The consumption-real exchange rate anomaly with extensive
margins✩
Masashige Hamanoa,aUniversity of Luxembourg-CREA

This draft: September 2011

Abstract
This paper revisits the consumption-real exchange rate anomaly known as the BackusSmith puzzle. Fundamentally the BS puzzle cannot be tested fully. This is because the"welfare-based" real exchange rate which captures variations in the number of varieties,
i.e. extensive margins and which is relevant for the risk sharing is only partially observed.
An expansion in extensive margins brings the "empirical-based" real exchange rate inan appreciation via a terms of trade appreciation while domestic consumption rises. In
our theoretical model, a realistic BS correlation possibly arises when the elasticity of
substitution between domestic and imported goods is high, consistent to the value of
micro founded estimations in the literature.
Keywords: terms of trade, real exchange rate, Backus-Smith puzzle, …rm entry
JEL classi…cation: F12, F41, F43
Perfect international risk sharing with complete asset markets predicts that consumption in one country should rise when the price of consumption in that country becomes
relatively cheap. However, this is not observed in reality. Table 1 reports correlations
between relative consumptions and real exchange rates (de…ned as the price of foreign in✩I gratefully acknowledge Phillippe Martin for his advice and encouragement. I would like to thankalso Nicolas Coeurdacier, Jan Kranich, Eiji Okano and Robert Kollmann for providing comments andinsightful discussions. I thank the all seminar participants at university of Rennes1, Paris1, Luxembourgand Hitotsubashi university and those in 13th T2M, XVth SMYE 2010 and 25th EEA meeting 2010.
The paper is based on the second chapter of my Ph.D. dissertation. All remaining errors are my own.
Corresponding author: University of Luxembourg, CREA, L-1511 Luxembourg, +352 4666446655
Email address: masashige.hamano@uni.lu (Masashige Hamano)
Preprint submitted to Elsevier

September 24, 2011

terms of domestic basket) among industrialized countries. They are close to zero or evennegative indicating that countries consume more when their price of consumption become relatively expensive. These consumption-real exchange rate anomalies are known as
the Backus-Smith (BS) puzzle in the literature (Backus and Smith (1993) and Kollmann(1995)).
Table 1: BS correlations

Country

U.S.

Austria-0.11 0.05

Italy

-0.28 -0.52

Belgium/Luxembourg -0.16 0.50

Japan

0.05

Canada-0.52 -0.31

Netherlands -0.45 -0.20

Denmark-0.14 -0.10

Portugal-0.61 -0.77

Finland-0.30 -0.49

Spain-0.63 -0.64

France-0.20 0.43

Sweden-0.56 -0.40

Germany-0.51 -0.27

U.K.

-0.51 -0.21

Greece-0.45 -0.35

U.S.

N/A

Ireland

ROW Country-0.39 0.72
MedianSource: Corsetti et al. (2008a)

U.S.

ROW

0.25

-0.71

-0.42 -0.27

Recent attempts to solve the BS puzzle question the existence of complete asset markets in the …rst place, a strong assumption on which the puzzle itself relies.1 Withoutcomplete markets a tight link between relative consumption and the real exchange rate is
broken. Consumption risk is insured only partially across countries raising the possibilityof reproducing a realistic BS correlation in theoretical models.
With such incomplete asset markets, Corsetti et al. (2008a) (henceforth CDL) show
how the puzzle is solved due to a wealth e¤ect induced by a low trade elasticity or combination of a high trade elasticity and highly persistent productivity shock. When the
trade elasticity is low, excessively supplied domestic goods due to a positive productivity
shock must be consumed by domestic agents who can be a¤orded by a terms of trade1

See for example Obstfeld and Rogo¤ (2000).

2

appreciation. When output takes a hump-shaped pattern over time due to a persistentshock, anticipation of future income rise raises demand for domestic goods well in excess
of supply creating a short-run terms of trade appreciation. In CDL the key is the termsof trade appreciation following a positive productivity shock, which is driven by either
mechanism in reproducing a realistic BS correlation. Also building upon incomplete markets, Benigno and Thoenissen (2008) discuss that a standard international real businesscycle model which includes non-traded sector could successfully provide a realistic BS
correlation through the well known Harrod-Balassa-Samuelson e¤ect.2
This paper presents a complementary view of the BS puzzle. We argue that a realistic BS correlation can be reproduced in a theoretical model once we take into account…rm entry, i.e. extensive margins. The theoretical model in this paper re‡ects recent
developments in international macroeconomics. Corsetti et al. (2007) and Ghironi andMelitz (2005) provide a two-country general equilibrium model in which the number of…rms is endogenously determined. In the …rst part of the paper we built a very simple
static general equilibrium model in the spirt of Corsetti et al. (2007). The mechanism
which provides a realistic BS correlation is shown analytically in this part. In the second
part of the paper dynamics are introduced following Ghironi and Melitz (2005) with someextensions. The purpose of the second part is to reproduce a realistic BS correlation
quantitatively using the standard calibration method in the international real businesscycle literature.
On top of the extensive margins, there are several arguments which found our results.
On the one hand, the key is the interaction between market incompleteness and the role
played by extensive margins. On the other hand, they are based on the observation thatstatistically relevant or empirically constructed price indices do not or only imperfectlyre‡ect ‡uctuations in extensive margins.
Recent empirical studies in the international trade literature emphasize the discrepancy between such "empirical" and "welfare-based" price indices in short and even in
2

Recently Kollmann (2009) and Devereux et al. (2009) argue a resolution of the puzzle relining on

another mechanism, hand-to-mouth behavior of subset households.

3

long run. For instance, Broda and Weinstein (2004, 2006) point out that large welfare
gains are stemming from the increased number of imported varieties and note equally that
empirical-based import price indices have an in‡ation bias. They report a 1.2% upwardbias per year for the U.S import price index. In Broda and Weinstein (2010) they also
report that in the U.S. an upward bias in CPI is around 0.8% per year from 1994 to
2003. Such discrepancy is important in considering the BS puzzle as well. This is becauseinternational risk sharing takes place on welfare basis not on empirical basis. Our mainargument is that fundamentally the BS correlation cannot be tested fully unless we know
the exact variations in extensive margins which are relevant to welfare.
The intuition of the paper can be summarized as follows: in our model with extensive
margins, home bias in consumption and monopolistic competition, a positive productivity
shock leads to entry of new varieties. Because of love for variety, this results in a higher
demand for them. The increase in demand leads to an appreciation of the terms of trade
and the "empirical-based" real exchange rate. Concurrently relative consumption rises
providing a realistic zero or even negative BS correlation. In this instance our result issimilar to CDL. Although it is generated by extensive margins, the key is the terms oftrade appreciation providing a positive wealth e¤ect.
Systematically, a realistic BS correlation might not be the case in "welfare-based"measure. The reason is that a rise in local extensive margins consumed with home bias
might be translated in a depreciation in welfare-based price index. Our model predicts
that a positive BS correlation could rest on welfare base.
In generating a realistic BS correlation with extensive margins, we claim the importance of a relatively high elasticity of substitution between local and imported goods.
Intuitively the higher the elasticity for a given preference for variety, the stronger thewage appreciation due to extensive margins becomes. Then it becomes more plausiblethat we observe a realistic BS correlation. This point is shown quantitatively as well as
analytically. In the second part of the paper we show that, under a mild love for variety,the value of elasticity around 7, which is in the range of micro founded estimations, canquantitatively provide a realistic BS correlation.

