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Trade Elasticity of Substitution and Equilibrium Dynamics*

Martin Bodenstein**

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publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.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


Abstract:

The empirical literature provides a wide range of estimates for trade elasticities at the aggregate level. Furthermore, recent contributions in international macroeconomics suggest that low (implied) values of the trade elasticity of substitution may play an important role in understanding the disconnect between international prices and real variables. However, a standard model of the international business cycle displays multiple locally isolated equilibria if the trade elasticity of substitution is sufficiently low. The main contribution of this paper is to compute and characterize some dynamic properties of these equilibria. While multiple steady states clearly signal equilibrium multiplicity in the dynamic setup, this is not a necessary condition. Solutions based on log-linearization around a deterministic steady state are of limited to no help in computing the true dynamics. However, the log-linear solution can hint at the presence of multiple dynamic equilibria.

Keywords: Multiple equilibria, linearization, DSGE models

JEL classification: C68, F41



1  Introduction

General equilibrium theory is plagued with the problem of equilibrium multiplicity.1 This paper analyzes a standard model of the international business cycle that is based on the seminal contribution of Backus, Kehoe, and Kydland (1995). If the elasticity of substitution between the home and the foreign good is sufficiently low, this model displays multiple locally isolated equilibria.2 The main contribution of this paper is to characterize the dynamic properties of these multiple equilibria in a model with endogenous capital accumulation and incomplete international financial markets for borrowing and lending. The two major computational issues that arise are to identify conditions that indicate the existence of multiple equilibria and to find a reliable method to compute these equilibria.

To build intuition, consider an endowment economy with two countries and two traded goods that are imperfect substitutes. The countries are mirroring each other with respect to preferences and endowments.3 There is always one equilibrium with the relative price of the traded goods equal to unity. However, there can be two more equilibria. Let the price of the domestic good be high relative to the price of the foreign good, so that domestic agents have high purchasing power relative to the foreign agents. If the elasticity of substitution is low, foreigners are willing to give up most of their good in order to consume at least some of the domestic good, and domestic agents end up consuming most of the domestic and the foreign good. The reverse is true as well. Foreign agents consume most of the two goods, if the foreign good is very expensive in relative terms. Of course, these last two scenarios cannot be an equilibrium for high values of the elasticity of substitution. In the limiting case of perfect substitutability the unique equilibrium features each country consuming its own endowment. Such a model of multiple locally unique equilibria allows for the construction of sunspot equilibria, meaning there are equilibria for which allocations are different across different states of nature, even though nothing fundamental has changed.4

In a model with endogenous state variables the computation of dynamic multiple equilibria cannot be separated from considering sunspot equilibria. To eliminate these complications I conduct the following experiment. In the first period the economy experiences an unforeseen shock to technology. This is the only period in which agents are free to coordinate on any of the possible equilibrium paths. Starting from the second period onwards, agents have perfect foresight and they keep coordinating on the equilibrium path that has been chosen in the first period.

Under these assumptions I can typically find three equilibria provided that the elasticity of substitution is sufficiently low. Along the first equilibrium path the dynamics are solely driven by the impact of the technology shock and all variables stay in the neighborhood of their pre-shock level. The dynamics of the other two equilibria are mostly driven by the shifts in relative purchasing power due to self-fulfilling changes in the relative price of the home and foreign good. The effects of the technology shock are negligible in these cases.

Whereas for a given calibration the existence of multiple equilibria mostly depends on the magnitude of the trade elasticity of substitution, it can also depend on the magnitude and the persistence of the shock. In some cases, two of the three equilibria cease to exists if the shock is sufficiently large or permanent. At least in the cases presented in this paper, the now unique equilibrium involves a large shift in relative purchasing power.

Although equilibrium multiplicity in the dynamic model mostly goes along with multiple steady states, one can also find multiple equilibria in cases with a unique steady state. In a model with a slow-moving capital stock the short-run excess demand function behaves differently from the long-run excess demand, as the economy is less flexible in the short run.

The major challenge in computing equilibria in an environment with multiple locally unique equilibria is to generate a good starting guess. Fortunately, it is possible to derive a starting guess for the model economy with capital and incomplete financial markets from an endowment economy with incomplete financial markets. Using a combination of backward and forward shooting algorithms the impulse response functions for a technology shock are derived under the aforementioned assumption that in the period of the shock agents coordinate once and for all on one equilibrium path.

Linearization or higher-order perturbation methods that approximate the equilibrium policy functions around a deterministic steady state are of limited use in an environment with low trade elasticities. First, these methods can only detect one of the three equilibria. In particular, the method cannot detect those equilibria that are associated with the large shifts in relative purchasing power. Second, in some cases of large shocks local approximation techniques are inappropriate. This issue is not just due to the declining accuracy of the approximation method with increasing distance from the deterministic steady state. The real problem is that the method searches for an equilibrium path where it can be shown that no such equilibrium path exists for a shock of the considered size. Hence, even if one is willing to abstract from equilibrium multiplicity and the possibility of sunspot equilibria, global non-linear methods are preferred to local approximation methods. However, the linear approximation to the policy functions can be put to use in detecting the presence multiple equilibria.

The literature provides a large range of estimates for the trade elasticity of substitution. Using aggregate data Whalley (1995) reports an elasticity of 1.5. In a recent study, Hooper, Johnson and Marquez (2000) estimate trade elasticities for the G7 countries. They report a short-run trade elasticity of 0.6 for the U.S. and values between 0 and 0.6 for the remaining G7 countries. Taylor (1993) estimates an import demand equation for the U.S. and finds a short-run elasticity of 0.22 and a long-run trade elasticity of 0.39. Obviously, these macro estimates are in sharp contrasts to the estimates from lower levels of aggregation. In Broda and Weinstein (2005), for example, the mean estimates are between 4 and 6. Most relevant for this paper, it is common in many applied macroeconomic models to choose values of the elasticity of substitution between 1 and 1.5. Examples include Backus et al (1995), Chari, Kehoe, and McGrattan (2003), and Heathcote and Perri (2002). Recently, however, models with (implied) low elasticities of substitution between home and foreign goods have received considerable attention - Corsetti, Dedola, and Leduc (2008), Collard and Dellas (2004), Benigno and Thoenissen (2008), Thoenissen (2008), and Enders and Mueller (2008) - as such models seem to provide a better fit to the international business cycle. In particular, this has been shown for the case of the puzzling negative correlation between the real exchange rate and relative consumption (Backus and Smith (1993)), cross-country consumption correlations, and the volatility of the real exchange rate. Furthermore, Rabanal and Tuesta (2005) and Lubik and Schorfheide (2005) show estimates for the elasticity of substitution well below unity in DSGE models using Bayesian techniques. In a model that is akin to Corsetti et al (2008), de Walque, Smets and Wouters (2005) find that the data speaks in favor of a low implied elasticity of substitution.

This paper does not take a stand on the value of the trade elasticity of substitution in macro models. However, as there seems to be considerable interest in models with low trade elasticities, it is important to search for tools that allow a complete analysis of models with multiple equilibria and to investigate the extent to which the results of these models and their support through the data hinge on the value of the elasticity of substitution between traded goods being so low that the model admits multiple equilibria.5

The remainder of the paper is organized as follows. Section 2 introduces the problem of multiple equilibria in a static endowment economy. Section 3 presents a dynamic model with endogenous capital accumulation and international borrowing and lending. Computational issues are addressed in Section 4. In Section 5, the equilibrium multiplicity in the dynamic economy is illustrated with the help of impulse response functions. Some sensitivity aspects of the results are discussed in Section 6. Section 7 concludes.


2  Multiple Equilibria

The analysis begins with the well-known example of a static exchange economy with two agents and two goods.6 Let agent $ i$ ($ i=1,2$) receive an endowment of $ y_{i}$ units of good $ i$. Agents can trade their endowments with each other and they have constant elasticity of substitution preferences over the two goods. The problem of agent $ i$ is given by

  $\displaystyle \max_{c_{i1}, c_{i2}}c_{i}=\left[ \left( \alpha_{i1}\right) ^{1-\rho }c_{i1}^{\rho}+\left( \alpha_{i2}\right) ^{1-\rho}c_{i2}^{\rho}\right] ^{ \frac{1}{\rho}}$ (1)
  s.t.    
  $\displaystyle \bar{P}_{1}c_{i1}+\bar{P}_{2}c_{i2}\leq\bar{P}_{i}y_{i},$ (2)

where $ \rho<1$ and $ \alpha_{ii}\geq\alpha_{ij}$, $ j\neq i$. $ \varepsilon \equiv\frac{1}{1- \rho}$ is the elasticity of substitution between the two goods. $ c_{ij}$ denotes the amount of good $ j$ that is consumed by agent $ i$. $ \bar{P}_{j}$ is the price of good $ j$. Absent trading frictions both agents face the same prices. Market clearing requires

$\displaystyle \sum_{j=1,2}c_{ji}\leq y_{i}, i=1,2.$ (3)

An equilibrium in this economy is defined as follows.




Definition 1   A competitive equilibrium is an allocation $ c_{ij},$ $ i=1,2,$ $ j=1,2$ and prices $ \bar{P}_{i},$ $ i=1,2$ such that $ \left( i\right) $ for every agent $ i $ the pair $ \left( c_{i1},c_{i2}\right) $ solves the problem stated in (1) at the given prices and $ \left( ii\right) $ all markets clear.


Define $ \bar{q}=\frac{\bar{P}_{2}}{\bar{P}_{1}}$, and let $ z_{2}$ denote the excess demand for good 2. A competitive equilibrium is then fully summarized by

$\displaystyle z_{2}\left( \bar{q}\right) =c_{12}\left( \bar{q}\right) +c_{22}\left( \bar{q}\right) -y_{2},$ (4)
$\displaystyle z_{2}\left( \bar{q}\right) \leq0$, $\displaystyle \bar{q}\geq0$ and $\displaystyle \bar {q}z_{2}\left( \bar{q}\right) =0.$ (5)

Since agents' preferences over goods in (1) are strictly monotone, the equilibrium price is strictly positive, and $ z_{2}\left( \bar{q}\right) =0$.

Standard theorems establish the existence of a competitive equilibrium. However, the equilibrium may not be unique. Let $ index\left( \bar{q}^{\ast}\right) =sign\left( \frac{\partial z_{2}\left( \bar{q}^{\ast }\right) }{\partial\bar{q}}\right) $ be the index of an equilibrium with the relative price $ \bar{q}^{\ast}$. If all equilibria are locally unique, the sum of the indices across equilibria equals $ +1$ by virtue of the index theorem. Hence, the number of equilibria is finite. If there is an equilibrium with $ \frac{\partial z_{2}\left( \bar{q}^{\ast}\right) }{\partial\bar{q}}>0$ , there are at least two more equilibria.7

Figure 1 plots the excess demand for good 2 as a function of a monotone transformation of the relative price, $ \frac{\bar{q}}{1+\bar{q}}$, for two different values of the elasticity of substitution, $ \varepsilon=2$ and $ \varepsilon=0.42$ .8 There is a unique equilibrium with $ \bar{q}=1$ $ \left( \frac{\bar{q}}{1+\bar{q}}=\frac{1}{2}\right) $ for $ \varepsilon=2$, but there are three equilibria with $ \bar{q}$ equal to 0.47 $ \left( \frac{\bar{q}}{1+\bar{q}}=0.32\right) $, 1, and 2.12 $ \left( \frac{\bar{q}}{1+\bar{q}}=0.68\right) $ for $ \varepsilon=0.42$. Notice, that in the latter case the slope of the excess demand function is positive for $ \bar{q}=1$ and all the equilibria are locally unique.9 To understand how multiple equilibria arise at low values of the elasticity of substitution consider the first equilibrium in the second panel of Figure 1. As the price of good 1 is high relative to the price of good 2, $ \bar{q}=0.47$, the value of the endowment of country 1 is high relative to country 2. Agents in country 2 are willing to pay the high price for good 1 and country 1 ends up consuming most of the two goods. The same logic applies in the third equilibrium, $ \bar{q}=2.12$, with the roles of country 1 and 2 being reversed. The second equilibrium is the symmetric equilibrium featuring $ \bar{q}=1$. If the elasticity of substitution is high, equilibria 1 and 3 cannot exist.

The dynamic extension of the static economy with a low elasticity of substitution delivers a simple example of an economy with sunspot equilibria. Let preferences admit an expected utility representation. Agents receive a fixed endowment every period and there are no international financial markets. The dynamic economy is then simply the repeated static economy. A sunspot equilibrium is given by a system of the three spot prices and a probability distribution $ \pi$ over the three spot prices: although the fundamentals of the economy, i.e., the endowments, are unaffected by the realization of the state, the equilibrium prices and allocations differ across states.10


3  General Model

The model is quite standard in the international business cycle literature and it is closely related to the seminal work of Backus et al (1995). There are two countries $ \left( i=1,2\right) $, each populated by an infinite number of households of measure one. Each country produces one good and the home and foreign good are imperfect substitutes in the households' utility functions. The two goods are produced under perfect competition using capital and labor. Agents have access to a non-contingent bond that pays one unit of country 1's currency.

