Heterogeneous Agent Models in Continuous Time

Heterogeneous Agent Models in Continuous Time∗
Yves Achdou
Jiequn Han
Jean-Michel Lasry
Princeton
Universit´e de Paris-Dauphine
Universit´e de Paris-Diderot
Pierre-Louis Lions
Benjamin Moll
Coll`ege de France
Princeton
October 22, 2014
preliminary and incomplete
download the latest draft from:
http://www.princeton.edu/~moll/HACT.pdf
Abstract
In continuous time, a wide class of heterogeneous agent models can be conveniently
solved as systems of partial differential equations. This paper develops tools for the
analysis and numerical solution of such systems. We illustrate our approach by studying a general equilibrium model with incomplete markets and uninsured idiosyncratic
labor income risk as in Aiyagari (1994), Bewley (1986) and Huggett (1993). We develop an efficient algorithm for computing both stationary and time-varying equilibria,
and obtain three theoretical results: a tight characterization of the speed at which
households hit the borrowing constraint, an intuitive characterization of the equilibrium interest rate in terms of the number of borrowing constraint households, and a
proof of existence of a stationary equilibrium.
∗
We are grateful to Dave Backus, Roland Benabou, Jess Benhabib, Lorenzo Caliendo, Dan Cao, Wouter
Den Haan, Mariacristina De Nardi, Greg Kaplan, Nobu Kiyotaki, Giuseppe Moscarini, Galo Nu˜
no-Barrau,
Jesse Perla, Tony Smith, Wei Xiong, Stan Zin and seminar participants at Yale, Princeton and NYU for
useful comments. We also thank D´eborah Sanchez for stimulating discussions in early stages of this project
and Xiaochen Feng and SeHyoun Ahn for outstanding research assistance.
1
Introduction
A number of important questions in macroeconomics concern not only aggregate variables
– GDP, employment, the general price level and so on – but also entire distributions of
variables, say the joint distribution of income and wealth or the size distribution of firms,
and how these may interact with and shape the aggregates macroeconomists are ultimately
interested in. Just to give one example, one may ask how an increase in the progressivity
of the tax system will affect aggregate savings, GDP, and income and wealth inequality.
Aiyagari (1994), Bewley (1986), Huggett (1993), and others have proposed theories designed
to answer exactly such questions. These theories, in which a large number of heterogeneous
individuals is subject to idiosyncratic income shocks and trades in incomplete asset markets,
have quickly become a workhorse of modern macroeconomics.1 But relatively little is known
about the theoretical properties of such models and they often prove difficult to compute.2
Heterogeneous agent models share a common structure: Individuals interact in markets
and make choices taking as given prices that are determined in general equilibrium and
therefore depend on the entire distribution of individuals in the economy as well as its evolution. Individuals’ choices together with idiosyncratic shocks in turn determine the evolution
of this distribution. In continuous time, such economies can be conveniently boiled down
to systems of two coupled partial differential equations: a Hamilton-Jacobi-Bellman (HJB)
equation for the optimal choices of a single atomistic individual who takes the evolution of
the distribution and hence prices as given. And a Kolmogorov Forward equation characterizing the evolution of the distribution, given the policy functions of individuals.3 In this paper,
we take advantage of this fact and develop tools for the analysis and numerical solution of
such systems. We build on earlier work by Lasry and Lions (2007) who have termed such
systems “Mean Field Games.”4
To explain the logic of our approach in the simplest possible fashion, we present it in a
context that should be very familiar to macroeconomists: a general equilibrium model with
incomplete markets and uninsured idiosyncratic income risk as in Aiyagari (1994), Bewley
(1986) and Huggett (1993). The HJB equation then characterizes workers’ saving behavior
given a stochastic process for income; and the Kolmogorov Forward equation the evolution
1
Hopenhayn (1992) and others have proposed similar theories featuring heterogeneous firms rather than
households.
2
See the “Related Literature” section at the end of this introduction for a more detailed discussion.
3
The “Kolmogorov Forward equation” is also often called “Fokker-Planck equation.” Because the term
“Kolmogorov Forward equation” seems to be somewhat more widely used in economics, we will use this
convention throughout the paper. But these are really two different names for the same equation.
4
In analogy to the continuum limit often taken in statistical mechanics and physics, e.g. when solving
the so-called Ising model.
1
of the joint distribution of income and wealth. We start with a particularly parsimonious
case: a version of the economy analyzed by Huggett (1993) in which idiosyncratic income
risk takes the form of exogenous endowment shocks that follow a two-state Poisson process
and in which individuals save in unproductive bonds that are in zero net supply. Later
in the paper, we extend all our results to the case in which individuals save in productive
capital and income is labor income with a more general stochastic process as in Aiyagari
(1994). While the present paper considers only such Aiyagari-Bewley-Huggett economies,
we hope to convey to the reader that the apparatus developed here – and particularly our
computational algorithm – is considerably more general and can handle many different types
of heterogeneous agent models. An important class of models we do not consider here are
theories with aggregate uncertainty in addition to idiosyncratic risk (Den Haan, 1996, 1997;
Krusell and Smith, 1998). These are instead the subject of a companion paper (Achdou
et al., 2014).
The first main contribution of our paper are three theoretical results characterizing the
stationary equilibrium of our economy. First, we show that individuals necessarily hit the
borrowing constraint in finite time after a long enough sequence of low income shocks, if either
the borrowing constraint is tighter than the natural borrowing constraint or the coefficient
of absolute risk aversion is bounded as consumption approaches zero. A finite hitting time
is reminiscent of optimal stopping time problems (see e.g. Stokey, 2009) for which the use of
continuous-time methods is well established.5 We then provide an intuitive characterization
of the speed at individuals hit the constraint: it is higher the lower is the interest rate
relative to the rate of time preference, the less risk averse individuals are, or the higher is the
likelihood of getting a high income draw. Second, we provide an intuitive characterization of
the equilibrium interest rate. We show that the gap between the stationary interest rate and
the rate of time preference measures the extra benefit from self-insurance due to the presence
of the borrowing constraint, and therefore depends on the number of borrowing constraint
individuals.6 Third, we prove existence of a stationary equilibrium for our continuous-time
5
This similarity suggests one of the themes of this paper, namely that continuous time methods are the
natural tool for analyzing optimal saving problems with borrowing constraints. In particular, continuous
time avoids a type of integer problem arising in discrete time: that the borrowing constraint can be reached
from wealth levels that are strictly great than this borrowing constraint.
6
An immediate corollary of our result is an important point made by Huggett and Ospina (2001): that
the interest rate is less than the rate of time preference in economies with idiosyncratic risk and incomplete
markets has nothing to do with a positive third derivative of the utility function which is the conventional
definition of precautionary savings at the individual level (Leland, 1968; Sandmo, 1970). Instead this statement coincides with the statement that a strictly positive number of individuals are borrowing constrained.
Intuitively, the borrowing constraint does introduce a form of precautionary savings: that driven by the
desire to stay away from the borrowing constraint. But this is quite different from the conventional view of
precautionary savings which is parameterized by the convexity of marginal utility.
2
version of an Aiyagari-Bewley-Huggett economy.7
The second main contribution of our paper is the development of a simple and efficient algorithm for computing both stationary and time-varying equilibria, based on a finite
difference method. Computing the stationary equilibrium of Huggett’s economy with two
income types and 1,000 grid points in the wealth dimension takes around half a second using
MATLAB R2012a on a Macbook Pro laptop computer. Similarly, computing the stationary
equilibrium of Aiyagari’s economy with forty grid points in the income dimension and 500
grid points in the wealth dimension takes around four seconds. All algorithms are available
as MATLAB codes from http://www.princeton.edu/~ moll/HACTproject.htm. The first
step of the algorithm is to solve individuals’ HJB equation for a given sequence of prices.
For this step, we take advantage of the fact that there is a well-developed theory concerning the numerical solution of HJB equations using finite difference schemes. In particular,
Barles and Souganidis (1991) have proven that under certain conditions the solution to a
finite difference scheme converges to the (unique viscosity) solution of the HJB equation. We
briefly review this theory in an easily accessible fashion and derive the appropriate conditions required for convergence in the context of the income fluctuation problem that is part
of Aiyagari-Bewley-Huggett models. We then provide some example computations. The
dynamics of the joint distribution of income and wealth can be conveniently visualized as
“movies” (see http://www.princeton.edu/~ moll/aiyagari.mov for an example).
A secondary contribution of our paper is to show how to handle borrowing constraints
in continuous time dynamic programming problems. Mathematically, these take the form
of “state constraints” (Soner, 1986a,b; Capuzzo-Dolcetta and Lions, 1990). One of the big
advantages of our continuous time approach is that the borrowing constraint never binds in
the interior of the state space.8 Instead, the borrowing constraint only shows up in the HJB
equation’s boundary conditions, and in particular gives rise to a so so-called state constraint
boundary condition. This condition is intuitive: it is a boundary condition on the first
derivative of the value function that ensures that the drift of wealth is non-negative at the
7
Our argument follows along the lines of existence proofs in discrete-time versions of the model, in
particular the graphical proof in the original paper by Aiyagari (1994) and that of Miao (2006). Also see
Cao (2011), Mertens and Judd (2013) and Acemoglu and Jensen (2012) for other existence results. We do
not consider our existence result a significant contribution relative to this discrete-time literature. But we
nevertheless find it useful to be able to guarantee existence in our continuous-time version. We have not been
able to prove uniqueness of a stationary equilibrium. In particular, one is faced with the same well-known
difficulty as in discrete time, namely that it is hard to show that the aggregate supply of savings is monotone
in the interest rate. We hope to make some progress on uniqueness in later versions of the paper.
8
This is in contrast to discrete-time formulations where there is typically a critical level of wealth, strictly
bigger than the borrowing constraint, such that the constraint binds for all lower level of wealth. See for
example Lemma 1 in Huggett and Ospina (2001).
3
borrowing constraint so that the constraint is never violated.9 It is also easily implemented
numerically.10 The fact that the borrowing constraint never binds in the interior of the
state space has a convenient implication: the density of wealth is smooth everywhere, except
possibly for a Dirac mass exactly at the borrowing constraint. Note that this is true even
though income follows a a process with discrete states (a two-state Poisson process). This is
in contrast to discrete-time Bewley models in which the wealth distribution tends to feature
“spikes” on the interior of the state space.11
Related Literature. A large theoretical and quantitative literature studies environments
in which heterogeneous households or producers are subject to uninsurable idiosyncratic
shocks. Early contributions are by Aiyagari (1994), Bewley (1986), Huggett (1993), and
Hopenhayn (1992). See Heathcote, Storesletten and Violante (2009) and Guvenen (2011)
for recent surveys, and the textbook treatment in Ljungqvist and Sargent (2004). All of
these are set in discrete time.
Much fewer papers have studied equilibrium models with heterogeneous agents in continuous time. Well-known examples include Jovanovic (1979), Moscarini (2005) and Luttmer
(2007) and Alvarez and Shimer (2011).12 All of these papers make “just the right assumptions” on the environment being studied that equilibria can be solved explicitly (or at least
characterized very tightly).13 In contrast, our aim is to develop tools for solving and ana9
The state constraint boundary condition can be derived more rigorously from the concept of a constrained
viscosity solution of the HJB equation (Soner, 1986a,b; Capuzzo-Dolcetta and Lions, 1990). However, given
the smoothness of the individual saving problem, this more complicated apparatus is not needed in our
application
10
A previous version of this paper used a penalty function method to enforce the state constraint. This
approach is redundant and our way of writing the state constraint boundary condition in the present version
is strictly superior.
11
See for example Figure 17.7.1 in Ljungqvist and Sargent (2004). Note that the spikes in Ljungqvist and
Sargent’s figure are not an artifact of the use of simulation to compute the stationary distribution. Instead,
they discretize the state space, form the corresponding transition matrix and solve the eigenvalue problem
for the stationary distribution.
12
Also see Wang (2007), Benhabib, Bisin and Zhu (2013), Moll (2014), Rocheteau, Weill and Wong (2014),
Stokey (2014) and Vindigni, Scotti and Tealdi (2014). These papers, like ours, all study economies with a
continuum of heterogeneous agents giving rise to a system of coupled PDEs: a HJB equation and a Kolmogorov Forward equation. In contrast, some other existing papers study environments with a finite number
of heterogeneous agents (typically equal to two). For example, see the early contribution by Scheinkman
and Weiss (1986) and applications of their framework by Conze, Lasry and Scheinkman (1993) and Lippi,
Ragni and Trachter (2013). Also see Brunnermeier and Sannikov (2011), He and Krishnamurthy (2012,
2013), Adrian and Boyarchenko (2012) and Di Tella (2013) who focus on aggregate fluctuations (in contrast
to our paper). There are also a number of papers that study environments with heterogeneous agents in
partial equilibrium (see e.g. Bentolila and Bertola, 1990; Bertola and Caballero, 1994; Lise, 2013; Bayer and
Waelde, 2013), an easier problem because the resulting HJB and Kolmogorov Forward equations are not
coupled. Yet other authors study these two building blocks of our theory separately in isolation (see e.g.
