Incomplete Markets, Optimal Portfolios, and International
Consumptions Correlations
Adam Gulan
∗
†
Rutgers University This version: November, 2010
Abstract In this paper, I revisit the consumption correlation as well as the BackusSmith puzzles by inspecting the role of nancial markets. Relative to the existing literature, I introduce explicit international trade in stocks and bonds in an otherwise standard model of international business cycles. The results show that markets with symmetric trade in stocks allow for a high degree of risk sharing and closely mimic the ArrowDebreu economy despite being formally incomplete. Risk sharing decreases in asymmetric stock and nominal bond markets, but is still higher than in a single commodity bond economy. The results, therefore, cast doubt on the explanation of the two puzzles based on highly restricted asset trade and large degree of market incompleteness. I also provide empirical evidence that output net of investment and government spending tends to be less correlated across countries than consumption, much less than output itself. This constitutes a new form of the consumption correlation puzzle. The puzzle can be accounted for in the presence of high home bias and low elasticities of substitution between domestic and foreign baskets.
Keywords : risk sharing; international comovements; optimal portfolios JEL Classi cation : E32; F41; F44; G11 ∗I
am very grateful to Roberto Chang, John Landon-Lane, Bruce Mizrach for their advice and support. I thank Andrés
Fernández, Cristina Fuentes-Albero, Gonçalo Pina and the participants of the seminars at ISNE 2010, Rutgers University and the National Bank of Poland for all comments and suggestions. All mistakes and omissions are mine. † Department
of Economics, Rutgers University, 75 Hamilton Street, New Brunswick NJ 08901-1248, USA. E-mail: agu-
[email protected]. 1
1
Introduction
The open economy macroeconomics literature has long attempted to explain some puzzling stylized facts in international comovements which are at odds with the predictions of standard business cycle models, as documented by Backus et al. (1992, 1995). and the BackusSmith puzzle.
Two particular issues are the consumption correlation puzzle
The former, also known as the quantity anomaly, is the observation that
consumption is only weakly correlated across countries, and less so than the respective correlations of output. According to the latter, relative consumption ratios are negatively correlated with the real exchange rate. Equivalently, consumption tends to be relatively high when the price level is relatively high.
Since these
standard models of the business cycle generally assume complete asset markets, the leading strategy in addressing these two puzzles involves signi cantly restricting possibilities of international risk sharing. This is most commonly done by allowing only for very basic international borrowing and lending (with the use of a single riskless commodity bond) or by assuming nancial autarky.
Nevertheless, modern developed
economies which are subject to these puzzles are characterized by sophisticated nancial markets, in which several types of assets are traded and capital controls have been absent for over two decades. Since early 1990s the world economy witnessed a remarkable growth of cross-country asset holdings and gross asset positions, as documented by Lane and Milesi-Ferretti (2001, 2007). In this paper I ask if the incomplete markets explanation of the consumption correlation and Backus Smith puzzles can be reconciled with more realistic modeling of nancial markets. I address this question by means of a two country dynamic stochastic general equilibrium (DSGE) model with incomplete asset markets economies which are hit by several types of shocks. The novelty of my paper is the introduction of a more realistic treatment of asset markets. In particular, I allow for explicit trade in dierent types of stocks and bonds and consider several asset market con gurations. The main nding of the paper is that nancial markets in which stocks of both countries are traded (symmetric trade) allow for similar allocations as in the ArrowDebreu economy, despite being formally incomplete. I use two proxy measures of the degree of eective market completeness which naturally arise from the business cycle models with complete markets. In an economy with no home bias in goods, completeness is measured by international consumption correlations
1
relative to the complete markets case in which those correlations are perfect . In an economy with a bias towards domestic goods, the appropriate measure becomes the relative correlation of consumption ratios with the real exchange rate. If only one country allows for international trade of its stocks (asymmetric trade), the risk sharing capability of the nancial market decreases signi cantly. Economies with bond trade only
1 Additional elements in the speci cation of the model make those correlations slightly, but negligibly less than 1 even for the
complete markets economy.
2
tend to perform similarly to asymmetric stock markets. This result is robust to speci cation modi cations which aect the dynamics of the model otherwise, i.e. set of shocks, changes in the elasticity of substitution, risk aversion, as well as the degree of home bias in goods. The results suggest that stock markets are e cient in eliminating uctuations in relative wealth. Yet, such eects are at the core of the incomplete markets explanations of the two puzzles. For example, in a recent study, Corsetti et al. (2008) show that a two-good model with a home bias, real bond trade and only productivity shocks can account for most of the problematic stylized facts. If elasticities between home and foreign goods are small enough, negative income eect of depreciating terms of trade is larger than the sum of positive substitution eects at home and abroad. This creates a disequilibrium because total quantity of home good supplied increases, whereas the quantity demanded falls. Therefore, in general equilibrium terms of trade have to actually improve after such shock in order for the markets to clear. Yet, this makes the home agent unambiguously better o and the foreign agent worse o. In eect, it breaks the comovement between consumptions and reverses the correlation between relative consumption and the real exchange rate. In my simulations I recover their result. I show, however, that it only holds for an extremely narrow value range of elasticities, and that even for those speci c values the result largely disappears once we allow for trade in more than one non-contingent asset. The hypothesis that market incompleteness may be the solution to the consumption correlation puzzle was rst studied by Kollmann (1991, 1996), who assumed that the nancial market trades with a real bond only in an environment with stationary productivity shocks. Baxter and Crucini (1995) compared the predictions of the real bond economy with the complete asset markets model. They concluded that the degree to which market (in)completeness aects international comovements depends on the nature of shocks. The more persistent and less (internationally) correlated the shocks are, the more market incompleteness hinders international risk sharing.
Arvanitis and Mikkola (1996), and, the forementioned Corsetti et al. (2008),
extended this framework to a two-good case to study the BackusSmith puzzle. Finally, Heathcote and Perri (2002) and Kollmann (2009) argue that the puzzles can be explained by models with full or partial nancial autarky. Several researchers have oered explanations for the two puzzles which do not rely on nancial markets
2
incompleteness . In an in uential paper, Obstfeld and Rogo (2001) argue that an introduction of transportation/shipping costs may solve the BackusSmith anomaly together with other existing puzzles of open
3
economy macroeconomics . They also claim that the quantity anomaly of Backus et al. (1992) should be
2 See Lewis (1996), Stockman and Tesar (1995) and Cole and Obstfeld (1991) for alternative approaches. 3 In their framework nontraded goods may be interpreted simply as goods with prohibitively high transportation costs.
3
modi ed by comparing consumption not with output, but rather with output net of investment and government spending, net output for short. This is because the latter variable constitutes a more precise measure of disposable income. They provide some evidence that international consumption correlations are assuringly larger than net output correlations. In section 2 I conduct a similar empirical exercise and show that their result is not robust and that the anomaly, even in the modi ed form, still persists in the data. I also show that consumption correlations lower than net output can be accounted for in a model with a home bias and
4
low elasticities of substitution . This result is independent of the asset structure of the model. One of the main objectives of this paper is to investigate the relative risk sharing performance of dierent con gurations of the nancial markets with trade of dierent types of assets. I apply the method developed
5
by Devereux and Sutherland (n.d.) and, in paralell, by Tille and van Wincoop (2008) . This method allows for an explicit optimal portfolio choice out of an exogenously given model-speci c set of tradable assets. It is applicable to models solved with perturbation techniques (log-linearization).
Algebraic solutions are
also possible if the model is not too extensive (see Devereux and Sutherland (2008) or Coeurdacier et al. (2010) for applications). The method has two main advantages. Namely, it allows for an introduction of an arbitrary set of assets and is straightforward to implement. Its disadvantage is that it treats the set of assets as exogenously given and is valid only locally. The development of optimal portfolio methodology has motivated several other papers which, in dierent contexts, ask the question about international risk sharing, although none of them addresses the consumption correlation puzzle.
Devereux and Sutherland (2008) develop a model with price stickiness and monetary
policy, in which they analyze a portfolio with two nominal bonds. They nd that an appropriate monetary policy targeted at strict price stability improves the degree of international risk sharing and from that point of view the traditional role of central banks should not be altered. In another recent article, Ghironi et al. (2009) analyze valuation eects in a model of a production economy with monopolistic competition, transaction frictions and technology shocks. In their setup, a trade in claims to pro t doesn't su ce to achieve perfect risk sharing in a decentralized economy. Devereux and Sutherland (2009) analyze the risk sharing properties of three types of portfolios in an asymmetric model of global imbalances. Finally, Coeurdacier et al. (2009) use the methodology to adress another bias in international macroeconomics a home bias in assets. The paper is organized as follows. Section 2 discusses the stylized facts about international comovements, and the quantity anomaly in particular. It revisits the argument of Obstfeld and Rogo and shows that puzzle
4 I also account for the fact that output is more correlated across countries than both consumption and net output. 5 Other work in this area include Evans and Hnatkovska (2005) and Pavlova and Rigobon (2008) who propose a closed-form
solution for asset holdings for a continous time model.
4
still exists, both in its old and the modi ed form. Section 3 describes the two country DSGE model used. Section 4 discusses parameterization, the steady state, as well as the portfolio solution method. Section 5 presents the results of the simulations. Section 6 discusses some caveats, possible extensions and provides nal remarks.
2
Facts about consumption correlations
The importance and mere existence of the consumption correlation puzzle has been challenged by Obstfeld and Rogo (2001). Those authors argue that the puzzle is virtually non-existent, because the comparison is made between a wrong pair of variables. Rather than using the output series, one should, as they argue, use output net of investment and government spending. The rationale behind this is the following. Government spending can be treated by the consumer both as a dead-weight loss if it is not productive, or as a form of public investment which would generate higher output in the future. In either case, it is exogenous from the point of view of households. The optimal level of investment is uniquely pinned down by the real interest rate and the marginal productivity of capital. In consequence, households have only
Y −I −G
at their disposal,
i.e. disposable income net of investment. For convenience, let me call this term net output, to distinguish it from the
Y,
the output itself. Bearing in mind the Obstfeld and Rogo 's argument, I work with
Y −I −G
Y
as well as
variable throughout this paper. Indeed, the statistics they report suggest that for the pairs
of G7 countries consumption is more correlated across countries (0.4 on average) than
Y −I−G
6
(0.17) .
It is con rmed that output itself is still more correlated than consumption. The authors conclude therefore that the consumption correlation puzzle in the original BKK form is ill-founded and that international data provides some evidence for risk sharing. However, the empirical moments provided by Obstfeld and Rogo are based on annual, rather than quarterly data, as it was the case in the original BKK paper, which, as I show below, is largery driving their puzzle dismissing argument. Using annual rather than quarterly data shuts down a large fraction of volatility at business cycles frequencies. To inspect the consequences of working with dierent types of data, I compute cross-correlations of output, net output and consumption using for both annual and quarterly series. I apply three cycle-isolating techniques: log-dierencing of the original series (the method used by Obstfeld and Rogo ), HP- ltering of the variables in logs, as well as the band-pass lter proposed by Baxter and King (1999). For annual data, I set the HP smoothing parameter starts in 1991.