4

We also explore the role played by market incompleteness to obtain our results. Even
under complete markets, the introduction of extensive margins is able to reproduce a
realistic BS correlation. However, such a realistic BS correlation requires unrealistic dynamics, in particular for relative consumption. Due to perfect risk sharing, it is not
usually optimal when a positive productivity shock occurs to experience a terms of trade
appreciation. When a realistic BS correlation is reproduced under complete markets, thisis only when foreign consumption rises relative to domestic consumption due to a strong
positive transmission via a terms of trade depreciation following a positive productivity
shock. We show that only the combination of market incompleteness and extensive margins can reproduce a realistic BS correlation in a plausible way. Therefore, the main focus
in this paper builds upon incomplete asset markets while we place detailed discussion
about complete markets in the appendix.
The remaining paper is organized as follows. In the next section a static general equilibrium model with endogenous entry is presented. The implication for the BS puzzle isdiscussed analytically in the following two sections. In section 4 dynamics are introduced.
The BS puzzle is explored quantitatively using the standard calibration method in theliterature in section 5. The …nal section o¤ers brief concluding remarks.
1. The model
We build a simple static general equilibrium model. There are two countries, "Home"and "Foreign". Each of which is populated by a unit mass of atomic households. Foreign
variables are denoted with asterisks. The number of …rms in each country is endogenously
determined. Each …rm is assumed to represent one product variety. In this static model,
there is no international borrowing and lending, trade is balanced. The main objective
here is to analytically show how extensive margins and market incompleteness interact
and might provide a realistic BS correlation.

5

1.1. Households
The Home representative household supplies inelastically one unit of labor. Her utility
U is speci…ed as
U=
where

(

C1
1

(1)

;

1) denotes relative risk aversion. C represents her consumption which isde…ned as

C=
where

(

h

1
!

1
CH

1
!

+ (1

)

1
!

1
CF

1
!

i 11
1

!

(2)

;

1=2) captures home bias in consumption. The parameter ! (> 0) is theelasticity of substitution between locally produced (CH ) and imported goods from Foreign(CF ). Each goods are composed by subset of N and N number of varieties respectivelyasCH = VH

Z N

Z N

1

ct (h)1

1

1

1

,

dh

CF = VF0where VHN1c (f )1

1

1

df

1

;

(3)

0

1
1

and VF

1

N

1. The parameterof substitution among varieties. We assume(> 1) denotes the elasticity!. c (h) (c (f )) is the demand forindividual Home (Foreign) variety indexed by h 2 [0; N ] (f 2 [0; N ]). The parameter(0) represents the marginal utility stemming from one additional increase in thenumber of varieties. This speci…cation follows Benassy (1996). In such a way …rms’
markup become distinct from love for variety. Speci…cally, the preference is Dixit andStiglitz (1977) when=11

. When= 0, there is no utility gains in consuming ahigher number of varieties.
The consumer price index P which minimizes spending on consumption basket C isfound asP =PH1 ! + (11

) PF1 ! 1 ! :(4)

In the above expression PH and PF denote the price of CH and CF respectively and eachof which is an index as well as follow
6

1
PH =
VH

Z N

Z N

1

1
PF =
VF

1

1

p (h)

,

dh

0

1
1

1

p (f )

df

:

(5)

0

where p (h) (p (f )) represents the domestic price of variety h produced in Home (Foreign).
Observe that these price indices are de…ned in "welfare-based": they decrease (increase)with a rise (decrease) in the number of varieties. Speci…cally, de‡ation (in‡ation) is large
when the love for variety

is high for a given change in extensive margins.

Finally, the optimal consumption for each basket are found as below:

CH =c (h) = VH 1

PH
P
p (h)
PH

!

C,

CF = (1

CH ,

PF
P

)

c (f ) = VF1!

p (f )
PF

(6)

C;

CF :

(7)

We choose the welfare-based consumer price index P as a numéraire and de…ne real
prices asPH
H = P ,PF
F = P ,)(h) = p(h), and (f ) = p(f.
PP

Similar expressions hold in Foreign.
1.2. Firms
Each …rm representing one product variety competes monopolistically with others. It
is assumed that upon entry a …rm h must pay sunk entry costs in terms of fE units ofe¤ective labor. The latter is de…ned as

fE = zE lEM (h) ;(8)where zE denotes a labor productivity on …rm setting up e¢ ciency. lEM (h) represents
the amount of labor demanded for …rm creation. With the above expression a rise in fE
is interpreted as an increasing regulation on …rm entry.
After entry the …rm produces output y (h) by the following technology

y (h) = zl (h) ;7(9)where z denotes a labor productivity in production which is symmetric across …rms. l (h)is the labor demand for the production of output y (h).
We now specify the …rm’s pricing behavior in below. Operational real pro…ts (dividends) of the …rm are expressed byd (h) =w
y (h) ;
z

(h)

(10)

where w denotes real wages. Goods market clearing condition implies that y (h) =c (h)+c (h). Thus using the optimal demands addressed to the …rm found in the previous
section, y (h) can be rewritten as

y (h) = N (

1) 1

(h)

!
H

) Q! C ] :[ C + (1where Q denotes the real exchange rate de…ned as Q

P
.
P

(11)

Knowing the demand, thepro…t maximization behavior by the …rm results in the following standard pricing:
w
:
1z

(h) =

(12)

Real price (h) is set to be equal to real marginal costs over markup. We denote the priceof exported goods by(h) = Q 1 (h), denominated in Foreign consumption basket.

Without any heterogeneity across …rms, they are symmetric in equilibrium. We denotethe price which holds in such a symmetric equilibrium as(h). The same type ofhnotation holds for other variables.
Finally using the optimal pricing (12) and the fact thatH=N

h from (5) andsymmetry, real dividends can be rewritten as

dh =

1 1 !
h

N (! 1) 1 [ C + (1) Q! C ] :(13)

The above expression highlights some important aspects. First, it shows how dividends
change along the price. The lower the price of individual variety ( h ), the higher the increase in dividends realizes when the elasticity of substitution between local and imported
goods is higher than unity (! > 1). Second, in the above expression N (! 1) 1 captures8

an additional competing e¤ect arising from domestic …rms (N ). When

> ! 1 1 , in thepresence of a strong preference for the variety, an increase in N induces further dividends
while it decreases with a relatively weak love for variety as
!=

and

=

1
1

< ! 1 1 . In particular when

, this term disappears.

