Heterogeneous Agent Models in Continuous Time∗

Yves AchdouUniversite de Paris-Diderot

Jean-Michel LasryUniversite de Paris-Dauphine

Pierre-Louis LionsCollege de France

Benjamin MollPrinceton

November 3, 2013

preliminary and incomplete

download the latest draft from:

http://www.princeton.edu/~moll/HACT.pdf

Abstract

We study a class of continuous time heterogeneous agent models with idiosyncratic

shocks and incomplete markets. This class can be boiled down to a system of two

coupled partial differential equations: a Hamilton-Jacobi-Bellman equation and a Kol-

mogorov Forward equation, a system that Lasry and Lions (2007) have termed a “Mean

Field Game.” We study two concrete model economies to show that continuous time

allows for both tighter theoretical results and more precise and efficient computations

as compared to traditional discrete time methods. The first one is an exact reformula-

tion of Aiyagari (1994) and we obtain three theoretical results: a tight characterization

of household savings behavior near the borrowing constraint, uniqueness of a station-

ary equilibrium (not yet in the current draft), and a tight link between the amount

of capital “overaccumulation” and the number of borrowing constrained households.

In our second economy, heterogeneous producers face collateral constraints and fixed

costs in production, creating the possibility of a “poverty trap,” i.e. multiple station-

ary equilibria. We find that such “poverty traps” arise only in extreme special cases.

Instead the economy typically features a unique but twin-peaked stationary distribu-

tions to which it converges extremely slowly. The precision of our algorithm is key for

this finding, and coarse, simulation-based discrete-time algorithms may have obtained

misleading results. We conclude by discussing an extension of our framework to the

case with both idiosyncratic and aggregate shocks as in Krusell and Smith (1998).

∗We are grateful to Jess Benhabib, Greg Kaplan, Nobu Kiyotaki, Jesse Perla and Wei Xiong for usefulcomments. We also thank Deborah Sanchez for stimulating discussions in early stages of this project andXiaochen Feng 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.

Bewley (1986), Huggett (1993), Aiyagari (1994), 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 labor 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. This is especially true when the question at hand requires an analysis of aggregate

dynamics out of steady state, of the circumstances under which aggregate “poverty traps”,

i.e. multiple stationary equilibria, may arise, or of the implications of aggregate shocks as

in Krusell and Smith (1998).2

In this paper we hope to make progress on these issues by studying a class of continuous

time heterogeneous agent models, that share many of the features of traditional discrete-

time versions of such theories. 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. Such

economies can be boiled down to a system of two coupled partial differential equations: a

Hamilton-Jacobi-Bellman 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 Lasry and Lions (2007) have termed such a system a “Mean Field Game”4

and we make use of many of the theoretical insights and the computational strategies they

and others have derived in the context of other applications.

1Hopenhayn (1992) and others have proposed similar theories featuring heterogeneous firms rather thanhouseholds.

2See the “Related Literature” section at the end of this introduction for a more detailed discussion.3The “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 thisconvention throughout the paper. But we did want to mention that these are the same equations.

4In analogy to the continuum limit often taken in Statistical Mechanics and Physics, e.g. when solvingthe so-called Ising model.

1

We study two concrete model economies to show that continuous time allows for both

tighter theoretical results and more precise and efficient computations as compared to tra-

ditional discrete time methods. The first one is an exact reformulation of the theories of

Aiyagari (1994), Bewley (1986) and Huggett (1993). The Hamilton-Jacobi-Bellman equation

then characterizes workers’ savings behavior given a stochastic process for labor productivity;

and the Kolmogorov Forward equation the evolution of the joint distribution of wealth and

labor productivity. Both are partial differential equations in three variables: wealth, labor

productivity and time. This model can be solved efficiently using discrete time methods and

its theoretical properties are relatively well understood, but we nevertheless obtain three

new theoretical results: a tight characterization of household savings behavior near the bor-

rowing constraint, uniqueness of a stationary equilibrium (not yet in the current draft), and

a tight link between the amount of capital “overaccumulation” and the number of borrowing

constrained households. While numerical techniques for solving Aiyagari-Bewley-Huggett

models are well developed, we develop corresponding continuous time methods, partly for

pedagogical reasons. In our numerical examples, the economy converges monotonically to a

stationary equilibrium. Depending on parameter values, a significant fraction of the popu-

lation piles up against the borrowing constraint over time, a feature that is easy to visualize

using pictures of the joint distribution of wealth and labor productivity.

Our second economy is motivated by the literature studying the role of financial market

imperfections in economic development (Banerjee and Newman, 1993; Galor and Zeira, 1993;

Aghion and Bolton, 1997; Piketty, 1997; Banerjee and Duflo, 2005; Buera, Kaboski and Shin,

2011; Buera and Shin, 2013; Midrigan and Xu, forthcoming).5 Heterogeneous entrepreneurs

face collateral constraints and fixed costs in production. The Hamilton-Jacobi-Bellman equa-

tion now characterizes an atomistic entrepreneur’s savings decisions that may now involve

him trying to save himself out of the collateral constraints through internal financing, and

the Kolmogorov Forward equation characterizes the evolution of the joint distribution of

productivity and wealth. The combination of financial frictions and fixed costs in produc-

tion introduces the possibility of a poverty trap both at the individual and the aggregate

level. Individuals with low wealth will choose to operate unproductive technologies because

the fixed costs are too high. If they are sufficiently poor, they will also never be able to

accumulate enough wealth to cover this fixed cost. We find that such “poverty traps” arise

only in extreme special cases. Instead the economy typically features a unique but twin-

peaked stationary distributions to which it converges extremely slowly. The precision of our

5Cagetti and De Nardi (2006) and Quadrini (2009) analyze similar frameworks to understand wealthconcentration in developed countries.

2

computational algorithm is key for this finding, and coarse, simulation-based discrete-time

algorithms may have obtained misleading results. Because the allocation of capital across

heterogeneous producers depends on the evolution of the joint distribution of productivity

and wealth, aggregate total factor productivity evolves endogenously over time and displays

rich dynamics.

While our analysis of these two concrete example economies serve the purpose of demon-

strating the theoretical and computational advantages of our continuous time methods, it

should be clear that the same apparatus can also handle many different types of heteroge-

neous agent models. This is particularly true for our efficient and robust computational al-

gorithm which can compute both stationary and time-varying equilibria. This efficiency and

robustness is due to the technical advantages of continuous-time methods over the traditional

discrete-time ones. As already mentioned, solving for an equilibrium boils down to solving a

system of two partial differential equations which can be accomplished very efficiently with a

simple and transparent finite difference method. For example, solving for the transition dy-

namics starting from an arbitrary initial distribution in our model with financial frictions fea-

turing a non-trivial market clearing condition, takes between five and ten minutes on a laptop

computer. This is in contrast to comparable discrete time methods for computing transition

dynamics in similar models which may run for several hours even when taking advantage of

parallel computing techniques.6 While the characterization of individual choices in terms of a

Bellman equation is relatively similar to the traditional discrete time analysis, a bigger differ-

ence lies in the characterization and computation of the evolution of the joint distribution of

wealth and productivity in our two example economies. Traditional discrete-time approaches

often simply simulate a large number of individuals and trace the resulting “histogram” over

time. We instead, solve a Kolmogorov Forward equation for the time path of this joint dis-

tribution.7 The resulting distributions move continuously over time and in space and can be

conveniently visualized as “movies” (see http://www.princeton.edu/~moll/aiyagari.mov

and http://www.princeton.edu/~moll/friction.mov for two examples).

