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Equilibrium portfolios in the
neoclassical growth model
Emilio Espino
Universidad Torcuato Di Tella, Department of Economics, Saenz Valiente 1010 (C1428BIJ), Buenos Aires, Argentina
Received 20 December 2005; final version received 7 February 2007
Abstract
This paper studies equilibrium portfolios in the standard neoclassical growth model under uncertainty with
heterogeneous agents and dynamically complete markets. Preferences are purposely restricted to be quasihomothetic. The main source of heterogeneity across agents is due to different endowments of shares of the
representative firm at date 0. Fixing portfolios is the optimal equilibrium strategy in stationary endowment
economies with dynamically complete markets. However, when the environment displays changing degrees
of heterogeneity across agents, the trading strategy of fixed portfolios cannot be optimal in equilibrium. Very
importantly, our framework can generate changing heterogeneity if and only if either minimum consumption
requirements are not zero or labor income is not zero and the value of human and non-human wealth are
linearly independent.
© 2007 Elsevier Inc. All rights reserved.
JEL classification: C61; D50; D90; E20; G11
Keywords: Neoclassical growth model; Equilibrium portfolios; Complete markets
1. Introduction
This paper studies equilibrium portfolios in the standard one-sector neoclassical growth model
under uncertainty with heterogeneous agents. There is an aggregate technology to produce the
unique consumption good which is either consumed or invested. This technology displays constant returns to scale, is subject to productivity shocks and is operated by a representative firm.
Preferences are purposely restricted such that momentary utility functions are quasi-homothetic.
This implies that Engel curves are affine linear in lifetime human and non-human wealth. Heterogeneity across agents can arise due to differences in initial wealth and preferences within the
E-mail address: [email protected].
0022-0531/$ - see front matter © 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.jet.2007.02.003
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class mentioned before. We abstract from any kind of friction and, in particular, we assume that
markets are dynamically complete. The environment is consequently an extension of Brock and
Mirman’s [4] seminal contribution in the spirit of Chatterjee [5].
Recently, the ability of the celebrated Lucas [11] tree model to generate non-trivial asset trading
has been under scrutiny. The original version is silent about the evolution of equilibrium portfolios
since they are kept fixed in the representative agent framework. However, Judd et al. [10] (JKS
from now on) study the Lucas tree model with complete markets under general assumptions
regarding heterogeneity across agents. In stationary pure exchange economies, their surprising
finding is that after some initial rebalancing in short- and long-lived assets, agents choose a fixed
equilibrium portfolio which is independent of the aggregate state of economy. Unarguably, the
Lucas tree model is one of the most popular asset pricing models in modern finance theory and
this equilibrium property questions its full further applicability. 1 JKS conclude that some friction
(informational, financial, etc.) must play a significant role in generating non-trivial asset trading
in that framework.
Espino and Hintermaier [7] cast some doubts on the necessity of those frictions by studying
the production economy proposed by Brock [3] with heterogeneous agents under dynamically
complete markets. Production of the consumption good can be carried out by several neoclassical
firms subject to idiosyncratic productivity shocks. They show that their environment can generate
a non-trivial amount of equilibrium asset trading resulting from portfolios that in general depend
upon the aggregate state of the economy. In other words, the trading strategy of fixed portfolios
is not optimal in equilibrium. In their economy, the time dependence of agents’ heterogeneity is
determined by the evolution of the distribution of capital across firms. This is mainly due to the
fact that agents can have different labor productivities across these different production units.
In a recent independent paper, Bossaerts and Zame [2] have also discussed JKS’s interpretation
of their result. In a stochastic endowment economy, they assume that individual endowment
processes are time-dependent while the aggregate endowment process is time-independent. In
that case, they construct examples where equilibrium portfolios are not kept fixed. But after
all, assuming away time-independence of individual endowments in that setting implies that the
distribution of the aggregate endowment across agents is time-dependent and consequently the
degree of heterogeneity across agents is naturally changing.
The crucial aspect pointed out by Espino and Hintermaier [7] and, implicitly, by Bossaerts and
Zame [2] is the following. Whenever the environment under study generates changing degrees
of heterogeneity across agents, the trading strategy of fixed portfolios will not be an equilibrium
outcome in general. The second of these papers simply assumes that a crucial dimension of
heterogeneity changes through time. On the other hand, one might suspect that the results in the
first of these papers rely on the particular assumptions regarding different labor productivities
across firms.
