1
Heterogeneity in the neoclassical growth model with
complete markets
1.1
Two preliminary results on aggregation of technologies and
preferences
In what follows we’ll talk about “aggregation”. What do we mean with this term? We
say that an economy admits aggregation if the behavior of the aggregate equilibrium
quantities (aggregate consumption, investment, wealth,...) and prices does not depend on the distribution of the individual quantities across agents. In other words, we can
aggregate whenever we can define a fictitious “representative agent” that behaves, in equilibrium, as the sum of all individual consumers.
1.1.1 Aggregating firms with the same technology
Consider an economy with M firms, indexed by i = 1,2, ..., M which produce a ho-mogeneous good with the same technology zF (ki, ni) where z is aggregate productivity.
Assume thatF is strictly increasing, strictly concave, differentiable in both arguments and
constant returns to scale. Can we aggregate these individual firms into a representative
firm?
Suppose inputs markets are competitive. Then, each price-takingfirm of typeisolves
max
{ki,ni}
©
zF ¡ki, ni¢−wni−(r+δ)kiª,
with first-order conditions
zFk
¡
ki, ni¢ = r+δ, (1)
zFn
¡
ki, ni¢ = w,
Recall that by CRS,Fk andFn are homogenous of degree zero, hence:
Fk(ki, ni)
Fn(ki, ni)
= fk(k
i/ni)
fn(ki/ni)
.
Dividing through the two first-order conditions, we obtain
fk(ki/ni)
fn(ki/ni)
= r+δ
and using the fact that the left-hand side is a strictly decreasing function of (ki/ni), we
obtain
ki
ni
=g
µr+δ
w ¶
= K
N, for every i= 1,2, ..., M
where capital letters denote averages: everyfirm chooses the same capital-labor ratio. Returning now to(1), we have that
zFk¡ki, ni¢=zfK(K/N) =r+δ ⇒zfK(K/N) =r+δ
where the last equality is obtained by averaging over alli0s. And similarly,
zfN(K/N) =w
which implies the existence of a “representative”firm with technology zF (K, N).
1.1.2 Aggregating consumers with the same preferences
Consider a version of the neoclassical growth model withN types of consumers indexed by
i= 1,2, ..., N with the same endowments of capitalki0 =κfor alli0sand same preferences
U¡ci0, c
i
1, ...
¢ =
∞ X
t=0
βtu¡cit¢.
Assume that markets are competitive, so that every consumer faces the same prices. Then,
one would think that since all theN consumers make the same decisions, we can aggregate
them into a representative agent, right? Not so quickly... Unless the utility function u
is strictly concave, agents may not make the same optimal choices of consumption and leisure.
Result 1.0: If every firm has the same CRS production function, consumers have
the same initial endowments and same preferences and their utility function is strictly
concave, then the neoclassical growth model admits a formulation with one representative
firm and one representative household.
1.2
Heterogeneity in endowments
We begin by studying a version of the neoclassical growth model with heterogeneity where
1.2.1 The economy
Demographics and preferences— The economy is inhabited by N types of infinitely lived agents, indexed byi= 1,2, ..., N. Denote byµi the measure of agentsiand normalize
the total measure of agents to one, i.e. PN
i=1
µi = 1. Preferences are time separable, defined
over streams of consumption, given by
U =
∞ X
t=0
βtu¡ci t
¢ ,
where the period utility function u belongs to one of the following three classes: log, power, exponential, i.e.
u(c) =
⎧ ⎨ ⎩
γln (¯c+c) with ¯c+c≥0 γ(¯c+c)σ with ¯c+c≥0
−c¯exp (−ηc)
(2)
Note that, for the moment, we allow c¯≤ 0 for log and power utility in order to be able
to model a subsistence level for consumption.
Technology and firm’s problem— The aggregate production technology is yt =
f(Kt) with f strictly increasing, strictly concave and differentiable. The representative
firm owns physical capital and makes the investment decision by solving the problem
At= max
{Iτ}
∞ X
τ=t
µ pτ
pt
¶
[f(Kτ)−Iτ]
s.t.
Kτ+1 = (1−δ)Kτ+Iτ,
wherept is the timet price of thefinal good. Let’s define real profitsπt ≡f(Kt)−It.
Then, it is easy to see that At is the value of the firm, i.e. the present value of future
profits discounted at rate (pt/pτ),the relative price of consumption between time t and
time τ.1 Recall that p
t/pt+1 = 1 +rt+1 wherer is the interest rate.
