Open economy business cycle models
Marcin Kolasa
Warsaw School of Economics
M. Kolasa (SGH)
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International linkages
Closed economy assumption inappropriate for most of countries
Important international linkages:
Trade in goods and services
Financial flows (cross-border lending)
Factor flows (migration, foreign direct investment)
(Common monetary policy)
Transmission channels:
Direct effect of foreign variables
Exchange rate
Terms of trade
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Modelling open economies
1
Multi-country models
All coutries / regions modelled explicitly
Relative size of countries in the parameter set
2
Small open economy models
Size of the modelled economy in the world implicitly set to zero
Rest of the world treated as exogenous, and usually modelled as a VAR
3
Hybrids
e.g. 2 countries and rest of the world treated as exogenous
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IRBC model - Main assumptions
Essentially: 2 country version of standard RBC model (Home and
Foreign)
Fully flexible prices
Homogeneous good -> law of one price (LOOP) holds
No frictions in trade and financial flows
Decentralized and social planner’s allocations coincide
Notation:
Variables with / without asterisk (*) indicate variables or parameters
related to Home / Foreign
World population normalized to 1, relative size of Home ω
All variables expressed per given country’s population
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Social planner’s problem
Maximize weighted utility in two countries
#
"
( "
#)
1+ϕ
∗1+ϕ∗
∞
1−θ
∗1−θ∗
X
ct+i
lt+i
ct+i
lt+i
t+i
Et
β
ω
−
+ (1 − ω)
−
1−θ 1+ϕ
1 − θ∗
1 + ϕ∗
i=0
subject to resource constraint
ωyt + (1 − ω)yt∗ = ω(ct + it ) + (1 − ω)(ct∗ + it∗ )
production functions
α
yt = zt kt−1
lt1−α
∗
∗α ∗1−α
yt∗ = zt∗ kt−1
lt
∗
and capital accumulation rules
kt = (1 − δ)kt−1 + it
∗
kt∗ = (1 − δ ∗ )kt−1
+ it∗
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Lagrangean
Substitute for output and investment from production function and
capital accumulation rules into resource constraint
Social planner’s problem reduces to choosing ct , lt and kt , as well as
their foreign counterparts ct∗ , lt∗ and kt∗
Lagrangean:
#
"
#
1+ϕ
∗1+ϕ∗
1−θ
∗1−θ∗
c
c
l
l
Lt =Et
β t+i ω t+i − t+i
+ β t+i (1 − ω) t+i ∗ − t+i ∗
1−θ 1+ϕ
1−θ
1+ϕ
i=0
"
1−α
α
+ λt+i ω zt+i kt+i−1
lt+i
+ (1 − δ)kt+i−1 − ct+i − kt+i
∞
X
(
+ (1 − ω)
M. Kolasa (SGH)
"
∗1−α∗
∗
∗α∗
zt+i
kt+i−1
lt+i
+ (1 − δ
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∗
∗
)kt+i−1
−
∗
ct+i
−
∗
kt+i
#)
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First order conditions
ct : β t ct−θ = λt
∗
ct∗ : β t ct∗−θ = λt
α l −α
lt : β t ltϕ = (1 − α)λt zt kt−1
t
∗
∗
∗α l ∗−α
lt∗ : β t lt∗ϕ = (1 − α∗ )λt zt∗ kt−1
t
∗
1−α
kt : λt = βαEt {λt+1 (zt+1 ktα−1 lt+1
+ 1 − δ)}
∗ k ∗α
kt∗ : λt = βα∗ Et {λt+1 (zt+1
t
∗ −1
∗
∗1−α
lt+1
+ 1 − δ ∗ )}
Note that additionally we have transversality condition
limt→∞ λt kt = 0
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Perfect risk sharing
First two FOCs imply
ctθ = ct∗θ
∗
This is special case of perfect risk sharing condition (see Chari, Kehoe
and McGrattan, RES 2002)
ctθ
∗ = qt
ct∗θ
(1)
where qt = et Pt∗ /Pt is real exchange rate (and et is nominal
exchange rate); note that in IRBC model LOOP holds (Pt = et Pt∗ )
and hence qt = 1
This condition implies very tight (with LOOP even perfect)
comovement between consumption in Home and Foreign
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Relative labor input
Labor supply choices imply
ltϕ+α
∗ϕ∗ +α∗
lt
=
α
1 − α zt kt−1
∗α∗
1 − α∗ zt∗ kt−1
Hence relative labor input very strongly and positively depends on
relative productivity
Similarly for relative output (since capital is predetermined)
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Capital flows
Investment choices imply
1−α
αEt {λt+1 (zt+1 ktα−1 lt+1
+ 1 − δ)}
∗
= α∗ Et {λt+1 (zt+1
kt∗α
∗ −1
∗
∗1−α
lt+1
+ 1 − δ ∗ )}
α l −α − δ is rate of return on capital in Home and
Note that zt kt−1
t
∗ ∗−α∗
∗
∗α
zt kt−1 lt
