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Debt Limits and Credit Bubbles
in General Equilibrium
WP 19-19
V. Filipe Martins-da-Rocha
Sao Paulo School of EconomicsFGV and CNRS
Toan Phan
Federal Reserve Bank of Richmond
Yiannis Vailakis
University of Glasgow
Debt Limits and Credit Bubbles in General Equilibrium∗
V. Filipe Martins-da-Rocha†
Toan Phan‡
Yiannis Vailakis§
October 2, 2019
Working Paper No. 19-19
Abstract
We provide a novel characterization of self-enforcing debt limits in a general equilibrium framework of risk sharing with limited commitment, where defaulters are subject
to recourse (a fractional loss of current and future endowments) and exclusion from future credit. We show that debt limits are exactly equal to the present value of recourse
plus a credit bubble component. We provide applications to models of sovereign debt,
private collateralized debt, and domestic public debt. Implications include an original
equivalence mapping among distinct institutional arrangements, thereby clarifying the
relationship between different enforcement mechanisms and the connection between
asset and credit bubbles.
Keywords: Limited commitment; general equilibrium; rational credit bubbles.
JEL codes: E00; E10; F00.
1
Introduction
Consider an environment where economic agents borrow and save in order to smooth their
consumption against fluctuating endowments but agents cannot commit to repaying debt.
∗
We thank Kartik Athreya, Bernardo Guimaraes, Guido Lorenzoni, Pietro Peretto, Nico Trachter, Jan
Werner and various seminar/conference participants at the Federal Reserve Bank of Richmond, Virginia
Tech, Duke University Triangle Dynamic Macro workshop, University of Tokyo, University of California
Santa Cruz, Santa Clara University, University of Virginia, the Economics School of Louvain, the Sao
Paulo School of Economics, the Canon Institute for Global Studies, and the 71st European Meeting of the
Econometric Society for useful comments and suggestions. The views expressed herein are those of the
authors and not those of the Federal Reserve Bank of Richmond or the Federal Reserve System.
†
Sao Paulo School of Economics–FGV and CNRS; filipe.econ@gmail.com.
‡
The Federal Reserve Bank of Richmond; toanvphan@gmail.com.
§
The University of Glasgow; yiannis.vailakis@glasgow.ac.uk.
1
Default induces both direct sanctions (loss of a fraction of current and future endowments)
and a reputation loss (no access to future credit). How much debt can these agents issue in
this setting?
This is an important and classic question with many relevant implications. When agents
are sovereign governments, the question maps to the sustainability of sovereign debt, which
is the topic of a large and growing body of research (Wright 2013, Aguiar and Amador
2014).1 When agents are individuals or corporations, the question maps to the sustainability of consumer debt (Livshits 2015) or corporate debt (Azariadis et al. 2015). Since one
agent’s liability is another agent’s asset in general equilibrium, the question also maps to
the availability of assets or liquidity. It is well-known that a scarcity of assets facilitates
bubbles (Tirole 1985; Farhi and Tirole 2012) and a growing body of papers has argued both
empirically and theoretically that asset and credit bubbles play an important role in the
observed phenomena of long recessions, low interest rates, liquidity traps, and global imbalances (Caballero et al. 2008; Jordà et al. 2015; Caballero and Farhi 2017; Barlevy 2018;
Biswas et al. 2018; Ikeda and Phan 2019).
In this paper, we revisit these issues with an emphasis on general equilibrium implications. We consider an endowment economy where agents can issue and trade debt contracts.
Agents cannot commit to honor their liabilities, but should they default, agents will face
two forms of punishment. First, they will lose a nonnegative fraction of their current and
future endowments. As mentioned, this loss can be mapped to output loss in the case of a
sovereign default or to recourse and seized collateral in the case of consumer and corporate
default. Second, agents will be excluded from future borrowing but they can still save. As in
Alvarez and Jermann (2000), agents face endogenous “not-too-tight” debt limits, which in
equilibrium are set at the largest possible levels such that repayment is always individually
rational. These debt limits determine the borrowing capacity and thus the availability of
assets in equilibrium and are the focus of this paper.
We provide an intuitive but powerful result that characterizes equilibrium debt limits: for
each agent, the debt limit in any contingency is exactly the sum of a “secured” component,
which is equal to the present value of the current and future endowment losses (discounted at
the endogenous equilibrium interest rates), and an “unsecured” credit bubble component. In
other words, the secured component of the debt limits is supported by the direct endowment
loss. The remaining component, supported by the threat of credit exclusion, is necessarily
1
Also related is the sustainability of public debt (D’Erasmo et al. 2016).
2
bubbly in the sense that it satisfies an exact rollover condition.
Despite its intuitive nature, proving this property is rather challenging. This is because
in our environment, unlike in Eaton and Gersovitz (1981) or Alvarez and Jermann (2000),
defaulting agents can still save, and hence the punishment value for defaulting involves equilibrium prices. Our proof strategy exploits a novel intermediate finding about the equilibrium
interest rates that has no analogue in the absence of output losses. Specifically, we show
that, in any equilibrium with self-enforcing debt, the present value of endowment losses upon
default must be finite, or, equivalently, the implied interest rates must be higher than the
growth rates of aggregate endowment losses. A standard argument then reveals that the
process of present values of endowment losses is itself not too tight. The result then follows
from the fact that the difference between two not-too-tight processes is necessarily a bubble.
We also show that a necessary condition for the emergence of credit bubbles is that
the economy’s total resources have an infinite present value. A necessary condition for this
is that the aggregate endowment losses are negligible with respect to the economy’s total
resources. Note that in the special case where the endowment losses are set to zero, our
characterization result implies as a corollary that the competitive equilibrium either features
no borrowing/lending or features trades where borrowers purely roll over their debt as credit
bubbles. Thus, our paper nests the well-known theorems in Bulow and Rogoff (1989) and
Hellwig and Lorenzoni (2009).
Our result has a broad set of applications and implications. First, it simplifies substantially the computation of equilibria, ruling out complications related to the fixed-point
determination of not-too-tight debt limits. This is important since in our setting the value
of default depends on prices, and thus equilibrium allocations cannot be obtained by solving
a planner’s problem as is the case when the default option is autarky (Alvarez and Jermann
2001). We illustrate this by looking at a simple deterministic economy with constant endowment loss and endowment growth. We fully characterize equilibrium outcomes and prove the
existence of competitive equilibria where a positive fundamental component in debt limits
coexists with a positive bubble component.
Another implication of our characterization theorem is an interesting equivalence result:
a competitive equilibrium with inside liquidity (i.e., tradable debt securities issued by private
agents) is isomorphic to a competitive equilibrium with outside liquidity in the form of public debt (Holmström and Tirole 2011). Outstanding debt in both environments reflects the
present value of overall resources used to back up liabilities plus a bubble component. How-
3
ever, at the self-enforcing equilibrium the privilege of issuing the credit bubble is attributed
to agents, whereas at the public debt equilibrium this privilege is entitled to the government.
A technical implication of this equivalence result is that it allows us to prove the existence of
a competitive equilibrium with limited commitment in a simpler way. It is well-known that
standard proofs based on truncation arguments are generally difficult to apply to models
with limited commitment due to the presence of the self-enforcing conditions. The equivalence allows us to employ a truncation technique to tackle an alternative problem that is free
of the not-too-tight restrictions. Nevertheless, an additional source of difficulty remains as
the supply of government debt is endogenous in the definition of the equilibrium with public
debt. We suggest a way to deal with this issue that can be of independent interest.
We subsequently argue, via a second equivalence result, that equilibria with not-too-tight
debt constraints can be mapped to equilibria with collateral constraints (Chien and Lustig
2010; Gottardi and Kubler 2015) and vice versa. In the latter environment, in addition to the
issuance of private debt, each agent is free to issue and trade a durable and collateralizable
asset (a Lucas tree) whose dividend process amounts to a fraction of her income. Upon
default, the borrower’s equity is confiscated and passed to the hands of creditors (as opposed
to the previous environment where endowment losses are not passed to creditors). Apart
from the seizure of equity holdings, there is no additional punishment for default: the agent
can trade on financial markets. The default option is endogenous and reflects not only the
loss of future investment income (the asset’s dividends), but also the loss of using the asset
as a source of liquidity. In equilibrium, equity prices reflect assets’ fundamental value and
a bubble component. Thus, the credit bubble component in the previous setting with selfenforcing debt can be mapped to the bubble component in asset prices in this setting with
collateral and vice versa.
The equivalence result challenges a common view that models with collateral constraints
are more in line with data at business cycle frequencies. For instance, Chien and Lustig
(2010) found that their model produces more volatile equity risk-premiums than the limited
commitment model of Alvarez and Jermann (2001). We show that, when debt enforcement
relies on recourse and one-sided exclusion, the self-enforcing mechanism is observationally
equivalent to the collateral mechanism.
Finally, it is well-known that Markovian (with respect to a simple state space) collateral
equilibria have a tractable representation, where Negishi’s method can be applied to prove
existence and even compute them in a fairly simple way (e.g., Gottardi and Kubler 2015).
4
Although this representation is often cited as a reason to prefer these types of models for
equilibrium analysis, our equivalence result shows that equilibria with self-enforcing debt
can also be cast in terms of solutions to a programming problem, thereby suggesting that
their computation is not of a different complexity. We prove existence and uniqueness of
a Markov equilibrium, thereby complementing the analysis in Gottardi and Kubler (2015)
along this line.
Related literature. The paper is related to the general equilibrium literature on risk
sharing with limited commitment and endogenous borrowing constraints. A nonexhaustive
list of contributions includes Kehoe and Levine (1993), Alvarez and Jermann (2000, 2001),
Kehoe and Perri (2002), Kehoe and Perri (2004), Ábrahám and Cárceles-Poveda (2006,
2010), Bloise and Reichlin (2011), Bloise et al. (2013), Werner (2014), Martins-da-Rocha and
Vailakis (2015), Bidian (2016), where default induces full exclusion from financial markets,
and Bulow and Rogoff (1989), Gul and Pesendorfer (2004), Hellwig and Lorenzoni (2009),
Werner (2014), Bidian and Bejan (2015) and Martins-da-Rocha and Vailakis (2017a,b), where
defaulters are only excluded from future credit. To the best of our knowledge, our paper is
the first that introduces the empirically relevant recourse feature of endowment losses from
default into this general equilibrium environment.
Our paper is also related to Woodford (1990), Holmström and Tirole (1998, 2011), and
Werner (2014) in exploring the relationship between private liquidity and public liquidity in
environments with financial frictions that lead to scarce collateral. The work is also related to
the rational asset price bubbles literature, which dates back to Samuelson (1958), Diamond
(1965), and Tirole (1985). Recent papers in this literature include Farhi and Tirole (2012),
Martin and Ventura (2012), Hirano and Yanagawa (2016), Miao and Wang (2018), and
Bengui and Phan (2018). To the best of our knowledge, our paper is the first to formalize a
mapping between the credit bubble component of debt limits and the bubble component of
asset prices.
