International Trade Dynamics with Sunk Costs and ...

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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 dynamics in a consistent manner. It explains “the international elasticity puzzle”, a low trade elasticity against 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 more general 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 both aggregate and idiosyncratic productivity shocks. A sharp analytical characterization of the equilibrium elucidates the microfoundations of trade dynamics linking a static trade model with heterogeneous firms and an international macroeconomic model. Due to the sunk costs and uncertainty, firms do not change their export status against small temporary shocks. Aggregate productivity shocks and export sunk costs explain the elasticity puzzle because of the diﬀerent adjustments on the extensive margin. If the productivity shocks are idiosyncratic, the economy is in a steady state, with individual firms moving around within a stationary distribution of productivities. Export hysteresis gives rise to a region of firm productivity where both exporters and non-exporters coexist given the same current productivity. The full model incorporates both types of shocks and oﬀers realistic microfoundations of trade dynamics including simultaneous export entry and exit, an evolving productivity density of exporters, and the sluggish trade response to aggregate shocks.

⇤

I am grateful to Samuel Kortum and Nancy Stokey for their advice and constant encouragements. I would also like to thank Fernando Alvarez, Costas Arkolakis, David Burk, Lorenzo Caliendo, Thomas Chaney, Robert Lucas, Robert Shimer, Serginio Sylvain and the seminar participants of International Trade Lunch at the University of Chicago and Yale University, and Capital Theory Workshop at the University of Chicago for helpful comments. All remaining errors are my own. † Email: [email protected] ‡ Latest version available at the author’s website: https://sites.google.com/site/fujii0622/research

1

1

Introduction

This paper develops a unified theoretical framework that captures both “export hysteresis,” the reluctance 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 trade elasticity against long-lived shocks. It incorporates export fixed costs and sunk costs, as well as productivity shocks, in a continuous-time framework with heterogeneous firms. Both aggregate and idiosyncratic productivity shocks are allowed. To develop intuition, two specialized versions of the model, each with only one type of shock, are considered. Then the full model is studied. A sharp analytical characterization of the equilibrium elucidates the microfoundations of trade dynamics linking 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 international real business cycle (IRBC) models, typically ranging between 0.5 and 2. On the other hand, in static general equilibrium models of trade that compare steady states, the elasticity with respect to 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”. He reconciles the contradictory findings by building a model in which business cycles are caused by temporary 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 firms start exporting when there is a tariﬀ reduction, which is considered to be permanent. The diﬀerent adjustments on the extensive margin explain the elasticity puzzle. This paper extends that model of export sunk costs and uncertainty to more general productivity shock processes and incorporates rich microfoundations for trade dynamics. The key economic mechanism is export hysteresis caused by sunk costs and uncertainty. Many empirical studies (Roberts and Tybout (1997), Bernard and Jensen (2004), and Das, Roberts and Tybout (2007)) find the existence of substantial up-front sunk costs for exporting and a high persistence of export participation. In an environment where an action of discrete choice requires a fixed cost and the future profit is uncertain, “hysteresis” behavior is prevalent. Hysteresis is defined as the failure of an eﬀect to revert itself when its underlying cause is removed. The export hysteresis was first investigated by Baldwin (1988) and many others. Dixit (1989) analyzes the optimal entry and exit decisions of a firm when the output price follows a random walk and the action requires a 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 goes back to the original level. The firm continues to export until the output price drops to a suﬃciently low level, lower than its continuation fixed cost, implying that the firm might run a temporary negative profit. The gap between the entry and exit triggers is called “the band of inaction”, which causes the export hysteresis behavior. The band of inaction is studied extensively in the literature of investment under uncertainty pioneered by Dixit and Pindyck (1994). The model in this paper 2

embeds the classical theory of export hysteresis into a continuous-time trade model to examine the eﬀects of both aggregate and idiosyncratic productivity shocks. By employing the continuous time setup and techniques developed in stochastic optimal control literature, the model generates sharp characterizations of equilibrium variables, and the export dynamics is described as a solution to a partial diﬀerential equation. Hence, the comparative statics or dynamics can be seen in a transparent way. The main contribution of this paper is to present a framework of trade dynamics, which is consistent with micro and macro empirical regularities at both short- and long-run time horizons. The aggregate productivity shocks and export sunk costs are the essential ingredients to reconcile the international elasticity puzzle. Introducing idiosyncratic productivity shocks that follow a Brownian motion leads to the endogenous Pareto distribution of cross-sectional productivity, and gives rise to a range of firm productivity 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 of trade dynamics such as simultaneous export entry and exit, an evolving productivity density of exporters, and the sluggish trade response to aggregate shocks. To build intuition, the model is presented step by step. After laying out the basic setup and explaining the mechanism of export hysteresis, I first focus on the case with aggregate productivity shocks only. The idiosyncratic firm productivity is fixed at a Pareto distribution. The aggregate shock hits all firms proportionally. At any date, exporters and non-exporters are separated by a cutoﬀ firm, which evolves stochastically. If the export profit of the cutoﬀ firm is in the inaction region, there is no firm entry nor exit, and the cutoﬀ productivity does not change. When the export profit of the cutoﬀ firm hits an upper bound, less productive firms enter the export market keeping the cutoﬀ profit constant. A similar result holds when the cutoﬀ profit hits the lower bound. Due to the inaction region, the extensive margin (set of exporters) of trade does not respond to a small temporary shock, so the trade elasticity only reflects the intensive margin in the short run. When a series of favorable shocks hits a country, less productive firms start entering and the extensive margin becomes operative. This results in a large trade elasticity in the long run. Therefore, the aggregate shock and sunk costs alone explain the international elasticity puzzle. Ruhl (2008) reviews the empirical evidence on the impact of the extensive margin adjustments in the long run in response to trade liberalizations. Kehoe and Ruhl (2013) also report that the extensive margin accounts for an important part of trade growth following tariﬀ reductions whereas it does not change in the absence of trade policy or structural change. After characterizing the equilibrium properties of the model with aggregate shocks only, I turn to the case where productivity shocks are idiosyncratic. A firm-level productivity evolves as a geometric Brownian motion which gives rise to the double-Pareto distribution of the cross-sectional productivity 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 variables and exporter distribution are constant in the stationary equilibrium. A range of firm productivity 3

in which both exporters and non-exporters coexist given the same productivity level emerges. The model explains the smooth increase in export participation as firm size becomes larger. The exporters in the “mixed region” are the ones whose productivity hit the entry margin, but dropped to a lower level. The non-exporters are those whose productivity has not reached the entry trigger yet since the last time they exited from the export market. This version of the model is essentially the same as the one in Impullitti et al. (2013), but simpler because I abstract from free entry and sunk costs for the domestic market. This allows a sharp characterization of the stationary equilibrium and clear comparative statics. The full model considers the eﬀects of both types of shocks. Both aggregate and idiosyncratic shocks 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, but the exporter distribution evolves stochastically. The exporter dynamics can be characterized as the solution to a partial diﬀerential equation, which enables us to analyze the dynamics in response to any aggregate shock paths without simulations. The full model exhibits simultaneous export entry and exit, “the mixed region” of exporters and non-exporters, and the sluggish transition dynamics of exporter’s distribution in response to aggregate shocks. It oﬀers an alternative explanation to the J-curve relationship between the terms of trade and exports as in Backus et al. (1994). Rather than the 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 this research. They develop an open-economy dynamic stochastic general equilibrium (DSGE) model with heterogeneous firms, export sunk costs and both idiosyncratic and aggregate productivity shocks to examine the aggregate eﬀects of trade liberalizations. Though it can incorporate more general features, the model is constructed in discrete time, and does not provide closed-form expressions for the firm distribution nor optimal entry/exit strategies. Ghironi and Melitz (2005) also construct a two-country DSGE model with heterogeneous firms to study the exporter dynamics and persistent deviations from PPP, but the model does not have sunk costs, and therefore, there is no hysteresis. This paper aims to elucidate the mechanism of the exporter dynamics in more transparent 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 covers the necessary mathematical tools for stochastic control models with fixed costs, and contains rich economic applications. As she suggests, those modeling tools and concepts may have a wide applicability in many economic fields. The model developed in this paper can be applied to other situations 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 with heterogeneous firms. Section 3 describes a firm’s dynamic problem of entry and exit, and analyzes the properties of the inaction region for a general diﬀusion process. Sections 4 characterizes the export dynamics with aggregate shocks only, which explains the international elasticity puzzle. Section 5 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. Section 8 concludes.

2

Setup

Time is continuous and indexed by t. Consumers have “love for variety”, and their preferences are modeled by a CES utility function as in many other trade models. A continuum of firms with measure M produces the diﬀerentiated varieties. There is a measure L of consumers, and each consumer supplies one unit of labor inelastically. Labor is the only factor of production. There are two countries, home and foreign. Foreign variables are denoted by an asterisk. To understand the diﬀerent eﬀects of aggregate and idiosyncratic productivity shocks, I first focus on the case with only aggregate shocks up through Section 4. Aggregate productivity shocks hit all firms proportionally. Due to the export sunk costs, firms solve not only a static profit maximization problem 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 described analogously with an asterisk. At any date t, the representative consumer maximizes the following expected discounted utility U t = Et

ˆ

1

t

e

⇢(s t) (1 ⇠) ⇠ qs Cs ds

,

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

ˆ

ct (!)

✓ 1 ✓

d!

(✓✓1)

,

⌦t

where ct (!) is the consumption of a variety !. The elasticity of substitution across varieties is denoted by ✓ and common for both countries . The set of available goods ⌦t will be characterized in a recursive general equilibrium evolving over time. Total expenditure in the home country (home GDP) is denoted by Yt . The demand for the homogenous good is qt = (1

5

⇠) Yt

and the demand for a diﬀerentiated variety ! at time t is ct (!) =

✓

pt (!) Pt

◆

✓

Ct =

✓

pt (!) Pt

◆

✓

⇠Yt , Pt

where the consumption-based price index is defined as Pt =

ˆ

pt (!)