Markets— there are spot markets for the final good and complete financial markets, i.e. there are no constraints on transfers of income across periods.
1We follow Chatterjee (1994) in assigning the property rights on capital to thefirm and the ownership
Household’s problem—Given complete markets, we can use the Arrow-Debreu
for-mulation of the household problem (with the time-zero lifetime budget constraint). The
maximization problem of household i can therefore be stated as:
max
{ci t}
∞ X
t=0
βtu¡cit¢
s.t. ∞
X
t=0
ptcit ≤p0ai0 (3)
where ai
0 is the initial wealth of agent i in term of consumption units. Note that, in
general, we can define the wealth of agent i at time t as
ptait=sit
∞ X
τ=t
pτπt, (4)
wheresi
tis the share of thefirm-value owned by consumeriat timet.Indeed, by summing
both sides of (4) overi and exploiting the fact thatPNi=1µ
isi
t = 1 for every t, we obtain
At as defined above. Hence, aggregate household wealth is the value of thefirm.
Solution—Consider the log-preferences case. From the FOC of the household problem,
we have:
βtu0¡cit¢ =λipt ⇒ βtγ
µ 1 ¯ c+ci
t
¶
=λipt ⇒ cit=
βtγ λipt −
¯
c, (5)
whereλi is the Lagrange multiplier on the budget constraint. Substituting this FOC into
the time-zero lifetime budget constraint(3), we can derive an expression for the multiplier
λi:
∞ X
t=0
pt
µ βtγ λipt −
¯ c ¶
= p0ai0
γ
λi(1−β) −¯c ∞ X
t=0
pt = p0ai0
µγ
λi ¶
= (1−β)p0ai0 + (1−β) ¯c
∞ X
t=0
pt (6)
t= 0 in order to solve explicitly for ci
0:
ci0 =
1 p0
"
(1−β)p0ai0+ (1−β) ¯c
∞ X
t=0
pt
#
−¯c
= ¯c "
(1−β) ∞ X
t=0
µ pt
p0
¶
−1 #
+ (1−β)ai0 (7)
= Θ¡p0
,c¯¢+ (1−β)ai0,
whereΘ(p0
,¯c)denotes a function of the subsistence level and of the whole price sequence
p0
={p0, p1, ..., pt, ...}.2 This derivation can be easily generalized for everyt >0(by using
the Arrow-Debreu constraint for timet) so that
cit =Θ¡pt,c¯¢+ (1−β)ait, (8)
which shows that the optimal consumption choice at timet is an affine function of asset
holdings at time t for each type i.
More in general, when period utility belongs to the families in (2), then preferences
share a common property. They are homothetic, i.e. have linear Engel curves in wealth:
any given change in wealth induces the same change in consumption, independently of
the wealth level (see below for a derivation).3
Even though we have only derived it for the log-case, it is easy to check that this
rep-resentation of the consumption function holds also for the other two classes of preferences (power and exponential utility).
Implications for aggregate dynamics— Denote aggregate variables with capital
letters. The first consequence of equation(8) is that to study the dynamics of aggregate variables in this model economy, we don’t need to keep track of the distribution of wealth.
From (8), we derive easily that aggregate consumption only depends on aggregate
variables (prices and aggregate wealth), i.e.
Ct =Θ¡pt¢+ (1−β)At, (9)
where At can be clearly be expressed only as a function of the sequence of prices{pt}∞t=0
2Without loss of generality, we can normalizep
0 to one.
3Technically, when¯c <0, preferences are quasi-homothetic because the Engel curves do not start at
and aggregate capital stocks{Kt}∞t=0. The competitive quantities can therefore be
recov-ered from the following standard single-agent planning problem
max
{Ct}
∞ P
t=0
βtu(Ct)
s.t.
Ct+Kt+1 ≤f(Kt) + (1−δ)Kt
K0 given
(PP)
with first order condition
u0(Ct) =βu0(Ct+1) [f0(Kt+1) + (1−δ)].
Note that, from(5) and the equation above, we can obtain equilibrium prices through
the recursion
pt
pt+1
=f0(Kt+1) + (1−δ), wherep0 ≡1, K0 given (10)
We can state our first important result:
Result 1.1: When preferences are homothetic and agents are heterogeneous only in
initial endowments, the aggregate dynamics of the neoclassical growth model with complete
markets admit a single-agent representation.
