Florence Goffette-Nagot

Bertrand Schmitt

AGGLOMERATION ECONOMIES AND SPATIAL CONFIGURATIONS IN RURAL AREAS 1

Keywords: economic geography, rural areas, labor-market areas, densities, land rents, urban spread, commuting, service location. JEL classification : R12, R23 économie géographique, espaces ruraux, bassins d'emploi, densités, rentes foncières, étalement urbain, migrations alternantes, localisation des services

The question to be addressed here is that of the agglomeration/dispersion forces that are likely to account for the location of people and jobs in rural areas and the way they explain spatial patterns in rural areas depending on urban influence. Economic geography models may provide suitable tools with which to investigate the organization of rural areas. We first review these models, focusing on dispersion forces, which rest basically on land consumption and transport costs. We suggest then a set of hypotheses concerning the main forces at work in rural areas. Intensity of agglomeration economies is hypothesized to be related to the urban size, which in turn induces increasing land rents and finally agglomeration diseconomies. Such diseconomies encourage population spread around the city and in a second stage a possible partial decentralization of population-serving firms, which seek the proximity to the households. The consequences in terms of spatial patterns are that beyond a certain threshold of the city size, decentralization of population-serving firms occurs, giving rise to secondary services centers, whereas services remain concentrated in the center for smaller cities. Empirical results concerning population densities, labor force exchanges and distribution of residentiary services in labor-market areas surrounding cities of more than 20,000 inhabitants in six French regions are presented.

INTRODUCTION

The pattern of development of rural areas in the Western countries has changed considerably in the last twenty-five years. A long tradition of movement away from the countryside has been superseded by net demographic gain in sparsely populated areas, primarily because of migration (Champion, 1989[START_REF] Champion A G | Urban and regional demographic trends in the developed world[END_REF][START_REF] Cochrane S G | Recent trends in migration between core and peripheral regions in developed and advanced developing countries[END_REF]. This population growth is not uniform across rural areas. For instance, in the United States, it was more important in adjacent non metro counties than in non adjacent ones [START_REF] Carlino G A | Declining Productivity and the Growth of Rural Regions: A Test of Alternative Explanations[END_REF]. In France, the migratory balance of rural districts ("communes") depends positively on the position relative to the urban network, that is, it improves with proximity to a city and with the size of that city [START_REF] Fanouillet C | L'influence des villes ne cesse de s'accroître[END_REF]Hilal et al. 9 1995). This means that rural areas are greatly influenced by urban spread. Actually, there are several reasons why metropolitan growth, which occurs due to growing agglomeration economies, results in decentralization toward rural areas [START_REF] Gaile G L | The spread-backwash concept[END_REF][START_REF] Parr | City size distributions and urban density functions : some interrelationships[END_REF][START_REF] Suarez-Villa L | Metropolitan Evolution, Sectoral Economic Change, and the City Size Distribution[END_REF].

Accordingly, rural development can be analyzed relative to metropolitan growth and depending on the distance to the urban core [START_REF] Barkley D | Identifying "spread" versus "backwash" effects in regional economic areas: a density functions approach[END_REF], although it is useful to account for local amenities as well [START_REF] Carlino G A | The determinants of county growth[END_REF][START_REF] Boarnet M G | An empirical model of intrametropolitan population and employment growth[END_REF]Schmitt, 1996;Henry et ai 9 1997). Urban influence on surrounding rural areas can be analyzed, not only on this dynamic viewpoint, but also in terms of spatial patterns, thus designating population and employment density curves and emergence of secondary centers in hinterlands of urbancores.

Namely, as urban spread spillovers with varying intensity depending on the distance from the core and the core size, one can suppose that this will result in specific spatial patterns around cities of different sizes. This is the question that is addressed here: which spatial configurations emerge in rural hinterlands of cities, and how can they be explained?

Economic geography [START_REF] Schmitt | Product differentiation, monopolistic competition and city size[END_REF][START_REF] Rivera-Batiz F L | Increasing returns, monopolistic competition, and agglomeration economies in consumption and production[END_REF]Krugman, 1991a and1991b; see Krugman, 1996 or Fujita and[START_REF] Fujita | Economics of agglomeration[END_REF] for a recent survey) aims at explaining spatial distribution of population and economic activity and emphasizes the forces that bring about particular spatial configurations. Although economic geography is usually used in order to explain formation of cities or more generally the emergence of spatial concentration, we make the assumption that it may provide suitable tools with which to investigate the organization of rural areas. Indeed, besides mechanisms of agglomeration, economic geography models include dispersive forces. These forces should be able to explain urban spillovers on rural areas. In other words, just as urban network emerges due to specific centripetal and centrifugal forces, we would like to show that the same forces generate specific spatial patterns in rural areas in proximity to the cities.

Our purpose is therefore to present the agglomeration/dispersion forces that are likely to account for the location of people and jobs in rural areas and the way they explain spatial patterns in rural areas depending on urban influence. Of course, there are a great many factors involved in answering these two questions and our aim is not to be exhaustive, but merely to explore the main mechanisms that operate in rural areas. This implies, first, a need to review economic geography models, focusing on dispersion forces, which rest basically on land consumption and transport costs. We suggest then a set of hypotheses concerning the main forces at work in rural areas. Intensity of agglomeration economies is hypothesized to be related to the urban size, which in turn induces increasing land rents and finally agglomeration diseconomies. Such diseconomies encourage population spread around the city and in a second stage a possible partial decentralization of population-serving firms, which seek the proximity to the households. The consequences in terms of spatial patterns are that beyond a certain threshold of the city size, decentralization of population-serving firms occurs, giving rise to secondary services centers, whereas services remain concentrated in the center for smaller cities. Empirical results are presented in order to test these hypotheses. Population densities, labor force exchanges and distribution of residentiary services are examined and show the existence of the hypothesized spatial patterns in labor-market areas surrounding cities of more than 20,000 inhabitants in six French regions.

Section 2 is a brief review of agglomeration models and focuses on the mechanisms of dispersion which are induced mainly by the consumption of land by economic agents. Section 3 sets out a series of hypotheses to explain the patterns of concentration/de-concentration of people and activities in rural areas. Emphasis is on the role played by forces that disperse population around the urban centers and on the effects of such de-concentration on the location of residentiary services. Section 4 is a empirical analysis of rural areas in six regions of France where we examine the distribution of population and economic activities within areas of urban influence and try to relate the patterns of distribution to the determinants referred to earlier.

THE PRINCIPLES OF AGGLOMERATION AND DISPERSION

There is in fact no general explanatory model of agglomeration and, as [START_REF] Fujita | Economics of agglomeration[END_REF] pointed out, "the forces in action, or at least their respective intensities, are not necessarily the same depending on the geographical entity selected'. We would like first to present an overview of economic geography models and the main assumptions developed in those models, in order to select and examine more closely those assumptions which are the most relevant to the analysis of population and activities' distribution in rural areas. Moreover, beyond recalling the different microeconomic mechanisms that explain agglomeration, we seek to show how the assumptions which are posited to integrate space in these models affect the way that the forces of concentration and dispersion are expressed.

Economic geography models seek to establish general spatial equilibrium on the basis of microeconomic mechanisms, usually by reference to imperfect competition principles. Four principal forces of agglomeration are highlighted to explain the formation of spatial equilibrium configurations [START_REF] Fujita M | Spatial interactions and agglomeration in urban economics[END_REF][START_REF] Fujita | Economics of agglomeration[END_REF].

Economic geography models with comparative advantages, dealing with industrial location, assume the existence of exogenous spatial heterogeneity within a framework of pure and perfect competition: the existence of location-related attributes giving rise to benefits from locating at particular points in geographical space leads to the spatial concentration of activities with the same "preferences" for those attributes of the area [START_REF] Arthur W B | Silicon Valley location clusters: when do increasing returns imply monopoly ?[END_REF].

At the intersection of industrial economics and urban economics, models featuring technological externalities (or externalities in the narrow sense) introduce non price interactions among agents, from which agents benefit and which induce them to seek mutual proximity; such externalities may be introduced through the household utility function [START_REF] Beckmann | Spatial equilibrium in a dispersed city[END_REF] or by altering costs (of transactions, information, etc.), or the productivity of firms (Ogawa and[START_REF] Ogawa | Equilibrium land use patterns in a nonmonocentric city[END_REF]1989).

Monopolistic competition models, initially developed to explain phenomena relating

to the international economics, use an industrial economics framework emphasizing price interactions ("pecuniary externalities") in order to examine the spatial consequences of the existence of increasing returns to scale in production, combined with preference for the variety of goods and the existence of a transport cost for manufactured goods (Krugman, 1991b); these models formalize a cumulative agglomeration process in which firms in monopolistic competition as described by [START_REF] Dixit | Monopolistic competition and optimum product diversity[END_REF] seek to be close to the most extensive markets, whereas households seek to be close to the places where firms agglomerate so as to benefit from a greater variety of goods.

Finally, in an endeavour to renew central place theories by providing them with microeconomic foundations, oligopolistic competition models look at the spatial consequences of the strategic location of firms which are in competition over prices and for market areas; price competition is a centrifugal force, whereas competition for market areas is a centripetal force (d'Aspremont et al. 9 1978).

The hypotheses posited to take space into consideration vary depending on the geographical scale of analysis of each model: inter-urban, inter-regional or intra-urban. These assumptions about space are important in that they generally represent the dispersive force that counteracts agglomeration. They relate on two major registers: land market and transport costs. Other dispersive forces may be envisaged (limited agglomeration economies, outcome of competition between firms), but it is often the consideration given to space as land area or distance that acts as a dispersive force for some agents, and it is the counterpart to concentration of activities which we feel it is essential to an understanding of the mechanisms at work in rural areas.