This paper abstracts from such “frictions’’ and generates changing degrees of heterogeneity
endogenously. We study the simplest extension of the Lucas tree model to a production economy
where we are able to analyze asset trading isolating the impact of capital accumulation on the
evolution of equilibrium portfolios. This setting is the stochastic one-sector optimal growth model
with heterogeneous agents. Preferences were purposely restricted to minimize both the incentives
and the possibilities for asset trading. Exploiting some features of this preference representation,
the analysis shows how to determine equilibrium portfolios explicitly as a function of preference
parameters, initial endowments and different sources of wealth.
1 Mehra and Prescott [12] point out another unsatisfactory feature of the model in relation to its asset pricing implications.
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3
At date 0 agents can be different due to two reasons in our setting. First, they may have
different preferences within the class mentioned before due to different minimum consumption
requirements. This kind of heterogeneity is kept fixed as time and uncertainty unfold. Secondly,
agents can own different shares of the representative firm at date 0 but they have the same labor
income profile. That is, they can differ in their initial non-human wealth but share the same human
wealth. If equilibrium evolves such that these shares are kept fixed at their initial levels, this crucial
second dimension of heterogeneity across agents also remains unchanged. Consequently when
both channels of heterogeneity remain unchanged, agents can trade away individual risk through
fixed portfolios.
Very importantly, we show under which conditions our framework can generate changing heterogeneity. That is, each agent’s participation in non-human wealth is not constant in equilibrium
if and only if either minimum consumption requirements are not zero or labor income is not zero
and human and non-human wealth are linearly independent. Consider the case where there are no
minimum consumption requirements. Under these preference representations, complete markets
will imply that each agent owns a fixed share of aggregate total wealth, which includes human
and non-human wealth. Both levels of wealth depend on the evolution of the stock of capital and
thus equilibrium portfolios will adjust accordingly to keep these shares fixed whenever human
and non-human wealth are not perfectly linearly correlated.
This paper then extends and complements the existing related literature mentioned above in
non-trivial dimensions. In particular, it is the natural framework to compare our results directly
with the literature studying asset trading volume, and thus to investigate the implications of capital
accumulation on the evolution of equilibrium portfolios. It shows that changing distributions of
aggregate wealth studied by Bossaerts and Zame [2] can be the natural consequence of capital
accumulation, even in simple environments like ours. Moreover, we provide a relatively simple method to solve for equilibrium portfolios in the standard neoclassical growth model under
alternative complete market structures.
The paper is organized as follows. In Section 2, we describe the economy and characterize the set
of Pareto optimal allocations. In Section 3, we study equilibrium portfolios in the corresponding
market economy with sequential trading and dynamically complete markets. Section 4 concludes.
All proofs are in the Appendix. 2
2. The economy
We consider an economy populated by I (types of) infinitely lived agents where i ∈ {1, . . . , I }.
Time is discrete and denoted by t = 0, 1, 2, . . . . Each agent is endowed with one unit of time every
period but for simplicity we assume that leisure is not valued. There is only one consumption
good which can be either consumed or invested. Goods invested transform one-to-one in new
capital next period. There is an aggregate technology with constant returns to scale to produce
the consumption good, which is operated by a representative firm. This technology is subject to
productivity shocks and st represents the realization of this shock at date t. We assume that {st } is
a first-order stationary Markov chain. Transition probabilities are denoted by (s, s ) > 0 where
st , s, s ∈ S = {1 , . . . , N } with N < ∞. Let s t = (s0 , . . . , st ) ∈ S t+1 represent the partial
history of aggregate shocks up to date t. These histories are observable by all the agents. The
probability of s t is constructed from in the standard way and x(s t ) denotes the value of x chosen
after s t has been observed.
2 Routine details are omitted. The interested reader is referred to the working paper version, Espino [6].
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t+1 →
A consumption bundle for agent i is a sequence of functions {Ct }∞
t=0 such that Ct : S
t
[i , +∞) for all t and sups t C(s ) < ∞. Ci is agent i’s consumption set with all these sequences
as elements. The parameter i 0 is interpreted as the minimum consumption level required by
agent i. Agent i’s preferences on Ci are represented by expected, time-separable, discounted
utility. That is , if Ci ∈ Ci then
U (Ci ) =
∞ t (s t )ui (Ci (s t )),
t=0 s t
where ∈ (0, 1) and the momentary utility function ui belongs to the following class:
⎧
⎨ (C − i )1−
if = 1 and > 0,
ui (c) =
1−
⎩
ln(C − i )
if = 1,
where (C − i )0 (with strict inequality if 1).
