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Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 867, August 2006, Revised: June 2009 --- Screen Reader Version*

Closing Large 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publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.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 with incomplete financial markets. I show that in a model with possibly multiple steady states, these methods can substantially impact the dynamic properties of the model. Convex portfolio costs, debt elastic interest rates, or an overlapping generations model allow for multiple steady state if the underlying static model has multiple locally isolated equilibria. Steady states for which the excess demand function is upward sloping are unstable. If the model is closed using Uzawa-type preferences (endogenous discount factor), the steady state is always unique and stable.

Keywords: Stationarity, incomplete markets, open economy

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 international bond holdings. In a stochastic environment the linearized model generates non-stationary variables as international bond holdings 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 for some of these 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 country produces one good. The 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 four approaches to obtain stationarity in the linearized economy: an endogenous discount factor, a debt elastic interest rate premium, a convex portfolio costs, and an overlapping generations structure as in Weil (1989).³

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 bond holdings. 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 international bond holdings is a steady state. All four approaches analyzed here, resolve this indeterminacy by construction. For example, with convex portfolio costs, agents face a non-zero costs for bond holdings that are different from a reference level for international bonds that is specified by the researcher. In the overlapping generations model, newly born agents do not own financial assets. In the endogenous discount factor framework, the steady state allocations imply that discount factors are equalized across countries thereby implicitly pinning down the international distribution of bond holdings. Stationarity of international bond holdings follows swiftly from here. In all four cases, agents finance additional consumption out of a positive net foreign asset position and the economy moves towards its steady state.

My main results concern the properties of these stationarity inducing devices if there are multiple price vectors that induce an equilibrium in the underlying static model of the model economy. In a two country model where foreign and domestic goods are imperfect substitutes there are multiple (locally isolated) price vectors that imply market clearing if the price elasticity of substitution between the foreign and domestic good is sufficiently low.⁴

If stationarity is induced by convex portfolio costs, there is a unique stable steady state only if the underlying static model has a unique equilibrium. If the static model has $n$ price equilibria, the model with convex portfolio costs has $n$ steady states. Steady states for which the excess demand for the foreign good is decreasing in its relative price are stable, but steady states for which the excess demand function is increasing in its relative price are unstable. Similar results obtain for the cases of the debt-elastic interest rate and the overlapping generations framework.

Following Uzawa (1968), when 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 Rather than pinning down the level of bond holdings like the other three methods, the endogenous discount factor methodology pins down uniquely the relative price of goods. Thus, the steady state is always a unique and stable irrespective of the number of equilibria in the underlying static setup.

In a two country model, stationarity inducing devices perform the role for which they have been introduced into the analysis of international macroeconomics: they remove the indeterminacy of the international bond distribution and turn the dynamics of the international bond stationary. All four devices in this paper induce similar stable dynamics around the steady state if the underlying static model has a unique equilibrium. However, if the underlying static model allows for multiple equilibria, substantial differences arise across devices. The model with endogenous discounting removes any equilibrium multiplicity and the model is stable around its now unique steady state. The remaining three approaches leave the equilibrium multiplicity intact and some steady states turn out to be unstable. These findings should not be interpreted as an endorsement of one method over the other, but rather as a warning that the choice of the stationarity inducing device is not innocuous.⁶

The remainder of the paper is organized as follows. Section 2 presents the static model that underlies the analysis. In section 3, the static model is extended to incorporate dynamics and I analyze the characteristics of the steady states and local dynamics under the different stationarity inducing approaches. Section 4 summarizes the results and provides intuition. Section 5 offers concluding remarks. A detailed Appendix is provided.

2  Model With Multiple Equilibria

Each country produces one good that can be traded internationally without frictions. The two goods are assumed to be imperfect substitutes in the household's utility function. Labor, which is supplied endogenously, is the sole factor of production and the population size is normalized to one. The model is static.

Households maximize utility subject to the budget constraint

$$\max_{\substack{c_{i1},c_{i2} \\ c_{i},l_{i}}}U\left( c_{i},l_{i}\right)$$ (1)

$$s.t.$$

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

where $c_{i}$ is given by a linear-homogeneous aggregator $H_{i}\left( c_{i1},c_{i2}\right)$.⁷ $H_{i}$ is assumed to satisfy

$$H_{ij}=\frac{\partial H_{i}}{\partial c_{ij}}>0$$ , $$H_{iii}=\frac {\partial^{2}H_{i}}{\partial c_{ii}^{2}}<0$$ , $$H_{iji}=\frac{\partial ^{2}H_{i}}{\partial c_{ij}\partial c_{ii}}>0,$$

and the Inada conditions

$$\lim_{c_{i1\rightarrow0}}H_{i1}\left( c_{i1},c_{i2}\right) =\lim _{c_{i2\rightarrow0}}H_{i2}\left( c_{i1},c_{i2}\right) =\infty,$$

$$\lim_{c_{i1\rightarrow\infty}}H_{i1}\left( c_{i1},c_{i2}\right) =\lim_{c_{i2\rightarrow\infty}}H_{i2}\left( c_{i1},c_{i2}\right) =0.$$

The strictly concave period utility function $U\left( c,l\right)$ satisfies

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

Utility functions that are commonly used in macroeconomics are in line with these assumptions. $c_{i}$ 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 exogenous lump sum transfer to agents in country $i$ with $dW_{1}+dW_{2}=0$.

The optimal choices for $c_{i1}$ and $c_{i2}$ can be found from the following optimization program:

$$\max_{c_{i1},c_{i2}}H_{i}\left( c_{i1},c_{i2}\right)$$ (3)

$$s.t.$$

$$\bar{P}_{1}c_{i1}+\bar{P}_{2}c_{i2}=\bar{P}_{i}w_{i}l_{i}+dW_{i}.$$ (4)

Linear homogeneity of $H_{i}$ implies that the first order conditions can be written as

$$H_{i1}\left( \frac{c_{i1}}{c_{i2}},1\right) =\lambda_{i}\bar{P} _{1},$$ (5)

$$H_{i2}\left( \frac{c_{i1}}{c_{i2}},1\right) =\lambda_{i}\bar{P} _{2}.$$ (6)

Given the properties of $H_{i1}$ and $H_{i2}$, this can be summarized as

$$\frac{c_{i1}\left( s^{t}\right) }{c_{i2}\left( s^{t}\right) }=\tilde {H}_{i}\left( \frac{1}{\bar{q}}\right) ,$$ (7)

where $\bar{q}$ is the relative price of good $2$ to good $1$, $\frac{\bar {P}_{2}}{\bar{P}_{1}}$. The aggregator $H_{i}$ is said to allow for home bias in goods if $\tilde{H}_{1}\left( \frac{1}{\bar{q}}\right) >\tilde{H} _{2}\left( \frac{1}{\bar{q}}\right)$ for all $\bar{q}$.⁸ Let $P_{i}$ denote the price of the final consumption basket and satisfies

$$\Phi_{1}\left( \bar{q}\right) \equiv\frac{\bar{P}_{1}}{P_{1}}=\frac {\bar{q}H_{1}\left( \tilde{H}_{1}\left( \frac{1}{\bar{q}}\right) ,1\right) }{\bar{q}\left[ \tilde{H}_{1}\left( \frac{1}{\bar{q}}\right) +\bar {q}\right] } \Phi_{1}^{\prime}\left( \bar{q}\right) =\frac{\partial\Phi_{1}\left( \bar{q}\right) }{\partial\bar{q}}<0$$ (8)

$$\Phi_{2}\left( \bar{q}\right) \equiv\frac{\bar{P}_{2}}{P_{2}}=\frac {\bar{q}H_{2}\left( \tilde{H}_{2}\left( \frac{1}{\bar{q}}\right) ,1\right) }{\left[ \tilde{H}_{2}\left( \frac{1}{\bar{q}}\right) +\bar{q}\right] } \Phi_{2}^{\prime}\left( \bar{q}\right) =\frac{\partial\Phi _{2}\left( \bar{q}\right) }{\partial\bar{q}}>0$$ (9)

I normalize the price of the consumption basket in country 1 to unity, $P_{1}=1$, and denote $P_{2}$ by $q$, the real exchange rate. $q$ and $\bar{q}$ are related as follows $q=\bar{q}\frac{\Phi_{1}\left( \bar{q}\right) }{\Phi_{2}\left( \bar{q}\right) }$ .

Using the budget constraint and $\frac{c_{i1}}{c_{i2}}=\tilde {H}_{i}\left( \frac{1}{\bar{q}}\right)$ , the demand functions for good 2 are

$$c_{12} =\frac{1}{\left( \tilde{H}_{1}\left( \frac{1}{\bar{q}}\right) +\bar{q}\right) }\left[ w_{1}l_{1}+\frac{1}{\Phi_{1}\left( \bar{q}\right) }dW_{1}\right] ,$$ (10)

$$c_{22} =\frac{1}{\left( \tilde{H}_{2}\left( \frac{1}{\bar{q}}\right) \frac{1}{\bar{q}}+1\right) }\left[ w_{2}l_{2}+\frac{1}{\bar{q}\Phi _{1}\left( \bar{q}\right) }dW_{2}\right] ,$$ (11)

Similar expressions can be derived for the demand of good 1 and an expression for $c_{i}=H_{i}\left( c_{i1},c_{i2}\right)$ can be provided:

$$c_{1} =\Phi_{1}\left( \bar{q}\right) w_{1}l_{1}+dW_{1} ,$$ (12)

$$c_{2} =\Phi_{2}\left( \bar{q}\right) w_{2}l_{2}+\frac{\Phi_{2}\left( \bar{q}\right) }{\Phi_{1}\left( \bar{q}\right) \bar{q}}dW_{2} .$$ (13)

Firms in country $i$ produce the traded good $i$ using a linear production technology,

$$y_{i}=A_{i}l_{i}.$$ (14)

Perfect competition and the linear technology imply that the equilibrium wage equals the productivity parameter, i.e., $w_{i}=A_{i}$.

Combining equations (12) to (14) with the intratemporal Euler equation for the consumption-leisure choice,

$$\frac{U_{l}\left( c_{i},l_{i}\right) }{U_{c}\left( c_{i},l_{i}\right) }=-\Phi_{i}\left( \bar{q}\right) w_{i},$$ (15)

allows to express $c_{i}$, $l_{i}$, and $q$ as functions of $\bar{q}$ (and $dW_{i}$).

Definition 1 (Competitive Equilibrium)   A competitive equilibrium is a collection of allocations $c_{i1}$, $c_{i2}$, $c_{i}$, $l_{i}$, $y_{i}$ and prices $\bar{q}$, $w_{i}$, $i=1,2$, 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.

As shown in Appendix A.1 the equilibrium conditions for the static model can be fully summarized by the excess demand function for good 2:

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

$$=\frac{A_{1}l_{1}\left( \bar{q},dW_{1}\right) +\frac{1}{\Phi_{1}\left( \bar{q}\right) }dW_{1}}{\tilde{H}_{1}\left( \frac{1}{\bar{q}}\right) +\bar{q}} +\frac{A_{2}l_{2}\left( \bar{q},dW_{1}\right) -\frac{1}{\bar {q}\Phi_{1}\left( \bar{q}\right) }dW_{1}}{\tilde{H}_{2}\left( \frac{1} {\bar{q}}\right) \frac{1}{\bar{q}}+1}$$

$$-A_{2}l_{2}\left( \bar{q},dW_{1}\right) .$$ (16)

An equilibrium is a relative price $\bar{q}^{\ast}$, s.t. $z_{2}\left( \bar{q}^{\ast},dW_{1}\right) =0$. Appendix A.1 proves the existence of the competitive equilibrium. Appendix A.2 shows that the sign of the slope of the excess demand function in equilibrium, $\frac{\partial z_{2}\left( \bar{q}^{\ast},dW_{1}\right) }{\partial\bar{q}}$ , gives information about the number of equilibria. If there is an equilibrium price vector $\bar{q}^{\ast}$such that $sign\left( \frac{\partial z_{2}\left( \bar{q}^{\ast} ,dW_{1}\right) }{\partial\bar{q}}\right) >0$ , i.e., the excess demand function is upward-sloping, then there must be at least two more equilibria for which the excess demand function is downward-sloping. If $sign\left( \frac{\partial z_{2}\left( \bar{q}^{\ast},dW_{1}\right) }{\partial\bar{q} }\right) <0$ for all equilibrium price vectors then the equilibrium must be unique.

As explained in more detail in Appendix A.2 this framework allows for multiple locally isolated equilibria provided that the elasticity of substitution between the home and foreign good is sufficiently low. To build intuition, consider an endowment economy with two countries and two traded goods that are imperfect substitutes. The countries are mirroring each other with respect to preferences and endowments.⁹ There is always one equilibrium with the relative price of the traded goods equal to unity. However, there can be two more equilibria. Let the price of the domestic good be high relative to the price of the foreign good, so that domestic agents have high purchasing power relative 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 two 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.¹⁰

3  Bond Economies

When extending the static model to a dynamic model with incomplete markets, the well-known stationarity problem of international bond holdings (or the net foreign asset position) emerges. If--as customary in the international business cycle literature--the model is approximated around its deterministic steady state, international bond holdings follow a unit root process.¹¹

Several approaches have been put forward to induce stationarity in the approximate models, among others are:

• convex portfolio costs as in Heathcote and Perri (2002), and Schmitt-Grohé and Uribe (2003),
• debt elastic interest rates as in Boileau and Normandin (2008), Devereux and Smith (2007), and Schmitt-Grohé and Uribe (2003),
• endogenous discounting as in Mendoza (1991), Corsetti, Dedola and Leduc (2005), and Schmitt-Grohé and Uribe (2003),
• overlapping generations as in Ghironi (2006).

In the dynamic extension of the model, 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. Intertemporal financial markets are exogenously incomplete in the sense that the only asset that is traded internationally is one non-state-contingent bond. The bond is in zero net-supply.

3.1  Bond Economy With Convex Portfolio Costs

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

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

$$s.t.$$

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

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

$P_{i}\left( s^{t}\right) c_{i}\left( s^{t}\right)$ are the household's total consumption expenditures which are equal to $\bar{P}_{1}\left( s^{t}\right) c_{i1}\left( s^{t}\right) +\bar{P}_{2}\left( s^{t}\right) c_{i2}\left( s^{t}\right)$ . $B_{i}\left( s^{t-1}\right)$ denotes the (nominal) bond holdings that agent $i$ has inherited from period $t-1$. $Q\left( s^{t}\right)$ is the price of the bond and $T_{i}\left( s^{t}\right)$ is the lump-sum reimbursement of the portfolio costs.