1 ✓

d!

1 1 ✓

⌦t

There is no borrowing and saving decisions by households because all consumers are identical and trade is assumed to be balanced. Thus, total expenditure must equal the sum of labor income and total 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 the diﬀerentiated good sector.

2.2

Trade Costs

There are three types of costs pertaining to export. The standard “iceberg” transportation cost of exporting goods from home to foreign is denoted by ⌧ . If one unit of good is shipped, only a fraction 1 ⌧

units arrives. In addition to this variable cost, a firm must pay a fixed cost f regardless of the

quantity it exports. This fixed cost must be paid at any given time for a firm to remain as an exporter. 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 starts exporting, 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 must pay  again. The examples of sunk costs include the costs for the initial marketing research and establishing distribution networks in the foreign market. All trade costs are the same across firms within 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 f Yt⇤ and Yt⇤ respectively. This is similar to the usage of foreign labor for trade costs if there is no freely traded homogeneous good and wage rates are determined in a general equilibrium. The proportionality of those costs to the foreign GDP simplifies the firms’ export entry and exit problem. Also, they are assumed to be iceberg costs, meaning that the incurred costs don’t go to labor income. This assumption simplifies the 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 the intensive margin, that is, how much each exporter exports. A permanent reduction in ⌧ also induces more firms to enter the export market aﬀecting the extensive margin (number of exporters) in the long run. The fixed cost f is responsible for the separation of exporters and non-exporters. Since f is assumed to be the same for all firms, only eﬃcient firms who are productive enough to recoup the fixed cost can export. The sunk cost  controls the sluggishness of trade adjustment on the extensive margin in a dynamic horizon. If  is large, the set of exporters is very persistent. The three 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 of labor in the home country produces w units of the homogeneous good. Since the homogeneous good is freely traded and its price is normalized to be 1, the wage in the home country is w if this good is produced. 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 diﬀerentiated by their idiosyncratic productivity

which is constant over time (the stochastic evolution of

will

be incorporated in Section 5 and 6). Also, a country is hit by an aggregate productivity shock Zt at any given time. This aggregate shock hits only the diﬀerentiated good sector. Hence, the economy-wide productivity is given by w whereas Zt is the relative productivity of the diﬀerentiated good sector, which follows a diﬀusion process specified in the next section. Firms have the following linear production function y (Zt ; ) = Zt l, where y is the output and l is the labor used. Each firm maximizes its expected discounted sum of profits. The firm’s optimization problem has two components: a static monopolist’s profit maximization 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 with productivity . Hereafter, I will use subscripts D and X to denote domestic and export variables. The prices charged by firm

are ✓

◆

w ✓ 1 Zt ✓ ◆ ✓ ⌧w pX,t ( ) = ⌧ pD,t ( ) = ✓ 1 Zt

pD,t ( ) =

✓

7

The variable profits (gross of fixed export cost) are ✓ ◆ ✓ ◆ ⇠ pD,t ( ) (1 ✓) Z t Pt ✓ 1 Yt = a1 Yt ✓ Pt w 8 ✓ ◆(1 ✓) ✓ ◆ > Zt Pt⇤ ✓ 1 ⇤ ⇤ < ⇠ pX,t ( ) Yt = a1 Yt if firm exports ✓ Pt⇤ ⌧w ⇡X,t ( ) = > : 0 otherwise

⇡D,t ( ) =

where a1 =

⇠ ✓

⇣

✓

✓ 1

⌘1

✓

. The net export profit is ⇡X,t ( )

f Yt⇤



= a1

⇣

Zt Pt⇤ ⌧w

⌘✓

1

(1)

(2)

f Yt⇤ if firm

exports. Due to the sunk export cost and productivity uncertainty, the decision of export entry and 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 shape parameter

and a lower bound

min

> 0, so its cumulative distribution and density functions are ✓

G( ) = 1

min ( +1)

◆

g( ) = min

for

2[

min , 1).

Assumption 1.

The following assumption ensures that aggregate variables are finite. >✓

1

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

ˆ "

1

1

✓ 1

dG ( )

1 ✓ 1

=⌫

min

1 G ¯t

ˆ

1 ¯t

✓ 1

dG ( )

min

#

1 ✓ 1

= ⌫ ¯t

⇣ ⌘ 1 ✓ 1 where ⌫ = . When productivity follows a Pareto distribution, the average productivity (✓ 1) ˜ is proportional to the lower bound. This result reflects the scale free property of the Pareto distribution, 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 ⇥

G ¯t

MX,t = M 1

2.5

⇤

✓ ¯ ◆ t

=M

min

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 foreign market as demonstrated in Melitz (2003). Let p˜D,t and p˜X,t be the average domestic and foreign prices of the home country. Using the previous results, we have p˜D,t p˜X,t

⇣ ⌘ ✓ ✓ ◆ w = pD,t ˜ = ✓ 1 Zt ⌫ min ⇣ ⌘ ✓ ✓ ◆ w⌧ = pX,t ˜X,t = ✓ 1 Zt ⌫ ¯t

The home and foreign price indices can be written as2 "

¯t ⇤ ✓ 1

Pt = a 2 M ⇤ "

Pt⇤ = a2 M ¯t

where a2 = Notice that

⇣

✓

✓ 1 Pt⇤ is

⌘

✓ 1

✓

✓

w⇤ ⌧ ⇤ Zt⇤

w⌧ Zt

◆1

◆1 ✓

✓

+M(

+ M⇤ (

min )

min )

✓ 1

✓ 1

✓

✓ w⇤ Zt⇤

w Zt

◆1

◆1

✓

✓

#

#

1 1 ✓

(3)

1 1 ✓

(4)

1 ✓ min

. The home export profits are determined by the price relative to Pt⇤ . decreasing in both Zt and Z ⇤ , and increasing in ¯t . A favorable technological ⌫

t

shock in either country reduces the price level and increases welfare. Conditional on Zt and Zt⇤ , high ¯t implies that the price of imported goods (from foreign’s perspective) is higher, so the price index will be high as well. ⇥

In the current model, aggregate productivity shocks Zt = [Zt , Zt⇤ ] and exporter cutoﬀs ¯ t = ⇤ ¯t , ¯⇤ are state variables. Unlike the Chaney-Melitz type trade models, the cutoﬀ productivity t 1

Alternatively, 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 extensive margins against diﬀerent kinds of shocks, I shut down the adjustment on M . 2 Using the expressions for the average prices, the home price index can be written as Pt =

h

M p˜1D,t✓

+

⇤ MX,t

1 ✓ p˜⇤X,t

i

1

1

✓

"

= M

✓✓

✓ ✓

1

◆

w Zt ⌫ min

9

◆1

✓

+M

⇤

✓ ¯⇤ ◆ t min

✓✓

✓ ✓

1

◆

w⇤ ⌧ ⇤ ⇤ Zt ⌫ min

◆1

✓

#

1

1

✓

¯t cannot be pinned down by the structural parameters and destination’s characteristics. Due to the sunk export cost and export hysteresis behavior, we need a complete history of the shocks Z to obtain ¯ t . Given Zt and ¯ t , the price indices are calculated according to (3) and (4). The average net profit in the home country is MX,t [˜ ⇡X,t f Yt⇤ ] M ⇣ ⌘ = ⇡X,t ˜X,t are the average profits of all firms and exporters ⇡ ˜t = ⇡ ˜D,t +

⇣ ⌘ where ⇡ ˜D,t = ⇡D,t ˜ and ⇡ ˜X,t

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 aﬀects both income and expenditure of a country and the vector [˜ ⇡t , ⇡ ˜t⇤ ] is the fixed point of the two excess demand functions. The derivations of the equilibrium average profits ⇡ ˜t and ⇡ ˜t⇤ are relegated to Appendix A.1. Trade is not necessarily balanced in the diﬀerentiated good sector, but I assume any imbalances in the diﬀerentiated good sector is made up for in the homogeneous good sector so that aggregate 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 domestic market. For the foreign market, they solve a dynamic optimal entry and exit problem due to the export sunk cost and profit uncertainty. Let ⇡ ˆX,t = be written as

⇡ ˆX,t

✓

⇡X,t Yt⇤

Zt⇤ ¯ , t, Zt

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

◆

= a3 ⇥

✓ 1

| {z } fixed, idiosyncratic

0

B ⇥@ |

M

⇣

¯t

⌘✓

1 1

⇣

Zt⇤ Zt

w + M⇤ w⇤ ⌧ min {z stochastic, aggregate

⌘✓

1 C

1A

(5)

}

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

a3 =

Zt⇤ Zt .

