Dynamics of structural change

Regional !kknce and Urban J3onomics 5 (1975) 41-57. :Q North-Holland

DYNAMICS

OF STRUCTURAL

CHANCE”

Gustav SCJHACHTER Nmheastrrn

Vttiwrsity, Boston, Mms., C’.S. A.

1. Introduction

Structural change in the sine qua non of true growth. jet models that predict structural change have not been developed. To have the capability ofincorporating structural change, one must begin with an intrinsically dynamical description. Many economists seem to recognize the need for a dynamic description but appear to be satisfied with incorporating or addending dynamical conditions to models that were originally developed for essentially static systems. In particular, traditional input-output analysis carries with it the burdens of (1) the assumption of linearity, ’ (2) the requirements of quasi-stationary tine series and of data, good in quantity and quality,2 (3) its intrinsically descriptive, nonprojective characteristic, i.e., it has no provision for predicting structural change,3 (4) its assumption of the U.S. econcmy as a normative goal of development,* and (5) its positivist orientation, that is, it provides no insight into mechanisms of the socio-economy - it is a black-box, operational method.5 *This general theory was presented at the n1eetin.gin connection with special probkms of under-developed arcas [Fox-Schachter (1970)]. ‘This holds for both the commonly used static analysis and for dynamic imalysis. ‘. . . instead of ordinary linear equations (dynamic analysis) leads to systems of linear differential equations’ ILcontief(l966, p. 8)]. 1‘. , , changes in the economy over periods of time are measured by co*7lparing before and after picthcs. Each is a static model, a cross-section of time [Leontief (19G6, p. 27)j. 3’Within each sector there is a relatively invariable connection between inputs it draws from other sectors and its contribution to the total output of the economy. This holds for an underdeveloped economy . . . us well as for highly developed economics’ [Lzontief (1966, p. a)]. JLeontief in his discussion 0:‘ the structure of development claims that, ‘. I . the PrOCeSS of development consists essentially in the installation and building of an approximation Of tj-le systcnlembodied in the advanced economics of t!w U.S. and Western Errrope and, more recently, of the U.S.S.R. . . .’ [Leontief (1963, p. 159)l. SFriedman states in reference to the equilibrium space economy that a model ‘. . . is udd for analysis, but it Ce,ses to be pertinent when it is converted into a normative rule for Plannit%. TO be nle;lninRful, every Social norm must be brought into concrete relation with the historical conditions of collective life. That static equilibrium model, valid only within a Parameter of carefully stated and artificial assumptions. is wholly inappropriate by this standard’ tt+iedman (I 963, p. 53)].

42

ML. Fox and G. Schachrer, Dynamics of strmwral change

Nonlinear relationships are a necessary characteristic of structurally changing systems. A given structure is a given relation&p between inputs and outputs. Only when the dependence of the structure itself on the variables is included can a model show structural change. To take structural parameters as time varying is teleological, it assumes that some invisible hand turns up the ?:nobs on the socio-economy. The history of development of socio-economic systems tells us that that is not true. On the contrary, without external influences, development occurs because of the response of structural relationships to the variables that the structure itself relates.” To be sure socio-economies undergo structural change because of external influences. Thus a complete theory must be capable of showing how the allocation of external resources will change the relationships among the socioeconomic variables. By incorporating relationships which describe self-develop ment and the influence of external resources one has a tool for normative analysis. Furthermore socio-economic development proceeds unevenly. Therefore, a correct theory will also include space variables. Then one has a tool for studying polarization, spread effect and cyclic behavior in an open economy. A theory that truly represents the internal dynamics - actual mechanisms of the so&-economy-can be used with a paucity of data. That conventional input-ottput analysis requires much data appears to be as much a consequence of its operational character - no consideration of mechanisms - as it is a consequence of statistical uncertainties. An important characteristic of the present theory is that, because it relies on a basically sound description of socioeconomic processes, it requires few data, it is simulative, it is predictive. The function of data is to determine parameters of equations of the theory that then describe the historic development of a socio-economy. Prediction is achieved by dynamic extrapolation and continued iteration. 2. l3mlamental definitions and relationships We describe a socio-economic syste. by a stale rector p. The elements of p are the relevant social and economic variables, each quantified in its common unit of measure. The spatial coordinate is indicated by a Greek subscript; thus the vector pa describes the socio-economic state of the ath zone. Usually we shall be studying a region, then we refer to p the regional state vector and the set (p,), the zonal state vectors. We shall be more precise about the meaning of zolzc and regiorzsubsequently. “A descriptive and linear theory cannot do any more than produce a coarse-grained time series of input-output coefficients. What is needed is a model that incorporates the quantitativequalitativechangemanifested in a dependence of the structural parameters on the levels themselves. True, Leontiefcan describe structural change by providing ‘. . . the possibility of using different sets of flow and capital coefficients for different years, thus incorporating into the dynamic system technological change’. However, a model of development must show how StWtUrd parameter such as ‘B, - the matrix #ofcapital coefficients’ are themselves altered by char:ges inlevelof outputs Keontief (1968, pp. 2-3)].

