Optimal Policy with Long Bonds Albert Marcet London School of Economics joint with Elisa Faraglia Institut d’An alisi Econ omica CSIC and London Business School Andrew Scott London Business School
Optimal Policy with Long Bonds
Albert Marcet
London School of Economics
joint with
Elisa Faraglia
Institut d'An�alisi Econ�omica CSIC and London Business School
Andrew Scott
London Business School
What we want to do
Formulate a model to study optimal �scal policy and government's optimal
portfolio choice jointly
What maturities?, nominal or indexed?
Fiscal Policy and Debt Management jointly determined:
Why we want to do it
Choice of long/short bonds often ignored in models of policy analysis
Very relevant in ... crisis.
Markets seem to appreciate a certain type of portfolio
What should be the aim of Debt Management?
minimize cost or �scal insurance?
It seemed easy at �rst ...
Just combine
� optimal policy with incomplete markets
� portfolio choice with heterogeneous agents
But the computer told us not so easy
Also, some in the profession thought there was no need to go away from
complete markets.
In this (long run) project
1. Look at data on portfolio of gov't debt
2. We �rst reexamine e�ectively complete markets approach
(a) justify using incomplete markets
(b) learn some properties of the model in an easier model
3. Set up a model with long bonds
4. Study optimal debt management under incomplete markets
Standard basic framework
Rational Expectations
Full Commitment
Benevolent government
Full Information
Flexible prices
Gov't knows mapping from �scal policy to equilibrium quantities
Debt Management with E�ectively Complete Mar-kets
Angeletos (2002) QJE
Buera and Nicolini (2004) JME
Assume government issues uncontingent debt at various maturities.
Same number of possible realizations of underlying shocks as number of
maturities.
So gov't can e�ectively complete the markets.
Their results:
� Govt't issue debt at long maturities (bMt > 0)
� Gov't save on private debt at short maturities (b1t < 0)
Barro (2004), Nosbusch (2008) extend these results.
Farhi (2007) argues gov't should hold private equity.
Problems with e�ectively completing markets
Buera and Nicolini (2004)
bMt ' 6 �GDP
b1t ' �5 �GDP
Figure 1: Money Market Instruments (% of Debt) 1993-2003
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Aus
tralia
Bel
gium
Can
ada
Den
mar
k
Finl
and
Fran
ce
Ger
man
y
Italy
Japa
n
Mex
ico
Net
herla
nds
New
Zea
land
Spa
in US
MaxMinAverage
Figure 2: Short Term Debt (% of Debt) 1993-2003
0
0.05
0.1
0.15
0.2
0.25
0.3
Aus
tral
ia
Bel
gium
Can
ada
Den
mar
k
Fin
land
Fran
ce
Ger
man
y
Italy
Japa
n
Mex
ico
Net
herla
nds
New
Zea
land
Spa
in
US
MaxMinAverage
Figure 3: Medium Term Debt (% of Debt) 1993-2003
0
0 .0 5
0 .1
0 .1 5
0 .2
0 .2 5
0 .3
0 .3 5
0 .4
0 .4 5
0 .5
Au
str
alia
Be
lgiu
m
Ca
na
da
De
nm
ark
Fin
lan
d
Fra
nc
e
Ge
rma
ny
Ita
ly
Ja
pa
n
Me
xic
o
Ne
the
rla
nd
s
Ne
w Z
ea
lan
d
Sp
ain
US
M a xM inA ve ra g e
Figure 4: Long Term Debt (% of Debt) 1993-2003
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Au
str
alia
Be
lgiu
m
Ca
na
da
De
nm
ark
Fin
lan
d
Fra
nc
e
Ge
rma
ny
Ita
ly
Ja
pa
n
Me
xic
o
Ne
the
rla
nd
s
Ne
w Z
ea
lan
d
Sp
ain
US
M axM inA verage
Figure 5: Average OECD: Debt Composition 1993-2003
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Money m
arket
Short term
Medium
Term
Long Term
Indexed
Variable
Other
NonM
arketab;eP
ort
foli
o S
har
e
Data
Data shows U-shaped portfolios
Positive holdings at most maturities.
