A dynamic model for regional and sectoral planning in the Federal Republic of Germany

ECONOMICS OF PLANNING Vol. 10, No. 1--2, 1970

Printed in Norway

A dynamic model lor regional and sectoral planning in the Federal Republic o! Germany 1

Rainer Thoss

Institut ftir Siedlungs- und Wohnungswesen University of Mtinster

I N T R O D U C T I O N

G e r m a n y has a l a rge n u m b e r of p l a n s a n d d e v e l o p m e n t r e p o r t s a t

h e r d i s p o s a l to g u i d e h e r r e g i o n a l po l i c i e s . T h e s e i n v e s t i g a t i o n s t o

assess r e g i o n a l g r o w t h p o t e n t i a l h a v e b e e n c a r r i e d o u t in t h e p a s t

f e w y e a r s f o r v a r i o u s d i s t r i c t s of t h e F e d e l a l R e p u b l i c . I t n o w s e e m s

r e a s o n a b l e t o s e t a b o u t t a c k l i n g t h e m e t h o d o l o g i c a l p r e p a r a t o r y w o r k fo r

G i e r s c h ' s p r o p o s a l fo r a " c o h e r e n c e t e s t ''~, a c c o r d i n g to w h i c h , a s e l e c t i o n

f r o m t h e v a r i o u s d e v e l o p m e n t a l t e r n a t i v e s is m a d e w h i c h l e ads t o a m a x i -

m i s a t i o n of t h e soc ia l p r o d u c t of t h e w h o l e e c o n o m y .

i This investigation was carried out at the "Institut fiir empirische Wirtschaftsforsch- ung" in the Division "Volkwlrtschaftslehre und Statistik" of the University of Mann- heim. Cooperation with Professor H. KSnig and his associates, particularly Messrs. W. Gries, H. Mannal, Dr. V. Timmerman and Mr. J. Wolters, led to much encour- agement and many critical appraisals. The writer also had the opportunity to submit parts of this work to Seminars of Professors E. von BSventer (Heidelberg), J. H. Mfiller (Freiburg) and R. Jochimsen (Kid). For all advice and proposals for improvement he remains grateful. ( I twasnot possible to include all of the many improve- ments suggested to him at the time but work is continuing on an improved version of the model incorporating many of these suggestions including an input-output model with 14 sectors. Further details of this later version may be obtained by direct application to the author.) Computations were made at the "Deutsche Rechenzentrum Darmstadt", under the supervision of Messrs. Faber and P6cker (Dipl. Math.), and financed by the "Gesellschaft der Freunde der Wirtschafts- hochschule Mannheim" and of the "Deutsche Forschungsgemeinschaft'. Its German version was published in "Jahrbficher ffir NationalSkonomie und Stat istik, 182 (1968/ 69), pp. 490 and following. Dr Paul A. Pellemans (Namur) and Mr. Malcolm Agnew have provided its English translation and the author expresses his thanks to them.

2 Cf. Giersch, "Das 5konomische Grundproblem der Regionalpolitik', in: H. Jiirgen- sen (ed.), Gestaltungsprobleme der Weltwirtschaft, G/Sttingen 1964, pp. 386 and following.

90 R. THOSS

The advantage of such a procedure is clear: on the one hand, it would be possible to utilize the resuks of the analysis of isolated economic areas for the formulation of concrete targets of economic and social policy and of possible combinations of objectives and means; on the other hand we would obtain from the preliminary results of the coher- ence test, a loose planning framework, along which plans for regional development could be developed. In this way, we would reduce the number of alternative plans to be elaborated, thus avoiding delay in the planning process. Still more important would be the possibility of avoiding the promotion of inappropriate investments, where several regions struggle for the attraction of the same economic sector (for example tourism), or introducing stimulating tools in the wrong sector or region.

Simultaneously with the detailed investigation into the development possibilities of the sub-regions, we should promote the setting up of a quantitative standard-guide for the regional development of the Federal Republic. This would permit the coordination of the regional economic policy actions, the values of which would be used as a test for the consistency of the regional plans.

Before we proceed to the formulation of a spatial standard-guide for the economy, it should be clear that it does not involve the establishment of a spatial structure which once and for all guarantees a maximisation of the welfare of the population or its social product. The growth of population and of capital, the evolution of new production processes and the appearance of new needs require a constant checking of the allocation of the factors concerned, with special reference to the time span and costs, which may require an eventual spatial modification of the production structure. If the standard-guide is to retain its validity, it should be con- stantly checked against reality and up-dated accordingly.

Thus, the determination of capacities recedes automatically into the background in favour of the determination for each period of desirable and possible changes from the initial state. A model by whose help the standard-guide is formulated, should meet the requirements set by Palander for a dynamic explicative spatial model: it should be developed from a given initial spatial allocation of resources and production, and it should contain relationships that take particular account of the time-lags and the market imperfections in the spatial economy. I "A static, or rather a comparative static analysis, under the hypothesis of an in-

1 Cf. T. Palander, "Beitr~ige zur Standortstheorie", Uppsala 1935, pp. 275 and following.

MODEL FOR PLANNING IN FRG 91

f i n i t e ly g rea t a d j u s t m e n t speed , is no t f i t for n o r m a t i v e proposa l s . I n -

d e e d we w o u l d no t expec t t ha t a s t imu lus for m o v e m e n t b r i ngs a b o u t

a n i m m e d i a t e r ea l loca t ion f r o m one r eg ion to ano the r b u t r a the r a fa i r ly

s low t rans fe r , where , p e r i o d af te r pe r iod , on ly pa r t of those t h ings tha t

fa l l u n d e r t h e in f luence of t he s t i m u l u s change the i r pos i t ion" . 1 S u c h

f r i c t ions shou ld be t aken into accoun t in t he d e t e r m i n a t i o n of t he s tan-

d a r d - g u i d e ; consequen t ly , i t is adv i sab le to focus a t t en t ion on the va r ia -

t i on of s tocks r a the r t h a n on the s tocks themse lves . T h e c o r r e s p o n d i n g

s tocks can r ead i ly be ca lcu la t ed if necessary .

T h e s e c o n d r e q u i r e m e n t of t he s t a n d a r d - g u i d e - b a s e d on the cohe r -

ence tes t - is t ha t i t r e su l t s in a se lec t ion of va r ious reg iona l d e v e l o p -

m e n t a l t e rna t ives w h i c h wil l rea l ize a col lect ive economic o p t i m u m . T h i s

i m p l i e s tha t a t t en t i on shou ld be f ixed on m o d e r n ve r s ions of t he t h e o r y

of spa t ia l e q u i l i b r i u m 2 in w h i c h t h e p r o b l e m of t he d e t e r m i n a t i o n of t he

e q u i l i b r i u m cond i t i ons is m a d e a m e n a b l e to n u m e r i c a l t r e a t m e n t b y

a p p l i c a t i o n of t he t e r m i n o l o g y of l inear and non - l i nea r p r o g r a m m i n g .

M e a n w h i l e , an ex tens ive n u m b e r of t heo re t i ca l p r c b l e m s have been

so lved and m a n y o p e r a t i o n a l m o d e l s have been d e v e l o p e d for the h a n d -

l ing of spa t ia l a l loca t ion p r o b l e m s . ~

1 Ibid. p. 281. E. yon B6venter, "Theorie des r/iumlichen Gleichgewichts", Tiibingen 1962; cf. also: L. Lefeber, "Allocation in Space; Production Transport and Industrial Location", Amsterdam 1958; J. Tinbergen, "The Spatial Dispersion of Production; A Hypo- thesis", in: Schweizerische Zeitschrift fiir Volkwirtschaft und Statistik, 97 (1961), pp. 412 and following; by the same author: "Sur un modele de la dispersion g6o- graphique, de l'activit6 6conomique", in Revue d'Economie Politique, 74 (1964), pp. 30 and following; H. C. Bos, "Spatial Dispersion of Economic Activity", Rotterdam 1964.

3 B. H. Stevens, "An Interregional Linear Programming Model", in: ~ournal of Re- gional Science, 1 (1958), pp. 60 and following; W. Isard, "Interregional Linear Pro- gramming: An Elementary Case and a General Model", in: ffournal of Regional Science, 1 (1958), pp. 1 and following; by the same author: "Methods of Regional Analysis", Cambridge, Massachusetts, 1963; L. N. Moses, "An Input-Output Linear Programming Approach to Interregional Analysis", in: Harvard Economic Research Project, Report on Research for 1956/57, Cambridge, Massachusetts, 1958, pp. 122 and following; R. E. Kuenne, "The Theory of General Equilibrium", Princeton, 1963, pp. 395 and following; H. K. Schneider, "Modelle ftir die Regionalpolitik", manuscript of a communication to the meeting of the Committee of Economic Policy of the "Gesellschaft f/Jr Wirtschafts- und Sozialwissenschaften", on April, 9th and 10th, 1965 at Unkel on Rhine. In addition, see the articles in "Papers and Proceedings of the Regional Science Association"; J. R. Boudeville, "An Operational Model of Regional Trade in France", 7 (1961), pp. 176 and following; J. Sebestyen, "Some

(cont. next page)

92 R. THOSS

Difficulties are, however, greater in practice if an optimization model is applied to the solution of actual problems of regional planning because the lack of statistical information generally involves a significant simplification of its handling. In this respect, the present proposal of a consistency test for planning of regional development should not be viewed as the ultimate quantitative conception of the author with regard to regional policies in the Federal Republic. There is no doubt that the spatial allocation of the economic activities proposed here rests on too many simplifying hypotheses to be regarded as a binding standard- guide. And this is even more so since most of the constraints used in the model contain subjective appraisals by the author, since no better information was available. These appraisals must be replaced by the quali- fied iudgement of experts or the resuks of political decisions, before we can speak of a binding frame of planning.

The purpose of this article is rather to present, with the aid of a sufficiently realistic example, a method that could be followed for the coordination of planning and regional economic policy. The author hopes that it will evoke constructive criticism which will lead to a synthesis resulting in a conceptualization of the spatial allocation of the Federal Republic.

