Monopolistic Competition with Big Firms∗ Sergey Kokovin† Mathieu Parenti‡

Jacques-François Thisse§ Evgeny Zhelobodko¶

October 23, 2011

Abstract We consider a differentiated market involving a handful of oligopolistic firms and a myriad of monopolistically competitive firms. Under various specifications of preferences, we show that the presence of small non-strategic firms is sufficient for the strategic interactions among oligopolistic firms to be diluted when monopolistically competitive firms can freely enter and exit the market in response to oligopolistic firms’ behavior. Furthermore, the oligopolistic firms adopt the same pricing rule as the monopolistically competitive firms. Our results also hold for heterogeneous firms. Keywords: monopolistic competition, oligopoly, product differentiation, entry, strategic behavior JEL Classification: D21, D43 and L10.

∗

We thank D. Encaoua, P. Fleckinger, L. Fontagné, J. Friedman, J. Hamilton, P. Neary, M. Riordan, J. Vogel and D.

Weinstein for comments and suggestions. We gratefully acknowledge the financial support from the Fonds de la Recherche Scientifique (Belgium), and the Economics Education and Research Consortium (EERC) under the grant No 08-036. † Novosibirsk State University and Higher School of Economics at Saint Petersburg. Email: [email protected] ‡ Université de Paris 1 - Paris School of Economics (France). Email: [email protected] § CORE-UCLouvain (Belgium), Higher School of Economics at Saint Petersburg and CEPR. Email: [email protected] ¶ Novosibirsk State University and Higher School of Economics at Saint Petersburg. Email: [email protected]

1

1

Introduction

As argued by many scholars ranging from Kaldor (1935) to Kreps (1990) through Stigler (1949), monopolistic competition would be a weird market structure which is unlikely to represent any real-world industry. For example, Stigler (1949) argues that this framework “has not been useful in the concrete analysis of economic problems.” Monopolistic competition has also been under the fire of economists working in industrial organization because it does not account for strategic interactions (Shubik, 1959; d’Aspremont et al., 1996; Neary, 2009). The dust is not settled down yet. Indeed, the monopolistic competition framework has proven to be extremely useful in various economic fields, both in modern economic theory and empirical studies (Matsuyama, 1995: Brakman and Heijdra, 2004; Eaton and Kortum, 2005; Bernard et al., 2007). For example, Eaton and Kortum (2005) maintain that monopolistic competition “has provided a more fertile ground for the quantitative analysis of trade flows.” Our aim is to show that the various criticisms raised against monopolistic competition need not be warranted. Specifically, we identify conditions under which monopolistic competition emerges as an equilibrium outcome in a setting involving a few large firms and a competitive fringe. This result runs against the conventional wisdom according to which big firms are expected to manipulate the market, while the small firms are unable to exert such an appreciable influence. To achieve our goal, we follow Aumann (1964, p.41) who suggests to combine a continuum of traders and atomic traders to study market power in an exchange economy. In measure-theoretic terms, this means that the market involves two kinds of firms: a finite number of atomic players/producers, which represent the big firms, and a continuum of nonatomic players/producers, which represent the competitive fringe. Such a setting thus blends oligopolistic and monopolistically competitive firms, which all supply varieties of a differentiated product. Specifically, the former supply a strictly positive measure subset of varieties, whereas the latter supply a single variety. Furthermore, in the wake of general equilibrium models with imperfect competition (Gabszewicz and Vial, 1972; Hart, 1985; Bonanno, 1990), we assume that big firms are aware that their choices may affect both consumers’ income and the size of the competitive fringe. As a result, it seems natural to describe the market process as a two-stage game. In the first stage, the big firms choose their prices (or outputs), anticipating the reactions of the small firms. In the second stage, the small businesses choose their prices (or outputs), treating the big firms’ choices parametrically. In doing so, they may enter or exit the market, whereas the big firms always stay in business.1 Although somewhat extreme, this difference in market 1

A more precise, but more cumbersome, description of the game would involve three stages In the first one, big firms

2

behavior aims to fit a fairly robust empirical fact, i.e. the survival probability of a firm is positively correlated with its size. Note also that this staging is similar to the one used in the dominant firm model (Markham, 1951; Kydland, 1979). Because the field of monopolistic competition has been very much dominated by the CES model of Dixit and Stiglitz (1977), we find it reasonable to begin our investigation with this utility function. Under these preferences, we show that the size of the competitive fringe acts as a buffer that leads the big firms to apply the same pricing rule as the monopolistic competitive firms, that is, a pricing rule unaffected by the other firms’ behavior. This has a far-reaching implication: the presence of a competitive fringe yields a market outcome in which strategic interactions among big firms are washed out. As a result, monopolistic competition à la Dixit-Stiglitz emerges as the actual market structure in an environment involving different kinds of firms. Because the CES model of monopolistic competition displays very special features, it is critical to investigate the robustness of those findings. First, we show that they hold true for heterogeneous firms à la Melitz (2003), be they big or small. Second, we turn to the case of general additive preferences, which have the merit of providing new impetus to empirical analysis (Houthakker, 1960). This is an important issue because the CES equilibrium price and output are independent of the market size, whereas alternative specifications of additive preferences generate either pro-competitive effects (equilibrium prices are lower in a bigger market) or anti-competitive effects (prices are higher in a bigger market), thus implying that the generalization of the above findings to general additive preferences is far from being straightforward (Zhelobodko et al., 2011). In addition, our preference specification is sufficiently flexible to allow for specific patterns of substitution between the varieties supplied by the big and small firms as well as between the varieties provided by the big firms, while the CES assumes the same degree of substitution among all varieties regardless of the type of firms supplying them. This extension is appealing from the empirical point of view because alternative preference specifications will typically provide theoretical predictions that are sufficiently simple to be tested, sufficiently general to make sense on an empirical level, and precise enough to allow one to discriminate between different explanations of firms’ behavior. Rather unexpectedly, under general additive preferences, all the results obtained under the CES remain valid in that the big firms mimic a non-strategic behavior. The reason is that small firms act again as a buffer that stabilizes competition among big firms. Indeed, the common thread of our approach is the existence of a single market statistic, the marginal utility of income, which subsumes firms’ behavior. To choose their output; in the second stage, small firms decide whether or not to enter and, in the third one, choose their best replies against the quantities chosen by the big firms.