The existence of land competition is a dispersive force for economic agents. It is not necessary to assume that all agents consume land and influence land market. But in order to take into account land as a dispersion force, there is a need for at least one agent category to be linked to land. For instance, in Krugman's model (1991b), two dimensionless regions are considered without any land market. Land is yet implicitly taken into account through nonmobile agricultural population producing a good that is consumed in the agglomeration, while at the same time it represents a dispersed population that must be supplied with industrial goods. This assumption forms a link with space as a land area. Other models assume more explicitly that land is consumed by agriculture, while continuing to consider dimensionless agglomerations [START_REF] Fujita | When is economy monocentric ? von Thunen and Chamberlin unified[END_REF]. In the case of intra-urban models, prominence is given to the occupation of land within the city, the existence of the agricultural sector being ignored. It is then generally considered that firms and households consume land [START_REF] Fujita | Multiple equilibria and structural transition of non-monocentric urban configurations[END_REF][START_REF] Fujita M | A monopolistic competition model of spatial agglomeration: a differentiated product approach[END_REF], whereas some "systems of city" models, where internal urban organization is less central, confine the consumption of land to households only [START_REF] Schmitt | Product differentiation, monopolistic competition and city size[END_REF][START_REF]Toward a general-equilibrium theory of a core-periphery system of cities[END_REF]. In this last instance, there is assumed to be a center (the Central Business District) as in the earliest urban economics models [START_REF] Alonso W | Location and land use[END_REF][START_REF] Muth R | Cities and housing[END_REF]. Finally, oligopolistic competition models, such as d'Aspremont et al (1979) assume most of the time that consumers are spread uniformly which, like the even distribution of farmers in Krugmaris model, comes down to introducing a degree of "attachment" to land, although one can also consider explicitly the households' consumption of land [START_REF] Fujita | Spatial competition with a land market: Hotelling and von ThQnen unified[END_REF].

The introduction of land consumption induces, generally, the consideration of transport costs. In particular, the fact that households use land and tend to spread out usually entails consideration of commuting costs, as is the case in [START_REF] Fujita | Multiple equilibria and structural transition of non-monocentric urban configurations[END_REF]. Symmetrically, the agglomeration force can be weakened by the existence of transport cost of agricultural commodities [START_REF] Fujita | When is economy monocentric ? von Thunen and Chamberlin unified[END_REF][START_REF] Calmette M-F | Localisation des activités : un modèle bi-sectoriel avec coûts de transport[END_REF]. Whether it be the cost of traveling from home to work or the transport cost of agricultural goods, in both cases we are faced with a cost generated by the consumption of a localized good: in the first case, it is consumption of land which compels households to disperse a certain distance away from firms and makes them bear -besides the actual land price -the travel costs; in the second case, it is the consumption of the agricultural good which is necessarily tied to the land and which is transported at a cost, which plays the same role.2 Finally in monopolistic competition models, the transport cost of industrial goods is introduced [START_REF] Fujita M | A monopolistic competition model of spatial agglomeration: a differentiated product approach[END_REF]Krugman, 1991b).

The cost of commuting is one reason why households cluster close to firms. But over time these costs act as a brake on the agglomeration of firms. For example, Abdel-Rahman and [START_REF]Specialization and diversification in a system of cities[END_REF] show how commuting costs that grow with the population cause wages to rise or, where firms are also assumed to consume land themselves, are passed on to firms through higher land prices. For the same reasons, the cost of transporting agricultural produce may, in monopolistic competition models, be thought prima facie to produce agglomeration diseconomies. But in some cases, increased transport costs for agricultural goods may induce greater concentration of firms and households [START_REF] Fujita | When is economy monocentric ? von Thunen and Chamberlin unified[END_REF]. The effects of the transport cost of industrial goods are also ambiguous. In a two-region model, the tendency to agglomerate declines with the increased cost of transport of industrial goods (Krugman, 1991b). Conversely, it can be shown in a system of cities model that increased transport costs of industrial goods lead to greater concentration [START_REF] Fujita | When is economy monocentric ? von Thunen and Chamberlin unified[END_REF]. In fact, the effect of transport costs of manufactured goods depends on the assumed mobility of workers between the agricultural and industrial sectors. More generally, it seems that very high and very low transport costs of industrial products lead to dispersion, whereas agglomeration occurs for medium values [START_REF] Fujita | Economics of agglomeration[END_REF].

Finally, it might be said that the purpose of agglomeration is to reduce distances between agents (to maximize the benefit derived from externalities, to increase sales, to reduce commuting costs, to cut the overall cost of goods consumed), but distance is not the only "dimension" of space: the necessary consumption of land imposes a limit on agglomeration and finally causes the agglomeration process to break up through the formation, in the case of systems of cities, of new agglomerations. These dispersive forces related to the direct or indirect consumption of land and to transport costs seem particularly important in analyzing the mechanisms governing the distribution of economic activities in rural areas.

3.FROM THE MECHANISMS OF ECONOMIC GEOGRAPHY TO SPATIAL CONFIGURATIONS IN RURAL AREAS

The economic mechanisms referred to previously (agglomeration externalities, transport costs, economies of scale, tastes for variety, etc.) and the historical development of the economic variables in play (reduced transport costs, greater economies of scale, increased share of spending on non-agricultural goods, increased importance of information exchanges, etc.) highlight the inevitability of the spatial concentration of activities and population through circular and cumulative phenomena. Such concentration is restricted, however, by agglomeration diseconomies generated, among other things, by congestion. 3 3 There are in addition a number of activities that do not follow the trend towards concentration: agriculture, forestry, mining and quarrying but also recreational activities and tourism as well as certain industrial activities that benefit from comparative advantages or externalities specific to rural areas (GofTette-Nagot, Schmitt, 1996)

The forces at work in rural areas

1. Clusters of activities spread over space and form a system of cities of various sizes and rank orders. We assume that the presence of cities of different sizes and functions has repercussions on the distribution of economic activities in rural areas. The intensity of agglomeration economies in the city which relates to its size is liable to influence land rents and ultimately the distribution of the population around the city. Hence, although agglomeration economies that account for the simultaneous concentration of firms and households in cities are of little direct concern to us, they nonetheless have repercussions on our subject matter.

2.

There is a trend of active population de-concentration towards the rural districts around cities. 4 The reduction in traveling costs through better roads and increased number of private cars, combined with the desire for homeownership and detached houses, and the demand for space growing with the household size and the search for rural amenities are all reflected by increased distances between work and residence places. Two assumptions may be made to explain why this dispersive movement affects households rather than firms: first, firms have more need than households to interact with other economic agents; second, land competition within cities bears more heavily on households than on firms because of the proportion of the household's spending devoted to housing. This "periurbanization" movement affects areas which are more and more remote (Goflfette-Nagot, 1997). It is nevertheless limited by the generalized commuting cost which has a "recoil effect" drawing households closer to firms. This component of population distribution is influenced by the city size, acting through land rents which increase, everything being equal, with the intensity of agglomeration. Thus the areas affected by population dispersion should increase more than proportionally to the city size, as do land rents.

Analyzing this population de-concentration involves reference to the consumption of land by households and the possibility of commuting. The phenomenon might be amenable to analysis with a model like that of [START_REF] Fujita | When is economy monocentric ? von Thunen and Chamberlin unified[END_REF] which deals with a system of 4 A tendency of retired households to move away from urban areas is observed [START_REF] Warnes | Housing aspirations and migration in later life: Developments during the 1980th[END_REF], although on the face of it nothing compels them to select a location on the edges of cities. It should be remembered though that the dispersion of this population is limited because of a degree of immobility with age and the need to be near facilities providing health care and shops. cities where the dispersion force is the consumption of agricultural produce. Such a model would involve replacing agricultural activity by residential activity: it is not so much agricultural produce as the labor force that is transported between cities and rural areas nowadays [START_REF] Jayet H | Flux et position des espaces ruraux[END_REF]. Not only does the location of crop systems no longer comply with von Thünen f s scheme but the transport costs for agricultural produce and the marketing channels for these goods are such that the need for agricultural areas to supply cities no longer constitutes an essential agglomeration diseconomy [START_REF] Duranton G | Economie géographique, urbanisation et développement[END_REF], as still appears in Krugman's models for instance. It could then be considered that agriculture is virtually independent of the proximity of the city but is an alternative land use, generating an opportunity cost for land (agricultural land rent) that is little influenced by distance from the city. Models featuring a system of cities emerging in a homogeneous space occupied by an agricultural sector which has no direct link with the cities seem therefore more relevant to our purpose.

3.

The dispersion of population we have referred to may alter the location of population-serving activities. Theoretical models of agglomeration seldom distinguish between the firms that produce the commodities required for household consumption and the firms that distribute those commodities. In such models it is as if either the manufacturing firms delivered their output to the household's home, or the households shopped directly at the firms when they commuted to the work places. This view of things raises few difficulties as long as households remain closely tied to their work place. But when the tie between work and home is substantially slackened, this assumption must be loosened to allow for possible differences in locations between manufacturing and population-serving firms. 5 The mechanisms governing the agglomeration of manufacturing firms among themselves and close to households are not the same as for population-serving firms. While manufacturing firms follow the general pattern of economic activity concentration for all of the reasons mentioned earlier, distributing firms may well remain closer to households because of the transport costs that households incur in traveling to them. It may be asserted that the 5 [START_REF] Rivera-Batiz F L | Increasing returns, monopolistic competition, and agglomeration economies in consumption and production[END_REF] and [START_REF] Schmitt | Product differentiation, monopolistic competition and city size[END_REF] introduce the household service sector into their models. But in so far as they are out to explain the size of the city and not its internal organization, they can overlook household traveling costs (commuting to work or shopping trips); services are then located either in the CBD or at the place of residence of the households. transport costs for these activities are comparatively higher than those related to productive activities: while transport costs between manufacturers and distributors are for the transport of goods, the costs between distributors and households are for carrying people. Such costs are higher overall both because it is more expensive to carry a person than a commodity and because these costs are borne by the household's budget, even if the possibility for households to combine shopping trips and commuting journeys limits the scope of this dispersion force. Of course, distributors also benefit from agglomeration economies because of internal scale economies as well as tastes for variety and range economies. Accordingly, it may be supposed that population-serving firms are comparatively closer to the population and therefore less concentrated than manufacturers for reasons of transport costs. At the same time, they may be concentrated in certain locations because of their own agglomeration economies.

