0. Introduction 1. Production Sets 1.1. The analytical framework 1.2. The set of all technically possible productions 1.3. The fuzzy subset of the efficient productions 2. The Producer's spatial Behaviour 2.1. The spatial price system 2.2. The producer's choice criterion 3. The Producer's spatial Equilibrium 3.1. Objective and constraint in a fuzzy context 3.2. The producer's fuzzy economic calculation 4. Conclusion 5.

0. INTRODUCTION 0 . 1 . The classical theory of the producer' s equilibrium rests on two sets of particularly restrictive hypotheses. First it is implicitely assumed that all inputs and outputs are located in a single place where the producer is also implan ted and where the production is carried out.

Next it is assumed that the producer follows a precise behaviour pattern, by this we mean that the producer has complete information concerning the conditions of his productive activity and he has perfect command over both the set of inputs and the set of outputs; he realises the maximum profit allowed by the technological constraint which limits his possible actions and by the given price system. 0.2. The aim of this study is to discard these two families of hypotheses relating first to the economic space of goods and of the agent and second to the precision of the economic agent's behaviour.

All inputs are not assumed to be available and accessible at the production place. They are scattered around the supply space and are transportable. If an input is both necessary and untransportable, then its location determines the implantation of the production unit. Similarly the outputs are not all sold at the place of production. If they are transportable,they are distributed in a demand space; if not,, they have to be used on the spot. The trans port costs of inputs and outputs are calculated from the The author wishes to thank Mrs Margaret CHEVAILLIER who translated this paper into English and Mr Michel PREVOT CIMh) whose suggestions enabled him to improve some demons trations, in particular those developed in paragraphs 3.2.4 and 3.2.5.01* course, the author alone is responsible for any shortcomings or possible mistakes in the text. (C.P.) location of the production unit,which is given.

Moreover the producer's spatial behaviour is impre cise. The economic agent has a utility function for profit such that for a given spatial price system, a greater or lesser degree of satisfaction is associated with any p r oduc tion. Furthermore, the producer has only imperfect control over the production process. Even if the physical combina tion of the inputs used to obtain one or several given o ut puts is optimal with respect to the state of technology, it usually yields a result which is short of the maximal product to a greater or lesser degree. 0.3. This leads us to elaborate a theory of the producer operating in an imprecise context, formalised with the help of instruments from the theory of fuzzy subsets [START_REF] Kaufmann | Introduction à la théorie des sous-ensem bles flous. 4 Tomes[END_REF] [START_REF] Prevot | Sous-ensembles flous. Une approche théorique[END_REF] . In order to avoid any ambiguity^ it must be pointed out that reliability is a particular case of imprecision-, it has been demonstrated rigorously that randomness is a particular case of fuzziness [START_REF] Kaufmann | Introduction à la théorie des sous-ensem bles flous. 4 Tomes[END_REF] [7] [START_REF] Prevot | Probability Calculation and Fuzzy Subsets Theory[END_REF] [START_REF] Zadeii | Fuzzy Sets as a Basis for a Theory of Possibility. Fuzzy Sets and Systems[END_REF] This study will obey the following plan. First the production sets will be described. Having defined the analytical framework, the main characteristics of the set of all possible productions and the fuzzy efficient productions subset will be given.

Then the producer's spatial behaviour will be studied. The environment in which he acts is described by a given spatial price system and his action obeys a choice criterion.

We will then have all the necessary elements to enable us to define the producer's spatial equilibrium in an imprecise context. The problem is to determine an optimal fuzzy supply, that is, the supply which maximises the fuzzy utility of profit under a fuzzy technological constraint and with rcspcct to a given spatial price system. 0.4. Notation: Ordinary mathematical concepts (non fuzzy) are underlined, whereas fuzzy concepts are not. For instance, H C Z is read: H is a fuzzy subset of the referential Z_. For simplification's sake,this convention is only applied to sets and not to their elements. No ambiguity is possible.

PRODUCTION SETS

1.1. The Analytical Framework 1.1.1. The producer is described in a general way: his type of activity and the organisational structure of his produc tion unit are disregarded. He is an individual or a collec tive agent who selects and carries out a production plan, ie. a combination of inputs with a view to obtaining a set of outputs. 1.1.2. The producer's space is characterized by the location of his production unit, by the inputs supply space and the outputs demand space.

(1.1.2.1) The location and the dimension of the production are given in one place indexed 1. By dimension we mean the existing production capacity. It is fixed. The analysis is limited to the short term.

(1.1.2.2) The inputs supply space peculiar to the producer is characterized by the datum of p places, indexed r, with r =1,... ,p where the inputs are accessible and available.

Y

The production process requires n inputs, denoted by with i =1, . ...,n and r =1,..,p. A place can supply one or several inputs.

Since this is a short term analysis, the set of inputs does not include fixed assets, their location and size are given.

(1.1.2.3) The outputs demand space is defined by the datum of (q-p) places, indexed r = p+1,,.,,q , where these outputs can be s o l d .

The production process generates (m-n) outputs, denoted by y T , with i = n+1,...,m and r = p+1,...,q. A placc can demand one or several outputs. 1.1.3. Hence there is a finite number of distinct goods, inputs and outputs, which define the set of located goods peculiar to the producer, denoted by Y , with Y = i = 1,...,m and r = 1,...,q.

If any one locality supplies (resp.demands) more than one input (resp.more than one output) it will be repre sented by the same number of points as inputs supplied (resp.outputs demanded). Similarly if any one locality is both an inputs supply place and an outputs demand place,, it is represented by the same number of points as supply or demand functions fulfilled.

The goods space peculiar to the producer thus has n.p + (m-n)(q-p) = n (2p-q) + m (q-p) = k coordinates and Y is included in lRk .

If IR^ , with 1 > k , designates the goods space peculiar to the whole economy, it is clear that Y is gene rally contained in a subspace of IR1 which has a small number of dimensions.

It is customary in modern producer analysis to represent input quantities by negative real numbers and output quantities by positive real numbers.

Using these notations, a production denoted by y k r is represented by a point of |R . We have y = (yi) with i 1,...,m and r -i,...,q.

1. These hypotheses do not all intervene at the same level of the analysis. They are not all independent. Finally, hypothesis (1.2.2.5) calls for greater attention and because of its specific application in spatial analysis the whole of paragraph 2.1.4 is devoted to it.

1.3. The fuzzy subset of the efficient productions lc v 1.3.1. In traditional theory, a preorder on (R denoted by f is defined by: V y = (y.) , y e Z , V y' = (y'T), y' G Z :

[ y ^ y' <r=^ (y£) > ly'\) , y i y' > V i = 1,... ,m, V r = 1,...,q ] (with at least one strict inequality), and a production y is said to be efficient in Z if and only if: v y-g lRk , (y' > y) = > (y* ^ Z) .

The efficient productions set in Z is denoted by H. We show that hypothesis (1.2.2.1) is a necessary condi tion and that hypotheses (1.2.2.2) and (1.2.2.3) are jointly a sufficient condition to the existence of an efficient production in Z [START_REF] Auray | Problèmes d'économie mathématique[END_REF] [ 2 ] . 1.3.2. The fundamental point under discussion here is that the traditional theory of the producer holds true for the limiting case in which (1) all the inputs have, in a p r o ductive combination, a maximal technical efficiency and (2) alone determine the quantities of outputs obtained.

However, this is not generally the case.

• * On the one hand, the efficiency of an input is relative and depends on several technical, institutional or other factors which are linked to its state and the conditions of its use. These factors cannot be weighted. Hence each input's contribu tion to the product only rarely reaches the maximal theoretical limits made possible by the prevailing state of technology.

On the other hand, it is generally impossible to take all the factors which influence the production into account such that, even if their technical efficiency is maximal, the inputs under consideration do not alone determine the quantity of outputs. This is true of free factors, incontrôlable factors and fixed factors. Their action on the production process is sufficient to make the technical efficiency of well-controlled inputs imprecise .

For all these reasons, the result of a production process is by nature imprecise. It follows that a technically possible production is more or less efficient. It is not advisable to partition the set of all possible productions into two classes: the efficient productions and the inefficient productions. As soon as at least one of the inputs does not have a maximal technical efficiency and/or as soon as at least one non-controlled input has an influence on the output quantities, the result of the productive combination is fuzzy. For a given combination the quantités of outputs obtained depend on the degree of efficiency of each of the controlled inputs and on the action of the factors which remain beyond the producer's control. 1.3.3. Formally, a set of more or less efficient productions is defined as a fuzzy subset of ^,denoted by H. Its elements have a membership function such that: ----------*

Mh : I -
M y ----------> ^H (y) e M
where M is a preordered and bounded membership set and where M^(y) is called the degree of membership of the element y of Z to the fuzzy subset H.

We choose M = [0,1 ] and H is thus a fuzzy subset in Zadeh's sense [START_REF] Zadeii | Fuzzy Sets[END_REF] Let us assume that: ¿^(y) = 1 if the technical efficiency of all the inputs is maximal and if all the inputs are well controlled, fiH (y) = 0 if y e iRk " C z ie. in the case of the production of wastes (hypothesis (1.2.2.4)) .

and Mpj(y) £ ] 0,1 [ if the technical efficiency of the production process lies between these extreme values.

Hence H is a fuzzy subset of Z such that:

H = ^ y, m h ; V y e z , mh (y) e [0,1]].
(1.3.3.1) Remark: In the particular case when the membership set M is the set ^0,1^ we come back to the classical theory of the producer.

(1.3.3.

2) The membership functions are determined by purely technical considerations. The fuzziness is objective.

2. THE PRODUCER'S SPATIAL BEHAVIOUR 2.1. The spatial price system 2.1.1.The spatial price system is given (competition hypothesis) and peculiar to the producer. Such a system is defined by the F.O.B. prices of inputs, the C.I.F. prices of inputs, the F.O.B. prices of outputs, the C.I.F. prices of outputs and the transport prices. A price system is spatial and peculiar to a producer in the sense that it depends on the location of the production unit and on the distances which separate the unit from the inputs supply places and the outputs demand p l a c e s . f r 2.1.2. Let p^ , with i = 1,...,n and r = 1,...,p , be the F.O.B. unit prices of inputs y^ supplied to the places indexed r .

For any i = 1,...,n and for any r = 1,...,p the unit transport prices of the inputs y^ from the places r to place 1 (the location of the production unit) equal the expenditures necessary for the delivery of those inputs units (transport costs proper, insurance premiums, costs of different opera tions) . These unit transport prices are denoted by t?"*"* .

The C.I.F. unit prices of the inputs y^ at the loca lity of the production unit are equal to: ^Pi + ti = CPl with i = 1,...,n and r = 1,...,p.

The outputs prices are defined in the same way. f. r Let p i , with i = n+1,...,m and r = 1 be the F.O.B. unit prices of the outputs at the production place. For any i = n+1,...,m and for any r = p+1,,..,q the unit transport prices of the outputs yT from the place indexed 1 to the places of demand, indexed rtare denoted by tl-^ . The C.I.F. unit priées of outputs.yT to the places of demand are equal to :

fpi + tr = Cpi with i = n+1 > • • • >m and r = p+1,...,q.