H.L. Fax and G. Schachter, Dyrtamicsqf structrrralchatgge

43

Tn general, a component of p say pi (we use Latin subscripts to label components) has a functional dependence on other elements (pi}, 011 exogenous variables {Q), and on the time t. It follows then that the total time-rate of change of the element pi is given by

n -_I dPl dPj c j+i

dP, -= dt

apj

dt

+ dPi ~‘~Zz-*

m ~PidEj

(1)

j

Note that the partial derivatives imply economic and social experiments not available to US since they are statements of the form, ‘given that everything else remains unchanged, what is the change in pi for a change in pj (in the case of api/JPj)‘, or ‘what is the change in ps for a change in time t if all endogenous variables remain constant (in the case of api/ptya Eq. (1) can be written in matrix form

where lis the unit matrix. Here Wis a square matrix that represents the infi.uence of each. state variable on the other state variables (in economic terms, the marginal productivity), By definition it has zeros along the diagonal and elements elsewhere given by wij

Z-,

6’ ’ i

11is a vector in the same vector space as p that represents the influence of ‘population level” on the state variables. It has elements

where Q, is the population. s is the source vector. It represents the flow rate out of the zone or region of all exogenous variables (except population),

“Populationlevel’here is not intended to nle;ln necessarily the nd3.f Of persons in the region but to represent a class of exogenous variables. In many problems the actual population he1 or birth rate are properly endogenous variables.

H.L. Fox and G. Schachter, Dynamics of structural change

44

K is a rectangular m x n matrix that represents the influence of exogenous variables on endogenous variables. Its elements are given by

The matrix W, representing marginal influences, also has a dynamic description. Since it has the same dependencies as the state variables, its dynamical equation is of similar form. For each element Wij one has 3 Wij dp, -m IT k-t1 i?pk dt

d Wij -= dt

f7)

As before we can rewrite this relationship in matrix form dW -=

‘Ydp dt +Gd,+

dt

aw-

-

at

xs.

(8)

Here Y is an n x 17x 11cubic matrix representing the influence of each state variable pk on the influence coellicient Wij (in economic terms, elasticity of marginal productivity). d2p, yy,jk

(9)

= apiapn ’

G is an Hx II square matrix representing the change induced in the influence coefficients Wik by a change in population Ed, Gik =

i3Wik

a2pi

z-=-*A

@kaEO

x is an nz x n x H rectangular-parallel-piped matrix representing the elasticity of each influence coefficient Wij with respect to each exogenous variable E&,

(11) Before proceeding, let us summarize what we have put down. Ba:kally everything thus far has been tautological. No assumptions have been made with regard to production functions and no approximations have been made. We have only defined variables and stated that they are interdependtnt. Eq. (I) simply states that the tota change in time of a state variable derives (a) from the

H.L. Fox and G. Schachter, Dynamics of structltral change

45

changes in time of the other state variables (endogenous factors) to which it is lrlnctionally related, (b) from the changes in time of exogenous variables to which it is also functionally related, and (c) from any intrinsic time dependence; i.e., structural change. Eq. (7) makes a similar statement about the influence factors We formally solve eq. (2) by using the inverse of I- Wand apply the solution to eq. (8) to eliminate dp/dt. We obtain apJi)pj*

ds= Y(I-

W)”

(

pie+

8P dl

-KS

>

+Gi,+

g

- )Is.

