Download Home-Bias in Consumption and Equities: Can Trade Costs Jointly

Home-Bias in Consumption and Equities: Can
Trade Costs Jointly Explain Both?
Håkon Tretvoll
New York University∗
This version: May, 2008
Abstract
This paper studies whether including trade costs to explain the home-bias
in consumption can explain the home-bias in equity portfolios in the model
provided by Heathcote and Perri (2008). Their model is a symmetric twocountry/two-good model with production and a home-bias in portfolios arises
since international relative price fluctuations make domestic stocks a good
hedge against non-diversifiable labor income risk. However, a crucial assumption is that preferences exhibit a bias toward domestic goods. Explaining this
bias using realistic trade costs requires a relatively high elasticity of substitution between traded goods. For high enough values the result from Heathcote
and Perri breaks down. For lower values we show that the effect of including
trade costs in the production economy differs from the effects in the endowment
economy considered by Coeurdacier (2008).
1
Introduction
This paper is concerned with two “puzzles” that feature prominently in international
economic data. First, there is the home-bias in consumption puzzle. This refers
to the fact that countries are not very open to trade. The classic reference on the
∗
I would like to thank Fransisco Barillas, Jonathan Eaton, Boyan Jovanovic, and Vivian Yue for
helpful discussions/comments. I would also like to thank Raquel Fernandez and other participants
at the 3rd-Year Paper Seminar at New York University for their questions/comments. The usual
disclaimer applies.
1
segmentation of international goods markets is McCallum (1995) who found that
trade among Canadian provinces was twenty times greater than between Canadian
provinces and US states. As stated in Obstfeld and Rogoff (2000) the subsequent
literature has found that the home-bias in consumption is less extreme than suggested
by McCallum, but there is still a significant degree of home-bias in international
trade. As an example, Coeurdacier (2008) states that in 2005 the openness to trade
ratio in the United States measured by the sum of exports and imports over GDP
was only 25%. Since the US accounts for about one third of world production, US
imports should amount to about two thirds of GDP if there were no frictions in goods
markets. The openness to trade ratio should then be higher than 130%.
Second, there is the home-bias in equity portfolios. This refers to the fact that
investors tend to hold a large fraction of their portfolios in domestic assets. French and
Poterba (1991) documented this puzzle and estimated that U.S. investors held about
94% of their equity portfolios in U.S. equities. Since their result was first published
the puzzle has become slightly less prominent, but according to Coeurdacier, U.S.
investors still held 82% of U.S. equities in 2005, despite the substantial financial
market liberalization during the last couple of decades.
There is quite a consensus that international trade costs1 can explain the homebias in consumption puzzle. However, there is less of a consensus in explaining the
home-bias in portfolios. As stated in Heathcote and Perri (2008) (from now on often
referred to as HP), there is a large literature that share the common conclusion that
there is too little diversification observed relative to the predictions of a frictionless
model. Hence, the observed low diversification should be explained by introducing
a friction into the model. HP provide citations to a list of papers focusing on different frictions, for example fixed costs of foreign equity holdings, liquidity or short
sales constraints, weak investor rights concentrating ownership among insiders, and
asymmetric information in financial markets, to name a few.
HP take a different approach however. They build a frictionless model in which
perfect risk sharing is consistent with relatively low levels of international diversification. The model builds on the international real business cycle model of Backus,
Kehoe and Kydland (1992), and consists of a two-country model with production.
The presence of production and particularly investment implies that the returns to
domestic and foreign equities are not automatically equated, so households face an
interesting portfolio problem. A home-bias in equity portfolios arises since relative
labor earnings and the relative returns on domestic stocks are negatively correlated.
1
“Trade costs” should be understood in a broad sense to include transport costs, tariffs, “border
effect”, etc.
2
This is due to the joint effects of international relative price movements and to the
presence of investment and capital. Since HP’s model is frictionless, it casts doubt on
the quantitative role of frictions in understanding the home-bias in portfolios puzzle.
A crucial assumption that HP make is that there is a home-bias in the production
of the final consumption good. Without this assumption the implications of their
model are similar to Cole and Obstfeld (1991). After a productivity shock, changes
in demand fall equally on domestic and foreign goods, and the changes in relative
quantities are canceled out by offsetting changes in the terms of trade. This implies
that stock returns are equated across countries, and as in Cole and Obstfeld’s endowment economy, any portfolio delivers perfect insurance against country-specific
risk.
In this paper we relax HP’s crucial assumption, and investigate whether introducing trade costs in the spirit of Obstfeld and Rogoff (2000), can explain both the
home-bias in consumption and the home-bias in portfolios. That is, instead of simply
assuming a home-bias in consumption as Heathcote and Perri do, we aim to explain
the home-bias in consumption using trade costs. The interesting question is then
whether the mechanisms of HP’s production economy still imply that a portfolio
biased toward domestic stocks delivers perfect risk sharing.
This idea is very similar to Coeurdacier (2008). In that paper the author studies a
two-country/two-good model with symmetric endowment economies. Coeurdacier’s
setup is essentially similar to Obstfeld and Rogoff, but he does not restrict attention
to the special case where the efficient Arrow-Debreu consumption allocation can be
perfectly replicated with equities. Instead he considers the general case, and finds
that in general trade costs worsen the home-bias in portfolios puzzle. Trade costs
explain the home-bias in consumption at the expense of a foreign-bias in portfolios.
Coeurdacier does not however, have production and investment in his model, so he
does not investigate how the mechanism that generates a home-bias in HP interacts
with trade costs. This paper aims to fill that gap.