2.1.3. Under competition, the transport price is borne by the buyer who, by assumption, has no means of passing the cost onto the seller. The unit prices paid by the producer for the purchase of the inputs y. are therefore equal to c r p^, with i = 1,...,n and r = 1,...,p.

On the other hand, the net unit prices received by X* the producer for the sale of the outputs y^ are equal to:

c v 1 -*r f r
Pj_ -t^ Pi , with i = n+1,...,m and r = p + 1,...,q.

Finally, a real number, its price for the producer, is associated to each element of the set of commodities peculiar to the producer Y. p denotes a spatial price system for the producer. Such a system is a

k-uple ^ c r c p f p+1 f r f q. P ( P 1 > • • • P » • • • > Pn+1 » * * * * P » • • • > Pjjj^ •
The price of commodity is positive (scarcity and utility) zero (gratuity) or negative (disutility). 2.1.4. We note that since the prices Cp^ with i = n+1,...,m and r = p+1,...,q, do not diminish with the increasing d is tances between the locality of the production unit and those of the buyers of the outputs, all other things being equal,the demand for these outputs tends towards zero when the prices tend towards sufficiently high values. Strictly, competition is never complete in space.

A quite common mistake consists in adding transport costs -when they are taken into account -to the production costs with out distinguishing their particuliar effects. Yet transport prices do have an effect on the opti mal volumes of production and on producer's profits such that hypothesis (1.2.2.5) proves to be less restrictive in the framework of producer's spatial theory than in that of classical theory [START_REF] Lantner | Une révision des conditions d'équilibre de la firme; l'intégration du facteur spatial[END_REF] 2.2. The producer's choice criterion 2.2.1. In classical theoryr the producer's choice criterion is the maximum profit where profit is defined as total receipts less total expenditure. According to the institutional characteristics of the production unit,the maximum profit objective corresponds to fulfillment of the producer's own particular self-interest or to the satisfaction of an indica tor of social efficiency. tion boils down to the maximisation of a profit function under a strict technological constraint. in Z, the producer's profit is by definition the interior product py, with: his profit is maximum, ie. a distribution of his located inputs and his located outputs which maximises py under the technological constraint . Such an optimal production is called an equilibrium production with respect to a spatial price system. 2.2.2. This classical conception holds true in the limiting case when the producer's behaviour is precise, that is when the maximum profit criterion dictates his behaviour and when, moreover, the technological constraint is rigid. These two assumptions imply perfect and complete information about the arguments of the profit function and the technological const rai n t . (2.2.2.I) Generally, the producer's behaviour is, like any human behaviour,imprecise.

view the producer has on the results of his productive acti vity. However these points of view are numerous and not necessarily In this framework, the producer's economic calcula-Given a spatial price system p and a production y

m q py = 2 2 i=n+1 r=p+1
The producer must choose a production y such that The concept of profit aggregates all the points of compatible. It follows that a given profit is of greater or lesser utility to the agent. In particular, the greatest profit may prove less useful to the producer than a lower profit if the former doesn't reconcile so well as the latter the conflicting points of view concerning the results of the concern.

Let P = p y . As p is given and constant:.;,? only depends on y.

We build the function, denoted by Mp> called imprecise profit utility function which makes an element of the interval [0,1 ] correspond to each element of the set Z.

¿ y z ------------* [0,1 ]

y -----------^ ^pCy) 6 [0.1 ]
where M_(y) is called the degree of membership of the element y of Z to the fuzzy subset P. Let us assume that: . ^( y ) = 0 if the inactivity assumption (1.2.2.3) is verified and : a: (y) £ ] 0,1 ] in the other cases. The function -up(y) depends directly on the result of the aggregation of the points of view expressed about the products of the concern.

At the limit, we have /;p(y) = 1 f°r a production y which reconciles the best these points of view. The amount of profit associated with it yields the maximal utility of profit. We are back to the classical hypothesis if the utility of profit is maximal for the maximum amount of profit.

Hence V is a fuzzy subset of Z such that:

p = [ y ' v V y G Z , ^p(y) e [0,1 ] j • (2.2,2.
2) The membership functions are determined by personal views which the producer synthesizes. The fuzziness is subj e c t i v e .

THE PRODUCER1S SPATIAL EQUILIBRIUM

3.1. Objective and constraint in a fuzzy context.

3.1.1. To solve the problem of the maximisation of the producer's profit under a technological constraint¿modern classical theory uses Minkowski's theorem which states that a hyperplane I exists which passes through y and leaves the ~ k whole of Z on one side, if and only if a point y of 1R is not interior to the convex subset Z of E [START_REF] Auray | Problèmes d'économie mathématique[END_REF] [ 3 ] For p f 0, y G H #the convex part 1 is contained in the closed half-space under the hyperplane I, passing through y, orthogonal to p. The intersection of 1^ and H gives the optimal productions set. An equilibrium p r o d u c tion is a point or a set of points in the efficient p r oduc tions set (subset of the frontier of the set of all possible productions) which depend on the price system [2 ] 3.1.2. Minkowski'stheorem does not apply when the production sets are fuzzy subsets. The latter are not disjointed and it is impossible to find a separating hyperplane.

The technological constraint and the objective function must therefore be defined in a fuzzy context, and the problem of the maximisation of the fuzzy utility of profit must be solved under the fuzzy technological constraint. We can then verify that the producer's fuzzy economic calcu lation comes down to a fuzzy mathematical programming. (3.1.2.1) In paragraphs 1.3.2. and 1.3.3. we showed that any point y of Z is a more or less efficient production and we defined the fuzzy subset H of Z each element of which is equipped with a function which expresses its degree of membership to this subset of more or less efficient p r o d u c tions .

This conception widens the sphere of search for the equilibrium production, that is for the producer's optimal supply,since that supply now belongs to any space bounded by the frontier of the set of all possible productions, and no longer, as in the limiting case examined in classical theory, to a subset of that frontier.

Formally, the fuzzy subset H therefore plays the role of a constraint in the producer's fuzzy economic calcu lation. (3.1.2.2.) In paragraph (2. 2. 2.1) we admitted that,placed in a imprecise universe>the producer's aim is to maximise the utility he gets from the profit realized. In a particular case, this objective coincides with that of maximum profit.

We have defined the fuzzy subset P of Z_ whose ele ments are equipped with functions which translate their respective degrees of membership to this productionssubset having profits associated with it which are more or less useful to the producer. 3.2. The producer's fuzzy economic calculation 3.2.1. The producer's equilibrium is attained for an optimal supply with respect to the spatial price system p and the fuzzy technological constraint H, with reference to its utility for profit, that is the fuzzy objective P.

S designates the producer's optimal supply which maximises the utility of profit for the producer over the fuzzy subset of the more or less efficient productions.

Hence, by definition, S is a fuzzy subset of Z such that: S = P n H and the producer's fuzzy economic c a l culation is an example of a Fuzzy Mathematical Programming (P.M.P.) [START_REF] Negoita | Applications of Fuzzy Sets to Systems Analysis[END_REF] [START_REF] Tanaka | On Fuzzy Mathematical Programming[END_REF] .

Indeed, the supply may be interpreted as a fuzzy decision. It is the intersection of the fuzzy subsets P ( the set of all productions which engender profits judged to be more or less useful) and H (the set of all the more or less technically efficient productions).

The optimal supply has to be determined such that:

sup Jis (y) = sup [ n (y) A M H (y) ] • y € _Z y G Z F
The solution employs the results of the F.M.P. theory, nevertheless certain technical details of its presentation have been modified and adapted to the economic problem under consideration. Indeed, ;.iH (y) = 1 i f .'-^(y) > -a . l .

(y) =o if (y) < a ±

Hence, the membership function of H is written:

v / w r X. i" i-ii«. 1 -v i«ii = a.[tvi'-c, .1 1 1 r u {y)> oti [ Q . 1 i. D . ] (3.2.2.3.) Demonstration of proposition 3.2.2.
According to the decomposition theorem we have:

V u ( y ) = V [<* • Mh (y) ] = v [a A .u" (y) ] H a E [0,1] -a a E [0,1]
where u" is the characteristic function of H ri -u -a By definition we have: (i) y has a fixed point, that is: 3 c' 6 [ 0,1 ] / <¿>(¿0 = u and a = sup u (y) . y S H_p UL Indeed, first we show that any continuous and decreasing application of a set in itself implies a fixed point. Let (<-<:) -a = f(a). This function is continuous. We have : f(0) = <p (0) > 0 f(1) = * ( 1) -1 < 0 and in order to verify the existence of a fixed point we merely have to apply the theoremof intermediate values.

i' s ( y ) = MpCy) a MH (y) = Mp(y) A [ V (a A (y)) ] (property of H ) v C x E [0,1] -a * = V [ a A ju" (y) A M p (y) ] (distributivity a E [0,1] -a property) Whence it follows that: sup /iq (y) = V V [u A /i" (y) A *:p (y)] y E Z y E Z « E [ 0,1 ] V V [a A ;iH (y) A ,up (y) ] a E [ 0 , 1 ] y E Z r (associativity and commutativity properties) V [a A V (m h (y) A M p (y))] a G [0,1] yEZ^a (because
Then, according to the continuity of <p: Furthemore we have:

V < • : G [ 0,1 ] , « ---» * n = $ lim [ sup ;x (y) ] = sup u (y) n-*» y G H y G H p n a [ Q.E.D.]
A = (A n supp H) U (A H ^ supp H) and sup M s (y) = sup (m a /ip) (y) > sup (MH AMp)(y) y E _z y E suppH y E A n suppH B u t , sup 0 H A M p)(y) = max [sup (m h A M p) (y) , sup O h A /zp) (y)] y E A A D suppH A n ^ suppH = sup (MH A ixp) (y)
A n suppH It follows that: Let us assume that a strict inequality exists. max ) < <¿>(<0 i > 1

sup n (y) > sup Cmh A M p) Cy) = sup M p (y) > sup M p (y) y E Z y E A y E A y E H_
According to this inequality, j y' G M a such that: According to the continuity of /¿p , we can choose Aa such that V y e ' ^pCyD t [Q.E.D.] 3.2.6. At the equilibrium determined in this way, the fuzzy supply provides the producer with the greatest fuzzy utility of profit with respect to a given spatial price system and under the fuzzy technological constraint.

In the outputs demand space, the selling places of the products and the respective quantities sold there, are determined simultaneously. Similarly, in the inputs supply space, the places where the factors of production are purchased and the respective quantities acquired are obtained simulta neously. We recall that an equilibrium production is an ele ment (y?) of the set Y, with i = 1,...,m and r = 1,...,q, where i = 1,...,n designate the inputs, and r = 1,...,p their supply points and where i = n + 1,...,m stand for the outputs and r = p + 1,...,q their demand points.

In general, the solution is not unique since the unicity conditions prove to be very restrictive and all these conditions can only be satisfied in a very particular case.