The plan for the rest of the paper is as follows. Section 2 sets up the model. Section 3 provides the main result. Section 4 provides applications and implications. Section 5
concludes.
5
2
Model
2.1
Fundamentals
Consider an infinite-horizon endowment economy with a single nonstorable consumption
good at each date. Time and uncertainty are both discrete. We use an event tree Σ to
describe the revelation of information over an infinite horizon. There is a unique initial
date-0 event s0 ∈ Σ and for each date t ∈ {0, 1, 2, . . .} there is a finite set S t ⊆ Σ of date-t
events st . Each st has a unique predecessor σ(st ) in S t−1 and a finite number of successors
st+1 in S t+1 for which σ(st+1 ) = st . The notation st+1 st specifies that st+1 is a successor
of st . The event st+τ is said to follow event st , also denoted st+τ st , if σ (τ ) (st+τ ) = st .
The set S t+τ (st ) := {st+τ ∈ S t+τ : st+τ st } denotes the collection of all date-(t+τ ) events
following st . Abusing notation, we let S t (st ) := {st }. The subtree starting at event st is
then given by
[
Σ(st ) :=
S t+τ (st ).
τ >0
We use the notation sτ st when sτ st or sτ = st . In particular, we have Σ(st ) = {sτ ∈
Σ : sτ st }.
There is a finite set I of household types, each consisting of a unit measure of identical,
infinitely lived agents who consume the single perishable good. Agents cannot commit to
future actions: at any event st , they can refuse to honor past promises and default.
Preferences over (nonnegative) consumption processes c = (c(st ))st s0 are represented by
the lifetime expected and discounted utility
X X
U (c) :=
βt
π(st )u(c(st ))
t>0
st ∈S t
where β ∈ (0, 1) is the discount factor, π(st ) is the unconditional probability of st , and
u : [0, ∞) → R is a utility function that is strictly increasing, concave, continuous on [0, ∞),
differentiable on (0, ∞), and satisfies Inada’s condition limε→0 [u(ε) − u(0)]/ε = ∞.2 As in
Hellwig and Lorenzoni (2009), we assume that u is bounded. The only role of this assumption
is to guarantee that the lifetime utility U is continuous and the demand set is non-empty.
The analysis can be extended and our results continue to hold even when u is unbounded,
2
For simplicity, we assume that agents’ preferences are homogeneous. All arguments can be adapted to
handle the case where the preferences differ among agents.
6
in particular, when u belongs to the class of constant relative risk aversion utility functions
u = c1−α /(1 − α) with α > 0. The example in Section 4.1 illustrates this point.3
Given an event st , we denote by U (c|st ) the lifetime continuation utility conditional to st
defined by
X
X
π(st+τ |st )u(c(st+τ ))
U (c|st ) := u(c(st )) +
βτ
τ >1
st+τ st
where π(st+τ |st ) := π(st+τ )/π(st ) is the conditional probability of st+τ given st .
Agents’ endowments are subject to random shocks. We denote by y i = (y i (st ))st s0 the
process of positive endowments y i (st ) > 0 of a representative agent of type i. A collection
(ci )i∈I of consumption processes is called a consumption allocation. It is said to be resource
P
P
feasible if i∈I ci = i∈I y i . We also fix an allocation (ai (s0 ))i∈I of initial financial claims
P
ai (s0 ) ∈ R that satisfies the usual market-clearing condition: i∈I ai (s0 ) = 0.
2.2
Markets
At any event st , agents can issue and trade a complete set of one-period contingent bonds,
each one promising to pay one unit of the consumption good contingent on the realization
of any successor event st+1 st . Let q(st+1 ) > 0 denote the price at event st of the st+1 contingent bond. Agent i’s bond holdings are ai = (ai (st ))st s0 , where ai (st ) 6 0 denotes a
liability and ai (st ) > 0 denotes a claim. Debt is observable and subject to state-contingent
(nonnegative and finite) debt limits Di = (Di (st ))st s0 . Given the initial financial claim
ai (s0 ), we denote by B i (Di , ai (s0 )|s0 ) the budget set of an agent who never defaults. It
consists of all pairs (ci , ai ) of consumption and bond holdings satisfying the following budget
and solvency constraints: for any event st s0 ,
ci (st ) +
X
q(st+1 )ai (st+1 ) 6 y i (st ) + ai (st )
(2.1)
st+1 st
and
ai (st+1 ) > −Di (st+1 ),
3
for all st+1 st .
(2.2)
A general treatment of unbounded utility functions requires some additional technical assumptions on
endowment processes together with a suitable modification of the utility function u outside a specific interval
such that the equilibrium outcomes remain unaffected. For a detailed discussion, see Martins-da-Rocha and
Santos (2019).
7
Fix an event st and some initial claim a ∈ R. We denote by V i (Di , a|st ) the value function
defined by
V i (Di , a|st ) := sup{U (ci |st ) : (ci , ai ) ∈ B i (Di , a|st )}
where B i (Di , a|st ) is the set of all plans (ci , ai ) satisfying ai (st ) = a together with restrictions (2.1) and (2.2) at every successor node sτ st .
Without any loss of generality, we restrict attention to debt limits (Di (st ))st s0 that are
consistent, meaning that at every event st , the maximal debt can be repaid out of the current
resources and the largest possible debt contingent on future events, i.e.,
X
Di (st ) 6 y i (st ) +
q(st+1 )Di (st+1 ), for all st s0 .
(2.3)
st+1 st
This condition is necessary for the budget set B i (Di , −Di (st )|st ) to be nonempty.
2.3
Default Costs
Agents might not honor their debt obligations and default if it is optimal for them.4
Following Bulow and Rogoff (1989), we assume that, upon default, debtors start with neither
assets nor liabilities, are excluded from future credit, but retain the ability to purchase bonds.
In addition, debt repudiation leads to a loss `i (st ) ∈ [0, y i (st )] of the endowment, where
`i (st ) may vary across agents and events.5 Formally, agent i’s default option at event st is
the largest continuation utility when starting with zero liabilities, cannot borrow, and her
income contracts by the amount `i (sτ ) at every sτ st :
V`ii (0, 0|st ) := sup{U (ci |st ) : (ci , ai ) ∈ B`ii (0, 0|st )},
(2.4)
where B`ii (0, 0|st ) denotes the budget set corresponding to B i (0, 0|st ) when the endowment
y i (sτ ) is replaced by y i (sτ ) − `i (sτ ) at any event sτ st .
4
Since the default punishment is independent of the default level, there will be no partial default in
equilibrium: agents either repay or default fully on their promises.
5
In their modeling of unsecured consumer credit, Chatterjee et al. (2007) assume that a defaulting
household cannot borrow and incurs a small reduction in its earning capability. Disruption of international
trade and of the domestic financial system can lead to a sovereign’s drop of if trade and/or credit are essential
for production. Among others, Mendoza and Yue (2012), Gennaioli et al. (2014), and Phan (2017) model
explicitly how sovereign default may lead to efficiency losses in production. We follow the tradition in the
sovereign debt literature (see for instance Cohen and Sachs 1986, Cole and Kehoe 2000, Aguiar and Gopinath
2006, Arellano 2008, Ábrahám and Cárceles-Poveda 2010, and Bai and Zhang 2010, 2012) and model the
negative implications on output as a loss of an exogenous fraction of income.
8
2.4
Not-Too-Tight Debt Limits
Since agents trade contingent bonds, potential lenders have no reason to provide credit
if they anticipate that debtors will default at an event st . Debt limits should reflect this
property. We say that debt limits are self enforcing if for every individual i, at every event st ,
V i (Di , −Di (st )|st ) > V`ii (0, 0|st ).
The left-hand side is the value of market participation beginning with the maximum sustainable debt, whereas the right-hand side is the value of default.
Competition among lenders naturally leads to consider the maximum self-enforcing debt
limit compatible with repayment. Following Alvarez and Jermann (2000) and Hellwig and
Lorenzoni (2009), we say that debt limits are not too tight if the individual is indifferent
between repaying and defaulting, i.e.,
V i (Di , −Di (st )|st ) = V`ii (0, 0|st ).
2.5
(2.5)
Competitive Equilibrium
A competitive equilibrium with self-enforcing private debt is defined as follows.
P
Definition 2.1. Given initial asset holdings (ai (s0 ))i∈I such that i∈I ai (s0 ) = 0, a competitive equilibrium with self-enforcing debt (q, (ci , ai , Di )i∈I ) is a collection of state-contingent
bond prices q, a resource feasible consumption allocation (ci )i∈I , a market clearing allocation of bond holdings (ai )i∈I , and a family of consistent, nonnegative, and finite debt limits
(Di )i∈I such that:
(a) for every agent i ∈ I, taking prices and the debt limits as given, the plan (ci , ai ) is
optimal among budget feasible plans in B i (Di , ai (s0 )|s0 );
(b) for every agent i ∈ I, the debt limits Di are not too tight, i.e., equation (2.5) is satisfied
at any event.
Throughout the paper, we restrict attention to competitive equilibria with self-enforcing
debt such that initial asset holdings are consistent with repayment incentives, i.e., ai (s0 ) >
−Di (s0 ) for each i.6
6
This is in particular the case when ai (s0 ) = 0 for each agent i.
9
We subsequently define some additional objects that are useful for the rest of the analysis.
Given bond prices q = (q(st ))st s0 , we denote by p(st ) the associated date-0 price of consumption at st defined recursively by p(s0 ) = 1 and p(st+1 ) = q(st+1 )p(st ) for all st+1 st .
Given date-0 prices, we define the present value of a process x conditional to an event st as
1 X
PV(x|st ) :=
p(sτ )x(sτ ).
p(st ) τ t
s s
The wealth of an agent at event st is defined as the present value of her endowments:
W i (st ) := PV(y i |st ).
Finally, a process (M i (st ))st s0 allows for exact rollover if it satisfies the following property:
X
M i (st ) =
q(st+1 )M i (st+1 ), for all st s0 .
(2.6)
st+1 st
Slightly abusing language, we will refer to exact rollover processes as discounted martingales
or bubbles.
Remark 2.1. If `i (st ) = y i (st ) at every event st , then agent i has no incentive to ever default.
In this case, the agent can effectively commit to her financial promises because the value of the
default option Vyii (0, 0|st ) equals U (0|st ), and, therefore, any process of consistent debt limits
is self-enforcing. Our specification of output costs thus encompasses a mixed environment
where some agents can perfectly commit to financial contracts while others have limited
commitment. The textbook model with full commitment obtains when `i (st ) = y i (st ) for
any agent i and every event st . The model nests both Bulow and Rogoff (1989) (unilateral
lack of commitment) and Hellwig and Lorenzoni (2009) (multilateral lack of commitment)
settings where ` ≡ 0.
