Dynamics of Firms and Tradein General Equilibrium(very preliminary)
Robert Dekle
University of Southern California
Hyeok Jeong
KDI School
Nobuhiro Kiyotaki
Princeton University

March 2012
Abstract
This paper develops a dynamic general equilibrium model that tries to reconcile theobservation that aggregate movements of exports and imports are "disconnected" fromreal exchange rate movements, while …rm-level exports co-move signi…cantly with thereal exchange rate. Firms are heterogenous, facing recurrent aggregate and …rm-productspeci…c productivity shocks, choose which goods to export, and decide to enter and exit
the business endogenously. We calibrate and estimate the model with both aggregate and…rm level data from Japan.

11Introduction

Figure 1 displays the series of aggregate exports and imports together with the real exchange
rate in Japan during the period of 1980-2009 in logarithmic scale. The real exchange rateis de…ned as the relative price between Japan’s trading partners and Japan.1 As the trading
partners’goods become relatively more expensive, we would expect that Japanese exports would
increase and imports would decrease. However, such a relationship between trade and the realexchange rate is not evident in Figure 1. As Japanese real exchange rate depreciates, exports do
not necessarily increase, and imports increase, which is not what we expect. During the entire
sample period, the elasticity of exports with respect to the real exchange rate is -0.17, and thatof imports is 0.08, although these estimates of elasticities are statistically insigni…cant. This
lack of correlation, or correlation contrary to what we expect is an example of the so called“exchange rate disconnect puzzle,” a long standing puzzle in international macroeconomics.2
This weak or opposite correlation between aggregate exports (or imports) and the exchange
rate is observed in many other countries as well (see Hooper, Johnson, and Marquez (2000),and Dekle, Jeong, and Ryoo (2007)).3 Obstfeld and Rogo¤ (2000) mention that the exchange
rate disconnect puzzle is one of the major puzzles in the international macroeconomics.4
Interestingly, after the year 2000, Figure 1 shows that aggregate exports positively comoved with the real exchange rate, but aggregate imports also positively co-moved with thereal exchange rate. These co-movements during this period suggests that a general equilibrium
linkage may be important in order to understand the dynamics of trade and exchange rates in
1

The real exchange rate is measured as the ratio of the weighted average of the prices of Japan’s major
trading partners (in yen term) to Japanese prices, where the weights are the trading shares. The four major
trading partner countries included here are the U.S., European Union, South Korea, and China and their trading
shares are 0.49, 0.366, 0.095, and 0.044, respectively. (Sources: OECD Statistics)
Aggregate exports and imports are measured in billions of year 2000 yen. (Source: Ministry of Finance Trade
Statistics: http://www.customs.go.jp/toukei/suii/html/time_e.htm)
2This empirical puzzle was …rst documented by Orcutt (1950).
3
The list of other countries showing such weak correlation is Canada, France, Germany, Italy, the U.K., andthe U.S.
4
Note that this “exchange rate disconnect puzzle”is di¤erent from the so called “J-curve e¤ect.”The exchange
rate disconnect puzzle is about the lack of association between the movements of exchange rates and gross export
quantities while the J-curve e¤ect is about the sluggish and J-shaped adjustment of trade balances (i.e., net
export sales) in response to an improvement in the terms of trade. See Backus, Kehoe, and Kydland (1994) for
the discussion of the J-curve e¤ect.

2

200001

40000

real exchange rate
1.5

260000 80000

Figure 1. Aggregate Exchange Rate Disconnect in Japan

1980

1990

2000

2010

year
real exchange rate
aggregate importaggregate export

Japan, where intermediate goods trade is dominant in imports, and increasingly more importantin exports.
Recent empirical studies using …rm-level data have found a more robust relationship between export and the exchange rate. In contrast to the results using aggregate data, estimatesusing …rm level tend to …nd a positive relationship between depreciating exchange rates andexport quantities. Among other studies, Verhoogen (2008) …nds that following the 1994 peso
devaluation, Mexican …rms increased their exports. Fitzgerald and Haller (2008), Dekle andRyoo (2007), and Tybout and Roberts (1997) …nd a positive relationship between exports andexchange rate depreciation for Irish, Japanese and Colombian …rms, respectively.
Some papers have tried to reconcile these aggregate and …rm level results, but mostly ina partial equilibrium framework. Dekle, Jeong, and Ryoo (2007) show that in the aggregate
export equation derived by consistently aggregating the …rm level export equations, where industry level productivity and …rm level (instrumented) export shares are controlled for, thedisconnect puzzle disappears. Berman, Martin, and Mayer (2009) use a model with heteroge3