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be precise, what drives our results is the fact that the value of this statistic is determined by the entry or exit of small firms. In other words, the marginal utility of income is independent of the big firms’ actions. Since the big firms correctly anticipate the second stage outcome, they accurately treat this statistic as a parameter. In doing so, they face demands that no longer depend on the other big firms’ strategies, which leads them to compete as if each of their variants were produced by a small firm! The market outcome with multi-product firms is thus formally identical to the one that a vast number of single-product firms would generate. Last, we show that the big firms price their varieties as if they were arbitrarily many. This result has a strong contestable market flavor: the entry and exit of small firms is sufficient for the big firms to price below the level they would choose in a purely oligopolistic environment. The prices selected by the big firms may be viewed as the differentiated product counterpart of the price charged by the incumbent producing an homogeneous good in the standard model of contestable markets. In other words, the presence of a competitive fringe disciplines the big firms. This contrasts with the standard viewpoint of antitrust policy where the emphasis is on the free entry of large firms. Though our whole analysis holds true when firms have variable marginal costs, for clarity we assume that they operate under constant marginal cost. Coming to the existing literature, we should mention Vives (1999) who shows that, under some preference structures, monopolistic competition is the limit of a sequence of oligopolistic markets. In a different strand of literature that relies on cooperative game theory, Gabszewicz and Mertens (1971) and Shitovitz (1973) show the following: when large traders are similar to each other, or when for each such large trader there are small traders similar to it, the core of an exchange economy involving a few large traders coincides with the set of competitive allocations. Under these circumstances, the market power of big traders is diluted (Gabszewicz, 2002). Our results have a similar flavor, but we focus on the inflow and outflow of small differentiated traders. Our paper is also related to the few contributions studying mixed market structure within the context of simultaneous games (Shimomura and Thisse, 2009; Parenti, 2010). Neary (2010) suggests a different setting in which firms choose to be big or small. Our approach is more specific because we assume that firms are borne big or small. However, our results show that differences in kind are immaterial under additive preferences since big and small firms all adopt the same pricing rule and non-strategic market behavior. Conversely, when preferences are non-additive, our paper suggests that more attention should be paid to the nature of the market structure. In other words, our analysis sheds light on a possible link between the structure of preferences and the nature of competition. We thus agree with Neary (2010) to say that there is room for additional and new modeling strategies of market behavior in the industrial organization and trade literature. 4

The remainder of the paper proceeds as follows. Section 2 presents our main results for the CES case. Section 3 extends the analysis to cope with heterogeneous firms à la Melitz. In the subsequent section, we show that the results of Section 2 can be generalized to the case of general additive preferences. However, when preferences are non-additive, large and small firms may interact in more convoluted ways, which could break the stabilizing impact of the competitive fringe. This will be illustrated in Section 5 where we study the quadratic utility to highlight the role of this assumption. Section 6 provides a general synthesis of previous results and identifies a stabilization condition that can be applied to more general preferences such as the translog (Feenstra, 2003). This point is worth stressing because the translog seems to work well from the empirical point of view (Feenstra and Weinstein, 2010). Section 7 concludes.

2

The CES utility

Describing the whole game used in this paper would be very cumbersome. For this reason, we ignore this step and proceed in a more intuitive way. As will become clear below, this has no implications for our analysis.

2.1

The model

The economy involves one horizontally differentiated good and one production factor - labor. As mentioned in the introduction, our aim is to study a mixed setting in which big and small firms adopt different market behaviors: big (oligopolistic) firms can manipulate the market aggregates; small (monopolistically competitive) firms are unable to influence market aggregates, but react to big firms’ behavior through entry or exit. Recall that our goal is to model big and small firms as firms different in kind. We capture this difference by assuming that the supply side of the economy involves (i) a continuum M of singleproduct (SP-) firms and (ii) N ≥ 2 (MP-) multi-product firms, where firm j = 1, .., N supplies a given range [0, nj ] of differentiated varieties with nj > 0. In this way, unlike SP-firms each MP-firm has the ability to manipulate the market outcome.2 Following the terminology used in the dominant firm model, the continuum of small firms will be referred to as the competitive fringe. On the demand side, there are L consumers who share the same CES preferences given by U= 2

ˆ

M

0

xρi di

+

N ˆ ! j=1

0

nj

ρ Xjk dk

(1)

We will see below that this assumption is not essential for our main results. Parenti (2010) studies the determination of

the product range under linear-quadratic preferences.

5

with 0 < ρ< 1. In (1), xi is the quantity of the variety provided by the SP-firm i, while Xjk is the quantity of the variety k provided by the MP-firm j.3 Alternatively, preferences could be described as follows (Shimomura and Thisse, 2009): U=

ˆ

M

0

xρi di

+

N !

Xjρ .

