Trade Liberalization and Firm Dynamics

In this paper, we analyze the transition dynamics associated with an economy's response to trade liberalization. We start by reviewing the recent literature that incorporates firm dynamics into models of international trade. We then build upon that literature to characterize the role of firm dynamics, export-market selection, firm-level innovation, sunk export costs, and firms' expectations regarding the time path of liberalization in generating those transition dynamics following trade liberalization. These modeling ingredients generate substantial aggregate transition dynamics as they shift and shape the endogenous distribution of firms over time. Our results show how the responses of trade volumes, innovation, and aggregate output can vary greatly over time depending on those modeling ingredients. This has important consequences for many issues in international economics that rely on predictions for the effects of globalization over time on those key aggregate outcomes.


Introduction
Firms operating in the same country and industry respond to globalization in very di¤erent ways. Empirical work using micro-level data on …rms or plants initially highlighted this contrast for export decisions and how this decision correlates with observable …rm performance measures such as size and productivity: only a subset of relatively bigger and more productive …rms export. 1 Subsequent work has documented a wide ranging set of other responses to globalization that consistently vary across …rms in the same country and industry, and are also strongly correlated with …rm-level performance measures: the number and location of export destinations, entry and exit from the domestic market, the range of products produced, the international organization of production (including but not limited to multinational status and outsourcing/o¤shoring decisions), and innovation activities such as R&D.
This empirical work has prompted the development of models with heterogeneous …rms in open economies that capture how changes in the extent of globalization (indexing frictions to trade and foreign investment, and the size of the global economy) in ‡uence those varied …rm-level responses.
These models highlight the composition e¤ects that are induced by the di¤erent …rm-level responses to globalization. In order to focus on those composition e¤ects, the majority of models examine cases where the …rms' responses to globalization (export decisions, organization of production, innovation) do not change over time. This assumes a stable aggregate environment with no …rm dynamics In this chapter, we focus on a relatively new direction in this literature examining how …rm responses evolve over time to changes in the extent of globalization (with an emphasis on trade liberalization). Recent empirical work has documented the dynamic interactions among those …rm responses, in particular between the export decision and innovation: trade liberalization increases the return to both …rm activities, while each activity additionally a¤ects the returns to the other.
Clearly, such interactions will induce the mapping between …rm characteristics and their export and innovation decisions to change over time. This provides one important rationale for the study of models that incorporate those dynamic interactions. Furthermore, the aggregation of those changing …rm decisions along with the evolution of entry and exit generate substantial di¤erences in the economy's overall response to globalization over time. This implies that an analysis of steady state outcomes may give a misleading summary of the overall e¤ects of liberalization. This is the second rationale for studying models that feature dynamic interactions in …rm responses to liberalization: they are able to contrast the response of key economic aggregates to trade liberalization at di¤erent time horizons.
In the next section, we review theoretical models that capture some of these dynamics, and summarize the associated empirical evidence. Our focus in the current paper is to explain and motivate in a uni…ed manner how key modeling ingredients of …rm dynamics interact to generate endogenous aggregate transition dynamics. We develop a range of variants of this type of model to isolate di¤erent interactions and mechanisms. We analyze some of the simpler variants analytically, and computationally derive the transition dynamics for the others. We show how the response of innovative activities magnify the productivity di¤erences between exporters and non-exporters. We examine how the responses of average productivity and trade ‡ows can be signi…cantly di¤erent to those in static trade models, and how these response can di¤er substantially at short and long term horizons, depending on the responses of entry, innovation, and the timing of trade liberalization.
We restrict our analysis to just one type of international market participation decision -export to a single destination -and one type of technology choice -innovation intensity -and focus on the key interactions between these two …rm decisions in response to trade liberalization. We incorporate …xed export costs (potentially sunk), but assume that trade liberalization takes the form of reductions in the per-unit export costs. We separately consider two cases for innovation: One where innovation is exogenous to the …rms, so that …rm productivity evolves stochastically, but independently from any …rm decisions; The other where innovation is endogenous and di¤ers across …rms and responds to changes in the aggregate trading environment. In all cases, incumbent …rms also make endogenous exit decisions. When trade is liberalized, the aggregate transition dynamics re ‡ect a combination of all those decisions by incumbent …rms as well as the aggregate response of entry. We highlight how the interaction of both …rm dynamics and endogenous export market selection jointly induce long lasting aggregate transition dynamics in response to a onetime unanticipated drop in that variable trade cost. We discuss how comparisons of consumption across steady-states can signi…cantly over-or under-state welfare measures that take into account the transition dynamics. We also show how expectations regarding future trade costs give rise to their own transition dynamics: either because the drop in the trade cost is anticipated ex-ante, or because it is not expected to last ex-post (the trade liberalization is temporary).
In order to highlight the importance of the interaction between …rm dynamics and export market selection, we develop analytical models that feature either one or the other, and show how, those models do not generate any endogenous transition dynamics in response to trade liberalization.
These analytical cases show that the entrants'expectation regarding future potential export pro…ts (relative to incumbent …rms) is a key factor generating aggregate transition dynamics. Endogenous innovation by …rms further ampli…es productivity di¤erences between exporters and non-exporters, and generates longer lasting transition dynamics. We show that trade liberalization even induces some non-exporters to increase their innovation activities (in spite of facing stronger competition in their domestic market), because they anticipate exporting in the future and respond to the associated higher returns post-liberalization. All of these dynamic e¤ects imply large di¤erences over time in the response of aggregate trade volumes to trade liberalization: long run trade elasticities (with respect to the fall in trade costs) are substantially higher than the corresponding short run elasticities.
We show that …rms'expectations regarding trade liberalization are particularly relevant when innovation is endogenous. If trade liberalization is only expected to be temporary, then the ampli-…cation e¤ect of endogenous innovation is weakened -and is no longer signi…cantly di¤erent than for the exogenous innovation case. On the other hand, when trade liberalization is anticipated ex-ante, endogenous innovation induces a signi…cant rise in innovation ahead of the drop in the trade cost. Di¤erences in the response of innovation between exporters and non-exporters also amplify (endogenously) productivity di¤erences between those two groups of …rms. Lastly, we incorporate sunk trade costs to examine how trade liberalization a¤ects the option value of becoming an exporter. This leads to additional anticipation e¤ects ahead of the change in trade costs.
The remainder of the chapter is organized as follows. Section 2 presents a brief overview of the recent literature on trade liberalization and …rm dynamics. Section 3 present our model and characterizes the equilibrium with two symmetric countries. Section 4 describes our model parameterization. Section 5 presents the theoretical and quantitative results for the di¤erent trade liberalization scenarios. The scenarios are designed to isolate the separate roles of export market participation, …rm dynamics, endogenous innovation, sunk export costs, and anticipation e¤ects regarding trade liberalization. Section 6 concludes. An Appendix presents proofs of our analytic results.

