Jean-Daniel Rinaudo
email: .jd.rinaudo@brgm.fr

Noémie Neverre

Marielle Montginoul

Simulating the impact of pricing policies on residential water demand: a Southern France case study

Keywords: demand elasticity, France, water pricing, residential water demand, simulation

The case study conducted in this paper looks at residential water pricing from three different points of view. It first describes existing urban water-pricing practices in Southern France, emphasizing that pricing is not yet being used as a tool for providing economic incentives to save water. It then looks at the observed impact of pricing on water consumption, through an econometric analysis of a cross-sectional data set. The analysis suggests that demand, with an estimated price elasticity of -0.2, is not yet very responsive to price variation. A regional water model (300 municipalities) is then developed and used to simulate the potential impact of various water-pricing scenarios on aggregate water demand, aggregate water sales revenue, and consumer surpluses. The results illustrate the trade-offs that have to be made between the search for environmental effectiveness, cost recovery, and equity when implementing complex water-pricing structures such as block rates or seasonal water pricing.

Introduction

The abundant literature on residential water demand, summarized by [START_REF] Espey | Price elasticity of residential demand for water: A meta-analysis[END_REF][START_REF] Arbues | Estimation of residential water demand: a state-of-the-art review[END_REF][START_REF] Worthington | An Empirical Survey of Residential Water Demand Modelling[END_REF] suggests that price is a key determinant of demand. In Europe, policy makers increasingly hear this message. Water pricing is progressively being considered as a cost-effective measure for reducing imbalances between demand and available resources in water-poor areas. This development is reflected in the legal framework, in particular the Water Framework Directive (2000/60) which requires that by 2010 water-pricing policies should provide adequate incentives for consumers to use water resources efficiently (Article 9). The 1992 and 2006 French Water laws confirm this trend, by compelling Water Utilities to design tariffs which provide higher conservation incentives (environmental objective) while allowing water utilities to cover their costs (economic sustainability). One of the specific requirements of the 2006 law is that fixed charges should not exceed 30% of the average bill (i.e. a bill for 120 m 3 /year).

To comply with this new regulatory framework, many Water Utilities will have to simultaneously reduce fixed charges and increase volumetric rates. Given that this change may have an undesirable social impact (Di Cosmo 2011), Water Utilities may consider using more complex tariff systems which, in theory, can simultaneously reduce demand, improve equity, and be revenue neutral. This is for instance the case for seasonal tariffs (ST), which consist of charging higher prices during the peak demand period, or increasing block tariffs (IBT), under which the marginal price increases with the quantity consumed [START_REF] Hanemann | Price and rate structure[END_REF]. In these two tariff systems, the rationale consists of imposing higher prices on users who generate a costly peak demand (e.g., users of second homes or owners of large homes with gardens and swimming pools). In the IBT case, social justice concerns can also be incorporated by making the initial volumes cheaper, the tariff being used to generate revenue-neutral cross-subsidies. Overall, both IBT and ST theoretically yield higher welfare benefits than the single-price policy [START_REF] Krause | The Demand for Water: Consumer Response to Scarcity[END_REF].

Many studies have however shown that the impacts of such complex tariff are relatively complex. [START_REF] Olmstead | Water demand under alternative price structures[END_REF][START_REF] Barberán | Equity in DomesticWater Rates Design[END_REF][START_REF] Barraque | Is individual metering socially sustainable? The case of multifamiliu Housing in France[END_REF]. Practical implementation of IBT (e.g., choosing the block width and water-price levels) is still considered by water utilities to be a challenge, in view of the risk of unexpected financial or social impacts. This may explain why few French Utilities currently use IBT or ST -or when they do so, do not vary marginal price significantly [START_REF] Montginoul | Analysing the Diversity of Water Pricing Structures: The Case of France[END_REF]. There is thus an obvious need for simulation tools that can be used to assess the impacts of various rate structures and levels on water demand, revenues, and certain equity indicators.

While relatively sophisticated approaches have been developed to simulate ex-ante the impacts of various tariff systems [START_REF] García-Valiñas | Efficiency and Equity in Natural Resources Pricing: A Proposal for Urban Water Distribution Service[END_REF], few of these approaches can easily be deployed by water professionals at the municipal, catchment, or regional level, without involving complex econometric analyses. Moreover, few simulation tools have been developed to simultaneously assess the impacts of tariff changes on water conservation, cost recovery, and equity indicators.

This paper presents an attempt to fill this gap. Using a methodological approach inspired by [START_REF] Jenkins | Using economic loss functions to value urban water scarcity in California[END_REF]), a model was developed which can easily be deployed at the catchment level. The originality of the approach lies in the scale at which the model is developed (i.e., several hundreds of municipalities). It also lies in its ability to account for the heterogeneity in municipal demand functions that result from differences in population income, housing characteristics, tourism-related activities, local climate, etc.

The research is based on a case study conducted in Southern France, in an area covering two river basins and one large aquifer (300 municipalities, area of 5000 km², 600,000 inhabitants). Data used include secondary data provided by the National Statistical Institute (INSEE) and primary data (pricing, water use) collected through a field survey. Using a cross-sectional data set, an econometric analysis is first conducted to identify factors that explain variations in waterconsumption levels, including price level and structure. Average price elasticity is estimated using various functional forms. A simulation model is then constructed to simulate changes in water demand under several scenarios. The model enables testing of the impact of changes in price level and tariff structure (increasing block rate pricing, seasonal pricing). The remainder of this paper is organised as follows. Section 2 briefly describes the case study and provides an overview of the methodology. Section 3 describes the tariffs currently used by water utilities, and Section 4 presents the results of the econometric analysis. Simulation results depicting the impacts of various tariff scenarios are presented in Section 5. The paper concludes with a discussion of its policy implications.

Case study and methodology

Case study presentation

The analysis presented in this paper is part of a broader research study carried out over several years in a large coastal zone located in the Languedoc Roussillon region of France. The case study area encompasses 300 municipalities that draw on three major water resources: the Orb and Hérault rivers; the alluvial aquifers of these two rivers; and the Astian sand captive aquifer. The region is characterized by growing water needs mainly due to urban population growth and, to a lesser extent, to agricultural water needs. In previous work conducted in this geographic area the authors have estimated the future water deficit at 28 million cubic metres per year by 2020 [START_REF] Rinaudo | Cost-effectiveness analysi of a water scarcity management plan: considering long term and climatic change[END_REF]. This future water deficit is mainly attributed to urban development and the types of dwelling constructed (mainly detached houses with gardens and swimming pools). In this context, local policy-makers are looking for new resources to meet the growing water demand. Feasibility studies are currently being conducted to develop inter-basin transfers, use deep groundwater, and to install desalination plants. In parallel, there is a growing interest in water-demand management solutions, including reduction of leakages in distribution systems, water conservation and the use of incentive water pricing. The present study was initiated in this context. Its main objective was to estimate the reduction in water demand which could be achieved with incentive pricing systems.

Methodology

The work was carried out in three steps: (1) a survey of existing water tariffs; (2) an econometric estimation of price elasticity; (3) the development of a water-demand model and its application to simulate the impact of changes in water tariffs.

The survey of existing water pricing practices was implemented by using a survey methodology developed by [START_REF] Montginoul | Analysing the Diversity of Water Pricing Structures: The Case of France[END_REF]. A detailed questionnaire was mailed to all the Public Water Utilities, to collect information on water abstraction and sales, and water-pricing structure (e.g., flat rate, volumetric pricing, increasing block). The information collected was then incorporated into a regional database containing socio-economic information (e.g., type of housing, population characteristics, income level), climatic information (temperature and rainfall), and land use (indicator of urban density). Using a cross-section of data for the year 2005, a statistical model was then constructed to explain the differences observed in terms of water consumption per inhabitant, using a sub-sample of 137 municipalities (out of the 300) for which complete information was available.

The statistical model was estimated using the following form: log(C)=ĮȕORJSȕ.log(I)ȕL.Xi, where C is the per capita annual water consumption 1 1 This average per capita water consumption is equal to the total volume of water sold to consumers divided by the population served. It therefore does not include water losses occurring in distribution system and which are on average equal to 30% in our case study area.

, p is the marginal price of water (for a 120 m 3 water use), I the average municipal household income, Xi represents other explanatory variables such as local weather, density of second homes, etc. (additional details are provided in the next section) Į LV WKH LQWHUFHSW, DQG DOO ȕL DUH HVWLPDWHG FRHIILFLHQWV. The choice of a log-linear functional form to describe the relationship between consumption on the one hand, and price and income on the other hand, is motivated by previous research where it was shown that elasticity can be considered to be constant and that it does not decrease along the demand curve [START_REF] Gaudin | Demand Specification for Municipal Water Management: Evaluation of the Stone-Geary Form[END_REF].

Also, as opposed to other functional forms, the log-log specification form facilitates interpretation of the coefficients as elasticity estimates, and enables direct comparisons with the existing literature.

In the third part of the work, a water-demand model is developed to simulate the impact of changes in price level and structures for the 300 municipalities in the case-study area. The methodology adopted is based on [START_REF] Jenkins | Using economic loss functions to value urban water scarcity in California[END_REF]. A simple water-demand function is assumed, such that C= k.P ȟ , where C is water consumption2 [START_REF] Gaudin | Demand Specification for Municipal Water Management: Evaluation of the Stone-Geary Form[END_REF] , P is the marginal ZDWHU SULFH ȟ LV WKH SULFH HODVWLFLW\, and k is a constant reflecting the structure of water use (e.g., differences in local climatic conditions and population income level). Our choice of this demand function is consistent with the choice of a loglinear functional form for the econometric analysis, and the assumption that price elasticity remains constant along the demand curve ( ).

