Welfare gains from international risksharing

JOURNALOF

Monetary ELSEVIER

ECONOMICS

Journal of Monetary Economics 34 (1994) 175-200

Welfare gains from international

risksharing

Eric van Wincoop

(Received

September

1992: final version

received July 1994)

Abstract

This paper uses consumption data to compute yet unexploited welfare gains that can be achieved through risksharing among twenty OECD countries. There is both aggregate national consumption risk and nondiversifiable individual-specific risk. Countries engage in an optimal risksharing arrangement to pool risk associated with the aggregate consumption endowment streams. Welfare gains are associated with reduced consumption variability and with the international pricing of the consumption streams of individual countries. The paper considers the standard VNM time-additive preferences, as well as nonexpected utility and habit formation preferences. Large unexploited gains are found under all sets of preferences, leading to an ‘international risksharing puzzle’. Keys lrord.7: Welfare gains; International JEL c/assjfication:

risksharing

F30; F41; G15

1. Introduction

The degree of international capital mobility is a key element in dynamic open economy models. Today many models assume perfect international capital mobility, justified by the elimination of binding restrictions on international

The final draft of this paper was written while I wasa research fellow at the Innocenzo Gasparini Institute for Economic Research in Milan. Funding through a Human Capital Mobility Grant by the European Community is gratefully acknowledged. I would like to thank the referee. Maurice Obstfeld, Paul Beaudry, Laurence Kotlikoff, Janathan Eaton, Douglas Gale, Russell Cooper. Robert Rosenthal, V.V. Chari, and seminar participants at Boston University. the Federal Reserve Bank of Minneapolis, the University of Lausanne. and the 1992 Econometric Soaety Meetings in Brussels and Seattle for helpful comments and suggestions.

0304-3932.94,PO7.00 (3 1994 Elsevler Science B.V. All rights reserved SSDI 0304393294Ol158 3

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capital flows by a majority of the industrialized countries. But two important and independent pieces of evidence indicate a very low degree of capital mobility, even among the developed countries. First, the Feldstein-Horioka puzzle points to the high correlation between savings and investment in both time-series and cross-country data. High savers, like Japan, are also high investors. And low savers, like the US, are also low investors. Second, there is very limited international portfolio diversification. At the end of 1989 US investors held 94% of their equity portfolio in US stock, and the Japanese held 98% of their equity portfolio in Japanese stock.’ Investors are apparently not attempting to reap the benefits of diversification available from holding a global portfolio. While an extensive literature has developed trying to explain the first puzzle, much less work has been done to account for the second. There are a number of explanations for the second puzzle, but none very convincing. First, there is exchange rate risk. But that risk can be covered through the forward market. Second, there may be tax differences. But domestic residents receive tax credits on foreign witholding taxes. There seems to be a greater tax advantage to holding foreign than domestic assets since it is easier to underreport foreign investment income. Third, there is a risk of expropriation associated with investing abroad. But when limiting the world to the industrialized countries, this risk does not seem big. Fourth, transaction costs may be large when investing in foreign markets. But in the US in 1989 gross foreign equity purchases were twenty times net purchases. Fifth, capital controls may be binding. But how can one then explain the fact that foreigners were on net selling US and Japanese assets in some years during the 198Os?’ Finally, it might be that investors are more optimistic about domestic assets. But French and Poterba (1991) show that, for example, British investors need to have a 5% higher expected return on British stock than US investors in order to account for portfolio differences. Such large differences in information sets seem unlikely. This paper will measure the unexploited potential welfare gains from international risksharing. If those gains are small, it might not be worthwhile to hold a globally diversified portfolio given small costs of investing abroad. Various authors have pointed out that investors can reap gains from holding a global portfolio, but have not attempted to measure such gains. In some early studies on this issue, Grubel(1968) and Levy and Sarnat (1970) find that the optimal US portfolio is very different than a portfolio consisting just of US stock, and Solnik (1974) finds that a global portfolio would be half as risky as a portfolio of only US stock. Looking at the economic fundamentals, Golub (1990) argues that there must be significant gains from international risksharing between the

‘See French and Poterba (1991). Golub evidence of the home bias in portfolios. ZSee French

and Poterba

(1990) and Tesar

(1990) for evidence

on the fourth

and

Werner

(1992) provide

and fifth arguments.

further

E. van Wincoop

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177

US and Japan based on a negative correlation between US and Japanese corporate profits. Cole and Obstfeld (1991) explicitly calculate the welfare gains from a risksharing arrangement between the US and Japan. They base their analysis on stochastic properties of production data.3 Assuming that both countries specialize in the production of one good and that consumers have a standard timeadditive utility function with a CES subutility index, they calculate potential gains from risksharing. These gains are small due to various factors. First, there are only two countries involved in risksharing. Second, endogenous terms of trade fluctuations pool some of the risk associated with productivity shocks. For example, a positive productivity shock in the US lowers the relative price of US goods, and therefore the US terms of trade. In the case of Cobb-Douglas utility there are no gains at all from risksharing. But, as mentioned by Golub (1990) a strong positive correlation between real national output and the terms of trade is observed in the data. Third, their parameterization implies a very high riskfree interest rate, leading to significant discounting of future risk. And finally, their stochastic setup leads to higher persistence of global than country-specific production shocks. An increase in persistence of production shocks reduces welfare. In this paper I use consumption data to measure potential gains from international risksharing for twenty OECD countries. This seems a natural approach since, after all, it is consumption variability that consumers are supposed to care about. The standard deviation of the growth rate of tradeables consumption of the average OECD country is twice as large as that for the aggregate of the twenty OECD countries. This indicates ample opportunities for risksharing. There are two types of risk: aggregate national consumption risk and nondiversifiable individual-specific risk (e.g., associated with a specific person’s career). Countries engage in an optimal risksharing arrangement to pool the risk associated with aggregate consumption endowment streams. Note that the stochastic properties of currently observed consumption streams may already reflect a certain degree of risksharing. In this paper, however, we are interested in potential additional gains that can be achieved through riskpooling. It is necessary to choose a particular type of preferences in order to evaluate the welfare gains from optimal international risksharing. Since the empirical relevance of the standard von NeumannMorgenstern (VNM) time-additive

“They do not dlstmguish versus nontradeables.

between

‘Even if there is no risksharing investment abroad or through assets.

the production

through current

of durables

versus nondurables

and tradeables

the stock market, there may be risksharing through direct account fluctuations financed by trade in relatively safe

preferences has been questioned, I also study two generalizations of that framework: Kreps-Porteus nonexpected utility preferences and habit formation preferences. They allow for nonseparability of consumption across, respectively, states (of the world) and dates. The main message of the paper is that the welfare gains are large for all types of preferences. For parameterizations with a realistic size of the riskfree rate and rate of relative riskaversion (l-4) the average welfare gain across countries is equivalent to a permanent increase in consumption by 1.8&5.6%. This amounts to one to three times the size of the entire securities industry in the US. These large unexploited gains lead to what we might call an ‘international risksharing puzzle’. The remainder of the paper is set up as follows. Section 2 derives the gains from risksharing for VNM time-additive preferences. Section 3 discusses the international consumption data, paying special attention to the treatment of durables. Sections 4, 5, and 6 discuss the welfare gains from international risksharing for, respectively, VNM time-additive preferences, nonexpected utility preferences and habit formation preferences. Section 7 concludes and discusses directions for future research.

