The Microeconomics of New Trade Models∗ Mart´ın Alfaro† November 21, 2017 PRELIMINARY For the latest version, click here
Abstract I provide a unified framework to analyze models of monopolistic competition with firm heterogeneity a` la Melitz under standard demands and any productivity distribution. By using this methodology, I disentangle the effects of export opportunities and import competition. I show that while the former entails pro-competitive effects for domestic firms, tougher import competition has no impact on either domestic firms’ prices or their survival productivity cutoff; it only affects the mass of incumbents. The results applied to the case of two big economies imply that a unilateral liberalization is equivalent for the liberalizing country to a reduction of its export opportunities, thus explaining the Metzler paradox found under this framework. I also show that when firms know their productivity and enter following a productivity order, the procompetitive effects of import competition are restored.
Several results of this paper have been included in the first chapter of my Ph.D. dissertation. I thank ¨ Anders Laugesen, Michael Koch, Omar Licandro, Peter Neary, Philipp Schroder, Allan Sørensen, Raymond Riezman and participants at various seminars for the helpful suggestions. I especially thank David Lander and Francisco Rold´an whose comments improved substantially the paper. All errors are my own. † University of Alberta, Department of Economics. 9-08 HM Tory Building, Edmonton, AB T6G 2H4, Canada. Link to personal website. Email:
[email protected]. ∗
1
Contents
Mart´ın Alfaro
Contents References
1
1
Introduction
3
2
An Illustration
7
2.1
Setup and Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.2
Analysis of the Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.3
Effects of the Exogenous Shocks . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3
The Model
11
3.1
Structure of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3.2
Demand System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.3
Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.4
Exogenous Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
4
Small-Economy Case
19
5
Trade Between Two Big Countries
21
6
Market Size and Income Effects
24
7
Restoring the Import-Competition Channel
26
7.1
Alternative Market Structures . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
7.2
A Model where Firms Know their Productivity . . . . . . . . . . . . . . . . .
29
8
Extensions
30
8.1
Non-Price Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
8.1.1
Nested Demands with Groups Defined by Own-Firm Varieties . . .
32
8.1.2
Nested Demands with Groups Defined by Country of Origin . . . .
32
Firm-Heterogeneity in Demand . . . . . . . . . . . . . . . . . . . . . . . . . .
33
8.2 9
Conclusion
33
Appendices
i
A
Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i
B
Demand Systems: Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
C
Differential Characterization of Demand Systems . . . . . . . . . . . . . . .
xi
2
1. Introduction
1
Mart´ın Alfaro
Introduction
In the last fifteen years, several frameworks to introduce firm heterogeneity have emerged in the International Trade literature. Following Melitz (2003), it has been standard to incorporate this feature in a monopolistic competition setting where firms make entry decisions without knowing their productivity and learn their efficiency ex post. This setup allows to capture the effects of trade liberalization on inter-firm reallocations towards more productive firms. In addition, several extensions which dispense with the CES demand assumption, are capable of simultaneously accounting for reductions of domestic firms’ markups triggered by trade. While the Melitz model has been successful in capturing these facts, little is known about the general conclusions that can be obtained from it as well as the mechanisms that are operating behind specific results. For one thing, conclusions of the model have revealed themselves sensitive to both the productivity distribution and the demand system assumed. For other thing, the effects of trade are usually analyzed by considering the experiment of liberalizing an economy, which entails variations in both import competition and the export opportunities. Policy recommendations might be quite different depending on which channel is under operation. In this paper, I provide a unified framework to analyze the canonical Melitz model under any productivity distribution and a class of demand systems that encompasses standard cases.1 By making use of this methodology, I analyze how a sector in isolation is affected by an exposure to new export opportunities and tougher import competition. The main conclusion I derive is that in this setup any pro-competitive effect in the domestic economy due to a trade liberalization comes from the export-opportunity channel. The import-competition channel only affects the mass of domestic incumbents but neither the survival productivity cutoff of domestic firms, or the prices/quantities set by them would 1
In terms of demands that account for an endogenous number of varieties, my setup covers (nonnecessarily symmetric) versions of: i) demands from an additive direct utility as in Krugman (1979), which includes Simonovska’s (2015) Stone-Geary, the generalized CES as in Jung, Simonovska, and Weinberger (2015) and Arkolakis, Costinot, Donaldson, and Rodr´ıguez-Clare (2015), and Behrens and Murata’s (2007) CARA ii) Melitz and Ottaviano’s (2008) linear demand, iii) Feenstra’s (2003) translog demand, iv) demands from an additive indirect utility as in Bertoletti and Etro (2015), including their version of the addilog demand, iv) attraction demand models as in Luce (1959) including the Logit, v) Nocke and Schutz’s (2015) demands from discrete-continuous choices vi) constant expenditure demand systems as in Vives (2001)), and vii) the nested Logit and nested CES with groups defined by country of origin or firm’s products bundle when firms are multiproduct.
3
1. Introduction
Mart´ın Alfaro
be affected.2 In fact, more generally, any domestic firm’s decision would not be affected by an exposure to tougher import competition, thus extending the conclusion to variables such as quality or number of products when firms are multiproduct. The underlying mechanisms are easy to grasp once it is noticed that the demands considered share the feature of collapsing the market conditions into a single sufficient statistic. I refer to this as an aggregator.3 Examples of an aggregator are a price index, a demand choke price or more complex functions which depend on different price moments. Intuitively, it can be interpreted as the level of competition in the industry. The scope of a demand system with this property is quite broad (see footnote 1): while it is is readily noticeable that some demands have this property, others that in principle depend on more than one aggregator end up collapsing into a single one by writing them properly. Exploiting the demand’s property, I set the equilibrium conditions in a specific way and show that they are composed of two set of conditions. The first set is given by the zero expected-profits and pins down the value of the aggregators. The other set of equations describe the equilibrium at the market stage and determines the composition of each aggregator. Crucial for the results, the system is decomposable, allowing to determine the aggregators’ values independently of their composition. This feature becomes relevant given that domestic firms’ optimal prices and their survival productivity cutoff are completely determined by the value of the aggregator, irrespective of its composition. Just like the usual analogy of variations in market size to mimic the effects of trade liberalization, new export opportunities and import competition can be understood, respectively, as positive shocks in a closed economy to expected profits and to the aggregator. The former gives rise to pro-competitive effects because competition has to become tougher, i.e. the value of aggregator has to be greater, in order to restore the zero expected-profits. Since domestic firms’ prices and the survival domestic productivity cutoff are completely determined by the aggregator’s value, the productivity cutoff increases and, if greater competition makes the demand more elastic, prices decrease. In addition, if the reduction in prices does not increase the value of the aggregator enough to validate the necessary increase in competition to reestablish the zero expected-profits, the system adjusts by also 2
The insight that it is the pull of the export opportunities what changes the survival productivity cutoff, rather than the push of import competition, was already noticed by Melitz’s (2002, 2003) for a CES demand. In Melitz (2002), he says: ”...the most obvious cause explaining the exit of the least productive domestic firms would be the new competition from the entry of the more productive firms into the domestic market. However, this intuition is incorrect... If the current model were amended to allow for the new import competition without introducing any export opportunities, then this trade opening would not induce any distributional changes among firms.” 3 Besides, by an application of a suitable law of large numbers and the continuum of firms, even when firms receive different draws of productivity, the aggregator is nonstochastic .
4
1. Introduction
Mart´ın Alfaro
fostering entry of incumbents. Regarding a shock to the aggregator, the system of equations is separable and only variations in expected profits can change the value of the aggregator. Thus, this shock cannot affect aggregators’ levels but only its composition. Moreover, prices and the survival productivity cutoff are completely determined by the aggregator’s value, so they do not vary either. Due to this, in terms of the aggregator composition, the masses of incumbents are the only remaining variable free to adjust, modifying its value until its ensured that market conditions remain at the same value previous to the shock. By making use of these intuitions, I analyze the effects of the export-opportunity and import-competition channels. I proceed to their study by providing a formalization of the concept of small economies and splitting the analysis for small and big countries.4 In the case of a small economy, the separation of channels can be accomplished through the study of unilateral liberalizations. I show that better export opportunities resemble a shock to expected profits, thus generating decreases in domestic firms’ prices and an increase in their productivity survival cutoff. On the other hand, tougher import competition acts like a shock to the aggregator. Thus, it has no impact on either domestic firms’ prices and their survival productivity cutoff, with all of the adjustment taking place through the mass of domestic incumbents. As far as the case of two big countries goes, unilateral liberalizations are not capable of isolating directly the channels. To understand why they are confounded, consider two countries A and B, with the former opening its economy to imports. Since countries are nonnegligible, new export opportunities in B would affect the market conditions of its own domestic market which, in turn, affects A’s export conditions. This generates feedback effects, making both channels operate simultaneously. I show that, accounting properly for each mechanism, A’s domestic economy is only affected by the changes in B’s export conditions; the effects stemming from the import-competition channel only affect domestic firms through their mass of incumbents. In fact, it can be shown that under this scenario opening an economy to imports is qualitatively similar to a decrease in the export opportunities of the liberalizing country. This provides an explanation for the result found by some authors under a Melitz framework (e1.g., Melitz and Ottaviano (2008)) where unilateral liberalizations were displaying anti-competitive effects I also use the methodology to inquire upon the effects of variations in market size and 4
Demidova and Rodrguez-Clare’s (2009,2013) define small economies as those that take as given the price of imports and the demand schedules for theirexports. Unlike them, I formalize the concept by endowing the set of countries with a measure.
5
1. Introduction
Mart´ın Alfaro
income in a closed economy. I show that, depending on the properties of the demand assumed, these shocks might entail effects similar to a shock to either expected profits or the aggregator. In the latter case, it rationalizes the insensitivity to market size and income under some commonly used demands.5 Finally, I provide modifications to the original framework in order to the importcompetition channel reemerge. To do this, I proceed in two steps. I begin by delving into the conditions of the market structure that determine its absence. Basically, this is due to the conjunction of free entry, symmetric entrants and the existence of a singlesufficient statistics for determining optimal prices and profits. Based on this, I show that, for instance, the result holds even under oligopoly or mixed market structures (i.e. big firms coexisting with a competitive fringe a` la Melitz) as long as those assumptions are met. Then, I show that by dispensing with the assumption of symmetric entrants, the import-competition channel is restored and it displays pro-competitive effects. Specifically, I consider a model where firms know their productivity at the moment of making their entry decisions and serve each market following a productivity order. The assumption is standard in oligopolistic models and it is use mainly to rule out equilibria where less efficient firms drive out the market more efficient ones. This paper makes several contributions. First, it is framed within the internationaltrade literature that aims at providing general results to models of monopolistic competition with heterogeneous firms.6 Since, in last instance, the majority of those papers inquire upon welfare effects, they make use of demands that can be derived from a representative consumer. On the contrary, since my focus is on the effects triggered by an exposure to trade, I take demand as a primitive, remaining agnostic about its microfoundation.7 The nature of the demand system turns the model into a large aggregative economy as in 5
For example, for a variation in income, this applies to demands from an additive separable utility. Regarding a shock to the market size, it applies to Luce’s demand system (including the Logit), discrete continuous choices as in Nocke and Schutz (2015) and demands from addivitive indirect utilities. For constant expenditure demand systems, it holds for both type of shocks. 6 See, for instance, Behrens and Murata (2007), Melitz and Ottaviano (2008), Feenstra (2010), Zhelobodko, Kokovin, Parenti, and Thisse (2012), Arkolakis, Costinot, Donaldson, and Rodr´ıguez-Clare (2015), Bertoletti and Etro (2015), Jung, Simonovska, and Weinberger (2015) and Neary and Mrazova (2017). 7 My choice of taking demands as primitives responds not only to the goal of analyzing the effects of trade but also to the generality of the results. By Sonnenschein-Mantel-Debreu theorem and its extension by Chiappori and Ekeland (1999) to aggregate market demands, there are almost no restrictions on a function to be an aggregate demand. Furthermore, in a context of differentiated products, welfare results always depend on the weights attached to each effect. Behrens, Kanemoto, and Murata (2014) show that, in fact, general results about welfare cannot be attained: given some effects of the channels (which this paper aims at identifiying), there are infinite welfare’s weights attached to them consistent with the same equilibrium outcome.
6
2. An Illustration
Mart´ın Alfaro
Acemoglu and Jensen (2010, 2015).8 I make use of some of the insights and techniques provided by these authors, and extend their model to account for an endogenous number of players to cope with monopolistic competition. By taking advantages of global monotone comparative statics, I am also able to identify results under a minimum set of assumptions on the primitives of the model. For instance, for small open economies, effects on domestic prices and the productivity cutoff hold either without further assumptions or under the mild assumption that a tougher competitive environment increases the price elasticity of the demand. In addition, this paper touches upon the so-called Metzler paradox. This refers to the existence of anti-competitive effects following a unilateral liberalization to imports: the prices set by active domestic-firms increase and, under firm heterogeneity, the domestic survival marginal-cost is greater. The paradox has been obtained under monopolistic and oligopolistic competition with symmetric firms (e.g. Venables, 1985, 1987; Bagwell and Staiger, 2012) and in a Melitz model (e.g. Melitz and Ottaviano, 2008; Demidova and Rodr´ıguez-Clare, 2013; Bagwell and Lee, 2015). To the best of my knowledge, by showing that this type of liberalization represents a decrease in the export opportunities of the liberalizing country, this paper is the first to provide an explanation of why the paradox arises in different models of imperfect competition. Moreover, I show that this holds for a larger class of demands than those considered by these authors and, in the case of the Melitz model, under any productivity distribution assumed.
2
An Illustration
Through this illustration, the goal is to shed some light on the mechanisms at work in a Melitz model. To show them as stark as possible, I appeal to a simple scenario where firms face different type of exogenous shocks in a closed economy. The effects triggered in the domestic economy are akin to different scenarios which are explored later, e.g. new export opportunities, tougher import competition and variations in market size or income. Given its prominence and adequacy for illustration purposes, the example makes use of the linear demand function of Melitz and Ottaviano (2008). 8
Large aggregative economies refer to models where: agents taken altogether determine the conditions of the market but individually are negligible to it, and the conditions of the market collapses into an aggregator. An important subset of these models are the so-called large aggregative games, which are aggregative games with a finite number of players that converge to a large aggregative economy (see Acemoglu and Jensen, 2010). Since I do not assume this property, the results of this paper hold for any large aggregative economy.