Let K(s t )0 denote the stock of capital chosen at the node s t and available in period t + 1 to
produce with the aggregate technology. Let sF(K, L) represent this technology, where K is the
stock of capital available, L is the level of labor and s is the productivity shock. F : R2+ → R+ is
homogeneous of degree 1, strictly concave, strictly increasing in K, increasing in L and continuously differentiable. We also assume that *F (0, L)/*K = ∞ and limK→∞ *F (K, L)/*K = 0
for all L > 0. The depreciation rate is given by ∈ (0, 1). Under these assumptions, there exist
a pair (K, K) ∈ R2+ such that 0 K < K where, without loss of generality, we can restrict
ourselves to K(s t ) ∈ X ≡ [K, K] for all s t . This holds because aggregate consumption must be
bounded from below. 3 K0 is the initial stock of capital and we assume that K0 ∈ int X. Since
leisure is not valued, note that L(s t ) = I for all s t and denote f (K) = F (K, I ). Below we will
refer to the special case of unproductive labor when F (K, L) = F (K) for all L and consequently
FL (K, L) = 0 for all (K, L). In this limit case, we still assume that F is strictly concave in K and
thus it is not homogeneous of degree 1 (see [5]).
To determine equilibrium portfolios we proceed as follows. We first characterize the set of Pareto
optimal allocations. Under our assumptions about preferences, the problem reduces to solving for
aggregate variables and then distributing aggregate consumption among agents through a fixed
sharing rule. Secondly, in the next section we proceed studying a competitive market arrangement
with sequential trading to analyze the evolution of equilibrium portfolios.
2.1. Pareto optimal allocations
Under the concavity assumptions on both utility and the production functions, the set of Pareto
optimal allocations can be parametrized by welfare weights. 4 Let i be the welfare weight
assigned by the planner to agent i. For a vector of welfare weights = (i )Ii=1 ∈ RI+ , the
recursive version of the planner’s problem is given by
I
(Ci − i )1−
i
(s, s )V (s , K , )
(RPP)
+
V (s, K, ) = max
1−
((C)i ,K )
i=1
s
3 More specifically, K solves min()F (K, I ) − K = I
i=1 i and K is the standard upper bound in the neoclassical
∈S
growth model.
4 That is, the utility possibility set is convex and closed and thus we can apply the supporting hyperplane theorem.
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5
s.t.
I
Ci + K − (1 − )K = sf (K),
i=1
K ∈ X,
Ci − i 0 for all i.
Under these preference representations, it is well-known that the solution to (RPP) is equivalent
to solve 5 :
(C − )1−
(s, s )V (s , K )
(RAPP)
+
V (s, K) = max
1−
(C,K )
s
s.t.
C + K − (1 − )K = sf (K),
K ∈ X, C − 0.
I
where = i=1 i and C = Ii=1 Ci is aggregate consumption. Given a vector , individual
consumption for each i is denoted by Ci () and determined as follows:
(i )1/
1/ [C − ] + i .
j =1 j
Ci () = I
(1)
The problem is then reduced to the traditional one-sector neoclassical growth model under uncertainty. The solution to the problem (RAPP) is a set of continuous policy functions
(C(s, K), K (s, K)). To fully characterize the set of Pareto optimal allocations, we construct
optimal individual consumption Ci (s, K, ) using (1) for each ∈ RI+ .
Next we will consider competitive decentralization with dynamically complete markets to study
asset trading and its corresponding equilibrium portfolios.
3. Equilibrium asset trading
We consider a market economy in which the following trading opportunities are available.
Every period agents meet in spot markets to trade the consumption good and a complete set of
fully enforceable Arrow securities in zero net supply. The Arrow security s pays one unit of
consumption if next period’s shock is s and 0 otherwise. Agent i s holdings of this security is
represented by ai , with initial endowment ai (s0 ) = 0 for all i at date 0.
There is a representative firm that chooses labor and accumulates capital to maximize its value
VF . Note that this means that the firm owns the aggregate stock of capital. Let w be the wage paid
by the firm per unit of labor. We assume that agent i is initially endowed with i (s−1 ) = 0i > 0
shares of this firm at date 0 where Ii=1 0i = 1. 6 To illustrate our results, we shut down stock
trading since a complete set of Arrow securities is sufficient to decentralize any PO allocation. 7
To rule out Ponzi schemes, we restrict agents to bounded trading strategies. 8
5 See Espino [6] for a discussion of this equivalence.
6 It is assumed that for each agent i this share is large enough such that the optimal consumption path is in the interior
of the consumption set (see Proposition 1).
7 With the analytical tools developed here, it is a relatively simple exercise to extend this setting to study asset trading
and equilibrium portfolios with any arbitrary complete market structure.