When solving HP’s model with trade costs included to fit the value of imports to
GDP, we find that in general, their effects are in some sense isomorphic to assuming
a home-bias in consumption. However, there is a crucial difference: using trade costs
imposes extra discipline on the parameterization of the model. In order to obtain a
realistic value of the trade cost that fits the import share, a high value of the elasticity
of substitution between intermediate goods is required. A high enough value implies
that the mechanism that generates a home-bias in portfolios in HP’s model no longer
operates, and the result breaks down. For lower elasticities the model generates
a home-bias that is far larger than that observed in the data. We also find some
interesting differences between the effects of trade costs in the production economy
3
of HP compared to the endowment economy of Coeurdacier.
This paper is structured as follows: In section 2 we describe the setup of Heathcote and Perri’s model with the introduction of trade costs in international trade of
intermediate goods. Section 3 provides a brief discussion of HP’s main result and
the intuition behind it. Section 4 then discusses our application of HP’s algorithm to
solve for optimal portfolio holdings with trade costs included in the model. Section 5
provides a discussion of our results, relates them to some of Coeurdacier’s results,
and outlines some directions for future work. Section 6 concludes.
2
The Model
The Heathcote and Perri model is based on the two-country real business cycle model
of Backus, Kehoe and Kydland (1992). We label the two countries H and F . In
each country there is the same measure of identical, infinitely lived households, a
representative intermediate goods producing firm and a final goods producing firm.
The intermediate goods firms in the two countries produce distinct intermediate goods
— country H produces good A and country F produces good B. They use the
same technology, but they are subject to country-specific productivity shocks. The
intermediate goods are traded, but we deviate from HP in that we make trade in
these goods costly. The final goods firms aggregate intermediate goods into a final
good which is used for consumption and investment. This final good is not traded.
Finally, there is international trade in assets. The assets traded are shares in the
intermediate goods producing firms. These firms make investment and employment
decisions and distribute any non-reinvested earnings to shareholders.
In each period t the economy experiences one event st ∈ S. The history of events
from date 0 to t is denoted st = (s0 , s1 , . . . , st ) ∈ S t . The probability at date 0 of any
particular history st is given by π(st ).
The notation used throughout is that x denotes a variable in country H, and x∗
denotes a variable in country F .
2.1
Households
Households derive utility from consumption and leisure. Their period utility function
is given by
n(st )1+φ
c(st )1−γ
−v
U (c(st ), n(st )) =
1−γ
1+φ
4
where c(st ) denotes consumption at date t after history st and n(st ) denotes labor supply. Relative risk aversion is denoted by γ.2 The parameters v ≥ 0 and φ determine
the disutility of labor.
At date 0 households in country H choose λH (st ), λF (st ), c(st ) ≥ 0 and n(st ) ∈ [0, 1]
for all st and for all t ≥ 0 to maximize the present discounted value of utility:
∞ X
X
π(st )β t U (c(st ), n(st ))
t=0 st
subject to their budget constraint. β denotes the households’ discount factor. The
budget constraint is given by (∀t ≥ 0, ∀st ):
c(st ) + P (st )(λH (st ) − λH (st−1 )) + e(st )P ∗ (st )(λF (st ) − λF (st−1 ))
= qa (st )w(st )n(st ) + λH (st−1 )d(st ) + λF (st−1 )e(st )d∗ (st )
(1)
Here P (st ) is the ex-dividend price of shares in the intermediate firm in country
H after history st . P (st ) is in units of the consumption good in country H. P ∗ (st )
is the ex-dividend price of shares in the intermediate firm in country F , in units of
the consumption good in country F . λH (λ∗H ) denotes the fraction that households in
country H (country F ) own of the intermediate firm in country H. λF (λ∗F ) denotes
the fraction that households in country H (country F ) own of the intermediate firm
in country F . d(st ) and d∗ (st ) denote the dividends per share from the intermediate
firms in country H and F respectively. w(st ) denotes the wage in country H in units
of good A, the good produced by the intermediate firm in country H, and qa denotes
the price of good A in units of the consumption good in country H. e denotes the
real exchange rate, the price of the consumption good in country F in terms of the
consumption good in country H.
Similarly, households in country F maximize
∞ X
X
π(st )β t U ∗ (c∗ (st ), n∗ (st ))
t=0 st
subject to their budget constraint (∀t ≥ 0, ∀st ):
1
P (st )(λ∗H (st ) − λ∗H (st−1 ))
t
e(s )
1
= qb∗ (st )w∗ (st )n∗ (st ) + λ∗F (st−1 )d∗ (st ) + λ∗H (st−1 ) t d(st )
e(s )
c∗ (st ) + P ∗ (st )(λ∗F (st ) − λ∗F (st−1 )) +
2
Note that for γ = 1 the consumption term in the utility function reduces to ln[c(st )].
5
(2)
Here w∗ (st ) denotes the wage in country F in units of good B, the good produced
by the intermediate firm in country F , and qb∗ denotes the price of good B in units
of the consumption good in country F .
By assumption, at the start of period 0, households in country H own the entire
intermediate goods firm in country H. Similarly for country F . Hence, λH (s−1 ) = 1,
λF (s−1 ) = 0, λ∗F (s−1 ) = 1 and λ∗H (s−1 ) = 0.
2.1.1
Households’ first order conditions
The first order conditions for households in country H with respect to the stock
purchases λH and λF , are given by
Uc (st )P (st ) = β
X
π(st+1 |st )Uc (st , st+1 )[d(st , st+1 ) + P (st , st+1 )]
(3)
st+1 ∈S
Uc (st )e(st )P ∗ (st ) = β
X
π(st+1 |st )Uc (st , st+1 )e(st , st+1 )[d∗ (st , st+1 ) + P ∗ (st , st+1 )]
st+1 ∈S
t
t
),n(s ))
where Uc (st ) denotes ∂U (c(s
and (st , st+1 ) denotes the history of length t + 1 of
∂c(st )
st followed by st+1 .