Thus it seems essential to separate manufacturing firms and population-serving firms.

The linkages that account for agglomeration in monopolistic competition models are of course relevant at the higher levels of systems of cities, but at our scale of analysis they must be split into two different mechanisms, that is, location of households close to manufacturing firms and location of population-serving firms close to households.

Consequences for the spatial organization of activities in rural areas

From agglomeration and dispersion forces which we have evoked above and the way they play in rural areas, one can build some assumptions about the spatial organization of economic activities and population around the places where activities are focused. First, it seems that the dispersive force ensuing from land rents affects households more strongly than firms, implying greater concentration of jobs than population. In addition, we have suggested that the periurbanization area (and therefore the range of urban influence) must increase more than proportionally with the size of the agglomeration, thus leading to higher population densities in the areas located on the outskirts of large agglomerations.

Beyond a certain threshold of city size, it is expected that shops and personal services become delocalized (i.e. develop outside the central city while forming secondary clusters because of their own agglomeration economies), as a result of the peripheral market attaining sufficient size. Hence, a nested pattern of the Christaller-Losch type is thought to develop around the largest agglomerations, concerning mainly shops and personal services. The existence of secondary clusters of shops and personal services on the periphery of large cities is liable to influence population densities, which on these peripheries cannot follow the exponential negative function expected in the monocentric model, but must rise at a certain distance from the center.

Conversely, on the outskirts of small agglomerations, the population density is low and the extent of urban influence area limited. Thus the local market is reduced (in population and in surface area). In this case, distributing firms do not find it advantageous to locate on the periphery where the demand they can drain is too low. It is therefore the scale economies that dominate in the location choices of these firms and it may be assumed that shops and personal services tend to be concentrated within the small agglomerations and to be absent around them. In this case, the spatial organization is close to the pattern of the monocentric model and we can assume that the population density can follow a exponential negative shape.

INTERNAL ORGANIZATION OF LABOR MARKET AREAS IN SIX FRENCH

REGIONS

In accordance with the previous section, the hypothesis to be examined here is that the areas of urban influence are organized differently depending on the size of the employment center. Such an analysis means detennining the places where activities are concentrated and the spheres of influence in terms of commuting patterns of these places. To this end, six regions (Alsace, Burgundy, Franche-Comte, Lorraine, Midi-Pyrenees and Rhone-Alpes) were divided up into labor market areas (in short LMAs) on the basis of 1990 inter-district commuting flows (see appendix 1). 429 LMAs of various sizes in terms of land area and population -this variability being related to the size of their employment center-were obtained (Map 1 and Table 1). Three aspects of the internal organization of LMAs will be considered. First, the degree of concentration of population and jobs and the shape of the population and employment density curves will be examined as functions of the center size. Second, the analysis of work force commuting between the employment center and the periphery will show that these flows are not symmetrical. Finally, examination of personal services' spatial distribution will demonstrate that beyond a certain threshold of population in the center, these activities are located not only at the center, but also on its outskirts.

Map 1 -Centres of employment, centres of services in the LMAs (Alsace, Burgundy, Franche-Comte, Lorraine, Midi-Pyrenees, Rhone-Alpes)

Population and jobs distribution and densities

Distributions of population and jobs between employment centers and their spheres of influence show that the degree of concentration of population and activities increases with the center size (Table 1). Thus, despite covering large areas, the spheres of influence of centers of more than 100,000 inhabitants account for only 30% of the population and 20% of jobs of the LMA to which they belong. Small centers concentrate less than half of the population and slightly more than 60% of the jobs of their LMA. These differences in the degree of concentration may be interpreted as the result of the greater presence in large LMAs of economic activities that are sensitive to agglomeration economies and carry a large proportion of population in their wake. Moreover, the fact that jobs are more intensely concentrated in centers than population is to be related to the idea that households are less sensitive than firms to agglomeration economies and more sensitive to the dispersive forces of competition for land use. Finally, we assume that the dispersive force of the active population around employment centers increases with the center size, since it results from competition for land use, which competition increases as the activities and population concentrate under the influence of more intensive agglomeration economies. (1951) and for two decades, the urban population densities could be described by the negative exponential function as D(x) = D^.e'"*, where D(x) is the population density at distance x from the center, Do the density in the center and a the density gradient [START_REF] Mcdonald | Econometric studies of urban population density : a survey[END_REF]). Yet this functional form is inadequate for capturing irregularities in density patterns [START_REF] Latham R | Population density in metropolitan Toronto[END_REF][START_REF] Mcdonald | Some tests of alternative urban population destiny functions[END_REF]. According to our hypotheses a flexible functional form has been chosen in order to investigate whether the curve shape is variable with the LMA central city size and whether some secondary agglomerations exist on the periphery of large cities. Following Anderson's suggestion (1982) and some recent research [START_REF] Anderson | The changing structure of a city : temporal changes in cubic spline urban density patterns[END_REF][START_REF] Skaburskis A | Inversions in urban density gradients: a brief look at the Vancouver metropolitan area's density profile[END_REF][START_REF] Zheng X P | Metropolitan spatial structure and its determinants: a case-study of Tokyo[END_REF]Barkley et al 9 1996), we estimated population density curves using a cubic spline specification (see Appendix 2 for a presentation of this function ; results are presented in Figure 1 andTable A.l appended).

It is worth noting that whatever the center size, population densities fall rapidly in the immediate periphery of the employment center (coefficients of the distance are negative and high) suggesting the existence of a discontinuity between densities in the center and on the periphery. Furthermore, the shapes of the density functions differ in relation to the center size. 6

There is a rise of density in the periphery of the employment centers having more than 20,000 inhabitants (coefficients of quadratic terms are positive), highlighting the existence of population (and probably jobs) agglomerations at a certain distance from the center (at 16 km for the LMA with center from 20,000 to 100,000 inhabitants; at 30 km where the center is greater than 100,000 inhabitants). 7 This « bump » is rather low on the periphery of centers of 5,000 to 20,000 inhabitants and doesn't appear around centers below 5,000 inhabitants. The shapes of density curves for the two latter categories are closer to the negative exponential function than for the others. This result is consistent with the hypothesis of emergence of secondary centers (agglomerations of population-serving jobs) on the periphery of large cities.

However, the coefficients of quadratic terms remain low, thus suggesting only small periphery centers which differs from some North-American results (Barkley et ai 9 1996; [START_REF] Garreau | Edge city. Life on the new frontier[END_REF].

6 Due to the existence of knot points located at different distances, structural parameter tests (such as Chow test) seem difficult to implement. Without one of them, we cannot have a real appreciation of the significance of coefficient differences between the LMAs classified by center size. LMAs with Center > 100,000 inhab.

(2 knot* : Kl -23 km, K2 -46 km) Density functions variations over time show that the density in the center increased between 1968 and 1975 but was stable through 1990. By contrast, the density on the periphery increased steadily during period (1968 to 1990) in accordance with our assumptions. Yet, one can speak of a real population spread only in the LMAs of which center is greater than 20,000 inhabitants. In these LMAs, the density rise is more important in the range between the employment center and the secondary centers than beyond these secondary centers. Therefore, the density "hollows" between the city edge and the secondary centers tend to disappear over time.

Finally, the population distribution patterns in LMAs differ in relation to the size of the employment center. When the center is large, the population is more concentrated in the center. But at the same time, its spread on the periphery is more intensive, with respect to the initial rural densities. Furthermore, we observe on the outskirts of large cities the presence of some density "bumps". We assume that they are indicative of places where population-serving jobs are maintained or developed. At the opposite, when the center is small, the population distribution pattern is closer than what is expected by the monocentric urban model.

Differentiated relations between employment centers and spheres of influence

There is an apparent paradox in saying, on the one hand, that concentration of jobs and population in the center is proportionally more intense when the employment center is larger and, on the other hand, that the population density on the outskirts increases with the center size. This paradox is reflected by differences in the respective roles of the centers and their peripheries in terms of supply and demand of labor.

The fact that the dispersion of population is more intense around large centers entails, greater reliance of the outskirts on the center in terms of employment. This is shown by the proportion of the active population who reside in the outskirts and work in the center, which is greater for large centers than for small ones (Table 2): a third of the active population living on the outskirts of centers of more than 100,000 inhabitants have their jobs in the center and only one quarter for centers of 5,000 to 20,000 inhabitants. Conversely, the greater concentration of population and jobs in the centers of large LMAs makes them less dependent on their periphery for labor supply. For instance, the proportion of jobs in the center occupied by the active population living in other districts of the same LMA stands at 13% for centers of more than 100,000 inhabitants and approaches 20% for poles of 5,000 to 100,000 inhabitants. Accordingly small centers have clearly greater need of their periphery than larger ones for the labor supply. Conversely, zones under the influence of large centers are clearly reliant on their center, whereas the outlying areas of small employment centers are more selfcontained. It is as if the relationship between center and sphere of influence were asymmetrical for large employment centers, with the center playing a fundamental role with regard to the labor supply of the periphery, and the periphery only being marginally involved in the workings of the center. This relationship is more evenly balanced in the small LMAs powered by small centers, the outskirts being as much in need of jobs at the center for their active population as the center needs the periphery to fill its jobs. We might speak in the first instance of a submission-dependence relationship and in the second of an interdependent relationship.