Steady-state—Clearly, the economy will converge to the steady-state values of capital
stock satisfying the modified golden rulef0(K∗) = 1/β−(1−δ). Note now that in
steady-statept/pt+1 = 1/β for all t, hence from the definition ofΘ(pt,¯c) in(7) we conclude that
ci = (1−β)ai. In other words, in steady-state, the average propensity to save is β,
independently of wealth, for every type of household.
1.2.2 Equilibrium dynamics of the wealth distribution
The result above implies that the dynamics of the aggregate variables are not affected by
the evolution of the wealth distribution, but the inverse statement is not true: in general,
the evolution of the wealth distribution across households (i.e. wealth inequality) depends
on the dynamics of aggregate variables (prices and quantities).
To see this, note that from the lifetime budget constraint of agenti at time t
ptcit+
∞ X
τ=t+1
pτciτ = ptait ⇒ ptcit+pt+1ait+1 =pta
i t
ci t
ai t
+ pt+1a
i t+1
ptait
= 1 ⇒ a
i t+1
ai t
= µ p
t
pt+1
¶ µ 1− c
i t
ai t
¶
which expresses the growth rate of wealth for type i as a function of her
consumption-wealth ratio. Note now that, from equations (8)and(9),
ci t
ai t
= Θ(p
t,¯c)
ai t
+ (1−β), and C
i t
At
= Θ(p
t,¯c)
At
+ (1−β)
Thus, putting together this last line and (11)
Θ¡pt,¯c¢ ¡ait−At¢≷0 ⇔
ci t
ai t
≶ Ct
At ⇔
ai t+1
ai t
≷ At+1
At
(12)
In other words,
Θ¡pt¢ ¡ait−At¢≷0 ⇔
si t+1
si t
≷1
which means that whether consumer’siwealth share is increasing or decreasing over time depends on the sign of the constant Θ (equal for everyone) and on her relative position
in the distribution.
In absence of subsistence level, ¯c= 0 andΘ= 0, the neoclassical growth model with
heterogeneous endowments has a sharp prediction for the evolution of inequality.
Result 1.2: In the neoclassical growth model with homothetic preferences, heterogeneous
endowments, but without subsistence level, the wealth distribution remains unchanged
along the transition path, i.e. initial conditions in endowments (and inequality) persist
forever.
In presence of a subsistence level, things change. For example, if Θ>0and ai t > At,
then consumer i wealth share will grow over time, hence the distribution will become
more unequal. We now determine the sign ofΘ, through:
Lemma 1.1 (Chatterjee, 1994): The common constant term of the consumption
function Θ(pt,¯c) is greater, equal or less than zero if and only if ¯c(K
t−K∗) is greater,
equal or less than zero.
Proof: Suppose the economy grows towards the steady-state, i.e. Kt < K∗. From
equation (10), the sequence {f0(K
t)} is decreasing and the sequence {pt+1/pt} will be
increasing towards β. Therefore, pτ+1/pτ ≤ β for all τ ≥ t where the strict inequality
holds at least for some t. It follows that
From the definition of Θ(pt,c)¯ in (7), use the above equation to obtain
Θ¡pt,c¯¢= ¯c "
(1−β) ∞ X
τ=t
µ pτ
pt
¶
−1 #
>c¯ "
(1−β) ∞ X
τ=t
βτ−t−1 #
= 0
where thefirst inequality follows from c <¯ 0. QED
The consequences for the evolution of the wealth distribution in an economy
grow-ing towards the steady-state are easy to determine, at this point. In the presence of a
subsistence level (¯c <0), Θ > 0 in a growing economy. Θ > 0 implies that the average propensity to consume (save) declines (increases) with wealth. In fact, from equation
(12), it is clear that agents with wealth above average will increase their wealth even
more relatively to the average. In other words:
Result 1.3: In the neoclassical growth model with homothetic preferences, heterogeneous
endowments and subsistence level ¯c <0: (i) the wealth distribution becomes more unequal
as the economy grows towards the steady-state, as rich agents accumulate more than poor
agents along the transition path, and (ii) there is no change in the ranking of households
in the wealth distribution, i.e., initial conditions in ranking persist forever.
Robustness—We now discuss how robust this result is to some of the key assumptions
made so far in the analysis: 1) all agents have same discount factor β, 2) markets are
complete, 3) absence of leisure and heterogeneity in efficiency units of labor.