7
2. An Illustration
2.1
Mart´ın Alfaro
Setup and Equilibrium
Consider a closed economy and a specific industry in isolation. Wages are determined exogenously to the sector and the economy comprises a mass L of agents. There is a pool of firms considering entry. They are symmetric ex ante and do not know their productivity. By paying a sunk cost F, they can get a draw of marginal cost c as well as an assignation of a variety ω. The draws of marginal costs come from a continuous cumulative distribution G with support [c, c] . Once a firm knows its costs, it decides on whether serving the market. Let M E be the measure of incumbents and M the subset of active firms that serve the market. Following Melitz and Ottaviano (2008), suppose the following demand function for the variety ω: qω :=
where α, γ, η > 0 and P :=
Z
1 η 1 α − pω + P, ηM + γ γ ηM + γ γ M
pω dω. 0
The demand displays a variable price elasticity and has a choke price given by, pmax :=
αγ + ηP . ηM + γ
The variable pmax represents a statistic which summarizes the conditions of the market in which the firm is operating. Consistent with a monopolistic competition structure, its value cannot be influenced by any firm unilaterally. The demand function can be reexpressed in term of the choke price: q (pmax , pω ) :=
pmax − pω . γ
Written in this way, it implies that, conditional on values of pmax and pω , the demand schedule is completely determined. As a result, the composition of pmax is irrelevant from the firm’s point of view: different combinations of M and P that result in the same value of pmax are considered equivalent. For a given value of pmax , an active firm with marginal costs c sets the following optimal prices, p (pmax ; c) :=
8
pmax + c 2
(1)
2. An Illustration
Mart´ın Alfaro
determining that its optimal domestic-profits are π (pmax ; c) :=
L max (p − c)2 . 4γ
The survival marginal-cost cutoff c∗ is determined by the cost that makes the firm indifferent between serving the market or not: c∗ = pmax
(2)
This defines a function c (pmax ) = pmax . Inspection of equations (1) and (2) reveal that optimal prices and the marginal-cost cutoff are completely determined by the value of pmax . Additionally, given a mass of domestic incumbents M E , the equilibrium at the market stage requires that all firms are optimizing simultaneously. Since for each firm with marginal cost c, optimal prices (1) and the decision of serving the market are determined by the value of pmax , the equilibrium can be characterized in the following way. Define a function Γ such that Γ p where P p
max
;M
E
:= M
E
max
Z
;M
E
αγ + ηP pmax ; M E := ηM (pmax ; M E ) + γ
pmax
p (pmax ; c) dG (c).
c The function Γ takes pmax as input, and provides a value Γ pmax ; M E as output. This
output is the choke price that would be self-generated by the agents’ decisions if they were faced to the input pmax . Thus, an equilibrium at the market stage is given by a fixed point pmax ∗ of Γ, which is the choke price consistent with the optimizing behavior of firms and the mass of firms that determines. Adding the possibility that Γ is subject to a shock α that decreases the function, I arrive to the following equilibrium condition, Γ pmax ∗ , M E∗ ; α − pmax ∗ = 0.
(3)
Finally, incorporating the fact that pmax ∗ = c∗ , the free-entry condition is, π e (pmax ∗ ) + = F where π e (p fits.
max ∗
Z ) :=
(4)
pmax ∗
π (pmax ∗ ; c) dG (c) and ε is a positive shock to expected pro-
c
9
2. An Illustration
2.2
Mart´ın Alfaro
Analysis of the Equilibrium
The equilibrium is defined by a tuple (p∗ (c))c≤c∗ , c∗ , pmax ∗ , M E∗ such that equations (1), (2), (3) and (4) hold. Equations (1) and (2) show that optimal prices and the marginal-cost cutoff are both determined by pmax and independently of M E∗ . Moreover, equations (3) and (4) reveal that the equilibrium of the economy can be completely characterized by a value of pmax ∗ , M E∗ such that conditions (3) and (4) hold. To determine pmax ∗ , M E∗ , it can be noticed that the equilibrium conditions are separable: the free-entry condition (4) pins down the value of pmax ∗ , independently of M E∗ . In other terms, irrespective of the choke-price composition, there is a unique value of pmax consistent with zero expected-profits. As a corollary, once that pmax ∗ is pinned down, the condition (3) establishes the value of M E∗ such that there is equilibrium in the market stage. Due to this, M E∗ plays the role of a residual variable which adjusts so that the good market clears. The first conclusion we get from the illustration is that, by writing the equilibrium conditions in this way, the free-entry conditions on their own determine the aggregate conditions of the market. They are represented by the choke price which, in turn, is a sufficient statistics for firms’ decisions, namely, prices and whether serving the market determined by the marginal-cost cutoff.
2.3
Effects of the Exogenous Shocks
Let’s consider the effects of variations in α and . They represent two different types of shocks, useful to develop intuitions of distinct experiments that might be conceived. α constitutes a shock to the market-clearing condition. Since this is given by the composition of the aggregate conditions, it can be interpreted as an exogenous shock to competition. For instance, it could reflect the case of tougher import competition, taking the form of either an exogenous import prices’ reduction or an increase in the mass of active foreign-firms. Other example could be the exogenous entry of firms. This would be the case of a closed economy where a given group of large firms coexists with a competitive fringe that behaves as in the illustration. The equilibrium can be characterized by a pair pmax ∗ , M E∗ such that conditions (3) and (4) hold, and pmax ∗ is completely determined by the latter. Optimal prices and the marginal-cost cutoff are completely determined by pmax ∗ . Since α only affects directly the condition (3), it cannot affect the value of pmax ∗ . It follows that tougher foreign competition 10
3. The Model
Mart´ın Alfaro
cannot affect either the domestic marginal-cost cutoff or the prices (or quantities) set by the domestic firms. It is only reflected through variations in the mass of domestic incumbents. From this result, we conclude that the mere introduction of a demand system with non-constant price elasticity in a Melitz model is not sufficient to display pro-competitive effects stemming from import competition. To get some intuition of the relevant assumptions for the result to hold, note the importance of the existence of a continuum of firms which are symmetric before entering. For one thing, it implies that, even though firms’ productivities are random, by a suitable application of the law of large numbers, the choke price collapses to a real number, thus vanishing any individual uncertainty. Furthermore, since firms are ex-ante symmetric, variations in the mass of incumbents have the property of exactly replicating the whole distribution of productivities. Regarding an increase in ε, it represents a positive shock to expected profits. Examples of cases that might be encompassed by the shock are new export opportunities and an increment in the domestic market size. Since pmax ∗ is completely determined by the freeentry condition, it can be shown that pmax ∗ has to decrease to restore the zero expectedprofits. By this, an increase in the expected profits reduces the price set by active firms and their marginal-cost cutoff. To provide an intuition behind the result, interpret the choke price as a measure of competition. An increase in the expected profits determine that competition has to become tougher in order to restore zero expected-profits. This is in part accomplished by a decrease of the price by active firms and. Under some regularity assumption, the expected profits are positive even accounting for the more aggresive pricing of active firms, so that there has to be an increase in the mass of incumbents that validates enough competition to drive the expected profits to zero. In summary, while a shock to the market-clearing condition cannot affect firms’ decisions, an increase of the expected profits has pro-competitive effects: it reduces both the prices of active firms and their marginal-cost cutoff to serve the market.
3
The Model
In this section, I proceed in several steps. First, I establish a framework in line with trade models of monopolistic competition and firm heterogeneity as in Melitz (2003). I extend the setup to account for the existence of small and big economies. Then, I add some struc-
11
3. The Model
Mart´ın Alfaro
ture to the demand in order to encompass demands that summarize market conditions through a single sufficient statistic. After this, I solve for the equilibrium and consider a closed economy facing different type of shocks. Just like the illustration of Section 2, this provides some intuitions of the mechanisms behinds liberalization experiments that I conduct after.
3.1
Structure of the Model
There is a world economy comprised of a set C of countries. Along the paper, when economies are open, I consider two different scenarios: one between a small economy and the rest of the world and other between two big economies. To avoid splitting the setup for each case, I incorporate the distinction in terms of a measure. C is partitioned into sets C B and C S of big and small countries, respectively, with C S ⊂
R++ and C B ⊂ N+ . The set C is endowed with a measure µ. In addition, there is a country-
specific measure µi for i’s trading partners. This measure allows to account for that, even if i is a small economy, it should always be considered nonnegligible relative to its own country9 . In the case of trade between two big countries, I assume that C := {i, j} with i, j ∈ C B . By making use of Lebesgue integrals, these measures allow to accommodate
sums and integrals in a unified notation.
To isolate the effects in a specific industry, different paths could be taken. I keep it simple and consider economies with a continuum of sectors, focusing on one of them h i 10 f in isolation. The sector consists of a differentiated good with a set Ωi := 0, Mi of fi is horizontally differentiated varieties (endowed with the Lebesgue measure) where M an arbitrary large mass of all conceivably varieties in i. This mass is taken fixed along the analysis. I denote Ωji := [0, Mji ] the set of varieties from j which are sold in i and Mi := Z Mji dµi (j) the total mass of varieties consumed in i. I take demands as a primitive of j∈C
the model, with qji (ω) a variety ω produced by a firm from j selling to i. The differentiated sector in i has a supply side as in Melitz (2003). The mass of profi and all the firms are ex-ante identical. To enter, they spective entrants is given by M 9
Formally, we endow each C S and C B with the Lebesgue λ and counting measure #, respectively and define the measure µ and µi by µ (·) := λ · ∩ C S + # · ∩ C B and µi (·) := λ · ∩ C S + # · ∩ C B ∪ {i} . Due to this, any integral should be understood as a Lebesgue integral. Thus, for instance, the total measure Z Z P of countries C is µ (C) = µ (dj) which is a compact way to express dj + j∈C B 1. 10
j∈C S
j∈C
Alternatively, it could be resorted to the usual assumption of an economy with two sectors, each consisting of a differentiated and homogeneous good and where the latter is freely traded and produced in each i in every equilibrium. When labor is perfectly mobile across sectors but not among countries and the homogenous good is supplied under a technology with constant returns to scale, wages are pinned down and so the differentiated sector can be analyzed separately from the rest of the economy.
12
3. The Model
Mart´ın Alfaro
consider paying a fixed (sunk) entry cost Fi > 0 which enables them to receive a productivity draw ϕ and an assignation of a unique variety ω. The random variable representing productivity has nonnegative support with bounds ϕi and ϕi , with ϕi ∈ R++ ∪ {∞}, and
a continuous cumulative distribution function Gi . Once that firms know their productivity, they make decisions regarding serving each country. They can choose not to sell in country j or do so by incurring in an overhead fixed cost fij ≥ 0 (with strict inequality if
the choke price is infinite). The production used to serve j entails constant marginal costs c (ϕ, τij ), where τij represents a trade cost that a firm in i has to incur to sell in j. I adopt the convention that τii := 1 and assume c (ϕ, τij ) is smooth, decreasing in ϕ and increasing in τij . Also, I suppose that in equilibrium every firm that exports also sells domestically. i h Each firm from i makes a decision in j over prices pij (ω) ∈ Pj := pj , pj with pj ∈ R+
and pj ∈ R+ ∪ {∞} , and pj such that it is equal or greater than demand’s choke price.
Markets are segmented with forbidden resale, so that firms can charge different prices in each country. To account for entry and exit of firms, I assume that unavailable varieties in j have a price set to pj . I denote by pji := (pji (ω))ω∈Ωj be the vector of prices of all varieties from j. The mass of incumbents in country i is denoted by MiE , with a mass of active firms in i
selling to j given by Mij := [1 − Gi (ϕij )] MiE where ϕij is the productivity cutoff of a firm
from i to break even in country j.
Definition 1. The market structure in i is a` la Melitz when it is given by the setup described above.
3.2
Demand System
I add some structure to the demand side to have demands which summarizes market conditions through a single sufficient statistic. The assumption is satisfied for a large group of demand systems used in the literature.11 I start with some definitions. Definition 2. A price aggregator for country i is a function Pi : ×Ωji Pi j∈C → R+ with h i (pji )j∈C 7→ Pi (pji )j∈C such that any measure-zero set of firms cannot influence its value.12 A price aggregate for country i is a value Pi ∈ range Pi where range Pi is a convex set. 11
Since it is not immediate to notice that some of the demand systems are actually covered, a list of demands can be found in Appendix B, along with the way in which they can be written properly to reflect the assumption. 12 Formally, let p0ji j∈C and p00ji j∈C be two vectors such that the set of prices with p0ji (ω) 6= p00ji (ω) are h i h i of measure zero, then P0i = P00i where P0i := Pi p0ji j∈C and P00i := Pi p00ji j∈C .
13
3. The Model
Mart´ın Alfaro
A price aggregator represents a statistic that summarizes information related to prices in the country. It could be, for instance, the average, variance, or a more complex function which depends on different price statistics. Some properties of the price aggregators are worthy to remark. First, by the general form in which I have defined it, it allows for asymmetries. Thus, firms’ prices might impact the aggregator in different ways. Furthemore, Pi takes as input the vector of prices, which in my framework will be random variables,
and provides a real number as an output. The intuition behind is that in large economies, under some appropiate law of large numbers, the price statistic vanishes any individual uncertainty establishing that there is no uncertainty in the aggregate.13 Standard demand systems used in the literature are functions of price aggregators which take the following form. Definition 3. ZA standard price aggregator for country i is a function h i hR i h (p (ω)) dω Pi (pji )j∈C := dµi (j) where (hji )j∈C are smooth monotone functiji ji ω∈Ωji ons.
j∈C
Along the paper I make use of these price aggregators since they simplify the exposition.14 Nonetheless, the conclusions of the paper do not hinge on this functional form and could be derived by making use of the general definition of a price aggregator. Notice also that relaxing the assumption of smoothness, the mass of varities would be also a price aggregator.
