«ABSTRACT Over the period 1972-1986, the correlations of GDP, employment and investment between the United States and an aggregate of Europe, Canada ...»
Several authors have considered international real business cycle models in which a single noncontingent bond is the only asset traded (see, for example, Baxter and Crucini 1995 and Arvanitis and Mikkola 1996). We choose not to directly follow this line of research mostly because with a single bond there is no distinction in the model between gross and net foreign assets, and thus the model cannot hope to capture the fact that the U.S. has accumulated more and more foreign assets while running large current account deﬁcits.13 Having decided that domestic and foreign stocks are the only assets traded, we still need to decide when stock trade occurs. Our benchmark assumption will be that all trade occurs in the ﬁrst period, and that portfolios remain ﬁxed (as long as the shock process remains unchanged) from that point on. After describing the details of the model, which we call the restricted stock trade economy, we will defend this assumption in some detail. In particular, we shall argue that the trade-only-at-date-zero assumption is much less restrictive than it at ﬁrst appears. We will show that for diﬀerent values for the tax on foreign dividends, the calibrated benchmark model nests the two extreme possibilities for international risk sharing. When the tax rate is zero, allocations are equivalent to those arising under a regime of unrestricted stock trade and also equivalent to those arising when international ﬁnancial markets are complete.
For intermediate values of the tax, partial international risk sharing is achieved.14 The nature of our main experiment is to ﬁrst estimate two shock processes corresponding to the ﬁrst and second halves of our data sample. We then assume that agents choose their portfolios at the start of the ﬁrst period and are unable to reoptimize until the stock process changes (unexpectedly) at the start of the second period. At this point, agents are given a single opportunity to costlessly re-optimize their portfolios. To provide additional intuition for our results, we repeat the experiment for an alternative economy with no tax on foreign dividends. The zero tax economy deOne could imagine a world with two non-contingent bonds, one denominated in units of domestic consumption, and one in units of foreign consumption. However, the observed increase in international diversiﬁcation mostly came through increases in foreign direct investment and foreign stock purchases. We therefore
from bonds in our analysis.
We will also argue that another advantage of the restricted stock trade economy relative to the unrestricted stock trade economy is that it is much easier to solve numerically.
livers allocations equivalent to a world with complete markets or unrestricted period by period stock trade. In both economies, the tax rate on foreign dividends is held constant across the simulations corresponding to the two sub periods.
In laying out the details of the model, we ﬁrst describe preferences and production technologies, which are taken directly from Backus, Kehoe and Kydland. Then we describe the benchmark restricted stock trade asset market structure, and deﬁne an equilibrium. Next, we discuss the unrestricted stock trade economy, and explain how alternative market structures are related.
A. The economy The world consists of two countries, each of which is populated by the same measure of identical, inﬁnitely-lived households. In each period t the economy experiences one event st ∈ S.
We denote by st the history of events up to and including date t. The probability at date 0 of any particular history st is given by π(st ).
where c(st ) denotes consumption and n(st ) labor supply at date t given history st. Households supply labor to domestically located perfectly competitive intermediate-goods-producing ﬁrms (i−ﬁrms).
I−ﬁrms in the domestic country produce good a, while those in the foreign country produce good b.
The i−ﬁrms’ production function is Cobb-Douglas in capital and labor:
where z(st ) is an exogenous technology shock.
The equations describing the foreign country are largely identical to those for the domestic country. We use star superscripts to denote foreign variables.
with variance-covariance matrix V.
Let w(st ) be the wage in terms of the domestically-produced intermediate good. The i−ﬁrm’s maximization problem after history st is given by
subject to k(st ), n(st ) ≥ 0, where Q(st ) is the price the ﬁrm uses to value dividends at st relative to consumption at date 0, and dividends (in units of the ﬁnal consumption / investment good) are given by
In this expression qa (st ) is the price of good a in units of the ﬁnal good, and δ is the depreciation rate for capital. The expression for proﬁts of the foreign ﬁrm is similar, except that
After trading in any active asset markets, households sell their holdings of intermediate goods to domestically located ﬁnal-goods-producing ﬁrms (f −ﬁrms). The f −ﬁrms are perfectly competitive and produce ﬁnal goods using intermediate goods a and b as inputs to a constant
returns to scale technology:
where σ is the elasticity of substitution between goods a and b, and ω 0.5 determines the size of the local input bias in the composition of domestically produced ﬁnal goods.
The f −ﬁrm’s static maximization problem in the domestic country after history st is given by
subject to a(st ), b(st ) ≥ 0.
Let rx(st ) denote the real exchange rate, deﬁned as the price of foreign relative to domestic consumption. Since the prices of traded intermediate goods in each country are deﬁned relative to local ﬁnal consumption, applying the law of one price to intermediate goods generates expressions
for rx(st ):
B. International asset market structures Restricted stock trade This is our benchmark asset market structure. All stock trade occurs in the initial period, which is period 0.16 Since ﬁrms are assumed to make the investment decisions, this means that in every period except the ﬁrst, the household simply consumes the sum of labor income and any dividend income from its shareholdings. Thus for t ≥ 1 the state by state budget constraint is given by
Here λ (λf ) denotes the fraction of the domestic (foreign) ﬁrm held by the domestic household.
Foreign dividend income is taxed locally at a constant rate τ, and revenue ψ(st ) is redistributed to domestic households in a lump-sum fashion.
