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This paper examines the dynamic implications of different preference formulations in open economy business cycle models with incomplete asset markets. In particular, we study two preference formulations: a time separable preference formulation with a fixed discount factor, and a time non-separable preference structure with an endogenous discount factor. We analyze the moment implications of two versions of an otherwise identical open economy modelone with a fixed discount factor and the other with an endogenous discount factorand study impulse responses to productivity and world real interest rate shocks. Our results suggest that business cycle implications of the two models are quite similar under conventional parameter values. We also find the approximation errors associated with the solutions of these two models are of the same magnitude.

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formulations in open economy business cycle models with incomplete asset markets. In

particular, we study two preference formulations: a time separable preference formulation

with a fixed discount factor, and a time non-separable preference structure with an

endogenous discount factor. We analyze the moment implications of two versions of an

otherwise identical open economy modelone with a fixed discount factor and the other

with an endogenous discount factorand study impulse responses to productivity and world

real interest rate shocks. Our results suggest that business cycle implications of the two

models are quite similar under conventional parameter values. We also find the

approximation errors associated with the solutions of these two models are of the same

magnitude.

* We would like to thank William A. Barnett, an anonymous Associate Editor, and two anonymous referees for

providing very detailed and helpful comments which significantly improved the paper. We are grateful to Marianne

Baxter, William Blankenau, Mario Crucini, Narayana Kocherlakota, Robert Kollmann, Jim Nason, Fabrizio Perri, and

Kei-Mu Yi for their helpful suggestions. An earlier version of this paper was presented at Tufts University, 1999

Computing in Economics and Finance Conference at Boston College, 2000 Midwest Macroeconomics Meetings at

University of Iowa. We would like to thank seminar and conference participants for their feedback. The usual

disclaimer applies.

** Graduate School of International Economics and Finance, Brandeis University, Waltham MA, 02454. E-mail (Kim):

hkim@brandeis.edu, e-mail (Kose): akose@brandeis.edu, phone (Kim): (781) 736-2268, phone (Kose): (781) 736-

2266.

There have been two popular preference formulations employed in infinite-horizon open economy

Real Business Cycle (RBC) models under incomplete asset markets: time separable preferences with a fixed

discount factor, and time non-separable preferences with an endogenous discount factor.1 The former

formulation is the “standard” one that is widely used in closed and open economy RBC models, under

complete and incomplete asset markets. However, it is by now well known that with this formulation, when

the models are solved using the usual linear approximation methods, it is not possible to generate stationary

state variables and a well-defined wealth distribution in an open economy setting under incomplete markets.2

This problem has led several researchers to employ the latter formulation, which generates well-defined

steady state dynamics in these models.

How can we justify the fact that a number of researchers have employed the fixed discount factor

formulation despite the well-known problems associated with steady state dynamics in open economy RBC

models under incomplete asset markets? Three potential reasons emerge for the widespread use of this

formulation. First, while the issue of the existence of a well-defined long-run wealth distribution might be

important for some economic experiments, there might be no reason to believe that these two preference

formulations generate significantly different model dynamics along the dimensions examined by typical

RBC studies. To be more specific, it might be the case that the models with these two preference

formulations produce very similar moments and impulse responses, which are the main interests of these

studies.

Second, considering that most models in this literature are solved using linear approximation

methods, potential additional accuracy gains from employing an endogenous discount factor might be

insignificant. In other words, while the true solution of the fixed discount factor model, which we call FDM

(exact), and the true solution of the endogenous discount factor model, which we call EDM (exact), might

exhibit some differences, it is not clear whether the approximate solutions of these two models, which we

call FDM (approximate) and EDM (approximate), are different under incomplete markets.

Third, some features of the endogenous discount factor formulation might not be desirable in a

representative agent business cycle model. For example, the endogenous discount factor formulation implies

that agents become more impatient as they become wealthier, and there is a steady state utility level that

relies on a predetermined saving target (see Senhadji (1995) and Daniel (1997)). Another undesirable

1 Baxter and Crucini (1995), Baxter (1995), Rebelo and Vegh (1995), Correia, Neves and Rebelo (1992, 1995), Kollman

(1996, 1998, 2000), Crucini (1999), Arvanitis and Mikkola (1996), van Wincoop (1996), van Wincoop and Marrinan

(1999), Kouparitsas (1997), Sadka and Yi (1996), Blankenau, Kose, and Yi (2001), and Kose and Riezman (2001)

employ the standard time separable preferences with a fixed discount factor in their models. Mendoza (1991, 1995),

Karayalcin (1995), Uribe (1997), Schmitt-Grohe (1998), and Cook and Devereux (2000) use the Uzawa-Epstein type

time nonseparable preferences with an endogenous discount factor.

