The welfare foundations of regional planning

Regional Science and Urban Economics 8 (1978) 175-1 q4. C North-Holland

THE

WELFARE

FOUNDATIONS

OF

REGIOXAL

General Equilibrium ad Pareto O@dity

PL4NNING

in a Spatial Economy

Michel MOUGEOT Dqwtrnent

of Econonrics. Vniwrsity

of Bcsanq-on, France

This paper is an attempt to define the conditions of a Pareto Optimum in a spatial economy. The conditions for an optimal allocation clf commodities and factors arc determined in either a centralized or a decentralized economic system.

I. Introduction If welfare economics has been applied in many different ways within the public economic field, ir has had little iufluence on the decisions of regional planning. Indeed, the latter seldom includes norms of a global economic efficiency. One may wonder why regional policies do not refer to a normative analysis. Two reasons may be put forward : First, most nations draw up regional policies that aim at a certain harmony between the different regions, so that disparities between developed and under-developed areas should be reduced. This objective is usually explicitly pursued. This leads to regional policies based on criteria of poverty when applied in backward areas, instead of norms of social welfare maximization. The second reason may be found in the evolution of econom;cs. When one compares the progression in space analysis wirh that in general economic theory, one may note the absence of relationship between their methods and scopes. Particularly, as far as welfare economics is concerned, one may notice that the theories influenced by Pareto’s and Walras’ studies have dealt with economies composed of a unique geographic ccntcr without taking into account a spatial extension of these economics [set Allais (1943)]. In turn, wclf:~c economics hns h:td little impact on space analysis [see Ponsard (l9SS)]. Common features do exist between thcsc NO spheres, but very few approaches IMVCfornially dsvcl~pcd a rc’;lllhcory of \Afarc toundrrtions of regional policy.

A?. Mougeot, WdJarrjkrndations of regional plamirlg

176

some measures qf economic policy likely to enable an optimal allocation of commodities and factors in space. In this first part, we develop a model that extends the general competitive and Pareto optimality Gonditions to a spatial economy by leaving out the traditional hypothesis of perfect mobility and of geographic unicity of markets. In so doing we have referred to Negishi’s (1960) model and adapted it to a spatial economy. Our approach [see Mougeot (1970, 1972, 1975)] resembles Takayama’s and Judge’s study but differs from it if we consider the variables dealt with and the tre;cment of the transport sector. It appears so as a generalization of Lefeber’s approach. Indeed our assumptions (concerning transportation factors mobility, production functions) are more general than Lcfebrian assumptions. Moreover, when Lefeber considers a given price system, \\e show the key role of the prices in a decentralised economy by proving an equivalence theorem. This one shows that a welfare maximum point satisfies the conditions of general competitive equilibrium into a spatial economy (and reciprocally) by means of tz price systems (instead of one in a spaceless economy). 2. The economy 2.J. Notations We consider an economy in which the locations of consumers and production units and the initial locations of production factors are given. The existence of a fixed number of discrete locations (or regions) pods is assumed (\\hich implies that the size of the regions is such as internal transport costs are negligible). 111each location (or region) /I = I, . . . , II, transportable goods i - I, . . ., I are produced by firms Ye = 1, . . . , R,,, supplying multiple commodities by the means of transportable inputs k = 1, . . . , K, available in the initial regions I = 1, . . %a, II, and fixed factors k’ = 1, . . . , K’. The endowments of factors in each region are given and respectively denoted by VA,and VAt,, . Transport services arc performed by spccialiscd firms sI, (s,, = 1. . . . , S,,) whose number is variable, located in every region 11, Final goods are sold on markets j = I, . . . , w to consumers ,cj = I, . . . , c;j, \vlWsC number Vi\riCS ill respect to the region. We assume that the commodii its move t’ron1the‘ l~roduccrs to the consumers. 2.2. Comurzrerpreferences Each consumer %

