Closing Large Open Economy Models

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 867R April 2008 Closing large open economy models Martin Bodenstein NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/.

∗ Closing large open economy models Martin Bodenstein∗∗ Federal Reserve Board First version: August 2006 This version: March 2011 Abstract A large class of international business cycle models admits multiple locally isolated deterministic steady states, if the elasticity of substitution between traded goods is sufficiently low. I explore the conditions under which such multiplicity occurs and characterize the dynamic properties in the neighborhood of each steady state. Models with standard incomplete markets, portfolio costs, a debt-elastic interest rate, or an overlapping generations framework allow for multiple steady states, if the model features multiple steady states under financial autarchy. If the excess demand for the foreign traded good is increasing in the good’s own price in a given steady state, the equilibrium dynamicsaroundthissteadystateareunbounded. Otherwise,thedynamicsareboundedandunique. By contrast, with Uzawa-type preferences, the steady state is always unique and the associated equilibrium dynamics are always bounded and unique. The same results obtain under complete markets. Keywords: stationarity, incomplete markets, open economy, multiple equilibria JEL Classification: D51 F41 ∗ I have benefited from conversationswith Roc Armenter, Fabio Ghironi, Sylvain Leduc, and Thomas Lubik. I am particularly grateful to two anonymous referees and Giancarlo Corsetti (the editor) whose suggestions have substantially improved the paper. All remaining errors are mine. 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 any other person associated with the Federal Reserve System. ∗∗ Contact information: Martin Bodenstein: phone (202) 452 3796, Martin.R.Bodenstein@frb.gov.

1 Introduction The class of international business cycle models nested in the framework of Corsetti, Dedola, and Leduc (2008) admits multiple steady states with zero net foreign asset holdings, if the elasticity of substitution between tradedgoodsis sufficiently low. This paper explores theconditions under which such multiplicity occurs andcharacterizes the models’ dynamic properties in the neighborhood of each steady state. Equilibrium multiplicity is a pervasive feature of models with heterogenous agents. To build intuition consider the case of a static two-country endowment economy with two traded goods that are imperfect substitutes as in Kehoe (1991) and Mas-Colell, Whinston, and Green (1995). For simplicity, let the countries be 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. With home bias in consumption and a low elasticity of substitution between the traded goods, two more equilibria occur. If the price of the domestic good is high relative to the price of the foreign good, domestic agents are wealthy compared to foreign agents. Under a low elasticity of substitution, 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 both goods. The reverse is true as well. Foreign agents consume most of the goods, if the foreign good is expensive in relative terms. This intuition carries over to richer models of the international business cycle that feature international borrowing and lending, endogenous production, intertemporal savings and investment decisions, or non-traded goods. For these models to feature multiple steady states with zero net foreign assets under an otherwise standard calibration the elasticity of substitution between traded goods has to lie below 0.5.1 Whereas in most studies the elasticity of substitution between traded goods and the trade (price) elasticity coincide, the two concepts differ in the model of Corsetti, Dedola, and Leduc (2008) due to the presence of non-traded distribution services. If the model is parameterized as in Corsetti, Dedola, and Leduc (2008), 1Prominent examples of such models are found in Backus, Kehoe, and Kydland (1995), Baxter and Crucini (1995), Heathcote and Perri (2002), and Stockman and Tesar (1995). However, none of these papers explores the possibility of multiple steady states, as the elasticity of substitution is generally assumed to be above unity. 3

multiple steady states occur for an elasticity of substitution between traded goods around unity, although the implied trade elasticity lies in the neighborhood of 0.5. For a symmetric parameterization of the model I generally find three steady states. However, the model by Corsetti, Dedola, and Leduc (2008) can be shown to have at least five steady states for some parameterizations.2 The multiplicity of the steady state price vector occurs whether international financial markets are absent from the model or one focuses on an incomplete financial markets framework with zero net foreign assets in steady state.3 This problem is unrelated to the issues about incomplete markets models addressed in Schmitt-Groh´e and Uribe (2003) and many others. In standard incomplete markets model with one non-state-contingent bond the deterministic steady state of the net foreign asset position is not determined and the dynamics of the net foreign asset position as derived from a linear approximation of the model around a deterministic steady state are not stationary.4 Absent arbitrage opportunities, the price of the non-state-contingent bond implies 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. In particular, any net foreign asset position is compatible with a steady state. To determine the steady state position of net foreign assets and to remove the nonstationarity of its dynamics, I modify the original model similar to Schmitt-Groh´e and Uribe (2003) by allowing for portfolio costs, a debt-elastic interest rate, or an endogenous discount factor. This list is augmented by the overlapping generations structure of Weil (1989) as implemented in Ghironi (2006) and Ghironi (2008). I show that the choice of a stationar- 2In the working paper version of Corsetti and Dedola (2005), the authors point out that for the case of an endowment economy a model with distribution costs admits multiple equilibria in the absence of international borrowing and lending. However, no systematic exploration of this feature is conducted. 3Multiplicity of the steady state price vector can also occur if the net foreign asset position differs from zero in steady state. However, the assumption of zero net foreign assets in steady state is widely made in the literature and implies that the steady states of the incomplete markets model coincide with those obtained in a model without financial markets. 4While throughout the analysis the only asset that trades internationally is one non-state-contingent bond, the issues raised carry over to environments with more assets such as those presented in Devereux and Sutherland (2008) as long as the available assets do not complete the market. 4

ity inducing device is not innocuous as it affects both the number of steady states in a low elasticity environment and the dynamics of the model around a steady state. With portfolio costs, agents face a non-zero cost for bond holdings that differs from a reference level for international bond holdings specified exogenously by the researcher. Furthermore, the steady state is unique and stable only if the model with financial autarchy (or equivalently with incomplete markets and a zero net foreign asset position) has a unique steady state. If the original model has N steady states, the model with portfolio costs has N steady states. Those steady states for which the excess demand of the foreign good is decreasing in its relative price are associated with unique and locally bounded equilibrium dynamics. If the excess demand function is increasing in its relative price in a given steady state, the local equilibrium dynamics are not bounded. Interestingly, if multiple steady states occur under a symmetric calibration, it is typically the symmetric steady state that is associated with unbounded dynamics. Similar results are obtained 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. With strictly concave preferences and technologies the relative price of traded goods is uniquely pinned down under endogenous discounting given the function of the discount factor. The net foreign asset position is merely a residual. The equilibrium dynamics in the neighborhood of the unique steady state arealways unique andlocally bounded irrespective of the number of steady states in the original model with incomplete markets.5 Schmitt-Groh´e and Uribe (2003) also analyze the case of complete markets. In this case, the net foreign asset position is a residual that does not enter the equilibrium dynamics as a state variable. The steady state of such a model is always unique and the associated equilibrium dynamics are unique and bounded. Both Schmitt-Groh´e and Uribe (2003) and Kim and Kose (2003) find that for the case of a small open economy the various approaches imply virtually identical dynamics. However, 5If the discount factorwas increasingin the agent’sutility level, the dynamics aroundanysteady state wouldalwaysbe unbounded. 5

generalizing this finding to richer models as made by many researchers, may not be appropriate. Boileau and Normandin (2008) extend the analysis in Schmitt-Groh´e and Uribe (2003) to a two-country model with one homogeneous good. Quantitative differences can occur in their setup depending on the persistence of technology shocks. This paper exclusively analyzes the local dynamics around a given steady state. However, in the presence of multiple steady states, global solution techniques may find richer dynamics than local solution techniques. Bodenstein (2010) presents an analysis of the global dynamics in the model of Backus, Kehoe, and Kydland (1995) under endogenous discounting when the elasticity of substitution between traded goods is sufficiently low. Remains to address the empirical relevance of models with low trade and substitution elasticities. For aggregate data Whalley (1984) reports a trade elasticity of 1.5. Hooper, Johnson, and Marquez (1998) report a short-run trade elasticity of 0.6 for the U.S. and values between 0 and 0.6 for the remaining G7 countries, while Taylor (1993) finds a short-run trade elasticity of 0.22. Using lower levels of aggregation, Broda and Weinstein (2006) report mean estimates for the elasticity of substitution for various pairs of traded goods between 4 and 6. Applied macroeconomic studies, have commonly parameterized the substitution elasticity between traded goods at a value between 1 and 1.5 (see e.g. Backus, Kehoe, and Kydland (1995), Chari, Kehoe, and McGrattan (2002), and Heathcote and Perri (2002)). However, in line with the macroeconometric evidence, various authors have recently argued in favor of low values of the trade elasticity which coincides with the elasticity of substitution between traded goods for these studies. Heathcote and Perri (2002), Benigno and Thoenissen (2008) and Collard and Dellas (2007) show improved model performance with respect to key features of the international business cycle when allowing for substitution elasticities below 0.5. Burstein, Neves, and Rebelo (2003), Corsetti and Dedola (2005), and Corsetti, Dedola, and Leduc (2008) refrain from assuming such low substitution elasticities directly, but instead introduce distribution costs in terms of non-traded goods to obtain a low implied value of the trade elasticity despite allowing the elasticity of substitution between traded goods to be around unity. The model in Corsetti, Dedola, and Leduc (2008) successfully addresses two 6

important puzzles in international economics: the high volatility of the real exchange rate relative to fundamentals and the observed negative correlation between the real exchange rate and relative consumption (Backus and Smith (1993)). As Kollmann (2006) shows, a low elasticity of substitution between traded goods may also be responsible for the apparent home bias in equity holdings. Rabanal and Tuesta (2010) estimate a DSGE model with sticky prices using a Bayesian approach. Their median estimates for the elasticity of substitution range from 0.01 to 0.91for different specifications of their model. Lubik and Schorfheide (2006) estimate the elasticity of substitution to be around 0.4. The remainder of the paper is structured as follows. Section 2 introduces the issues considered in this paper in a simple model. Section 3 lays out the general model, which is parameterized in Section 4. Steady state multiplicity is discussed in Section 5, while the local dynamics of the model are discussed in Section 6. Concluding remarks are offered in Section 7. The paper is accompanied by a separate Technical Appendix. 2 Simple example Consideratwo-country, two-goodendowment economywithincompleteinternationalfinancial markets. The two countries are each inhabited by a continuum of identical agents of measure 1. This model extends the introductory example in Corsetti, Dedola, and Leduc (2008) to a dynamic environment. I first illustrate under what restrictions on the value of the elasticity of substitution between traded goods the model admits multiple deterministic steady states. Second, I relate steady state multiplicity to the question when the local dynamics around a given steady state are locally bounded or unbounded. Each period t + j, agents in country i obtain yT units of the traded good i. Agents i,t+j consume both the home and the foreign goodand maximize their expected discounted lifetime utility. The only asset that trades internationally is one non-state-contingent bond. More 7

formally, a representative agent faces: (cid:3) (cid:7) (cid:4)∞ (cid:5) (cid:6) max E(cid:2) βjln cT (1) ci1,t+j,ci2,t+j, t i,t+j bi,t+j j=0 s.t. P c +P c ≤ P yT +b −Q b . (2) 1,t+j i1,t+j 2,t+j i2,t+j i,t+j i,t+j i,t−1+j i,t+j i,t+j All variables are expressed in per capita terms. c is the consumption of good m by agents im of country i. P is the price of good i. b denotes holdings of a non-state-contingent bond with i i price Q . To rule out Ponzi-schemes, I assume that agents face an upper bound for borrowing i ˜b that is large enough to never bind in this application. The aggregate consumption good cT i i satisfies: (cid:8) (cid:5) (cid:6) (cid:5) (cid:6) (cid:9) εT 1 εT−1 1 εT−1 εT−1 cT i,t+j = αT i1 εT (c i1,t+j ) εT + αT i2 εT (c i2,t+j ) εT , (3) with 0 < αT < 1 for i = 1,2 and m = 1,2. The elasticity of substitution between the two im traded goods is denoted by εT. Market clearing requires: c +c = yT , (4) 11,t+j 21,t+j 1,t+j c +c = yT , (5) 12,t+j 22,t+j 2,t+j b +b = 0. (6) 1,t+j 2,t+j The price of the aggregate consumption good in country 1 is taken to be the num´eraire. 2.1 Steady state multiplicity Under the assumption that the net foreign asset position is zero in steady state (b = 0), i this model can display multiple distinct steady states if the elasticity of substitution between the two traded goods is sufficiently low. The relative price q = P2 constitutes a steady state P1 equilibrium, if the excess demand function for good 2 satisfies: z (q) ≡ c (q)+c (q)−yT (q) 2 12 22 2 z (q) ≤ 0, ∞ ≥ q ≥ 0 and qz (q) = 0. (7) 2 2 8

Applying standard fixed-point theorems, it can be shown that a steady state exists for this model. Furthermore, by virtue of the index theorem, see Kehoe (1991) and Mas-Colell, Whinston, and Green (1995), the number of steady state equilibria needs to be odd. Furthermore, if the slope of the excess demand function is positive for a given steady state value of q with zero net foreign assets, there must be at least two more values of q for which z (q) = 0. 2 Figure 1: Excess demand function and steady state multiplicity 0.02 0.01 0 −0.01 −0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z 2 doog dnamed ssecxe 2 elasticity of substitution εT = 0.60 normalized relative price good 2 q/(1+q) 0.02 0.01 0 −0.01 −0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z 2 doog dnamed ssecxe 2 elasticity of substitution εT = 0.42 normalized relative price good 2 q/(1+q) Notes: The figure plots the excess demand function for good 2 as a function of the normalized relative price q/(1+q) with αT 11 = αT 22 = 0.9, αT im = 1−αT ii, and y1 T = y2 T = 1. The solid linedepicts theexcess demandfunction. Thedotted vertical linesindicatethezeros oftheexcess demand function. For εT = 0.6, the unique zero features q/(1+q) = 0.500. For εT = 0.42, the threezerosoccuratq/(1+q)equal to0.043,0.500,and0.957,respectively. To advance the analysis, I set αT = αT , αT = 1−αT , for i = 1,2 and m = 1,2 and fix ii mm ii im yT = yT in steady state. Under these assumptions, one steady state always features q = 1, 1 2 i.e., q = 0.5, irrespective of the value of εT. Figure 1 plots the excess demand for good 1+q 2 as a function the normalized price q for a high and a low value of εT. The remaining 1+q 9

parameter values are αT = αT = 0.9, αT = 1−αT, and yT = yT = 1. In the case of εT = 0.6, 11 22 im ii 1 2 the excess demand function equals zero only for q = 1. Furthermore, the slope of the excess demand function is negative at q = 1. For the lower value of εT of 0.42, the slope of the excess demand function is positive at the steady state equilibrium with q = 1. Consequently, there are two more price equilibria: one for q = 0.043 and the other one for q = 0.957. 1+q 1+q As shown inAppendix A, theslope of theexcess demand function ina deterministic steady state is given by: (cid:10) (cid:11) (cid:12) (cid:13) ∂z (cid:5) (cid:6) c (q) c (q) 2q = c (q) 1−εT 11 + 22 −1 . ∂q 12 yT yT 1 2 Balanced trade in steady state implies cii(q) = αT and the slope of the excess demand function yi T ii for good 2 is then positive at q = 1, if: 1 εT ≤ 1− . (8) 2αT 11 Thus, the model features multiple steady states, whenever condition (8) is satisfied. For the parameterization underlying Figure 1 the threshold value for the elasticity of substitution at which steady state multiplicity occurs is 0.4444. The associated change inthe sign of the slope of the excess demand function from negative to positive at a low value of the substitution elasticity liesbehindwhatCorsetti, Dedola,andLeduc(2008)term“thenegativetransmission mechanism” and allows their model to be a candidate solution to the Backus-Smith puzzle.6 2.2 Local equilibrium dynamics The incomplete markets model outlined in the previous section features two well-known problems if the decision rules are derived from a linear approximation around a deterministic steady state. First, any value of the net foreign asset position is compatible with a steady state. Second, the model is not stationary as the net foreign asset position follows a unit root process. 6Corsetti, Dedola, and Leduc (2008) argue that a positive supply shock to the home country can cause an appreciation of the home country’s terms of trade when the excess demand function is upwardsloping in the steady state aroundwhich the model is linearized. Absent “the negative transmission mechanism” their model can resolve the Backus-Smith puzzle only for very persistent shock processes. 10

The first order conditions with respect to bond holdings imply, that expected marginal utility growth expressed in a common good needs to be equalized across countries: ⎡ ⎤ ⎡ ⎤ (cid:16) (cid:17) (cid:16) (cid:17) cT −1 cT −1 rer E ⎣β 1,t+1+j ⎦ = E ⎣β 2,t+1+j t+j ⎦. (9) t+j cT t+j cT rer 1,t+j 2,t+j t+1+j In steady state, equation (9) reduces to an identity, leading to a system of N unknown endogenous variables, but N −1 equations. It is common practice to exogenously specify the steady state value of the net foreign asset position. For the remainder of the paper, I will assume that the net foreign position is zero in steady state. Thus, the model with incomplete markets has the same steady states as a model without internationally traded assets. However, imposing the steady state level of net foreign assets exogenously does not remove the unit root of the net foreign asset position in the approximate model. Following among others Schmitt-Groh´e and Uribe (2003), I introduce portfolio costs to render the net foreign (cid:20) (cid:21) 2 asset position stationary.7 Households incur the cost 1γ bi,t+j P for holding or issuing 2 Pi,t+j i,t+j international debt. These costs are rebated lump-sum to the households. As portfolio costs are zero whenever the net foreign asset position is zero, the steady states in the model with portfolio costs are the same as in the model described in the previous subsection. To study the dynamics around a given steady state, Appendix A shows that the unique state variable Δb satisfies the system:8 1,t−1+j ⎛ ⎞ ⎛ (cid:11) (cid:12) ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ −∂ ∂ d z W 2 1β 0 ⎟ Δb ⎜ 1+β +β ∂ ∂ z q 2qγˇ ∂ ∂ d z W 2 1 −∂ ∂ d z W 2 1 ⎟ Δb ⎝ ∂ ∂ z q 2q ⎠ ⎝ 1,t+1+j ⎠ +⎝ dz2 ∂ ∂ z q 2q ∂ ∂ z q 2q ⎠ ⎝ 1,t+j ⎠ = 0. 0 1 Δb 1,t+j −1 0 Δb 1,t−1+j (10) ∂z2q represents the slope of the excess demand function for good 2. The term dz is always ∂q 2 negative in steady state. and the sign of ∂z2 is immaterial for the subsequent analysis. The ∂dW1 7The netforeignassetpositionbecomesnon-stationaryasaresultofusinglocalsolutiontechniques. While higherorder perturbationmethods imply the sameproblemsasalinearsolutionapproach,globalsolutionmethods donot. Ifthe model was solved using global methods, stationarity would be preserved due to the presence of the borrowing constraints ˜b i. In addition, the concept of a deterministic steady state would need to be replaced by that of a stationary distribution. 8Δb i,t+j is the absolute deviation of real bond holdings b i,t+j /P i,t+j from zero. 11

portfolio costs imposed on the agents are represented by the parameter: (cid:8) (cid:9) γ 1 γˇ = 1+ > 0. (11) β2Φ (q) q 1 The term ∂ ∂ z q 2qγˇ is positive (negative), whenever ∂z2q is negative (positive) in the deterministic dz2 ∂q steady state around which the model is approximated. The following theorem shows that steady states for which the excess demand function is upward-sloping display unbounded dynamics under the portfolio cost approach. Theorem 1 Around a given steady state, the eigenvalues of the dynamic system associated with the portfolio cost model λ and λ satisfy: 1 2 1. For γ = 0, λ = 1 and λ = 1. Thus, bond holdings follow a unit-root process (restating 1 2 β the non-stationarity problem under incomplete markets). 2. For γ > 0, and ∂z2q < 0 (a downward sloping excess demand function), λ < 1 and ∂q 1 λ > 1. Thus, the dynamics around the point of approximation are locally unique and 2 β bounded. 3. For γ > 0, and ∂z2q > 0 (an upward sloping excess demand function), λ > 1 and ∂q 1 1 < λ < 1. Thus, the dynamics around the point of approximation are not bounded. 2 β Proof. The proof follows directly from the characteristic equation associated with the approximate system: (cid:16) (cid:17) 1+β ∂z2q 1 λ2 −λ + ∂q γˇ + = 0. (12) β dz β 2 If the number of eigenvalues larger than one in absolute value exceeds (falls short of) the number of state variables, the equilibrium dynamics are not bounded (indeterminate). If the number of such eigenvalues coincides with the number of state variables, the dynamics are unique and bounded. An alternative approach of rendering the net foreign asset position stationary is due to Uzawa (1968). As shown in Section 5, the steady state under endogenous discounting is always unique for a given parameterization of the model. Replacing βj in equation (1) by θ = β (cT )θ with β(cid:4) < 0 and, to ease exposition, assuming that agents do not i,t+1+j i i,t+j i,t+j i internalize the effects of their current choices on future discount factors, one obtains: 12

