Regional Science and Urban Economics I1 (1981) 351-361. North-Holland

STRUCTURAL CHANGE AND TECHNOLOGICAL DEVELOPMENT

Ake E. ANDERSSQN

Urned Uniuerstty, Umeci, Sweden

It is important for our understanding of sectoral and regional structural change to analyze the R & D-activities. In this paper we show that R &D should be analyzed as an endogenows,

intermediate, public investment variable rather than as ordinary capital investment. The allocation of resources for R&D cannot be decentralized regionally or sectorally but should preferably be decided on as a problem of optimal taxation. It is shown that the optimal rate of taxation for R&D is determined by possibilities of substitution between ordinary capital investments and R&D, by the propensity to invest and by the productivity of the R&D- producing sector. A dynamic model with accessibility representation of knowledge is finally formulated. This interregional R&D-model also has equilibrium growth properties. The equilibrium rate of growth of all regions of this model CEF t.c raised by decreasing any interregional distanc’e or by raising any regional propensity to iT!li!:Gt.

1. Intraduction

There are three fundamental factors determining structural change:

(1) Reallocation of resources or products between processes. (2) Quantitative expansion of the resource base (labor, capital, human

capital, natural resources). (3) Technological development of outputs, inputs or processes.

The relative importance of these factors of structural change is hard to assess. It is fairly clear that reallocation of resources can be a structural change factor already in the short run, while quantitative expansion of resources and technological development is of relevance in the longer run, only. It should be pointed out that even Keynesian ‘pump priming’ can be seen as a structural change policy of type 1.

In this paper I will put the emphasis on struct:\ral change analysis in the long run stressing the problem of balancing R&D for technological development against capital expanding investments. Bergman (this issue) stresses the need to develop models to analyze structural change as an issue of reallocation of resources between sectors as well as products in the short run (model of type 1). In his case this implies that the model is built as an hybrid-model, blending neo-classical features such as substitution with

0166--0462/81/ OOOO-@M/%O2.50 0 1981 North-Holland

352 AE. Andersson, Structural change ud technological development

statical input-output features such as interdependency between commodities.

intermediate

Other models, stress thlle quantitative expansion : problems and consequently they are forced to emphasize dynamics proper at the expense of certain important aspects like substitution of inputs and outputs.

In this paper a number of important aspects of a dynamic analysis also have to be suppressed. Instances of simplification are

economic

(a) lack of basic resource considerations, (b) simplified analysis of consumer behavior, (c) suppression of trading interdependencies between regions, and (d) no analysis of the labor allacation problem.

In this respect this paper should be regarded as a complement to papers of this issue.

2. KaowIedge as an endogenous input of tbe growtb process

the other

Traditionally economists have treated the stock of knowledge and the flow counterpart, R&D, as an exogenous factor influencing the growth possibilities of an economy.

In some cases it is reflected in some macroeconomic growth model as a shift parameter only [see any standard textbook on growth economics. A typical formulation might be

Q(t)=e’.‘F(K(t),L(t)).

where

Q(r ) = rate of production of a malleable standard commodity in time t, K (t ) = rate of capital services in time t, L(t) = rate of labor senices in time t, i. = technological expansion factor, exogenously determined.

F(K (1). L(t )) is often assumed to be an at least twice differentiable production function, homogeneous of degree one.

With assumptions about savings being a constant fraction of output, S, savings equilibrating capital growth, and labor growing at some exogenous rate, n, one can deduce that the equilibrium rate of growth equals the sum of labor growth and technological expansion, i.e., g = u -t- il.

T’he ge;leral result of this class of models is that if knowledge expansion is exogenous, the economic s?jtern will in principle be capable of automatic adaption to this fact. Technological development will act as if it was instead

A.8. AnderssoR, Structural change and technological development 353

a quantitative expansion of the resource base. In view of the policy debate on the role of R&D in the economic system it seems more fruitful to formulate the question how knowledge creation interacts with capital- formation in the economy. In Order to do that we introduce of knowledge as the variable G and the change of it as Gr R, indicates the volume of R&D.

standard the stock where R

We furthermore assume that the allocation of resources between capital investments and R &D is determined by the parameter z which must be between or on the values zero and one. A simple variant of this approach would then be

lt=s(l -r)Q(K,G,Z,),

~=R(~~Qt?W,hd,hd, (1 1 a

where

R =rate of capital investment, Q = neo-classical production function, K = capital service capacity, G = knowledge service capacity,

EK = preallocated labor services to commodity production, G =rate of research and development (R dill), R = production function of the research sector (universities, consultant

firms, etc.),

El4 = preallocated research labor, S = total savings or investment ratio, 7 =research investments as a percentage share of total investments.

