Quantitative Macroeconomic Models with
Heterogeneous AgentsPer Krusell∗
Princeton University
Anthony A. Smith, Jr.∗
Yale University
March 2006∗

We would like to thank Torsten Persson for valuable comments, Rafael Lopes de Melo forresearch assistance, and the National Science Foundation for financial support.

1

Introduction

The present paper reviews recent research aimed at constructing a theoretical model ofthe macroeconomy with five key elements: (i) it is based on rational decision making byconsumers and firms using standard microeconomic theory beginning with assumptions on
preferences and technology; (ii) it is dynamic, so that savings and investment decisions
are determined by intertemporal decisions; (iii) it has stochastic aggregate shocks whichlead to macroeconomic upswings and downswings; (iv) it considers general equilibrium,
so that factor prices and interest rates are endogenous; and (v) it has a heterogeneous
population structure where consumers differ in wealth and face idiosyncratic income shocks
against which they cannot fully insure. As argued by Lucas (1976), the four first elements
seem necessary in any model that aims to evaluate the effects and desirability of economic
policy, and they are by now standard and rather broadly accepted as very important if notindispensable. What is new in the present work is the fifth element: heterogeneity.
The incorporation of a cross-section of consumers, with an accompanying nontrivial determination of the wealth distribution and of individual wealth transitions, is important
for at least two reasons. First, it constitutes a robustness check on representative-agent
macroeconomic models. Wealth is very unevenly distributed in actual economies. Moreover,a wide range of applied microeconomic studies suggests that, because of the incompletenessof insurance markets, wealth aggregation—equal propensities to save, hedge against risk,
and work—fails.1 Indeed, an important goal of the work discussed here is to establish asolid connection with the applied microeconomic literature studying consumption and labor
supply decisions. So, put differently, the first reason to worry about inequality is that it may
influence the macroeconomy.
Second, there is widespread interest in the determination of inequality per se, and in thepossibility that macroeconomic forces influence inequality. Specifically, business cycles may
affect the rich and the poor differently, and macroeconomic policy may also have impor1

For surveys, see Attanasio (2005), Blundell and Stoker (2003), and Browning, Hansen, and Heckman(2005).

1

tant distributional implications. If so, they ought to be taken into account in the welfare
evaluation of policy.
The purpose of the present paper is not, however, to answer the two questions above:
we will not provide a detailed assessment of the extent to which inequality influences themacroeconomy, and neither will we explore in any generality how inequality is determined.
We will, instead, focus on the methodological aspect of these questions. The task of solving macroeconomic models with nontrivial heterogeneity appears daunting. In short, thedetermination of the equilibrium dynamics of the cross-section of wealth requires both solving dynamic optimization problems with large-dimensional state variables and finding fixed
points in large-dimensional function spaces. Because of the so-called “curse of dimensionality”, computer speed alone cannot suffice as a remedy to the problem of high dimensionality.
Recent progress, however, now permits such analysis to take place. Thus, we will review indetail how and why the new methods work. In so doing, we will also provide some illustrative
results touching on the two motivating questions.
The illustrations we use allow the reader to obtain insights into how to make this kind oftheory operational. They also constitute examples of two important points. First, for someissues, the incorporation of inequality does not seem essential. The aggregate behavior ofa model where the only friction—indeed the only reason why exact aggregation would not
apply—is the absence of insurance markets for idiosyncratic risk is almost identical to that ofa representative-agent model. Thus, for purely aggregate issues where this model is deemed
appropriate, one could simply analyze a representative-agent economy: the representativeagent model is robust.
Second, for other issues and other setups, the representative-agent model is not robust.
Though we do not claim generality, we suspect that the addition of other frictions is key
here. As an illustration we add a certain form of credit-market constraints to the setup
with missing insurance markets. We show here that for asset pricing—in particular, thedetermination of the risk-free interest rate—it can make a big difference whether one uses a
representative-agent model or not.
The key insight to solving the model with consumer heterogeneity using numerical methods is “approximate aggregation” in wealth. Exact aggregation in wealth means that all2

aggregates, such as prices and aggregate capital, depend only on average capital (or, moregenerally, wealth) in the economy. Thus, under aggregation, propensities to undertake different activities (such as saving, portfolio allocations, and working) are equalized among
consumers of all wealth levels, so that redistribution of capital among agents does not influence totals. Approximate aggregation means that aggregates almost do not depend onanything but average capital. The implication of approximate aggregation therefore is that
individual decision makers make very small mistakes by ignoring how higher-than-first moments of the wealth distribution influence future prices.
If, in contrast, aggregation fails, such moments by definition do influence savings, portfolio
decisions, and so on, thus affecting not only the future distribution of wealth, but also average
resources available in the future, and hence also future prices relevant to the agent’s current
decisions. Thus, approximate aggregation allows one to solve the problems of forwardlooking agents with a very small set of state variables—aggregate capital only—and attainnonetheless a very high degree of accuracy. This is the key insight. The specific numerical
procedure we outline here is the natural one, given this insight, but it does not constitutethe only method that could be used to exploit this insight.
The computational algorithm has two key features. First, it is based on bounded rationality in the sense that we endow agents with boundedly rational perceptions of how theaggregate state evolves: they think that no other moments than the first moment of thewealth distribution matters. Second, we use solution by simulation, which works as follows:(i) given the boundedly rational perceptions, we solve the individuals’ problems using standard dynamic-programming methods; (ii) we draw individual and aggregate shocks over time
for a large population of individuals; (iii) we evaluate the decision rules for an initial wealthdistribution and the simulated shocks and, using a method for clearing markets in every
time period along the simulation, we generate a time series for all aggregates; and finally(iv) we compare the perceptions about the aggregates to those in the actual simulations, andthese perceptions are then updated. We think that this approach—bounded rationality andsolution by simulation—can be a productive also for other applications, and we therefore
devote space to a careful description of it.2
2

We note that it has been employed in a variety of other contexts, e.g., Cooley, Marimon, and Quadrini(2004), Cooley and Quadrini (2006), Khan and Thomas (2003, 2005), and Zhang (2005).

3

The present paper consists of two parts. In the first part we outline and show how tosolve the “big model”—the infinite-horizon, stochastic general-equilibrium model with idiosyncratic risk and incomplete insurance. For ease of exposition, we focus in this part on
idiosyncratic employment risk. We point out that the baseline model without heterogeneity
in anything but wealth and employment status generates much less wealth inequality thanthat seen in the data. Based on this insight, we discuss what kinds of (richer) model frameworks might generate more wealth dispersion. Moreover, we argue that it is important toanalyze this issue further, because the evaluation of policy aimed at improving the allocationof risk likely critically depends on what determines the wealth dispersion.
The second part of the paper contains a two-period version of the big model. The twoperiod model is useful because it is a convenient laboratory for illustrating, and testing therobustness of, approximate aggregation. In particular, in the two-period model, one cansolve for an equilibrium with arbitrary precision: because the wealth distribution in the first
period is exogenous we can vary it to trace out how aggregates move.
In contrast, in the infinite-horizon model the wealth distribution evolves in a fundamentally endogenous manner, and equilibria are constructed in a guess-and-verify manner:assuming that individuals perceive only one moment of the wealth distribution to matter,
one proceeds to find a(n almost) fixed point. In this model, it is not clear if this is the
only (approximate) equilibrium or, perhaps more fundamentally, whether the approximate
equilibrium is actually close to an exact equilibrium.
In the two-period model, we can examine such doubts. We find, for the two-period model,that the equilibrium with approximate aggregation, constructed as for the corresponding
infinite-horizon economy, is indeed very close to the exact (and unique) equilibrium. To ourknowledge, this kind of two-period model has not been analyzed before, and we think it isa very valuable tool to use for “pilot studies” of more complex economies. We also use thetwo-period model to illustrate how aggregate policy can have differing effects on the welfare
of different kinds of consumers. Finally, we look at a version of the 2-period model where
there is another friction—entrepreneurs have to fund their investment projects out of own
funds or borrowed money and cannot sell equity—and demonstrate that the determination of
the risk-free rate is quite different in this model than in its representative-agent counterpart.

4

The organization of the paper may look a little backward. It starts with two sections on
the big model: Section 2 discusses the representative-agent model and Section 3 discusses
the model with partially insurable idiosyncratic risk. Section 4 then discusses the two-period
model. A reverse order—thus starting with the simple setup and adding complication later—
might for some purposes be appropriate, and indeed it is possible to read the paper in suchan order. We begin with the big model for two reasons. First, we find it useful to state thegoal of the work at the outset, which is to analyze the infinite-horizon problem with all its
features. In doing this, we also motivate the work in the sense of explaining the problem thatneeds to be overcome—the dynamic determination of a large-dimensional object. Second, thespecific computational method we use in the infinite-horizon model—one based on boundedly
rational perceptions—can be applied in the two-period model. Though the two-period model
does not require this method—it can be solved exactly—using the method in this context
illustrates how our infinite-horizon method works, while at the same time showing that it
works well also in the two-period model. What is less unusual is that our concluding section,
Section 5, comes last.

2

The representative-agent economy

We describe our main framework in this section and in Section 3. It is, in essence, a typical“real business-cycle” model—aggregate fluctuations have their origin in technology shockd
and there is no money—with consumer heterogeneity and incomplete insurance against idiosyncratic unemployment risk.3 In this section we lay out the representative-agent model—
in which consumers can insure fully against idiosyncratic risk—and then in Section 3 we layout the heterogeneous-agent model in which consumers cannot.
The representative-agent economy, our baseline setup, has preferences given by
E

"∞
X

#

t

β u(ct , 1 − nt ) ,t=0

where consumption, ct , has to be non-negative and labor, nt , has to be in [0, 1]. The resource
constraint is
ct + kt+1 = F (zt , kt , nt ) + (1 − δ)kt + a,
3

The reliance on a real model and on technology shocks is mainly chosen for convenience. We suspectthat the key findings are robust to the exact mechanism driving business cycles, though such a robustness
check has not been performed.

5

where kt also has to be non-negative. Standard assumptions on primitives, which we will
adopt here, are that β and δ are in [0, 1], that u is strictly concave, increasing in eachargument, and twice continuously differentiable, and that F is strictly increasing in allarguments, homogenous of degree in one and concave in the second two arguments, andtwice continuously differentiable. Assumptions are also needed on the stochastic process for
zt , but we defer description of this process until later. The nonnegative constant a is a formof endowment, perhaps interpreted as an exogenous amount of home production.

2.1

Sequential competitive equilibrium

A decentralized equilibrium for this economy can be described as follows. Consumers solve
max{ct ,kt+1 ,nt ,bt+1 }∞
t=0

E

"∞
X
t=0

t

#

β u(ct , 1 − nt ) s.t.

ct + qt bt+1 + kt+1 = bt + (1 + rt − δ)kt + wt nt + a ∀t;here, we use bt to be the units of riskless bonds held, qt the time-t price of a bond that pays
one unit at time t+1 in all states of nature, and rt and wt the rental rates of capital and labor,
respectively. The consumer takes prices as given when solving this problem. We suppress
the dependence of all variables—prices as well as choice variables—on the uncertainty: in
effect, we are dealing with stochastic processes. Finally, the maximization problem above
presumes another condition, namely a constraint that prevents running Ponzi schemes.
Similarly, firms maximize profits at each date and state, implying that rt = Fk (zt , kt , nt )and wt = Fn (zt , kt , nt ). A sequential competitive equilibrium is a set of stochastic sequences
for all quantities and prices such that (i) the quantities solve the consumer problem; (ii)quantities and prices satisfy the firm’s first-order conditions stated above; (iii) the resource
constraint is satisfied; and (iv) the bond market clears, i.e., bt equals zero for all t (it isassumed that b0 = 0).
Note that this economy can be interpreted as one with a continuum of agents withidentical preferences and identical initial wealth where, in equilibrium, since each consumer’s
maximization problem has a unique solution, all consumers choose the same savings, portfolio
allocation, and hours worked.