Theorem 2 Around the unique steady state, the eigenvalues of the dynamic system with an endogenous discount factor satisfy λ < 1 and λ = 1. Thus, the equilibrium dynamics 1 2 β around a steady state are always locally unique and bounded irrespective of the slope of the excess demand function in this steady state. Proof. The proof follows directly from the characteristic equation associated with the approximate system: (cid:16) (cid:17) (cid:16) (cid:17) (cid:28) (cid:28) 1+β dz dz 1 λ2 − + 2 λ+ 1+ 2 = 0, (13) β dz dz β 2 2 (cid:28) where the additional term dz is always positive. 2 2.3 Intuition Multiplicity of steady states In the model with portfolio costs, the steady state interest rate equals 1. Hence, no country has an incentive to borrow or lend in steady state. β All equilibria of the financial autarchy case are therefore valid steady states since they are compatible with b = 0. 1 The model of endogenous discounting, however, dictates that for a given functional choice of the discount factor β (q) = β (q) in the steady state. Uniqueness of the steady state 1 2 price q∗ follows promptly: suppose that another price vector q∗ that constitutes a steady endog state with b = b = 0 is also a steady state of the model with endogenous discounting. 1 2 Let q < q∗ , which would imply that overall consumption in country 1 exceeds overall endog consumption in country 2, i.e., cT(q∗) > cT(q∗). Thus, country 1 agents are willing to borrow 1 2 resources at an interest rate of 1 while country 2 agents only demand 1 < 1 . β1(q∗) β2(q∗) β1(q∗) Hence, country 1 finds it optimal to borrow from country 2 violating b = 0. 1 Stability of steady states The logic behind the stability of the unique steady state in themodelwithendogenousdiscountingiscloselyrelatedtotheargumentaboutitsuniqueness. Assume that 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 13

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 q is below its steady state value at time t, q must rise in t + 1 and the economy converges to its unique steady state. Consider the case of a steady state with ∂z2q > 0 in the bond economy with portfolio costs. ∂q The price of bonds consists of two pieces: the intertemporal marginal rate of substitution and the derivative of the portfolio costs. First, if q is slightly below its steady state value, overall consumption in country 1 (2) is above (below) its corresponding steady state value. Stability of the steady state requires q to rise and cT to fall over time. As a result, the intertemporal 1 marginal rate of substitution in country 1 (2) rises (falls), which leads to a divergence of bond prices. Second, when q rises, bond holdings and the derivative of the portfolio costs fall. This effect onbondpricesisnegativeinbothcountries, butitisstrongerincountry2sinceportfolio 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 such a steady (cid:20) (cid:21) state. Hence, bond prices would drift apart if a steady state with sign ∂z2q > 0 was stable. ∂q Hence, such a steady must be associated by unbounded dynamics. Using the model by Corsetti, Dedola, and Leduc (2008), the remainder of this paper explores the conditions for steady state multiplicity and the local dynamics under different stationarity inducing devices. 3 General model Each of the two countries (i = 1,2) produces two goods. The first good is traded internationally without trade frictions (traded good), while the other good is not traded (non-traded good). The home and foreign traded goods are imperfect substitutes in a household’s utility function. The traded and the non-traded goods are produced by perfectly competitive firms 14

and production requires the use of labor and capital. As an important deviation from earlier work, Corsetti, Dedola, and Leduc (2008) assume that consumption of traded goods requires non-traded distribution services as input. 3.1 Standard incomplete markets economy Thesubsequent analysisexplores theimplications ofdifferent assumptions aboutinternational financial markets in the presence of multiple steady states. In laying out the full model, I assume that international financial markets are incomplete in the sense that the only asset that trades internationally is one non-state-contingent bond. 3.1.1 Households The representative household in country i maximizes its expected discounted lifetime utility subject to its budget constraint. All variables are expressed in per household units: (cid:3) (cid:7) (cid:4)∞ max E(cid:2) βjU (c ,l ) (14) t i i,t+j i,t+j ci,t+j,li,t+j ki,t+j,xi,t+j,bi,t+j j=0 s.t. PC c +PI x ≤ P w l +P r k i,t+j i,t+j i,t+j i,t+j i,t+j i,t+j i,t+j i,t+j i,t+j i,t−1+j +P v +b −Q b , (15) i,t+j i,t+j i,t−1+j i,t+j i,t+j k ≤ (1−δ )k +x , (16) i,t+j i i,t−1+j i,t+j where c , x , l and k denote final consumption, final investment, labor, and capital, respeci i i i tively. PC is the price of the final consumption good, PI is the price of the final investment i i good, P is the price of the traded good produced by country i, P r and P w are the nominal i i i i i rental rate of capital and the nominal wage. Assuming local ownership of firms, P v are the i i profits of the traded and non-traded goods producers that are located in country i. b denotes i holdings of the non-state-contingent bond. Q is the price of this bond. To rule out Ponzii schemes, I assume that agents face an upper bound for borrowing ˜b that is large enough to i never bind in this application. 15

3.1.2 Firms Both the producers of traded and non-traded goods utilize labor and capital and act in perfectly competitive factor and product markets. Technologies are of the constant elasticity of substitution type. Since capital is owned by households and rented out to firms, the maximization problem of a traded goods producer from country i can be written as: (cid:5) (cid:6) max FT lT ,kT −w lT −r kT , (17) i i,t+j i,t+j i,t+j i,t+j i,t+j i,t+j li T ,t+j,ki T ,t+j where the production technology satisfies: (cid:29) (cid:30) (cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6) (cid:5) (cid:6) 1 FT lT ,kT = ωT 1−κT i AT lT κT i + ωT 1−κT i kT κT i κT i (18) i i,t+j i,t+j li i,t+j i,t+j ki i,t+j if κT < 1, and i (cid:16) (cid:17) (cid:16) (cid:17) (cid:5) (cid:6) AT lT ωT li kT ωT ki FT lT ,kT = i,t+j i,t+j i,t+j (19) i i,t+j i,t+j ωT ωT li ki if κT = 0, i with ωT +ωT = 1. Let yT denote the supply of the traded good. The problem of non-traded li ki i goods producers is analogous with variables and parameters carrying the superscript N rather than T. Because of the normalization choice for factor prices, the objective function of such firms is given by: PN (cid:5) (cid:6) i,t+jFN lN ,kN −w lN −r kN . (20) P i i,t+j i,t+j i,t+j i,t+j i,t+j i,t+j i,t+j 3.1.3 Trade and aggregation Households in country i consume the final consumption good c and purchase the final investi ment good x . Shipping of the traded goods is costless. i Consumption good A household in country i purchases c units of the traded good of i1 country 1, c units of the traded good of country 2, and cN units of the non-traded good. i2 i 16

The three types of goods are combined to form the final consumption good according to: (cid:31) εT i (cid:5) (cid:6) 1 εT i −1 (cid:5) (cid:6) 1 εT i −1 εT i −1 cT i,t+j = αT i1 εT i (c i1,t+j ) εT i + αT i2 εT i (c i2,t+j ) εT i , (21) (cid:31) εN i (cid:5) (cid:6) 1 (cid:5) (cid:6)εN i −1 (cid:5) (cid:6) 1 (cid:5) (cid:6)εN i −1 εN i −1 c i,t+j = αN i1 εN i cT i,t+j εN i + αN i2 εN i cN i,t+j εN i , (22) with 0 < αT < 1 and 0 < αN < 1 for i = 1,2, m = 1,2. These quasi shares in the im im aggregators are assumed to sum up to 1. εN is the substitution elasticity between the traded i goodsaggregateandthenon-tradedgoodincountryi. εT isthesubstitutionelasticitybetween i the home and foreign traded goods. As in Burstein, Neves, and Rebelo (2003) and Corsetti, Dedola, and Leduc (2008), the consumption of traded goods requires distribution services. Consuming one unit of any traded good goes along with a cost of η units of the country’s i non-traded good. Thus, minimizing the costs for c units of the final consumption good i requires solving the problem: (cid:5) (cid:6) (cid:5) (cid:6) min P +η PN c + P +η PN c +PN cN (23) ci1,t+j,ci2,t+j, 1,t+j i i,t+j i1,t+j 2,t+j i i,t+j i2,t+j i,t+j i,t+j cT i,t+j,cN i,t+j subject to (21) and (22). Investment good The final investment good is an aggregate of traded goods only. In combining the two traded goods, the household solves: min P xT +P xT (24) 1,t+j i1,t+j 2,t+j i2,t+j xi1,t+j,xi2,t+j s.t. (cid:31) εI i (cid:5) (cid:6) 1 εI i−1 (cid:5) (cid:6) 1 εI i−1 εI i −1 x i,t+j = αI i1 εI i (x i1,t+j ) εI i + αI i2 εI i (x i2,t+j ) εI i , (25) with 0 < αI < 1 for i = 1,2, m = 1,2 and αI +αI = 1. The substitution elasticity between im i1 i2 traded goods for investment is denoted by εI. i Relative prices For later reference, define the following relative prices: the relative price of country 2’s good to country 1’s good q ≡ P2,t+j, the relative price of the non-traded t+j P1,t+j good qN ≡ Pi N ,t+j, the inverse relative price of final consumption Φ ≡ Pi,t+j, the inverse i,t+j Pi,t+j i,t+j Pi C ,t+j 17

relative price of final investment Π ≡ Pi,t+j. Thus, the consumption real exchange rate is i,t+j Pi I ,t+j given by rer = Φ2,t+jq . In solving the model, aggregate consumption in country 1 is set t+j Φ1,t+j t+j to be the num´eraire, i.e., PC = 1. 1,t+j 3.1.4 Market clearing and equilibrium Goods market clearing requires: (cid:4)2 ν yT = (c +x ) l, (26) i,t+j li,t+j li,t+j ν i l=1 yN = cN +η (c +c ), (27) i,t+j i,t+j i i1,t+j i2,t+j for i = 1,2 at all t+j. With ν = 1, differences in relative population size are captured by 1 ν . Factor market clearing requires: 2 lT +lN = l , (28) i,t+j i,t+j i,t+j kT +kN = k , (29) i,t+j i,t+j i,t−1+j for i = 1,2 at all j. The bond market clears if: (cid:4)2 b ν = 0. (30) i,t+j i i=1 A competitive equilibrium is consequently defined as: Definition 1 A competitive equilibrium is a collection of allocations c , c , cT , i1,t+j i2,t+j i,t+j cN , c , x , x , x , k , l , yT , kT , lT , yN , kN , lN , b , i,t+j i,t+j i1,t+j i2,t+j i,t+j i,t−1+j i,t+j i,t+j i,t+j i,t+j i,t+j i,t+j i,t+j i,t+j prices q , qN , rer , w , r , Q , Φ , Π and profits v , for i = 1,2 t+j i,t+j t+j i,t+j i,t+j i,t+j i,t+j i,t+j i,t+j and all j, such that (i) for every household the allocations solve a household’s maximization problem for given prices, (ii) for every firm profits are maximized, and (iii) the markets for labor, capital, goods, and bonds clear. 3.1.5 Trade elasticity Absent distribution services (η = 0 for i = 1,2), the elasticity of substitution between traded i goods coincides with the trade (price) elasticity. However, this is not the case for η > 0; the i price of the home good relative to the foreign good at the consumer level no longer matches 18

the terms of trade, i.e., the relative price between traded goods. The first order condition of the consumer problem implies: (cid:16) (cid:17) αT 1+η qN εT 1 c = 12 1 1,t+j c , (31) 12,t+j αT q +η qN 11,t+j 11 t+j 1 1,t+j or after linearizing around a deterministic steady state: (cid:8) (cid:9) εT 1 1 cˆ −cˆ = 1 qˆ − − εTη qNqˆN , (32) 11,t+j 12,t+j 1+η q1 N t+j 1+η 1 q 1 N q +η 1 q 1 N 1 1 1 1,t+j 1 q where xˆ denotes the log-linear deviation of variable x from its steady state value. Corsetti, Dedola, and Leduc (2008) refer to εT 1 as the trade elasticity. In addition to qN 1+η1 1 q the elasticity of substitution between traded goods εT, the trade elasticity depends on the 1 endogenous steady state values of both the relative price of traded goods q and the relative price of non-traded goods qN. 1 3.2 Assumptions on international financial markets I investigate the following deviations fromthe standard incomplete markets model that render the net foreign asset position stationary and determine its steady state: 1. Agents face portfolio costs for holding/issuing bonds as in Heathcote and Perri (2002) and Schmitt-Groh´e and Uribe (2003). The budget constraint (15) changes to: PC c +PI x ≤ P w l +P r k i,t+j i,t+j i,t+j i,t+j i,t+j i,t+j i,t+j i,t+j i,t+j i,t−1+j +P v +b −Q b −P Γ(b /P )+T . (33) i,t+j i,t+j i,t−1+j i,t+j i,t+j i,t+j i,t+j i,t+j i,t+j The portfolio cost function Γ satisfies Γ(0) = 0, Γ(cid:4)(0) = 0, and Γ(cid:4) > 0 for b > 0 and i Γ(cid:4) < 0 for b < 0, and Γ(cid:4)(cid:4)(0) > 0. T is a lump-sum transfer in the amount of the i i,t+j collected fees. Consequently, condition (9) changes to: (cid:8) (cid:9) (cid:8) (cid:9) U (c ,l ) U (c ,l ) rer E β c 1,t+1+j 1,t+1+j = E β c 2,t+1+j 2,t+1+j t+j t+j U (c ,l ) t+j U (c ,l ) rer (cid:11) c(cid:12) 1,t+j (cid:11)1,t+j (cid:12) c 2,t+j 2,t+j t+1+j b b +Γ (cid:4) 1,t+j −Γ (cid:4) 2,t+j . (34) P P 1,t+j 2,t+j 19

2. Theinterest rateisdebt-elasticasinBoileauandNormandin(2008),DevereuxandSmith (2007), and Schmitt-Groh´e and Uribe (2003). The interest rate differential between the two countries is given by: (cid:11) (cid:12) b R = R Ψ 1,t+1+j −¯b , (35) 1,t+j 2,t+j P 1 1,t+j with R = 1/Q . The function Ψ satisfies Ψ(0) = 1 and Ψ(cid:4) < 0. ¯b is a reference i,t+j i,t+j 1 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. Condition (9) changes to: (cid:8) (cid:9) (cid:5) (cid:6) U (c ,l ) Ψ b −¯b E β c 1,t+1+j 1,t+1+j 1,t+1+j 1 t+j U (c ,l ) (cid:8) c 1,t+j 1,t(cid:9)+j U (c ,l ) rer = E β c 2,t+1+j 2,t+1+j t+j . (36) t+j U (c ,l ) rer c 2,t+j 2,t+j t+1+j 3. To break Ricardian equivalence, the representative household assumption is replaced by an overlapping generations framework based on Weil (1989) and Ghironi (2006). Each period, the population in country i grows at the constant growth rate n. As newborn households do not hold assets, the net foreign asset position is zero in the deterministic steady state. More details are provided in Appendix B. 4. The discount factorisendogenousasinUzawa (1968),Mendoza (1991),Corsetti, Dedola, and Leduc (2008), and Schmitt-Groh´e and Uribe (2003). The problem of a household is given by: (cid:3) (cid:7) (cid:4)∞ (cid:2) max E θ U (c ,l ) (37) t i,t+j i i,t+j i,t+j ci,t+j,li,t+j, ki,t+j,xi,t+j,bi,t+j j=0 s.t. θ = β [U (c ,l )]θ , (38) i,t+1+j i i,t+j i,t+j i,t+j and equations (15) and (16). The discount factor is assumed to be decreasing with the level of utility, i.e., β(cid:4) (U ) < 0. As in Schmitt-Groh´e and Uribe (2003), I lay out the case i i when agents do not internalize the effects of time t+j choices on the discount factor in 20

future periods. Condition (9) changes to: (cid:8) (cid:9) U (c ,l ) E β [U (c ,l )] c 1,t+1+j 1,t+1+j t+j 1 1,t+j 1,t+j U (c ,l ) (cid:8) c 1,t+j 1,t+j (cid:9) U (c ,l ) rer = E β [U (c ,l )] c 2,t+1+j 2,t+1+j t+j . (39) t+j 2 2,t+j 2,t+j U (c ,l ) rer c 2,t+j 2,t+j t+1+j The subsequent analysis also considers the case when agents internalize the effects of their choices on the discount factor. 5. For completeness, I also investigate the case of complete international financial markets, i.e., agents have access to a full set of state-contingent claims. Efficient risk sharing implies that the marginal utilities expressed in a common good are equalized across countries subject to a constant μ : t U (c ,l ) c 2,t+j 2,t+j = rer μ , (40) U (c ,l ) t+j t c 1,t+j 1,t+j where μ is determined by the allocations and prices at time t: t rer U (c ,l ) μ = t c 2,t 2,t . (41) t U (c ,l ) c 1,t 1,t In the first three approaches, the wedge between expected marginal utility growth in the two countries is eliminated in a deterministic steady state only for b = 0. Under endogenous 1 discounting, condition(39)requiresthatβ [U ] = β [U ]insteadystate. Withconcaveutility 1 1 2 2 and technology functions and β being strictly increasing in U , the allocations and prices are i i directly pinned down by this condition. The net foreign asset position is then determined from the goods market clearing condition in order to be compatible with these steady state allocations and prices. As, in principle, the steady state net foreign asset position may differ from zero in this model, I restrict attention to parameterizations of the endogenous discount factor such that the model implies the same steady states as the original model, i.e., net foreign assets are zero in steady state. Similarly, under complete markets the unique steady state for a given value of μ may not imply zero net foreign assets. Thus, I restrict attention t to those value of μ that imply the same steady states as the original model. t 21

4 Parameterizations Overall, I consider five models. Model I is a simple production economy without capital and traded goods only. Model II introduces endogenous capital formation. Model III allows for capital and non-traded goods. Table 1: Calibration of Baseline Model Description Symbol Value 1. Parameters constant across models elasticity between k and l non-traded goods (Cobb-Douglas) κN 0.00 i elasticity between k and l traded goods (Cobb-Douglas) κT 0.00 i depreciation rate of capital δ i 0.025 elasticity between non-traded goods εN 0.74 i elasticity between traded goods consumption εT 0.85 i elasticity between traded goods investment εI 0.85 i relative population size ν i 1.00 discount factor β 0.99 i intertemporal consumption elasticity σ i 2.00 (a) Cobb-Douglas preferences weight on consumption ξ 0.34 i weight on leisure 1−ξ 0.66 i (b) Additive separable preferences labor supply elasticity χ -5.00 i weight labor disutility χ 5.00 0i 2. Parameters varying across models model I II III IV V weight on labor in production of non-traded good ωN n.a. n.a. n.a. 0.560 0.560 li weight on labor in production of traded good ωT 1.000 0.610 0.610 0.610 0.610 li distribution cost in terms of non-traded good η n.a. n.a. 0.000 1.090 1.090 i (a) Cobb-Douglas preferences weight own traded good in consumption αT 0.938 0.938 0.910 0.720 0.910 ii weight own traded good in investment αI 1.000 0.938 0.910 1.000 0.855 ii weight overall traded good αN 1.000 1.000 0.550 0.550 0.550 i1 (b) Additive separable preferences weight own traded good in consumption αT 0.945 0.945 0.924 0.720 0.905 ii weight own traded good in investment αI 1.000 0.945 0.924 1.000 0.905 ii weight overall traded good αN 1.000 1.000 0.550 0.550 0.550 i1 The fourth model is identical to the model presented in Corsetti, Dedola, and Leduc (2008), i.e., consumption of traded goods requires the use of non-traded distribution services. In contrast to models II and III, investment features complete home bias. Relative to model IV, model V relaxes the assumption of complete home bias in investment. As models IV 22

and V feature non-zero distribution costs, the trade elasticity is not fully determined by the elasticity of substitution between traded goods. Table 1 summarizes the parameterizations for the five different models. To the extent possible, parameters are set equal across models. Parameter differences reflect the differences in model features. For each model, the quasi shares in equation (21) and (22) are adjusted to obtain an import to GDP ratio of 6.2 percent as in Corsetti, Dedola, and Leduc (2008). Household preferences are described either by a Cobb-Douglas function: (cid:20) (cid:21) cξi (1−l )1−ξi 1−σi i,t+j i,t+j U (c ,l ) = , (42) i i,t+j i,t+j 1−σ i or an additive separable utility function: c1−σi l1−χi U (c ,l ) = i,t+j −χ i,t+j . (43) i i,t+j i,t+j 1−σ 0i1−χ i i The functional forms for the stationarity inducing devices are: • for the portfolio costs approach (cid:11) (cid:12) 1 b 2 Γ(b ) = γ i,t+j P , (44) i,t+j 2 i P i,t+j i,t+j • for a debt-elastic interest rate (cid:5) (cid:5) (cid:6)(cid:6) Ψ(b ) = exp ψ b −¯b , (45) i,t+j d i,t+j • for endogenous discounting under Cobb-Douglas preferences (cid:20) (cid:21) −1 β = 1+ψ cξi (1−l )1−ξi , (46) i,t+j i i,t+j i,t+j under additive separable preferences (cid:16) (cid:17) c1−σi l1−χi β = 1−ψ exp i,t+j −χ i,t+j . (47) i,t+j i 1−σ 0i1−χ i i 23

