International Trade Dynamics with Sunk Costs and Productivity

Shocks

⇤

Daisuke Fujii

†

The University of Chicago

Job Market Paper

‡

November 25, 2013

Abstract

This paper offers a unified framework to analyze both short- and long-run trade dynamicsin a consistent manner. It explains “the international elasticity puzzle”, a low trade elasticityagainst temporary shocks and a high trade elasticity against permanent shocks, studied in Ruhl(2008). The model in this paper extends the idea of export sunk costs and uncertainty to moregeneral productivity shock processes by embedding the classical theory of “export hysteresis”into a continuous-time trade model with heterogeneous firms, and considers the effects of bothaggregate and idiosyncratic productivity shocks. A sharp analytical characterization of theequilibrium elucidates the microfoundations of trade dynamics linking a static trade model withheterogeneous firms and an international macroeconomic model. Due to the sunk costs anduncertainty, firms do not change their export status against small temporary shocks. Aggregateproductivity shocks and export sunk costs explain the elasticity puzzle because of the differentadjustments on the extensive margin. If the productivity shocks are idiosyncratic, the economyis in a steady state, with individual firms moving around within a stationary distribution ofproductivities. Export hysteresis gives rise to a region of firm productivity where both exportersand non-exporters coexist given the same current productivity. The full model incorporates bothtypes of shocks and offers realistic microfoundations of trade dynamics including simultaneousexport entry and exit, an evolving productivity density of exporters, and the sluggish traderesponse to aggregate shocks.

⇤I am grateful to Samuel Kortum and Nancy Stokey for their advice and constant encouragements. I would also liketo thank Fernando Alvarez, Costas Arkolakis, David Burk, Lorenzo Caliendo, Thomas Chaney, Robert Lucas, RobertShimer, Serginio Sylvain and the seminar participants of International Trade Lunch at the University of Chicago andYale University, and Capital Theory Workshop at the University of Chicago for helpful comments. All remainingerrors are my own.

†Email: fujii@uchicago.edu‡Latest version available at the author’s website: https://sites.google.com/site/fujii0622/research

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1 Introduction

This paper develops a unified theoretical framework that captures both “export hysteresis,” thereluctance of firms to enter or exit from export markets in response to cyclical trade shocks, and the“international elasticity puzzle,” a low trade elasticity against temporary shocks and a high tradeelasticity against long-lived shocks. It incorporates export fixed costs and sunk costs, as well asproductivity shocks, in a continuous-time framework with heterogeneous firms. Both aggregate andidiosyncratic productivity shocks are allowed. To develop intuition, two specialized versions of themodel, each with only one type of shock, are considered. Then the full model is studied. A sharpanalytical characterization of the equilibrium elucidates the microfoundations of trade dynamicslinking a static trade model with heterogeneous firms and an international macroeconomic model.

The elasticity of exports with respect to the terms of trade is estimated to be low in internationalreal business cycle (IRBC) models, typically ranging between 0.5 and 2. On the other hand, instatic general equilibrium models of trade that compare steady states, the elasticity with respectto trade libralizations is estimated to be quite high, typically ranging between 5 and 10. Ruhl(2008) recognizes this discrepancy, and coins the term “the international elasticity puzzle”. Hereconciles the contradictory findings by building a model in which business cycles are caused bytemporary productivity shocks, and exporting entails an up-front sunk cost. Due to the sunk cost,few firms enter the export market in response to temporary shocks while a larger number of firmsstart exporting when there is a tariff reduction, which is considered to be permanent. The differentadjustments on the extensive margin explain the elasticity puzzle. This paper extends that modelof export sunk costs and uncertainty to more general productivity shock processes and incorporatesrich microfoundations for trade dynamics.

The key economic mechanism is export hysteresis caused by sunk costs and uncertainty. Manyempirical studies (Roberts and Tybout (1997), Bernard and Jensen (2004), and Das, Roberts andTybout (2007)) find the existence of substantial up-front sunk costs for exporting and a high persis-tence of export participation. In an environment where an action of discrete choice requires a fixedcost and the future profit is uncertain, “hysteresis” behavior is prevalent. Hysteresis is defined asthe failure of an effect to revert itself when its underlying cause is removed. The export hysteresiswas first investigated by Baldwin (1988) and many others. Dixit (1989) analyzes the optimal entryand exit decisions of a firm when the output price follows a random walk and the action requiresa sunk cost. A firm waits to enter the export market until the output price rises sufficiently high,higher than the amortized sunk cost. After entry, the firm will not exit if the output price goesback to the original level. The firm continues to export until the output price drops to a sufficientlylow level, lower than its continuation fixed cost, implying that the firm might run a temporarynegative profit. The gap between the entry and exit triggers is called “the band of inaction”, whichcauses the export hysteresis behavior. The band of inaction is studied extensively in the literatureof investment under uncertainty pioneered by Dixit and Pindyck (1994). The model in this paper

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embeds the classical theory of export hysteresis into a continuous-time trade model to examine theeffects of both aggregate and idiosyncratic productivity shocks.

By employing the continuous time setup and techniques developed in stochastic optimal controlliterature, the model generates sharp characterizations of equilibrium variables, and the exportdynamics is described as a solution to a partial differential equation. Hence, the comparative staticsor dynamics can be seen in a transparent way. The main contribution of this paper is to presenta framework of trade dynamics, which is consistent with micro and macro empirical regularities atboth short- and long-run time horizons. The aggregate productivity shocks and export sunk costsare the essential ingredients to reconcile the international elasticity puzzle. Introducing idiosyncraticproductivity shocks that follow a Brownian motion leads to the endogenous Pareto distribution ofcross-sectional productivity, and gives rise to a range of firm productivity where both exporters andnon-exporters coexist given the same current productivity. The full model captures the effects ofboth types of shocks and offers realistic microfoundations of trade dynamics such as simultaneousexport entry and exit, an evolving productivity density of exporters, and the sluggish trade responseto aggregate shocks.

To build intuition, the model is presented step by step. After laying out the basic setup andexplaining the mechanism of export hysteresis, I first focus on the case with aggregate productivityshocks only. The idiosyncratic firm productivity is fixed at a Pareto distribution. The aggregateshock hits all firms proportionally. At any date, exporters and non-exporters are separated bya cutoff firm, which evolves stochastically. If the export profit of the cutoff firm is in the inactionregion, there is no firm entry nor exit, and the cutoff productivity does not change. When the exportprofit of the cutoff firm hits an upper bound, less productive firms enter the export market keepingthe cutoff profit constant. A similar result holds when the cutoff profit hits the lower bound. Dueto the inaction region, the extensive margin (set of exporters) of trade does not respond to a smalltemporary shock, so the trade elasticity only reflects the intensive margin in the short run. Whena series of favorable shocks hits a country, less productive firms start entering and the extensivemargin becomes operative. This results in a large trade elasticity in the long run. Therefore, theaggregate shock and sunk costs alone explain the international elasticity puzzle. Ruhl (2008) reviewsthe empirical evidence on the impact of the extensive margin adjustments in the long run in responseto trade liberalizations. Kehoe and Ruhl (2013) also report that the extensive margin accounts foran important part of trade growth following tariff reductions whereas it does not change in theabsence of trade policy or structural change.

After characterizing the equilibrium properties of the model with aggregate shocks only, I turnto the case where productivity shocks are idiosyncratic. A firm-level productivity evolves as ageometric Brownian motion which gives rise to the double-Pareto distribution of the cross-sectionalproductivity as in Luttmer (2007). The idiosyncratic shocks make the export profit stochastic,and hence, we observe the export hysteresis at the firm level. Nonetheless, the aggregate variablesand exporter distribution are constant in the stationary equilibrium. A range of firm productivity

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in which both exporters and non-exporters coexist given the same productivity level emerges. Themodel explains the smooth increase in export participation as firm size becomes larger. The exportersin the “mixed region” are the ones whose productivity hit the entry margin, but dropped to a lowerlevel. The non-exporters are those whose productivity has not reached the entry trigger yet since thelast time they exited from the export market. This version of the model is essentially the same asthe one in Impullitti et al. (2013), but simpler because I abstract from free entry and sunk costs forthe domestic market. This allows a sharp characterization of the stationary equilibrium and clearcomparative statics.

The full model considers the effects of both types of shocks. Both aggregate and idiosyncraticshocks follow a Brownian motion. Firms derive the optimal strategies for export entry and exit,which characterizes the inaction region. The distribution of relative firm productivity is Pareto, butthe exporter distribution evolves stochastically. The exporter dynamics can be characterized as thesolution to a partial differential equation, which enables us to analyze the dynamics in response toany aggregate shock paths without simulations. The full model exhibits simultaneous export entryand exit, “the mixed region” of exporters and non-exporters, and the sluggish transition dynamicsof exporter’s distribution in response to aggregate shocks. It offers an alternative explanation to theJ-curve relationship between the terms of trade and exports as in Backus et al. (1994). Rather thanthe time to build capital, the time to “build” the set of exporters explains the J-curve.

Besides Ruhl (2008), the papers by Alessandria and Choi (2007, 2012) are closely related to thisresearch. They develop an open-economy dynamic stochastic general equilibrium (DSGE) modelwith heterogeneous firms, export sunk costs and both idiosyncratic and aggregate productivityshocks to examine the aggregate effects of trade liberalizations. Though it can incorporate moregeneral features, the model is constructed in discrete time, and does not provide closed-form ex-pressions for the firm distribution nor optimal entry/exit strategies. Ghironi and Melitz (2005) alsoconstruct a two-country DSGE model with heterogeneous firms to study the exporter dynamicsand persistent deviations from PPP, but the model does not have sunk costs, and therefore, thereis no hysteresis. This paper aims to elucidate the mechanism of the exporter dynamics in moretransparent way using a continuos-time model. In addition to Dixit (1989) and Dixit and Pindyck(1994), many analytical techniques are borrowed from Stokey (2009). The self-contained book cov-ers the necessary mathematical tools for stochastic control models with fixed costs, and containsrich economic applications. As she suggests, those modeling tools and concepts may have a wideapplicability in many economic fields. The model developed in this paper can be applied to othersituations in which heterogeneous agents face sunk costs and uncertainty.

The paper proceeds as follows. The next section lays out the basic structure of the economy withheterogeneous firms. Section 3 describes a firm’s dynamic problem of entry and exit, and analyzes theproperties of the inaction region for a general diffusion process. Sections 4 characterizes the exportdynamics with aggregate shocks only, which explains the international elasticity puzzle. Section5 characterizes the stationary equilibrium in which only idiosyncratic shocks hit firms. Section 6

4

presents the full model which incorporates both aggregate and idiosyncratic productivity shocks.Section 7 calibrates the full model to match the U.S. data and performs numerical analyses. Section8 concludes.

2 Setup

Time is continuous and indexed by t. Consumers have “love for variety”, and their preferences aremodeled by a CES utility function as in many other trade models. A continuum of firms withmeasure M produces the differentiated varieties. There is a measure L of consumers, and eachconsumer supplies one unit of labor inelastically. Labor is the only factor of production. There aretwo countries, home and foreign. Foreign variables are denoted by an asterisk.

To understand the different effects of aggregate and idiosyncratic productivity shocks, I first focuson the case with only aggregate shocks up through Section 4. Aggregate productivity shocks hit allfirms proportionally. Due to the export sunk costs, firms solve not only a static profit maximizationproblem but a dynamic entry and exit problem for exporting.

2.1 Demand

Within a country, all consumers are identical and have the same homothetic risk-neutral preference.In what follows, I will focus on variables in the home country. Foreign variables are describedanalogously with an asterisk. At any date t, the representative consumer maximizes the followingexpected discounted utility

Ut = Et

ˆ 1

te�⇢(s�t)q(1�⇠)

s C⇠sds

�,

where ⇢ is the preference discounting rate, and qt is the consumption of freely traded homogeneousgood. The homogeneous good serves as a numeraire with its price normalized to one. The compositegood Ct is produced by combining differentiated goods according to the following CES aggregator

Ct =

ˆ⌦t

ct (!)✓�1✓ d!

�(

✓✓�1)

,

where ct (!) is the consumption of a variety !. The elasticity of substitution across varieties isdenoted by ✓ and common for both countries . The set of available goods ⌦t will be characterizedin a recursive general equilibrium evolving over time.

Total expenditure in the home country (home GDP) is denoted by Yt. The demand for thehomogenous good is

qt = (1� ⇠)Yt

5

and the demand for a differentiated variety ! at time t is

ct (!) =

✓pt (!)

Pt

◆�✓

Ct =

✓pt (!)

Pt

◆�✓ ⇠YtPt

,

where the consumption-based price index is defined as

Pt =

ˆ⌦t

pt (!)1�✓ d!

� 11�✓

There is no borrowing and saving decisions by households because all consumers are identical andtrade is assumed to be balanced. Thus, total expenditure must equal the sum of labor income andtotal profits of the home firms at any time t, giving the following budget constraint

wL+⇧t = Yt

where ⇧t is the total profits (both from domestic and foreign markets) of the home firms in thedifferentiated good sector.