(2)

j=1

Under this specification, a big firm supplies a single variety that affects positively consumers’ behavior, while small firms remain negligible. In what follows, we will focus on (1). It is worth stressing, however, that the both specifications lead to the same qualitative results. Each consumer supplies inelastically one unit of labor and owns 1/L of each firm.4 The labor market is perfectly competitive and labor is the numéraire. Each consumer maximizes her utility (1) subject to the budget constraint where the wage is equal to 1: ˆ

M

pi xi di +

0

N ˆ ! j=1

0

nj

N

Pjk Xjk dk ≤ y ≡ 1 +

1 1! Πj + L L j=1

ˆ

M

πi di

(3)

0

where pi denotes the price of the variety produced by the SP-firm i and Pjk the price of variety k produced by the MP-firm j, while Πj stands for the profit earned by the MP-firm j and πi for the profit made by the SP-firm i. Observe that the income (or expenditure) y is endogenous and determined at the market equilibrium through the redistribution of profits. Let σ = 1/(1 − ρ) be the elasticity of substitution across varieties. First-order conditions for utility maximization with respect to each variety i ∈ [0, M ] and each variety k ∈ [0, nj ] of firm j = 1, ..., N yields the following consumer demand functions: " # λpi −σ dc (pi ) = ρ

Dc (Pjk ) =

"

λPjk ρ

#−σ

where λ=

ρ . y 1−ρ Pρ

(4)

is the marginal utility of income, which is endogenous, and P the total price index that satisfies P

1−σ

≡

ˆ

0

M

p1−σ di i

+

N ˆ ! j=1

0

nj

1−σ 1−σ Pjk dk ≡ P1−σ sp + Pmp

(5)

in which Psp stands for the price index in the SP-subsector (with the subscript sp) and Pmp for the price index in the MP-subsector (with the subscript mp). Because any big firm is able to manipulate the 3

Throughout the paper, variables associated with MP-firms are described by capital letters and those corresponding to

SP-firms by lower case letters. 4 This assumption involves no loss of generality because the structure of firms’ ownership does not matter for our results.

6

price index as well as the income, each such firm understands that its action also affects the value of the marginal utility of income (4). In contrast, every small firm treats λ parametrically because its behavior has no impact on P and y. Observe that all our results hold true if we assume that big firms behave like income-takers. Plugging (4) into the demand functions, we obtain the CES market demand functions: Y $ pi %−σ d(pi ; P, Y ) = Ldc (pi ) = P P

Y D(Pjk ; P, Y ) = LDc (Pjk ) = P

"

Pjk P

#−σ

(6)

where Y ≡ Ly is the total income.

Let c be the marginal cost and f the fixed cost of a SP-firm. Small firm i’s profits are given by πi = (pi − c)d(pi ; P, Y ) − f. Letting Cj be the marginal cost and Fj the fixed cost of the MP-firm j, its profits are: Πj =

ˆ

0

nj

(Pjk − Cj )D(Pjk ; P, Y )dk − Fj .

Hence, MP-firms are heterogeneous, whereas SP-firms are homogeneous. We will see in Section 3 how this assumption may be relaxed, whereas the general argument of Section 4 holds true when marginal costs are variable. The strategy of a SP-firm is its price level pi , while the strategy of a MP-firm is a price schedule Pjk defined on the interval [0, nj ]. The SP-firm i being negligible, its behavior does not affect the market. In contrast, since the MP-firm j is an atom of mass nj , it influences the market when it changes its prices on a positive measure set of varieties, even though any single variety provided by this firm has a zero impact. Unlike small firms, each big firm has enough market power to manipulate its demand function through the redistribution of its own profits - what is called the Ford effect (d’Aspremont et al., 1996). As discussed in the introduction, the second feature that distinguishes big and small firms is the order in which they move. Specifically, our setting adopt the staging of the dominant firm model in which one large firm chooses its output, anticipating the reaction of a given competitive fringe. Besides the emphasis put on product differentiation, our setting extends this model in fundamental ways that yield new and unexpected (at least to us) results.5 5

Although we believe that a sequential game does capture an essential difference between firms that differ in kind, it is

worth stressing that several of our results are changed when all firms move simultaneously. Since the small firms’ equilibrium strategies of the simultaneous game must solve the equilibrium conditions of the second stage, the equilibrium marginal utility of income is the same in both the sequential and simultaneous games. However, the big firms do not treat this statistic

7

2.2

The market outcome

Because the game is sequential, we seek a subgame perfect Nash equilibrium and solve the game by backward induction. Stage 2. Observe that Pmp encapsulates the price choices made by all MP-firms in the first stage of the game. Since each SP-firm is negligible, it accurately treats the overall price index P and the total income Y as parameters. This yields the equilibrium price: p∗i ≡ p∗ =

σ c. σ−1

Therefore, for a given mass M of small firms, (5) may be rewritten as follows: P

1−σ

=M

"

#1−σ σ c + P1−σ mp . σ−1

(7)

By definition of monopolistic competition, free entry and exit prevails in the SP-subsector to reflect the high turnover of small firms. As usual, the resulting zero-profit condition allows one to determine the equilibrium output of a SP-firm:

f (σ − 1) . c

qi∗ ≡ q ∗ =

Hence, the small firms’ equilibrium price and output are independent of the big firms’ choices, which implies that the MP-firms influence the competitive fringe through the mass of SP-firms only. It remains to determine the equilibrium mass of SP-firms, M ∗ , conditional upon Pmp and Y . To this end, we consider the market clearing condition for variety i ∈ [0, M ]: Y P1−σ

"

#−σ σ f (σ − 1) c = d(p∗ ) = q ∗ = σ−1 c

which yields P

1−σ

Y = σf

"

#1−σ σ c . σ−1

(8)

#σ−1 σ 1−σ c Pmp . σ−1

(9)

Using (7) and (8), the equilibrium mass of SP-firms is given by Y M = − σf ∗

"

When profits are not redistributed to the workers, (8) shows that the SP-subsector acts as a buffer stabilizing the total price index at a constant value through a change in the equilibrium mass M ∗ of parametrically when they choose their strategies in the simultaneous game. Everything else being equal, they try to reduce the marginal utility of money, which allows them to have higher mark-ups. Consequently, big firms behave strategically, as in Shimomura and Thisse (2009) and Parenti (2010).