2 As we discuss in detail in section 2, there are some methods to deal with these problems in open economy models

under incomplete markets.

1

property of the EDM is that it is not possible to examine the state dependent steady state level of net foreign

assets because the EDM forces consumption to behave in such a way that the net foreign asset position goes

back to its initial steady state. Moreover, since a number of macroeconomic time series are nonstationary, it

might not be advantageous to have a model generating stationary variables (see Correia, et al. 1995).3

Despite the wide use of these two preference formulations, there has yet been no rigorous

examination of their impact on the cyclical dynamics of open economy RBC models. The objective of this

paper is to provide a comprehensive comparison of the behavior of an open economy RBC model with

incomplete asset markets under the fixed discount factor with that under the endogenous discount factor. In

particular, we consider two versions of an otherwise identical small open economy RBC model, one with a

fixed discount factor (FDM), and the other with an endogenous discount factor (EDM), and study the

moment implications and impulse responses of these models. We provide empirical support for the three

reasons above and argue for the similarity between FDM (exact) and EDM (exact) based on the similarity

between FDM (approximate) and EDM (approximate), which we systematically study in this paper. 4

Section 2 starts with a brief discussion of the problem associated with the steady state dynamics in

open economy RBC models with incomplete asset markets. Then, we present the two models and their

calibration and parameterization. In section 3, we provide empirical support for the three justifications

described above. We analyze the policy functions generated by the two models in section 3.1. Our

discussion in sections 2 and 3.1 makes transparent the arguments regarding the deterministic and stochastic

steady state dynamics of the two models. We find that while the net foreign asset series in the FDM follows

a unit root process, this series exhibits a near unit root behavior in the EDM. More importantly, the

coefficients in the policy functions generated by the two models display minimal differences under

conventional parameter values. In section 3.2, we examine the second moment implications pertaining to

business cycles and find that the two models generate almost identical business cycle moments. Section 3.3

presents the impulse responses to productivity and the world interest rate shocks. While some small

quantitative differences exist between the impulse responses produced by the two models, qualitative

implications are identical. In section 3.4, we compare the distributions of variables produced by the two

models using the Kolmogorov-Smirnov test statistic. The test results suggest that these empirical

distributions are not statistically different.

3 Daniel (1997) notes that it is “intuitively unappealing” to have wealthier agents who are more impatient than less

wealthy agents in this utility formulation. Senhadji (1995) criticizes the use of endogenous discount factor since the

representative agent must save enough to reach a saving target in order to attain a fixed utility level determined by the

steady state. Correia, et al. (1995), after noting that consumption, the net exports, and the net foreign assets are

nonstationary in the data, claim that the stationarity of these variables induced by the non-separable preference

formulation is not “necessarily a desirable property.” (p. 1100)

4 To be more specific, our ultimate objective is to argue for the similarity between the exact solutions of these two

models based on the similarity between their approximate solutions. Our findings suggest that the approximation errors

are quite small implying that the results from these approximate solutions can potentially apply to the exact solutions of

these models.

2

with the solutions of these models. We find that while the solution of the FDM results in slightly larger

approximation errors than that of the EDM, the errors are quite small. We provide a brief conclusion in

section 4.

Before getting into the details of the model economies, we provide a brief discussion of the problem

associated with the steady state. Consider the standard time separable preference formulation in an infinite-

horizon open economy model with incomplete asset markets. In a deterministic setting, the steady state or

the long-run wealth depends on the initial conditions of the economy and the steady state is compatible with

any level of net foreign assets.5 In a stochastic environment, since the net foreign asset series follows a unit

root process, the model generates nonstationary variables implying the absence of a well-defined stochastic

steady state.6 In other words, certain model variables do not return to their initial steady state values when

the model is subjected to a temporary shock. Hence, the long-run wealth of the economy changes with the

state of nature, i.e. the long-run wealth distribution is not well defined.