neous …rms in the spirit of Melitz (2003) to show that high productivity …rms (who are heavily
involved in exports) will raise their prices–that is, increase their markups–instead of increasing
their export quantities in response to an exchange rate depreciation. The authors show that
this selection e¤ect of low quantity response of high productivity …rms can explain the weak
impact of exchange rate movements in aggregate data. There are some other recent papers
that have tried to reconcile the discrepancy in a general equilibrium. Imbs and Majean (2009)
and Feenstra, Russ, and Obstfeld (2010) show that the aggregation of heterogeneous industrial
sectors can result in an aggregation bias in the elasticity of exports and imports with respect
to exchange rates changes. Both of these papers examine only the steady-state.
In this paper, we develop a dynamic general equilibrium model with heterogeneous …rms
that attempts to reconcile the di¤erent responses of trade (exports and imports) to exchange
rates at the aggregate- and at the …rm-levels. Our model is a real business cycle model of asmall open economy with a rich production structure. Firms are heterogeneous, facing recurrent
aggregate and …rm-product speci…c productivity shocks: they choose which varieties of goods
to produce and export and decide to enter and exit endogenously. We calibrate and estimate
our model with both aggregate and …rm level data. We then carry out quantitative exercises
regarding the impact of shocks to productivity and preferences on aggregate and …rm-level
exports and other variables of interest.5
We make a few choices to model heterogeneous …rms to re‡ect our panel data of Japanese…rms listed on the stock exchanges of Japan.6 In a well-known paper, Melitz (2003) provides a5

One distinguishing feature of our work is the inclusion of heterogeneous …rm dynamics that is actuallyestimated from …rm level data. In the estimation of the …rm-level responses, in addition to the …rm level data,
we rely on the aggregate variables and moments generated from the general equilibrium model. Thus, in asense, we provide a general equilibrium model that is integrated with a structural model of heterogeneous …rm
dynamics that is estimated from actual …rm level data.
6
The raw data used here and in our paper are from almost all of the …rms listed on the stock exchanges ofJapan.The particular data set that we use were compiled by the Development Bank of Japan (or "Kaigin," inJapanese prior to the 2008 re-organization of government-owned enterprises, when the name of the bank waschanged). Japanese listed …rms cover a fairly respectable portion of the entire Japanese economy in terms of
output (Fukao, et. al., 2008). In 2000, the gross sales of all the …rms listed on the stock exchanges of Japan
were 81 percent of Japanese nominal GDP, and 60 percent of total sales in the Japanese economy. However,listed …rms are larger than the average …rm in the economy. Thus, the number of listed …rms account for lessthan 12 percent of the total number of Japanese …rms, and the number of employees in listed …rms are only 40
percent of all employees (Fukao, et. al., 2008).

4

framework where …rms with di¤erent …rm speci…c total factor productivity subject to …xed costs
can generate heterogeneous exporting behavior. Das, Roberts, and Tybout (2007) provide anempirical study showing that this heterogeneity in total factor productivity among producers
explains entry into and exit out of the export market, the so-called extensive margin of trade.
In our Japanese panel data, there is a strong relationship between …rm size and exporting
status, as in Melitz (2003). The average total sales of the incumbent exporting …rms is about
double of the non-exporting …rms. When …rm level productivity is determined by a single
factor of productivity, the Melitz type of trade model implies that the export share at theintensive margin (in addition to the extensive margin) be strongly correlated with …rm size.
Our Japanese …rm level data show that this prediction is not true. The correlation between
the export share and total sales is rather weak. The average correlation coe¢ cient is only 0.08
among all …rms. Among exporting …rms, the correlation coe¢ cient becomes even lower at 0.05.
This weak correlation remains robust even after controlling for the industry and year e¤ects.
Another interesting observation from Japanese …rm level data is the signi…cant presenceof …rms with negative pro…ts staying in the market. About 8 percent of Japanese …rms in our
sample report negative pro…ts. This fraction becomes even bigger at 11 percent among thealways exporting …rms, the biggest …rms. Despite this presence of negative pro…ts, Japaneselisted …rms do not easily exit from the business, although entry into and exit from the export
market are a little more frequent.
Given these empirical observations, we choose …rms to produce multiple products and areheterogeneous in terms of the productivity distribution as well as the number of the products.
Firms choose which products to produce and which products to export. Thus Melitz styleextensive margin adjustment is mainly at the product level (even though there are endogenous
entries and exits of …rms). This multi-dimensional heterogeneity helps explain the weak relationships among size, the export share and pro…tability in our …rm-level data. Our …rms also
face recurrent idiosyncratic productivity shocks, and thus they may not exit with temporary5

negative pro…ts in order to enjoy the option value of continuing production.7,8 This explains
our empirical …nding why Japanese …rms with recurring negative pro…ts resist to exit from theirbusiness.9
In Section 2, we present a model of small open economy and study the equilibrium dynamics
and steady state of the model economy. In Section 3, we calibrate the model. In Section 4,
aggregate dynamics is simulated. Section 5 concludes.