(12)

Consider the practical problems in using eq. (12). First it contains ap/df and 2 IV/at. A measure of the time-rate of change of a variable while all other variables are held constant is certainly not an accessible measurement in socio-economics. On the other hand, if Wean be measured then a finite-difference approximation to d W/dt could be obtained, since d W/dt is the total observed change per unit time. But W itself as well as W, cc,and G are partial derivatives requiring that, in a measurement, all other variables are held constant. The latter problem we shall handle by introducing an entire set of production functions as constitutive equations. The problem of evaluating ap/i.?1and d W/&requires a basic assumption. 3. Rate-equationassumption represents that time rate of change of the socio-economic state that can be attributed to structural changes. That the source of structural changes results from internal growth and from outside inff uences is obvious. Our assumption, one which is primary to our approach, is to assume that the explicit time rate of change of each state variable is a linear cumbiuationof endogenous and exogenous factors, It is proportional to the present level of all state variables and to the present level of rates of change of all exogenous variables. Mathematically one has8 3p/iTt

(13) The n x II square matrix hY is, in general, time dependent and a function of the state variables. Consequently, eq. (13) is deceptively simple looking. Clearly, knowledge of the elements of 114is the key to understanding and predicting development and is thus the immediate object of our study. sNote that this equation is similar in form to Leontief’s dynamic equation if his eq. (1) is written X,, *-X, = -B- ‘(I- A),$‘~+ B- *C’, [Leontief (19&S)]. Subsequent developments show that the similarity is only in form. D

H.L. Fox and G.

46

Schachter, Qvnarnicsof strurtrrralchange

4. Determinationof M

A4is a rate matrix. Each element ~,

describes the :atio of the change per unit time in the state variables pi to the level of state variable Pj (in economic terms, the average growth rate per unit input). It also is not directly measurable because its measurement would require controlled experiments which are not possible in socio-economic systems. Hence, we seek ways of relating the matrix M to quantities that are measurable in situ. We accomplish this by the following manipulations. We take the partial derivative of eq. (13) with respect to each state variable pb and obtain i?W =

M(W+I)+@p+

at

p,

in which

(14) The only assumptions made to obtain eq. (14) are that the functions are analytic and that s the rate of exogenous input is independent of each state variable pi. By introducingeqs. (13) and (14) into eq. (12), we obtain Y(I-

wpMp+M(W+I)

=

-7

-@p-fY(I-

W)‘‘lp+G]i**

(15) Description of the method of inversion of eq. (15) appears in appendix A. The troublesome term in eq. (15) is @. Since one must solve eq. (15) to determine M and Since each @ijk iS a partial derivative On h!f (@ijk = ahfij/apk), eq. (15) is not closed. Several methods of closure are possible. Here we present one suitable to underdeveloped regions,g where one can assume that for short periods of observation the matrix of growth rates per unit input M is virtually a null matrix. That is, in zeroth order M = 0 and #p satisfies the right side of eq. (15) set equal to zero. If we now make the further approximation that for I’M= 0 and short periods of observation the contribution of the rate of population growth can be ignored, we obtain

(@p)(O)

t= !Pt,

116)

‘An equalaliyreasowbie and more general assumption is Q, = 0 in zeroth order, but computations are then much more complex. Only applications can determine the most etkctive zeroth app:oximatiow.

H.L. Fox and G. Schachter, Dyttatnics of stnrc~rrrulchange

47

where we have used eq. (8). Using eq. (16) with eqs. (2), (13) and (151,we obtain an equation for M in first order,

Further iterations would proceed by differentiating eq. (17) on p to obtain a first order estimate of Q,then soIving eq. (15) for second order in M, etc. Eq. (15) [or its first approximation eq. (17)] is the fundamental equation For the growth matrix M. Eq. (13) is the growth equation itself. The procedure is to use eq. (15) to obtain an N and to solve eq. (13) to obtain projected values of p. These values are then used in eq. (15) to obtain a new M, and thus one bootstraps the dynamical projections. Since the inhomogeneous term of eq. (13) changes the levels in the socio-economic state vector and these levels are then used to determine the growth-rate matrix M, the growth rate is a functional of the resourceallocation plan. 5. Constantgrowth-ratematrix approximation To make the evaluation of p, Wetc. practicable, weintroduce static functional relationships between the elements of p. For any time interval over which the structure of the socio-economic system ren;ains constant, each state variable can be expressed in terms of some other state variables through a production function

