ISSN (Print) 0473-453X

Discussion Paper No. 1177

ISSN (Online) 2435-0982

WHEN IS THE TREND
THE CYCLE?

Dan CaoJean-Paul L’Huillier
Donghoon Yoo

June 2022

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

When is the Trend the Cycle?
Dan Cao, Jean-Paul L’Huillier, Donghoon Yoo*
December 8, 2021

Abstract
A literature debates the explanations for the cyclical properties of emerging markets using either trend shocks (Aguiar and Gopinath 2007) or financial frictions(Neumeyer and Perri 2004; Garcia-Cicco, Pancrazi, and Uribe 2010). We state aformal proposition that makes explicit the parametric assumptions needed for consumption to behave (exactly) as in a random-walk, permanent income model. The
result is general and applies to economies with endogenous investment and production. The proposition offers a fresh perspective on the debate regarding the sourcesof emerging market fluctuations, and reconciles diverging findings in the literature.
Moreover, we quantitatively explore the business cycle properties of the RBC model
when one moves away from the parametric assumptions suggested by the proposition.
Keywords: Aggregate productivity, permanent income, trend.
JEL codes: E21, E27, E32.

* Georgetown University; Brandeis University; Osaka University.

We thank Daniele Terlizzese for detailed comments on an earlier draft. We are also grateful to Mark Aguiar, Luigi Bocola, Pascal Michaillat,
Claudio Michelacci, Juan Passadore, Fabrizio Perri, Facundo Piguillem, Raphael Schoenle, Kirill Shakhnov
and Anton Tsoy for their reactions.

11Introduction

A large literature seeks to explain a set of important and salient features of emerging market business cycles based on a class of baseline RBC open economy models.
Different from developed countries, emerging markets tend to exhibit a larger volatility of consumption than output, and a volatile and countercyclical current account.
These features of the data represent a challenge to standard, frictionless, models.
There are two strands of this literature. One strand of the literature, following theseminal contribution by Aguiar and Gopinath (2007), asserts that a standard RBC
small open economy à la Schmitt-Grohe and Uribe (2003) subject to permanent
productivity shocks (called also nonstationary technology shocks or trend shocks) is
capable of accounting for the aforementioned features of emerging economies. Thisappealing economic insight has proven quite influential. The logic is that, due tothe permanent income assumption, permanent shocks induce large movements in
consumption and a volatile and countercyclical current account. This point has been
extended in work by Boz, Daude, and Durdu (2011), Naoussi and Tripier (2013),
among others.1 A related strand of the literature, initiated by Neumeyer and Perri(2004) and Uribe and Yue (2006), asserts instead that financial frictions are theexplanation to this large volatility. More recent work by Garcia-Cicco, Pancrazi, andUribe (2010), Alvarez-Parra, Brandao-Marques, and Toledo (2013), and Chang andFernandez (2013), among others, has also provided quantitative models of financial
frictions. Most often used are purely exogenous shocks to the interest rate facedby the domestic economy, which are in fact able to generate realistic business cycle
dynamics.
Our main contribution to this debate is analytical. We prove a proposition thatmakes explicit the conditions under which the permanent income hypothesis holds
exactly in the specification by Aguiar and Gopinath (2007) (henceforth AG). Thisis important, because as AG emphasize, the permanent income hypothesis forms thebasis of an explanation grounded on trend shocks.
Our proposition states that, when the sensitivity of the interest rate to movementsin the stock of debt goes to zero and preferences are separable, consumption dynamics
are only determined by the long-run of level of productivity (up to a constant):
ct = constant · long-run level of productivityt
Somewhat surprisingly, in this parameter region consumption is completely disconnected from the rest of the model, and, it is highly sensitive to trend shocks. To
see this, consider a positive trend shock. Because such a shock raises income at in1

See Durdu (2013) for a recent survey.

2

finity, consumption reacts strongly on impact. Thus, not only consumption tends
to be volatile, but the small open economy finances this increase in consumption byborrowing from the rest of the world, i.e., the net exports are countercyclical. So
long the interest rate does not increase following this increase in debt, this effect isquantitatively powerful. Instead, if the interest rate increases, this effect is muted
through the consumption Euler equation, and the model may not be able to fit thefacts.
The proposition offers a fresh perspective on the debate regarding the sources of
emerging market fluctuations.
First, it clarifies the key economic role of the interest rate sensitivity. It is important that, after an accumulation of debt, the interest rate does not increase by much,or not at all. So long this is true, the effect of trend shocks is quantitatively powerful. Notice that AG (and several other following up papers) indeed fix the sensitivityof the interest rate using a single parameter ψ, calibrated to a very small value ofψ = 0.001. So far, this strand of the literature has attributed to ψ just the technicalrole of delivering stationarity.2 Our proposition assigns it an economic role, the oneof delivering the random-walk permanent income behavior of consumption.
From a quantitative point of view, the analytical condition on the behavior ofinterest rate is an invitation to explore how insensitive the interest rate ought to be
for trend shocks to have traction. We thoroughly study the implications of increasing
the value of ψ, while leaving all other parameters used by AG unchanged (and keeping
their exact specification.) We show that when this parameter has a higher value, but
still rather small (say 0.1), consumption already features excess smoothness, thevolatility of output being higher than the one of consumption. Also, the ratio ofthe variances of net exports to output goes down to 0.19, whereas in the benchmark
results it is 0.71. So, net exports volatility is reduced considerably. In addition, thecorrelation of net exports and output is also reduced (although to a lesser extent).
Thus, assigning a moderate value to ψ overturns most of the quantitative insights.
Using the value ψ = 1 delivers excess smoothness, a ratio of the variances of net
exports to output of 0.10, and correlation of net exports to output more than 2 times
smaller, overturning the results in AG. To sum up, the permanent shocks explanation
is overturned when the sensitivity of the interest rate in the AG model is increased.
An influential paper by Garcia-Cicco, Pancrazi, and Uribe (2010) (henceforth
GPU), appearing after AG, estimates ψ to 2.8 using Bayesian methods. However,2

Our reading of this literature is that, tacitly, it is comfortable with assigning a very low value to ψ inorder to mimic the behavior of the nonstationary model in which the interest rate is simply fixed. Moving
away from the nonstationary model to a stationary one provides computational advantages and allows
researchers to match second moments (of endogenous model variables in levels) as in the standard RBC
closed-economy model. See for instance the discussion in Mendoza (1991) or Schmitt-Grohe and Uribe(2003).

3the model used by GPU is not exactly the same as in AG, because it is augmented
with interest rate shocks, preference shocks, and spending shocks. To clarify thedifferent findings in AG and GPU, in our numerical exercises, we also consider thisestimate (ψ = 2.8) in exactly the same model as in AG. It delivers similar results asusing ψ = 1, overturning the findings in AG. Thus, our proposition underlines thepivotal role played by the behavior of the interest rate.
To the best of our knowledge, this is the first paper to take a theoretical approachto analyze the performance of the RBC model in mimicking the economies of emerging
markets. In addition, the paper generalizes previous theoretical results (for instanceby Campbell and Deaton 1989, or Galı́ 1991 for endowment economies) by showingthat the random-walk permanent income hypothesis for consumption holds exactly,
and quite generally, in economies with endogenous investment and production.
Our proposition is closely related to the result by Engel and West (2005) in thecontext of asset prices. In fact, Engel and West (2005) showcase the condition thatthe discount factor approaches 1 to guarantee that asset prices manifest a random
walk behavior. Our theorem below also requires this condition to hold.
Our paper is related to a large body of existing literature on emerging market
business cycles. In addition to those already mentioned, for more recent discussion,see Chen and Crucini (2016), Rothert (2020), Hevia (2014), Dogan (2019), Seoane(2016), Drechsel and Tenreyro (2018), Akinci (2021), among others.
Chen and Crucini (2016) stress that the main limitation of the small open economy model is the lack of attention paid to the role of TFP spillovers from the large
economies and instead propose a large aggregate economic region in general equilibrium with a small open economy to capture the international correlation of businesscycles. In a similar general equilibrium framework, Rothert (2020) allows the domestic and foreign tradable goods to be imperfect substitutes to account for the behaviorof the real exchange rates. There, the impact of the trend shocks on aggregate consumption expenditure becomes smaller in emerging economies as expansionary productivity shocks reduce the relative price of domestic goods, dampening the impact
on the country’s income. Extending the ‘Business Cycle Accounting’ methodology
by Chari et al. (2007) to an open economy setting, Hevia (2014) suggests that RBC
models with just productivity shocks do not provide a successful benchmark to understand emerging market business cycles as productivity shocks in RBC models do
not distort the labor-consumption margin.
In a two-country international real business cycle model with investment and consumption goods, Dogan (2019) shows that investment-specific technology shocks play
an important role in our understanding of emerging market business cycles. Seoane(2016) proposes a small open economy model with tradable and non-tradable sectors, endogenously accounting for the real exchange rate, and shows that stationary