6

2.2

Recursive competitive equilibrium

To solve dynamic equilibrium models numerically, it is useful to use recursive methods; inour economy with idiosyncratic risk, these methods are especially helpful if not indispensable. Using recursive language involves, first and foremost, expressing behavior and prices asa function of individual and aggregate state variables. The aggregate state variable here is(k̄, z), where by k̄ we now mean aggregate (mean, or total) capital: this is what is predetermined in any given period, and relevant, because it is what determines prices and therefore
any behavior.
A recursive competitive equilibrium for the representative-agent model is defined as functions V , hk , hn , hb , H k , H n , R, W , and Q such that
1. V (ω, k̄, z) solves
V (ω, k̄, z) = maxu(ω + a + nW (k̄, z) − k ′ − Q(k̄, z)b′ )+′′k ,n,b

E[V (b′ + k ′ (1 − δ + R(H k (k̄, z), z ′ )), H k (k̄, z), z ′ )|z]
for all (ω, k̄, z).
2. (hk (ω, k̄, z), hn (ω, k̄, z), hb (ω, k̄, z)) attains the maximum, for all (ω, k̄, z), in the above
maximization problem.
3. The input pricing functions satisfy, for all (k̄, z),R(k̄, z) = Fk (z, k̄, H n (k̄, z)) and W (k̄, z) = Fn (z, k̄, H n (k̄, z)).
4. Consistency: for all (k̄, z),
H k (k̄, z) = hk (k̄(1 − δ + R(k̄, z)), k̄, z),
H n (k̄, z) = hn (k̄(1 − δ + R(k̄, z)), k̄, z),and
0 = hb (k̄(1 − δ + R(k̄, z)), k̄, z).
Thus, we use ω to denote individual asset wealth in the beginning of any period, a variablewhich in equilibrium must equal k̄(1 − δ + R(k̄, z)), since bonds are in zero net supply. The
7

consistency conditions simply require that, at the equilibrium value for ω, the individual
behaves as the aggregate behaves, with h representing the individual and H the aggregate.
Notice that (k̄, z) in the consumer’s problem are not directly influencing the consumer;
if consumers knew current prices and perceived a distribution for future prices, (k̄, z) would
be superfluous. However, (k̄, z) determines current prices directly, and by determining therepresentative agent’s behavior, it determines the distribution of future prices as well.
Our market structure assumes that there are two assets: capital and bonds. If thedomain for z ′ has only two values given any current z, then two assets suffice for completing
markets. What if the domain for uncertainty is larger? A complete-markets structure would
then require more assets, and it is often convenient to introduce, in addition to capital andbonds, “Arrow securities” or “contingent claims”, which pay 1 unit of consumption in oneof the states next period and 0 otherwise. But it is not necessary to introduce these assets
in a representative-agent economy. Because consumers are identical in this model, eachconsumer’s holdings of each of the added contingent claims have to be zero in equilibrium,
just as holdings of bonds have to be zero. Thus, the market structure does not matter in a
representative-agent economy, and for this reason we do not need to include the additional
assets in the equilibrium definition. For the same reason, bonds do not have to be included
either. We include bonds here because doing so offers a convenient parallel with the modelin the next section, where bonds are present and play a nontrivial role.

2.3

Aggregation

Suppose now that we were to permit differences in individual wealth. It is well known
that, if u is in a certain class (e.g., if it equals α log c + (1 − α) log(1 − n)), then there isaggregation in wealth, presuming that the market structure is complete (for the purpose ofthis discussion, suppose that capital and bonds complete the asset markets).4 Aggregation
in wealth means, in terms of the recursive equilibrium definition, that hk (ω, k̄, z) is linear inω (and similarly for hn and hb ), i.e., that it can be written hk (ω, k̄, z) = µ(k̄, z) + λ(k̄, z)ω.
The key point is that µ and λ are functions that depend only on k̄ and z, so that thefunction hk is linear in individual wealth, i.e., consumers with different wealth levels have
4Here, we have to we presume interior solutions for leisure. For a description of the class of preferencesthat delivers aggregation, see Altug and Labadie (1994).

8

equal marginal propensities to save out of individual wealth. In this case, with heterogeneity
in the distribution of ωs across people, total capital savings and total hours worked do notdepend on anything but the mean of the distribution of ωs, i.e., on k̄(1 − δ)R(k̄, z)).
In the economy with idiosyncratic, imperfectly insurable risk decision rules will not belinear in individual wealth. However, as we shall see, decision rules will be almost linear in
wealth.

2.4

Computation

How is a recursive competitive equilibrium of the representative-agent economy computed
numerically? There are two ways to proceed. One is to solve the planning problem for allocations directly, since the equilibrium is Pareto optimal in the absence of frictions. Given
the allocations one would then simply construct R and W using the marginal-product expressions. For Q, one would just use the first-order condition for bonds from the equilibrium
definition, evaluated using the allocation obtained in the previous step.
The other approach is to only use conditions from the equilibrium definition. If one isinterested in aggregate allocations and prices only (and not in value functions and individual
decision rules), one can also compute the equilibrium easily, without the need to find either
value functions or individual decision rules. This is achieved by first deriving first-orderconditions using the consumer’s dynamic-programming problem and using the envelope condition to eliminate the value-function derivative from these first-order conditions. In those
first-order conditions—one for savings and one for leisure—one then replaces all individual
choice variables with the equilibrium functions, and one replaces the pricing functions R
and W with their marginal-product versions. Thus, one arrives at two functional first-orderconditions: the two equations have to hold for all k̄ and z. The unknowns in the two functional equations are the two functions H k (k̄, z) and H n (k̄, z). Since the equilibrium, again,
is Pareto optimal, these functional first-order conditions are identical to those implied by
the planning problem. However, the approach of solving the functional first-order conditionsof the representative agent for the equilibrium decision rules works also in the case where
there are taxes or other reasons why the equilibrium is not optimal (e.g., in the presence ofmonopolistic competition), in which case the equilibrium cannot be found just by solving a9planning problem.
Aa variety of numerical methods can be employed for solving the dynamic-programming
problem, either in its value-function version or its first-order-condition version. We will not
discuss these methods here, since there is an ample literature on this topic.

3

The economy with idiosyncratic unemployment risk

In this section, we incorporate imperfectly insurable unemployment risk into the baselinemodel described in Section 2.

3.1

Preliminaries: a steady state without aggregate shocks

We first study the model without aggregate shocks. The classic references here are Bewley (undated) and Aiyagari (1994); Huggett (1997) studies steady states and transitional
dynamics.5
Suppose that ǫ ∈ {ǫℓ , ǫh } denotes the employment status of a particular consumer, with ǫhand ǫℓ denoting the number of “employed” and “unemployed” labor input units, respectively(we will mostly assume (ǫℓ , ǫh ) = (0, 1)). ǫ is random and statistically independent acrossconsumers.6 Let the transitions between employment states for an individual be governedby a two-state Markov chain whose probability transition matrix has typical element πǫ|ǫ .
We assume that ǫ satisfies a law of large numbers: at any point in time, the total fractionof consumers with ǫ = 1 is known with certainty.
Suppose moreover that there are no other assets in the economy than those mentioned
above, i.e., bonds and capital. Finally, suppose that there are constraints on the agentsability to borrow, such as separate lower bounds on capital and bonds or a lower bound on
net next-period asset wealth.7 Then the consumer’s budget constraint in period t can be
5

For further characterization and comparative statics, see Miao (2002).
A common alternative assumption is that ǫ is individual-specific productivity, which can take on many
more values than two: there is wage risk. We adopt our simpler setup here simply for ease of exposition.
7
The assumption of exogenously incomplete asset markets of this sort, along with restrictions on borrowing, is made in order to mimic real-world arrangements. Clearly, it would be preferable to make assumptions
on a more primitive level so that the market structure would (i) be derived endogenously and (ii) maintainthe key realistic features. Fully satisfactory such setups do not, to our knowledge, yet exist, but Allen (1985)
and Cole and Kocherlakota (2001) are promising attempts. For an “in-between” approach, where someinstitutional features are given exogenously but others derived, see Chatterjee et al (2002).
6

10

written as
ct + qt bt+1 + kt+1 = bt + rt kt + ǫt wt nt + a,with bt+1 ≥ b and kt+1 ≥ k.8
These individual-level assumptions are the key assumptions on which the model with
idiosyncratic risk rests. The equilibrium is otherwise defined as in the previous sections,apart from the special attention that needs to be paid to exactly how the distribution of
wealth evolves over time.
To this end, let Γ denote the current measure over wealth and employment status. We
need to define a joint measure, since these two variables will be related in equilibrium:
employment outcomes will influence wealth. So Γ(B, ǫ) reports how many consumers haveω ∈ B and this value for ǫ, for any interval B and ǫ.
The idea now is that, though for any initial distribution Γ0 there will be a nontrivial
transition path for the distribution of wealth, one can imagine a stationary equilibrium,or steady state, where Γ has settled down to a time-invariant function. In such a case,individuals continue to experience shocks over time, but for every interval B and value for ǫ,Γ(B, ǫ) is constant: people move around within the distribution, but the number of employedwith less than a given amount of wealth is the same in every period. In a steady state, total
assets are constant, so that the interest rate and the wage rate are constant. In a steadystate, moreover, and indeed also during any transition, the return on capital must equal the
return on the riskless bond, so the two assets are identical from the consumer’s perspective,
and we can ignore bonds. So the consumer solves
V (ω, ǫ) = ′ max

k ≥k,n∈[0,1]u(ω + a + nǫw − k ′ , 1 − n) + βE[V (k ′ (1 − δ + r), ǫ′ )|ǫ]for all (ω, ǫ). This leads to decision rules hk (ω, ǫ) and hn (ω, ǫ).
Thus, as in Aiyagari (1994) and Huggett (1993), we can define a stationary equilibriumby prices r and w, decision rules hk and hn , and a stationary distribution Γ such that
1. hk (ω, ǫ) and hn (ω, ǫ) attain the maximum in the consumer’s problem for all (ω, ǫ).
8

The constraint could also be imposed in other ways, such as in terms of a lower bound on total savings.
What is key is that one insures that the agent can pay back—while maintaining non-negative consumption—no matter what shocks are realized. The loosest possible borrowing constraint, thus, is that which just ensuressolvency; see Aiyagari (1994) for a detailed discussion.

11

2. r = Fk (k̄, n̄) and w = F2 (k̄, n̄), where k̄ ≡ (n
ω h (ω, 1)Γ(dω, 1).

R

3. Γ(B, ǫ) =P

P Rǫ ω ωΓ(dω, ǫ))/(1 − δ + r) and n̄≡Rǫ̃ πǫ|ǫ̃ ω:hk (ω,ǫ̃)∈B Γ(dω, ǫ̃).

The last condition is new: given the decision rule for saving, it is a fixed-point problem
determining the function Γ. The condition counts up, on the right-hand side and overwealth and employment statuses, all the consumers who save so that their next-period-wealthbelongs to the interval B, which is a deterministic event, and multiply by the probabilityof ending up in state ǫ, thus using the law of large numbers to obtain the actual size of thegroup of agents ending up with this value for ǫ, and wealth in B.
It is straightforward to compute a stationary distribution. Given that F has constantreturns, one can guess on r, which implies a value for the capital-labor ratio and therefore
for w, and then the consumer’s problem can be solved using standard dynamic-programming
techniques: in this dynamic-programming problem there is one endogenous state variable,ω, and one exogenous variable, ǫ, with two states. The obtained decision rules can thenbe used to find the implied fixed point for Γ. The fixed point can be computed either by
iterating on the fixed-point condition giving a starting value for Γ, or by simulating a single
individual’s decisions over time. For the latter procedure, one simulates a very long timeseries for the shock ǫ and then, given an initial condition for wealth ω, using the decision rule
hk to generate a simulated time series for wealth. Using ergodicity (the state (ω, ǫ) follows
a stationary process now), it must be the case that the average value for ω in the long timeseries equals the average value in the stationary cross-section: i.e., it equals k̄(1 − δ + r).
Similarly, the average number of hours worked in the time series equals n̄. Given theseaggregates, one can check condition 2 of the stationary equilibrium: one can check whether
the implied interest rate is equal to the initially conjectured interest rate. If it is, a stationary
equilibrium is obtained; if it is not, r can be adjusted and the procedure can be repeated,until the initial guess is close enough to the implied value.99

A specific algorithm for a similar problem was described in Huggett (1993)) and, for something closer
to the present model, in Aiyagari (1994).

