Multiproduct-Firm Oligopoly: an Aggregative Games Approach∗

Multiproduct-Firm Oligopoly: An Aggregative Games Approach∗

Volker Nocke† Nicolas Schutz‡

August 14, 2015

Abstract

This paper introduces a new class of demand systems to study oligopolistic pricing games with multiproduct firms. We prove that this class of demand systems coincides with the set of demand systems that can be derived from discrete/continuous choice with iid type 1 extreme-value taste shocks. We also show that these demand systems are integrable with quasi-linear preferences. The pricing game is aggregative and payoff functions are uni-modal, although not necessarily quasi-concave. Firms’ fitting-in and best-response functions can be entirely summarized by a uni-dimensional sufficient statistic, called the iota-markup. This allows us to show that, under fairly weak conditions, the pricing game has a Nash equilibrium. Under stronger conditions, this equilibrium is unique. We also provide an algorithm which exploits the aggregative nature of the game to compute the pricing equilibrium with multiproduct firms and CES demands. The algorithm always converges. As an application, we derive a number of results on the dynamic optimality of myopic merger policy under differentiated Bertrand competition.

1 Introduction

We introduce a new class of demand systems that nests the standard multinomial logit and CES demand systems. Using this demand system, we analyze an oligopoly pricing model

∗We thank Andreas Kleiner, Tim Lee, Michael Riordan as well as seminar and conference participants at SFB TR/15 2015, MaCCI Summer Institute 2015, Humboldt University and the University of Vienna for helpful suggestions. †University of Mannheim and CEPR. Email: [email protected]. ‡University of Mannheim. Email: [email protected].

1 with multiproduct firms. Exploiting the aggregative games structure of the model, we prove existence of equilibrium under fairly general conditions and provide sufficient conditions for uniqueness of equilibrium. We apply the model to static and dynamic merger analysis. Analyzing the behavior of multiproduct firms in oligopolistic markets appears to be of a first-order importance. Multiproduct firms are endemic and play an important role in the economy. Even when defining products quite broadly at the XXX-digit level, multiproduct firms account for XXX percent of sales and XXX percent of exports (XXXSOURCE). Similarly, many markets are characterized by oligopolistic competition: Even at the 5-digit industry level, concentration ratios are fairly high (XXXEVIDENCE). While there has been a lot of interest in multiproduct firms in the industrial organization and international trade literatures, researchers have generally shied away from dealing with the theoretical difficulties arising in oligopolistic models with multiproduct firms. The first source of difficulties is the high dimensionality of firms’ strategy sets. The second source is that even with “well behaved” demand systems such as the multinomial logit demand system, firms’ payoff functions are typically not quasi-concave when firms offer multiple products (Spady, 1984; Hanson and Martin, 1996). The third is that action sets are not bounded, and that it is often difficult to find natural upper bounds on prices.1 The fourth is that payoff functions typically fail to be supermodular or log-supermodular. In light of these technical difficulties, it is perhaps not surprising that the burgeoning literature on multiproduct firms in international trade has focused almost exclusively on models of monopolistic competition (Bernard, Redding and Schott, 2011; Dhingra, 2013; Mayer, Melitz and Ottaviano, 2014; Nocke and Yeaple, 2014).2 In industrial organization, multiproducts firms are at the heart of the literature on bundling but the existing models are highly stylized.3 Multiproduct firms feature very prominently in the empirical industrial organization literature on demand estimation where marginal costs are backed out under the assumption that the pricing equilibrium exists and that first-order conditions are sufficient (Berry, 1994; Berry, Levinsohn and Pakes, 1995; Nevo, 2001).4 In the first part of this paper, we introduce a new class of quasi-linear demand systems which nests standard CES and multinomial logit demand systems with heterogeneous quality and price-sensitivity parameters. We provide necessary and sufficient conditions for the

1For example, in the case of multinomial logit demand without an outside option, even a single-product firm’s best-response price goes to infinity when the prices of rivals’ offerings become large. 2An exception is Eckel and Neary (2010) who study (identical) multiproduct firms in a Cournot model with linear demand. 3Much of the bundling literature focuses on monopoly and/or two goods only (XXXREFERENCES). 4A theoretical foundation of these assumptions is missing so far, and the assumptions are likely to be violated in applications.

2 demand system to be integrable, i.e., to be derivable from a representative consumer choice problem. We show that, under the same conditions, the demand system is also derivable from a discrete-continuous choice problem with random utility.5 As the demand system satisfies the Independence of Irrelevant Alternatives (IIA) axiom, the representative consumer’s indirect (sub-)utility function can be written as as a function of a single-dimensional aggregator, which is given by the sum (over all products) of transforms of product prices. This property implies that the pricing game between multiproduct firms is aggregative; that is, each firm’s profit can be written as a function of its own prices and the single-dimensional aggregator. In the second part of the paper, we study a pricing game between multiproduct firms with arbitrary product portfolios and product-level heterogeneity in marginal costs and qualities (price-sensitivity parameters). The dimensionality of the problem is reduced, first, because the pricing game is aggregative, and, second, by showing that a firm’s multidimensional pricing strategy can be fully summarized by a unidimensional sufficient statistic. This allows us to establish equilibrium existence under mild conditions, and equilibrium uniqueness under more stringent conditions. In case equilibrium is not unique, equilibria can be Pareto-ranked for the players (firms), with firms’ ranking being the inverse of consumers’ ranking of equilibria. The reduction in the dimensionality of relevant strategy sets not only helps proving existence and uniqueness but also computing equilibria efficiently, as we show. In the special cases of CES and multinomial logit demands, an additional aggregation property obtains: A firm’s product portfolio with associated qualities and marginal costs can be fully summarized by a unidimensional sufficient statistic. In the third part of the paper, we apply the pricing game to static and dynamic merger analysis. For the special cases of CES and multinomial logit demands, we extend Nocke and Whinston’s (2010) result on the dynamic optimality of myopic merger approval policy to mergers between arbitrary multiproduct firms.

2 The Demand System: Quasi-Linear Integrability and Discrete-Continuous Choice

2.1 The Demand System Deﬁnition 1. A demand component is a triple N , (hk)k∈N , (gk)k∈N , where N is a ﬁnite and non-empty set, gk, hk are functions from R++ to R++ for every k ∈ N , and hk is strictly 5See Novshek and Sonnenschein (1979), Hanemann (1984), Dubin and McFadden (1984), Smith (2004) and Chan (2006) for references on discrete-continuous choice.

3 decreasing for every k ∈ N . 1 2 If, in addition, gk is C and hk is C for every k ∈ N , then N , (hk)k∈N , (gk)k∈N is called a diﬀerentiable demand component.

Deﬁnition 2. The demand system associated with demand component N , (hk)k∈N , (gk)k∈N N is deﬁned as follows: for every (pj)j∈N ∈ R++, for every k ∈ N ,

gk (pk) Dk N , (hj)j∈N , (gj)j∈N (pj)j∈N = P . (1) j∈N hj (pj)

Product k ∈ N is a variety of the differentiated product we are interested in. The assumption that hk is strictly decreasing for every k ∈ N means that varieties are strict substitutes. This class of demand systems nests multinomial logit (with gi(pi) = hi(pi) = ai−pi 1−σ −σ exp µ , µ > 0) and CES (with hi(pi) = pi , gi(pi) = pi , σ > 1) demands as special cases. Notice that demand is aggregative in the following sense: the demand for variety i de- P pends only on price pi and on aggregator H = j∈N hj(pj). That is, demand satisfies the Independence of Irrelevant Alternatives (IIA) axiom. As we will see later on, this property implies that the multiproduct-firm pricing game we will analyze is aggregative, in the sense that the profit of a firm depends on the vector of prices it sets and on the aggregator H. This will allow us to establish existence and uniqueness of a pricing equilibrium. Before turning our attention to the pricing game, we start by imposing consistency re- quirements on our class of demand systems. We first derive necessary and sufficient conditions under which demand system (1) can be derived from quasi-linear utility maximization. We will then show that these conditions are also necessary and sufficient for this demand system to be derivable from discrete-continuous choice.

2.2 Integrability

We recall the following deﬁnition from Nocke and Schutz (2015):

n n Deﬁnition 3. Let n ≥ 1 and D : R++ −→ R+. We say that D is quasi-linearly integrable if n n there exists a function u : R+ → R ∪ {−∞} such that for every (p0, p, y) ∈ R++ × R++ × R+ such that p · D p ≤ y, vector 1 y − p · D p ,D p is the unique solution of p0 p0 p0 p0

max {q0 + u(q)} s.t. p0q0 + p · q ≤ y, q0 ≥ 0 and q ≥ 0. (q0,q)

4 n When this is the case, we say that u (resp. v : p ∈ R++ 7→ u (D(p))) is a direct (resp. indirect) subutility function for demand system D.

In words, D is quasi-linearly integrable if it can be derived from quasi-linear utility maximization. We now adapt the deﬁnition of quasi-linear integrability to demand components:

Definition 4. A demand component is quasi-linearly integrable if the demand system associated with it is quasi-linearly integrable. Let N , (hk)k∈N , (gk)k∈N be a differentiable demand component. For every k ∈ N , we 0 2 (hk) put γk ≡ 00 (whenever this expression is well-defined). hk Theorem 1. Let N , (hk)k∈N , (gk)k∈N be a differentiable demand component. The following statements are equivalent: (i) N , (hk)k∈N , (gk)k∈N is quasi-linearly integrable.

0 (ii) There exists a strictly positive scalar α such that, for every k ∈ N , gk = −αhk. 00 P P Moreover, hk > 0 for every k ∈ N , and k∈N γk ≤ k∈N hk. P When this is the case, function v(p) = α log j∈N hj(pj) + β is an indirect subutility function for the associated demand system.

Proof. The proof relies on Nocke and Schutz (2015)’s Theorem 1, which states that a continuously differentiable demand system is quasi-linearly integrable if and only if its substitution matrix is symmetric and negative semi-definite at every price vector. In Appendix A.1, we show that symmetry of the substitution matrix is equivalent to the existence of an α > 0 0 such that gk = −αhk for every k ∈ N , and that negative semi-definiteness is equivalent to 00 P P hk > 0 for every k ∈ N and k∈N γk ≤ k∈N hk.

Notice that the representative consumer’s indirect utility is aggregative as well, in that it P only depends on the value of aggregator H = j∈N hj(pj). In the remainder of the paper, we will only work with integrable demand components. 3 We will also normalize α to 1, assume that hk is C for all k, and drop the gk’s from the definition of a demand component, since these functions can be readily derived from the hk’s. We therefore redefine demand components as follows: Definition 5. A smooth and integrable demand component is a pair N , (hk)k∈N such that 3 N is a finite and non-empty set, for every k ∈ N , hk is a C function from R++ to R++, 0 00 P P hk < 0 and hk > 0, and k∈N γk ≤ k∈N hk.

5 The demand system associated with smooth and integrable demand component N , (hk)k∈N is: 0 −hk (pk) N Dk N , (hj)j∈N (pj)j∈N = P , ∀k ∈ N , ∀ (pj)j∈N ∈ R++. (2) j∈N hj (pj) We denote the set of demand systems that can be generated by a smooth and integrable demand component by D.

1 1 Deﬁnition 6. We say that smooth and integrable demand components N , (hk)k∈N 1 and 2 2 1 2 N , (hk)k∈N 2 are equivalent if there exists a bijection σ : N −→ N such that for every 1 k ∈ N and (pj)j∈N 1 ,

1 1 2 2 D N , h (p ) = D N , h p −1 . k j j∈N 1 j j∈N 1 σ(k) j j∈N 2 σ (j) j∈N 2

In words, two smooth and integrable demand components are equivalent if they induce the same demand system (up to a relabelling of products). The following lemma will be useful in the next subsection:

Lemma 1. Let N , (hk)k∈N be a smooth and integrable demand component. There exists ˜ ˜ an equivalent smooth and integrable demand component N , hk such that hk ≥ γ˜k for k∈N every k ∈ N .

Proof. See Appendix A.2.

2.3 Discrete-Continuous Choice Deﬁnition 7. A discrete-continuous choice model of consumer demand is a pair N , (vj)j∈N , 3 where N is a ﬁnite and non-empty set, and, for every j ∈ N , vj is a C function from R++ 0 00 to R such that vj < 0 and vj ≥ 0.

For every j, vj is an indirect subutility function in a quasi-linear economy in which only 0 00 variety j and the outside good are available. Conditions vj < 0 and vj ≥ 0 are necessary and suﬃcient for vj to be an indirect subutility function (see Nocke and Schutz, 2015). Let y be the consumer’s income. We normalize the price of the outside good to 1. The consumer makes discrete-continuous choices as follows. He ﬁrst observes all varieties’ prices

(pj)j∈N , and a vector of taste shocks (εj)j∈N . If he chooses variety k ∈ N , then he consumes 0 qk(pk) = −vk(pk) units of product k (by Roy’s identity), uses the rest of his income to consume the outside good, and receives utility y + vk(pk) + εk. Therefore, the consumer

6 chooses variety k only if

y + vk(pk) + εk ≥ y + vj(pj) + εj ∀j ∈ N .

We assume that the components of vector (εj)j∈N are identically and independently drawn from a type-1 extreme value distribution. Therefore, by Holman and Marley’s theorem, variety k is chosen with probability

k(p) = Pr vk(pk) + εk = max (vj(pj) + εj) , P j∈N evk(pk) = . P vj (pj ) j∈N e

It follows that the expected demand for product k is given by

vk(pk) e 0 k(p)qk(pk) = (−v (pk)) . P P vj (pj ) k j∈N e

This motivates the following deﬁnition: Deﬁnition 8. The demand system generated by discrete-continuous choice model N , (vj)j∈N is:

vk(pk) e 0 N ∆k N , (vj) (pj) = (−v (pk)) , ∀k ∈ N , ∀ (pj) ∈ . (3) j∈N j∈N P vj (pj ) k j∈N R++ j∈N e

We denote the set of demand systems that can be generated by a discrete-continuous choice model by D.

Upon inspection of equations (2) and (3), it seems clear that, given a smooth and integrable demand component N , (hj)j∈N , one can construct a discrete-continuous choice model N , (log(hj))j∈N such that

D N , (hj)j∈N = ∆ N , (log(hj))j∈N ,

although we still need to check that N , (log(hj))j∈N is indeed a discrete-continuous choice model. Conversely, given a discrete-continuous choice model N , (vj)j∈N , one can construct

7 vj a smooth and integrable demand component N , (e )j∈N such that

vj ∆ N , (vj)j∈N = D N , (e )j∈N ,

vj although, again, we still need to check that N , (e )j∈N is indeed a smooth and integrable demand component. We state this result formally below, and prove it in the appendix:

Theorem 2. D = D: the set of demand systems that can be generated by smooth and integrable demand components and the set of demand systems that can be generated by discrete- continuous choice coincide.

Proof. See Appendix A.3.

3 The Pricing Game: Existence of an Equilibrium

3.1 Deﬁnitions Deﬁnition 9. A pricing structure is a tuple N , (hk)k∈N , F , where: • N , (hk)k∈N is a smooth and integrable demand component such that |N | ≥ 2,

•F , the set of ﬁrms, is a partition of N such that |F| ≥ 2.

Deﬁnition 10. A pricing game is a tuple N , (hk)k∈N , F, (ck)k∈N , where: • N , (hk)k∈N , F is a pricing structure,

N • and (ck)k∈N ∈ R++ is a proﬁle of marginal costs. Deﬁnition 11. The normal-form game associated with pricing game N , (hk) , F, (ck) k∈N k∈N f f is triple F, R++ , π , where for every f ∈ F, f∈F f∈F

0 f X −h (p ) N\f πf (pf , p−f ) = (pf − c ) k k , ∀pf ∈ f , ∀p−f ∈ . k k P f P −f R++ R++ k∈f j∈f hj(pj ) + j∈N \f hj(pj )

A pricing equilibrium of pricing game N , (hk) , F, (ck) is a Nash equilibrium of game k∈N k∈N f f F, R++ , π . f∈F f∈F

8 3.2 Main Idea for the Existence Proof Let N , (hk)k∈N , F, (ck)k∈N be a pricing game. There are three main difficulties associated with the equilibrium existence problem. First, nothing guarantees that payoff functions are quasi-concave. In fact, it is well known that, with standard multinomial logit demands, profit functions can fail to be quasi-concave (Spady, 1984; Hanson and Martin, 1996). Second, firms’ action sets are unbounded, and therefore not compact. Third, payoff functions are not necessarily supermodular. With single-product firms, it is possible to get around this issue by showing that profit functions are log-supermodular (see Milgrom and Shannon, 1994; Vives, 2000).6 Unfortunately, we are not aware of a similar trick for multi-product firms. The first two difficulties imply that standard existence theorems for compact games (such as Nash or Glicksberg’s theorem) based on Kakutani’s fixed-point theorem cannot be applied. The last two difficulties imply that existence theorems based on supermodularity theory and Tarski’s fixed-point theorem (see Milgrom and Shannon, 1994; Topkis, 1998) have no bite. The idea behind our existence proof is to exploit the fact that the pricing game is aggregative (Selten, 1970). In the following semi-formal discussion, we assume that first-order conditions are sufficient for optimality. The first-order conditions for each firm’s profit max- N imization problem hold at price vector p ∈ R++ if and only if for every f ∈ F and k ∈ f,

f 0 00 0 ! ∂π −hk(pk) pk − ck −hk(pk) X −hj(pj) = 1 − pk 0 + (pj − cj) = 0, ∂pk H pk −h (pk) H k j∈f

00 P −hk (pk) where H = j∈N hj(pj) is the aggregator. Put ιk(pk) = pk 0 for every k ∈ N and −hk(pk) pk > 0. Then, proﬁle of prices p is a pricing equilibrium if and only if

0 pk − ck X −hj(pj) ιk(pk) = 1 + (pj − cj) , ∀f ∈ F, ∀k ∈ f (4) pk H j∈f X and H = hj(pj). (5) j∈N

Notice that, for a given f ∈ F, the right-hand side does not depend on k ∈ f. It follows

6If f = {k} is a single-product ﬁrm, then, for every j ∈ N such that j 6= k,

2 f 0 0 ∂ log π hj(pj)hk(pk) = P > 0. ∂pk∂pj i∈N hi(pi)

9 that, in any Nash equilibrium, for any f ∈ F, and for all k, l ∈ f,

pk − ck pl − cl ιk(pk) = ιl(pl). pk pl

f pk−ck f Put diﬀerently, there exists a real number µ such that ιk(pk) = µ for every k ∈ f. pk pk−ck Assume that function pk 7→ ιk(pk) is one-to-one for every k ∈ N , and denote its pk inverse function by rk(.). Then, ﬁrm f’s strategy can be fully described by a uni-dimensional variable, µf , such that

0 f X −hj rj(µ ) µf = 1 + r (µf ) − c . j j H j∈f

Assume that the above equation has a unique solution in µf , denoted mf (H). mf is called firm f’s fitting-in function. Then, the equilibrium existence problem boils down to finding an H such that X X f H = hj rj m (H) . f∈F j∈f | {z } ≡Γ(H) In the parlance of aggregative games, Γ is called the aggregate fitting-in function. The equilibrium existence problem reduces to finding a fixed point of the aggregate fitting-in function. As we will see later on, the aggregative games approach is also useful to establish equilibrium uniqueness: the pricing game has a unique equilibrium if the following index condition is satisfied: Γ0(H) < 1 whenever Γ(H) = H. This informal presentation leaves a number of questions open. Are first-order conditions

pk−ck sufficient for optimality? Is function pk 7→ ιk(pk) one-to-one for every k? Are fitting-in pk functions well-defined? Does the aggregate fitting-in function have a fixed point? We need two more assumptions to answer all these questions in the affirmative.

3.3 Assumptions In the following, we ﬁx a pricing structure N , (hk)k∈N , F . We make the following assumption:

00 hk (x) Assumption 1. For every k ∈ N , function ιk : x ∈ R++ 7→ x 0 is non-decreasing. −hk(x) This assumption holds under multinomial logit and CES demands. Under logit demands ai−xi 1−σ (hi(xi) = e λ , λ > 0), ιk(x) = λx for every k. Under CES demands (hi(xi) = xi , σ > 1),

10 ιk(x) = σ for every k. To gain intuition, consider the monopolistic competition benchmark with single-product firms. Under monopolistic competition, firm k does not internalize the 0 −hk(pk) impact of its behavior on aggregator H. It therefore maximizes (pk −ck) H . The absolute value of the (perceived) price elasticity of firm k’s demand at price pk is therefore equal to

ιk(pk). Assumption 1 states that the price elasticity of the monopolistic competition demand for product k is non-decreasing. If this assumption were violated, then the monopolistic competition game would have undesirable properties: an equilibrium may fail to exist; or multiple equilibria may arise; the equilibrium level of the aggregator could be discontinuous in ﬁrms’ marginal costs.7 All these issues are ruled out under Assumption 1. We will see that Assumption 1 also makes the multiproduct-ﬁrm pricing game well behaved. Under Assumption 1, we can prove the following lemma:

Lemma 2. For every k ∈ N , µ¯k ≡ lim∞ ιk > 1.