8 These (implicit) bounds are assumed to be sufficiently large such that they do not bind in equilibrium.
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The recursive structure of this economy allows us to study recursive competitive equilibria
(RCE) directly. Consider the set of state variables. At the consumer level, it is described by
individual non-human wealth, denoted by i and defined below. At the firm level, k describes
the firm’s stock of capital. Let and K describe the distribution of wealth and the aggregate
stock of capital, respectively. Therefore, the set of aggregate state variables is fully described by
(s, , K).
Consider now the price
Let q(s, , K)(s ) denote the price of the Arrow security s ∈ S
system.
at (s, , K) where q = q(s ) s ∈S . Similarly, define (p(s, , K), w(s, , K)) as the ex-dividend
price of one share and the wage rate at (s, , K), respectively. Thus, the price system is a vector
given by (p, q, w) : S × RI+ × X → R+ × RN
+ × R+ .
Let (ci , ai ) be the agent i s policy functions representing optimal individual consumption
and next period Arrow security holdings, respectively. Similarly, let (k , l) be the firm’s policy
functions denoting capital accumulation and labor demand, respectively. Since agents do not
value leisure, they supply inelastically their time every period to obtain w as labor income. The
definition of a RCE in this setting is the following.
Definition 1. A RCE is a set of value functions for the individuals (vi )i∈I , a value function
for the firm (vF ), a set of policy functions for the individuals (ci , ai )i∈I , policy functions for
the firm (k , l), a price system (p, q, w) and laws of motion for the aggregate state variables
= G(s, , K) and K = H (s, , K) such that
(RCE 1) Given (p, q, w), for each agent i (vi , ci , ai ) solves
vi (
i , s, , K) = max
(ci ,ai )
u(ci ) + (s, s
s
)vi (
i , s , , K )
subject to
ci + p(s, , K)0i +
s
q(s, , K)(s )ai (s ) = i + w(s, , K),
i = p(s , , K ) + d(s , , K ) 0i + ai (s ),
ci i ,
where = G(s, , K) and K = H (s, , K).
(RCE 2) Given (p, Q, w), (vF , k , l) solves
vF (s, k, , K) = max
(k ,l) 0
d+
q(s, , K)(s )vF (s , k , , K )
(FP)
s
subject to
d = sF(k, l) + (1 − )k − k − w(s, , K)l,
where = G(s, , K) and K = H (s, , K).
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(RCE 3) All markets clear. For all (
i , k, s, , K)
ci (
i , s, , K) + [k (k, s, , K) − (1 − )k] = sf (k),
i
ai (
i , s, , K)(s ) = 0
for all s ,
i
l(s, k, , K) = I.
(RCE 4) Consistency. For all (s, , K) and each i:
K = H (s, , K) = k (s, K, , K),
i = Gi (s, , K) = i (i , s, , K).
We impose the consistency conditions (RCE 4) and thus we avoid writing individual policy
functions depending on individual state variables. For instance, we directly write vF (s, , K) as
the equilibrium value of the firm at (s, , K).
We are ready now to decentralize a Pareto optimal allocation as a RCE with zero initial transfers,
conditional on the assumptions about the initial distribution of shares 0 = (0i )Ii=1 and the state
of the economy at date 0, (s0 , K0 ). We will follow Negishi’s [13] approach and show that given
(s0 , K0 , 0 ) there exists a vector 0 = (s0 , K0 , 0 ) such that the corresponding Pareto optimal
allocation can be decentralized as a RCE.
With that purpose, we proceed constructively. Consider first the policy functions (C, K ) for
the problem (RAPP). Define the stochastic discount factors as follows:
−
C(s , K (s, K)) − Q(s, K)(s ) = (s, s )
,
(2)
(C(s, K) − )−
for all (s, K, s ). The value of human wealth is expressed as follows:
Vw (s, K) = W (s, K) +
Q(s, K)(s )Vw (s , K (s, K)),
(3)
s
where
W (s, K) = sF 2 (K, I ).
(4)
The shadow value of a perpetual bond that pays 1 unit of the consumption good in each period
and in each state is given by
VP (s, K) = 1 +
Q(s, K)(s )VP (s , K (s, K)).
(5)
s
The value of the firm will be given by
Q(s, K)(s )VF (s , K (s, K)),
VF (s, K) = D(s, K) +
(6)
s
where D(s, K) = sF(K, I ) + (1 − )K − K (s, K) − W (s, K)I, and
P (s, K) = VF (s, K) − D(s, K),
(7)
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for all (s, K). Notice that (2)–(7) are independent of welfare weights since aggregate variables
are independent of this distributional parameter.
