Large firms and heterogeneity: the structure of trade and industry under oligopoly

Transcription

1 Large firms and heterogeneity: the structure of trade and industry under oligooly Eddy Bekkers University of Linz Joseh Francois University of Linz & CEPR (London) ABSTRACT: We develo a model of trade with endogeneity in key features of industry structure linked to heterogeneous cost structures under Cournot cometition. As such we address firm heterogeneity in international trade in a setting with strategic interaction between firms as oosed to existing models where firms are atomistic. Entry/exit dynamics are therefore more comlex but also richer. The equilibrium value of the main variables like market rice and total sales wander around as firms enter and exit. The model nests two workhorse trade models, the Brander & Krugman recirocal duming model and the Ricardian technology-based trade model, as secial cases. We examine both free entry and limited entry (free exit) cases. The model generates testable redictions on the robability of zero trade flows and the attern of exort rices and generates endogenous moves from zero to non-zero trade flows due to entry and exit. As two countries move from autarky to free trade, market rices fall, but along the ath from autarky to free trade, market rices might increase at certain oints. Also with lower trade costs the least roductive firms get squeezed out of the market, exorting firms gain market share, and more firms become trade oriented. Keywords: Firm heterogeneity, Oligooly, Comosition effects of trade liberalization JEL codes: L11, L13, F12 rintdate: Setember 17, 2009 Thanks are due to articiants in Tinbergen Institute and CEPR workshos and the ETSG annual meetings and to anonymous referees. Address for corresondence: Univ. Prof. Dr. J. Francois, Johannes Keler University Linz, Deartment of Economics, Altenbergerstraße 69, A Linz, AUSTRIA. joseh.francois@jku.at, web:

2 Heterogeneous Firms, the Structure of Industry, & Trade under Oligooly ABSTRACT: We develo a model of trade with endogeneity in key features of industry structure linked to heterogeneous cost structures under Cournot cometition. As such we address firm heterogeneity in international trade in a setting with strategic interaction between firms as oosed to existing models where firms are atomistic. Entry/exit dynamics are therefore more comlex but also richer. The equilibrium value of the main variables like market rice and total sales wander around as firms enter and exit. The model nests two workhorse trade models, the Brander & Krugman recirocal duming model and the Ricardian technology-based trade model, as secial cases. We examine both free entry and limited entry (free exit) cases. The model generates testable redictions on the robability of zero trade flows and the attern of exort rices and generates endogenous moves from zero to non-zero trade flows due to entry and exit. As two countries move from autarky to free trade, market rices fall, but along the ath from autarky to free trade, market rices might increase at certain oints. Also with lower trade costs the least roductive firms get squeezed out of the market, exorting firms gain market share, and more firms become trade oriented. keywords: Firm heterogeneity, Oligooly, Comosition effects of trade liberalization JEL codes: L11, L13, F12. 1 Introduction Research on the imact of globalization on firms has shown that the reallocation effects of trade are an imortant mechanism linking oenness to roductivity. Bernard and Jensen (2004) find that almost half of the rise of manufacturing total factor roductivity in the USA between 1983 and 1992 is linked to a reallocation effect of resources towards more roductive and trade oriented firms. Eisodes of liberalization in develoing countries also show the imortance of changes in firm comosition comosition effects (Tybout 2001). A number of models of heterogeneous roductivity have been ut forward in the recent theoretical trade literature to exlain comosition effects of trade. A standard result is that rocesses of firm formation involving heterogeneous cost structures across firms imly beneficial reallocation effects from trade liberalization linked to rationalization of the oulation of firms. Less efficient firms roducing for the domestic market are squeezed out by more efficient trading firms. For examle, Melitz (2003) introduces heterogeneous roductivity in a monoolistic cometition framework with CES-references, while Bernard, et al (2003) include heterogeneous roductivity in a model with Bertrand cometition. Working with monoolistic cometition models, Baldwin and Robert-Nicoud (2005) examine linkages between trade and growth given firm heterogeneity, while Ghironi and Melitz (2004) examine macroeconomic dynamics. 1

3 An imortant feature of the monoolistic cometition models of Melitz (2003) and Melitz and Ottaviano (2008) is that there is no strategic interaction between firms. Firms are atomistic in the market, which enables the definition of a smooth entry and exit rocess. When firms exit due to an exogenous shock and are relaced by new entering firms, nothing changes in the market equilibrium. In the real world, trade seems to be dominated by a small number of very large firms that interact strategically. Also, the relacement of dying firms by new ones is likely to affect the market equilibrium. To account for this fact, we exlore heterogeneous roductivity in a model with oligooly characterized by Cournot cometition. Basic asects of market structure markus, industrial concentration, relative firm ositions, and rices for domestic and exort markets are endogenous. They deend on the interaction between the technology set, market size, and trade oenness. We do not assume a secific distribution of initial roductivities, as is the case with Bernard, et al (2003) and Melitz and Ottaviano (2008), to generate our results. Preferences are assumed to be CES across different sectors. For entry and exit dynamics we exose the model in two different ways. We start with firms drawing a new roductivity arameter each eriod uon aying a sunk entry cost. This can be interreted as firms making annual roduction lans. Later on we model entry/exit like in Melitz (2003) with firms having the same roductivity over their whole life time and exiting with a fixed death robability δ. We show that all the results for the easier entry/exit rocess carry through for the more comlicated entry/exit rocess. As mentioned, in Melitz (2003) and Melitz and Ottaviano (2008) this leads to a steady state of entry and exit where secific draws of entering firms or secific firms dying does not affect the equilibrium value of variables like market rice and cutoff cost level, because firms are atomistic. In our model this is different: the market rice wanders around from eriod to eriod. The model we develo is a two-country, multi-sector model of trade under Cournot cometition. 1 The value added of this aroach is threefold. First, the model accounts for strategic interaction between firms in a firm heterogeneity setting with endogenous dynamics in equilibrium values due to entry and exit, while generating a rich set of results linked to comosition effects. In articular, we find that a larger market size generates a lower market rice by inducing more entry. Interestingly, as two countries integrate their markets and move from autarky to free trade, the market rice does not go down monotonically and can jum u at certain levels 1 Van Long et al. (2007) also address firm heterogeneity in an oligooly model. Their aer is focused on a different set of issues however, the interaction of trade and R&D. 2