Proof. See Appendix B.1.

Notice that some of theµ ¯k’s may be inﬁnite. We impose the following constraint on the

µ¯k’s:

Assumption 2. f ∀f ∈ F, ∀i, j ∈ f, µ¯i =µ ¯j ≡ µ¯ .

Assumption 2 mainly plays a technical role. We conjecture that our existence proof would still go through without it. See the discussion after Theorem 3.

3.4 Preliminary Technical Lemmas Fix a pricing structure N , (hk)k∈N , F , and assume that Assumptions 1 and 2 hold. For every k ∈ N , deﬁne

p = inf {x > 0 : ι (x) > 1} . k k

By Lemma 2, lim ι > 1. Therefore, p is a well-deﬁned a non-negative real number. In ∞ k k addition, ι (x) > 1 if and only if x > p . k k

Lemma 3. For every k ∈ N , for every x > p , γ0 (x) < 0. k k 7Notice that, under monopolistic competition, a ﬁrm always sets the same price, no matter what the competitive environment is. This comes from the fact that the indirect utility function is additively separable in (transforms of) prices, as in Bertoletti and Etro (2015).

11 Proof. See Appendix B.2.

Lemma 4. For every k ∈ N , lim∞ γk = 0. Proof. See Appendix B.3.

0 Lemma 5. For every k ∈ N , limx→∞ xhk(x) = 0. Proof. See Appendix B.4.

For every k ∈ N and x > p , deﬁne k

hk(x) ρk(x) ≡ . γk(x)

µ¯k Lemma 6. Let k ∈ N , and assume that µ¯k < ∞ and lim∞ hk = 0. Then, lim∞ ρk = . µ¯k−1 Proof. See Appendix B.5.

3.5 The Common ι-Markup Property Fix a pricing game N , (hk)k∈N , F, (ck)k∈N , and assume that Assumptions 1 and 2 hold.

Definition 12. Let f ∈ F. A profile of prices for firm f, (pk)k∈f , satisfies the common ι-markup property if there exists µf ≥ 0 such that

f pk − ck ∀k ∈ f, µ = ιk(pk). pk

Lemma 7. Let f ∈ F and k ∈ f. Deﬁne

2 pk − ck νk :(pk, ck) ∈ (pk, ck) ∈ R++ : pk > ck 7→ ιk(pk). pk

1 For every ck > 0, function νk(., ck) is a strictly increasing C -diﬀeomorphism from (ck, ∞) to f f f (0, µ¯ ). Denote its inverse function by rk(., ck). Then, for all µ ∈ (0, µ¯ ),

f ∂rk γk(rk(µ , ck)) f = f 0 f f 0 f > 0. ∂µ µ (−γk (rk(µ , ck))) − (µ − 1) (−hk (rk(µ , ck)))

In addition, r is strictly increasing in µf and c , and r (µf , c ) > p whenever µf ≥ 1. k k k k k Proof. See Appendix B.6.

0 In the following, we will often omit argument ck from functions rk and νk, and write rk f instead of ∂rk/∂µ when there is no risk of confusion.

12 3.6 The Firm’s Proﬁt-Maximization Problem Fix a pricing game N , (hk)k∈N , F, (ck)k∈N , and assume that Assumptions 1 and 2 hold. Let f ∈ F, and ﬁx H−f > 0. In this subsection, we study the following maximization problem: f −f max Π (pk) ,H , (6) f k∈f (pk)k∈f ∈R++ where 0 X −h (pk) Πf (p ) ,H−f ≡ (p − c ) k . k k∈f k k −f P H + hj(pj) k∈f j∈f

−f N\f −f P −f Notice that, if p ∈ R++ and H = k∈N \f hk(pk ), then solving maximization problem (6) gives us ﬁrm f’s best response to p−f . Let us ﬁrst rule out below-cost pricing:

f Lemma 8. Let (pk)k∈f ∈ R++, and assume that there exists k ∈ f such that pk < ck. Then,

(pk)k∈f is not a solution of maximization problem (6). Proof. See Appendix B.7.

Therefore, maximization problem (6) is equivalent to the following maximization problem (in the sense that the sets of solutions of both problems coincide):

max Πf (p ) ,H−f , (7) Q k k∈f (pk)k∈f ∈ k∈f [ck,∞)

We therefore focus on maximization problem (7) from now on. In the following two lemmas, we show that if a profile of prices satisfies the first-order conditions of problem (7), then it has the common ι-markup property, and that there exists a unique profile of prices satisfying these first-order conditions. Q Lemma 9. Let (pk)k∈f ∈ k∈f [ck, ∞). The following statements are equivalent:

(i) The ﬁrst-order conditions for maximization problem (7) hold at price proﬁle (pk)k∈f .

f (ii) Price proﬁle (pk)k∈f satisﬁes the common ι-markup property. The corresponding µ solves the following equation on interval (1, µ¯f ):

P f γj rj(µ ) µf = 1 + µf j∈f . (8) P f H−f + j∈f hj (rj(µ ))

Proof. See Appendix B.8.

13 Lemma 10. Equation (8) has a unique solution on interval (1, µ¯f ), which we denote by µˆf . Therefore, there exists a unique profile of prices that satisfies the first-order conditions of maximization problem (7). In addition,

f f −f f Π rk(ˆµ ) j∈f ,H =µ ˆ − 1.

Proof. See Appendix B.9.

Next, we show that ﬁrst-order conditions are necessary for optimality:

Lemma 11. If (pk)k∈f solves maximization problem (7), then the first-order conditions are satisfied at price profile (pk)k∈f .

Proof. See Appendix B.10.

Lemma 12. Maximization problem (7) has a solution.

Proof. See Appendix B.11.

Combining these lemmas, we get:

−f f Proposition 1. There exists a unique profile of prices pk(H ) k∈f ∈ R++ which solves maximization problem (6). At this price profile, the first order conditions hold, and condition (ii) in Lemma 9 is satisfied. Conversely, if the first order conditions of maximization problem (6) hold at price profile Q Q (pk)k∈f ∈ k∈f [ck, ∞), or, equivalently, if price profile (pk)k∈f ∈ k∈f [ck, ∞) satisfies con- −f dition (ii) in Lemma 9, then (pk)k∈f = pk(H ) k∈f . f −f −f f f Moreover, Π pk(H ) k∈f ,H =µ ˆ − 1, where µˆ is defined in Lemma 10.

3.7 Existence of a Pricing Equilibrium Fix a pricing game N , (hk)k∈N , F, (ck)k∈N , and assume that Assumptions 1 and 2 hold.

∗ N ∗ Lemma 13. If (pk)k∈N ∈ R++ is a pricing equilibrium, then pk ≥ ck for every k in N . ∗ Q Let (pk)k∈N ∈ k∈N [ck, ∞). The following statements are equivalent:

∗ (i) (pk)k∈N is a pricing equilibrium.

f∗ Q f ∗ (ii) There exist µ f∈F ∈ f∈F (1, µ¯ ) and H > 0 such that

∗ f∗ pk = rk µ , ∀f ∈ F, ∀k ∈ f,

14 ∗ X X f∗ H = hk rk µ , f∈F k∈f P f∗ γk rk(µ ) µf∗ = 1 + µf∗ k∈f , ∀f ∈ F. H∗

Proof. See Appendix B.12.

f Therefore, looking for a pricing equilibrium boils down to looking for a pair H, (µ )f∈F ∈ Q f P P f R++ × f∈F (1, µ¯ ) such that H = f∈F k∈f hk rk µ , and, for all f ∈ F,

P f ! γk rk(µ ) µf 1 − k∈f = 1. (9) H

Lemma 14. For every f ∈ F, for every H > 0, equation (9) has a unique solution in µf on interval (1, µ¯f ). Denote this solution by mf (H). Function mf (.) is C1. Its derivative is given by

1 mf (H)(mf (H) − 1) mf0(H) = − < 0. (10) P r0 mf (H) −γ0 r mf (H) H f f k∈f k( )( k( k( ))) 1 + m (H)(m (H) − 1) P f k∈N f γk(rk(m (H)))

f f f Moreover, lim0+ m =µ ¯ and lim∞ m = 1.

Proof. See Appendix B.13.

For every H > 0, put

P P f hk rk m (H) Ω(H) = f∈F k∈f . H

By Lemmas 13 and 14, the equilibrium existence problem boils down to ﬁnding an H ∈ R++ such that Ω(H) = 1.

Lemma 15. There exists H∗ > 0 such that Ω(H∗) = 1.

Proof. See Appendix B.14.

We can conclude: Theorem 3. Let N , (hk)k∈N , F be a pricing structure. If Assumptions 1 and 2 hold, then, N for every (ck)k∈N ∈ R++, pricing game N , (hk)k∈N , F, (ck)k∈N has an equilibrium.

15 In any equilibrium, the common ι-markup property is satisfied for every firm. Moreover, a firm’s equilibrium profit is equal to its common ι-markup minus 1.

Therefore, if we know that H∗ is an equilibrium aggregator level, then we can compute the indirect subutility of the representative consumer (log (H∗)), the profit of firm f ∈ F f ∗ f ∗ (m (H ) − 1) and the price of product k ∈ f (rk m (H ) ). If there are multiple equilibria, then these equilibria can be ranked: consumers prefer equilibria with higher H, whereas all firms prefer those with lower H. Our existence theorem relies on Assumption 2, but we think the result would still go through without this assumption.8 To fix ideas, suppose f = {1, 2}, i.e., firm f has two products, and assume thatµ ¯1 < µ¯2, so that Assumption 2 does not hold. The problem is f that firm f might end up setting a markup µ ∈ (¯µ1, µ¯2), so that the price of product 1 would have to go to infinity. We therefore allow firms to set infinite prices for some of their products. We extend firm f’s profit function by continuity to points (p1, ∞), (∞, p2) and −f N\f 9 (∞, ∞) for every p1, p2 > 0: for every p ∈ R++ ,

0 f −f −h1(p1) π (p1, ∞), p = (p1 − c1) P , h1(p1) + lim∞ h2 + j∈ /f hj(pj) 0 f −f −h2(p2) π (∞, p2), p = (p2 − c2) P , lim∞ h1 + h2(p1) + j∈ /f hj(pj) πf (∞, ∞), p−f = 0.

f Then, firm f’s fitting-in function, m , varies continuously fromµ ¯2 to 1 as H goes from 0 to +∞. Therefore the aggregate fitting-in function is still continuous. When H is in the + f neighborhood of 0 , m is strictly greater thanµ ¯1, so the argument developed in the proof of Lemma 15 should readily extend. This guarantees the existence of a fixed point for the aggregate fitting-in function. The monotonicity argument used in the proof of Lemma 10 should then ensure that first-order conditions are sufficient for optimality.

3.8 Comparing Equilibria

Before providing conditions under which the pricing equilibrium is unique, we brieﬂy compare equilibria. We assume throughout that Assumptions 1 and 2 hold.

8There are a number of technical details we still have to check, so this paragraph should not be taken as a full-ﬂedged formal proof. 9 By Lemma 5, the proﬁt function is indeed continuous at points (p1, ∞), (∞, p2) and (∞, ∞).

16 The following proposition states that equilibria can be Pareto-ranked among players (ﬁrms), with this ranking being the inverse of consumers’ ranking of equilibria.

∗ ∗ Proposition 2. Suppose that there are two pricing equilibria with aggregators H1 and H2 > ∗ H1 , respectively. Then, each firm f ∈ F makes a strictly larger profit in the first equilibrium ∗ (with aggregator H1 ), whereas consumers’ indirect utility is higher in the second equilibrium ∗ (with aggregator H2 ).

Proof. To see that firm f’s profit is higher in the first equilibrium, recall first from Lemma 14 that the firm’s ι−markup mf (H) is decreasing in the aggregator H, and second that, by ∗ ∗ Theorem 3, the firm’s equilibrium profit is equal to its ι−markup minus 1. As H1 < H2 , the result follows. To see that consumer welfare is higher in the second equilibrium, recall from Theorem 1 ∗ ∗ ∗ that indirect subutility is of the form log(Hi ) + β in equilibrium i ∈ {1, 2}. As H1 < H2 , the result follows.

In the following proposition we investigate the comparative statics with respect to marginal costs.

Proposition 3. Consider an increase in the marginal cost of product k ∈ f, offered by firm f. In both the equilibrium with the smallest and largest value of the aggregator H, this induces (i) an increase in the profit of any rival firm g 6= f, (ii) an increase in the prices of all goods, and (iii) a reduction in consumer surplus (indirect utility).

Proof. See Appendix B.15.

4 Equilibrium Uniqueness

4.1 Preliminaries Let N , (hk)k∈N , F be a pricing structure satisfying Assumptions 1 and 2. We introduce the following notations. For every f ∈ F, for every µf ∈ (1, µ¯f ),

µf − 1 ωf = , µf µf − 1 ω¯f = lim , µf →µ¯f µf

17 and for every k ∈ N , for every x > p , k

ιk(x) − 1 χk(x) = , ιk(x) 0 −hk(x) θk(x) = 0 . −γk(x)

The following lemma is useful to understand our uniqueness conditions: Lemma 16. Let N , (hk)k∈N , F be a pricing structure satisfying Assumptions 1 and 2. For every f ∈ F, ωf ∈ (0, ω¯f ) and k ∈ f:

f f • For every x such that χk(x) > ω , 1 − ω θk(x) > 0.

1 f • In particular, for every ck > 0, for every x ≥ rk 1−ωf , ck , 1 − ω θk(x) > 0. • In particular, for every x > p , χ (x)θ (x) ≤ 1. k k k Proof. See Appendix C.1.

We establish equilibrium uniqueness by showing that function Ω (the aggregate ﬁtting-in function divided by the aggregator) is strictly decreasing. The following lemma provides a suﬃcient condition: Lemma 17. Let N , (hk)k∈N , F be a pricing structure satisfying Assumptions 1 and 2. If, for every f ∈ F,

f f n f f o ∀ω ∈ (0, ω¯ ), ∀ (xk)k∈f ∈ (xk)k∈f ∈ R++ : ∀k ∈ f, χk(xk) > ω , f ! f ! (11) X ω θk(xk) 1 ω γ (x ) − < 1, f k k P P 1 − ω θk(xk) hk(xk) γk(xk) k∈f k∈f k∈f or, equivalently, if10

f f n f f o ∀ω ∈ (0, ω¯ ), ∀ (xk)k∈f ∈ (xk)k∈f ∈ R++ : ∀k ∈ f, χk(xk) > ω , f (12) X f 1 − ω ρj(xj) γi(xi)γj(xj) ω θi(xi) f − ρi(xi) < 0, 1 − ω θi(xi) i,j∈f

then pricing game N , (hk)k∈N , F, (ck)k∈N has a unique equilibrium for every (ck)k∈N ∈ N R++. Proof. See Appendix C.2.

10 hk Recall that ρk = . γk

18 4.2 A First Uniqueness Theorem

As can be seen by inspecting conditions (11) and (12), whether or not the equilibrium is unique depends on whether the ρi’s are big or small compared to the θj’s. Let f ∈ F. We introduce two new assumptions: Assumption 3. For every k ∈ f, ρ is non-decreasing on p , ∞ , or, equivalently, k k

ρ (x) ≥ θ (x), ∀x > p . k k k

Since 0 0 −γk ρk = (ρk − θk) , γk both wordings are indeed equivalent.

Assumption 4.

max sup θk(x) ≤ min inf ρk(x). k∈f x>p k∈f x>p k k Notice that Assumption 4 implies Assumption 3. Both assumptions hold under multinomial logit and CES demands. Under multinomial logit demands, ρk = θk = 1; under CES σ demands, ρk = θk = σ−1 . Equilibrium uniqueness follows almost immediately under Assumption 4: Theorem 4. Let N , (hk)k∈N , F be a pricing structure. If Assumptions 1 and 2 hold, and if Assumption 4 holds for every f ∈ F, then pricing game N , (hk)k∈N , F, (ck)k∈N has a N unique equilibrium for every (ck)k∈N ∈ R++. Proof. See Appendix C.3.

Although Assumption 4 is quite convenient for uniqueness, it is also rather strong. Start with the following deﬁnition:

R++ Deﬁnition 13. Let h ∈ R++ . We say that function h is logit on interval I ⊆ R++ if there exists (α, λ) ∈ R++ such that h(x) = α e−λx for all x ∈ I.

We say that function h is (non-homothetic) CES on interval I ⊆ R++ if there exists (α, β, σ) ∈ 1−σ R++ × R+ × (1, ∞) such that h(x) = (αx + β) for all x ∈ I. Proposition 4. Let N , (hk)k∈N , F be a pricing structure satisfying Assumptions 1 and 2. Let f ∈ F, and assume that Assumption 4 holds for ﬁrm f. Let k ∈ f. Assume that lim h = 0, and that lim θ exists. Then, h is logit or CES on (p , ∞). ∞ k ∞ k k k

19 Proof. See Appendix C.4. Proposition 5. Let N , (hk)k∈N , F be a pricing structure satisfying Assumptions 1 and 2. Let f ∈ F, and assume that Assumption 3 holds for ﬁrm f. Let k ∈ f. Assume that µ¯f = ∞, lim∞ hk = 0, ρk ≥ 1 and lim∞ θk exists. Then, hk is logit on (ck, ∞).

Proof. See Appendix C.5.

These propositions tell us two things. First, multinomial logit and CES demands are quite special. Second, Assumption 4 (and, to some extent, Assumption 3) is quite strong when lim∞ hk = 0. To improve on this, we will prove a uniqueness theorem which does not rely on Assumption 4.

4.3 Other Uniqueness Results Fix a pricing structure N , (hk)k∈N , F satisfying Assumptions 1 and 2. We start by proving two technical lemmas, which will be useful to ﬁnd upper bounds for the left-hand side of expression 12.

Lemma 18. Let f ∈ F. Assume that Assumption 3 holds for ﬁrm f, that lim∞ hk = 0 for all k ∈ f, and that µ¯f < ∞. Then, for every ωf ∈ (0, ω¯f ), for every k ∈ f, for every x > 0 f such that χk(x) > ω , 1 − ω¯f 1 1 ≤ ρ (x) ≤ . ω¯f 1 − ωf k ω¯f Proof. See Appendix C.6.

Lemma 19. For every ω¯ ∈ (0, 1], for every ω ∈ (0, ω¯), deﬁne

1 − ω¯ 1 1 2 1 − ωz 1 − ωy φ :(y, z) ∈ , 7→ ωy + ωz − y − z. ω,ω¯ ω¯ 1 − ω ω¯ 1 − ωy 1 − ωz

∗ ∗ ∗ There exists a threshold ω ∈ (0, 1) (ω ' 0.64) such that if ω¯ ≤ ω , then φω,ω¯ ≤ 0 for all ω ∈ (0, ω¯).

Proof. See Appendix C.7. Theorem 5. Let N , (hk)k∈N , F be a pricing structure satisfying Assumptions 1 and 2.

Let f ∈ F. Assume that Assumption 3 is satisfied for firm f, that lim∞ hk = 0 for every k ∈ f, and that ω¯f ≤ ω∗. Then, condition (12) holds for firm f.

Proof. See Appendix C.8.

20 1 Consider the following function: h(x) = log(1+ex) . This function satisﬁes the integrability conditions and Assumptions 1 and 3. Assumption 4 does not hold, butµ ¯ = 2 (i.e.,ω ¯ = ∗ 1/2 < ω ) and lim∞ h = 0. Therefore, equilibrium uniqueness follows from Theorem 5, but not from Theorem 4.