Substituting $B_{i}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{i}\left( s^{t}\right)$ for $dW_{i}\left( s^{t}\right)$ into $\left( \ref{excess demand}\right)$ , the equilibrium dynamics are fully summarized by the zero excess demand condition for good 2, i.e.,

$$z_{2}\left( \bar{q}\left( s^{t}\right) ,B_{1}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{1}\left( s^{t}\right) \right) =0,$$ (17)

where

$$Q\left( s^{t}\right) =\sum_{s^{t+1}\vert s^{t}}\beta\frac{U_{c}\left( c_{1}\left( s^{t+1}\right) ,l_{1}\left( s^{t+1}\right) \right) } {U_{c}\left( c_{1}\left( s^{t}\right) ,l_{1}\left( s^{t}\right) \right) }\pi\left( s^{t+1}\vert s^{t}\right) -\Gamma^{\prime}\left( \frac{B_{1}\left( s^{t}\right) }{\bar{P}_{1}\left( s^{t}\right) }\right) ,$$ (18)

and the risk sharing condition

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

$$=\Gamma^{\prime}\left( \frac{B_{1}\left( s^{t}\right) }{\bar{P} _{1}\left( s^{t}\right) }\right) -\Gamma^{\prime}\left( \frac{B_{2}\left( s^{t}\right) }{\bar{P}_{2}\left( s^{t}\right) }\right) .$$ (19)

Using equations (12) to (15) $c_{i}$, $l_{i}$ and $q$ can be expressed as functions of $\bar{q}$ and $B_{1}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{1}\left( s^{t}\right)$ .

As $\Gamma^{\prime}=0$ for $B_{i}=0$ and larger than zero otherwise, equation (19) implies that in any deterministic steady state $B_{1}=B_{2}=0$. Consequently, if the excess demand function for good 2 has $n$ zeros in $\bar{q}$, then the model with incomplete asset markets and convex portfolio costs has $n$ deterministic steady states.

Turning to the local dynamic properties around a specific steady state, note that the equilibrium dynamics for the relative price $\bar{q}$ and bond holdings $B_{1}$ can be approximated by the system

$$\hat{q}_{t+1} =\hat{q}_{t}-\frac{d_{p}}{d_{q}}\left[ b_{t}-\beta b_{t+1}\right] +\frac{d_{p}}{d_{q}}\left[ b_{t-1}-\beta b_{t}\right] +\check{\Gamma}\frac{\beta}{d_{q}}b_{t},$$ (20)

$$b_{t} =\frac{1}{\beta}\frac{\frac{\partial z_{2}}{\partial\bar{q}}\bar{q} }{\frac{\partial z_{2}}{\partial dW_{1}}}\bar{q}_{t}+\frac{1}{\beta} b_{t-1},$$ (21)

where $\bar{q}_{t}$ denotes the log-deviation of the relative price from its steady state value, and $b_{1,t}$ is the linear deviation of bond holdings from 0. Appendix B provides details on the derivations. In a nutshell, equation (20) is the log-linear approximation to equation (19), which is the risk-sharing condition under incomplete markets. Equation (21) is the log-linear approximation to the zero excess demand condition for good 2. Written more compactly, let $x_{t}^{\prime}=\left( \bar{q}_{t},b_{t-1}\right)$ and $x_{t+1}=M_{P}x_{t}$, where

$$M_{P}=\left( \begin{array}[c]{cc} 1+\frac{\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}}{\frac{\partial z_{2} }{\partial dW_{1}}d_{q}-\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}d_{b} }\check{\Gamma} & \frac{\frac{\partial z_{2}}{\partial dW_{1}}}{\frac{\partial z_{2}}{\partial dW_{1}}d_{q}-\frac{\partial z_{2}}{\partial\bar{q}}\bar {q}d_{b}}\check{\Gamma}\ \frac{1}{\beta}\frac{\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}} {\frac{\partial z_{2}}{\partial dW_{1}}} & \frac{1}{\beta} \end{array}\right) .$$

$\frac{\partial z_{2}}{\partial\bar{q}}$ denotes the slope of the excess demand function with respect to the relative price in the steady state under consideration, which can be positive or negative depending on the steady state around which the model is approximated. As shown in the Appendix, $\frac{\partial z_{2}}{\partial dW_{1}}d_{q}-\frac{\partial z_{2}} {\partial\bar{q}}\bar{q}d_{b}$ negative for all parameter values and independent of the sign of $\frac{\partial z_{2}}{\partial\bar{q}}$. $\check{\Gamma}=\frac{\Gamma^{\prime\prime}\left( 0\right) }{\beta^{2} \Phi_{1}\left( \bar{q}\right) }\left[ 1+\frac{1}{\bar{q}}\right]$ measures the importance of the convex portfolio cost. For $\check{\Gamma}=0$ one obtains the standard linearized international business cycle model with incomplete markets and non-stationary bond holdings. The following theorem summarizes the dynamic properties of the model.

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.

Proof 1 The determinacy and the trace associated with $M_{P}$ are $\det\left( M_{P}\right) =\frac{1}{\beta}$ and $tr \left( M_{P}\right) =\frac{\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}}{\frac{\partial z_{2} }{\partial dW_{1}}d_{q}-\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}d_{b} }\check{\Gamma}+\left( 1+\frac{1}{\beta}\right)$ . Since $\beta<1$, $\vert\det\left( M_{P}\right) \vert>1$. Remember that $\frac{\partial z_{2} }{\partial dW_{1}}d_{q}-\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}d_{b} <0$ .

If $\frac{\partial z_{2}}{\partial\bar{q}}<0$, tr $\left( M_{P}\right) >1+\frac{1}{\beta}>0$ and the modulus of one eigenvalue is larger than 1, while the other one is smaller than 1. Given that bond holdings are the only state variable, the system is saddle-path stable.

If $\frac{\partial z_{2}}{\partial\bar{q}}>0$, the modulus of each eigenvalue is larger than 1 for $\vert tr\left( M_{P}\right) \vert < 1+\det\left( M_{P}\right)$ , requiring that

$$\Delta_{P} \equiv-2\beta\frac{\left( 1+\beta\right) \Phi_{1}\left( \bar{q}\right) }{\bar{q}\left[ 1+\frac{1}{\bar{q}}\right] }\frac {\frac{\partial z_{2}}{\partial dW_{1}}d_{q}-\frac{\partial z_{2}} {\partial\bar{q}}\bar{q}d_{b}}{\frac{\partial z_{2}}{\partial\bar{q}}} >\Gamma^{\prime\prime}\left( 0\right) >0.$$

Otherwise, the modulus of exactly one of the eigenvalues is larger than 1, while the other one is smaller than 1. Hence for $\Gamma^{\prime\prime }\left( 0\right) <\Delta_{P}$ the system is unstable whenever $\frac {\partial z_{2}}{\partial\bar{q}}>0$.¹² $\qedsymbol$

$\Gamma^{\prime\prime}\left( 0\right)$ measures the sensitivity of the portfolio costs in the neighborhood of the steady state. In most applications, this sensitivity is low. If $\Gamma^{\prime\prime}\left( 0\right)$ is assumed to be very large, the economy behaves similarly to an economy without international financial markets. In the latter, any steady state is saddle-path stable. Hence, any steady state can be turned into a saddle point in the model with portfolio costs if the marginal costs of portfolio holdings increase strongly enough as the economy deviates from the steady state. However, given that the model with convex portfolio costs is supposed to behave closely to the original (non-stationary) model, it is common practice to specify portfolio costs that are small and that do not change dramatically in the neighborhood of the steady state. Such specifications are also in line with actual portfolio costs.

3.2  Bond Economy With Debt Elastic Interest Rate

Consumers in the two countries face different prices for the bond, and the spread between the prices is a function of international bond holdings. The households budget constraint is given by

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

Following Devereux and Smith (2007), the interest rate differential is of the form

$$R_{1}\left( s^{t}\right) =R_{2}\left( s^{t}\right) \Psi\left( B_{1}\left( s^{t+1}\right) -\bar{B}_{1}\right) ,$$ (22)

where the function $\Psi\left( B_{1}\left( s^{t+1}\right) \right)$ satisfies $\Psi\left( 0\right) =1$ and $\Psi^{\prime}<0$. $\bar{B}_{1}$ is a reference level of debt for country 1, which is set to zero. 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

$$\frac{1}{R_{1}\left( s^{t}\right) } =Q_{1}\left( s^{t}\right) =\sum_{s^{t+1}\vert s^{t}}\beta\frac{U_{c}\left( c_{1}\left( s^{t+1}\right) ,l_{1}\left( s^{t+1}\right) \right) }{U_{c}\left( c_{1}\left( s^{t}\right) ,l_{1}\left( s^{t}\right) \right) }\pi\left( s^{t+1} \vert s^{t}\right) ,$$ (23)

$$\frac{1}{R_{2}\left( s^{t}\right) } =Q_{2}\left( s^{t}\right) =\sum_{s^{t+1}\vert s^{t}}\beta\frac{U_{c}\left( c_{2}\left( s^{t+1}\right) ,l_{2}\left( s^{t+1}\right) \right) }{U_{c}\left( c_{2}\left( s^{t}\right) ,l_{2}\left( s^{t}\right) \right) }\frac{q\left( s^{t}\right) }{q\left( s^{t+1}\right) }\pi\left( s^{t+1}\vert s^{t}\right) .$$ (24)

Combined with equation (22) this implies

$$\frac{R_{1}\left( s^{t}\right) }{R_{2}\left( s^{t}\right) }=\frac {\sum_{s^{t+1}\vert s^{t}}\beta\frac{U_{c}\left( c_{2}\left( s^{t+1}\right) ,l_{2}\left( s^{t+1}\right) \right) }{U_{c}\left( c_{2}\left( s^{t}\right) ,l_{2}\left( s^{t}\right) \right) }\frac{q\left( s^{t}\right) }{q\left( s^{t+1}\right) }\pi\left( s^{t+1}\vert s^{t}\right) }{\sum_{s^{t+1}\vert s^{t}}\beta\frac{U_{c}\left( c_{1}\left( s^{t+1}\right) ,l_{1}\left( s^{t+1}\right) \right) }{U_{c}\left( c_{1}\left( s^{t}\right) ,l_{1}\left( s^{t}\right) \right) }\pi\left( s^{t+1} \vert s^{t}\right) }=\Psi\left( B_{1}\left( s^{t+1}\right) -\bar{B}_{1}\right) .$$ (25)

The dynamics of the economy are described by (25) and the condition that the excess demand for good 2 is zero, i.e., $z_{2}\left( \bar{q},B_{1}\left( s^{t-1}\right) -Q_{1}\left( s^{t}\right) B_{1}\left( s^{t}\right) \right) =0$ .

In a steady state equation $\left( \ref{DE risk sharing equations}\right)$ implies $B_{1}=B_{2}=0$ given the assumption $\Psi\left( 0\right) =1$. Hence, values of $\bar{q}$ that are solutions to $z_{2}\left( \bar{q},0\right) =0$ in the static model, are steady states in the model with a debt elastic interest rate just like in the model with convex portfolio costs.

The linear dynamics of the model with a debt elastic interest rate are given by $x_{t+1}=M_{B}x_{t}$, where

$$M_{B}=\left( \begin{array}[c]{cc} 1-\frac{\Psi^{\prime}\left( 0\right) }{\beta}\frac{\frac{\partial z_{2} }{\partial\bar{q}}\bar{q}}{\frac{\partial z_{2}}{\partial dW_{1}}d_{q} -\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}d_{b}} & -\frac{\Psi^{\prime }\left( 0\right) }{\beta}\frac{\frac{\partial z_{2}}{\partial dW_{1}}} {\frac{\partial z_{2}}{\partial dW_{1}}d_{q}-\frac{\partial z_{2}} {\partial\bar{q}}\bar{q}d_{b}}\ \frac{1}{\beta}\frac{\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}} {\frac{\partial z_{2}}{\partial dW_{1}}} & \frac{1}{\beta} \end{array}\right) .$$

As in the model with convex portfolio costs the stability of a steady state is 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.

Proof 2. The proof follows the same steps as for Theorem 1 with the difference that $\Delta_{P}$ is replaced by $\Delta_{D}\,$ where

$$\Delta_{D}\equiv2\left( 1+\beta\right) \frac{\frac{\partial z_{2}}{\partial dW_{1}}d_{q}-\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}d_{b}} {\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}}<\Psi^{\prime}\left( 0\right) <0.$$

$\qedsymbol$

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 for which the excess demand function is upward sloping is unstable.

3.3  Bond Economy With Endogenous Discounting

This concept of preferences with intertemporal dependencies 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. I consider two specifications. In the first case agents do not take into account the effects of their choices on the discount factor, in the second case they do.

No internalization  The problem of the representative household is given by

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

$$s.t.$$

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

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

with $\beta_{i}^{\prime}\left( U_{i}\right) <0$.

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

$$Q\left( s^{t}\right) =\sum_{s^{t+1}\vert s^{t}}\beta_{1}\left[ U\left( c_{1}\left( s^{t}\right) ,l_{1}\left( s^{t}\right) \right) \right] \frac{U_{c,1}\left( c_{1}\left( s^{t+1}\right) ,l_{1}\left( s^{t+1} \right) \right) }{U_{c,1}\left( c_{1}\left( s^{t}\right) ,l_{1}\left( s^{t}\right) \right) }\pi\left( s^{t+1}\vert s^{t}\right)$$ (26)

and the risk sharing condition

$$\sum_{s^{t+1}\vert s^{t}}\left[ \beta_{1}\left[ U\left( s^{t}\right) \right] \frac{U_{c}\left( c_{1}\left( s^{t+1}\right) ,l_{1}\left( s^{t+1}\right) \right) }{U_{c}\left( c_{1}\left( s^{t}\right) ,l_{1}\left( s^{t}\right) \right) }-\beta_{2}\left[ U\left( s^{t}\right) \right] \frac{U_{c}\left( c_{2}\left( s^{t+1}\right) ,l_{2}\left( s^{t+1}\right) \right) } {U_{c}\left( c_{2}\left( s^{t}\right) ,l_{2}\left( s^{t}\right) \right) }\frac{q\left( s^{t}\right) }{q\left( s^{t+1}\right) }\right] \pi\left( s^{t+1}\vert s^{t}\right) =0.$$ (27)

Equation (27) implies that in a steady state the discount factors are equalized across countries

$$\beta_{1}\left[ U\left( c_{1}\left( s^{t}\right) ,l_{1}\left( s^{t}\right) \right) \right] =\beta_{2}\left[ U\left( c_{2}\left( s^{t}\right) ,l_{2}\left( s^{t}\right) \right) \right] .$$ (28)

As $\beta_{i}$ is strictly decreasing in $U_{i}$, the utility function is strictly concave, and the technology is concave, there is a unique allocation and a unique price $\bar{q}$ that solves (28). The initial allocation of bond holdings is determined from the zero excess demand condition for good 2. In contrast to the two models discussed previously the steady state of the model with endogenous discounting is unique irrespective of the sign $\frac{\partial z_{2}}{\partial\bar{q}}$. Furthermore, this steady state does not necessarily feature zero bond holdings. However, the functional forms of $\beta_{1}$ and $\beta_{2}$ can always be calibrated such that the unique steady state features $B_{1} =B_{2}=0$.

