Skip to: [Printable Version (PDF)]

Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 867, August 2006-Screen Reader Version*

Closing Open Economy Models^*

Martin Bodenstein^†

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/.

Abstract:

Several methods have been proposed to obtain stationarity in open economy models. I find substantial qualitative and quantitative differences between these methods in a two-country framework, in contrast to the results of Schmitt-Grohé and Uribe (2003). In models with a debt elastic interest rate premium or a convex portfolio cost, both the steady state and the equilibrium dynamics are unique if the elasticity of substitution between the domestic and the foreign traded good is high. However, there are three steady states if the elasticity of substitution is sufficiently low. With endogenous discounting, there is always a unique and stable steady state irrespective of the magnitude of the elasticity of substitution. Similar to the model with convex portfolio costs or a debt elastic interest rate premium, though, there can be multiple convergence paths for low values of the elasticity in response to shocks.

Keywords: multiple equilibria, stationarity, incomplete markets

JEL Classification: D51, F41

1  Introduction

In open economy models with incomplete asset markets the deterministic steady state depends on the initial conditions of the economy and the steady state is compatible with any level of net foreign assets. In a stochastic environment the model generates non-stationary variables as net foreign assets follow a unit root process.¹

Several modifications of the standard model have been proposed in order to induce stationarity among which are an endogenous discount factor (Uzawa-type preferences), a debt elastic interest rate premium or convex portfolio costs. Schmitt-Grohé and Uribe (2003) present quantitative comparisons of these alternative approaches and find that all of them deliver virtually identical dynamics. However, their analysis is restricted to the case of a small open economy, and therefore further scrutiny is justified. Nevertheless, their work has been cited extensively by others to claim irrelevance of the chosen approach that induces stationarity in a specific model even in multi-country setups.²

In this paper, I investigate the theoretical differences between several stationarity inducing approaches in a standard two-country model with limited substitutability between traded goods. If goods are highly substitutable across countries, the stationarity inducing approaches that I investigate have very similar properties. However, for low values of the elasticity of substitution between traded goods there are important nonlinearities which give rise to substantial differences across methods.

Each of the two countries produces one good. These imperfectly substitutable goods are traded in a frictionless goods market. International financial markets are incomplete as the only asset that is traded between countries is one non-state-contingent bond. I consider three approaches to obtain stationarity: an endogenous discount factor, a debt elastic interest rate premium and convex portfolio costs. While I focus on these three most popular approaches, there are other approaches. Ghironi (2003) solves the stationarity problem by introducing an overlapping generations structure.³ Huggett (1993) solves the stationarity problem by introducing explicit limits on the level of asset holdings.⁴

In the standard model with incomplete markets the steady state is undetermined since the growth rate of marginal utility does not depend on the allocation of net foreign assets. Absent arbitrage opportunities, the price of the non-state-contingent bond is equalized across countries implying that expected marginal utility growth is equalized across countries. In the deterministic steady state, this condition contains no information about the steady state values of the system and the system of equilibrium conditions becomes underdetermined. Any level of net foreign asset holdings is a steady state.

If stationarity is induced by convex portfolio costs, there is a unique stable steady state only if the elasticity of substitution between the domestic and foreign traded goods $\varepsilon$ is sufficiently large, i.e., $\varepsilon$ is above some threshold level $\bar{\varepsilon}$. For lower values of the elasticity of substitution, however, I find three steady states two of which are locally stable, but the third one is not. It is important to note that this multiplicity of steady states is unrelated to the aforementioned indeterminacy in the non-stationary model. I also analyze the dynamic implications of shocks under different values of the elasticity of substitution $\varepsilon$. For a high value of the elasticity of substitution, there is a unique impulse response function for a small technology shock. If $\varepsilon<\bar{\varepsilon}$ this finding no longer holds true. Assume that the economy is in one of the two stable steady states. For a small technology shock there are two paths that lead the economy back into the original steady state. However, for the same shock, the economy can also converge to the other stable steady state. For example, if the shock improves country $1$'s technology, the real exchange rate may either depreciate on impact by a small, an intermediate or a large amount relative to the original steady state. The model with a debt elastic interest rate shares these features with the model of convex portfolio costs.

If, following Uzawa (1968), the discount factor is assumed to be endogenous, an agent's rate of time preference is strictly decreasing in the agent's utility level.⁵ In this setup there is always a unique and stable steady state irrespective of the value of the elasticity of substitution between foreign and domestic goods $\varepsilon$. Ironically, the unique and stable steady state in the model with endogenous discounting features the same allocations as the unstable steady state in the model with portfolio costs for $\varepsilon\leq \bar{\varepsilon}$. In response to a technology shock a high elasticity of substitution $\varepsilon$ implies a unique adjustment path of the economy. However, if $\varepsilon$ is below the critical value $\bar{\varepsilon}$ I find three different impulse response functions for a given small technology shock. If the shock raises country $1$'s technology the real exchange rate may either appreciate on impact by a small or a large amount relative to the magnitude of the shock or it may depreciate by a large amount on impact.

The reason for the striking differences between the models lies in the nonlinearities that arise for low values of the elasticity of substitution. Absent international financial markets, there are multiple equilibria if $\varepsilon$ is below $\bar{\varepsilon}$. Consider an endowment economy with two countries and two traded goods that are imperfect substitutes.⁶Assume that the countries are just mirroring each other with respect to preferences and endowments.⁷ Then, there is always one equilibrium with the relative price of the traded goods equal to unity. However, there can be two more equilibria. If the price of the domestic good is very high relative to the price of the foreign good, domestic agents are very wealthy compared to the foreign agents. If the elasticity of substitution is low, foreigners are willing to give up most of their good in order to consume at least some of the domestic good, and domestic agents end up consuming most of the domestic and the foreign good. The reverse is true as well. Foreign agents consume most of the goods, if the foreign good is very expensive in relative terms. Of course, these last two scenarios cannot be an equilibrium for high values of the elasticity of substitution. In the limiting case of perfect substitutability the unique equilibrium features each country consuming its own endowment.

In the dynamic economy with incomplete asset markets, the equilibria of the economy without international financial markets are the candidate steady states. Consider the case of a low elasticity of substitution, i.e., $\varepsilon<\bar{\varepsilon}$. Under the assumption that portfolio costs are zero if and only if net foreign assets are zero, all three candidate steady states are in fact steady states of the bond economy with convex portfolio costs. Similarly, if the debt elastic interest rate premium is zero if and only if net foreign assets are zero, there are three steady states. However, if the discount factor is endogenous, absence of arbitrage requires that the discount factors are equalized across countries in any steady state. As the discount factor is assumed to be strictly decreasing in the agent's utility level, this condition uniquely determines the steady state allocations (provided a strictly concave utility function and a convex technology).

In the simple model presented in this paper, the critical value of the elasticity of substitution $\bar{\varepsilon}$ lies in the range of $0.4-0.7$ for reasonable choices of the remaining parameters. However, the value of $\bar{\varepsilon}$ is sensitive to changes in the model. For example, if the model is extended along the lines of Corsetti and Dedola (2005) to allow for non-traded goods and a strong complementarity between traded and non-traded goods, $\bar{\varepsilon}$ can easily assume values larger than $1$ for reasonable parameterizations of the model.

The empirical literature reports a wide range of trade elasticities at the aggregate level from 0 to $1.5$.⁸ Whalley (1985) reports an elasticity of $1.5$. In a recent study, Hooper, Johnson and Marquez (2000) estimate trade elasticities for the G7 countries. They report a short-run trade elasticity of $0.6$ for the U.S., and values ranging between 0 and $0.6$ for the remaining G7 countries. Earlier studies by Houthakker and Magee (1969) and Marquez (1990) also suggest trade elasticities between 0 and $1$. In his study, Taylor (1993) estimates an import demand equation for the US and finds a short-run trade elasticity of $0.22$ and a long-run trade elasticity of $0.39$.⁹

The remainder of the paper is organized as follows. Section 2 presents the model and analyses it under the assumption that there are no international financial markets. In section 3, agents have access to one non-state-contingent bond. I analyze the characteristics of the steady states under the different stationarity inducing approaches. The impulse response functions for the model with endogenous discounting and convex portfolio costs are investigated in section 4. Finally, section 5 offers concluding remarks.