2.2 Trade Costs

There are three types of costs pertaining to export. The standard “iceberg” transportation cost ofexporting goods from home to foreign is denoted by ⌧ . If one unit of good is shipped, only a fraction1⌧ units arrives. In addition to this variable cost, a firm must pay a fixed cost f regardless of thequantity it exports. This fixed cost must be paid at any given time for a firm to remain as anexporter. There is no fixed cost for the domestic sales, so firms never exit from the domestic market.The third type of trade costs is a sunk cost for export entry denoted by . When a firm startsexporting, it must pay this sunk cost, which is assumed to be larger than the fixed cost: f < .This is a one-time sunk cost so that the firm does not have to pay it again if it continues to export.However, if a firm exits from exporting and wishes to resume exporting at a later date, it mustpay again. The examples of sunk costs include the costs for the initial marketing research andestablishing distribution networks in the foreign market. All trade costs are the same across firmswithin a country.

The fixed and sunk export costs f and are measured as a portion of the destination’s GDP.Thus, in the unit of numeraire, the fixed and sunk export costs are fY ⇤

t and Y ⇤t respectively. This

is similar to the usage of foreign labor for trade costs if there is no freely traded homogeneous goodand wage rates are determined in a general equilibrium. The proportionality of those costs to theforeign GDP simplifies the firms’ export entry and exit problem. Also, they are assumed to beiceberg costs, meaning that the incurred costs don’t go to labor income. This assumption simplifiesthe equilibrium conditions, but does not change the nature of the derived results.

6

The three types of trade costs have distinct effects on trade. The per-unit cost ⌧ affects theintensive margin, that is, how much each exporter exports. A permanent reduction in ⌧ also inducesmore firms to enter the export market affecting the extensive margin (number of exporters) in thelong run. The fixed cost f is responsible for the separation of exporters and non-exporters. Sincef is assumed to be the same for all firms, only efficient firms who are productive enough to recoupthe fixed cost can export. The sunk cost controls the sluggishness of trade adjustment on theextensive margin in a dynamic horizon. If is large, the set of exporters is very persistent. Thethree different roles played by each trade cost will become clear in Section 4.

2.3 Production

The homogenous and differentiated goods are produced under constant returns to scale. One unit oflabor in the home country produces w units of the homogeneous good. Since the homogeneous goodis freely traded and its price is normalized to be 1, the wage in the home country is w if this good isproduced. I only consider the equilibrium in which both countries produce some of the numeraire.This allows the countries to differ both in size L and in productivity w as in Chaney (2008).

There is a continuum of differentiated good producers in both countries. Firms are differentiatedby their idiosyncratic productivity � which is constant over time (the stochastic evolution of � willbe incorporated in Section 5 and 6). Also, a country is hit by an aggregate productivity shockZt at any given time. This aggregate shock hits only the differentiated good sector. Hence, theeconomy-wide productivity is given by w whereas Zt is the relative productivity of the differentiatedgood sector, which follows a diffusion process specified in the next section. Firms have the followinglinear production function

y (Zt;�) = Zt�l,

where y is the output and l is the labor used. Each firm maximizes its expected discounted sumof profits. The firm’s optimization problem has two components: a static monopolist’s profit maxi-mization problem in each period and a dynamic discrete choice problem of export entry and exit.

Let pD,t (�) and pX,t (�) denote the domestic and export prices respectively of a home firm withproductivity �. Hereafter, I will use subscripts D and X to denote domestic and export variables.The prices charged by firm � are

pD,t (�) =

✓✓

✓ � 1

◆w

Zt�

pX,t (�) = ⌧pD,t (�) =

✓✓

✓ � 1

◆⌧w

Zt�

7

The variable profits (gross of fixed export cost) are

⇡D,t (�) =

⇠

✓

✓pD,t (�)

Pt

◆(1�✓)

Yt = a1

✓Zt�Pt

w

◆✓�1

Yt (1)

⇡X,t (�) =

8><

>:

⇠

✓

✓pX,t (�)

P ⇤t

◆(1�✓)

Y ⇤t = a1

✓Zt�P

⇤t

⌧w

◆✓�1

Y ⇤t if firm � exports

0 otherwise(2)

where a1 =

⇠✓

⇣✓

✓�1

⌘1�✓. The net export profit is ⇡X,t (�) � fY ⇤

t =

a1

⇣Zt�P ⇤

t⌧w

⌘✓�1� f

�Y ⇤t if firm

� exports. Due to the sunk export cost and productivity uncertainty, the decision of export entryand exit will be optimally determined by the firm’s dynamic discrete choice problem.

2.4 Productivity Distribution

For both countries, the relative productivity distribution is assumed to be Pareto with a shapeparameter � and a lower bound �min > 0, so its cumulative distribution and density functions are

G (�) = 1�✓

�

�min

◆��

g (�) = ���(�+1)

���min

for � 2 [�min,1). The following assumption ensures that aggregate variables are finite.

Assumption 1. � > ✓ � 1

At any given time, there is a cutoff firm which separates exporters and non-exporters. Denotethe cutoff by ¯�t. Any firm with � > ¯�t exports at time t and the rest sell only in the domesticmarket. ¯�t is a key variable in the recursive equilibrium and evolves over time depending on thehistory of aggregate shocks. Define the following average productivities

˜� =

ˆ 1

�min

�✓�1dG (�)

� 1✓�1

= ⌫�min

˜�X,t =

"1

1�G�¯�t�ˆ 1

�̄t

�✓�1dG (�)

# 1✓�1

= ⌫ ¯�t

where ⌫ =

⇣�

��(✓�1)

⌘ 1✓�1 . When productivity follows a Pareto distribution, the average productivity

˜� is proportional to the lower bound. This result reflects the scale free property of the Paretodistribution, and simplifies the expressions of aggregate variables.

The measure of firms in the home country is fixed at M . There is no firm entry or exit for the

8

domestic market, so M is constant over time.1 Then the measure of exporting firms is

MX,t = M⇥1�G

�¯�t�⇤

= M

✓¯�t

�min

◆��

2.5 Aggregate Variables and Equilibrium

In terms of aggregate implications, this model is isomorphic to one where M firms of productivity˜� produce only for the domestic market and MX,t firms of productivity ˜�X,t export to the foreignmarket as demonstrated in Melitz (2003).

Let p̃D,t and p̃X,t be the average domestic and foreign prices of the home country. Using theprevious results, we have

p̃D,t = pD,t

⇣˜�⌘=

✓✓

✓ � 1

◆w

Zt⌫�min

p̃X,t = pX,t

⇣˜�X,t

⌘=

✓✓

✓ � 1

◆w⌧

Zt⌫ ¯�t

The home and foreign price indices can be written as2

Pt = a2

"M⇤ �

¯�t⇤�✓�1��

✓w⇤⌧⇤

Z⇤t

◆1�✓

+M (�min)✓�1��

✓w

Zt

◆1�✓# 1

1�✓

(3)

P ⇤t = a2

"M�¯�t�✓�1��

✓w⌧

Zt

◆1�✓

+M⇤(�min)

✓�1��

✓w⇤

Z⇤t

◆1�✓# 1

1�✓

(4)

where a2 =

⇣✓

✓�1

⌘�

�1�✓min⌫ . The home export profits are determined by the price relative to P ⇤

t .Notice that P ⇤

t is decreasing in both Zt and Z⇤t , and increasing in ¯�t. A favorable technological

shock in either country reduces the price level and increases welfare. Conditional on Zt and Z⇤t , high

¯�t implies that the price of imported goods (from foreign’s perspective) is higher, so the price indexwill be high as well.

In the current model, aggregate productivity shocks Zt = [Zt, Z⇤t ] and exporter cutoffs ¯

�t =

⇥¯�t, ¯�

⇤t

⇤are state variables. Unlike the Chaney-Melitz type trade models, the cutoff productivity

1Alternatively, we can incorporate free entry with fixed cost and exogenous firm death shock to allow firm entry.In that case, any aggregate productivity shocks are absorbed by the adjustment in M , and the average total (domestic+ foreign) profit is constant over time. To explicitly illustrate the different adjustments of intensive and extensivemargins against different kinds of shocks, I shut down the adjustment on M .

2Using the expressions for the average prices, the home price index can be written as

Pt =

hMp̃1�✓

D,t +M⇤X,t

�p̃⇤X,t

�1�✓i 1

1�✓=

"M

✓✓✓

✓ � 1

◆w

Zt⌫�min

◆1�✓

+M⇤✓

¯�⇤t

�min

◆�� ✓✓✓

✓ � 1

◆w⇤⌧⇤

Z⇤t ⌫�min

◆1�✓# 1

1�✓

9

¯�t cannot be pinned down by the structural parameters and destination’s characteristics. Due tothe sunk export cost and export hysteresis behavior, we need a complete history of the shocks Z toobtain ¯

�t. Given Zt and ¯

�t, the price indices are calculated according to (3) and (4). The averagenet profit in the home country is

⇡̃t = ⇡̃D,t +MX,t

M[⇡̃X,t � fY ⇤

t ]

where ⇡̃D,t = ⇡D,t

⇣˜�⌘

and ⇡̃X,t = ⇡X,t

⇣˜�X,t

⌘are the average profits of all firms and exporters

respectively. The total home expenditure is

Yt = wL+M ⇡̃t

In this model, ⇡̃t acts like wage in standard general equilibrium models. The average profit ⇡̃t affectsboth income and expenditure of a country and the vector [⇡̃t, ⇡̃⇤t ] is the fixed point of the two excessdemand functions. The derivations of the equilibrium average profits ⇡̃t and ⇡̃⇤t are relegated toAppendix A.1. Trade is not necessarily balanced in the differentiated good sector, but I assume anyimbalances in the differentiated good sector is made up for in the homogeneous good sector so thataggregate trade is balanced.

3 Export Hysteresis

3.1 Firms’s Dynamic Problem

Since there is no fixed or sunk entry costs for the domestic market, all firms sell in the domesticmarket. For the foreign market, they solve a dynamic optimal entry and exit problem due to theexport sunk cost and profit uncertainty.

Let ⇡̂X,t =⇡X,t

Y ⇤t

be the export profit normalized by the foreign GDP. Using (2) and (4), it canbe written as

⇡̂X,t

✓Z⇤t

Zt, ¯�t,�

◆= a3 ⇥ �✓�1

|{z}fixed, idiosyncratic

⇥

0

B@1

M⇣

�̄t

�min

⌘✓�1��+M⇤

⇣Z⇤t

Zt

ww⇤ ⌧

⌘✓�1

1

CA

| {z }stochastic, aggregate

(5)

where a3 is a constant.3 If we take the log of (5), it is clear that the stochastic movement of theexport profit is the same for all firms. Only the level is different due to the idiosyncratic productivityheterogeneity. Notice that the normalized export profit depends on the ratio of the productivityshocks Z⇤

tZt

. If aggregate productivities rise by the same magnitude in both countries leaving Z⇤t

Zt

3a3 =

a1

a1�✓2 (�min)✓�1��

=

⇠✓

⇣��(✓�1)

�

⌘�(1�✓)min

10

constant, export profit is not affected. In that situation, the export price pX,t will decrease due tothe home productivity shock, but the foreign price index will also decrease by the same proportiondue to the foreign shock, which offsets the extra profit. Also, the home exporter cutoff ¯�t is theaggregate state variable which evolves over time depending on the history of productivity shocksas shown below. Since � > ✓ � 1, when ¯�t increases, ⇡X,t (�) increases. For exporters, an increasein ¯�t implies a higher foreign price index implying milder competition there. In the dynamic entryand exit problem, firms take the stochastic evolution of ⇡̂X as exogenous, and compare the expectedvalue of being an exporter to the sunk costs to determine the optimal entry and exit thresholds.Conditional on exporting, each firm follows the monopolist’s profit maximization rule described inSection 2.3. Given that result, they solve a dynamic stochastic control problem of export entry andexit.

Recall that the fixed and sunk export costs fY ⇤t and Y ⇤

t are proportional to the foreign GDP.Normalizing the export profit and the associated costs by Y ⇤

t , we can analyze the stochastic evolutionof ⇡̂X,t in relation to f and to derive the optimal entry and exit policies. The proportionality ofexport costs to Y ⇤

t prove useful in this analysis since any effect of Y ⇤t on the export profit is offset by

the same change in export costs, and hence, we can focus on the effect of the relative price pX,t(�)P ⇤t

on

trade. Because ⇡̂X,t in (5) only depends on Z⇤t

Zt, ¯� and �, it is convenient to define the log of relative

productivity as

zt = ln

✓Zt

Z⇤t

◆

Then, the state variables of ⇡̂X,t

�zt, ¯�t,�

�are zt and ¯�t. In this section, it is assumed that the

effect of ¯�t on export profit is negligible so that the only stochastic variable of ⇡̂X,t (zt,�) is zt. Thisassumption corresponds to the case of a small home country discussed in Section 4. The generalcase where the export profit is a function of both zt and ¯�t is discussed in Appendix A.3.