1.3. Free entry, labor market clearing and balanced trade
In this subsection we fully characterize the general equilibrium by considering freeentry, labor market clearing and the balanced traded conditions.
We start with the free entry condition. All dividends eared by each …rm are assumed
to …nance her entry costs.3 Thus each …rm’s dividends must be equal to her entry costs
in equilibrium:
fE w
:
(14)
zE
Next in the labor market one unit of total labor supplied is demanded in goods prodh =duction and …rm creation by N number of …rms in equilibrium: 1 = N lh + N lEM;h . Notethat we have yh = (1) dwh z by (10) and (12) and lEM = dwh by (8) and (14). Using

these expressions the above labor market clearing condition can be rewritten as4

1=

N dh
:
w

(15)

Similar expressions hold in Foreign.
Finally the balanced trade implies that the value exported is equal to that imported:
N h ch = QN

f cf : Rewriting with the optimal demands found in the previous section,

the balanced trade condition becomesQ2! 1 N (! 1) 1h ! C = N
3

1 !

(! 1)
f

C:

(16)

In this static version of model there is no investment choice by households, i.e. no arbitrage between

current and future consumption by investment. This is a distinct feature from a full dynamic model that
we will discuss in the following section.
4
It is easily shown that the labor market clearing condition is basically identical to the aggregatedidentity which can be obtained from aggregating budget constraints among households: C = w.

9

The equilibrium has characterized by these equations. Because of the non-linearity ofthe model, in what follows we solve its log-linearized version in order to explore the role
played by extensive margins in the BS puzzle.
2. Extensive margins, relative wages and the terms of trade
We let express percentage deviations from the steady state level with sans-serif fonts.
Relative wages, relative number of varieties in log-deviations are de…ned as wR
and NRNw (Q + w )N . By log-linearizing, the model is reduced to the system of two equations,labor market clearing and free entry conditions, and two unknowns, wR and NR . We assume no regulation shock as fE = f E =0 without loss of generality. By solving the system,wR and NR are expressed with relative exogenous shocks in log-deviations as zR
and zR
E

zE

z

z

zE :5
NR = zR
E:

wR =

(20)

2 (! 1) R
2 (! 1) Rz +z ;1 + 2 (! 1)
1 + 2 (! 1) E

(21)

The number of varieties NR changes one for one with a labor productivity shock on…rm setting up e¢ ciency zR
E while it remains unchanged with a productivity shock on
5

The log-linearized version of labor market clearing condition is
wR = NR + dR :(17)

Free entry condition is expressed in log-deviations by
dR = wR

zR
E:

(18)

And using the log-linearized balanced trade condition, we can write log-linearized dividends as
dR =

2 (!

1) wR

zR + [2(!

1)

1] NR :

(19)

Plugging this expression in the above two equations, the system becomes such that with two equations
and two unknowns.

10marginal costs zR .
R
Contrary, relative wages wR change with both shocks, zR and zR
E . A zE shock, thus

induced extensive margins, has impact on them because of love for variety ( > 0).
Whether wages appreciate or depreciate depends on the elasticity of substitution between local and imported goods ! and home bias . Speci…cally, it shows a non-linearity
in terms of ! provided a size of home bias . Under zR > 0 and zR
E > 0, wages appreciate for Home when 0 < ! < 1

1
2

while depreciate when 1

1
2< ! < 1 and again

appreciate when 1 < !. What is important to observe is the fact that extensive margins
add a further wage appreciation when the elasticity is high (! > 1) under love for variety( > 0).6
pf, the relative price of Foreign goods in termsphWe de…ne the terms of trade as T OT

of Home goods. Their …rst order variations are de…ned as TOT = Q +fh.

Provided

the above expression of relative wages, TOT is expressed as

TOT =

1
1 + 2 (!

1)

zR

2 (! 1) Rz :1 + 2 (! 1) E

(23)

The …rst term is the same one discussed in CDL with an endowment economy. CDL
argue a possibility of the terms of trade appreciation (TOT < 0) following a positive
productivity shock on marginal costs (zR > 0). The terms of trade appreciate when theelasticity of substitution is very low ranging 0 < ! < 1elasticity, ! > 1
6

1
.
2

With a relatively high1
, the terms of trade depreciate from the …rst term.7
2

There is a restriction on parameters’values so that the number of varieties in one country does notincrease in…nitely, i.e. full agglomeration in one country. This kind of thing happens when dividends
increase in one country more proportionally than the cost (wage) appreciation following an increase in
the number of …rms. To avoid it must be21>

4 2 (! 1):
1 + 2 (! 1)

The above condition may not be respected under a very high love for variety(22)and elasticity of substitution!. Figure 1 and Figure 2 satisfy this parameter restriction.
7
In addition to the low trade elasticity, CDL analytically show that a short-run terms of trade appreciation hence a negative BS correlation takes place when a persistence of productivity shock and the11Di¤erent from CDL, however, under love for variety (> 0) and a su¢ ciently highelasticity of substitution (! > 1), the second term prevents the terms of trade from depreciating when zR
E > 0. The higher the love for variety and the elasticity of substitution,
the higher the appreciation from the second term becomes. Intuitively when the love forvariety and the elasticity of substitution are high, a given number of varieties is further
demanded and makes wages further appreciated. This cost appreciation works to counteract the …rst term depreciation due to more e¢ cient production technology (zR > 0).
The mechanism is in essence exactly the point discussed in Krugman (1989) where heargues how it is possible for a country to prevent from a terms of trade depreciation by
providing higher extensive margins along her economic growth.8
In Figure 1 we provide a numerical example about percentage deviations in the terms oftrade for di¤erent values of !. The …gure produces both with and without love for variety
cases ( = 0:2 and= 0). In calibrating, the two shocks, zR and zR
E , are supposed to be

perfectly correlated as zR = zR
E . Home bias in consumption

is set to 0.72. The …gurewell captures how a high elasticity of substitution might contribute to a terms of trade
appreciation under love for variety.
Based on the result found here we next discuss the implication for the BS puzzle.
3. The BS puzzle with extensive margins
Using the balanced trade condition, we can …nd the …rst-order relationship between
the real exchange rate and relative consumption in our model. This iselasticity of substitution are high in a bond economy. Such a short-run appreciation due to a high persistence of productivity shock and a high elasticity is also true in our model but it provides a mitigate resulton the behavior of the terms of trade with extensive margins. As we will see in a full DSGE model, what
is typically hump-shaped in the model are extensive margins. When the persistence of shock rises, thecurrent wealth induced by extensive margins appears only gradually over time reducing the possibility ofan impact terms of trade appreciation.
8
For example, Hummels and Klenow (2005) and Galstyan and Lane (2008) document the empiricalvalidity of the terms of trade appreciation due to higher extensive margins.