Literature Overview
Workhorse general equilibrium models of heterogeneous …rm dynamics in closed economies (such   as Hopenhayn 1992, Atkeson and Kehoe 2005, and Luttmer 2007) recently have been extended to open economy settings. 2 For example, Arkolakis (2008) and Irarrazabal and Opromolla (2008) consider dynamics extensions of Melitz (2003), in which …rms experience exogenous random shocks to their productivity, to account for salient features of the data on …rm dynamics by domestic and exporting …rms. While these papers focus on stable aggregate environments, Alessandria and Choi (2007) and Ruhl (2008) examine the transition dynamics to trade liberalizations, focusing on the role of entry into domestic and export markets. 3 Alessandria and Choi (2007) show, as we do below, that welfare calculations based on steady-state consumption comparisons can give signi…cantly di¤erent answers than welfare calculations taking into account transition dynamics across steady-states. They also consider physical capital, which we abstract from, as another source of endogenous dynamics as in the neoclassical growth model. Other papers examine more speci…cally how …rms make joint decisions regarding both export status and technology choice.
For example, Bustos (2007), and Yeaple (2005) consider static models in which there is a binary technology choice, and highlight how …rms jointly decide to both enter export markets and adopt the new technology (or do neither). 4 Costantini and Melitz (2008) extend this type of joint decision to a dynamic framework where …rms face both idiosyncratic uncertainty and sunk costs for both exporting and technology adoption. The sunk costs and uncertainty combine to generate option values for export and exit decisions. We explore the e¤ects of those option values in the current chapter. Our model of innovation follows closely the model developed by Atkeson and Burstein (2010), which builds on Griliches' (1979) model of knowledge capital. As in Ericson and Pakes (1995) and more recently in Doraszelski and Jaumandreu (2008), the fruits of innovative activity are stochastic, so the model can account for simultaneous growth and decline, and entry and exit of …rms in steady state. While Atkeson and Burstein (2010) focus on the o¤setting responses of exit, export, innovation, and entry decisions to permanent trade liberalizations and the o¤setting e¤ects of changes in these decisions on aggregate productivity and welfare, we focus on the e¤ects of changes in these decisions on transition dynamics of average productivity, trade ‡ows, and output, allowing for sunk export costs, considering both temporary and anticipated liberalizations.
While in this chapter we model …rm dynamics arising from changes over time in productivity or product quality that a¤ect …rm's production for all markets simultaneously, a number of recent papers have focused on demand dynamics as …rms accumulate customers in foreign markets. 5 Ruhl and Willis (2008) introduce demand shifters that grow over time to explain the slow growth of exporters as they enter new markets. Eaton et al (2008) model …rms' investments directed at increasing foreign demand (such as searches for foreign buyers and maintaining existing relationships with current buyers), in response to export market entry. Such demand-related investments have very similar e¤ects to innovation, except that their returns only a¤ect export pro…ts, as opposed to overall pro…ts. Chaney (2011) models the growth of exporters as they meet foreign importers in international social networks. 6 There are also a number of recent papers featuring models of …rm dynamics driven by frictional labor markets. See, e.g., Cosar  Empirical work using micro-level data has con…rmed the importance of dynamics in explaining …rm export behavior. This behavior is also re ‡ected in aggregate export patterns: new exporters initially account for a small proportion of aggregate exports, but those exporters grow faster than both established exporters and non-exporters, and they account for a substantial portion of aggregate export growth over longer periods of time (Over 40% for the U.S. export growth from 1987 to 1992, and over 50% for the export growth in both Colombia and Morocco from 1984 to 1991; See Bernard and Jensen (2004a) and Roberts and Tybout (1997).) Hysteresis e¤ects are one major driver of those aggregate export dynamics: past export experience explains a very large proportion of a …rm's current and future export performance, even after controlling for all observable …rm performance indicators. 7 This type of hysteresis behavior is explained by a combination of sunk export costs and …rm level uncertainty (some form of stochastic …rm dynamics, which could just be generated by shocks exogenous to the …rm). Das, Roberts, and Tybout (2007) econometrically measure sizable sunk export costs for Colombian exporters, which induce e¤ects of …rms' expec- 5 Foster, Haltiwanger, and Syverson (2010) document the importance of demand accumulation in accounting for U.S. plant dynamics in a number of U.S. manufacturing sectors. 6 Drozd and Nosal (Forthcoming) present a macroeconomic model of customer accumulation to account for salient features of international relative prices and the dynamics of aggregate trade ‡ows. Alessandria, Kaboski, and Midrigan (2010) present a model of trade and inventory management to account for the dynamics of aggregate trade ‡ows and prices in the aftermath of loarge devaluations. 7 See Roberts and Tybout (1997) for evidence in Colombia, Bernard and Wagner (2001) for Germany, and Bernard and Jensen (2004b) for the U.S. tations regarding future export market conditions on current export behavior. Ruhl (2008) shows that this combination of sunk export costs and idiosyncratic …rm uncertainty also explains how the elasticity of trade with respect to changes in trade costs or aggregate productivity can vary substantially depending on the perceived persistence of those changes. Bergin and Lin (2010) document another example where …rms'expectations about the future aggregate trading environment induces noticeable changes in the …rms'current export market entry decisions. They show that European …rms enter export markets prior to the implementation of EMU (and the associated decreases in trading frictions).
Another driver of the rapid export growth by new exporters is the link between export market entry and …rm innovation. Many recent papers using micro-level have documented this link: Lileeva  8 They all …nd that a …rm's export market entry (driven by reductions in trade costs) is associated with increased innovation. Bloom, Draca and Van Reenen (2008) also document a similar relationship between changes in the trading environment and …rm innovation and skill upgrading, but on the import competition side: Firms in European industries most exposed to increased import competition from China respond by increasing their innovation and information technology intensity. 9

