Financial Frictions and Equity Home Bias

Financial Frictions and Equity Home Bias Kyu-Chul Jung∗ September 10, 2010

A BSTRACT This paper studies equity home bias. Since productivity is persistent, people want to produce more capital goods in good times. If it is hard to borrow to finance capital-good production, people want their net worth to move with home productivity to ease the financial frictions. Therefore, people would like to hold home-biased equity positions. In a dynamic stochastic general equilibrium model, we show that the equity home bias is substantial under the financial frictions. We also provide empirical evidence that supports the claim.

1.

INTRODUCTION

This paper studies equity home bias, that is, holding domestic equities more than foreign equities after controlling for countries’ equity-market sizes. Economists have regarded the home bias as a puzzle since people do not seem to exploit the risk sharing opportunity. Researches about the puzzle have focused on consumption smoothing across states as a main motive for equity diversification.1 In contrast, this paper tries to find a motive for the equity non-diversification and to explain the home bias quantitatively in a dynamic stochastic general equilibrium model. Figure 1 shows that the home bias of U.S. investors. As Ahearne, Griever, and Warnock (2004), we define homebias as (1- foreign share in country’s portfolio / foreign share in the global portfolio). According to capital asset pricing model, every investor should hold the market portfolio, which implies that the hombias should be zero. Figure 1 shows that the home bias is decreasing but still substantial. ∗ Department

of Economics, University of Wisconsin, 1180 Observatory Drive, Madison, WI 53706, U.S.A. (e-mail: [email protected]). 1 French and Poterba (1991) and Tesar and Werner (1995) reported the evidence of the equity home bias. For recent evidence, see Baele, Pungulescu, and Ter Horst (2007). See Lewis (1999) and Sercu and Vanp´ee (2007) for a review.

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1 0.9 0.8

Homebias

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1988

1990

1992

1994

1996

1998 2000 Years

2002

2004

2006

2008

F IGURE 1.— The U.S. home bias We claim that financial frictions in physical capital investment help to explain the equity home bias. When the productivity of the home economy is high, it is also a good time to accumulate physical capital since the productivity is persistent and thus the productivity in the near future is also expected to be high. If it is hard to borrow to finance the capital-good production due to frictions, people need to reduce consumption in order to increase investment. People do not want to reduce consumption especially when they expect their future income to be high. On the other hand, when the productivity of the home economy is low, they want to reduce the physical capital investment and thus they will increase consumption. In order to reduce undesirable consumption fluctuation while maintaining the desirable investment movement, people may want to have more worth in good times and vise versa. If people can access the international equity market, they will choose their equity positions such that the returns to their equities are relatively high when the home economy is relatively in a good state. That is, they should hold home-biased equity positions. Previous studies have focused on the correlation between the returns to home equities and nonmarketable human capital. The seminal paper of Baxter and Jermann (1997) claimed that, since the returns to labor and capital in a country are highly correlated and labor’s share in income is large, people should short sell the domestic equities to hedge the human capital risks. In contrast, Bottazzi, Pesenti, and van Wincoop (1996) reported that the returns to human capital are more correlated with those on foreign equities than those on home equities and thus the paper obtained equity home bias

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for most countries. Similarly, Julliard (2002) reported that including human capital helps to explain equity home bias for some countries. Coeurdacier, Kollmann, and Martin (2010) and Engel and Matsumoto (2009) emphasized the role of bonds in risk sharing. They showed that what matters in the choice of equity positions is not unconditional correlation of returns to physical and human capital but their conditional correlation given the returns to bonds. They claimed that the conditional correlation implies the equity home bias. This paper considers a situation in which people have country-specific capital-production technologies, which are non-marketable due to financial frictions. Since productivity is persistent, home capital is more valuable when the home economy is in a good state. Thus, the return to the nonmarketable technology moves together with the domestic-equity return. In contrast to previous studies, this comovement does not necessarily imply that people want to hold foreign-biased equities. We will claim that people do not want to be fully insured from country-specific shocks since it hurts the efficiency of the capital production in the incomplete market. Like this paper, Coeurdacier, Kollmann, and Martin (2010) and Heathcote and Perri (2009) also emphasized the role of physical capital investment. They assumed a local bias in the production of country-specific capital goods. If capital production increases for some reasons, the demand of the local goods increases and so does labor income. Firms reduce dividends when they increase capital formation. The negative correlation between labor income and dividends implies the equity home bias in the papers. The key parameter is the local bias in the capital-good production and the reason why the capital production increases plays a minor role. Therefore, the main results in the papers do not depend on the persistence of technology shocks. In contrast, our result does not rely on the local bias in the capital-good production and we assume that all corporate firms produce identical goods, which are used in consumption or capital-good production in both countries, although the assumption that capital goods are country specific is common. In their models, markets are complete and thus the allocation is efficient to a first-order approximation. In contrast, our main results come from a tradeoff between insurance and efficiency due to market incompleteness. This paper is based on the literature on financial frictions. For example, Bernanke and Gertler (1989) and Kiyotaki and Moore (1997) showed that financial frictions help explain large and persistent fluctuations of an economy. Quadrini (2000) and Cagetti and Nardi (2006) studied the importance of agency costs in explaining the U.S. wealth distribution. Buera and Shin (2008) reported that the financial frictions help explain economic transitions. Like these papers, resources are allocated inefficiently due to financial frictions in our model. When home productivity is higher, home capital goods should be produced more in order to increase the expected consumption-good production. Since home people produce the home capital, it is efficient for home people to have more resources. 3

If ex-post resource reallocation is hard due to financial frictions, people may try to use stochastic equity returns such that the home people have more resources when the home productivity is higher. That is, people hold home-biased equities to reduce the need of resource reallocation. We abstract from the resource misallocations within a country and focus on those across countries. We claim that people have an incentive to hold home-biased equities to reduce the inefficiency. The paper is organized as follows. Section 2 presents the model and equilibrium conditions. Section 3 illustrates the main mechanism in a three-period model with a closed-form solution. Section 4 presents how to solve the model and to obtain the long-run equity position. Section 5 shows the quantitative results. Section 6 presents extensions and discussions. Section 7 presents empirical analysis and Section 7 concludes.

2.

THE MODEL

Our model is motivated by Carlstrom and Fuerst (1997), which was built on Bernanke and Gertler (1989). In Carlstrom and Fuerst (1997), risk-neutral entrepreneurs have a stochastic technology that transforms consumption goods into capital goods. Entrepreneurs suffer from agency cost due to asymmetric information. Risk neutrality makes aggregation easy in their paper, but may not be appropriate to study equity positions. In this paper, instead, we do not distinguish entrepreneurs from households and assume that every investor is risk averse while maintaining the financial frictions in capital production. In a closed-economy model with financial frictions, heterogeneity among agents are essential since otherwise there is no need for resource reallocation. In contrast, in an openeconomy model, agents in the two countries are different and the heterogeneity among agents within a country is unnecessary. Thus, we assume that there is a representative investor in each country. We, of course, introduce corporate firms’ equities, which can be traded internationally without any frictions. The model is a symmetric two-country model with three goods. A unitary measure of investors live in each of the two countries, Home and Foreign. All investors consume the same kind of perishable consumption goods but firms use country-specific capital goods in production. Preferences " (1)

E0

∞

The preference of Home and Foreign investors is, respectively, represented by #

∑ β t u(ct )

t=0

" and E0

∞

#

∑ β t u(ct∗)

,

t=0

where ct and ct∗ are consumption in period t. We assume that u(c) = (c1−σ − 1)/(1 − σ ) for σ > 0 and β ∈ (0, 1). 4

Technology

There are two sectors in each country. The two sectors are different in production

technology and financial frictions. Cole and Obstfeld (1991) claimed that the terms-of-trade movement provides insurance for productivity shock. That is, the terms-of-trade improves when a country experience negative productivity shock in the paper. This prediction, however, is not supported empirically, as Backus, Kehoe, and Kydland (1994) showed. Thus, in order not to rely on the counterfactual terms-of-trade movement, we assume that corporate firms in both countries produce the same kind of consumption good. The firms use country-specific capital and labor inputs. That is, Home firms use Home capital and labor while Foreign firms use Foreign capital and labor. Their stochastic productivities may differ. Explicitly, their production functions are yt = zt Ktα Lt1−α and yt∗ = zt∗ Kt∗α Lt∗1−α , where Kt and Kt∗ are, respectively, Home and Foreign capital used in production and Lt and Lt∗ are, respectively, Home and Foreign labor. The country-specific technology shocks, zt and zt∗ , follow a first-order autoregressive process, (2)

à ! à !à ! à ! log zt+1 π p πs log zt εt+1 = + ∗ , ∗ log zt+1 πs π p log zt∗ εt+1 Ã

! 2 ρε 2 ε where (εt , εt∗ ) are independent and identically distributed with mean zero and covariance ρε 2 ε 2 for ρ ∈ [−1, 1) and a small constant ε > 0. π p and πs are persistence and spillover parameters and we assume that π p > πs as in the data. For stationarity we assume that π p + |πs | < 1. Each investor conducts entrepreneurial activity. Home investors produce Home capital and Foreign investors Foreign capital. The capital-production technology is constant returns to scale and transforms a unit of consumption good into a unit of Home or Foreign capital goods depending on those who produce the capital. Since investors in each country are identical, they produce the same amount of capital goods in equilibrium. Let it and it∗ denote the production of Home and Foreign capital in period t. Then the aggregate Home and Foreign capital stocks, Kt and Kt∗ , evolve according to (3)

∗ Kt+1 = it + (1 − δ )Kt and Kt+1 = it∗ + (1 − δ )Kt∗ ,

where δ is the depreciation rate. Firms’ objective Let Rt and Rt∗ be the rents of Home and Foreign capital and Wt and Wt∗ be the rents of Home and Foreign labor in period t, respectively. The rents are measured and paid in terms

5

of consumption goods in period t. Given the rents, the Home and Foreign firms solve, respectively, (4) (5)

max yt − Rt Kt −Wt Lt ,

Kt ,Lt ≥0

max

Kt∗ ,Lt∗ ≥0

yt∗ − Rt∗ Kt∗ −Wt∗ Lt∗ .