This is particularly relevant for the non-economic constraints (nature protection, social rights, etc . . . . ). Precisely in the field of regional allocation policy, where the number of variables that are conceptual- ized in terms of utility only is especially high, the lack of an adequate social preference function is felt acutely? As before, we must be con- tent with the maximization of social product or consumption. We can, however, in the quantitative approach, take in a large number of non- economic variables as constraints if their value can be considered

(cont. from page 91) Thoughts on a Spatial Model for Development Purposes", 12 (1964), pp. 119 and following; A. P. Hur ter and L. N. Moses, "Regional Investment and Interregional Programming", 13 (1964), pp. 105 and following; E. O. Heady, "Discussion of Some Particular Programming Models", 13 (1964), pp. 121 and following; V. S. Dadayan, "A Model of Interregional Relationships in a Single-System Opt imum Plan of the Economy", 14 (1965), pp. 53 and following; T. KronsjS, "Iterative Price and Quan- tity Determination for Short-Run Production and Foreign Trade Planning", 14 (1965), pp. 63 and following; J. G. Waardenburg, "Space in Development Planning", 18 (1967), pp. 91 and following.

1 For discussions in that field, see, for example: G. Giifgen, "Zur Theorie kollektiver Entscheidungen in der Wirtschaft" in: Jahrbucher fiir National6konomie und Sta- tistik, 173 (1961), pp. 1 and following.

MODEL FOt~ PLANNING IN FRG 93

as given "by the general demand of the population" or "by political decision".

To demonstrate this, a series of these conditions were introduced in the model with the implication that, in their provisional form, they considerably limit the explicative value of the results obtained so far.

The long-term approach, which is the basis for the concept of development potential, corresponds to the maximization of the social product (its utility) over a longer period (10 to 20 years), or else at a given time in the future. This problem can be considered as solved to-day from the point of view of the theory of economic policy, since we may link it to (or with) the results of growth theory, in particular to considerations about the optimum rate of savings with a limited planning horizon. 1 If we had time series on factor input and the corresponding output in a

�9 satisfactory regional and sectoral disaggregation at our disposal, then by analogy to current growth theory procedure, we could determine the production function of every single region taking into account explicitly its peculiarities in terms of advantages in location and industrial structure. From these functions, we could determine the optimum volume of investment and the optimum production programme - taking into account the constraints required by each region, e.g. the optimum population density. Sadly enough, the German regional statistics are, as yet, in- adequate to permit this precise procedure to be followed. Figures such as the gross domestic product of towns "Kreisfreien Stfidte" and rural districts, "Landkreise" of the FedeIal Republic are only available for three years, and there is only a coarse disaggregation into four sectors measured for the requirements of state planning.

For the moment, in the absence of long time series, we are dependent on cross-sectional methods for the estimation of production functions. This implies, for example, the hypothesis that from the input-output relations of cities, we can project the same relations for small towns; in other words, that the relation between factor input and output for cities may be representive for all regions of the Federal Republic. It implies a sub-optimization of the procedure and a considerable reduction in the validity of the results. The author hopes that the growing awareness of

1 Cf. F. P. Ramsey, "A Mathematical Theory of Saving", in: Economic ffournal, 38 (1928), p. 543; J. Tinbergen, "The Opt imum Rate of Saving", in: Economic ffournal, 66 (1956), pp. 603 and following; S. Chakravarly, "Optimal savings with Finite Planning Horizon", in: International Economic Review, 3 (1962), pp. 338 and following. Numerous methodological examples are found in K. A. Fox, J. K. Sengupta and E. Thorbecke, "The theory of Quantitative Economic Policy", Amsterdam, 1966.

94 R. THOSS

the value of regional statistics for the formulation of economic policy will lead to the development of this field of official statistics and thus make possible more refined analyses.

On the basis of the above-mentioned hypotheses, the data on the use of capital and labour, and the domestic product of one period in different "Kreise" - town and country - will allow the determination of the most likely relationship between these three quantities with the help of regres- sion analysis. Differentiation of the function thus calculated gives an estimate of additional product caused by a factor increment of one unit in the different regions, i.e. from a general production function one derives marginal products of the factors in each region, on the basis of the quantities of factors which are available there.

If in addition, two or more periods are compared, elements for the estimation of the influence of technical progress may be obtained. The marginal productivity of the factors of the different "Kreise" are used as coefficients in the objective function of an optimization model that enable the computation of the modification to the quantities of factors in the various regions that maximize the growth of the social product during a given period. From period to period, the optimal regional distribution of gross investment and the optimal migration of the factor labour can be determined if the model is so extended as to establish its relationship to growth theory.

In each individual period, the total available stocks of labour and capital should be allocated between regions and sectors so that social product (taking into account other socio-economic goals) increases. If several periods are observed consecutively, population growth and, ceteris paribus, the growth of labour potential must also be taken into account. Thus the number of workers for which the optimal location of workplace and residence have to be found, will increase from one period to the next. The same is true for the allocation of capital investment. Given that in a growing economy a part of output is used for the public and private formation of capital, the stock of capital will increase during each period by the amount of net investment. For both new investment and replacement of material depletion, decisions on location must be made. Even though a re-location of existing means of production will be disregarded due to its high cost, a fixed percentage of output for private and public investment means that a growing amount of gross investment will have to be taken care of by the national planning agencies. Further, in a dynamic analysis we must take account of the fact that the marginal productivities of the factors of production may change with time. Firstly,


the influence of technical progress, which results in a general increase in the productivity of factors, has to be estimated. If it is assumed that the production functions are non-linear, the ranking of marginal productivities will also alter if the regional and sectoral structure of factors change. If for a period t in region r it is shown by the optimization process of the model that the use of a factor should be increased, the marginal productivity of this factor will then diminish in that region according to the usual assumptions and thus marginal productivities in period t + 1 must be calculated anew.

A model constructed along these lines includes all the main elements of a simple model of growth: the growth of labour potential, the accumulation of capital and the pace of technical progress. They will be re- cognized in part in the objective function [13], in part in the constraints [12] and [23] and partially in the definitions [28] to [37]. With the help of the latter, through an iterative process, we can make step by step adjustments of the constraints and the objective function which become necessary through the growth of the social product and labour potential in the following periods. 1 A sequence of optimization models is thus obtained, i.e. one for each unitary period t, that describes the growth process and the required structural modifications. The resuking system of equations and inequalities can be considered as a combined model of growth and allocation. In the following, we shall first describe the mech- anism of allocation for one period (objective function and constraints), and then the development of the system over time.

2. AN E X A M P L E F O R T H E F E D E R A L R E P U B L I C

2.1. Regions As yon B6venter has demonstrated, many difficulties are to be ex-

pected if a decision is made to treat space on a continuous basis when the spatial equilibrium of the plan is established, i.e. when "from the point of view of spatial adjustment, infinitely small locational variations are possible". 2 It is thus advisable to start with a discrete model with respect to space. Subsequently, the number of possible locations could be increased so that it tends to a continuum. This does not exclude the possibility that in the discrete model, each of the points under consideration may be looked upon as representative of its immediate environment,

1 For the method, cf. R. H. Day, "Recursive Programming and Production Response", Amsterdam, 1963; K. A. Fox, J. K. Sengupta and E. Thorbecke, ol). cit. pp. 185-196.

2 Cf. E. yon B6venter, op. cit. , p. 96; cf. also pp. 108 and following.

~6 R. THOSS

its "region". In this way it becomes possible to structure, f rom the be- ginning, the areas under review, so as to cover the entire space. A subsequent increase of the number of possible locations may be obtained simply by an additional subdivision of the regions initially examined.

The model described here contains one hundred possible locations; due to their size they will be called "regions", although they rather have the features of "planning areas", for which the optimal population and product ion means is to be calculated.

For the definition of regions, it was aimed to follow the boundaries of the economic regions of the Federal Republic and, where possible, of the regions of the operative state planning programmes. 1 Th e reason for this is that, from the start, with the formulation of a model which is later planned to be used as a tool of regional planning, all its possible appli- cations should be born in mind. Modifications of the boundaries of the economic regions can be interpreted as a revision of population and of the production means of the neighbouring planning units if they are small enough, otherwise an adjustment of the boundaries of the programme regions is necessitated.

In addition, it was aimed to build regions with equivalent intra- regional transportations costs, which allows us to delete them in many cases. Some regions have been combined with neighbouring regions, others were subdivided; map 1 shows the result of this selection. 2

2.2. Sectors

In the model, the economy of the Federal Republic was subdivided into the agricultural sector (I), productive industry, trade and transportation (II), and Government and other service industries (III) . In detail, this delimination of the sectors corresponds to that of the "Ar- beitskreis Sozialproduktsberechnungen der Lfinder ''a with the exception that the whole employment groups falling in the domain of "productive industries" and "trade and transportation" were grouped together. This

1 The author is grateful to Messrs D. Bartels and O. Boustedt for their advice, and also the Home Ministry of the "L~inder Schleswig-Holstein, Nordrheim-Westfalen, Hessen, Rheinland-Pfalz and Baden-Wfirtemburg". After completion of the map, the number of regions had to be reduced to 100 for computations. This required a regrouping of some regions (cf. Table 2). Cf. "Arbeitskreis Sozialproduktsberechnungen der LSnder", H. 1. Das Bruttoinlands- produkt der kreistseien St~dte und Landkreise in der Bundesrepublik Deutschland 1957 und 1961. Wiesbaden 1964.


aggregation was necessary since the information about the stocks of capital for each region is only available for this collection of economic activities.

These three sectors together produce the gross domestic product of the Federal Republic in which are inserted the factors Labour, Capital and Land. To the level of production of each sector, flows income to the owners of these factors, which is utilized for the consumption of each sector, as well as the private and public formation of capital.