CONCLUSION

4.1. Turning to the theory of fuzzy subsets makes it possible to determine the producer's spatial equilibrium in an imprecise context by enabling us to formalize, in a rigorous way, the economic calculation of the maximisation of the fuzzy utility of profit under a fuzzy technological constraint and with respect to a given spatial price system. 4 .2. This analysis should not be contrasted with the classical theory of the producer, for the former contains the latter as a particular case: profit maximisation producing the maximal utility of profit under a rigid technological constraint. 4.3. This generalisation is not the only outcome of the applica tion of the theory of fuzzy subsets. New results are also achieved. Equilibrium is no longer necessarily determined at a point or at a set of points belonging to a subset of the frontier of the set of all possible productions. Furthermore, the convexity assumption proves to be less restrictive on the production sets in this theory than in the classical theory. Indeed, on the one hand, in the theory of fuzzy subsets, the convexity is by definition a weakened convexity, and, on the other hand, in the framework of spatial analysis, the convexity hypothesis often seems non cons training, owing to the specific effect of transport prices. 4.4. This type of model requires developments. The dynamic aspect deserves to be integrated in terms of the maximisation of a fuzzy expected utility of profit under a fuzzy random constraint (imprecise reliability).

Finally, the problem of the optimal location of the production unit in a fuzzy universe remains to be solved (the passage from the short to the long period).

  (y) A /ip(y)] y £ H* 2a = v [ 1 A ;ip(y)] V V [ 0 A M p (y)] This proposition reduces the problem of finding the fuzzy optimal supply to that of finding the extremum of a scalar function [ 6 ] 3.2.3. Let us now introduce a function y such that: * : [0,1 ] , ---------> [0,1 ] a I ---------> *(«) = sup u(y) 2.3.1.) Properties of the function <¿>1 (3.2.3.1.1.) : *(0) = sup u (y) y g z v This is obvious, since HQ = Z (3 . 2 . 3 .1 . 2.) : a < p <p (a) > ? (i* ) • Indeed, according to the decomposition theorem:

  to proposition 3.2.2.: sup ¡x (y) = sup [a A sup ¿t_(y)] y € Z a e[0,1] y 6 H / = sup 'l'(a) [from the definition of 'l'(a) ] a G[ 0,1 ] Moreover, 'l'(a) = a A >p(a) = a A â [from (i)] » â Let us show that : 'i'(a) < a If a < a f we have : ^ (a) > <p (â) [ from (3 . 2 . 3 .1 . 2.) ] <p(a) ><^(a) = a > a [from (i)] and 'i'(a) = a A <¿>(<-0 = a < â = 'l'(â) . ' If a > a , we have : «¿>(a) < <p (â)zâ < a and 'I-'(u) = a A v?(a) = <p(a) < a = 'l'(a) = â A (a) = a In every case : '£(a) < 'i(â) 2. 3. 3,) Proposition (3. 2. 3. 2.) shows that the solution to the problem of finding sup (x (y) amounts to finding a = sup ;x (y) y e z y e M_ p a We know that : sup ;i (y) = sup [MD (y) A ;*H (y) ] = sup [ / _ (y) A ^H (y)] where supp H stands for the support of H, that is the non fuzzy subset such that: supp H = [ y ; V y e z , ;*H (y) > 0 J . Furthermore, by virtue of propositions (ii) and (i) , we have : sup [¿'(y) A ¿..(y) ] = sup ; . i (y) y G suppM 1 y G H p a a nd U_ G supp H , since ü ? 0. â In order to calculate this supremum we have to find an appropriate set, denoted by A,such that: H C A C supp H . { y; y E Z , MR (y) > Mp (y)J Indeed, first we demonstrate that H_ C A We h a v e : a V y° e h_ ==> n H (y°) > o i = sup /ip (y) > M p (y°) H (y°) > M p (y°) y° e a .

  y) = sup M p (y) .

4 .

 4 We then study the continuity conditions of the function *(&■) = sup M p (y

5 .

 5 the strict convexity of P, we have: V y G z , V y * G z , y t y* : t o -^y ' + h y 1 > M p(y'^ A M p (y) = % [ (1 y ' + i yl > % and I C i y * + ^ y ] -> y ! M p [ C1-^)y' + i y ] < sup IXp(y) < Mp(y') y G h , J -a . l According to the continuity of /ip > we have: M p ( y ' ) < sup Mp(y) < M p ( y ' ) Finally, it remains to study the unicity conditions of the solution.

( 3 .

 3 2.5.1.) Lemma 1. If closed, if jup is strongly quasi-concave and if,tyy e H , M p (y) t 1» the point y* , ★ such that Mp(y ) = sup M p (y) is an element of the frontier y e H * _0: of H , denoted by R . ★ © Indeed, let us suppose that: y G ft , where ft is the --C x interior of H , and we shall demonstrate that we end up with -a ' a contradiction. Let us suppose y^ such that Mp(y-|) = 1 . By assumption y 1 £ H . Since Z is convex, and according to the"passingI --(X through customs* theorem, J y' such that y* = X y^ + (1-X) y and y* e ft C H because H is closedfi'l. is strongly quasi-concave, we have: M p ( y ' ) > min [Mp (y*), M p (y-|) ] = M p (y*), which contradicts the 1 definition of y* . [ Q.E.D.] (3.2.5.2.) Lemma 2: Any element y of H belongs to the interior o£ ^a -£ u ' denoted by A a , V A a > 0. In other terms: C ^ ,V Aa > 0 . Indeed, since M^(y) is continuous on Z, y e , V e > 0 , r? exists such that: distance ( y , y p = d(y,y^) < i? lMH (y) -,«H (y1) I < e Let us put A a = e We have the following inequalities: M H (y ) -A a < M H (y1) < M H (y) + A a a n d : a -A a < Mjj(y) -A a , because by definition M n (y) > a in . T h e n : a -Au < M H (y) -Aa < M H (y-j) He n c e : e L j « (1) "Passing through customs" Theorem: If a connected subset A included into E intersects with a subset B and its complementary set, then it intersects the frontier of B.

1 -a i -¿mx 3 -a 2

 132 (a* -Aa) A sup Mp(y) < « * A sup Mp(y) = a* y G H * A y G H ★ Since (a* -Aa) > a* and (a* -Aa) A sup Mp(y) < a* y e ^'a c in order to realize these two conditions, it is necessary that sup M p (y) < " 2 = SUP Mp(y


The technical production structure will be characterized by substitution possi bilities ex ante, fixed factor proportions ex post. This means that in construc ting a factory which shall be able to produce a given maximal output level, one may choose whether this output level shall be produced by using much capital and little labour, or vice versa. Once a factory is established with a given capital equipment, we shall assume that an output smaller than the maximal output may be produced with a use of labour which is proportional to the out put.

We shall make a number of simplifying assumptions. All industries will be as sumed to be identical in the following sense. They will all be assumed to have the same given maximal output, measured in money terms, the same for small and large towns. Further, the set of corresponding quantities of labour and capital which can produce this maximal output (the isoquant) will also be the same in all industries. The demand for output of any industry is stochas tic, and with the same distribution function for all industries. The demands for outputs of the various industries are stochastically independent.

The size of a town, measured by the number of industries, is exogenously given.

Part of the reason for this may be that the space available for industrial plants at the locations of the towns is given. If we further assume that the size of each plant measured by maximal output is given, and that there is a unique relationship between maximal output and requirement for space, we ob tain an upper limit on the number of industries at each location. The assump tion of given maximal output level in each industry would be a reasonable ap proximation if average cost as a function of output had a clear minimum for a small range of output volumes, independent of the factor proportions used.

This assumption is, however, hardly very realistic. The assumptions of a given number of industries and a given maximal output level should be considered as assumptions made for the sake of convenience in deriving results concerning the optimal factor proportions -and not as assumptions which are considered to be very realistic.

The problems which will be discussed below have features in common with inven tory models : First, in both types of models demand is stochastic, and an op timal solution will in general imply a positive probability for not being able to satisfy demandThe problem of substitution between production factors is not, however, taken up in inventory models, as far as I know. A second simila rity is that in our model and in many inventory models, the probability of be ing able to satisfy demand is determined so as to maximize the value of an ob jective function of an egoistic type : possible losses for those who do not get their demand satisfied is ignored.

(1) cf. for example [START_REF] Hochstadter | Stochastische Lagerhaltungsmodelle[END_REF].

(2 ) applied to, for example, the supply of electricity . These models do also (3) determine inputs of production factors . But to my knowledge, the distinc tion between mobile and immobile production factors is not taken up. The ob jective function of the electricity producer will often differ from the one we shall use, as the electricity producer, at least for government agencies like Electricité de France, exists in order to benefit the country. An often used heuristic way of taking into account the interests of those who are supplied is to determine exogenously the probability that all demand shall be satis fied.

The problem which will be taken up below is the problem of a central planner of a country who shall decide on certain characteristics of a town. We shall describe the problem he faces and discuss properties of its solution. The pro blem of a possible decentralization of some decisions will not be taken up.

However, the knowledge of the properties of the optimal solutions found in this paper may serve as a starting point for further studies of how to implement an optimal town structure when demands for the outputs of the industries are sto chastic .

We Obviously, the fact we impose the condition that the choice of technique be the same in all industries reduces the interest of the paper. One may perhaps find it plausible that, for institutional reasons, equal industries should be treated equally. In that case the solution discussed below may be considered as a second-best solution.But even if one finds this farfet ched, it may be hoped that the discussion below will illustrate some ef fects which will also exist in less restricitive models.

The calculations which are carried out below can also be interpreted as a discussion of effects in the following model-world.

We shall establish a factory whose output demand is stochastic. The output capacity is given. The factory shall choose a) a labour stock N which is permanently paid a given salary, even when there is not work for all of them ; b) the ratio between capital and labour in use. This ratio is the same in all states of the world.

The results obtained below are relevant for the discussion of effects on the optimal choice of the factory of changes of the wage rate and of the varian ce of the demand for its output, given that the expected demand is kept cons tant .

THE MODEL. OPTIMALITY CONDITIONS

We introduce the following notation, usually omitting index for industry. We assume the following relationships.

( The second order condition D > O will not be satisfied if the isoquant K = f(L) is only weakly curved against the origin. This contrasts with the result from "usual1 1 production theory for a profit maximizing producer who is quantity adapter and shall produce a given output, using two production factors.An inte rior ooint satisfying the first order conditions for profit maximum will then be a local profit maximum if the isoquant is curved towards the origin at this point, no matter how weakly curved.

The system of equations ( 7) and ( 8 2) The variance of y decreases with k.

3) The range of values of y for which g(y;k) is positive is independent of k.

The lower limit is zmin, the upper limit is 1.

4) The density function g(y;k) will approach a normal distribution when k tends to infinity.

. 10.

The forms of

It is clear that the sign of the parenthesis { } in (19

) can be negative. This will be the case if f" is very large, which means that the curve for -f 1 (l) in figure 1 is very steep. In this case we know that the solution given by ( 7) and (8) satisfies the second order conditions.

In order to show that the parenthesis { } is not always negative when the se cond order conditions (9) are satisfied, we shall make use of ( 11 clearly the value of l will be larger and the value of B/kL will be smaller the larger is k. Consequently, the value of K will be smaller the larger is k. Whether B/k will be smaller or larger depends on the elasticity of f ' (l ) .