4

productivity shocks and the country premium explain a large share of the variability
observed in the data for emerging economies. Drechsel and Tenreyro (2018) consider
a second sector to capture the separate role of commodities in the economy. There,
commodity price and stationary productivity shocks are the most important sourceof fluctuations for output, consumption, and investment, while the contribution of
nonstationary productivity shocks remains non-negligible. In a model with an endogenously evolving time-varying country risk premium, Akinci (2021) shows that
nonstationary productivity shocks are non-negligible but not dominant in explaining
the economic fluctuations in output, consumption, and investment.
The rest of the paper is organized as follows. We setup the model and the log-linear
equilibrium in Section 2. We present the proposition and provide an interpretation in
Section 3. We explore several quantitative implications of the proposition in Section
4. Section 5 concludes. The proof of our proposition is quite lengthy and it is therefore
relegated to the appendix.

2

The Setup and Solution Method

For convenience, it seems natural to use exactly the same model as AG. Thus, we
reproduce here the published setup of the model. Then, we reproduce the normalization and log-linearization reported by Aguiar and Gopinath (undated) (henceforth
AGb). We use the notation adopted there.

2.1

Aguiar and Gopinath’s 2007 Model

This is a single-good, single-asset, small open economy. Technology is characterized
by a Cobb-Douglas production function that uses capital Kt and labor Lt as inputs:
Yt = ezt Kt1−α (Γt Lt )α
where α ∈ (0, 1) represents labor’s share of output, and zt and Γt are productivity
processes. Specifically, level productivity zt follows
zt = ρz zt−1 + ϵzt
with ρz < 1, and ϵzt is a stationary technology shock, labeled also cycle or transitory
shock, and modeled as an i.i.d. draw from a normal distribution with zero mean andstandard deviation σz . Trend productivity is modeled with a nonstationary process
Γt = egt Γt−15

wheregt = ρg gt−1 + ϵgt
and ρg < 1. ϵgt is a nonstationary technology shock, labeled also trend or permanent
shock, and modeled as an i.i.d. draw from a normal distribution with zero mean andstandard deviation σg . AG allow for a deterministic trend in gt (denoted µg , see AG
p. 80.), but to simplify the algebra in the proof of our main result below, we set thisdeterministic trend to zero (µg = 0). Our result extends easily to the general casewith unrestricted µg .
Period utility is Cobb-Douglas
ut =[Ctγ (1 − Lt )1−γ ]1−σ1−σwhere 0 < γ < 1. The resource constraint is
ϕ
Ct + Kt+1 = Yt + (1 − δ)Kt −
2



Kt+1
−1
Kt

2
Kt − Bt + Qt Bt+1

where Kt+1 is capital, δ is the capital depreciation rate, Bt represents debt due inperiod t, qt is the time t price of debt due in period t + 1 and adjustment costs incapital are captured by
ϕ
2



Kt+1
−1
Kt

2
Kt

where ϕ > 0 is a parameter.
The price of debt is sensitive to the level of outstanding debt, taking the form
used by Schmitt-Grohe and Uribe (2003):




Bt+1
1
∗
= 1 + rt = 1 + r + ψ exp
−b −1
Qt
Γt

(1)

where r∗ is the world interest rate, b represent an exogenous steady-state level ofnormalized debt, and ψ > 0 governs the elasticity of the interest rate to debt.

Normalization and Recursive Formulation. For a variable Xt , we write
its detrended counterpart by normalizing the variable using previous period’s trend
productivity:
X̂t =

Xt
Γt−1

In normalized form, the representative agent’s problem is written recursively as(
V (K̂, B̂, z, g) =max
{Ĉ,L,K̂ ′ ,B̂ ′ })[Ĉ γ (1 − L)1−γ ]1−σgγ(1−σ)
′
′ ′ ′
+ βe
EV (K̂ , B̂ , z , g )1−σ6subject toϕĈ + e K̂ = Ŷ + (1 − δ)K̂ −
2
g

′

e

g K̂

!2

′

K̂

−1

K̂ − B̂ + eg q B̂ ′where a prime on a variable ′ denotes the value of the variable at t + 1.

Log-linearization. For nonstationary variables, we define the following log-deviations
from stationary steady state quantities:ĉt ≡ log(Ct /Γt−1 ) − log(C̄/Γ̄)ŷt ≡ log(Yt /Γt−1 ) − log(Ȳ /Γ̄)x̂t ≡ log(Xt /Γt−1 ) − log(X̄/Γ̄)k̂t+1 ≡ log(Kt+1 /Γt ) − log(K̄/Γ̄)For variables that are already stationary, we define the following log-deviationsn̂t ≡ log(Nt ) − log(N̄ )ˆlt ≡ log(Lt ) − log(L̄)
q̂t ≡ log(Qt ) − log(Q̄)
We then define the absolute deviation of the net exports-to-output rationxˆ t = N Xt /Yt − N¯X/ȲWe also define the absolute deviation ofb̂t+1 =

Bt+1 B̄−ΓtΓ̄

These definitions for steady state deviations are identical to the ones used in AGb,
with the exception of the last one. There, we use an absolute deviation the relative(log) deviation used in the original paper, in order to allow for B̄/Γ̄ = 0. This allows
us to obtain general expressions in the proposition below, but our results do not relyon this specification.
The resulting log-linearized model is fully characterized by the following set of
equations:

7

The dynamics of productivity, including the cycle shocks and the trend shocks:z ′ = ρz z + ϵzg ′ = ρg g + ϵg
The first-order condition in k ′ , which corresponds to equation 12 in AGb:
0 = (γ(1 − σ) − 1)Eĉ′ + (1 − γ)(1 − σ)Eˆl′ + βϕEg ′Ȳ
+ β(1 − α) Eŷ ′ + βϕEk̂ ′′
K̄
 


Ȳ
− β (1 − α) + ϕ + ϕ k̂ ′ − (γ(1 − σ) − 1)ĉK̄
− (1 − γ)(1 − σ)ˆl + (γ(1 − σ) − 1 − ϕ)g + ϕk̂

(AGb12)

The first-order condition in b′ , which corresponds to equation 17 in AGb:
0 = (γ(1 − σ) − 1)Eĉ′ + (1 − γ)(1 − σ)Eˆl′ + (γ(1 − σ) − 1)g− (γ(1 − σ) − 1)ĉ − (1 − γ)(1 − σ)ˆl − q̂

(AGb17)

Other equations that describe the dynamics of the log-linearized model are:0 = ŷ − n̂ − ĉ + ˆl
0=

C̄
B̄
X̄
Ȳ
ŷ + Q̄b̂′ + Q̄ (g + q̂) − b̂ − x̂ − ĉ
Γ
Γ̄
Γ̄
Γ̄


X̄
K̄ ′
x̂ =
k̂ − (1 − δ)k̂ + gΓ̄Γ̄ŷ = z + (1 − α)k̂ + α(g + n̂)(AGb20)(AGb21)(AGb22)L̄ˆl = −N̄ n̂q̂ = −ψ Q̄b̂′∆nx = (1 − N¯X/Ȳ )ŷ −(AGb24)X̄C̄
x̂ − ĉȲȲ

where we have followed the notation in AGb. The model is exactly the same as in
AGb except for (AGb20) and (AGb24). They are different because of the way wenormalize the level of debt.