2  The model

In the remainder of this section, I analyze the simple two-country model under the assumption of balanced trade, i.e., there are no international financial markets. Absent capital accumulation this assumption allows me to present the issues of multiple equilibria without the additional complications that occur in a dynamic model. Furthermore, under financial autarchy the model is stationary.¹⁰

In Section $3$ the simple model is augmented by the assumption that agents have access to international financial markets. I first present the standard model with incomplete markets in order to illustrate the stationarity problem. Three different approaches to induce stationarity are studied: convex portfolio costs, debt elastic interest rate, and endogenous discount factor. The analysis is guided by two questions. How does the number of steady states in the closed model relate to the number of equilibria in the model with financial autarchy? How are the dynamic properties of a steady state related to the slope of the excess demand function for the different stationarity inducing approaches?

2.1  Financial autarchy

There are two countries, each populated by an infinite number of households with a total measure of one. Each country produces only one good that can be traded without frictions in the international goods market. The two goods are assumed to be imperfect substitutes in the household's utility function. Labor, which is supplied endogenously, is the sole input into the production process.

Time is discrete and each period the economy experiences one of finitely many events $s_{t}$. $s^{t}=\left( s_{0},...,s_{t}\right)$ denotes the history of events up through and including period $t$. The probability, as of period 0, of any particular history $s^{t}$ is $\pi\left( s^{t}\right)$. The initial realization $s_{0}$ is given.

Households maximize their expected lifetime utility subject to the budget constraint

$$\max_{\substack{c_{i1}\left( s^{t}\right) ,c_{i2}\left( s^{t}\right) \\ c_{i}\left( s^{t}\right) ,l_{i}\left( s^{t}\right) }}\sum_{t=0}^{\infty }\sum_{s^{t}}\beta^{t}U\left( c_{i}\left( s^{t}\right) ,l_{i}\left( s^{t}\right) \right) \pi\left( s^{t}\right)$$ (1)

$$s.t.$$

$$\bar{P}_{1}\left( s^{t}\right) c_{i1}\left( s^{t}\right) +\bar{P} _{2}\left( s^{t}\right) c_{i2}\left( s^{t}\right) \leq\bar{P}_{i}\left( s^{t}\right) w_{i}\left( s^{t}\right) l_{i}\left( s^{t}\right) +\bar {P}_{i}\left( s^{t}\right) \Pi_{i}\left( s^{t}\right) +dW_{i}\left( s^{t}\right) ,$$ (2)

where $c_{i}$ is given by the $CES$ aggregator $c_{i}\left( s^{t}\right) =\left[ \alpha_{i1}^{1-\rho}c_{i1}^{\rho}\left( s^{t}\right) +\alpha _{i2}^{1-\rho}c_{i2}^{\rho}\left( s^{t}\right) \right] ^{\frac{1}{\rho}}$ with $\alpha_{ii}\geq\alpha_{ij}>0$ for $j\neq i$ and $\rho<1$. $\frac {1}{1-\rho}$ is the elasticity of substitution between traded goods. In appendix $A$, I generalize the model to allow for any linear-homogeneous aggregator of the form $c_{i}\left( s^{t}\right) =H_{i}\left( c_{i1}\left( s^{t}\right) ,c_{i2}\left( s^{t}\right) \right)$ .

The strictly concave period utility function $U\left( c,l\right)$ is assumed to satisfy the following sign conditions:

$$U_{c}>0,U_{l}<0$$ and $$U_{cc}<0,U_{ll}<0,U_{cl}\leq0.$$

These assumptions are satisfied by almost all utility functions that are commonly used in macroeconomics. $c_{i}$ denotes final consumption, $l_{i}$ labor, $c_{ij}$ is the consumption of good $j$ by a household located in country $i$, $\bar{P}_{i}$ is the price at which good $i$ is traded and $w_{i}$ is the wage in country $i$ denoted in units of country $i$'s traded good. Real profits are $\Pi_{i}$. $dW_{i}$ is an arbitrary lump sum transfer to agents in country $i$, with $dW_{1}\left( s^{t}\right) +dW_{2}\left( s^{t}\right) =0$ for all $s^{t}$. I introduce this transfer to make the following derivations general enough to be of use for the case with international financial markets.

Agent $i$ chooses consumption of the two traded goods such that

$$\frac{c_{i1}\left( s^{t}\right) }{c_{i2}\left( s^{t}\right) }=\frac {\alpha_{i1}}{\alpha_{i2}}\left( \frac{1}{\bar{q}\left( s^{t}\right) }\right) ^{\frac{1}{\rho-1}},$$ (3)

where $\bar{q}$ is the relative price of good $2$ to good $1$, $\frac{\bar {P}_{2}}{\bar{P}_{1}}$. Let $P_{i}$ denote the price of the final consumption basket, which is related to $\bar{q}$ by

$$\Phi_{1}\left( \bar{q}\left( s^{t}\right) \right) \equiv\frac{\bar {P}_{1}\left( s^{t}\right) }{P_{1}\left( s^{t}\right) }=\left[ \alpha_{11}\left( \frac{1}{\bar{q}\left( s^{t}\right) }\right) ^{\frac{\rho}{\rho-1}}+\alpha_{12}\right] ^{\frac{1-\rho}{\rho}}\frac{1} {\bar{q}\left( s^{t}\right) } \Phi_{1}^{\prime}\left( \bar {q}\left( s^{t}\right) \right) <0$$

$$\Phi_{2}\left( \bar{q}\left( s^{t}\right) \right) \equiv\frac{\bar {P}_{2}\left( s^{t}\right) }{P_{2}\left( s^{t}\right) }=\left[ \alpha_{21}\left( \frac{1}{\bar{q}\left( s^{t}\right) }\right) ^{\frac{\rho}{\rho-1}}+\alpha_{22}\right] ^{\frac{1-\rho}{\rho}} \Phi_{2}^{\prime}\left( \bar{q}\left( s^{t}\right) \right) >0$$

I normalize the price of the consumption basket in country $1$ to unity, $P_{1}=1$. Therefore, $P_{2}$ is equal to the consumption-based real exchange rate, $q$. Obviously, $q$ and $\bar{q}$ are related as follows

$$q\left( s^{t}\right) =\bar{q}\left( s^{t}\right) \frac{\Phi_{1}\left( \bar{q}\left( s^{t}\right) \right) }{\Phi_{2}\left( \bar{q}\left( s^{t}\right) \right) }.$$

Using the budget constraint, (2) and equation (3), the demand functions for good $2$ are

$$c_{12}\left( s^{t}\right) =\frac{1}{\left( \frac{\alpha_{11}} {\alpha_{12}}\left( \frac{1}{\bar{q}\left( s^{t}\right) }\right) ^{\frac{1}{\rho-1}}+\bar{q}\left( s^{t}\right) \right) }\left[ w_{1}\left( s^{t}\right) l_{1}\left( s^{t}\right) +\frac{1}{\Phi _{1}\left( \bar{q}\left( s^{t}\right) \right) }dW_{1}\left( s^{t}\right) \right] ,$$ (4)

$$c_{22}\left( s^{t}\right) =\frac{1}{\left( \frac{\alpha_{21}} {\alpha_{22}}\left( \frac{1}{\bar{q}\left( s^{t}\right) }\right) ^{\frac{\rho}{\rho-1}}+1\right) }\left[ w_{2}\left( s^{t}\right) l_{2}\left( s^{t}\right) +\frac{1}{\bar{q}\left( s^{t}\right) \Phi _{1}\left( \bar{q}\left( s^{t}\right) \right) }dW_{2}\left( s^{t}\right) \right] .$$ (5)