Investors’ income Although each investor can produce only one kind of capital good, he can own both kinds of capital goods by trading them in international capital markets. Let xh,t and x f ,t denote the units of Home and Foreign capital goods that the Home investor owns at the end of period t − 1. ∗ Home capital and x∗ Foreign capital. Then the Home Similarly, the Foreign investor owns xh,t f ,t investor’ income of consumption goods from the capital and labor rents is (6)

Rt xh,t + Rt∗ x f ,t +Wt .

After production, he owns the undepreciated capital, (1− δ )xh,t Home capital and (1− δ )x f ,t Foreign capital. Similarly, the Foreign investor’s income is (7)

∗ Rt xh,t + Rt∗ x∗f ,t +Wt∗ .

∗ Home capital and (1 − δ )x∗ Foreign capital Similarly, he owns the undepreciated capital, (1 − δ )xh,t f ,t

after the production. Labor income modified Previous studies focused on the returns to physical and human capital. For example, Baxter and Jermann (1997) assumed that human capital is non-marketable and claimed that investors’ equity positions should be biased toward foreign equities to hedge the labor income risk. The argument depends crucially on the comovement of capital income and labor income. In our model, the two rents, Rt and Wt , in equilibrium are perfectly correlated since (8)

Corrt−1 [Rt ,Wt ] = Corrt−1 [α zt Ktα −1 , (1 − α )zt Ktα ] = Corrt−1 [zt , zt ] = 1,

where we used that Kt is in the information set of period t − 1. In contrast, Rt∗ and Wt are not perfectly correlated since (9)

Corrt−1 [Rt∗ ,Wt ] = Corrt−1 [α zt∗ Kt∗α −1 , (1 − α )zt Ktα ] = Corrt−1 [zt∗ , zt ] < 1.

The empirical evidence about the correlation of the two returns is mixed. Our strategy is to assume that investors’ labor incomes are not exposed to the country-specific risks. In particular, we assume

6

that the labor incomes of both Home and Foreign investors are (10) wt =

Wt +Wt∗ . 2

We neither claim that labor incomes are marketable nor present evidence or economic theory about the labor income. We make the assumption partly because the evidence about the correlation is mixed and partly because there have been a lot of studies about equity position with non-marketable labor income and we do not have much to add. Without country-specific labor-income risks, all investors are identical in a one-good economy and thus they should hold identical equity positions if their initial wealth is the same. In our model, investors are different not because their labor incomes are different but because they have different capital-production technologies. This technology is non-marketable because of financial frictions, which we will present later in detail. Then the relation between returns to the non-marketable capital-good production and those to firms’ consumption-good production would be crucial. We will claim that investors may be biased toward domestic equities even though the returns to domestic firms’ equities and investors’ non-marketable technology are positively correlated. In subsection 6.2, we will discuss about the equilibrium equity position when investors face country-specific labor-income risks. Financial Frictions

We assume that an investor as an entrepreneur suffers from financial frictions

while corporate firms do not. Financial frictions prevent investors from borrowing the consumption good. In other words, investors cannot promise to deliver the capital good in return for their borrowed consumption goods. In order to produce it units of Home capital goods, the Home investor needs to use it units of the consumption goods. If he consume ct consumption goods, his total expenditure of consumption goods is ct + it , which must be less than or equal to his total income of consumption goods. That is, (11) ct + it ≤ Rt xh,t + Rt∗ x f ,t + wt , where the income is given in (6) with the modified labor income. Similarly, the counterpart of the Foreign investor is ∗ (12) ct∗ + it∗ ≤ Rt xh,t + Rt∗ x∗f ,t + wt .

Equity markets

At the end of each period, investors trade capital ownership in the international

capital market. Let pt denote the relative price of Home to Foreign capital in period t. After produc7

F IGURE 2.— Timeline productivity shocks realized dividends and wages paid consumption/capital-good production

Thus, the firms’ profit is always zero in equilibrium. Since investors do not value leisure, they supply labor inelastically. That is, Lt = Lt∗ = 1. Since we focus on the case of Kt , Kt∗ > 0, we can obtain equilibrium rents from the first-order conditions, (21)

Rt = α zt Ktα −1 ,

(22)

Wt = (1 − α )zt Ktα ,

(23)

Rt∗ = α zt∗ Kt∗α −1 ,

(24) Wt∗ = (1 − α )zt∗ Kt∗α . The Lagrangian for the Home investor’s problem is " ½ ³ ´ ∞ t (25) L = E0 ∑ β u(ct ) + λt α zt Ktα −1 xh,t + α zt∗ Kt∗α −1 x f ,t + wt − ct − it t=0

# h i¾ + µt it + (1 − δ )xh,t + (1 − δ )x f ,t /pt − xh,t+1 − x f ,t+1 /pt .

The first-order conditions include (26)

ct : u0 (ct ) = λt ,

(27)

it : λt = µt ,

α −1 (28) xh,t+1 : µt = β Et [λt+1 α zt+1 Kt+1 + µt+1 (1 − δ )], ∗ ∗α −1 (29) x f ,t+1 : µt /pt = β Et [λt+1 α zt+1 Kt+1 + µt+1 (1 − δ )/pt+1 ],

(30)

λt : ct + it = α zt Ktα −1 xh,t + α zt∗ Kt∗α −1 x f ,t + wt ,

(31)

µt : xh,t+1 + x f ,t+1 /pt = it + (1 − δ )xh,t + (1 − δ )x f ,t /pt .

Eliminating the Lagrangian multipliers, λt and µt , the six equations are reduced to four equations, (32)

α −1 u0 (ct ) = β Et [u0 (ct+1 ){α zt+1 Kt+1 + (1 − δ )}],

(33)

∗ ∗α −1 u0 (ct ) = β pt Et [u0 (ct+1 ){α zt+1 Kt+1 + (1 − δ )/pt+1 }],

(34)

ct + it = α zt Ktα −1 xh,t + α zt∗ Kt∗α −1 x f ,t + wt ,

(35) xh,t+1 + x f ,t+1 /pt = it + (1 − δ )xh,t + (1 − δ )x f ,t /pt .

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The Foreign counterpart is (36)

∗ α −1 u0 (ct∗ ) = β /pt · Et [u0 (ct+1 ){α zt+1 Kt+1 + pt+1 (1 − δ )}],

(37)

∗ ∗ ∗α −1 u0 (ct∗ ) = β Et [u0 (ct+1 ){α zt+1 Kt+1 + (1 − δ )}],

(38)

∗ ct∗ + it∗ = α zt Ktα −1 xh,t + α zt∗ Kt∗α −1 x∗f ,t + wt ,

∗ ∗ (39) pt xh,t+1 + x∗f ,t+1 = it∗ + (1 − δ )pt xh,t + (1 − δ )x∗f ,t .

These eight equations plus two capital evolution equations, (3), plus two market clearing conditions, (15) and (16), with transversality conditions will determine a sequence of eleven variables ∗ ∗ , p } while one condition is redundant due to Wal, x f ,t+1 , x∗f ,t+1 , Kt+1 , Kt+1 {ct , ct∗ , it , it∗ , xh,t+1 , xh,t+1 t ras’s law. Now we will reduce the twelve equations to nine equations by eliminating it , it∗ , and one redundant equation. From (34) and (35), we obtain (40) ct + xh,t+1 + x f ,t+1 /pt = [α zt Ktα −1 + (1 − δ )]xh,t + [α zt∗ Kt∗α −1 + (1 − δ )/pt ]x f ,t + wt . The Foreign counterpart is ∗ ∗ (41) ct∗ + pt xh,t+1 + x∗f ,t+1 = [α zt Ktα −1 + pt (1 − δ )]xh,t + [α zt∗ Kt∗α −1 + (1 − δ )]x∗f ,t + wt .

From the equation of motion for Kt+1 (3), and Home-capital-market clearing, (15), we obtain (42) it = Kt+1 − (1 − δ )Kt (43)

∗ ∗ = xh,t+1 + xh,t+1 − (1 − δ )(xh,t + xh,t ).

Plugging this equation into it in (35), we obtain ∗ ∗ (44) xh,t+1 + x f ,t+1 /pt = xh,t+1 + xh,t+1 − (1 − δ )(xh,t + xh,t ) + (1 − δ )xh,t + (1 − δ )x f ,t /pt .

