The Role of Nonseparable Utility and Nontradables in International Portfolio Choice∗ Akito Matsumoto International Monetary Fund February 22, 2012

Abstract This paper analyzes the role of nonseparable utility and nontradables in portfolio choice using a two-country, two-sector production economy model. I find that nonseparability in utility can change the optimal portfolio choice significantly. Unlike existing results, the optimal portfolio of the traded-good sector equities is no longer a well-diversified portfolio under nonseparability. When two sectors’ equities are combined into a single “all-sector equity fund” of each country and forward contracts are traded, the portfolio does not depend on the elasticity of substitution between traded and nontraded goods, but nonseparability between leisure and consumption still plays an important role. JEL Classification: E32, F30, F40, G11. Keywords: international portfolio choice; nonseparability in utility; nontraded goods; nontraded factors; asset market structures Author(s) E-Mail Address: [email protected];

∗ I thank Mick Devereux, Charles Engel, Fabio Ghironi, Urban Jermann, Karen Lewis, Enrique Mendoza, Eric van Wincoop, Randy Wright, participants at the NBER IFM meeting, and my IMF colleagues, especially Julian di Giovanni and Anna Ivanova, for comments and suggestions. The views expressed in this paper are those of the author and should not be attributed to the International Monetary Fund, its Executive Board, or its management. All remaining errors are my own.

1

1

Introduction

What explains equity home bias? This is one of the most frequently asked questions in international finance. While there are several candidate explanations, the existing literature focuses on a particular factors such as the existence of nontraded goods or labor income. This paper attempts to evaluate several important candidates simultaneously. In particular, I examine the optimal portfolio in a model that nests several important models of international risk sharing and portfolio allocation, such as Stockman and Dellas (1989), Cole and Obstfeld (1991), Baxter and Jermann (1997), Baxter et al. (1998), Jermann (2002), and Coeurdacier (2009). My model features a twocountry production economy, nontraded goods, a nontradable factor (labor/leisure), differentiated traded goods, nonseparable utility function, and different asset market structures in a complete asset market setting. The primary contribution of this paper is a closed-form solution for the optimal portfolio. Most importantly, I show that the optimal portfolio of the traded good sector equities is no longer a welldiversified world portfolio – the result of Stockman and Dellas (1989) or Baxter et al. (1998) – once nonseparability between leisure and consumption is introduced. Moreover, under an alternative asset market structure with forward contracts, the foreign equity share is identical to Jermann (2002), even in the presence of nontraded goods. Nonseparability is motivated by the insight of Lewis (1996), who hypothesizes that perfect international risk sharing cannot be rejected when there is nonseparability in utility and capital controls exist in some countries. Nonseparability between leisure and consumption also helps explain the Backus-Smith puzzle by breaking the perfect correlation between relative consumption across countries with their real exchange rates.1 Jermann (2002) finds that nonseparability can be a potential solution for the home bias puzzle in his analysis of a production economy model with a single traded good. I find that in a production economy without nonseparability between leisure and consumption – a simple extension of Baxter et al.’s model – the optimal traded good equity portfolio is still well diversified. That is, nonseparability is necessary to overturn Baxter et al.’s results, while nontradable goods may or may not affect the results of Jermann. Even in the presence of nonseparability, an important result of Cole and Obstfeld (1991) – any portfolio can support the optimal allocation as far as the elasticity of substitution between home and foreign traded goods is unity – is still true for the traded good sector equity portfolio when the nontraded 1 Backus and Smith (1993) build a two-country endowment economy model with nontraded goods, which can explain some of the puzzles in international finance but the model introduces another puzzle known as the BackusSmith puzzle; a perfect correlation between relative consumption across countries and real exchange rates, which is not observed in the data.

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sector equity portfolio is optimal. However, their result is not robust because it holds only when the elasticity is unity as previously pointed out. Another novelty of this paper is the examination of different asset market structures with identical business cycle properties. Specifically, in addition to the traditional setup of asset markets where equities of traded- and nontraded-good sectors are traded separately with two sectoral shocks in each country, I also consider two different market structures: (1) one national equity mutual fund that combines equities of traded- and nontraded-good sectors, called the “all-sector equity fund,” with a country specific shock in each country; and (2) one all-sector equity fund and forward contracts with two sectoral shocks. The introduction of forward contracts, which can be replicated by two national currency denominated bonds, is realistic as emphasized in Engel and Matsumoto (2008, 2009), and Coeurdacier and Gourinchas (2009).2 Since asset menus in reality are much richer than a simple model, studying several settings is a first step toward understanding optimal portfolio choice problems. Under a traditional asset market structure where the equities of two sectors are traded separately, the characteristics of portfolios of both nontraded and traded good sector equities in my model are similar to the portfolio of nontraded good sector equities in Baxter et al. (1998). That is, the optimal equity portfolios of both sectors are sensitive to the elasticity of substitution between traded and nontraded goods. In addition, the optimal portfolios in my model are sensitive to the coefficient of relative risk aversion and the elasticity of substitution between leisure and consumption once I consider nonseparability. I find that the elasticity of substitution between home and foreign traded goods does not affect the optimal equity portfolio weight unless the elasticity is unity under a traditional asset market structure.3 The share of total foreign equities is very similar to the data given plausible parameter values. However, this does not mean that the home bias puzzle disappears under this asset market structure. Depending on the parameter values and asset market structures, the optimal portfolio can be extremely biased towards either home or foreign equities. Under this asset market structure, the existence of nontradables may weaken the results of Jermann (2002), though nonseparability is still a potential solution. Under an alternative asset market structure, where only the “all-sector equity fund” is traded, the elasticity of substitution between home and foreign traded goods affects the optimal equity 2 The real exchange rate fluctuation is caused by nominal rigidity in the cases of Engel and Matsumoto (2008, 2009) and by home bias in consumption in Coeurdacier and Gourinchas (2009). In my model, it is caused by existence of nontradables. 3 In the case of unit elasticity of substitution, the portfolio weight for the traded good sector equities is indeterminate, similar to the results of Cole and Obstfeld (1991). This indeterminacy may be eliminated when one introduces sticky prices as presented in Engel and Matsumoto (2009), who show that a slight degree of price stickiness can generate home bias if the elasticity is unity.

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portfolio weight unlike the other cases. In this setup, my model can nest a model of home bias in consumption (or trade costs). Obstfeld and Rogoff (2001) suggest that the existence of trade costs can be a potential solution for major puzzles including the home bias puzzle, but Coeurdacier (2009) shows that their conjecture was incorrect.4 When nonseparability is considered, home bias in consumption (or trade costs) can help explain home bias in equity. In this sense, the existence of nonseparability overturns the results of Coeurdacier (2009) as well.5 Under the other asset market structure, where forward contracts and a single all-sector mutual fund are traded, the equity portfolio coincides with the one in Jermann (2002), which tends to be more stable and exhibits home bias under a relatively reasonable range of parameter values. Under this market structure, Jermann’s result is robust to the addition of differentiated traded goods and nontraded goods unlike the traditional asset market case. With forward contracts, the optimal portfolio is independent of the elasticity of substitution between nontraded and traded goods, which plays an important role in Baxter et al., but nonseparability is still a key element. The forward contract can be either long or short in foreign currency depending on the parameter values.6 Under this asset market structure, assuming that home bias in equity is optimal, one can conclude that international risk sharing is optimal as in Lewis (1996), or that forward positions hinder international risk sharing. In addition to the home bias puzzle in equity, nontraded goods have been used to explain other puzzles in international finance.7 Obviously, researchers have studied the role of nontraded goods for different questions and/or in different environments.8 Indeed, Collard et al. (2007) revisit the role of nontraded goods with consumption home bias in traded goods on portfolio allocation.9 Though researchers have studied the role of nontraded goods, nontraded factors and 4 Heathcote and Perri (2004) use simple utility but included capital accumulation with home bias in consumption to study portfolio allocation. Kollmann (2006b) studies the effects of home bias in consumption on portfolio. van Wincoop and Warnock (2006) find that home bias in consumption does not help explain home bias in portfolio in a partial equilibrium setup. 5 See Obstfeld (2007) for another extension; the combination of nontraded goods and home bias in traded goods. 6 As discussed in Engel and Matsumoto (2008), we do not have good understanding of foreign currency positions as they are affected by derivatives such as options and forward contracts. 7 See Lewis (1995) and Obstfeld and Rogoff (2001) for various puzzles. 8 For example, Stockman and Tesar (1995) build a two-country production model with nontraded goods and investment to replicate many features of both cross-country and within-country correlations. Devereux et al. (1992) utilize nonseparable utility to explain cross-country consumption with a single good model. Tesar (1993) adopt a production economy to explain these puzzles in her model with a single good. For a small open economy, see Engel and Kletzer (1989) and Balsam and Eckstein (2001). Obstfeld and Rogoff (2005a,b) emphasize the role of nontraded goods in current account adjustments. Evans and Hnatkovska (2005) study capital flows under different asset market settings with nontraded goods. Burstein et al. (2003), Burstein et al. (2005, 2006), Benigno and Toenissen (2006), and Corsetti et al. (2006) try to explain real exchange rate behavior by including nontraded goods. Most of these authors use numerical solutions. 9 Pesenti and van Wincoop (2002) also study optimal portfolio choice with nontraded goods in a partial equilibrium model. Hnatkovska (2009) studies asset allocation under incomplete market setup. Another important work in this area is Kollmann (2006a), who corrects the solution of portfolio choice problem with nontraded goods by Serrat (2001). Kollmann shows that the optimal portfolio of traded goods equities is still well diversified under Serrat’s assumption.