8

SP-firms. In this case, an external shock (e.g. the entry of a big firm) that makes a subsector more (less) aggressive translates into less (more) competition in the other subsector. More generally, the equilibrium mass of SP-firms is such that the decisions made by the MP-firms affect the total price index through the income effect only. Stage 1. The MP-firms choose their price schedules, anticipating the SP-firms’ optimal responses. In particular, the MP-firms know that the mass of SP-firms will be adjusted for the total price index to remain equal to (8). Substituting (8) into D(Pjk ), we obtain what we call the adjusted demand for firm j’s variety k: ¯ jk ) = σf D(P

"

#σ−1 σ −σ c Pjk . σ−1

(10)

Hence, because the MP-firms anticipate the adjustment in the SP-subsector, in the first stage these firms face the adjusted demand (10), which is independent of P and Y unlike the market demand (23). This is because the former encapsulates the reactions of the SP-firms to the MP-firms’ actions, whereas the latter does not. Since firm j’s marginal cost is the same across its varieties and each variety is negligible within its product line, in equilibrium it must be that Pjk = Pj .6 Therefore, this firm’s profits may be rewritten as follows: ¯ jk ) − Fj = nj (Pj − Cj )σf Πj = nj (Pj − Cj )D(P

"

#σ−1 σ c Pj−σ − Fj σ−1

which depends only upon the common price Pj set by firm j. Hence, this firm’s equilibrium price is given by Pj∗ =

σ Cj . σ−1

This expression has the following far-reaching implication: the MP-firms adopt the same pricing rule as the SP-firms, the reason being that each big firm chooses its price schedule as if it were a monopolist facing its varieties’ adjusted demands. Furthermore, when choosing its price, a MP-firm does not have to ¯ jk ) is independent of Y . Accordingly, care about the equilibrium value of the total income because D(P despite the fact that big firms are atomic players, the presence of a competitive fringe conceals all forms of strategic interactions through the adjustment of the mass of small firms. This effect includes the interactions among big firms that affect the total price index, as well as the indirect interactions channeled by the redistribution of profits. Consequently, as long as the market accommodates a competitive fringe, we may conclude that both big and small firms adopt the same market behavior. In contrast, once the 6

Hence, (2) may be viewed as a special case of (1) in which nj = 1 for all j.

9

entire competitive fringe has vanished, (M ∗ = 0), the big firms become strategic because both the price index and the total income vary with their price choices. To put it differently, we are back to the world of oligopoly theory. Plugging Pj∗ into (10) and substituting the resulting expression into Πj shows that the firm j’s equilibrium profits are equal to Π∗j

= nj f

"

c Cj

#σ−1

− Fj

which are positive as long as Fj is not too large, c is much larger than Cj , or both. In what follows, we assume that Π∗j is positive for all j = 1, ...N ., and thus all big firms remain in business at the equilibrium. Using (9), it is readily verified that both big and small firms coexist at the equilibrium outcome (M ∗ > 0) if L>

N !

nj

i=1

"

c Cj

#σ−1

.

This holds when the market size is sufficiently large or when the degree of product differentiation is high enough to relax competition once the large firms are more productive than the small ones (c > Cj ). The above discussion is summarized as follows. Proposition 1 When the market involves a fixed number of big firms and a competitive fringe governed by free entry and exit, all active firms adopt the same pricing rule, i.e. the constant markup σ/(σ − 1). Furthermore, in the game among big firms, each one has a dominant strategy once they anticipate small firms’ reactions. To sum-up, when the size of the competitive fringe is endogenous, the strategic interdependence among big firms is dissolved in an ocean of firms that all adopt the pricing behavior of small firms. In other words, each big firm may be viewed as a divisional firm, where each division does not play strategically against the other insider divisions and the outsiders. This provides a strong justification for the use of monopolistic competition in modeling imperfect competition since in the real world few sectors have a competitive fringe. Observe that increasing returns and product differentiation are needed for that to be true. Indeed, under constant or decreasing returns, the mass of small firms would be infinite when entry is free. Furthermore, firms’ operating profits must be positive to cover their fixed costs. This shows how our setting differs from that of the dominant firm where small firms, the total mass of which is fixed, operate under decreasing returns and supply the same homogeneous good as the big firm. It is worth comparing mixed and different forms of oligopolistic markets. For simplicity, we assume that big firms share the same marginal cost C. In the oligopoly case, the common equilibrium price is 10

given by P∗ =

C 1 − ε1∗

where ε∗ is the equilibrium value of the price-elasticity. The value of this elasticity depends on the nature of competition. In a partial equilibrium, we have (Baldwin et al., 2003): σ−1 <σ N 1 1σ−1 1 + > σ σ N σ