Large failure to match data
This could be because
� micro fundamentals (utility, technology) are wrong
� �nancial frictions play a role
� some other failure of the model (gov't is not benevolent, or lazy...)
Faraglia, Marcet and Scott (2009) "In Search of a Theory of Debt Man-
agement", working paper
consider more general technology and utility
� Add capital
� habits
� no buyback of gov't bonds
Outcome: it gets worse all the time
� even larger positions
� no longer long bonds should be issued
� huge changes in sign from one period to next
Figure 2: Policy Functions for Debt Issuance - Capital Accumulation and Persistent
Technology Shocks
b_1
-3000
-2000
-1000
0
1000
2000
3000
4000
5000
6000
7000
8000
1115 1165 1215 1265 1315 1365 1415
k
b1_L b1_H
b_16
-8000
-7000
-6000
-5000
-4000
-3000
-2000
-1000
0
1000
2000
3000
1115 1165 1215 1265 1315 1365 1415
k
b_16_L b_16_H
44
Reason: for any properly calibrated model yield curve does not move too
much
Di�cult to implement complete markets with bonds.
ONE PROBLEM DETECTED: very non-linear laws of motion
Debt Management under INcomplete Markets
The Model
Technology
ct + gt � 1� xt
xt leisure
ct private consumption
gt government expenditure shock
Consumer
Preferences
E0
1Xt=0
�t [u (ct) + v (xt)]
Government
Levies taxes at a tax rate � t:
Chooses taxes.
Issues uncontingent (real) bonds at several maturities.
Ramsey Equilibrium
A Peek at Debt Management
Government issues bonds of maturity 1 and of maturity M
bMt government bonds, pay one unit of consumption at t+M .
pMt competitive price of long bonds
b1t short bonds
p1t price short bonds
primary de�cit dt = gt � � twt (1� xt)
Gov't budget constraint,
p1t b1t + p
Mt bMt = dt + p
M�1t bMt�1 + b
1t�1
Uncertainty in only one period.
Consider special case where
eg = g0 = g2 = g3 = :::
g1 random
Add transaction costs
p1t b1t + p
Mt bMt + TC(b1t ; b
Mt ) = dt + p
M�1t bMt�1 + b
1t�1
3 bonds, mat 1 5 30
� b1
yb5
yb30
ytot TCy Welf loss (rel to no TC)
0 -0.9455 -9.3157 10.2612 0 0.10�7 -0.8833 0.6233 0.2599 10�6 5�10�810�6 -0.1763 0.1243 0.0519 10�7 10�7
10�5 -0.0196 0.0138 0.0057 10�8 10�7
ANOTHER PROBLEM DETECTED: Still huge and sensitive positions.
With very small transaction costs reasonable positions.
So frictions a big part of the story if we are to get reasonable positions
Simplify the model for now and look only at long bonds
Long Bonds
Government issues bonds of maturity M:
pMt bMt = dt + pM�1t bMt�1
Aiyagari, Marcet, Sargent and Seppala (2002), assumed M = 1.
Useful because
� We can address some technical di�culties that will also be present indebt management
� of interest to compare with M = 1
� We can gain some intuition about the role of commitment in optimaltaxes with incomplete markets
Two technical di�culties:
� Recursive formulation non-standard
Apply recursive Contracts as in Marcet and Marimon.
� Many state variables.
Design a technique to reduce dimension of state space.
A recursive formulation
pMt bMt = dt + pM�1t bMt�1
Plugging equilibrium prices
�MEtu
0(ct+M)u0(ct)
bMt = dt + �M�1Etu
0(ct+M�1)u0(ct)
bMt�1
�t lagrange multiplier of this constraint.
