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

Free entry equilibria are usually characterized by the zero proﬁt condition. We plead instead fora strict application of the Nashequilibrium concept to a symmetric simultaneous game played by actual and potential entrants, producing under decreasing average cost. Equilibrium is then typically indeterminate, with a number of active ﬁrms varying between an upper bound imposed by proﬁtability and a lower bound required by sustainability. We use a canonical model with strategies represented by prices, although covering standard regimes of quantity and price competition, to show that in equilibrium the critical (proﬁt maximizing) price must lie between the break-even and the limit prices.


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 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 Baumol et al. (1982), which implicitly refers to Bertrand competition, we may think of a "quantity sustainability" referring to Cournot competition, with different implications (Brock and Scheinkman 1983;Brock 1983). 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 Cooper and John 1988). 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 (Novshek 1980;d'Aspremont et al. 2000;Corchón and Fradera 2002;Yano 2005Yano , 2006. 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 N − n 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 n − δ elements equal to s n (with δ = 1i ffi r mi is active and δ = 0 if it is inactive) and N − 1 − (n − δ) 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 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: (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 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,whereasN − n inactive firms all choose y = 0.Theresidualdemandatpricep foranyfirm,whetheractive 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: where α(n) is a positive aggregating factor to be used when the product is a composite good (α(n) ≡ 1, otherwise). 5 We 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 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 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.
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)).
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: 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 (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. 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.

Lemma 2 (Profitability II)
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 (Modigliani 1958), 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).
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: −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). 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. 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
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 Cooper and John (1988). 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 (Yano 2005(Yano , 2006. We might for instance use the well-known Dixit and Stiglitz (1977) 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' Aspremont et al. (2000), we shall devote our analysis of price competition to the Salop (1979) 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 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 where p isthemarketpriceaimedatbythefirmand p n = b/ny n the price representing the strategy y n expected from each one of its n − δ 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 /(1 − δ/n)) in which individual demand is positive and finite) given by 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 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): Profit non-negativity ( p * (n) ≥ p(n)) requires, by (10), 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: 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 10 Using (11) with δ = 0, we get from (8) py = b(1 − p/ 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 . 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 Weitzman (1982), who introduced a macroeconomic version of the Salop (1979) 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 (x − a j ) < p j +1 + τ (a j +1 − x), 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 +1 − p j )/2τ , so that the market area of firm j is 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): with p ∈ (0, (1 + δ) τ/2n + p n ]. The partial elasticity of d( p, p n , n, δ)withrespecttop is: 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: Profit non-negativity ( p * (n) ≥ p(n)) imposes, by this equation and equation (10), an upper bound on the number n of active firms: 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 price 11 11 Using (18) with δ = 0, we obtain from (8) py = (b/τ )(τ/2n + p n − p) = φ + y. Also, using (19) with δ = 0, we get from (9): p/( p n − p + τ/2n) = y/φ. These equations together entail: y = √ bφ/τ and p = 1 + √ τφ/b, and then p n = (1 + √ τφ/b) 2 − τ/2n. and, after a straightforward computation, to the lower bound imposed by sustainability ( p * (n) ≤ p(n)): 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, 12 Imposing either the same upper bound n or the same corresponding critical price p * (n) = p(n)leadstothe relation: The choice of the numerical value φ/b = 0.04 then implies τ = 25/16.
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 (Dos Santos Ferreira and Dufourt 2006;Dos Santos Ferreira and Lloyd-Braga 2003). Also, taking into account this particular type of indeterminacy enlarges the scope for coordination failures, which cease in particular to depend upon strategic complementarity. and replace α(n) P by p * n , using the first order condition that defines a critical price: ∂ 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 .

Proof of Lemma 3 (Sustainability)
As d( p, ·, n, 0) is increasing (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 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)).