Toward a theory of monopolistic competition

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Sep 12, 2016 - Keywords: monopolistic competition, general equilibrium, additive preferences, .... that differ in their degree of product differentiation. In addition ...
Toward a theory of monopolistic competition∗ Mathieu Parenti



Philip Ushchev



Ÿ

Jacques-François Thisse

September 12, 2016

Abstract We propose a general model of monopolistic competition, which encompasses existing models while being exible enough to take into account new demand and competition features. Even though preferences need not be additive and/or homothetic, the market outcome is still driven by the sole variable elasticity of substitution. We impose elementary conditions on this function to guarantee empirically relevant properties of a free-entry equilibrium. Comparative statics with respect to market size and productivity shocks are characterized through necessary and sucient conditions. Furthermore, we show that the attention to the CES based on its normative implications was misguided: we propose a new class of preferences, which express consumers' uncertainty about their love for variety, that yield variable markups and may sustain the optimum. Last, we show how our approach can cope with heterogeneous rms once it is recognized that the elasticity of substitution is rm-specic.

Keywords:

monopolistic competition, general equilibrium, additive preferences, homothetic pref-

erences

JEL classication:

D43, L11, L13.

We are grateful to two referees, C. d'Aspremont, K. Behrens, A. Costinot, F. Di Comite, R. Dos Santos, S. Kichko, H. Konishi, V. Ivanova, P. Legros, M.-A. López Garcia, O. Loginova , Y. Murata, G. Ottaviano, J. Tharakan and seminar audience at Bologna U., Brown, CEFIM, Columbia, ECORES, E.I.E.F., Lille 1, NES, Princeton, Stockholm U., UABarcelona, and UQAMontréal for comments and suggestions. We owe special thanks to A. Savvateev for discussions about the application of functional analysis to the consumer problem. The study has been funded by the Russian Academic Excellence Project '5-100'. † ECARES, Université Libre de Bruxelles (Belgium) and CEPR. Email: [email protected]. ∗



NRU-Higher School of Economics (Russia). Email: [email protected]

Ÿ CORE

(Belgium), NRU-Higher School of Economics (Russia) and CEPR. Email: [email protected] 1

1 Introduction The constant elasticity of substitution (CES) model of monopolistic competition, developed by Dixit and Stiglitz (1977), has been used in so many economic elds that a large number of scholars view it as virtually

the

model of monopolistic competition. For example, Head and Mayer (2014)

observe that the CES is nearly ubiquitous in the trade literature. However, owing to its extreme simplicity, this model ignores several important eects that contradict basic ndings in economic theory. For example, unlike what the CES predicts, prices and rm sizes are aected by entry and market size, markups vary with costs and consumer income, while the pass-through is incomplete. Recent empirical studies conducted at the rm level provide direct evidence for these ndings (De Loecker and Goldberg, 2014). In addition, tweaking the CES or using other specic models in the hope of getting around those diculties unfortunately prevents a direct comparison between results. We realize that such a research strategy is motivated by its tractability, but one takes the chance of ignoring the fragility of the results. For example, by nesting quadratic preferences into a quasi-linear utility, Melitz and Ottaviano (2008) show that prices depend on market size but suppress the per capita income eect. Markups depend on per capita income under the linear expenditure system in an open economy (Simonovska, 2015), but this eect disappears in a closed economy under additive preferences (Zhelobodko

et al.,

2012).

Under indirectly additive preferences, there is an income eect but

market size has no impact on prices (Bertoletti and Etro, 2016). Prices are independent of the number of competitors in the CES model of monopolistic competition, but not in oligopolistic competition (d'Aspremont

et al.,

1996). Therefore, the absence of a general framework makes it

hard to deal with the implications of dierent specications of preferences. The supply side of monopolistic competition models has attracted a great deal of attention ever since the work of Melitz (2003). In a very recent work, Hottman to

75%

et al.

(2016) nd that

50

of the variance in U.S. rm size can be attributed to dierences in what these authors call

rms' appeal, that is, the demand side, and less than

20%

to average marginal cost dierences.

Thus, we may safely conclude that it is time to pay more attention to the

demand side.

This is

where we hope to contribute by showing that working with general preferences is both doable and desirable. Our aim is to develop

a general equilibrium model of monopolistic competition

that leads to

clear-cut predictions regarding the impact of various types of shocks on the market outcome. Working at a high level of generality is desirable because it buys enough exibility to capture a much wider range of industrial patterns than specic models. For example, working with nonadditive and nonhomothetic preferences allows us to uncover how markups may react to both population size and income eects, and under which circumstances the pass-through is incomplete. As our set-up encompasses all existing models of monopolistic competition, including those with CES, quadratic, CARA, and translog preferences, we may determine when a result is, or is not, specic

2

to a functional form. Furthermore, despite its generality, our set-up leads to a neat comparative statics in market size, per capita income, and production eciency, which is economical enough to be summarized in a simple table (Table 1 in Section 3.2.4). From the applied viewpoint, our approach suggests the following research strategy.

Using

stylized facts permits one to restrict the class of admissible preferences to work with. The next step is to nd a functional form of utility that belongs to this class while displaying enough exibility to capture a wide range of possible eects. One can then estimate the structural parameters of the model and check whether its predictions are consistent with the theory. It is worth noting that this approach has been taken recently by Mayer

et al.

(2016) who focus on additive preferences.

Finally, the versatility of our model shows that it can be used as a building block in full-edged general equilibrium models that aim to cope with various economic issues. Although common wisdom holds that the elasticity of substitution is relevant only in the CES case, we show that this view is not justied  a

variable

elasticity of substitution can be used

as the primitive of a general model of monopolistic competition.

What makes this approach

appealing is that an increasing (decreasing) elasticity of substitution means that varieties are less (more) dierentiated.

This vastly facilitates the interpretation of our results and permits

a neat comparison with what we know from industrial organization about the impact of product dierentiation on the market outcome. When the elasticity of substitution is variable, the key issue is to determine the relevant variables. To this end, we start by studying the case of symmetric rms because predictions are governed by the sole properties of preferences.

More specically,

we nd necessary and sucient conditions to be imposed on the elasticity of substitution for the market outcome to react in a specic way to shocks on market size, individual income and rm productivity.

Dealing with heterogeneous rms amounts to adding one more dimension to the

problem, that is, the elasticity of substitution becomes type-specic. Having sketched our road map, we may summarize our main ndings as follows. (i) Using the concept of Fréchet dierentiability, which applies to a case where the unknowns are functions rather than vectors, we determine a general demand system that includes the functional forms used in the literature. By focussing on symmetric consumption proles, we are able to develop a parsimonious approach in which the elasticity of substitution depends on just

two

variables, namely, the consumption level and the number of available varieties. We will show that additive and homothetic preferences, which are used in various models of monopolistic competition, may be visualized as the horizontal and vertical axes of a two-dimensional space whose origin is the CES. It is easier to insulate the sole impact of dierent types of preferences on the market outcome

the properties of an equilibrium depend on how the elasticity of substitution behaves when the per variety consumption and the number of rms vary. By imposing plausible and empirically relevant conditions on this function, we are able to by considering symmetric rms. We rst show that

disentangle the various determinants of rms' equilibrium behavior. Hence, the applied economist

3

who uses a particular functional form may determine how the market outcome behaves simply by inspecting the corresponding elasticity of substitution. (ii) Our set-up is especially well suited for conducting comparative static analyses in that we can determine the necessary and sucient conditions for various thought experiments undertaken in the literature. The most common experiment is to study the impact of market size on the market outcome. What market size signies is not always clear because it compounds two variables, that is, the number of consumers as well as their willingness to pay for the product under consideration. The impact of population size and income level on prices, output, and the number of rms does not need be the same because these two parameters (population size and income level) aect rms' demand in dierent ways. An increase in population or income raises demand, thereby fostering entry and lower prices. But an income hike also increases consumers' willingness to pay, which tends to push prices upward. The nal impact is thus a priori ambiguous. We show that a larger population results in a lower market price and bigger rms if, and only if, the elasticity of substitution responds more to a change in the number of varieties than to a change in the per variety consumption.

This case arises when the entry of new rms makes

varieties less dierentiated. As for the number of varieties, it increases with the population size if varieties do not become too similar when their number rises. Thus, as in most oligopoly models, monopolistic competition exhibits the standard pro-competitive eects associated with market size and entry. An increase in individual income generates eects that are similar, but not identical, to a population hike if, and only if, varieties become closer substitutes when their range widens. The CES is the only utility in which price and output are independent of both income and population size. However, even with identical consumers and non-strategic rms, standard assumptions about preferences are not sucient to rule out anti-competitive eects.

For this, we need additional

assumptions. Our model also allows us to study the impact of a productivity or trade liberalization shock on markups. The existing empirical evidence collected in trade models will guide us in choosing which assumption to work with. In sum, studying the elasticity of substitution is sucient to predict how the market reacts to various shocks. (iii) Ever since Chamberlin (1933), the question of whether the market under- or over-provides diversity is one of the most studied issues in the theory of imperfect competition. It is well known that the CES is the only additive utility for which the market achieved the optimum (Dixit and Stiglitz, 1977; Dhingra and Morrow, 2016).