Given ∈ RI+ , individual consumption Ci (s, K, ) is constructed using (1) and thus individual
non-human wealth will be determined by
Q(s, K)(s )i (s , K (s, K), ).
(8)
i (s, K, ) = Ci (s, K, ) − W (s, K) +
s
It can be shown that the operators (3), (5), (6) and (8) have unique continuous solutions Vw , VP ,
VF and i , respectively. 9 Below, VF will also be referred as the value of aggregate non-human
wealth.
Now we show that there exists a welfare weight 0 = (s0 , K0 , 0 ) such that i (s0 , K0 , 0 ) =
0
i VF (s0 , K0 ). That is, we identify a particular PO allocation such that agent i s (implicit) individual non-human wealth equals his corresponding initial endowment. Then, we show how this
allocation can be decentralized as a RCE with zero initial transfers.
To obtain equilibrium Arrow security holdings, we proceed as follows. First, we fix 0 and
define
Ai (s, K, 0 ) = i (s, K, 0 ) − 0i VF (s, K),
(9)
for each agent i. Then, equilibrium portfolios will be given by
ai (s, , K)(s ) = Ai (s , K (s, K), 0 ),
(10)
where i = i (s, K, 0 ) is uniquely determined by (s, K, 0 ) for each i. We can observe from
(10) that equilibrium portfolios depend upon (s, K) because they determine next period’s stock
of capital and thus the evolution of relative individual non-human wealth.
We can now summarize this discussion with the following result where we show how to express
the welfare weight 0 and the function (9) as explicit functions of VF , Vw , VP , 0 and i ’s.
Proposition 1. There exists a unique welfare weight 0 = (s0 , K0 , 0 ) given by
1/
i (s0 , K0 , 0 )
=
0i VF (s0 , K0 ) + Vw (s0 , K0 ) − VP (s0 , K0 )i
,
VF (s0 , K0 ) + Vw (s0 , K0 )I − VP (s0 , K0 )
(11)
such that the corresponding Pareto optimal allocation can be decentralized as a RCE.
Fix 0 given by (11) and define i = i (s, K, 0 ) for each i using (8). The corresponding
price system is constructed using (2), (4) and (7) such that
q(s, , K)(s ) = Q(s, K)(s ) for all s ,
p(s, , K) = P (s, K),
w(s, , K) = W (s, K).
Equilibrium individual consumption is given by (1) where
ci (s, , K) = Ci (s, K, 0 ).
9 See Espino [6] for details.
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Finally, equilibrium portfolios are given by (10) where
1/ 1/
Ai (s, K, 0 ) = 0i
− 0i VF (s, K) + i − 0i
VP (s, K)
1/
+ I 0i
− 1 Vw (s, K).
9
(12)
We can establish the affine linearity of individual consumption with respect to total wealth.
This is a direct consequence of the particular preference representation that we have purposely
assumed to minimize asset trading incentives and it will be an important aspect to simplify the
analysis of our main result. 10
Lemma 1. There exist b(s, K) and z(s, K) such that agent i’s consumption can be expressed by
ci (s, , 0 ) = i (1 − b(s, K)) + z(s, K) i (s, K, 0 ) + Vw (s, K) ,
(13)
for all (s, K).
We say that the fixed equilibrium portfolio property is satisfied if for each s , for all (s, , K)
and for all i, it follows that ai (s, , K)(s ) = ai (s ). That is, individual portfolios are kept fixed,
independently of the aggregate state of the economy.
We first show that this property holds if we reduce this framework to an endowment economy.
That is, when capital is unproductive and = 1, we get a version of the stationary Lucas tree
model with heterogeneous agents which is a particular case of JKS [10]. Observe that in this case
K does not appear as a state variable in (10).
Proposition 2 (JKS). In the endowment version of this economy, equilibrium portfolios are fixed
such that
ai (s, )(s ) = ai (s )
for all (s, ) and for all s ,
where i = i (s, 0 ) for each i.
It is important to notice that the demand for Arrow securities is independent of the unique
aggregate state variable, s. That is, independently of the state s today (and thus independent of the
uniquely determined wealth distribution given by (s)), agents choose to construct fixed equilibrium portfolios of Arrow securities. JKS show how to extend this result in several dimensions for
stationary pure exchange economies.
It turns out that if minimum consumption requirements and labor income are both assumed
away, this environment is still unable to generate changing equilibrium portfolios and thus the
fixed equilibrium portfolio property is preserved. The next result formalizes this claim.
Proposition 3. If i = 0 for all i and W (s, K) = 0 for all (s, K), then the equilibrium portfolio
is kept fixed and given by
ai (s, , K)(s ) = 0
for all s and all (s, , K).