3
Main Result
At the core of the analysis is a novel characterization of equilibrium debt limits when
default entails both credit exclusion and a loss of a fraction of the private endowment. Nottoo-tight debt limits are decomposed into a “fundamental” component, PV(`i ), associated
to direct default costs, and a “credit bubble” component, M i , that captures the possibility of
rolling over a fraction of debt indefinitely. That is, credit beyond the fundamental component
is sustainable only if debtors can exactly refinance past liabilities by issuing new debt. The
following theorem provides the formal statement.
10
Theorem 3.1. Not-too-tight debt limits equal the sum of the present value of endowment
losses upon default and a bubble component, i.e.,
Di = PV(`i ) + M i ,
where M i is a nonnegative discounted martingale process.
Before we proceed to the proof of Theorem 3.1, we present two intermediate results. The
first and crucial observation, that has no analogue in the absence of output contraction, is to
show that the present value of foregone endowment imposes a lower bound on not-too-tight
debt limits. A direct implication of this property is that the process PV(`i ) is finite. This is
summarized in the following lemma.
Lemma 3.1. Not-too-tight debt limits are at least as large as the present value of endowment
losses: for each agent i, Di (st ) > PV(`i |st ) at any event st .
e i (st ), where
A natural approach to prove this result is to show that Di (st ) > `i (st ) + D
e i (st ) := P t+1 t q(st+1 )Di (st+1 ) is the present value of next period’s debt limits, and then
D
s
s
use a standard iteration argument. Because, in equilibrium, debt limits are not too tight,
this is equivalent to proving that agent i does not have an incentive to default when her net
e i (st ), i.e.,
asset position is `i (st ) + D
e i (st )|st ) > V ii (0, 0|st ).
V i (Di , −`i (st ) − D
`
By definition, the default value function V`ii satisfies:
X
π(st+1 |st )V`ii (0, 0|st+1 ).
V`ii (0, 0|st ) > u(y i (st ) − `i (st )) + β
(3.1)
(3.2)
st+1 st
If we had an equality in (3.2), then inequality (3.1) would be straightforward. Indeed,
consuming y i (st )−`i (st ) and borrowing up to each debt limit Di (st+1 ) at event st leads to the
right-hand side continuation utility in (3.2) and satisfies the solvency constraint at event st
in the budget set defining the left-hand side of (3.1). However, in our environment where
an agent can save upon default, (3.2) need not be satisfied with an equality.7 Overcoming
this problem is the technical challenge in the proof of Lemma 3.1. The formal argument is
presented in Appendix A.1.
7
In the simpler environment where saving is not possible after default (as it is the case in Alvarez and
Jermann 2000) we always have an equality in (3.2).
11
A second observation is that the process PV(`i ) of present values of endowment losses,
when it is finite, is itself not too tight. The following lemma provides the formal statement. The proof follows from a simple translation invariance of the flow budget constraints
presented in Appendix A.2.
Lemma 3.2. If PV(`i |s0 ) is finite, then the process PV(`i ) is not too tight, i.e.,
V i (PV(`i ), − PV(`i |st )|st ) = V`ii (0, 0|st ),
for all st s0 .
Equipped with Lemma 3.1 and Lemma 3.2, we can provide a simple proof of Theorem 3.1.
Proof of Theorem 3.1. Fix a process Di of not-too-tight debt limits. Lemma 3.1 implies
that PV(`i |s0 ) is finite. From Lemma 3.2 we also deduce that the process Di := PV(`i )
is not too tight. Martins-da-Rocha and Santos (2019) proved that the difference between
two processes of not-too-tight debt limits must be a discounted martingale. Therefore, there
exists a process M i satisfying (2.6) such that Di = Di + M i . By Lemma 3.1, Di > Di , in
which case the process M i must be nonnegative.
We next show that a necessary condition for the emergence of credit bubbles is that the
direct sanctions against defaulters become negligible (relative to aggregate resources) in the
distant future. Indeed, as argued below, when the fraction of endowment losses to aggregate
endowment is uniformly bounded away from zero, the bubble component disappears and
debt can never exceed the present value of direct default costs.
Specifically, we say that endowment losses are nonnegligible if there exists ε > 0 such
that:
X
X
`i (st ) > ε
y i (st ), for all st s0 .
(3.3)
i∈I
i∈I
The condition implies that, in equilibrium, interest rates should be sufficiently high, in the
sense that the aggregate wealth of the economy is finite. A direct consequence is that the
determination of not-too-tight debt limits is akin to the pricing of long-lived assets in a
competitive equilibrium with full commitment. On the one hand, high interest rates rule
out the possibility of rolling over debt indefinitely and forces equilibrium debt to not exceed
the present value of endowment losses exactly the same way asset bubbles are ruled out in
Santos and Woodford (1997). On the other hand, debtors would always choose to honor any
debt that is at least as large as the present value of the endowment losses (see Lemma 3.1).
Together, those forces imply that equilibrium debt limits should exactly reflect the present
value of foregone endowment upon default. All this is summarized in the following corollary.
12
Corollary 3.1. If the endowment losses are nonnegligible, then the not-too-tight debt limits
equal exactly the present value of foregone endowment upon default, i.e.,
Di (st ) = PV(`i |st ),
for all st s0 .
(3.4)
Proof. Let (q, (ci , ai , Di )i∈I ) be a competitive equilibrium with self-enforcing debt. Since
endowment losses are nonnegligible, we must have
X
W i (s0 ) =
i∈I
X
PV(y i |s0 ) 6
i∈I
1X
PV(`i |s0 ).
ε i∈I
Lemma 3.1 then implies that the aggregate wealth of the economy must be finite. Since
consumption markets clear, we obtain that the present value of optimal consumption is
finite for all agents. In addition, due to the Inada’s condition, the optimal consumption is
strictly positive.8 Lemma A.1 in Martins-da-Rocha and Vailakis (2017a) then implies that
the following market transversality condition holds true:9
X
p(st )[ai (st ) + Di (st )] = 0.
(3.5)
lim
t→∞
st ∈S t
We know from Theorem 3.1 that for each i, there exists a nonnegative discounted martingale
process M i satisfying Di = PV(`i ) + M i . The market transversality condition can then be
written as follows
X
p(st )ai (st ) = −p(s0 )M i (s0 ).
lim
t→∞
st ∈S t
Since bond markets clear, we deduce that
M i = 0 for each i.
P
i∈I
M i (s0 ) = 0 and we get the desired result:
Corollary 3.1 provides a general equilibrium foundation for Bulow and Rogoff (1989)’s
no-trade result. Indeed, in their framework, upon default, the sovereign can take the saved
repayments to Swiss bankers who are committed to honoring contracts made with governments. This can be translated in our setting by assuming that a subset I c ⊂ I of agents
(the Swiss bankers) face the maximum endowment loss, i.e., `i = y i for all i ∈ I c while the
remaining agents (the debtors) face no endowment loss, i.e., `i = 0 for all i ∈ I nc := I \ I c .
8
See the Supplemental Material of Martins-da-Rocha and Santos (2019) for a detailed proof.
The market transversality condition differs from the individual transversality condition. Indeed, due
to the lack of commitment, agent i’s debt limits may bind, in which case we do not necessarily have that
p(st ) = β t π(st )u0 (ci (st ))/u0 (ci (s0 )).
9
13
As long as the endowment of committed agents is large relative to the one of noncommitted
agents, i.e, there exists α > 0 such that
X
X
y i (st ) > α
y i (st ),
(3.6)
i∈I c
i∈I nc
then condition (3.3) holds true (for ε 6 α/(1 + α)) and Corollary 3.1 applies: Di (st ) = 0
for any i ∈ I nc and all st s0 . Instead of assuming that the endogenously determined
interest rates are high enough (as in Bulow and Rogoff 1989 or Proposition 3.1 in Hellwig
and Lorenzoni 2009), we show this is always true when endowments satisfy condition (3.6).
4
Applications
4.1
Equilibrium Computation: An Example
We illustrate the applicability of Theorem 3.1 by analyzing a simple deterministic economy with identical agents whose endowments switch between a high and low value and
default entails the loss of a fixed amount ` > 0. Our characterization result simplifies substantially the computation of equilibria. It also enables us to overcome the complications
that arise in constructing and characterizing bubbly equilibria where a positive fundamental
component coexists with a positive bubble component.10
There are two agents I := {i1 , i2 } with a constant relative risk aversion utility function
u(c) =
c1−α
,
1−α
α > 0.
At every date t, each agent i’s income yti alternates between a high value yh,t and a low value
yl,t . Agent i1 starts with the high income. Incomes grow at a constant gross rate ρ > 1:
(yh,t , yl,t ) = ρt (yh , yl ),
with yh > yl > ` > 0.
We focus on symmetric equilibria and denote by xt the not-too-tight debt limit at date t,
i.e., Dti = xt for each i.11 It follows from our general characterization result that there exists
M0 > 0 such that
1
xt = [`(pt + pt+1 + . . .) + M0 ],
pt
10
For details of the difficulty in establishing the coexistence of the two components in asset price bubbles,
see Miao and Wang (2018).
11
Symmetry means that the debt limit is the same for both agents.
14
where pt := q1 · · · qt . We restrict attention to equilibria where the high-income agent at
date t purchases the amount xt+1 of the one-period bond and the low-income agent issues
the largest debt xt+1 . To support this allocation as a competitive equilibrium, we assume
an initial positive transfer x0 for the low-income agent and an initial debt level x0 for the
high-income agent. Then, for all t, the consumption of the high-income agent is
ch,t = yh,t − (xt + qt+1 xt+1 )
while the consumption of the low-income agent is
cl,t = yl,t + (xt + qt+1 xt+1 ).
Let zt denote the net trade position, i.e.,
zt := xt + qt+1 xt+1 .
Since there is growth, we let ẑt := ρ−t zt represent the detrended trade position at period t.
Observe that
1
ẑt := t [`(pt + 2pt+1 + 2pt+2 + . . .) + 2M0 ].
(4.1)
ρ pt
The Euler equation associated to the high-income agent’s saving decision is
α
u0 (cl,t+1 )
yh − ẑt
,
qt+1 = β 0
=β
u (ch,t )
ρ(yl + ẑt+1 )
(4.2)
while the Euler equation associated to the low-income agent’s borrowing decision is
α
u0 (ch,t+1 )
yl + ẑt
qt+1 > β 0
.
=β
u (cl,t )
ρ(yh − ẑt+1 )
The last inequality follows from (4.2) when ẑt 6 (yh − yl )/2. Therefore, to get the existence
of a competitive equilibrium it is sufficient to find a sequence (qt+1 )t>0 of positive bond prices
and a sequence (ẑt )t>0 of detrended net trades such that
α
yh − ẑt
qt+1 = β
(4.3)
ρ(yl + ẑt+1 )
and
`
`
ẑt = t + ρqt+1 ẑt+1 + t+1 ,
ρ
ρ
15
(4.4)
where each ẑt belongs to [0, (yh − yl )/2].