We are referring here only to explicit dependencies. That is, the entire gamut of interconnectedness is expressed in the equations thus far presented; in eq. (18) we express only direct dependencies. For example, the output of a given sector ultimately depends on virtually all the region’s social and economic activity, but most of these dependencies are one step removed - not explicit. Typically eq. (IS) would express the relationship of the output to each of that sector’s inputs without explicitly referring to the connectedness of those inputs to other inputs. Suppose now we write out a set of production functions. Then for each contiguous time interval ever;’ variable in Y, W, p, ,u, and G could be expressed in terms of averages of the values of the state variables and the parameters of th? production functions. The parameters themselves could also be obtained for CX~ time interval, The result would be a time series of rate matrices. But this requires an abundance of data which is never available. For most practical problen:s w can assume that for the observation interval the average growth rate (change rate) is approximately constant. Hence we solve eq. (15) [or (17)] by avern$ng over the interval, The result is a constant grotith-rate matrix M.

45

H.L. Fox and G. Schachter, Dynamics of structural change

,

6. Leakages For constant growth-rate matrix M, the formal solution of eq. (13) is

PW = e”‘p(0) + jb dt’ e”“-“)K(t’)s(t’).

(19)

Since M is averaged over the interval and p(O) is chosen as the observed initial state (also generally coarse-grain averaged) p(t) must agree with the observed values at t = T, the end of the interval of observation. This will not generally be true because of leakages. We introduce, therefore, a leakage matrix L, a diagonal matrix that accounts for the decay of capital, inventories, skills, land, etc., as well as for our ignorance of hidden flows. We assume the leakage of a given component of the state vector to be groportional to that element. Hence, the corrected eq. (13) is

zp -dt = iWp+Lp+K(t)S(t),

(20)

where L is obtained by comparing the solution p(t) of eq. (19) at time t = TV the end of the observation interval, with the measured values p(t) at time T. Each diagonal element is [e-“rjj(T)-

5: dt’ eeM”K(t’)s(t’)]ii P(O)i

.

(21)

Let us summarize the mathematical developments to this point. A set of production functions Pi = Pi[(PkIl

are chosen {according to criteria we have yet to discuss). These parameters are estimated or evaluated for an observation interval T. The derivate quantities CV, fy, G, p are expressed analytically from the production functions. These are then used with the difference approximation to the derivative d M//dl in the following [eq. f I 5)] : !#‘(I- FV)-1A4p+M(W+I) = y

-Q-[!$‘(I-

W)-lp+G]&,,

which by an iterative or approximation method is used to obtain the growth rate matr.x M. h4 is assumed constant and evaluated by averaging over the observation period. Finally, tie diagonal leakage matrix L is obtained by matching the

H.L. Fox and G. Schachter, Dynamics of strtrcturalchange

49

end point of the interval [eq. (21J] 1

Lii = T In

[ewMTp(T)- {l dt’ e-MT’K(t’)s(t’)]it

I

. PC”Ji

One then evaluates alternate resource-allocation plans through solutions of eq. (20)9 8P 5 = Mp+Lp+K(t)s(t).

However, this equation does not explicitly show the spatial coordinates. We turn next, therefore, to a restatement of the basic equation with the spatial coordinate a explicit. 7. Spatial analysis TC begin, we define a region within which there is an essentially closed economy; that is, the flows of products, people, raw materials, books, etc. across the boundaries of the region are well defined. Within this region we define v zones. The simplest zones are defined by a geographical grid; but other criteria for zone determination may be more suitable. For example, an industrial center may be one zone; another might be the countryside around the city. The ath zone has a socio-economic state vector pa. The totality of the state vectors comprise the regional state vector p. (N.B. pa and pg for cc+ p are in different vector spaces so that the regional state vector p is the direct sum.) To obtain the dynamic equation for pa we proceed as before. The development equation for the ccthzone is thus

(221 Note that iMaPis a matrix that couples the jth zone to the ccthzone and KZis the Alocation matrix that relates the external resources s to the ath zone. Since the mathematics :$ould not be able to tell the difference between a zone and a region, eq. (22) should be representable in the form of eq. (20). That this is indeed the case is apparent when we recognize that some of what is endogenous to the entire region is exogenous to a given zone. Thus by considering all pp exogenous to zone 3 for /II+ a, we obtain the canonical form

50

HAL Fox and G. Schachter, Dyrtanzics of structural change

in which (24) Then in analogy to the regional solution eq. (19) we have the zonal solution

p,(O = expKKu +L.l~lpu(0) + ji dt’ exP IWuu+4&t - 01 x K;(t’)s;(t’) ,

(25)

where the first term on the right represents intrazonal development and the second term represents the influence on the zonal development of factors exogenous to the zone. These extrazonal factors are of two types, intraregional

and extraregional

55dt’ expE(~,,+LJ(t-

t’)]~,(t’)~(t’).