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nonseparability in utility, this is the first paper to solve the optimal portfolio choice problem with both nontraded goods and nontraded factors in a general equilibrium setting with a nonseparable utility function.10 As a byproduct of the analysis of nonseparability, this paper also investigates the role of the nontradable factor, human capital, which turns out to be another significant factor in portfolio allocation. This is not surprising as the labor share in national income is about two thirds in industrial countries.11 The relation between international portfolio choice and human capital has been considered in several papers.12 Empirical support for nontraded goods as an explanation of home bias in equity is generally weak, while support for nontraded factors is relatively favorable. Pesenti and van Wincoop (2002) and van Wincoop and Warnock (2006) find that nontraded goods or home bias in consumption cannot explain home bias in equity. Meanwhile, Bottazzi et al. (1996), Palacios-Huerta (2001), and Julliard (2002) find that human capital may explain home bias to a certain degree. In the finance literature, Boyd et al. (2005) find that bad news about employment is usually good for equity returns. Furthermore, Lustig and Van Nieuwerburgh (2008) find that innovations in human capital returns are negatively correlated with innovations in financial asset returns. Di Giovanni and Matsumoto (2010) find that human capital returns are more negatively correlated with returns on domestic equity than foreign equity. I explain the model setup more precisely in the next section. Then, I solve for real allocations and prices in section 3, discuss the optimal portfolio in section 4, and conclude in section 5.

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The Model

I consider a completely-technology-shock-driven two-country, two-sector production stochastic general equilibrium model. There are two countries, called Home and Foreign, with populations “n” and “1 − n,” respectively. Prices are assumed to be flexible. Given this, the model features a standard international real business cycle model setup with nontraded goods except for endogenous portfolio choice. However, as I assume that asset markets are such that can replicate complete market allocation up to a linear approximation, business cycle properties are not different from 10 The recent studies on international portfolio includes Evans and Hnatkovska (2005, 2006, 2007), Devereux and Sutherland (2006, 2010, 2008, 2007), Ghironi et al. (2007), Tille and van Wincoop (2007). See Obstfeld (2007) for the recent literature. 11 Jorgenson and Fraumeni (1989) and Lustig et al. (2008) claim that the share of human capital in total wealth is more than 90 percent, whereas Di Giovanni and Matsumoto (2010) claim that it is about three quarters. Empirical work finds that the stock (value) share of human capital tends to be higher than the flow (income) share. 12 Using a simple model, Baxter and Jermann (1997) predict that a foreign bias in equity is needed when labor income is considered as it is perfectly correlated with equity returns. Engel and Matsumoto (2009) show that home bias may be optimal to hedge labor income risks in a sticky price model.

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these exhibited in a complete asset market model.

2.1

Firms and Technology

Firms are producing differentiated goods ` a la Blanchard and Kiyotaki (1987) using labor as sole input. The production functions for firms producing traded (T) and nontraded (N) good are

YT,t (i) = AT,t LT,t (i),

YN,t (i) = AN,t LN,t (i),

(1)

where A.,t is the technology level in each sector, and L.,t (i) is labor hours used in each firm. I use a dot, “.”, to denote either the nontraded good sector “N” or the traded good sector “T” hereafter. For example, above equations can be written Y.,t (i) = A.,t L.,t (i). Also, Foreign variables are denoted with an superscript asterisk, “*”. Technology is assumed to be country and sector specific. I assume that labor is mobile between the two sectors within a country. Therefore, the wage rate will be the same across the two sectors. Note that A.,t s are the only exogenous variables in the model. I assume that technology levels in two sectors in each country are i.i.d. except when only an all-sector equity fund is traded. In this case, I assume log AT,t = log AN,t so that there is no sector specific shock. Firms set prices in each period to maximize profits after the realization of shocks. Home firms’ profits, Π.,t , in each period are simply:

Π.,t = P.,t Y.,t − W.,t L.,t .

(2)

Because firms in each sector are identical, I omit index i when it is not necessary.

2.2

Households

The representative household j in Home country solves

max Et−1

~ γt (j),···

max Ct (j),Lt (j),···

  ∞ X U Cs (j), Ls (j) ,

s.t. budget constraint,

s=t

where U is a well-defined twice-differentiable utility function with UC > 0, and UL < 0. Ct (j) denotes the consumption basket of Home agent j, and Lt (j), the labor supply. Ct (j) is a consumption

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basket of a representative Home household defined as iθ/(θ−1) h Ct (j) ≡ η 1/θ CN,t (j)(θ−1)/θ + (1 − η)1/θ CT,t (j)(θ−1)/θ h iω/(ω−1) CT,t (j) ≡ n1/ω Ch,t (j)(ω−1)/ω + (1 − n)1/ω Cf,t (j)(ω−1)/ω ,

(3) (4)

where θ > 0 is the elasticity of substitution between traded and nontraded goods and ω > 0 is the elasticity of substitution between Home and Foreign produced traded goods. I assume an identical utility function for Foreign households to avoid home bias in traded goods consumption except in the special case of ω = θ, and both nontraded and Home traded-goods are bundled as Home goods. The consumption basket looks like n iθ/(θ−1) . Ct (j) = [η 1/θ + [(1 − η)n]1/θ ]Ch,t (j)(θ−1)/θ + [(1 − η)(1 − n)]1/θ Cf,t (j)(θ−1)/θ

This formulation allows for home bias in consumption. That is, when η = 0, there is no home bias in consumption and when η = 1, there is complete home bias. In addition to a relatively general form of utility, it is also important to have general CES aggregation instead of Cobb-Douglas aggregation in order to examine asset allocation and the transmission mechanism. Ch,t is the consumption basket of Home produced traded goods, Cf,t is that of Foreign produced traded goods, and CN,t is that of nontraded goods defined as follows:  Z −1/λ Ch,t (j) ≡ n

λ/(λ−1)

n

Ch,t (j, i)

(λ−1)/λ

di

,

(5)

0

 λ/(λ−1) Z 1 −1/λ (λ−1)/λ Cf,t (j) ≡ (1 − n) , Cf,t (j, i) di ,

(6)

n

Z CN,t (j) ≡

λ/(λ−1)

1

CN,t (j, i)

(λ−1)/λ

di

,

(7)

0

where λ denotes the elasticity of substitution among varieties, with λ > 1.13 The CPI can be written as  1/(1−θ) 1−θ 1−θ Pt = ηPN,t + (1 − η)PT,t ,

(8)

13 I use monopolistic competition in this model, which is isomorphic to having firms with fixed capital and a Cobb-Douglas production function.

7

where

PT,t =

h

1−ω nPh,t



Z

+ (1 −

1−ω n)Pf,t

Ph,t (i)

Ph,t = 1/n

Z ,

PN,t =

di

1/(1−λ)

1

PN,t (i)

1−λ

di

,

(9)

0

1/(1−λ)

n 1−λ

i1/(1−ω)

,

Pf,t

 Z = 1/(1 − n)

1/(1−λ)

1

Pf,t (i)

1−λ

di

.

(10)

n

0

PN,t (i) is the nominal price of Home nontraded good, Ph,t (i) is the price of Home traded good i sold in Home, and Pf,t (i) is the price of Foreign traded good i sold in Home. I use asterisks to denote foreign prices and quantities. Let St be the Home currency price of Foreign currency. Then the real exchange rate is

Qt ≡

St Pt∗ . Pt

(11)

Since all prices are flexible, nominal prices and the nominal exchange rate are indeterminate without specifying the conduct of monetary policy, but relative prices can be determined. Without loss of generality, I assume that monetary policy in each country is set to keep the CPI level, Pt , at unity, so that nominal prices are equal to relative prices with respect to CPI in each country.