Rearranging yields ∗ ∗ (45) x f ,t+1 − (1 − δ )x f ,t = pt [xh,t+1 − (1 − δ )xh,t ],

which we will call price conditions. In words, the value of Foreign capital acquired in period t must be equal to that of Home capital sold in period t in equilibrium. If we calculate the Foreign counterpart, we obtain the same equation since there is a redundant equation. Now we have nine equations, (15), (16), (32), (33), (36), (37), (40), (41), and (45) for nine 10

∗ ∗ , p }. variables {ct , ct∗ , xh,t+1 , xh,t+1 , x f ,t+1 , x∗f ,t+1 , Kt+1 , Kt+1 t ∗ , c , c∗ } can be expressed with control variables Note that {pt , Kt+1 , Kt+1 t t ∗ (46) Xt ≡ (xh,t+1 , x f ,t+1 , xh,t+1 , x∗f ,t+1 )

and state variables ∗ (47) St ≡ (xh,t , x f ,t , xh,t , x∗f ,t , zt , zt∗ ). ∗ That is, pt is from (45), Kt+1 from (15), Kt+1 from (16), ct from (40), ct∗ from (41). Then the

remaining four equations, (32), (33), (36), and (37), will determine Xt . Since (zt , zt∗ ) follows a firstorder autoregressive process, we can define a time-invariant policy function g(·) such that (48) g(St ) = Xt Equity returns

for all t. From (32), (33), (36), and (37), we can see that the returns to equity are different

across investors. We define the gross returns of equities as α −1 (49) rh,t+1 ≡ α zt+1 Kt+1 +1−δ, ¶ µ 1−δ ∗ ∗α −1 (50) r f ,t+1 ≡ pt α zt+1 Kt+1 + , pt+1 α −1 α zt+1 Kt+1 + pt+1 (1 − δ ) (51) ≡ , pt ∗ ∗α −1 +1−δ. (52) r∗f ,t+1 ≡ α zt+1 Kt+1 ∗ rh,t+1

∗ The returns are different, rh,t+1 6= rh,t+1 and r f ,t+1 6= r∗f ,t+1 , mainly because investors have different

technologies. One unit of the consumption good has the same value as one unit of the capital good that the investor can produce. If the values of the two capital goods are different, pt 6= 1 or pt+1 6= 1, then so are those of consumption goods to each investor although all the consumption goods are identical. If investors did not face the financial frictions, the values of the two capital goods are the same and pt = 1 for all t. Thus, the return differentials or capital price differentials will play an important role in our model with financial frictions. Non-stochastic steady state By a symmetric non-stochastic steady state or simply a steady state, ¯ pt = p¯ = 1, ct = ct∗ = c, ¯ wt = w. ¯ for all t. we mean that zt = zt∗ = 1, Kt = Kt∗ = K,

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From (32), we obtain (53) u0 (c) ¯ = β u0 (c)( ¯ α K¯ α −1 + 1 − δ ). Therefore, the steady-state capital stock is µ (54) K¯ =

α 1/β − 1 + δ

¶1/(1−α ) .

From (10), the steady-state labor income is (55) w¯ =

(1 − α )K¯ α + (1 − α )K¯ α = (1 − α )K¯ α . 2

∗ ∗ = K, ¯ and x f ,t+1 + x∗f ,t+1 = Kt+1 ¯ we Adding (40) and (41) with pt = 1, xh,t+1 + xh,t+1 = Kt+1 = K, obtain

(56) 2c¯ + 2K¯ = 2α K¯ α + 2(1 − δ )K¯ + 2w. ¯ Thus, the steady-state consumption is ¯ (57) c¯ = K¯ α − δ K. From (49)–(52), the steady state equity returns are (58) r¯h = r¯ f = r¯h∗ = r¯∗f = 1/β . Therefore, in the non-stochastic steady state, equity returns are all the same and equity positions Xt are undetermined. The zeroth-order equity position

The zeroth-order equity positions (x¯ f , x¯∗f , x¯h∗ , x¯∗f ) are defined as

∗ (59) xh,t+1 = x¯h + O(ε ), x f ,t+1 = x¯ f + O(ε ), xh,t+1 = x¯h∗ + O(ε ), x∗f ,t+1 = x¯∗f + O(ε ),

where we write Q = O(ε n ) for a positive number n if there exists a positive number M such that |Q| ≤ M ε n as ε goes to zero. Equity home bias Corporate firms have a consumption-good production technology. Since the technology has constant returns to scale, firms’ profits are always zero in equilibrium. In this paper, 12

ownership of Home capital is regarded as ownership of Home firms. Home firms pay rents for using Home capital in each period and we regard the rents as dividends. Firms increase the capital that they use in production by issuing stocks. That is, the total amount of stocks of a firm is the same with that of the capital it uses. If an investor is holding xh,t amounts of Home capital goods and the aggregate capital is Kt , then his share for the Home firm is xh,t /Kt . Likewise, if his holdings of Foreign capital is x f ,t , then his share for the Foreign firm is x f ,t /Kt∗ . We say that the Home investor is biased toward the Home equity if pt xh,t > x f ,t . That is, the value of the Home equities in his financial wealth is greater than that of the Foreign equities. We do not use in this paper but the relative price of the two shares is pt Kt /Kt∗ . Equity home bias means that the value of investors’ Home shares is greater than that of their Foreign shares. That is, pt Kt /Kt∗ · xh,t /Kt > x f ,t /Kt∗ , which is the same condition as pt xh,t > x f ,t . In a symmetric equilibrium, x¯h = x¯∗f and x¯ f = x¯h∗ . From the capital-market clearing conditions, (15) and (16), we obtain (60) x¯h∗ = K¯ − x¯h , x¯ f = K¯ − x¯∗f . We define ¯ (61) x ≡ x¯h /K. ¯ Since p¯ = 1, the investors are biased toward home equities Then x¯ f = (1 − x)K¯ = x¯h∗ and x¯∗f = xK. to a zeroth-order approximation, p¯x¯h > x¯ f , when (62) x > 1/2. In this paper, we do not study the fluctuation of equity holdings. Instead, we care mostly about the zeroth-order equity position, x.

3.

AN ILLUSTRATIVE THREE - PERIOD MODEL

In this section we present a simple model, which has a closed-form solution, as an approximation of the long-run infinite horizon model. Investors live three periods. They consume in period 1 and 2. We assume full depreciation of ¯ In period 0, a Home investor chooses his capital, δ = 1. Then K¯ = (αβ )1/(1−α ) and c¯ = K¯ α − K. ¯ Technology shocks are (log z1 , log z∗1 ) = (ε1 , ε1∗ ) equity position (xh,1 , x f ,1 ) subject to xh,1 + x f ,1 = K. and (log z2 , log z∗2 ) = (π p ε1 + πs ε1∗ , πs ε1 + π p ε1∗ ). That is, all uncertainty is resolved in period one. 13

Therefore, investors are indifferent between holding Home and Foreign equities at the end of period one since the returns to the two equities are the same in equilibrium. That is, the equity positions in equilibrium are undetermined. Without loss of generality, we assume that at the end of period one all investors hold only domestic equities. Since we want consumption and investment to be in their steady state levels if there are no shocks, we assume that investors produce K¯ capital goods in period ¯ The Home two although investors will not use the capital goods in period three. Define x ≡ xh,1 /K. investor solves (63)

max

x,c1 ,c2 ,xh,2

E[u(c1 ) + β u(c2 )]

(64) subject to c1 + xh,2 = α z1 K¯ α x + α z∗1 K¯ α (1 − x) + w1 , (65)

¯ c2 = α z2 K2α −1 xh,2 + w2 − K.

The first-order conditions are (66)

σ ∗ 0 = E[c− 1 (z1 − z1 )],

(67)

σ −σ α −1 c− , 1 = β c2 α z2 K2

(68) c1 + K2 = α z1 K¯ α x + α z∗1 K¯ α (1 − x) + w1 , (69)

¯ c2 = α z2 K2α + w2 − K,

where we used that K2 = xh,2 in equilibrium. If (z1 , z∗1 ) = (1, 1) almost surely, then c1 = c2 = c¯ and K2 = K¯ is the equilibrium. We approximate the first-order conditions for non-portfolio choices to the order of ε while we do those for portfolio choice to the order of ε 2 . This is because all equities have the same expected returns to a first-order approximation. Portfolio decisions are affected mainly by the covariance of the marginal utility and equity returns. Thus, we need to consider the first-order condition to the second orders of ε . Approximations of the first-order conditions yield (70) (71)

3 0 = E[ˆz1 − zˆ∗1 − σ cˆ1 (ˆz1 − zˆ∗1 ) + (ˆz21 − zˆ∗2 1 )/2] + O(ε ),

−σ cˆ1 = −σ cˆ2 + zˆ2 + (α − 1)Kˆ 2 + O(ε 2 ),

(72) c¯cˆ1 + K¯ Kˆ 2 = α K¯ α [ˆz1 x + zˆ∗1 (1 − x)] + w¯ wˆ 1 + O(ε 2 ), (73)

c¯cˆ2 = α K¯ α (ˆz2 + α Kˆ 2 ) + w¯ wˆ 2 + O(ε 2 ),

where ˆ denotes a log deviation from the non-stochastic equilibrium. For example, cˆt = log ct − log c. ¯

14

The Foreign counterparts are (74) (75)