If aggregate productivities rise by the same magnitude in both countries leaving

a1 ✓ a1 ( min )✓ 1 2

=

⇠ ✓

⇣

(✓ 1)

⌘

(1 ✓) min

10

Zt⇤ Zt

constant, export profit is not aﬀected. In that situation, the export price pX,t will decrease due to the home productivity shock, but the foreign price index will also decrease by the same proportion due to the foreign shock, which oﬀsets the extra profit. Also, the home exporter cutoﬀ ¯t is the aggregate state variable which evolves over time depending on the history of productivity shocks as shown below. Since > ✓ 1, when ¯t increases, ⇡X,t ( ) increases. For exporters, an increase in ¯t implies a higher foreign price index implying milder competition there. In the dynamic entry and exit problem, firms take the stochastic evolution of ⇡ ˆX as exogenous, and compare the expected value 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 in Section 2.3. Given that result, they solve a dynamic stochastic control problem of export entry and exit. Recall that the fixed and sunk export costs f Yt⇤ and Yt⇤ are proportional to the foreign GDP. Normalizing the export profit and the associated costs by Yt⇤ , we can analyze the stochastic evolution of ⇡ ˆX,t in relation to f and  to derive the optimal entry and exit policies. The proportionality of export costs to Yt⇤ prove useful in this analysis since any eﬀect of Yt⇤ on the export profit is oﬀset by p

( )

the same change in export costs, and hence, we can focus on the eﬀect of the relative price X,t on Pt⇤ Zt⇤ ¯ trade. Because ⇡ ˆX,t in (5) only depends on , and , it is convenient to define the log of relative productivity as

Zt

zt = ln

✓

Zt Zt⇤

◆

Then, the state variables of ⇡ ˆX,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 ⇡ ˆX,t (zt , ) is zt . This assumption corresponds to the case of a small home country discussed in Section 4. The general case 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 optimally choosing the timings of entry and exit. Due to the aggregate shock, the export profit follows a diﬀusion process. This is a classical problem of investment under uncertainty (e.g. Dixit (1989) and Dixit and Pindyck (1994)). An exporter and a non-exporter can be viewed as assets that are call options on each other, and the optimal strategy is characterized by a pair of threshold productivity levels. The export sunk cost and uncertainty coming from aggregate productivity shocks give rise to 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 ⇡ ˆX,t (z, ) to simplify notations. The subscript

is dropped since the structure of the problem is identical for all firms. Yet, it should

be understood that all value functions and derived triggers depend on

11

. Consider the following

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 infinitesimal

drift and variance parameters respectively. They are time invariant, but can depend on the current status of z. Let V0 (z) to be the value function for a non-exporter when the current aggregate shock is z, and similarly V1 (z) for an exporter. By Ito’s lemma, we obtain 1 dV0 = V00 (z) dz + V000 (z) 2 Therefore,



0

E (dV0 ) = µ (z) V0 (z) +

1 2

2

2

(z) dt

00

(z) V0 (z) dt

The asset equilibrium condition gives the following Hamilton-Jacobi-Bellman (HJB) equations for V0 and V1 4 ⇢V0 (z) = µ (z) V00 (z) +

1 2

2

(z) V000 (z)

1 2 ⇢V1 (z) = ⇡ (z) f + µ (z) V10 (z) + (z) V100 (z) | {z } | 2 {z } dividend capital gain

(6) (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 dividend profit 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 value

of exiting from export market. Both equations are second-order linear ODE, but (6) is homogeneous whereas (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-order linear ODE, two boundary conditions are required to characterize the particular solution. Since we have 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 threshold levels of z: zL and zH with zL < zH . zH is the threshold at which a non-exporter becomes an exporter, and zL is the threshold at which an exporter becomes a non-exporter. Since zL < zH , this is an inaction region. If the current aggregate shock z is in this range [zL , zH ], the optimal policy is to remain the status quo; exporters should continue exporting and non-exporters should not start exporting. Thus, the current aggregate productivity z is not a suﬃcient statistic to determine if a particular firm is an exporter or not if z falls in the inaction band for the firm. The definition of zL 4

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

12

and zH suggests the following value-matching conditions5 (8)

V0 (zL ) = V1 (zL ) V0 (zH ) = V1 (zH )

(9)



These boundary conditions pin down the form of V0 and V1 . For optimality, we need the following smooth pasting conditions6 V00 (zL ) = V10 (zL )

(10)

V00 (zH )

(11)

=

V10 (zH )

Above four conditions (8) - (11) characterize the particular solutions for V0 and V1 , and the optimal thresholds zL and zH . Because the two HJB equations (6) and (7) share the same homogeneous part, it is convenient to define the following function as in Dixit (1989) U (z) = V1 (z)

V0 (z)

Then, we can collapse the two HJB equations into f + µ (z) U 0 (z) +

1 2

(z) U 00 (z)

(12)

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

(13)

⇢U (z) = ⇡ (z)

2

The value-matching and smooth pasting conditions become

Any two of the above boundary conditions determine the particular solution for U (z) and the other two 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 the export market, it must have limz!0 V0 (z) = 0 since when z is very small, the probability of rising to zH in any given finite time is almost zero. Also V1 (z) consists of the profit flow which is increasing 5

The non-exporter value function can be expressed as ⇢  0 1 V0 (z) = max V1 (z) , V0 (z) + µ (z) V0 (z) + 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 exporter and non-exporter implying V0 (zH ) = V1 (zH ) . A similar argument holds for the case of zL . 6 See Chapter 4 of Dixit and Pindyck (1994) and Stokey (2009) for more detailed treatments of this condition. If the smooth pasting conditions are not satisfied, two value functions are connected at a kink. It can be shown that the 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 is small, the option value dominates, and when z is large, the profit flow dominates. When z follows a Brownian motion, closed-form solutions for V0 (z) and V1 (z) are available, but in general, one must solve the ODEs numerically.

3.3

Band of Inaction

Because zL < zH , the band of inaction [zL , zH ] causes the export hysteresis behavior. We can convert the band of inaction of z to that of the export profit, [⇡L , ⇡H ]. Consider a firm whose export 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 the band 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, a firm 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 uncertainty expands the Marshallian inaction band [f, f + ⇢]. Thus, it is possible that an exporter receives negative 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 to the 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 the aggregate productivity shocks. I focus on the case of small home country to elucidate the workings of export hysteresis on the international elasticity puzzle. The analysis of the general case is presented in Appendix A.3 because the full model also uses the small home country assumption. The following assumption 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 Pt⇤ nor ⇡X,t . The normalized export profit (5) can be expressed as ⇡ ˆX,t (z, ) = S ( ) e(✓

14

1)z

,

(14)

value V0 HzL

30

V1 HzL

25

k

20 15 10 5

zL 0.2

zH 0.4

0.6

0.8

1.0

z

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

value 6

5

4

U@zD

3

2

1

zL 0.2

zH 0.4

0.6

0.8

(b) U (z)

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

15

1.0

z

dont' export

inaction region

π_L

export

f

f+ρκ

π_H

negative profits π

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 ⇡ ˆX,t is z alone, we can apply the results of the previous section to derive the inaction region. 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 variance parameters µ (z) and

(z) don’t depend on the current level of z. The HJB equations for exporters

and non-exporters become ⇢V0 (z) = µV00 (z) + ⇢V1 (z) = Se(✓

1 2

2

V000 (z)

f + µV10 (z) +

1)z

(15) 1 2

2

V100 (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 V1 (z) = Be 7

a4 =

a3 M⇤

⇣

w⇤ ⌧w

⌘✓

1

(

min )

(✓ 1)

16

(17) z

+

⇡ (z) ⌘

f ⇢

(18)

where ↵ = =

µ+

p p

µ

µ2 + 2⇢ 2

µ2 + 2⇢

2

⌘ = ⇢

2

⇡ (z) = Se(✓

2

(✓

2

2

1)2

µ (✓

1)

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’s

value 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 characterizes A, B, zL and zH . 2

F1 (A, B, zL , zH )

6 6 F2 (A, B, zL , zH ) F =6 6 F (A, B, z , z ) L H 4 3 F4 (A, B, zL , zH )

3

2

Ae↵zL

Be

zL

7 6 7 6 Ae↵zH Be zH 7=6 7 6 ↵Ae↵zL + Be zL 5 4 ↵Ae↵zH + Be zH

Se(✓

1)zL

⌘

Se(✓

(✓ (✓

f ⇢

3

2

0

3

7 6 7 6 7 + 7 7=6 0 7 (✓ 1)z 6 7 1) Se ⌘ L 7 5 4 0 5 (✓ 1)z 0 1) Se ⌘ H

1)zH

⌘

+

+

f ⇢

(19)

We do not obtain the explicit analytical expressions for zL and zH , but we can do comparative statics analytically using the implicit function theorem. Figure 1 plots the value functions when z follows a Brownian motion with the parameters summarized in Table 1. The exit and entry triggers are zL = 0.311 and zH = 0.98, that imply ⇡L = 0.638 and ⇡H = 1.74. We confirm that 0.638 = ⇡L < f = 1 and 1.35 = f + ⇢ < ⇡H = 1.74. Thus, it is possible that an exporter receives negative profit when ⇡ falls in [f, ⇡L ] hoping ⇡ to rise in the near future. Similarly, the profit level 8 The second part of equation (18) has a nice interpretation. Since z follows a Brownian motion with µ and 2 , the aggregate productivity ZZ⇤ = ez follows a geometric Brownian motion with a drift µ + 12 2 and variance 2 . The additional 12 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 . So the following relationship holds ˆ 1 ⇡ f ⇡ f = =E e ⇢t (⇡t f ) dt ⌘ ⇢ ⇢ µ⇡ ⇢ 0

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

17

value

µ 0.01

2

0.01

✓

1 1.5

⇢ 0.07

S 0.4

f 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 aggregate

shock [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 the Brownian 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. The above result can be found in Dixit (1989) and Dixit and Pindyck (1994). Since ⇡X , f and  are measured in the same unit, any rescaling by f and  does not change the nature of the problem when ⇡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 aggregate

shock: 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 profit levels are the same for all firms. This is evident by considering the parallel movements of ln ⇡ ˆX, of diﬀerent ’s as pointed out in 3.1. We can also see the result by considering the movement of the export profit. When z follows a Brownian motion with µ and geometric Brownian motion with a drift µ⇡ = (✓

1) µ+ 12

(✓

1)

2

2, 2

the export profit ⇡ follows a

and variance

2 ⇡

= (✓

1)2

2.