The local dynamics around the unique steady state are approximated by $x_{t+1}=M_{E}x_{t}$ with

$$M_{E}=\left( \begin{array}[c]{cc} 1+\frac{\frac{\partial z_{2}}{\partial dW_{1}}g_{q}-\frac{\partial z_{2} }{\partial\bar{q}}\bar{q}g_{b}}{\frac{\partial z_{2}}{\partial dW_{1}} d_{q}-\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}d_{b}} & 0\ \frac{1}{\beta}\frac{\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}} {\frac{\partial z_{2}}{\partial dW_{1}}} & \frac{1}{\beta} \end{array}\right)$$

where $\frac{\partial z_{2}}{\partial dW_{1}}g_{q}-\frac{\partial z_{2} }{\partial\bar{q}}\bar{q}g_{b}>0$ .

Theorem 3   Assume that the agents' discount factors are endogenous and strictly decreasing in the current utility level. Furthermore, agents do not internalize the effects of their choices on their discount factors. Then the unique steady state is a saddle point irrespective of the sign of the slope of the excess demand function.

Proof 3. The determinacy and the trace are $\det\left( M_{E}\right) =\frac{1}{\beta}\left[ 1+\frac{\frac{\partial z_{2}}{\partial dW_{1}} g_{q}-\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}g_{b}}{\frac{\partial z_{2}}{\partial dW_{1}}d_{q}-\frac{\partial z_{2}}{\partial\bar{q}}\bar {q}d_{b}}\right]$ and tr $\left( M_{E}\right) =-\frac{1-\beta}{\beta} \frac{\frac{\partial z_{2}}{\partial dW_{1}}g_{q}-\frac{\partial z_{2} }{\partial\bar{q}}\bar{q}g_{b}}{\frac{\partial z_{2}}{\partial dW_{1}} d_{q}-\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}d_{b}}+1+\det\left( M_{E}\right)$ , respectively. Since $\frac{\frac{\partial z_{2}}{\partial dW_{1}}g_{q}-\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}g_{b}} {\frac{\partial z_{2}}{\partial dW_{1}}d_{q}-\frac{\partial z_{2}} {\partial\bar{q}}\bar{q}d_{b}}<0$ irrespective of the sign of the slope of the excess demand function, the modulus of exactly one eigenvalue is smaller than 1. With bond holdings being the only state variable, the dynamic system is saddle-path stable. $\qedsymbol$

Notice that it is crucial to assume that the endogenous discount factor is decreasing in the utility level. Otherwise it is $\frac{\partial z_{2}}{\partial dW_{1}}g_{q}-\frac{\partial z_{2}}{\partial\bar{q}}\bar {q}g_{b}<0$ and $\vert$tr $\left( M_{E}\right) \vert<1+\det\left( M_{E}\right)$ irrespective of the slope of the excess demand function. In this case, both eigenvalues would be larger than 1.

With internalization  If agents internalize the effects of their consumption and labor decisions on the discount factor, the risk sharing condition is given by

$$\sum_{s^{t+1}\vert s^{t}}\left[ \beta_{1}\left( s^{t}\right) \frac{U_{c,1}\left( s^{t+1}\right) -\eta _{1}\left( s^{t+1}\right) \beta_{c,1}\left( s^{t+1}\right) } {U_{c,1}\left( s^{t}\right) -\eta_{1}\left( s^{t}\right) \beta _{c,1}\left( s^{t}\right) }\right] \pi\left( s^{t+1}\vert s^{t}\right)$$

$$=\sum_{s^{t+1}\vert s^{t}}\left[ \beta_{2}\left( s^{t}\right) \frac {U_{c,2}\left( s^{t+1}\right) -\eta_{2}\left( s^{t+1}\right) \beta _{c,2}\left( s^{t+1}\right) }{U_{c,2}\left( s^{t}\right) -\eta_{2}\left( s^{t}\right) \beta_{c,2}\left( s^{t}\right) }\frac{\bar{q}\left( s^{t}\right) }{\bar{q}\left( s^{t+1}\right) }\frac{\Phi_{1}\left( \bar {q}\left( s^{t}\right) \right) }{\Phi_{1}\left( \bar{q}\left( s^{t+1}\right) \right) }\frac{\Phi_{2}\left( \bar{q}\left( s^{t+1}\right) \right) }{\Phi_{2}\left( \bar{q}\left( s^{t}\right) \right) }\right] \pi\left( s^{t+1}\vert s^{t}\right) .$$ (29)  

$\eta_{i}$ is the Lagrangian multiplier on the law of motion for the discount factor in country $i$ and it evolves according to

$$\eta_{i}\left( s^{t}\right) =\sum_{s^{t+1}\vert s^{t}}\left[ -U_{i}\left( s^{t+1}\right) +\beta_{i}\left( s^{t+1}\right) \eta\left( s^{t+1}\right) \right] \pi\left( s^{t+1}\vert s^{t}\right) .$$ (30)

Again, a steady state requires that the discount factors are equalized across countries, i.e., $\beta_{1}\left( U\left( s^{t}\right) \right) =\beta _{2}\left( U\left( s^{t}\right) \right)$ . Therefore, the model with internalization always has a unique steady state. A weaker version of Theorem 3 applies if agents internalize the effects of their choices on the discount factor.

Theorem 4   Assume that the agents' discount factors are endogenous and that agents internalize the effects of their choices on their discount factors. Irrespective of the sign of the slope of the excess demand function, any steady state is a saddle point if the discount factor does not react too strongly to changes in bond holdings.

To the extent that the model with endogenous discounting is supposed to be close to the original model, the discount factor should not change excessively as the utility level deviates from its steady state level. Note, that the (in-)stability of the steady state is not at all related to the slope of the excess demand function, but merely to the parameterization of the endogenous discount factor itself. Appendix B provides a proof of Theorem 4.

3.4  Overlapping Generations

Ghironi (2006) develops a model with an overlapping generations structure to overcome the non-stationarity problem. Following his work, I assume that each country is populated by a continuum of infinitely lived households of measure $N_{i}\left( s^{t}\right)$ which grows at the exogenous and constant rate $n$. The key departure from the standard representative agent framework lies in the assumption that newly born households come into being without financial assets.¹³ More specifically, at time $t_{0}$, the representative consumer in country $i$ born in period $v\in\left( -\infty,t_{0}\right)$ maximizes the intertemporal utility function

$$U_{t_{0}}^{v}=\sum_{t=t_{0}}^{\infty}\beta^{t-t_{0}}\left[ \zeta\log c_{i}^{v}\left( s^{t}\right) +(1-\zeta)\log\left( 1-l_{i}^{v}\left( s^{t}\right) \right) \right]$$

subject to the intertemporal budget constraint

$$P_{i}\left( s^{t}\right) c_{i}^{v}\left( s^{t}\right) \leq\bar{P} _{i}\left( s^{t}\right) w_{i}\left( s^{t}\right) l_{i}^{v}\left( s^{t}\right) +\bar{P}_{i}\left( s^{t}\right) \Pi_{i}^{v}\left( s^{t}\right) +B_{i}^{v}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{i}^{v}\left( s^{t}\right) .$$ (31)

The consumption index of a household of generation v is given by $c_{i}^{v}\left( s^{t}\right) =H_{i}\left( c_{i1}^{v}\left( s^{t}\right) ,c_{i2}^{v}\left( s^{t}\right) \right)$ . $c_{ij}^{v}$ $\left( s^{t}\right)$ is the amount of good $j$ that the representative household of country i born in period v consumes in time t. All other variables are defined analogously. In order to be able to aggregate consumption across generations, the period utility function for households is the log of the Cobb-Douglas function.

While details of the analysis are relegated to Appendix C, the equilibrium conditions of the model are given by the market clearing condition for good 2, i.e., $z_{2}\left( \bar{q}\left( s^{t}\right) ,B_{1}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{1}\left( s^{t}\right) \right) =0$ , and a condition that relates cross county consumption dynamics, which is the analogue of the risk sharing condition in the previous models,

$$\left( \frac{c_{1}\left( s^{t+1}\right) }{c_{1}\left( s^{t}\right) }-\frac{\frac{n}{1+n}c_{1}^{t+1}\left( s^{t+1}\right) }{c_{1}\left( s^{t}\right) }\right) ^{-1}=\left( \frac{c_{2}\left( s^{t+1}\right) }{c_{2}\left( s^{t}\right) }-\frac{\frac{n}{1+n}c_{2}^{t+1}\left( s^{t+1}\right) }{c_{2}\left( s^{t}\right) }\right) ^{-1}\frac{q\left( s^{t}\right) }{q\left( s^{t+1}\right) }.$$ (32)

$c_{i}$ denotes average consumption of households in country $i$ and $c_{i}^{t}$ denotes consumption of newly born households. Consumption of newly born households is a (constant) fraction of their human wealth, i.e., $c_{i}^{t}\left( s^{t}\right) =\zeta\left( 1-\beta\right) h_{i}$ $\left( s^{t}\right)$ where

$$h_{i}\left( s^{t}\right) =\Phi_{i}\left( \bar{q}\left( s^{t}\right) \right) A_{i}\left( s^{t}\right) +Q\left( s^{t}\right) \frac{P_{i}\left( s^{t+1}\right) }{P_{i}\left( s^{t}\right) }h_{i}\left( s^{t+1}\right) ,$$ (33)

$$Q\left( s^{t}\right) =\beta\left( \frac{c_{1}\left( s^{t+1}\right) -\frac{n}{1+n}c_{1}^{t+1}\left( s^{t+1}\right) }{\frac{1}{1+n}c_{1}\left( s^{t}\right) }\right) ^{-1}.$$ (34)

Intuitively, as newly born households have no financial assets a deterministic steady state requires that international bond holdings are zero. Hence, $c_{i}\left( s^{t}\right) =c_{i}^{t}$ $\left( s^{t}\right)$, leaving the steady state value of the real exchange rate undetermined. The dynamic properties of the model around a deterministic steady state are summarized in Theorem 5.

Theorem 5   Assume an overlapping generations structure in which newly born agents have no financial assets. 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 provided that agents are sufficiently patient. Otherwise such a steady state is a saddle point.

Thus, the overlapping generations model displays similar properties as the model with convex portfolio costs or a debt elastic interest rate.

4  Discussion and Intuition

Table 1 summarizes the above results:

Table 1:  Summary of Results

Model	convex portfolio cost	debt elastic interest rate	endog. dcf. (no int.)	endog. dcf. (int.)	overlapping generations
#steady states	equal to static model	equal to static model	unique	unique	equal to static model
sign$\frac{\partial z_{2}}{\partial\bar{q}}<0$	saddle stable	saddle stable	saddle stable	saddle stable	saddle stable
sign$\frac{\partial z_{2}}{\partial\bar{q}}>0$	unstable	unstable	saddle stable	saddle stable	unstable

Multiplicity of steady states  In the models with convex portfolio costs, endogenous interest rates, and the overlapping generations structure, the steady state interest rate equals $\frac{1}{\beta}$. Hence, no country has an incentive to borrow or to lend in any steady states. All equilibria of the static model are therefore valid steady states since they are compatible with $B_{1}=0$.

The models of endogenous discounting, however, dictate that for a given functional choice of the discount factor $\beta_{1}=\beta_{2}$ in the steady state. Uniqueness of the equilibrium price vector $\bar{q} _{endog}^{\ast}$ follows promptly: suppose that another price vector $\bar {q}$ that constitutes an equilibrium in the underlying static setup is also a steady state of the model with endogenous discounting. Let $\bar{q}<\bar {q}_{endog}^{\ast}$ implying $U_{1}>U_{2}$. Thus, country 1 agents are willing to borrow resources at an interest rate of $\frac{1}{\beta_{1}}$ while country 2 agents only demand $\frac{1}{\beta_{2}}$. Hence, country 1 finds it optimal to borrow from country 2 violating $B_{1}=0$.¹⁴

Stability of steady states  Theorems 1, 2, and 5 show that under reasonable parameterizations of the convex portfolio cost, the debt elastic interest rate, and the overlapping generations structure the stability of the dynamic system in the neighborhood of a steady state depends on the sign of the slope of the excess demand function in this steady state. Whenever the excess demand function is upward-sloping in a steady state, the steady state is locally unstable.

Under endogenous discounting (Theorems 3 and 4) the stability of the system in the neighborhood of a steady state does not depend on the slope of the excess demand function in the steady state. The stability depends solely on the parameterization of the endogenous discount factor.

The logic behind the stability of the unique steady state in the model with endogenous discounting is closely related to the argument about its uniqueness. Assume that $\bar{q}$ is below its steady state value. This implies that consumption in country 1 (2) is above (below) its steady state value. Suppose, that the relative price is even lower in the next period, suggesting that the economy moves away from the steady state. This implies an increasing (decreasing) consumption profile in country 1 (2). In addition, the discount factor in country 1 (2) falls (rises). Hence, the price of the non-state-contingent bond falls in country 1 but rises in country 2. Obviously, the opposite movement of bond prices is inconsistent with the absence of arbitrage dictated by the risk sharing condition. Hence, if $\bar{q}$ is below its steady state value at time $t$, $\bar{q}$ must rise in $t+1$ and the economy converges to its unique steady state.

Consider the case of a steady state with sign $\left( \frac{\partial z_{2}}{\partial\bar{q}}\right) >0$ in the bond economy with convex portfolio costs. The price of bonds consists of two pieces: the intertemporal marginal rate of substitution and the derivative of the portfolio costs. If $\bar{q}$ is slightly below its steady state value, consumption in country 1 (2) is above (below) its corresponding steady state value. Stability of a certain steady state requires $\bar{q}$ to rise and $c_{1}$ to fall over time. As a result, the intertemporal marginal rate of substitution in country 1 (2) rises (falls), which leads to a divergence of bond prices. However, when $\bar{q}$ rises, bond holdings and, due to the convexity of the portfolio costs, the derivative of the portfolio costs fall. The effect on bond prices is negative in both countries. However, it is stronger in country 2 since portfolio costs are measured in terms of each country's good. This second effect operates towards a rise of the bond price in country 2 relative to country 1. However, the change in bond holdings is small owing to the fact that the excess demand function is fairly flat around this steady state. Hence, bond prices drift apart and a steady state with sign $\left( \frac{\partial z_{2}}{\partial\bar{q}}\right) >0$ is unstable.