3 0 = E[ˆz1 − zˆ∗1 − σ cˆ∗1 (ˆz1 − zˆ∗1 ) + (ˆz21 − zˆ∗2 1 )/2] + O(ε ),

−σ cˆ∗1 = −σ cˆ∗2 + zˆ∗2 + (α − 1)Kˆ 2∗ + O(ε 2 ),

(76) c¯cˆ∗1 + K¯ Kˆ 2∗ = α K¯ α [ˆz1 (1 − x) + zˆ∗1 x] + w¯ wˆ 1 + O(ε 2 ), (77)

c¯cˆ∗2 = α K¯ α (ˆz∗2 + α Kˆ 2∗ ) + w¯ wˆ 2 + O(ε 2 ),

where we used x∗f ,1 /K¯ = xh,1 /K¯ = x in a symmetric equilibrium. Subtracting the equations of the Foreign investor from the corresponding equations of the Home investor, we obtain 0 = E[(cˆ1 − cˆ∗1 )(ˆz1 − zˆ∗1 )] + O(ε 3 ),

(78)

−σ (cˆ1 − cˆ∗1 ) = −σ (cˆ2 − cˆ∗2 ) + (ˆz2 − zˆ∗2 ) + (α − 1)(Kˆ 2 − Kˆ 2∗ ) + O(ε 2 ),

(79)

¯ Kˆ 2 − Kˆ 2∗ ) = α K¯ α (ˆz1 − zˆ∗1 )(2x − 1) + O(ε 2 ), (80) c( ¯ cˆ1 − cˆ∗1 ) + K( c( ¯ cˆ2 − cˆ∗2 ) = α K¯ α [(ˆz2 − zˆ∗2 ) + α (Kˆ 2 − Kˆ 2∗ )] + O(ε 2 ).

(81)

Solving these four equations simultaneously, we obtain 1 β (1 − αβ − ασ )(π p − πs ) + , 2 2[(1 − α )(1 − αβ ) + α 2 σ ] (83) cˆ1 − cˆ∗1 = O(ε 2 ), (1 − αβ − ασ )(π p − πs ) (84) Kˆ 2 − Kˆ 2∗ = (ˆz1 − zˆ∗1 ) + O(ε 2 ), (1 − α )(1 − αβ ) + σ α 2 α (π p − πs ) (85) cˆ2 − cˆ∗2 = (ˆz1 − zˆ∗1 ) + O(ε 2 ). 2 (1 − α )(1 − αβ ) + σ α (82)

x=

Therefore, x is decreasing in σ and greater than a half if σ < σc ≡ 1/α − β . To first-order approximations, consumption in period 1 is fully insured from country-specific shocks whereas that in period 2 are not. The degree of insurance is increasing in the risk-aversion parameter σ . Figure 3 shows the equity position x as a function of σ . The relative price of Home to Foreign capital is (86) pˆ1 = rˆh,2 − rˆ f ,2 + O(ε 2 ) (87) (88)

= zˆ2 + (α − 1)Kˆ 2 − zˆ∗2 − (α − 1)Kˆ 2∗ + O(ε 2 ) ασ (π p − πs ) = (ˆz1 − zˆ∗1 ) + O(ε 2 ). 2 (1 − α )(1 − αβ ) + σ α

Thus, in good times Home capital is more valuable even after controlling for capital adjustment. If investors are risk averse enough, σ > σc , then the relative investment, Kˆ 2 − Kˆ 2∗ , is reduced when 15

Equity Positions

x(σ=0)

W/ Frictions W/O Frictions

.5

x(σ=∞) 0 Risk Aversion Parameters

F IGURE 3.— The equity positions and risk aversion parameters. the country is in a good state, zˆ1 > zˆ∗1 , and vice versa if σ < σc . On the one hand, since the cross derivative of the production function with respect to the productivity and investment is positive, they have complementarity and expected income increases when productivity and the investment move together. On the other hand, the comovement of productivity and investment increases the volatility of the income. Thus, if investors are risk averse enough, then they do not want the productivity and investment to move together. In contrast, if investors are not very risk averse, they will care more about the expected consumption. The tradeoff between the expectation and volatility of consumption determines the optimal equity position. The size of the bias is increasing in the persistence π p − πs . Full insurance of consumption in period one is mainly because we assumed full depreciation. Since ∗ and r ∗ ∗ p0 = 1 and δ = 1, rh,1 = rh,1 f ,1 = r f ,1 . In contrast, p1 6= 1 unless z1 6= z1 and thus the returns in period two are different across investors. (87) implies that pˆ1 is the difference of marginal products in period two. Thus, pˆ1 represents how inefficient the resource allocation is. Figure 4 shows the degree of the inefficiency as a function of σ . As σ is bigger, the insurance is more important and hence the efficiency of production is lower. Note that limσ →0 pˆ1 = O(ε 2 ). When investors are risk neutral, there is no tradeoff between insurance and efficiency and thus the marginal products are the same in the two countries. Previous studies have focused on what equity position is a hedge for the return risks of nonmarketable human capital state by state because they assumed that there were enough assets that

16

Degrees of Resource Misallocation

pˆ1(σ =∞)

W/ Frictions W/O Frictions

0 0 Risk Aversion Parameters

F IGURE 4.— The inefficiency of resource allocations and risk aversion parameters. span the risks. In our incomplete economy, investors want full insurance only if σ is infinite. In that case, the equity position is (89)

lim x =

σ →∞

1 β − (π p − πs ), 2 2α

Thus, the Home investor must be biased toward Foreign equities as in Baxter and Jermann (1997). Again, the bias is increasing in the persistence π p − πs .

4.

THE ZEROTH - ORDER COMPONENTS OF EQUITY POSITIONS

In this paper we focus on the zeroth-order components of equity positions. That is, we want to find x defined in (61). As well known, in a non-stochastic steady state all equities that some investors hold have the same returns and thus the equity positions are undetermined. We use the method recently developed by Devereux and Sutherland (2009) and Tille and van Wincoop (2010). To get equity positions, we need to find covariance between the marginal utility and equity returns. That is, we need to expand the euler equations to a second-order approximation. The major challenge in solving the problem is that the price sequence of pt complicates equity returns and investors’ wealth. We simplify our calculation using the symmetry assumption.

17

Let S¯ and X¯ be the zeroth-order approximation of state variables St and control variables Xt , respectively. That is, St = S¯ + O(ε ) and Xt = X¯ + O(ε ). Then we obtain ¯ since X¯ = g(S) Steady state in period zero

¯ + O(ε ), (90) Xt = g(St ) = g(S) where we have defined the time-invariant policy function in (48). That is, if (εt , εt∗ ) = (0, 0) for all ¯ In words, if the economy is in a steady state in period zero, t ≤ 0, then S0 = S¯ and thus X0 = X. then the control variables in period zero have only the zeroth-order components. That is, to get zeroth-order control variables, it is enough to focus on the realization where the economy is in a non-stochastic state in period zero. The economy, of course, may not be in the steady state from period one. We are not claiming that the economy must be in the steady state in period zero. It is just for ¯ we care mostly about the choice the calculation of the steady-state equity positions. Since X0 = X, in period zero, which, of course, may depend on the plan about future controls contingent on the realization of future shocks. Main equations to be calculated In period 0, the economy is in the non-stochastic steady state, ¯ p0 = 1, xh,1 = x∗f ,1 = xK, ¯ and x f ,1 = x∗ = (1 − x)K¯ in equilibrium. The first-order K1 = K1∗ = K, h,1 equations, (32) and (33), imply that £ ¤ (91) 0 = E0 u0 (c1 )(rh,1 − r f ,1 ) £ σ ¤ ∗ ¯α (92) = E0 c− 1 {α K (z1 − z1 ) + (1 − 1/p1 )(1 − δ )} £ ¤ 2 3 (93) = c¯−σ E0 (1 − σ cˆ1 ){α K¯ α (ˆz1 + zˆ21 /2 − zˆ∗1 − zˆ∗2 1 /2) + (1 − δ )( pˆ1 − pˆ1 /2)} + O(ε ). Similarly, the Foreign counterpart is £ ¤ ∗ (94) 0 = E0 u0 (c∗1 )(rh,1 − r∗f ,1 ) £ σ ¤ (95) = E0 c∗− {α K¯ α (z1 − z∗1 ) + (p1 − 1)(1 − δ )} 1 ¤ £ 2 3 (96) = c¯−σ E0 (1 − σ cˆ∗1 ){α K¯ α (ˆz1 + zˆ21 /2 − zˆ∗1 − zˆ∗2 1 /2) + (1 − δ )( pˆ1 + pˆ1 /2)} + O(ε ). Subtracting (93) from (96), we obtain £ ¤ (97) E0 (1 − δ ) pˆ21 + σ (cˆ1 − cˆ∗1 ){α K¯ α (ˆz1 − zˆ∗1 ) + (1 − δ ) pˆ1 } = O(ε 3 ), which is the main equation we need to calculate. In order to calculate the equation, we need to find cˆ1 − cˆ∗1 and pˆ1 to a first-order approximation. 18

Symmetry

We focus on a symmetric equilibrium. The components of the policy function are

∗ ∗ , and g4 (St ) = x∗f ,t+1 . Note that x∗f ,t+1 and xh,t+1 are, g1 (St ) = xh,t+1 , g2 (St ) = x f ,t+1 , g3 (St ) = xh,t+1 respectively, Foreign counterparts for xh,t+1 and x f ,t+1 . In the symmetric equilibrium, ∗ ∗ (98) g1 (xh,t , x f ,t , xh,t , x∗f ,t , zt , zt∗ ) = g4 (x∗f ,t , xh,t , x f ,t , xh,t , zt∗ , zt ) ∗ ∗ (99) g2 (xh,t , x f ,t , xh,t , x∗f ,t , zt , zt∗ ) = g3 (x∗f ,t , xh,t , x f ,t , xh,t , zt∗ , zt ).