The first order condition for households in country H with respect to hours worked
n(st ), is given by
Uc (st )qa (st )w(st ) + Un (st ) ≥ 0
= if n(st ) ≥ 0
(4)
Analogously, the first order conditions for households in country F with respect
to stock purchases λ∗H and λ∗F , are
P (st )
Uc∗ (st )
t
e(s )
"
= β
X
t
π(st+1 |s
)Uc∗ (st , st+1 )
st+1 ∈S
Uc∗ (st )P ∗ (st ) = β
X
d(st , st+1 ) + P (st , st+1 )
e(st , st+1 )
#
(5)
π(st+1 |st )Uc∗ (st , st+1 )[d∗ (st , st+1 ) + P ∗ (st , st+1 )]
st+1 ∈S
and with respect to hours worked n∗ (st )
Uc∗ (st )qb∗ (st )w∗ (st ) + Un∗ (st ) ≥ 0
= if n∗ (st ) ≥ 0
6
(6)
2.2
Intermediate goods firms
Households supply labor to the perfectly competitive intermediate goods firms in the
same country (labor is not mobile across countries). The intermediate goods firm in
country H produces good A using the production function
t
F (z(st ), k(st−1 ), n(st )) = ez(s ) k(st−1 )θ n(st )1−θ ,
(7)
where z(st ) is an exogenous productivity shock, k(st−1 ) is the capital stock at date t
which was determined in the previous period, and θ denotes capital’s share in production.
The firm chooses k(st ) ≥ 0 and n(st ) ≥ 0 for all st and t ≥ 0 to maximize the
value of the stream of dividends from the firm:
∞ X
X
Q(st )d(st )
t=0 st
taking k(s−1 ) as given. Q(st ) is the price the firm uses to value the dividend at st
relative to consumption at date 0.
By assumption the intermediate goods firms in country H use the discount factor
of the household in country H to price dividends. Hence,
Q(st ) =
π(st )β t Uc (st )
.
Uc (s0 )
(8)
The dividend d(st ) is given by
d(st ) = qa (st )[F (z(st ), k(st−1 ), n(st )) − w(st )n(st )]
− [k(st ) − (1 − δ)k(st−1 )]
(9)
where δ is the rate of depreciation of capital. Note that the dividend, capital stock
and investment are given in units of the final consumption good in country H.
The intermediate goods firm in country F produces good B using the same technology as in (7), but it is subject to the productivity shock z ∗ (st ) which is uncorrelated
with z(st ). The vector of shocks [z(st ), z ∗ (st )] evolves stochastically. This firm thus
solves an analogous problem to the firm in country H, where the dividends are priced
using the discount factor of the households in country F . Hence,
Q∗ (st ) =
π(st )β t Uc∗ (st )
.
Uc∗ (s0 )
7
(10)
The dividends in country F are given by
d∗ (st ) = qb∗ (st )[F (z ∗ (st ), k ∗ (st−1 ), n∗ (st )) − w∗ (st )n∗ (st )]
− [k ∗ (st ) − (1 − δ)k ∗ (st−1 )],
2.2.1
(11)
Intermediate goods firms’ first order conditions
The first order conditions with respect to n(st ) for firms in country H and n∗ (st ) for
firms in country F determine wages in each country:
F (z(st ), k(st−1 ), n(st ))
n(st )
(12)
F (z ∗ (st ), k ∗ (st−1 ), n∗ (st ))
n∗ (st )
(13)
w(st ) = (1 − θ)
w∗ (st ) = (1 − θ)
These wages are in terms of the intermediate good produced in the respective country,
so that w(st ) is in units of good A and w∗ (st ) is in units of good B.
The corresponding first order conditions for k(st ) and k ∗ (st ) are
F (z(st , st+1 ), k(st ), n(st , st+1 ))
Q(s , st+1 ) qa (s , st+1 )θ
+1−δ
Q(s ) =
k(st )
st+1 ∈S
"
t
t
X
t
"
∗
t
Q (s ) =
X
∗
t
Q (s , st+1 )
F (z
qb∗ (st , st+1 )θ
st+1 ∈S
2.3
∗
#
(14)
(st , st+1 ), k ∗ (st ), n∗ (st , st+1 ))
+1−δ
k ∗ (st )
(15)
#
Final goods firms
The final goods firms are perfectly competitive and produce the final consumption
good using intermediate goods A and B as inputs. The technology used in production
of the final good in country H is given by
G(a(st ), b(st )) =


a(st )ω b(st )1−ω

t
ωa(s )
σ−1
σ
if σ = 1
t
+ (1 − ω)b(s )
8
σ−1
σ
σ
σ−1
if σ 6= 1
(16)
where a and b denote the quantities demanded of goods A and B in country H
and σ denotes the elasticity of substitution between intermediate goods. Heathcote
and Perri use the parameter ω to fit the import share in each country. ω > 0.5
means that the production of final goods are biased toward the domestically produced
intermediate good. Essentially this means assuming a home-bias in consumption, and
this is crucial for HP’s result. We leave ω in the production function since we will
refer to it in the discussion later on, but in this paper the interesting case to consider
is ω = 0.5 which implies that it plays no role.
The technology used in production of the final good in country F is given by
∗
∗
t
∗
t
G (a (s ), b (s )) =


a∗ (st )1−ω b∗ (st )ω

(1 − ω)a∗ (st )
σ−1
σ
+ ωb∗ (st )
if σ = 1
σ−1
σ
σ
σ−1
(17)
if σ 6= 1
where a∗ and b∗ denote the quantities demanded of goods A and B in country F .
Note that while this expression is different from equation (16), the two functions
are equal for ω = 0.5 when the production technologies are identical in the two
countries.