This difference in the relations between centers and their spheres of influence may be interpreted in the context of our analysis as the outcome of agglomeration economies which increase with the increased size of the city -explaining the high concentration of jobs and population at the center of the largest LMAs -and of a dispersive movement of the population with jobs at the center which is greater when the center is large, which accounts for the greater density of zones influenced by large centers. The first set of factors accounts for the greater independence of large centers relative to their outskirts; the second, explains the greater dependence of their outskirts relative to the employment center. The combination of the two may account for the interdependence between the center and the periphery within small LMAs.

Distribution of shops and personal services

The enhanced density of the peripheral areas increases local demand for goods and services, which may entail relative derealization of distribution activities. The trade-off between economies of scale and transport costs to which these activities are subject, results

either in maintaining their concentration at the center, or in causing relative dispersion in the periphery with secondary clustering in the service centers, depending on the size of the local markets. Analysis of the spatial distribution of population-serving activities has been conducted on the basis of districts' endowment in shops and services as reported in the 1988 District Inventory (cf. appendix 3). 8 The level of residentiary services in LMAs is, of course, related to their population (Table 3), but this relationship is not linear and LMAs powered by small centers tend to have higher proportion relative to their population of basic household services, than those influenced by large centers. Their small size notwithstanding, employment centers with 5,000 to 20,000 inhabitants have extensive facilities, that is, they have almost all types of residentiary services. They are distinctive from the large centers not in the range of available facilities but in having a smaller number of facilities of the same type. The greater number of retail stores in large centers increases the range of products available and diversifies their quality, which may favor the cumulative agglomeration of consumers because of their taste for variety. Conversely, employment centers of less than 5,000 inhabitants do not have the full range of facilities. They lack higher ranking facilities and even certain intermediate level facilities.

Not only employment centers supply goods and services for local populations. 55% of the weighted frequency of residentiary services in LMAs powered by centers of more than 20,000 inhabitants comes from residentiary services located outside the employment center, 40% for centers of 5,000 to 20,000 inhabitants and 30% for centers under 5,000 inhabitants.

Thus, while shops and personal services are numerous in zones influenced by large centers, they are rare on the outskirts of small centers. This tendency toward differentiated dispersion Analysis of the level of facilities in districts has been preferred to the spatial distribution of jobs in commerce and personal services. The latter indicator is less specific about the goods and service available locally. of population-serving activities reveals the role both of the size of the local market which is known to increase with the center size and the long distances that some of the population of large centers' outskirts have to cover to reach the center. It is as if there were a level of population in the center (and therefore of spatial and demographic dimension of the periphery) beyond which shops and personal services tended to de-centralize. Below this level, they remain clearly grouped in the employment center. The residentiary services located on the outskirts of the large centers are not uniformly distributed though. A secondary concentration of these activities can be observed in a limited number of places that we have termed service centers (cf. appendix 3). There are more than ten of them per LMA with centers of 100,000 inhabitants, more than three per LMA with centers of 20,000 to 100,000 inhabitants and almost non existent on the outskirts of small centers. Naturally their mean level of residentiary services is very low compared with that of employment centers and they can only provide goods and services of an intermediate level.

Such re-concentration reveals the role of internal scale economies, and above all range economies, which off-set the influence of shopping trip costs for households. Moreover, the fact that these service centers are often small urban units tends to show that, under the play of cumulative agglomeration mechanisms and market size effects, these facilities are grouped and localized where there is a population cluster, albeit a limited cluster.

There is then a fabric of service centers on the periphery of the large employment centers providing a large dispersed population with a range of low and intermediate ranking facilities. Facility areas driven by these service centers are rather small and higher-ranking services and shops are available in the employment center. Conversely, the same mechanisms entail a tendency of shops and personal services to cluster in the center when it is small. Small employment centers provide inhabitants of the districts they influence with all the goods and services the population usually requires. It is as if the space under the influence of large agglomerations was organized along the lines described by Christaller, whereas it was organized as a monocentric space around small centers.

CONCLUSION

The theoretical considerations presented above, supported by the empirical analysis, highlight the value of looking at contemporary rural areas with the results of the economic geography models developed over recent years. While preserving the mechanisms that explain agglomeration, the task here is to implement them by examining the dispersive forces involved more closely. The inevitable and self-sustaining character of the agglomeration is emphasized, as is therefore the existence of a system of cities of different sizes around which the rural areas are structured. The influence of competition for land use is a force that disperses population around these points of concentration of activities and population. As land rent increases with the size of the city, the movement is more intense and more extented for larger employment centers. The denser demographic fabric on the outskirts of employment centers may, when the local market becomes sufficiently large, lead to delocation of those activities most closely related to households some of which, where applicable and under the combined effect of household traveling costs and of their own agglomeration economies, concentrate on the outskirts of the employment center.

The densification of population on the periphery of employment centers may, when the local market becomes sufficiently large -that is, near big cities-lead to a primary decentralization to the periphery of those activities most closely related to households.

Moreover, because of households' transport costs and their own agglomeration economies, some of these activities will concentrate within the periphery, thus forming secondary services centers.

This set of assumptions is used to show that LMAs are structured in two different ways and that the size of the employment center is responsible for this difference. The large and medium-sized centers exert their influence over large land areas. These centers which, under the effect of agglomeration economies, are strong foci for the area's labor supply and for a large part of its population exert great drawing power over their entire zone of influence whereas the area has little impact on the workings of the center. In parallel, the overflow of the urban residential fabric to the now remote outskirts of these centers, under the effect of competition for land use, seems to entail a tendency towards delocation of services to households, which activities are concentrated outside of the central agglomeration in small peripheral urban units under the combined effect of the cost of transport for households, economies of scale of these sectors, and agglomeration economies.

By contrast, the LMAs with small employment centers display internal consistency that is less dependent on the central focus. They are spread over smaller land areas, have more widely dispersed population and jobs because of the impact, among other things, of farming and have a less attractive centers in terms of employment although their economy depends largely on the active population living outside the center. The employment center also has the specific feature of concentrating most shops and personal services which reinforces the reciprocal links that unite the centers to their surrounding areas.

This is a return to the idea of H. Jayet (1996) that there are "several modes of operation of centrality, fluctuating between two extremes corresponding in the first case to the primacy of the city which dominates the rural area and in the second case to the primacy of the organized rural area which generates, at a more limited scale, its own forms of centrality" (p. 391). This view suggests that there are, on the one hand, rural areas that are urbanized or under urban influence and, on the other hand, ruralizing agglomerations that fit into their surrounding areas.

This work, which must be considered exploratory, requires further investigation. First a more formal approach to the spatial configuration of LMAs could show under what circumstances the interplay of agglomeration forces and dispersive forces is reflected by the spatial forms observed. In addition, the set of assumptions proposed, which may be considered as made up of "urban" agglomeration and dispersion forces, may be supplemented by a series of other factors for explaining, at another scale, the distribution of activities and populations among LMAs. There are arguably truly "rural" agglomeration forces where agriculture has only a residual role and is gradually superseded by other activities. These escape from the general trend towards agglomeration in places where there is already substantial concentration for reasons related either to the geographical distribution of the production factors they use or to the existence of special externalities which might develop in sparsely populated areas.

7

  Estimation results made for several specific LMAs are available from the authors on request. They shown that the shapes are very often similar. Only the distance where the "bump" occurs distinguishes the different LMAs.

Figure 1 -

 1 Figure 1 -Estimated Population Density Functions by Employment Centre Size of LMAs (1968 & 1990)

Table 1 -Population and jobs within Labor Market Areas (LMAs)

 1

		LMAs with LMAs with LMAs with LMAs with	All
		center	center of	center of	center	categories of
		< 5,000	5,000 to	20,000 to	> 100,000	LMAs
		inhab.	20,000	100,000	inhab.
	Number of LMAs	227	127	59	16	429
	Land area by LMA (km 2 )	164	372	938	2,502	419
	Population
	Mean by LMA	4,619	18,101	76,723	440,722	34,792
	% pop. outside center	52.0	47.7	44.5	31.7	39.4
	Density in center (/km 2 )	76	188	448	1199	399
	Density outside center (/km 2 )	18	27	40	62	37
	Jobs
	Mean by LMA	1,699	6,603	28,290	178,185	13,390
	% jobs outside center	37.4	31.8	30.0	20.2	25.9
	Density in center (/km 2 )	37	89	208	566	188
	Density outside center (/km 2 )	5	7	10	16	9
				Source : 1990 Population Census

Another way to examine the population spread is to estimate the population density functions around cities and their changes over time. After the pioneering research by Clark

Table 2 -Flows between center and periphery by LMAs' size and category of district

 2

		Employment rate: Percentage Percentage LMA internal stability
		ratio of jobs to	of jobs at	of active	level
		active pop.	center held	Pop-
		residents	by	commuting
		at center outside commuters	to center Stable pop/	Stable
			center			active pop. pop./Jobs
	LMAs with Center < 5,000	1.29	0.71	15.6	18.5	75.4	76.6
	LMAs with Center 5 to 20,000	1.24	0.63	18.6	25.3	77.1	80.8
	LMAs with Center 20 to 100,000	1.17	0.63	19.6	29.0	80.9	86.8
	LMAs with Center > 100,000	1.16	0.64	13.2	33.2	80.2	80.4
	All LMAs	1.18	0.64	15.8	29.1	79.6	82.0
	* Stable pop. = active population living and working in same LMA	Source : 1990 Population Census