12

3.2

Assumptions on z and ǫ

We now introduce an aggregate shock, z, which is assumed to take on one of two values,zg (good times) or z = zb (bad times), with typical transition probability πz ′ |z , i.e., with afirst-order Markov structure given by (with slight abuse of notation)


πg|g
πb|g

πg|b
.
πb|b


As in the model without aggregate shocks, the individual employment shock, ǫ, is identically
distributed across consumers but is serially correlated. Moreover, it satisfies a law of large
numbers: conditional on knowing the aggregate shock, the total fraction of consumers with
ǫ = 1 is known with certainty. More precisely, if z = zg , then the number of unemployed
always equals ug , and if z = zb , a fraction ub of the consumers are unemployed, with ug < ub .
That is, individual and aggregate shocks are correlated, but controlling for z, individual
shocks are independently distributed.
We implement these assumptions in the easiest possible way: we employ a Markov structure on (z, ǫ):πg1|g1πΠ′ =  b1|g1
 πg0|g1
πb0|g1

πg1|b1πb1|b1
πg0|b1πb0|b1πg1|g0πb1|g0πg0|g0πb0|g0πg1|b0πb1|b0 .
πg0|b0 
πb0|b0For example, this means that the probability that an unemployed consumer will be employed
tomorrow will depend not just on his own status but on the current aggregate state as well:
it is πg1|j0 + πb1|j0 , where j is the current aggregate state.

3.3

Recursive competitive equilibrium

In the presence of aggregate shocks, the measure Γ evolves over time stochastically: givena Γ0 we can, in principle, compute all agents’ asset accumulation decisions, and therefore
determine how much wealth each consumer starts with tomorrow. Does this mean that Γ1
is known? No, because we know only the marginal wealth distribution tomorrow: we know
how many of those employed in period 0 will be in different wealth groups, but we do notknow how many of those consumers will be employed: this depends on the total number of
employed tomorrow, which in turn depends on the realization of the random variable z1 .

13

The difficulty in the model with aggregate shocks is precisely to determine the stochastic
evolution of Γ. To analyze this model, we need to employ recursive methods. Thus, we needto specify the aggregate and individual state variables. The aggregate state variable, unlikein the representative-agent economy, now contains more than just the total capital stockand the current value for the productivity shock: since different consumers have different
amounts of wealth and their propensities to save are not equal, the distribution of a given
amount of total capital will influence total savings. Thus, the distribution of wealth is astate variable. That is, the relevant aggregate state is (Γ, z).
The individual state, relevant in the individual’s maximization problem, is then(ω, ǫ; Γ, z); ω and ǫ are directly budget-relevant, and Γ and z are relevant for determining prices. Here, thus, for the individual to know k̄ ′ , it is not sufficient to know k̄: it isnecessary to know Γ. As in the representative-agent economy, the individual predicts k̄ ′with a law of motion, but here the consumer needs to predict the entire Γ′ too in orderto predict k̄ in periods beyond the next one. We let H denote the equilibrium transition
function for Γ:Γ′ = H(Γ, z, z ′ ).
The transition function contains z ′ because Γ also describes how many agents are unemployed(for each set of wealth levels), and it is not possible to know that for tomorrow until z ′ , andthus u′, is known.
A recursive competitive equilibrium for the model with idiosyncratic shocks is now defined
as functions V , hk , hn , hb , H, H n , R, W , and Q such that
1. V (ω, ǫ, Γ, z) solves
V (ω, ǫ, Γ, z) = maxu(ω + a + nǫW (Γ, z) − k ′ − Q(Γ, z)b′ )+′′k ,n,b

E[V (b′ + k ′ (1 − δ + R(H(Γ, z, z ′ ), z ′ )), ǫ′ , H(Γ, z, z ′ ), z ′ )|z]
for all (ω, ǫ, Γ, z).
2. (hk (ω, ǫ, Γ, z), hn (ω, ǫ, Γ, z), hb(ω, ǫ, Γ, z)) attains the maximum, for all (ω, ǫ, Γ, z), in
the above maximization problem.

14

3. The input pricing functions satisfy, for all (Γ, z),R(Γ, z) = Fk (z, k̄, n̄) and W (Γ, z) = Fn (z, k̄, n̄),
P R

where now k̄ = (ǫωΓ(dω))/(1 − δ + R(Γ, z)) and n̄ = H n (Γ, z).

4. Consistency: for all (Γ, z) and (when applicable) (B, ǫ) and z ′ ,′H(Γ, z, z )(B, ǫ) =Xπz ′ ,ǫ|z,ǫ̃ǫ̃Zω:hk (ω,ǫ̃,Γ,z)(1−δ+R(H(Γ,z,z ′ ),z ′ ))+hb (ω,ǫ̃,Γ,z)∈BnH (Γ, z) =ZωΓ(dω, ǫ̃).

hn (ω, 1, Γ, z)Γ(dω, 1).

and
0=

XZ
ǫ

ω

hb (ω, ǫ, Γ, z)Γ(dω, ǫ).

Though the notation is more intense, this definition is a straightforward generalizationof the equilibrium definition in the representative-agent case.
The main difficulty in finding an equilibrium for this economy is readily noted by inspecting the consumer’s dynamic optimization problem: the state variable contains Γ, avariable of, in principle, infinite dimension (it is a function!). This is not only more difficult
to handle than having the one-dimensional state variable k̄: for a general solution to thisproblem, this state variable is simply too large, no matter how fast one’s computer. Thus,to the extent that there really is nontrivial dependence of consumer decisions on the entire
function Γ, there is little hope for finding accurate solutions even to the consumer’s problem.
Furthermore, the consistency conditions need to met for every (Γ, z): a very large set of
equations. In sum, how could one ever hope to numerically compute this equilibrium?
The answer lies not in computational methods, but in properties of the economy justdescribed. It turns out that the dependence of consumer decisions on Γ is almost degenerate
when this economy is calibrated: consumers’ decisions depend on Γ only in a very, very
limited way. One indication of this fact could be obtained from studying the steady state,because by solving for a steady state (i.e., an equilibrium without aggregate uncertainty, as
outlined above) for a calibrated economy, one can inspect the obtained decision rules. They
are almost linear in individual wealth. Thus, redistribution of capital among consumers,
subject to a given total, almost would not change aggregate savings or aggregate hours
15

worked. So at least if in an economy with aggregate shocks the fluctuations in Γ—subject
to a given total—are not too large, it would seem plausible that the finding from the steadystate economy would carry over. That is, one would (almost) need to know no more than k̄
in order to know aggregate decisions.
We will now explore this line of thinking in detail. In particular, we will employ acomputational algorithm that makes heavy use of the idea that only the first moment of Γ
matters for aggregates.10

3.4

Algorithm I: trivial price determination within the period

In order to describe the algorithm and the basic finding that there is approximation aggregation in this economy, it is convenient to focus on a special case of the model where (i) leisure
is not valued and (ii) bonds are not traded. In this case, price determination is significantly
easier: as for (i), wages and interest rates are given immediately from knowing k̄, since hours
worked are exogenous, and as for (ii), there is no bond-price determination. The resulting
model has the feature that current prices are pinned down from knowing just the aggregate
capital stock and the aggregate productivity shock. This would not be true for wages andrental rates if hours were endogenous, since then the distribution of wealth would influence
total hours: in the absence of aggregation consumers with different amounts of wealth havedifferent marginal propensities to work.11 Similarly, the bond price would depend on the
distribution of wealth to the extent that marginal propensities to take on risk differ acrosspeople. The fact that the level of savings does not change any current prices in this versionof the model also reflects the perfect substitutability between consumption and investment;
if inputs were costly to move across these two activities, or if production technologies weredifferent for these two goods (either of which would mean less than perfect substitutability),
then the relative price of investment would be endogenous and depend on the total amountof saving: it would depend on the demand for investment.
10

As we describe in more detail below, should the approach based on first moments fail to produce
sufficiently accurate results, one could keep track of additional information about the distribution Γ, such as
second-order moments, in the hope of increasing accuracy. The computational algorithms that we describein Sections 3.4 and 3.5 generalize easily to setups in which consumers keep track of more features of the
distribution. For a survey and a discussion of different algorithms, see Rı́os-Rull (2001).
11
In a version of the model where there are no wealth effects in labor supply, propensities to work out of
more wealth would also be equalized (at zero) among agents.

16

The algorithm we use specifies individual perceptions of how Γ evolves that are boundedly
rational : individuals perceive a very simple law of motion for it (even though in actuality
the law of motion is very complex). Then optimal individual behavior implied by theseperceptions is derived. Based on the obtained decision rules and an initial wealth distribution
Γ0 , aggregate and individual shocks are then drawn and the resulting behavior is simulated
for an economy with a large number of agents. This means that in period zero, shocks
are drawn for the aggregate and for many agents, savings levels across the population arecomputed and, based on draws of shocks in the next period, the distribution Γ1 is obtained.
Based on the value for k1 implied by Γ1 and the new shocks, new savings decisions arecomputed across the population, date-2 shocks are drawn, and Γ2 is obtained. Thus, avery long sequence for Γt is simulated, and it can be used for assessing the accuracy of
individuals’ perceptions. Finally, perceptions are updated, and the process is repeated until
the perceptions are good enough in a sense to be made precise, at which point the algorithm
has reached its end. The finding, as we will see, is that if k̄t+1 is believed to depend only on k̄t(aside from its dependence on the aggregate shocks), this belief is almost exactly confirmedin the simulation.
We call this kind of algorithm solution by simulation since Γ is updated from period toperiod using simulation of a large number of agents, rather than using integration according
to the equilibrium definition of how the updating of Γ occurs.12 An advantage with simulationis that no functional form for Γ needs to be known, or approximated. In addition, when there
is nontrivial determination of within-period prices there are other advantages from solution
by simulation. Simulation has been used in other contexts (see, e.g., Den Haan and Marcet(1990) on parameterized expectations) but then in a context of solving a decision problem;
here, simulation is not used for solving the decision problems, but rather for solving for theequilibrium, since the key unknown is the equilibrium law of motion.13
An example serves to illustrate the workings of the algorithm. Suppose that perceptions
12

Den Haan (1997) develops a related method that does not involve simulation.
In a very similar context to that here, Obiols-Homs (2003) uses parameterized expectations for solving
the consumer’s problem and solution by simulation in order to determine the law of motion for aggregate
capital.
1317are given byz = zg : k̄ ′ = a0g + a1g k̄
z = zb : k̄ ′ = a0b + a1b k̄.