Lagrangean
L = E0
1Xt=0
�tfu (ct) + v (xt)
+�th�MEtu
0(ct+M) bMt � dtu0(ct)� �M�1Etu0(ct+M�1) b
Mt�1
ig
L = E0
1Xt=0
�tfu (ct) + v (xt)
+�th�MEtu
0(ct+M) bMt � dtu0(ct)� �M�1Etu0(ct+M�1) b
Mt�1
ig
L = E0
1Xt=0
�tfu (ct) + v (xt)
+�th�Mu0(ct+M) b
Mt � dtu0(ct)� �M�1u0(ct+M�1) b
Mt�1
ig
L = E0
1Xt=0
�tfu (ct) + v (xt)
+�th�Mu0(ct+M) b
Mt � dtu0(ct)� �M�1u0(ct+M�1) b
Mt�1
ig
L = E0
1Xt=0
�tfu (ct) + v (xt)� �tdtu0(ct)
+�t�Mu0(ct) bMt�M � �t�M+1u
0(ct) bMt�Mg
L = E0
1Xt=0
�tfu (ct) + v (xt)� �tdtu0(ct)
+�t�Mu0(ct) bMt�M � �t�M+1u
0(ct) bMt�Mg
L = E0
1Xt=0
�tfu (ct) + v (xt)� �tdtu0(ct)
+��t�M � �t�M+1
�bMt�M u0(ct)g
for
��1 = ::: = ��M = 0
L = E0
1Xt=0
�tfu (ct) + v (xt)� �tdtu0(ct)
+st�M u0(ct)g
For
st = (�t�1 � �t) bMt�1
As long as
s�1 = ::: = s�M = 0
So, optimal choice26664� tbMt�tct
37775 = F (gt; st�1; :::; st�M ; bMt�1)
s�1 = ::: = s�M = 0; given bM�1
So M + 2 state variables
In Aiyagari et al. (2002) case M = 1 state variables are
(gt; �t; �t�1; b1t�1)
Promised utility approach (APS):
For M = 1 APS can be used as follows:
assume g can take two values gH ; gL
Budget constraint of gov't
dt + b1t�1 = �
Et(u0(ct+1))u0(ct)
b1t
dt + b1t�1 = �
�u0(cHt+1) + (1� �)u0(cLt+1)u0(ct)
b1t
Su�cient state variables decided at t are
cHt+1; cLt+1
These variables are decided at t
Realized shock is a state at t+ 1:
With one-period bonds this means ct is a state variable at t:2664b1tcHt+1cLt+1
3775 = F (gt; ct; b1t�1)Same number of state variables as lagrangean approach for M = 1.
Standard (Big) Problem with APS:
In fact these are sets of feasibla future consumptions CH(b1t ); CL(b1t ):
Then impose additional constraints
cHt+1 2 CH(b1t ); cLt+1 2 CL(b1t )
Very di�cult.
No such problem with Lagrangean approach
continuation problem always well de�ned
Recall for M = 1 recursive formulation with lagrangean approach264 btct�t
375 = F (gt; �t�1; bt�1)
All we need is to impose
�t�1 2 R+
The "continuation problem" is always well de�ned
Recall
L = E0
1Xt=0
�tfu (ct) + v (xt)� �tdtu0(ct)
+(�t�1 � �t)u0(ct) b1t�1g
In new version of Marcet Marimon (2008) we show full commitment amounts
to re-solving at t
maxnb1t+j;ct+j
oEt 1Xj=0
�jhu�ct+j
�+ v
�xt+j
�i+ �t�1u
0(ct) b1t�1s.t. CE constraints
given optimal �t�1:
So continuation problem changes objective function
Well de�ned for all �t�1:
APS unfeasible for large M
dt + bMt�M = �M
Et(u0(ct+M))u0(ct)
bMt
dt + bMt�1 = �
Peg2M u0(ct+M(gt; eg)) P (gt+M = (gt; eg))u0(ct)
bMt
Even if gt 2 (gH ; gL) there are 2M + 3 state variables!.
Recall with Lagragean approach only M + 3 state variables.
Second Technical Di�culty
M + 3 state variables, still many variables
In many economic models, many state variables are nearly redundant
Try to introduce only "relevant" combinations of state variables
Related to Smolyak polynomials
Related to Sims' reduction of state space.
We could start with few variables,
then add variables one by one,
and claim victory when one new variable makes little di�erence
But this could easily overlook that globally the remaining variables may be
relevant.