2.3

Asset Markets

I consider three different asset market structures where agents can replicate complete market allocation up to a first-order approximation to focus on portfolio allocations rather than business cycle properties. I use complete asset market allocation to solve for portfolio positions. This is not because I believe that asset markets are complete. The purpose of finding explanation for home bias in equity under complete market is to find out the reason for the lack of international risk sharing. If home bias is optimal, then the under-diversification is not the reason but something else. For example, currency positions may not be optimal. In general, budget constraint of household j can be written as

Ct (j) + Vt+1 + Ht = Vt−1 Rt + Ht−1 RtH ,

(12)

where Ht is the time t value of human capital wealth and RtH is the gross return on it from time t − 1 to t. Vt and Rt are the value of the financial wealth and its return. Generally, the value of

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human capital wealth and its return can be written as

Ht =

∞ X

β s Dt,t+s Wt+s Lt+s

(13)

s=1

Ht + Wt Lt Ht−1 UC (Ct+s+k , Lt+s+k ) = . UC (Ct+s , Lt+s )

RtH = Dt+s,t+s+k

(14) (15)

Assuming complete asset markets, I consider three different asset market structures. The first one is a traditional asset market structure, as in Stockman and Dellas (1989) and Baxter et al. (1998), where both traded- and nontraded-good equities are traded separately.14 My setup nests their models. In the second structure, all domestic equities are combined into a single “allsector mutual fund” with one country specific productivity shock, which affects both traded- and nontraded-good sectors. Using this setup, I also consider the case of home bias in consumption assuming no distinction between traded and nontraded goods. This formulation nests endowment economy models studied in Kollmann (2006b), Obstfeld (2007), and Coeurdacier (2009) since my model includes production.15 In the third structure, the asset menu includes both an all-sector equity fund and forward contracts while keeping both nontraded- and traded sector technology shocks in each country. This allows me to evaluate the claim by Engel and Matsumoto (2009) that the lack of proper exchange rate hedging could be the source of incomplete international risk sharing in a sticky price model.

2.4

First Order Conditions

Since households are identical in each country, I will suppress household index j from now on. Given prices and the total consumption basket Ct , the optimal consumption allocations are −θ

−θ

CT,t = (1 − η) (PT,t ) Ct ,  −ω Pf,t Cf,t = (1 − n) CT,t , PT,t  −λ 1 Pf,t (i) Cf,t (i) = Cf,t , 1−n Pf,t

CN,t = η (PN,t ) Ct ,  −ω Ph,t Ch,t = n CT,t , PT,t  −λ 1 Ph,t (i) Ch,t (i) = Ch,t , n Ph,t −λ  PN,t (i) CN,t . CN,t (i) = PN,t

(16) (17) (18) (19)

14 Note that nontraded-good equites can be traded internationally and they are in reality. The dividends of nontraded good equities paid out in nontraded goods can also be exchanged to traded goods in its relative price. 15 When the labor shares is zero, the model becomes an endowment model.

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Labor supply conditions, and Euler equations are also standard: UL (Ct , Lt ) , UC (Ct , Lt )   βUC (Ct+1 , Lt+1 ) (Ri,t+1 ) , 1 = Et UC (Ct , Lt )

Wt = −

(20) (21)

where i denotes any asset. Home firm pricing decisions are

Ph,t (i) =

λ Wt , λ − 1 AT,t

PN,t (i) =

λ Wt , λ − 1 AN T,t

(22)

where λ is elasticity of substitutions among variety of goods as defined before. Note that the prices in foreign market are simply multiplied by St = Qt .

2.5

Market Clearing Conditions

The goods market clearing conditions are

∗ nAT,t LT,t = nCh,t + (1 − n)Ch,t ,

nAN,t LN,T = nCN,t .

(23)

The labor market clearing condition is

Lt = LN,t + LT,t ,

where LN,t =

3

R1 0

LN,t (i)di, and LT,t =

Rn 0

(24)

LT,t (i)di.

Solution

I first solve for the real allocation that replicates the complete asset market allocation and then find the supporting portfolio for this allocation in the next section. The complete market assumption implies Qt =

UC (Ct∗ , L∗t ) St Pt∗ =κ . Pt UC (Ct , Lt )

(25)

While κ is a part of the solution and depends on initial conditions, it is not important for portfolio shares; therefore, I assume κ = 1 for simplicity.16 In the initial period, t = 0, I assume AN,t = 16 This is equivalent to setting arbitrary weights for Home and Foreign in the social welfare function. This equilibrium can be supported by the wealth transfer in the initial period.

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AT,t = A∗N,t = A∗T,t = 1. I use a log approximation to solve for an equilibrium.

3.1

the Complete Asset Market Allocation

This subsection discusses key aspects of the solution. Details are in the Appendix. Lower-case letters refer to log deviations from the initial state. World variables are defined as xW t ≡ nxt + ∗ W R (1 − n)x∗t and relative variables as xR t ≡ xt − xt . This in turn means that xt = xt + (1 − n)xt . R R For example, the total consumption in Home is ct = cW t + (1 − n)ct , and ct can be regarded as

the country specific component. Let ψ =

¯ L) ¯ L ¯ ULL (C, ¯ L) ¯ , UL (C,

¯ ¯ ¯

CL (C,L)L φC = − UU ¯ ¯ , φL = C (C,L)

initial state value of X, and UXY =

∂2U ∂X∂Y

¯ L) ¯ C ¯ UCL (C, , ¯ L) ¯ UL (C,

¯ L) ¯ C ¯ (C, ¯ is the and ρ = − UCC , where X ¯ L) ¯ UC (C,

.

For example, if the utility function is CES:  µ(1−σ) µ−1 1 1  µ1 µ−1 µ−1 µ µ µ + (1 − γ) (1 − L) U (C, L) = , γ C 1−σ

(26)

then  1 1 − σ = (1 − sc ) + sc σ, µ µ  ¯  L 1 −σ φC =(1 − sc ) ¯, µ 1−L   1 −σ , φL =sc µ    ¯   ¯ 1 L L 1 1 ψ= − (1 − sc ) −σ = sc + (1 − sc )σ ¯ ¯, µ µ µ 1−L 1−L 1 ρ = − sc µ



where 1

sc =

γ µ C¯ 1 γ µ C¯

µ−1 µ

µ−1 µ

1 ¯ + (1 − γ) µ (1 − L)

µ−1 µ

is the consumption share in utility. Note that in this example, the sign of



1 µ

 − σ determines the

sign of cross term φC , and φL and other parameters should be nonnegative. The estimated values of these parameters are discussed in Appendix.

11

The following set of equations describes the solutions for key relative variables: κCT κCN R ηaN,t + (ω − 1)(1 − η)aR T,t , K K κLN R κLT ltR = ηaN,t + (1 − η)(ω − 1)aR T,t , K K K κLT + ω−1 κLN + K R ytR = ηaN,t + (1 − η)(ω − 1)aR T,t , K K κQT κQN R ηaN,t + (1 − η)(ω − 1)aR qt = T,t , K K κ κ κQN  R κQT  CN CT cR ηaN,t − (1 − η)(ω − 1)aR −θ +θ T,t = T,t , K K K K cR t =

(27) (28) (29) (30) (31)

where 1 φC ψ κCN ≡ + [η − (θ − 1)(1 − η)] + [1 + (ω − 1)(1 − η)] , ρ ρ ρ 1 φL κLN ≡η − [η − (θ − 1)(1 − η)] − [1 + (ω − 1)(1 − η)] , ρ ρ

ψ φC + ), ρ ρ φL ≡1+η ρ

κCT ≡ −(η κLT

K ≡ρ(κLT κCN − κCT κLN ) φC φL + η(ηψ + ) + (1 − η)[η(θ − 1) + (ω − 1) + 1 + η](1/εl,w ) ρ ρ φC ≡ρκCN + φC κLN = 1 + η + [1 + (ω − 1)(1 − η)](1/εl,w ) ρ =1 + η

κQN

κQT ≡ρκCT + φC κLT = −(1/εl,w )η ρ is a Frisch elasticity of labor supply. The above allocations are independent ψρ − φC φL of asset market structures up to a first-order linear approximation as far as they are complete asset and εl,w =

markets. By examining the above equations, I investigate the implications of the general CES specification in traded goods aggregation and of nonseparability in utility instead of Cobb-Douglas aggregation.17 With Cobb-Douglas, ω = 1, country-specific productivity shocks in the traded good sector, aR T,t , will not affect total consumption unlike nontraded good sector productivity shocks, aR N,t . This is true not only for total consumption but also for other variables including traded good consumption and real exchange rates. Stockman and Tesar (1995) find that their model is missing some source of nation-specific variation in the consumption of traded goods. However, this is partly because they assume that the elasticity of substitution between Home and Foreign traded goods is unity. As the value of ω is commonly believed to lie between about 0.8 and 6, the 17 Assuming the other extreme case, namely perfect substitution between Home and Foreign traded goods, is becoming less common since two-way trade contradicts the assumption of ω = ∞.

12

Cobb-Douglas specification seems reasonable.18 However, as shown here, if the productivity level in the traded good sector is more volatile than in the nontraded good sector, the Cobb-Douglas assumption eliminates variation in consumptions resulting from the productivity difference in the Home and Foreign traded good sectors. In addition, there is no Balassa-Samuelson effect if ω = 1. With typical values for other parameters, ω > 1 is a necessary condition for real exchange rates to appreciate in response to positive productivity shocks in the traded good sector. Both relative consumption and real exchange rates are linear functions of the relative nontraded good sector productivity if ω = 1. This leads to a perfect correlation between relative consumption and real exchange rates even with nonseparable utility function. It should be also noted that nonseparability between leisure and consumption, or nonzero φC and φL , is necessary in order to break the perfect correlation between relative consumption and real exchange rates in this class of models because otherwise qt = ρcR t .