A secondary contribution of our paper is to clarify how to handle borrowing constraints

in continuous time dynamic programming problems. Mathematically, these typically take

the form of “state constraints” (Soner, 1986a,b; Capuzzo-Dolcetta and Lions, 1990). Theo-

6For a state-of-the-art discrete-time algorithm, see Appendix B in the working paper version of][]buera-shin.

7This is essentially the same approach as in Luttmer (2007) who presents a model economy in which thestationary equilibrium can be solved in closed form. Our algorithm is instead intended to work in setupswhere closed form solutions cannot be obtained, and for computing time-varying equilibria from arbitraryinitial conditions. Our approach is also related to the “explicit aggregation” approach of Den Haan andRendahl (2010).

3

retically, the appropriate solution concept of Hamilton-Jacobi-Bellman equations with state

constraints is that of a “viscosity solution” (Crandall and Lions, 1983; Crandall, Ishii and Li-

ons, 1992). Hamilton-Jacobi-Bellman equations have potentially multiple solutions and only

the viscosity solution is the one that corresponds to the value obtained from the “sequence

problem” of maximizing a utility function subject to budget and borrowing constraints. But

it turns out that things are much simpler when computing solutions in practice. This is

for two reasons. First, state constraints can be well approximated with a penalty method

(Capuzzo-Dolcetta and Lions, 1990). Second, Hamilton-Jacobi-Bellman equations can be

efficiently solved using finite difference methods and, importantly, one can prove that (un-

der certain conditions) the finite different scheme converges to the viscosity solution of the

Hamilton-Jacobi-Bellman equation (Barles and Souganidis, 1991).

Finally, we discuss briefly in how far our framework can be extended to the case with

both idiosyncratic and aggregate shocks. We do this for the case of an Aiyagari-Bewley-

Huggett economy with shocks to aggregate total factor productivity, exactly as in Krusell

and Smith (1998). We begin by spelling out the functional equation characterizing the

problem of a household. As in the original paper by Krusell and Smith, it becomes necessary

to include in the state space the entire joint distribution of wealth and labor productivity,

an infinite-dimensional object. Even though this equation cannot be solved due to the

dimensionality problem, we nevertheless consider this a useful starting point: for example,

we hope that this will later allow us to judge the quality of various approximation methods.

This is especially true given that, to the best of our knowledge, none of the papers analyzing

similar environments in discrete time spell out these equations precisely.8 Next, we outline a

possible solution algorithm for a simplified special case of the model: finitely many shocks hit

our continuous time economy at discrete time intervals and follow a discrete space Markov

process.9 Developing more satisfactory solution strategies is left for future research.

Related Literature (Incomplete) A large theoretical and quantitative literature stud-

ies environments in which heterogeneous households or producers are subject to uninsurable

8For example, the original paper by Krusell and Smith (1998) spells out the value function for householdstaking as given an abstract law of motion for the wealth distribution. However, they never spell out thatlaw of motion.

9In between the arrival of two shocks, the dynamics of the economy are governed by the same transitiondynamics as in the economy without shocks above. Handling aggregate shocks then boils down to findingappropriate boundary conditions for the two partial differential equations at the time the shocks hit. Com-pared to the existing literature, this implies that we can solve equilibria exactly in the sense that we do notneed to resort to approximating the distribution of state variables with a finite set of moments or assumingthat individuals are boundedly rational. So far we are able to solve equilibria for the case where aggregateshocks are “infrequent” (hitting at, say, five year intervals).

4

idiosyncratic shocks. Early contributions are by Bewley (1986), Huggett (1993), Aiyagari

(1994), and Hopenhayn (1992). See Heathcote, Storesletten and Violante (2009) and Guve-

nen (2011) for recent surveys. We contribute to this literature by developing a continuous

time framework, that is both relatively tractable theoretically and in which transition dy-

namics, multiple stationary equilibria and dynamics in the presence of aggregate shocks are

relatively easy to compute.

Some existing papers analyze heterogeneous agent models in continuous time, and in

particular the Hamilton-Jacobi-Bellman and Kolmogorov Forward equations arising in such

frameworks. Examples include Luttmer (2007, 2012), Alvarez and Shimer (2011), Benhabib,

Bisin and Zhu (2013) and Moll (2012). In all of these the right assumptions are made so that

equilibria can be solved in closed form (or at least characterized very tightly). Our approach

is instead intended to work in setups where closed form solutions cannot be obtained, and

for computing time-varying equilibria from arbitrary initial conditions. Some preliminary

work also indicates that our computational algorithm can comfortably handle setups with

multiple (say, two or three) non-trivial market clearing clearing conditions as for example in

Guvenen (2009), Buera and Shin (2013), Buera, Kaboski and Shin (2011) or Rıos-Rull and

Sanchez-Marcos (2012). This has proved difficult with traditional discrete-time methods.

Following Krusell and Smith (1998), a large literature has sought to improve computational

algorithms for computing equilibria in heterogeneous agent economies with aggregate shocks.

See the papers summarized by Den Haan, Judd and Juillard (2010) and Den Haan (2010),

among others. Our approach shares some similarities with some of these (in particular the

“explicit aggregation” scheme of Den Haan and Rendahl (2010)), but additionally makes use

of some technical advantages of continuous time.

Much fewer papers characterize the theoretical properties of heterogeneous agent economies.

A notable exception is by Acemoglu and Jensen (2012) who focus on stationary equilibria and

present a method for conducting comparative statics. Finally, Miao (2006), Cao (2011) and

Mertens and Judd (2013), among others present existence results for heterogeneous agent

economies. In future versions of the paper, we hope to contribute to this literature by proving

the existence of a unique stationary equilibrium in Aiyagari-Bewley-Huggett economy.

Section 2 presents our continuous-time version of an Aiyagari, Bewley and Huggett econ-

omy. We characterize it theoretically, present a computational algorithm and some compu-

tational results. Section 3 presents a second example economy that can be analyzed with our

apparatus: an economy with financial frictions and a non-convexity in production. Section

4 briefly discusses the case with aggregate shocks as in the work by Krusell and Smith, and

section 5 is a conclusion.

5

2 An Aiyagari-Bewley-Huggett Economy

To explain the logic of our approach in the simplest possible fashion, we first 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 Bewley (1986),

Huggett (1993) and Aiyagari (1994). Once this is accomplished, we show in section 3 how

our framework can be extended to handle more general setups. As an example, we present a

model in which heterogeneous producers face collateral constraints, uninsured productivity

shocks and fixed costs in production.

2.1 Setup

Workers There is a continuum of workers that are heterogeneous in their wealth a and

labor productivity z. The state of the economy is the joint distribution g(a, z, t). Workers

have standard preferences over utility flows from future consumption ct discounted at rate

ρ ≥ 0:

E0

ˆ ∞

0

e−ρtu(ct)dt. (1)

The function u is strictly increasing and strictly concave. A worker supplies zt efficiency

units of labor to the labor market and these get valued at wage wt. A worker’s wealth

evolves according to

dat = [wtzt + rtat − ct]dt, (2)

where rt is the interest rate. Workers also face a borrowing limit

at ≥ a, (3)

where a ≤ 0. Finally, a worker’s efficiency units evolve stochastically over time on a bounded

interval [z, z] with z ≥ 0, according to a stationary diffusion process10

dzt = µz(zt)dt+ σz(zt)dWt. (4)

10The process (4) either stays in the interval [z, z] by itself or is reflected at z and z. From a theoreticalperspective there is no need for restricting the process to a bounded interval, and unbounded processescan be easily analyzed (to guarantee existence of a stationary equilibrium, we would then assume that theunbounded process is ergodic). Instead the motivation for this assumption is purely practical: we ultimatelysolve the problem numerically and any computations necessarily require efficiency units to lie in a boundedinterval.