5 Steady state multiplicity I first compute the set of steady states under the assumption that financial markets are absent at the international level, i.e., b = 0 for all j, before commenting on the remaining market 1,t+j arrangements outlined in Section 3.2. Absent dynamics all the endogenous variables can be expressed as functions of the relative price q = P2. An equilibrium with zero net foreign asset holdings is then characterized by any P1 value of q that implies zero excess demand z for the traded good of country 2: 2 (cid:5) (cid:6) z (q) ≡ (c (q)+x (q))ν + c (q)+x (q)−yT (q) ν (48) 2 12 12 1 22 22 2 2 z (q) ≤ 0, ∞ ≥ q ≥ 0 and qz (q) = 0. (49) 2 2 Thus, the steady states can be computed reliably by searching over a one-dimensional grid on the interval [0,1] for the normalized relative price q . 1+q 5.1 Steady states under financial autarchy Figure 2 plots the set of steady state equilibrium prices for different values of the elasticity of substitution between traded goods in the five models for both Cobb-Douglas preferences and additive separable preferences. Each model displays a unique price equilibrium with q = 1 (cid:20) (cid:21) q = 1 , if the substitution elasticity is high. When the two traded goods are assumed to 1+q 2 be less substitutable, multiple steady states arise. The two asymmetric steady states feature a relative price q close to zero or one, respectively, and are hard to detect in practice. 1+q To understand how multiple equilibria arise at low values of the elasticity of substitution consider the case of the model without capital and traded goods only (model I). If the price of good 1 is high relative to the price of good 2 (lower branch), the value of country 1’s production is high relative to country 2’s. As labor supply is assumed to be fairly inelastic, overall production is price inelastic, as well. Agents in country 2 are willing to pay the high price for good1 and country 1 ends up consuming most of the two goods. Similar logic applies when q ismuch larger than1(upper branch) with theroles ofcountries 1 and2being reversed. The middle equilibrium, which always exists, is the symmetric equilibrium with q = 1. 24

Figure 2: Steady state multiplicity for different values of the elasticity of substitution Cobb−Douglas preferences model I: no capital, traded goods only 1 0.5 0 0 0.2 0.4 0.6 0.8 1 )q+1(/q model II: traded goods only 1 0.5 0 0 0.2 0.4 0.6 0.8 1 )q+1(/q model III: traded and nontraded goods 1 0.5 0 0 0.2 0.4 0.6 0.8 1 )q+1(/q model IV: traded, nontraded goods, and distribution costs 1 0.5 0 0 0.2 0.4 0.6 0.8 1 )q+1(/q model V: model IV with trade in investment 1 0.5 0 0 0.2 0.4 0.6 0.8 1 elasticity of substitution εT = εT 1 2 )q+1(/q Additive separable preferences model I: no capital, traded goods only 1 0.5 0 0 0.2 0.4 0.6 0.8 1 )q+1(/q model II: traded goods only 1 0.5 0 0 0.2 0.4 0.6 0.8 1 )q+1(/q model III: traded and nontraded goods 1 0.5 0 0 0.2 0.4 0.6 0.8 1 )q+1(/q model IV: traded, nontraded goods, and distribution costs 1 0.5 0 0 0.2 0.4 0.6 0.8 1 )q+1(/q model V: model V with trade in investment 1 0.5 0 0 0.2 0.4 0.6 0.8 1 elasticity of substitution εT = εT 1 2 )q+1(/q Notes: Set of normalized relative prices q/(1+q) that constitute a steady state absent financial markets as a function of the elasticity of substitution between traded goods. The (symmetric) parameterizations ofmodelsItoVaregiveninTable1. As can be seen in Figure 2 and Table 2 the threshold value of the substitution elasticity ¯εT 1 forwhich equilibrium multiplicity occursismodel dependent, but it doesnot varymuch across the two specifications of household preferences for standard parameterizations. Keeping trade shares constant at an import to GDP ratio of 6.2 percent across models, ¯εT falls when capital 1 accumulation and non-traded goods are introduced. For model II, with an endogenous capital stock, a country carries a lower capital stock and reduces production, if the relative price for its good is low. Given reduced supply of the country’s good, the low relative price may turn out not to be an equilibrium. Hence, for equilibrium multiplicity to occur, ¯εT has to be lower 1 in the model with an endogenous capital stock than in model I. In model III, the presence of 25

non-traded goods opens up an additional channel for a country to react to a low relative price of its traded good. By shifting labor and capital towards the production of the non-traded good, production of the traded good is lowered, making it even less likely that the low relative price constitutes an equilibrium for given εT. 1 The two bottom panels of Figure 2 introduce distribution costs as in Corsetti, Dedola, and Leduc (2008). For modelIV, which is parameterized exactly asin Corsetti, Dedola, andLeduc (2008), the threshold value ¯εT is well above its value in models I-III. However, as consumption 1 of traded goods requires non-traded goods as input, the implied trade elasticity is 0.5359 for Cobb-Douglas preferences and 0.5532 for additive separable preferences.9 Furthermore, there are five steady states under Cobb-Douglas preferences for εT ∈ [0.9189;0.9798] and three 1 steady states for εT ∈ [0;0.9189]. Under additive separable preferences, there are five steady 1 states for εT ∈ [0.9516;1.0115] and three steady states for εT ∈ [0;0.9516]. 1 1 Table 2: Model dependent threshold levels for ¯εT = ¯εT 1 2 Model I II III IV V Cobb-Douglas pref. 0.4670 0.4248 0.3930 0.9798 0.3968 Add. sep. pref. 0.4793 0.4474 0.4255 1.0115 0.5205 Whereas the baseline model in Corsetti, Dedola, and Leduc (2008) assumes complete home bias in investment, the bottom panel of the figure assumes incomplete home bias in investment. Unlike for consumption, no distribution services are required for the investment good. In model V, the threshold value ¯εT falls well below unity again. Furthermore, the maxi- 1 mum number of equilibria is three. If investment was subject to sufficiently high distribution services, however, a situation similar to model IV would reemerge. Figure 3 further explores the role of distribution costs in generating multiple steady states. Starting from the parameterization of model IV, the figure shows how the steady state values of the relative price q change as distribution costs are raised for a fixed value of εT. For low 1 9Numerically, the steady state value of the prices of the non-traded goods do not vary with εT. While in equation (32) 1 the measuredtrade elasticity in the symmetric steady state is proportionalto the elasticity of substitution between traded goods, this is not true for the asymmetric steady states. 26

values of εT, such as in the top panel, multiple steady states exist even absent distribution 1 costs. Irrespective of the value of εT assumed in the figure, multiple steady states arise once 1 distribution costs are sufficiently high. Figure 3: Steady state multiplicity in model IV (CDL) for changing distribution costs Cobb−Douglas preferences εT 1 = εT 2 = 0.3 1 0.5 0 0 0.5 1 1.5 2 )q+1(/q εT 1 = εT 2 = 0.85 1 0.5 0 0 0.5 1 1.5 2 )q+1(/q εT 1 = εT 2 = 1.2 1 0.5 0 0 0.5 1 1.5 2 )q+1(/q Additive separable preferences εT 1 = εT 2 = 0.3 1 0.5 0 0 0.5 1 1.5 2 distribution costs η 1 = η 2 )q+1(/q εT 1 = εT 2 = 0.85 1 0.5 0 0 0.5 1 1.5 2 )q+1(/q εT 1 = εT 2 = 1.2 1 0.5 0 0 0.5 1 1.5 2 )q+1(/q distribution costs η 1 = η 2 Notes: Set of normalized relative prices q/(1+q) that constitute a steady state absent financial marketsasafunctionofthedistributioncostparameters. Exceptfortheelasticityofsubstitution between traded goods and the distribution cost parameters, parameters are chosen to match the calibrationinCorsetti,Dedola,andLeduc(2008). 5.2 Steady states under different assumptions on international financial markets The choice of the stationarity inducing device affects the number of steady state equilibria. In a steady state with zero net foreign assets, the standard incomplete markets model, the portfolio cost model, and the debt-elastic interest rate model imply the same steady state restrictions as the model without international financial markets. If under financial autarchy multiple values of the relative price q induce market clearing, then this is also the case for thesethreeframeworks. Similarly, theoverlapping generationsmodelwithincompletemarkets displays the same steady states as its analogue without financial markets.10 10AsshowninAppendixB,thesteadystatesintheoverlappinggenerationsmodeldifferfromthesteadystatesinmodels for which each country’s agents can be represented by a single agent. However, for the parameterizations adopted in this paper, the quantitative differences are small. 27

By contrast, for a given parameterization of the discount factor functions, the model with endogenous discounting admits a unique steady state irrespective of the number of steady states under financial autarchy. Condition (39) determines uniquely the steady state allocations and prices including the relative price q. It is not guaranteed that the resulting steady state coincides with any of the steady states obtained under financial autarchy. I restrict the set discount factor functions such that the unique steady state under endogenous discounting coincides with one of the steady states obtained under financial autarchy and thus features a zero net foreign asset position. Similarly, under complete markets, for a given initial value of μ , condition (40) admits t only one steady state irrespective of the number of steady states in the model with financial autarchy. As each value of μ gives rise to a different steady state, that may not coincide with t any of those obtained under financial autarchy, I restrict attention to those values of μ , that t imply a steady state that can also be obtained under financial autarchy. 6 Local equilibrium dynamics Turning to the dynamics of the different models in the neighborhood of a given deterministic steady state, the model is (log-)linearized around this steady state. Reducing the system of equilibrium conditions to its state variables (log deviations of the capital stocks kˆ and kˆ , 1,t 2,t and the absolute deviation of the net foreign asset position Δb ), one obtains the difference 1,t equation: ⎛ ⎞ ⎛ ⎞ z z a⎝ t+1 ⎠ +b⎝ t ⎠ = 0, z z t t−1 (cid:20) (cid:21) (cid:4) with the 6×6 coefficient matrices a and b and the 3×1 state vector z = Δb ,kˆ ,kˆ . t 1,t 1,t 2,t With a being an invertible matrix for the models considered in this paper, a simple count of the eigenvalues of the matrix −a−1b suffices to characterize the stability of the local dynamics as described in Christiano (2002).11 Let N (here equal 6) denote the number of eigenvalues of 11Christiano (2002) discusses also the case if a is not invertible. 28

−a−1b and N∗ ≤ N be the number of eigenvalues larger than one in absolute value. n (here equal to 3) is the number of predetermined variables. The equilibrium dynamics identified by the minimum state variable solution are: 1. bounded and unique, if N∗ = N −n, 2. bounded, but not unique, if N∗ < N −n, 3. unbounded, if N∗ > N −n. Thus, the equilibrium dynamics can be characterized by the absolute value of the nth and n + 1st smallest eigenvalue, |λ | and |λ |. If |λ | > 1, the equilibrium dynamics are n n+1 n unbounded. If |λ | < 1 and |λ | > 1, the dynamics are bounded and unique. Otherwise, n n+1 they are bounded, but not unique. As |λ | > 1 for all cases analyzed below, I will reduce n+1 attention to |λ |. The subsequent analysis also links |λ | to the slope of the excess demand n n function for good 2 with respect to the relative price ∂z2q in a steady state. ∂q Tostudytherelationshipbetweenthestabilityoftheequilibriumdynamicsaroundasteady state and the multiplicity of steady states, I analyze the model with endogenous capital and tradedgoods(modelII)aswellasthemodelwithdistributioncosts(modelIV)under different stationarity inducing devices. In the following, εT and εT are always set to be equal. 1 2 6.1 Results for model II Figures 4 and 5 perform this stability analysis for model II. Figure 4 examines the cases of standard incomplete markets, portfolio costs, and overlapping generations. For each value of the elasticity of substitution εT ∈ [0.4,0.45], the model is linearized around each of the steady 1 states that exist for εT. The value of the nth smallest eigenvalue of the system |λ | (left scale) 1 n and the slope of the excess demand function for good 2 in the steady state (right scale) are recorded. In accordance with the second panel in Figure 2, there are three steady states for εT < ¯εT = 0.4248 in the standard incomplete markets model and the portfolio cost model, but 1 1 a unique one otherwise. For the overlapping generations model the value is ¯εT = 0.4247. The 1 label “low q steady state” refers to steady states represented by the lower branch in Figure 2 29

for model II with Cobb-Douglas preferences. “High q steady state” and “q = 1 steady state” refer to the top and middle branches, respectively. Under standard incomplete markets (left column) |λ | equals unity irrespective of both n the steady state around which the model is approximated and the value of εT. This finding 1 simply restates the non-stationarity of the net foreign asset position. Figure 4: Local dynamics in model II (BKK) unstable models 1.001 1 0.999 0.4 0.41 0.42 0.43 0.44 0.45 eulavnegie tsellams htn nonstationary model low q steady state 0 −0.9 0.4 0.41 0.42 0.43 0.44 0.45 1.001 1 0.999 0.4 0.41 0.42 0.43 0.44 0.45 eulavnegie tsellams htn q=1 steady state 0.02 0 −0.02 0.4 0.41 0.42 0.43 0.44 0.45 1.001 1 0.999 0.4 0.41 0.42 0.43 0.44 0.45 eulavnegie tsellams htn portfolio costs olg framework low q steady state low q steady state 1.01 0 1.0001 0 1 1 0.9745 −0.90.9998 −0.08 00..44 00..4411 00..4422 00..4433 00..4444 00..4455 00..44 00..4411 00..4422 00..4433 00..4444 00..4455 q=1 steady state 1.0065 0.02 1 0 0.9935 −0.02 00..44 00..4411 00..4422 00..4433 00..4444 00..4455 high q steady state x 10−4 high q steady state x 10−4 0 1.01 0 1 −9 0.9745 −9 0.4 0.41 0.42 0.43 0.44 0.45 00..44 00..4411 00..4422 00..4433 00..4444 00..4455 elasticity of substitution εT = εT elasticity of substitution εT = εT 1 2 1 2 .cnuf .med ssecxe epols q=1 steady state 1.0001 0.02 1 0 0.9999 −0.02 00..44 00..4411 00..4422 00..4433 00..4444 00..4455 .cnuf .med ssecxe epols high q steady state x 10−3 1.0001 0 1 0.9998 −7 00..44 00..4411 00..4422 00..4433 00..4444 00..4455 .cnuf .med ssecxe epols slope eigenvalue mult. eq unique. eq elasticity of substitution εT = εT 1 2 Notes: “Low q steady state” refers to steady states represented by the lower branch in Figure 2 for modelIIwithCobb-Douglas preferences. “Highqsteady state”and“q=1steadystate” refertothe top and middle branch, respectively. Each panel shows the nth largest eigenvalue (left axis) and the slope of the excess demand function (right axis) associated with the steady state around which the model is approximated as a function of the elasticity of substitution between traded goods. For all threemodelsnequals3andN =6. Ifasteadystatedoesnotexistforagivenrangeofthesubstitution elasticity,novaluesareplotted. For the portfolio cost model (middle column), in the top and bottom panels, |λ | is always n smaller than 1 and the equilibrium dynamics are therefore locally unique and bounded. In line with Section 2, the slope of the excess demand function is always negative for these steady 30

states. However, in the middle panel, that shows the results for the symmetric steady state of the model, it is |λ | < 1 with ∂z2q > 0 for εT > ¯εT, and it is |λ | ≥ 1 with ∂z2q ≤ 0 for n ∂q 1 1 n ∂q εT ≤ ¯εT. Hence, for εT ≤ ¯εT the equilibrium dynamics around the symmetric steady state 1 1 1 1 are not bounded. Although not shown, the same findings apply for the case of a debt-elastic interest rate. Figure 5: Local dynamics in model II (BKK) stable models 1.0094 1 0.9889 0.4 0.41 0.42 0.43 0.44 0.45 eulavnegie tsellams htn endogenous discounting low q steady state 0 −0.9 0.4 0.41 0.42 0.43 0.44 0.45 1.0016 1 0.9984 0.4 0.41 0.42 0.43 0.44 0.45 eulavnegie tsellams htn q=1 steady state 0.02 0 −0.02 0.4 0.41 0.42 0.43 0.44 0.45 1.0094 1 0.9889 0.4 0.41 0.42 0.43 0.44 0.45 ,eulavnegie tsellams htn endogenous discounting with internalization complete markets low q steady state low q steady state 1.0094 0 0.9775 0 1 0.9889 −0.90.9575 −0.9 00..44 00..4411 00..4422 00..4433 00..4444 00..4455 00..44 00..4411 00..4422 00..4433 00..4444 00..4455 q=1 steady state 1.0016 0.02 1 0 0.9984 −0.02 00..44 00..4411 00..4422 00..4433 00..4444 00..4455 high q steady state x 10−4 high q steady state x 10−4 0 1.0094 0 1 −9 0.9889 −9 0.4 0.41 0.42 0.43 0.44 0.45 00..44 00..4411 00..4422 00..4433 00..4444 00..4455 elasticity of substitution εT = εT elasticity of substitution εT = εT 1 2 1 2 .cnuf .med ssecxe epols q=1 steady state 1.0335 0.02 1 0 0.9665 −0.02 00..44 00..4411 00..4422 00..4433 00..4444 00..4455 .cnuf .med ssecxe epols high q steady state x 10−4 0.9775 0 0.9575 −9 00..44 00..4411 00..4422 00..4433 00..4444 00..4455 .cnuf .med ssecxe epols slope eigenvalue mult. eq unique. eq elasticity of substitution εT = εT 1 2 Notes: “Low q steady state” refers to steady states represented by the lower branch in Figure 2 for modelIIwithCobb-Douglas preferences. “Highqsteady state”and“q=1steadystate” refertothe top and middle branch, respectively. Each panel shows the nth largest eigenvalue (left axis) and the slope of the excess demand function (right axis) associated with the steady state around which the model is approximated as a function of the elasticity of substitution between traded goods. For the firsttwo models nequals 3 and N =6, for the complete markets model n=2 and N =4. If a steady statedoesnotexistforagivenrangeofthesubstitutionelasticity,novaluesareplotted. In the overlapping generations framework (right column) the same patterns arise: steady states for which the excess demand function is downward sloping are associated with |λ | < 1 n and locally unique and bounded dynamics. Otherwise, the equilibrium dynamics are un- 31

bounded. The local dynamics under endogenous discounting – both with and without internalization – and under complete markets are studied in Figure 5. As pointed out earlier, for a given parameterization ofthe discount factor function orthe choice of μ , these three settings always t imply a unique steady state. The figure, though, shows results for each of the steady states that occur under financial autarchy by appropriately adjusting the discount factor function or the weight μ , respectively. The first two columns reveal that |λ | is always less than unity t n irrespective of the steady state that is analyzed and the associated sign of ∂z2q. Thus, under ∂q endogenous discounting, the equilibrium dynamics are always unique and bounded. The same findings apply under complete markets. 6.2 Results for model IV The dynamic characteristics of model IV under the calibration of Corsetti, Dedola, and Leduc (2008) with Cobb-Douglas preferences are examined in Figure 6 for the case of portfolio costs and endogenous discounting. Each of the five panels per column corresponds to one of the steady states depicted in the fourth panel of Figure 2. For the portfolio cost model, |λ | is always less than unity in the first and fifth panels n and ∂z2q < 0. To the extent that these steady states exist, which is the case for εT < 0.979, ∂q 1 these steady states are associated with unique and bounded equilibrium dynamics. For the second and fourth panels, |λ | is always larger than unity and the equilibrium dynamics are n unbounded. Consistent with all previous findings, the sign of ∂z2q is positive. The case of ∂q q = 1 (third panel) is more interesting. For εT > 0.9189, ∂z2q is negative and |λ | < 1. 1 ∂q n It is only for εT < 0.9189 that the excess demand function switches sign, |λ | turns larger 1 n than one, and the equilibrium dynamics become unbounded. Under endogenous discounting |λ | is always smaller than one irrespective of the steady state around which the model is n approximated and the sign of ∂z2q. The equilibrium dynamics are unique and bounded. ∂q 32

Figure 6: Local dynamics in model IV (CDL) with Cobb-Douglas preferences 0.9998 0.9707 0.92 0.94 0.96 0.98 1 eulavnegie tsellams htn portfolio costs very low q steady state 0 −4000 0.92 0.94 0.96 0.98 1 1.0049 1 0.92 0.94 0.96 0.98 1 eulavnegie tsellams htn low q steady state 1.2 0 0.92 0.94 0.96 0.98 1 1.0062 1 0.9938 0.92 0.94 0.96 0.98 1 eulavnegie tsellams htn q=1 steady state 0.02 0 −0.02 0.92 0.94 0.96 0.98 1 1.0049 1 0.92 0.94 0.96 0.98 1 eulavnegie tsellams htn high q steady state x 10−4 7 0 0.92 0.94 0.96 0.98 1 0.9993 0.9712 0.92 0.94 0.96 0.98 1 eulavnegie tsellams htn endogenous discounting very low q steady state 1 0 0.9981 −4000 00..9922 00..9944 00..9966 00..9988 11 very high q steady state x 10−7 0 −1.3 0.92 0.94 0.96 0.98 1 elasticity of substitution εT = εT 1 2 .cnuf .med ssecxe epols low q steady state 1 1.2 0.9994 0 00..9922 00..9944 00..9966 00..9988 11 .cnuf .med ssecxe epols q=1 steady state 1.0007 0.02 1 0 0.9993 −0.02 00..9922 00..9944 00..9966 00..9988 11 .cnuf .med ssecxe epols high q steady state x 10−4 1 7 0.9994 0 00..9922 00..9944 00..9966 00..9988 11 .cnuf .med ssecxe epols very high q steady state x 10−7 1 0 0.9986 −1.3 00..9922 00..9944 00..9966 00..9988 11 elasticity of substitution εT = εT 1 2 .cnuf .med ssecxe epols 3. ss. 5. ss. 1. ss. slope eigenvalue Notes: The labels “very low q steady state”, “low q steady state”, “q =1 steady state”, “high q steady state”, and “very high qsteady state” refer to the steady states represented by the bottom, second from bottom,middle,secondfromtop,andtopbranchinFigure2formodelIVwithCobb-Douglaspreferences, respectively. Each panel shows the nth largest eigenvalue (left axis) and the slope of the excess demand function(rightaxis)associatedwiththesteadystatearoundwhichthemodelisapproximatedasafunction oftheelasticityofsubstitutionbetweentradedgoods. Forbothmodelsnequals3andN =6. Ifasteady statedoesnotexistforagivenrangeofthesubstitutionelasticity,novaluesareplotted. Thesteadystate isuniqueforεT 1 >0.9798. TherearefivesteadystatesforεT 1 ∈[0.9189;0.9798]andthreesteadystatesfor εT 1 <0.9189. 33

