We present some complementary results to Bidian and Bejan (2012). Part 1 provides necessary and sufficient transversality conditions for an agent's optimization problem. They are extensions to stochastic environments of the conditions given by Kocherlakota (1992), or alternatively, extensions to nonzero debt constraints of the corresponding conditions in Forno2003}. Part \ref{ap:B} presents an elementary proof of the characterization of NTT debt limits (Theorem 3.5 in the main paper) for the case when debt constraints bind in bounded time, that requires no martingale techniques or boundedness assumptions on the discounted debt limits. Part \ref{ap:C} complements results in Section 5.1 (in the main paper), showing that all the equilibria that can sustain bubbles under an interdiction to trade can be achieved from fixed, zero initial wealth for the agents. Thus endogeneity of debt limits causes multiplicity of not only asset prices (through bubbles), but also of real equilibrium allocations.rational bubbles, transversality conditions, endogenous debt limits, not-too-tight constraints, self-enforcing debt, limited enforcement

Bejan, Camelia

Bidian, Florin

English

Research Papers in Economics

Supplement to \Martingale properties of
self-enforcing debt"
Florin Bidian  Camelia Bejany
February 12, 2012
Abstract
We present some complementary results to Bidian and Bejan (2012). Part
1 provides necessary and sucient transversality conditions for an agent's opti-
mization problem. They are extensions to stochastic environments of the con-
ditions given by Kocherlakota (1992), or alternatively, extensions to nonzero
debt constraints of the corresponding conditions in Forno and Montrucchio
(2003). Part 2 presents an elementary proof of the characterization of NTT
debt limits (Theorem 3.5 in the main paper) for the case when debt constraints
bind in bounded time, that requires no martingale techniques or boundedness
assumptions on the discounted debt limits. Part 3 complements results in Sec-
tion 5.1 (in the main paper), showing that all the equilibria that can sustain
bubbles under an interdiction to trade can be achieved from xed, zero initial
wealth for the agents. Thus endogeneity of debt limits causes multiplicity of
not only asset prices (through bubbles), but also of real equilibrium allocations.
1 Transversality conditions
We analyze the problem Pt(^ at;;p) of a consumer that faces debt bounds , pricing
kernel p and starts with wealth ^ at (Ft-measurable) at period t (see Section 3 in the
Corresponding author. Robinson College of Business, Georgia State University, PO Box 4036,
Atlanta GA 30302-4036. E-mail: fbidian@gsu.edu
yRice University, MS 22, PO Box 1892, Houston, TX 77251-1892. E-mail: camelia@rice.edu
1main text). Let ( c; a) 2 Ct(^ at;;p) be the optimal consumption (assumed positive)
and asset holdings for the agent. Familiar variational arguments show that ( c; a)
satises the following Kuhn-Tucker necessary conditions, for all s  t:
u
0
s( cs)   u
0
s+1( cs+1)
ps
ps+1
 0; (1.1)