4 of trade costs. This result is due to ossible exit induced by the fact that lower trade costs drive down rofits for domestic sales, an effect that ossibly outweighs the increased rofits from exorting sales. Another imortant result is that we can generate endogenous variation in the robability of zero trade flows and the fob exort rice due to entry and exit of firms and the variation in roductivities drawn. Second, the model nests two workhorse trade models, the Brander and Krugman (1983) recirocal duming model and the Ricardian model, as secial cases. Third, the model generates testable redictions on the robability of zero trade flows and exort rices. The attern of zeros and unit values has emerged as a articularly imortant issue in the recent emirical trade literature. (See Baldwin and Harrigan 2007, Baldwin and Taglioni 2006). In the model develoed here, the effect of a larger distance between countries on the robability of zero trade flows and fob exort rices is ambiguous. When the number of entrants does not change, the robability of zero trade flows declines with a lower distance, when the number of entrants induced by a smaller distance declines, the robability of zero trade flows rises and when the number of entrants due to a msaller distance rises, the effect is ambiguous. A larger market size of the imorter country increases the robability of zero trade flows and increases the fob exort rice. The model in this aer is related to two strands of literature in international trade. The first one is the firm heterogeneity literature, which is already discussed above. The aer differentiates itself from that literature by focusing on oligooly and thereby allowing for strategic interaction, which generates entry and exit dynamics not resent in the existing models. The second related strand of literature is the one on oligooly in international trade. This literature can be characterized by the following three dimensions. First, whether firms are homogeneous or heteregeneous, second whether there is free entry or not and third whether the model is embedded in a general equilibrium framework. Brander and Krugman (1983) exlore a model with homogeneous firms in a artial equilibrium setting both with and without free entry. The main innovation of their aer is to show that recirocal duming can emerge in an oligoolistic market. Van Long and Soubeyran (1997) study a model with heterogeneous firms in a artial equilibrium setting without free entry. Their main innovation is to show that the Herfindahl index of concentration is related to the variance of marginal costs of the different firms with imlications for otimal strategic trade olicy. A third aer in the literature on trade and oligooly is Neary 3

5 (2009), featuring homogenous firms in a general equilibrium setting without free entry imosed. His main innovation is to show how oligooly can be modeled in a general equilibrium setting by assuming a continuum of sectors. The current aer is different from the other aers in the oligooly and trade literature, as it combines firm heterogeneity with free entry in a general equilibrium setting. This allows the study of reallocation effects of trade in a market setting with strategic interaction. The aer is organized as follows. In Section 2 we lay out the basic model. Section 3 introduces international trade and contains four subsections. The first subsection abstracts from free entry, the second subsection adds a free entry condition, the third subsection addresses the effect of distance and market size on zeros and fob rices and the fourth subsection oints out how the Brander&Krugman model and the Ricardian model can be seen as secial cases of our model. Section 4 concludes. 2 The Basic Model This section lays out the basics of the model without trade (or identically for an integrated or single global economy without trade costs). Industrial concentration emerges endogenously as a function of the degree of firm heterogeneity and market size, while the relationshi of concentration to rice deends on the cost structure. We start by assuming that there are Q + 1 sectors in the economy, Q oligoolistic sectors roducing q j and 1 sector roducing z under conditions of erfect cometition. In the first sections it is assumed that the Cournot sectors are symmetric. Later on this assumtion is relaxed when asymmetries in national technology sets, country size, and olicy are exlored. Throughout it is assumed that there are sufficient sectors in the economy so that the effect of a rice change on demand through the rice index is negligible for firms. (There is no numeraire roblem). There are L equal agents each sulying 1 unit of labor. All rofit income from the Cournot sectors goes to the economic agents. The utility function of each agent is CES. The otimization roblem of the consumer generates the following market demand functions in the Cournot sectors q j and in the erfect cometition sector z: q j = IP σ 1 U σ j (1) z = IP σ 1 U (2) 4