Stokey, 2009, and the references cited therein).
13
Similar tricks for retaining tractability have also been developed in discrete-time heterogeneous agent
4
lyzing models for which closed form solutions are not available. In principle, our methods
apply as long as the model under consideration can be boiled down to an HJB equations and
a Kolmogorov Forward equation, a feature shared by a wide class of heterogeneous agent
models. To our minds, these two approaches are clearly complimentary: on one hand, having
explicit solutions is often extremely valuable for gaining intuition; on the other hand, for
certain applications, restricting attention to environments for which these can be found may
represent a sort of “analytic straitjacket.” The particular application pursued in detail in
this paper is the incomplete markets model of Aiyagari (1994), Bewley (1986) and Huggett
(1993), an environment for which it is well-known that explicit solutions are not available
under standard assumptions. One other paper by Bayer and Waelde (2010) also studies
a continuous-time version of an Aiyagari-Bewley-Huggett model, and constitutes a notable
exception from the continuous-time literature focusing on explicit solutions cited above.14
Our paper differs from theirs in a number of dimensions. Most importantly, we develop an
efficient algorithm for solving both stationary and time-varying equilibria, thereby actually
“operationalizing” the analysis of this class of models.15
Relative to existing discrete-time methods for solving Aiyagari-Bewley-Huggett models
(see citations above), there are two main conceptual differences. The first is lies in the
characterization and computation of the evolution of the joint distribution of income and
wealth. Traditional discrete-time approaches either simulate a large number of individuals
and trace the resulting “histogram” over time or they discretize the state space and work with
the corresponding transition matrix. In continuous time, the evolution of this distribution
can instead be written in terms of a Kolmogorov Forward equation, a relatively parsimonious
partial differential equation that can be analyzed tightly and solved efficiently.16 The second
models (B´enabou, 1996, 2002; Krebs, 2003; Angeletos, 2007; Kiyotaki and Moore, 2012).
14
In contrast, Wang (2007) shows that a continuous-time Aiyagari-Bewley-Huggett model can be solved
in closed form under two non-standard assumptions: that individuals have CARA utility and that current
discount rates are increasing in past consumption (in the absence of the second assumption, CARA utility
implies exploding wealth inequality).
15
Our paper additionally differs from Bayer and Waelde’s in the following three respects. First, they focus
on stationary problems only whereas we also analyze transition dynamics. Second, they study a version of the
model with the “natural borrowing constraint” implying that households never actually hit that constraint.
In contrast, we also study tighter constraints and obtain two theoretical results concerning the speed at
which this constraint is hit and how it affects the equilibrium interest rate. Third and on a more technical
level, they characterize households’ saving behavior in terms of a differential equation for its consumption
policy function. In contrast, we do this in terms of the HJB equation for the value function, the standard
formulation in optimal control theory, allowing us to take advantage of advances made in this field. Of
course the two approaches are related and their equation for consumption can be derived from the envelope
condition of the HJB equation.
16
That being said, the actual numerical solution of the Kolmogorov Forward equation using a finite
difference method is tightly related to the discrete-time transition-matrix approach. The main difference is
the extreme sparseness of the transition matrix from the discretized Kolmogorov Forward equation.
5
difference lies in individuals’ saving behavior in the vicinity of the borrowing constraint.
As discussed above, continuous time implies that the borrowing constraint never binds in
the interior of the state space. This technical advantage of continuous time has a number
of convenient implications, for instance that the wealth distribution is smooth everywhere
except exactly at the borrowing constraint.
Section 2 presents our continuous-time version of Huggett (1993). Section 3 contains
theoretical results regarding the stationary equilibrium and section 5 describes our computational algorithm for both stationary and time-varying equilibria. Section 6 describes an
extension of our benchmark model to the case where wealth takes the form of productive
capital and income is labor income as in Aiyagari (1994). Finally, section 7 concludes.
2
A Huggett Economy
To explain the logic of our approach in the simplest possible fashion, we present it in a
context that should be very familiar to many economists: a general equilibrium model with
incomplete markets and uninsured idiosyncratic labor income risk as in Aiyagari (1994),
Bewley (1986) and Huggett (1993). We first focus on the simplest of these: a Huggett (1993)
economy with wealth taking the form of unproductive bonds and idiosyncratic income risk
in the form of exogenous endowment shocks. Once this environment has been set up and
analyzed, it can be very easily generalized in a number of directions. For example section 6
generalizes it to the case where individuals save in productive capital and face idiosyncratic
labor income risk as in Aiyagari (1994).
2.1
Setup
Individuals
There is a continuum of individuals that are heterogeneous in their wealth a
and income z. The state of the economy is the joint distribution of income and wealth. Individuals have standard preferences over utility flows from future consumption ct discounted
at rate ρ ≥ 0:
E0
ˆ
∞
e−ρt u(ct )dt.
(1)
0
The function u is strictly increasing and strictly concave. An individual has an income zt
which is simply an endowment of the economy’s final good. His wealth takes the form of
bonds and evolves according to
dat = (zt + rt at − ct )dt,
6
(2)
where rt is the interest rate. Individuals also face a borrowing limit
at ≥ a,
(3)
where −∞ < a < 0.17 Finally, an individual’s income evolves stochastically over time. In
particular, we assume that income follows a two-state Poisson process zt ∈ {z1 , z2 }, with
z2 > z1 . The process jumps from state 1 to state 2 with intensity λ1 and vice versa with
intensity λ2 . The two-state Poisson process is chosen for simplicity and in Section 6 we show
how the setup can be extended to the case where aggregate productivity follows a general
diffusion process with a continuum of aggregate productivity types.
Individuals maximize (1) subject to (2), (3) and the process for zt , taking as given the
evolution of the equilibrium interest rate rt for t ≥ 0. For future reference, we denote by
gi (a, t), i = 1, 2 the joint distribution of income zi and wealth a.18
Equilibrium The only price in this economy is the interest rate and it is determined by
the fact that, in equilibrium, bonds must be in zero net supply:
0=
ˆ
∞
ag1 (a, t)da +
a
2.2
ˆ
∞
ag2 (a, t)da.
(4)
a
Stationary Equilibrium
Individuals’ consumption-saving decision and the evolution of the joint distribution of their
income and wealth can be summarized with two differential equations: a Hamilton-JacobiBellman (HJB) equation and a Kolmogorov Forward (or Fokker-Planck) equation. In a
17
As discussed in detail in Aiyagari (1994), if the borrowing limit a is less tight than the so-called “natural
borrowing limit”, the constraint at ≥ a will never bind and the “natural borrowing limit” will be the effective
borrowing limit. In a stationary equilibrium with r > 0, the “natural borrowing limit” is at ≥ −z1 /r where
z1 is the lowest income.
equilibrium with a time-varying interest rate rt , t ≥ 0 the natural borrowing
´ ∞In ´an
s
constraint is at ≥ −z1 t e 0 rτ dτ dt. That is, the natural borrowing constraint imposes that at should never
become so negative that the individual
debt even if it chooses zero consumption thereafter.
´ ∞cannot repay ´its
∞
18
This joint distribution satisfies 0 g1 (a, t)da + 0 g2 (a, t)da = 1, that is gi (a, t) is the unconditional
distribution of wealth for a given productivity type i = 1, 2.
7
stationary equilibrium these take the form:19
ρvi (a) = max u(c) + vi′ (a)(zi + ra − c) + λi (vj (a) − vi (a))
c
0=−
d
[si (a)gi (a)] − λi gi (a) + λj gj (a)
da
(5)
(6)
for i = 1, 2 and j 6= i. The derivations of the HJB equation (5) and the Kolmogorov Forward
equation (6) can be found in Appendix A. The function s is the saving policy function, i.e.
the optimally chosen drift of wealth
si (a) = zi + ra − ci (a),
where ci (a) = (u′ )−1 (vi′ (a)).
(7)
The domain of the two ordinary differential equations (5) and (6) is (a, ∞) where a is the
borrowing limit in (3). Finally, g1 and g2 have to integrate to the population fraction of the
relevant productivity types:
ˆ
a
∞
λ2
,
g1 (a)da =
λ1 + λ2
ˆ
∞
g2 (a)da =
a
λ1
.
λ1 + λ2
(8)
The reader may wonder why the borrowing constraint (3) does not feature in the HJB
equation (5). The reason is that, in our continuous-time formulation, the borrowing constraint never binds on the interior of the state space, i.e. for a > a and as a result an
undistorted first-order condition u′ (ci (a)) = vi′ (a) holds everywhere.20 Instead, the borrowing constraint gives rise to a so-called state constraint boundary condition which can be
written as
u′ (zi + ra) ≥ vi′ (a),
i = 1, 2.
(9)
To see why this is the appropriate boundary condition, note that the first order condition
u′ (ci (a)) = vi′ (a) still holds at a = a. The boundary condition (9) therefore implies si (a) =
zi + ra − ci (a) ≥ 0, i.e. it ensures that the borrowing constraint is never violated. The
19
A more compact way of writing this is to define the Hamiltonian H(p) = maxc u(c) − pc and to write
the savings policy function as si (a) = zi + ra + H ′ (vi′ (a)). The Hamilton-Jacobi-Bellman and Kolmogorov
Forward Equations (5) and (6) can then be expressed as two non-linear partial differential equations in vi
and gi only, that do not involve a max operator:
ρvi (a) = H(vi′ (a)) + vi′ (a)(zi + ra) + λi (vj (a) − vi (a)),
0 = −∂a [(zi + ra + H ′ (vi′ (a)))gi (a)] − λi gi (a) + λj gj (a).
20
This is in contrast to discrete-time formulations where there is a set B = [a, a∗ ) with a∗ > a such that
type 1’s borrowing constraint binds for all a ∈ B and hence the first-order condition is distorted. See for
example Lemma 1 in Huggett and Ospina (2001).
8
state constraint boundary condition (9) can be derived more rigorously from the concept of
a constrained viscosity solution of the HJB equation (5) (Soner, 1986a,b; Capuzzo-Dolcetta
and Lions, 1990). However, given the smoothness of the individual saving problem, this
more complicated apparatus is not needed in our application.
Finally, the stationary interest rate r must satisfy the analogue of the market clearing
condition (4)
0=
ˆ
∞
ag1 (a)da +
a
ˆ
∞
a
ag2 (a)da ≡ S(r).
(10)
The two ordinary differential equations (5) and (6) together with (7), and the equilibrium
relationship (10) fully characterize the stationary equilibrium of our economy. This system
is an instance of what Lasry and Lions (2007) have called a “Mean Field Game,” here in
its stationary form. We next turn to its time-dependent analogue, and discuss more of the
properties of such systems.
2.3
Transition Dynamics
Many interesting questions require studying transition dynamics, that is the evolution of the
economy when the initial distribution of income and wealth does not equal the stationary
distribution. The time-dependent analogue of the stationary system (5) to (10) is21
ρvi (a, t) = max u(c) + ∂a vi (a, t)[zi + r(t)a − c] + λi [vj (a, t) − vi (a, t)] + ∂t vi (a, t),
c
∂t gi (a, t) = − ∂a [si (a, t)gi (a, t)] − λi gi (a, t) + λj gj (a, t)
si (a, t) = zi + r(t)a − ci (a, t),
ci (a, t) = (u′ )−1 (∂a vi (a, t))
(11)
(12)
(13)
for i = 1, 2, j 6= i, and together with the equilibrium condition (4). We here use the short-
hand notation ∂a v = ∂v/∂a and so on, and as before si denotes the optimal saving policy
function. The domain of the two partial differential equations (11) and (12) is (a, ∞) × R+
(though more on the time domain momentarily). The function vi again satisfies a state
constraint boundary condition similar to (9)
u′ (zi + r(t)a) ≥ ∂a vi (a, t),
i = 1, 2.
21
(14)
As in the stationary case, there is again a more compact way of writing this system as two non-linear
partial differential equations in vi and gi only, that do not involve a max operator:
ρvi (a, t) = H(∂a vi (a, t)) + ∂a vi (a, t)(zi + r(t)a) + λi (vj (a, t) − vi (a, t)) + ∂t vi (a, t),
∂t gi (a, t) = −∂a [(zi + r(t)a + H ′ (∂a vi (a, t)))gi (a, t)] − λi gi (a, t) + λj gj (a, t),
with the Hamiltonian H given by H(p) = maxc u(c) − pc.
9
The density gi satisfies the initial condition
gi (a, 0) = gi,0(a).