4

The Optimal Portfolio

I next demonstrate the existence of the supporting equity portfolio in this economy for the allocation derived under the complete asset market assumption. I also show that the nonexistence of the supporting portfolio with some combination of parameter values. The literature has not paid much attention to the case of nonexistence. However, this case might be important in explaining the lack of international risk sharing. Before going to the discussion of nonexistence of the supporting portfolio, I first show the supporting portfolio when it exists.

4.1

Two Sector Mutual Funds - Traded and Nontraded

I first consider the traditional case, where equities are the only assets traded. The asset market structure with 4 mutual funds (2 mutual funds in each country), which pay the profit of Home or Foreign firms in the traded or nontraded good sectors, would support this allocation when the portfolio holdings are optimal. The detailed derivation is in Appendix. Asset market clearing conditions are

18 See

∗ nγT,h,t + (1 − n)γT,h,t = n,

∗ nγT,f,t + (1 − n)γT,f,t = 1 − n,

(32)

∗ nγN,h,t + (1 − n)γN,h,t = 1,

∗ nγN,f,t + (1 − n)γN,f,t = 1.

(33)

Imbs and Mejean (2011) and Feenstra et al. (2011) for more discussion about the value.

13

Let R ∗ ∗ r.,t ≡ r.,h,t − r.,f,t = r.,h,t − r.,f,t

be the excess return of Home equities in each sector relative to that of Foreign equities. Then,

R rT,t =(1 − β)

R rN,t =(1 − β)

∞ X s=0 ∞ X

β s Et [−(ω − 1)(ψlt+s + φL ct+s − aR T,t+s ]

(34)

R R β s Et [−(θ − 1)(ψlt+s + φL ct+s + aR N,t+s − θqt+s + ct+s ]

(35)

s=0 R Note that when ω = 1, rT,t = 0 for any realization of stochastic variables. This plays a key role in

generating indeterminacy of the traded good equities portfolio. Households first allocate a portion of equity portfolio, η, to the nontraded good sector portfolio and the rest, 1 − η, to the traded good sector portfolio. Note that η is also the weight of nontraded goods in total consumption. This allocation is obvious because the value of firms depends on the future sales and profit margin, but the margin in each sector is identical and the future sales share of each sector is the same as the consumption share. Then, Home households allocate a portion, δN , of nontraded good sector equity portfolio to Foreign equities and δT of traded good sector equity portfolio to Foreign.19 For example, the optimal weight on Home traded good sector equities in the total equity portfolio of the Home residence is η × (1 − δT ). Using the above notation, the relative budget constraint becomes [cR t − η(pN,t + yN,t ) − (1 − η)(pT,t + yT,t )] =

1 ∗ δN η(1 − ζ)[(p∗N,t + yN,t + qt ) − (pN,t + yN,t )] 1−n 1 ∗ δT (1 − η)(1 − ζ)[(p∗T,t + yT,t + qt ) − (pT,t + yT,t )]. + 1−n

(36)

The left-hand side of this equation is the difference between the consumption expenditure and income of Home households if there were no assets traded under the optimal allocation. In order to achieve the optimal allocation, assets trade should offset this difference. The right hand side is the value of Home households’ gain from asset trade as Home households exchange δN of Home firm equities in nontraded good sector to acquire δN of Foreign firm equities in the nontraded good ∗ sector and δT for the traded sector. For example, p∗T,t + yT,t + qt is the revenue of foreign firms

in the traded good sector in terms of Home consumption goods, and 1 − ζ is the capital share.20 ∗ = n δ . order for equity market to clear, δT 1−n T this model, the capital is related to monopolistic rent. As said, this model is isomorphic to a model with fixed capital. 19 In

20 In

14

Hence, the profit is the product of these two.21 Then, the equity portfolio weights δT and δN can R be determined from equation (36) because it must hold ∀ aR N,t and aT,t .

Rewriting the above equation in terms of ltR , cR t , and the relative return (excess return) of Home equity in each sector gives R R R R R (1 − n)(ρcR t + φC lt + ψlt + φL ct + lt − ct )

(37)

R R =δN η(1 − ζ)rN,t + δT (1 − η)(1 − ζ)rT,t )

Since ltR and cR t are given by equations (27) and (28), it is straightforward to solve for δN and δT except for the case with ω = 1, for which δT , the portfolio weight for the traded good sector equities, cannot be determined.   1−n A 1+ for ω 6= 1, 1−ζ B + ηC   1−n A−C δN = 1+ , 1−ζ B + ηC   A − ηC 1−n 1+ , δ ≡(1 − η)δT + ηδN = 1−ζ B + ηC δT =

(38) (39) (40)

where

A =(1 − η)(θ − 1)

φC ρ

(41)

B =(1 − η)(θ − 1) (1 + 1/εl,w )

(42)

ψ+1 − (1 + 1/εl,w ) ρ

(43)

C=

and δ is the total share of foreign equities. When the denominator is zero, there is no supporting portfolio. This solution nests several important results as special cases. In the case of the separable 1−n utility case, where φC = 0, then δT = . That is, “The International Diversification Puzzle Is 1−ζ Worse Than You Think” situation. In the case of an endowment economy, δT = 1 − n and δN =   1 − 1/ρ (1 − n) 1 + , which coincide with Baxter et al. (1998).22 Without (1 − η)(θ − 1) − η(1 − 1/ρ)   1−n φC /ρ nontraded goods, η = 0, δ = δT = 1+ , for ω 6= 1, which is a general result of 1−ζ 1 + 1/εl,w Jermann (2002).23 21 While there are capital gains from the portfolio, the flexible price assumption allows me to focus on income from dividends. As the current budget constraint is satisfied in each period, the solution from this equation leads to the supporting portfolio. In Appendix B, I describe the budget constraint in terms of the total return. 22 Baxter et al. (1998) nest Stockman and Dellas (1989). 23 Jermann (2002) looks at the case where all traded goods are homogeneous, i.e., ω = ∞.

15

There are five important implications regarding the portfolio allocation. First, when the elasticity of substitution between home and foreign traded goods is unity, or ω = 1, the result is similar to Cole and Obstfeld (1991), where they find that there is no gain from equity trade. However, the existence of the nontraded good sector with ω = 1 has an interesting implication. If there is only one mutual fund for each country, which implies δ T = δ N , then the equity portfolio weight for foreign equity as a whole is δ N . If θ = 1 for example, then the portfolio weight is ‘super’ home biased, meaning that home would go short in foreign equity. Also, as shown in Engel and Matsumoto (2009), if ω is close to unity, then price rigidity is an important factor in determining equity portfolio allocation. Since empirical estimates of ω are often close to one, short-run effects of price rigidity deserve closer attention. Second, ω has quite an important role in terms of determinacy of the traded good sector equity portfolio but does not have any further effect on the portfolio weight in the case ω 6= 1. This is because the shocks from the relative productivity in the traded good sector are transmitted to consumption and labor with the common coefficient, ω − 1, as in equations (27) and (28). In R other words, by defining a˙ R t = (ω − 1)at , ω becomes a part of an exogenous variable. Because

the complete market supporting portfolio offsets the effect from relative shocks, ω does not enter into the portfolio function itself. The traded goods equity portfolio, δT , without a nontraded good sector, η = 0, is then identical to that in Jermann (2002) who assumes homogeneous traded goods, i.e., ω = ∞. However, my finding shows that his result is robust to variations in the elasticity of substitution between Home and Foreign goods except for the Cobb-Douglas case. Third, this particular portfolio is constant and markets are complete up to a linear approximation. This constant portfolio is similar to the finding of Judd et al. (2003), who find that the portfolio of all securities is constant over time and states in a Lucas asset pricing model with heterogeneous agents and complete asset markets. Finally, I should note that the denominator in equations (38) and (39),  B + ηC = (1 − η)(θ − 1) (1 + 1/εl,w ) + η

 ψ+1 − (1 + 1/εl,w ) ρ

(44)

can be zero given reasonable parameter values.24 As εl,w does not depend on θ, this denominator is a linear function of θ and can become zero as θ changes its value. It is easy to see with separable utility that the zero denominator case is empirically relevant. The denominator can be rewritten as (ψ + 1)[(1 − η)(θ − 1) + η(1/ρ − 1)] under separable utility. What are the reasonable parameter 24 See

Appendix for reasonable ranges of parameter values.