3

The Proposition

We consider the case with γ = 1, which corresponds to labor supply being exogenously
given. One can trivially restate the arguments for the case γ ̸= 1 but σ = 1, which8corresponds to endogenous labor supply, but additively separable from consumption(log-log preferences). Our result holds when preferences belong to one of these twocases. (Numerical simulations also show that the result does not hold outside thesetwo cases.)If γ = 1, nt = 0. Under this assumption, the linearization of the production
technology (AGb22) becomesŷt = zt + (1 − α) k̂t + αgt(2)

Weh will solve for a log-linearized
solution of the system using the state state spacei′
Xt = bt k̂t ζt ζt−1 zt , where ζt = log(Γt ) andbt = b̂t +

B̄
ζt−1
Γ̄

h
i′
h
i′
For further use we let X0t = bt k̂t and X1t = ζt ζt−1 zt . It is also important to notice thatgt = ζt − ζt−1
Using the definition of log-consumption, we havect = ĉt + ζt−1
Following standard log-linearization techniques, for example as presented in Blanchard and Kahn (1980) and Uhlig (1999), the solution to the log-linearized model(AGb12)-(AGb24) takes the form:

ct = Dc Xt
bt+1 = Db Xt
k̂t+1 = Dk Xt(3)

In particular,
ct = Dc Xt = Dc0 X0t + Dc1 X1t= Dc,b bt + Dc,k k̂t + Dc,ζ1 ζt + Dc,ζ2 ζt−1 + Dc,z zt
Denote by ζt+∞ the expected long-run level of productivity, i.e.
ζt+∞ = lim E[ζt+j ] =j→∞9ζt − ρg ζt−1
1 − ρg(4)

We claim that as Q̄ → 1 and ψ → 0, consumption is only a function of long-run
productivity. Specifically,ct =1 − X̄/Ȳ
C̄/Ȳ


ζt+∞

(5)

The result is expressed formally as follows.
Proposition 1
lim lim Dc,k = 0

Q̄→1 ψ→0

lim lim Dc,b = 0
1 − X̄/Ȳ1
lim lim Dc,ζ1 =1 − ρgC̄/Ȳ
Q̄→1 ψ→0−ρg
1 − X̄/Ȳ
lim lim Dc,ζ2 =1 − ρgC̄/Ȳ
Q̄→1 ψ→0
Q̄→1 ψ→0

lim lim Dc,z = 0

Q̄→1 ψ→0

3.1

Interpretation

The proposition states that when the interest rate becomes insensitive to changes indebt holdings, consumption is only determined by the long-run level of productivity,
as expressed by equation (5). This means, at this limit, the level of debt holdings or
the stock of capital do not matter for the determination of consumption. The result
also requires that in the steady-state the world interest rate goes to zero (Q̄ → 1,
following from β → 1), which allows the agent to roll-over any existing stock of debt
to infinity, thereby allowing to maintain consumption at the long-run level of income(determined by long-run output).3
A corollary is that consumption only reacts to permanent shocks, and does notreact to transitory shocks. Although not immediately obvious by looking at the
proposition, after a permanent shock to productivity consumption jumps to its longrun level γt+∞ and stays there. This can be seen in a probably more transparent
way by considering the case of zero steady-state debt holdings (or zero steady-state
net exports). The following Corollary considers this case and derives the resulting
behavior of ĉt , the normalized log-deviation of consumption.
3

Campbell and Deaton (1989) and Galı́ (1991) considered a version of this result in the case of endowment economies. They also focus on the limit of zero interest rate to study the empirical relationshipbetween the variance of consumption changes and of permanent income shocks (transitory income shocks
will be negligible in this limit) in a simple setting with constant interest rate and without capital accumulation.

10

Corollary 1 If C̄/Ȳ + X̄/Ȳ = 1,
lim lim ĉt =Q̄→1 ψ→0

1
gt
1 − ρg

Proof. Proposition 1 shows that in the limitct =1 − X̄/Ȳ
C̄/Ȳ



1
1 − ρgζt −1 − X̄/Ȳ
C̄/Ȳ



ρg
1 − ρgζt−1

from which, given C̄/Ȳ + X̄/Ȳ = 1,
ct =

ρg
1
ζt −
ζt−1
1 − ρg1 − ρgUsing the definition of ĉt :ĉt = ct − ζt−1
ρg
1
=
ζt −
ζt−1 − ζt−1
1 − ρg1 − ρg1
1(ζt − ζt−1 ) =gt=
1 − ρg1 − ρg

The constant (1− X̄/Ȳ )/(C̄/Ȳ ) in front of equation (5) is simply a factor that adjusts the size of deviations to the value of steady-state variables, which depend on thesteady-state capital-to-output and consumption-to-output ratios (both exogenous).
When C̄/Ȳ + X̄/Ȳ = 1, this constant is equal to 1. Using this and expressing the result in terms of normalized log-deviations of consumption instead of log-consumption
allows to obtain a simple intuitive expression for the behavior of consumption in the
AG model, where consumption is equal to the expected cumulated sum of permanent
productivity increases (1/(1 − ρg )) · gt .
Even though our proposition focuses on consumption, it has indirect implications
for the behavior of net exports. The key point is that persistent permanent shocks
embed a large wealth effect that generates large short-run volatility on consumption,
while they have relatively small effects on output. Thus, the implication is countercyclical and volatile net exports.4
4

Notice that transitory shocks are sort of a nuisance in the model because they generate extra output
volatility (and little consumption volatility), competing with the main channel emphasized here. However,they turn out to be useful to match output volatility.

11

4

Quantitative Explorations

In this section, we explore the quantitative implications of the proposition. We proceed in three steps. First, we explore the robustness of the ability of the AG model
to match three key moments: the relative variance of consumption to output, therelative variance of net exports with respect to output, and the correlation of net exports and output. We look at what happens when the interest rate is more sensitive
than the usual calibration in the literature. Second, we look at the implication ofusing separable preferences. Third, we explore an endowment economy and check its
ability to match the three moments just mentioned.

4.1

The Importance of the Sensitivity of the Interest

Rate ψ
Proposition 1 requires ψ −→ 0 in order for consumption to react strongly to changes
in the trend of productivity, thereby causing net exports to be highly volatile andcountercyclical. We now quantitatively explore this point, and study what happens
when ψ is assigned a higher value than the one used in AG (0.001).
All values of the rest of the parameters in this section are the ones used in AG, withthe exception of ψ (sensitivity of the interest rate) and γ, as required by Proposition
1. The other benchmark parameter values are presented in Table 1.

Table 1: Benchmark Parameter Values
Parameter

Value

Intertemporal elasticity of substitution
Steady state level of normalized debt
Labor’s share of output
Depreciation rate
Capital adjustment costs

2
0.10
0.68
0.05
4.00

Persistence permanent shock
Persistence transitory shockStandard dev. permanent shockStandard dev. transitory shock

0.01
0.95
2.81
0.48

Non-productivity Parameters

σ
B/Y
α
δ
ϕ
Productivity Processes

ρg
ρz
σg
σz

Notes: These parameters for shock processes are the same used for Table 4 of AG, Specification 1,
Mexico.

We first check our theoretical result numerically. As shown in Figure 1, imposing
ψ very small (and β close to 1 together with ψ/ (1 − β) very small5 ) ensures the5

These are equivalent to limQ̄→1 limψ→0 (in that order).

12random walk behavior of consumption. Consumption jumps on impact following apermanent shock, but does not move at all following a transitory shock. These sharp
results are quite striking given the complex structure of the rest of the model.6

Figure 1: Impulse Responses: Consumption and Net Exports
Permanent Shock: Consumption

Transitory Shock: Consumption

1.5

1.5

1

1

0.5

0.5

0
0

5

10

15

0
0

20

Permanent Shock: Net Exports

10

15

20

Transitory Shock: Net Exports
1.5

0

1

−0.5

0.5

−1

−1.5
0

5

5

10

15

0
0

20

5

10

15

20

Notes: The lines depict responses of the model in AG in the limiting case of Proposition 1 where we set β = 0.99999 andψ = 10−12 . We also set γ = 1 which implies that labor supply is exogenously given. The standard deviations of technology
shocks are normalized to 1, similar to Figure 3 in AG. The other parameters are from Table 1.