Firms maximize the expected discounted sum of export profits net of trade costs by optimallychoosing the timings of entry and exit. Due to the aggregate shock, the export profit follows adiffusion process. This is a classical problem of investment under uncertainty (e.g. Dixit (1989) andDixit and Pindyck (1994)). An exporter and a non-exporter can be viewed as assets that are calloptions on each other, and the optimal strategy is characterized by a pair of threshold productivitylevels. The export sunk cost and uncertainty coming from aggregate productivity shocks give riseto export hysteresis studied by Baldwin (1988) and Baldwin and Krugman (1989).

3.2 Hamilton-Jacobi-Bellman (HJB) Equations

Hereafter, I will use ⇡ (z) to denote the normalized export profit ⇡̂X,t (z,�) to simplify notations.The subscript � is dropped since the structure of the problem is identical for all firms. Yet, it shouldbe understood that all value functions and derived triggers depend on �. Consider the following

11

general diffusion process for z

dz = µ (z) dt+ � (z) dW

where W is a standard Brownian motion (Wiener process) and µ (z) and � (z) are the infinitesimaldrift and variance parameters respectively. They are time invariant, but can depend on the currentstatus of z. Let V0 (z) to be the value function for a non-exporter when the current aggregate shockis z, and similarly V1 (z) for an exporter. By Ito’s lemma, we obtain

dV0 = V 00 (z) dz +

1

2

V 000 (z)�2 (z) dt

Therefore,

E (dV0) =

µ (z)V

00 (z) +

1

2

�2 (z)V000 (z)

�dt

The asset equilibrium condition gives the following Hamilton-Jacobi-Bellman (HJB) equations forV0 and V1

4

⇢V0 (z) = µ (z)V 00 (z) +

1

2

�2 (z)V 000 (z) (6)

⇢V1 (z) = ⇡ (z)� f| {z }dividend

+µ (z)V 01 (z) +

1

2

�2 (z)V 001 (z)

| {z }capital gain

(7)

The interpretation of the above HJB equations is straightforward. The return on the asset (LHS)must equal the sum of dividend and the capital gain (RHS). For non-exporters, there is no dividendprofit flow, but V0 is still positive due to the option value of entering the export market in the future.For exporters, the dividend is the net export profit ⇡ (z)� f and the capital gain is the option valueof exiting from export market. Both equations are second-order linear ODE, but (6) is homogeneouswhereas (7) is nonhomogeneous. The ODE’s (6) and (7) characterize the general solutions for V0

and V1. To pin down the particular solutions, we need boundary conditions. For a second-orderlinear ODE, two boundary conditions are required to characterize the particular solution. Since wehave two of those ODE’s, we need four boundary conditions in total.

The optimal solution to this stochastic dynamic control problem is characterized by two thresholdlevels of z: zL and zH with zL < zH . zH is the threshold at which a non-exporter becomes anexporter, and zL is the threshold at which an exporter becomes a non-exporter. Since zL < zH , thisis an inaction region. If the current aggregate shock z is in this range [zL, zH ], the optimal policy isto remain the status quo; exporters should continue exporting and non-exporters should not startexporting. Thus, the current aggregate productivity z is not a sufficient statistic to determine if aparticular firm is an exporter or not if z falls in the inaction band for the firm. The definition of zL

4See Chapter 3 of Stokey (2009) for the detailed treatment of HJB equations. Basically, HJB equations can bederived by considering a small change �t, applying E (dV ) and taking the limit �t ! 0.

12

and zH suggests the following value-matching conditions5

V0 (zL) = V1 (zL) (8)

V0 (zH) = V1 (zH)� (9)

These boundary conditions pin down the form of V0 and V1. For optimality, we need the followingsmooth pasting conditions6

V 00 (zL) = V 0

1 (zL) (10)

V 00 (zH) = V 0

1 (zH) (11)

Above four conditions (8) - (11) characterize the particular solutions for V0 and V1, and the optimalthresholds zL and zH .

Because the two HJB equations (6) and (7) share the same homogeneous part, it is convenientto define the following function as in Dixit (1989)

U (z) = V1 (z)� V0 (z)

Then, we can collapse the two HJB equations into

⇢U (z) = ⇡ (z)� f + µ (z)U 0(z) +

1

2

�2 (z)U 00(z) (12)

The value-matching and smooth pasting conditions become

U (zL) = 0, U (zH) = , U 0(zL) = 0, U 0

(zH) = 0 (13)

Any two of the above boundary conditions determine the particular solution for U (z) and the othertwo conditions are required to solve zL and zH . Therefore, for any general diffusion process of z

with µ (z) and � (z), above expressions (12) and (13) determine zL and zH . An example of V0, V1

and U is illustrated in Figure 1. Because V0 (z) only consists of the option value of entering theexport market, it must have limz!0 V0 (z) = 0 since when z is very small, the probability of rising tozH in any given finite time is almost zero. Also V1 (z) consists of the profit flow which is increasing

5The non-exporter value function can be expressed as

V0 (z) = max

⇢V1 (z)� , V0 (z) +

µ (z)V

00 (z) +

1

2

�2(z)V

00(z)� ⇢V0 (z)

�dt

�

for the region of z where being a non-exporter is optimal. At zH , the firm is indifferent between being an exporterand non-exporter implying V0 (zH) = V1 (zH)� . A similar argument holds for the case of zL.

6See Chapter 4 of Dixit and Pindyck (1994) and Stokey (2009) for more detailed treatments of this condition. Ifthe smooth pasting conditions are not satisfied, two value functions are connected at a kink. It can be shown thatthe optimal switching strategies are no longer valid due to the local convexity around the kink.

13

in z and the option value of exiting which is decreasing in z. V1 (z) exhibits a U-shape: when z issmall, the option value dominates, and when z is large, the profit flow dominates. When z follows aBrownian motion, closed-form solutions for V0 (z) and V1 (z) are available, but in general, one mustsolve the ODEs numerically.

3.3 Band of Inaction

Because zL < zH , the band of inaction [zL, zH ] causes the export hysteresis behavior. We canconvert the band of inaction of z to that of the export profit, [⇡L,⇡H ]. Consider a firm whoseexport profit ⇡X is in the inaction region [⇡L,⇡H ]. The firm starts exporting when the export profit⇡ goes above ⇡H but does not stop exporting when ⇡ comes back to the original level. When ⇡

decreases to low enough, which is ⇡L, the firm will exit. Dixit (1989) analytically characterizes theband of inaction, and performs comparative statics when ⇡ follows a geometric Brownian motion.He demonstrates that ⇡L < f and f + ⇢ < ⇡H . The inaction region is illustrated in Figure 2.The standard Marshallian trigger profit levels are f and f + ⇢. When there is no uncertainty, afirm should start exporting when ⇡X rises to f + ⇢, which is the amortized full cost of exporting.Similarly, an exporter should exit when its export profit drops to f . The presence of uncertaintyexpands the Marshallian inaction band [f, f + ⇢]. Thus, it is possible that an exporter receivesnegative profit when ⇡ falls in [f,⇡L] since the firm hopes ⇡ to rise in the near future. Similarly,the profit level of f + ⇢ is not enough to attract a non-exporter to become an exporter. Due tothe uncertainty, ⇡ must rise more than f + ⇢ to attract a non-exporter. As �2⇡ ! 0, ⇡H ! f + ⇢

and ⇡L ! f . Also, as becomes smaller, the inaction region [⇡L,⇡H ] shrinks but doesn’t vanish.In essence, sunk cost creates the inaction band and the uncertainty parameter �2 amplifies it.Export hysteresis is the key ingredient to derive the results in the following sections.

4 Model with Aggregate Shocks

This section analyzes the trade dynamics in which the source of uncertainty only comes from theaggregate productivity shocks. I focus on the case of small home country to elucidate the workings ofexport hysteresis on the international elasticity puzzle. The analysis of the general case is presentedin Appendix A.3 because the full model also uses the small home country assumption. The followingassumption is placed in this section.

Assumption 2. The home mass of firms is small: M ⇡ 0.

With this assumption, the share of home firms in the foreign price index is negligible so that ¯�t

does not affect P ⇤t nor ⇡X,t. The normalized export profit (5) can be expressed as

⇡̂X,t (z,�) = S (�) e(✓�1)z, (14)

14

V0HzLV1HzL

0.2 0.4 0.6 0.8 1.0z

5

10

15

20

25

30

value

zL zH

k

(a) V0 (z) and V1 (z)

U@zD

0.2 0.4 0.6 0.8 1.0z

1

2

3

4

5

6

value

zL zH

(b) U (z)

Figure 1: An example of V0 (z) , V1 (z) and U (z)

15

π_L f f+ρκ π_H

π

inaction region exportdont' export

negative profits

Figure 2: Entry and exit decisions

where S (�) = a4�✓�1.7 The scaling constant S (�) does not change over time. Because the state

variable of ⇡̂X,t is z alone, we can apply the results of the previous section to derive the inactionregion. Another ingredient for the problem is the diffusion process of z, which is described below.

4.1 Band of Inaction: Brownian Motion

Suppose z follows a Brownian motion with the following law of motion

dz = µdt+ �dW

where W is a Wiener process (standard Brownian motion). The infinitesimal drift and varianceparameters µ (z) and � (z) don’t depend on the current level of z. The HJB equations for exportersand non-exporters become

⇢V0 (z) = µV 00 (z) +

1

2

�2V 000 (z) (15)

⇢V1 (z) = Se(✓�1)z � f + µV 01 (z) +

1

2

�2V 001 (z) (16)

In this case, we obtain closed-form solutions for V0 and V1.

Lemma 1. The value functions that solve the HJB equations (15) and (16) take the following form

V0 (z) = Ae↵z (17)

V1 (z) = Be��z+

⇡ (z)

⌘� f

⇢(18)

7a4 =

a3M⇤

⇣w⇤

⌧w

⌘✓�1(�min)

��(✓�1)

16

where

↵ =

�µ+

pµ2

+ 2⇢�2

�2

�� =

�µ�p

µ2+ 2⇢�2

�2

⌘ = ⇢� �2

2

(✓ � 1)

2 � µ (✓ � 1)

⇡ (z) = Se(✓�1)z

and A and B are constants determined by associated boundary conditions. 8

Proof. See Appendix A.2.

The non-exporter’s value function V0 (z) is valid in the interval (�1, zH) and the exporter’svalue function V1 (z) is valid in (zL,1). The U (z) function becomes

U (z) = Be��z �Ae↵z +⇡ (z)

⌘� f

⇢,

and the associated boundary conditions provide the following system of four equations which char-acterizes A,B, zL and zH .

F =

2

66664

F1 (A,B, zL, zH)

F2 (A,B, zL, zH)

F3 (A,B, zL, zH)

F4 (A,B, zL, zH)

3

77775=

2

66664

Ae↵zL �Be��zL � Se(✓�1)zL

⌘ +

f⇢

Ae↵zH �Be��zH � Se(✓�1)zH

⌘ +

f⇢ +

↵Ae↵zL + �Be��zL � (✓ � 1)

Se(✓�1)zL

⌘

↵Ae↵zH + �Be��zH � (✓ � 1)

Se(✓�1)zH

⌘

3

77775=

2

66664

0

0

0

0

3

77775(19)

We do not obtain the explicit analytical expressions for zL and zH , but we can do comparativestatics analytically using the implicit function theorem. Figure 1 plots the value functions whenz follows a Brownian motion with the parameters summarized in Table 1. The exit and entrytriggers are zL = 0.311 and zH = 0.98, that imply ⇡L = 0.638 and ⇡H = 1.74. We confirm that0.638 = ⇡L < f = 1 and 1.35 = f + ⇢ < ⇡H = 1.74. Thus, it is possible that an exporter receivesnegative profit when ⇡ falls in [f,⇡L] hoping ⇡ to rise in the near future. Similarly, the profit level

8The second part of equation (18) has a nice interpretation. Since z follows a Brownian motion with µ and �2,the aggregate productivity Z

Z⇤ = ez follows a geometric Brownian motion with a drift µ +

12�

2 and variance �2.The additional 1

2�2 term comes from the convexity of exponential function combined with Ito’s lemma. Then, ⇡ (z)

follows a geometric Brownian motion with a drift µ⇡ = (✓ � 1)µ +

12 (✓ � 1)

2 �2 and variance �2⇡ = (✓ � 1)

2 �2. Sothe following relationship holds

⇡⌘� f

⇢=

⇡⇢� µ⇡

� f⇢= E

ˆ 1

0

e�⇢t(⇡t � f) dt

�

Thus, the second part of (18) is the expected present value of exporting if the firm exports forever given the initialexport profit ⇡. This is explained in Dixit (1989). The first part of (18) Be��s corresponds to the option value ofexiting from the export market optimally. Similarly, V0 (s) = Ae↵s is the option value of entering the export market.