5  Conclusions

If the static model that underlies a model of the international macroeconomy admits multiple equilibria the stationarity inducing device that is used to close the model impacts the dynamic properties of the model fundamentally. If the excess demand function in a steady state is upward sloping, the model is stable in the neighborhood of this model if stationarity is induced using Uzawa-type preferences. However, the model dynamics are unstable around such a steady state if the stationarity is induced using convex portfolio cost, a debt elastic interest rate or an overlapping generations structure. For applied general equilibrium modeling the recommendation is therefore to try different ways of closing the model - otherwise there is little guarantee that results are robust across model specifications. While I abstract from endogenous capital accumulation to obtain analytical results, the same results apply in a model with investment. In Bodenstein (2008) I analyze the global dynamics in a model with capital. This analysis reveals that sunspot fluctuations can lead to multiple equilibria even if the steady state is unique under endogenous discounting provided that there are multiple equilibria in the underlying static model.

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A  A Formal Analysis of Existence and Multiplicity

This appendix shows the existence of the equilibrium in the static model of section 2. I also discuss conditions under which multiple equilibria arise.

A.1  Existence of the Equilibrium

The following analysis is based on Kehoe (1980, 1985 and 1991). Let $L_{i}$ be the time endowment of agents in country $i$. Kehoe defines the excess demand for a good as the difference between the demand for a specific good and the aggregate endowment with this good. The economy's endowment with goods 1 and 2 is zero, while the leisure endowments are $L_{1}$ and $L_{2}$. I denote the excess demand for goods 1 and 2 by $d_{c,i} =c_{1i}+c_{2i}$, $i=1,2$. The excess demand for leisure is given by $d_{l,i}=\left( L_{i}-l_{i}\right) -L_{i}=-l_{i}$ and let $d=\left( d_{c,1},d_{c,2},d_{l,1},d_{l,2}\right)$.

The production side of the economy is given by a 4x6 activity analysis matrix A. Each column of A represents an activity, which transforms inputs taken from the vector of aggregate initial endowments or from the outputs of other activities into outputs, which are either consumed or further used as inputs. Positive entries in an activity denote quantities of outputs produced by the activity; negative entries denote quantities of inputs consumed. Aggregate production is denoted by $Ay^{\prime}$, where $y$ is a 6x1 vector of nonnegative activity levels:

$$A=\left[ \begin{array}[c]{rrrrrr} -1 & 0 & 0 & 0 & A_{1} & 0\ 0 & -1 & 0 & 0 & 0 & A_{2}\ 0 & 0 & -1 & 0 & -1 & 0\ 0 & 0 & 0 & -1 & 0 & -1 \end{array}\right] ,$$

with the first 4 columns of this matrix being free disposal activities.

An equilibrium of this economy is a price vector $\hat{p}=\left( \bar{P}_{1},\bar{P}_{2},w_{1},w_{2}\right)$ that satisfies the following three properties: first $\hat{p}^{\prime}A\leq0$; second there exists a nonnegative vector of activity levels $\hat{y}$ such that $A\hat{y}^{\prime }=d\left( \hat{p}\right)$; and third $\bar{P}_{1}=1$. The first condition requires that there be no excess profits available. The second one requires that supply equals demand. The third one is simply a price normalization.

Existence of an equilibrium follows directly from Theorem 1 in Kehoe (1985). Notice, how Kehoe's presentation of the problem can be reduced to the presentation in the main text. Let the activity vector be $y^{\prime }=\left( 0,0,0,0,l_{1},l_{2}\right)$. Then

$$d\left( p\right) -Ay^{\prime}=\left( \begin{array}[c]{c} c_{11}+c_{21}\ c_{12}+c_{22}\ -l_{1}\ -l_{2} \end{array}\right) -\left( \begin{array}[c]{c} A_{1}l_{1}\ A_{2}l_{2}\ -l_{1}\ -l_{2} \end{array}\right) \text{.}$$

Using Walras' Law, an equilibrium is a price vector such that $z_{2}\left( p\right) =c_{12}\left( p\right) +c_{22}\left( p\right) -A_{2}l_{2}\left( p\right) =0$ . As profit maximization implies $w_{i}=A_{i}$, all that needs to be found is the relative price $\bar{q}=\bar{P}_{2}$.

A.2  Multiplicity of Equilibria

If all the equilibria of an economy are locally unique, the economy is referred to as regular. Kehoe (1980) provides general conditions for a production economy that ensure regularity. In addition, he shows that the number of equilibria in a production economy is odd. Let the index of an equilibrium $\hat{p}$ be defined as

$$index\left( \hat{p}\right) =sgn\left( \det\left[ \begin{array}[c]{cc} -\bar{J} & \bar{B}\ -\bar{B}^{\prime} & 0 \end{array}\right] \right) .$$

$\bar{J}$ is formed by deleting the first row and$B\left( \hat{p}\right)$ the first column from $Dd\left( \hat{p}\right)$, the matrix of derivatives of the excess demand functions with respect to each price, if good 1 is the numeraire. $\bar{B}$ is formed by deleting the first row from , where $B\left( \hat{p}\right)$ is the submatrix of A whose columns are all those activities that earn zero profits at $\hat{p}$.

Theorem 2 in Kehoe (1985) states that the sum of the indices across all equilibria equals +1, i.e., $\sum_{j}index\left( \hat{p} ^{j}\right) =+1$. Hence the number of equilibria in a regular economy is finite and odd. If it cannot be proven that there is a unique equilibrium, this is usually all that can be said about the number of equilibria. Although there has been substantial progress in the development of fixed point algorithms, it is in general impossible to find all the equilibria of an economy if there is no guarantee that there is only one.

What can be said about the equilibria in the model presented in this paper? Using Kehoe's approach,

$$\bar{J}=\left[ \begin{array}[c]{ccc} \partial d_{c,2}/\partial\bar{q} & 0 & 0\ \partial d_{l,1}/\partial\bar{q} & 0 & 0\ \partial d_{l,2}/\partial\bar{q} & 0 & 0 \end{array}\right] ,\bar{B}=\left[ \begin{array}[c]{rr} 0 & A_{2}\ -1 & 0\ 0 & -1 \end{array}\right] ,$$

since $w_{i}=A_{i}$. It turns out that

$$\det\left[ \begin{array}[c]{cc} -\bar{J} & \bar{B}\ -\bar{B}^{\prime} & 0 \end{array}\right] =-\frac{\partial d_{c,2}}{\partial\bar{q}}+A_{2}\frac{\partial d_{l,1}}{\partial\bar{q}}=-\frac{\partial z_{2}}{\partial\bar{q}}\text{.}$$

If the excess demand function as defined in the main text, $z_{2}$, is downward sloping in each equilibrium, the equilibrium is unique. However, if an equilibrium with $\frac{\partial z_{2}}{\partial\bar{q}}>0$ is found then there must be at least two more equilibria.

In order to find calibrated economies with multiple equilibria for the model presented in this paper, I search for parameters such that the slope of the excess demand function is zero in equilibrium. Totally differentiating equation (16) delivers

$$\frac{\partial z_{2}}{\partial\bar{q}} =\frac{A_{1}l_{1}}{\bar{q}\left[ \tilde{H}_{1}\left( \frac{1}{\bar{q}}\right) +\bar{q}\right] }\left[ \frac{\tilde{H}_{1}^{\prime}\left( \frac{1}{\bar{q}}\right) \frac{1}{\bar {q}}-\bar{q}}{\tilde{H}_{1}\left( \frac{1}{\bar{q}}\right) +\bar{q}} +\frac{\partial l_{1}}{\partial\bar{q}}\frac{\bar{q}}{l_{1}}\right]$$

$$+\frac{\tilde{H}_{2}\left( \frac{1}{\bar{q}}\right) A_{2}l_{2}}{\bar {q}\left[ \tilde{H}_{2}\left( \frac{1}{\bar{q}}\right) +\bar{q}\right] }\left[ \frac{\frac{\tilde{H}_{2}^{\prime}\left( \frac{1}{\bar{q}}\right) }{\tilde{H}_{2}\left( \frac{1}{\bar{q}}\right) }+\bar{q}}{\left[ \tilde {H}_{2}\left( \frac{1}{\bar{q}}\right) +\bar{q}\right] }-\frac{\partial l_{2}}{\partial\bar{q}}\frac{\bar{q}}{l_{2}}\right] .$$

Let the (possibly variable) elasticity of substitution between traded goods in country $i$ be denoted by $\varepsilon_{i}\left( \bar{q}\right) .$ Hence, equation (5) to (7) imply

$$\varepsilon_{i}\left( \bar{q}\right) =\frac{\partial\left( \frac{c_{i1} }{c_{i2}}\right) \left( \bar{q}\right) }{\partial\left( \bar{q}\right) \frac{c_{i1}}{c_{i2}}}=-\frac{\tilde{H}_{i}^{\prime}\left( \frac{1}{\bar{q} }\right) \frac{1}{\bar{q}}}{\tilde{H}_{i}\left( \frac{1}{\bar{q}}\right) }$$ (35)

The slope of $\tilde{z}_{2}$ in equilibrium $\left( \tilde{z}_{2}\left( \bar{q}^{\ast}\right) =0\right)$ can then be expressed as

$$\frac{\partial z_{2}}{\partial\bar{q}}\vert _{\bar{q}=\bar{q}^{\ast}}=-\frac {A_{1}l_{1}}{\bar{q}\left[ \tilde{H}_{1}\left( \frac{1}{\bar{q}}\right) +\bar{q}\right] }\left[ \frac{\varepsilon_{1}\left( \bar{q}\right) \tilde{H}_{1}\left( \frac{1}{\bar{q}}\right) \frac{1}{\bar{q}}+1}{\tilde {H}_{1}\left( \frac{1}{\bar{q}}\right) \frac{1}{\bar{q}}+1}+\frac {\varepsilon_{2}\left( \bar{q}\right) -1}{\tilde{H}_{2}\left( \frac{1} {\bar{q}}\right) \frac{1}{\bar{q}}+1}-\frac{\partial l_{1}}{\partial\bar{q} }\frac{\bar{q}}{l_{1}}+\frac{\partial l_{2}}{\partial\bar{q}}\frac{\bar{q} }{l_{2}}\right] .$$

$\frac{\partial l_{i}}{\partial\bar{q}}\frac{\bar{q}}{l_{i}}$ is the general equilibrium elasticity of labor with respect to the relative price $\bar{q}$.

To find an expression for $\frac{\partial l_{i}}{\partial\bar{q} }\frac{\bar{q}}{l_{i}}$ notice that equations (5), (10), (11), and the consumption aggregators, $H_{i}\left( c_{i1} ,c_{i2}\right)$, imply $c_{i}=\Phi_{i}\left( \bar{q}\right) A_{i}l_{i}$. Total differentiation of $c_{i}=\Phi_{i}\left( \bar{q}\right) A_{i}l_{i}$ and (15) yields

$$\frac{\partial l_{i}}{\partial\bar{q}}\frac{\bar{q}}{l_{i}}=-\left[ \frac{-\eta_{i}\left[ \frac{U_{lc,i}^{2}}{U_{cc,i}}\frac{U_{c,i}}{U_{l,i} }-U_{ll,i}\frac{U_{c,i}}{U_{l,i}}\right] +\left[ U_{lc,i}-\frac{U_{l,i} }{U_{c,i}}U_{cc,i}\right] }{\left[ U_{lc,i}-U_{ll,i}\frac{U_{c,i}}{U_{l,i} }\right] +\left[ U_{lc,i}-\frac{U_{l,i}}{U_{c,i}}U_{cc,i}\right] }\right] \frac{\Phi_{i}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{i}\left( \bar{q}\right) },$$

where $\eta_{i}$ is the Frisch labor supply elasticity.¹⁵ From the definition of $\Phi_{i}\left( \bar{q}\right)$,

$$\bar{q}\frac{\Phi_{1}^{\prime}\left( \bar{q}\right) }{\Phi_{1}\left( \bar{q}\right) } =-\frac{1}{\tilde{H}_{1}\left( \frac{1}{\bar{q}}\right) \frac{1}{\bar{q}}+1},$$ (36)

$$\bar{q}\frac{\Phi_{2}^{\prime}\left( \bar{q}\right) }{\Phi_{2}\left( \bar{q}\right) } =\frac{\tilde{H}_{2}\left( \frac{1}{\bar{q}}\right) \frac{1}{\bar{q}}}{\tilde{H}_{2}\left( \frac{1}{\bar{q}}\right) \frac {1}{\bar{q}}+1}.$$ (37)

To gain additional insights, I define the share of imports in GDP of country 1 to be $1-\alpha_{1}=\frac{c_{12}}{\frac{1}{\bar{q}}A_{1}l_{1}}$. Since trade is balanced in the model with financial autarchy, $dW_{i}=0$, it is $\frac{c_{11}}{A_{1}l_{1}}=\alpha_{1}$. From the definition of $\tilde{H}_{1}$ follows $\tilde{H}_{1}\left( \frac{1}{\bar{q}}\right) \frac{1}{\bar{q} }=\frac{\alpha_{1}}{1-\alpha_{1}}$ . Analogously, it is $\tilde{H}_{2}\left( \frac{1}{\bar{q}}\right) \frac{1}{\bar{q}}=\frac{1-\alpha_{2}}{\alpha_{2}}$ . Let the relative country size, $\frac{A_{1}l_{1}}{A_{2}l_{2}}$, be denoted by $\theta$. The relative price is then given by $\bar{q}=\theta\frac {1-\alpha_{1}}{1-\alpha_{2}}$. With these definitions at hand

$$\frac{\partial z_{2}}{\partial\bar{q}}\vert _{\bar{q}=\bar{q}^{\ast}}=-\frac {A_{1}l_{1}}{\bar{q}\left[ \tilde{H}_{1}\left( \frac{1}{\bar{q}}\right) +\bar{q}\right] }\left[ \varepsilon_{1}\left( \bar{q}\right) \alpha _{1}+\varepsilon_{2}\left( \bar{q}\right) \alpha_{2}+\left( 1-\alpha _{1}-\alpha_{2}\right) -\frac{\partial l_{1}}{\partial\bar{q}}\frac{\bar{q} }{l_{1}}+\frac{\partial l_{2}}{\partial\bar{q}}\frac{\bar{q}}{l_{2}}\right] ,$$

and $\frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }=-\left( 1-\alpha_{1}\right)$ , $\frac{\Phi_{2}^{\prime }\left( \bar{q}\right) \bar{q}}{\Phi_{2}\left( \bar{q}\right) } =1-\alpha_{2}$ .