Figure 2 examines the role played by the parameter ψ in determining the sensitivity of consumption to trend shocks by depicting impulse responses of consumption
using different values of ψ. We consider the value used in AG (0.001) along withsome larger values (0.01, 2.8). 2.8 is the value estimated by Garcia-Cicco, Pancrazi,
and Uribe (2010) (henceforth GPU). Consumption does not immediately reach its
long-run level with a permanent shock when the parameter ψ takes the values wechoose here, these values being substantially larger than the one we used previously
in Figure 1 (ψ = 10−12 ). Notice that, crucially, the larger ψ, the smaller the responseof consumption following a permanent shock, and the larger the response following atransitory shock. This tends to overturn the results.
Further examining the role played by the parameter ψ, we reproduce Figure 3 of
AG with different values of ψ. Again, we consider ψ = 0.001, ψ = 0.01, and ψ = 2.8
and obtain impulse responses of (A) the ratio of net exports to GDP, (B) the ratio ofconsumption to GDP, and (C) the ratio of investment to GDP following a 1 percent
6

For this Figure, we also set β = 0.99999 and γ = 1 (see p. 10.) We have also verified that setting
σ = 1 instead delivers exactly the same results for consumption, and similar results for net exports.

13

Figure 2: Impulse Responses: Consumption (Varying ψ)Permanent Shock: Consumption

Transitory Shock: Consumption

0.95

0.85

0.9

0.8

0.85

0.75

0.8

0.7

0.75

0.65

0.7

0.6

0.65

0.55

0.6
0

5

10

15

0.5
0

20

ψ=0.001ψ=0.01
ψ=2.8

5

10

15

20

Notes: For all specifications, we set β = 0.98 and σ = 2. Also, the standard deviations of technology
shocks are normalized to one. Other parameter values are given in Table 1.

shock on ϵg and ϵz . We clearly observe that these ratios vary substantially with theparameter ψ. Most importantly, the response of net exports to a permanent g shockis muted with large values of ψ. The reason is the muted response of consumption.
These numerical results reveal a similar behavior of investment, which also features
a muted response.
Finally, Table 2 reports a set of moments7 using different parameter values of ψ,
ranging from 10−12 to 2.8, the value estimated by Garcia-Cicco, Pancrazi, and Uribe(2010). We use exactly same parameters used in AG, except for ψ.8 As shown in
the table, the volatility of consumption σ(c), the relative volatility of consumption to
output σ(c)/σ(y), and that of net exports to output σ(N X)/σ(y) depend greatly on
the parameter ψ. Also, similar to the results shown in Figure 3, the relative volatilityof investment respect to output σ(I)/σ(y) is decreasing with ψ.
To sum up, consistent with Proposition 1, this subsection has numerically illustrated the role played by the interest rate sensitivity parameter ψ for the results in
AG. The main point is that a small value of ψ ensures the random walk behavior of
consumption and leads the model to generate the key moments emphasized in AG.
Larger values of ψ tend to reverse this, the model losing the ability to generate thosemoments. A value of ψ of around 0.1 already generates important difficulties at thistask. Larger values overturn the results.
7

The same shown in Table 5 in AG, plus σ(c).
This means that we no longer set σg and σz to one. Instead, following Table 4 in AG, we set σg = 2.81
and σz = 0.48.
8

14

15
5

10

15

Ratio of investment to GDP (g shock)

20

−0.5
0

0

0.5

1

1.5

10

15

10

15

5

10

15

ψ=0.001ψ=0.01ψ=2.8

Ratio of investment to GDP (z shock)5

Ratio of consumption to GDP (z shock)5

Ratio of net exports to GDP (z shock)

20

20

20

Notes: For all specifications, we set σ = 2, β = 0.98, and the standard deviations of technology shocks are normalized to one. All other parameters as shown in Table 1.
We then produce the impulse responses of (A) ratio of net exports to GDP, (B) ratio of consumption to GDP, and (C) ratio of investment to GDP with different values of ψ:ψ = {0.001, 0.01, 2.8}.

Figure 3: Impulse Responses: (A) Ratio of net exports to GDP, (B) Ratio of consumption to GDP, (C) Ratio of investment to
GDP

0
0

0.5

1

1.5

−0.8
0

−0.2
0
20

−0.6

0
15

−0.4

0.2

10

−0.2

0.4

5

0

0.6

Ratio of consumption to GDP (g shock)

−0.2
0

−0.6
0
20

0

−0.4
15

0.2

−0.2

10

0.4

0

5

0.6

Ratio of net exports to GDP (g shock)0.2

Table 2: Moments - Emerging Market: Mexico

ψ

AG

(1)

(2)

(3)

(4)

(5)

(6)

GPU

0.001

10−12

0.00001

0.0001

0.01

0.1

1

2.8

2.76
2.05
0.88
1.62
0.10
0.75
0.07
-0.35
1.00
0.97
2.43

2.76
2.05
0.88
1.61
0.10
0.75
0.07
-0.31
1.00
0.98
2.43

Emerging Market: Mexico
σ(y)
σ(∆y)
σ(c)/σ(y)σ(I)/σ(y)σ(N X)/σ(y)
ρ(y)
ρ(∆y)
ρ(y, N X)
ρ(y, c)ρ(y, I)
σ(c)

2.40
1.73
1.26
2.60
0.71
0.78
0.13
-0.66
0.94
0.92
3.03

2.29
1.65
1.39
2.67
0.84
0.78
0.14
-0.65
0.92
0.92
3.18

2.29
1.66
1.39
2.67
0.84
0.78
0.14
-0.65
0.92
0.92
3.18

2.31
1.67
1.37
2.67
0.82
0.78
0.13
-0.66
0.93
0.92
3.16

2.59
1.86
1.06
2.26
0.45
0.77
0.10
-0.63
0.98
0.93
2.75

2.73
2.00
0.91
1.80
0.19
0.76
0.08
-0.49
0.99
0.96
2.48

Notes: AG refers to specification 1 in Table 5 of AG and GPU refers to Garcia-Cicco,
Pancrazi, and Uribe (2010). In GPU, ψ is estimated to be 2.8. Thus, we keep all otherparameters as same as in AG and choose ψ to be 2.8. Similarly, for our specifications (1)to (6) are obtained by using different parameter values of ψ.

4.2

Implications of Separable Preferences

Second, we investigate whether alternative preferences to those originally used by
AG do a better job at generating consumption volatility. This is indeed suggested
by Proposition 1 because only when preferences are separable consumption has arandom-walk behavior and thus jumps on impact to the long-run level of productivity
implied by the trend shock. The parametrization in AG sets σ = 2, which implies
Cobb-Douglass, non-separable preferences, which can possibly dampen the reactionof consumption on impact.

Table 3: Moments - Separable Preferences Rather than Cobb-Douglas

σ(c)/σ(y)σ(N X)/σ(y)ρ(N X, y)AG (σ = 2)

AG: Separable preferences (σ = 1)

1.26
0.71
-0.66

1.35
0.82
-0.62

Notes: All parameters are those from AG, except for σ in the second column.

To investigate this point we simulate moments with σ = 1 (and letting all otherfeatures and parameter values in the original AG model, including the productivity

16

parameters). Table 3 shows that using separable preferences generates an increasein the volatility of consumption to output by 9%. This finding confirm the intuition
provided by Proposition 1. Also, this produces an increase in the volatility of net
exports to output by 11%. There is slight fall in the correlation between net exports
and output from -0.66 to -0.62.
Should one then advocate the use of separable preference in this type of exercises?
Actually, one important caveat of separable preferences is the negative comovementof labor supply and trend productivity due to the wealth effect. Avoiding this issue
is one reason to resort to other preferences, as in AG. Figure 4 shows that, after apermanent shock, labor supply falls. The next subsection discusses this issue further.
Permanent Shock: Labor Supply
0.1

0

−0.1

−0.2
0

5

10

15

20

Figure 4: Impulse Responses: Labor Supply
Notes: All parameters are those from AG, except for σ, which is set to 1.