15
K
k K with P := P , K < ∞, where each Pik is a price agi i k=1 k=1 gregator as in Definition 3 with functions hkji j∈C . An aggregator for country i is a smooth
Definition 4. Let P i := Pik
Pi , Mi ) 7→ Ai (P Pi , Mi ) which is decreasing in (pji )j∈C when defined real-valued function Ai with (P directly through P i and increasing in Mi . An aggregate for country i is a value Ai ∈ range Ai . 16
The aggregator plays the role of a single sufficient statistic that sums up the overall conditions of the sector in i. Since reductions of prices or increases in the mass of active firms increases the aggregate, it can be interpreted as a measure of toughness of the competitive 13
Under some assumptions, this real number corresponds to a degenerate random variable that takes a value Pi with probability 1. For further details, see Uhlig (1996) and Acemoglu and Jensen (2010). 14 As in the general definition of the price aggregator, the standard price aggregator can also be defined more generally through functions (hω )ω∈Ωji such that asymmetries, not only among countries but between varieties, are incorporated.h i R R 15 Formally, Mi := j∈C ω∈Ωji 1 (pji (ω) < ∞) dω dµi (j) . 16
Notice that range Ai is convex, which is important for the determination of equilibrium conditions. The result follows becauseAi is continuous, and range Pi and Mi are convex sets.
14
3. The Model
Mart´ın Alfaro
environment. Also, the aggregator is not uniquely defined. Any monotone transformation of the original aggregator defines a new aggregator. Making use of these definitions, the demand is specified in the following way. Definition DEM. The demand of a variety ω produced by a firm from i in j is given by, qij (ω) := max {0, qj [Aj , pij (ω)]} where Aj is as in Definition 4 and qj is a smooth function such that qj is decreasing in Aj and pij (ω) and has a choke price belonging to R++ ∪ {∞} . Mathematically, Definition DEM states the demand satisfies weak separability of
K Pki k=1
from pij (ω).17 This determines that, from the firm’s point of view, the only relevant piece of information regarding the environment in j is Aj . Given a value Aj , its composition becomes irrelevant.
3.3
Equilibrium Conditions
Given demands consistent with DEM and assuming that each maximization has a unique and interior solution, the first-order condition gives an implicit characterization of optimal prices set by a ϕ-type firm from i which is active in j: pij (Aj , ϕ, τij ) := mj (Aj , ϕ, τij ) ci (ϕ, τij ) where
mj (Aj , ϕ, τij )
εj (Aj , ϕ, τij ) := −
∂ ln qj (Aj ,p) ∂ ln p
:= k p=pij (Aj ,ϕ,τij )
εj (Aj ,ϕ,τij ) εj (Aj ,ϕ,τij )−1
is
ϕ-firm’s
markup,
with
. Taking into account that firms also decide whet-
her they are active in each market, optimal prices are, p (A , ϕ, τ ) if ϕ ≥ ϕ ij j ij ij p∗ij (Aj , ϕij , ϕ, τij ) := pj otherwise
(PRC)
where ϕij represents a productivity-cutoff of a firm from i in j. Conditional on entry, the optimal gross-profits of an active ϕ-type firm from i in market j are, πij (Aj , ϕ, τij ) := qj [Aj , pij (Aj , ϕ, τij )] [pij (Aj , ϕ, τij ) − ci (ϕ, τij )] 17
Exploiting this feature and based on the differential characterization of weak separability due to Leontief (1947) and Sono (1961), in the Appendix I provide conditions to check whether a demand satisfies definition DEM.
15
, Mi
3. The Model
Mart´ın Alfaro
Thus, optimal profits are, πij∗ (Aj , ϕij , ϕ, τij ) := 1(ϕ≥ϕij ) [πij (Aj , ϕ, τij ) − fij ]
(PROF)
Given a value of Ai , the zero-profit condition of a firm from i selling in j determines the productivity cutoff ϕ∗ij , the infimum productivity such that profits in j are zero. This determines the following zero-profits conditions around the world, πij Aj , ϕ∗ij , τij = fij for µ-almost all i ∈ C and µi -almost all j ∈ C
(ZCP)
I denote the implicit solution by ϕ∗ij := ϕij (Aj , τij , fij ) . As far as the market-clearing conditions for i go, given a mass of incumbents ME := MjE j∈C , variety-markets clear when, up to a set of measure zero, all firm choose prices optimally. By exploiting the structure of the problem, this condition can be characterized in a straightforward way. First, note that the function P i requires firms’ optimal prices as inputs. They are com pletely characterized by the price decision p∗ji Ai , ϕ∗ji , ϕ, τji and the density of firms’ mass for each level productivity. Thus, in equilibrium, the price aggregator can be described by a function P ∗i Ai , ME , (τji , fji )j∈C . Similarly, given Mji := 1 − Gj ϕ∗ji MiE and how ϕ∗ij is determined, the measure of firms in i is a function Mi∗ Ai , ME , (τji , fji )j∈C . Since Pi , Mi ) , both facts imply that variety-markets clear around the aggregator is a function Ai (P
the world if and only if,
A∗i = A∗i A∗i , ME∗ ; (τji , fji )j∈C for µ-almost all i ∈ C
(ME)
where A∗i (·) := Ai [P ∗i (·) , Mi∗ (·)] with (·) := A∗i , ME∗ ; (τji , fji )j∈C
Finally, incorporating ϕij (Aj ; τij , fij ) ,the free-entry conditions in the world are, Z
π eij (Aj ; τij , fij ) dµi (j) = Fi for µ-almost all i ∈ C
j∈C
where π eij (Aj ; τij , fij ) :=
Z
ϕi
ϕij (Aj ;τij ,fij )
(FE)
[πij (Aj ; ϕ, τij ) − fij ] dGi (ϕ).18
Once the equilibrium conditions are written in this specific way, the following result follows by simple observation. 18
Notice that each term
Z j∈C
π eij (·) dµi (j) in (FE) is equivalent to
Z j∈C S
π eij (·) dµi (j) +
P
j∈C B \{i}
π eij (·) +
π eii (·). This definition applies irrespective if i is a small or a big country since in the former case, the integral over C S \ {i} or C S gives the same result and C B \ {i} = C B .
16
3. The Model
Mart´ın Alfaro
Lemma 3.1. Suppose the structure in each country i ∈ C is a` la Melitz with demands for each variety given by DEM. Then, the equilibrium can be completely characterized by finding a solu tion A∗i , MiE∗ i∈C such that conditions (ME) and (FE) hold. Furthermore, (A∗i )i∈C is completely determined by (FE) and independently of MiE i∈C . Lemma 3.1 reveals how the model identifies the equilibrium. In first place, equations (FE) pin down the aggregates around the world by ensuring that zero expected-profits are achieved. Remarkably, the aggregates are identified independently of their composition. Once the aggregates are determined, their compositions follow by the market-clearing conditions (ME). With the prices and productivity cutoffs already obtained since they are determined by values of the aggregator, only the mass of incumbents is free to adjust. Hence, the mass of incumbents in each country adjust until their levels are consistent with an equilibrium at the market stage.
3.4
Exogenous Shocks
In this section, I present results for a closed economy that faces two types of shock: α that hits the aggregator and which affects the expected profits. Just like the usual analogy of a trade liberalization with an increase in the market size, the effects of these shocks resemble the impact in the domestic economy stemming from tougher import competition and better export opportunities, respectively. Incorporating these shocks, the equilibrium conditions (ME) and (FE) for a closed economy are: A∗ = A∗ A∗ , M E∗ ; α
π e (A∗ ) + = F where π e (A) :=
Z
ϕi
ϕ(A)
[π (A; ϕ) − f ] dG (ϕ). I refer to the case where A∗ is increasing in α
and analyze variations in α and separately.
Proposition 3.1: A Shock to the Aggregator Suppose a closed economy with a market structure a` la Melitz, and demands for varieties given by (DEM). If there is a positive shock to the aggregator (an increase in α) then: • the aggregate remains the same, i.e. the level of competition does not vary,. • the prices, quantities, and markups of active firms remain the same, • the productivity cutoff remains the same, and • the mass of incumbents decreases. 17
3. The Model
Mart´ın Alfaro
The result holds regardless of if the shock is infinitesimal or discrete. The consequences of a shock to the aggregator resembles the effects of variations in the import competition. This could take the form of a variations of foreign firms’ prices, which would affect P ∗ , or entry/exit of foreign firms, which would vary M ∗ . More generally, the shock to the aggregator could be representing a variation in any outside option. Thus, in addition to tougher import competition, it also encompasses other cases, such as, for instance, the entry of a group of nonzero measure firms in a market structure a` la Melitz. The proposition states that none of these experiments would affect the pricing of active domestic firms or their productivity cutoff. To consider an exogenous change in firms’ expected profits, I add an assumption. Assumption STB. For any Ai ,
∂A∗i (Ai ;ME ,(τji ,fji )j∈C ) ∂Ai
< 1.
Since Assumption STB only affects conditions (ME) , it is only necessary when we are after comparative statics results for the mass of incumbents. While it might look less transparent than other assumptions, it has some specific interpretations. First, in case the condition holds only at the equilibrium value A∗i , it constitutes an ”almost” if and only if condition for a unique equilibrium at the market stage.19 In this sense, arguably, it constitutes a necessary assumption to get meaningful comparative statics. In addition, when stated globally, it establishes that Ai is a contraction along any op-
timal path. Hence, the assumption can be justified in relation to a global stability condition in line with Samuelson’s Correspondence Principle. To get unambiguous global comparative-statics results, stability of some sort is usually required.20 Proposition 3.2: Shock to the Expected Profits Suppose a closed economy with a market structure a` la Melitz, and demands for varieties given by (DEM). If there is a positive shock to firms’ expected profits (an increase in ) then: • the aggregate increases, i.e. competition becomes tougher, • the survival productivity cutoff increases, 19
Formally, the local condition is
∂Ai (Ai ;ME ,(τji ,fji )j∈C ) ∂Ai
< 1 which is the Poincar´e-Hopf index con-
Ai =A∗ i
dition. Under compactness and smoothness conditions, the Poincar´e-Hopf condition is an ”almost” if and ∂Ai (Ai ;ME ,(τji ,fji )j∈C ) only if in the sense that if > 1 for some A∗i , there are necessarily multiple equi∂Ai Ai =A∗ i
libria. Hence, if the possibility that 6= 1 can be ruled out, then the condition is also necessary. 20 Stability to get unambiguous results is needed even in models with complementarities. Monotone comparative statics as in Milgrom and Shannon (1994) can only provide information for extremal equilibria but it is silent for the rest of equilibria. Under this framework, Echenique (2002) shows that stability conditions are required if our goal is to obtain define comparative statics for any equilibrium.
18
4. Small-Economy Case
Mart´ın Alfaro
> 0 the prices, markups and quantities of active domestic firms decrease, and • if ∂ε(·,A) ∂A • if assumption (STB) holds then the mass of incumbents is higher. The proposition indicates that higher expected profits need of tougher competitive conditions to restore the zero expected-profits. The effects in the country are triggered by this: the survival productivity cutoff increases and, under the plausible assumption that a tougher competitive environment increases the price elasticity of demand, prices decrease. Furthermore, Assumption STB implies that the reduction of prices are not enough to restore the zero expected-profits, thus requiring entry of new incumbents to validate the increase in competition. Experiments with effects akin to the ones from a shock to expected profits are, for instance, new export opportunities and a increase in the market size.21 Figure 1: Exogenous Shocks in a Closed Economy
4
Shock to the Aggregator
Shock to the Expected Profits
=A
↑A =ϕ
↑ϕ
=p
↓ p if
=m
↓ m if
=q
↓ q if
↓ M E (↓ M )
↑ M E if (STB)
∂ε(·,A) ∂A ∂ε(·,A) ∂A ∂ε(·,A) ∂A
>0 >0 >0
Small-Economy Case
In this section, I consider the effects of a trade liberalization in a small country i, accounting for the equilibrium in the world economy. Formally, I consider a country i ∈ C S and
proceed by distinguishing between the effects on the domestic market stemming from tougher import competition and better export opportunities. To isolate each channel, and 21
Notice that the effect on prices do not hinge on the sign of ∂ε(p,A) ∂p . This might be puzzling given that, since Krugman (1979), it is been common to generalize results in monopolistic competition under demands derived from additive separable utility functions. In that case, the sign of ∂ε(p,A) determines the effects on ∂p the economy of increases in the market size and trade liberalizations. Nonetheless, those demands satisfy the property that sgn ∂ε(p,A) = sgn ∂ε(p,A) ∂A ∂p .
19
4. Small-Economy Case
Mart´ın Alfaro
since i is a small economy, it is enough to consider unilateral liberalizations. I begin by considering the effects of import competition. Proposition 4.1: Import Competition in a Small Economy Suppose that the structure of any k ∈ C is a` la Melitz and the domestic demand for each variety is given by (DEM). Consider a small economy i ∈ C S and suppose there is a reduction of τji or fji for each j ∈ C\ {i}. Then, • (Ak )k∈C remains the same, • Regarding country i, – the productivity cutoffs, quantities and prices of active firms in each country remain the same, – the mass of incumbents decreases, and – the mass of active firms selling in each country k ∈ C decreases. • For any country j 6= i: – the productivity cutoff to sell in i decreases, – the mass of active firms selling in i increases, – if there is a reduction in τji , the prices in i of active firms from j decrease and if ∂εi (pji ,·) > 0, they set a higher markup, and ∂pji – if fji varies, the prices of active firms in i remain the same.
Figure 2: Tougher Import Competition in a Small Country i = Ai (↓ τji for all j 6= i)
Domestic firms
Firms from j 6= i
= ϕii
↓ ϕji
= pii
↓ pji
= mii
↑ mji if
= qii
↓ qji
↓ MiE (↓ Mik ∀k)
= MiE (↑ Mji )
20
∂εi (pji ,·) ∂pji
>0
5. Trade Between Two Big Countries
Mart´ın Alfaro
Regarding the effects of better export opportunities, the results are the following.