2. Welfare gains with time-additive Assume that there identical time-additive

expected utility preferences

are N countries. Consumers from each von Neuman-Morgenstern preferences:

country

have

(14

(lb) where E is the expectations operator. Aggregate consumption is a CES index of J consumption categories. When the consumption good is durable, cj refers to the services per unit of time provided by the durable good. It is assumed that utility from nontradeables is separable from utility of tradeables. Since risksharing with respect to nontradeables is not feasible, utility from nontradeables is omitted from (1). All J consumption catagories are therefore tradeable. In a separate paper (van Wincoop, 1994b), I consider the implications of nonseparability between tradeables and nontradeables consumption. All N economies receive stochastic endowments of the J consumption categories.’ In general it is possible that some degree of risksharing is already taking

“It is shown in Appendix

A that allowing for investment does not change the results much

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place today, so that domestic residents own claims on foreign endowments. For country n the sum of these domestic and foreign claims yields $ units of good j (or services of the durablej) per unit of time; ?J follows some stochastic growth process. There are I, individuals in country n, with individual i receiving a stochastic share iai of aggregate country n consumption. This share follows the stochastic process dvi = ri6 dw;, d$

(2)

dw; = c(vr, .

, v,,) dt

if

i # h,

dt

if

i = h.

=

Here WYis a standard Brownian motion, and E < 0 is defined in such a way that the sum of shares is constant at one. With perfect capital markets, the risks associated with these shares could be pooled so that everyone’s share would be constant. But I will assume that, for example due to moral hazard, there are no domestic or international markets to pool this type of risk. One can think specifically of risk associated with a particular individual’s career. For simplicity it will be assumed that the allocation of the consumption stream across the individuals of a country, based on the process (2) is done by the national governments of the N countries. Let pj be the price of good j. At each time individual i from country n maximizes consumption index (1b) subject to the budget constraint p, Ck” + .” + PJCt;” = vi(p,;‘;

+ “.

+ pJi’;),

so that Cn (.i.n

= Xj( pi/p<) =

ViCn

=

l,Qci, n,

r~i(p,

r; +

(34 “’

+

pJy3)/pc,

(3b)

(P - 1 VP

(3c) where pc is the consumer price index. Here ci-n refers to the consumption index of individual i from country n. cn is the aggregate country n consumption index, of which individual i receives a share vi. Since all goods are traded, there is equilibrium in the market for good j if world demand for good j equals world supply of good j. This leads to the following equilibrium relative prices (good J is the numeraire): Pi = .L(Y)> y = (yi,

‘..

j= -7’

2 ,J’

1, .” >J-

1,

(4)

N “.

The form of the functi0n.f;

>Y,

3 ...

> $7.

depends

on the parameters

of the utility

function.

180

E. uan Wincoop

It follows consumption assumed that matches this

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from (3) and (4) that we can write c” = c”(y). In most countries is reasonably well described by a random walk. It is therefore the stochastic process of y is such that the consumption process observation:

(5)

dc” = PC” dt + c,c” dw”,

where w”(t) is a standard Brownian motion. dw(t) = (dw’(t), . . . , dwN(t)) has a correlation matrix adt, with dt on the diagonal elements. It should be noted that Lucas (1987) found very small welfare gains from eliminating all consumption risk in the US based on a trend-stationary consumption process with innovations that are uncorrelated over time. Obstfeld (1991) finds that the welfare gains from completely eliminating all business cycle risk are still quite small when consumption is assumed to follow a stationary AR process. While empirically it is hard to say whether consumption is stationary or nonstationary, theory tells us that optimizing behaviour leads to a random walk in consumption.6 Using (2) and (5), the stochastic differential equation for consumption by individual i from country n is &i.

n

= /xi.”

dt + ci,“(~,

dw” + sdw;).

(6)

It is assumed that dwl dw” = 0, so that individual-specific risk is uncorrelated with aggregate risk. The stochastic consumption process (6) then leads to the following expected utility: 1 u i,n = _ ,l..(o)l ‘Y

-y/(1 _ ?),

(7)

2, = /I + (y - l)(p - OSy(cr,2 + 8’)). Now imagine that, starting at time 0, risksharing takes place. Only risk associated with the aggregate national consumption streams can be diversified. This is done by the governments of the individual countries on behalf of their citizens. After risksharing the government of country n receives some aggregate stochastic consumption stream, which is allocated among the individuals based on the same stochastic process for vi as in (2). Divide up the stochastic consumption stream of country n into c”(O) units, each with the same stochastic process (5). The price of one such unit is pi (p,” = 1). There is a competitive market of global mutual funds to which

‘One might argue that some individuals arc liquidity-constrained, which may lead to a stationary component in aggregate consumption. In a related paper (van Wincoop, 1993), I consider a model with two types of agents: workers, who are possibly liquidity-constrained, and capital owners. Potential welfare gains from international risksharing for the capital owners are similar to those reported in this paper.

E. oan Winroop ;I Journal of Morvzlar_r Economics 34 ilY94J

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governments can sell their domestic consumption assets, and from which shares can be purchased. Assume that a global mutual fund F buys 4°F country n stochastic consumption claims (n = 1, . . . , N). Now divide up the fund into qk + . . . + q; shares. The payoff of one share in fund F is the real stream of outputs

ey.cl(t)

RF(f) =

1

+

. . . +

ON-

F CN(0)’

c (0)

where e”, = qF/(qk

+ ...

d&(t)/&(t)

cN(t)

+ 4;).

Therefore,

= pddt + cl $’ dw’(t) + ... + .N$N dwN(t),

(9)

where $” =

HF?c”(W”(O) 1

R, follows a complicated stochastic process because the ratios c”(t)/c”(t) are themselves stochastic processes, so that the G’s are stochastic as well. However, numerical analysis indicates that when we hold the $‘s constant, the stochastic process of RF(t) is practically the same. In particular, the difference in expected utility from those two processes turns out to be negligible, which is what really matters. Since it is analytically much easier to work with constant rc/‘s,I will adopt this approximation, which implies Ic/” = e”,. One may refer to constant $‘s as implying a constant country exposure of the fund. Theoretically this may be justified if one thinks of the endowment streams as generated by stochastic growing capital stocks, which can be marginally reallocated between countries in order to keep country exposure constant. The government of country n can buy pic”(O)/8j-p; + ... + @pi) shares of fund F, the denominator being the price of one share. Individual i from

n

i from u,,i n * _ -7’ ’

i,n.*(o)l~y/(1

_ ?I,

w 2, = #8+ (y - l)(P - osy(0:

+ P)),

now becomes: (114 (1 lb)

(114

182

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where CJ~is the standard deviation of the return of one share of the mutual fund. Maximization of the utility function (1 la) over Ob, . . , 19~ ‘, subject to the constraints (1 lb), (1 lc), and (lOa), yields the following first-order conditions:

(12) n = 1,

,N -

1.