2

Small Open Economy Model

2.1

Set-upThere is a continuum of home …rms h 2 Ht each of which produces Iht number of di¤erentiatedproducts for the home and export markets at date t. Firm h produces a di¤erentiated product
H
H
H
qhit
for the home market from labor lhit
and imported input mhit
; according to a constantreturns to scale technologyHqhit= ahit Zt

H
lhit
L

L

H
mhit
1
L

1

L

; for i = 1; 2; ::; Iht ;where ahit is the productivity of …rm h to produce the di¤erentiated product (h; i) at date t
and Zt is the aggregate productivity shock andL 2 (0; 1) is the labor share. Because no two…rms produce the same product, each di¤erentiated product is indexed by (h; i). Producing a
7

Ghironi and Melitz (2005) analyze the dynamic e¤ects of an aggregate productivity shock on the realexchange rate in a general equilibrium model with heterogeneous …rms. But they concentrate on the extensive
margin of products for export. Because there are no further idiosyncratic shocks after entry, there is no
endogenous exit nor negative pro…ts in their model.
8More broadly, our paper is related to the recent policy literature that examines how much of a real exchange
rate depreciation is necessary to close a nation’s current account imbalances. Obstfeld and Rogo¤ (2004) use
a three-country model to calculate how much of a depreciation in the real exchange rate is needed to set the
U.S. current account to zero. Dekle, Eaton, and Kortum (2008) …t their model to bilateral trade ‡ows for 42
countries and solve for the new equilibrium in real exchange rates to eliminate all current account imbalances.
9
Strictly speaking, in our sample of Japanese listed …rms, …rms that drop out of the sample are "delisted."
Of the 2386 …rms in our sample that we examine between 1985 and 1999, 104 …rms became "delisted." We
examined the circumstances surrounding the de-listing of all of these 104 …rms and the vast majority weredelisted because of bankruptcy or "ceasing to do business." A small number disappeared as independent …rms
because of mergers with stronger …rms. Thus, we are on reasonably …rm ground when we equate a …rm thathas been "delisted" as essentially "exiting" from production.

6

di¤erentiated product for the export market has the same marginal productivity as producing
for the home market, but requires a …xed cost for each variety,
#
"
1 L
L
F
F
m
l
hit
hit
F
; for i = 1; 2; ::Iht :qhit= ahit Zt
1
L
L
We assume that the …xed cost for exporting a di¤erentiated product is constant in term of
input composite - Cobb-Douglas function of labor and imported input.
Home …nal goods are produced from a variety of di¤erentiated products according to a
constant returns to scale CES production function
"Z
Iht
X
H
H
Qt =
qhit
h2Ht

where

1

i=1

!

#

1

dh

;

> 1 is the elasticity of substitution between among goods. Home output for export QFt

is produced from home di¤erentiated goods
"Z
QFt =

h2Ht

Any new entrant who pays a sunk cost

Iht
X
i=1
E

F
qhit

1

!

#

1

dh

:

at date t draws an opportunity of producing

b 2 f1; 2g number of new products from date t+1, whereb=2; with probability (wp:) ;1; wp: 1:

Thus the average number of new products drawn is equal to 1 + : The productivity of eachnew product is independently and identically distributed: ahit = 0 with probability 10

andahit is distributed according to a Pareto distribution with parameter (1; ) with probability

0:That isahit =

2 [1; a] ; wp: 0 F (a) = 0 (100; wp: 1:a);

The probability density function of the Pareto distribution isf (a)F 0 (a) = a ( +1) ; for a 2 [1; 1):

Any one can enter by paying the sunk-cost, and new entrants are heterogeneous in terms ofnumber of di¤erentiated products (width b) and the distribution of productivity of products(height ahit+1 ).
7

We assume(1 + ) 0 < 1;
> 1 and>(Assumption 1)(Assumption 2)

1:

Assumption 1 implies that the average number of products with positive productivity is less thanunity per new draw, and that some new entrants fail to obtain the productive new products.
Assumption 2 says that the fraction of products with high productivity is relatively small, which
later guarantees that CES production function of …nal goods is well behaved.
The …rm who has existing products must pay the …xed maintenance cost(in terms ofhome …nal goods) for each product in order to produce and maintain its productivity. (The…rm who wants to maintain Iht number of products must pay

Iht ). The product which

the …rm pays the …xed cost has the same productivity in the next period (ahit+1 = ahit ) withprobability 1; and receives a new productivity draw according to the same distribution

as a new entrant with probability : Thus the number of products each …rm produces may
increase or decrease depending upon the new draw of b; and the distribution of productivityof producible di¤erentiated products changes depending upon the new draw of ahit+1 . Because…rms are heterogeneous in terms of the number and the productivity of their products, we
can show that there are only weak relationships among size, the export share and pro…tability
across …rms - a feature of our Japanese …rm-level data. If the …rm does not pay the …xed cost
for an existing product; it loses the technology for this product for sure and forever.
Home …nal goods are either consumed by households and government, or used for new
entry and maintenance of the existing technology as
QH
t = Ct + Gt +E NEt +

Nt ;where Ct and Gt are consumption of households and government, NEt is the measure of entering…rms and Nt is the measure of existing di¤erentiated products which …rms try to maintain. We
can think the cost of drawing new technology and maintaining old technology as investment inintangible capital. On the other hand, we abstract from tangible capital and tangible capital
investment.
8

The representative household supplies labor Lt to earn wage income, consumes …nal goods
Ct and holds home and foreign real bonds DtH and Dt H to maximize its expected utility,
!
1
1+1=
X
L
t
t
ln Ct
U0 = E0+ t H ln Dt H ;
0
1
+
1=
t=0
subject to the budget constraintCt +E NEt +

Nt + DtH + t Dt H = wLt Lt +H
H
t + Rt 1 Dt 1 + t Rt 1 Dt 1

Tt :

(1)

(We index goods by the origin and user countries. For the origin, we label goods by naught
for home and by * for foreign. For the user, we label goods by H for home and by F forforeign. For example, Dt H is foreign bond that are held in the home country.) wLt is real
wage rate,t is the real exchange rate (the relative price of foreign and home …nal goods), andRt and Rt are home and foreign real gross interest rates. Tt is lump-sum tax andgross pro…ts distribution from businesses, while the net pro…t is

tt is the real

Nt . Home and foreign

bonds are used as means of saving. In addition we assume that the holding of foreign bonds
facilitates transactions and provides utility. The utility from holding foreign bonds is subject
to the "liquidity" shock,

t

H 10

.

We assume that all home imports are inputs to production, and that the imported input
price is normalized to be one in terms of foreign …nal goods. We assume that foreign aggregate
demand for home exports are given by
QFt = pFt'

Yt ;where Yt is an exogenous foreign demand parameter, pFt is an endogenous export price in termsof foreign …nal goods, and ' > 0 is the constant elasticity of demand for home exports. We
assume that foreigners do not hold home bond. Then foreign bond holdings Dt H of the home
representative household evolves along with exports and imports as
Dt H = Rt 1 Dt H1 + pFt QFt
10

Mt H ;(2)

The idea is similar to money in utility function. Section 5.3.8 of Obstsfeld and Rogo¤ (1998) presents amodel with both home and foreign money in the utility function to analyze the phenomenon of dollarization.
We ignore the utility of home bonds for simplicity.

9where Mt H =

R

h2Ht

i

hP

Iht
F
H
i=1 (mhit + mhit )dh are total imported input of the home country.

The government budget constraint is given by
DtH = Rt 1 DtH 1 + Gt

(3)

Tt :

Here, because the foreigners do not hold home bond, the home bond holding of the representative household is equal to the government bond supply. Government adjusts tax to stabilize
the outstanding debt
Tt

T =

T

RDH ;

Rt 1 DtH 1(4)

for an autonomous choice of the expenditure Gt ; where T and RDH are tax and government
debt at the beginning of period in the steady state.