2.3.1
Final goods firms’ first order conditions
The maximization problems of the final goods firm in country H is given by
max {G(a(st ), b(st )) − qa (st )a(st ) − qb (st )b(st )}
(18)
a(st ),b(st )
subject to a(st ), b(st ) ≥ 0, where qb denotes the price of good B in country H in units
of the final consumption good in country H.
The corresponding problem in country F is given by
{G∗ (a∗ (st ), b∗ (st )) − qa∗ (st )a∗ (st ) − qb∗ (st )b∗ (st )}
max
(19)
a∗ (st ),b∗ (st )
subject to a∗ (st ), b∗ (st ) ≥ 0, where qa∗ denotes the price of good A in country F in
units of the final consumption good in country F .
The first order conditions for these firms can then be written
qa (st ) = ω
G(a(st ),b(st ))
a(st )
1
σ
qb (st ) = (1 − ω)
,
G(a(st ),b(st ))
b(st )
1
σ
,
(20)
qa∗ (st )
= (1 − ω)
G∗ (a∗ (st ),b∗ (st ))
a∗ (st )
1
σ
qb∗ (st )
,
9
=ω
G∗ (a∗ (st ),b∗ (st ))
b∗ (st )
1
σ
.
2.4
International linkages
So far we have considered all prices in country H as denoted in terms of the final
consumption good in country H, and all prices in country F as denoted in terms of
the final consumption good in country F . This follows Heathcote and Perri’s setup
and has the benefit of making the expressions for maximization problems, first-order
conditions etc. completely analogous between the two countries. However, now we
want to think about the links between the two countries and how these links are
affected by making trade in intermediate goods costly.
The costly trade in intermediate goods is modeled in the standard way by introducing iceberg trade costs. That is, to ensure that one unit of a good that is shipped
arrives in the destination country, an amount τ ≥ 1 of the good has to be shipped
from the source country. To derive how this affects the prices denoted in terms of
the final consumption goods (qa , qb , qa∗ , qb∗ ), it is useful to think in terms of nominal
prices for a moment.
Let pa and pb be the nominal prices in some world currency of goods A and B
in country H, and let p∗a and p∗b be the nominal prices in the same currency in
country F . The trade costs and a no-arbitrage argument then implies that prices in
the two countries are related by
τ pa = p∗a and pb = τ p∗b ,
(21)
so that the law of one price does not hold.
Now, given the production function for final goods, the prices of intermediate
goods imply price levels PH in country H and PF in country F . These can be derived
in the usual way from the CES production function, but the expressions for PH and
PF are not relevant to this discussion. The relevant point is the following expressions
for the prices we’re interested in:
qa =
pa
PH
qb =
pb
PH
qa∗
p∗a
PF
qb∗
p∗b
PF
=
=
(22)
Combining equations (21) and (22) we see that we can write the real exchange
rate as a function of the trade costs and the relative price of either of the intermediate
goods in the following way:
e=
PF
qa
1 qb
=τ ∗ =
PH
qa
τ qb∗
10
(23)
2.5
Definition of an equilibrium
An equilibrium in this economy is a set of quantities c(st ), c∗ (st ), n(st ), n∗ (st ), k(st ),
k ∗ (st ), a(st ), a∗ (st ), b(st ), b∗ (st ), λH (st ), λ∗H (st ), λF (st ), λ∗F (st ), a set of prices P (st ),
P ∗ (st ), w(st ), w∗ (st ), Q(st ), Q∗ (st ), qa (st ), qa∗ (st ), qb (st ), qb∗ (st ), productivity shocks
z(st ), z ∗ (st ) and probabilities π(st ) for all t ≥ 0 and st which satisfy the following
conditions:
1. The first-order conditions for intermediate-goods purchases by final-goods firms
(equation (20)).
2. The first-order conditions for labor demand by intermediate-goods firms (equations (12) and (13)).
3. The first-order conditions for labor supply by households (equations (4) and (6)).
4. The first-order conditions for capital accumulation (equations (14) and (15)).
5. The market clearing conditions for intermediate goods A and B:
a(st ) + τ a∗ (st ) = F (z(st ), k(st−1 ), n(st ))
τ b(st ) + b∗ (st ) = F (z ∗ (st ), k ∗ (st−1 ), n∗ (st ))
(24)
6. The market clearing conditions for final goods:
c(st ) + k(st ) − (1 − δ)k(st−1 ) = G(a(st ), b(st ))
c∗ (st ) + k ∗ (st ) − (1 − δ)k ∗ (st−1 ) = G∗ (a∗ (st ), b∗ (st ))
(25)
7. The market clearing conditions for stocks:
λH (st ) + λ∗H (st ) = 1
λF (st ) + λ∗F (st ) = 1
(26)
8. The households’ budget constraints (equations (1) and (2)).
9. The households’ first-order conditions for stock purchases (equations (3) and (5)).
10. The probabilities π(st ) are consistent with the stochastic process for [z(st ), z ∗ (st )].
11
3
Heathcote and Perri’s main result
HP are able to derive an analytical expression for a constant equilibrium portfolio in
their model. Three main assumptions are necessary for this derivation:
• that preferences are logarithmic in consumption (γ = 1),
• that the elasticity of substitution between intermediate goods in the production
of final goods equals unity (Cobb-Douglas production function, σ = 1),
• and that there is a home-bias in preferences that explains the observed volume
of trade relative to GDP. That is, in the model in section 2 τ is fixed at τ = 1
and ω is set to fit the import share.
The expression HP derive for the fraction invested in foreign shares (proposition
1 of their paper) given these assumptions is3
1 − λ = λF (st ) = λ∗H (st ) =
1−ω
1 + θ − 2ωθ
∀t, st
(27)
The equilibrium stock prices are given by
P (st ) = k(st ), P ∗ (st ) = k ∗ (st ) ∀t, st .