The conventional wisdom holds that the constant

markup associated with the CES is necessary for this result to hold. Non-CES homothetic preferences typically generate markups that vary with the number of rms. We introduce an intuitive measure of the social benet generated by a new variety and nd a necessary and sucient condition for the market equilibrium and the optimum to coincide. We propose a new class of preferences, which express consumers' uncertainty about their love for variety, that yield variable markups and may sustain the optimum. Therefore, the attention that the CES has received regarding its

4

normative implications was not warranted; constant markups, additivity, or homotheticitythree properties veried by the CESare neither necessary nor sucient for the market to deliver the optimum. (iv) Our nal step is to gure out how the approach developed in this paper can cope with heterogeneous rms à la Melitz. What renders the analysis much more complex here is the fact that a rm's prot-maximizing strategy now depends directly on the strategies chosen by all the other rms.

To be precise, a rm's equilibrium output cannot be determined pointwise, like in

Melitz (2003) and Melitz and Ottaviano (2008), or through a market aggregate, like in Zhelobodko

et al.

(2012). Despite this, our model displays enough structure for us to show the existence of

a quantity equilibrium when the mass of entrants and cuto cost are given.

This being done,

regardless of the distribution of marginal costs, we show that the elasticity of substitution across varieties produced by rms that enjoy the same productivity level depends on these rms' marginal cost, as well as upon the total number of entrants and the cuto cost, that is, the two variables that determine the mass of operating rms. rises from

2

to

In other words, the dimensionality of the problem

3.

Even though the pricing rule remains the same as in the symmetric rm case,

of substitution is now cost-specic.

the elasticity

In other words, rms that dier in productivity sell varieties

that dier in their degree of product dierentiation. In addition, whereas the CES cuto cost is independent of market size in a closed economy, the cuto cost is not constant anymore. It may decrease or increase with market size, the reason being that the intensity of competition varies with the number of entrants. Thus, more work is called for before taking for granted that trade liberalization is always productivity-enhancing. The following remark is in order. Modeling monopolistic competition as a non-cooperative game with a continuum of rms yields a framework easier to handle than standard oligopoly models while coping with general equilibrium eects, a task which is hard to accomplish in oligopoly theory (Hart, 1985). Even though rms do not compete strategically, our model of monopolistic competition mimics the behavior of oligopolistic markets and oers a possible solution to the diculties encountered in general equilibrium with imperfectly competitive markets. Our model thus builds a link between two bodies of the literature on imperfect competition which were perceived so far as totally separate. This is in accordance with Mas-Colell (1984, p.19) for whom the theory of monopolistic competition derives its theoretical importance not from being a realistic complication of the theory of perfect competition, but from being a simplied, tractable limit of oligopoly theory. To accomplish this, however, we need a richer description of the demand side than the CES model of monopolistic competition.

Related literature.

Dierent alternatives have been proposed to avoid the main pitfalls of the

CES model. Behrens and Murata (2007) propose the CARA utility that captures price competition eects, while Zhelobodko

et al.

(2012) use general additive preferences to work with a variable

5

elasticity of substitution, and thus variable markups, while Bertoletti and Etro (2016) focus on indirectly additive preferences. Vives (1999) and Ottaviano

et al.

(2002) show how the quadratic

utility model avoids some of the diculties associated with the CES model, while delivering a full

et al. (2012) use symmetric homothetic preferences in a real business et al. (2015) propose a symmetric demand system that includes both

analytical solution. Bilbiie cycle model.

Arkolakis

additive preferences and Kimball's exible aggregator homothetic preferences (Kimball, 1995). Last, Mrázová and Neary (2013), pursuing a dierent, but related, objective, study a class of demands that implies an invariant relationship between the demand elasticity and the curvature of demand schedules. In the next section, we describe the demand and supply sides of our set-up. In particular, the primitive of the model being the elasticity of substitution function, we discuss how this function may vary with the per variety consumption and the number of varieties. In Section 3, we prove the existence and uniqueness of a free entry equilibrium and characterize its various properties when rms are symmetric. Section 4 focuses on the optimality of the market outcome. Heterogeneous rms are studied in Section 5. Section 6 concludes.

2 The model and preliminary results In this section, we set out a simple model of Chamberlinian monopolistic competition. Our main purpose is to develop sound micro-foundations for what will be the key tool of our whole analysis  the

elasticity of substitution function.

To keep things simple, we consider an economy with

L

identical consumers, one sector and one production factor  labor, which is used as the numéraire. Each consumer is endowed with

y

units of labor. On the supply side, there is a continuum of rms

producing each a horizontally dierentiated good under increasing returns. Each rm supplies a single variety and each variety is supplied by a single rm.

2.1 Consumers Let

N,

an arbitrarily large number, be the mass of

denition of arbitrarily large). consumers, we denote by

N ≤N

potential

varieties (see below for a precise

As all potential varieties are not necessarily made available to the endogenous mass of

available

varieties.

Since we work with a continuum of varieties, we cannot use the standard tools of calculus. Rather, we must work in a functional space whose elements are functions, and not vectors.

A

consumption prole x ≥ 0 is a Lebesgue-measurable mapping from the space of potential varieties [0, N] to R+ . We denote by xi the consumption of variety i; for i ∈]N, N] we set xi = 0. We assume that x belongs to L2 ([0, N]). The benet of using this space is that it is fairly straightforward to impart precise content to the assumption that the utility functional is dierentiable for Hilbert spaces (see (5) below). This space is rich enough for our purpose. For example, it embraces all

6

consumption proles that are bounded. To start with, we give examples of preferences used in models of monopolistic competition.

Examples. 1. Additive preferences (Spence, 1976; Dixit and Stiglitz, 1977; Kühn and Vives, 1999). Preferences are additive over the set of varieties if ˆ

N

U(x) ≡

u(xi )di,

(1)

0 where

u

is strictly increasing, strictly concave, and such that

u(0) = 0.

The CES, which has been

used extensively in many elds and the CARA (Behrens and Murata, 2007) are special cases of

indirectly additive

(1). Bertoletti and Etro (2016) consider the case of

ˆ

preferences:

N

V(p; Y ) ≡

v(Y /pi )di, 0

where

v

is strictly increasing, strictly convex, and homogeneous of degree zero,

income and

pi

Y

the consumer's

i.

the price of variety

Mixed additive preferences.

Additive preferences can be generalized to cope with consumers

who are unsure about their taste for product dierentiation. More specically, consider a family

{U (·, ρ)}ρ∈[0,1]

of utility functions and assume that

ρ

is a random variable distributed over

[0, 1].

Dene

 ˆ U(x) ≡ E ln

N

 U (xi , ρ)di ,

(2)

0 where

E(·)

is the expectation operator.

It is readily veried that, when all

U (·, ρ)

are positive,

strictly increasing, and strictly concave functions, then (2) describes a well-behaved utility. Under a Dirac distribution, we fall back on additive utilities.

2. Homothetic preferences.

A tractable example of non-CES homothetic preferences used

in the macroeconomic and trade literature is the

al.,

2012; Feenstra and Weinstein, 2016).

translog

(Bergin and Feenstra, 2009; Bilbiie

et

There is no closed-form expression for the translog

utility function. Nevertheless, by appealing to the duality principle in consumption theory, these preferences can be described by the following expenditure function:

1 ln E(p) = ln U0 + N

ˆ 0

N

β ln pi di − 2N

"ˆ 0

N

1 (ln pi )2 di − N



N

2 # ln pi di .

0

A broad class of homothetic preferences is given by what is known as

gregator,

Kimball's exible ag-

which has been introduced by Kimball (1995) as a production function used in the

macroeconomic literature (Chari

et al., 2000; Smets and Wouters, 2007).

A utility functional

U(x)

is said to be described by Kimball's exible aggregator if there exists a strictly increasing and

7

strictly concave function

θ(·)

U(x)

such that

ˆ



N

θ 0 for any consumption bundle

x.

satises

Whenever

xi U(x)

U(x)

 di = 1

(3)

satises (3), it is single-valued, continuous, in-

creasing, strictly quasi-concave, and linear homogeneous.

The CES is the only utility function

which is additive, indirectly additive and homogeneous (Samuelson, 1965).

3. Quadratic preferences. etic is the quadratic utility: ˆ U(x) ≡ α 0 where

α, β,and γ

N

An example of preferences that are neither additive nor homoth-

β xi di − 2

ˆ

N

x2i di

0

are positive constants and

X0

γ − 2

ˆ

N

0



N

 xi di xj dj + X0 ,

0

the quantity of an outside good (Dixit, 1979; Singh

et al., 2002; Melitz and Ottaviano, 2008). 4. The almost ideal demand system (Deaton and Muellbauer,

and Vives, 1984; Ottaviano

1980; Fajgelbaum and

Khandelwal, 2016). This demand system is associated with preferences generated by the following indirect utility function:

" V(p; Y ) ≡ F where

a(p)

and

b(p)

are two price aggregates and

Since we work with unspecied preferences,

Y a(p) F

any

1/b(p) #

is an increasing function. symmetric demand system may be cast into

our framework. In particular, our approach encompasses Arkolakis (2016) and Behrens

et al.

et al.

(2015), Bertoletti

et al.

(2016).

This incomplete list of examples should be sucient to show that the authors who use monopolistic competition appeal to a range of models that display very dierent properties. Since it is unclear how these functional forms relate to each other and, more importantly, it is hard to assess the robustness of the theoretical predictions derived for specic demand systems and to match them to the empirical results obtained with other demand systems.