In Appendix C, we prove a number of additional uniqueness results. We first show that, under certain symmetry or monotonicity conditions, Theorem 5 extends to settings where lim∞ hk > 0 for some k’s (Propositions 15 and 16; Corollaries 1 and 2 in Section C.9). Next, under Assumption 3, we prove that the equilibrium is unique if firms’ products are symmetric (Proposition 17), or if firms are single-product firms (Corollary 3). When Assumption 3 does not hold, we show that the equilibrium is unique, provided that marginal costs are high enough and that lim∞ hk > 0 for at least one k (Proposition 18). Similarly, the equilibrium P is unique, provided that lim∞ k∈N hk is high enough (Proposition 19); put differently, the equilibrium is unique if the outside option is attractive enough.

4.4 A Cookbook for Constructing Pricing Structures that Yield a Unique Pricing Equilibrium Proposition 6. Let N , (hk)k∈N , F be a pricing structure satisfying Assumptions 1, 2, and 4 for every f ∈ F. Assume that ρk ≥ 1 for every k ∈ N . Let Ne, ehk , Fe be such that Ne is a ﬁnite set containing at least two elements, Fe is a k∈Ne partition of Ne containing at least two elements, and

2 2 ∀g ∈ Fe, ∃f ∈ F, ∀j ∈ g, ∃k ∈ f, ∃ (αj, βj, δj, j) ∈ R++ × R+,

∀x > 0, ehj(x) = αjhk(βjx + δj) + j.

Then, Ne, ehk , Fe is a pricing structure satisfying Assumptions 1, 2, and 4 for every k∈Ne f ∈ Fe. In addition, ρek ≥ 1 for every k ∈ Ne. Proof. See Appendix C.11.

If we know that function h satisfies h > 0, h0 < 0, h00 > 0, ι0 ≥ 0, ρ ≥ 1 and sup θ ≤ inf ρ, then we can construct a firm’s set of products by defining

hk(x) = αkh(βkx + δk) + k,

21 2 2 f where (αk, βk, δk, k)k∈{ ∈ R++ × R+ . By Proposition 6, (hk)k∈{ satisfies all the assumptions we need for uniqueness. For instance, we can construct a multiproduct-firm pricing game, where firm f has CES products (firm f’s base h is h(x) = x1−σf ) with heterogeneous qualities, firm f 0 also has CES products, potentially with σf 0 6= σf , and heterogeneous qualities, firm f 00 has logit products (firm f 00’s base h is h(x) = e−x) with heterogeneous qualities and price sensitivity parameters, etc. Other candidates for the base h include −x 1 1 h(x) = exp (e ), h(x) = 1 + 1+e1+x and h(x) = 1 + cosh(2+x) ,...

5 Multinomial Logit and CES Demands

5.1 The CES case

In this section, we study a multiproduct-ﬁrm pricing game with CES demands and heterogeneous qualities and productivities. Let N be a ﬁnite set containing at least two elements. 1−σ For every k ∈ N , for every x > 0, put hk(x) = akx , where ak > 0 is the quality of product k, and σ > 1 is the elasticity of substitution. We have already shown that demand component N , (hk)k∈N is smooth and integrable. Let N , (hk)k∈N , F, (ck)k∈N be a pricing 1−σ game based on CES demand component N , (hk)k∈N . Put h(x) = x for all x > 0. As discussed earlier, the corresponding ι is ι(x) = σ, and the corresponding ρ and θ are both σ equal to σ−1 . By Theorem 4 and Proposition 6, pricing game N , (hk)k∈N , F, (ck)k∈N has a unique equilibrium.

Firm f’s fitting-in function is pinned down by equation (9), which involves functions γk σ−1 1−σ and rk. With CES demands, γk(x) = σ akx . Recall that rk is the inverse function of pk−ck νk : pk 7→ ιk(pk). With CES demands, ιk = σ, so that νk is just σ times the Lerner pk f f f ck f µ index. Therefore, for every µ ∈ [1, σ), rk(µ ) = µf . From now on, we redefine µ as σ , 1− σ so that µf is firm f’s Lerner index, and takes values between 1/σ and 1. Equation (9) can then be rewritten as follows:

 1−σ  P σ−1 ck k∈f σ ak 1−µf σµf 1 −  = 1. (13)  H 

f P 1−σ Put T = k∈f akck . Simplifying and rearranging terms in (13), we get:

1 µf = . (14) T f f σ−1 σ − (σ − 1) H (1 − µ )

22 It follows from Lemma 14 that equation (14) has a unique solution. This implicitly defines a T f T f function m H . Firm f’s fitting-in function is H 7→ m H . An immediate implication is that firms f and g have the same type (T f = T g) if and only if they share the same fitting-in function. Next, we claim, that if firms f and g have the same type, then their contributions to the aggregator are the same. To see this, we introduce the following notations: for a given level 1−σ akpk f P of aggregator H, sk = H is the market share of product k ∈ N , and s = k∈f sk is the market share of firm f ∈ F. Then, for every f ∈ F and k ∈ f,

a r µf 1−σ 1 − µf σ−1 a c1−σ s = k k = k k . k H H

Therefore,

f σ−1 1−σ f f σ−1! f X 1 − µ akc T T T sf = k = 1 − m ≡ S . H H H H k∈f

T f Firm f’s market share function is H 7→ S H . Therefore, firms f and g share the same market share function if and only if T f = T g. Put differently, firm f and g’s contributions to the aggregator are identical if and only if they have the same type. Recall that H is an equilibrium aggregator level if and only if Ω(H) = 1, where

1 X X Ω(H) = h r mf (H) , H k k f∈F k∈f X X = sk, f∈F k∈f X T f = S . H f∈F

In words, H is an equilibrium aggregator level if and only if firms’ market shares add up to 1. Last, we claim that, if firms f and g have the same type, then they earn the same profit: for every f ∈ F,

−σ X akp πf = (p − c ) k , k k H k∈f

23 1−σ X pk − ck akp = k , pk H k∈f T f T f = m S , H H T f ≡ π . H

We summarize these ﬁndings in the following proposition:

f P 1−σ Proposition 7. Let f and g be two CES multiproduct ﬁrms. Put T = k∈f akck and g P 1−σ T = k∈g akck . The following assertions are equivalent:

(i) T f = T g.

(ii) Firms f and g have the same markup ﬁtting-in function.

(iii) Firms f and g have the same market share ﬁtting-in function.

(iv) Firms f and g have the same proﬁt ﬁtting-in function.

Under CES demands, firms’ types are aggregative as well. Firms f and g may differ widely in terms of product portfolios, productivity and product qualities, but if their types are the same, i.e., if T f = T g, then they share the same fitting-in functions. This implies that, no matter what the competitive environment is, these two firms will always behave in the exact same way. If we replace firm f by firm g, then the equilibrium aggregator level will not change, the behavior of firm f’s rivals will not be affected, and firm g will end up charging the same markup, having the same market share, and earning the same profit as firm f. Interestingly, given a multiproduct firm f, there always exists an equivalent single- product firm. To see this, define firm fˆ as a firm selling only one product with quality P 1−σ f fˆ ˆ aˆ = k∈f akck and marginal costc ˆ = 1. Then, T = T , and firms f and f are therefore equivalent. We also obtain the following comparative statics results:

Proposition 8. In a multiproduct-ﬁrm pricing game with CES demands,

(i) m0,S0, π0 > 0.

dH∗ dµf∗ dsf∗ dπf∗ (ii) For every f ∈ F, dT f , dT f , dT f , dT f > 0, where superscript ∗ indicates equilibrium values, and d/dT f is the total derivative with respect to T f (taking into account the impact of T f on the equilibrium aggregator level).

24 dµg∗ dsg∗ dπg∗ (iii) For every f, g ∈ F, f 6= g, dT f , dT f , dT f < 0.

Proof. See Appendix D.1.

Point (i) says that a firm charges a high markup, has a high market share, and makes high profits if it has many products, if it is highly productive, if it sells high-quality products (high T f ), or if it operates in a less competitive environment (low H). Points (ii) and (iii) say that if firm f’s type increases, then consumers benefit, firm f’s markup, market share and profit increase, to the detriment of its rivals.

5.2 The Logit Case

In this section, we study a multiproduct-firm pricing game with logit demands and heterogeneous qualities and productivities. Let N be a finite set containing at least two elements. ak−x For every k ∈ N , for every x > 0, put hk(x) = e λ , where ak ∈ R is the quality of product k, and λ > 0 is a substitutability parameter. We have already shown that demand component N , (hk)k∈N is smooth and integrable. Let N , (hk)k∈N , F, (ck)k∈N be a pricing −x game based on logit demand component N , (hk)k∈N . Put h(x) = e for all x > 0. As discussed earlier, the corresponding ι is ι(x) = x, and the corresponding ρ and θ are both equal to 1. By Theorem 4 and Proposition 6, pricing game N , (hk)k∈N , F, (ck)k∈N has a unique equilibrium. As in the previous section, we want to reexpress firm f’s fitting-in function. With logit

pk−ck pk−ck f demands, γk(x) = hk(x), and νk(pk) = ιk(pk) = . Therefore, for every µ ∈ [1, ∞), pk λ f f rk(µ ) = λµ + ck. Equation (9) can then be rewritten as follows:

f ! 1 X ak − ck − λµ µf 1 − exp = 1. (15) H λ k∈f

f P ak−ck Put T = k∈f exp λ . Simplifying and rearranging terms in (15), we get:

f T f µf 1 − e−µ = 1. (16) H

f T f This uniquely pins down a function m(.) such that µ = m H . As before, deﬁne sk =

25 ak−pk λ f P e /H and s = k∈f sk. Then,

a −c −λµf k k f f f X e λ T T T sf = = exp −m ≡ S . H H H H k∈f

P T f We can then rewrite equilibrium condition Ω(H) = 1 as f∈F S H = 1. In addition, it is straightforward to check that

T f T f T f πf = µf sf = m S ≡ π . H H H

Therefore, ﬁrms’ types are still aggregative under logit demands:

f P ak−ck Proposition 9. Let f and g be two logit multiproduct ﬁrms. Put T = k∈f exp λ g P ak−ck and T = k∈g exp λ . The following assertions are equivalent:

(i) T f = T g.

(ii) Firms f and g have the same markup ﬁtting-in function.

(iii) Firms f and g have the same market share ﬁtting-in function.

(iv) Firms f and g have the same proﬁt ﬁtting-in function.

We also derive the following comparative statics:

Proposition 10. In a multiproduct-ﬁrm pricing game with logit demands,

(i) m0,S0, π0 > 0.

dH∗ dµf∗ dsf∗ dπf∗ (ii) For every f ∈ F, dT f , dT f , dT f , dT f > 0.

dµg∗ dsg∗ dπg∗ (iii) For every f, g ∈ F, f 6= g, dT f , dT f , dT f < 0.

Proof. See Appendix D.2.

To summarize, we obtain type aggregation both under CES and logit demands. With CES (resp. logit) demands, the relevant ι-markup is the Lerner index (resp. the unnormalized markup), and the relevant sf is ﬁrm f’s market share in value (resp. in volume).

26 5.3 An Algorithm to Solve Multiproduct Pricing Games with CES Demands

Numerically solving for the equilibrium of a multiproduct-firm pricing game in an industry with many firms and products can be a daunting task with standard methods, due to the high dimensionality of the problem. Exploiting the aggregative structure of the pricing game allows us to reduce this dimensionality tremendously: instead of solving a system of |N | non- linear equations in |N |, we only need to look for and H > 0 such that Ω(H) = 1. Of course, there usually will not be a closed-form expression for Ω(.), so we still need to approximate this function numerically. But Ω(H) is simple to compute as well, since all we need to do is solve for |F| separate equations, each with one unknown. Below, we describe how this general approach can be implemented to solve a multiproduct-firm pricing game with CES demands. The algorithm use two nested loops. The inner loop computes Ω(H) for a given H. The outer loop iterates on H. We start by describing the inner loop. Fix some H > 0. As argued in Section 5.1, we need to compute

f T σ−1 sf = 1 − µf , H where µf is the unique solution of

f 1 µ = , T f f (σ−1) σ − (σ − 1) H (1 − µ ) or, equivalently, T f µ σ − (σ − 1) (1 − µf )σ−1 − 1 = 0. (17) f H | {z } ≡φf (µf ) To do so, we solve equation (9) numerically using the Newton-Raphson method. The derivative of φf can be computed analytically:

T f T f φf0(µf ) = σ − (σ − 1) (1 − µf )σ−1 + µf (σ − 1)2 (1 − µf )σ−2, H H so we do not need to take ﬁnite diﬀerences to compute the Newton step. The usual problem with the Newton-Raphson method is that it may fail to converge if starting values are not good enough. This is a potentially major issue, because the value of Ω(H) used by the

27 outer loop would then be inaccurate. Fortunately, the following starting value guarantees convergence: 1 ! 1 H σ−1 µf = max , 1 − . 0 σ T f

In fact, the Newton method converges extremely fast (usually fewer than 5 steps). Notice, in addition, that this method can be easily vectorized by stacking up the µf ’s in a vector. The outer loop iterates on H to solve equation Ω(Γ) − 1 = 0. This can be done by using standard derivative-based methods (we currently use Matlab’s implementation of the trust-region dogleg algorithm). The Jacobian can be computed analytically:

X T g T g Ω0(H) = − S0 , H2 H g∈F g σ−1 −1 X T (1 − µg) = H . H 2 T g g σ−2 g 2 g∈F 1 + (σ − 1) H (1 − µ ) (µ )

We use the value of H that would prevail under monopolistic competition as starting value P f 1 σ−1 11 (H0 = f∈F T 1 − σ ), and we always get convergence (usually in about 20 steps).

6 Application to Merger Policy

We first provide a simple static merger analysis under the assumption that the merged firm continues to produce the same set of products as the merger partners did before the merger. We allow for merger-specific synergies in marginal cost, under the restriction that the merger- induced fractional change in marginal cost is the same for all products offered by the merged firm. More formally, we consider a merger between firms f and g. Let H∗ (resp., Hˆ ∗) denote the equilibrium value of the aggregator before (resp., after) the merger. As indirect utility is increasing in the value of that aggregator, we say that the merger is CS-increasing (resp., CS-decreasing) if Hˆ ∗ > H∗ (resp., Hˆ ∗ < H∗); it is CS-neutral if Hˆ ∗ = H∗. For simplicity, we assume for now that the merged firm M continues to offer each product k ∈ f ∪ g (i.e., M = f ∪ g), and that the post-merger marginal cost of product k ∈ f ∪ g is cˆk = (1 − η)ck, where η ≤ 1 is a (possibly negative) synergy parameter.

11In Breinlich, Nocke and Schutz (2015), we use this algorithm to calibrate an international trade model with two countries, 160 manufacturing industries, CES demands and oligopolistic competition.

28 Proposition 11. In the equilibrium with the smallest (resp., largest) value of the aggregator H, there exists a cutoff ηˆ > 0 (which may differ across equilibria) such that the merger between firms f and g is CS-increasing if and only if η > ηˆ. In the absence of synergies (η = 0), the merger raises the prices of all goods as well as the joint profit of the merger partners.

Proof. See Appendix E.1

That is, for a merger to be CS-increasing, the merger has to involve synergies.

6.1 Dynamic Merger Review with CES/Logit Demand

In the remainder of this section, we assume that demand is either of the CES or multinomial logit forms. As shown in Section 5, in this case an additional aggregation property obtains: a firm’s product portolio (with heterogeneous qualities and marginal costs) can be fully f f P 1−σ summarized by its one-dimensional type T , where T = k∈f akck in the case of CES f P ak−ck demand and T = k∈f exp λ . For the following merger analysis, this allows us to expense with any restriction on merger-specific synergies. Consider a merger M between firms f and g. As before, we denote by H∗ and Hˆ ∗ the pre- and post-merger values of the aggregator, respectively. Assume that the merger partners’ pre-merger types are T f and T g, respectively. Let T M denote the merged firm’s post-merger type. We do not impose any restriction on the merged firm’s product portfolio: the set of products may or may not coincide with those previously offered by the merger partners, and for those products that the merged firm continues to produce, the qualities and marginal costs may differ arbitrarily from those pre-merger. We have:

Lemma 20. There exists a cutoﬀ TˆM > 0 such that merger M is CS-neutral if T M = TˆM , CS-increasing if T M > TˆM , and CS-decreasing if T M < TˆM . Moreover, if merger M is CS-nondecreasing (i.e., either CS-neutral or CS-increasing), then it is (strictly) proﬁtable.

Proof. See Appendix E.2.

We now turn to studying the interaction between mergers. Consider two mergers, M1 and M2, and assume that these mergers are disjoint, i.e., no firm takes part in more than one merger. In the context of a homogeneous goods Cournot model, Nocke and Whinston (2010) have established that that there is a sign-preserving complementarity in the consumer surplus effect of (disjoint) mergers that share the same sign in terms of their consumer surplus effect.

29 The following proposition shows that this result carries over to mergers between arbitrary multiproduct ﬁrms, provided demand takes the CES or multinomial logit forms.

Proposition 12. There is a sign-preserving complementarity in the consumer surplus effect of disjoint mergers that share the same sign in terms of their consumer surplus effect.. If merger Mi is CS-nondecreasing (and hence proﬁtable) in isolation, it remains CS- nondecreasing (and hence proﬁtable) if another merger Mj, j 6=, that is CS-nondecreasing in isolation takes place. If merger Mi is CS-decreasing in isolation, it remains CS-decreasing if another merger Mj, j 6=, that is CS-decreasing in isolation takes place.

Proof. See Appendix E.3.

Nocke and Whinston (2010)’s result on the interaction of CS-increasing and CS-decreasing mergers also carries over to our setting with multiproduct ﬁrms:

Proposition 13. Suppose that mergers M1 and M2 are CS-nondecreasing and CS-decreasing, respectively, in isolation. Then, merger M1 is CS-increasing (and hence profitable), conditional on merger M2 taking place. Moreover, the joint profit of the firms involved in M1 is strictly larger if both mergers take place than if neither does.

Proof. The proof is identical to that of Proposition 2 in Nocke and Whinston (2010). It involves inverting the order of the two mergers: at the ﬁrst step, merger M2 and, at the second step, merger M1. As consumer surplus must, by assumption, be (weakly) higher after both mergers have taken place than before, and because consumer surplus (strictly) falls at step 1 (again, by assumption), consumer surplus must (strictly) increase at step 2. That is,

M1 is CS-increasing, conditional on M2 taking place. By Lemma 20, this implies that the joint profit of the firms in M1 must go up at step 2. Finally, we assert that the joint profit of the firms in M1 must go up at step 1 as well. This follows from an argument identical to that used in the proof of Proposition 2, as the CS-decreasing merger at step 1 induces a reduction in the equilibrium value of the aggregator H∗, which benefits all outsiders to that merger, including the firms involved in M1.

We now embed our pricing game in a dynamic model with endogenous mergers and merger policy, as in Nocke and Whinston (2010). There are T periods, and a set {M1,M2, ..., MK } of disjoint potential mergers. Merger Mk becomes feasible at the beginning of period t with P probability pkt ∈ [0, 1], where t pkt ≤ 1. Conditional on becoming feasible, the post-merger 12 type of the merged ﬁrm Mk is drawn from some distribution Ckt. The feasibility of a 12The main result would not change if we assumed instead an arbitrary stochastic process governing (i) when mergers become feasible, and (ii) the merged ﬁrms’ post-merger type.

30 particular merger (including its efficiency) is publicly observed by all firms. In each period, the firms involved in a feasible and not-yet-approved merger decide whether or not to propose their merger to the antitrust authority. We assume that bargaining is efficient so that the merger partners propose the merger if and only if it is in their joint interest to do so. Given a set of proposed mergers, the antitrust authority then decides which mergers to approve (if any). An approved merger is consummated immediately. Finally, at the end of each period, the firms play the pricing game, given current market structure. All firms as well as the antitrust authority discount payoffs with factor δ ≤ 1. Following Nocke and Whinston (2010), we define a myopically CS-maximizing merger policy as an approval policy, where in each period, given the set of proposed mergers and current market structure, the antitrust authority approves a set of mergers that maximizes consumer surplus in the current period. The most lenient myopically CS-maximizing merger policy is a myopically CS-maximizing merger policy that approves the largest such set (i.e., including CS-neutral mergers). (As shown in Nocke and Whinston (2010) such a policy is well-defined.) The following proposition shows that Nocke and Whinston (2010)’s result on the dynamic optimality of a myopic merger approval policy carries over to our multiproduct firm setting:

Proposition 14. Suppose the antitrust authority adopts the most lenient myopically CS- maximizing merger policy. Then, all feasible mergers being proposed in each period after any history is a subgame-perfect Nash equilibrium for the ﬁrms. The equilibrium outcome maximizes discounted consumer surplus (indirect utility) for any realized sequence of feasible mergers. Moreover, for each such sequence, every subgame-perfect Nash equilibrium results in the same optimal sequence of period-by-period consumer surpluses.