Figure 7: Local dynamics in model IV (CDL) with additive separable preferences 0.9994 0.9755 0.94 0.96 0.98 1 1.02 eulavnegie tsellams htn portfolio costs very low q steady state 0 −8000 0.94 0.96 0.98 1 1.02 1.0058 1 0.94 0.96 0.98 1 1.02 eulavnegie tsellams htn low q steady state 2 0 0.94 0.96 0.98 1 1.02 1.0104 1 0.9896 0.94 0.96 0.98 1 1.02 eulavnegie tsellams htn q=1 steady state 0.03 0 −0.03 0.94 0.96 0.98 1 1.02 1.0058 1 0.94 0.96 0.98 1 1.02 eulavnegie tsellams htn high q steady state x 10−3 1.4 0 0.94 0.96 0.98 1 1.02 0.9994 0.9755 0.94 0.96 0.98 1 1.02 eulavnegie tsellams htn endogenous discounting very low q steady state 1 0 0.9962 −8000 00..9944 00..9966 00..9988 11 11..0022 very high q steady state x 10−7 0 −3.5 0.94 0.96 0.98 1 1.02 elasticity of substitution εT = εT 1 2 .cnuf .med ssecxe epols low q steady state 1 2 0.9993 0 00..9944 00..9966 00..9988 11 11..0022 .cnuf .med ssecxe epols q=1 steady state 1.0005 0.03 1 0 0.9995 −0.03 00..9944 00..9966 00..9988 11 11..0022 .cnuf .med ssecxe epols high q steady state x 10−3 1 1.4 0.9993 0 00..9944 00..9966 00..9988 11 11..0022 .cnuf .med ssecxe epols very high q steady state x 10−7 1 0 0.9967 −3.5 00..9944 00..9966 00..9988 11 11..0022 elasticity of substitution εT = εT 1 2 .cnuf .med ssecxe epols 3. ss. 5. ss. 1. ss. eigenvalue slope Notes: The labels “very low q steady state”, “low q steady state”, “q =1 steady state”, “high q steady state”, and “very high qsteady state” refer to the steady states represented by the bottom, second from bottom, middle, second from top, and top branch in Figure2 for model IV with additive separable preferences, respectively. Each panel shows the nth largest eigenvalue (left axis) and the slope of the excess demandfunction(rightaxis)associatedwiththesteadystatearoundwhichthemodelisapproximatedas afunctionoftheelasticityofsubstitution between tradedgoods. Forbothmodelsnequals 3and N =6. Ifasteady state doesnot existforagivenrangeofthe substitutionelasticity, novalues areplotted. The steady state is unique for εT 1 > 1.0115. There are five steady states for εT 1 ∈ [0.9516;1.0115] and three steadystates forεT 1 <0.9516. 34

Figure 8: Local dynamics in model IV (CDL) with additive separable preferences, real and nominal rigidities 0.9999 0.988 0.94 0.96 0.98 1 1.02 eulavnegie tsellams htn portfolio costs very low q steady state 0 −8000 0.94 0.96 0.98 1 1.02 1.0053 1 0.94 0.96 0.98 1 1.02 eulavnegie tsellams htn low q steady state 2 0 0.94 0.96 0.98 1 1.02 1.0074 1 0.9926 0.94 0.96 0.98 1 1.02 eulavnegie tsellams htn q=1 steady state 0.03 0 −0.03 0.94 0.96 0.98 1 1.02 1.0053 1 0.94 0.96 0.98 1 1.02 eulavnegie tsellams htn high q steady state x 10−3 1.4 0 0.94 0.96 0.98 1 1.02 0.9996 0.9861 0.94 0.96 0.98 1 1.02 eulavnegie tsellams htn endogenous discounting very low q steady state 1 0 0.9962 −8000 00..9944 00..9966 00..9988 11 11..0022 very high q steady state x 10−7 0 −3.5 0.94 0.96 0.98 1 1.02 elasticity of substitution εT = εT 1 2 .cnuf .med ssecxe epols low q steady state 1 2 0.9993 0 00..9944 00..9966 00..9988 11 11..0022 .cnuf .med ssecxe epols q=1 steady state 1.0005 0.03 1 0 0.9995 −0.03 00..9944 00..9966 00..9988 11 11..0022 .cnuf .med ssecxe epols high q steady state x 10−3 1 1.4 0.9993 0 00..9944 00..9966 00..9988 11 11..0022 .cnuf .med ssecxe epols very high q steady state x 10−7 1 0 0.9966 −3.5 00..9944 00..9966 00..9988 11 11..0022 elasticity of substitution εT = εT 1 2 .cnuf .med ssecxe epols 3. ss. 5. ss. 1. ss. slope eigenvalue Notes: The labels “very low q steady state”, “low q steady state”, “q =1 steady state”, “high q steady state”, and “very high qsteady state” refer to the steady states represented by the bottom, second from bottom,middle,secondfromtop,andtopbranchinFigure2formodelIVwithadditiveseparablepreferences, respectively. Panels show the nth largesteigenvalue (leftaxis)and theslopeof the excess demand function(rightaxis)associatedwiththesteadystatearoundwhichthemodelisapproximatedasafunction oftheelasticityofsubstitutionbetweentradedgoods. Duetotheadditionalstatevariablesnequals40and N =78. ThesteadystateisuniqueforεT 1 >1.0115. Therearefivesteadystates forεT 1 ∈[0.9516;1.0115] andthreesteadystatesforεT 1 <0.9516. MonetarypolicyfollowsaTaylor-typerulewithalong-runweight on consumer price inflation of 1.5, an interest rate smoothing coefficient of 0.7, and zero weight on the outputgap. 35

Allowing for additive separable preferences does not overturn these findings as shown in Figure 7. Introducing real and nominal rigidities under additive separable preferences does not affect the findings either. Figure 8 confirms this claim in a model with Calvo sticky wage and price contracts, consumption habits, investment adjustment costs, and local currency pricing. Monetary policy follows an interest rate rule with the policy rate being a function of consumer price inflation only. A detailed model description is provided in the Technical Appendix accompanying this paper. 7 Conclusions Widely used international business cycle models admit multiple steady states absent international financial markets for low substitutability between traded goods. Once a limited set of internationally traded assets is introduced, the steady state multiplicity may or may not be preserved. If the incomplete markets model is augmented with portfolio costs, a debt-elastic interest rate, or an overlapping generations framework as in Weil (1989) the number of steady states remains unchanged relative to the model without financial markets. If Uzawa-type preferences are introduced (Uzawa (1968)), the steady state of the model is unique. The choice of stationarity inducing device also affects the stability of the dynamics around a given steady state. Under portfolio costs, debt-elastic interest rates, or overlapping generations, steady states that are associated with a downward-sloping excess demand function of the foreign good display locally unique and bounded dynamics. For steady states with an upward-sloping excess demand function, the local dynamics are unbounded. Under endogenous discounting or complete markets, the local dynamics are always unique and bounded. The results stressed in this paper prevail under (local) higher order perturbation methods. The findings also extend to environments with a larger set of available assets. If country portfolios are determined as in Devereux and Sutherland (2008) and Tille and van Wincoop (2010), the net foreign asset position is not determined in steady state and displays unit root behavior unless stationarity is induced with one of the devices discussed in this paper. Absent trade adjustment costs short- and long-run substitution elasticities are identical in 36

this paper. If trade adjustment costs were to affect the model’s allocations away froma steady state, the analysis would not change given the use of local solution techniques. Steady state multiplicity would only occur if the long-run substitution elasticity between traded goods fell below its threshold level ¯εT in the model without trade adjustment costs. However, if i one were to employ global solution techniques, differences across models could be detected. Similar tothe findings presented inBodenstein (2010)for the case of capital, a lower short-run substitution elasticity may lead to multiple equilibrium paths despite a unique steady state for the given value of the long-run substitution elasticity. 37

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A Appendix: Details on the simple model This appendix describes the equations underlying the discussion in Section 2. The first order conditions of the households imply: (cid:8) (cid:9) αT q−εT b −Q b c = 12 t yT + 1,t−1 1,t 1,t , (50) 12,t αT +αT q1−εT 1,t Φ (q ) 11 12 t (cid:8) 1 t (cid:9) αT b −Q b c = (cid:20) (cid:21) 22 yT + 2,t−1 2,t 2,t , (51) 22,t 1−εT 2,t Φ (q )q αT 1 +αT 1 t t 21 qt 22 cT = Φ (q )yT +b −Q b , (52) 1,t 1 t 1,t 1,t−1 1,t 1,t Φ (q ) cT = Φ (q )yT + 2 t (b −Q b ), (53) 2,t 2 t 2,t q Φ (q ) 2,t−1 2,t 2,t t 1 t where (cid:31) P (cid:11) 1 (cid:12) 1−εT − 1− 1 εT 1 Φ (q ) ≡ 1,t = αT +αT , (54) 1 t P c +P c 11 q 12 q 1,t 11,t 2,t 12,t t t (cid:31) P (cid:11) 1 (cid:12) 1−εT − 1− 1 εT Φ (q ) ≡ 2,t = αT +αT , (55) 2 t P c +P c 21 q 22 1,t 21,t 2,t 22,t t Φ (q ) rer = q 1 t . (56) t t Φ (q ) 2 t The price of the aggregate consumption good in country 1 is taken to be the num´eraire. Loglinearizing these equations around a steady state with zero net foreign assets and setting the exogenous variables yT and yT to be constant (“xˆ” for log-linearized and “Δx” for linearized 1,t 2,t variables x). Variables without time index indicate steady state values: ! " 1 cˆ = − (1−εT)(1−αT )+εT qˆ + [Δb −βΔb ], (57) 12,t 11 t y Φ (q) 1,t−1 1,t 1 1 1 cˆ = (1−εT)(1−αT )qˆ − [Δb −βΔb ], (58) 22,t 22 t qΦ (q)y 1,t−1 1,t 1 2 r#er = −(1−αT −αT )qˆ, (59) t 11 22 t where I have used the fact that in steady state Φ(cid:3) 1(q)q = −(1−αT ) and Φ(cid:3) 2(q)q = (1−αT ). Φ1(q) 11 Φ2(q) 22 Using (57) and (58) in the approximation of the market clearing condition for good 2: ∂z ∂z 2qqˆ + 2 [Δb −βΔb ] = 0, (60) ∂q t ∂dW 1t−1 1t 1 with the coefficients (cid:10) (cid:11) (cid:12) (cid:13) ∂z (cid:5) (cid:6) c (q) c (q) 2q = c (q) 1−εT 11 + 22 −1 , (61) ∂q 12 yT yT (cid:10)(cid:11) 1 (cid:12) 2 (cid:13) ∂z 1 c (q) c (q) 2 = − 11 + 22 −1 . (62) ∂dW qΦ (q) yT yT 1 1 1 2 40

Note that in a steady state with balanced trade α = cii(q). The slope of the excess demand ii yi T function in steady state with respect to q is given by equation (61). Aggregate consumption is determined by: 1 cˆT = −(1−αT )qˆ + (Δb −βΔb ), (63) 1,t 11 t c (q) 1,t−1 1,t 1 Φ (q) 1 cˆT = (1−αT )qˆ − 2 (Δb −βΔb ), (64) 2,t 22 t qΦ (q)c (q) 1,t−1 1,t 1 2 while the risk sharing condition is given by: cˆT −cˆT = cˆT −cˆT −(1−αT −αT )(qˆ −qˆ ) 1,t 1,t+1 2,t 2,t+1 11 (cid:8)22 t(cid:9) t+1 γ 1 + 1+ Δb . (65) βΦ (q) q 1,t 1 Thus, the dynamic system under portfolio costs can be written as: ⎛ ⎞ ⎛ (cid:11) (cid:12) ⎞ ⎝ −∂ ∂ ∂ ∂ d z z q W 2 2 q 1β 0 ⎠ (cid:11) Δ Δ b 1 b ,t+1 (cid:12) + ⎝ 1+β +β ∂ d ∂ z z q 2 2 qγˇ ∂ ∂ ∂ ∂ d z z q W 2 2 q 1 −∂ ∂ ∂ ∂ d z z q W 2 2 q 1 ⎠ (cid:11) Δ Δ b b 1,t (cid:12) = 0, (66) 0 1 1,t −1 0 1,t−1 with the coefficients (cid:10)(cid:11) (cid:12) (cid:13) 1 c (q) c (q) 2 dz = − 11 + 22 −1 2 qΦ (q) yT yT 1 (cid:10) (cid:11)1 2 (cid:12)(cid:13)(cid:10) (cid:13) 1 c (q) c (q) c (q) c (q) − 2− 11 + 22 11 + 22 εT < 0, (67) qΦ (q) yT yT yT yT 1 (cid:8) (cid:9) 1 2 1 2 γ 1 γˇ = 1+ > 0. (68) β2Φ (q) q 1 Hence, the characteristic equation associated with the dynamic system (66) under the portfolio cost approach satisfies: (cid:16) (cid:17) 1+β ∂z2q 1 λ2 −λ + ∂q γˇ + = 0. (69) β dz β 2 Alternatively, assume that households have Uzawa-type preferences. The time discount (cid:5) (cid:6) factor βj is replaced by θ = β (cT )θ with β (cT ) = 1+ψ cT −1 . ψ is chosen i,t+1+j i i,t+j i,t+j i i,t+j i i,t+j i suchthatβ (cT) = β insteadystate. Absentinternalization,therisksharingconditionimplies: i i βcˆT −cˆT = βcˆT −cˆT −(1−αT −αT )(qˆ −qˆ ). (70) 1,t 1,t+1 2,t 2,t+1 11 22 t t+1 Hence, the characteristic equation associated with the dynamic system under endogenous discounting without internalization satisfies: (cid:16) (cid:17) (cid:16) (cid:17) (cid:28) (cid:28) 1+β dz dz 1 λ2 − + 2 λ+ 1+ 2 = 0, (71) β dz dz β 2 2 41

with (cid:8) (cid:11) (cid:12) (cid:11) (cid:12)(cid:9) (cid:10) (cid:13) d (cid:28) z = − 1 β(cid:4) 1 (q) 1− c 21 (q) + β(cid:4) 2 (q) 1− c 22 (q) εT c 11 (q) + c 22 (q) > 0, (72) 2 qΦ (q) β (q) yT β (q) yT yT yT 1 1 1 2 2 1 2 as the discount factors are assumed to be decreasing in the level of consumption.12 B Appendix: Overlapping generations model This appendix merges the framework with overlapping generations of infinitely lived households by Weil (1989) and Ghironi (2006) with the model presented in the main text. Absent Ricardian equivalence, the steady state level of the net foreign asset position can be shown to be unique. Furthermore, the local dynamics of the net foreign asset position are stationary. More details on the overlapping generations framework are provided in the Technical Appendix accompanying this paper. Each household consumes, supplies labor, and holds financial assets. Households are born on different dates owning no assets, but they own the present discounted value of their labor income. The number of households in country i N grows over time at the exogenous rate i,t n , i.e., N = (1+n )N . The utility function of each household is assumed to be of the i i,t+1 i i,t Cobb-Douglas type. Capital is no longer accumulated by households directly, but through capital producers. Households can purchase shares in these and all other firms that produce in their country of residence. B.1 Individual Households Consumers have identical preferences over the real consumption index and leisure. At time t , the representative consumer in country i born in period v ∈ (−∞,t ) maximizes the 0 0 intertemporal utility function: (cid:29) (cid:30) (cid:5) (cid:6) (cid:5) (cid:6) (cid:4)∞ cv ξi 1−lv 1−ξi 1−σi −1 Uv = βt−t0 i,t i,t (73) t0 1−σ i t=t0 s.t. (cid:5) (cid:6) PCcv ≤ P w lv +bv −Q bv + Q(cid:12) +P d (cid:17)v −Q(cid:12)(cid:17)v . (74) i,t i,t i,t i,t i,t i,t−1 i,t i,t i,t i,t i,t i,t−1 i,t i,t The final consumption of a representative household of country i born in period v at time t is denoted by cv . The production of the final consumption good follows the same process as i,t outlined in the main text. lv is the labor supply of such a household. The bond holdings are i,t denoted by bv . (cid:17)v are the household’s holdings of the domestic stock, that sells at the price i,t i,t Q(cid:12) and pays the dividend P d . The supply of the stock is normalized to 1. i,t i,t i,t 12Ifthe discountfactorswereincreasinginthe levelofconsumption,d(cid:28)z 2 wouldbenegativeandthe dynamicsaroundany deterministic steady state would be explosive. 42

B.2 Firms WhileIassumethattheproductionstructureoftheeconomyisunchangedrelativetothemain text, capital is held by capital producers. These firms buy the investment good, augment the existing capital stock, and rent out capital to the producers of traded and non-traded goods. They also pay a dividend to the stockholders. The optimization problem of the capital producers is given by: (cid:4)∞ (cid:5) (cid:6) max Q (1+n )t P r k −PI x i,t0 |t i i,t i,t i,t−1 i,t i,t ki,t,xi,t t=t0 s.t. (1+n )k ≤ (1−δ)k +x . (75) i i,t i,t−1 i,t Q is the pricing kernel used by the capital producers to discount future profits. The i,t0 |t capital used in time t production, k , and investment, x , are expressed as per capita i,t−1 i,t averages. TheaveragepercapitaldividendpaymentisdenotedbyP d andsatisfiesP d = i,t i,t i,t i,t P r k −PI x . i,t i,t i,t−1 i,t i,t 43

Closing large open economy models: Technical Appendix Martin Bodenstein∗∗ Federal Reserve Board Abstract This Technical Appendix describes the details of the following topics in the paper “Closing large open economy models:” the simple example of Section 2, the linearized model equations of the model in the main text, the overlapping generations framework, and the model with real and nominal rigidities.