Model Economy
In this Section we present our model of trade liberalization and …rm dynamics. Time is discrete, and each period is labeled t = 0; 1; 2; : : : : We do not model any aggregate uncertainty. The economy has two symmetric countries: home and foreign; foreign variables are denoted with an asterisk superscript. Given the symmetry across countries, we only focus on the variables relevant for the home country. Households inelastically supply L units of labor and derive utility from consumption only. Production in each country is structured as follows. There is a …nal non-traded consumption good that is produced using a continuum of di¤erentiated intermediate goods with a constant returns to scale C.E.S. technology with elasticity of substitution > 1. 10 These intermediate 8 There is also another strand of the empirical literature that has documented a reduced form relationship between export market entry and subsequent …rm productivity growth (often labeled as learning by exporting). See Greenaway and Kneller (2007) and Lopez (2005) for recent surveys. 9 Bernard, Jensen, and Schott (2006) document a similar e¤ect for import competition on the capital and skill intensity of the a¤ected U.S. …rms (increases in competition from low wage exporters associated with increases in both capital and skill intensity).
1 0 By assuming a demand structure with constant markups, we do not allowed for strategic considerations in pricing. This could generate some interesting implications for di¤erent returns to innovation across …rms (see e.g. Aghion  A …rm in the home country with state z has productivity equal to exp(z) 1=( 1) and produces output y t (z) with labor l t (z) according to the constant returns to scale production technology: 11 In addition, every operating …rm must pay an overhead …xed cost equal to f units of labor in every period. We re-scale …rm productivity using the exponent 1= ( 1) for expositional convenience: As we explain below, this re-scales a …rm's size, variable pro…ts and production employment (net of the overhead cost) to be proportional to exp (z).
Di¤erentiated intermediate goods produced in home can be used for domestic production of the …nal good, or exported to foreign for use in foreign's …nal good production. Let a t (z) denote the domestic absorption of …rm z's production. Similarly, let a t (z) denote the quantity of …rm z's production used in foreign's …nal good production. Exports by …rm z incur both a …xed cost f X (measured in units of domestic labor) as well as a per-unit cost. The latter takes the form of an iceberg cost equal to 1 units for each unit of the good exported (with 1). Due to the …xed export cost, it is unpro…table for some …rms to export. Let x t (z) 2 f0; 1g denote the export indicator for …rm z at home (x t = 1 if the …rm exports and 0 otherwise). Feasibility requires A …rm in the foreign country with state z has the same production technology as the home …rm, but with output denoted y t (z); production labor l t (z); and domestic absorption b t (z): Exports to the home country, b t (z), are subject to both …xed and per-unit costs; hence, feasibility requires that x t (z) b t (z) + b t (z) = y t (z) and that f X units of foreign labor be used to pay the …xed export al 2005). 1 1 Since we do not make any assumptions regarding physical quantity units for the di¤erentiated intermediate goods, …rm productivity in that sector can be directly re-interpreted as product quality. In this alternative variant, …rms innovate to improve product quality rather than productivity. This re-interpretation does not change any of our …ndings. Our model can also be extended to include other forms of physical and human capital as variable factors of production. Consideration of these forms of capital would lead to the standard ampli…cation of the impact of a change in productivity on aggregate output. cost for all foreign exporting …rms (with x t (z) = 1). 12 The constant returns to scale C.E.S. production technology for the …nal good implies that the total quantity of the good produced at home is given by where M t (z) is the distribution of operating …rms in the home country over the productivity index z, and M t the corresponding distribution in the foreign country. The total measure of operating …rms in the home country is given by R M t (z) dz. Production of the …nal good in the foreign country, Y t ; is de…ned analogously.
The …nal good sector in both countries is competitive. Let P t denote the …nal good price in home, and p at (z) and p bt (z) the prices of the domestic and imported intermediate goods in the home country. We pick labor as the numeraire good, and normalize the wage to 1. Final good producers take these prices as given; they also take all the decisions by the intermediate good …rms ) as given. The …nal good price is thus given by its unit cost: The demand for the intermediate goods at home is then given by: Analogous equations hold for prices and quantities in the foreign country.
Intermediate good …rms in each country are monopolistically competitive. A home …rm with productivity index z faces a static pro…t maximization problem involving the choice of labor input l t (z); prices p at (z); p at (z); quantities a t (z); a t (z); and an export decision x t (z) : In so doing, the …rms take as given the prices and quantities of the …nal good in both countries (recall that the wage is normalized to 1). Firm z's pro…t maximization problem is: t (z) = max yt(z);lt(z);pat(z);p at (z); at(z);a t (z);xt(z) subject to (1), (2), and (5). We de…ne z Xt as the export productivity cuto¤: z Xt arg min fx t (z) = 1g. 1 2 Symmetry between home and foreign implies that yt(z) = y t (z); xt(z) = x t (z), at(z) = b t (z); a t (z) = bt(z).
We now describe the productivity dynamics of individual …rms. At the beginning of each period t; every existing …rm faces an exogenous probability of incurring a "death"shock that would induce exit (exogenous exit unconditional on productivity). The remaining 1 portion of …rms can still choose to exit (endogenous exit conditional on productivity) or to continue to operate and pay the overhead cost f . Productivity for those …rms then evolves over time depending on the their investment in innovation, which stochastically improves productivity. We model the evolution of productivity for producing …rms as follows: In the following period (t + 1), a …rm with productivity index z has a probability q of having productivity exp(z + z ) 1=( 1) and a probability 1 q of having productivity exp(z z ) 1=( 1) : Hence, the expected growth of z is given by (2q 1) z . 13 We will examine a special case where z = 0, so that productivity is constant throughout a …rm's life; this is the case with no productivity dynamics.
In the speci…cation of our model with endogenous innovation, each …rm invests in innovation by choosing the expected growth of their current productivity z (determined by q). A …rm with productivity z choosing an innovation intensity q must hire exp(z)c(q) units of labor as its investment in innovation. We assume that c (q) is increasing and convex in q: With this evolution of …rm productivity, the expected, discounted present value of pro…ts (abstracting from aggregate uncertainty) for a …rm with initial productivity index z is where the value for operating …rms is given by the following Bellman equation: R t is the world interest rate in period t (in units of labor). Let q t (z) denote the optimal innovation intensity of the …rm referenced in (8). This choice of innovation intensity must satisfy the …rst order condition: Note that with our scaling of the innovation cost function, exp (z), we are assuming that the innovation cost required to increase the size of the …rm by a …xed percentage scales with the size of the …rm. This will imply that, for su¢ ciently large …rms, the innovation decision and derived growth rate is independent of size, consistent with Gibrat's law, and the …rm size distribution has a Pareto right-tail. We denote by q t the innovation intensity for such very large …rms, i.e.
In our quantitative analysis, we assume that the innovation cost function has the form c (q) = h exp(bq), so that the curvature of this function is indexed by the parameter b: If this curvature parameter b is high (or low), then innovation is highly inelastic (or elastic) to changes in the incentives to innovate. With a very high curvature parameter b, innovation decisions of …rms are e¤ectively constant both across …rms and over time, q t (z) = q. This is the exogenous innovation case.
Since the value function of operating …rms V 0 t (z) is strictly increasing in z; the endogenous exit or operate decision (7) must follow a cuto¤ rule: …rms with productivity at or above a cuto¤ z t choose to operate, while the remaining …rms choose the exit option. Note that if f = 0, then New …rms are created with an investment of f E units of labor, which yields a new …rm in the following period with initial productivity index z drawn from a distribution G (z) : In any period in which new …rms enter, free entry requires that Let M Et denote the measure of new …rms entering in period t that start producing in period t + 1.
Households in the home country have preferences of the form where C t is their consumption of the home …nal good in period t and 1 is their discount factor. Each household in the home country faces an intertemporal budget constraint of the form with R 1 = 1. In writing the budget constraint (11) we have assumed that the free-entry condition holds with equality in every period and that the representative consumer in each country owns only the …rms in their respective countries. The right hand side thus represents the initial home consumer wealth from the existing stock of home …rms in period 0.
With balanced trade (which is immediate with symmetric countries), production and consumption of the …nal good are equalized in every period: C t = Y t . Without loss of generality, we normalize aggregate labor supply to 1. This labor is used for production of intermediate goods , and to cover the …xed costs for entry, overhead production, and export: 14 where The evolution of the distribution of operating …rms M t over time is given by the exogenous probability of exit , the decisions of operating …rms to invest in their productivity q t (z), and the measure of entering …rms in period t, M Et : The distribution of operating …rms M t+1 (z 0 ) in the home country in period t + 1 is equal to the sum of three in ‡ows of …rms: new …rms that entered in period t; …rms continuing from period t that draw positive productivity shocks (and, hence, had productivity equal to z 0 z in period t); and …rms continuing from period t that draw negative productivity shocks (and, hence, had productivity equal to z 0 + z in period t). We write this as follows: An equilibrium in this economy is a collection of sequences of aggregate prices fR t ; P t ; P t g and prices for intermediate goods fp at (z); p at (z); p bt (z); p bt (z)g, a collection of sequences of aggregate quantities fY t ; Y t ; C t ; C t ; L P t ; L P t ; L It ; L It g and quantities of the intermediate goods fa t (z); a t (z); b t (z); b t (z); l t (z); l t (z)g; and a collection of sequences of …rm value functions and pro…t, exit, export, and innovation decisions fV per-period pro…ts, …nal good …rms in each country maximize pro…ts, all of the feasibility constraints are satis…ed, and the distribution of operating …rms evolve as described above.
As we previously mentioned, we focus our analysis on symmetric equilibria across countries. To ensure this symmetry, we assume that the initial distribution of …rms is identical across countries: This also ensures equality of initial wealth for home and foreign consumers. The symmetry between home and foreign will then be preserved in every subsequent period. Thus, all the home country variables that we have introduced will be equal to their foreign country counterpart.
A steady state of our model is an equilibrium in which all of the aggregate variables are constant. In what follows, we omit time subscripts when discussing the steady state. 15 Depending on parameter values, there are two types of steady states in our model: one with entry and one without entry. The parameter restrictions required to have a steady-state with entry imply that the equilibrium innovation decision of large …rms leads them to shrink in expectation. 16 Our results are derived in the case in which every period there is positive …rm entry.