Proposition 4.2: Export Opportunity in a Small Economy Consider a small economy i ∈ C S and suppose there is a reduction of τij or fij for some j ∈ C\ {i} . Then, • A∗i increases and (A∗k )k∈C\{i} remains the same. • Regarding firms from i in their domestic market: (·,Ai ) – the domestic productivity cutoff increases and if ∂εi∂A > 0 the prices and mari kups of active domestic firms decrease, and – if assumption (STB) holds, the mass of incumbents from i increases. • Regarding exporters from i and destination j 6= i: – the productivity cutoff decreases and the mass of active firms increases, ∂ε (p ,·) – if there is a reduction in τji the prices of active firms decrease and if j∂pijij > 0, they set a higher markup, and – if fij varies, prices and markups of active firms do not vary. • Regarding exporters from k 6= i and destination i: – the productivity cutoff increases and the mass of active firms in i decreases, and (·,Ai ) – if ∂εi∂A > 0, the prices and markups of active firms decrease. i
Figure 3: Better Export Opportunities in a Small Country i Effects on country i
i’s exporters in j
↑ Ai
= Aj (↓ τij )
↑ ϕki ↓ pki if
↓ ϕij ∂εi (·,Ai ) ∂Ai
↓ mki if ↓ qki if
>0
∂εi (·,Ai ) ∂Ai
∂εi (·,Ai ) ∂Ai
↓ pij
>0
↑ mij if
>0
>0
↑ qij
↑ MiE if (STB) (↓ Mki k 6= i)
5
∂εj (·,pij ) ∂pij
↑ Mij if (STB)
Trade Between Two Big Countries
In this section I consider the case of trade between two big economies and study the effects of trade liberalizations on the prices and productivity cutoff of domestic firms. When 21
5. Trade Between Two Big Countries
Mart´ın Alfaro
a small economy is under analysis, the experiment of a unilateral liberalization allows to account for export-opportunities and import-competition effects directly. Nevertheless, when countries are nonnegligible, isolating the effects of each mechanism is a more complex task due to the presence of feedback effects. To see why this is the case, consider two big economies i and j, and suppose there is a reduction of the trade costs from j to i. Regarding i, for one thing, the country is subject to tougher import competition due to the change in behavior of firms from j that the fall in trade costs cause. For other thing, unlike the case of small economies, country j represents for i the conditions of its export market. The variation in trade costs trigger changes in j’s sector, and so different export conditions for i. As a consequence, i faces more import competition and different access conditions to j, confounding the export and import channels. By analogous reasons, j has more export opportunities and a change in the access conditions to i. I present the effects of a unilateral liberalization and then proceed to its decomposition. Depending on the country under analysis, this type of experiment accounts for the importcompetition or the export-opportunity channels. Proposition 5.1: Unilateral Liberalization between Two Big Economies Suppose a world economy with two countries i, j ∈ C B which have a structure a` la Melitz and demands for each variety given by (DEM). Suppose there is an infinitesimal variation of τji with no changes in τij . Suppose that |JF E | > 0 and |JM S | > 0 at the equilibrium, where JF E and JM S refer to the Jacobians of the free-entry and market-stage conditions. Then: • In country i: – the aggregate decreases, i.e. competition becomes less tough, – the productivity cutoffs of domestic firms and firms from j selling in i decrease, (·,Ai ) – if ∂εi∂A > 0 the prices and markups of active domestic-firms increase, i ∂ε (p ,·)
(·,Ai ) – if ∂εi∂A > 0 and i∂pjiji > 0, the markups of firms from j selling in i increase, i and – the measure of incumbents from i decreases. • In country j: – the aggregate increases, i.e. competition becomes tougher, – the productivity cutoffs of domestic firms and firms from i selling in j increase, ∂ε (·,A ) – if j∂Aj j > 0, both active domestic-firms and firms from i selling in j decrease their prices and markups, and – the measure of incumbents from j increases.
Notice that the effect on prices of j’s firms selling in i is indeterminate, since the re22
5. Trade Between Two Big Countries
Mart´ın Alfaro
duction of trade costs gives incentives to decrease the price while a more lenient competition does the opposite. The effects on domestic firms’ prices and their the productivity cutoff are completely determined by the changes in the aggregate’s value. Hence, to split the effects, it is enough to consider the impact of each of them on each country’s aggregate. Figure 4 provides a visual aid for the effect on each aggregate. Regarding i’s aggregate, its total variation is split in two. I define the import-competition channel as the total effects on Ai (direct and indirect) triggered by the direct impact of trade costs on Ai . The exportconditions channel is the total effects on Ai triggered by the direct impact of trade costs on Aj . Concerning j’s aggregate, the export-opportunities channel are the total effects (direct and indirect) on Aj triggered by the direct impact of trade costs on Aj , while the changes in export-conditions are all the effects on Aj triggered by the direct impact of trade costs on Ai . Figure 4: Unilateral trade liberalization in i (b) Effects on Aj
(a) Effects on Ai ∆τji
Direct Effect on Ai
Direct Effect on Aj
Direct Effect on Ai
∆Aj
∆Ai Indirect Effects of Aj on Ai
∆Ai
∆Aj Indirect Effects of Ai on Aj
∆Aj
∆Ai
Import-Competition Channel
Direct Effect on Aj
Indirect Effects of Aj on Ai
Indirect Effects of Ai on Aj ∆Aj
∆τji
Export-Conditions Channel
Export-Conditions Channel
∆Ai Export-Opportunity Channel
In order to get an explicit characterization of the channels, the free-entry conditions in i and j are: Z
ϕi
ϕii (Ai )
Z
[πii (Ai ; ϕ) − fii ] dGi (ϕ) +
ϕj
ϕjj (Aj )
[πjj (Aj ; ϕ) − fjj ] dGj (ϕ) +
Z
Z
ϕi
ϕij (Aj )
[πij (Aj ; ϕ) − fij ] dGi (ϕ) = Fi
ϕj
ϕji (Aj ,τji )
[πji (Ai ; τji , ϕ) − fji ] dGj (ϕ) = Fj .
From each equation, it can be defined implicit solutions Ai (Aj ; τji ) and Aj (Ai ; τji ), re-
23
6. Market Size and Income Effects
Mart´ın Alfaro
spectively. Let A∗i , A∗j be the pair of equilibrium values. Then, dA∗i = dτji
∂Ai A∗j ; τji κ ∂τji | {z }
∂Ai A∗j ; τji ∂Aj (A∗i ; τji ) + κ ∂Aj ∂τji | {z }
∂Aj (A∗i ; τji ) κ ∂τji | {z }
∂Aj (A∗i ; τji ) ∂Ai A∗j ; τji + κ ∂Ai ∂τji | {z }
import-competition channel (=0)
dA∗j = dτji
export-opportunity channel (<0)
where κ := 1 −
∂Ai (A∗j ;τji ) ∂Aj (A∗i ;τji ) ∂Aj ∂Ai
−1
export-conditions channel (>0)
export-conditions channel (=0)
is a multiplier of effects capturing all the indi-
rect ones.22
To grasp some intuition of the signs of each effect, consider a decrease in τji . Notice that Ai does not depend directly on τji since the expected profits of i’s firms are independent of it. Thus, any effect on A∗i must necessarily come indirectly through changes in the export conditions, captured by variations in A∗j . By using the results of a small economy, it is known that the new export opportunities in j have pro-competitive effects in its domestic economy so that A∗j increases. At the same time, this decreases the expected profits of i’s firms and so, to restore the equilibrium, competition in i has to be more lenient to counterbalance the tougher conditions to export to j.
6
Market Size and Income Effects
A common analogy used in the literature to illustrate the effects of trade liberalizations is given by an increase of market size in a closed economy. I proceed to consider this case. Given a mass L of consumers, the aggregate demand of variety ω, denoted Q (ω) , can be written as, (5)
Q (ω) := max {0, Lq [A, p (ω)]} .
L does not affect optimal prices or markups which are still given by (PRC). However, it does affect optimal profits and so the productivity cutoff. Conditional on entry, the optimal gross-profits of a ϕ-type firm are, Lπ (A; ϕ) := Lq [A, p (A; ϕ)] [p (A; ϕ) − c (ϕ)] . 22
The expression is derived by use of
dAi dτji
=
∂Ai ∂τji
+
∂Ai dAj ∂Aj dτji
24
and
dAj dτji
=
∂Aj ∂τji
+
∂Aj dAi ∂Ai dτji .
6. Market Size and Income Effects
Mart´ın Alfaro
resulting in the following zero-profits condition, Lπ (A, ϕ∗ ) − f = 0. The implicit solution for the productivity cutoff is now ϕ∗ := ϕ (A, L) . Regarding the free-entry and market-clearing conditions, they are given by, Z
ϕ
ϕ(A∗ ,L)
[Lπ (A∗ , ϕ) − f ] dG (ϕ) − F = 0, A∗ = A∗ A∗ , L; M E∗
where A∗ A, L; M E := A P ∗ A, L; M E , M ∗ A, L; M E .
The following proposition shows that, under some circumstances, variations in the market size might entail only a variation in the mass of incumbents without affecting pricing decisions or the productivity cutoff. In fact, it can be shown that this result is more general and it could be extended to any parameter that affects the aggregate demand. Proposition 6.1: Effects of Market Size in a Closed Economy Consider a closed economy with a market structure a` la Melitz, and aggregate demand for each variety Qω := max {0, Q [θ, A, p (ω)]} where θ is a parameter. Suppose the aggregate demand is weakly separable in (A, θ) from p (ω). Then, variations of θ represents a shock to the aggregator and so its effects are the ones given by Proposition 3.1. The intuition behind is that if the aggregate demand is weakly separable in (A, θ) from p (ω) , then a new aggregator dependent on (A, θ) can be defined. Thus, all the results follow by considering θ as a shock to the new aggregator. This determines that variations in the demand parameter have no effects on optimal prices and quantities of active firms or the productivity cutoff. Proposition 6.1 explains the neutrality of market size found by Bertoletti and Etro (2015) when demands are derived from an additively separable indirect utility. Furthermore, it shows that the demand system of Nocke and Schutz (2015) , which includes the Logit as a special case, as well as the constant expenditure demand display a similar behavior. Under the interpretation that θ is the expenditure allocated to the sector, it also explains the neutrality of this variable obtained by Zhelobodko, Kokovin, Parenti, and Thisse (2012) when demands are derived from an additively separable direct utility as in Krug25
7. Restoring the Import-Competition Channel
Mart´ın Alfaro
man (1979). In addition, it implies that the neutrality also prevails for constant expenditure demands. For the cases where the conditions of Proposition 6.1 cannot be applied, variations in market size have the following effects. Proposition 6.2: Effects of Market Size in a Closed Economy Consider a closed economy with a market structure a` la Melitz, and demand for each variety given by (5). Then, if L increases: • the aggregate increases, i.e. competition becomes tougher, > 0 , the survival productivity cutoff increases, and active firms charge lower • if ∂ε(·,A) ∂A prices and markups.a Notice that since L enters multiplicatively in the demand, the assumption the aggregate demand is not weakly separable in (A, L) from p (ω). a
∂ε(·,A) ∂A
> 0 implies that
When demands are homothetic in income, the following corollary holds. Corollary 6.1. Consider a closed economy with a market structure a` la Melitz, and demand for variety ω given by q (ω) := max {0, q [A, y, p (ω)]}. Suppose the demand is linearly homogeneous in income. Then, an increase of income is isomorphic to an increase in market size and so the effects are the same as in Proposition 6.2..
7
Restoring the Import-Competition Channel
To propose an alternative where the import-competition channel reemerges, first it is necessary to understand why its absence arises. Due to this, I start by considering different market structures where the insensitivity of domestic decisions to import competition is also present. The cases analyzed unveil that three assumptions are key: free entry, homogeneity of the firms at the moment of making an entry choice, and the existence of a non-stochastic aggregator that is a sufficient statistic to determine optimal firms’ choices and profits. In particular, it is illustrated that even with market structures that feature oligopolistic characteristics the insensitivity would still be present if those conditions are satisfied. After this, I consider a scenario where firms know their productivity and enter following a productivity order. The assumption is standard in oligopoly models and, in an international trade context, has been the rule when that market structure was assumed, such as in Feenstra and Ma (2007), Atkeson and Burstein (2008), Eaton, Kortum, and Sotelo 26
7. Restoring the Import-Competition Channel
Mart´ın Alfaro
(2012), Edmond, Midrigan, and Xu (2015) and Gaubert and Itskhoki (2016). Incorporating that firms enter through a productiviy ranking is in order to rule out equilibria where inefficient firms drive more productive ones out of the market. Importantly, although the assumption turns the model isomorphic to the short-run version of a Melitz model where the mass of incumbents is given23 , the framework also corresponds to the long-run version of a well-defined model. In last instance, whether assuming that firms are ex ante symmetric or know their productivity at the moment of making an entry choice, relies on which description is more accurate for the phenomenon under study.
7.1
Alternative Market Structures
The fact that import competition has no impact on domestic firms’ decisions might consider peculiar to the assumptions of the Melitz model. In particular, it might be suspected that assuming zero-measure firms is crucial for the result to emerge. The goal of this section is considering a market structure with nonnegligible firms and show that the intuition is misdirected. To keep interpretations as transparent as possible, I consider a closed economy which faces a shock to the competitive environment. The sector under analysis consists of a differentiated good with an arbitrary large measure of conceivably varieties. Let Ω be the set of varieties which are produced in equilibrium. I partition the set Ω into sets ΩE and ΩI where the subscripts are mnemotechnics for ”entrants” and ”insiders”, consistent with assumptions I establish below. ΩL is a discrete set (endowed with the counting measure) of nonnegligible heterogeneous firms which have different varieties assigned and know their productivity. As for ΩE , I suppose that the minimum productivity of firms belonging to ΩL is greater than the maximum (potential or certain) productivity of firms in ΩE . Moreover, I assume that firms enter sequentially according to a productivity order, so that entry and exit of firms occur within firms belonging to ΩE . Each prospective entrants in ΩE considers paying an entry cost F > 0 that would allow it to get a productivity draw ϕ and the assignation of a unique variety ω. A firm producing variety ω makes a decision on a vector of variables xω ∈ X := RN + ∪ x0 , with N < ∞ and x0 representing inaction (exit of the market).