(14)
10 Expression (13) is a generalization of Chatterjee [5] for a setting that includes aggregate uncertainty and labor income.
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Now we show that this result might not be robust in the more general setting introduced above.
For the next two results below, we assume that there exist at least two agents i, h such that 0i = 0h .
First, Proposition 4 shows that if minimum consumption requirements differ from 0 for at least one
agent, then the trading strategy of fixed portfolios cannot be optimal in equilibrium even if there
is no labor income. Secondly, Proposition 5 shows that under certain conditions the introduction
of labor income (productive labor) is sufficient to generate changing equilibrium portfolios.
Proposition 4. Suppose that labor is unproductive with W (s, K) = 0 for all (s, K). If i > 0
for some agent i, then fixed portfolios cannot be optimal in equilibrium.
In Proposition 4 it is important to notice that the fixed portfolio trading strategy will not be
optimal in equilibrium even if the initial heterogeneity in shares is assumed away (i.e., 0i = 1/I
for all i) whenever i = h for some i, h.
Proposition 5. Assume away minimum consumption requirements where i = 0 for all i. If
W (s, K) > 0 for all (s, K), then the fixed equilibrium portfolio property holds only if human
wealth and non-human wealth are perfectly linearly correlated.
Observe that in this case if the initial heterogeneity in share holdings is assumed away, agents
are ex ante identical and naturally there is no asset trading in equilibrium.
3.1. Discussion of the main result
To simplify, suppose that i = for all i. Hence, at any state (s, K) the degree of heterogeneity
across agents is represented by their relative participation in aggregate non-human wealth. Whenever this participation changes with the current state (s, K), the associated equilibrium portfolios
will in general adjust accordingly.
To see this more clearly, consider the evolution of individual non-human wealth. It follows
from Lemma 1 that
i (s , K (s, K))
= (s, K, s ) +
VP (s , K (s, K)) − VP (s, K)(s, K, s )
i (s, K)
i (s, K)
1
+
Vw (s, K)(s, K, s ) − Vw (s , K (s, K)) ,
i (s, K)
(s,s ) 1/
where (s, K, s ) = z(s z(s,K)
.
,K (s,K)) Q(s,K)(s )
This says that agent i s stochastic growth of non-human wealth in general depends on his
individual wealth. This holds except in the case in which minimum consumption requirements
are 0 for all agents and labor is unproductive. In that particular case, the growth rate of non-human
wealth is the same for each agent. Consequently, at any aggregate state (s, K), individual wealth
will be perfectly correlated with aggregate wealth. Thus, individual risk coincides with aggregate
risk and there is no trade in equilibrium.
When minimum consumption requirements are zero but there is labor income, it is interesting to
notice that each agent i’s participation in aggregate total wealth, given by VF (s, K) + Vw (s, K)I,
1/
is kept constant at 0i
for all (s, K). Equilibrium portfolios must then adjust such that this
property holds if VF (s, K) and Vw (s, K) are not perfectly linearly correlated. In this case, changes
in the stock of capital will impact differently on non-human wealth among agents generating
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changes in individual portfolios. On the other hand, if VF (s, K) and Vw (s, K) are perfectly
linearly correlated, the evolution of the aggregate stock of capital will have no impact on the
individual composition of wealth. Consequently, the equilibrium level of trading is reduced to
zero since all agents share the same kind of risk (i.e., individual risk and aggregate risk coincide
again).
An important issue here is to determine which classes of economies display perfectly linearly
correlated levels of human and non-human wealth. Consider the fundamental case where the
production function is Cobb–Douglas. Since in this case the value of human wealth is always a
constant fraction of the value of aggregate output, VF (s, K) and Vw (s, K) are perfectly correlated
if and only if preferences are logarithmic and capital fully depreciates. That is, dividends are also
a constant fraction of output only if those two assumptions are satisfied. Thus, in that case the
value of non-human wealth is perfectly correlated with the value of aggregate output. On the
other hand, suppose that capital does not fully depreciate, Vw (s, K) is still a linear function of
the value of aggregate output. But it is known that investment is not a constant fraction of output
in this case. This implies that dividends are not a linear function of output and thus VF (s, K) and
Vw (s, K) cannot be perfectly correlated. We can summarize this discussion as follows.
Corollary 6. Suppose that F (K, L) = K L1− where ∈ (0, 1). If i = 0 for all i, then the
fixed equilibrium portfolio property holds if and only if = = 1.