Notice that the optimal mutual fund is the same for all countries. Equilibrium prices are such that demand and supply of the national consumption assets are equalized: u;. = q%/(qk + ‘.’ + q;) = c”(O)/(c’(O) + “’ + CN(0)) = fl”, n=l, Solving (12) for equilibrium

prices, and rewriting,

. . ..N-

I.

yields (13)

n =

1, . . . , N - 1,

where g: is the growth rate of country n aggregate consumption before risksharing, and an its standard deviation. g,” is the growth rate of world consumption. It is clear that the price of one unit of the country n stochastic consumption stream is high if the growth rate of country n consumption has a low standard deviation and low correlation with the world consumption growth rate. Given these equilibrium prices, we can use (1 la) and (10a) to obtain the new utility level. In Appendix A it is shown that the gains from risksharing are practically the same if we allow for investment and restrict ourselves to global funds with a constant country exposure.’ The difference is that then the reduction in uncertainty leads to an immediate rise (drop) in the consumption level and a drop (rise) in the growth rate of consumption if ‘4 > 1 (y < 1). But that intertemporal reallocation of consumption only has a small second-order impact on welfare. Before applying the analysis of this section to twenty OECD countries, I will now first turn to a description of the data. ‘The gains can be a bit larger with investment if through investment and disinvestment we allow the global portfolio to gradually change over time (e.g., Invest a lot in countries with low-risk capital).

183

3. The data The System of National Accounts developed by the United Nations distinguishes eight consumption categories: (1) food and beverages, (2) clothing and footwear, (3) rent, fuel, and power, (4) furniture, household equipment, and operation, (5) medical care and health expenses, (6) transport and communication, (7) recreation, entertainment, education, cultural services, (8) miscellaneous. The last category includes restaurants, hotels, financial services, and some other services. We first need to single out the tradeables. Categories (3) (5) (7) (8) are clearly mostly nontradeables. Categories (l), (2), (4) are mostly tradeables. Category (6) is less clear. It has two subcategories: (6a) personal transportation and (6b) other. The latter includes telephone and postal services, public transportation services, and operational spending on personal transport, which are mostly nontradeable. This leaves us with four categories of tradeables: (11, (2), (4), and (6a). I will use annual data from the United Nations National Accounts Statistics on private consumption spending at constant prices, from 1970 to 1988, for each of those four categories. Consumption is divided by the population. The sample includes twenty countries. For the four remaining OECD countries - Portugal, New Zealand, Luxembourg, and Turkey - no data are available at this disaggregated level.’ Categories (2) (4), and (6a) are all durables. Since the services produced by durables are proportional to the stock of durables, the latter needs to be measured. Durables are assumed to depreciate at a constant rate: o(t + 1) = (1 - c)D(t) + E(r + 1) where c’is the rate of depreciation and E is new spending on durables. Given estimate of the stock of durables at time 0, we can then calculate the path of stock of durables over time using data on E(l), , E(T). If E growths a steady rate gE, the ratio D/E approaches l/(sE + v). I use this in order approximate D(0) as follows:

an the at to

1 E(1) + .‘. + (1 _

SE)T

where gE is the observed average growth rate of E. Finally, we need to find estimates for the parameters aj of the utility function in order to aggregate the various consumption categories. Choose quantities

“For Japan, the UN does not publish data on the subcategory were obtained from the Annual Report on National Accounts Government of Japan.

(6a), personal transport. Those data by the Economic Planning Agency,

184

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such that prices of nondurable goods, as well as prices of services from durables, are normalized to 1 in 1980. Then aj equals the ratio of category j consumption (of nondurables or durables services), divided by the aggregate consumption of nondurables and durables services in 1980. I will use u + /l + gE times the stock of durables at 1980 prices as a proxy for the value of 1980 durables services.’ The rate of depreciation is assumed to be 0.5 for clothing and footwear, 0.4 for furniture and household equipment, and 0.3 for personal transportation. This implies that clothing and footwear depreciate to 10% of their original value in 3.3 years, household equipment in 4.5 years, and personal transportation in 6.5 years.” Lowering or increasing these depreciation rates by 0.1 or less does not change the results reported below very much. While in Section 2 we assumed that each country has the same utility function, the estimated aj’s are somewhat different for different countries. But the standard deviations of consumption in these countries, using the estimated aj’s for each country, turn out to be quite similar to those when the average of the aj’s over all countries is used. I will therefore use averages, consistent with the assumption in Section 2.’ ’ I calculated the ratios 0” of a country’s consumption to world consumption using 1988 nominal spending on the sum of the four categories, converted into dollars. Table 1 shows the standard deviations of per capita private consumption growth rates of each country, as well as its correlation with the world consumption growth rate for values of the intratemporal substitution elasticity l/p of 0.5, 1, and 2. Here the world is the sum of the twenty OECD countries in the sample. A higher substitution elasticity tends to raise both the standard deviation of the consumption growth rate and the correlation with the growth rate of world consumption somewhat, although the differences are very small. Ignoring durability would lead to big errors (see last column). Standard deviations would on average be 58% higher, and correlations would be smaller, with the exception of the US and Denmark. Both of these elements would lead to a significant overestimation of the gains from risksharing.

‘If for example consumption of nondurables and all durables services growth at the same stochastic rate with expectation p and standard deviation 0, the value of a durable’s services can be shown to be c + p + ;j(p + 0.5(1 - ;~)a’) times the value of the durable itself. This is close to c + /r’ + p for parameters chosen in this paper. “One should keep in mind that the category household equipment limited durability, such as electrical bulbs and cleaning materials, household equipment and furnitures, and household services.

includes household goods of as well as repairs of durable

’ ‘With the exception of Greece and Iceland, the difference between these two standard deviations is less than 0.001 for all countries. In Greece and Iceland the standard deviation based on their own spending shares is, respectively, 0.006 lower and 0.006 higher than the standard deviation based on average spending shares. This is because Greece spends relatively little on the volatile household equipment and personal transportation categories, while Iceland spends a relatively large fraction of its budget on those categories.