I consider two different cases depending on the nature of firms in ΩE . In the first one, 23
fi = M E where M fi is the mass of all conceivably firms in Formally, it would correspond to the case of M i
i.
27
7. Restoring the Import-Competition Channel
Mart´ın Alfaro
firms are characterized as in the Melitz model. In the other, firms are of nonzero measure and have the same level of productivity.24 Depending on how ΩE is defined, I refer to each model as oligopoly with a fringe a` la Melitz and oligopoly with symmetric entrants. As is common in the literature, in order to avoid dealing with bounds of equilibrium’s values, I ignore the integer constraint. This makes the zero-profit condition hold with equality. Given that nonnegligible firms can influence the sector’s conditions, more stringent conditions over the aggregator are needed to get an aggregate that is a sufficient statistic to determine optimal choices and profits.25 I accomplish this by assuming that the aggregator can be expressed as a transformation of a standard price aggregator. This ensures that the aggregator is strongly separable and, hence, the game fully aggregative in the sense of Cornes and Hartley (2012). To keep notation simple and do not split between the possibilities of having zero- or nonzero-measure firms, I endow the set ΩE with a general measure λ. Assumption DEM’. The demand of variety ω can be expressed as q (ω) := q (A, xω ) where R P A (X) := H (X) with H monotone, X (xω )ω∈Ω := ω∈ΩL hω (xω ) + ω∈ΩE hE (xω ) dλ (ω), X ∈ range X and A ∈ range A. Alternatively, the inverse demand of ω is p (ω) := p (A, xω ) with an aggregator defined in a similar fashion. A demand system with these features encompasses an important subset of the demands included in Definition DEM.26 Regarding costs, I assume a general function that could depend on x (ω) . In this way, I am allowing for the vector of choices either affecting marginal costs or entailing fixed costs (or both). This characteristic is important for choices such as quality which could affect both type of costs. Notice that under an oligopoly with symmetric entrants, the survival productivity cutoff invariant to market shocks as long as some firm in ΩL is active. As a consequence, in that case the the insensitivity of import competition refers only to firms’ choices. 24
Anderson, Erkal, and Piccinin (2016) use this market structure to reflect a scenario where firms are heterogeneous but marginal entrants are homogeneous. In principle, this market structure might be considered as a simplifying assumption where the goal is focusing on the behavior of large firms but allowing for entry and exit. 25 With the aim of providing some insight for the assumption, consider an oligopoly with N active firms choosing prices and each facing a demand q (ω) := q [A, p (ω)]. Firms set their optimal price by analyzing ∂q(ω) dq(ω) ∂A := ∂q(ω) how it totally affects its demand. Thus, the relevant term to analyze is dp(ω) ∂A ∂p(ω) + ∂p(ω) . Note ∂q(ω) ∂p(ω)
and ∂q(ω) ∂A are functions of the pair (A, p (ω)) . On the contrary, if we do not impose any structure on the ∂A aggregator and merely assume that A := H (P) , then ∂p(ω) is possibly dependent on a subset of the prices and/or the mass of varieties available. 26 For instance, the CES, attraction demand models including the Logit, Nocke and Schutz’s (2015) demands, constant expenditure demand systems, and demands derived from an additive indirect utility.
28
7. Restoring the Import-Competition Channel
Mart´ın Alfaro
Proposition 7.1: Import-Competition Channel with Big Firms Consider a closed economy where the market structure is oligopoly with an active fringe a` la Melitz or with symmetric entrants (and heterogeneous insiders). Suppose the demands for varieties are given by (DEM’) and that when firms know their productivity, they enter sequentially. Then, if there is a shock to the aggregator, • the aggregate remains the same, i.e. the level of competition does not vary,. • all active firms’ choices remain the same, • the productivity cutoff remains the same, and • the mass of entrants decreases.
7.2
A Model where Firms Know their Productivity
To restore the import-competition channel, I modify the Melitz model in the following f of potential firms which are not homogeneous ex ante. Rather, way. There is a mass M each has a unique variety ω and productivity ϕ assigned. Firms are indexed from the most productive to the lowest with productivity distribution Gi and entry decisions in market j are made sequentially according to a productivity order. Firms keep entering in each market j until the marginal firm breaks even. The equilibrium conditions in this scenario can be summarized as in the Melitz model by dispensing with the free-entry conditions. Specifically, given optimal prices (PRC) and profits (PROF) , the equilibrium is described by similar conditions (ZCP) and (ME) as in the Melitz model: A∗i
=
A∗i
A∗i ;
ϕ∗ji , τji j∈C
for µ-almost all i ∈ C
πij A∗j ; ϕ∗ij , τij = fij for µ-almost all i ∈ C and µi -almost all j ∈ C.
Proposition 7.2: Trade Liberalization when Firms Know their Productivity Consider economies described as in the Melitz model but where firms know their productivity and enter sequentially following a productivity order. Suppose the demands for varieties are given by (DEM) and assumption (STB) holds. Then, if there is a reduction of τji for all j ∈ C\ {i} in country i: • the aggregate increases, i.e. competition becomes tougher • the productivity cutoffs of domestic firms increase, (·,Ai ) • if ∂εi∂A > 0, the prices, markups and quantities of active domestic-firms decrease, and i
29
8. Extensions
Mart´ın Alfaro
• the mass of active domestic-firms in i decrease. If i is a small economy and there is a reduction of τij for some j ∈ C\ {i} : • exporters from i to j reduces their prices, increase their quantities and, if markups increase.
8
∂εj (·,Aj ) ∂pj
> 0,
Extensions
In this section, I modify the baseline model in order to derive conclusions under some alternative assumptions. In particular, I consider three extensions. First, I show that the insensitivity to import competition holds for any firm’s countryspecific decision, thus encompassing widespread choices considered in the literature such as quality and number of products when firms are multiproduct. After this, I study alternative demands with common nested structures used in the literature. Specifically, I deal with two type of varieties’ partitions which are pervasive in the literature: by country of origin and by varieties belonging to the same (multiproduct) firm. The main conclusion is that with both demands import competition still has no impact on domestic prices/quantites and the survival productivity cutoff. Finally, I analyze the case of firm heterogeneity in terms of the demand by assuming that the demand DEM is firm specific. In that case, all the results of the baseline case would still hold.
8.1
Non-Price Choices
In the baseline model, firms were only choosing prices at the market stage. In this subsection, I consider the possibility that they make decisions on a vector of variables which are country specific. This extends the Melitz model to cases where firms decide on variables such as quality or the number of products when they are multiproduct. The main conclusion derived is that the result of the baseline case in terms of prices also holds for any choice variable: import competition does not affect any firm’s decision. Thus, only the export-opportunity channel can affect firms’ decisions.27 To show the result as stark as possible, I consider a closed economy which faces a shock to the aggregator. The framework is exactly as the Melitz model but with each firm that produces variety ω making a decision on a vector of variables xω ∈ X := RN + ∪ {x0 } with
N < ∞ and where x0 represents inaction (exit of the market). I assume that xω includes 27
How export opportunities affect firms’ decisions depend on the nature of variables considered.
30
8. Extensions
Mart´ın Alfaro
ω’s prices. Given this feature, it is necessary to redefine the aggregator and specify how demands are defined accordingly. K K Assumption 1. Let X := X k k=1 with X := Xk k=1 , K < ∞, where each X k is X k (xω )ω∈Ω := R hk (xω ) dω and Xk ∈ range X k . An aggregator is a smooth real-valued function A with ω∈Ω ω X, M ) 7→ A (X X, M ). An aggregate is a value A ∈ range A. (X
Assumption 2. The demand of variety ω can be expressed as q (ω) := q (A, xω ) with A defined as in Assumption 1. Notice I do not impose any restriction on how the choice vector is incorporated in the demand so that the result is quite general in terms of functional forms considered. Moreover, I assume a general cost function C [q (ω) , xω ; ϕ] . This allows for the possibility of non-constant returns to scale, existence of investments costs and non-prices variables affecting marginal costs. The latter allows to encompass the case where higher quality products entail greater unitary costs. Proposition 8.1: Import Competition with Country-Specific Choices Suppose a closed economy with a market structure a` la Melitz extended to a vector of decision variables and with demands for varieties given by (2). If there is a positive shock to the aggregator then: • the aggregate remains the same, i.e. the level of competition does not vary,. • active firms’ decisions remain the same, and • the productivity cutoff remains the same. I proceed to analyze usual demand systems with a nested structure when groups are defined by country of origin and by varieties produced by the same multiproduct firm. Before studying these cases, I define generically demands with nested structures. To do it in a transparent way, I consider the case of a closed economy. The set of total varieties L are partitioned into L nests such that Ω := Ωl l=1 . The demand of a firm producing a variety ω belonging to nest l is defined by, (6)
q l (ω) := q l P, Pl , pω , with P
h
L Pk k=1
i
Z
L
:=
H k=1
k
P
k
l
dk, P (pω0 )ω0 ∈Ωl
Z :=
hl (p (ω 0 )) dω 0 where hl
ω 0 ∈Ωl
and H l are monotone functions, P ∈ range P and Pl ∈ range P l . For the case of demands
with groups by country of origin, one additional property I assume is that that q l (ω) is weakly separable in P, Pl from pω . 31
8. Extensions
Mart´ın Alfaro
Two demands consistent with 6 that satisfy all the assumptions stated are the nested CES and nested Logit. 8.1.1
Nested Demands with Groups Defined by Own-Firm Varieties
As for multiproduct firms with nests defined in terms of varieties produced by the same firm, the group l corresponds to a specific firm. In this case, the model becomes a particular case of the extension where firms choosing a vector of variables.Z Formally, let firm hl (p (ω 0 )) dω 0 . l choose xl := (p (ω))ω∈Ωl . Then, its demand 6 by defining P l xl := ω 0 ∈Ωl
Thus, Pl is a real-valued function which depends on the prices of all the varieties produced by firm l and the optimization problem would determine an optimal choice xlii (Ai ) . 8.1.2
Nested Demands with Groups Defined by Country of Origin
Define each nest l by the country of origin. Then, demands can be expressed by, qji (ω) := qj [Pi , Pji , pji (ω)] Z
i
h
:= H j (Pji ) with Pi (Pji )j∈C j∈C Z hji (pji (ω)) dω. Pji (pji ) :=
dµi (j) where Pji
∈
range
Pji and
ω∈Ωji
Since it is assumed that qj is weakly separable from Pi and Pji in the function qj , the demand can be expressed by, qji (ω) := qj [Aji , pji (ω)] where Aji is a country-specific aggregate with Aji ∈ range Aji and Aji is a smooth
real-valued function (Pji , Pi ) 7→ A (Pji , Pi ).
In a similar fashion to the baseline case, the optimal prices and productivity cutoff are
pji (Aji ; τji ) and ϕji (Aji ; τji , fji ) , respectively. Moreover, the free-entry conditions are: Z j∈C
π eij A∗ij ; τij dµi (j) = Fi for µ-almost all i ∈ C
From this, it can be observed that any reduction of (τji , fji )j∈C\{i} does not affect directly the free-entry conditions. Also, assuming that i is a small economy, the profits garnered by a firm from k 6= i in i are negligible, determining that the free-entry condition of any k 6= i is not affected either. Hence, A∗ij j,i∈C do not vary, and neither p∗ii (A∗ii ) nor 32
9. Conclusion
Mart´ın Alfaro
ϕii (A∗ii ; fii ) .
8.2
Firm-Heterogeneity in Demand
In the main part of the text, I have assumed that firms are heterogeneous only in terms of their productivity. Regarding the demand side, country-specific differentiation was allowed. Here, I show that in fact the model can account for heterogeneity in both the demand and productivity side. Assume that each firm ω from i receives a draw of productivity ϕ (ω) and a countryspecific demand-parameter σij (ω). The demand is then, qij (ω) := max {0, qω [Aj , pij (ω) , σij (ω)]} In this case, the analysis of each firm would be in terms of the random vector (ϕ (ω) , σij (ω)). Conditional on it, firms’ decisions still depend on Aj and so the framework can encompass multidimensional heterogeneity. Notice that, in particular, import competition would still only affect the mass of incumbents under this framework. To fix ideas, assume that firms in the domestic market of country i can be ordered in such a way that (ϕ (ω) , σi (ω)) is increasing in ω. Then, this defines a ω-cutoff that depends on Ai . Moreover, conditional on ω, firms make decisions dependent on Ai . All this determines that expected domestic-profits are a function of Ai . Thus, following a same procedure as in the baseline case, it can be shown that import competition represents a shock to the aggregator.
9
Conclusion
TBW.