A particularly simple example of this model would be

c;= (stK”fiaG’-“-@&)P, or (lb)

For p= 1 this reduces to the equilibrium condition that the knowledge- capital ratio should tend to

G/K=&/(l-7).

354 kE. Anderson. Structural change and technological deuebpment

This simple condition says that on an equilibrium ray knowledge expands at the same rate ;as capita!l so that the knowledge-capital ratio is kept at a constant. It is positively determined by the research priority ratio z and the size of the research labor force L,. With a PC 1 an increasing savings ratio decreases the equilibrium ratio between knowIedge and capital. Returning to equation system (la) we see that it can be rewritten as: Ax = H(x), where x = (K. G) and where H(x) is a positive mapping. For such a system of equations an existence theorem is available [Nikaido (1968)].

Theorem. Assume the following conditions to hold:

M’ix)= (Hi(x)) is defined for all non-negative x in R”,, with its values EL ing also non-negative vectors in R: , H(x) 2 0. H(x ) is continuous us a mapping H : R”, + R’!,., except possibly at x = 0. H(x) is positively homogeneous of order m, 1 ZrnzO in the sense that H(rx)=x”H(x)for rzo, x20.

A={il]H(x)=l.xfor some x0,), where

is the standard simplex. Then A i( H ). Furthermore, if m = 1, l-(H) H.

We can thus conclude that the

contains a maximum which is denoted by is the greatest among all the eigenvalues of

stock of material capital and knowledge should expand at the same rate, which is determined by

(a) the propensity to save (or invest), lb) the propensity to allocate investment funds to R&D, (c) the commodity production technology, tdi the rewarch sector productivity, and (e) the allocation of labor to the commodity and research sectors.

3. Knowledge as a public good

Knowledge should be looked upon as a public good. This is eq-livchlent to regarding it as a catalyzer. This means that knowledge partici- pates in the production process as an input without being used up by the process. The role of the dose is probably non-linear, for instance like in fig. 1.

Being a public good, knowledge can be used without conflict between all commodity producing activities in a sufficiently small region.

kE. Antlersson, Structural change und technological development 355

Productivity contribution

. .

CzzaI increase

Knowledge

Fig. 1

This implies that disaggregation of the decisions on production and allocation of resources becomes critical for aggregate efficiency.

The problem can be illustrated in a simple diagram, in which firms are alternatively free to determine their R&D requirements or to have it determined in a joint optimization.

Joint centralized maximization gives a clear indication of the nature of this problem, first shown in a consumer framework by Samuelson (1954),

max V=CPiQi(G,,Ri)-rl(rG-R),

with the optimality conditions:

W -= c p aQi ----lJr=o, dG i 'dG

W -qf&.&(). aul

(2)

The essential characteristic of this problem is the collective interest in the size of G. This requires some collective decision on the level of G, that is how

356 AS. hdersson, Structuraf change and teclrnofogicaf deuefoptnent

AG; AG;

Fig. 2

to split the cost, qr, of providing the optimal would be to split the cost in the proportions

AG* -

amount of G. One candidate

p ZQi / ‘SGI c p dQi

‘dG l 1

This procedure, howeve r, requires centralized information about the marginal productivities of the intermediate public good, G. This problem has no obvious resolution with decentralized decisionmakers which can be seen by reformulating it as a game-theoretic problem.

Trying to optimize jointly would cause a profit and cost sharing problem with no obvious solution (see fig. 2). Simple 50/!;0 allocation of costs or profits are unstable, as can be seen from table 1. If participant 1 would be uncoqxrative the maximum profit for 2 would be 6.5. Similarly, if participant 2 would be uncooperative the maximum profit for 1 would be 0.5. The two potential fesses from counteractions are thus (2, 3), which in some procedures are also taken into consideration in the distribution of profit&

h, Andersson, Structural change and technological development 357

Inoptimality of. investmqnts in R&D for these reasons always means too little R &D in relation to material investments. Social planning is also (possibly as a consequence of this assymetry) common in developed countries.

Table 1

Total revenue of AC* = 19 Total cost of AG” = 7

Total profit of AG” = i-i:

SO/SO cost sharing: 3.5 from each implies

Profit allocation to 1 = 2.5 Profit allocation to 2 =9.5

4. Optimal investments in research and development

It is argued above that decisions on the rate of research and development must be made in cooperation by the users of knowledge. The most common way of acting collectively is to have some possibility of taxing output or profits for the purpose of financing collective or public goods. It is now assumed that an agreement has been reached on how to distribute costs of R&D but that the level has not been decided on. This is now approached as a problem of determining a rate of taxation (7) of output. I furthermore assume that the level of production is determined in a putty-clay fashion. This means that the growth of production is determined by the equation

AQ = $ (1, RI, (3

AQ=Q-Q-1 =growth of production bet.ween two consecutive lberiods of time,

I = rate of material capital accumulation (net investment) R = AG = rate of knowledge accumulation (research and dc ,.elopment

investment) f = a continuously differentiable strictly concave growth functjd\l’

The rate of investment is determined by a Keynesian investmelt 1, 1 l.1 ion :

I=s(l -z)Q-1,

where

(4)

I = rate of investments, S = savings (investment) ratio, ? = R & D-taxation-ratio, Q -1 =rate of production in the preceding period.