I consider the following three classes of utility functions:

1. Additive separable in consumption and leisure (labor)
   $$U\left( c,l\right) =v_{1}\left( c\right) -v_{2}\left( l\right) ,$$
   and
   $$U\left( c,l\right) =v_{1}\left( c\right) -v_{2}\left( 1-l\right) .$$
   Since $U_{cl}=0$,
   $$\frac{\partial l_{i}}{\partial\bar{q}}\frac{\bar{q}}{l_{i}}=\frac{1-\sigma }{\frac{1}{\eta}+\sigma}\frac{\Phi_{i}^{\prime}\left( \bar{q}\right) \bar {q}}{\Phi_{i}\left( \bar{q}\right) },$$
   where $\sigma=-\frac{U_{cc}c}{U_{c}}$, the relative risk aversion, and $\eta$ is the Frisch labor supply elasticity.
2. Preferences without wealth effects
   $$U\left( c,l\right) =v_{1}\left( c-v_{2}\left( l\right) \right) ,$$
   and
   $$\frac{\partial l_{i}}{\partial\bar{q}}\frac{\bar{q}}{l_{i}}=\eta\frac{\Phi _{i}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{i}\left( \bar{q}\right) }.$$
3. Cobb-Douglas aggregator
   $$U\left( c,l\right) =V\left( c^{\xi}\left( 1-l\right) ^{1-\xi}\right) ,$$
   where $V\left( \cdot\right)$ is strictly monotone in its argument. In this case
   $$\frac{\partial l_{i}}{\partial\bar{q}}\frac{\bar{q}}{l_{i}}=0.$$

B  Stability and the Slope of the Excess Demand Function

This appendix provides the linear approximation of the first four models in the main text and gives the details for Theorem 4.

B.1  Preliminaries

In this section I derive a log-linear approximation of the model's dynamics solely in terms of the relative price $\bar{q}$ and bond holdings $B_{1}$. I assume that the model is parameterized such that in any steady state bond holdings are zero.

Consumption and labor  With constant technology, equations (12) - (15) imply

$$\hat{l}_{1,t} =\omega_{q,1}\frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }\bar{q}_{t}-\omega_{b,1}\frac {1}{c_{1}}\left[ b_{1,t-1}-\beta b_{1,t}\right] ,$$ (38)

$$\hat{l}_{2,t} =\omega_{q,2}\frac{\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{2}\left( \bar{q}\right) }\bar{q}_{t}+\omega_{b,2}\frac {\Phi_{2}\left( \bar{q}\right) }{\bar{q}\Phi_{1}\left( \bar{q}\right) }\frac{1}{c_{2}}\left[ b_{1,t-1}-\beta b_{1,t}\right] ,$$ (39)

$$\hat{c}_{1,t} =\left[ 1+\omega_{q,1}\right] \frac{\Phi_{1}^{\prime }\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }\bar{q} _{t}+\left[ 1-\omega_{b,1}\right] \frac{1}{c_{1}}\left[ b_{1,t-1}-\beta b_{1,t}\right] ,$$ (40)

$$\hat{c}_{2,t} =\left[ 1+\omega_{q,2}\right] \frac{\Phi_{2}^{\prime }\left( \bar{q}\right) \bar{q}}{\Phi_{2}\left( \bar{q}\right) }\bar{q} _{t}-\left[ 1-\omega_{b,2}\right] \frac{\Phi_{2}\left( \bar{q}\right) }{\bar{q}\Phi_{1}\left( \bar{q}\right) }\frac{1}{c_{2}}\left[ b_{1,t-1}-\beta b_{1,t}\right] ,$$ (41)

where $\bar{q}_{t}$ denotes the percentage deviation of the relative price $\bar{q}$ from its steady state value at time $t$. $b_{1,t}$ is the absolute deviation of country $1$'s bond holdings. If $b_{1,t}>0$, country 1 is lending to country 2 in period $t$. $\omega_{b,i}$ and $w_{q,i}$ are given by

$$\omega_{b,i} =\frac{\left[ U_{lc,i}-\frac{U_{l,i}}{U_{c,i}}U_{cc,i} \right] }{\left[ U_{lc,i}-\frac{U_{l,i}}{U_{c,i}}U_{cc,i}\right] +\left[ U_{cl,i}-\frac{U_{c,i}}{U_{l,i}}U_{ll,i}\right] },$$

$$\tau_{i} =\frac{-U_{c,i}\frac{1}{l_{i}}}{\left[ U_{lc,i}-\frac{U_{l,i} }{U_{c,i}}U_{cc,i}\right] +\left[ U_{cl,i}-\frac{U_{c,i}}{U_{l,i}} U_{ll,i}\right] },$$

$$\omega_{q,i} =-\omega_{b,i}+\tau_{i}.$$

With the assumptions on the utility function $U\left( c,l\right)$, which are satisfied by almost all utility functions that are commonly used in macroeconomics, one obtains $0<\omega_{b,i}<1$ and $\tau_{i}>0$.

Excess demand function  Using equations (38) - (41) the log-linear approximation of the excess demand function in equilibrium, $z_{2}\left( \bar{q},dW_{1}\right) =0$, can be written as

$$\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}\bar{q}_{t}+\frac{\partial z_{2} }{\partial dW_{1}}\left[ b_{1,t-1}-\beta b_{1,t}\right] =0$$ (42)

with

$$\frac{\partial z_{2}}{\partial\bar{q}}\bar{q} =\frac{A_{1}l_{1}}{\left[ \tilde{H}_{1}\left( \frac{1}{\bar{q}}\right) +\bar{q}\right] }\left[ \frac{\tilde{H}_{1}^{\prime}\left( \frac{1}{\bar{q}}\right) \frac{1}{\bar {q}}-\bar{q}}{\tilde{H}_{1}\left( \frac{1}{\bar{q}}\right) +\bar{q}} +\frac{\partial l_{1}}{\partial\bar{q}}\frac{\bar{q}}{l_{1}}\right] +\frac{\tilde{H}_{2}\left( \frac{1}{\bar{q}}\right) A_{2}l_{2}}{\bar {q}\left[ \tilde{H}_{2}\left( \frac{1}{\bar{q}}\right) +\bar{q}\right] }\left[ \frac{\frac{\tilde{H}_{2}^{\prime}\left( \frac{1}{\bar{q}}\right) }{\tilde{H}_{2}\left( \frac{1}{\bar{q}}\right) }+\bar{q}}{\tilde{H} _{2}\left( \frac{1}{\bar{q}}\right) +\bar{q}}-\frac{\partial l_{2}} {\partial\bar{q}}\frac{\bar{q}}{l_{2}}\right]$$

$$=c_{12}\left\{ \left[ \varepsilon_{1}\left( \bar{q}\right) \tilde{H} _{1}\left( \frac{1}{\bar{q}}\right) \frac{1}{\bar{q}}+1+\omega_{q,1}\right] \frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }+\left[ \frac{1-\varepsilon_{2}\left( \bar{q}\right) }{\tilde{H}_{2}\left( \frac{1}{\bar{q}}\right) \frac{1}{\bar{q}}} -\omega_{q,2}\right] \frac{\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q} }{\Phi_{2}\left( \bar{q}\right) }\right\} ,$$

$$\frac{\partial z_{2}}{\partial dW_{1}} =\frac{A_{1}l_{1}}{\left[ \tilde {H}_{1}\left( \frac{1}{\bar{q}}\right) +\bar{q}\right] }\frac{\partial l_{1}}{\partial dW_{1}}\frac{1}{l_{1}}+\frac{\tilde{H}_{2}\left( \frac {1}{\bar{q}}\right) \frac{1}{\bar{q}}A_{2}l_{2}}{\tilde{H}_{2}\left( \frac{1}{\bar{q}}\right) \frac{1}{\bar{q}}+1}\frac{\partial l_{2}}{\partial dW_{2}}\frac{1}{l_{2}}+\frac{\frac{1}{\Phi_{1}\left( \bar{q}\right) } }{\left[ \tilde{H}_{1}\left( \frac{1}{\bar{q}}\right) +\bar{q}\right] }-\frac{\frac{1}{q\Phi_{2}\left( \bar{q}\right) }}{\tilde{H}_{2}\left( \frac{1}{\bar{q}}\right) \frac{1}{\bar{q}}+1}$$

$$=-\frac{1}{\bar{q}\Phi_{1}\left( \bar{q}\right) }\left\{ 1+\left[ 1-\omega_{b,1}\right] \frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q} }{\Phi_{1}\left( \bar{q}\right) }-\left[ 1-\omega_{b,2}\right] \frac {\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{2}\left( \bar {q}\right) }\right\} .$$

In simplifying the expressions for $\frac{\partial z_{2}}{\partial\bar{q}} \bar{q}$ and $\frac{\partial z_{2}}{\partial dW_{1}}$, I use the definitions of $\frac{\Phi_{i}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{i}\left( \bar{q}\right) }$ , $i=1,2$ (equations (36) and (37) ) and the country demand functions for good 2. Furthermore, $\varepsilon_{i}\left( \bar{q}\right)$ denotes the elasticity of substitution between the two traded goods in country $i$ as defined in equation (35).

B.2  Linearized Models

Convex portfolio costs  Using equations (38) - (41) and the log-linearized risk sharing condition, that is derived from equation (19), delivers the following system of linear difference equations

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

where

$$\check{\Gamma} =\frac{\Gamma^{\prime\prime}\left( 0\right) }{\beta ^{2}\Phi_{1}\left( \bar{q}\right) }\left[ 1+\frac{1}{\bar{q}}\right] ,$$

$$d_{b} =-\left[ \frac{U_{cc,1}c_{1}}{U_{c,1}}-\left( \frac{U_{cc,1}c_{1} }{U_{c,1}}+\frac{U_{cl,1}l_{1}}{U_{c,1}}\right) \omega_{b,1}\right] \frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }\frac{1}{\bar{q}\Phi_{1}\left( \bar{q}\right) c_{12}}$$

$$+\left[ \frac{U_{cc,2}c_{2}}{U_{c,2}}-\left( \frac{U_{cc,2}c_{2}}{U_{c,2} }+\frac{U_{cl,2}l_{2}}{U_{c,2}}\right) \omega_{b,2}\right] \frac{\Phi _{2}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{2}\left( \bar{q}\right) }\frac{1}{\bar{q}\Phi_{1}\left( \bar{q}\right) c_{12}},$$

$$d_{q} =-\bar{q}\Phi_{1}\left( \bar{q}\right) c_{12}d_{b}+1+\left[ 1-\omega_{b,1}\right] \frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q} }{\Phi_{1}\left( \bar{q}\right) }-\left[ 1-\omega_{b,2}\right] \frac {\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{2}\left( \bar {q}\right) }.$$

Note that $d_{b}<0$ as $-\frac{U_{cc,i}c_{i}}{U_{c,i}}+\left( \frac {U_{cc,i}c_{i}}{U_{c,i}}+\frac{U_{cl,i}l_{i}}{U_{c,i}}\right) \omega _{b,i}=-\frac{U_{ll,i}U_{cc,i}-U_{lc,i}^{2}}{\left[ U_{lc,i}-\frac{U_{l,i} }{U_{c,i}}U_{cc,i}\right] +\left[ U_{cl,i}-\frac{U_{c,i}}{U_{l,i}} U_{ll,i}\right] }\frac{l_{i}}{U_{c,i}}>0$ .

Debt elastic interest rate  The model with a debt elastic interest rate is very similar to the model with portfolio costs. Following the standard assumption that agents do not internalize the effects of their decisions on the interest rate, it is

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

Endogenous discounting without internalization  If agents do not internalize the effects of their consumption and leisure choices on the discount factor, the risk sharing condition, equation (27), implies the following system of difference equations

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

where

$$g_{b} =\left[ U_{c,1}c_{1}\frac{\beta_{1}^{\prime}}{\beta_{1}}\frac {\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar {q}\right) }-U_{c,2}c_{2}\frac{\beta_{2}^{\prime}}{\beta_{2}}\frac{\Phi _{2}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{2}\left( \bar{q}\right) }\right] \frac{1}{\bar{q}\Phi_{1}\left( \bar{q}\right) c_{12}},$$

$$g_{q} =-\bar{q}\Phi_{1}\left( \bar{q}\right) c_{12}g_{b}.$$

$d_{b}$ and $d_{q}$ are as defined above and $g_{b}>0$ as $\beta_{i}^{\prime }<0$ by assumption.

Endogenous discounting with internalization  In this last model, agents take into account the effects of their consumption and leisure choices on the discount factor. As equations (29) and (30) reveal this implies two additional state variables. In addition to $b_{t}$, $\hat{\eta}_{1,t}$ and $\hat{\eta}_{2,t}$ are also state variables of the linearized system:

$$\left( \begin{array}[c]{c} \bar{q}_{t+1}\ \hat{\eta}_{1,t+1}\ \hat{\eta}_{2,t+1}\ b_{t} \end{array}\right) =\left( \begin{array}[c]{cccc} 1 & -zm_{1} & zm_{2} & 0\ \frac{a\phi_{1}}{m_{1}} & \left[ \frac{1}{\beta_{1}}-za\phi_{1}\right] & za\phi_{1}\frac{m_{2}}{m_{1}} & 0\ \frac{a\phi_{2}}{m_{2}} & -za\phi_{2}\frac{m_{1}}{m_{2}} & \left[ \frac {1}{\beta_{2}}+za\phi_{2}\right] & 0\ \frac{1}{\beta}\frac{\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}} {\frac{\partial z_{2}}{\partial dW_{1}}} & 0 & 0 & \frac{1}{\beta} \end{array}\right) \left( \begin{array}[c]{c} \bar{q}_{t}\ \hat{\eta}_{1,t}\ \hat{\eta}_{2,t}\ b_{t-1} \end{array}\right) ,$$

where

$$z =\frac{\frac{\partial z_{2}}{\partial dW_{1}}\left( 1-\frac{1}{\beta }\right) }{\frac{\partial z_{2}}{\partial dW_{1}}\left( d_{q}+g_{q} +h_{q}\right) -\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}\left( d_{b}+g_{b}+h_{b}\right) },$$

$$a =\left[ 1+\frac{\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}} {\frac{\partial z_{2}}{\partial dW_{1}}}\frac{1}{\bar{q}\Phi_{1}\left( \bar{q}\right) c_{12}}\right] ,$$

$$\phi_{i} =U_{c,i}c_{i}\frac{\beta_{i}^{\prime}}{\beta_{i}}\frac{\Phi _{i}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{i}\left( \bar{q}\right) },$$

$$m_{i} =\frac{\eta_{i}\beta_{i}^{\prime}}{1-\eta_{i}\beta_{i}^{\prime}},$$

for $i=1,2$ and

$$h_{b} =\left\{ U_{c,1}c_{1}\frac{\eta_{1}\beta_{1}^{\prime}}{1-\eta _{1}\beta_{1}^{\prime}}\frac{\beta_{1}^{\prime\prime}}{\beta_{1}^{\prime} }\frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }-U_{c,2}c_{2}\frac{\eta_{2}\beta_{2}^{\prime}}{1-\eta _{2}\beta_{2}^{\prime}}\frac{\beta_{2}^{\prime\prime}}{\beta_{2}^{\prime} }\frac{\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{2}\left( \bar{q}\right) }\right\} \frac{1}{\bar{q}\Phi_{1}\left( \bar{q}\right) c_{12}},$$

$$h_{q} =-\bar{q}\Phi_{1}\left( \bar{q}\right) c_{12}h_{b}.$$

Important sign restrictions  Before I study the local stability in the next section, it is useful to find the signs of the following expressions:

$$\frac{\partial z_{2}}{\partial dW_{1}}d_{q}-\frac{\partial z_{2}}{\partial \bar{q}}\bar{q}d_{b} =-\frac{1}{\bar{q}\Phi_{1}\left( \bar{q}\right) }\left( 1+\left[ 1-\omega_{b,1}\right] \frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }-\left[ 1-\omega_{b,2}\right] \frac{\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q} }{\Phi_{2}\left( \bar{q}\right) }\right) ^{2}+\Upsilon c_{12}d_{b}<0,$$

$$\frac{\partial z_{2}}{\partial dW_{1}}g_{q}-\frac{\partial z_{2}}{\partial \bar{q}}\bar{q}g_{b} =\Upsilon c_{12}g_{b}>0,$$

where $\Upsilon=\left\{ -\tau_{1}\frac{\Phi_{1}^{\prime}\left( \bar {q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }+\tau_{2}\frac{\Phi _{2}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{2}\left( \bar{q}\right) }-\varepsilon_{1}\left( \bar{q}\right) \tilde{H}_{1}\left( \frac{1}{\bar {q}}\right) \frac{1}{\bar{q}}\frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }+\frac{\varepsilon_{2}\left( \bar{q}\right) }{\tilde{H}_{2}\left( \frac{1}{\bar{q}}\right) \frac{1} {\bar{q}}}\frac{\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi _{2}\left( \bar{q}\right) }\right\} >0$ .