E. wn

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/ Journal

Table I Standard deviations of consumption consumption growth

of MoneturyEconomics

growth

and correlations

34 (1994)

of consumption

175- 200

growth

185

with world

The sample period is 1970-1988. Data on four tradeables consumption categories are aggregated into a CES consumption index, with p the elasticity of substitution among the four categories. The table shows the standard deviation of the per capita growth rate of this consumption index for each country (first four columns) and the correlation of each country’s per capita consumption growth with that of the world (sum of the twenty OECD countries). With the exception of columns 4 and 8, for durables consumption categories consumption is computed by approximating the associated stock of durables. 0’ (%)

Australia Austria Belgium Canada Denmark Finland France Germany Greece Iceland Ireland Italy Japan Netherlands Norway Spain Sweden Switzerland UK us “Depreciation

corr(d, Y”)

p=os

p=l

p=2

p=

1.3 1.8 1.7 2.3 2.1 2.1 0.9 1.4 2.9 4.7 2.7 1.5 2.2 1.6 2.0 2.3 2.2 1.6 1.8 1.2

1.3 1.9 1.7 2.3 2.1 2.1 1.0 1.4 3.3 4.6 2.7 1.5 2.3 1.7 2.0 2.3 2.1 1.6 1.8 1.2

1.4 2.1 1.7 2.4 2.1 2.2 1.0 1.4 4.8 4.4 2.7 1.6 2.4 1.7 1.9 2.4 2.1 1.6 1.8 1.3

2.0 3.6 2.4 3.0 4.3 3.4 1.8 1.9 5.0 7.5 4.1 2.5 3.5 2.3 3.6 2.8 3.2 2.5 2.9 2.4

rates =

1”

p=o.5

p=l

p=2

0.13 0.69 0.48 0.58 0.38 0.37 0.64 0.53 0.57 0.49 0.44 0.35 0.88 0.66 0.35 0.67 0.20 0.65 0.70 0.58

0.16 0.72 0.50 0.60 0.38 0.38 0.67 0.53 0.60 0.51 0.46 0.36 0.88 0.65 0.35 0.68 0.19 0.66 0.70 0.59

0.23 0.77 0.55 0.63 0.36 0.40 0.73 0.53 0.66 0.55 0.50 0.40 0.90 0.60 0.36 0.71 0.18 0.66 0.72 0.62

p=

1”

0.05 0.52 0.38 0.47 0.52 0.3 1 0.62 0.52 0.48 0.27 0.46 0.35 0.87 0.54 0.15 0.60 _ 0.02 0.67 0.68 0.75

1

4. Gains from risksharing with time-additive

expected utility preferences

Before applying the data to the analysis of Section 2, I need to make one qualification with respect to mean growth rate of consumption. In Section 2 it was assumed that each country has the same mean growth rate of consumption. This assumption is at odds with the facts. Average per capita consumption growth ranges from 0.3% in Ireland to 4.1% in Iceland. Assuming different consumption growth rates implies that the global mutual fund will eventually be dominated by the capita1 of the highest growing country. In order to avoid that,

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Table 2 Gains from risksharing

with VNM time-additive

Economics

34 11994)

175

NO

preferences

;’ is the rate of relative riskaversion and fi the standard deviation of the component of consumption growth associated with nondiversifiable individual-specific risk. The first four columns show welfare gains from perfect riskpooling among the twenty OECD countries, measured as the percentage increase in the expected consumption path that generates an equivalent rise in welfare. The last four columns show to what extent welfare rises or falls as a result of the immediate change in the consumptton level related to the pricing of the country-specific consumption streams. Welfare gains (%)

Price gains (%)

y=] 6=0

.,=4 &=O

y=lO

;‘=4

?I= ,

ii=0

6 = 7.5”/0

k = 0

Austrilia Austria Belgium Canada Denmark Finland France Germany Greece Iceland Ireland Italy Japan Netherlands Norway Spain Sweden Switzerland UK US

1.2 0.9 1.1 1.8 2.0 2.0 0.3 0.7 3.9 8.9 2.8 1.2 1.0 0.8 1.8 1.6 2.4 0.7 0.9 0.6

0.8 0.6 0.7 1.2 1.3 1.4 0.2 0.5 2.8 6.7 2.0 0.8 0.7 0.5 1.2 1.1 1.7 0.5 0.6 0.4

0.8 0.6 0.7 1.2 1.3 1.4 0.2 0.5 3.0 9.0 2.0 0.8 0.7 0.5 1.2 1.1 1.6 0.5 0.6 0.4

1.9 1.5 1.8 3.0 3.2 3.3 0.5 1.2 7.2 2.0 4.8 1.8 1.7 1.3 2.9 2.7 4.0 1.1 1.4 0.9

Average

1.8

1.3

1.4

3.3

0.9 ~ 0.3 0.3 - 0.3 0.3 0.3 0.5 0.4 ~ 0.9 ~ 1.3 ~ 0.2 0.6 ~ 1.0 0.0 0.4 - 0.6 0.7 0.1 - 0.2 0.4

., =4 b=O

7: 6=0

6 = 7.5%

0.6 ~ 0.2 0.2 - 0.2 0.2 0.2 0.3 0.3 - 0.6 ~ 0.9 ~ 0.1 0.4 ~ 0.7 0.0 0.3 - 0.4 0.5 0.0 ~ 0.1 0.3

0.6 ~ 0.2 0.2 - 0.2 0.2 0.2 0.3 0.2 - 0.6 ~ 0.9 ~ 0.1 0.4 - 0.6 0.0 0.3 - 0.4 0.5 0.0 ~ 0.1 0.2

1.5 ~ 0.5 0.4 - 0.5 0.5 0.5 0.7 0.6 - 1.5 ~ 2.1 - 0.2 0.9 - 1.5 0.0 0.7 - 0.9 1.1 0.1 ~ 0.3 0.6

]O

;.=4

one needs to assume that in the long run expected consumption growth rates revert to some average, which is consistent with the conditional convergence finding for GDP data by Barro (1991). That kind of assumption, while maybe more realistic, is much too difficult to handle analytically. Here the unweighted average per capita consumption growth rate is assumed to be the future mean growth rate for each country (p = 0.017). Table 2 shows the welfare gains from risksharing, assuming p = 1 and p = 0.01. The welfare gains are measured as the permanent relative increase in the expected level of the consumption path that would lead to the same welfare improvement as can be achieved through international risksharing. Welfare

