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Large and small firms in a global market: David vs. Goliath
Large and small firms in a global market: David vs. Goliath Mathieu Parenti PII: DOI: Reference:
S0022-1996(17)30116-2 doi:10.1016/j.jinteco.2017.09.001 INEC 3081
To appear in:
Journal of International Economics
Received date: Revised date: Accepted date:
9 July 2016 11 September 2017 11 September 2017
Please cite this article as: Parenti, Mathieu, Large and small firms in a global market: David vs. Goliath, Journal of International Economics (2017), doi:10.1016/j.jinteco.2017.09.001
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Mathieu Parenti†
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Large and small firms in a global market: David vs. Goliath∗
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Abstract
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This version: September 2017
This paper studies the impact of trade liberalization when large and small firms coexist in the same market. I develop a model of imperfect competition where a few oligopolistic firms coexist with a monopolistically competitive fringe. The
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former have a direct impact on the industry average price level, which depends on the breadth of their product scope. Oligopolists produce higher output per worker, charge higher markups, but also sell at higher prices than monopolistic single-product firms. Under linear demand, trade liberalization triggers the exit of small firms and the entry of foreign
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large firms, leading to more product variety but also to a higher average price. For a broad class of demand functions, the entry of an oligopolist induced by trade liberalization has a negative impact on consumer surplus. Producer surplus rises, but the total effect on welfare is ambiguous.
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JEL Classifications: D4, L10, F11
Keywords: Monopolistic Competition, Oligopolistic Market Structure, Multi-product firms, International
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Trade.
∗ I am very grateful to Lionel Fontagné, Peter Neary and Jacques Thisse for many challeging interactions on this topic. I would also like to thank Aya Ahmed, Claude d’Aspremont, Kristian Behrens, Paola Conconi, Arnaud Costinot, Mathieu Couttenier, Meredith Crowley, Matthieu Crozet, Ron Davies, Don Davis, Swati Dhingra, Francesco Di Comite, Jonathan Dingle, Martin Hellwig, Gianmarco Ottaviano, Jean Imbs, Sergey Kokovin, Sebastien Krautheim, Helene Latzer, Florian Mayneris, Julien Martin, Thierry Mayer, Joan Monras, John Morrow, Volker Nocke, Vincent Rebeyrol, Nicolas Roux, Nicolas Schmitt, Nicholas Schutz, Farid Toubal, Federico Trionfetti, Thierry Verdier, Helene Windish, Stephen Yeaple as well as two anonymous referees for their helpful comments and suggestions. † ECARES (Université Libre de Bruxelles) - C.E.P.R.. E-mail: [email protected].
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1
Introduction
It is well documented that firms are heterogeneous within an industry. Although the exact shape of this heterogeneity
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is still debated, it is widely acknowledged that the firm size distribution is highly skewed, featuring a large number
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of small firms and a handful of large firms (Axtell, 2001). This implies that the aggregate trade patterns of an industry can be shaped by the individual behavior of a few firms. For instance, Bernard et al. (2007) report that 96% of US exports in 2000 were made up by 0.4% of US firms.1
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In the theoretical literature, however, the assumption of a continuum of firms implies that the actions of a large firm are inconsequential. In this paper, I follow a different approach and assume that those happy few large firms
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behave differently in the marketplace. I rule out differences in marginal costs of production, focusing on differences in individual firm behavior and how they shape the impact of trade liberalization. To be precise, I build a model of a “mixed” market structure in which a few multi-product oligopolists and a
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continuum of single-product monopolistically competitive firms coexist. In doing so, I depart from the standard models of monopolistic competition along two dimensions.
First, I assume away the Chamberlinian large-group assumption, considering a setting in which a few large firms
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behave strategically while competing with a large number of monopolistically competitive firms. Large firms have a direct impact on the market outcome while a large number (formally, a continuum) of negligible monopolistically
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competitive firms take the market aggregates (total output here) as given. The second assumption I depart from is the free-entry condition. To capture the persistence of a few large
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firms over time,2 I follow Friedman (1993) for whom free entry is “untenable for oligopolistic markets”. The setting features an exogenous number of potentially large firms which may earn positive profits in equilibrium. On the contrary, monopolistically competitive firms adjust to the intensity of competition through entry and exit until their profits are driven down to zero. Compared to a standard oligopolistic market structure, the present model takes the importance of the turnover of small firms into account. Starting with a closed economy, I take a step towards understanding the endogenous formation of large firms. Heterogeneity in productive efficiency emerges as a necessary condition for both large and small firms to coexist in equilibrium. I assume therefore that large firms benefit from economies of scope which allows them to generate more physical output per worker. How do the pricing strategies of large and small firms differ? I show that even if the marginal cost of production of a variety is the same for both kinds of firms, large firms charge higher markups. This is because large firms have a strategic market power: under linear demand and Cournot competition, they internalize the impact of their production on total output. On the contrary, the market power of a small firm is solely monopolistic: each variety 1 See also Mayer and Ottaviano (2008), Berthou and Fontagné (2013), Bernard et al. (2007) and Manova and Zhang (2009) for respectively Belgium, France, the U.S.A. and China. Using firm-level data on 32 countries, Freund and Pierola (2015) find that the export superstars “critically shape trade patterns”. Importantly enough, these patterns hold both for the manufacturing and the services sector (Crozet et al. 2010 and Breinlich and Criscuolo 2011). 2 Many papers have found that firm survival, firm size and firm age are positively correlated; see, e.g., Thompson (2005).
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is an imperfect substitute for its competitors’. To assess how these differences matter in a global market, I consider an open economy and carry out three exercises. First, I analyze the impact of the entry of a large (foreign) firm. The increase in the competitive pressure
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exerted by this entrant leads to the exit of small firms. The average price rises through the proliferation of new
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higher-priced varieties at the expense of consumer welfare.
Second, I consider the free trade benchmark when many identical countries - i.e., with the same number of
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oligopolists and the same population size - trade with each other. The entry of large foreign firms fosters the exit of small firms again. Similarly to Krugman (1979) however, the access to foreign markets makes the monopolistically competitive fringe more competitive, both through the intensive margin (firms increase their scale, thereby decreas-
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ing their average cost) and the extensive margin (new firms enter the market). Under this scenario, consumers benefit unambiguously from trade openness.
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Third, I show that the presence of trade frictions, including variable and fixed costs to reach the foreign market, alters these predictions significantly. A drop in variable trade costs increases import penetration, but in the presence of fixed costs, small firms cannot compensate for their domestic losses with their exports to the foreign market, as
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they can in the free trade scenario. Their scale and markup remain unchanged and the size of the monopolistically competitive fringe shrinks. By contrast, a large firm expands its market share abroad, hence increasing its market power. In contrast with Brander (1981), bilateral trade liberalization leads to a monotonic decrease in consumer
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welfare and a monotonic increase in producer surplus. Central to these results is the anti-competitive effect following the entry of a large firm considered in the first
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exercise. I show that this effect holds for a large class of preferences. Specifically, the prediction holds under the class of additively separable preferences featuring an increasing preference for variety as in Kuhn and Vives (1999). The remainder of the paper is organized as follows. Section 2 briefly describes the related literature. Section 3 presents the baseline model and determines the conditions under which both large and small firms coexist in equilibrium. Section 4 deals with the properties of the market outcome. Section 5 determines the effects of trade liberalization on consumer welfare. In Section 6, the robustness of the main findings against alternative specifications of preferences is discussed. Section 7 concludes.
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Related Literature
This paper lies at the intersection of two strands of literature. The first one deals with international trade under oligopoly and the seminal contributions of Dixit and Norman (1980), Brander (1981) and Markusen (1981). In contrast with the industrial organization literature, strategic interactions have not received much attention from trade economists. A reason which Neary (2010, 2016) has pointed out to explain the theoretical bias against oligopoly, favoring monopolistic competition is the lack of tractability of the former in general equilibrium (Roberts
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and Sonnenschein, 1977). However, Neary (2016) shows that the problem becomes tractable when considering a market structure in which firms are “large in the small but small in the large,” meaning that each firm impacts competition in its own industry, but not consumer income which is determined at the economy-wide level. Instead,
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I stick to partial equilibrium here, as the focus is placed on the differences in firms’ market behavior within an
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industry.
More recently, observing that large firms “depart substantially” from the monopolistic competition benchmark
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(Hottman et al., 2016), a few papers have considered a model featuring heterogeneous firms with non-zero mass as in Bernard et al. (2016). In their model, differences in firm size stem from differences in their marginal cost of production. Therefore, quality differences put aside, a firm’s market power always translates into a higher markup
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and a lower price. On the contrary, a large firm in my model not only charges a higher markup over marginal cost, it also sells at a higher price. This distinction allows me to isolate the importance of firm’s strategic market
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power when discussing the distribution of the gains from trade. The main insight is that the source of firms’ market power matters for consumer surplus. This echoes also Eckel and Yeaple (2016) who find that the source of firms’ performance may alter the welfare conclusions following the reallocation of labor towards larger firms.
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This paper also builds on the vast literature in industrial organization combining large and small firms within the same setting, starting from the dominant firm model developed by Markham (1951). This approach has also been applied to vertical relationships by Kuhn and Vives (1999) in a setting where large upstream firms trade with
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small downstream firms. Yet, the coexistence of both types of behavior in a given industry has been neglected. In the conclusion of his celebrated Chamberlinian model of monopolistic competition, Hart (1985) explains the need
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to invoke oligopoly theory to study a market which “contains a mix of significant and negligible firms.” To the best of my knowledge, the first paper to deal with competition between big, strategic firms, and small, monopolistically competitive firms is Shimomura and Thisse (2012). Besides the motivation for the study of trade liberalization, the closed-economy section of my model also differs from Shimomura and Thisse (2012) in several respects. The first one concerns the interpretation of large and small firms. Assuming large single-product firms is hard to reconcile with the idea that a consumer displaying a love for variety smooths its consumption across all varieties. Instead, this paper considers that each large firm supplies a continuum of varieties. An appealing feature of this approach is the endogenous determination of the product range so firms choose to become large or not in equilibrium. Second, the present paper uncovers the free-riding of non-strategic small firms on large firms. This explains why large firms must necessarily be more efficient than small firms to coexist with them in equilibrium. Third, the paper disentangles the implications of the assumptions made in Shimomura and Thisse (2012) about the market structure. For instance, the fact that large firms charge a higher markup than small firms survives to a series of alternative specifications of demand. On the contrary, the result according to which the entry of a large firm is necessary beneficial to consumers does not. For a large class of preferences, consumer surplus always decreases with the entry of a large firm while it remains constant for CES preferences. Instead, Shimomura and Thisse (2012)’s
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result is driven by the assumption that firms’ profits feedback into consumers’ demand for differentiated varieties.
The Baseline Model
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3
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The economy involves an homogeneous good produced under constant returns and perfect competition, a horizontally differentiated good produced under increasing returns and monopolistic competition, and one production factor, labor. The differentiated sector involves both a discrete number Ω of multi-product (MP) firms ω = 1 . . . Ω,
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which decide on the breadth of their product scope Nω ≥ 0, as well as a continuum i ∈ [0, M ] of single-product
Preferences
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3.1
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(SP) firms. In what follows, I refer to Nω as the mass of firm ω. Thus, the total mass of varieties provided by this PΩ sector is equal N := ω=1 Nω + M .
On the demand side, there is a continuum of L identical consumers sharing the same preferences. Denote by xωk the individual consumption of variety k produced by the MP-firm ω, by xi the consumption of variety i produced
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by the SP-firm i, and by X the aggregate consumption of all varieties by one consumer: Z
N
xν dν =
0
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X :=
Ω Z X
ω=1
Nω
xωk dk + 0
Z
M
xi di. 0
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The utility of consuming the differentiated good is given by the following quadratic utility function:3
U (¯ x) :=
Z
N
0
β αxν − x2ν 2
γ dν − 2
Z
N
xν dν 0
!2
.
(1)
¯ := xν≤N is the consumption profile and α, β and γ positive parameters. The upper-tier utility function where x is quasi-linear:
¯ ) := A + U (¯ U (A, x x) where A is the consumption of the homogeneous good; it is used as a numéraire. An increase in α or a decrease in γ shifts expenditure away from the numéraire good towards the differentiated good. Consumers’ love for variety is captured by β. When β = 0, consumers care about the overall consumption level X only. The utility function is symmetric (i.e., renumbering varieties has no impact on the utility level) so at a symmetric consumption profile, whether a variety is produced by a MP- or SP-firm is irrelevant from the consumer’s perspective. 3 This functional form has been used in Industrial Organization by Dixit (1979) and Singh and Vives (1984) over a finite a number of varieties. For applications to a continuum of goods see Ottaviano et al. (2002) and Melitz and Ottaviano (2008).
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Each consumer supplies one unit of labor inelastically. Thus, she faces the following budget constraint:
0 1 L
pν xν dν ≤ R
(2)
of each firm profit.4
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where R is equal to her wage income plus a fraction
N
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A+
Z
Under the assumption of positive consumption of the homogenous good A > 0, the inverse demand functions
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for varieties i and k are
=
p(xi , X ) := α − βxi − γX
(3)
pωk
=
p(xωk , X ) := α − βxωk − γX .
(4)
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pi
The variable X has the nature of a market aggregate, which can be interpreted as a proxy for the intensity of
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competition as it shifts up or down the demand schedule.5 For any given X , the price elasticity increases with the price along the demand curve, as in Krugman (1979).
Firms
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3.2
In order to isolate differences in market power between large and small firms I assume that they share the same
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constant marginal cost c. Two types of fixed costs are introduced. There is an overhead cost F > 0 to be paid before starting production, which is incurred by all firms. Following a widely-used simplification in the multi-product firm
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literature,6 adding a new variety requires an additional fixed cost f > 0. Therefore, the total fixed costs incurred by an SP-firm are F + f . On the other hand, an MP firms of mass N > 0 incurs a fixed cost equal to N f . If this firm were a single-product firm (N = 0), its fixed cost would be F + f . That F > 0 implies that there are scope economies.7 In what follows, all firms are assumed to be quantity-setters. A multi-product firm ω understands that its decision has a non-negligible impact on the economy because Nω > 0. Specifically, each MP-firm behaves like an oligopolistic firm that manipulates the market aggregate X . By contrast, because any particular SP-firm is negligible to the market, its decision to enter the market and to produce a certain output, or to exit the market, has no impact on the economy. As a consequence, a SP-firm may accurately treat X as a parameter. Since the inverse demands (3) and (4) depend only upon a firm’s own strategy and the market aggregate X , the 4 This
assumption involves no loss of generality. Since preferences are quasi-linear, firms’ ownership does not play any role in this model. 5 Alternatively, X can be expressed as a function of the average price and the overall mass of varieties N (see Melitz and Ottaviano, 2008). 6 See Baldwin and Gu (2009), Dhingra (2013), Feenstra and Ma (2008), Minniti and Turino (2013), Ottaviano and Thisse (2011). Admittedly, it is a crude simplification from reality, but it should be clear that product adjustments within multi-product firms are not of direct interest here. Instead, the focus is placed on firm heterogeneity. See Eckel and Neary (2010) and Mayer et al. (2014) for models of multi-product firms with a core competence. R 7 Formally, the fixed costs incurred by an MP-firm with N > 0 amount to Nω (f + 1 ω k=0 F ) dk. 0
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profits earned by an MP-firm are given by Nω
[p(xωk , X ) − c] xωk dk − Nω f
while the profits made by a SP-firm are as follows:
(5)
i ∈ [0, M ]
(6)
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πi (xi , X ) := L · [p(xi , X ) − c] xi − (F + f ) where M is the number of operating SP-firms.
The Game
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3.3
ω = 1, ..., Ω
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0
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Πω (xωk , Nω , X ) := L ·
Z
The number of MP-firms is exogenously given, but their respective mass is endogenous. On the other hand, each
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SP-firm is negligible, but their number is determined by free entry and exit. The MP-firms choose their breadth and outputs and the SP-firms choose to enter the market and their outputs simultaneously. As shown by (5) and (6), the game is aggregative in nature where X is the aggregate i.e. it is one-dimensional
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and additively separable in the strategies of the players. A Nash equilibrium is then defined by x∗ωk , Nω∗ , x∗i , M ∗ ,
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and X ∗ such that (up to a zero-measure set of SP-firms) the following conditions hold: Πω (x∗ωk , Nω∗ , X ∗ ) P
R N ∗′
x∗ω′ k dk +
R M∗
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∗ ≡ where X−ω
≥
∗ Πω (xωk , Nω , X−ω
ω ′ 6=ω
ω
0
0
+
Z
Nω
xωk dk) 0
for all xωk ≥ 0 and Nω > 0
x∗i di.
πi (x∗i , X ∗ ) ≥ πi (xi , X ∗ )
for all xi > 0 and i ∈ [0, M ∗ ]
πi (xi , X ∗ ) < 0 for all xi > 0 and i > M ∗
Both types of firms coexist if Nω∗ > 0 for at least one MP-firm and M ∗ > 0. I show below that there exists a unique equilibrium featuring both types of firms, which is symmetric within each type and features symmetry within MP-firms’ product lines.
4 4.1
The Market Outcome Equilibrium
Single-product firms A SP-firm i faces a downward-sloping inverse demand (3) from each consumer. Maximizing its operating profit with respect to xi and treating X as a parameter yields this firm’s best reply as a function of 7
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X only. Since SP-firms share the same marginal cost, (7) implies that all active SP-firms produce the same output ∗ qsp (X ) := L · x∗sp where
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α − c − γX 2β
(7)
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x∗sp (X ) :=
Multi-product firms A multi-product firm ω chooses simultaneously its product scope Nω from an exclusive continuum of varieties and the output of all its varieties xωk for k ≤ Nω .
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The symmetry of profits and the strict concavity of per-variety profits given total output implies that the choice of an MP-firm boils down to choosing the breadth of its product range Nω and the common level of output per
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variety xω (see Appendix A.1). In this event, an MP-firm maximizes its profits
max Nω [(p(xω , X ) − c) L · xω − f ] .
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Nω ,xω
(8)
The first-order8 condition with respect to xω is given by
(9)
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α − c − (2β + γNω ) xω − γX = 0.
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which leads to total output Qω = L · Xω of firm ω where: α − c − γX 2β + γNω
with density x∗ω :=
α − c − γX . 2β + γNω
(10)
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Xω := Nω
Note that x∗ω = x∗sp when Nω = 0. I assume for the moment that α − c − γX > 0. From (10), the relationship between an MP-firm’s total output and its mass, given X , is an increasing concave function. In other words, were two MP-firms with different product breadth to coexist at equilibrium, the one with the larger product breadth would have a larger, yet less than proportional total output. This is a direct consequence of the cannibalization effect. As emphasized by Baldwin and Gu (2006) , Feenstra and Ma (2008), Eckel and Neary (2010), Ottaviano and Thisse (2010) and Minniti and Turino (2013), multi-product firms internalize demand linkages between varieties. When γ = 0, Xω becomes linear in Nω because there is no linkage effect anymore. Turning next to the determination of the product breadth, we get
(α − c − βxω − γNω xω − γX ) L · xω = f.
(11)
This expression shows that an MP-firm faces the following trade-off. The marginal profit gained by introducing an additional variety is a decreasing function of Nω , i.e., its operating profits are a concave function of Nω . Profits 8 I determine the candidate global maximizer of this function by inspecting of the first-order conditions with respect to x and N . ω ω In Appendix A.2, I show that the Hessian matrix at (9) and (11) is negative definite and I derive the condition under which this local maximum is also global.
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are maximized when the marginal operating profit equals the fixed cost per variety, as shown by (11). Solving (9) for Nω and substituting into (11), we obtain the common per-variety output of an active MP-firm
:=
Lx∗ω
=
s
fL β
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:=
Lx∗mp
(12)
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∗ qmp
Repeating the same argument shows that N ∗ can be expressed as a function of X only: s
βL f
α − c − 2β
s
f − γX Lβ
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1 N (X ) = γ ∗
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which is positive as long as
α − c − γX − 2
r
βf L
>
!