(28)
The result is derived by considering the problem of a social planner with equal
weights on households in the two countries. After solving for the planner’s allocation,
HP find a set of candidate prices such that the portfolio in equation (27) decentralizes
that allocation.
To fit an import share of 15%, HP set ω = 0.85. They also set θ = 0.34 and
obtain an optimal portfolio where the fraction invested in foreign stocks is about
20%. Hence, given the key assumptions above, their model can explain a large homebias in portfolios.
3.1
Intuition for the result
The result that a home-bias in portfolios arises in a frictionless model where agents
are optimally hedging the risks they face may seem counterintuitive at first. In
section 3 of their paper HP therefore provide a detailed discussion of the intuition
behind their result both from a macroeconomic general equilibrium perspective, and a
3
See appendix A of Heathcote and Perri (2008) for a proof.
12
microeconomic perspective based on a price-taking individual’s point of view. Below
we briefly summarize this discussion.4
3.1.1
Macroeconomic intuition
Three key equations are helpful for understanding the macroeconomics of how the
portfolios defined in equation (27) deliver perfect risk-sharing. The first equation
follows from the standard condition for perfect international risk-sharing5
PH (st )
Uc (st )
=
.
Uc∗ (st )
PF (st )
When preferences are log-separable in consumption, this reduces to
c(st ) = e(st )c∗ (st )
∀st
(29)
which can be written more compactly as ∆c(st ) = 0, where ∆c(st ) denotes the difference between consumption in countries H and F in units of the consumption good in
country H.
The second equation expresses relative consumption as a function of relative investment and relative GDP. It is derived from the budget constraints (1) and (2) with
constant portfolios. Let λ = λH = λ∗F , then consumption in country H is given by
c(st ) = qa (st )w(st )n(st ) + λd(st ) + (1 − λ)e(st )d∗ (st )
= (1 − θ)y(st ) + λ(θy(st ) − x(st )) + (1 − λ)e(st )(θy ∗ (st ) − x∗ (st ))
where the second equality follows from the expressions for dividends in equations (9)
and (11), the expressions for wages in equations (12) and (13), from writing investment
as x(st ) = k(st ) − (1 − δ)k(st−1 ), and from writing GDP in country H (in units of
the consumption good in country H) as y(st ).
Using a similar expression for consumption in country F , relative consumption
can be expressed as
∆c(st ) = (1 − 2(1 − λ)θ)∆y(st ) + (1 − 2λ)∆x(st )
4
(30)
HP go into more detail, and also relate their work to the earlier literature. They discuss how
their model relates to Lucas (1982), Baxter and Jermann (1997) and Cole and Obstfeld (1991) among
others.
5
That is, the standard expression given that we have assumed that there are no asset market
frictions.
13
With complete home-bias (λ = 1) this expression says that the relative consumption between countries is simply given by the difference between relative output and
investment. With λ < 1 however, some fraction of changes in relative output and
investment are financed by foreign shareholders.
The third key equation is an additional relationship between ∆y(st ), ∆c(st ) and
∆x(st ), which follows from equations (20), (23), and (24) (with τ = σ = 1 as in HP).
Together they imply the following expression for GDP in country H in units of the
consumption good in country H:
y(st ) = qa (st )(a(st ) + a∗ (st )) = qa (st )a(st ) + e(st )qa∗ (st )a∗ (st )
= ωG(st ) + e(st )(1 − ω)G∗ (st )
where G(st ) stands for G(a(st ), b(st )). Similarly, foreign GDP is given by
y(st ) =
1
(1 − ω)G(st ) + ωG∗ (st ).
t
e(s )
Combining the above expressions we get the third key equation
∆y(st ) = (2ω − 1)(G(st ) − e(st )G∗ (st ))
= (2ω − 1)(∆c(st ) + ∆x(st )).
(31)
This equation indicates that changes in relative domestic demand for consumption
or investment changes the relative value of intermediate output. In HP’s setup with
a Cobb-Douglas production function for the final good (σ = 1), countries devote a
constant fraction of expenditure to each intermediate good. Then the size of the
effect on relative GDP is proportional to the change in demand with the constant of
proportionality given by 2ω − 1. With no home-bias in production of the final good
(ω = 0.5), changes in demand do not affect the relative value of output of goods A
and B. For ω > 0.5 the effect is stronger the stronger the preference for home goods is.
Substituting equation (31) into equation (30) gives an expression for relative consumption as a function of relative investment:
∆c(st ) = (1 − 2(1 − λ)θ)(2ω − 1)(∆c(st ) + ∆x(st )) + (1 − 2λ)∆x(st )
which implies that
µ∆c(st ) =
(1 − 2λ)
|
{z
∆x(st ) + (2ω − 1)(1 − 2(1 − λ)θ) ∆x(st ) (32)
}
|
direct foreign financing
{z
}
indirect foreign financing
14
where µ is a constant. The right hand side of (32) is always equal to zero for λ given
by equation (27), which shows that HP’s optimal portfolio delivers the condition for
perfect risk-sharing as implied by equation (29).
One way to think about the role of portfolio diversification is to ensure that the
cost of funding changes in investment is efficiently split between investors in the
two countries. For λ < 1, some of the cost of additional domestic investment is
paid for by foreign shareholders. The direct foreign financing of investment depends
on the difference between the fraction of domestic stock held by foreigners relative
to domestic agents ((1 − λ) − λ). The indirect foreign financing works as follows:
an increase in relative domestic investment increases the relative value of domestic
output in proportion to (2ω − 1) (equation (31)). The fraction of additional output
that goes to domestic households is given by (1 − 2(1 − λ)θ). That is, labor’s share
in income (1 − θ) plus the difference between domestic and foreign shareholder’s
claim to domestic capital income (θ(λ − (1 − λ))). The equilibrium value for λ from
equation (27) makes the direct and indirect effects exactly offset.