λ = 6.25,
as suggested by Ravn and Uhlig (2002). The time interval chosen
This allows to avoid problems with the treatment of Germany before its uni cation and
is also homogenous in terms of no of restrictions to international capital ows across developed countries.
6 Tables 5 and 6 of the Obstfeld and Rogo (2001) paper. Time interval is 1973-92.
5
BP3 (2, 8) (log(Xt ))
log(Xt ) − log(Xt−1 )
HP (log Xt ), λ = 6.25
ρ(C)
0.112 (0.352)
0.108 (0.361)
0.131 (0.361)
ρ(Y )
0.275 (0.509)
0.291 (0.458)
0.287 (0.505)
ρ(Y − I − G)
0.083 (0.251)
0.085 (0.328)
0.109 (0.258)
ρ(C) − ρ(Y )
-0.163 (-0.157)
-0.183 (-0.097)
-0.156 (-0.144)
ρ(C) − ρ(Y − I − G)
0.029 (0.102)
0.023 (0.033)
0.022 (0.104)
Table 1: Average correlations between G7 countries for annual data.
Source: Penn World Tables v.
6.3,
1991-2007 (1971-2007 in brackets, 1973-2004 for BP lter).
The annual results are reported in table 1.
Most strikingly, average correlations of all variables are very
low, for consumption slightly above 0.1, for output below 0.3. Hence, the BKK stylized fact is con rmed. Most importantly, however, consumption correlations are on average virtually the same as net output correlations, the dierence being a negligible 0.02. The correlation results are largely robust to the ltering technique applied. Hence, the positive result that Obstfeld and Rogo reported is not con rmed in the more
7
recent data interval despite using the same source of data . Turning into quarterly data, the approach taken by BKK, makes things even worse.
Log-dierencing
results in lower correlation numbers than the HP and BP lters, which is in line with the commonly known properties of those lters. The overall picture, however, is clear. Output correlations reach 0.4. Strikingly, net output is now even more correlated across countries than consumption, let alone output itself. Again, comparing output net of investment and government expenditure improves the overall result relative to output itself, but this is clearly insu cient to dismiss the existence of the puzzle. I also make a few robustness checks
8
for the quarterly data . In particular, I repeat the computations for quarterly data using the Gross National Income rather than the GDP, the rationale being that consumption should be a function of income rather
9
than output . I also use the CPI rather than the GDP de ator to de ate the nominal series. Further, I HP lter the GDP series in levels rather than in logs. Finally, I extend the sample back in time, up to 1971, the year when the convertibility of the US dollar was suspended and the Bretton Woods system eectively broke down. None them change the picture qualitatively. Only when the CPI is used instead of the GDP de ator, does the
ρ(C) − ρ(Y − I − G)
dierence slightly exceed zero in the HP cycle. Usually, the dierence is in the
6
BP12 (6, 32) (log(Xt ))
log(Xt ) − log(Xt−1 )
HP (log Xt ), λ = 1600
ρ(C)
0.188 (0.174)
0.045 (0.074)
0.111 (0.137)
ρ(Y )
0.391 (0.464)
0.223 (0.202)
0.395 (0.407)
ρ(Y − I − G)
0.244 (0.367)
0.107 (0.146)
0.234 (0.286)
ρ(C) − ρ(Y )
-0.24 (-0.289)
-0.179 (-0.129)
-0.202 (-0.269)
ρ(C) − ρ(Y − I − G)
-0.056 (-0.183)
-0.062 (-0.072)
-0.041 (-0.149)
Table 2: Average correlations between G7 countries for quarterly data. Source: author's computations based on International Financial Statistics data, 1:1992-2:2005 (1:1971-2:2005 in brackets)
range
(−0.1, −0.05).
Correlations of output itself are always larger than correlations of consumption.
For the sake of completeness, I also report analogous correlations between the 4 major industrialized
10 and the BRIC (Brazil, Russia, India, China) countries using PWT annual data11 . The results
economies
are reported in table 3. On average, both consumption and output correlations are much lower than for both of the previous two tables describing industrialized countries only. Average consumption correlations (including France as well) is 0.21, of consumption 0.05 and of net output, zero. They are also statistically insigni cant. BRIC countries are, among other things, characterized by higher degree of market incompleteness, which is due to several factors. E.g. for Brazil and Russia the default history may play a signi cant role. China, on ther other hand, has highly underdeveloped nancial markets and is still, at least partly, a centrally planned economy. Therefore the dierences between this and the former tables provides another justi cation to the presumption that dierences in international comovements may be explained by the structure of the nancial market and the degree of market incompleteness. Concluding, there seems to be no strong evidence that consumption is more correlated across countries than output net of investment and government spending, although the comparison looks much better than a comparison with output, as was done in the consumption correlations literature. However, the data reported by Obstfeld and Rogo (2001) are not robust to data time range, frequency and ltering techniques. In fact, for quarterly data consumption seems to be always slightly
less
correlated across economies than net output.
7 Admittedly, I use the 6.3 revision of PWT , whereas OR used revision 5.6. 8 The robustness checks are not reported, but are available upon request. 9 In fact, BKK do not make a distinction. They use GDP for some countries and GNP/GNI for others. 10 Germany, Japan, UK and the U.S. 11 The 6.3 PWT data does not include the 2005 International Comparison Program (ICP) revisions of international price data,
which were substantial for countries like China and India. Therefore those numbers should be treated with some caution.
7
Germany Japan UK USA Table 3:
Brazil
China
India
Russia
0.13 (0.29)
-0.19 (-0.05)
0.09 (0.02)
-0.33 (0.19)
0.24 (0.20)
-0.28 (0.04)
-0.15 (0.06)
0.18 (0.58)
0.36 (0.31)
-0.31 (0.19)
-0.06 (0.09)
-0.23 (0.05)
0.22 (0.18)
-0.21 (0.16)
0.22 (0.27)
-0.43 (-0.08)
Correlations of consumption and (output) between industrialized and BRIC countries.
Source:
author's computations based on Penn World Tables v. 6.3, 1973-2004, for Russia since 1993
Therefore, to consider the puzzle as solved, one has to account for these data patterns. Let me now move to the next section which introduces the model and discusses the assumptions about the nancial market.
3
Assumptions of the model
The world consists of two perfectly symmetric countries of equal size: home, and foreign countries is inhabited by a continuum of households indexed by
j,
12 . Each of the
nal goods producing rms indexed by
z,
and capital producing rms.
3.1 Consumers Consumers consume a mix of domestic and foreign goods baskets:
Ctj
γ γ−1 γ−1 γ−1 1 1 γ γ j j γ + (1 − a) CF,t = a γ CH,t
(3.1)
where
j CH,t
ω " 1 Z # ω−1 ω−1 1 ω a j ω = ct (z) dz a 0
and
j CF,t =
"
1 1−a
ω1 Z
1
ω−1 ω ctj (z ∗ ) dz ∗
ω # ω−1
(3.2)
a
It is assumed, as usual, that the elasticity between dierent varieties within the same country is
ω > 1.
ω The higher this elasticity, the lower is the gross markup of a rm ω−1 because the varieties become closer substitutes. that
γ > 0.
γ,
in turn, measures the elasticity between home and foreign baskets of goods and it is assumed
Finally, the parameter
a
measures the degree of openess of the economy. If
preference bias towards domestically produced goods. If
a = 0.5,
there's a
there's no bias and consumers discriminate
between goods only with respect to their relative prices.
12 All variables relating to the foreign country are denoted with an asterisk *.
8
a > 0.5,
Since I assume perfect competition on the labor and capital markets, the households don't exert any monopolistic power and can be therefore treated as average per-capita units - one domestic and one foreign. Households maximize their expected discounted lifetime utility by optimally choosing consumption, labor supply and a portfolio of assets. The instantenous utility of the representative household is given by
U Ctj , Htj =
1−φ 1−σ φ j j −1 Ct 1 − Ht (3.3)
1−σ
Using the CobbDouglas aggregate of consumption and labor enables to treat argument.
U
as a function of a single
This in turn allows to unambiguously implement the Epstein (1983) conditions for the proper
behavior of intertemporal utility in the presence of an endogenous discount factor, because those conditions are provided for single argument functions only
13 . It is also important to note that utility functions which
are not additively separable in consumption and labor do not, in general, generate perfect consumption correlations even in a central planner's problem of allocating a single good. In fact, Devereux et al. (1992) show that for a complete markets model and GHH preferences with
σ = 1
one can obtain consumption
correlations reduced su ciently to match those observed in the data. However, those preferences are tailored speci cally to capture the business cycle properties of developing countries. Importantly, their result doesn't hold for the CobbDouglas aggregate in which the optimal labor supply depends on the level of consumption. As will be shown later in the paper, in the complete markets case consumption correlations are still very high for any feasible parameterization of
σ
and
φ
and in several cases it is almost 1. Therefore, this utility
function allows to continue treating the perfect consumption correlations as a proxy measure for risk sharing and allows to focus on dierent than consumption-labor interactions sources of imperfect risk sharing. Incomplete asset markets, which I investigate, pose a well known problem in open economy macro models. Namely, they may generate a random walk in the key variables of the model, e.g. consumption and net foreign wealth. This not only makes the steady state level of those variables indeterminate, but also precludes computing the unconditional moments of the model, which is essential to the analysis of international comovements. One solution of this problem is to endogenize the subjective discount factor. The idea, due to Uzawa (1968) is that the higher the consumption level, the more impatient home agents become. This mechanism not only induces stationarity in all variables of the models but also allows to partly capture variations and dierences in time preference between countries which aect consumption and savings patterns
14 .
13 Compare e.g. Mendoza (1991). 14 Those variations are, however, negligible in the model. For the benchmark quarterly calibration with real bond trade the
average deviation from the steady state value of the one period subjective discount factor ξ = 0.99 was 0.0000555%, and of a
9
For computational convenience, leisure
1 − Ht 15 .
β
is assumed to depend on the average level of country consumption
Ct
and
The discount factor evolves according to:
h i 1−φ βt+1 = βt ξ Ctφ (1 − Ht ) ,
β0 = 1
(3.4)
where
h i h i−η 1−φ 1−φ ξ Ctφ (1 − Ht ) = 1 + Ctφ (1 − Ht ) ≡ ξt
(3.5)
The home household's budget constraint at t, expressed in terms of the home goods basket, can be written as
Ctj + Gt where
αtn
N N X PH,t X n n + αt ≤ wt Ht + rtn αt−1 Pt n=1 n=1
denotes the holdings of asset
t times the quantity held between periods t t−1
and
t.
Finally,
w
(3.6)
n in terms of the domestic consumption basket,
denotes the domestic wage and
H
is the amount of hours worked. All variables are
real, measured in terms of the domestic consumption bundle. From now on I also de ne be the portfolio of assets held from
t
I introduce government spending
until
Gt
i.e. payo at time
n and t+1. Also, rt is the interest rate on this asset earned between
t + 1.