Similar expressions can be derived for the demand of good $1$. This provides expressions for aggregate consumption $c_{i}$, $i=1,2$:

$$c_{1}\left( s^{t}\right) =\Phi_{1}\left( \bar{q}\left( s^{t}\right) \right) w_{1}\left( s^{t}\right) l_{1}\left( s^{t}\right) +dW_{1}\left( s^{t}\right) ,$$ (6)

$$c_{2}\left( s^{t}\right) =\Phi_{2}\left( \bar{q}\left( s^{t}\right) \right) w_{2}\left( s^{t}\right) l_{2}\left( s^{t}\right) +\frac{\Phi _{2}\left( \bar{q}\left( s^{t}\right) \right) }{\bar{q}\left( s^{t}\right) \Phi_{1}\left( \bar{q}\left( s^{t}\right) \right) } dW_{2}\left( s^{t}\right) .$$ (7)

The optimal allocation of labor relative to consumption is determined from

$$\frac{U_{l}\left( c_{i}\left( s^{t}\right) ,l_{i}\left( s^{t}\right) \right) }{U_{c}\left( c_{i}\left( s^{t}\right) ,l_{i}\left( s^{t}\right) \right) }=-\Phi_{i}\left( \bar{q}\left( s^{t}\right) \right) w_{i}\left( s^{t}\right) .$$ (8)

Firms in country $i$ produce the traded good $i$ using a linear production technology, $y_{i}\left( s^{t}\right) =A_{i}\left( s^{t}\right) l_{i}\left( s^{t}\right)$ . Appendix $A.1$ shows that equations (6) - (8) can be used to express $c_{i}$ and $l_{i}$ as functions of the prices $w_{1}$, $w_{2}$ and $\bar{q}$ (and $dW_{1}$) only.

Definition 1 (Competitive Equilibrium in Financial Autarchy)   A competitive equilibrium is a collection of allocations $c_{i1}\left( s^{t}\right)$, $c_{i2}\left( s^{t}\right)$, $c_{i}\left( s^{t}\right)$, $l_{i}\left( s^{t}\right)$, $y_{i}\left( s^{t}\right)$ and prices $\bar{q}\left( s^{t}\right)$, $w_{i}\left( s^{t}\right)$, $i=1,2$, such that $\left( i\right)$ for every household the allocations solve the household's maximization problem for given prices, $\left( ii\right)$ for every firm profits are maximized and $\left( iii\right)$ the markets for labor and for the two traded goods clear.

Perfect competition and the linear technology imply that the equilibrium wage equals the productivity parameter, i.e. $w_{i}\left( s^{t}\right) =A_{i}\left( s^{t}\right)$. As shown in appendix $A.2$ the equilibrium conditions for this model can be fully summarized by the excess demand function for good $2$:

$$z_{2}\left( \bar{q}\left( s^{t}\right) ,dW_{1}\left( s^{t}\right) ,dW_{2}\left( s^{t}\right) \right) =c_{12}\left( s^{t}\right) +c_{22}\left( s^{t}\right) -y_{2}\left( s^{t}\right)$$

$$=\frac{A_{1}\left( s^{t}\right) l_{1}\left( \bar{q}\left( s^{t}\right) ,dW_{1}\left( s^{t}\right) \right) +\frac{1}{\Phi_{1}\left( \bar{q}\left( s^{t}\right) \right) }dW_{1}\left( s^{t}\right) }{\frac{\alpha_{11} }{\alpha_{12}}\left( \frac{1}{\bar{q}\left( s^{t}\right) }\right) ^{\frac{1}{\rho-1}}+\bar{q}\left( s^{t}\right) }$$

$$+\frac{A_{2}\left( s^{t}\right) l_{2}\left( \bar{q}\left( s^{t}\right) ,dW_{2}\left( s^{t}\right) \right) +\frac{1}{\bar{q}\Phi_{1}\left( \bar {q}\left( s^{t}\right) \right) }dW_{2}\left( s^{t}\right) }{\frac {\alpha_{21}}{\alpha_{22}}\left( \frac{1}{\bar{q}\left( s^{t}\right) }\right) ^{\frac{\rho}{\rho-1}}+1}$$

$$-A_{2}\left( s^{t}\right) l_{2}\left( \bar{q}\left( s^{t}\right) ,dW_{2}\left( s^{t}\right) \right) .$$ (9)

An equilibrium is a relative price $\bar{q}^{\ast}\left( s^{t}\right)$, s.t. $z_{2}\left( \bar{q}^{\ast}\left( s^{t}\right) ,dW_{1}\left( s^{t}\right) ,dW_{2}\left( s^{t}\right) \right) =0$ . Appendix $A.2$ proves the existence of the competitive equilibrium.

Figure 1 plots the excess demand for good $2$ as a function of $\frac{\bar{q}}{1+\bar{q}}$ for different values of the elasticity of substitution $\varepsilon$. In plotting $z_{2}$, I assume the following utility function

$$U\left( c,l\right) =\frac{c^{1-\gamma}}{1-\gamma}-\chi_{0}\frac{l^{1-\chi} }{1-\chi}.$$ (10)

The parameter values are

Table 1: parameterization

parameter	value	explanation of the parameter
$\chi$	-2.75	$-\frac{1}{\chi}$is the Frisch labor supply elasticity
$\chi_{0}$	7.00
$\gamma$	3.00	coefficient of relative risk aversion
$\alpha_{11}=\alpha_{22}$	0.90	weight on domestic good in CES aggregator
$\alpha_{21}=\alpha_{12}$	0.10	weight on foreign good in CES aggregator
$A_{1}=A_{2}$	1.00	technology level

Unless noted otherwise these are the parameters for all figures in this paper. Furthermore, I assume $dW_{1}=dW_{2}=0$.

Figure 1: Excess demand for good 2 for different values of $\varepsilon .$

Figure 1 description:

Figure 1 consists of three graphs presented in a 3-by-1 matrix. The caption for figure 1 reads “excess demand for good 2 for different values of epsilon”. The x-axis of each graph is labeled q-bar over 1 plus q-bar, and ranges in values from 0 to 1 in increments of 0.2. The top graph is titled epsilon equals 1. The y-axis is labeled Z sub 2, and ranges in values from -0.4 to 4 in increments of 0.4. A dashed horizontal line exists in the graph where Z sub 2 equals 0. The curve begins at (0.11,0.4), it steadily decreases to (0.5,0), and continues to decrease, albeit more slowly, until (1,-0.05). The middle graph is titled epsilon equals 0.48. The y-axis is labeled Z sub 2, and ranges in values from -0.002 to 0.002 in increments of 0.002. A dashed horizontal line exists in the graph where Z sub 2 equals 0. The curve begins at (0.08,0.002). It decreases quickly to (0.2,-0.0005). It then very slowly increases, crosses the dashed horizontal line at (0.5,0) and continues to (0.7,0.0005). Afterwards, it slowly decreases until (1,-0.0005). The bottom graph is titled epsilon equals 0.48309. The y-axis is labeled Z sub 2, and ranges in values from -0.02 to 0.02 in increments of 0.02. A dashed horizontal line exists in the graph where Z sub 2 equals 0. The curve begins at (0.05,0.02). It quickly decreases until converging to (0.2, 0). It remains at this value for the rest of q-bar over 1 plus q-bar.