33
References
Mart´ın Alfaro
References A CEMOGLU , D.,
AND
M. J ENSEN (2010): “Robust comparative statics in large static ga-
mes,” in 49th IEEE Conference on Decision and Control (CDC), pp. 3133–3139. A CEMOGLU , D.,
AND
M. K. J ENSEN (2015): “Robust Comparative Statics in Large Dyna-
mic Economies,” Journal of Political Economy, 123(3), 587–640. A NDERSON ,
S.,
N. E RKAL ,
AND
D. P ICCININ (2016):
“Aggregative Oligo-
poly Games with Entry,” http://www.tse-fr.eu/sites/default/files/TSE/ documents/sem2016/io/anderson.pdf. Accessed: 2016-10-01. A RKOLAKIS , C., A. C OSTINOT, D. D ONALDSON ,
AND
A. R ODR´I GUEZ -C LARE (2015):
“The Elusive Pro-Competitive Effects of Trade,” Working Paper 21370, National Bureau of Economic Research. ATKESON , A.,
A. B URSTEIN (2008): “Pricing-to-Market, Trade Costs, and Internatio-
AND
nal Relative Prices,” American Economic Review, 98(5), 1998–2031. B AGWELL , K., AND S. H. L EE (2015): “Trade Policy Under Monopolistic Competition with Firm Selection,” Mimeo. B AGWELL , K.,
AND
R. W. S TAIGER (2012): “The Economics of Trade Agreements in the
Linear Cournot Delocation Model,” Journal of International Economics, 88(1), 32–46. B EHRENS , K., Y. K ANEMOTO ,
AND
Y. M URATA (2014): “New Trade Models, Elusive Wel-
fare Gains,” . B EHRENS , K.,
AND
Y. M URATA (2007): “General Equilibrium Models of Monopolistic
Competition: A New Approach,” Journal of Economic Theory, 136(1), 776–787. B ERTOLETTI , P., AND F. E TRO (2015): “Monopolistic Competition When Income Matters,” The Economic Journal. C HIAPPORI , P. A.,
AND
I. E KELAND (1999): “Aggregation and Market Demand: An Exte-
rior Differential Calculus Viewpoint,” Econometrica, 67(6), 1435–1457. C ORNES , R.,
AND
R. H ARTLEY (2012): “Fully Aggregative Games,” Economics Letters,
116(3), 631–633. D EMIDOVA , S.,
AND
A. R ODRGUEZ -C LARE (2009): “Trade Policy under Firm-Level Hete-
rogeneity in a Small Economy,” Journal of International Economics, 78(1), 100–112. 34
References
Mart´ın Alfaro
D EMIDOVA , S.,
AND
A. R ODR´I GUEZ -C LARE (2013): “The Simple Analytics of the Melitz
Model in a Small Economy,” Journal of International Economics, 90(2), 266–272. E ATON , J., S. S. K ORTUM ,
AND
S. S OTELO (2012): “International Trade: Linking Micro
and Macro,” Working Paper 17864, National Bureau of Economic Research. E CHENIQUE , F. (2002): “Comparative Statics by Adaptive Dynamics and the Correspondence Principle,” Econometrica, 70(2), 833–844. E DMOND , C., V. M IDRIGAN , AND D. Y. X U (2015): “Competition, Markups, and the Gains from International Trade,” American Economic Review, 105(10), 3183–3221. F EENSTRA , R. (2010): “Measuring the Gains from Trade under Monopolistic Competition,” Canadian Journal of Economics, 43(1), 1–28. F EENSTRA , R.,
AND
H. M A (2007): “Optimal Choice of Product Scope for Multiproduct
Firms under Monopolistic Competition,” Working Paper 13703, National Bureau of Economic Research. F EENSTRA , R. C. (2003): “A Homothetic Utility Function for Monopolistic Competition Models, Without Constant Price Elasticity,” Economics Letters, 78(1), 79–86. G AUBERT, C.,
AND
O. I TSKHOKI (2016): “Granular Comparative Advantage,” http://
www.princeton.edu/˜itskhoki/papers/GranularCA.pdf. Accessed: 2016-0601. J UNG , J. W., I. S IMONOVSKA ,
AND
A. W EINBERGER (2015): “Exporter Heterogeneity and
Price Discrimination: A Quantitative View,” Working Paper 21408, National Bureau of Economic Research. K RUGMAN , P. R. (1979): “Increasing Returns, Monopolistic Competition, and International Trade,” Journal of international Economics, 9(4), 469–479. L EONTIEF, W. (1947): “Introduction to a Theory of The Internal Structure of Functional Relationships,” Econometrica, pp. 361–373. L UCE , R. (1959): Individual Choice Behavior: A Theoretical Analysis. Wiley. M ELITZ , M. J. (2002): “The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity,” Working Paper 8881, National Bureau of Economic Research. 35
References
Mart´ın Alfaro
M ELITZ , M. J. (2003): “The Impact of Trade on Intra-industry Reallocations and Aggregate Industry Productivity,” Econometrica, 71(6), 1695–1725. M ELITZ , M. J.,
AND
G. I. P. O TTAVIANO (2008): “Market Size, Trade, and Productivity,”
The Review of Economic Studies, 75(1), 295–316. M ILGROM , P.,
AND
C. S HANNON (1994): “Monotone Comparative Statics,” Econometrica,
62(1), 157–180. N EARY, P.,
AND
M. M RAZOVA (2017): “Not So Demanding: Preference Structure, Firm
Behavior, and Welfare,” American Economic Review, 107(12). N OCKE , V.,
AND
N. S CHUTZ (2015): “Multiproduct-Firm Oligopoly: An Aggregative Ga-
mes Approach,” http://economics.mit.edu/files/11285. Accessed: 2016-0601. S IMONOVSKA , I. (2015): “Income Differences and Prices of Tradables: Insights from an Online Retailer,” The Review of Economic Studies, 82(4), 1612–1656. S ONO , M. (1961): “The Effect of Price Changes on The Demand and Supply of Separable Goods,” International Economic Review, 2(3), 239–271. U HLIG , H. (1996): “A Law of Large Numbers for Large Economies,” Economic Theory, 8(1), 41–50. V ENABLES , A. (1985): “Trade and Trade Policy with Imperfect Competition: The Case of Identical Products and Free Entry,” Journal of International Economics, 19(1-2), 1–19. (1987): “Trade and Trade Policy with Differentiated Products: A ChamberlinianRicardian Model,” Economic Journal, 97(387), 700–717. V IVES , X. (2001): Oligopoly Pricing: Old Ideas and New Tools, vol. 1. The MIT Press, 1 edn. Z HELOBODKO , E., S. K OKOVIN , M. PARENTI ,
AND
J.-F. T HISSE (2012): “Monopolistic
Competition: Beyond the Constant Elasticity of Substitution,” Econometrica, 80(6), 2765– 2784.
36
Mart´ın Alfaro
Appendices A
Proofs
Conventions: I useb· to define the natural logarithm of any function or variable ·. To avoid
cumbersome notation, if a parameter remains fixed along the analysis, I omit it from the arguments of the functions. I start by proving some lemmas which will come in handy for different proofs. Lemma A.1. πij (Aj , ϕ, τij ) is decreasing in Aj and τij , and increasing in ϕ. Moreover, ϕij (Aj , τij ) is increasing in Aj and τij . Lemma A.2. If for all Aj ,
∂εj (Aj ,·) ∂Aj
> 0 then prices pij (Aj ; ϕ), markups mij (Aj ; ϕ) and quantities
qij (Aj ; ϕ) are decreasing in Aj and πij (Aj , ϕ) is log-supermodular in (Aj , ϕ) . Lemma A.3. pij (Aj , τij , ϕ) is decreasing in ϕ and increasing in τij , and qij (Aj , τij , ϕ) is increasing in ϕ and decreasing in τij . If for all p,
∂εj (·,p) ∂p
> 0 then markups mij (Aj , τij , ϕ) are increasing
in ϕ and decreasing in τij . Lemma A.4. A∗i A∗i ; ME , (τji , fji )j∈C is increasing in MiE . Lemma A.5. A∗i A∗i , ME , (τji , fji )j∈C\{i} is decreasing in τji and fji when j ∈ C B . Proof of Lemma A.1. Gross profits are πij (pij , Aj , ϕ, τij ) := qj [Aj , pij (ω)] [pij − ci (ϕ, τij )] .
Since the prices’ domain does not depend on any of the parameters, by the value-function
theorem (i.e. a revealed preference argument) the result follows because qj is decreasing in Aj , and ci (ϕ, τij ) increasing in ϕ and decreasing in τij . Thus, the optimal gross profits for active firms πij (Aj , ϕ, τij ) are decreasing in Aj and τij , and increasing in ϕ. Regarding ϕij (Aj , τij ), we have that πij Aj , ϕ∗ij , τji = fij . Since πij (Aj , ϕ, τji ) is decreasing in Aj and τij , and increasing in ϕ, then ϕij (Aj , τij ) is increasing in Aj and τij since otherwise the equality of gross profits to fixed costs would not hold. Proof of Lemma A.2. Let Bj := −Aj and β := −ϕ. The result follows by showing that
πij (Aj , pij , ϕ) is quasisupermodular in (pij , Bj ). In particular, we show that the function
is log-supermodular in (pij , Bj ). Since πij (Aj , pij , ϕ) = qj (Aj , pij ) [pij − ci (ϕ, τij )] − fj , ∂2π bij (Bj ,pij ,ϕ) ∂ 2 qb (A ,p ) ∂ε (A ,·) = ∂Bj j ∂pj ijij which is positive if and only if j∂Ajj > 0. ∂Bj ∂pij ∂mij (Aj ,·) −∂εj (Aj ,·) = sgn . Then, by applying Topkis’ theorem, pij (Aj ; ϕ) and ∂Aj ∂Aj
then
Moreover,
sgn
mij (Aj ; ϕ)
are decreasing in Aj . i
A. Proofs
Mart´ın Alfaro
To show that qij (Aj ; ϕ) is decreasing in Aj , take the inverse function pj [Aj , qij (ω)] and let εej (Aj , q) :=
∂ pbj (Aj ,q) ∂ qb
(notice this is not the inverse demand function). Since εj (A0 , p0 ) εej (A0 , q0 ) =
1 when q0 = qj (A0 , p0 ), sgn
∂εj (A0 ,p0 ) ∂A
= sgn (−1)
∂e εj (A0 ,q0 ) ∂A
which implies that
∂e εj (A0 ,q0 ) ∂A
<
0. I show that πij (Bj , qij , ϕ) := qij [pj (Bj , qij ) − ci (ϕ, τij )] − fj is quasisupermodular in
∂e ε (A ,q ) ∂ 2 p (A ,q ) ∂e εj (A0 ,q0 ) < 0. The condition j ∂A0 0 < 0 is equivalent to ∂Aj j ∂qj ijij < ∂A ∂pj (Aj ,qij ) ∂pj (Aj ,qij ) ∂2π bij (Bj ,qij ,ϕ) ∂ 2 p (A ,q ) ∂p (A ,q ) ∂p (A ,q ) 1 . Moreover, ∂B > 0 iff ∂Aj j ∂qj ijij [pj (Aj , qij ) − c]− j ∂qijj ij j ∂Ajj ij ∂qij ∂Aj pj (Aj ,qij ) j ∂qij ∂p (A ,q ) ∂p (A ,q ) ∂ 2 p (A ,q ) p (A ,q )−c 0, or what is same, ∂Aj j ∂qj ijij jpj (Aj j ,qijij ) − pj (A1j ,qij ) j ∂qijj ij j ∂Ajj ij < 0. Since at any optimal p (A ,q )−c ∂e ε (A ,q ) point pj (Aj , qij ) ≥ c, then jpj (Aj j ,qijij ) ∈ [0, 1] and the result holds when j ∂A0 0 < 0.
(qij , Bj ) when
To show that πij (Aj , ϕ) is log-supermodular in (Aj , ϕ), I show that πij (Aj , pij , ϕ) is log-
supermodular in each pair of variables. I have already shown that πij (Aj , pij , ϕ) is logsupermodular in (pij , −Aj ) since
∂2π bij (Bj ,pij ,ϕ) ∂Bj ∂pij
is log-supermodular in (pij , β) , irrespective of the ∂c(β,τij ) 1 ∂β (p−c(ϕ,τij ))2
> 0. Finally,
∂2π bij (Aj ,pij ,ϕ) ∂Aj ∂ϕ
∂εj (Aj ,·) > 0. Also, πij (Aj , pij , ϕ) ∂Aj ∂ε (A ,·) ∂2π bij (Aj ,pij ,ϕ) sign of j∂Ajj , since = ∂β∂pij
> 0 when
= 0. Hence, πij (Aj , ϕ) is log-supermodular in
(Bj , β) and so in (Aj , ϕ). Proof of Lemma A.3. Let β := −ϕ. Since
∂2π bij (Aj ,pij ,ϕ) ∂β∂pij
=
∂c(β,τij ) 1 ∂β (p−c(ϕ,τij ))2
> 0, optimal
prices are decreasing in ϕ due to quasisupermodularity. By the same token, optimal prices are increasing in τij . Regarding quantities, expressing the profits in terms of the inverse function as we did for the proof of Lemma A.2,
∂2π bij (Aj ,qij ,ϕ) ∂ϕ∂qij
=
∂c(ϕ,τij ) ∂pj (Aj ,qij ) 1 ∂ϕ ∂qij (p−c(ϕ,τij ))2
>
0. This implies that optimal quantities are increasing in ϕ and, in a similar fashion, it can be shown that they are decreasing in τij . Reexpressing the profits as function of markups, πij (Aj , mij , ϕ, τij ) = qj [Aj , mij ci (ϕ, τij )] mij . Then ∂qj (Aj ,mij ci (ϕ)) ∂ci (ϕ) ∂pij ∂ϕ
∂b πij (Aj ,mij ,ϕ) ∂mij ∂ϕ
=
∂ 2 qj (Aj ,mij ci (ϕ)) mij ci ∂p2ij
i (ϕ) (ϕ) ∂c∂ϕ +
> 0. Hence mij (Aj , ϕ) is increasing in ϕ. By the same token, markups
are decreasing in τij . Pi , Mi ) and MiE affects both P i and Mi . EvaProof of Lemma A.4. We know that Ai (P
luated at A∗i , the price aggregator is in particular a function of MiE and domestic prices p∗ii (A∗i , ϕ∗ii ; ϕ) with ϕ∗ii := ϕii (A∗i , fii ). When MiE increases, the additional firms that were setting pi are now either still setting pi or pii (A∗i , ϕ) < pi . Thus, the increase of MiE represents a decrease of prices for P i . Since P i is decreasing in prices for nonnegligible firms and i is a big country when it is its own country, Ai increases through P i . Moreover,
Mii = [1 − Gi (ϕii )] MiE . Since ϕ∗ii is independent of MiE , Mii is increasing in MiE . Thus, Ai increases through Mi too. Then A∗i is increasing in MiE .