u
0
s( cs)   u
0
s+1( cs+1)
ps
ps+1

( as+1   s+1) = 0: (1.2)
Let  es := es + s   Es
ps+1
ps s+1, for all s  t. Adapting the arguments of Forno and
Montrucchio (2003), we obtain the following necessary transversality condition:1
Lemma 1.1 (Necessary transversality condition). The optimal path ( c; a) satises
lim
t!1Etu
0
s( cs)( as   s) = 0: (1.3)
Proof. Fix an  " > 0 a period s > t. Concavity implies that for any 0 < " <  " and
n  t,
un( cn)   un( cn + "( en    cn)) 
"
 "
(un( cn)   un( cn +  "( en    cn))):
We construct the alternative asset holdings process (an("))
1
n=t where an(") =  an if t 
n  s, and an(") = (1 ") an +"n if n  s+1. It sustains the feasible consumption
process (cn("))
1
n=s dened by cn(") =  cn if t  n < s, cs(") =  cs+Es
ps+1
ps ( as+1 s+1),
and cn(") =  cn + "( en    cn) for n > s. Optimality of  c implies that
0  Et (us( cs)   us(cs("))) + limsup
T!1
Et
T X
n=s+1
(un( cn)   un(cn("))): (1.4)
Notice that
T X
n=s+1
1
"
(un( cn)   un(cn("))) 
1 X
n=s+1
1
 "
(un( cn)   un(cn( ")))
+ ;
1 The proof works for general period utilities ut(), not necessarily of the discounted and bounded
variety assumed in the text, if one uses a weak optimality criterion (Forno and Montrucchio 2003)
and if there exists  " > 0 such that E
P1
s=t (us( cs)   us( cs +  "( es    cs)))
+ < 1:
2and the term
P1
n=s+1
1
 " (un( cn)   un(cn( ")))
+ is integrable, by hypothesis. Fatou's
lemma gives
limsup
T!1
Et
T X
n=s+1
1
"
(un( cn)   un(cn(")))  Et limsup
T!1
T X
n=s+1
1
"
(un( cn)   un(cn(")))
 Et
1 X
n=s+1
1
 "
(un( cn)   un(cn( ")))
+ : (1.5)
Dividing both sides of (1.4) by " and using (1.5),
 Et
1
"
(us( cs)   us(cs(")))  Et
1 X
n=s+1
1
 "
(un( cn)   un(cn( ")))
+ < 1:
By the monotone convergence theorem, when " & 0, the left hand side of the above
equation converges to Etu0
s( cs)
ps+1
ps ( as+1   s+1), which equals Etu0
s+1( cs+1)( as+1  
s+1), due to the Kuhn-Tucker equations (1.1),(1.2). The conclusion follows by
letting s ! 1.
We include for completeness the standard proof of suciency of the Kuhn-Tucker
and transversality conditions for the optimality of a path.
Lemma 1.2 (Sucient transversality condition). If a feasible path ( c; a) 2 Bt(^ at;;p)
satises the Kuhn-Tucker conditions (1.1) and (1.2), then for any other feasible path
(c;a) 2 Bt(^ at;;p) and any bounded stopping time T  t,
Et
T X
s=t
(us(cs)   us( cs))  Etu
0
T+1( cT+1)( aT+1   T+1): (1.6)
Thus a sucient condition for ( c; a) to be optimal for problem Pt(^ at;;p) is
liminf
s!1 Etu
0
s( cs)( as   s) = 0; (1.7)
Proof. Let s+1 := u0
s( cs)   u0
s+1( cs+1)
ps
ps+1. Consider an arbitrary feasible path
3(c;a) 2 Bt(^ at;;p). Using concavity of us() and the budget constraints,
Et
T X
s=t
(us(cs)   us( cs))  Et
T X
s=t
u
0
s( cs)(cs    cs) =
= Et
T X
s=t
u
0
s( cs)

as   s   Es
ps+1
ps
(as+1   s+1)

 
 Et
T X
s=t
u
0
s( cs)

 as   s   Es
ps+1
ps
( as+1   s+1)

:
We analyze separately the last two terms. Using the Kuhn-Tucker conditions
(1.1) and (1.2), which show that s+1  0 for all s  t, it follows that
Et
T X
s=t
u
0
s( cs)

(as   s)   Es
ps+1
ps
(as+1   s+1)

= Et
T X
s=t

u
0
s( cs)(as   s)  

u
0
s+1( cs+1) +
ps+1
ps
s+1

(as+1   s+1)

 u
0
t( ct)(at   t)   Etu
0
T+1( cT+1)(aT+1   T+1)  Etu
0
t( ct)(at   t):
Similarly, using both (1.1) and (1.2),
Et
T X
s=t
u
0
s( cs)

(as   s)   Es
ps+1
ps
(as+1   s+1)