After calibrating the k parameter for each of the 300 municipalities, the impact of changes in waterprice levels and structures are simulated, taking into account their specific socio-economic characteristics (e.g., initial price level and proportion of detached houses). Simulations are carried out for price increase ranging from 0 to 2.5 €/m3, using various tariff structures (simple proportional, increasing block rate, seasonal pricing). The model calculates three main output variables: variation in total water demand; change in consumer surplus; and variation in revenue from water charges for the Utility operating the distribution system. Note that calculations are performed separately for each municipality before aggregating the results and that the model only simulates water demand by consumers, without considering losses occurring in the distribution system which are assumed to be independent from tariffs implemented.

Existing water-pricing practices

In the area studied, the average price is 2.2 €/m 3 , drinking water and sanitation charges included.

This average price is calculated for a consumption of 120 m 3 /year. It corresponds to the total bill for 120 m3 divided by 120. This value is significantly lower than the national average (3.1 €/m 3 ), which can be explained by the fact that 17% of the municipalities do not have any wastewater treatment facility (no charge). As in in most French Water Utilities, bills are issued every six or twelve months. Water is charged on a volumetric basis in 99% of the municipalities (2 flat-rate cases only). On average, the fixed portion of the water bill represents 26% of the total water bill (for a 120 m 3 consumption), but it exceeds 40% of the bill in 19% of the municipalities. A simple twopart tariff is used by 80% of the municipalities.

Increasing block pricing is used by 14% of the municipalities. The first block corresponds to a consumption of 80 m 3 (median) -which is less than the average 120 m 3 consumed by most households in a year. Other block widths are highly variable, but they are often quite large and designed to discriminate between residential, commercial, and industrial users. Table 1 below shows the prices charged for the various blocks, for the 14% of municipalities using increasing block tariffs. Block prices are significantly different: one cubic metre consumed in the 4 th block costs five times more than in the first block (drinking-water charge); however, the difference in price is less significant for wastewater treatment tariffs.

At the time of the survey (2007), decreasing block pricing was still used in 5% of the municipalities. However, the differences between prices charged for the various blocks are not significant: the average price is 0.82 €/m 3 for the first block and 0.58 for the last block. Moreover, the price generally decreases for very high consumption levels (250 to 500 m 3 per year). This suggests that decreasing block pricing targets commercial activities, not households. It is mainly implemented where public utilities want to maximize water sales to recover heavy investment costs.

Insert table 1 about here.

Factors determining the level of water consumption

A statistical analysis was next conducted to estimate the impact of price on water demand, using cross-sectional aggregate data at the municipal level. The explained variable is the total volume of water sold in each municipality divided by the permanent population of the municipality. Industrial water sales are excluded, but sales to commercial activities, offices, hotels, and camping sites are included in the explained variable. The model was developed on the assumption that water consumption is determined by the following groups of factors:

-Population socio-economic characteristics (including income level, household size, proportion of households owning their housing, etc.).

-Urban characteristics (number of second homes, and proportion of population living in apartment buildings versus detached houses).

-Structure and level of water and sanitation tariffs (marginal price level, andpricing structure).

-Climate, which determines the intensity of garden-water needs: number of consecutive hot days (heat wave phenomena) and number of consecutive dry days (drought conditions).

-Whether consumers are allowed to drill a low-cost private well in order to substitute cheap untreated groundwater for municipal drinking water; the less expensive drilling is, the lower drinking-water consumption should be. For a description of the phenomenon in this part of France, see [START_REF] Montginoul | Controlling households' drilling fever in France: An economic modeling approach[END_REF].

The primary data employed relate to price, water use, and estimates of the cost of drilling. The secondary data come from the national population census (INSEE, the National Statistical Institute)

and the French Meteorological Institute (MeteoFrance).

The results of the econometric analysis (Table 3) show that the marginal price of water is a significant determining factor. Price elasticity is estimated at -0.18 which is well within the range of values reported in case studies in France (Nauges andThomas 2003, Reynaud and[START_REF] Reynaud | How much water do residential users really need ? An Author-produced version of the article[END_REF] or elsewhere in Europe [START_REF] Espey | Price elasticity of residential demand for water: A meta-analysis[END_REF][START_REF] Arbues | Estimation of residential water demand: a state-of-the-art review[END_REF][START_REF] Worthington | An Empirical Survey of Residential Water Demand Modelling[END_REF][START_REF] Schleich | Determinants of residential water demand in Germany[END_REF]. This means that a 10% increase in price would result in a 1.8%

decrease in water consumption. Other factors have the expected signs. As anticipated, water use is positively correlated with the cost of drilling a private borehole (the more expensive a borehole, the less people drill, and the more water they take from the mains). Water use is positively correlated with heat waves and drought indicators, and this relationship is statistically significant. The presence of second homes is also a key determining factor, and water consumption is higher for municipalities characterized by high income populations (income elasticity estimated at 0.42).

Unexpectedly, the use of IBT does not have a statistically significant impact on water use. Also, the indicators do not provide evidence of a statistical relationship between water use and the ratio of single to multifamily housing units.

Insert table 2 about here.

Insert table 3 about here.

Simulating the impact of changes in water pricing

The simulation model presented in the second section was then implemented and used to simulate the impact of various water-pricing scenarios. The originality of the model developed resides in the scale at which it operates (calculation conducted at the municipal level, considering their specific characteristics) and the types of results it delivers. It not only estimates changes in water consumption but also calculates changes in revenues for the public water utility, and variations of consumer surplus for four main types of households. The information obtained can thus be used to assess various scenarios in terms of environmental effectiveness, cost recovery, and equity.

Several water-pricing scenarios are simulated. The various curves in Figure 1 show the simulated variation of total water demand (in m 3 ) in the 300 municipalities for various price level increases ranging from 0 to +2.5 €/m 3 (horizontal axis) and for various pricing structures (separate curves).

The first scenario (curve 2PARTS on Figure 1) simply consists of increasing the volumetric price by a fixed amount (ranging from 0 to 2.5 €/m 3 ), which consumers have to pay on top of the existing price in each municipality. Given that the objective is solely the reduction of water consumption, the extra revenue generated by the price increase is redistributed to consumers through a reduction in the fixed charge. Figure 1 shows that an increase of 1€/m 3 (corresponding to a 50% increase in the current marginal price) would reduce total regional demand by approximately 3 million cubic meters per year -the equivalent of 20,000 households or 48,000 inhabitants. Given that approximately 8,400 new inhabitants migrate into the case-study area each year, this would allow a 6-year postponement of the costly investments needed to mobilise new water resources. This clearly illustrates an additional benefit of water conservation which is rarely highlighted in the literature [START_REF] Kanakoudis | Basic principles for urban water value assessment and price setting towards full-cost recovery -pinpointing the role of the water losses[END_REF].

The second scenario consists of increasing the volumetric water price during the summer period only (curve SEASONAL). Note that although the reduction in water demand is smaller than in the first scenario, it occurs in summer when water resources are limited and demand is at its maximum level. Moreover, this seasonal pricing system is theoretically more cost-effective, since it does not impact the consumer surplus in winter when water is plentiful, and operates only when water demand is at a maximum and water resources are limited. Moreover it improves equity by increasing the relative contribution of seasonal residents (owners of second homes, tourists), who are responsible for additional investment costs but rarely pay for them under a uniform annual water

Author-produced version of the article published in Water Resources Management, 2012, 26 (7),2057-2068. The original publication is available at http://springer.com 10.1007/s11269-012-9998-z block from 51 to 100 m 3 and the third block more than 100 m 3 . Each water-pricing scenario is described by three values (a,b,c), where a is the price increase for the first block, b the price increase for the second block, and c the price increase for the third block. The three block prices are then varied to achieve different objectives:

-In scenario (0,0,X), the price is unchanged for the first two blocks and increased by X for the third block (X being varied from 0 to 2.5 €/m 3 ). The objective of this scenario is to penalise water uses above100 m3/year, which mainly correspond to outdoor uses (garden irrigation and swimming pools).

-In scenario (0,X,X), the price is unchanged for the first block (for social reasons) but progressively increased by X €/m 3 for the second and third blocks. This scenario aims to promote water conservation indoors (toilet flushes and appliances) as well as outdoors.

-Scenario (0,X/2,X) is a progressive scenario: the first block is unchanged, the second increases by X/2 €/m 3 (targeting indoor uses), and the third by X €/m 3 (targeting outdoor uses).

-In the last scenario (-X/5,0,X), the price of the first block is reduced by X/5 €/m 3 (i.e., first block subsidised at a 20% rate), the second block is unchanged, and the third block is increased by X €/m 3 (collection of additional revenue to offset the first-block subsidy).

The resulting effect on aggregate water demand is depicted in Figure 4. The curves shown on this figure are constructed by varying X from 0 to 2.5 €/m 3 . They show, for instance, that if an increase of 1 €/m 3 is applied with a (0,0,X) structure, water demand will decrease by 2.3 million cubic metres. If a (0,X,X) structure is applied, this 1 €/m 3 price increase will reduce total water demand by 3 million m 3 . Overall, Figure 4 shows that for a reasonable price increase (X=1€/m 3 ), the various scenarios will yield relatively similar water conservation effects (between 2.7 and 3.5 million m 3 per year). For a higher price increase (X= 2 €/m 3 ), the difference becomes significant (from 3 to 4.5

Author-produced version of the article published in Water Resources Management, 2012, 26 (7),2057-2068. The original publication is available at http://springer.com 10.1007/s11269-012-9998-z million m3).

Insert figure 4 about here.