Proof of Lemma A.5. Ai is increasing in PiZand Mi . The result follows if we show that Pi and Mi are decreasing in τji . First, Mi = [1 − Gk (ϕ∗ki )] MkE dµk . Since Gj ϕ∗ji is k∈C
increasing and by Lemma A.1 ϕ∗ji is increasing in τji , then Mi is decreasing in τji since j
is a big country. Regarding prices, since ϕ∗ji is greater when τji increases, firms that were ii
<
A. Proofs
Mart´ın Alfaro
setting pji (Ai , ϕ, τji ) , now set pi > pji (Ai , ϕ, τji ). Firms that are active before and after the change in τji , set a price pji (Ai ; ϕ, τji ) which is increasing in τji by Lemma A.3. Thus, prices increase and P i decreases since j is by assumption a big country. The proof for fji follows verbatim with the difference that active firms set the same price before and after the change in fji . Proof of Proposition 3.1. Let α00 > α0 . Applying Lemma 3.1, the equilibrium can be completely characterized by determining A∗ and M E∗ through the conditions conditions π e (A∗ ) = F and A∗ = A∗ A∗ , M E∗ ; α . Moreover, by the same lemma, A∗ is completely determined by the former equation and independently of α and M E∗ . Thus, A∗ does not vary and the shock α only affects M E∗ . By Lemma A.4, A∗ A, M E is M E∗ is . Moreover, since p (A∗ ; ϕ) and ϕ (A∗ ) , and A∗ has not changed, the variables do not change either.
Proof of Proposition 3.2. Assume that the shock is such that ε0 increases to ε00 . Applying Lemma 3.1, the equilibrium can be completely characterized by A∗ and M E∗ . The equilibrium conditions are π e (A∗ ) + ε = F and A∗ = A A∗ , M E∗ . Let A0i , MiE0 and A00i , MiE00 be the equilibrium values when shocks are αi0 and αi00 , respectively. Z
Since by Lemma ?? π (A; ϕ) is decreasing in A and ϕ (A) increasing in A, then π e (A) :=
ϕi
ϕ(A)
[π (A; ϕ) − f ] dG (ϕ) is decreasing in Ai . Thus, A00 > A0 and ϕ (A00 ) > ϕ (A0 ). Markups
and prices of active firms decrease by Lemma A.3.
Regarding the mass of domestic incumbents, A∗ is determined independently of M E∗ which, in turn, is determined through A∗ = A∗ A∗ , M E∗ . By assumption (STB), A − A∗ A, M E is increasing in A. Also, by Lemma A.4, A∗ A∗ , M E is increasing in M E . Hence, M E has to increase to restore the equilibrium and the result follows.
Proof of Proposition 4.1. Consider a reduction of τji . The proof for a decrease in fji is similar. By Lemma 3.1, the conditions (FE) pin down the equilibrium value (A∗k )k∈C . The parameter (τji )j∈C\{i} does not affect directly the condition (FE) of i. Moreover, since i is a small economy, the profits garnered by a firm from k 6= i in i are negligible. Hence, the conditions (FE) for any k 6= i are not affected either. Hence, (A∗k )k∈C do not vary.
Consider firms from country i. Since (τik , A∗k )k∈C does not vary, neither p∗ik (·) nor ϕ∗ik
vary for any k ∈ C.
Consider firms from a country j 6= i. Since (τjk , A∗k )k∈C\{i} does not vary, again ϕ∗jk and
p∗jk (·) do not vary for any k 6= i. In country i, A∗i does not vary but there is a reduction of
τji . Thus, ϕ∗ji (·) decreases by Lemma A.1. Moreover, firms with productivity ϕ that remain
active after the shock set pji (A∗i , ϕ, τji ) which decreases by Lemma A.3 (if the shock is in fji , prices remain the same). In case
∂εi (pji ,·;·) ∂pji
> 0, the reduction of prices entails an increase
iii
A. Proofs
Mart´ın Alfaro
of markup by these firms. Regarding the measure of firms, consider the system of equations of (ME) for countries j ∈ C\ {i} . Since i is a small economy, changes in the measure of their incumbents do not affect those conditions. Moreover, A∗j , (τkj , fkj )k∈C j∈C\{i} do not vary. This determines that MjE∗ does not vary for any j 6= i. This implies that for condition (ME) in
i to hold, MiE∗ has to adjust. In equilibrium, condition (ME) in i can be expressed by A∗i = A∗i A∗i , MiE∗ , (τji )j∈C\{i} . By Lemma A.5, A∗i is decreasing in (τji )j∈C B and A∗i is the same before and after the shock. Thus, by the positive relation between Mi and MiE , there
is a reduction in the mass of incumbents from i. Since ϕ∗ii has not changed, then the mass of active firms from i decreases. Regarding a country j 6= i serving i, since ϕ∗ji decreases and its mass of incumbents remains the same, then Mji∗ increases.
Proof of Proposition 4.2. Consider a reduction of τji . The proof for a decrease in fji is similar except for prices. By Lemma 3.1, the conditions (FE) pin down the equilibrium value (A∗k )k∈C . The parameter τij does not affect directly the condition (FE) of any country k 6= i. Moreover, since i is a small economy, the profits garnered by a firm from k 6= i in i are negligible. Hence,
the conditions (FE) for any k 6= i do not change through this channel either, determining that A∗j j∈C\{i} do not vary. Moreover, since i is a small economy, it has no impact on the system of equations of (ME) for countries k 6= i, determining that MkE∗ is the same for each
k 6= i. Since (A∗k )k∈C\{i} does not change, p∗jk (·) and ϕ∗jk for any j, k 6= {i} do not change
∗ either. This also implies that Mjk does not change for any j, k 6= i.
Regarding country i, the reduction of τij represents a positive shock to the expected
profits of a firm from i. By proposition 3.2, this increases A∗i .Since ϕki (Ai , τki ) is increasing in Ai and τki does not vary for any k, ϕ∗ki increases by Lemma A.1 for any k ∈ C. Besides,
∗ Mki decreases for any k 6= i since MkE∗ does not vary and ϕ∗ki increases.
Regarding prices of active firms, when
∂εi (·,Ai ) ∂Ai
> 0, then pki (Ai ; ϕ) and mki (Ai ; ϕ) de-
crease by Lemma A.2 for any k ∈ C.
Moreover, regarding ϕij (Aj , τij ), A∗j does not vary and ϕ∗ij is increasing in τij , so ϕ∗ij de-
creases by Lemma A.1. Moreover, by Lemma A.3, pij (Aj , τij , ϕ) decreases for each ϕ ≥ ϕ∗ij
∂εj (pij ,·) > 0. ∂pij A∗i A∗i , MiE∗ since
since A∗j does not vary and τij decreases, and mij (Aj , τij ; ϕ) decreases too since Concerning MiE∗ , condition (ME) in i can be expressed by A∗i =
MkE∗ is the same for each k 6= i and τki does not vary for any k. When Assumption STB holds, A∗i − A∗i A∗i , MiE∗ is increasing in A∗i and, by Lemma A.4, it implies that MiE has to increase. Thus, the mass of active firms from i selling in j increases too. Proof of Propositon 5.1. iv
A. Proofs
Mart´ın Alfaro
I first show that A∗i is increasing in τji , and A∗j is decreasing in τji . The system of equations that describe the zero expected profits is: Z
ϕi
[πii (Ai ; ϕ) − fii ] dGi (ϕ) +
ϕii (Ai )
Z
ϕj
ϕjj (Aj )
[πjj (Aj ; ϕ) − fjj ] dGj (ϕ) +
Differentiating the system, ∂e πii (A∗i ) ∂e πij (A∗j ) ∂Ai ∂Aj ∂e πji (A∗i ;τji ) ∂e πjj (A∗j ) ∂Ai
∂Aj dA∗i dτji
From this, we get
dA∗i dτji dA∗j dτji
∂π eji A∗ i ;τji ∂Ai
)
ϕi
[πij (Aj ; ϕ) − fij ] dGi (ϕ) = Fi
ϕij (Aj )
ϕj
ϕji (Aj ,τji )
[πji (Ai ; τji , ϕ) − fji ] dGj (ϕ) = Fj
0 . ∂e πji (A∗i ;τji )
=
(
=
Z
Z
− ∂τji ( ) dA∗ and dτjij =
∂π eij A∗ j ∂Aj
|JF E |
−
∂π eji A∗ i ;τji ∂Ai
(
) ∂ πeii (A∗i ) ∂Ai
|JF E |
. By Lemma A.1,
π eji (Ai , τji ) is decreasing in τji and π ekl (Al , ·) is decreasing in Al for k, l ∈ {i, j}. Hence, since |JF E | > 0, then
dA∗i dτji
> 0 and
dA∗j dτji
< 0.
Regarding the effects on country i. For domestic firms, the results are determined
through the decrease of A∗i . By Lemma A.2, pii (A∗i ) and mii (A∗i ) increase since
∂εi (·,Ai ) ∂Ai
>
0 and by Lemma A.1, ϕ∗ii decreases. Regarding exporters from j selling in i, they are affected by both the decrease of A∗i and τji . By Lemmas A.2 and A.3, mji (A∗i ) increases if > 0 and
∂εi (·,Ai ) ∂Ai
∂εi (p,·) ∂p
> 0. Moreover, by Lemma A.1, ϕ∗ji decreases.
Concerning the masses of incumbents, given values of A∗i and A∗j , they are obtained through the system of equations, A∗i = A∗i MiE∗ , MjE∗ ; A∗i , τji A∗j = A∗j MiE∗ , MjE∗ ; A∗j
By the zero expected profits, A∗i = Ai (τji ) and A∗j = Aj (τji ) . Differentiating the system taking account into this, ∗ ∗
∂Ai ∂MiE ∂A∗j
∂Ai ∂MjE ∂A∗j
∂MiE
∂MjE
dMiE∗ dτji dMjE∗ dτji
=
dA∗i dτji
∂A∗i ∂A∗i
1− 1−
dA∗j dτji
∂A∗i ∂τji
−
∂A∗j ∂A∗j
Let JM S be the Jacobian matrix and assume |JM S | > 0. Let ∆1 := dA∗j ∂A∗j and ∆2 := dτji 1 − ∂A∗ .
dA∗i dτji
1−
∂A∗i ∂A∗i
−
∂A∗i ∂τji
j
Notice that ∆1 > 0 and ∆2 < 0. Moreover all the entries in JM S are positive. Hence, dMiE∗ dτji dMjE∗ dτji
∆1
= =
∂A∗ j ∂M E j
−∆2
|JM S | ∂A∗ i ∆2 ∂M E i
∂A∗ i ∂M E j
∂A∗ j −∆1 ∂M E i
|JM S |
> 0, and < 0. v
A. Proofs
Mart´ın Alfaro
Derivation of channels in Section 5. Using the zero expected profits equation for country i and j, we get Ai (Aj ; τji ) and Aj (Ai ; τji ) , respectively. An equilibrium is a pair A∗i , A∗j such that A∗i = Ai A∗j ; τji and A∗j = Aj (A∗i ; τji ) . Differentiating Ai (Aj ; τji ) and Aj (Ai ; τji ) , and evaluating them at the equilibrium, we get ∂Ai A∗j ; τji ∂Ai A∗j ; τji dA∗j dA∗i = + dτji ∂τji ∂Aj dτji ∗ ∗ ∗ dAj ∂Aj (Ai ; τji ) ∂Aj (Ai ; τji ) dA∗i = + . dτji ∂τji ∂Ai dτji Working out the expressions, we get the equations stated in the main text: ∂Ai A∗j ; τji ∂Ai A∗j ; τji ∂Aj (A∗i ; τji ) dA∗i = κ+ κ dτji ∂τji ∂Aj ∂τji dA∗j ∂Aj (A∗i ; τji ) ∂Aj (A∗i ; τji ) ∂Ai A∗j ; τji = κ+ κ dτji ∂τji ∂Ai ∂τji with κ := 1 −
∂Ai (A∗j ;τji ) ∂Aj (A∗i ;τji ) ∂Aj ∂Ai
−1
.
∂Ai (A∗j ;τji ) By Lemma A.1, π ekl (Al , ·) is decreasing in Al for k, l ∈ {i, j}. Hence, < 0 and ∂Aj ∗ ∗ ∗ ∂Aj (Ai ;τji ) ∂Ai (Aj ;τji ) ∂Aj (Ai ;τji ) < 0. Since |JF E | > 0, then < 1. Furthermore, since neither ∂Ai ∂Aj ∂Ai ∗ ∂Ai (Aj ;τji ) π eii or π eij depends on τji directly, then = 0. Finally, by Lemma A.1, π eji (Ai , τji ) ∂τji
is decreasing in τji and π ekl (Al , ·) is decreasing in Al for k, l ∈ {i, j}. This determines that ∂Aj (A∗i ;τji ) < 0. ∂τji Proof of 6.1. If the aggregate demand Q (ω) := Q [θ, A, p (ω)] is weakly separable in h i e e := Ae (A, θ). Then A e is (A, θ) from p (ω) , then it can be represented by Q A, p (ω) where A a new aggregate for the demand and θ represents a shock to it.
Proof of 6.2. By Lemma A.1, expected profits are decreasing in A. Moreover, optimal profits of an active firm are increasing in L. This also implies that ϕ (A, L) is decreasing in L. Thus, the free-entry condition determines that A∗ is increasing in L. By Lemma A.2, when
∂ε(·,A) ∂A
> 0, then p (A∗ ; ϕ) and m (A∗ ; ϕ) decrease due to the re-
duction of A∗ . Regarding ϕ∗ , there are two countervailing effects: ϕ∗ decreases due to the increase of L but it is greater since A∗ increases. I show that when
∂ε(·,A) ∂A
> 0, ϕ∗ increases.
Using the zero-profits equation and the free-entry condition, we get
vi
dϕ∗ d ln L
=
∂π (A∗ ,ϕ∗ ) dA∗ ∂A d ln L ∂π(A∗ ,ϕ∗ ) − ∂ϕ
π(A∗ ,ϕ∗ )+
A. Proofs
Mart´ın Alfaro Z
ϕ π(A∗ ,ϕ)dG(ϕ)
and
dA∗ d ln L
= (−1)
∗ Z ϕϕ
ϕ∗
. Define the numerator of
by κ. Then κ := π (A∗ , ϕ∗ ) +
∂π(A∗ ,ϕ) dG(ϕ) ∂A
∂π(A∗ ,ϕ∗ ) dA ∂A d ln L
dϕ∗ d ln L
Z
ϕ
π (A∗ ,ϕ) dG(ϕ) π(A∗ ,ϕ∗ )
∗ ⇒ κ := π (A , ϕ ) 1 − Z ϕϕ ∂π(A∗ ,ϕ) ∂A ∗
∗
ϕ∗
∂π(A∗ ,ϕ∗ ) ∂A
dG(ϕ)
.