Wt =
PN
n=1
αtn
to
The composition of the portfolio will be discussed later.
to the model in the form of an exogenous AR(1) process:
¯ + G,t ln Gt = ρG ln Gt−1 + (1 − ρG ) ln G
(3.7)
where
ρG ∈ (0, 1) ,
2 iG,t ∼ N IID 0, σG ∀i
The government consumes only home produced goods. This spending is unproductive and the government has always a balanced budget, so lump-sum taxes are always equal to the expenditure. Introduction of the government serves several purposes. Most obviously, it allows to analyze
Y − I − G,
which is one of the key
objectives of this paper. It also increases the spectrum of shocks, which is a convenient way to generate market incompleteness without unrealistic reduction of the menu of internationally traded assets. Finally, a shock in government spending may be treated as a form of a demand shock. Demand shocks play an important role in many models attempting to explain some of the major puzzles in international macroeconomics (e.g. Stockman and Tesar (1995) or Coeurdacier et al. (2009)). However, contrary to the government spending shock, they are di cult to identify from the data.
similar order for other models. 15 See Schmitt-Grohé and Uribe (2003) for a discount factor without internalization. Although those authors analyze small
open economies models with exogenously given world interest rate only, the problem persists in two-country models as well, even if the interest rate is allowed to deviate from the nonstochastic steady state. Other stationarity inducing techniques for incomplete markets models include portfolio adjustment costs, interest rate premia as well as overlapping generations. 10
Each household optimally chooses consumption, working hours as well as portfolio holdings. The Euler equation becomes
o n 1−φ λt = Et ξ Ctφ (1 − Ht ) λt+1 int+1 where
φ
λt =
Ctj
Ctj
φ
1−
Htj
1−φ 1−σ
/Pt
(3.8)
(3.9)
is the shadow price of a domestic currency unit in terms of utility. International risk sharing condition can be written as:
h h i i ∗ 1−φ λt+1 n φ ∗ φ ∗ 1−φ λt+1 et n Et ξ Ct (1 − Ht ) i = Et ξ (Ct ) (1 − Ht ) i λt t+1 λ∗t et+1 t+1
(3.10)
Finally, the domestic household's labor supply is given by
1−φ = φ
wt 1 − Htj (3.11)
Ctj
which, given a exible labor market, implies that the real wage is equal to the marginal rate of substitution between leisure and consumption.
3.2 Capital producing rms Each country has a market for xed capital, which is owned and produced by perfectly competitive rms. Those rms make investment decisions and use a basket of domestically produced nal goods to produce new capital. Fixed capital, however, is homogenous in each country. Then, the capital is rented to nal goods producing rms at the real rental rate
rtK ,
again measured in terms of the domestic aggregate consumption
basket. The appropriate maximization problem may be summarized as follows:
max
It+i ,Kt+i+1
where
qt+i
is the standard Tobin's
consumption bundle.
Φ( ),
∞ X
PH,t+i K ∆CP t,t+i rt+i Kt+i − It+i Pt+i i=0 It+i −qt+i Kt+i+1 − (1 − δ) Kt+i − Φ Kt+i χt+i Kt+i
L = Et
which satis es
q , i.e.
(3.12)
shadow price of capital as measured in real unit of the composite
By adjusting investment, the rm faces investment costs, captured by the function
I I Φ( K ) = δ , Φ0 ( K )=1
The larger the parameter
and
I Φ00 ( K ) = − ϕδ < 0,
as suggested by Bernanke et al. (1999).
ϕ, the higher the concavity of the adjustment cost function and hence the higher
the cost of changing the amount of investment by
∆It .
11
Introduction of capital adjustment costs serves two purposes. more realistic volatility of investment.
First, it allows the model to generate
Second, it creates a dierence between the returns on investment
and commodity bonds in the benchmark real bond economy with incomplete asset markets. those costs, the price of capital changes over time and becomes another variable in the model,
Because of
qt . χt
is an
investment-speci c technology shock (IST), which follows a stationary AR(1) process. It will be discussed further in subsection 5.3. Finally,
∆CPH t,t+i
is the stochastic discount factor between
t
and
t + i.
Following Devereux and Sutherland
(2009), the domestic capital producing rm uses a discount factor which is a linear combination of the factors of the domestic and foreign households. The linear weights are determined by the ownership structure of the rm:
∆CPH t,t+1 where
=
StCPH
StCP H
h h i ∗ i 1−φ λt+1 φ CPH ∗ φ ∗ 1−φ λt+1 et + 1 − St ξ (Ct ) (1 − Ht ) ξ Ct (1 − Ht ) λt λ∗t et+1
(3.13)
is the percentage of shares of the capital producing rm held by domestic households. In a
world, in which claims to capital producing rms are not traded internationally,
StCP H
becomes
1.
The computational problem which arises here comes from the fact that the discount factor has to be loglinearized around its steady state. However, the steady state value of
StCP H
is unknown until the log-linearized
model is solved and the optimal portfolio structure is found. Therefore I apply an iterative algorithm. I start with 0.5 weights, solve the model and the optimal portfolio structure. Then I log-linearize the model using steady state relative asset ownership obtained in the rst loop. This procedure is repeated until convergence is achieved. I also experiment with a few other initial guesses to ascertain the uniqueness of the convergence
W,
value. If the world is symmetric, the steady state value of
i.e. the gross portfolio holdings, is just equal
16 .
to the total supply of a country's assets (times their price) The derivative with respect to capital yields
qt =
Et ∆CPH t,t+1
K rt+1
+ qt+1
It+i It+i It+i 0 (1 − δ) + Φ −Φ χt+i Kt+i Kt+i Kt+i
(3.14)
and the optimal investment choice is given by
PH,t = q t Φ0 Pt
It Kt
χt
16 If the supply of assets is asymmetric, however, W has to be obtained by a similar iterative procedure.
12
(3.15)
3.3 Final goods producing rms Each country is also inhabited by a continuum of monopolistically competitive rms, every one of them producing a unique good variety
z
of a nal good, using the standard CobbDouglas production technology
yt (z) = At Ktα Ht1−α
(3.16)
with a common log-stationary stochastic process of the total factor productivity
ln At = ρA ln At−1 + (1 − ρA ) ln A¯ + A,t
(3.17)
where
ρA ∈ (0, 1) ,
2 iA,t ∼ N IID 0, σA ∀i
Each rm optimally chooses the amount of capital and labor for production. Since nal goods producing rms are assumed to be monopolistically competitive, each period they also choose the price
pt+i (z)
for the unique good variety that they produce, conditional on the upward-sloping demand function.
The
maximization problem is therefore summed up as:
max Kt+i ,Ht+i ,pt+i (z)
L = Et
∞ X
∆F t,t+i
i=0
pt (z) yt (z) − wt Ht − rtK Kt − mct yt (z) − At Ktα Ht1−α Pt
where
yt (z) = is the demand for variety
z
and
YtW
pt (z) Pt
−ω
PH,t Pt
is the global output.
ω−γ
YtW
(3.18)
(3.19)
The Lagrange multiplier associated with this
problem is at the same time the real marginal cost. The optimality conditions are given by
mct =
w t Ht (1 − α) Yt
(3.20)
rtK Kt αYt
(3.21)
and
mct = The optimal pricing rule is
pt (z) ω = mct Pt ω−1 by which every rm charges a constant markup
ω ω−1
>1
(3.22)
over the marginal cost.
3.4 Financial market The nancial market is assumed to be frictionless. Throughout this paper, I consider several arrangements of the international nancial market, in which dierent sets of assets are traded. Those sets will be exogenously prede ned. One of the innovations is the introduction of two types of rms. The goal here is to be able to
13
analyze the relative, possibly dierent, risk sharing properties of various types stocks. For example, in the model of Ghironi et al. (2007), full risk sharing is not achieved in a market equilibrium (where the number of assets is equal to the number of shocks) because the stock value is limited to the economic pro t of monopolistically competitive rms. Two types of bonds will be considered, real and nominal. All bonds are one-period only. For simplicity, the net supply of nominal bonds is zero: ∗
Btn + Btn = 0 ∀t ∀n ∈ {BH , BF } where
∗
Btn
denotes the quantity of a bond denominated in the currency of country
the other country (e.g. foreign). If then If
B αtB = Bt QB t /Pt = Bt qt
αtB
(3.23)
QB t
n
(e.g. home), held by
is the nominal price (in terms of the domestic currency) of bond at
are the domestic holdings of this bond. Foreign holdings become simply
t
−αtB .
is a negative number, home country has a debt relative to the foreign country in its own currency. A
positive number means that the country is long in bonds of its own currency (and the foreign country is short in bonds of the home country). If we additionally assume that a country issues only debt denominated in its own currency,
αB
has also the interpretation of net debt in that currency.
A riskless real bond is worth one unit of domestic consumption today and pays consumption tomorrow. The quantity purchased is denoted with
bt .
RB rt+1
units of (domestic)
Nominal bonds pay one unit of domestic
17 . They are not riskless since
or foreign currency next period and in that sense they are not state contingent
their real value is aected by monetary shocks. The gross real return (in terms of the domestic consumption basket) on the nominal home currency bond can be expressed as
BH rt+1 =
1/Pt+1 QBH /Pt t
(3.24)
The total supply of any type of equity is positive and equals one: ∗
Stn + Stn = 1 ∀n ∈ {CPH , CPF , FGH , FGF }
(3.25)
Total value of a home capital producing rm are then, for example,
∗ ∗ αtCPH + αtCPH = StCPH + StCPH qtCPH where
qtCPH = QCPH /Pt t
is the real price at
t and αtCPH
∗
are foreign holdings of
(3.26)
CPH
rm. This assumption
may seem to be a redundant complication, and in fact it invalidates the convenient interpretation of a net foreign asset position.
Wt
as
However, it is needed to obtain weights in rms' stochastic discount factors,
17 Note that the bond is riskless only from the point of view of the home country, because the t + 1 real exchange rate is
unknown as of time t. However, this is irrelevant if a=0.5.
14
as discussed in subsection 3.2.
This modi cation also means that the total supply of assets
Wt
changes,
depending on the world asset market con guration. Shares of both types of rms are perpetual, i.e. they can be resold in the next period. They are claims to pro ts generated by rms. For the domestic capital producing rm, the pro t is given by
ΠCPH = rtK Kt − It t
PH,t Pt
and, for the foreign rm by
ΠCPF = rtK∗ Kt∗ − It∗ t
(3.27)
∗ PF,t Pt∗
(3.28)
The returns on claims to those pro ts are given by:
rtCPH =
ΠCPH + qtCPH t CPH qt−1
rtCPF =
and
Note that although both pro ts are expressed in real terms, sumption whereas terms of
home
ΠCPF t
is in terms of aggregate
consumption basket and
rtCPF
foreign
ΠCPF + qtCPF t CPF qt−1
ΠCPH t
(3.29)
is in terms of aggregate
consumption. Therefore,
is the real return in terms of the
rtCPH
foreign
domestic
con-
is the real return in
basket.
The pro ts of monopolistically competitive domestic nal goods producing rms are given by
ΠFGH = z,t where
rtK
pt (z) yt (z) − wt Ht − rtK Kt Pt
is the real rental rate of capital rented at date
are made in advance, at date
t.The
rtFGH =
t
until
t + 1.