2.2

Competitive Equilibrium

All markets for the factors of production and outputs are perfectly competitive, except thatthe market for di¤erentiated products are monopolistically competitive.
Consistent with the usual CES production function of …nal goods manufactured from differentiated products, each …rm faces a downward sloping demand curve for its product in homeFand foreign markets as a function of prices pH
hit and phit ; such that

H
qhit
=

pH
hit
pH
t

QH
t ;

F
qhit
=

pFhit
pFt

QFt ;

F
where pHt and pt are the price indices of home …nal output for the home and export markets

1 = pH
t =

"Z

h2Ht

pFt =

"Z

h2Ht

Iht
X

Iht
X

!

dh

#11

:

1
pH
hit

j=1

pFhit 1

j=1

!

dh

#11

;

(5)

We use home …nal goods as the numeraire in the home market, and foreign …nal goods as thenumeraire in the foreign market.
10Note that the production functions of di¤erentiated products all have a common component:
the Cobb-Douglas function of input composite. Moreover the ratio of labor to imported inputs
is equal across …rms when …rms minimize the costs under perfectly competitive factor market.
F
Let xH
hit and xhit be input composite used for producing a given di¤erentiated product for thehome and export markets. Then the production functions can be simpli…ed asH= ahit Zt xH
qhit
hit ;
F
qhit
= ahit Zt

xFhit

:

Aggregate production of the input composite is equal to the sum of input composite use
!
Z
Iht
1 L
X
Lt L M t H
F
Xt =
=
(xH
hit + xhit ) dh:
1
h2Ht
L
L
i=1
Because the price of imported inputs at home is equal to the real exchange rate (due to our
choice of numeraire), cost minimization implies that the unit cost of the input composite wt ;and the demand for imported inputs satisfy
1

wt = (wLt ) L t
Mt H = (1

L)

L

;

wt Xt

:

(6)

t

After maximizing current pro…ts, each …rm sets price as a mark-up over their unit production costpHhit =pFhit =

wt
1 ahit Zt
wt = t1 ahit ZtpHt (ahit );(7)

pFt (ahit );for all products produced. The quantity of each product produced for home and foreign market
depends only upon the productivity (aside from the aggregate variables).
H
qhit
=

pH
t (ahit )
pH
t

QH
t

qtH (ahit );Fqhit

pFt (ahit )
pFt

QFt

qtF (ahit );=11

Although each …rm may produce multiple di¤erentiated products, the …rm can decide how to
produce and whether to continue producing each product independently from their choice ofthe other products.11
We conjecture that in equilibrium, all …rms choose to pay the …xed maintenance cost withpositive productivity. Then the total measure of products with positive productivity evolves
with the maintenance and new entries as
Nt+1 = (1+(8)) Nt + NEt :Because of Assumption 1, Nt does not grow without limit. Let Nt (a) be the measure of productswith productivity a: Then. from the speci…c feature of our idiosyncratic productivity evolution,
Nt (a) is a proportional to Nt as:
Nt (a) = f (a)Nt :Then from (5) and (7), the price index for home …nal goods for the home market becomes
Z 1

1 = pHt =1

1
pHNt f (a)da
t (a)

wt
; where
1 AH
t

1

=

1

1

AH
t

aNt Zt and
Z 1
a 1 f (a)da
1

a

1

1
1

1

=

1

+1

AH
t is the aggregate productivity of home …rms for home market, and a is the average productivity of products that are produced. Thus the unit cost of input composite is given by
wt =

1

AH
t = w (Nt ; Zt ) :(9)

We conjecture that there is a lowest productivity level at > 1 at which the product makes
zero pro…t for export:Ft (at )=

qtF (at ) = (FFt pt (at )qt (at )wt

qtF (at )+at Z t= wt

1

qtF (at )1 at Z t= 0; or1) at Zt .

11

The founder of Kyocera, Mr Kazuo Inamori, proposes an "amoeba" management style, in which each production unit makes relatively independent production decisions, while the number of production units multiply
and shrink like "amoebas." Our technology can be seen as a justi…cation for the "amoeba" management sytle.
See also Bernard, Redding and Schott. (2010, 2011).

12

Thus only a fraction
nFt = P rob(aat ) = (at )

(10)

<1

of products are exported.
Price index of home …nal output for foreign market is
wt = t

pFt =
where
aFt

"Z

1

a

1

1 aFt Nt

# 11

1

1 a
;
F
t at

=
Zt

1

f (a)da

at

1

=

(at )

+1

1

1

= a (at )+1
1:This home …rms’productivity measure for export aFt is a decreasing function of the lower bondof productivity for export at .
The export quantity index becomes
QFt

=

"Z

1

1

qtF (a)#f (a)Nt daat=qtF (at ) Nt

1

"Z

1

at

= (

1) Zt Nt

1

1

#

1

a
at

1

f (a)da+11

a (at ):

To derive the second equation, we use the property qtF (a) =qtF (at ) = pFt (a) =pFt (at )=(a=at ) : From the export demand condition, we also have
QFt = (pFt ) ' Yt = (at )