(13)
0.
equilibrium if and only if
1 4β
2
(α − c − γX ) ≥
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Last I solve for X ∗ using the free-entry condition for small firms. Plugging (7) into (6), SP-firms produce at f +F L
or equivalently
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α−c F (f + F ) ≥ 0 where F(y) := q −1 2 βy L
(14)
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At equilibrium, the equality holds so that
=0
with
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∗ (x∗sp , X ∗ ) πsp
x∗sp (X ∗ )
=
s
f +F βL
(15)
When F > 0, condition (14) implies that the inequality (13) holds. This means that when they benefit from economies of scope, each MP-firm finds it optimal to be large at equilibrium in the presence of a monopolistically competitive fringe.9 On the contrary, when F = 0 we have F(f + 0) = 0 and (13) implies that N ∗ = 0. In this case MP-firms choose to be negligible in the presence of a monopolistically competitive fringe and the market structure is characterized by monopolistic competition. It is worth noting that the condition F > 0 uncovers a free-rider problem: SP-firms free-ride off the strategic behavior of MP-firms.10 Since an MP-firm internalizes that its varieties cannibalize each other, it charges higher prices and sells less of each variety to relax competition within its product line. Everything else being equal, the strategic behavior of an MP-firm increases the average price for the differentiated good, which, in turn, relaxes competition for all the other firms belonging to the competitive fringe. Now, this positive externality imposes that there exists some further asymmetry between MP and SP-firms for them to coexist in equilibrium.11 This is true, 9 Note that this implies that a strict subset of the Ω firms being active in the presence of a monopolistically competitive fringe is not an equilibrium. 10 This is a positive externality, however the “free-riding” term has been extensively used in the literature dealing with the merger paradox (Salant et al., 1983). 11 This result justifies the title chosen for the present paper: unless there are asymmetries in efficiency, large firms can not coexist
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Figure 1: Typology of the market structure according to (F, Ω)
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for instance, when MP-firms benefit from economies of scope F > 0. To sum-up, in the event where F > 0 the
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presence of a monopolistically competitive fringe ensures the existence of MP-firms.
To derive the conditions under which there exists a monopolistically competitive fringe, I use the identity ∗
∗
or equivalently F < F (Ω) :=
F(f ) 1+ Ω+1
2
!
−1 f
(16)
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¯ := F(fq+ F ) Ω<Ω f 1 − f +F
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ΩN ∗ xmp (X ∗ ) + M ∗ xsp (X ∗ ) = X ∗ so that M ∗ > 0 if and only if
Conditions (16) states that the market must be large enough (F is large for high values of α or L) and the number of big firms Ω not too large for the competitive fringe to exist in equilibrium. Note that under the event that F = 0,
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¯ = +∞ i.e. the presence of MP-firms is immaterial as they choose to be negligible. we have Ω The equilibrium values for the mass of SP-firms and the product range of MP-firms can be simplified as follows:
1−
s
ED
" 2β F(f + F ) − M ∗ (Ω) = γ
f f +F
! # Ω
2β N∗ = γ
s
! F 1+ −1 . f
(17)
PT
Last, I have assumed throughout that there is positive consumption of the homogeneous good at equilibrium. This will be true when γ is large enough (see Appendix B for details), a condition that is assumed to hold throughout. 0 < F < F¯ . There exists a unique equilibrium featuring both types of firms which
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Proposition 1. Assume that is symmetric within each type. Under
0 < F < F¯ the variables M ∗ , N ∗ , x∗mp , x∗sp are uniquely defined and strictly positive, which proves
proposition 1.
The pair (F, Ω) defines a market structure that evolves from monopolistic competition, to a mixed market structure, and then on to oligopolistic competition, as represented in Figure 1. Figure 1 shows that for a given number of oligopolists Ω, when barriers to entry are prohibitive (F ≥ F¯ ) the market structure is purely oligopolistic. Likewise, for given barriers to entry, if oligopolistic incumbents are too many, small firms do not enter and pure oligopoly is preserved.
4.2
Prices and Markups: a Tale of Two Market Powers.
From an aggregative-game perspective, the inverse demand for any variety p(x, X ) can be written as a function of each firm’s decision variable x and the aggregate statistic X . Applying the standard Lerner pricing rule shows that a firm’s markup reflects two kinds of market power: with a monopolistically competitive fringe.
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p−c = Ex (p) + Ex (X )EX (p) p
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where Ex (g) denotes the elasticity of a function g w.r.t. x.
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The first term on the right-hand side describes a firm’s monopolistic market power, i.e. the fact that each variety is perceived as an imperfect substitute for other varieties. On the contrary, the second term describes the fact that large firms internalize their impact on the market aggregate X , it is a strategic source of market power. It is a
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combination of two terms: Ex (X ) is the extent to which a firm alone impacts the market aggregate X , while EX (p) captures the impact of the aggregate on the price p.
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In the case of monopolistic competition, firms are negligible, so that Ex (X ) = 0 and only the first term is non-zero. On the contrary, in Cournot competition with a homogenous good (β = 0), the market power of firms stems from strategic interactions only, so that Ex (p) = 0.
12
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The market power of small firms is solely monopolistic, while large firms have both types of market power. Combining (3) with (7) and (9) respectively yield a direct comparison of the pricing behavior of SP and MP
p∗sp − c =
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firms:
1 (α − γX ∗ − c) 2
p∗mp − c =
1 ∗ (α + γXmp − γX ∗ − c) 2
(18)
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Proposition 2. MP-firms charge higher markups than SP-firms.
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This contrasts sharply with monopolistic competition models for two reasons. First, for monopolistically competitive firms that have the same marginal costs to charge different markups, it would be necessary to assume that consumers’ willingness to pay for their product is different (see Di Comite et al. 2014). Here instead, a consumer’s willingness to pay for a variety does not depend on whether it has been produced by an MP-firm or an SP-firm. Consequently, since a higher price does not reflect a higher (perceived) quality, a reallocation of market shares towards the largest firms may be detrimental to consumer welfare. Second, this proposition derived under linear demand is general and holds true whether the price-elasticity of demand is increasing, constant or even decreasing (see Section 6). The simple fact that large firms strategically manipulate the aggregate X implies that, everything else being equal, larger firms charge higher markups. The same result would also hold in a model with single-product firms having a positive mass as in Bernard et al. (2016). Last, it is noteworthy that not only MP-firms have more market power, but they also produce more physical output (and thereby a larger value-added) per worker.13 In this sense, their production is more efficient than SP-firms’. 12 Note
that if large fims produce a homogenous good, then Ex (X ) = mass becomes irrelevant, as long as it remains positive. f 13 This can be directly observed since c + < c + fq+F ∗ . q ∗ mp
sp
11
α−c−γX γX
> 0 does not depend on Nω . In this case, a large firm’s
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5
Open Economy
Turning to the open-economy case, I assume that the homogeneous good is freely traded between countries so that
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changes in the goods market do not affect the relative wage between countries.
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In my model, international trade may increase market size for domestic firms if they start exporting but also increase the competitive pressure on their domestic market. This increase in competition may result from the entry of either SP or MP-firms. Since the impact of SP-firms entry in models of monopolistic competition is well
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understood, the first scenario below focuses on the impact of the costless entry of a large (foreign) MP-firm. The second exercise is the free-trade benchmark when all firms can export costlessly to all countries. Last, I introduce
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trade frictions between two symmetric countries and explore the impact of a reduction in trade costs when only the largest firms export.
The Impact of the Entry of a Large Foreign Firm
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5.1
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I start by considering the exogenous and costless entry of a large foreign firm with the same characteristics.14 I $ ¯ + 1 so that both types of firms continue to coexist in equilibrium after the entry of an assume that F < F¯ Ω MP-firm.
We know from (17) that the entry of a large firm leads to the exit of small firms. However, it has no impact on
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the market aggregate X ∗ . This is because the mass of the competitive fringe adjusts so that the free entry condition holds. Therefore the breadth of the product range given by equation (17) does not depend on Ω. In other words, a
firms.
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mixed-market structure puts the emphasis on the persistence of large firms despite the important turnover of small
Since X ∗ is stabilized by the free-entry condition, the impact on producer surplus is given by 4βf ∆P S := Π∗mp (x∗mp , N ∗ , X ∗ ) = γ
s
F 1+ −1 f
!2
(19)
The impact on consumer welfare however is less straightforward. Using that X ∗ does not change following the entry of the oligopolist, it can be shown that the overall number of varieties has increased:
∆N = N
∗
1−
s
f f +F
!
>0
This variety effect is enabled because an MP-firm internalizes the consumers’ love for variety. Thus, a multiproduct firm trades off a smaller output of each of its varieties against a broader product range. Since these varieties are supplied at a higher price, we have the following result: 14 Assuming
that the foreign oligopolist faces trade costs does not change the propositions below (see also 4.3).
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Proposition 3. As long as both types of firms exist in equilibrium, the entry of a foreign oligopolist increases the total number of varieties but raises the average price.
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All varieties produced by SP-firms which disappear are replaced with higher-priced (proposition 2) from the
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new oligopolist. The average price unambiguously increases with Ω through a composition effect. To conclude on the net effect of firm entry on consumer surplus, I re-express it as follows: β CS := L 2
Z
N
x2ν dν
γ + X2 2
#
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"
0
Contrary to producer surplus, two aggregates are necessary to describe consumer surplus as consumers care
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about both the total consumption level X and its composition since they have a love for variety. Since I have shown that X ∗ , x∗mp , x∗sp and N ∗ do not depend on Ω, the change in consumer surplus ∆CS following the entry of one
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large firm is constant, negative, and equal to:
!2 F 1+ −1 . f
s
(20)
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βf ∆CS := − γ
Proposition 4. Consumer (producer) surplus decreases (increases) with the entry of a foreign oligopolist.
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The above proposition states that the variety effect on consumer welfare is dominated by the increase in the average price. I show in section 6 that this is not an artefact of the assumed quadratic utility but holds generally
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for standard alternatives to CES preferences. Under this particular specification of preferences, summing (19) and (20) shows that the entry of a foreign large firm increases social welfare:15
3βf ∆SW := γ
s
F 1+ −1 f
!2
The consumers’ expenditures shift from the homogenous to the differentiated good, while at the same time the level of utility increases. The robustness of this result is discussed in Section 6.
5.2
The Free-Trade Benchmark
I assume here that κ symmetric countries are freely trading with each other. Following Eckel and Neary (2010), the openness to trade of each country is then measured by κ. Compared to the previous subsection, increasing κ also leads to an increase in market size. I assume that (16) is fulfilled for the total market size κL and the total number of oligopolists κΩ.16 15 This corresponds to the sum of social welfare worldwide, including the profits of the foreign firm. Were its profits entirely repatriated to its origin country, the change in social welfare of the country where the large firm enters boils down to the negative variation of consumer surplus. 16 Formally,
equation (16) becomes Ω <
F f +F 1 qκ f κ 1− f +F
.
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Augmenting market size and the number of oligopolists by a factor κ in (14) and (17), the mass of monopolistic firms in each country can be derived: f f +F
!
κΩ
#
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1−
s
(21)
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" 2β f +F ∗ F − M (κ) = γκ κ
Setting Ω = 0 in (21) leads to Krugman (1979)’s well-known result: under an increase in market size, monopo-
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listically competitive firms enjoy economies of scale and consumers spread their consumption over a large set of
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varieties, thereby increasing the elasticity of demand so that firms reduce their markups. The number of varieties f +F F provided in each country decreases, but consumers enjoy a higher number of varieties overall: κM ∗ = 2β · γ κ
In other words, the mass of SP-firms is a concave and increasing function of κ.
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On the contrary, given Ω ≥ 1, the openness to trade also increases the number of large firms κΩ which triggers the exit of small firms as shown in 4.1.
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These two opposite effects lead to an inverted U-shaped curve for κM : the market size effect itself decreases in is a concave function of κ), while the increase in κ (consumers value new varieties less and less, so that F f +F κ q F the number of large firms is constant and proportional to 1 − f +F Ω, so that it eventually dominates. When
economies of scope are large for MP-firms (high F ), the latter effect is stronger. Regarding the total number
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of varieties, Proposition 3 guarantees that the overall number of varieties (SP- and MP-firms altogether) is an
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increasing function of the number of countries despite the exit of small firms. Pro-competitive effects While the entry of foreign oligopolists on the domestic market would raise prices, the increase in market size has a pro-competitive effect. Indeed, because SP-firms maintain their average-cost pricing rule, a higher market size translates into economies of scale, which leads to lower prices. This pro-competitive effect also operates for large firms: equation (12) implies that an increase in market size increases the output of large firms, yet their profit remains constant by (19). Thus, large firms have reduced their markup. To sum up, we have: Proposition 5. As long as both types of firms exist in equilibrium, an increase in the number of trading partners under free trade decreases the prices of all firms and increases the number of varieties available to consumers. Welfare Using equation (19), we know that changes in social welfare are driven by changes in consumer surplus only. While pro-competitive effects are typically welfare-enhancing, the fact that SP-firms exit the market increases the share of overall consumption devoted to the higher-priced varieties supplied by MP-firms. Under free trade, this anti-competitive selection effect is more than offset by the pro-competitive effect (see Appendix C). In this section, I have shown that free trade opens the domestic market to foreign exporters, which triggers the exit of domestic small firms. In the absence of fixed costs to export, trade has a pro-competitive effect. Free trade allows small firms to export: in doing so, they maintain a tough competitive environment for big firms. As shown 14
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in the next section, there is a sharp contrast when firms self-select in the exporting activity in the presence of fixed costs.
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Reduction in Trade Costs
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5.3
I now assume that, in order to export a product to the foreign market, a firm must incur variable and fixed costs. Compared to the previous scenario, trade frictions imply that small and large firms will respond differently to a
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reduction in trade costs. In the spirit of Melitz (2003), I consider that the foreign market has the same size as the domestic market, the same distribution of firms (i.e. the same number of large firms Ω), and symmetric trade costs. Last, I assume that the condition for both types of firms to co-exist holds for any τ i.e. F < F¯ (2Ω).
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The variable component of trade costs here is a per-unit additive trade cost τ , so that the marginal cost of
5.3.1
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exporting a variety is c + τ . Comparison of MP- and SP-Firms
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The inverse demand can be written again for any variety sold on the domestic market in quantity x to each consumer:
p(x, X ) = α − βx − γX
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By definition, it is equal to:
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where X is, by symmetry, the per-capita aggregated consumption across products in any of the two countries.
H H F F F F X = ΩN H xH mp + M xsp + ΩN xmp + M xsp | {z } Imported
varieties
The upperscripts H and F stand respectively for home and foreign. Production
The foreign and domestic markets are segmented. Both small and large firms have a constant marginal cost of production. Furthermore, large firms have a constant marginal cost of adding a product, so both SP- and MP-firms make their decision independently on each market. The profits of a multi-product firm (based in Home) on the domestic and foreign markets are then:
H H ΠH mp (xmp , N , X ) := L ·
F F ΠF mp (xmp , N , X ; τ ) := L ·
Z
Z
NH H ((p(xH mp , X ) − c)xmp − f )dk
0
NF 0
F ((p(xF mp , X ) − c − τ )xmp − f )dk
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In the same way, the profits of small firms are given by:
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H H H (xH πsp sp , X ; τ ) := ((p(xsp , X ) − c)Lxsp − F − f
F F F πsp (xF sp , X ; τ ) := ((p(xsp , X ) − c − τ )Lxsp − Fx − f
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Fixed costs
To export, all firms face an overhead fixed cost Fx . This fixed cost can be interpreted as in the standard Melitz
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model: firms need to establish a network or more generally look for buyers on the foreign market. Any variety exported requires an additional cost which can be interpreted as a packaging cost, or more generally as any other
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market-product-specific fixed cost. This cost is assumed to be equal to f only for the sake of simplicity.17 Again, any form of heterogeneity within MP-firms is ruled out.
Turning to SP-firms, they need to incur a total fixed cost equal to f + Fx in order to export their product. Large
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firms spread Fx on a continuum of varieties which they supply, so that the exporting activity fosters economies of scope. At the margin, it is easier for them to export a new product than it is for an SP-firm. The cost of introducing
of finite mass N F ∗ f .
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a new product is f for an MP-firm, and f + Fx for an SP-firm. Each large firm, however, incurs a total fixed cost
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Prices, Markups and Output in Home and Foreign Taking the equilibrium value of X ∗ as given for now, cross-section comparisons between firms are derived, assuming both MP- and SP-firms export. Considering the first-order conditions of large firms respectively w.r.t. xH mp and ∗H ∗ ∗F ∗ xF mp , their total output Qmp (X ) resp. Qmp (X ) supplied on the domestic market can be derived:
∗ H Q∗H mp (X ) = L · N
α − c − γX ∗ 2β + γN H
∗ F Q∗F mp (X ; τ ) = L · N
∗H (X ∗ ) = L qmp
α − c − γX ∗ 2β + γN H
∗F qmp (X ∗ ; τ ) = L
α − c − τ − γX ∗ 2β + γN F
with density: α − c − τ − γX ∗ 2β + γN F
(22)
∗H ∗H ∗F ∗F The output of an SP-firm is obtained for N F = N H = 0 in which case qmp = qsp and qmp = qsp . s∗ ∗ I denote by ps∗ r := p(xr , X ), where r = {sp; mp} and s = {H; F }, the prices set on the market by domestic
and foreign MP and SP firms. When set by a foreign firm, this price is inclusive of trade costs. At equilibrium, the absolute markups on the domestic and foreign market are given by the following expressions: 17 None of the results below rely on this assumption: a higher fixed cost per product to export would only further decrease the breadth of the exported product range.
16
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pH∗ mp − c =
γ 1 (α + Q∗H − γX ∗ − c) 2 L mp
∗ pF mp − c =
γ 1 (α + Q∗F − γX ∗ − c + τ ) 2 L mp
pH∗ sp − c =
1 (α − γX ∗ − c) 2
∗ pF sp − c =
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which yield a direct comparison with SP-firm markups: 1 (α − γX ∗ − c + τ ) 2
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Comparing prices vertically: firms with a higher mass set higher prices by charging higher markups. This also implies that when both large and small firms export, MP-firms set higher (f.o.b. or c.i.f.) prices than SP- firms. Market Outcome
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5.3.2
Selection
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A firm is an exporter if its profits abroad can cover the fixed cost of exporting. In the spirit of the Melitz model, I focus here on the case featuring selection into the exporting activity. This is the case when fixed costs are high enough. For the sake of simplicity, I assume that Fx = F .18 When there is no prospect of economies of scale, small
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firms choose not to export no matter what the extent of trade liberalization τ > 0 is. However, the prospect of
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capturing rents abroad is a sufficient reason for MP-firms to export. Comparative statics w.r.t. to τ
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Comparing the market outcome under trade with the closed economy section, the market aggregate X ∗ at equilibrium is the same and determined by the free-entry condition on the domestic market for SP-firms.
∗ (x∗sp , X ∗ ) = πsp
L 2 (α − c − γX ∗ ) − (f + F ) = 0 4β
M ∗ > 0 ⇒ X ∗ (τ ) = X ∗ (τmax ) = X ∗
where X ∗ is the equilibrium value of X in the absence of trade. Large firms start exporting as soon as trade q q 1 + Ff − 1 . costs fall below τmax := 2 βf L
Writing that the inverse demand is met by firms’ prices at equilibrium shows that imported varieties are sold
at the same price as the ones produced domestically:
F∗ ∗ ∗ pH∗ mp = pmp = α − βxmp − γX 18 A sufficient condition is F ≥ F . When F = F , a positive level of trade costs τ guarantees that exporting is less profitable x x (including fixed costs) than domestic sales. Since small firms are free to enter and choose to export only if it is profitable, they never choose to do so. In other words small firms have no reason to export in the absence of economies of scale.