(3.30)
Note that payments for rented capital
associated real gross returns are de ned as
ΠFGH + qtFGH t FGH qt−1
and
rtFGF =
ΠFGF + qtFGF t FGF qt−1
(3.31)
3.5 Market clearing conditions and the remaining speci cation The goods market clearing condition states that
∗ Yt = CH,t + CH,t + It + Gt where
CH,t
and
∗ CH,t
(3.32)
denote consumption of domestic goods by domestic and foreign households respec-
tively. Output net of investment and government spending, which is our variable of interest, is simply
NY t = Yt − It − Gt
(3.33)
Demand functions are given by
CH,t = a
PH,t Pt
−γ
Ct
and
CF,t = (1 − a)
15
∗ PF,t et Pt
−γ Ct
(3.34)
where the price indices are
PH,t
1 Z a 1−ω 1 1−ω = (pt (z) dz a 0
and
PF,t
1 = 1−a
Z
1
(p∗t
∗
1−ω
(z ) et )
dz
∗
1 1−ω
(3.35)
a
as well as
1 h 1−γ i 1−γ 1−γ ∗ Pt = aPH,t + (1 − a) PF,t et
(3.36)
Finally, the model is closed by specifying the monetary policy, which is described by the simple quantity theory of money equation with constant volatility
Mt Vt = Pt Yt
(3.37)
Money supply, in turn, follows a simple AR(1) process:
¯ + M,t ln Mt = ρM ln Mt−1 + (1 − ρM ) ln M
(3.38)
where
ρM ∈ (0, 1) ,
2 iM,t ∼ N IID 0, σM ∀i
Recently, the nancial frictions literature has pointed to the fact that a vast majority of debt contracts in countries of low in ation (e.g. the G7 countries) are speci ed in nominal terms, even for very long time horizons, i.e. they are not indexed by in ation. Such arrangement of the nancial market, as ine cient as it can initially appear, may play a signi cant role in the ampli cation of movement of real economic variables at business cycle frequencies (e.g. Iacoviello (2005) or Christiano et al. (2010)). Therefore, it may also be an important element which reduces the degree of international risk sharing. A money supply shock is the simplest way to create ex post return dierentials on such nominal assets. It will be shown later that in a world where only nominal bonds are traded, the degree of risk sharing may in fact be signi cantly decreased.
Before summarizing the whole system note that since they charge the same prices, i.e.
pt (z) =
PH,t and p∗t (z ∗ )
aggregates will be expressed using the 3 variables
ex post
=
PH , PF∗
all producers within a country are equal,
∗ PF,t . Also, in what follows, all price and in ation and
e.
The system consists of the following 32
variables:
∗ ∗ ∗ ∗ ∗ ∗ ∗ Ct , Ct∗ , Yt , Yt∗ , Gt , G∗t , It , It∗ , Kt+1 , Kt+1 , PH,t , PF,t , et , Wt , qt , qt , λt , λt , ξt , ξt , Ht , Ht , mct , mct , wt ,
wt∗ , rtK , rtK
∗
,
At , A∗t , Mt , Mt∗
as well as asset-related variables (describing asset prices, pro ts and returns),
the number of which depends of the nancial market speci cation. The 32 equations are: (3.5), (3.7), (3.9), (3.11), motion of capital, i.e. the constraint of (3.12), (3.14), (3.15), (3.17), (3.20), (3.21), (3.22), (3.32), (3.37), (3.38) and the CobbDouglas production function (3.16),
16
all with their foreign counterparts, as well as the domestic budget constraint equation (3.6) and the risk sharing condition (3.10). In an economy which trades with a single real bond, the home budget constraint (3.6) is replaced by
Ctj + Gt
PH,t + bt = ΠFGH + ΠCPH + wt Ht + bt−1 rtRB t t Pt
In the ArrowDebreu contingent claims economy, the budget constraint and sharing condition holds also
Wt
(3.39) drop out whereas the risk
ex post :
h i h i ∗ 1−φ λt+1 φ 1−φ λt+1 et ξ Ctφ (1 − Ht ) = ξ (Ct∗ ) (1 − Ht∗ ) λt λ∗t et+1
4
(3.40)
Solving the model
4.1 Steady state and calibration 18 , with quarterly frequency. The basic parameters take values standard for
Calibration is based on U.S. data
the international business cycles and new open economy macroeconomics literature (e.g. Backus et al. (1992) or Lubik and Schorfheide (2006)). The remaining parameters (including steady state ratios of the main GDP components and parameters of the AR processes) were computed for the time interval 1970-2008. All series cast in per capita terms and ltered with the Baxter and King (1999) band-pass lter. Following Burns and Mitchell (1946), business cycles are uctuations of frequency between 6 and 32 months. The series for total factor productivity (TFP) has been computed from the Solow residual. Data for (U.S.) hours were taken from the Groningen Growth and Development Centre Total Economy Database and capital series was constructed using the perpetual inventory method. To obtain the initial guess for the capital level, I assumed that capital and GDP grew at the same rate in the pre-sample period of 1950-1959
19 . The parameters for the monetary
process were computed using the Federal Reserve Board seasonally adjusted monthly M2 aggregate. 3-month average per capita was taken. The steady state value of the total factor productivity level
A¯
was rescaled to
0.01, so that the steady state value of contemporaneous utility and the endogenous discount factor function satisfy the Epstein (1983) conditions for any
σ
between 0.1 and 5.
variables are derived using this common level of technology
A.
The steady state levels of the other
The investment cost parameter
so that investment-output volatility ratio be feasible, between 3 and 3.5. Finally, steady state value of the discount factor
ξ
η
ϕ
was chosen
was chosen so that the
equals 0.99. Table 4 summarizes the values of parameters used,
18 In fact, because the model needs to be perfectly symmetric, both countries have the parameters of the U.S. 19 See the appendix of Aguiar and Gopinath (2007) who I follow in this matter, for details.
17
together with their source references.
Table 4: Benchmark calibrations
Parameter Description
Value Reference
α
capital share
0.36
Backus et al. (1992)
ξ¯
discount rate
0.99
Backus et al. (1992) & annualized
δ
depreciation rate
0.025
Backus et al. (1992) & annualized
σ
relative risk aversion
3.84
Lubik and Schorfheide (2006)
γ
elasticity btw. baskets
0.35
Lubik and Schorfheide (2006)
φ
cons. weight in utility
0.296
φ=
ψ
investment cost
0.5
chosen to pin down a feasible
cons. output ratio
0.647
NIPA
inv. output ratio
0.154
NIPA
govt output ratio
0.199
NIPA
r¯K
rental rate of capital
0.035
r¯K =
ω
elasticity btw. varieties
2.504
computed as
ω=
η
EDF parameter
0.139
computed as
ξ η = − ln 1+C¯ φln ¯ 1−φ ) (1−H) (
ρA
TFP persistence
0.909
OECD & Groningen
ρM
M persistence
0.928
FRB
ρG
G persistence
0.935
NIPA
σA
std. dev. of TFP
0.0039
OECD & Groningen
σM
std. dev. of M
0.0023
FRB
σG
std. dev. of G
0.0036
NIPA
¯ C Y¯ I¯ Y¯ ¯ G Y¯
¯ C ¯ w(1− ¯ C+ ¯ H) σI σY
¯ 1−ξ(1−δ) from (3.14) ξ¯ αY¯ ¯ αY¯ −¯ rK K ¯
4.2 Solving for optimal portfolios Linearized rational expectation models are known to have an undesirable property of certainty equivalence. All assets are equivalent not only in the nonstochastic steady state, but also up to
rst order
approximation,
because they have to yield the same expected return. Their properties dier only when we move to higher moments.
Following this presumption, Devereux and Sutherland (n.d.)
the optimal steady state portfolio holdings, which I employ. observation that the
rst order
developed a method to solve for
More speci cally, the method bases on the
dynamics of portfolios aects real variables only up to the
obtaining steady state asset holdings is enough for a
rst order
18
second order,
so
analysis of the other model variables (e.g.
consumption, output), as well as for simulation and impulse response analysis. Asset returns dier only ex post, once the unexpected shocks are realized.
Since the shocks which
determine the ex post dierences in returns are i.i.d. (e.g. white noise), those dierences, or excess returns, are i.i.d. as well. Therefore, to apply the DS method, it is necessary to de ne and compute the ex post excess returns on
N −1
assets using asset
N
as benchmark (where the choice of the benchmark asset is a matter of
convenience). Before the portfolio is found, those excess returns are treated as another shock an unknown linear combination of the other structural shocks of the model. Ex ante, or in expectations, all assets have to yield the same rate of return, but they dier in volatility, which is described by second moments.
Given the functional form of the assumed utility function (3.3),
the optimal portfolio choice must therefore satisfy the two conditions which hold up to the
second order
approximation:
Et
nh i o ∗ x ˆ t+1 − H ˆ∗ Γ Cˆt+1 − Cˆt+1 +Θ H εt+1 rˆt+1 =0 t+1 − Λˆ
(4.1)
and
h i o 1 n x2 ∗ ∗ x x ˆ t+1 + H ˆ t+1 rt+1 + Γ Cˆt+1 + Cˆt+1 +Θ H + Λˆ εt+1 rˆt+1 =0 Et rˆt+1 = − Et Ωˆ 2
(4.2)
where the coe cients are given by
¯ Ω = φC¯ φ(1−σ)−1 1 − H
(1−φ)(1−σ)
Θ = − (1 − θ) (1 − σ) and the
ε¯ = 1
,
Γ = [φ (1 − σ) − 1] Ω
¯ H ¯ Ω, 1−H
Λ = −Ω
term has been supressed.
It proves convenient to solve the linearized model using the method proposed by Sims (2002)
20 which
directly casts the model in the following handy form
yt = Θ1 yt−1 + Φzt + ΘC +
∞ X
Θi−1 S ΘZ Et zt+i
(4.3)
i=1 where
Θ1
is the transition matrix,
Φ
is the shock impact matrix,
ΘC
is the vector of constants and the
nal term is the expected value of all future shocks. The last two two terms drop out in the particular model discussed. Because all variables in this output form depend only on their lagged values, the
D and R matrices
for the DS method are constructed by directly taking the appropriate rows from the impact matrix
Φ.
It is also important to note, that the dierences in returns have to be expressed in the same units. Therefore, one has to convert returns on all foreign assets from foreign consumption bundle units to domestic
20 In general however, the linearized model may be solved using any solution method. See Anderson (2008) for review and
comparison.
19
consumption bundle units. This is simply done by premultiplying those returns by the real exchange rate ratio:
rtn∗
εt−1 εt
(4.4)
where
εt = et
Pt∗ Pt
(4.5)
Log-linearized excess returns on nominal bonds, for example, become
∗ ˆ BH ˆ BF rˆtB,x = rˆtBH − rˆtBF − εˆt−1 + εˆt = −Pˆt + Pˆt∗ + Pˆt−1 − Pˆt−1 −Q ˆt + εˆt−1 t−1 + Qt−1 − ε
where the foreign bond terms known at
5
t−1
BF
is used as benchmark. Note that when constructing the
R
and
will eectively drop because they don't constitute a shock as of time
D
(4.6)
matrices all
t.