That is, if we exchange the labels of Home and Foreign in the state variables, then we can get the equilibrium in the new state by exchanging the labels of Home and Foreign in the control variables. ¡ ¢ ¯ (1 − x)K, ¯ (1 − x)K, ¯ xK, ¯ z1 , z∗1 . Suppose that in For example, in period one the state variable is xK, equilibrium ∗ ∗ (100) cˆ1 = γc1 zˆ1 + γc1 zˆ1 + O(ε 2 ) ∗ . We can write the equilibrium cˆ in this form since zˆ = zˆ∗ = 0 for all t ≤ 0. By for some γc1 and γc1 t 1 t ∗ exchanging zˆ1 for zˆ1 , we obtain that ∗ (101) cˆ∗1 = γc1 zˆ∗1 + γc1 zˆ1 + O(ε 2 ).

Therefore, we obtain (102) cˆ1 − cˆ∗1 = γc (ˆz1 − zˆ∗1 ) + O(ε 2 ), ∗. where γc ≡ γc1 − γc1 We will apply the same logic for pˆ1 . We can write the relative price in period one as

(103) pˆ1 = log p1 − log p¯ (104)

∗ ∗ = log(x f ,2 − (1 − δ )x f ,1 ) − log(xh,2 − (1 − δ )xh,1 )

(105)

∗ ¯ − log(xh,2 ¯ = log(x f ,2 − (1 − δ )(1 − x)K) − (1 − δ )(1 − x)K),

where the first equality is from the definition of pˆ1 , the second is from (45) and p¯ = 1, and the third ∗ = (1 − x)K. ¯ Suppose that in equilibrium is from x f ,1 = xh,1 ∗ ∗ ¯ = γ p1 zˆ1 + γ p1 (106) log(x f ,2 − (1 − δ )(1 − x)K) zˆ1 + O(ε 2 ),

19

∗ . By exchanging zˆ for zˆ∗ , we obtain that for some γ p1 and γ p1 1 1 ∗ ∗ ¯ = γ p1 zˆ∗1 + γ p1 (107) log(xh,2 − (1 − δ )(1 − x)K) zˆ1 + O(ε 2 ).

Therefore, we obtain (108) pˆ1 = γ p (ˆz1 − zˆ∗1 ) + O(ε 2 ), ∗ . where γ p ≡ γ p1 − γ p1 Plugging (102) and (108) into (97), we obtain

(109) (1 − δ )γ p2 + σ γc {α K¯ α + (1 − δ )γ p } = 0. Thus, we need to find cˆ1 − cˆ∗1 and pˆ1 to a first-order approximation as a function of the model parameters, zˆ1 − zˆ∗1 , and x. Then (109) is an equation with model parameters and x and thus it will determine x. In the Appendix we show the calculation method in details.

5.

QUANTITATIVE EXPLORATION

In the model there are five parameters, α , β , δ , σ , and π p − πs . We set the period length to be a quarter. We use the standard values for the first three parameters, α = .36, β = .99, and δ = .025. Since σ and π p − πs are the key parameters in our model, we will compute the equilibrium equity positions for several values of the parameters. More precisely, we experiment for a wide range of σ for three values of π p − πs . The first value follows Backus, Kehoe, and Kydland (1992) (BKK), π p − πs = .906 − .088 = .818, which is widely used in the international real business cycle models. The second one follows Baxter and Crucini (1995) and Kollmann (1996). They did not find the international spillover of productivity shocks and focused on π p − πs = .95 − .0 = .95, which we call High Persistence. The last one is π p − πs = .7, Low Persistence. Figure 5 shows the equity positions, x, as a function of σ . We see that for a wide range of σ and for all the three values of π p − πs , the model predicts substantial equity home bias. The more persistent the technology process, the larger the bias. Compared with the U.S. equity holdings, the experiment is over-predicting the home bias unless the risk aversion parameter is very large. To investigate the role of the infinite horizon, we calculate the zeroth-order equity positions in a T -period model by setting Kˆ 1r = E1 [Kˆ Tr ] = 0. Following BKK, we used σ = 2 and π p − πs = .818. Figure 6 shows the equity position as a function of T . We can see that the longer the horizon, the more the equity home bias. When many periods remain, investors can smooth their consumption

20

5 High Persistence BKK Low Persistence

4.5 4

Equity Positions

3.5 3 2.5 2 1.5 1 0.5 0

0

2

4 6 Risk Aversion Parameters

8

10

F IGURE 5.— The equity position and risk aversion.

1.5

Equity Positions

W/ Frictions W/O Frictions

1

0.5

0

50

100 Numbers of Periods

F IGURE 6.— The equity position in a T -period model.

21

150

1.5 Output Response

Consumption Response

0.06 0.04 0.02 0

W/ Frictions W/O Frictions 1

0.5

0 10

20 30 Quarters

40

50

10

20 30 Quarters

40

50

40

50

Price Response

Capital Response

0.1 1

0.5

0 10

20 30 Quarters

40

0.05 0 −0.05 −0.1

50

10

20 30 Quarters

F IGURE 7.— The response of the difference between Home and Foreign variables to 1% increase of ε1 − ε1∗ . On the vertical axis are corresponding % deviation from the non-stochastic steady state. intertemporally and thus the consumption does not fluctuate much. That is, they suffer less from the income fluctuation. Therefore, investors care less about risk sharing and hold more home equities in order to increase the expected income. Figure 7 shows the impulse response functions of the differences between Home and Foreign variables when σ = 2 and π p − πs = .818. In contrast to the model without the financial frictions, capital stocks do not respond much. This is because investors have to reduce their consumption to increase the investment due to the financial frictions. And the consumption response is hump-shaped. At the beginning, the Home investor increases physical capital investment due to high productivity and thus the consumption increases only slowly. Later, as productivity decreases, investors do not invest much. In this model, investors need to produce capital goods to reallocate their consumption intertemporally. With a good shock, the Home investor becomes rich but the consumption difference across countries diminishes. This is because the saving through capital production is inefficient in the long run and thus the Home investor reduces the saving or the capital production. This explains why the consumption difference is hump-shaped and why the capital stocks decreases slowly. At the beginning, the price of the Home capital is higher than that of the Foreign capital because the Home capital stock does not increase fast enough compared with the productivity difference.

22

Later, the Home capital stock is too much compared with the productivity since the Home investor is saving through capital production to smooth his consumption intertemporally. Thus, the Foreign capital becomes more expensive.

6.

EXTENSIONS AND DISCUSSIONS

6.1. An implication for the composition of the foreign equities Michaelides (2003) reported that, although a model in the paper might explain the degree of home bias for some parameter values, it creates another puzzle about the composition of the foreign equities. For example, the model implies that U.S. investors should not prefer Canadian equities to European or Japanese ones mainly because Canadian equity market is more correlated with the U.S. market than other markets are. Tesar and Werner (1995) reported that people in the U.S. were biased toward Canadian equities against European ones, controlling for the equity-market sizes. We investigate our model’s prediction about the composition of the foreign equities. We will examine it with a three-period full-depreciation model as in Section 3. Backus, Kehoe, and Kydland (1992) reported the differences between persistence and spillover of the U.S. productivity are .904 − .052 = .852 with European aggregate and .796 − .131 = .665 with Canadian. If the logic in the two-country model continues to hold, the U.S. should be less biased toward home equities against Canadian equities than against European equities. This story, however, may not be obvious because if the Home investor is holding lots of Home equities, he may want to focus on consumption smoothing with foreign equities. That is, the Home investor may not want to hold an equity whose returns are correlated with the returns to Home equities. The model is a symmetric four-country model with five goods, one consumption good and four country-specific capital goods. Countries are labeled as Home, a, e, and e0 . As in Section 3, investors live in three periods. Home and country a are far from countries e and e0 in terms of productivity 0 spillover. The technology shock in period one (log z1 , log za1 , log ze1 , log ze1 ) follows a distribution whose mean is zero and covariance is   ε 2 ρn ε 2 ρ f ε 2 ρ f ε 2   ρn ε 2 ε 2 ρ f ε 2 ρ f ε 2   (110)  ρ ε 2 ρ ε 2 ε 2 ρ ε 2  , n f  f  2 2 2 2 ρ f ε ρ f ε ρn ε ε where ρn , ρ f ∈ [−1, 1). Here n and f denote, respectively, “near” and “far.” As before, we assume

23

that all uncertainty is resolved in period one. Thus, the productivity in period two is 

  π p πn π f π f log z1       log za2  πn π p π f π f   log za1      (111)   log ze  = π π π π   log ze  , p n   f f  2 1 0 0 e e log z2 π f π f πn π p log z1 log z2





where we are focusing on the case that the spillover from a near country is greater than that from a far country, π p > πn > π f . We choose this four-country model so that the countries are symmetric. a + xe + xe0 = K. ¯ Due In period zero, a Home investor chooses equity position such that xh,1 + xh,1 h,1 h,1 0 e = xe in equilibrium. Define x ≡ x /K, n a ¯ and x f ≡ xe /K. ¯ Then to the symmetry, xh,1 h,1 ¯ x ≡ xh,1 /K, h,1 h,1 x = 1 − xn − 2x f . As before, we assume, without loss of generality, that in period one the Home investor chooses to hold only Home equities. Then the Home investor solves (112)

max 0

a ,xe ,xe ,c ,c ,x xh,1 h,1 h,1 1 2 h,2

(113)

E[u(c1 ) + β u(c2 )]

a e e0 a subject to c1 + xh,2 = α z1 K¯ α −1 (K¯ − xh,1 − xh,1 − xh,1 ) + α za1 K¯ α −1 xh,1 0

e e +α za1 K¯ α −1 xh,1 + α za1 K¯ α −1 xh,1 + w1 ,

(114)

¯ c2 = α z2 K2α −1 xh,2 + w2 − K.