By contrast, the effects on aggregate revenues differ drastically. Figure 5 clearly shows, for instance, that the choice of a (0,X,X) pricing structure will increase revenue. The same conclusion applies to (0,X/2,X). Interpreting the results for the two other scenarios is more complex. For instance, the reason why (0,0,X) leads to a reduction of income is because a number of large users are shifting from the third to the second block. In this case the loss of volume sold in the last block is not offset by the increase in price of this last block. Similarly, the (-X/5, 0, X) scenario generates huge financial losses for the Water Utilities. Indeed, while all consumers are being subsidised for their first block (50 m 3 /year/household), only a few large consumers generate additional income due to the increase in the last block. Moreover, many large consumers shift from the last to the second block -whose price is unchanged. Overall, these results show the complexity of predicting the revenue impact of using IBT, at the aggregate regional level.

Insert Figure 5 about here.

The various water-pricing scenarios simulated above are compared, assuming an increase in X of 1 €/m 3 . Table 4 shows that an increase of 1 €/m 3 can yield a reduction of water demand ranging from 1.6 to 3 million m 3 per year. Keeping in mind that within our case-study area the anticipated 2020 water deficit will be about 28 million m 3 , these simulation results show that water pricing is probably part of the solution -but not a complete solution-for the future water crisis. Other actions will in any event have to be undertaken, including reduction of leakages in distribution networks, the promotion of water-conservation actions, and the mobilization of new water resources.

Table 4 also shows that increasing X by 1 €/m 3 may have contrary effects on the budgets of water utilities, depending on which tariff structure is selected. Two block tariffs (0, 0, X and (-X/5, 0, X)

Author-produced version of the article published in Water Resources Management, 2012, 26 (7),2057-2068. The original publication is available at http://springer.com 10.1007/s11269-012-9998-z lead to budget deficit situations. All the other scenarios will generate excess revenues which could either be used to reduce the fixed portion of the water bill for all consumers, or reinvested in waterconservation programs.

Insert Table 4 about here.

Discussion and conclusion

The results of this study illustrate that the choice of a water-pricing structure is a complex exercise, involving trade-offs between cost recovery, equity, and environmental efficiency. The simulation tool presented in this paper can be helpful in understanding this trade-off and in exploring the various impacts of alternative scenarios. The main advantage of this model lies in the scale at which it can be implemented (river basin) and its ease of implementation by water managers and planners, without involving the use of complex econometric techniques.

From a policy perspective, the results of our simulations confirm previous findings concerning the use of IBT as a redistributive tool. Cross subsidies between blocks will most likely cause cost recovery problems. To avoid such problems, more sophisticated IBTs have been considered by scholars and practitioners, including for instance option tariffs specifically designed to reduce the bills of large families [START_REF] Barberán | Equity in DomesticWater Rates Design[END_REF] or the use of per capita IBTs as already practiced by many water utilities in Southern California. It is however often recommended that social issues should be addressed "outside the water bill", as already suggested by both academic authors [START_REF] Barraque | Is individual metering socially sustainable? The case of multifamiliu Housing in France[END_REF]practitioners (AWWA 2004).

From a methodological perspective, our simple modelling approach is characterised by a number of assumptions which can be debated. The first problematic assumption is that price elasticity values remain valid outside the price range within which they were estimated. Results from empirical studies suggest that this may not be the case. In their meta-analysis of price elasticity, [START_REF] Espey | Price elasticity of residential demand for water: A meta-analysis[END_REF] However, given the difficulty of predicting future changes in price elasticity, the authors recommend the adoption of a conservative assumption, e.g.,. keeping elasticity constant, as in our own case study.

The second debatable assumption is that households perfectly understand their rate structure and respond to marginal price changes. This has been challenged by [START_REF] Nieswiadomy | Estimating urban residential water demand: Effects of price structure, conservation, and education[END_REF] and by later studies showing that the mere existence of IBT can reduce demand, irrespective of the change in price level [START_REF] Kenney | Residential water demand management: lessons learnt from AUrora, Colorado[END_REF]. Other studies have also shown that price response was positively correlated with households' level of information concerning tariffs. As an illustration, one

American study found that a utility that provides detailed tariff information on its water bill can attain the same level of conservation with a 30 to 40% lower rate increase [START_REF] Gaudin | Efect of price information on residential water demand[END_REF]). The model presented in this study can clearly not account for this effect of information and awarenessraising programs, thus probably underestimating the water-conservation effect of water-pricing policies.

The third caveat regarding our modelling approach is that it does not account for the possibility that households may develop alternative water sources, such as gray water recycling, rainwater harvesting, or private borehole drilling in response to tariff increases. This trend is already being observed in many industrialised countries such as France [START_REF] Montginoul | Controlling households' drilling fever in France: An economic modeling approach[END_REF], Australia [START_REF] Hurlimann | Household use of and satisfaction with alternative water sources in Victoria Australia[END_REF], and Belgium (AQUAWAL 2010) among other countries. Indeed, an increased marginal price is likely to provide significant incentives for large consumers to invest in such alternative, decentralized water-supply systems, as shown by [START_REF] Montginoul | Controlling households' drilling fever in France: An economic modeling approach[END_REF] in the case of private boreholes in France. However, if some consumers quit or use less, then others will have to pay more, since all consumers are bound together by the heavy fixed cost of infrastructure (Barraque). This shows the need for more comprehensive modelling approaches that will account for the possibility of resource substitution-a new area of research which the authors Figure 4: Impact of various block pricing scenarios on aggregate water demand (300 municipalities). Curve corresponding to scenario (-X/5, 0, X) is identical to (0, 0, X).

Figure 1 :Figure 2 :Figure 3

 123 Figures

Figure 5 :

 5 Figure 5: Variation of water sales revenue under various block pricing structures.

  have shown that elasticity values reported in case studies are positively correlated with priceAuthor-produced version of the article published in Water ResourcesManagement, 2012, 26 (7),2057-2068. The original publication is available at http://springer.com 10.1007/s11269-012-9998-z levels and that they can be influenced by the pricing structure used, climate, and other factors.

Table 1 :

 1 Drinking Water (DW) and Wastewater Treatment (WWT) price levels for municipalities using Increasing Block Tariff (in €/m 3 , all taxes included).

	Author-produced version of the article published in Water Resources Management, 2012, 26 (7),2057-2068.
	The original publication is available at http://springer.com
	10.1007/s11269-012-9998-z

Table 2

 2 Main dependent variables included in the statistical analysis of water demand.

	Name	Description
	MARGINAL_PRICE	Marginal price paid by the consumer (for a consumption of 120 m 3 )
		including drinking-water and sewerage charges
	INCOME	Average taxable income per inhabitant (calculated at municipal level)
	DRILL_COST	Estimated investment cost (in €) of drilling a well for domestic water
		(considering geological characteristics of each municipality)
	TEMPERATURE	Number of days from May to August when temperature exceeds 28°C .
	DRY_DAYS	Total number of dry days in a year
	SECOND_HOMES	Number of second homes (used only for holidays and weekends) in the
		municipality, divided by number of permanent inhabitants
	DETACHED_HOUSES Proportion of detached houses in total housing

Table 3

 3 Results of the best reduced econometric model (linear regression). Explained variable = log (consumption in m 3 per inhabitant). ** : significant at the 99% confidence level.

	Variable	Coefficient	Model statistics
	Log MARGINAL_PRICE	-0.18	***
	Log (INCOME)	0.42	***	Number of
	DRILL_COST	0.00002 ***	observations:	137
	TEMPERATURE	0.12	***	Adjusted R²	0.4521
	DRY_DAYS	0.009	***	Fisher	0.0000
	SECOND_HOMES	0.4	***	Link Test	0.28
	DETACHED_HOUSES	-0.45	***	White Test	0.6825
	Constant	0.4

*

Table 4 :

 4 Effect of increasing X by 1 €/m 3 for 6 different tariff structures.Author-produced version of the article published in Water ResourcesManagement, 2012, 26 (7),2057-2068. The original publication is available at http://springer.com 10.1007/s11269-012-9998-z

	Water-pricing scenario	Seasonal	Block	Block	Block	Block	Block
		pricing	(X,X,X)	(0,0,X)	(0,X,X)	(0,X/2,X)	(-X/5,0,X)
	Water demand reduction (M m 3 )	1.6	3	2.3	3	2.6	2.3

Same definition as in foot-note 1.Author-produced version of the article published in Water ResourcesManagement, 2012, 26 (7),2057-2068. The original publication is available at http://springer.com 10.1007/s11269-012-9998-z

price.

Insert figure 1 about here

Figure 2 below illustrates how the model can be used to assess the impact of water-pricing scenarios for various types of household separately. The main types considered are temporary residents (second homes); households in apartment buildings (with and without individual water meters); and households in detached houses (with or without gardens). Figure 2 shows how an increase in the price level (with a two-part tariff) affects the total demand of consumers in detached houses with and without gardens. The curves shown in Figure 2 are based on calculations performed at the municipal level, taking into account the proportion of each type of housing, before regional aggregation. Figure 2 clearly shows that households occupying single-family houses (with gardens) are the most affected. Similarly, welfare changes are estimated for various types of households, using the consumer economic surplus as a measure of well-being. Figure 3 shows the variation of economic surplus for several types of consumers, assuming a seasonal tariff is implemented and progressively raised (summer price surcharge of 0 to 2.5 €/m 3 , compensated by a reduction of the price in winter). For instance, adding a surcharge of 1€/m 3 in summer would reduce by 40 €/household per year the economic surplus of households occupying single family houses, versus only 15 to 20 € for other types of consumers. One may consider this to be a desirable social impact, since it leads to an increased financial contribution from households generating extra investment costs for the peak-demand period.