16

values? The nontraded good sector weight, η, is typically 0.5 to 0.8; the estimate of the elasticity of substitution between traded and nontraded goods, θ, ranges from 0.44 to 1.44 as discussed in Appendix, and the inverse of the elasticity of intertemporal substitution of consumption, ρ, ranges from .5 to 10. With these parameter values, the sign of denominator can be either positive or negative. When the denominator is close to zero then the optimal portfolio is sometimes extremely biased towards either Home or Foreign under nonseparability as depicted in Figures 1-3.25 Baxter et al. (1998) find that the optimal equity portfolio in the traded good sector is a world diversified portfolio. In my setting with a production economy and separable utility with two sector equities separately traded, the portfolio allocation of the traded good sector is similar to that of Baxter and Jermann (1997), where home owns more foreign equity. However, once I introduce nonseparability, then the equity portfolio in the traded good sector behaves similarly to the nontraded good sector, though either one of them tends to foreign bias portfolio. As shown, the existence of a nontraded good sector affects equity portfolio in the traded good sector. Most important, with nonseparability, I can overturn the previous result that the optimal portfolio of the traded good sector equities is well diversified. I can no longer dismiss the claim that existence of nontraded goods explains the home bias puzzle, since the validity of the claim depends on the model parameters. Baxter et al. (1998) find that the equity portfolio in the nontraded good sector is extremely sensitive to the elasticity of substitution between traded and nontraded goods.26 However, in the case of nonseparability, this sensitivity is also true of the portfolios of both traded and nontraded good sector equities as depicted in Figure 1. The portfolio is also sensitive to changes in other parameters as demonstrated in Figure 2 with respect to the elasticity of substitution between leisure and consumption, µ, and in Figure 3 with respect to the coefficient of relative risk aversion, σ. The denominator becomes zero when the relative return of traded good sector equities and that of nontraded good sector equities are linearly dependent. In the case of separability, this happens when the relative return of nontraded good sector equities is constant. When the denominator is zero, then there exists no supporting equity portfolio. What is going on in this case? The relative return of Home equity in each sector can be rewritten as

R rN,t =(1 − β)(ψlt + φL ct + lN,t )

(45)

R rT,t =(1 − β)(ψlt + φL ct + lT,t ).

(46)

25 The parameter values used in the figures are in Table 1 unless otherwise noted in figures. The choices of values are discussed in Appendix. 26 If ζ = 0, then the portfolio will coincide with that in Baxter et al. (1998).

17

When B + ηC = 0, they are linearly dependent.27 When ω 6= 1 optimal allocations of relative R variables depend on both aR N,t and aT,t . Thus, equity returns fail to span the allocation plane.

While depending on the parameter values, the total portfolio exhibits home bias (i.e., the point lies between zero and no bias in the figures), either traded or nontraded-good sector equity portfolios tends to be unrealistic. For most parameter values, at least one of them is in extremely biased either towards home or foreign equities, though parameters values can be such that both equity portfolios are reasonable, e.g., σ = 10 with other parameters at the benchmark values. The extremely biased portfolio implies that even if the number of assets is sufficient to span all the shocks, it may not be possible to achieve the complete market allocation since small market frictions such as short-selling constraints can prevent agents from holding the complete market allocation supporting portfolio. In fact, there are many different assets in reality, but asset returns might be highly correlated with each other, and the optimal portfolio without market frictions might require an extremely biased position in one of the assets. In this case, transaction costs or some other frictions could explain imperfect risk sharing. This suggests that in building a model with incomplete asset markets, it is more realistic to assume certain frictions rather than arbitrarily missing assets as has been common practice.28

4.2

All-Sector Equity Fund

In this subsection, I consider the optimal portfolio if a single national “mutual fund” which combines traded- and nontraded-good sector equites, called “all-sector equity fund,” is traded. The all-sector equity fund mimics the Home equity market by setting the portfolio weight on nontraded good sector equity at its steady state GDP share, η. Home households can own the Foreign allsector equity fund, and let δT +N be the share of foreign mutual fund in the equity portfolio of Home households. R I consider two special cases, (i) always aR N,t = aT,t , so that trading a single “mutual fund” can

replicate the complete market allocation up to a linear approximation, and (ii) forward contracts R are traded, so that the complete market allocation is still replicated even with aR N,t 6= aT,t .

In either case, the return on the Home all-sector equity fund can be written as a weighted sum of equity returns of two sectors in Home:

rT +N,h,t ≡ ηrN,h,t + (1 − η)rT,h,t .

(47)

27 If ω = 1 but B + ηC 6= 0, then r R is still linear function of aR (recall that r R = 0 when ω = 1. However, if N,t N,t T,t R = r R = 0. both ω = 1 and B + ηC = 0, rT,t N,t 28 Recently, Tille and van Wincoop (2007) and Coeurdacier (2009) use market friction to generate home bias.

18

Then, returns on equity portfolio owned by Home and Foreign households given the portfolio weight are

rt =(1 − η)[(1 − δT +N )rT,h,t + δT +N rT,f,t ] + η[(1 − δT +N )rN,h,t + δT +N rN,f,t ]

(48)

∗ ∗ ∗ ∗ rt∗ =(1 − η)[δT∗ +N rT,h,t + (1 − δT∗ +N )rT,f,t ] + η[δT∗ +N rN,h,t + (1 − δT∗ +N )rN,f,t ].

(49)

Using the individual stock returns, the relative returns on equity portfolios are    1 1 R R δT +N rT,t ] + η[∆qt + 1 − δT +N rN,t ] 1−n 1−n  ∞ X ˆt (ψlR + φL cR + lR ). δT +N (1 − β) βsE t+s t+s t+s

 rtR =(1 − η)[∆qt + 1 − 

1 =∆qt + 1 − 1−n

(50)

s=0

Next, I examine two cases in detail.

4.2.1

No Forward Contract

R In the special case where aR N,t = aT,t with probability one, this asset market structure, where

all-sector equity funds are traded, still replicates the complete market allocation up to a linear approximation as we have enough assets traded. The budget constraint can be simplified: ( R δT +N (1 − η)(1 − ω)(ψltR + φL cR t − aT,t )

) R R R R + η[(1 − θ)(ψltR + φL cR t − aN,t ) − θ(ρct + φC lt ) + ct ]

=

(51)

1−n R R R (ρct + φC ltR + ψltR + φL cR t + lt − ct ). 1−ζ

Using the same portfolio solution method as in the previous section, the optimal portfolio can be written as

δT +N =

1−n 1−ζ

 1+

A0 + η(A − C) B 0 + η(A + ηC)

 (52)

where

A0 =(1 − η)(ω − 1)

φC ρ

B 0 =(1 − η)(ω − 1)(1 + 1/εl,w ).

19

(53) (54)

When ω = 1, the share of the Foreign all-sector equity fund is the same as the share of Foreign nontraded sector equities in the previous section. This is because when ω = 1, any equity portfolio in the traded sector can support the complete market except for the zero denominator case, and what matters is the nontraded sector. In general, the elasticity of substitution between Home and Foreign traded goods influences the portfolio in the case of a single mutual fund without forward contracts. Another important case is ω = θ, so that A = A0 and B = B 0 . This case is isomorphic to the model where every good is traded with η determining the degree of home bias in consumption. The specification is also isomorphic to an iceberg trade cost model. A higher η implies more home bias in consumption or higher trade costs. η = 0 implies no home bias in consumption. The share of Foreign all-sector equity fund in equity portfolio of Home households is

δT +N

1−n = 1−ζ



(1 + η)A − ηB 1+ (1 + η)C + η 2 B

 .

(55)

When the utility function is separable in an endowment economy; that is φC = 0 (A = 0) and ζ = 0, then the optimal weight on the Foreign national mutual fund:

δT +N = (1 − n)

      

 η

 1 −1 ρ

η 2 )(ω

η2

1−

   

 (1 −

− 1) +

 .  1  −1  ρ

(56)

This coincides to Coeurdacier (2009), who considers portfolio allocation with iceberg trade costs. Unless the coefficient of relative risk aversion is less than one, trade costs (or home bias in consumption) generate either foreign bias or a short foreign position, either of which does not match the data. The result of Coeurdacier does not hold in the case of nonseparability as shown in Figure 4. My model can generate home bias in equity with reasonable ranges of parameter values (e.g., µ = 2.5 with other parameter values at benchmark.). Of course, depending on the other parameter values, the portfolio can be extreme like the cases in the previous section, but the role of nonseparability is still important in the case of home bias in consumption.

4.2.2

With Forward Contract

R Now, I introduce forward contracts and allow aR N,t and aT,t to again move independently. Let δS

be the amount of forward contracts owned by a representative Home household in terms of Home consumption unit. This implies that a Foreign household owns δS∗ =

20

n 1−n δS Qt

of forward contracts.

Since the monetary policy is conducted so that CPI is unity, the real exchange rate coincides the nominal exchange rate (St = Qt ); thus, the payoff from a unit of contract is

Qt−1 Qt .