(15)
The value function satisfies a terminal condition. In principle, the time domain is R+ but in
practice we work with (0, T ) for T “large” and impose
vi (a, T ) = vi,∞ (a),
(16)
where vi,∞ is the stationary value function, i.e. the solution to the stationary problem (5)
to (10).
The two partial differential equations (11) and (12) together with (13), the equilibrium
relationship (4) and the boundary conditions (14) to (16) fully characterize the evolution of
our economy. As already noted, this system is an instance of what Lasry and Lions (2007)
have called a “Mean Field Game.” Two more observations are worth making. First, the
two equations (11) and (12) are coupled: on one hand, an individual’s consumption-saving
decision depends on the evolution of the interest rate which is in turn determined by the
evolution of the distribution; on the other hand, the evolution of the distribution depends
on individuals’ saving decisions. Second, the two equations run in opposite directions in
time: as indicated by its name, the Kolmogorov Forward equation runs forward and looks
backwards – it answers the question “given the wealth distribution today, savings decisions
and the random evolution of income, what is the wealth distribution tomorrow?” In contrast,
the Hamilton-Jacobi-Bellman equation (11) runs backwards and looks forward – it answers
the question “given an individual’s valuation of income and wealth tomorrow, how much will
he save today and what is the corresponding value function today?”
3
Theoretical Results for Stationary Equilibrium
In this section we present four theoretical results. Two of these are about the saving behavior and wealth distribution in partial equilibrium: a tight characterization of the speed at
which households hit the borrowing constraint, an analytic solution for the stationary wealth
distribution for the case with two income types, and a complete characterization of its left
and right tails. The other two results are about the economy’s general equilibrium: an intuitive characterization of the equilibrium interest rate in terms of the number of borrowing
constraint households, and a proof of existence of a stationary equilibrium. The tools de-
10
veloped here should be applicable much more generally, in particular in environments where
heterogeneous individuals face borrowing constraints.
3.1
Speed of Hitting the Borrowing Constraint
Our first theoretical result concerns the saving behavior of individuals near the borrowing
constraint. As in discrete-time versions of income fluctuation problems, a long enough sequence of bad shocks, may lead individuals to decumulate wealth until they hit the borrowing
constraint. We here provide conditions under which the borrowing constraint is reached in
finite time and, for the case in which it is, a tight characterization of the speed at which
this happens. Our result is a characterization of the saving behavior of individuals in partial
equilibrium, i.e. for a given fixed stationary interest rate r. We assume that r < ρ which
will be the equilibrium outcome (see Proposition 4 below).
In the following Proposition, the saving behavior of individuals near the borrowing constraint depends crucially on two factors: the tightness of the borrowing constraint a, and the
properties of the utility function at low levels of consumption. More precisely, what matters
is whether
u′′ (z1 + ra)
a→a u′ (z1 + ra)
R = − lim
(17)
is finite. The interpretation R is the behavior of the coefficient of absolute risk aversion
R(c) = −u′′ (c)/u′(c) when wealth a approaches the borrowing limit a and hence consumption
approaches c1 (a) = z1 + ra. For example with CRRA utility u′ (c) = c−γ , γ > 0, we have
R = γ/(z1 + ra) and R is finite whenever the borrowing limit is tighter than the natural
borrowing limit, a > −z1 /r.
Proposition 1 Assume that r < ρ and z1 < z2 . Then the solution to the HJB equation (5)
and the corresponding saving policy function (7) have the following properties. Type z1 with
wealth a is constrained and hence s1 (a) = 0, but type z1 with wealth a > a is unconstrained
and decumulates assets s1 (a) < 0, for a > a. Furthermore:
1. If R < ∞, the Taylor expansion of (s1 (a))2 around the borrowing constraint a is
√
s1 (a) ≈ −ν a − a,
ν=
s
2
(ρ − r)u′(c1 ) + λ1 (u′ (c1 ) − u′ (c2 ))
> 0,
−u′′ (c1 )
(18)
where ci = ci (a), i = 1, 2 is consumption of the two types at the borrowing constraint.22
22
Type 1’s consumption at the borrowing constraint is given by c1 = z1 + ra and type 2’s consumption
c2 > c1 is a more complicated object determined by the HJB equation (5) for i = 2.
11
This implies that the derivative of s1 becomes unbounded at the borrowing constraint,
s′1 (a) → −∞ as a → a. Therefore the wealth of an individual with initial wealth a0 and
successive low income draws z1 converges to the borrowing constraint in finite time at
speed governed by ν:
√
a(t) − a ≈ ( a0 − a − νt)2 .
(19)
2. If R = ∞, the Taylor expansion of s1 (a) around the borrowing constraint a is
s1 (a) ≈ −η(a − a)
(20)
for a constant η > 0 defined in (77) in the Appendix. This implies that the derivative
of s1 stays bounded at the borrowing constraint, s′1 (a) < ∞. Therefore the wealth of an
individual with initial wealth a0 and successive low income draws z1 converges to the
borrowing constraint asymptotically, i.e. it never reaches it
a(t) − a ≈ e−ηt (a0 − a).
(21)
The proof of the Proposition, like that of all others, is in the Appendix. The Proposition
first states what we have already alluded to when discussing the setup of the model: in
our continuous time formulation, the borrowing constraint only binds for individuals whose
wealth exactly equals the lowest admissible wealth level a. At all other levels of wealth, the
low income type is unconstrained and decumulates wealth.
The saving behavior at the constraint depends on which of the two parts of the Proposition applies. For standard utility functions, part 1 applies whenever the borrowing constraint
is tighter than then natural borrowing constraint a > −z1 /r and it also applies if the bor-
rowing constraint equals the natural borrowing constraint a = −z1 /r and the coefficient of
absolute risk aversion −u′′ (c)/u′(c) is bounded at when consumption equals zero. Part 2 only
applies if the borrowing constraint equals the natural borrowing constraint and risk aversion
becomes unbounded at as consumption goes to zero. Part 1 should therefore be viewed as
the “standard case” and is illustrated in panel (a) of Figure 1. Importantly, the derivative of
type z1 ’s saving policy function becomes unbounded at the borrowing constraint, a property
that can be seen clearly in Figure 1(a). This property implies the third part of the Proposition, namely that individuals hit the borrowing constraint in finite time. To understand this,
it is useful to contrast part 1 with part 2 of the Proposition in which case the saving policy
function of type z1 is linear at the borrowing constraint as in (20). In that case (part 2), as
wealth decumulates towards the borrowing constraint, the rate of decumulation slows down
12
g1(a)
s2(a)
g2(a)
Savings , s i ( a)
D e ns it ie s , g i ( a)
s1(a)
a
a
We alt h, a
We alt h, a
(a) Stationary Saving Policy Function
(b) Stationary Densities
Figure 1: Saving Behavior and Wealth Distribution in Stationary Equilibrium
more and more and individuals never actually hit the borrowing constraint in finite time. But
in part 1, with the saving behavior implied by the unbounded derivative in (18), the rate of
decumulation stays large even for individuals with wealth arbitrarily close to the borrowing
constraint and only drops to zero at the very moment the borrowing constraint is reached
(and does so infinitely quickly). The result that the borrowing constraint is reached in finite
time bears some similarity to optimal stopping time problems (see e.g. Stokey, 2009). This
similarity suggests one of the themes of this paper, namely that continuous time methods
are the natural tool for analyzing optimal saving problems with borrowing constraints. In
particular, continuous time avoids a type of integer problem arising in discrete time: that
the borrowing constraint can be reached from wealth levels that are strictly great than this
borrowing constraint.
The formula ν in (18) also provides an intuitive characterization of the speed at which
individuals hit the borrowing constraint. Individuals hit the borrowing constraint faster, the
lower is the interest rate r relative to the rate of rate of time preference ρ, and the higher
is the probability λ1 of getting a high income draw (so that the probability of getting stuck
at the constraint is lower). Similarly, individuals approach a faster the lower is coefficient of
absolute risk aversion −u′′ (c1 )/u′(c1 ), and the larger is the consumption gain from getting a
high income draw (the larger is c2 /c1 or equivalently u′ (c1 )/u′ (c2 )). This last point can be
seen even more clearly for the special case of CRRA utility, u′ (c) = c−γ , γ > 0, so that (18)
becomes
v
u
u 2c1
ν=t
γ
ρ − r + λ1
13
1−
c2
c1
−γ !!
> 0.
In particular, the speed of decumulation ν is decreasing in risk aversion γ and increasing in
the gap c2 /c1 .
Finally, the following Lemma characterizes saving behavior for large a and in particular
the question whether individuals eventually decumulate assets. This will be useful below,
when we characterize the upper tail of the wealth distribution.
Lemma 1 Assume that r < ρ and that relative risk aversion −cu′′ (c)/u′ (c) is bounded
above for all c. Then there exists amax < ∞ such that s2 (a) < 0 for all a > amax , and
s2 (a) ∼ η2 (amax − a) close to a = amax for some constant η2 .
This result is the analogue of Proposition 4 in Aiyagari (1993). The condition that −cu′′ (c)/u′(c)
is bounded above for all c for example rules out exponential utility u(c) = −(1/γ)e−γc in
which case −cu′′ (c)/u′(c) = γc.
3.2
The Shape of Inequality
We now present one of the main results of the paper: an analytic solution to the Kolmogorov
Forward equation characterizing the stationary distribution with two income types (6) for any
given individual saving policy functions. We then use this analytic solution to characterize
tightly the distribution’s behavior at the borrowing constraint and in the right tail.
The derivation of this analytic solution is constructive and straightforward and we therefore present it in the main text. Summing the Kolmogorov Forward equation (6) for the two
income types, we have
d
[s1 (a)g1 (a) + s2 (a)g2 (a)] = 0
da
for all a, which implies that s1 (a)g1 (a)+s2 (a)g2 (a) equals a constant. Because any stationary
distribution must be bounded, it must be the case that s1 (a)g1 (a)+s2 (a)g2 (a) = 0 identically
for all a. Substituting this relationship into (6), we have
d
si (a)
[si (a)gi (a)] − λi gi (a) + λj
gi (a) ⇔
da
sj (a)
′
λi
λj
si (a)
′
gi (a), i = 1, 2, j 6= i
+
+
gi (a) = −
si (a) si (a) sj (a)
0=
(22)
Importantly, (22) are two independent ODEs for g1 and g2 rather than the coupled system
of two ODEs (6) we started out with, and they can be solved separately. The solution to
14
(22) is
si (a0 )gi (a0 )
gi (a) =
exp −
si (a)
ˆ
a
a
λ2
λ1
+
s1 (x) s2 (x)
!
dx ,
i = 1, 2, j 6= i
(23)
for some fixed a0 with gi (a0 ), i = 1, 2 pinned down by (8), i.e. the requirement that the
densities integrate to one. The expression in (23) holds for any individual saving policy
function, which could in principle even be sub optimal. Using our characterization of the
optimal saving policy functions from Section 3.1 we can say more about the behavior of the
distribution close to the borrowing constraint and in the right tail.
Proposition 2 Assume that relative risk aversion −cu′′ (c)/u′ (c) is bounded above for all c.
Then for any given interest rate r < ρ there exists a stationary distribution which has the
following properties close to the borrowing constraint and in the right tail.
1. If R < ∞, the stationary distribution g1 (a) has a Dirac point mass of type z1 individuals
at the borrowing constraint a. If instead R = ∞, there is no Dirac mass and the density
at the borrowing constraint equals zero, g1 (a) = 0.
2. The stationary wealth distribution is bounded above at some amax < ∞, with g1 (amax ) =
g2 (amax ) = 0. For a close to amax , the behavior of the marginal distribution of wealth
g(a) = g1 (a) + g2 (a) is given by
g(a) ∼ ξ(amax − a)λ2 /η2 −1
(24)
where η2 = |s′2 (amax )| < λ2 and ξ is a constant.
Part 1 of the Proposition is illustrated in panel (b) of Figure 1 for the case R < ∞. In par-
ticular, note the spike in the density g1 (a) at a = a. This property follows immediately from
Proposition 1 and in particular the conditions under which individuals reach the borrowing
constraint in finite time. In particular, the proof shows that if s1 (a) satisfies (18), then g1
in (23) explodes as a → a, i.e. there is a Dirac mass at a. In contrast, if s1 (a) is as in (20),
then g1 (a) = 0.
Part 2 of the Proposition states that the stationary wealth distribution in our economy
is bounded above. Like discrete-time versions of Aiyagairi-Bewley-Huggett economies with
idiosyncratic labor income risk only, our model therefore has difficulties explaining the high
observed wealth concentration in developed economies like the United States (e.g. that
the top one percent of the population own roughly thirty-five percent of total wealth). In
15
particular, the wealth distribution in the data appears to feature a fat Pareto tail. Motivated
by this observation, we show in Section 4 how to extend the model to feature a stationary
distribution with a Pareto tail by introducing a second, risky asset.