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In line with Brander and Krugman (1983), there is reciprocal dumping in the sense that large firms set a f.o.b. price lower than the price at which varieties are sold domestically. However and contrary to Brander and Krugman (1983), MP-firms’ market power in the foreign market increases with trade liberalization as they absorb any decrease
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in trade costs by increasing their markups. While they show that oligopolistic rents are driven down by a decrease in
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trade costs, a mixed market structure emphasizes the persistence of the market power of large firms so that, in the long run (i.e. after the monopolistically fringe adjusts), there is no pro-competitive effect hurting the oligopolists.
and in F leads to
N
F∗
s
2β = γ =N
! F 1+ −1 f
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N
H∗
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Turning to the product range, the two first-order conditions of large firms w.r.t. their product breadth in H
H∗
· 1−
τ
τmax
(23)
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which shows that large firms export only a subset of their varieties abroad. When τ decreases, the marginal profit of exporting an additional variety increases so N F ∗ −→ N H∗ . τ →0
To sum-up trade liberalization allows large firms to increase their market share abroad: they expand their
ED
product scope and raise their markups on all their varieties.
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Last, to see how MP-firms gain from firm selection, their profits at equilibrium can be derived:
4βf = γ
s
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ΠH∗ mp
!2 F 1+ −1 f
∗ F ∗ F∗ ΠF , X ∗; τ ) = mp (τ ) := Πmp (xmp , N
L 2 (τmax − τ ) γ
Because the exporting activity is now costly, only large firms benefit from trade liberalization. Furthermore, small firms do not benefit from any efficiency gain as in the free-trade scenario. Economies of scope realized on the exporting fixed costs protect large firms to the extent that the competitive fringe cannot export. A decrease in trade costs does not increase competition: new varieties that are imported replace the exit of SP-firms. From then on, any further drop in trade costs τ only makes oligopolists more profitable. By contrast, SP-firms cannot compensate the increase in competitive pressure on the domestic market with their exporting activity since the latter is too costly. SP-firms are forced to exit the domestic market as trade liberalization occurs: there is a reallocation of their market shares towards the surviving firms. In that sense, the market structure becomes more oligopolistic. The distribution of the gains from trade MP-firms make positive profits in equilibrium that are a decreasing function of τ so trade raises producer surplus unambiguously. Turning to the impact on consumer surplus, a drop in trade costs does not impact prices consumers pay but increase the overall number of varieties. Following the logic of section 5.1, the change in consumer surplus 18
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in one country following a drop in τ is given by s
! F 1 + − 1 ∆τ N F f
N H∗ · ∆τ τmax
so that the change in consumer surplus can be simplified into s
F 1+ −1 f
!2
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βf ∆τ CS = Ω γ
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∆τ N F = −
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where ∆τ N F is the change in product scope w.r.t. τ . From (23), it is equal to
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f ∆τ CS = Ω 2
∆τ τmax
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Proposition 6. Consumer (producer) surplus decreases (increases) when variable trade costs decrease. The increase in the number of varieties does not increase consumer surplus because new varieties are supplied at
ED
a higher price: large foreign firms gain more market power. It is worth stressing that this result departs both from a pure monopolistic (Krugman, 1979) and from an oligopolistic (Brander, 1981) market structure. Here, although there are gains from trade liberalization, there are absorbed by an increase in producer surplus. This shows how
Robustness
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6
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important it is to tackle the distribution of the gains from trade between producers and consumers.
Central to the results obtained in the previous section is the anti-competitive effect following the entry of a large firm derived in subsection 5.1. In order to study the robustness of this result, I proceed in two steps. First, I show that under mild assumptions, the impact of entry of a foreign oligopolist on the average price holds true when the market aggregate is additively separable in firms’ strategies. The robustness of the variety effect is weaker, but holds under an additional yet reasonable assumption. Second, I turn to consumer surplus and consider the class of additively separable preferences which are consistent with these assumptions and nest a large class of utilities considered in international trade. I show that the negative impact on consumer surplus typically holds in models featuring an increasing price-elasticity of demand. Specifically, I consider below that the inverse demand for variety ν can be written as p(xν , Λ) where Λ is an aggregate statistic of total consumption. The profits of firm ω with respect to a given variety are assumed to be strictly quasi-concave so the output profile of an MP-firm will be necessary symmetric at equilibrium. The choice of the multi-product firm ω boils down to two variables xω and Nω so that the strategy set of each player is twodimensional. Last, I consider the monopolistically competitive fringe as a Ω + 1th player e.g. a multi-divisional
19
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firm whose product range breadth is governed by the free-entry condition.19 Therefore, the output profile in the
6.1
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¯ can be replaced with the (2Ω + 2)−uplet (x,N). industry x " Ω+1 −→ R. Under the assumptions of the baseline The term “aggregate” is defined here as a function Λ : R2 P model Λ(x, N) ≡ X = ω≤Ω Nω xω + M xsp . In that case the aggregator is additively separable.20
Variety vs. Prices.
differentiable, strictly decreasing resp. w.r.t. x i.e. px :=
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Assumptions The aggregate is assumed to be a scalar, i.e. Y = R. The inverse demand is twice continuously ∂p ∂s (x, Λ)
x < 0 and not “too convex” i.e. − pxx px < 2 (A1).
An increase in the aggregate shifts the inverse demand curve downwards i.e. pΛ :=
∂p ∂Λ (x, Λ)
<0
(A2).
The
∂Λ ∂xω (x, N)
= 0 if Nω = 0 and ΛNω (x, N) :=
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aggregate Λ(x, N) is twice differentiable w.r.t. each of its argument. For each ω ∈ [[1, Ω]] we have Λxω (x, N) := ∂Λ ∂Nω (x, N)
= 0 if xω = 0. Otherwise, Λxω (x, N) > 0 if Nω >0 and
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ΛNω (x, N) > 0 if xω >0 (A3). Last I assume that Λxω Nω (x, N) :=
∂2Λ ∂xω Nω (x, N)
> 0 ∀(x, N ) ∈ R∗2 + (A4).
Assumption (A1) implies that the marginal revenue curve of an SP-firm is decreasing in output. Combining (A2) and (A3) implies that an MP-firm may shift down the demand schedule i.e. raise competition toughness
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either by increasing its output or its product range. Its output per-product xω has no impact on Λ when it’s negligible Nω = 0 and its product range has no impact on Λ if it supplies zero output xω = 0. Last, (A4) implies
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that the impact of the product breadth of an MP-firm on competition toughness is all the more important that it has a large output per-product.
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Separable market aggregates I show in Appendix D that under these conditions and as long as an equilibrium exists, Λ∗ is uniquely determined. As a consequence, under (A1)-(A2) the output and prices chosen by SP-firms do not depend upon any shock on Λ. When the best-reply function of large firms can also be expressed as a function of Λ only, i.e. when the game is fully aggregative (Cornes and Hartley, 2012), the entry of a foreign oligopolist, even if it faces variable trade costs τ , does not affect the equilibrium among MP-firms. When is the game fully aggregative? Cornes and Hartley (2012) have shown that under strict monotonicity assumptions with at least three players, a game is fully aggregative if and only if the aggregate is additively separable (up to a monotonic transformation). I show in Appendix E that under (A3)-(A4) with at least three players, Cornes and Hartley (2012)’ result can be extended to the case of two-dimensional strategy sets. In what Pω=Ω follows, I assume therefore that Λ is a monotonic transformation of ω=1 gω (xω , Nω ) + gsp (xsp , M ) where gω (.)
and gsp (.) are continuously differentiable functions. 19 See
for instance Shimomura and Thisse (2012). is a special case of the class of aggregative games considered by Acemoglu and Jensen (2013) who consider finite-dimensional strategy sets. In the event where MP-firms have an exogenous unitary mass, the aggregate X in the baseline model boils down to the sum of the player’s actions as in Anderson et al. (2013). 20 This
20
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Entry of a large firm
I explore here the robustness of the impact of an MP-firm entry on prices and product
variety when the inverse demand can be written as p(x, Λ) where x is still the consumption of a variety and PΩ Λ(x, N) = ω=1 g(xω , Nω )+g(xsp , M ) is an aggregate statistic of a consumer’s consumption of all other varieties.21
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Under this functional form, assumption (A3) requires that g(.) is twice differentiable with g(0, N ) = g(x, 0) =
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0 ∀(x, N ) ∈ R2+ with gx > 0, gN > 0 and (A4) requires that gxN > 0 ∀(x, N ) ∈ R∗2 + . Assumptions (A1)-(A4) R are verified in the baseline model under which Λ = X = N xν dν and g(x, N ) = N x.
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The pair (x∗ω , Nω∗ ) verifies the following first-order conditions at equilibrium
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px (x∗ω , Λ∗ )x∗ω + N ∗ gx (x∗ω , Nω∗ )pΛ (x∗ω , Λ∗ )x∗ω + p(x∗ω , Λ∗ ) − c = 0 gN (x∗ω , Nω∗ )pΛ (x∗ω , Λ∗ )N ∗ x∗ω + (p(x∗ω , Λ∗ ) − c)x∗ω = f
(24)
(25)
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which is to be compared with the first-order condition for an SP-firm:
(26)
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px (x∗sp , Λ)x∗sp + p(x∗sp , Λ∗ ) − c = 0
Two results can be derived from these equations. Under (A1), (A2) and (A3), (30) and (26) imply that x∗ω < x∗sp and thus p(x∗ω , Λ∗ ) > p(x∗sp , Λ∗ ). Hence larger firms charge higher markups.
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Second, if there exists an optimal and finite product range Nω∗ , equation (31) implies that it depends on Ω only through Λ∗ .
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Now, Λ∗ is pinned down by the free-entry condition and Λ is additively separable so x∗ω and x∗sp do not depend upon firm-entry. Writing that the aggregate Λ∗ does not change with the entry of a large firm ω leads to the following equality
g(x∗sp , M ∗ (Ω + 1)) − g(x∗sp , M ∗ (Ω)) = g(x∗ω , Nω∗ )
so the entry of a large firms triggers again the exit of SP-firms. From (A4), we get that g(x∗ω , Nω∗ ) > g(x∗ω , M ∗ (Ω + 1)) − g(x∗ω , M ∗ (Ω)) Making the further assumption that gN N ≥ 0 (A5), and keeping in mind that g(x, 0) = 0, we have the following inequality g(x∗ω , Nω∗ ) > g(x∗ω , ∆M ) and therefore the total number of varieties increases again with the entry of a large firm. Assumption (A5) is 21 I assume for the sake of simplicity that g (.) = g (.) ≡ g(.) ∀ω ∈ [[1, Ω]]. This is to be considered the standard case under ω sp symmetric utility functions. Note also that the same exercice may be conducted under the assumption of price competition when the direct demand depends on an additively separable aggregate. Interestingly enough, Bertrand competition in the baseline model falls outside this case. Nevertheless, I show in Appendix F that even in that case, the entry of a large firm raises the average price and decreases consumer surplus.
21
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far from being necessary for this result. Note that it holds strictly even when gN N = 0 as in the baseline model. Because the marginal cost of adding a variety is constant, the only reason for a firm not to supply an infinite number of varieties is cannibalization, which is magnified by the convexity of g(.) w.r.t. N . From the perspective of an
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MP-firm, adding a variety should decrease its marginal profit w.r.t. N . Inspecting the second partial derivative of
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the profit shows that the convexity of g only magnifies the cannibalization effect.22
Last, since the firms’ payoff depends on Λ∗ only, the total producer surplus increases with the entry of a large
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firm as in the baseline model.
To sum-up: under a mixed-market structure where assumptions (A1)-(A4) hold, MP-firms charge higher markups, the exogenous entry of a foreign oligopolist leads to the exit of monopolistic firms and raises the average
Consumer Welfare
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6.2
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price. The total number of varieties increases as soon as (A5) holds.
In this subsection, I aim to check that my results are not driven by the assumption of quadratic preferences. Two effects conflict in the baseline model: the variety effect and the anti-competitive effect (i.e. the average price goes
ED
up). While the anti-competitive effect has been shown to be quite robust, the variety effect requires an extra assumption (A5). However, since (A5) appears quite reasonable, I focus on the case where the variety effects go
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against the anti-competitive effect. To do so, I consider the class of additively separable preferences (see Kuhn and Vives (1999), Vives, 1999 and Zhelobodko et al. (2012) which nest CES preferences) for which (A5) holds.
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Preferences First, it is important to note that if the relevant market aggregate from a firm’s perspective is Λ, this aggregate is generally not relevant to analyze consumer surplus.23 For instance, in the baseline model, the R RN N expression of consumer surplus depends on a two-dimensional aggregate 0 xν dν, 0 x2ν dν . RV I now consider an additively separable lower-tier utility 0 u(xν )dν where u(.) is increasing, concave and thrice differentiable on R+ . The upper-tier utility is
U (A, x) := A + U (x)
where
U (x) := W
Z
V
u(xν )dν 0
!
(27)
with W being twice continuously differentiable. Inspecting the first-order condition implies that the inverse demand for a variety ν features an additively separable aggregate:
′
′
where
p(xν , Λ) := u (xν )W (Λ) 22 The
Λ :=
Z
V
u(xν )dν
(28)
0
expression of the second derivative of per-product profits w.r.t. N is given by gN N (x∗ω , N )pΛ (x∗ω , Λ∗ )N ∗ x∗ω + gN (x∗ω , N ∗ )2 pΛΛ (x∗ω , Λ∗ )N ∗ x∗ω + 2gN (x∗ω , N ∗ )pΛ (x∗ω , Λ∗ )x∗ω < 0 | {z } <0
Note that the first term drops out in the baseline model. This condition is more likely to be met when (A5) holds. exception is when the direct demand verifies the Independence of Irrelevant Alternatives as in Anderson et al. (2013). Note that the only additively separable preferences which verify this property feature a constant elasticity of substitution. 23 An
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so assumptions (A1)-(A5) hold. Consumer surplus can be expressed as follows: N
p(x∗ν (Λ), Λ)x∗ν (Λ)dν
0
T
Z
RI P
CS := U (Λ) −
Variations in the consumer surplus are therefore driven by variations in expenditure. Similarly to the baseline model, consumer surplus requires one additional aggregate compared to producer surplus - the expenditure on the
SC
differentiated good.
Using that Λ∗ does not depend on Ω, the change in consumer surplus with the entry of a large firm is
u(x∗mp ) ∗ ∗ p x − u(x∗sp ) sp sp
NU
∆CS = −N
∗
p∗mp x∗mp
MA
Now, plugging (28) into the above expression leads to:
u′ (x)x u(x)
(29)
ED
where Eu =
′ ∆CS = N ∗ u(x∗mp )W (Λ∗ ) Eu (x∗sp ) − Eu (x∗mp ) is the elasticity of the sub-utility u(.) w.r.t. consumption.
The change in consumer surplus depends crucially on whether Eu increases or decreases with individual conas the most relevant.
PT
sumption. As argued by Kuhn and Vives (1999) and Dhingra and Morrow (2016), the latter should be considered
AC CE
Furthermore, under the assumption that u(0) = 0, it is possible to show that the monotonicity of Eu is disciplined by the monotonicity of the demand price-elasticity. In fact, based on Kuhn-Vivès (1999) results, I show in Appendix G.1 that an increasing price-elasticity of demand implies that Eu′ < 0. This happens to be the standard case in trade models featuring a variable elasticity of demand.24 Therefore the change in consumer surplus is generally negative, showing thereby that the insights from the baseline model are not an artefact of the quadratic preferences considered. Interestingly enough, ∆CS = 0 when Eu is constant, i.e., when preferences feature a constant elasticity of substitution. Last, it is important to note that (29) implies that the marginal cost of the foreign oligopolist is irrelevant as long as it yields x∗mp < x∗sp . Consequently, the entry of an oligopolist which faces variable trade costs τ - so that the marginal cost of exporting a variety is c + τ - impacts negatively consumer surplus. 24 In the trade literature, this assumption goes back at least to Krugman (1979). This is also equivalent to assuming an increasing relative love for variety in Zhelobodkho et al. (2012) or a decreasing elasticity of substitution as in Bertoletti and Epifani (2014). Mrázová and Neary (2013) have also found out that this assumption can be traced back to Marshall and is known as the second Marshall law of demand. An increasing price-elasticity of demand is also directly related to firms’ pass-through rate. Empirical work e.g. Berman Martin Mayer (2012) and Amiti, Itskhoki, and Konings (2014) find support for a pass-through rate below 1 (see also De Loecker et al. 2014. for additional empirical evidence) which requires again an increasing price-elasticity of demand under monopolistic competition (Mayer Melitz and Ottaviano, 2016).
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Social welfare The change in social welfare is ambiguous. Indeed, new varieties come at the cost of the multiproduct technology, i.e. the fixed cost f which is incurred by large firms for each product. Even if producing one variety is more efficient when undertaken by a large firm rather than by a small firm, large firms may still produce
T
too many varieties, i.e. the marginal gain from selling a new variety for an MP-firm may be smaller than the
RI P
decrease it entails for consumer surplus.
Whether one or the other dominates is generally ambiguous and depends on the functional forms assumed for
SC
the utility function: even if the level of social welfare increases in the baseline model or in the CES case, it need not be so, even for well-behaved preferences.25
Note however that if large firms behaved like multi-divisional firms, they would set a price equal to the one
NU
charged by small firms. In that case, social welfare would always increase with a large firm’s entry thanks to economies of scope. To sum up, the distribution of the surplus following the entry of a large firm between consumers
MA
and producers is robust to a large class of functional forms. In spite of the ambiguity of the results regarding social welfare,26 it should be noted that were large firms not behaving strategically, social welfare would always increase
ED
with the entry of a foreign large firm through an increase in productive efficiency.
7
Concluding Remarks
PT
Consistent with the empirical evidence that most industries feature a small number of large firms and a large number of small firms, I have built a model with a “mixed” market structure in which a few multi-product oligopolists and
AC CE
a continuum of single-product monopolistically competitive firms coexist. Large firms charge higher markups than monopolistically competitive firms, even if they have the same marginal cost of production. This is because large firms are able to manipulate the market aggregate which is relevant to all their competitors’ decisions.
When a reduction in trade barriers selects the largest firms into exporting, the overall number of varieties available to consumers increases, but so does the average price in the industry. This results in a decrease of consumer surplus, while large firms absorb the decrease in trade costs by increasing their markups. I have shown that the main insights obtained in the baseline model under linear demand hold true for a wide class of demand functions. This shows the importance of considering heterogeneity in firm size, together with strategic interactions. Finally, this paper leaves at least two avenues for future research. First, this paper has been silent on policy. In the setting I have considered, trade liberalization may hurt consumers at the expense of producers. This calls for an extension in which governments may manipulate trade barriers and domestic redistributive policies to affect the degree of income inequality. 25 Counter-examples
include functional forms which belong to the Hyperbolic Absolute Risk Aversion (HARA) family. some special cases, it is possible to show that social welfare increases e.g. when economies of scope are very low so F = 0+ . I illustrate this case in Appendix G.2. Intuitively, this is because MP-firms behave like multi-divisional firms again. 26 In
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Second, I have assumed that the number of firms that can be large is exogenous. Since the entry of a large firm impacts the distribution of the gains from trade, this is an important assumption to relax. In the respect, the
AC CE
PT
ED
MA
NU
SC
RI P
T
contribution of Eckel and Yeaple (2016) is an important first step in this direction.