ε∗bertrand = σ − 1

ε∗cournot

=

whereas, in a setting where firms manipulate consumers’ income, the price-elasticity is even smaller than ε∗bertrand (d’Aspremont et al., 1996). In all these cases, firms charge a price exceeding the price they set in the presence of a competitive fringe. This is because the MP-firms behave here like SP-firms, while the above expressions show that all oligopolistic elasticities converge toward σ when the number of firms is arbitrarily large. Therefore, a mixed market works “as if” the market were to operate under monopolistic competition. Or, to put it differently, the competitive fringe disciplines the big firms, which results in lower market prices. Furthermore, the mixed market outcome makes consumers better-off. Indeed, they enjoy lower prices on the varieties provided by the MP-firms as well as a wider range of varieties due to the SP-firms’ supply. As a result, the presence of a competitive fringe is beneficial to consumers. Last, note that the merger of some big firms, or of an array of small firms into a multidivisional large firm, does not change the market outcome, thus yielding what we may call a “new” merger paradox (Salant et al., 1983). A few more comments are in order. First, the market outcome is the same regardless of the strategy quantity or price - used by big firms. In other words, Bertrand and Cournot competition yields the same market outcome because large firms behave as if they were in monopolistic competition. This strikingly differs from what we know in oligopoly theory and shows once more how a mixed market obeys very different rules. Second, the entry or exit of a big firm, or how big is the range of varieties supplied by big firms, does not affect these firms’ pricing behavior. Last, one may wonder why large firms accommodate the presence of a competitive fringe since they would earn higher profits in a pure oligopoly. Because Pj∗ maximizes firm j’s profits regardless of the value of M ∗ > 0, firm j has no incentive to lower its price for M ∗ to be equal to zero. In other words, no firm has an incentive to charge unilaterally a sufficiently low price for the competitive fringe to vanish. Furthermore, a pure oligopoly is sustainable only when large firms find it profitable to keep the small firms out of business. This is unlikely to be true. To illustrate, consider the case in which Cj = C so that the large firms charge the same limit price PL . This strategy is

11

never profitable here because N ! i=1

Πj (PL ; M ∗ = 0) ≤ max ≤

N ! i=1

&

N !

'

Πj (·; M ∗ = 0)

i=1

max Πj (·; M ∗ = 0)] ≤

N !

Πj (Pj∗ )

i=1

which shows that no large firm wants to enter a coalition that would blockade the entry of the competitive fringe. Having done this, it is tempting to speculate that the foregoing is an artefact of the CES, which is known to yield fairly peculiar results. In addition, even though the big firms are heterogeneous, the small firms have been assumed to share the same marginal cost. This runs against the empirical literature, which stresses the importance of firms’ heterogeneity. We show below that the homogeneity assumption does not drive our main results, while we will turn to the case of more general preferences in Section 4. There, we will show what are the mechanisms driving the above results.

3

Heterogeneous competitive fringe

In this section, we assume that the SP-firms are heterogeneous in the sense of Melitz (2003). For conciseness, we use the one-period timing proposed by Melitz and Ottaviano (2008). What makes our analysis potentially richer than those papers is that the cutoff could be affected by the behavior of the MP-firms through the redistribution of profits earned by the N big firms. Prior to entry the SP-firms face uncertainty about their marginal cost. To enter the market, the SPfirms must bear a sunk cost fe . After entry, the SP-firm i observes its marginal cost ci , which is drawn randomly from the distribution Γ(c), the density of which is denoted by γ(c). When this firm decides to produce, it also incurs a fixed production cost f , so that its production cost function is given by f + ci qi . Under monopolistic competition, firm i sets a price equal to p∗i =

σ ci . σ−1

Since the equilibrium output is given by qi∗

=

Y P1−σ

"

σci σ−1

#−σ

(11)

the equilibrium profits are as follows: p∗ q ∗ Y π (ci ) = i i − f = σ σ ∗

12

"

σ ci σ−1P

#1−σ

− f.

(12)

Because π(ci ) is strictly decreasing in ci , there is a unique cutoff c¯ such that all firms having a marginal cost exceeding c¯ decide not to produce and exit the market without incurring the fixed production cost f . Therefore, we may set π(c) = 0 for c > c¯. Using π(¯ c) = 0, we have Y σ

"

#1−σ

=f

σ¯ c σ−1

#1−σ

σ c¯ σ−1P

which implies that the total price index is given by Y σf

P1−σ =

"

.

which is identical to (8) but in which c is replaced by c¯. Substituting this expression into (12) yields ∗

π (ci , c¯) = f

($ %σ−1 c¯ c

)

−1 .

(13)

Last, a firm enters when its expected profit net of entry cost is nonnegative. Therefore, if ME firms enter, there will be Γ(¯ c)ME active firms. Because all SP-firms face the same uncertainty, free entry implies that the expected profit is equal to zero in equilibrium: Eγ (π) =

ˆ

∞

π(c)γ(c)dc = fe .

0

After substitution of (13), this expression becomes ∗

Eγ (π ) = f

ˆ c¯ ($ %σ−1 c¯ 0

c

) − 1 γ(c)dc = fe

(14)

which shows existence and uniqueness of the cutoff cost c¯ for any distribution Γ. Furthermore, the same expression reveals that the cutoff cost is independent of the big firms’ behavior. Even though the total income depends on big firms’ behavior through the redistribution of profits, we obtain a result identical to Melitz (2003). To sum up, Proposition 2 Under the CES, the cutoff cost is independent of the big firms’ behavior when small firms are heterogeneous. Furthermore, Proposition 1 still holds. Note that the equilibrium mass ME∗ of SP-firms that enter the market is determined by the behavior of the MP-firms, very much as M ∗ is determined in the homogeneous firm case. Consequently, small heterogeneous firms keep playing the role of a buffer. This highlights how free entry and exit among SP-firms is the key-assumption in our analysis, as discussed in the preceding section.