Figure 1 also illustrates an important difference between our continuous-time formulation
and the traditional discrete-time one: except for the Dirac mass exactly at the borrowing
constraint a, the wealth distribution is smooth for all a > a. Note that this is true even
though income follows a a process with discrete states (a two-state Poisson process). In contrast, the wealth distribution in discrete-time versions of Aiyagari-Bewley-Huggett models
with discrete-state income processes, tends to feature “spikes” on the interior of the state
space. See for example Figure 17.7.1 in Ljungqvist and Sargent (2004).23 To see why this
must happen in discrete time, consider a discrete-time Huggett economy with two income
states. All individuals with wealth a = a who get the high income draw choose the same
wealth level a′ > a. So if there is a Dirac mass at a, there must also be a Dirac mass at
a′ . But all individuals with wealth a′ who get the high income draw, also choose the same
wealth level a′′ . So there must also be a Dirac mass at a′′ > a′ . And so on. Through this
mechanism the Dirac mass at the borrowing constraint “spreads into the rest of the state
space.” In continuous time this does not happen because individuals leave the borrowing
constraint in a smooth fashion (here according to a Poisson process, i.e. a process with a
random and continuously distributed arrival time of shocks).
3.3
Existence of Stationary Equilibrium
We construct stationary equilibria along the same lines as in Aiyagari (1994). That is, we fix
an interest rate r < ρ, solve the households optimization problem (5), find the corresponding
stationary distribution from (6), and then find the interest rate such that bonds are in zero
net supply, i.e. the r such that S(r) = 0 defined in (10).
Figure 2 illustrates the typical effect of an increase in r on the solutions to the HJB
equation (5) and the Kolmogorov Forward equation(6) (we explain the computational algorithm used to construct this figure in more detail in section 5). An increase in r leads to an
increase in saving at the individual level and the stationary distribution shifts to the right.
Aggregate savings S(r) as a function of the interest rate r typically look like in Figure 3,
23
Note that the spikes in Ljungqvist and Sargent’s figure are not an artifact of the use of simulation
to compute the stationary distribution. Instead, they discretize the state space, form the corresponding
transition matrix and solve the eigenvalue problem for the stationary distribution. See their MATLAB codes
that are available at https://files.nyu.edu/ts43/public/source_code/mitbook.zip in the subfolder
“bewley.”
16
g2(a;rL)
s1(a;rH)
g1(a;rH)
s2(a;rH)
g2(a;rH)
a
D e ns it ie s , g i ( a)
g1(a;rL)
s2(a;rL)
Savings , s i ( a)
s1(a;rL)
a
We alt h, a
We alt h, a
Figure 2: Effect of an Increase in r on Saving Behavior and Stationary Distribution
i.e. it is increasing. A stationary equilibrium is then an interest rate r such that S(r) = 0.
0.05
r =ρ
0.04
0.03
0.02
a=a
r
0.01
S(r)
0
−0.01
−0.02
−0.03
−0.04
−0.05
−0.15
−0.1
−0.05
0
0.05
S(r)
Figure 3: Aggregate Net Supply of Bonds
Using an argument similar to the graphical proof in the original paper by Aiyagari (1994)
and along the lines of Miao (2006), one can proof existence of a stationary equilibrium.
Proposition 3 (Existence of Stationary Equilibrium) Assume R < ∞. Then there
exists a stationary equilibrium in our continuous time version of Huggett’s economy.
The logic behind the proof is simple. One can show that the function S(r) defined in (10)
is continuous. To guarantee that there is at least one r such that S(r) = 0, it then suffices
17
to show that
lim S(r) = ∞,
lim S(r) = a.
r↑ρ
r↓−∞
We have not been able to prove uniqueness of a stationary equilibrium. In particular, one is
faced with the same well-known difficulty as in discrete time, namely that it is hard to show
that S is monotone in r. We hope to make some progress on uniqueness in later versions of
the paper.
3.4
The Stationary Interest Rate and the Borrowing Constraint
A well-known feature of Aiyagari-Bewley-Huggett economies with uninsured idiosyncratic
labor income risk is that the stationary interest rate is smaller than the rate of time preferences. This has important implications for a number of issues. For example, in the Aiyagari
(1994) version of the model studied in Section 6 in which savings take the form of productive capital, there is “overaccumulation” of capital in the sense that aggregate steady state
savings are higher than in the analogous representative agent economy (or equivalently with
full insurance).
While this qualitative point is well understood, some related questions are not: How
large is the gap between the stationary interest rate and rate of time preference, e.g. is the
interest rate negative? And what parameters or equilibrium objects does this depend on?
Understanding the circumstances under which negative interest rates arise may, for example,
be of interest in an extension with nominal rigidities in which a negative “natural” interest
rate triggers a binding zero lower bound (Guerrieri and Lorenzoni, 2011). Our continuous
time formulation allows us to make some progress on these questions. Using the tight
characterization of individual saving behavior near the borrowing constraint in Proposition
1, we can derive a useful formula linking the gap between the stationary interest rate and
the rate of time preference to the number of borrowing constrained individuals.
Proposition 4 Assume R < ∞. The stationary interest rate r satisfies
u′ (c1 ) − u′ (c2 )
r =ρ−
λ1 M1 ,
(25)
U′
´∞
where ci = ci (a), i = 1, 2, U ′ = a [u′ (c1 (a))g1 (a)+u′ (c2 (a))g2 (a)]da and M1 > 0 is the Dirac
mass of type z1 at the borrowing constraint.
The formula in (25) says that the gap between the stationary interest rate and the rate
of time preference depends on the ratio of the gap in marginal utility of consumption of
18
borrowing constrained households u′ (c1 ) − u′ (c2 ) to the average marginal utility of U ′ , the
likelihood of obtaining a high income draw λ1 and the number of borrowing constrained
individuals M1 .
The intuition for the formula in (25) comes straight from individuals’ Euler equations.
In our continuous-time framework, these can be written as
u′ (ci (a)) = (1 + (r − ρ)dt)u′ (ci (a)) + Et [du′(ci (a))]
(26)
Carrying over an additional unit of wealth to the next time period involves giving up some
consumption at a marginal utility cost u′ (ci (a)), in return for increased consumption tomorrow, the marginal benefit of which crucially depends the realization of the income shock
tomorrow as captured by the term Et [du′ (ci (a))]. An individual therefore chooses his savings such that the marginal benefit of self-insurance on the right-hand side of (26) equals the
marginal cost of self-insurance on the left-hand side. Formula (25) is then derived from aggregating the individual Euler equations (26). In particular, averaging (26) over the stationary
cross-sectional distribution of income and wealth yields
U ′ = (1 + (r − ρ)dt)U ′ + (u′(c1 ) − u′ (c2 ))λ1 M1 dt,
which is a rearranged version of (25). The intuition for this aggregate Euler equation is
similar to that for the individual Euler equation (26). The left-hand side still measures the
marginal cost of self-insurance and the right-hand side measures the marginal benefit. The
non-standard term (u′ (c1 ) − u′ (c2 ))λ1 M1 measures the additional benefit from self-insurance
due to the presence of the borrowing constraint. Intuitively, an extra unit of wealth is more
valuable in the presence of a borrowing constraint because it decreases the chance of being
constrained and hence not able to smooth consumption after a long enough sequence of bad
shocks. This extra benefit depends on “hunger” at the constraint u′(c1 ) − u′(c2 ) and the
likelihood of hitting it which can be expressed as λ1 M1 .24 Summarizing, the gap between
r and ρ in (25) measures the extra benefit from self-insurance due to the presence of the
borrowing constraint.
Formula (25) also illustrates clearly an important point made by Huggett and Ospina
(2001): that r < ρ in economies with idiosyncratic risk and incomplete markets has nothing
to do with a positive third derivative of the utility function which is the conventional definition of precautionary savings at the individual level (Leland, 1968; Sandmo, 1970). As can
24
The term λ1 M1 is really the flow of individuals out of the borrowing constraint. But in a stationary
equilibrium this outflow must equal the inflow. See equation (81) in the proof of the Proposition.
19
be seen from (25), the third derivative u′′′ does not enter the formula and its sign is therefore
irrelevant. Instead the statement r < ρ coincides with the statement that a strictly positive
number of individuals are borrowing constrained. Intuitively, the borrowing constraint does
introduce a form of precautionary savings: that driven by the desire to stay away from the
borrowing constraint. But this is quite different from the conventional view of precautionary
savings which is parameterized by the convexity of marginal utility.25 We add to the result of
Huggett and Ospina (2001) by providing an easily interpretable expression for the stationary
interest rate in terms of the number of borrowing constrained individuals, the likelihood of a
high income draw, and marginal utilities of consumption.26 Finally, it should be noted that
most variables on both the left- and right-hand sides of (25) are endogenous objects. We
discuss some applications of the formula below for which this is not an issue.
4
Fat Tails in a Huggett Model with Two Assets
As shown in Proposition 2, the stationary wealth distribution in the Huggett economy with
two income types is bounded above. In Section 6 we show that the same is true for any
thin-tailed income process (including unbounded processes). This property of the model is,
of course, problematic vis-`a-vis wealth distributions observed in the data which are typically
heavily skewed and feature fat upper tails. In this section we show how to extend the
Huggett model of Section 2 to feature a fat-tailed stationary wealth distribution. We do
this by introducing a risky asset in addition to the riskless bond. The insight that the
introduction of “investment risk” into a Bewley model generates a Pareto tail for the wealth
distribution is due to Benhabib, Bisin and Zhu (2014) and our argument mimics several of
their steps.27 Our result differs from theirs in three regards. First, Benhabib, Bisin and Zhu
25
Huggett and Ospina (2001) study an economy with productive capital like the one we study in Section
6 in which r < ρ results in a steady state capital stock that is higher than in the absence of idiocycratic risk.
They terms this “aggregate precautionary savings” (as opposed to individual precautionary savings) and
note that “third derivative conditions are irrelevant for the issue of when aggregate precautionary savings
occurs within this model in the precise sense that any steady-state capital stock is larger with idiosyncratic
uncertainty than without, regardless of the sign of the third derivative.”
26
It is useful to compare our formula (25) to Huggett and Ospina’s Theorem 2. Using our notation, their
Theorem 2 states that r < ρ if and only if M1 > 0, a statement (25) makes more precise. Interestingly, the
steps by which we arrive at (25) are almost identical to those followed in Huggett and Ospina’s Theorem
2. But working in continuous time simplifies things considerably, especially because of the fact that the
borrowing constraint only binds at the boundary of the state space.
27
Also see Benhabib, Bisin and Zhu (2011) and Benhabib, Bisin and Zhu (2013). Quadrini (2009) and
Cagetti and De Nardi (2006) argue for the importance of entrepreneurial risk in explaining the right tail of
the wealth distribution, which is one particular form of investment risk. Also see Krusell and Smith (1998),
Castaneda, Diaz-Gimenez and Rios-Rull (2003) and Carroll, Slacalek and Tokuoka (2014b) and Carroll,
Slacalek and Tokuoka (2014a) for alternative mechanisms accounting for skewed wealth distributions in the
data.
20
make their argument in an environment with a risky asset only and no market for bonds,
i.e. no borrowing and lending. In contrast, we analyze a framework with two assets, a
risky asset and a riskless bond that is in zero net supply. Our framework therefore nests
both the standard Aiyagari-Bewley-Huggett model and the framework of Benhabib, Bisin
and Zhu as special cases. Conveniently, in continuous time, analyzing a model with two
assets poses no extra difficulty relative to the one-asset case.28 Second, we obtain an easily
interpretable analytic solution for the tail exponent of the wealth distribution and we show
that, somewhat counterintuitively, top wealth inequality is decreasing in the riskiness of the
risky asset. Finally, we explore the effects of both linear and progressive capital income
taxation on top wealth inequality and macroeconomic performance.
4.1
Setup
The setup is similar to that described in Section 2. We here keep the description as short
as possible and focus on highlighting the differences between the two setups. The main
difference to the previous setup is that individuals now have access to a real risky asset kt
in addition to the riskless bond which we now denote by bt . With this additional asset, the
budget constraint (2) now becomes
˜ t kt + rt bt − ct )dt
dkt + dbt = (zt + R
(27)
˜ t is the
where rt is the return on the riskless bond, i.e. the real interest rate, as before and R
return on the risky asset. The return of the risky asset is stochastic and given by
˜ t = R + σdWt
R
√
where R is a parameter and Wt is a standard Brownian motion, that is dWt ≡ lim∆t→0 εt ∆t,
˜ t kt units
with εt ∼ N (0, 1). The risky asset is a real asset in the sense that kt units produce R
of physical output, and only positive asset positions are possible kt ≥ 0. One particularly
appealing interpretation of the risky asset is that Rt is the return from owning and running a
˜ t captures strong enough depreciation. But other interpretations
private firm.29 A negative R
are possible as well. Finally, there is still a borrowing constraint which we now write as
28
In particular, the two assets can still be summarized by a single state variable, “net worth.”
