# «ABSTRACT Over the period 1972-1986, the correlations of GDP, employment and investment between the United States and an aggregate of Europe, Canada ...»

Proposition 1 highlights the properties that we used to motivate our assumed asset market structure. First, if the tax rate on foreign dividends is zero (and the other assumptions required for the proposition are satisﬁed), then by property (ii) allocations in the model with stock trade only at date zero are identical to those in the model with stock trade period by period. Since individuals do not wish to trade after date zero, assuming that they cannot does not aﬀect equilibrium allocations.

Second, the model nests the extreme possibilities for international risk sharing for diﬀerent values for the tax rate τ on foreign dividend income. If τ = 0, then by property (iii) allocations are identical to complete markets and perfect risk sharing is achieved. If τ is large enough to ensure 100 percent home bias, then there is no international asset trade at any date, including date zero.18 In this case, no international risk sharing is provided through asset trade.

This proposition may be viewed as extending some of the results in Lucas (1982), Cantor and Mark (1988) and Cole and Obstfeld (1991). Lucas shows that in an endowment economy with common preferences across countries, perfect risk pooling is achieved when agents hold 50 percent of both domestic and foreign shares in each period, where shares are claims to future endowment streams. For ω = 0.5,we reproduce this prediction. Cantor and Mark extend Lucas’ analysis to a simple environment with production. As in Lucas’ paper, equities are suﬃcient to complete markets, and both domestic and foreign agents should hold identical and constant portfolios through time.19 Cole and Obstfeld consider a economy similar to ours, although they abstract from labor supply, and, as in Cantor and Mark, assume 100 percent depreciation for capital. They show that with symmetric (logarithmic) tastes and technologies, a regime of portfolio autarky (100 percent home bias or λ = 1) delivers complete markets (i.e. eﬃcient) allocations irrespective of the value for ω. By contrast, property (i) of our proposition indicates that in the economy considered here, portfolio autarky will only be eﬃcient for the case when there is complete specialization in tastes;

This is what Cole and Obstfeld label ‘portfolio autarky’. The logic is simply that if foreign dividend income is taxed at a high enough rate, then domestic agents will not want to purchases a positive quantity of foreign stock in any possible date or state. By assumption (the short-selling constraint) they can never purchase a negative quantity of foreign stock. Thus in equilibrium it must be the case that domestic agents always own 100% of the domestic ﬁrm.

In their model both countries produce the same good, and domestic and foreign agents have the same log separable preferences over consumption and leisure. Productivity shocks are assumed to be iid through time.

i.e. ω = 1. The reason for the diﬀerence is that with log separable preferences and full depreciation, consumption, investment and dividends are all ﬁxed fractions of output, and the real exchange rate is equal to the ratio of output across countries. Thus total dividend income in any given period is independent of the initial portfolio split. In this sense changes in the real exchange provide automatic insurance against country-speciﬁc income changes. In our economy with partial depreciation, investment is no longer a ﬁxed fraction of output, and changing the initial portfolio changes the properties of the stream of asset income. However, eﬃciency can still be achieved for ω 1 provided the initial portfolio contains an appropriately weighted mix of both domestic and foreign stock.

Notice that the result of proposition 1 might also be of interest for understanding the observed home bias in asset holdings. Home bias of domestic portfolios is a fact that is hard to explain in models in which agents receive a country-speciﬁc labor income stream (see, for example, Baxter and Jermann, 1997). The proposition shows that there is an equilibrium in which domestic agents diversify away all country speciﬁc risk by holding a fraction of foreign stocks that depends on the import share and on the capital share. Reasonable values of these two shares (see the calibration section below) yield a portfolio share for foreign stocks of around 20%, which is not very far from the observed level of diversiﬁcation (see the data section). Part of the intuition for the result is that in a model with two traded goods, relative price movements provide a signiﬁcant amount of insurance, and hence a small share of foreign stocks is suﬃcient to diversify away all country speciﬁc risk.

At this point, two caveats are probably in order. The ﬁrst is that, for intermediate values for τ, allocations in the restricted stock trade economy may diﬀer from those in the unrestricted stock trade economy. The second is that Proposition 1 only applies when preferences are log separable between consumption and leisure, and when all production functions are Cobb Douglas. However, by continuity one would expect that allocations across the economies with and without trade after date zero should look similar for values for γ and σ close to zero and one respectively.20 Moreover, these particular parameter values are standard choices, and within the existing range of estimates.

A ﬁnal attractive feature of our asset market relative to the unrestricted stock trade alternative is that it is much easier to characterize equilibrium allocations numerically. Allowing stocks to be traded period by period adds two new continuous state variables to the problem (for example, the current holdings of both stocks by the domestic agent) in addition to the stocks of capital in the two countries and the values for the shocks. Moreover, to accurately solve portfolio problems linearization techniques will not work; a global solution across a very ﬁne grid is required, and a suﬃciently accurate approximation may not be currently feasible.

5. Parameter values Our benchmark parameter values are reported in table 10. In the calibration process we identify country 1 as the United States and country 2 as the same aggregate of major trading partners described in the data section of this paper. Most parameter values are standard for this class of models. The steady state share of imports in production of the ﬁnal good is set to 15%, which is approximately the ratio of imports or exports to GDP in the United States. We assume that preferences are log-separable in consumption and leisure (γ = 1) and set the elasticity of substitution between the domestic and foreign intermediate good to 1. Note that this value for σ is close to the value of 0.9 estimated by Heathcote and Perri (2002).