From now on, we impose the following restrictions on primitives:
α
α
yh
1
1
> >β
1>β
.
ρyl
ρ
ρ
| {z }
=: q̄
This condition implies that there exists a unique M0? such that
α
yh − 2M0?
1
?
2M0 < (yh − yl )/2 and β
= .
?
ρ(yl + 2M0 )
ρ
(4.5)
(4.6)
It is straightforward to check that if ` = 0, then there exists a competitive equilibrium with
time-invariant interest rate characterized by
pt =
1
ρt
and xt = ρt M0? .
When the endowment loss ` is positive but small enough, in the sense that `/(1− q̄) < M0? ,
we show in the online Supplemental Material of this paper that for any arbitrary value M0 (`)
satisfying
`
+ M0 (`) 6 M0? ,
1 − q̄
there exists a competitive equilibrium (qt+1 (`), ẑt (`))t>0 such that pt (`) > ρ−t .12 Note that
there is an indeterminacy of equilibria parametrized by the size of the initial bubble M0 (`).
We also show in the online Supplemental Material of this paper that the detrended net trade
vanishes (i.e., limt→∞ ẑt (`) = 0). This property follows from the fact that the initial net
trade satisfies ẑ0 (`) < 2M0? .
Nevertheless, because of the difference equation (4.4), we may look for other type of
equilibria with an initial net trade satisfying ẑ0 > 2M0? . In fact, depending on the choice of
the initial value ẑ0 , an equilibrium can display very different properties. To illustrate this,
we analyze numerically two types of possible equilibrium outcomes for a given endowment
loss `: one where the initial trade is fixed and equal to 2M0? (Figure 1) and another one
where the asymptotic detrended trade is fixed and equal to 2M0? (Figure 2). We use the
following parameters: α = 1 (that is, the utility function is u(c) = ln(c)), yL = 75, yH = 140,
β = 0.55, ρ = 1.05, and set various values of `.
12
Recall that pt (`) := q1 (`) . . . qt (`) is the Arrow–Debreu price at t = 0 of one unit of consumption at
date t.
16
In Figure 1, setting ẑ0 = 2M0? , we solve equations (4.3) and (4.4) for ` equal to: `0 = 0,
`1 = 0.01, `2 = 0.015, and `3 = 0.020. We appeal to Theorem 3.1 to get that Dt (`) = (zt (`)+
`)/2. The initial value M0 (`) of the bubble component can then be identified by computing
limt→∞ pt (`)Dt (`).13 Bond prices increase and net trade decreases with the endowment loss.
Since the initial trade is fixed and satisfies
2M0? = ẑ0 = `(1 + 2p1 (`) + 2p2 (`) + . . .) + 2M0 (`),
the initial size of the bubble must decrease with the endowment loss. Moreover, since bond
−1
prices converge to ρ−1
0 := q̄ > ρ , the (detrended) debt limit converges to zero.
In Figure 2, instead of fixing the initial value of the detrended trade, we look for a
solution involving the largest ẑ0 compatible with equilibrium for each of the following values
for `: `0 = 0, `1 = 0.05, `2 = 0.12, and `3 = 0.40.14 Observe that, given ` > 0, along any
equilibrium path we now have pt (`) 6 ρ−t for every t > 0 (as opposed to pt (`) > ρ−t in
Figure 1). We still have that the level of initial bubble M0 (`) decreases with the endowment
loss, but now trade opportunities increase as ` increases. Moreover, debt does not vanish
asymptotically but rather converges to the pure bubble debt level M0? (that is independent of
the level of endowment loss), so trade persists in the limit. Bond prices also have a different
behavior. The larger the endowment loss `, the higher the implied interest rate, the lower
the asset price.
13
The values for `1 , `2 , and `3 have been chosen to get a nice distribution of M0 (`) between 0 and M0? .
We refer to the online Supplemental Material of this paper where we justify the existence of such a
solution. We have modified the values for `1 , `2 , and `3 to keep a nice distribution of M0 (`) between 0 and
M0? .
14
17
Figure 1: ẑ0 = 2M0?
18
Figure 2: limt→∞ ẑt = 2M0?
Figure 3: Races defining the debt level
(a) ρ−t Ct (`)
(b) ρ−t `Ct (`)
(c) ρ−t M0 (`)/pt (`)
We now demonstrate how our characterization result can help disentangle the forces that
drive the pattern of the sustained debt shown in Figure 2. Theorem 3.1 implies the following
decomposition:
M0 (`)
Dt (`) = ` (1 + qt+1 (`) + qt+1 qt+2 (`) + . . .) +
|
{z
} pt (`)
Ct (`)
where `Ct (`) is the fundamental component with Ct (`) be the (cum-dividend) price of a
console at date t and M0 (`)/pt (`) is the bubble component. There is a first race between
the endowment loss ` and the price of the console Ct (`) since the later decreases when the
endowment loss increases (Figure 3(a)). The first effect dominates, so a larger ` implies
a larger fundamental component (Figure 3(b)). There is also a second race that dictates
the way the bubble component changes as ` increases. We have seen that the initial value
of the bubble M0 (`) decreases with ` while the long-term gross return pt (`) increases with
` (Figure 2). The first effect is stronger, so a larger ` implies a lower bubble component
M0 (`)/pt (`) (Figure 3(c)). The overall effect of a change of the endowment loss to the debt
limits, and therefore to liquidity, is therefore determined by the competition between the rise
in the fundamental component and the decrease in the bubble component. For the chosen
specification of the economy’s fundamentals and the type of equilibria analyzed in Figure 2,
the rise in the fundamental component outweighs the drop in the bubble component, so risk
sharing is enhanced as direct sanctions upon default become more severe.
19
4.2
Public Debt
This section shows that the consumption allocation of an equilibrium with self-enforcing
debt can be implemented as an equilibrium allocation of an economy in which debt is only
issued by a government, and vice versa. In that respect, we establish a one-to-one mapping
between an environment with private liquidity (debt issued by private agents) and public
liquidity (debt issued by the government).
This result might come as a surprise as it is generally thought that public debt provides a
superior instrument for allocating risk efficiently: using its tax power, a government can enlarge agents’ insurance opportunities by transferring additional resources to the debt holders
who are hit by adverse shocks (see, for instance, Holmström and Tirole 2011). Furthermore,
there are differences that relate to institutional arrangements. Specifically, in an equilibrium
with backed public debt, the tax revenue is seized along the equilibrium path and transferred to the government. Implicitly, it is assumed that the government is endowed with an
enforcement technology. At the self-enforcing equilibrium, instead, the loss of resources only
occurs on out-of-equilibrium paths and nothing is transferred to creditors upon default (i.e.,
they represent dead-weight losses).
We argue that these differences are immaterial in a setting with complete markets and linear pricing. The proof exploits the characterization of equilibrium private debt (Theorem 3.1)
together with a straightforward decomposition of equilibrium public debt (Proposition 4.1
below). By securing a fraction of debt, tax revenue is a source of liquidity as it is the case
with endowment losses in a setting with self-enforcing debt. Any amount of public debt in
excess of backed resources is valued in the market as a speculative bubble, exactly as private
debt (in excess of the present value of income losses) is valued at the competitive equilibrium
with self-enforcing debt.
4.2.1
Institutional Arrangements
The environment is as before, but now we assume that individual agents can no longer
issue debt, i.e., Di (st ) = 0 for every agent i and any event st . Instead, debt is only issued
by a government, which backs its liabilities by taxing income according to a tax schedule
(τ i (st ))st s0 for each type i ∈ I.
b i (θi (s0 )|s0 ) denote the budget set of agent
Given an initial asset position θi (s0 ) > 0, let B
i ∈ I in this economy. It contains all pairs (ci , θi ) of consumption ci = (ci (st ))st s0 and public
debt holdings θi = (θi (st ))st s0 satisfying the (after-tax) budget constraint at any event st ,
20
i.e.,
ci (st ) +
X
st+1 s
q(st+1 )θi (st+1 ) 6 (1 − τ i (st ))y i (st ) +θi (st ),
|
{z
}
t
(4.7)
after-tax endowment
together with the no-borrowing restrictions
θi (st+1 ) > 0 at all successors st+1 st .
(4.8)
At any event st , the government issues public debt d(st+1 ) > 0 contingent to every successor
event st+1 . The outstanding debt d(st ) at an event st is financed partially by tax revenues
while the rest is rolled over across next period’s contingencies. The government’s budget
restriction is then
X
X
q(st+1 )d(st+1 ).
(4.9)
d(st ) 6
τ i (st )y i (st ) +
st+1 st
i∈I
Definition 4.1. Given a tax schedule (τ i (st ))st s0 for each agent i ∈ I and an allocation
(θi (s0 ))i∈I of initial asset positions, a competitive equilibrium with public debt (q, d, (ci , θi )i∈I )
consists of state-contingent bond prices q, a resource feasible consumption allocation (ci )i∈I ,
an allocation of government bond holdings (θi )i∈I , and the government’s net liability positions
d such that:
(i) for each agent i ∈ I, taking prices as given, the plan (ci , θi ) is optimal among budget
b i (θi (s0 )|s0 );
feasible plans in B
(ii) the debt market clears
X
θi (st ) = d(st ),
for all st s0 ;
(4.10)
i∈I
(iii) the government’s budget constraint (4.9) is satisfied at all contingencies st s0 .
4.2.2
Characterization of Public Debt
We show that, given the individual tax schedules, equilibrium public debt is decomposed
to a fundamental component reflecting the present value of total tax revenue and a speculative bubble. As opposed to an equilibrium with self-enforcing debt, the privilege of issuing
the speculative bubble is now attributed to the government that rolls over part of its debt
forever.
21
Proposition 4.1. At any competitive equilibrium with public debt,
d(st ) =
X
PV(τ i y i |st ) + M (st ),
for all st s0 ,
i∈I
where M is a nonnegative discounted martingale process. Equivalently, the government’s
debt level at any contingency is decomposed into the present value of total tax revenue and a
bubble component.
Proof. Fix a competitive equilibrium (q, d, (ci , θi )i∈I ) with public debt. To simplify the preP
sentation, let δ := i∈I τ i y i represent the taxed aggregate income process. Combining resource feasibility of (ci )i∈I , the government’s budget constraint, and the debt market clearing
condition, we deduce that the government’s budget constraints is satisfied with equality, i.e.,
d(st ) = δ(st ) +
X
q(st+1 )d(st+1 ).
st+1 st
Consolidating the above equations from date 0 to any arbitrary date T > 0, we get
p(s0 )d(s0 ) =
T X
X
p(st )δ(st ) +
t=0 st ∈S t
X
p(sT +1 )d(sT +1 ).
sT +1 ∈S T +1
Since d(sT +1 ) > 0, we deduce that the process δ has finite present value, i.e., PV(δ|s0 ) < ∞.