= u
0
t( ct)( at   t)   Etu
0
T+1( cT+1)( aT+1   T+1):
Moreover at =  at since they equal the initial period t wealth of the consumer, ^ at.
Thus
liminf
T!1
Et
T X
s=t
(us(cs)   us( cs))  liminf
T!1
Etu
0
T+1( cT+1)( aT+1   T+1) = 0;
and therefore ( c; a) is optimal for Pt(^ at;t;p).
42 The case when debt limits bind in bounded time
We give an elementary proof of Theorem 3.5 (in the paper) that does not use results
from the theory of martingales, for the case when (t) is bounded, for any t 2 N.
Assumption 3.1 and Proposition 3.4 are not needed in this case. As in the paper,
continuation utilities after default are the same under the two debt limits ;  . We
assume that for any period t, there exists a natural number n(t) such that the bounds
 bind before period n(t) along the optimal path for the problem Pt(t;;p).
The process ( ^ Ms)
(t)
s=t from STEP 1 can be simply chosen to be ^ Ms = EsM(t) (for
any stopping time s such that t  s  (t)), and shown that ^ M  M by backward
induction. This proof is given in Proposition 2.1 below. Thus (3.12) (in the main
text) follows therefore directly from the construction of ^ M. In STEP 2, ( ^ Ms)1
s=t is
obtained as before by letting ^ Ms := EsMk+1(t) for each k  1 natural and each nite
stopping time s such that k(t)+1  s  k+1(t). The optimal solution ( c; a) to the
problem Pt(t;;p) is also an optimal solution for the \relaxed" problem Pt(t; ^ ;p)
(with ^  =   + ^ M=p  ), since for any feasible (c;a) 2 Bt(t; ^ ;p),
Ut(c) Ut( c) = liminf
k!1
Et
k(t) 1 X
s=t
(us(cs) us( cs))  liminf
k!1
Etu
0
k(t)( ck(t))( ak(t) ^ k(t)) = 0:
The inequality above follows from  ak(t)   ^ k(t) =  ak(t)   k(t) = 0: An identical
argument shows that Mt = ^ Mt, hence Mt = EtM(t). STEP 3 is unchanged.
Proposition 2.1. Assume that  are NTT. Let T 2 N and ! 2 
 such that there
exists n 2 N with the property that (T)  T + n on FT(!). Then
MT  ETM(T) on FT(!): (2.1)
Proof. We prove the Proposition by induction on n.
We show, rst, that the claim in the Proposition is true for n = 1. When not
explicit, all equalities and inequalities that follow are understood to hold on F(!).
Since T + 1  (T)  T + n = T + 1, it follows that (T) = T + 1. Assume by
contradiction that MT < ETMT+1. Let (c;a) 2 CT(T;;p). As  are binding at
5T + 1,
cT = eT + T   ET
pT+1
pT
T+1 < eT +  T   ET
pT+1
pT
 T+1 =:  cT:
Let  aT :=  T and t > T, choose  ct; at such that ( c; a) 2 BT+1( T+1;  ;p). It is
immediate to check that ( c; a) 2 BT( T;  ;p). We reached a contradiction, as the
continuation utility after T of path  c is less or equal to V d
T and strictly dominates
the continuation utility after T over c, which equals V d
T :
V
d
T (p) = VT(T;;p) = UT(c) = uT(cT)+ETV
d
T+1 < uT( cT)+ETV
d
T+1 = UT( c)  V
d
T :
Suppose now that the claim in the proposition is true for arbitrary T and ! such
that (T)  T + n on FT(!) for some n = 1;:::;k. Choose T and ! such that
(T)  T + k + 1 on FT(!). We need to show that MT  ETM(T) on FT(!).
Assume, by contradiction, that MT < ETM(T).
For any t satisfying T+1  t  (T), using the induction hypothesis and applying
the law of iterated expectations a nite number of times, we get Mt  EtM(T). Let
(c;a) 2 CT(T;;p). Let  at :=  T and for t satisfying T + 1  t < (T), construct
 at := at  
1
pt
EtM(T)  t  
1
pt
EtM(T) =  t +
1
pt
(Mt   EtM(T))   t:
For t  (T), let ( c; a) 2 B(T)( (T);  ;p). For t 2 [T;(T)), let
 ct := et +  at  
1
pt
Etpt+1 at+1:
Notice that  ct = ct for T < t < (T), and
 cT = eT +  T  
1
pT
ETpT+1 aT+1 = cT  
1
pT
(MT   ETM(T)) > cT:
Moreover, ( c; a) 2 BT( T;  ;p). It follows that the path  c dominates c in terms of
utility after T, and a contradiction is obtained in exactly the same manner as for the
case n = 1 treated above.
63 One period transition to a cyclical equilibrium
All cyclical AJ-equilibrium allocations described in Proposition 5.3 in the main text
(where agents cannot borrow after default) can be achieved with zero initial wealth by
the agents. In contrast, each non-autarchic cyclical equilibrium described in Propo-
sition 5.1 in the paper (where agents cannot trade after default) requires specic
non-zero initial wealth for the agents. However, we show here that all such cyclical
equilibrium paths can be reached after a one period transition, when all agents start
with zero wealth. With zero initial wealth, there exists an equilibrium in which the
transfers from the high-type to low-type agents are constant after the rst period and
an innite number of equilibria converging to autarchy. Therefore the endogeneity
of debt limits causes multiplicity of not only asset prices (through bubbles), but also
of real equilibrium allocations, for both types of punishment for default.
For the rest of this section we assume that the penalty for default is the interdic-
tion to trade. We add an extra period and assume that time starts at  1 and that,
in agreement with our convention, the even agent has low endowment at  1 while
the odd agent has high endowment at  1. We investigate the equilibria where the
high-type agent at period  1 (the odd agent) is a saver, transferring an amount x 1
to the the low-type at period  1, and the transfers (xt) from high-type to low type
agents for periods greater or equal to zero (and consumption, asset holdings, debt
limits and bond prices) are described in Proposition 5.1.
Proposition 3.1. Let