Thus, the

linearized relative budget constraint is as follows: ( R δT +N (1 − η)(1 − ω)(ψltR + φL cR t − aT,t )

) + η[(1 − =

θ)(ψltR

+

φL cR t

−

aR N,t )

−

θ(ρcR t

+

φC ltR )

+

cR t ]

+

1 δS ∆qt 1−ζ

(57)

1−n R R R (ρct + φC ltR + ψltR + φL cR t + lt − ct ). 1−ζ

This yields

δT +N δS

( ) 1−n φC /ρ 1+ = 1−ζ 1 + 1/εl,w ( ) (ψ + 1)/ρ = (1 − n) −1 + . 1 + 1/εl,w

(58)

(59)

Figure 5 shows how the share of foreign equity changes with respect to θ, µ and σ. Obviously, it does not change with the elasticity of substitution between nontraded and traded goods, θ. Note that the share of foreign equity δT +N coincides to the traded good equity portfolio when the nontraded good sector does not exist, η = 0, i.e., the equity portfolio found in Jermann (2002). Surprisingly, with forward contracts the share of nontraded goods or elasticity of substitution between nontraded and traded goods does not affect portfolio positions, though existence of nontraded goods plays a role in terms of determinacy of the portfolio positions. In addition, this portfolio (either forward or equity) is relatively stable compared to the traditional two sector mutual fund setup. This is because the denominator, 1 + 1/εl,w , is always positive. This stable equity portfolio with forward contracts is in line with Engel and Matsumoto (2009) and Coeurdacier and Gourinchas (2009). In order to explain equity home bias, we need φC /ρ 1+1/εl,w

< −ζ. As discussed in Jermann (2002), this is feasible but I think that the estimates of

the parameters are still inconclusive. But one can immediately see that nonseparability, whether ¯ ¯ ¯

CL (C,L)L leisure and consumption are complements or substitutes matters a lot. That is φc = − UU ¯ ¯ C (C,L)

plays a key role.29 When φc is positive, there is a definite foreign bias. It should be noted that portfolio allocations (equities or forward contracts) do not depend on the elasticity of substitution between nontraded and traded goods, that of Home and Foreign traded goods, or the share of   ¯ 1 L = (1 − sc ) µ − σ 1− ¯ ,, under CES. Where µ is intra temporal elasticity of substitution between leisure L and consumption and σ is risk aversion or inverse of infratemporal substitution of utility. 29 φ

C

21

traded and nontraded goods. Now suppose that parameter values were such that the optimal portfolio has a home bias in equity. Does it mean that risk sharing is perfect? On the one hand, one cannot dismiss the possibility of perfect risk sharing as this is one of the cases that Lewis (1996) could not reject the hypothesis among countries with no capital controls. Moreover, when φC 6= 0, the real exchange rate does not have to be perfectly correlated with relative consumption. On the other hand, one cannot still claim that the risk sharing is perfect as forward contracts or currency positions have to be optimal. Thus, home bias in equity may be indeed optimal, and home bias in equity itself is not the cause of perfect risk sharing; however, perfect risk sharing may not be achieved as a result of non-optimal currency portfolio.

5

Conclusion

This paper presents a two-country, two-sector production economy model with flexible prices. It nests the models of Stockman and Dellas (1989), Baxter et al. (1998), and Jermann (2002) as special cases. It also nests a simple version of home bias in consumption model (e.g., Coeurdacier (2009)). I find that the presence of nontraded goods and nontraded factors with nonseparable utility can be a potential solution to the equity home bias puzzle. The optimal portfolio of traded good sector equities is no longer a well diversified world portfolio, overturning the finding of Stockman and Dellas (1989), and Baxter et al. (1998). More realistically, when equity portfolios are combined into a single national mutual fund and forward contracts are traded, then the equity portfolio replicates that of Jermann (2002). While parameter values for nonseparable utility functions are still under discussion, the equity portfolio can exhibit home bias under “reasonable” values. Thus, further research is needed to determine whether nontraded goods can explain the equity home bias puzzle. Nonseparability can also helps to explain the equity home bias with trade costs or home bias in consumption. Another important finding regarding international portfolio choice is that the elasticity of substitution between home and foreign traded goods is an important factor for determinacy of the traded good sector equity portfolio. If the elasticity is unity, then the terms of trade plays the role of insurance and any traded good sector equity portfolio will support the optimal allocation.30 This issue is discussed in Engel and Matsumoto (2009), who find that if the elasticity between home and foreign traded goods is close to unity then price stickiness plays an important role for 30 However, there is an exception for this: when nontraded good sector equity portfolio cannot support the optimal allocation then no portfolio can support the optimal allocation.

22

the optimal portfolio choice. This finding calls for introducing price rigidity into the model. On the other hand, when it is not unity, it does not affect portfolio allocation when two different assets are traded. Note that even the role of the elasticity of substitution between nontraded and traded goods diminishes with introduction of forward contracts in asset markets. In contrast, nonseparability always affects portfolio allocation with nontraded goods or home bias in consumption. This is the key finding of the paper. Since separability of leisure and consumptions are often assumed for simplicity, adding more realistic feature helps explain home bias in equity. With nonseparability, one can interpret the observed home bias in equity and international risk sharing in three ways. Home bias in equity is not optimal and international risk sharing is imperfect. Or while equity portfolios are optimal, suboptimal currency positions hinder perfect risk sharing. Alternatively, it may be that international risk sharing is close to perfect as shown in Lewis (1996) and portfolios are supporting the optimal allocation.

23

Figure 1: Equity Portfolio θ

Portfolio Share of Foreien Equity Benchmark Case 600%

300%

No Bias 0

−300%

δT δN Total

−600%

0

0.5

1 1.5 θ: elasticity of substitution btw. traded and nontraded goods

2

2.5

Portfolio Share of Foreien Equity σ=1 Case 600%

300%

No Bias 0

−300%

δT δN Total

−600%

0

0.5

1 1.5 θ: elasticity of substitution btw. traded and nontraded goods

2

2.5

Portfolio Share of Foreien Equity µ=1 Case 600%

300%

No Bias 0

−300%

δT δN Total

−600%

0

0.5

1 1.5 θ: elasticity of substitution btw. traded and nontraded goods

24

2

2.5

Figure 2: Equity Portfolio µ

Portfolio Share of Foreien Equity Benchmark Case 600%

300%

No Bias 0

−300%

δT δN Total

−600%

0

1

2

3 4 5 6 7 µ: elasticity of substitution btw. consumption and leisure

8

9

10

Portfolio Share of Foreien Equity σ=1 Case 600%

300%

No Bias 0

−300%

δT δN Total

−600%

0

1

2

3 4 5 6 7 µ: elasticity of substitution btw. consumption and leisure

8

9

10

Portfolio Share of Foreien Equity θ=0.7 Case 600%

300%

No Bias 0

−300%

δT δN Total

−600%

0

1

2

3 4 5 6 7 µ: elasticity of substitution btw. consumption and leisure

25

8

9

10

Figure 3: Equity Portfolio σ

Portfolio Share of Foreien Equity Benchmark Case 600%

300%

No Bias 0

−300%

δT δN Total

−600%

0

1

2

3

4 5 6 σ: coefficient of relative risk aversion

7

8

9

10

Portfolio Share of Foreien Equity µ=1 Case 600%

300%

No Bias 0

−300%

δT δN Total

−600%

0

1

2

3

4 5 6 σ: coefficient of relative risk aversion

7

8

9

10

Portfolio Share of Foreien Equity θ=0.7 Case 600%

300%

No Bias 0

−300%

δT δN Total

−600%

0

1

2

3

4 5 6 σ: coefficient of relative risk aversion

26

7

8

9

10

Figure 4: Equity Portfolio (Home Bias in Consumption)

600%

300%

No Bias 0

−300% δT+N −600%

0

0.5

1 1.5 θ: elasticity of substitution btw. Home and Foreign traded goods

2

2.5

600%

300%

No Bias 0

−300% δT+N −600%

0

1

2

0

1

2

3 4 5 6 7 µ: elasticity of substitution btw. consumption and leisure

8

9

8

9

10

600%

300%

No Bias 0

−300% δT+N −600%

3

4 5 6 σ: coefficient of relative risk aversion

27

7

10

Figure 5: Equity Portfolio (With Forward)

100%

No Bias

0

δT+n δS −100%

0

0.5

1 1.5 θ: elasticity of substitution btw. Home and Foreign traded goods

2

2.5

100%

No Bias

0

δT+n δS −100%

0

1

2

3 4 5 6 7 µ: elasticity of substitution btw. consumption and leisure

8

9

10

100%

No Bias

0

δT+n δS −100%

0

1

2

3

4 5 6 σ: coefficient of relative risk aversion

7

8

9

10

Notes: Negative δS implies Home households take a long position in home currency and a short position in foreign currency. That is, they lend in bonds denominated in home currency and borrow in bonds denominated in foreign currency.

28

Appendix A

The Choice of Parameter Values in Figures

The choice of parameter values in this model is not easy as perturbations in these values can sometimes result in significant changes in moments and portfolio shares. I try to use established values as often as possible. The share of the Home country is assumed to be 0.5. I set the share of consumption in utility ¯ is 0.33 assuming one sc to be 0.33, which implies γ = 0.33 so that the steady state value of C¯ = L third of non-sleeping time is allocated to work. The share of nontraded goods in consumption is assumed to be 0.75. I use the parameter values of Jermann (2002) for the elasticity of substitution between leisure and consumption, µ, which is set to be 5.