− δ ∗ is rate of return on capital in Foreign
Hence, to first-order, in equilibrium we have equalization of rates of
return on capital across countries
This in turn implies very strong adjustments in relative investment
following asymmetric productivity shocks
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Stylized facts
Source: Backus, Kehoe and Kydland (1995)
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IRBC model and stylized facts
Source: Backus, Kehoe and Kydland (1995)
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IRBC and data - summary
Implications of IRBC model:
Tight cross-country comovement in consumption
Negative cross-country correlation in labor, output and investment
Constant real exchange rate
Cyclical position of current account depends on inertia and
cross-correlation of productivity shocks
Data:
Cross-country correlation of output positive and higher that
cross-country correlation of consumption (Backus-Kehoe-Kydland
puzzle)
Real exchange rates are very volatile
Current account balance is countercyclical
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Preference shocks
Suppose households utility in Home is given by
)
(
1+ϕ
∞
1−θ
X
l
c
t+i
t+i
−
Et
β t+i Γt+i
1−θ 1+ϕ
i=0
where Γt is preference shock
Similarly defined preference shock Γ∗t applies also to Foreign
Then perfect risk sharing condition (1) becomes
Γt
ctθ
∗ = qt ∗
Γt
ct∗θ
And relative consumption may fluctuate even if LOOP holds (and
qt = 1)
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Imperfect substitution between home and foreign goods
Final consumption good is made of home and foreign goods according
to CES function
1 φ−1
φ
φ−1 φ−1
1
φ
φ
φ
φ
ct = ωH cH,t + (1 − ωH ) cF ,t
where φ > 0 is elasticity of substitution between consumption goods
produced in Home cH,t and goods imported from Foreign cF ,t
Special cases:
φ → ∞: perfect substitution (as in IRBC model)
φ → 0: no substitution (Leontief aggregator)
φ → 1: Cobb-Douglas preferences (convenient benchmark)
Similarly defined consumption basket for consumption in Foreign
1 φ−1
φ
φ−1 φ−1
1
∗φ ∗ φ
∗
∗ φ ∗ φ
ct = ωH cH,t + (1 − ωH ) cF ,t
Similar aggregators for other (e.g. investment) goods
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Optimal choice of home and foreign goods
Consider for simplicity Cobb-Douglas preferences
ct =
ωH 1−ωH
cH,t
cF ,t
(2)
ωH
ωH
(1 − ωH )1−ωH
Perfectly competitive final goods aggregators maximize profits
Pt ct − PH,t cH,t − PF ,t cF ,t
subject to (2)
First order conditions
PH,t −1
ct
cH,t = ωH
Pt
cF ,t = (1 − ωH )
PF ,t
Pt
−1
ct
where Pt is price of consumption that can be written as (after
plugging FOCs into (2)):
ωH 1−ωH
Pt = PH,t
PF ,t
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Implications for real exchange rate
Decompose real exchange rate
∗ω ∗
qt
∗1−ω ∗
et PH,tH PF ,t H
et Pt∗
=
=
ωH 1−ωH
Pt
PH,t
PF ,t
∗ ∗
ωH −ωH∗ ∗ ωH
et PH,t
et PF∗ ,t 1−ωH
PF ,t
=
PH,t
PH,t
PF ,t
(3)
∗ and
Even when LOOP holds for the same goods, i.e. PH,t = et PH,t
∗
PF ,t = et PF ,t , real exchange rate might not be constant as long as
∗
ωH 6= ωH
∗ - preferences exhibit ’home bias’
Typically we have ωH > ωH
Note: home bias may arise endogenously as a result of transport costs
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Incomplete financial markets
Perfect risk sharing condition (1) is derived from assumption that
international financial markets are complete, i.e. agents have access
to a full set of Arrow-Debreu securities that allow them to insure
against any country-specific risk
Assume instead that households have only access to risk-free bonds so
that their budget constraint in Home is
∗
Bt + et Ft + ... = Rt−1 Bt−1 + Rt−1
et Ft−1 + ...
(4)
where Bt is nominal bond denominated in Home currency, Ft is
nominal bond denominated in Foreign currency, while Rt and Rt∗ are
(gross) nominal risk-free interest rates in Home and Foreign
Similar constraint holds for Foreign agents except that we do not
need them to have access to bonds denominated in Home currency
∗
∗
Ft∗ + ... = Rt−1
Ft−1
+ ...