Now when most of income goes to labor (θ < 0.5) and preferences are biased
toward domestic goods (ω > 0.5), the indirect effect is positive. The reason is that
the change in the terms of trade that follows an increase in domestic demand favors
domestic agents. Because the relative value of domestic earnings increases, domestic
residents can afford to finance most of the increase in investment. They do so by
holding most of domestic equity (λ > 0.5), and hence, portfolios exhibit a home
bias.
3.1.2
Microeconomic intuition
The optimal portfolio choice of an individual agent depends on how the returns to
domestic and foreign stocks co-vary with non-diversifiable labor income. Two key
features of the BKK environment affect this covariance: durable capital and relative
price dynamics.
The difference between labor earnings in country H and country F (in units of
the consumption good in country H) is given by
qa (st )w(st )n(st ) − e(st )qb∗ (st )w∗ (st )n∗ (st ) = qa (st )(1 − θ)(F (st ) − t(st )F ∗ (st )) (33)
where F (st ) = F (z(st ), k(st−1 ), n(st )), F ∗ (st ) = F (z ∗ (st ), k ∗ (st−1 ), n∗ (st )), and
t)
t(st ) = qqab (s
denotes the terms of trade. The relative value of labor earnings in
(st )
country H thus increases in response to a positive productivity shock if and only if
the increase of production of good A relative to good B exceeds the increase in the
terms of trade. In HP’s economy this is satisfied.
15
Period t returns on stocks in country H and country F (in units of the consumption
good in country H) are given by
r(st ) =
d(st ) + P (st )
,
P (st−1 )
r∗ (st ) =
e(st ) d∗ (st ) + P ∗ (st )
.
e(st−1 )
P ∗ (st−1 )
Combining this with the expressions for dividends (equations (9) and (11)) and stock
prices (equation (28)) implies that these returns can be expressed as
θqa (st )F (st )
+ 1 − δ,
r(s ) =
k(st−1 )
t
e(st )
r (s ) =
e(st−1 )
∗
t
θqb∗ (st )F ∗ (st )
+1−δ .
k ∗ (st−1 )
!
The difference between the aggregate returns to stocks in country H and country F
is then
r(st )P (st−1 ) − r∗ (st )e(st−1 )P (st−1 ) =
θqa (st )[F (st ) − t(st )F (st )] + (1 − δ)[k(st−1 ) − e(st )k ∗ (st−1 )].
(34)
The first term in this equation captures the change in relative income from capital, and clearly it will covary positively with the change in relative earnings in equation (33). However, partial depreciation of capital implies that there is a second
term in the expression for relative returns. Part of the return to buying a stock is the
change in its price, and since a positive productivity shock drives up the real exchange
rate e(st ), it drives down the relative value of undepreciated capital. The overall effect
on the relative returns to stocks depends on which effect dominates. In HP’s model,
the second term dominates, and the covariance between relative labor earnings and
relative returns to shares is therefore negative. Hence, the optimal hedging strategy
for investors is to bias their portfolios toward domestic shares.
3.1.3
Impulse response
Figure (1) shows an impulse response of the economy using HP’s baseline calibration.
That is, the parameters in table (1) and those in line 1 of table (2). We see that a
productivity shock in country H increases relative labor earnings in country H, and
this effect persists over time as labor is immobile internationally. The shock leads to
a jump in the real exchange rate which increases the relative return on foreign stocks
through it’s effect on the value of undepreciated capital (equation (34)). Subsequently
returns on shares are equalized however. Since agents bias their portfolios toward
domestic assets, the increase in the relative return on shares in country F increases
the relative financial wealth of households in country F . This compensates them for
16
0.6
4.6
0.4
0.2
0
Domestic
Foreign
4.4
4.2
0
10
Quarters
4
20
0
0.2
0.18
0.16
0.14
0.12
0
10
Quarters
20
0.4
0.6
0.4
0.2
0
0
0.3
0.2
0.1
0
0
10
Quarters
20
10
Quarters
20
(f) Stock prices
Percentage deviation from ss
0.22
20
0.8
(e) Financial Wealth
Percentage deviation from ss
Percentage deviation from ss
(d) Real Exchange Rate
0.24
10
Quarters
(c) Labor Earnings
Percentage deviation from ss
(b) Stock Returns
4.8
Percent
Percentage deviation from ss
(a) Productivity
0.8
0.4
0.3
0.2
0.1
0
0
10
Quarters
20
Figure 1: Impulse response for HP’s baseline case (table (2), line 1)
the difference in relative labor earnings. Over time as the shock dissipates, relative
labor earnings decline and become negative as the real exchange rate remains above its
steady state level. However, capital accumulation in country H implies that financial
wealth in country H eventually exceeds wealth in country F . Hence, households in
country H are compensated for the expected lower labor earnings in the future. This
mechanism delivers perfect risk-sharing with a home-bias in equity portfolios in HP’s
model.
4
Solving the model with trade costs
With trade costs included in the model, the aim is to set ω = 0.5 and let τ adjust
to fit the import share. For several reasons, the most interesting cases then involve
letting σ and γ deviate from 1. First, the case of σ = 1 and ω = 0.5 corresponds to
Cobb-Douglas preferences with an equal weight on each good. In that case agents’
expenditures on each intermediate good will be the same independent of the value
of τ . To be able to fit the import share using trade costs, σ must therefore, differ
from 1.
17
Second, to fit an import share of 15% the expression for the trade costs is
1−ω
τ=
ω
σ
σ−1
1 − 0.15
0.15
1
σ−1
(35)
where the first term cancels out when ω = 0.5. It is easy to see that to obtain a
realistic value for τ we need a value of σ substantially different from 1. In Coeurdacier
(2008)’s favored parameterization for example, σ = 5 which implies that τ = 1.54.