For a given parameterization of the remainder of the model I distinguish three cases for the elasticity of substitution, $\varepsilon=\frac{1}{1-\rho}$:

1. If $\varepsilon$ is sufficiently high, as in the first panel, the excess demand function has exactly one zero. Hence, the equilibrium is unique and it features $\frac{\bar{q}}{1+\bar{q}}=\frac{1}{2}$. Furthermore, the slope of the excess demand function at $\frac{\bar{q}}{1+\bar{q}}=\frac{1}{2}$ is negative.
2. If $\varepsilon$ is sufficiently low, as in the second panel, the excess demand function has three zeros. While the excess demand function is downward sloping at the first and third equilibrium, it is upward sloping in the second one (counted from left to right).
3. There is a critical value $\bar{\varepsilon}$ (here $\bar{\varepsilon }=0.48309$) of the elasticity of substitution such that there is a continuum of equilibria because the slope of the excess demand function is zero around $\frac{\bar{q}}{1+\bar{q}}=\frac{1}{2}$.

Multiple equilibria arise at low values of the elasticity of substitution for the following reason. Consider the first equilibrium in the second panel. If the price of good $1$ is high relative to the price of good $2$, $\frac{\bar {q}}{1+\bar{q}}<<\frac{1}{2}$, agents of country $2$ produce more of their good than agents of country $1$. As the elasticity of substitution between the goods is very low, country $2$ agents are willing to pay the high price for good $1$ and country $1$ ends up consuming most of the two goods. The same logic applied in the third equilibrium, $\frac{\bar{q}}{1+\bar{q}}>>\frac {1}{2}$, with the roles of country $1$ and $2$ being reversed. The second equilibrium is the symmetric equilibrium featuring $\frac{\bar{q}}{1+\bar{q}}=\frac{1}{2}$. If the elasticity of substitution is high, equilibria $1$ and $3$ cannot exist. In fact, if the elasticity of substitution is infinite, agents in both countries only consume their own goods and the relative price in the unique equilibrium has to be equal to $1$.

2.2  More about multiple equilibria

The above findings are not surprising. The analogous endowment economy with a CES aggregator has been used extensively in general equilibrium theory to study equilibrium multiplicity. For instance see Mas-Colell (1991), Kehoe (1991), Gjerstad (1996) and Bela (1997). In appendix A.3, I summarize some of the findings of general equilibrium theory in the context of the model presented here in this paper. In general, the number of equilibria is odd. If the excess demand function is upward sloping in an equilibrium, there have to be at least two more. Unfortunately, nothing can be said with certainty about the number of equilibria unless one can prove that the equilibrium is unique.

To gain an idea about the relationship between the critical value $\bar{\varepsilon}$ and the remaining model parameters, consider the case of $\alpha_{11}=\alpha_{22}\geq\frac{1}{2}$, $\alpha_{12}=\alpha_{21}$ and identical preferences over consumption and leisure in the two countries. Irrespective of the value of the elasticity of substitution, $\bar{q}=1$ is an equilibrium. As shown more generally in appendix $A.3$, the critical value $\bar{\varepsilon}$ is given by

$$2\alpha_{11}\bar{\varepsilon}+\left( 1-2\alpha_{11}\right) -\frac{\partial l_{1}}{\partial\bar{q}}\frac{\bar{q}}{l_{1}}+\frac{\partial l_{2}} {\partial\bar{q}}\frac{\bar{q}}{l_{2}}=0\text{,}$$

where $\frac{\partial l_{i}}{\partial\bar{q}}\frac{\bar{q}}{l_{i}}$ is the general equilibrium elasticity of labor with respect to $\bar{q}$. With additive separable preferences as in (10), $\frac{\partial l_{i}}{\partial\bar{q}}\frac{\bar{q}}{l_{i}}=\frac {1-\gamma}{\gamma-\chi}\left( 1-\alpha_{ii}\right) \left( -1\right) ^{i}$ , $i=1,2$, ($\gamma>0$, $\chi<0$) and

$$\bar{\varepsilon}=\frac{2\alpha_{11}-1}{2\alpha_{11}}+\frac{\gamma-1} {\gamma-\chi}\frac{1-\alpha_{11}}{\alpha_{11}}.$$

This implies:

• Even without home bias in consumption, i.e., $\alpha_{11}=\frac{1}{2}$, $\bar{\varepsilon}>0$ if agents are sufficiently risk averse ($\gamma>1$).
• $\bar{\varepsilon}$ is strictly increasing in $\gamma$.
• If $\gamma+\chi<2$, $\bar{\varepsilon}$ is increasing in the home bias. Otherwise, $\bar{\varepsilon}$ is decreasing in $\alpha_{11}$.
• If $\gamma>1$, $\bar{\varepsilon}$ is increasing in the Frish labor supply elasticity $\eta=-\frac{1}{\chi}$. Otherwise, $\bar{\varepsilon}$ is decreasing in $\alpha_{11}$.
• If $\gamma\rightarrow\infty$ and $\chi\rightarrow0$, $\bar{\varepsilon }=\frac{1}{2\alpha_{11}}$.

If the household's preferences over consumption and leisure are Cobb-Douglas the appendix shows that $\frac{\partial l_{i}}{\partial\bar{q}}\frac{\bar{q} }{l_{i}}=0$ , $i=1,2$ and

$$\bar{\varepsilon}=\frac{2\alpha_{11}-1}{2\alpha_{11}}.$$

In this case, $\bar{\varepsilon}$ does not depend on the preference parameters over $c$ and $l$.

As these examples show, the critical value of $\bar{\varepsilon}$ (and therefore the presence of multiple equilibria) is greatly affected by certain model choices. In the technical appendix to this paper, which is available upon request, I also show the following:

• In a model with endogenous capital formation, $\bar{\varepsilon}$ also depends on the elasticity of substitution between labor and capital. Relative to the model without capital, $\bar{\varepsilon}$ can be lower or higher for otherwise identical parameters.
• In a model with non-traded goods, $\bar{\varepsilon}$ can be very high (much larger than unity) for reasonable parameter choices if there is some complementarity between traded and non-traded goods as in Corsetti, Dedola and Leduc (2005).

The value of $\varepsilon$ also has an important impact on the comparative static properties of the model. Consider a small increase in the productivity level of country $1$. Such a change deforms the excess demand function and shifts it upwards. Figure 2 shows the excess demand function for $\varepsilon=0.5>\bar{\varepsilon}$ (upper panel) and $\varepsilon=0.48<\bar{\varepsilon}$ (lower panel) for two technology shocks of different magnitude. The solid line depicts the original excess demand function with $A_{1}/A_{2}=1.00$. The dashed line shows the case of $A_{1}/A_{2}=1.0025$ and the dashed-dotted line is the case of $A_{1} /A_{2}=1.005$. If the elasticity of substitution is large ( $\varepsilon >\bar{\varepsilon}$) the increase in $A_{1}$ leads to a small increase in the equilibrium value of the relative price of traded goods irrespective of the magnitude of the shock.

The situation is quite different if the elasticity is low ( $\varepsilon<\bar{\varepsilon}$). For a small relative increase in $A_{1}$ all three equilibria are preserved. While the price of traded goods rises in the first and third equilibrium relative to the original equilibrium, $A_{1}/A_{2}=1.00$, $\bar{q}$ drops in the second equilibrium. However, if the technology shock is sufficiently large, the first two equilibria disappear. Only the third equilibrium survives. The dashed-dotted line in panel 2 has only one zero, which occurs around $\frac{\bar{q}}{1+\bar{q}}=0.92$.

Figure 2: Equilibria for small changes in $A_{1}/A_{2}$.