If the curve -f 1 (L) is very steep, the effect of k on L and on K will be small, B/k will decrease, and the percentage effect on B/k will be larger than the percentage effect on K. This again implies that kK/B will increase when k increases. In the case that the curve -f1(L) is fairly flat, it is possible, however, that an increase of k will induce a larger relative decrea se of K than of B/k, in which case dR/dk is negative.

THE EFFECTS OF w

The effect of w on the values of B and l in an interior maximum of t t is gi ven by Figure 4 If we have more than one industry, and therefore a density function g(y;k) more similar to a normal distribution than on figures 3 and 4, one will find that dL/dw will be negative if B/kL is not so much smaller than the g central value z* that g (-; k) becomes "very small". And from figure 1 we may conclude that such solutions can only occur if f"(L) is very large.

Thus, the normal case seems to be that dL/dw is negative. We see that this derivative is positive -which is the sign one would ex pect -if f" (L) is "sufficiently" large , and therefore certainly positi ve if the interior solution is a very sharply determined local maximum of t t .

In order to find out whether the derivative is always positive in a local interior maximum of t t , we may use the same approach as in the discussion of dR/dk given by ( 19), cf (20). We obtain the following inequality

.20.

*

For L > L , an increase of l will not reduce the required amount of capital, and f 1(L) is therefore zero.

If the isoquant approaches the axes asymptotically, if for example the pro duction function is of Cobb-Douglas type, the functions f(L) and -f'(L) will be as in figure 6.

.21.

We shall now discuss what is the optimal solution under some alternative forms of the functions f (L) and f' ( l ) , for a given form of the function wh(L,k,w). We shall assume the form of the latter function to be of the form in figure 1, which is the form we get if g(y;k) is unimodal.

Case A . Suppose first that -f'(L) > wh(L,k,w) for all values of L smaller than Y/w. In this case t t will increase by increasing L for L < Y/w. For wB L > Y/w, the optimal value of -is 0 and tt will also increase by increa-RL sing L in this interval until we reach a value of L for which either f 1 (L)=0 or f (L) = 0. The optimal solution in this case is to have no population in the town, to produce nothing and to have K as small as possible. The optimal value of K is zero if we have isoquants as in figure 5 or if we have iso quants which go asymptotically towards the L-axis, as in figure 6. The op timal value of K is positive if we have an isoquant as in figure 7. The ar tificial character of this solution is due to our assumptions : we require that we shall install enough capital to be able to produce the quantity Y, but we make no requirement about actual production. 1) An interior solution with B > 0, 0 < L < Y/w, K > 0. This is the solu tion determined by t t = t t = 0 (equations ( 7) and ( 8)) .

2) A corner solution with L>0, K>0, B = 0.

In this case B > Y/w, f 1 (L) = 0, t t < 0, t t = 0.

3) A corner solution with B=L=0, K>0.

In this case both t t and t t become 0/0. But for all positive values of L, B L no matter how small, t t is negative if t t = 0. .29.

In studies of this type, two types of capital/labour ratios may be of interest.

One has to do with the choice of technique in a plant

  PREVOT : Théorème du point fixe. Une étude topologique générale (juin 1974) N° 2 Daniel LEBLANC : L'introduction des consommations intermédiaires dans le modèle de LEFEBER (juin 1974) N° 3 Colette BOUBON : Spatial Equilibrium of the Sector in Quasi-Perfect Compétition (september 1974) N° 4 Claude PONSARD : L'imprécision et son traitement en analyse économique (septembre 1974) N° 5 Claude PONSARD : Economie urbaine et espaces métriques (septembre 1974) N° 6 Michel PREVOT : Convexité (mars 1975) N° 7 Claude PONSARD : Contribution à une théorie des espaces économiques imprécis (avril 1975) N° 8 Aimé VOGT : Analyse factorielle en composantes principales d'un caractère de dimension-n (juin 1975) N° 9 Jacques THISSE et Jacky PERREUR : Relation between the Point of Maximum Profit and the Point of Minimum Total Transportation Cost : A Restatement (juillet 1975) N° 10 Bernard FUSTIER : L'attraction des points de vente dans des espaces précis et imprécis (juillet 1975) N° 11 Régis DELOCHE : Théorie des sous-ensembles flous et classification en analyse économique spatiale (juillet 1975) N° 12 Gérard LASSIBILLE et Catherine PARRON : Analyse multicritère dans un con texte imprécis (juillet 1975) N° 13 Claude PONSARD : On the Axiomatization of Fuzzy Subsets Theory (july 1975) N° 14 Michel PREVOT : Probability Calculation and Fuzzy Subsets Theory (august 1975) N° 15 Claude PONSARD : Hiérarchie des places centrales et graphes -flous (avril 1976) N° 16 Jean-Pierre AURAY et Gérard DURU : Introduction à la théorie des espaces multiflous (avril 1976) N° 17 Roland LANTNER, Bernard PETITJEAN et Marie-Claude PICHERY : Jeu de simulation du circuit économique (août 1976) N° 18 Claude PONSARD : Esquisse de simulation d'une économie régionale : l'apport de la théorie des systèmes flous (septembre 1976) N° 19 Marie-Claude PICHERY : Les systèmes complets de fonctions de demande (avril 1977) N° 20 Gérard LASSIBILLE et Alain MINGAT : L'estimation de modèles à variation dépendante dichotomique -La sélection universitaire et la réussite en première année d'économie (avril 1977) N° 21 Claude PONSARD : La région en analyse spatiale (mai 1977) N° 22 Dan RALESCU : Abstract Models for Systems Identification (juin 1977) N° 23 J. MARCHAL et F. POULON : Multiplicateur, graphes et chaînes de Markov (décembre 1977) N° 24 Pietro BALESTRA : Determinant and Inverse of a Sum of Matrices with Applications in Economics and Statistics (avril 1978) N° Bernard FUSTIER : Etude empirique sur la notion de région homogène (avril 1978) N° Claude PONSARD : On the Imprecision of Consumer's Spatial Preferences (avril 1978) N° Roland LANTNER : L'apport de la théorie des graphes aux représentations de l'espace économique (avril 1978) N° Emmanuel JOLLES : La théorie des sous-ensembles flous au service de la décision : deux exemples d'application (mai 1978) N° Michel PREVOT : Algorithme pour la résolution des systèmes flous (mai 1978) N° Bernard FUSTIER : Contribution à l'analyse spatiale de l'attraction impré cise (juin 1978) N° TRAN QUI Phuoc : Régionalisation de l'économie française par une méthode de taxinomie numérique floue (juin 1978) N° Louis De MESNARD : La dominance régionale et son imprécision, traitement dans le type général de structure (juin 1978) N° Max PINHAS : Investissement et taux d'intérêt. Un modèle stochastique d'analyse conjoncturelle (octobre 1978) N° Bernard FUSTIER , Bernard ROUGET : La nouvelle théorie du consommateur estelle testable ? (janvier 1979) N° Didier DUBOIS : Notes sur l'intérêt des sous-ensembles flous en analyse de l'attraction de points de vente (février 1979) N° Heinz SCHLEICHER : Equity Analysis of Public Investments : Pure and Mixed Game-Theoretic Solutions (april 1979) N° Jean JASKOLD GABSZHWICZ : Théories de la concurrence imparfaite : illustra tions récentes de thèmes anciens (juin 1979) N° 38 Bernard FUSTIER : Contribution à l'étude d'un caractère statistique flou (janvier 1980) N° 39 Pietro BALESTRA : Modèles de régression avec variables muettes explicatives (janvier .1980) N° Jean-Jacques LAFFONT : Théorie des incitations un exemple introductif (février 1980) N° Claude PONSARD : L'équilibre spatial du consommateur dans un contexte im précis (février 1980) I. INTRODUCTION One of the advantages of larger towns with a diversified industrial structure compared to smaller towns with less diversified industries, is that larger towns may utilize more efficiently production factors which are mobile within the town but immobile between the town and the outside world. If demand for outputs of the various industries is stochastic and not too strongly correla ted, the corresponding demand for mobile production factors may in a certain sense even out to a larger extent in larger than in smaller towns. This is a consequence of the law of large numbers. In larger towns one may avoid in an average sense both that a large relative part of the mobile production factors are unemployed because of lack of demand and that there is a large surplus demand for these factors in relation to the supply. Due to the latter proper ty, one may also avoid that a large part of the immobile factors like specia lized machinery is unutilized due to lack of supply of mobile factors. It is obvious that a central planner who shall decide on the quantities of the (internally) mobile production factors and of the immobile factors ought to take into account the degree to which the factors will be used, or how effec tively they will be used. The degree of use of any factor will depend on the degree of mobility of all factors and on the degree to which different factors may substitute one another in production. In this note, we are going to determine the optimal use of mobile and of immo bile production factors in towns of different size. With size of a town we shall mean the number of different industries existing in the town. All indus tries are export industries. The only relationship which exists between the va rious industries in a town is that they get their inputs of a common mobile pro duction factor, labour, from the same common stock of this factor. In addition to labour, each industry uses one immobile production factor, capital.

  shall impose the condition that the choice of production technique, defined by the value of the capital equipment of an industry, is the same for all in dustries. One might perhaps think that this condition would automatically be satisfied in a first-best solution, since all industries are identical as con cerns their exogenously given characteristics, production function, capacity output and demand distribution. And this would indeed be the case if the demand for output in the industries were non-stochastic, under usual conditions. It can be shown, however, that it is better to let industries with identical exo genous characteristics have different capital/labour ratios -provided that both capital and labour is used. If there are two industries, one industry should have more capital and use less labour for a given production volume than the other (cf. appendix). I shall not try to give any intuitive explanation of this result, but the ef fect can be demonstrated in simple examples. One may consider it as being due to a kind of economies of scale, which is, however, quite different from the kind of economics of scale due to the law of large numbers when the number of production units is increased (4). (2) There are numerous references in Drèze (1964) (3) cf. Massé : Les investissements électriques in Morlat and Bessières (1971) (4) This effect is mentioned in Essay 4 in Arrow (1974). Comments on the inter pretation of this type of economies of scale are given in Guesnerie and Montbrial (1974).