Time is discrete and each period the economy experiences one of finitely many events $ s_{t}$. $ s^{t}=\left( s_{0},s_{1},...,s_{t}\right) $ denotes the history of events up through and including period $ t$. The probability, as of period 0, of any particular history $ s^{t}$ is $ \pi\left( s^{t}\right) $. The initial realization $ s_{0}$ is given.

3.1  Households

Households in country $ i$ maximize their expected discounted lifetime utility subject to their budget constraint. All variables are expressed in per household units

  $\displaystyle \max_{\substack{c_{i}\left( s^{t}\right) ,l_{i}\left( s^{t}\right) , c_{i1}\left( s^{t}\right) ,c_{i2}\left( s^{t}\right) \\ k_{i}\left( s^{t}\right) ,i_{i}\left( s^{t}\right) ,b_{i}\left( s^{t}\right) }} \sum_{t=0}^{\infty}\sum_{s^{t}}\beta^{t}\pi\left( s^{t}\right) U\left( c_{i}\left( s^{t}\right) ,l_{i}\left( s^{t}\right) \right)$ (6)
  s.t.    
  $\displaystyle P_{i}\left( s^{t}\right) \left( c_{i}\left( s^{t}\right) +i_{i}\left( s^{t}\right) \right) \leq\bar{P}_{i}\left( s^{t}\right) w_{i}\left( s^{t}\right) l_{i}\left( s^{t}\right) +\bar{P}_{i}\left( s^{t}\right) r_{i}\left( s^{t}\right) k_{i}\left( s^{t-1}\right)$    
  $\displaystyle +\bar{P}_{i}\left( s^{t}\right) p_{i}\left( s^{t}\right) +b_{i}\left( s^{t-1}\right) -Q_{i}\left( s^{t}\right) b_{i}\left( s^{t}\right) ,$ (7)
  $\displaystyle k_{i}\left( s^{t}\right) \leq\left( 1-\delta\right) k_{i}\left( s^{t-1}\right) +i_{i}\left( s^{t}\right) .$ (8)

where $ c_{i}$, $ i_{i}$, $ l_{i}$ and $ k_{i}$ denote consumption, investment, labor, and capital, respectively. $ P_{i}$ is the price of the final consumption-investment good, $ \bar{P}_{i}$ is the price of good $ i$, $ \bar {P}_{i}r_{i}$ and $ \bar{P}_{i}w_{i}$ are the nominal rental rate of capital and the nominal wage, $ \bar{P}_{i}p_{i}$ denotes nominal profits, and $ b_{i}$ denotes holdings of a non-contingent bond. $ Q_{i}$ is the price of the bond. To rule out Ponzi-schemes, I assume that agents face an upper bound for borrowing $ \bar{b}_{i}$ that is large enough to never bind in this application.

3.2  Firms

Firms utilize labor and capital in order to produce the traded good under perfect competition. The technology is assumed to be of the constant elasticity of substitution type. Since capital is owned by households and rented out to firms, the solution to the firms' maximization problem can be found from

  $\displaystyle \max_{l_{i},k_{i}}F\left( l_{i}\left( s^{t}\right) ,k_{i}\left( s^{t-1}\right) \right) -w_{i}\left( s^{t}\right) l_{i}\left( s^{t}\right) -r_{i}\left( s^{t}\right) k_{i}\left( s^{t-1}\right)$ (9)
  s.t.    
  $\displaystyle F_{i}\left( l_{i},k_{i}\right) =\left[ \omega_{li}^{1-\kappa}\left( A_{i}\left( s^{t}\right) l_{i}\left( s^{t}\right) \right) ^{\kappa }+\omega_{ki}^{1-\kappa}k_{i}\left( s^{t-1}\right) ^{\kappa}\right] ^{\frac{1}{\kappa}},$ (10)
  if $\displaystyle \kappa<1$ and    
  $\displaystyle F_{i}\left( l_{i},k_{i}\right) =\left( \frac{A_{i}\left( s^{t}\right) l_{i}\left( s^{t}\right) }{\omega_{li}}\right) ^{\omega_{li}}\left( \frac{k_{i}\left( s^{t-1}\right) }{\omega_{ki}}\right) ^{\omega_{ki}},$ (11)
  if $\displaystyle \kappa=0\ .$    

3.3  International trade

Households in country $ i$ demand a consumption-investment good $ \tilde{c}_{i} $ $ \left( =c_{i}+i_{i}\right) $. $ \tilde{c}_{i}$ is an aggregate of the domestically produced good and the imports of the foreign good according to

$\displaystyle \tilde{c}_{i}\left( s^{t}\right) =c_{i}\left( s^{t}\right) +i_{i}\left( s^{t}\right) =\left[ \alpha_{i1}^{1-\rho}c_{i1}\left( s^{t}\right) ^{\rho }+\alpha_{i2}^{1-\rho}c_{i2}\left( s^{t}\right) ^{\rho}\right] ^{\frac{1}{ \rho}},$ (12)

with $ 0<\alpha_{ij}<1$. The parameters $ \alpha_{ij}$ will be calibrated to match aggregate trade shares in the data.11 $ \rho<1$ and $ \varepsilon=\frac{1}{1-\rho}$ measures the elasticity of substitution between traded goods.

A household's optimal choices for consumption of the home and the foreign good are determined from the cost minimization problem

  $\displaystyle \min_{c_{i1}\left( s^{t}\right) ,c_{i2}\left( s^{t}\right) }\bar{P} _{1}\left( s^{t}\right) c_{i1}\left( s^{t}\right) +\bar{P}_{2}\left( s^{t}\right) c_{i2}\left( s^{t}\right)$ (13)
  s.t.    
  $\displaystyle \tilde{c}_{i}\left( s^{t}\right) = \left[ \alpha_{i1}^{1-\rho} c_{i1}\left( s^{t}\right) ^{\rho}+\alpha_{i2}^{1-\rho}c_{i2}\left( s^{t}\right) ^{\rho}\right] ^{\frac{1}{\rho}}.$ (14)

The first order conditions for country $ i$'s households imply

$\displaystyle \frac{c_{i1}\left( s^{t}\right) }{c_{i2}\left( s^{t}\right) }=\frac {\alpha_{i1}}{\alpha_{i2}}\left( \frac{1}{\bar{q}\left( s^{t}\right) }\right) ^{ \frac{1}{\rho-1}},$ (15)

where I have defined the relative price to be

$\displaystyle \bar{q}\left( s^{t}\right) =\frac{\bar{P}_{2}\left( s^{t}\right) }{\bar {P}_{1}\left( s^{t}\right) }.$ (16)

This relative price relates to the real exchange rate as follows

$\displaystyle q\left( s^{t}\right) =\frac{P_{2}\left( s^{t}\right) }{P_{1}\left( s^{t}\right) }=\frac{\tau_{22}}{\tau_{11}}\left[ \frac{\alpha_{11} +\alpha_{12}\left( \frac{\tau_{11}}{\tau_{12}}\bar{q}\left( s^{t}\right) \right) ^{\frac{\rho}{1-\rho}}}{\alpha_{21}\left( \frac{\tau_{21}}{\tau _{22}} \right) ^{\frac{\rho}{\rho-1}}+\alpha_{22}\left( \bar{q}\left( s^{t}\right) \right) ^{\frac{\rho}{\rho-1}}}\right] ^{\frac{1-\rho}{\rho} }.$ (17)

In what follows $ P_{1}\left( s^{t}\right) $ is normalized to unity. Using goods market clearing, budget constraints, and optimality conditions the demand for good 2 in countries 1 and 2 can be expressed as

$\displaystyle c_{12}\left( s^{t}\right) =\frac{\alpha_{12}\bar{q}\left( s^{t}\right) ^{ \frac{1}{\rho-1}}}{\alpha_{11}+\alpha_{12}\bar{q}\left( s^{t}\right) ^{ \frac{\rho}{\rho-1}}}\left[ y_{1}\left( s^{t}\right) +\frac{ b_{1}\left( s^{t-1}\right) -Q_{1}\left( s^{t}\right) b_{1}\left( s^{t}\right) } {\Phi_{1}\left( s^{t}\right) } \right] ,$ (18)
$\displaystyle c_{22}\left( s^{t}\right) =\frac{\alpha_{22}}{\alpha_{21}\left( \frac{1}{ \bar{q}\left( s^{t}\right) }\right) ^{\frac{\rho}{\rho-1}}+\alpha_{22}} \left[ y_{2}\left( s^{t}\right) +\frac{ b_{2}\left( s^{t-1}\right) -Q_{2}\left( s^{t}\right) b_{2}\left( s^{t}\right) }{\Phi_{1}\left( s^{t}\right) \bar{q }\left( s^{t}\right) } \right] ,$ (19)

and $ \Phi_{1}\left( s^{t}\right) =\frac{\bar{P}_{1}\left( s^{t}\right) }{ P_{1}\left( s^{t}\right) }$ .

3.4  Definition of Equilibrium

A competitive equilibrium in the dynamic model is defined as follows:



Definition 2   A competitive equilibrium is a collection of allocations $ c_{i1}\left( s^{t}\right) $, $ c_{i2}\left( s^{t}\right) $, $ c_{i}\left( s^{t}\right) $, $ i_{i}\left( s^{t}\right) $, $ y_{i}\left( s^{t}\right) $, $ k_{i}\left( s^{t}\right) $, $ l_{i}\left( s^{t}\right) $, prices $ q\left( s^{t}\right) $, $ \bar{q}\left( s^{t}\right) $, $ w\left( s^{t}\right) $, $ r\left( s^{t}\right) $, $ Q_{i}\left( s^{t}\right) $, and profits $ p_{i}\left( s^{t}\right) $, $ i=1,2$, such that $ \left( i\right) $ for every household the allocations solve the household's maximization problem for given prices, $ \left( ii\right) $ for every firm profits are maximized, and $ \left( iii\right) $ the markets for labor, capital, goods, and bonds ( $ b_{1}\left( s^{t}\right) + b_{2}\left( s^{t}\right) = 0$) clear.


4  Computation and Calibration

In a static model, the computation of the possibly multiple equilibria can be separated from the consideration of sunspot equilibria. However, in a dynamic model this is no longer possible. Each equilibrium path depends on the probability distribution over all the possible equilibria. To impose more discipline on the analysis, the following assumptions are made in the subsequent computations.

4.1  More Assumptions

As argued in Section 2, equilibrium multiplicity occurs because changes in the relative price lead to a shift in purchasing power that can be supported as equilibrium if the trade elasticity is low. In a dynamic economy with capital, labor, and internationally traded bonds the same intuition applies provided that markets are not complete.12

In order to characterize the equilibrium multiplicity in the general model, I impose two assumptions:

  1. At time 1 country 2 experiences an unexpected rise in its technology. The shock follows an AR(1) process with known persistence. Once the shock is realized agents perfectly foresee the future path of technology.

  2. The agents' problem is modified to yield stationarity of the net foreign asset position under the first assumption. Two alternatives are considered:

  3. (a)  Agents face a convex cost for holding/issuing bonds as in Heathcote and Perri (2002), and Schmitt-Grohé and Uribe (2003). The collected fees are reimbursed to the agents by a lump-sum transfer. $ \Gamma\left( \frac{B_{i}\left( s^{t}\right) }{\bar{P}_{i}\left( s^{t}\right) }\right) $ denotes the portfolio costs in terms of country $ i$'s traded good, where $ \Gamma\left( 0\right) =0$, $ \Gamma^{\prime}\left( 0\right) =0$, and $ \Gamma^{\prime}>0$ otherwise. The budget constraint of a household is now given by


      $\displaystyle P_{i}\left( s^{t}\right) \left( c_{i}\left( s^{t}\right) +i_{i}\left( s^{t}\right) \right) \leq\bar{P}_{i}\left( s^{t}\right) w_{i}\left( s^{t}\right) l_{i}\left( s^{t}\right)$    
      $\displaystyle +\bar{P}_{i}\left( s^{t}\right) r_{i}\left( s^{t}\right) k_{i}\left( s^{t-1}\right) +\bar{P}_{i}\left( s^{t}\right) p_{i}\left( s^{t}\right)$    
      $\displaystyle +b_{i}\left( s^{t-1}\right) -Q_{i}\left( s^{t}\right) b_{i}\left( s^{t}\right) -\bar{P}_{i}\left( s^{t}\right) \Gamma\left( \frac {B_{i}\left( s^{t}\right) }{\bar{P}_{i}\left( s^{t}\right) }\right) +T_{i}\left( s^{t}\right) .$ (20)

    (b)  Agents' intertemporal discount factors are endogenous as in Uzawa (1968).13 More specifically, the problem of the representative household is given by


      $\displaystyle \max_{\substack{c_{i}\left( s^{t}\right) ,l_{i}\left( s^{t}\right) c_{i1}\left( s^{t}\right) ,c_{i2}\left( s^{t}\right) \\ k_{i}\left( s^{t}\right) , i_{i}\left( s^{t}\right) ,b_{i}\left( s^{t}\right) }} \sum_{t=0}^{\infty}\sum_{s^{t}}\theta_{i}\left( s^{t}\right) \pi\left( s^{t}\right) U\left( c_{i}\left( s^{t}\right) ,l_{i}\left( s^{t}\right) \right)$ (21)
      s.t.    
      $\displaystyle \theta_{i}\left( s^{t+1}\right) =\beta_{i}\left[ U\left( c_{i}\left( s^{t}\right) ,l_{i}\left( s^{t}\right) \right) \right] \theta_{i}\left( s^{t}\right)$ (22)

    and equations (7) and (8).