358 A.E. Andersson, Structural’ rirange and technologkd dewlopment

The rate of knowledge accumulation is determined by a simplified function

R=gTQ-1, (9

where g is the productivity of the R&D-producing sector. Optimizing the rate of knowledge- and capital-accumulation can be done by formulating the Lagrangian

maxA=f(I,R)-1,(1-s(l-t)Q_,)-A,(R-gtQ_,). t&I,rI

(6)

Conditions for a maximum are

2/l ?f ---~--;

(‘I 3 .I =0,

2/l z=l-s(l -t)Q-1 =Q

an -=R-g7Q_,=O.

I dA

R

This implies that at the optimum, the condition

dR ?f ?f S R, -+_-=_ dl (‘I ?R ,g 4 ’

must hold. We can formulate a golden rule of taxation for R&D:

The rate of taxation for accumulation of knowledge should be adjusted so that the ratio between the rate of savings and the marginal productivity of the R&D-sector equals the ratio betb;teen marginal growth eflects of investments and of R&D.

The model of optimal R&D can be illustrated in a pedagogical diagram. Fig. 3 illustrates the propensity to save, the productivity of the R&D-sector and the growth function properties together determine the optimal rate of taxation.

It is clear that any increase in the savings ratio or the productivity of the R&D sector must increase the rate of growth. What will happen to the optimal rate of taxation is much less clear. In order to show the importance of the possibilities of substitution we assume the growth-function to be of the

h3. Andersson, Structural change ad technological dewdopmnt 359

Rrl

4

/ .L

/

)

/

?

f

Fig. 3

CES-form,

f(l,R)=(~~I-~f (1 -+)R-P)-~IP,

g=, I--” +q dl l

El 1--P+ (1 __E )p)-‘l+ l/p)

1 9

as -= -&@ 8R (I

-(* +p)(EII-p+ (1 -Q)R-P)-(I+ l/P),

(8)

(9)

(10)

360 A.&. Andersson, Structmal change and technological dcdopm;ent

If growth is regulated by a CES-function, the optimal rate of taxation is determined by the equation

The following proposition can then be formulated:

Proposition. Given Q CES growth function, investments function (4) and R & Dflitnction (5) the optimal rate of taxation z will decrease with an increasing (s/g) if the elasticity of substitution is larger than one. z is independent of s/g for tr = 1.

5. Regional generalization of tbe model of R&D and growtb

The regional dimension is extremely important in all studies of public goods. Collectivity without spatial restriction must be extremely rare (like world peace and other ‘goods’ of this kind). Knowledge is quite different. The availability of technological information is clearly much better on smaller than on larger distances. Accessibility is a reasonable way of representing the economic effects of a spatial distribution of public good like technological knowledge. A general representation is given by

where

cr, = accessibility from region t, &I,,) = a non-increasing function of the distance from region r to region s

id,, zo), &‘, = level of knowledge in region s.

A regional growth process (with an optimized or non-optimal z j can now be tc:rmulated :

&=s,(l -dQ,(K,.a,),

d, = WgTQ, 1,

~1, = c h(d,, P,. s

(12)

The formerly formulated theorem can be used to prove the existence of a hsrluncvd growth solution also for this system.

A&. Andersson, Stwtural ehonge and technologicuf development 361

Simulations with this model have shown that it is relatively stable for any set of neo43iassicd production functions.

After linearising the system it has also been shown [Andersson and Mantsinen (198U)$ that

(a) a decrease of any distance, increases the balanced rate of growth of ah regions, and

(b) an increase of any regional rate of savings increases the rate of growth of all regions,

(c) the structure of production generally changes with changing research priority and changes in regional investment priorities.

The most interesting property of this model is the tendency to transform spatial distances into time lags.

Growth rates of the periphery regions thus generally increase at a later stage than growth rates of centrally located regions. This implies that increasing the general research priority ratio (or tax ratio), 2, normally leads to increasing regional disparities, if not counteracted by regionally defined instruments.

References

Andersson, I(.E. and J. Mantsinen, 1980, Mobility of resources, accessibility of knowledge, and economic growth, Behavioral Sciences 25 (5).

Nikaido, H., 1968, Convex structures and economic theory (Academic Press, New York). Samuelson, P.A., 1954, The pure theory of public goods, Review of Economics and Statistics 36,

Nov., 387-389.