There is strong evidence that the international correlation of productivity has declined over the past thirty years. To estimate productivity processes we follow essentially the same procedure We were able to partially verify this conjecture as follows. We computed a linear numerical approximation to the system of equations characterizing equilibrium in the economy with stocks traded freely period by period. As a portfolio to linearize around, we used the value for λ predicted by Proposition 1. We were then able to conﬁrm that for values for γ and σ close to 0 and 1 respectively the allocations in simulations of linearized economies with and without trade after date zero look close to identical. In particular, when trade is allowed after date zero, very little such trade is in fact observed. Of course, this experiment is only suggestive, since without solving the unrestricted stock trade economy globally, it is not possible to determine the true average equilibrium portfolio weights.

as Backus, Kehoe and Kydland (1992). Since quarterly data on the capital stock are not available for all countries, we rely on employment data, and identify productivity at date t as21

the rest of the world. We assume that labor’s share of income, 1 − θ, is the same across regions and equal to 0.64. Our total sample period is 1972 : 1 to 2000 : 4.

We ﬁrst eliminate secular growth in productivity by linearly detrending the series for the vector z (st ). We then assume that detrended productivity evolves according to the law of motion b

**described in the model section:**

mean zero, standard deviation (common across regions) σ ε and correlation coeﬃcient ρt. We will assume that ρt is the only model parameter that changes over time.22 Moreover we model this time variation in a very simple fashion by assuming that in the ﬁrst half of our sample period this parameter takes on one value (ρ1 ) while in the second half of the sample it takes a diﬀerent value (ρ2 ). This corresponds to the same sample split we used to document changes in the data.

To obtain estimates for the elements of the matrix A we estimate eq. 26 using SUR on data for the entire sample period, which gives 0.91 0.00 (27) A= .

0.00 0.91 Cooley and Prescott (1995) note that the capital stock varies very little over the business cycle, so omitting capital should not greatly aﬀect the time series properties of z at business cycle frequencies.

Another possibility would have been to allow the oﬀ-diagonal element (determining the degree of spillover) to change through time. However, this has the eﬀect of changing the persistence of shocks to relative productivity, which makes results more diﬃcult to interpret. Moreover, given a relatively short sample period, it is diﬃcult to separately identify changes in the correlation of innovations from changes in the spill-over terms.

These estimates are similar to those found by Backus et. al. (1992) for the United States versus Europe, though our process displays no spill-overs.23 Given the estimated A matrix we then

will also simulate a version of the model in which we keep the correlation of the shocks constant, so we also compute the innovation correlation for the entire sample, which we denote ρ.

The only remaining parameter is the tax rate τ that applies to foreign dividend income. We pick this parameter so that in the ﬁrst sub-period, equilibrium diversiﬁcation in the model matches that in the data. In particular we set τ such that 1−λ is exactly equal to the average ratio (averaging across assets and liabilities and through time over the 1972 to 1985 period) of the gross foreign direct investment position plus the foreign equity portfolio stock relative to the U.S. capital stock. This ratio is 0.055 (see the US vs EU+CA+JP columns in table 7).

To summarize, all parameter values except the innovation correlation ρ, are held constant across simulations of the model for the two sample periods.

One reason for this diﬀerence is that we subtract linear trends from the productivity series prior to estimation while Backus and al. estimate A directly on the raw data.

6. Solution method In order to accurately compute the equilibrium value for λ, we use a global solution method which is designed to generate close approximations to equilibrium allocations across the entire state space. We ﬁrst approximate the joint process for productivity with a nine state Markov chain. Each state deﬁnes particular values for the productivities of the domestic and foreign representative ﬁrms.

The values for the states and the transition probabilities are such that the implied Markov process exhibits the same persistence, variance and cross-country correlation as the analogous continuous process estimated via a VAR (see the calibration section). The states and transitions are assumed to be symmetric across the two countries, so that over a long simulation, business cycles will have the same statistical properties in both countries.

We assume that at date zero, both the productivity levels and the capital stocks are equal across countries, and equal to their values in the non-stochastic steady state for this economy.

Computing a competitive equilibrium given a particular calibration (which includes a choice for τ ) amounts to ﬁnding a value for λ such that at the allocations corresponding to that particular value for λ, the optimal portfolio condition (eq. 24) is satisﬁed.

In practice we proceed as follows. First we create a ﬁne grid over values for λ from λ = 0 (indicating perfect foreign bias in stock holding) to λ = 1 (indicating perfect home bias). For each value for λ in this grid, we solve for equilibrium allocations. Given a discrete representation for the productivity process, we can solve for equilibrium allocations given a particular value for λ using standard Euler equation iteration. Since tax revenues are rebated lump-sum to the representative agents in each country, eq. 24 is the only equation in the set of equations characterizing equilibrium that references τ. Thus for each value for λ in our grid we can back out the tax rate τ that supports this equilibrium using eq. 24. Once we have a value for τ for each point in the grid on λ, we can count how many interior equilibria, if any, are supported by any particular choice for τ. For example, for suﬃciently high values for τ we should expect that there will be no values for λ satisfying equation 24, and that the only equilibrium is a corner solution in which λ = 1 and there is 100 percent home bias. However, for small but positive values for τ we might expect to ﬁnd (possibly non-unique) equilibria with some diversiﬁcation. In the results section we discuss a numerical example in which two positive diversiﬁcation equilibria emerge for certain tax rates.

Note that if the optimal portfolio condition is satisﬁed for the domestic agent, then it is easy to verify that the corresponding condition is also satisﬁed for the foreign agent.

7. Results We now use the model to answer two key questions. First can an exogenous fall in the correlation of productivity shocks account for the magnitude of the observed increase in diversiﬁcation? Second, is increased diversiﬁcation important in accounting for the magnitude of the observed decline in business cycle correlations?

Our results are summarized in tables 11, 12 and 13 and ﬁgures 7 and 8.