+1

' '

1

Yt :

Thus the export market clearing condition implies
at =

(
1) AH
t Nt
'
+1
Yt

1
(

') (

1)(1 ')

a

'
t Yt
AH
t Nt

; where a0 ( ) < 0:(11)

For at > 1 so that the extensive margin for export to exist, we need
'

Yt
<
H
At Nt

(
1)
:
+1

13

(Condition 1)

If this condition were violated, then all home …rms would export all the products, but this is notconsistent with the observation. Thus we restricts the attention to the case in which Condition
1 is satis…ed in the following.
The export value in terms of home …nal goods is
F F
t pt Qt

=
=

2.3

(

F 1 ' '
Yt
t pt

( +1

= at

(
1)
AH
t Nt
+1

)(1 ')
1

( +1

'

Yt

)(1 ')( 't Yt ) (

1)

) (

1
') (

1)(1 ')(12)

Disconnect Between the Aggregate and Firm Level Responses

Because the number of products produced Nt is a state variable, the extensive margin at reacts
to an exogenous shift in foreign demand and the real exchange rate. When the elasticity offoreign demand for home products is relatively small compared to the elasticity of substitution
among di¤erentiated products (' is small relative to ), then the export quantity QFt and export
value (in terms of home …nal goods) are relatively insensitive to the real exchange rate shift.
In contrast, because both intensive and extensive margins adjust in the short run, the exportsof the products whose productivities are close to the lower bound for export is sensitive to thereal exchange rate shift. As in Green (2009), the exports of the low productivity products drop
like "‡ies" when there is an adverse shock such as a real exchange rate appreciation.
Our Japanese …rm-level data (Kaigin data) are mostly of relatively large …rms, which
typically produce multiple products - possibly after a number of good new draws of b = 2. If
a majority of products of some …rm is close to the lower bound for export, then the export ofthis …rm is sensitive to the real exchange rate shifts (which moves the lower bound). Becausesuch …rms are common under Assumption 2, the …rm-level export tends to react signi…cantly
to the real exchange rate. In contrast, the products with considerably higher productivity thanthe lower bound is not very sensitive to the real exchange rate shifts, and their share in theaggregate export is large. Thus the aggregate exports are less sensitive contemporaneously tothe real exchange rate shift. This heterogeneous reaction of exports to the real exchange rate14

shift across di¤erent products explains why …rm level exports co-move signi…cantly with the realexchange rate, while aggregate exports appear "disconnected" from the real exchange rate.12

2.4

Equilibrium Dynamics

The aggregate composite input use for export is
Z 1 F
qt (a)FXt =+ f (a)Nt da
aZt
at
#
Z 1 "
1
a
(
1) + 1 f (a)Nt da
=
at
at
+1
+1

=

(at )

(13)

Nt :

Aggregate composite input use for home …nal goods is proportional to the home …nal goods
output as
XtH

Z 1

=

qtH (a)f (a)Nt da
aZt

1

QH
t
:
=
AH
t
The labor supply condition of the household is given by
1

wLt =

0 Lt Ct :Together with (6), we have
Lt =1
( 0 Ct )
1

Xt =

wt

L
L

t

=

1
t

L
L

;

Lt
L

1
L(

wt

0 Ct )= XtH + XtF :
12

"

wt 1
(1
t

L+
L )(1+

#1

L

)

(14)
(15)

Our explanation of the extensive margin adjustment at product level is consistent with Dekle, Jeong and
Ryoo ( 2007), which …nd that the apparent lack of relationship between the exchange rate and aggregate exports
occur through the intensive margin of export sales within …rms, rather than through the extensive margin ofentry and exit of …rms in the export market.

15

The pro…t arising from selling a product with productivity ahit = a in the home market is
H
t (a)

H
= pH
t (a)qt (a)

1

=

1

wt xH
t (a)

wt xH
t (a);

i.e., the net mark-up times the input costs. The pro…ts arising from exporting a product withproductivity ahit = a

at is
F
t (a)

F
F
t pt (a)qt (a)

=

1

= wt

wt xFt (a)xF (a)
1 t:

Let Vt (a) be the values of the products with productivity a at the beginning of this period(who decides to pay the …xed cost of maintenance). The Bellman equation isHFt (a) + M ax[ t (a); 0]

Vt (a) =

+Et

t;t+1

(1

Z 1

)Vt+1 (a) +Vt+1 (a0 )f (a0 )da0 ;

1

where

t;t+1 =

Ct =Ct+1 : The average value of the products produced is given as
Z 1
Vt
Vt (a)f (a)da
1