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A. MP-firms’ Program A.1 Reducing the Dimensionality of the Maximization Problem
T
First, one can notice that the output profile x ¯ω of firm ω can be defined as a positive-valued function up to a threshold
RI P
N (potentially infinite) and zero above without loss of generality. Second, if x ¯∗ω = (x∗ωk )k≤Nω is a maximizing output profile for firm ω, then there exist x∗ω > 0 and Nω > 0 (potentially zero or infinite) such that x∗ωk = x∗ω almost
SC
everywhere ∀k ∈ [0, Nω ]. To see this, assume this is not true and that there exist two convex subsets [0, N1 ] and
NU
]N1 , Nω ] of R+ with 0 < N1 < N2 such that xωk = x1 ∀k ∈ [0, N1 ] and xωk = x2 ∀k ∈ [0, Nω ]. Now, since RN ∗ z − f /L is strictly concave, the symmetric output the function π ˜ (z) := z −→ α − c − βz − γ 0 ω xωk dk − γX−ω RN N1 N1 ˜ωk , generates higher profits: profile x ˜ωk = N x1 + 1 − N x2 ∀k ∈ [0, N2 ], which is such that 0 ω xωk dk ≡ Nω x ω ω
MA
N1 N1 N1 N1 π ˜ (x1 ) + 1 − ˜ x1 + 1 − π ˜ (x2 ) < π x2 Nω Nω Nω Nω
ED
Therefore, the maximization program of firm ω comes down to choosing a pair (x∗ω , Nω∗ ) such that:
PT
(x∗ω , Nω∗ ) = Argmax(xω ,Nω ) Nω
"
∗ α − c − βxω − γNω xω − γX−ω xω − f /L
A.2 Determining the Global Maximum P
ω6=ω ′
Nω∗ x∗ω − βM x∗sp − c. The objective function of firm ω is:
AC CE
Let us denote α ˜ := α − β
max [α ˜ − (β + γNω ) xω ] Nω xω − Nω f /L
xω ,Nω
The first-order conditions w.r.t. xω and Nω are respectively
[˜ α − 2 (β + γNω ) xω ] Nω = 0 and
[˜ α − (β + 2γNω ) xω ] xω = f /L so the Hessian is given by
−2 (β + γNω ) Nω H := α ˜ − 2 (β + 2γNω ) xω
26
α ˜ − 2 (β + 2γNω ) xω −2γx2ω
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with determinant | H |= 4γ [(β + γNω )] N x2ω − [˜ α − 2 (β + 2γNω ) xω ]
x∗ω
=
s
f βL
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leads to
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and
(30)
SC
2 (γNω∗ + β) x∗ω = α ˜
T
Using the first-order conditions, it is evaluated at (x∗ω , Nω∗ ) where
2
∗ ∗ | H∗ |= 4γ [(β + γNω∗ )] Nω∗ x∗2 ω − [2γNω xω ]
(31)
2
MA
| H∗ |= 4γβNω∗ x∗2 ω >0 so (x∗ω , Nω∗ ) is a local maximum. Since this is a two-dimensional problem, the fact that this local maximum is the only critical point of the profit function on R+ × R+ is not enough to conclude it is a global one. However, the
ED
maximization problem can be reformulated on a compact without loss of generality by ruling out cases in which
PT
profits are negative. Indeed, for [˜ α − (β + γNω ) xω ] Nω xω − Nω f /L to be non-negative, a necessary and sufficient √ 2 α+ α ˜ −4(β+γNω )f /L with α ˜ 2 ≥ 4(γ + βNω )f /L. This expression shows that there is an condition is that xω ≤ 2(β+γNω ) upper bound on the output per variety: xω ≤
α ˜ β.
These last two inequalities define a support for the maximization
AC CE
problem which is a compact subset of R+ × R+ . On the boundary of this support, firm ω makes at most zero profits. Therefore any strictly positive solution (x∗ω , Nω∗ ) to (30) and (31) (which guarantees strictly positive profits) is the unique global maximum.
B. Derivation of γ¯
Writing that consumer income is not entirely spent on the differentiated good leads to 1 + ΩΠ∗mp > which can be rearranged into 1 > p∗sp X ∗ + ΩN ∗ f − (p∗sp − c)x∗mp
R N∗ 0
p∗ν x∗ν dν
Using the expressions for equilibrium output and the inverse demand evaluated at x∗mp shows that the right-hand side is decreasing in Ω. A necessary and sufficient condition for the consumption of the homogenous good to be positive for any Ω is 1 > p∗sp X ∗ which is equivalent to γ > γ¯ :=
!s
! " " p α−c f +F p −1 + c 2β f + F β 2 β (f + F )
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Note that the other equilibrium conditions do not depend upon γ so the parameter space is non-empty.
The market aggregate X∗ = varies with market size as follows:
q +F ) α − c − 2 β(fκL γ
NU
βx∗sp dX ∗ = dκ γκ
SC
q β(f +F ) dX ∗ L = dκ γ · κ3/2
RI P
T
C. Welfare and Trade Openness
MA
Using that κx2sp does not depend on κ, differentiating consumer surplus CS := L
h R β N 2
0
i x2ν dν + γ2 X 2 leads to (32)
ED
dM ∗ β dCS dX := + γLX ∗ κLx∗2 sp dκ 2 dκ dκ To assess the impact of κ on the mass of small firms per country, notice that
∗ X ∗ κx∗ α−c dM ∗ 2β sp where X =d = − ∗ ∗ dκ dκ κxsp γκxsp γκ
AC CE
and thus
PT
X ∗ −κΩN ∗ x∗ mp κxsp∗ dM ∗ =d dκ dκ
After simplication we have
βxsp − dM = dκ κ2
X∗ 2
1 xsp
Plugging the above expresssion into (32), we get that
dCS β := dκ 2
βxsp −
X∗ 2
κ2
1 xsp κLx2sp + βLX xsp κ
The total change in consumer surplus w.r.t. the number of countries is unambiguously positive βqsp∗ dCS := dκ κ
3 βxsp + X ∗ 4
>0
Since we have already shown that firms’ profits remain constant, we may conclude that social welfare increases with trade openness.
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D. Uniqueness of the Equilibrium Market Aggregate Under (A1), the maximizing output of an SP-firm x∗sp (Λ) is a well-defined function of Λ. The free-entry condition
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T
reads as
πi∗ (Λ) := (p(xi , Λ) − c) xi = πi (x∗ (Λ), Λ) = 0.
(33)
SC
Using (A2), the envelop theorem implies that πi∗ (Λ) is decreasing and that this equation has at most one solution Λ∗ . Therefore, as long as an equilibrium exists, Λ∗ is uniquely determined. As a consequence, under (A1)-(A2) the output and prices chosen by SP-firms do not depend upon any shock
NU
on Λ. This is the case if a large firm enters the market or if one (or many) incumbents face an exogenous shock on their variable or per-product fixed costs. The dimensionality of the aggregate X in the baseline model is therefore
MA
crucial to explain the neutrality of SP-firms response.
E. Conditions for the Game to be Fully Aggregative
ED
The uniqueness of Λ∗ is not enough to explain the neutrality of MP-firms’. Inspecting the first-order conditions of an MP-firm ω at equilibrium shows that a best reply function generally depends both on Λ and on its derivatives
PT
with respect to xω (resp. Nω ). Eventhough Λ∗ is determined by free-entry, the equilibrium values of Λ∗xω and Λ∗Nω generally depend upon other-firms’ equilibrium strategies. For the neutrality of firm entry (or a cost shock) to hold
AC CE
it is necessary and sufficient that the marginal profits w.r.t. xω and Nω (and thereby Λxω and ΛNω ) depend upon xω , Nω and Λ only. In this event, the game admits backward reply correspondences27 x∗ω (Λ) and Nω∗ (Λ) for each firm ω .
This is precisely the class of aggregative games considered by Cornes and Hartley (2012) which they refer to as fully aggregative games. In the case of fully aggregative games with one-dimensional strategy sets, they show that under strict monotonicity assumptions with at least three players, one can assume additively separable aggregates (up to a monotonic transformation) without loss of generality. I show below that under (A3)-(A4) with at least three players, Cornes and Hartley (2012)’ result can be extended to the case of two-dimensional strategy sets. Cornes and Hartley (2012) Extended to Two-Dimensional Strategy Sets The proof closely follows Cornes and Hartley’s (2012) in which the regularity assumptions are given by (A3)-(A4). By definition, the game is fully aggregative if and only if there exist functions φ and φ˜ such that
Λxω := 27 This
∂Λ = φω (xω , Nω , Λ) ∂xω
notion has been refered to under various names and goes back at least to Selten (1970).
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ΛNω :=
∂Λ = φ˜ω (xω , Nω , Λ) ∂Nω
T
The cross-derivatives w.r.t. xω and xω′ are given by:
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∂φω (xω , Nω , Λ) ∂2Λ ∂Λ ∂φω (xi , Ni , Λ) = φω′ (xω′ , Nω′ , Λ) = ∂xω′ ∂xω ∂xω′ ∂Λ ∂Λ
SC
∂Λ ∂φω′ (xω′ , Nω′ , Λ) ∂2Λ = ∂xω ∂xω′ ∂xω ∂Λ Denoting by
∂ ln [φω (xω Nω , Λ)] ∂Λ h i ∂ ln φ˜ω (xω Nω , Λ) ψ˜ω (xω , Nω , Λ) = ∂Λ
NU
ψω (xω , Nω , Λ) =
Λxω
MA
we get that
∂φω′ (xω′ , Nω′ , Λ) ∂φω (xω , Nω , Λ) = Λx ω ′ ∂Λ ∂Λ
Or equivalently
ED
ψω (xω , Nω , Λ) = ψω’ (xω′ , Nω′ , Λ)
(34)
PT
Applying the same reasoning using the cross-derivatives of Λ w.r.t. Nω and Nω′ , we get:
(35)
AC CE
ψ˜ω (xω , Nω , Λ) = ψ˜ω′ (xω′ , Nω′ , Λ)
When Ω ≥ 3, ω ′′ 6= ω and ω ′′ 6= ω ′ can be chosen and (34) can be differentiated with respect to xω and xω′′ . Under (A3), this implies that ψω does not depend upon xω . Now, differentiating (35) with respect to Nω and Nω′′ , we get: ∂ψω ∂ψω ∂ψω ˜ ∂ψω′ ˜ ∂ψω + + ΛN ω ≡ φω (xω , Nω , Λ) = φω (xω , Nω , Λ) ∂Nω ∂Λ ∂Nω ∂Λ ∂Λ ∂ψω′ ˜ ∂ψω ˜ φω” (xω′′ , Nω′′ , Λ) = φω′′ (xω′′ , Nω′′ , Λ) ∂Λ ∂Λ Similarly, under (A4) and Ω ≥ 3, ψω does not depend upon Nω either. Therefore, ln φω is additively separable in Λ and (xω , Nω ). This implies in turn that there exist functions fˆω and hω such that Λxω = φω (xω , Nω , Λ) = fˆω (xω , Nω )hω (Λ)
30
(36)
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˜ Similarly, there exist functions ˆfω and h˜ω such that ˜ ˜ ω (Λ) ΛNω = φ˜ω (xω , Nω , Λ) = ˆfω(xω , Nω )h
T
(37)
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Now substituting (36) into (34) yields
SC
fˆω (xω , Nω )hω (Λ(x,N))h′ω′ (Λ(x,N))fˆω′ (xω′ , Nω′ ) = fˆω′ (xω′ , Nω′ )hω′ (Λ(x,N))h′ω (Λ(x,N))fˆω (xω , Nω )
which can be simplified into
NU
hω (Λ(x,N))h′ω′ (Λ(x,N)) = hω′ (Λ(x,N))h′ω (Λ(x,N))
∀(x,N) ∈ R2
∀(x,N) ∈ R2
Ω+1
Ω+1
MA
This means that hω is a linear transformation of hω′ . Therefore, (36) can be rewritten as
φω (xω , Nω , Λ) = fω (xω , Nω )h1 (Λ)
∀ω ∈ [[1, Ω]]
(38)
∀ω ∈ [[1, Ω]]
(39)
PT
Similarly there exists f˜ω such that
ED
where fω is a linear transformation of fˆω .
˜ 1 (Λ) φ˜ω (xω , Nω , Λ) = f˜ω (xω , Nω )h
AC CE
Last, taking the derivative of (38) w.r.t. Nω and the derivative of (39) w.r.t. xω , we get: ∂ ˜ ∂ ˜ 1 (Λ) fω (xω , Nω )h1 (Λ) = fω (xω , Nω )h ∂Nω ∂xω
Under the additional regularity assumption (A4), (39) can be rewritten as follows φ˜ω (xω , Nω , Λ) = f˘ω (xω , Nω )h1 (Λ)
∀ω ∈ [[1, Ω]]
∂
fω (xω ,Nω ) where f˘ω (xω , Nω ) := f˜ω (xω , Nω ) ∂N∂ ω f˜ (x ,N ) . ∂xω
ω
ω
ω
Now, the end of the proof follows Cornes and Hartley (2012). Denote by
Γ(Λ) =
Z
Λ Λ1
1 dΛ′ h1 (Λ′ )
We write T (x, N) = Γ[Λ(x,N)] and we have ∂T (x, N) = Γ′ [Λ] Λxω = fω (xω , Nω ) ∂xω
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∂T (x, N) = Γ′ [Λ] ΛNω = f˘ω (xω , Nω ) ∂Nω So
∂ 2 T (x,N) ∂xω ∂xω′
=
∂ 2 T (x,N) ∂Nω ∂Nω′
=
∂ 2 T (x,N) ∂xω ∂Nω′ =0
for ω 6= ω ′
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T
T is therefore additively separable in (xω , Nω ) for all ω ∈[[1,Ω]] up to a monotonic transformation.
F Non-separable market aggregates: the case of Bertrand competition
SC
and linear demand
NU
Under Bertrand competition, the direct demand can be expressed as ˜ =Λ ˜− d(pi , Λ)
MA
˜ is a market aggregate defined as follows: where Λ
pi β
α γN p + β + γN β (β + γN )
(40)
ED
˜ := Λ
PT
The equilibrium value of the aggregate is determined in the same fashion through the free-entry condition for q β(f +F ) 1 ∗ ˜ small firms which leads to Λ = β 2 +c . L Rearranging the first-order conditions w.r.t. to product scope and price for MP-firms lead respectively to ˜∗ − Λ
AC CE
p∗mp β
p∗mp − c
= p∗mp − c
p∗mp β
˜∗ − Λ
β + γ (N ∗ − N ∗ ) β (β + γN ∗ )
β + γ (N ∗ − N ∗ ) =f β + γN ∗
(41)
So the equilibrium price of a variety supplied by a large firm is now given by p∗mp
=
p∗sp
+
r
β (f + F ) − L
r
βf L
where p∗sp is the price of a variety supplied by a small firm: p∗sp =
˜∗ + c βΛ 2
˜ is no longer additively separable in firms’ strategies and falls therefore outside the class of Note however that Λ ˜ ∗ , we preferences 5.1.. The entry of a large firm is no longer neutral on MP-firms’ product scope. By definition of Λ
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know that ∗ ˜ β − p∗sp (β + γN ∗ ) α − p∗sp β + γΩN ∗ p∗mp − p∗sp = Λ γN ∗ β+γN ∗
does not depend on Ω. Consequently, it must be that the breadth of the
T
Now from (41), we also know that
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product range of a N ∗ increases with Ω.
Now using (40) again, the entry of a large firm leads to a change in the product scope of MP-firms as well as in
SC
the mass of SP-firms:
˜ ∗ βγ (Ω∆N + N ∗ + ∆M ) γN ∗ p∗mp + γΩ∆N p∗mp + γ∆M p∗sp = Λ
NU
Rearraging the last expression shows that the mass of SP-firms that exit the market following the entry of a large firm is smaller than the increase in the overall number of varieties supplied by large firms:
1− 1−
p∗ mp ˜∗ βΛ
MA
p∗ sp ˜∗ βΛ
(Ω∆N + N ∗ ) = −∆M
ED
The overall mass of varieties N increases.
Now the change in consumer surplus following the entry of a large firm is still given by a change in the aggregate 0
x2ν dν. Using the same identities as above, one can show that the change in consumer surplus is negatively
PT
RN
AC CE
related to the mass of SP-firms:
∆CS ∝ 1 −
1− 1−
p∗ mp ˜∗ βΛ p∗ sp ˜∗ βΛ
∆M < 0
To sum-up: under Bertrand competition, MP-firms charge higher mark-ups, the exogenous entry of a foreign oligopolist leads to the exit of monopolistic firms and raises the average price. The total number of varieties increases but consumer surplus decreases.
G Additively Separable Preferences with an Additive Upper-Tier Utility Notation: I denote the elasticity of θ w.r.t. y by Eθ (y) :=
θ ′ (y)y θ(y) .
G.1 Consumer Surplus I aim at showing that when u(0) = 0, an increasing price-elasticity of demand (or equivalently a decreasing Eu′ ) implies Eu′ < 0. To do so, we use the two following results.
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(R1) Following the appendix of Kuhn and Vivès (1999), 1 − Eu (x) and −Eu′ (x) are equal at 0 and have the same monotonicity in its neighborhood. (R2) When Eu′ (x) < 0, we have 1 − Eu (x) < −Eu′ (x) while the reverse holds under Eu′ (x) > 0. For the
T
monotonicity of Eu to change at an interior point x ¯ it has to be that 1 − Eu (¯ x) = −Eu′ (¯ x).28
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Now, to prove our result, we proceed by contradiction. Let us assume that Eu′ is a decreasing function while ˆ such that Eu′ (ˆ Eu′ (x0 )′ > 0 holds for some x0 > 0. By continuity, (R1) implies that there exist x x) = 0 with
SC
x) with 1 − Eu (x) < −Eu′ (x) for any x < x ˆ and x) = −Eu′ (ˆ Eu′ ′ (x) < 0 for any x < x ˆ. Then (R2) implies 1 − Eu (ˆ x, x ˆ + ε[, which proves our result. that there exists ε > 0 so that 1 − Eu (x) > −Eu′ (x) and Eu (x)′ > 0 when x ∈]ˆ
NU
G.2 Social Welfare
MA
The following preferences are considered:
U (A, x) = A + U (x)
U (x) = W
where
!Z
V
u(xν )dν 0
"
ED
I have already shown that the change in consumer surplus is given by
PT
′ ∆CS = N ∗ u(x∗mp )W (Λ∗ ) Eu (x∗sp ) − Eu (x∗mp )
The change in total producer surplus following the entry of one large firm equals its profits so
that is
AC CE
′ ∆P S = N ∗ u′ (x∗mp )x∗mp W (Λ∗ ) − cx∗mp − f
∆P S = N
∗
′ u(x∗mp )W
∗
(Λ )
Eu (x∗mp )
cx∗mp + f − u(x∗mp )W ′ (Λ∗ )
cx∗mp + f − u(x∗mp )W ′ (Λ∗ )
Thus, the change in social welfare is:
∆SW = N
∗
′ u(x∗mp )W
∗
(Λ )
Eu (x∗sp )
The first-order condition w.r.t. N evaluated at a symmetric equilibrium is given by:
′′
(p∗mp − c)x∗mp + ΛN u′ (x∗mp )W (Λ∗ )N ∗ x∗mp = f so 28 This
is easily obtained by differentiating Eu (x).
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#
′′
′
u′ (x∗mp )W (Λ∗ )x∗mp + u(x∗mp )u′ (x∗mp )W (Λ∗ )N ∗ x∗mp ∆CS + ∆P S = N ∗ u(x∗mp )W (Λ∗ ) Eu (x∗sp ) − u(x∗mp )W ′ (Λ∗ )
$
T
′
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′ ∆CS + ∆P S = N ∗ u(x∗mp )W (Λ∗ ) Eu (x∗sp ) − Eu (x∗mp ) − EW ′ (x∗mp )
where EW ′ (x∗mp ) reflects the change in expenditure.
SC
This expression can be further simplified using the ratio of the first-order conditions for both firms:
so that the change in social welfare can be written:
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1 + Eu′ (x∗sp ) u′ (x∗mp ) = >1 u′ (x∗sp ) 1 + Eu′ (x∗sp ) + EW ′ (x∗mp )
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u′ (x∗sp ) ′ ∆CS + ∆P S = N ∗ u(x∗mp )W (Λ∗ ) Eu (x∗sp ) − Eu (x∗mp ) + 1 − Eu′ (x∗mp ) + 1 − Eu′ (x∗sp ) ′ ∗ u (xmp )
(42)
And I have already shown that x∗mp < x∗sp . This expression can be positive or negative. For instance, if
welfare.
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Social welfare: the case F = 0+
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u(x) = (1 + x)α − 1 with α < 1 and if F is not “too small”, import penetration by a foreign firm decreases social
Note that: ′
(Eu ) =
u′ [1 − Eu′ − Eu ] u ′
In the limit: x∗mp − x∗sp −→ 0. In the standard case where preference for variety is increasing, i.e. (Eu ) < 0, F →0 ≥
(42) implies that ∆CS + ∆P S > 0.
