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A standard SOE-RBC model Closing standard models Eleni Iliopulos PSE, University of Paris 1, CEPREMAP Lecture 4 E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 1 / 32
A standard SOE-RBC modelClosing standard models
Eleni Iliopulos
PSE, University of Paris 1, CEPREMAP
Lecture 4
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 1 / 32
Aim of this lecture
Introduce a standard framework for studying transmissionmechansisms in a SOE.
Settle a standard framework for models in a SOE.
Discuss the role of external debt accumulation.
Discuss issues of indeterminacy.
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 2 / 32
References
Schmitt-Grohé, S. and M. Uribe (2003), "Closing small open economymodels", Journal of International Economics, 61, 163-185.Additional:
General RBC: Romer, Advanced Macroeconomics; M. Wickens(2008), Macroeconomic Theory.
General (but technical) on complete markets and risksharing:Ljungquvist and Sargent (2004), Recursive MacroeconomicTheory.
Lecture notes of Martin Uribe Open Economy Macroeconomics:http://www.columbia.edu/~mu2166/GIM/lecture_notes.pdf
My lecture notes on complete markets in open economy (see my webpage)
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 3 / 32
RBC models
to match uctuations in aggregate output and employment (considerstochastic real shocks)
to match CA counter-cycles.
stemming from standad Walrasian models.
distinction between developed and emerging economies.
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 4 / 32
Emerging vs developed countries.
Aguiar and Gopinath(2004)
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 5 / 32
The standard model
stochastic setting
complete markets: state contingent Arrow-Debreu assets.
capital and adjustment costs
productivity shocks
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 6 / 32
Complete markets
L = U (C (s)) + β ∑s 0js
π (s 0 j s)U (C (s 0 j s))
�λ (s)"∑s 0jsQ (s 0 j s)B (s 0) + C (s)� Y (s)
#�β ∑
s 0jsλ (s 0 j s) [C (s 0 j s)� B (s 0)� Y (s 0 j s)]
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 7 / 32
Complete markets: FOCs
C : Uc (s) = λ (s)C 0 : βπ (s 0 j s)Uc (s 0 j s) = βλ (s 0 j s)
B (s 0) : �λ (s)Q (s 0 j s) = �λ (s 0 j s) βthus:
Q (s 0 j s) = β π (s 0 j s)Uc (s 0 j s)Uc (s)
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 8 / 32
RBC complete markets
νt+1,t � ν (st+1 j st ) is the period-t price of a claim to one unit ofdomestic unit of account currency in state st+1 divided by theprobability of occurrence of that state.
Each asset in the portfolio Bt+1 pays one unit of domestic currency attime t + 1 and in state st+1.
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 9 / 32
RBC complete markets
Utility
E0∞
∑t=0
�βtU (Ct ,Nt )
BC:
∑st+1
νt+1,tBt+1 = Bt + Yt � Ct � It �Φ (kt+1 � kt )
No-Ponzi game condition:
limj!∞
Et fνt+j ,tBt+jg � 0
where Et f.g � ∑st+1 π�st+1 j st
�
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 10 / 32
Production
Yt = Atkαt N1�αt
kt+1 = kt (1� δ) + It
lnAt+1 = ρ lnAt + et+1et : iid N(0, σ2).
rt = r �
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 11 / 32
FOCs
L = Et∞
∑t=0
βt fU (Ct ,Nt )
�λt
"∑st+1
νt+1,tBt+1 � Bt � Yt + Ct
+kt+1 � kt (1� δ) +Φ (kt+1 � kt )]g
Ct : Uc ,t = λt (1)
Nt : UN ,t = �λtAtFN ,t (2)Bt+1 : λtνt+1,t = βλt+1 (3)
Euler eq. is holding at all contingencies!
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 12 / 32
kt+1 : (4)βEtλt+1
�At+1Fk (kt+1,Nt+1) + (1� δ) +Φ0 (kt+2 � kt+1)
�= λt
�1+Φ0 (kt+1 � kt )
��λ�t νt+1,t + βλ�t+1 = 0
Uc ,t+1Uc ,t
=U�c ,t+1U�c ,t
iterate backward and obtain risk sharing condition:
Uc ,t = ξU�c ,t (5)
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 13 / 32
Structure of assets allows to price one-period riskless bonds!!Arbitrage condition
Rt = Et fνt+1,tg�1
Obtain Euler:
Rt =�
βEt
�Uc ,t+1Uc ,t
���1
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 14 / 32
Incomplete markets
Interest rates are risk-free returns on securities.
Whats the steady state? Is it unique?
Indeterminacy:
of the steady stateof dynamics
Stationarity inducing methods: SGU (2003); Cole and Obstfeld,Corsetti and Pesenti; Ghironi.
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 15 / 32
Endogenous discount factor
E0∞
∑t=0
θtU (Ct ,Nt )
θ0 = 1 (6)
θt+1 = β (Ct ,Nt ) θt (7)
t � 0; βc < 0; βN > 0
Dt = Dt�1 (1+ rt�1) + Ct + It � Yt +Φ (kt+1 � kt )
limDt+j
Πjs=0 (1+ rs )� 0
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 16 / 32
FOCs
L = Et∞
∑t=0
θt fU (Ct ,Nt )
�λt (�Dt +Dt�1 (1+ rt�1) + Ct + It � Yt +Φ (kt+1 � kt ))g
�Et∞
∑t=0
ηt (β (Ct ,Nt ) θt � θt+1)
Ct : λt = Uc � ηtβc (8)Dt : λt = β (Ct ,Nt )Etλt+1 (1+ rt ) (9)Nt : UN ,t = ηtβN � λtAtFN ,t (10)
θt+1 : ηt = �EtU (Ct+1,Nt+1) + Etηt+1β (Ct+1,Nt+1) (11)kt+1 : 0 = β (Ct ,Nt )Etλt+1 [At+1Fk (kt+1,Nt+1) (12)
+ (1� δ) +Φ0 (kt+2 � kt+1)�� λt
�1+Φ0 (kt+1 � kt )
�E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 17 / 32
Steady state
Let β (Ct ,Nt ) =�1+ Ct � N
ωt
ω
��ψ1Capital, output and labor are uniquely pinned down and independentfrom external debt.
