) , ) ' i

respectively. But t h i s l a s t s e t m u s t possess c e r t a i n properties in o r d e r for replication t o be possible a n d this pro- p e r t y is by no m e a n s implied by t h e replicability of t h e original s y s t e m .

€3. Prospects and Conclusions

The development of (M,R)-systems as a theoretical framework for t h e s t u d y of industrial growth a n d re-structuring h a s only been t e n t a - tively sketched in t h e preceding pages t o t h e degree n e c e s s a r y t o d e m o n s t r a t e feasibility of t h e idea. To transform t h e basic idea i n t o a working tool to study, for i n s t a n c e , t h e evolution and development of t h e world automotive industry, r e q u i r e s a substantial r e s e a r c h effort on both t h e theoretical, a s well a s applied fronts. I t will be n e c e s s a r y to give con- c r e t e meaning and s t r u c t u r e t o t h e various a b s t r a c t components com- posing t h e (M,R)-network ( t h e e l e m e n t s

R

, I',H(R, I').@f. e t ~ . ) , a s well a s work out t h e various connectivity s t r u c t u r e s t h a t link t h e individuai f i r m s comprising a n i n d u s t r y . S u c h activities form t h e basis of t h e applied component of a n y implementation of the (M,R)-framework for a specific industry. Some c o m p l e m e n t a r y evolutionary i d e a s a r e given for t h e a u t o industry by ~ u s i n a r o ' in [ 9 ] and their connection with

(M,R)-

s y s t e m s m e r i t m u c h f u r t h e r study. See also t h e g e n e r a l evolutionary

ideas in

[lo,

^111.

B u t t h e r e a r e also a n u m b e r of purely t h e o r e t i c a l aspects of t h e (M,R)-formalism t h a t need f u r t h e r study if t h e overall s t r u c t u r e is to bear the weight of providing the foundation for s u c h an investigation of individual dynamics. We have already touched upon some of these issues in passing, b u t i t is worthwhile t o re-examine t h e m again a s t h e basis for a f u t u r e r e s e a r c h agenda.

i) L a m a r c k i c n changes - we have seen t h a t changes in the firm's r e p a i r m e c h a n i s m c a n n o t c o m e about by environmental alterations alone, a s long a s c e r t a i n invertibility assumptions on t h e replication pro- cedure hold. This assumption, a n d its r e s u l t a n t conclusion, a r e quite acceptable in t h e biological context but r e s t upon m u c h shakier ground in o u r i n d u s t r i a l setting. It certainly seems plausible t h a t a t least cer- tain types of e n v i r o n m e n t a l changes could give rise to a change of t h e firm's "genotype." At t h i s stage i t is g n c l e a r exactly how t o modify t h e m a t h e m a t i c a l s e t t i n g given above t o accommodate s u c h "Lamarckian"

changes.

iz) Networks and Time-Lags - t h e m a n n e r i n which time-lags, in both firm operations a n d in transport from one firm t o a n o t h e r , affect t h e overall behavior of a n industry is critical for determination of t h e long-term growth or decline of given f i r r ~ s within t h e industry. We have already s e e n s i m p l e examples in which time-lags can r e s u l t in either t h e p e r m a n e n t e x t i n c t i o n of a firm or, conversely, in i t s "resurrection" after being t h e o r e t i c a l l y " d e a d " The interdependencies of lags of different types a n d l e n g t h s i s a topic t h a t cannot be ignored if t h e (M,R)- framework is to be used t o gain insight i n t o t h e behavior of real

industries.

Dynamics - t h e procedure outlined in the t e x t for regarding an (M,R)-system as a sequential machine is one w a y t o i n t ~ o d u c e d y ~ a r n i c a l considerations i n t o t h e overall formalism. There may be m a n y o t h e r non-equivalent approaches, each leading t o a different view of the dynarnical behavior of a firm. Even accepting t h e approach given h e r e for a single firm, t h e r e still arises t h e question of what will be t h e dynarnical behavior of a c o l l e c t k n of such firms, i.e. an industry. Obvi- ously, t h e answer t o such a question depends upon t h e connective and dependency s t r u c t u r e of t h e network, which in t u r n t a k e s u s back t o some of t h e t i n e - l a g considerations discussed earlier.

iv) a d a p t a t i o n and s e l e c t i o n - if a n (M.R)-network is t o provide a m a t h e m a t i c a l m e t a p h o r for the e v o l u t i o n of an industry, t h e n i t m u s t possess s o m e m e a n s t o accommodate t h e concepts of genetic variability and adaptive selection. We have already spoken of t h e need t o be able t o incorporate genetic changes in t h e repair m a p

19f

into t h e m a t h e m a t i c a l m a c h i n e r y of (M,R)-systems. A n a t u r a l candidate for the selection mechanism is t o impose some s o r t of optimality criterion upon t h e possi- ble a b s t r a c t firms t h a t m a y result from genetic "mutations." Production efficiency, profitability, survivability a r e logical possibilities, but so also a r e less econornic-oriented c r i t e r i a like degree of re-establishability and level of centrality, c r i t e r i a suggested more by t h e functional role of a firm in a network t h a n by i t s economic performance as a n isolated unit.