12

Q=

2
1
(C
2 ! 1

C ):

(24)

The above expression is exactly identical to the one found in CDL. Under the balanced
trade (incomplete markets), it is di¢ cult to reproduce a realistic BS correlation for abroad range of the elasticity of substitution. Households in Home consume more when
their basket becomes relatively cheap under a su¢ ciently high elasticity, ! > 21 . With
home bias in consumption (> 1=2), this is the case roughly with ! > 1. A way togenerate a negative BS correlation is to suppose a very low elasticity as CDL.9
The above relationship holds at the "welfare-based" real exchange rate and consumption which fully capture variations in extensive margins. However, the BS puzzle is the
puzzle about the empirically observed real exchange rate and consumption. In contrast
to the welfare-based, such empirical-based real exchange rate does not contain (or poorly
measures) variations in extensive margins. Let we suppose that the price indices do notre‡ect at all such variations for simplicity. Denoting the variables without ‡uctuations inextensive margins withe, the welfare based-real exchange rate is broken in two parts:
e+
Q=Q

where

(2

e = (2Q

1) NR :

1) TOT:

(25)

(26)

Observe in welfare basis the real exchange rate depreciates for relatively higher extensive
margins (NR > 0) under home bias ( > 1=2) and love for variety ( > 0).
Consumption is also poorly measured using such price indices. For instance, total nominal consumption spending is given by P C. Statistical agencies divide this amount by Pe

e As a result, ‡uctuations in empirical-based consumption
to measure "consumption" (C).
are given by
9

It is worth noting that when

= 1=2 and ! = 1 it is possible to achieve the complete markets alloca-

tion without any …nancial assets. This is the case discussed in Cole and Obstfeld (1991). The movementof the terms of trade transmits the productivity gains so that the relative consumption (empirical as wellas welfare-based) remains unchanged.

13

e
C

e C
P + C P=

[ N+ (1

)N ]:

(27)

Provided the above de…nition, the welfare-based relative consumption is also broken in
two parts as

C

e
C =C

e +
C

1) NR :

(2(28)

Plugging (25) and (28) in (24), we …nally …nd the BS correlation with extensive margins
as follows
1 e
e= 2
C
Q
2 ! 1

2 (2

e
C

1) (!
2 ! 1

1)

NR .

(29)

The tight relationship which holds together the welfare-based real exchange rate andconsumptions is now broken. And it is possible to have a negative BS correlation betweene and CeQ

e without relying on a low elasticity.
C

We see the above point more in detail. Supposing a perfect correlation between two

types of shocks as zR = zR
E , we can rewrite (29) ase=Q(2
1) [1 22 (! 1) [1 + 2 (1(! 1)])] + 2

1

e
C

e
C

:

(30)

In the above expression not only for a low range of elasticity of substitution (0 < ! <1

2
1
2 [1+2 (1)]), but also for a high range of elasticity (1 + 2 1 < !) the BS correlation

becomes negative.
The reason why a negative BS correlation appears for a low range of elasticity is almost
the same as discussed in CDL: when Home is hit by a positive productivity shock, Home
provides more goods with intensive as well as extensive margins. However, with a very
low elasticity of substitution the goods market clearing requires an appreciation of theterms of trade (expensive Home goods), therefore providing a positive wealth e¤ect for
Home agents in order to absorb their own production with home bias.
Following a positive shock, extensive margins appear in Home (NR > 0). When theelasticity of substitution is high as 1 + 2 1 < !, because of the demand addressed to thesenew varieties, wages appreciate further. As we have seen in the previous section, the higher
14the elasticity of substitution, the stronger such a wage appreciation is. Counteracting thedepreciation due to a higher productivity on marginal costs, the terms of trade tend to beeappreciated with extensive margins which brings also the observed real exchange rate Qeinto an appreciation. Concurrently, relative consumption C
BS correlation.

Note that without love for variety (e rise providing a realisticC= 0) the expression collapses to (24) which isagain identical to CDL. In Figure 2 we give a numerical example with the same parameters
as in Figure 1. It is shown that a high elasticity of substitution can contribute to theresolution of the puzzle under love for variety.

Figure 1: the terms of trade

Figure 2: the BS correlation

4. Quantitative investigation
In what follows, we investigate whether the intuition described analytically in the previous sections quantitatively holds. For that purpose, we construct a two-country DSGE
model in which the number of …rms is endogenously determined and the internationalborrowing and lending is allowed using non-contingent bonds. Our model is considered
as a simpli…ed version of Ghironi and Melitz (2005) without …rm heterogeneity.
All variables now have time index t. A productivity shock on marginal costs of production zt and …rm creation zE;t are assumed to be perfectly correlated for the sakeof simplicity. Investment takes place in terms of new …rm creation whose number is15

represented by NE;t . Other than the investment dynamics we add three more realistic
extensions compared to the static model: endogenous labor supply, internationally held
non-contingent bonds and entry cost paid in terms of capital goods as well as labor. Only
these modi…ed points are discussed below.
4.1. Households
The Home representative household maximizes Et

P1

s=t

s t

Ut at time period t. Her

utility now depends on labor supply as well as consumption as follows:
1+ 1

C1
Ut = t1
The parameterLt ':1 + '1(31)(> 0) represents the degree of non satisfaction stemming from supplying

labor service Lt . ' is the Frisch elasticity of labor supply.10 With this speci…cation themarginal disutility in providing one additional labor service is increasing.
4.2. Budget constraint with non-contingent bonds
In this dynamic version of the model, the budget constraint is given by## 2+ Qt B 2;t+1 + sh;t+1 (Nt + NE;t ) xh;t + Ct
Bt+1 + Qt B ;t+1 + Bt+1
2
2
= (1 + rt ) Bt + Qt (1 + rt ) B ;t + sh;t Nt (dh;t + xh;t ) + Ttf + wt Lt ; (32)where Bt+1 (B ;t+1 ) is real holdings of Home (Foreign) bonds into t + 1. To ensure
zero bond holdings at the steady state, quadratic adjusting costs of bond holdings # are
introduced. Ttf is the free rebate of adjusting costs which is exogenous for households. rt(rt ) denotes real interest rate. It is assumed that investment takes place only domesticallyby purchasing mutual funds among domestic existing …rms Nt and new entrants NE;t .
sh;t+1 denotes real holdings of shares into t + 1. xh;t represents real share price of Homemutual fund.
10

With ' = 1 the marginal disutility in supplying one additional unit of labor becomes constant, .

When ' = 0 the marginal disutility becomes in…nite and the labor supply becomes inelastic.