13

4

General additive preferences

The purpose of this section is to generalize Proposition 1 to the case of general additive preferences: U=

ˆ

M

u(xi )di +

0

N ˆ ! j=1

nj

Uj (Xjk )dk

(15)

0

where u and Uj are thrice continuously differentiable, strictly increasing and concave, with u(0) = Uj (0) = 0. It is worth stressing that varieties do not enter symmetrically into individual preferences. In particular, (15) allows for the utility of a SP-variety to differ from that of an MP-variety, whereas varieties supplied by different big firms need not have the same elasticity of substitution. Our sole restrictive assumption is that varieties supplied by a big firm are symmetric (Allanson and Montagna, 2005). Despite of this, our preference structure is general enough to permit the elasticity of substitution within a big firm’s product line to be higher or lower than the elasticity of substitution between any two firms. Furthermore, besides the CES discussed in Section 2, another special case of (15) is given by the CARA utility proposed by Behrens and Murata (2007) to model preferences in monopolistic competition: u(x) = 1 − exp(−αx)

α > 0.

Apart from the generalization of preferences, the model is the same as the one described in Section 2. Let y be the consumer income defined in (3), Pj the price function set by the MP-firm j and pi the price chosen by the SP-firm i. Denoting by λ the Lagrange multiplier, the utility-maximizing conditions yield the inverse demand functions:

pi (xi ) =

u$ (xi ) λ

Pjk (Xjk ) =

Uj$ (Xjk ) . λ

(16)

Therefore, the marginal utility of income aggregates all the information that matters to consumers and firms.7 Observe that the demand functions differ between firms’ types and across big firms. In addition, the strict concavity of u implies that the conditions (16) are sufficient for the consumer optimization problem to have a unique solution. To illustrate, observe that in the CES case of Section 2 the expressions (16) can be obtained as follows: using (4) and solving (23) for prices, we obtain the inverse demands as a function of its own price and of 7

The same point is made by Neary (2009) in a different, but related, context. Our setting thus may be viewed as

an aggregative game in which the value of the marginal utility of income is the aggregate of big and small firms’ actions (Acemoglu and Jensen, 2009).

14

the marginal utility of income: pi (xi ) =

ρxρ−1 i λ

Pjk (Xjk ) =

ρ−1 ρXjk

λ

.

which highlight the fact that the statistic λ encompasses all the market information a firm needs to make its profit-maximizing decision. The same expressions can be applied to heterogeneous firms studied in Section 3. As in the foregoing, the game is solved by backward induction. Stage 2. To simplify algebra, we assume that the SP-firms share the same marginal cost c and fixed cost f . Note, however, that the results below still hold when these firms are heterogeneous in the sense of Section 3. Therefore, firm i’s profits are given by ) ( $ u (xi ) − c Lxi − f. πi = λ Using the“relative love for variety”ru (x) introduced by Zhelobodko et al. (2011): ru (x) ≡ −

xi u$$ (x) u$ (x)

we may state the equilibrium conditions in a very simple way. At a symmetric equilibrium, the first-order condition for profit-maximization may be written as follows: p∗ =

c . 1 − ru (x∗ )

(17)

Combining this condition with the zero-profit condition yields the equilibrium output q ∗ = Lx∗ , where x∗ is the solution to

x∗ ru (x∗ ) f = . ∗ 1 − ru (x ) cL

(18)

As shown by Zhelobodko et al. (2011), the equation (18) has a unique solution when (i) profits are concave (ru" < 2), (ii) varieties are gross substitutes (0 < ru < 1) and (iii) the marginal revenue ((xi u$$ (xi ) + u$ (xi ))/λ) intersects the horizontal line c for all admissible λ. Moreover, since the equations (17) and (18) do not depend on λ, the price or output decisions made by the SP-firms are independent of what the MP-firms do. Using (16) and (17), it is readily verified that the equilibrium value of the statistic λ is given by λ∗ =

u$ (x∗ )[1 − ru (x∗ )] c

(19)

which is independent of the MP-firms’ actions. In other words, even though big firms have the ability to change the marginal utility of income through their price or output choices, the equilibrium value of λ is 15

affected only by small firms’ entry and exit decisions. this a priori surprising result stems from the fact that small firms’ profit-maximizing prices and outputs depend on big firms’ actions only through λ. Stage 1. Since the MP-firm j anticipates the equilibrium values of p∗ and q ∗ , this firm is able to compute the resulting value of λ∗ and to determine the adjusted inverse demand for its variety k: Uj$ (Xjk ) Pjk (Xjk ) = λ∗ which depends only upon the individual consumption of this variety since λ∗ is already determined. In other words, all the properties shown in Section 3 remain valid. Consequently, Proposition 3 Under general additive preferences, all active firms adopt the same pricing rule as the SP-firms, which is independent of the strategies chosen by competitors. Furthermore, an exogenous shock affecting a big firm’s cost and demand, as well as the entry of a new big firm, does not change the behavior of its competitors. In particular, when preferences are additive, the presence of a competitive fringe operating under free entry and exit is sufficient to yield a market outcome in which big firms mimic non-strategic behavior. Note also that Proposition 3 still holds when firms are heterogeneous, while the cutoff is independent of the big firms’ behavior as in Proposition 2. To better understand the role of the competitive fringe in stabilizing competition, consider the symmetric equilibrium (¯ p, x ¯) of the SP-subsector prevailing before a shock on big firms’ costs and/or demands, and let (˜ p, x ˜) be the equilibrium prevailing after the shock. These two equilibria must satisfy the zero-profit condition: π = [p(x) − c]Lx − f = 0. Since the inverse demand p(x) = u$ (x)/λ must be met in equilibrium, the shock to the MP-subsector may influence SP-firms’ profits only through an upward (downward) shift of the inverse demand caused by a change in the value of λ. Hence, operating profits are increased (decreased) after the shock, which implies that the zero-profit condition cannot hold at (˜ p, x ˜). This means that λ must the same before and after the shock, which implies that the inverse demands for the MP-varieties are the same. When the mass of small firms remains the same after and before the shock, the value of the marginal utility of income is increased or decreased depending on the 16