Insert figure 2 about here

Insert figure 3 about here

The model is then used to simulate the complex impacts of IBT. Several IBT scenarios are considered, all assuming a three-block system. The first block ranges from 0 to 50 m 3 , the second


Rodolphe Dos

Santos Ferreira

Frédéric Dufourt

Free entry equilibria with positive profits: A unified approach to quantity and price competition games

Keywords: Free entry equilibrium, zero-profit condition, oligopolistic competition JEL classification D43, L13

Introduction

Free entry is commonly associated with zero profits. Under free entry and exit, positive profits are supposed to stimulate creation of new firms and negative profits to induce destruction of existent firms. A free entry equilibrium might, therefore, be seen as a stationary state, characterized by the zero profit condition, of a dynamic process of net business formation. This view is implicit in the concept of long-run perfectly competitive equilibrium, and is naturally extensive to monopolistic competition, where the relevant scales of individual firms also appear as negligible with respect to market size. As long as profits remain positive, any entrant is then able to reproduce in an unreactive environment the operating conditions and the proceeds of a high number of successful incumbents.

This line of argument ceases to hold, however, when a potential entrant has to compete with a few incumbents only, all producing under internal economies of scale. In this context, a simple replication of the incumbents' performance cannot guarantee identical success to the entrant, whose environment might be seriously perturbed by that replication. Entry must now be examined as a strategic decision, generally in a complex context, where timing and information considerations are at stake. Faced with this difficulty, a common attitude is to take the zero profit condition as an acceptable approximation, in particular in macroeconomic -or, more widely, general equilibrium -modeling, and to leave more sophisticated approaches, resorting to sequential games with incomplete information, to industrial organization theory. This was already the position adopted in one of the first macroeconomic papers emphasizing the role of increasing returns and imperfect competition, and treating the number of producers as endogenous: "The story [behind the solution concept based on the zero profit condition] can only be defended as an approximation. Entry and exit are complicated phenomena, involving difficult game theoretic issues that defy neat analytic formulation" (Weitzman 1982, p. 797). So, we seem to be trapped in a dilemma: either to force a solution concept devised for non-strategic forms of competition into the domain of oligopoly, or to resort to industrial organization tools that might prove too complex and also too specific for an accurate macroeconomic or general equilibrium use. The point we want to make in the present paper is that we are by no means doomed to that dilemma. A straightforward application of the concept of Nash equilibrium to static symmetric games reproducing standard regimes of oligopolistic competition offers in fact a simple way out.

We owe to [START_REF] Shubik | Strategy and Market Structure[END_REF][START_REF] Shubik | A Game-Theoretic Approach to Political Economy[END_REF] the idea that entry can be modeled as a simultaneous game between actual and potential entrants, depicted as "firms-in-being". At an equilibrium of such a game, along with active profit maximizing firms, there might be inactive firms that optimally decide not to produce, on the basis of correct conjectures about the actions of the former. This asymmetry might be the consequence of some advantage of incumbents over potential entrants, creating a barrier to entry. However, it can also prevail in a completely symmetric game where all players are a priori indistinguishable. As a matter of fact, ex ante symmetry is required for an accurate representation of a perfectly contestable market, characterized by costless entry and exit and no disadvantage for potential entrants relative to incumbents [START_REF] Baumol | Contestable Markets and the Theory of Industry Structure[END_REF].

For a market to be perfectly contestable any observable profile of incumbents' strategies must be sustainable; that is, no potential entrant may be able to make a profit by becoming active. This requirement might suggest that, as in the dynamic story of business formation, e q u i l i b r i u mp r o fi t sa r en e c e s s a r i l yc l o s et oz e r oa ss o o na se n t r yi sf r e e ,o rt h em a r k e t perfectly contestable. This is indeed true if incumbents' capacity to earn positive profits extends to any potential entrant, always in a position to attract enough customers, either by simply imitating incumbents' behavior when firms are small relative to the market, or by slightly undercutting incumbents' prices when competition takes place in the undifferentiated Bertrand oligopoly. An example of the former alternative is competition within the Chamberlinian "large group", where an individual price decision has no sensible repercussions on the industry price level, as in [START_REF] Dixit | Monopolistic competition and optimum product diversity[END_REF]. Another example is Cournot competition in an undifferentiated industry, provided the optimal scale of each firm is small with respect to market size, so that the Cournot equilibrium outcome approximates the long-run perfectly competitive outcome, characterized by efficiency and zero profits, as in [START_REF] Novshek | Cournot equilibrium with free entry[END_REF]. [START_REF] Yano | A price competition game under free entry[END_REF] offers an example of the latter alternative. He uses the same type of free entry equilibrium concept as Novshek (i.e. a Nash equilibrium of a single stage game played in a perfectly contestable market), but, by assuming Bertrand-like competition in an undifferentiated oligopoly,1 he obtains the long-run perfectly competitive outcome without assuming a small firm to market ratio. At the same time, he provides a solid game theoretic foundation to the theory of contestable markets.

These examples suggest that sustainability is not independent of the regime of competition. Instead of the "price sustainability" appearing in the original definition of [START_REF] Baumol | Contestable Markets and the Theory of Industry Structure[END_REF], which implicitly refers to Bertrand competition, we may think of a "quantity sustainability" referring to Cournot competition, with different implications (Brock and Scheinkman 1983;[START_REF] Brock | Contestable markets and the theory of industry structure: A review article[END_REF]). However, preferably, rather than simply opposing price and quantity sustainability, we may completely divorce the concept of sustainability from the specific regime of competition to which it applies (d'Aspremont et al. 2000). Indeed, along with Cournot competition, price competition within a "small group" producing differentiated goods is another situation where the zero profit condition is not implied by sustainability, and where multiple free entry equilibria may exist in addition to the one at break-even prices. In such equilibria, the strategies of active firms entail positive profits and are nevertheless sustainable because potential entrants, taking those strategies as given, realize that, whatever they do, demand will be insufficient for attaining the scale at which production becomes profitable. Also, under these circumstances, there is no sensible reason for the incumbents to accommodate entry.

The objective of the present paper is to provide a unified conceptual and analytical framework for the study of free entry equilibria, covering different regimes of oligopolistic competition and different specifications of internal increasing returns. Our starting point isthesameasind 'Aspremontet al. (2000): we take as the relevant concept of free entry equilibrium the standard Nash solution of a simultaneous symmetric non-cooperative game, such that active and inactive firms coexist. Sustainability is then just the optimizing condition that applies to inactive firms, given their correct conjectures about the decisions of active firms. However, in the present paper, we go a step further in the way of unification: we build a canonical model, where firms strategies are always represented by prices, but which covers different regimes of both price and quantity competition. One advantage of this comprehensive representation is that free entry equilibria can then be generally characterized by the interval between the break-even and the limit prices, to which the critical price maximizing the incumbent profit should belong. This interval induces a range of numbers of active firms that are compatible with a free entry equilibrium: a type of indeterminacy appearing as a robust property of oligopolistic competition in contestable markets.

Our paper is organized as follows. We present our conceptual framework in Section 2, by: (i) defining the concept of free entry equilibrium; (ii) introducing the canonical model where strategies are represented by prices; and (iii) establishing equilibrium conditions on incumbents' prices under general specifications of cost and demand functions. These conditions are then applied in Section 3 to standard regimes of quantity and price competition in homogeneous and differentiated markets, respectively, and the corresponding outcomes are compared. We conclude in Section 4.

Oligopolistic competition with free entry

The analysis carried out in this section comprises three steps. In Subsection 2.1, we introduce a fairly general, although simple, game theoretic framework applying to perfectly contestable oligopolistic markets, and exploiting symmetry of strategy profiles (in the spirit of [START_REF] Cooper | Coordinating coordination failures in Keynesian models[END_REF]. In this context, we use in a standard way the equilibrium solution of a simultaneous and symmetric game to define the concept of free entry equilibrium, under the additional requirement that some firms optimally decide to remain inactive. Two conditions characterize equilibrium: profitability and sustainability. In Subsection 2.2, we propose a canonical model covering different regimes of competition in which incumbents' strategies are always represented by prices, even when quantities, or locations in some characteristics space, are involved (as in Cournotian or spatial competition, respectively). We further formulate general assumptions on the cost and demand functions. In Subsection 2.3, we translate the two conditions characterizing a free entry equilibrium into the requirement that the common price chosen by the incumbents be a critical point of their profit function above the break-even price (for profitability) and below the limit price (for sustainability). These bounds on incumbents' prices translate in turn into a non-degenerate admissible interval to which the number of active firms should belong.

Concept of free entry equilibrium

Free entry means absence of any entry barrier accounting for some advantage of incumbents over potential entrants. Under free entry all firms, whether established or not, are supposed to benefit from full equality of opportunities. However, this does not imply that they are assured of equality of results. In game theoretic terms, firms are assumed to play a symmetric game (equality of opportunities), the equilibria of which need, however, not be symmetric (possible inequality of results). Equilibria might display a primary kind of asymmetry (the one which concerns us here) involving the distinction between active and inactive firms. A free entry equilibrium is just a Nash equilibrium of the symmetric game, such that some firms are active and some inactive. In other words, there is at least a potential entrant optimally deciding not to actually enter .

This free entry equilibrium is usually viewed as a sub-game perfect equilibrium of a two-stage game, with firms deciding at the first stage either to enter or not, and then with entrants competing at the second stage according to some specified regime (typically, quantity or price competition). The first stage entry decision, possibly implying the same sunk cost for any entrant, does, however, not give the right to an equal treatment at the second stage, as implicitly assumed. Under internal increasing returns, there may be active and inactive entrants at a second stage equilibrium. Consequently, we might as well resort to a simultaneous game, and take entry/exit decisions as implicit in quantity and/or price decisions [START_REF] Novshek | Cournot equilibrium with free entry[END_REF]d'Aspremont et al. 2000;[START_REF] Corch Ón | Comparative statics in Cournot free entry equilibrium[END_REF][START_REF] Yano | Coexistence of large firms and less efficient small firms under price competition with free entry[END_REF][START_REF] Yano | A price competition game under free entry[END_REF].