A Appendix: Details on the simple model This appendix describes the equations underlying the discussion in Section 2. The first order conditions of the households imply: (cid:2) (cid:3) αT q−εT b −Q b c = 12 t yT + 1,t−1 1,t 1,t , (1) 12,t αT 11 +αT 12 q t 1−εT 1,t (cid:2) Φ 1 (q t ) (cid:3) αT b −Q b c = (cid:4) (cid:5) 22 yT + 2,t−1 2,t 2,t , (2) 22,t 1−εT 2,t Φ (q )q αT 1 +αT 1 t t 21 qt 22 cT = Φ (q )yT +b −Q b , (3) 1,t 1 t 1,t 1,t−1 1,t 1,t Φ (q ) cT = Φ (q )yT + 2 t (b −Q b ), (4) 2,t 2 t 2,t q Φ (q ) 2,t−1 2,t 2,t t 1 t where (cid:6) (cid:9) P (cid:7) 1 (cid:8) 1−εT − 1− 1 εT 1 Φ (q ) ≡ 1,t = αT +αT , (5) 1 t P c +P c 11 q 12 q 1,t 11,t 2,t 12,t t t (cid:6) (cid:9) P (cid:7) 1 (cid:8) 1−εT − 1− 1 εT Φ (q ) ≡ 2,t = αT +αT , (6) 2 t P c +P c 21 q 22 1,t 21,t 2,t 22,t t Φ (q ) rer = q 1 t . (7) t t Φ (q ) 2 t The price of the aggregate consumption good in country 1 is taken to be the num´eraire. Loglinearizing these equations around a steady state with zero net foreign assets and setting the exogenous variables yT and yT to be constant (“xˆ” for log-linearized and “Δx” for linearized 1,t 2,t variables x). Variables without time index indicate steady state values: (cid:10) (cid:11) 1 cˆ = − (1−εT)(1−αT )+εT qˆ + [Δb −βΔb ], (8) 12,t 11 t y Φ (q) 1,t−1 1,t 1 1 1 cˆ = (1−εT)(1−αT )qˆ − [Δb −βΔb ], (9) 22,t 22 t qΦ (q)y 1,t−1 1,t 1 2 r(cid:12)er = −(1−αT −αT )qˆ, (10) t 11 22 t where I have used the fact that in steady state Φ(cid:2) 1 (q)q = −(1−αT ) and Φ(cid:2) 2 (q)q = (1−αT ). Φ 1 (q) 11 Φ 2 (q) 22 Using (8) and (9) in the approximation of the market clearing condition for good 2: ∂z ∂z 2qqˆ + 2 [Δb −βΔb ] = 0, (11) ∂q t ∂dW 1t−1 1t 1 2

with the coefficients (cid:13) (cid:7) (cid:8) (cid:16) ∂z (cid:14) (cid:15) c (q) c (q) 2q = c (q) 1−εT 11 + 22 −1 , (12) ∂q 12 yT yT (cid:13)(cid:7) 1 (cid:8) 2 (cid:16) ∂z 1 c (q) c (q) 2 = − 11 + 22 −1 . (13) ∂dW qΦ (q) yT yT 1 1 1 2 Note that in a steady state with balanced trade α = cii (q) . The slope of the excess demand ii yi T function in steady state with respect to q is given by equation (12). Aggregate consumption is determined by: 1 cˆT = −(1−αT )qˆ + (Δb −βΔb ), (14) 1,t 11 t c (q) 1,t−1 1,t 1 Φ (q) 1 cˆT = (1−αT )qˆ − 2 (Δb −βΔb ), (15) 2,t 22 t qΦ (q)c (q) 1,t−1 1,t 1 2 while the risk sharing condition is given by: cˆT −cˆT = cˆT −cˆT −(1−αT −αT )(qˆ −qˆ ) 1,t 1,t+1 2,t 2,t+1 11 (cid:2)22 t(cid:3) t+1 γ 1 + 1+ Δb . (16) βΦ (q) q 1,t 1 Thus, the dynamic system under portfolio costs can be written as: ⎛ ⎞ ⎛ (cid:7) (cid:8) ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎝ −∂ ∂ ∂ ∂ d z z q W 2 2 q 1β 0 ⎟ ⎠ ⎝ Δb 1,t+1 ⎠ + ⎜ ⎝ 1+β +β ∂ d ∂ z z q 2 2 qγˇ ∂ ∂ ∂ ∂ d z z q W 2 2 q 1 −∂ ∂ ∂ ∂ d z z q W 2 2 q 1 ⎟ ⎠ ⎝ Δb 1,t ⎠ = 0, 0 1 Δb 1,t −1 0 Δb 1,t−1 (17) with the coefficients (cid:13)(cid:7) (cid:8) (cid:16) 1 c (q) c (q) 2 dz = − 11 + 22 −1 2 qΦ (q) yT yT 1 (cid:13) (cid:7)1 2 (cid:8)(cid:16)(cid:13) (cid:16) 1 c (q) c (q) c (q) c (q) − 2− 11 + 22 11 + 22 εT < 0, (18) qΦ (q) yT yT yT yT 1 (cid:2) (cid:3) 1 2 1 2 γ 1 γˇ = 1+ > 0. (19) β2 Φ (q) q 1 Hence, the characteristic equation associated with the dynamic system (17) under the portfolio cost approach satisfies: (cid:23) (cid:24) 1+β ∂z2q 1 λ2 −λ + ∂q γˇ + = 0. (20) β dz β 2 3

Alternatively, assume that households have Uzawa-type preferences. The time discount (cid:14) (cid:15) factor βj is replaced by θ = β (cT )θ with β (cT ) = 1+ψ cT −1 . ψ is chosen i,t+1+j i i,t+j i,t+j i i,t+j i i,t+j i suchthatβ (cT) = β insteadystate. Absentinternalization,therisksharingconditionimplies: i i βcˆT −cˆT = βcˆT −cˆT −(1−αT −αT )(qˆ −qˆ ). (21) 1,t 1,t+1 2,t 2,t+1 11 22 t t+1 Hence, the characteristic equation associated with the dynamic system under endogenous discounting without internalization satisfies: (cid:23) (cid:24) (cid:23) (cid:24) (cid:25) (cid:25) 1+β dz dz 1 λ2 − + 2 λ+ 1+ 2 = 0, (22) β dz dz β 2 2 with (cid:2) (cid:7) (cid:8) (cid:7) (cid:8)(cid:3) (cid:13) (cid:16) d (cid:25) z = − 1 β(cid:3) 1 (q) 1− c 21 (q) + β(cid:3) 2 (q) 1− c 22 (q) εT c 11 (q) + c 22 (q) > 0, (23) 2 qΦ (q) β (q) yT β (q) yT yT yT 1 1 1 2 2 1 2 as the discount factors are assumed to be decreasing in the level of consumption.1 B Appendix: Linearized model This appendix describes the nonlinear first order conditions together with their linear approximations. xˆ denotes the log-linear approximation of variable x from its steady state value, Δx denotes its deviation in absolute value: 1. choice of leisure in country i: • with additive separable preferences χ lχ = Φ w c−σ 0 i,t i,t i,t i,t is approximated by χˆl = Φˆ +wˆ −σcˆ i,t i,t i,t i,t • with Cobb-Douglas preferences 1 1 (1−ξ) = Φ w ξ (1−l ) i,t i,t c i,t i,t 1Ifthe discountfactorswereincreasinginthe levelofconsumption,d(cid:25)z 2 wouldbenegativeandthe dynamicsaroundany deterministic steady state would be explosive. 4

is approximated by l i ˆl = Φˆ +wˆ −cˆ 1−l i,t i,t i,t i,t i 2. choice of capital in country i: • with additive separable preferences (cid:7) (cid:8) c −σ Φ Π β i,t+1 i,t+1 i,t [Π r +(1−δ)] = 1 c Φ Π i,t+1 i,t+1 i,t i,t i,t+1 is approximated by (cid:26) (cid:27) βΠ r Πˆ +rˆ −σ(cˆ −cˆ ) i i i,t+1 i,t+1 i,t+1 i,t (cid:4) (cid:5) (cid:4) (cid:5) + Φˆ −Φˆ − Πˆ −Πˆ = 0 i,t+1 i,t i,t+1 i,t which is augmented by βˆ on the left side under endogenous discounting i,t • with Cobb-Douglas preferences (cid:23) (cid:24) β cξ i,t+1 (1−l i,t+1 ) 1−ξ 1−σ c i,t Φ i,t+1 Π i,t [Π r +(1−δ)] = 1 cξ i,t (1−l i,t ) 1−ξ c i,t+1 Φ i,t Π i,t+1 i,t+1 i,t+1 is approximated by {(1−σ)ξ −1}(cˆ −cˆ )−(1−σ)(1−ξ)l ( ˆl −ˆl ) i,t+1 i,t i i,t+1 i,t (cid:4) (cid:5) +Φ ˆ −Φ ˆ +Π ˆ −Π ˆ +βΠ r Π ˆ +rˆ = 0 i,t+1 i,t i,t i,t+1 i i i,t+1 i,t+1 which is augmented by βˆ on the left side under endogenous discounting i,t 3. capital accumulation in country i: k ≤ (1−δ)k +x i,t i,t−1 i,t is approximated by kˆ ≤ (1−δ)kˆ +δxˆ i,t i,t−1 i,t 4. consumer budget constraint • for country 1: (cid:14) (cid:15) Φ c = Φ yT +qNyN − 1,tx +dW 1,t 1,t 1,t 1,t 1,t Π 1,t 1,t 1,t is approximated by (cid:14) (cid:15) (cid:14) (cid:15) c cˆ = yT +qNyN Φ Φˆ +Φ yTyˆT +qNyNyˆN +qNyNqˆN 1 1,t 1 1 (cid:4) 1 1 1,t (cid:5) 1 1 1,t 1 1 1,t 1 1 1,t Φ Φ − 1x Φ ˆ −Π ˆ − 1x xˆ +ΔdW 1 1,t 1,t 1 1,t 1,t Π Π 1 1 5

• for country 2: (cid:14) (cid:15) Φ 1 1 c = Φ yT +qNyN − 2,tx − dW 2,t 2,t 2,t 2,t 2,t Π 2,t rer ν 1,t 2,t t 2 is approximated by (cid:14) (cid:15) (cid:14) (cid:15) c cˆ = yT +qNyN Φ Φ ˆ +Φ yTyˆT +qNyNyˆN +qNyNqˆN 2 2,t 2 2 (cid:4) 2 2 2,t (cid:5) 2 2 2,t 2 2 2,t 2 2 2,t Φ Φ 1 1 − 2x Φˆ −Πˆ − 2x xˆ − ΔdW Π 2 2,t 2,t Π 2 2,t rerν 1,t 2 2 2 and under complete markets, the two budget constraints are aggregated to deliver Φ c +ν rer c = Φ (w l +r k )− 1,tx 1,t 2 t 2,t 1,t 1,t 1,t 1,t 1,t−1 1,t Π (cid:7) 1,t (cid:8) Φ +ν rer Φ (w l +r k )− 2,tx 2 t 2,t 2,t 2,t 2,t 2,t−1 2,t Π 2,t 5. risk sharing condition with additive separable preferences • with adjustment costs: (cid:7) (cid:8) (cid:7) (cid:8) (cid:7) (cid:8) (cid:7) (cid:8) U U rer b b β c1,t+1 −β c2,t+1 t = Γ (cid:3) 1,t −Γ (cid:3) 2,t 1 U 2 U rer P P c1,t c2,t t+1 1,t 2,t under additive separable preferences is approximated by (cid:7) (cid:8) 1 1 −σ (cˆ −cˆ )+σ (cˆ −cˆ )−r(cid:12)er +r(cid:12)er = −φ + Δb 1 1,t+1 1,t 2 2,t+1 2,t t t+1 b Φ rerΦ t 1 2 • with debt-elastic interest rate: E Uc2,t+1 rert (cid:14) (cid:15) t Uc2,t rert+1 = Ψ b −¯b E Uc1,t+1 1,t+1 t Uc1,t is approximated by −σ (cˆ −cˆ )+σ (cˆ −cˆ )−r(cid:12)er +r(cid:12)er = −Ψ (cid:3) (0)Δb 1 1,t+1 1,t 2 2,t+1 2,t t t+1 1,t+1 • with endogenous discounting, but no internalization (cid:7) (cid:8) (cid:7) (cid:8) U U rer β c1,t+1 −β c2,t+1 t = 0 1,t U 2,t U rer c1,t c2,t t+1 under additive separable preferences is approximated by −σ (cˆ −cˆ )+σ (cˆ −cˆ )−r(cid:12)er +r(cid:12)er = −βˆ +βˆ 1 1,t+1 1,t 2 2,t+1 2,t t t+1 1,t 2,t • with endogenous discounting and internalization: U −η β(cid:3) U −η β(cid:3) rer β c1,t+1 1,t+1 1,t+1 = β c2,t+1 2,t+1 2,t+1 t 1,t U −η β(cid:3) 2,t U −η β(cid:3) rer c(cid:14) 1,t 1,t 1,t (cid:15) c2,t 2,t 2,t t+1 η = E −U +β η i,t t i,t+1 i,t+1 i,t+1 6

is approximated by η β(cid:3) (cid:14) (cid:15) β(cid:3)U c cˆ +β(cid:3)U l ˆl −β 1 1 ηˆ −ηˆ 1 c1 1 1,t 1 l1 1 1,t 1U −η β(cid:3) 1,t+1 1,t c1 1 1 (cid:4) (cid:5) U −η β(cid:3)(cid:3)U U −η β(cid:3)(cid:3)U +β cc1 1 1 c1c (cˆ −cˆ )+β cl1 1 1 l1l ˆl −ˆl 1 U −η β(cid:3) 1 1,t+1 1,t 1 U −η β(cid:3) 1 1,t+1 1,t c1 1 1 c1 1 1 η β(cid:3) (cid:14) (cid:15) = β(cid:3)U c cˆ +β(cid:3)U l ˆl −β 2 2 ηˆ −ηˆ 2 c2 2 2,t 2 l2 2 2,t 2U −η β(cid:3) 2,t+1 2,t c2 2 2 (cid:4) (cid:5) U −η β(cid:3)(cid:3)U U −η β(cid:3)(cid:3)U +β cc2 2 2 c2c (cˆ −cˆ )+β cl2 2 2 l2l ˆl −ˆl 2 U −η β(cid:3) 2 2,t+1 2,t 2 U −η β(cid:3) 2 2,t+1 2,t c2 2 2 c2 2 2 −r(cid:12)er +r(cid:12)er t t+1 and (cid:4) (cid:5) 1−η β(cid:3) ηˆ = − i i U c cˆ +U l ˆl +β ηˆ i,t η ci i i,t+1 li i i,t+1 i i,t+1 i 6. endogenous discount factors • with additive separable preferences: (cid:23) (cid:24) c1−σi l1+χi β = 1−ψ exp i,t −χ i,t i,t i 1−σ 0 1+χ i i is approximated by (cid:26) (cid:27) β −1 βˆ = c1−σicˆ −χ l1+χiˆl i,t β i i,t 0 i i,t • with Cobb Douglas preferences (cid:4) (cid:26) (cid:27)(cid:5) −1 β = 1+ψ cξ (1−l ) 1−ξ i,t i,t i,t is approximated by l βˆ = −(1−β )ξcˆ +(1−β ) i (1−ξ) ˆl i,t i i,t i 1−l i,t i 7. wealth evolution dW = b −Q b 1,t 1,t−1 1,t 1,t where Q = β Uc1,t+1 implies the approximation 1,t 1,t Uc1,t ΔdW = Δb −βΔb 1,t 1,t−1 1,t 8. investment aggregate for country i (cid:6) (cid:9) εI i (cid:14) (cid:15) 1 εI i−1 (cid:14) (cid:15) 1 εI i−1 εI i −1 x i,t = αI i1 εI i (x i1,t ) εI i + αI i2 εI i (x i2,t ) εI i 7

is approximated by xˆ = αI (cid:7) x i1 (cid:8) εI i ε − I i 1 xˆ +αI (cid:7) x i2 (cid:8) εI i ε − I i 1 xˆ i,t i1 αI x i1,t i2 αI x i2,t i1 i i2 i 9. investment optimal choices • for country 1: (cid:7) (cid:8) αI 1 εI 1 x = 12 x 12,t αI q 11,t 11 t is approximated by xˆ = xˆ −εTqˆ 12,t 11,t 1 t • for country 2: (cid:7) (cid:8) αI 1 εI 2 x = 22 x 22,t αI q 21,t 21 t is approximated by xˆ = xˆ −εTqˆ 22,t 21,t 2 t 10. consumption aggregate over traded goods for country i: (cid:6) (cid:9) εT i (cid:14) (cid:15) 1 εT i −1 (cid:14) (cid:15) 1 εT i −1 εT i −1 cT i,t = αT i1 εT i (c i1,t ) εT i + αT i2 εT i (c i2,t ) εT i is approximated by cˆT = αT (cid:7) c i1 (cid:8) εT i εT i −1 cˆ +αT (cid:7) c i2 (cid:8) εT i εT i −1 cˆ i,t i1 αTcT i1,t i2 αTcT i2,t i1 i i2 i 11. consumption aggregate over traded and non traded goods for country i (cid:6) (cid:9) εN i (cid:14) (cid:15) 1 (cid:14) (cid:15)εN i −1 (cid:14) (cid:15) 1 (cid:14) (cid:15)εN i −1 εN i −1 c i,t = αN i1 εN i cT i,t εN i + αN i2 εN i cN i,t εN i is approximated by (cid:7) (cid:8)εN i −1 (cid:7) (cid:8)εN i −1 cˆ = αN cT i εN i cˆT +αN cN i εN i cˆN i,t i1 αNc i,t i2 αNc i,t i1 i i2 i 12. consumption optimal choices non-traded goods 8

• for country 1: cN 1,t = c 12,t α α N 1 N 2 ⎡ ⎣ (cid:14) αT 11 (cid:15) ε 1 T 1 (cid:7) c c 11,t (cid:8) εT 1 εT 1 −1 + (cid:14) αT 12 (cid:15) ε 1 T 1 ⎤ ⎦ ε ε T 1 T 1 − − ε 1 N 1 (cid:23) q + q 1 N η ,t qN (cid:24) −εN 1 (cid:7) α 1 T (cid:8) ε ε N 1 T 1 11 12,t t 1,t 12 is approximated by (cid:14) (cid:15) 1 (cid:4) (cid:5) εT 1−1 cˆN = cˆ + εT 1 −εN 1 αT 11 εT 1 c c 1 1 1 2 εT 1 (cˆ −cˆ ) 1,t 12,t εT 1 1 (cid:4) (cid:5) εT 1−1 1 11,t 12,t (αT 11 )εT 1 c c 1 1 1 2 εT 1 +(αT 12 )εT 1 q (cid:14) (cid:15) −εN qˆN −qˆ 1 q+ηqN 1,t t 1 • for country 2: cN 2,t = c 21,t α α N 2 N 2 2 1 ⎡ ⎣ (cid:14) αT 21 (cid:15) ε 1 T 2 + (cid:14) αT 22 (cid:15) ε 1 T 2 (cid:7) c c 2 2 2 1 , , t t (cid:8) εT 2 εT 2 −1 ⎤ ⎦ ε ε T 2 T 2 − − ε 1 N 2 (cid:23) q 1 t + q 2 N η ,t q 2 N ,t (cid:24) −εN 2 (cid:7) α 1 T 21 (cid:8) ε ε N 2 T 2 is approximated by (cid:14) (cid:15) 1 (cid:4) (cid:5) εT 2−1 cNcˆN = cˆ + εT 2 −εN 2 αT 22 εT 2 c c 2 2 2 1 εT 2 (cˆ −cˆ ) 2 2,t 21,t εT 2 1 1 (cid:4) (cid:5) εT 2−1 22,t 21,t (αT 21 )εT 2 +(αT 22 )εT 2 c c 2 2 2 1 εT 2 1 (cid:14) (cid:15) −εN q qˆN +qˆ 2 1 +ηqN 2,t t q 2 13. consumption optimal choice traded goods • for country 1: (cid:23) (cid:24) αT 1+ηqN εT 1 c = 12 1,t c 12,t αT q +ηqN 11,t 11 t 1,t (cid:2) (cid:3) 1 1 εT cˆ = cˆ + − εTηqNqˆN − 1 qqˆ 12,t 11,t 1+ηqN q +ηqN 1 1 1,t q +ηqN t 1 1 1 • for country 2: (cid:23) (cid:24) αT 1 +ηqN εT 2 c = 22 qt 2,t c 22,t αT 1+ηqN 21,t 21 (cid:6) 2,t (cid:9) 1 1 εT 1 cˆ = cˆ + − εTηqNqˆN − 2 qˆ 22,t 21,t 1 +ηqN 1+ηqN 2 2 2,t 1 +ηqN q t q 2 2 q 2 9

14. definition Φ i (cid:14) (cid:15) Φ = Φ q ,qN i,t i t i,t ∂Φ q ∂Φ qN Φˆ = i qˆ + i i qˆN i,t ∂q Φ t ∂qN Φ i,t i i i • for country 1: ∂Φ 1 q = −(cid:26) c c 1 N 1 2q (cid:27) and ∂Φ 1 q 1 N = ∂Φ 1 q y 1 Nq 1 N ∂q Φ 1 (1+ηqN) c11 +(q +ηqN) c12 +qN ∂q 1 N Φ 1 ∂q Φ 1 c 12 q 1 c12 1 cN 1 1 • for country 2: ∂Φ 2 q = (cid:26)(cid:4) (cid:5) c c 2 N 2 11 q (cid:27) and ∂Φ 2 q 2 N = − y 2 Nq 2 N ∂Φ 2 q ∂q Φ 2 1 +ηqN c21 +(1+ηqN) c22 +qN ∂q 2 N Φ 2 c 21 1 ∂q Φ 2 q 2 c22 2 cN 2 2 q 15. definition Π i Π = Π (q ) i,t i t ∂Π q Πˆ = i qˆ i,t ∂q Π t i • for country 1: ∂Π q q 1 = − ∂q Π 1 i11 +q i12 • for country 2: ∂Π q i21 2 = i22 ∂q Π 2 i21 +q i22 16. real exchange rate Φ rer = 1,tq t t Φ 2,t is approximated by r(cid:12)er = Φˆ −Φˆ +qˆ t 1,t 2,t t 17. Cobb-Douglas production function with traded goods for country i: (cid:23) (cid:24) (cid:23) (cid:24) AT lT ωT li kT ωT ki yT = i,t i,t i,t i,t ωT ωT li ki 10

is approximated by yˆT = ωTAˆT +ωTˆlT +ωT kˆT i,t li i,t li i,t ki i,t 18. optimal labor input choices traded goods for country i: ωTyT w = li i,t i,t lT i,t is approximated by wˆ = yˆT −ˆlT i,t i,t i,t 19. optimal capital input choices traded goods for country i: ωT yT r = kli i,t i,t kT i,t is approximated by rˆ = yˆT −kˆT i,t i,t i,t 20. Cobb-Douglas production function with non-traded goods for country i: (cid:23) (cid:24) (cid:23) (cid:24) ANlN ωN li kN ωN ki yN = i,t i,t i,t i,t ωN ωN li ki is approximated by yˆN = ωNAˆN +ωNˆlN +ωNkˆN i,t li i,t li i,t ki i,t 21. optimal labor input choices non-traded goods for country i: (cid:23) (cid:24) PN ωNyN w = i,t li i,t i,t P lN i,t i,t is approximated by wˆ = qˆN +yˆN −ˆlN i,t i,t i,t i,t 22. optimal capital input choices non-traded goods for country i: (cid:23) (cid:24) PN ωN yN r = i,t kli i,t i,t P kN i,t i,t 11

is approximated by rˆ = qˆN +yˆN −kˆN i,t i,t i,t i,t 23. labor market clearing for country i: lT +lN = l i,t i,t i,t is approximated by lT lN i ˆlT + i ˆlN = ˆl l i,t l i,t i,t i i 24. capital market clearing for country i: kT +kN = k i,t i,t i,t−1 is approximated by kT kN i kˆT + i kˆN = kˆ k i,t k i,t i,t−1 i i 25. goods market clearing non-traded goods for country i: yN = cN +η(c +c ) i,t i,t i1,t i2,t is approximated by cN ηc ηc yˆN = i cˆN + i1cˆ + i2cˆ i,t yN i,t yN i1,t yN i2,t i i i 26. goods market clearing for good 2: yT ν = c +x +(c +x )ν 2,t 2 12,t 12,t 22,t 22,t 2 is approximated by c c x x yˆT = 12 cˆ + 22cˆ + 12 xˆ + 22xˆ 2,t yTν 12,t yT 22,t yTν 12,t yT 22,t 2 2 2 2 2 2 These equations can be rewritten in terms of the three state variables Δb , kˆ , and kˆ . 1,t 1,t 2,t 12

