Limited Enforcement of Debt Contracts and Macroeconomic Implications for Asset Prices

INTRODUCTION 2
The predictions of this model are not consistent with several empirical
findings. In particular, as extensively discussed in the final part of this work,
it requires a high level of risk aversion in order to explain the observed level
of risk premia. In light of this, several modifications to this canonical model
have been proposed, mainly due to the resolution of empirical puzzles; among
such modifications, the formulation of alternative preference structures, the
consideration of incomplete markets models and the introduction of mar-
ket frictions, such as transaction costs, imperfect information and imperfect
enforceability of financial contracts (or limited commitment).
Limited commitment results in the imposition of material constraints
on borrowing. Consumption-based asset pricing models with endogenous
debt constraints under limited commitment are the main focus of this work.
The main contributions to this topic include Kehoe and Levine [25], Kocher-
lakota [27] and Alvarez and Jermann [4]. In Kehoe and Levine’s competitive
equilibrium model, contracts are supposed to be imperfectly enforceable and
endogenous debt limits have the form of individual rationality constraints.
Their model of partial insurance is used to explain the imperfect correlation
between individual and aggregate consumption. Kocherlakota develops a no-
commitment environment in order to analyze the imperfect diversification of
individual consumption risk; differently from Kehoe and Levine, he assumes
that agents can interact strategically. Finally, Alvarez and Jermann build
their competitive equilibrium on both Kehoe and Levine and Kocherlakota’s
works, but, differently from them, they consider a sequential formulation and
not-too-tight borrowing constraints.
INTRODUCTION 3
This work is aimed at analyzing Alvarez and Jermann’s model with en-
dogenous (not-too-tight) borrowing constraints, its asset pricing implications
and thus its contribution to the explanation of some empirical findings. For
this purpose, I will first introduce a model with natural debt limits and full
commitment and then Alvarez and Jermann’s model with endogenous debt
limits and limited commitment, emphasizing their different features and im-
plications.
Both these models are not representative agent models, such as that of
Lucas, rather they consider a finite set of consumers. In each period of trade,
such consumers must allocate their wealth between current consumption and
the purchase of an asset. Since borrowing is allowed, in order for solvency
to be guaranteed, material constraints on borrowing need to be imposed.
Otherwise consumers would be allowed to finance unbounded levels of con-
sumption by always rolling over existing debt. In particular, the two models
under discussion impose natural and endogenous debt limits, respectively.
This work aims at exploring the role and the features of both such debt
constraints, focusing on the way they are determined, their implications and
their effectiveness. Natural debt limits (among others, Levine and Zame
[34]) require that in every date the amount of debt does not exceed the
current value of future endowments. This allows consumers to repay their
debt by converting all future endowments into current wealth. Therefore,
solvency merely depends on the material availability of incomes. Agents are
not allowed to repudiate their debt (full commitment). Differently, in case
of endogenous debt limits by Alvarez and Jermann, they are allowed to re-
nege on their debt and default (limited commitment). The option to default
INTRODUCTION 4
results in agent-specific debt constraints, that is, debt constraints that pre-
vent highly indebted agents, hence agents who have an higher incentive to
default, from borrowing again. Such constraints are said to be endogenous
as they depend on risk of default. Default is punished by the exclusion from
financial markets and by the seizure of assets. Alvarez and Jermann define
their debt limits not too tight because they must be tight enough to pre-
vent default, but, at the same time, allow as much risk sharing as possible.
Strictly speaking, they allow agents to borrow up to the amount in which
they are indifferent between repaying and defaulting. It is worth noting that
the option to default makes insurance partial. In fact, in case of natural debt
limits, if there is no uncertainty and the aggregate endowment is constant,
individual consumption is stationary at equilibrium; hence, agents can fully
insure against fluctuations in their individual income. Conversely, in case of
Alvarez and Jermann’s endogenous debt limits, agents are unable to fully
insure against idiosyncratic volatility of income, because default is allowed.
Several scholars have built their works on the literature on endogenous
debt constraints. Alvarez and Jermann [5] and Krueger and Perri [33] have
developed a model with endogenous solvency constraints, which builds on
earlier works by Kehoe and Levine [25] and Alvarez and Jermann [4], to ad-
dress some empirical questions (see chapter 4 for a detailed discussion of this
topic). Hellwig and Lorenzoni [24] have considered the same environment
as in Kehoe and Levine [25] and Alvarez and Jermann [4], but with weaker
consequences of default, in order to explore debt sustainability under limited
contract enforcement. They suppose that, since agents’ default is publicly
INTRODUCTION 5
observable, the agents who repudiate debt lose the ability to borrow in fu-
ture periods and their assets can be seized by creditors. Hence, in this case
agents who default do not revert to autarchy. Hellwig and Lorenzoni’s self-
enforcing debt constraints, as Kehoe and Levine’s participation constraints
and Alvarez and Jermann’s not-too-tight debt constraints, relies on an incen-
tive mechanism: agents will repay their debt only if it is convenient for them
to do so. Thus, debt limits are endogenously determined so as to provide
agents with the incentive to repay their debt; to be more precise, they are
determined so that agents are indifferent between repaying and defaulting.