Then, the HJB equation (12) can be written in terms of ⇡ ⇢U (⇡) = ⇡

f + µ⇡ ⇡U 0 (⇡) +

1 2

2 2 00 ⇡ ⇡ U (⇡)

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 and non-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 of exporters) changes, the entry or exit must start from the cutoﬀ firm ¯t . Figure 3 illustrates the evolution of ⇡ ¯ and ¯ in this environment. The aggregate productivity shock z (the top line plot of Figure 3) starts rising at t1 which raises ⇡ ¯ (the middle plot) as well. At t2 , ⇡ ¯ hits the upper ¯ threshold ⇡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 adjustment in ¯ 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 profit is not low enough to induce the threshold firm to exit. The aggregate shock z bumps up between t4 and t5 but this does not change ¯ either because ⇡ ¯ does not hit the upper threshold. From t5 , z and ⇡ ¯ 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 results in this section. The inaction band emerges from the sunk export cost and uncertainty. As long as the cutoﬀ profit ⇡ ¯ remains within the inaction band, the set of exporters does not change. When the cutoﬀ profit is at one of the thresholds, any further shocks are completely absorbed by the updating of the cutoﬀ firm ¯. The current model generates a low trade elasticity in the short run and a high elasticity 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 aggregate shock

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 Yt⇤ ) by xt ( ) and Xt

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

⇣ ⌘ MX,t ⇥ xt ˜X,t = a4 ⇥ ¯t | {z } | {z } set of exporters average export

19

⇥ e(✓

1)z ¯✓ 1 t

t1

t2

t3

t4

t5

t6

Time !!!!!!

Figure 3: Dynamics of ln ⇡ ¯ and ln ¯

20

t7

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

where a4 is a! constant. Taking the log of the above expression yields ln Xt = 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 on ✓ 1 Z one of the boundaries, ¯ is being updated to keep ⇡ ¯ / ¯ Z⇤ 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 slopes multiplied by ✓

1 and

1 as shown in Figure 3. The log of the cutoﬀ profit ln ⇡ ¯ follows a regulated

Brownian motion with the two-sided regulators ln ⇡L and ln ⇡H . The long-run stationary probability density of x = ln ⇡ ¯ is (x) = where

=

2(✓ 1)µ (✓ 1)2 2

e

xH

e

xL

e

x

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 ⇡H . 2

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 extensive margin 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 productivity evolves as a Brownian motion. The growth process of firm productivities is borrowed from Gabaix (1999) and Luttmer (2007). I abstract from the balanced growth path analysis, but incorporate export sunk costs in the model. Due to the sunk costs and idiosyncratic productivity uncertainty, each firm exhibits the export hysteresis. Yet, a stationary equilibrium emerges in which the productivity distribution of all firms and exporters are both modeled by a double Pareto distribution as in Luttmer (2007). The same idea is investigated in Impullitti et al. (2013), but their analysis is restricted for the case of two symmetric countries. The current model can be extended to 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 endogenously emerges from the random growth of idiosyncratic productivities. The log of idiosyncratic productivity

is assumed to evolve as a Brownian motion. Define s = ln . The law of motion of s

is (20)

ds = µdt + dW

where W is a standard Brownian motion. The entry point of productivity is fixed at s⇤ . When new firms enter the economy, they all start from productivity s⇤ . After that, each idea follows a diﬀusion process 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. The following Kolmogorov Forward Equation (henceforth KFE) describes the evolution of productivity density f (a, s)9 @f (a, s) = @a

f (a, s)

µ

@f (a, s) 1 + @s 2

2@

2f

(a, s) @s2

(21)

The first term on the RHS reflects the reduction of density due to the exogenous death shock. The second term reflects the outflow of the density due to the drift of s. The third term reflects the inflow of density due to randomness. The solution to the above second-order PDE is the following 9

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

22

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

f (a, s) = e

p

1

e

2

2⇡a

2

[s (s⇤ +µa)]2

(22)

2a 2

The derivation of the above solution can be found in Harrison (1985) and Luttmer (2007). The intuition for the above density is clear. All ideas starts from s⇤ and follow a Brownian motion. Since the drift and variance are proportional to the elapsed time a, the distribution of productivities of age cohort a has a mean s⇤ + µa and variance a

2.

Due to the death shock , the density of age

cohort a must be discounted by a. This implies that old ideas have a small weight since most of their cohort have died out up until time a. The first term e

a

reflects this discounting. Conditional

on surviving, they have larger variance. Let f (s) be the stationary probability density of s. This is the marginal density of f (a, s) with ´1 respect to s: f (s) = 0 f (a, s) da. By the definition of stationarity, we have @f@a(s) = 0. Then, we obtain the following KFE which describes f (s) 0=

f (s)

µ

@f (s) 1 + @s 2

2@

2f

(s) @s2

(23)

This is a homogeneous second-order linear ODE (not PDE). The general solution to this ODE takes the following form 1s

f (s) = C1 e where

1

and

2

2 2

µ

Quadratic formula gives 1

2 1, 2

(24)

are the two solutions to the following characteristic equation 1 2

so that

2s

+ C2 e

> 0. Since

> 0,

1

>

2µ 2

= =

µ+

=0

p µ2 + 2

2

2

p µ2 + 2

µ

and

(25)

2

2

2

> 0. The constants C1 and C2 are determined by

additional conditions. The stationary probability density f (s) must satisfy the following conditions as well (this is from Arkolakis (2011)) lim f (s) = 0

s! 1

ˆ µ [f (s⇤ )

f (s⇤ +)] +

s⇤

f (s) ds + 1

1 2

2

⇥

ˆ

1

f (s) ds = 1

s⇤

f 0 (s⇤ ) 23

0 8s 2 ( 1, 1)

f (s)

f 0 (s⇤ +)

⇤

=

(26) (27) (28) (29)

The first condition is the lower boundary condition. For the shock process of s,

1 is the absorbing

barrier. The second and third conditions ensure that f (s) is a probability density. The last condition

emerges by integrating the KFE (23). This states that the net inflows to the density are equal to the net outflows. The particular solution of (24) which satisfies the above four conditions is given by f (s) =

8 > < > :

1 2 1+ 1 2 1

+

e

1 (s

s⇤ )

for s < s⇤

2 (s

s⇤ )

for s

2

e 2

(30)

s⇤

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 is assumed that

> 0. The resulting stationary distribution of

g( ) =

where

⇤

⇤

8 > > > < > > > :

1 2 1+

2(

1 2 1

+

2

(

1

1

⇤)

1

2

1

⇤)

2

for

= es is the following11 <

⇤

(31) for

⇤

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

power law behavior. The entry point 2

⇤

is a scale parameter which controls the absolute advantage whereas

1

and

are shape parameters that govern comparative advantage. They are similar to the location and

shape parameters of the Frechet distribution introduced in Eaton and Kortum (2002). The entering rate of new firms to the domestic market is fixed to keep M constant. Since there is no sunk or fixed cost for the domestic sales, the above stationary distribution arises independent of the exporter’s distribution.

5.2

Band of Inaction

Because of the export sunk cost and idiosyncratic productivity uncertainty, firms solve the same dynamic entry and exit problem as in subsection 4.1. In this section, the foreign GDP, Y ⇤ is constant in the stationary equilibrium, and thus, the fixed and export costs are not tied to the foreign GDP (they are just f and ). Define the discounting rate which augments the death shock rate as r = ⇢+ . The export profit ⇡X follows a geometric Brownian motion, and the HJB equations 10

Reed (2001) decomposes the expression (22) into exponential and normal distributions, and uses their moment generating functions (MGF) to characterize the MGF of the resulting distribution which can be verified as the ´1 asymmetric Laplace distribution (30). Arkolakis (2011) explicitly calculates the marginal density f (s) = 0 f (a, s) da to show the same result. 11 Let = h (s) = es so that h 1 ( ) = ln . The PDF of is given by the following equation g( ) =

d h d

1

( ) f h

24

1

( )

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

H

L

= e sL

= esH as the common exit and entry thresholds. We obtain the following system which

characterizes A, B, 2

L

and

F1 (A, B,

6 6 F2 (A, B, F =6 6 F (A, B, 4 3 F4 (A, B,

H.

L,

H)

falls below

L.

L,

A

↵ L

B

S L

7 6 6 A ↵ B ) H 7 H H 7=6 7 6 6 ↵ , ) L H 5 4 ↵A L + B L , ) L H ↵A ↵H + B H L,

When a firm’s productivity reaches but still in the range [

2

3

H ],

H,

✓ 1 L

⌘

S

✓ 1 H

⌘

(✓ (✓

+

f r

f r + S ✓ 1 1) ⌘L S ✓ 1 1) ⌘H

+

3

2

0

3

7 6 7 7 6 0 7 7 6 7 7=6 7 7 4 0 5 5 0

the firm starts exporting. When the productivity falls,

the firm keeps exporting. It exits from the export market once

Unlike the model with aggregate shocks, firms’ export entry and exit decisions are

not uniform, and both exporters and non-exporters coexist in the range [

L,

H ].

The stationary

exporter density including this “mixed region” of productivity is derived in the next subsection.

5.3

12

Exporter’s Distribution

To obtain aggregate variables such as price index and trade shares, we need to characterize the distribution of exporters to each destination. We know that all firms with

>

all firms with

H ],

<

relationship between

L

don’t export. For firms in the inaction range

2[

L,

H

export and

the monotonic

and export status breaks down. Two firms can be exporters and non-exporters

given the same current productivity . Exporters in this range are those who overcame the entry threshold

H

once, but the productivity declined below

H

(yet still above

L ).