C Appendix: Overlapping generations model This appendix merges the framework with overlapping generations of infinitely lived households by Weil (1989) and Ghironi (2006) with the model presented in the main text. Absent Ricardian equivalence, the steady state level of the net foreign asset position can be shown to be unique. Furthermore, the local dynamics of the net foreign asset position are stationary. C.1 The model Each household consumes, supplies labor, and holds financial assets. Households are born on different dates owning no assets, but they own the present discounted value of their labor income. The number of households in country i N grows over time at the exogenous rate i,t n , i.e., N = (1+n )N . The utility function of each household is assumed to be of the i i,t+1 i i,t Cobb-Douglas type. Capital is no longer accumulated by households directly, but through capital producers. Households can purchase shares in these and all other firms that produce in their country of residence. To allow the intertemporal elasticity of substitution σ to differ i from 1 as in Ghironi (2006), aggregation requires perfect foresight. C.1.1 Individual Households Consumers have identical preferences over the real consumption index and leisure. At time t , the representative consumer in country i born in period v ∈ (−∞,t ) maximizes the 0 0 intertemporal utility function: (cid:26) (cid:27) (cid:14) (cid:15) (cid:14) (cid:15) ∞ cv ξi 1−lv 1−ξi 1−σi −1 Uv = βt−t0 i,t i,t (24) t0 1−σ t=t0 i s.t. (cid:14) (cid:15) PCcv ≤ P w lv +bv −Q bv + Q(cid:8) +P d (cid:18)v −Q(cid:8)(cid:18)v . (25) i,t i,t i,t i,t i,t i,t−1 i,t i,t i,t i,t i,t i,t−1 i,t i,t The final consumption of a representative household of country i born in period v at time t is denoted by cv . The production of the final consumption good follows the same process as i,t outlined in the main text. lv is the labor supply of such a household. The bond holdings are i,t 13

denoted by bv . (cid:18)v are the household’s holdings of the domestic stock, that sells at the price i,t i,t Q(cid:8) and pays the dividend P d . The supply of the stock is normalized to 1. i,t i,t i,t Household choices for consumption, leisure, and asset holdings must satisfy the following first order conditions: 1−ξ 1 lv = 1− i cv , (26) i,t ξ Φ w i,t ! i i,t i,t " (cid:7) (cid:8) (cid:7) (cid:8) cv −σi PC Φ w (1−ξi )(1−σi ) Q = β i,t+1 i,t i,t i,t , (27) i,t cv PC Φ w ! i,t i,t+1 i,t+1 i,t+1 " (cid:7) (cid:8) (cid:7) (cid:8) cv −σi PC Φ w (1−ξi )(1−σi ) 1 = β R(cid:8) i,t+1 i,t i,t i,t , (28) i,t+1 cv PC Φ w i,t i,t+1 i,t+1 i,t+1 where the return of the stock is given by R(cid:8) = Q(cid:4) i,t+1 +Pi,t+1di,t+1. i,t+1 Q(cid:4) i,t C.1.2 Firms WhileIassumethattheproductionstructureoftheeconomyisunchangedrelativetothemain text, capital is held by capital producers. These firms buy the investment good, augment the existing capital stock, and rent out capital to the producers of traded and non-traded goods. They also pay a dividend to the stockholders. The optimization problem of the capital producers is given by: ∞ (cid:14) (cid:15) max Q (1+n )t P r k −PI x i,t0 |t i i,t i,t i,t−1 i,t i,t ki,t,xi,t t=t0 s.t. (1+n )k ≤ (1−δ)k +x . (29) i i,t i,t−1 i,t Q is the pricing kernel used by the capital producers to discount future profits. The i,t0 |t capital used in time t production, k , and investment, x , are expressed as per capita i,t−1 i,t averages. TheaveragepercapitaldividendpaymentisdenotedbyP d andsatisfiesP d = i,t i,t i,t i,t P r k −PI x . The optimality condition for the capital producer is therefore given by: i,t i,t i,t−1 i,t i,t Q i,t0 |t+1 P i c ,t+1 Φ i,t+1 Π i,t [Π r +(1−δ)] = 1. (30) Q Pc Φ ΠI i,t+1 i,t+1 i,t0 |t i,t i,t i,t+1 14

C.1.3 Aggregation If the population at time t is (1+n )t, aggregate per capita consumption in country i satisfies i ⎡ ⎤ 1 ni c−t +...+ ni c−1 + ni c0 c = ⎣ (1+ni )t+1 i,t (1+ni )2 i,t 1+ni i,t ⎦. i,t (1+n i )t +nc1 i,t +n i (1+n i )c2 i,t +...+n i (1+n i )t−1ct i,t 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 1−ζ 1 1−l = i c . (31) i,t ζ Φ w i,t i i,t i,t From (27) and (31) the price of the bond can be shown to satisfy (cid:23) (cid:24) (cid:7) (cid:8) Q = β Φ i,t w i,t (1−ξi )(1−σi ) c i,t+1 − 1+ ni ni ct i, + t+ 1 1 −σi P i C ,t (32) i,t Φ w 1 c PC i,t+1 i,t+1 1+ni i,t i,t+1 under perfect foresight. To obtainthe aggregateper capita budget constraint, notethat thestock isin fixed supply. Thus, 1 c = Φ w l +Φ r k −Π x + {b −(1+n )Q b }. (33) i,t i,t i,t i,t i,t i,t i,t−1 i,t i,t PC i,t−1 i i,t i,t i,t Remains to determine the consumption of the newly born generation in period t. Defining total asset holdings of generation v in period t by 1 1 Q(cid:8) av = bv + i,t(cid:18)v , (34) i,t PC i,t Q PC i,t i,t i,t i,t the budget constraint of a generation v household is written as Q av ≤ av +Φ w lv −cv . (35) i,t i,t i,t−1 i,t i,t i,t i,t Iterating forward on equation (35) and invoking the appropriate transversality condition, the budget constraint can be shown to satisfy (cid:10) (cid:14) (cid:15)(cid:11) 0 = av +Φ w Θ 2 − cv +Φ w 1−lv Θ 1 (36) i,t−1 i,t i,t i,t i,t i,t i,t i,t i,t 15

with (cid:23) (cid:24) Θ 1 = 1+Q Q i,t P i C ,t+1 − σ 1 i (cid:7) Φ i,t w i,t (cid:8)(1−ξi σ )( i 1−σi ) Θ 1 , (37) i,t i,t β PC Φ w i,t+1 i,t i,t+1 i,t+1 Φ w Θ 2 = 1+Q i,t+1 i,t+1 Θ 2 , (38) i,t i,t Φ w i,t+1 i,t i,t using the fact that (cid:23) (cid:24) Q i,t+s−1 ...Q i,t+1 Q i,t P i C ,t+s − σ 1 i (cid:7) Φ i,t w i,t (cid:8)(1−ξi σ )( i 1−σi ) (cid:10) cv +Φ w (cid:14) 1−lv (cid:15)(cid:11) βs PC Φ w i,t i,t i,t i,t (cid:14) i,t (cid:15) i,t+s i,t+s = cv +Φ w 1−lv . (39) i,t+s i,t+s i,t+s i,t+s As newly born agents do not hold financial assets, their consumption is given by Θ2 ct = ξ Φ w i,t. (40) i,t i i,t i,t Θ1 i,t Note, in general the consumption of newly born households differs from the aggregate per capita consumption. Hence, the steady states of the model with overlapping generations differ from those obtained in the representative agents model. D Appendix: nominal and real rigidities This appendix lays out the model with nominal and real rigidities. Following the influential work of Christiano, Eichenbaum, and Evans (1999) and Smets and Wouters (2003), the model features sticky wages and prices, consumption habits, and investment adjustment costs. Wage and price contracts are modeled as in Calvo (1983) and Yun (1996). Unless noted otherwise, the modeling features introduced in the main text remain unchanged. As in the main text, variables are expressed in per capita terms. D.1 Additional modeling features Wages A continuum of monopolistically competitive households (indexed on the unit interval) supplies differentiated labor services to the intermediate goods-producing sector. A 16

representative labor aggregator combines the households’ labor hours in the same proportions as firms would choose. This labor index l has the Dixit-Stiglitz form: t+j (cid:2)# (cid:3) 1 1 1+θw i l t+j = l t+j (h)1+θw i dh , (41) 0 where θw > 0 and l (h) is hours worked by a typical member of household h. The aggregator i t minimizes the cost of producing a given amount of the aggregate labor index, taking each household’s wage rate W (h) as given. One unit of the labor index sells at the unit cost t+j W : t+j (cid:2)# (cid:3) 1 −1 −θw i W t+j = W t+j (h)θw i dh . (42) 0 W is referred to as the aggregate wage index. The aggregator’s demand for the labor t+j services of household h satisfies: l (h) = (cid:2) W i,t+j (h) (cid:3) − 1+ θw i θw i l . (43) i,t+j W i,t+j i,t+j The utility functional of a typical member of household h is: (cid:13) ∞ E $ βj 1 (c (h)−κ c −ν ) 1−σi t i 1−σ i,t+j i i,t+j−1 i,t+j j=0 i (cid:7) (cid:8)(cid:16) χ mb (h) − 1− 0i χ (l i,t+j (h)) 1−χi +V i, P t+ C j+1 , (44) i i,t+j where the discount factor β satisfies 0 < β < 1. As in Smets and Wouters (2003), I allow i i for the possibility of external habits. At date t + j, a member of household h cares about consumption relative to lagged per capita consumption c . ν is a preference shock i,t+j−1 i,t+j that follows an AR(1) process. Furthermore, the period utility function depends on labor l (h) and the end-of-period real money balances, mbi,t+j+1 (h) . The budget constraint of a i,t+j Pi C ,t+j 17

member of household h is given by: 1 (x −x (h)) 2 PC c (h)+PI x (h)+PI φI i,t+j i,t−1+j i,t+j i,t+j i,t+j i,t+j i,t+j2 i x (h) # i,t−1+j + ζ bD (h)−bD (h)+PG BG −BG t+j,t+1+j i,t+j,t+1+j i,t−1+j,t+j i,t+j i,t+j i,t−1+j S (cid:7) (cid:8) B +Q B (h)−B (h)+PC Γ i,t+j i,t+j i,t+j i,t−1+j i,t+j PC i,t+j ≤ (1−τw)W (h)l (h)+(1−τr)R k (h) i i,t+j i,t+j i i,t+j i,t−1+j +Pr +Tr . (45) i,t+j i,t+j The left hand side of the budget constraint summarizes the expenditures for consumption, investment (plus investment adjustment costs), expenditure and income from domesticallyissued and traded bonds, government bonds, and the internationally-traded bond (plus portfolio cost). The right hand side provides information about the agent’s labor and capital income, profits, and transfers. Household wages are determined by Calvo-style staggered contracts subject to static wage indexation. In particular, with probability 1 − ξw, each household is allowed to reoptimize i its wage contract. If a household is not allowed to reoptimize its wage rate, it sets its wage according to W i,t+j (h) = ω¯ i,t−1+j W i,t−1+j (h) with ω¯ i,t+j = ωγ i, w i t+j π∗1−γw i . ω i,t+j is the actual wage inflation rate and π∗ is the steady state inflation rate. γw is referred to as the wage i indexation parameter. A household who is resetting its wage in period t chooses W∗(h) such i,t that: ⎡ ⎤ ∞ ⎢ (cid:7) 1+θw (cid:8) (cid:6) VW W∗ (h) (cid:9) − 1+ θw i θw i l ⎥ 0 = −E t (β i ξw i )j⎣χ 0i (l i,t+j (h)) −χi − θw i i,t W ,j i,t W i ∗ ,t+ ( j h) ⎦ j=0 i i,t+j i,t ⎡ ⎤ ∞ ⎢ (cid:7) 1+θw (cid:8) (cid:6) VW W∗ (h) (cid:9) − 1+ θw i θw i l ⎥ +E (β ξw)j⎣(1−τw)λ (h)VW W∗ (h) − i i,t,j i,t i,t+j ⎦ t i i i i,t+j i,t,j i,t θw W W∗ (h) j=0 i i,t+j i,t ∞ +E (β ξw)jλ (h)(1−τw)VW l (h), t i i i,t+j i i,t,j i,t+j j=0 ’j where VW = 1 and VW = ω¯ for j > 0. Since households have access to complete i,t,0 i,t,j i,t+k−1 k=1 18

insurance markets domestically, the marginal utility of wealth λ (h) is constant across i,t+j households. Thus, I drop the dependence of this variable on h. Furthermore, the assumption of complete markets implies that all households that reoptimize in period t choose the same wage rate W∗ (h) = W∗ Hence: i,t i,t HW (cid:7) W∗ (cid:8) 1− 1+ θw θw i χi i,t = i,t i (46) GW P i,t ii,t ⎛ ⎞ HW = E ∞ (β ξw)jχ ⎝ (cid:2) VW P ii,t+j P ii,t (cid:3) − 1+ θw i θw i l ⎠ 1−χi i,t t i i 0i i,t,jW P i,t+j j=0 i,t+j ii,t+j ⎛ ⎞ = χ ⎝ (cid:2) P ii,t (cid:3) − 1+ θw i θw i l ⎠ 1−χi +(β ξw) 1E (cid:7) ω¯ i,t (cid:8) − 1+ θw i θw i (1−χi ) HW 0i W i,t i i t π i,t+1 i,t ii,t+1 GW = E ∞ (β ξw)j 1−τw i λ PC Φ (cid:7) P ii,t VW (cid:8)(cid:2) VW P ii,t+j P ii,t (cid:3) − 1+ θw i θw i l i,t t i i 1+θw i,t+j i,t+j i,t+j P i,t,j i,t,jW P i,t+j j=0 i ii,t+j i,t+j ii,t+j = 1−τw i λ PCΦ (cid:2) P ii,t (cid:3) − 1+ θw i θw i l +β ξwE (cid:7) ω¯ i,t (cid:8) θ − w i 1 GW . 1+θw i,t i,t i,t W i,t i i t π i,t+1 i i,t ii,t+1 The real wage expressed in terms of country i’s traded good evolves according to (cid:6) (cid:9) W i,t+j = (1−ξw) (cid:7) W i ∗ ,t+j (cid:8) θ − w i 1 +ξw (cid:7) ω¯ i,t+j−1 W i,t+j−1 (cid:8) θ − w i 1 −θw i (47) P i P i π P ii,t+j ii,t+j ii,t+j ii,t+j−1 with π = Pii,t+j . ii,t+j Pii,t+j−1 Prices Both for traded and non-traded goods, there is a continuum of differentiated intermediate goods in country i indexed by h ∈ [0,1], each produced by a single monopolistically competitive firm. Deviating from the main text, I assume that individual varieties are traded internationally rather than the final aggregate. Goods with index h ∈ [0,h¯ ] are traded. i Consumption and investment varieties are interchangeable. The use of the index h in the description of prices is unrelated to its use in the description of wages. Let yT (h) bethe overall quantity producedby firm hofthe tradedgoodsproducers using a i CES technology. Firms operate in perfectly competitive factor markets for aggregate capital and the labor aggregate and thus operate with the same capital to labor ratio. Denoting 19

individual labor and capital demands by lT (h) and kT (h), respectively, and aggregate i,t+j i,t+j ( ( labor and capital demand by lT = 1lT (h)dh and kT = 1kT (h)dh, I define i,t+j 0 i,t+j i,t+j 0 i,t+j aggregate output as: ⎡ ⎤ yT = # 1 yT (h)dh = ⎣ (cid:14) ωT (cid:15) 1−κT (cid:7) ωT ki (cid:8) κT (cid:7) w i,t+j (cid:8) κT κT −1 + (cid:14) ωT (cid:15) 1−κT⎦ κ 1 T kT i,t+j i,t+j li ωT AT r ki i,t+j 0 li i,t+j i,t+j if κT < 1, and i # (cid:7) (cid:8) 1 1 r ωT li yT = yT (h)dh = AT i,t+j kT i,t+j i,t+j 1−ωT i,t+jw i,t+j 0 li i,t+j if κT = 0. i Home andforeignprices of theintermediate goodsaredetermined by Calvo-style staggered contracts. Each period, a firm of country 1’s traded goods sector faces a constant probability 1−ξpT of being able to reoptimize its price at home and 1−ξpT probability of being able to 11 21 reoptimize its price abroad. These probabilities are assumed to be independent across firms, time, and countries. For country 2 the respective probabilities are 1−ξpT and 1−ξpT. Similar 12 22 to the behavior of wages, firms that are not allowed to reoptimize their price in the current period, adjust their price by a geometric average of the actual inflation rate of the previous period (with weight γpT, i = 1,2 and m = 1,2) and the steady state inflation rate π∗ (with im weight 1−γpT, i = 1,2 and m = 1,2). I denote this average by π¯pT = πpTγ p im T π∗1−γ p im T . im im im I define the following aggregates of consumer and investment demand for the varieties produced in country 1: (cid:6) (cid:9) c 11,t+j = (cid:2)# 1 c 11,t+j (h)1+θ 1 p 1 T dh (cid:3) 1+θ p 1 T and c 21,t+j = h¯− 1 θ p 1 T # h ¯ 1 c 21,t+j (h)1+θ 1 p 1 T dh 1+θ p 1 T , 0 0 (cid:6) (cid:9) x 11,t+j = (cid:2)# 1 x 11,t+j (h)1+θ 1 p 1 T dh (cid:3) 1+θ p 1 T and x 21,t+j = h¯− 1 θ p 1 T # h ¯ 1 x 21,t+j (h)1+θ 1 p 1 T dh 1+θ p 1 T , 0 0 and the associated demand equations: (cid:2) P (h) (cid:3)−(1+ pT θ p 1 T) 1 (cid:7) P (h) (cid:8) − 1+ p θ T p 1 T c (h) = 11,t+j θ 1 c and c (h) = 21,t+j θ 1 c , 11,t+j P 11,t+j 21,t+j h¯ P 21,t+j 11,t+j 1 21,t+j (cid:2) P (h) (cid:3)−(1+ p θ T p 1 T) 1 (cid:7) P (h) (cid:8) − 1+ p θ T p 1 T x (h) = 11,t+j θ 1 x and x (h) = 21,t+j θ 1 x . 11,t+j P 11,t+j 21,t+j h¯ P 21,t+j 11,t+j 1 21,t+j 20