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gains can be broken up in two parts. The first is the gain from the drop in the standard deviation of consumption to cw = 1.1%. The second is the change in welfare associated with the pricing of the endowment streams. The latter equals [p”,/(pk 8’ + . .. + pi HN)] - 1 for country n. Those gains (or losses) are shown in Table 2 under ‘price gains’. Countries with a low standard deviation of consumption growth and a low correlation with world consumption growth have positive price gains, but may not gain much in the form of a drop in the standard deviation of consumption.’ * In the first column, where >’= 1 and 6 = 0, the average welfare gain is 1.8%. This is a very large number if one realizes that in the United States the average size of the securities industry as a fraction of GNP was 0.6% in the 1980s which is about 1.8% of tradeables consumption.13 This means that if we were to double the size of the entire securities industry in order to reap the gains from international risksharing, we would break even. It is hard to imagine that doubling of the securities industry is what is necessary in order to achieve international diversification. Note that, the gains range from 0.3% for France to 8.9% for Iceland. The endowment stream of Iceland is unattractive, because it leads to a 4.6% standard deviation of per capita consumption before risksharing. This generates a price loss of 1.3%. But the welfare gain from a drop in the standard deviation of consumption growth from 4.6% to 1.l % is as much as 10.2%. The large gain for Iceland, and similar for other high-risk countries, is related to the fact that the welfare gain from a marginal drop in the standard deviation of consumption rises rapidly as the standard deviation itself rises. The price loss is not so large since other countries evaluate how buying one unit of the Icelandic endowment stream affects their standard deviation of consumption when it has already been reduced to only 1.l %. At that point the welfare impact of a marginal change in the standard deviation is small. For the same reason low-risk countries such as the US, France, and Germany show relatively small gains. The correlation of a country’s consumption with world consumption is the second factor affecting a country’s gains. Denmark and Japan have a standard deviation of consumption of respectively 2.1% and 2.2%, but Japan shows a correlation with world consumption growth of 0.88 against only 0.38 for Denmark. This leads to a welfare gain twice as big in Denmark as in Japan (2.0% versus 1.0%). Why is the average welfare gain of 1.8% so much higher than the gains reported in Cole and Obstfeld (1991) (from here on CO)? CO consider a

“In France the standard deviation of consumption growth actually rises from 1.0% to l.l%, itself leading to a drop in welfare. But this is more than offset by the price gain.

by

“I would like to thank Terry Fitzgerald for providing me with data on the size of the US securities industry, which are used in Diaz-Gimenez, Prescott. Fitzgerald, and Alvarez (1992).

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two-good world, where significant risksharing takes place automatically through endogenous terms of trade adjustments. But even the gains they report under the assumption that the two goods are perfect substitutes are still much lower than those reported here. The difference can be understood as follows. Using (7) we can write the welfare gain associated with a drop in the standard deviation of consumption growth by do,2 as - 0.57 dc’c = /I + (y - l)(p - 0.5ya,2)

- 0.51: do2 da,2 = ~ r-p c’

(14)

where b = p - 0.5ga: is the riskadjusted growth rate and Y = b + ;jj is the (implicit) riskfree interest rate. The appropriate ‘discount factor’ in measuring the welfare gains is the difference between the riskfree interest rate and the riskadjusted growth rate. Under our assumptions so far, fi is approximately 0.017 and r = 0.027. This is only a bit higher than the average real interest rate of 1.3% for Treasury bills, found in van Wincoop (1993) for the period 1970-1989 for ten of the countries studied here. Under those assumptions a drop in the variance of consumption growth by 0.00036, the average for the twenty OECD countries, leads to a welfare gain of 1.8%, precisely the average reported under the first column of Table 1. While the 1.8% welfare gain is based on 7 = 1, CO only report results for ‘J 2 2. Holding p = 1.7%, which is close to the 1.8% in CO, this implies a higher riskfree rate. For example, when y = 4, the riskfree rate is 7.8%. This is much higher than what is commonly observed. From (14) we find that the welfare gain from a 0.00036 drop in the variance of consumption growth drops to 1.2%. For a given riskless rate, a higher rate of relative riskaversion raises welfare gains. But here the riskless rate rises enough to lead to a net reduction in gains. The second and third columns of Table 2 show the welfare gains for individual countries for y = 4 and y = 10. A second difference is that CO assume a slightly higher time discount rate of 2%. Holding y = 4, this raises the riskless rate even more, to 8.8%. The welfare gain from da: = 0.00036 then drops to 1.0%. Third, CO allow for risksharing among only two countries. This leads to a lower drop in daf, by only 0.000228. The welfare gain will then fall further to 0.64%. Finally, CO base their results on a finite horizon of fifty periods. When the horizon in T, the welfare gain in (14) needs to be multiplied by 1 - [T(r - ~)e-“-~‘T/(l - e -cr-G)T)]. Maintaining the other CO assumptions mentioned above, this leads to a small further drop in the welfare gain, to 0.57%. This is still more than double the 0.276% they report. This last difference is a result of the specific Markov growth process they assume for the home and foreign endowments. While the transition matrix is chosen to generate a 0.1 autocorrelation of the growth rate of national output, it also implies an autocorrelation of world output growth of 0.3. The resulting higher

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persistence of consumption growth after risksharing further dampens the welfare gains. We may conclude from the discussion above that the very low welfare gains reported in Cole and Obstfeld, even without endogenous terms of trade adjustment, are particularly due to (i) a very high riskless rate, (ii) risksharing among only two countries, (iii) a much higher persistence of shocks after risksharing. In contrast, the average welfare gain of 1.8% reported in the first column of Table 2 does not suffer from any of these problems. The impact of individual specific risk is shown in column 4 of Table 2. It is assumed that y = 4, and the standard deviation of individual risk is 7.5%. This leads to a standard deviation of consumption ten years into the future of only 24%. This is not very high given the considerable uncertainty most people face about their individual careers. Unfortunately there are no good empirical estimates of individual specific risk. The standard deviation of annual consumption growth based on PSID data is about 30%.14 Abowd and Card (1987) find that the standard deviation of annual earnings growth in the US is 40% based on PSID data and 35% based on the National Longitudinal Survey. But those numbers are probably much too high due to significant measurement error at the micro level. With y = 4 and 6 = 0.075, the average welfare gain rises to 3.3%. In general the presence of individual-specific risk raises welfare gains as long as y r 1. In that case, holding other parameters constant, the riskless rate drops more than the riskadjusted growth rate. It should be pointed out that, even though y = 4, as a result of the individual-specific risk the riskless rate, Y = /I + y(p - 0.5 ?a:), is only 3.3%, not much higher than what is generally observed. The gains are now even higher than in the first column of Table 2 since the rate of relative risk aversion has increased, while the riskless rate is still low.

5. Welfare gains with Kreps-Porteus

preferences

I will now study the welfare gains under Kreps-Porteus preferences, which allow for a separate rate of relative riskaversion and intertemporal substitution elasticity. Epstein and Zin (1989) and Weil (1990) developed the following parametric class of Kreps-Porteus preferences: l5 l/(1 -+I

L((E,J(t + l)l-Y)li(l-Y))l-ti

U(t) =

.

1

‘%ee for example “For

a discussion

Altonji

and Siow (1987).

of Kreps-Porteus

preferences,

see Weil (1990) and Farmer

(1990).