Non-exporters in

the same range are the ones who haven’t received enough favorable shocks to cross the threshold since 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 history of this firm. Nonetheless, the cross-sectional distribution of exporters in this region converges to a stationary 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 . Once productivity hits sL , a firm exits from exporter’s distribution. Let f1 (a, s) be the density of exporters with eﬃciency s and age a. We have the same KFE as in (21) and conditions (26) - (29) except that 12

As a limiting case, if  goes infinity, the entry option becomes worthless. In that case, A = 0 and sH = 1. The exit threshold is  1 f ⌘ sL = ln (✓ 1) + (✓ 1) S r which means

⇡L =

+ (✓

25

⌘ f 1) r

the lower bound

1 is now replaced by sL . Then, the solution to the KFE becomes the following a

f1 (a, s) = e

(a, s|sH ) for a > 0 and s > sL

where 1 (a, s|sH ) = p

a

"

N

✓

s

sH p

µa a

◆

µ(sH sL ) 2 /2

e

N

✓

s + sH

2s p L a

µa

◆#

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 expression

becomes equation (22) derived earlier. The size density of exporters f1 (s) is given by f1 (s) =

ˆ

1

1

0

where

1

and

2

min e(

1 2

f1 (a, s) da =

+

2

1 + 2 )(s

⇥

sL ) , e(

1 (sH

e

sL )

1 + 2 )(sH

⇤

2 (s

1 e

sL )

1

sL )

for s > sL

were computed in subsection 5.1. Since the entry barrier sH is a point mass, above

expression is also a probability density. Thus,

f1 (s) =

8 > > > <

1 2

+

1

> > > :

2

⇥

h ⇤ e 1

1 e

1 (sH

sL )

1 (s

sL )

2 (s

e

sL )

e( 1 + 2 )(sH sL ) 1 ⇥ ⇤ e e 1 (sH sL ) 1 1+ 2 1 2

i

for s 2 (sL , sH )

2 (s

sL )

for s > sH

This was introduced in Luttmer (2007) and used in Impullitti et al. (2012) as well. The corresponding probability density for

= es is

g1 ( ) =

where (

2

L

= esL and

H

8 > > > > > > > > < > > > > > > > > :

1 2 1

+

2

2⇣

⌘

6 L 4 ⇣ ⌘ +

1

H L

1 2 1

2

3

1+ 2

2⇣

H L

6 4 ⇣

17 5

1 ⌘

H L

(

1

= esH . All firms with

for

2

L

1+ 2

⌘

2 +1)

1 >

3 17 5 H

2(

L,

H)

(32) (

2 +1) 2

for

>

H

L

export and their Pareto tail index is

+ 1) which is equal to the tail index of the overall stationary distribution. This confirms the

scale free property of the Pareto distribution. Figure 5 depicts the overall stationary density of firms and exporter’s density. From (31), we can easily show that the share of firms with 1 1+ 2

<

.

26

⇤

is

2 1+ 2

and the share of firms with

>

⇤

is

0.6

0.5

0.4

0.3

0.2

Non-exporters 0.1

Exporters

*

fH

f

0.0

1

fL

2

3

4

5

Figure 5: Stationary Density of Firms As in Meliz (2003), define the following average productivity for exporters ˜=

ˆ

1

✓ 1

dG1 ( )

1 ✓ 1

L

We have the following result. Lemma 4. The average productivity for exporters ˜ is given by the following expression 2

where k1 =

˜=6 4

1

1) and k2 =

+ (✓

2

✓

1 2

k1 k2

(✓

◆

⇣

⇣

H L H L

3✓11

⌘ k1 ⌘

17 5 1

1

(33)

L

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 = lemma 2, we can see that ⇡X ( implies that q =

H L

L)

and ⇡X (

H)

H L

. Using

are homogeneous of degree one in f and . This

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, the results in this section can be extended to asymmetric multi-country general equilibrium of trade extending Chaney (2008)13 . 13

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

27

ij fij

is common for all

6

Full Model

This section presents a unified model which incorporates both aggregate and idiosyncratic productivity shocks. The basic setup is the same as before, but now there are two types of productivity shocks st (idiosyncratic) and zt (aggregate) that follow a Brownian motion. Assumptions in this section are summarized below. 1. (small home country): The home mass of firms relative to foreign is small: 2. (aggregate shock): The log of relative productivity z = ln

Z Z⇤

motion dz =

M M⇤

⇡ 0.

follows a drift-less Brownian

z dW

3. (idiosyncratic shock): The log of idiosyncratic productivity s = ln ˆ evolves as a Brownian motion ds = µs dt +

s dW

4. (independence): Idiosyncratic shocks are i.i.d, and independent from aggregate shocks. 5. (parameters):

>✓

1,

1)2

> (✓

2 s

2

Again, the small country assumption simplifies the diﬀusion process of the export profit, and is a good approximation for most of the countries. Because there is no fixed or sunk entry costs for domestic 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 fixed at M to keep the mass of operating firms constant. New firms are born at productivity s⇤ as nonexporters. The fixed and sunk export costs are proportional to the foreign GDP, and the diﬀusion process of the normalized export profit ⇡ ˆX is analyzed as in Section 4. Unlike Section 5, the density of 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 and the 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 define the average price for exporters as p˜ij = p ˜ij and the expression of the price index of country j becomes Pj =

✓ ✓

1

N X

Lkj

⌫kj (wk ⌧kj )

1 ✓

k=1

where ⌫kj =

Mk

k2,k

2,k

✓

+ 2,k k1,k k2,k

1,k

Other comparative statics can be conducted as well.

28

◆

k

qkj1,k 1,k + 2,k

qkj

!11✓

1 1

(34)

6.1

Band of Inaction

By the small country assumption, a firm’s export profit can be written as ⇡ ˆX (zt , st ) = Se(✓

1)(zt +st )

where S is a scaling constant. The aggregate and idiosyncratic productivity shocks enter the export profit function symmetrically. Because of these two shocks, the export profit also follows a diﬀusion process. The associated HJB equation is a second-order PDE (not ODE): rU (z, s) = ⇡ (z, s)

f

µs U s +

1 2

2 s Uss

+

1 2

2 z Uzz

(35)

where r = ⇢ + . In the zs plane, two curves separate the domain into three regions of diﬀerent optimal strategies: export, inaction, and do not export. We can also express the threshold levels of s as functions of z: sL (z) and sH (z). The curves sL (z) and sH (z) characterize the lower and upper 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 quite

diﬃcult to analyze because not only the surface function, but the boundary conditions are also unknown. Nevertheless, when both shocks follow a Brownian motion, we can reduce the problem to an ODE with an analytical solution. Proposition 2. Given the assumptions in this section, the value function U (z, s) has the following form U (z, s) = Be

(z+s)

Ae↵(z+s) +

⇡ (z + s) ⌘

f , r

where where p

µ2s + 2r ( z2 + s2 ) ( 2 + s2 ) p z µs µ2s + 2r ( z2 + s2 ) = ( z2 + s2 ) 2 2 z + s ⌘ = r (✓ 1)2 µs (✓ 2

↵ =

µs +

1)

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

29

s 1.2 1.0 10

UHz,sL

0.8

Export

5 0.2

0 -5

0.6 0.4

0.0

0.0

0.5

s

0.0

-0.2

z

Inaction region

0.2

-0.2

1.0

(a) Plot of U (z, s)

Do not export 0.0

0.2

0.4

0.6

0.8

1.0

1.2

z

(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

2 v

=

2 z

+

2 s

since z and s are

independent. Then, we can solve the lower and upper thresholds vL and vH as in subsection 4.1, that characterize the two curves sL (z) = vL

z and sH (z) = vH

z. Along these lines, the boundary

conditions 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 any

z. This result is used below to transform the KFE. The graph of U (z, s) and the contour lines at U (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 any date 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. Because the 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) 1 + @s 2

2@ s

2 m (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 1 2

m (h (t) , t) = 1

+

e

2 (h(t)

s⇤ )

2

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

l (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 function n (s, t) must satisfy the following p.d.e @n (s, t) = @t

n (s, t)

z 0 (t)

@n (s, t) @s

µs

@n (s, t) 1 + @s 2

2@ s

2 n (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 boundary

problem to the one with fixed domains. The new function n (s, t) can be numerically solved in Mathematica. Note the new term z 0 (t) @n(s,t) @s . The growth rate of the aggregate shock amplifies the eﬀect 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 obtain the analytical expression of n (s, t) (See Appendix A.6). 15 If this happens, new firms are all exporters. The calculations of the exporter dynamics and its stationary density are complicated. 16 In 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 death shock rate elasticity of substitution variable trade cost sunk export cost relative to the fixed cost drift parameter (idiosyncratic) standard deviation (idiosyncratic) standard deviation (aggregate)

⇢

0.05 0.09 2 0.3 20.5 -0.0163 0.081 0.036

✓ ⌧  f

µs s z

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 distribution and 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 5 percent. For the benchmark case, ⌧ = 1.3 to match the results in Obstfeld and Rogoﬀ (2000). I choose 2 for the elasticity of substitution between varieties ✓ to reflect the low trade elasticity in the short run. Following Impullitti et al. (2012), I set the death shock rate

= 0.09. This is based on

the 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 is governed by the upper Pareto tail index which is

2.

I target the estimate from Eaton and Kortum (2002),

= 8.28. I borrow the drift p parameter of the idiosyncratic shocks µs = 1.63% from µs + µ2s +2 s2 Arkolakis (2011). Since 2 = , this gives s = 8.1%. As in Ruhl (2008), z is set 2 2

s

to 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. exporter

distribution. I normalize s⇤ = 0 and the scaling constant for the export profit to be S = 1. Although the 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 stationary equilibrium with the absence of the aggregate shock. Bernard and Jensen (2004b) find that firms enter and exit from the export market at diﬀerent productivity levels using U.S. data for the period of 1983-1992. According to their statistics, the exit trigger productivity is two-thirds of the entry trigger 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 the current 17

model17 .