Proof. The result follows from Lemma 20 and Propositions 12 and 13, which are the anologs of Corollary 1 and Proposition 1 and 2 in Nocke and Whinston (2010). See Nocke and Whinston (2010) for details.

7 Concluding Remarks

TBW

31 A Proofs for Section 2

A.1 Proof of Theorem 1

We ﬁrst state and prove two preliminary technical lemmas, which will be useful to prove Theorem 1:

n Lemma 21. For every n ≥ 1, for every (αi)1≤i≤n ∈ R , deﬁne

  1 − α1 1 ··· 1    1 1 − α2 ··· 1  M (α ) =   i 1≤i≤n  . . . .   . . .. .    1 1 ··· 1 − αn

Then,13    n ! n n  Y X  Y  det M (αi)1≤i≤n = (−1)  αk −  αk  k=1 j=1 1≤k≤n  k6=j Moreover, matrix M (αi)1≤i≤n is negative semi-deﬁnite if and only if αi ≥ 1 for all 1 ≤ i ≤ n and n X 1 ≤ 1. α i=1 i Proof. We prove the ﬁrst part of the lemma by induction on n ≥ 1. Start with n = 1. Then,

1 det M (αi)1≤i≤n = 1 − α1 = (−1) (α1 − 1), so the property is true for n = 1. Next, let n ≥ 2, and assume the property holds for all 1 ≤ m < n. By n-linearity of the determinant,

1 1 ··· 1 1 1 ··· 1

0 1 − α2 ··· 1 1 1 − α2 ··· 1 det M (α ) = (−α ) + . i 1≤i≤n 1 ......

0 1 ··· 1 − αn 1 1 ··· 1 − αn

Applying Laplace’s formula to the ﬁrst column, we can see that the ﬁrst determinant is,

13We adopt the convention that the product of an empty collection of real numbers is equal to 1.

32 in fact, equal to det M (αi)2≤i≤n . The second determinant can be simpliﬁed by using n-linearity one more time:

1 1 ··· 1 1 0 ··· 1 1 1 ··· 1

1 1 − α ··· 1 1 1 ··· 1 1 1 ··· 1 2 . . . . = −α2 . . . . + . . . . , ......

1 1 ··· 1 − αn 1 0 ··· 1 − αn 1 1 ··· 1 − αn

= −α2 det M 0, (αi)3≤i≤n + 0, where the second line follows again from Laplace’s formula and from the fact that the ﬁrst two rows of the second matrix in the ﬁrst-line’s right-hand side are colinear. Therefore,

det M (αi)1≤i≤n = − α1 det M (αi)2≤i≤n − α2 det M 0, (αi)3≤i≤n ,    n ! n n−1  Y X  Y  = − α1(−1)  αk −  αk  k=2 j=2 2≤k≤n  k6=j n ! n−1 Y − α2(−1) 0 − αk , k=3     n ! n n n  Y X  Y  Y  = (−1)  αk −  αk − αk ,  k=1 j=2 1≤k≤n  k=2  k6=j    n ! n n  Y X  Y  = (−1)  αk −  αk .  k=1 j=1 1≤k≤n  k6=j

We now turn our attention to the second part of the lemma. Assume first that matrix M (αi)1≤i≤n is negative semi-definite. Then, all its diagonal terms have to be non-positive, i.e., αi ≥ 1 for all i. Besides, the determinant of this matrix should be non-negative (resp. non-positive) if n is even (resp. odd). Put differently, the sign of the determinant should be (−1)n or 0. Since the α’s are all different from zero, this determinant can be simplified as follows: n ! n ! n Y X 1 det M (αi)1≤i≤n = (−1) αk 1 − . αk k=1 k=1

33 This expression has sign (−1)n or 0 if and only if Pn 1 ≤ 1. k=1 αk Conversely, assume that the α’s are all ≥ 1, and that Pn 1 ≤ 1. The characteristic k=1 αk polynomial of matrix M (αi)1≤i≤n is deﬁned as

1 − α1 − X 1 ··· 1

1 1 − α − X ··· 1 2 P (X) = ......

1 1 ··· 1 − αn − X

This determinant can be calculated using the ﬁrst part of the lemma. For every X > 0,

n ! n ! n Y X 1 (−1) P (X) = (αk + X) 1 − , αk + X k=1 k=1 | {z } >0 n ! n ! Y X 1 > (αk + X) 1 − , αk k=1 k=1 | {z } | {z } >0 ≥0 > 0.

Therefore, P (X) has no strictly positive root, matrix M (αi)1≤i≤n has no strictly positive eigenvalue, and this matrix is therefore negative semi-deﬁnite.

Lemma 22. Let M be a symmetric n-by-n matrix, λ 6= 0, and 1 ≤ k ≤ n. Let Ak be the matrix obtained by dividing the k-th line and the k-th column of M by λ. Then, M is negative semi-deﬁnite if and only if Ak is negative semi-deﬁnite.

n k Proof. Suppose M is negative semi-deﬁnite, and let X ∈ R . Write A as (aij)1≤i,j≤n and M as (mij)1≤i,j≤n. Finally, deﬁne Y as the n-dimensional vector obtained by dividing X’s k-th component by λ. Then,

n n 0 k X X X A X = aijxixj, i=1 j=1  

 X X  X 2 =  aijxixj + 2xk aikxi + xkakk, 1≤i≤n 1≤j≤n  1≤i≤n i6=k j6=k i6=k

34   2  X X  xk X xk =  mijxixj + 2 mikxi + mkk, λ λ 1≤i≤n 1≤j≤n  1≤i≤n i6=k j6=k i6=k  

 X X  X 2 =  mijyiyj + 2yk mikyi + y mkk, 1≤i≤n 1≤j≤n  1≤i≤n i6=k j6=k i6=k = Y 0MY, ≤ 0, since M is negative semi-deﬁnite.

Therefore, Ak is negative semi-deﬁnite. The other direction is now immediate, since M can be obtained by dividing the k-th line and the k-th column of matrix Ak by 1/λ.

We now have all we need to prove Theorem 1:

Proof. To simplify notation, assume without loss of generality that N = {1, . . . , n}, and let D(.) be the demand system associated with the demand component under consideration. For ∂Di every p >> 0, put J(p) = ∂p (p) . Theorem 1 in Nocke and Schutz (2015) states j 1≤i,j≤n that D is quasi-linearly integrable if and only if J(p) is symmetric and negative semi-deﬁnite for every p >> 0. We ﬁrst show that matrix J(p) is symmetric for every p if and only if there exists a strictly 0 positive scalar α such that, for every k ∈ N , gk = −αhk. If J(p) is symmetric for every p, then, for every 1 ≤ i, j ≤ n such that i 6= j, for every p >> 0,

0 0 h (pj)gi(pi) h (p )g (p ) − j = J (p) = J (p) = − i i j j . P 2 i,j j,i P 2 k∈N hk(pk) k∈N hk(pk)

It follows that, for every 1 ≤ i ≤ n, for every x > 0,

h0 (x) h0 (1) i = 1 ≡ −β (18) gi(x) g1(1)

0 If β = 0, then hi = 0 for every i, which violates the assumption that hi is strictly decreasing. 0 Therefore, β 6= 0, and we can deﬁne α ≡ 1/β. It follows that gi = −αhi. Since gi > 0 and 0 hi ≤ 0, we can conclude that α > 0. Conversely, if there exists a strictly positive scalar α

35 0 such that, for every k ∈ N , gk = −αhk, then, for every 1 ≤ i, j ≤ n, i 6= j, for every p >> 0,

0 0 0 h (pj)gi(pi) h (pj)h (pi) J (p) = − j = α j i = J (p), i,j P 2 P 2 j,i k∈N hk(pk) k∈N hk(pk) and matrix J(p) is therefore symmetric for every p.

0 Next, suppose that there exists α > 0 such that, for every 1 ≤ k ≤ n, gk = −αhk. We 00 want to show that J(p) is negative semi-deﬁnite for every p >> 0 if and only if hk > 0 for Pn Pn every 1 ≤ k ≤ n, and k=1 γk ≤ k=1 hk. P Fix p >> 0. To ease notation, we write hk = hk(pk) for every k, and deﬁne H ≡ k∈N hk. We obtain the following expression for matrix J(p):

 0 2 00 0 0 0 0  (h1) − h1H h1h2 ··· h1hn  2  α  h0 h0 (h0 ) − h00H ··· h0 h0  J(p) =  2 1 2 2 2 n  . 2  . . . .  H  . . .. .    0 0 0 0 0 2 00 hnh1 hnh2 ··· (hn) − hnH

J(p) is negative semi-deﬁnite if and only if

 0 2 00 0 0 0 0  (h1) − h1H h1h2 ··· h1hn  2   h0 h0 (h0 ) − h00H ··· h0 h0   2 1 2 2 2 n   . . . .   . . .. .    0 0 0 0 0 2 00 hnh1 hnh2 ··· (hn) − hnH is negative semi-deﬁnite. Applying Lemma 22 n times (by dividing row k and column k by 0 hk, 1 ≤ k ≤ n), this is equivalent to matrix

 h00  1 − 1 H 1 ··· 1 0 2 (h1)  00  h2  1 1 − 2 H ··· 1   h0   ( 2)   . . . .   . . .. .   00  hn 1 1 ··· 1 − 0 2 H (hn)

h00 being negative semi-deﬁnite. By Lemma 21, this holds if and only if k H ≥ 1 for all 1 ≤ 0 2 (hk) 2 h0 1 Pn ( k) 00 Pn Pn k ≤ n, and k=1 00 ≤ 1. This is equivalent to hk > 0 for all k, and k=1 γk ≤ k=1 hk. H hk

36 Finally, Nocke and Schutz (2015) show that, if v is such that ∇v = −D, then v is an P indirect subutility function for demand system D. Put v(p) = α log j∈N hj(pj) . Clearly, for every i ∈ N , 0 ∂v hi(pi) = P = −Di(p). ∂pi j∈N hj(pj) Therefore, v is indeed an indirect subutility function for demand system D.

A.2 Proof of Lemma 1 P Proof. For every i in N , let αi = infR++ (hi − γi). Clearly, i∈N αi ≥ 0, otherwise it would P be possible to ﬁnd a price vector p such that i∈N (hi(pi) − γi(pi)) < 0, which would violate integrability. Let i0 ∈ N . For every i in N , let

( h − α if i 6= i , h˜ = i i 0 i P hi0 − αi0 + j∈N αj if i = i0.

˜ 3 ˜0 0 ˜00 00 Then, for every i, hi is C , hi = hi < 0, hi = hi > 0, andγ ˜i = γi. In addition, for every i ∈ N , for every x > 0,

˜ hi(x) − γ˜i(x) ≥ (hi(x) − γi(x)) − inf (hi − γi) ≥ 0. R++

˜ ˜ Therefore, hi ≥ γ˜i for every i. Notice, in addition, that hi ≥ γ˜i > 0. ˜ Therefore, N , (hi)i∈N is a smooth and integrable demand component which satisﬁes ˜ P ˜ P hi ≥ γ˜i for every i. In addition, i∈N hi = i∈N hi. Therefore, for every k ∈ N and p >> 0,

0 −hk(pk) D N , (hj)j∈N (pj)j∈N ) = P , j∈N hj(pj) −h˜0 (p ) = k k , P ˜ j∈N hj(pj) ˜ = D N , hj (pj)j∈N ) . j∈N

˜ Therefore, (N , (hi)i∈N ) and N , (hi)i∈N are equivalent. This concludes the proof.

37 A.3 Proof of Theorem 2 Proof. Let D ∈ D. There exists a smooth and integrable demand component N , (hj)j∈N such that D = D N , (hj)j∈N . ˜ By Lemma 1, there exists a smooth and integrable demand component N , hj such j∈N that ˜ D = D N , (hj)j∈N = D N , hj , j∈N ˜ ˜ and for every k ∈ N , hk ≥ γ˜k. For every j ∈ N , put vj = log hj . Then, since ˜ 3 N , hj is a smooth and integrable demand component, for every j ∈ N , vj is C j∈N ˜0 0 hj from ++ to , vj = < 0, and R R h˜j

2 ˜00˜ ˜0 00 hj hj − hj h˜ v00 = , = j h˜ − γ˜ ≥ 0. j ˜2 ˜2 j j hj hj

Therefore, N , (vj)j∈N is a discrete continuous choice model. Moreover, for every k ∈ N ,

vk(pk) e 0 ∆k N , (vj) (pj) = (−v (pk)) , j∈N j∈N P vj (pj ) k j∈N e −h˜0 (p ) = k k , P ˜ j∈N hj(pj) ˜ = Dk N , hj (pj)j∈N , j∈N = Dk (pj)j∈N

Therefore, D ∈ D. Conversely, let D ∈ D. There exists a discrete-continuous choice model N , (vj)j∈N such that D = ∆ N , (vj)j∈N .

vj 3 0 0 vj For every j ∈ N , put hj = e . Then, hj is C from R++ to R++, hj = vj e < 0, 00 0 2 00 vj hj = vj + vj e > 0,

38 and 0 2 hj 1 γj = 00 = v00 hj ≤ hj. hj 1 + j 0 2 (vj ) Therefore, N , (hj)j∈N is a smooth and integrable demand component. Moreover, for every k ∈ N ,

0 −hj(pk) Dk N , (hj)j∈N (pj)j∈N = P , j∈N hj(pj)

vk(pk) e 0 = (−v (pk)) , P vj (pj ) k j∈N e = Dk (pj)j∈N .

Therefore, D ∈ D.

B Proofs for Section 3

B.1 Proof of Lemma 2

Proof. Since ιk is monotone,µ ¯k exists. Assume for a contradiction thatµ ¯k ≤ 1. In the following, drop index k to ease notation. Then, for all x > 0,

h00(x) x ≤ 1, i.e., xh00(x) + h0(x) ≤ 0. −h0(x)

It follows that d (xh0(x)) ≤ 0, ∀x > 0. dx

Let x0 > 0. Then, for every x ≥ x0,

0 0 xh (x) ≤ x0h (x0).

Therefore, x h0(x ) h0(x) ≤ 0 0 , ∀x ≥ x . x 0

Integrating both sides of the inequality between x0 and y ≥ x0, we get:

Z y 0 dx 0 h(y) − h(x0) ≤ x0h (x0) −→ −∞, since h < 0. y→∞ x0 x

39 Therefore, lim∞ h = −∞. This contradicts the assumption that h > 0.

B.2 Proof of Lemma 3

Proof. Let k ∈ N and x > p . We drop subscript k from now on to ease notation. Notice k that

−h0(x) γ(x) = (x(−h0(x))) , xh00(x) −xh0(x) = . ι(x)

Therefore,

1 γ0(x) = (− (xh00(x) + h0(x)) × ι(x) + ι0(x) × xh0(x)) , (ι(x))2 1 = (−h0(x) (1 − ι(x)) ι(x) + ι0(x)xh0(x)) < 0, (ι(x))2 as ι0 ≥ 0 and ι(x) > 1 for all x > p.

B.3 Proof of Lemma 4

Proof. Fix k ∈ N . We drop subscript k to ease notation. Notice ﬁrst that

1 h00 1 0 = = . γ h02 −h0

γ is strictly positive and, by Lemma 3, strictly decreasing on (p, ∞). Therefore, lim∞ γ ≡ l exists, and l ∈ [0, ∞). Assume for a contradiction that l > 0, and let ε > 0. There exists x0 > p such that 1 1 ∀t ≥ x , ≤ + ε. 0 γ(t) l

Integrating this inequality between x0 and x ≥ x0, we get:

1 Z x dt 1 1 + ε (x − x0) ≥ = 0 − 0 . l x0 γ(t) −h (x) −h (x0) | {z } | {z } ≡M>0 ≡A>0

Therefore, 0 1 −h (x) ≥ , ∀x ≥ x0. A + M(x − x0)

40 Integrating this inequality between x0 and y ≥ x0, we get:

Z y dx ≤ −h(y) + h(x0). x0 A + M(x − x0) | {z } −→ ∞ y→∞

Therefore, lim∞ h = −∞, which contradicts the assumption that h > 0.

B.4 Proof of Lemma 5

Proof. Fix k ∈ N . We drop subscript k to ease notation. Notice that

d (−xh0(x)) = −h0(x) − xh00(x), dx = −h0(x) (1 − ι(x)) , which is strictly negative for all x > p. Therefore, −xh0(x) is strictly decreasing for x 0 high enough, and limx→∞ −xh (x) is a non-negative real number. Call this number l, and 0 assume for a contradiction that l > 0. Let x0 > p. Since −xh (x) is decreasing on (x0, ∞), 0 −xh (x) ≥ l for all x ≥ x0. It follows that

Z x Z x 0 l h(x0) − h(x) = −h (t)dt ≥ dt . x0 x0 t | {z } −→ ∞ x→∞

Therefore, lim∞ h = −∞, which is a contradiction.

B.5 Proof of Lemma 6

Proof. Let k ∈ N such thatµ ¯k < ∞ and lim∞ hk = 0. We drop subscript k from now on. For all x > p,

h(x)h00(x) ρ(x) = , (h0(x))2 xh00(x) h(x) = , −h0(x) −xh0(x) h(x) = ι(x) . −xh0(x)

41 0 By assumption, lim∞ h = 0. By Lemma 5, limx→∞ −xh (x) = 0. Moreover,

d h(x) h0(x) lim dx = lim , x→∞ d 0 x→∞ −h0(x) − xh00(x) dx (−xh (x)) 1 = lim , x→∞ ι(x) − 1 1 = . µ¯ − 1

Therefore, by L’Hospital’s rule,

h(x) 1 lim = , x→∞ −xh0(x) µ¯ − 1

µ¯ and lim∞ ρ = µ¯−1 .

B.6 Proof of Lemma 7

Proof. Let f ∈ F and k ∈ f. We drop indices k and f from now on. Let p > c. Then, ν(., c) is C1, and ∂ν c p − c = ι(p) + ι0(p) > 0. ∂p p2 p | {z } | {z } >0 ≥0 It follows from the inverse function theorem that ν(., c) is a C1-diﬀeomorphism from (c, ∞) to ν ((c, ∞)). Since ν(., c) is strictly increasing,

ν ((c, ∞), c) = lim ν(p, c), lim ν(p, c) , p→c p→∞ = (0, µ¯).

Notice, in addition, that −h0(p) ν(p, c) = (p − c) . γ(p) Therefore,

∂ν −h0(p) −h00(p)γ(p) + h0(p)γ0(p) = + (p − c) , ∂p γ(p) γ(p)2 −h0(p) h00(p) −γ0(p) = + ν(p, c) − + , γ(p) −h0(p) γ(p) −h0(p) −h0(p) −γ0(p) = + ν(p, c) − + . γ(p) γ(p) γ(p)

42 Therefore, for all µ ∈ (0, µ¯),

!−1 ∂r ∂ν = , ∂µ ∂p (r(µ,c),c) γ(r(µ, c)) = > 0. µ (−γ0 (r(µ, c))) − (µ − 1) (−h0 (r(µ, c)))

The fact that r is increasing in c follows immediately from the fact that ν is decreasing in c and increasing in p. Finally, let us show that r(µ, c) > p whenever µ ≥ 1. Let µ ≥ 1. If c ≥ p, then this is trivial. Assume instead that c < p. Then,

ν(p, c) < ι(p) = 1 ≤ µ.

Since r is increasing in µ, it follows that r(µ, c) > p.

B.7 Proof of Lemma 8

Proof. Put g ≡ {k ∈ f : pk < ck}, and deﬁne (˜pk)k∈f as follows:

( ck if k ∈ g, p˜k = pk otherwise.

P P Since the hks are decreasing, k∈f hk(pk) > k∈f hk(˜pk). Therefore,

0 0 X −h (pk) X −h (pk) Πf (p ) ,H−f = (p − c ) k + (p − c ) k , k k∈f k k −f P k k −f P H + hj(pj) H + hj(pj) k∈g j∈f k∈f\g j∈f 0 X −h (pk) < 0 + (p − c ) k , k k −f P H + hj(pj) k∈f\g j∈f 0 X −h (˜pk) ≤ 0 + (˜p − c ) k , k k −f P H + hj(˜pj) k∈f\g j∈f

f −f = Π (˜pk)k∈f ,H .

Therefore, (pk)k∈f is not a solution of maximization problem (6).