4.3

A Simple Specification

We showed previously that even though separable preferences allow to more easily
obtain consumption and net exports volatility, labor supply falls. Recalling Jaimovich
and Rebelo (2009), one immediate reaction is that the model requires more frictions
to perform well on this dimension. However, an easier and interesting fix is to removelabor supply altogether, i.e., to consider an endowment economy. Notice however that
this would also require removing investment from the model. From the point of viewof Proposition 1 this is not a problem because its proof does not use any particularform of the economy’s supply side.
Table 4 shows the results. It compares the moments generated by the baseline
emerging market calibration in AG to a calibration of the simple model we propose.
The calibration is in Table 5. The calibrated simple moments does well in replicating
the moments in AG, with a larger ratio of the volatility of consumption to the volatility of output σ(c)/σ(y), a slightly smaller ratio of the volatility of net exports to thevolatility of output σ(N X)/σ(y), and the same correlation between net exports andoutput ρ(N X, y). So, one could recur to the simple model to match these momentsin the data.

17

Table 4: Moments - Simple Model

σ(c)/σ(y)σ(N X)/σ(y)ρ(N X, y)AG

Simple Model

1.26
0.71
-0.66

1.55
0.66
-0.66

Notes: The first column (AG) is from Column of Table 4 in AG (equivalently, Specification 1, Mexico,Table 5). For the simple model, we use the following parameter values ρg = 0.40, ρz = 0.95, β = 0.99,
ψ = 0.0001. Other parameter values are from Table 1. The simple model does not include labor supply
nor capital (endowment economy).

Table 5: Calibration
Parameter

Value

Steady state level of normalized debt

0.10

Persistence permanent shock
Persistence transitory shockStandard dev. permanent shockStandard dev. transitory shock

0.40
0.95
2.81
0.48

Non-productivity Parameters

B/Y
Productivity Processes

ρg
ρz
σg
σz

Notes: All other parameters are from AG. Specifically, we set β = 0.98 and ψ = 0.001.

5

Conclusions

In our view, the main conclusions can be drawn from these exercises is the importance of efforts towards a precise and well-identified estimation of the sensitivity of
the interest rate ψ. In this direction, recently Miyamoto and Nguyen (2017) estimate using data for 17 developing and developed countries and find that ψ features acredible interval that is bounded away from 0, but the point estimate greatly varies
across countries. So far, these estimates have been obtained via structural estimation.
Therefore, identification has remained dependent on the exact specification and details of the model. Finding complementary approaches to identifying the sensitivityof the interest rate is a fruitful research avenue.
We finalize this discussion by highlighting the importance of trend shocks to fit
the data in other contexts. We emphasize that trend shocks have an importantconceptual advantage, the one of resting on a well-established economic mechanism(the permanent income hypothesis) in order to (at least qualitatively) generate highly
volatile consumption and net exports, and countercyclical net exports. Trend shocks
have been successfully used in “cousin” literatures to match interesting facts implied18

by consumption dynamics.9 Thus, we remain under the impression that efforts inthe direction of improving the propagation of trend shocks with the help of extra
frictions constitute a fruitful research avenue. We look forward to developments inthis direction.

9

See, for instance, Blanchard and Quah (1989), Gali (1999), or Blanchard, L’Huillier, and Lorenzoni(2013).

19

A

Main Proof

The following equations help determine the steady state values and will be important
for the proof of Proposition 1:Q̄ =(1 − α)

Ȳ
K̄
C̄
Ȳ
N̄
K̄
Γ̄
Ȳ
Γ̄
C̄
Γ̄
X̄
Γ̄

N¯X/Ȳ

Proof of Proposition 1.

=
=
=
=
=
=
=
=

1
=β
1 + r∗
1
−1+δ
Q̄
K̄
B̄
1 − δ + (Q̄ − 1)
Ȳ
Ȳ


−1
C̄ 1 − γ1+αγȲ  α1K̄N̄Ȳ  
ȲK̄K̄Γ̄  
C̄
Ȳ
Ȳ
Γ̄
K̄
δ
Γ̄
X̄
C̄
1− −
Ȳ
Ȳ

Substituting ŷt from (2) and substituting x̂t from

(AGb21) in the linearization of the budget constraint (AGb20) implies
0 =



Ȳ 
B̄ 
zt + (1 − α) k̂t + αgt + Q̄gt − ψ Q̄b̂t+1
Γ̄
Γ̄
 C̄
K̄ 
+Q̄b̂t+1 − b̂t −k̂t+1 − (1 − δ) k̂t + gt − ĉt
Γ̄
Γ̄

20

Combining with the definition of bt+1 and ct , we obtainB̄Q̄ 1 − ψ Q̄
Γ̄







Ȳ
K̄
B̄
Ȳ
K̄
= bt − α −
gt −
+ Q̄
(1 − α) + (1 − δ) k̂t
Γ̄
Γ̄
Γ̄
Γ̄
Γ̄
#
"


h
i c
B̄
B̄
Ȳ
C̄
B̄
t
1 − ψ Q̄γt − γt−1 + C̄/Γ̄ K̄/Γ̄− zt − γt−1 + Q̄
Γ̄
Γ̄
Γ̄
Γ̄
Γ̄
k̂t+1

bt+1

(6)
h

i
= 1 − ȲΓ̄ (1 − α) + K̄
(1 − δ) X0t
Γ̄
 


2  

Ȳ
K̄
B̄
Ȳ
K̄
B̄
C̄
B̄
Ȳ
+ − α Γ̄ − Γ̄ + ψ Q̄ Γ̄
α Γ̄ − Γ̄ + Q̄ Γ̄ − Γ̄ − Γ̄− Γ̄ X1t+

C̄ 0 0 K̄ 0 0 C̄ 1 1 K̄ 1 1
D X + Dk Xt + Dc Xt + Dk Xt
Γ̄ c tΓ̄Γ̄Γ̄We regroup the coefficients on X0t and X1t to obtain:
Db0 =



1


Q̄ 1 − ψ Q̄ B̄
Γ̄



C̄
K̄
D̄b + Dc0 + Dk0
Γ̄
Γ̄


(7)

i
h

(1
−
δ)
and
where D̄b = 1 − ȲΓ̄ (1 − α) + K̄
Γ̄
Db1 =

1


Q̄ 1 − ψ Q̄ B̄
Γ̄




h 
 

i C̄
K̄ 1
1
K̄
B̄ 2
Ȳ
K̄
B̄
C̄
Ȳ
Ȳ
α Γ̄ − Γ̄ + (Q̄ − 1) Γ̄ − Γ̄− Γ̄ + Dc + Dk
− α Γ̄ − Γ̄ + ψ(Q̄ Γ̄ )Γ̄Γ̄More explicitly:
Db,b =

Db,k =


C̄K̄ 1 + Dk,b + Dc,bΓ̄Γ̄Q̄ 1 − ψ Q̄ B̄
Γ̄



1
K̄
C̄
Ȳ
K̄


Dk,k + Dc,k −(1 − α) + (1 − δ)
Γ̄
Γ̄
Γ̄
Γ̄
Q̄ 1 − ψ Q̄ B̄
Γ̄
1

and
Db,z =

Db,ζ1 =

Db,ζ2 =1


Q̄ 1 − ψ Q̄ B̄Γ̄1
Q̄ 1 − ψ Q̄ B̄Γ̄1
Q̄ 1 − ψ Q̄ B̄
Γ̄



K̄
C̄
Ȳ
Dk,z + Dc,z −
Γ̄
Γ̄
Γ̄



K̄
C̄
Dk,ζ1 + Dc,ζ1 −
Γ̄
Γ̄


2 !!
Ȳ
K̄
B̄
α −
+ ψ Q̄
Γ̄
Γ̄
Γ̄



K̄
C̄
Ȳ
K̄
B̄ C̄
B̄
Dk,ζ2 + Dc,ζ2 + α −+ Q̄ − −
Γ̄
Γ̄
Γ̄
Γ̄
Γ̄
Γ̄
Γ̄

21

From the log-linearizing equation of the Euler equation with respect to k ′ (AGb12),
using the assumption that γ = 1 and ˆl ≡ 0, we have
0 = −σEt [ct+1 − ct ] + βϕEt [gt+1 ]
h
i
h
i
Ȳ
+β (1 − α) Et zt+1 + (1 − α) k̂t+1 + αgt+1 + βϕEt k̂t+2
K̄
 