Aggregation in a Symmetric Equilibrium
The equilibrium of our model with heterogeneous …rms and endogenous innovation cannot be fully solved in closed-form. Appendix A describes a simple algorithm to solve for the symmetric steadystate. The online appendix of Atkeson and Burstein (2010) describes an algorithm to solve for the transition dynamics across steady-states. We now present some aggregate relations that hold in a symmetric equilibrium of our model that we use in our analytic results. We also de…ne aggregate statistics that we report from our experiments.
The solution to the static pro…t maximization problem (6) in a symmetric equilibrium is where Dt is a domestic market demand index given by 1 5 See Atkeson and Burstein (2011) for a detailed discussion of how a similar model of innovation by …rms in a closed economy can be extended to allow for endogenous and semi-endogenous growth. 1 6 A su¢ cient condition to obtain a steady state with entry is that In a steady state without entry, this condition is satis…ed with equality, and the expected growth of continuing …rms is exactly o¤set by exit. In this case, aggregate variables are constant but the distribution of …rms by size is not as production becomes concentrated in a shrinking number of large …rms.
This market demand index also proportionally scales production employment for all …rms: Given that …rm revenues are proportional to …rm employment, the ratio of exports to the aggregate value of production (for the intermediate goods sector) is given by Similarly, the ratio of exporters'revenues in the domestic market to total domestic revenues (the market share of exporters in their domestic market) is given by This share of exporters in the domestic market di¤ers from the share of exporters in total production in that it does not capture the direct impact of changes in trade costs on the size of exporters, but instead just re ‡ects the reallocation of production between non-exporters and exporters in the domestic market.
Aggregating (16) across …rms, and using the expression for the aggregate price (4), we can express aggregate output Y t as where is an index of aggregate productivity. Average productivity for domestic production is given by Since …rm size on the domestic market is proportional to exp(z), this average productivity index is also proportional to the average …rm size on the domestic market. Put another way, for a given level of market demand Dt , changes in average productivity Z t are proportional to changes in average …rm size on the domestic market.  We consider two choices of the curvature parameter b for the innovation cost function: one where b is su¢ ciently high that the innovation decision is e¤ectively inelastic across …rms and over time (q t (z) = q), and another where b = 10 so that innovation decisions are quite elastic across …rms and over time. We set such that the steady-state interest rate (annualized) is 5 percent, and = 5 consistent with estimates by Broda and Weinstein (2005). We choose the distribution G of productivity draws of entrants so that all …rms enter with a common productivity index z = 0 (the cross-sectional distribution of z is then determined by the ergodic distribution generated by the productivity dynamics). We choose z to match the standard deviation of the growth rate of employment of large US …rms. We choose the exogenous exit rate to match the annual employment-weighted exit rate of large US …rms. We normalize entry costs f E = 1, and we set the …xed costs of operation f = 0:1. We choose the parameters f X ; 1 , and h to match three observations in the U.S.: (1) the fraction of exports in gross output; (2) the fraction of total production employment accounted for by exporting …rms; and (3) the shape of the right tail of the …rm size distribution.

Trade Liberalization Scenarios
We restrict our analysis of trade liberalization to decreases in the per-unit trade cost . Throughout all scenarios, we consider the e¤ects of a 3:5% reduction in , starting from the calibrated steady state described above. 17 We consider the following experiments. We …rst consider the e¤ects of a permanent unanticipated reduction in the trade cost, and describe how …rm productivity dynamics interact with endogenous export market selection to generate endogenous aggregate transition dynamics. We then examine the role of anticipated changes in trade policy. We …rst look at the case where that same unanticipated drop in the trade cost is anticipated to revert back to the old steady state (so the trade liberalization is only temporary). We then look at the case where the initial trade liberalization is anticipated (and permanent thereafter). We present our computational results graphically, using …gures that show the responses over time of key variables for the economy.
In several cases, we also complement these numerical results with analytical propositions regarding those dynamic responses (the proofs are relegated to the appendix).
In order to highlight the key interactions between …rm productivity dynamics and endogenous export market selection, we start with two scenarios that exhibit only one of those features at a time (eliminating the other). Our …rst scenario eliminates endogenous market selection (all …rms export), while our second one eliminates …rm productivity dynamics (…rm productivity remains constant post entry). In both those cases, we show that permanent trade liberalization does not induce any endogenous transition dynamics. In order to motivate how …rm productivity dynamics and endogenous export market selection interact to generate transition dynamics, we develop a simpli…ed analytical variant of our model that nevertheless exhibits both of those features. This analytical model clari…es how di¤erences in current and future export market pro…tability (relative to the pro…tability of domestic sales) drives transition dynamics in response to a sudden, permanent trade liberalization (via its e¤ect on …rm entry). In our third scenario we show that these analytic results carry over to our full model with exogenous innovation. In our fourth scenario, we add endogenous innovation and describe its e¤ect on the transition dynamics -contrasting them to our previous scenario where innovation is exogenous. In our …fth scenario, we examine the case of a temporary trade liberalization. We show how expectations regarding the permanence of trade liberalization a¤ect the response of innovation. Lastly, we study the e¤ects of an anticipated trade liberalization. We return to the case of a permanent decrease in trade costs, but we now assume that this decrease in trade cost is anticipated (ahead of any change in the trade cost). Scenario six captures those anticipation e¤ects and shows how they are much more pronounced when innovation is endogenous (relative to the exogenous innovation case). Our seventh and eight scenarios add sunk export market costs, thus introducing partial irreversibility and option values associated to the export entry decisions. We show how the perceived permanence of the trade liberalization shapes the strength of entry into exporting, and how anticipated trade liberalization a¤ects the option values from entry into exporting and induces transition dynamics ahead of the actual drop in the trade cost.