As shown in the Appendix, the equilibrium equity positions are 1 β (1 − αβ − ασ )[π p − πn + 2(π p − π f )] + , 4 4[(1 − α )(1 − αβ ) + α 2 σ ] 1 β (1 − αβ − ασ )[πn − π p + 2(πn − π f )] (116) xn = + , 4 4[(1 − α )(1 − αβ ) + α 2 σ ] 1 β (1 − αβ − ασ )[π f − π p + π f − πn ] (117) x f = + . 4 4[(1 − α )(1 − αβ ) + α 2 σ ] (115) x =

Therefore, the relative equity positions are

β (1 − αβ − ασ )(π p − πn ) , [(1 − α )(1 − αβ ) + α 2 σ ] β (1 − αβ − ασ )(π p − π f ) , (119) x − x f = [(1 − α )(1 − αβ ) + α 2 σ ] β (1 − αβ − ασ )(πn − π f ) . (120) xn − x f = [(1 − α )(1 − αβ ) + α 2 σ ] (118) x − xn =

24

Note that their signs are the same, (121) sign(x − xn ) = sign(x − x f ) = sign(xn − x f ). That is, when the Home investor is biased toward Home equities against foreign equities, he is also biased toward equities of the near country against those of far countries, which is consistent with the evidence that Tesar and Werner (1995) reported. Julliard (2002) reported that the home bias predicted in a partial equilibrium model is greater than that predicted in Baxter and Jermann (1997). The paper, however, found that the ranking of U.S. investors’ shares for foreign equities should be Japanese, U.K., and German equities whereas the ranking is U.K., Japanese, and German in Coordinated Portfolio Investment Survey data in all years of 2001–2007. We measured the persistence and spillover of U.S. productivity shocks2 and obtained Germany

US (122) log zt+1 = .779 log ztUS + .172 log ztUK + .052 log zt

(.048)

(.045)

Japan

− .024 log zt

(.023)

+ εt+1 ,

(.011)

where the standard errors are in the parentheses. Ahearne, Griever, and Warnock (2004) reported that relative weights in US portfolio relative to market capitalization are 1.86 for U.S., .21 for U.K., .15 for Germany, .12 for Japan. Thus, our model predicts the same raking of the relative weights with that in the data.

6.2. Labor income risk We have not modeled about the labor income and assumed that it dose not depend on countryspecific shocks. In the standard one-good model, Cobb-Douglas production technology implies that the dividends and labor incomes are perfectly correlated. Baxter and Jermann (1997) reported that investors should short sell the home equities to hedge the labor income risk. To investigate, we set that the labor incomes of the Home and Foreign investor are (123) wt = (1 − α )[(1 + φ )zt Ktα + (1 − φ )zt∗ Kt∗α ]/2, (124) wt∗ = (1 − α )[(1 − φ )zt Ktα + (1 + φ )zt∗ Kt∗α ]/2, 2 The

seasonally-adjusted-quarterly data are from 1976Q1–2009Q1, 133 periods. The civilian employment data are from OECD Main Economic Indicators: Labour Force Survey based Statistics and the GDP data are from International Financial Statistics of the International Monetary Fund. The productivity is measured as log zt = log yt − .64 log nt , where yt is GDP and nt employment. We subtract linear trends from the productivity series before the estimation.

25

1.5 W/ Frictions W/O Frictions

Equity Positions

1

0.5

0

−0.5

0

0.2

0.4 0.6 Degrees of Labor Income Risks

0.8

1

F IGURE 8.— The equity positions and the degrees of the labor income risk, φ . where φ ∈ [0, 1] represents the country-specific labor income risk. Previously in this paper, we assumed that φ = 0. Without the financial frictions investors will hold equities that insure the countryspecific risks in the total income. That is, (125) xα − (1 − x)α = φ (1 − α ). Therefore, the optimal equity positions are (126) x =

1 φ (1 − α ) − . 2 2α

If φ > 0, to hedge the labor income risks investors hold Foreign-biased equities without the financial frictions. Figure 8 shows the differences between equity positions with and without the financial frictions. The gap of the two graphs represents how much the financial frictions help to explain the equity home bias.

26

6.3. Monitoring costs We assumed that investors should self-finance the capital-good production although the investors do not pay any costs for holding equities. We may relax the self-finance constraint and assume that a portion of the borrowing is used as a monitoring cost. For example, suppose that the Home investor lend bt > 0 to the Foreign investor. Then the Foreign investor can produce the Foreign capital as an amount of (1 − τ )bt , where the monitoring cost is τ bt for τ ∈ [0, 1]. No arbitrage condition is that the Foreign investor has to repay (1 − τ )bt Foreign capital goods to the Home investor. Then the budget constraint of the Home investor is (127) ct + xh,t+1 + x f ,t+1 /pt ≤ [α zt Ktα −1 + (1 − δ )]xh,t + [α zt∗ Kt∗α −1 + (1 − δ )/pt ]x f ,t + w +[(1 − τ )/pt − 1]bt . Similarly, the budget constraint of the Foreign investor is ∗ ∗ (128) ct∗ + pt xh,t+1 + x∗f ,t+1 ≤ [α zt Ktα −1 + pt (1 − δ )]xh,t + [α zt∗ Kt∗α −1 + (1 − δ )]x∗f ,t + w

+[(1 − τ )pt − 1]bt∗ . The lending must be non-negative bt ≥ 0 and bt∗ ≥ 0 and thus in equilibrium (129) (1 − τ )/pt − 1 ≤ 0,

with equality if bt > 0,

(130) (1 − τ )pt − 1 ≤ 0,

with equality if bt∗ > 0.

That is, [(1 − τ )/pt − 1]bt = 0 and [(1 − τ )pt − 1]bt∗ = 0 in equilibrium. Thus, in equilibrium, the prices must be (131) 1 − τ ≤ pt ≤ 1/(1 − τ ) since, otherwise, the investors will choose an arbitrary large amount of bt or bt∗ . Note that log pt = O(ε ). Unless τ = O(ε ), 1 − τ < pt < 1/(1 − τ ) and thus bt = bt∗ = 0. That is, investors will not borrow or lend since the cost is too high. Since bt = O(ε ) and bt∗ = O(ε ), τ = O(ε ) implies the monitoring cost is O(ε 2 ). Therefore, the equilibrium dose not change with the moderate reduction of the monitoring costs.

27

8

ARGTUR

RUS

6

IDN

IND PHL THA

4

BRA

MYS

POL

ISR

KOR GRC JPN

0

2

CAN ESP ZAF CHL AUS USA SGP HKG FRA GBR FIN PRT DNK SWE DEU ITA BEL CHE AUT NLD

3

4

5

6

ln(credit to gdp) −ln(1−homebias)

Fitted values

F IGURE 9.— The relationship between homebias and credit-to-GDP ratios. The fitted equation is HB = −2.740 × ln(credit to GDP) + 15.318. The three letters next to the points are the ISO 3166-1 alpha-3 codes in Appendix.

7.

EMPIRICAL ANALYSIS

This section investigates how important the financial frictions are in explaining the home bias. We regress a variable for homebias on the easiness to access to credit. As Ahearne, Griever, and Warnock (2004), we define homebias as (1- foreign share in country’s portfolio / foreign share in the global portfolio). Since homebias is, technically, bounded above by one, we transform monotonically into HB = − ln(1 − homebias). To measure the easiness to access to credit, we use the domestic credit provided by banking sector to GDP ratios. To eliminate short-term fluctuation, we average the sevenyear values from 2001 to 2007. The regression coefficient is significantly negative with t-value -5.88. Figure 9 shows the relationship. Since countries in the sample may be different in accessing the global equity market, we include the financial openness. We use the indices developed in Chinn and Ito (2008). The indices are updated to 2008 and publicly available. As before, we averaged the values from 2001 to 2007. As expected, the financial openness is significantly associated with the home bias. Controlling for the financial openness, the credit ratios still matter in the home bias. In the model, we distinguish corporate firms from entrepreneurs. One major difference between them is that it is easier for corporate firms to access credit than for entrepreneurs. For example,

28

TABLE I R EGRESSION R ESULTS Dependent variable: -ln(1-homebias) Variables (1) (2) (3) (4) ∗∗∗ ∗∗∗ ∗∗∗ ln(credit to GDP) -2.740 -1.418 -1.277 -1.101∗∗∗ (0.466)

financial openness

(0.379)

(0.396)

(0.351)

-0.973∗∗∗

-0.969∗∗∗

-0.539∗∗

(0.155)

(0.155)

(0.197)

market capitalization to GDP

-.003 (0.003)

-0.721∗∗∗

ln(GDP per capita)

(0.234)

constant

15.318∗∗∗

10.600∗∗∗

10.229∗∗∗

15.360∗∗∗

(2.174)

(1.649)

(1.673)

(2.127)

Number of observations 34 2 R 0.520 Note: Standard errors are in parentheses.