6

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 µz and σz are called the

drift and the diffusion of the process. Finally, we impose an additional restriction on the

borrowing limit, −a ≤ wz/r, that is the borrowing limit is at least as tight as the “natural

borrowing limit.” Workers maximize (1) subject to (2), (3) and (4), taking as given the

evolution of the equilibrium interest rate rt and the wage wt for t ≥ 0.

Firms There is representative firm with a constant returns to scale production function

Y = F (K,L). The total amount of capital supplied in the economy equals the total amount

of wealth

K(t) =

ˆ

ag(a, z, t)dadz, (5)

and we normalize the total amount of labor supplied in the economy to one. Capital depre-

ciates at rate δ. Since factor markets are competitive, the wage and the interest rate are

given by

r(t) = ∂KF (K(t), 1)− δ, w(t) = ∂LF (K(t), 1), (6)

where we use the short-hand notation ∂KF = ∂F/∂K and ∂LF = ∂F/∂L (this notation will

be useful below). This completes the description of the economy.

Recursive Representation The consumption-savings decision of workers and the evolu-

tion of the joint distribution of their wealth and labor productivity can be summarized with

two partial differential equations: a Hamilton-Jacobi-Bellman equation and a Kolmogorov

Forward (or Fokker-Planck) equation

ρv(a, z, t) = maxc

u(c) + ∂av(a, z, t)[w(t)z + r(t)a− c]

+ ∂zv(a, z, t)µz(z) +1

2∂zzv(a, z, t)σ

2z(z) + ∂tv(a, z, t),

(7)

∂tg(a, z, t) = −∂a[µa(a, z, t)g(a, z, t)]− ∂z[µz(z)g(a, z, t)] +1

2∂zz[σ

2z(z)g(a, z, t)]. (8)

As above we use the short-hand notation ∂zv = ∂v/∂z, ∂zzv = ∂2v/∂z2 and so on. The

function µa is the optimally chosen drift of wealth, i.e. the savings policy function

µa(a, z, t) = w(t)z + r(t)a− c(a, z, t), where c(a, z, t) = (u′)−1(∂av(a, z, t)) (9)

7

is the optimal consumption policy function.11 The domain of the two partial differential

equations (7) and (8) is (a, a)×(z, z)×R+. The lower bound on wealth a equals the borrowing

limit in (3) and we set the upper bound a equal to infinity in our theoretical analysis

and equal to some large finite number when we perform computations. The borrowing

constraint (3) forms a state constraint (Soner, 1986a,b; Capuzzo-Dolcetta and Lions, 1990).

Theoretically, the appropriate solution concept of Hamilton-Jacobi-Bellman equations with

state constraints is that of a “viscosity solution” (Crandall and Lions, 1983; Crandall, Ishii

and Lions, 1992).12 The partial differential equation (11) has potentially multiple solutions

v(a, z, t) and only the viscosity solution is the one that corresponds to the value obtained

from the “sequence problem” of maximizing (1) subject to (2), (3) and (4). When actually

computing solutions in practice, things are much simpler because state constraints can be

well approximated with a penalty method (Capuzzo-Dolcetta and Lions, 1990). There are

also boundary conditions corresponding to reflecting barriers at z and z. We discuss all of

these issues in more detail when we describe our computational algorithm below. The initial

distribution g0(a, z) is exogenously given so that the evolution of g(a, z, t) must satisfy an

initial condition:

g(a, z, 0) = g0(a, z). (10)

In practice, there is also a terminal condition on v(a, z, t) which we again describe in more

detail below.

Given an initial condition (10) and appropriate boundary conditions, the two partial

differential equations (7) and (8) together with the equilibrium relationships (5) and (6) fully

characterize the evolution of our economy. 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 (7) and (8) are coupled: on one hand, a worker’s consumption-savings

decision depends on the evolution of prices which are in turn determined by the evolution

11A more compact way of writing this is to define the Hamiltonian H(p) = maxc u(c) − pc and to writethe savings policy function as

µa(a, z, t) = w(t)z + r(t)a +H ′(∂av(a, z, t)).

The Hamilton-Jacobi-Bellman and Kolmogorov Forward Equations (7) and (8) can then be expressed as twonon-linear partial differential equations in v and g only, that do not involve a max operator:

ρv(a, z, t) = H(∂av(a, z, t)) + ∂av(a, z, t)[w(t)z + r(t)a] + ∂zv(a, z, t)µz(z) +1

2∂zzv(a, z, t)σ

2

z(z) + ∂tv(a, z, t),

∂tg(a, z, t) = −∂a[(w(t)z + r(t)a +H ′(∂av(a, z, t)))g(a, z, t)]− ∂z [µz(z)g(a, z, t)] +1

2∂zz[σ

2

z(z)g(a, z, t)].

12More precisely, one says that v is a viscosity supersolution on the closed domain [a, a] × [z, z] and asubsolution on the open domain (a, a)× (z, z).

8

of the distribution; on the other hand, the evolution of the distribution depends on workers’

savings 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 labor productivity, what is the wealth distribution tomorrow?” In contrast, the Hamilton-

Jacobi-Bellman equation (7) runs backwards and looks forward – it answers the question

“given a worker’s valuation of assets and labor productivity tomorrow, how much will he

save today and what is the corresponding value function today?”

Stationary Equilibrium A stationary equilibrium is a time-invariant solution to the

Hamilton-Jacobi-Bellman and Kolmogorov Forward equations (7) and (8) (and the equi-

librium relationships (5) and (6)):

ρv(a, z) =maxc

u(c) + ∂av(a, z)[wz + ra− c] + ∂zv(a, z)µz(z) +1

2∂zzv(a, z)σ

2z(z), (11)

0 =− ∂a[µa(a, z)g(a, z)]− ∂z[µz(z)g(a, z)] +1

2∂zz[σ

2z(z)g(a, z)], (12)

µa(a, z) = wz + ra− c(a, z), where c(a, z) = (u′)−1(∂av(a, z)). (13)

Now, the domain of the partial differential equations is simply the state space (a, a)× (z, z),

with appropriate boundary conditions at a,a, z and z, capturing the savings policy function

and density at the borrowing limit among other things.13

2.2 Theoretical Results for Stationary Equilibrium

Characterization of Worker’s Savings Problem. Consider a worker’s savings problem

in a stationary equilibrium, that is (11) with a constant interest rate r and wage w.

Proposition 1 If r < ρ, then the solution to (11) has the following properties:

1. There is a cutoff z such that households with z ≤ z are borrowing constrained, and

those with z > z are not. Therefore µa(a, z) = 0 for all z ≤ z and µa(a, z) > 0 for all

z > z.

2. For z ≤ z, the Taylor expansion of µa(a, z) around the borrowing constraint a is

µa(a, z) ∼ −κ(z)√a− a where κ(z) > 0 is a constant. This implies that:

13More precisely, the value function v is a viscosity supersolution of the HJB equation on the closed domain[a, a]× [z, z] and a subsolution on the open domain (a, a)× (z, z). The Kolmogorov Forward equation holdson the open domain and satisfies appropriate boundary conditions.

9

(a) the first derivative is unbounded lima→a ∂aµa(a, z) = −∞ for z ≤ z

(b) if a worker has constant productivity z ≤ z, his wealth evolves as a(t) − a ∼(√a0 − a−κ(z)t)2 and hence converges to the borrowing constraint in finite time.

Figure 1 illustrates part 3 of the Proposition by plotting the savings policy function µa(a, z)

for a constrained and an unconstrained labor productivity type z1 < z < z2. This property

Wealth, a

Savings,

µa(a

,z)

Constrained type z1

Unconstrained type z2

Figure 1: Savings Behavior at Borrowing Constraint

of the savings policy function is extremely important because, as we show below, it implies

that there is a Dirac point mass of workers at the borrowing constraint.