nature of the shock. Once small firms are free to enter or exit the market, their mass is determined by the zero-profit condition which restores the pre-shock value of λ. It is worth stressing here that the value of λ changes during the entry and exit process. It is determined endogenously, which accounts for general equilibrium effects. Before proceeding, it is worth stressing that the above argument, hence Proposition 4, holds true when big and small firms’ marginal costs are variable. This is because the equilibrium value of the marginal utility of income remains independent of big firms’ actions. For the same reason, big firms still mimic non-strategic behavior when each one is free to select its own range of varieties.

5

Non-additive preferences: the quadratic utility

Our purpose in this section is to figure out whether or not the stabilization effect identified in the above section remains valid when preferences are not additively separable. To this end, we use the model presented in Section 2, except that individual preferences are now described by the quadratic utility with cross-effects:

  ˆ N ˆ nj ! X2 γ M 2 2 U =X− xi di + Xjk dk  − 2 2 0 0

(20)

j=1

where the market statistic

X=

ˆ

M

xi di +

0

N ˆ ! j=1

nj

Xjk dk

0

is the aggregate consumption index, γ > 0 measuring the intensity of consumers’ love for variety.8 First-order conditions for utility maximization with respect to each variety i ∈ [0, M ] and each variety k ∈ [0, nj ] of firm j = 1, ..., N yield the individual inverse demand functions: λpi = 1 − γxi − X

λPjk = 1 − γXjk − X.

where λ is again the marginal utility of income. Using these conditions and the budget constraint, we can determine the equilibrium value of λ as follows:

. γE(P) − γ (γ + M + nj ) y . λ = (γ + M + nj ) E(P2 ) − E2 (P) ∗

where P is the price profile of all varieties. Therefore, λ∗ depends on the price moments E(P) and E(P2 ) as well as on the income y. Since the MP-firms produce a strictly positive mass of varieties, λ∗ can 8

Without loss of generality, the coefficients of X and X2 have been normalized to 1.

17

be impacted by the MP-firms’ strategies chosen in the first stage through the prices’ moments. On the contrary, the SP-firms will keep treating λ∗ as given because they are negligible to the market. After rearrangement, this yields the following expressions for the demand functions: pi = (1 − X − γxi )/λ∗

Pjk = (1 − X − γXjk )/λ∗ .

(21)

Stage 2. Since each SP-firm is negligible, it accurately treats both statistics X and λ∗ as given. Therefore, the equilibrium output is given by qi∗ ≡ q ∗ =

L (1 − X − λ∗ c) 2γ

and the equilibrium price by p∗i ≡ p∗ =

1 − X + λ∗ c . 2λ∗

Conditional upon λ∗ , the zero-profit condition determines the equilibrium value of the total consumption index X∗ :

Lλ∗ 4γ

"

/ 1 0 #2 1 − X∗ γf ∗ ∗ ∗ − c − f = 0 ⇒ X (λ ) = 1 − λ c + 2 λ∗ λ∗ L

(22)

which depends on λ∗ . It is worth stressing the difference between this setting and the one presented in the previous sections. The marginal utility of income is not anymore determined by the SP-firms only. It now depends upon the actions chosen by all (big and small) firms through the aggregate output X∗ . Stage 1. The MP-firms face the inverse demand given by Pjk [Xjk ; X∗ (λ∗ )] =

1 − X∗ (λ∗ ) − γXjk . λ∗

Because the MP-firm j accurately anticipates the value of X∗ , the adjusted demand for its variety k is obtained as follows: ∗

Pjk (Xjk ; λ ) = 2

0

γf γ − ∗ Xjk . ∗ λ L λ

Since every MP-firm j influences the value of λ∗ through the price moments, this firm understands it can manipulate λ∗ in the first stage game. Therefore, big firms’ actions affect their adjusted demands, which implies that these firms compete strategically. In this case, their actions also affect the small firms’ choices, not just their mass. This example shows that the distinction between atomic and non-atomic firms may become crucial under non-additive preferences, thus highlighting the role of additive preferences in stabilizing competition in mixed markets. 18

However, things become very different when (20) is embedded into a quasi-linear upper-tier utility (Ottaviano et al., 2002):   ˆ N ˆ nj ! X2 γ M 2 2 Xjk dk  − xi di + U =X− +Y 2 2 0 0 j=1

where Y is the numéraire, very much like in several partial equilibrium analyses. If each consumer is endowed with a quantity of the numéraire sufficiently large for the equilibrium consumption of this good to be positive, the marginal utility of income is fixed and equal to 1. Thus, the statistic that drives here the market is the aggregate consumption index X, not the marginal utility of income. As shown by (22), the equilibrium value X∗ is constant and independent of the choices made by the MP-firms. Consequently, the stabilization effect uncovered in Proposition 1 may still hold under nonadditive preferences such as the linear-quadratic utility because the marginal utility of income is fixed (λ = 1), while a firm’s demand is shifted upward or downward with the aggregate consumption index.