In particular, consolidating the government’s budget constraints along the sub-tree Σ(st ),
we get that
X
1
d(st ) = PV(δ|st ) +
lim
p(sT )d(sT ) .
p(st ) T →∞ T T
s ∈S
|
{z
}
=: M (st )
The desired result follows from the fact that M satisfies exact rollover.
Remark 4.1. Notice that characterizing public debt (i.e., proving Proposition 4.1) is far
simpler than characterizing private debt (i.e., proving Theorem 3.1). In fact, showing that
the total tax revenue has finite present value is straightforward, as the government’s budget
restriction (4.9) and market clearing imply the recursive property
d(st ) = δ(st ) +
X
st+1 st
22
q(st+1 )d(st+1 ).
Proving the same claim for the private endowment losses is more involved, as Lemma 3.1
reveals. This is because we do not know a priori whether the not-too-tight debt limits satisfy
the recursive property
X
Di (st ) = `i (st ) +
q(st+1 )Di (st+1 ),
st+1 st
so this recursive equation is obtained after applying a fixed-point theorem to a suitably
defined operator.
4.2.3
Equivalence Result
We now have all elements to establish an equivalence mapping between a competitive
equilibrium with public debt and a self-enforcing equilibrium. The implication is that public
liquidity is akin to private liquidity. Who is issuing the debt does not affect the economy’s
risk-sharing opportunities. Formally, using the characterization results established in Theorem 3.1 and Proposition 4.1, we obtain the following equivalence result.15
Theorem 4.1. A consumption allocation is the outcome of a competitive equilibrium with
public debt backed by taxes (τ i )i∈I if, and only if, it is the outcome of a competitive equilibrium
with self-enforcing debt and endowment losses (`i )i∈I such that `i = τ i y i for any i ∈ I.
Formally, we have the following properties:
(a) Let (τ i )i∈I be a family of tax schedules. If (q, d, (ci , θi )i∈I ) is a competitive equilibrium
with public debt where M is the bubble component of the government debt, i.e.,
X
d=
PV(τ i y i ) + M,
i∈I
then the family of income loss processes (`i )i∈I satisfying `i = τ i y i supports (q, (ci , ai , Di )i∈I )
as a competitive equilibrium with self-enforcing debt with
Di := PV(`i ) + M i
and
ai := θi − Di ,
where (M i )i∈I is any family of nonnegative discounted martingale processes satisfying
X
M i = M.
i∈I
15
Note that by setting ` = τ ≡ 0, Theorem 4.1 nests the equivalence result (Theorem 2) in Hellwig and
Lorenzoni (2009).
23
(b) Reciprocally, if (q, (ci , ai , Di )i∈I ) is a competitive equilibrium with self-enforcing debt associated with a family (`i )i∈I of income losses and M i is the bubble component of agent i’s
debt limits, i.e.,
Di = PV(`i ) + M i ,
then the family of tax schedules (τ i )i∈I satisfying `i = τ i y i supports (q, d, (ci , θi )i∈I ) as a
competitive equilibrium with public debt with
X
d :=
Di and θi := ai + Di .
i∈I
Proof. We only provide the proof of Part (a). Part (b) follows in the same spirit. The
following claim formalizes an observation that is a consequence of a translation invariance
property of the flow budget constraints.
Claim 4.1. If the plan (ci , ai ) belongs to the budget set B i (Di , ai (s0 )|s0 ), then the plan
b i (θi (s0 )|s0 ) where θi := ai + Di . Reciprocally, if the plan
(ci , θi ) belongs to the budget set B
b i (θi (s0 )|s0 ), then the plan (ci , ai ) belongs to the budget set
(ci , θi ) belongs to the budget set B
B i (Di , ai (s0 )|s0 ) where ai := θi − Di .
Fix a competitive equilibrium (q, d, (ci , θi )i∈I ) with public debt backed by taxes (τ i )i∈I .
It follows from Proposition 4.1 that PV(τ i y i |s0 ) is finite for each i. Moreover, there exists a
nonnegative process M satisfying exact roll-over such that
X
d=M+
PV(τ i y i ).
i∈I
Fix an arbitrary family (M i )i∈I of nonnegative processes satisfying exact roll-over and such
that
X
M i = M.
i∈I
i
i
i
i
i
Pose D := PV(` ) + M , a := θ − Di and `i := τ i y i . We claim that (q, (ci , ai , Di )i∈I )
constitutes a competitive equilibrium with self-enforcing debt and endowment losses (`i )i∈I .
Indeed, we first observe that the market clearing conditions are satisfied:
X
X
X
ai =
θi −
Di
i∈I
i∈I
i∈I
= d−
X
i∈I
= 0.
24
PV(`i ) − M
It follows from Theorem 3.1 that debt limits are not too tight. The debt constraints are
satisfied since θi > 0 for each i. To conclude the proof we only have to prove that (ci , ai ) is
optimal in the budget set B i (Di , ai (s0 )|s0 ). This is, however, true given Claim 4.1.
4.3
Collateralized Debt
We next focus on economies in which all borrowing and lending operations are fully
secured by collateral and default carries no other consequences than the loss of the collateral.
Following Chien and Lustig (2010), we assume that agents back their promises by means
of issuing and trading a long-lived asset whose dividend process accrues a fraction of their
income. The default option is endogenous and reflects the continuation value associated
with losing the privilege of the stream of liquidity services collateral provides to its owner.
In contrast to the economy with self-enforcing debt, there is no reputational effect: collateral
secures the full face value of the debt contract as opposed to endowment losses that have no
value to creditors.
We show that despite the differences in institutional arrangements, an economy with
self-enforcing debt can support as much risk-sharing as an economy with collateralized debt.
The established equivalence unravels an interesting link between credit and asset bubbles.
By securing debt, collateral is a source of liquidity in two ways. First, collateralized debt
reflects the assets’ fundamental value (i.e., the investment income due to dividend payments).
This is equivalent to endowment losses in economies with self-enforcing debt. Second, any
debt level in excess of the fundamental value reflects a bubble in the asset price, the same
way credit beyond the present value of income losses reflects a speculative bubble at the
self-enforcing equilibrium.
The finding has potential implications for the assessment of models with market exclusion.
Chien and Lustig (2010) find that their model supports more volatile risk-premiums and
interest rates compared to the setting in Alvarez and Jermann (2001), where autarky is
enforced upon default. Our equivalence result suggests that, for a different default option,
economies with self-enforcing debt might produce the same variation in equity risk-premiums
as in collateralized economies.
25
4.3.1
Institutional Arrangements
Assume that each agent i can pledge at the initial event s0 a part `i (st ) of the endowment
y i (st ) at any event st s0 . This is the equivalent of issuing a long-lived asset (Lucas tree)
whose dividend process is (`i (st ))st s0 . We now refer to the process `i as the collateralizable
income.
Shares of Lucas trees can be traded at every event st . We denote by P j (st ) the exdividend price of agent j’s tree, and we let αji (st ) > 0 represent agent i’s share on agent j’s
equity at event st .
Agents can also trade at every event st one-period-ahead contingent claims bi (st+1 ) ∈ R
for each successor event st+1 at a price q(st+1 ) (expressed in units of st -consumption). Agents
can default and file for bankruptcy. In this case, all assets (equity holdings) and current
period dividends are seized and transferred to lenders to redeem their debt. However, the
nonpledged endowment cannot be seized and agents still maintain access to financial markets.
This specification of the default punishment leads to the following debt constraints:
X
bi (st ) > −
αji (σ(st )) P j (st ) + `j (st ) , for all st s0 .
(4.11)
j∈I
Constraint (4.11) states that no agent can promise to deliver more than the value of her
equity holdings in any state. Following Chien and Lustig (2010), we refer to liquidity risk as
the risk associated with the constraint (4.11) be binding.
e i (bi (s0 )|s0 ) denote the
For each agent i, given an initial financial claim bi (s0 ), we let B
budget set consisting of all triples (ci , αi , bi ) of consumption processes (ci (st ))st s0 , equity
holdings (αi (st ))st s0 , and contingent claims (bi (st ))st s0 such that:
X
X
ci (s0 ) +
P j (s0 )αji (s0 ) +
q(s1 )bi (s1 ) 6 y i (s0 ) + bi (s0 ) + P i (s0 ),
(4.12)
s1 s0
j∈I
and at any event st s0 ,
X
X
ci (st ) +
P j (st )αji (st ) +
q(st+1 )bi (st+1 ) 6
j∈I
st+1 st
y i (st ) − `i (st ) + bi (st ) +
X
αji (σ(st )) `j (st ) + P j (st ) , (4.13)
j∈I
where b(st ) is subject to the liquidity constraint (4.11). Notice that at any event st s0 ,
agent i’s available income is only y i (st ) − `i (st ) since the rest has been sold at t = 0 in the
equity markets.
26
P
Definition 4.2. Given initial financial claims (bi (s0 ))i∈I such that i∈I bi (s0 ) = 0, a competitive equilibrium with collateralized debt (q, (P i )i∈I , (ci , αi , bi )i∈I ) consists of state-contingent
bond prices q, equity prices (P i )i∈I , a resource feasible consumption (ci )i∈I , an allocation of
nonnegative equity holdings (αi )i∈I , and an allocation of contingent claims (bi )i∈I such that:
(a) for each agent i ∈ I, taking prices as given, the plan (ci , αi , bi ) is optimal among budget
e i (bi (s0 )|s0 );
feasible plans in B
(b) all equity markets clear:
∀i ∈ I,
X
αij (st ) = 1,
for all st s0 ;
(4.14)
j∈I
(c) the bond market clears:
X
bi (st ) = 0,
for all st s0 .
(4.15)
i∈I
4.3.2
Characterization of Asset Prices
We derive a well-understood asset pricing equation that turns out to be useful for our
equivalence result. At any contingency, equity prices are decomposed into a fundamental
component related to dividend payments and a bubble component. Similarly to the equilibrium with self-enforcing debt, asset price bubbles emerge when dividends are small relative
to aggregate resources (see Santos and Woodford 1997).
Proposition 4.2. At any competitive equilibrium with collateralizable wealth, the cumdividend price of agent j’s equity is given by:
`j + P j = PV(`j ) + M j
(4.16)
for some nonnegative exact roll-over process M j .
Proof. Fix an agent j and an event st s0 . Market clearing implies that there exists at
least one agent i ∈ I who is holding a positive amount αji (st ) > 0 of the agent’s j equity.
Fix ε ∈ R such that ε > −αji (st ). The following changes in contingent claims and equity j’s
holding are admissible
α̃ji (st ) := αji (st ) + ε and b̃i (st+1 ) := bi (st+1 ) − ε[P j (st+1 ) + `j (st+1 )].
27
Since agent’s j welfare cannot improve after these changes, we must have
X
P j (st ) =
q(st+1 )[P j (st+1 ) + `j (st+1 )].