The first assumption eliminates all uncertainty including sunspots from period 2 onwards. This assumption allows the characterization of the equilibrium multiplicity without the additional complications that arise in a fully stochastic model with incomplete markets, borrowing constraints and endogenous capital accumulation. Identifying multiple equilibria is a tedious task, but without this assumption, one does not even know where to look for the equilibrium multiplicity.

The second assumption is a direct consequence of the first. Since I assume perfect foresight from period 2 onwards, the long-run equilibrium value of the net foreign asset position changes in response to the temporary shock.14 Unfortunately, this long-run value and therefore the long-run equilibrium are unknown thereby making the computational procedure suggested below inapplicable. Under the second assumption, however, net foreign assets are known to return to their pre-shock level.15

4.2  Solution Method

I describe the solution algorithm for finding the equilibrium path of the endogenous variables for the (perfect-foresight) experiment described above.

The major step in accounting for equilibrium multiplicity is to generate a good starting guess. Unfortunately, local approximation methods such as linearization or second-order perturbation methods are of little help. These methods can only find one equilibrium. While the suggested algorithm does not guarantee the detection of all equilibria, it does detect some.16

To find the equilibrium paths I use a combination of forward and backward shooting algorithms. Since shooting algorithms with many state variable are computationally intensive, the algorithm starts under the assumption of fixed capital. Endogenous capital accumulation is then reintroduced once a starting guess is found. Without loss in generality, the stationarity inducing devices introduced under assumption 2 are parameterized to induce zero net-foreign asset positions in any steady state.




Algorithm 1  
  1. Choose the parametrization of the model, set $ \omega _{ki}\approx0$, and assume that capital is fixed at its steady level, i.e., remove the Euler equation for capital from the model. Assume that there are no shocks.

  2. For each locally stable steady state compute the stable manifold using a reverse shooting algorithm as described in Judd (1998). I reduce the dimension of the problem to the relative price $ \bar{q}$, and bond holdings $ b_{1}$. The resulting manifold shows the path along which the economy converges to a steady state from the initial net foreign asset position $ b_{1}\left( s_{-1}\right) $ and the initial value of the relative price $ \bar{q}\left( s_{0}\right) $.

  3. Compute the equilibrium path of the endogenous variables to a purely transitory shock in time 1, $ A_{2}\left( s_{1}\right) $. Prior to the shock the economy is assumed to be in steady state. Since bond holdings can freely adjust from one period to the other, the economy must move along the stable manifold(s) computed under step 2 starting from the second period on. To find a candidate equilibrium path:
    (a)  choose a pair $ \left( b_{1}\left( s^{1}\right) ,\bar {q}\left( s^{2}\right) \right) $ on the manifold as a guess for the position of the economy in the second period, the period right after the shock,
    (b)  use the first order conditions of the model to compute the implied value of $ b_{1}\left( s_{0}\right) $ if $ A_{2}\left( s_{1}\right) >0$. Since bond holdings are 0 in the steady state, a bond-price pair $ \left( b_{1}\left( s^{1}\right) ,\bar{q}\left( s^{2}\right) \right) $ is an equilibrium only if the implied bond holdings for state $ s_{0}$ satisfy $ b_{1}\left( s_{0}\right) =0$.

  4. This candidate impulse response is used as starting guess in a code that solves non-linear perfect foresight problems using a Newton method. The solution algorithm for this step borrows heavily from the DYNARE code "simul" and is implemented in FORTRAN.

  5. Once an equilibrium path has been computed, I reintroduce the dynamic investment decision and increase $ \omega_{ki}$ to its desired value. At this stage it is also convenient to increase the persistence of the shock if desired.


Figure 2 shows the stable manifolds computed under step 2 for the case of convex portfolio costs (top panel) and endogenous discounting (lower panel) with $ \varepsilon\approx0.44$ and $ \omega_{ki} =0.01$. In the case of convex portfolio costs the two stable steady states which are marked by the filled in circles feature different capital stocks and therefore the two manifolds do not "connect". As argued in Bodenstein (2007), the third steady state with $ \bar{q}=1$ is unstable. In the case of endogenous discounting the steady state is always unique and stable as indicated by the black arrows.

4.3  Calibration

Most parameter choices are taken straight from the international business cycle literature and are summarized in Table 1. The utility function is assumed to be additive separable between labor and consumption

$\displaystyle U\left( c,l\right) =\frac{c^{1-\sigma}}{1-\sigma}-\chi_{0}\frac{l^{1+\chi} }{1+\chi}.$ (23)

Following assumption 2a the convex portfolio costs are chosen to be quadratic

$\displaystyle \Gamma\left( \frac{B_{i}}{\bar{P}_{i}}\right) =\frac{1}{2}\phi_{b}\left( \frac{B_{i} }{\bar{P}_{i}}\right) ^{2},$ (24)

and under assumption 2b the endogenous discount factor is given by

$\displaystyle \beta\left( c_{i},l_{i}\right) =\left[ 1+\exp\left( \frac{c^{1-\sigma} }{1-\sigma}-\chi_{0}\frac{l^{1+\chi}}{1+\chi}\right) \right] ^{-\psi_{i}}.$ (25)

The $ \psi_{i}$'s are chosen such that $ \beta\left( c_{i},l_{i}\right) $ takes on the value of $ \beta$ at the steady state. The parameters $ \alpha _{11}$ and $ \alpha_{12}$ determine the home bias in consumption. Following the applied DSGE literature, these consumption weights are chosen to match the import to GDP ratio for the U.S., which is about 13% .17


5  Results

Before discussing the results for the dynamic economy a look at the steady state equilibria for this economy seems instructive. Similar to the endowment economy in Section 2, there are three steady states absent international financial markets provided that the elasticity of substitution between the traded goods $ \varepsilon$ is sufficiently low. For the above calibration this is true for $ \rho=-1.6$.18 The same steady states arise with incomplete markets under the additional restriction that steady state bond holdings are zero.

For an elasticity of substitution $ \varepsilon\approx0.38$ $ \left( \rho=-1.65\right) $ the three different values of the relative price that are consistent with an equilibrium are given in the first column of Table 2. Furthermore, Table 2 summarizes how capital, employment and output in country 1 differ across steady states. The value of variable $ x_{2}\left( \bar{q}^{\ast}\right) $ for country 2 coincides with $ x_{1}\left( \frac{1}{\bar{q}^{\ast}}\right) $. In contrast to the endowment economy, the differences in purchasing power across equilibria also have an impact on the supply side of the economy. For example, if the relative price is in favor of country 1, i.e., $ \bar{q}=0.35$, country 1 accumulates higher capital, enjoys higher output and uses less labor in the production process.

5.1  Convex Portfolio Costs

Under the above specification of the portfolio cost function the steady state value of bond holdings is uniquely determined to be zero. Thus, the steady states of the model with portfolio costs are the same as in the economy without international financial markets.

Similar to Bodenstein (2007), the steady state with $ \bar{q}=1$ is dynamically unstable, while the remaining two are stable. Prior to the realization of the positive technology shock in the foreign country, the economy is assumed to be in the steady state with $ \bar{q}=0.35$. The size of the shock is set equal to 0.01%.

5.1.1  Impulse Response Functions

Figure 3 plots the impulse response functions for selected variables in percentage deviations from the original steady state. Paths 1 (solid line) and 2 (dashed line) lead back to the original steady state, whereas path 3 (dotted line) converges to the other stable steady state with $ \bar{q}=2.86$. As the magnitude of the responses is considerably smaller for path 1 than for the other two paths, Figure 4 plots the responses of the variables for the non-linear solution of path 1 (solid line).

The reasoning behind the equilibrium multiplicity in the dynamic economy is the same as in the static endowment economy: given the predetermined capital stock, there are three locally isolated price equilibria each associated with a different distribution of relative purchasing power. If the capital stock adjusted quickly, one would expect that there is one path in the neighborhood of each of the three steady states. However, the capital stock is predetermined relative to the shock and adjusts slowly afterwards. Hence, only path 1 starts in the neighborhood of a steady state - the original steady state in the case of path 1. In particular, path 3, which converges to the steady state with $ \bar{q}=2.86$, starts with allocations and prices that are far away from their long run values. This fact is driven by the differences in the capital stock in the original steady state ($ \bar {q}=0.35$) and the new one ($ \bar{q}=2.86$) as reported in Table 2.

Path 1 is the sole path that linearization around the original steady state detects. In response to the transitory technology shock output rises in the foreign country for a few periods. Simultaneously, the real exchange rate appreciates for the home country reflecting a change of the relative price $ \bar{q}$ against the foreign country. Even more purchasing power is shifted to the home country and the home country's consumption rises relative to the foreign country's.

While the dynamics for path 1 are solely driven by the increase in foreign technology, the dynamics for paths 2 and 3 are driven mostly by the large shifts in relative purchasing power towards the foreign country that are associated with the rise in the relative price of the foreign good. The effects of the technology shock are negligible in these two cases. In fact, the same responses would be obtained if the technology shock was replaced by a pure sunspot shock. Demand in the home country (consumption plus investment) falls considerably. Home output falls as well, but by less than home consumption, as the foreign country raises its demand for the home good due to generally stronger foreign demand. Furthermore, the trade balance to GDP ratio falls on impact, indicating that the home country borrows funds to smooth the consequences of the shock.

5.1.2  Local Approximation

As shown in Figure 4, path 1 can be reliably approximated using (log-)linearization techniques around the model's non-stochastic steady state. However, this is not the case for paths 2 and 3.

Consider path 3 first. In general, one should be able to compute an approximate path using local approximation techniques around the new steady state with $ \bar{q}=2.86$. The values of the predetermined variables (capital stocks and bonds) in the original steady state are then taken as starting values of the system. While this approach works reasonably well if the share of capital in production is low, Figure 5 shows that this is not true for the chosen calibration. The linear approximation to path 3 (solid line) and the actual path 3 (dotted line) differ considerably. These differences are most obvious in the exaggerated response of the real exchange rate and the relative price $ \bar{q}$ in the linearized framework. A second order perturbation approach yields results that are even further away from the true path. Equally disappointing results are obtained with a starting guess that uses knowledge about the actual path 3, namely the values that the state variables assume for the period right after the shock. These observations suggest, that for a realistic calibration of the model the deviations of the starting values from their new steady state values are simply too large for path 3 to be correctly approximated by local approximation methods.

In the case of path 2 the attempt of using local approximation techniques is even less successful. Prior to computing path 2 it is not even known whether path 2 converges back to the original steady state or to the other stable steady state. Depending on the magnitude of the technology shock and the share of capital in production $ \omega_{ki}$, one can find either behavior. Even if the convergence properties of path 2 are known, one faces the question how to approximate this path by a first- or second-order approximation technique that expands the equilibrium system around the steady state to which the system will converge. For a given realization of the state variables (technology, home and foreign capital, international bond holdings) these methods prescribe a unique adjustment path. As the economy is in steady state prior to the shock, the adjustment path that is recovered using perturbation methods is path 1. Using the values of the state variables of the first period after the shock along the true path 2 rather than those of the period of the shock as starting values in the approximated decision rules does not recover path 2 either.

5.2  Endogenous Discount Factor

As discussed in more detail in Bodenstein (2007), there is always a unique steady state irrespective of the value of the trade elasticity if the agents' discount factors are endogenous. In the following I calibrate the free parameters in the functional form of the endogenous discount factor such that the unique stable steady state features $ \bar{q}=1$. Obviously, this uniqueness is somewhat artificial, since the underlying economy without endogenous discounting still has multiple steady states with zero bond holdings if $ \rho<-1.60$ for the above calibration.

However, the analysis with an endogenous discount factor provides insights that go beyond those gained in the model with convex portfolio costs. First, the version of the model allows for the construction of equilibria around the symmetric steady state as in Corsetti et al (2008) and Thoenissen (2008). Second, it turns out that there can be multiple equilibria even if $ \rho>-1.60$, i.e., for calibrations that never admit multiple steady states.

5.2.1  Impulse Response Functions

Figure 6 plots the impulse response functions for selected endogenous variables for $ \rho\approx-1.469$ to a purely transitory shock to the foreign country's technology level of 0.01%. Similar to the analysis under convex portfolio costs, the technology shock is the major determinant of the equilibrium dynamics for only one of the three paths. The other two paths are associated with large shifts in relative purchasing power and the effects of the technology shock are negligible. However, all these paths lead the economy back to the original steady state with $ \bar{q} =1$.

Path 1 (solid line) implies a persistent decline of the relative price $ \bar{q}$ that increases the purchasing power of the home country. Consequently, home consumption, investment, and output show a strong rise. The foreign country is willing to trade at the low relative price in order to receive at least some of the home good from which it can hardly substitute away. The roles are reversed for path 3 (dotted line) which features a strong rise in the relative price.