17

µ �2 ✓ � 1 ⇢ S f

value 0.01 0.01 1.5 0.07 0.4 1 5

Table 1: Baseline parameters for the Brownian motion

of f + ⇢ is not enough to attract a non-exporter to become an exporter. Due to the uncertainty, ⇡must rise more than f + ⇢ to attract the non-exporter.

Again, it should be noted that each firm � derives different inaction regions of the aggregateshock [zL,�, zH,�] because the scaling constant S = a4�

✓�1 is different. Let ⇡L,� = ⇡X,� (zL,�) and⇡H,� = ⇡X,� (zH,�) be the lower and upper threshold levels of export profit for firm �. With theBrownian motion assumption, we have the following lemma.

Lemma 2. When the aggregate shock z follows a Brownian motion, the entry and exit trigger profits⇡L,� and ⇡H,� are homogeneous of degree one in f and jointly.

When z follows a Brownian motion, the export profit follows a geometric Brownian motion. Theabove result can be found in Dixit (1989) and Dixit and Pindyck (1994). Since ⇡X , f and aremeasured in the same unit, any rescaling by f and does not change the nature of the problemwhen ⇡X follows a geometric Brownian motion.

4.2 Evolution of the Cutoff Firm

Each firm � solves the above problem to determine the optimal threshold levels of the aggregateshock: zL,� and zH,�. The following lemma can be proved.

Lemma 3. When z follows a Brownian motion, the threshold levels of export profit ⇡L = ⇡X,� (zL,�)

and ⇡H = ⇡X,� (zH,�) do not depend on �.

Even though each firm derives different threshold productivity levels, the threshold export profitlevels are the same for all firms. This is evident by considering the parallel movements of ln ⇡̂X,�

of different �’s as pointed out in 3.1. We can also see the result by considering the movement ofthe export profit. When z follows a Brownian motion with µ and �2, the export profit ⇡ follows ageometric Brownian motion with a drift µ⇡ = (✓ � 1)µ+ 1

2 (✓ � 1)

2 �2 and variance �2⇡ = (✓ � 1)

2 �2.Then, the HJB equation (12) can be written in terms of ⇡

⇢U (⇡) = ⇡ � f + µ⇡⇡U0(⇡) +

1

2

�2⇡⇡2U 00

(⇡)

All firms face the same problem and derive the common thresholds ⇡L and ⇡H .Because firm’s idiosyncratic productivity � is constant over time and the aggregate shock affects

the export profit of all firms proportionally, there is a cutoff firm ¯�t which separates exporters andnon-exporters at any given time. Firms with � > ¯�t export and firms with � < ¯�t are non-exporters.

18

Denote the export profit of the cutoff firm by ⇡̄t = ⇡X�¯�t�. The aggregate shock influences all firm’s

export profit including that of the cutoff firm ⇡̄t. Whenever the extensive margin of trade (set ofexporters) changes, the entry or exit must start from the cutoff firm ¯�t. Figure 3 illustrates theevolution of ⇡̄ and ¯� in this environment. The aggregate productivity shock z (the top line plotof Figure 3) starts rising at t1 which raises ⇡̄ (the middle plot) as well. At t2, ⇡̄ hits the upperthreshold ⇡H . At this point, firms below ¯� enters the export market, which implies a decrease in ¯�

(the bottom plot). When ⇡̄ is at ⇡H , any further positive shocks will be absorbed by the adjustmentin ¯� keeping ⇡̄ constant at ⇡H . At t3, z starts falling, and hence, ⇡̄ falls as well. Yet, this decline in⇡̄ does not affect ¯� because ⇡̄ is still above ⇡L which is the exit trigger. So the current export profitis not low enough to induce the threshold firm to exit. The aggregate shock z bumps up betweent4 and t5 but this does not change ¯� either because ⇡̄ does not hit the upper threshold. From t5, zand ⇡̄ start falling again, and ⇡̄ hits the lower threshold ⇡L at t6. Exporters start exiting from firm¯� first. The negative shock continues until t7, so ¯� rises between t6 and t7 making ⇡̄ constant at ⇡L.At t7, the shock turns to positive, and ¯� stops rising.

4.3 Short Run vs. Long Run

The interaction between ⇡̄ and ¯� along with the inaction band [⇡L,⇡H ] generates the main resultsin this section. The inaction band emerges from the sunk export cost and uncertainty. As long asthe cutoff profit ⇡̄ remains within the inaction band, the set of exporters does not change. When thecutoff profit is at one of the thresholds, any further shocks are completely absorbed by the updatingof the cutoff firm ¯�. The current model generates a low trade elasticity in the short run and a highelasticity in the long run as summarized in the next proposition.

Proposition 1. (Short run vs. Long run) Let ✏xz be the trade elasticity with respect to the aggregateshock Z

Z⇤ . With Assumptions 1 and 2,

✏xz =

(✓ � 1 if ⇡̄ 2 (⇡L,⇡H) (short run)

� if ⇡̄ = ⇡L or ⇡H (long run)

From Assumption 1, the long-run trade elasticity � is larger than the short-run elasticity ✓ � 1.I will use ✏XY to denote the elasticity of X with respect to Y throughout the paper. Denote

the export sales of firm � and the aggregate export sales (normalized by Y ⇤t ) by xt (�) and Xt

respectively. Using the results in Section 2, it is informative to express the aggregate export as

Xt = MX,t| {z }set of exporters

⇥ xt

⇣˜�X,t

⌘

| {z }average export

= a4 ⇥�¯�t��� ⇥ e(✓�1)z

¯�✓�1t

19

!!!!!!

t1 t2 t3 t4 t5 t7

t6

Time

Figure 3: Dynamics of ln ⇡̄ and ln

¯�

20

!!

Figure 4: The cutoff exporter ¯� and export profit ⇡̄

where a4 is a constant. Taking the log of the above expression yields

lnXt = ln a4 + (✓ � 1) zt � (� � (✓ � 1)) ln

¯�t

The trade elasticity with respect to z is

✏xz = (✓ � 1) + (✓ � 1� �) ✏�̄z

where ✏�̄z is the elasticity of ¯� with respect to z. When ⇡̄ 2 (⇡L,⇡H), ✏�̄z = 0 and when ⇡̄ is onone of the boundaries, ¯� is being updated to keep ⇡̄ /

�¯� ZZ⇤�✓�1 constant which implies ✏�̄z = �1

confirming Proposition 1. Because the elasticities of ⇡̄ and ¯� with respect to z are ✓ � 1 and �1

respectively, when there is a linear movement in z, ⇡̄ and ¯� also have linear movements with slopesmultiplied by ✓�1 and �1 as shown in Figure 3. The log of the cutoff profit ln ⇡̄ follows a regulatedBrownian motion with the two-sided regulators ln⇡L and ln⇡H . The long-run stationary probabilitydensity of x = ln ⇡̄ is

(x) =�

e�xH � e�xLe�x

where � =

2(✓�1)µ

(✓�1)2�2 and xi = ln⇡i for i = L,H (see Chapter 3 of Dixit and Pindyck (1994) for

derivation). If µ = 0, the long-run average of ln ⇡̄ is ln⇡L+ln⇡H2 .

In most of the time, the cutoff profit ⇡̄ is in the inaction band, so only the intensive margin(average export sales of the current exporters) adjust to the movement of z (short run). When

21

either positive or negative shocks keep hitting the economy, the extensive margin (change in ¯�)will be operative, which generates a larger trade elasticity. When = 0, the inaction band vanishes(⇡L = ⇡H) and the result of Chaney (2008) is recovered. Because there is no sunk cost, the extensivemargin always adjusts to a change in z generating the trade elasticity �.

5 Model with Idiosyncratic Shocks

This section presents a trade model in which there is no aggregate shock, but idiosyncratic pro-ductivity evolves as a Brownian motion. The growth process of firm productivities is borrowedfrom Gabaix (1999) and Luttmer (2007). I abstract from the balanced growth path analysis, butincorporate export sunk costs in the model. Due to the sunk costs and idiosyncratic productivityuncertainty, each firm exhibits the export hysteresis. Yet, a stationary equilibrium emerges in whichthe productivity distribution of all firms and exporters are both modeled by a double Pareto dis-tribution as in Luttmer (2007). The same idea is investigated in Impullitti et al. (2013), but theiranalysis is restricted for the case of two symmetric countries. The current model can be extendedto asymmetric multi-country case.

5.1 Productivity Shocks and Stationary Distribution

The basic setup is the same as in Section 2, but the Pareto distribution of productivity endoge-nously emerges from the random growth of idiosyncratic productivities. The log of idiosyncraticproductivity � is assumed to evolve as a Brownian motion. Define s = ln�. The law of motion of sis

ds = µdt+ �dW (20)

where W is a standard Brownian motion. The entry point of productivity is fixed at s⇤. When newfirms enter the economy, they all start from productivity s⇤. After that, each idea follows a diffusionprocess as specified in (20). There is an exogenous death shock rate � which hits all ideas uniformly.Let a be the age of a firm (time elapsed after birth) and f (a, s) be the density of s of age a. Thefollowing Kolmogorov Forward Equation (henceforth KFE) describes the evolution of productivitydensity f (a, s)9

@f (a, s)

@a= ��f (a, s)� µ

@f (a, s)

@s+

1

2

�2@2f (a, s)

@s2(21)

The first term on the RHS reflects the reduction of density due to the exogenous death shock. Thesecond term reflects the outflow of the density due to the drift of s. The third term reflects theinflow of density due to randomness. The solution to the above second-order PDE is the following

9See Chapter 3 of Dixit and Pindyck (1994) for more details. The KFE is “the balance equation”, which links thedensities of evolving particles at different points in time.

22

discounted normal density with mean s⇤ + µa and variance a�2

f (a, s) = e��a 1p2⇡a�2

e�[s�(s⇤+µa)]2

2a�2 (22)

The derivation of the above solution can be found in Harrison (1985) and Luttmer (2007). Theintuition for the above density is clear. All ideas starts from s⇤ and follow a Brownian motion. Sincethe drift and variance are proportional to the elapsed time a, the distribution of productivities ofage cohort a has a mean s⇤ + µa and variance a�2. Due to the death shock �, the density of agecohort a must be discounted by �a. This implies that old ideas have a small weight since most oftheir cohort have died out up until time a. The first term e��a reflects this discounting. Conditionalon surviving, they have larger variance.

Let f (s) be the stationary probability density of s. This is the marginal density of f (a, s) withrespect to s: f (s) =

´10 f (a, s) da. By the definition of stationarity, we have @f(s)

@a = 0. Then, weobtain the following KFE which describes f (s)

0 = ��f (s)� µ@f (s)

@s+

1

2

�2@2f (s)

@s2(23)

This is a homogeneous second-order linear ODE (not PDE). The general solution to this ODE takesthe following form

f (s) = C1e�1s

+ C2e��2s (24)

where �1 and ��2 are the two solutions to the following characteristic equation

1

2

�2�2 � µ� � � = 0

Quadratic formula gives

�1 =µ+

pµ2

+ 2��2

�2

��2 =µ�

pµ2

+ 2��2

�2

(25)

so that �1, �2 > 0. Since � > 0, �1 > 2µ�2 and �2 > 0. The constants C1 and C2 are determined by

additional conditions. The stationary probability density f (s) must satisfy the following conditionsas well (this is from Arkolakis (2011))

lim

s!�1f (s) = 0 (26)

f (s) � 0 8s 2 (�1,1) (27)ˆ s⇤

�1f (s) ds+

ˆ 1

s⇤f (s) ds = 1 (28)

�µ [f (s⇤�)� f (s⇤+)] +

1

2

�2⇥f 0

(s⇤�)� f 0(s⇤+)

⇤= � (29)

23

The first condition is the lower boundary condition. For the shock process of s, �1 is the absorbingbarrier. The second and third conditions ensure that f (s) is a probability density. The last conditionemerges by integrating the KFE (23). This states that the net inflows to the density are equal tothe net outflows. The particular solution of (24) which satisfies the above four conditions is givenby

f (s) =

8><

>:

�1�2�1 + �2

e�1(s�s⇤) for s < s⇤

�1�2�1 + �2

e��2(s�s⇤) for s � s⇤(30)

where derivations can be found in Reed (2001) and Arkolakis (2011).10 This is a skewed (asymmetric)Laplace distribution with a mode at s⇤. For the convergence of the stationary distribution, it isassumed that � > 0. The resulting stationary distribution of � = es is the following11

g (�) =

8>>><

>>>:

�1�2�1 + �2

��1�1

(�⇤)�1for � < �⇤

�1�2�1 + �2

���2�1

(�⇤)��2for � � �⇤

(31)

where �⇤ = es⇤ . This is called double Pareto distribution since both lower and upper tails exhibit

power law behavior.The entry point �⇤ is a scale parameter which controls the absolute advantage whereas �1 and

�2 are shape parameters that govern comparative advantage. They are similar to the location andshape parameters of the Frechet distribution introduced in Eaton and Kortum (2002). The enteringrate of new firms to the domestic market is fixed to keep M constant. Since there is no sunk or fixedcost for the domestic sales, the above stationary distribution arises independent of the exporter’sdistribution.