Eqs. (23)--(27)are equivalent to the regional equation in that solutions to the regional eq. (20) can be broken down by standard procet:ures into the solution for each zone including all the coupling effects. However, in eqs. (23)-(27), we have a form that is better suited for discussion and b:tter adapted to computation under certain circumstances. Methods of iterating solutions for coupled zones are in appendix B and a second order differential equation for two coupled zones is developed in appendix C. The zonal matrices Mplsare submatrices of the regional growth-rate matrix M. But the decomposition of the regional matrix is not unique. Thus it may be possible to find alternate decompositions which are more ‘natural’ in that they ISescribezones that are less coupled than the original. The method is discussed in appendix D. 8.

Role of prochcti~on

factions

We consider now the introduction of production functions. A production function expresses a relaticnship between a variable, say pi and the other variables on which it is explicitly dependent. The relationship has parameters in the form of coefficients, exponents and additive constants; designate these {Ojj. FormaIly one has fc.f. eq. (I 8)]

In a structurally unchanging economy the i?,, may be functions of time but the are constants. Purely quantitative development of an economy occurs t+hen the inputs are increased leading to increased outputs. Qualitative development changes the structure, i.e., the parameters (Oj>, so that the same input produces more output. A traditional input-output analysis provides a set of parameters - the inputoutput coefficients - so that one can either (I) assume unchanging structure and determine the inputs required for an increase or redistributed output, or (2) point to the elements of the structure that need to be changed in order to achieve an economy more like some norm (traditionally the U.S.). However, historically real socio-economic systems have developed by quantitative-qualitative processes; i.e., increases in resources such as population, new-found materials, etc. have led EOincreased activity which has in turn led to qualitative changes in input/output relationships. lo In terms of the formal expressior! eq. (28): the {ei) are functionals of the fpk}. The problem is to determine these functionals. Our method is to postulate production functions, to evaluate the parameters (0,j over the observation period as a zeroth approximation and to use the dynamical equations to alter the functions as sbcio-economic change occurs. In many cases the production functions characteristic of an underdeveloped economy and of a developed system are not the same. Then we postulate a production function that is a linear combination of the two. In this way, we provide a framework within which development can shift the production function from one form to another. (Oj)

9. Method of application’

’

We summarize: (1) Postulate the explicit dependencies of each state variable on the other state variables by qualitative analysis of the nature of the socio-economic system. (2) Devise a set of production functions that give explicit form to the dependencies. l”Quantitative-qualitativc processes arc often not continuous, * hence, the analyticity that we have assumed in our mathematical developments must be considered as a choice of mathematical convenience.This choice will not cause usanyditkultyas long as we confine application of the methodology to socio-economic growth in regions where many small discontinuous processes are contributing to growth and thus, on the average, growth appears continuous. WC must, however, be prepared to ihrow away all theories, including this one, when historical forces synthesize a substantial quantum leap. Social revolution, large-scale introduction of capital-intensi .e farming methods into a mono-agricultural economy or unexpected large-scale drought are outside the domain of any theory. Yet these are the processes which pulse history. What then is the role of a method for analysis and prediction. 7 It is the tool for carryir;g the swell put into motion by a historic discontinuity -- for ordering the social energy released into a coherent pattern for growth and development. 1 1Details of the n~et]~odology are in a forthcoming article ( DJWNIGCSof strvctrrrd ~hir/~gc?11: :IhJfiimfhlogy).