Scenario 1: No Export Market Selection
In this scenario, we start with our baseline model but eliminate export market selection by setting the …xed export cost to zero, f X = 0 (all …rms export regardless of their productivity z). The Panel F depicts the response of the innovation intensity q t for large …rms (technically, lim z!1 q t (z)).
As is apparent from those responses in Figure 1, analytically, and shows that the proportional increase in …nal output is This is the same rise in …nal output as in an Armington world where every country produces a single good with exogenous unit labor requirements. The main intuition for the absence of any inter-…rm reallocation or change in innovation is that, in a world with C.E.S. demand and constant markups, the increased export opportunities from lower trade costs are exactly o¤set by the reductions in domestic sales (driven by increased imports). All …rms face those same exact trade-o¤s, and there is thus no scope for any inter-…rm reallocations, and no motive for any change in innovation decisions.
The expected value of an entering …rm relative to the entry cost is unchanged, and there is hence no change in entry behavior. 18

Scenario 2: No Firm Productivity Dynamics
We now allow for export market selection (the …xed cost f X > 0 is calibrated to match the aggregate export patterns described above), but eliminate …rm productivity dynamics. This is equivalent to setting z = 0. In order to match the right tail of the …rm-size distribution of the model with …rm dynamics, we assume that the initial …rm productivity is drawn at entry from a Pareto distribution G with shape parameter > 1.
The responses of the same key variables are depicted in Figure 2. Here too, we see that the permanent trade liberalization does not induce any endogenous transition dynamics. As in the previous scenario, there is no response in entry, and …nal output (consumption) immediately jumps to its new steady state level. However, this scenario does feature some inter-…rm reallocation. The lower trade cost induces high productivity …rms to become exporters. Firms with low productivity cannot take advantage of the increased export opportunities, and thus disproportionately su¤er from the reductions in domestic sales. This induces the exit of the least productive …rms, and labor is reallocated from non-exporters and exiters towards exporters. The expansion of the exporters relative to the non-exporters and exiters induces an increase in average …rm productivity (Panel D). Panel E shows how this expansion of exporters relative to non-exporters is re ‡ected in the increased share of exporters in domestic revenues (which was ‡at in the previous scenario with no inter-…rm reallocations). The response of the share of exporters in total revenues is now higher (relative to the previous scenario) due to the e¤ects of the lower trade cost on the extensive margin of trade (new exporters). 19 1 8 If the cost of entry falls (for instance, if entry cost required both labor and …nal output), then entry rises, giving rise to endogenous transition dynamics across steady-states. All other results still hold (e.g. innovation and exit thresholds remain unchanged). Similarly, if innovation requires both labor and …nal output, innovation rises for all …rms, giving rise to endogenous transition dynamics. 1 9 In an alternative calibration, in which is chosen to be equal to The exact o¤setting of these two forces is naturally very speci…c to the assumed parametrization (especially, C.E.S. demand and Pareto distribution for …rm productivity).

Firm Dynamics and the E¤ect of Trade Liberalization on Entry: Building Intuition
We now show that, adding …rm productivity dynamics completely changes the trade-o¤s described above for the e¤ects of trade liberalization on entry. We …rst build a simpli…ed (and slightly modi…ed) version of our model that motivates a simple analytic rule governing this trade-o¤. This rule is not tied to a speci…c parametrization, and allows us to predict the response of entry to trade liberalization for more general cases. We then show how it applies to our full model with both endogenous export market selection and …rm productivity dynamics.
Consider the following version of our model that allows for time variation in the …xed export cost f X . In particular, f X = 1 if the …rm has entered T 0 or less periods ago, and f X = 0 if the …rm has entered T + 1 or more periods ago. Hence, conditional on survival, …rms start exporting after T periods of operation. Note that if T = 0, all …rms export. We also assume that all entering …rms have productivity index z = 0, there are no productivity dynamics ( z = 0), and there are no …xed overhead costs, f = 0 (so exit takes place due to exogenous death only).
We denote bys X the ratio of discounted time-series revenues from exports in total discounted time-series revenues of entering …rms. That is, while s X measures the export share in the crosssection of …rms,s X measures the export share in the discounted revenues of entering …rms. 20 If = 1, thens X = s X ; If < 1 and T > 0, then exporters are back-loaded sos X < s X . More generally, if < 1, then as T rises (so that it takes longer for an entrant to become an exporter), would be equal, while our other qualitative results remain unchanged.

0s
X is calculated analogously to the export share sX using (17), with a "discounted" distribution of …rmsM (z) that is de…ned analogously to M (z) in (13) -except that (1 ) replaces (1 ) and G (z) replaces G (z).
pro…ts from exporting become a less important component of a …rm's value upon entry, ands X falls relative to s X .

Proposition 3 in Appendix D states that in this economy, a permanent reduction in marginal
trade costs, , leads to a decline (increase) in the steady-state mass of entering …rms, M E , if and only ifs X is strictly lower (higher) than s X , and the steady-state mass of entering …rms remains unchanged if and only ifs X = s X . The intuition for this result is straightforward. Whens X < s X , incumbent exporting …rms bene…t proportionally more than entering non-exporting …rms from lower trade costs. Hence, trade liberalization makes entry less pro…table, and entry falls. This result suggests that the relative size of entering …rms plays an important role in shaping the change in entry and the aggregate transition dynamics in response to a trade liberalization. 21 We now apply this logic to understand the response of entry and aggregate transition dynamics in our full model with productivity dynamics.

Scenario 3: Export Market Selection and Firm Dynamics with Exogenous Innovation
We now return to our baseline model with both endogenous export market selection and …rm productivity dynamics. In this scenario, we assume that the parameter b of the innovation cost function is su¢ ciently high so that innovation intensity is constant across …rms and over time: q t (z) = q.
This is the exogenous innovation case. The Markov evolution process for …rm productivity is such that entering …rms are, on average, smaller than incumbents -and are thus less likely to export than incumbents. 22 This implies that, with < 1,s X < s X from our previous analytical model.
The responses of the same set of variables for this scenario are depicted in Figure 3. A permanent trade liberalization makes entry less pro…table as incumbent exporters bene…t proportionally more than non-exporting entrants. As we see from panel C, entry responds as our simple analytic model suggests: it drops upon the reduction of the trade costs. Because the new steady state with lower trade costs features a smaller mass of producing …rms (due to the smaller mass of entrants in steady-state), there is an "overhang" of incumbent …rms immediately after trade liberalization.
This leads to an overshooting of the entry response as the mass of producing slowly decreases to its new lower steady-state level. The labor resources that were previously allocated to entry are available for production employment. There is a spike in production employment (inversely 2 1 Atkeson and Burstein (2010) extend this result to a more general setting for the structure of productivity and export participation dynamics. Fattal-Jaef (2010) presents a related result in a model of …rm dynamics and misallocation distortions. 2 2 This also matches the empirical regularity that most …rms do not export immediately after entry.
related to the downward spike in entry) and a similar spike in …nal output. Due to these transition dynamics, we see that comparing consumption across steady-states understates the welfare gains from liberalization that accrue during the transition as the mass of …rms declines.
The reallocation of market share towards exporting …rms leads to an increase in average …rm productivity in the new steady state. Average productivity overshoots this new steady-state level over the transition due to the overshooting of entry (since entrants are relatively less productive).
The transition dynamics for the fraction of exporters' revenues in total and domestic revenues are not very pronounced. They behave in a similar way as the previous scenario with no …rm productivity dynamics.
In order to emphasize how the di¤erence between the export pro…tability sharess X and s X drive the response of entry, Figure 4 contrasts the response of entry and …nal output under our baseline interest rate and a zero interest rate ( = 1). In the latter case, the delay for entrants to grow and start exporting becomes inconsequential; their future expected revenues from exporting s X is equal to the cross-sectional export share s X . Figure 4 shows how, in this case, the response of entry becomes very muted, as do the transition dynamics for …nal output. Similarly, if the model is calibrated so that entering …rms grow faster (and hence become exporters earlier), the decline in entry is also muted relative to our baseline parameterization.