5.2 Band of Inaction

Because of the export sunk cost and idiosyncratic productivity uncertainty, firms solve the samedynamic entry and exit problem as in subsection 4.1. In this section, the foreign GDP, Y ⇤ isconstant in the stationary equilibrium, and thus, the fixed and export costs are not tied to theforeign GDP (they are just f and ). Define the discounting rate which augments the death shockrate as r = ⇢+�. The export profit ⇡X follows a geometric Brownian motion, and the HJB equations

10Reed (2001) decomposes the expression (22) into exponential and normal distributions, and uses their momentgenerating functions (MGF) to characterize the MGF of the resulting distribution which can be verified as theasymmetric Laplace distribution (30). Arkolakis (2011) explicitly calculates the marginal density f (s) =

´10

f (a, s) dato show the same result.

11Let � = h (s) = es so that h�1(�) = ln�. The PDF of � is given by the following equation

g (�) =

����dd�

h�1(�)

���� f�h�1

(�)�

24

are the same except that ⇢ is now replaced by r. Lemma 1 holds in this case as well. Define �L = esL

and �H = esH as the common exit and entry thresholds. We obtain the following system whichcharacterizes A,B,�L and �H .

F =

2

66664

F1 (A,B,�L,�H)

F2 (A,B,�L,�H)

F3 (A,B,�L,�H)

F4 (A,B,�L,�H)

3

77775=

2

666664

A�↵L �B���L � S�✓�1

L⌘ +

fr

A�↵H �B���H � S�✓�1

H⌘ +

fr +

↵A�↵L + �B���L � (✓ � 1)

S�✓�1L⌘

↵A�↵H + �B���H � (✓ � 1)

S�✓�1H⌘

3

777775=

2

66664

0

0

0

0

3

77775

When a firm’s productivity reaches �H , the firm starts exporting. When the productivity falls,but still in the range [�L,�H ], the firm keeps exporting. It exits from the export market once �falls below �L. Unlike the model with aggregate shocks, firms’ export entry and exit decisions arenot uniform, and both exporters and non-exporters coexist in the range [�L,�H ]. The stationaryexporter density including this “mixed region” of productivity is derived in the next subsection. 12

5.3 Exporter’s Distribution

To obtain aggregate variables such as price index and trade shares, we need to characterize thedistribution of exporters to each destination. We know that all firms with � > �H export andall firms with � < �L don’t export. For firms in the inaction range � 2 [�L,�H ], the monotonicrelationship between � and export status breaks down. Two firms can be exporters and non-exportersgiven the same current productivity �. Exporters in this range are those who overcame the entrythreshold �H once, but the productivity declined below �H (yet still above �L). Non-exporters inthe same range are the ones who haven’t received enough favorable shocks to cross the thresholdsince the last time they stopped exporting. If a firm is in this mixed region, its current productivity� is not enough to determine the firm is an exporter or not. We need information on the past historyof this firm. Nonetheless, the cross-sectional distribution of exporters in this region converges to astationary distribution, and we can characterize it in the similar manner as in subsection 5.1.

Let’s keep our attention only on exporters. In any given time, exporters are “born” at sH .Thus, sH is the entry point for the exporters. There is one absorbing barrier, namely sL. Onceproductivity hits sL, a firm exits from exporter’s distribution. Let f1 (a, s) be the density of exporterswith efficiency s and age a. We have the same KFE as in (21) and conditions (26) - (29) except that

12As a limiting case, if goes infinity, the entry option becomes worthless. In that case, A = 0 and sH = 1. Theexit threshold is

sL =

1

(✓ � 1)

ln

�

� + (✓ � 1)

fS⌘r

�

which means⇡L =

�� + (✓ � 1)

⌘rf

25

the lower bound �1 is now replaced by sL. Then, the solution to the KFE becomes the following

f1 (a, s) = e��a (a, s|sH) for a > 0 and s > sL

where

(a, s|sH) =

1

�pa

"N

✓s� sH � µa

�pa

◆� e

�µ(sH�sL)�2/2 N

✓s+ sH � 2sL � µa

�pa

◆#

and N (·) a standard normal density. More details about this solution can be found in Harrison(1985). As a tends to zero, the first term in the square brackets approaches to a point mass as sH

and the second term becomes zero since s+sH > 2sL. Also, if sL tends to �1, the above expressionbecomes equation (22) derived earlier. The size density of exporters f1 (s) is given by

f1 (s) =

ˆ 1

0f1 (a, s) da =

�1�2�1 + �2

min

�e(�1+�2)(s�sL), e(�1+�2)(sH�sL)

� 1⇥

e�1(sH�sL) � 1

⇤e�2(s�sL)

for s > sL

where �1 and �2 were computed in subsection 5.1. Since the entry barrier sH is a point mass, aboveexpression is also a probability density. Thus,

f1 (s) =

8>>><

>>>:

�1�2�1 + �2

1⇥e�1(sH�sL) � 1

⇤he�1(s�sL) � e��2(s�sL)

ifor s 2 (sL, sH)

�1�2�1 + �2

e(�1+�2)(sH�sL) � 1⇥e�1(sH�sL) � 1

⇤ e��2(s�sL) for s > sH

This was introduced in Luttmer (2007) and used in Impullitti et al. (2012) as well. The correspondingprobability density for � = es is

g1 (�) =

8>>>>>>>><

>>>>>>>>:

�1�2�1 + �2

2

64

⇣��L

⌘�1+�2� 1

⇣�H�L

⌘�1� 1

3

75��(�2+1)

���2L

for � 2 (�L,�H)

�1�2�1 + �2

2

64

⇣�H�L

⌘�1+�2� 1

⇣�H�L

⌘�1� 1

3

75��(�2+1)

���2L

for � > �H

(32)

where �L = esL and �H = esH . All firms with � > �H export and their Pareto tail index is� (�2 + 1) which is equal to the tail index of the overall stationary distribution. This confirms thescale free property of the Pareto distribution.

Figure 5 depicts the overall stationary density of firms and exporter’s density. From (31), wecan easily show that the share of firms with � < �⇤ is �2

�1+�2and the share of firms with � > �⇤ is

�1�1+�2

.

26

1 2 3 4 50.0

0.1

0.2

0.3

0.4

0.5

0.6

Non-exporters

Exportersf*

fL

fH

Figure 5: Stationary Density of Firms

As in Meliz (2003), define the following average productivity for exporters

˜� =

ˆ 1

�L

�✓�1dG1 (�)

� 1✓�1

We have the following result.

Lemma 4. The average productivity for exporters ˜� is given by the following expression

˜� =

2

64✓�1�2k1k2

◆⇣�H�L

⌘k1� 1

⇣�H�L

⌘�1� 1

3

75

1✓�1

�L (33)

where k1 = �1 + (✓ � 1) and k2 = �2 � (✓ � 1).

Proof. See Appendix A.4.

By using L’Hopital’s Rule, we can confirm that lim�L!�H˜� =

h�2

�2�(✓�1)

i 1✓�1

�H which is also

obtained in Ghironi and Melitz (2005). For further analyses, it is useful to define q =

�H�L

. Usinglemma 2, we can see that ⇡X (�L) and ⇡X (�H) are homogeneous of degree one in f and . Thisimplies that q =

�H�L

is determined by the ratio f . The average productivity of exporters (33)

summarizes relevant information for all aggregate variables. Though not pursued in this paper, theresults in this section can be extended to asymmetric multi-country general equilibrium of tradeextending Chaney (2008)13.

13Let i and j be the subscripts for the source and destination countries. If we assume that ij

fijis common for all

27

6 Full Model

This section presents a unified model which incorporates both aggregate and idiosyncratic produc-tivity shocks. The basic setup is the same as before, but now there are two types of productivityshocks st (idiosyncratic) and zt (aggregate) that follow a Brownian motion. Assumptions in thissection are summarized below.

1. (small home country): The home mass of firms relative to foreign is small: MM⇤ ⇡ 0.

2. (aggregate shock): The log of relative productivity z = ln

�ZZ⇤�

follows a drift-less Brownianmotion

dz = �zdW

3. (idiosyncratic shock): The log of idiosyncratic productivity s = ln

ˆ� evolves as a Brownianmotion

ds = µsdt+ �sdW

4. (independence): Idiosyncratic shocks are i.i.d, and independent from aggregate shocks.

5. (parameters): � > ✓ � 1, � > (✓ � 1)

2 �2s2

Again, the small country assumption simplifies the diffusion process of the export profit, and is agood approximation for most of the countries. Because there is no fixed or sunk entry costs fordomestic sales, firms never exit from the domestic market unless they are hit by the death shock.The aggregate shock does not affect the stationary distribution of the relative firm productivities.The stationary density of all firms is given by equation (31). The entry rate of new firms is fixedat �M to keep the mass of operating firms constant. New firms are born at productivity s⇤ as non-exporters. The fixed and sunk export costs are proportional to the foreign GDP, and the diffusionprocess of the normalized export profit ⇡̂X is analyzed as in Section 4. Unlike Section 5, the densityof the cross-sectional productivities of exporters is not constant due to the aggregate fluctuations.The dynamics of the exporter’s density depends on the optimal entry and exit strategy of firms andthe aggregate shock path, which is analyzed below.

country pairs but allow variation in fij , then qij is equal but fij varies across countries. Using (33), we can definethe average price for exporters as p̃ij = p

⇣˜�ij

⌘and the expression of the price index of country j becomes

Pj =

✓✓ � 1

NX

k=1

Mk��k2,kLkj

⌫kj (wk⌧kj)1�✓

! 11�✓

(34)

where

⌫kj = �2,k

✓�1,k + �2,kk1,kk2,k

◆qk1,k

kj � 1

q�1,k+�2,k

kj � 1

Other comparative statics can be conducted as well.

28

6.1 Band of Inaction

By the small country assumption, a firm’s export profit can be written as

⇡̂X (zt, st) = Se(✓�1)(zt+st)

where S is a scaling constant. The aggregate and idiosyncratic productivity shocks enter the exportprofit function symmetrically. Because of these two shocks, the export profit also follows a diffusionprocess.

The associated HJB equation is a second-order PDE (not ODE):

rU (z, s) = ⇡ (z, s)� f � µsUs +1

2

�2sUss +1

2

�2zUzz (35)

where r = ⇢ + �. In the zs plane, two curves separate the domain into three regions of differentoptimal strategies: export, inaction, and do not export. We can also express the threshold levelsof s as functions of z: sL (z) and sH (z). The curves sL (z) and sH (z) characterize the lower andupper thresholds of the export hysteresis. The following boundary conditions must be satisfied

U (z, sL (z)) = 0, U (z, sH (z)) = , Uz (z, sL (z)) = Us (z, sL (z)) = Uz (z, sH (z)) = Us (z, sH (z)) = 0

(36)This is a free boundary problem of a partial differential equation14. In general, the problem is quitedifficult to analyze because not only the surface function, but the boundary conditions are alsounknown. Nevertheless, when both shocks follow a Brownian motion, we can reduce the problem toan ODE with an analytical solution.

Proposition 2. Given the assumptions in this section, the value function U (z, s) has the followingform

U (z, s) = Be��(z+s) �Ae↵(z+s)+

⇡ (z + s)

⌘� f

r,

where where

↵ =

�µs +p

µ2s + 2r (�2z + �2s)

(�2z + �2s)

�� =

�µs �p

µ2s + 2r (�2z + �2s)

(�2z + �2s)

⌘ = r � �2z + �2s2

(✓ � 1)

2 � µs (✓ � 1)

and A and B are constants determined by associated boundary conditions.14An example of this type of problem is called Stefan problem in physics, which analyzes the heat evolution of ice

passing to water.

29

0.0

0.5

1.0

z -0.2

0.0

0.2

s

-5

0

5

10

UHz,sL

(a) Plot of U (z, s)

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2z

0.0

0.2

0.4

0.6

0.8

1.0

1.2s

Do not export

Inaction region

Export

(b) Contour lines of U (z, s) = 0 and U (z, s) =

Figure 6: Value function U (z, s) and its contour sets

Proof. See Appendix A.5.