6 The rice of good z is normalized at 1 and I is the endogenous income of all agents, the sum of labor and rofit income. P U is the consumer rice index, corresonding to one unit of utility: Q P U = j=1 j σ 1 1 σ = [ Q 1 σ + 1 ] 1 1 σ (3) Until we relax out symmetry assumtions, we will focus on one reresentative Cournot sector. (We will warn the reader when we dro these assumtions.) This means we can dro the sector index j for now. Labor is the only factor of roduction and there is a labor force of size L. One unit of labor is needed to roduce one unit of the erfect cometition good y. This means the wage is equal to 1. In the q sectors roductivity is heterogeneous. One unit of labor can be transformed into 1/c i units of q for the i-th firm which has marginal cost of roduction c i. There are no fixed costs of roduction. Therefore the cost function of firm i is given by C i (q i ) = c i q i (4) There is Cournot cometition between the different firms in the q-sectors. So, firms maximize rofits towards quantity sulied, taking the quantity sulied by other firms as given. Profit of firm i is given by: π i = q i c i q i (5) The first order condition is defined as: With q = n [ π i = 1 1 q i σ ] q i c i = 0 (6) q q i. n is the number of firms in the market. Using the first order condition, the second order condition can be written as follows (derivation in aendix A): 1 σ q [ ] (σ + 1) ci (σ 1) < 0 (7) Using the definition for market share, θ i = q i q, the first order condition can be rewritten as: ( 1 θ ) i = c i (8) σ 5

7 θ i = σ c i (9) The marginal revenues on the LHS of equation (9) should be at least as large as the marginal costs on the RHS. The larger is market share θ i, the lower is marginal revenue. So, for ositive sales (θ i 0) which are imlicitly imosed, a firm can satisfy the FOC by just reducing its market share as long as its marginal cost is smaller than the market rice. The cutoff cost level c is defined as the cost level with which a firm would just stay in the market. This cutoff cost level c is equal to the market rice. The highest cost firm staying in the market has a cost level equal or just below the cutoff cost level and selling an amount just above zero. In the actual market equilibrium, the highest cost firm that can stay in the market can have a cost lower than c. This deends uon the secific samle of firms drawn from an initial cost distribution at a secific oint in time. Hence, the market rice can vary over time with firms entering and exiting. Still, the exression for the market rice and total sales q can be determined as a function of average costs c and the number of firms n at an arbitrary oint in time. And also the free entry condition can be determined, although it deends uon variables that vary over time. First, let us focus on the equilibrium exressions without imosing a free entry condition. Aendix A shows that the reaction functions are stable when σ > 3/2. Although it is ossible that a very efficient firm dominating the market with a very large market share has a reaction function with ositive sloe, the condition σ > 3/2 guarantees that the equilibrium is stable, see for discussion the aendix. The equilibrium rice and quantities sold can be found for a given number of firms. Suose for now there are n firms, endogenizing this later on with a free entry condition. Combining the demand equation in (1) with n first order conditions in equation (6) and with the equation for the sum of market shares, one can find the following solutions for the market rice, total sector sales q and sales of an individual firm q i : = σ σn 1 n c i = σn c (10) σn 1 q = IP σ 1 U c σ ( ) σn 1 σ (11) σn q i = σip σ 1 U σn σn 1 c c i c σ+1 ( ) σn 1 σ+1 (12) σn 6

8 with c the unweighted average cost of firms, c = 1 n n c i. Using the fact that the rice is equal to the cutoff cost level, the rice equation (10) can be rewritten to solve for the number of firms as a function of the cutoff cost level and average cost: n = 1 σ c c c = 1 m σ m 1 (13) With m an average marku measure defined as rice divided by unweighted average cost, m = /c. On the basis of equation (13) we make the following Proosition. Proosition 1 The market structures and average markus of industries are related to the degree of heterogeneity. In articular, the less the degree of cost heterogeneity, the more cometitive the structure of the industry and the smaller the average marku. Equation (13) shows that an increase in the number of firms imlies that the firm with the highest cost needs to have a cost arameter ever closer to average cost. Therefore, the cost levels of firms become ever closer to each other with more firms in the market. Proosition 1 highlights how the market structure and the cost structure in the model are interrelated. When we observe more cometitive industries with more firms in equilibrium, this means the cost levels of firms should be closer to each other. 2 Next, we add free entry to the model. This will endogenise the number of entrants n e. Firms make roduction lans for each eriod. Basically, we interret the entry and exit mechanism for firms each eriod as reflecting a mechanism where firms make new roduction lans each year. This interretation imlies that firms also have to ay sunk entry costs and draw a new cost arameter each eriod. Oerationally, we assume that firms ay a sunk entry cost f e each eriod to draw a cost arameter c randomly from a set of initial costs C with a certain discrete distribution of costs F (c) with lower bound c and uer bound c. Hence, uncertainty about roductivity is a barrier to entry for firms. They start to roduce when they can make ositive oerating rofits. Production lans are good for only one eriod. Every eriod all roducing firms have to ay again sunk entry costs to draw again from the cost distribution for lans for the next eriod. We start with a model where firms have to draw a new roductivity arameter each eriod 2 Working with a more restricted model without entry, Van Long and Soubeyran (1997) find similar results. They show that the variance of the cost distribution and the Herfindahl index of industry concentration are ositively related in a model with Cournot cometition: a larger variance leads to more industry concentration. From equation (13) above, it is clear that this result is more general, and holds with entry and exit as well. 7