34 34 34 0.788 0.797 0.839 ∗∗∗ p < .01, ∗∗ p < .05.

corporate firms can issue equities. This paper emphasizes that the financial frictions on entrepreneurs imply the home bias. To see this point, we include market capitalization to GDP ratios, which measures how easy for corporate firms to access the credit. Its coefficient, shown in regression (3), is not significant. The regressors considered so far may capture only partially how easy to invest domestic and foreign equity. Thus, we include the GDP per capita to capture the degree of easiness to access financial markets. Since the credit ratios are a component of the degree, we expect that the regression underestimates the importance of the credit ratios. We still get the significant coefficient for the credit ratios with t-value -3.14. In sum, although these regressions do not necessarily imply that the financial frictions on entrepreneurs cause the home bias, they are in line with the model’s prediction.

8.

CONCLUSION

We have studied how the financial frictions affect the resource allocation and the equity positions. While most researches have focused on the consumption risk sharing, we claim that there is a tradeoff between the consumption insurance and production efficiency if the market is incomplete due to the financial frictions. For standard parameter values, we have shown that the financial frictions help to explain the equity home bias in a dynamic stochastic general equilibrium model. This claim is also supported in cross-country regressions. 29

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32

APPENDIX

Data 1. TFP process and U.S. foreign-equity positions • Countries: United States, United Kingdom, Germany, and Japan. • Seasonally-adjusted real GDP from 1976Q1 to 2009Q1 are from the International Financial Statistics of the International Monetary Fund. • The civilian employment data from 1976Q1 to 2009Q1 are from OECD Main Economic Indicators: Labour Force Survey based Statistics. • U.S. foreign-equity positions from 2001 to 2007 are from the Coordinated Portfolio Investment Survey. 2. Regressions • Countries (codes): Argentina (ARG), Australia (AUS), Austria (AUT), Belgium (BEL), Brazil (BRA), Canada (CAN), Chile (CHL), Denmark (DNK), Finland (FIN), France (FRA), Germany (DEU), Greece (GRC), Hong Kong (HKG), India (IND), Indonesia (IDN), Israel (ISR), Italy (ITA), Japan (JPN), Korea (KOR), Malaysia (MYS), Netherlands (NLD), Philippines (PHL), Poland (POL), Portugal (PRT), Russian Federation (RUS), Singapore (SGP), South Africa (ZAF), Spain (ESP), Sweden (SWE), Switzerland (CHE), Thailand (THA), Turkey (TUR), United Kingdom (GBR), United States (USA) • The equity asset and liability positions from 2001 to 2007 are from the International Financial Statistics of the International Monetary Fund. • Domestic credit provided by banking sector (% of GDP), Market capitalization of listed companies (current US$), and GDP per capita (current US$) from 2001 to 2007 are from the World Bank. • Financial openness data developed by Chinn and Ito (2008) from 2001 to 2007 are from http://www.ssc.wisc.edu/~mchinn/research.html.

The equilibrium equity position in the dynamic model Calculation method Here we will show how to find cˆ1 − cˆ∗1 and pˆ1 to a first-order approximation for a given x. Since c1 , c∗1 , and p1 are in information set (ˆz1 , zˆ∗1 ), we only consider expected values 33

conditional on (ˆz1 , zˆ∗1 ). First, we will calculate cˆ1 − cˆ∗1 to a first order approximation. Note that first-order conditions include (132) Et [u0 (ct+1 )(rh,t+1 − r f ,t+1 )] = 0, ∗ ∗ (133) Et [u0 (ct+1 )(rh,t+1 − r∗f ,t+1 )] = 0.

Thus, we obtain (134) E1 [ˆrh,t+1 − rˆ f ,t+1 ] = O(ε 2 ), ∗ (135) E1 [ˆrh,t+1 − rˆ∗f ,t+1 ] = O(ε 2 ),

for t ≥ 1. That is, we can ignore investors’ choice between Home and Foreign equities since the expected returns are the same to a first-order approximation. In other words, the portfolio choice dose not affect the conditional expectation of budget constraints to a first-order approximation. Formally, ∗ we can choose a sequence of {xh,t+1 , x f ,t+1 , xh,t+1 , x∗f ,t+1 } for t ≥ 1 subject to (136)

∗ xh,t+1 + xh,t+1 = Kt+1 ,

(137)

∗ x f ,t+1 + x∗f ,t+1 = Kt+1 ,

∗ ∗ (138) x f ,t+1 − (1 − δ )x f ,t = pt [xh,t+1 − (1 − δ )xh,t ].

That is, they are consistent with market-clearing conditions and price conditions.3 In this paper ¯ Other equity positions are determined by the we choose x f ,t+1 = (1 − δ )x f ,t = (1 − δ )t (1 − x)K. constraints (136)–(138). That is, ¯ (139) xh,t+1 = Kt+1 − (1 − δ )t (1 − x)K, ∗ ¯ (140) xh,t+1 = (1 − δ )t (1 − x)K, ∗ ¯ (141) x∗f ,t+1 = Kt+1 − (1 − δ )t (1 − x)K.

Here we have chosen a sequence of equity positions where investors do not trade equities from period one. These equations for the equity positions also hold in period zero. The budget constraints, (40) 3 For

our purpose, these conditions are stricter than needed. The conditions need to be hold up to a first-order approx-

imation.

34

and (41), become ¯ (142) ct + Kt+1 − (1 − δ )t (1 − x)K¯ = (α zt Ktα −1 + 1 − δ )[Kt − (1 − δ )t−1 (1 − x)K] +α zt∗ Kt∗α −1 (1 − δ )t−1 (1 − x)K¯ + wt , ∗ ¯ (143) ct∗ + Kt+1 − (1 − δ )t (1 − x)K¯ = (α zt∗ Kt∗α −1 + 1 − δ )[Kt∗ − (1 − δ )t−1 (1 − x)K]

+α zt Ktα −1 (1 − δ )t−1 (1 − x)K¯ + wt . Since investors do not trade from period one, the relative prices disappear in the budget constraints. Taking conditional expectations on the first-order conditions yields £ ¤ E1 [−σ cˆt ] = E1 − σ cˆt+1 + αβ K¯ α −1 {ˆzt+1 + (α − 1)Kˆt+1 } + O(ε 2 ), £ (145) E1 [c¯cˆt + K¯ Kˆt+1 ] = E1 α K¯ α {1 − (1 − δ )t−1 (1 − x)}{ˆzt + (α − 1)Kˆt } + (α K¯ α −1 + 1 − δ )K¯ Kˆt ¤ +α K¯ α (1 − δ )t−1 (1 − x){ˆzt∗ + (α − 1)Kˆt∗ } + w¯ wˆ t + O(ε 2 ), £ ¤ ∗ ∗ ∗ (146) E1 [−σ cˆt∗ ] = E1 − σ cˆt+1 + αβ K¯ α −1 {ˆzt+1 + (α − 1)Kˆt+1 } + O(ε 2 ), £ ∗ (147)E1 [c¯cˆt∗ + K¯ Kˆt+1 ] = E1 α K¯ α {1 − (1 − δ )t−1 (1 − x)}{ˆzt∗ + (α − 1)Kˆt∗ } + (α K¯ α −1 + 1 − δ )K¯ Kˆt∗ ¤ +α K¯ α (1 − δ )t−1 (1 − x){ˆzt + (α − 1)Kˆt } + w¯ wˆ t + O(ε 2 ). (144)

Subtracting the first-order conditions for the Foreign investor from those for the Home investor, we obtain (148)

r r r ] + (α − 1)E1 [Kˆt+1 − σ E1 [cˆtr ] = −σ E1 [cˆt+1 ] + αβ K¯ α −1 {E1 [ˆzt+1 ]} + O(ε 2 ),

r ¯ 1 [Kˆt+1 ] = α K¯ α {1 − 2(1 − δ )t−1 (1 − x)}{E1 [ˆztr ] + (α − 1)E1 [Kˆtr ]} (149) cE ¯ 1 [cˆtr ] + KE

¯ 1 [Kˆtr ] + O(ε 2 ), +(α K¯ α −1 + 1 − δ )KE where variables with superscript r are differences between Home and Foreign variables. The transversality condition implies that the system should converge to the non-stochastic steady state along the r ] = 0. Solving the difference equations stable saddle path in the long run. That is, limt→∞ E1 [Kˆt+1 r ] = 0, we can obtain cˆr as a function of model pawith boundary conditions Kˆ 1r = limt→∞ E1 [Kˆt+1 1 r rameters, zˆ1 , and x. Now, we will show how to find pˆ1 . Since r r (150) O(ε 2 ) = E1 [ˆrh,t+1 − rˆ f ,t+1 ] = E1 [β α K¯ α −1 {zt+1 }− pˆt +(1− δ )β pˆt+1 ]+O(ε 2 ), +(α −1)Kˆt+1