The result is intuitive. A firm’s export profit depends on the combined productivity v = z + s,which follows a Brownian motion with drift µv = µs and variance �2v = �2z + �2s since z and s areindependent. Then, we can solve the lower and upper thresholds vL and vH as in subsection 4.1, thatcharacterize the two curves sL (z) = vL � z and sH (z) = vH � z. Along these lines, the boundaryconditions are satisfied. Note that the value function U (z, s) are symmetric about the line z = s.The two curves that separate the regions of optimal strategies are both linear with a slope of -1.An important result is that the distance between sH (z) and sL (z) are constant at vH � vL for anyz. This result is used below to transform the KFE. The graph of U (z, s) and the contour lines atU (z, s) = 0 and U (z, s) = are shown in Figure 6.

6.2 Dynamics of the Exporter’s Density

The above result gives the optimal entry and exit strategies for firms given the aggregate shock z.Consider a continuous path of z (t), which determines the paths of sL (z (t)) and sH (z (t)). At anydate t, all firms with s > sH (z (t)) are exporters whereas firms with s < sL (z (t)) are non-exporters.Write l (t) = sL (z (t)) and h (t) = sH (z (t)). We can characterize the dynamics of the exporters’density in the inaction range [l (t) , h (t)] as follows.

Let m (s, t) be the density of exporters s in the inaction region [l (t) , h (t)] at time t. Becausethe productivities follow a Brownian motion, m (s, t) must satisfy the following KFE as in Section 5

@m (s, t)

@t= ��m (s, t)� µs

@m (s, t)

@s+

1

2

�2s@2m (s, t)

@s2(37)

Let m0 (s) be the initial distribution of exporters in the inaction range [l (0) , h (0)] . The boundary

30

conditions are

m (s, 0) = m0 (s)

m (l (t) , t) = 0

m (h (t) , t) =

�1�2�1 + �2

e��2(h(t)�s⇤)

The first condition specifies the initial distribution. The second condition reflects that l (t) is anabsorbing barrier for exporters at time t. The third condition smoothly matches the density ofexporters at h (t), where the expression is derived in (30). I only consider the aggregate shockpaths by which h (t) does not go below s⇤15. This is a moving boundary problem for a p.d.e. Thedomain changes over time, which makes the problem difficult to analyze16. In our case, the problemcan be reduced to a simpler one using the fact that the length of the inaction range is h (t) � l (t)

is constant. Let d = h (t) � l (t). Consider the following transformation of the density function;n (s, t) = m (s+ l (t) , t). The domain of the function n (s, t) is fixed in s 2 [0, d]. The new functionn (s, t) must satisfy the following p.d.e

@n (s, t)

@t= ��n (s, t)� z0 (t)

@n (s, t)

@s� µs

@n (s, t)

@s+

1

2

�2s@2n (s, t)

@s2(38)

The boundary conditions become

n (s, 0) = n0 (s)

n (0, t) = 0

n (d, t) = b (t)

where n0 (s) = m0 (s� l (0)) and b (t) = �1�2�1+�2

e��2(h(t)�s⇤). We transformed the moving boundaryproblem to the one with fixed domains. The new function n (s, t) can be numerically solved inMathematica. Note the new term z0 (t) @n(s,t)

@s . The growth rate of the aggregate shock amplifies theeffect of µs in the KFE because the entry and exit decisions are based on the export profit ⇡ (z + s)

in which z and s are perfect substitutes. When the aggregate shock path is linear in time, we obtainthe analytical expression of n (s, t) (See Appendix A.6).

15If this happens, new firms are all exporters. The calculations of the exporter dynamics and its stationary densityare complicated.

16In physics, this type of problem appears in the analysis of the heat equation with moving insulating barriers.Various numerical, and some analytical solutions can be found in the literature.

31

Description Parameter Value

preference discount rate ⇢ 0.05death shock rate � 0.09

elasticity of substitution ✓ 2variable trade cost ⌧ 0.3

sunk export cost relative to the fixed cost f 20.5

drift parameter (idiosyncratic) µs -0.0163standard deviation (idiosyncratic) �s 0.081

standard deviation (aggregate) �z 0.036

Table 2: Parameters for calibration

7 Calibration and Numerical Analysis

7.1 Parameters

This section calibrates the full model to match the relevant U.S. data on the firm size distributionand exporter dynamics. With the calibrated model, I perform several comparative dynamics.

The preference discounting rate ⇢ is set to 0.05, which implies the annual interest rate of 5percent. For the benchmark case, ⌧ = 1.3 to match the results in Obstfeld and Rogoff (2000). Ichoose 2 for the elasticity of substitution between varieties ✓ to reflect the low trade elasticity in theshort run. Following Impullitti et al. (2012), I set the death shock rate � = 0.09. This is based onthe average annual death rate for manufacturing firms in the 1998-2004 period from the 2004 U.S.Census data. The aggregate trade elasticity which augments the extensive margin adjustment isgoverned by the upper Pareto tail index �2. I target the estimate from Eaton and Kortum (2002),which is �2 = 8.28. I borrow the drift parameter of the idiosyncratic shocks µs = �1.63% fromArkolakis (2011). Since �2 =

�µs+p

µ2s+2��2

s

�2s

, this gives �s = 8.1%. As in Ruhl (2008), �z is setto match the volatility of the logged output in the U.S. for the period of 1950-2000, which gives�z = 3.6%.

The fixed and sunk costs for exporting are calculated to match two moments of the U.S. exporterdistribution. I normalize s⇤ = 0 and the scaling constant for the export profit to be S = 1. Althoughthe model does not have a stationary equilibrium, I set z (0) = 0 and interpret this as a benchmark.The moments of the U.S. firm distribution are matched to their counterparts in the stationaryequilibrium with the absence of the aggregate shock. Bernard and Jensen (2004b) find that firmsenter and exit from the export market at different productivity levels using U.S. data for the periodof 1983-1992. According to their statistics, the exit trigger productivity is two-thirds of the entrytrigger productivity on average. The implied inaction band is sH�sL = 0.41. Impullitti et al. (2012)calibrate the ratio of the sunk and fixed export costs as

f = 91, which leads to sH � sL = 1.64 inthe current model17. I target sH � sL = 1 and obtain

f = 20.5. Next, Bernard et al. (2003) report

17From lemma 2, we know that the inaction band is determined by the ratio of the sunk and fixed export costs.

32

that 21 percent of U.S. manufacturing firms are exporters in 1992. Given sL and sH , the share ofexporters is18

m1 =e��2sH

⇥e�1(sH�sL) � 1

⇤

e�1(sH�sL) � e��2(sH�sL),

which gives sH = 0.184 and sL = �0.816.

7.2 Trade Dynamics

Given an aggregate shock path and initial exporter’s distribution, the full model has the ability togenerate nonlinear export dynamics. The dynamics of the exporter distribution is characterized bythe p.d.e (38), which enables us to compute aggregate export for any t. Two scenarios are investigatedbelow with the implications on the short- and long-run trade elasticities. Those aggregate shockpaths are particular realizations of z, which all firms expect to follow a Brownian motion. For bothscenarios, the economy starts from the stationary distribution, and the KFE (38) is numericallysolved to characterize n (s, t), which is converted back to m (s, t). The corresponding density functionof � = es, m̂ (�, t), is then computed to obtain ˜�t (the average productivity of exporters), whichsummarizes the model’s implications for aggregate export. Log of export (normalized by Y ⇤

t ) is

lnXt = ln

"ˆ 1

�L(t)x (�) m̂ (�, t) d�

#= a5 + (✓ � 1)

hln

˜�t + zt

i,

where a5 is a constant and ˜�t =h´1

�L(t)�✓�1m̂ (�, t) d�

i 1✓�1 . The intensive and extensive margins of

trade adjustments are represented by (✓ � 1) zt and (✓ � 1) ln

˜�t respectively. The log of export inthe initial period lnX0 is normalized to be 0.

7.2.1 Regime Switching

Consider the following regime-switching scenario. The aggregate shock z is initially 0, jumps by10% within a quarter, and stays at the new level for 8 years as shown in Figure 7a. In the initialquarter, the total export increases by 31% implying the trade elasticity of 3.1. This number is acomposite of the intensive margin elasticity of ✓�1 = 1 and the initial extensive margin adjustmentof 2.1. Let {�L,0,�H,0} be the original exit and entry thresholds for firms with z = 0, and similarly{�L,1,�H,1} for the new regime with z = 0.1. Because the aggregate shock is positive, �L,1 < �L,0

and �H,1 < �H,0. Due to the jump in z, non-exporters in the range [�H,1,�H,0] become exportersimmediately. This is the initial adjustment on the extensive margin. The exporter density in thenew inaction region [�L,1,�H,1] is lower compared to the new steady state. Starting from the newexport entry point �H,1, few firms have “diffused” into the inaction region. As time passes, newexporters enter and some incumbent exporters exit from this region, but the entry force is dominant

18See Appendix for derivation.

33

accumulating the exporter’s density. The sluggish trade adjustment on the extensive margin iscaused by the evolution of exporter’s density in the inaction region. By the seventh quarter, thetotal export increases by 50%. As t ! 1, the trade elasticity tends to �2 = 8.28 as we saw inSection 5.

7.2.2 Business Cycles

Next, consider a business cycle scenario with z (t) = 0.05 ⇥ sin (t) as illustrated in Figure 7b. Theaggregate shock displays a cyclical movement with a period of 2⇡ ⇡ 6.28 years. In this analysis, Ipick a minimum time interval of ⇡

16 ⇡ 2.4 months and call it “quarter”. The peaks of z occur at 8th,40th, 72nd quarters whereas the bottoms occur at 32nd, 64th quarters and so on. The peaks andbottoms of the log export occur with two or three quarters of a positive lag. As z is on an upwardtrend, the exporter’s density adjusts toward the new steady state. When z changes its directionfrom positive to negative (at a peak), the exporter’s density in the inaction region still carries thepositive effect from previous periods. Thus, the peaks of the extensive margin occur with some lags.In the 16th quarters, the shock z goes back to the original state of 0, but the log export is 0.122exhibiting the path dependence of the export dynamics.

The current model offers an alternative explanation for “the S-curve” phenomena examined byBackus, Kehoe and Kydland (1994). They document that, for many countries, the trade balance (netexports) is negatively correlated with the terms of trade (the relative price of imports to exports) inthe current and future periods, but positively correlated with that in the past periods. The shape ofthe lagged correlations exhibits the S-curve. They develop an IRBC model with time-to-build lag forcapital formation to explain the S-curve shape of the lagged correlations. In the current model, ztcorresponds to the terms of trade in their definition since a high zt implies a low export price whichmeans the high terms of trade. Figure 8 depicts the lagged correlations Corr (zt, lnXt+k). We canconfirm the S-curve in the plot. The lag of k = 3 gives the highest correlation coefficient. Exportis negatively correlated with z in the future but positively correlated with z in the past. Becausephysical investment and the international flow of capital are not featured in the current model, itdoes not explain the countercyclical movements of exports (a negative sign of Corr (zt, lnXt)), butit captures the S-curve shape. In this model, the mass of exporters can be interpreted as capital,which requires the time to build due to the sluggish adjustments of the exporter’s density.

8 Conclusion

This paper offers a unified theoretical framework to analyze both short- and long-run trade dynamicsin a consistent manner. The model incorporates export fixed costs and sunk costs, as well astwo types of productivity shocks, in a continuous-time framework with heterogeneous firms. Theexport sunk costs and uncertainty give rise to export hysteresis, the reluctance of firms to enter

34

0 5 10 15 20 25 300.00

0.02

0.04

0.06

0.08

0.10

quarter

aggregateshockzHblu

edashedlineL

0.00.10.20.30.40.50.60.7

logoftotalexportHre

dlineL

(a) Regime switching

0 20 40 60 80 100 120

-0.04

-0.02

0.00

0.02

0.04

quarter

aggregateshockzHblu

edashedlineL

-0.05

0.00

0.05

0.10

0.15

0.20logoftotalexportHre

dlineL

(b) Business cycle

Figure 7: Export dynamics of the full model

35

-10 -5 0 5 10

-0.5

0.0

0.5

1.0

Lag k of z behind the log export

CorrHz t

,lnX

t+kL

Figure 8: Lagged correlations between the aggregate shock and export

or exit from export markets in response to temporary trade shocks. If the economy is hit byaggregate productivity shocks only and the firm-level efficiency is fixed over time, the cutoff firmwhich separates exporters and non-exporters evolves stochastically. The cutoff does not respondagainst small temporary shocks, which results in a small trade elasticity in short run due to theinoperative extensive margin. When the shock is long-lived, new firms start entering or exiting fromthe export market inducing a large trade elasticity in the long run. This mechanism explains the“international elasticity puzzle,” a low trade elasticity against temporary shocks and a high elasticityagainst long-lived shocks studied by Ruhl (2008). If the productivity shocks are idiosyncratic andthere is no aggregate shock, the economy converges to a steady state, with individual firms movingaround within a stationary distribution of productivities. Export hysteresis at the micro level leadsto a region where both exporters and non-exporters coexist given the same current productivity.The full model captures the effects of both types of shocks and offers realistic microfoundations oftrade dynamics. It generates simultaneous export entry and exit, “the mixed region” of exportersand non-exporters, and the sluggish transition dynamics of exporter’s distribution in response toaggregate shocks, which provides an alternative explanation for the S-curve of the lagged correlationsbetween exports and the terms of trade.