16

Euler equations for share and bond holdings are given by respectively

xh;t =

(1

) Et

Ct+1
Ct

Ct (1 + #Bt+1 ) =(xh;t+1 + dh;t+1 ) ;(1 + rt+1 ) Et Ct+1 ;

(33)

(34)

and

Ct (1 + #B ;t+1 ) =1 + rt+1 Et

Qt+1
C :
Qt t+1

Finally, the optimal labor supply decision gives
1(35)(Lt ) = wt Ct :Similar conditions hold in Foreign.
4.3. Firms
4.3.1. EntryDi¤erent from the static model and original speci…cation in Ghironi and Melitz (2005),
we assume that new entrants need capital goods as well as labor in order to set up theirproduction site. This type of speci…cation is proposed in Bilbiie et al. (2007a) as a realistic
extension. One …rm/variety creation is assumed to need an amount of …rms setting up
goods fE . The production of such goods is supposed to be done by the following CobbDouglas technology using capital goods Kt as well as labor lEM;t as inputs:

fE =
where

(1

Kt

zt lEM;t

1

1

(36)

;

) is the share of labor (capital) in total costs. For simplicity we assume

that capital goods Kt has the same composition as consumption goods Ct . The cost
minimization problem by …rms yields following factor demands

lEM;t =

wt

t fE ;

Kt = (117)

t fE ;(37)

wheret =wt
zt

is the real cost of …rm creation. In particular when

= 1 only labor isused for entry.
It is assumed that production takes place only one period after the entry. The motionof …rms is de…ned as

Nt+1 = (1
where

(38)

) (Nt + NE;t ) ;denotes the "death shock" which takes place at the very end of each period

after investment has been completed. As a result, a fraction of new entrants die without
producing.
Similar conditions hold in Foreign.
4.3.2. Production
The production of intensive margins is almost identical to the previous static model.
However, because capital goods are required for entry, the demand addressed for each…rm now includes these terms:

yh;t = ch;t + ch;t + NE;t kh;t + NE;t kh;t ;(39)where kh;t (kh;t ) denotes the capital demand from Home (Foreign) new entrants. The
expression of dividends also changes provided the above speci…cation as follows

dh;t =

1 1 !
h;t

Nt

(! 1) 1

) Q!t Mt ] ;[ Mt + (1(40)

where Mt (Mt ) is consumption and investment goods demand in each country de…ned as

Mt = Ct + NE;t Kt ,Mt = Ct + NE;t Kt :Note especially, using factor demand (37), when

= 1 (only labor is used as input)

the expression of dividends is similar to the previous one.
Similar expressions hold in Foreign.

18

(41)

4.4. Characterizing the general equilibrium
As it is the case for the static model, we characterize the general equilibrium by
free entry and labor market clearing and net foreign asset dynamics which replaced thebalanced trade condition.
In the dynamic model, free entry condition equates real share price to real entry costs
asxh;t = fE

wt
zt

(42)

:

The labor market clearing condition gives Lt = Nt lt + NE;t lEM;t because now the labor
supply which is demanded for production by Nt number of …rms and …rm creation by
NE;t number of …rms is endogenous. Noting yh;t = (d1) wh;tt zt and lEM;t =

xh;t
, the
wt

above condition can be rewritten as

Lt = (

1)

Nt dh;tNE;t xh;t
+
:
wt
wt

(43)

Similar expressions hold in Foreign.
Because of international borrowing and lending the trade is no more balanced and thefollowing net foreign asset dynamics hold:11
11Aggregation implies the following net foreign assets accumulation for each country:

Bt+1 + Qt B ;t+1 = (1 + rt ) Bt + Qt (1 + rt ) B ;t + Lt wt + Nt dh;t
Bt+1
(1 + rt ) Bt
+ B ;t+1 =+ (1 + rt ) B ;t + Lt wt + Nt df;t
Qt
Qt

NE;t xt

NE;t xt

Ct :

(44)

Ct :

(45)

The above two equations (eliminating bonds position by Foreign using bond market clearings (47))
yield (46).

19Bt+1 + Qt B ;t+1 = (1 + rt ) Bt + Qt (1 + rt ) B ;t+1
Lt wt + Nt dh;t2

Qt Lt wt + Nt df;t
1
NE;t xt + Ct
2

Qt NE;t xt + Ct

: (46)

Finally in addition to the above three equations, bond markets should be clear inequilibrium:

Bt+1 + Bt+1 = 0;

B ;t+1 + B ;t+1 = 0:(47)

The dynamic model contains 31 equations and 31 variables among which 8 are endogenous state variables (Nt ; Nt ; Bt ; Bt ; B ;t ; B ;t ; rt and rt ) and 2 are exogenous shocks (ztand zt ). Table 2 summarizes the system. Details about the steady state are in the appendix. In what follows we calibrate the linearized version of the model and quantitatively
explore the mechanism in generating a realistic BS correlation.

5. Calibration
The dynamic model is calibrated with parameters in Table 3. The value of constantrisk aversion ( ), discount factor ( ), Frisch elasticity of labor supply (') come from Bilbiie
et al. (2007b) who choose them based on the standard RBC literature. The value of deathshock ( ) is selected such that it matches to the U.S. empirical level of 10 percent job
destruction per year as in Ghironi and Melitz (2005). The cost of adjusting bond holdings(#) is set also following them.12
The elasticity of substitution between Home and Foreign goods (!) is set to 6 in thebenchmark calibration. Given this value, the elasticity of substitution among varieties ( )is set to 7. These values may be considered too high compared to the lower value used in
the open macroeconomics literature, which typically range from 0.5 to 2. However, this12

% deviations of bond positions are de…ned relative to the steady state consumption C in the linearizedversion of the model.

20

Table 2: The model1 !
H;t + (1Price indices) 1F;t! = 1

H;t = Nt

h;t ;

1 !
+ (1
F;t

f;t

1 !
) H;t
=1

F;t = Nt

Pricing

F;t = Nt

f;t ;

h;t =

wt
1 zt

h;t = Qt

1

f;t =

wt
1 zt

H;t = Nth;th;tf;t = Qt f;t

Pro…ts

De…nition of M

Free entry

Optimal labor supply

dh;t = 1 Nt(! 1) 1 1 !
h;t [df;t = 1 Nt(! 1) 1

1 !
f;t

Mt + (1Mt = Ct + (1) NE;t xh;t

Mt = Ct + (1) NE;t xf;t

xh;t = fE

wt
zt

xf;t = fE

wt
zt) Q!t Mt ]Mt + (1) Qt ! Mt

1(Lt ) = wt Ct
1(Lt ) = wt Ct
Labor Market clearing

Number of …rms

Lt = (

Nt dh;tN xh;t+ E;twt
wt
Nt df;tNE;t xf;t1) w +
wtt

Nt = (1) (Nt 1 + NE;t 1 )Nt = (1) Nt 1 + NE;t 1

Lt = (

1)

Euler equation (shares) xh;t =

(1

) Et

Ct+1
Ct

(xh;t+1 + dh;t+1 )

xf;t =

(1

) Et

Ct+1
Ct

xf;t+1 + df;t+1

Euler equation (bonds)

Net foreign AssetCt (1 + #Bt+1 ) =(1 + rt+1 ) Et Ct+1

Ct (1 + #B ;t+1 ) =1 + rt+1 Et QQt+1
Ct+1
t

Ct

1 + #B ;t+1 =1 + rt+1 Et Ct+1

Ct

1 + #Bt+1 =t
Ct+1(1 + rt+1 ) Et QQt+1

Bt+1(1 + rt ) Bt + Qt [B ;t+1

= 12 Lt wt + Nt dh;t12Bond market clearing

NE;t xt + Ct
21
Bt+1 + Bt+1 = 0(1 + rt ) B ;t ]Qt Lt wt + Nt df;tQt NE;t xt + Ct

B ;t+1 + B ;t+1 = 0:

Table 3: Baseline parametrization

'

!