This points to the need of

a more general set-up in which we can cast all these special cases and compare their properties. Having this in mind, we choose to describe individual preferences by a general utility functional

U(x)

dened over

L2 ([0, N]).

Assumptions. In this paper, we make two main assumptions about U . First, the functional U is symmetric over [0, N] in the sense that any Lebesgue measure-preserving mapping from [0, N] into itself does not change the value of

U.

Intuitively, this means that renumbering varieties has

no impact on the utility level. Second, the utility function exhibits say, for any

[0, N ]

N ≤ N,

love for variety.

Than is to

a consumer strictly prefers to consume the whole range of available varieties

than any subinterval

[0, n]

of

[0, N ],

that is,

8

 U

X I[0,n] k



 0 is the consumer's total consumption of the dierentiated good and IA is the indicator of A v [0, N ]. Consumers love variety whenever U(x) is continuous and strictly quasi-concave. We also impose that U has well-behaved marginal utilities in the following sense. For any given N , the utility functional U is said to be Fréchet dierentiable in x ∈ L2 ([0, N]) when there exists a unique function D(xi , x) from [0, N] × L2 ([0, N]) to R such that, for all h ∈ L2 , the equality

where

ˆ

N

D(xi , x) hi di + ◦ (||h||2 )

U(x + h) = U(x) +

(5)

0

||·||2

L2 -norm marginal utility of variety i.1

holds,

being the

(Dunford and Schwartz, 1988). The function That

D(xi , x)

i

is not indexed by

D(xi , x)

follows from the symmetry of

preferences. From now on, we focus on utility functionals that satisfy (5) for all

D(xi , x) is decreasing and D(xi , x) (strictly) decreases

that the marginal utility to

xi ∈ R + .

Evidently,

is called the

x≥0

and such

twice continuously dierentiable with respect with

xi

if

U

is (strictly) concave. The reason

for restricting ourselves to decreasing marginal utilities is that this property allows us to work directly with well-behaved inverse demand functions.

p ≥ 0 be a market price prole which is a Lebesgue-measurable mapping from [0, N] to R+ . We denote by pi the price of variety i when this variety is available; otherwise pi is arbitrarily large but nite. Bewley (1972) has shown that a market price prole p ≥ 0 must belong to the dual of the space of consumption proles. This condition holds here when p also belongs to L2 ([0, N]) Let

because this space is its own dual. In this case, the total expenditure, which is given by the inner product

p · x,

is nite.

Under (5), the rst-order conditions in

L2 ([0, N])

are similar to the corresponding conditions

in a nite-dimensional space. Therefore, maximizing the utility functional budget constraint

ˆ

U(x)

subject to (i) the

N

p·x=

pi xi di = Y,

(6)

0 and (ii) the availability constraint

xi ≥ 0

for all

i ∈ [0, N ]

and

xi = 0

for all

i ∈]N, N]

yields the following inverse demand function for the available variety

pi =

D(xi , x) λ

for all

1 The

i ∈ [0, N ],

i: (7)

denition of a Fréchet-dierentiable function must be changed when the marginal utility grows very fast in the neighborhood of xi = 0. The proof is available upon request from the authors.

9

where

λ

is the Lagrange multiplier of the consumer's optimization problem.

function of

Y

and

x,

Expressing

λ

as a

we obtain

´N 0

λ(Y, x) =

xi D(xi , x) di , Y

(8)

which is the marginal utility of income at the consumption prole

x

under income

Y.

2.2 Firms: rst- and second-order conditions We model monopolistic competition as a non-cooperative game with a continuum of rms. The total mass

N

of rms is endogenously determined by the zero-prot condition. There are increasing

returns at the rm level, but no scope economies that would induce a rm to produce several varieties. The continuum assumption distinguishes monopolistic competition from other market structures in that it is the formal counterpart of the basic idea that a rm's action has no impact on the others.

As a result, by being negligible to the market, each rm may choose its output

(or price) while accurately treating market variables as given. However, for the market to be in equilibrium, rms must accurately guess what these variables will be. Firms share the same xed cost above by

yL/F .

F

In other words, to produce

units of labor. Hence, rm

i's

c, so that N is bounded i needs F + cqi eciency

and the same constant marginal cost

qi

units of its variety, rm

prot is given by

π(qi ) = (pi − c)qi − F.

(9)

Since consumers share the same preferences, the consumption of each variety is the same across consumers. Therefore, product market clearing implies to its output

qi

qi = Lxi .

Firm

i maximizes (9) with respect

while the market outcome is given by a Nash equilibrium. The Nash equilibrium

distribution of rms' actions is encapsulated in

x

and

λ.

In the CES case, this comes down to

treating the price-index parametrically, while under additive preferences the only payo-relevant market statistic is

λ.

Combining (7) with (9), the program of rm

i

is given by

 D (xi , x) − c L xi − F. max πi (xi , x) ≡ xi λ 

Setting

Di ≡ D(xi , x)

Di0 ≡

∂D(xi , x) ∂xi

Di00 ≡

∂D2 (xi , x) , ∂x2i

for notational simplicity, the rst-order condition for prot-maximization can be cast as

Di + xi Di0 = [1 − η¯(xi , x)] Di = λc,

10

(10)

where

η¯(xi , x) ≡ − is the elasticity of the inverse demand for variety standard monopoly-pricing rule:

Since

λ

xi 0 D Di i i.

(11)

Using (7), (10) may be rewritten as the

pi − c = η¯(xi , x). pi

(12)

is endogenous, we seek necessary and sucient conditions for a unique (interior or

corner) prot-maximizer to exist regardless of the value of

λc > 0.

Lemma 1. Assume that, for any x, the following conditions hold: lim Di = ∞

lim Di = 0.

xi →0

(13)

xi →∞

Then, (10) has at least one positive solution xi (x).2 Proof. See Appendix A. The rst-order condition (10) is sucient if the prot function

πi

is strictly quasi-concave in

xi , that is, the second derivative of πi is negative at any solution to the rst-order condition. rm i treats λ parametrically, the second-order condition is given by xi Di00 + 2Di0 < 0. In other words, the prot function

πi

is strictly quasi-concave in

Since

(14)

xi

for all values of

λc

if and only

if the following consition holds:

A) rm i's marginal revenue decreases in xi . Observe that (A) is equivalent to the well-known condition obtained by Caplin and Nalebu (

(1991) for a rm's prots to be quasi-concave in its own price: the Marshallian demand for variety

i

is

(−1)-convex

in

pi .

Thus, (

A) may be viewed as the least demanding condition to impose on

demands to show the existence of an equilibrium for all

λc.

2.3 The elasticity of substitution In what follows, we treat the

elasticity of substitution

as the primitive of our approach to monop-

olistic competition. The merit of this approach is that the elasticity of substitution is an inverse measure of product dierentiation, which in turn allows us to appeal to industrial organization to back up some assumptions. In addition, we will show below that choosing the elasticity of substitution as a primitive is a reasonable research strategy since our comparative static results are all stated as necessary and sucient conditions to be imposed on the elasticity of substitution. The elasticity of substitution is, therefore, a sucient statistic to study the property of the free-entry

2 The

condition (13) does not hold when Di /λ displays a nite choke price. However, it is readily veried that (10) has at least one positive solution when the choke price exceeds λc. 11

equilibrium. It is well known that there are dierent denitions of the elasticity of substitution when the number of goods exceeds

2

(Blackorby and Russell, 1989). We consider below an innite-dimensional

version of the elasticity of substitution between two varieties conditional on the consumption prole

x.

Since

x

be that the cross-price elasticity between varieties substitution between

Under

i

xi = xj = x,

and

j

η¯ and σ ¯

given by Nadiri (1982, p.442)

is dened up to a zero measure set, it must

i

and

j

is zero.

x

is dened as follows:

σ ¯ (xi , xj , x) = −

Di Dj (xi Dj + xj Di ) . xi xj Di0 Dj2 + Dj0 Di2

conditional on

Therefore, the elasticity of

this expression boils down to

σ ¯ (x, x, x) = Evaluating

i and j

at the consumption prole

η(x, N ) ≡ η¯(x, xI[0,N ] )

1 . η¯(x, x)

(15)

x = xI[0,N ]

yields

σ(x, N ) ≡ σ ¯ (x, x, xI[0,N ] ) = 1/η(x, N ).

(16)

By symmetrizing, we control for the impact of the other varieties, thus implying that the expression chosen for the elasticity of substitution is inconsequential. The expressions (16) show that, for any symmetric preferences, along the diagonal of the

the elasticity of substitution, as well as the elasticity of the inverse demand, depend only upon the individual consumption and the mass of varieties. This means that the consumption space

dimensionality of the problem is reduced to two variables. From now on, we consider the function

σ(x, N )

as the

primitive

of the model. There are several reasons for making this choice. First,

the expression (16) allows for a parsimonious and simple description of preferences in the case of symmetric rms. When rms are heterogeneous, the consumption pattern is no longer symmetric, but we will see in Section 5 how the elasticity of substitution keeps its relevance. Second, we will see that what matters for the properties of the symmetric equilibrium is how

σ(x, N )

varies with the variables

x

and

N.

More precisely, we will show that the behavior of the

market outcome can be characterized by necessary and sucient conditions stating how with

x

and

N.