One limitation of the model is the assumption of a Brownian motion for the aggregate shockprocess. It provides a sharp analytical characterization of the optimal strategies, but prevents us fromthe analysis of long-run equilibria. A model in which the aggregate shock follows a mean-revertingprocess is currently under investigation. A mean-reverting process has a long-run stationary density,and is more suitable to model the economy-wide shock processes. This complicates the problems ofpartial differential equations in Section 6 because they become free-boundary and moving-boundaryproblems that require more sophisticated numerical methods.

36

References

Alessandria, George and Horag Choi (2007) “Do Sunk Costs of Exporting Matter for Net ExportDynamics?.,” Quarterly Journal of Economics, Vol. 122, No. 1, pp. 289 – 336.

(2011) “Establishment heterogeneity, exporter dynamics, and the effects of trade liberaliza-tion.,” Working Paper.

Arkolakis, Costas (2011) “A Unified Theory of Firm Selection and Growth.,” NBER Working Paper17553.

Backus, David K., Patrick J. Kehoe, and Finn E. Kydland (1994) “Dynamics of the Trade Balanceand the Terms of Trade: The J-Curve?.,” American Economic Review, Vol. 84, No. 1, pp. 84– 103.

Baldwin, Richard (1988) “Hysteresis in Import Prices: The Beachhead Effect.,” American EconomicReview, Vol. 78, No. 4, pp. 773 – 785.

Baldwin, Richard and Paul Krugman (1989) “Persistent Trade Effects of Large Exchange RateShocks.,” Quarterly Journal of Economics, Vol. 104, No. 4, pp. 635 – 654.

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Bernard, Andrew B. and J. Bradford Jensen (2004) “Exporting and Productivity in the USA.,”Oxford Review of Economic Policy, Vol. 20, No. 3, pp. 343 – 357.

Chaney, Thomas (2008) “Distorted Gravity: The Intensive and Extensive Margins of InternationalTrade.,” American Economic Review, Vol. 98, No. 4, pp. 1707 – 1721.

Das, Sanghamitra, Mark J. Roberts, and James R. Tybout (2007) “Market Entry Costs, ProducerHeterogeneity, and Export Dynamics.,” Econometrica, Vol. 75, No. 3, pp. 837 – 873.

Dixit, Avinash K. (1989) “Entry and Exit Decisions under Uncertainty.,” Journal of Political Econ-omy, Vol. 97, No. 3, pp. 620 – 638.

Dixit, Avinash K. and Robert S. Pindyck (1994) Investment under uncertainty: Princeton UniversityPress.

Eaton, Jonathan and Samuel Kortum (2002) “Technology, Geography, and Trade.,” Econometrica,Vol. 70, No. 5, pp. 1741 – 1779.

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37

Ghironi, Fabio and Marc J. Melitz (2005) “International Trade and Macroeconomic Dynamics withHeterogeneous Firms.,” Quarterly Journal of Economics, Vol. 120, No. 3, pp. 865 – 915.

Harrison, Michael J. (1985) Brownian Motion and Stochastic Flow Systems: John Wiley and Sons.

Impullitti, Giammario, Alfonso A. Irarrazabal, and Luca David Opromolla (2013) “A Theory ofEntry into and Exit from Export Markets.,” Journal of International Economics, Vol. 90, No.1, pp. 75 – 90.

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Roberts, Mark J. and James R. Tybout (1997) “The Decision to Export in Colombia: An EmpiricalModel of Entry with Sunk Costs.,” American Economic Review, Vol. 87, No. 4, pp. 545 – 564.

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38

Appendix

A Proofs and Derivations

A.1 Derivation of ⇡̃t and ⇡̃⇤t

The home expenditure shares on domestic and imported goods can be expressed as

�D,t =

1

1 +

M⇤

M

⇣�̄⇤t

�min

⌘✓�1�� ⇣Z⇤

t

Zt

ww⇤

1⌧

⌘✓�1

�X,t =

M⇤

M

⇣�̄⇤t

�min

⌘✓�1�� ⇣Z⇤

t

Zt

ww⇤

1⌧

⌘✓�1

1 +

M⇤

M

⇣�̄⇤t

�min

⌘✓�1�� ⇣Z⇤

t

Zt

ww⇤

1⌧

⌘✓�1

It is clear from the above expressions that the relative variables M⇤

M , w⇤

w ,Z⇤

t

Ztand the export cutoff �̄⇤

t

�minare

the sufficient statistics to compute the expenditure shares.The average total profit is

⇡̃t = ⇡D,t

⇣˜�⌘+

MX,t

M

h⇡X,t

⇣˜�X,t

⌘� fY ⇤

t

i

= a1

✓Zt⌫�minPt

w

◆✓�1

| {z }�D,tM

Yt +

2

666664

✓¯�t

�min

◆��

a1

✓Zt⌫ ¯�tP

⇤t

⌧w

◆✓�1

| {z }�⇤X,tM

Y ⇤t �

✓¯�t

�min

◆��

fY ⇤t

3

777775

=

�D,t

M[wL+M ⇡̃t] +

"�⇤X,t

M� f

✓¯�t

�min

◆��#[w⇤L⇤

+M⇤⇡̃⇤t ]

= �D,twL

M+ �D,t⇡̃t + bt [w

⇤L⇤+M⇤⇡̃⇤

t ]

where bt =�⇤X,t

M � f⇣

�̄t

�min

⌘��. Since the average net export profit is nonnegative, we have bt > 0. Given

the state variables, we can compute �’s and bt. By rearranging, we obtain

⇡̃t =�D,t

1� �D,t

wL

M+

bt1� �D,t

[w⇤L⇤+M⇤⇡̃⇤

t ]

Analogously for the foreign country,

⇡̃⇤t =

�⇤D,t

1� �⇤D,t

w⇤L⇤

M⇤ +

b⇤t1� �⇤D,t

[wL+M ⇡̃t]

The above two equations determine the two unknowns ⇡̃t and ⇡̃⇤t . Plugging the second equation into the first

39

equation yields

⇡̃t =

�D,t

�X,t

wL

M+

bt�X,t

"w⇤L⇤

+M⇤

�⇤D,t

�⇤X,t

w⇤L⇤

M⇤ +

b⇤t�⇤X,t

[wL+M ⇡̃t]

!#

=

�D,t

�X,t

wL

M+

bt�X,t

"w⇤L⇤

+

�⇤D,t

�⇤X,t

w⇤L⇤+

b⇤t�⇤X,t

M⇤wL+

b⇤t�⇤X,t

M⇤M ⇡̃t

#

where I used 1� �D,t = �X,t. Hence, we obtain

⇡̃t =

⇣�⇤X,t�D,t

M + b⇤t btM⇤⌘wL+

��X,t + �⇤D,t

�btw

⇤L⇤

�X,t

⇣�⇤X,t � b⇤tM

⇤M⌘

⇡̃⇤t =

⇣�⇤X,t�D,t

M⇤ + b⇤t btM⌘w⇤L⇤

+

��⇤X,t + �D,t

�b⇤twL

�⇤X,t (�X,t � btM⇤M)

A.2 Proof of Lemma 1

The differential equations (15) and (16) have the same homogeneous part. Let’s focus on V0. Equation (15)has a general solution of the following form

V0 (z) = A0e↵z

+B0e��z

where ↵ and �� are the two roots for the characteristic equation

�2

2

x2+ µx� ⇢ = 0

Then, we obtain

↵ =

�µ+

pµ2

+ 2⇢�2

�2

�� =

�µ�p

µ2+ 2⇢�2

�2

Both ↵ and � are positive but the lower bound for ↵ is 0 whereas that for � is 2µ�2 .

Equation (16) is a nonhomogeneous differential equation due to the net export profit flow. The equationis

�2

2

V 001 + µV 0

1 � ⇢V1 = f � Se(✓�1)z

| {z }u(z)

Since this is a linear equation, the solution takes the following form

V1 (z) = V0 (z) + w (z)

We obtain the following relationship between u (z) and w (z)

�2

2

w00(z) + µw0

(z)� ⇢w (z) = u (z)

40

Guess w (z) = Ce(✓�1)z � f⇢ for some constant C. Then,

�2

2 (✓ � 1)

2Ce(✓�1)z

+ µ (✓ � 1)Ce(✓�1)z � ⇢Ce(✓�1)z+ f = f � Se(✓�1)z

) e(✓�1)zhC⇣�2

2 (✓ � 1)

2+ µ (✓ � 1)� ⇢

⌘+ S

i= 0

) C⇣�2

2 (✓ � 1)

2+ µ (✓ � 1)� ⇢

⌘+ S = 0

which results inC =

S

⇢� �2

2 (✓ � 1)

2 � µ (✓ � 1)

Thus,

V1 (z) = A1e↵z

+B1e��z

+

S

⇢� �2

2 (✓ � 1)

2 � µ (✓ � 1)

e(✓�1)z � f

⇢

If z is very small, it is very unlikely that it rises to zH in any given finite amount of time. The option ofentering the export market is nearly worthless. This implies B0 = 0. A similar argument implies A1 = 0.Combining with the above results, we obtain

V0 (z) = Ae↵z

V1 (z) = Be��z +S

⇢� �2

2 (✓ � 1)

2 � µ (✓ � 1)

e(✓�1)z � f

⇢

which concludes the proof.

A.3 Model with Aggregate Shocks Only: General Case

Results in Section 4 are based on the small home country assumption. Here, we consider a general case whereboth countries have substantial presence in the price index of the partner country. The optimal behavior ofindividual firms is represented by the two threshold levels of export profits at which entry and exit occurs.However, setting up the HJB equation is more complex in this case since ¯�t evolves stochastically over timeaffecting the profits. For simplicity, let’s assume two symmetric countries.

Assumption 3. The two countries have the same wage rate,

ww⇤ = 1, and mass of firms equal one, M =

M⇤= 1.

In this case, the expression for the export profit becomes

⇡̂X,t

�zt, ¯�t;�

�= a3 ⇥ �✓�1 ⇥

0

B@1

⇣�̄t

�min

⌘✓�1��+

�ez

⌧

��(✓�1)

1

CA

Again, the stochastic part is common for all firms. Two differences should be noted. First, the elasticity of ⇡with respect to z is not constant (it converges to ✓� 1 as the home country becomes smaller). Second, ⇡ (z)does not have a strong Markov property since ¯�t also affects ⇡ and evolves over time. The cutoff productivity¯�t reflects the cumulative control the regulators have exercised up to t, which depend on the entire historyof z.

41

If we take both z and ¯� as state variables, then ⇡�z, ¯��

has a strong Markov property, and we can applythe HJB equations to characterize the inaction region. The modified HJB equation for U

�z, ¯��

is

⇢U�z, ¯��= ⇡

�z, ¯��� f +

1

dtE

2

664Uzdz +1

2

Uzz (dz)2+ U�̄d

¯�+

1

2

U�̄�d¯��2

+ Uz�̄dzd¯�

| {z }dU

3

775

Since we have two state variables, this is a second-order linear nonhomogeneous PDE.

A.3.1 Transformation of the State Variables

Rather than working with the state variables z and ¯�, I transform z to ⇡̄, and use ⇡̄ and ¯� as new statevariables. The new HJB equation is

⇢U�⇡̄, ¯�

�= ⇡

�⇡̄, ¯�

�� f +

1

dtE

2

664U⇡̄d⇡̄ +

1

2

U⇡̄⇡̄ (d⇡̄)2++U�̄d

¯�+

1

2

U�̄�̄�d¯��2

+ U⇡̄�̄d⇡̄d¯�

| {z }dU

3

775

If d¯� 6= 0, the cutoff firm is being updated. During that period, ⇡̄ remains constant so d⇡̄ = 0. If d¯� = 0, thecutoff firm is in the inaction region and ⇡̄ changes due to the productivity shock z so d⇡̄ 6= 0. At any pointin time, there is a complementary slackness between ⇡̄ and ¯� as d⇡̄d¯� = 0. This complementary slackness isborrowed from Caplin and Leahy (1997), and simplifies the HJB equation as

dU =

8><

>:

U⇡̄d⇡̄ +

1

2

U⇡̄⇡̄ (d⇡̄)2 if d¯� = 0

U�̄d¯�+

1

2

U�̄�̄�d¯��2 if d⇡̄ = 0

(39)

Given dz, we can compute d⇡̄ and d¯�, and solve the differential equation numerically.