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S0022-1996(17)30116-2 doi:10.1016/j.jinteco.2017.09.001 INEC 3081
To appear in:
Journal of International Economics
Received date: Revised date: Accepted date:
9 July 2016 11 September 2017 11 September 2017
Please cite this article as: Parenti, Mathieu, Large and small firms in a global market: David vs. Goliath, Journal of International Economics (2017), doi:10.1016/j.jinteco.2017.09.001
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Mathieu Parenti†
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Large and small firms in a global market: David vs. Goliath∗
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Abstract
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This version: September 2017
This paper studies the impact of trade liberalization when large and small firms coexist in the same market. I develop a model of imperfect competition where a few oligopolistic firms coexist with a monopolistically competitive fringe. The
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former have a direct impact on the industry average price level, which depends on the breadth of their product scope. Oligopolists produce higher output per worker, charge higher markups, but also sell at higher prices than monopolistic single-product firms. Under linear demand, trade liberalization triggers the exit of small firms and the entry of foreign
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large firms, leading to more product variety but also to a higher average price. For a broad class of demand functions, the entry of an oligopolist induced by trade liberalization has a negative impact on consumer surplus. Producer surplus rises, but the total effect on welfare is ambiguous.
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JEL Classifications: D4, L10, F11
Keywords: Monopolistic Competition, Oligopolistic Market Structure, Multi-product firms, International
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Trade.
∗ I am very grateful to Lionel Fontagné, Peter Neary and Jacques Thisse for many challeging interactions on this topic. I would also like to thank Aya Ahmed, Claude d’Aspremont, Kristian Behrens, Paola Conconi, Arnaud Costinot, Mathieu Couttenier, Meredith Crowley, Matthieu Crozet, Ron Davies, Don Davis, Swati Dhingra, Francesco Di Comite, Jonathan Dingle, Martin Hellwig, Gianmarco Ottaviano, Jean Imbs, Sergey Kokovin, Sebastien Krautheim, Helene Latzer, Florian Mayneris, Julien Martin, Thierry Mayer, Joan Monras, John Morrow, Volker Nocke, Vincent Rebeyrol, Nicolas Roux, Nicolas Schmitt, Nicholas Schutz, Farid Toubal, Federico Trionfetti, Thierry Verdier, Helene Windish, Stephen Yeaple as well as two anonymous referees for their helpful comments and suggestions. † ECARES (Université Libre de Bruxelles) - C.E.P.R.. E-mail: [email protected].
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1
Introduction
It is well documented that firms are heterogeneous within an industry. Although the exact shape of this heterogeneity
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is still debated, it is widely acknowledged that the firm size distribution is highly skewed, featuring a large number
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of small firms and a handful of large firms (Axtell, 2001). This implies that the aggregate trade patterns of an industry can be shaped by the individual behavior of a few firms. For instance, Bernard et al. (2007) report that 96% of US exports in 2000 were made up by 0.4% of US firms.1
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In the theoretical literature, however, the assumption of a continuum of firms implies that the actions of a large firm are inconsequential. In this paper, I follow a different approach and assume that those happy few large firms
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behave differently in the marketplace. I rule out differences in marginal costs of production, focusing on differences in individual firm behavior and how they shape the impact of trade liberalization. To be precise, I build a model of a “mixed” market structure in which a few multi-product oligopolists and a
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continuum of single-product monopolistically competitive firms coexist. In doing so, I depart from the standard models of monopolistic competition along two dimensions.
First, I assume away the Chamberlinian large-group assumption, considering a setting in which a few large firms
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behave strategically while competing with a large number of monopolistically competitive firms. Large firms have a direct impact on the market outcome while a large number (formally, a continuum) of negligible monopolistically
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competitive firms take the market aggregates (total output here) as given. The second assumption I depart from is the free-entry condition. To capture the persistence of a few large
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firms over time,2 I follow Friedman (1993) for whom free entry is “untenable for oligopolistic markets”. The setting features an exogenous number of potentially large firms which may earn positive profits in equilibrium. On the contrary, monopolistically competitive firms adjust to the intensity of competition through entry and exit until their profits are driven down to zero. Compared to a standard oligopolistic market structure, the present model takes the importance of the turnover of small firms into account. Starting with a closed economy, I take a step towards understanding the endogenous formation of large firms. Heterogeneity in productive efficiency emerges as a necessary condition for both large and small firms to coexist in equilibrium. I assume therefore that large firms benefit from economies of scope which allows them to generate more physical output per worker. How do the pricing strategies of large and small firms differ? I show that even if the marginal cost of production of a variety is the same for both kinds of firms, large firms charge higher markups. This is because large firms have a strategic market power: under linear demand and Cournot competition, they internalize the impact of their production on total output. On the contrary, the market power of a small firm is solely monopolistic: each variety 1 See also Mayer and Ottaviano (2008), Berthou and Fontagné (2013), Bernard et al. (2007) and Manova and Zhang (2009) for respectively Belgium, France, the U.S.A. and China. Using firm-level data on 32 countries, Freund and Pierola (2015) find that the export superstars “critically shape trade patterns”. Importantly enough, these patterns hold both for the manufacturing and the services sector (Crozet et al. 2010 and Breinlich and Criscuolo 2011). 2 Many papers have found that firm survival, firm size and firm age are positively correlated; see, e.g., Thompson (2005).
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is an imperfect substitute for its competitors’. To assess how these differences matter in a global market, I consider an open economy and carry out three exercises. First, I analyze the impact of the entry of a large (foreign) firm. The increase in the competitive pressure
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exerted by this entrant leads to the exit of small firms. The average price rises through the proliferation of new
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higher-priced varieties at the expense of consumer welfare.
Second, I consider the free trade benchmark when many identical countries - i.e., with the same number of
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oligopolists and the same population size - trade with each other. The entry of large foreign firms fosters the exit of small firms again. Similarly to Krugman (1979) however, the access to foreign markets makes the monopolistically competitive fringe more competitive, both through the intensive margin (firms increase their scale, thereby decreas-
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ing their average cost) and the extensive margin (new firms enter the market). Under this scenario, consumers benefit unambiguously from trade openness.
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Third, I show that the presence of trade frictions, including variable and fixed costs to reach the foreign market, alters these predictions significantly. A drop in variable trade costs increases import penetration, but in the presence of fixed costs, small firms cannot compensate for their domestic losses with their exports to the foreign market, as
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they can in the free trade scenario. Their scale and markup remain unchanged and the size of the monopolistically competitive fringe shrinks. By contrast, a large firm expands its market share abroad, hence increasing its market power. In contrast with Brander (1981), bilateral trade liberalization leads to a monotonic decrease in consumer
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welfare and a monotonic increase in producer surplus. Central to these results is the anti-competitive effect following the entry of a large firm considered in the first
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exercise. I show that this effect holds for a large class of preferences. Specifically, the prediction holds under the class of additively separable preferences featuring an increasing preference for variety as in Kuhn and Vives (1999). The remainder of the paper is organized as follows. Section 2 briefly describes the related literature. Section 3 presents the baseline model and determines the conditions under which both large and small firms coexist in equilibrium. Section 4 deals with the properties of the market outcome. Section 5 determines the effects of trade liberalization on consumer welfare. In Section 6, the robustness of the main findings against alternative specifications of preferences is discussed. Section 7 concludes.
2
Related Literature
This paper lies at the intersection of two strands of literature. The first one deals with international trade under oligopoly and the seminal contributions of Dixit and Norman (1980), Brander (1981) and Markusen (1981). In contrast with the industrial organization literature, strategic interactions have not received much attention from trade economists. A reason which Neary (2010, 2016) has pointed out to explain the theoretical bias against oligopoly, favoring monopolistic competition is the lack of tractability of the former in general equilibrium (Roberts
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and Sonnenschein, 1977). However, Neary (2016) shows that the problem becomes tractable when considering a market structure in which firms are “large in the small but small in the large,” meaning that each firm impacts competition in its own industry, but not consumer income which is determined at the economy-wide level. Instead,
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I stick to partial equilibrium here, as the focus is placed on the differences in firms’ market behavior within an
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industry.
More recently, observing that large firms “depart substantially” from the monopolistic competition benchmark
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(Hottman et al., 2016), a few papers have considered a model featuring heterogeneous firms with non-zero mass as in Bernard et al. (2016). In their model, differences in firm size stem from differences in their marginal cost of production. Therefore, quality differences put aside, a firm’s market power always translates into a higher markup
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and a lower price. On the contrary, a large firm in my model not only charges a higher markup over marginal cost, it also sells at a higher price. This distinction allows me to isolate the importance of firm’s strategic market
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power when discussing the distribution of the gains from trade. The main insight is that the source of firms’ market power matters for consumer surplus. This echoes also Eckel and Yeaple (2016) who find that the source of firms’ performance may alter the welfare conclusions following the reallocation of labor towards larger firms.
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This paper also builds on the vast literature in industrial organization combining large and small firms within the same setting, starting from the dominant firm model developed by Markham (1951). This approach has also been applied to vertical relationships by Kuhn and Vives (1999) in a setting where large upstream firms trade with
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small downstream firms. Yet, the coexistence of both types of behavior in a given industry has been neglected. In the conclusion of his celebrated Chamberlinian model of monopolistic competition, Hart (1985) explains the need
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to invoke oligopoly theory to study a market which “contains a mix of significant and negligible firms.” To the best of my knowledge, the first paper to deal with competition between big, strategic firms, and small, monopolistically competitive firms is Shimomura and Thisse (2012). Besides the motivation for the study of trade liberalization, the closed-economy section of my model also differs from Shimomura and Thisse (2012) in several respects. The first one concerns the interpretation of large and small firms. Assuming large single-product firms is hard to reconcile with the idea that a consumer displaying a love for variety smooths its consumption across all varieties. Instead, this paper considers that each large firm supplies a continuum of varieties. An appealing feature of this approach is the endogenous determination of the product range so firms choose to become large or not in equilibrium. Second, the present paper uncovers the free-riding of non-strategic small firms on large firms. This explains why large firms must necessarily be more efficient than small firms to coexist with them in equilibrium. Third, the paper disentangles the implications of the assumptions made in Shimomura and Thisse (2012) about the market structure. For instance, the fact that large firms charge a higher markup than small firms survives to a series of alternative specifications of demand. On the contrary, the result according to which the entry of a large firm is necessary beneficial to consumers does not. For a large class of preferences, consumer surplus always decreases with the entry of a large firm while it remains constant for CES preferences. Instead, Shimomura and Thisse (2012)’s
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result is driven by the assumption that firms’ profits feedback into consumers’ demand for differentiated varieties.
The Baseline Model
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3
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The economy involves an homogeneous good produced under constant returns and perfect competition, a horizontally differentiated good produced under increasing returns and monopolistic competition, and one production factor, labor. The differentiated sector involves both a discrete number Ω of multi-product (MP) firms ω = 1 . . . Ω,
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which decide on the breadth of their product scope Nω ≥ 0, as well as a continuum i ∈ [0, M ] of single-product
Preferences
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3.1
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(SP) firms. In what follows, I refer to Nω as the mass of firm ω. Thus, the total mass of varieties provided by this PΩ sector is equal N := ω=1 Nω + M .
On the demand side, there is a continuum of L identical consumers sharing the same preferences. Denote by xωk the individual consumption of variety k produced by the MP-firm ω, by xi the consumption of variety i produced
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by the SP-firm i, and by X the aggregate consumption of all varieties by one consumer: Z
N
xν dν =
0
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X :=
Ω Z X
ω=1
Nω
xωk dk + 0
Z
M
xi di. 0
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The utility of consuming the differentiated good is given by the following quadratic utility function:3
U (¯ x) :=
Z
N
0
β αxν − x2ν 2
γ dν − 2
Z
N
xν dν 0
!2
.
(1)
¯ := xν≤N is the consumption profile and α, β and γ positive parameters. The upper-tier utility function where x is quasi-linear:
¯ ) := A + U (¯ U (A, x x) where A is the consumption of the homogeneous good; it is used as a numéraire. An increase in α or a decrease in γ shifts expenditure away from the numéraire good towards the differentiated good. Consumers’ love for variety is captured by β. When β = 0, consumers care about the overall consumption level X only. The utility function is symmetric (i.e., renumbering varieties has no impact on the utility level) so at a symmetric consumption profile, whether a variety is produced by a MP- or SP-firm is irrelevant from the consumer’s perspective. 3 This functional form has been used in Industrial Organization by Dixit (1979) and Singh and Vives (1984) over a finite a number of varieties. For applications to a continuum of goods see Ottaviano et al. (2002) and Melitz and Ottaviano (2008).
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Each consumer supplies one unit of labor inelastically. Thus, she faces the following budget constraint:
0 1 L
pν xν dν ≤ R
(2)
of each firm profit.4
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where R is equal to her wage income plus a fraction
N
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A+
Z
Under the assumption of positive consumption of the homogenous good A > 0, the inverse demand functions
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for varieties i and k are
=
p(xi , X ) := α − βxi − γX
(3)
pωk
=
p(xωk , X ) := α − βxωk − γX .
(4)
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pi
The variable X has the nature of a market aggregate, which can be interpreted as a proxy for the intensity of
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competition as it shifts up or down the demand schedule.5 For any given X , the price elasticity increases with the price along the demand curve, as in Krugman (1979).
Firms
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3.2
In order to isolate differences in market power between large and small firms I assume that they share the same
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constant marginal cost c. Two types of fixed costs are introduced. There is an overhead cost F > 0 to be paid before starting production, which is incurred by all firms. Following a widely-used simplification in the multi-product firm
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literature,6 adding a new variety requires an additional fixed cost f > 0. Therefore, the total fixed costs incurred by an SP-firm are F + f . On the other hand, an MP firms of mass N > 0 incurs a fixed cost equal to N f . If this firm were a single-product firm (N = 0), its fixed cost would be F + f . That F > 0 implies that there are scope economies.7 In what follows, all firms are assumed to be quantity-setters. A multi-product firm ω understands that its decision has a non-negligible impact on the economy because Nω > 0. Specifically, each MP-firm behaves like an oligopolistic firm that manipulates the market aggregate X . By contrast, because any particular SP-firm is negligible to the market, its decision to enter the market and to produce a certain output, or to exit the market, has no impact on the economy. As a consequence, a SP-firm may accurately treat X as a parameter. Since the inverse demands (3) and (4) depend only upon a firm’s own strategy and the market aggregate X , the 4 This
assumption involves no loss of generality. Since preferences are quasi-linear, firms’ ownership does not play any role in this model. 5 Alternatively, X can be expressed as a function of the average price and the overall mass of varieties N (see Melitz and Ottaviano, 2008). 6 See Baldwin and Gu (2009), Dhingra (2013), Feenstra and Ma (2008), Minniti and Turino (2013), Ottaviano and Thisse (2011). Admittedly, it is a crude simplification from reality, but it should be clear that product adjustments within multi-product firms are not of direct interest here. Instead, the focus is placed on firm heterogeneity. See Eckel and Neary (2010) and Mayer et al. (2014) for models of multi-product firms with a core competence. R 7 Formally, the fixed costs incurred by an MP-firm with N > 0 amount to Nω (f + 1 ω k=0 F ) dk. 0
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profits earned by an MP-firm are given by Nω
[p(xωk , X ) − c] xωk dk − Nω f
while the profits made by a SP-firm are as follows:
(5)
i ∈ [0, M ]
(6)
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πi (xi , X ) := L · [p(xi , X ) − c] xi − (F + f ) where M is the number of operating SP-firms.
The Game
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3.3
ω = 1, ..., Ω
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0
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Πω (xωk , Nω , X ) := L ·
Z
The number of MP-firms is exogenously given, but their respective mass is endogenous. On the other hand, each
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SP-firm is negligible, but their number is determined by free entry and exit. The MP-firms choose their breadth and outputs and the SP-firms choose to enter the market and their outputs simultaneously. As shown by (5) and (6), the game is aggregative in nature where X is the aggregate i.e. it is one-dimensional
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and additively separable in the strategies of the players. A Nash equilibrium is then defined by x∗ωk , Nω∗ , x∗i , M ∗ ,
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and X ∗ such that (up to a zero-measure set of SP-firms) the following conditions hold: Πω (x∗ωk , Nω∗ , X ∗ ) P
R N ∗′
x∗ω′ k dk +
R M∗
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∗ ≡ where X−ω
≥
∗ Πω (xωk , Nω , X−ω
ω ′ 6=ω
ω
0
0
+
Z
Nω
xωk dk) 0
for all xωk ≥ 0 and Nω > 0
x∗i di.
πi (x∗i , X ∗ ) ≥ πi (xi , X ∗ )
for all xi > 0 and i ∈ [0, M ∗ ]
πi (xi , X ∗ ) < 0 for all xi > 0 and i > M ∗
Both types of firms coexist if Nω∗ > 0 for at least one MP-firm and M ∗ > 0. I show below that there exists a unique equilibrium featuring both types of firms, which is symmetric within each type and features symmetry within MP-firms’ product lines.
4 4.1
The Market Outcome Equilibrium
Single-product firms A SP-firm i faces a downward-sloping inverse demand (3) from each consumer. Maximizing its operating profit with respect to xi and treating X as a parameter yields this firm’s best reply as a function of 7
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X only. Since SP-firms share the same marginal cost, (7) implies that all active SP-firms produce the same output ∗ qsp (X ) := L · x∗sp where
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α − c − γX 2β
(7)
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x∗sp (X ) :=
Multi-product firms A multi-product firm ω chooses simultaneously its product scope Nω from an exclusive continuum of varieties and the output of all its varieties xωk for k ≤ Nω .
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The symmetry of profits and the strict concavity of per-variety profits given total output implies that the choice of an MP-firm boils down to choosing the breadth of its product range Nω and the common level of output per
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variety xω (see Appendix A.1). In this event, an MP-firm maximizes its profits
max Nω [(p(xω , X ) − c) L · xω − f ] .
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Nω ,xω
(8)
The first-order8 condition with respect to xω is given by
(9)
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α − c − (2β + γNω ) xω − γX = 0.
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which leads to total output Qω = L · Xω of firm ω where: α − c − γX 2β + γNω
with density x∗ω :=
α − c − γX . 2β + γNω
(10)
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Xω := Nω
Note that x∗ω = x∗sp when Nω = 0. I assume for the moment that α − c − γX > 0. From (10), the relationship between an MP-firm’s total output and its mass, given X , is an increasing concave function. In other words, were two MP-firms with different product breadth to coexist at equilibrium, the one with the larger product breadth would have a larger, yet less than proportional total output. This is a direct consequence of the cannibalization effect. As emphasized by Baldwin and Gu (2006) , Feenstra and Ma (2008), Eckel and Neary (2010), Ottaviano and Thisse (2010) and Minniti and Turino (2013), multi-product firms internalize demand linkages between varieties. When γ = 0, Xω becomes linear in Nω because there is no linkage effect anymore. Turning next to the determination of the product breadth, we get
(α − c − βxω − γNω xω − γX ) L · xω = f.
(11)
This expression shows that an MP-firm faces the following trade-off. The marginal profit gained by introducing an additional variety is a decreasing function of Nω , i.e., its operating profits are a concave function of Nω . Profits 8 I determine the candidate global maximizer of this function by inspecting of the first-order conditions with respect to x and N . ω ω In Appendix A.2, I show that the Hessian matrix at (9) and (11) is negative definite and I derive the condition under which this local maximum is also global.
8
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are maximized when the marginal operating profit equals the fixed cost per variety, as shown by (11). Solving (9) for Nω and substituting into (11), we obtain the common per-variety output of an active MP-firm
:=
Lx∗ω
=
s
fL β
T
:=
Lx∗mp
(12)
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∗ qmp
Repeating the same argument shows that N ∗ can be expressed as a function of X only: s
βL f
α − c − 2β
s
f − γX Lβ
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1 N (X ) = γ ∗
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which is positive as long as
α − c − γX − 2
r
βf L
>
!