Path 2 (dashed line) has received considerable attention in the recent literature, see, e.g., the work of Corsetti et al (2008) and Thoenissen (2008). Subject to some caveats discussed below, this is the unique path that is recovered by linearization around the model's steady state.19 The increase in foreign technology raises foreign output. In contrast to the dynamics under a high elasticity of substitution the price of the foreign good does not need to fall under a low elasticity of substitution: if $ \bar{q}$ rises, the value of the foreign country's production increases, while it falls in the home country. In both countries there is a tendency to substitute away from the more expensive foreign good. This situation can only be an equilibrium if the increase in purchasing power in the foreign country offsets the negative impact that the higher price and lower home income exert on overall demand for the foreign good.

Corsetti et al (2008) refer to the appreciation of the terms of trade in light of a positive supply shock as "negative transmission mechanism".20 From an applied point of view, path 2 has certain appealing features. In a richer model with distribution costs and nontraded goods, Corsetti et al (2008) invoke this mechanism to address two important puzzles in the international macroeconomics literature, namely the Backus-Smith puzzle (Backus and Smith (1993)) and the real exchange rate volatility puzzle. If the implied elasticity of substitution between foreign and domestic goods is low enough, an appreciation of the relative price and the real exchange rate for the home country goes along with an increase in the home country's consumption relative to the foreign country's. In addition, the real exchange rate is about as volatile as in the data. Furthermore, just as the model presented in Thoenissen (2008) and in line with the data their model predicts a negative cross-country correlation for consumption and a negative correlation of output with the real exchange rate - a major step forward relative to previous attempts in the literature. The model presented here can replicate all of these findings if $ \rho$ is sufficiently close to but less than a critical value of -1.4685. The next section sheds more light on this critical value of the trade elasticity of substitution.

5.2.2  Multiple Equilibria and the Short-Run Excess Demand Function

In Section 5.1 the analysis of the multiple equilibrium paths relies on the presence of multiple steady state equilibria, i.e., $ \rho<-1.60$. However, as just shown equilibrium multiplicity can also occurs if there is a unique steady state with $ \rho>-1.60$. The reason for this finding lies in the behavior of the economy in the short-run: the capital stock is not fully flexible. Consequently, if the relative price moves against country $ i$ its agents' ability to lower the production of good $ i$ and offset some of the adverse movements in the relative price is diminished. Hence, multiple equilibria can exist in the short run but may be absent in the long run.

More formally, the short-run excess demand function can be upward-sloping while the long-run excess demand function is downward-sloping. Figure 7 plots the slope of the excess demand function around $ \bar{q} =1$, i.e., the partial derivative of the excess demand for good 2 with respect to the relative price, denoted by $ \frac{\partial z_{2t}}{\partial\bar{q}_{t}}$, for the two horizons as a function of $ \rho$. The underlying derivations are provided in Appendix A. The scale for the short-run and the long-run slopes are depicted on the left and the right vertical axis, respectively. A positive value of the slope indicates the presence of multiple equilibria. The curve labeled "long run excess demand'' intersects the horizontal axis at $ \rho= -1.60$, whereas the curve labeled "short run excess demand'' intersects with the horizontal axis for the higher value of $ \rho=-1.4685$.

The change in the sign of the slope of the excess demand function, $ sign\left( \frac{\partial z_{2t}}{\partial\bar{q}_{t}}\right) $ , can be easily detected by looking at the policy function for the relative price $ \bar{q}$ that is obtained from linearization around the steady state with $ \bar{q} = 1$. If $ sign\left( \frac{\partial z_{2t}}{\partial\bar{q}_{t} }\right) $ changes sign so do the coefficients on the endogenous state variables in the policy function for the relative price.

To offer an economic interpretation, consider the coefficient on current bond holdings which is denoted by $ \frac{\partial\bar{q}_{t}}{\partial b_{1t-1}}$. This coefficient is positive if $ sign\left( \frac{\partial z_{2t} }{\partial\bar{q}_{t}}\right) $ is positive and negative otherwise. Suppose $ b_{1t-1}>0$, i.e., the home country lends funds to the foreign country in period t - 1. In order to pay back its debt obligations in time t, the foreign country has to increase its total export revenue which implies that the price and/or the number of exported units have to rise. If $ \frac {\partial\bar{q}_{t}}{\partial b_{1t-1}} <0$, the foreign country increases the number of units determined for exports and the relative price of the foreign good falls. Since the elasticity of substitution is high, a small decline in the price leads to a relatively large increase in exports and the total export revenue rises. If $ \frac{\partial\bar{q}_{t}}{\partial b_{1t-1} }>0$, the relative price of the foreign good increases and the number of exported units falls. However, since the substitutability between the traded goods is low, a large price increase leads to a relatively small decline in the number of exported units and the total export revenue rises.21

Remember, that the slope of the short-run excess demand function is close to zero in the neighborhood of $ \rho=-1.4685$ for the equilibrium with $ \bar{q} =1$. When deriving the linear approximation to the law of motion of the relative price one divides by a number that is arbitrarily close to zero. This finding explains why Thoenissen (2008), Corsetti et al (2008), de Walque et al (2005), and Benigno and Thoenissen (2008) find that in each of their models there is a critical value of the trade elasticity of substitution for which the volatility of the real exchange rate is infinite. Such behavior of the model is again indicative of equilibrium multiplicity.22

5.2.3  Local Approximation

The existence of multiple equilibria poses some challenges to applying the model with a low trade elasticity of substitution. For example, it turns out that for a sufficiently large technology shock only one of the three equilibrium paths exists. For $ \rho\approx-1.469$ the maximum size of the technology shock in the foreign country for which all three paths exists is around 0.1%. For shocks larger than 0.1%, only path 1 exists.23

In particular, this finding poses a problem if relying on local approximation techniques. Log-linearization around the model's unique steady state delivers an approximation of path 2 as the sole equilibrium path. Figures 8 and 9 show the possible computational errors for a calibration of the technology shock that follows Backus et al (1995). The shock to the foreign country's technology is 0.85% with persistence of 0.95. The solid line in Figure 8 shows the unique equilibrium path for this technology shock. The dashed line is the suggested solution path derived from log-linearization around the steady state. The discrepancy between the two paths is obviously large. To convince the reader of the (in-)accuracy of the two suggested solution paths Figure 9 plots the approximation errors in the three dynamic equations of the model: the two Euler equations for capital and the risk sharing condition. The systematic nature of the approximation error indicates, that the dashed line (path 2) is not a solution of the model for a technology shock of size 0.85%.

The closer the value of the elasticity of substitution is to the threshold level of $ \varepsilon$ (meaning $ \rho$ close to -1.4685, the smaller is the size of the technology shock for which path 2 exists. For a calibration of $ \rho=-1.65$ as in Section 5.1, all three impulse responses exist for a 0.85% technology shock.


6  Sensitivity

The results are sensitive to three types of changes: parameters, model specification and the assumptions about financial markets. If financial markets are complete with respect to all states of nature (fundamental and non-fundamental), the equilibrium is always unique. However, if agents cannot insure against self-fulfilling fluctuations, sunspot equilibria can be constructed and there will be multiple equilibria provided that the trade elasticity is sufficiently low. Therefore, multiple equilibria will also be present in models with a richer set of assets, unless the assets span the complete market. As noted earlier, fundamental shocks are not really needed in the analysis, but they facilitate the computations and allow for comparisons of my results to the literature.

6.1  Sensitivity to Parametrization

Changes in the underlying model parameters affect the threshold level of the trade elasticity $ \bar{\varepsilon}$ for which the model displays multiple equilibria as follows:

$\displaystyle \begin{tabular}[c]{lllll} $\omega_{ki}$\ & $\downarrow$\ & $\Longrightarrow$\ & $\bar{\varepsilon}$\ & $\uparrow$\\ $\kappa$\ & $\downarrow$\ & $\Longrightarrow$\ & $\bar{\varepsilon}$\ & $\uparrow $\\ $\alpha_{ii}$\ & $\uparrow$\ & $\Longrightarrow$\ & $\bar{\varepsilon}$\ & $\uparrow$\\ $\chi$\ & $\uparrow$\ & $\Longrightarrow$\ & $\bar{\varepsilon}$\ & $\downarrow$ \end{tabular}$ (26)

Both lowering the share of capital in production $ \omega_{ki}$ or the elasticity of substitution between capital and labor $ \kappa$ makes the economy look more like an endowment economy and raises the threshold level $ \bar{\varepsilon}$ for which multiple equilibria are observed. Increasing home bias, i.e., a rise in $ \alpha_{ii}$, also leads to a higher value of $ \bar{\varepsilon}$ as shifts in relative purchasing power become less relevant with each country consuming mostly its own good.

The effect of lowering the labor supply elasticity (raising $ \chi$) depends on the value of $ \sigma$. If $ \sigma>1$, the threshold level $ \bar{\varepsilon}$ falls in response to a rise in $ \chi$. However, if $ \sigma<1$, a rise in $ \chi$ lowers $ \bar{\varepsilon}$. All these results can be obtained more formally from the derivations of the long- and short-run excess demand function that are provided in Appendix A.

6.2  Sensitivity to Model Specification

Multiple equilibria of the type discussed here can also occur in richer models with a low trade elasticity of substitution. A prominent example is the model with nontraded goods and distribution costs that is at the core of Corsetti et al (2008). In that model the implied elasticity of substitution between the home and foreign good at the consumer level is endogenous. Corsetti and Dedola (2005) mention the possibility of multiple steady states in such a model but do not analyze the dynamic case.

The following example serves to highlight some additional challenges that arise in models with distribution costs. Assume that agents in country $ i$ are endowed with $ y_{i}$ units of the traded good $ i$ and $ y_{iN}$ units of a nontraded good. Agents consume the two traded goods and their own nontraded good. However, in order to consume one unit of a traded good, agents have to forgive $ \eta$ units of their nontraded good. In the static economy agent $ i $ solves

  $\displaystyle \max_{{ c_{iT},c_{iN},c_{i1},c_{i2}}}c_{i}=\left[ \left( \alpha _{iT}\right) ^{1-\phi}c_{iT}^{\phi}+\left( \alpha_{iN}\right) ^{1-\phi }c_{iN}^{\phi}\right] ^{\frac{1}{\phi}}$ (27)
  s.t.    
  $\displaystyle P_{iN}c_{iN}+\left( \bar{P}_{1}+\eta P_{iN}\right) c_{i1}+\left( \bar{P} _{2}+\eta P_{iN}\right) c_{i2}\leq\bar{P}_{i}y_{i}+P_{iN}y_{iN},$ (28)

where $ c_{iT}=\left[ \left( \alpha_{i1}\right) ^{1-\rho}c_{i1}^{\rho }+\left( \alpha_{i2}\right) ^{1-\rho}c_{i2}^{\rho}\right] ^{\frac{1}{\rho} }$ . The market clearing conditions for this economy are

$\displaystyle c_{11}+c_{21}$ $\displaystyle \leq y_{1},$ (29)
$\displaystyle c_{12}+c_{22}$ $\displaystyle \leq y_{2},$ (30)
$\displaystyle c_{iN}+\eta\left( c_{i1}+c_{i2}\right)$ $\displaystyle \leq y_{iN},$ (31)

with $ i=1,2$. Following Corsetti et al (2008), let $ \alpha_{iT}=0.55$, $ \alpha_{iN}=0.45$, $ \alpha_{11} =$ $ \alpha_{22}=0.72$, $ \alpha_{12}=$ $ \alpha_{21}=0.28$, $ \eta=1.09$, $ \rho=-0.17$, and $ \phi=-0.35$. I set the ratio of nontraded to traded goods $ \frac{y_{iN}}{y_{i}}$ equal to 2.25, which is of similar magnitude as the ratio of service to manufacturing output in the U.S., the common proxies for traded and nontraded goods. For this parametrization, the trade elasticity of substitution between the foreign and the domestic good is around 0.27, the producer price elasticity of traded goods is around 0.68, and the consumer price elasticity of traded goods is $ \frac{1}{1- \rho}=0.85$.

This endowment economy features a unique equilibrium with the relative price $ \bar{q} = \frac{ \bar{P_{1}}}{ \bar{P_{2}}} = 1$. Consider, however, a 3 percent decline of the endowment with both traded goods while keeping the endowment with nontraded goods constant. Two new equilibria arise, one with $ \bar{q} = 0.82$ and the other one with $ \bar{q} = 1.22$. However, the equilibrium remains unique if the endowment with traded goods increases in both countries by the same amount. Hence, with an endogenous trade elasticity of substitution symmetric shocks can have very different implications for the number of equilibria and the resulting dynamics of the model.


7  Conclusions

Dynamic models with multiple equilibria can give rise to complex dynamics. As a first step, I characterize the dynamics of three equilibrium paths in a standard model of the international business cycle under the assumption that the elasticity of substitution between traded goods is low.