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Equilibrium bond holdings
Denote Lagrange multipliers on (4) and (5) by Λt and Λ∗t , respectively
First-order conditions derived by solving for optimal bond holdings
Bt : Λt = βEt {Λt+1 Rt }
Ft : Λt et = βEt {Λt+1 et+1 Rt∗ }
Ft∗ : Λ∗t = βEt Λ∗t+1 Rt∗
Note that Λt Pt = ct−θ and Λ∗t Pt∗ = ct∗−θ
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Uncovered interest rate parity (UIP)
Combine FOCs for Bt and Ft :
Et {Λt+1 Rt } = Et
et+1 ∗
Λt+1
R
et t
This is the UIP condition
After log-linearization it becomes
R̂t = R̂t∗ + Et {∆êt+1 }
and states that expected return on both types of bonds should be
equal in equilibrium
Note that the perfect risk sharing condition (1) also implies the UIP
condition (but not the other way round!)
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Stationarity issues
Use FOCs for Ft and Ft∗
∗ Λt+1
Λt+1 et+1
= Et
Et
Λt et
Λ∗t
∗
Or in real terms (using Λt Pt = ct−θ and Λ∗t Pt∗ = ct∗−θ )
)
(
)
(
∗−θ∗
−θ
ct+1
ct+1
qt+1
= Et
Et
∗
ct−θ qt
ct∗−θ
For simplicity, assume LOOP and consider a small open economy.
Then, after log-linearization:
ĉt = Et {ĉt+1 } + ât
where ât is exogenous
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Stationarity issues ctd.
Conclusion: if markets are incomplete, SOE model has a unit root,
i.e. temporary shocks have permanent effects
How to render the model stationary? Several solutions available, see
Schmitt-Grohe and Uribe (JIE 2003)
Most popular - risk premium on foreign bond holdings:
Modify budget constraint (4)
∗
Bt + et Ft + ... = Rt−1 Bt−1 + Rt−1
Υt−1 et Ft−1 + ...
where Υt = Υ(Ft ) such that Υ(0) = 1, Υ0 < 0
If households accumulate more foreign assets (debt), interest they
obtain (pay) decreases (increases)
This drives foreign assets to zero after shocks, bringing stationarity
back to the model
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New Keynesian open economy model: Overview
Usually referred to as New Open Economy Macroeconomics (NOEM)
For simplicity abstract away from capital accumulation
Two countries with symmetric structure
As in the closed economy New Keynesian model: 2 stages of
production to introduce price stickiness
Presentation concentrates on open economy extension
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Households
Households in Home maximise their expected lifetime utility
!
1+ϕ
1−θ
∞
X
ct+j
lt+j
t+j
Ut = Et
β
−ϑ
1−θ
1+ϕ
j=0
subject to budget constraint (Lagrange multiplier λt )
ct +
Rt−1 Bt−1
Bt
= wt lt +
− Tt + Divt
Pt
Pt
Consumption basket defined as before:
ct =
M. Kolasa (SGH)
ωH 1−ωH
cH,t
cF ,t
ωH
ωH
(1 − ωH )1−ωH
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Households
First-order conditions:
ct :
β t ct−θ = λt
lt :
β t ϑltϕ = λt wt
λt+1
λt
= Et Rt
Pt
Pt+1
Bt :
Optimal composition of ct
cH,t = ωH
PH,t
Pt
−1
ct
cF ,t = (1 − ωH )
PF ,t
Pt
−1
ct
Consumer price index
ωH 1−ωH
Pt = PH,t
PF ,t
Similar set of equilibrium conditions for Foreign
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Final goods production
Final-goods firms in Home produce consumption goods according to
the CES production function (Dixit-Stiglitz aggregator):
Z
1
cH,t (i)
cH,t =
1
1+µ
1+µ
di
0
∗
cH,t
Z
1
=
1
∗
cH,t
(i) 1+µ di
1+µ
0
Similarly in Foreign
Z
1
cF ,t =
cF ,t (i)
1
1+µ
1+µ
di
0
cF∗ ,t =
Z
1
1
cF∗ ,t (i) 1+µ di
1+µ
0
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Final goods production - solution
Solving final goods producers’ problem leads to the following
first-order conditions (demand functions for intermediate goods):
cH,t (i) =
∗
cH,t
(i)
=
cF ,t (i) =
M. Kolasa (SGH)
=
∗ (i)
PH,t
−(1+µ)
µ
cH,t
−(1+µ)
µ
Pt∗
cF∗ ,t (i)
PH,t (i)
Pt
PF ,t (i)
Pt
PF∗ ,t (i)
Pt∗
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∗
cH,t
−(1+µ)
µ
cF ,t
−(1+µ)
µ
cF∗ ,t
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Intermediate goods production
Firm i production function:
∗
cH,t (i) + cH,t
(i) = zt lt (i)
cF ,t (i) + cF∗ ,t (i) = zt∗ lt∗ (i)
Prices are set according to the Calvo (1983) mechanism with
probabilities γ for Home firms and γ ∗ for Foreign firms
Two possible pricing schemes:
Producer currency pricing (PCP)
Local currency pricing (LCP)
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Producer currency price setting
Firms set one price for Home and Foreign markets such that LOOP
holds for goods produced by the same firm
∗
PH,t (i) = et PH,t
(i) PF ,t (i) = et PF∗ ,t (i)
new (i) to maximise:
Intermediate-good firm i in Home chooses PH,t
Et
∞
X
(βγ)j
j=0
Λt+j
new
∗
PH,t
(i) − MCt+j cH,t+j (i) + cH,t+j
(i)
Λt
subject to the demand functions from final goods producers.