(4.17)
st+1 st
Given this recursive equation, it follows that PV(τ j y j |s0 ) is finite. Moreover, for every
event st , the following limit
X
1
M j (st ) = lim
p(sτ )P j (sτ )
τ →∞ p(st )
τ
τ t
s ∈S (s )
is well-defined, so we obtain equation (4.16).
4.3.3
Equivalence Result
We now have all elements to establish an equivalence mapping between a competitive
equilibrium with collateralized debt and a competitive equilibrium with public debt. Given
Theorem 4.1, the coincidence extends to a competitive equilibrium with self-enforcing debt.
More precisely, starting from an equilibrium in an economy where debt is self-enforcing, we
can always support the same consumption allocation in an economy that is subject to liquidity risks for some appropriately chosen levels of pledgeable resources. In that respect, there
is no difference in terms of trade between equilibria with self-enforcing debt and equilibria
with debt secured by collateral.
Theorem 4.2. Given a family (`i )i∈I representing individual pledgeable resources, a consumption allocation is the outcome of a competitive equilibrium with collaterized debt if, and
only if, it is the outcome of a competitive equilibrium with public debt backed by taxes (τ i )i∈I
satisfying `i = τ i y i (or, equivalently, a competitive equilibrium with self-enforcing debt and
endowment losses (`i )i∈I ). Formally, we have the following properties:
(a) Let (q, (P i )i∈I , (ci , αi , bi )i∈I ) be a competitive equilibrium with collateralizable income represented by (`i )i∈I and denote by M i the bubble component of agent i’s Lucas tree. Then,
(q, d, (ci , θi )i∈I ) is a competitive equilibrium with public debt backed by the family of taxes
(τ i )i∈I satisfying `i = τ i y i , where
X
θi (st ) := bi (st ) +
αji (σ(st ))[P j (st ) + τ j (st )y j (st )]
(4.18)
j∈I
and
d :=
X
PV(τ i y i ) + M i .
i∈I
28
(4.19)
(b) Let (q, d, (ci , θi )i∈I ) be a competitive equilibrium with public debt backed by the family
of taxes (τ i )i∈I . Denote by M the nonnegative process satisfying exact roll-over such
P
that d = i∈I PV(τ i y i ) + M . Fix any family (M i )i∈I of nonnegative processes satisfying
P
exact roll-over and such that i∈I M i = M . Fix also a family (αi )i∈I of shares satisfying
market clearing.16 Then, (q, (P i )i∈I , (ci , αi , bi )i∈I ) constitutes a competitive equilibrium
with collateralized debt (associated to individual pledgeable resources `i = τ i y i ) where
P i := PV(`i ) − `i + M i
(4.20)
and
bi (st ) := θi (st ) −
X
αji (σ(st ))[P j (st ) + `j (st )].
(4.21)
j∈I
Proof. We only prove part (a). Part (b) follows in the same spirit. The debt constraints (4.11)
imply that θi (st ), as defined by (4.18), is nonnegative. Combining the market clearing conditions (4.14) and (4.15), we get that
X
X
X
θi (st ) =
[P j (st ) + τ j (st )y j (st )] =
PV(τ i y i ) + M i ,
i∈I
j∈I
i∈I
where the last equality follows form the asset pricing equation (4.16). The market clearing
condition (4.10) then follows from our choice of the government debt. Combining the flow
budget constraints (4.12) and (4.13) together with the asset pricing equation (4.16), we get
b i (θi (s0 )|s0 ). Reciprocally, let (c̃i , θ̃i ) be a
that the pair (ci , θi ) belongs to the budget set B
b i (θi (s0 )|s0 ). Let b̃i be the process of bond holdings defined by
plan in the budget set B
X
b̃i (st ) := θ̃i (st ) −
αji (σ(st ))[P j (st ) + τ j (st )y j (st )].
j∈I
Using the asset pricing equation (4.16) and the fact that b̃i (s0 ) = bi (s0 ), we can show that
e i (s0 )|s0 ). Since (ci , αi , bi ) is optimal in
the plan (c̃i , αi , b̃i ) belongs to the budget set B(b
e i (s0 )|s0 ), we must have U (c̃i |s0 ) 6 U (ci |s0 ). This proves that the plan (ci , θi ) is optimal
B(b
b i (θi (s0 )|s0 ). We can conclude that (q, d, (ci , θi )i∈I ) is a competitive equilibrium with
in B
public debt backed by the family of taxes (τ i )i∈I .
Remark 4.2. The following observation is a direct consequence of our equivalence result.
Assume, as in Chien and Lustig (2010), that there exists ` > 0 such that `i (st ) = ` for all i
16
In the sense that
P
j∈I
αji (st ) = 1 for all i ∈ I and all st s0 .
29
and st and that there is no endowment growth. If ` > 0, then Corollary 3.1 implies that
the present value of pledgeable resources is finite and assets are priced at their fundamental
value, so they are bubble-free. One may think that for ` = 0 (i.e., assets pay no dividends),
asset prices must equal zero, so autarky is the only equilibrium outcome. However, this claim
is true provided that the aggregate wealth is still finite, or equivalently, when the implied
interest rates are higher than growth rates. Nevertheless, as documented by Hellwig and
Lorenzoni (2009), when ` = 0, equilibrium interest rates can be sufficiently low so that the
economy’s aggregate wealth is infinite. The implication for the collateral equilibrium, is that,
even if the trees pay no dividend, assets may be priced as a speculative bubble. Indeed, it is
sufficient to appeal to Theorem 4.1 and Theorem 4.2 and translate the bubbly equilibrium
of Hellwig and Lorenzoni (2009) in the environment of Chien and Lustig (2010).
The intuition for this discrepancy relies on the dual role of collateral as a source of
liquidity. As dividends become negligible (i.e., ` approaches zero), the value of the asset
increases to compensate for the decreased investment value. In the limit, the value of the
collateral asset is still positive, reflecting purely a bubble, even though there is no collateral
in the market anymore.
4.4
Existence of a Competitive Equilibrium
It is well-known that proving the existence of a competitive equilibrium with self-enforcing
debt is difficult. The presence of self-enforcing conditions does not permit a direct adaptation
of a standard truncation technique, as is the case in models with exogenous debt limits. We
show below how to bypass these complications by proving the existence of a competitive
equilibrium with public debt and then appealing to our equivalence result (Theorem 4.1).
Our insight derives from the observation that applying a truncation argument to the economy
with public debt, though it requires taking into account the endogeneity of the supply side
(condition 4.9), is far more simple than dealing directly with the endogeneity of individual
debt limits in the economy with self-enforcing debt.
Formally, the idea of proof strategy goes as follows. We can apply standard arguments
based on continuity, convexity, and compactness to prove the existence of a competitive equilibrium (qT , dT , (ciT , θTi )i∈I ) with public debt in a truncated economy with finite horizon T .
Passing to a subsequence if necessary, we can assume that the sequence of consumption
allocation (ciT (st ))T >1 converges to a feasible allocation (ci (st ))i∈I . Convergence of bond
prices (qT (st ))T >1 to some price q(st ) follows then from the Euler equation. The flow budget
30
constraints remain valid in the limit. The only difficult part is to show that bond holdings converge (or admit a converging subsequence. Since markets clear, it is sufficient to
show that outstanding public debt is bounded. Intuitively, if the sequence (dT (s0 ))T >1 of
initial public debt is unbounded, then there must be at least one agent with an arbitrary
large amount θTi (s0 ) of initial resources as T goes to infinite. This agent could support an
arbitrary large consumption plan that would necessarily violate optimallity.
Theorem 4.3. Fix a family of tax schedules (τ i )i∈I and consider an arbitrary decomposition
of tax revenues
X
X
τ iyi =
δi
i∈I
i∈I
where δ i is a nonnegative process. If aggregate tax revenues are nonzero at any contingency,
i.e.,
X
τ i (st )y i (st ) > 0, for all event st ,
i∈I
then there exists a competitive equilibrium with backed public debt (q, d, (ci , θi )i∈I ) where the
allocation of initial bond holdings satisfies θi (s0 ) > PV(δ i |s0 ) for each agent i.
A formal proof is presented in Appendix A.3.17 Existence of a competitive equilibrium
with self-enforcing debt is then a direct corollary of the last theorem and Theorem 4.1.
Corollary 4.1. Fix a family of income loss processes (`i )i∈I and consider an arbitrary decomposition
X
X
δi
`i =
i∈I
i∈I
where δ i is a nonnegative process. There exists a competitive equilibrium with self-enforcing
debt (q, (ci , ai , Di )i∈I ) where the allocation of initial bond holdings satisfies ai (s0 ) = PV(δ i −
`i |s0 ) for each agent i.
Proof. Fix tax schedules (τ i )i∈I satisfying `i = τ i y i for each i. From Theorem 4.3 there
exists a competitive equilibrium with backed public debt (q, d, (ci , θi )i∈I ) where the allocation of initial bond holdings satisfies θi (s0 ) > PV(δ i |s0 ) for each agent i. We can apply
17
Assuming that aggregate tax revenues are nonzero at any contingency facilitates proving that the
limiting bond prices are nonzero. This requirement could be relaxed, but the argument becomes more
involved.
31
Proposition 4.1 to get the existence of a nonnegative discounted martingale process M such
that
X
d=M+
PV(τ i y i ).
i∈I
i
0
i
0
i
0
We pose M (s ) := θ (s ) − PV(δ |s ), for each i. Observe that M i (s0 ) > 0 and
X
X
M i (s0 ) = d(s0 ) −
PV(τ i y i |s0 ) = M (s0 ).
i∈I
i∈I
We can then extend the definition of M i (s0 ) to the whole tree Σ such that each M i is a
P
nonnegative discounted martingale and M = i∈I M i .18 We can now apply Theorem 4.1
to deduce that (q, (ci , ai , Di )i∈I ) is a competitive equilibrium with self-enforcing debt where
Di := PV(`i ) + M i and ai := θi − Di for each i ∈ I. The way we define each M i (s0 ), implies
that ai (s0 ) = PV(δ i − `i |s0 ).
Remark 4.3. If we choose the decomposition such that δ i = `i for each i, we then get the
existence of a competitive equilibrium with zero initial financial endowments (ai (s0 ) = 0).
Existence is also guaranteed for many other allocations of initial financial endowments. For
P
instance, we may have an agent j starting with the initial financial endowment i∈I PV(`i |s0 )
while all other agents i 6= j start with (self-enforcing) debt PV(`i |s0 ).
Existence of a competitive equilibrium with self-enforcing debt as presented in Corollary 4.1 is an important result. However, equilibria are only tractable if they are Markov for
some simple endogenous state variable. For the rest of the section we assume that uncertainty is governed by a Markov process on a finite space S with strictly positive transitions,
i.e., there exists a vector
(π(s0 |s))(s,s0 )∈S 2
with π(s0 |s) > 0
such that
(a) s0 =: s0 ∈ S and for every t > 1, an event st is a history (s0 , s1 , . . . , st ) where each
sr ∈ S;
(b) conditional probabilities satisfy π(st+1 |st ) = π(st+1 |st ) for all st+1 st .