The empirical literature reports a wide range of trade elasticities at the aggregate level from 0 to 1.5. Recent macroeconomic research has pointed towards low trade elasticities: Corsetti, Dedola, and Leduc (2008), Thoenissen (2008) and others have shown that business cycle models with low trade elasticities may explain several puzzles in international macroeconomics, such as the Backus-Smith puzzle, the real exchange rate volatility puzzle, or the cross country correlation puzzles for consumption and output (see Thoenissen (2008) for a comprehensive list). In estimated DSGE models Rabanal and Tuesta (2005), Lubik and Schorfheide (2005), and de Walque, Smets, and Wouters (2005) show estimates for the trade elasticity that are very low and in some cases close to zero. At least in some of these cases the results hinge on values of the trade elasticity of substitution that imply the presence of multiple equilibria. However, no study has discussed the presence multiple equilibria in these DSGE models in depth, let alone the possible dynamics.24

Going forward the literature faces two major possibilities to address the aforementioned aspects of the data. Using the insights of this paper, one can proceed by building fully stochastic general equilibrium models with low trade elasticities that explicitly allow for sunspot equilibria. Such work requires the use of global approximation methods and is computationally intensive. It also seems important to continue the search for alternative models. Oil shocks as in Backus and Crucini (1998) and Bodenstein, Erceg, Guerrieri (2008), or investment-specific shocks as in Raffo (2008) are potentially promising. In the meantime trade adjustment costs as in Erceg, Guerrieri and Gust (2006) and taste shocks as in Corsetti et al (2008) can serve as simple fixes in the estimation of DSGE models to prevent the estimation algorithm from forcing the elasticity of substitution towards very low values in the attempt of fitting the model to the data.


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Table 1: Calibration of Baseline Model - Panel A: Parameters Governing Households' Behavior

Parameter Used to Determine
β = 0.99 discount factor
σ = 1 intertemporal consumption elasticity
α11 = 0.87 weight on good 1 country 1's cons. basket
α21 = 0.13 weight on good 1 country 2's cons. basket
χ = 5 labor supply elasticitya
ρ = -1.65 elasticity between traded goods
α12 = 0.13 weight on good 2 country 1's cons. basket
α22 = 0.87 weight on good 2 country 2's cons. basket

a  The Frisch elasticity is 1/χ = 0.2.


Table 1: Calibration of Baseline Model - Panel B: Parameters Governing Firms' Behavior

Parameter Used to Determine
ωk = 0.33 share of capital in production
δ = 0.025 depreciation rate of capital
ωl = 0.67 share of capital in production

Table 1: Calibration of Baseline Model - Panel C: Parameters Governing Asset Dynamics

Parameter Used to Determine
ψ1 = 0.0275 endogenous discounting
φb = 0.0025 adjustment costs
ψ2 = 0.0275 endogenous discounting

Table 2:  Steady State Values of Selected Prices and Allocations

Relative Price:
$ \bar{q}^{\ast}$
Domestic Output:
$ y_{1}^{\ast}$
Domestic Capital:
$ k_{1}^{\ast} $
Domestic Labor:
$ l_{1}^{\ast}$
0.35
1.87
19.5
0.22
1.00
1.81
17.0
0.23
2.86
1.72
13.5
0.24

Steady state values for country 2 are mirroring the values for country 1. The value for variable x2, at the equilibrium with $ \bar{q}^{\ast}$ coincides with the equilibrium value for variable x1 at the equilibrium $ \frac{1}{\bar{q}^{\ast}}$.

Figure 1:  Excess Demand Function in the Endowment Economy

Data for Figure 1 immediately.

Data for Figure 1

Relative Price:
epsilon = 2
Excess Demand:
epsilon = 2
Relative Price:
epsilon = 0.42
Excess Demand:
epsilon = 0.42
0.010000
92.729934
0.010000
0.357704
0.014945
59.834305
0.014945
0.244339
0.019890
43.381403
0.019890
0.181651
0.024835
33.544174
0.024835
0.141842
0.029780
27.023144
0.029780
0.114399
0.034725
22.398591
0.034725
0.094418
0.039670
18.958927
0.039670
0.079287
0.044615
16.308235
0.044615
0.067483
0.049560
14.208765
0.049560
0.058059
0.054505
12.509170
0.054505
0.050394
0.059450
11.108562
0.059450
0.044062
0.064395
9.937125
0.064395
0.038764
0.069340
8.945029
0.069340
0.034284
0.074285
8.095772
0.074285
0.030458
0.079230
7.362013
0.079230
0.027165
0.084175
6.722880
0.084175
0.024311
0.089120
6.162169
0.089120
0.021821
0.094065
5.667114
0.094065
0.019637
0.099010
5.227529
0.099010
0.017711
0.103955
4.835186
0.103955
0.016007
0.108900
4.483376
0.108900
0.014491
0.113845
4.166570
0.113845
0.013139
0.118790
3.880177
0.118790
0.011928
0.123735
3.620353
0.123735
0.010841
0.128680
3.383859
0.128680
0.009863
0.133625
3.167944
0.133625
0.008980
0.138570
2.970261
0.138570
0.008182
0.143515
2.788793
0.143515
0.007458
0.148460
2.621802
0.148460
0.006801
0.153405
2.467779
0.153405
0.006203
0.158350
2.325411
0.158350
0.005659
0.163295
2.193548
0.163295
0.005162
0.168240
2.071181
0.168240
0.004709
0.173185
1.957421
0.173185
0.004294
0.178130
1.851481
0.178130
0.003915
0.183075
1.752664
0.183075
0.003567
0.188020
1.660349
0.188020
0.003248
0.192965
1.573980
0.192965
0.002956
0.197910
1.493063
0.197910
0.002687
0.202855
1.417150
0.202855
0.002440
0.207800
1.345843
0.207800
0.002214
0.212745
1.278779
0.212745
0.002005
0.217690
1.215631
0.217690
0.001814
0.222635
1.156105
0.222635
0.001638
0.227580
1.099931
0.227580
0.001476
0.232525
1.046865
0.232525
0.001327
0.237470
0.996685
0.237470
0.001190
0.242415
0.949188
0.242415
0.001064
0.247360
0.904188
0.247360
0.000949
0.252305
0.861514
0.252305
0.000842
0.257250
0.821012
0.257250
0.000745
0.262195
0.782537
0.262195
0.000656
0.267140
0.745958
0.267140
0.000574
0.272085
0.711152
0.272085
0.000499
0.277030
0.678009
0.277030
0.000430
0.281975
0.646425
0.281975
0.000368
0.286920
0.616303
0.286920
0.000311
0.291865
0.587554
0.291865
0.000259
0.296810
0.560097
0.296810
0.000211
0.301755
0.533856
0.301755
0.000169
0.306700
0.508758
0.306700
0.000130
0.311645
0.484738
0.311645
0.000095
0.316590
0.461735
0.316590
0.000064
0.321535
0.439690
0.321535
0.000036
0.326480
0.418551
0.326480
0.000011
0.331425
0.398267
0.331425
-0.000012
0.336370
0.378792
0.336370
-0.000031
0.341315
0.360081
0.341315
-0.000048
0.346260
0.342094
0.346260
-0.000063
0.351205
0.324792
0.351205
-0.000076
0.356150
0.308139
0.356150
-0.000087
0.361095
0.292101
0.361095
-0.000096
0.366040
0.276646
0.366040
-0.000103
0.370985
0.261744
0.370985
-0.000109
0.375930
0.247367
0.375930
-0.000113
0.380875
0.233488
0.380875
-0.000117
0.385820
0.220082
0.385820
-0.000118
0.390765
0.207126
0.390765
-0.000119
0.395710
0.194596
0.395710
-0.000119
0.400655
0.182472
0.400655
-0.000118
0.405600
0.170734
0.405600
-0.000116
0.410545
0.159362
0.410545
-0.000113
0.415490
0.148340
0.415490
-0.000110
0.420435
0.137648
0.420435
-0.000106
0.425380
0.127273
0.425380
-0.000101
0.430325
0.117198
0.430325
-0.000096
0.435270
0.107408
0.435270
-0.000090
0.440215
0.097890
0.440215
-0.000085
0.445160
0.088631
0.445160
-0.000078
0.450105
0.079617
0.450105
-0.000072
0.455050
0.070838
0.455050
-0.000065
0.459995
0.062282
0.459995
-0.000058
0.464940
0.053937
0.464940
-0.000051
0.469885
0.045794
0.469885
-0.000044
0.474830
0.037842
0.474830
-0.000037
0.479775
0.030072
0.479775
-0.000029
0.484720
0.022474
0.484720
-0.000022
0.489665
0.015041
0.489665
-0.000015
0.494610
0.007764
0.494610
-0.000008
0.499555
0.000635
0.499555
-0.000001
0.504500
-0.006355
0.504500
0.000006
0.509445
-0.013212
0.509445
0.000013
0.514390
-0.019943
0.514390
0.000020
0.519335
-0.026555
0.519335
0.000026
0.524280
-0.033056
0.524280
0.000032
0.529225
-0.039450
0.529225
0.000038
0.534170
-0.045746
0.534170
0.000043
0.539115
-0.051947
0.539115
0.000049
0.544060
-0.058061
0.544060
0.000053
0.549005
-0.064092
0.549005
0.000058
0.553950
-0.070046
0.553950
0.000062
0.558895
-0.075929
0.558895
0.000066
0.563840
-0.081745
0.563840
0.000069
0.568785
-0.087500
0.568785
0.000072
0.573730
-0.093198
0.573730
0.000074
0.578675
-0.098844
0.578675
0.000076
0.583620
-0.104442
0.583620
0.000078
0.588565
-0.109997
0.588565
0.000079
0.593510
-0.115514
0.593510
0.000079
0.598455
-0.120997
0.598455
0.000079
0.603400
-0.126450
0.603400
0.000078
0.608345
-0.131877
0.608345
0.000077
0.613290
-0.137282
0.613290
0.000075
0.618235
-0.142670
0.618235
0.000072
0.623180
-0.148044
0.623180
0.000069
0.628125
-0.153409
0.628125
0.000065
0.633070
-0.158768
0.633070
0.000060
0.638015
-0.164125
0.638015
0.000055
0.642960
-0.169484
0.642960
0.000049
0.647905
-0.174850
0.647905
0.000042
0.652850
-0.180225
0.652850
0.000035
0.657795
-0.185614
0.657795
0.000027
0.662740
-0.191021
0.662740
0.000018
0.667685
-0.196449
0.667685
0.000008
0.672630
-0.201903
0.672630
-0.000003
0.677575
-0.207386
0.677575
-0.000015
0.682520
-0.212902
0.682520
-0.000027
0.687465
-0.218456
0.687465
-0.000041
0.692410
-0.224051
0.692410
-0.000055
0.697355
-0.229692
0.697355
-0.000070
0.702300
-0.235382
0.702300
-0.000086
0.707245
-0.241127
0.707245
-0.000103
0.712190
-0.246929
0.712190
-0.000122
0.717135
-0.252794
0.717135
-0.000141
0.722080
-0.258727
0.722080
-0.000161
0.727025
-0.264731
0.727025
-0.000182
0.731970
-0.270811
0.731970
-0.000205
0.736915
-0.276973
0.736915
-0.000229
0.741860
-0.283222
0.741860
-0.000253
0.746805
-0.289561
0.746805
-0.000279
0.751750
-0.295998
0.751750
-0.000307
0.756695
-0.302537
0.756695
-0.000335
0.761640
-0.309183
0.761640
-0.000365
0.766585
-0.315944
0.766585
-0.000396
0.771530
-0.322824
0.771530
-0.000429
0.776475
-0.329830
0.776475
-0.000463
0.781420
-0.336969
0.781420
-0.000498
0.786365
-0.344247
0.786365
-0.000535
0.791310
-0.351672
0.791310
-0.000574
0.796255
-0.359251
0.796255
-0.000614
0.801200
-0.366992
0.801200
-0.000655
0.806145
-0.374902
0.806145
-0.000699
0.811090
-0.382991
0.811090
-0.000744
0.816035
-0.391267
0.816035
-0.000791
0.820980
-0.399739
0.820980
-0.000839
0.825925
-0.408417
0.825925
-0.000890
0.830870
-0.417313
0.830870
-0.000943
0.835815
-0.426435
0.835815
-0.000997
0.840760
-0.435797
0.840760
-0.001054
0.845705
-0.445410
0.845705
-0.001113
0.850650
-0.455286
0.850650
-0.001174
0.855595
-0.465440
0.855595
-0.001238
0.860540
-0.475885
0.860540
-0.001304
0.865485
-0.486638
0.865485
-0.001372
0.870430
-0.497713
0.870430
-0.001444
0.875375
-0.509128
0.875375
-0.001517
0.880320
-0.520902
0.880320
-0.001594
0.885265
-0.533054
0.885265
-0.001673
0.890210
-0.545604
0.890210
-0.001756
0.895155
-0.558575
0.895155
-0.001841
0.900100
-0.571991
0.900100
-0.001930
0.905045
-0.585876
0.905045
-0.002022
0.909990
-0.600259
0.909990
-0.002117
0.914935
-0.615169
0.914935
-0.002216
0.919880
-0.630637
0.919880
-0.002319
0.924825
-0.646698
0.924825
-0.002425
0.929770
-0.663387
0.929770
-0.002534
0.934715
-0.680745
0.934715
-0.002647
0.939660
-0.698815
0.939660
-0.002764
0.944605
-0.717644
0.944605
-0.002883
0.949550
-0.737281
0.949550
-0.003005
0.954495
-0.757783
0.954495
-0.003129
0.959440
-0.779209
0.959440
-0.003253
0.964385
-0.801626
0.964385
-0.003375
0.969330
-0.825106
0.969330
-0.003491
0.974275
-0.849727
0.974275
-0.003596
0.979220
-0.875576
0.979220
-0.003676
0.984165
-0.902751
0.984165
-0.003709
0.989110
-0.931357
0.989110
-0.003644
0.994055
-0.961511
0.994055
-0.003330

Figure 2:  Stable Manifolds

Figure 2 shows the stable manifolds computed under step 2 for the case of convex portfolio costs (top panel) and endogenous discounting (lower panel) with epsilon almost equal to 0.44 and omega subscript ki equal to 0.01. In the case of convex portfolio costs the two stable steady states which are marked by the filled in circles feature different capital stocks and therefore the two manifolds do not connect. The third steady state with q bar equals 1 is unstable. In the case of endogenous discounting the steady state is always unique and stable as indicated by the black arrows.