Figure 2 description:

Figure 2 consists of two graphs presented in a 2-by-1 matrix. The caption reads “Equilibria for small changes in A sub 1 over A sub 2.” The x-axis of each graph is labeled q-bar over 1 plus q-bar, and ranges in values from 0 to 1 in increments of 0.2. The top graph is titled epsilon equals 0.5. The y-axis is labeled Z sub 2, and ranges in values from -0.002 to 0.002 in increments of 0.001. A dashed horizontal line exists in the graph where Z sub 2 equals 0. The first curve is labeled A sub 1 over A sub 2 equals 1. The second curve is labeled A sub 1 over A sub 2 equals 1.0025. The third curve is labeled A sub 1 over A sub 2 equals 1.005. The curves all take the same shape. The first and third curves lie on both sides of the second curve. The second curve begins at the point (0.35, 0.002). It decreases to (0.95,-0.0015) before increasing to (1,0). The difference between the second curve and the first curve (below it) and the difference between the second curve and the third curve (above it) varies. Initially, it is 0.0001 on each side of curve 2. This difference decreases, and by the end, all three curves end at the same point (1,0). The bottom graph is titled epsilon equals 0.48. The y-axis is labeled Z sub 2, and ranges in values from -0.0005 to 0.001 in increments of 0.0005. A dashed horizontal line exists in the graph where Z sub 2 equals 0. The first curve is labeled A sub 1 over A sub 2 equals 1. The second curve is labeled A sub 1 over A sub 2 equals 1.0025. The third curve is labeled A sub 1 over A sub 2 equals 1.005. The curves all take the same shape. The first and third curves lie on both sides of the second curve. The second curve begins at (0.1,0.001). It quickly decreases to (0.2,-0.0005), then increases slowly to (0.8,0.0001) before falling back down again to (1,-0.0005). The difference between the second curve and the first curve (below it) and the difference between the second curve and the third curve (above it) varies. Initially, it appears to be 0.0001 on each side of curve 2. By the time, q-bar over 1 plus q-bar equals 0.2, this difference has increased to 0.0003. Afterwards, this difference decreases, and by the end, all three curves end at the same point (1,-0.0005).

3  Bond economies

In contrast to the last section agents are now assumed to have access to international financial markets. Following the standard assumption in international macroeconomics, financial markets are exogenously incomplete in the sense that the only asset that is traded internationally is one non-state-contingent bond. This bond is in zero net supply, i.e., $B_{1}\left( s^{t}\right) +B_{2}\left( s^{t}\right) =0$.

In order to illustrate the stationarity problem, I begin with the standard incomplete markets setup without stationarity-inducing features.

3.1  The non-stationary model

In the standard two-country model a household faces the following maximization problem

$$\max_{\substack{c_{i}\left( s^{t}\right) ,l_{i}\left( s^{t}\right) , \\ c_{i1}\left( s^{t}\right) ,c_{i2}\left( s^{t}\right) , \\ B_{i}\left( s^{t}\right) }}\sum_{t=0}^{\infty}\sum_{s^{t}}\beta^{t}U\left( c_{i}\left( s^{t}\right) ,l_{i}\left( s^{t}\right) \right) \pi\left( s^{t}\right)$$

$$s.t.$$

$$P_{i}\left( s^{t}\right) c_{i}\left( s^{t}\right) \leq\bar{P}_{i}\left( s^{t}\right) w_{i}\left( s^{t}\right) l_{i}\left( s^{t}\right) +\bar {P}_{i}\left( s^{t}\right) \Pi_{i}\left( s^{t}\right) +B_{i}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{i}\left( s^{t}\right) .$$

$P_{i}c_{i}$ are the household's total consumption expenditures which are equal to $\bar{P}_{1}c_{i1}+\bar{P}_{2}c_{i2}$. In order to make use of the above derivations, notice that I have replaced the lump-sum transfer between countries $dW_{i}\left( s^{t}\right)$ by $B_{i}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{i}\left( s^{t}\right)$ . $B_{i}\left( s^{t-1}\right)$ denotes the (nominal) bond holdings that agent $i$ has inherited from period $t-1$. $Q$ is the price of bonds.

Given the assumptions on technology, preferences, and trade stated in the previous section the equilibrium dynamics are fully summarized by

1. the excess demand function for good $2$

   $$z_{2}\left( \bar{q}\left( s^{t}\right) ,B_{1}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{1}\left( s^{t}\right) \right) \smallskip$$

   $$=\frac{A_{1}l_{1}\left( \bar{q}\left( s^{t}\right) ,B_{1}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{1}\left( s^{t}\right) \right) }{\frac{\alpha_{11}}{\alpha_{12}}\left( \frac{1}{\bar{q}\left( s^{t}\right) }\right) ^{\frac{1}{\rho-1}}+\bar{q}\left( s^{t}\right) }$$

   $$-\frac{\frac{\alpha_{21}}{\alpha_{22}}\left( \frac{1}{\bar{q}\left( s^{t}\right) }\right) ^{\frac{\rho}{\rho-1}}A_{2}l_{2}\left( \bar{q}\left( s^{t}\right) ,-B_{1}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{1}\left( s^{t}\right) \right) }{\frac{\alpha_{21}}{\alpha_{22}}\left( \frac{1}{\bar{q}\left( s^{t}\right) }\right) ^{\frac{\rho}{\rho-1}} +1}$$

   $$+\left[ \frac{1}{\frac{\alpha_{11}}{\alpha_{12}}\left( \frac{1}{\bar {q}\left( s^{t}\right) }\right) ^{\frac{1}{\rho-1}}+\bar{q}\left( s^{t}\right) }-\frac{1}{\frac{\alpha_{21}}{\alpha_{22}}\left( \frac{1} {\bar{q}\left( s^{t}\right) }\right) ^{\frac{\rho}{\rho-1}}+1}\right] \frac{1}{\Phi_{1}\left( \bar{q}\left( s^{t}\right) \right) }\left[ B_{1}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{1}\left( s^{t}\right) \right] ,$$ (11)

   
   where
   $$Q\left( s^{t}\right) =\sum_{s^{t+1}\vert s^{t}}\beta\frac{U_{c}\left( c_{1}\left( s^{t+1}\right) ,l_{1}\left( s^{t+1}\right) \right) } {U_{c}\left( c_{1}\left( s^{t}\right) ,l_{1}\left( s^{t}\right) \right) }\pi\left( s^{t+1}\vert s^{t}\right) ,$$
   and $z_{2}\left( \bar{q}\left( s^{t}\right) ,B_{1}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{1}\left( s^{t}\right) \right) =0$ in equilibrium,
2. the familiar risk sharing equation, states that expected marginal utility growth is equalized across countries:

   $$\sum_{s^{t+1}\vert s^{t}}\beta\left[ \frac{U_{c}\left( c_{1}\left( s^{t+1}\right) ,l_{1}\left( s^{t+1}\right) \right) }{U_{c}\left( c_{1}\left( s^{t}\right) ,l_{1}\left( s^{t}\right) \right) }-\frac {U_{c}\left( c_{2}\left( s^{t+1}\right) ,l_{2}\left( s^{t+1}\right) \right) }{U_{c}\left( c_{2}\left( s^{t}\right) ,l_{2}\left( s^{t}\right) \right) }\frac{q\left( s^{t}\right) }{q\left( s^{t+1}\right) }\right] \pi\left( s^{t+1}\vert s^{t}\right) =0,$$ (12)

   
   where $q\left( s^{t}\right) =\bar{q}\left( s^{t}\right) \frac{\Phi _{1}\left( \bar{q}\left( s^{t}\right) \right) }{\Phi_{2}\left( \bar {q}\left( s^{t}\right) \right) }$ .

I have used the assumption that bonds are in zero net supply. As shown in appendix $A.1$, consumption and labor choices can be expressed as functions of the relative price $\bar{q}$. Moreover, this system of difference equations has to satisfy the appropriate initial and transversality conditions.

Unfortunately, the deterministic steady state of this model is not unique. With $c_{i}=c_{i}\left( \bar{q}\right)$ and $l_{i}=l_{i}\left( \bar {q}\right)$ as shown in Appendix $A.1$, equations (11) and (12) have to solve for the steady state values of $\bar{q}$ and $B_{1}$. However, in a steady state, equation (12) collapses to an identity and contains no information about the endogenous variables. Hence, there is one equations but two unknowns.