hlLs a utility function

=S,,(.Ti ,,.. .,.I$,),

gj=

l,...,Gj,

j=

l,...,

II,

where S,, is a utility index for consumer Cqjlocated in j and A-:~, ihc quantity of

by him. Assume thx S, ib wntinuous, monotone incrxGy. difkrcntiablc and concave. The concavity allows us to USC the thcorcms of non-linear pwgramming. Howcwr. our results could be extended to the case in which utility function is explicitly quasi-concave rather than concave [Takyama (1974)]. WC may note that the concavity implies the convexity of indifference curves and the decrease of marginal utilities. good

i consunmJ

2.3. Productiorr sets

Let p

be the production possibility set of firm I’ located in region 11and

dcfincd 3s follows :

where

Mougeot, Welfare foundations of regional planning

178

factor k b:J firm r in region h (delivered by all the regions)and used in h for the production

total use of transportable of final goods:

total use of fixed factor k’ =

1, . . . , K’

by firm r,,.

All the information concerning the production can be summariteu by the function :

following production

[j 1 1

f rh c qy”, 1 y;:, yh+,, 2 0,

I’/, = 1, . . . , R,,,

h =

I....,11. (3)

Function f’” is supposed to be continuous, concave, monotone and differentiable. In consequence, sets eC”are convex, which excludes increasing returns to scale, u hereas the concavity ofp implies non-increasing marginal returns.

2.3.2. Transport$rms Transport services are performed by specialized firms s,, = 1, . . . , S,, , in any region h. These firms utilize final goods in quantity qi,, and fixed transportation means (such as roads, railroads, etc.) denoted by 3$&. Let q:,, be the supply of transport services ensured by firm s,, and letf”” be its production function,

f”” is supposed to be continuous, concave, monotone and differentiable.

2.3.3. Production possibilityset of the entire eco/lomy This set denoted by Q is defined by the vector sum

Q=

QS" +c 1

r;

*

/I

I

Qr"

l

sj,

We assume that the sets Q, Q”“, QAhverify bility of production processes, negation of inactivity. Therefore, f”‘(O) = 0

and

f”‘*(O) = 0.

.\I. .Wougcot, We&me foundations of regional plaming

179

2.4. hcornc distribution

Each consumer gi gets an income derived from two sources. 2.4. I. Jrrcomede&Ted from tlrc selling of factors Each individual owns a finite-positive-limited endowment of production factors that he may sell on the market. Let us assume the following notations:

4i = quantity of factor k in region 1. owned by consumer g,;

agJ A’h

= quantity of fixed factor k’ in region h owned by consumer gj; c&* = quantity of factors used by the transport firms and owned by consumer gj.

If PAI,Pk., and Pth-. respectively, denote the prices of these three factors, the income derived from their sale may be written as

2.4.2. Profits distrihrtiorr Let x,,,and II,, be the profits respectively obtained by firms rh and s,, and defined as the difference between the benefits and the production and transport costs (see (37) and (41)]; R,, and it,,, are distributed to the consumers according to the following rules:

z

Jr@’ max (0, yshgJ 3~,,). SII

cc

Ysh@J

I

II)

l

=

1, Vll, Sh.

(7)

2.5. Space aid trcmport

madict

The concept of space appears very simplified in this model. Only the imperfections ‘of spatial mobility of goods and the existence ot additiljnal costs due to differences in location are taken into account. As a conscquencc, the transport market plays a key role in the formalisation of general cquilib-ium. In each region, the total supply of transport is represented by the output of the transport firms. Referring to the demand, we assume for the simplification of the model that each firm is able to estimate its need of transport services, and each firm Ye utilizes transport services provided my the specific firms located in h, both for the transportation of its output to the market location and the delivery of the factors entering into its production process. Let Trh(qirh, y$) be fitm rh’s demand for transport scrviccs, and Cssume that T’” is concave. This function differs from Lefebrisn demand function which was a macroeconomic one. Tfh plays in our model the same role as a technological constraint for the production units.