Figure 3:  Impulse Responses for Selected Variables: Convex Portfolio Costs

Data for Figure 3 immediately follows.

Data for Figure 3 - Panel A

Quarters
Output Home:
Path 3
Output Home:
Path 2
Output Home:
Path 1
Consumption Home:
Path 3
Consumption Home:
Path 2
Consumption Home:
Path 1
Investment Home:
Path 3
Investment Home:
Path 2
Investment Home:
Path 1
1
-1.984
0.089
0.000
-33.624
-3.923
0.049
-136.725
-13.470
-0.103
2
-2.544
0.007
-0.002
-34.328
-4.033
0.051
-129.735
-12.792
0.190
3
-3.085
-0.068
0.000
-34.996
-4.138
0.052
-124.097
-12.518
0.186
4
-3.616
-0.140
0.001
-35.629
-4.237
0.054
-118.835
-12.256
0.182
5
-4.134
-0.208
0.002
-36.228
-4.332
0.055
-113.926
-12.006
0.177
6
-4.638
-0.274
0.003
-36.793
-4.421
0.056
-109.348
-11.767
0.173
7
-5.126
-0.337
0.004
-37.327
-4.506
0.058
-105.081
-11.538
0.170
8
-5.599
-0.397
0.005
-37.830
-4.587
0.059
-101.102
-11.319
0.166
9
-6.055
-0.454
0.006
-38.303
-4.663
0.060
-97.393
-11.109
0.162
10
-6.493
-0.509
0.006
-38.747
-4.735
0.061
-93.935
-10.907
0.159
11
-6.913
-0.561
0.007
-39.163
-4.803
0.062
-90.710
-10.714
0.156
12
-7.316
-0.611
0.008
-39.553
-4.867
0.063
-87.703
-10.529
0.152
13
-7.701
-0.659
0.009
-39.918
-4.928
0.064
-84.897
-10.352
0.149
14
-8.067
-0.704
0.009
-40.258
-4.985
0.065
-82.278
-10.181
0.146
15
-8.416
-0.748
0.010
-40.575
-5.039
0.065
-79.834
-10.018
0.144
16
-8.748
-0.789
0.011
-40.869
-5.090
0.066
-77.551
-9.860
0.141
17
-9.062
-0.828
0.011
-41.142
-5.138
0.067
-75.417
-9.709
0.138
18
-9.360
-0.866
0.012
-41.395
-5.183
0.067
-73.423
-9.564
0.136
19
-9.641
-0.902
0.012
-41.628
-5.225
0.068
-71.557
-9.424
0.133
20
-9.906
-0.936
0.013
-41.843
-5.265
0.068
-69.811
-9.289
0.131
21
-10.156
-0.968
0.013
-42.041
-5.301
0.069
-68.177
-9.159
0.128
22
-10.390
-0.999
0.014
-42.221
-5.336
0.069
-66.645
-9.034
0.126
23
-10.611
-1.029
0.014
-42.386
-5.368
0.070
-65.209
-8.914
0.124
24
-10.817
-1.057
0.014
-42.536
-5.398
0.070
-63.863
-8.797
0.122
25
-11.010
-1.083
0.015
-42.671
-5.426
0.070
-62.598
-8.685
0.120
26
-11.190
-1.109
0.015
-42.793
-5.451
0.071
-61.411
-8.577
0.118
27
-11.358
-1.133
0.015
-42.903
-5.475
0.071
-60.295
-8.473
0.116
28
-11.514
-1.155
0.016
-43.000
-5.497
0.071
-59.246
-8.372
0.114
29
-11.659
-1.177
0.016
-43.086
-5.517
0.071
-58.259
-8.274
0.113
30
-11.793
-1.197
0.016
-43.161
-5.535
0.072
-57.329
-8.180
0.111
31
-11.917
-1.217
0.016
-43.225
-5.551
0.072
-56.452
-8.089
0.109
32
-12.032
-1.235
0.017
-43.281
-5.566
0.072
-55.626
-8.001
0.108
33
-12.137
-1.252
0.017
-43.327
-5.580
0.072
-54.847
-7.915
0.106
34
-12.233
-1.269
0.017
-43.365
-5.592
0.072
-54.110
-7.833
0.105
35
-12.322
-1.284
0.017
-43.394
-5.603
0.072
-53.415
-7.752
0.103
36
-12.402
-1.299
0.017
-43.416
-5.612
0.072
-52.757
-7.675
0.102
37
-12.475
-1.312
0.018
-43.432
-5.620
0.072
-52.135
-7.600
0.100
38
-12.541
-1.325
0.018
-43.440
-5.627
0.072
-51.545
-7.526
0.099
39
-12.600
-1.337
0.018
-43.442
-5.632
0.072
-50.987
-7.456
0.098
40
-12.653
-1.349
0.018
-43.439
-5.637
0.072
-50.458
-7.387
0.097
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-0.776
0.007
490
-8.015
-0.228
0.002
-31.526
-0.788
0.007
-31.499
-0.771
0.007
491
-8.013
-0.227
0.002
-31.522
-0.784
0.007
-31.494
-0.767
0.007
492
-8.011
-0.226
0.002
-31.517
-0.780
0.007
-31.490
-0.763
0.007
493
-8.009
-0.225
0.002
-31.513
-0.776
0.007
-31.486
-0.759
0.006
494
-8.008
-0.223
0.002
-31.508
-0.772
0.007
-31.482
-0.755
0.006
495
-8.006
-0.222
0.002
-31.504
-0.768
0.007
-31.477
-0.751
0.006
496
-8.004
-0.221
0.002
-31.500
-0.764
0.007
-31.473
-0.747
0.006
497
-8.002
-0.220
0.002
-31.495
-0.760
0.007
-31.469
-0.743
0.006
498
-8.001
-0.219
0.002
-31.491
-0.756
0.006
-31.465
-0.740
0.006
499
-7.999
-0.218
0.002
-31.487
-0.752
0.006
-31.461
-0.736
0.006