A.3.2 Derivation of d⇡̄ and d¯�

The cutoff export profit is

⇡̄ = a4

0

B@

⇣�̄t

�min

⌘✓�1

⇣�̄t

�min

⌘✓�1��+

�ez

⌧

��(✓�1)

1

CA

where a4 is a constant. The shock z can be expressed by ⇡̄ and ¯� as

z =

1

1� ✓ln

2

64a4⌧

⇣�̄t

�min

⌘✓�1

⇡̄� ⌧

✓¯�t

�min

◆✓�1��3

75

42

Then,

⇡̄z =

(✓ � 1)

⇣e(1�✓)z

⌧✓

⌘

⇣�̄t

�min

⌘✓�1��+

�ez

⌧

��(✓�1)⇡̄ =

(✓ � 1)

⇣e(1�✓)z

⌧✓

⌘

a4

⇣�̄t

�min

⌘✓�1⇡̄2

= (✓ � 1) ⌧1�✓

⇡̄ � 1

a4

✓¯�t

�min

◆��

⇡̄2

!

⇡̄zz =

� (✓ � 1)

2⇣

e(1�✓)z

⌧✓

⌘

a4

⇣�̄t

�min

⌘✓�1⇡̄2

+

(✓ � 1)

⇣e(1�✓)z

⌧✓

⌘

a4

⇣�̄t

�min

⌘✓�12⇡̄⇡̄z

=

� (✓ � 1)

2⇣

e(1�✓)z

⌧✓

⌘

a4

⇣�̄t

�min

⌘✓�1⇡̄2

+

(✓ � 1)

⇣e(1�✓)z

⌧✓

⌘

a4

⇣�̄t

�min

⌘✓�12⇡̄

(✓ � 1)

⇣e(1�✓)z

⌧✓

⌘

a4

⇣�̄t

�min

⌘✓�1⇡̄2

=

(✓ � 1)

⇣e(1�✓)z

⌧✓

⌘

a4

⇣�̄t

�min

⌘✓�1⇡̄2

2

64(1� ✓) +(✓ � 1)

⇣e(1�✓)z

⌧✓

⌘

a4

⇣�̄t

�min

⌘✓�12⇡̄

3

75

By Ito’s lemma,

d⇡̄ = ⇡̄zdz +1

2

⇡̄zz�2dt = ⇡̄z (�↵zdt+ �dW ) +

1

2

⇡̄zz�2dt

=

✓�⇡̄z↵z +

1

2

⇡̄zz�2

◆dt+ ⇡̄z�dW

We can apply the implicit function theorem to derive d¯�.

A.4 Proof of Lemma 4

We have

˜� =

2

6664

ˆ 1

�L

�✓�1dG1 (�)

| {z }X

3

7775

1✓�1

The expression in the bracket is

X = C1

ˆ �H

�L

�✓�1

"✓�

�L

◆�1+�2� 1

#��(�2+1)

���2L

d�

| {z }X1

+C2

ˆ 1

�H

�✓�1��(�2+1)

���2L

d�

| {z }X2

43

where C1 =

�1�2�1+�2

h⇣�H

�L

⌘�1� 1

i�1

and C2 = C1

⇣�H

�L

⌘�1+�2� 1

�. Then

X1 = C1

ˆ �H

�L

�✓+�1�2

��1L� �✓��2�2

���2L

d� = C1

"1

✓ � 1 + �1

�✓�1+�1

��1L� 1

✓ � 1� �2

�✓�1��2

���2L

#�H

�L

= C1

"1

k1

�✓�1+�1H

��1L� �✓�1+�1

L

��1L

!+

1

k2

�✓�1��2H

���2L

� �✓�1��2L

���2L

!#

= C1

"1

k1

✓✓�H�L

◆�1�✓�1H � �✓�1

L

◆+

1

k2

✓�H�L

◆��2�✓�1H � �✓�1

L

!#

where k1 = �1 + (✓ � 1) and k2 = �2 � (✓ � 1). Also,

X2 = C2

ˆ 1

�H

�✓�1�(�2+1)

���2L

d� = C2

"�k2

�✓�1��2

���2L

!#1

�H

= C2k2�✓�1��2H

���2L

= C2k2

✓�H�L

◆��2�✓�1H

Then,

X = X1 +X2

= C1

"1

k1

✓✓�H�L

◆�1�✓�1H � �✓�1

L

◆+

1

k2

✓�H�L

◆��2�✓�1H � �✓�1

L

!#+ C2

1

k2

✓�H�L

◆��2�✓�1H

= C1

"1

k1

✓✓�H�L

◆�1�✓�1H � �✓�1

L

◆+

1

k2

✓�H�L

◆��2�✓�1H � �✓�1

L

!#

+C1

"✓�H�L

◆�1+�2� 1

#1

k2

✓�H�L

◆��2�✓�1H

= C1

1

k1

✓✓�H�L

◆�1�✓�1H � �✓�1

L

◆� 1

k2�✓�1L

�+ C1

✓�H�L

◆�11

k2�✓�1H

= C1

✓1

k1+

1

k2

◆✓�H�L

◆�1�✓�1H �

✓1

k1+

1

k2

◆�✓�1L

�= C1

✓1

k1+

1

k2

◆�✓�1L

"✓�H�L

◆✓�1+�1

� 1

#

=

�1�2�1 + �2

✓�H�L

◆�1� 1

��1✓k1 + k2k1k2

◆�✓�1L

"✓�H�L

◆k1

� 1

#=

�1�2k1k2

✓k1 + k2�1 + �2

◆⇣�H

�L

⌘k1

� 1

⇣�H

�L

⌘�1� 1

�✓�1L

=

✓�1�2k1k2

◆⇣�H

�L

⌘k1

� 1

⇣�H

�L

⌘�1� 1

�✓�1L

Thus,

˜� = X1

✓�1=

2

64✓�1�2k1k2

◆⇣�H

�L

⌘k1

� 1

⇣�H

�L

⌘�1� 1

3

75

1✓�1

�L

44

A.5 Proof of Proposition 2

I will guess and verify the solution. Since z and s enter the profit function symmetrically, guess the functionalform of U (z, s) = F (z + s). Then, the HJB equation becomes an ODE

rF (z + s) = a4e(✓�1)(z+s) � f + µsF

0(z + s) +

1

2

�2zF

00(z + s) +

1

2

�2sF

00(z + s)

= a4e(✓�1)(z+s) � f + µsF

0(z + s) +

1

2

��2z + �2

s

�F 00

(z + s)

By letting v = z + s and �2v = �2

z + �2s , we obtain

rF (v) = a4e(✓�1)v � f + µsF

0(v) +

1

2

�2vF

00(v)

Then, we can apply lemma 1 to obtain

F (v) = Be��v �Ae↵v +⇡ (v)

⌘� f

r,

where

↵ =

�µs +p

µ2s + 2r�2

v

�2v

�� =

�µs �p

µ2s + 2r�2

v

�2v

⌘ = r � �2v

2

(✓ � 1)

2 � µs (✓ � 1)

⇡ (v) = a4e(✓�1)v

The boundary conditions are F (vL) = 0, F (vH) = , F 0(vL) = 0, F 0

(vH) = 0. We can solve for vL andvH using (19). The contour lines of F (z + s) in the zs plane are linear with a slope of -1. Thus, along theline z + s = vL, the boundary conditions for the lower threshold are satisfied, and similarly for the upperthreshold.

A.6 Further Analysis of the KFE (37)

The transformed function n (s, t) satisfies the p.d.e

@n (s, t)

@t= ��n (s, t)� l0 (t)

@n (s, t)

@s� µs

@n (s, t)

@s+

1

2

�2s

@2n (s, t)

@s2(40)

with the following boundary conditions

n (s, 0) = n0 (s)

n (0, t) = 0

n (d, t) = b (t)

45

where n0 (s) = m0 (s� l (0)) and b (t) = �1�2�1+�2

e��2(h(t)�s⇤). We transformed the moving boundary problemto the one with fixed domains. The upper boundary condition is inhomogeneous since b (t) depends on t.Now, apply the following decomposition

n (s, t) = n̄ (s, t) + v (s, t) ,

where n̄ (s, t) satisfies the inhomogenous boundary conditions and v (s, t) is a remainder. In particular, n̄ (s, t)

must satisfy

n̄ (0, t) = 0

n̄ (d, t) = b (t)

The p.d.e and the initial condition are satisfied by v (s, t), so n̄ (s, t) does not have to satisfy either of them.Let’s choose the following linear functional form

n̄ (s, t) =b (t)

ds (41)

This satisfies the inhomogeneous boundary conditions. The p.d.e (40) gives

@n̄ (s, t)

@t+

@v (s, t)

@t= �� [n̄ (s, t) + v (s, t)]�(l0 (t) + µs)

@n̄ (s, t)

@s+

@v (s, t)

@s

�+

1

2

�2s

@2n̄ (s, t)

@s2+

@2v (s, t)

@s2

�

Substituting (41) yields

b0 (t)

ds+

@v (s, t)

@t= ��

b (t)

ds+ v (s, t)

�� (l0 (t) + µs)

b0 (t)

d+

@v (s, t)

@s

�+

1

2

�2s

@2v (s, t)

@s2

or@v (s, t)

@t= ��v (s, t)� (l0 (t) + µs)

@v (s, t)

@s+

1

2

�2s

@2v (s, t)

@s2� p (s, t)

where p (s, t) =⇣

b0(t)d + � b(t)

d

⌘s+(l0 (t) + µs)

b0(t)d . v (s, t) must satisfy the following initial and homogeneous

boundary conditions

v (s, 0) = v0 (s)

v (0, t) = 0

v (d, t) = 0

where v0 (s) = n0 (s)� b(0)d s.

A.6.1 Linear shock path

Consider the case in which the aggregate shock is linear in time: z (t) = kt. Then, l (t) = vL � kt andh (t) = vH � kt. We have b (t) = c1e

�2kt where c1 =

�1�2�1+�2

e��2(vH�s⇤) is a constant. We obtain

n̄ (s, t) =c1de�2kts

46

We also havep (s, t) =

✓�2k + �

d

◆s+

(µs � k) �2k

d

�b (t)

Then v (s, t) must satisfy

@v (s, t)

@t= ��v (s, t)� (µs � k)

@v (s, t)

@s+

1

2

�2s

@2v (s, t)

@s2� p (s, t)

Let’s focus on the homogeneous part ignoring p (s, t) (this will be taken care of later). Since v (s, t) hasthe homogeneous boundary conditions, decompose v as v (s, t) = S (s)T (t) to find the eigenfunctions. Theseparation of variables yields

T 0

T= �� � (µs � k)

S0

S+

1

2

�2s

S00

S= constant =

The general solutions of T and S are

T (t) = cT e t

S (s) = K1ek1s

+K2ek2s

where cT ,K1 and K2are constants and

k1 =

(µ� k) +

q(µ� k)

2+ 2�2

s ( + �)

�2s

, k2 =

(µ� k)�q

(µ� k)2+ 2�2

s ( + �)

�2s

From the boundary conditions of v, we obtain S (0) = 0 and S (d) = 0. It can be verified that if (µ� k)2+

2�2s ( + �) � 0, there are only trivial solutions of K1 = K2 = 0. Thus, we need (µ� k)

2+ 2�2

s ( + �) < 0

which implies < �⇣

(µ�k)2

2�2s

+ �⌘. Then, S has the following form

S (s) = K1 sin (k1s) +K2 cos (k2s)

Using the boundary conditions, we obtain K2 = 0 and 0 = K1 sin (k1d). For a nonzero solution of K1, weneed sin (k1d) to be zero. The eigenvalues are

k1 =

2⇡

d, k2 =

4⇡

d, k3 =

6⇡

d, ...

and the corresponding eigenfunctions are

S1 = sin

✓2⇡

ds

◆, S2 = sin

✓4⇡

ds

◆, S3 = sin

✓6⇡

ds

◆, ...

We can compute the corresponding n. The general solution of v is

v (s, t)

1X

n=1

un (t)Sn (s)

where un (t) are Fourier coefficients.

47

A.7 Share of exporters in the stationary equilibrium

In the steady state, we know that the share of firms with s < s⇤ is �2�1+�2

and the share of firms with s > s⇤

is �1�1+�2

. Let SL and SH be the shares of exporters with s 2 [sL, sH ] and with s 2 (sH ,1) respectively. Byintegrating the density f1 (s), we obtain

SL =

�2�1 + �2

+

�1�1 + �2

e��2(sH�sL) � 1

e�1(sH�sL) � 1

�

SH =

�1�1 + �2

e�1(sH�sL) � e��2(sH�sL)

e�1(sH�sL) � 1

�

with SL + SH = 1. Assuming s⇤ < sH , the share of firms (among all operating firms) above sH isˆ 1

sH

�1�2�1 + �2

e��2(s�s⇤)ds =�1

�1 + �2e��2(sH�s⇤)

This corresponds to the share SH of all exporters. Thus, the share of exporters in the total mass of operatingfirms is

�1�1 + �2

e��2(sH�s⇤) ⇥ 1

SH=

e��2(sH�s⇤)⇥e�1(sH�sL) � 1

⇤

e�1(sH�sL) � e��2(sH�sL)

48