With unproductive labor but non-zero minimum consumption requirements, the intuition is very
similar. In that case, these i ’s work as “negative endowment’’ for each agent and thus its value
is one of the determinants to explain the evolution of individual wealth relative to the aggregate
level. Again, note that the participation of each agent i in net aggregate wealth, now given by
VF (s, K) − VP (s, K), is kept constant in this case as well. Thus, equilibrium portfolios will
adjust conditional upon (s, K) such that this holds. Under these assumptions, the coefficient of
relative risk aversion is decreasing in wealth and consequently the wealthy should hold a relatively
higher share of aggregate risk.
4. Conclusions
This paper studies equilibrium portfolios in the traditional one-sector stochastic neoclassical
growth model with heterogeneous agents and dynamically complete markets. Preferences are
purposely restricted such that Engel curves are affine linear in lifetime human and non-human
wealth. Heterogeneity across agents can arise due to differences in initial non-human wealth and
minimum consumption requirements. The analysis shows how to determine equilibrium portfolios
as a function of preference parameters, initial endowments and different sources of wealth.
JKS [10] have questioned the ability of the stationary Lucas tree model to generate nontrivial asset trading under alternative complete market structures and fairly general patterns of
heterogeneity across agents. They show that agents choose a fixed equilibrium portfolio which
is independent of the state of nature. They conclude that some friction (informational, financial,
etc.) must play a significant role in generating non-trivial asset trading.
In this paper, a crucial aspect independently pointed out by Bossaerts and Zame [2] and Espino
and Hintermaier [7] has been isolated and studied in more detail. That is, whenever the environment
under study generates changing degrees of heterogeneity across agents, the trading strategy of
fixed portfolios cannot be optimal in equilibrium.
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This paper purposely abstracts from all kind of frictions to study the impact of capital accumulation on the evolution of equilibrium portfolios. We have shown that our environment can
generate changing heterogeneity if and only if either minimum consumption requirements are
not zero or labor income is not zero and the value of human and non-human wealth are linearly
independent.
The theoretical framework analyzed here can be extended to study more general market structures. In particular, the discussion about alternative dynamically complete market structures in
Espino and Hintermaier [7] can be easily adapted to deal with our framework. Also, it is immediate to extend the analysis to include individual endowments as an additional source of wealth,
stochastic minimum consumption requirements, labor-leisure choices and more consumption
goods. Furthermore, since the literature has shown that accumulation technologies allowing for
adjustment costs were important to study quantitatively asset returns in production economies, this
setting can also be adapted to deal with that extension (see Jermann [8,9] and also the discussion
in Boldrin et al. [1]).
We understand that the framework presented here is the natural benchmark to perform quantitative experiments to test the predictions of the neoclassical growth model in terms of the evolution
of asset holdings and the stock trading volume. We do not claim that these crucial quantitative
aspects will be fully explained by the simple setting presented here. Some other candidates to
generate additional trading might be required. However, we do consider important to disentangle
the specific contribution of each of those alternative sources and thus the environment described
here seems a natural first step.
Appendix
Proofs of Lemma 1, Proposition 2 and Corollary 6 are relatively standard and the interested
reader is referred to Espino [6] for additional details.
1/
Proof of Proposition 1. We normalize Ij =1 j
= 1. Compute first the value of aggregate
consumption as the solution of the following functional equation:
VC (s, K) = C(s, K) +
Q(s, K)(s )VC (s , K (s, K)).
(15)
s
Since W (s, K) = sF 2 (K, I ), note that
VF (s, K) = D(s, K) +
Q(s, K)(s )VF (s , K (s, K)),
s
Vw (s, K) = W (s, K) +
Q(s, K)(s )Vw (s , K (s, K)),
s
are both independent of . Since C(s, K) = D(s, K) + W (s, K)I , it follows that VC (s, K) =
VF (s, K)+Vw (s, K)I for all (s, K). Espino [6] shows that there exist unique continuous functions
VC , VF and Vw solving these functional equations.
The value of individual consumption corresponding to the Pareto optimal allocation is given
by
VCi (s, K, ) = Ci (s, K, ) +
Q(s, K)(s )VCi (s , K (s, K)),
(16)
s
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and thus individual consumption (1) implies that
VCi (s, K, ) = (i )1/ VC (s, K) + VP (s, K)[i − (i )1/ ]
= (i )1/ [VF (s, K) + Vw (s, K)I ] + VP (s, K)[i − (i )1/ ],
solves (16) by definition of VC and VP . Also notice that i (s, K, ) = VCi (s, K, ) − Vw (s, K)
and consequently it follows that
i (s, K, )= (i )1/ VF (s, K) + VP (s, K)[i − (i )1/ ] + Vw (s, K)[I (i )1/ − 1].