Simulation results
The main task is to analyze the risk sharing properties of international nancial markets in which several types of assets are traded, as it is the case with modern developed countries. International business cycle models with complete asset markets provide a natural proxy metric for measuring risk sharing. If there is a single good or at least no home bias in consumption, optimal risk sharing requires that
ξt where
UC
denotes the rst derivative of
UC ∗ ,t+1 UC,t+1 = ξt∗ UC,t UC ∗ ,t U ()
(5.1)
with respect to
C.
If the discount factor was constant
and utility was separable in consumption and labor, this condition would translate into perfectly correlated growth rates of consumption over time.
Although it is not the case in my model, the eects of discount
factor endogeneity and labor inseparability are negligible so the consumption correlations measure can be
21 . In particular, the higher the consumption correlations, the higher the degree of risk sharing
used directly
and the closer is the allocation to the complete markets economy. If there's a bias in preferences towards home goods (a
> 0.5)
the perfect risk sharing prediction changes.
Now the real exchange rate is no longer equal to 1 so the condition becomes
εt ξt UC,t+1 UC∗ ∗ ,t+1 / ∗ = ξt∗ UC,t UC ∗ ,t εt+1
(5.2)
This is the standard BackusSmith condition (see Backus and Smith (1993)), derived also independently by Kollmann (1991). It implies, loosely speaking, that consumption should be relatively high (i.e. marginal
21 This is valid for a CobbDouglas aggregate between consumption and labor. If the utility was GHH, consumption correlations
would drop signi cantly, as showed by Devereux et al. (1992).
20
utility will be relatively low) when it is relatively cheap, i.e. when the real exchange rate is high. Therefore, the risk sharing capacity is measured by the correlation between relative consumption and the exchange rate relative to the ArrowDebreu economy. I consider several arrangements of the international nancial market and look at their impact on the degree of eective market completeness. I compare their performance to the arrangements used in the literature, i.e. ArrowDebreu and real bond economies, which serve as the upper and lower limit benchmarks respectively. The following sets of internationally traded assets are considered:
•
2CP - shares in CP rms of both countries
•
2FG - shares in FG goods producing rms of both countries
•
2NB-2CP - shares in CP rms and nominal bonds
•
6 - shares in CP and FG rms, as well as nominal bonds
•
2NB-CP - both countries allow for bond trade, but only home country allows for CP rm trade
•
NB-CP - home country allows for trade in CP rm, foreign country allows for bond trade
•
NB-FG - home country allows for trade in FG rm, foreign country allows for bond trade
•
2NB - nominal bonds only, both home and foreign
A common feature of the rst 4 markets is the fact that they all allow for a symmetric trade in stocks of dierent kinds. In the next 3 cases stock markets are asymmetric, i.e. only one (home) country allows for an international trade in its stocks of some sort. In the last case only nominal bonds are traded internationally (stock trade is nonexistent). I do not report cases in which there is a symmetric stock trade combined with asymmetric bond trade because, as will be clear below, once symmetric stock trade is allowed for, the results become eectively indistinguishable from a complete markets economy. In all cases the number of independently traded assets is at most equal to the number of shocks in the economy. In general, markets in which only bonds and/or stocks are traded are incomplete in the sense that
22 . Yet, in some cases such markets may still be
not all market allocations are attainable through asset trade
eectively (see e.g.
complete, if trade in those assets is able to achieve all those allocations which are Pareto optimal LeRoy and Werner (2001)).
A classic result due to Rubinstein (1974) shows that under HARA
preferences, one may achieve, up to rst order approximation, eective market completeness as long as the
23 .
number of shocks in the economy is no greater than the number of assets with independent returns
22 This is because the number of possible states of the world is higher (with normally distributed shocks it is in fact in nite)
than the number of independently priced assets. A trade in ArrowDebreu contingent claims is able to generate any allocation. 23 A utility function falls within the HARA class if its ArrowPratt measure of absolute risk aversion is a hyperbolic function:
21
The analysis is done by tracking the behavior of key model variables depending on the parameterization. In the main set of simulations, I keep the relative risk aversion coe cient moments of interest for
γ
elasticity of substitution
varying on the interval
γ
constant and equal to
[0.1; 5]. 0.35
σ
xed at 3.84 and compute the
I also perform an opposite exercise, by keeping the
and varying
σ.
All parameter combinations generate a
deterioration in terms of trade after a positive technology shock, regardless of the nancial market structure. The choice of
σ
and especially
γ
as parameters subject to manipulation has a considerable justi cation.
First, the economic literature is far from reaching a conclusion on what their values are. In practice, several numbers for
σ
are chosen, and the nance literature suggests a higher risk aversion than it is commonly
assumed in macroeconomics.
Intra-temporal elasticity
γ
is known not to be stable.
It tends to be small
24 (below 1) in the short run, but much larger (above 1) in the long run . Also, the values generally found in the trade literature are much larger (in particular, much larger than 1), than in macroeconomic studies. Secondly, the role of consumption elasticites in international risk sharing has been stressed since the seminal contribution of Cole and Obstfeld (1991). Those authors point to the role of terms of trade uctuations in which achieving complete markets allocation. If the elasticity of intra-temporal substitution is equal to 1, terms of trade uctuations exactly oset all endowment risks in an exchange economy, despite nancial autarky. In a production economy, a similar eect is obtained for logarithmic preferences, i.e. for
σ = 1.
Their result shows that one is not able to generate signi cant dierences in allocations for dierent nancial market arrangements for elasticities close to 1. Such mechanism implies that after a positive technology shock terms of trade should depreciate In eect, the correlation between relative output
T oT =
PF PH should be strongly positive. Since the correlation observed in the data is in fact negative, this
casts some doubt on unitary values of
γ
and
σ
parameters. In fact, most studies which attempt to explain the
BackusSmith puzzle and the quantity anomaly, rely on
greater
YH YF and the terms of trade
than 1. In my simulations, I use
γ = 0.35
and
γ
either signi cantly lower than 1, or signi cantly
σ = 3.84
as benchmark values, which were obtained
by Lubik and Schorfheide (2006) in a Bayesian estimation procedure. Finally, the value of these parameters tends to play, in dierent contexts, a crucial role in international macro models' ability to replicate some of the stylized facts. For example, Coeurdacier et al. (2009) analyze their role in generating the home bias in assets. Before commenting on the results, it is important to stress that all simulated data are post- ltered using the Baxter and King (1999) (BK) band-pass lter. Although the model is stationary by construction, the u00 (c)
ARA (c) = − u0 (c) =
1 . ac+b
Note that because of the endogeneity of the subjecive discount factor this condition is not satis ed
in this model except for the case when σ = η + 1. However, the results of the simulation show that this discrepancy doesn't play any role in practice. 24 See e.g. Drozd and Nosal (2008) for a recent analysis of this eect.
22
lter is supposed to pick up only business cycle frequencies (6 to 32 periods for quarterly data),to make the model-generated results comparable with the BK- ltered empirical moments from the data, as summarized in table 2.
5.1 No home bias I start with simulations of the economy with no home bias. The economy is hit by three types of shocks: in total factor productivity (TFP), money supply (M) and government spending (G). 6 assets are therefore traded in total. For the time being, the investment speci c technology shock (IST) is assumed to be zero. Figure 1 presents the consumption correlations for a range of values of the intra-temporal elasticity risk aversion
σ.
For
γ,
γ = 1.5
the results are truncated at
γ
and
for clearer exposition. There is no noticeable
changes for higher values. Consider rst the ArrowDebreu economy (red line). Consumption correlation are persistently very high, almost perfect. For
γ=1
they are exactly equal to 1, which replicates the result
of Cole and Obstfeld (1991). Away from this point the correlations are slightly lower, which is due to the lack of separability between consumption and labor in the utility function as well as the endogeneity of the discount factor. Yet those factors play an almost negligible role. The most important observation is that any portfolio with symmetric trade in stocks generate eectively
regardless
of the parameterizations in gure 1(a) all
correlation lines overlap with the red complete markets line.
This is despite the fact that all but one of
the same results as the complete markets economy,
those markets (i.e. the market with all 6 assets traded) are formally incomplete, i.e. despite the fact that the number of shocks is lower than the number of shocks and despite the fact that the utility function is not HARA for this parameterization. This suggests, to the extent that this result is robust to other sets of frictions, shocks and portfolio adjustment costs, that the complete markets paradigm is a reasonable approximation of asset markets wherever symmetric equity markets are traded. Confront now this result with the real bond economy. If
γ
is less than 1, income of the home country
falls after a postive TFP shock, because demand is relatively inelastic. Home country is worse o, since the substitution eect is to weak to oset the negative income eect of falling terms of trade.
On the other
hand, the foreign economy witnesses only a positive income eect. In eect, consumption starts moving in opposite directions. For
γ
close to 1, the terms of trade hedging mechanism is strong. For
γ
greater than 1,
the postive substitution eect in the home country dominates the negative income eect, so the comovement with the foreign country becomes positive. Because there is no home bias in consumption, total dierences in the consumption baskets between home and foreign countries are relatively very small and it takes very low elasticities to drive a signi cant wedge between them.
23
The result described above breaks down as soon as either of the countries restricts asset trade or when the trade in stocks is absent whatsoever, as shown in gure 1(b). The fact that only one stock type is traded means that only one TFP shock can be hedged. To the extent that the second shock is not correlated with the rst, it remains unhedged. Symmetric bond trade improves risk sharing relative to e.g. NB-FG or NB-CP markets (except for marginally small
γ ),
which closely mimic the real bond economy. Also, adding one more
asset unambiguously improves risk sharing opportunities, as the 2NB-CP portfolios con rms. The degree of risk sharing changes only slightly with the intertemporal risk aversion, as gures 1(c) and 1(d) show. This factor plays a much smaller role than the intra-temporal elasticity peaks at
σ = 1,
the case of intra-temporally additive utility.
Here, risk sharing
Most importantly, the relative risk sharing
performance of all asset portfolios is robust to changes in the parameterization of stocks markets mimic the complete markets economy for any
γ.
σ.
In particular, symmetric
σ.
The analysis of impulse response functions, summed up in gures 5 through 8 and conducted for the benchmark parameterization (γ
= 0.35
and
symmetric stocks markets ( gures 5 and 6).
σ = 3.84),
tells, in general, a similar story.
First, consider
In most cases they mimic the complete markets model very
closely, especially in the responses of output, investment and in ation (recall that prices are exible in the model). The dierences between portfolios start to emerge in consumption, exchange rate and net exports. The 6 assets model (green diamonds) tells exactly the same story as the complete markets model (red line). The eect of Rubinstein (1974) is still present in practice despite the aforementioned caveats about the consistency of the utility function with the HARA class. All other dierences in responses are in terms of the size of the initial deviation, rather than the shapes of the transition paths. Interestingly, they do not necessarily fall between the real bond (black line) and the complete markets case. Consider the reaction to a
1% increase of government spending.
On impact, consumption falls in the 2CP, 2FG and 2NB-2CP economies
by more than in the CM case and remains lower for at least 20 quarters ( gure 5(d)).