With regard to the determination of revenue in the sectors I and II on the one hand and sector III on the other hand, there exists a major distinction. Agriculture, Trade, Industry and Transportation produce a directly measurable product whose level may be related directly (through the production functions) to the corresponding factor inputs. The public sector supplies services which cannot easily be measured. When the social product is calculated we use as an indicator for the contribution to the gross domestic product, the sum of wages and salaries paid to the person- nel employed in the public sector. The level of the contribution of the Government thus becomes a function of the wage and salary level and of the number of people employed in the public sector; this again de- pends upon the level of salaries in the private sector and the growth of population. Consequently, the model described calculates the level of the contribution of the sectors I and II to the gross national product (= revenues of the factors) by using the production functions; income and employment in sector III depend in turn upon the income and employment of the two other sectors.

2.3. The objective function

For each period, the optimum allocation of the factors is determined by a linear programme. As the dynamic character of our statement of the problem leads to certain special features in the formulation of this programme, we shall first of all explain the basis of our procedure.

Let us assume that in sector I the production is the result of a com- bination of the factors Labour (L), Capital (C) and Land (N), in sector II of the factors Labour (A) and capital (K). Sector II is, moreover, dominated by neutral technical progress. The production in region r at time t is then described by the production function

urt=F(Lt, Ct, Nt) for sector I, and [1]

V~t = G(A~t, K't, t) for sector II. [2]

98 R. THOSS

If h~ is the relationship of the prices of the goods produced in sectors I OF

and II and ~-~ (r, t) is the partial derivative of the function F w.r.t. L in

region r at time t (etc.), the growth in production is given by the total differential w.r.t, t ime

�9 �9 �9 O F ( r , t ) L ~ + O F t) C~ OF yr= U~+h; V;=bZ yS(r, + b-~(r, t)N~+

, rOa (r,t)A~ + ,OC t)K~ + .rOa +nt3- ~ h,o-~ (r, hi-O- t- [3]

Summation over all regions and after transition to discrete variables gives the total increase of production - in as much as it is not imputable to technical progress - as a function of the variation of the factor inputs in all regions

loo OG 1~176 OF OF A Y , - ,=2E h~t -Ot ~ r~l~-OZ_, (r, t )Ar t + -~ (r, t)AC~t + ~ (r, t)ANt +

OG o c + ht-z-7~ (r, t) AA~ + h; (r, t) A K~J [4]

OK o ~ /

For small variations in the quantities of factors, the partial derivatives of [4] may be considered as constant, although [1] and [2] are not ne- cessarily linear. We want, after some modifications, to use [4] as an objective function of a linear programme that will enable us to calculate from one point in time to another the modifications of the quantities of factors which, taking into account certain constraints of the economy as a whole, will ensure a growth of production as high as possible.

2.4�9 Activities Similarly to the use of marginal productivities as coefficients of the

objective function, the activities of the model have to be defined as in- crements or diminutions of the quantities of factors in their different uses (sectors) and regions. We will examine each of the factors Labour (unit: 10,000 workers), Land (unit: 1,000 ha) and Capital (billions of DM), and for each factor identify its type of use: for Labour and Capi- tal all three sectors, for Land the use of land for building or for agriculture. In addition comes the income of the people employed in Sec- tor III .

MODEL FOR P L A N N I N G IN FRG 99

At first sight, it seems almost impossible in an optimization model to calculate the optimum variation in the factors rather than their absolute level, because under normal conditions negative alterations of the quantities of factors must also be expected, for example emigration from some regions, whereas linear programming only allows positive levels of activities. This difficulty can be avoided when we introduce alongside the increase of a variable, its decrease as separate activity, if we want to investigate these negative variations. In the objective function, these decreases will be represented by negative coefficients. For the variables handled in our problem, only Labour in all three sectors, Land for agricultural purposes and the income of sector III need be treated this way. For building land and stock of capital of the three sectors, it seems reasonable not to accept decreases, since it would imply high costs for demolition.

If one designates a unitary increase or decrease by ~X~, we get according to the above, for each region at time t, 13 activities in total i.e.

1X~ and 2X~ for increase or decrease of employment in industry (AA~) 3X~ and 4Xt for increase or decrease of employment in agriculture (AL'~) sX~ and 6X~ for increase or decrease of employment in sector I I I (ADt) 7X~ for increase of constructed area (ABt) sX~ for decrease of agricultural land (ANt) 9X~ for investment in industry (AK~)

10Xt for investment in agriculture (ACt) 11Xt for investment in sector III (AQ~)

12X~ and ,aX~ for the increases or decreases of incomes of people employed in sector III (AE't)

In a model with 100 regions, this leads to a total of 1,300 activities without slack variables. No negative values are retained, whence

kX~ > 0, r = 1 , . . .100

k = 1 . . . . 13

t > O .

2.5. Marginal productivity of factors The objective function that has to be maximized - here the increase

of the contribution of sectors I and II t o the gross domestic product for

100 R. THOS$

period t - is expressed as a linear function of the levels of activities. The activities must thus be priced by the contribution that a unit of each factor realizes in the growth of the gross domestic product of those sectors in which they are applied. The marginal productivities of these production factors have to be calculated now for each use and region considered.

When the functional relations are specified, we may assume that the factors can be partially substituted one with each, but also that limiting relationships can be observed. We wish to assume that the production function of agriculture is substitutive in labour, capital and agricultural land, that of sector II in capital and labour. For the use of land for building and for personal and material infrastructure, it will be assumed that the limiting condition is reached. This implies that an isolated increase of the built-up area or of capital and labour in sector I I I has no effect on the magnitude of the objective function - the output of agriculture and industry - but that an increase of the use of these factors (particularly the number of people employed) should always be accom- panied by an increase of the limiting factors. Under these circumstances we can dispense with the introduction of a special production function for sector I I I since in the maximizing operation this sector is simply not included as it consists predominantly of public services and a maximization of the gross domestic product, including the services of the ad- ministration, cannot be a target of an optimization problem in economic policy.

When the marginal productivities are calculated, we will start from the functions where the substitution-elasticity is not fixed at 1. For sector 1, it proves to be the Cobb-Douglas function

lg U t = - 1.2714 + 0.413 lgL t + 0.118 lgC t + 0.441 lgNt [5]

R =0.8266 (0.085) (0.107) (0.112) r = 1 , . . . 100

which is the best approximation. Here Ut, Lt, C t and N~ represent the contribution of agriculture to gross domestic product and the use of labour, capital and agricultural land in 1961. Attempts to estimate parameters of a CES function with three factors, as proposed by Scheper:, have failed so far. The relationships of another year have not been analyzed either since the figures for the stock of capital and employment

1 Cf. W. Scheper, "Produktionsfunktionen mit konstanten Substitutionselastizit~iten", in: ffahrbiicher fiir Nationalb'konomie und Xtatistik, 177 (1965), pp. 1 and following.


were not available. For sector II , we have analyzed a CES funct ion with neutral technical progress

V't=z~eXt[~(K~t)-P +(1--6) (A't)-~] p r = l , . . . 1 0 0 [6]

t = 0 for 1957

t = 4 for 1961

V~, K~ and A t represent again production, capital and labour. According to the procedure of J. Kmenta 1 we obtain the relationship

lnV', = - 2 . 5 8 1 +0.061 t+O.2231nK~+O.8571nArt [7]

R=0 .9918 (0.005) (0.028) (0.034) r r 2

- - 0.119 [lnK t - lnAt] ,

(0.033)

f rom which the following values of the parameters were derived

x = 0 . 0 7 6 4=0.061 6=0 .207 ~o=1.339 #1.080

We should notice that the parameter # is significantly higher than 1, which indicates clearly that for the period under review in the industrial and trade sectors, internal and/or external economies were realized. Similar results are obtained when two years are analyzed separately (after elimination of the "technical progress" component), even if we take, in- stead of the 100 regions analyzed here, the disaggregated data for each "Stadt and Landkreise" of the Federal Republic. Similar results were obtained by Hildebrand and Liu for the U.S. 2 These authors have estimated product ion functions for 17 sectors of the American Industry in 1957 on the basis of cross sectional data for production and factor inputs in the various states of the U.S. For almost all sectors, they were able to deduce returns to scale which are often considerably higher than those calculated for the Federal Republic. A separation of this effect between internal and agglomeration economies is not possible there either, due to aggregation. We may, however, assume that agglomeration economies contribute significantly to this result.

1 Cf. J. Kmenta, "On Estimation of the CES Production Function", in: International Economic Review, 8, 1967, pp. 180 and following.

2 G. H. Hildebrand and Ta-Chung Liu, "Manufacturing Production Functions in the United States, 1957. An interindustry and Interstate Comparison of Productivity", Geneva and New York, 1965.

102 R. THOSS

If equation [5] is partially differentiated w.r.t. L, C and N and equation [6] w.r.t. A and K, we get the marginal productivities of the three factors in sectors I and II of region r at time t, as a function of the factor inputs and time, for example:

U r 0 118 r - 0 587 r 0 441 from [5]: ~-~(r,t)=O.O22(Ct)" (Lt) �9 (Nt)" r = l , . . . , 1 0 0 [8]

OV from [6]: ~-~ (r, t)=0,065 e~176 (Kt)-l-~9 +

+ 0.793 (At) -1.~9] -1.o~o (A~)-2.~39, r = l , . . . ,100 [9]

The marginal productivities of the other factors are obtained similarly. If we substitute the observed factor quantities for each region and the

corresponding value of t into these partial derivatives, we obtain esti- mations for the production forecast of one incremental unit of the factors in the different regions. Through the current calculation of the derivatives and the corresponding change of value of t the effect of the modifications in the relations of factors and the improvement of the efficiency of these factors is taken into account. Step by step, the coefficients of the objective function are adjusted to accord with the factor resources of the regions - which vary during the execution of the plan - and to the respective position of technical progress. The error made, when non-linear expressions of productivity are used as constant parameters of the objective function, will be minimal if the variations are small.