Hence, the key role here belongs to the two elasticities

Ex (σ)

and

EN (σ):

σ

varies

knowing

only the signs of these elasticities and the relationship between them (i.e. which one exceeds the other) allows characterizing completely the market outcome. Third, since the elasticity of substitution is an inverse measure of the degree of product dierentiation across varieties, we are able to appeal to the theory of product dierentiation to choose the most convincing assumptions regarding the behavior of

σ(x, N )

with respect to

x

and

N

and

to check whether the resulting predictions are consistent with empirical evidence. Last, given (15),

12

our results could be equivalently reformulated in terms of demand elasticity, as in Mrázová and Neary (2013), by using

η(x, N ).

2.3.1 Examples To develop more insights about the behavior of

σ

as a function of

x

and

N,

we give below the

elasticity of substitution for the dierent types of preferences discussed in the foregoing. (i) When the utility is additive, we have:

xu00 (x) 1 = r(x) ≡ − 0 , σ(x, N ) u (x) which means that

σ

(17)

depends only upon the per variety consumption when preferences are additive.

(ii) When preferences are homothetic, depends solely on the mass

N

η(x, x)

evaluated at a symmetric consumption prole

of available varieties. Setting

ϕ(N ) ≡ η(1, N )

and using (15) yields

1 = ϕ(N ). σ(x, N )

(18)

ϕ(N ) = 1/(1 + βN ).

For example, under translog preferences, we have

Since the CES is additive, the elasticity of substitution is independent of

N.

Furthermore,

since the CES is also homothetic, it must be that

r(x) = ϕ(N ) = It is, therefore, no surprise that the constant

1 = constant. σ

σ

is the only demand parameter that drives the

market outcome under CES preferences.

EN (σ) = 0 when preferences are additive, Ex (σ) = 0 when preferences are homothetic. Furthermore, we have Ex (σ) = EN (σ) under indirectly additive preferences. The CES satises Ex (σ) = EN (σ) = 0. Figure 1 shows the relationship between these Using (17) and (18), it is readily veried that

three families of preferences by by highlighting how the market price responds to various shocks.

13

Fig.1 The space of preferences

2.3.2 How does σ(x, N ) vary with x and N ? While our framework allows for various patterns of

σ,

it should be clear that they do not lead

to equally plausible results. This is why most applications of monopolistic competition focus on subclasses of utilities to cope with particular eects. For instance, Bilbiie

et al.

(2012) use the

translog expenditure function to capture the pro-competitive impact of entry on markups, since

σ(x, N ) = 1 + βN

increases with the number of varieties.

On the same grounds, working with

additive preferences Krugman (1979) assumes without apology that with

σ(x, N ) = 1/r(x)

decreases

x.

Admittedly, making realistic assumptions on how the elasticity of substitution varies with

x

and

N

is not an easy task.

That said, it is worth recalling Stigler (1969) that it is often

impossible to determine whether assumption A is more or less realistic than assumption B, except by comparing the agreement between their implications and the observable course of events. This is what we will do below. Spatial and discrete choice models of product dierentiation suggest that varieties become closer substitutes when the number of competing varieties rises (Tirole, 1988; Anderson 1995). Therefore,

x

EN (σ) ≥ 0

seems to be a reasonable assumption. In contrast, how

σ

et al.,

varies with

is even less clear. For this reason, we will follow Stigler's suggestion to uncover what the most

plausible assumption about the sign of

Ex (σ).

14

The empirical evidence strongly suggests the pass-through of a cost change triggered by a trade liberalization or productivity shock is smaller than De Loecker

et al., p/c

percent (Amiti

et al.,

2014, 2016;

σ leads to this result? The intuition is easy to m(x) = 1/σ(x). Incomplete pass-through amounts to

2016). Which assumption on

grasp when preferences are additive, that is, saying that

100

increases when

c

decreases, which means that rms have more market power or,

equivalently, varieties are more dierentiated. As rms facing a lower marginal cost produce more, the per capita consumption increases. Therefore, it must be that

σ(x)

decreases with

x.

In the

case of general symmetric preferences, we will show below that there is incomplete pass-through

Ex (σ) < 0 holds. Because Ex (σ) = 0 equal to 100 per cent.

if and only if must be

under homothetic preferences, the pass-through

All of this suggests the following conditions:

Ex (σ) ≤ 0 ≤ EN (σ).

(19)

However, a wider range of market patterns can be captured under a weaker condition:

Ex (σ) ≤ EN (σ),

(20)

which is appealing because it captures the pro-competitive eect of a larger market. In particular, it boils down to

Ex (σ) ≤ 0

(respectively,

EN (σ) ≥ 0)

for additive (respectively, homothetic)

preferences. However, (20) also holds for preferences that are neither additive nor homothetic (e.g. quadratic preferences), capturing a variety of eects that are elusive under additive or homothetic preferences. This is what we show in the next section.

3 Market equilibrium A), we rst

The purpose of this section is to characterize the symmetric market outcome. Given (

determine a necessary and sucient condition to be satised by the elasticity of substitution for the existence and uniqueness of a symmetric free-entry equilibrium. We then conduct three thought experiments:

a larger population size, a higher individual income and a positive productivity

shock. The responses of the market to these shocks are completely characterized by necessary and sucient conditions imposed on the elasticity of substitution.

3.1 Existence and uniqueness of a symmetric free-entry equilibrium We determine prices, outputs and prots when the mass zero-prot condition to pin down

(i) When N and

N

of rms is xed, and then use the

N.

is exogenously given, the market equilibrium is given by the proles

x ¯(N ) dened over [0, N ], which satisfy the following four conditions: 15

(i) no rm

q ¯(N ), p ¯ (N )

i can increase

its prot by changing its output, (ii) each consumer maximizes her utility subject to her budget constraint, (iii) the product market clearing condition

q¯i = L¯ xi

for all

i ∈ [0, N ]

and (iv) the labor market balance

ˆ

N

qi di + N F = yL.

c

(21)

0

q ¯(N ), p ¯ (N )

Since we focus here on symmetric equilibria, the functions

q¯(N ), p¯(N )

the scalars

and

x¯(N ).

and

x ¯(N )

boil down to

For this to hold, consumers must have the same income. This

occurs when prots are uniformly distributed across consumers. Hence, the budget constraint (6) takes the form:

ˆ

N

0

1 pi xi di = Y ≡ y + L

where the unit wage has been normalized to treats

Y

1.

ˆ

N

πi di,

(22)

0

Being negligible to the market, each rm accurately

as a given.

π(xi , x) is the for any x. As a

Each rm facing the same demand schedule, the function

A)

addition, (

implies that

π(xi , x)

has a unique maximizer

equilibrium must be symmetric. Using

πi ≡ (pi − c)Lxi − F ,

same for all

i.

In

result, the market

(22) boils down to the labor market

balance (21), which yields the only candidate symmetric equilibrium value for the per variety consumption:

y F − . cN cL N ≤ Ly/F . The

x¯(N ) = Therefore,

x¯(N )

is positive if and only if

(23) product market clearing condition

implies that the candidate equilibrium output is

q¯(N ) =

yL F − . cN c

Using (12) and (15) shows that there is a unique candidate equilibrium price given by

p¯(N ) = c

σ (¯ x(N ), N ) . σ (¯ x(N ), N ) − 1

N > Ly/F , there exists no interior equilibrium. Accordingly, we have the following If (A) holds and N ≤ Ly/F , then there exists a unique market equilibrium. Furthermore,

Clearly, if result:

(24)

this equilibrium is symmetric.

16

The pricing rule (24) may be rewritten as

m(N ¯ )≡ which shows that, for any given

1 p¯(N ) − c = , p¯(N ) σ(¯ x(N ), N )

(25)

N , the equilibrium markup m(N ¯ ) varies inversely with the elasticity

of substitution. Combining (23) with (24), the equilibrium operating prots are

N

π ¯ (N ) earned by a rm when there

rms are given by

π ¯ (N ) = How does

π ¯ (N )

vary with

N?

c L¯ x(N ). σ (¯ x(N ), N ) − 1

(26)

There is no immediate answer to this question. However, what

we learn from (26) is that the impact of rm entry on the market outcome depends on how the elasticity of substitution varies with

(ii)

x

and

N.

We now pin down the equilibrium value of

N

by using the zero-prot condition.

Since

Y = y . A symmetric free(q ∗ , p∗ , x∗ , N ∗ ), where N ∗ solves the zero-prot

prots are zero, a consumer's income is equal to her sole labor income:

entry equilibrium

(SFE) is described by the vector

condition

π ¯ (N ) = F, while

(q ∗ , p∗ , x∗ )

deliver an

N -rm

(27)

equilibrium dened above for

q ∗ = q¯(N ∗ ),

p∗ = p¯(N ∗ ),

N = N ∗:

x∗ = x¯(N ∗ ).

The Walras Law implies that the budget constraint

N ∗ p ∗ x∗ = y

is satised. Without loss of

generality, we restrict ourselves to the domain of parameters for which Equation (27) has a unique solution decreases with

N.