To be explicit, consider a symmetric simultaneous game played by N competing oligopolistic firms, each one with the same strategy space S and the same payoff function : S N → R.Afirmisinactive if it chooses an element of the subset S 0 of strategies that lead to zero output, and it is active if it chooses a strategy in the complementary subset. The nature of the subset S 0 results from the particular specification of the model, S 0 being for instance equal to {0} in quantity competition games, or to the set of prices higher than any customer's reservation price in price competition games. We admit that the payoff function is constant with respect to any of its arguments over S 0 , if this set has more than one element. Now consider strategy profiles s ∈ S N that are symmetric within the class of n active firms (0 < n < N),2 all choosing s n ∈ S S 0 while Nn inactive firms indifferently choose some element of S 0 . It is clear that the relevant information in s is completely contained in the pair (s n , n). Similarly, as the vector s -i ∈ S N-1 of strategies of the N -1 competitors of any firm i has nδ elements equal to s n (with δ = 1i ffi r mi is active and δ = 0 if it is inactive) and N -1 -(nδ) elements belonging to S 0 , it can be fully characterized by the triplet (s n , n, δ). The profit (s i , s -i )ofanyfirmi, choosing strategy s i = s and facing a profile s -i of its competitors' strategies with such characterization, can then be denoted accordingly by (s , s n , n, δ).

If we apply the Nash equilibrium concept to this framework, for a pair (s n , n)t o characterize an equilibrium, the profit (s , s n , n,1)ofanactivefirmmustreachamaximum at s = s n ,andthepr ofit (s , s n , n, 0) of an inactive firm must reach a maximum at any s 0 ∈ S 0 . Furthermore, if we take free entry as comprehending free exit, so that sunk costs are excluded,3 inactivity always results in zero profits, so that any equilibrium (s n , n)must verify (s n , s n , n,1)≥ 0and (s 0 , s n , n,0)= 0.

In the usual approach,4 if n is the number of firms having chosen to enter at the first stage, the two conditions for a free entry equilibrium (s n , n) without sunk costs are: first, that (s , s n , n, 1) reach a maximum non-negative value on S S 0 at s = s n , and second, that whenever this value is positive there be no equilibrium with n + 1 entrants (symmetric with respect to all of them). The second condition means that, for any strategy

s n + 1 ∈ S S 0 , if (s , s n+1 , n + 1, 1) is maximized on S S 0 at s = s n+1 then (s n+1 , s n+1 , n + 1, 1) < 0.
Putting together these two conditions one obtains the zero profit condition commonly seen as implied by free entry ( (s n , s n , n,1)= max s ∈ S S 0 (s , s n , n,1)= 0), provided one neglects the so-called "non-integer problem" (as

n belongs to N * ,n o tt o R + , (s n+1 , s n+1 , n + 1, 1) < 0 < (s n , s n , n, 1) is in fact the generic case).
Instead, we use the standard definition of a Nash equilibrium of a single stage game, just characterizing separately, as best responses, the decisions of active and inactive firms:

Definition 1 A non-trivial symmetric free entry equilibrium is a pair (s n , n)in(S S 0 ) × {1, ..., N -1} satisfying two conditions: max s ∈ S (s , s n , n,1) = (s n , s n , n,1) ≥ 0(profitability)and max s ∈ S (s , s n , n,0) = max s ∈ S 0 (s , s n , n,0) = 0 (sustainability).
For a strategy profile characterized by the pair (s n , n) to be an equilibrium, it must be profitable for any active firm to choose the strategy s n , meaning that no higher profit is attainable either while staying active ( (•, s n , n, 1) is maximized at s n ) or through becoming inactive ( (s n , s n , n, 1) is non-negative). The strategy profile must also be sustainable with respect to inactive firms, meaning that these firms should not be able to obtain a positive profit by becoming active.

A canonical pricing model

To be more specific and still cover different regimes of competition with a unified framework, we now introduce a simple model where strategies, whatever their nature, can be represented by prices. Take an industry producing either a homogeneous or a composite good sold at price P. Demand for this good is given by a twice differentiable decreasing function D :(0, P ) → (0, ∞), with P ∈ (0, ∞], continuously extended for P ≥ P as D(P ) = 0. The good is potentially produced by N firms, under internal increasing returns, with the same twice differentiable increasing cost function C :(0,∞) → (0, ∞), extended to C (0) = 0 (no sunk costs) and such that average cost C(y)/y is decreasing on (0, ∞). As in the preceding subsection, we restrict our analysis to equilibria that are symmetric with respect to n active firms (0 < n < N), all choosing the same strategy s n ∈ S S 0 .This strategy will always be represented by the price p n at which any active firm intends to sell its output. Therefore, any firm deciding to supply quantity y at price p, and facing demand d( p, p n , n, δ), has to solve a problem that can be stated as follows:

max ( p, y) ∈ R 2 + { py -C (y) : y ≤ d ( p, p n , n, δ)} .
(1)

Clearly ,apair(p, y)suchthat0< y < d( p, p n , n, δ) cannot be a solution to this problem, because the profit is increasing in y if C (y)/y ≤ p. Therefore, the firm will always decide either to produce y = d( p, p n , n, δ) or to stay inactive (i.e. to choose y = 0), so that we can directly refer to the canonical program (CP) in the single decision variable p

max p ∈ R + { pd ( p, p n , n, δ) -C (d ( p, p n , n, δ))} , (CP)
and then check that the maximum profit is non-negative, taking otherwise y = 0a st h e optimal decision. One sees immediately that the CP covers the case where firms produce differentiated goods and compete in prices. It is less evident yet true that it also covers for instance the case of a homogeneous oligopoly with Cournotian firms. Indeed, given symmetry with respect to n active firms, each one of these firms chooses p n = P and y n = D(P )/n,whereasNn inactive firms all choose y = 0.Theresidualdemandatpricep foranyfirm,whetheractive

(δ = 1) or inactive (δ = 0) is D( p) -(n -δ)y n = D( p) -(1 -δ/n)D( p n ) ≡ d( p, p n , n, δ). As y = d( p, p n , n, δ)ifandonlyifD( p) = (n -δ) y n + y, the CP is indeed equivalent to the standard program of the Cournotian firm; namely, max y ∈ R + {D -1 ((n -δ)y n + y)y - C (y)}.
The specific form of contingent demand to the firm d(•, p n , n, δ), of which the Cournotian contingent demand is just an example, depends on the assumed regime of competition. However, we can establish a general relation between the demand to the active firm d( p n , p n , n, 1) as a function of p n (the price set by all the n active firms) and the demand to the industry:

nα(n)d( p n , p n , n,1)= D(P ), with P = p n /α(n), (2)
where α(n) is a positive aggregating factor to be used when the product is a composite good (α(n) ≡ 1, otherwise). 5We now introduce two general assumptions on the cost and demand functions. The first expresses existence of internal increasing returns to scale (yet not at an increasing rate). Formally:6

Assumption 1 The function C (y)/y is twice differentiable and has a negative, non-decreasing elasticity (ǫ y C (y) -1 < 0, ǫ 2 yy C (y) ≥ 0) in the interval (0, ∞). The second assumption is introduced to ensure that the profit function is well-behaved and has adequate boundary properties, namely, that it takes negative values when the scale of production is either too small or too large. It further ensures that the demand to any firm, hence its average revenue, is larger for the incumbent than for the potential entrant (as the former competes with only n -1 producing firms, whereas the latter faces one more competitor). Formally: Assumption 2 For any triplet ( p n , n, δ), the function d(•, p n , n, δ) is twice differentiable in the interval (0, p( p n , n, δ)) in which it is positive (where p( p n , n, δ) ∈ (0, ∞] is the supremum of customers' reservation prices), and has in this interval a negative, decreasing elasticity Furthermore, for any p, the function d( p, •) is increasing in p n and δ, and non-increasing in n, as long as its value remains positive.

(ǫ p d(•, p n , n, δ) < 0, ǫ 2 pp d(•, p n , n, δ) > 0), such that 1 lim p→0 ǫ p d ( p, p n , n, δ) < lim y→∞ ǫ y C (y) -1and (3) 1 lim p→ p( p n ,n,δ) ǫ p d ( p, p n , n, δ) > lim y→0 ǫ y C (y) -1. ( 4
Geometrically, as illustrated by Figure 1, Assumption 1 states that the average cost curve, C (y)/y, is decreasing and convex when represented on a diagram with logarithmic scales (see curve AC, with slope ǫ y C (y) -1). Assumption 2 states that the average revenue curve (given by the inverse of the function d(•, p n , n, δ)) is decreasing, strictly concave, and always lower than the average cost curve for an output either close to zero or close to infinity (see curves AR, with slopes 1/ǫ p d( p, p n , n, δ)).

In this diagram, the profitability condition requires that the average revenue curve of an active firm AR (δ = 1) be higher than the average cost curve, AC, for intermediate values of y, whereas the sustainability condition requires that the average revenue curve of an inactive firm, AR (δ = 0), be lower than the average cost curve, AC, for all values of y.The two conditions are compatible because average revenue is increasing in δ by Assumption 2. The curve that is tangent to the average cost curve corresponds to the limit case of a price p n such that a potential entrant can at most make a zero profit if it decides to become active. We will come back to this limit price in the following subsection.