18
A possible way (among infinitely many) to construct the family (M i )i∈I is as follows: if M (s0 ) = 0,
then M i (s0 ) = 0 for all i, and we can pose M i (st ) := 0 for every st s0 . If M (s0 ) > 0, then let
αi := M i (s0 )/M (s0 ) and pose M i (st ) := αi M (st ) for every i and every st s0 .
32
Individual endowments y i and income losses `i oscillate according to this Markov process
and, hence, all fundamentals are measurable with respect to the finite Markov state space
S.19 This, in particular, implies that the economy cannot grow or decline over time, that is,
the aggregate endowment is bounded.
In general, at a competitive equilibrium, prices and debt limits would be affected by
the distribution of wealth, as well as possibly by future expectations, and would not be
measurable with respect to the Markov state space S. Following the recent contributions
of Chien and Lustig (2010) and Gottardi and Kubler (2015), we consider the instantaneous
Negishi weights as a candidate for the endogenous state variable. Indeed, fix an arbitrary
event st and observe that an allocation (ci (st ))i∈I of strictly positive consumption satisfies
P
P
i t
i
i∈I c (s ) = ȳ(st ) where ȳ :=
i∈I y if, and only if, there exists a unique (up to positive
scaling) family λ(st ) = (λi (st ))i∈I ∈ RI++ of strictly positive instantaneous Negishi weights
such that
(
)
X
X
(ci (st ))i∈I ∈ argmax
λi (st )u(xi ) : (xi )i∈I ∈ RI++ and
xi = ȳ(st ) .
(4.22)
i∈I
i∈I
The unique feasible allocation that solves the maximization problem (4.22) can be expressed
as a continuous function (λ(st ), st ) 7−→ c̃(λ(st ), st ) = (c̃i (λ(st ), st ))i∈I from RI++ × S to RI+ .
It is then natural to introduce the following definition.
Definition 4.3. A stationary Markov equilibrium with instantaneous Negishi weigths is a
family of policy functions ((q̃(s0 )s0 ∈S ), (c̃i , ãi , D̃i )i∈I ) defined on RI++ × S and a transition
function L : RI++ × S → RI++ satisfying the following properties.
(1) For each state s, the set Λ(s) ⊆ RI++ of instantaneous Negishi weights λ such that
ãi (λ, s) > −D̃i (λ, s) for each i is nonempty and L(λ, s) ⊆ Λ(s) for all instantaneous
Negishi weights λ ∈ RI++ .
(2) For any initial λ0 ∈ Λ(s0 ), the family (q, (ci , ai , Di )i∈I ) defined by
(ci (st ), ai (st ), Di (st )) := (c̃i (λ(st ), st ), ãi (λ(st ), st ), D̃i (λ(st ), st ))
(4.23)
and for any st+1 st
q(st+1 ) := q̃(λ(st ), st )(st+1 )
19
(4.24)
In other words, y i (st ) and `i (st ) only depend on the current shock st . We abuse notations by writing
y (st ) and `i (st ).
i
33
is a competitive equilibrium with self-enforcing debt where the process (λ(st ))st s0 is
defined recursively by
λ(s0 ) := λ0
and λ(st ) := L(λ(σ(st )), st ),
for all st s0 .
(4.25)
We can now combine Theorem 5 in Gottardi and Kubler (2015) with our equivalence
results to prove the existence and uniqueness of a stationary Markov equilibrium under
specific conditions on primitives.
Proposition 4.3. Assume that endowment losses are nonnegligible and that the map c 7→
u0 (c)c is increasing over R++ . Then there exists a unique stationary Markov equilibrium with
self-enforcing debt.
The proof of this result is presented in Appendix A.4. In the spirit of Theorem 6 in
Gottardi and Kubler (2015), if there are only two agents, we can identify further conditions
on primitives under which the unique stationary Markov equilibrium has finite support. This
is important since such an equilibrium can be characterized by a finite system of equations
and can typically be computed easily. One can also conduct local comparative statics using
the implicit function theorem.
5
Conclusion
Since the global financial crisis, there has been a revival in the literature that aims to
understand credit and asset bubbles. In this paper, we have shown that credit bubbles can
naturally arise in various general equilibrium environments with limited commitment, in a
similar manner to how asset price bubbles can naturally arise in environments with limited
asset supply. We also provided several applications of our theory to the context of sovereign
debt, domestic public debt, and consumer debt. While we have focused on equilibria where
bubbles are safe, it is possible to extend our framework to incorporate sunspot shocks (as
in Calvo 1988, Cole and Kehoe 2000, Azariadis et al. 2015) to allow for the possibility of
stochastically bursting credit bubbles. It would be interesting for future research to study the
general equilibrium effects of the booms and busts of credit bubbles in the various economic
environments that we have studied.
34
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A
A.1
Appendix: Omitted Proofs
Proof of Lemma 3.1
Since we are exclusively concerned with the single-agent problem, we simplify notation
throughout this section by dropping the superscript i.
Let D be a process of not-too-tight bounds. We first show that there exists a nonnegative
process D satisfying
X
q(st+1 ) min{D(st+1 ), D(st+1 )}, for all st s0 .
(A.1)
D(st ) = `(st ) +
st+1 st
Indeed, let Φ be the mapping B ∈ RΣ 7−→ ΦB ∈ RΣ defined by
X
(ΦB)(st ) := `(st ) +
q(st+1 ) min{D(st+1 ), B(st+1 )},
for all st s0 .
st+1 st
Denote by [0, D̄] the set of all processes B ∈ RΣ satisfying 0 6 B 6 D̄ where
X
D̄(st ) := `(st ) +
q(st+1 )D(st+1 ), for all st s0 .
st+1 st
The mapping Φ is continuous (for the product topology), and we have Φ[0, D̄] ⊆ [0, D̄].
Since [0, D̄] is convex and compact (for the product topology), it follows that Φ admits a
fixed point D in [0, D̄].
39
Claim A.1. The process D is tighter than the process D, i.e., D 6 D.
Proof of Claim A.1. Fix a node st . Since V` (0, 0|st ) = V (D, −D(st )|st ) and V (D, ·|st ) is
strictly increasing, it is sufficient to show that V (D, −D(st )|st ) > V` (0, 0|st ). Denote by
(c̃, ã) the optimal consumption and bond holdings associated with the default option at st ,
b be the process defined by D(s
b t ) := min{D(st ), D(st )} for
i.e., (c̃, ã) ∈ d` (0, 0|st ).20 We let D
all st . Observe that
y(st ) − D(st ) = y(st ) − `(st ) −
X
b t+1 )
q(st+1 )D(s
st+1 st
X
= c̃(st ) +
b t+1 )]
q(st+1 )[ã(st+1 ) − D(s
st+1 st
X
t
= c̃(s ) +
q(st+1 )a(st+1 )
st+1 st
b t+1 ). Since D
b 6 D, we have a(st+1 ) > −D(st+1 ). At any
where a(st+1 ) := ã(st+1 ) − D(s
successor event st+1 st , we have
b t+1 )
y(st+1 ) + a(st+1 ) = y(st+1 ) + ã(st+1 ) − D(s
> y(st+1 ) + ã(st+1 ) − D(st+1 )
> y(st+1 ) − `(st+1 ) + ã(st+1 ) −
X
b t+2 )
q(st+2 )D(s
st+2 st+1
> c̃(st+2 ) +
X
b t+2 )]
q(st+2 )[ã(st+2 ) − D(s
st+2 st+1
> c̃(st+2 ) +
X
q(st+2 )a(st+2 )
st+2 st+1
b t+2 ).21 Observe that a(st+2 ) > −D(st+2 ) (since D
b 6 D).
where a(st+2 ) := ã(st+2 ) − D(s
b τ ) for any successor sτ st and iterating the above arguDefining a(sτ ) := ã(sτ ) − D(s
ment, we can show that (c̃, a) belongs to the budget set B(D, −D(st )|st ). It follows that
V (D, −D(st )|st ) > U (c̃|st ) = V` (0, 0|st )
implying the desired result: D(st ) 6 D(st ).
20
21
Equivalently, (c̃, ã) satisfies U (c̃|st ) := V` (0, 0|st ) and belongs to B` (0, 0|st ).
To get the second weak inequality, we use equation (A.1).
40
It follows from Claim A.1 that D satisfies
X
D(st ) = `(st ) +
q(st+1 )D(st+1 ),
for all st s0 .
(A.2)
st+1 st
Applying equation (A.2) recursively, we get
p(st )D(st ) = p(st )`(st ) +
X
p(st+1 )`(st+1 ) + . . .
st+1 ∈S t+1 (st )
X
... +
X
p(sT )`(sT ) +
sT ∈S T (st )
p(sT +1 )D(sT +1 )
sT +1 ∈S T +1 (st )
for any T > t. Since D is nonnegative, it follows that
p(st )D(st ) >
T
X
X
p(sτ )`(sτ ).
τ =t sτ ∈S τ (st )
Passing to the limit when T goes to infinity, we get that PV(`|st ) is finite for any event st
(in particular for s0 ). Recalling that D > D, we also get that D(st ) > PV(`|st ).
A.2
Proof of Lemma 3.2
Denote by (ci , ai ) the optimal consumption and bond holdings in the budget set B`ii (0, 0|sτ )
for some arbitrary event sτ . We pose Di := PV(`i ) and observe that
X
Di (st ) = `i (st ) +
q(st+1 )Di (st+1 ).
st+1 st
We can easily show that (ci , ai ) is optimal in the budget set B i (Di , −Di (s0 )|sτ ) where ai :=
ai − Di . We then deduce that V i (Di , −Di (sτ )|sτ ) = V`ii (0, 0|sτ ).
A.3
Proof of Theorem 4.3
P
To simplify the presentation, we let δ := i∈I τ i y i . Fix an arbitrary decomposition of δ
as follows:
X
δ=
δ i where δ i > 0 for each i.
i∈I
41
We claim that there exists a competitive equilibrium (q, d, (ci , θi )i∈I ) with public debt backed
by the tax schedules (τ i )i∈I and such that the initial bond holdings satisfy
θi (s0 ) > PV(δ i |s0 ),
for each i ∈ I.
To prove existence, we follow the standard truncation approach. Given an arbitrary date
ξ > 1, for every event st with t 6 ξ, the ξ-truncated present value PVξ (x|st ; q) of some
process x under the price process q is defined by
ξ
PVξ (x|st ; q) =
1 X
p(st ) r=t
X
p(sr )x(sr )
sr ∈S r (st )
where p is the process of Arrow–Debreu prices associated with q.