Consumption and debt are pinned down thanks to ψ1
More functional forms: U (Ct ,Nt ) =
�Ct�
Nωtω
�1�γ�1
1�γ ; Yt = Atkαt N
1�αt ;
Φ = φ2 (kt+1 � kt )
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 18 / 32
Steady state
Try to pin down the steady state. Hints:
1 Use (8)+(10) to nd: (Uc � ηtβc )AtFN ,t = ηtβN � UN ,t !AtFN ,t = Nω�1t
2 Use (12) in ss to obtain: [Fk + (1� δ)] = 1β !kN =
�αr+δ
�1/(1�α) )3 1+2 ! N =
h(1� α)
�αr+δ
�α/(1�α)i1/(ω�1) Labor depends onparameters only.
4 !SS capital labor ratio:h(1� α)
� kN
�αi1/(ω�1)= N + SS output
from production function
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 19 / 32
Steady state
Trade balance: From (9): 1 = β (Ct ,Nt ) (1+ rt )
Moreover: β (Ct ,Nt ) =�1+ C � Nωω
��ψ1and: C = Y � TB � I
tb = 1� IY �h
1(1+r )
i�1/ψ1+ N
ω
ω �1Y is uniquely pinned down. It depends
on the value of ψ1.
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 20 / 32
Models performance
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 21 / 32
Calibration
SGU, (2003)
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 22 / 32
IRFsResponse to a 1% productivity shock
SGU, (2003)
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 23 / 32
The persistence of the productivity shock and the adj costs play afundamental role in triggering TB conter-cycles.
SGU, (2003)
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 24 / 32
External discount factor (EDF)
Same as before but the discount factor funcion changes in the followingway:
θt+1 = β (c̄t , n̄t ) θtt � 0
Ct : Uc ,t = λt
externalities are not internalized.
Dt : λt = β (c̄t , n̄t )Etλt+1 (1+ rt )
Nt : UN ,t = �λtAtFN ,t
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 25 / 32
External discount factor (EDF)
kt+1 : β (c̄t , n̄t )Etλt+1�At+1Fk (kt+1,Nt+1) + (1� δ) +Φ0 (kt+2 � kt+1)
�= λt
�1+Φ0 (kt+1 � kt )
�Notice that in equilibrium:
Ct = c̄tNt = n̄t
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 26 / 32
Debt-elastic interest rate
E0∞
∑t=0
βtU (Ct ,Nt )
BC does not change:
Dt = Dt�1 (1+ rt�1) + Ct + It � Yt +Φ (kt+1 � kt )
the discount rate is exogenous and equal to β.Stationarity is insured by thefollowing assumption:
rt = r � + p (d̄t ) (13)
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 27 / 32
Functional forms
p (d̄t ) = ψ2�edt�d̄ � 1
�Calibration:
SGU, (2003)
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 28 / 32
Steady state
r = r � + p (d̄)
from Euler eq:
Dt : λt = βλt+1 (1+ rt )!
β =1
1+ r=
11+ r �
!
where substituting for:r = r � + ψ2�ed�d̄ � 1
�, we obtain:
1 = β�1+ r � + ψ2
�ed�d̄ � 1
��thus, in ss, d = d̄ !!, debt is pinned down uniquely by the parameter d̄!!
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 29 / 32
Portfolio adjustment costs
Same utility function:
E0∞
∑t=0
βtU (Ct ,Nt )
no interest rate premia, discount rate is exogenous, β.Stationarity is insured by portfolio adjustment costs: ψ32 (Dt � D̄)
2
Dt = Dt�1 (1+ rt�1) + Ct + It � Yt +Φ (kt+1 � kt ) +ψ32(Dt � D̄)2
In, the Euler equation becomes:
[1� ψ3 (D � D̄)] = β (1+ r)
Moreover:β = 11+r ! D = D̄ !!, debt is pinned down uniquely by theparameter D̄.
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 30 / 32
Performance
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 31 / 32
To resume:
Small open economy models generally present unit roots !debt andconsumption are not pinned down uniquely.Stationarity inducing methods: allow to pin down steady-statedebt/consumption
Complete markets allow to pin down consumption through risksharing (eq 5)! you can then pin down uniquely the external debt.Endogenous discount factor. They allow to pin dow the tradebalance, thus consumption, and thus debt (see above).
EDF endogenous discount factor allow to pin down consumptionand labor (which are equal to the population average)! pin downuniquely debt.
Debt elastic interest rate. It allows to pin down uniquely debt (viaeq 13 in ss)
Portofolio adjustment costs. Allow to pin down uniquely debt (viaadj. costs).
E. ILIOPULOS (PSE, University of Paris 1, CEPREMAP) RBC, SoE Lecture 4 32 / 32
FOCs