(15)

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Here /I is the time discount rate, y the coefficient of relative risk aversion, and l/$ the intertemporal substitution elasticity. If I/I = y, this is the discrete time version of (1). Duffie and Epstein (1989) show that a continuous time version of (15) can be written in the form of the following stochastic integral for U: dU=

&(bU”

- c’-‘),U-~

+ yo;U,2

where wLiis a standard Brownian motion. If the stochastic is represented by dc = pcdt + rsCcdwC,with w, a standard follows immediately that

u= K =

Kc,

Gel =

(16) consumption process Brownian motion, it

ocr

[fl + (l,b - l)(,U - o.5~0,z)]1’~~‘1’.

(17)

Notice that when y = $, and we apply the transformation U1 -‘/(l - y) to utility, this is the same equation as (7). Let K, be K evaluated at of = c,’ + a2, and let K, be K evaluated at CJ,’= G:, + d2. Using a similar analysis as in Section 2, the utility of individual i from country n before and after risksharing can be written respectively as U i,n = ~c~v”(O) and n

U i,n,*

=

K,Ci.n,

*lo)

=

Pq

K,

U’p,‘+...

2*“(O), + UN

with equilibrium prices as in (13) K$- ' replacing &_. Table 3 shows the welfare gains for all countries for p = 1, /I = 0.01, 6 = O%, and various $ and 7. Holding the rate of relative risk aversion constant at 3, the gains from risksharing become much smaller if we lower the intertemporal substitution elasticity from i to & (average gains 0.4%). But they become very large if we either raise the rate of relative risk aversion from 3 to 10 (average gains 5.2%) or raise the intertemporal substitution elasticity from $ to 1 (average gains 5.6%). To understand the role of the intertemporal substitution elasticity, consider again the percentage increase in the expected consumption path leading to the same rise in utility as a drop in the variance of consumption growth by da:: de/c =

- 0.51/ /I + ($ - l)(p - 0.5ya?)

dof =

- 0.5? -do;, r-ji

(19)

where r = /I + $j and ji the same as defined before. For a given riskless rate, riskadjusted growth rate, and rate of relative riskaversion, the gains are the same

E. vm

Table 3 Gains from risksharing

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preferences

;’ is the rate of relative riskaversion and I/$ the intertemporal substitution elasticity. There is assumed to be no individual-specific risk (6 = 0). The first four columns show welfare gains from perfect riskpooling among the twenty OECD countries, measured as the percentage increase in the expected consumption path that generates an equivalent rise in welfare. The last four columns show to what extent welfare rises or falls as a result of the immediate change in the consumption level related to the pricing of the country-specific consumption streams. Welfare gains (X)

Price gains

,_3

,-3

?=I0

;‘=3

;‘=3

$=3

cl/=10

@=3

$=l

$=3

0.9 0.7 0.8 1.3

3.0 2.3 2.7 4.5 4.9 5.0 0.8 1.8 32.6 1.4 2.8 2.6 2.0 4.4 4.1 6.1 1.7 2.1 1.3

3.1 2.8 3.3 5.4 6.0 6.2 1.0 2.2 12.0 28.X 8.7 3.5 2.9 2.4 5.4 4.8 7.4 2.1 2.6 1.7

0.7 - 0.2 0.2 - 0.2 0.2 0.2 0.3 0.3 - 0.7 - 0.9 - 0.1 0.4 - 0.7 0.0 0.3 - 0.4 0.5 0.0 - 0.1 0.3

5.2

5.6

Australia Austria Belgium Canada Denmark Finland France Germany Greece Iceland Ireland Italy Japan Netherlands Norway Spain Sweden Switzerland UK us

1.4 0.2 0.5 2.8 6.7 2.0 0.8 0.7 0.6 1.2 1.1 1.7 0.5 0.6 0.4

0.2 0.2 0.2 0.3 0.4 0.4 0.1 0. I 0.8 1.8 0.5 0.2 0.2 0.2 0.3 0.3 0.5 0.1 0.2 0. I

Average

1.3

0.4

I .4

11.1

(%)

i’=3 $=I0 0.2 ~ 0.1 0.0 - 0.1 0.1 0.1 0.1

0.1 ~ 0.2 - 0.3 ~ 0.0 0.1 ~ 0.2 0.0 0.1 - 0.1 0.1 0.0 - 0.0 0. I

;>=I0 $=3 2.2 - 0.8 0.6 - 0.8 0.7 0.7 1.1 0.9 - 2.2 - 3.1 - 0.4 1.3 - 2.3 0.0 1.0 ~ 1.3 1.7 0.1 ~ 0.5 0.9

7=3 l/j=1 2.8 - 1.0 0.8 - 1.0 0.9 0.9 1.4 1.1 - 2.8 ~ 3.9 - 0.5 1.7 - 2.9 0.0 1.3 - 1.6 2.2 0.2 - 0.6 1.1

as under the standard expected utility preferences. However, for a given time discount rate and rate of relative riskaversion, the riskless rate is now smaller than under expected utility preferences as long as $ < y, that is, the intertemporal elasticity of substitution is larger than the reciprocal of the rate of relative riskaversion. The combination of significant riskaversion and relatively low riskless rates is what generates the high welfare gains in columns 3 and 4 of Table 3. The empirical evidence shows that the rate of relative riskaversion is somewhere in the range 2 to 4 (for example, Friend and Blume, 1975). Beaudry and van Wincoop (1992) find that estimates of the intertemporal substitution elasticity are very imprecise when based on aggregate US data. Using consumption

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data for nineteen US states a point estimate of the intertemporal substitution elasticity of 1 is found, with a relatively small standard error of 0.3. Based on these empirical findings, the best estimate of welfare gains with KrepssPorteus preferences are those in column 4, where ‘/ = 3 and $ = 1. In that case the average riskless rate is a reasonable 2.6%, while the welfare gains are 5.6%. Adding individual-specific risk does not alter the results when $ = 1. It will only raise (lower) the welfare gains when $ is larger (smaller) than 1.

6. Welfare gains with habit formation breferences I will adopt the following specific form of habit formation used by Constantinides (1990), and Sundaresan (1989): I6 U = E,,

J0 z=c-x = 0.01

e-PrLdr l-y if

preferences,

also

’

c>x+O.Ol,

(20)

otherwise,

i = - ax + bc. Here x is the experience level, an exponentially weighted sum of past consumption. Therefore the utility function is not time-separable. As long as c - x > 0.01, the marginal subutility of consumption rises if the experience level rises, leading to habit persistence. As the parameter a approaches infinity, so that the consumer forgets infinitely fast, x becomes zero, and we again have the standard time-separable VNM preferences. If consumption follows the stochastic process dc = ,ucdt + occdw,, utility can be written as a function of c(O), x(O), and the other parameters. The problem with the specification (20) is that this function V(c(O), x(O), p, of, a, b, y) cannot be derived analytically. I therefore approximate it numerically, using a discrete-time 50-period horizon version of (20). For each period a random growth rate of consumption with expectation p and standard deviation cr is drawn from a normal distribution. Expected utility is computed based on 10,000 independent histories. Constantinides (1990) shows that the short-run intertemporal substitution elasticity (ISE), defined as the immediate change in the growth rate of consumption relative to a marginal permanent rise in the riskfree rate, is (1 - x/c)/v.