At the start of period 0, the domestic household owns the entire domestic ﬁrm. In this period alone the household chooses purchases of domestic and foreign stocks subject to the budget constraint
consumption. Note that the timing convention is that in period 0 the household receives dividend income from his initial portfolio, and then trades shares in the two representative ﬁrms ex-dividend.
The household also faces a constraint that precludes short positions in foreign stocks: λf ≥ 0.17 A (rather trivial) alternative interpretation is that an international stock market is open at each date, that stocks may be freely traded at date 0, and that at subsequent dates trading costs are large enough to make it optimal for representative agents to do no international asset trade.
We allow short positions in domestic stocks.
At date 0, domestic households choose λ, λf ≥ 0, c(st ) ≥ 0 and n(st ) ∈ [0, 1] for all st and
subject to 14 and 13. Let γ be the multiplier associated with 14, µ(st ) be the multiplier associated with 13, and χ be the multiplier associated with the short-selling constraint for foreign stocks. The ﬁrst order conditions characterizing the solution to the domestic household’s problem are (with
Note that the value the household assigns to an additional unit of consumption (dividend income) in state st relative to additional unit of consumption at date 0 (the household’s stochastic
the marginal value for the domestic agent of the stream of dividend income to which the share is a claim.
Note that if the domestic and foreign economies are perfectly symmetric at date 0, then in equilibrium
where λ∗ (λh∗ ) denotes the fraction of the foreign (domestic) ﬁrm held by the foreign household.
What state contingent consumption prices Q(st ) should the domestic ﬁrm use in this economy when making state contingent investment and employment decisions, which determine state by state dividend payments? If domestic and foreign agents are unable to perfectly insure against country speciﬁc shocks, they will use diﬀerent shadow prices to discount dividends in any particular state.
If a domestic ﬁrm has both domestic and foreign shareholders, these shareholders may therefore disagree on the ﬁrms optimal strategy for reinvesting earnings versus paying out dividends. For example, in some states agents in country 1 may have a low marginal utility of consumption and thus prefer the ﬁrm to reinvest rather than make dividend payments. At the same time agents country 2 may attach much higher marginal value to dividend payments from the domestic ﬁrm, and would therefore prefer a larger dividend payment in the current period rather than the promise of higher dividends in the future.
We assume that ﬁrms price state-contingent dividends using a weighted sum of values of the
where ω is the weight the domestic ﬁrm assigns to domestic shareholders. Moreover, we will focus on the particular case in which ω = Λ, where Λ is the aggregate weight of domestic ﬁrms in the portfolios of domestic agents. In this case ﬁrms weight the preferences of domestic and foreign agents in strict proportion to the average fractions of the ﬁrm they hold. If a required property of equilibrium is that ﬁrms cannot choose Pareto-improving investment policies even when sidepayments between shareholders are possible, then ﬁrms eﬀectively maximize proﬁts with respect to a system of shadow prices that average the idiosyncratic shadow prices of all shareholders (see Diamond 1967, Drèze 1974, and Grossman and Hart 1979). This corresponds precisely to the objective function assumed here.
To assess how sensitive equilibrium allocations are to alternative weighting schemes, we also consider an speciﬁcation in which ω = 1, implying that the ﬁrm ignores foreign shareholders (irrespective of the quantity of stock they hold), and maximizes the value of the ﬁrm for domestic shareholders.
Using equations 16, 17 and 22, the condition deﬁning the optimal portfolio split between domestic and foreign stocks for the domestic household is
Thus in equilibria featuring some degree of international diversiﬁcation, the equilibrium marginal value for the domestic agent of the stream of dividend income associated with an additional domestic share is equal to the value to the domestic agent of the stream of after-tax foreign dividends associated with an additional share of the foreign ﬁrm. Both these marginal values are equal to the market price (equal across countries) of buying shares at date 0. This equation is used later to determine the equilibrium value for λ = λ∗ given a tax rate τ.
Note that λ = 1 corresponds to the case of complete home bias in asset holding.
Deﬁnition of equilibrium An equilibrium is a set of prices for all st and for all t ≥ 0 such that when households and ﬁrms solve their problems taking these prices as given all markets clear.
Unrestricted stock trade Consider now the alternative unrestricted stock trade economy in which stocks may be freely traded period by period. The description of this economy is similar to our benchmark model, except that after receiving dividends each period, individuals can buy and sell shares in the domestic and foreign ﬁrms. These ﬁrms make investment decisions using a weighted sum of marginal utilities of existing shareholders to price the cost of foregoing current dividends, and a weighted sum of marginal utilities of next period shareholders to price the return to investment.
It is possible to show that under certain conditions, allocations in our benchmark economy, which restricts asset trade to date zero, are identical to allocations in this alternative economy with unrestricted trade. This claim is formalized in Proposition 1.
Assume that preferences are log-separable in consumption and leisure (γ = 1), and that the elasticity of substitution between domestic and foreign intermediate goods in ﬁnal goods production (σ) is one. Assume also that no taxes are levied on foreign dividend income (τ = 0).
Then there exists an equilibrium for the unrestricted stock trade economy with the following
(i) The optimal portfolio choice at date zero is for the domestic (foreign) agent to buy a fraction
of the shares in the foreign (domestic) ﬁrm, where ω is the share of domestically produced intermediate goods in ﬁnal goods production, and θ is capital’s share in intermediate goods production.
(ii) The optimal trading rule for both stocks after date zero is ‘hold’ for all future dates and states, irrespective of whether or not there are costs associated with trading stocks after date zero.
(iii) Allocations are identical to those for an economy in which asset markets are complete.