31

The coefficient of relative risk aversion with respect

to utility, σ, is assumed to be 3, which falls between Jermann’s σ = 5 and Stockman and Tesar’s σ = 2. As these values have a wide range of estimates, the alternative parameter values of σ = 1 and µ = 1 are also examined. The elasticity of substitution between nontraded and traded goods, θ, is set to be 1.2. Ostry and Reinhart (1992) estimate θ in the range 1.22-1.28 for all regions and 0.66-1.44 for each individual region. Stockman and Tesar (1995) find that θ = 0.44 and claim that θ tends to be low among industrialized countries. Mendoza (1995) estimates θ = 0.74 for industrialized countries. While θ can potentially alter the moments in general, given other parameter values in this section, important correlations hardly respond to changes in θ. The elasticity of substitution between Home and Foreign traded goods, ω, is assumed to be 2 following Ruhl (2005). As the share of traded goods in consumption is 0.25, this may be relatively low.32 In order to compensate for the low share of the traded sector, I use ω = 5 as an alternative so that expenditure switching effect in the total consumption is not too low. 31 Jermann

(2002) picked µ = 5 as his baseline because empirical studies find µ in the range (0, 5), but others including Stockman and Tesar, Backus et al. (1992), and Benigno and Toenissen (2006) use a Cobb-Douglas specification, which implies µ = 1. 32 Chari et al. (2002) and Engel and Matsumoto (2009) use ω = 1.5 in models without nontraded goods.

29

B

Equilibrium Conditions

First, it turns out to be convenient to rewrite the budget constraint:

Ct (j) + Vt+1 + Ht = Vt−1 Rt + Ht−1 RtH

(60)

where,

Ht =

∞ X

β s Dt,t+s Wt+s Lt+s

(61)

s=1

Ht + Wt Lt Ht−1 −−−−→ − → Vt =γt+1 (j)0 (Xt )

RtH =

X.,.,t + Π.,.,t Xt−1 γ.,.,t+1 X.,.,t = ~ .,.,t ~γt+1 X

R.,.t = γ¯.,.,t+1

(62) (63) (=

Qt ∗ R ) Qt−1 t

→ − −−→ Rt = γ¯t 0 R.,.,t Dt+s,t+s+k =

UC (Ct+s+k , Lt+s+k ) . UC (Ct+s , Lt+s )

(64) (65) (66) (67)

denote human capital, return on human capital, financial wealth, return on equity, return on portfolio, and the stochastic discount factor.

B.1

Linearization

Prices can be expressed as

pN,t =wt − aN,t

(68)

ph,t =wt − aT,t

(69)

pf,t =wt∗ − a∗T,t + qt

(70)

pT,t =nph,t + (1 − n)pf,t

(71)

pt =ηpN,t + (1 − η)pT,t ≡ 0, (normalization).

30

(72)

Optimal consumption allocation can be expressed as

ch,t = − ω(ph,t − pT,t ) + cT,t

(73)

cf,t = − ω(pf,t − pT,t ) + cT,t

(74)

cT,t = − θpT,t + ct

(75)

c∗T,t = − θp∗T,t + ct = −θ(pT,t − qt ) + ct cN,t =

−θpN,t + ct .

(76)

Goods market clearing conditions can be expressed as

yN,t = aN,t + lN,t =cN,t

(77)

yh,t = aT,t + lT,t =nch,t + (1 − n)c∗h,t .

(78)

Home labor market clearing condition is

lt =ηlN,t + (1 − η)lT,t .

Let ψ =

¯ L) ¯ L ¯ ULL (C, ¯ L) ¯ , UL (C,

¯ ¯ ¯

CL (C,L)L φC = − UU ¯ ¯ , φL = C (C,L)

¯ L) ¯ C ¯ UCL (C, ¯ L) ¯ UL (C,

(79) ¯ ¯ ¯

(C,L)C and ρ = − UCC . The first order ¯ L) ¯ UC (C,

condition for labor supply is wt = ψlt + φC lt + φL ct + ρct

(80)

The optimal risk sharing condition implies

qt = ρ(ct − c∗t ) + φC (lt − lt∗ )

(81)

Using 14 equations (68)-(81) and their foreign counterparts, I can solve for 13 unknown home variables {ct , wt , lt , ch,t , cf,t , cT,t , cN,t , lN,t , lT,t , ph,t , pf,t , pT,t , pN,t }, their foreign counterparts and qt given aT,t , aN,t and their foreign counterparts. By Walras’s Law, the foreign counter part of equation (81) is redundant.

31

B.2

Solution: World Variables

Combining the goods market clearing condition, the world labor hours and world consumption can be solved as a function of world technology level: 1 − φL − ρ [ηaW + (1 − η)aW T,t ] ψ + φC + φL + ρ N,t 1 + ψ + φC [ηaW + (1 − η)aW = T,t ]. ψ + φC + φL + ρ N,t

ltW =

(82)

cW t

(83)

W Note that ηaW N,t + (1 − η)aT,t is the weighted average of world productivity level.

Combining each country’s traded goods market clearing condition I get

W W W W aW T,t + lT,t = −θ(wt − aT,t ) + ct .

Using the above world traded goods market clearing conditions, I get labor hours in traded good sectors and consumption of traded goods: 1 − φL − ρ [ηaW + (1 − η)aW T,t ] ψ + φC + φL + ρ N,t 1 + ψ + φC W =θη(aW [ηaW + (1 − η)aW T,t − aN,t ) + T,t ]. ψ + φC + φL + ρ N,t

W W lT,t =(θ − 1)η(aW T,t − aN,t ) +

(84)

cW T,t

(85)

Nontraded goods can be solved in a similar way:

B.3

1 − φL − ρ [ηaW + (1 − η)aW T,t ] ψ + φC + φL + ρ N,t 1 + ψ + φC W =θ(1 − η)(aW [ηaW + (1 − η)aW N,t − aT,t ) + T,t ]. ψ + φC + φL + ρ N,t

W W lN,t =(θ − 1)(1 − η)(aW N,t − aT,t ) +

(86)

cW N,t

(87)

Solution: Relative Variables

I make use of the complete market condition:

R qt = ρcR t + φC lt .

(88)

By taking the difference of variables, I can solve for the allocation: κCN R κCT ηaN,t + (ω − 1)(1 − η)aR T,t K K κLN R κLT ltR = ηaN,t + (ω − 1)(1 − η)aR T,t , K K

cR t =

32

(89) (90)

where

κCN κLN

1 φC ψ ≡ + [η − (θ − 1)(1 − η)] + [1 + (ω − 1)(1 − η)] , ρ ρ ρ 1 1 φL ≡η − [η − (θ − 1)(1 − η)] − [1 + (ω − 1)(1 − η)] , ρ ρ ρ



φC ψ κCT ≡ − +η ρ ρ φL κLT ≡ 1 + η ρ

 ,

K ≡ρ(κLT κCN − κCT κLN ) =1 + η

φL ψ φC + η(η + ) + (1 − η)[η(θ − 1) + (ω − 1) + 1 + η](1/εl,w ) ρ ρ ρ

Then, substituting these into equation (88)

R qt =ρcR t + φc lt =

κQN R κQT ηaN,t + (ω − 1)(1 − η)aR T,t K K

(91)

where

κQN ≡ρκCN + φC κLN = 1 + η

φC + [1 + (ω − 1)(1 − η)](1/εl,w ) ρ

κQT ≡ρκCT + φC κLT = −(1/εl,w )η.

For consumption of traded goods and nontraded goods: R cR T,t =ct − θqt κ κ κQN  R κQT  CN CT = −θ +θ ηaN,t − (1 − η)(ω − 1)aR T,t K K K K 1−η R 1 R 1−η R c = cR θqt cR t + N,t = ct − η η T,t η     1 − η κQN κCT 1 − η κQT κCN ηaR + (1 − η)(ω − 1)aR = + θ + θ N,t T,t . K η K K η K

(92)

(93)

The relative labor hours of both sectors expressed using other endogenous variables are

R R lT,t = − ω(ψltR + φL cR t ) + (ω − 1)aT,t

(94)

R R R R lN,t = − θ(ψltR + φC ltR + φL cR t + ρct ) + ct − (1 − θ)aN,t .

(95)

33

C C.1

Supporting Portfolio Returns on Assets

So far I solved real allocation using the complete market condition. I will find a supporting portfolio for the complete market allocation. I first solve for the returns on assets and then show the supporting portfolio.