Standard arguments (continuity, compacity, and convexity) can be applied to prove the
existence of a ξ-truncated equilibrium with public debt defined as a family (qξ , dξ , (ciξ , θξi )i∈I )
where
(i) for each agent i ∈ I, taking prices as given, the plan (ciξ , θξi ) maximizes the utility U (c)
among all budget feasible plans (ci , θi ) satisfying the initial condition
θi (s0 ) = PVξ (δ i |s0 ; qξ ),
for every t 6 ξ and every event st ∈ S t ,
X
ci (st ) +
qξ (st+1 )θi (st+1 ) = (1 − τ i (st ))y i (st ) + θi (st ) and θi (st ) > 0
st+1 st
and for every t > ξ and every st ∈ S t
ci (st ) = y i (st ) and θi (st ) = 0;
(ii) the government debt market clears
X
θξi (st ) = dξ (st ),
for all t 6 ξ and st ∈ S t ;
i∈I
(iii) the government budget constraint is satisfied
dξ (st ) = PVξ (δ|st ; qξ ),
42
for all t 6 ξ and st ∈ S t .
For each ξ, the consumption allocation (ciξ )i∈I satisfies market clearing. Passing to a
subsequence if necessary, we can assume that there exists a consumption allocation (ci )i∈I ,
also satisfying market clearing, such that, for each i, the sequence (ciξ )ξ>1 converges (for the
product topology) to ci . We can also assume that for each event st , the sequence of bond
prices (qξ (st ))ξ>1 converges to some number q(st ) ∈ [0, ∞].
Claim A.2. For every event st , we have q(st ) < ∞.
Proof. Fix an event st and assume, by way of contradiction, that the sequence (qξ (st ))ξ>1
converges to ∞. Fix an arbitrary agent i. If there exists some large enough ξ¯ such that
¯ then we have
θξi (st ) = 0 for every ξ > ξ,
ciξ (st ) 6 (1 − τ i (st ))y i (st )
and passing to the limit, we get that ci (st ) 6 (1 − τ i (st ))y i (st ). Assume now that there
i
(st ) > 0 for every ξ. From the
exists a strictly increasing function ϕ : N → N such that θϕ(ξ)
first-order condition associated with agent i’s maximization problem, we have that
t
t
u0 (ciϕ(ξ) (st ))
t
qϕ(ξ) (s ) = βπ(s |σ(s ))
u0 (ciϕ(ξ) (σ(st )))
.
Passing to the limit when ξ approaches infinity, we deduce that ci (st ) = 0.22 We have then
proved that either ci (st ) 6 (1 − τ i (st ))y i (st ) or ci (st ) = 0. This contradicts the fact that the
P
allocation (ci )i∈I satisfies market clearing since we assumed that i∈I τ i (st )y i (st ) > 0.
We claim that the sequence (dξ (s0 ))ξ>1 is bounded. Indeed, first observe that there exists
a date T such that UT (2y|s0 ) > U (y|s0 ) where
UT (c|s0 ) =
T
X
t=0
βt
X
π(st )u(c(st )).
st ∈S t
Since the sequence (qξ )ξ>1 of bond prices converges to q, there exists ξ0 large enough such
that
PVT (2y|s0 ; qξ ) 6 PVT (2y|s0 ; q) + 1, for all ξ > ξ0 .
Now, assume by way of contradiction that (dξ (s0 ))ξ>1 is unbounded. There must exist ξ,
say ξ > ξ0 , such that dξ (s0 ) > (#I)(PVT (2y|s0 ; q) + 1). In particular, there must exist an
22
Recall that ciϕ(ξ) (σ(st )) is bounded from above by aggregate endowments.
43
agent i such that θξi (s0 ) > PVT (2y|s0 ; q) + 1. With this initial bond holding, agent i can
finance the consumption 2y(st ) for any t 6 T . By optimality, we deduce that
U (ciξ |s0 ) > UT (2y|s0 ) > U (y|s0 ),
which contradicts the feasibility of the allocation (ciξ )i∈I .
We have thus proved that the sequence (dξ (s0 ))ξ>1 is bounded. This implies that for
each event st , the sequence (dξ (st ))ξ>1 is bounded. Since the markets for bonds clear, we
deduce that for each i and each event st , the sequence (θξi (st ))ξ>1 is bounded. Passing to
a subsequence if necessary, we can assume that for each i, there exists a process of bond
holdings θi such that the sequence (θξi )ξ>1 converges for the product topology to θi . It follows
from the flow budget constraints satisfied by (ciξ , θξi ) that
ci (st ) +
X
q(st+1 )θi (st+1 ) 6 (1 − τ i (st ))y i (st ) + θi (st ).
st+1 st
b i (θi (s0 )|s0 ). We omit the
In other words, the plan (ci , θi ) is a budget feasible plan in B
standard arguments to show that the plan (ci , ai ) is optimal among budget feasible plans in
b i (θi (s0 )|s0 ).
B
Fix an event st . Since the government debt market clears for every ξ-truncated equilibrium, the sequence (dξ (st ))ξ>1 of government debts converges to some d(st ) satisfying
X
d(st ) =
θi (st ).
i∈I
We still have to prove that the government budget restriction is satisfied. Recall that for
any ξ-truncated economy (with ξ > t), the government budget constraint is satisfied
dξ (st ) = δ(st ) +
X
qξ (st+1 )dξ (st+1 ).
st+1 st
Passing to the limit, we get that
d(st ) = δ(st ) +
X
q(st+1 )d(st+1 ).
st+1 st
We have thus proved that (q, d, (ci , θi )i∈I ) is a competitive equilibrium with public debt
backed by the tax schedule (τ i )i∈I .
44
Fix now an agent i and an arbitrary T > 0. Recall that for any ξ > T , we have
θξi (s0 ) = PVξ (δ i |s0 ; qξ ) > PVT (δ i |s0 ; qξ ).
Passing to the limit when ξ approaches infinity, we get that23
θi (s0 ) > PVT (δ i |s0 ; q) = PVT (δ i |s0 ).
Since the sequence (PVT (δ i |s0 ))T >0 is increasing, we deduce that it converges and satisfies
θi (s0 ) > lim PVT (δ i |s0 ) = PV(δ i |st ).
T →∞
A.4
Proof of Proposition 4.3
To simplify the presentation, we let ei (s) := y i (s) − `i (s) denote agent i’s endowment
after default. It follows from Theorem 5 in Gottardi and Kubler (2015) that there exists a
family (V (·, s), L(·, s))s∈S of functions
V (·, s) : RI++ −→ RI
and L(·, s) : RI++ −→ RI++
satisfying the following properties: for every λ ∈ RI++ , for every s ∈ S, and for every α > 0,
(a) L(λ, s) > λ and L(αλ, s) = αL(λ, s);
(b) V (L(λ, s), s) > 0 and V (αλ, s) = V (λ, s);
(c) for each i,
V i (L(λ, s), s) > 0 =⇒ Li (λ, s) = λi ;
(d) for each i,
V i (λ, s) = u0 (c̃i (λ, s))[c̃i (λ, s) − ei (s)] + β
X
π(s0 |s)V i (L(λ, s0 ), s0 );
s0 ∈S
where we recall that c̃(λ, s) = (c̃i (λ, s))i∈I is the unique solution of the following maximization
problem
(
)
X
X
max
λi u(xi ) : (xi )i∈I ∈ RI++ and
xi = ȳ(s) .
i∈I
i∈I
23
When the present value is computed with the limit price process q, we omit to specify the price in the
definition.
45
We let θ̃i (·, s) : RI++ → R be defined by
θ̃i (λ, s) :=
V i (λ, s)
,
u0 (c̃i (λ, s))
for all λ ∈ RI++
and for each current state s and contingent to each possible future state s0 , we let q̃(·, s)(s0 )
be defined by
q̃(λ, s)(s0 ) := βπ(s0 |s) max
i∈I
u0 (c̃i (L(λ, s0 ), s0 ))
,
u0 (c̃i (λ, s))
for all λ ∈ RI++ .
Combining conditions (a), (c), and (d), we get that for each i,
X
c̃i (λ, s) +
q(λ, s)(s0 )θ̃i (L(λ, s0 ), s0 ) = ei (s) + θ̃i (λ, s).
s0 ∈S
Condition (b) implies that
θ̃(L(λ, s0 ), s0 ) > 0,
for all s0 .
If λ ∈ Λ(s), we also have that θ̃(λ, s) > 0.
Fix an arbitrary λ0 ∈ Λ(s0 ). Define the process (λ(st ))st s0 according to (4.25), the
process (q(st ))st s0 according to (4.24), the process (ci (st ))st s0 according to (4.23) and let
b i (θi (s0 )|s0 ). Moreover, following the
θi (st ) := θ̃i (λ(st ), st ). We have proved that (ci , θi ) ∈ B
b i (θi (s0 )|s0 ).
arguments in Gottardi and Kubler (2015), the plan (ci , θi ) is actually optimal in B
Fix an arbitrary date T . Consolidating the budget restrictions from date 0 to date T and
summing over the agents, we get
T X
X
X
p(st )δ(st ) +
t=0 st ∈S t
p(sT +1 )
X
θi (sT +1 ) =
i∈I
sT +1 ∈S T +1
X
θi (s0 )
i∈I
where δ := i∈I `i . The above inequality implies that PV(δ|s0 ) is finite. Since endowment
losses are nonnegligible, we deduce that PV(ȳ|s0 ) is finite. In particular, for each i the
consumption plan ci has finite present value. We can then apply Lemma B.1 in Martins-daRocha and Vailakis (2017a) to deduce the following market transversality condition
X
lim
p(st )θi (st ) = 0.
P
t→∞
st ∈S t
The above condition implies that
X
θi (s0 ) = PV(δ|s0 ).
i∈I
46
This is exactly the budget restriction of the government at the initial event s0 . Reproducing
the above argument at any event st , we can show that
X
θi (st ) = PV(δ|st ).
i∈I
We have thus proved that (q, d, (ci , θi )i∈I ) is a competitive equilibrium with public debt where
d := PV(δ).
Observe that for any arbitrary s0 ∈ S and any λ0 ∈ Λ(s0 ), we have constructed a
competitive equilibrium with public debt
(qλ0 ,s0 , dλ0 ,s0 , (ciλ0 ,s0 , θλi 0 ,s0 )i∈I )
such that PV(`i |s0 ; qλ0 ,s0 ) < ∞. For any s ∈ S and any λ ∈ Λ(s), we pose
D̃i (λ, s) := PV(`i |s; qλ,s ).
By construction, we have
D̃i (λ, s) = `i (s) +
X
q̃(λ, s)(s0 )D̃i (L(λ, s0 ), s0 ).
s0 ∈S
Since L(λ, s0 ) ∈ Λ(s0 ) for any instantaneous Negishi weights λ ∈ RI++ , we can use the above
recursive formula to extend the definition of D̃i (·, s) to the whole set RI++ .
B
Online Supplemental Material
Please visit https://www.dropbox.com/s/7rwl9m0yutfpo28/BR_GE_supplement.pdf?
dl=1
47
@FedFRASER