(1)Then the agent solves the following problem:
V (ω, ǫ, k̄, z) = maxk′ ≥k u(ω+F2(k̄, uz , z)ǫ+a−k ′ )+βE[V (k ′ (1−δ+F1 (k̄ ′ , uz ′ , z ′ )), ǫ′ , k̄ ′ , z ′ )|z, ǫ]subject tok̄ ′ = a0g + a1g k̄ if z = zgk̄ ′ = a0b + a1b k̄ if z = zb
This implies an optimal decision rule k ′ = hk (ω, ǫ, k̄, z), and this rule is then used to simulate
the economy with a sample of N agents (with N large). Then the “stationary region” ofthe simulated data is used to estimate—using least-squares regression—the parameters of
the linear law of motion for k̄. Using these estimates, the law of motion is updated, andthe procedure is repeated until a fixed point in these parameters is found. Thus, the chief
computational task is to find the fixed point (a⋆0 , a⋆1 , b⋆0 , b⋆1 ): individual perceptions haveto be consistent with the aggregate economy’s simulated evolution, where these parameters
represent the best goodness-of-fit to the simulated data. At this stage, the maximal goodnessof-fit represents a measure of how close approximate aggregation is to exact. Krusell andSmith (1998) discuss different goodness-of-fit measures, but the simplest one is R2 . Thus, if
the R2 is very close to 1, we say that we have approximate aggregation.14
Using a calibration of the model which is standard in the macroeconomic literature—1−γbased on the parameterizations u(c) = c 1−γ−1 and F (z, k, n) = zk α n1−α —and using a loglinear law of motion for aggregate capital, one obtains R2 measures in each of the two
equations (i.e., for good times as well as for bad times) of around 0.999998. Thus, thefit is not perfect, but close to perfect. As for the “not perfect” part, one can show thata significantly (in a statistical sense) better fit can be obtained using higher moments of
Γ in explaining the simulated series of aggregate capital stocks, but the improved fit has
negligible impact on the role of k̄ in predicting k̄ ′ , and the improvements in price forecasts
are negligible as well.
14

It is conceivable that a poor goodness of fit is obtained not due to a lack of approximate aggregation but
because the mapping from k̄ (and shocks) to k̄ ′ is nonlinear. Then, one could parameterize such a nonlinear
relationship and another goodness-of-fit could be used. This has not turned out to be a problem in any
existing applications, however.

18

The algorithm leaves open whether the finding of approximate aggregation is a case of
a “self-fulfilling prophecy”: given that agents believe in a certain simplified law of motion,
this simplified law of motion is (almost) confirmed. Does the model with boundedly rational
agents have multiple equilibria? There are several ways of addressing this question.
First, the present algorithm naturally lends itself to a generalization which is to include more moments in agents’ perceptions. The question then is whether including more
moments alters the computed equilibrium in a quantitatively significant way. Such analgorithm proceeds, first, by postulating that agents only think future prices depend ona finite set of current moments of Γ (in addition to the dependence on the aggregate
shocks): m ≡ (m1 , m2 , . . . , mM ), where this dependence is expressed using a function HM :m′ = HM (m, z, z ′ ). Second, one derives the aggregate behavior implied by these perceptions
and assesses the extent to which the agents’ perceptions differ from how the economy behaves. Thus, one (i) selects M; (ii) guesses on HM in the form of some given parameterizedfunctional form, and guesses on parameter values; (iii) solves the consumer’s problem given
HM and obtains the implied decision rules fM ; (iv) uses these decision rules to simulate thebehavior of an economy with many agents; and (v) evaluates the fit and updates until thebest possible fit within this class. If the fit is satisfactory, stop. If it is not satisfactory,
increase M, or, as a less ambitious step, try a different functional form for HM . For the calibrated economies studied, the addition of more moments does not alter the initial, simpler
law of motion based on one moment in any economically significant way.
Second, one can try to isolate and investigate those properties of decision rules andmovements in Γ—and their determinants—that seem to underlie approximate aggregation.
This analysis points to important economic mechanisms about which we elaborate below.
But one relevant observation has already been made: decision rules are almost linear in
the steady-state equilibrium of this kind of economy (when reasonably calibrated), and thisnear-linearity thus makes it difficult for higher moments to influence aggregates. Steady-state
equilibria can be solved with arbitrary accuracy, and they therefore offer reliable information
about the shapes of decision rules when the aggregate shocks are not too large.
Third, and this is a route we will explore below, one can obtain important insights fromstudying a two-period model with otherwise the same assumptions on preference, technology,

19and market structure. In particular, in a two-period model the initial wealth distribution isexogenous and can therefore be altered in order to examine the effects of its higher moments
on aggregates. Because the two-period model can be solved to any degree of accuracy, one
can directly verify whether approximate aggregation, to the extent that it holds, is due to aself-fulfilling prophecy.

3.5

Algorithm II: nontrivial price determination within the period

Consider now the model with nontrivial determination of prices within the period, because
the bond is reintroduced and because leisure is valued. A straightforward extension of
the algorithm above would postulate, alongside the law of motion for k̄, a pricing function
for bonds, Q, and a function for aggregate hours worked, H n , both with k̄ as argument(and no higher moments of Γ), and these functions would then be used in the consumer’s
dynamic programming problem to derive behavioral rules for savings, hours worked, andbond demand. These rules could then, as in the previous algorithm, be simulated. However,in this simulation, how would market clearing for hours worked and for bonds be guaranteed
at every point in time? Unless the functions are the exactly the right ones—and we know thatlack of perfect aggregation implies that these functions need to include Γ as an argument,
and not just its first moment—market clearing will never hold exactly. Moreover, in thecase of bonds, it turns out, as described in more detail in Krusell and Smith (1997), that analgorithm that does not pay attention explicitly to market clearing yields larger and larger
deviations from zero excess demand as the simulation goes on, no matter how Q is chosen
Here, one could accept deviations from market clearing—if they are not large—but a more
attractive alternative is to insist on market clearing at each point along a simulation so thatthe only deviation from full satisfaction of the equilibrium conditions is in the perceptions
that agents have about future prices. Thus, as before, we will insist on boundedly rational
perceptions as a way of computing equilibria while demanding that all markets clear andthat consumers act rationally conditionally on the assumed perceptions.
Implementation of these principles follows a two-stage procedure. Consumers view future
prices as being given by the aggregate laws Q and H n (and, of course, H k , as before, sothat w and r can be computed), but they observe directly all current prices while making

20decisions. That is, current prices are parameters in a consumer’s problem, and decision rules
for current savings, bonds, and hours worked therefore depend explicitly on these prices.
In the simulation, then, these prices can be varied to ensure market clearing at all points
in time, and the simulated price outcomes can then be compared to those implied by theperceived price functions.
Thus, we derive the decision rules as follows. Extending the example from the previous
section, we first specify H k as in equation (1), but with more compact notation:k̄ ′ = H k (k̄, z) = a0z + a1z k̄
but then add similar perceptions for Q and H n :q = Q(k̄, z) = b0z + b1z k̄
n̄ = H n (k̄, z) = c0z + c1z k̄.
Then the agent solves the following problem:
V (ω, ǫ, k̄, z) = maxk′≥k,n∈[0,1],b′≥b u(ω + F2 (k̄, H n (k̄, z), z)nǫ + a − k ′ − Q(k̄, z)b′ , 1 − n)+βE[V (k ′ (1 − δ + F1 (H k (k̄, z), H n (H k (k̄, z), z ′ ), z ′ )), ǫ′ , H k (k̄, z), z ′ )|z, ǫ].
Thus, in this problem, k̄ ′ , q, and n̄ are all given as their representative functions of aggregate
capital and the aggregate shock.
Equipped with the key output of this problem—the value function—we can then specify
the problem of an agent at any point in time in a simulation:maxk′ ≥k,n∈[0,1],b′≥b u(ω + F2 (k̄, n̄, z)nǫ + a − k ′ − qb′ , 1 − n)+βE[V (k ′ (1 − δ + F1 (H k (k̄, z), H n (H k (k̄, z), z ′ ), z ′ )), ǫ′ , H k (k̄, z), z ′ )|z, ǫ],
where current q and n̄ are now treated as parameters —note that they appear in the current
payoffs—but their future values are given by the functions Q and H n , respectively, as isk̄ ′ , which is also replaced by H k in the agent’s assessment of the future. This maximization
problem gives rise to decision rules hk (ω, ǫ, k̄, z; q, n̄), hn (ω, ǫ, k̄, z; q, n̄), and hb (ω, ǫ, k̄, z; q, n̄),and it is these rules that are used in the simulations, because they now have an explicit
dependence on current q and n̄, the latter pinning down w and r. Thus, all markets can bemade clear at every point in the simulation.
21

In summary, there are two differences compared to the simpler case in Section 3.4. First,
decision rules require a two-state derivation. Second, once these rules are obtained, thesimulation requires an additional step at each point in time, which is to vary (q, n̄) so as toclear markets for bonds and labor. Notice, therefore, that the use of the simulation offers
a way of clearing markets at each date here. Thus, solution by simulation offers a way of
dealing with an otherwise nontrivial task.

3.6

Origins of approximate aggregation

Why is there approximate aggregation? Inspection of the decision rules, both in the case of
the steady-state equilibrium and in the case of aggregate uncertainty, reveal near linearity inindividual wealth. More precisely, for all agents but the very poorest, marginal propensities
are almost all identical. Figure 1 below displays typical decision rules: the line above the
45-degree line (i.e., the middle line) is the decision rule of an employed agent (holding fixed
the aggregate state variables), and the line below is the decision rule of an unemployed agent.
Savings Decision Rules in the Infinite-Horizon Model(top line = employed; bottom line = unemployed)

Tomorrow’s Individual Capital

10.06

0
0

2

4
6
Today’s Individual Capital

8

10

Figure 1
As the graph shows, for the poorest (and especially the poorest unemployed) agents,
there are large differences in slopes of decision rules: the slope of the savings rule is, in fact,
22

zero for those who are borrowing-constrained, but there is also a small region of values forω where slopes are positive but noticeably lower than for richer agents.
For several reasons, however, these differences in marginal propensities play only a very
minor role. First, the region where there are different marginal propensities to save is verysmall. Second, the distribution of wealth is very thin in the tails and so it contains few
consumers in the region where marginal propensities are different. Third, the consumers inleft tail have, by definition, very little wealth and simply do not matter in the determinationof aggregate savings: those with the bulk of savings have wealth, and therefore (almost) havethe same marginal savings propensity. Fourth and finally, for deviations from aggregation to
be important, one needs large redistributions involving this subsegment of population, andΓ does not seem to vary enough to make such effects quantitatively visible. In conclusion,a number of conditions need to be satisfied to cause significant deviations from aggregation,and none of these conditions are present in an important way.
What causes these features? In the case of the steady-state equilibrium, this near linearity
is clearly not due to boundedly rational perceptions (and it will be found to be present in
the two-period model below as well, where an equilibrium is also computed with controlled
accuracy), and the addition of price uncertainty does not change the finding. In fact, thefinding of linearity is remarkably robust to changes in the utility function: very large amountsof risk aversion are needed to cause precautionary savings motives to make savers differ
significantly in marginal propensities. The finding seems related to other observations: theequity premium puzzle (see Mehra and Prescott (1985)) and the low estimates for the welfare
costs of business cycles (see Lucas (1987)) both suggest that the utility functions that seemconsistent with micro data do not give large costs of bearing moderate risks.
In addition, though the presence of incomplete insurance opportunities makes agents defacto more sensitive to shocks, precautionary savings are allowed here and they seem to beeffective in providing (partial) insurance against shocks. When an agent is allowed to save,for given fluctuations in ǫ (labor income), the wealthier an agent is in terms of initial asset
wealth, the less important are these fluctuations to the agent. As a result, the marginal
propensity to consume ends up being determined by permanent-income considerations: forevery additional dollar of asset income, the agent consumes only the interest component and23keeps the principal intact, thus making the savings propensity independent of wealth forhigh enough wealth levels.15 It simply turns out, then, that very little wealth is needed forthis (approximate) constancy to start to hold.
It is also possible to understand why there are few agents in the region with the very lowest
asset levels: agents do suffer from a lack of assets, because marginal utility of consumption
is very high there. Thus, with a long enough time horizon, agents save ex ante so as to avoid
ending up in this region. Preferences with higher risk aversion make being in this region
even more costly, and the distribution of wealth thus moves to the right endogenously. Inmodels with short enough life spans for consumers and no bequests, it is easier to findequilibria with more agents in the left tail of the distribution; see, e.g., Gourinchas (2000)
and Storesletten, Telmer, and Yaron (2004a). However, in these cases it is still the casethat the small amounts of wealth held by these agents still make their different marginal
propensities not matter much for aggregate savings and other aggregates.
The approximate aggregation result has surprising generality for all the above reasons.
For studies that investigate the limits of approximate aggregation, see, for example, Young(2004a), and for a discussion of a set of circumstances in which approximate aggregation
may fail in the context of overlapping-generations economies, Krueger and Kubler (2004).
The preliminary findings suggest that models that violate approximate aggregation needparameterizations that are quite different than those in standard macroeconomic calibrations.