Admittedly, it is possible to choose a particular steady state amongst the set of feasible solutions to (11). It is common practice see, e.g., Baxter and Crucini (1995) to assume $B_{1}=B_{2} =0$. Although this choice pins down the original steady state, the dynamic system that describes the behavior of the economy in the neighborhood of the steady state is not stationary. Even a completely temporary shock has long lasting effects on the economy: whatever the level of bond holdings materializes in the period immediately following a shock becomes the new long-run position until a new shock occurs.

This problem is easily seen by looking at the linear approximation of the dynamic system around a candidate steady state with $B_{1}=B_{2} =0$. For simplicity assume that preferences are additive-separable in consumption and leisure, i.e.,

$$U\left( c,l\right) =\frac{c^{1-\gamma}}{1-\gamma}-\chi_{0}\frac{l^{1-\chi} }{1-\chi}.$$

Using equations (6) - (8) with $dW_{1}=B_{1}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{1}\left( s^{t}\right)$ , consumption in country $1$ can be expressed as

$$c_{1}\left( s^{t}\right) =\chi_{0}^{\frac{1}{\chi}}\left[ \Phi_{1}\left( \bar{q}\left( s^{t}\right) \right) A_{1}\left( s^{t}\right) \right] ^{\frac{\chi-1}{\chi}}\left[ c_{1}\left( s^{t}\right) \right] ^{\frac{\gamma}{\chi}}+B_{1}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{1}\left( s^{t}\right)$$    .

Log-linearization around a steady state delivers

$$\left( 1-\frac{\gamma}{\chi}\right) \hat{c}_{1,t}=\frac{\chi-1}{\chi} \frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }\bar{q}_{t}+\frac{1}{c_{1}}\left( b_{1,t-1}-\beta b_{1,t}\right) ,$$

where $\hat{c}_{1,t}$ and $\bar{q}_{t}$ are the percentage deviation of consumption and the relative price from their respective steady state values. $b_{1,t}$ is the absolute deviations of bond holdings from 0. Notice that the log-linearized excess demand function implies

$$\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}\bar{q}_{t}+\frac{\partial z_{2} }{\partial dW_{1}}\left[ b_{1,t-1}-\beta b_{1,t}\right] =0\text{.}$$

Hence, $\hat{c}_{1,t}$ can simply be expressed as a function of $\bar{q}_{t}$. Similar reasoning applies for $\hat{c}_{2,t}$, which can also be expressed as a function of $\bar{q}_{t}$ only.

Finally, the log-linear approximation of equation (12) is given by

$$-\gamma\left( \left( \hat{c}_{1,t+1}-\hat{c}_{2,t+1}\right) -\left( \hat{c}_{1,t}-\hat{c}_{2,t}\right) \right) =\left[ 1+\frac{\Phi_{1} ^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }-\frac{\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{2}\left( \bar{q}\right) }\right] \left( \hat{q}_{t}-\hat{q}_{t+1}\right) .$$

The findings for $\hat{c}_{i,t}$ (and $\hat{c}_{i,t+1}$) imply

$$\hat{q}_{t}=\hat{q}_{t+1}$$.

The dynamics of the system can therefore be approximated by

$$\left( \begin{array}[c]{c} \bar{q}_{t+1}\ b_{t} \end{array}\right) =\left( \begin{array}[c]{cc} 1 & 0\ \frac{1}{\beta}\frac{\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}} {\frac{\partial z_{2}}{\partial dW_{1}}} & \frac{1}{\beta} \end{array}\right) \left( \begin{array}[c]{c} \bar{q}_{t}\ b_{t-1} \end{array}\right)$$

The two eigenvalues that are associated with this system are $1$ and $\frac {1}{\beta}$. This implies that a purely transitory shock to technology in country $1$ at time $t$ permanently raises the relative price $\bar{q}$, and turns country $1$ into a borrower. There is no internal mechanism that leads $\bar{q}$ and $B_{1}$ back to the original steady state values. $\bar{q}$ and $B_{1}$ only change if new shocks occur.

3.2  Bond economy with convex portfolio costs

Similar to Heathcote and Perri (2002), and Schmitt-Grohé and Uribe (2003) let agents face a convex cost for holding/issuing bonds. The collected fees are reimbursed to the agents by a lump-sum transfer. $\Phi\left( B_{i} /\bar{P}_{i}\right)$ denotes the portfolio costs in terms of country $i$'s traded good, where $\Phi^{\prime}\left( 0\right) =0$ and $\Phi^{\prime}>0$ otherwise. The representative household in country $i$ solves

$$\max_{\substack{c_{i}\left( s^{t}\right) ,l_{i}\left( s^{t}\right) , \\ c_{i1}\left( s^{t}\right) ,c_{i2}\left( s^{t}\right) , \\ B_{i}\left( s^{t}\right) }}\sum_{t=0}^{\infty}\sum_{s^{t}}\beta^{t}U\left( c_{i}\left( s^{t}\right) ,l_{i}\left( s^{t}\right) \right) \pi\left( s^{t}\right)$$

$$s.t.$$

$$P_{i}\left( s^{t}\right) c_{i}\left( s^{t}\right) \leq\bar{P}_{i}\left( s^{t}\right) w_{i}\left( s^{t}\right) l_{i}\left( s^{t}\right) +\bar {P}_{i}\left( s^{t}\right) \Pi_{i}\left( s^{t}\right) +B_{i}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{i}\left( s^{t}\right)$$

$$-\bar{P}_{i}\left( s^{t}\right) \Gamma\left( \frac{B_{i}\left( s^{t}\right) }{\bar{P} _{i}\left( s^{t}\right) }\right) +T_{i}\left( s^{t}\right)$$

where $T_{i}$ is the lump-sum reimbursement of the portfolio costs.

The equilibrium dynamics are fully summarized by $z_{2}\left( \bar{q}\left( s^{t}\right) ,B_{1}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{1}\left( s^{t}\right) \right) =0$ with $z_{2}$ given by equation (11),

$$Q\left( s^{t}\right) =\sum_{s^{t+1}\vert s^{t}}\beta\frac{U_{c}\left( c_{1}\left( s^{t+1}\right) ,l_{1}\left( s^{t+1}\right) \right) } {U_{c}\left( c_{1}\left( s^{t}\right) ,l_{1}\left( s^{t}\right) \right) }\pi\left( s^{t+1}\vert s^{t}\right) -\Gamma^{\prime}\left( \frac{B_{1}\left( s^{t}\right) }{\bar{P}_{1}\left( s^{t}\right) }\right) ,$$

and the risk sharing condition

$$\sum_{s^{t+1}\vert s^{t}}\beta\left[ \frac{U_{c}\left( c_{1}\left( s^{t+1}\right) ,l_{1}\left( s^{t+1}\right) \right) }{U_{c}\left( c_{1}\left( s^{t}\right) ,l_{1}\left( s^{t}\right) \right) }-\frac {U_{c}\left( c_{2}\left( s^{t+1}\right) ,l_{2}\left( s^{t+1}\right) \right) }{U_{c}\left( c_{2}\left( s^{t}\right) ,l_{2}\left( s^{t}\right) \right) }\frac{q\left( s^{t}\right) }{q\left( s^{t+1}\right) }\right] \pi\left( s^{t+1}\vert s^{t}\right)$$ (13)

$$=\Gamma^{\prime}\left( \frac{B_{1}\left( s^{t}\right) }{\bar{P} _{1}\left( s^{t}\right) }\right) -\Gamma^{\prime}\left( \frac{B_{2}\left( s^{t}\right) }{\bar{P}_{2}\left( s^{t}\right) }\right) .$$

As $\Gamma^{\prime}=0$ for $B_{i}=0$ and larger than zero otherwise, equation (13) implies that in a steady state $B_{1}=B_{2} =0$. Although the steady state value of bond holdings is uniquely determined, there can still be multiple steady states if $z_{2}\left( \bar{q},0\right) =0$ has multiple solutions in $\bar{q}$. Hence, any steady state of the financial autarchy model is also a steady state of the model with convex portfolio costs. Consequently, for $\varepsilon<\bar{\varepsilon}$ the portfolio cost model has multiple steady states.