9 to kee the exosition of the model straightforward and simle. After this exosition we will set u the model where firms live for more than one eriod, until they die according to a fixed death robability, the entry/exit rocess of Melitz (2003), and show that an equilibrium exists. The sunk entry costs use labor. As free entry leads to almost zero exected rofits, all rofit income on average is used to ay labor in the entry sector. 3 Therefore, total income in the economy is fixed and equal to the amount of labor (with wages normalized at 1). The major change in this model relative to models without strategic interaction is that there is no steady state of entry and exit. The market rice will tyically be different in each eriod, deendent uon the cost draws of the entering firms. Still we can characterize the equilibrium based uon a zero cutoff rofit condition (ZCP) and a free entry condition (FE). Also we will be able to determine the sign of the effect of a larger market and (later in the oen economy) of lower trade costs uon the exected market rice. We roceed in two stes. First we show that a unique Nash equilibrium emerges for a given number of entrants n e and a given cost draw for the entering firms. Second, we show that for each initial state in the revious eriod there is a number of entrants n e that leads to satisfaction of the free entry condition. The ZCP follows from the fact that without fixed costs zero rofit imlies that rice should be equal or just above marginal cost for the cutoff firm. The FOC in equation (6) shows that this firm will reduce market share to (just above) zero, to satisfy the first order condition and make non-negative rofit. As the distribution of initial costs is discrete, the cutoff cost level c is equal or smaller than the market rice : c (14) For a secific cost draw of the n e entrants, a unique market rice will emerge as stated in the following roosition: Proosition 2 For a given number of entrants n e and a given set of cost draws {c 1, c 2,..., c n e}, a unique Nash equilibrium of number of roducing firms, n, sales {q 1, q 2,..., q n } and market rice will emerge. The roof of roosition 2 roceeds as follows. We can rank the marginal cost draws from 3 Profit can be slightly ositive, when the free entry condition is not exactly satisfied, as we work with a discrete number of entrants. 8

10 small to large as c 1 c 2... c n e. The number of roducing firms n is then determined by the following conditions: n c n ; n+1 < c n+1 (15) n is the market rice with the n most efficient cost drawing firms roducing. Hence, with n firms the highest cost firm can still roduce rofitably and with n + 1 firms the highest cost firm could not survive. Inequalities (15) imly that n 1 c n 1 and n+2 > c n+2 and hence imly a unique n that satisfies these inequalities, because n decreases in n as we show now. We start from equation (10) and add n firms with average cost c j. This generates a change in rice of: = = σ σ (n + n) 1 ( c i + c j ) σ n σ (n + n) 1 (c j ) < 0 when c j < and = 0 when c j =, i.e. when all entering firms would have a cost equal to the ruling market rice. Entry of firms with a cost higher than the ruling market rice, i.e. c j >, would lead to an increase of the market rice, but is not ossible as these entering firms would not survive, as we have that < c j, hence the new market rice would be below the cost of the entered firms c j. Or in other words, if a firm with marginal cost higher than the ruling market rice comes in, the market rice increases insufficiently for this firm to make ositive rofit and hence it will not enter. Hence, for a given set of cost draws there is a unique market rice, a unique number of roducing firms n and the sales of each firm q i follows from equation (12). To determine the number of entrants n e, we add a free entry condition. We start with a model where firms make annual roduction lans and draw a new marginal cost arameter each eriod, that is indeendent uon cost draws in other eriods. We show now that the FE will lead to a unique number of entrants n e in this case. The FE is given by equalizing the ex ante exected rofits from entry with the sunk entry cost. The FE exression is more comlicated than in models without strategic interaction, like Melitz (2003) and Melitz and Ottaviano (2008). First, as the market rice deends uon the marginal costs drawn, the exected rofit is written as an exected value over the market rice given the cost arameter drawn and the number of entrants, j ( c, n e ) and the cost distribution, f (c). The distribution 9

11 of is conditional uon the number of entering firms n e, as the number of entering firms affects robability distribution of the market rice. Second, the free entry condition will in general not hold exactly and is therefore written in inequality terms: π (n e + 1) f e π (n e ) π is the rofit unconditional uon entry. We can elaborate uon this FE as follows: P P π (, c) I [ c] j ( c, n e + 1) f (c) f e P P π (, c) I [ c] j ( c, n e ) f (c) (16) c C M(c,n e +1) c C M(c,n e ) To generate exected rofit we sum over all ossible cost distributions and conditional uon cost c for one firm and number of entrants n e as well over all ossible rices, exressed by the conditional rice density j ( c, n e ). Note that density of costs f (c) is indeendent of the number of entrants: the robability to draw a certain cost for an entrant is indeendent of the number of other entrants. The set M (c, y) is the set of all ossible market rices when 1 firm has drawn cost c and the number of entrants is y. π (, c) is the rofit of a firm facing market rice and having marginal cost c: π (, c) = σipu σ 1 ( c) 2 σ+1 I [ c] (17) I [ c] is the indicator function for rice larger than marginal cost. It features in the FE, as we sum over all ossible cost draws. Therefore, we have to add that only firms with costs lower than the market rice make ositive rofits. The following roosition claims that the free entry condition generates a unique equilibrium for the number of entering firms n e, i.e. a unique long run equilibrium. Proosition 3 There is a unique number of entering firms n e that leads to satisfaction of the free entry condition in equation (16). The roof of roosition 3 roceeds in three stes. We start by showing that the robability density function J ( c, n e ) first order stochastically dominates (FOSD) J ( c, n e + 1) for any n e and for any c. From the roof of roosition 2 it follows that an increase in the number of roducing firms has a non-increasing effect on the market rice. Using this result we know that the entry of one additional firm leads to an uward shift of the suort of robability function j ( c, n e ). All oints in the suort of the robability function either do not change 10