3.7

A sample of positive results

For illustration, we will review some properties of the basic model described above. The
purpose here is thus not to discuss the ultimate value of specific models, but rather to makesome remarks about quantitative work in this area and about what approximate aggregation
does and does not imply.
Aggregates The time series for aggregates generated by the model can be compared tothose generated by the corresponding representative-agent model. Here, a striking findingis the robustness of the representative-agent setting: the aggregate time series are almost
identical in the two models. This fact follows rather directly from approximate aggregation:
15

See Bewley (1977).

24

one can construct a representative agent—with the same preferences as those of all agentsand the same kind of budget constraint, though with labor income being the mean labor
income of all agents—and endow this agent with the aggregate (mean) capital stock, andthis agent will then save (almost exactly) as in the aggregate economy with heterogeneity.
However, as we will discuss in the context of a few examples, further extensions of themodel give different time series properties than the basic representative-agent model of Section 2, even when approximate aggregation holds. In particular, in the extended models one
cannot identify a(n approximate) representative agent with the representative agent of themodel in Section 2.
1. Heterogeneity in discount rates

The easiest way to illustrate this is perhaps the exten-

sion in Krusell and Smith (1998), where consumers are assumed to have stochastic discountfactors, and where the movements in these discount factors, like employment risk, are notdirectly insurable. Thus, at any point in time there is a nontrivial distribution of discountfactors in the population. This model displays approximate aggregation, for much the same
reasons as discussed above, even though there is heterogeneity in attitudes toward savingin the population. The added heterogeneity generates larger differences in the marginal
propensities to consume in the population, but now there is in addition an offsetting mechanism: more patient agents accumulate more wealth, and therefore aggregate savings are
more concentrated. Since savings com (almost) entirely from this this group, the deviations
from perfect aggregation are again extremely small. Thus, interest rates are largely pinned
down by the marginal rates of substitution of the rich, patient agents.
At the same time, however, poor and impatient agents do command a large fraction of
total income, since they work and therefore receive labor income, and this implies that aggregate consumption is influenced more by the poor than are aggregate savings. In particular,
the poor look more like hand-to-mouth consumers, since their discount rates tend to be significantly higher than the interest rate: unlike the rich, who display permanent-income-likebehavior, poor agents who receive positive income shocks consume most of the added income.
Thus, aggregate consumption and aggregate income comove more strongly than in the modelof Section 2: we tend to see more of a traditional “consumption function” here. Since thereis approximate aggregation, though, should there not be a representative-agent counterpart25

of this setup? Perhaps, but the question then is what the preferences of such an agent would
look like.16 Instead, it seems more appropriate to think of a two-agent “shortcut” of thismodel: a model with rich savers and poor workers, who do not save at all.
2. Individual vs. aggregate labor supply

In an effort to make the above macroeconomic

model consistent with the estimates of labor supply in the applied labor literature, Hansen(1985) and Rogerson (1988) assume that labor supply is indivisible: consumers can choose
either to work or not to work, so that labor supply is highly inelastic on the individuallevel. Using a lottery mechanism with complete consumption insurance, they showed that
aggregate labor supply in this case would be infinitely elastic, with adjustments of total hours
taking place on the extensive margin only. Clearly, though, their assumption of complete
insurance markets is questionable (and furthermore leads to unemployed workers to havehigher utility than employed workers). Chang and Kim (2004, 2006), however, pursue thethe idea that the interaction of partial insurance and indivisible labor could generate different
aggregate implications. Incorporating indivisible labor into the kind of model developedin the present paper, they find that indeed aggregate labor supply is significantly moreelastic in the aggregate than one might guess based on the inelastic labor supply of any
given individual. Approximate aggregation holds, but at the same time the model produces
aggregate time series behavior that is different from that in either a standard representativeagent model with highly inelastic labor supply or in a Hansen-Rogerson economy (whichfeatures a representative agent with quasi-linear preferences).17
3. Idiosyncratic risk and asset prices

Imperfectly insurable individual risk as an explana-tion of asset prices has been explored within the same class of models discussed here. One of
the first to explore this idea was Mankiw (1986), who studied a two-period model. Constantinides and Duffie (1996), Heaton and Lucas (1996), Krusell and Smith (1997), Storesletten,
Telmer, and Yaron (2004b), and others have built infinite-horizon models to examine this possibility. Whereas Constantinides and Duffie explore assumptions under which complete an16

One might think that the answer should be the complete-markets version of the model, but this turns
out not to be correct. For simplicity, consider the case with permanent differences in discount factors, which
is a model with much the same properties. There, the complete-markets version leads to concentration of
wealth over time among the agent with the highest discount factor—all other agents “disappear” from theeconomy in the long run. In the long run, therefore, the economy would look like a representative-agent
economy like the one in Section 2, which we know displays very different behavior.
17
For another labor-market application, see Gomes, Greenwood, and Rebelo (2001).

26

alytical characterization is possible (e.g., only permanent—fully uninsurable—idiosyncratic
shocks), the latter two papers explore quantitatively restricted settings of the sort describedin this paper, and approximate aggregation applies in each case. There are two generalfindings here: the “market price of risk” increases and the risk-free rate falls. Underlying
the higher market price of risk is the assumption that idiosyncratic risk is larger in bad
times than in good times. This qualitative point, which was made in Mankiw’s early paper,
appears in Krusell and Smith’s setting in the form of unemployment risk, which is naturally
countercyclical, and in Storesletten, Telmer, and Yaron (2004c), who document and explore
the implications of countercyclical idiosyncratic wage risk.
To understand why the risk-free rate decreases, it is useful to consider a variant of thesteady-state model in Section 3.1 developed by Huggett (1993). Huggett’s model is a pureexchange economy without aggregate uncertainty in which the only asset is a risk-free bondin zero net supply.18 In this model, consider setting the borrowing constraint on bonds to 0:
b = 0. In such a case, the equilibrium is by necessity autarkic: the bond price must adjust
so that no consumer wants to hold positive amounts of bonds, i.e., it must be determinedby the intertemporal marginal rate of substitution of the agent who is willing to pay themost for the safe asset.19 If the idiosyncratic shock follows a two-state Markov chain, thenit turns out that consumers with the high shock—who face the possibility of an income loss,
unlike consumers with the low shock—determine the bond price. Such a consumer’s Euler
equation can be written−γπh|h ǫ−γh + (1 − πh|h )ǫℓ
q≥β
,
ǫ−γ
h

which exceeds β whenever ǫh > ǫℓ , πh|h < 1, and γ > 0. Intuitively, this consumer would
like to accumulate bonds to protect himself against the possible income loss, but he cannot
because the borrowing constraint precludes consumers with low shocks from lending to him,
so to restore equilibrium in the bond market, the price of the bond must rise relative to valueof the discount factor (which equals the bond price under complete markets). Thus the riskfree return falls. The expression above also allows us to show that as ǫh /ǫℓ grows large(i.e., as the gap between rich and poor increases), the interest rate declines monotonically18

Huggett’s setup is obtained by setting F (z, k, n) ≡ n.
For another paper that uses this kind of modeling in an analysis of foreign exchange risk premia, see
Leduc (2002).
19

27

from 1/β—its value in the economy without idiosyncratic risk—to zero. Second, the lesspersistent is the high endowment shock—the more likely is the income loss—the lower is therisk-free rate. Third and finally, the risk-free rate is lower, the more risk-averse is the agent(the higher is γ). Depending on primitives, thus, the gross risk-free return can be anywherein (0, 1/β).20
The effect of consumer heterogeneity on the risk-free rate can also be examined in othercontexts, such as the economies with entrepreneurial risks considered by Angeletos (2005),
Angeletos and Calvet (2004), and Covas (2005). There, entrepreneurs make investment
decisions that due to market incompleteness will be tied with their consumption decisions,and thus any borrowing using state-uncontingent bonds will deliver a risk-free interest ratethat is disproportionately influenced by the less wealthy lucky entrepreneurs, who are moreworried about risk. We will explore this kind of setup in the context of the two-period model
below.
Inequality The second main output of this model is time series for inequality, both interms of wealth and consumption. Here, we only briefly note the main findings for wealthinequality.
One can use the time-series for inequality to compute unconditional moments for theGini coefficient or some other measure of inequality in wealth and in consumption. The
variations in Γ are not so large that a model with aggregate shocks is really necessary forthe analysis of long-run properties, however: analysis of steady state suffices, along the linesof the discussion in Section 3.1. There is quite a large number of papers in this generalvein, and it is broadly recognized that the present model has a difficult time generating
wealth dispersion to the extent observed in U.S. (and other) data. As an example, the Gini
coefficient for wealth in the model calibrated in Aiyagari (1994) is 0.3, whereas in the data it
is 0.8; the fraction of wealth held by the 1% richest is much less than its value in the data of
30% (see Dı́az-Giménez et al (1997) and Rodrı́guez et al (2002) for a documentation of facts
about inequality in the U.S.), and there are few agents that have very low levels of wealth.
20
In a Huggett-style endowment economy with aggregate shocks, the bond price chiefly depends on theaggregate shock, and not on the higher moments of the asset distribution (recall that its first moment always
has to be zero); in the case with a zero-borrowing constraint, this result holds exactly. For an analysis, see
Young (2004b).

28

A number of ways of altering the basic framework have been suggested and evaluated.
To generate a large mass of agents at the lowest levels of wealth, one can introduce a featurecommon to many modern economies, namely specific welfare benefits to the very poorest(this idea was considered in Hubbard, Skinner, and Zeldes (1995) and put into an equilibrium
model by Huggett (1996)). Thus, poor agents are given disincentives to save. To generate
extreme wealth concentration, one can follow Krusell and Smith (1998) who hypothesize
discount-rate heterogeneity and show that a small amount of such heterogeneity is sufficientfor generating realistic wealth Ginis and a large concentration of wealth among the richest;
this mechanism also helps create a class of very poor agents.
As a somewhat related mechanism, one can consider some form credit-market imperfection that implies that the rates of return on savings earned by wealthy agents is higher thanthose for poor agents. It remains to be seen whether a fully microfounded setting can be constructed that delivers this result as an equilibrium outcome (models of costly participationin different markets can perhaps be developed, or models where information is asymmetricand costly to acquire); investigations with versions of increasing returns to saving can befound in Quadrini (2000), Campanale (2005), and Cagetti and De Nardi (2004, 2005).21
Castañeda et al (2003b) pursue another approach. They formulate a steady-state modelwith idiosyncratic wage-rate shocks and find that it is possible to generate large wealthinequality without any other form of heterogeneity (i.e., without preference or rate-of-return
heterogeneity). The required wage process has drastically higher dispersion: it features avery small probability of entering a state with enormous wages, while having strong regression
to the mean for this group.
Thus, in short, several different stories have been proposed to account for the stark
inequality in wealth, and each has shown some success. Does it matter, then, which story ismost relevant quantitatively? It does. Models with preference heterogeneity suggest that thepoor are poor because they choose to be poor, and thus welfare policy aimed at distribution
toward the poor is hard to defend on the grounds of efficiency: it is not that the lack of
21

In principle, different risk attitudes between the rich and the poor can help: if wealth lowers de-facto risk
aversion, then the rich earn a higher return on average, and thus become even richer. However, it is difficult
for such a mechanism to be potent quantitatively unless one departs radically from what is considered to bereasonable specifications for individual risk attitudes: a very high degree of risk aversion is needed to generate
a substantial equity premium, even in the presence of wealth heterogeneity and incomplete insurance against
idiosyncratic shocks.