Assume that A; and K~ are the input quantities of region r at time 0 for the calculation of the marginal productivities, that A g~ is the estimated output change obtained by application of marginal productivity calculated at time 0, that A g ~ is the actual output increase due to the quantities of factors d~o + A A ~, K~ + A K ~, and further let us limit A A ~ and A K r to e% of the final values; then we obtain as an expression of the relative error of the marginal product

A N - - A N , =1 e# [10] A V (1 + s) ~ - 1

In an estimation using the marginal productivities of equation [6], an error of 0.4% due to linearization would be made for a variation of factors of 5 %. The fixing of an upper limit for the variation of factors for


each period would guarantee that the errors caused by linearization of the objective function will be strictly limited. It is advisable, however, when changes of the allocation of factors is planned, to take into account the "inertia of obsolete spatial structures". 1 This will be done most ap- propriately if, from the start, flows of factors to their most efficient place of use, are limited to specified realistic values.

Regional dispersion of marginal products is reproduced in Table 1. It shows that the marginal productivity of labour in 1961 was highest in the Ruhr, that of capital in the "S/idpfalz". If we had no other relations to consider at the time of planning, the policy of the optimum development of the Federal Republic would have consisted of stimulating the immigration of workers and the erection of additional productive facilities in these two regions, since the increase of expected revenue would be highest there, Only if, by intense use of labour or capital, the marginal productivity of these two factors were reduced, should we stimu- late other regions.

2.6. Transportation costs

Each modification of the spatial structure of the economy has con- sequences for the level of resources required for transportation. A heavier concentration of economic activity at a few points would probably lead in total to a saving of transportation costs, a more uniform dispersion of production to an increase in these costs.

In general, interregional optimization models take these facts into account by defining the possible transportation flows between regions explicitly as activities. The levels of optimal production within the regions and the optimal traffic between them are determined simultaneously, thus also determining the factor requirements for these transports. The disadvantage of this procedure is that the number of possible supply lines for each product increases by the square of the number of regions. This implies that for most research, very large regions are built in order to reduce their number. The finer precision of results obtained by the introduction of transportation activities is neutralized by the elimination of greater regional differentiation.

If we want neither to excessively increase the level of spatial aggregation nor to renounce consideration of the advantages and disadvantages

1 H. J/irgensen, "Antinomien der Regionalpolitik", in: H. Jtirgensen (ed.), Gestalt- ungsprobleme der Weltwirtschaft, op. cit., pp. 405-406.

1 0 4 R. THOSS

Table 1. Marginal productivity of factors

1 2 3 4 5 6 7 8 9 + 1 0

11 12 13 1 4 +1 5 16 17 18 19 20+21 2 2 + 2 8 23 2 4 +2 5 26 27 29 30+31 32 33 34 3 5 +3 6 37 38 39 40 41 42 4 3 + 4 4 4 5 + 4 6 47 48 49 50 51 52+53 54+55 56 57 58 59 6O 61+63

69 77 84 89 96 76 76

133 96 82 83 56 64 76 77

112 65 84

116 101

69 84

122 82 81 78 82 95 93

102 137 100

70 129

95 75 95

133 75 64 96 81 73 69

122 62 86 94 92 64

277 87 256 80 182 80 232 81 128 81 286 73 257 80

32 48 95 66

269 69 179 68 432 66 387 71 205 67 285 65

88 64 306 72 141 70

54 65 45 67

392 61 148 64

46 68 250 60 230 66 320 54 237 63 173 64 240 56 116 62 49 64

135 79 374 61

73 63 240 61 306 62 160 62

30 58 364 44 371 56 164 54 209 51 370 48 331 56

86 53 440 50 254 52 138 48 181 49 391 50

cq~ Nq~ r

258 0.87 6 2 + 6 4 221 0.98 65 231 0.95 66 228 0.93 67 287 0.92 68 260 1.02 69 259 0.94 70 128 1.90 71 323 1.07 72+73 268 1.06 74 260 1.09 75+76 264 1.15 77 249 1.08 78 254 1.13 79 254 1.12 80+85 228 1.16 81 234 1.07 82 267 1.10 83 266 1.13 84+86 238 1.15 87 258 1.23 88 223 1.18 89+90 244 1.11 91 257 1.25 92+93 259 1.18 94 285 1.31 95+96 245 1.10 97 260 1.12 98 248 1.30 99 217 1.22 100 289 1.14 101 240 0.97 102 292 1.16 103 229 1.19 104 235 1.29 105 278 1.22 106 251 1.23 107 261 1.23 108 337 1.59 109 252 1.35 110 270 1.33 111 281 1.42 112 316 1.48 113 248 1.35 114 267 1.33 115 303 1.42 116 263 1.41 117 296 1.48 118 295 1.40 119 316 1.39 120

Kq;

59 492 90 184 73 331

111 61 63 518

118 59 106 129

65 421 81 333

112 95 107 141

86 231 77 340 85 282 83 200 88 185 95 131 82 286 81 241 66 360 85 292 89 152 77 325 86 213 66 288

105 134 75 296

104 29 62 438 56 470 58 400 97 127 56 444 62 485 98 53 72 292 81 223 74 282

108 18 89 93 70 338 94 171

132 20 77 217 92 147 84 173 86 157 78 96 97 75 82 195

Lq; cq;

58 233 49 308 57 308 48 316 45 345 55 423 51 321 52 270 54 212 40 372 48 248 61 197 44 296 54 265 65 190 46 309 53 249 56 230 60 195 54 322 44 363 56 272 56 247 62 231 62 210 53 240 64 220 63 218 60 190 63 222 63 239 63 228 59 239 56 248 54 291 62 170 62 191 66 177 65 191 59 215 61 167 64 180 70 197 69 180 66 226 68 181 70 222 73 282 64 198 64 219

1.36 1.42 1.24 1.48 1.49 1.17 1.42 1.35 1.37 1.60 1.53 1.29 1.54 1.30 1.24 1.55 1.40 1.33 1.35 1.31 1.50 1.31 1.31 1.26 1.29 1.42 1.22 1.27 1.31 1.25 1.21 1.23 1.29 1.29 1.34 1.31 1.27 1.22 1.23 1.30 1.38 1.27 1.13 1.18 1.15 1.20 1.11 1.06 1.25 1.22


of the regions from the point of view of transportation costs, it is worth- while looking at the work of Harris, Dunn a.o. who have attempted to classify the different sub-regions of an economy in relation to their market proximity and their probable transportation advantages. 1

If we attempt to estimate the loadings of transportation costs tied to a given production at different possible locations, we may assume, in the absence of more precise data for the locations of supplies and the quantities to be transported, that the allocation of deliveries between the different potential customer-regions is directly proportional to their economic activity (measured by the level of their gross domestic product). If y s + E s represents the gross domestic product of the customer-region s, d rs its distance from the region under study r, we may expect that the transportation costs in r are proportional to

IOO

pr= E (Y'+E~) d" [11] S--|

If this value is calculated for the different possible locations, the regions may be compared from the point of view of their relative transportation advantages.

Map 2, in the appendix, summarizes the value of P" in the different parts of the Federal Republic. 2 It shows that if a location on the Rhine Plain between Darmstadt and the Dutch border is chosen we can expect approximately equally low transportation costs, and that with increasing distance from the Rhine (Pr~98), the probable costs rise until, for example in the Northern part of Schleswig-Holstein (Pr=190), they are almost twice as high as in the "Regierungsbezirk" of K61n.

A transfer of production to regions with higher productivity would only be justified if the expected increase in production were larger than the eventual growth of the corresponding transportation costs of the entire economy. An adequate formulation of the model must consequently take care that the optimization in fact guarantees an improved

* Cf. Ch. D. Harr is , " T h e marke t as a Fac tor in the Locat ion of I n d u s t r y in the U n i t e d States" , in : Annals of the Association of American Geographers, 44 (1954), pp . 315 a n d fol lowing; E. S. D u n n , " T h e M a r k e t Potent ia l Concep t and the Analys is of Locat ion, in : Papers and Proceedings of the Regional Science Association, 2 (1956), pp. 183 a n d following.

T h e values pr have been calculated s imply on the basis of the "Regie rungsbez i rke" . T h e regions wi th 95 ~ pr < 100 have been d r awn clearly. T h e s u r r o u n d i n g reg ions were classified into five g roups according to the level of pr.

106 R. THOSS

supply to the population, wkhout an extension of the transportation sector equal or superior to the increase of output. This is obtained either by constraints - in which the different consumption of factors for freight transport in the individual regions is specified -- or by a correction of the expected social product per factor-unit, for the part taken by freight transport. 1

We have preferred the second solution which allows a simplification of the computation. The share of freight transport of the domestic product may be estimated by the ratio of the turnover of freight transport

to the total turnover of transportation ~v in 1962 and by the share

of transportation business in the production of sector II ~ ; the al-

location of this share for the regions is made with the help of the relation

/ 100

By this method, we obtain the share that is to be subtracted from the expected output of region r if a maximization of the production of goods alone is aimed at. ~ The marginal productivities of labour and capital in the sectors I and II are thus to be multiplied by a factor

ff = l Ug Yv P~ r = l , . . . l O 0 [12] Uv Y }2p ~

r

The objective function to be maximized for each period is

r r r "-''-" *X * [131 YAV' t+ ~,A U~ + (1 -2 . )2 V~= ~, ~ p iq t iX t - 2. 2.flti t, r r r r i r ]

r = l . . . . 100 i=1 ,3 ,9 ,10 j = 2 , 4 , 8

where the expressions kq~ represent the marginal productivities of the factors at time t, evaluated in the industry by the price relation h~. 3

1 I am grateful to Professor yon BSventer for t h i s p o i n t . Modif ica t ions in the spat ial s t ruc tu re of the o u t p u t alter the advantages of location

of regions. T h a t effect was not taken into account . Cf. equat ions [8] and [9]. For the variable wi th index j , it per ta ins to activities in decline for wh i ch the posit ive levels represen t a p roduc t ion decrease.

M O D E L F O R P L A N N I N G I N F R G 107

2.7. Constraints

The maximization of the function [13] will be made under a number of constraints, in which the targets of regional planning will be specified as well as the initial state from which we start.