12 when the additional entering firm draws a cost higher than the ruling market rice without this firm or the oints in the suort shift to the left when the entering firm draws a cost lower than the ruling market rice. In formal terms, for any market rice given n e entrants, we have that the market rice z given n e + 1 is z = x with x having a distribution function with I x (0) = 0. Define the distribution function of the market rice with n e + 1 entrants as J ( c, n e + 1). Then we have that for any non-decreasing function v : R R, v (z) j (z c, n e + 1) = ( v ( x) i (x)) j ( c, n e ) v () j ( c, n e ) Hence, J ( c, n e ) FOSD J ( c, n e + 1) for any level of n e and c. Second, we need to show that rofit, π (, c), rises in. This is done in Aendix A. FOSD of J ( c, n e ) over J ( c, n e + 1) imlies that the exected value of any function rising in taken over the distribution function J ( c, n e ) is larger than when it is taken over J ( c, n e + 1). Third, we note that the density function of costs conditional uon the number of entrants n e is indeendent of the number of entrants n e. This imlies that the exected rofit exression in equation (16) declines in n e and hence there is a unique n e that satisfies the FE. Now we generalize the entry/exit dynamics and assume that firms stay in the market until they leave with a fixed death robability δ. As there are no fixed costs, firms will always stay in the market until they die, either roducing when their marginal costs c is smaller than market rice or not roducing (lingering) in a eriod where < c, waiting for the rice to go u. The imlication is that the market rice is stochastic and varies over time. When firms die this has a non-negligible imact on rofits of the remaining firms. The stochastic market rice imlies that ossible entrants have to form exectations about ossible rofits when they enter conditional uon the state of the economy before entry. This imlies that there will be a FE condition with corresonding number of entrants n e (s) for each initial state s. To exress the free entry condition, we have to define the Markov rocess describing the dynamics of the different variables. A state s t in eriod t is characterized by a tule d (s t ) = { (st ), n (s t ), n L (s t ) }. is the market rice, n the number of roducing firms, n L is the number of lingering firms. Based uon these variables an entering firm has sufficient information to calculate exected rofits and the dynamics are described by a stationary Markov rocess. The state sace is given by the set S. The sequence of events is as follows. A firm considering entrance in eriod t observes s t 1 11

13 and calculates the exected rofit from entry which deends uon the rofits exected in eriod t + 1, t + 2,.... As we will show the exected rofit declines in the number of entrants, hence given initial state s t 1 a certain number of firms ays the sunk entry costs and decides to enter. After entrance firms (both existing firms and just entered ones) die according to a fixed death robability δ. 4 Next, the firms that can make ositive rofits start to roduce, the others linger and wait for later eriods to start to roduce. As firms do not have to incur fixed costs, firms will never exit definitively from the market unless they die, i.e. there is no otimal stoing rule. As a consequence, the absence of fixed costs makes the model substantially less comlicated. An imlication of the discussed sequence of events is that there is a ositive robability that there is no suly in a certain eriod, as all firms could have died. For each initial state we can write a free entry condition that determines the number of entrants for that initial state. As a result, the state variables in eriod t + 1 are a function of the state variables in eriod t and the number of entrants in eriod t + 1 which are solved from the FE conditions. A comlication is that we have to define the state in eriod t + 1 conditional uon one firm having marginal cost c. This is the firm whose exected rofits are under examination in the FE. 5 Defining states in this way allows us to sum the exected rofit of the firm under consideration over all ossible cost draws and all eriods (with for each eriod a fixed death robability). If we would sum first over ossible states unconditional of the cost level and then for each state over ossible cost draws, we do not get the correct exected rofit as firms draw costs only once and forever. 6 Hence, we have transition robabilities Pr ( u t, n e t+1 (u t), c s t 1, n e t (s t 1 ), c ). An equilibrium is defined as a rule characterizing for each initial state s S the number of entrants n e (s) leading to satisfaction of all the free entry conditions. As imlied by the revious aragrah on information we assume that firms do observe the market rice in the revious eriod and the number of roducing and lingering firms, but they 4 This timing seems weird, but is necessary to guarantee that the Markov rocess is stationary. The transition robabilities should not deend uon whether the firm under consideration for the free entry condition has just entered or is already roducing for some eriods. Also, Hoenhayn (1992) assumes that firms face a shock to roductivity immediately after entrance and before roduction starts. In the Melitz entry/exit rocess that we use and that builds on Hoenhayn (1992), the shock facing firms is dying. Hence, the timing is consistent with Hoenhayn (1992). 5 The firm we are considering has either entered in eriod t+1 or in a revious eriod. Hence, the market rice in eriod t + 1 is conditional uon one firm having cost c, be it an entrant or an already roducing firm. This is necessary for the Markov rocess to be stationary and not deendent uon the one firm having just entered or being in the market already. 6 To hrase it differently, we cannot reverse the order of conditioning and summation over states and marginal costs. 12