• When agents have different discount factors, then none of the results hold any longer.
Suppose that ¯c= 0to simplify the analysis. Then, from(8)
cit =¡1−βi¢ait,
therefore the average propensity to save out of wealth is higher the more patient is
the individual and from(12), wealth grows faster for the more patient individuals.
In the limit, in steady-state, the most patient type holds all the wealth, and the
distribution is degenerate.
• In absence of markets (autarky), every consumer has access to her own technology,
but there is no trade. Each agent will solve a problem like (P P), with different
initial conditions Ki
0. It is easy to see that, independently of the initial conditions,
distribution of wealth is perfectly equal. Interestingly, we conclude that a less
developed financial market induces less wealth inequality, in the long-run. Also, there is an important parallel with the convergence literature in growth theory: just
think of consumers as countries. If every country has access to the same world technology and capital is perfectly mobile across countries, then the neoclassical
growth model doesnot predict convergence anymore.4
• When preferences are also a function of hours worked h, and households differ by
their (fixed) endowment of efficiency units, aggregation can also occur, under certain restrictions on preferences, for example when
u(c, h) = ¡
cα(1−h)1−α¢1−γ
1−γ . (13)
We will study exactly this case in detail later.
1.2.3 Indeterminacy of the wealth distribution in steady-state
One very important implication of the aggregation Result 1.1 is that in steady-state the
wealth distribution is indeterminate. Suppose again that ¯c= 0 to simplify the analysis.
From(8) and(4), the set of equations characterizing the steady-state is
ci = (1−β)ai, i= 1,2, ..., N
ai = si 1
1−β [f(K
∗)−δK∗], i= 1,2, ..., N
f0(K∗) = 1/β−(1−δ),
N
X
i=1
µisi = 1,
We therefore have (2N + 2) equations and (3N + 1) unknowns ³{ci, ai, si}N i=1, K∗
´
. In other words, the multiplicity of the steady-state wealth distributions is of order N −1.5
However, suppose we start from a given wealth distribution at time 0 when the
econ-omy has not yet reached its steady-state, then the dynamics of the model are uniquely
determined by Results 1.2 and 1.3 and the final steady-state distribution is determined as well. So, let’s restate ourfinding in:
4More precisely, f0¡Ki¢ would be equalized across countries, hence countries would have the same
capital stock and produce the same output. Thus, there would be convergence in GDP, but not in GNP.
5This means that, ifN = 1 (representative agent), the steady-state is unique. If N = 2, there is a
Result 1.4: In the steady-state of the neoclassical growth model with N agents
hetero-geneous in initial endowments and homothetic preferences, there is a continuum of
steady-state wealth distributions, with dimension (N −1). However, given an initial wealth
dis-tribution {si0}N
i=1 at t = 0, the equilibrium wealth distribution {sit} N
i=1 in every period t is uniquely determined, and so is thefinal steady-state distribution.
Example— Consider an economy where N = 2, where the production technology is
zf(K). Then, the set of steady-state equations is
zf0(K∗) = 1
z (1/β−(1−δ)),
A∗ = 1
1−β [zf(K
∗)−δK∗],
µ1
a1
+µ2
a2
= A∗
ci = (1−β)ai, i= 1,2,
So we have 5 equations, but6 unknowns {a1
, a2
, K∗, A∗, c1
, c2
}. We can represent
graph-ically all the possible equilibrium paths between two steady states that differ for their
level of technological progress z , say (zL, zH). The figure shows that the model has a
continuum of steady-state distributions of wealth of dimension one, all consistent with the
uniquely determined aggregate capital stock K∗. If we pick an initial distribution in the initial steady-state with productivity zL, the equilibrium path to the final steady-state
with productivity levelzH is uniquely determined.
Finally, in terms of language, this whole section shows that it important to distinguish
“steady-state” from “equilibrium path”. In this economy, the equilibrium path is always
unique (given initial conditions), but the steady-state is not.