We try to give the best chance to all remaining variables when we re�ne
the solution.
Recall state variables are
(gt; �t; st�1; :::; st�M�1; bMt�1) � Xt
Split state variables in "core" variables and "remaining" variables.
Xcoret � XtXoutt remaining variables in Xt
Replace Xoutt by linear combinations
(�1 �Xoutt ; :::; �N �Xoutt )
where each �j 2 RM+3 is chosen so as to maximize "relevance" of this
linear combination relative to previous solution.
More precisely.
Step 1. Choose Xcoret � Xt
Find approximate solution with only Xcoret :
In our case we set Xcoret � (gt; �t; st�1; bMt�1)
Step 2. Solve the model with a time-invariant function of Xcoret
In our case we use PEA
Approximate
Etnu0�ct+M
�o= � �Xcoret :
Converge on �: Call it �1:
Step 3. Add linear combinations of Xoutt :
Find �1 by �tting the Euler equation residual on all remaining state
variables
In our case
Run a regression of Xoutt on the core variables:
Xoutt = B1 �Xcoret
Find the residuals
Xres;1t = Xoutt �B1 �Xcoret :
Given solution for c;X found with core variables, �nd �1 such that
�1 = argmin�
TXt=1
�u0 (ct+1)� �1 �Xcoret � � �Xres;1t
�2
If
�1 �Xcoret + �1 �Xres;1t�= �1 �Xcoret
for every t and every realization stop here.
Solve the model using as state variables (Xcoret ; �1 �Xres;1t )
Notice, new �xed point problem is well conditioned, initial condition (�1; 1)
If the solution changes very little, stop here.
If not, �nd Xres;2t and so on.
In our problem we only need one linear combination and it changes very,
very little the solution.
Comments:
� We can also use this to add higher-order terms in non-linear approxi-mation
� Do not revise past B or � at each iteration.
Role of commitment �scal policy under incompletemarkets
First look at certainty, Lucas and Stokey (M = 1)
1Xt=0
�tu0(ct)u0(c0)
dt = �b1�11Xt=0
�tu0(ct)dt = �b1�1 u0(c0)
Then optimal �scal policy is to lower interest rates:
ct < c0
� t > �0
With a long bond
1Xt=0
�tu0(ct)u0(c0)
dt = �bM�1 pM�1t
1Xt=0
�tu0(ct)u0(c0)
dt = �bM�1 �M�1u0(cM�1)u0(c0)
1Xt=0
�tu0(ct)dt = �bM�1 �M�1 u0(cM�1)
Here the government sets
ct < cM�1� t > �M�1
so as to lower interest rates
Uncertainty in only one period.
If g1 high
� increases taxes at �1
� commit to lowering taxes in M periods to reduce today's higher debt
burden
Time Inconsistency: at t = M + 1 government would prefer to smooth
taxes.
In Aiyagari et al. both e�ects take e�ect simultaneously, less apparent
long bondtax
0,2800
0,2820
0,2840
0,2860
0,2880
0,2900
0,2920
0,2940
0,2960
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Uncertainty in all periods. ONE long bond.
The algorithm for state space reduction works very well.
Only one linear combination needs to be added.
long bondtax t>=2
0,28720,28720,28720,28720,28720,28720,28720,28720,2872
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
long bondb and MV
0,0000
1,0000
2,0000
3,0000
4,0000
5,0000
6,0000
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
b MV
long bonddef
1,0000
0,0000
1,0000
2,0000
3,0000
4,0000
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
long bonddef: t>=2
0,0797
0,0797
0,0797
0,0797
0,0797
0,0797
0,07972 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
With only one bond
� Debt is used as a bu�er stock, as in Aiyagari et al. (2002)
� gov't promises to lower interest rates if debt becomes high (spike)
� payo� of long bond closer to complete markets.
� reasonable positions.