29insurance markets explains poverty, but rather that some consumers choose poverty.22 In
contrast, the model where inequality is due to a wage process with very large variance predicts
that the poor are poor because they were unlucky. Based on such a model, hence, it seems
easier to argue for redistribution on efficiency grounds.23 Thus, it is important for future
research to sort out and compare these different mechanisms; we are still far from a stage
where all the properties, especially in the time-series dimension but also cross-sectionally,
have been explored for these models.
Turning now to the cyclical properties of wealth inequality, the baseline model, with orwithout preference heterogeneity, predicts that measures of wealth inequality are countercyclical. However, whereas the data on income inequality is quite good (for recent studies of
U.S. data, based on different sources, see Castañeda et al (2003a) on short-run fluctuations
and Piketty and Saez (2003) for a longer-run perspective), there are too few observations on
the wealth distribution for a meaningful time-series analysis of it.

3.8

Policy evaluation

Approximate aggregation does not mean that there is close to full consumption insurance;
in fact, the models considered in this paper feature substantial consumption inequality.24
Furthermore, macroeconomic policies, such as stabilization policy, influence consumption
inequality. Because the models considered in this paper allow for macroeconomic variation,
they can be used to evaluate the distributional effects of macroeconomic policies. Macroeconomic policy affects prices (such as wages and interest rates) through general equilibrium
channels, thereby changing the allocation of risk across consumers with different compositions of financial and human wealth; we will revisit this idea formally in Section 4.1.3 below.
So far, there are only a few papers exploring these issues; one example is a set of papers exploring how the elimination of business cycle risk affects the welfare of different groups in theeconomy (Atkeson and Phelan (1994), İmrohoroğlu (1989), Krebs (2003), Krusell and Smith
22

It should be noted that for such a stance one must assume that “discounting”, or “impatience”, is aprimitive, and not an outcome of a social or cultural process. This is not a foregone conclusion.
23
Of course, the extent to which efficiency can be used as an argument in these models is somewhat
unclear, since the market incompleteness has not been modelled from first principles. That is: why can thegovernment provide valuable insurance when the markets cannot?
24
Moreover, Cordóba and Verdier (2005) show that the potential welfare gains from eliminating U.S.
consumption inequality, relative to those from eliminating suboptimal growth and business cycles, can bevery large.

30

(1999, 2002), Mukoyama and Şahin (2005), and Storesletten, Telmer, and Yaron (2001); seealso Lucas (2003)). Additional examples include Heathcote (2005), who studies the distributional effects of shocks to taxes in an economy of the kind discussed here (where Ricardianequivalence fails to the extent that borrowing constraints bind), and Gomes (2002), who
studies the welfare effects of countercyclical unemployment insurance.

3.9

Summing up: implications of approximate aggregation

Does approximate aggregation mean that macroeconomists might as well limit attention torepresentative-agent models? For many issues, the answer is “no”: both aggregate quantities
and prices behave quite differently in many of the models discussed above than in the standard representative-agent model. We think, moreover, that more radical departures fromthe representative-agent model will occur when idiosyncratic uninsurable risk is combinedwith other frictions or other elements of heterogeneity. But it is premature to speculate inthis direction.25
Another point to stress is that there is significant consumption inequality in the modelwith idiosyncratic risk, even in the baseline version where the wealth distribution has muchless variance than in the data. That is, approximate aggregation does not say that inequality
in consumption and wealth are eliminated by means of precautionary savings; after all, all the
consumption and wealth inequality in the present model are due to market incompleteness.
A generalization of the steady-state models of Bewley, Huggett, and Aiyagari, the present
model is also stationary: the distribution of wealth is (in all the applications in the literature,at least) unique, so any effects of initial wealth differences among agents disappear overtime, and the long-run differences in consumption and wealth among agents are thus due toidiosyncratic shocks and the absence of full insurance.
In addition, this class of models does deliver nontrivial and interesting implications for theevolution of wealth and consumption inequality that are still far from completely understood.
The tools are now available to use general-equilibrium models of inequality to tackle theempirical challenges in a serious way.
25

Carroll (2000) draws a more radical conclusion than we do on this issue and pronounces a “requiem” forthe representative-agent model.

31

4

Using 2-period models

In this section we develop and analyze a series of two-period models with the aim of (i)illustrating and further analyzing approximate aggregation; (ii) evaluating the method ofsolution by simulation in a case where alternative methods are feasible; (iii) investigating
the robustness of the representative-agent model; and (iv) examining the social insurancerole of macroeconomic policy. The first model we describe shares many features with the“big model” above but ends after two periods. The second model introduces entrepreneurs,
following Angeletos (2005), Angeletos and Calvet (2004), and Covas (2005), to study approximate aggregation in a slightly different context with additional credit-market frictions.
It also serves to illustrate a mechanism for generating a lower risk-free rate.

4.1

1998 setup, baby version

We focus on a version of the model without valued leisure. Preferences are given by u(c1 ) +βE(u(c2)); we assume that u(c) has constant relative risk aversion. The technology is thestandard neoclassical one: consumption and investment are perfect substitutes in the first
period, and an aggregate investment of k̄2 units in period 1 delivers output k̄2α + (1 − δ)k̄2
in period 2. There is no aggregate technology shock in the second period, but there areidiosyncratic wage shocks (i.e., we depart somewhat from the model above in focusing on
income uncertainty due to wage, not unemployment, risk). Finally, in the first period theresources are given exogenously; there is no production. In the second period, each consumer
is endowed with one unit of time, which he supplies inelastically.
The decentralized economy has individual budgets that readc1 + k2 = ω and c2 = (1 + F2 (k̄2 , 1) − δ)k2 + F (k̄2 , 1)ǫ + a.
We assume that ǫ is lognormally distributed (with mean normalized to 1), and that shocks
are independently and identically distributed across agents so that, in the second period,
the distribution of ex-post realizations of ǫs is lognormal. Moreover, we assume that the(exogenous) distribution of ωs is lognormal; we denote this distribution Γ1 in line with thenotation in the big model.
For any initial wealth distribution Γ1 , this model can be solved with arbitrary accuracy
32using a simple guess-and-verify procedure: one guesses on k̄2 , which implies a rental rate
and a wage rate in the second period. Based on these prices, it is straightforward to solve
the consumer’s problem, which is strictly concave, by solving a first-order condition with
one unknown: the individual’s capital holdings. One then sums up, over a large number of
agents (or integrates numerically), the savings and then checks whether the obtained number
equals k̄2 . It is easy to show that there is a unique value for k̄2 that constitutes a fixed point,
and a standard Newton-Raphson algorithm converges securely to it, thus generating anequilibrium decision rule k2 = hk (ω). Then one can repeat this for any alternative initial
wealth distribution, Γ1 , and thereby obtain k2 = hk (ω, Γ1 ). This decision rule can now becompared to the dynamic decision rule for savings in Section 3: it has the entire wealthdistribution as a state variable, though this distribution is exogenous here. The task, thus,
is to vary the wealth distribution Γ1 and to explore how it impacts on the macroeconomic
equilibrium. As above, the lack of insurance markets for idiosyncratic risk makes the answer
nontrivial.
4.1.1

Results for the baseline setup

We assume that utility has a relative risk aversion coefficient of 3, and the other parameterssatisfy: β (discount factor) = 0.99, α (exponent on capital in the production function) =0.36, and δ (rate of depreciation) = 0.025. Since Γ1 is lognormal, the key variable of interestfor us is σω : the standard deviation of the initial distribution of income. Given a value
for mean wealth, does σω matter for equilibrium savings? If it (almost) does not, we haveapproximate aggregation.
Table 1 summarizes the key findings.
Table 1

σω

σǫ
0
0.5
1
k̄2
0 1.000 1.003 1.012
0.5 1.000 1.003 1.012
1 1.000 1.003 1.012

1/q
0
0.5
1

0
5.41%
5.41%
5.41%

σǫ
0.5
1
5.40% 5.34%
5.40% 5.34%
5.40% 5.34%

The left panel in Table 1 tabulates the equilibrium capital stock, k̄2 , as a function of theamount of idiosyncratic risk (σǫ ), horizontally, and the amount of initial wealth dispersion(σω ), vertically, and the right panel tabulates the interest rate as a function of the same33

variables. We normalize k̄2 so that it is 1 in the case of no idiosyncratic risk (the firstcolumn): in this case, since preferences admit exact aggregation, and we indeed see thatthe dispersion of initial wealth does not influence total savings. In the second column, totalsavings are higher since there is now idiosyncratic risk: there is some precautionary savings.
However, looking down rows in the second column, we see identical numbers: the amountof initial wealth dispersion does not seem to influence total savings, even though there areincomplete markets. Here, in fact, it is possible to detect small differences—if one examines
more significant digits. But there is, clearly, approximate aggregation. In fact, all columns
have identical numbers up to 4 significant digits, simply revealing no economically significant
aggregate effects of wealth inequality on total savings or, as seen in the right panel, on interest
rates.
Table 1 also shows that the representative-agent model is rather robust to the introduction of idiosyncratic risk in this case. The top left corners of the two panels show the
representative-agent model: the model with no idiosyncratic risk and no wealth dispersion.
But in fact all four entries in the first column are identical because preferences admit exact
aggregation over initial wealth in the absence of idiosyncratic shocks. In terms of the interest
rate, we thus see that the presence of uninsurable risk, and wealth inequality, does not lower
the interest rate by more than 7 basis points (i.e., seven hundredths of one percent).
Under the surface of the aggregates in Table 1, there is significant consumption inequality
and, in addition, consumption risk. Thus, approximate aggregation does not imply an
elimination of fluctuations in consumption: “self-insurance” using capital is not capable ofdoing this. Nonetheless, self-insurance, at least given the preferences considered here andin most of the macroeconomic literature, is very effective in insuring consumers in utility
terms. That is, the remaining consumption risk just does not hurt consumers enough tochange their overall attitudes toward saving much, and hence aggregate savings and interest
rates are hardly influenced at all by idiosyncratic risk.
The effectiveness of self-insurance is reflected in consumers’ decision rules, as depicted
in Figure 2. The top line is a typical decision rule with partial insurance. The bottom line
provides a point of comparison: it is the decision rule of a consumer who faces the equilibrium
prices from the model with partial insurance but has a complete set of assets to insure fully

34

against idiosyncratic risk.
Savings Decision Rules in the Two-Period Model(top line = partial insurance; bottom line = full insurance)

Tomorrow’s individual capital

1.78

-.72
0

1

2
3
Individual wealth45Figure 2
The decision rule under full insurance has a constant marginal propensity to save: for theclass of preferences we use, it is exactly linear. By contrast, the marginal propensity to save(or slope) of the decision rule under partial insurance increases with wealth. In addition,
as wealth increases, the decision rule becomes more and more like the full-insurance decision rule for total savings: not only marginal savings propensities but average propensities
become identical. In other words, with wealth sufficiently large, idiosyncratic risk plays anegligible role in the decision-making of consumers; these statements are proved in detail in
the Appendix. However, what is important here is quantitative: the variations in the slope
are very small and concentrated among poor consumers. This is the key finding underlying
approximate aggregation in this model.
Given that there is consumption inequality due to uninsurable shocks, one can investigate
how macroeconomic policy—i.e., aggregate policy—influences the distribution of consumption and, possibly, may improve the market’s imperfect ability to insure agents. We will
examine this issue in Section 4.1.3 below.