16These authors do not restrict z to be larger than 0.01 since they allow for investment, consumers automatically choose a positive z.

in which case

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However, the consumption growth rate rises more in the long run, leading to a long-run intertemporal substitution elasticity of l/y. I will assume in the rest of this section that x(0)/c(O) = b/(p + a). This is the steady-state ratio x/c if consumption growths at a steady rate p. Then ISE = (1 - b/(p + a))/r. The rate of relative riskaversion is the Arrow-Pratt riskaversion coefficient for an atemporal consumption gamble:’ 7 RRA

=

-

c(o)a2u/ac(o)2 au/at(o) .

(21)

Numerical analysis shows that RRA is now the reciprocal of some ‘intermediate-term’ intertemporal substitution elasticity, the latter lying somewhere in between ISE and l/y. Therefore RRA > y and ISE < IIRRA. Since no empirical estimates of the short-run versus the long-run intertemporal substitution elasticity exist, it is not possible to evaluate the implication of habit formation preferences that the reciprocal of the RRA lies in between the short- and long-run substitution elasticities. A rise in b or y and a drop in a, all lead to a lower ISE and higher RRA. Since there is one degree of freedom left, there is a multitude of combinations (y, a, b) that yield a certain RRA and ISE. The numerical calculations show that a drop in y, combined with an appropriate drop in both a and b can keep both RRA and ISE constant. Different countries will now choose different portfolios after risksharing. This is because consumption at time 0 rises in some countries and drops in others, depending on the pricing of their endowment streams. Under habit formation, consumers become more riskaverse if consumption drops relative to the experience variable, and the opposite if consumption rises. Since this makes it very hard to repeat the analysis of the previous sections here, I will limit myself to the more modest goal of studying the welfare gains from a reduction in the standard deviation of consumption growth from respectively 2%, 3%, and 4% to l%, with no change in the consumption level to time 0. Those numbers are within the range we saw in Section 3. The same average growth rate of consumption as used in Sections 4 and 5 is assumed.

“Usually RRA is measured as - WU,,/U,, where W is an individual’s wealth. Since there is no investment, wealth is proportional to consumption, so that this is the same as (21). If we add investment to the model, consumption is not proportional to wealth any more with habit formation. Unfortunately, it is not possible to analytically derive expected utility with investment in one risky asset. Constantinides (1990) allowed for investment, with a capital stock that can be reallocated at zero cost between a risky and safe technology. The problem with that framework in the context of this paper is that a lower standard deviation of the return on the risky asset leads to a higher standard deviation of consumption growth because of a portfolio switch to the risky asset.

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preferences

Parameters have been chosen such that the rate of relative riskaversion is 3 for both types of preferences. Under time-additive preferences the intertemporal substitution elasticity is lj3. Under habit formation preferences parameters have been chosen such that the short-run intertemporal substitution elasticity is 0.2. In the long run, it is I/‘?. Welfare gains are shown from lowering the standard deviation of consumption growth from, respectively, rr, = 0.02, 0.03, 0.04 to 0, = 0.01, assuming a fifty-year horizon. Welfare gains are expressed as the permanent percentage increase in consumption that generates an equivalent rise in welfare as achieved through the riskreduction. The numbers in brackets express the welfare gain as a permanent percentage increase in consumption after consumers have ‘become used’ to the higher consumption level (their consumption ‘experience’ level has increased by the same percentage).

Habit

formation

Time-additive preferences

preferences

7 = 0.7

v=l

.?’= 2

a = 0.029

a = 0.038

a = 0.109

0,

h = 0.039

h = 0.044

h = 0.075

0.02 0.03 0.04

1.8 (6.8) 4.3 (16.6) 6.8 (25.2)

1.5 (4.0) 4.8 (12.8) 9.6 (25.7)

1.0 (1.4) 2.1 (3.8) 5.5 (7.5)

.j z 3

0.75 2.04 3.9

Table 4 shows those welfare gains for various parameter pairs. The gains are again measured by the percentage increase in the expected consumption path that leads to the same improvement in welfare as the reduction in the standard deviation of consumption growth. Since it takes time for the consumer to get used to higher consumption levels, in parentheses the welfare gains are also shown in terms of an increase in the expected consumption path after consumers have become used to the higher consumption level. The latter is an equal percentage change in both x(0) and c(0). Measured this way the gains are bigger since more consumption experience by itself lowers utility. All parameter pairs are chosen so that RRA = 3, when measured at zero risk. The combination of parameters a, h, and y is chosen so that the short-run intertemporal substitution elasticity is 0.2. The last column shows welfare gains in the time-additive case, where y = RRA = 3 and ISE = i. The message from Table 4 is clear. The welfare gains under habit formation are significantly larger than under time additive preferences. For ‘/ = 1, the welfare gains in the table are more than double those under time-additive VNM preferences. The gains in parentheses are even much larger than this. In order to understand these results, in Appendix B I derive for the infinite-horizon continuous-time case the following percentage increase in c(0) that would lead to the same rise in utility as a change in the variance of consumption

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by dof:” dc(O)/c(O) =

RRA’ = RI-U Y =

p +

- OSRRA

[1 + aln RRA’/aln r-j

a,Z] do2 c>

(22)

c UC (I

_

y)

u,

y/T,

,ti = p - OSRRA’o;. Here I* is the riskadjusted growth rate of c - X. The term in brackets in the numerator of (22) is numerically very close to one. When that term is one, welfare gains are the same as for the other types of preferences for a given rate of relative riskaversion, riskless interest rate, and riskadjusted growth rate. But for a given time discount rate, rate of relative riskaversion, and riskadjusted growth rate, the riskfree rate is now smaller than under time-additive VNM preferences. For all types of preferences considered in this paper, the riskfree rate can be written as the time discount rate plus the product of the riskadjusted growth rate and the reciprocal of the intertemporal substitution elasticity. For habit formation preferences it is the long-run intertemporal substitution elasticity l/l) that determines the steady-state riskfree rate. Since y < RRA, for a given rate of relative riskaversion the riskless rate is smaller than under time-additive VNM preferences. The same result was found under KrepssPorteus preferences when t/j > 7. Finally, the fact that RRA’ > RRA leads by itself to both a lower riskadjusted growth rate and riskfree rate at a given rate of relative riskaversion. This will increase (lower) the gains when y > 1 (y < 1).