The global equilibrium dynamics in an economy with perfect foresight are depicted in figures 3 and 4 in a phase diagram. The dashed lines are the $B_{1,t}-B_{1,t-1}=0$ locus and the $\bar{q}_{t+1}-\bar{q}_{t}=0$ locus of the dynamic system, respectively. Each intersection of the two loci corresponds to a steady state. The manifold which has been computed by a reverse shooting algorithm is depicted by the solid line.¹¹ If the elasticity of substitution is high $\left( \varepsilon>\bar{\varepsilon}\right)$, there is a unique and stable steady state, see figure 4.

However, there are three steady states if the elasticity of substitution is low $\left( \varepsilon<\bar{\varepsilon}\right)$ as depicted in figure 5. As indicated by the arrows the first and the third steady state are locally stable, but the second one $\left( \frac{\bar{q}}{1+\bar{q}}=\frac{1} {2},B_{1}=0\right)$ is not. For intermediate values of initial bond holdings the economy converges either to the first or to the third steady state. Convergence to a steady state is unique only if the initial bond holdings are sufficiently high in absolute value.

The (local) dynamic properties of the model with convex portfolio costs are summarized in the following theorem.

Theorem 1   Assume that agents face convex portfolio costs for holding/issuing bonds as described above. If the slope of the excess demand function is negative in a steady state, then this steady state is a saddle point. If the slope of the excess demand function is positive in a steady state, then such a steady state is unstable if $\Gamma^{\prime\prime}\left( 0\right)$ is sufficiently small, i.e., $\Gamma^{\prime\prime}\left( 0\right) <\Delta_{P}$. Otherwise this steady state is a saddle point.

Appendix $B.3$ provides an exact definition of $\Delta_{P}$ and a proof of the above theorem. $\Gamma^{\prime\prime}\left( 0\right)$ measures how sensitive the portfolio costs are in the steady state with respect to changes in the bond distribution. To keep the model with convex portfolio costs close to the original model, the portfolio costs need to be small and quite insensitive to changes in the allocation of assets. In fact, if $\Gamma$ is quadratic as in Heathcote and Perri (2002), $\Gamma^{\prime\prime}\left( 0\right)$ is sufficiently small for portfolio costs that are of realistic magnitude.

If portfolio costs are chosen to be very large, the model becomes similar to the model with financial autarchy. Under financial autarchy, any steady state is locally stable in this set-up.

Figure 3: Stability of the steady state for $\varepsilon =1$ with convex portfolio costs.

Figure 3 description:

Figure 3 consists of one graph. The caption reads “stability of the steady state for epsilon equals one with convex portfolio costs.” The x-axis is labeled q-bar over 1 minus q-bar, and ranges in values from 0 to 1 in increments of 0.2. The y-axis is labeled B sub 1, and ranges in values from -1.2 to 1 in increments of 0.2. A dashed horizontal line exists where B sub 1 equals 0. The line is labeled q-bar sub (t plus 1) minus q-bar sub t equals 0. A dashed diagonal line exists starting at about (0.5, 1) and running to (0.6,-1.2). The line is labeled B sub 1 comma t minus B sub 1 comma (t minus 1) equals 0. Note that the dashed lines create four quadrants. The dashed lines intersect at (0.5,0). A single parabolic-shaped curve exists starting at about (0.2, 1) running through (0.5, 0) and then to (0.7, -1.2). Recall that the two dashed lines intersect to create four quadrants. The portion of the curve in the upper left quadrant contains an arrow pointing southeast. The portion of the curve in the lower right quadrant contains an arrow pointing northwest.

Figure 4: Stability of the steady state for $\varepsilon =0.48$ with convex portfolio costs.

Figure 4 description:

Figure 4 consists of one graph. The caption reads “stability of the steady state for epsilon equals 0.48 with convex portfolio costs.” The x-axis is labeled q-bar over 1 plus q-bar, and ranges in values from 0 to 1 in increments of 0.2. The y-axis is labeled B sub 1, and ranges in values from -0.02 to 0.035 in increments of 0.01. A dashed horizontal line exists where B sub 1 equals 0. The line is labeled q-bar sub (t plus 1) minus q-bar sub t equals 0. A dashed curve and a solid curve exist with similar polynomial shapes. Each curve crosses the dashed horizontal line at (0.1,0), (0.5,0) and (0.85,0). The dashed line is labeled B sub 1 comma t minus B sub 1 comma (t minus 1) equals 0. It begins at (0.08,0.015). The solid curve begins at (0.08,0.01) and decreases, crosses the dashed line at (0.1,0) and decreases until (0.35,-0.015). It then increases, crosses the dashed horizontal line at (0.5,0) and reaches a high point of (0.7, 0.035). It then decreases, crosses the dashed horizontal line at (0.85,0), and reaches the value (0.9,-0.02). In the part of the curve where q-bar over 1 plus q-bar is less than 0.1, there exists an arrow pointing southeast. In the part of the curve where q-bar over 1 plus q-bar is between 0.1 and 0.3, there exists an arrow pointing northwest. In the part of the curve where q-bar over 1 plus q-bar is between 0.3 and 0.5, there exists an arrow pointing southwest. In the part of the curve where q-bar over 1 plus q-bar is between 0.5 and 0.7, there exists an arrow pointing northeast. In the part of the curve where q-bar over 1 plus q-bar is between 0.7 and 0.85, there exists an arrow point southeast. In the part of the curve where q-bar over 1 plus q-bar is between 0.85 and 0.9, there exists an arrow pointing north.

3.3  Bond economy with debt elastic interest rate

In the setup of this paper the portfolio cost approach is very similar to a model with a debt elastic interest rate. The latter approach assumes that the consumers in countries $1$ and $2$ face different prices for the bond, and that the spread between the prices is a function of the net foreign asset position. This approach appears among others in Boileau and Normandin (2005), Devereux and Smith (2003), and Schmitt-Grohé and Uribe (2003). The households budget constraint is given by

$$P_{i}\left( s^{t}\right) c_{i}\left( s^{t}\right) \leq\bar{P}_{i}\left( s^{t}\right) w_{i}\left( s^{t}\right) l_{i}\left( s^{t}\right) +\bar {P}_{i}\left( s^{t}\right) \Pi_{i}\left( s^{t}\right) +B_{i}\left( s^{t-1}\right) -Q_{i}\left( s^{t}\right) B_{i}\left( s^{t}\right) ,$$

where $Q_{i}\left( s^{t}\right)$ is the price of the bond in country $i$. Similar to Devereux and Smith (2003), the interest rate differential is of the form

$$R_{1}\left( s^{t}\right) =R_{2}\left( s^{t}\right) \Psi\left( B_{1,t+1}-\bar{B}_{1}\right) ,$$ (14)

where the function $\Psi\left( B_{1,t+1}\right)$ satisfies $\Psi\left( 0\right) =1$ and $\Psi^{\prime}<0$. $\bar{B}_{1}$ is a reference level of debt for country $1$. For simplicity, I assume $\bar{B}_{1}=0$. When country $1$ is a net borrower, it faces an interest rate that is higher than the interest rate in country $2$. When country $1$ is a lender, it receives an interest rate that is lower. In equilibrium, interest rates and bond prices satisfy

$$\frac{1}{R_{1}\left( s^{t}\right) } =Q_{1}\left( s^{t}\right) =\sum_{s^{t+1}\vert s^{t}}\beta\frac{U_{c}\left( c_{1}\left( s^{t+1}\right) ,l_{1}\left( s^{t+1}\right) \right) }{U_{c}\left( c_{1}\left( s^{t}\right) ,l_{1}\left( s^{t}\right) \right) }\pi\left( s^{t+1} \vert s^{t}\right) ,$$

$$\frac{1}{R_{2}\left( s^{t}\right) } =Q_{2}\left( s^{t}\right) =\sum_{s^{t+1}\vert s^{t}}\beta\frac{U_{c}\left( c_{2}\left( s^{t+1}\right) ,l_{2}\left( s^{t+1}\right) \right) }{U_{c}\left( c_{2}\left( s^{t}\right) ,l_{2}\left( s^{t}\right) \right) }\frac{q\left( s^{t}\right) }{q\left( s^{t+1}\right) }\pi\left( s^{t+1}\vert s^{t}\right) .$$