Scenario 4: Endogenous Export Market Selection and Endogenous Innovation
We now switch to the more elastic parametrization for innovation. The economy's responses are reported in Figure 5. This is the …rst scenario to feature a response in …rm innovation intensity to trade liberalization. This response is exhibited in Panel F for the innovation intensity of large exporting …rms (all …rms above a given large size threshold). As in the case with exogenous innovation, the permanently lower trade costs increase the value of exporters relative to nonexporters. In the current case, those exporters respond by innovating relatively more. In Appendix E we derive analytically this reallocation of innovation from non-exporters to exporters, and discuss how it depends on key model parameters.
The immediate response of entry is similar to the case of exogenous innovation (and for the same reasons: trade liberalization makes entry less pro…table as entrants' exporting pro…ts are pushed back into the future). Average …rm size and productivity steadily increases (Panel D) from the combination of selection (reduced entry by smaller, less productive …rms), and increased innovation by exporters. Although the response in increased innovation intensity is immediate, its e¤ects on the productivity and size of exporters takes a long time to unfold. As the average size and productivity of exporters grow relative to those of entrants, the expected pro…tability for that latter group further falls, inducing a second dip in entry (Panel C). The increase in relative size and productivity of exporters also leads to a steadily increasing share of exporters in domestic and total revenue (Panel E). Hence, the elasticity of trade volumes to changes in trade costs is higher when it is calculated over longer time horizons.
The labor resources used for entry are reallocated to production (and some to the increased innovation activity). Production labor steadily rises, mirroring the decrease in entry (Panel B).
Those changes in production labor are also re ‡ected in a rise in …nal output. As opposed to the exogenous innovation case, steady state consumption now overstates the welfare gains from trade liberalization because the increase in average productivity takes many periods to materialize. The main message for welfare comparative statics remains that transition dynamics strongly a¤ect any comparison based on steady state consumption.
We note that the increase of innovation to trade liberalization is not limited to current exporters: trade liberalization also spurs the innovation response of non-exporters who anticipate exporting in the future (given the lower trade costs). In fact, from equation (9), …rms close to the export threshold have the biggest incentive to innovate, because they have the highest marginal bene…t from successful innovation (i.e. the di¤erence between pro…ts as an exporter relative to pro…ts from only serving the domestic market). We highlight this result in Figure 6, which shows the di¤ erence in innovation intensity, q(z), across the two steady states (pre-and post-liberalization) as a function of z. In the …gure, z is reported as the di¤erence relative to the exit threshold z X . The …gure also shows the change across the two steady states in the export threshold. All exporters increase their innovation intensity ( q(z) > 0), but …rms around the export thresholdboth current exporters and non-exporters that anticipate to be exporters in the future -have the highest increase in innovation activity. Due to the increased competition from foreign producers and more productive domestic producers, a subset of …rms with the lowest productivity (who have the smallest probability of becoming an exporters) innovate less in response to trade liberalization.  The anticipated response of innovation in this scenario is driven by the …rms'desire to smooth their innovation activities over time in response to the higher bene…ts to innovate: As the cost of innovation is convex, …rms do not want to cluster their innovation activities right before the drop in the trade cost, and prefer to spread them out ahead of the anticipated liberalization.

Scenario 7: Temporary Trade Liberalization and Sunk Export Costs
In our last two scenarios, we examine the consequences for the dynamics of trade liberalization that relate to the export entry decision in the presence of sunk export costs. We introduce (partial) irreversibility of the export decision by assuming that some of the …xed export costs are sunk. In particular, in addition to the …xed export costs f X , we assume that a non-exporter that becomes an exporter must hire f XS units of labor during the …rst period as an exporter. An exporter that stops exporting must incur these sunk costs to re-start exporting.
With sunk export costs, the choice to enter the export market is no longer a static decision as in expression (14). Instead, current export entry decisions are shaped by expected future exporting pro…ts, subject to …rms' expectations regarding their idiosyncratic productivity and the path of trade costs. This represents an additional channel that generates endogenous aggregate transition dynamics from trade liberalization. Since this channel operates independently of the endogenous innovation response, we highlight its e¤ect using the case of exogenous innovation. We …rst examine how export entry decisions under sunk export costs interact with the perceived persistence of trade liberalization. Figure 9, Panel A, depicts the path of trade costs under a temporary (i.e. lasting 4 years) and a permanent trade liberalization. In both cases, the liberalization is initially unanticipated.
As a reference point, we …rst consider our baseline model calibrated with only …xed export costs (and exogenous innovation). In this case the perceived permanence of the reduction in trade costs has little bearing on export decisions (because entry into exporting is a static decision based on the level of current pro…ts from exporting). This implies, as shown in Panel B, that the increase in the share of exporters in domestic revenues over the …rst 4 years of the trade liberalization is very similar in both the temporary and permanent cases.
Next, we designate a proportion of the …xed export cost to be sunk. In particular, we set the …xed export cost at a low level (f X = 0:2), and choose f XS to match our baseline calibration targets. In this case, entry of non-exporters into exporting is stronger when the reduction in trade costs is perceived to be permanent. This can be seen in Panel C: Over the …rst 4 years, the rise in the share of exporters in domestic revenues (and hence also the overall response of trade volumes) is larger in the permanent case than in the temporary case. That is, trade liberalization must be perceived to be long-lasting in order to provide incentives for …rms to pay the sunk export cost. 2425 2 4 This result is similar to the one in Ruhl (2008). The latter shows how sunk export costs induce a larger response of exports to permanent shocks (i.e. trade liberalization) than to temporary shocks (i.e. business cycles). See also the related work by Albuquerque and Rebelo (2000). 2 5 Note also that the nature of …xed export costs shapes the strength of the short run response of entry into export markets. In particular, comparing Panels B and C for either the temporary or the permanent trade liberalization case, we see that the short-run increase in the share of exporters in their domestic market (and hence the overall response of trade volumes) is much larger under …xed export costs than under sunk export costs. In contrast, the long-run responses of the trade volumes are very similar under …xed or sunk export costs (projecting the responses beyond the 10 years displayed in the …gures). The hysteresis band in export participation (that we describe in the next Scenario) plays an important role in generating this di¤erence in the short run responses.