Ȳ
− β (1 − α)
+ ϕ + ϕ k̂t+1 − (σ + ϕ) gt + ϕk̂t
K̄

(8)

We use the fact thatEt [gt+1 ] = ρg gt
Et [zt+1 ] = ρz zt
and gt = ζt − ζt−1 , together withhi
Et k̂t+2 = Et [Dk Xt+1 ]= Et [Dk,b Db Xt + Dk,k Dk Xt + Dk,z zt+1 + Dk,ζ1 ζt+1 ] + Dk,ζ2 ζt= (Dk,b Db + Dk,k Dk ) Xt + Dk,z ρz zt + ((1 + ρg )Dk,ζ1 + Dk,ζ2 )ζt − ρg Dk,ζ1 ζt−1h

Et k̂t+1i= Dk Xt

Et [ct+1 ] = Et [Dc Xt+1 ]= (Dc,b Db + Dc,k Dk ) Xt + Dc,z ρz zt + ((1 + ρg )Dc,ζ1 + Dc,ζ2 )ζt − ρg Dc,ζ1 ζt−1
Et [ct ] = Dc Xt
to simplify (8) to
0 = −σ ((Dc,b Db + Dc,k Dk ) Xt + Dc,z ρz zt + ((1 + ρg )Dc,ζ1 + Dc,ζ2 )ζt − ρg Dc,ζ1 ζt−1 − Dc Xt )ȲȲ+βϕρg (ζt − ζt−1 ) + β (1 − α) (ρz zt + αρg (ζt − ζt−1 )) + β (1 − α) (1 − α) Dk XtK̄K̄+βϕ ((Dk,b Db + Dk,k Dk ) Xt + Dk,z ρz zt + ((1 + ρg )Dk,ζ1 + Dk,ζ2 )ζt − ρg Dk,ζ1 ζt−1 )


 
Ȳ
− β (1 − α)
+ ϕ + ϕ Dk Xt − (σ + ϕ) (ζt − ζt−1 ) + ϕk̂t
K̄22

Now, extracting the components related to X0t from this equation, we have
0 = −σDc,b Db0 + Dc,k Dk0 X0t − Dc0 X0t Ȳ+β (1 − α) (1 − α) Dk0 X0t + βϕ Dk,b Db0 + Dk,k Dk0 X0t
K̄
 


Ȳ
− β (1 − α)
+ ϕ + ϕ Dk0 X0t + ϕk̂t
K̄for all X0t . This implies
0 = −σDc,b Db0 + Dc,k Dk0 − Dc0Ȳ+β (1 − α) (1 − α) Dk0 + βϕ Dk,b Db0 + Dk,k Dk0
K̄
 


Ȳ
− β (1 − α)
+ ϕ + ϕ Dk0 + ϕDk
K̄

(9)

h
i
0 1 . This equation helps determines Dk0 , i.e., Dk,b and Dk,k as
functions of Dc,b and Dc,k . In particular when Dc,b and Dc,k are close to zero, we

where Dk =have the Taylor expansion:
Dk,b = α1 Dc,b + β1 Dc,k + o (Dc,b ) + o (Dc,k )∗+ α2 Dc,b + β2 Dc,k + o (Dc,b ) + o (Dc,k )Dk,k = Dk,k(10)

∗ is the solution ofwhere Dk,k

 


Ȳ
∗
∗ 2
0 = − β (1 − α) α + ϕ + ϕ Dk,k+ βϕ(Dk,k) +ϕK̄

0 when D 0 = 0.
i.e. equation (9) for Dk,kc

Armed with the solution (10), we now use the first order condition for b′ (AGb17),
again with γ = 1 and ˆl ≡ 0:
0 = σEt [ct+1 − ct ] − ψ Q̄bt+1 + ψ Q̄

B̄
ζt
Γ̄

and extract the coefficients on X0t to obtain
0=σDc,b Db0 + Dc,k Dk0 − Dc0 − ψ Q̄Db0

Substituting Db0 from (7) into this equation, we arrive at

23

(11)





ψ
1
C̄ 0 K̄ 0
0 =Dc,b − Q̄
D b + Dc + DkσΓ̄Γ̄Q̄ 1 − ψ Q̄ B̄Γ̄+Dc,k Dk0 − Dc0

We separate the equations for Dc,b and Dc,k to obtain





ψ
C̄
1

 Db,b + Dc,b − Dc,b
0 = Dc,b − Q̄σΓ̄Q̄ 1 − ψ Q̄ B̄
Γ̄





K̄
ψ
1

 + Dc,k Dk,b+
Dc,b − Q̄σQ̄ 1 − ψ Q̄ B̄ Γ̄

(12)

Γ̄

and




1
C̄
ψ
 Db,k + Dc,k − Dc,k
0 = Dc,b − Q̄σΓ̄Q̄ 1 − ψ Q̄ B̄
Γ̄





1
K̄
ψ

 + Dc,k Dk,k+Dc,b − Q̄σQ̄ 1 − ψ Q̄ B̄ Γ̄

(13)

Γ̄

Now, we use the results above to show that as Q̄ → 1 and ψ → 0, Dc,b → 0.
Indeed, we first solve for Dc,k from the second equation (13):Dc,b − ψσ Q̄  1 B̄  Db,k + K̄DΓ̄ k,kQ̄ 1−ψ Q̄ Γ̄
Dc,k =
1 − Dc,b − ψσ Q̄  1 B̄  C̄Γ̄ − Dk,kQ̄ 1−ψ Q̄ Γ̄

As ψ → 0, this equation simplifies to

Dc,k =


Dc,b Q̄1 Db,k + K̄DΓ̄ k,k
1 − Dc,b Q̄1 C̄Γ̄ − Dk,k(14)

In addition, equation (12) simplifies to
0 = Dc,b

1
Q̄





C̄
1 K̄
Db,b + Dc,b − Dc,b + Dc,b
+ Dc,k Dk,b
Γ̄
Q̄ Γ̄

(15)

Plugging (14) into (15) and grouping by Dc,b (also by definition Db,b = 1), we

24

have



1
K̄




D
+
D
b,k
k,k
C̄
1
1 K̄
Q̄
Γ̄
1 + Dc,b − 1 +
Dk,b
0=
+
 Q Γ̄
Q̄Γ̄
1 − Dc,b 1 C̄ − Dk,k Q̄ Γ̄Equivalently,


 K̄

Db,k + K̄
D
C̄
k,k
Γ̄
Q̄ − 1 = Dc,b +
+
Dk,b
 Γ̄
Γ̄
1 − Dc,b 1 C̄ − Dk,k Q̄ Γ̄As Q̄ → 1 and ψ → 0, Lemma 1 below shows that (10) holds with α1 = β1 = 0,
∗ < 1. Therefore,
and 0 < Dk,kQ̄ − 1 =C̄
Dc,b + o(Dc,b )Γ̄Therefore, as Q̄ → 1, Dc,b → 0.
Then, (10) and (14) imply that
lim lim Dc,k = 0

Q̄→1 ψ→0

lim lim Dk,b = 0

Q̄→1 ψ→0

lim lim Dk,k =Q̄→1 ψ→0∗
lim Dk,kQ̄→1q(αδ/ϕ + 2) − (αδ/ϕ + 2)2 − 4=2where the last limit is given in Lemma 1.
We now move on to compute Dc1 .
Rearranging equation (11) and using the conjecture for ct+1 , we obtain:ψ Q̄ψ Q̄B̄/Γ̄
0 = Dc,b −bt+1 + Dc,k k̂t+1 + Dc1 AX1t − ct +ζtσσ
and substituting bt+1 from (6), we have
(1 − x̄) ct

"
#

2 !


x̄
Ȳ
K̄
B̄
Ȳ
K̄
B̄ C̄
B̄
Ȳ
=
bt − α −+ ψ Q̄
ζt + α −+ Q̄ − −ζt−1 − zt
C̄/Γ̄
Γ̄
Γ̄
Γ̄
Γ̄
Γ̄
Γ̄
Γ̄
Γ̄
Γ̄



x̄
Ȳ
K̄
K̄
−
(1 − α) + (1 − δ) k̂t − k̂t+1
C̄/Γ̄
Γ̄
Γ̄
Γ̄
ψ Q̄B̄/Γ̄+Dc1 AX1t + Dc,k k̂t+1 +
ζt
(16)
σ

25

where
1 + ρg −ρg

A= 1
0


0

0

ρz

and
x̄ =

0

0



(Dc,b − ψσQ̄ )C̄/Γ̄
Q̄ 1 − ψ Q̄ B̄Γ̄

Collecting the terms for ζt from (16), we have
ψ Q̄B̄/Γ̄(1 − x̄) Dc,ζ1 = (1 + ρg )Dc,ζ1 + Dc,ζ2 +
σ !