Furthermore, equation (14) implies

$$\frac{R_{1}\left( s^{t}\right) }{R_{2}\left( s^{t}\right) }=\frac {\sum_{s^{t+1}\vert s^{t}}\beta\frac{U_{c}\left( c_{2}\left( s^{t+1}\right) ,l_{2}\left( s^{t+1}\right) \right) }{U_{c}\left( c_{2}\left( s^{t}\right) ,l_{2}\left( s^{t}\right) \right) }\frac{q\left( s^{t}\right) }{q\left( s^{t+1}\right) }\pi\left( s^{t+1}\vert s^{t}\right) }{\sum_{s^{t+1}\vert s^{t}}\beta\frac{U_{c}\left( c_{1}\left( s^{t+1}\right) ,l_{1}\left( s^{t+1}\right) \right) }{U_{c}\left( c_{1}\left( s^{t}\right) ,l_{1}\left( s^{t}\right) \right) }\pi\left( s^{t+1} \vert s^{t}\right) }=\Psi\left( B_{1,t+1}-\bar{B}_{1}\right) .$$ (15)

As $c_{i}$, $l_{i}$ and $q$ can be expressed as functions of $\bar{q}$ only, the dynamics of the economy are fully described by (15) and the condition that the excess demand for good $2$ has to be zero, i.e., $z_{2}\left( \bar{q},B_{1}\left( s^{t-1}\right) -Q_{1}\left( s^{t}\right) B_{1}\left( s^{t}\right) ,B_{2}\left( s^{t-1}\right) -Q_{2}\left( s^{t}\right) B_{2}\left( s^{t}\right) \right) =0$ .

In a steady state equation (15) implies $B_{1}=B_{2} =0$ given the assumption $\Psi\left( 0\right) =1$. Hence, in the model with a debt elastic interest rate all the steady states of the financial autarchy model are preserved.

As in the model with convex portfolio costs the stability of a steady state can be linked to the slope of the excess demand function.

Theorem 2   Assume that the interest rate differential between the two countries is debt elastic as described above. If the slope of the excess demand function is negative in a steady state, then this steady state is a saddle point. If the slope of the excess demand function is positive in a steady state, then this steady state is unstable if $\Psi^{\prime}\left( 0\right)$ is sufficiently large, i.e., $0>\Psi^{\prime}\left( 0\right) >\Delta_{D}$. Otherwise this steady state is a saddle point.

Appendix $B.3$ provides an exact definition of $\Delta_{D}$. Similar to the model with convex portfolio costs, the condition $\Psi^{\prime}\left( 0\right) >\Delta_{D}$ implies that the interest rate does not react too strongly to changes in the bond holdings. Hence, to the extent that the model with a debt elastic interest rate is supposed to behave close to the original model, any steady state with an upward-sloping excess demand function is unstable.¹²

3.4  Bond economy with endogenous discounting

In this section agents' discount factors are assumed to be endogenous as in Mendoza (1991), Corsetti, Dedola and Leduc (2005), and Schmitt-Grohé and Uribe (2003).¹³ This concept of preferences with intertemporal dependences was introduced by Uzawa (1968) and it has been extended and clarified by Epstein (1983, 1987). Uzawa-Epstein preferences fall into the broader class of recursive preferences. The subjective discount factor is assumed to be a decreasing function of the period utility level, i.e., agents become more impatient as current utility rises. For most of the analysis, I assume that agents do not internalize the effect that their current consumption and labor choices have on their discount factor. As the model is solved by a backward shooting algorithm, this assumption simplifies the analysis drastically because it reduces the number of state variables from three to just one.

The problem of the representative household is given by

$$\max_{_{\substack{c_{i}\left( s^{t}\right) ,l_{i}\left( s^{t}\right) , \\ c_{i1}\left( s^{t}\right) ,c_{i2}\left( s^{t}\right) , \\ B_{i}\left( s^{t}\right) }}}\sum_{t=0}^{\infty}\sum_{s^{t}}\theta_{i}\left( s^{t}\right) U\left( c_{i}\left( s^{t}\right) ,l_{i}\left( s^{t}\right) \right) \pi\left( s^{t}\right)$$

$$s.t.$$

$$\theta_{i}\left( s^{t+1}\right) =\beta_{i}\left[ U\left( c_{i}\left( s^{t}\right) ,l_{i}\left( s^{t}\right) \right) \right] \theta_{i}\left( s^{t}\right)$$

$$P_{i}\left( s^{t}\right) c_{i}\left( s^{t}\right) \leq\bar{P}_{i}\left( s^{t}\right) w_{i}\left( s^{t}\right) l_{i}\left( s^{t}\right) +\bar {P}_{i}\left( s^{t}\right) \Pi_{i}\left( s^{t}\right) +B_{i}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{i}\left( s^{t}\right) .$$

The equilibrium dynamics are fully summarized by $z_{2}\left( \bar{q}\left( s^{t}\right) ,B_{1}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{1}\left( s^{t}\right) \right) =0$ with $z_{2}$ given by equation (11), where

$$Q\left( s^{t}\right) =\sum_{s^{t+1}\vert s^{t}}\beta_{1}\left[ U\left( c_{1}\left( s^{t}\right) ,l_{1}\left( s^{t}\right) \right) \right] \frac{U_{c,1}\left( c_{1}\left( s^{t+1}\right) ,l_{1}\left( s^{t+1} \right) \right) }{U_{c,1}\left( c_{1}\left( s^{t}\right) ,l_{1}\left( s^{t}\right) \right) }\pi\left( s^{t+1}\vert s^{t}\right)$$

and the risk sharing condition

$$\sum_{s^{t+1}\vert s^{t}}\left[ \beta_{1}\left[ U\left( s^{t}\right) \right] \frac{U_{c}\left( c_{1}\left( s^{t+1}\right) ,l_{1}\left( s^{t+1}\right) \right) }{U_{c}\left( c_{1}\left( s^{t}\right) ,l_{1}\left( s^{t}\right) \right) }-\beta_{2}\left[ U\left( s^{t}\right) \right] \frac{U_{c}\left( c_{2}\left( s^{t+1}\right) ,l_{2}\left( s^{t+1}\right) \right) } {U_{c}\left( c_{2}\left( s^{t}\right) ,l_{2}\left( s^{t}\right) \right) }\frac{q\left( s^{t}\right) }{q\left( s^{t+1}\right) }\right] \pi\left( s^{t+1}\vert s^{t}\right) =0,$$ (16)

where $c_{i}=c_{i}\left( \bar{q}\right)$ and $l_{i}=l_{i}\left( \bar {q}\right)$ as shown earlier. This system of difference equations has to satisfy the appropriate initial and transversality conditions.

Equation (16) implies that in a steady state the discount factors are equalized across countries

<