For example, assume that private firms produce using capital and labor using a constant returns to scale
production functions Zt f (kt , ℓt ) as in Angeletos (2007), and define
29
˜ t kt = max {Zt f (kt , ℓt ) − wt ℓt + (1 − δt )kt } .
R
ℓt
21
bt ≥ −φ with φ ≥ 0.
The problem of an individual can be simplified by writing the budget constraint in terms
of wealth or net worth at = bt + kt :
dat = (zt + rat + (R − r)kt − ct )dt + σkt dWt
(28)
Because capital satisfies kt ≥ 0, there is a state constraint at ≥ a = −φ as before. Similarly,
the borrowing constraint bt ≥ −φ can be written as
kt ≤ at + φ
(29)
˜ t , taking as
Individuals maximize (1) subject to (28), (29) and the processes for zt and R
given the evolution of the equilibrium interest rate rt for t ≥ 0.
4.2
Stationary Equilibrium
As before, individuals’ saving decisions and the joint distribution of income and wealth can
be summarized by means of a Hamilton-Jacobi-Bellman equation and a Kolmogorov Forward
equation
ρvi (a) =
max
c,0≤k≤a+φ
u(c) + vi′ (a)(zi + ra + (R − r)k − c)
(30)
1
+ vi′′ (a)σ 2 k 2 + λi (vj (a) − vi (a))
2
0=−
d
1 d2 2
[si (a)gi (a)] +
[σ ki (a)2 gi (a)] − λi gi (a) + λj gj (a).
2
da
2 da
(31)
for i = 1, 2 and j 6= i. As before, si (a) is the optimal saving policy function and ki (a) is the
optimal choice of the risky asset. It can be seen that (30) is an optimal portfolio allocation
problem as in Merton (1969) and ki (a)/a is the share of the individual’s portfolio invested
in risky asset. For example, ki (a) > a means that the individual borrows in riskless bonds
so as to invest into the risky asset. The interest rate r is determined in equilibrium by the
fact that bonds are in zero net supply. The bond market clearing condition can be written
as:
ˆ
∞
k1 (a)g1 (a)da +
a
ˆ
∞
k2 (a)g2 (a)da =
a
ˆ
∞
ag1 (a)da +
a
ˆ
∞
ag2 (a)da
a
˜ t inherits the properties of the process for Zt . Also see Quadrini (2009) and Cagetti
Then the process for R
and De Nardi (2006) for related models of private firms.
22
4.3
The Tail of the Wealth Distribution
We now show that if individuals have CRRA utility
u(c) =
c1−γ
,
1−γ
γ>0
(32)
the stationary wealth distribution has a Pareto tail, and derive an analytic expression for
the tail parameter. They key to this result is the following Lemma.
Lemma 2 With CRRA utility (32), individual policy functions are asymptotically linear in
a (as a → ∞) and given by
ρ − (1 − γ)r 1 (R − r)2 1 − γ
−
ci (a) ∼
γ
2
σ2
γ2
r − ρ 1 + γ (R − r)2
si (a) ∼
a
+
γ
2γ
γσ 2
R−r
a.
ki (a) ∼
γσ 2
a
(33)
(34)
(35)
The key idea of this result is that for large enough wealth a, labor income and the borrowing
constraint become irrelevant, and so individual behavior will be like in a problem without
labor income and without a borrowing constraint. And with CRRA utility, this problem is
the portfolio allocation problem of Merton (1969) which can be solved analytically with the
policy functions in Lemma 2.
Before proceeding to the main result of this section, we make one additional assumption
on parameter values.
Assumption 1 (R − r)2 < 2σ 2 (ρ − r).
This assumption states the excess return on the risky asset cannot be too large relative to
the riskiness of assets and the gap between the interest rate and the rate of time preference.
With this assumption in hand, we obtain the following analytic solution for the fatness of
the stationary wealth distribution.
Proposition 5 With CRRA utility (32) and under Assumption 1, there is a unique stationary wealth distribution which follows an asymptotic power law, that is 1 − G(a) ∼ ma−ζ
with tail exponent
ζ=γ
2σ 2 (ρ − r)
−1 .
(R − r)2
23
(36)
Therefore top wealth inequality 1/ζ is decreasing in volatility σ, risk aversion γ and the rate
of time preference ρ, and increasing in the stationary interest rate r, and the excess return
of risky assets R − r.
Somewhat counterintuitively, top wealth inequality is decreasing in the volatility of the risky
asset. The reason for this is that there are two offsetting effects. On one hand, a higher σ
has a direct effect in that more randomness in the risky asset leads to higher inequality. On
the other hand, if σ increases, risk averse individuals optimally choose a smaller portfolio
share of risky assets (see (35)) which is a force towards lower top wealth inequality. The
formula (36) shows that the latter effect always dominates so that top wealth inequality 1/ζ
is unambiguously decreasing in volatility σ. Another way of stating this is that what matters
for top wealth inequality is the volatility of wealth σ 2 ki (a)2 and from (35) we have
σ 2 ki (a)2 ∼
(R − r)2 2
a
γ 2σ2
(37)
which is decreasing in σ. The behavior of top wealth inequality with respect to the other
parameter values is more intuitive: individuals invest a large share of their assets into risky
assets when they are not too risk averse, or when the excess return of risky assets is high,
and this also implies that wealth inequality is high. Also note that the fatness of the tail
parameter does not depend in any way on the properties of the stochastic process for labor
income (the income levels z1 , z2 or the Poisson intensities λ1 , λ2 ). This property of models
with investment risk was first pointed out by Benhabib, Bisin and Zhu (2011) using the
theory of “Kesten processes” in a discrete-time model with investment risk.
Proposition 5 provides a powerful formula for calibrating models with investment risk.
Empirically, wealth distributions for developed countries like the United States feature a
high degree of concentration with a tail exponent of ζ ≈ 1.5. From (36) it can be seen that
the model can generate such high wealth concentration quite easily. For example with a
standard risk aversion parameter of γ = 2, an excess return of four percent, R − r = 0.04,
a gap between interest rate and rate of time preference of ρ − r = 0.035, and a standard
deviation of returns of twenty percent, σ = 0.2, we get ζ = 1.5 just like in the data.
Figure 4 plots individuals’ optimal choices and the resulting wealth distribution for the
model with both a risky and a riskless asset. Panel (d) in particular shows that the distribution behaves asymptotically like a Pareto distribution by showing that the logarithm of the
density of log wealth fi (x) is asymptotically linear in the logarithm of wealth x = log(a).30
30
We here use the fact that if a variable a follows a Pareto distribution g(a) ∝ a−ζ−1 , then x = log a
follows an exponential distribution f (x) ∝ e−ζx and hence log f (x) is a linear function of x where the slope
24
k1(a)
s2(a)
k2(a)
Savings , s i ( a)
Ris ky As s e t s , k i ( a)
s1(a)
a
a
We alt h, a
(a) Saving Policy Function
We alt h, a
(b) Investment in Risky Assets
g1(a)
log(f1(x))
g (a)
log(f2(x))
log ( f i ( x ) )
D e ns it ie s , g i ( a)
2
a
Log We alt h, x = log ( a)
We alt h, a
(c) Stationary Densities
(d) Pareto Tail of Stationary Distribution
Figure 4: Optimal Choices and Pareto Tail of Wealth Distribution in Two-Asset Model
25
4.4
Effect of Capital Income Taxation on Top Wealth Inequality
We briefly examine the question how a tax on capital income affects top wealth inequality.
To this end, we introduce a linear tax on capital income into our version of Huggett’s model
with both a risky and a riskless asset. We modify the budget constraint (27) to
˜ t kt + rt bt ) + Tt − ct )dt
dkt + dbt = (zt + (1 − τ )(R
where τ is the linear tax on capital income and Tt are lump-sum transfers. We assume
that the government balances its budget each period and redistributes revenues from capital
income taxation equally to all households. It is not hard to show that the formula for top
wealth inequality (36) becomes
ζ=γ
2σ 2 (ρ − r(1 − τ ))
−1
(R − r)2
A higher capital income tax rate τ lowers top wealth inequality. Interestingly, capital taxation affects top wealth inequality only through its effect on the return of the riskless asset.
This is because a linear capital income tax does not affect the volatility of wealth in (37).
On one hand, a high tax rate directly lowers the effective variance of the risky asset σ(1 − τ ).
On the other hand, this reduced riskiness implies that individuals invest a larger fraction of
their wealth into risky assets. The two effects exactly offset each other as can be seen from
(37).
5
Computing Equilibria
In this section we describe our algorithm for numerically computing equilibria of our continuoustime version of Huggett’s economy. We use a finite difference method based on work by
Achdou, Camilli and Capuzzo Dolcetta (2012) which is simple, robust and easily extended
to other environments. It is also extremely efficient: for example, computing the stationary
equilibrium of Huggett’s economy from section 2.2 with two income types and 1,000 grid
points in the wealth dimension takes around half a second using MATLAB R2012a on a
Macbook Pro laptop computer. Similarly, computing the stationary equilibrium of Aiyagari’s economy described later in section 6 with forty grid points in the income dimension
and 500 grid points in the wealth dimension takes around four seconds. All algorithms are
available as MATLAB codes from http://www.princeton.edu/~ moll/HACTproject.htm.
equals the tail exponent ζ.
26
5.1
Computational Advantages relative to Discrete Time
Before we explain our algorithm, we briefly outline four computational advantages of continuoustime Bellman equations relative to their discrete-time counterparts. The first three of these
can be easily explained by considering the optimization problem (5) for the case of CRRA
utility (32) and the corresponding first-order condition for consumption
c−γ = ∂a v(a, z)
(38)
and by contrasting it with its discrete-time analogue
c
−γ
≥
ˆ
∂a v(a′ , z ′ )dF (z ′ |z),
a′ = z + (1 + r)a − c,
(39)
where F (z ′ |z) encodes the Markov process for income in the discrete-time problem. The first
advantage of our continuous-time approach is that, given a guess for the value function, firstorder conditions like (38) can usually be solved by hand (e.g. above c(a, z) = (∂a v(a, z))−1/γ ).
In contrast, in discrete time, the first order condition typically defines the optimal choice as
an implicit function of the value function (and its derivatives) as in (39), and one therefore
usually resorts to costly root-finding operations for solving it. Our continuous-time approach
sidesteps this difficulty completely. In this regard, it shares some similarities with the “endogenous grid method” of Carroll (2006). The difference is that in continuous-time this also
works with “exogenous grids.” Intuitively, discrete time distinguishes between “today” and
“tomorrow” but in continuous time, “tomorrow” is the same thing as “today.” The second
advantage of our continuous-time approach is that, as can again be seen from comparing
(38) and (39), there is never a need for computing integrals. Instead the evolution of the
stochastic process for zt is captured by additive terms that do not affect the first order con-
dition, e.g. the term involving λi in (5). The third advantage is that in continuous time,
there is no need to worry about the binding pattern of borrowing constraints (“occasionally binding constraints” and so on). In contrast to discrete time (39) and as explained
in section 2.2, the first-order condition (38) always holds with equality. The fourth advantage of continuous time is a form of “sparsity.” To solve the HJB and Kolmogorov Forward
equations, we discretize these so that their solution boils down to solving a system of linear equations, that is inverting a matrix (possibly repeatedly). Conveniently, the resulting
matrices are typically extremely sparse. In particular they are typically “tridiagonal” or at
least “block-tridiagonal.” And there are extremely well-developed routines for solving sparse
linear systems (either implemented as part of commercial software packages like MATLAB
27
or open-source libraries like SuiteSparse). The reason that tridiagonal matrices arise is that
a discretized continuous-time process either stays at the current grid point, takes one step
to left or one step to right. But it never jumps.31
5.2
Computing Stationary Equilibria: Overview
In this section we describe how we calculate stationary equilibria – functions v1 , v2 and g1 , g2
and a scalar r satisfying (4), (5) and (6) – given a specified functions u, and values for the
parameters ρ, λ1 , λ2 and a.
We use a bisection algorithm on the stationary interest rate. We begin an iteration with
an initial guess r0 . Then for ℓ = 0, 1, 2, ... we follow
1. Given rℓ , solve the stationary HJB equation (5) using a finite difference method. Calculate the saving policy function si,ℓ (a), i = 1, 2.
2. Given si,ℓ (a), i = 1, 2, solve the stationary Kolmogorov Forward equation (6) for gi,ℓ (a)
using a finite difference method.