Y

  output value (endogenous) k = member of industries (exogenous) w = production value of any person not included in B (exogenous).

  the probability that the statement in the parenthesis is true. The summation is taken over all industries. G(y;k) is thus the probability that the demand for output, as an average per industry, does not exceed Y~ y. Assuming that the density function for G(y,k) exists for all values of yfull mobility of labour between industries in the town. The sum is the employment which is necessary if all output demand shall be satisfied. This equals actual employment if this demand for employ ment does not exceed the available manpower in the town. g(y;k) .The first expression in (5) is the weighted integral of output va lues over all values of Ez V k for which there is excess supply of labour. The weights are the probability densities for the different values of Ez./k. B B When there is excess demand for labour, actual outout is k Y . -= Y -. kL L The probability that this occurs is given by the second integral in (5) . Let us now consider the problem of a central planner who wants to maximize total expected output in the country to which a town belongs. The planner takes the number of industries in the town as given and has the total popula tion B of the town and the capital/labour ratio of the industries in the town, represented by L , as his policy variables. He seeks the values of B

  ) is non-linear in the two endogenous va riables B and L. In addition to the question whether a set of values of B and l satisfying (7) and (8) yields a maximum of tt, one may also raise the question of uniqueness of the solutions of the equations. For given values of L, k and w, (7) will yield a unique positive solution for B provided that w is positive and so small that Y -wL is positive. The re quirement that Y > wL implies that production yields a surplus after paying for labour when labour is fully employed. The uniqueness of B follows from the fact that when B is so small that B/kL < zmin, tt = Y /L-w which was as-B sumed to be positive. When B increases so that B/kL > z , tt will decrease B since ttb b < 0 for z ' < B/kL < 1. For B/kL > 1, tt = -w. (7) can also be considered as determining B/kL as a function of L,k and w. In the remainder of this section, we shall discuss the determination of the endogenous variables B and L following from (7) and (8) by determining B/k.L as a function of L in the two equations. Let h(L,k,w) be B/k.L as a function of .L,k and w determined by (7) . Inserting this in (7) and differentiating with respect to L, we obtain The result that h (L,k,w) is negative is explained intuitively in the following i_i " way. If it is given that we shall use a very capital intensive technique, i.e. that L is small and K is large, then it costs relatively little, for a given wage level, to have so large work force in the town that one is almost sure that the supply of labour shall not limit the production possibilities. In this case B/kL will be close to 1. If on the other hand we use a production techni que which is so labour intensive that there is almost no surplus in production after paying the "wage" w for the labour actually used, i.e. if w.L is almost as then one ought to have so low population that one is almost sure that there will be no unemployment due to lack of demand for the products. In the case when w l = Y , the optimal value of B/kL is any value between zero and zmin. By convention we shall assume that it is zmin. For wL > Y , the optimal value of B/kL is zero. The value of the function wh(L,k,w) for alternative values of l and given va lues of k and w is the S-shaped line in figure 1. Its form is explained in the following way. We have already found that for L = 0 , h(L',k,w) = 1 and that for L = Y /w, h(L,k,w) = zmln. For the intervening values of Lr the form of the curve is determined by the form of the function g(y;k). About this function we know the following 1) The expected value of y is independent of k.

  the density function g(y;k) for one smaller and one larger value of k are sketched in figure 2. From the above properties of g(y;k) it follows that at least for large vai lues of k, the absolute value of h (L,k,w) cf. figure 1, will be large when L is close to 0 or to Y /w. The curve for wh(Lfk,w), in figure 1 will be relatively flat for central values of l , for which B/kL is close to the value for which g(B/kL;k) has its maximum. In figure 1, wh(L,k,w) is drawn for only one value of k. From figure 2 and (12) we may conclude that a cur ve for a higher value of k will be steeper close to L = o and L = Y /w,but flatter for central values of L . Having found how B/kL depends on L when (7) is satisfied, we may use this in formation together with (8) to find the value (values) of L which satisfies (satisfy) both equations.By inserting from (7) in the expression for t t _ in (8) , we obtain the derivasuch a way that t t is maximized with respect to B for any value of L.Let 7i(L)be Tr/k as a function of L when B/kL = h(L,k,w). We can then write the derivative of tt(l) as.11.

  that an increase in L will increase (decrease) profit. For the three values of L for which the two curves intersects will be unchan ged as a first approximation when L is increased. From figure1, we now see that the central one of the three values of L for which the curves intersect yields a local maximum of i t with respect to L when B/kL is given by (7) . The two other intersections yield local minima for t t . By taking the second deri vative of tt(L), we will find that the condition D > 0 in (9) is satisfied in the central intersection in figure 1, and that the condition D > 0 can be gi ven an interpretation in terms of the slopes of the two curves in the figu re : the curve for wh(L,k,w) should intersect the curve for -f'(L) from below in a local maximum for tt. From this result we easily see, what follows from (11), that we can never have an interior maximum of t t if f 1 (L) is a constant. For in this case, dL/dk is therefore the opposite of the sign of the second integral in (15). Concerning the value of this integral, we shall, cf. figure 2, assume that the interval for y from zmin to 1 can be divided into three subintervals : I a lower and an upper interval in which g^(y;k) is negative, and a central interval in which g2 (y;k) is positive. I have not, however, been able to pro ve that this property is a necessary consequence of the way in which the func tion g(y;k) is constructed -no matter what is the density function gthat the integral over g^iy;^:) in (15) is positive if B/kL is between zmin and a central value z*,negative if B/kL is between z* and one. For B/kL equal to zmin, z* and 1, the integral is zero. The central value z* equals the expected value and the median value of y if the common density function g(y?l) for all z is symmetric. In this case, and assuming the second order conditions (9) to be satisfied, the conclusion ob tainable from (15) can be stated in the following way :The optimal capital/labour ratio K/L -which we shall call micro capital/ labour ratio -for an industry producing at full capacity will decrease (in crease) with increasing size of the town in which it is located if the proba bility of having surplus demand for labour is between zero and 0,5 (between 0,5 and one) when the town population is optimally determined. The optimal micro capital /labour ratio is independent of the size of the town if the pro bability of having surplus demand for labour is either zero, 0.5 or one.5. INTERIOR SOLUTION. EFFECT OF k ON kK/BIn our model world, two types of capital/labour ratios may be relevant. One is the micro capital/labour ratio in the sense K/L of a plant which is produ cing at full capacity. This has to do with the choice of technique of the in dividual plants. The other capital/labour ratio -which we shall call macro capital/labour ratio -is the ratio between the value of the total capital stock in the town and the total town population.In the previous section, we found a relatively simple rule concerning the way the number of industries influenced the optimal micro capital/labour ratio K/L. We shall now discuss the effect on kK/B of increasing the number of plants. zero, the macro capital/labour ratio will be unchanged if k is gi ven a small change. This corresponds to the result we obtained concerning the effect of k on k/L above. But in order to determine the sign of the derivati ve dR/dk when the integral over g^(y;k) is not zero, we have to determine the sign of the parenthesis { } in (19). We shall see that the sign of the coeffi cient of the integral over g^iyfk) in (19) is ambiguous -in contrast to what was the case for the coefficient of that integral in (15).

  figure 1 at the intersection do not differ much.

  increasing w on total work force in the town, B, is unambiguous and negative, as one should expect. The effect on L is less obvious. Let us discuss the sign of dL/dw using figures 3 and 4. Figure 3 is of the same type as figure 1. In figure 3 we have, for the sake of simplicity, assumed that there is only one industry in the town ; that gLet us now interprete the expression for dL/dw, using figure 4. The value of the square bracket equals the difference between the right shadowed area and the left shadowed area in the figure. This difference is certainly negative if B/L is close to l.If zmin < l-z*11111, it will be positive for so me value of B/L close to zmin. Thus it seems likely that L will increase by increasing w in the solution given by the line I in figure 3, l will decrea se by increasing w in the solution given by the line II. (This conclusion can be confirmed by a graphic analysis based on figure 3).

  Since K = f(L) with f'(L)< o, we have obtained the somewhat surprising con clusion that the optimal micro capital/labour ratio of the plants, k /L, may decrease with increasing cost of labour, w. The normal case is, howe ver, that k /L will increase when labour becomes more expensive,

  figure 5.

Figure 6 It

 6 Figure 6

Case

  Figure 1 is consistent with the isoquant in figure 5, but not with those in figures 6 and 7.

  Figure 8

  : the ratio between the value of the capital stock and the number of workers employed when the plant produces at full capacity. Assuming fixed production coefficients ex post this ratio also equals the ratio between capital in use and employment in the plant at any point of time. This ratio is called the micro capital/labour ratio. The second type of capital/labour ratio is the ratio between the total stock of production capital in a town and the total work force. This ratio is called the macro capital/labour ratio. Even if the optimal micro capital/ labour ratio is the same in all industries, it will usually not equal the macro ratio. The macro capital/labour ratio should in this case be larger than the common micro ratio since it is usually not optimal to have a labour force in the town which is so large that one can simultaneously produce at full capacity in all industries -the probability being very small that de mand conditions should permit production at full capacity in all industries at the same time.One question which arises naturally in a modelof this type is this : Since the two production factors capital and labour enter unsymmetrically in the assumptions, will there be a tendency towards using systematically more of one in relation to the other in larger than in smaller towns ? The answer to this question turns out ot be negative, no matter which of the two types of capital/labour ratios one considers. A result which at first sight may seem surprising is that in the "normal"case, the two types of capital/labour ratio change in opposite direction when the size of a town increases : if the micro ratio increases, the macro ratio decreases, and vice versa. The only type of asymmetry in the way town size affects the use of capital and labour may occur if the substitution possibilities between labour and capital are so important that production either with use only of labour or only of capital is possible and economically feasible. In this case it may happen that it is optimal to choose a production technique in plants which uses only capital in a small town, a technique which uses only labour in a larger town. The opposite sequence cannot occur. A second general type of questions one may ask is this : do the conclusions obtained differ from those one should expect by analogy with deterministic models. This is, of course, a very unprecise question. But one may perhaps mention two conclusions which may apply in small towns and which differ from what one may call corresponding conclusions in deterministic models. The first conclusion is that even if the isoquant of each industry is curved towards the origin (but not too much) , local optima in which both production factors are used may not exist, no matter what is the ratio between the pri ces of the production factors. The second conclusion which is different from what one would expect from de terministic models is that an increase of the price of labour in relation to capital may decrease the optimal micro as well as macro capital/labour ratio. It should be noted, however, that these "perverse" results may not occur in large towns. It is clear that this paper no more than touches the surface of the problems connected with the creation of optimal towns when demand for output of the various industries is stochastic. Problems which have not been discussed in the present paper and whose answers do not seem obvious to me are the follo wing -just to give a few examples.It has been asserted above that the optimal capital/labour ratios should not be the same in two industries whose exogenously given characteristics are identical. Suppose now that tnere are more than two industries. Should then all of these have different capital/labour ratios, or would there be some kind of "clustering". And would the range of optimal capital/labour ratios be larger in smaller or in bigger towns ?The number of questions of this kind which one may ask will of course in crease if one admits differences between industries in exogenously given characteristics, such as the form of production functions, if one admits stochastic dependence between demand for products in various industries etc. A second type of problems which has not been discussed in the present paper concerns the possibility for decentralizing decisions about choice of tech nique. The very fact that the optimal capital/labour ratios differ in indus tries with identical exogenous characteristics indicate that the degree of decentralization cannot be as high as in markets with non-stochastic demand for output -if one shall obtain a first-best solution. On the other hand, it is obvious that in the real world some degree of decentralization has to take place. By assuming different ways of decentralizing some decisions, while letting other decisions be taken by a central authority, so-called second-best problems may be discussed which may be relevant for the capacity, form of demand density function g(z^) and ca pacity output value Y. Let us assume that L^ < L^. Assuming constant produc tion coefficients ex post, output value per person employed in industry 1 is Y /l^ which is greater than output value per person employed in industry 2, Y/L . This means that industry 1 ought to be given priority over indus try 2 when allocating labour to industries in states of the world with sur plus demand for labour. In order to simplify the interpretation of the expression for expected out put value when industry 1 is given priority, let us consider figure Al.

w and wzmin, while the opposite applies for central values of wh(L,k,w) .