14 do not know the marginal costs drawn by the other firms. This information is sufficient to calculate transition robabilities and thus exected rofit. Firms calculate transition robabilities, conditional uon the observed state in the last eriod to determine whether it ays off to enter or not. As firms live for more than one eriod, we also have to take into account exected rofits in future eriods, which deend uon transition robabilities towards states in future eriods. As such we can write down the free entry condition in eriod t for initial state s t 1 as follows: (1 δ) i+1 { π ( (u t+i ), c) Pr (u t+i, n e (u t+i ), c s t 1, n e (s t 1 ) + 1, c) (18) i=0 c C u s } +π ( (s t+i ), c) Pr (s t+i, n e (s t+i ) + 1, c s t 1, n e (s t 1 ) + 1, c) f (c) f e ( ) π ( (u t+i ), c) Pr (u t+i, n e (u t+i ), c s t 1, n e (s t 1 ), c) f (c) c C u S As mentioned before, as the number of firms is discrete we have to write the FE in terms of inequalities. The FE for state s t 1 is written such that with n e (s t+i ) + 1 entrants in state s t+i, rofits are lower than the sunk entry costs and with n e (s t+i ) entrants, rofits are larger than sunk entry costs. Hence, the number of entrants on which we condition on the LHS is n e (s t 1 ) + 1 for state s in eriod t 1, but also in all other eriods. 7 The number of free entry conditions is equal to the number of states, by slight abuse of notation also defined as S. An equilibrium of the model solves the number of entrants corresonding to each initial state using all free entry conditions. Hence, there are S inequalities in S unknowns. We can rewrite the conditional robability in terms of one eriod transition robabilities as follows using the Markovroerty: = Pr (u t+i, n e (u t+i ), c s t 1, n e (s t 1 ), c) v t+i 1 S... v t 1 S Pr (u t+i, n e (u t+i ), c v t+i 1, n e (v t+i 1 ), c)... Pr (v t, n e (v t ), c s t 1, n e (s t 1 ), c) 7 A technical detail is that formally we should have exressed exected rofit with the robability that a firm dies within the summation terms, as states are defined unconditional uon one firm not dying. That means that the rofit exression for each state should have been multilied by the robability that one firm dies given the state. But it is easily shown (derivation available uon request) that the robability that one firm dies is indeendent of the state, i.e. of market rice and number of firms. Therefore, we can take this robability out of the summation term. 13

15 We omit the exression for Pr (u t+i, n e (u t+i ) s t 1, n e (s t 1 ) + 1), it is straightforward, but very cumbersome. We therefore also dislay only the greater than art of the FE in the remainder, as it is sufficient for our uroses. We have S FEs defined for each initial state s t 1 : f e (1 δ) i { i=0 c K u t+i S Pr (u t, n e (u t ) s t 1, n e (s t 1 )) u t S π ( (u t+i ), c) Pr (u t+i, n e (u t+i ) u t+i 1, n e (u t+i 1 )) } f (c) (19) We notice that the S FEs are stationary, i.e. it does not matter for which eriod they are defined. This imlies that the otimal number of entrants are stationary, n e (s t ) = n e (s). We define an equilibrium of the free entry model as follows: Definition An equilibrium of the free entry model consists of a maing n e (s) : S N such that for all s S (i) If n e (s) 0, then inequality (19) is satisfied (ii) If n e (s) = 0, then the LHS of inequality (19) is smaller than 0 Hence, definition 2 states that a solution of the model requires a maing from the state sace S to a natural number n e solving the FE-inequality for each inequality for which n e is larger than 0 and having an exected rofit smaller than 0 in state s when the number of entrants for that state, n e (s) is 0. We now state the following roosition: Proosition 4 There exists an equilibrium maing n e (s) from initial state s to number of entrants n e leading to an equilibrium as defined in definition 2. To rove existence of equilibrium, we will rove, using first order stochastic dominance (FOSD), that exected rofit for each FE given initial state s is rising in the number of entrants for each initial state u, n e (u) with u equal or unequal to s, for given number of entrants for other initial states, v, n e (v), v s. We will show that this is sufficient for the existence of equilibrium. As we have a Markov rocess secifying transition robabilities over a set, we define FOSD as follows in this case, following the definition roosed by Green, McKelvey and Packel (1983). Consider a state sace X with transition robabilities from state i to j defined by the function g (j i). Define a function L : X R +. Define for any c R the subset S c as a subset 14

16 of the state sace X that satisfies: S c = {x X L (x) c} (20) g y = g ( y) : X R is the measure defining the robability to go from state y to some secific state in X. We say that the Markov rocess g first order stochastically dominates (FOSD) h with resect to L iff for all c R and for all y X: h (S c y) g (S c y) (21) Hence, stochastic dominance in this case says that the robability to go to any subset S c of X from any initial value y is larger for the rocess described by h than for the rocess described by g. The following definition of FOSD is equivalent. The Markov rocess g FOSD h with resect to L iff for any non-decreasing function w : R R and for any initial state y: x X w (L (x)) g (x y) x X w (L (x)) h (x y) (22) We will rove now that the Markov rocess Pr (u, n e, c s, n e, c) FOSD Pr (u, n e, c s, n e + 1, c) with resect to the function L (S) = (S) with the second Markov rocess defined such that the number of entrants n e is larger for the second Markov rocess for one or more initial states s. We roceed as follows. We note from the roof of roosition 2 that an increase in the number of roducing firms has a non-increasing effect on the market rice. Using this result we know that the entry of one additional firm leads to a leftward shift of the suort of robability function Pr (u, n e, c s, n e, c) for any initial state s. All oints in the suort of the robability function either do not change when the additional entering firm draws a cost higher than the ruling market rice without this firm or the oints in the suort shift to the left when the entering firm draws a cost lower than the ruling market rice. In formal terms for any market rice given n e entrants, we have that the market rice z given n e + 1 is z = q with q having a distribution function with I q (0) = 0, a state sace Q and a robability function i q. 15