´∞
3. Given gi,ℓ (a), i = 1, 2, compute the net supply of bonds S(rℓ ) = a ag1,ℓ (a)da +
´∞
ag2,ℓ (a)da and update the interest rate: if S(rℓ ) > 0, decrease the interest rate
a
rℓ+1 < rℓ and vice versa.
When rℓ+1 is close enough to rℓ , we call (rℓ , v1,ℓ , v2,ℓ , g1,ℓ , g2,ℓ ) a stationary equilibrium.
5.3
Solving the Stationary HJB Equation
For step 1, we solve the HJB equation (5) using a finite difference method taking advantage
of the fact that there is a well-developed theory concerning the numerical solution of HJB
equations using finite difference schemes. In particular, Barles and Souganidis (1991) have
proven that under certain conditions the solution to a finite difference scheme converges to
the (unique viscosity) solution of the HJB equation. We here briefly explain what a finite
difference scheme is in the context of our particular (5) application and review the relevant
theory. The online Appendix contains a more detailed explanation.
We here explain the finite difference method for solving the stationary HJB equation for
a special case of (5), z1 = z2 = z so that there is no income uncertainty and v1 = v2 = v.
31
Except of course if the process is a Poisson process, i.e. if jumps are “built in.”
28
The method then immediately generalizes to the case z2 > z1 in an obvious fashion. The
HJB equation in this special case is
ρv(a) = max u(c) + v ′ (a)(z + ra − c).
(40)
c
The finite difference method approximates the function v at J discrete points in the space
dimension, aj , j = 1, ..., J. We use equispaced grids, denote by ∆a the distance between
grid points, and use the short-hand notation vj ≡ v(aj ). The derivative vj′ = v ′ (aj ) is
approximated with either a forward or a backward difference approximation
vj+1 − vj
′
≡ vj,F
∆a
vj − vj−1
′
v ′ (aj ) ≈
≡ vj,B
.
∆a
v ′ (aj ) ≈
(41)
The finite difference approximation to (40) is
ρvj = u(cj ) + vj′ (w + raj − cj ),
cj = (u′ )−1 (vj′ ),
j = 1, ..., J
(42)
where vj′ is either the forward or backward difference approximation. Which of the two
approximations is used and where in the state space is extremely important, an issue we
turn to momentarily.
5.3.1
Conditions for Convergence of Finite Difference Scheme
We here briefly review the theory developed by Barles and Souganidis (1991), drawing on
the relatively accessible version of Tourin (2013). The goal is to give the reader a flavor
of the results and required conditions, rather than a rigorous statement of the theory. In
its general form, a stationary, one-dimensional HJB equation like (40) is an equation of the
form
0 = F (a, v(a), v ′(a), v ′′ (a))
(43)
and a corresponding finite difference scheme like (42) is an equation of the form
0 = S (∆a, a, vj ; vj−1, vj+1 ) = S˜ (∆a, a, vj ; vj − vj−1 , vj+1 − vj )
29
(44)
For instance, (40) is (43) with F (a, v(a), v ′(a), v ′′ (a)) = ρv(a) − maxc u(c) − v ′ (a)(w + ra − c),
and (42) is (44) with
S(∆a, a, vj ; vj−1 , vj+1) = ρvj − u(cj ) − vj′ (z + raj − cj )
and where vj′ is given by either the forward or backward difference approximation in (41). The
following three conditions are required for the convergence result of Barles and Souganidis.
Condition 1 (Monotonicity) The numerical scheme (44) is monotone, that is S is nonincreasing in both vj−1 and vj+1 .
Condition 2 (Consistency) The numerical scheme (44) is consistent, that is for every
smooth function v with bounded derivatives
S˜ (∆a, v(aj ); v(aj ) − v(aj−1), v(aj+1) − v(aj )) → F (v(a), v ′ (a), v ′′ (a))
as ∆a → 0 and aj → a.
Condition 3 (Stability) The numerical scheme (44) is stable, that is for every ∆a > 0,
it has a solution vj , j = 1, .., J which is uniformly bounded independently of ∆a.
Theorem 1 (Barles and Souganidis (1991)) If the scheme (44) satisfies the monotonicity, consistency and stability conditions 1 to 3, its solution vj , j = 1, ..., J converges locally
uniformly to the unique viscosity solution of (43).
As mentioned this theorem generalizes immediately to the case with two income types z2 >
z1 . It also generalizes to the case of a continuum of income types and the case where the
value function v is time-dependent which will be useful when computing transition dynamics.
In fact, it even generalizes to the case where the state variable is N-dimensional and applies
to other classes of partial differential equations than HJB equations (see e.g. Tourin, 2013).
In practice, the main difficulty for the numerical solution of HJB equations is to design a
finite difference scheme that satisfies the monotonicity condition 1. In particular, it can be
seen from Condition 1 that it matters whether and when a forward or a backward difference
approximation is used. The other two conditions are typically relatively easy to satisfy.
5.3.2
Upwinding
As just mentioned, it is important whether and when a forward or a backward difference
approximation is used, so as to satisfy the monotonicity condition 1. The ideal solution to this
30
problem is to use a so-called “upwind scheme.” The rough idea is to use a forward difference
approximation whenever the drift of the state variable (here, saving sj = z + raj − cj )
is positive and to use a backward difference whenever it is negative. This is intuitive: if
savings are positive, what matters is how the value function changes when wealth increases
by a small amount; and vice versa when savings are negative. The right thing to do is
therefore to approximate the derivative in the direction of the movement of the state. To
+
this end use the notation s+
j = max{sj , 0}, i.e. sj is “the positive part of sj ” and analogously
s−
j = min{sj , 0}. The upwind version of (42) is then
ρvj = u(cj ) +
vj+1 − vj + vj − vj−1 −
sj +
sj
∆a
∆a
(45)
or (44) with
S(∆a, a, vj ; vj−1 , vj+1) = ρvj − u(cj ) −
vj+1 − vj + vj − vj−1 −
sj −
sj
∆a
∆a
(46)
and where also the finite difference approximation vj′ used to compute cj depends on the
sign of sj in the same way. Importantly, examining (46), it can be seen that S is decreasing
in both vj−1 and vj+1 and hence the scheme satisfies the monotonicity condition 1.32 The
simplified exposition here ignores two important and related issues. First, that the HJB
equation (40) is highly non-linear due to the presence of the max operator, and therefore so
is its finite difference approximation. It therefore has to be solved using an iterative scheme
(rather than just inverting a matrix) and one faces a choice between using so-called “explicit”
or “implicit” schemes. Related, from the first order condition cj = (u′)−1 (vj′ ), saving sj and
consumption cj themselves depend on whether the forward or backward approximation is
used so the upwind scheme (46) has a circular element to it. The solution to both these
issues is described in detail in the Online Appendix. Finally, it is relatively easy to see that
the upwind scheme (46) is not the only scheme satisfying the monotonicity condition 1 and,
in principle, one could use one of these other schemes.33 That being said, the upwind scheme
v
32
−v
v −v
j
and j ∆aj−1 .
Our way of writing the upwind scheme (46) suppresses the dependence of cj on j+1
∆a
However, due to the envelope theorem, this dependence does not affect whether the scheme is monotone. In
particular
s−
∂S(∆a, a, vj ; vj−1 , vj+1 )
∂
j
=
−
∂vj−1
∆a ∂cj
s−
vj − vj−1
∂cj
j
u(cj ) −
=
cj
≤ 0,
∆a
∂vj−1
∆a
where the second equality follows from the envelope theorem. A similar argument shows that ∂S/∂vj+1 ≤ 0.
33
For example, consider the following scheme:
S(∆a, vj ; vj−1 , vj+1 ) = ρvj − u(cj ) +
vj+1 − vj
vj − vj−1
cj,B −
(w + raj ).
∆a
∆a
31
(46) has one additional advantage having to do with boundary conditions. This is what we
turn to next.
5.3.3
Boundary Conditions and Handling the Borrowing Constraint
Besides being monotone, an upwind scheme like (46) has an additional great advantage: the
handling of boundary conditions. First, consider the upper end of the state space aJ . If this
34
upper end is large enough, saving will be negative sJ < 0 so that s+
It can then be
J = 0.
seen from (45) that the forward difference is never used at the end of the state space. As
a result no boundary condition needs to be imposed. Next, consider the lower end of the
state space and in particular the question how to impose the borrowing constraint (3) and
when the state constraint boundary condition (9) holds with equality and when not. This
can again be taken care of by using the special structure of the upwind scheme as follows:
impose the boundary condition
′
v1,B
≡
v1 − v0
= u′(z + ra1 )
∆a
but only for the backward difference approximation. Then let the upwind scheme itself select
whether this boundary condition is used. From (45) we can again see that the boundary
condition is only imposed if it would be the case that s1 < 0; but it is not used if s1 > 0.
This ensures that the borrowing constraint is never violated.35
5.4
Solving the Stationary Kolmogorov Forward Equation
For step 2, consider the stationary Kolmogorov Forward Equation (6). The equation is
discretized using a finite-difference scheme, similar to that for the stationary HJB equation.
In contrast to the HJB equation, the Kolmogorov Forward equation is linear in g and hence
solving it is easy. In particular, solving its discretized counterpart means solving a system
of J linear equations, i.e. inverting a matrix. This matrix is simply the transition matrix of
the discretized stochastic process in (a, zi )-space. See the online Appendix for details.
S is decreasing in both vj−1 and vj+1 and hence the scheme satisfies the monotonicity condition 1. It can
also be shown to satisfy the consistency and stability conditions 2 and 3.
34
In fact, in the special case without uncertainty analyzed in the present section and under the assumption
r < ρ, saving is negative everywhere in the state space. The condition that aJ needs to be large enough
really only matters for the case with uncertainty.
35
See again the discussion in footnote 34 and note that in the special case without uncertainty studied
here, the state constraint boundary condition is always imposed with equality, but this is not true in the
more general case.
32
5.5
Computing Transition Dynamics
We now briefly describe the algorithm we use to calculate time-varying equilibria – functions
v1 , v2 , g1 , g2 and r satisfying (4) (11) and (12) – given an initial condition (15) and a terminal condition (16). The algorithm is relatively similar to that used to compute stationary
equilibria. We again use a bisection method, this time on the entire function r(t). We begin
an iteration with an initial guess r0 (t), t ∈ (0, T ). Then for ℓ = 0, 1, 2, ... we follow
1. Given rℓ (t) and the terminal condition (16), solve the HJB equation (11), marching
backward in time. Calculate the savings policy function si (a, t).
2. Given si (a, t) and the initial condition (15), solve the Kolmogorov Forward equation,
marching forward in time, for gi,ℓ (a, t).
3. Given gi,ℓ (a, t), i = 1, 2, compute the net supply of bonds Sℓ (t) =
´∞
ag2,ℓ (a, t)da and update the interest rate according to
a
rℓ+1 (t) = rℓ (t) − ξ
´∞
a
ag1,ℓ (a, t)da +
dSℓ (t)
dt
where ξ > 0.
When rℓ+1 (t) is close enough to rℓ (t) for all t, we call (rℓ , v1,ℓ , v2,ℓ , g1,ℓ , g2,ℓ ) an equilibrium.
The algorithms used for computing the time-dependent HJB and Kolmogorov Forward
equations for a given time path rℓ (t) are quite similar to those used to compute their stationary counterparts. As already mentioned the convergence theory of Barles and Souganidis
(1991) still applies in the time-dependent case. All this is explained in more detail in the
online Appendix.
6
Extension: An Aiyagari Economy with a Continuum
of Productivity Types
In this section we extend our benchmark economy from section 2 in two directions. First,
we allow for a continuum of productivity types rather than just two, by working with a
continuous diffusion process for productivity types z. Second, rather than assuming that
individuals hold wealth in the form of unproductive bonds, we now analyze an economy in
which wealth takes the form of productive capital as in Aiyagari (1994).
33
6.1
Setup
Workers As in section 2.1, there is still a continuum of individuals that are heterogeneous
in their wealth a and income z. The state of the economy is the joint distribution of income
and wealth g(a, z, t). Relative to the problem of an individual in section 2.1, only two things
are different. First an individual’s budget constraint (2) now becomes
dat = (wt zt + rt at − ct )dt
(47)
where in particular wt is the economy-wide wage and zt now has the interpretation of an
individual’s efficiency units of labor. Second, the stochastic process for income zt is different.
In particular, we assume that a a worker’s efficiency units evolve stochastically over time on
a bounded interval [z, z¯] with z ≥ 0, according to a stationary diffusion process36
dzt = µ(zt )dt + σ(zt )dWt .
(48)
This is simply the continuous time analogue of a Markov process (without jumps). Wt is a
Wiener process or standard Brownian motion and the functions µ and σ are called the drift
and the diffusion of the process. Individuals also still face a borrowing constraint (3).37 The
problem of an individual is now to maximize (1) subject to (47), (3) and (48), taking as
given the evolution of equilibrium prices rt , wt for t ≥ 0.