1.3
The Negishi Approach
Negishi (1960) suggested a method to calculate the competitive equilibrium (CE) prices
and allocations of complete markets economies (in particular, economies for which the
first welfare theorem holds) with heterogeneous households. This method proves to be particularly useful for those economies where aggregation does not go through and, hence
we cannot use the representative agent.6
6In his original paper, Negishi (1960) used this equivalence to propose a simple way to show existence
From the first welfare theorem, we know that any CE is a Pareto optimum (PO), hence it can be found as the solution to a social planner problem with “some” Pareto
weights given to each agent. Suppose we want to compute a particular CE of an economy
where agents are initially endowed with heterogeneous shares {si0}N
i=1 of the aggregate
wealth. Can we use the planner problem for this purpose? Negishi showed that the key is
to search for the “right” weights given to each type of agent in the social welfare function
of the planner. Each set of weights corresponds to a Pareto efficient allocation, the key is
tofind the set of weights which correspond to our desired CE allocation.
1.3.1 An Example
Consider our economy of section(1.2)with N = 2 (i.e., two consumers). Write down the
following planner problem
max
{c1
t,c2t}
∞ X
t=0
βt£α1
u¡c1
t
¢ +α2
u¡c2
t
¢¤
(NP)
s.t.
c1
t +c
2
t +Kt+1 ≤f(Kt) + (1−δ)Kt
K0 given
where (α1
, α2
) are the planner’s weights for each type of household in the social welfare
function. The FOC’s for this problem are
αiβtu0¡cti¢ = θt, i= 1,2 (14)
u0¡c1
t
¢
= βu0¡c1
t+1
¢
[f0(Kt+1) + (1−δ)] (15)
whereθtis the Lagrange multiplier on the planner’s resource constraint. Note that putting
together thefirst-order conditions for consumption for the two agents we arrive at
u0(c1
t)
u0(c2
t)
= α
2
α1,
which tells us that the planner allocates consumption proportionately to the weight it
gives to each consumer (with strictly concave utility). Note also that the ratio of marginal
utility across agents is kept constant in every period (a key features of complete markets
Recall equation (5) from the household optimization in the CE:
βtu0¡cit¢=λipt.
If we want the NP to deliver the same solution as the CE, we need the PO allocations and the CE allocations to be the same. Given strict concavity of preferences, this implies
that, putting together(5) and(14):
θt
αi =λ ip
t⇒
α2
α1 =
λ1 λ2
Hence, the relative weights of the planner must correspond to the inverse of the ratio of the Lagrange multipliers on the time-zero Arrow-Debreu budget constraint for the two
agents in the CE.
In particular, for log preferences withc¯= 0, from(6) µ
γ λi
¶
= (1−β)p0ai0 →λ
i =
∙
γ (1−β)p0A0
¸ 1 si
0
,
we obtain that
α2
α1 =
λ1
λ2 =
s2 0
s1 0
.
Imposing the natural and innocuous normalization α1
+α2
= 1, we can solve explicitly
for the two weights: α1
=s1
0 andα 2
=s2
0, i.e., the weights are exactly equal to the initial
endowments. The higher is the initial wealth share si
0, the lower is the multiplierλ
i and
the larger is the Pareto weightαi on the Negishi problem: the planner must deliver more
to consumption to the agent who has a large initial share of wealth.
Result 1.5: Take an economy with agents heterogeneous in endowments where the First
Welfare Theorem holds. Then, the competitive equilibrium allocations can be computed
through an appropriate planner’s problem where the relative weights on each agent in
the social welfare function are proportional to the relative individual endowments: those
agents who initially have more will get a higher weight in the planner’s problem.
Moreover, note that using equation(14) into(15), we arrive at a relationship between
the sequence of capital stocks and the sequence of Lagrange multipliers on the resource
constraint
θt
θt+1
=f0(K
which compared to equation(10) in the CE implies that
θt
θt+1
= pt pt+1
,
in other words, the Arrow-Debreu equilibrium prices can be uncovered as the sequence
of Lagrange multipliers in the Pareto problem: intuitively, the multipliers gives us the
shadow value of an extra unit of consumption and, in the CE, prices signal exactly this
type of scarcity.7
1.3.2 General application of the Negishi method
In general, without restrictions on preferences, one may not have closed form solutions for
theλ’s in the CE, so the algorithm is a little more involved. The objective is to compute
the CE allocations for an economy with N types of agents and endowment distribution
{ai0}N
i=1. We can describe it in three steps:
1. In the social planner problem (NP), guess a vector of weightsα=©α1
, α2
, ..., αNª.