(cid:2) (cid:3) ( − 1 −θ p 1 T The aggregator P 11,t+j = 0 1 [P 11,t+j (h)] θ p 1 T dh for domestically produced varieties implies: (cid:2) (cid:3) (cid:2) (cid:3) (cid:4) (cid:5) P∗ − p 1 T π¯ − p 1 T 1 = 1−ξpT 11,t+j θ1 +ξpT 11,t+j−1 θ1 , 11 P 11 π 11,t+j 11,t+j whereP∗ isthepricesetbythosefirmsthatreoptimizeinperiodt+j andπ = P11,t+j . 11,t+j 11,t+j P11,t+j−1 Reoptimizing firms choose to set the same price since marginal costs are identical across firms. With respect to the export price index, P denotes the price index in the currency of 21,t+j country 2 (local currency) and P˜ denotes the price index in the currency of country 1 21,t+j (exporter currency): (cid:6) (cid:9) 1 # h ¯ 1 − 1 −θ p 1 T P 21,t+j = h¯ (P 21,t+j (h)) θ p 1 T dh , 1 0 (cid:6) (cid:9) (cid:4) (cid:5) (cid:2) P∗ (cid:3) − p 1 T (cid:7) π¯ (cid:8) − p 1 T −θ p 1 T 1 = 1−ξpT 21,t+j θ 1 +ξpT 21,t+j−1 θ 1 , 21 P 21 π 21,t+j 21,t+j (cid:6) (cid:9) 1 # h ¯ 1 (cid:4) (cid:5) − 1 −θ p 1 T P˜ 21,t+j = h¯ P˜ 21,t+j (h) θ p 1 T dh , 1 0 ⎡ ⎤ 1 = ⎣ (cid:4) 1−ξpT (cid:5) (cid:6) P˜ 2 ∗ 1,t+j (cid:9) − θ p 1 1 T +ξpT (cid:7) π¯˜ 21,t+j−1 (cid:8) − θ p 1 1 T ⎦ −θ p 1 T , 21 P˜ 21 π˜ 21,t+j 21,t+j ˜ where π = P21,t+j and π˜ = P21,t+j . In addition, I define the price dispersion 21,t+j P21,t+j−1 21,t+j P ˜ 21,t+j−1 functions: # (cid:2) (cid:3)−(1+θ pT) 1 1 P (h) pT Δ = 11,t+j θ 1 dh, 11,t+j P 0 11,t+j (cid:4) (cid:5) (cid:2) P∗ (cid:3)−(1+ p θ T p 1 T) (cid:2) π¯ (cid:3)−(1+ pT θ p 1 T) Δ = 1−ξpT 11,t+j θ 1 +ξpT 11,t+j−1 θ 1 Δ , 11,t+j 11 P 11 π 11,t+j−1 11,t+j 11,t+j 1 # ¯ h1 (cid:7) P (h) (cid:8) − 1+ p θ T p 1 T Δ = 21,t+j θ1 dh, 21,t+j h¯ P 1 0 21,t+j (cid:2) (cid:3) pT (cid:2) (cid:3) pT (cid:4) (cid:5) P∗ − 1+ p θ T 1 π¯ − 1+ p θ T 1 Δ = 1−ξpT 21,t+j θ1 +ξpT 21,t+j−1 θ1 Δ , 21,t+j 21 P 21 π 21,t+j−1 21,t+j 21,t+j 21

and equivalently for Δ ˜ . As the market clearing condition for each variety h satisfies: 21,t+j yT (h) = [c (h)+x (h)]+[c (h)+x (h)]υ , 1,t+j 11,t+j 11,t+j 21,t+j 21,t+j 2 in the aggregate one obtains under the assumption of local currency pricing: ⎛ ⎞ # (cid:2) (cid:3)−(1+θ p 1 T) ⎜ 1 P (h) pT ⎟ yT = ⎝ 11,t+j θ1 dh⎠[c +x ] 1,t+j P 11,t+j 11,t+j 0 11,t+j ⎛ ⎞ ⎜ 1 # h ¯ 1 (cid:7) P (h) (cid:8) − 1+ p θ T p 1 T ⎟ +⎝ 21,t+j θ 1 dh⎠[c +x ] h¯ P 21,t+j 21,t+j 1 0 21,t+j = Δ [c +x ]+Δ [c +x ]. 11,t+j 11,t+j 11,t+j 21,t+j 21,t+j 21,t+j The parameter υ adjusts for the differences in country size. 2 As before the consumption of traded goods requires distribution services in the form of the non-traded goods. Market clearing for the non-traded good is derived from: yN (h) = cN (h)+η(c (h)+c (h)), i,t+j #i,t+j i1,t+j i2,t+j yN = yN (h)dh = ΔNcN +η(c +c ), i,t+j i,t+j i,t i,t i1,t+j i2,t+j where # (cid:2) (cid:3)−(1+θ p n N) P (h) pN ΔN = i,t θn dh i,t P i,t which follows a similar law of motion as the dispersion measure of traded goods prices. Remainstodefinetheoptimizationproblem offirms. Atradedgoodsproducer firmlocated in country 1 that sells domestically chooses its price P (h) to maximize: 11,t ∞ (cid:26) (cid:27) E ξpTψ VpT (1+τ )P (h)yT (h)−MC yT (h) , (48) t 11 1,t,t+j 11,t,j 11 11,t 11,t+j 1,t+j 11,t+j j=0 22

where yT (h) = c (h)+x (h), (49) 11,t+j 11,t+j 11,t+j ’j VpT = π¯ , (50) 11,t,j 11,t+k−1 k=1 (cid:6) (cid:9)−(1+θ pT) 1 yT (h) = V 1 p 1 T ,t,j P 11,t (h) θ p 1 T yT . (51) 11,t+j P 11,t+j 11,t+j τ is a price subsidy. ψ is the stochastic discount factor that relates to the marginal 11 1,t,t+j utility ofwealthasdefined below. Using ∂y1 T 1,t+j (h) = −(1+θ p 1 T)y1 T 1,t+j (h) , thefirst order condition ∂P11,t (h) θ p 1 T P11,t (h) implies: (cid:2) (cid:3) E ∞ (cid:4) ξpT (cid:5) j ψ VpT 1+τ 11P∗ −MC y 1 T 1,t+j (h) = 0, (52) t j=0 11 1,t,t+j 11,t,j 1+θp 1 T 11,t 1,t+j P 1 ∗ 1,t which can also be written as: P∗ HpT 11,t = 11,t, (53) P GpT 11,t 11,t (cid:23) (cid:24)−(1+θ pT) HpT = E ∞ (cid:4) ξpT (cid:5) j ψ P 11,t+j MC 1,t+j V 1 p 1 T ,t,j P 11,t θ p 1 T 1 yT 11,t t 11 1,t,t+j P P P 11,t+j j=0 ⎧ 11,t 11,t+j 11,t+j ⎫ ⎪ ⎨ (cid:7) (cid:8)−(1+θ p 1 T) ⎪ ⎬ MC π¯ pT = 1,tyT +ξpTE ψ π 11,t θ 1 HpT , (54) P 11,t 11,t 11 t⎪ ⎩ 1,t,t+1 11,t+1 π 11,t+1 11,t+1⎪ ⎭ GpT = E ∞ (cid:4) ξpT (cid:5) j ψ VpT 1+τ 11 (cid:23) V 1 p 1 T ,t,j P 11,t (cid:24)−(1 θ + p 1 θ T p 1 T) yT 11,t t j=0 11 1,t,t+j ⎧ 11,t,j 1+θp 1 T P 11,t+j 11,t+j ⎫ 1+τ ⎪ ⎨ (cid:7) π¯ (cid:8) 1− (1+ p θ T p 1 T) ⎪ ⎬ = 11yT +ξpTE ψ π 11,t θ 1 GpT , (55) 1+θp 1 T 11,t 11 t⎪ ⎩ 1,t,t+1 11,t+1 π 11,t+1 11,t+1⎪ ⎭ λ PC PC ψ = β t+1 t+1 t . (56) 1,t,t+1 λ PC PC t t t+1 With respect to export pricing, I distinguish possibilities: producer currency pricing and 23

local currency pricing. If country 1’s exporters engage in producer currency pricing: ∞ (cid:26) (cid:27) E ξpTψ (1+τ )V˜pT P˜ (h)−MC yT (h), (57) t 21 1,t,t+j 21 21,t,j 21,t 1,t+j 21,t+j j=0 where yT (h) = c (h)+x (h), (58) 21,t+j 21,t+j 21,t+j ’j V˜pT = π¯˜ , (59) 21,t,j 21,t+k−1 k=1 (cid:23) (cid:24) pT −(1+θ ) 1 yT (h) = 1 V˜ 2 p 1 T ,t,j P˜ 21,t (h) θ p 1 T yT . (60) 21,t+j h¯ e P 21,t+j 1 1,t+j 21,t+j Note,thatIhavedefinedπ˜ = P ˜ 21,t and P ˜ 21,t (h) = P (h). Using ∂y2 T 1,t+j (h) = −1+θ p 1 T y2 T 1,t+j (h) , 21,t P ˜ 21,t−1 e1,t+j 21,t ∂P ˜ 21,t (h) θ p 1 T P ˜ 21,t (h) the first order condition implies: (cid:2) (cid:3) E ∞ (cid:4) ξpT (cid:5) j ψ 1+τ 21V˜pT P˜∗ −MC y 2 T 1,t+j (h) = 0, t 21 1,t,t+j 1+θpT 21,t,j 21,t 1,t+j P˜∗ j=0 1 21,t or P˜∗ H˜pT 21,t 21,t = P˜ G˜pT 21,t 21,t ⎧ ⎫ H˜pT = MC 1,t P 11,t yT +ξpTE ⎪ ⎨ ψ π˜ (cid:7) π¯˜ 21,t (cid:8)−(1 θ + p 1 T θ p 1 T ) H˜pT ⎪ ⎬ , 21,t P 11,t e 1,t P 21,t 21,t 21 t⎪ ⎩ 1,t,t+1 21,t+1 π˜ 21,t+1 21,t+1⎪ ⎭ ⎧ ⎫ 1+τ ⎪ ⎨ (cid:7) π¯˜ (cid:8) 1− (1+ p θ T p 1 T ) ⎪ ⎬ G˜pT = 21yT +ξpTE ψ π˜ 21,t θ1 G˜pT . 21,t 1+θp 1 T 21,t 21 t⎪ ⎩ 1,t,t+1 21,t+1 π˜ 21,t+1 21,t+1⎪ ⎭ Under local currency pricing, a firm maximizes: ∞ (cid:4) (cid:5) (cid:26) (cid:27) j E ξpT ψ (1+τ )VpT P (h)e −MC y (h), (61) t 21 1,t,t+j 21 21,t,j 21,t 1,t+j 1,t+j 21,t+j j=0 24

where ’j VpT = π¯ , (62) 21,t,j 21,t+k−1 k=1 ’j Ve = πe , (63) 1,t,j 1,t+k−1 k=1 (cid:23) (cid:24) pT − 1+θ1 y (h) = 1 V 2 p 1 T ,t,j P 21,t (h) θ p 1 T y . (64) 21,t+j h¯ P 21,t+j 1 21,t+j Note, that I have used the definitions π = P21,t , P ˜ 21,t (h) = P (h), πe = e1,t . Using 21,t P21,t−1 e1,t+j 21,t 1,t e1,t−1 ∂y21,t+j (h) = −1+θ p 1 T y21,t+j (h) , the first order condition implies: ∂P21,t (h) θ p 1 T P21,t (h) (cid:2) (cid:3) E ∞ (cid:4) ξpT (cid:5) j ψ 1+τ 21VpT P∗ e −MC y 21,t+j (h) = 0, (65) t j=0 21 1,t,t+j 1+θp 1 T 21,t,j 21,t 1,t+j 1,t+j P 2 ∗ 1,t or P∗ HpT 21,t 21,t = P GpT 21,t 21,t ⎧ ⎫ ⎪ (cid:7) (cid:8) pT ⎪ MC P ⎨ π¯ −(1+ pT θ1 ) ⎬ HpT = 1,t 11,t yT +ξpTE ψ π 21,t θ 1 HpT , 21,t P 11,t e 1,t P 21,t 21,t 21 t⎪ ⎩ 1,t,t+1 21,t+1 π 21,t+1 21,t+1⎪ ⎭ ⎧ ⎫ ⎪ (cid:7) (cid:8) pT ⎪ 1+τ ⎨ π¯ 1− (1+ p θ T 1 ) ⎬ GpT = 21yT +ξpTE ψ π 21,t θ1 GpT . 21,t 1+θp 1 T 21,t 21 t⎪ ⎩ 1,t,t+1 21,t+1 π 21,t+1 21,t+1⎪ ⎭ Similar expressions apply for country 2 and the non-traded goods sector. D.2 Model equations D.2.1 Definitions The following definitions are adopted. 25

1. Real relative prices: P q = 12,t+j, 1,t+j P 11,t+j P q = 22,t+j, 2,t+j P 21,t+j e P q = 1,t+j 22,t+j, t+j P 11,t+j e PC rer = 1,t+j 2,t+j, t+j PC 1,t+j PN∗ qN∗ = i,t+j , i,t+j P ii,t+j P Φ = ii,t, i,t+j PC i,t P Π = ii,t. i,t+j PI i,t 2. Optimized relative prices: P˜∗ p˜ ∗ = 21,t , 21,t e P 1,t 22,t P∗ p∗ = 21,t, 21,t P 22,t e P˜∗ p˜ ∗ = 1,t 12,t, 12,t P 11,t P∗ p∗ = 12,t. 12,t P 11,t 26

3. Inflation terms: P q π = 12,t = 1,t π , 12,t P q 11,t 12,t−1 1,t−1 P˜ q π π˜ = 12,t = 1,t 11,t, 12,t P˜ q πe 12,t−1 1,t−1 t P q π = 21,t = 2,t−1π , 21,t P q 22,t 21,t−1 2,t P˜ q π˜ = 21,t = 2,t−1π πe, 21,t P˜ q 22,t t 21,t−1 2,t PC/P P Φ πc = i,t ii,t ii,t = i,t−1π , i,t PC /P P Φ ii,t i,t−1 ii,t−1 ii,t−1 i,t PN qN πN = i,t = i,t π , i,t PN qN ii,t i,t−1 i,t−1 e rer πc πe = 1,t = t 1,t. t e rer πc 1,t−1 t−1 2,t 4. Price updating: π¯pT = πpTγ p 11 T π∗1−γ p 11 T, 11 11 π¯pT = πpTγ p 12 T π∗1−γ p 12 T, 12 12 π¯pT = πpTγ p 21 T π∗1−γ p 21 T, 21 21 π¯pT = πpTγ p 22 T π∗1−γ p 22 T, 22 22 π¯pN = πpNγ p 1 N π∗1−γ p 1 N, 1 1 π¯pN = πpNγ p 2 N π∗1−γ p 2 N. 2 2 5. Wages and wage inflation: W w = i,t, i,t P ii,t W∗ w∗ = i,t, i,t P ii,t W w ω = i,t = i,t π , i,t W w ii,t i,t−1 i,t−1 ω¯ i,t+j = ωγ i, w i t+j π∗1−γw i . 27

6. Real marginal costs: MCT mcT = i,t+j, i,t+j P ii,t MCN mcN = i,t+j. i,t+j PN i,t+j 7. Profits and assets: Pr pr = i,t+j, i,t+j P ii,t+j BG bG = i,t+j, i,t+j PC i,t+j B b = i,t+j, i,t+j PC i,t+j B b = 1,t+j, 1,t+j PC 1,t+j B b = 2,t+j . 2,t+j e PC 1,t+j 2,t+j D.2.2 Model equations The following conditions need to be satisfied in the model with real and nominal rigidities. 1. Labor choice: HW (cid:7) W∗ (cid:8) 1− 1+ θw θw i χi i,t i,t i = GW P i,t ⎛ ii,t ⎞ HW = χ ⎝ (cid:2) P ii,t (cid:3) − 1+ θw i θw i l ⎠ 1−χi +(β ξw) 1E (cid:7) ω¯ i,t (cid:8) − 1+ θw i θw i (1−χi ) HW i,t 0i W i,t i i t π i,t+1 i,t ii,t+1 GW = 1−τw i λ PCΦ (cid:2) P ii,t (cid:3) − 1+ θw i θw i l +β ξwE (cid:7) ω¯ i,t (cid:8) θ − w i 1 GW i,t 1+θw i,t i,t i,t W i,t i i t π i,t+1 i i,t ii,t+1 2. Aggregate wage evolution (cid:6) (cid:9) W i,t+j = (1−ξw) (cid:7) W i ∗ ,t+j (cid:8) θ − w i 1 +ξw (cid:7) ω¯ i,t+j−1 W i,t+j−1 (cid:8) θ − w i 1 −θw i P i P i π P ii,t+j ii,t+j ii,t+j ii,t+j−1 3. Marginal utility of consumption (cid:14) (cid:15) c (h)−κcc −νc −σ = λ (h)PC i,t+j i i,t+j−1 i,t+j i,t+j i,t+j 28

4. Investment choice Φ Φ (x (h)−x (h)) 0 = −λ (h)PC i,t+j −λ PC i,t+jφI i,t+j i,t−1+j i,t+j i,t+jΠ i,t+j i,t+jΠ i x (h) i,t+j i,t+j i,t−1+j Φ 1 x2 (h)−x2 (h) −β λ (h)PC i,t+1+j φI i,t+j i,t+1+j i i,t+1+j i,t+1+jΠ 2 i x2 (h) i,t+1+j i,t+j +μ (h) i,t+j 5. Capital choice (cid:14) (cid:15) R E β λ (h)PC Φ 1−τR i,t+1+j +E β μ (h)(1−δ) = μ (h) t i i,t+1+j i,t+1+j i,t+1+j i,t+1+j P t i i,t+1+j i,t+j ii,t+1+j Absent investment adjustment costs, i.e., φI = 0, it is λ (h)PC Φ i,t+j = μ (h) i i,t+j i,t+jΠ i,t+j i,t+j and thus (cid:13) (cid:16) E β λ i,t+1+j (h)P i C ,t+1+j Φ i,t+1+j Π i,t+j (cid:14) 1−τR (cid:15) Π R i,t+1+j +(1−δ) = 1. t λ (h)PC Φ Π i,t+1+j i,t+1+jP i,t+j i,t+j i,t+j i,t+1+j ii,t+1+j 6. Capital accumulation k ≤ (1−δ)k +x i,t+j i,t−1+j i,t+j 7. Consumption budget constraint 1 (cid:14)(cid:14) (cid:15) (cid:14) (cid:15) (cid:15) c = Φ 1−τW w l + 1−τR r k 1,t+j 1,t+j (cid:23) 1,t+j 1,t+j 1,t+j 1,t+(cid:24)j 1,t+j 1i,t−1+j Φ 1 (x −x ) 2 − 1,t+j x + φI 1,t+j 1,t−1+j Π 1,t+j 2 1 x 1,t+j 1,t−1+j +Φ pr 1,t+j (cid:7)1,t+j (cid:8) 1 −Φ QG bG −bG 1,t+j 1,t+j 1,t+j 1,t−1+jπ (cid:7) (cid:8)11,t+j 1 − QB b −b 1,t+j 1,t+j 1,t−1+jπc (cid:7) (cid:8) 1,t+j B (h) −Γ 1,t+j +Φ tr Pc 1,t+j 1,t+j 1,t+j 29

(cid:23) (cid:24) Φ 1 (x −x ) 2 c = − 1,t+j x + φI 1,t+j 1,t−1+j 1,t+j Π 1,t+j 2 1 x 1,t+j 1,t−1+j (cid:7) (cid:8) 1 −Φ QG bG −bG 1,t+j 1,t+j 1,t+j 1,t−1+jπ (cid:7) (cid:8)11,t+j 1 − QB b −b 1,t+j 1,t+j 1,t−1+jπc 1,t+j +Φ (c +x ) 1,t 11,t 11,t q +Φ t (c +x )ζ 1,tq 21,t 21,t 2 2,t(cid:14) (cid:15) +Φ qN cN +η(c +c ) 1,t 1,t 1,t 11,t 12,t for country 2 (cid:14)(cid:14) (cid:15) (cid:14) (cid:15) (cid:15) c = Φ 1−τW w l + 1−τR r k 2,t+j 2,t+j (cid:23) 2,t+j 2,t+j 2,t+j 2,t+(cid:24)j 2,t+j 2,t−1+j Φ 1 (x −x ) 2 − 2,t+j x − φI 2,t+j 2,t−1+j Π 2,t+j 2 2 x 2,t+j 2,t−1+j +Φ pr 2,t+j (cid:7)2,t+j (cid:8) 1 −Φ QG bG −bG 2,t+j 2,t+j 2,t+j 2,t−1+jπ (cid:7) 22,t+(cid:8)j 1 − QB b −b 2,t+j 2,t+j 2,t−1+jπe πc (cid:7) (cid:8) 1,t+j 2,t+j B (h) −Γ 2,t+j +Φ tr e PC 2,t+j 2,t+j 1,t+j 2,t+j (cid:23) (cid:24) Φ 1 (x −x ) 2 c = − 2,t+j x − φI 2,t+j 2,t−1+j 2,t+j Π 2,t+j 2 2 x 2,t+j 2,t−1+j (cid:7) (cid:8) 1 −Φ QG bG −bG 2,t+j 2,t+j 2,t+j 2,t−1+jπ (cid:7) 22,t+j (cid:8) 1 −Φ QB b −b 2,t+j 2,t+j 2,t+j 2,t−1+jπe π 1,t+j 22,t+j +Φ (c +x ) 2,t 22,t 22,t q 1 +Φ 2,t (c +x ) 2,t q 12,t 12,t ζ t (cid:14) 2 (cid:15) +Φ qN cN +η(c +c ) 2,t 2,t 2,t 22,t 21,t 30

and 1 1 b = b 2,t ζ rer 1,t 2 t 8. Risk sharing condition: with adjustment costs (cid:23) (cid:24) β λ 1,t+1+j P 1 C ,t+1+j −β λ 2,t+1+j P 2 C ,t+1+j rer 1,t+j 1 = Γ (cid:3) (b )−Γ (cid:3) (b ) 1 λ PC 2 λ PC rer πc 1,t+j 2,t+j 1,t+j 1,t+j 2,t+j 2,t+j 1,t+1+j 1,t+1+j 9. Price of bond derived from country 1 (cid:7) (cid:8) λ B Q = β 1,t+1+j −Γ (cid:3) 1,t+j 1,t+j 1 λ PC 1,t+j 1,t+j = β λ 1,t+1+j P 1 C ,t+1+j 1 −Γ (cid:3) (b ) 1 λ PC πc 1,t+j 1,t+j 1,t+j 1,t+j Q = β λ 2,t+1+j P 2 C ,t+1+j rer t+j 1 −Γ (cid:3) (b ) 2,t+j 2 λ PC rer πc 2,t+j 2,t+j 2,t+j t+1+j 1,t+j 10. Bond market clearing b +ν b q = 0 1,t+j 2 2,t+j t+j 11. Investment aggregate: for country i (cid:6) (cid:9) εI i (cid:14) (cid:15) 1 εI i−1 (cid:14) (cid:15) 1 εI i−1 εI i −1 x i,t+j = αI i1 εI i (x i1,t+j ) εI i + αI i2 εI i (x i2,t+j ) εI i 12. Investment optimal choices: (cid:7) (cid:8) αI 1 εI i x = 12 x 12,t+j αI q 11,t+j 11 1,t+j (cid:7) (cid:8) αI 1 εI i x = 22 x 22,t+j αI q 21,t+j 21 2,t+j 13. Consumption aggregate over traded goods: for country i (cid:6) (cid:9) εT i (cid:14) (cid:15) 1 εT i −1 (cid:14) (cid:15) 1 εT i −1 εT i −1 cT i,t+j = αT i1 εT i (c i1,t+j ) εT i + αT i2 εT i (c i2,t+j ) εT i 14. Consumption aggregate over traded and non traded goods: for country i (cid:6) (cid:9) εN i (cid:14) (cid:15) 1 (cid:14) (cid:15)εN i −1 (cid:14) (cid:15) 1 (cid:14) (cid:15)εN i −1 εN i −1 c i,t+j = αN i1 εN i cT i,t+j εN i + αN i2 εN i cN i,t+j εN i 31