It involves basically: - s e c u r i n g sufficient open space for leisure and for regeneration of

water and air; - p r o v i d i n g the population with a minimum material and personal

infrastructure; -p reven t ing excessive income differentiation between the different

regions and sectors; - determination of the means available for output and infrastructure. The designation of the equalities and inequalities in which these ob-

jectives and means are specified, as constraints gives rise to frequent mis- interpretation because it implies that it handles secondary things. Con- cretely, this would mean that the maximization of the social product would be looked upon as the main issue, the accomplishment of the other targets being secondary.

In fact, exactly the opposite is true: the direct fullfilment of the wishes of the planners, formulated as constraints, is a binding prescription of the programme; it is only after satisfaction of these conditions, and if means remain available for allocation decisions, that a coordination from the point of view of the maximization of production is followed. The solution of the system does not only give a spatial distribution of the factors leading to the most rapid possible growth, but first of all that one which meets the objectives set in the constraints. From this point of view, the constraints described in the sub-sections that follow are particularly relevant.

2.8. Labour forces

For an optimal distribution of production, we should keep in mind that the annual growth of labour potential is limited. Certainly the labour forces are mobile at the interregional and intersectoral levels, but in total the increase of labour potential is determined by the activity level, the population growth rate, the initial population and the immigration at time t. The immigrations in industry, agriculture and services may at most exceed emigrations to the extent of the total annual growth of labour potential (Rt).

108 R. THOSS

100 100 i=1, 3, 5 2 2~X~--~, ~jX~R~+I--R, j = 2 , 4 , 6 [14J

r = l i r = l j

With a new distribution of population, it must be taken for granted that the requirements of Sector III are sufficiently guaranteed in each region. This means that the modification of industrial and agricultural population entails a proportional modification of employment in the tertiary sector, i.e.

yr Yr>bt[~iXrt__Ejyt r] i=1 ,3 ,5 j = 2 , 4 , 6 [15] 5~xt 6 ~ t __ i S"

where we establish b t =0.213 (average 1962 for the Federal Republic). Another inequality establishes an upper limit to the modification of

population per region at a maximum of 3 per cent, i.e. three times the national average. Regions with large protected areas for nature, landscape or water resources as well as overcrowded regions - are only allowed a growth equal to the average of the Federal Republic. With the introduction of these constraints two objectives are realized. First, we are able to demonstrate how easily, even in a complex model, conditions specific to each region may be respected by the introduction of different limits whose choice is placed in the care of the expertise of the planners. Second, we must take into account, in the sense of recursive programming, the fact that a great deal of resistance stands in the way of the realization of planning objectives, especially the fact that the movements of necessary factors to achieve the optimization require time and cannot be realized in the necessary quantities within a single period 1.

1xrt--2xrt-]-3xrt--4Xrt~krt(Art-]-Lrt), Y = l , . . . 1 0 0 [ 1 6 ]

The limitation of the increase of workers in industry and agriculture at a fixed percentage of the potential of the preceding period (At_i, or Lt_l) suffices to limit the population density, since employment in the tertiary sector and the remainder of the population are in constant relation with the workers in industry and agriculture.

We thus have

4Xt ~ 0.005 L~, r = l , . . . 1 0 0 [173

This corresponds to the effective rate of agricultural emigration.

1 F o r t he d a t a kt , see t he A p p e n d i x .


2.9. Land requirements

Computations by Winter have established that the expansion of a com- muni ty of 25,000 to 50,000 inhabitants requires an area of 505.1 ha. 1 T h e rise of this value and of the activity rate of 0.47 gives the following rela- t ion for building space for a modification in population:

X r r 7 X ~ 0 . 4 2 9 [ ] ~ i t-]~jXt] r = 1 , . . . 1 0 0 , i = 1 , 3 , 5 , j = 2 , 4 , 6 [18] i j

On the other hand, as the space of a region is fixed and as the forest areas of the Federal Republic according to current opinion should not decrease significantly, an increase of construction in a region implies a decrease of the agricultural areas

7Xt-- 8Xt = 0, /~=1 . . . . 100 [191

In any case, building in a region should at most be pushed forward t o the point where a specific minimum of open space is preserved. For ou r purpose, it seems appropriate to make the required area of open space depend on the size of the population, with an absolute minimum of open space per head such as that which lies at the disposal for the inhabitants of the city-states. Here also, the possibility of making a larger regional differentation should be considered since the standard of the city- states are already deemed insufficient. In the first place, we will have to ~llow in some parts of the Federal Republic a common utilization of open space by the inhabitants of several regions. The re is, however, at the moment little clarity concerning the distances that can reasonably be accepted for this purpose. Temporari ly, we will accept the requirement that, in each region, after a population increase - brought about in the course of planning - the forest area (W ~) and the agricultural area (N~+I) :may not fall short of a minimum relation to the number of workers in :the sectors I I (A~+a) and I I I (D~+I):

Wr+N~t+l ~ 0.4 (A~+I + Dtt+,), r = l . . . . 100 [20]

T h e value 0.4 comes from the relation of open space per inhabitant in the Hanseatic towns in 1960, taking into account their rate of activity.

1 Among them transportation spaces 88.4; open spaces 63.2; spaces for collective equipment 85.8; industrial and other activities spaces 101.0, housing spaces 166.6. Cf. C. Winter, "Was kostet der St~idtebau ?", in: Geminniitgiges Wohnungswesen, 11 (1963), p. 355.

110 R. THOSS

2.10. Capital

As with demand for labour, the demand for private capital is determined by the objective function. For investments that do not contribute directly to the private product, a constraint is required that assures a sufficient material infrastructure, as was already the case for the per- sonnel infrastructure.

A statement on the probable requirements of infrastructure for capital in connection with an increase of population is exceedingly difficult. On the one hand, in sparsely populated areas we must reckon with considerable expenditure per head due to indivisibility of capital goods which implies that numerous facilities must be created that, initially, cannot be utilized at full capacity. On the other hand, it is indeed clear that in the overcrowded areas there are extensible and, in part, not fully utilized stocks available for public facilities but, in particular, the increasing scarcity of the factor land leads to significantly increasing expenditures with population increase. 1 Recently J. Dahlhaus and D. Marx have pre- sented a statistical research paper that supports our last argument. 2

Until the ultimate solution of the debate, i t appears reasonable to work with a "need-per-inhabitant" which would be the same for each region. Modifications with the lapse of time and regional differentiation may be handled without difficulty. From this, and by a differentiation of the various components of the material infrastructure, we can dedttce criteria for the decision between different public investment policies from the point of view of their efficiency. ~

On the quantitative research undertaken, the calculations of Winter appear to be the most adequate for our purpose since they refer especially to the re-settlement of labour. 4 Together, we must reckon with a

1 A summary of the arguments is found in B. Dietrichs, "Aktive oder passive Sanier- ung ?", in: Mitteilungen des Deutschen Verbandes fiir Wohnungsbau, Stiidtebau und Raumplanung, 1965, Volume 4, p. 5.

2 Cf. J. Dahlhaus and D. Marx, "Fl~ichenbedarf und Kosten, yon Wohnbauland, Meindebedarfseinrichtungen, Verkehrsanlagen und Arbeitsst~itten", Hannover, 1968; cf. also, from the same author, "Raumordnungsprobleme bei wirtschafttichen Wachs- rum", in: Zeitschrift fiir die gesamte Staatswissenschaft, 121 (1965), pp. 151 and following; D. Marx, "Regionale Produktivit~itsmessung als Ansatzpunkt tiberregionaler Raumordnungspolitik", in: ~ahrbuch fiir Sozialwissenschaft, 14 (1963), pp. 414 and following.

3 Cf. J. Stohler, "Zur Methode und Technik der Cost-Benefit-Analyse', in: Kyklos, 20 (1967), pp. 240 and following.

4 Cf. Winter and others, p. 35. The study includes the sectors of Stohler, except the (cont. next page)


level of 25,000 D M per head for 1963. Given the rate of activity, we get the condition

,1X' ,>O.59[]~gX',-XiX~] , r = l . . . . 100, i=1 ,3 ,5 , j = 2 , 4 , 6 [21] i j

The public means available for the investment in infrastructure and the remuneration of the people employed in Sector III (Et~+l) depend on the "taxation rate" and on the domestic product:

E (nX~+E;+I)=f[X X iq~tiX't - X Xiq~jX~t+ r r i r j

+ ]~ U~+ 1 +(1 +~t) X Vt+l + X Et+l] r r 7'

r = l . . . . 100 i = 1, 3, 9, 10 j = 2 , 4 , 8

[22]

The "taxation rate" f=0 .307 was chosen in order to obtain an actual domestic product for 1962 exactly equal to the sum of public investments of that year and the income of people employed in Sector III. Eventual increases of output that result from the model also lead - due to the constant "taxation rate" - to an improvement of the material and personal infrastructure in relation to 1962.

Similarly to [22], we obtain private capital formation by using the investment rate observed in 1962, g = 0.183.

X: +,oX9 = g [Z X Z Z qbX; + r r r i r j

+ Z U~+(1 +2) Z V~+ Z E~'+,], T r r

r = l . . . . 100 i = 1, 3, 9, 10 j = 2 , 4 , 8

[23]

Capital input in agriculture and industry increases also as a consequence of the production increases resuking from re-allocation of factors. As flexibility constraints, the following inequalities were introduced

9X~ < 0.20 Kt, r = 1 . . . . 100 [24]

This is a little bit more than double the average increase of the capital stock observed in 1962 in the German Federal Republic for Sector II. For agriculture, we would have

defence, the hydraulic investment and long distance transportation, b u t in addition investments in housing and local transportation, e l . J. Stohler, "Zur Rationalen Planung der Infrastruktur", in: K o n j u n k t u r p o l i t i k , 11 (1965), p. 288. Cf. also J. Kraft, "Die erforderliche Grundausrt is tung l~indlicher R~iume", K61n und Opladen, 1961; B. Weinberger and H. Elsner, "Investit ionsbedarf der Gemeinden (GV) 1966 bis 1975", Stuttgart und K61n, 1967.