H.L. FOXand G. Schachter, Dynamics of structural change

52 W

(3b) (4)

Evaluate the parameters of the production function by statistical averages over the observed data, or estimate the parameters if the data are too sparse. Using the production functions, express analytically the functions Wil, yij,, Mi, Gkjk These are functions of the variables (pk) and of the parameters (0i) evaluated in step 3. Solve eq. (15) for each sub-interval in the observation period and average the resulting Ms. Use the averaged M and data from the beginning and end of the observation interval in eq. (21) to obtain the leakages Lii. Solve eq. (20) using the calculated M, L, and data from the end of the observation period as initial data. The result is a projection. Substitute the projection and the data at the end of the observation period in the production function to estimate new parameters. Evaluate new IV, Y, 14,G. Use new IV, Y, 9, G in eq. (15) to obtain new M. Solve eq. (20) using the new M (but, old L) and previous projection as initial data. Proceed to step 7 and reiterate until the projection is extended over planning interval. l

(5) (6) (7) (8) (9) (10) (11) (12)

From the above procedure, one obtains the expected self-development (or the continued stimulated development if there already is an external-resource plan in action). To evaluate alternate resource-allocation plans one must repeat the procedure for each plan. The result - the projected socio-economic state vector is compared with a goal, generally a set of objectives and constraints. If the goal is a socio-economic state ,g then Ip(t)--g12 is the mean-square deviation of +he projection from the goal. Minimization of this mean-square deviation with respect to adjustable parameters in the allocation plan K(t) or the resource composition s(t) is one way of applying the criterion g to the projection p(t). In general, the considerable accumulated technique of optimal-control theory are avaiiable to tind optimal 1-~ with constraints: Eq. (10) is in the canonical form of the optimal control problem. [See Athans and Falb (1966).] AppendixA: Inversion of eq. (15)

Eq. (15) in component form is

By uji:Ig the Kronecker delta (liiii -- 1 if

i = I

and ;cro if i # !), we can

H.L. FOXand G. Schachter, Dyttanticsof strrrcturul change

53

rewrite the second term on the left side as follows: C C 6,iM,fnt w+Gnjm

642)

1

The right side of eq. (Al) is simply a matrix which we denote Qij. By rewriting eq. (Al) and using eq. (A2), we obtain

in which the four-subscript symbol /1 ij,lm is A ij,lm=

Ck {yijklI(r-W)‘llklPm+ 6,i(w+ 0 ml

l

Eq. (A3) is nothing more than the matrix equation AM=Q,

with solution M = A-‘Q.

6-W

Computer manipulations with equations of this type are facilitated by mapping the subscripts onto a new set

i,j-v, I,m

+q.

M, an N x N matrix, is represented by an N* component vector and solution eq. (A6) requires inversion ofthe N2 x IV2 matrix, Ap,i,j,,qfm,f).

Appendix B: Iterated solutions for coupled zones Zonal equations in the form eqs. (23)-(27) are primarily of use when the regional economy is weakly coupled to a particular zone in which the socioeconomy is expected to develop rapidly for a period without strongly affecting the neighboring zone’s socio-economy. Then a solution generated in a perturbation series is useful. The zeroth order term p$)(t) is based on the assumption that initially the reaction of one zone on another depends only on the level of the state vector at the initial time,’ 2

d”(t) = exPi?fa,~h(O~

+ 8+z cpjf,dt’ exp

[M&t-

t’)]M,,p&O).

(~1)

’ zIn this appendix, we leave out the leakage matrix L,, for notational convenience. One can consider that each diagonal term of AI,, has an additive term.

54

H.L. Fax atid G. ScSachtcr, Dynamics of structural change

The next approximation phi’(t) is obtained by iterating. We replace the inizial values by the zerath order solution and obtain

u-m

-C-J;dt’ exp [M,,(d- t’)]&(t’)s(t’). The general form for the pith iterate is PL%) = exp PW+

f)lp,(O)

& & dt’ exp [M,,(t-t’Wmppj;’‘)W JI

+

5; dt ’ exp [M,,(t - t ‘)]&(t’)s(t’) .