35

we obtain ∞

r r (151) pˆ1 = β α K¯ α −1 ∑ [β (1 − δ )]t−1 {E1 [ˆzt+1 ] + (α − 1)E1 [Kˆt+1 ]} + O(ε 2 ), t=1

r ] when we calculated cˆr . where we obtained E1 [Kˆt+1 1

A special case with a closed-form solution Consider the case of δ = 1 and σ → ∞ in a T -period model. In a T -period model, investors live from period zero to period T − 1 and they pay lump-sum tax K¯ in period T − 1. Alternatively, we may assume that investors produce capital goods as amount ¯ Since δ = 1, the main of iT −1 = i∗T −1 = K¯ and do not pay tax, which implies that KT = KT∗ = K. equation (109) becomes σ γc = 0 or σ cˆr1 = O(ε 2 ). r ]. (149) implies that if E1 [Kˆtr ] and E1 [cˆtr ] are finite to a first order approximation, then so is E1 [Kˆt+1 r ] are finite, then so is σ E [cˆr ]. The finiteness of σ E [cˆr ] (148) implies that if σ E1 [cˆtr ] and E1 [Kˆt+1 1 t+1 1 t+1 r 2 r r r ] ˆ implies that limσ →∞ E1 [cˆt+1 ] = O(ε ). In sum, if E1 [Kt ] and σ E1 [cˆt ] are finite, then so are E1 [Kˆt+1 r ] and lim r 2 ˆ 1r and σ E1 [cˆt+1 σ →∞ E1 [cˆt+1 ] = O(ε ). Applying this statement forward, we obtain that if K r ] = O(ε 2 ) for all t ≥ 2. Since K ˆ 1r = σ cˆr1 = O(ε 2 ), we obtain and σ cˆr1 are finite, then limσ →∞ E1 [cˆt+1 r ] = O(ε 2 ) for all t ≥ 1. That is, investors are fully insured from country-specific that limσ →∞ E1 [cˆt+1 shocks. Therefore, we can rewrite (149) as

(152)

E1 [Kˆ 2r ] = α K¯ α −1 (2x − 1)ˆzr1 + O(ε 2 ),

r (153) E1 [Kˆt+1 ] = α K¯ α −1 {E1 [ˆztr ] + α E1 [Kˆtr ]} + O(ε 2 ), for 2 ≤ t ≤ T − 2,

(154) E1 [Kˆ Tr ] = 0, where we used that Kˆ 1r = 0 and δ = 1. From (153) and (154), we obtain µ

µ ¶t−1 1 T −2 K¯ 1−α r r ˆ E1 [KT ] − ∑ E1 [ˆzt+1 ] 2 α t=1 α µ ¶ π p − πs T −2 β (π p − πs ) t−1 r =− zˆ1 , ∑ α α t=1

(155) E1 [Kˆ 2r ] = (156)

K¯ 1−α α2

¶T −2

r ] = (π − π )t zˆr . Solving this equation and (152) where we used that K¯ = (αβ )1/(1−α ) and E1 [ˆzt+1 p s 1 simultaneously, we obtain that the equity position is

· ¸ 1 1 T −2 β (π p − πs ) t (157) lim x = − ∑ . σ →∞ 2 2 t=1 α Thus, to be insured fully, investors should be biased toward Foreign equities and the bias is increasing 36

in π p − πs as we have seen in Section 3. If the shock is highly persistent such that π p − πs ≥ α /β , x goes to negative infinity as T goes to infinity.

Equilibrium in the four-country model The first-order conditions are (158) (159)

0

σ a −σ e −σ e 0 = E[c− 1 (z1 − z1 )] = E[c1 (z1 − z1 )] = E[c1 (z1 − z1 )], −σ α −1 σ c− , 1 = β c2 α z2 K2

0 (160) c1 + K2 = α z1 K¯ α (1 − xn − 2x f ) + α za1 K¯ α xn + α (ze1 + ze1 )K¯ α x f + w1 ,

(161)

¯ c2 = α z2 K2α + w2 − K,

where we used that K2 = xh,2 in equilibrium. Second-order approximations of (158) include 3 (162) E[ˆz1 − zˆa1 − σ cˆ1 (ˆz1 − zˆa1 ) + (ˆz21 − zˆa2 1 )/2] = O(ε ), 3 (163) E[ˆz1 − zˆe1 − σ cˆ1 (ˆz1 − zˆe1 ) + (ˆz21 − zˆe2 1 )/2] = O(ε ).

First-order approximations of other first-order conditions are − σ cˆ1 = −σ cˆ2 + zˆ2 + (α − 1)Kˆ 2 + O(ε 2 ), £ ¤ 0 (165) c¯cˆ1 + K¯ Kˆ 2 = α K¯ α zˆ1 (1 − xn − 2x f ) + zˆa1 xn + (ˆze1 + zˆe1 )x f + w¯ wˆ 1 + O(ε 2 ), (164) (166)

c¯cˆ2 = α K¯ α (ˆz2 + α Kˆ 2 ) + w¯ wˆ 2 + O(ε 2 ).

The Foreign counterparts include the first-order conditions for portfolio choices, 3 (167) E[ˆz1 − zˆa1 − σ cˆa1 (ˆz1 − zˆa1 ) + (ˆz21 − zˆa2 1 )/2] = O(ε ), 3 (168) E[ˆz1 − zˆe1 − σ cˆe1 (ˆz1 − zˆe1 ) + (ˆz21 − zˆe2 1 )/2] = O(ε ),

intertemporal consumption allocations, (169) − σ cˆa1 = −σ cˆa2 + zˆa2 + (α − 1)Kˆ 2a + O(ε 2 ), (170) −σ cˆe1 = −σ cˆe2 + zˆe2 + (α − 1)Kˆ 2e + O(ε 2 ), budget constraints in period one, £ ¤ 0 (171) c¯cˆa1 + K¯ Kˆ 2a = α K¯ α zˆa1 (1 − xn − 2x f ) + zˆ1 xn + (ˆze1 + zˆe1 )x f + w¯ wˆ 1 + O(ε 2 ), £ ¤ 0 (172) c¯cˆe1 + K¯ Kˆ 2e = α K¯ α zˆe1 (1 − xn − 2x f ) + zˆe1 xn + (ˆz1 + zˆa1 )x f + w¯ wˆ 1 + O(ε 2 ), 37

budget constraints in period two, (173) c¯cˆa2 = α K¯ α (ˆza2 + α Kˆ 2a ) + w¯ wˆ 2 + O(ε 2 ), (174) c¯cˆe2 = α K¯ α (ˆze2 + α Kˆ 2e ) + w¯ wˆ 2 + O(ε 2 ). Note that we do not need to use the first-order conditions of the investor in e0 country to calculate e = xe0 . Subtracting the first-order conditions of the Foreign equity positions since we know that xh,1 h,1 investors from those of the Home investor, we obtain (175)

0 = E[(cˆ1 − cˆa1 )(ˆz1 − zˆa1 )] + O(ε 3 ),

(176)

0 = E[(cˆ1 − cˆe1 )(ˆz1 − zˆe1 )] + O(ε 3 ),

(177)

−σ (cˆ1 − cˆa1 ) = −σ (cˆ2 − cˆa2 ) + (ˆz2 − zˆa2 ) + (α − 1)(Kˆ 2 − Kˆ 2a ) + O(ε 2 ),

(178)

−σ (cˆ1 − cˆe1 ) = −σ (cˆ2 − cˆe2 ) + (ˆz2 − zˆe2 ) + (α − 1)(Kˆ 2 − Kˆ 2e ) + O(ε 2 ),

¯ Kˆ 2 − Kˆ 2a ) = α K¯ α (ˆz1 − zˆa1 )(1 − 2xn − 2x f ) + O(ε 2 ), (179) c( ¯ cˆ1 − cˆa1 ) + K( 0

¯ Kˆ 2 − Kˆ 2e ) = α K¯ α [(ˆz1 − zˆe1 )(1 − xn − 3x f ) + (ˆza1 − zˆe1 )(xn − x f )] + O(ε 2 ), (180) c( ¯ cˆ1 − cˆe1 ) + K( (181)

c( ¯ cˆ2 − cˆa2 ) = α K¯ α [(ˆz2 − zˆa2 ) + α (Kˆ 2 − Kˆ 2a )] + O(ε 2 ),

(182)

c( ¯ cˆ2 − cˆe2 ) = α K¯ α [(ˆz2 − zˆe2 ) + α (Kˆ 2 − Kˆ 2e )] + O(ε 2 ).

Solving the eight equations simultaneously, we obtain the equilibrium equity positions, 1 β (1 − αβ − ασ )[π p − πn + 2(π p − π f )] + , 4 4[(1 − α )(1 − αβ ) + α 2 σ ] 1 β (1 − αβ − ασ )[πn − π p + 2(πn − π f )] (184) xn = + , 4 4[(1 − α )(1 − αβ ) + α 2 σ ] 1 β (1 − αβ − ασ )[π f − π p + π f − πn ] (185) x f = + . 4 4[(1 − α )(1 − αβ ) + α 2 σ ] (183) x =

38