Table 3 -Level of residentiary services by LMAs and service centers

 3

		Weighted frequency of		Service centers
		residentiary services by LMA
		mean	per 1,000	Number of	Mean by LMAs Mean frequency
			inhabitants	service centers		by center
	Center < 5,000	44	9.5	196	0.86	35.9
	In empi, center	31	14.1	177	0.78	37.2
	Outside empi, center	13	5.3	19	0.08	23.9
	Center 5-20,000	103	5.7	180	1.42	52.9
	In empi, center	63	6.7	127	1.00	63.3
	Outside empi, center	40	4.6	53	0.42	28.8
	Center 20-100,000	306	4.0	191	3.24	62.7
	In empi, center	129	3.0	59	1.00	129.4
	Outside empi, center	177	5.2	132	2.24	32.6
	Center > 100,000	1307	3.0	181	11.31	86.5
	In empi, center	588	2.0	16	1.00	587.8
	Outside empi, center	719	5.2	165	10.31	37.8
	All LMAs	145	4.2	748	1.74	59.1
	In empi, center	75	3.6	379	0.88	83.8
	Outside empi, center	70	5.1	369	0.86	33.7
			Source : 1988 District Inventory and 1990 Population Census

In the first version ofKrugman's model (1991b), the cost of transport of agricultural goods is assumed to be zero. Krugman mentions competition on the local country market (which finally has the same effect on agglomeration as a transport cost), which acts as a dispersive force but which is not specified in the formalization.

Dutanc* (km)

Appendix 1: Labor market areas (LMAs)

The definition of labor market areas was based on commuter flows between districts measured from the 1990 Population Census. The method used (MIRABELLE) was developed by the National Institute for Statistics and Economic Surveys (INSEE) and is similar to increasing rank ordering. It consists in aggregating districts step by step depending on the intensity of the relations between them and minimizing at each stage the flows outside the clusters that are formed [START_REF] Terrier C | Une méthode d'analyse des relations intercommunales (ou la classification d'un graphe value)[END_REF]. The result is a breakdown by Commuting areas between which total commuter flows are minimized. The areas are made up of one or more clusters of districts which form labor market areas (LMAs). The LMAs are organized around an employment center which is usually the most populated district or urban unit of the LMA.

Appendix 2: Cubic spline functions A spline function is a system of piece-wise polynomial approximations. The function is continuous as well as its derivatives. The spline estimation begins by dividing the x axis (in our case, the distance axis) into segments where the dividing points are called « knots ». Three segments of equal length are generally suggested (Suits et al. 9 1978). In the case where the cubic form is retained and where the number of knots is fixed to three, the function can be written as:

where D f is the density at a distance x t from the center, Di, D2, D3 are dummy variables corresponding to the different intervals {xo,Xi}, {xi,x 2 }, {x2,X3}, xo is the distance zero,

X2

the distances between the origin and the two interior knots, X3 the maximum distance. This function can be estimated by the following multiple regression equation [START_REF] Suits | Spline functions fitted by standard regression methods[END_REF]:

where 1 and 2 are dummy variables which equals 1 if and only if x t > x, , i=(l, 2) and 0 otherwise. Source : 1968, 1975, 1982, 1990Population Census In this paper, we analyze the district population density in the LMAs by the cubic spline function, distinguishing four categories of LMAs depending on the size of their employment center. The radius of the different categories of LMAs is divided into three equal segments with two interior knots. For the two categories of largest cities, we use dummy variables by LMA in order to control for the differences in city size and characteristics. Following Skaburskis (1989), the OLS regression was chosen instead of the weighted-least squares. The results presented in Neighborhood services were too widespread and finally of little structural value. Each facility was then weighted by its frequency in the district (1 for one facility; 1.8 for two; 2.4 for three; 2.8 for four; and 3 for five facilities). The district frequency is the total score obtained for each facility. The LMA frequency of residentiary services is made up of the total facility scores for the districts composing the LMA; the urban unit frequency is calculated in the same way by totaling the scores for the districts making up the urban unit. Finally, districts with facility scores of 20 or more were considered as service centers. If one district of an urban unit made up of several districts exceeded this level the entire urban unit was considered as a service center. The score of 20 was set on the basis of a basket of facilities systematically found above this level.


Maureen Clerc

Juliette Leblond

Jean-Paul Marmorat

Christos Papageorgakis

Inverse conductivity recovery problem in a spherical geometry from EEG data: uniqueness, reconstruction and stability results

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INVERSE CONDUCTIVITY RECOVERY PROBLEM IN A SPHERICAL GEOMETRY FROM EEG DATA: UNIQUENESS, RECONSTRUCTION AND STABILITY RESULTS

MAUREEN CLERC 1,3 , JULIETTE LEBLOND 1,3 , JEAN-PAUL MARMORAT 2 , CHRISTOS PAPAGEORGAKIS 1,3 .

1 INRIA Sophia Antipolis Méditerranée, Équipes Apics et Athena, France. 2 Ecole des Mines ParisTech, Centre de Mathématiques Appliquées, Sophia Antipolis, France.

3 Université Côte d'Azur, France.

Abstract

Electroencephalography (EEG) is a non invasive imaging technique that measures the effect of the electric activity of active brain regions, called sources, through values of the electric potential furnished by a set of electrodes placed at the surface of the scalp. A fundamental problem there is the inverse problem of source localization which aims at locating the sources of the electric activity using the acquired EEG measurements [START_REF] Clerc | Source localization in EEG using rational approximation on plane sections[END_REF]. The quality of the source estimation depends on the accuracy of the conductivity model used to solve the problem. Among the head tissues, the skull conductivity is the one that influences most the accuracy of EEG source localization [START_REF] Vallaghé | A Global Sensitivity Analysis of Three-and Four-Layer EEG Conductivity Models[END_REF]. Often, conductivity estimation is performed prior to the source estimation to determine the unknown conductivity using either supplementary EEG measurements or even measurements acquired by other imaging techniques. Such a technique is electrical impedance tomography (EIT) where current is injected through a pair of EEG electrodes while the unknown conductivities can be estimated by the resulting measurements at the rest electrode locations.

We examine the inverse skull conductivity estimation problem, which aims at recovering the electrical conductivity properties of the skull from measurements given at the surface of the head by EEG measurements. Our goal is to show uniqueness and a constructive scheme for the inverse skull conductivity estimation problem using partial boundary EEG data from a single experiment, in the preliminary case of an homogeneous skull conductivity. This is a version of the many inverse conductivity issues still under study nowadays [START_REF] Alessandrini | Generic uniqueness and size estimates in the inverse conductivity problem with one measurement[END_REF].

The head is assumed to be an isotropic piecewise homogeneous medium and we examine a layered spherical head model made of three concentric nested spheres, each of them modelling scalp, skull and brain tissues (from the outermost to the innermost layer). Each of the three layers is supposed to have a constant conductivity. We also assume that the conductivities of the brain and the scalp are known, while the conductivity to be recovered is the one of the intermediate spherical layer (skull).

We solve the above conductivity estimation problem from the available EEG partial boundary data, expanded on the spherical harmonics basis, and transmitted over the spherical interfaces by transfer functions, while we consider that the source term is already estimated (through a number of coefficients of its spherical harmonics expansion). Linear algebra computations then allow us to find polynomials that possess a root which should coincide with the unknown skull conductivity, thus solving the estimation problem. We derive uniqueness properties and a reconstruction algorithm for the skull conductivity. A numerical study shows that the algorithm is able to accurately estimate the skull conductivity, with good robustness properties with respect to various levels of noise. The properties of the inverse conductivity estimation problem are also examined with various source configurations (partially known sources) and EIT measurements.

This work was supported by the Région Provence-Alpes-Côte d'Azur, France, and BESA GmbH, Germany.


Sylvain Chevillard

Juliette Leblond

Konstantinos Mavreas

Dipole

Dipole localization in Moon rocks from sparse magnetic data

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Moon today has no global magnetic field, nevertheless there are evidences showing that it used to have one in the past [START_REF] Weiss | The lunar dynamo[END_REF]. Paleomagnetic scientists try to understand the reasons of the creation and the evolution of this ancient magnetic field, by studying the remanent magnetism of surface Moon rocks.

In our study, we consider dipole recovery issues from sparse magnetic data, with the use of best quadratic rational approximation techniques, together with geometrical and algebraic properties of the poles of the approximants.

The data acquisition process generates the main "cylindrical" geometry of our study. A special magnetometer has been constructed by scientists at CEREGE * which they call "lunometer". Lunometer provides measurements of the components of the magnetic field B, on 3 circular sections i = {1, 2, 3}.

The underlying magnetic phenomenon is modeled by Maxwell equations in the magnetostatic and macroscopic framework [START_REF] Jackson | Classical electrodynamics[END_REF]. We preliminary assume that the sample contains a unique pointwise dipolar unknown magnetic source located at X d , with moment M d . The expression of the magnetic field at X = X d (where µ 0 is the permeability of the free space) has the form:

- 4π µ 0 B(X) = |X -X d | 2 M d -3 [M d • (X -X d )] (X -X d ) |X -X d | 5 (1)
From measurements of the magnetic field B at sensors positions, we want to recover the moment M d and the location X d of the dipole. Our strategy is to decompose this inverse problem into two subproblems and search firstly for the location X d of the dipole (which is a nonlinear problem) and secondly for its moment M d (which becomes a linear problem when the location is known).