#

constant risk aversion

2

discount factor

0:99

Frisch elasticity of labor supply

2

elasticity of substitution among varieties

7

between Home and Foreign goods

6home bias in consumption

0:72

death shock

0:025

bond holding adjusting costs 0:0025
share of labor in entry costs

0:64

love for variety

0:2

is well in the range of micro founded estimations in the trade literature. For instance,
Romalis (2007) estimates elasticities which range from 4 to 13. For the purposes ofcomparison, we also consider the standard value of elasticity in open macroeconomics(! = 2) as it is in Benigno and Thoenissen (2008). Provided the above value of elasticity
we set= 3:8 following Ghironi and Melitz (2005).13

We set , love for variety, to 0:2 arbitrary. This is slightly higher than the implied value
under the Dixit-Stiglitz preference for the benchmark elasticity (1/(7-1)=0.17). Becauseof the ambiguity which surrounds this parameter, the role played by the love for variety
in the BS puzzle is explored with a sensitivity analysis in the following section.
The value of home bias in consumption ( ) is taken from CDL. We set the share oflabor ( ) in entry costs as 0.64 based on Heathcote and Perri (2002). This is the standard
value in the DSGE model including capital and labor in its production function.
13

A low elasticity in the open macroeconomics literature is controversial: recently Imbs and Mejean(2009) argue the conventional estimation in the open macroeconomics literature about the elasticity of
substitution has a downward bias without considering the heterogeneity of these values among sectors.
They propose the value of elasticity around 7 in calibrating DSGE models and support the view of the"elasticity optimism".

22

Productivity process is selected from Backus et al. (1992) such that Zt+1 = Zt + t
h
i;
h
i;
where Zt = zt ; zt , t = t ; t and where the correlation of shocks and error terms
are given by23
2
30:906 0:088
0:73 0:19
5 , and V ( ) = 4
5:=4
0:088 0:906
0:19 0:73(48)

5.1. Intuition by impulse responses

Impulse responses for the real exchange rate, the terms of trade and relative consumption are reported in Figure 3. For the real exchange rate and relative consumption both
welfare and empirical-based are documented. The shock used is one percent increase inHome productivity. For simplicity we omit spillover terms in the correlation matrix (48)in this exercise.
On impact of the shock new entry takes place in Home. The relative number of varieties increases steadily showing a hump-shaped pattern. Over time the terms of trade
appreciate for Home in spite of persistent e¢ ciency gains due to a positive productivity shock. The reason is identical to the one explained in the …rst part of the paper.
A combination of a relatively high elasticity of substitution and a mild love for variety
are su¢ cient to change the direction of the terms of trade from a depreciation into an
appreciation (dotted lines on upper and lower panels). Re‡ecting these terms of trade appreciation, the empirical-based real exchange rate appreciates as well (solid line on upper
panel). However, the welfare-based real exchange rate remains depreciated re‡ecting ahigher Home originated extensive margins (solid line on lower panel).
At the same time of the empirical-based real exchange rate appreciation, relative consumption rises (crossed line on upper panel). It is well known that with non-contingent
bonds the condition which holds under complete markets between the real exchange rate
and relative consumption is only veri…ed in expected …rst di¤erence. Home and Foreign
households stabilize their consumption using such bonds only in the aftermath, not exanté of a shock. As a result, a tight link between the real exchange rate and relative
consumption under complete markets is broken. However, in our model the market in-23

completeness alone is not su¢ cient to reproduce a realistic BS correlation as they remain
positively correlated in welfare-based (crossed and solid line on lower panel). We need awealth e¤ect induced by higher extensive margins which brings the terms of trade into
an appreciation and the fact that such ‡uctuations in extensive margins are not fullymeasured in observed real exchange rates as argued in Broda and Weinstein (2004, 2006,2010).

Figure 3: IRFs under incomplete markets.

5.2. Characteristics of the theoretical model
Table 4 reports second moments of the dynamic model. The US data comes fromeither Backus et al. (1992) or Heathcote and Perri (2002) except for the BS correlation(-0.27) drawn from CDL, the median among OECD countries relative to the ROW. Allvariables are empirical-based denoted withesuch that for simplicity they do not containany variations in extensive margins. GDP in this model is Yet = wt Lt + Nt deh;t (labor +…nancial income). Investment value is NE;t x
eh;t , share price multiplied by the number ofget
new entrants. Net export (trade balance) is de…ned as T
Bt = Xg t =Yet where XetIM

g t denote export and import value respectively. For the purpose of comparison weand IM24

report second moments obtained with the lower elasticity ( = 3:8 and ! = 2) and thoseunder complete markets as well.14
Under incomplete markets with a high elasticity of substitution between Home andForeign goods (! = 6), a high investment volatility (15.25) and a strong negative crosscountry correlation of investment (…rm entry) (-0.91) appear. Because …rm creation requires labor services, employment is internationally correlated negatively (-0.91) and output as well (-0.52). This high elasticity is what is needed for providing a realistic BS
correlation (0.17) along the mechanism discussed in the …rst part of the paper. With analternative elasticity (! = 2) investment volatility declines (8.38) and Home and Foreign
investment becomes less correlated (-0.77) while the BS correlation remains in the puzzle(0.95).
Second moments under complete markets are quite similar to those under incomplete
markets. The result is reminiscent of Heathcote and Perri (2002) who discuss that only
the balanced trade case is very di¤erent and close to the reality. Although they are
similar, under complete markets, investment volatility increases (18.93) and it becomescorrelated more negatively (-0.94). This can be interpreted based on the result of Corsetti
et al. (2007). Although they argue with a static model, it is shown that under complete
markets there is more entry compared to incomplete markets in a more e¢ cient country. Intuitively because consumption is perfectly insured under complete markets, theequilibrium allocation of …rms becomes such that the world has the maximum number ofvarieties.
The BS correlation becomes negative (-0.19) under complete markets but the reason
is quite di¤erent from the one discussed for incomplete markets. Under complete markets
when it becomes negative there is a very strong positive transmission via the terms oftrade depreciation following a positive shock. As a result empirical-based consumption in
Foreign rises more than Home while the empirical-based real exchange rate is depreciated.
However, as it is reported in Corsetti et al. (2008b) using VAR model, empirical-based
14

Second moments of the theoretical model are calculated using the frequency domain technique pre-sented in Uhlig (1998) for HP …ltered series. The smoothing parameter is set to 1600.