=

where

+ (1

t

+

)Et ( t;t+1 V t+1 );(16)

t is the average pro…t of the products with positive productivity:

t

=

Z 1

H
F
t (a) + M ax[ t (a); 0]

f (a)da

1

= wt

(

Xt
1) Nt

nFt :

(17)

The free entry condition for a potential entrant isE =Et

t;t+1 V t+1

(18)

:

The necessary and su¢ cient condition that the …rm strictly prefers to maintain a product withthe lowest productivity by paying the …xed cost is
0 < Vt (1) =

H
t (1)

+ Et (116)Vt+1 (1) +V t+1 ; for all t

or
0<

H
t (1)

+

(Condition 2)

E:

Notice that this condition is satis…ed even if realized current net pro…ts of each product isnegative ( Ht (1) <), because there is an option value for the low productivity product tobecome a high productivity product.13 This helps explain why …rms often record negative
pro…ts after paying their …xed costs of maintaining the business. In addition, because …rms
may have a large number of low productivity products, there is only a weak correlation between
size and pro…tability across …rms - another curious aspect of Japanese …rms.
The …nal goods market clearing implies
Ct + Gt +E NEt +

H
Nt = AHt Xt :(19)

From (2), (6) and (12), net foreign assets evolve as
Dt H = Rt 1 Dt H1
+

(

(1

L)

wt Xt

(
1)
AH
t Nt
+1

t
( +1

)(1 ')

Yt
(
t

+1

(

1)
)(1 ')

) (

1
') (

1)(1 '):(20)

From the utility maximization of the household, we have(21)

1 = Rt Et ( t;t+1 )andtRt Et ( t;t+1 t+1 ) =

H
t

Ct
:
Dt H

(22)

The the …rst equation is a standard Euler equation for home bond. The left hand side (LHS)of the second equation is the opportunity cost of holding one unit of the foreign bond. The
right hand side (RHS) is the marginal rate of substitution between foreign bond holdings andconsumption.
13

The option value due to the idiosyncratic productivity shock cannot be too large, because we conjecturethat the …rm will not maintain the product with zero productivity. The condition for …rms not to maintain zeroproductivity product is
0>
+
E:

17

The aggregate state of our small open economy is described by the state variables Mt =Nt ; Dt H1 ; DtH 1 ; Zt ; Gt ; t H ; Yt ; Rt , where the …rst three are endogenous and the last …ve areexogenous. The market equilibrium condition (3), (4), (8) - (11) and (13) - (22) determine(wt ; at ; nFt ; Xt ; XtH ; XtF ; Ct ; t ; Rt ; V t ;t ; NEt , Nt+1 ; Dt

H

; DtH ; Tt ) as a function of the state vari-ables. The budget constraint (1) is automatically satis…ed once all the market clearing conditions
are satis…ed (by a variant of Walras’Law), noting that aggregate pro…t is equal to the average
pro…t multiplied by the number of products produced (

2.5

t =

t Nt ).

Dynamics of the "Shrunk" ModelWe …rst examine the market clearing condition for net foreign assets (22) to …x intuition.
Suppose as it is likely, that a liquidity shock to foreign bonds

Htis very volatile in the short-run. The supply of net foreign assets changes sluggishly over time through the current account.
Consumption is relatively smooth by permanent income theory if labor supply is relatively
inelastic and the investment on technology ( E NEt + Nt ) serves as a bu¤er to absorb shocks(which we have to verify later). Then since the liquidity shock to foreign bonds appears only inthe market clearing condition for net foreign assets, the real exchange rate has to adjust quickly
to the volatile liquidity shock at high frequency - even though at low frequency, the adjustmentof the current account and consumption are as important as the real exchange rate adjustment.
Thus in our economy, the high frequency movement of the real exchange rate is dominatedby the liquidity shock. Empirically, we can treat the short-run movement of the real exchange
rate as almost "exogenous", because we can always …nd a liquidity shock to justify the observed movement of real exchange rates as long as our boundary conditions (Condition 1) and(Condition 2) are satis…ed and the evolution of net foreign assets is stable in the long-run. In
addition, net foreign assets appear only in the equation for the evolution of net foreign assets(20).
Therefore, below we consider the following abbreviated or "shrunk" model.
M0t = Nt ; DtH 1 ; Zt ; Gt ; Yt ; t