35

We will also demonstrate more clearly that the approximate aggregation is not exact, i.e.,that there is no “theorem” here, even though Table 1, with its 4 significant digits, does notreveal any effects of wealth inequality on the equilibrium. So we will consider two changes:
we will impose strict borrowing constraints—consumers cannot borrow at all—and we willconsider much larger changes in wealth inequality than those in the above tables, which werebased on empirically plausible changes in inequality. The results are contained in Table 2
below:
Table 2

Giniω

σǫ
k̄2
0
0.5
1
0
1.000 1.004 1.014
0.32 1.000 1.004 1.015
0.68 1.026 1.032 1.043

1/q
0
0.5
1

0
6.05%
6.05%
5.91%

σǫ
0.5
1
6.03% 5.98%
6.03% 5.97%
5.88% 5.83%

On the vertical axis, we now display the Gini coefficient of wealth instead, which ranges
from 0 to 0.68. Such wide variation in this coefficient is rarely (if ever) observed in U.S.
data, especially not in the short run. When the Gini coefficient is 0.32, a little less than
one percent of the consumers are borrowing-constrained, whereas when the Gini coefficientis 0.68, roughly 30% of the consumers are constrained. Thus, moving from row 2 to row 3
in Table 2 involves huge amounts of redistribution, and this redistribution occurs in largepart between a group with zero marginal savings propensity and a group with much higher
propensity. The result, indeed, is a more radical departure from aggregation: we now see
aggregate capital increase—when doubling the Gini from 0.32 to 0.64 and 30% of the consumers become constrained—an increase in the total capital stock by around 3 percentage
points and an intereste-rate decrease of around 15 basis points. Also, note that even theeconomy without idiosyncratic risk fails to meet aggregation: in column 1 (and 4), rows 2
and 3 differ. This is because some consumers want to borrow here and cannot: markets,
thus, are not complete, and this restriction binds for a group of agents. In sum, this economy
clearly does not feature exact aggregation: the previous findings of approximate aggregation
is a quantitative result.
In the big, quantitative model, the two channels just combined will not operate in any
important way. The reason is twofold. First, in the quantitative model, most consumers are
not constrained. Of course, there are no direct empirical observations of whether consumers
36

have “binding” borrowing constraints, so it is possible that indeed a large fraction of consumers are constrained; however, though there are disagreements, most empirical researchers
would probably set the fraction of households with binding constraints at a number much
lower than 30%. Second, in Table 2, we exogenously change the wealth distribution fromone with a Gini of 0.32 to one of 0.64, thus effectuating a huge redistribution across the
population. Such huge redistributions simply do not occur in the data, and they clearly
do not occur in the big model either; the model roughly, at least, reproduces the wealthdistribution dynamics.
4.1.2

Assessing solution by simulation

The results in the previous section are obtained with controlled—and for the numbers wereport, very high—accuracy. We now use the knowledge of what the “exact” equilibrium isin order to assess how solution by simulation works in the 2-period model. For this purpose,
let the mean and variance of the (lognormal) wealth distribution be random variables drawn
from known distributions. Consumers are now assumed to be boundedly rational: theirperceptions are given by a view of what k̄2 is that depends only on mean wealth. We thus
specify, as in the big model, a mapping from mean wealth today to future wealth that is
simple; it is parameterized by θ, and we choose θ so that, given the parametric form, it gives
the best fit for consumers. To pin down the parameters θ of this “forecasting rule”, solution
by simulation is employed: (i) we draw simulated values for the mean and variance of thewealth distribution, thus for each draw obtaining the value of k̄2 implied by the initial guesson θ; (ii) we use the simulated data to estimate a new value for θ; and (iii) we iterate toconvergence and assess the fit of the forecasting rule.
Notice that the solution by simulation, by construction, allows the possibility of a selffulfilling prophecy. That is, agents’ behavior are based on certain (boundedly rational)
perceptions, and then one could imagine that their implied behavior, due to the imperfectperceptions, somehow, would (almost) confirm these perceptions. Fully rational expectations, at least in models with complete markets, eliminate self-fulfilling equilibria, and we do
not expect there to be multiple expectational equilibria in the big model; in the two-period
model considered in this section, we can prove that the equilibrium is unique. The point,
thus, is that without rational expectations, little is known regarding uniqueness, which is37

why it is important to examine the issue here, where we know the exact equilibrium and cancompare it to the equilibrium we find using boundedly rational perceptions and solution bysimulation.
We let the mean and standard deviation of the wealth distribution vary by 10%; theaverage wealth Gini is 0.51. The results can be summarized in two observations. First, in allcases, the fit of the forecasting rule is nearly perfect: R2 ≈ 0.99999, with a residual standard
deviation of about 0.04%.26 Second, the solution by simulation was extremely close to theexact solution: the deviations are on the order of 0.001%!
Thus, we conclude that the solution found by solution by simulation is not produced due
to a self-fulfilling prophecy, as expected. The basic reason why our algorithm works reallyis the fundamental workings of this model: decision rules are extremely close to linear, andthey are linear because very little wealth suffices to insure very well in utility terms.
4.1.3

Illustrating the social insurance role of macro policy

One of the interesting aspects of macroeconomic models with heterogeneous consumers isthat they allow us to trace effects of policy across different consumer groups. Thus, one canuse these models to examine the effects—positive and normative—of government insurance
policy, aimed explicitly at improving on market outcomes from the insurance perspective.
Perhaps more interestingly, the model also makes clear how purely aggregate policy actually
may have nontrivial and interesting effects on inequality. Here we will illustrate the effects
of a tax on investment using a simple example from Davila, Hong, Krusell, and Rı́os-Rull(2005).
For the sake of the argument, assume that there is no initial wealth inequality in the
two-period model: ω is the same for all consumers. Moreover, suppose that ǫ can only
take on two values (as in the employed/unemployed case): ǫ1 occurs with probability π. A
competitive equilibrium is now simply a vector (k̄2 , r, w) such that (i) k̄2 solves
max u(ω − k) + β (πu(rk + we1 ) + (1 − π)u(rk + we2 ))k∈[0,y]and (ii) r = F2 (k̄2 , n̄2 ) and w = F2 (k̄2 , n̄2 ), with n̄2 = πe1 + (1 − π)e2 .
26

When we use the extreme changes in wealth inequality, the lowest recorded R2 was 0.99919.

38

By changing prices (wages and interest rates), a tax or subsidy on initial investment
would influence the allocation of risk in this economy, and thus it would influence ex-post
inequality. Could it in fact improve welfare? We will reproduce the analysis of Davila et al.
here and demonstrate that a tax on investment is welfare-improving. We will though simply
look at how welfare is changed when k̄2 is changed away from the laissez-faire equilibriumby a small amount. The government balances its budget, returning the proceeds of the tax
in the form of lump-sum subsidies.
Differentiating the indirect equilibrium utility with respect to aggregates and using thefact that individuals are making an optimal savings decision when the tax is zero, we obtain:
dU = βπuc (r k̄2 + we1 ) + (1 − π)uc (r k̄2 + we2 ) k̄2 dr







+ πuc (r k̄2 + we1 )e1 + (1 − π)uc (r k̄2 + we2 )e2 dw .
This expression shows that changes in factor prices are what is key here; these prices influencede-facto insurance in this economy with incomplete markets.
How are factor prices affected? We have dr = F11 (k̄2 , n̄2 )dk̄2 and dw = F21 (k̄2 , n̄2 )dk̄2 . A
tax on investment decreases aggregate savings (i.e., dk̄2 < 0), so the interest rate falls and thewage rate rises provided that the marginal product of capital is strictly decreasing and that
capital and labor are complements. Then using Euler’s theorem one can show (see Davila
et al (2005) for all the algebra) that dU > 0 if (uc (r k̄2 + we1 ) − uc (r k̄2 + we2 ))F11 k̄2 dk̄2 > 0.
This inequality follows from the strict concavity of u and F and the fact that the tax
on investment decreases aggregate savings. Thus, even though aggregate policy does notredistribute directly, it has an impact on social insurance through its effects on prices. In
a complete-markets environment, such an effect would still be present, but it could not be
welfare-improving, whereas here it is. We see, from the logic of the proof, specifically, that alower capital stock alters the prices so as to reduce de-facto risk: lucky consumers are lucky
through labor income, and an overall decrease in the wage thus has the effect of lowering
wage risk. This is what the investment tax accomplishes: it lowers investment, thus raising
the return to capital and lowering the wage rate in the second period.
If one considers a three-period model, one can see that an investment tax need not
be appropriate at all times. For in the second period, lucky consumers will save part of39

their proceeds from having a high wage outcome. This means that a subsidy to investment
may be appropriate at that point in time: it will raise w3 and lower r3 , thus reducing thede-facto risk in period-2 wages. Of course, if period-3 wages are also random, a period-2
investment subsidy would increase risk from that perspective, so whether a subsidy or a tax
is appropriate is a quantitative matter in a longer-horizon model. Davila et al. indeed showthat steady-state capital stocks are too high for some calibrations and too low for others.
Quantitatively, they argue, based on a calibration for the wealth distribution based on luck,
not discount-factor heterogeneity, that a subsidy looks most appropriate.
Finally, let us comment briefly on a methodological aspect of the kind of welfare analysis
undertaken here. In our analysis, markets are incomplete by assumption: the restrictions
consumers face in being able to insure against the idiosyncratic shocks are not derived fromfirst principles based on explicit frictions, such as asymmetric information or enforcement
problems. This means that welfare experiments are hazardous, because one might imagine
that a change in policy would also change the market structure: economic agents wouldchange their behavior in response to the new policy. That is, one could raise a form of“Lucas critique” here.
Indeed, the recent literature on dynamic contracting is in large part motivated by suchconcerns; see Kocherlakota (2005). One of the goals of that literature is to “endogenously
derive” what we perceive as empirically plausible market imperfections, and also to try to
explain observed taxation patterns in terms of optimal insurance policy for these dynamic
environments (thus building intertemporal versions of the work of Mirrlees (1971)). This
work is promising and very interesting from the present perspective, since it delivers implications for inequality. However, because it is still in its infancy, its most successful attempts
are not that close to what we perceive as a realistic setting from the perspective of typicalhouseholds in the data. In contrast, the model we use here seems much more descriptively
accurate—it is what empirical microeconomic researchers use, and also seems appealing fromthe (admittedly limited) perspective of our own lives. Therefore, it does seem reasonable toexamine its welfare properties. We also think that the mechanisms emphasized here can beevaluated on their own terms to some extent; for example, the price manipulations a government would undertake according to the results discussed above are intuitive. Hopefully,40

a more complete analysis will be possible in the future, and it is far from impossible thatthe mechanisms we discuss here also play an important role in such an analysis.

4.2

The risk-free rate: an economy with entrepreneurs

Finally, we consider a model which is a step further from the neoclassical growth model:
production does not take place using a ubiquitous technology. Put differently, factors of
production are not costlessly mobile across production sites. We will use this setup to
demonstrate that, with otherwise unchanged assumptions—idiosyncratic shocks with no direct insurance available, and initial wealth inequality—some new features materialize: theasset-pricing implications are far from those of the neoclassical economy.
We look at a two-period version of Covas (2005), which itself can be viewed as a versionof Angeletos (2005) where entrepreneurs cannot hire labor. More precisely, entrepreneurs
invest in capital in period 1 and face idiosyncratic productivity risk in the second period:
if an entrepreneur invests k, his second-period output is ǫk α , with ǫ lognormal and iid.27
As in the previous two-period model, there is then no aggregate risk in the second period.
Capital is immobile ex post: two entrepreneurs with different realizations for their ǫs cannot
increase output by letting capital move towards the more productive entrepreneur. Moreover,the production technology makes clear that the amount of labor input is also fixed (at anormalized value of one); this could reflect an assumption that the entrepreneur himself isan unsubstitutable input, or it could reflect an assumption that the labor input takes evenlonger to reallocate across production sites than does the capital input. The productivity risk
cannot be insured against with any direct insurance markets; moreover, there is a “financing
friction” in that the only individual capable of investing in an entrepreneur’s technology isthe entrepreneur himself. That is, equity cannot be traded.
The budget of a typical entrepreneur reads:
c1 + k2 + qb2 = ω and c2 = ǫk2α + b2 + a.
Thus, the entrepreneur can borrow, or lend, at a risk-free interest rate 1/q.
27
Notice that risk in some sense is different here: it is complementary with the investment choice. Thatis, higher investment increases individual risk. Conversely, zero investment, which is feasible, allows risk tobe eliminated entirely.