Existence and Uniqueness of Stationary Equilibrium.

Definition 1 A stationary equilibrium in the Aiyagari-Bewley-Huggett economy 2 are scalars

(K, r, w) satisfying (5) and (6), and a triple of functions (v, g, µa) on (a, a)×(z, z) satisfying

(11), (12), (13) and appropriate boundary conditions.

Proposition 2 (Existence of Stationary Equilibrium) There exists a stationary equi-

librium in the Aiyagari-Bewley-Huggett economy if 0 < σz(z) < ∞ for all z.

The proof will be added later. The gist of the proof is similar to the graphical proof in the

original paper by Aiyagari (1994) and that in Miao (2006).

[TO BE COMPLETED]

10

Borrowing Constraints and “Overaccumulation.” A well-known feature of Aiyagari-

Bewley-Huggett economies with uninsured idiosyncratic labor income risk is that the station-

ary interest rate is smaller than the rate of time preferences, and that therefore aggregate

steady state savings are higher than in the analogous representative agent economy (or

equivalently with full insurance). Not surprisingly the same is true in our version. But our

continuous time formulation allows for a more precise statement than just the qualitative

point that there will be such “overaccumulation” of capital. Using the tight characterization

of household savings behavior near the borrowing constraint in 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 households. The size of this gap then maps directly

to the amount of capital “overaccumulation.”

Proposition 3 In a stationary equilibrium:

1. there is a Dirac point mass of individuals at the borrowing constraint a = 0

2. there is a cutoff z such that the z-type-specific Dirac mass satisfies M(z) > 0 for z < z

and M(z) = 0 for z ≥ z

3. the interest rate r satisfies

(ρ− r)

ˆ ∞

0

ˆ z

z

u′(c(a, z))g(a, z)dzda =

ˆ z

z

u′(c(a, z))p(z)dz > 0, (14)

where 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.

The formula in (14) says that the product of the gap ρ−r and the average marginal utility of

consumption´∞

0

´ z

zu′(c(a, z))g(a, z)dzda equals a weighted average of the marginal utilities

of those households who are borrowing constrained. The weights p(z) are observable, at

least in principle: given an estimate of the stochastic process for labor productivity, µz(z)

and σz(z), and knowledge of the number of borrowing constrained households M(z), they

can be readily computed. The intuition for the formula in (14) comes straight from the

household Euler equation. In particular, in our continuous-time framework an undistorted

Euler equation holds for all households except for those directly at the borrowing constraint

dE[u′(ct(a, z))]

dt= (ρ− r)u′(ct(a, z)), a > a

dE[u′(ct(a, z))]

dt< (ρ− r)u′(ct(a, z))

(15)

11

In a stationary equilibrium, growth in the marginal utility of consumption – the left-hand

side of (15) – averaged over the entire population equals zero. Therefore, it must be true

that 0 < ρ− r. [EXPAND]

2.3 Computational Algorithm

We solve equilibria numerically. In this section we describe our algorithm for solving the

Aiyagari-Bewley-Huggett economy. The algorithms used to solve the model with financial

frictions in the following section 3 is very similar. We use a finite difference method based on

work by Achdou, Camilli and Capuzzo Dolcetta (2012) which is simple, robust and flexible.

At the moment, we are unable to prove that this method converges, but we believe that a

convergence result should be feasible.14

Algorithm for Computing Stationary Equilibria. In this section we describe how we

calculate stationary equilibria – functions v and g satisfying (11) and (12) – given specified

functions u, F , µz and σz, and values for the parameters ρ, δ and a.

We use a fixed point algorithm on the steady state capital stock, K. We choose a

relaxation parameter θ ∈ (0, 1] and begin an iteration with an initial guess K0. Then for

ℓ = 0, 1, 2, ... we follow

1. Given Kℓ, calculate the factor prices wℓ and rℓ from (6).

2. Given wℓ and rℓ, solve the stationary HJB equation (11) using a finite difference

method. Calculate the savings policy function µa,ℓ(a, z).

3. Given µa,ℓ(a, z), solve the stationary Kolmogorov Forward equation (12) for gℓ(a, z)

using a finite difference method.

4. Generate a new guess using a relaxed and stationary version of (5)

Kℓ+1 = (1− θ)Kℓ + θ

ˆ

agℓ(a, z)dadz.

When Kℓ+1 is close enough to Kℓ, we call (Kℓ, vℓ, gℓ, wℓ, rℓ) a stationary equilibrium.

For step 2, consider the stationary HJB equation (11). As already mentioned, the appro-

priate solution concept is that of a “viscosity solution” (Crandall and Lions, 1983; Crandall,

14If we were able to prove existence and uniqueness, we could simply apply the results of Achdou, Camilliand Capuzzo Dolcetta (2012) to establish convergence. Even in the absence of such results, it should bepossible to prove convergence.

12

Ishii and Lions, 1992). We solve the HJB equation (11) using a finite difference method.

One major advantage of this approach is that one can prove that (under certain conditions)

the finite different scheme converges to the viscosity solution of the HJB equation (Barles

and Souganidis, 1991).15 A difficulty arises at the lower bound of the state space for wealth

a, that is how to impose the borrowing limit or “state constraint” (3). In practice, we

approximate the borrowing constraint with a penalty method. We enlarge the domain by

taking amin < a but introduce a penalty function for violating the borrowing constraint. In

particular, for a < a we replace the utility function u(c) in the HJB equation (11) by

u(c)− ξ|a− a|,

where ξ is the penalty parameter, a large positive number. For a ≥ a, the HJB equation is

unchanged. Capuzzo-Dolcetta and Lions (1990) have proved that as ξ → ∞ the solution to

this modified HJB equation converges to the viscosity solution corresponding to the solution

of the sequence problem with state constraint (3). This method works very well in practice

and ensures that workers do not violate the borrowing constraint. Finally, at z and z, we

impose the boundary conditions corresponding to reflecting barriers:

∂zv(a, z) = ∂zv(a, z) = 0 all a.

For step 3, consider the Kolmogorov Forward Equation (12). The main things to note are

that our scheme is monotone like the one for solving the HJB equation; and that it conserves

mass, that is if the initial density g0(a, z) integrates to one, so will the densities in all future

time periods, g(a, z, t), t > 0. The finite difference methods that we use in steps 2 and 3 are

described in more detail in the Appendix.

Algorithm for Computing Transition Dynamics. We now briefly describe the algo-

rithm we use to calculate time-varying equilibria – functions v and g satisfying (7) and (8)

– given an initial distribution g0(a, z). In practice, it is of course impossible to compute

transition dynamics for an infinite time horizon. We therefore choose a large finite time

horizon T and take the domain of the PDEs to be (a, a)× (z, z)× (0, T ). We impose as a ter-

minal condition that the value function at time T equals the value function in the stationary

15In particular, Barles and Souganidis (1991) provide the following criteria for the convergence of approx-imation schemes: they need to be “monotone”, “consistent”, and “stable”. These properties are satisfied byour scheme. See also Oberman (2006) among others.

13

equilibrium which we here denote by v∗ to avoid confusion:

v(a, z, T ) = v∗(a, z) all (a, z). (16)

Given this terminal condition, the algorithm is relatively similar to that used to compute

stationary equilibria. We again use a fixed point algorithm, this time on the entire function

K(t). We choose a relaxation parameter θ ∈ (0, 1]. We begin an iteration with an initial

guess K0(t), t ∈ (0, T ). Then for ℓ = 0, 1, 2, ... we follow

1. Given Kℓ(t), calculate the factor prices wℓ(t) and rℓ(t) from (6).

2. Given wℓ(t) and rℓ(t) and the terminal condition (16), solve the HJB equation (7),

marching backward in time. Calculate the savings policy function µa,ℓ(a, z, t).