112 R. THOSS

toXt ~ 0.05 Ct, r = 1 . . . . 100 [25]

This corresponds to the actual increase 1.

Here, we should notice that the equations [22] and [23] assume a complete mobility of the factor capital, as long as it refers to gross investment during period t, although the preceding investments cannot be modified any more by the optimization. This is based on the consideration that the displacement of capacities already settled induces generally higher costs than the possible savings of production and transportation cos t s .

2.11. Social legislation

Undesirable discrepancies of income - sectoral and regional - can be corrected either subsequently by transfer payments and subsidies, or initially by so controlling the economy that subsequent corrections are unnecessary. Either possibility may be integrated in an optimization model, but we shall follow the second path here. 2

If the share of employment in Sector I I I is not to decrease, but rather to increase slightly, it is necessary to offer an adequate income incentive in each region. As, according to [15], the variation of employment in the service sector always amounts to approximately 27 per cent of the variation in the two other sectors, it seems justified to allow for about a 10 per cent stronger variation of income i.e. 30 per cent

12x;- 13x; = 0.3 [X X ,q;,X:- X X r i r j

r=l , . . .100 i = 1, 3, 9, 10 j = 2 , 4 , 8

[26]

Once the relationship of income in the tertiary sector is established, the problem of interregional balance is reduced to fixing interregional

1 The depth and the structure of the capital stock were communicated to the author by Dr. W. Kirner, who has already published computations of this type in the past. Cf. W. Kirner, "Struktur und Strukturver~inderungen des Anlageverm6gens in der Bundesrepublik von 1950 bis 1960", in: H. K6nig (ed.), Wandlungen der Wirtsehafts- struktur in der Bundesrepublik Deutschland, Berlin, 1962, pp. 129 and following.

2 On the possibility of integrating the distribution criteria of revenue in objective function, cf. A. Maass, "Benefit-Cost-Analysis: Its Relevance to Public Investments De- cisions", in: The Quarterly Journal of Economics, 80 (1966), pp. 208 and following.


relations of income in the two sectors. Similar to Rahman, 1 we require that income per head in any region shall not be more than 20 per cent below the average. In a linear model, this requires that future employment is known; for this, we may refer to constraints [14], [15] and [16], which give information on the growth of labour potential and on the maximum population increase. In this way, we may formulate for each region the parity condition:

Z ~q~ ~x~- Z jq~ jx~ + u~ + v~ i j

(1 + k~) (At + Lt)

Z E , q ' , , x T - z Z jq'.X',+ Z u',+ E v7 ~ 0 , 8 r i r i r

(1 +n) ]~ (At+L'~) r

r = l , . . . 1 0 0 , i=1 , 3,9, 10 [27]

The revenue of factors per head obtained in region r should thus, even if a maximum increase of the labour force is assumed, be equal to at least 80 per cent of the average revenue.

2.12. Economic growth

We turn now to the problems that result from the temporal aspect of our theme. Evidently, no theoretical difficulty should result from setting up the system [13] to [27] for any arbitrary number of periods - for example 10 or 20 years. The model already contains aspects essential to a growth model, accumulation of capital and (neutral) technical progress, so that we now need only bring into consideration population growth to introduce the change in the quantities of factors that occur from one period to the other in the various regions. For this, the following definitions are used

A t + l = A t + 1X~-- 2X7

L~t+I=L~ + 3Xt-- 4X7 D r + 1 = D t -am 5Xt - 6Xt

N7+1 = N t + -- 8Xt K;+, = K~ + 9X;

C', + , = C', + , oX', r = 1 . . . . 100

[28] [29]

[30] [31] [32] [33]

a A Rahrnan, "Regional Allocation of Investment", in: The Quarterly ffournal of Eco- nomics, 77 (1965), pp. 27 and following.

114 R. THOSS

E +I [34] UL =G+ G [35] V~+ 1 = l/~ +A V~ [36] Rt +I=Ro en(t+ l) [37]

For each period, the marginal productivities are recalculated from the base of the quantities of factors, which have been modified by the optimization process in the preceding period. The constraints are similarly corrected. So then we again determine the optimum flow of factors. The development of the model over time may be sketched out as follows:

- - in each period, population increases independently at a given rate. The additional workers are allocated between regions according to their marginal productivity, taking into account the given constraints. At the same time, savings are built up that increase the stock of capital which is distributed between regions and sectors on the basis of a maximization of production. The total increase in production given by the increased input of factors, the pace of technical progress and the interregional reallocation of factors leads, in the following period, to increasing investments, etc.

One of the advantages of this procedure is the fact that new information may be introduced in the system at any time. The coefficients and the quantities of factors may be adjusted again and again as required if, in the course of the development process, it appears that they do not correspond with reality.

It remains to be demonstrated that this procedure leads to an optimal allocation of capital and labour over time, and that deviations from the optimum are avoided. We will thus formulate our system as a growth model, neglecting a few minor restrictions.

According to the objective function [13], we get for the total output of Sectors I and II at time t

~z

Y, = Z {~(L~) ~ (C;) a (N~) v + h~ ~e x' [O(A~) -~ + (1 - ~) (K~)-~]g}, r

In addition, we have the relations

100

Ro ent = ~. (A'~ + L t + D',) r ~

D t = D o + a ( A t - - A o + L t - -Lo)

r=1,...100 [38]

[392 (corresponding to [142)

[40] (corresponding to [15])

M O D E L F O R P L A N N I N G I N F R G 115

N't---N~) - b(A;--A~o + L~--L~o + D~-D~o) [41] (corresponding to [18]

+ [19])

1 0 0

g ( V , + E t ) = ~, (/s + ~t) r = l

[42] (corresponding to [23])

E t = E o + c ( Y t - Yo) [43] (corresponding to [26])

The equations [39] to [43] are summarized as

r r r . r r r N t = N O - b(1 + a) (At--'_//o + L t - L o ) [44]

Roe't=X[D~o--a(A~o + Lg ) +( I+a) (A~+L~)] , r = l , . . . l O 0 [45] r

g(1 + c) Y, +g(Eo- cYo) = E (K: + ~ ) [46] r

Given the condensed constraints [44] to [46], we get the Lagrangian

Zt= y t +9; [N~ _ b(1 + a) (A ' , -A 'o + L ' t - U o ) - N t ] +

+ vt{Ro ent -- 2F. [O~o -- a(A'o +Uo) + (1 + a) (A~ +L~)]} + r

+cot[g(l + c ) Y t + g ( E o - c Y o ) - ~ ( I s r = l , . . . 1 0 0 r

[47]

or if we collect the relations including Yt

Z t = [ 1 +%g(1 +c)]Yt+qz~[N ~ -- b(1 +a)(A't--A~o + L~-L~o)-N' , ]

+ vt{RO e ' t - Z [Do --a(A~o + L'o ) + (1 + a) (A~ +L~)]} r

+ oot[g(Eo--cYo)- }2 (Kt + C~t )] , r = l , . . . l O 0 r

[48]

We are now able to determine the relationships of factors in different sectors and regions, when revenue is maximized, not only for each period, but also over all planning periods from t = 0 to t = T, fulfilling the constraints.

T

For this, we have to maximize the integral I = f Ztdt defining, in the 0

116 R. THOSS

different locations and utilizations, the respective quantities of factors. We have thus a variation of the following kind of problem~:

X2

f F(X, Y1, Y2 , ' " , Y,,, Y'I Y ' z , ' ' . , Y',,)dX = Extremum X l

which is solved by the following Euler equations

oF __a( OF~=o 0 Y1 dX[O Y' l ) '

OF a____( OF )=0 OY2 dX~OY'2} '

oF a__( OF ) OY. dX~,OY'fl =0"

Using [38] we have as necessary conditions the Euler differential

equations

OZt = [1 +r + c)]=ex~[O(A~)-o + (1 + 8)(K~)-~]-~ - ' 5(At) -~-~ OA~

-(p~b(1 + a ) - v,(1 + a ) = O [49]

OZt O Lrt

[1 +o~tg(1 +c)]z~(L~) ~'-~ r ~ r v = (Ct) (Nt) --~otb(1 +a)

--vt(1 + a ) = O [50]

OZt OKt

gr = [1 + ~otg(1 + c)]z~eX'#[b(A~)-~+ ( 1 - 6) (K~)-~]-; - '

( l - - ~) (K~) -~ + o~ t = 0 [51]

OZt = [1 +o~tg(1 +c)]afl(L~t) ~-I (N~) v + o~t=O [52]

i I. N. Bronstein and K. A. Semendjajew, "Taschenbuch der Mathematik", 7th Ger- man edition, Ziirich and Frankfurt 1957, pp. 497 and following.

M O D E L F O R P L A N N I N G I N F R G 1 1 7

O Z t

ON',

OZ t oG

= c)]ay(Lt) (C t) (N,) - % =0 [1 +wtg(1 + " ~ " ~ r •-1 [53]

= N'o--b(1 +a) ( A ~ - A " o + L ' t - - L ' o ) - N t =0,

r = 1 , . . . 100 [54]

OZt = Ro en ' - X [D'o --a(A'o + L'o) + (1 +a ) (A~ +Lt) ] = 0 [55]

OV t r

OZt = g(1 + c) Yt + g ( E o - cYo) -- ~2 (/~t + Ct) = 0 [56]

f-O t r

The equations [54] to [56] establish that, to obtain the maximum, the factors should always be fully utilized. The equations [49] to [53] describe the relationships in which the factors have to be distributed between the two sectors considered. As these relationships may be worked out for each region, we can derive from them the intersectoral as well as the interregional distribution of factors, for the maximization of output. Next we relate the equations [49] to [52] for each region

~ ( K t ) I+p aCt , 1- -8 A--~t = f lL-~ ' r = l , . . . 1 0 0 [571

as a condition for the intersectoral balance of the marginal productivities. If the relations of the factors in the region s is chosen as a reference

measure, the conditions for an opt imum are obtained, which include the interregional equality

,KtAt I ,+o CtL~ A t K t ] L t C t , r = l , . . . 1 0 0 [58]

For land, we obtain from [53]

r r ~ r [3 r y - 1 % (L,) (Ct) (N , )

9"t (Lt)~(Ct)~(N~) ~-~' r = l , . . . 100 [591

[54] to [56] and [58] to [59] thus contain the necessary conditions whose observation Ln each point in time assure that both for this point in time, as well as for the full planning horizon, the maximum output is reached.