N∗

for

all

values of

Dierentiating (26) with respect to

N

F > 0

N ∗ < Ly/F .

if and only if

π ¯ (N )

strictly

and using (23) and (27), we obtain

  σ(x, N ) y Ex (σ(x, N )) . π ¯ (N ) = − 2 [σ(x, N ) − 1] 1 + EN (σ(x, N )) − cN σ(x, N ) − 1 x=¯ x(N ) 0

The second term in the right-hand side of this expression is positive if and only if

Ex (σ(x, N )) < Therefore,

π ¯ (N )

strictly decreases with

σ(x, N ) − 1 [1 + EN (σ(x, N ))] . σ(x, N ) N

for all

L , y , c,

and

F

(28)

if and only if (28) holds. This

implies the following proposition.

Proposition 1. Assume (A). Then, there exists a unique SFE for all c > 0 and F > 0 if and 17

only if

(28)

holds for all x > 0 and N > 0. Furthermore, no asymmetric equilibria exist.

Thus, we may safely conclude that the set of assumptions required to bring into play monopolistic competition must include (28).

This condition allows one to work with preferences that

display a great deal of exibility. Indeed,

σ

may decrease or increase with

x

and/or

N.

Evidently,

(28) is satised when the folk wisdom conditions (19) hold. More generally, the conditions (20) and (28) dene a range of possibilities which is broader than the one dened by (19). Under additive preferences, (28) amounts to assuming that boils down to

Ex (σ) < σ − 1

Ex (σ) < (σ − 1)/σ ,

while (28)

when preferences are indirectly additive. Both conditions mean that

σ cannot increase too fast with x. When preferences are homothetic, (28) holds if and only if EN (σ) exceeds −1, which means that varieties cannot become too dierentiated when their number increases. All these inequalities will play a critical role in the static comparative analysis performed below, which demonstrates their relevance (see Table 1). When (28) does not hold for all signs of

Ex (σ)

and

EN (σ)

x>0

and

N > 0,

there may exist several SFEs. Since the

need not be the same at all equilibria, the market outcome may react

dierently to a particular shock at dierent SFEs.

3.2 Comparative statics We are now equipped to study the impact of shocks in population

L,

per capita income

y,

and

productivity of rms on the SFE. Combining (25) and (27), we obtain a single equilibrium condition given by

m(N ¯ )=

NF . Ly

(29)

It proves to be convenient to work with the markup as the endogenous variable.

m ≡ N F/(Ly)

and using (25), we may rewrite the equilibrium condition (29) in terms of

 mσ

F 1 − m Ly , m cL m F

Setting

m

only:

 = 1.

Note that (30) involves the four structural parameters of the economy: thermore, it is readily veried that the left-hand side of (30) increases with

(30)

L, y , c and F . Furm if and only if (28)

holds. Therefore, to study the impact of a specic parameter, we only have to nd out how the left-hand side of (30) is shifted.

3.2.1 The impact of population size The following proposition spells out the implications of a larger population for the market outcome.

Proposition 2. If Ex (σ) is smaller than EN (σ) at the SFE, then a higher population size results in a lower markup and larger rms. Furthermore, the mass of varieties increases with L if and only if the condition 18

Ex (σ) < σ − 1

(31)

holds in equilibrium. Proof. Consider rst the impact of L on the market price p∗ . L,

to

Dierentiating (30) with respect

we nd that the left-hand side of (30) is shifted upwards under an increase in

if (20) holds. decreases with

As a consequence, the equilibrium markup

L.



m

L

if and only

, whence the equilibrium price

p∗ ,

This is in accordance with Handbury and Weinstein (2015) who observe that

the price level for food products falls with city size. Second, the zero-prot condition implies that

L

always shifts

p∗

and

q∗

in opposite directions. Therefore, rm sizes are larger in bigger markets,

as suggested by the empirical evidence provided by Manning (2010). To nd out how

N∗

changes with

L,

we dierentiate (26) with respect to

L

and obtain

  cxσ ∂π ¯ (σ − 1 − E (σ)) = . x ∗ ∂L N =N ∗ (σ − 1)3 x=x ,N =N ∗

(32)

Since the rst term in the right-hand side of this expression is positive, (32) is positive if and only if (31) holds. In this case, a population hike invites more rms to enter. Q.E.D. What happens when

Ex (σ) > EN (σ)

at the SFE? In this event, a higher population size results

in a higher markup, smaller rms, a more than proportional rise in the mass of varieties, and a lower per variety consumption. In other words, a larger market would generate anti-competitive eects, which do not seem plausible.

Finally,

Ex (σ) = EN (σ)

is necessary and sucient for the

3

market price not to response to a population shock.

3.2.2 The impact of individual income We now come to the impact of the per capita income on the SFE. One expects a positive shock on

y

to trigger the entry of new rms because more labor is available for production. However,

consumers have a higher willingness-to-pay for the incumbent varieties and can aord to buy each of them in a larger volume. Therefore, the impact of

y

on the SFE is a priori ambiguous.

Proposition 3. If EN (σ) > 0 at the SFE, then a higher per capita income results in a lower markup and bigger rms. Furthermore, the mass of varieties increases with y if and only if the condition Ex (σ)
0.

3 As

In this event, the equilibrium markup

shown by Bertoletti and Etro (2016), there is no population size eect when preferences are indirectly additive, whence Ex (σ) = EN (σ) must hold under such preferences. 19

decreases with

y.

To check the impact of

y

on

N ∗,

we dierentiate (26) with respect to

y

and get

after simplication:

L σ − 1 − σEx (σ) ∂π ¯ (N ) = . ∗ ∂y N =N ∗ N (σ − 1)2 x=x ,N =N ∗ ∂π ¯ (N ∗ )/∂y > 0 if and only if (33) EN (σ) > 0, then (33) implies (28). Q.E.D. Hence,

holds, a condition more stringent than (31). Thus, if

3.2.3 The impact of rm productivity Finally, to uncover the impact on the market outcome of a productivity shock common to all rms, we conduct a comparative static analysis of the SFE with respect to

nature of preferences determines the extent of the pass-through. downwards under a decrease in

c

c

and show that

the

The left-hand side of (30) is shifted

if and only if

Ex (σ) < 0

(34)

m∗ and the equilibrium mass of rms N ∗ = (Ly/F ) · m∗ increases with c. In other words, when Ex (σ) < 0, the pass-through is smaller than 100 percent. The intuition is easy to grasp when preferences are additive, that is, m(x) = ru (x) = σ(x). Incomplete pass-through amounts to saying that p/c increases when c decreases, which means that holds.

In this case, both the equilibrium markup

rms have more market power or, equivalently, varieties are more dierentiated. As rms facing a lower marginal cost produce more, the per capita consumption increases. Therefore, it must be that

σ(x)

decreases with

x.

It must be kept in mind that the price change occurs through the following three channels. First, when facing a change in its own marginal cost, a rm changes its price more or less proportionally by balancing the impact of the cost change on its markup and market share. Second, since all rms face the same cost change, they all change their prices, which aects the toughness of competition and, thereby, the prices set by the incumbents (Amiti

et al.,

2016). Third, as rms change their

pricing behavior, the number of rms in the market changes, changing the markup of the active rms. Under homothetic preferences, the markup remains the same regardless of the productivity shock, implying that the pass-through is equal to markup function

m(·)

depends only upon

N,

100

percent.

Indeed, we have seen that the

and thus (29) does not involve

c

as a parameter.

Proposition 4. If rms face a drop in their marginal production cost, (i) the market price decreases, (ii) the markup and number of rms increase if and only if Ex (σ) < 0 holds in equilibrium, and (iii) rms are larger if and only if EN (σ) > −1 in equilibrium. Proof. Rewriting (30) as follows:  mσ

 q Ly , m =1 L F 20

and totally dierentiating this expression yields

Ex (σ) Since

dm∗

and

Ex (σ)

dm∗ dq ∗ + [1 + E (σ)] = 0. N q∗ m∗

have opposite signs under a positive productivity shock,

must have the same sign for (35) to hold. In other words, a drop in and only if

EN (σ) > −1

100

c

dq ∗

leads to an

1 + EN (σ) ∗ increase in q if and

holds. Q.E.D.

Proposition 4 has an important implication. than

(35)

percent, then it must be that

Since the data suggest a pass-through smaller

Ex (σ) < 0.

In this case, (31) and (33) must hold, thereby

a bigger or richer economy displays more diversity than a smaller or poorer one.

3.2.4 Summary We have found a necessary and sucient condition for the existence and uniqueness of a SFE (Proposition 1) and provided a complete characterization of the eect of a market size or productivity shock (Propositions 2 to 4). Given that (19) implies (28), (20) and (33), we may conclude

if (19) holds, a unique SFE exists (Proposition 1), a larger market or a higher income leads to lower markups, bigger rms and a larger number of varieties (Propositions 2 and 3), while the pass-through is incomplete (Proposition 4). Note, however, that Propositions 1-4 may still hold

as follows:

under assumptions weaker than (19). Although we consider general symmetric preferences, the market outcome is governed by the behavior of the elasticity of substitution

σ(x, N ),

namely, by

Ex (σ)

and

EN (σ).

summarizes the main results of this section.