(17)
Given some initial state (s0 , K0 ) and some initial distribution 0 , we are ready to compute
(s0 , K0 , 0 ) such that
i (s0 , K0 , (s0 , K0 , 0 )) = 0i [P (s0 , K0 ) + D(s0 , K0 )] = 0i VF (s0 , K0 ),
for all i. Note that
1/
0i [P (s0 , K0 ) + D(s0 , K0 )] = i (s0 , K0 , 0 )
VF (s0 , K0 )
1/ +VP (s0 , K0 ) i − i (s0 , K0 , 0 )
1/
0
+Vw (s0 , K0 ) I i (s0 , K0 , )
−1 ,
and this implies that
1/
i (s0 , K0 , 0 )
=
0i VF (s0 , K0 ) + Vw (s0 , K0 ) − VP (s0 , K0 )i
,
VF (s0 , K0 ) + Vw (s0 , K0 )I − VP (s0 , K0 )
for i = 1, . . . , (I − 1). The assumption that the initial distribution of shares are “large enough’’
means that 0i VF (s0 , K0 ) + Vw (s0 , K0 ) − VP (s0 , K0 )i > 0 for all i. Espino [6] presents the
details to decentralize the candidate recursive allocation and the price system proposed as a
RCE. Proof of Proposition 3. Consider 0 = 0 (s0 , K0 , 0i ) given by (11) and note that (17) implies
that equilibrium portfolios will be determined by
1/ 1/
Ai (s, K, 0 ) = 0i
− 0i VF (s, K) + i − 0i
VP (s, K)
1/
+ I 0i
− 1 Vw (s, K).
If we assume that i = Vw (s, K) = 0 for all i and for all (s, K), then it follows from (11)
1/
that 0i
= 0i . Therefore, (12) implies that Ai (s, K, 0 ) = 0 for all (s, K) and for all i.
Consequently, (14) holds and the fixed equilibrium portfolio property is satisfied by
definition. Proof of Proposition 4. Suppose that i > 0 for some agent i and Vw (s, K) = 0 for all (s, K).
It follows from (12) that
1/ 1/
0
0
0
Ai (s, K, ) = i
− i VF (s, K) + i − 0i
VP (s, K).
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The fixed equilibrium portfolio property means that Ai (s, K, 0 ) = Ai (s, 0 ) for all i and for all
(s, K). In particular, it implies that Ai (s0 , K, 0 ) = 0 for all K by definition of 0 . Then,
1/
1/
0
0
0
i
− i VF (s0 , K) = i
− i VP (s0 , K),
(18)
for all K and thus it follows from (11) that 0 is independent of K (i.e., (s0 , 0 ) = (s0 , K, 0 )
for all K). Since there exist at least two agents i, h such that 0i = 0h , note that (18) implies
that there exists x > 0 such that VF (s0 , K) = xVP (s0 , K) for all K. Espino [6] shows that this
implies that C(s0 , K) = x solves the recursive planner’s problem (RAPP) for all K ∈ X. This
leads to a contradiction since it would violate well-known standard properties of the one-sector
neoclassical growth model. Proof of Proposition 5. Suppose that i = 0 for all i and Vw (s, K) > 0 for all (s, K). It follows
from (12) that
1/
1/
− 0i VF (s, K) + I 0i
− 1 Vw (s, K).
Ai (s, K, 0 ) = 0i
Suppose that the fixed equilibrium portfolio property holds where Ai (s, K, 0 ) = Ai (s, 0 )
for all i and for all (s, K) Fix some i and note that this implies that for all (s, K):
1/
I 0i
−1
Ai (s, 0 )
− Vw (s, K)
VF (s, K) = 1/
1/
− 0i
− 0i
0i
0i
and thus VF (s, K) and Vw (s, K) need to be perfectly linearly correlated. Also notice that Ai (s0 , 0 )
= 0 for all i implies that 0 is again independent of K0 . Acknowledgments
I would like to thank Fernando Alvarez, David DeJong, Huberto Ennis and, in particular,
an anonymous referee for comments and suggestions. I also thank seminar participants at the
Cornell/Penn State Macro Workshop (2005), the University of Virginia, Universidad Di Tella,
Universidad de San Andrés, the Central Bank of Argentina (BCRA) and SED Meetings 2006
(Vancouver). The first drafts of this paper were written while I was visiting the Department of
Economics at Universidad de San Andrés (Argentina) whose hospitality is greatly appreciated.
All the remaining errors are my responsibility.
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