2. Compute the sequence of allocations {ci
t(αi), Kt}∞t=0 and the implied sequence of
multipliers{θt}∞t=0 on the resource constraint in each period t. In practice, at every
t, one needs to solve the four equations
αiβtu0¡cit¢ = θt, i= 1, .., N N
X
i=1
cit+Kt+1 = f(Kt) + (1−δ)Kt
θt
θt+1
= f0(Kt+1) + (1−δ)
in N+2 unknowns ³{ci t}
N
i=1, Kt+1, θt+1
´
— recall that (θt, Kt) are given, e.g. θ0 can
be normalized to 1, in the same wayp0 can be used as a numeraire and normalized.
This is why the Negishi method simplifies the computation of the equilibrium: the solution requires solving, for every timet, a small system of equations. Recall that to
solve for the CE allocations, at every timetone must set the excess demand function
to zero and the excess demand function depends on the entire price sequence–an
infinitely dimensional object.
3. Use the equivalence between pricespt and multipliersθtto verify whether the
time-zero Arrow-Debreu budget constraint of each agent holds exactly at the guessed
vector of weight α. For each agent, compute the implicit transfer function τi(α)
associated with the assumed structure of weights
τi(α) = ∞ X
t=0
θtcit
¡
αi¢−θ0ai0, for everyi= 1,2, ..., N
and look for the vector α that sets every transfer function τi(α) to zero. This
vector corresponds to the PO allocations that are affordable by each agent in the
CE, given their initial endowment, without the need for any transfer. Thus, we are
computing exactly the CE associated to initial conditions{ai0}.
See also Ljungqvist-Sargent, section 8.5.3, for a discussion of the Negishi algorithm.
1.4
Heterogeneity in labor endowments
Maliar and Maliar (2001) make use of the Negishi approach to prove a more general ag-gregation result, in the presence of labor supply and heterogeneity in labor endowments,
with preferences as in(13). They prove that the neoclassical growth model with
heteroge-neous endowments of wealth and time-invariant heterogeheteroge-neous endowments of efficiency
units of labor admits a single-agent representation.
In Maliar and Maliar (2003), they show that if idiosyncratic efficiency units of labor
are time-varying but they are perfectly insurable (i.e., markets are still complete) then
the strong aggregation result (the dynamics of aggregate variables do not depend on
the distribution) fails, but one can obtain a weaker aggregation result. The aggregate dynamics of the model now do depend on the distribution of the shocks and on the
distribution of the wealth endowments, but they do in a very simple way: one particular
moment of the distribution of productivity shocks act like a preference shifter for the
representative agent.
Finally, note that the assumption that agents can trade a full set of claims contingent
on all possible realizations of idiosyncratic labor productivity shocks is not very realistic,
1.5
Notes
Gorman (1953) developed the theory of aggregation of individual preferences. His main
result is that when preferences are homothetic and households differ in initial wealth levels
only, social preferences do not depend on the distribution of individual wealth. Stiglitz
(1969) discusses the income and wealth distribution dynamics in the context of a Solow
growth model, where individual savings are assumed to be linear in capital. Our
discus-sion in sections 1.2 is based on Chatterjee (1994). Caselli and Ventura (2000) extend the
Gorman result to economies where agents also differ in their endowments of efficiency
units of labor. They work in continuous time and apply their results to the transitional dynamics of the neoclassical growth model and to an economy with Arrow-style
exter-nalities. Maliar and Maliar (2001, 2003) are examples of how the aggregation theorems
apply to economies with both idiosyncratic and aggregate uncertainty when markets are
complete. In particular, they show that even when preferences are non-homothetic, in
certain cases one can obtain closed-form aggregation, although the aggregate preferences
depend on the distribution of individual shocks. See also chapter 4.d in Mas Colell for a
more abstract discussion of the existence of a representative consumer.
References
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Distribution”, American Economic Review.
[2] Chatterjee, Satyajit (1994) "Transitional dynamics and the distribution of wealth in
a neoclassical growth model", Journal of Public Economics
[3] Gorman, “Community preference field,” Econometrica
[4] Maliar, Lilia and Serguei Maliar (2001) "Heterogeneity in capital and skills in a
neo-classical stochastic growth model", Journal of Economic Dynamics and Control
[5] Maliar Lilia and Serguei Maliar (2003) "The representative consumer in the
neoclas-sical growth model with idiosyncratic shocks", Review of Economic Dynamics
[6] Negishi Takashi (1960), "Welfare Economics and Existence of an Equilibrium for a
[7] Stiglitz, Joseph (1969), “Distribution of Income and Wealth among Individuals”,