43 B.8 Proof of Lemma 9

Proof. Notice ﬁrst that equation (8) has no solution on interval [0, 1], and that the right-hand side is not well-deﬁned for µf ≥ µ¯f .

The ﬁrst-order conditions hold at price vector (pk)k∈f if and only if, for every k ∈ f,

0 00 0 0 −h (pk) −h (pk) X h (pk)h (pi) k + (p − c ) k + (p − c ) k i = 0. −f P k k −f P i i 2 H + j∈f hj(pj) H + j∈f hj(pj) −f P i∈f H + j∈f hj(pj)

−f P H + j∈f hj (pj ) Multiplying both sides of the equality by 0 , simplifying, and rearranging terms, −hk(pk) we see that this is equivalent to:

0 X −h (pi) ν (p ) = 1 + (p − c ) i , (19) k k i i −f P H + hj(pj) i∈f j∈f

0 2 (hi(pi)) 00 00 X pi − ci pih (pi) h (pi) = 1 + i i , 0 −f P pi −h (pi) H + hj(pj) i∈f i j∈f

X γi(pi) = 1 + ν (p ) , ∀k ∈ f. (20) i i −f P H + hj(pj) i∈f j∈f

Since the right-hand side of equation (20) does not depend on k, this is equivalent to:

P γj(pj) ν (p ) = 1 + ν (p ) j∈f , ∀k ∈ f. k k k k −f P H + j∈f hj(pj)

Using the inverse functions deﬁned in Lemma 7, this is equivalent to:

f pk = rk(µ ), ∀k ∈ f, P f γj rj(µ ) and µf = 1 + µf j∈f . −f P f H + j∈f hj (rj(µ ))

B.9 Proof of Lemma 10

Proof. We deﬁne the following function:

! f f f −f X f f X f φ : µ ∈ (1, µ¯ ) 7→ (µ − 1) H + hj rj(µ ) − µ γj rj(µ ) . j∈f j∈f

44 Notice that µf solves equation (8) if and only if φ(µf ) = 0. φ is continuous, and

X lim φ = − γj (rj(1)) < 0. 1+ j∈f

If we show that limµ¯f φ > 0, then we can apply the intermediate value theorem to obtain f f f 14 the existence of a µ ∈ (1, µ¯ ) such that φ(µ ) = 0. Recall that limµ¯f rj = ∞, and that lim∞ γj = 0 (Lemma 4), for all j ∈ f. Notice also that, since hj is decreasing and non- negative, lim∞ hj exists, and it is non-negative. We distinguish two cases. Assume ﬁrst that µ¯f is ﬁnite. Then, ! f −f X lim φ = (¯µ − 1) H + lim hj > 0. µ¯f ∞ j∈f Next, assumeµ ¯f = ∞. Then,

f ! ! f µ − 1 −f X f X f lim φ = lim µ H + hj rj(µ ) − γj rj(µ ) , µ¯f µf →∞ µf j∈f j∈f f ! !! f µ − 1 −f X f X f = lim µ lim H + hj rj(µ ) − γj rj(µ ) , µf →∞ µf →∞ µf j∈f j∈f ! f −f X = lim µ H + lim hj , µf →∞ ∞ j∈f = ∞.

Therefore, φ has a zero. Next, we show that φ is strictly increasing. By Lemma 7, φ is C1, and its derivative is given by (we omit the arguments of functions to save space):

! ! 0 f −f X f X 0 0 f X 0 0 φ (µ ) = H + (hj − γj) + (µ − 1) rjhj − µ rjγj , j∈f j∈f j∈f −f X X 0 f 0 f 0 = H + (hj − γj) + rj µ (−γj) − (µ − 1)(−hj) , j∈f j∈f | {z } =γj by Lemma 7 −f X = H + hj > 0. j∈f

Therefore, equation (8) has a unique solution. The fact that

f f −f f Π rk(ˆµ ) j∈f ,H =µ ˆ − 1

14 This follows from the fact that rj is strictly increasing and has range (ck, ∞); see Lemma 7.

45 follows immediately from equation (19).

B.10 Proof of Lemma 11

Proof. Assume for a contradiction that (pk)k∈f solves maximization problem (7), and that there exists j ∈ f such that f −f ∂Π (pk)k∈f ,H 6= 0. ∂pj

Then, pj = cj, but

f −f ∂Π (pk) ,H 0 0 0 k∈f −hj(pj) X hj(pj)hi(pi) = + (p − c ) > 0. −f P i i 2 ∂pj H + hk(pk) k∈f i∈f H−f + P h (p ) | {z } k∈f k k >0 | {z } ≥0

Therefore, (pk)k∈f does not solve maximization problem (7), a contradiction.

B.11 Proof of Lemma 12

Proof. Put ¯ f −f Π = sup Π (pk)k∈f ,H . Q (pk)k∈f ∈ k∈f [ck,∞)

N n Q There exists a sequence of price proﬁles (pk )k∈f ∈ k∈f [ck, ∞) such that n∈N f n −f ¯ Π (pk ) ,H −→ Π. k∈f n→∞

To ﬁx ideas, assume f = {1,...,N}. We extract subsequences as follows:

• Start with k = 1. Either sequence (pn) is bounded, or it is not. If it is bounded, 1 n∈N ϕ1(n) then extract a convergent subsequence p1 , and denote its limit byp ˆ1(≥ c1). n∈N ϕ1(n) ϕ1(n) If it is unbounded, then extract a subsequence p1 such that p1 −→ ∞. n∈N n→∞

ϕ1(n) • Continue with k = 2. Either p2 is bounded, or it is not. If it is bounded, then n∈N ϕ1◦ϕ2(n) extract a convergent subsequence p2 , and denote its limit byp ˆ2(≥ c2). If n∈N ϕ1◦ϕ2(n) ϕ1◦ϕ2(n) it is unbounded, then extract a subsequence p2 such that p2 −→ ∞. n∈N n→∞ • Continue this process up to k = N.

46 We are left with a subsequence of vectors pϕ1◦...◦ϕN (n) , which, for brevity, we relabel as n∈N n n (p )n∈N. By construction, for every k,(pk )n∈N either has a finite limit (ˆpk), or goes to infinity as n goes to infinity. n If (pk )n∈N has a finite limit for every k, then we are done. By continuity of the profit function, ¯ f n −f f −f Π = lim Π (p ,H ) = Π (ˆpk) ,H , n→∞ 1≤j≤N f −f and (ˆpk)1≤j≤N therefore maximizes Π ., H . Next, assume for a contradiction that (pn) goes to infinity as n goes to infinity for at k n∈N n least one k. Assume without loss of generality that pk −→ ∞ for 1 ≤ k ≤ K, and that n→∞ (pn) has a finite limit for K + 1 ≤ k ≤ N. In other words, k k∈N

n p −→ (∞,..., ∞, pˆK+1,..., pˆN ) ≡ p.˜ n→∞

Let lk = lim∞ hk for every k in {1,...,N}. Since the numerator and the denominator f −f of Π (pk)k∈f ,H are additively separable in the components of price proﬁle (pk)k∈f , lim Πf (p ) ,H−f exists and is equal to: (pk)k∈f →p˜ k k∈f

PK 0 PN 0 k=1 limpk→∞(pk − ck)(−hk(pk)) + k=K+1 limpk→pˆk (pk − ck)(−hk(pk)) −f PK PN H + k=1 limpk→∞ h(pk) + k=K+1 limpk→pˆk h(pk) PN (ˆp − c )(−h0 (ˆp )) = k=K+1 k k k k , −f PK PN H + k=1 lk + k=K+1 h(ˆpk)

0 0 since, by Lemma 5, limx→∞ xhk(x) = limx→∞ hk(x) = 0 for every k. It follows that

Π¯ = lim Πf (pn,H−f ), n→∞ = lim Πf (p, H−f ), p→p˜ PN (ˆp − c )(−h0 (ˆp )) = k=K+1 k k k k . −f PK PN H + k=1 lk + k=K+1 h(ˆpk)

¯ If K = N, then Π = 0. This is clearly a contradiction, since any price proﬁle (pk)k∈f such that pk > ck for all k yields strictly positive proﬁts. Next, assume that 1 < K < N.

Claim 1: There existsµ ˆf ∈ (1, µ¯f ) such that Π¯ =µ ˆf − 1.

47 Define a new firm, called firm g = {K + 1,...,N}, let

0 X −h (pk) Πg (p ) ,H−g = (p − c ) k , k k∈g k k −g P H + h(pj) k∈g j∈g and notice that K ! g −f X ¯ Π (ˆpK+1,..., pˆN ) ,H + lk = Π. k=1

We claim that price vector (ˆpK+1,..., pˆN ) solves maximization problem

K ! X max Πg (p ) ,H−f + l . (21) Q k k∈g k (pk) ∈ [ck,∞) k∈g k∈g k=1

0 0 Assume this is not the case. Then, there exists pK+1, . . . , pN such that

K ! g 0 0 −f X ¯ Π pK+1, . . . , pN ,H + lk > Π. k=1

Since K ! g 0 0 −f X f −f Π pK+1, . . . , pN ,H + lk = lim Π p, H , p→ ∞,...,∞,p0 ,...,p0 k=1 ( K+1 N ) 0 0 there exists (p1, . . . , pK ) such that

f 0 0 0 0 −f ¯ Π p1, . . . , pK , pK+1, . . . , pN ,H > Π,

which is a contradiction. Therefore, (ˆpK+1,..., pˆN ) solves maximization problem (21), which is the counterpart of maximization problem (7) for ﬁrm g. By Lemmas 9, 10 and 11, there existsµ ˆf ∈ (1, µ¯g) = (1, µ¯f ) such that Π¯ =µ ˆf − 1.

¯ Claim 2: There existsp ˆK > cK such that φ(ˆpK ) > Π, where

0 PN 0 (x − cK )(−hK (x)) + k=K+1(ˆpk − ck)(−hk(ˆpk)) φ(x) = , ∀x > cK . −f PK−1 PN H + k=1 lk + hK (x) + k=K+1 h(ˆpk)

48 ¯ f Notice that lim∞ φ = Π =µ ˆ − 1. Let us show that φ is strictly decreasing when x is high enough. The derivative of φ with respect to x is equal to:

( K−1 ! N ! 0 0 00 −f X X φ (x) = (−hK (x) − (x − cK )hK (x)) H + lk + hK (x) + hk(ˆpk) k=1 k=K+1 N !) 0 0 X 0 − hK (x) (x − cK )(−hK (x)) + (ˆpk − ck)(−hk(ˆpk)) k=K+1 K−1 ! N !2 −f X X / H + lk + hK (x) + hk(ˆpk) . k=1 k=K+1 0 −hK (x) = (φ(x) − (νK (x) − 1)) . −f PK−1 PN H + k=1 lk + hK (x) + k=K+1 hk(ˆpk) | {z } >0

Since ( µˆf − µ¯f < 0 ifµ ¯f < ∞, (φ(x) − (νK (x) − 1)) −→ x→∞ −∞ ifµ ¯f = ∞,

0 ¯ it follows that φ (x) < 0 for x high enough. Therefore, φ(x) > Π = lim∞ φ for x high enough, which concludes the proof of Claim 2.

We can conclude: by Claim 2,

f −f ¯ lim Π p, H = φ(ˆpK ) > Π. p→(∞,...,∞,pˆK ,pˆK+1,...,pˆN )

Therefore, there exists (ˆp1,..., pˆK−1) such that

f −f ¯ Π (ˆp1,..., pˆK−1, pˆK , pˆK+1,..., pˆN ) ,H > Π, which is a contradiction.

B.12 Proof of Lemma 13

Proof. The fact that ﬁrms do not price below cost in any pricing equilibrium follows immediately from Lemma 8.

49 ∗ Q (pk)k∈N ∈ k∈N [ck, ∞) is a pricing equilibrium if and only if for every f ∈ F,   X X (p ) ∈ arg max Πf (p ) , h (p∗) . k k∈f Q  k k∈f k k  (pk) ∈ [ck,∞) k∈f k∈f  g∈F k∈g  g6=f

By Proposition 1, this holds if and only if for every f ∈ F, there exists µf∗ ∈ 1, µ¯f such that

f∗ ∗ µ = νk (pk) , ∀k ∈ f, P f∗ γk rk(µ ) and µf∗ = 1 + µf∗ k∈f , P f∗ P P ∗ k∈f hk (rk (µ )) + g∈F k∈g hk (pk) g6=f which is clearly equivalent to point (ii) in the statement of the lemma.

B.13 Proof of Lemma 14

Proof. Let H > 0 and f ∈ F. The left-hand side of equation (9) is continuous in µf , converges to P γk (rk(1)) 1 − k∈f < 1 H when µf goes to 1 (by Lemma 7), and converges toµ ¯f > 1 when µf goes toµ ¯f (by Lemmas 4 and 7). Therefore, by the intermediate value theorem, equation (9) has a solution. In addition, by Lemmas 3 and 7, the left-hand side is strictly increasing in µf . Therefore, this solution is unique. Totally diﬀerentiating equation (9), we get (we omit the arguments of functions to save space):

P P 0 0 P γk r (−γ ) γk 0 = 1 − k∈f + mf k∈f k k dmf + mf k∈f dH, H H H2 ! 1 P r0 (−γ0 ) dH = + (mf − 1) k∈f k k dmf + (mf − 1) , f P m k∈f γk H | {z } >0

50 mf P as H = mf −1 k∈f γk from (9). By the implicit function theorem,

1 mf (mf − 1) mf0(H) = − , P r0 −γ0 H f f k∈f k( k) 1 + m (m − 1) P k∈N f γk

f f which is indeed strictly negative. It follows that m is strictly decreasing, and that lim0+ m f f f and lim∞ m exist. Assume for a contradiction that lim0+ m ∈ [1, µ¯ ). Then, the left- hand side of equation (9) goes to −∞ as H goes to 0, which is a contradiction. Therefore, f f P f f lim0+ m =µ ¯ . Next, notice that k∈f γk rk(µ ) is bounded when µ ≥ 1. Therefore, the f f left-hand side of equation (9) goes to µ when H goes to ∞, and lim∞ m = 1.

B.14 Proof of Lemma 15 ˜ Proof. By Lemma 1, there exists a smooth and integrable demand component N , hk k∈N ˜ ˜ such that N , (hk)k∈N and N , hk are equivalent and hk ≥ γ˜k for every k ∈ N . k∈N ˜ Notice, in addition, that N , hk satisﬁes Assumptions 1 and 2. In the following, k∈N ˜ we use N , hk instead of N , (hk)k∈N , which is not a problem, since these demand k∈N components are equivalent. We drop the tildes to ease notation.

Ω is continuous on R++. By Lemmas 7 and 14, when H goes to ∞, the numerator of Ω goes to X X hk (rk(1)) , f∈F k∈f which is ﬁnite. Therefore, lim∞ Ω = 0. If we show that Ω is strictly greater than 1 in the neighborhood of 0+, then we can apply the intermediate value theorem to obtain the existence of H∗.

Assume ﬁrst that there exists j ∈ N such that lim∞ hj = l > 0. Since hj is decreasing, hj(x) ≥ l for all x > 0. Let f ∈ F such that j ∈ f. Then, for all H > 0,

h r mf (H) Ω(H) ≥ j j , H l ≥ −→ ∞. H H→0+

Therefore, lim0+ Ω = ∞ > 1.

51 Next, assume that hk(x) −→ 0 for all k ∈ N . Put x→∞

F 0 = f ∈ F :µ ¯f = ∞

00 0 00 µ¯f and F = F\F . Let f ∈ F . Then, by Lemma 6, lim∞ ρk = µ¯f −1 for every k ∈ f. In f addition, by Lemmas 7 and 14, for every k ∈ f, rk m (H) −→ ∞. Therefore, there exists H→0+ ηf > 0 such that for every H < ηf ,

µ¯f 1 ρ r mf (H) ≥ 1 − . k k µ¯f − 1 2|F|

f 00 Let η = minf∈F 00 η (or any strictly positive real number if F is empty). For every H < η,

P P f P P f f∈F 0 k∈f hk rk m (H) + f∈F 00 k∈f hk rk m (H) Ω(H) = , H P P f P P f f f∈F 0 k∈f γk rk m (H) + f∈F 00 k∈f γk rk m (H) ρk rk m (H) ≥ , H P f P X k∈f γk X µ¯ 1 k∈f γk ≥ + 1 − , since H < η, H µ¯f − 1 2|F| H f∈F 0 f∈F 00 X mf (H) − 1 X mf (H) − 1 µ¯f 1 = + 1 − , using equation (9). mf (H) mf (H) µ¯f − 1 2|F| f∈F 0 f∈F 00

When H goes to 0+, the right-hand side term on the last line goes to

1 1 |F 0| + |F 00| 1 − ≥ |F| − , 2|F| 2 which is strictly greater than 1. Therefore, Ω(H) > 1 when H is small enough. This concludes the proof.

B.15 Proof of Lemma 3

Consider an increase in marginal cost ck, k ∈ f. f g First, recall from Lemma 7 that ∂rk(µ , ck)/∂ck > 0 whereas ∂rj(µ , cj)/∂ck = 0 for all g ∈ F and j ∈ g, j 6= k. Next, by Lemma 14, ﬁrm f’s ﬁtting in function mf (H) is uniquely determined by P f ! γj rj(m , cj) mf 1 − j∈f = 1. H

52 Applying the implicit funtion theorem, we obtain

f mf − γ0 ∂rk dm (H) k ∂ck = ∂r < 0, dck P f P 0 j H − j∈f γj − m j∈f γj ∂mf

f f f 0 where the inequality follows from ∂rk(µ , ck)/∂ck > 0, ∂rj(m , cj)/∂m > 0, and γj(rj) < 0. P f g Hence, j∈f hj(rj(m (H))) decreases with ck. For any firm g 6= f, dm (H)/dck = 0, so that P P g g∈F,g6=f j∈g hj(rj(m (H))) remains unchanged. This implies that, for any H > 0, Ω(H) is decreasing with ck. Recall from the proof of Lemma 15 that Ω(H) > 1 for H sufficiently small and Ω(H) < 1 for H sufficiently large. Hence, both the smallest and the largest solution in

H to Ω(H) = 1 are decreasing with ck, and so is therefore indirect utility and the proﬁt of any ﬁrm g 6= f, in both the smallest and largest equilibrium. As dmg(H)/dH < 0 by Lemma 14, it also follows that all prices increase in both the smallest and largest equilibrium.

C Proofs for Section 4

C.1 Proof of Lemma 16

f f f f 1 Proof. Let f ∈ F, k ∈ f, ω ∈ 0, ω¯ , and x such that χk(x) > ω . Put µ = 1−ωf . Then, f f ιk(x) > µ . Therefore, there exists c > 0 such that νk(x, c) = µ . We know from Lemma 7 that

f ∂rk f γk(rk(µ , ck)) f (µ , c) = f 0 f f 0 f , ∂µ µ (−γk (rk(µ , ck))) − (µ − 1) (−hk (rk(µ , ck))) γk(x) 1 = 0 f f > 0. −γk(x)µ 1 − ω θk(x)

0 f In addition, by Lemmas 3 and 7, γk(x) < 0. Therefore, 1 − ω θk(x) > 0. This establishes the ﬁrst bullet point in the statement of the lemma. f f Next, let c > 0 and x ≥ rk(µ , c). Then, since νk(., c) is increasing, νk(x, c) ≥ µ . Since f f c > 0, it follows that ιk(x) > µ , and that χk(x) > ω . It follows from the ﬁrst part of the f lemma that 1 − ω θk(x) > 0. Finally, let x > p . Put ωf = χ (x). Then, for every y such that χ (y) > ωf , 1−ωf θ (y) > k k k k 0. By monotonicity of χk, this implies that, for every y > x, 1 − χk(x)θk(y) > 0. Therefore, by continuity of θk, χk(x)θk(x) ≤ 1.

53 C.2 Proof of Lemma 17

We start by proving two intermediate lemmas:

Lemma 23. Let N , (hk)k∈N , F, (ck)k∈N be a pricing game satisfying Assumptions 1 and 2. If, for every f ∈ F,

f ! f ! X ω θk 1 ω ∀ωf ∈ (0, ω¯f ), γ − < 1, (22) f k P P 1 − ω θk hk γk k∈f k∈f k∈f

1 where, for every k, functions θk, γk and hk are all evaluated at point pk = rk 1−ωf , then the pricing game has a unique equilibrium.