v ) c a f e g o r i e s and the compa*on of industrial s t m r t u r e s

-

^{a basic}

question in t h e s t u d y of industrial evolution and change is t o ask if t h e processes a t work modifying one industry c a n be used i n any way to infer

information about t h e forces influencing a n o t h e r ; if we u n d e r s t a n d t h e dynamics t h a t shape, say, t h e chemical industry, c a n t h a t h o w l e d g e be used t o a n d e r s t a n d , for i n s t a n c e , t h e evolution of t h e a u t o m o t i v e indus- t r y ? In o r d e r t o answer s u c h a question, we m u s t have a s y s t e m a t i c pro- cedure for compari~g t h e industries a n d a m e a n s for deciding w h e t h e r t h e y a r e a b s t r a c t l y equivalent. The (M.R)-system framework provides a m e a n s for making s u c h comparisons t h r o u g h t h e m a t h e m a t i c a l a p p a r a t u s t e r m e d "category theory" [7. 81. Briefly, a n y collection of s e t s A , B , C , . . , , s u c h t h a t t o each ordered p a i r ( A . B ) we have a n o t h e r s e t H(A , B ) , t h e mappings from A to B, i s called a category provided c e r t a i n primitive assumptions a r e satisfied lor t h e s e t of mappings H(A ,B). We will defer any technical discussion of t h e s e m a t t e r s t o a n o t h e r paper, b u t i t is i m p o r t a n t h e r e t o observe t h a t every (M,R)-system is a category.

i n which t h e objects

R .r

a r e t h e s e t s a n d t h e metabolic m a p s H(R , r ) a r e t h e mappings of t h e category. Thus, every firm c a n be regarded a s a category, and by e x t e n h n g t h e sets a n d t h e mappings, s o c a n every industry. If we change t h e s e t s

R , r

a n d / o r t h e mappings H ( R , r ) obtaining a different firm, t h e n we have a new category, a n d t h e m a c h i n e r y of category t h e o r y allows u s t o c o m p a r e t h e s t r u c t u r e s of t h e two c a t e g o r i e s by m e a n s of mappings called f u n c t o r s . Roughly spealang, a f u n c t o r is a s o r t of dictionary allowing u s t o t r a n s l a t e t h e s t r u c t u r e of one c a t e g o r y i n t o t h a t of a n o t h e r , a n d conversely. This is exactly t h e t y p e of tool t h a t is n e e d e d t o c o m p a r e o n e f i r m o r one i n d u s t r y with a n o t h e r . The systematic exploitation of t h i s idea in t h e c o n t e x t of indus- t r i a l s t r u c t u r a l change within t h e above (M,R)-framework offers t h e promise of unlocking m a n y key f e a t u r e s responsible for t h e dynamics

underlying t h e evolution a n d development of m o d e r n global industries.

A c k n o w l e d g e m e n t . I t is a pleasure to acknowledge t h e benefit of n u m e r o u s conversations with R. Rosen on t h e subject of (M,R)-systems.

Most of t h e ideas p r e s e n t e d h e r e have t h e i r origins in e a r l i e r papers of h i s written f r o m a biological vantage point. I have only added a few embellishments h e r e a n d t h e r e a n d t r a n s l a t e d t h e r e s u l t s i n t o a n indus- trial setting.

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2. Rosen,

R.,

"The Representatior? of Biological Systems from t h e Stand- point of the Theory Categories, " Bull, Math-Biophysics, 20(1958), 317-341.

3. Casti, J . , Dzfnamical a s t e r n s and t h e b Applications: Linear ;rPLeory, Academic Press, New York, 1977.

4. Kalman, R., P. Falb and

M.

Arbib, Topics in Mathernafical * s t e m llzeory, McGraw Hill, New York, 1969.

5. Casti, J., Nonlinear S y s t e m l'heory, Academic Press, New York, 1985.

6. Eilenberg.

S.

Automata, Languages and Machinss, v o l . A, Academic Press, New York. 1974.

7. Eilenberg, S., and S. MacLane, "General Theory of Natural Equivalence," T r a n . A m e r . ,%~th.Soc., 58(1945), 231-234.

8. MacLane, S. C a t e g o r i e s f o r t h ~ Working M a t h e m a t i c a n , Springer, New York, 1972.

9. Businaro, U. "Comparing Natural Evolution a n d Technological Innova- tion," F L t u r e s , April 1982

10. Nelson, R., a n d S. Winter, An E ' u o l u t i o n q i%eory of E c o n o m i c Change, Harvard U. Press, Cambridge, 1982.

11. Businaro, U. "Applying t h e Biological Evolution Metaphor t o Techno- logical Innovation," F'utures, December 1983.