Data for Figure 3 - Panel B

Quarters
Consumption Home Relative to Foreign:
Path 3
Consumption Home Relative to Foreign
Path 2
Consumption Home Relative to Foreign
Path 1
Real Exchange Rate Home
Path 3
Real Exchange Rate Home
Path 2
Real Exchange Rate Home
Path 1
Trade Balance to GDP Ratio Home
Path 3
Trade Balance to GDP Ratio Home
Path 2
Trade Balance to GDP Ratio Home
Path 1
1
-48.876
-12.234
0.169
1432.772
78.605
-0.959
-0.144592
-0.008430
0.000848
2
-49.765
-12.555
0.177
1378.950
77.281
-0.941
-0.139647
-0.008916
0.000114
3
-50.599
-12.860
0.185
1328.396
76.014
-0.924
-0.129174
-0.008403
0.000106
4
-51.381
-13.148
0.192
1280.942
74.801
-0.908
-0.119334
-0.007916
0.000100
5
-52.115
-13.421
0.199
1236.422
73.639
-0.893
-0.110104
-0.007453
0.000093
6
-52.804
-13.679
0.205
1194.669
72.525
-0.878
-0.101458
-0.007013
0.000087
7
-53.449
-13.923
0.211
1155.520
71.455
-0.863
-0.093368
-0.006593
0.000081
8
-54.054
-14.154
0.217
1118.817
70.429
-0.849
-0.085803
-0.006194
0.000075
9
-54.621
-14.373
0.222
1084.408
69.442
-0.835
-0.078734
-0.005814
0.000070
10
-55.152
-14.579
0.227
1052.147
68.493
-0.822
-0.072132
-0.005452
0.000065
11
-55.649
-14.774
0.232
1021.897
67.581
-0.809
-0.065969
-0.005107
0.000060
12
-56.114
-14.959
0.237
993.527
66.702
-0.797
-0.060218
-0.004778
0.000055
13
-56.549
-15.133
0.241
966.913
65.856
-0.785
-0.054851
-0.004465
0.000050
14
-56.955
-15.297
0.245
941.939
65.041
-0.774
-0.049845
-0.004166
0.000046
15
-57.334
-15.452
0.249
918.495
64.254
-0.762
-0.045176
-0.003881
0.000042
16
-57.689
-15.598
0.252
896.481
63.495
-0.752
-0.040822
-0.003609
0.000038
17
-58.019
-15.736
0.255
875.799
62.762
-0.741
-0.036761
-0.003349
0.000034
18
-58.326
-15.865
0.259
856.361
62.054
-0.731
-0.032975
-0.003101
0.000030
19
-58.613
-15.987
0.261
838.083
61.369
-0.721
-0.029445
-0.002865
0.000027
20
-58.879
-16.101
0.264
820.887
60.707
-0.711
-0.026155
-0.002640
0.000024
21
-59.126
-16.209
0.267
804.702
60.066
-0.702
-0.023087
-0.002424
0.000021
22
-59.356
-16.310
0.269
789.459
59.446
-0.693
-0.020229
-0.002218
0.000018
23
-59.568
-16.404
0.271
775.096
58.845
-0.684
-0.017564
-0.002022
0.000015
24
-59.765
-16.492
0.273
761.554
58.262
-0.676
-0.015081
-0.001834
0.000012
25
-59.947
-16.575
0.275
748.779
57.697
-0.668
-0.012768
-0.001655
0.000009
26
-60.115
-16.652
0.277
736.720
57.148
-0.660
-0.010613
-0.001484
0.000007
27
-60.269
-16.723
0.278
725.330
56.616
-0.652
-0.008606
-0.001320
0.000005
28
-60.411
-16.790
0.280
714.565
56.099
-0.644
-0.006737
-0.001164
0.000002
29
-60.541
-16.852
0.281
704.385
55.596
-0.637
-0.004997
-0.001015
0.000000
30
-60.660
-16.909
0.282
694.751
55.107
-0.630
-0.003377
-0.000872
-0.000002
31
-60.769
-16.961
0.283
685.628
54.631
-0.623
-0.001870
-0.000736
-0.000004
32
-60.867
-17.009
0.284
676.984
54.168
-0.616
-0.000468
-0.000605
-0.000006
33
-60.957
-17.054
0.285
668.788
53.717
-0.610
0.000835
-0.000481
-0.000008
34
-61.037
-17.094
0.285
661.010
53.278
-0.603
0.002047
-0.000362
-0.000009
35
-61.109
-17.131
0.286
653.626
52.849
-0.597
0.003172
-0.000248
-0.000011
36
-61.174
-17.164
0.287
646.610
52.431
-0.591
0.004218
-0.000139
-0.000012
37
-61.231
-17.193
0.287
639.939
52.024
-0.585
0.005187
-0.000035
-0.000014
38
-61.282
-17.219
0.287
633.591
51.625
-0.579
0.006087
0.000064
-0.000015
39
-61.325
-17.243
0.288
627.548
51.237
-0.573
0.006921
0.000159
-0.000017
40
-61.363
-17.263
0.288
621.789
50.857
-0.568
0.007693
0.000250
-0.000018
41
-61.395
-17.280
0.288
616.299
50.485
-0.562
0.008407
0.000337
-0.000019
42
-61.422
-17.295
0.288
611.060
50.122
-0.557
0.009067
0.000420
-0.000020
43
-61.444
-17.307
0.288
606.057
49.766
-0.552
0.009677
0.000499
-0.000021
44
-61.461
-17.316
0.288
601.276
49.418
-0.547
0.010240
0.000575
-0.000022
45
-61.473
-17.323
0.288
596.705
49.077
-0.542
0.010758
0.000647
-0.000023
46
-61.481
-17.328
0.287
592.331
48.743
-0.537
0.011235
0.000717
-0.000024
47
-61.485
-17.330
0.287
588.142
48.416
-0.533
0.011673
0.000783
-0.000025
48
-61.486
-17.330
0.287
584.127
48.095
-0.528
0.012075
0.000846
-0.000026
49
-61.483
-17.329
0.286
580.277
47.780
-0.523
0.012443
0.000906
-0.000027
50
-61.477
-17.325
0.286
576.582
47.471
-0.519
0.012779
0.000964
-0.000027
51
-61.468
-17.319
0.286
573.034
47.168
-0.515
0.013085
0.001019
-0.000028
52
-61.455
-17.312
0.285
569.623
46.869
-0.510
0.013363
0.001072
-0.000029
53
-61.441
-17.303
0.284
566.343
46.576
-0.506
0.013615
0.001122
-0.000029
54
-61.423
-17.292
0.284
563.186
46.289
-0.502
0.013843
0.001170
-0.000030
55
-61.403
-17.280
0.283
560.146
46.005
-0.498
0.014047
0.001216
-0.000030
56
-61.381
-17.266
0.283
557.215
45.727
-0.494
0.014230
0.001259
-0.000031
57
-61.357
-17.250
0.282
554.389
45.453
-0.490
0.014393
0.001301
-0.000031
58
-61.331
-17.234
0.281
551.661
45.183
-0.487
0.014538
0.001341
-0.000032
59
-61.304
-17.216
0.280
549.027
44.917
-0.483
0.014664
0.001379
-0.000032
60
-61.274
-17.196
0.280
546.481
44.656
-0.479
0.014774
0.001415
-0.000033
61
-61.243
-17.176
0.279
544.018
44.398
-0.476
0.014869
0.001450
-0.000033
62
-61.211
-17.154
0.278
541.635
44.144
-0.472
0.014949
0.001483
-0.000033
63
-61.177
-17.131
0.277
539.328
43.893
-0.469
0.015016
0.001514
-0.000034
64
-61.142
-17.107
0.276
537.092
43.646
-0.465
0.015071
0.001544
-0.000034
65
-61.105
-17.082
0.275
534.924
43.402
-0.462
0.015113
0.001572
-0.000034
66
-61.068
-17.056
0.274
532.821
43.161
-0.459
0.015144
0.001600
-0.000034
67
-61.029
-17.029
0.273
530.779
42.923
-0.455
0.015166
0.001625
-0.000035
68
-60.990
-17.001
0.272
528.796
42.689
-0.452
0.015177
0.001650
-0.000035
69
-60.950
-16.973
0.272
526.868
42.457
-0.449
0.015180
0.001673
-0.000035
70
-60.909
-16.943
0.271
524.993
42.228
-0.446
0.015174
0.001696
-0.000035
71
-60.867
-16.913
0.269
523.169
42.001
-0.443
0.015160
0.001717
-0.000035
72
-60.824
-16.881
0.268
521.392
41.778
-0.440
0.015140
0.001737
-0.000036
73
-60.781
-16.850
0.267
519.662
41.556
-0.437
0.015112
0.001756
-0.000036
74
-60.738
-16.817
0.266
517.975
41.337
-0.434
0.015078
0.001774
-0.000036
75
-60.693
-16.784
0.265
516.331
41.121
-0.431
0.015039
0.001791
-0.000036
76
-60.649
-16.750
0.264
514.726
40.907
-0.428
0.014993
0.001807
-0.000036
77
-60.603
-16.715
0.263
513.159
40.695
-0.425
0.014943
0.001823
-0.000036
78
-60.558
-16.680
0.262
511.629
40.485
-0.422
0.014888
0.001837
-0.000036
79
-60.512
-16.644
0.261
510.133
40.277
-0.420
0.014829
0.001851
-0.000036
80
-60.466
-16.608
0.260
508.672
40.071
-0.417
0.014766
0.001864
-0.000036
81
-60.420
-16.571
0.259
507.242
39.867
-0.414
0.014699
0.001876
-0.000036
82
-60.373
-16.534
0.258
505.843
39.665
-0.412
0.014629
0.001888
-0.000036
83
-60.326
-16.496
0.256
504.473
39.465
-0.409
0.014555
0.001899
-0.000036
84
-60.279
-16.458
0.255
503.132
39.266
-0.406
0.014479
0.001909
-0.000036
85
-60.232
-16.419
0.254
501.818
39.070
-0.404
0.014400
0.001918
-0.000036
86
-60.185
-16.380
0.253
500.530
38.875
-0.401
0.014318
0.001928
-0.000036
87
-60.137
-16.341
0.252
499.267
38.681
-0.399
0.014235
0.001936
-0.000036
88
-60.090
-16.301
0.251
498.028
38.490
-0.396
0.014149
0.001944
-0.000036
89
-60.042
-16.261
0.250
496.812
38.299
-0.394
0.014061
0.001951
-0.000036
90
-59.995
-16.220
0.248
495.618
38.111
-0.391
0.013972
0.001958
-0.000036
91
-59.947
-16.179
0.247
494.446
37.923
-0.389
0.013881
0.001965
-0.000036
92
-59.899
-16.138
0.246
493.295
37.737
-0.387
0.013789
0.001971
-0.000036
93
-59.852
-16.097
0.245
492.164
37.553
-0.384
0.013696
0.001976
-0.000036
94
-59.804
-16.055
0.244
491.052
37.370
-0.382
0.013601
0.001981
-0.000036
95
-59.757
-16.013
0.243
489.959
37.188
-0.380
0.013506
0.001986
-0.000036
96
-59.710
-15.971
0.241
488.883
37.008
-0.377
0.013409
0.001990
-0.000036
97
-59.662
-15.928
0.240
487.825
36.828
-0.375
0.013312
0.001994
-0.000036
98
-59.615
-15.885
0.239
486.784
36.650
-0.373
0.013215
0.001997
-0.000036
99
-59.568
-15.842
0.238
485.759
36.474
-0.371
0.013117
0.002001
-0.000036
100
-59.521
-15.799
0.237
484.750
36.298
-0.368
0.013018
0.002003
-0.000036
101
-59.474
-15.755
0.236
483.756
36.124
-0.366
0.012919
0.002006
-0.000036
102
-59.428
-15.712
0.234
482.777
35.950
-0.364
0.012820
0.002008
-0.000035
103
-59.381
-15.668
0.233
481.812
35.778
-0.362
0.012721
0.002010
-0.000035
104
-59.335
-15.624
0.232
480.861
35.607
-0.360
0.012621
0.002012
-0.000035
105
-59.289
-15.580
0.231
479.923
35.437
-0.358
0.012522
0.002013
-0.000035
106
-59.243
-15.536
0.230
478.999
35.268
-0.355
0.012422
0.002014
-0.000035
107
-59.197
-15.491
0.229
478.087
35.100
-0.353
0.012323
0.002015
-0.000035
108
-59.151
-15.446
0.228
477.187
34.933
-0.351
0.012223
0.002015
-0.000035
109
-59.106
-15.402
0.226
476.300
34.767
-0.349
0.012124
0.002016
-0.000035
110
-59.061
-15.357
0.225
475.424
34.601
-0.347
0.012025
0.002016
-0.000035
111
-59.016
-15.312
0.224
474.559
34.437
-0.345
0.011927
0.002016
-0.000034
112
-58.971
-15.267
0.223
473.706
34.274
-0.343
0.011828
0.002016
-0.000034
113
-58.926
-15.222
0.222
472.863
34.112
-0.341
0.011730
0.002015
-0.000034
114
-58.882
-15.176
0.221
472.031
33.950
-0.339
0.011633
0.002014
-0.000034
115
-58.838
-15.131
0.219
471.209
33.790
-0.337
0.011536
0.002013
-0.000034
116
-58.794
-15.086
0.218
470.397
33.630
-0.335
0.011439
0.002012
-0.000034
117
-58.750
-15.040
0.217
469.595
33.471
-0.334
0.011343
0.002011
-0.000034
118
-58.707
-14.995
0.216
468.802
33.313
-0.332
0.011247
0.002010
-0.000034
119
-58.664
-14.949
0.215
468.019
33.156
-0.330
0.011151
0.002008
-0.000033
120
-58.621
-14.903
0.214
467.244
33.000
-0.328
0.011057
0.002007
-0.000033
121
-58.578
-14.857
0.213
466.478
32.844
-0.326
0.010963
0.002005
-0.000033
122
-58.536
-14.812
0.212
465.721
32.689
-0.324
0.010869
0.002003
-0.000033
123
-58.493
-14.766
0.211
464.973
32.535
-0.322
0.010776
0.002001
-0.000033
124
-58.451
-14.720
0.209
464.233
32.382
-0.320
0.010684
0.001998
-0.000033
125
-58.410
-14.674
0.208
463.501
32.230
-0.319
0.010592
0.001996
-0.000033
126
-58.368
-14.628
0.207
462.776
32.078
-0.317
0.010501
0.001994
-0.000032
127
-58.327
-14.582
0.206
462.060
31.927
-0.315
0.010410
0.001991
-0.000032
128
-58.286
-14.536
0.205
461.351
31.777
-0.313
0.010320
0.001988
-0.000032
129
-58.245
-14.490
0.204
460.650
31.627
-0.311
0.010231
0.001985
-0.000032
130
-58.205
-14.444
0.203
459.955
31.478
-0.310
0.010143
0.001982
-0.000032
131
-58.165
-14.398
0.202
459.268
31.330
-0.308
0.010055
0.001979
-0.000032
132
-58.125
-14.352
0.201
458.589
31.183
-0.306
0.009968
0.001976
-0.000032
133
-58.085
-14.306
0.200
457.915
31.036
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0.000667
-0.000006
428
-53.040
-4.199
0.039
380.038
7.007
-0.060
0.001168
0.000664
-0.000006
429
-53.035
-4.178
0.039
379.951
6.970
-0.059
0.001161
0.000660
-0.000006
430
-53.029
-4.158
0.039
379.865
6.933
-0.059
0.001154
0.000657
-0.000006
431
-53.023
-4.138
0.039
379.780
6.897
-0.059
0.001147
0.000654
-0.000006
432
-53.017
-4.118
0.039
379.695
6.860
-0.058
0.001140
0.000651
-0.000006
433
-53.012
-4.098
0.038
379.611
6.824
-0.058
0.001133
0.000648
-0.000006
434
-53.006
-4.078
0.038
379.527
6.788
-0.058
0.001126
0.000645
-0.000006
435
-53.001
-4.059
0.038
379.443
6.752
-0.057
0.001120
0.000642
-0.000006
436
-52.995
-4.039
0.038
379.360
6.716
-0.057
0.001113
0.000639
-0.000006
437
-52.989
-4.020
0.038
379.278
6.680
-0.057
0.001106
0.000636
-0.000006
438
-52.984
-4.000
0.037
379.196
6.645
-0.056
0.001099
0.000633
-0.000006
439
-52.978
-3.981
0.037
379.115
6.609
-0.056
0.001093
0.000630
-0.000006
440
-52.973
-3.961
0.037
379.034
6.574
-0.056
0.001086
0.000627
-0.000006
441
-52.968
-3.942
0.037
378.953
6.540
-0.055
0.001079
0.000624
-0.000006
442
-52.962
-3.923
0.036
378.873
6.505
-0.055
0.001073
0.000621
-0.000006
443
-52.957
-3.904
0.036
378.794
6.470
-0.055
0.001066
0.000618
-0.000006
444
-52.952
-3.885
0.036
378.715
6.436
-0.055
0.001060
0.000615
-0.000006
445
-52.946
-3.866
0.036
378.636
6.402
-0.054
0.001054
0.000612
-0.000006
446
-52.941
-3.847
0.036
378.558
6.368
-0.054
0.001047
0.000610
-0.000006
447
-52.936
-3.828
0.035
378.480
6.334
-0.054
0.001041
0.000607
-0.000006
448
-52.931
-3.810
0.035
378.403
6.300
-0.053
0.001035
0.000604
-0.000006
449
-52.925
-3.791
0.035
378.326
6.267
-0.053
0.001028
0.000601
-0.000006
450
-52.920
-3.773
0.035
378.250
6.233
-0.053
0.001022
0.000598
-0.000006
451
-52.915
-3.754
0.035
378.174
6.200
-0.052
0.001016
0.000595
-0.000006
452
-52.910
-3.736
0.034
378.098
6.167
-0.052
0.001010
0.000592
-0.000006
453
-52.905
-3.718
0.034
378.023
6.135
-0.052
0.001004
0.000590
-0.000006
454
-52.900
-3.699
0.034
377.949
6.102
-0.052
0.000998
0.000587
-0.000006
455
-52.895
-3.681
0.034
377.875
6.069
-0.051
0.000992
0.000584
-0.000006
456
-52.890
-3.663
0.034
377.801
6.037
-0.051
0.000986
0.000581
-0.000006
457
-52.885
-3.645
0.034
377.728
6.005
-0.051
0.000980
0.000578
-0.000005
458
-52.880
-3.627
0.033
377.655
5.973
-0.050
0.000974
0.000576
-0.000005
459
-52.875
-3.610
0.033
377.582
5.941
-0.050
0.000968
0.000573
-0.000005
460
-52.871
-3.592
0.033
377.510
5.910
-0.050
0.000963
0.000570
-0.000005
461
-52.866
-3.574
0.033
377.439
5.878
-0.050
0.000957
0.000567
-0.000005
462
-52.861
-3.557
0.033
377.368
5.847
-0.049
0.000951
0.000565
-0.000005
463
-52.856
-3.539
0.032
377.297
5.816
-0.049
0.000945
0.000562
-0.000005
464
-52.851
-3.522
0.032
377.227
5.785
-0.049
0.000940
0.000559
-0.000005
465
-52.847
-3.504
0.032
377.157
5.754
-0.048
0.000934
0.000557
-0.000005
466
-52.842
-3.487
0.032
377.087
5.723
-0.048
0.000929
0.000554
-0.000005
467
-52.837
-3.470
0.032
377.018
5.693
-0.048
0.000923
0.000551
-0.000005
468
-52.833
-3.453
0.032
376.949
5.662
-0.048
0.000918
0.000549
-0.000005
469
-52.828
-3.436
0.031
376.881
5.632
-0.047
0.000912
0.000546
-0.000005
470
-52.824
-3.419
0.031
376.813
5.602
-0.047
0.000907
0.000543
-0.000005
471
-52.819
-3.402
0.031
376.746
5.572
-0.047
0.000901
0.000541
-0.000005
472
-52.815
-3.385
0.031
376.678
5.542
-0.047
0.000896
0.000538
-0.000005
473
-52.810
-3.368
0.031
376.612
5.513
-0.046
0.000891
0.000536
-0.000005
474
-52.806
-3.352
0.031
376.545
5.483
-0.046
0.000885
0.000533
-0.000005
475
-52.801
-3.335
0.030
376.479
5.454
-0.046
0.000880
0.000530
-0.000005
476
-52.797
-3.319
0.030
376.414
5.425
-0.046
0.000875
0.000528
-0.000005
477
-52.792
-3.302
0.030
376.349
5.396
-0.045
0.000870
0.000525
-0.000005
478
-52.788
-3.286
0.030
376.284
5.367
-0.045
0.000865
0.000523
-0.000005
479
-52.784
-3.270
0.030
376.219
5.339
-0.045
0.000859
0.000520
-0.000005
480
-52.779
-3.253
0.030
376.155
5.310
-0.045
0.000854
0.000518
-0.000005
481
-52.775
-3.237
0.029
376.092
5.282
-0.044
0.000849
0.000515
-0.000005
482
-52.771
-3.221
0.029
376.028
5.254
-0.044
0.000844
0.000513
-0.000005
483
-52.767
-3.205
0.029
375.965
5.225
-0.044
0.000839
0.000510
-0.000005
484
-52.762
-3.189
0.029
375.903
5.198
-0.044
0.000834
0.000508
-0.000005
485
-52.758
-3.173
0.029
375.840
5.170
-0.043
0.000829
0.000505
-0.000005
486
-52.754
-3.157
0.029
375.779
5.142
-0.043
0.000825
0.000503
-0.000005
487
-52.750
-3.142
0.028
375.717
5.115
-0.043
0.000820
0.000500
-0.000005
488
-52.746
-3.126
0.028
375.656
5.087
-0.043
0.000815
0.000498
-0.000005
489
-52.742
-3.110
0.028
375.595
5.060
-0.042
0.000810
0.000495
-0.000005
490
-52.738
-3.095
0.028
375.535
5.033
-0.042
0.000805
0.000493
-0.000005
491
-52.733
-3.080
0.028
375.474
5.006
-0.042
0.000800
0.000491
-0.000005
492
-52.729
-3.064
0.028
375.415
4.979
-0.042
0.000796
0.000488
-0.000005
493
-52.725
-3.049
0.027
375.355
4.953
-0.041
0.000791
0.000486
-0.000004
494
-52.721
-3.034
0.027
375.296
4.926
-0.041
0.000786
0.000483
-0.000004
495
-52.717
-3.018
0.027
375.237
4.900
-0.041
0.000782
0.000481
-0.000004
496
-52.714
-3.003
0.027
375.179
4.874
-0.041
0.000777
0.000479
-0.000004
497
-52.710
-2.988
0.027
375.121
4.847
-0.041
0.000773
0.000476
-0.000004
498
-52.706
-2.973
0.027
375.063
4.821
-0.040
0.000768
0.000474
-0.000004
499
-52.702
-2.958
0.027
375.006
4.796
-0.040
0.000764
0.000472
-0.000004