(pt);(ci
t);(ai
t);(i
t);(V
i;d
t )

t0
be a cyclical non-autarchic AJ-
equilibrium associated to a sequence of transfers (xt)t0, as in Proposition 5.1 (in
the main text). Let  := minf x 1;yH   yL   x0g, where  x 1 is chosen such that
f( x 1;x0) = 0 (see (5.8) in the paper). Let x 1 2 [0;]. Then (xt)t 1 are transfers
from high-type to low-type agents in an AJ-equilibrium

(pt);(ci
t);(ai
t);(i
t);(V
i;d
t )

t 1
with initial wealth levels ai
 1 for the agents, where2
p 1 :=
u0(yH   x 1)
u0(yL + x0)
; a
o
 1 = a
o
0=p 1   x 1; a
e
 1 = a
e
 1 :=  a
o
 1:
2The outside option at period  1 is autarchy, that is V
i;d
 1 :=
P
t 1 tu(ei
t), and i
 1 is chosen
to be NTT, that is V i
 1(i
 1;(i
t)t0;(pt)t 1) = V
i;d
 1.
7The initial wealth ae
 1 of the even agent is strictly increasing in x 1, and ae
 1 < 0 for
x 1 = 0 and ae
 1 > 0 for x 1 = .
Proof. The rst order condition of the high-type (odd) agent at  1 are satised, by
the construction of p 1. By (5.5) (in the paper), the rst order condition for the low-
type (even) agent at  1 is satised since x 1+x0  yH yL x0+x0  yH yL. The
participation constraint at  1 of the even agent is satised, since x 1  0, therefore
he receives a positive transfer at  1 and at 0 his continuation utility is equal to that
provided by autarchy. The participation constraint at  1 if the odd agent is also
satised, since f(x 1;x0)  f( x 1;x0) = 0. Agents' budget constraints at  1 hold
by construction, and their transversality conditions hold, being the same as in the
equilibrium starting at 0.
Finally, ae
 1 = x 1 ao
0=p 1 and is therefore strictly increasing in x 1, since p 1 is
strictly decreasing in x 1 and ao
0 > 0. Indeed, by (5.12) (in the paper), the sequence
(ptxt)t0 is strictly decreasing, and therefore
a
o
0  L1 := lim
t!1
2t 1 X
s=0
( 1)
spsxs > p0x0   p1x1 > 0:
If x 1 = 0, then clearly
a
e
 1 =  a
o
0
u0(yL + x0)
u0(yH)
< 0:
Assume now that x 1 = . If  x 1  yH yL x0, then x 1 =  x 1 and f(x 1;x0) =
0. It follows that
u
0(y
H  x 1)x 1 > u(y
H) u(y
H  x 1) = 
 
u(y
L + x0)   u(y
L)

> u
0(y
L +x0)x0;
and thus p 1 > x0=x 1. Therefore
a
e
 1 =
p 1x 1   ao
0
p 1
>
p 1x 1   x0
p 1
> 0;
since
a
o
0  L2 := lim
t!1
2t X
s=0
( 1)
spsxs < p0x0 = x0:
8If  x 1 > yH   yL   x0, then x 1 = yH   yL   x0. Hence p 1 = 1=, and
a
e
 1 = y
H   y
L   x0   a
o
0 > y
H   y
L   x0   x0 > y
H   y
L   (1 + )
yH   yL
2
> 0;
where we used the inequality x0  (yH   yL)=2.
Proposition 3.1 implies that all the cyclical equilibrium paths described in Propo-
sition 5.1 in the main text (including the autarchic one) can be achieved after a one
period transition if agents start with zero wealth.
References
Bidian, F., and C. Bejan (2012): \Martingale Properties of Self-Enforcing Debt,"
mimeo, Rice University and Georgia State University.
Forno, A. D., and L. Montrucchio (2003): \Optimality Conditions and Bub-
bles in Sequential Economies and Bounded Relative Risk-Aversion," Decisions in
Economics and Finance, 26(1), 53{80.
Kocherlakota, N. R. (1992): \Bubbles and Constraints on Debt Accumulation,"
Journal of Economic Theory, 57(1), 245{256.
9

Supplement to ``Martingale properties of self-enforcing debt''

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