033)

Appendii C: Second order differential equation for two mupled zones with no external intiuence The equations of two zones Qand /I with no external influence are

= 2,p = 1. oneforg = 1,/I = 2andonefora Differertiate eq. (Cl) with respect to time,

Now substitute for kYpZ ‘dt from eq. (Cl) and for changing z and ,8,

@,/at

from eq. (Cl) by inter,

(C3) and substitute from the inverted form of eq. (Cl),

55 to

obtain the closed second order differential

equation,

ah dt2 -([(M,,+L,)M,,+~~,(,M,,+~~)][Me~~]-l)g + NK, +4zmf,,+ +tl~&I +L~p)l[wJ ’[M,,SL,] - wfa, +a2 + 4~~/?,IIP,

Eq.(C5) is solved with initial values for pa and time t = 0 in terms of p,(O) and p&O),

aA -z

t=o

(C5)

l

= (MSz+L)RAO) + K,P,@)

i?p:JZt. 2p,/dt

*

can be expressed at

K6)

The solution of eq. (C5) is of the form Pa =: exp [@]a+

exp+[Q*t]b,

in which vectors a and b are determined by the initial conditions eq. (C6jLrrzd Q, Q* are solutions of the matrix equation Q2-BBQ+C

= 0.

!C7)

Here B and C are the matrices that multiply into vectors @,,‘dt and ps, respectively [in eq. (CS)]. Hence the matrices Q and Q* must satisfy .a

Q2+QQ*=BQ, QQ*+(Q*)2= BQ*, QQ* = C.

(0

Eqs, (c8) yield one of four types of behavior accol-ding to the Composition of

the matrices Band C:

(1) decay, (2) unstable growth, (3) either of the above with cyclic behavior, or (4) pure cyclic behavior.

56

H.L. Fox and G. Schachter, Dynamics of srructural change

Appendix D : Redefining zones by obtaining imxhcible representations of M

An ancillary feature of our description is the potential for redefining zones in terms of natural, virtually autonomIous, subdivisions of region’s socioeconomy. The regional rate matrix M is, of course, the direct sum of the zonal matrices,

@

M,l

@ M”2 0

------“-I-

@

m

M,,.

One seeks irreducible representations which would then define a set of mathematical subspaces different from those that represent the v zones but which also comprise the space of M, only such that the subspaces are uncoupled. The transformation that accomplishes this also defines a transformation on the total state vector p. The transformed state vector p’ can then be decomposed to new irreducible zonal definitions particularly suitable to various problems such as application #ofthe techniques of transportation analysis, etc., because the zonal definitions, usual as they may be, are the maximally decoupled set. We demonstrate the idea graphically. Consider M for four zones of different compositions,

Ml3 ---M21

;

M22

----:---M3,

M23

:.

M32

M41 ;

M34

M’33

..----

MG2

hf24 -_--

_---

_-_-:-_--

Ml4 __--

j

_---

M,3

VW

M44 J

Assume that t.ypically IMiiI is of the order of IMill for i # j; that is, in this choice of zones c !erything is strongly coupled, Now by the well-developed methods of group theory, one can obtain the +transformation matrix S that produces an M ‘. M’ = S-‘MS

03)

H. L. Fox and G. Schachter, Dynatttics s f’ structural change

57

such that, in our example, M’might look like

___-I__-_-_--

M’

=

-_

'Mix;

in which the submatrices Mij for i #i are virtually (or maybe even actually) null. Note that the decomposition does not necessarily result in the same number of zones. Apply S- * to the dynamical equation [eq. (13) 1,

where we have inserted the unit matrix in the form SS- ’ = 1. Eq. (IX) is stil1 in the canonical form provided we consider as the new rate matrix M’ = S- ‘MS’, the new state vector p’ = S- ‘p and the new allocation matrix K’ = S- ‘K. But the zonal equations are now decrupled,

aP;

WJ)

and correspond to the new set of zones. These zones may be unusual : They generally will not be solely geographically defined; they will tend to relate industries and populations that are strongly interdependent. References Athans, M. and P.L. Falb, 1966, Optimal control (McGraw-Hill, New York). Fox, H.L. and G. Schachter, 1970, Dynamics of development, polarization and spread effect in underdeveloped regions, Presented at the Italian Section of the Regional Science Association, Rome, Italy, 2 Sept., Tech. Memo. (Bolt Beranek and Newman Inc., Cambridge, Mass.). Friedman, J., 1963, Regional economic policy for developing areas, Regional Science Association Papers XI. Leontief, W., 1963, The structure of development, Scientific American 2C?, Sept. Leontief, W., 1966, Input-output economics (Oxford University Press, New York). Leontief, W., 1968, The dynamical inverse, Presented at the Fourth bternational Conference on Input-Output Techniques, mkleo., S -I 2 Jan. (Geneva).