The sign of

$$\frac{\partial z_{2}}{\partial dW_{1}}\left( g_{q}+h_{q}\right) -\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}\left( g_{b}+h_{b}\right) =\Upsilon c_{12}\left( g_{b}+h_{b}\right)$$

depends on the sign of $\frac{\beta_{1}^{\prime}}{\beta_{1}}+\frac{\eta _{1}\beta_{1}^{\prime}}{1-\eta_{1}\beta_{1}^{\prime}}\frac{\beta_{1} ^{\prime\prime}}{\beta_{1}^{\prime}}$ . Without imposing more structure on the functional form of the discount factor nothing can be said about the sign of this expression.

B.3  Theorem 4

If agents internalize the effects of their choices on the endogenous discount factor, only a weaker theorem can be proven since the sign of $d_{b}+h_{b}+g_{b}$ cannot be determined. In preparation for this theorem, consider an increase in the wealth of agents in country 1. I am interested in the change of the intertemporal marginal rate of substitution under the assumption that current and future prices as well as future allocations remain unchanged. The only variables that are allowed to change are current consumption and leisure and therefore also utility in the current period. I refer to this experiment as the direct impact of a wealth increase.

The intertemporal marginal rate of substitution in country 1 is given by

$$IMRS\left( s^{t+1}\right) =\beta\left( U_{1}\left( s^{t}\right) \right) \frac{1-\eta_{1}\left( s^{t+1}\right) \beta^{\prime}\left( U_{1}\left( s^{t+1}\right) \right) }{1-\eta_{1}\left( s^{t}\right) \beta^{\prime }\left( U_{1}\left( s^{t}\right) \right) }\frac{U_{c,1}\left( s^{t+1}\right) }{U_{c,1}\left( s^{t}\right) }\text{.}$$

Equation (30) in the main text reveals that $\eta_{1}\left( s^{t}\right)$ is nothing but the negative of the expected discounted lifetime utility of agents of country $1$ from $t+1$ onwards. Therefore $\eta_{1}\left( s^{t}\right)$ does not depend on any time $t$ variables. Under the assumption that future allocations and prices are held constant, the following equations are relevant for the experiment:

$$\frac{U_{l}\left( c_{1}\left( s^{t}\right) ,l_{1}\left( s^{t}\right) \right) }{U_{c}\left( c_{1}\left( s^{t}\right) ,l_{1}\left( s^{t}\right) \right) } =-A_{1}\Phi_{1}\left( \bar{q}\left( s^{t}\right) \right)$$

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

The first equation states the familiar equilibrium condition that the marginal rate of substitution between labor and consumption equals the real wage. As shown earlier, the second equation can be derived straight from the intertemporal budget constraint of the agents. $dW_{1}\left( s^{t}\right)$ denotes the wealth transfer to country 1. Under the assumption that prices are kept unchanged for all $s^{t+j}$, $j\geq0$, the marginal rate of substitution between labor and consumption is held constant.

The direct impact of a wealth increase on the intertemporal marginal rate of substitution is given by

$$\frac{\partial IMRS\left( s^{t+1}\right) }{\partial dW_{1}\left( s^{t}\right) }\vert _{direct}$$

$$=\left[ -\frac{U_{cc,1}\left( s^{t}\right) c_{1}\left( s^{t}\right) }{U_{c,1}\left( s^{t}\right) }+\left( \frac{U_{cc,1}\left( s^{t}\right) c_{1}\left( s^{t}\right) }{U_{c,1}\left( s^{t}\right) }+\frac {U_{cl,1}\left( s^{t}\right) l_{1}\left( s^{t}\right) }{U_{c,1}\left( s^{t}\right) }\right) \omega_{1b}\left( s^{t}\right) \right] \frac {1}{c_{1}\left( s^{t}\right) }IMRS\left( s^{t+1}\right)$$

$$+\left( \frac{\beta^{\prime}\left( U_{1}\left( s^{t}\right) \right) }{\beta\left( U_{1}\left( s^{t}\right) \right) }+\frac{\eta_{1}\left( s^{t}\right) \beta^{\prime}\left( U_{1}\left( s^{t}\right) \right) }{1-\eta_{1}\left( s^{t}\right) \beta^{\prime}\left( U_{1}\left( s^{t}\right) \right) }\frac{\beta^{\prime\prime}\left( U_{1}\left( s^{t}\right) \right) }{\beta^{\prime}\left( U_{1}\left( s^{t}\right) \right) }\right) U_{c,1}\left( s^{t}\right) IMRS\left( s^{t+1}\right) .$$ (43)

The first term in equation (43) measures the direct impact of the wealth transfer on the marginal utility of consumption under the assumption that the marginal rate of substitution between leisure and consumption is held constant. Under the assumptions on the utility function the first term is positive. In the experiment the increase in the wealth of the agents of country 1 lowers the labor supply and increases consumption. Consequently, the marginal utility of consumption, $U_{c,1} \left( s^{t}\right)$, rises. This effect operates towards a rise of $IMRS_{1}$.

The second term measures the effect of the wealth increase on $IMRS_{1}$ through the endogeneity of the discount factor. There are two effects. First, as consumption and leisure rise in the current period, so does utility $U_{1}\left( s^{t}\right)$. As the discount factor is decreasing in the utility level this effect operates towards a decline of the $IMRS_{1}$. Furthermore, the change in the discount factor effects the $IMRS_{1}$ also through its impact on the discounted future utility summarized in $\eta _{1}\left( s^{t}\right)$. Absent assumptions on $\beta^{\prime\prime}$ this expression cannot be signed.

If the discount factor is constant, $\frac{\partial IMRS_{1}\left( s^{t+1}\right) }{\partial dW_{1}\left( s^{t}\right) }\vert _{direct}>0$ . Hence, if the discount factor $\beta_{i}$ does not react too strongly to changes in $U_{i}$, the effect will still be positive.

Given the original questions this restriction is not too restrictive and $\frac{\partial IMRS_{1}\left( s^{t+1}\right) }{\partial dW_{1}\left( s^{t}\right) }\vert _{direct}$ is most likely to be positive. Endogenous discounting is introduced to obtain stationarity in the model with incomplete asset markets. To the extent that the stationary model is supposed to behave closely to the original non-stationary model it is desirable that the discount factor does not move around too much.

Under the assumption that $\frac{\partial IMRS_{i}\left( s^{t+1}\right) }{\partial dW_{i}\left( s^{t}\right) }\vert _{direct}>0$ , $i=1,2$, theorem 4 can be proven.

Proof 4 (Proof of Theorem 4)  The linearized dynamic system for the model with endogenous discounting and internalization is given by $x_{t+1} =M_{I}x_{t}$ and $x_{t}^{\prime}=\left( \bar{q}_{t},\hat{\eta}_{1,t} ,\hat{\eta}_{2,t},b_{t-1}\right) ^{\prime}$ . The 4x4 coefficient matrix $M_{I}$ is given by

$$M_{I}=\left( \begin{array}[c]{cccc} 1 & -zm_{1} & zm_{2} & 0\ \frac{a\phi_{1}}{m_{1}} & \left[ \frac{1}{\beta_{1}}-za\phi_{1}\right] & za\phi_{1}\frac{m_{2}}{m_{1}} & 0\ \frac{a\phi_{2}}{m_{2}} & -za\phi_{2}\frac{m_{1}}{m_{2}} & \left[ \frac {1}{\beta_{2}}+za\phi_{2}\right] & 0\ \frac{1}{\beta}\frac{\frac{\partial z_{2}}{\partial\bar{q}}\bar{q}} {\frac{\partial z_{2}}{\partial dW_{1}}} & 0 & 0 & \frac{1}{\beta} \end{array}\right) \text{.}$$

The characteristic equations that is associated with $M_{I}$ simplifies to

$$-\left( \lambda-\frac{1}{\beta}\right) ^{2}\left( \lambda^{2}-\left[ \frac{1}{\beta}+1+za\left[ \phi_{2}-\phi_{1}\right] \right] \lambda +\frac{1}{\beta}\right) =0$$ ,

with

$$za\left[ \phi_{2}-\phi_{1}\right] =\frac{\frac{1}{\beta}\left( 1-\beta\right) \Upsilon\bar{q}\Phi_{1}\left( \bar{q}\right) c_{12}g_{b} }{\left( 1+\left[ 1-\omega_{b,1}\right] \frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }-\left[ 1-\omega_{b,2}\right] \frac{\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q} }{\Phi_{2}\left( \bar{q}\right) }\right) ^{2}-\Upsilon\bar{q}\Phi _{1}\left( \bar{q}\right) c_{12}\left( d_{b}+h_{b}+g_{b}\right) }\text{,}$$

where $\Upsilon>0$ irrespective of the sign of the slope of $z_{2}$ as shown above. With three state variables, $\hat{\eta}_{1,t}$, $\hat{\eta}_{2,t}$ and $b_{t-1}$, the dynamic system is saddle-path stable if the modulus of exactly three eigenvalues is larger than 1. Since two of the four eigenvalues are equal to $\frac{1}{\beta}$, stability of the system requires:

$$za\left[ \phi_{2}-\phi_{1}\right] >0$$

or

$$za\left[ \phi_{2}-\phi_{1}\right] <-2\left( 1+\frac{1}{\beta}\right)$$    .

A sufficient condition for stability is $\Phi_{1}\left( \bar{q}\right) c_{12}\left( d_{b}+h_{b}+g_{b}\right) <0$ as it implies that $za\left[ \phi_{2}-\phi_{1}\right] >0$:

$$\left( d_{b}+h_{b}+g_{b}\right) \bar{q}\Phi_{1}\left( \bar{q}\right) c_{12}$$

$$=\left[ -\frac{U_{cc,1}c_{1}}{U_{c,1}}+\left( \frac{U_{cc,1}c_{1}} {U_{c,1}}+\frac{U_{cl,1}l_{1}}{U_{c,1}}\right) \omega_{b,1}+\left( \frac{\beta_{1}^{\prime}}{\beta_{1}}+\frac{\eta_{1}\beta_{1}^{\prime}} {1-\eta_{1}\beta_{1}^{\prime}}\frac{\beta_{1}^{\prime\prime}}{\beta _{1}^{\prime}}\right) U_{c,1}c_{1}\right] \frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }$$

$$-\left[ -\frac{U_{cc,2}c_{2}}{U_{c,2}}+\left( \frac{U_{cc,2}c_{2}} {U_{c,2}}+\frac{U_{cl,2}l_{2}}{U_{c,2}}\right) \omega_{b,2}+\left( \frac{\beta_{2}^{\prime}}{\beta_{2}}+\frac{\eta_{2}\beta_{2}^{\prime}} {1-\eta_{2}\beta_{2}^{\prime}}\frac{\beta_{2}^{\prime\prime}}{\beta _{2}^{\prime}}\right) U_{c,2}c_{2}\right] \frac{\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{2}\left( \bar{q}\right) }$$

Under the assumption that $\frac{\partial IMRS_{i}\left( s^{t+1}\right) }{\partial dW_{i}\left( s^{t}\right) }\vert _{direct}>0$ , $i=1,2$, each of two the expressions in brackets is positive and therefore $\Phi_{1}\left( \bar {q}\right) c_{12}\left( d_{b}+h_{b}+g_{b}\right) <0$ . $\qedsymbol$

Obviously the assumption that $\frac{\partial IMRS_{i}\left( s^{t+1}\right) }{\partial dW_{i}\left( s^{t}\right) }\vert _{direct}>0$ for each $i=1,2$ is unnecessarily strong. However, this expression is somewhat more intuitive than other possible restrictions.

C  Analysis of the Overlapping Generations Model

In this Appendix, I outline the analysis in the overlapping generations model, derive a log-linear approximation of the model, and prove Theorem 5.

C.1  The Model

The world consists of two countries. In each period, the world economy is populated by a continuum of infinitely lived households between 0 and $N_{t}$. Each household consumes, supplies labor and holds financial assets. Newly born households own no financial assets, but they own the present discounted value of their labor income. The number of households in country $i$, $N_{i,t}$, grows over time at the exogenous rate $n$, i.e., $N_{i,t+1}=\left( 1+n\right) N_{i,t}$.

C.1.1  Individual Behavior

Consumers have identical preferences over the real consumption and leisure. At time $t_{0}$, the representative consumer in country $i$ born in period $v\in\left( -\infty,t_{0}\right)$ maximizes the intertemporal utility function

$$U_{t_{0}}^{v}=\sum_{t=t_{0}}^{\infty}\beta^{t-t_{0}}\left[ \zeta\log c_{i}^{v}\left( s^{t}\right) +(1-\zeta)\log\left( 1-l_{i}^{v}\left( s^{t}\right) \right) \right]$$

subject to the intertemporal budget constraint

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

The consumption index is given by the CES aggregator $c_{i}^{v}=\left[ \alpha_{i1}^{1-\rho}\left( c_{i1}^{v}\right) ^{\rho}+\alpha_{i2}^{1-\rho }\left( c_{i2}^{v}\right) ^{\rho}\right] ^{\frac{1}{\rho}}$ . $c_{ij}^{v}$ $\left( s^{t}\right)$ is the amount of good $j$ that the representative household of country $i$ born in period $v$ consumes in time $t$. Producers of the traded goods face perfect competition. Hence firm profits are zero in equilibrium and the distribution of stock ownership has no impact on allocations.