More precisely, the idea is to use the available data of a component of B denoted as B i . Then, B 2 i can be seen as the trace on the circle of a rational function of the complex variable in the corresponding plane [START_REF] Baratchart | Inverse source problem in 3D ball from best meromorphic approximation on 2D slices[END_REF]. This function has one pole inside the disk and one outside. From the values of this function, we use the best quadratic rational algorithm called RARL2 ‡ which provides us with an approximation of the pole inside. Vieta's formula allows us to estimate the pole outside, hence to fully recover the denominator of the rational function. We establish relations between the denominator and the unknown actual dipole position X d .

Observe that a variety of factors can affect the moment recovery process. Indeed, the data is subject to measurement errors and the condition number of the problem depends on the actual dipole location/orientation. Geometrical methods could work as supplementary to the algebraic methods. During the talk, these techniques will be explained and numerical results will be discussed.

* Centre Européen de Recherche et d'Enseignement des Géosciences de l'Environnement, CNRS. Aix en Provence, France. ‡ https://project.inria.fr/rarl2/ * * http://maglune.cerege.fr/

Acknowledgments

This research is part of the ANR project MagLune * * .


Louis De Mesnard
email: lmesnard@satie.u-bourgogne.fr

Coordination and profit rate maximization

Keywords: Coordination, Profit, Profit rate, Maximization, Microeconomics

We study the effects over the coordination between firms of the choice of another objective function for the firm: the profit rate maximization which seems to be closer to business practice than the classical pure profit maximization. After recalling the effect of the new objective function over the optimum (the optimal production is lower), two cases are considered : perfect competition (the price signal disappear), and duopoly-oligopoly (the functions of reactions may vanish).

RESUME.

Nous étudions les effets sur la coordination inter-firmes du choix d'une autre fonction objectif pour la firme: la maximisation du taux de profit qui semble être plus proche de la pratique des affaires que la classique maximisation du profit pur. Après avoir rappelé l'effet de la nouvelle fonction objectif sur l'optimum (la production optimale est plus faible), deux cas sont considéré : la concurrence parfaite (le signal prix disparaît), et le duopole-oligopole (les fonctions de réactions peuvent disparaître).

I.

Introduction

Coordination between firms is one of the major theme of economics : it is the central idea of both signal theory and games theory.

• In perfect competition, the firm equalize price to marginal cost in order to maximize its pure profit. The price signal allows coordination between the firm and the market.

• In imperfect competition with a little number of firms, like duopoly-oligopy, firms dp 1 build reactions functions where the conjectural variations, the terms v' = with " d Q' Q l as the production of firm / and Q* as the production of other firms, play the central role, indicating the reaction of the firm(s) other than / regarding to the action of the firm / . The intersect of the functions of reaction determine the optimal production of both firms. It is one of the starting point of games theory.

We shall explore the consequences over the coordination between firms of the abandoning of the neoclassical objective function, pure profit maximization, to the benefit of another objective function, profit rate maximization, which looks to be closer to business practice.

II. Preliminaries

A.

The objective functions of the Firm

Two alternative functions

Leaving aside deliberately the improvements made to the neoclassical theory of the firm, like incomplete information, theory of the bounded rationality, theory of the agency, theory of contracts and incentive, managerial and behaviorist theory of the firm, etc., there are two possibilities for a firm : to maximize the absolute satisfaction or to maximize the relative satisfaction.

In the following,

• The words accounting profit denotes the accounting profit able to remunerate the shareholders, that is to say before payment of dividends. It is not used to pay the eventual interest (yet paid). If accounting profit is positive, a dividend is optionally paid, equally to every shareholders, in order to remunerate this authorized capital. In this paper, the costs corresponding to the accounting profits are the accounting costs.

• The words authorized capital refers to the funds advanced by the shareholders to the firm: there is nor obligatory pay back, nor obligatory payment of a remuneration, but it is the usual counterpart of a property right. This remuneration is free ex ante as ex post, and varies between zero and the amount of accounting profit (less the funds putted in reserve and taxes), depending on the policy of dividend of the firm.

Authorized capital remains in the balance sheet of the firm and remains the property of the shareholder (to the contrary of a debt, which disappear of the balance sheet of the firm when paid back).

The cost of the authorized capital is an opportunity cost, because these free funds may be invested in another place at a normal interest rate: the authorized capital looks to come from an operation of renting similar to a borrowing, and its opportunity cost looks like an interest of which the rate is the average rate of the market (calculated with a standard alternative investment).

In the opposite, borrowing are paid back obligatorily at the settlement date (except if bankruptcy or at the dissolution of the firm) with obligatory payment of an interest to the moneylender, without giving to him a property right (the question of the control of the shareholders over the firm differs): interest is a cost. Self-financing correspond to a withdraw over a reserve (a past profit or over an actual profit): it is comparable to a withdraw over the shareholder and then it is similar to an increase of authorized capital.

Remark. The reselling of shares is only a transfer between shareholder and does not concern the firm directly (it determines only the future success of the operations of increasing the authorized capital); it may constitute a remuneration of shareholders, but only of the ancient one, the seller, not of the new one, the purchaser.

Entry in the sector

The conditions of entry in the sector for the firm are the following :

• if the pure profit is positive or nil, that is to say if:

(accounting profit) -[ (opportunity costs per unit of authorized capital) x (authorized capital) ] > 0 (1)

• or if the ratio of profit is superior or equal to the opportunity costs per unit of capital, i.e. if:

(accounting profit) , .

• ^ « . i • .X -:-:--> (opportunity costs per unit of authorized capital) .

(I ) (authorized capital) v

These two conditions are obviously equivalent to decide if the firm must or must not produce. However, the classical microeconomic theory choose the first possibility. Moreover, we shall see that the results diverge when we want to maximize the pure profit or the profit rate. pure profit profit rate i i Entry in the sector : same conditions Maximisation : not the same results

3.

Historical aspects

So, the question is: why the subtractive form of the function of profit (profit minus opportunity cost of the capital) prevailed over the "divisive" form of the function of profit (profit over capital)? Yet, nor the first nor the second possibility of calculation indicate anything about the question: what is the optimal production? We shall see that these two objective functions differ on this point.

Why did one historically choose to maximize the pure profit? We may propose the following conjecture.

In the pre-industrial age and from the beginning of the industrial revolution, capitalists (or landowners) often had no alternative possibility of investment than their own business (or property). The modern legal form of firms, like company, was not widespread and the financial market was non-existent. Thus the opportunity costs were disregarded and one choose the maximisation of the simple profit. That is why the pre-classic authors, like Bacon or Hobes. have adopted naturally the principle the more in accordance with the observed reality of their era, the maximization of the satisfaction, that is to say the maximization of the earnings, later modernized as the maximization of the profit.

Even if the classic authors insist on the profit rate (Smith, Ricardo, etc.), they consider, like Marx later, the social profit rate and the profit rate of the sectors (see the law of equalization of the profit rate of sectors), more than the profit rate of the firm, and they do not think in terms of profit rate maximization.

Later, the neoclassic authors have conserved the principle of profit maximization, introducing the opportunity costs and the pure profit. In the General Equilibrium, the profit of each firm is the revenue of this productive agent. Thus, it seems natural to maximize the revenue of the agent. The notion of pure profit was introduced to consider the opportunity costs of the social capital, however keeping the profit maximisation and rejecting other forms as the maximisation of a ratio, in order to not modify too much habits of reasonning. Now, it is clear that, in the real world, firm maximizes a ratio of profitability, that is to say a profit rate. And the financial theory does not allows to leave out profit rate maximization (see

Annex).

This proves that it is necessary to explore the alternative theory of the profit rate maximization. The microeconomic consequences of it has not been studied, even if it is an old idea in the business world.

Remarks.

• Many theories think into terms of profitability: the theory of the choice of investment (with the rate of actualization), the theory of demand of capital (Keynes, among others, with the marginal efficiency of the capital), etc.

• This behavior, take into account the available means to reach the objective, is observed in any human activities, when a problem of choice exists. In some cases, human people do not respect this principle, for example when the objective is vital (high level sports competitions, military fight, etc.).

6.

Variables and notations

The variables that we shall introduce are:

• P{Q) •> the inverse of the curve of demand. • R(Q) = p (Q) Q the receipt.
• Cp(Q) , the cost of production. It includes all the costs (direct costs and opportunity costs) of variable factors, like labor, plus renting costs and writing off of the fixed capital). It excludes the interest of borrowings and the remuneration of the authorized capital. The opportunity cost associated to the remuneration of the authorized capital will be introduced explicitly.

• Kr(Q) , the positive capital (it is monetary, not physical) used in the production.

Obviously, Kr(Q) is linked to Cp(Q), because it serves to pay the costs, and generally K r (Q) > Cp(Q).

However, the link between the cost of production and the capital funds is not so important because we find the costs of production both in the maximization of profit and in the maximization of a profit rate.

• /, the constant and positive interest rate in the financial market, valid for the borrower and the moneylender: as the financial market is assumed to be perfect, the same rate / is used for the borrowing of the firm and for the opportunity cost of the funds.

• K(0) , the positive authorized capital used in the production

We confuse the authorized capital with the totality of the own funds of the firm. It is given by the shareholders and recovered by them only at the end of the process of production. There is an opportunity cost endured at each period at the rate / when the firm uses it.

The authorized capital is remunerated at the fixed interest rate /, giving an opportunity cost, / K(Q) .

• KB(Q) •> the borrowed capital used in the production.

An interest is paid by the firm at each period : / KB(Q) .

We have by hypothesis :

K r (0) = K(Q) + K B (Q) . • (\Q) = C P (Q) +1 K R (Q) = ( >(0 -1 [ K r(Q) ~ K(Q)]
, the accounting cost.

• n(0 = tf(0 -C\Q) , the accounting profit.