6

Stability in competition: a synthesis

This section shows that the above arguments can be applied to more general settings. To be precise, consider an general preference model that generates small firms’ demand schedules di (pi ; Λ), where Λ is the vector of all market statistics that matter to small firms. For example, under additive preferences, we have Λ = λ, while Λ = X in the linear-quadratic case. Let a be the vector of exogenous parameters affecting big firms’ behavior, thus making Λ potentially dependent on a. Examples include parameters affecting big firms’ cost (Cj ), their number (N ) or their demand through the utility Uj . Consider a shock on a which changes the vector Λ(a) and assume that the shift of any small firm’s demand is such that, everything else being equal, the pre-shock demand di (pi ; Λ(a)) does not intersect the after-shock demand di (pi ; Λ(a$ )). Formally, this no-crossing condition may be stated as follows: Λ(a$ ) (= Λ(a) => di (pi ; Λ(a)) > di (pi ; Λ(a$ )) or di (pi ; Λ(a)) < di (pi ; Λ(a$ )) for all pi > c. Slightly extending the above argument shows that the stabilization effect remains valid in this broader setting. Indeed, if Λ(a) (= Λ(a$ ) under the no-crossing condition, then the SP-demand functions would be shifted upward or downward. This in turn is incompatible with the fact that both the pre- and after-shock operating profits must remain equal to f . As a result, we have:9 9

The same argument may be applied to heterogeneous small firms once operating profits are replaced by expected operating

profits.

19

Figure 1: Equilibria without and with price stabilization Proposition 4 Consider an exogenous shock affecting big firms only such that the SP-demand schedules satisfy the no-crossing condition. Then, the entry or exit of small firms keeps the after-shock values of the market statistics Λ equal to their pre-shock values. Furthermore, small firms’ prices also remain the same. This proposition extends our main findings to some non-additive preferences compatible with the nocrossing condition. In particular, it shows why big firms mimic non-strategic behavior. In the quadratic case with income effects, we have seen that λ and X are interdependent so that Λ =( λ, X). Inverting (21) and using (22) shows that L di = γ

/

∗

0

λ c+2

λ∗ γf − λ ∗ pi L

1

and thus the no-crossing condition is violated. In this event, the distinction between atomic and non-atomic firms matters under (20). By way of contrast, the linear-quadratic utility case highlights the meaning of Proposition 4 by showing how it can cope with settings in which firms’ demands a priori depend on several statistics. The left and right panels of Figure 1 illustrate the differences between the quadratic and linearquadratic cases, respectively. In the former one, (21) shows that a simultaneous change in λ and X modifies demands through a rotation and a shift of the corresponding curves. This changes both the slope and the intercept of the demand schedules, thus violating the no-crossing property. In the latter one, λ is fixed so that a change in X triggers an upward or downward parallel shift of the demand schedule. Hence, only the chock-off price is modified, which prevents the pre-shock and post-shock curves to intersect. 20

Therefore, Proposition 4 brings price stabilization. To illustrate the interest of Proposition 4, we consider the symmetric translog expenditure function which is used increasingly in empirical works (Feenstra and Weinstein, 2010). The demand functions for the variety produced by the SP-firm i and the variety k supplied by the MP-firm j are, respectively, as follows:

d(pi ; Λ0 , Y ) =

Y (Λ0 − β ln pi ) pi

D(Pjk ; Λ0 , Y ) =

Y (Λ0 − β ln Pjk ) Pjk

(23)

where β is a positive constant and Λ0 ≡

β 1 . + . M + nj M + nj

 ˆ 

M

ln pi di +

0

N ˆ ! j=1

0

nj



ln Pjk dk  .

In this setting, a firm’s competitive environment is described by the following two statistics: Λ = (Λ0 , Y ). When firms treat Y parametrically, any shock on Λ shifts upward or downward the trans-log demand function. In this event, d(pi , Λ) satisfies the no-crossing condition. However, the no-crossing condition is violated when firms manipulate the total income (d’Aspremont et al., 1996).

7

Concluding remarks

We have identified simple conditions on preferences for a market with big and small firms to mimic monopolistic competition. In other words, within our framework monopolistic competition emerges as an equilibrium outcome. That said, we would be the last to claim that oligopoly theory is irrelevant. Instead, our aim was to show that differentiated markets involving a few big firms endowed with market power as well as a competitive fringe may be accurately described by monopolistic competition settings. In particular, when the no-crossing condition holds, small firms’ equilibrium price and output are stable in response to shocks affecting the big firms. Furthermore, when preferences for the differentiated good are additive, (i) the marginal utility of income is endogenous and captures all the market ingredients that matter to consumers and firms; (ii) any firm, either big or small, either homogeneous or heterogeneous, adopts a non-strategic pricing rule; and (iii) our results hold true under heterogeneous SP-firms. Therefore, our paper has an important methodological implication: modeling competition in an industrial sector that has a competitive fringe by means of a pure oligopoly setting may yield dubious results.

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By showing that big and small firms behave alike, our paper provides new and sound theoretical foundations to the use of monopolistic competition in modeling imperfectly competitive markets. In particular, as long as specific preferences used in empirical studies satisfy the no-crossing property, appealing to monopolistic competition could well be a sensible modeling strategy in the trade literature, even when some markets involve large firms which are expected to behave strategically. Furthermore, it can be shown that the pure monopolistically competitive equilibrium is the limit of pure oligopolistic economies in which firms compete either in quantity (Cournot) or prices (Bertrand). All these results build links between the two main approaches used in imperfect competition. A word, to conclude: our analysis highlights the fact that small firms enter or exit the market in response to an external shock in a way such that big firms have no incentives to deviate from the strategies adopted before the shock. This sheds new light on the well-documented stickiness of prices on several markets.

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