3. Given µa,ℓ(a, z, t) and the initial condition (10), solve the Kolmogorov Forward equa-

tion, marching forward in time, for gℓ(a, z, t).

4. Generate a new guess using a relaxed version of (5)

Kℓ+1(t) = (1− θ)Kℓ(t) + θ

ˆ

agℓ(a, z, t)dadz

When Kℓ+1(t) is close enough to Kℓ(t), we call (Kℓ, vℓ, gℓ, wℓ, rℓ) an equilibrium.

2.4 Parameterization

We use standard CRRA utility and Cobb-Douglas production functions

u(c) =c1−γ

1− γ, γ > 0, (17)

F (K,L) = AKαL1−α, α ∈ (0, 1). (18)

We further assume that the stochastic process for labor productivity (4) is a simple Brownian

motion

dzt = σzdWt

reflected at z and z. We have also tried different stochastic processes, for example an

Ornstein-Uhlenbeck process for the logarithm of labor productivity, and results are very

similar. We use the following parameters

α = 1/3, γ = 2, δ = 0.05, ρ = 0.05, A = 1, σz = 0.02,

14

and choose the state space and the length of the time interval over which we compute

transition dynamics (the domain for our differential equations) to be

a = 0, a = 22, z = 0.5, z = 1.5, T = 50.

Note that a is also the borrowing limit. That is, in our numerical experiments workers are

not allowed to borrow at all. For our finite difference method, we discretize wealth a, labor

productivity z and time t. Our grid for a has 60 nodes, the one for z has 21 nodes, and we

use 200 time steps.

Practical Considerations The algorithms described in sections 2.3 and 2.3 can be im-

plemented in a variety of different languages. Our current implementation is in C++ which

is extremely efficient: all our computations take between one and five minutes on a laptop

computer, even with relatively dense grids. We are planning on having a MATLAB version of

the code.

2.5 Results

Figure 2 plots the time path for the aggregate capital stock, GDP, the wage and the interest

rate for our transition experiment. We start the economy with an aggregate capital stock

that is below its steady state level. As expected, the economy converges monotonically to

its steady state. The dynamics of GDP, the wage and the interest rate are those typically

observed in the neoclassical growth model and Aiyagari-Bewley-Huggett economies.

Figure 3 shows the evolution of the joint distribution of wealth and labor productivity,

by plotting it at t = 0, 15, 30, 50. For a video version, see http://www.princeton.edu/

~moll/aiyagari.mov. Because our parameterization features a relatively large amount of

income risk (as captured by the parameter σz), the wealth distribution diverges over time.

Over time, the distribution also displays considerable positive correlation between labor

productivity and wealth (see the contour plots below the distribution). This is because

labor productivity shocks are very persistent. Finally, a large fraction of individuals piles

up at the borrowing limit a = 0. Figure 4 plot consumption and savings policy functions at

the initial date t = 0 and the terminal date t = 50. The figures look as expected.

15

3

3.5

4

4.5

5

5.5

6

6.5

0 5 10 15 20 25 30 35 40 45 50

Capital, K(t)

Years, t

(a) Capital

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

0 5 10 15 20 25 30 35 40 45 50

GDP, Y(t)

Years, t

(b) GDP

0.95

1

1.05

1.1

1.15

1.2

1.25

0 5 10 15 20 25 30 35 40 45 50

Wage, w(t)

Years, t

(c) Wage

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0 5 10 15 20 25 30 35 40 45 50

Interest Rate, r(t)

Years, t

(d) Interest Rate

Figure 2: Time Paths of Aggregates in Aiyagari-Bewley-Huggett Economy

16

Labor Prod.,z

Density,g(a,z,t)

Wealth,a

(a) t = 0

Labor Prod.,z

Density,g(a,z,t)

Wealth,a

(b) t = 15

Labor Prod.,z

Density,g(a,z,t)

Wealth,a

(c) t = 30

Labor Prod.,z

Density,g(a,z,t)

Wealth,a

(d) t = ∞

Figure 3: Evolution of Joint Distribution of Labor Productivity and Wealth in Aiyagari-Bewley-Huggett Economy

17

Labor Prod.,z

Wealth,a

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

(a) Consumption Policy Function, c(a, z, t) at t = 0

Labor Prod.,z

Wealth,a

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

(b) Consumption Policy Function, c(a, z, t) at t = 50

Labor Prod.,z

Wealth,a

-0.5

0

0.5

1

(c) Savings Policy Function, µa(a, z, t) at t = 0

Labor Prod.,z

Wealth,a

-0.5

0

0.5

1

(d) Savings Policy Function, µa(a, z, t) at t = 50

Figure 4: Consumption and Savings Policy Functions in Aiyagari-Bewley-Huggett Economy

18

3 Poverty Traps in an Economy with Financial Fric-

tions

The description of an equilibrium in terms of a Hamilton-Jacobi-Bellman and a Kolmogorov

Forward equation provides a highly flexible apparatus that can handle a variety of alternative

heterogeneous agent models with idiosyncratic uncertainty. We now provide an example

based on work by Cagetti and De Nardi (2006), Buera, Kaboski and Shin (2011), Buera

and Shin (2013) and Midrigan and Xu (forthcoming) in which heterogeneous producers as

in Hopenhayn (1992) face collateral constraints and fixed costs in production.

3.1 Setup

Entrepreneurs There is a continuum of entrepreneurs that are heterogeneous in their

wealth a and their productivity or ability z. The state of the economy is the joint distribution

g(a, z, t). Entrepreneurs have the same preferences (1) as workers in section 2.

Each entrepreneur owns a private firm which uses capital k to produce output y. En-

trepreneurs have access to two technologies, a productive one and an unproductive one. The

unproductive technology is

y = fu(z, k) = zAuf(k),

where f is strictly increasing and strictly concave and f(0) = 0 and Au ≥ 0. Using the more

productive technology requires paying a per-period fixed cost in units of capital, κ:

y = fp(z, k) =

zApf(k − κ), k ≥ κ

0, k < κ,(19)

where Ap > Au and κ > 0. Capital depreciates at the rate δ. Entrepreneurs rent capital

from other entrepreneurs in a competitive capital rental market at a rental rate R(t). This

rental rate equals the user cost of capital, that is R(t) = r(t) + δ where r(t) is the interest

rate and δ the depreciation rate. Entrepreneurs face collateral constraints

k ≤ λa. (20)

The parameter λ (maximum leverage ratio) captures the quality of financial markets; λ = ∞corresponds to a perfect capital market, and λ = 1 to the case where it is completely shut

19

down. An entrepreneur’s profits from operating technology j = p, u are

Πj(a, z; r) = maxk

fj(z, k)− (r + δ)k, s.t. k ≤ λa. (21)

An entrepreneur chooses to operate the more profitable of the two technologies so that his

profits are

Π(a, z; r) = max{Πu(a, z; r),Πp(a, z; r)}.

His wealth, denoted by at, then evolves according to

dat = [Π(at, zt; rt) + rtat − ct]dt (22)

Analogous to workers’ labor productivity in section 2, entrepreneurial productivity follows

a diffusion process (4) on some bounded interval [z, z] with z ≥ 0.