• n(0

= n A (Q) ~ t K(Q) , the pure profit. • *(Q) = ^71T ' the P ro f lt rate 2 • M0 2
In fact, we may define many other types of profit ratios. This one correspond to a financial profit rate. However, it is the more in accordance with the business practice, and we use it, calling it "profit rate" in order to lighter the notations. In the short period, the classical microeconomics assume that the capital is fixed. In this case, obviously, {Maxprofit} and JMax are equivalent. However it is not these formulas that we compare. We compare the classical pure profit maximization {Max U A -/ K }to the profit rate maximization j Max-^r-j : at once in the pure profit maximization and in the profit rate maximisation, the authorized capital is considered, by its opportunity costs in the first case, directly by a ratio in the second case.

Anyway, the classical theory says . at short-term, the capital is fixed, at long-term the capital varies. It is not correct, because, at short-term, if the capital is fixed, the scale of production is given; thus it is necessary to introduce this constraint in the classical calculation of the short-term equilibrium : the program is Max 11(0 under Q e Q,Q , where Q and ()are the bounds of Q due to the physical capital invested, which generates the fixed costs. This is why, in the following, we will not consider a long period where the capital is variable (as in [de Mesnard, 1992 ) because it is confusing, but a short period with an ex ante calculation, where the capital is freely fixed (while ex post, the capital is fixed).

D. General case : monopoly

Proposition 1. Consider the monopoly with function of capital : it is the more general case.

The firm produces up to the point where the marginal profit is equal to the average profit multiplied by the relative elasticity of the capital to the production.

Proof. Consider the profit rate 7i(0 = ^A^ . Its maximization gives: Proof. We have : K(Q) = kQ with k > 0 , and then e m = 1. Hence, 11^(0

= n A (Q) • •
As said in [ de MESNARD 1992 ], this corresponds to an observed behavior of the firms: a firm produces and invest as much as the profit brought by a supplementary unit is superior or at least equal to the average profit brought by other units.

Denoting O u as the classical equilibrium and O n is the profit rate equilibrium, we obtain the following graph : Remark. The equilibrium is not the point of long period equilibrium of the traditional monopoly (that is to say R'(Q) = ( */j{0 + / K'IAQ) ) at a long period) and do not correspond to the point of the minimum of the average cost.

E. Profit rate versus pure profit

We may compare the optimum of the profit rate with the classical microeconomic optimum (pure profit); we obtain the following fundamental theorem.

Theorem. Consider the more probable case K F (Q K ) > 0 , where Q N is the optimal quantity which maximizes the profit rate n(Q).

Compared to the pure profit maximization, the profit rate maximization gives a lower optimal production (respectively higher) when the profit rate is higher (respectively lower) than the interest rate at the optimum of the profit rate. The results are the

opposite if K'(O n )<o. • n(Q*) > t => (max n) is before (max IT) • n(Q n ) < t => (max n) is after (max IT)
This is the condition of [ de MESNARD 1992 ].

Note that these results are made at the optimum: it is necessary to calculate optimum of a profit rate to know if it will be before or after the pure profit.

Proof. The solution for the maximization of the pure profit 11(0 is obtained for :

^(0=^,(0-/^ = 0.

The solution for the maximization of the profit rate TT(0 is obtained for : n'j(0 -tt(0 K'(Q) = 0 . Consider the curve f w (0 = n > / ,(0 -tc(0 K\Q) .

Let us compare the respective positions of n'(0 and of f*(0 Suppose that n / (0 is decreasing ( n"(0 < 0 ), so that the optimum of the pure profit 11(0 would be a maximum.

Let us write U\Q) as a function of f ff (0 : W{Q) =f*(0 + [tt(0 -/] #'(0 .

Denote Q n the point of maximization of the pure profit, such as U\Q U ) = 0 .

Denote Q n the solution of the maximization of the profit rate, such as fn(Q n ) = 0 .

1) Suppose that ^'(0 > 0 and

n(Q n ) > t. Then, Yl'(Q n ) >fn(Q n ) = 0 .
Near to O u , the curve f*(0 is under the curve n / (0 . Thus, O u must be higher than Q n .

At the limit, if n / (0 is vertical near to Q n , FI / (0 will cut the horizontal axis in O n and then

Q n = Q n . " n,7l A n ri(Q n ) f 71 Figure 2.

Mathematical comparison of optimums

2) Suppose that K'{Q) > 0 and 7t(0*) < /. Then,

n / (Q n ) <U(Q) = 0 .
Then, near to Q* , the curve fjr(0 is over the curve fl'i

Thus Q li must be lower than Q n . •

III. Coordination

A. Perfect competition

1.

Short-term

Consider one firm, in short-run perfect competition among a market (the number of firms is given). The above result obtained for monopoly with function of capital is applied here, assuming that the price is independent of Q: p(Q) = p . At the optimum, we have: Proposition 3. In perfect competition with fixed coefficient of capital, the firm produces up to the point of minimum cost.

Proof. We have :

e K io = 1 . Thus C'(Q) = C(Q) at optimum. •
The optimum does not depend on the price; thus, the production does not vary if the price vary. It corresponds to an observed behavior.

C +• t K' P C Q Q ,n
Figure 3. Short-term equilibrium of the firm in perfect competion

• Moreover, there is no individual supply curve, even in this case of perfect competition. However, there is a global supply curve, completely inelastic and there exists a global equilibrium of the market, following a classic scheme 3 :

J > Q* Figure 4.
Global equilibrium in short-term perfect competition

• In perfect competition, firms are smaller, there could be more firms; thus interaction between firms could be lower and competition could be greater.

Remark. The optimum of pure profit is given by p = C f {Q) + / K'(Q) .

Remark. The optimum seems to be identical to the classical point of long period equilibrium of the firm. However, do not confuse : we are not in long-term, but only ex ante, as said before. The number of firms remains fixed in our calculation, whereas the number of firms is variable in the long-run.

In [ de MESNARD 1992 ], we assert that, as the price is no more a signal for the volume of production of the firm, the optimum does not automatically clears the market in the short period as supposed by the neoclassical theory. It is false : there is a clearing of the market in the sense that price is fully adapting, the quantity remaining fixed.

2.

Long-term

Denote Q% as the long-term optimum of pure profit and Q\ T as the long-term optimum of profit rate. These points are confused :

QIT = Q\T-
To prove it, denote KIJ(Q) as the mean capital at long-term. At long-term,

• the pure profit maximizing firm moves to the point Q^T such as

nLiiQh) =P~ [CIAQLT) +1 KijiQVr)] = 0 ;
• the profit rate maximizing firm moves to the point Q\ T such as

KMVJIT) -= ; = t •
These formulas are obviously identical : the conditions of entry in the sector are the same.

B. Imperfect competition

1.

Duopoly and functions of reactions

The demand for the sector with two firms 1 and

2 is D(Q l + Q 2 ) .
The inverse demand is p = D~{(0 [ + Q 2 ) , assumed to be decreasing.

The total revenue of each firm are:

R l (Q l +Q 2 ) = Q l D-\Q X +Q 2 ) and R 2 (Q l +Q 2 ) = Q 2 D'\Q l +Q 2 ) .
The cost of each firm are C ] (Q ] ) and ( 2 (Q 2 ) .

The profits of each firm are :

n l A (Q ] ,Q 2 ) = Q l ir\Q*+Q 2 )-C\Q*) *nd Tl 2 l (Q ] 9 Q 2 ) = Q 2 D'\Q' + Q 2 )-C 2 (Q l ) .
/.

The classical duopoly

For the classical maximisation of the pure profit, we solve the system an ^',-g') = o^a n '(g:-g') = o,

6Q { dQ 2
That is to say, for the firm 1 :

dir\Q l +Q 2 ) dQ Q*+D-*(Q*+Q 2 )- C 1 (Q 1 )-t K\Q 1 ) = 0
Even for a very simple linear function of demand, the system has two functions of reaction the solution is found by the intersect of two curves (look to the example later).

ÔQ 2 , dQ x -^-and v = -^- ÔQ 2
Denote v 1 = and = -^=-7 as the conjectural variations.

In the Cournot-Nash duopoly, when each firm modifies its production, it is supposed that the other firm would not react and change its own production. Thus v 1 = 0 and v 2 = 0 and the

• dD-\Q x +Q 2 ) . , JU dD'\O x +Q 2 ) TJ c , quantity -is replaced by jq\--L
. However, functions or reaction remain.

//. The profit rate maximizing dupoly

Now, we assume that each firm maximizes its profit rate :

K (Q )
The necessary conditions of optimality are,

dQ 1
For the firm 1, we obtain :

dR ] (Q x ,Q 2 ) d€\Q x ) BQ X dQ x dR\Q\Q 2 ) dC-jQ 1 ) dQ 2 dQ 2 dir\Q x +Q 2 ) K\Q ] )~[Q l D~\Q X + Q 2 )-C x (O x )] C -^^ = 0 K 2 (Q 2 )-[Q 2 ir\Q x +Q 2 )-C 2 (Q 2 )] JK d ^2'Ko dQ x Q 1 + IT 1 (Q 1 + Q 2 ) (1 -evw) = C" ] (Q X )-C X (Q 1 ) e k . /o 1
where CW) -^ and (? <(£><) = ^>, and = «0-.

To solve the system, the clasical process is : the firm 1 annouces that it will produce the quantity Q\ t . X) .

The firm 2 calculates the level of its own production

Q 2 t)
which maximizes its profit, taking the production of the firm 1 as fixed, i.e. d Q\ t -\) = 0 .

Then, the firm 1 recalculates £?//+!) maximizing its profit, with 0} t) fixed, i.e.J Q\ t ) = 0 .