As a last remark, it should be noticed that twice differentiability of the function d(•,p n ,n,δ) is more than a technical assumption. Indeed, it excludes the case of Bertrand competition (price competition in a homogeneous oligopoly) because d(•, p n , n, δ)h a s then a discontinuity at p = p n .

Equilibrium conditions

Under the assumptions of the preceding subsection, we can reformulate the profitability and the sustainability conditions in terms of the price p n set (or targeted) by all the n active firms. Profitability requires that this price be an interior solution to the CP (hence, a critical point of the corresponding profit function (•, p n , n, 1)), and that this critical price be at least equal to the break-even price, entailing zero profits. By contrast, sustainability requires that the critical price be at most equal to the limit price deterring entry. In the following, we are going to give formal definitions of these three reference prices, and examine sufficiency of the above conditions. Definition 2 A critical price p * n is a positive price that, when simultaneously set by n active firms, satisfies the first order condition necessary for an interior solution of CP; that is, solves the equation of marginal revenue with marginal cost:

p n (1 + 1/ǫ p d( p n , p n , n,1))= C ′ (d( p n , p n , n,1)).
(FOC)

If the critical price p * n entails non-negative profits, this first order condition is in fact sufficient, under our assumptions, for an interior solution of the CP, in spite of the possible lack of quasi-concavity of (•, p n , n, 1), because of the non-convexity of the cost function. Profitability is then satisfied at p * n , as stated in the following lemma: Lemma 1 (Profitability I) Under Assumptions 1 and 2, the symmetric strategy profile represented by the pair ( p * n , n) ∈ R ++ ×{1, ..., N -1} satisfies the profitability condition if and only if p * n is a critical price entailing non-negative profits or, equivalently, leading to a revenue-cost ratio at least equal to one:

g p * n , n ≡ p * n d p * n , p * n , n,1 C d p * n , p * n , n,1 ≥ 1. (PNNC)
PROOF: See Appendix. Now, notice that for an increasing function g (•, n), condition (PNNC) can equivalently be expressed by requiring that the critical price be at least equal to the break-even price p(n), which entails zero profits, or equivalently leads to a unit value of the revenue-cost ratio. We formalize and develop this idea in the following. To begin with, we give a general definition of the break-even price, independently of g (•, n) being an increasing function. To understand this definition, recall that P is the price, possibly infinite, at which demand becomes nil, and that the price P of the composite good is equal to p n deflated by the aggregating factor α(n).

Definition 3

The break-even price p(n), is the lowest price p n that, when set by all the n active firms, allows them to get non-negative profits: p(n) ≡ inf P(n), with

P(n) ≡{p n ∈ (0, α(n) P ):g ( p n , n) ≥ 1} (5) (by convention, p(n) =∞if P(n) = ∅).
As just observed, the profitability requirement can be equivalently expressed by the inequality p * n ≥ p(n)( w h e r e p * n is a critical price), provided g (•, n) is an increasing function, a property that is verified when the demand to the industry has an elasticity that is always at least equal to -1. Otherwise, the market revenue, PD(P), may decrease with P, imposing an upper bound on the set P(n), so that the preceding inequality is only a necessary (but in general not sufficient) condition for profitability. However, under the following additional assumption, implying in particular quasi-concavity of the function g (•, n), that inequality remains a sufficient condition for profitability, as stated in Lemma 2.

Assumption 3 Demand to the industry has an elasticity ǫ P D(•) that is non-increasing whenever smaller than -1. The elasticity ǫ y C ′ (y) of marginal cost is larger, for any y, than the least upper bound 1/ lim P → P ǫ P D(P ) of the elasticity of inverse demand.

Lemma 2 (Profitability II) Under Assumptions 1, 2 and 3, the symmetric strategy profile represented by the pair ( p * n , n) ∈ R ++ ×{1, ..., N -1} satisfies the profitability condition if and only if p * n is a critical price at least equal to the break-even price: p * n ≥ p(n). PROOF: See Appendix.

Finally, we introduce the concept of limit price, to be taken as un upper bound imposed on the critical price so as to ensure sustainability. This price is defined as "the highest common price which the established seller(s) believe they can charge without inducing at least one increment to entry" (Bain 1949, p. 454). This is the price leading to an average revenue curve of the potential entrant that is just below the average cost curve [START_REF] Modigliani | New developments on the oligopoly front[END_REF], as represented by the curve that is tangent to curve AC in Figure 1. Formally: Definition 4 The limit price, p(n), is the highest price p n that, when set by all the n active firms, prevents an inactive firm from getting positive profits: p(n) ≡ sup P(n), with

P(n) ≡ p n ∈ (0, α(n) P ): max p ∈ (0, p( p n ,n,0)) G ( p, p n , n) ≤ 1 , ( 6 )
with G ( p, p n , n) ≡ pd( p, p n ,n,0) C (d( p, p n ,n,0)) and p( p n , n, 0) as defined in Assumption 2. Observe that the elasticity with respect to p of the revenue-cost ratio G is

ǫ p G ( p, p n , n) = 1 + (1 -ǫ y C (d( p, p n , n,0))) ǫ p d( p, p n , n,0). ( 7
)
Hence, by inequalities (3) and (4) in Assumption 2, this elasticity is positive for p close to zero and negative for p close to p( p n , n, 0), implying that G (•, p n , n) has indeed an interior maximum. Therefore, we can determine the limit price, p(n), as the solution in p n to equations:

pd ( p, p n , n,0) = C (d ( p, p n , n,0)) (8) -ǫ p d( p, p n , n,0)= 1 1 -ǫ y C (d ( p, p n , n,0)) ;( 9 )
namely, the zero profit condition and the first order condition for maximization of G (•, p n , n), respectively.

Given Definition 4, we can now reformulate the sustainability condition by reference to the limit price, p(n).

Lemma 3 (Sustainability) Under Assumptions 1 and 2, the condition p n ≤ p(n) is necessary and sufficient for the pair ( p n , n) ∈ R ++ ×{1, ..., N -1} to satisfy the sustainability condition. PROOF: See Appendix.

We summarize in the following proposition the results stated in the two last lemmata.

Proposition 1 Under Assumptions 1, 2 and 3, a symmetric profile represented by the pair

( p * n , n) ∈ R ++ ×{1, .
.., N -1} is a free entry equilibrium if and only if p * n is a critical price between the break-even price and the limit price: p(n) ≤ p * n ≤ p(n). Figure 2 provides a representation of this equilibrium condition in the space (n, p n ), where the critical, break-even and limit prices appear as functions p * (•), p(•)and p(•)of a real (instead of an integer) number n. 7 Observe how the condition p(n) ≤ p * n ≤ p(n), resulting in the thick segment of the critical price curve, translates into a restriction on the number n of active firms. This number should belong to the interval [n, n], with endpoints defined by p * (n) = p(n)and p * (n) = p(n).

Obviously, as soon as the interval [n, n] contains more than one integer, the profitability and sustainability conditions are compatible with existence of free entry equilibria with positive profits, along with the one determined by the zero profit condition (corresponding to a number of active firms equal to the highest integer in the interval, precisely equal to n in this particular example). It is worthwhile emphasizing that this source of equilibrium multiplicity differs from the one usually considered in the coordination failures literature and stemming from the seminal paper of [START_REF] Cooper | Coordinating coordination failures in Keynesian models[END_REF]. In this literature, multiple symmetric equilibria are associated with the same exogenous number n = N of players (thus resulting, in our framework, in a multi-valued function p * (•)). Such multiplicity relies on strategic complementarity, resulting in the requirement that the best response of p be an increasing function of p n . No such condition is necessary in our approach, because symmetry is now imposed only within each class of active and inactive firms. This explains why, as we are going to show, equilibrium multiplicity can prevail even under strategic substitutability, as it is typically the case under Cournot competition.

Regimes of competition

To illustrate the potential of our analytical framework, we now apply it to two standard regimes of quantity and price competition. As for quantity competition, we limit our analysis to the Cournot homogeneous oligopoly. As for price competition, we must refer to a market for differentiated products, because the Bertrand homogeneous oligopoly is outside our scope and leads, at free entry equilibrium, to the competitive outcome [START_REF] Yano | Coexistence of large firms and less efficient small firms under price competition with free entry[END_REF][START_REF] Yano | A price competition game under free entry[END_REF]. We might for instance use the well-known [START_REF] Dixit | Monopolistic competition and optimum product diversity[END_REF] model, modified so as to account for manipulability of the industry price index by each one of a "small group" of active firms (d'Aspremont et al. 1996). However, because this model has already been treated in this perspective by d [START_REF] Aspremont | Contestability and the indeterminacy of free entry equilibria[END_REF], we shall devote our analysis of price competition to the [START_REF] Salop | Monopolistic competition with outside goods[END_REF] spatial model, where the strategy variables include, besides prices, locations in the characteristics space.