I target sH

sL = 1 and obtain

 f

 f

= 91, which leads to sH

sL = 1.64 in

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

From 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 of exporters is18

which gives sH

7.2

⇥ e 2 sH e m1 = e 1 (sH sL ) = 0.184 and sL = 0.816.

1 (sH

sL ) 2 (sH

e

⇤ 1

sL )

,

Trade Dynamics

Given an aggregate shock path and initial exporter’s distribution, the full model has the ability to generate nonlinear export dynamics. The dynamics of the exporter distribution is characterized by the p.d.e (38), which enables us to compute aggregate export for any t. Two scenarios are investigated below with the implications on the short- and long-run trade elasticities. Those aggregate shock paths are particular realizations of z, which all firms expect to follow a Brownian motion. For both scenarios, the economy starts from the stationary distribution, and the KFE (38) is numerically solved to characterize n (s, t), which is converted back to m (s, t). The corresponding density function of = es , m ˆ ( , t), is then computed to obtain ˜t (the average productivity of exporters), which summarizes the model’s implications for aggregate export. Log of export (normalized by Yt⇤ ) is ln Xt = ln

"ˆ

where a5 is a constant and ˜t =

1

x( )m ˆ ( , t) d

L (t)

h´

1

L (t)

trade adjustments are represented by (✓

✓ 1m ˆ(

#

, t) d

= a5 + (✓ i

1) zt and (✓

h i 1) ln ˜t + zt ,

1 ✓ 1

. The intensive and extensive margins of 1) ln ˜t respectively. The log of export in

the initial period ln X0 is normalized to be 0. 7.2.1

Regime Switching

Consider the following regime-switching scenario. The aggregate shock z is initially 0, jumps by 10% within a quarter, and stays at the new level for 8 years as shown in Figure 7a. In the initial quarter, the total export increases by 31% implying the trade elasticity of 3.1. This number is a composite of the intensive margin elasticity of ✓ of 2.1. Let { {

L,1 ,

and

H,1 }

H,1

<

H,0 }

L,0 ,

1 = 1 and the initial extensive margin adjustment

be the original exit and entry thresholds for firms with z = 0, and similarly

for the new regime with z = 0.1. Because the aggregate shock is positive, H,0 .

Due to the jump in z, non-exporters in the range [

H,1 ,

H,0 ]

L,1

<

L,0

become exporters

immediately. This is the initial adjustment on the extensive margin. The exporter density in the new inaction region [ export entry point

L,1 ,

H,1 ,

H,1 ]

is lower compared to the new steady state. Starting from the new

few firms have “diﬀused” into the inaction region. As time passes, new

exporters enter and some incumbent exporters exit from this region, but the entry force is dominant 18

See Appendix for derivation.

33

accumulating the exporter’s density. The sluggish trade adjustment on the extensive margin is caused by the evolution of exporter’s density in the inaction region. By the seventh quarter, the total export increases by 50%. As t ! 1, the trade elasticity tends to

2

= 8.28 as we saw in

Section 5. 7.2.2

Business Cycles

Next, consider a business cycle scenario with z (t) = 0.05 ⇥ sin (t) as illustrated in Figure 7b. The

aggregate shock displays a cyclical movement with a period of 2⇡ ⇡ 6.28 years. In this analysis, I pick 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 and bottoms of the log export occur with two or three quarters of a positive lag. As z is on an upward trend, the exporter’s density adjusts toward the new steady state. When z changes its direction from positive to negative (at a peak), the exporter’s density in the inaction region still carries the positive 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 0.122 exhibiting the path dependence of the export dynamics. The current model oﬀers an alternative explanation for “the S-curve” phenomena examined by Backus, Kehoe and Kydland (1994). They document that, for many countries, the trade balance (net exports) is negatively correlated with the terms of trade (the relative price of imports to exports) in the current and future periods, but positively correlated with that in the past periods. The shape of the lagged correlations exhibits the S-curve. They develop an IRBC model with time-to-build lag for capital formation to explain the S-curve shape of the lagged correlations. In the current model, zt corresponds to the terms of trade in their definition since a high zt implies a low export price which means the high terms of trade. Figure 8 depicts the lagged correlations Corr (zt , ln Xt+k ). We can confirm the S-curve in the plot. The lag of k = 3 gives the highest correlation coeﬃcient. Export is negatively correlated with z in the future but positively correlated with z in the past. Because physical investment and the international flow of capital are not featured in the current model, it does not explain the countercyclical movements of exports (a negative sign of Corr (zt , ln Xt )), but it 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 dynamics in a consistent manner. The model incorporates export fixed costs and sunk costs, as well as two types of productivity shocks, in a continuous-time framework with heterogeneous firms. The export sunk costs and uncertainty give rise to export hysteresis, the reluctance of firms to enter

34

0.7

0.08

0.6

log of total export Hred lineL

aggregate shock z Hblue dashed lineL

0.10

0.5 0.06

0.4

0.04

0.3 0.2

0.02 0.00

0.1 0

5

10

15

20

25

0.0

30

quarter

log of total export Hred lineL

aggregate shock z Hblue dashed lineL

(a) Regime switching

0.04

0.20

0.02

0.15 0.10

0.00

0.05

-0.02

0.00

-0.04

-0.05 0

20

40

60

80

100

120

quarter (b) Business cycle

Figure 7: Export dynamics of the full model

35

CorrHzt , lnXt+k L

1.0

0.5

0.0

-0.5 -10

-5 0 5 Lag k of z behind the log export

10

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 by aggregate productivity shocks only and the firm-level eﬃciency is fixed over time, the cutoﬀ firm which separates exporters and non-exporters evolves stochastically. The cutoﬀ does not respond against small temporary shocks, which results in a small trade elasticity in short run due to the inoperative extensive margin. When the shock is long-lived, new firms start entering or exiting from the 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 elasticity against long-lived shocks studied by Ruhl (2008). If the productivity shocks are idiosyncratic and there is no aggregate shock, the economy converges to a steady state, with individual firms moving around within a stationary distribution of productivities. Export hysteresis at the micro level leads to 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 of trade dynamics. It generates simultaneous export entry and exit, “the mixed region” of exporters and non-exporters, and the sluggish transition dynamics of exporter’s distribution in response to aggregate shocks, which provides an alternative explanation for the S-curve of the lagged correlations between exports and the terms of trade. One limitation of the model is the assumption of a Brownian motion for the aggregate shock process. It provides a sharp analytical characterization of the optimal strategies, but prevents us from the analysis of long-run equilibria. A model in which the aggregate shock follows a mean-reverting process 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 of partial diﬀerential equations in Section 6 because they become free-boundary and moving-boundary problems that require more sophisticated numerical methods.

36

References Alessandria, George and Horag Choi (2007) “Do Sunk Costs of Exporting Matter for Net Export Dynamics?.,” Quarterly Journal of Economics, Vol. 122, No. 1, pp. 289 – 336. (2011) “Establishment heterogeneity, exporter dynamics, and the eﬀects of trade liberalization.,” Working Paper. Arkolakis, Costas (2011) “A Unified Theory of Firm Selection and Growth.,” NBER Working Paper 17553. Backus, David K., Patrick J. Kehoe, and Finn E. Kydland (1994) “Dynamics of the Trade Balance and 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 Economic Review, Vol. 78, No. 4, pp. 773 – 785. Baldwin, Richard and Paul Krugman (1989) “Persistent Trade Eﬀects of Large Exchange Rate Shocks.,” Quarterly Journal of Economics, Vol. 104, No. 4, pp. 635 – 654. Bernard, Andrew B., Jonathan Eaton, J. Bradford Jensen, and Samuel Kortum (2003) “Plants and Productivity in International Trade.,” American Economic Review, Vol. 93, No. 4, pp. 1268 – 1290. 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 International Trade.,” American Economic Review, Vol. 98, No. 4, pp. 1707 – 1721. Das, Sanghamitra, Mark J. Roberts, and James R. Tybout (2007) “Market Entry Costs, Producer Heterogeneity, and Export Dynamics.,” Econometrica, Vol. 75, No. 3, pp. 837 – 873. Dixit, Avinash K. (1989) “Entry and Exit Decisions under Uncertainty.,” Journal of Political Economy, Vol. 97, No. 3, pp. 620 – 638. Dixit, Avinash K. and Robert S. Pindyck (1994) Investment under uncertainty: Princeton University Press. Eaton, Jonathan and Samuel Kortum (2002) “Technology, Geography, and Trade.,” Econometrica, Vol. 70, No. 5, pp. 1741 – 1779. Gabaix, Xavier (1999) “Zipf’s Law for Cities: An Explanation.,” Quarterly Journal of Economics, Vol. 114, No. 3, pp. 739 – 767. 37

Ghironi, Fabio and Marc J. Melitz (2005) “International Trade and Macroeconomic Dynamics with Heterogeneous 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 of Entry into and Exit from Export Markets.,” Journal of International Economics, Vol. 90, No. 1, pp. 75 – 90. Luttmer, Erzo G. J. (2007) “Selection, Growth, and the Size Distribution of Firms.,” Quarterly Journal of Economics, Vol. 122, No. 3, pp. 1103 – 1144. Obstfeld, Maurice and Kenneth Rogoﬀ (2000) “The Six Major Puzzles in International Macroeconomics: Is There a Common Cause?”. Reed, William J. (2001) “The Pareto, Zipf and Other Power Laws.,” Economics Letters, Vol. 74, No. 1, pp. 15 – 19. Roberts, Mark J. and James R. Tybout (1997) “The Decision to Export in Colombia: An Empirical Model of Entry with Sunk Costs.,” American Economic Review, Vol. 87, No. 4, pp. 545 – 564. Ruhl, Kim J. (2008) “The International Elasticity Puzzle.,” Working Paper. Stokey, Nancy L. (2009) The Economics of Inaction: Stochastic Control Models with Fixed Costs: Princeton Universiy Press.