2
Ȳ
K̄
x̄
B̄
α −
+ ψ Q̄
−
C̄/Γ̄
Γ̄
Γ̄
Γ̄


x̄ K̄
+ Dc,k Dk,ζ1+
C̄/Γ̄ Γ̄
which leads to

2 !
Ȳ
x̄
K̄
B̄
α −
=
+ ψ Q̄
C̄/Γ̄
Γ̄
Γ̄
Γ̄
 


K̄
ψ Q̄B̄/Γ̄x̄
+ Dc,k Dk,ζ1 −−σC̄/Γ̄ Γ̄(ρg + x̄)Dc,ζ1 + Dc,ζ2(17)

Similarly, collecting the terms for ζt−1 from (16), we have
(1 − x̄) Dc,ζ2



x̄
Ȳ
K̄
B̄ C̄
B̄
= −ρg Dc,ζ1 +α −+ Q̄ − −
C̄/Γ̄
Γ̄
Γ̄
Γ̄
Γ̄
Γ̄

 

x̄
K̄
+ Dc,k Dk,ζ2+
C̄/Γ̄ Γ̄

which leads toρgDc,ζ1 + Dc,ζ2 =1 − x̄




x̄
1
Ȳ
K̄
B̄ C̄
B̄
α −
+ Q̄ − −1 − x̄ C̄/Γ̄
Γ̄
Γ̄
Γ̄
Γ̄
Γ̄

 

1
x̄
K̄
+
+ Dc,k Dk,ζ2
1 − x̄ C̄/Γ̄ Γ̄

(18)

Substituting Dc,ζ2 from (18) into (17) and using the following steady state relations:

C̄
X̄
B̄
Ȳ
B̄
+
+
=
+ Q̄
Γ̄
Γ̄
Γ̄
Γ̄
Γ̄

26

we obtain


2 !

1
Ȳ
1 − x̄
K̄B̄
α −=+ ψ Q̄
1 − x̄ − ρg
C̄/Γ̄
Γ̄
Γ̄
Γ̄




Ȳ
1
K̄
X̄
Ȳ
1
α −
+
−
−
1 − x̄ − ρg
C̄/Γ̄
Γ̄
Γ̄
Γ̄
Γ̄




1 − x̄
1
1−(Dk,ζ1 + Dk,ζ2 )1 − x̄ − ρg
C̄/Γ̄
C̄/Γ̄




1
1 − x̄
1
Dc,k (Dk,ζ1 + Dk,ζ2 )−1 − x̄ − ρg
x̄
C̄/Γ̄

 
 

x̄
K̄
1
+ Dc,k Dk,ζ2−
1 − x̄ − ρg
C̄/Γ̄
Γ̄


Dc,ζ1

(19)

Now as ψ goes to zero,Dc,b /Q̄ C̄/Γ̄lim x̄ =ψ→0Then, as Q̄ goes to one, as shown above, we have
lim lim x̄ =Q̄→1 ψ→0

lim lim Dc,b /Q̄Q̄→1 ψ→0
C̄/Γ̄ = 0

In addition, Lemma 2 shows that as Q̄ → 1 and ψ → 0, Dk,ζ1 + Dk,ζ2 = 0.
Therefore (19) implies that
lim lim Dc,ζ1 =Q̄→1 ψ→0

1 − X̄/Ȳ
C̄/Ȳ



1
1 − ρg

Similarly, from (18),
lim lim Dc,ζ2 = −ρg lim lim Dc,ζ1
Q̄→1 ψ→0−ρg
1 − X̄/Ȳ
=
1 − ρgC̄/Ȳ

Q̄→1 ψ→0

Finally, collecting the terms for zt from (16), we have
(1 − x̄) Dc,zx̄= ρz Dc,z −
C̄/Γ̄



Ȳ
Γ̄

27




+

x̄
C̄/Γ̄



K̄
Γ̄




+ Dc,k Dk,zand rearranging the equations, we haveDc,z =

1(1 − x̄ − ρz )


  
 


x̄
Ȳ
x̄
K̄
−
+
+ Dc,k Dk,zC̄/Γ̄ Γ̄C̄/Γ̄ Γ̄As as ψ goes to zero and Q̄ goes to one, we have already shown that x̄ goes tozero and that Dc,k goes to zero such that in the limit Dc,z becomes
lim lim Dc,zQ̄→1 ψ→0

1= lim lim
Q̄→1 ψ→0 (1 − x̄ − ρz )1×0=0=
1 − ρg


  
 


x̄
Ȳ
x̄
K̄
−
+
+ Dc,k Dk,zC̄/Γ̄ Γ̄C̄/Γ̄ Γ̄This completes the proof.

Lemma 1 Consider the Taylor expansion in (10). As Q̄ → 1 and ψ → 0, α1 , β1 → 0,
∗ < 1.
and 0 < Dk,k∗ in (10), as Q̄ → 1 andProof. First of all, from the equation that determines Dk,k

ψ → 0, this equation becomes∗ 2) −0 = (Dk,k




αδ
∗
+1
+ 2 Dk,kϕȲsince (1 − α) K̄= δ. This equation gives

∗
=
Dk,k

q
(αδ/ϕ + 2) − (αδ/ϕ + 2)2 − 4
2

=

2
q
∈ (0, 1)2(αδ/ϕ + 2) + (αδ/ϕ + 2) − 4

Now, we can use the solution for Dk,b and Dk,k from (10) to obtain the constant
α1 and β1 . We collect the terms for bt from (9):0 = −σ ((Dc,b Db,b ) − Dc,b ) + βδ(1 − α)Dk,b+βϕ(Dk,b Db,b + Dk,k Dk,b ) − (β(δ + ϕ) + ϕ) Dk,b

∗ , we rewrite the last equation
Substituting Db,b from (7) and let D̃k,k = Dk,k −Dk,k

28

as







K̄
C̄
 1 + Dk,b + Dc,b  − Dc,b  + βδ(1 − α)Dk,b
0 = −σ Dc,b  Γ̄Γ̄Q̄ 1 − ψ Q̄ B̄
Γ̄








1
K̄
C̄
∗
 1 + Dk,b + Dc,b  + D̃k,k + Dk,k
Dk,b 
+βϕ Dk,b  B̄Γ̄Γ̄Q̄ 1 − ψ Q̄


1

Γ̄

−(β(δ + ϕ) + ϕ)Dk,b
2 , D 2 , and D̃
Ignoring the second order terms such as Dc,b Dk,b , Dc,bk,k Dk,b in thek,bfirst order approximation, the last equation becomes
0 = −σ Dc,b1
Q̄ 1 − ψ Q̄ B̄Γ̄ − Dc,b  + βδ(1 − α)Dk,b+βϕ Dk,b1Q̄

1 − ψ Q̄ B̄Γ̄∗
 + Dk,kDk,b  − (β(δ + ϕ) + ϕ)Dk,b

Therefore, we have

β(δ + ϕ) + ϕ − βδ(1 − α) − βϕ 

1
Q̄



1 − ψ Q̄ B̄
Γ̄







∗
  − Dk,kβϕ Dk,b = −σ 

1
Q̄



1 − ψ Q̄ B̄Γ̄or equivalently,βδα + ϕ 1 −


β
Q̄



1 − ψ Q̄ B̄
Γ̄





∗ 
  + βϕ(1 − Dk,k) Dk,b = −σ 1Q̄Then, as ψ goes to zero, since β = Q̄, we have∗βδα + βϕ(1 − Dk,k) Dk,b = −σ1
− 1 Dc,bQ̄

Also, Q̄ → 1, the coefficient on Dk,b on the right-hand side is∗δα + ϕ 1 − Dk,k>0
and on the left-hand side
1
−1→0
Q̄

29



1 − ψ Q̄ B̄Γ̄ − 1 Dc,b

 − 1 Dc,b

Therefore, α1 , β1 → 0.