41

Table 3 summarizes the equilibrium behavior of this model for a range of parametersettings.

σωTable 3
σǫ
k̄2
0
0.5
1
1/q
0 1.000 1.006 1.020
0
0.5 1.000 1.006 1.020 0.5
1 1.000 1.006 1.020
1

0
5.41%
5.41%
5.41%

σǫ
0.5
1
4.57% 2.74%
4.57% 2.74%
4.57% 2.74%

Table 3 illustrates two points quite clearly: (i) approximate aggregation obtains again;and (ii) the model has very different implications than its representative-agent counterpart.
As for the first point, again all the numbers in any given column are the same: differences
appear only if more significant digits are included. As for the effects of entrepreneurial risk on
asset prices, we now see very large effects. With higher idiosyncratic risk, interest rates fall
drastically—Table 3 reveals a fall of over 2.5 percentage points. Even though decision rules
for investment/savings are quite linear in wealth here in the relevant range, the demand forbonds is influenced significantly by the poorer entrepreneurs who value insurance more. The
rich entrepreneurs invest up to an amount that roughly equalizes the expected return frominvestment with the risk-free rate, whereas the poorer entrepreneurs do not, thus stopping
short: for them, the marginal resources otherwise invested are better saved at a safe rate.
Thus, they value consumption in the future more on the margin: they are willing to pay
more for a bond, every thing else equal.

5

Conclusions and final remarks

We have reviewed what has become a rather active quantitative research area in the intersection of macroeconomics and the applied consumption-savings, labor, and finance/assetpricing literatures. The overall idea is to use available insights from these applied literatures
to construct aggregate models with realistic underpinnings. Besides performing an important “robustness check” on the standard representative-agent macroeconomic model, thisresearch allows an understanding of the equilibrium determination of inequality. Moreover,it opens up an interesting avenue for exploring effects of macroeconomic policy—namely,on the cross-section of consumers, and on their ability to insure risk effectively—that have
hitherto been ignored, or abstracted from.
42

Much of the work reviewed here emphasizes the methodological advances. In particular,
we illustrate and explain the finding of approximate aggregation, a feature of economies
with heterogeneous agents that has greatly expanded the class of models that can feasibly
be studied. The boundaries of the applicability of approximate aggregation are far fromknown; so far, we know of no quantitatively convincing models with large departures fromaggregation, but our imagination is admittedly limited and we foresee important examples ofsuch phenomena to be discovered in future research. Indeed, we think that the explorationof this boundary is an important task for the future work in this area.
Finally, we develop a two-period model, in Section 4, which not only is useful for illustration, we think, but also for conducting “pilot studies” within this area. Thus, one likely
obtains important insights about whether approximate aggregation will hold, and about
what interesting features a model might have, by first investigating a two-period model ofthe sort we examine here.
Almost by necessity, the work on dynamic macroeconomic equilibrium models requires
heavy use of numerical methods, but computers are powerful now and the numerical methods
available are becoming standard fare in many graduate programs all over the world. The
original model in Krusell and Smith (1998) can now be solved in a matter of minutes using
our own code, and that code was not written for the purposes of maximizing speed.
It seems obvious to us that many exciting issues are now open for exploration using thekinds of theoretical structures discussed here. There is ongoing work that attempts to integrate the present model with risk-averse individuals and incomplete asset markets with more
explicit models of labor market frictions, along the lines of Mortensen and Pissarides (1994),
where wage/employment shocks are endogenous. The work on policy—examining macroeconomic policy from a social insurance perspective, and comparing it with, and looking at how
it complements, more explicit social insurance/welfare policy—has also begun. Similarly,
there is some work that incorporates firm heterogeneity, credit-market frictions, and immobility of input factors—which appear to be plausible sources of idiosyncratic shocks both toworkers and firm owners—but there is much to do. Here, our two-period example economy
of Section 4.2 is an example: it has not yet been solved in an infinite-horizon version. Thisillustrates the value of using the two-period model for pilot studies: we have identified a43feasible paper to be written, namely the examination of aggregate risk in environments likethose studied by Covas (2005) and Angeletos (2005): approximate aggregation will almost
certainly hold there, and the asset-pricing implications of such a model seem quite promising.
In contrast, there is very little work examining further elements of heterogeneity, suchas the heterogeneity of consumption goods. Here, one may not expect that distinguishing
between different kinds of cereal to be an important step forward for macroeconomics, but it
may be important to makes distinctions between broad categories: nondurable consumption,
durable consumption, services, and housing consumption. Further, preference heterogeneity
seem very important to explore: if preferences for cereal are as different as they are (Tonylikes Grape Nuts and Per Mueslix), then isn’t it reasonable to expect differences between
consumers also in dimensions that appear to be of greater macroeconomic relevance, such as
their attitudes toward risk and saving?
Finally, we expect that work on asymmetric information—different views among consumers and firms regarding future events—to be an important element of future macroeconomic models; after all, it seems that most of real-world asset-market trade and the associated investments are not just explained by risk-sharing or liquidity needs. Another areawith which the quantitative literature on consumer heterogeneity has had relatively little
overlap so far is monetary economics; recent work by Doepke and Schneider (2004) suggeststhat monetary policy shocks may have important consequences for inequality.
A very final point is that, with computer speed still growing at a significant rate, we
foresee the possibility that the class of models studied here can be estimated structurally.
Here as well, methodological work, both in numerical analysis and statistics, is expected to
be an important component for the next several years.

44

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Appendix: approximate linearity in the 2-period model
Consider first the case with full insurance; let the price of a state-contingent bond paying
1 in state ǫ and zero otherwise be qf (ǫ), where f (ǫ) is the probability density associated tostate ǫ (the distribution function is denoted F (ǫ), with an expected value of ǭ); thus, forcomparison with the incomplete-markets economy, we assume actuarially fair insurance andthat the price of a riskless bond is q. It is straightforward in this case to show that a(ω, ǫ),the optimal holding of this state-contingent bond, must satisfya(ω, ǫ) =ω + q(ǭ − ǫ) − ǫ
q+

 1
q
β

γ

 1
q
β

γ

.

Thus, this amount is linear in ω, and therefore so is the total amount of bonds:
Za(ω, ǫ)F (dǫ) =ω − ǭ
q+

 1
q
β

γ

.

 1
q
β

γ

Thus, the marginal propensity to buy bonds is 1/(q+(q/β)1/γ ), and as wealth goes to infinity,
this also equals the average propensity (i.e., bond holdings over total initial wealth).
In the economy with only a riskless bond, the first-order condition readsZq−γ(ω − qa(ω)) = (ǫ + a(ω))−γ F (dǫ)β
for all ω. This allows us to see that a(ω) must be increasing. Take the derivative with respectto ω and obtainZq(ω − qa(ω))−γ−1 (1 − a′ (ω)) =(ǫ + a(ω))−γ−1 F (dǫ) a′ (ω).
β
These two equations can be rewritten as
q
=
β
and

Z

q 1 − a′ (ω)=β a′ (ω)ω − qa(ω)ǫ + a(ω)
Z!γF (dǫ)

(2)

!γ+1

(3)

ω − qa(ω)ǫ + a(ω)

F (dǫ).

Let s(ω, ǫ) ≡ ω−qa(ω); s is the ratio of current to future consumption. It is positive andǫ+a(ω)
decreasing in ǫ. One can show that there exists an ǫ∗ such that s1 (ω, ǫ∗ ) = 0, s1 (ω, ǫ) > 0
for all ǫ > ǫ∗ , and s1 (ω, ǫ) < 0 for all ǫ < ǫ∗ .28 Now we can write equations (2) and (3) as
q
=
β

Z

s(ω, ǫ)γ F (dǫ)

(4)
′

′

(ω)
To see this, first note that the derivative of ω−qa(ω)
equals (1−qa (ω))(ǫ+a)−(ω−qa(ω))a=(ǫ+a)2 ǫ+a(ω)
qǫ+ω
qǫ+ω
1
1
′
′
= ǫ+a(ω) 1 − qa (ω) qǫ+qa(ω) . This expression cannot be zero for more than
ǫ+a(ω) 1 − a (ω) ǫ+a(ω)
28qǫ+ωone value of ǫ; call that value ǫ∗ . Moreover, since ω > qa(ω), qǫ+qa(ω)
is decreasing in ǫ and therefore


qǫ+ω
1
′
∗
ǫ+a(ω) 1 − qa (ω) qǫ+qa(ω) > (<) 0 for all ǫ > (<) ǫ .

50

and

q 1 − qa′ (ω)=β a′ (ω)

Z

s(ω, ǫ)γ+1 F (dǫ).

(5)

The left-hand side of equation (4) is constant. Therefore, the right-hand side of the equation
does not depend on ω either. This means thatZs(ω, ǫ)γ−1 s1 (ω, ǫ)F (dǫ) = 0.

(6)

To find out whether the right-hand side of equation (5) is increasing or decreasing, we needto signZs(ω, ǫ)γ s1 (ω, ǫ)F (dǫ).
The sign of the latter is the same as the sign of∗s(ω, ǫ )Z ǫ∗−∞s(ω, ǫ)γ−1!

∞s(ω, ǫ)s(ω, ǫ)F(dǫ)+
s(ω, ǫ)γ−1 s1 (ω, ǫ)F (dǫ) .
s1 (ω, ǫ)∗∗s(ω, ǫ )s(ω, ǫ∗)ǫ

Z

The first of these integrals is negative, and the second one is positive, since s1 (ω, ǫ) > (<) 0
for all ǫ > (<) ǫ∗ . So the sign of the overall expression must be negative, since comparing tos(ω,ǫ)∗
equation (6) and recalling that s(ω,ǫ∗ ) < (>) 1 when ǫ > (<) ǫ , the positive integral must
become smaller and the negative integral must become larger (in absolute value). Thus,
we conclude that the right-hand side of equation (3) is decreasing in ω and, thus, that
(1 − qa′ (ω))/a′(ω) must be decreasing in ω. This implies that a′ (ω) is increasing.29
Equation (3) also implies that a′ (ω) ∈ (0, 1/q). So we know that a′ (ω) is increasing andis bounded above. Therefore, it has a limit; let it be denoted a′ . Also, let ā(ω) ≡ a(ω)/ω,so that equation (3) can be writtenZq=β

1 − qā(ω)ǫ+ ā(ω)ω!γF (dǫ).

This equation implies that ā(ω) must be increasing. Since it is bounded above by 1/q, it
must have a limit, which we denote ā. Thus, taking limits, we have two equations in two
unknowns:q
1 − qā γ
=
β
ā
and

q 1 − qa′
1 − qā
=
′
β a
ā


It follows that

γ+1

.

1

a′ = ā =q+ 1 ,qβγi.e., that the marginal and average propensities to save converge to the same value. Moreover,this is the value that obtains with full insurance.
29

The convexity of the savings function in a long-horizon model is demonstrated in Carroll and Kimball(1996).

51