Equilibrium In equilibrium, the interest rate rt is such that aggregate capital demand

equals aggregate capital supply

ˆ

[

ku(a, z; rt)1{Πu(a,z,rt)>Πp(a,z,rt)} + kp(a, z; rt)1{Πu(a,z,rt)<Πp(a,z,rt)}

]

g(a, z, t)dadz

=

ˆ

ag(a, z, t)dadz

(23)

where ku(a, z; rt) and kp(a, z; rt) are the optimal capital demands in (21) conditional on oper-

ating the unproductive and productive technologies, and 1{·} is the indicator function. Note

that capital demands depends on wealth through the collateral constraint. The equilibrium

price (the interest rate) is now a non-trivial function of the entire distribution g(a, z, t), that

is it does not only depend on its first moment as in the Bewley-Aiyagari-Hugget economy in

section 2.

Recursive Representation As in section 2, we can summarize our economy as a “mean

field game,” that is in terms of a Hamilton-Jacobi-Bellman and a Kolmogorov Forward

Equation:

ρv(a, z, t) = maxc

u(c) + ∂av(a, z, t)[Π(a, z; r(t)) + r(t)a− c]

+ ∂zv(a, z, t)µz(z) +1

2∂zzv(a, z, t)σ

2z(z) + ∂tv(a, z, t),

(24)

∂tg(a, z, t) = −∂a[µa(a, z, t)g(a, z, t)]− ∂z[µz(z)g(a, z, t)] +1

2∂zz[σ

2z(z)g(a, z, t)]. (25)

20

The function µa is again the optimal savings policy function, that is

µa(a, z, t) = Π(a, z; r(t)) + r(t)a− c(a, z, t), where c(a, z, t) = (u′)−1(∂av(a, z, t))

is the optimal consumption policy function. The remaining boundary conditions, in par-

ticular that corresponding to the borrowing constraint are the same as above. Given these

and an initial condition g(a, z, 0), the two partial differential equations (7) and (8) together

with the equilibrium relationship (23) fully characterize the evolution of our economy. A

stationary equilibrium is again a time-invariant solution to this system of equations.

3.2 Parameterization

We again use a CRRA utility function (17). The production function we use is f(k − κ) =

(k − κ)α with α ∈ (0, 1) and we assume that the logarithm of entrepreneurial productivity

follows an Ornstein-Uhlenbeck process

d log zt = −ν log zt + σzdWt

The parameter values and the state space we use are

Ap = 1, α = 0.6, γ = 2, δ = 0.05, ρ = 0.05, ν = 0.05, σz = 0.1, λ = 2,

a = 0, a = 60, z = 0, z = 3, T = 50

The parameter Au

For our finite difference method, we discretize wealth a, productivity z and time t. Our

grid for a has 500 nodes, the one for z has 60 nodes, and we use 200 time steps.

3.3 Results: Does the model feature a “poverty trap”?

We now present two sets of numerical results for our economy. In the first set of results

we assume that the productivity of the unproductive technology is zero, Au = 0. That is,

without paying the fixed cost an entrepreneur can never produce. In the second set of result,

we instead assume that the productivity of the unproductive technology is strictly positive.

We show that the special case Au = 0 features an aggregate poverty trap: there are multiple

stationary equilibria and the initial wealth distribution in the economy determines where it

ends up. In contrast, the case Au > 0 features a unique stationary distribution. However,

21

this unique stationary distribution has two peaks, one populated by entrepreneurs operating

the unproductive technology and the other by those operating the productive technology.

Mobility between the two classes of entrepreneurs is extremely low, and convergence to the

stationary distribution is very slow.

A Poverty Trap in the Special Case Au = 0. When the productivity of the unpro-

ductive technology is zero, entrepreneurs who cannot cover the fixed cost do not have any

source of income. They therefore cannot accumulate any funds and are perpetually stuck

in poverty. Figures 5 and 6 compare the development paths of two economies: a relatively

poor one (Figure 5) and a relatively rich one (Figure 6). The initial joint distribution of

productivity and wealth determines where the economies end up. There is no convergence at

the aggregate level and the initially poor economy remains stuck in poverty forever. Note

Productivity,z

Density,g(a,z,t)

Wealth,a

(a) t = 0

Productivity,z

Density,g(a,z,t)

Wealth,a

(b) t = 30

Productivity,z

Density,g(a,z,t)

Wealth,a

(c) t = 100

Productivity,z

Density,g(a,z,t)

Wealth,a

(d) t = ∞

Figure 5: Poverty trap in special case Au = 0 – country is poor at t = 0

also that convergence to the stationary distribution is extremely slow. The reason for this

can be seen by examining the savings policy functions [CONTINUE HERE].

22

Productivity,z

Density,g(a,z,t)

Wealth,a

(a) t = 0

Productivity,z

Density,g(a,z,t)

Wealth,a

(b) t = 30

Productivity,z

Density,g(a,z,t)

Wealth,a

(c) t = 100

Productivity,z

Density,g(a,z,t)

Wealth,a

(d) t = ∞

Figure 6: Poverty trap in special case Au = 0 – country is rich at t = 0

Unique but Twin-Peaked Stationary Distribution when Au > 0. In contrast to the

case Au = 0 analyzed in the preceding section, entrepreneurs who cannot cover the fixed

cost now do have a source of income: the output they can produce with the unproductive

technology. Furthermore, this output is stochastic because it depends on their productivity

z. Figures 7 and 8 again plot the evolution of the joint distribution for productivity and

wealth over time for two economies: a relatively poor one and a relatively rich one. In

contrast to the case Au = 0 in the preceding section, both economies now converge to the

same stationary distribution. Hence there is no poverty trap and long-run outcomes are

independent of the initial condition. However, there are two noteworthy features: first,

the unique stationary distribution is twin peaked. This reflects the fact that entrepreneurs

operating the unproductive technology are extremely unlikely to obtain a sufficiently long

series of high enough productivity shocks to save enough to be able to cover the fixed

cost. Vice versa, entrepreneurs operating the productive technology stay wealthy even when

experiencing a sequence of low productivity shocks. Second and related, convergence to the

unique stationary distribution is extremely slow. This is especially true for the initially poor

23

Productivity,z

Density,g(a,z,t)

Wealth,a

(a) t = 0

Productivity,z

Density,g(a,z,t)

Wealth,a

(b) t = 2

Productivity,z

Density,g(a,z,t)

Wealth,a

(c) t = 50

Productivity,z

Density,g(a,z,t)

Wealth,a

(d) t = ∞

Figure 7: Unique but Twin-Peaked Stationary Distribution Au > 0 – country is poor att = 0

country in Figure 7: because mobility is low it takes a long time for the entrepreneurs in the

lower half of the distribution to escape poverty.

4 Aggregate Shocks

We extend our approach to the case where heterogeneous agent economies are hit by aggre-

gate shocks. We first formulate our approach in the context of an exact reformulation of

the framework considered in Krusell and Smith (1998), that is an Aiyagari-Bewley-Huggett

economy hit by aggregate shocks. We then present some results for an economy with financial

frictions that is hit by aggregate shocks.

24

Productivity,z

Density,g(a,z,t)

Wealth,a

(a) t = 0

Productivity,z

Density,g(a,z,t)

Wealth,a

(b) t = 2

Productivity,z

Density,g(a,z,t)

Wealth,a

(c) t = 50

Productivity,z

Density,g(a,z,t)

Wealth,a

(d) t = ∞

Figure 8: Unique but Twin-Peaked Stationary Distribution Au > 0 – country is rich at t = 0

4.1 A Krusell-Smith Economy

Consider the economy of section 2 but assume that the production is Yt = AtF (Kt, Lt) where

A follows a diffusion process

dAt = µA(At)dt+ σA(At)dZt.

As in the original paper by Krusell and Smith (1998), it becomes necessary to include in the

state space the entire distribution g, an infinite-dimensional object. In this section, we first

write down the equations characterizing the competitive equilibrium of this economy. Even

though these equations cannot be solved due to the dimensionality problem, we nevertheless

consider this a useful starting point: for example, we hope that this will later allow us to

judge the quality of various approximation methods. This is especially true given that, to

the best of our knowledge, none of the papers analyzing similar environments in discrete

25

time spell out these equations precisely.16 In the next section, we then outline a possible

solution algorithm for a simplified special case of the model.