In the real bond
economy, on the other hand, the initial response is actually positive and consumption falls below the steady state value only after 10 quarters. In consequence, the drop in net exports is lower in those economies than in the CM case and they recover balanced trade faster than CM, let alone the real bond economy ( gure 6(f )). Accordingly, all economies witness a nominal appreciation. In sum, the only dierences between the CM and symmetric asset economies emerge after government spending shocks. Impulse responses are virtually the same as in complete markets after technological and monetary shocks. In the case of asymmetric and bond portfolios the dierences are larger ( gures 7 and 8), only the behavior of in ation and investment remains similar in all asset variants. The responses of consumption get quite dierent, and they do not necessarily fall between the real bond and the complete markets cases. This is especially the case for the government spending and the monetary shock. In fact, for symmetric bond markets (2NB and 2NB-CP) which tend to
24
25
γ
and
σ , a = 0.5.
(d) Asymmetric and bond trade
(b) Asymmetric and bond trade
Figure 1: Consumption correlations under dierent
(c) Symmetric stocks trade
(a) Symmetric stocks trade
generate similar responses, the increase in consumption is even greater (after a government spending shock) than in the real economy case. NB-CP and NB-FG economies tend to behave similarly not only to each other but also to the real bond economy. Especially in the former case, there is no dierence. In sum, the model's behavior depends on whether there's symmetricity in the asset markets. Symmetric stock markets go a long way towards mimicking the ArrowDebreu economy. On the other hand, nominal bond economies tend to display some idiosyncratic characteristics, not necessarily similar to the real bond or CM benchmarks. Next, I ask whether the model is able to capture the stylized facts described in section 2, i.e. the fact that net output tends to be less correlated across countries than output itelf, and the fact that it may also be more correlated across countries than consumption. Figure 9 shows the cross-country correlations of consumption (green line), now combined with the respective data for
Y
(blue line) and
Y −I −G
(red line). I restrict
myself to reporting only symmetric asset market con gurations, which seems to be reasonable in the case of G7 country pairs
25 . Capital ows among developed economies have been eectively free since at latest,
the beginning of the great moderation period. Nevertheless, I also show the 2NB case, because the data shows that cross-country bond holdings are on average larger than holdings of stocks
26 . Asymmetric market
simulations would in general be useful in analyzing risk sharing between developed and developing countries where factors like capital ow restrictions, original sin and the like may eectively eliminate trade in some types of assets. It is clear that the model fails miserably, regardless of the asset market con guration. This is for two reasons. First, whenever symmetric trade in stocks takes place, consumption correlations are very high and closely resemble the ArrowDebreu economy. Only the 2NB and real bond markets are able, and only for very low
γ,
to generate su ently low consumption correlations.
parameterizations for which
Y −I −G
is more correlated than
Y
But these are exactly the same
itself, which is counterfactual.
5.2 Home bias Consider now a model in which consumption is biased towards home produced goods. This is an alternative to introducing explicitly non-tradable goods.
As argued by Obstfeld and Rogo (2001) the bias not only
expresses dierences in tastes, but mainly objective factors which restrict international trade in goods like transportation costs
27 . I follow Lubik and Schorfheide (2006) and set
a = 0.93,
which is a Bayesian estimate
and corresponds to their model with perfect pass-through. Before asking if the modi ed model replicates the empirical moments, i.e. the new form of the consumption
25 Results for asymmetric markets are available upon request. See Devereux and Sutherland (2009) for another example of
asymmetric markets. 26 Compare Lane and Milesi-Ferretti (2007). The average cross-country and cross-time proportion is approximately 2:1. 27 For iceberg transportation costs, those two eects are in fact isomorphic in a CES aggregate.
26
correlation puzzle, I check if the risk sharing properties of portfolios have changed. Recall that the risk sharing criterion is now measured by the correlation of the relative consumption with the real exchange rate, as in equation 5.2. Figure 2 presents the results for varying intra-temporal elasticity the results on a single graph.
γ.
For compactness, I present
It is clear from this picture that eectively full risk sharing is achieved for
most parameter values, regardless of the portfolio (all lines overlap with the red line for complete markets). The most important exception is the the point around which correlations fall signi cantly. This is the point derived by Corsetti et al. (2008) and given by the formula
γ˜ = which gives the breakpoint value of positive domestic output shock. For
γ˜ = 0.46 γ
for
2a − 1 2a
a = 0.93.
(5.3)
Consider the reaction of both economies after a
above the threshold terms of trade depreciate, which is the standard
case. Four eects are at work here: the positive domestic substitution eect, negative domestic home eect as well as two foreign eects, which are both positive. weak to dominate the negative home income eect.
For
γ < 1,
the domestic substitution eect is too
In result, home consumption of the home good falls.
Yet, foreign consumption of home good grows enough to make up for this fall, i.e. Yet, if
γ
gets very low (i.e.
global demand grows.
is below the threshold), the negative home income eect is so strong, that it
dominates all positive eects, home and foreign. In result, global demand for the home good falls. Hence a deterioration of the terms of trade would imply that quantity supplied goes up, but the quantity demanded falls, which would prevent markets from clearing. Therefore, for those very low elasticities terms of trade have to improve, rather than deteriorate. Corsetti et al. (2008) derive this theoretical result in the context of an endowment economy. for nancial autarky as well as in a real bond economy.
It holds
Further, they calibrate an international business
cycle model with production, capital accumulation and single bond trade. They are able to replicate several stylized facts in international comovements. This includes the consumption correlation puzzle, BackusSmith puzzle as well as the international comovements puzzle, i.e. the strong international positive comovement of investment and labor (see Baxter (1995)). The result relies on generating strong relative wealth eects. Yet, gure 2 shows that strong relative wealth eects can arise only in a very close proximity of the point given by equation 5.3. As
γ
drops further, then although terms of trade still appreciate, the strength of the
reaction quickly dies out. In eect, the key mechanism which breaks the comovement of consumptions and the real exchange rates (which comove with the terms of trade) disappears. However, the most important result are the dierences between dierent nancial markets. In particular, whenever more than one asset is traded, the drop around
γ˜
largely disappears. The eect disappears entirely for a portfolio with symmetric
stock trade, here represented by the 2CP case (the violet line fully overlaps with the red line, even around
27
Figure 2: Correlations of relative consumption with the real exchange rate under dierent
γ , a = 0.93
γ˜ ). Next, I turn to the problem of cross country comovements. All simulations, summed up in gure 10, show a very dierent picture than for the model without a home goods consumption bias ( gure 9). First, note
28 . Output gets
that the correlations of output and net output do not change with asset market con gurations
more correlated internationally than net output for risk aversion parameter above, approximately, Consumption correlations are lower than correlations of output, unless risk aversion
σ
σ = 1.4.
is low (below 1.5 for
economies with symmetric stock trade). Also, any version of the model with symmetric international stock markets is able to generate consumption correlations which are similar or slightly lower than net output
29 . There is somewhat less consistency with the data for benchmark parametrization. According
correlations
to the model with symmetric stock markets, consumption is somewhat more correlated than net output. However, simulations of
σ
suggest that a slightly lower risk aversion should make the match very close to
actual numbers. Correlations of output, on the other hand tend to be too low in the model relative to the data by around 10 percentage points, i.e. the dierences
ρ (Y ) − ρ (C)
and
ρ (Y ) − ρ (Y − I − G)
should be
28 This is also the case for assymetric markets not reported in gure 10. 29 The results of the moments simulations generated by models with the bias are sensitive to the choice of a. In particular,
international consumption correlations rise very quickly as a drops. For example, setting a = 0.85 generates counterfactually high correlations of consumption.
28
larger. Yet, the results are satisfactory given that the model doesn't assume correlations between shocks.
5.3 Risk sharing with investment speci c shocks In this subsection I ask if the risk sharing properties of dierent portfolios are robust to a change in the set of shocks that hit the economy. First, I drop the government spending and the monetary shocks. As we have seen in previous subsections, these shocks are not able to reduce risk sharing in models with symmetric trade in stocks. However, these shocks do not explicitly enter any of the optimal behavior rules (Euler equations of the consumer and optimal investment equations of the capital producing rm). The only way in which the government spending shock can potentially aect optimal economic decisions is by changing relative prices of home versus foreign goods because of the full bias in government spending. Otherwise, it just constitutes a random loss of the households' disposable income (growth in domestic output). Monetary shock, on the other hand, is not able to generate signi cant real eects without introducing price stickiness. Therefore, I replace the G and M shocks with an investment-speci c technology shock (IST). The importance of the shock has been stressed by several studies, e.g. Fisher (2006) or Greenwood et al. (1997). and has been used in dierent contexts (e.g. Justiniano et al. (2010) or Coeurdacier et al. (2010)). The main appeal of the shock is that it is a type of a demand side shock, which to some extent works in a similar way as a taste shock. The advantage of the IST shock, relative to a taste/preference shock is that the former is much better identi able in the data. Secondly, models driven by supply side shocks only have long been known to have problems in replicating many of the business cycle stylized facts. Additionally, the IST shock may drive a dierence between risk sharing properties of dierent types of stocks, because it constitutes a pro t for one type of rm, but a loss for the other type of rm at the same time. For the purpose of this exercise, I loosely follow Rao (2010)
30 and set the IST innovation variance 6 times higher than the TFP variance.
The persistence and spillovers are the same as in the TFP shock, equal to
0.909
and
0.0096
respectively. IST
innovations are assumed to be uncorrelated with the TFP shocks. Consider rst the model with no home bias. varying elasticity of intra-temporal substitution
γ
For brevity of exposition, I report only the results for and for a choice of portfolios. The results are presented
in gure 3. It is apparent that the main results remain unchanged. It is worth to mention the case of the 2CP economy. It is an incomplete markets case, because there is only 2 assets and 4 sources of uncertainty. The presumption that IST shock might have an impact on the risk sharing properties, relative to G and M shocks, is right. In particular, for low
γ
consumption correlations are lower than in the complete markets
case. Yet the dierences are much too small to account for the consumption anomaly that we observe in the
30 He sets the parameters of the shock so that he achieves optimal matching of the business cycle moments.
29
Figure 3: Consumption correlations under dierent
γ
in a model with IST and TFP shocks.
data. Finally, gure 4 shows the correlations of relative consumption with the real exchange rate and con rms that similar conclusion holds for the model with a home bias in consumption.
6
Conclusions
This paper revisits the argument that the consumption correlations and BackusSmith puzzles are generated by incompleteness of international asset markets.
This line of research has stressed that incompleteness
may generate substantial dierences in relative wealth and therefore break the international comovements of consumption as well as the correlation between relative consumption and the real exchange rate.
In
that literature, incompleteness is generated by either allowing for trade in a real bond only, or by assuming nancial autarky.
In this paper, I show that once one allows for more realistically modeled international
nancial markets, the relative wealth eects eectively disappear. To this end, I apply the newly developed methods of Devereux and Sutherland (n.d.) for solving for optimal portfolios in the DSGE framework. The key result of this paper is that once both countries allow for trade in their stocks, market allocations mimic the ArrowDebreu economy very closely, despite formal market incompleteness. Yet, modern nancial markets are characterized by free capital ows and international cross holdings of assets have increased several times
30
Figure 4: Correlations of relative consumption with the real exchange rate under dierent IST and TFP shocks,
γ,
in a model with
a = 0.93.
over the last two decades. Therefore, my result casts some doubt on this line of argument. Following the argument of Obstfeld and Rogo (2001) I also compare the international comovements of consumption with those of output net of investment and government spending, or net output. I show that their result is not robust and that consumption is no more correlated than net output for the G7 group of developed countries.