, ��� $�� � ��� �; @�� �� /���� ������� #������� 6 * 6 ���� ������
��
Optimal Model w ith Buy Back 1 and 10 period bond
CONSUMPTION
-0.0300
-0.0250
-0.0200
-0.0150
-0.0100
-0.0050
0.00001 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
c_mat1_gH c_mat10_gH
Optimal Model w ith Buy Back 1 and 10 period bond
TAXES
0.0000
0.0050
0.0100
0.0150
0.0200
0.0250
0.0300
0.0350
0.0400
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
tax_mat1_gH tax_mat10_gH
Optimal Model w ith Buy Back 1 and 10 period bond
DEFICIT
-0.2000
0.0000
0.2000
0.4000
0.6000
0.8000
1.0000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
def_mat1_gH def_mat10_gH
Optimal Model w ith Buy Back 1 and 10 period bond
BOND PRICE
-0.0140
-0.0120
-0.0100
-0.0080
-0.0060
-0.0040
-0.0020
0.00001 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
pN_mat1_gH pN_mat10_gH
Optimal Model w ith Buy Back 1 and 10 period bond
NUMBER OF NEW BONDS
0.0000
2.0000
4.0000
6.0000
8.0000
10.0000
12.0000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
b_mat1_gH b_mat10_gH
Optimal Model w ith Buy Back 1 and 10 period bond
MARKET VALUE OF TOTAL DEBT
0.00001.00002.00003.00004.00005.00006.00007.00008.00009.0000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
MV_mat1_gH MV_mat10_gH
Optimal Model w ith Buy Back 1 and 10 period bond
LAMDA
0.0000
0.0100
0.0200
0.0300
0.0400
0.0500
0.0600
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
lam_mat1_gH lam_mat10_gH
?
No buyback
pMt bMt = dt + bMt�M
Now if g high, promise lower taxes for next M periods
No spike.
, ��� A� %������� 5���� � �� ��� � ����� /�� /���� ������� #������� 6 *6
���� ������
����� ��� � ����� � ��� �
���� � �����
��
���
��
Optimal Model w ith and w ithout Buy Back 10 period bond CONSUMPTION
-0.0300-0.0250-0.0200-0.0150
-0.0100-0.00500.0000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
c_gH c_nbb_gH
Optimal Model w ith and w ithout Buy Back 10 period bond
TAXES
0.0000
0.0050
0.0100
0.0150
0.0200
0.0250
0.0300
0.0350
0.0400
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
t ax_gH t ax_nbb_gH
Optimal Model with and without Buy Back 10 period bond
DEFICIT
-0.1000
0.0000
0.1000
0.2000
0.3000
0.4000
0.5000
0.6000
0.7000
0.8000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
def _gH def _nbb_gH
Optimal Model with and w ithout Buy Back 10 period bond
BOND PRICE
-0.0140
-0.0120
-0.0100
-0.0080
-0.0060
-0.0040
-0.0020
0.0000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
pN_gH pN_nbb_gH
Optimal Model w ith and w ithout Buy Back 10 period bond
NUMBER OF NEW BONDS
0.0000
2.0000
4.0000
6.0000
8.0000
10.0000
12.0000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
b_gH b_nbb_gH
Optimal Model with and without Buy Back 10 period bond
MARKET VALUE OF DEBT
0.0000
2.0000
4.0000
6.0000
8.0000
10.0000
12.0000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
MV_gH MV_nbb_gH
Optimal Model with and w ithout Buy Back 10 period bond
LAMDA
0.0000
0.0100
0.0200
0.0300
0.0400
0.0500
0.0600
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
lam_gH lam_nbb_gH
Optimal Model w ith and without Buy Back 10 period bond
LEISURE
-0.0070
-0.0060
-0.0050
-0.0040
-0.0030
-0.0020
-0.0010
0.0000
0.0010
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
x_gH x_nbb_gH
Op t imal M o d el wit h and wit ho ut Buy Back 10 p er io d b o nd
OUT PU T
-0.0005
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
y_gH y_nbb_gH
4>
For the future
Finish this paper
Model with two bonds, add "relevant" frictions
1. potentially large welfare gain from using debt optimally under incom-
plete markets.
2. Add private default risk.
3. Add rollover risk