17 Hence, we have that for any non-decreasing function w : R R and and any initial state s, w (z (u)) Pr (u, n e, c s, n e + 1, c) u S = ( ) w ( (u) q) i q (q) Pr (u, n e, c s, n e, c) u S q Q w () Pr (u, n e, c s, n e, c) (23) u S Hence, Pr (u, n e, c s, n e, c) FOSD Pr (u, n e, c s, n e + 1, c). In Aendix A it is shown that rofit decreases in market rice. Hence, when the number of entrants in eriod t increases for any initial state s t 1, exected rofit in eriod t will decline. Still, we also need to know the imact of more entrants in eriod t on rofits in future eriods. To show that rofits in other eriods (for other states) also decrease, we use that the Markov rocess Pr (u, n e, c s, n e, c) is stochastically increasing with resect to L (S) = (S) for given number of entrants in other states. Alying the definition for X and S c (equation (20)) above, we say that a Markov rocess g is stochastically increasing with resect to L if for all y, x L (y) L (x) g (S c y) g (S c x) In our alication this definition says that starting from a higher rice, we will go to a higher exected rice in the next eriod. To rove that our Markov rocess is stochastically increasing for given number of entrants, we can use the same line of reasoning as in the roof of FOSD above. Suose (s t ) = (u t ) + b, b > 0. It is easy to show that market rice given s t, z t+1 (s t ) is z t+1 = t+1 (u t ) + q with q having a a distribution function with I q (0) = 0, a state sace Q and a robability function i q. The roof is now analogue to the roof of FOSD above. As an increase in n e (s) leads to a lower rice in the next eriod and by the roerty of stochastic increase also in future eriods, we know that an increase in n e (s) generates also a lower exected rofit given initial state u s. Hence, we have shown that in the FEs defined in equation (19) for each given initial state s, the exected rofit declines in the number of entrants n e (u) with u = s or u s for given number of entrants for other initial states n e (v), v s. So, we have S inequality constraints in S variables with each function defining the inequality constraint declining in all variables. The roof of existence of equilibrium entry levels requires showing the existence of equilib- 16

18 rium of the following system of S equations and inequalities in S unknowns: f s (x (s), x ( s)) = 0, s with x (s) > 0 f s (x (s), x ( s)) < 0, s with x (s) = 0 In the resent context this is relatively straightforward. Exected rofit is downward sloing in the number of new entrants, while we assume there are some states where exected rofits are non-negative with entry. As such, we can focus on the exected rofit of a reresentative entrant, which is declining in the number of entrants, and so solve for the maximum number of entrants such that the exected rofit of each entrant is non-negative. Existence follows from the Arrow-Enthoven theorem. 3 International Trade International trade is introduced in a setu with two countries, not necessarily equal. We make a distinction between a situation where a free entry condition does not aly and the situation where a free entry condition does aly. We start with the first aroach, that can be interreted in two ways. First, we can consider it as a short-run aroach, where new firms do not come in yet to drive exected rofits to zero. Second, it can be interreted as an analysis of the effect of changes in variables like market size and trade costs, when these changes are sufficiently small that they do not induce any change in the number of entering firms for the different initial states. As we work with a discrete number of firms, free entry conditions are exressed in terms of inequalities and sufficiently small changes in exogenous variables will ossibly not change the number of entering firms. After the no free entry analysis, we introduce a free entry condition and analyse the effects of unilateral and bilateral liberalization. In a third subsection we do additional comarative statics analyses on the effect of distance and market size on the robability of zero trade flows and on fob rices. In the last subsection it is shown that the Brander&Krugman model and the Ricardian comarative advantage model are nested cases of our model. 3.1 International Trade without Free Entry Before introducing trade, we exlore the tyical exeriment in the trade literature of doubling the market size to mimic for the effects of a move from autarky to free trade. In the no-free-entry 17

19 case the effect of an increase in market size is straightforward as ointed out in the following roosition: Proosition 5 An increase in market size increases the sales of all firms roortionally in the no-free-entry case without a change in market rice. Proosition 5 can be roved as follows. Rewrite the market share equation (9) as follows: q i = σ c i q = σ c i (L + Π) P σ 1 u σ Π is total rofit income in the economy. It is easy to show (see the derivation of equation (B.7) in the aendix) that I = L + Π is given by the following exression: L + Π = 1 L σ 1 QPU ( c) σ (24) c = n c i θ i is the market share weighted average cost. From equation (9) and (10) it follows that market rice and market share θ i do not deend uon L. Hence, an increase in L leads to a roortional increase in total income L + Π, a roortional increase in demand in one Cournot sector q and also a roortional increase in roduction of each roducer, q i. In the next subsection we will see that with a free entry condition, the market rice does change. Now we discuss the finding that lower trade costs do affect the market rice, also without imosing a free entry condition. The reason is that the market rice deends uon the costs of the firms in the market and they change when trade costs change. We introduce international trade between two countries a, b = H, F with markets effectively segmented by trading costs. In articular we now introduce iceberg trade costs τ ba in the Cournot sectors, meaning that marginal cost for delivery into country a is increased at the rate τ relative to roduction and delivery for the domestic market. There are no fixed or beachhead trade costs, and the trading costs reclude return exorts. We focus on the imact of increased globalization (i.e. falling trade costs). Under our assumtions about trade costs, the equilibrium market rice in the reresentative Cournot sector becomes: a = σn a σn a 1 c a (25) 18