Shocks

Variables Markups

Firm sizes

Increase in

L

Increase in

y

Decrease in

c



Ex (σ) < EN (σ)

EN (σ) > 0

Ex (σ) > 0



Ex (σ) < EN (σ)

EN (σ) > 0

EN (σ) > −1

Ex (σ) < σ − 1

Ex (σ) < (σ − 1)/σ

Ex (σ) < 0

Number of rms



Table 1. Market equilibrium: the role of σ(x, N )

21

only

Table 1

4 Variable markups and optimum product diversity In this section, we apply our approach to some issues discussed in the literature on optimum product diversity (Spence, 1976; Dixit and Stiglitz, 1977; Kühn and Vives, 1999; Dhingra and Morrow, 2016). We show that, contrary to the common wisdom, constant markups and homothetic preferences are not necessary for the equilibrium to be optimal. Building on the class (2) of mixed additive preferences, we provide an example of non-homothetic preferences which yields variable markups and sustains the social optimum.

4.1 Over- or under-provision of varieties The social planner faces the following optimization problem:

max U(x) x

subject to the labor balance condition:

ˆ

N

xi di + N F.

Ly = cL 0 and (ii) the availability constraints:

xi = 0

for all

i ∈]N, N].

Using the symmetry of preferences, we set:

 U xI[0,N ] ≡ φ(x, N ). Since

φ

is increasing in both its arguments, a consumer is willing to reduce its per-variety con-

sumption against more product diversity, and vice versa. Love for variety (4) means here that

  N φ(x, N ) ≥ φ αx, α

for all

α ≥ 1.

Dierentiating the right-hand side of this expression with respect to

(36)

α

at

α = 1

shows that

consumers have a love variety if and only if

EN (φ) > Ex (φ) for any An

(x, N ) ∈ R2+ . increase in N

for a given per variety consumption

x

makes the individuals better o for

the following two reasons: variety-loving consumers face a broader product range and their total

22

consumption

xN

increases. We purge

EN (φ)

from the increased consumption eect by using the

relative gain from diversity : ν(x, N ) ≡

EN (φ) − Ex (φ) . EN (φ)

The social planner's program boils down to

max φ(x, N )

N (cLx + F ) = Ly.

s.t.

x,N

The ratio of the rst-order conditions with respect to

x and N

leads to the following expression:

F EN (φ) − Ex (φ) = , cLx + F EN (φ)

(37)

while the SFE condition (29) is equivalent to

F 1 = . cLx + F σ(x, N )

(38)

Since the labor balance must hold in both the equilibrium and the social optimum, the two outcomes coincide if and only if the right-hand sides of (37) and (38) are equal. In other words, the market delivers the socially optimal outcome if and only if the markup is equal to the relative gain from diversity. For this to hold regardless of the admissible parameter values, we have the following condition:

for all

1 = ν(x, N ) σ(x, N )

(39)

(x, N ) ∈ R2+ .

Proposition 5. Assume that (28) and φ(x, N ) > φ(0, Ly/F ) for all x > 0 and N > 0 hold. Then, the social optimum is unique and identical to the market equilibrium if and only if (39) is satised. Proof. To guarantee that the solution to the rst-order conditions is a maximizer of φ(x, N ), φ(x, N ) > φ(0, Ly/F )

implies that the optimum always involves a positive consumption

Furthermore, it follows from (36) in which

α → ∞ that φ(x, N ) > φ(∞, 0).

x > 0.

Therefore, the solution

of the social planner's program must be interior. Last, when (28) holds, the equilibrium outcome is unique (Proposition 1). Therefore, (39) implies that the solution to (37) is unique too. This solution is the optimum because corner solutions are ruled out. Q.E.D. The two expressions in (39) are easy to compare for any given functional form of preferences. Since the markup measures the private relative gain generated by introducing an additional variety while

ν(x, N ) is the social relative gain generated by a broader product range, the intuition behind

(39) is straightforward: the private and social relative gains are aligned. Otherwise, there is excess

23

(insucient) entry if and only if the equilibrium markup is larger (smaller) than the percentage gain from diversity evaluated at the SFE. This observation will be used below as a simple test of the SFE optimality. When preferences are homothetic,

V(N )/(V(N ) + 1), under additive

ν(x, N ) does not depend on x and is given by 1 − 1/EN (φ) =

V(N ) is what Benassy (1996) preferences, ν(x, N ) does not depend on N where

calls taste for variety. Furthermore, and boils down to the social markup

1 − Ex (u) discussed by Kühn and Vives (1999) and, more recently, by Dhingra and Morrow (2016). The necessary and sucient condition (39) covers the general case where the relative gain from diversity depends on

both x

N.

and

Note also that (39) holds if and only if the elasticity of

substitution is constant under additive preferences. We show below that this ceases to hold when we consider more general preferences.

4.2 Reconciling variable markups with optimality Under additive preferences, the CES is the only utility for which the SFE is socially optimal. This result suggests that constant markups and homotheticity are critical for the market to deliver the optimal product diversity. We now show that this belief is an artefact of additive preferences: the elasticity of substitution need not be constant for the market equilibrium to be optimal. To show this, consider the following special case of (2):

 ˆ U(x) = E ln

N

 (r ln xi + 1) di ρ r

where

0

r≡

Var(ρ) . E(ρ)

(40)

ρ

U (x, ρ) = (r ln x + 1) r associated with (40) is well-dened for all x exceeding the subsistence level exp(−1/r). Moreover, this function is strictly increasing and strictly concave over this domain. Last, when the distribution of ρ is ρ degenerate (r → 0), (40) boils down to the power function x . Here

Var(·)

is the variance operator.

The function

Proposition 6. Assume that preferences are given by (40). Then (i) the equilibrium markup decreases with the mass of rms and (ii) the market equilibrium is socially optimal. Proof. (i) Evaluating the elasticity of substitution associated with (40) at a symmetric consumption pattern with

N

available varieties yields (see Appendix B for a proof ):

σ(x, N ) =

r ln x + 1 . r ln x + 1 − E(ρ)

(41)

Combining (41) with (25), we obtain the markup as a function of the number of rms:

m(N ¯ )=1− Since the individual consumption

E(ρ) . r ln x(N ) + 1 − E(ρ)

x¯(N ) decreases with N , the markup falls in response to entry,

meaning that the market displays pro-competitive behavior.

24

(ii) As discussed in Section 4.1, comparing the right-hand sides of (37) and (38) yields a simple test of the relationship between the SFE and the social optimum. Under (40), the utility level at a symmetric consumption pattern is given by

φ(x, N ) = ln N +

E(ρ) ln(r ln x + 1). r

(42)

Using (41) and (42), we get:

ν(x, N ) = 1 −

1 E(ρ) = . r ln x + 1 σ(x, N )

Therefore, the equilibrium condition (38) coincides with the optimality condition (37).

For the

φ(x, N ), it remains to show that (42) satises the (28) follows from (41), while φ(x, N ) = −∞ at the corner solutions.

solution to (37) to be unique and a maximizer of assumptions of Proposition 5: Q.E.D.

the eciency of equilibrium can be obtained in set-ups featuring variable mark-ups and non-homothetic preferences. In particular, Proposition 6 has the following interesting implication:

pro-competitive eects driven by variable markups need not yield excessive entry.

r is larger (smaller) than Var(ρ)/E(ρ) (see ApIn other words, a mean-preserving increase in Var(ρ) means that consumers

Note also that there is over- (under)-entry when pendix B for a proof ).

are even more unsure about their love for variety, which invites more entry.

5 Heterogeneous rms We now extend our approach to heterogeneous rms à la Melitz. In this case, the consumption pattern ceases to be symmetric, making the problem innite dimensional.

Although all rms

sharing the same cost supply the same equilibrium output level, nding this level for each type of rms is a hard task because it can no longer be determined pointwise.

However, when the

mass of entrants and cuto cost are exogenously given, we are able to show the existence of an equilibrium output prole using Tarski's theorem. Conditional on this equilibrium, the elasticity of substitution becomes type-specic, so that we can still use this concept for studying heterogeneous rms at the cost of only one additional dimension, i.e. the rm's type. We refer to rms sharing the same marginal cost as

c-rms.

Our strategy builds on Asplund and Nocke (2006) and uses a

one-period framework à la Melitz and Ottaviano (2008). There is a continuum of potential risk-neutral rms of mass

N.

uncertainty about their marginal cost and entry requires a sunk cost

Prior to entry, rms face

Fe .

Once this cost is paid,

rms observe their marginal cost drawn randomly from the continuous probability distribution

Γ(c)

dened over

its type

c,

R+ ,

with a density

γ(c)

and a support bounded away from

0.

After observing

each entrant decides whether to produce or not, given that an active rm must incur a

25

xed production cost

F.

Under such circumstances, the mass

Ne

of entrants exceeds the mass

N

of operating rms. The equilibrium conditions are as follows: (i) the prot-maximization condition for

c-type

rms:

 D (xc , x) − c Lxc − F ; max Πc (xc , x) ≡ xc λ 

(43)

(ii) the zero-prot condition for the cuto rm:

(pcˆ − cˆ)qcˆ = F, where

cˆ is

the cuto cost. At the equilibrium, rms are sorted out by decreasing order of produc-

tivity, which implies that the mass of active rms is equal to

N ≡ Ne Γ(ˆ c) ;

(iii) the product market clearing condition:

qc = Lxc for all

c ∈ [0, cˆ];

(iv) the labor market clearing condition:

ˆ Ne Fe + Ne



(F + cqc )dΓ(c) = yL; 0

(v) rms enter the market until their expected prots net of the entry cost

ˆ

Fe

are zero:



Πc (xc , x)dΓ(c) − Fe = 0. 0

Γ is given, the prot-maximization condition implies that the equilibrium entirely determined by the set of active rms, which is fully described by c ˆ

Since the distribution consumption prole is and

Ne .