25

consumption should rise in a country receiving a positive shock relative to the rest of theworld. Hence a realistic BS correlation which appears under complete markets would not
be considered as plausible because it bases on unrealistic pattern of consumption. We
place details about complete markets in the appendix.
Obviously the model shares principle (bad) characteristics of the standard two-country
real business cycle model, such discussed in Heathcote and Perri (2002): lower volatility in the terms of trade and the real exchange rate, higher cross country correlationof consumption than output and negative cross country correlation of investment andemployment. In summary we can say that adding extensive margins with a high elasticity of substitution brings the BS correlation into a realistic range. On the other hand,
other puzzles in international real business cycle remain without signi…cant quantitative
improvement.

5.3. Sensitivity analysis
Figure 4 and Figure 5 examine the BS correlation with di¤erent values of the elasticity
of substitution (!) and love for variety ( ) with the baseline parameters. It becomessteadily weaker and …nally becomes negative as the elasticity increases. For ! = 7, a
proposed value in Imbs and Mejean (2009), the BS correlation is

0:23. For love forvariety, almost the same pattern is observed. The BS correlation becomes weaker andarrives in negative range as love for variety increases. Again the intuition is shed by the
analytical part of the paper where the interaction between the elasticity of substitution
and the love for variety has been emphasized.

26

Table 4: Second moments% std.dev.
relative to Ye

US data

Incomplete= 3:8 ! = 2

Complete= 3:8 ! = 2

Ye

e
C

1:71

0:84

(1:00)(0:49)

1:46

NE x
eh

L

g
T
B=Ye

5:38

0:66

0:45

(3:15)(0:34 )

0:48

15:25

0:83

(1:00)(0:33)(10:42)(0:56)

1:40

0:45

8:38

0:82

(1:00)(0:32)(5:98)(0:58)

1:76

0:47

18:93

1:19

(1:00)(0:27)(10:76)(0:68)

1:47

0:44

8:90

0:92

(1:00)(0:30)(6:07)(0:63)

e
C

NE x
eh

L

g
T
B=Ye

T OT

0:46

0:19

0:74

0:17

T OT
2:99

e
Q

3:73

(1:79 )

(2:23 )

0:09

0:04

(0:06)(0:03)

0:33

0:15(0:24)(0:10)

0:09

0:04(0:05)(0:02)

0:42

0:19(0:29)(0:13)

Correlation
with outputeQ

US data

0:76

0:90

0:87

Incomplete

0:64

0:89

0:78

0:18

0:55

0:55

0:68

0:96

0:81

0:67

0:67

0:67

0:39

0:87

0:86

0:91

0:91

0:61

0:96

0:83

0:65

0:65

Ye

e
C

NE x
eh

e e
C
e ;Q
C

C
;Q
C

= 3:8 ! = 2
Complete= 3:8 ! = 2

0:28

0:06
0:65

0:24

0:13

Cross country
correlation
US data
Incomplete= 3:8 ! = 2
Complete

0:58

0:36

0:52 0:90

0:30

L
0:42

0:27CDL

0:91

0:91

0:17

0:85

0:33 0:93

0:77

0:95

0:95

0:97

0:67 0:98

0:94

0:96

0:19

1:00

= 3:8 ! = 2
0:39 0:99
0:80
0:96 0:84
1:00
Data with * from Heathcote and Perri (2002). Others from Backus et al. (1992).

27

Figure 4: the BS correlation and the Figure 5: the BS correlation and love
elasticity of substitution.

for variety.

6. Conclusion
This paper revisits the consumption-real exchange rate anomaly known as the BackusSmith puzzle. We examine how extensive margins can contribute to the resolution of the
puzzle. Our argument is based on the observation that the BS correlation cannot be
tested fully because of unobservable ‡uctuations in extensive margins.
Higher extensive margins, following a positive productivity shock, work as a demand
shock under love for variety and bring the terms of trade in an appreciation. Re‡ecting
such terms of trade movement, the empirical-based real exchange rate appreciates aswell while consumption rises resolving the puzzle. Along the story the asset market
incompleteness and a relatively high trade elasticity which is consistent to micro founded
estimations play important roles.
In contrast the welfare-based real exchange rate depreciates because of higher extensive
margins consumed with home bias. Thus the puzzling positive correlation might remain
in welfare basis. One of challenges in the future research would be to test this prediction.

28

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Appendix A. Complete markets
We discuss the implication of extensive margins under complete asset markets. As it
is explained in the paper when a realistic BS correlation appears under complete markets
it requires implausible pattern of consumption. Here we look this point more in depth
analytically and quantitatively.
Appendix A.1. Static model
Under complete markets, the marginal utility which stems from an additional nominal
wealth is the same across countries:

Q=

C
C

(A.1)

:

Using the above perfect risk sharing condition, we can write relative dividends in …rst
order deviations as dR = (1) R + [ (1)

1] NR . Plugging this expression in thelabor market clearing and free entry conditions, we have the solution for wR and NR as
follows

wR =

1 R
(
z +

31

1) R
zE ;

(A.2)

NR = zR
E:

(A.3)

where

! 1
The parameter

1)2 + (2

(2

1
1)2 :

roughly represents the elasticity of substitution between local and im-

ported goods. Relative wages wR increase (decrease) with a positive productivity shockon marginal costs zR when

> 1 ( < 1). Extensive margins NR change one by one withan investment shock zRE as it is in the balanced trade case. With love for variety ( > 0)
and a su¢ ciently high elasticity of substitution ( > 1) wages appreciates with higher
extensive margins. When the elasticity is low ( < 1) they depreciate. As it is the case forincomplete markets, we can …nd a restriction on parameters as 1 >(1)2

for complete

markets.
Provided the above expression of relative wages, the terms of trade in …rst orderdeviations are given by
1
TOT = zR

(

1) R
zE :

(A.4)

In contrast to the terms of trade ‡uctuations analyzed in the paper, under complete markets following a positive productivity shock on marginal costs zR , they never appreciatefrom the …rst term. As it is under incomplete markets, however, following a positive shock
on …rm creation e¢ ciency zR
E , the second term adds a terms of trade appreciation when( > 1) with love for variety ( > 0). Because of the …rst term, with whatever correlationof the shocks, the terms of trade less likely appreciate compared to those under incomplete
markets.
Appendix A.1.1. Implication for the BS puzzle
By constructing empirical-based measure, the BS correlation with extensive margins
under complete markets become:
e
Q=

e C
e
C

+(

1)
32

(2

1) NR :