Third, Coeurdacier argues that with γ = 1 agents are indifferent to fluctuations
in the real exchange rate, and will have no incentive to bias portfolios toward either
domestic or foreign stocks. We will investigate both whether this holds in an economy
with production, and the effect of setting γ > 1.
With σ, γ 6= 1 the model must be solved numerically. HP provide an algorithm
to do so, that they use to consider the robustness of their analytical result. However,
they only consider changing one of the two parameters at a time in a narrow range
around 1, and they do not consider the effect of using trade costs instead of the
home-bias in preferences to fit the import share.
4.1
Algorithm
HP’s algorithm to solve for equilibrium portfolio holdings is based on Schmitt-Grohe
and Uribe (2004). They build on the work of Klein (2000), and provide a method
for computing a second-order approximation to the policy functions around a steady
state using the Symbolic Toolbox in Matlab. The steps in HP’s algorithm are as
follows:6
1. Pick a non-stochastic steady state equilibrium (i.e. one where it is known
that productivities z(st ) and z ∗ (st ) are constant and equal to 0). The firstorder conditions plus symmetry pins down the non-portfolio state variables (i.e.
productivities and capital stocks) and the non-portfolio control variables (i.e.
consumption, hours, investment etc.), but any value of λ0 = λH = λ∗F is a
non-stochastic symmetric steady state equilibrium.7
2. Compute decision rules that characterize the solution to a second-order approximation around the steady state following Schmitt-Grohe and Uribe. Note that
6
See Appendix B in Heathcote and Perri (2008) for a more detailed discussion of the algorithm.
To see this, consider the households’ budget constraints (equations (1) and (2)) when portfolio
holdings are constant, the real exchange rate is 1 (which it must be in a symmetric non-stochastic
steady state), and dividends are equated across countries.
7
18
Parameter Value
β
0.99
Preferences
φ
1
n̄
0.3
v
fits n̄
Technology
θ
0.34
δ
0.025
Productivity ρ
0.91
νε
0.006
Table 1: Calibration of the fixed parameters.
to apply their method it is necessary to add a small adjustment cost for changing portfolios from the current guess for the steady state value. This step yields
decision rules for all variables that are correct up to a second-order approximation, but we do not know whether the initial steady state portfolio λ0 is equal
to the average equilibrium portfolio in the true stochastic economy.
3. Starting from the guess λ0 , simulate the model for a large number of periods,
and compute the average portfolio share along the simulation. If this average
is different from λ0 , update the guess for steady state portfolio holdings to λ1
given by the average simulated share, and return to step 1. Iterate until the
simulated average equals the guess.
4.2
Calibration
The set of parameters in the model is divided into one set that is kept fixed through
all the calibrations, and another set for which various combinations are considered.
We follow HP in setting the values for the fixed set of parameters. The values used
are listed in table (1). Note that the processes for the productivities z(st ) and z ∗ (st )
are written as
z(st ) = ρz(st−1 ) + ε(st ),
z ∗ (st ) = ρz ∗ (st−1 ) + ε∗ (st ).
(36)
where the productivity shocks ε and ε∗ are distributed with mean zero and variance νε2 .
Table (2) contains the various sets of values for the remaining parameters γ, σ,
τ , and ω, for which the model is solved. The table also shows the import share that
the parameters are calibrated to fit, and fraction of the portfolio invested in foreign
shares in equilibrium.
19
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
γ
1
1
1
1
1
1
2
2
2
2
σ
1
1
1.5
1.5
3
3
3
3
5
5
τ
1
any
1
32
1
2.4
1
2.4
1
1.5
ω
0.85
0.50
0.76
0.50
0.64
0.50
0.64
0.50
0.59
0.50
i.s.
0.15
0.5
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
λF = λ∗H
19.7%
any
−57.9%
−57.9%
−716%
−716%
−602%
−602%
no convergence
no convergence
Table 2: Results with either trade costs or home-bias in preferences.
5
Discussion
Table (2) indicates that Obstfeld and Rogoff’s statement that “the effects of home
bias in preferences . . . can be isomorphic to the effects of trade costs“8 applies to the
general case of HP’s model. Whenever σ > 1 the equilibrium portfolio holding is not
affected by whether we choose to fit the import share by assuming a home-bias in
preferences (ω > 0.5) or by introducing trade costs (τ > 1).
In the special case of σ = 1 however, the difference is extreme. In that case
using ω to fit the import share allows HP to derive an analytical expression for
the equilibrium portfolio holding, and this analytical expression yields a value for the
fraction invested in foreign shares that corresponds well with reality. However using τ
to fit the import share is impossible! With ω = 0.5 the expenditure on each good
is the same, so that expenditure on foreign goods equals half of income independent
of τ . Furthermore, in this case changes in demand fall equally on domestic and foreign
intermediate goods. From equation (31) we see that output is equated across countries
(∆y(st ) = 0), which reflects that changes in relative quantities are exactly canceled
out by offsetting changes in the terms of trade, as in Cole and Obstfeld (1991). Stock
returns are then equated across countries, so any portfolio share is consistent with
perfect risk-sharing.
Going back to the general case, there are some interesting points to note. First,
there is the point made by Obstfeld and Rogoff that in order to explain the home-bias
in consumption it is not only the size of trade costs that is relevant. Instead it is
8
Obstfeld and Rogoff (2000), page 348.
20
the combination of trade costs and the elasticity of substitution that determines the
import share. Using a realistic value for trade costs to fit the import share therefore
imposes an additional constraint on how we parameterize the model.