3. Optimol allocation A position in which one apent ‘s welfare cannot be increased without reducing another’s welfare is sought. This implies the existence of a partial order among the vectors of individual preferences, under the assumption that a vector is greater than another when its components arc greater than or equal to components of the other, so long as at least one component is strictly greater. In consequence, let us find the maximum of vector

over the consumption

ptissibility set which satisfies the conditions

concerning:

- market equilibrium [( 12), (13)], - the technological constraints [( 14), (15)], - the endowments of production factors [( 16), (17), (IQ]. Hence it become; possible to consider the problem of optimal allocation as a non-linear progranrming, and deduce the conditions for reaching the optimum point, by the means of the Kuhn-Tucker thcorcm. 3.1. The program According to the Kuhn-Tucker equivalence theorem [SW Ku1111and Tucker (1951) and Karlin (1959)], the problem of maximizing a vector may be replaced by that of maximizing the weighted sum of the vector’s components. Let OZ,, be the weighi coeficicnts of utility functions Sgj.

A< Ural in ~clfarc economics, the xg, can be interpretated as distriburional weights of a linear additive social welfare function IV. Therefore they represent the marginal social importance of each consumer and they have to be related to income distribution in a private property cconomq- [hfougeot (1976a, 1976b)]. WC will ascumc that the socid wclfarc function is Paretian, that is: rg, > O,Vg,.Vj

and

xxr,, = 1. 91 j

As an implication, the optimum following program I [( 1I)-( 18)]:

z&+&f, 81

‘I

=
can be found by the means of solutions of

i= I,...,I,

.j=

1, . . . . n,

(12) (13)

If we assume that the Slatcr conchion is verified, the optimum can bc obtained by differentiating the Lagrangian, v hich has previously been found in ;mociltting dual varhb!cs \Gth progrtlm 1. Lc: Pj, Pi, PA,, Pk.,,., PI,,, i.‘“, P, be the dual variables respcctivcly related to constrtiints ( 12) -( IS). t.

182

M. Mougeot, Welfare foundations

of regional plarmirlg

3.2.1. Consumptiolz optimum

By using the Kuhn-Tucker theorem, WCobtain

(19)

These relations allow us to draw several conclusions:

(1) Contrary to the spaceless Paretian model, prices change in respect to locations, but in each place there remains one price system. Actually we obtain as many systems as markets, that is to say M price systems. The existence of one price system therefore depends only on there being one market. As soon as a geographic distance is taken into account, the existence of a unique market: is unrealistic and, consequently, prices vary from location tc location. (2) From an intraregional point of’view, it is clear that the consequences to be noted in the spaceless theor:r of Pareto optimum are still valid: in each location the marginal rates of’ substitution between two goods are identical for any individual. (3) From an interregional point of view, one agent’s marginal rate of substitution cannot equal another’s as long as the agents arc differently located, because of disparities in the price systems. 3.2.2. Production optimum By differentiating the Lagrnngian with respect t.3qpl, we obtain

i=l,...,

1, j=l,...,rl,

11-l

. . . . . II, J-,,= 1, **-. R,,.

According to (20), when a commodit:! is produced ut the location where it is consumed, its selling price equals the marginal cost of production.