This work is organized as follows.
Chapter 1 describes the optimization problem of a single consumer over
an infinite horizon, with no uncertainty and sequentially complete markets;
the consumer is subject to debt constraints and sequential budget con-
straints. The sequential formulation of the consumer problem is definitely
more realistic than an intertemporal formulation, but the latter is much
more tractable. In light of this, I will demonstrate that, in the particular
case of natural debt limits, the sequence of budget constraints is equivalent
to a single intertemporal budget constraint. The latter does not require the
imposition of debt limits. Subsequently, I will determine an interior solution
to the consumer’s problem, first in case of natural debt limits, then in case of
arbitrary debt limits, emphasizing the differences between the corresponding
first order conditions.
Chapter 2 discusses the competitive equilibrium analysis of an full com-
mitment economy with natural debt limits under no uncertainty. The first
INTRODUCTION 6
part of the chapter describes the economy and introduces the notion of equi-
librium. The infinite-horizon, endowment economy has sequentially complete
markets and consists of a finite set of individuals, who use their accumulated
wealth to finance consumption in excess to current income and invest in a
one-period bond. Since there is no uncertainty, consumers do not need to
operate in financial markets in order to insure against risk, but they still need
to do so because their individual endowments fluctuate over time. Aggregate
endowment is assumed to be constant. I will first introduce the sequential
markets equilibrium; subsequently, in light of the equivalence demonstrated
in the first chapter, I will argue that such equilibrium can be implemented as
an Arrow-Debreu equilibrium and I will specify the latter. The second part
of the chapter aims at determining equilibrium prices and allocation. I will
show that, in the particular scenario of no uncertainty and constant aggregate
endowment, each individual consumes a constant share of aggregate income
at equilibrium. On the contrary, in a more general case, since equilibrium
is considerably more difficult to compute, it is computed by using Negishi’s
method, that is, by solving a social planning problem. In order for Negishi’s
method to be used, the two fundamental welfare theorems must hold; in fact,
establishing a link between competitive equilibrium and Pareto efficient allo-
cation, they provide a theoretical justification for Negishi’s method. Finally,
I will show that, once equilibrium prices have been computed, any redundant
arbitrary asset can be priced.
Chapter 3 presents the competitive equilibrium analysis of a limited com-
mitment economy, the economy with endogenous debt limits by Alvarez and
Jermann, under the simplifying assumption of no uncertainty. The structure
INTRODUCTION 7
of this chapter is specular to the structure of the second chapter. The first
part introduces the economy and the notion of equilibrium. The economy is
the same as the one of the previous chapter, except for the nature of debt
constraints. I will first specify the sequential equilibrium with not-too-tight
debt limits; then, I will show that, as emphasized by Alvarez and Jermann,
such equilibrium is equivalent to Kehoe and Levine’s Arrow-Debreu equilib-
rium, provided that the high implied interest rates condition is satisfied. The
second part of the chapter discusses the determination of equilibrium. Differ-
ently from the full commitment economy with natural debt limits discussed
in the second chapter, the assumptions of certainty and constant aggregate
endowment do not make individual consumption stationary at equilibrium;
hence, a closed form solution cannot even be attained in this particular case.
As in the previous chapter, I will show that Negishi’s method allows for a
simplification of the computation of equilibrium. Again, in order for it to
be used, the two fundamental welfare theorems will be stated. Differently
from the second chapter, efficiency cannot be reached, because agents have
different marginal rates of substitution; hence, the notion of efficiency will
be substituted by that of constrained efficiency. Finally, I will show that the
prices of Arrow securities are given by the marginal rate of substitution of
the agents whose solvency constraints do not bind; since such agents have the
highest marginal valuation of the security, Arrow prices equals the highest
marginal rate of substitution across agents.
Chapter 4 discusses two applications of Alvarez and Jermann’s model
with endogenous solvency constraints. The first of such applications con-
cerns the evolution of consumption inequality in the U.S. I will first describe
INTRODUCTION 8
the evolution of consumption inequality across time and its implications on
the availability of insurance to consumers; I will then review the literature
on this topic; finally, I will present the contribution by Krueger and Perri,
which builds on Alvarez and Jermann’s model with not-too-tight solvency
constraints. The second application concerns the evolution in equity and
bond returns. Again, I will first describe the equity premium puzzle and
the risk free rate puzzle; I will then review the main contributions of the
literature on this topic; finally, I will present the contribution by Alvarez
and Jermann [5].