From the above also follows that the integrals under the two curves I and II from L = 0 to L = Y /w is the same ; which again implies that the sum of the areas 2,3 and 5 in figure 8 Let us consider each of these separately.

The first function is drawn in figure A2. We note that it is symmetric 'Xj around the point (0,B/2L). We may therefore conclude that the difference (L,L) -X2 (L,L) in (A5) is negative, and that, therefore, a solution with equal capital/output ratio in the two industries cannot be optimal.


This information has varying degrees of credibility, thus its interpretation calls for caution. The more complex a space (for instance an urban space) the less well it is known. These approaches have used a F.M.P. (Fuzzy M a t h e matica l Programming), whose the theory was first elaborated by H.Tanaka, T.Okuda and K.Asai [ 14] (1) A fuzzy subset A of a referential E = [ x ^ is formed by the images of the elements x w hich take their values in a preordered set M, w i t h Card M > 2, b y an application /'. defined on E and with values in M. The application ,"^(x) expresses the degree of membership of the element x to the fuzzy subset A of E. It is often assumed that M = [0,1] . The theory of fuzzy subsets was first presented by L. A.Zadeh [16 ] . A ny reader wishing to go into the subject further might consult the follo w ing works given in the bibliography [2] [3] [4] [6] [11] [17] as well as the international journal under reference number [15] 0.4. In certain cases, only the constraint is fuzzy, whereas the objective is precise. The consumer maximises a traditional utility function under a fuzzy budget constraint . The pro ducer maximises a profit function under an elastic technological c o n s t r a i n t .

The aim of the present paper is to provide a solution to this particular type of fuzzy economic calculation by applying a-specific method -to it.Indeed, D. where i e I, _l finite or not, designates the nature of the actions and where j e J, J finite or not, stands for the places where these actions can be carried out.

In the description of the general model that follows, the notation can be simplified without causing any ambiguity by assuming:

The economic agent formulates a precise objective on the elements of E and must take into account a fuzzy constraint, ie, a constraint which is "more or less" strict, on the elements of the same set E (1).

An objective is a subset of E, denoted by F, defined by a function f on E such that:

f: E | -------------------0, + [ V x €■ E , x |___________^ f (x) € [ 0, + 00 [
where f(x) measures the objective under consideration.

A fuzzy constraint is a fuzzy subset of E,denoted by C, defined by its m e mbership function such that:

/tc : E , ----------------> [ 0,1 ] V x e E, x | ^ /'c (x) e [ 0,1 ]
where ^( x ) expresses the degree of satisfaction wi t h the fuzzy constraint.

1.2. The problem consists in maximising the function f on the fuzzy subset C, that is to determine:

sup f (x) = sup [ f (x) A M r (x) ] x e C
x E E This relation means that the best possible alternative is the maximal element of the intersection of the subset F and the fuzzy subset C (2).

We prove that this problem can be put in the following form (3):

sup f(x) = sup [a A sup f(x) ] x £ C c:G[ 0,1] x S
where C^, with a G [0,1] , designates the a -cut of the fuzzy subset C, that is to say the (non fuzzy) set such that:

(1) We prove that there is no loss of generality by handling this simple case in w h ich there is only one constraint. When there are several constraints, all one has to do is to consider their intersections The solution to this problem can be reduced to the solution of a Sugeno's fuzzy integral [1] [12] [13] .

For that purpose, we first need a definition of a fuzzy meas ure (or v a l u a t i o n ) .

Let f be a monotone family of (non-fuzzy) subsets of E with the following properties: (P.

v : f c q (E) | _________ ^[ 0 , + °° [ such that (1): (P.3.) v(0) = 0 ; v(E) = 1 (P.4.) ( F . e i , Fk G £ , F. C Fk ) =*> vfFj) < v ( F k ) (P.5.) (If F i e 3; , F. C F ) s» v( if F ^ = lim v(F.) i=o i -> »
We now define the function M£ on the monotone family such that:

K f : J ( E ) j ^ [0,+ oo[ V F G 2 , F |--------------> M f (F) = sup f(x) x S F with sup f(x) = 0 and sup f(x) = 1 0 x e E
It can be v e r ified that the function is a fuzzy me a s u r e in Sugeno's sense of the word (2).

(1) We note that is not a a-algebra on E. Indeed does not have the property of complementation, but only that of monotonicity (P.2.)* M o r e o v e r , v is not a measure in the traditional sense of the word. The valuation v does not possess the property of additivity, only that of monotonicity in the inclusion (P.4J. A measure is a particular cise of a fuzzy measure.

(2) Cf. Appendix 5.3.

1.3.

£ is a Sugeno's fuzzy measure, the fuzzy integral of a measurable function f can be defined:

f : E | ------------------> [ 0, + oo [ f f d M f = sup [ a A m f f f > a jf 1 J E 1 aE[ 0 , 1] with = x E E / f (x) > a ^ .
This is Sugeno's definition of a fuzzy integral [5] [13] 1.3.1. We therefore establish the following proposition:

T h e o r e m : If the function F(<*) = /¿£ ^ f 5* ^ is continuous, being a fuzzy measure, then a. > 0 exists such that:

f d ^f > "} / .
'E Indeed let F be a function such that:

F : [ 0,+ oo [ | --------^ [0,+ oo [ First we show that we cannot have F(a) > a t a G [0,1]
Indeed, let a ^ ^ > 0. Then:

£ x; x e E / f ( x ) > a2J 3 ^ x ; x e E / f(x) > a ând M f £ f > a £ ^ ^ p { f ^ " 1 i » because the function M is increasing (1). Let F (a: 2) > F («-j ) • It is therefore not possible to have at once, \f « € [ 0,1 ] : F(g£2) > a 2 F (« 1) > a 1 a 1 > a 2 F (a 2) > F(a -|)
(1) The function ju is increasing since it is a fuzzy measure [Property (P.4.) of a fuzzy measure ] .

which proves that F(a) > a , a E [ 0 , H is impossible. Consequently, a Q E [0,1] / F ( « 0) < <xQ exists.

Let us now assume that IfK") = F (a ) " a • This function is continuous since F is by assumption continuous. We have:

<p(0) = F(0) > 0 <fi(a0 ) = F (a0 ) -a Q < 0
From the theorem of intermediate values applied to <p , there exists "a > 0 such that <¿>(«0 = 0, let F(&) = a . Thus a is a fixed point.

It remains to prove that: T h u s , in both cases : a A F (a) < c? , G [0,1] C o n s e q u e n t l y :

f d ji£ = cT
sup [ a A /x r f f > at] < a CfG[ 0,1] Since we have: a = a A F(a) and a A F(a) < sup [ a A fir ff > « a G [ 0 , 1 ] 1 C J
it can be verified that the only possible case is:

sup [a A ju-c (f > «1] = & aG[ 0,1] 1 1 J [ Q.E.D.] 1.5.
sup [a A jUr { f > a \ ] = p f f f > â > = sup f (x) a€[0,1] * ^ 1 L ^ x é ç - Now, we have: sup [a A / f > a 1 ] = sup [ f (x) A ¡xp (x) ] = sup f (x) ae[ 0,1 ] 1 v > x G E L x £ Ç -
Hence the solution: The same conditions are, of course, valid in the present study.

sup f(x) = sup f(x) x G C x G C-a [Q.E.D.]

Illustration

Let us take a simple example. Let us adopt:

-a set of alternatives : E = [0,3]

-an objective function : f(x) = e , 0 < x < 3 , w h ich is monotone and decreasing.

2

-a fuzzy constraint : M r (x) = -which is monotone and

L 10
increasing.

We want to find the value of S u g e n o 's fuzzy integral:

f f d = sup [ OL A { f > <*\] J E «£[ 0,1]
First we need to know the sets } f > a } for 0 .30259 / f(x) >0.2) = [ 0, -in 0.2] = [ 0, 1 .60943 / f(x) > 0 .3] = [ 0, -In 0.3] = [ 0, 1.20397 i* / f (x) > 0 . 4 } = [ 0, -u 0.4] = [ 0, 0.91629 t* / f(x) > 0.5^= [ 0, -(n 0.5] = [ 0, 0.69314 i* / f(x) > 0.6] = [ 0, -in 0.6] = [ 0, 0. 51082 t* / f (x) > 0.7] = [ 0, -in 0.7] = [ 0, 0.35667 There is no need to go on to a = 0.4, since the value required lies between 0.2 and 0.1449543.

I I We have * / f(x) \v o = [ 0, 3.0 ] fx / f(x) > 0 . 1 } = [ 0, -in 0.1 ] = [ 0, 2
i* / f(x) > 0.8} = [ 0, -in 0.8] = [ 0, 0.22314 / f(x) > 0.93 = [ 0, -tn 0.9] = [ 0, 0.10536 f* / f(x) > i.
Let us make an interpolation between a = 0.2 and a = 0.3. This operation is justified since f(x) and Mq(x) are monotone and have variations in opposite directions. We obtain:

[x / f(x) > 0.21 j = [ 0, -fn 0.21 ] = [ 0, 1.56065 ] (x / f(x) > 0 . 2 2 ] = [ 0, -fn 0.22 ] = [ 0, 1.51413 ] {x / f(x) > 0 . 2 3 j = [ 0, -fn 0.23 ] = [ 0, 1 .46968 ]
And so on. In the latter case we consider that at least one product is supplied at this locality, at a price which might be zero, and enters into consumption (a tourist site, for instance, which supplies one or more market or non-market goods, such as a nature reserve, a picturesque l a n d s c a p e ,e t c .) . If a product is not transportable, the locality of consumption is n e c e s s a rily determined at its supply point.

The set of all the located consumer goods is denoted by X = | x | ^ where the index i stands for the nature of the good, with i = 1,...,n;and where the index j designates its supply point, with j = 1 , . . . ,m. The quantities of goods X^ are exp r essed by positive real numbers, or zero, denoted by x^. We have _X_ = |Rm n and a consumption, at a point in the consumer's space, is an element of |Rm n .

2.1.2.

From the set X, we define the set of all the possibl located consumptions, denoted by K,and assumed to be countable.

A vector of K describes a complex of located goods. It is denoted by k, with h 1,.««,g,.

Thus h k = [ h x{ ] and K C |Rm n .

Using these notations, it is possible to check that K is a specification of the set of all possible actions de s c ribed in paragraph 1.1. Let J i b e a monotone family of subsets of K. Let us assume that it has the properties (P.1.) and (P.2.) describ e d in paragraph 1.2. Indeed K is the set on which the consumer must make his choice. Property (P. 1.) implies that a m o notone family of subsets of K necessarily has as elements the empty set (the consumer intends not to consume at a given moment) and the full set, that is, K itself (the consumer effects a trade-off be tween all the possible located consumptions). Property (P.2.) implies that if a sequence of subsets of possible located consumpt i o n goods which is monotone is an element of , then the reunion and intersection of its components are the elements of X and the limit of this sequence belongs to .