Scenario 8: Anticipated Trade Liberalization and Sunk Export Costs
We now illustrate a second implication of export entry decisions in the presence of export sunk costs for the aggregate dynamics following trade liberalization. Here we focus on an anticipated trade liberalization, as described in Scenario 6. In the presence of productivity dynamics, sunk export costs generate an option value associated with the export market entry decision, and an associated hysteresis band: There is a set of …rms (associated with productivity z in a given range) whose export decision is determined by their prior export status. This option value is sensitive to the …rms' expectations regarding future trade costs, and thus responds to announcements regarding those future costs.
As a useful starting point, consider …rst a version of our model with no productivity dynamics ( z = 0). In the initial equilibrium before trade liberalization is announced, the static pro…t gain from exporting for any non-exporting …rm z must be lower than the sum of the …xed export cost, f X , and the per-period ‡ow value of the sunk cost, R+ 1 R f XS : Consider now the periods after trade liberalization is announced but ahead of the actual reduction in trade costs . Abstracting from general equilibrium changes in the domestic market demand index D , and in the interest rate R, condition (20) remains unchanged. That is, before the reduction in trade costs, it is still the case that the static pro…t gains from exporting are lower than the annual …xed and sunk costs of export. Therefore, …rms that were non-exporters before the announced trade liberalization will not …nd it optimal to start exporting ahead of the reduction in trade costs.
Once the reduction in trade costs is materialized, condition (20) will be reversed for some …rms, who will then start exporting. Without productivity dynamics (and abstracting from some general equilibrium considerations before the reduction in trade costs), sunk export costs do not give rise to anticipated changes in export decisions. A similar argument can be applied to the innovation decision. Abstracting from general equilibrium changes in D and R, there would be no incentive to innovate ahead of the reduction in trade costs (in contrast to the results under Scenario 6). Figure 10 shows how the combination of sunk export costs and productivity dynamics does result in changes in export decisions ahead of the reduction in trade costs. The path and announcement regarding the trade costs are identical to Scenario 6 (depicted again in Panel A). We run one simulation using our previously calibrated model with only …xed export costs, and then another one with sunk export costs (parameterized as described for Scenario 7). Both cases include productivity dynamics ( z > 0) but exogenous innovation, and take into account the general equilibrium changes in the domestic market demand index D , and in interest rate R.
Panel B shows how the option value associated with the sunk export cost induces the entry of …rms into the export market -captured by the rising share of exporters in domestic revenuesahead of the decrease in the trade cost. In the pre-liberalization steady state, some …rms just below the export cost cuto¤ do not export due to the option value associated with waiting (to export): they want to mitigate potential losses from an adverse productivity shock following export market entry. The announcement of future trade liberalization then substantially reduces this option value of waiting to export for those …rms (the consequences of a future adverse shock are not as dire, as those …rms would still choose to export so long as the per-unit trading costs are reduced); and those …rms enter the export market ahead of the reduction in trade costs. This is re ‡ected in the increased share of exporters in domestic revenues ahead of the trade liberalization. This increase is substantially larger in the model with sunk export costs than in the model with only …xed export costs.
Productivity dynamics are key for this result. Without productivity dynamics ( z = 0) there are no noticeable di¤erences in the aggregate response of the economies with …xed and sunk export costs (even considering the general equilibrium e¤ects on D and R that we abstracted from in our analytic derivation above). 26

Conclusions
In this chapter, we have reviewed the recent literature that incorporates …rm dynamics in models of international trade. We have characterized key model ingredients that generate substantial aggregate transition dynamics stemming from endogenous shifts in the …rm-size distribution in response to trade liberalization. We showed how two ingredients, export market selection and …rm dynamics, are jointly needed to generate any endogenous transition dynamics. We then examined the aggregate e¤ects resulting from endogenous innovation responses and the anticipation e¤ects regarding the timing of trade liberalization (when it is expected to begin, and whether it is temporary or permanent). Our computational results show how the responses of trade volumes, innovation, and aggregate output vary greatly over time depending on the details assumed for …rm dynamics, en-dogenous innovation, and the expected time path of the trade liberalization. This has important consequences for many issues in international economics that rely on predictions for the e¤ects of globalization over time on those key aggregate outcomes such as trade volumes, innovation, and aggregate productivity (an endogenous source of comparative advantage). 27 For empirical work, it is important to recognize that a measured response to globalization at a given point in time represents just one snapshot for the e¤ects of globalization: one that may evolve considerably in the future (without any subsequent changes to economic fundamentals), and one that is sensitive to many unobservables regarding …rms'expectations and anticipations.
In order to keep our analysis tractable, we have limited our modeling exercise to just one form of international market participation (exports) and one form of innovation. As we noted in our introduction, the literature has examined many other forms of international market participation.
Our choice should not be construed to imply that dynamics are more important for exports than for those other forms of market participation. We picked exports as an example (one that is very well understood) to highlight the importance of dynamic considerations. Similarly, we recognize that we modeled innovation in a coarse, reduced form way. First, we have assumed that only labor is used for innovative activities, and thus have not captured how globalization may a¤ect the cost of innovation via changes in other input costs or spillovers. Second, we have not explicitly speci…ed the channels through which innovation/investment activities generate performance improvements -other than through a direct e¤ect on labor productivity. Such channels could include management quality, o¤shoring that breaks-up the production chain, development of a …rm's product range, changes in input usage due to the availability of imported intermediates, and marketing/relationship building with foreign buyers. Many of these factors apply di¤erentially across a …rm's destination markets, and could hence result in destination-speci…c dynamics that di¤er from those of our model in which productivity improvements apply world-wide. These are all fruitful areas for future work. Overhead production cost f where L P =M E = R l (z)M (z) dz is determined using (16). We can then solve aggregate output and productivity using (18) and (19), and the price level as P = 1 1 Z .

B No export market selection
We state our main result in the following proposition, which is straightforward to extend to asymmetric countries (see Atkeson and Burstein 2010). transits immediately to a new steady-state in which changes in the export share, s X , aggregate productivity, Z and output, Y , are: Proof: We …rst solve for the equilibrium in a symmetric steady-state. With f X = 0 for all …rms, (A.1) implies that all …rms export and the variable pro…ts of a …rm with productivity z are Hence, under the assumption that all …rms export, the Bellman equation in the steady state, (7), can be written with~ exp (z) replacing t (z), where~ = D 1 + 1 .
Our arguments in the previous section imply that a unique level of~ exists which satis…es the free-entry condition (10), independent of the parameter . The corresponding exit thresholds, z, innovation decisions, q (z), that solve the Bellman equation at this level of~ are the equilibrium exit and innovation decisions. Hence, these are also independent of . From (16), variable employment l(z) for each …rm is independent of , and so is s D and, from (A.2), the mass of entrants. The change in the other aggregate variables immediate follow from (17), (18), and (19). Given that …rms'productivity distribution M (z) is unchanged across steady-state, the adjustment to the new steady-state is immediate.

C No productivity dynamics
Proposition 2: Consider a world economy with positive …xed costs of trade (f X > 0), no productivity dynamics ( z = 0), and the distribution of entering …rms G is such that the distribution of exp (z) is Pareto, where > 1. Suppose that both the exit and export cuto¤s are interior, z D > z 0 , z X > z 0 . In this economy, a permanent reduction in marginal trade costs, , leads to an increase in the steady-state exit cuto¤, z, a decline in the export cuto¤, z X , and has no impact on the steady-state mass of entering …rms, M E . The economy transits immediately to a new steady-state.
Proof: We …rst solve for the steady-state. With the Pareto distribution of entering …rms and no productivity dynamics, the free-entry condition (10) can be written as or using the cuto¤ de…nitions in the model without productivity dynamics, exp ( z) = f = D and Note from this equation that a reduction in international trade costs results in a decline in pro…tability D and a rise in D (1 ) . Hence, z D rises and z X falls.
The labor market clearing condition (A.2) can be written, using (16) and the de…nition of the cuto¤s, as Combining (C.1) and (C.2), we obtain the result that in steady-state M E is independent of . In response to a permanent reduction in , with the mass of entering …rms unchanged and the exit cuto¤ z D increasing, the transition to a new steady-state is immediate. Note that in response to an increase in , even though the mass of entrants is unchanged, the exit cuto¤ falls, so the transition to the new steady-state is not immediate unless = 1.