Furthor,

,\I. Muugeot, We.&& foundations of regional plmnir~g

183

in each region. if any tuo goods are effectively delivered on a market, their

price ratio equals their marginal rate of substitution. In turn, if the price of a commodity is smaller than its marginal cost, this commodity will not be delivered. If two firms produce and deliver the same commodity on the local market, their marginal costs of production are equal. The marginal rates of technical substitution for two commodities are equal for all the firms which produce and sell these commodities within their own regions. When a commodity is not produced at the place where it is consumed, we obtain some results similar to Lefeber’s optimum conditions: The equality between the producer’s marginal rate of substitution and the consumer’s rate of substitution for any pair of products is no longer ensured. Indeed at the optimum, the consumer’s marginal rate of substitution equals the sum of the producer’s marginal rate of transformation and the relative increases-ratio of the demand for transport services, induced by transportation of two goods from h to j. Two firms in region h which export to j will therefore have the same marginal costs, whereas two firms located in two different regions but delivering on the same market will have the same costs only if their marginal costs of transportation are equal. Eqs. (20) define the conditions for holding a regional market at the optimum. In each place, the price equals the sum of marginal production costs and marginal transportation costs. If the commodities are produced within a certain region, new firms will hold the market only if the difference between the prices prevailing in the export region and those prevailing in the import region is greater than the marginal transportation costs. 3.2.3. Optimm conditionsirt the sector of trunsport in each region, we obtain

f lerc

\VChave

an equality

betwen

I~C unil

the m:~rgind cost of various means 01’ tr;tnsport:ttiol~ wtilitcd. The price of tr:tnsp~vtiitioll wn ices \I ill h! urliquc iI1 c;lcll pl;ic-c, but 3s ~0~1~ irl the various regions cunnot bc the same, prices will vary from region to region. If the price of one unit of transportation service prevailing in one point in space is smaller ;han the’ marginal cost of its production by it firm. this firm will not provide the transport service. price of‘ t;*&nsport scrviccs ;d

184

M. Morrgeot, Welfare foundations of regional plannirlg

Final goods utilized by transport firms are determined such that

which implies the equality between their marginal imputed value and their price. 3.2.4. Optimaluse of transportableinputs We obtain three sets of conditions, according to whether USPSare local, external or in the transport sector. The local use:

(23)

t’h =

1

9

l .*,

R,,

h=

k=l,...,K.

l,r..,ll,

According to (23), tl,: price of a locally employed factor equals the m@nnl value of one additional unit of this factor iri the production process, that is equals its marginal product in value. If the latter is smaller than the former price, the factor will not be employed. In the region, factors will be used so that the marginal product ratio equals the local price ratio. The external use :

(24) h=

l,...,II,

I’/, = 1I..,, I?,,,

li = 1,...I

K,

1 =

l,...,rr,

so that: (1) If the marginal product of a factor used by a firm located in I1 is sllxiller than the sum of the price in k and the marginal cos: of transpartlrtion. this factor will not be utilized by this firm.

M. Mougeot, We&re foundations of regional planning

185

(2) If the factor is transported from place I to place h, its marginal product, in the latter place, must equal its C.I.F. price, at the optimum. (3) The marginal rate of technical substitution between two factors, in each firm, equals the C.I.F. price ratio. Consequently, a factor will be used in the place where its marginal product in value equals its local marginal product plus the marginal cost of transportation. If WCassume that a firm performing the transThe use iti the tramport swtor: portation utilizes only local factors, the condition for reaching the optimum may be denoted by

(25)

[I’&P*+i=

0,

k = I,...,

bi-, h=

I,...,

tl,

sh=

According to (25), if a factor is used as an input by a transport value equals the factor price.

I,.,.,

sh.

firm $,, its

marginal imputed

3.2.5. Optimal USCof fixed j’uctars As far as the two kinds of fixed factors arc concerned, we obtain the two conditions (26) and (27) of an optimum,

k’ = I, , . . , K’,

rh

=

I, . . . ,

a?,,, /I =

I, . , .

(

12,

186

M. Mougeot, Welfarefoundations of regional planning

These relations show that marginal products equal prices in each region but the former differ from the latter from place to place. This inequality in the productivity to fixed factors prevents the usual conditions of a Pareto optimum from being satisfied, but allows us to determine an optimal allocation, accounting for the constraints. 3.2.6.

Conditions of optimum in respect to dual t(ariables

They may be defined by differentiating the Lagrangian with respect to the Kuhn-Tucker multipliers. They are easily understood.

(28)

i=

l,...,

I,

J

=

l,...,

n,

(29)

11= I

)

.

.

.

