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The small open economy model with incomplete asset markets features a steady-state that depends on initial conditions and equilibrium dynamics that possess a random walk component. A number of modifications to the standard model have been proposed to induce stationarity. This paper presents a quantitative comparison of these alternative approaches. Five different specifications are considered: (1) A model with an endogenous discount factor (Uzawa-type preferences); (2) a model with a debt-elastic interest-rate premium; (3) a model with convex portfolio adjustment costs; (4) a model with complete asset markets; and (5) a model without stationarity-inducing features. The main finding of the paper is that all models deliver virtually identical dynamics at business-cycle frequencies, as measured by unconditional second moments and impulse response functions. The only noticeable difference among the alternative specifications is that the complete-asset-market model induces smoother consumption dynamics.

Computing business-cycle dynamics in the standard small open economy model is problematic. In this model, domestic residents have only access to a risk-free bond whose rate of return is exogenously determined abroad. As a consequence, the steady-state of the model depends on initial conditions. In particular, it depends upon the country’s initial net foreign asset position.1 Put differently, transient shocks have long-run effects on the state of the economy. That is, the equilibrium dynamics posses a random walk component. The random walk property of the dynamics implies that the unconditional variance of variables such as asset holdings and consumption is infinite. Thus, endogenous variables in general wonder around an infinitely large region in response to bounded shocks. This introduces serious computational difficulties because all available techniques are valid locally around a given stationary path. To resolve this problem, researchers resort to a number of modifications to the standard model that have no other purpose than to induce stationarity of the equilibrium dynamics. Obviously, because these modifications basically remove the built-in random walk property of the canonical model, they all necessarily alter the low-frequency properties of the model. The focus of the present study is to assess the extent to which these stationarity-inducing techniques affect the equilibrium dynamics at business-cycle frequencies. We compare the business-cycle properties of five variations of the small open economy. In Section 2 we consider a model with an endogenous discount factor (Uzawa, 1968 type preferences). Recent papers using this type of preferences include Obstfeld (1990), Mendoza (1991), Schmitt-Grohé (1998), and Uribe (1997). In this model, the subjective discount factor, typically denoted by β, is assumed to be decreasing in consumption. Agents become more impatient the more they consume. The reason why this modification makes the steady-state independent of initial conditions becomes clear from inspection of the Euler equation λt=β(ct)(1+r)λt+1. Here, λt denotes the marginal utility of wealth, and r denotes the world interest rate. In the steady-state, this equation reduces to β(c)(1+r)=1, which pins down the steady-state level of consumption solely as a function of r and the parameters defining the function β(·). Kim and Kose (2001) compare the business-cycle implications of this model to those implied by a model with a constant discount factor. They find that both models feature similar comovements of macroeconomic aggregates. We also consider a simplified specification of Uzawa preferences where the discount factor is assumed to be a function of aggregate per capita consumption rather than individual consumption. This specification is arguably no more arbitrary than the original Uzawa specification and has a number of advantages. First, it also induces stationarity since the above Euler equation still holds. Second, the modified Uzawa preferences result in a model that is computationally much simpler than the standard Uzawa model, for it contains one less Euler equation and one less Lagrange multiplier. Finally, the quantitative predictions of the modified Uzawa model are not significantly different from those of the original model. In Section 3 we study a model with a debt-elastic interest-rate premium. This stationarity inducing technique has been used, among others, in recent papers by Senhadji (1994), Mendoza and Uribe (2000), and Schmitt-Grohé and Uribe (2001). In this model, domestic agents are assumed to face an interest rate that is increasing in the country’s net foreign debt. To see why this device induces stationarity, let p(dt) denote the premium over the world interest rate paid by domestic residents, and dt the stock of foreign debt. Then in the steady-state the Euler equation implies that β[1+r+p(d)]=1. This expression determines the steady-state net foreign asset position as a function of r and the parameters that define the premium function p(·) only. Section 4 features a model with convex portfolio adjustment costs. This way of ensuring stationarity has recently been used by Neumeyer and Perri (2001). In this model, the cost of increasing asset holdings by one unit is greater than one because it includes the marginal cost of adjusting the size of the portfolio. The Euler equation thus becomes λt[1+ψ′(dt)]=β(1+r)λt+1, where ψ(·) is the portfolio adjustment cost. In the steady-state, this expression simplifies to 1+ψ′(d)=β(1+r), which implies a steady-state level of foreign debt that depends only on parameters of the model. The models discussed thus far all feature incomplete asset markets. Section 5 presents a model of a small open economy with complete asset markets. Under complete asset markets, the marginal utility of consumption is proportional across countries. So one equilibrium condition states that Uc(ct)=αU*(ct*), where U denotes the period utility function and stars are used to denote foreign variables. Because the domestic economy is small, ct* is determined exogenously. Thus, stationarity of ct* implies stationarity of ct. For the purpose of comparison, in Section 6 we also study the dynamics of the standard small open economy model without any type of stationarity-inducing features, such as the economy analyzed in Correia et al. (1995). In this economy, the equilibrium levels of consumption and net foreign assets display a unit root. As a result unconditional second moments are not well defined. For this reason, we limit the numerical characterization of this model to impulse response functions. All models are calibrated in such a way that they predict identical steady-states. The functional forms of preferences and technologies are also identical across models. The basic calibration and parameterization is taken from Mendoza (1991). The business-cycle implications of the alternative models are measured by second moments and impulse responses. The central result of the paper is that all models with incomplete asset markets deliver virtually identical dynamics at business-cycle frequencies. The complete-asset-market model induces smoother consumption dynamics but similar implications for hours and investment. Section 8 presents a sensitivity analysis. It shows that the main results of the paper are robust to alternative preference specifications. In addition, it explores the relationship between the magnitude of the parameters determining stationarity and the speed of convergence to the long-run equilibrium.

In this paper, we present five alternative ways of making the small open economy real business cycle model stationary: two versions of an endogenous discount factor, a debt-contingent interest rate premium, portfolio adjustment costs, and complete asset markets. The main finding of the paper is that once all five models are made to share the same calibration, their quantitative predictions regarding the behavior of key macroeconomic variables, as measured by unconditional second moments and impulse response functions, is virtually identical. We conclude that if the reason for modifying the canonical non-stationary small open economy model in any of the ways presented in the present study is simply technical, that is, solely aimed at introducing stationarity so that the most commonly used numerical approximation methods can be applied and unconditional second moments can be computed, then in choosing a particular modification of the model the researcher should be guided by computational convenience. In other words, the researcher should choose the variant of the model he finds easiest to approximate numerically. In this respect Model 1, featuring an endogenous discount factor a la Uzawa (1968) is in disadvantage vis a vis the other models. For its equilibrium conditions contain an additional state variable. A second result of our paper is that, in line with results previously obtained in the context of two-country real-business-cycle models by Kollmann (1996) and Baxter and Crucini (1995), whether asset markets are complete or incomplete makes no significant quantitative difference. This paper could be extended in several dimensions. One would be to allow for additional sources of uncertainty, such as domestic demand shocks (i.e. government purchases and preference shocks) and external shocks (i.e. terms-of-trade and world-interest-rate shocks). A second possible extension is to consider other characteristics of the business cycle along which to compare the various models, such as frequency decompositions. Finally, one could study additional stationarity induces variations of the small open economy model. For example, Cardia (1991) and Ghironi (2001) among others bring about stationarity by introducing overlapping generations with perpetually young agents á la Blanchard (1985).