D Not all …rms export, with time variation in …xed export costs
Consider the modi…ed version of our economy described in Section 5. Proof: The free-entry condition in this economy is The share of discounted time-series revenues of exports in total discounted time-series revenues of entering …rms is: Given a mass of entrants M E , the mass of non-exporters, M D , and exporters, M X , is The share of revenues accounted for by exports is: Note that if = 1, then s X =s X . Di¤erentiating the free-entry condition, Aggregate output and the price level are given by When trade costs fall, log 1 + 1 > 0, and we obtain the result stated in the proposition.

E Endogenous innovation
Here we provide some analytics to understand the response of innovation decisions q (z) to changes in trade costs. To simplify our analysis, we set …xed overhead costs to zero, f = 0, so that there is no endogenous exit, and we focus on steady-states.
We …rst use the Bellman equation (7) and free-entry condition (10) to determine the impact of changes in trade costs on …rm pro…tability D in equilibrium. Consider the impact of a decline in the marginal cost of trade, , on D . Since this raises 1 , D has to fall in equilibrium to restore the free entry condition. Note as well that D 1 must rise if the free entry condition is to be satis…ed. Hence, for …rms that do not switch export status, the pro…ts of exporters -proportional to D 1 + 1 , must rise relative to the pro…ts of non exporters. The export threshold falls so that some …rms that previously did not export now start to export.
Note that the magnitude of the decline in D in response to a decline in international trade costs is determined in large part by the distribution G of productivities of newly entering …rms. If newly created …rms tend to be small non-exporters, then free entry requires that the discounted expected value of pro…ts of these …rms remain roughly constant. In this case, D remains roughly constant and the pro…ts of for large exporting …rms, D 1 + 1 rises by roughly the change in A-5 1 + 1 . In contrast, if newly created …rms tend to be large exporting …rms, then free entry requires that D 1 + 1 remains roughly constant, and D falls.
We now examine the impact of these equilibrium changes in …rm pro…tability on the level of innovation. To do so, we solve for the innovation decisions q(z) in (7) as a function of variable pro…ts D and the other parameters of the model. The Bellman equation (7) is a standard problem of valuing the pro…ts of the …rm together with an option: the option to start exporting (recall that in this section we are abstracting from the option to exit by assuming f = 0, so V (z) = V o (z)).
The following Proposition characterizes the shape of the value function V (z) and the innovation decision q(z) in steady-state.
Proposition 4: Suppose that the …xed overhead costs is set to zero, f = 0. Then, the value function V (z) that solves (7) in steady-state has the form V (z) = A(z) exp(z) with lim z!1 A(z) = A X and lim z! 1 A(z) = A D , and the optimal q(z) has lim z!1 q(z) = q X and lim z! 1 q(z) = q D where A D and q D solve , (E.1) and A X and q X solve these two equations with the term D in (E.1) replaced with D 1 + 1 : These solutions have A X > A D and q X > q D : Moreover, A D ; A X and q D and q X are increasing in D , while A X and q X are decreasing in : Proof: The …rst part of the Proposition follows by construction. The term A D exp(z) with A D given as above represents the expected discounted present value of variable pro…ts of a …rm that sets its innovation decision q(z) to the constant q D and which never exports. Likewise, the term A X exp(z) with A X given as above represents the expected discounted present value of variable pro…ts of a …rm that sets its process innovation decision q(z) to the constant q X ; which always exports, and which is so large that the …xed cost of exporting f X is a negligible portion of its variable pro…ts.
For the second part of the Proposition, we show that A D is increasing in D . Di¤erentiating where we used the fact that innovation choice is optimal, @A D =@ q D = 0, and (1 ) [ q D exp( z ) + (1 q D ) exp( z )] < 1 to guarantee that the discounted value of pro…ts is …nite. This same logic can be used to show that A X > A D , A X is increasing in D , and A X is decreasing in : For the third part of the Proposition, we show that q D is increasing in A D . Di¤erentiating where we used @A D @ D > 0, and the assumption that c 00 ( q D ) > 0. The same logic is used to show that q X is increasing in D and decreasing in .
This Proposition implies that for very small …rms, the innovation decision q(z) is constant at q D across …rms. These …rms do not export and all grow at the constant rate [ q D exp( z ) + (1 q D ) exp( z )] in expectation. Likewise, for very large …rms, the innovation decision q(z) is constant at q X across …rms: These …rms do export and all grow at the constant rate [ q X exp( z ) + (1 q X ) exp( z )] in expectation. The intuition for how A i and q i change with changes in pro…tability is then straightforward. If D exp(z) or D 1 + 1 exp(z) rise, this raises the spread between the value of a …rm that successfully innovates to z + z relative to the same …rm that fails to innovate and falls to z z : From equation (9), this increased spread in pro…ts raises the incentives to engage in innovation.
Note that the responsiveness of very large and very small …rms' innovation decisions q X and q D to changes in pro…tability and marginal trade costs is determined by the curvature of the innovation cost function as indexed by c 00 (q)=c 0 (q): In particular, because the innovation choice is optimal, @A i =@ q i = 0; and hence the change in steady state innovation with a change in pro…ts is given by and d q X = c 0 ( q X ) c 00 ( q X ) : If c 00 ( )=c 0 ( ) is very large, then innovation decisions and …rm growth rates are not very responsive to changes in …rm pro…tability, while if this curvature is small, then innovation decisions and …rm growth rates are very responsive to changes in pro…tability. By a similar argument, this curvature of the innovation cost function c 00 ( )=c 0 ( ) also controls the di¤erence in the innovation decisions and implied growth rates of very large …rms ( q X ) and very small …rms ( q D ) in a steady-state.

A-7
With this Proposition, we have the following results regarding the impact of permanent changes in trade costs on the innovation decisions of very large …rms (exporters) and very small …rms (nonexporters). A reduction in the marginal costs of trade leads to a reduction in the innovation of very small …rms and an increase in the innovation of very large …rms relative to very small …rms. This result follows directly from the fact that a reduction in the marginal costs of trade reduces D and increases D 1 + 1 relative to D : The extent of reallocation of innovation from non-exporters to exporters depends in part on the size distribution of newly created …rms. If newly created …rms are small, then D 1 + 1 increases and innovation in very large …rms rises in absolute terms while D remains roughly constant leaving innovation in small …rms roughly unchanged. Conversely, if newly created …rms are large exporting …rms, then pro…ts D 1 + 1 and innovation in these …rms remains roughly unchanged, while for small …rms pro…ts and innovation falls.
Note that a reduction in the …xed costs of trade f X , by lowering the equilibrium level of pro…tability D , leads to a reduction in innovation in both very large and very small …rms (note that these …rms do not switch export status). Similar arguments give that a decline in the entry cost f E results in a decline in innovation for both very large and very small …rms. A-8