,

II,

(30)

lc=

rh

l,...,K,

1=

I,,,,,

n,

(31)

h sh

11’= 1, . . ..n.

3 I. Alougeot. We ffarc foundations of rceiortnlp/arming

187

(33)

4. General spatial equilibrium

As is well-known in classic welfare economics, the optimum underlies the existence of a competitive equilibrium under a certain price system (and reciprocally). If all the economic agents behave,‘nt_,rmally’, that is maximize their utility functions or their profits, subject to the s;me price vector, their individual decisions comply with Pareto optimum. In turn, at a maximum welfare position, there are prices such that optimal quantities maximize satisfactions, under income constraints, and maximize the profits under technological constraints. Can this equivalence relationship be sustained when distances are taken into account? In other words do the price systems associated with the previously defined optimal position lead to a decentralization of the decisions? The equilibrium is defined according to three propositions: (i) all the consumers gj maximize their utility indexes under their budget constraints: (ii) all the producers maximize their profits under their technological constraints; and (iii) the markets are in equilibrium.

Consumer gj located in j maximizes its utility function Se,($) under his income constraint

where R,, denotes the consumer’s income as defined in the pwious assumptions and PJ’the price of commodity i on market j, at the optimum. If we call /I,, the multiplier assocititcd with the income constraint, the problem of the corresponding Lagrtingian extrcnsum enables us to state the following conditions :

(36)

M. Morrgeot, W&are foundations of regional plarvting

188

which reveals that the price ratios equal the marginal utility ratios for all the consumers. &,, the dual variable associated with the income constraint may be interpreted as the marginal utility of this income, since it expresses the increase of the objective function caused by the relaxing of the budget constraint. Moreover, according to Kuhn-Tucker theorem,

with #I, 2 0. &, is necessarily > 0 for if it were equal to zero, ihat is if the income were not all spent,

%J = 0 when

g

Pf < 0,

which is impossible because the utility function is increasing and the consumption possibility set is bounded (since initial endowments are limited). Therefore &, > 0 at the equilibrium position, and the budget constraint is met with equality. 4.2. Production equilibrium 4.2.1. The productionfirms

Each firm is assumed to maximize its profit over its production set (defined by its production function), under a given price system. A decentralization of spatial policy may especially lead to an optimum if the price system remains the one defined at the optimum in program I. The profit of a firm rlr expresses the difference between the benefits and the production and transport costs supported by the firm. In consequence, it may be written as

I

J

k

1

k’

The four successive terms of this formulation

respectively express: the total

income derived by firm r,, , the total cost of factors’ employment estimated by the prices prevailing in the factor’s original region, the total cost of the fixed factors utilized and the total cost. of the inputs and outputs transport (if we assume that the producers are aware of 7’J The conditions of profit maximization may

189

At. Mougeoi, Welfare foundations of regional planniqg

therefore bc stated by maximizing n, under the tecIlnofogica1 constraint. By associating a dual variable lrh with this constraint, we obtain l

(39)

V’”

-Pk’*+;Jh [

igj&l 1

l)j(rh

Pw =

0.

h

4.2.2. The transport Jjrms A similar process reveals that transport

firms s,, maximize their profits,

(41)

z ‘h h‘

subject to their technological constraintsf”h 2 0. By associating a dual variable 2% with these constraints, differentitlting the Lagrangiun

we obtain

by

(43)

190

M. Mougeot, Welfare foundations of regional planniw

?f”” <

-Pkh+I.Sh a_r)f”h = ’

5t

-Prh’+P

0

5 0, ’

4.3. Market equilibrium On each market, supply equals demand for non-free commodities. If the former is smaller than the latter, the commodity price is zero. Consequently, the conditions of equilibrium are similar to conditions (25) to (32). :clrthermore regional balances of payment must be at an equilibrium. Hence the following additional conditions may be stated, for all regions j (if we note lt the external regions) :