Lemma 2 As Q̄ → 1 and ψ → 0, Dk,ζ1 + Dk,ζ2 = 0.
Proof. Combining the Euler equation with respect to k ′ (8) and the Euler equation
with respect to b′ (11), we have
B̄
ζt
Γ̄

Ȳ 
ρz zt + (1 − α)k̂t+1 + αρ(ζt − ζt−1 )+βϕρg (ζt − ζt−1 ) + β(1 − α)K̄+βϕE[k̂t+2 − k̂t+1 ]Ȳ− β(1 − α) + ϕ k̂t+1 − (σ + ϕ)(ζt − ζt−1 ) + ϕk̂tK̄0 = σ(ζt − ζt−1 ) + ψ Q̄bt+1 − ψ Q̄As ψ goes to zero, this equation simplifies to

Ȳ
Ȳ
0 =
βϕρg + β(1 − α) αρg − ϕ (ζt − ζt−1 ) + β(1 − α) ρz zt
K̄
K̄
h
i
+βϕE k̂t+2 − k̂t+1 + ϕk̂t − (αδ + ϕ)k̂t+1
(20)

We use the conjecture for k̂hi
E k̂t+2 = Dk,b Db Xt + Dk,k Dk Xt + ρz Dk,z zt + ((1 + ρg )Dk,ζ1 + Dk,ζ2 )ζt − ρg Dk,ζ1 ζt−1hi
E k̂t+1 = Dk,b bt + Dk,k k̂t + Dk,z zt + Dk,ζ1 ζt + Dk,ζ2 ζt−1
to collect the terms for ζt from (20):Ȳ
βϕρg + β(1 − α) αρg − ϕ −(αδ+ϕ)Dk,ζ1 +βϕ(Dk,b Db,ζ1 +Dk,k Dk,ζ1 +ρg Dk,ζ1 +Dk,ζ2 ) = 0
K̄(21)

Similarly, collecting the terms for ζt−1 :Ȳ− βϕρg + β(1 − α) αρg − ϕ −(αδ+ϕ)Dk,ζ2 +βϕ(Dk,b Db,ζ2 +Dk,k Dk,ζ2 −ρg Dk,ζ1 −Dk,ζ2 ) = 0
K̄
(22)
Combining (21) and (22), as Q̄ → 1, and consequently β → 1 , we have(αδ + ϕ)(Dk,ζ1 + Dk,ζ2 ) − ϕDk,k (Dk,ζ1 + Dk,ζ2 ) = 0
which leads to(Dk,ζ1 + Dk,ζ2 )(αδ + ϕ(1 − Dk,k )) = 030

∗ < 1, therefore
As Q̄ → 1 and ψ → 0, Dk,k → Dk,kDk,ζ1 + Dk,ζ2 = 0

31

References
Aguiar, M. and G. Gopinath. Log-linearization notes.
http://scholar.harvard.edu/files/gopinath/files/program_notes.pdf.
Aguiar, M. and G. Gopinath (2007). Emerging market business cycles: The cycleis the trend. Journal of Political Economy 115 (1), 69–102.
Akinci, O. (2021). Financial frictions and macro-economic fluctuations in emerging
economies. Journal of Money, Credit and Banking 53 (6), 1267–1312.
Alvarez-Parra, F., L. Brandao-Marques, and M. Toledo (2013). Durable goods,
financial frictions, and business cycles in emerging economies. Journal of Monetary Economics 60 (6), 720–736.
Blanchard, O. and D. Quah (1989). The dynamic effects of aggregate demand andsupply disturbances. American Economic Review 79 (4), 655–673.
Blanchard, O. J. and C. M. Kahn (1980). The solution of linear difference models
under rational expectations. Econometrica 48 (5), 1305–1311.
Blanchard, O. J., J.-P. L’Huillier, and G. Lorenzoni (2013). News, noise, and fluctuations: An empirical exploration. American Economic Review 103 (7), 3045–
3070.
Boz, E., C. Daude, and C. B. Durdu (2011). Emerging market business cycles:
Learning about the trend. Journal of Monetary Economics 58 (6), 616–631.
Campbell, J. and A. Deaton (1989). Why is consumption so smooth? Review of
Economic Studies 56 (3), 357–373.
Chang, R. and A. Fernandez (2013). On the sources of aggregate fluctuations inemerging economies. International Economic Review 54 (4), 1265–1293.
Chari, V., P. J. Kehoe, and E. R. McGrattan (2007). Business cycle accounting.
Econometrica 75 (3), 781–836.
Chen, K. and M. J. Crucini (2016). Trends and cycles in small open economies:
Making the case for a general equilibrium approach. Journal of Economic Dynamics and Control 72, 159–168.
Dogan, A. (2019). Investment specific technology shocks and emerging market
business cycle dynamics. Review of Economic Dynamics 34, 202–220.
Drechsel, T. and S. Tenreyro (2018). Commodity booms and busts in emerging
economies. Journal of International Economies 112, 200–218.
Durdu, C. B. (2013). Emerging market business cycles: Recent advances. International Economic Journal 27 (2), 183–199.
Engel, C. and K. D. West (2005). Exchange rates and fundamentals. Journal of
Political Economy 113 (3), 485–517.

32

Galı́, J. (1991). Budget constraints and time-series evidence on consumption. American Economic Review 81 (5), 1238–1253.
Gali, J. (1999). Technology, employment, and the business cycle: Do technology
shocks explain aggregate fluctuations? American Economic Review 89 (1), 246–
271.
Garcia-Cicco, J., R. Pancrazi, and M. Uribe (2010). Real business cycles in emerging countries? American Economic Review 100 (5), 2510–2531.
Hevia, C. (2014). Emerging market fluctuations: What makes the difference? Journal of International Economics 94 (1), 33–49.
Jaimovich, N. and S. Rebelo (2009). Can news about the future drive the businesscycle? American Economic Review 99 (4), 1097–1118.
Mendoza, E. G. (1991). Real business cycles in a small open economy. American
Economic Review 81 (4), 797–818.
Miyamoto, W. and T. L. Nguyen (2017). Business cycles in small open economies:
evidence from panel data between 1900 and 2013. International Economic Review 58 (3), 1007–1044.
Naoussi, C. F. and F. Tripier (2013). Trend shocks and economic development.
Journal of Development Economics 103, 29–42.
Neumeyer, P. A. and F. Perri (2004). Business cycles in emerging economies: therole of interest rates. Journal of Monetary Economics 52 (2), 345–380.
Rothert, J. (2020). International business cycles in emerging markets. International
Economic Review 61 (2), 753–781.
Schmitt-Grohe, S. and M. Uribe (2003). Closing small open economy models. Journal of International Economics 61, 163–185.
Seoane, H. D. (2016). Parameter drifts, misspecification and the real exchange ratein emerging countries. Journal of International Economics 98, 204–215.
Uhlig, H. (1999). A toolkit for analysing nonlinear dynamic stochastic models
easily. In R. Marimon and A. Scott (Eds.), Computational Methods for the
Study of Dynamic Economies, pp. 30–61. Oxford: Oxford University Press.
Uribe, M. and V. Z. Yue (2006). Country spreads and emerging countries: Who
drives whom? Journal of International Economics 69 (1), 6–36.

33