20 with c a = 1 n a [ nda c ida + n xb τ ba c ixb ] and n a = n da + n xb. In equation (25), there is a direct effect of lower trade costs from country b to country a, τ ba, on the market rice: exorting firms have lower costs and therefore average costs decline. And there is an indirect effect, because firms roducing for the domestic market can disaear and exorting firms can aear on the market. It can be shown that this indirect effect is 0 at the margin (see aendix B). Therefore, the relative change in the market rice is equal to: P a = n da n xb τ ba c ixb c ida + n τ xb ba (26) τ ba c ixb The elasticity of the market rice with resect to trade costs, ε a,τ ba, is between 0 and 1: ε a,τ ba = n da n xb τ ba c ixb From equations (26) and (27) we make the following roosition: c ida + n (27) xb τ ba c ixb Proosition 6 Without imosing a free entry condition, a decline in trade cost τ ba into country a leads to a lower market rice a in country a. The elasticity of the market rice a with resect to trade costs τ ba is between 0 and 1. Equation (26) shows that a decline of trade costs τ drives down the market rice. The domestic cutoff marginal cost is equal to the market rice, so it also declines. Proosition 6 alies both to unilateral and bilateral liberalization. Several other Proositions can be made on the effect of trade liberalization in the short run. Proosition 7 Some of the least roductive firms are squeezed out of the market with a decline in trade cost τ. How many firms are squeezed out of the market deends on the rice distribution of the firms, i.e. it deends on how far the highest cost firms are from the old market rice. Proosition 8 More of the remaining firms exort with a decline in trade cost τ. 19

21 More firms can enter the exort market, as the exorting cutoff marginal cost rises when τ declines: c xb = P a τ ab (28) ĉ xb = P a τ ba = (1 ε a,τ ba ) τ ba (29) Proosition 9 For all firms in the market, markus from domestic sales decline and markus from exorting sales rise with a decline in trade cost τ. Markus of all domestic sales decline, as the costs of the firms remain equal, whereas the market rice declines. Markus of the exorting firms rise with trade liberalization, as the effect of the declining trade costs dominates the effect of the decrease in market rice in the exorting market. Using the letter m to indicate marku, the following can be derived: m ixa = P b τ ab c ixa (30) ˆm ixa = b τ ab = (ε b,τ ab 1) τ ab (31) As in almost any model of international trade (for examle Armington) firms increase their market share on the exorting market and their market share is reduced in domestic markets. But the relative gain and loss of exorters and domestic roducers dislays an interesting attern: Proosition 10 Large low cost firms lose less market share on the domestic market than small high cost firms and small high cost exorting firms gain more market share on the exort market than large low cost firms Proosition 10 follows from totally differentiating the exressions for market shares: dθ ida = σ c ida a ε a,τ ba τ ba (32) dθ ixa = σ c ixb b (ε b,τ ab 1) τ ab (33) Therefore small firms lose relatively more market share on the domestic market and small firms gain relatively more market share on the exorting market than large firms. So, more efficient big firms do not gain more from imroved market access abroad than less efficient small firms. 20

22 Essentially, big firms already have a strong osition in an exorting market, so they cannot grow as much as a result of trade liberalization as small firms. 8 Consider next the welfare effect of trade liberalization assuming equal countries. This is comlicated by the fact that income is endogenous as it deends on rofit income in the imerfect cometition sector. With free entry rofit income is driven to zero, but in the no free entry case rofit income is non-zero and varies. Welfare er worker in country a is equal to utility er worker in that country: W a = U a = I a P Ua = L a + Π a L a P Ua (34) Π a is total rofit income in the economy. Elaborating uon this equation (see aendix B) assuming that both countries are equal, one arrives at the following exression: W = (Q + σ ) σ + Q c 1 (35) P U c are the market share weighted average costs, c = n d c i θ id + nx τc i θ ix. Log-linearizing welfare towards trade costs τ from equation (35) and treating the rice and the market share weighted average costs as endogenous, one finds (derivation in aendix B): [ Ŵ = σ σ Q + σ ] σσ σ ˆ + Q c Q σ d c (36) + Q c The first term in (36) is the welfare gain through a decline in rice. As exected the gain for the consumer from lower rices outweighs the loss of a lower rofit income with lower rices. The second term measures the ossible gain from trade liberalization of lower costs leading to a higher rofit income. Elaborating on the cost effect, d c, one gets: [ Ŵ = ] σσ σ ˆ + Q c σ σ Q + σ [ Q nd ] n x n x σ c i θ idˆθid + τc i θ ixˆθix + τc i θ ixˆτ + Q c (37) 8 This set of results, related in articular to Proosition 10, has interesting olitical economy imlications beyond the scoe of this aer. As trade liberalization rogresses, the dominant domestic firms gain relative domestic osition (known as standing in the antiduming and trade safeguards literature). Assuming that lobbying efficiency is a function of industry concentration, increased concentration of firms with standing (i.e. the domestic industry) may increase their ability to organize and seek rotection or relief against further dros in trade costs and foreign cometition. 21