118 R. THOSS

If we look again at the sequence of linear programmes referred to at the start, in the light of the conditions developed above for the optimum structure of factor inputs, we note - as well as a few normative restrictions, the distribution of income and the maximum population density. not mentioned here to simplify matters - two important deviations; on the one hand, the relations [58] and [59] are not contained in the constraints [14] to [27]; on the other hand, the model contains a series of recursive relations (flexibility constraints) which limit the maximum possible modification for each region and time period to a given precentage of the stock of the preceding period.

This means that the system, in the course of time from one programme to another, moves from different initial conditions, step by step to the optimum sectoral and regional distribution of factors, until after a more or less large number of steps (programmes) the marginal productivities of the factors are approximately equal. This effect, however, is intended, because it would be unrealistic to admit that the necessary movements of factors could be made during a single period.

It is only near the optimum that neglect of [58] and [59] is felt significantly. As long as there exists between regions and/or sectors small differences in the productivities of factors, we will increase (decrease) - according to the rules of linear programming - the quantities of factors in the region or the sector where the marginal productivities of the factors are still relatively high (low). Near the optimum, the length of the steps are probably too long to reach precisely the opt imum described in [58] and [59]. In fact, a re-calculation of the marginal productivities will show that the productivities will be relatively small in exactly those regions and that sector where they were relatively high in the preceding period, and vice-versa. The consequence will be that the optimum will not be reached exactly, but that the quantities of factors oscillate around their optimum value.

One way to avoid this problem would be to diminish the length of each step with the lapse of time, making the flexibility constraints simply a function of t ime or of the dual prices of the factors 1.

In any case, there is no clear idea of the type of functional relations to be assumed. In addition, we should no longer guarantee full employment of the enhanced factors of production. For that reason, it seems more reasonable to linearize the conditions [58] and [59] so that they can be adjusted in each programme of the sequence.

I am grateful to Dr. W. Henrichsmeyer and Professor R. H. Day for these proposals.

M O D E L F O R P L A N N I N G I N F R G 1 1 9

If we define according to [58]

r s 1 (K:A:I +~ C~tL: Ot = kA~ K~J Lt Ci , r = 1 , . . . 100 [60]

it is required that the quantity O t does not change sign with the solution of the problem of optimization for each period t. According to the sign of O I in the initial period, we have to introduce the constraints

O[ + O~ > 0 for 0 D > 0 [61a] r = l , . . . 1 0 0

O~+Ot < 0 for O ~ < 0 [61b]

From [59], we deduce similarly

qort (Lr t )a (Cr t ) f 3 N t ) V - 1

ap[ 9t (Lt) (Ct) N~) ~-'' "" = --7 = , ~ ~ ~ r = 1, .100 [62]

For the value qb~ it is required that it reaches at most the value 1 during the growth process; depending on whether it was initially higher or lower than 1, we will have

qb t + @~ > 1 for q~o > 1 [63a] r=l,. . .100

q~t + @t <-- 1 for q~; < 1 [63b]

This guarantees that the opt imum relation of factors will indeed be reached but' not exeeded and that the system, from this time on, moves on the same growth path which is an opt imum from the point of view of regional and sectoral structure of factor inputs. We only need carry out the linearization of [61a] and [63a]. Under application of the total differential w.r.t, time we obtain from [61a] and [59]

Cr, LI

L;Cl

r s 1 " K t + __ ( K , A t I +~(K~ A'~ "" At I A7 K A',J

for O~ > O,

r =1 . . . . 100

r " / g r A s \ l + ~ r s d; Lt - - - [ - - - - - - [ - - - - -L/ 4" C t L t [64a] C~ + L t] \ AtKt / u-L~' c7 L CI r

120 R. THOSS

From [63a] and [62] we obtain

" $ " $ t L~ C~ ' 1" 2~ L~ ~ Ct + Nt for > 1, - ~-77 ( ~ - 1 ) - - O0 ~ - - + [ 3 +tY-- ) -N-~t - ~ - ~ t Nt L~ ~ Lt ' r = 1 , . . . 1 0 0

s a s ~

> (L,) (C,) ( N ; ) ' - ' - 1 [65a] (Lt) (C,) (N~) "-1

[64b] and [65b] are obtained similarly. They are linear in the derivatives w.r.t, time. We find in this way, after changing to discrete variables, that in addition to constraints [14] to [27], for the example with 100 regions, 99 inequalities of the type [64] and [65], the sign of the inequality depending upon the initial conditions.

Together with [14] to [27], these restrictions take care that after a limited number of steps, the distribution of the factors between sectors and locations corresponding to the equilibrium will be reached but not overstepped and that the system, from there on, moves on an optimum growth path.

3. R E S U L T S F OR 1962.

In the presentation of the results, we shall limit ourselves to the variations of the quantities of factors employed in sectors I and II. 1 Table 2 and Maps 3 to 7 show the modifications that would be obtained by observation of the constraints [14] to [27] under maximization of the product of the economy of the German Federal Republic for 1962. Further computations are planned.

Whereas the tables give the absolute modifications, the maps show the regions to which two thirds of the aggregated change should fall. The darkest areas on the maps show those regions iepresenting more than 2 per cent of the total variations; next, we have those representing 1.5 to 2 percent, while the lightest shaded areas represent 1 to 1.5 per cent.

As expected, the input of capital and land in agriculture is to be reduced in total to reach the global economic optimum. Five regions only call for neither a reduction of labour nor of land; in five other regions, we should reduce agricultural land alone; in one, labour only. In all other parts of the Federal Republic, the differences of productivity require an increased input of labour in the secondary and tertiary sectors at the

1 The other levels of activity may be calculated on the basis of [14] to [27].


Table 2. Optimal variation of the quantities of factors

r AA" t AK" t -AL~ AC t -AN~ (1000) (mill. DM) (1000) (mill. DM) (100 ha)

1 2 3 4 5 6 7 8 9 + 1 0

11 12 13 14+15 16 17 18 19 20+21 2 2 +2 8 23 2 4 +2 5 26 27 29 30+31 32 33 34 3 5 + 3 6 37 38 39 40 41 42 43 + 44 4 5 + 4 6 47 48 49 50 51 52+53 54+55 56 57 58 59 6O 6 1 + 6 3

1.259 2.233 3.447 5.633 4.347 3.686 2.045 8.299 5.993 5.457 3.433

-0.306 2.543 1.514 5.340 8.848 1 . 2 0 1

3.705 5.772 2.766

-5.439 3.272 5.172 2.365

�9 6.112 4.720 3.606 4.007 8.196 7.504 8.741 4.500 3.733

13.817 4.446 3.320 8.123

38.3 89.4

109.9 321.1 265.1 105.8

84.7

195.9 97.5 20.1 70.5 47.2

122.7

28.6 147.8

1215.6

183.9 75.2

167.1 250.3 181.8 154.6 373.4 827.4 446.9

399.7 106.1

675.4 214.3 623.1

m

1.113 1.508 0.676 1.030

0.721 2.446 1.191 1.223

1.043 1.873 1.828

1.076 2.167 0.899

1 . 3 9 1

1.223 1.156 1.175 1.643 1.315 1.877 3.062 1.463 0.951 0.847 1.539 1.536 0.657 1.583 1.790

26.0 33.0 35.3 49.2 17.5 26.4 23.3 25.0 45.8 28.0 29.0 14.9 38.4 25.0 43.6 47.2 28.0 25.8 48.6 23.1 44.0 36.8 31.1 24.5 27.6 28.5 30.8 42.2 63.4 38.5 19.2 25.6 29.4 38.7 15.6 32.1 41.0

9.909 3.876 2.600 5.896 3.287 8.551 4.665

10.591 3.268 3.239 2.884 7.042 5.652

183.0 43.5

369.5 102.8 304.4 117.2

54.6 277.9 245.6 275.5 104.4

2.326 0.901 0.996 1.792 1.004 2.950 2.139 2.109 1.180 1.360 1.527 1.890 2.863

47.0 10.8 20.2 32.5 16.8 41.3 44.1 38.4 17.8 24.5 22.5 29.0 41.2

0.686 1.217 1.272 2.248 2.001 1.448 1.115 4.130 1.934 2.325 1.205

1.386 0.257 1.890 3.826 0.655 1.433 1.966 1.018

1.025 2.153 0.659 2.691 1.677 1.249 1.161 2.799 3.293 4.246 1.992 1.196 6.693 2.065 0.947 3.453 4.133 1.621 0.874 2.238 1.244 3.054 1.377 4.625 1.138 1.024 0.740 2.808 1.520

122 R. THOSS

Table 2. (Continued)

r A A t A K t - A L ~ A C t - A N t (1000) (mill. DM) (1000) (mill. DM) (100 ha)

6 2 + 6 4 65 66 67 68 69 70 71 7 2 +7 3 74 7 5 + 7 6 77 78 79 8 0 +8 5 81 82 85 8 4 + 8 6 87 88 8 9 +9 0 91 9 2 +9 3 94 9 5 +9 6 97 98 99

100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

-3.974 6.300 4.165 4.851 3.132 5.081 3.922 5.055 8.983 6.416

12.809 4.353 8.418 9.176 5.350 3.506 3.789 7.088 4.276 3.162 2.871

102.8 248.2 111.5

127.4

778.8 80.8

267.2 1004.0 2115.6

236.5 205.2 317.3 245.8 184.8 235.1 246.1 270.6

49.3 197.3

1.991 1.647 1.154 0.930 1.663 1.419 0.698 1.950 2.774 2.450 3.641 1.238 2.628 2.123 1.967 1.184 1.303 1.573 2.257 1.280 1.480