Household choices must satisfy the following set of first order conditions

$$l_{i}^{v}\left( s^{t}\right) =1-\frac{1-\zeta}{\zeta}\frac{P_{i}\left( s^{t}\right) }{\bar{P}_{i}\left( s^{t}\right) w_{i}\left( s^{t}\right) }c_{i}^{v}\left( s^{t}\right)$$

$$c_{i}^{v}\left( s^{t}\right) =\frac{Q\left( s^{t}\right) }{\beta} c_{i}^{v}\left( s^{t+1}\right) \frac{P_{i}\left( s^{t+1}\right) } {P_{i}\left( s^{t}\right) }$$

where

$$Q\left( s^{t}\right) =\beta\frac{c_{i}^{v}\left( s^{t}\right) }{c_{i} ^{v}\left( s^{t+1}\right) }\frac{P_{i}\left( s^{t}\right) }{P_{i}\left( s^{t+1}\right) }=\frac{1}{1+r\left( s^{t}\right) }.$$

The demand for goods 1 and 2 are

$$c_{i1}^{v}\left( s^{t}\right) =\alpha_{i1}\left( \frac{\bar{P} _{1}\left( s^{t}\right) }{P_{i}\left( s^{t}\right) }\right) ^{\frac {1}{\rho-1}}c_{i}^{v}\left( s^{t}\right) ,$$ (44)

$$c_{i2}^{v}\left( s^{t}\right) =\alpha_{i2}\left( \frac{\bar{P} _{2}\left( s^{t}\right) }{P_{i}\left( s^{t}\right) }\right) ^{\frac {1}{\rho-1}}c_{i}^{v}\left( s^{t}\right) .$$ (45)

If the population at time $t$ is $N_{t}$, total demand for good $j$ in country $i$ by all consumers of country $i$ satisfies

$$c_{i,j}^{T}\left( s^{t}\right) =a\left( 1+n\right) ^{t}\alpha_{ij}\left( \frac{P_{j}^{v}\left( s^{t}\right) }{P_{i}\left( s^{t}\right) }\right) ^{\frac{1}{\rho-1}}c_{i}\left( s^{t}\right) ,$$

where

$$c_{i}\left( s^{t}\right) =\frac{a}{N_{i}\left( s^{t}\right) \left( 1+n\right) ^{t}}\left[ \begin{array}[c]{c} \frac{n}{\left( 1+n\right) ^{t+1}}c_{i}^{-t}\left( s^{t}\right) +...+\frac{n}{\left( 1+n\right) ^{2}}c_{i}^{-1}\left( s^{t}\right) +\frac{n}{1+n}c_{i}^{0}\left( s^{t}\right) \ +nc_{i}^{1}\left( s^{t}\right) +n\left( 1+n\right) c_{i}^{2}\left( s^{t}\right) +...+n\left( 1+n\right) ^{t-1}c_{i}^{t}\left( s^{t}\right) \end{array}\right] .$$

$c_{i}$ denotes aggregate per capita consumption of the composite consumption basket in country 1.

Firms in country $i$ produce the traded good $i$ and operate using the linear technology, i.e., $y_{i}\left( s^{t}\right) =A_{i}\left( s^{t}\right) l_{i}$ $\left( s^{t}\right)$. Profit maximization requires $A_{i}\left( s^{t}\right) =w_{i}$ $\left( s^{t}\right)$.

C.1.2  Aggregation

Aggregate per capita labor supply equations are obtained by aggregating labor-leisure tradeoff equations across generations and dividing by the total population at each point in time. The aggregate per capita labor-leisure tradeoff satisfies

$$l_{i}\left( s^{t}\right) =1-\frac{1-\zeta}{\zeta}\frac{P_{i}\left( s^{t}\right) }{\bar{P}_{i}\left( s^{t}\right) w_{i}\left( s^{t}\right) }c_{i}\left( s^{t}\right) .$$ (46)

Consumption Euler equations in aggregate per capita terms contain an adjustment for consumption by the newborn generation at time $t+1$. It is

$$c_{i}\left( s^{t}\right) =\frac{1+n}{\beta\left( 1+r\left( s^{t}\right) \right) }\frac{P_{i}\left( s^{t+1}\right) }{P_{i}\left( s^{t}\right) }\left( c_{i}\left( s^{t+1}\right) -\frac{n}{1+n}c_{i}^{t+1}\left( s^{t+1}\right) \right) .$$ (47)

Newborn housholds hold no assets, but they own the present discounted value of their labor income. Using the Euler equations for individual agents and the intertemporal budget constraint of newborn agents, it is possible to show that the household's consumption in the first period of life is a fraction of human wealth, $h_{i}$:

$$c_{i}^{t}\left( s^{t}\right) =\zeta\left( 1-\beta\right) h_{i}\left( s^{t}\right)$$ (48)

with

$$h_{i}\left( s^{t}\right) =\frac{\bar{P}_{i}\left( s^{t}\right) } {P_{i}\left( s^{t}\right) }w_{i}\left( s^{t}\right) +\frac{1}{1+r\left( s^{t}\right) }\frac{P_{i}\left( s^{t+1}\right) }{P_{i}\left( s^{t}\right) }h_{i}\left( s^{t+1}\right) .$$ (49)

The law of motion of aggregate per capita bond holdings is obtained by aggregating the intertemporal budget constraints across living generations,

$$\left( 1+n\right) B_{i}\left( s^{t}\right) =\left( 1+r\left( s^{t-1}\right) \right) B_{i}\left( s^{t-1}\right) +\bar{P}_{i}\left( s^{t}\right) w_{i}\left( s^{t}\right) l_{i}\left( s^{t}\right) -P_{i}\left( s^{t}\right) c_{i}\left( s^{t}\right) .$$ (50)

Per capita output is simply $y_{i}\left( s^{t}\right) =A_{i}\left( s^{t}\right) l_{i}$ $\left( s^{t}\right)$. The wage rate is found to be $A_{i}\left( s^{t}\right) =w_{i}$ $\left( s^{t}\right)$.

Using the per capita demand equations for good 2, (45), the consumption-leisure choice, (46), and the demand for per capita final consumption, one finds

$$c_{1}\left( s^{t}\right) =\Phi_{1}\left( \bar {q}\left( s^{t}\right) \right) A_{1}\left( s^{t}\right) l_{1}\left( s^{t}\right) +\left( B_{1}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{1}\left( s^{t}\right) \right) ,$$ (51)

$$c_{2}\left( s^{t}\right) =\Phi_{2}\left( \bar{q}\left( s^{t}\right) \right) A_{2}\left( s^{t}\right) l_{2}\left( s^{t}\right) +\frac{\Phi _{2}\left( \bar{q}\left( s^{t}\right) \right) }{\Phi_{1}\left( \bar {q}\left( s^{t}\right) \right) \bar{q}\left( s^{t}\right) }\left( B_{2}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{2}\left( s^{t}\right) \right) ,$$  (52)

and the excess demand function for good 2

$$z_{2}\left( s^{t}\right) =a\frac{A_{1,t}\zeta}{\left( \frac{\alpha_{11} }{\alpha_{12}}\left( \frac{1}{\bar{q}\left( s^{t}\right) }\right) ^{\frac{1}{\rho-1}}+\bar{q}\left( s^{t}\right) \right) }+\left( 1-a\right) \frac{-\frac{\alpha_{21}}{\alpha_{22}}\left( \frac{1}{\bar {q}\left( s^{t}\right) }\right) ^{\frac{\rho}{\rho-1}}A_{2}\left( s^{t}\right) \zeta}{\left( \frac{\alpha_{21}}{\alpha_{22}}\left( \frac {1}{\bar{q}\left( s^{t}\right) }\right) ^{\frac{\rho}{\rho-1}}+1\right) }$$

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

$$\left( B_{1}\left( s^{t-1}\right) -Q\left( s^{t}\right) B_{1}\left( s^{t}\right) \right) .$$ (53)

The consumption Euler equations (47) imply a risk sharing condition of the form

$$\frac{c_{1}\left( s^{t}\right) }{c_{1}\left( s^{t+1}\right) -\frac{n} {1+n}c_{1}^{t+1}\left( s^{t+1}\right) }=\frac{c_{2}\left( s^{t}\right) }{c_{2}\left( s^{t+1}\right) -\frac{n}{1+n}c_{2}^{t+1}\left( s^{t+1} \right) }\frac{\bar{q}\left( s^{t}\right) }{\bar{q}\left( s^{t+1}\right) }\frac{\Phi_{1}\left( \bar{q}\left( s^{t}\right) \right) }{\Phi_{2}\left( \bar{q}\left( s^{t}\right) \right) }\frac{\Phi_{2}\left( \bar{q}\left( s^{t+1}\right) \right) }{\Phi_{1}\left( \bar{q}\left( s^{t+1}\right) \right) }.$$ (54)

The price of bonds is given by

$$Q\left( s^{t}\right) =\beta\frac{\frac{1}{1+n}c_{i}\left( s^{t}\right) }{c_{i}\left( s^{t+1}\right) -\frac{n}{1+n}c_{i}^{t+1}\left( s^{t+1} \right) }\frac{P_{i}\left( s^{t}\right) }{P_{i}\left( s^{t+1}\right) }.$$ (55)

Bond market clearing requires

$$aB_{1}\left( s^{t}\right) +\left( 1-a\right) B_{2}\left( s^{t}\right) =0.$$ (56)

C.2  The Log-Linear System

The main feature of the steady state is that $B_{1}=B_{2}=0$ and $Q=\beta$. To conserve space, I present an already simplified system of log-linear equilibrium conditions. Most of the equations differ from the ones in Appendix $B$ only because of the less general assumptions on the utility function in this section.

Consumption and non-financial wealth  Equations (46), (49), (51), (52), and (55) imply

$$\hat{c}_{1,t} =\frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}} {\Phi_{1}\left( \bar{q}\right) }\hat{q}_{t}+\frac{\zeta}{c_{1}}\left( b_{1,t-1}-\beta b_{1,t}\right) ,$$ (57)

$$\hat{c}_{2,t} =\frac{\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q}} {\Phi_{2}\left( \bar{q}\right) }\hat{q}_{t}-\frac{\zeta}{c_{2}}\frac{a} {1-a}\frac{\Phi_{2}\left( \bar{q}\right) }{\Phi_{1}\left( \bar{q}\right) \bar{q}}\left( b_{1,t-1}-\beta b_{1,t}\right) ,$$ (58)

$$\hat{h}_{1,t} =\frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}} {\Phi_{1}\left( \bar{q}\right) }\hat{q}_{t}+\frac{\beta}{1-\beta}\hat {c}_{1,t}-\frac{\beta\left( 1+n\right) }{1-\beta}\hat{c}_{1,t+1}+\frac{\beta n}{1-\beta}n\hat{h}_{1,t+1},$$ (59)

$$\hat{h}_{2,t} =\frac{\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q}} {\Phi_{2}\left( \bar{q}\right) }\hat{q}_{t}+\frac{\beta}{1-\beta}\hat {c}_{2,t}-\frac{\beta\left( 1+n\right) }{1-\beta}\hat{c}_{2,t+1}+\frac{\beta n}{1-\beta}\hat{h}_{2,t+1}.$$ (60)

Risk sharing condition  Using equation (48) in (54),

$$\hat{c}_{1,t}-\frac{c_{1}}{c_{1}-\frac{n}{1+n}c_{1}^{+1}}\hat{c} _{1,t+1}+\frac{\frac{n}{1+n}c_{1}^{+1}}{c_{1}-\frac{n}{1+n}c_{1}^{+1}}\hat {h}_{1,t+1}$$

$$=\hat{c}_{2,t}-\frac{c_{2}}{c_{2}-\frac{n}{1+n}c_{2}^{+1}}\hat{c} _{2,t+1}+\frac{\frac{n}{1+n}c_{2}^{+1}}{c_{2}-\frac{n}{1+n}c_{2}^{+1}}\hat {h}_{2,t+1}$$

$$-\left[ 1+\frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi _{1}\left( \bar{q}\right) }-\frac{\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{2}\left( \bar{q}\right) }\right] \left( \hat{q}_{t+1} -\hat{q}_{t}\right) .$$ (61)

Excess demand function  Equation (53) implies

$$\frac{\partial z_{2}\bar{q}}{\partial\bar{q}}\hat{q}_{t}+\frac{\partial z_{2} }{\partial dW_{1}}\left( \bar{b}_{1,t-1}-\beta\bar{b}_{1,t}\right) =0,$$ (62)

where

$$\frac{\partial z_{2}\bar{q}}{\partial\bar{q}} =ac_{12}\left[ \left[ \frac{1}{1-\rho}\frac{\alpha_{11}}{\alpha_{12}}\left( \frac{1}{\bar{q} }\right) ^{\frac{\rho}{\rho-1}}+1\right] \frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }+\frac{\left( 1-\frac{1}{1-\rho}\right) }{\frac{\alpha_{21}}{\alpha_{22}}\left( \frac {1}{\bar{q}}\right) ^{\frac{\rho}{\rho-1}}}\frac{\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{2}\left( \bar{q}\right) }\right] ,$$

$$\frac{\partial z_{2}}{\partial dW_{1}} =-\frac{a}{\bar{q}\Phi_{1}\left( \bar{q}\right) }\left[ 1+\zeta\frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }-\zeta\frac{\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{2}\left( \bar{q}\right) }\right] ,$$

with

$$\frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) } =-\frac{\bar{q}}{\frac{\alpha_{11}}{\alpha_{12}}\left( \frac{1}{\bar{q}}\right) ^{\frac{1}{\rho-1}}+\bar{q}},$$

$$\frac{\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{2}\left( \bar{q}\right) } =\frac{\frac{\alpha_{21}}{\alpha_{22}}\left( \frac {1}{\bar{q}}\right) ^{\frac{\rho}{\rho-1}}}{\frac{\alpha_{21}}{\alpha_{22} }\left( \frac{1}{\bar{q}}\right) ^{\frac{\rho}{\rho-1}}+1},$$

and

$$0<-\frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }+\frac{\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q}} {\Phi_{2}\left( \bar{q}\right) }<1$$

if $\alpha_{12}\alpha_{21}<\alpha_{11}\alpha_{22}$. As $0<\zeta<1$, $\frac{\partial z_{2}}{\partial dW_{1}}<0$ irrespective of the value of $\bar{q}$.