Proof. By Theorem 3, N , (hk)k∈N , F, (ck)k∈N has a pricing equilibrium. To prove that there is only one equilibrium, we will show that Ω(.) is strictly decreasing.15 Let H > 0, and, f f f µf −1 for every f ∈ F, µ = m (H) and ω = µf . Then,

2 0 X f0(H) X 0 f 0 f X f H Ω (H) = H m rk µ hk rk µ − hk rk µ , f∈F k∈f k∈N   f f ! X µ (µ − 1) X 0 0 X =  r (−h ) − hk , by Lemma 14.  P r0 −γ0 k k  f f k∈f k( k) f∈F 1 + µ (µ − 1) P k∈f k∈f k∈f γk

Therefore, a suﬃcient condition for this derivative to be strictly negative is that, for all f ∈ F, P 0 0 f f k∈f rk(−hk) µ (µ − 1) P h k∈f k < 1. (23) P r0 −γ0 f f k∈f k( k) 1 + µ (µ − 1) P k∈f γk Let f ∈ F. Then,

P µf r0 (−h0 ) P µf r0 (−γ0 )! f k∈f k k k∈f k k (23) ⇐⇒ (µ − 1) P − P < 1, k∈f hk k∈f γk

0 0  γk(−h ) γk(−γ )  P µf k P µf k k∈f µf (−γ0 )−(µf −1)(−h0 ) k∈f µf (−γ0 )−(µf −1)(−h0 ) f  k k k k  ⇐⇒ (µ − 1)  P − P  < 1, k∈f hk k∈f γk

15An a priori less restrictive way of establishing uniqueness would be to show that Ω0(H) < 1 whenever Ω(H) = 1. Unfortunately, this is not particularly useful, because we simply do not know what these equilibrium H’s could be. See also Appendix C.12, where we use an index approach to derive uniqueness conditions.

54 P θk P 1 ! f γk f γk f k∈f 1−ω θk k∈f 1−ω θk ⇐⇒ (µ − 1) P − P < 1, k∈f hk k∈f γk f P θk P ω θk ! f γk f γk f k∈f 1−ω θk k∈f 1−ω θk ⇐⇒ (µ − 1) P − 1 − P < 1, k∈f hk k∈f γk f !! X θk 1 ω ⇐⇒ (µf − 1) −1 + γ − < 1, f k P P 1 − ω θk hk γk k∈f k∈f k∈f f ! f ! X ω θk 1 ω ⇐⇒ γ − < 1, f k P P 1 − ω θk hk γk k∈f k∈f k∈f

f where, for every k ∈ f, functions θk, γk and hk are evaluated at point pk = rk(µ ) = 1 rk 1−ωf . Since condition (22) holds by assumption, Ω is strictly decreasing. Therefore, the pricing game has a unique equilibrium. Lemma 24. Let N , (hk)k∈N , F be a pricing structure satisfying Assumptions 1 and 2. Let N (ck)k∈N ∈ R++. If, for every f ∈ F,

Y 1 ∀ωf ∈ (0, ω¯f ), ∀ (x ) ∈ r , c , ∞ , k k∈f k 1 − ωf k k∈f (24) f ! f ! X ω θk(xk) 1 ω γ (x ) − < 1, f k k P P 1 − ω θk(xk) hk(xk) γk(xk) k∈f k∈f k∈f or, equivalently, if

Y 1 ∀ωf ∈ (0, ω¯f ), ∀ (x ) ∈ r , c , ∞ , k k∈f k 1 − ωf k k∈f (25) f X f 1 − ω ρj(xj) γi(xi)γj(xj) ω θi(xi) f − ρi(xi) < 0, 1 − ω θi(xi) i,j∈f

then pricing game N , (hk)k∈N , F, (ck)k∈N has a unique equilibrium for every (ck)k∈N ∈ Q k∈N [ck, ∞). Q Proof. Assume that condition (24) holds, and let (ck)k∈N ∈ k∈N [ck, ∞). We want to show f f f 1 that condition (22) holds, so let f ∈ F, ω ∈ (0, ω¯ ) and µ = 1−ωf . Let k ∈ f and f pk = rk(µ , ck). Since ck ≥ ck and rk is increasing in its second argument, it follows that f Q f pk ≥ rk(µ , ck). Therefore, (pk)k∈f ∈ k∈f rk µ , ck , ∞ . It follows that condition (22) holds. By Lemma 23, pricing game N , (hk)k∈N , F, (ck)k∈N has a unique equilibrium. Finally, we show that conditions (24) and (25) are equivalent. Let f ∈ F, ωf ∈ (0, ω¯f ),

55 Q h 1 and (xk)k∈f ∈ k∈f rk 1−ωf , ck , ∞ . Then,

f ! f ! X ω θk 1 ω γ − < 1 f k P P 1 − ω θk hk γk k∈f k∈f k∈f f ! ! ! ! X ω θi X f X X ⇐⇒ f γi γj − ω hj − hi γj < 0, 1 − ω θi i∈f j∈f i∈f j∈f f ! ! ! ! X ω θi X f X X ⇐⇒ f γi γj 1 − ω ρj − ρiγi γj < 0, 1 − ω θi i∈f j∈f i∈f j∈f f X f 1 − ω ρj ⇐⇒ γiγj ω θi f − ρi < 0. 1 − ω θi i,j∈f

We can now prove Lemma 16:

N f f Proof. Let (ck)k∈N ∈ R++, and assume that condition (11) holds. Let f ∈ F, ω ∈ (0, ω¯ ) f f Q f and µ = 1/(1 − ω ). Let (xk)k∈f ∈ k∈f rk(µ , ck), ∞ . Then, for every k ∈ f,

f ιk(xk) > νk(xk, ck) = µ .

Therefore, n f f o (xk)k∈f ∈ (xk)k∈f ∈ R++ : ∀k ∈ f, χk(xk) > ω , and, by condition (11), condition (24) holds for (ck)k∈N = (ck)k∈N . By Lemma 24, pricing game N , (hk)k∈N , F, (ck)k∈N has a unique equilibrium. In addition, as shown in the proof of Lemma 24, conditions (11) and (12) are equivalent.

C.3 Proof of Theorem 4

Proof. Let us show that condition (12) holds for every ﬁrm. Let f ∈ F, ωf ∈ (0, ω¯f ), and

n f f o (xk)k∈f ∈ (xk)k∈f ∈ R++ : ∀k ∈ f, χk(xk) > ω .

Since for every k ∈ f, χ (x ) > ωf , it follows that ι (x ) > 1. Therefore, x > p for every k k k k k k k, and, by Assumption 4,

max θk(xk) ≤ min ρk(xk). k∈f k∈f

56 Therefore,

f X f 1 − ω ρj X f γiγj ω θi f − ρi ≤ γiγj ω ρi − ρi , 1 − ω θi i,j∈f i,j∈f f X ≤ (ω − 1) γiγjρi < 0, i,j∈f where the ﬁrst inequality follows by Lemma 16 and maxk∈f θk(xk) ≤ mink∈f ρk(xk). By Lemma 17, pricing game N , (hk)k∈N , F, (ck)k∈N has a unique equilibrium for every (ck)k∈N ∈ N R++.

C.4 Proof of Proposition 4

Proof. We focus on ﬁrm k and drop subscript k from now on. We know that lim∞ h = h0 lim∞ γ = 0 and that lim∞ γ0 exists. By l’Hospital’s rule,

h h0 lim ρ = lim = lim = lim θ. ∞ ∞ γ ∞ γ0 ∞

By Assumption 4, ρ is non-decreasing. This implies that lim∞ ρ = sup(p,∞) ρ. Therefore,

sup ρ ≤ sup θ ≤ inf ρ, (p,∞) (p,∞) (p,∞) where the second inequality follows from Assumption 4. Therefore, there exists K > 0 such that ρ(x) = K for all x > p. This is equivalent to:

−h00(x) h0(x) = K ∀x > p. −h0(x) h(x)

Therefore, there exists L ∈ R such that

log (−h0(x)) = K log (h(x)) + L, ∀x > p.

−h0(x) L Taking exponentials, we get: h(x)K = e . We distinguish two cases. Assume ﬁrst that K = 1. Then, there exists M > 0 such that h(x) = M exp − eL x for all x > p, i.e., h is logit on (p, ∞).

57 Next, assume K 6= 1. Then, there exists M ∈ R such that

h(x)1−K = eL x + M, ∀x > p. K − 1

If K < 1, then h(x)1−K < 0 for x high enough, which is impossible. Therefore, K > 1. Similarly, if M < 0, then, h −M e−L1−K = 0, which cannot be since h > 0. Therefore, M ≥ 0, 1 h(x) = (K − 1) eL x + (K − 1)M 1−K , ∀x > p, and h is CES on (p, ∞).

C.5 Proof of Proposition 5

Proof. We focus on product k and drop subscript k from now on. Sinceµ ¯f = 1, by deﬁnition f f ofω ¯ , ω = 1. As in the proof of Proposition 4, it follows from l’Hospital’s rule that lim∞ ρ = f f f lim∞ θ. We have shown in the proof of Lemma 24 that 1 − ω θ(x) > 0 if ω ∈ (0, ω¯ ) = (0, 1) f f f and χ(x) > ω . Therefore, lim∞ θ ≤ 1/ω for all ω ∈ (0, 1), and lim∞ θ ≤ 1. It follows that lim∞ ρ ≤ 1. Since ρ ≥ 1 and ρ is non-decreasing, this implies that ρ = 1. Integrating this diﬀerential equation as we did in the proof of Proposition 4, we can conclude that ρ is logit.

C.6 Proof of Lemma 18

f f µ¯f 1 Proof. Let k ∈ f and ω ∈ (0, ω¯ ). By Lemma 6, lim∞ ρk = µ¯f −1 = ω¯f . By Assumption 3, ρ is non-decreasing. Therefore, ρ (x) ≤ 1 for all x > p . In particular, this inequality is k k ω¯f k f also satisﬁed if x is such that χk(x) > ω . In addition, as shown in the proof of Lemma 6,

hk(x) ρk(x) = ιk(x) 0 . −xhk(x)

Therefore,

0 0 00 d log ρk(x) ιk(x) hk(x) 1 hk(x) = + − + 0 , dx ιk(x) hk(x) x −hk(x) 0 ιk(x) 1 ιk(x) = + − − 1 + ιk(x) , ιk(x) x ρk(x) 0 ιk(x) ιk(x) = + (ρk(x)χk(x) − 1) , ιk(x) xρk(x)

58 ι0 (x) ≤ k , ιk(x)

f 1 where the last inequality follows from the fact that χk(x) ≤ ω¯ and ρk(x) ≤ ω¯f . Therefore, for all x > p , k

Z ∞ 0 1 ρk(t) log f = dt, ω¯ ρk(x) x ρk(t) Z ∞ ι0 (t) ≤ k dt, x ιk(t) µ¯f = log , ιk(x) 1 − χ (x) = log k . 1 − ω¯f

Therefore, 1 − ω¯f 1 ρ (x) ≥ , ∀x > p . k f k ω¯ 1 − χk(x)

f In particular, if χk(x) > ω , then

1 − ω¯f 1 ρ (x) ≥ . k ω¯f 1 − ωf

C.7 Proof of Lemma 19

Proof. Letω ¯ ∈ (0, 1) and ω ∈ (0, ω¯). Deﬁne

M (ω, ω¯) = max φω,ω¯ (y, z). 1−ω¯ 1 1 2 (y,z)∈[ ω¯ 1−ω , ω¯ ]

Notice that φω,ω¯ (y, z) = φω,ω¯ (z, y) for every y and z. It follows that

M (ω, ω¯) = max φω,ω¯ (y, z). 1−ω¯ 1 1 2 (y,z)∈[ ω¯ 1−ω , ω¯ ] y≤z

1−ω¯ 1 1 Let ω¯ 1−ω ≤ y ≤ z ≤ ω¯ . Then,

∂φ ω(1 − ωz) ω2z ω,ω¯ = − − 1, ∂y (1 − ωy)2 1 − ωz

59 ! 1 1 − ωz 2 = ω − ω2z − (1 − ωz) , 1 − ωz 1 − ωy 1 ≤ ω − ω2z − (1 − ωz) , since y ≤ z, 1 − ωz = ω − 1 < 0.

1−ω¯ 1 1 2 It follows that, for every (y, z) ∈ ω¯ 1−ω , ω¯ such that y ≤ z,

1 − ω¯ 1 φ (y, z) ≤ φ , z ≡ ψ (z). ω ω ω¯ 1 − ω ω,ω¯

Therefore,

M (ω, ω¯) = max ψω,ω¯ (z). 1−ω¯ 1 1 z∈[ ω¯ 1−ω , ω¯ ] Since ω 1 − ω¯ 2ω2 ψ00 (z) = 1 − > 0, ω,ω¯ 1 − ω ω¯ (1 − ωz)3 function ψω,ω¯ (.) is strictly convex. Therefore,

1 − ω¯ 1 1 − ω¯ 1 1 − ω¯ 1 1 M (ω, ω¯) = max φ , , φ , . ω,ω¯ ω¯ 1 − ω ω¯ 1 − ω ω,ω¯ ω¯ 1 − ω ω¯

Since φω,ω¯ (z, z) = 2(ω − 1)z < 0 for every z, it follows that M (ω, ω¯) ≤ 0 if and only if ζ(ω, ω¯) ≤ 0, where

1 − ω¯ 1 1 ζ(ω, ω¯) ≡ φ , , ω¯ 1 − ω ω¯ ω 1−ω¯ ω 1−ω ω¯ ω ω 1 − ω¯ 1 − ω¯ 1 1 = 1 − ω 1−ω¯ + 1 − − − , ω¯ 1 − 1−ω ω¯ ω¯ − ω 1 − ω ω¯ ω¯ 1 − ω ω¯ ω(1 − ω¯) ω 1 − ω¯ 1 1 = + − − , ω¯ (1 − ω)¯ω ω¯ 1 − ω ω¯ 1 ω − 2 = + − ω. 1 − ω ω¯

For every ω ∈ (0, ω¯), ∂ζ 1 1 = + − 1 > 0. ∂ω (1 − ω)2 ω¯

60 Therefore, ζ is strictly increasing in ω on interval (0, ω¯). It follows that M (ω, ω¯) ≤ 0 for every ω ∈ (0, ω¯) if and only if ξ (¯ω) ≤ 0, where

ξ (¯ω) ≡ ζ (¯ω, ω¯) , 1 2 = + 1 − ω¯ − . 1 − ω¯ ω¯

For everyω ¯ ∈ (0, 1), 1 2 ξ0(¯ω) = + − 1 > 0. (1 − ω¯)2 (¯ω)2

Therefore, ξ is strictly increasing on (0, 1). Since lim0+ ξ = −∞ and lim1− = +∞, there exists a unique threshold ω∗ ∈ (0, 1) such that ξ(¯ω) ≤ 0 if and only ifω ¯ ≤ ω∗. Numerically, we ﬁnd that ω∗ ' 0.64. This concludes the proof.

C.8 Proof of Theorem 5

Proof. Assume thatω ¯f < ω∗. Splitting the sum in two terms, condition (12) can be rewritten as follows:

f f n f f o ∀ω ∈ (0, ω¯ ), ∀ (xk)k∈f ∈ (xk)k∈f ∈ R++ : ∀k ∈ f, χk(xk) > ω , f f 1 X f 1 − ω ρj(xj) f 1 − ω ρi(xi) γi(xi)γj(xj) ω θi(xi) f + ω θj(xj) f − ρi(xi) − ρj(xj) 2 1 − ω θi(xi) 1 − ω θj(xj) i,j∈f (26) i6=j f ! X 2 f 1 − ω ρi(xi) + γi(xi) ω θi(xi) f − ρi(xi) < 0. 1 − ω θi(xi) i∈f

f f Let us ﬁrst show that the second sum is strictly negative. Let ω ∈ (0, ω¯ ), i ∈ f and xi such f that χi(xi) > ω . Therefore,

f f 1 − ω ρi(xi) f ω θi(xi) f − ρi(xi) ≤ ω θi(xi) − ρi(xi) < 0, 1 − ω θi(xi) where the ﬁrst inequality follows from Assumption 3 (θi(xi) ≤ ρi(xi)) and Lemma 16 (1 − f ω θi(xi) > 0). f f Next, we turn our attention to the ﬁrst sum. Let ω ∈ (0, ω¯ ) and (xk)k∈f such that f χk(xk) > ω for every k ∈ f. By Lemma 18,

1 − ω¯f 1 1 ∀k ∈ f, ρ (x) ∈ , . k ω¯f 1 − ωf ω¯f

61 1 In addition, as shown above, for every k ∈ f, θk(xk) ≤ ρk(xk) < ωf . Therefore,

f f 1 X f 1 − ω ρj(xj) f 1 − ω ρi(xi) γi(xi)γj(xj) ω θi(xi) f + ω θj(xj) f − ρi(xi) − ρj(xj) 2 1 − ω θi(xi) 1 − ω θj(xj) i,j∈f i6=j 1 X ≤ γ (x )γ (x )φ f f (ρ (x ), ρ (x )) , 2 i i j j ω ,ω¯ i i j j i,j∈f i6=j ≤ 0, by Lemma 19.

This concludes the proof.

C.9 Uniqueness when µ¯f is Small

In this section, we extend Theorem 5 to cases where lim∞ hk is not necessarily equal to zero. We start with the following technical lemma:

Lemma 25. Let N , (hk)k∈N , F be a pricing structure satisfying Assumptions 1 and 2. Let f ∈ F. Assume that Assumption 3 is satisﬁed for ﬁrm f. Then, for every k ∈ f,

1 S = ω ∈ (0, ω¯f ): ∃x > p , ω = χ (x) = k k k ρk(x) contains at most one element. If S is empty, then, either χ (x)ρ (x) > 1 for every x > p , k k k k or χ (x)ρ (x) < 1 for every x > p . If, instead, S = {ωˆ}, then, for every x > p , k k k k k

1 • θk(x) ≤ ωˆ , and

1 1−ωˆ 1 • if ρk(x) < , then ρk(x) ≥ . ωˆ ωˆ 1−χk(x)

Proof. Let k ∈ f, and assume for a contradiction that Sk contains two distinct elements. There exist x, y > p such that χ (x)ρ (x) = 1, χ (y)ρ (y) = 1 and χ (x) 6= χ (y). To ﬁx k k k k k k k ideas, assume χk(y) > χk(x). Then, since χk is non-decreasing, y > x. Since ρk is non- decreasing, ρk(x) ≤ ρk(y). Therefore, χk(x)ρk(x) < χk(y)ρk(y) = 1, which is a contradiction. Let κ : x ∈ (p , ∞) 7→ ρ (x)χ (x), and notice that κ is continuous and non-decreasing. If k k k Sk = ∅, then, there is no x such that κ(x) = 1. Since κ is continuous, either κ > 1, or κ < 1. Next, let x > p . If ρ (x) ≤ 1 , then, θ (x) ≤ ρ (x) ≤ 1 . Assume instead that ρ (x) > 1 . k k ωˆ k k ωˆ k ωˆ 1 1 Letx ˆ such that χk(ˆx) =ω ˆ = . Then, ρk(x) > ρk(ˆx) = and, by monotonicity, x > xˆ. ρk(ˆx) ωˆ 1 Therefore, χk(x) ≥ χk(ˆx) =ω ˆ. Next, we claim that θk(x) ≤ . To see this, notice that χk(x)

62 0 −hk(x) ιk(x) = x . Therefore, γk(x)

0 00 0 ιk(x) 1 hk(x) γk(x) = + 0 − , ιk(x) x hk(x) γk(x) 0 0 1 γk(x) −hk(x) = 1 − ιk(x) + 0 x , x hk(x) γk(x) 1 ιk(x) = 1 − ιk(x) + . x θk(x)

Therefore, ιk(x) ιk(x) 1 θk(x) = ι0 (x) ≤ = . k ι (x) − 1 χ (x) ιk(x) − 1 + x k k ιk(x) 1 1 Therefore, θk(x) ≤ ≤ . χk(x) ωˆ 1 Next, assume that ρk(x) < ωˆ . We know from the proof of Lemma 18 that for every t ∈ [x, xˆ],

0 0 ρk(t) ιk(t) ιk(t) = + (ρk(t)χk(t) − 1) , ρk(t) ιk(t) tρk(t) 0 ιk(t) ιk(t) ≤ + (ρk(ˆx)χk(ˆx) − 1) , by monotonicity, ιk(t) tρk(t) 0 ιk(t) = , since ρk(ˆx)χk(ˆx) = 1. ιk(t)

Integrating this inequality between x andx ˆ, we obtain that ρk(ˆx) ≤ ιk(ˆx) . Therefore, ρk(x) ιk(x)

ιk(x) ρk(x) ≥ ρk(ˆx) , ιk(ˆx) 1 − ωˆ 1 = . ωˆ 1 − χk(x)

Proposition 15. Let N , (hk)k∈N , F be a pricing structure satisfying Assumptions 1 and 2. Let f ∈ F. Assume that Assumption 3 is satisfied for firm f and that ω¯f ≤ ω∗. Assume also, using the notations introduced in Lemma 25 that, for every i ∈ f, Si = {ωˆ} . Then, condition (12) holds for firm f.