+C 1 h

Ptj'a,BJ

gh

(46) 5. Existence of a competitive spatial equilibrium and price dcccutraliaation If the economic agents are aware of the price syst.ems resolving program I [(l l)-(IQ], will their decentralized decisions lcrtd to an optimal allocation ? In order to verify this equivalence relation connecting equilibrium with optimunl, we have to compare the conditions of the latter with the Kuhn-Tucker conditions to be found in program I. Let us first recall that the market conditions and the technological constraints must be satisfied in both problems. Moreover, WCcan verify that conditions (20)-(22) and (24)-(27) are respectively similar to conditions (38)-(40) and (42)-(45), as far as the firms are concerned. Finally, by

Ai. Molcgcol, Wclfirrc~foan&tio

* 3 of regional

planning

191

comparing the conditions of the consumer with the consumption Pareto optimum, we may note that (9) is equivalent to (36), if o[~,*B, = 1. We now have to prove that such coefficients do exist and lead to the budget constraints and the balances of payments to be in equilibrium. 5.1. Existence proof The following reasoning is based on Negishi’s existence proof (1960), Judge and Takayama (1971)]. It has two parts: first, we the existence of a Pareto optimum for a given vector of weights prove the possibility of finding such a vector for which optimum competitive equilibrium.

[see Negishi demonstrate dcQ,,then we would imply

5. I. I. Existetrce of a welfare nmuintm point Optimum does exist if, for any vector cx = (x,, , . . . , Q,) belonging to a simplex SCjGi- ’ , we can define a solution of program I. If the Slater condition is fulfilled, the possible solution set is not empty, and if resources’ endowments are limited and production functions concave, production sets are bounded. Accounting for market equilibria, the possible consumption sets is closed. Therefore, HP,a continuous function, is maximized over a compact set. From the Weierstrass theorem, we know that IV has a maximum point for any c1E S~JG~-i. From the concavity of sgl, IV is concave. j;, ,Jsk and Tr,,being concave, the possible solution domain of program I is convex. In addition, if the Slater condition is verified, the Kuhn-Tucker conditions are necessary and sufficient to define the welfare maximum point.

To prove the possibility of finding a vector z = (z,, , . . . , rGn) for which optimum would bc a competitive equilibrium, we construct the following mapping [see Mougeot ( 1975)] : I)-dimensional simplex, we get a welfare (a) For any point 3con the (cjG,mr\simum point (X, Q, Y, P, 1.) as a solution of program I. (b) From homogeneity of supply and danan; functions, P can bc normalized and therefore bclougs to a simplex. The wll’;lre maximum point is contained in a wnvcx compwt set A’.So WC‘can take a positive number A such that

[IR,,-Ci

pi’ Q+~,# 0

for some gi

1.

M. Mougeo t, Welfare foundations of regional plarmirg

192

Then, the Pareto optimum differs from the general equilibrium and we have to seek another set of weights E such that -0

ag, and by normalization,

(d) Combining (a) and (c), we have a mapping from a convex compact set Kx Sx:lGf’l into itself

(a, X, Q, Y,P, 4 -+ (a-,y, t?, r,p,3. We can establish that the mapping has a fixed point: PGJ--l, c artesian product of compact, convex and non-void sets is a compact, convex, non-void set. (ii) The mapping a --) (X, Q, Y, P, A) is upper semi-continuous [see Berge (1959)] and the image is non-void and convex. (iii) The mapping a --) Z is continuous. (iv) The mapping E --) (Z, 0, F, Jj, ;i) is an upper semi-continuous mapping whose image is non-void and convex.