38

Appendix A A.1

Proofs and Derivations Derivation of ⇡ ˜t and ⇡ ˜t⇤

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

= 1+

M⇤ M

⇤

X,t

M M

=

1+

⇣

M⇤ M

⇣

¯⇤ t

min

¯⇤ t

min

⇣

⌘✓

⌘✓

¯⇤ t

min

1 1

⇣

1

⌘✓

1

⇣

Zt⇤ w 1 Zt w ⇤ ⌧

⇣

Zt⇤ w 1 Zt w ⇤ ⌧

Zt⇤ w 1 Zt w ⇤ ⌧

It is clear from the above expressions that the relative variables the suﬃcient statistics to compute the expenditure shares. The average total profit is ⇡ ˜t

=

=

= =

⇣ ⌘ M h ⇣ ⌘ X,t ⇡D,t ˜ + ⇡X,t ˜X,t M 2 a1 |

✓

D,t

M D,t

Zt ⌫

min Pt

w {z

◆✓

D,t M

[wL + M ⇡ ˜t ] + wL + M

˜t D,t ⇡

f Yt⇤

6✓ 6 ¯t ◆ 6 Yt + 6 6 min 4| } ⇤ X,t

M

f

⌘✓

1

1

⌘✓

1

⇤ M ⇤ w ⇤ Zt M , w , Zt

and the export cutoﬀ

i

1

"

⌘✓

✓ ¯ ◆ t min

a1

✓

{z

Zt ⌫ ¯t Pt⇤ ⌧w

⇤ X,t M

#

◆✓

1

Yt⇤ }

✓ ¯ ◆ t min

¯⇤ t

min

are

3 7 7

7 f Yt⇤ 7 7 5

[w⇤ L⇤ + M ⇤ ⇡ ˜t⇤ ]

+ bt [w⇤ L⇤ + M ⇤ ⇡ ˜t⇤ ]

⇣ ¯ ⌘ ⇤ X,t t where bt = M f 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 =

wL bt + [w⇤ L⇤ + M ⇤ ⇡ ˜t⇤ ] 1 D,t M D,t

D,t

1

Analogously for the foreign country, ⇡ ˜t⇤ =

1

⇤ D,t ⇤ D,t

w ⇤ L⇤ b⇤t + [wL + M ⇡ ˜t ] ⇤ M⇤ 1 D,t

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

39

equation yields ⇡ ˜t

" !# ⇤ ⇤ ⇤ ⇤ wL bt w L b D,t + w ⇤ L⇤ + M ⇤ + ⇤t [wL + M ⇡ ˜t ] ⇤ ⇤ X,t M X,t X,t M X,t " # ⇤ bt b⇤t b⇤t D,t wL D,t ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ + w L + ⇤ w L + ⇤ M wL + ⇤ M M ⇡ ˜t X,t M X,t X,t X,t X,t D,t

= =

where I used 1

D,t

=

X,t .

⇡ ˜t

⇡ ˜t⇤

A.2

Hence, we obtain

=

=

⇣ ⇣

⇤ X,t

D,t

M

⇤ X,t

D,t

M⇤

⌘ + b⇤t bt M ⇤ wL + X,t + ⇣ ⌘ ⇤ ⇤M ⇤M b X,t t X,t ⌘ + b⇤t bt M w⇤ L⇤ + ⇤X,t + ⇤ X,t

(

⇤ D,t

D,t

bt w ⇤ L ⇤

b⇤t wL

bt M ⇤ M )

X,t

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) = A0 e↵z + B0 e where ↵ and

z

are the two roots for the characteristic equation 2

2

x2 + µx

⇢=0

Then, we obtain ↵

= =

µ+ µ

p p

µ2 + 2⇢

2

2

µ2 + 2⇢

2

2

Both ↵ and are positive but the lower bound for ↵ is 0 whereas that for is 2µ2 . Equation (16) is a nonhomogeneous diﬀerential equation due to the net export profit flow. The equation is 2

2

V100 + µV10

⇢V1 = f |

Se(✓ {z u(z)

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

1)z

}

V1 (z) = V0 (z) + w (z) We obtain the following relationship between u (z) and w (z) 2

2

w00 (z) + µw0 (z)

40

⇢w (z) = u (z)

Guess w (z) = Ce(✓

2

2

) ) which results in

f ⇢

1)z

(✓

for some constant C. Then, 2

1) Ce(✓ 1)z + µ (✓ 1) Ce(✓ 1)z ⇢Ce(✓ 1)z + f h ⇣ 2 ⌘ i 2 e(✓ 1)z C 2 (✓ 1) + µ (✓ 1) ⇢ + S ⇣ 2 ⌘ 2 C 2 (✓ 1) + µ (✓ 1) ⇢ + S C=

1)z

=0 =0

S 2

⇢

2

(✓

1)

2

µ (✓

Thus, V1 (z) = A1 e↵z + B1 e

Se(✓

=f

z

+

1)

S 2

⇢

2

(✓

1)

2

µ (✓

1)

e(✓

f ⇢

1)z

If z is very small, it is very unlikely that it rises to zH in any given finite amount of time. The option of entering 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 where both countries have substantial presence in the price index of the partner country. The optimal behavior of individual 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 time aﬀecting the profits. For simplicity, let’s assume two symmetric countries. Assumption 3. The two countries have the same wage rate, M ⇤ = 1.

w w⇤

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

In this case, the expression for the export profit becomes

⇡ ˆX,t zt , ¯t ;

= a3 ⇥

✓ 1

0

B ⇥ @⇣

¯t min

⌘✓

1 1

+

ez ⌧

(✓ 1)

1 C A

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 history of z.

41

If we take both z and ¯ as state variables, then ⇡ z, ¯ has a strong Markov property, and we can apply the HJB equations to characterize the inaction region. The modified HJB equation for U z, ¯ is

⇢U z, ¯ = ⇡ z, ¯

f+

2

1 6 1 1 2 E 6Uz dz + Uzz (dz) + U ¯d ¯ + U ¯ d ¯ dt 4| 2 {z 2

2

dU

3

7 + Uz ¯dzd ¯7 5 }

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 state variables. The new HJB equation is

⇢U ⇡ ¯, ¯ = ⇡ ⇡ ¯, ¯

f+

2

1 6 1 1 2 E 6U⇡¯ d¯ ⇡ + U⇡¯ ⇡¯ (d¯ ⇡ ) + +U ¯d ¯ + U ¯ ¯ d ¯ dt 4| 2 2 {z

2

dU

3

7 + U⇡¯ ¯d¯ ⇡ d ¯7 5 }

If d ¯ 6= 0, the cutoﬀ firm is being updated. During that period, ⇡ ¯ remains constant so d¯ ⇡ = 0. If d ¯ = 0, the cutoﬀ firm is in the inaction region and ⇡ ¯ changes due to the productivity shock z so d¯ ⇡ 6= 0. At any point ¯ ¯ in time, there is a complementary slackness between ⇡ ¯ and as d¯ ⇡ d = 0. This complementary slackness is borrowed from Caplin and Leahy (1997), and simplifies the HJB equation as 8 1 2 > < U⇡¯ d¯ ⇡ + U⇡¯ ⇡¯ (d¯ ⇡ ) if d ¯ = 0 2 dU = > : U ¯d ¯ + 1 U ¯ ¯ d ¯ 2 if d¯ ⇡=0 2

(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

0

B ⇡ ¯ = a4 @ ⇣

¯t min

⇣

⌘✓

¯t min

1

⌘✓

+

1

ez ⌧

(✓ 1)

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

z=

1 1

✓

2

6 a4 ⌧ ln 4

⇣

¯t min

⇡ ¯

⌘✓

42

1

⌧

1 C A

✓ ¯ ◆✓ t min

1

3 7 5

Then,

⇡ ¯z

=

=

⇣

¯t

(✓ 1) ⌘✓ 1

min

(✓

1) ⌧ (✓

⇡ ¯zz

= a4 (✓ = a4 (✓ = a4

By Ito’s lemma, d¯ ⇡

⇣

e(1 ✓)z ⌧✓ ez ⌧

+

⌘

1 ⇡ ¯ a4 ⇣ (1 ✓)z ⌘

1 ✓

2

1) e ⌧ ✓ ⇣ ¯ ⌘✓ 1 t

min

2

⇣

(1

1) e ⌧ ✓ ⇣ ¯ ⌘✓ 1 t

(✓ 1)

⇡ ¯=

a4 !