In other words, a variety supplied by an active rm can be viewed as a point in the set

Ω ≡ {(c, ν) ∈ R2+ | c ≤ cˆ; ν ≤ Ne γ(c)}. In the case of homogeneous rms, the variable so that

N

is sucient to describe the set of active rms,

Ω = [0, N ].

It follows from (43) and the envelope theorem that rms with a higher productivity earn higher prots.

so that there is perfect sorting across rms' types at any equilibrium.

conditions for any

ci

and

cj

The rst-order

imply

ci D(xi , x) [1 − η¯(xi , x)] = . D(xj , x) [1 − η¯(xj , x)] cj 26

(44)

A)

The condition (

of Section 2.2 implies that, for any given

x,

a rm's marginal revenue

D(x, x) [1 − η¯(x, x)] decreases with x regardless of its marginal cost. Therefore, it ensues from (44) that xi > xj if and only if ci < cj . In other words, more ecient rms produce more than less ecient rms. Furthermore, since pi = D(xi , x)/λ and D(·, x) decreases for any given x, more

ecient rms sell at lower prices than less ecient rms.

As for the markups, (44) yields

1 − η¯(xj , x) pi /ci = . pj /cj 1 − η¯(xi , x) Consequently, more ecient rms enjoy higher markups  as argued by De Loecker and Warzynski (2012)  if and only if the following condition holds:

(AA) the elasticity of the inverse demand η¯(·, x) increases for any given x. Using (14), we can show that (AA) implies (A). Very much like in 3.1 where

Ne

N

is treated parametrically, we assume for the moment that

cˆ and

are given and consider the game in which the corresponding active rms compete in quantities.

Because the quantity game cannot be solved pointwise, we have to prove that an equilibrium

q ¯(ˆ c, Ne ) of this game exists. Since all the c-type rms are symmetric, they sell the same quantity q¯c (ˆ c, Ne ) of their varieties. The following lemma states a sucient condition for an equilibrium of the quantity game to exist. This condition is more transparent when it is expressed in terms of price competition, rather than quantity competition.

Lemma 2. Assume (AA). If the demand price elasticity of the c-type rms ε¯(pc , p, y) ≡ −

∂D pc ∂pc D

decreases with the price p charged by the other rms, then a Nash equilibrium x¯ (¯c, Ne ) ∈ L(Ω) of the quantity game exists. Proof. See Appendix C. The operating prots of a

c-type

rm made at an equilibrium

x∗ (¯ c, Ne )

of the quantity game

are as follows:



 D (xc , x ¯(ˆ c, Ne )) − c Lxc , π ¯c (ˆ c, Ne ) ≡ max xc λ (¯ x(ˆ c, Ne )) π ¯ (N ) in the case of heterogeneous π ¯c (ˆ c, Ne ) decreases with c.

which is the counterpart of of rms implies that

rms. Note that the perfect sorting

A free-entry equilibrium with heterogeneous rms is dened by a pair

(ˆ c∗ , Ne∗ ) that satises the

zero expected prot condition for each rm:

ˆ



[¯ πc (ˆ c, Ne ) − F ]dΓ(c) − Fe = 0, 0 as well as the cuto condition

27

(45)

π ¯cˆ(ˆ c, Ne ) − F = 0.

(46)

Thus, regardless of the nature of preferences and the distribution of marginal costs, the heterogeneity of rms amounts to replacing the variable

N = Γ(ˆ c)Ne .

N

by the two variables

As a consequence, compared to the case of symmetric rms,

problem increases from one to two dimensions.



and

Ne

because

the complexity of the

Dividing (45) by (46) yields the following new equilibrium condition:

ˆ cˆ  0

 Fe π ¯c (ˆ c, Ne ) − 1 dΓ(c) = . π ¯cˆ(ˆ c, Ne ) F

(47)

5.1 Making the elasticity of substitution cost-specic When rms are symmetric, we have seen that the sign of

EN (σ) plays a critical role in comparative

statics. Since rms are symmetric within types, the same holds here. The dierence is that the mass of active rms is now determined by the two endogenous variables

cˆ and Ne .

As a consequence,

understanding how the mass of active rms responds to a population hike requires studying the way the left-hand side of (47) varies with

cˆ and Ne .

elasticity of substitution between any two varieties

σc (ˆ c, Ne ) be the equilibrium supplied by c-type rms: Let

value of the

σc (ˆ c, Ne ) ≡ σ ¯ [¯ xc (ˆ c, Ne ), x ¯(ˆ c, Ne )]. In this case, we may rewrite

π ¯c (ˆ c, Ne )

as follows:

π ¯c (ˆ c, Ne ) =

c L¯ xc (ˆ c, Ne ), σc (ˆ c, Ne ) − 1

which is the counter-part of (26). The markup of a

c, Ne ) = m∗c (ˆ Hence, by making

σ

c-type

(48)

rm is thus given by

1 . σc (ˆ c, Ne )

cost-specic, we are able to use the elasticity of substitution for studying the

market outcomes with heterogeneous rms. Using the envelope theorem and the prot-maximization condition (43), we obtain:

Ec (¯ πc (ˆ c, Ne )) = 1 − σc (ˆ c, Ne ).

(49)

Combining this with (48) allows us to rewrite the equilibrium conditions (46) and (47) as follows:

cˆ L¯ xcˆ(ˆ c, Ne ) = F, σcˆ(ˆ c, Ne ) − 1 28

(50)

and

ˆ cˆ 





σz (ˆ c, Ne ) − 1 dz z

exp 0

c



 Fe − 1 dΓ(c) = . F

Note that the integral in (51) converges because the support of

Γ

(51)

is bounded away from

0.

5.2 Properties of the free-entry equilibrium As in Section 3, we now impose simple conditions on The left-hand side of (51) increases with to assuming that, for any given cuto

cˆ,

Ne

σc

to predict the impact of a larger market.

if and only if

σc (ˆ c, Ne )

increases in

Ne .

This amounts

the relative impact of entry on the low-productivity rms

(i.e., the small rms) is larger than the impact on the high-productivity rms. As implied by (49),

Ec (¯ πc (ˆ c, Ne ))

decreases in

Ne

if and only if

σc (ˆ c, Ne )

increases in

Ne .

This leads us to impose the

following additional condition.

B) The equilibrium prot of each rm's type decreases in cˆ and Ne . Intuitively, (B) means that a larger number of entrants, a higher cuto, or both lead to lower prots, for the mass of active rms rises. Using Zhelobodko et al. (2012), it is readily veried that 4 any additive preference satisfying (A) also satises (B). (

We now develop a simple geometric argument that allows us to study the consequence of population size.

Let

cˆ = g(Ne )

be the locus of solutions to the equilibrium condition (50) and

cˆ = h(Ne ) the locus of solutions to the equilibrium condition (51). A free-entry equilibrium c, Ne )-plane. The (ˆ c∗ , Ne∗ ) is an intersection point of the two loci cˆ = g(Ne ) and cˆ = h(Ne ) in the (ˆ ∗ ∗ properties of the equilibrium (ˆ c , Ne ) depend only upon the slopes of these two curves, which in turn are determined by the behavior of σc (ˆ c, Ne ). In particular, if σc (ˆ c, Ne ) increases with cˆ, the left-hand side of (51) increases with c ˆ. Intuitively, when cˆ increases, the mass of rms rises as less ecient rms stay in business, which intensies competition and lowers markups. In this case, the selection process is tougher. This is not the end of the story, however. Indeed, the competitiveness of the market also depends on how

Ne

aects the degree of dierentiation across varieties.

B), g(Ne ) is downward-sloping in the (ˆc, Ne )-plane.

As implied by (

h(Ne ), it is independent of L but its slope is a priori undetermined. Three cases may arise. First, if h(Ne ) is upward-sloping, there exists a unique free-entry equi∗ librium. Therefore, both Ne and c ˆ∗ increase with L (see Figure 2a). Second, under the CES ∗ preferences, h(Ne ) is horizontal, which implies that Ne rises with L while c ˆ∗ remains constant. upward when

L

Furthermore, it is shifted

rises. As for

4 Note

that when rms are symmetric, the equilibrium operating prots depend only upon the number N of active rms (see (26)). Thus, (B) amounts to assuming that these prots decrease with N .

29

Figure 2. Cuto and market size Last, when

g(Ne )

h(Ne )

is steeper than

is downward-sloping, two subcases must be distinguished. In the rst one,

h(Ne ).

In this case,

Ne∗

increases with

L,

while

cˆ∗

decreases with

L

(see

Figure 2b). A sucient, but not necessary, condition for this to arise is as follows:

(C) Expected prots

(45)

increase with cˆ and (weakly) decrease with Ne .

The argument is as follows. Plugging (46) into (45) and taking the logarithm yields the two expressions:

− ln π ¯cˆ(ˆ c, Ne ) = ln ˆ

(52)



ln 0 When

L , F

 Fe (¯ πc (ˆ c, Ne ) − π ¯cˆ(ˆ c, Ne ))dΓ(c) − ln π ¯cˆ(ˆ c, Ne ) = ln . F

(53)

(C) holds, the gradient of the left-hand side of (52) is steeper in the (Ne , cˆ)-plane than

that of the right-hand side of (53), which yields the desired property. The intuition behind

(C) is easy to grasp.