Researchers have developed several methods that can resolve the steady state issues associated with

the fixed discount factor formulation.7 As mentioned above, one popular alternative is to employ the Uzawa-

Epstein type time non-separable preferences with an endogenous discount factor. Unlike the fixed discount

factor model, the endogenous discount factor model generates stationary net foreign asset series and, in turn,

other state variables become stationary. Moreover, the endogenous discount factor model generates a well-

defined stochastic steady state at which, under certain conditions, a unique level of net foreign assets is

attained implying that the long-run wealth distribution of the economy is also well defined.

5 We consider the case when the time preference is equal to the world interest rate. Otherwise, no deterministic steady

state exists. If the world interest rate is greater (smaller) than the time preference rate, agents permanently accumulate

(deplete) foreign assets. These issues are first discussed in Helpman and Razin (1982). Mendoza and Tesar (1998)

provide a detailed discussion of the steady state issues in a deterministic model with incomplete asset markets.

features a unit root in asset holdings does not necessarily imply that the exact solution of the model also has a unit root.

The stochastic steady state of the model might be well defined, but it is not possible to capture this steady state by the

approximate solution method. It is not possible to provide an exact solution of the incomplete market models with the

fixed discount factor formulation unless the state space of asset holdings is bounded.

7 Heathcote and Perri (2000) impose a quadratic adjustment cost on bond holdings to produce stationary asset series.

Cardia (1991) uses the uncertain lifetime approach advanced by Blanchard (1985) in her small open economy model.

Harjes (1997) assumes that the world real interest rate depends on the net foreign assets, and Bruno and Portier (1995)

assume that net foreign assets negatively affect the households’ utility in their small open economy models. Senhadji

(1998) considers a set-up with a downward sloping export demand function. It is also possible to produce stationary

equilibria in models with incomplete asset markets by introducing limits on the level of asset holdings (explicit bounds

as in Huggett (1993) or implicit bounds as in Levine and Zame (1996, 1999) and Kubler and Schmedders (2000)), or by

introducing endogenous solvency constraints (see Alvarez and Jermann (1999)), or enforcement constraints (see Kehoe

and Perri (2000)).

3

To evaluate the impact of the discount factor on the dynamics of business cycles, we study a small

open economy RBC model in which agents produce an internationally tradable good using labor and capital.8

There are two shocks in the model: a productivity shock and a world real interest rate shock. Agents have

access to world financial markets in which they can buy and sell one-period risk free bonds at a stochastic

world real interest rate. We solve the model using a linear approximation method and choose the parameter

values that provide both versions of the model have the same steady state. We calibrate the model to

Canada, an economy that has been extensively studied in the small open economy RBC literature. In section

2.1, we present the model with a fixed discount factor. In section 2.2, we analyze the same model with an

endogenous discount factor. Section 2.3 explains the derivation of the steady state and calibration of the

model.

The optimization problem of the representative agent is the following:

1 σ

−

−

∞

1

max

=

0

β

, σ > 1,

,

,

,

∑

1

−σ

(1)

.

1

(

+

),

1

−

−

α

−

(2)

1

α

(3)

( − δ )

in the momentary utility function

substitution in labor supply, β is the fixed discount factor, and σ is the household’s coefficient of relative risk

aversion. Our momentary utility formulation implies that the elasticity of substitution associated with leisure

is zero.9 In resource constraint (1),

the stochastic world real interest rate from period

stock at the beginning of the period

8 See Baxter (1995) and Backus, Kehoe, and Kydland (1995) for a survey of open economy RBC models and their use

in studying the sources and transmission of international business cycles.

9 This utility function is introduced by Greenwood, Hercowitz, and Huffman (1988) and is widely used in the open

economy RBC models. Correia, et al. (1995) and Crucini (1999) compare the dynamic implications of this utility

function with those of the Cobb-Douglas utility function, both with a fixed discount factor. We do not adopt the Cobb-

Douglas preference structure because as shown by Correia, et al. (1995) a small open economy model with the Cobb-

Douglas utility function is unable to match the volatility of consumption and the countercyclical behavior of net exports

in real data.

4

labor in constraint (2) that describes the production function.10 In constraint (3), δ denotes depreciation rate

and φ (.) represents the standard adjustment cost function with φ (.) > 0 , φ )

(. ′ > 0 , and φ )

(. ′ < 0 (see

Baxter and Crucini (1993)).