(13)
0.
equilibrium if and only if
1 4β
2
(α − c − γX ) ≥
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Last I solve for X ∗ using the free-entry condition for small firms. Plugging (7) into (6), SP-firms produce at f +F L
or equivalently
ED
α−c F (f + F ) ≥ 0 where F(y) := q −1 2 βy L
(14)
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At equilibrium, the equality holds so that
=0
with
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∗ (x∗sp , X ∗ ) πsp
x∗sp (X ∗ )
=
s
f +F βL
(15)
When F > 0, condition (14) implies that the inequality (13) holds. This means that when they benefit from economies of scope, each MP-firm finds it optimal to be large at equilibrium in the presence of a monopolistically competitive fringe.9 On the contrary, when F = 0 we have F(f + 0) = 0 and (13) implies that N ∗ = 0. In this case MP-firms choose to be negligible in the presence of a monopolistically competitive fringe and the market structure is characterized by monopolistic competition. It is worth noting that the condition F > 0 uncovers a free-rider problem: SP-firms free-ride off the strategic behavior of MP-firms.10 Since an MP-firm internalizes that its varieties cannibalize each other, it charges higher prices and sells less of each variety to relax competition within its product line. Everything else being equal, the strategic behavior of an MP-firm increases the average price for the differentiated good, which, in turn, relaxes competition for all the other firms belonging to the competitive fringe. Now, this positive externality imposes that there exists some further asymmetry between MP and SP-firms for them to coexist in equilibrium.11 This is true, 9 Note that this implies that a strict subset of the Ω firms being active in the presence of a monopolistically competitive fringe is not an equilibrium. 10 This is a positive externality, however the “free-riding” term has been extensively used in the literature dealing with the merger paradox (Salant et al., 1983). 11 This result justifies the title chosen for the present paper: unless there are asymmetries in efficiency, large firms can not coexist
9
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Figure 1: Typology of the market structure according to (F, Ω)
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for instance, when MP-firms benefit from economies of scope F > 0. To sum-up, in the event where F > 0 the
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presence of a monopolistically competitive fringe ensures the existence of MP-firms.
To derive the conditions under which there exists a monopolistically competitive fringe, I use the identity ∗
∗
or equivalently F < F (Ω) :=
F(f ) 1+ Ω+1
2
!
−1 f
(16)
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¯ := F(fq+ F ) Ω<Ω f 1 − f +F
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ΩN ∗ xmp (X ∗ ) + M ∗ xsp (X ∗ ) = X ∗ so that M ∗ > 0 if and only if
Conditions (16) states that the market must be large enough (F is large for high values of α or L) and the number of big firms Ω not too large for the competitive fringe to exist in equilibrium. Note that under the event that F = 0,
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¯ = +∞ i.e. the presence of MP-firms is immaterial as they choose to be negligible. we have Ω The equilibrium values for the mass of SP-firms and the product range of MP-firms can be simplified as follows:
1−
s
ED
" 2β F(f + F ) − M ∗ (Ω) = γ
f f +F
! # Ω
2β N∗ = γ
s
! F 1+ −1 . f
(17)
PT
Last, I have assumed throughout that there is positive consumption of the homogeneous good at equilibrium. This will be true when γ is large enough (see Appendix B for details), a condition that is assumed to hold throughout. 0 < F < F¯ . There exists a unique equilibrium featuring both types of firms which
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Proposition 1. Assume that is symmetric within each type. Under
0 < F < F¯ the variables M ∗ , N ∗ , x∗mp , x∗sp are uniquely defined and strictly positive, which proves
proposition 1.
The pair (F, Ω) defines a market structure that evolves from monopolistic competition, to a mixed market structure, and then on to oligopolistic competition, as represented in Figure 1. Figure 1 shows that for a given number of oligopolists Ω, when barriers to entry are prohibitive (F ≥ F¯ ) the market structure is purely oligopolistic. Likewise, for given barriers to entry, if oligopolistic incumbents are too many, small firms do not enter and pure oligopoly is preserved.
4.2
Prices and Markups: a Tale of Two Market Powers.
From an aggregative-game perspective, the inverse demand for any variety p(x, X ) can be written as a function of each firm’s decision variable x and the aggregate statistic X . Applying the standard Lerner pricing rule shows that a firm’s markup reflects two kinds of market power: with a monopolistically competitive fringe.
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p−c = Ex (p) + Ex (X )EX (p) p
T
where Ex (g) denotes the elasticity of a function g w.r.t. x.
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The first term on the right-hand side describes a firm’s monopolistic market power, i.e. the fact that each variety is perceived as an imperfect substitute for other varieties. On the contrary, the second term describes the fact that large firms internalize their impact on the market aggregate X , it is a strategic source of market power. It is a
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combination of two terms: Ex (X ) is the extent to which a firm alone impacts the market aggregate X , while EX (p) captures the impact of the aggregate on the price p.
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In the case of monopolistic competition, firms are negligible, so that Ex (X ) = 0 and only the first term is non-zero. On the contrary, in Cournot competition with a homogenous good (β = 0), the market power of firms stems from strategic interactions only, so that Ex (p) = 0.
12
MA
The market power of small firms is solely monopolistic, while large firms have both types of market power. Combining (3) with (7) and (9) respectively yield a direct comparison of the pricing behavior of SP and MP
p∗sp − c =
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firms:
1 (α − γX ∗ − c) 2
p∗mp − c =
1 ∗ (α + γXmp − γX ∗ − c) 2
(18)
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Proposition 2. MP-firms charge higher markups than SP-firms.
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This contrasts sharply with monopolistic competition models for two reasons. First, for monopolistically competitive firms that have the same marginal costs to charge different markups, it would be necessary to assume that consumers’ willingness to pay for their product is different (see Di Comite et al. 2014). Here instead, a consumer’s willingness to pay for a variety does not depend on whether it has been produced by an MP-firm or an SP-firm. Consequently, since a higher price does not reflect a higher (perceived) quality, a reallocation of market shares towards the largest firms may be detrimental to consumer welfare. Second, this proposition derived under linear demand is general and holds true whether the price-elasticity of demand is increasing, constant or even decreasing (see Section 6). The simple fact that large firms strategically manipulate the aggregate X implies that, everything else being equal, larger firms charge higher markups. The same result would also hold in a model with single-product firms having a positive mass as in Bernard et al. (2016). Last, it is noteworthy that not only MP-firms have more market power, but they also produce more physical output (and thereby a larger value-added) per worker.13 In this sense, their production is more efficient than SP-firms’. 12 Note
that if large fims produce a homogenous good, then Ex (X ) = mass becomes irrelevant, as long as it remains positive. f 13 This can be directly observed since c + < c + fq+F ∗ . q ∗ mp
sp
11
α−c−γX γX
> 0 does not depend on Nω . In this case, a large firm’s
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5
Open Economy
Turning to the open-economy case, I assume that the homogeneous good is freely traded between countries so that
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changes in the goods market do not affect the relative wage between countries.
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In my model, international trade may increase market size for domestic firms if they start exporting but also increase the competitive pressure on their domestic market. This increase in competition may result from the entry of either SP or MP-firms. Since the impact of SP-firms entry in models of monopolistic competition is well
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understood, the first scenario below focuses on the impact of the costless entry of a large (foreign) MP-firm. The second exercise is the free-trade benchmark when all firms can export costlessly to all countries. Last, I introduce
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trade frictions between two symmetric countries and explore the impact of a reduction in trade costs when only the largest firms export.
The Impact of the Entry of a Large Foreign Firm
MA
5.1
ED
I start by considering the exogenous and costless entry of a large foreign firm with the same characteristics.14 I $ ¯ + 1 so that both types of firms continue to coexist in equilibrium after the entry of an assume that F < F¯ Ω MP-firm.
We know from (17) that the entry of a large firm leads to the exit of small firms. However, it has no impact on
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the market aggregate X ∗ . This is because the mass of the competitive fringe adjusts so that the free entry condition holds. Therefore the breadth of the product range given by equation (17) does not depend on Ω. In other words, a
firms.
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mixed-market structure puts the emphasis on the persistence of large firms despite the important turnover of small
Since X ∗ is stabilized by the free-entry condition, the impact on producer surplus is given by 4βf ∆P S := Π∗mp (x∗mp , N ∗ , X ∗ ) = γ
s
F 1+ −1 f
!2
(19)
The impact on consumer welfare however is less straightforward. Using that X ∗ does not change following the entry of the oligopolist, it can be shown that the overall number of varieties has increased:
∆N = N
∗
1−
s
f f +F
!
>0
This variety effect is enabled because an MP-firm internalizes the consumers’ love for variety. Thus, a multiproduct firm trades off a smaller output of each of its varieties against a broader product range. Since these varieties are supplied at a higher price, we have the following result: 14 Assuming
that the foreign oligopolist faces trade costs does not change the propositions below (see also 4.3).
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Proposition 3. As long as both types of firms exist in equilibrium, the entry of a foreign oligopolist increases the total number of varieties but raises the average price.
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All varieties produced by SP-firms which disappear are replaced with higher-priced (proposition 2) from the
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new oligopolist. The average price unambiguously increases with Ω through a composition effect. To conclude on the net effect of firm entry on consumer surplus, I re-express it as follows: β CS := L 2
Z
N
x2ν dν
γ + X2 2
#
SC
"
0
Contrary to producer surplus, two aggregates are necessary to describe consumer surplus as consumers care
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about both the total consumption level X and its composition since they have a love for variety. Since I have shown that X ∗ , x∗mp , x∗sp and N ∗ do not depend on Ω, the change in consumer surplus ∆CS following the entry of one
MA
large firm is constant, negative, and equal to:
!2 F 1+ −1 . f
s
(20)
ED
βf ∆CS := − γ
Proposition 4. Consumer (producer) surplus decreases (increases) with the entry of a foreign oligopolist.
PT
The above proposition states that the variety effect on consumer welfare is dominated by the increase in the average price. I show in section 6 that this is not an artefact of the assumed quadratic utility but holds generally
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for standard alternatives to CES preferences. Under this particular specification of preferences, summing (19) and (20) shows that the entry of a foreign large firm increases social welfare:15
3βf ∆SW := γ
s
F 1+ −1 f
!2
The consumers’ expenditures shift from the homogenous to the differentiated good, while at the same time the level of utility increases. The robustness of this result is discussed in Section 6.
5.2
The Free-Trade Benchmark
I assume here that κ symmetric countries are freely trading with each other. Following Eckel and Neary (2010), the openness to trade of each country is then measured by κ. Compared to the previous subsection, increasing κ also leads to an increase in market size. I assume that (16) is fulfilled for the total market size κL and the total number of oligopolists κΩ.16 15 This corresponds to the sum of social welfare worldwide, including the profits of the foreign firm. Were its profits entirely repatriated to its origin country, the change in social welfare of the country where the large firm enters boils down to the negative variation of consumer surplus. 16 Formally,
equation (16) becomes Ω <
F f +F 1 qκ f κ 1− f +F
.
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Augmenting market size and the number of oligopolists by a factor κ in (14) and (17), the mass of monopolistic firms in each country can be derived: f f +F
!
κΩ
#
T
1−
s
(21)
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" 2β f +F ∗ F − M (κ) = γκ κ
Setting Ω = 0 in (21) leads to Krugman (1979)’s well-known result: under an increase in market size, monopo-
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listically competitive firms enjoy economies of scale and consumers spread their consumption over a large set of
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varieties, thereby increasing the elasticity of demand so that firms reduce their markups. The number of varieties f +F F provided in each country decreases, but consumers enjoy a higher number of varieties overall: κM ∗ = 2β · γ κ
In other words, the mass of SP-firms is a concave and increasing function of κ.
MA
On the contrary, given Ω ≥ 1, the openness to trade also increases the number of large firms κΩ which triggers the exit of small firms as shown in 4.1.
ED
These two opposite effects lead to an inverted U-shaped curve for κM : the market size effect itself decreases in is a concave function of κ), while the increase in κ (consumers value new varieties less and less, so that F f +F κ q F the number of large firms is constant and proportional to 1 − f +F Ω, so that it eventually dominates. When
economies of scope are large for MP-firms (high F ), the latter effect is stronger. Regarding the total number
PT
of varieties, Proposition 3 guarantees that the overall number of varieties (SP- and MP-firms altogether) is an
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increasing function of the number of countries despite the exit of small firms. Pro-competitive effects While the entry of foreign oligopolists on the domestic market would raise prices, the increase in market size has a pro-competitive effect. Indeed, because SP-firms maintain their average-cost pricing rule, a higher market size translates into economies of scale, which leads to lower prices. This pro-competitive effect also operates for large firms: equation (12) implies that an increase in market size increases the output of large firms, yet their profit remains constant by (19). Thus, large firms have reduced their markup. To sum up, we have: Proposition 5. As long as both types of firms exist in equilibrium, an increase in the number of trading partners under free trade decreases the prices of all firms and increases the number of varieties available to consumers. Welfare Using equation (19), we know that changes in social welfare are driven by changes in consumer surplus only. While pro-competitive effects are typically welfare-enhancing, the fact that SP-firms exit the market increases the share of overall consumption devoted to the higher-priced varieties supplied by MP-firms. Under free trade, this anti-competitive selection effect is more than offset by the pro-competitive effect (see Appendix C). In this section, I have shown that free trade opens the domestic market to foreign exporters, which triggers the exit of domestic small firms. In the absence of fixed costs to export, trade has a pro-competitive effect. Free trade allows small firms to export: in doing so, they maintain a tough competitive environment for big firms. As shown 14
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in the next section, there is a sharp contrast when firms self-select in the exporting activity in the presence of fixed costs.
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Reduction in Trade Costs
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5.3
I now assume that, in order to export a product to the foreign market, a firm must incur variable and fixed costs. Compared to the previous scenario, trade frictions imply that small and large firms will respond differently to a
SC
reduction in trade costs. In the spirit of Melitz (2003), I consider that the foreign market has the same size as the domestic market, the same distribution of firms (i.e. the same number of large firms Ω), and symmetric trade costs. Last, I assume that the condition for both types of firms to co-exist holds for any τ i.e. F < F¯ (2Ω).
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The variable component of trade costs here is a per-unit additive trade cost τ , so that the marginal cost of
5.3.1
MA
exporting a variety is c + τ . Comparison of MP- and SP-Firms
ED
The inverse demand can be written again for any variety sold on the domestic market in quantity x to each consumer:
p(x, X ) = α − βx − γX
AC CE
By definition, it is equal to:
PT
where X is, by symmetry, the per-capita aggregated consumption across products in any of the two countries.
H H F F F F X = ΩN H xH mp + M xsp + ΩN xmp + M xsp | {z } Imported
varieties
The upperscripts H and F stand respectively for home and foreign. Production
The foreign and domestic markets are segmented. Both small and large firms have a constant marginal cost of production. Furthermore, large firms have a constant marginal cost of adding a product, so both SP- and MP-firms make their decision independently on each market. The profits of a multi-product firm (based in Home) on the domestic and foreign markets are then:
H H ΠH mp (xmp , N , X ) := L ·
F F ΠF mp (xmp , N , X ; τ ) := L ·
Z
Z
NH H ((p(xH mp , X ) − c)xmp − f )dk
0
NF 0
F ((p(xF mp , X ) − c − τ )xmp − f )dk
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In the same way, the profits of small firms are given by:
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H H H (xH πsp sp , X ; τ ) := ((p(xsp , X ) − c)Lxsp − F − f
F F F πsp (xF sp , X ; τ ) := ((p(xsp , X ) − c − τ )Lxsp − Fx − f
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Fixed costs
To export, all firms face an overhead fixed cost Fx . This fixed cost can be interpreted as in the standard Melitz
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model: firms need to establish a network or more generally look for buyers on the foreign market. Any variety exported requires an additional cost which can be interpreted as a packaging cost, or more generally as any other
MA
market-product-specific fixed cost. This cost is assumed to be equal to f only for the sake of simplicity.17 Again, any form of heterogeneity within MP-firms is ruled out.
Turning to SP-firms, they need to incur a total fixed cost equal to f + Fx in order to export their product. Large
ED
firms spread Fx on a continuum of varieties which they supply, so that the exporting activity fosters economies of scope. At the margin, it is easier for them to export a new product than it is for an SP-firm. The cost of introducing
of finite mass N F ∗ f .
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a new product is f for an MP-firm, and f + Fx for an SP-firm. Each large firm, however, incurs a total fixed cost
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Prices, Markups and Output in Home and Foreign Taking the equilibrium value of X ∗ as given for now, cross-section comparisons between firms are derived, assuming both MP- and SP-firms export. Considering the first-order conditions of large firms respectively w.r.t. xH mp and ∗H ∗ ∗F ∗ xF mp , their total output Qmp (X ) resp. Qmp (X ) supplied on the domestic market can be derived:
∗ H Q∗H mp (X ) = L · N
α − c − γX ∗ 2β + γN H
∗ F Q∗F mp (X ; τ ) = L · N
∗H (X ∗ ) = L qmp
α − c − γX ∗ 2β + γN H
∗F qmp (X ∗ ; τ ) = L
α − c − τ − γX ∗ 2β + γN F
with density: α − c − τ − γX ∗ 2β + γN F
(22)
∗H ∗H ∗F ∗F The output of an SP-firm is obtained for N F = N H = 0 in which case qmp = qsp and qmp = qsp . s∗ ∗ I denote by ps∗ r := p(xr , X ), where r = {sp; mp} and s = {H; F }, the prices set on the market by domestic
and foreign MP and SP firms. When set by a foreign firm, this price is inclusive of trade costs. At equilibrium, the absolute markups on the domestic and foreign market are given by the following expressions: 17 None of the results below rely on this assumption: a higher fixed cost per product to export would only further decrease the breadth of the exported product range.
16
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pH∗ mp − c =
γ 1 (α + Q∗H − γX ∗ − c) 2 L mp
∗ pF mp − c =
γ 1 (α + Q∗F − γX ∗ − c + τ ) 2 L mp
pH∗ sp − c =
1 (α − γX ∗ − c) 2
∗ pF sp − c =
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T
which yield a direct comparison with SP-firm markups: 1 (α − γX ∗ − c + τ ) 2
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Comparing prices vertically: firms with a higher mass set higher prices by charging higher markups. This also implies that when both large and small firms export, MP-firms set higher (f.o.b. or c.i.f.) prices than SP- firms. Market Outcome
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5.3.2
Selection
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A firm is an exporter if its profits abroad can cover the fixed cost of exporting. In the spirit of the Melitz model, I focus here on the case featuring selection into the exporting activity. This is the case when fixed costs are high enough. For the sake of simplicity, I assume that Fx = F .18 When there is no prospect of economies of scale, small
ED
firms choose not to export no matter what the extent of trade liberalization τ > 0 is. However, the prospect of
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capturing rents abroad is a sufficient reason for MP-firms to export. Comparative statics w.r.t. to τ
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Comparing the market outcome under trade with the closed economy section, the market aggregate X ∗ at equilibrium is the same and determined by the free-entry condition on the domestic market for SP-firms.
∗ (x∗sp , X ∗ ) = πsp
L 2 (α − c − γX ∗ ) − (f + F ) = 0 4β
M ∗ > 0 ⇒ X ∗ (τ ) = X ∗ (τmax ) = X ∗
where X ∗ is the equilibrium value of X in the absence of trade. Large firms start exporting as soon as trade q q 1 + Ff − 1 . costs fall below τmax := 2 βf L
Writing that the inverse demand is met by firms’ prices at equilibrium shows that imported varieties are sold
at the same price as the ones produced domestically:
F∗ ∗ ∗ pH∗ mp = pmp = α − βxmp − γX 18 A sufficient condition is F ≥ F . When F = F , a positive level of trade costs τ guarantees that exporting is less profitable x x (including fixed costs) than domestic sales. Since small firms are free to enter and choose to export only if it is profitable, they never choose to do so. In other words small firms have no reason to export in the absence of economies of scale.