Reduced linear system  Using equations (57) to (62) the equilibrium system can be summaized even more compactly as

$$\left( \begin{array}[c]{c} \hat{q}_{t+1}\ \hat{h}_{1,t+1}\ \hat{h}_{2,t+1} \end{array}\right) =\left( \begin{array}[c]{ccc} \frac{1}{\beta}-\frac{1-\beta}{u\beta} & -\frac{1-\beta}{u\beta} & \frac{1-\beta}{u\beta}\ e_{1}\left( \frac{u}{1-\beta}-1\right) +f_{1} & -e_{1}+\frac{1}{n} \frac{1-\beta}{\beta} & e_{1}\ -e_{2}\left( \frac{u}{1-\beta}-1\right) -f_{2} & e_{2} & -e_{2}+\frac{1} {n}\frac{1-\beta}{\beta} \end{array}\right) \left( \begin{array}[c]{c} \hat{q}_{t}\ \hat{h}_{1,t}\ \hat{h}_{2,t} \end{array}\right)$$

with

$$u =1+\frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi _{1}\left( \bar{q}\right) }-\frac{\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{2}\left( \bar{q}\right) },$$

$$e_{1} =\frac{1+n}{n}\frac{1-\beta}{\beta}\frac{1}{u}\left[ \frac{\Phi _{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }-\frac{\zeta}{c_{1}}\frac{\frac{\partial z_{2}\bar{q}}{\partial\bar{q}} }{\frac{\partial z_{2}}{\partial dW_{1}}}\right] ,$$

$$e_{2} =\frac{1+n}{n}\frac{1-\beta}{\beta}\frac{1}{u}\left[ -\frac{\Phi _{2}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{2}\left( \bar{q}\right) }-\frac{\zeta}{c_{2}}\frac{\frac{a}{1-a}\Phi_{2}\left( \bar{q}\right) } {\Phi_{1}\left( \bar{q}\right) \bar{q}}\left( \frac{\frac{\partial z_{2}\bar{q}}{\partial\bar{q}}}{\frac{\partial z_{2}}{\partial dW_{1}} }\right) \right] ,$$

$$f_{1} =-\frac{1}{\beta n}\left[ \frac{\Phi_{1}^{\prime}\left( \bar {q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }-\beta\frac{\zeta} {c_{1}}\frac{\frac{\partial z_{2}\bar{q}}{\partial\bar{q}}}{\frac{\partial z_{2}}{\partial dW_{1}}}\right] ,$$

$$f_{2} =-\frac{1}{\beta n}\left[ -\frac{\Phi_{2}^{\prime}\left( \bar {q}\right) \bar{q}}{\Phi_{2}\left( \bar{q}\right) }-\beta\frac{\zeta} {c_{2}}\frac{\frac{a}{1-a}\Phi_{2}\left( \bar{q}\right) }{\Phi_{1}\left( \bar{q}\right) \bar{q}}\frac{\frac{\partial z_{2}\bar{q}}{\partial\bar{q}} }{\frac{\partial z_{2}}{\partial dW_{1}}}\right] .$$

With some algebra effort, it can be shown that in any steady state

$$\frac{\zeta}{c_{1}}+\frac{\zeta}{c_{2}}\frac{\frac{a}{1-a}\Phi_{2}\left( \bar{q}\right) }{\Phi_{1}\left( \bar{q}\right) \bar{q}}=-\left[ \zeta \frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q}}{\Phi_{1}\left( \bar{q}\right) }-\zeta\frac{\Phi_{2}^{\prime}\left( \bar{q}\right) \bar{q} }{\Phi_{2}\left( \bar{q}\right) }\right] \frac{1}{\Phi_{1}\left( \bar {q}\right) \bar{q}c_{12}},$$

and

$$\left( \frac{\zeta}{c_{1}}+\frac{\zeta}{c_{2}}\frac{\frac{a}{1-a}\Phi _{2}\left( \bar{q}\right) }{\Phi_{1}\left( \bar{q}\right) \bar{q}}\right) \frac{\frac{\partial z_{2}\bar{q}}{\partial\bar{q}}}{\frac{\partial z_{2} }{\partial dW_{1}}}>-1.$$

C.3  Theorem 5

Let $M_{O}$ denote the coefficient matrix of the reduced linear system. The eigenvalues of $M_{O}$ are found as the solution to the characteristic equations

$$=-\lambda^{3}$$ 0

$$+\left\{ \frac{2}{n}\frac{1-\beta}{\beta}+\left( \frac{1}{\beta} -\frac{1-\beta}{u\beta}\right) -\left( e_{1}+e_{2}\right) \right\} \lambda^{2}$$

$$+\frac{1-\beta}{\beta}\left\{ \left[ \frac{1}{n}\left( e_{1} +e_{2}\right) -\frac{1}{u}\left( f_{1}+f_{2}\right) \right] -\frac {1-\beta}{\beta}\left( \frac{1}{n}\right) ^{2}-\frac{2}{nu}\frac{1-\beta }{\beta}\left( \frac{u}{1-\beta}-1\right) \right\} \lambda$$

$$+\left( \frac{1}{n}\frac{1-\beta}{\beta}\right) ^{2}\left[ \frac{1} {\beta}-\frac{1-\beta}{u\beta}\right] +\frac{1}{nu}\left( \frac{1-\beta }{\beta}\right) ^{2}\left( f_{1}+f_{2}\right) .$$ (63)

It is easy to show that $\lambda=\frac{1-\beta}{\beta}\frac{1}{n}$ is one solution to the characteristic equation. Thus, right hand side of $\left( \ref{OV char 1}\right)$ can be written as the product of $\left( \lambda-\frac{1-\beta}{\beta}\frac{1}{n}\right)$ and

$$\left( \lambda^{2}-\left\{ \frac{1-\beta}{n\beta}+\left( \frac{1}{\beta }-\frac{1-\beta}{u\beta}\right) -\left( e_{1}+e_{2}\right) \right\} \lambda+\left\{ \frac{1-\beta}{n\beta}\left[ \frac{1}{\beta}-\frac{1-\beta }{u\beta}\right] +\frac{1-\beta}{u\beta}\left( f_{1}+f_{2}\right) \right\} \right) .$$

Determinacy and trace of the matrix $\tilde{M}_{O}$ that goes along with the second term are

$$\left( \tilde{M}_{O}\right) =\frac{1-\beta}{nu\beta}\left[ 1+\left( \frac{\zeta}{c_{1}}+\frac{\zeta}{c_{2}}\frac{\frac{a}{1-a}\Phi _{2}\left( \bar{q}\right) }{\Phi_{1}\left( \bar{q}\right) \bar{q}}\right) \frac{\frac{\partial z_{2}\bar{q}}{\partial\bar{q}}}{\frac{\partial z_{2} }{\partial dW_{1}}}\right] ,$$

$$\left( \tilde{M}_{O}\right) =\frac{1-\beta}{u\beta}\left( \frac{\zeta}{c_{1}}+\frac{\zeta}{c_{2}}\frac{\frac{a}{1-a}\Phi_{2}\left( \bar{q}\right) }{\Phi_{1}\left( \bar{q}\right) \bar{q}}\right) \frac {\frac{\partial z_{2}\bar{q}}{\partial\bar{q}}}{\frac{\partial z_{2}}{\partial dW_{1}}}+1+ \left( \tilde{M}_{O}\right) .$$

As $\left( \frac{\zeta}{c_{1}}+\frac{\zeta}{c_{2}}\frac{\frac{a}{1-a}\Phi _{2}\left( \bar{q}\right) }{\Phi_{1}\left( \bar{q}\right) \bar{q}}\right) \frac{\frac{\partial z_{2}\bar{q}}{\partial\bar{q}}}{\frac{\partial z_{2} }{\partial dW_{1}}}>-1$ in any steady state det $\left( \tilde{M}_{O}\right) >0$.

If the slope of the excess demand function is negative in the steady state, i.e., $\frac{\frac{\partial z_{2}\bar{q}}{\partial\bar{q}} }{\frac{\partial z_{2}}{\partial dW_{1}}}>0$ , tr $\left( \tilde{M}_{O}\right) >0$ and tr$\left( \tilde{M}_{O}\right) >1+$det$\left( \tilde{M}_{O}\right)$. If the excess demand function is downward sloping in the steady state, exactly one of the remaining eigenvalues is larger than one in absolute value.

If the excess demand function is upward sloping in the steady state, i.e., $\frac{\frac{\partial z_{2}\bar{q}}{\partial\bar{q}}} {\frac{\partial z_{2}}{\partial dW_{1}}}<0$ , both eigenvalues are outside the unit circle unless the following holds (implying that $x=\left( \frac{\zeta }{c_{1}}+\frac{\zeta}{c_{2}}\frac{\frac{a}{1-a}\Phi_{2}\left( \bar{q}\right) }{\Phi_{1}\left( \bar{q}\right) \bar{q}}\right) \frac{\frac{\partial z_{2}\bar{q}}{\partial\bar{q}}}{\frac{\partial z_{2}}{\partial dW_{1}}}$ is sufficiently negative):

$$-\left\{ \frac{1-\beta}{u\beta}x+1+\frac{1-\beta}{nu\beta}\left[ 1+x\right] \right\} >1+\frac{1-\beta}{nu\beta}\left[ 1+x\right] ,$$

$$x <-2\frac{nu\beta+\left( 1-\beta\right) }{\left( 2+n\right) \left( 1-\beta\right) }.$$

As $x>-1$, a sufficient condition for both eigenvalues to be outside the unit circle is given by $1+\frac{\Phi_{1}^{\prime}\left( \bar{q}\right) \bar{q} }{\Phi_{1}\left( \bar{q}\right) }-\frac{\Phi_{2}^{\prime}\left( \bar {q}\right) \bar{q}}{\Phi_{2}\left( \bar{q}\right) }>\frac{1}{2} \frac{1-\beta}{\beta}$ . Hence, unless agents are very impatient steady states for which the excess demand function is upward sloping are unstable.

Footnotes

**  I am grateful to Roc Armenter, Fabio Ghironi, Sylvain Leduc, and Thomas Lubik for insightful comments. All remaining errors are mine. The content of this paper used to be part of a paper circulated under the name "Closing Open Economy Models". The views expressed in this paper are solely the responsibility of the author and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. Return to text

***  Telephone (202) 452 3796. E-mail Martin.R.Bodenstein@frb.gov. Return to text

1.  Obviously, this problem is not unique to international economics. The same issues occur in models with heterogeneous agents and incomplete asset markets unless the model is solved using global methods as in Hugget (1993). Return to text

2.  See also Kim and Kose (2003) for a related study in a small open economy framework. Lubik (2007) analyses some additional approaches that induce stationarity and finds substantial qualitative differences. Hence, the implicit generalization of the results in Schmitt-Grohé and Uribe (2003) by many researchers is not even appropriate for the case of a small open economy. Boileau and Normandin (2008) extend the analysis to a two-country model with one homogeneous good. Interesting quantitative differences can occur in their setup depending on the persistence of technology shocks. Return to text

3.  Ghironi (2006) is the first to point out the stationarity inducing feature of the overlapping generations framework of Weil (1989) in models of incomplete markets. Return to text

4.  Consider an endowment economy with two countries and two traded goods that are imperfect substitutes as in Kehoe (1991) and Mas-Colell et al (1995). The countries are mirror images of each other with respect to preferences and endowments. One equilibrium always features a relative price of the traded goods equal to unity. However, there can be two more equilibria if there is home bias in consumption and the price elasticity of substitution between goods is low. If the price of the domestic good is high relative to the price of the foreign good, domestic agents are 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 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. Return to text

5.  If the discount factor is increasing in the agent's utility level, the dynamics around any steady state are always explosive. Return to text

6.  In a model with a low implied price elasticity of substitution between traded goods, Corsetti, Dedola and Leduc (2008) linearize their model economy around a steady state with an upward sloping excess demand function to illustrate what they refer to as the negative transmission mechanism. Using the endogenous discount factor to close the model they find stable equilibrium dynamics. However, if the same model had been closed using convex portfolio costs, the model would have displayed explosive dynamics around this (non-unique) steady state. Return to text

7.  An aggregator that satisfies the restrictions imposed on $H_{i}$ is given by the generalization of the CES aggregator as suggested by Dotsey and King (2005):

$$\frac{\alpha_{i1}}{\left( 1+\eta\right) \rho}\left[ \frac{\left( 1+\eta\right) }{\alpha_{i1}}\left( \frac{c_{i1}}{c_{i}}\right) -\eta\right] ^{\rho}+\frac{\alpha_{i2}}{\left( 1+\eta\right) \rho}\left[ \frac{\left( 1+\eta\right) }{\alpha_{i2}}\left( \frac{c_{i2}}{c_{i} }\right) -\eta\right] ^{\rho}=\frac{1}{\left( 1+\eta\right) \rho}.$$

This aggregator allows for the elasticity of substitution to be non-constant. For $\eta=0$, one obtains the standard CES aggregator. Return to text

8.  An equivalent approach to specifying home bias in consumption preferences is to introduce iceberg transportation costs. Return to text

9.  By mirroring I mean that good 1 (2) enters the utility function of agents in country 1 the same way that good 2 (1) enters the utility function of agents in country 2. The same holds true for agents' endowments with goods 1 and 2. Return to text

10.  Models with multiple static equilibria give rise to sunspot equilibria when extended to incorporate dynamics. See the aforementioned companion paper, Bodenstein (2008), for at least a partial analysis of this issue. Return to text

11.  It is important to keep in mind, that international bond holdings become non-stationary as a result of the solution technique. If the model was solved using global methods stationarity would be preserved. Return to text

12.  The roots of the characteristic equation that is associated with $M$ satisfy $P\left( \lambda\right) =\lambda^{2}-\lambda$tr$\left( M\right) +\det\left( M\right) .$ For convenience, I summarize the necessary and sufficient conditions such that none, exactly one or both eigenvalues $\lambda$ lie in the unit circle:

i.

if $\vert\det\left( M\right) \vert<1$ and $\vert$tr $\left( M\right) \vert<1+\det\left( M\right)$, the modulus of all eigenvalue is smaller than 1,

ii.

if $\vert\det\left( M\right) \vert>1$ and $\vert$tr $\left( M\right) \vert<1+\det\left( M\right)$, the modulus of all eigenvalues is larger than 1,

iii.

if $\vert\det\left( M\right) \vert<1$ and $\vert$tr $\left( M\right) \vert>1+\det\left( M\right)$ or $\vert\det\left( M\right) \vert>1$ and $\vert$tr $\left( M\right) \vert>1+\det\left( M\right)$, the modulus of one eigenvalue is larger than 1, while the other one is smaller than 1. Return to text

13.  Prominent predecessors to Ghironi's approach are Yari (1965), Blanchard (1985), and Weil (1989). Return to text

14.  Although the steady state in the model with endogenous discounting is unique, I show in Bodenstein (2008) that there can be multiple equilibria away from the steady state. Return to text

15.  The Frisch (or constant marginal utility of wealth) labor supply elasticity is defined as

$$\eta=\frac{dl}{dw}\frac{w}{l}\vert _{\lambda}=\frac{U_{l}}{lU_{ll}-\frac {lU_{lc}^{2}}{U_{cc}}}.$$

 Return to text

This version is optimized for use by screen readers. Descriptions for all mathematical expressions are provided in LaTex format. A printable pdf version is available. Return to text

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