We takeas the state variables, and solve for the fourteen endogenous

variables (wt ; at ; nFt ; Xt ; XtH ; XtF ; Ct ; Rt ; Tt ; V t ;Ht ; NEt ; Nt+1 ; Dt ) as functions of the state vari-18

ables which satisfy the fourteen equilibrium conditions (3), (4), (8) - (11), (13) - (19) and (21).
Here there are only two endogenous state variables, Nt and DtH 1 , which greatly simpli…es ouranalysis.
Equation (12) provides the expression for the value of total exports as explicit functionsof the state variables. Our discussion of the adjustment of exports through the extensive andintensive margins can then be made clearer, because we can now take the real exchange rate as
exogenous. Moreover from (16) and (18), we have
Vt =

+ (1

t

+

Multiplying both sides of this expression at date t+1 byE= Et

t;t+1 V t+1 = Et

n

t;t+1

h

E

)

:

t;t+1 ; we have

+ (1

t+1

+

E

)

io

;

or
E [1

(1

+

)Et ( t;t+1 )] = Et [ t;t+1 ( t+1)]:

The LHS is the cost of entering now instead of the next period, and the RHS is the expected
net pro…ts of the next period which the …rm would lose by delaying entry. (Remember theentering …rm can only produce from the next period). Using (17), this can be written as
E

=

Et

where AH
t

aNt

1

(1

+

Ct
Ct+1
1
1

(

Ct
Ct+1

)Et
1

+

AH
t+1

"

(

Xt+1
1)Nt+1't+1 Yt+1
AH
t+1 Nt+1

a

#)!

;

(23)

Zt : Supply condition for aggregate composite input is
Xt =

1
L(

0 Ct )

"

1

1

L+

L )(1+

)

AH
t

(1
t

#1

L

(24)

:

The goods market clearing condition can be written as
Ct + Nt +
(

= AH
t

Xt

E

[Nt+1

+1
+1

(1
a

19

+
'
t Yt
AH
t Nt

)Nt ] + Gt
)
Nt

(25)

(from (13)).
(Ct ; Xt ; Nt+1 ) solves (23, 24, 25) as a function of (Nt ; Zt ; Gt ; Yt ; t ) : Then we can …nd
Rt ; DtH ; Tt which satis…es (21, 3, 4). We have to verify that the conditions (Condition 1) and(Condition 2) are satis…ed in equilibrium. (Condition 2) can be written as
(
)
'
Xt
+1
+1
t Yt
H
At
a
:
E <
Nt
+1
AH
t Nt
Using (6), (13) and (15), home real GDP is given by
F F
Yt = QHt + t pt Qt
(

= wt

1

= aNt

1

Zt

H

t Mt

Xt
1
"

'
t+1 Yt+1
AH
t+1 Nt+1

a
1

L

Nt

1
Xt + Xt

a

(1

L )Xt

)

#

'
t+1 Yt+1
AH
t+1 Nt+1

Nt :The …rst term of RHS is wage, the second is gross pro…t and the last is the …xed cost for export.

2.6

Steady State of the Full Model

From the free entry condition withE [1in the steady state, we havet;t+1 =

(1

+

) ]=

E

1

(

);

or
=

+

=

wX
(
1) N

1+
(26)

w a

From (22) ; we have
DH=

1
1

H

R

C:

Together with (20), we have(R1) D H =

R
1

= (1

1
R

H

L )wX

20

C
a

+1
1

(1 ') '

Y

Then
(1

L)

'
+1
Y
R
wX
= a 1 (1 ')
+
N
N
1

1
R

HC

N

(27)

:

From (11) ; we get(aFrom (26)') (1)(1 ')
1

=

wN
:
'Y

+1

(28)

(28), we have+1

w = a a

+1

1

(1

= a

(1 ')

'

Y
N

L) (

1)

+w a

R
1

1
R

1C
=
N

HC

1
R

N

or
(1

+1

L)

1 wX
R
+
N
1

(1

)+

H

(
1)
:
+1

(29)

From (25), we have
C
+
N

+

E

G
X
= AH
N
N

+1
+1

AH a

;

or using (26)(+11)( + 1

)

1

wX
+
N

1C
=
N

+1
+1

1

+

1

E

1

1

g ;
(30)

where g = G=N is government purchase per measure of products.
C
We can solve (29) and (30) simultaneously with respect to N
and wX
as a function ofNparameters. Also from (24) ; we know
L(

0

C wX
)
=
N
N

"

1

1

aZN 1
1 LN L

#1+

:

(31)

Thus we can …nd the steady state value of N: Then all the endogenous variables in steady state
are determined as functions of exogenous parameters.

3

Calibration

21

4

Simulation

5

Conclusion

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23

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