✓ ¯ ◆ t min

(✓

⇡ ¯2 +

⌘ ✓)z

⇣ (1 ✓)z ⌘ 1) e ⌧ ✓ ¯2 ⇣ ¯ ⌘✓ 1 ⇡

(✓

a4 (✓

⇡ ¯2 +

a4

min

⇣ (1 ✓)z ⌘ 2 1) e ⌧ ✓ 6 ¯ 2 4(1 ⇣ ¯ ⌘✓ 1 ⇡

t

min

⇡ ¯2

⇣ (1 ✓)z ⌘ 1) e ⌧ ✓ ⇡⇡ ¯z ⇣ ¯ ⌘✓ 1 2¯ t

min

⇣

(1

✓)z

1) e ⌧ ✓ ⇣ ¯ ⌘✓

1

t

⌘

min

(✓

✓) + a4

t

min

2¯ ⇡

t a4 min ⌘ 3 ✓)z

⇣ (1 1) e ⌧ ✓ ⇣ ¯ ⌘✓ t

⇣ (1 ✓)z ⌘ 1) e ⌧ ✓ ¯2 ⇣ ¯ ⌘✓ 1 ⇡

(✓

7 2¯ ⇡5

1

min

1 1 ⇡ ¯z dz + ⇡ ¯zz 2 dt = ⇡ ¯z ( ↵zdt + dW ) + ⇡ ¯zz 2 2 ✓ ◆ 1 ⇡ ¯z ↵z + ⇡ ¯zz 2 dt + ⇡ ¯z dW 2

= =

2

dt

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

A.4

Proof of Lemma 4

We have

2

6ˆ ˜=6 6 4 |

The expression in the bracket is X = C1 |

ˆ

H

L

✓ 1

"✓

L

◆

1+ 2

{z

3✓11

7 7 dG1 ( )7 5 {z }

1

✓ 1

L

X

#

(

2 +1)

1

X1

43

2

L

d + C2 }

|

ˆ

1

✓ 1

(

2 +1) 2

H

{z

X2

L

d }

where C1 =

1 2 1+ 2

X1

=

C1

=

C1

= where k1 =

1

h⇣

C1

H L

⌘

✓+

H

ˆ

i 1

1

1

2

1

✓

1

" "

1 k1 1 k1

✓ 1+ L

1

1

✓✓

H L

L

◆

1

✓ 1 H

1) and k2 =

+ (✓ X2

=

C2

=

ˆ

C2 k2

!

1

1

L

1

✓ 1 L

✓ 1 (

+

◆

◆

H L

1 . Then ✓ 1+

1 1+

✓

✓ 1 H

1

1

L ✓ 1 L

2

✓

L H

2

L

2

L

◆

1 1

1

2

1 + k2

2 +1)

d = C2

2

✓

"

1+ 2

1 k2

L

H

⌘

✓ !#

2

✓ 1 H

✓ 1 L

✓ 1

2 2

2

L

#

H

L

!#

1). Also,

(✓

2

L

d = C1

2

L

✓ 1+ H

H

2

2

L

L

⇣

and C2 = C1

"

✓ 1

k2

2 2

L

!#1

= C2 k2

✓ 1 H

2 2

L

H

2

✓ 1 H

Then, X

= = =

= =

X1 + X2 " ✓✓ ◆ 1 H C1 k1 L " ✓✓ ◆ 1 H C1 k1 L "✓ ◆ 1+ H +C1 C1 C1

1

Thus,



✓✓

1 k1 ✓

✓

H L

◆

1 2

k1 k2

◆

⇣

⇣

L

H L H L

⌘k1 ⌘

1

✓ 1 L

#

1 k2

✓

✓ 1 H H L

◆

✓ 1 L 1

✓ 1 H

1

1

1

1 1

✓

◆

1 + k2

◆

H

1

◆

2

✓ 1 H

1

✓

+

✓ 1 L

2

◆✓

H

✓ 1 H 1

1 1 + k1 k2

1 2

=

=

L

1

L

◆

1 + k2 ◆ 2

✓

1 k2

✓

H L

✓

H L

◆ ◆

✓ 1 H

✓ 1 L

✓ 1 H

✓ 1 L

2

!#

✓ 1 L

+ C1 ◆

◆

"✓

!#

✓ 1 L

✓

✓ 1 L

H L

H L

◆

= C1

◆k1

1

✓ #

1 k2

˜ = X ✓11 = 6 4

✓

1 k2

✓

H L

◆

2

✓ 1 H

✓ 1 H

1 1 + k1 k2

1 =

1 2

k1 k2

✓ 1 L

2

+ C2

✓ 1 H

1 1 + k1 k2

k1 + k2 k1 k2

2

1 2

k1 k2

◆

44

⇣

⇣

H L H L

⌘k1 ⌘

1

3✓11

17 5 1

L

◆ ✓

✓ 1 L

"✓

k1 + k2 1+ 2

◆

H L

⇣

⇣

◆✓ H L H L

1+

1

⌘k1 ⌘

#

1

1

1 1

✓ 1 L

A.5

Proof of Proposition 2

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

By letting v = z + s and

=

a4 e(✓

1)(z+s)

=

a4 e(✓

1)(z+s)

2 v

2 z

=

+

2 s,

1 2 1 f + µs F 0 (z + s) + 2 f + µs F 0 (z + s) +

2 00 zF 2 z

(z + s) +

+

1 2

2 00 sF

(z + s)

F 00 (z + s)

2 s

we obtain

rF (v) = a4 e(✓

f + µs F 0 (v) +

1)v

1 2

2 00 vF

(v)

Then, we can apply lemma 1 to obtain v

F (v) = Be

Ae↵v +

⇡ (v) ⌘

f , r

where ↵

µs +

=

µs

=

2 v

⌘

=

r

⇡ (v)

=

a4 e(✓

2

p

µ2s + 2r

2 pv µ2s 2 v

+ 2r

(✓

1)

2 v 2 v

2

µs (✓

1)

1)v

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

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) 1 + @s 2

with the following boundary conditions n (s, 0)

=

n0 (s)

n (0, t)

=

0

n (d, t)

=

b (t)

45

2@ s

2

n (s, t) @s2

(40)

⇤

where n0 (s) = m0 (s l (0)) and b (t) = 11+ 22 e 2 (h(t) s ) . We transformed the moving boundary problem to 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) = n ¯ (s, t) + v (s, t) , where n ¯ (s, t) satisfies the inhomogenous boundary conditions and v (s, t) is a remainder. In particular, n ¯ (s, t) must satisfy n ¯ (0, t)

=

0

n ¯ (d, t)

=

b (t)

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

b (t) s d

(41)

This satisfies the inhomogeneous boundary conditions. The p.d.e (40) gives @n ¯ (s, t) @v (s, t) + = @t @t



@n ¯ (s, t) @v (s, t) 1 [¯ n (s, t) + v (s, t)] (l (t) + µs ) + + @s @s 2 0

2 s



@2n ¯ (s, t) @ 2 v (s, t) + @s2 @s2

Substituting (41) yields b0 (t) @v (s, t) s+ = d @t



b (t) s + v (s, t) d

0

(l (t) + µs )



b0 (t) @v (s, t) 1 + + d @s 2

2@ s

2

v (s, t) @s2

or

@v (s, t) @v (s, t) 1 2 @ 2 v (s, t) = v (s, t) (l0 (t) + µs ) + p (s, t) @t @s 2 s @s2 ⇣ 0 ⌘ 0 where p (s, t) = b d(t) + b(t) s + (l0 (t) + µs ) b d(t) . v (s, t) must satisfy the following initial and homogeneous d boundary conditions

where v0 (s) = n0 (s)

A.6.1

v (s, 0)

=

v0 (s)

v (0, t)

=

0

v (d, t)

=

0

b(0) d s.

Linear shock path

Consider the case in which the aggregate shock is linear in time: z (t) = kt. Then, l (t) = vL ⇤ h (t) = vH kt. We have b (t) = c1 e 2 kt where c1 = 11+ 22 e 2 (vH s ) is a constant. We obtain n ¯ (s, t) =

c1 e d

46

2 kt

s

kt and

We also have

✓

p (s, t) =

2k

+ d

◆

s+

(µs

k) d

2k

b (t)

Then v (s, t) must satisfy @v (s, t) = @t

v (s, t)

(µs

k)

@v (s, t) 1 + @s 2

2@ s

2

v (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) has the homogeneous boundary conditions, decompose v as v (s, t) = S (s) T (t) to find the eigenfunctions. The separation of variables yields T0 = T

(µs

k)

S0 1 + S 2

T (t)

=

cT e

S (s)

=

K 1 e k1 s + K 2 e k2 s

2S s

00

S

= constant =

The general solutions of T and S are t

where cT , K1 and K2 are constants and

k1 =

(µ

k) +

q (µ

2

2 s

k) + 2

( + )

, k2 =

2 s

(µ

k)

q

(µ

2

k) + 2

2 s

( + )

2 s 2

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 s2 ( + ) 0, there⇣ are only trivial solutions of K1 = K2 = 0. Thus, we need (µ k) + 2 s2 ( + ) < 0 ⌘ 2 (µ k) which implies < . Then, S has the following form 2 2 + s

S (s) = K1 sin (k1 s) + K2 cos (k2 s) Using the boundary conditions, we obtain K2 = 0 and 0 = K1 sin (k1 d). For a nonzero solution of K1 , we need sin (k1 d) to be zero. The eigenvalues are k1 =

2⇡ 4⇡ 6⇡ , k2 = , k3 = , ... d d d

and the corresponding eigenfunctions are S1 = sin

✓

We can compute the corresponding

◆ ✓ ◆ ✓ ◆ 2⇡ 4⇡ 6⇡ s , S2 = sin s , S3 = sin s , ... d d d n.

The general solution of v is v (s, t)

1 X

un (t) Sn (s)

n=1

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 1 +2 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. By integrating the density f1 (s), we obtain SL SH

2

=

1+

+ 2

1

= 1

+

2





1 1+

e

1 (sH

e

2 sL )

e e

1 (sH

2 (sH 1 (sH

sL ) sL )

2 (sH

e sL )

1 1

sL )

1

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

1

sH

1 2 1

+

e

2 (s

s⇤ )

1

ds =

2

1

+

e

2 (sH

s⇤ )

2

This corresponds to the share SH of all exporters. Thus, the share of exporters in the total mass of operating firms is ⇤ ⇤ ⇥ e 2 (sH s ) e 1 (sH sL ) 1 ⇤ 1 1 e 2 (sH s ) ⇥ = SH e 1 (sH sL ) e 2 (sH sL ) 1+ 2

48

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