15. Consumption optimal choices non-traded goods: for country 1 cN 1,t+j = c 12,t+j α α N 1 N 2 ⎡ ⎣ (cid:14) αT 11 (cid:15) ε 1 T i (cid:7) c c 11,t+j (cid:8) εT i εT i −1 + (cid:14) αT 12 (cid:15) ε 1 T i ⎤ ⎦ ε ε T i T i − − ε 1 N i (cid:23) q q 1 + N ,t+ η j qN (cid:24) −εN i (cid:7) α 1 T (cid:8) ε ε N i T i 11 12,t+j 1,t+j 1,t+j 12 and for country 2 cN 2,t+j = c 21,t+j α α N 2 N 2 2 1 ⎡ ⎣ (cid:14) αT 21 (cid:15) ε 1 T i + (cid:14) αT 22 (cid:15) ε 1 T i (cid:7) c c 2 2 2 1 , , t t + + j j (cid:8) εT i εT i −1 ⎤ ⎦ ε ε T i T i − − ε 1 N i (cid:23) q2, 1 t+j q + 2 N ,t+ η j q 2 N ,t+j (cid:24) −εN i (cid:7) α 1 T 21 (cid:8) ε ε N i T i 16. Consumption optimal choice traded goods: for country 1 (cid:23) (cid:24) αT 1+ηqN εT 1 c = 12 1,t+j c 12,t+j αT q +ηqN 11,t+j 11 1,t+j 1,t+j for country 2 (cid:23) (cid:24) c = αT 22 q2, 1 t+j +ηq 2 N ,t+j εT 2 c 22,t+j αT 1+ηqN 21,t+j 21 2,t+j 17. Price definition Φ : for country 1 i c1,t+j Φ = (cid:26) (cid:14) (cid:15) (cid:14) cN 1,t+j (cid:15) (cid:27) 1,t+j 1+ηqN c11,t+j + q +ηqN c12,t+j +qN 1,t+j c12,t+j 1,t+j 1,t+j cN 1,t+j 1,t+j for country 2 c2,t+j Φ = (cid:26)(cid:4) (cid:5) cN 2,(cid:14)t+j (cid:15) (cid:27) 2,t+j 1 +ηqN c21,t+j + 1+ηqN c22,t+j +qN q2,t+j 2,t+j c22,t+j 2,t+j cN 2,t+j 2,t+j 18. Price definition Π : for country 1 i P 1 x 1,t+j 1,t+j Π = = 1,t+j P 1 I ,t+j x x T 1 T i2 1 , , t t + + j j +q 1,t+j x 12,t+j for country 2 P 1 x 2,t+j 2,t+j Π = = 2,t+j P 2 I ,t+j xT 21,t+j + 1 x 22,t+j xT 22,t+j q2,t+j 32

19. Consumption real exchange rate rer = P 2 C ,t+j e 1,t+j = Φ 1,t+jq t+j PC Φ t+j 1,t+j 2,t+j e P q = 1,t+j 22,t+j t+j P 11,t+j 20. Production traded goods: for country i (cid:26) (cid:27) (cid:14) (cid:15) (cid:14) (cid:15) (cid:14) (cid:15) (cid:14) (cid:15) 1 yT = ωT 1−κT i AT lT κT i + ωT 1−κT i kT κT i κT i i,t+j li i,t+j i,t+j ki i,t+j 21. Production non-traded goods: for country i (cid:26) (cid:27) (cid:14) (cid:15) (cid:14) (cid:15) (cid:14) (cid:15) (cid:14) (cid:15) 1 yN = ωN 1−κN i AN lN κN i + ωN 1−κN i kN κN i κN i i,t+j li i,t+j i,t+j ki i,t+j 22. Optimal labor input choices traded goods: for country i (cid:23) (cid:24) ωTyT 1−κT i w = mcT AT li i,t+j i,t+j i,t+j i,t+j AT lT i,t+j i,t+j 23. Optimal capital input choices traded goods: for country i (cid:23) (cid:24) ωT yT 1−κT i r = mcT ki i,t+j i,t+j i,t+j kT i,t+j 24. Optimal labor input choices traded goods: for country i (cid:23) (cid:24) ωNyN 1−κN i w = qN mcN AN li i,t+j i,t+j i,t+j i,t+j i,t+j AN lN i,t+j i,t+j 25. Optimal capital input choices traded goods: for country i (cid:23) (cid:24) ωNyN 1−κN i r = qN mcN ki i,t+j i,t+j i,t+j i,t+j kN i,t+j 26. Labor market clearing lT +lN = l i,t+j i,t+j i,t+j 27. Capital market clearing kT +kN = k i,t+j i,t+j i,t−1+j 33

28. Goods market clearing for good 1: yT = Δ [c +x ]+Δ [c +x ]ν 1 11,t 11,t 11,t 21,t 21,t 21,t 2 29. Evolution of dispersion index good 1: country 1 (cid:4) (cid:5) (cid:14) (cid:15)−(1+θ p 1 T) (cid:7) π¯ (cid:8)−(1+ p θ T p 1 T) ΔT 11,t = 1−ξp 11 T p∗ 11,t θ p 1 T +ξp 11 T π 11,t−1 θ 1 ΔT 11,t−1 11,t country 2 under producer currency pricing (cid:4) (cid:5) (cid:14) (cid:15) − 1+θ p 1 T (cid:7) π¯˜ (cid:8) − 1+ p θ T p 1 T ΔT 21,t = 1−ξp 21 T p˜ ∗ 21,t q 2,t θ p 1 T +ξp 21 T π˜ 21,t−1 θ 1 ΔT 21,t−1 21,t q π˜ = 2,t−1π πe 21,t q 22,t 1,t 2,t under local currency pricing (cid:7) (cid:8) pT (cid:4) (cid:5) (cid:14) (cid:15) − 1+θ p 1 T π¯ − 1+ p θ T 1 ΔT 21,t = 1−ξp 21 T p∗ 21,t q 2,t θ p 1 T +ξp 21 T π 21,t−1 θ1 ΔT 21,t−1 21,t 30. Goods market clearing for good 2: ν yT = Δ [c +x ]+Δ [c +x ]ν 2 2 12,t 12,t 12,t 22,t 22,t 22,t 2 31. Evolution of dispersion index good 2: country 2 (cid:4) (cid:5) (cid:14) (cid:15)−(1+θ p 2 T) (cid:7) π¯ (cid:8)−(1+ p θ T p 2 T) ΔT 22,t = 1−ξp 22 T p∗ 22,t θ p 2 T +ξp 22 T π 22,t−1 θ2 ΔT 22,t−1 22,t country 1 under producer currency pricing ΔT = (cid:4) 1−ξpT (cid:5) (cid:7) p˜∗ 12,t (cid:8) − 1+ θ p 2 θ T p 2 T +ξpT (cid:7) π¯˜ 12,t−1 (cid:8) − 1+ θ p 2 θ T p 2 T ΔT 12,t 12 q 12 π˜ 12,t−1 1,t 12,t q π π˜ = 1,t 11,t 12,t q πe 1,t−1 1,t under local currency pricing (cid:7) (cid:8) pT (cid:7) (cid:8) pT (cid:4) (cid:5) p∗ − 1+ p θ T 2 π¯ − 1+ p θ T 2 ΔT = 1−ξpT 12,t θ2 +ξpT 12,t−1 θ2 ΔT 12,t 12 q 12 π 12,t−1 1,t 12,t 34

32. Goods market clearing non-traded goods: for country i 0 1 yN = ΔN cN +η(c +c ) i,t+j i,t+j i,t+j i1,t+j i2,t+j 33. Evolution of dispersion index non-traded good i: (cid:23) (cid:24)−(1+θ pN) (cid:23) (cid:24)−(1+θ pN) (cid:4) (cid:5) i i qN∗ θ pN π¯N θ pN ΔN = 1−ξpN i,t i +ξpN i,t−1 i ΔN i,t i qN i πN i,t−1 i,t i,t qN πN = i,t π i,t qN ii,t i,t−1 34. Price indices good 1: country 1 (cid:7) (cid:8) (cid:4) (cid:5) (cid:14) (cid:15) − 1 π¯ − p 1 T 1 = 1−ξp 11 T p∗ 11,t θ p 1 T +ξp 11 T π 11,t−1 θ 1 11,t for country 2: if producer currency pricing (cid:7) (cid:8) 1 = (cid:4) 1−ξp 21 T (cid:5) (cid:14) p˜ ∗ 21,t q 2,t (cid:15) − θ p 1 1 T +ξp 21 T π¯˜ π˜ 21,t−1 − θ p 1 1 T 21,t if local currency pricing (cid:7) (cid:8) (cid:4) (cid:5) (cid:14) (cid:15) − 1 π¯ − p 1 T 1 = 1−ξp 21 T p∗ 21,t q 2,t θ p 1 T +ξp 21 T π 21,t−1 θ1 21,t 35. Price indices good 2: country 2 (cid:7) (cid:8) (cid:4) (cid:5) (cid:14) (cid:15) − 1 π¯ − p 1 T 1 = 1−ξp 22 T p∗ 22,t θ p 2 T +ξp 22 T π 22,t−1 θ2 22,t for country 1: if producer currency pricing (cid:7) (cid:8) (cid:7) (cid:8) 1 = (cid:4) 1−ξpT (cid:5) p˜∗ 12,t − θ p 2 1 T +ξpT π¯˜ 12,t−1 − θ p 2 1 T 12 q 12 π˜ 1,t 12,t if local currency pricing (cid:7) (cid:8) (cid:7) (cid:8) (cid:4) (cid:5) p∗ − p 1 T π¯ − p 1 T 1 = 1−ξpT 12,t θ2 +ξpT 12,t−1 θ2 12 q 12 π 1,t 12,t 36. Price index non-traded good (cid:23) (cid:24) (cid:23) (cid:24) (cid:4) (cid:5) −1 −1 qN∗ θ pN π¯N θ pN 1 = 1−ξpN i,t i +ξpN i,t−1 i i qN i πN i,t i,t 35

37. Stochastic discount factors λ PC PC ψ = β 1,t+1 1,t+1 1,t 1,t,t+1 1 λ PC PC 1,t 1,t 1,t+1 λ PC PC ψ = β 2,t+1 2,t+1 2,t 2,t,t+1 2 λ PC PC 2,t 2,t 2,t+1 38. Price for traded good country 1 P∗ HpT 11,t = 11,t, P GpT 11,t 11,t ⎧ ⎫ ⎪ ⎨ (cid:7) (cid:8)−(1+θ p 1 T) ⎪ ⎬ MC π¯ pT HpT = 1,tyT +ξpTE ψ π 11,t θ1 HpT 11,t P 11,t 11,t 11 t⎪ ⎩ 1,t,t+1 11,t+1 π 11,t+1 11,t+1⎪ ⎭ ⎧ ⎫ 1+τ ⎪ ⎨ (cid:7) π¯ (cid:8) 1− (1+ p θ T p 1 T) ⎪ ⎬ GpT = 11yT +ξpTE ψ π 11,t θ 1 GpT 11,t 1+θp 1 T 11,t 11 t⎪ ⎩ 1,t,t+1 11,t+1 π 11,t+1 11,t+1⎪ ⎭ 39. Export price for traded good country 1: producer currency pricing P˜∗ H˜pT 21,t 21,t = P˜ G˜pT 21,t 21,t ⎧ ⎫ MC P ⎪ ⎨ (cid:7) π¯˜ (cid:8)−(1+ p θ T p 1 T ) ⎪ ⎬ H˜pT = 1,t 11,t yT +ξpTE ψ π˜ 21,t θ1 H˜pT 21,t P 11,t e 1,t P 21,t 21,t 21 t⎪ ⎩ 1,t,t+1 21,t+1 π˜ 21,t+1 21,t+1⎪ ⎭ ⎧ ⎫ G˜pT = 1+τ 21yT +ξpTE ⎪ ⎨ ψ π˜ (cid:7) π¯˜ 21,t (cid:8) 1− (1+ θ p 1 θ T p 1 T ) G˜pT ⎪ ⎬ 21,t 1+θp 1 T 21,t 21 t⎪ ⎩ 1,t,t+1 21,t+1 π˜ 21,t+1 21,t+1⎪ ⎭ local currency pricing P∗ HpT 21,t 21,t = P GpT 21,t 21,t ⎧ ⎫ ⎪ (cid:7) (cid:8) pT ⎪ MC P ⎨ π¯ −(1+ p θ T 1 ) ⎬ HpT = 1,t 11,t yT +ξpTE ψ π 21,t θ1 HpT 21,t P 11,t e 1,t P 21,t 21,t 21 t⎪ ⎩ 1,t,t+1 21,t+1 π 21,t+1 21,t+1⎪ ⎭ ⎧ ⎫ ⎪ (cid:7) (cid:8) pT ⎪ ⎨ (1+θ ) ⎬ 1+τ π¯ 1− pT 1 GpT = 21yT +ξpTE ψ π 21,t θ 1 GpT 21,t 1+θp 1 T 21,t 21 t⎪ ⎩ 1,t,t+1 21,t+1 π 21,t+1 21,t+1⎪ ⎭ 36

40. Price for traded good country 2 P∗ HpT 22,t = 22,t, P GpT 22,t 22,t ⎧ ⎫ ⎪ ⎨ (cid:7) (cid:8)−(1+θ p 1 T) ⎪ ⎬ MC π¯ pT HpT = 2,tyT +ξpTE ψ π 22,t θ 2 HpT 22,t P 22,t 22,t 22 t⎪ ⎩ 2,t,t+1 22,t+1 π 22,t+1 22,t+1⎪ ⎭ ⎧ ⎫ 1+τ ⎪ ⎨ (cid:7) π¯ (cid:8) 1− (1+ p θ T p 1 T) ⎪ ⎬ GpT = 22yT +ξpTE ψ π 22,t θ 2 GpT 22,t 1+θp 2 T 22,t 22 t⎪ ⎩ 2,t,t+1 22,t+1 π 22,t+1 22,t+1⎪ ⎭ 41. Export price for traded good country 2: producer currency pricing P˜∗ H˜pT 12,t = 12,t P˜ G˜pT 12,t 12,t ⎧ ⎫ MC e P ⎪ ⎨ (cid:7) π¯˜ (cid:8)−(1+ p θ T p 2 T ) ⎪ ⎬ H˜pT = 2,t 1,t 22,tyT +ξpTE ψ π˜ 12,t θ 2 H˜pT 12,t P 22,t P 12,t 12,t 12 t⎪ ⎩ 2,t,t+1 12,t+1 π˜ 12,t+1 12,t+1⎪ ⎭ ⎧ ⎫ 1+τ ⎪ ⎨ (cid:7) π¯˜ (cid:8) 1− (1+ p θ T p 2 T ) ⎪ ⎬ G˜pT = 12yT +ξpTE ψ π˜ 12,t θ2 G˜pT 12,t 1+θp 2 T 12,t 12 t⎪ ⎩ 2,t,t+1 12,t+1 π˜ 12,t+1 12,t+1⎪ ⎭ local currency pricing P∗ HpT 12,t 12,t = P GpT 12,t 12,t ⎧ ⎫ ⎪ (cid:7) (cid:8) pT ⎪ MC e P ⎨ π¯ −(1+ p θ T 2 ) ⎬ HpT = 2,t 1,t 22,tyT +ξpTE ψ π 12,t θ 2 HpT 12,t P 22,t P 12,t 12,t 12 t⎪ ⎩ 2,t,t+1 12,t+1 π 12,t+1 12,t+1⎪ ⎭ ⎧ ⎫ ⎪ (cid:7) (cid:8) pT ⎪ 1+τ ⎨ π¯ 1− (1+ p θ T 2 ) ⎬ GpT = 12yT +ξpTE ψ π 12,t θ2 GpT 12,t 1+θp 2 T 12,t 12 t⎪ ⎩ 2,t,t+1 12,t+1 π 12,t+1 12,t+1⎪ ⎭ 37

42. Price for non-traded good country i: PN∗ HpN i,t = i,t , PN GpN i,t i,t ⎧ ⎫ ⎪ ⎪ ⎨ (cid:23) (cid:24)−(1+θ p i N) ⎪ ⎪ ⎬ MC P π¯N θ pN HpN = i,t ii,tyN +ξpNE ψ πN i,t i HpN i,t P PN i,t i t⎪ ⎪ i,t,t+1 i,t+1 πN i,t+1⎪ ⎪ ii,t i,t ⎩ i,t+1 ⎭ ⎧ ⎫ ⎪ ⎪ ⎨ (cid:23) (cid:24) 1− (1+θ p i N) ⎪ ⎪ ⎬ GpN = 1+τN i yN +ξpNE ψ πN π¯N i,t θ p i N GpN i,t 1+θp i N i,t i t⎪ ⎪ ⎩ i,t,t+1 i,t+1 πN i,t+1 i,t+1⎪ ⎪ ⎭ 43. Definition of real profits Pri,t+j : profits from home activities Pii,t+j # 1 Pr = ((1+τ )P (h)−MC )(c (h)+x (h))dh 11,t 11 11,t 1,t 11,t 11,t 00 1 pr = (1+τ )−mc ΔT (c +x ) 11,t 11 1,t 11,t 11,t 11,t from exporting with producer currency pricing # ¯ (cid:4) (cid:5) h1 Pr = (1+τ )P˜ (h)−MC (c (h)+x (h))ν dh 21,t 21 21,t 1,t 21,t 21,t 2 (cid:2)0 (cid:3) Pr q 21,t = (1+τ ) t −mc ΔT (c +x )ν P 21 q 1,t 21,t 21,t 21,t 2 11,t 2,t with local currency pricing # ¯ h1 Pr = ((1+τ )e P (h)−MC )(c (h)+x (h))ν dh 21,t 21 1,t 21,t 1,t 21,t 21,t 2 (cid:2)0 (cid:3) Pr q 21,t = (1+τ ) t −mc ΔT (c +x )ν P 21 q 1,t 21,t 21,t 21,t 2 11,t 2,t and the non-traded goods PrN 0(cid:14) (cid:15) 1(cid:14) (cid:15) 1,t = qN 1+τN −mcN ΔN cN +η(c +c ) P 1,t 1,t 1,t 1,t 1,t 11,t 12,t 11,t 38

Consolidate household budget constraint (cid:14)(cid:14) (cid:15) (cid:14) (cid:15) (cid:15) Φ 1−τW w l + 1−τR r k +Φ (pr +tr ) 1,t 1 1,t 1,t 1 1,t 1i,t−1 1,t 1,t 1,t 0 1 = Φ w l +Φ r k +Φ 1−mc ΔT (c +x ) 1,t 1 (cid:2),t 1,t 1,t 1,t 1i,t− (cid:3) 1 1,t 1,t 11,t 11,t 11,t q 0 1(cid:14) (cid:15) +Φ t −mc ΔT (c +x )ν +Φ qN 1−mcN ΔN cN +η(c +c ) 1,t q 1,t 21,t 21,t 21,t 2 1,t 1,t 1,t 1,t 1,t 11,t 12,t 2,t q = +Φ (c +x )+Φ t (c +x )ν 1,t 11,t 11,t 1,tq 21,t 21,t 2 (cid:14) 2,t (cid:15) +Φ qN cN +η(c +c ) 1,t 1,t 1,t 11,t 12,t 44. Price of government debt λC λC PC 1 QG = β t+1+j = β t+1+j t+1+j i,t+j i λC i λC PC πC t+j t+j t+j t+j 45. Government budget constraint tr = τrr k +τww l i,t+j i i,t+j i,t−1+j i i,t+j i,t+j −τ (c +x ) 11 11,t 11,t q −τ t (c +x )ν 21q 21,t 21,t 2 2,t(cid:14) (cid:15) −qN τN cN +η(c +c ) 1,t 1 1,t 11,t 12,t 46. Nominal interest rate 1 = QG Rs i,t+j i,t+j 47. Monetary policy (cid:7) (cid:8) (cid:7) (cid:8) Rs φr i s πc βiφc i (1−φr i s) Rs = Rs i,t−1+j i,t+j i,t+j i Rs πc i i withthesteadystateinflationrateπc andthesteadystatenominalinterestrateRs = πc i. i i βi References Calvo, G. (1983). Staggered prices in a utility maximizing framework. Journal of Monetary Economics 12(3), 383–398. Christiano, L. J., M. Eichenbaum, and C. L. Evans (1999). Monetary policy shocks: What have we learned and to what end? 39

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