C.1.1

Return on Human Capital

Recall Ht =

∞ X

β s+1 Et Dt,t+s+1 Wt+s+1 Lt+s+1 .

s=0

In linear form:

ht =

∞ 1 − β X s+1 β Et (dt,t+s+1 + wt+s+1 + lt+s+1 ) β s=0

where dt,t+s+1 = ρ(ct − ct+s+1 ) + φC (lt − lt+s+1 ); therefore,

ht =

∞ 1 − β X s+1 β [(ρct + φc lt ) + Et (ψlt+s+1 + φL ct+s+1 + lt+s+1 )] . β s=0

(96)

Return on the Human capital is defined as follows:

RtH = β

Ht + Wt Lt . Ht−1

(97)

In linear form, rtH =βht + (1 − β)(wt + lt ) − ht−1 =ρ∆ct + φc ∆lt + (1 − β)

∞ X

(98) ˆt (ψlt+s + φL ct+s + lt+s ), βsE

s=0

ˆt Xt+s ≡ Et Xt+s − Et−1 Xt+s . where, E C.1.2

Returns on Equities

˜ .,h,t denote the ex-dividend equity price of a Home firm in Home currency where dot “.” is Let X used as “T” or “NT” sector indicator. The second subscript, h or f , describes the location of the

34

firm.

X.,h,t =

˜ .,h,t ˜ .,h,t X St Pt∗ X ∗ = = Qt X.,h,t Pt Pt St Pt∗

∗ X.,f,t =

˜∗ X X.,f,t .,f,t = Pt∗ Qt

(99)

∗ X.,.,t denotes real equity price in Home consumption basket, while X.,.,t denotes real equity price in

Foreign. Let γ.,h,t (j) denote the number of shares owned by individual j of Home firms producing − → ˜ t be defined analogously traded or nontraded goods. Let γ~t ≡ (γT,h,t , γT,f,t , γN,h,t , γN,f,t )0 , and Π ˜ T,h,t is the nominal profit (dividend) of Home firms producing traded goods. Therefore, where Π real returns on the home equities are

R.,.t =

X.,.,t + Π.,.,t Xt−1

(100)

Now in linear approximation, the dividend of each sector can be written

πT,h,t =wt + lT,t ,

(101)

∗ πT,f,t =qt + wt∗ + lT,t ,

(102)

πN,h,t =wt + lN,t .

(103)

The stock price of the home firms, for example, can be written,

x.,h,t =

∞ 1 − β X s+1 β [(ρct + φc lt ) + Et (ψlt+s+1 + φL ct+s+1 + l.,t+s+1 )] . β s=0

(104)

Therefore, I can write returns on equities as

r.,h,t =ρ∆ct + φc ∆lt + (1 − β)

∞ X

β s Et (ψlt+s + φL ct+s + l.,t+s )

s=0 ∞ X

∗ r.,f,t =ρ∆c∗t + φc ∆lt∗ + (1 − β)

r.,f,t =ρ∆ct + φc ∆lt + (1 − β)

∗ ∗ β s Et (ψlt+s + φL c∗t+s + l.,t+s )

s=0 ∞ X

∗ ∗ β s Et (ψlt+s + φL c∗t+s + l.,t+s )

s=0

Let R ∗ ∗ r.,t ≡ r.,h,t − r.,f,t = r.,h,t − r.,f,t

35

(105) (106) (107)

be the excess return of Home equities in each sector relative to that of Foreign equities. Then,

R rT,t =(1 − β)

∞ X

R R β s Et (ψlt+s + φL cR t+s + lT,t+s )

s=0

=(1 − β) R rN,t =(1 − β)

∞ X s=0 ∞ X

(108) s

β Et [−(ω −

R 1)(ψlt+s

+

φ L cR t+s

−

aR T,t+s )]

R R R R β s Et [−(θ − 1)(ψlt+s + φL cR t+s + aN,t+s − θqt+s + ct+s ]

(109)

s=0

Note that there is no ex-ante excess returns. Expressing them only with exogenous variables, κRN T κRN N (1 − β)aR (1 − β)aR N,t + T,t K K κRT N κRT T =− (1 − β)(ω − 1)aR (1 − β)(ω − 1)aR N,t + T,t K K

R rN,t =−

(110)

R rT,t

(111)

where 

κRN N

κRN T κRT N κRT T

 ψ+1 =(θ − 1)(1 − η)(1 + 1/εl,w ) + η − (1 + 1/εl,w ) + ρ   ψ [(θ − 1)(1 − η) − η](1/εl,w ) + η (ω − 1)(1 − η) ρ   ψ φC ) − [(θ − 1)(1 − η) − η](1/εl,w ) (1 − η)(ω − 1) = −(η + ρ ρ 1 φL =ηψ + + [(θ − 1)(1 − η) − η](1/εl,w ) ρ ρ     1 φL ψ φC =− (1 + 1/εl,w ) + + η + + [(θ − 1)(1 − η) − η](1/εl,w ) η ρ ρ ρ

When B + ηC = (θ − 1)(1 − η)(1 + 1/εl,w ) + η

h

ψ+1 ρ

(112)

(113) (114) (115)

i − (1 + 1/εl,w ) = 0, then one can show that

R R are linearly dependent as κRN N κRT T − κRT N κRN T = 0. rN,t and rT,t

C.1.3

Solution: Portfolio Allocation

Given the same markup rate across sectors, the share of the firms in each sector should equal the consumption weight. In addition, as the consumption basket and ex-ante value of each firm in each sector are identical between Home and Foreign, the optimal share should be symmetric. The portfolio weight of each sector must be η and 1 − η. Let δT be the time invariant portfolio weight of Foreign equities in the traded good sector equities of Home household and δN be the portfolio weight of Foreign equities in the nontraded good sector. In order for equity market to clear, we need δ.∗ =

n 1−n δ. .

I will find this class of portfolio which satisfies the budget constraint

with complete market allocation. Then, this portfolio supports complete market allocation. 36

First, portfolio returns of Home and Foreign household are

rt =(1 − η)[(1 − δT )rT,h,t + δT rT,f,t ] + η[(1 − δN )rN,h,t + δN rN,f,t ]

(116)

∗ ∗ ∗ ∗ ∗ ∗ rt∗ =(1 − η)[δT∗ rT,h,t + (1 − δT∗ )rT,f,t ] + η[δN rN,h,t + (1 − δN )rN,f,t ].

(117)

The budget constraint can be linearized as follows: 1 β [(1 − ζ)vt + ζht ] = [(1 − ζ)(rt + vt−1 ) + ζ(rtH + ht−1 )]. 1−β 1−β

ct +

(118)

Then, the relative budget constraint is

cR t + With δ.∗ =

 1 β  R R (1 − ζ)vtR + ζhR [(1 − ζ)(rtR + vt−1 ) + ζ(rtH + hR t = t−1 )]. 1−β 1−β

n 1−n δ.

(119)

and the sector weight of η and 1 − η, it is trivial to see that world budget

constraint holds. Note that ∞  1 − β X s+1  R R R β (ρct + φc ltR ) + Et (ψlt+s+1 + φL cR t+s+1 + lt+s+1 ) β s=0 ( ∞ 1 − β X s+1 R R vt = β (ρcR t + φ c lt ) β s=0   1 R R + (1 − η) 1 − δT Et (ψlt+s+1 + φL cR t+s+1 + lT,t+s+1 ) 1−n )   1 R R R δN Et (ψlt+s+1 + φL ct+s+1 + lN,t+s+1 ) . +η 1− 1−n

hR t =

(120)

(121)

By substituting these and rearranging, I get R δT (1 − η)(1 − ω)(ψltR + φL cR t − aT,t ) R R R R + δN η[(1 − θ)(ψltR + φL cR t − aN,t ) − θ(ρct + φC lt ) + ct ]

=

(122)

1−n R R R (ρct + φC ltR + ψltR + φL cR t + lt − ct ). 1−ζ

R R Substituting cR t , and lt using equations (89) and (90), we can express the above only by aN,t and

aR T,t . If ω = 1, then we cannot determine the traded good sector equities portfolio. However, if

37

ω 6= 1, then there exists the unique portfolio that satisfies the budget constraint:    

 φC    1−n ρ   if, ω 6= 1 δT = 1+  ψρ − φC φL  η 1−ζ     (ψ + 1) + [(θ − 1)(1 − η) − η] 1 + ρ ρ     ψ+1 ψρ − φC φL  φC    − + 1 + (1 − η)(θ − 1)  1−n ρ ρ ρ   δN = 1+ η ψρ − φC φL  1−ζ      (ψ + 1) + [(θ − 1)(1 − η) − η] 1 + ρ ρ (1 − η)(θ − 1)

(123)

(124)

with exception when denominator becomes zero. Otherwise, this portfolio supports the complete market allocation.

38

Table 1: List of Parameters and Benchmark Values variables

n β σ µ γ ¯ L θ η ω ζ

home population discount factor coefficient of relative risk aversion and inverse of the intertemporal substitution with respect to contemporaneous utility elasticity of substitution between consumption and leisure utility weight on consumption steady state share of working hours in non-sleeping hours elasticity of substitution between traded and nontraded goods share of nontraded goods in consumption basket elasticity of substitution between Home and Foreign traded goods the labor share in the national income;ζ ≈ λ−1 λ

39

0.5 3 5 0.33 0.33 1.2 0.75 2 0.6

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