Quantity competition with product homogeneity: the Cournot model

There are two sources of decreasing average cost; namely, the presence of a fixed cost and the existence of internal economies of scale accounting for decreasing marginal cost. These two sources are not equivalent, as attested by their specific effects in the Dixit-Stiglitz model, where multiple free entry equilibria generally exist when marginal cost is an isoelastic decreasing function, but never when it is constant, in the presence of a fixed cost (d'Aspremont et al. 2000).8 However, in the two regimes of competition we are going to analyze, the weaker source of decreasing average cost is sufficient to ensure multiplicity of free entry equilibria and, hence, to illustrate the use of our framework. For simplicity, we shall accordingly assume a positive fixed (non sunk) cost, φ, and a constant positive marginal cost, normalized to one: C (y) = φ + y if y > 0andC (0) = 0, a function that clearly satisfies Assumption 1. On the demand side we assume, also in the analysis of both regimes, a unitelastic demand to the industry, D(P ) = b/P ,wi t hb > 0, satisfying Assumption 3.9 One advantage of this specification is that the break-even price becomes independent of the particular regime of competition we are considering, because expenditure b in the industry is not affected by price changes. The break-even price is then given by

p(n) = 1 1 -nφ/b , (10)
an increasing function of the number n of active firms and of the share φ/b of individual fixed cost in aggregate expenditure. With this specification of the demand to the industry and according to what has already been shown in Subsection 2.2, the Cournotian contingent demand can be expressed in a symmetric configuration as

d ( p, p n , n, δ) = b 1 p - 1 -δ/n p n , (11)
where p isthemarketpriceaimedatbythefirmand p n = b/ny n the price representing the strategy y n expected from each one of its nδ active competitors (with δ = 1ifthefirmis itself active, δ = 0 otherwise). This function has a partial elasticity with respect to p (for p in the interval (0, p n /(1δ/n)) in which individual demand is positive and finite) given by

ǫ p d( p, p n , n, δ) =- 1 1 -(1 -δ/n)( p/ p n ) . ( 12
)
It is easy to check that it satisfies Assumption 2, and that ǫ p d( p n , p n , n,1)=-n. From ( 11), we see that the contingent demand to the inactive firm is independent of the number n of active firms, so that the limit price is itself constant in n:10

p(n) = 1 (1 - √ φ/b) 2 , (13)
again an increasing function of the share φ/b of individual fixed cost in aggregate expenditure. As to the critical price, it is simply equal to the markup factor µ (n) = n/(n -1) on marginal cost (multiplied by 1, the normalized marginal cost):

p * (n) = n n -1 . ( 14
)
Profit non-negativity ( p * (n) ≥ p(n)) requires, by (10),

n ≤ b/φ ≡ n, ( 15
)
an upper bound that must be at least equal to 2, for a free entry equilibrium to exist, so that φ/b ≤ 1/4 is a necessary condition for existence. Sustainability ( p * (n) ≤ p(n)) in turn requires:

n ≥ n 2 -1/n ≡ n. ( 16
)
Notice that the admissible interval [n, n] contains more than one integer for n ≥ 3; that is, for a small enough degree of economies of scale, as determined by the share φ/b of individual fixed cost in aggregate expenditure, which should not exceed 1/9. The degree of indeterminacy increases with n; that is, it is larger the smaller the degree of economies of scale. However, a low degree of economies of scale results in a relatively large number of active firms, reducing the impact of variations in n on the markup factor (and, hence, on the equilibrium price). Figure 3 gives an illustration (for φ/b = 0.04 and similarly to Figure 2) of the conditions demanded from the critical, break-even and limit prices to ensure profitability and sustainability. For the sake of a comparison with the price competition regime, notice that the set of candidate numbers of active firms in a free entry equilibrium is {3, 4, 5} in this particular case.

Price competition with strategic product differentiation: the Salop model

In the industrial organization literature, spatial competition is a popular alternative to nonaddress models relying on constant elasticity of substitution or quadratic consumers' utility functions. Although less frequent in macroeconomic modeling, it has, for instance, already been used by [START_REF] Weitzman | Increasing returns and the foundations of unemployment theory[END_REF], who introduced a macroeconomic version of the [START_REF] Salop | Monopolistic competition with outside goods[END_REF] model of the circular city. The space of characteristics of the industry good is represented by a circle with perimeter equal to 1, on which consumers' locations are uniformly distributed with unit density. A consumer devoting a positive budget b to the purchase of that good and located at point x between two firms j and j + 1, which are themselves located at a j and a j +1 ,respectively ,willbuyfromfirmj if p j + τ (xa j ) < p j +1 + τ (a j +1x), where p j and p j +1 are the prices set by the two firms and τ is the subjective transportation rate in money equivalent units. The marginal consumer who is indifferent between the two suppliers is the one located at point x ( j, j + 1) = (a j + a j +1 )/2 + ( p j +1p j )/2τ , so that the market area of firm j is

x ( j , j + 1) -x ( j -1, j ) = a j + 1 -a j -1 2 + ( p j -1 + p j + 1 )/2 -p j τ , (17)
which is independent upon its own location a j . However, although indifferent about its precise location within its market area, firm j is assumed to set its price, p j , on the basis of its conjectures not only about prices p j -1 and p j +1 but also about the locations a j -1 and a j +1 simultaneously chosen by its neighbors. This implies in particular that, when inactive at the strategy profile taken as reference, a deviating firm does not conjecture that the locations of the two competitors between which it decides to locate are going to be benevolently accommodated in response to its decision to deviate into activity. As a consequence, any deviating firm is able to manipulate its market area through its pricing decision, within the segment separating its two neighbors, but the length of this segment is 2/n if the firm is active and only 1/n if it is inactive (assuming that locations are symmetric with respect to the n active firms).

On the basis of symmetry with respect to both locations and prices, we obtain from (17) the following expression for contingent demand to the representative firm (with δ = 1ifit is active, δ = 0otherwise):

d( p, p n , n, δ) = (b/τ p)((1 + δ)τ/2n + p n -p), ( 18
)
with p ∈ (0, (1 + δ) τ/2n + p n ]. The partial elasticity of d( p, p n , n, δ)withrespecttop is:

ǫ p d ( p, p n , n, δ) =- (1 + δ) τ/2n + p n (1 + δ) τ/2n + p n -p . ( 19
)
All the conditions of Assumption 2 are again satisfied. Because ǫ p d( p n , p n , n,1)=-(1 + np n /τ ), the markup factor on the marginal cost is µ = 1 + τ/np n , so that we obtain the following expression for the critical price as a (decreasing) function of n:

p * (n) = µ * = 1 2 + 1 4 + τ n . ( 20
)
Profit non-negativity ( p * (n) ≥ p(n)) imposes, by this equation and equation ( 10), an upper bound on the number n of active firms:

n ≤ τ √ τφ/b(1 + √ τφ/b) ≡ n, (21)
which must be at least equal to 2 for existence of a free entry equilibrium, so that the ratio φ/b cannotbetoolarge.Theupperboundn on the number of active firms is increasing in the transportation rate, τ (representing the degree of product differentiation), and decreasing in the share φ/b of individual fixed cost in aggregate expenditure (determining the degree of internal economies of scale). Finally, equations ( 8) and (9) lead to the limit price11

p(n) = (1 + τφ/b) 2 -τ/2n, (22)
( p * (n) ≤ p(n)): n ≥ τ ( 2(1 + √ τφ/b) 2 + 1/4 -1) 2 -1/4 ≡ n. ( 23
)
It can easily be checked that the ratio n/n between the endpoints of the admissible interval [n, n] is a decreasing function of the variable √ τφ/b, which tends to 3/4asthis variable tends to zero. As a consequence, the admissible interval contains at least two integers if n ≥ 4, which requires a small enough share of individual fixed cost in aggregate expenditure (φ/b < 1/4) and a high enough degree of product differentiation as measured by τ . We represent in Figure 4 (for the same value 0.04 of the ratio φ/b and for τ = 25/16, so as to get the same n = 5andthesame p * (n) = µ(n) = 1.25 as in Figure 3)12 the free entry equilibrium conditions in terms of the critical, break-even and limit prices. Observe that thesetofnumbersofactivefirmscompatiblewithfreeentryequilibriumisnow{4, 5},a proper subset of the corresponding set under quantity competition.

We can now formulate a tentative conclusion of the analysis performed in this section. The degree of internal economies of scale (here determined by the share φ/b of the individual fixed cost in the aggregate expenditure) must be low enough to ensure existence, and even more so to entail multiplicity, of free entry equilibria. However, as the share φ/b becomes smaller, the amplitude of price variations across equilibria becomes smaller too. Under price competition, existence and especially multiplicity also require a high enough degree of product differentiation (here represented by the rate of transportation τ )a n d a correspondingly high degree of market power. Sustainability (as indicated by the level of the lower bound n on the admissible number of active firms, given the same upper bound n) is harder to ensure under price competition than under quantity competition, where potential entrants expect incumbents to stick to specific output levels. However, price sustainability remains compatible with positive profits under strategic product differentiation, where potential entrants at least expect incumbents to stick to specific locations in the characteristics space. In the non-address model of Dixit and Stiglitz, where potential entrants have no disadvantage relative to incumbents as regards the capacity to benefit from product differentiation, the presence of a fixed cost is, however, not enough to ensure sustainability along with positive profits.

Conclusion

We have argued in this paper that, in spite of an almost universal convention, zero profits should not be imposed as an equilibrium condition under free entry, beyond the realm of non-strategic forms of competition. A straightforward application of the Nash equilibrium concept to standard simultaneous symmetric games, portraying diverse regimes of oligopolistic competition, typically entails multiple free entry equilibria with various adjacent numbers of active firms and different levels of positive profits. These equilibria are characterized by two conditions: profitability (the price should be no smaller than the break-even price) and sustainability (the price should be no larger than the limit price). These conditions define a non-degenerate interval of admissible numbers of active firms, that typically contains more than one integer. The zero profit condition then appears as no more than a particular selection criterion, picking up the least profitable equilibrium, associated with the highest integer in this interval. Our analytical framework and the indeterminacy results that it allows us to establish are quite robust, and apply to both quantity and price competition; although the latter regime, in particular under non-strategic forms of product differentiation, makes sustainability harder to attain when profits are positive.