The following system to be solved is the expression of the two reaction functions .

(?(»i)

= 0 (/)
It is easy to find some counter-examples where the reaction functions disappear with the profit rate maximisation, while they exists with the pure profit maximisation. Generally, the reaction function disappear if the first equation does not include Q 2 , and the second equation does not include Q l . Proposition 4. The deletion of reaction functions depends only on the parameters of the function of demand and on the elasticity of authorized capital regarding to the production, but it does not depend on the function of costs. Proof In the last system of equations, the right member of the equation of equilibrium of firm / does not depend on the production Qi of the other firm j , but the left member depends on it .• Proposition 5. In the duopoly in prices, the functions of reactions are deleted if:

• the inverse function of demand is linear :

D~[(Q 1 , Q 2 ) -d-aQ x -b Q 1
• and the elasticity of the authorized capital according to the production is equal to 1 for each firm : e K ,/Qi = 1 for all / (this occurs when there is a coefficient of capital:

= k Q l for all /).
Proof With these conditions, the system of equations becomes :

-aQi-bv l Q l =C i(Q*)-C l (Q { )evw 1 -a Q 2 -b v 2 Q 2 = C \Q 2 ) -C 2 (Q 2 ) e K 2 /Q 2 J ' Thus, Q x does not depend on Q 2 . Reciprocally, Q 2 is independent from Q x . •
The solution is not found by the intersect of two functions of reaction: in fact, each firm acts independently to the other and finds immediately the good solution; each firm considers only its cost to determine its production of equilibrium: only the parameters of cost and play a role; and are not equal. It is known that the duopoly have not always a stable solution.

However, there are reaction function in all other models of duopoly even if dynamics may be complicated or chaotic [ RAND 1978] [ PIATECKI 1994 ].

Example. The above proposition corresponds to a very popular case, as the example of J.-C.

Milleron [ MILLERON 1979, pp. 117-122 ], adapted to introduce the pure profit.

P = d-(Q X +Q 2 ) R l (Q x +Q 2 ) = dQ x -(Q X ) 2 -Q X Q 2 andR 2 (Q x +Q 2 ) = dQ 2 -Q X Q 2 -(Q 2 ) 2 and, C X (Q X ) = c x + (Q x f , C\Q 2 ) = c 2 + (Q 2 ) 2 md,K\Q l ) = kQ l ,K 2 (Q 2 ) = kQ 2
Then,

nkQ l ,Q 2 ) = -c l +JQ l -2(Q ] ) 2 -Q i Q 2 ,n\{Q\Q 2 ) = -c 2 +dQ 2 -Q' Q 2 -2(Q 2 ) 2
Thus, with the profit rate maximization :

For the general duopoly in prices, the system to be solved is :

Q(t+\) i _ 2 + v 1
which could be compare to the equivalent solution of the pure profit maximizing general duopoly:

2 _ 3(d-tk) + (d-tk)v l 15-f 4v l + 4v 2 + v ! v 2 15 + 4V 1 + 4v 2 + v*v 2
• For the Cournot-Nash duopoly, the system to be solved is 4 .

-

Q(i) -Qu) --r Q 2 =
which could be compare to the equivalent solution of the pure profit maximizing duopoly of Cournot-Nash :

)i _ n2 _ d-tk 4
This result was initially set up in : "The new theory of the microeconomic maximization of profit rate : a typological and applied study", roneo, Dijon, march 1994. Proposition 6. The functions of reactions disappear in the duopoly in prices if :

• the inverse function of demand is simply quadratic : D~](Q\Q 2 ) -d-aQ x Q 2

• and the elasticity of the authorized capital according to the production is equal to 2 for each firm : e K , /Ql = 2 for all /

Proof. With the first condition, the left member of the equation of the firm 1 becomes :

It is false to say that the condition about the elasticity seems to be linked strictly with the degree of the demand function.

Proposition 7. The functions of reaction does not disappear if the quadratic inverse function of demand includes a linear term :

D~l (Q\Q 2 ) -d-a Q x -b Q 2 -fQ x Q 1 .
Proof. With the first condition, the terms including Q 2 in the left member of the equation of the firm 1 are :

This does not vanish whatever e K , /0)i . •

Oligopoly

The "model of Mason" reflects the paradigm Structure-Conduct-Performance. However ancient, it serves as a start point to the new economics of Industrial Organization. K. Cowling and M. Waterson present a theoretical version of this classical model [ COWLING and WATERSON 1976] .

-2aQ x Q 2 -av { (Q 1 ) 2 + d-de + e a Q x Q 2
The terms Q 2 are off if e K , IQl = 2 .• Consider N firms producing at the same unique price p , the quantity Q l at the cost C' ; the number of firms is given (there is no entry in the sector).

.V

( n \ We have Q = £ 0'". The demand is p = D" 1 E £>>' .

Assume that the financial capital is proportional to the production for each firm. K' A = k' Q 1 where the coefficient of capital kj is constant.

We assume that each firm maximizes the profit rate .

IV. Conclusion

We proved that, when the function of profit of the firm is changed, maximisation of the rate of profit instead of the pure profit, these may be no coordination because each firm acts independently to the other, in some special cases :

• in competition with fixed coefficient of capital, the price signal plays no role : firms are not coordinated even if they are smaller than in the classical model; the market may not clear in the short-term.

• in imperfect competition, duopoly or oligopoly, the functions of reaction may disappear in some popular cases, like a linear inverse function of demand with a coefficient of capital: there is no more game between firms. These simple cases may be seen as counter-examples of no coordination (in the classical model, there are functions of reactions even in these cases).

These results are not a simple curiosity : the profit rate maximization is a credible alternative to the pure profit maximization because its looks not so far to practice and financial theory (see Annex).

V. Annex : profit rate and financial theory

A. Static model

1.

Value of the firm

We will study a simple version of the financial theory. The simple financial theory consider a financial ratio of profit n:

IT ^
where YIA is the accounting profit after taxes and KA is the authorized capital (plus accumulated funds).

When they control the firm, shareholders want to maximize their wealth. Shareholders must maximize the (Stock Exchange) value Vof the firm (at this step, the rate n is not supposed to be maximized):

FC 7i IT V--Ar-= -jjTT-where n* is the minimal profitability required by shareholders.

The required minimal profitability n* tends to be equal to the interest rate (without risk) of the financial market, /, if the financial market is perfect (each firm has no individual impact on this market). Thus: V-

  ,(0 = n i(0 o rr,<0 = e K ,o n.i(0 o n / l (0 = n(Q) K\Q) M0 The monopoly with fixed coefficient of capital produces up to the point where the marginal profit is equal to the average profit (or to the point where the average profit is maximal).

Figure 1 .

 1 Figure 1. Short-term monopoly

  right member of this equation does not depend on the production Q l of the other firms, but the left member depends on it: functions of reactions vanish depending on the the parameters of the function of demand and on the elasticity of the authorized capital.

As t is independent of the firm, it is constant for it, then the maximization of V is identical to the maximization of UA .

Vbecomes homogeneous to an absolute value. At this point, the maximization of Vis near to the maximization of another absolute value, the authorized profit (even if the authorized capital KA performs no role in the maximization)

2.

Profit by dollar

For the financial theory, the justification of that behavior (maximize the value of the firm) is.

the shareholders want to maximize the profit by share in order to see if a supplementary investment of the firm is opportune. If shareholders have only one firm to choose, it is acceptable to say that they do it. However, if there are many firms to choose, the shareholders cannot think in terms of profit by share. They must think in terms of profit by dollar, dividing the profit by share by the value of the share.

Unfortunately, it is easy to prove that, in this model with no time, this rate is fixed and equal to the interest rate (N is the number of shares):

Thus this rate, despite its homogeneity with 71 , is trivial when there is no time.

The objective function in the static model rest homogeneous to the maximisation of UA .

That is why it is necessary to introduce the time and an actualization rate.

B. Dynamic model

In this case, classically the firm maximizes its present actualized value, that is to say the actualized sum of the future dividends D(T)'

V=YJ -, where a is the actualization rate.

T=\ (1 +a) [ In the Lerner-Carleton model, under a constraint of financing, which indicates the relation the rate of profit and the value of the capital [ LERNER and CARLETON 1966 ], the future dividends DT depends on g, the growth rate of the firm:

where YIA is the profit and s is the share of benefits kept by the firm.

(1-v) 0 4 (1) This gives the Gordon-Shapiro formula (when « -> oo): Va-g

V is homogeneous to an absolute value; ceteris paribus, maximize V is identical to maximize the profit OKI) .

However, as the rate of growth depends on the profit rate n (supposed to be invariable) by the trivial relation g = s n , V become a function of TX .

(1-^) n 4 (i) a -s n Ifs and a are constant, Vis an increasing function of the profit O-iO) and of the profit rate n.

Finally, introducing the Lerner-Carleton model, we can say that the maximization of the present value V is close to the maximization of the financial profit rate 7i. These developments show that, if firm maximizes profitability, in fact it maximizes the rate of growth, that is to say the financial profit rate, if the ratio of retention of profits is fixed.

Remark. A rational evaluation of a is a = n* . If s = 0 we retrieve the simple formula :

V-^r-. If the financial market is perfect, a = / . Thus, the profitability to be maximized becomes: rui)

Remark. The Price Earning Ratio (ratio of quotation of the share over profit by share) is the inverse of this rate PER = . We must have t > g to obtain a positive profitability (or a positive PER). In the time model this ratio shows only the possibilities of growth and the policy of retention of profits, even if in the no-time model the profitability (or the PER) is trivial. If there are some alternative opportunities of investment, the financial theory says that the shareholders choose the firm with the best possibility of growth.

VI. Bibliographical references