Proof. As in the proof of Theorem 5, the expression in condition (12) can be split in two terms (see equation (26)). Given Assumption 3 and Lemma 16, the second sum is strictly f f negative. Next, we turn our attention to the ﬁrst sum. Let ω ∈ (0, ω¯ ), i, j ∈ f, and xi, xj

63 f f such that χi(xi) > ω and χj(xj) > ω . We want to show that

f f f 1 − ω ρj(xj) f 1 − ω ρi(xi) Ψ = ω θi(xi) f + ω θj(xj) f − ρi(xi) − ρj(xj) ≤ 0. (27) 1 − ω θi(xi) 1 − ω θj(xj)

1 To fix ideas, assume that ρi(xi) ≤ ρj(xj). If ρi(xi) ≥ ωf , then condition (27) is clearly f f satisfied, since, by Lemma 16, 1 − ω θi(xi) and 1 − ω θj(xi) are strictly positive. Assume 1 f instead that ρi(xi) < ωf . Then, we claim that ω < ωˆ. Assume for a contradiction that f 1 ωˆ ≤ ω . Since Si = {ωˆ}, there existsx î > p such that χi(ˆxi) =ω ˆ = . Therefore, i ρi(ˆxi) ρi(xi) < ρi(ˆxi) and, by monotonicity, xi < xî. Since χi is non-decreasing, it follows that

f ω < χi(xi) ≤ χi(ˆxi) =ω, ˆ which is a contradiction. Therefore, ωf < ωˆ. 1 We distinguish three cases. Assume ﬁrst that ρj(xj) < ωˆ . Then, by Lemma 25,

1 − ωˆ 1 1 − ωˆ 1 ρk(xk) ≥ ≥ f , ωˆ 1 − χk(xk) ωˆ 1 − ω

θi(xi) ρi(xi) θj (xj ) ρj (xj ) for k ∈ {i, j}. In addition, f ≤ f and f ≤ f . Therefore, 1−ω θi(xi) 1−ω ρi(xi) 1−ω θj (xj ) 1−ω ρj (xj )

Ψ ≤ φωf ,ωˆ (ρi(xi), ρj(xj)) , which, by Lemma 19, is non-positive, sinceω ˆ < ω¯f ≤ ω∗. 1 Next, assume that ρi(xi) < ωˆ ≤ ρj(xj). Then, by Lemma 25,

1 − ωˆ 1 1 − ωˆ 1 ρi(xi) ≥ ≥ f , ωˆ 1 − χi(xi) ωˆ 1 − ω

1 and θj(xj) ≤ ωˆ . Therefore,

f ωf θ (x ) ωf ω 1 Ψ ≤ i i 1 − + ωˆ (1 − ωf ρ (x )) − ρ (x ) − , 1 − ωf θ (x ) ωˆ ωf i i i i ωˆ i i 1 − ωˆ f ωf ρ (x ) ωf ω 1 ≤ i i 1 − + ωˆ (1 − ωf ρ (x )) − ρ (x ) − , 1 − ωf ρ (x ) ωˆ ωf i i i i ωˆ i i 1 − ωˆ 1 = φ f ρ (x ), , ω ,ωˆ i i ωˆ ≤ 0 by Lemma 19.

64 1 1 1 Finally, assume that ρi(xi) ≥ ωˆ . By Lemma 25, θi(xi) ≤ ωˆ and θj(xj) ≤ ωˆ . Therefore,

f f f f ω θi(xi) ω ω θj(xj) ω 1 1 Ψ ≤ f 1 − + f 1 − − − , 1 − ω θi(xi) ωˆ 1 − ω θj(xj) ωˆ ωˆ ωˆ f f ω ωf ω ωf 1 1 ≤ ωˆ 1 − + ωˆ 1 − − − , ωf ωˆ ωf ωˆ ωˆ ωˆ 1 − ωˆ 1 − ωˆ 1 1 = φ f , , ω ,ωˆ ωˆ ωˆ ≤ 0 by Lemma 19.

This concludes the proof.

Condition Si = {ωˆ} ∀i in Proposition 15 may look a little bit arcane. The following corollary is easier to understand:

Corollary 1. Let N , (hk)k∈N , F be a pricing structure satisfying Assumptions 1 and 2. Let f ∈ F. Assume that Assumption 3 is satisfied for firm f and that ω¯f ≤ ω∗. Assume R++ 2 f also that there exist h ∈ R++ and (αk, βk)k∈f ∈ R++ such that for every k ∈ f, for every x > 0, hk(x) = αkh(βkx). Then, condition (12) holds for firm f.

Proof. Let us ﬁrst show that Si ⊆ Sj for all i, j ∈ f. Let i, j ∈ f. If Si is empty, then, trivially, S ⊆ S . Assume instead that S 6= ∅, and letω ˆ ∈ S . There existsx ˆ > p such that i j i i i i

1 χi(ˆxi) =ω ˆ = . ρi(ˆxi)

Since hi(xi) = αih(βix), we also know from the proof of Proposition 6 that ρi(ˆxi) = ρ(βixˆi)

βi and χi(ˆxi) = χ(βixˆi). Letx ˆj = xˆi. Then, βj

βi 1 1 1 χj (ˆxj) = χ βj xˆi = χi(ˆxi) =ω ˆ = = = . βj ρi(ˆxi) ρ(βixˆi) ρj(ˆxj)

Therefore,ω ˆ ∈ Sj, and Si ⊆ Sj. It follows that Si = Sj for all i, j ∈ f.

If Si 6= ∅, then, by Proposition 15, condition (12) holds for ﬁrm f. Assume instead that

Si = ∅ for all i. Let i ∈ f. By Lemma 25, either χi(xi)ρi(xi) < 1 for all xi, or χi(xi)ρi(xi) > 1 for all x . Assume ﬁrst that χ (x )ρ (x ) < 1 for all x . Let j ∈ f and x > p . Then, i i i i i i j j

βj βj χj(xj)ρj(xj) = χi xj ρi xj < 1. βi βi

65 Therefore, χjρj < 1 for every j in f. It follows that

1 1 lim ρj ≤ lim = f < ∞. ∞ ∞ χj ω¯

Therefore, lim∞ hj = 0 for every j ∈ f (if lim∞ hj were strictly positive, then ρj(xj) would go to ∞ as xj goes to ∞). By Theorem 5, condition (12) holds for ﬁrm f.

Finally, assume that χi(xi)ρi(xi) > 1 for all xi. Then, using the same argument as above, χ ρ > 1 for every j ∈ f. Let i ∈ f, and assume for a contradiction that p > 0. Since 1/χ is j j i i 1 non-increasing, and since, by continuity, ιi(p ) = 1, it follows that lim + = ∞. Therefore, i pi χi lim + ρi = ∞, which is a contradiction, since ρi is non-decreasing. Therefore, p = 0. pi i Assume for a contradiction that lim0+ ιi = 1. Then, using the same reasoning as in the previous paragraph, lim0+ ρi = ∞, which is again a contradiction, since ρi is non-decreasing.

Therefore, lim0+ ιi > 1, andω ˆ ≡ lim0+ χi is strictly positive. In addition, since βj χj(x) = χi x , βi lim0+ χj =ω ˆ for every j ∈ f. Notice that, for every j ∈ f, for every x > 0,

1 1 ρj(x) ≥ lim ρj ≥ lim = , 0+ 0+ χj ωˆ and that, by Lemma 16, 1 1 1 θj(x) ≤ ≤ lim = . χj(x) 0+ χj ωˆ It follows that 1 max sup θi ≤ ≤ min inf ρi, i∈f ωˆ i∈f i.e., Assumption 4 holds. As shown in the proof of Theorem 4, condition (12) is therefore satisﬁed for ﬁrm f.

Proposition 16. Let N , (hk)k∈N , F be a pricing structure satisfying Assumptions 1 and 2. f ∗ 1 Let f ∈ F. Assume that Assumption 3 is satisfied for firm f, that ω¯ ≤ ω , and that θk ≤ ω¯f for every k in f. Then, condition (12) holds for firm f.

f f f f Proof. Let i, j ∈ f, ω ∈ (0, ω¯ ) and xi, xj > 0 such that χi(xi) > ω and χj(xj) > ω . Deﬁne

f f ω θi(xi) f ω θj(xj) f Ψ = f 1 − ω ρj(xj) + f 1 − ω ρi(xi) − ρi(xi) − ρj(xj). 1 − ω θi(xi) 1 − ω θj(xj)

66 1 As in the previous proofs, all we need to do is show that Ψ ≤ 0. Assume ﬁrst that ρi(xi) ≥ ω¯f 1 and ρj(xj) ≥ ω¯f . Then,

max (θi(xi), θj(xj)) ≤ min (ρi(xi), ρj(xj)) .

Therefore, Ψ < 0. 1 1 Next, assume that ρi(xi) < ω¯f and ρj(xj) ≥ ω¯f . Then, we claim that

1 − ω¯f 1 ρ (x ) ≥ . (28) i i ω¯f 1 − ωf

f 1 1 To see this, assume first that Si = {ωî}, whereω î ∈ (0, ω¯ ). Since ρi(xi) < ω¯f < ωî , by Lemma 25, f 1 − ωî 1 1 − ω¯ 1 ρi(xi) ≥ ≥ f f . ωî 1 − χi(xi) ω¯ 1 − ω

Assume instead that Si = ∅. By Lemma 25, either χiρi < 1 or χiρi > 1. If χiρi > 1, then we know from the proof of Corollary 1 that

1 1 ρi ≥ sup ≥ f . χi ω¯

1 This contradicts our assumption that ρi(xi) < ω¯f . If, instead, χiρi < 1, then we know from the proof of Corollary 1 that lim∞ hi = 0. Therefore, by Lemma 18, inequality (28) holds. Therefore,

f f ωf ω θ (x ) ω f 1 Ψ ≤ i i 1 − + ω¯ 1 − ωf ρ (x ) − ρ (x ) − , 1 − ωf θ (x ) ω¯f ωf i i i i ω¯f i i 1 − ω¯f f f ωf ω ρ (x ) ω f 1 ≤ i i 1 − + ω¯ 1 − ωf ρ (x ) − ρ (x ) − , 1 − ωf ρ (x ) ω¯f ωf i i i i ω¯f i i 1 − ω¯f 1 = φ f f ρ (x ), , ω ,ω¯ i i ω¯f ≤ 0 by Lemma 19.

1 1 Finally, assume that ρi(xi) < ω¯f and ρj(xj) < ω¯f . Then, as above,

1 − ω¯f 1 ρ (x ) ≥ k k ω¯f 1 − ωf

67 for k ∈ {i, j}. Therefore,

Ψ ≤ φωf ,ω¯f (ρi(xi), ρj(xj)) , which is non-positive by Lemma 19. Corollary 2. Let N , (hk)k∈N , F be a pricing structure satisfying Assumptions 1 and 2. f ∗ Let f ∈ F. Assume that Assumption 3 is satisfied for firm f, that ω¯ ≤ ω , and that θk is non-decreasing for every k in f. Then, condition (12) holds for firm f.

Proof. Let k ∈ f. Since θ is non-increasing, for every x > p , k k

1 1 θk(x) ≤ sup θk = lim θk ≤ lim = f , ∞ ∞ χk ω¯ where the second inequality follows from Lemma 16. Therefore, by Proposition 16, condition (12) holds for ﬁrm f.

C.10 Other Uniqueness Results Proposition 17. Let N , (hk)k∈N , F, (ck)k∈N be a pricing game satisfying Assumptions 1 and 2. Let f ∈ F. Assume that Assumption 3 is satisﬁed for ﬁrm f. Assume also that there R++ f exist h ∈ R++ , c > 0 and (αk)k∈f ∈ R++ such that for every k in f, ck = c, and for every x > 0, hk(x) = αkh (x). Then, condition (22) holds.

Proof. Let k ∈ f. It follows immediately from the proof of Proposition 6 that θk = θ, ρk = ρ,

γk = αkγ, ιk = ι, and χk = χ. In addition, νk = ν. Therefore, rk = r. Condition (22) is equivalent to

! ! X ωf θ 1 ωf ∀ωf ∈ (0, ω¯f ), α γ − < 1, f k P P 1 − ω θ αkh αkγ k∈f k∈f k∈f

1 where all functions are evaluated at r 1−ωf . This is equivalent to

1 − ωf ρ ωf θ < 1, 1 − ωf θ ρ which clearly holds, since θ ≤ ρ. Corollary 3. Let N , (hk)k∈N , F, (ck)k∈N be a pricing game satisfying Assumptions 1 and 2. Let f ∈ F. Assume that |f| = 1 and that assumption 3 is satisﬁed for ﬁrm f. Then, condition (22) holds.

68 Proof. This follows immediately from Proposition 17.

Proposition 18. Let N , (hk)k∈N , F be a pricing structure satisfying Assumptions 1 and 2. N Assume that there exists k ∈ N such that lim∞ hk > 0. Then, there exists (ck)k∈N ∈ R++ such that pricing game N , (hk)k∈N , F, (ck)k∈N has a unique equilibrium for every (ck)k∈N ∈ Q k∈N [ck, ∞). P Proof. Let L = k∈N lim∞ hk(> 0). For every k ∈ N , let

˜ L hk = hk − lim hk + . ∞ |N |

˜ By Theorem 1, N , hk is still a smooth and integrable demand component. In ad- k∈N dition, it is easy to see that the demand system associated with smooth and integrable ˜ demand components N , (hk)k∈N and N , hk are identical. Therefore, for ev- k∈N N ery (ck)k∈N ∈ R++, the sets of equilibria of pricing games N , (hk)k∈N , F, (ck)k∈N and ˜ ˜ N , hk , F, (ck)k∈N coincide. In the following, we work with hk , and drop the k∈N k∈N tilde from the notations. N We want to show that condition (25) holds for every f ∈ F for some (ck) ∈ R++. Let k∈N f ∈ F. Let (c0 ) ∈ Q p , ∞ . Then, for every k in f, for every x ≥ c0 , k k∈f k k

1 1 θk(x) ≤ ≤ 0 , χk(x) χk (ck) where the ﬁrst inequality follows by Lemma 16. Put

1 θ¯f = max . k∈f 0 χk (ck)

For every every k ∈ f, lim∞ hk > 0, and therefore, lim∞ ρk = ∞. It follows that there exists Q 0 ¯f (ck)k∈f ∈ k∈f (ck, ∞) such that for every k ∈ f, ρk(x) ≥ θ whenever x ≥ ck. Therefore, for every i, j ∈ f, for every xi ≥ ci and xj ≥ cj, ρi(xi) ≥ θj(xj), and, in particular,

f ω θi(xi) f f 1 − ω ρj(xj) − ρi(xi) < 0. 1 − ω θi(xi)

Therefore condition (25) holds for firm f and minimum cost profile (ck)k∈f . Constructing such a minimum cost profile for every firm in F and applying Lemma 24 allows us to conclude.

Proposition 19. Let N , (hk)k∈N , F be a pricing structure satisfying Assumptions 1 and 2.

69 For every c ∈ R++, there exists α ∈ R+ such that pricing game N , (hk + αk)k∈N , F, (ck)k∈N N N has a unique equilibrium for every (αk)k∈N ∈ [α, ∞) and (ck)k∈N ∈ [c, ∞) . Proof. Notice ﬁrst that pricing structure N , (hk + αk)k∈N , F still satisﬁes Assumptions 1 and 2. Therefore, every pricing game based on this pricing structure has an equilibrium. f f 16 Let c > 0. For every ck ≥ c, for every µ ∈ (1, µ¯ ),

r (µf , c ) > r (1, c ) ≥ r (1, c) ≡ pmin > p . k k k k k k k

min Next, notice that θk does not depend on (αk)k∈N . For every x ≥ pk ,

1 1 ¯ θk(x) ≤ ≤ min ≡ θk, χk(x) χk (pk )

¯ ¯ where the ﬁrst inequality follows from Lemma 16. Put θ ≡ maxk∈N θk. On the other hand, α ρk does depend on (αk)k∈N . We make this dependence explicit by writing ρk . For every min x ≥ pk ,

α hk(x) + αk ρk (x) = , γk(x) α ≥ k , γk(x) αk ≥ min . γk (pk )

N Therefore, there exists α ≥ 0 such that for every (αk)k∈N ∈ [α, ∞) , for every k ∈ N , for min α ¯ every x ≥ pk , ρk (x) ≥ θ. N Next, ﬁx a pricing structure N , (hk + αk)k∈N , F , where (αk)k∈N ∈ [α, ∞) . Let f ∈ F, ωf ∈ (0, ω¯f ) and µf = 1/(1 − ωf ). We want to show that condition (25) holds for ﬁrm f and f f min lower bound c. Let i, j ∈ f, xi ≥ ri µ , c and xj ≥ rj µ , c . Notice that xi ≥ pi and min α ¯ xj ≥ pj . Therefore, ρj (xj) ≥ θ ≥ θi(xi), and

f α f 1 − ω ρj (xj) α f α ω θi(xi) f − ρi (xi) ≤ ω θi(xi) − ρi (xi), 1 − ω θi(xi) ≤ θ¯(ωf − 1) < 0.

Therefore, condition (25) holds for ﬁrm f and lower bound c. By Lemma 24, pricing game N , (hk + αk)k∈N , F, (ck)k∈N has a unique pricing equilibrium for every (ck)k∈N ∈ [c, ∞).

16 Notice that rk does not depend on (αk)k∈N , since ιk is not aﬀected by (αk)k∈N .

70 Proposition 20. Let N , (hk)k∈N , F be a pricing structure satisfying Assumptions 1 and 2. f Assume that there exists a function h and scalars (αk, βk, δk, k)k∈f ∈ (R++ × R++ × R+ × R+) such that for every x > 0, for every k ∈ f,

hk(x) = αkh (βkx + δk) + k.

Assume also that θ is non-increasing and ρ is non-decreasing on (p, ∞). Then, Assumption 4 holds for ﬁrm f.

Proof. Since ρ is non-decreasing and and θ is non-increasing, ρ(x) ≥ θ(x) for every x > p.

Therefore, limp θ ≤ limp ρ, and for every x > p,

θ(x) ≤ lim θ ≤ lim ρ ≤ ρ(x). p p

Let i and j in l, x > p and x > p . Let k ∈ {i, j}. Then, i i j j

βkxk ιk(xk) = ι (βkxk + δk) , βkxk + δk

≤ ι (βkxk + δk) .

Since ιk(xk) > 1, it follows that ι (βkxk + δk) > 1. Therefore,

θk(xk) = θ (βkxk + δk) ≤ lim θ ≤ lim ρ ≤ ρ (βkxk + δk) = ρk(xk). p p

Therefore,

θi(xi) ≤ ρj(xj).

Therefore, Assumption 4 holds for ﬁrm f. Proposition 21. Let N , (hk)k∈N , F be a pricing structure satisfying Assumptions 1 and 2.

Assume that condition (11) holds for every ﬁrm in F, and let α > 0 and (βk, δk, k)k∈N ∈ N (R++ × R+ × R+) . Let Ne, hek , Fe , where Ne = N , Fe = F, and k∈Ne

hek(x) = αhk (βkx + δk) + k, ∀x > 0, ∀k ∈ N .

Then, Ne, hek , Fe is a pricing structure satisfying Assumptions 1 and 2, and condi- k∈Ne tion (11) for every ﬁrm in Fe.

71 Proof. We already know from Proposition 6 that Ne, hek , Fe is a pricing structure k∈Ne f f f satisfying Assumptions 1 and 2. Let f ∈ Fe, ω ∈ (0, ω¯ ) and (xek)k∈f such that χfk (xek) > ω for every k ∈ f. For every k in f, deﬁne xk = βkxek + δk. As shown in the proof of Proposition 6, ιek (xek) ≤ ιk(xk). Therefore,

f χk(xk) ≥ χfk (xek) > ω .