2.1.3.

The c o n s u m e r ' s space is bestowed wit h a given price system which is peculiar to the consumer since it is a C.I.F.

price system.

The consumer's location (location of his residence) serves as a point of reference, wherever the consumption is actually physically carried out. If the consumer t located at a point denoted by j*, consumes at that point, the transport costs equal the costs of bringing the goods from their respective supply points to point j* . If the consumer prefers to go to a supply point to consume a complex of goods, the transport costs equal the sum of the costs of moving the agent and the costs of bringing the goods to that place. Some of these costs ma y be z e r o .

For any goods i, with i = 1 ,...,n,and for any place j, with j=1,...,m, we denote by t| the transport costs associa t e d with the consumption of a unit of the good i made in place j which is effected either at the place j* or at the place j , w i t h The consumer compares the complexes of located goods and evaluates them with respect to his preference system. His preferences are assumed to be expressed numerically. More precisely, it is assumed that the conditions of existence and of continuity of a utility function are satisfied.

We have an objective function denoted by f, such that:

f : K |-----------------^ [0, + oo [ V h k G K, hk \--------> f (h k) G [0, + oo [
with the usual properties:

V ( m k,n k) GK2, m k >-n k <£r=-~-.> f(m k) > f (n k) m k 'v n k < £=-* f(m k) = f(n k) .

2.2.2.

In a static analysis, a consumer has a given budget which corresponds to the value of all the goods in his possession, whatever their respective locations. This wealth is represe n t e d by the real number w G|R.

For a set of possible located consumptions K C|Rm n , for a C.I.F. price system cp G |Rm n and a wealth w G|R, we define a budget set, denoted by B, B C K, by: B = £ hk; V h k G K : cp . h k < w j where c p . ^k < w designates the budget constraint and implies that the value of a consumption cannot exceed the consumer's wealth.

In classical theory, the optimal c o nsumption ma x i mises the consumer's utility on this budget set. The set of all the possible consumptions is partitioned into two classes: that of the consumptions said to be efficient and that of the non-e f f i c i e n t consumptions. For a given budget constraint, the class of the efficient consumptions is the set of the consumer's technical optima (or efficiency boundary) and the consumer's choice is directed towards the elementsof this set.

However this theory only holds true in the part i c u l a r case of a precise behaviour. In general, the budget constraint is "more or less" limiting. If we admit that there exists an imprecise frontier between "a little more" consumption and "a little less" saving, or the contrary, according to the consumer's p r eferences and his associated utility function, then we admit that the set of efficient located consumptions is not reduced to the classical efficiency frontier, but that it is a fuzzy subset of the set of all the possible located consumptions. Indeed, any complex of located goods belongs "more or less" to the set of consumptions compatible wit h the budget constraint. This idea is translated formally by defining a m e m b e r ship function on the elements of K which takes its values in the interval [0,1] . This function, denoted by M^yis such that:

M c : K , ___________ >[0,1] V h k G K, h k , -----^ juc (h k) G [0,1] and = 1 if c p . ^k = w M c (h k) = 0 if cp.h k > w M c (h k) G J O , 1C if cp . h k < w .
The function depends on w. It is mo n o t o n e and decreasing for the increasing values of w:

(w' > w) ^r> [ ( W -Cp . hk) > (w -cp. h k)] ^ /.¿(h k) < Mc (h k)
It enables us to construct a fuzzy subset of K, denoted by C, such that:

C = { h k; H h k e K, M c (h k) e [0,1] ] .
The fuzzy subset C of K plays the role of a fuzzy constraint (or elastic constraint) in the consumer's economic calculation.

. . 3 .

Let us define the function on the m o notone family tfC defined in paragraph 2.1.2. such that:

M.f : li c t f (K) , -----^ [0, + «> [ \/K. e X , K. | ------M f (K.) = sup f(x) _1 - "1 r 1 x G K i
with sup f(x) = 0 and sup f(x) = 1 0 x e K We have verified that the function n £ is a S u g e n o 's fuzzy measure on $C (1). --------the utility f( k) by satisfying as well as possible the f u z z y h. h budget constraint k) for any k of K.

2.2.4.-Finally the consumer's choice criterion is to m a x imise

Finding the optimal demand therefore comes down to d e t e r m i n i n g :

sup f (^k) = sup [f(^k) A ju^,(^k) ] h k G C h k G K
It suffices to apply the solution d e veloped in p a r a graphs 1.2. and 1.3.

(1) Cf.Appendix 5.3.

At equilibrium, the optimal demand is determined and, in the consumer's space, the places to which the consumer directs his particular demands for located goods are known h. ŝince the optimal complex k is equal to the optimal vector

[^x^ ]* where j is the index of the supply points of the goods indexed i entering the complex indexed h. Finally, the consumer's global volume of saving is determined since he does not saturate his wealth constraint, apart from in particular cases ( bare living wage or voluntary zero s a v i n g ) .

3

. A pplication to the analysis of the producer's fuzzy spatial equilibrium 3.1. Similarly, the analysis of the producer's spatial e q u i l i b r i u m with a fuzzy technological constraint is a particular s p e c i f i c a tion of the general model of imprecise behaviour.

3.1.1.

The producer's space is characterized by the locatio of his production unit, by the inputs supply space and the outputs demand space.

The location and technical dimension of the pr o d u c t i o n unit are given in one place indexed j which is the point of reference from which the spatial price system is determined.

The production process requires a set of inputs, denoted by , where the index i designates their nature, w i t h i = 1,...,n, and the index j stands for the places where they are supplied,with j = 1,...,p. This process generates a set of outputs, denoted by with i=n+1,...,m, which is the index designa t i n g the nature of the goods produced, and with j=p+1,...,q, w h i c h is the index standing for the places where they are d e m a nded . U s i n g these notations, this set Y-? has as elements (m-n) outputs demanded at (q-p) places.

The set of located goods peculiar to the producer (inputs and outputs) is denoted by Y, with Y = ^ , i=1,...,m and j=1,...,q. Hence the goods space peculiar to the producer has: np + (m-n)(q-p) = n(2p-q) + m(q-p) = k coordinates; by convention let us assume that a single place supplying (respec tively demanding) more than one input (resp.more than one L e t oj^be a monotone family of subsets of Z. It has properties (P.1.) and (P.2.) described in paragraph 1. 2. Property (P.1.) implies firstly the possibility of inaction (0 e £ ) and secondly the existence of a trade off b e tween all the possible productions (Z e ^) . Property (P.2.) implies that if a monotone sequence of the subsets of all the possible productions is an element of then the limit of this sequence belongs tooj^. A real number representing its price to the producer is associated with each element of the set of located goods peculiar to the producer. cp denotes a C.I.F. price system in the producer's space. 5.2. The producer's aim is to make the maximum profit. 5.2.1. Given a production y in Z and a spatial price r system p peculiar to the producer, then his profit, denoted by P, is by definition the internal product py such that:

c m q f i i n p c i i P = cpy = 2 2 V d ~ 2 2 V y] i = n + 1 j = p + 1 i=1 j =1
The producer must choose a distribution of his located inputs and outputs which maximises P under a fuzzy technological constraint. This production, called e q u i l ibri u m production, is the producer's optimal supply wit h respect to the spatial price system. c Ŝince p is given and constant (for given j ) the profit P only depends on y.

We have an objective function, denoted by f, such t h a t : V y e Z, y j----------------^ M c (y) G [0,1] and M^(y) = 0 in the assumption of the production of waste M^iy) = 1 if the efficiency of inputs is maximal and if all inputs are well-controlled. At equilibrium, the optimal supply is determine d and, in the producer's space, the selling places of the outputs and the quantities supplied are known, as well as the places where inputs are purchased and the quantities demanded. Indeed an equilibrium supply is an element (y^)* of the set Y, with i=1,...,m, and j=1,...,q, where i=1,...,n, designates the inputs and j=1,...,p, their supply points, and where i=n+1,...,m, stands for the outputs and j=p+1,...,q, their demand points.

(1) Cf.Appendix 5.3. desirable and what is "more or less" possible, means that the properties of their optimality are put into perspective.

In addition, the theoretical analysis gains in realism without any loss of rigour. Its operational character is obvious. By partitioning , we obtain:

V [h q (x) A f(x)] = V [ m c (x) A f(x)] V V [fiQ (x) A f(x)] X G E UG[ 0,1] x G E V [ 1 A f(x)] V V [0 A f(x)]
x G C x £ C a. Indeed:

(1) M £ (0) = 0 ; M £ (E) = 1 by construction

(2) Let Fj c Fk ; \j x G F . , therefore x e Fk Therefore: sup f(x) must be such that the element x belongs to n *eF.