Substituting (2) into (1), the first order conditions of this optimization problem are given as

−σ

(4)

= λ ,

(5)

α ,

(6)

′

),

(7)

: µ

β λ

α

µ

{

1

(

− )

(

+ )},

+

1

+

+

+

+

where

+ ) = 1

( − δ ) + φ (

+ ) − φ (

′

)

+ ,

+

+

+

+

(8)

1

(

+

+

where λ

Since this problem cannot be solved analytically, we find an approximate solution using the

approximation method of King, Plosser and Rebelo (1988).11

This part describes the same model with a time non-separable preference formulation in which the

discount factor is endogenous.12 The optimization problem of the representative agent is

1 σ

−

− 1

∞

max

, σ > 1,

=

∑

1 − σ

10 We do not consider other shocks such as government spending shocks, or net foreign transfers shocks since Correia,

et al. (1995) convincingly argue that these shocks are not able to generate significant business cycle dynamics in a small

open economy model.

11 Detailed derivation of the steady state and the linearized first order conditions of the model are available upon

request.

12 This preference formulation was first introduced by Uzawa (1968), and further developed by Epstein (1983, 1987).

See Obstfeld (1990) for a theoretical analysis of dynamics of this type of small open economy model.

5

subject to the same constraints (1) - (3) above. The endogenous discount factor, γ , is defined as

−

(9)

γ

θ

θ

1

ln( + τ −

) ,

∑

τ =

0

The discount factor depends on the level of consumption and labor input in the previous periods. θ

denotes the elasticity of the discount factor with respect to utility. The functional form of the endogenous

discount factor implies that an increase (decrease) in current consumption (labor input) decreases the weights

assigned to all future utility, and, in turn, the agent becomes more impatient.13

Using (9), we define an auxiliary variable ψ , the time

(

)1 σ

−

−1

∞

γ

(10)

ψ =

.

∑

= +

− σ

γ

1

The first order conditions of the optimization problem above are:

−σ

1

−

(11)

λ ψ θ

=

1

+

,

(12)

(13)

′

),

−

θ

(14)

: µ

α

µ

1 +

{

1

(

− )

(

+ )},

+

1

+

+

+

+

−θ

(15)

(

+

)

1

(

+

+

Using equation (10), we write the following law of motion for the auxiliary variable ψ :

13 Epstein (1983) shows that under certain conditions this preference formulation generates “a unique invariant limiting

distribution of the state variables.” Our parameterization also meets those conditions.

6

−

1 σ

(

)

−1

θ

(16)

.

(

)

1

(

)

ψ

+

= + −

1 − σ

We solve the model using the same approximation method.

We choose the parameter values to ensure that we have the same steady state for both models. The

variables without time subscripts refer to the steady state values of the corresponding variables. The three

important steady state parameters to be discussed here are

the same steady state by using the same values for

θ in the EDM.

In the FDM, given

Therefore, the number of endogenous variables is less than the number of steady state equations by one.

This implies that any level of foreign assets is compatible with the initial steady state of the model.

Following the standard approach in the literature (see for example, Baxter and Crucini (1993), and Correia, et

al. (1995)), we draw the value of

foreign assets/output) is uniquely determined by

Unlike the FDM, the EDM does not suffer from the indeterminacy problem because the number of

endogenous variables is equal to the number of steady state equations: i.e. the model generates a unique

steady state level of the net foreign assets. However, this hinges on the assumption that the value of θ is

known in advance. Using the value of

endogenously using the steady state version of equation (15). This guarantees that all the variables including

the discount rates in the two models have the same steady state values. This is also the standard calibration

method used in the studies employing the endogenous discounting factor.

We calibrate the structural parameters to correspond to the existing RBC literature and to be consistent

with the long-run features of Canadian economy. Table 1 presents the calibrated values of parameters. We

set the quarterly steady state world real interest rate at 1.21 % which is the average rate calculated using the

U.S. 3-month T-Bill rate deflated with the CPI inflation. Following Mendoza (1991), the elasticity of

substitution,

between the commonly used values of 2 and 1 (see Schmitt-Grohe (1998)). Following Mendoza (1991) and

Schmitt-Grohe (1998), the share of labor income in the production, α, is set to 0.68. The quarterly

depreciation rate, δ, is set to 0.025, a widely used value in the RBC literature.

