Toward a theory of monopolistic competition∗ Mathieu Parenti
†
Philip Ushchev
‡
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 su cient 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-speci c.
Keywords:
monopolistic competition, general equilibrium, additive preferences, homothetic pref-
erences
JEL classi cation:
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]
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 speci c models in the hope of getting around those di culties 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 speci cations 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 speci c 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, speci c
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 e ciency, 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 justi ed 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 speci cally,
we nd necessary and su cient conditions to be imposed on the elasticity of substitution for the market outcome to react in a speci c 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-speci c. 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 pro les, 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 su cient 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 signi es 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 su cient 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 su cient 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 bene t generated by a new variety and nd a necessary and su cient 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 veri ed by the CESare neither necessary nor su cient 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 pro t-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-speci c.
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 di culties 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 simpli ed, 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 di culties 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
de nition 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 pro le 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 bene t 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 pro les 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 speci cally, consider a family
{U (·, ρ)}ρ∈[0,1]
of utility functions and assume that
ρ
is a random variable distributed over
[0, 1].
De ne
ˆ U(x) ≡ E ln
N
U (xi , ρ)di ,
(2)
0 where
E(·)
is the expectation operator.
It is readily veri ed 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
ˆ
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.
satis es
Whenever
xi U(x)
U(x)
di = 1
(3)
satis es (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
ˆ
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 unspeci ed 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 su cient 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 speci c 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)
de ned 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 pro le 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 pro le p ≥ 0 must belong to the dual of the space of consumption pro les. 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)
de nition 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 pro le
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-pro t 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 e ciency
and the same constant marginal cost
qi
units of its variety, rm
pro t 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 pro t-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 su cient conditions for a unique (interior or
corner) pro t-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 su cient if the pro t 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 pro t 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 pro ts 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 su cient conditions to be imposed on the elasticity of substitution. The elasticity of substitution is, therefore, a su cient 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 veri ed that (10) has at least one positive solution when the choke price exceeds λc. 11
equilibrium. It is well known that there are dierent de nitions of the elasticity of substitution when the number of goods exceeds
2
(Blackorby and Russell, 1989). We consider below an in nite-dimensional
version of the elasticity of substitution between two varieties conditional on the consumption pro le
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 de ned up to a zero measure set, it must
i
and
j
is zero.
x
is de ned 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 pro le
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 su cient 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 pro le
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 satis es Ex (σ) = EN (σ) = 0. Figure 1 shows the relationship between these Using (17) and (18), it is readily veri ed 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 su cient condition to be satis ed 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 su cient conditions imposed on the elasticity of substitution.
3.1 Existence and uniqueness of a symmetric free-entry equilibrium We determine prices, outputs and pro ts when the mass zero-pro t 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 pro les
x ¯(N ) de ned over [0, N ], which satisfy the following four conditions: 15
(i) no rm
q ¯(N ), p ¯ (N )
i can increase
its pro t 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 pro ts 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 pro ts 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-pro t condition.
Since
Y = y . A symmetric free(q ∗ , p∗ , x∗ , N ∗ ), where N ∗ solves the zero-pro t
pro ts 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 de ned 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 satis ed. 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 satis ed when the folk wisdom conditions (19) hold. More generally, the conditions (20) and (28) de ne a range of possibilities which is broader than the one de ned 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 veri ed 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 speci c 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-pro t 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 su cient 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 simpli cation:
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 su cient 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 satis ed. 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
(insu cient) 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 su cient 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-de ned 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) satis es 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 e ciency 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 in nite 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 pro le using Tarski's theorem. Conditional on this equilibrium, the elasticity of substitution becomes type-speci c, 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)
de ned 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 pro t-maximization condition for
c-type
rms:
D (xc , x) − c Lxc − F ; max Πc (xc , x) ≡ xc λ
(43)
(ii) the zero-pro t 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
cˆ
(F + cqc )dΓ(c) = yL; 0
(v) rms enter the market until their expected pro ts net of the entry cost
ˆ
Fe
are zero:
cˆ
Πc (xc , x)dΓ(c) − Fe = 0. 0
Γ is given, the pro t-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 pro le 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 su cient 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 pro ts.
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 e cient rms produce more than less e cient rms. Furthermore, since pi = D(xi , x)/λ and D(·, x) decreases for any given x, more
e cient rms sell at lower prices than less e cient rms.
As for the markups, (44) yields
1 − η¯(xj , x) pi /ci = . pj /cj 1 − η¯(xi , x) Consequently, more e cient 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 su cient 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 pro ts 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 de ned by a pair
(ˆ c∗ , Ne∗ ) that satis es the
zero expected pro t condition for each rm:
ˆ
cˆ
[¯ 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.
cˆ
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-speci c 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-speci c, we are able to use the elasticity of substitution for studying the
market outcomes with heterogeneous rms. Using the envelope theorem and the pro t-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
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 pro t 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 pro ts, for the mass of active rms rises. Using Zhelobodko et al. (2012), it is readily veri ed that 4 any additive preference satisfying (A) also satis es (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 e cient rms stay in business, which intensi es 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 pro ts depend only upon the number N of active rms (see (26)). Thus, (B) amounts to assuming that these pro ts 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 su cient, but not necessary, condition for this to arise is as follows:
(C) Expected pro ts
(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)
cˆ
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
e cient 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 e cient 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 con rming 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 su cient 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 pro le
x ¯(p, Y ).
Recall that (i) a quasi-concave Fréchet dierentiable
functional is weakly upper-semicontinuous, and (ii) an upper-semicontinuous functional de ned 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 pro le 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 de ned 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 su ces 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 in mum that belong S . This is so because pointwise supremum and pointwise Denote by
in mum 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 pro le 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