Second, from equation (35) we see that when ω = 0.5 an elasticity of substitution
substantially higher than 1 is needed to obtain a realistic value of the trade cost. This
is most starkly illustrated in line 4 of table (2), where a value of σ = 1.5 implies that
we have to set τ = 32 to fit the import share. Such a high value for trade costs is not
at all plausible. A more realistic example is Coeurdacier’s favored parameterization
which is σ = 5 (and γ = 2) which implies that τ = 1.54. However, lines 9 and 10 in
table (2) show that for those values HP’s algorithm fails to converge to an equilibrium
value for the portfolio share. In fact, it diverges toward a very large foreign bias in
portfolios, and this does not depend on whether ω or τ is set to fit the import share.
Clearly this failure to converge is not simply due to the introduction of trade costs.
HP conduct some sensitivity analysis of their main result by letting first σ and
then γ deviate from 1. When doing so they only plot the portfolio share for values
of σ up to 2.5. They state: “For very high elasticities (for values of σ exceeding 4),
price movements become so small that, following a positive domestic shock, returns
to domestic stocks exceed returns to foreign stocks, and the correlation between relative labor income and relative domestic stock returns turns positive. For such high
elasticities, the two-good model is sufficiently close to the one-good model that its
portfolio implications are similar.”9 The one-good model they refer to is that of Baxter and Jermann (1997) in which the implication is that investors should hedge by
shorting domestic assets — that is, there is a foreign bias in portfolios.
In conclusion it seems that although the effects of trade costs and a home-bias in
preferences may be isomorphic, there is one crucial difference between them: the use
of trade costs rather than the preference parameter imposes an extra constraint on the
elasticity of substitution to be able to use a realistic value for the parameter governing
the frictions in goods trade. As we have seen, a realistic value for τ requires a high
elasticity of substitution. A high elasticity of substitution implies that the special
result derived by Heathcote and Perri breaks down.
5.1
Comparison with Coeurdacier’s results
Having solved for the implications of including trade costs in HP’s model with production, it is interesting to compare our results to the results in Coeurdacier (2008).
There the effect of trade costs on portfolio investment is considered in an endow9
Heathcote and Perri (2008), page 24.
21
1.
2.
γ
1
2
σ
3
3
τ
2.4
2.4
θ∗
1.00
0.89
θ(τ )
0.70
0.70
Table 3: Coeurdacier’s cutoff value for trade costs
ment economy. In Coeurdacier’s derivations, the effect of trade costs depends on the
function
1 − τ 1−σ
θ(τ ) =
.
(37)
1 + τ 1−σ
In particular, trade costs can only generate a home bias in portfolios in his model
when they are large enough to imply that θ(τ ) > θ∗ , where θ∗ is a cutoff value defined
by
!1
2
σ
−
1
.
(38)
θ∗ =
σ − 1/γ
Table (3) contains values for θ(τ ) and the cutoff value θ∗ that arise with the
main parameterizations in table (2). We see that introducing trade costs in HP’s
production economy generates a home-bias in portfolios (although it is too strong to
be realistic) without satisfying Coeurdacier’s condition.
In addition, lines 5 and 6 in table (2) show that in the production economy trade
costs can generate a home-bias in portfolios even when γ = 1. This is counter
to Coeurdacier’s result that for γ = 1 trade costs will have no effect on investor’s
incentives to invest across countries. He states that “a logarithmic investor (γ = 1) is
not affected by fluctuations in the real exchange rate [in an endowment economy]”10 .
In HP’s model however, the presence of physical capital implies that the effect of
real exchange rate fluctuations affect the correlation between stock returns and labor
earnings and gives an incentive to bias portfolios toward domestic shares even with
γ = 1.
5.2
Future work
Some issues remain unresolved and would be interesting to explore further. First,
there is the issue of not obtaining convergence for certain parameterizations of the
model. One option could be to consider whether this issue is particular to the solution
method applied. Other approaches for solving the portfolio problem are suggested
in the literature, for example by Tille and van Wincoop (2007) and Devereux and
10
Coeurdacier (2008), page 10.
22
Sutherland (2006). An interesting exercise would be to compare the robustness of
these approaches to the one proposed by HP.
Second, it is relatively straightforward to reduce the model into a model of an
endowment economy. All that is necessary is to take out the labor supply decision,
and to replace the intermediate goods producing firms with an exogenous endowment
process. With such a model we can apply HP’s algorithm (or one of those mentioned
above) to calculate equilibrium portfolio holdings and compare these to the results of
Coeurdacier (2008) and Obstfeld and Rogoff (2000). This would enable us to isolate
the effect of including production in the economy.11
6
Concluding remarks
The explanation for the home-bias in portfolios puzzle provided by Heathcote and
Perri (2008) relies on the assumption that preferences are biased toward domestically
produced intermediate goods. A home-bias in consumption can be explained by trade
costs, and it is therefore tempting to conclude that the segmentation of both goods
markets and financial markets are due to frictions in goods trade. Furthermore, it
would seem plausible that the decrease in the portfolio puzzle observed over the last
few decades could be connected with the observed decrease in trade costs.
A careful examination of this connection suggests otherwise. Including trade costs
to explain the consumption home-bias in Heathcote and Perri’s model imposes an extra constraint on the parameters. To be consistent with a realistic value of both trade
costs and the observed ratio of the value of imports to GDP a relatively high value for
the elasticity of substitution between intermediate goods is required. However, with a
high elasticity of substitution, the mechanism that generates a portfolio home-bias in
Heathcote and Perri’s model no longer operates. With a lower elasticity, trade costs
and preferences biased toward domestic goods are in some sense isomorphic, but very
high trade costs are necessary to fit the import share, and the resulting portfolio home
bias is far stronger than observed portfolios suggest.
11
We have attempted to do this, but in the model without production there was also some issues
with obtaining convergence. It may therefore be worth applying some of the other approaches to
solving the portfolio problem.
23
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25