  cette collection est de fiffuser rapidement une première version de travaux poursuivis dans le cadre de l'I.M.E. afin de provoquer des discussions scientifiques. Les lecteurs désirant entrer en rapport avec un auteur sont priés d'écrire à l'adresse suivante : Institut de Mathématiques Economiques 4, boulevard Gabriel -21000 DIJON (France) N° 1 Michel PREVOT : Théorème du point fixe. Une étude topologique générale (juin 1974) N° 2 Daniel LEBLANC : L'introduction des consommations intermédiaires dans le modèle de LEFEBER (juin 1974) N° 3 Colette BOUBON : Spatial Equilibrium of the Sector in Quasi-Perfect Compétition (september 1974) N° 4 Claude PONSARD : L'imprécision et son traitement en analyse économique (septembre 1974) N° 5 Claude PONSARD : Economie urbaine et espaces métriques (septembre 1974) N° 6 Michel PREVOT : Convexité (mars 1975) .N0 7 Claude PONSARD : Contribution à une théorie des espaces économiques imprécis (avril 1975) N° 8 Aimé VOGT : Analyse factorielle en composantes principales d'un caractère de dimension-n (juin 1975) N° 9 Jacques THISSE et Jacky PERREUR : Relation between the Point of Maximum Profit and the Point of Minimum Total Transportation Cost : A Restatement (juillet 1975) N° 10 Bernard FUSTIER : L'attraction des points de vente dans des espaces précis et imprécis (juillet 1975) N° 11 Régis DELOCHE : Théorie des sous-ensembles flous et classification en analyse économique spatiale (juillet 1975) N° 12 Gérard LASSIBILLE et Catherine PARRON : Analyse multicritère dans un con texte imprécis (juillet 1975) N° 13 Claude PONSARD : On the Axiomatization of Fuzzy Subsets Theory (july 1975) N° 14 Michel PREVOT : Probability Calculation and Fuzzy Subsets Theory (august 1975) N° 15 Claude PONSARD : Hiérarchie des places centrales et graphes -flous (avril 1976) N° 16 Jean-Pierre AURAY et Gérard DURU : Introduction à la théorie des espaces multiflous (avril 1976) N° 17 Roland LANTNER, Bernard PETITJEAN et Marie-Claude PICHERY : Jeu de simulation du circuit économique (août 1976) N° 18 Claude PONSARD : Esquisse de simulation d'une économie régionale : l'apport de la théorie des systèmes flous (septembre 1976) N° 19 Marie-Claude PICHERY : Les systèmes complets de fonctions de demande (avril 1977) N° 20 Gérard LASSIBILLE et Alain MINGAT : L'estimation de modèles à variation dépendante dichotomique -La sélection universitaire et la réussite en première année d'économie (avril 1977) N° 21 Claude PONSARD : La région en analyse spatiale (mai 1977) N° 22 Dan RALESCU : Abstract Models for Systems Identification (juin 1977) N° 23 J. MARCHAL et F. POULON : Multiplicateur, graphes et chaînes de Markov (décembre 1977) N° 24 Pietro BALESTRA : Déterminant and Inverse of a Sum of Matrices with Applications in Economies and Statistics (avril 1978) N° Bernard FUSTIER : Etude empirique sur la notion de région homogène (avril 1978) N° Claude PONSARD : On the Imprécision of Consumer's Spatial Preferences (avril 1978) N° Roland LANTNER : L'apport de la théorie des graphes aux représentations de l'espace économique (avril 1978) N° Emmanuel JOLLES : La théorie des sous-ensembles flous au service de la décision : deux exemples d'application (mai 1978) N° Michel PREVOT : Algorithme pour la résolution des systèmes flous (mai 1978) N° Bernard FUSTIER : Contribution â l'analyse spatiale de l'attraction impré cise (juin 1978) N° TRAN QUI Phuoc : Régionalisation de l'économie française par une méthode de taxinomie numérique floue (juin 1978) N° Louis De MESNARD : La dominance régionale et son imprécision, traitement dans le type général de structure (juin 1978) N° Max PINHAS : Investissement et taux d'intérêt. Un modèle stochastique d'analyse conjoncturelle j[octobre 1978) N° Bernard FUSTIER , Bernard ROUGET : La nouvelle théorie du consommateur estelle testable ? (janvier 1979) N° Didier DUBOIS : Notes sur l'intérêt des sous-ensembles flous en analyse de l'attraction de points de vente (février 1979) N° Heinz SCHLEICHER : Equity Analysis of Public Investments : Pure and Mixed Game-Theoretic Solutions (april 1979) N° Jean JASKOLD GABSZHWICZ : Théories de la concurrence imparfaite : illustra tions récentes de thèmes anciens (juin 1979) N° Bernard FUSTIER : Contribution à l'étude d'un caractère statistique flou (janvier 1980) N° Pietro BALESTRA : Modèles de régression avec variables muettes explicatives (janvier 1980) N° Jean-Jacques LAFFONT : Théorie des incitations un exemple introductif (février 1980) N° Claude PONSARD : L'équilibre spatial du consommateur dans un contexte im précis (février 1980) N° Claude PONSARD : Fuzzy Economic Spaces (aprii 1980) N° Alan KIRMAN : Imperfect Communication in Markets -A big world problem (aprii 1980) N° Michel PREVOT : Variétés différentielles (mai 1980) N° Claude PONSARD : Producer's spatial equilibrium with a fuzzy constraint (may 1980) N° Michel PREVOT : Théorie des catastrophes (mai 1980) N° Bernadette MARECHAL : Recherche de la forme d'un modèle à retards échelonnés application à la fonction d'investissement, (novembre 1980) N° Bernard FUSTIER : Une méthode d'analyse multicritère, SPARTE, (mars 1981) PARTIAL SPATIAL EQUILIBRIA WITH FUZZY CONSTRAINTS 0 -Introduction 0.1. It is implicitly accepted by spatial economic analysis that the economic behaviour of agents located in given spaces(market areas, regions, etc.) is precise, that is to say, their behaviour is such that a possible action (consumption, production) is, or is not, preferable to another. In other words, economic agents are assumed to make accurate economic calculations and optimise the objective functions under strict constraints of resource limitation. These objective functions have clearly defined arguments and well-controlled parameters. 0.2. Generally, however, the economic spaces in w h i c h agents live are not transparent and the information available to agents is incomplete, imperfect and more or less accurate.

  [ 6 ] . (2) In fuzzy algebra the intersection operator (n ) is the m i n (denoted by A) and that of the union (U) is the masc (denoted by V ) . (3) Cf.Appendix 5.1. and 5.2. Ca = | x ; x e E : Mc(x) > a j .

  (a) > F(aT) . But , F(a) = a and 7x > a Hence: F(a) > a and a A F(a) = a a A F (a) < oT -Let a > cT Then: F(a) < F(a) . But,F(a) = a and a < a Hence: F(ct) < a. and a A F(a) = F(a) a A F(a) < a.

  2. It only remains to apply this fundamental theorem to the function f omde.r consideration, i.e.: f : JE. j > [0,+ ~ [ and to the function /*£• We derive the following proposition: C o r r o l a r y : If the function F (a) = sup f(x) is continuous, then there exists a € [0,1] x e -a such that sup f(x) = sup f(x) x £ C x e C--a Indeed, from the previous theorem, ôc G [0,1] exists such that : fE f d "f * "f [ f or by definition of a Sugeno's integral:

1 . 5

 5 . 5 . The question of which conditions are ne c e s s a r y for a function F to be continuous has been examined thoroughly in the framework of the models using F.M.P. [9] [10] [14]

  <x included between 0.223 and 0.224. We h a v e : j x / f(x) > 0.2231] = [ 0, -in 0.2231 ] = ■ [ 0of the fuzzy integral is equal to 0.22395 with an error of 10-4 . 2. Application to the analysis of the consumer's fuzzy spatial e quilibrium 2.1. The analysis of the consumer's spatial equil i b r i u m with a fuzzy budget constraint is a particular specification of the general model developed in Section 1.2.1.1. The consumer's space is characterized by a finite set of localities where consumer goods are s&pplied. One of these localities where the consumer is implanted is called the demand spot. If the consumer is mobile or if the goods are transportable he can consume, according to his preferences, either where he lives, or at a supply point, or at any locality of his choice.

•

  Let p^ be the F.O.B. price of a unit of the good X^. The C.I.F. unit price of this good at the consu m p t i o n place, denoted by cp^ , is equal to: Cp{ = + > with i=1,...,n, and j=1,...,m.A C.I.F. price system in the consumer's space,d e n o t e d by cp, is therefore a point of lRm n such that, for any good i and any place j, the real number cp| is the C.I.F. price of the good at its place of consumption. For a complex of goods hk e K and for the spatial C.I.F. price system cp, the value of this consumption is by definition equal to: The consumer's aim is the best possible s a tisfaction of his n e e d s .2.2.1.

  output) is represented by the same number of points as distinct goods being supplied (demanded). We have Y = IR . As is customary, input quantities are represented by negative real numbers and output quantities by positive real numbers. Thus a production, denoted by y, wit h y = (y^) , i = 1 , . . . ,m and j = 1,...,q, is represented by a point of 3.1.2. The set Y is partitioned into a subset of all the technically possible productions, denoted by Z, and its c o m plement in Y. An element y of Z is called a producer's supply. This set Z is a specification of the set of all possible actions described in paragraph 1.1.

3. 1

 1 .3. The spatial price system peculiar to the producer is given. It depends on the location of the prod u c t i o n unit and the distances separating it from the input supply places and the output demand places. r • Let p-? , with i=1 , . . . ,n, and j = 1 , . . . ,p, be the F.O.B.unit prices of the inputs y^ supplied at the places indexed j .The unit transport prices of these inputs to the pr o d u c t i o n place j* are denoted by t^ -T h e C.I.F. unit prices of ★ c i inputs sent back to the point j , denoted by p^ , with i=1,...,n, and j=1,...,p, are therefore equal to:

  p^ , with i = n + 1 ,...,m, and j = j , be the F.O.B. unit prices of the outputs at the place of production. The unit transport prices of taking these outputs from the place indexed j* to the demand places indexed j are denoted by t|* . The C.I.F. unit prices of the outputs delivered to points j, denoted by °p^, are equal to: c ^ = p " ? + tand j = p + 1 ,...,q.

  f : Z , , -----------» [0, + °o [ tyy G Z, y | ------^ £ (y) e [0, + °° [ which is monotone and increasing for the increasing values of y.3.2.2.The classical theory of the producer p a r t i t i o n s the set of all possible productions 1 into two classes (for a given technological c o n s t r a i n t ) : (a) production said to be efficient and (b) production said to be inefficient. Howev e r this theory only holds in the special case where all inputs display maximal technical efficiency and alone determine the quantities of outputs obtained. Generally however, the efficiency of an input is relative and depends on several factors which are linked to its state and the conditions of its use. These factors are not always measurable. Hence each input's contribution to a product only corresponds to maximal technical norms in very exceptional circumstances. Moreover, free (non-economic) factors, u n c o n t r o l l a ble and fixed factors also influence production so that wellcontrolled inputs do not alone determine the quantity of o u t p u t s . It follows that any technically possible produc t i o n is more or less efficient. Instead of partitioning the set of all possible productions, we define the set of "more or less" efficient productions as a fuzzy subset of Z, denoted by C. Its elements have a membership function, denoted by such that : M c : Z |----------------^ [0,1]

  M c (y) e ]0,1 [ in all other cases. This function enables us to construct a fuzzy subset C of Z which is such that: C = £ y; V y e z, M c (y) e [0 , 1] This fuzzy subset C of Z plays the role of a fuzzy constraint in the p r o d u c e r ' s economic calculation. 3.2.3. We define the function on the monotone family defined in paragraph 3.1.2. such that: M£ • & C 2 (Z) I------------» [0 , + °° [

4. 3 .

 3 Research should now turn to the elaboration of a theory of general spatial equilibrium in a fuzzy context. Such a task seems difficult. Not only must the convenient assumptions of perfect competition be discarded as soon as the spatial factor is introduced, but also the introduction of fuzziness implies prior creation of new mathematical tools which have proved so necessary to the elaboration of this theory.

  of a fuzzy subset of a referential. 5.1.1. Definition. Let a referential E = £ x} , and let C be a fuzzy subset of E. We call a-cut of C, denoted by with a G [0,1] the non fuzzy set such that: = | x ; \|x 6 E : M^tx) > a ŵhere ^(-¡(x) is the characteristic membership function of the element x to the fuzzy subset C which takes its values in the interval Decomposition Theorem. Let C e i ( E ) , (j (E) being the fuzzy power set of E, and ^^a 3 a e [ 0 1] *ts a _ c u t s. Hence: C = U a . C aeTheorem. The following relation holds true: sup f(x) = sup [a A sup f(x)]

  Theorem. Let m £ denote the function defined in paragraph 1.2. This function is a Sugeno's fuzzy measure.

  : M £ (F.,) < M f ( F 2 ) < • • • < *l£ (F) •The sequence m £ (F^) is a monotone increasing and upper ly bounded sequence. It is therefore converging to the limit sup F.) and sup M-r(Fi) < M-r(F) from the properties of limits iX) sup Mr(F.) < f(x ) i > 0 * ~X 0 Now F = ?f F^ . Therefore, 3 j e I such that x Q £ F^ .Consequently 3 j e J such that M£(Fj) > f(xQ)Now sup (i,(F.) < f (x ) . Whence the contradiction, i > 0 * -1 0Thus we have: lim M jt(F-) = sup ju _(F. ) = M.p(