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