(9

Kx

Therefore, from Kakutani’s (1948) theorem, there is a fiscd point (3, x, D, 7, F, 1) in the mapping (x, X, Q, Y, P, 3.) + (G,x, 0. y, p, 2). As the fixed point is a Pareto optimum, to prove that it is an equilibrium point, it is sufficient to prove that consumer’s equilibrium is achieved at the fixed point. First we have to demonstrate that the budget constraints are met with equality. Let B be the sum of all income constraints:

As the fixed point is a

Pareto optimum. WCcan use the solutions of program I

to reduce B and it can be easily praved that B =f 0. As from (35),

therefore, at the ~qui~~briun~p&t,

A similar reasoning could prove that the regional balances of payments are met with equality. Second, we have to demonstrate that each consumer maximizes his utility at the fixed point. From (19),

Because of the assumption R, > 0, d,, > 0. Therefore, we get

and if we replace l/i&,

by Ii,,, we obtain the equilibrium condition.

QED.

194

M. Mougeot, Werfare foundations of regional planning

(2) The optimal allocation is also connected with a given distribution of regional endowments of production factors, the latter being likely to give rise to a structural policy aiming at reducing the wide spatial disparities. (3) At least, there exist two choices for carrying out an efficient policy of resource allocation: either a centralized policy in which the produced quantities would represent the variables, or a decentralized policy based on prices. But these are expected to fill their purpose only if they are considered as given by t le agents, which implies the existence of pure competition (which is most of the time unrealistic in a regional analysis) or the setting of a decentralized planning. We deal with these measures of regional policy in a second paper [Mougeot (forthcoming)].

References Andersson, A.E. and B. Marksjii, 1972, General equilibrium models for allocation in space under interdependency and increasing returns to scale, Regional and Urban Economics 2, no. 2. Arrow, K.J. and G. Debreu, 1954, Existence of an equilibrium for a competitive economy, Econometrica 22, no. 3. Allais, M., 1943, A la recherche d’une discipline &onomique (Tmprimerie Nationale, Paris). Berge, C., 1959, Espaces topologiques, fonctions multivoques (Durlod. Paris). Boventer, E. von, 1962, Theorie des rtiumlichen Gleichgewichts (J.T.B. Mohr, Tiibingen). Forsund, F.R., 1973, Externalities, environmental pollution and allocation in space: A genera1 equilibrium approach, Regional and Urban Economics 3. no. 1. Isard, W., 1956, Location and space economy (M.I.T. Press, New York). Isard. W. et al., 1969, General theory: Social, political, economic and regional (M.I.T. Prc~c, New York). Karlin, S., 1959, Mathematical methods and theory in games, programming and economics, vol. I (Pergamon Press, Oxford). Kakutani, S., 1948, A generalization of Brouwer’s fixed point theorem, Duke Mathematical Journal 8. Kuhn, H. and A. Tucker, 1951, Non linear programming, in: Proceedings of the second Berkeley symposium on mathematical statistics and probability (University of California Press, Berkeley, CA). Lefeber, L., 1958, Allocation in space; Production, transport and industrial location (NorthHolland, Amsterdam). Mougeot, M., 1970, L’optimum en tconomie spatiale, Working paper (I.M.E., Dijon). Mougeot,M., 1972, 3ptimumCconomique et annlyse spatialc, Doctoral dishcrtution (University of Dijon). Mougeot, M., 1975, ThCorie et politique hconomiques rkgionalcs (Economica, Paris). Mougeot, M., 1976a, Peut-on &parer I’Cconomique et le social? Rcvuc d’Economic Politiquc, no. 4. Mougeot, M., 1976b, Analyst de surplus et repartition des rc\‘cnus Revue Economiquc, no. 6. Mougeot, M., forthcoming, The welfare found;ltions of regional l?Ii\nning: Ol>tin~;llrcgioniil planning, Regional Science and Urban Economics. Negishi, T., 1960, Welfare economics and existence of an equilibrium for a competitive economy, Metroeconomica 12. Ponsard, C., 1955, Economic et espace (S.E.D.E.I.S., Paris). Takayama, A., 1974, Mathematical economics (Dryden Press, Hinsdale). Takayama, T. and G.G. Judge, 1971, Spatial and temporal price and allocr~tion models (North-Holland, Amsterdam).