The aggregate state is (A, g) and the individual state is (a, z) so that the value function

of an individual is V (a, z, A, g). The aggregate capital stock and corresponding prices are

K(g) =

ˆ

ag(a, z)dadz, w(A, g) = A∂LF (K(g), 1), r(A, g) = A∂KF (K(g), 1)− δ,

The optimal savings policy function will be some function µa(a, z, A, g). For a given A, the

evolution of the joint distribution of wealth and productivity g(a, z, t) is characterized by a

Kolmogorov Forward Equation. It will momentarily be useful to write this law of motion as

∂tg(a, z, t) = T [g(·, t), µa(·, A)](a, z),

where T is the “Kolmogorov Forward operator” that maps functions g and µa to the time

derivative of g:

T [g, µa(·, A)](a, z) = −∂a[µa(a, z, g, A)g(a, z)]− ∂z[µz(z)g(a, z)] +1

2∂zz[σ

2z(z)g(a, z)] (26)

The optimization problem of a household can be written as:

ρV (a, z, A, g) = maxc

u(c) + ∂aV (a, z, A, g)[w(A, g)z + r(A, g)a− c]

+ ∂zV (a, z, A, g)µz(z) +1

2∂zzV (a, z, A, g)σ2

z(z)

+ ∂AV (a, z, A, g)µA(A) +1

2∂AAV (a, z, A, g)σ2

A(A)

+

ˆ

δV (a, z, A, g)

δg(a, z)T [g, µa(·, A)](a, z)dadz.

(27)

Here δV (a, z, A, g)/δg(a, z) denotes the functional derivative of V with respect to g at point

(a, z). For completeness, the savings policy function µa is

µa(a, z, A, g) = w(A, g)z + r(A, g)a− c(a, z, A, g)

= w(A, g)z + r(A, g)a− (u′)−1(∂aV (a, z, A, g)).(28)

A recursive equilibrium is a solution to (26) to(28). Note that the equation for the value

function (27) is not an ordinary HJB equation because of the presence of g in the state space.

16For example, the original paper by Krusell and Smith (1998) spells out the value function for householdstaking as given an abstract law of motion for the wealth distribution. However, they never spell out thatlaw of motion.

26

The problem, of course, is that g is an infinite-dimensional object.

4.2 Exact Solution: Finitely Many and Infrequent Aggregate Shocks

This section outlines a solution algorithm for a simplified special case of the model presented

in the previous section. In particular, we assume that finitely many shocks hit our continuous

time economy at discrete time intervals and follow a discrete space Markov process. We first

present the simplest possible formulation of aggregate uncertainty: the economy starts out

with some initial joint distribution of wealth and productivity and converges deterministically

towards its steady state. But in the middle of these transition dynamics, the economy is

hit by a single and permanent aggregate total factor productivity shock which can take

two possible values. After having analyzed this simple benchmark case, it is relatively

easy to extend our framework to arbitrarily many aggregate productivity shocks (at least

theoretically).

One Shock in the Middle of the Transition We now assume that productivity A in

the time interval [T/2, T ] is a random variable which may take two values Aℓ and Ah > Aℓ.

That is A(t) = A0 for t = [0, T/2] and A(t) = As, s = ℓ, h with probability ps. Let us define

for s = 0, ℓ, h

rs(t) = As∂KF (Ks(t), 1)− δ, ws(t) = As∂LF (Ks(t), 1), (29)

Ks(t) =

ˆ

ags(a, z, t)dadz. (30)

For s = 0, 1, 2, the value functions vs and densities gs are found by solving the natural

analogues of (7) and (8):

ρvs(a, z, t) = maxc

u(c) + ∂avs(a, z, t)[ws(t)z + rs(t)a− c]

+ ∂zvs(a, z, t)µz(z) +1

2∂zzvs(a, z, t)σ

2z(z) + ∂tvs(a, z, t),

(31)

∂tgs(a, z, t) = −∂a[µa,s(a, z, t)gs(a, z, t)]− ∂z[µz(z)gs(a, z, t)] +1

2∂zz[σ

2z(z)gs(a, z, t)], (32)

for t ∈ [0, T/2) if s = 0 and for t ∈ [T/2, T ) if s > 0. Analogously to above

µa,s(a, z, t) = ws(t)z + rs(t)a− cs(a, z, t), where cs(a, z, t) = (u′)−1(vs(a, z, t)) (33)

27

is the optimal savings policy function. In between the arrival of two shocks, the dynamics of

the economy are governed by the same transition dynamics as in the economy without shocks

in section 2. It turns out that handling aggregate shocks boils down to finding appropriate

boundary conditions for the two partial differential equations at the time the shocks hit. In

particular, there is a terminal condition for v0 at T/2:

v0 (a, z, (T/2)−) = p1v1 (a, z, (T/2)+) + p2v2 (a, z, (T/2)+) , (34)

where (T/2)− and (T/2)+ are the times just before and just after the aggregate shock hits.

Because workers are forward-looking, they take a simple expectation over the two branches

that the aggregate economy can take. Mathematically, the Hamilton-Jacobi-Bellman equa-

tion (31) is a forward-looking equation and therefore we impose the corresponding terminal

condition and solve the equation backwards in time from it.

Similarly, there is an initial conditions for gs, s > 0 at T/2:

gs (a, z, (T/2)+) = g0 (a, z, (T/2)−) , s = 1, 2. (35)

This equation simply states that because the density is the state variable of the economy

and therefore cannot jump, its time path needs to be continuous. The condition is an initial

condition because the Kolmogorov Forward equation (32) is a backward-looking equation

that is solved marching forward in time.

Compared to the existing literature, we can solve equilibria exactly in the sense that we

do not need to resort to approximating the distribution of state variables with a finite set of

moments or assuming that individuals are boundedly rational.

Multiple Shocks Our approach can be extended to handle multiple shocks. As long as

there are finitely many of them, it is possible to keep track of the history of productivity

shocks and the evolution of the wealth distribution corresponding to each history. Suppose

a finite number N of shocks hit at times τ = τ0, τ1, ..., τN , and denote the history of shocks

up to τn by sn = (s0, s1, ..., sn). Productivity can take S possible values and A1, ..., AS and

we denote productivity after n shocks have hit by A(sn) = Asn . Finally, we assume that

shocks are Markov with transition probabilities pss′ = Pr(sn+1 = s′|sn = s).

28

The appropriate generalizations of (29) to (35) are:

r(t, sn) = A(sn)∂KF (K(t, sn), 1)− δ, w(t, sn) = A(sn)∂LF (K(t, sn), 1),

K(t, sn) =

ˆ

ag(a, z, t, sn)dadz,

ρv(a, z, t, sn) = maxc

u(c) + ∂av(a, z, t, sn)[w(t, sn)z + r(t, sn)a− c]

+ ∂zv(a, z, t, sn)µz(z) +

1

2∂zzv(a, z, t, s

n)σ2z(z) + ∂tv(a, z, t, s

n),

∂tg(a, z, t, sn) = −∂a[µa(a, z, t, s

n)g(a, z, t, sn)]− ∂z[µz(z)g(a, z, t, sn)] +

1

2∂zz[σ

2z(z)g(a, z, t, s

n)],

with boundary conditions

v (a, z, τn+1, sn) =

∑

sn+1

psn,sn+1v(

a, z, τn+1, sn+1

)

g(

a, z, τn+1, sn+1

)

= g (a, z, τn+1, sn) , all sn.