7. Conclusion and discussion To summarize the results, this paper has shown that unexploited welfare gains from international risksharing among a set of twenty OECD countries are large for a wide variety of preferences. For parameterizations with a realistic size of the riskfree rate and rate of relative riskaversion (l-4) the average welfare gain across countries is equivalent to a permanent increase in consumption by 1.885.6%. This amounts to one to three times the size of the entire securities industry in the US. These large unexploited gains lead to what we might call an ‘international risksharing puzzle’. The work of this paper has been extended along different lines in van Wincoop (1993, 1994b). Van Wincoop (1994b) allows for nonseparability of ‘!‘ln this expression RRA in the numerator is replaced by RRA’ if the welfare gains are measured terms of the percentage change of both c(0) and x(O) that yields the same rise in utility.

in

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preferences between tradeables and nontradeables consumption. This is found to lead to welfare gains that are generally even much higher than under separable preferences, thus only strengthening the risksharing puzzle. Van Wincoop (1993) adds heterogeneity of consumers by considering a multicountry real business cycle model with two types of agents, workers and capital owners. Workers can only trade international bonds and are possibly liquidity-constrained. Capital owners do not work and hold all claims on the physical capital stocks. The potential welfare gains from international portfolio diversification for capital owners are similar to those reported in this paper. Future work should establish a connection between this ‘international risksharing puzzle’ and the ‘home bias puzzle’ of equity portfolios that has been documented by French and Poterba (1991) Golub (1990). and Tesar and Werner (1992). Bottazzi, Pesenti, and van Wincoop (1994) find that nontradeability of human capital may lead to significant endogenous ‘home bias’ in portfolios as a result of a lower correlation between domestic wages and the domestic capital return than between domestic wages and foreign capital returns. This means that in the presence of human capital further portfolio diversification can actually lead to welfare losses. The observed significant volatility of consumption growth and the low cross-country correlations of consumption growth documented in this paper may be a result of nondiversifiable shocks to human capital returns. Further work needs to be done to evaluate the empirical validity of this potential explanation.

Appendix A Consider an extension without individual risk. only a limited number portfolio diversification. optimal portfolio has an the consumer maximizes

allowing for investment in capital for the one good case Presently a consumer from country n is restricted to of foreign assets, leading to imperfect international Given the limited number of foreign assets held, the expected return r and standard deviation 0,. Therefore (l), subject to

Kn = (rK” - c)dt + K”a,dw”.

64.1)

This yields c”, i = A,,K”, with 2, as in (7), leading to the same expected utility as in (7), with an expected growth rate of consumption p = r - A,,, which now depends on the riskyness of the portfolio. Now allow for risksharing. Think about trading the mutual funds held by individual countries, where the country n mutual fund consists of c”(O) shares. Let pi be the price of a dollar in fund n in terms of the price of a dollar in fund N. Since the capital stocks of different countries are fixed in the short run, those relative prices are generally not one. Theoretically it is possible to optimally reallocate the portfolio over time by, for example, investing more in the capital

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of relatively low-risk countries. In order to prevent strange long-run results (such as close to complete decapitalization of some countries), one should have at least decreasing returns to scale in capital. Here I will restrict myself to global mutual funds that maintain a constant country exposure, ignoring potential additional gains in the long run. For fund F, the constant exposure to country n is 9;, which is K;/(Kk + ... + KF). K% is the number of shares of the country n mutual fund bought by fund F. This leads to the same utility (11) after risksharing and the same FOCs for an optimal portfolio. But now the expected growth rate of consumption has changed to p’ = r - /1,, while the consumption level at time 0 has increased i,,,,/& - 1% more than that in the no-investment case. This intertemporal reallocation of consumption is the result of a substitution and income effect associated with a rise in the certainty equivalent of the real return on capital. Utility can now be written as u,,*

- CP;/(P;@+ ... + P,"@')I'-~K"(O)~-~/(~ CB+ (Y- l)(P’ - 0.5Yd)lY

_

-Y)

64.2)

The change in utility is associated with a lower variance of consumption growth and the international pricing of the mutual funds. Differentiating (A.2) with respect to (0;, . . . , OF- ‘) leads to the same equilibrium prices as in (13) in the text. If oc = 8-,‘, the welfare gain of a marginal reduction in the variance of consumption growth by do: < 0, measured in terms of a percentage rise in the expected consumption path, is - OSyda,2 (A.3)

fi + (i - l)(fi - OSyr?f)

where fi is the optimal growth rate of consumption if the variance of consumption growth is 6:. For a given expected consumption growth rate, this is exactly the same as without investment. The only difference is that for a larger than marginal reduction in do,2 the denominator of (A.3) is lowered somewhat a dp = O.S(y - l)do:. But this is a second-order gain associated with the optimal intertemporal reallocation of consumption in response to the reduced uncertainty. For realistic parameter values it is quite small.

Appendix B Proqf ?f Eq. (22) Let dt be an infinitesimal

U(c(O),x(O), d, =

period

length. Then

(c(O)- x(O))’-y 1-Y

dt

+

1 1 +Bdt

6, U(c(dr), x(dr), of).

(B.1)

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Using a Taylor expansion and then taking expectations, ignoring secondhigher-order terms in dt, we can write [assume x(0)/c(O) = b/(p + a)]:

and

Eo U (c (dt), x (dt), 0:) = U(c(O), x(0),6:)

+ U,pc(O)dt

+ U,px(O)dt

+ U&T,

- 5;) + U,,;(a,z - o;)pc(O)dt

+ [U,:.:

+ ) U,++~cdf

+ ) U,,,,(a,2

+ &,;(a,2

+ f U,++~xdt]

- af)px(O)dt

[CC’- F,‘]’

- C,2)o,‘c(0)2 dt + U,f+f[af

Here U,; = aE U(c(O), x(O), CJ:)/~D:, evaluated tive of (B.2) with respect to a: and evaluating

+ + UCi,,o;c(0)’ dt

- c?,‘]“.

at 0,’ = Cf. Taking at Ol, yields

(B.2) the deriva-

aE,, U(c(dt), x(dt), CJ,‘)/&, = +U,,C(O)~~~ + U,:(l

+ (1 - r)pdt)

+ fU,,,,o,Z~(O)~dt.

(B.3)

Here I used the fact that U,c(O) + U,x(O) = (1 - y)U, since utility is homogeneous of degree 1 - y in (c(O), x(0)). Let RRA’ = RRAc U,/[( 1 - y) U] = - c2 Ucc/[( 1 - y) U]. Then aln RRA‘/aln

CJ~= u,’ Uc,,:/Uci,, - of U&.

Now take the derivative u

= 0.5UC,c2[I O’

03.4)

of (B.l) with respect to a:, using (B.3) and (B.4), + alnRRA’/alnoz]

p + (y - l)[/~ - 0.5RRA’af]

(B.5)

’

Therefore the relative increase in c(0) that leads to the same increase in utility as a drop in the variance of consumption growth by da,2 is dc(O)/c(O) = -da: “: U,c(O)

=

- OSRRA[l

+ alnRRA’/alna,?]

B + (i’ - l)[p - OSRRA’af]

d,$.

(B.6)

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