Firms Wealth in this economy takes the form of productive capital as in Aiyagari (1994).
The total amount of capital supplied in the economy equals the total amount of wealth
K(t) =
ˆ
z
z¯ ˆ ∞
ag(a, z, t)dadz
(49)
a
Capital depreciates at rate δ. There is a representative firm with a constant returns to scale
production function Y = F (K, L). Since factor markets are competitive, the wage and the
36
The process (48) either stays in the interval [z, z¯] by itself or is reflected at z and z¯. From a theoretical
perspective there is no need for restricting the process to a bounded interval, and unbounded processes
can be easily analyzed (to guarantee existence of a stationary equilibrium, we would then assume that the
unbounded process is ergodic). Instead the motivation for this assumption is purely practical: we ultimately
solve the problem numerically and any computations necessarily require efficiency units to lie in a bounded
interval.
´s
´∞
37
The corresponding “natural borrowing constraint” is now at ≥ −z t wt e 0 rτ dτ dt. As before, the
borrowing constraint a only binds if it is tighter than this “natural” borrowing limit.
34
interest rate are given by
r(t) = ∂K F (K(t), 1) − δ,
w(t) = ∂L F (K(t), 1)
(50)
where again ∂K F means ∂F/∂K and similarly for ∂L F .
Stationary Equilibrium Similarly to section 2, a stationary equilibrium can be written
as a system of partial differential equations. The problem of individuals and the joint distribution of labor productivity and wealth satisfy stationary HJB and Kolmogorov Forward
equations.
1
ρv(a, z) = max u(c) + ∂a v(a, z)(wz + ra − c) + ∂z v(a, z)µ(z) + ∂zz v(a, z)σ 2 (z)
c
2
1
2
0 =∂a (s(a, z)g(a, z)) − ∂z (µ(z)g(a, z)) + ∂zz (σ (z)g(a, z))
2
(51)
(52)
on (a, ∞) × (z, z¯). The function s is the optimally chosen drift of wealth, i.e. the savings
policy function
s(a, z) = wz + ra − c(a, z),
where c(a, z) = (u′ )−1 (∂a v(a, z))
(53)
The function v again satisfies a state constraint boundary condition at a = a which is now
u′ (wz + ra) ≥ ∂a v(a, z),
all z.
(54)
Because the diffusion is reflected at z and z¯ v also satisfies the boundary conditions
∂z v(a, z) = 0,
∂z v(a, z¯) = 0,
all a
(55)
Finally the interest rate and the wage satisfy stationary versions of (49) and (50)
r = ∂K F (K, 1) − δ,
w = ∂L F (K, 1),
K=
ˆ
z
z¯ ˆ ∞
ag(a, z)dadz
(56)
a
A stationary equilibrium are scalars r and w and functions v and g satisfying the PDEs
(51) and (52) with s given by (53), boundary conditions (54), (55) and together with the
equilibrium condition (56).
35
Transition Dynamics Transition dynamics again satisfy a system of PDEs analogous to
that in section 2. In particular, a time-dependent equilibrium are functions w and r that
satisfy (49) and (50) and functions v, g satisfying
ρv(a, z, t) = max u(c) + ∂a v(a, z, t)(w(t)z + r(t)a − c) + ∂z v(a, z, t)µ(z)
c
1
+ ∂zz v(a, z, t)σ 2 (z) + ∂t v(a, z, t)
2
1
∂t g(a, z, t) =∂a (s(a, z, t)g(a, z, t)) − ∂z (µ(z)g(a, z, t)) + ∂zz (σ 2 (z)g(a, z, t))
2
(57)
(58)
on (a, ∞) × (z, z¯) × (0, T ) and where as before s is the optimal saving policy function. The
density g satisfies the initial condition g(a, z, 0) = g0 (a, z). For the terminal condition for
the value function v we generally take T large and impose v(a, z, T ) = v∞ (a, z) where v∞
is the stationary value function, i.e. the solution to the stationary problem (51) to (56).
The function v also still satisfies the state constraint boundary condition (14) and boundary
conditions corresponding to reflecting barriers at z = z and z = z¯.
6.2
Generalization of Theoretical Results from Section 3
Analogously to (17) define
u′′ (z + ra)
a→a u′ (z + ra)
R = − lim
(59)
With this definition in hand, our theoretical results from Section 3 generalize as follows.
Proposition 6 Assume that r < ρ. Then the solution to the HJB equation (5) and the
corresponding saving policy function (7) have the following properties. There is a cutoff zˆ
such that households with z ≤ zˆ and a = a are borrowing constrained, and those with z > zˆ
are not. Therefore s(a, z) = 0 for all z ≤ zˆ, s(a, z) < 0 for all z ≤ zˆ, a > a and s(a, z) > 0
for all z > zˆ. Furthermore:
1. If R < ∞, the Taylor expansion of (s(a, z))2 , z ≤ zˆ around the borrowing constraint a
is
√
s(a, z) ≈ −ν(z) a − a,
s
(r − ρ)u′ (c(z)) + ∂z u′ (c(z))µ(z) + 21 ∂zz u′(c(z))σ 2 (z)
ν(z) = 2
> 0,
u′′ (c(z))
where c(z) = c(a, z) is consumption at the borrowing constraint. This implies that
the derivative of s becomes unbounded at the borrowing constraint, ∂a s(a, z) → −∞
36
as a → a. Therefore the wealth of an individual with initial wealth a0 and successive
low income draws z ≤ zˆ converges to the borrowing constraint in finite time at speed
governed by ν:
√
a(t) − a ≈ ( a0 − a − ν(z)t)2 .
2. If R = ∞, the Taylor expansion of s(a, z), z ≤ zˆ around the borrowing constraint a is
s(a, z) ≈ −η(z)(a − a)
for η(z) > 0. This implies that the derivative of s stays bounded at the borrowing
constraint, ∂a s(a, z) < ∞. Therefore the wealth of an individual with initial wealth a0
and successive low income draws z ≤ zˆ converges to the borrowing constraint asymptotically, i.e. it never reaches it
a(t) − a ≈ e−η(z)t (a0 − a).
Proposition 7 (Existence of Stationary Equilibrium) Assume R < ∞. Then there
exists a stationary equilibrium in our continuous time version of Aiyagari’s economy with a
continuum of productivity types.
Proposition 8 Assume R < ∞. The stationary interest rate r satisfies
1
r =ρ− ′
U
where c(z) = c(a, z), U ′ =
´ z¯ ´ ∞
z
a
ˆ
zˆ
u′ (c(a, z))p(z)dz
z
u′ (c(a, z))g(a, z)dadz, p(z) ≡ ∂z (µz (z)M(z))− 12 ∂zz (σ 2 (z)M(z)) ≥
0 is the probability that type z runs into the borrowing constraint, and M(z) is the Dirac
mass of type z at the borrowing constraint.
6.3
Application and Numerical Examples
Parameterization. In this section we briefly summarize some numerical examples that
are computed under the assumption that the logarithm of labor productivity follows an
Ornstein-Uhlenbeck process
d log zt = −θ log zt dt + σ¯ dWt .
37
(60)
The parameter θ captures the speed of mean reversion of the process and we set it equal
to 0.15 corresponding to an annual autocorrelation of e−0.15 ≈ 0.86.38 We further assume
that the process is reflected at z = 0.5 and z¯ = 1.5. The other parameter values and
the functional form of the utility function are the same as above. We further assume a
Cobb-Douglas aggregate production function F (K, L) = K α L1−α with parameter α = 1/3.
Stationary Equilibrium. Figure 5 plots the stationary saving policy function and wealth
distribution.39 In particular note that the saving policy function has an unbounded derivative
0.35
0.5
De ns ity f ( a , z )
Sav ings s( a , z )
0.3
0
0.25
0.2
0.15
0.1
0.05
0
−0.5
1.5
0
5
1.5
0
5
1
1
10
10
0.5
We alt h, a
0.5
Pr oduc t iv ity, z
We alt h, a
(a) Saving as Function of Income and Wealth
P r oduc t iv ity, z
(b) Distribution of Income and Wealth
Figure 5: Stationary Equilibrium of Aiyagari’s model, (51) to (56)
at a = a for low enough z as shown in part 1 of Proposition 1. Similarly, a positive mass of
individuals is borrowing constrained for these productivity types as witnessed by the spike
in the wealth distribution.
7
Conclusion
We have tried to make some progress in terms of developing a deeper understanding of the
theoretical properties of a class of heterogeneous agent models, and in terms of developing
better computational algorithms to compute them. Our methods should be amenable to
more complicated setups, in particular with regard to the interactions between individuals.
In the class of theories we have considered, the only interactions between individuals are
38
The Ornstein-Uhlenbeck process (60) has a stationary distribution which is normal with mean zero and
¯2
and an autocorrelation function Corr(zt+s , zt ) = e−θs .
variance σ2θ
39
I am extremely grateful to Galo Nu˜
no-Barrau for sharing his code of Aiyagari’s model with a continuum
of productivity types, and to SeHyoun Ahn for optimizing it for speed.
38
through prices. But in many other questions of interest, more “local” interactions may be
important. For a recent example in macroeconomics see La’O (2013). The extension of our
methods to environments with such local interactions could be an exciting avenue for future
research.
Appendix
A
Derivation of Hamilton-Jacobi-Bellman and Kolmogorov
Forward Equations with Poisson Shocks
This Appendix shows how to derive the HJB equations (5) and (51) and the Kolmogorov
Forward or Fokker-Planck equation (12) and its stationary version (6) from a discrete time
environment with time periods of length ∆ and then taking the limit as ∆ → 0.
A.1
Hamilton-Jacobi-Bellman Equation with Poisson Process
Consider the following income fluctuation problem in discrete time. Periods are of length
∆, individuals discount the future with discount factor β(∆) = e−ρ∆ , and individuals with
income zi keep their income with probability pi (∆) = e−λi ∆ and switch to state zj , j 6= i
with probability 1 − pi (∆). The Bellman equation for this problem is:
vi (at ) = max u(c)∆ + β(∆) (pi (∆)vi (at+∆ ) + (1 − pi (∆))vj (at+∆ ))
c
s.t.
(61)
at+∆ = ∆(zi + rat − c) + at
(62)
at+∆ ≥ a
(63)
for i = 1, 2, j 6= i. We will momentarily take ∆ → 0 so we can use that for ∆ small
β(∆) = e−ρ∆ ≈ 1 − ρ∆,
pi (∆) = e−λi ∆ ≈ 1 − λi ∆.
Substituting these into (61) we have
vi (at ) = max u(c)∆ + (1 − ρ∆) ((1 − ∆λi )vi (at+∆ ) + ∆λi vj (at+∆ ))
c
39
subject to (65) and (66). Subtracting (1 − ρ∆)vi (a) from both sides and rearranging, we get
∆ρvi (at ) = max u(c)∆ + (1 − ρ∆) (vi (at+∆ ) − vi (a) + ∆λi (vj (at+∆ ) − vi (at+∆ )))
c
subject to (65) and (66). Dividing by ∆, taking ∆ → 0 and using that
vi (at+∆ ) − vi (a)
vi (∆(zi + rat − c) + at ) − vi (at )
= lim
= vi′ (at )(zi + rat − c)
∆→0
∆→0
∆
∆
lim
yields (5) where we drop the t-subscripts on at for notational simplicity. Note also that
the borrowing constraint (66) never binds in the interior of the state space because with ∆
arbitrarily small at > a implies at+∆ > a. The time-dependent case (11) can be derived in
an analogous fashion, and the derivation can be generalized to any number of income states
N > 2.
A.2
Hamilton-Jacobi-Bellman Equation with Diffusion Process
Consider a more general version of the discrete-time Bellman equation in Section A.1 but
where now zt is assumed to follow any general Markov process
v(at , zt ) = max u(c)∆ + β(∆)E[v(at+∆ , zt+∆ )] s.t.
(64)
at+∆ = ∆(wzt + rat − c) + at
(65)
at+∆ ≥ a
(66)
c
where E[·] is the appropriate expectation over zt+∆ . Then, following similar steps as in
Section A.1, we get
ρv(at , zt ) = max u(c) +
c
E[dv(at , zt )]
dt
(67)
Now assume zt follows a diffusion process (48) and wealth is as in (48). Then by Ito’s Lemma
1
2
dv(at , zt ) = ∂a v(at , zt )(wzt + rat − ct ) + ∂z v(at , zt )µ(zt ) + ∂zz v(at , zt )σ (zt ) dt
2
+ ∂z v(at , zt )σ(zt )dWt .