Parameterization of Stochastic Process In our numerical examples, we assume that

initial productivity is A0 = 0.5 and thereafter it fluctuates between Aℓ = 0.4 and Ah = 0.6.

The transition probabilities are given by:

P =

[

pℓℓ pℓh

phℓ phh

]

=

[

0.7 0.3

0.3 0.7

]

.

4.3 Preliminary Results

We present an economy in which aggregate total factor productivity shocks hit every five

years over a twenty-five year horizon. Figure 9 plots four sample paths corresponding to

four particular histories of shocks (out of a total of 31 histories) for the aggregate economy.

For example, the lowest of the four capital paths in panel (a) corresponds to continuously

drawing the lower productivity A1 = 0.4. Capital in panel (a) moves continuously regardless

of the arrival of new productivity shocks. In contrast, GDP, the wage and the interest rate

in panels (c) to (d) jump discontinuously if productivity switches from low to high or vice

versa.

29

1.5

2

2.5

3

3.5

4

0 5 10 15 20 25

Capital, K(t)

Years, t

(a) Capital

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

0 5 10 15 20 25

GDP, Y(t)

Years, t

(b) GDP

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0 5 10 15 20 25

Wage, w(t)

Years, t

(c) Wage

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 5 10 15 20 25

Interest Rate, r(t)

Years, t

(d) Interest Rate

Figure 9: Four Sample Paths of Aggregates

30

5 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

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.

A The Numerical Method

The algorithm is as described in sections 2.3 and 2.3. This Appendix describes in more detail

the finite difference method we use to compute equilibria, in particular how we approximate

the two partial differential equations that summarize an equilibrium: the Hamilton-Jacobi-

Bellman equation and the Kolmogorov Forward equation.

We use a uniform grid with steps (ha, hz,∆t) such that amax−amin = Naha, zmax−zmin =

Nzhz, T = M∆t. The grid nodes are (ai, zj, tn) with ai = amin + iha and zj = zmin + jhz,

tn = nδt.

A.1 The discrete Hamilton-Jacobi-Bellman equation

For stability, we use a Euler scheme, implicit w.r.t. v and explicit w.r.t. g, and an upwind

method. Such a scheme is extremely stable, but only first order accurate.

For i = 1, . . . , Na − 1, j = 1, . . . , Nz − 1 and n = 0, . . . ,M − 1,

0 =vn+1i,j − vni,j

∆t− ρvni,j

− σ2a(ai)

2

2vni,j − vni+1,j − vni−1,j

h2a

− σ2z

2

2vni,j − vni,j+1 − vni,j+1

h2z

− G(

vni+1,j − vni,jha

,vni,j − vni−1,j

ha

, wn+1(zj) + airn+1

)

(36)

where G(p1, p2, d) is defined on R× R× R+ by, see also Figure A.1:

31

• If p2 < p1 then G(p1, p2, d) = H(p2)− dp1

• Else

– If Hp(p2) > d then G(p1, p2, d) = H(p2)− dp2

– Else if Hp(p1) < d then G(p1, p2, d) = H(p1)− dp1

– Else G(p1, p2, d) = − 11−γ

d1−γ

We see that G(p1, p2, d) is continuous, non increasing with respect to p1 and non decreasing

with respect to p2. The situation p2 < p1 should not occur in our cases.

Here,

rn+1 = A∂KF (Kn+1, 1)− δ, (37)

wn+1 = A∂LF (Kn+1, 1), (38)

Kn+1 =

ˆ amax

amin

ˆ zmax

zmin

agn+1h (a, z)dadz, (39)

where gn+1h is the piecewise bilinear function taking the values gn+1

i,j at the grid node (ai, zj).

The parameter A is the productivity of the firms.

p1

H(p2)− dp2

H(p1)− dp1

d−γ

p2

H(p2)− dp1

−

1

1−γd1−γ

d−γ

Figure 10: The Numerical Hamiltonian

Boundary conditions The domain is enlarged by taking amin < 0, σa(a) = 0 if a < 0, and

the state constraint a ≥ 0 can be approximated by a penalty method: let ǫ be the penalty

32

parameter; for a < 0, (36) is modified as follows

vn+1i,j − vni,j

∆t− ρvni,j

− σ2z

2

2vni,j − vni,j+1 − vni,j+1

h2z

− G(

vni+1,j − vni,jha

,vni,j − vni−1,j

ha

, wn+1(zj) + airn+1

)

=a

ǫif a < 0.

(40)

On the parts of the boundary given by z = zmin and z = zmax, there is a Neumann

condition. We use ghost nodes and a first order approximation, namely vi,Nz+1 = vi,Nzand

vi,0 = vi,1, and we plug these values in (36) or (40).

At a = amax, depending on σa, we have the following conditions:

• if σa = 0, we do not impose any boundary conditions, (transparent conditions). Then

amax has to be chosen large enough in such a way that the density g stays 0 near

k = amax.

• if σa > 0, we impose a Neumann condition, i.e. we vNa+1,j = vNa,j: this yields

0 =vn+1Na,j

− vnNa,j

∆t− ρvnNa,j

− σ2a(aNa

)

2

vnNa,j− vnNa−1,j

h2a

− σ2z

2

2vnNa,j− vnNa,j+1 − vnNa,j+1

h2z

− G(

0,vnNa,j

− vnNa−1,j

ha

, wn+1(zj) + amaxrn+1

)

(41)

A.2 The discrete Kolmogorov Forward equations

The way of finding the discrete Kolmogorov Forward equations is

• differentiate the discrete HJB equation in which we have taken ρ = 0 and no penaliza-

tion term. This yields a system of linear equations of the type

W n+1 −W n +∆tA(V n, Gn+1)W n = 0

where V n is the vector containing all the unknowns vni,j, Gn+1 is the vector containing

all the unknowns gn+1i,j , and A(V n, Gn+1) is a square matrix of order (Na+1)×(Nz+1).

Note that the conservative form of the scheme yields that A(V n, Gn+1)1 = 0 where 1

is the vector whose coefficients are all 1.

33

• The discrete Kolmogorov Forward equation is then

Gn+1 −Gn −∆tAT (V n, Gn+1)Gn+1 = 0.

The conservativity of the scheme automatically yields that (Gn+1, 1)2 = (Gn, 1)2 (mass

conservation). Note also that AT (V n, Gn+1) has positive diagonal entries and nonposi-

tive extradiagonal entries; since the graph of the matrix is connected, this implies that

Gn+1 has positive coefficients if Gn has nonnegative coefficients.

Let us write the equation resulting from this approximation of the Fokker-Planck equa-

tions:

for i = 1, . . . , Na − 1, j = 1, . . . , Nz − 1 and n = 0, . . . ,M − 1,

0 =gn+1i,j − gni,j

∆t

+1

2

2σ2a(ai)g

n+1i,j − σ2

a(ai+1)gn+1i+1,j − σ2

a(ai−1)gn+1i−1,j

h2a

+σ2z

2

2gn+1i,j − gn+1

i,j+1 − gn+1i,j+1

h2z

+1

h2a

−gn+1i,j Gp1

(

vni+1,j−vni,jha

,vni,j−vni−1,j

ha, wn+1zj + air

n+1)

+gn+1i−1,jGp1

(

vni,j−vni−1,j

ha,vni−1,j−vni−2,j

ha, wn+1zj + ai−1r

n+1)

+1

h2a

gn+1i,j Gp2

(

vni+1,j

−vni,j

ha,vni,j

−vni−1,j

ha, wn+1zj + air

n+1)

−gn+1i+1,jGp2

(

vni+2,j−vni+1,j

ha,vni+1,j−vni,j

ha, wn+1zj + ai+1r

n+1)

.

(42)

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