17
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In line with Brander and Krugman (1983), there is reciprocal dumping in the sense that large firms set a f.o.b. price lower than the price at which varieties are sold domestically. However and contrary to Brander and Krugman (1983), MP-firms’ market power in the foreign market increases with trade liberalization as they absorb any decrease
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in trade costs by increasing their markups. While they show that oligopolistic rents are driven down by a decrease in
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trade costs, a mixed market structure emphasizes the persistence of the market power of large firms so that, in the long run (i.e. after the monopolistically fringe adjusts), there is no pro-competitive effect hurting the oligopolists.
and in F leads to
N
F∗
s
2β = γ =N
! F 1+ −1 f
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N
H∗
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Turning to the product range, the two first-order conditions of large firms w.r.t. their product breadth in H
H∗
· 1−
τ
τmax
(23)
MA
which shows that large firms export only a subset of their varieties abroad. When τ decreases, the marginal profit of exporting an additional variety increases so N F ∗ −→ N H∗ . τ →0
To sum-up trade liberalization allows large firms to increase their market share abroad: they expand their
ED
product scope and raise their markups on all their varieties.
PT
Last, to see how MP-firms gain from firm selection, their profits at equilibrium can be derived:
4βf = γ
s
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ΠH∗ mp
!2 F 1+ −1 f
∗ F ∗ F∗ ΠF , X ∗; τ ) = mp (τ ) := Πmp (xmp , N
L 2 (τmax − τ ) γ
Because the exporting activity is now costly, only large firms benefit from trade liberalization. Furthermore, small firms do not benefit from any efficiency gain as in the free-trade scenario. Economies of scope realized on the exporting fixed costs protect large firms to the extent that the competitive fringe cannot export. A decrease in trade costs does not increase competition: new varieties that are imported replace the exit of SP-firms. From then on, any further drop in trade costs τ only makes oligopolists more profitable. By contrast, SP-firms cannot compensate the increase in competitive pressure on the domestic market with their exporting activity since the latter is too costly. SP-firms are forced to exit the domestic market as trade liberalization occurs: there is a reallocation of their market shares towards the surviving firms. In that sense, the market structure becomes more oligopolistic. The distribution of the gains from trade MP-firms make positive profits in equilibrium that are a decreasing function of τ so trade raises producer surplus unambiguously. Turning to the impact on consumer surplus, a drop in trade costs does not impact prices consumers pay but increase the overall number of varieties. Following the logic of section 5.1, the change in consumer surplus 18
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in one country following a drop in τ is given by s
! F 1 + − 1 ∆τ N F f
N H∗ · ∆τ τmax
so that the change in consumer surplus can be simplified into s
F 1+ −1 f
!2
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βf ∆τ CS = Ω γ
SC
∆τ N F = −
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where ∆τ N F is the change in product scope w.r.t. τ . From (23), it is equal to
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f ∆τ CS = Ω 2
∆τ τmax
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Proposition 6. Consumer (producer) surplus decreases (increases) when variable trade costs decrease. The increase in the number of varieties does not increase consumer surplus because new varieties are supplied at
ED
a higher price: large foreign firms gain more market power. It is worth stressing that this result departs both from a pure monopolistic (Krugman, 1979) and from an oligopolistic (Brander, 1981) market structure. Here, although there are gains from trade liberalization, there are absorbed by an increase in producer surplus. This shows how
Robustness
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6
PT
important it is to tackle the distribution of the gains from trade between producers and consumers.
Central to the results obtained in the previous section is the anti-competitive effect following the entry of a large firm derived in subsection 5.1. In order to study the robustness of this result, I proceed in two steps. First, I show that under mild assumptions, the impact of entry of a foreign oligopolist on the average price holds true when the market aggregate is additively separable in firms’ strategies. The robustness of the variety effect is weaker, but holds under an additional yet reasonable assumption. Second, I turn to consumer surplus and consider the class of additively separable preferences which are consistent with these assumptions and nest a large class of utilities considered in international trade. I show that the negative impact on consumer surplus typically holds in models featuring an increasing price-elasticity of demand. Specifically, I consider below that the inverse demand for variety ν can be written as p(xν , Λ) where Λ is an aggregate statistic of total consumption. The profits of firm ω with respect to a given variety are assumed to be strictly quasi-concave so the output profile of an MP-firm will be necessary symmetric at equilibrium. The choice of the multi-product firm ω boils down to two variables xω and Nω so that the strategy set of each player is twodimensional. Last, I consider the monopolistically competitive fringe as a Ω + 1th player e.g. a multi-divisional
19
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firm whose product range breadth is governed by the free-entry condition.19 Therefore, the output profile in the
6.1
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¯ can be replaced with the (2Ω + 2)−uplet (x,N). industry x " Ω+1 −→ R. Under the assumptions of the baseline The term “aggregate” is defined here as a function Λ : R2 P model Λ(x, N) ≡ X = ω≤Ω Nω xω + M xsp . In that case the aggregator is additively separable.20
Variety vs. Prices.
differentiable, strictly decreasing resp. w.r.t. x i.e. px :=
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Assumptions The aggregate is assumed to be a scalar, i.e. Y = R. The inverse demand is twice continuously ∂p ∂s (x, Λ)
x < 0 and not “too convex” i.e. − pxx px < 2 (A1).
An increase in the aggregate shifts the inverse demand curve downwards i.e. pΛ :=
∂p ∂Λ (x, Λ)
<0
(A2).
The
∂Λ ∂xω (x, N)
= 0 if Nω = 0 and ΛNω (x, N) :=
NU
aggregate Λ(x, N) is twice differentiable w.r.t. each of its argument. For each ω ∈ [[1, Ω]] we have Λxω (x, N) := ∂Λ ∂Nω (x, N)
= 0 if xω = 0. Otherwise, Λxω (x, N) > 0 if Nω >0 and
MA
ΛNω (x, N) > 0 if xω >0 (A3). Last I assume that Λxω Nω (x, N) :=
∂2Λ ∂xω Nω (x, N)
> 0 ∀(x, N ) ∈ R∗2 + (A4).
Assumption (A1) implies that the marginal revenue curve of an SP-firm is decreasing in output. Combining (A2) and (A3) implies that an MP-firm may shift down the demand schedule i.e. raise competition toughness
ED
either by increasing its output or its product range. Its output per-product xω has no impact on Λ when it’s negligible Nω = 0 and its product range has no impact on Λ if it supplies zero output xω = 0. Last, (A4) implies
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that the impact of the product breadth of an MP-firm on competition toughness is all the more important that it has a large output per-product.
AC CE
Separable market aggregates I show in Appendix D that under these conditions and as long as an equilibrium exists, Λ∗ is uniquely determined. As a consequence, under (A1)-(A2) the output and prices chosen by SP-firms do not depend upon any shock on Λ. When the best-reply function of large firms can also be expressed as a function of Λ only, i.e. when the game is fully aggregative (Cornes and Hartley, 2012), the entry of a foreign oligopolist, even if it faces variable trade costs τ , does not affect the equilibrium among MP-firms. When is the game fully aggregative? Cornes and Hartley (2012) have shown that under strict monotonicity assumptions with at least three players, a game is fully aggregative if and only if the aggregate is additively separable (up to a monotonic transformation). I show in Appendix E that under (A3)-(A4) with at least three players, Cornes and Hartley (2012)’ result can be extended to the case of two-dimensional strategy sets. In what Pω=Ω follows, I assume therefore that Λ is a monotonic transformation of ω=1 gω (xω , Nω ) + gsp (xsp , M ) where gω (.)
and gsp (.) are continuously differentiable functions. 19 See
for instance Shimomura and Thisse (2012). is a special case of the class of aggregative games considered by Acemoglu and Jensen (2013) who consider finite-dimensional strategy sets. In the event where MP-firms have an exogenous unitary mass, the aggregate X in the baseline model boils down to the sum of the player’s actions as in Anderson et al. (2013). 20 This
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Entry of a large firm
I explore here the robustness of the impact of an MP-firm entry on prices and product
variety when the inverse demand can be written as p(x, Λ) where x is still the consumption of a variety and PΩ Λ(x, N) = ω=1 g(xω , Nω )+g(xsp , M ) is an aggregate statistic of a consumer’s consumption of all other varieties.21
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Under this functional form, assumption (A3) requires that g(.) is twice differentiable with g(0, N ) = g(x, 0) =
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0 ∀(x, N ) ∈ R2+ with gx > 0, gN > 0 and (A4) requires that gxN > 0 ∀(x, N ) ∈ R∗2 + . Assumptions (A1)-(A4) R are verified in the baseline model under which Λ = X = N xν dν and g(x, N ) = N x.
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The pair (x∗ω , Nω∗ ) verifies the following first-order conditions at equilibrium
NU
px (x∗ω , Λ∗ )x∗ω + N ∗ gx (x∗ω , Nω∗ )pΛ (x∗ω , Λ∗ )x∗ω + p(x∗ω , Λ∗ ) − c = 0 gN (x∗ω , Nω∗ )pΛ (x∗ω , Λ∗ )N ∗ x∗ω + (p(x∗ω , Λ∗ ) − c)x∗ω = f
(24)
(25)
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which is to be compared with the first-order condition for an SP-firm:
(26)
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px (x∗sp , Λ)x∗sp + p(x∗sp , Λ∗ ) − c = 0
Two results can be derived from these equations. Under (A1), (A2) and (A3), (30) and (26) imply that x∗ω < x∗sp and thus p(x∗ω , Λ∗ ) > p(x∗sp , Λ∗ ). Hence larger firms charge higher markups.
PT
Second, if there exists an optimal and finite product range Nω∗ , equation (31) implies that it depends on Ω only through Λ∗ .
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Now, Λ∗ is pinned down by the free-entry condition and Λ is additively separable so x∗ω and x∗sp do not depend upon firm-entry. Writing that the aggregate Λ∗ does not change with the entry of a large firm ω leads to the following equality
g(x∗sp , M ∗ (Ω + 1)) − g(x∗sp , M ∗ (Ω)) = g(x∗ω , Nω∗ )
so the entry of a large firms triggers again the exit of SP-firms. From (A4), we get that g(x∗ω , Nω∗ ) > g(x∗ω , M ∗ (Ω + 1)) − g(x∗ω , M ∗ (Ω)) Making the further assumption that gN N ≥ 0 (A5), and keeping in mind that g(x, 0) = 0, we have the following inequality g(x∗ω , Nω∗ ) > g(x∗ω , ∆M ) and therefore the total number of varieties increases again with the entry of a large firm. Assumption (A5) is 21 I assume for the sake of simplicity that g (.) = g (.) ≡ g(.) ∀ω ∈ [[1, Ω]]. This is to be considered the standard case under ω sp symmetric utility functions. Note also that the same exercice may be conducted under the assumption of price competition when the direct demand depends on an additively separable aggregate. Interestingly enough, Bertrand competition in the baseline model falls outside this case. Nevertheless, I show in Appendix F that even in that case, the entry of a large firm raises the average price and decreases consumer surplus.
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far from being necessary for this result. Note that it holds strictly even when gN N = 0 as in the baseline model. Because the marginal cost of adding a variety is constant, the only reason for a firm not to supply an infinite number of varieties is cannibalization, which is magnified by the convexity of g(.) w.r.t. N . From the perspective of an
T
MP-firm, adding a variety should decrease its marginal profit w.r.t. N . Inspecting the second partial derivative of
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the profit shows that the convexity of g only magnifies the cannibalization effect.22
Last, since the firms’ payoff depends on Λ∗ only, the total producer surplus increases with the entry of a large
SC
firm as in the baseline model.
To sum-up: under a mixed-market structure where assumptions (A1)-(A4) hold, MP-firms charge higher markups, the exogenous entry of a foreign oligopolist leads to the exit of monopolistic firms and raises the average
Consumer Welfare
MA
6.2
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price. The total number of varieties increases as soon as (A5) holds.
In this subsection, I aim to check that my results are not driven by the assumption of quadratic preferences. Two effects conflict in the baseline model: the variety effect and the anti-competitive effect (i.e. the average price goes
ED
up). While the anti-competitive effect has been shown to be quite robust, the variety effect requires an extra assumption (A5). However, since (A5) appears quite reasonable, I focus on the case where the variety effects go
PT
against the anti-competitive effect. To do so, I consider the class of additively separable preferences (see Kuhn and Vives (1999), Vives, 1999 and Zhelobodko et al. (2012) which nest CES preferences) for which (A5) holds.
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Preferences First, it is important to note that if the relevant market aggregate from a firm’s perspective is Λ, this aggregate is generally not relevant to analyze consumer surplus.23 For instance, in the baseline model, the R RN N expression of consumer surplus depends on a two-dimensional aggregate 0 xν dν, 0 x2ν dν . RV I now consider an additively separable lower-tier utility 0 u(xν )dν where u(.) is increasing, concave and thrice differentiable on R+ . The upper-tier utility is
U (A, x) := A + U (x)
where
U (x) := W
Z
V
u(xν )dν 0
!
(27)
with W being twice continuously differentiable. Inspecting the first-order condition implies that the inverse demand for a variety ν features an additively separable aggregate:
′
′
where
p(xν , Λ) := u (xν )W (Λ) 22 The
Λ :=
Z
V
u(xν )dν
(28)
0
expression of the second derivative of per-product profits w.r.t. N is given by gN N (x∗ω , N )pΛ (x∗ω , Λ∗ )N ∗ x∗ω + gN (x∗ω , N ∗ )2 pΛΛ (x∗ω , Λ∗ )N ∗ x∗ω + 2gN (x∗ω , N ∗ )pΛ (x∗ω , Λ∗ )x∗ω < 0 | {z } <0
Note that the first term drops out in the baseline model. This condition is more likely to be met when (A5) holds. exception is when the direct demand verifies the Independence of Irrelevant Alternatives as in Anderson et al. (2013). Note that the only additively separable preferences which verify this property feature a constant elasticity of substitution. 23 An
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so assumptions (A1)-(A5) hold. Consumer surplus can be expressed as follows: N
p(x∗ν (Λ), Λ)x∗ν (Λ)dν
0
T
Z
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CS := U (Λ) −
Variations in the consumer surplus are therefore driven by variations in expenditure. Similarly to the baseline model, consumer surplus requires one additional aggregate compared to producer surplus - the expenditure on the
SC
differentiated good.
Using that Λ∗ does not depend on Ω, the change in consumer surplus with the entry of a large firm is
u(x∗mp ) ∗ ∗ p x − u(x∗sp ) sp sp
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∆CS = −N
∗
p∗mp x∗mp
MA
Now, plugging (28) into the above expression leads to:
u′ (x)x u(x)
(29)
ED
where Eu =
′ ∆CS = N ∗ u(x∗mp )W (Λ∗ ) Eu (x∗sp ) − Eu (x∗mp ) is the elasticity of the sub-utility u(.) w.r.t. consumption.
The change in consumer surplus depends crucially on whether Eu increases or decreases with individual conas the most relevant.
PT
sumption. As argued by Kuhn and Vives (1999) and Dhingra and Morrow (2016), the latter should be considered
AC CE
Furthermore, under the assumption that u(0) = 0, it is possible to show that the monotonicity of Eu is disciplined by the monotonicity of the demand price-elasticity. In fact, based on Kuhn-Vivès (1999) results, I show in Appendix G.1 that an increasing price-elasticity of demand implies that Eu′ < 0. This happens to be the standard case in trade models featuring a variable elasticity of demand.24 Therefore the change in consumer surplus is generally negative, showing thereby that the insights from the baseline model are not an artefact of the quadratic preferences considered. Interestingly enough, ∆CS = 0 when Eu is constant, i.e., when preferences feature a constant elasticity of substitution. Last, it is important to note that (29) implies that the marginal cost of the foreign oligopolist is irrelevant as long as it yields x∗mp < x∗sp . Consequently, the entry of an oligopolist which faces variable trade costs τ - so that the marginal cost of exporting a variety is c + τ - impacts negatively consumer surplus. 24 In the trade literature, this assumption goes back at least to Krugman (1979). This is also equivalent to assuming an increasing relative love for variety in Zhelobodkho et al. (2012) or a decreasing elasticity of substitution as in Bertoletti and Epifani (2014). Mrázová and Neary (2013) have also found out that this assumption can be traced back to Marshall and is known as the second Marshall law of demand. An increasing price-elasticity of demand is also directly related to firms’ pass-through rate. Empirical work e.g. Berman Martin Mayer (2012) and Amiti, Itskhoki, and Konings (2014) find support for a pass-through rate below 1 (see also De Loecker et al. 2014. for additional empirical evidence) which requires again an increasing price-elasticity of demand under monopolistic competition (Mayer Melitz and Ottaviano, 2016).
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Social welfare The change in social welfare is ambiguous. Indeed, new varieties come at the cost of the multiproduct technology, i.e. the fixed cost f which is incurred by large firms for each product. Even if producing one variety is more efficient when undertaken by a large firm rather than by a small firm, large firms may still produce
T
too many varieties, i.e. the marginal gain from selling a new variety for an MP-firm may be smaller than the
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decrease it entails for consumer surplus.
Whether one or the other dominates is generally ambiguous and depends on the functional forms assumed for
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the utility function: even if the level of social welfare increases in the baseline model or in the CES case, it need not be so, even for well-behaved preferences.25
Note however that if large firms behaved like multi-divisional firms, they would set a price equal to the one
NU
charged by small firms. In that case, social welfare would always increase with a large firm’s entry thanks to economies of scope. To sum up, the distribution of the surplus following the entry of a large firm between consumers
MA
and producers is robust to a large class of functional forms. In spite of the ambiguity of the results regarding social welfare,26 it should be noted that were large firms not behaving strategically, social welfare would always increase
ED
with the entry of a foreign large firm through an increase in productive efficiency.
7
Concluding Remarks
PT
Consistent with the empirical evidence that most industries feature a small number of large firms and a large number of small firms, I have built a model with a “mixed” market structure in which a few multi-product oligopolists and
AC CE
a continuum of single-product monopolistically competitive firms coexist. Large firms charge higher markups than monopolistically competitive firms, even if they have the same marginal cost of production. This is because large firms are able to manipulate the market aggregate which is relevant to all their competitors’ decisions.
When a reduction in trade barriers selects the largest firms into exporting, the overall number of varieties available to consumers increases, but so does the average price in the industry. This results in a decrease of consumer surplus, while large firms absorb the decrease in trade costs by increasing their markups. I have shown that the main insights obtained in the baseline model under linear demand hold true for a wide class of demand functions. This shows the importance of considering heterogeneity in firm size, together with strategic interactions. Finally, this paper leaves at least two avenues for future research. First, this paper has been silent on policy. In the setting I have considered, trade liberalization may hurt consumers at the expense of producers. This calls for an extension in which governments may manipulate trade barriers and domestic redistributive policies to affect the degree of income inequality. 25 Counter-examples
include functional forms which belong to the Hyperbolic Absolute Risk Aversion (HARA) family. some special cases, it is possible to show that social welfare increases e.g. when economies of scope are very low so F = 0+ . I illustrate this case in Appendix G.2. Intuitively, this is because MP-firms behave like multi-divisional firms again. 26 In
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Second, I have assumed that the number of firms that can be large is exogenous. Since the entry of a large firm impacts the distribution of the gains from trade, this is an important assumption to relax. In the respect, the
AC CE
PT
ED
MA
NU
SC
RI P
T
contribution of Eckel and Yeaple (2016) is an important first step in this direction.