The standard model with complete asset markets, similar to Backus et al. (1992)
and no home bias in consumption is not able to replicate those facts, which constitutes a new form of the consumption correlation puzzle. The puzzle can be largely accounted for once a large bias in consumption is introduced and as long as the elasticity of intra-temporal substitution is su ciently low.
References Aguiar, Mark and Gita Gopinath, Emerging Market Business Cycles: of Political Economy, 2007, 115 (1), 69102.
31
The Cycle Is the Trend,
Journal
Anderson, Gary,
Solving Linear Rational Expectations Models:
A Horse Race,
Computational Eco-
nomics, March 2008, 31 (2), 95113.
Arvanitis, Athanasios V. and Anne Mikkola, namics,
Asset-Market Structure and International Trade Dy-
The American Economic Review, 1996, 86 (2), pp. 6770.
Backus, David K. and Gregor W. Smith, Consumption and real exchange rates in dynamic economies with non-traded goods,
Journal of International Economics, November 1993, 35 (3-4), 297316.
, Patrick J. Kehoe, and Finn E. Kydland, International Real Business Cycles, Journal of Political Economy, August 1992, 100 (4), 74575.
,
, and
, International Real Business Cycles: Theory and Evidence, in Cooley Thomas F., ed.,
Frontiers of Business Cycle Research, Prin, 1995, chapter 11, pp. 331356.
Baxter, Marianne,
International trade and business cycles, in G. M. Grossman and K. Rogo, eds.,
Handbook of International Economics,
Vol. 3 of
Handbook of International Economics,
Elsevier, January
1995, chapter 35, pp. 18011864.
and Mario J. Crucini,
Business Cycles and the Asset Structure of Foreign Trade,
International
Economic Review, 1995, 36 (4), 821854.
and Robert G. King, Measuring Business Cycles: Series,
Approximate Band-Pass Filters for Economic Time
The Review of Economics and Statistics, 1999, 81 (4), 575593.
Bernanke, Ben S., Mark Gertler, and Simon Gilchrist, The nancial accelerator in a quantitative business cycle framework, in J. B. Taylor and M. Woodford, eds.,
Handbook of Macroeconomics, Vol. 1 of
Handbook of Macroeconomics, Elsevier, 1999, chapter 21, pp. 13411393.
Burns, Arthur F. and Wesley C. Mitchell, Measuring Business Cycles NBER Books, National Bureau of Economic Research, Inc, 1946.
Christiano, Lawrence, Roberto Motto, and Massimo Rostagno,
Financial Factors in Economic
Fluctuations, Technical Report 1192, European Central Bank May 2010.
Coeurdacier, Nicolas, Robert Kollmann, and Philippe Martin, International Portfolios with Supply, Demand and Redistributive Shocks, in Richard Clarida and Francesco Giavazzi, eds.,
NBER International
Seminar on Macroeconomics 2007, University of Chicago Press January 2009, pp. 231263.
32
,
, and
, International portfolios, capital accumulation and foreign assets dynamics,
International Economics, 2010, 80 (1), 100112.
Journal of
Special Issue: JIE Special Issue on International Macro-
Finance.
Cole, Harold L. and Maurice Obstfeld, Commodity trade and international risk sharing : do nancial markets matter?,
How much
Journal of Monetary Economics, August 1991, 28 (1), 324.
Corsetti, Giancarlo, Luca Dedola, and Sylvain Leduc, International Risk Sharing and the Transmission of Productivity Shocks,
The Review of Economic Studies, 2008, 75 (2), pp. 443473.
Devereux, Michael B., Allan W. Gregory, and Gregor W. Smith, Realistic cross-country consumption correlations in a two-country, equilibrium, business cycle model,
Journal of International Money and
Finance, February 1992, 11 (1), 316.
and Alan Sutherland, Financial globalization and monetary policy, Journal of Monetary Economics, November 2008,
and 2009,
58 (8), 13631375.
, A portfolio model of capital ows to emerging markets,
89 (2), 181193.
and ciation.
Journal of Development Economics,
New Approaches to Financial Globalization.
, Country Portfolios in Open Economy Macro Models,
Journal of the European Economic Asso-
forthcoming.
Drozd, Lukasz A. and Jaromir B. Nosal, Long Run Price Elasticity of Trade and the Trade-Comovement Puzzle, Working Paper 2008.
Epstein, Larry G.,
Stationary Cardinal Utility and Optimal Growth under Uncertainty,
Journal of
Economic Theory, 1983, 31, 133152.
Evans, Martin D. D. and Viktoria Hnatkovska, Solving General Equilibrium Models with Incomplete Markets and Many Assets, NBER Technical Working Papers 0318, National Bureau of Economic Research, Inc October 2005.
Fisher, Jonas D. M., The Dynamic Eects of Neutral and Investment-Speci c Technology Shocks, Journal of Political Economy, June 2006, 114 (3), 413451.
Ghironi, Fabio, Jaewoo Lee, and Alessandro Rebucci, The Valuation Channel of External Adjustment, NBER Working Papers 12937, National Bureau of Economic Research, Inc February 2007.
,
, and
, The Valuation Channel of External Adjustment, October 2009. working paper.
33
Greenwood, Jeremy, Zvi Hercowitz, and Per Krusell, Long-Run Implications of Investment-Speci c Technological Change,
The American Economic Review, 1997, 87 (3), pp. 342362.
Heathcote, Jonathan and Fabrizio Perri, Financial autarky and international business cycles, Journal of Monetary Economics, 2002, 49 (3), 601 627.
Iacoviello, Matteo,
House Prices, Borrowing Constraints, and Monetary Policy in the Business Cycle,
The American Economic Review, 2005, 95 (3), 739764.
Justiniano, Alejandro, Giorgio E. Primiceri, and Andrea Tambalotti, Investment shocks and business cycles,
Journal of Monetary Economics, 2010, 57 (2), 132 145.
Kollmann, Robert,
Essays on International Business Cycles. PhD dissertation, University of Chicago
1991.
, Incomplete asset markets and the cross-country consumption correlation puzzle,
Journal of Economic
Dynamics and Control, 1996, 20 (5), 945 961. , Limited Asset Market Participation and the Consumption-Real Exchange Rate Anomaly, CEPR Discussion Papers 7452, C.E.P.R. Discussion Papers September 2009.
Lane, Philip R. and Gian Maria Milesi-Ferretti, The external wealth of nations: assets and liabilities for industrial and developing countries,
measures of foreign
Journal of International Economics,
2001,
55 (2), 263 294.
and
, The external wealth of nations mark II: Revised and extended estimates of foreign assets and
liabilities, 1970-2004,
Journal of International Economics, 2007, 73 (2), 223 250.
LeRoy, Stephen F. and Jan Werner, Principles of Financial Economics, Cambridge University Press, 2001.
Lewis, Karen K,
What Can Explain the Apparent Lack of International Consumption Risk Sharing?,
Journal of Political Economy, April 1996, 104 (2), 26797.
Lubik, Thomas and Frank Schorfheide, A Bayesian Look at New Open Economy Macroeconomics, in Mark Gertler and Kenneth Rogo, eds.,
NBER Macroeconomics Annual 2005, Vol. 20, MIT Press, April
2006, chapter 5, pp. 313366 (382).
Mendoza, Enrique G, September 1991,
Real Business Cycles in a Small Open Economy,
81 (4), 797818.
34
American Economic Review,
Obstfeld, Maurice and Kenneth Rogo, The Six Major Puzzles in International Macroeconomics: There a Common Cause?, in Ben S. Bernanke and Kenneth Rogo, eds.,
Is
NBER Macroeconomics Annual
2000, Vol. 15, MIT Press, January 2001, chapter 6, pp. 339412.
Pavlova, Anna and Roberto Rigobon, Equilibrium Portfolios and External Adjustment under Incomplete Markets, March 2008. working paper.
Rao, Andrea,
Technology Shocks:
Novel Implications for International Business Cycles, Discussion
Paper 7980, Centre for Economic Policy Research 2010.
Ravn, Morten O. and Harald Uhlig, Observations,
On Adjusting the Hodrick-Prescott Filter for the Frequency of
The Review of Economics and Statistics, 2002, 84 (2), 371376.
Rubinstein, Mark, An aggregation theorem for securities markets, Journal of Financial Economics, 1974, 1 (3), 225 244.
Schmitt-Grohé, Stephanie and Martín Uribe, Closing small open economy models, Journal of International Economics, 2003, 61 (1), 163 185.
Sims, Christopher A, Solving Linear Rational Expectations Models, Computational Economics, October 2002,
20 (1-2), 120.
Stockman, Alan C. and Linda L. Tesar, Tastes and Technology in a Two-Country Model of the Business Cycle: Explaining International Comovements,
Tille, Cédric and Eric van Wincoop,
The American Economic Review, 1995, 85 (1), 168185.
International Capital Flows, CEPR Discussion Papers 6705,
C.E.P.R. Discussion Papers February 2008.
Uzawa, Hirofumi, Time Preference, the Consumption Function, and Optimum Asset Holdings, in James N Wolfe, ed.,
Value, capital and Growth. Papers in honour of Sir John Hicks,
Edinburgh University Press,
Edinburgh, 1968, pp. 485504.
Appendices A
The linearized system
For the economy trading with nominal bonds and/or stocks, the log-linearized counterparts of the 28 model equations are: (A.1), (A.3), (A.4), (A.5), (A.6), (A.7), (A.12), (A.13), (A.14), (A.15), (A.16), (A.17), (A.18),
35
(A.19), (A.20), (A.29), (A.30), (A.31), (A.32), (A.33), (A.34), (A.35), (A.36), (A.37), (A.38), (A.39), (A.40) and (A.41). In the economy which trades with one real bond only,
ˆ W
is replaced by
ˆb,
(A.2). In the ArrowDebreu contingent claims economy, (A.1) and the variable
so that (A.1) is replaced by
ˆ W
drop out, and the arbitrage
equation (A.20) is replaced by (A.21). Note that because the steady state value of net exports log-linearized version is de ned as a deviation from the steady state value of output itself: All price and in ation aggregates are expressed using just the 3 variables
PˆH , PˆF∗
and
¯ = 0, NX
dt = NX
eˆ,
its
NX t . Y¯
so only they are
counted as variables. The system with nominal bonds and/or stocks consists of the following 28 variables:
ˆ , PˆH , Pˆ ∗ , eˆ, w, ˆ H ˆ ∗ , C, ˆ Cˆ ∗ , K, ˆ K ˆ ∗ , I, ˆ Iˆ∗ , qˆ, qˆ∗ , rˆK , rˆK ∗ , A, ˆ Aˆ∗ , Yˆ , Yˆ ∗ , mc ˆ,M ˆ ∗, G ˆ W ˆ w ˆ ∗ , H, ˆ , mc ˆ ∗, M F
and
ˆ∗. G
A.1 Home country equations, cast in Sims's form The budget constraint is expressed as
w ¯ ¯ ¯ ¯ ¯ ¯H ∗ ˆ t + G (1 − a) PˆH,t − PˆF,t ˆt ˆ t − W rˆtN + C Cˆt + G G − e ˆ − w ˆ + H W t t Y¯ Y¯ Y¯ Y¯ ξ¯Y¯ N −1 1 ˆ 1 X n n = ¯W α ¯ rˆt − rˆtN t−1 + ¯ ¯ ξ ξY
(A.1)
n=1
where the second term on the RHS is the real value of realized excess returns (relative to asset
N ).