Therefore, using again the formulas derived in the proof of Proposition 6,

f ! f ! X ω θek (xek) 1 ω γek (xek) − P f P γk (xk) k∈f 1 − ω θek (xek) k∈f hek (xek) k∈f e e f ! f ! X ω θk (xk) 1 ω = αγ (x ) − , f k k P P 1 − ω θk (xk) αhk (xk) + k αγk (xk) k∈f k∈f k∈f f ! f ! X ω θk (xk) 1 ω ≤ γ (x ) − , f k k P P 1 − ω θk (xk) hk (xk) γk (xk) k∈f k∈f k∈f < 0,

f since χk(xk) > ω for every k ∈ f and condition (11) holds for ﬁrm f in pricing structure N , (hk)k∈N , F .

C.11 Proof of Proposition 6.

Let g ∈ Fe. There exists f(g) ∈ F such that for all j ∈ g, there exists k(j) ∈ f(g) and 2 (αj, βj, δj, j) ∈ R++ × R+ such that

∀x > 0, ehj(x) = αjhk(j)(βjx + δj) + j.

Then, for all x > 0,

0 0 ehj(x) = αjβjhk(j)(βjx + δj) < 0, 00 2 00 ehj (x) = αjβj hk(j)(βjx + δj) > 0,

γej(x) = αjγk(j)(βjx + δj), 0 0 γej(x) = αjβjγk(j)(βjx + δj), j ρej(x) = ρk(j)(βjx + δj) + ≥ ρk(j)(βjx + δj), αjγk(βjx + δj)

72 θej(x) = θk(j)(βjx + δj),

βjx eιj(x) = ιk(j)(βjx + δj). βjx + δj

Therefore, Ne, ehk is smooth and integrable, ρej ≥ 1 for every j in Ne, and Assump- k∈Ne tion 1 holds for Ne, ehk , Fe . Moreover, for every j ∈ g, k∈Ne

f(g) lim ιj = lim ιk(j) =µ ¯ . ∞ e ∞ Therefore, Assumption 2 also holds for Ne, ehk , Fe . Finally, we show that Assump- k∈Ne tion 4 also holds for ﬁrm g ∈ F. Let j ∈ g and x > p . Then, e ej

1 < eιj(x) ≤ ιk(j)(βjx + δj).

Therefore, β x + δ > p , and j j k(j)

θej(x) ≤ sup θk(j)(y). y>p k(j)

It follows that supy>p θej(y) ≤ supy>p θk(j)(y). Using the same reasoning, we also obtain ej k(j) that infy>p ρj(y) ≥ infy>p ρk(j)(y). Therefore, ej e k(j)

max sup θej(x) ≤ max sup θk(j)(x), j∈g x>p j∈g x>p ej k(j)

≤ max sup θk(x), k∈f(g) x>p k

≤ min inf ρk(x), x>p k∈f(g) k

≤ min inf ρk(j)(x), j∈g x>p k(j)

≤ min inf ρj(x). j∈g x>p e ej

Therefore, Assumption 4 also holds for every g in Fe.

73 C.12 Establishing Equilibrium Uniqueness using Poincar´e-Hopf

The reader may wonder whether we could obtain weaker uniqueness conditions by using more standard approaches. Uniqueness of a fixed point is usually established by using the contraction mapping approach, the univalence approach or the index (Poincaré-Hopf) approach. It is well known that the index approach is more general than the others, and that it provides an “almost if and only if” condition for uniqueness. We will therefore focus on the index approach. Since we will be working with matrices, we will sometimes assume that F = {1,...,F }, and that firm f’s set of products is N f . We know that establishing uniqueness in the pricing game is equivalent to establishing uniqueness in the auxiliary game in which firms are simultaneously choosing their µf ’s. We f also know that a profile µ = (µ )f∈F is an equilibrium of the auxiliary game if and only if for every f ∈ F,

   ! f f  X X X  f X φ (µ) ≡ (µ − 1)  hk +  hk − µ γk = 0.    k∈N f g∈F k∈N f k∈N f g6=f

Therefore, all we need to do is show that map φ has a unique zero. By the index theorem, this holds if the determinant of the Jacobian matrix of φ evaluated at µ is strictly positive whenever φ(µ) = 0. We have shown in the proof of Lemma 10 that

∂φf X X = h ≡ H(µ). ∂µf k f∈F k∈N f

Moreover, if g 6= f, then ∂φf X = (µf − 1) r0 h0 . ∂µg k k k∈N g Therefore,

P 0 0 P 0 0 H(µ)(µ1 − 1) k∈N 2 rkhk ··· (µ1 − 1) k∈N F rkhk P 0 0 P 0 0 (µ − 1) 1 r h H(µ) ··· (µ − 1) F r h 2 k∈N k k 2 k∈N k k det J(φ) = . . . . , . . .. .

F P 0 0 F P 0 0 (µ − 1) k∈N 1 rkhk (µ − 1) k∈N 2 rkhk ··· H(µ) ! ! Y X H(µ) = (µf − 1) r0 h0 det M 1 + , k k f P 0 0 (µ − 1) f r (−h ) f∈F k∈N f k∈N k k 1≤f≤F

74 where the second line has been obtained by dividing row f by µf − 1 and dividing column P 0 0 f by k∈N f rkhk for every f in {1,...,F } and by using the F-linearity of the determinant. By Lemma 21,

! ! Y X Y H(µ) det (J(φ)) = (µf − 1) r0 h0 (−1)F 1 + k k f P 0 0 (µ − 1) f r (−h ) f∈F k∈N f f∈F k∈N k k ! X Y H(µ) − 1 + , f P 0 0 (µ − 1) f r (−h ) g∈F f6=g k∈N k k ! ! Y X Y H(µ) = (µf − 1) r0 h0 (−1)F 1 + k k f P 0 0 (µ − 1) f r (−h ) f∈F k∈N f f∈F k∈N k k   X 1 × 1 − H(µ)  , f∈F 1 + f P 0 0 (µ −1) k∈N f rk(−hk) !!   Y f X 0 0 X 1 = H(µ) + (µ − 1) rk(−hk) 1 − H(µ)  . f∈F f f∈F 1 + f P 0 0 k∈N (µ −1) k∈N f rk(−hk) | {z } >0

Therefore, we need to show that

µf −1 P 0 0 X H(µ) k∈N f rk(−hk) < 1 (29) µf −1 P 0 0 f∈F 1 + H(µ) k∈N f rk(−hk) whenever φ(µ) = 0. Notice that

f P 0 0 ! X (µ − 1) k∈N f rk(−hk) X (29) ⇐⇒ − hk < 0 µf −1 P 0 0 f∈F 1 + H(µ) k∈N f rk(−hk) k∈N f   f P 0 0 X (µ − 1) f r (−h ) X ⇐⇒ k∈N k k − h < 0, since φ(µ) = 0,  f 2 P 0 0 k (µ −1) k∈N f rk(−hk) f∈F 1 + f P k∈N f µ k∈N f γk   f P 0 0 X (µ − 1) f r (−h ) X ⇐⇒  k∈N k k − h  < 0, P 0 f 0 f 0 f 0 k  µf −1 f rk((µ −1)(−hk)−µ (−γk)+µ (−γk))  k∈N f f∈F 1 + f P k∈N µ k∈N f γk   f P 0 0 X (µ − 1) f r (−h ) X ⇐⇒ k∈N k k − h < 0, by Lemma 7,  f P r0 (−γ0 ) k µ −1 f k∈N f k k f∈F 1 − f + (µ − 1) P k∈N f µ k∈N f γk

75   f f P 0 0 X µ (µ − 1) k∈N f rk(−hk) X ⇐⇒  P r0 (−γ0 ) − hk < 0, f f k∈N f k k f∈F 1 + µ (µ − 1) P k∈N f k∈N f γk ⇐⇒ Ω0 (H(µ)) < 0 (see the proof of Lemma 23).

Therefore, the index approach gives us the exact same condition as the aggregative game approach.

D Proofs for Section 5

D.1 Proof of Proposition D.1

Using the deﬁnition of function S and equation (14), it is easy to see that m(x) and S(x) are jointly pinned down by:

1 m(x) = , σ − (σ − 1)S(x) S(x) = x (1 − m(x))σ−1 .

Diﬀerentiating wrt x, we get:

m0(x) = (σ − 1) (m(x))2 S0(x), S0(x) = (1 − m(x))σ−1 − (σ − 1)xm0(x) (1 − m(x))σ−2 .

Solving out for S0 and m0 yields:

(1 − m(x))σ−1 S0(x) = > 0, 1 + (σ − 1)2x (1 − m(x))σ−2 (m(x))2 (σ − 1) (m(x))2 (1 − m(x))σ−1 m0(x) = > 0. 1 + (σ − 1)2x (1 − m(x))σ−2 (m(x))2

Since π = mS, it follows that π0 > 0. Applying the implicit function theorem to equation Ω(H) = 1 yields:

0 T f ∗ dH S H∗ = > 0. dT f P T g 0 T g g∈F H∗ S H∗

76 Next, notice that

T f f ∗ d H∗ 1 T dH = 1 − , dθf H∗ H∗ dT f  T f 0 T f  1 H∗ S H∗ = 1 − g g  > 0, H∗ P T 0 T g∈F H∗ S H∗ and that, for g 6= f, T g g ∗ d ∗ T dH H = − < 0. dθf H∗2 dT f Therefore, points (ii) and (iii) follow immediately by applying the chain rule.

D.2 Proof of Proposition 10

Proof. Applying the implicit function theorem to equation (16), we see that, for every x > 0,

m(x)e−m(x) m0 (x) = > 0. 1 −m(x) m(x) + m (x) xe

−m(x) 1 0 0 Notice also that S(x) = x e = 1 − m(x) . Therefore, S > 0 and π > 0. Points (ii) and (iii) follow by applying the implicit function theorem to equation

X T f S = 1 H f∈F as we did in D.1.

E Proofs for Section 6

E.1 Proof of Proposition 11

Proof. Suppose ﬁrst that there are no synergies, i.e., η = 0. We claim that the ﬁtting- in functions satisfym ˆ f∪g(H) > max mf (H), mg(H) for all H > 0. We now show that mˆ f∪g(H) > mf (H). (The argument as to whym ˆ f∪g(H) > mg(H) is the same.) We have

P f∪g ! γj rj(m ˆ , cj) 1 =m ˆ f∪g 1 − j∈f∪g H

77 P f ! γj rj(m , cj) = mf 1 − j∈f H P f ! γj rj(m , cj) > mf 1 − j∈f∪g H where the equalities follow from Lemma 14. As the right-hand side is increasing in the ι- markup µ, it follows thatm ˆ f∪g(H) > mf (H). Hence, for any H, the contribution of the merged ﬁrm to the aggregator is smaller than the sum of the contributions of the merger partners pre-merger:

X f∪g X f X g hj(rj(m ˆ (H))) < hj(rj(m (H))) + hj(rj(m (H))) j∈f∪g j∈f j∈g

As the fitting-in function of any firm not participating in the merger (and thus its contribution to the aggregator) is unaffected, this implies that Ω(ˆ H) < Ω(H) for all H > 0. Recall from the proof of Lemma 15 that Ω(H) > 1 for H sufficiently small and Ω(H) < 1 for H sufficiently large. Hence, in both the smallest and largest equilibrium, Hˆ ∗ < H∗. That is, in the absence of synergies (η = 0), the merger is CS-decreasing. As a decrease in marginal cost induces an increase in consumers’ indirect utility by Propo- sition 3, it follows that there exists a cutoffη ˆ > 0 such that the merger is CS-increasing if and only if η > ηˆ. (Ifη ˆ = 1, then even maximal synergies do not suffice to make the merger CS-increasing.) We also claim that, in the absence of synergies (η = 0), the merger is privately profitable ˆ ∗ ∗ 0 and induces an increase in all prices. To see this, note that as H < H , rk(·) > 0, and m0(·) < 0, the prices of all products of outsiders to the merger are higher after the merger than before, implying that Hˆ ∗−(f∪g) < H∗−(f∪g). So, even if the merged firm were to charge the same prices as the merger partners did before the merger, the merger would be privately profitable. But, in fact, the merged firm choses to charge higher prices than its partners did before the merger. This is for two reasons. First, because in the absence of synergies, mˆ f∪g(H) > max mf (H), mg(H) for all H > 0. Second, because in the absence of synergies, the merger induces an increase in the equilibrium level of the aggregator.

78 E.2 Proof of Lemma 20

Proof. Using the notation introduced in Section 5, let

T f T g TˆM ≡ H∗S−1 S + S . (30) H∗ H∗

If T M = TˆM , we have:

X T l 1 = S H∗ l∈F T M X T l = S + S , H∗ H∗ l∈F\(f∪g) where the ﬁrst equality is the pre-merger equilibrium condition whereas the second equality follows from T M = TˆM . As equilibrium is unique when demand is CES or multinomial logit, we have Hˆ ∗ = H∗. That is, the merger is CS-neutral if T M = TˆM . As S0(·) > 0, if T M > TˆM , we have T M X T l S + S > 1, H∗ H∗ l∈F\(f∪g) implying that Hˆ ∗ > H∗, so the merger is CS-increasing. Similarly, if T M < TˆM , then Hˆ ∗ < H∗, so the merger is CS-decreasing. Next, we note that a CS-neutral merger involves synergies in that TˆM > T f +T g. Suppose otherwise that TˆM ≤ T f + T g. Then,

! TˆM T f + T g S ≤ S H∗ H∗ T T g < S + S , H∗ H∗ where the first inequality follows from S0(·) > 0 and the second from S00(·) < 0, as can be verified to hold under both CES and multinomial logit demands. But then the merger would be CS-decreasing, a contradiction. Hence, TˆM > T f + T g. To see that a CS-neutral merger is profitable, note that:

! ! ! TˆM TˆM TˆM π > m S H∗ H∗ H∗

79 ! TˆM T f T g = m S + S H∗ H∗ H∗ T f T f T g T g > m S + m S H∗ H∗ H∗ H∗ T T g = π + π , H∗ H∗ where the second equality follows because the merger is CS-neutral, and the second inequality follows because TˆM > T f + T g > max T f ,T g and m0(·) > 0, both under CES and multinomial logit demands. Hence, merger M is profitable if T M = TˆM . Next, consider the effect of an increase in firm type T M on the equilibrium level of the ∗ ∗ P T u aggregator H . Applying the implicit function theorem to Ω(H ) ≡ u∈F S H∗ = 1, we obtain: 0 T M ∗ dH S H∗ = > 0, dT M P T u 0 T u u∈F H∗ S H∗ where the inequality follows as S0(·) > 0. The effect of an increase in T M on M’s equilibrium profit is thus given by

TˆM 0 T M M ∗ dπ H∗ π H∗ T dH = 1 − T M H∗ H∗ dT M 0 T M  0 T M  M π H∗ T S H∗ = 1 − u u  > 0, H∗ H∗ P T 0 T u∈F H∗ S H∗ where the inequality follows as π(·) = m(·)S(·), m0(·) > 0 and S0(·) > 0, implying that π0(·) > 0. That is, an increase in a firm’s type induces an increase in that firm’s equilibrium profit. It follows that merger M is profitable if T M ≥ TˆM , i.e., it is CS-nondecreasing.

E.3 Proof of Proposition 12

We ﬁrst prove the following lemma:

Lemma 26. In a multiproduct-ﬁrm pricing game with CES or Logit demands, ε0 < 0, where S0(x) ε(x) = x S(x) for all x > 0.

Proof. Under CES demands,

1 ε(x) = , 1 + (σ − 1)2x(1 − m(x))σ−2m(x)2

80 1 = , 2 m(x)2 1 + (σ − 1) S(x) 1−m(x) which is indeed decreasing in x, since m and S are decreasing. Under Logit demands, 1 ε(x) = , 1 + m(x)2S(x) which is also decreasing in x, since m and S are decreasing.

We can now prove Proposition 12:

Proof. We ﬁrst show that dTˆM /dH∗ < 0. Diﬀerentiating equation (30), we obtain

! ! TˆM dTˆM TˆM TˆM T f T f T g T g S0 = S0 − S0 − S0 , H∗ dH∗ H∗ H∗ H∗ H∗ H∗ H∗ ! ! TˆM TˆM T f T f T g T g = ε S − ε S − ε S , H∗ H∗ H∗ H∗ H∗ H∗ ! TˆM T f T g T f T f T g T g = ε S + S − ε S − ε S , H∗ H∗ H∗ H∗ H∗ H∗ H∗ < 0, where the third line follows by deﬁnition of TˆM and the last line follows from Lemma 26 and from the fact that TˆM > T f + T g.

M1 ˆM1 Suppose Mi is CS-nondecreasing in isolation, which means that T ≥ T . If the CS- ∗ nondecreasing merger Mj takes place, the equilibrium value of the aggregator H weakly increases, and so the cutoﬀ TˆM1 weakly decreases. As T M1 was initially above the cutoﬀ, it therefore remains so after Mj has taken place, i.e., Mi is still CS-nondecreasing. A similar argument can be used to show the sign-preserving complementarity for mergers that are CS-decreasing in isolation.

81 References

[1] Bernard, A. B., S. J. Redding, and P. K. Schott (2011). “Multiproduct Firms and Trade Liberalization.” Quarterly Journal of Economics, vol. 126, 1271–1318.

[2] Berry, S. T. (1994). “Estimating Discrete-Choice Models of Product Diﬀerentiation.” The RAND Journal of Economics, vol. 25, 242–262.

[3] Berry, S.T., J. Levinsohn and A. Pakes (1995). “Automobile Prices in Market Equilib- rium.” Econometrica, vol. 63, 841–890.

[4] Bertoletti, P. and F. Etro (2015). “Monopolistic Competition when Income Matters.” Unpublished Manuscript.

[5] Breinlich, H., V. Nocke and N. Schutz (2015).“Merger Policy in a Quantitative Model of International Trade.” Unpublished Manuscript.

[6] Chan, T. Y. (2006). “Estimating a Continuous Hedonic-Choice Model with an Applica- tion to Demand for Soft Drinks.” The RAND Journal of Economics, vol. 37, 1–17.

[7] Dhingra, S. (2013). “Trading Away Wide Brands for Cheap Brands.” American Eco- nomic Review, vol. 103, 2554–2584.

[8] Dubin, J. A. and D. L. McFadden (1984). “An Econometric Analysis of Residential Electric Appliance Holdings and Consumption ” Econometrica, vol. 52, 345–362.

[9] Eckel, C. and J. P. Neary (2010). “Multi-Product Firms and Flexible Manufacturing in the Global Economy.” Review of Economic Studies, vol. 77, 188–217.

[10] Hanemann, W. M. (1984). “Discrete/Continuous Models of Consumer Demand.” Econo- metrica, vol. 52, 541–561.

[11] Hanson, W., and K. Martin (1996). “Optimizing Multinomial Logit Proﬁt Functions.” Management Science, vol. 42, 992–1003.

[12] Mayer, T., M. Melitz, and G. I. P. Ottaviano (2014). “Market Size, Competition, and the Product Mix of Exporters” American Economic Review, vol. 104, 495–536.

[13] Milgrom, P. and C. Shannon (1994). “Monotone comparative statics.” Econometrica, vol. 62, 157–180.”

82 [14] Nevo, A. (2001). “Measuring Market Power in the Ready-to-Eat Cereal Industry. Econo- metrica, vol. 69, 307–342.

[15] Nocke, V., and N. Schutz (2015). “Quasilinear Integrability.” Unpublished manuscript.

[16] Nocke, V., and M. D. Whinston (2010). “Dynamic Merger Review.” Journal of Political Economy, vol. 118(6), 1200–1251.

[17] Nocke, V. and S. Yeaple (2014). “Globalization and Multiproduct Firms.” International Economic Review, vol. 55, 993–1018.

[18] Novshek, W. and H. Sonnenschein (1979). “Marginal Consumers and Neoclassical De- mand Theory.” Journal of Political Economy, vol. 1368–1376.

[19] Selten, R. (1970). “Preispolitik der Mehrproduktenunternehmung in der Statischen The- orie.” Springer Verlag: Berlin, Germany.

[20] Spady, R. H. (1984). “Non-Cooperative Price-Setting by Asymmetric Multiproduct Firms.” Unpublished manuscript.

[21] Smith, H. (2004). “Supermarket Choice and Supermarket Competition in Market Equi- librium.” Review of Economic Studies, vol. 71, 235–263.

[22] Topkis, D. M. (1998). “Supermodularity and Complementarity.” Princeton University Press.

[23] Vives, X. (2000). “Oligopoly Pricing: Old Ideas and New Tools.” MIT Press.