Therefore, using that the expectation of the increment of a standard Brownian motion is
zero, E[dWt ] = 0,
1
2
E[dv(at , zt )] = ∂a v(at , zt )(wzt + rat − ct ) + ∂z v(at , zt )µ(zt ) + ∂zz v(at , zt )σ (zt ) dt
2
40
Substituting into (67) yields (51). For completeness, note that the connection to the Poisson
HJB equation (5) is that with a two-state Poisson process with states zi and intensities λi
E[dvi (at )] = (vi′ (at )(zi + rat − ct ) + λi (vj (at ) − vi (at ))) dt,
i = 1, 2 j 6= i
and hence substituting into (64) yields (5).
A.3
Kolmogorov Forward Equation with Poisson Process
First recall the continuous time economy. There is a continuum of individuals who are
heterogeneous in their wealth a and their income z. To avoid confusion, we here adopt the
convention that variables with tilde superscripts, a˜t and z˜t , denote stochastic variables and
variables without superscripts denote the values these can take. Income takes two values
z˜t ∈ {z1 , z2 } and follows a two-state Poisson process with intensities λ1 and λ2 . Wealth
evolves as
d˜at = si (˜at , t)dt
(68)
where the optimal saving policy function si is derived from individuals’ utility maximization
problem. The state of the economy is the density gi (a, t), i = 1, 2.
Now consider the discrete time analogue. The timing of events over a time period of length
∆ is as follows: individuals of type i = 1, 2 first make their savings decisions according to
the discrete time analogue of (68)
a˜t+∆ = a
˜t + ∆si (˜at )
(69)
where we suppress the dependence of si on t for notational simplicity. After saving decision
are made, next period’s income z˜t+∆ is realized: it switches from zi to zj with probability
∆λi .
It turns out to be easiest to work with the CDF (in the wealth dimension)
Gi (a, t) = Pr(˜at ≤ a, z˜t = zi ).
(70)
This is the fraction of people with income zi and wealth below a. It satisfies G1 (a, t) +
G2 (a, t) = 0 and lima→∞ (G1 (a, t)+G2 (a, t)) = 1. The density gi satisfies gi (a, t) = ∂a Gi (a, t).
41
The fraction of individuals with wealth below a evolves as follows:
Pr(˜at+∆ ≤ a, z˜t+∆ = zi ) = (1 − ∆λi ) Pr(˜at ≤ a − ∆si (a), z˜t = zi )
+∆λj Pr(˜at ≤ a − ∆sj (a), z˜t = zj )
(71)
The intuition for the term Pr(˜at ≤ a − ∆si (a), z˜t = zi ) is that for wealth to end up below a
at time t + ∆, it must have been below a − ∆si (a) at time t. For example, if si (a) < 0 such
that type i individuals dissave, all individuals with wealth below a − ∆si (a) > a at time t
will have wealth below a at time t + ∆. Using the definition of Gi in (70), we have
Gi (a, t + ∆) = (1 − ∆λi )Gi (a − ∆si (a), t) + ∆λj Gj (a − ∆sj (a), t)
Subtracting Gi (a, t) from both sides and dividing by ∆
Gi (a, t + ∆) − Gi (a, t)
Gi (a − ∆si (a), t) − Gi (a, t)
=
−λi Gi (a−∆si (a), t)+λj Gj (a−∆sj (a), t)
∆
∆
Taking the limit as ∆ → 0 gives
∂t Gi (a, t) = −si (a)∂a Gi (a, t) − λi Gi (a, t) + λj Gj (a, t),
where we have used that
Gi (a − ∆si (a), t) − G(a, t)
G(a − x, t) − G(a, t)
= lim
si (a) = −si (a)∂a Gi (a, t),
x→0
∆→0
∆
x
lim
Differentiating with respect to a and using the definition of the density as gi (a, t) = ∂a Gi (a, t),
we obtain (12).
A.4
Kolmogorov Forward Equation with Diffusion Process
We are not aware of any intuitive derivations of the Kolmogorov Forward (Fokker-Planck)
equation with a diffusion process (52). One relatively accessible derivation is provided by
Kredler (2014).
42
B
B.1
Proofs
Proof of Proposition 1
The envelope condition of the HJB equation (5) is
(ρ − r)v1′ (a) = v1′′ (a)s1 (a) + λ1 (v2′ (a) − v1′ (a))
(72)
The first order condition is u′ (c1 (a)) = v1′ (a) and so u′′ (c1 (a))c′1 (a) = v1′′ (a). Combining,
using s′1 (a) = r − c′1 (a), and rearranging gives
(s′1 (a) − r)s1 (a) =
(r − ρ)u′ (c1 (a)) + λ1 [u′ (c2 (a)) − u′ (c1 (a))]
u′′ (c1 (a))
(73)
Also recall that at the borrowing constraint s1 (a) so that c1 (a) = z1 + ra. Which part of
the Proposition applies depends on the behavior of R = − lima→a
u′′ (z1 +ra)
u′ (z1 +ra)
which in turn
determines the behavior of the right hand side of (73) as a → a.
Part 1 Consider first part 1, i.e. R < ∞. As a → a, we have that s1 (a) → 0, c1 (a) → c1 =
z1 + ra, c2 (a) → c2 and −u′ (c1 (a))/u′′ (c1 (a)) → 1/R > 0. Therefore
s1 (a)s′1 (a) → f (a) =
(r − ρ)u′ (c1 ) + λ1 [u′ (c2 ) − u′ (c1 )]
u′′ (c1 )
(74)
The Taylor series approximation of (s1 (a))2 around a is
(s1 (a))2 ≈ 2s1 (a)s′1 (a)(a − a)
Substituting in from (82) and taking the square root yields (18).
Part 2 Next consider part 2, i.e. R = ∞. From (73), we have
s′1 (a)
u′ (c2 (a))
u′(c1 (a)) c1 (a)
− r = r − ρ − λ1 + λ1 ′
u (c1 (a)) u′′ (c1 (a))c1 (a) s1 (a)
u′ (c1 (a))
s1 (a) − ra − z
u′ (c2 (a))
= r − ρ − λ1 + λ1 ′
u (c1 (a)) −u′′ (c1 (a))c1 (a)
s1 (a)
Define
η = − lim s′1 (a),
a→a
u′′ (c1 (a))c1 (a)
,
a→a
u′ (c1 (a))
γ = − lim
43
u′ (c2 (a))
a→a u′ (c1 (a))
ξ = lim
(75)
(76)
Taking a → a in (76), we have
−(η + r) = (r − ρ − λ1 + λ1 ξ)
where we have used l’Hopital’s rule for the term
η=
1η+r
γ η
s1 (a)−ra−z
.
s1 (a)
Hence
ρ − r + λ1 (1 − ξ)
γ
(77)
Hence the Taylor series expansion of s1 (a) around a is (20) with η defined in (77).
B.2
Proof of Proposition 2
TO BE COMPLETED
B.3
Proof of Proposition 3
TO BE COMPLETED
B.4
Proof of Proposition 4
The envelope condition of (5) is
(ρ − r)vi′ (a) = si (a)vi′′ (a) + λi (vj′ (a) − vi′ (a)),
i = 1, 2, j 6= i
Multiplying the two equations by g1 and g2 and integrating between a = a + ε and a = ∞:
(ρ − r)
ˆ
=
ˆ
∞
a+ε
∞
a+ε
+
ˆ
(v1′ (a)g1 (a) + v2′ (a)g2 (a))da
[v1′′ (a)s1 (a)g1 (a) + v2′′ (a)s2 (a)g2 (a)]da
∞
a+ε
[v2′ (a)(−λ2 g2 (a) + λ1 g1 (a)) + v1′ (a)(−λ1 g1 (a) + λ2 g2 (a))]da
44
(78)
Using and integration by parts we have
ˆ
∞
a+ε
v1′′ (a)s1 (a)g1 (a)da
∞
′
v1 (a)s1 (a)g1 (a)
a+ε
∞
d
[s1 (a)g1 (a)]da
da
a+ε
ˆ ∞
d
′
= −v1 (a + ε)s1 (a + ε)g1 (a + ε) −
v1′ (a) [s1 (a)g1 (a)]da
da
a+ε
=
−
ˆ
v1′ (a)
and similarly for the term involving v2′′ . Therefore (78) is
(ρ − r)
ˆ
∞
a+ε
(v1′ (a)g1 (a) + v2′ (a)g2 (a))da
= − v1′ (a + ε)s1 (a + ε)g1 (a + ε) − v2′ (a + ε)s2 (a + ε)g2(a + ε)
ˆ ∞
d
′
+
v1 (a) − (s1 (a)g1 (a)) − λ1 g1 (a) + λ2 g2 (a) da
da
a+ε
ˆ ∞
d
′
v2 (a) − (s2 (a)g2 (a)) − λ2 g2 (a) + λ1 g1 (a) da
+
da
a+ε
(79)
From the stationary Kolmogorov Forward equation (6), the second and third terms equal
zero for all a. Furthermore from the stationary Kolmogorov Forward equation (6) we have
d
0 = − da
(s1 (a)g1 (a) + s2 (a)g2 (a)), i.e. s1 (a)g1 (a) + s2 (a)g2 (a) is constant. Since it has to
equal zero as a → ∞, we have s1 (a)g1 (a) + s2 (a)g2 (a) = 0 for all a and therefore taking
ε → 0 in (79)
(ρ − r)
ˆ
a
∞
(v1′ (a)g1 (a) + v2′ (a)g2 (a))da = − lim(v1′ (a + ε) − v2′ (a + ε))s1(a + ε)g1(a + ε) (80)
ε→0
It remains to evaluate the limit on the right-hand side. As ε → 0, v1′ and v2′ remain bounded
and strictly positive but s1 → 0 and g1 → ∞. We can nevertheless compute the right-hand
side as follows. Integrate the stationary Kolmogorov Forward equation (6) between a = a+ ε
and a = ∞ yields
0 = s1 (a + ε)g1(a + ε) − λ1
ˆ
∞
g1 (a)da + λ2
a+ε
ˆ
∞
g2 (a)da
a+ε
Recalling that we denote by M1 the Dirac mass of individuals at a = a, we need
M1 +
ˆ
∞
a
λ2
,
g1 (a)da =
λ1 + λ2
ˆ
∞
g2 (a)da =
a
λ1
λ1 + λ2
Therefore
0 = − lim s1 (a + ε)g1 (a + ε) − λ1 M1
ε→0
45
(81)
(81) states that the inflow of type 1 individuals into the borrowing constraint equals the
outflow out of the constraint. Substituting into (80), we have
(ρ − r)
ˆ
∞
a
(v1′ (a)g1 (a) + v2′ (a)g2 (a))da = (v1′ (a) − v2′ (a))λ1 M1
Finally, using the first order condition of (5) u′ (ci (a)) = vi′ (a), we arrive at (25).
B.5
Proof of Lemma 2
[TO BE COMPLETED]
B.6
Proof of Proposition 5
[TO BE COMPLETED]
B.7
Proof of Proposition 6
The envelope condition is
1
(ρ − r)∂a v(a, z) = ∂aa v(a, z)s(a, z) + ∂az v(a, z)µ(z) + ∂azz v(a, z)σ 2 (z)
2
The first order condition is u′ (c(a, z)) = ∂a v(a, z) and so ∂aa v(a, z) = u′′ (c(a, z))∂a c(a, z).
Combining, using ∂a s(a, z) = r − ∂a c(a, z), and rearranging gives
(∂a s(a, z) − r)s(a, z) =
(r − ρ)u′ (c(a, z)) + ∂z u′(c(a, z))µ(z) + 21 ∂zz u′ (c(a, z))σ 2 (z)
u′′ (c(a, z))
If s(a, z) → 0, c(a, z) → c(z) as a → a, then
s(a, z)∂a s(a, z) →
(r − ρ)u′ (c(z)) + ∂z u′ (c(z))µ(z) + 21 ∂zz u′ (c(z))σ 2 (z)
u′′ (c(z))
The Taylor series approximation of (s(a, z))2 around a is
(s(a, z))2 ∼ 2s(a, z)(∂a s(a, z))(a − a)
Substitute in from (82) and take the square root.
46
(82)
B.8
Proof of Proposition 7
B.9
Proof of Proposition 8
Following similar steps as in the proof of Proposition 8, we have
(ρ − r)
ˆ z¯ ˆ
z
∞
a+ε
∂a v(a, z)g(a, z)dadz = −
ˆ
z¯
∂a v(a + ε, z)s(a + ε, z)g(a + ε, z)dz
(83)
z
Similarly, integrating (52) yields
1
0 = − lim s(a + ε, z)g(a + ε, z) − ∂z (µ(z)M(z)) + ∂zz σ 2 (z)M(z)
ε→0
2
(84)
Taking ε → 0 in (83), using (84) and rearranging yields the expression in the Proposition.
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