A higher cuto relaxes competition by allowing less

ecient to survive, whereas a higher number of entrants strengthens potential competition. For

(C) holds under quadratic preferences (Melitz and Ottaviano, 2008). Finally, when (C) does not hold, h(Ne ) could intersect g(Ne ) from below, which implies that a

example,

hike in

L would lead to a higher cuto and a smaller number of entrants.

This does not seem very

plausible because a higher cuto softens the selection of rms, which should invite more entry. In sum, we have:

Proposition 7. Assume (B) and (C). Then, the equilibrium mass of entrants always increases with L. When when

Ne

σc (ˆ c, Ne )

increases both with

cˆ and Ne ,

rises, so does the left-hand side of (51). Hence,

must be that

cˆ decreases

h(Ne ) is downward-sloping. Indeed, since σc (ˆ c, Ne ) also increases with cˆ, it

the locus

for (51) to hold. As a consequence, we have:

Proposition 8. Assume (B) and (C). If σc (ˆc, Ne ) increases with cˆ and Ne , then the equilibrium cuto decreases with L. If σc (ˆc, Ne ) increases with cˆ and decreases with Ne , then cˆ∗ increases with L. 30

cˆ, the number of active rms N is proportional to the number of entrants Ne . Therefore, assuming that σc (ˆ c, Ne ) increases with Ne is the counterpart of (20) under rm heterogeneity. The Given

pro-competitive eect generated by entry exacerbates the selection eect across rms. Therefore, in response to a hike in

L, the two eects combine to induce the exit of the least ecient active rms.

This echoes Melitz and Ottaviano (2008) who show that a trade liberalization shock gives rise to a similar eect under quadratic preferences. However, the impact of population size on the cuto cost is ambiguous because it now depends on how the elasticity of substitution with

Ne .

σc (ˆ c, Ne )

varies

In other words, even under preferences generating pro-competitive eects, predictions

regarding the direction of the rms' selection are fragile. Furthermore, when

σc (ˆ c, Ne )

decreases with

cˆ,

that is, varieties become more dierentiated,

the left-hand side of (51) need not be monotone in Nevertheless, the behavior of

cˆ with

respect to

L

cˆ,

which may result in multiple equilibria.

still depends solely on whether

h(Ne )

is locally

upward-sloping or downward-sloping.

6 Concluding remarks We have shown that monopolistic competition can be considered as the marriage between oligopoly theory and the negligibility hypothesis, thus conrming Mas-Colell's (1984) observation.

Using

the concept of elasticity of substitution, we have provided a complete characterization of the market outcome and of the comparative statics implications in terms prices, rm size, and mass of rms/varieties. Somewhat ironically, the concept of elasticity of substitution, which has vastly contributed to the success of the CES model of monopolistic competition, keeps its relevance in the case of general preferences, both for symmetric and heterogeneous rms.

The fundamental

dierence is that the elasticity of substitution ceases to be constant and now varies with the keyvariables of the setting under study. We take leverage on this to make clear-cut predictions about the impact of market size and productivity shocks on the market outcome. Furthermore, we have singled out our most preferred set of assumptions and given a disarmingly simple necessary and sucient conditions for the standard comparative statics eects to hold true. But we have also shown that relaxing these assumptions does not jeopardize the tractability of the model. Future empirical studies should shed light on the plausibility of the assumptions discussed in this paper by checking their respective implications.

It would be unreasonable, however, to

expect a single set of conditions to be universally valid. We would be the last to say that monopolistic competition is able to replicate the rich array of ndings obtained in industrial organization. However, it is our contention that models such as those presented in this paper may help avoiding several of the limitations imposed by the partial equilibrium analyses of oligopoly theory.

31

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Appendix A. Since η¯(0, x) < 1, (13) implies that limxi →0 (1 − η¯)Di = ∞. it ensues from (13) that

limxi →∞ (1 − η¯)Di = 0.

Because

Similarly, since

(1 − η¯)Di

0 < (1 − η¯)Di < Di ,

is continuous, the Lemma

follows from the intermediate value theorem.

B. Deriving

.

(41)

The marginal utility of a variety

i ∈ [0, N ] "

D(xi , x) = E

is given by

ρ ρ (ln xri + 1) r −1 ρ xi ´ N ln xr + 1 r dj j 0

# .

The corresponding inverse demand elasticity (11) can be expressed as follows:

 E η(xi , x) =

ρ r −1

ρ (ln xi +1) ρ xi ´ N r r 0 (ln xj +1) dj r

 E



1− ρ

−1

ρ−r ln xri +1

r r ρ (ln xi +1) ρ xi ´ N r r 0 (ln xj +1) dj



 .

Combining this with (15) and evaluating the elasticity of substitution at a symmetric consumption pattern

x = xI[0,N ]

yields

35

σ(x, N ) =

r ln x + 1 h i, r ln x + 1 − E(ρ)− Var(ρ) − r E(ρ)

which is equivalent to (41) if and only if

Var(ρ).

it also decreases with

r = Var(ρ)/E(ρ).

Since the RHS of (38) decreases with

As the RHS of (37) is independent of

spread, there is over- (under-)entry when

Var(ρ)

σ,

Var(ρ) under a mean-preserving

increases. Q.E.D.

C. For any cˆ and Ne , there exists an equilibrium x¯ (¯c, Ne ) of the quantity game. We rst show that for any set of active rms given by c ˆ and Ne , there exists a unique utilitymaximizing consumption prole

x ¯(p, Y ).

Recall that (i) a quasi-concave Fréchet dierentiable

functional is weakly upper-semicontinuous, and (ii) an upper-semicontinuous functional dened over a bounded weakly closed subset of a Hilbert space has a maximizer. Since the budget set is

L2 (Ω) is a Hilbert space, existence is proven. Uniqueness follows from the strict quasi-concavity of U and the convexity of the budget set. Hence, there exists a ∗ unique utility-maximizing consumption prole x (p, Y ) (Dunford and Schwartz, 1988). Plugging x∗ (p, Y ) into (7)  (8) and solving (7) for xi yields bounded and weakly closed, while

xi = D(pi , p, Y ), which is weakly decreasing in its own price.

c

Assume now that

and

Ne

5

By construction,

x ¯(p, Y )

belongs to

L2 (Ω).

are given and consider the game in which rms compete in prices.

We show below that a Nash equilibrium of the price game exists if the demand price elasticity

ε¯(pc , p, Y ) ≡ − increases in

pc

∂D pc ∂pc D

AA) holds) and decreases in the common price p charged by the other rms.

(i.e. (

Since each rm behaves as if it were a monopolist on the residual demand, the quantities sold at the price equilibrium

¯ p

determine a Nash equilibrium

The rst-order condition for a

c-type rm,

¯ c ≡ D(¯ ¯, Y ) x pc , p

of the quantity game.

conditional on the vector of prices

p charged by the

other rms, is given by

pc − c 1 = . pc ε¯(pc , p, Y )

(C.1)

pc for any given c ∈ [0, c]. Since ε¯

Since the left (right) -hand side of (C.1) continuously increases (decreases) with

p,

there exists a unique

decreases with

p, pˆc (p)

pˆc (p)

that solves (C.1) for any given

increases both in

p.

p ∈ L2 ([0, c])

and

Finally, since the left-hand side of (C.1) decreases in

5 Since

D is continuously decreasing in xi , there exists at most one solution of (7) with respect to xi . If there is a nite choke price (D(0, x∗ )/λ < ∞), there may be no solution. To encompass this case, the Marshallian demand should be formally dened by D(pi , p, y) ≡ inf{xi ≥ 0 | D(xi , x∗ )/λ(y, x∗ ) ≤ pi }.

36

c, pˆc (p) also increases in c. Let P be the best-reply

mapping from

L2 ([0, c])

into itself:

P(p; c) ≡ pˆc (p). Since

pˆc (p)

increases in

p, P

is an increasing operator. To check the assumption of Tarski's

xed-point theorem, it suces to construct a set the lattice

S

into itself and (ii)

S

S ⊂ L2 ([0, c])

such that (i)

PS ⊆ S ,

i.e.

P

maps

is a complete lattice.

p¯ the unique symmetric equilibrium price when all rms share the marginal cost c ¯ ≡ p¯1[0,c] . Since pˆc (p) increases in c, we have p¯ ≥ pˆc (¯ and observe that p ¯ = pˆc (¯ p), where p p) for ¯ ≥ Pp all c ∈ [0, c] or, equivalently, p ¯ . Furthermore, because P is an increasing operator, p ≤ p ¯ implies Pp ≤ P p ¯≤p ¯ . In addition, Pp is an increasing function of c because pˆc (p) increases in c. In other words, P maps the set S of all non-negative weakly increasing functions bounded above by p ¯ into itself. It remains to show that S is a complete lattice, i.e. any non-empty subset of S has a supremum and an inmum that belong S . This is so because pointwise supremum and pointwise Denote by

inmum of a family of increasing functions are also increasing.

P has a xed point, which is a Nash equilibrium of the price game. Plugging this price prole into x ¯(p, Y ) yields the quantity equilibrium, which belongs to L2 (Ω) by construction. Q.E.D. To sum-up, since

S

is a complete lattice and

PS ⊆ S ,

37

Tarski's theorem implies that