Note:
Profit maximisation is dynamic: firms take into account that they may
not have a chance to reset their prices in the future
Firms are owned by households, so they discount their future profits by
the discount factor β j Λt+j /Λt , where Λt is Lagrange multiplier on
households’ nominal budget constraint
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Producer currency price setting - ctd.
First-order condition:
Et
X
(βγ)j
j=0
Λt+j
new
∗
PH,t
(i) − (1 + µ) MCt+j cH,t+j (i)+cH,t+j
(i) = 0
Λt
This first-order condition is the same for each firm allowed to reset its
price
Therefore, all firms allowed to reoptimise at time t choose the same
new
price, which we denote by PH,t
Similar problem and its solution can be written for Foreign firms
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PPI inflation
After log-linearization we obtain the Phillips curve for Home
π̂H,t = βEt {π̂H,t+1 } +
(1 − γ)(1 − βγ)
mc
ˆ H,t
γ
ˆ t − P̂H,t = MC
ˆ t − P̂t + P̂t − P̂H,t = mc
where mc
ˆ H,t = MC
ˆ t − p̂H,t
And for Foreign
(1 − γ ∗ )(1 − βγ ∗ ) ∗
π̂F∗ ,t = βEt π̂F∗ ,t+1 +
mc
ˆ F ,t
γ∗
ˆ ∗t − P̂ ∗ = MC
ˆ ∗t − P̂t∗ + P̂t∗ − P̂ ∗ = mc
where mc
ˆ ∗F ,t = MC
ˆ ∗t − p̂F∗ ,t
F ,t
F ,t
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CPI inflation
CPI inflation in Home country is
π̂t = ωH π̂H,t + (1 − ωH )π̂F ,t = ωH π̂H,t + (1 − ωH )(π̂F∗ ,t + ∆êt )
where second equality follows from LOOP holding for the same goods
Note that movements in the nominal exchange rate pass through CPI
immediately and effect can be strong if economy is very open (ωH is
small)
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Local currency price setting
Firms can discriminate prices between Home and Foreign (market
segmentation) so that LOOP does not hold and we have additional
source of real exchange rate volatility (see (3))
Price is set in the currency of destination country.
new (i) and P new ∗ to
Intermediate-good firm i in Home chooses PH,t
H,t
maximise:
Et
∞
X
(βγ)j
j=0
Et
∞
X
j=0
(βγ)j
Λt+j
new
PH,t
(i) − MCt+j cH,t+j (i)
Λt
∗
Λt+j
new ∗
PH,t
(i) et+j − MCt+j cH,t+j
(i)
Λt
subject to demand functions from final goods producers.
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Local currency price setting - ctd.
First-order conditions:
Et
X
(βγ)j
j=0
Et
X
(βγ)j
j=0
Λt+j
new
PH,t
(i) − (1 + µ) MCt+j cH,t+j (i) = 0
Λt
∗
Λt+j
new ∗
PH,t
(i) et+j − (1 + µ) MCt+j cH,t+j
(i) = 0
Λt
This first-order condition is the same for each firm allowed to reset its
price
Therefore, all firms allowed to reoptimise at time t choose the same
new and P new ∗
prices, which we denote by PH,t
H,t
Similar problem and its solution can be written for Foreign firms
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PPI inflation
Now we have 4 Phillips curves
π̂H,t = βEt {π̂H,t+1 } +
(1 − γ)(1 − βγ)
mc
ˆ H,t
γ
∗
(1 − γ)(1 − βγ) ∗
∗
+
π̂H,t
= βEt π̂H,t+1
mc
ˆ H,t
γ
(1 − γ ∗ )(1 − βγ ∗ ) ∗
π̂F∗ ,t = βEt π̂F∗ ,t+1 +
mc
ˆ F ,t
γ∗
π̂F ,t = βEt {π̂F ,t+1 } +
M. Kolasa (SGH)
(1 − γ ∗ )(1 − βγ ∗ )
mc
ˆ F ,t
γ∗
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CPI inflation
CPI inflation in Home country is
π̂t = ωH π̂H,t + (1 − ωH )π̂F ,t
Since both components of CPI are subject to stickiness, pass through
of exchange rate is limited
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