42.0 23.7 20.2 12.9 19.8 16.9 10.1 34.1 69.3 24.1 64.0 35.2 35.8 39.7 61.8 16.0 25.3 35.3 63.2 19.5 16.5

8.493 6.243 8.960 2.078 7.323 3.847 2.343 4.251

-1.356 -1.243

5.291 -1.265

4.634 2.654 3.732 4.582 3.960 3.280 2.872 3.297 6.559 6.752 3.142 3.667 3.486 3.055 1.371 3.309 3.036

323.6 181.7 380.8

27.8 1098.6

104.2

57.6 30.7 24.1

286.4 25.3 81.0

68.0 117.0

82.3

81.3 337.2

66.6 201.2 105.3 105.6

64.1

95.5

2.570 1.616 1.794 0.944 2.428 1.127 0.675 1.751

1.119

2.296 0.937 1.489 1.635 1.483 1.214 1.279 0.976 1.133 1.167 1.257 1.342 1.203 0.978 0.402 1.048 0.953

48.0 33.5 44.5 25.4 49.4 29.8 18.0 50.2 22.8 25.7 28.2 22.5 47.1 16.0 49.9 48.6 51.1 37.8 32.4 32.6 37.1 38.2 43.9 36.0 41.1 28.3

9.5 30.8 25.5

2.536 1.641 2.138 0.801 1.996 1.758 1.693 3.385 2.162 4.998 1.698 3.156 3.845 1.844 1.266 1.355 3.006 1.101 1.026 0.758 3.229 2.522 3.906 1.618 2.669 1.483 0.909 1.363

2.274

1.274 0.936 1.223 1.606 1.350 1.126 0.868 1.265 2 958 3.045 1.028 1.267 1.244 1.132 0.528 1.232 1.135


cost of agriculture and by a corresponding decrease of agricultural land. The expected decrease of production is partially compensated by an increase of the use of capital. At a later time, the relation between agricultural and industrial production will be determined more precisely by the introduction of the demand function in the model, similar to the ones that have been used by Weinschenck and Henrichsmeyer.

Map 6 shows that economically active people freed from agriculture or created by population growth are essentially to be distributed in already industrialized regions. Without the comparatively severe flexibility constraints for the overcrowded regions, concentration on the existing centres would no doubt have been heavier since the productivity differences are substantial and can only be compensated for by a strong, over-proportional rise in investment in the infrastructure.

For investment, Map 6, we obtain a considerably different picture. In contrast to population movements, the increase in capital stock is not, for the most part, concentrated in the principal overcrowded regions. Of these, only the regions of Mannheim-Karlsruhe-Stuttgart and Ntirn- berg prove to be favourable locations from the point of view of the total economy, whereas most of the others should already be considered as over-capitalized. It seems more reasonable to favour investments in centres of average size.

Naturally, such a policy would rapidly increase the marginal productivity of labour in the less densely populated areas whereas immigration would cause it to sink in the overcrowded regions. Soon a sufficient sti- mulus to productivity would be built up to act against a further concentration of popuiation in the overcrowded regions.

1 Cf. W. Henrichsmeyer and G. Weinschenk, "Spatial Equilibrium and Prediction of Structural Change in Agriculture", Conference at the European Meeting of the "Econometric Society", Warsaw, September 1966. Summary in Econometrica, 35 (1967), Volume 5, p. 45 and following.

124

Concerning inequality [16]:

Region

4 8 9 + 1 0

16 18 2 2 + 2 8 27 32

k t Region [

0.02 33 0.01 34 0.02 35+36 0.01 38 0.02 39 0.01 40 0.01 41 0.02 5 4 + 4 6

R. THOSS

Appendix.

k~

0.02 0 . 0 1 O.Ol

0 005 10.02 10.02

0.01 0.02

Region

47 48 50 52+53 54+55 56 61 + 63 67

k~

0.01 0.02 0.02 0.02 0.02 0.01 0.02 0.02

Region

69 70 74 7 5 + 7 6 9 5 + 9 6

104 113 115

k~

0.01 0.01 0.01 0.01 0.01 0.02 0.01 0.02

For all the other regions k~ =0.03.

SOURCES

Employment and Gross Domestic Product:

Sozialproduktsberechnungen der L~inder, H. 1., Das Bruttoinlandsprodukt der kreisfreien St~idte und Landkreise in der Bundesrepublik Deutschland 1957 und 1961.

Capital in Sector I:

Communication of Dr. W. Kirner; "Regional Structure according to the agricultural census of 1960", (Statistisches Bundesamt, Landwirtschaftsz~ihlung vom 31. Mai 1960, H. 5 (Viehhaltung), H. 6 (Maschinenverwendung), Faseherie B), valued at average prices based on the llst of the output prices.

Capital in Sector I I :

Communication by Dr. Kirner; "Regional Structure according to the stock of capital (Statistisches Bundesamt, Einheitswerte der gewerblichen Betriebe 1957 und 1960, Tabelle 8, Spalte 4 + 5 , Fachserie L, Reihe 6, IV) validated against analysis of tax payments (Statistisches Bundesamt, Fachserie L, Reihe 9, II, S. 260 ff.).

Agricultural lands and forests:

Statistisches Bundesamt, Bodenniitzung und Ernte, Fachserie B, Reihe 1, I.


L I S T O F R E G I O N S

1. Husum 41. Dfisseldorf 81. Freiburg 2. Flensburg 42. Iserlohn 82. L6rrach 3. Steinburg 43. Arnsberg 83. Villingen 4. Kiel 44. Meschede 84. Konstanz 5. L~beek 45. Aachen 85. Biberaeh 6. Segeberg 46. D~iren 86. Ravensburg 7. Hzgt. Lauenburg 47. K61n 87. Bad Kissingen 8. Hamburg 48. Siegen 88. Aschaffenburg 9. Emden 49. Frankenberg 89. Wfirzburg

10. Wilhelmshaven 50. Kassel 90. Schweinfurt 11. Bremerhaveu 51. Rotenburg (Fulda) 91. Bamberg 12. Cuxhaven 52. Wetzlar 92. Bayreuth 13. Harburg 53. Giessen 93. Hof 14. Lt ineburg 54. Alsfeld 94. Uffenheim 15. Lt ichow-Dannenberg 55. Fulda 95. Nfirnberg 16. Meppen 56. Frankfurt 96. Forchheim 17. Gidenburg 57. Geinhausen 97. Weiden 18. Bremen 58. Darmstadt 98. Amberg 19. Rotenburg (Hannover) 59. Montabaru 99. Ansbach 20. Fallingsbostel 60. Koblenz 100. Weissenburg i. Bay. 21. Celle 61. Bitburg 101. Neumarkt i. d. OPf. 22. Gifhorn 62. Bernkastel 102. Regensburg 23. Lingen 63. Tr ier 103. Chain 24. Vechta 64. Kreuznach 104. Straubing 25. Osnabrtick 65. Mainz 105. Passau 26. Nienburg 66. Kaiserslautem 106. N6rdingen 27. Hannover 67. Ludwigshafen (Rhein) 107. Ingolstadt 28. Braunschweig 68. Pirmasens 108. Landshut 29. Hameln 69. Saarbrticken 109. Alt6ttingen 30. Hildesheim 70. Mannhe im 110. Vilshofen 31. Goslar 71. Buchen 111. Neu-Ulm 32. G6tt ingen 72. Heilbronn 112. Augsburg 33. Bocholdt 73. Schw~ibisch Hall 113. Mtinchen 34. Mfinster 74. Karlsruhe 114. Freising 35. BieIefeld 75. Stuttgart 115. Kempten 36. Detmold 76. G6pplngen 116. Kaufbeuren 37. Geldern 77. Aalen 117. Landsberg a. Lech 38. Essen 78. Freudenstadt 118. Bad T61z 39. Hamm 79. Tfibingen 119. Rosenheim 40. Paderborn 80. U l m 120. Traustein

126 R. THOSS

Map 1. Regional division of the Federal Republic.

. y .

- Re~ -S~, -6,e.=~

" k,eJsf,~ie. S~adt


Map 2. Relative position of the regions in relation to the centres of economic activities (pr).

U

k')

~ s t ~ s g r e . z e

- - L~desg,enze

- - Reg..Be=.-Grenze

. . . . . Landkrelsgrenze G,enze er~t kr~isfreisn Stad=

128

e

R. THOSS

Map 3. Regional allocation of the decrease of employment in agriculture ( - - AL).

i �84 , ~ . .

I , i .

I

~Landesg fenze

Rog,Bez.Grenze

~relsireien Stadt

MODEL FOR PLANNINO IN FRG 129

�9 +.++i Map 4. Regional distribution of the decrease of cultivated areas ( - -AN) .

5 0

f , s t=+tsg,enze I

- - Landesgfeoze

- - R+g..Ber.-Orenze

. . . . . . t.and~reTsgrenze

k,e[s~re}en ~ t a ~

1 3 0 R. T~ioss

Map 5. Regional distribution of agricul-

tural investments (AC))

\

13

Staatsg,e.ze

Landesgrenze

Landkreisg~nze Gro.ze einef kreisfroien S~dr


Map 6. Regional distribution of the increase of labour in industry, trade and transport ( A A).

�9 S

s~a~,tsore.ze

L;md~reisgrente

km~rm~nS lad t

132 R. THOS,q

Map 7. Regional distribution of investment in industry, trade and transportation ( A K).

j i r

I ~ ~" .

. ' . : . . . j . ' "

~d~sa renz~ ,

- - Reg -Bez , -G renze ,

" \ ". " " " ~re~s=re~e. Stadt

s , .~d : , J . ~o , , ns

Documents

A dynamic model for regional and sectoral planning in the Federal Republic of Germany