7

Our benchmark value of

with different values of

-0.05 to 0.05 per quarter.15

0.0074 to 0.0088.16 The corresponding value of θ is 0.008 when

The adjustment cost parameters are chosen so that the steady state of the model is same as the one

without adjustment costs. This implies that φ

( /

is equal to the depreciation rate, δ. The elasticity of the marginal adjustment cost function,

1

η = −(φ′/φ )(

′

, is set to 10, to match the volatility of investment in the data (see Baxter and Crucini

(1993)).

The exogenous shocks,

(17)

1 + ε

(18)

1 + ε

where ε

2

z and σr . We set the

standard deviations of the productivity and interest rate shocks at 0.615 % and 0.1 % to match the volatility

of output in the data. The persistence parameters of shocks, ρ

0.95 and 0.7.

We study the dynamic implications of the FDM and the EDM on five dimensions: first, we compare

the policy functions generated by the two models. Second, we study their second moment implications.

Third, we analyze the impulse responses to productivity and interest rate shocks. Fourth, we compare the

distributions of the three variables, which are nonstationary in the FDM, in the two models. We also

formally test whether the distributions produced by these two models are statistically different. Finally, we

estimate the approximation errors generated by the two models considering that the FDM generates

nonstationary variables that might induce larger approximation errors than those in the EDM.

Table 2 presents the coefficients of policy functions for consumption, asset holdings and the net

exports of the two models. We concentrate on the coefficients of the endogenous state variables considering

that impulse responses in the next section illustrate the differences in the coefficients associated with shocks.

14 The average value of

15 This range is wide enough to cover most realistic cases. The observed trade balance rarely exceeds ±5% of output per

quarter for the OECD countries.

8

Asset holdings follow a unit root process in the FDM, i.e. the coefficient of asset holdings,

The unit root property implies that a shock in the current period has a permanent impact and the long-run

wealth of the economy depends on the shock realizations.

to be nonstationary. While the coefficients of the policy function for consumption (

larger in the EDM than those in the FDM, differences between the coefficients are very small. In other

words, consumption is more responsive to the state variables in the EDM compared to the FDM.

As the elasticity of the endogenous discount factor θ decreases, differences in the coefficients of the

policy functions disappear. As θ

utility, therefore the variables in the EDM behave as if the discount factor is almost fixed. For example,

when θ

identical and the net foreign asset series in the EDM follows a near unit root process (

EDM). In other words, our findings suggest that the moments produced by the EDM converge to those

produced by the FDM when the elasticity of the endogenous discount factor, θ

The policy functions of output, labor hours, investment, and capital stock generated by the two

models are identical and invariant to the value of θ. These variables do not depend on the asset holdings, i.e.

the coefficients associated with asset holdings are zero in both models. Therefore, even in the FDM, these

four variables, unlike consumption and the net exports that depend on the previous period's net foreign asset

holdings, become stationary. This results from the momentary preference structure where labor hours

depend only on the current output and there is no intertemporal substitution involving labor. This implies

that labor hours become stationary, which induces output, investment, and capital stock to be stationary as

well. We also examine the sensitivity of policy functions’ coefficients to changes in other parameters of the

model such as σ and

One of the important objectives of the RBC research program is to construct models that are able to

replicate certain moments of the data. In this section, we compare the business cycle moments generated by

each model. If the discrepancies between the moments generated by the two models are small, then this

suggests that the FDM constitutes a reasonable alternative to the EDM for business cycle analysis. We

simulate the model for 100 periods with our benchmark parameterization and report the average moments

over 300 simulations. All results refer to the moments of Hodrick-Prescott (HP(1600)) filtered variables (see

Hodrick and Prescott (1997)).

Panel A in table 3 reports the second moments generated by productivity shocks. The results suggest

that there is no statistically significant difference between the moments produced by the two models. As

expected, investment is the most volatile variable, and output is more volatile than consumption. All model

16 The value of θ used in the literature with the EDM ranges from 0.001 to 0.1 depending on the model specification.

9

- Dynamics of Open Economy Business Cycle Models:
- Understanding the Role of the Discount FactorŽ *
- Sunghyun Henry Kim** and M. Ayhan Kose**
- Keywords: business cycle dynamics, fixed discount factor, endogenous discount factor, nonstationarity.
- JEL classification: F4, E3

- 2.1. The Fixed Discount Factor Model
- 2.3. Calibration

- Shocks
- All cases
- Business cycle moments

- Panel A: Productivity shocks
- Output
- Output

- Variable
- NFA/Y
- Productivity and interest rate shocks

- FDM

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