In the
real bond economy, the constraint is simply
C¯ ∗ − eˆt + −Yˆt − ¯ (1 − a) PˆH,t − PˆF,t Y where
ˆbt = bt /Y¯
C¯ ˆ Ct + Y¯
I¯ ˆ It + Y¯
¯ G ˆ t + ˆbt = 1 ˆbt−1 G ¯ Y ξ¯
(A.2)
and the government expenditure evolves as
ˆ t = ρG G ˆ t−1 + G,t G
(A.3)
Market clearing
C¯ C¯ Yˆt − a ¯ Cˆt − (1 − a) ¯ Cˆt∗ − Y Y
I¯ ˆ It − Y¯
i ¯ ¯h G ∗ ˆ t + 2a (1 − a) γ C PˆH,t − PˆF,t G − e ˆ =0 t Y¯ Y¯
(A.4)
Investment
where
ϕ = −Φ00
∗ ˆt + χ (1 − a) PˆH,t − PˆF,t − eˆt − qˆt + ϕIˆt = ϕK ˆt
(A.5)
ˆ t+1 − δ Iˆt = (1 − δ)K ˆ t + δχ K ˆt
(A.6)
n h io H K ˆ CP ¯ CP H r¯K rˆt+1 ˆ t+1 = 0 qˆt − Et ∆ + (1 − δ) qˆt+1 + ϕδ Iˆt+1 − ϕδ K t,t+1 + ∆
(A.7)
I K
δ.
Motion of capital
Shadow price of capital
36
where
¯ CP H ∆
is the one-period stochastic discount factor of the capital producing rm in the steady state,
de ned as:
¯ CP H = ξ¯ = 1 + C¯ φ (1 − H) ¯ 1−φ ∆
−η
(A.8)
and where
CP H
α H ˆ q ˆ CP ∆ t−1,t − CP H λt + q
αCP H q CP H − αCP H − αCP H ∗ ∗ ∗ ˆ ˆ ˆ ˆ ˆ = e ˆ − λ ξ − λ + ξ − λ + e ˆ t t−1 t−1 t−1 t t−1 t−1 q CP H q CP H q CP H
CP H
(A.9) with
ξˆt = −η
¯ 1−φ ¯ C¯ φ 1 − H H ˆ ˆ φCt − (1 − φ) ¯ Ht 1−H ¯ 1−φ 1 + C¯ φ 1 − H
(A.10)
and
ˆ t + aPˆH,t + (1 − a) Pˆ ∗ + (1 − a) eˆt − [(1 − σ) φ − 1] Cˆt + (1 − σ) (1 − φ) λ F,t
¯ H ˆ ¯ Ht = 0 1−H
(A.11)
Pricing decisions
h i ∗ (1 − a) PˆH,t − PˆF,t − eˆt − mc ˆ t=0
(A.12)
ˆ t + Yˆt = 0 mc ˆ t−w ˆt − H
(A.13)
ˆt mc ˆ t − rˆtK + Yˆt = K
(A.14)
Aˆt = ρA Aˆt−1 + A,t
(A.15)
where
as well as
The technology evolves according to
Labor supply
w ˆt −
¯ H ˆ ˆ ¯ Ht − Ct = 0 1−H
(A.16)
Production function
ˆ t = αK ˆt Yˆt − Aˆt − (1 − α)H
(A.17)
ˆ t − aPˆH,t − (1 − a) Pˆ ∗ − (1 − a) eˆt − Yˆt = 0 M F,t
(A.18)
Monetary policy
where the money supply evolves according to
ˆ t = ρM M ˆ t−1 + M,t M
(A.19)
ˆ t+1 − Et λ ˆ ∗ + Et eˆt+1 = λ ˆt − λ ˆ ∗ − ξˆt + ξˆ∗ + eˆt Et λ t+1 t t
(A.20)
Finally, the risk sharing condition implies:
37
In the complete markets case, this equation holds
ex post
as well, which gives
ˆt − λ ˆ ∗ + eˆt = λ ˆ t−1 − λ ˆ ∗ + ξˆ∗ − ξˆt−1 + eˆt−1 λ t t−1 t−1
(A.21)
After appropriate substitutions for the price index, the real return (in domestic terms) on the domestic nominal bond becomes
∗ ˆ BH ˆ ˆ∗ rˆtBH = −aPˆH,t − (1 − a) PˆF,t − (1 − a) eˆt − Q ˆt−1 t−1 + aPH,t−1 + (1 − a) PF,t−1 + (1 − a) e
(A.22)
Returns on nal goods producing rms shares are given by
GH F GH ¯Π ˆF rˆtF GH = (1 − ξ) + ξ¯qˆtF GH − qˆt−1 t
where the time
GH ˆF Π = t
t
pro t
GH ˆF Π t
(A.23)
is given by
wH r¯K K Y K ˆH,t + Yˆt − Pˆt − ˆ ˆ P w ˆ + H r ˆ + K − t t t Y − wH − r¯K K Y − wH − r¯K K Y − wH − r¯K K t (A.24)
Using the optimal pricing conditions and showing that the in non-stochastic steady state well as
rK K/wH = α/(1 − α),
ΠF GH = Y /ω
as
this simpli es to
∗ GH ˆF ˆ t + (ω − 1) αˆ ˆt − eˆt − ω Yˆt + (ω − 1) (1 − α) w Π − ω (1 − a) PˆH,t − PˆF,t ˆt + H rtK = − (ω − 1) αK t (A.25) Similarly, the returns on capital producing rms are
CP H CP H ˆ rˆtCP H = 1 − ξ¯ Π + ξ¯qˆtCP H − qˆt−1 t
(A.26)
and the linearized pro t is given by
ˆ CP H + Π t
I¯ Y¯ ω−1 ω α
−
I¯ Y¯
∗ Iˆt + (1 − a) PˆH,t − (1 − a) PˆF,t − (1 − a) eˆt −
where, again, I use the fact that
rK K/Y =
ω−1 ω α ω−1 ω α−
I¯ Y¯
rˆtK =
ω−1 ω α ω−1 ω α−
I¯ Y¯
ˆt K
(A.27)
ω−1 ω α in the steady state. Finally, net output is de ned as
d t − Y Yˆt + I Iˆt + G G ˆt = 0 NY C C C
(A.28)
A.2 Foreign country equations Similarly, the equations for the foreign country are:
i ¯ ¯h C¯ C¯ I¯ G ∗ ˆ ∗t + 2a (1 − a) γ C PˆF,t ˆH,t − eˆt = 0 Yˆt∗ − a ¯ Cˆt∗ − (1 − a) ¯ Cˆt − ¯ Iˆt∗ − ¯ G − P Y Y Y Y Y¯ ∗ ˆ t∗ + χ (1 − a) −PˆH,t + PˆF,t + eˆt − qˆt∗ + ϕIˆt∗ = ϕK ˆ∗t
(A.30)
∗ ˆ t+1 ˆ t∗ + δ χ K − δ Iˆt∗ = (1 − δ)K ˆ∗t
(A.31)
38
(A.29)
h n io F K∗ ∗ ∗ ∗ ˆ CP ¯ CP F r¯K ∗ rˆt+1 ˆ t+1 qˆt∗ − Et ∆ + (1 − δ) qˆt+1 + ϕδ Iˆt+1 − ϕδ K =0 t,t+1 + ∆
(A.32)
∗ mc ˆ ∗t + (1 − a) PˆH,t − (1 − a) PˆF,t − (1 − a) eˆt = 0
(A.33)
ˆ t∗ + Yˆt∗ = 0 mc ˆ ∗t − w ˆt∗ − H
(A.34)
∗
ˆ t∗ mc ˆ ∗t − rˆtK + Yˆt∗ = K
(A.35)
Aˆ∗t = ρA∗ Aˆ∗t−1 + A∗ ,t
(A.36)
w ˆt∗ −
¯ H ˆ ∗ ˆ∗ ¯ Ht − Ct = 0 1−H
(A.37)
ˆ t∗ = αK ˆ t∗ Yˆt∗ − Aˆ∗t − (1 − α)H
(A.38)
∗ ˆ t∗ − aPˆH,t − (1 − a) PˆF,t M + (1 − a) eˆt − Yˆt∗ = 0
(A.39)
∗ ˆ t∗ = ρM ∗ M ˆ t−1 M + M ∗ ,t
(A.40)
ˆ ∗t = ρG∗ G ˆ ∗t−1 + G∗ ,t G
(A.41)
∗ ˆ∗ = 0 d t − Y Yˆt∗ + I Iˆt∗ + G G NY C C C t
(A.42)
39
40
(h) M shock, output
(g) M shock, consumption
Figure 5: Impulse responses of symmetric stock portfolios,
(e) G shock, output
(b) A shock, output
(d) G shock, consumption
(a) A shock, consumption
a = 0.5.
(i) M shock, investment
(f) G shock, investment
(c) A shock, investment
41
(h) M shock, exchange rate
(g) M shock, in ation
Figure 6: Impulse responses of symmetric stock portfolios,
(e) G shock, exchange rate
(b) A shock, exchange rate
(d) G shock, in ation
(a) A shock, in ation
a = 0.5,
cntd.
(i) M shock, net exports
(f) G shock, net exports
(c) A shock, net exports
42
(h) M shock, output
(g) M shock, consumption
a = 0.5.
(i) M shock, investment
(f) G shock, investment
(c) A shock, investment
Figure 7: Impulse responses of bond and asymmetric stock portfolios,
(e) G shock, output
(b) A shock, output
(d) G shock, consumption
(a) A shock, consumption
43
(h) M shock, exchange rate
(g) M shock, in ation
a = 0.5,
cntd.
(i) M shock, net exports
(f) G shock, net exports
(c) A shock, net exports
Figure 8: Impulse responses of bond and asymmetric stock portfolios,
(e) G shock, exchange rate
(b) A shock, exchange rate
(d) G shock, in ation
(a) A shock, in ation
44
(j) Complete markets, gamma
(i) 6, gamma
(k) 6, sigma
(g) 2CP, sigma
(c) Real bond, sigma
Figure 9: International correlations for a model without a home bias.
(f) 2NB2CP, gamma
(b) 2NB, gamma
(e) 2CP, gamma
(a) Real bond, gamma
(l) Complete markets, sigma
(h) 2NB2CP, sigma
(d) 2NB, sigma
45
(j) Complete markets, gamma
(i) 6, gamma
(k) 6, sigma
(g) 2CP, sigma
(c) Real bond, sigma
Figure 10: International correlations for a model with home bias
(f) 2NB2CP, gamma
(b) 2NB, gamma
(e) 2CP, gamma
(a) Real bond, gamma
(l) Complete markets, sigma
(h) 2NB2CP, sigma
(d) 2NB, sigma