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A. MP-firms’ Program A.1 Reducing the Dimensionality of the Maximization Problem
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First, one can notice that the output profile x ¯ω of firm ω can be defined as a positive-valued function up to a threshold
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N (potentially infinite) and zero above without loss of generality. Second, if x ¯∗ω = (x∗ωk )k≤Nω is a maximizing output profile for firm ω, then there exist x∗ω > 0 and Nω > 0 (potentially zero or infinite) such that x∗ωk = x∗ω almost
SC
everywhere ∀k ∈ [0, Nω ]. To see this, assume this is not true and that there exist two convex subsets [0, N1 ] and
NU
]N1 , Nω ] of R+ with 0 < N1 < N2 such that xωk = x1 ∀k ∈ [0, N1 ] and xωk = x2 ∀k ∈ [0, Nω ]. Now, since RN ∗ z − f /L is strictly concave, the symmetric output the function π ˜ (z) := z −→ α − c − βz − γ 0 ω xωk dk − γX−ω RN N1 N1 ˜ωk , generates higher profits: profile x ˜ωk = N x1 + 1 − N x2 ∀k ∈ [0, N2 ], which is such that 0 ω xωk dk ≡ Nω x ω ω
MA
N1 N1 N1 N1 π ˜ (x1 ) + 1 − ˜ x1 + 1 − π ˜ (x2 ) < π x2 Nω Nω Nω Nω
ED
Therefore, the maximization program of firm ω comes down to choosing a pair (x∗ω , Nω∗ ) such that:
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(x∗ω , Nω∗ ) = Argmax(xω ,Nω ) Nω
"
∗ α − c − βxω − γNω xω − γX−ω xω − f /L
A.2 Determining the Global Maximum P
ω6=ω ′
Nω∗ x∗ω − βM x∗sp − c. The objective function of firm ω is:
AC CE
Let us denote α ˜ := α − β
max [α ˜ − (β + γNω ) xω ] Nω xω − Nω f /L
xω ,Nω
The first-order conditions w.r.t. xω and Nω are respectively
[˜ α − 2 (β + γNω ) xω ] Nω = 0 and
[˜ α − (β + 2γNω ) xω ] xω = f /L so the Hessian is given by
−2 (β + γNω ) Nω H := α ˜ − 2 (β + 2γNω ) xω
26
α ˜ − 2 (β + 2γNω ) xω −2γx2ω
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with determinant | H |= 4γ [(β + γNω )] N x2ω − [˜ α − 2 (β + 2γNω ) xω ]
x∗ω
=
s
f βL
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leads to
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and
(30)
SC
2 (γNω∗ + β) x∗ω = α ˜
T
Using the first-order conditions, it is evaluated at (x∗ω , Nω∗ ) where
2
∗ ∗ | H∗ |= 4γ [(β + γNω∗ )] Nω∗ x∗2 ω − [2γNω xω ]
(31)
2
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| H∗ |= 4γβNω∗ x∗2 ω >0 so (x∗ω , Nω∗ ) is a local maximum. Since this is a two-dimensional problem, the fact that this local maximum is the only critical point of the profit function on R+ × R+ is not enough to conclude it is a global one. However, the
ED
maximization problem can be reformulated on a compact without loss of generality by ruling out cases in which
PT
profits are negative. Indeed, for [˜ α − (β + γNω ) xω ] Nω xω − Nω f /L to be non-negative, a necessary and sufficient √ 2 α+ α ˜ −4(β+γNω )f /L with α ˜ 2 ≥ 4(γ + βNω )f /L. This expression shows that there is an condition is that xω ≤ 2(β+γNω ) upper bound on the output per variety: xω ≤
α ˜ β.
These last two inequalities define a support for the maximization
AC CE
problem which is a compact subset of R+ × R+ . On the boundary of this support, firm ω makes at most zero profits. Therefore any strictly positive solution (x∗ω , Nω∗ ) to (30) and (31) (which guarantees strictly positive profits) is the unique global maximum.
B. Derivation of γ¯
Writing that consumer income is not entirely spent on the differentiated good leads to 1 + ΩΠ∗mp > which can be rearranged into 1 > p∗sp X ∗ + ΩN ∗ f − (p∗sp − c)x∗mp
R N∗ 0
p∗ν x∗ν dν
Using the expressions for equilibrium output and the inverse demand evaluated at x∗mp shows that the right-hand side is decreasing in Ω. A necessary and sufficient condition for the consumption of the homogenous good to be positive for any Ω is 1 > p∗sp X ∗ which is equivalent to γ > γ¯ :=
!s
! " " p α−c f +F p −1 + c 2β f + F β 2 β (f + F )
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Note that the other equilibrium conditions do not depend upon γ so the parameter space is non-empty.
The market aggregate X∗ = varies with market size as follows:
q +F ) α − c − 2 β(fκL γ
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βx∗sp dX ∗ = dκ γκ
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q β(f +F ) dX ∗ L = dκ γ · κ3/2
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C. Welfare and Trade Openness
MA
Using that κx2sp does not depend on κ, differentiating consumer surplus CS := L
h R β N 2
0
i x2ν dν + γ2 X 2 leads to (32)
ED
dM ∗ β dCS dX := + γLX ∗ κLx∗2 sp dκ 2 dκ dκ To assess the impact of κ on the mass of small firms per country, notice that
∗ X ∗ κx∗ α−c dM ∗ 2β sp where X =d = − ∗ ∗ dκ dκ κxsp γκxsp γκ
AC CE
and thus
PT
X ∗ −κΩN ∗ x∗ mp κxsp∗ dM ∗ =d dκ dκ
After simplication we have
βxsp − dM = dκ κ2
X∗ 2
1 xsp
Plugging the above expresssion into (32), we get that
dCS β := dκ 2
βxsp −
X∗ 2
κ2
1 xsp κLx2sp + βLX xsp κ
The total change in consumer surplus w.r.t. the number of countries is unambiguously positive βqsp∗ dCS := dκ κ
3 βxsp + X ∗ 4
>0
Since we have already shown that firms’ profits remain constant, we may conclude that social welfare increases with trade openness.
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D. Uniqueness of the Equilibrium Market Aggregate Under (A1), the maximizing output of an SP-firm x∗sp (Λ) is a well-defined function of Λ. The free-entry condition
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T
reads as
πi∗ (Λ) := (p(xi , Λ) − c) xi = πi (x∗ (Λ), Λ) = 0.
(33)
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Using (A2), the envelop theorem implies that πi∗ (Λ) is decreasing and that this equation has at most one solution Λ∗ . Therefore, as long as an equilibrium exists, Λ∗ is uniquely determined. As a consequence, under (A1)-(A2) the output and prices chosen by SP-firms do not depend upon any shock
NU
on Λ. This is the case if a large firm enters the market or if one (or many) incumbents face an exogenous shock on their variable or per-product fixed costs. The dimensionality of the aggregate X in the baseline model is therefore
MA
crucial to explain the neutrality of SP-firms response.
E. Conditions for the Game to be Fully Aggregative
ED
The uniqueness of Λ∗ is not enough to explain the neutrality of MP-firms’. Inspecting the first-order conditions of an MP-firm ω at equilibrium shows that a best reply function generally depends both on Λ and on its derivatives
PT
with respect to xω (resp. Nω ). Eventhough Λ∗ is determined by free-entry, the equilibrium values of Λ∗xω and Λ∗Nω generally depend upon other-firms’ equilibrium strategies. For the neutrality of firm entry (or a cost shock) to hold
AC CE
it is necessary and sufficient that the marginal profits w.r.t. xω and Nω (and thereby Λxω and ΛNω ) depend upon xω , Nω and Λ only. In this event, the game admits backward reply correspondences27 x∗ω (Λ) and Nω∗ (Λ) for each firm ω .
This is precisely the class of aggregative games considered by Cornes and Hartley (2012) which they refer to as fully aggregative games. In the case of fully aggregative games with one-dimensional strategy sets, they show that under strict monotonicity assumptions with at least three players, one can assume additively separable aggregates (up to a monotonic transformation) without loss of generality. I show below that under (A3)-(A4) with at least three players, Cornes and Hartley (2012)’ result can be extended to the case of two-dimensional strategy sets. Cornes and Hartley (2012) Extended to Two-Dimensional Strategy Sets The proof closely follows Cornes and Hartley’s (2012) in which the regularity assumptions are given by (A3)-(A4). By definition, the game is fully aggregative if and only if there exist functions φ and φ˜ such that
Λxω := 27 This
∂Λ = φω (xω , Nω , Λ) ∂xω
notion has been refered to under various names and goes back at least to Selten (1970).
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ΛNω :=
∂Λ = φ˜ω (xω , Nω , Λ) ∂Nω
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The cross-derivatives w.r.t. xω and xω′ are given by:
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∂φω (xω , Nω , Λ) ∂2Λ ∂Λ ∂φω (xi , Ni , Λ) = φω′ (xω′ , Nω′ , Λ) = ∂xω′ ∂xω ∂xω′ ∂Λ ∂Λ
SC
∂Λ ∂φω′ (xω′ , Nω′ , Λ) ∂2Λ = ∂xω ∂xω′ ∂xω ∂Λ Denoting by
∂ ln [φω (xω Nω , Λ)] ∂Λ h i ∂ ln φ˜ω (xω Nω , Λ) ψ˜ω (xω , Nω , Λ) = ∂Λ
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ψω (xω , Nω , Λ) =
Λxω
MA
we get that
∂φω′ (xω′ , Nω′ , Λ) ∂φω (xω , Nω , Λ) = Λx ω ′ ∂Λ ∂Λ
Or equivalently
ED
ψω (xω , Nω , Λ) = ψω’ (xω′ , Nω′ , Λ)
(34)
PT
Applying the same reasoning using the cross-derivatives of Λ w.r.t. Nω and Nω′ , we get:
(35)
AC CE
ψ˜ω (xω , Nω , Λ) = ψ˜ω′ (xω′ , Nω′ , Λ)
When Ω ≥ 3, ω ′′ 6= ω and ω ′′ 6= ω ′ can be chosen and (34) can be differentiated with respect to xω and xω′′ . Under (A3), this implies that ψω does not depend upon xω . Now, differentiating (35) with respect to Nω and Nω′′ , we get: ∂ψω ∂ψω ∂ψω ˜ ∂ψω′ ˜ ∂ψω + + ΛN ω ≡ φω (xω , Nω , Λ) = φω (xω , Nω , Λ) ∂Nω ∂Λ ∂Nω ∂Λ ∂Λ ∂ψω′ ˜ ∂ψω ˜ φω” (xω′′ , Nω′′ , Λ) = φω′′ (xω′′ , Nω′′ , Λ) ∂Λ ∂Λ Similarly, under (A4) and Ω ≥ 3, ψω does not depend upon Nω either. Therefore, ln φω is additively separable in Λ and (xω , Nω ). This implies in turn that there exist functions fˆω and hω such that Λxω = φω (xω , Nω , Λ) = fˆω (xω , Nω )hω (Λ)
30
(36)
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˜ Similarly, there exist functions ˆfω and h˜ω such that ˜ ˜ ω (Λ) ΛNω = φ˜ω (xω , Nω , Λ) = ˆfω(xω , Nω )h
T
(37)
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Now substituting (36) into (34) yields
SC
fˆω (xω , Nω )hω (Λ(x,N))h′ω′ (Λ(x,N))fˆω′ (xω′ , Nω′ ) = fˆω′ (xω′ , Nω′ )hω′ (Λ(x,N))h′ω (Λ(x,N))fˆω (xω , Nω )
which can be simplified into
NU
hω (Λ(x,N))h′ω′ (Λ(x,N)) = hω′ (Λ(x,N))h′ω (Λ(x,N))
∀(x,N) ∈ R2
∀(x,N) ∈ R2
Ω+1
Ω+1
MA
This means that hω is a linear transformation of hω′ . Therefore, (36) can be rewritten as
φω (xω , Nω , Λ) = fω (xω , Nω )h1 (Λ)
∀ω ∈ [[1, Ω]]
(38)
∀ω ∈ [[1, Ω]]
(39)
PT
Similarly there exists f˜ω such that
ED
where fω is a linear transformation of fˆω .
˜ 1 (Λ) φ˜ω (xω , Nω , Λ) = f˜ω (xω , Nω )h
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Last, taking the derivative of (38) w.r.t. Nω and the derivative of (39) w.r.t. xω , we get: ∂ ˜ ∂ ˜ 1 (Λ) fω (xω , Nω )h1 (Λ) = fω (xω , Nω )h ∂Nω ∂xω
Under the additional regularity assumption (A4), (39) can be rewritten as follows φ˜ω (xω , Nω , Λ) = f˘ω (xω , Nω )h1 (Λ)
∀ω ∈ [[1, Ω]]
∂
fω (xω ,Nω ) where f˘ω (xω , Nω ) := f˜ω (xω , Nω ) ∂N∂ ω f˜ (x ,N ) . ∂xω
ω
ω
ω
Now, the end of the proof follows Cornes and Hartley (2012). Denote by
Γ(Λ) =
Z
Λ Λ1
1 dΛ′ h1 (Λ′ )
We write T (x, N) = Γ[Λ(x,N)] and we have ∂T (x, N) = Γ′ [Λ] Λxω = fω (xω , Nω ) ∂xω
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∂T (x, N) = Γ′ [Λ] ΛNω = f˘ω (xω , Nω ) ∂Nω So
∂ 2 T (x,N) ∂xω ∂xω′
=
∂ 2 T (x,N) ∂Nω ∂Nω′
=
∂ 2 T (x,N) ∂xω ∂Nω′ =0
for ω 6= ω ′
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T
T is therefore additively separable in (xω , Nω ) for all ω ∈[[1,Ω]] up to a monotonic transformation.
F Non-separable market aggregates: the case of Bertrand competition
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and linear demand
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Under Bertrand competition, the direct demand can be expressed as ˜ =Λ ˜− d(pi , Λ)
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˜ is a market aggregate defined as follows: where Λ
pi β
α γN p + β + γN β (β + γN )
(40)
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˜ := Λ
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The equilibrium value of the aggregate is determined in the same fashion through the free-entry condition for q β(f +F ) 1 ∗ ˜ small firms which leads to Λ = β 2 +c . L Rearranging the first-order conditions w.r.t. to product scope and price for MP-firms lead respectively to ˜∗ − Λ
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p∗mp β
p∗mp − c
= p∗mp − c
p∗mp β
˜∗ − Λ
β + γ (N ∗ − N ∗ ) β (β + γN ∗ )
β + γ (N ∗ − N ∗ ) =f β + γN ∗
(41)
So the equilibrium price of a variety supplied by a large firm is now given by p∗mp
=
p∗sp
+
r
β (f + F ) − L
r
βf L
where p∗sp is the price of a variety supplied by a small firm: p∗sp =
˜∗ + c βΛ 2
˜ is no longer additively separable in firms’ strategies and falls therefore outside the class of Note however that Λ ˜ ∗ , we preferences 5.1.. The entry of a large firm is no longer neutral on MP-firms’ product scope. By definition of Λ
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know that ∗ ˜ β − p∗sp (β + γN ∗ ) α − p∗sp β + γΩN ∗ p∗mp − p∗sp = Λ γN ∗ β+γN ∗
does not depend on Ω. Consequently, it must be that the breadth of the
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Now from (41), we also know that
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product range of a N ∗ increases with Ω.
Now using (40) again, the entry of a large firm leads to a change in the product scope of MP-firms as well as in
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the mass of SP-firms:
˜ ∗ βγ (Ω∆N + N ∗ + ∆M ) γN ∗ p∗mp + γΩ∆N p∗mp + γ∆M p∗sp = Λ
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Rearraging the last expression shows that the mass of SP-firms that exit the market following the entry of a large firm is smaller than the increase in the overall number of varieties supplied by large firms:
1− 1−
p∗ mp ˜∗ βΛ
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p∗ sp ˜∗ βΛ
(Ω∆N + N ∗ ) = −∆M
ED
The overall mass of varieties N increases.
Now the change in consumer surplus following the entry of a large firm is still given by a change in the aggregate 0
x2ν dν. Using the same identities as above, one can show that the change in consumer surplus is negatively
PT
RN
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related to the mass of SP-firms:
∆CS ∝ 1 −
1− 1−
p∗ mp ˜∗ βΛ p∗ sp ˜∗ βΛ
∆M < 0
To sum-up: under Bertrand competition, MP-firms charge higher mark-ups, the exogenous entry of a foreign oligopolist leads to the exit of monopolistic firms and raises the average price. The total number of varieties increases but consumer surplus decreases.
G Additively Separable Preferences with an Additive Upper-Tier Utility Notation: I denote the elasticity of θ w.r.t. y by Eθ (y) :=
θ ′ (y)y θ(y) .
G.1 Consumer Surplus I aim at showing that when u(0) = 0, an increasing price-elasticity of demand (or equivalently a decreasing Eu′ ) implies Eu′ < 0. To do so, we use the two following results.
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(R1) Following the appendix of Kuhn and Vivès (1999), 1 − Eu (x) and −Eu′ (x) are equal at 0 and have the same monotonicity in its neighborhood. (R2) When Eu′ (x) < 0, we have 1 − Eu (x) < −Eu′ (x) while the reverse holds under Eu′ (x) > 0. For the
T
monotonicity of Eu to change at an interior point x ¯ it has to be that 1 − Eu (¯ x) = −Eu′ (¯ x).28
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Now, to prove our result, we proceed by contradiction. Let us assume that Eu′ is a decreasing function while ˆ such that Eu′ (ˆ Eu′ (x0 )′ > 0 holds for some x0 > 0. By continuity, (R1) implies that there exist x x) = 0 with
SC
x) with 1 − Eu (x) < −Eu′ (x) for any x < x ˆ and x) = −Eu′ (ˆ Eu′ ′ (x) < 0 for any x < x ˆ. Then (R2) implies 1 − Eu (ˆ x, x ˆ + ε[, which proves our result. that there exists ε > 0 so that 1 − Eu (x) > −Eu′ (x) and Eu (x)′ > 0 when x ∈]ˆ
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G.2 Social Welfare
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The following preferences are considered:
U (A, x) = A + U (x)
U (x) = W
where
!Z
V
u(xν )dν 0
"
ED
I have already shown that the change in consumer surplus is given by
PT
′ ∆CS = N ∗ u(x∗mp )W (Λ∗ ) Eu (x∗sp ) − Eu (x∗mp )
The change in total producer surplus following the entry of one large firm equals its profits so
that is
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′ ∆P S = N ∗ u′ (x∗mp )x∗mp W (Λ∗ ) − cx∗mp − f
∆P S = N
∗
′ u(x∗mp )W
∗
(Λ )
Eu (x∗mp )
cx∗mp + f − u(x∗mp )W ′ (Λ∗ )
cx∗mp + f − u(x∗mp )W ′ (Λ∗ )
Thus, the change in social welfare is:
∆SW = N
∗
′ u(x∗mp )W
∗
(Λ )
Eu (x∗sp )
The first-order condition w.r.t. N evaluated at a symmetric equilibrium is given by:
′′
(p∗mp − c)x∗mp + ΛN u′ (x∗mp )W (Λ∗ )N ∗ x∗mp = f so 28 This
is easily obtained by differentiating Eu (x).
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#
′′
′
u′ (x∗mp )W (Λ∗ )x∗mp + u(x∗mp )u′ (x∗mp )W (Λ∗ )N ∗ x∗mp ∆CS + ∆P S = N ∗ u(x∗mp )W (Λ∗ ) Eu (x∗sp ) − u(x∗mp )W ′ (Λ∗ )
$
T
′
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′ ∆CS + ∆P S = N ∗ u(x∗mp )W (Λ∗ ) Eu (x∗sp ) − Eu (x∗mp ) − EW ′ (x∗mp )
where EW ′ (x∗mp ) reflects the change in expenditure.
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This expression can be further simplified using the ratio of the first-order conditions for both firms:
so that the change in social welfare can be written:
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1 + Eu′ (x∗sp ) u′ (x∗mp ) = >1 u′ (x∗sp ) 1 + Eu′ (x∗sp ) + EW ′ (x∗mp )
MA
u′ (x∗sp ) ′ ∆CS + ∆P S = N ∗ u(x∗mp )W (Λ∗ ) Eu (x∗sp ) − Eu (x∗mp ) + 1 − Eu′ (x∗mp ) + 1 − Eu′ (x∗sp ) ′ ∗ u (xmp )
(42)
And I have already shown that x∗mp < x∗sp . This expression can be positive or negative. For instance, if
welfare.
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PT
Social welfare: the case F = 0+
ED
u(x) = (1 + x)α − 1 with α < 1 and if F is not “too small”, import penetration by a foreign firm decreases social
Note that: ′
(Eu ) =
u′ [1 − Eu′ − Eu ] u ′
In the limit: x∗mp − x∗sp −→ 0. In the standard case where preference for variety is increasing, i.e. (Eu ) < 0, F →0 ≥
(42) implies that ∆CS + ∆P S > 0.
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