Hence, the sign depends on the term in brackets. Suppose that it is positive which ∗
< 0. Working out the term in brackets and using that, for any function f, implies that ddϕ h ln L i b ∂f (x) = exp fb(x) ∂ f∂x(x) , then ∂x Z ϕ i h ∗ ,ϕ) ∂b π (A∗ ,ϕ∗ ) − dG (ϕ) < 0 exp [b π (A∗ , ϕ) − π b (A∗ , ϕ∗ )] ∂bπ(A ∂A ∂A ϕ∗
∂b π (A∗ ,ϕ) ∂b π (A∗ ,ϕ∗ ) − < 0. By the fundamental theorem of the ∂A Z∂Aϕ ∗ ,ϕ) ∗ ∗ ∂b π (A∗ ,t) calculus, ∂bπ(A − ∂bπ(A∂A,ϕ ) = dG (t) . By Lemma A.2, when ∂ε(·,A) > 0, π b (A, ϕ) ∂A ∂A∂t ∂A ϕ∗ (A∗ ,ϕ) is supermodular in (A, ϕ) , so for all (A, ϕ) , ∂bπ∂A∂ϕ > 0 which is a contradiction.
This is negative only if
Proof of Proposition 7.1. For the proof, I proceed in several steps. First, I show that the aggregate is a sufficient statistic for optimal choices and profits. This is trivial for the case of negligible firms since they take the aggregate as a parameter. Thus, consider a nonzero-measure firm. Since the demand is assumed symmetric across firms, all firms with the same productivity behave in a similar fashion. Hence, the optimization problem of a firm with productivity ϕ is max π (x,A; ϕ) . The vector of first-order conditions is ∇x π (xω ,A; ϕ) =
∂π(xω ,A;ϕ) ∂xω
Since the aggregator
x ∂π(xω ,A;ϕ) ∂H(X) ∂X [(xω )ω∈Ω ] + = 0. ∂A ∂X ∂xω ∂X [(xω )ω∈Ω ] is strongly separable, ∂xω
=
∂hω (xω ) . ∂xω
Thus, it is only a
function of its own vector of choices. Moreover, since H is monotone, we can express X =H −1 (A) . All this determines that optimal choices and profits can be expressed as x (A; ϕ) and π (A; ϕ), and the result follows. Suppose there is a shock α to the aggregator such that R P X (xω )ω∈Ω , α := ω∈ΩL hω (xω ) + ω∈ΩE hE (xω ) dλ (ω) + α. I want to show that the
aggregate is not affected by it. If ΩE consists of firms as characterized in Melitz, the result follows as a corollary of Proposition 3.1. By the same proposition, the active firms’choices as well as the productivity cutoff do not vary. Only the mass of incumbents decreases. Consider ΩE comprised of symmetric non-zero measure firms. Then, A is nonstochastic and the zero-profits condition pin down its value independently of its composition: π (A; ϕ) = F . Thus, given A∗ , the shock α does not affect A∗ , implying that the choices of active firms remain the same. Moreover, since marginal entrants are symmetric and we vii
A. Proofs
Mart´ın Alfaro
are assuming that entry and exit takes place through that set, the productivity cutoff does not vary. Ignoring the integer constraint, the equilibrium is restored by a decrease in the number of marginal entrants. Proof of 7.2. Consider a variation dτji for j ∈ C B \ {i} . Assume that the cardinality of ∂A∗ (Ai ;ME ,(τji ,fji )j∈C ) ∂A∗i (Ai ,(ϕji )j∈C ;(τji )j∈C ) C B \ {i} is N. Notice that if for any Ai , i < 1.then < ∂Ai ∂Ai 1. In form, a matrix ∂A∗i (·) ∂A∗ (·) ∂A∗ (·) ∂A∗ (·) ∂A∗ (·) 1 − ∂Ai − ∂ϕi ii − ∂ϕi1i − ∂ϕi2i · · · − ∂ϕi∗ dA∗i Ni dϕ∗ − ∂πii (·) − ∂πii (·) 0 0 ··· 0 ii ∂Ai ∂ϕii ∂π (·) ∂π (·) dϕ∗ − 1i 0 − ∂ϕ1i1i 0 ··· 0 1i ∂Ai ∂π (·) ∂π (·) dϕ∗2i − 2i 0 0 − ∂ϕ2i2i · · · 0 ∂Ai .. .. .. .. .. .. .. . . . . . . . ∂πN i (·) ∂πN i (·) dϕ∗N i 0 0 0 · · · − ∂ϕN i − ∂Ai Let’s the and partition the denote as J · dx system: = dτ P system ∂A∗i (·) ∂A∗i (·) 1 − ∂Ai B dτ dAi = j6=i ∂τji ji dx−1 dτ −1 C D with ∂A∗ (·) A := 1 − ∂Ai i > 0 ∂A∗ (·) BT := − ∂ϕiki >0 k∈{i}∪C B ki (·) >0 C := − ∂π∂A i k∈{i}∪C B D := diag − ∂π∂ϕkiki(·) <0 k∈{i}∪C B P ∂A∗i (·) dτ 1 := j6=i ∂τji dτji < 0 T ∂πki (·) (dτ −1 ) := ∂τki <0 dx−1 :=
P ∂A∗i (·) j6=i ∂τji dτji 0 ∂π 1i (·) ∂τ1i = ∂π2i (·) ∂τ2i .. . ∂πN i (·) ∂τN i
k∈{i}∪C B ∗ (dϕki )k∈{i}∪C B
Working out the expressions, it can be shown that −1 A − BD−1 C dA i = dτ 1 − BD ·dτ−1 −1 −1 P ∂A∗i (·) ∂πki (·) ∂πki (·) −1 −1 Also, D = diag − ∂ϕki , BD C = k∈{i}∪C B (−1) ∂ϕki ∂ϕki k∈{i}∪C B −1 P ∂A∗ (·) ∂πki (·) BD−1 ·dτ−1 = k∈{i}∪C B ∂ϕiki ∂π∂ϕkiki(·) dµi (k) > 0. ∂τki
∂πki (·) . ∂Ai
Using these expressions, now we account for variations of trade costs in small econo-
mies too. Formally, consider a variation dτji for j ∈ C\ {i} , −1 R ∂A∗ (·) ∂πki (·) Then, BD−1 C = − k∈C ∂ϕiki ∂π∂ϕkiki(·) dµi (k) < 0 ∂Ai −1 R ∂A∗ (·) ∂πki (·) dµi (k) > 0 BD−1 ·dτ−1 = k∈C ∂ϕiki ∂π∂ϕkiki(·) ∂τki ∗ ∂A (·) Moreover, since 1 − ∂Ai i > 0, hence A − BD−1 C > 0. viii
A. Proofs
Mart´ın Alfaro
Regarding dτ 1 − BD−1 ·dτ−1 , it can be shown that dτ 1 − BD−1 ·dτ−1 < 0 Thus, if dτji < 0 for j ∈ C\ {i} then A∗i increases.
By πii (A∗i ; ϕ∗ii ) = fii and using Lemma A.1, ϕ∗ii increases, which implies that the mass f decreases. of active firms in the domestic market Mii := [1 − Gi (ϕ∗ )] M ii
Finally, by Lemma A.2 and
(·,Ai ) since ∂εi∂A i
> 0, the prices, markups and quantities of
active domestic-firms decrease. Proof of Proposition 8.1. The optimization problem of each active firm determines the following vector of optimal choices:
x (A, ϕ) if ϕ ≥ ϕ∗ ∗ ∗ x (A, ϕ , ϕ) := x0 otherwise
where ϕ∗ represents the productivity-cutoff to serve the market. Conditional on entry, the optimal gross-profits of an active ϕ-type firm is a function π (A, ϕ). From this, the zero-profits condition determines ϕ∗ : (ZCP-CH)
π (A, ϕ∗ ) = f where ϕ∗ := ϕ (A) . Optimal profits are: π ∗ (A, ϕ∗ , ϕ) := 1(ϕ≥ϕ∗ ) [π (A, ϕ) − f ] .
As in the baseline case of price decisions, the equilibrium at the market stage can be determined by the following equation A∗ = A∗ A∗ , M E∗ ; α
(NE-CH)
where α is a shock to the aggregator and A∗ (·) := A [X ∗ (·) , M ∗ (·)] with (·) := A∗ , M E∗ . Finally, the free-entry condition can be expressed as, π e (A) = F where π e (A) :=
Z
ϕ
ϕ(Aj )
[π (Aj ; ϕ) − f ] dG (ϕ).
Then, the proof follows in the same fashion as the baseline case.
ix
(FE-CH)
B. Demand Systems: Examples
B
Mart´ın Alfaro
Demand Systems: Examples
I provide some examples of standard demand consistent with DEM. I restrict to derivations that Although some of these systems have particular cases which overlap with other families considered, it is illustrative to show how they can be explicitly rewritten in terms of an aggregator. Consider the following demand per capita qω of a variety ω given a total measure of varieties M and price pω . Also, E is an exogenous demand shifter (e.g. income). Any greek letter refers to a positive parameter. • Z Krugman’s (1979) demands from an additive direct utility Given utility U (qω0 )ω0 ∈Ω := u (qω0 ) dω 0 with u monotone. Let g := (u0 )−1 . Then, qω := g pAω where A is the inω0
verse of the marginal utility of income. This demand includes as special cases: – Behrens and Murata’s (2007) demands derived from an Zexponential utility qω := A− α1 ln pω where A := PE1 + lnαP1 + Pa2 with P1 := pω0 dω 0 and P2 := 0 ω Z pω 0 pω 0 0 ln P1 P1 dω . ω0
– Generalized CES (Arkolakis, Costinot, Donaldson, and Rodr´ıguez-Clare, 2015; Jung, Simonovska, and Weinberger, 2015): qω := Ap−σ ω − α with A := Z Z P1 := pω0 dω 0 and P2 := (pω0 )1−σ dω 0 ω0
ω0
A pω
−α
α+ηP ηM +γ
and
– Simonovska’s (2015) Stone-Geary (Generalized CES with σ → 1): qω := Z E+αP with A := M and P := pω0 dω 0 . ω0
• Melitz and Ottaviano’s (2008) linear demand qω := A − Z 1 P := γ pω0 dω 0 . ω0
• Feenstra’s (2003) translog demand qω := Z ln pω0 dω 0 .
E pω
pω γ
with A :=
[A − ln (pω )] where A :=
ω0
• Luce’s (1959) attraction demand models: qω :=
hω (pω ) A
with A := H
– Multinomial Logit demand: hω (pω ) := exp (α − βpω )
1+γP M
E+αP1 , P2
and P :=
Z hω0 (pω0 ) dω
0
.
ω0
– Multiplicative Competitive Interaction demand: hω (pω ) := α (pω )−β . • Nocke and Schutz’s (2015) demands with discrete-continuous choices: qω := Z with A := H hω0 (pω0 ) dω 0 . It includes the Logit as special case.
∂hω (pω )/∂pω A
ω0
• Constant expenditure demand systems (Vives, 2001): qω := Z 0 H hω0 (pω0 ) dω . It includes: ω0
x
E h(pω ) pω A
with A :=
C. Differential Characterization of Demand Systems
Mart´ın Alfaro
– CES: h (pω ) := α (pω )−β . – Exponential demand: by defining h (pω ) := exp (α − βpω ).
• Bertoletti and Etro’s (2015) demands from an additive indirect utility Given an Z p vω0 Eω0 dω 0 which determines demands, qω := indirect utility V (pω0 )ω0 ∈Ω , E := ω0 Z pω 0 vω ( ) p p 0 E vω0 0 Eω0 Eω0 dω 0 . with A := A ω0
In Section 8, it is mentioned that the setup covers the case of nested CES and Nested Logit • Nested CES
ε−σ −ε q l (ω) := αPσ Pl pω Z 1−σ RL l 1−ε dl. (pω0 )1−ε dω and P1−σ := 1 Pl where α is a demand shifter, P := ω 0 ∈Ωl
• Nested Logit
p Pl λl −1 ω q (ω) := α exp − l λ P Z λl RL p exp − λωl0 dω 0 and P := 1 Pl with α a demand shifter, Pl := dl. l
ω 0 ∈Ωl
C
Differential Characterization of Demand Systems
Checking whether a demand system satisfies Definition DEM requires verifying that there exists an aggregate A such that the demand can be expressed as qω := max {0, q (A, pω )}. K Mathematically, this is equivalent to ask for weak separability of Pk k=1 , M from pω in qω . Using results from the separability literature, there is a characterization of weak separability that allows us to identify if a demand is consistent with Definition DEM without constructing the aggregate A. I start by stating the differential characterization of weak separability. Lemma C.1. (Leontief, 1947; Sono, 1961). Let f : X → R with X ⊆ RN + with N < ∞. 1 2 r Consider a partition of the N variables into R groups I , I , ..., I R so that X := ×R r=1 X .
Denote generic elements by x ∈ X and xr ∈ X r . We say that each group r = 1, ...R is weakly
separable from all other variables in f if there exist real-valued functions H and (hr )R r=1 such that 1 1 f (x) = H h (x ) , ..., hR xR . If f ∈ C 1 , group r is weakly separable from the rest of the variables in f if and only if the marginal rate of substitution between any two variables belonging to the group r are independent of any variable which does not belong to r. Formally, ∂
∂f (x)/∂xi0 ∂f (x)/∂xi00
∂xj
= 0 for i0 , i00 ∈ r and j ∈ / r. xi
C. Differential Characterization of Demand Systems
Mart´ın Alfaro
Applying this Lemma to the demand function, the following corollary can be applied to identify whether a particular demand satisfies Definition DEM. Corollary C.2. pω is weakly separable from Pk
K
conditions hold simultaneously: 0 ∂qω /∂Pk 00 ∂qω /∂Pk
∂
•
∂pω ∂
•
∂qω /∂Pk ∂qω /∂M
∂pω
= 0 for all k 0 , k 00 = 1, ..., K.
= 0 for all k = 1, ..., K.
xii
k=1
and M in qω if and only the following two