Figure 4:  Linear vs Nonlinear Impulse Responses: Path 1

Data for Figure 4 immediately follows.

Data for Figure 4

Quarters
Output Home:
Path 1
(nonlinear)
Output Home:
Path 1
(linear)
Consumption Home:
Path 1
(nonlinear)
Consumption Home:
Path 1
(linear)
Investment Home:
Path 1
(nonlinear)
Investment Home:
Path 1
(linear)
Real Exchange Rate Home:
Path 1
(nonlinear)
Real Exchange Rate Home:
Path 1
(linear)
Real Exchange Rate Home:
Path 1
(nonlinear)
Real Exchange Rate Home:
Path 1
(linear)
Trade Balance to GDP Ratio Home:
Path 1
(nonlinear)
Trade Balance to GDP Ratio Home:
Path 1
(linear)
1
0.0000
0.0000
0.0497
0.0489
-0.1003
-0.1025
0.1719
0.1692
-0.9719
-0.9585
0.00085272
0.00084798
2
-0.0016
-0.0016
0.0514
0.0506
0.1926
0.1904
0.1799
0.1772
-0.9543
-0.9412
0.00011781
0.00011357
3
-0.0004
-0.0004
0.0530
0.0522
0.1881
0.1859
0.1875
0.1847
-0.9374
-0.9245
0.00011051
0.00010644
4
0.0007
0.0007
0.0545
0.0537
0.1837
0.1816
0.1947
0.1918
-0.9210
-0.9083
0.00010353
0.00009964
5
0.0018
0.0018
0.0560
0.0552
0.1795
0.1774
0.2016
0.1986
-0.9052
-0.8927
0.00009686
0.00009314
6
0.0028
0.0028
0.0573
0.0565
0.1755
0.1734
0.2080
0.2050
-0.8899
-0.8776
0.00009048
0.00008691
7
0.0038
0.0037
0.0586
0.0578
0.1716
0.1696
0.2141
0.2110
-0.8752
-0.8630
0.00008438
0.00008095
8
0.0047
0.0047
0.0598
0.0589
0.1679
0.1659
0.2199
0.2167
-0.8609
-0.8489
0.00007855
0.00007526
9
0.0056
0.0055
0.0609
0.0601
0.1643
0.1623
0.2254
0.2221
-0.8471
-0.8353
0.00007297
0.00006983
10
0.0064
0.0064
0.0620
0.0611
0.1608
0.1589
0.2305
0.2272
-0.8338
-0.8221
0.00006765
0.00006461
11
0.0072
0.0071
0.0630
0.0621
0.1575
0.1556
0.2354
0.2320
-0.8209
-0.8094
0.00006255
0.00005964
12
0.0080
0.0079
0.0639
0.0630
0.1542
0.1524
0.2399
0.2366
-0.8085
-0.7970
0.00005769
0.00005492
13
0.0087
0.0086
0.0648
0.0639
0.1511
0.1493
0.2442
0.2408
-0.7964
-0.7851
0.00005304
0.00005035
14
0.0094
0.0093
0.0656
0.0647
0.1482
0.1464
0.2483
0.2448
-0.7847
-0.7735
0.00004860
0.00004599
15
0.0100
0.0099
0.0664
0.0654
0.1453
0.1435
0.2521
0.2486
-0.7734
-0.7624
0.00004435
0.00004186
16
0.0106
0.0105
0.0671
0.0661
0.1425
0.1408
0.2556
0.2521
-0.7625
-0.7515
0.00004030
0.00003789
17
0.0112
0.0111
0.0677
0.0668
0.1398
0.1381
0.2590
0.2554
-0.7519
-0.7410
0.00003642
0.00003411
18
0.0118
0.0116
0.0683
0.0674
0.1372
0.1356
0.2621
0.2585
-0.7416
-0.7309
0.00003273
0.00003050
19
0.0123
0.0121
0.0689
0.0679
0.1347
0.1331
0.2650
0.2614
-0.7316
-0.7210
0.00002919
0.00002705
20
0.0128
0.0126
0.0694
0.0684
0.1323
0.1307
0.2677
0.2641
-0.7220
-0.7114
0.00002582
0.00002376
21
0.0132
0.0131
0.0699
0.0689
0.1300
0.1284
0.2703
0.2666
-0.7126
-0.7022
0.00002260
0.00002061
22
0.0137
0.0135
0.0703
0.0694
0.1278
0.1262
0.2726
0.2689
-0.7035
-0.6932
0.00001953
0.00001761
23
0.0141
0.0139
0.0707
0.0697
0.1256
0.1240
0.2748
0.2711
-0.6947
-0.6844
0.00001660
0.00001474
24
0.0145
0.0143
0.0711
0.0701
0.1235
0.1220
0.2768
0.2731
-0.6861
-0.6759
0.00001380
0.00001203
25
0.0148
0.0147
0.0714
0.0704
0.1215
0.1200
0.2787
0.2749
-0.6778
-0.6677
0.00001113
0.00000938
26
0.0152
0.0150
0.0717
0.0707
0.1196
0.1180
0.2804
0.2766
-0.6697
-0.6597
0.00000858
0.00000690
27
0.0155
0.0153
0.0720
0.0710
0.1177
0.1162
0.2820
0.2781
-0.6619
-0.6519
0.00000615
0.00000454
28
0.0158
0.0156
0.0723
0.0712
0.1158
0.1144
0.2834
0.2795
-0.6542
-0.6444
0.00000383
0.00000227
29
0.0161
0.0159
0.0725
0.0714
0.1141
0.1126
0.2847
0.2808
-0.6468
-0.6370
0.00000163
0.00000011
30
0.0164
0.0162
0.0726
0.0716
0.1124
0.1109
0.2859
0.2820
-0.6396
-0.6299
-0.00000048
-0.00000192
31
0.0166
0.0164
0.0728
0.0718
0.1107
0.1093
0.2869
0.2830
-0.6326
-0.6229
-0.00000249
-0.00000390
32
0.0169
0.0167
0.0729
0.0719
0.1091
0.1077
0.2878
0.2839
-0.6257
-0.6162
-0.00000440
-0.00000577
33
0.0171
0.0169
0.0730
0.0720
0.1076
0.1062
0.2887
0.2847
-0.6191
-0.6096
-0.00000622
-0.00000753
34
0.0173
0.0171
0.0731
0.0721
0.1061
0.1047
0.2894
0.2854
-0.6126
-0.6031
-0.00000795
-0.00000922
35
0.0175
0.0173
0.0732
0.0721
0.1046
0.1032
0.2900
0.2860
-0.6063
-0.5969
-0.00000960
-0.00001084
36
0.0177
0.0175
0.0732
0.0722
0.1032
0.1018
0.2905
0.2866
-0.6001
-0.5908
-0.00001117
-0.00001234
37
0.0178
0.0176
0.0733
0.0722
0.1019
0.1005
0.2910
0.2870
-0.5941
-0.5848
-0.00001267
-0.00001383
38
0.0180
0.0178
0.0733
0.0722
0.1005
0.0992
0.2913
0.2873
-0.5883
-0.5790
-0.00001409
-0.00001520
39
0.0181
0.0179
0.0733
0.0722
0.0993
0.0979
0.2916
0.2876
-0.5826
-0.5734
-0.00001544
-0.00001653
40
0.0183
0.0180
0.0732
0.0722
0.0980
0.0967
0.2918
0.2878
-0.5770
-0.5679
-0.00001673
-0.00001776
41
0.0184
0.0182
0.0732
0.0721
0.0968
0.0955
0.2919
0.2879
-0.5715
-0.5625
-0.00001795
-0.00001897
42
0.0185
0.0183
0.0731
0.0721
0.0956
0.0943
0.2920
0.2879
-0.5662
-0.5572
-0.00001911
-0.00002009
43
0.0186
0.0184
0.0730
0.0720
0.0945
0.0932
0.2919
0.2879
-0.5610
-0.5520
-0.00002021
-0.00002117
44
0.0187
0.0184
0.0730
0.0719
0.0934
0.0921
0.2918
0.2878
-0.5560
-0.5470
-0.00002126
-0.00002217
45
0.0188
0.0185
0.0729
0.0718
0.0923
0.0910
0.2917
0.2876
-0.5510
-0.5421
-0.00002225
-0.00002311
46
0.0188
0.0186
0.0727
0.0717
0.0913
0.0900
0.2915
0.2874
-0.5461
-0.5373
-0.00002319
-0.00002404
47
0.0189
0.0187
0.0726
0.0715
0.0903
0.0890
0.2912
0.2871
-0.5414
-0.5326
-0.00002408
-0.00002492
48
0.0190
0.0187
0.0725
0.0714
0.0893
0.0880
0.2909
0.2868
-0.5367
-0.5280
-0.00002492
-0.00002573
49
0.0190
0.0188
0.0723
0.0713
0.0883
0.0871
0.2905
0.2865
-0.5322
-0.5234
-0.00002572
-0.00002652
50
0.0191
0.0188
0.0722
0.0711
0.0874
0.0861
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