(A.5)

The tight link between the empirical-based relative consumption and real exchange ratethat we …nd ordinary under complete markets is broken because of extensive margins.
And it is possible even under complete markets to generate a negative BS correlation.
This happens when there is a relatively strong positive transmission via the terms ofe > 0) (in spite of the higher number of varieties NR > 0 which
trade depreciation (Qadds them an appreciation) to the extent that the relative empirical-based consumption
e C
e < 0).
decreases (CIn particular, with perfect correlation between two shocks, zR = zR
E , the BS correlation

is described as

Without love for variety (

e 1
Q=
1

(
(

1)
1)

e C
e
C

:

(A.6)

= 0) or with log utility case ( = 1), we are in the original

puzzle. But depending on the value of parameters the BS correlation would become
negative or positive.
However, we consider a realistic BS correlation under complete markets is implausible
e C
e <0because it requires unrealistic empirical-based consumption pattern such that C
combined with a terms of trade depreciation following a positive shock.
Appendix A.2. Dynamic modelWith complete markets the model becomes simpler. We discuss only these modi…ed
points compared to the dynamic model in the paper. The real budget constraint for theHome representative household now contains the state-contingents securities in the placeof non-contingent bonds. This is

Ct + sh;t+1 xh;t (Nt + NE;t )X
X
bt+1 (St+1 ) qt (St+1 j St ) + Qt
bt+1 (St+1 ) qt (St+1 j St )
+
st+1

st+1

= wt Lt + sh;t Nt (xh;t + dh;t ) + bt (St ) + Qt bt (St ) : (A.7)

33

The household has now access to the full set of Arrow-Debreu securities which give one unitof Home or Foreign goods in the next period. bt+1 (St+1 ) (bt+1 (St+1 )) is holdings of such
assets into t + 1 indexed by the future state of nature St+1 . qt (St+1 j St ) (qt (St+1 j St ))denotes its real price which is conditional on the current state of nature St .
First order conditions about these state-contingent securities yield the well known
perfect risk sharing condition as (A.1) with time indices. Under complete markets, Euler
equations about non-contingent bonds, bonds markets clearing conditions and the evolution of net foreign asset are no more needed. Other …rst order conditions remain thesame as they are in the paper. Finally the model contains 25 equations and 25 variables
among which 2 are endogenous state variables (Nt and Nt ) and 2 are exogenous shocks(zt and zt ).
Appendix A.3. (Unrealistic) characteristics under complete markets
Here we quantitatively show how it is implausible a realistic BS correlation obtained
under complete markets. For this purpose we look impulse responses as in the paper withthe same baseline parameters and shocks (Figure A-1). Following a rise in Home laborproductivity the terms of trade always depreciate in spite of a relatively high number ofHome originated …rms. The welfare-based real exchange rate strongly depreciates (solidline on lower panel) mainly due to higher Home extensive margins consumed with homebias. Elimination of such ‡uctuations in extensive margins from consumption baskets
makes the empirical-based relative consumption decreased (crossed line on upper panel)
while the empirical-based real exchange rate remains depreciated (solid line on upper
panel). In other words, households in Home consume less only with intensive margins
whilst more with extensive margins in order to achieve a perfect risk sharing. Again,
although it is possible to create a realistic BS correlation, it must be based on implausible
consumption pattern.
Next we perform sensitivity analysis with benchmark parameters (Figure A-2 andFigure A-3). Under complete markets as the elasticity of substitution increases the BS
correlation declines and afterwards starts to increase. The analytical solution again sheds
light on this non-linearity. As it can be seen from (A.4), when the elasticity of substitution34

or love for variety approaches to a su¢ ciently high value, the terms of trade change theirdirection, from depreciation to appreciation. Because variations in the empirical-based
relative consumption remain negative along such a rise in elasticity or love for variety, thenon-linear pattern of the BS correlation as in Figure A-2 and A-3 appears.

Figure A-1: IRFs under complete
markets.

Figure A-2: the BS correlation and

Figure A-3: the BS correlation andthe elasticity of substitution.

love for variety.

Appendix B. Steady state
Steady state values are expressed without time indices. At the symmetric steadystate, it must be Q = 1, ph = ph = pf = pf , N = N and PH = PH = PF , = PF . Then
35

H =

H =

F

=

F

= 1 and

h =

h =

f

=

f

= N . Also C = C , NE = NE and

K = K . Then M = M . Also dh = df and xh = xh = xf = xf .
First we start to …nd steady state ratios relative to M denoted with SiM where i = D(dividends), I (investment), W (wages) and C (consumption). The steady state share of
real dividends relative to the demand addressed to each …rm becomes

M
SD

N dh
1
= :
M

With this condition, Euler equation about share holdings and the motion of …rms thesteady state share of investment becomesSIM

NE x h
=
M
1

1
(1

)

:

Using the above two steady state shares, from the labor market clearing condition thesteady state share of labor income becomes

M
SW

Lw
M
= SD
(
M

1) + SID :Finally at the symmetric steady state, aggregated demand must be equal to aggregatedincome: C + NE xh = Lw + N dh , using this identity, we have
SCM

Noting M = C + (1

C
M
M
= SW
+ SD
M

SIM :

(B.1)

) NE xh ; with the above steady state ratios de…ned relative to

M , the steady state ratios relative to the consumption C denoted with Si are expressed
as follows
SIM
;
(1
) SIM

SI

NE x h
=
C
1

SD

N dh
M
= SD
[1 + (1C

36) SI ] ;

(B.2)

SW

Lw
= SD (
C1) + SI :Based on the above ratios, we can …nd explicit solutions about variables at the steadystate. From the de…nition of price index we have,
h = N

(B.3)

:

From the optimal pricing, steady state wages are expressed as
1w=
We choose

(B.4)

N ;

so that the steady state labor supply becomes the unity: L = 1. From theoptimal labor supply condition this implies

= wC

(B.5)

:

Also we have NCE x = SI . Using the steady state low of motion, N = (1) NE and thefree entry condition, x = w , SI can be rewritten as
Nw
1
=
C

(B.6)

SI :

Because L = 1, the steady state ratio of wage is given byw= SW :C(B.7)

In the above two equations eliminating C and plugging B.4 the unknown is N . Solving
this equation, the steady state number of variety is given by

N1

(1

)

1

=

1

1

SI
:
SW

(B.8)

The remaining variables are easily found as follows
1

w=

37

N ;

(B.9)

C=

w
;
SW

(B.10)

dh =

CSD
;
N

(B.11)NE =1

xh = w :Especially, from (B.5) the value ofN;(B.12)(B.13)

is found. With baseline parametrization it must be= 0:39:38