Beyond industrial organization, the indeterminacy of free entry equilibrium has potentially significant macroeconomic implications. In particular, it raises a coordination problem and, consequently, favors the emergence, even under dynamic determinacy, of sunspot fluctuations. These might be induced by some extrinsic, potentially varying, public signal on which firms in each industry need to coordinate [START_REF] Santos Ferreira | Free entry and business cycles under the influence of animal spirits[END_REF][START_REF] Santos Ferreira | Business cycles with free entry ruled by animal spirits[END_REF]. Also, taking into account this particular type of indeterminacy enlarges the scope for coordination failures, which cease in particular to depend upon strategic complementarity.

Figure 1

 1 Figure 1 Average cost (AC) and revenue (AR) curves.

Figure 2

 2 Figure2Critical, limit and break-even prices.

Figure 3

 3 Figure 3 Equilibrium conditions in the Cournot model.

Figure 4

 4 Figure 4 Equilibrium conditions in the Salop model.

  and replace α(n) P by p * n , using the first order condition that defines a critical price:By Assumption 2, this derivative is positive. Also, ∂ (P , n)/∂ P becomes negative at zero by Assumption 3, because ǫ P D(P ) is non-increasing and since ǫ y C ′ (D(P )/nα(n)) ǫ P D(P ) < 1foranyP. Hence, the joint profit function (•, n) is (strictly) quasi-concave, so that sup P ∈ (0, P ) (P , n) > p * n /α(n). Also, sup P (P , n), unless infinite, must belong to the set P(n) of prices entailing non-negative profits. In any case, as this set is an interval,p(n) ≤ p * n <α(n)sup P (P , n)implies: p * n ∈ P(n). 2 As d( p, •, n, 0) isincreasing (by Assumption 2), and C (y)/y is decreasing in y (by Assumption 1), the revenue-cost ratio G ( p, p n , n) is increasing in p n . By definition of the limit price p(n), G ( p, p(n), n) = 1a t p maximizing

	∂	p * n /α(n), n ∂ P	= D p * n /α(n) 1 -	ǫ P D p * n /α(n) ǫ p d p * n , p * n , n,1
			= D p * n /α(n) -	ǫ pn d p * n , p * n , n,1 ǫ p d p * n , p * n , n,1	.	(26)
		Proof of Lemma 3 (Sustainability)

G (•, p(n), n). Hence, p n ≤ p(n) is clearly a necessary condition for sustainability (G ( p, p n , n) > 1if p n > p(n)).

For sufficiency, consider

p n < p(n)andG ( p, p n , n) > 1forsomep.Then,G ( p, p(n), n) > 1, contradicting the definition of p(n).

2

International Journal of Economic Theory 3 (2007) 75-94 C IAET

As a matter of fact, in the game designed by[START_REF] Yano | A price competition game under free entry[END_REF], firms simultaneously choose a price and a set of quantities that are equivalent from the point of view of the profits they generate. Together with free entry, this device allows him to extend Bertrand equilibrium to the case where the average cost function is not constant, without incurring the inexistence problem pointed out by Edgeworth.[START_REF] Yano | Coexistence of large firms and less efficient small firms under price competition with free entry[END_REF] extends this analysis to encompass strategic technological choices.International Journal of Economic Theory 3 (2007) 75-94 C IAET

We a d m it t h a t n > 0 in order to eliminate trivial equilibria, and that n < N to put aside the case where entry is impossible for lack of further participants.

Exclusion of sunk costs is consistent with our assumption of a simultaneous game, but is not crucial. In a two-stage game, if all (actual) entrants incur the same positive sunk cost k at a first stage, the profit of an inactive entrant -k is negative, instead of 0, but this translation does not fundamentally alter the argument.

See Götz (2005) for a recent example.International Journal of Economic Theory 3 (2007) 75-94 C IAET

Deflating the price p n by the aggregating factor in the definition of the price index, P (the price of the composite good), ensures that the aggregate expenditure, PD(P), in the industry is indeed equal to the sum np n d( p n , p n , n,1) of firms revenues. A well-known example of the use of such price and quantity indices for industries producing a composite commodity is afforded by the[START_REF] Dixit | Monopolistic competition and optimum product diversity[END_REF] framework (where α(n) = n 1/(σ -1) ,withσ the constant elasticity of substitution between elementary goods). For a more general analysis, see d[START_REF] Aspremont | Competition for market share and for market size: Oligopolistic competition with varying competitive toughness[END_REF].

We deno te ǫ x f (x, y) ≡ (∂ f (x, y)/∂ x) x/ f (x, y) the partial elasticity of f at (x, y)withrespecttox.Allrelated elasticity notations are self-explanatory.International Journal of Economic Theory 3(2007)

75-94 C IAET

The case represented in Figure2corresponds to price competition in a differentiated oligopoly with a constant elasticity of substitution aggregator(Dixit-Stiglitz), where we have assumed a unit-elastic demand to the industry, with constant expenditure normalized to unity, a constant elasticity of substitution within the industry equal to 6, and a constant elasticity of the marginal cost function equal to -0.2. To facilitate comparisons between regimes of competition, we have chosen the parameter values so as to ensure that the critical and break-even price functions intersect at the same integer value n.International Journal of Economic Theory 3 (2007) 75-94 C IAET

In the latter case, the admissible interval [n , n] is still non-degenerate, but it contains at most one integer, the one determined by the zero profit condition.

Equilibrium with a single active firm will consequently be excluded.

Using (11) with δ = 0, we get from (8) py = b(1p/ p n ) = φ + y,sothat p n = [b/(b -(φ + y))] [(φ + y)/y].From (9), we get p/ p n = y/(φ + y). Hence, φ + y = √ φb,and,finally,p n = 1/(1 -√ φ/b) 2 .International Journal of Economic Theory 3 (2007) 75-94 C IAET

Using (18) with δ = 0, we obtain from (8) py = (b/τ )(τ/2n + p np) = φ + y. Also, using (19) with δ = 0, we get from (9): p/( p np + τ/2n) = y/φ. These equations together entail: y = √ bφ/τ and p = 1 + √ τφ/b, and then p n = (1 + √ τφ/b) 2τ/2n. International Journal of Economic Theory 3 (2007) 75-94 C IAET

Imposing either the same upper bound n or the same corresponding critical price p * (n) = p(n)leadstothe relation: √ τ = 1/(1 -√ φ/b). The choice of the numerical value φ/b = 0.04 then implies τ = 25/16.

thank an anonymous referee for very useful suggestions, as well as Stefano Bosi and Bertrand Wigniolle for their discussions of our paper. We gratefully acknowledge financial support by the Agence Nationale de la Recherche.

Appendix

Proof of Lemma 1 (Profitability I)

Ta ke a p a i r ( p * n , n) ∈ R ++ ×{1, ..., N -1}. Equation (FOC) in Definition 2 (i.e. the equality of marginal revenue and marginal cost) is the necessary first order condition for an interior solution of the CP. Hence, the condition that p * n be a critical price entailing non-negative profits is clearly necessary for profitability. Let us examine if it is sufficient. A sufficient second order condition for a local maximum is that the marginal revenue decrease faster with y (increase faster with p) than the marginal cost; that is, that the elasticity with respect to p of International Journal of Economic Theory 3 (2007) 75-94 C IAET the left-hand side of (FOC) be larger than the corresponding elasticity of the right-hand side:

By Assumption 2 and given (FOC) (implying 1 + ǫ p d( p * n , p * n , n,1)< 0), the left-hand side of inequality (SOC) is larger than one, so that the inequality is satisfied if the right-hand side is not larger than one. This is always the case if marginal cost is non-decreasing. Otherwise, if ǫ y C ′ (y * ) < 0( w i t hy * = d( p * n , p * n , n, 1)), the profit non-negativity condition (PNNC), by (FOC) and Assumption 1, can be expressed for a critical price as

This inequality implies that the right-hand side of (SOC) is indeed at most equal to

by Assumption 1, thus verifying condition (SOC). Hence, the profit function has a local interior non-negative maximum at any critical price p * n satisfying (PNNC). This maximum is in fact a global maximum. Indeed, by differntiability of the profit function, if there were two maxima, they would be separated by a minimum satisfying (FOC) and violating (SOC), hence (PNNC) (and (PNNC*)). However, if profit has a negative minimum at some price, then it cannot have a positive maximum at a higher price, because ǫ p d(•, p * n , n, 1) (in the left-hand side of (PNNC*)) is decreasing and ǫ y C (d(•, p * n , n, 1)) (in the right-hand side) is non-increasing, so that (PNNC*) cannot be satisfied at this higher price.

2

Proof of Lemma 2 (Profitability II)

Using equation ( 2) relating demand to the active firm and demand to the industry, we can compute the elasticity with respect to p n of the revenue-cost ratio g ( p n , n) defined in Lemma 1:

There are two cases. If ǫ P D(P )∈[-1, 0) for any P ∈ (0, P ), this elasticity is positive by Assumption 1. The function g (•, n) is then increasing, so that the condition g ( p n , n) ≥ 1 in Lemma 1 can be immediately replaced by the condition p n ≥ p(n), with g ( p(n), n) = 1. If ǫ P D(P ) < -1forsomeP,theelasticityǫ pn g ( p n , n)might be non-positive. The right-hand side of ( 24) is then non-increasing in p n , either through the elasticity of the cost function (by Assumption 1), or through ǫ P D( p n /α(n)) (by Assumption 3). Hence, once non-positive, the elasticity ǫ pn g ( p n , n) can never become positive again. In other words, the function g (•, n) is quasi-concave, and the set P(n) of prices entailing non-negative profits, defined by (5), is an interval. Next, we show that the joint profit (P , n) ≡ PD(P ) -nC(D(P )/nα(n)), computed under the constraint that every firm produce the same quantity, is increasing at a price P = p * n /α(n), such that p * n is a critical price. Consider the derivative with respect to P of the joint profit function: