Notes on Inequality in the
Neoclassical
Model of
Growth
F. Obiols
Barcelona GSE
2
1. Introduction
Casual observation suggests that there is substantial
heterogeneity across households in dimensions such as
human capital and education, age (or position in the
life-cycle), income and/or wealth, marital status, labor
market status, health, etc.
It seems obvious that for a sensible design of policies
related to health care, unemployment benefits, and education (amongst others), it would be necessary to take
into account these relevant dimensions of heterogeneity.
The simple version of the NMG we developed in the previous notes abstracts from any source of heterogeneity,
and thus, it would be hard to try to use that plain version of the model to obtain policy prescriptions along
these lines.
In these notes we develop a preliminary version of the
model to incorporate heterogeneity among households,
and we study the evolution of inequality over transitions
toward the steady state.
The model we use is an extended version of the NMG,
and although we mainly concentrate on heterogeneity
of initial assets, the model could easily encompass heterogeneity in preferences, and in labor income (which
could appear as the result of differences in education
across workers). The purpose of the model is to shed
light on the competitive dynamics of inequality in a
particular measure of wealth.
The version of the model we develop can also be used
to guide policies aiming at reducing inequality in wealth:
informally, it is often claimed that more unequal societies tend to “do worse” than more equal ones (i.e.,
beyond moral/philosophical concerns, informal considerations regarding growth and efficiency seem to advocate for less inequality). The model we develop here
will conclude that the trade off between inequality and
efficiency is in fact not a very favorable one: less inequality is possible, but at the cost loosing efficiency.
1. Basic model
• In the economy there is a large number of agents
(consumer-workers) choosing consumption and saving over an infinite horizon. The “name” of the
agents is given by i = 1, 2, 3..., N .
• Agents are identical in every respect (preferences,
endowments, available technologies...) except in
their initial amount of assets ai0.
• There are competitive firms producing a consumption/investment good. The available technology
uses capital and labor, and it displays CRS in these
two factors of production. The problem of the
firm/s, therefore, is exactly the same one we studied in the notes about the NMG and it wont be
repeated here.
2. The problem of one of the agents
The problem of a given agent i can be written as:
max
∞
X
{cit ,ait+1 } t=0
β t log(cit − c̄), β ∈ (0, 1),
subject to the budget constraint in every period t:
cit + ait+1 = wt + (1 + Rt)ait,
non negativity constraints for consumption and assets
(ct, at+1 ≥ 0), and taking as given all prices ({Rt, wt}∞
t=0 ),
and ai0.
There are a few comments that is worth emphasizing
before we continue:
1. The log-utility assumption is convenient but not restrictive: the results to be presented below also hold under more general preferences. Furthermore, log-utility
is very often used in quantitative studies (e.g., the Real
Business Cycle literature).
2. The usual case in the literature is c̄ = 0. If c̄ > 0,
then we need to restrict consumption so that cit ≥ c̄
for all t. In this case c̄ > 0 is interpreted as a minimum consumption requirement. It is also possible to
consider the case of c̄ < 0, in which the agent derives
positive utility even if cit = 0 (a possible interpretation
in this case is that c̄ is an endowment of consumption
goods, an transfer from the government, which we do
not model explicitly).
3. In the budget constraint we are assuming that all
agents supply the same amount of labor in efficiency
units, and that all agents are equally productive (1 unit
of efficient labor). Also, we take Rt = rt − δ, that is, Rt
is the return to assets net of depreciation. That is, we
have already substituted the law of motion for capital
in the budget constraint.
All the equilibrium action in the model comes from the
consumer’s side, hence it will be convenient to introduce a notion of CE, which we do next.
Definition 1: A Competitive Equilibrium in sequence
i , ai
∞ for every i = 1, ..., N , a
form is a list xh
=
{c
}
t t+1 t=0
i
P
∞
list xf = {Kt, Ht}∞
t=0 , and a list x = {rt , wt }t=0 , such
that:
1) For each i = 1, ..., N , given xp and ai0, xh
i solves the
utility maximization problem of agent i.
2) Given xp, xf solves the profit maximization problem
of the representative firm. In particular, the choices of
the firm satisfy:
∂F (Kt, Ht)
∂F (Kt, Ht)
= rt, and
= wt.
∂Kt
∂Ht
3) All markets clear in all periods:
- labor market: N = Ht,
P
- capital market: i ait = Kt,
P
- goods market: i cit + Kt+1 = F (Kt, Ht) + (1 − δ)Kt.
(notice that as usual by Walras’ Law, one of the above
three conditions is redundant).
We have in mind that essentially there is only capital
in the economy, and so, the supply of capital is given
P
P
by i ait = i kti. However, we could also assume that
in addition to capital, there are other assets in zero net
supply.
One instance of such an asset would be a one period
bond: in period t agents are able to buy/sell bit+1 units
of bonds at a price qt (in units of period t consumption
goods), and each of these bonds will return.
• For the “buyers” of bonds, bit+1 > 0, and it represents an additional means of saving.
• For the “sellers” of bonds, bit+1 < 0, and it represents an obligation to deliver consumption goods in
t + 1.
The fact that these bonds are in zero net supply simply
means that the initial aggregate amount of bonds is
zero (hence the economy starts out with no debts nor
P i
additional sources of wealth than capital, thus i a0 =
P i
i k0 ).
The market clearing condition for bonds in any period
P
i
= 0, ∀t. Hence, we would
is of course that N
b
i
t+1
P
P i
P
P i
have i ait+1 = i kt+1
+ i bit+1 = i kt+1
.
Notice, therefore, that the market clearing condition in
the definition above still holds true.
Finally, for these bonds to coexist with capital its return
must be identical to the return to capital (or agents
would only hold the asset with the highest return). This
is known as a no arbitrage condition and it requires that
1 − qt
= Rt+1,
qt
(the left hand side is the return on bonds, and the right
hand side is the return on capital), which we finally write
as qt = 1/(1 + Rt+1). Clearly, the portfolio composition
is irrelevant when the no arbitrage condition holds.
We conclude that restricting assets to only capital, or to
assume a large number of assets including assets in zero
net supply, has no economic consequences when the no
arbitrage condition holds and there are no frictions.
3. Intermediate results
In this section we introduce two intermediate results
that will be useful later.
1. Welfare depends on life-time wealth.
2. Perfect aggregation holds.
3.1 Welfare depends on Life-time wealth
We will show that consumption in present value is a
function of life-time wealth. We will also show that in
every period, consumption itself is also a function of
life-time wealth. Since welfare depends only on consumption, these facts justify our interest on life-time
wealth.
Proceeding as in class, take the budget constraint of
an agent i in period t + 1 and isolate ait+1 to obtain
ait+1 =
cit+1 + ait+2 − wt+1
1 + Rt+1
,
and insert the resulting expression in the period−t budget constraint, so that after rearranging, we end up
with:
cit +
cit+1
1 + Rt+1
+
ait+2
1 + Rt+1
= ait(1 + Rt) + wt +
wt+1
.
1 + Rt+1
Repeat the same strategy (get ait+2 from the period
t + 2 budget constraint), to get that:
cit+1
cit+2
ait+3
i
ct + 1+R
+ (1+R )(1+R ) (1+R )(1+R )
t+1
t+1
t+2
t+1
t+2
wt+1
wt+2
i
+ (1+R )(1+R ) .
= at(1 + Rt) + wt + 1+R
t+1
t+1
t+2
If we continue this recursion up to infinity then on the
right hand side we get:

RHS = (1 + Rt) ait +
∞
X
wt+j

.
Qj
j=0 s=0 1 + Rt+s
Notice the RHS is, in fact, the definition we introduced
in class for life-time wealth: the value of assets (nonhuman assets) today, plus the present value of all future
labor income (i.e., of human assets). To be precise, we
define life-time wealth as

ωti = ait +
∞
X
wt+j
Qj
j=0 s=0 1 + Rt+s
Notice that ωti is linear in ait.


(1)
On the left hand side of the budget constraint we simply
have:
LHS
cit+1
cit+2
i
= ct + 1+R
+ (1+R )(1+R )
t+1
t+1
t+2
i
ct+3
+ (1+R )(1+R )(1+R ) + ...
t+1
t+2
t+3
where we have used a version of the transversality condition: the present value of assets that are arbitrarily
away in the future is zero:
aT +1
lim Q
= 0.
T →∞ T
t=0 1 + Rt
Hence, the LHS is simply the present value of the infinite sequence of consumptions.
Hence, we write the following inter-temporal budget
constraint:
cit +
∞
X
cit+j
= (1 + Rt)ωti.
Qj
j=1 s=1 (1 + Rt+s )
(2)
This is our first result: present value of consumption is
a function of life-time wealth.
We will now use the FOC of the consumer problem
to get rid of the consumptions dated t + 1, t + 2,...
and show that in every period t, ct is a linear function
of life-time wealth (that is, Engel curves (also called
expansion paths) are linear in life-time wealth).
With log-utility the Euler Equation in period t reads:
(cit − c̄)−1 = β(1 + Rt+1)(cit+1 − c̄)−1,
and after rearranging:
cit+1 = β(1 + Rt+1)cit + c̄(1 − β(1 + Rt+1)),
or
cit+1
1 − β(1 + Rt+1)
i
= βct + c̄
.
1 + Rt+1
1 + Rt+1
This expression can be substituted in the LHS of the
inter-temporal budget constraint we obtained above. In
fact, we can repeat the same strategy in t + 1 to get
cit+2
1 + Rt+2
= βcit+1 + c̄
1 − (β(1 + Rt+2)
,
1 + Rt+2
and combining with the previous expression, we get:
cit+2
(1+Rt+1 )(1+Rt+2 )
1−β(1+Rt+1 )
1+Rt+1
1−β(1+Rt+2 )
+c̄ (1+R )(1+R
.
t+1
t+2 )
= β 2cit + βc̄
Repeating this procedure, and substituting the resulting
expressions in the equation for LHS, we finally write:
∞ 1 − β(1 + R
X
cit
c̄
t+1+j )
LHS =
+
.
Qj
1−β
1 − β j=0
1 + Rt+1+s
s=0
Writing down together the LHS and the RHS, we finally
obtain
cit = (1 − β)(1 + Rt)ωti − c̄Pt,
(3)
with
∞ 1 − β(1 + R
X
t+1+j )
Pt =
.
Qj
j=0
s=0 1 + Rt+1+s
Notice that Pt is the same for all i (hence it is independent of i). Equation (3) above states our second result
and shows that consumption in period t is a function
of life-time wealth in period t. Hence, it is justified to
focus on the evolution of inequality in life-time wealth.
To follow the evolution of ωti over time and over agents
in general is a cumbersome task. The reason is that
ωti depends on all future prices as well as on the asset
level brought from the precedent period, which in turn
depends on consumption on that period, which in turn
depends on all future prices from that period onwards.
That is, prices depend on aggregate assets, and aggregate assets in t + 1 may depend in a complicated way
of the period t distribution of assets among agents.
That is, in general it is true that in order to know
aggregate assets we need to know equilibrium prices,
and in order to know equilibrium prices we need to know
aggregate assets and its distribution.
Fortunately, however, in our model there is a couple
of simple facts that help us to find a way out of this
perverse loop:
1. Equilibrium prices depend only on aggregate assets,
and not on it’s distribution over agents (remember that
in the FOC of the firm -which must be satisfied- all that
matters is to know aggregate capital and aggregate
labor (hence the way they are actually distributed is in
fact irrelevant). We conclude that, at least to compute
prices, the distribution of wealth is irrelevant.∗
∗ This
conclusion would not be true if in addition to aggregate
assets, prices were a function of the standard deviation of assets
(the first moment of the distribution of assets over agents), or
other higher order moments of that distribution.
2. The second interesting observation follows from Eq.
(3) above: consumption per capita is a linear function
of life-time wealth per capita:
X ci
X ωi
t = (1 − β)(1 + R )
t − c̄P .
t
t
N
N
i
i
P i
Furthermore, ωti is linear in assets, hence
i ωt /N is
linear in per capita assets.
The crucial implication of this fact is that consumption
per capita in period t is independent of the distribution
of assets in period t: again, it only depends on the
amount of assets per capita in that period. In particular, if in two economies the term Pt is the same, per
capita assets are also the same, and the only difference
is in the way these assets are distributed among the
agents, then consumption per capita in both economies
will necessarily be the same.
It follows from the statement above that all we need to
be able to follow consumption and assets in per capita
terms is the equilibrium sequence of prices, which in
turn depends only on aggregate assets and not in its
distribution.
In oder words, if we were given the equilibrium allocation in per capita terms (say {ct, at+1}∞
t=0 ), then we
would be able to compute prices, and hence, we would
be also able to compute consumption, assets, and lifetime wealth for every agent in the economy and over
all time periods.
3.2 Perfect aggregation holds
The previous reasoning boils down to the fact that,
in per capita terms, it is observationally equivalent to
study an economy with lots of heterogeneous agents or
an economy with a single representative agent whose
initial endowment of assets is the per capita assets of
the multi agent economy.
That is, a perfect aggregation result holds such that the
heterogeneous agents economy admits a representative
agent representation.
Of course, the problem of the representative agent is
identical to the one described by the NMG and of which
we already have a number of results. In particular,
the CE is efficient, which means that we can solve the
much simpler planner’s problem, obtain the efficient allocation (which is in fact the competitive allocation),
and use the FOC of the firm to obtain the sequence of
equilibrium prices corresponding to the transition starting from the given k0.
Since we derive all these results using the artificially
constructed representative agent, we are entitled to use
all the results on convergence we obtained in the NMG.
In particular, it should be clear, that
1. In per capita terms, the steady state of the economy is independent of the distribution of initial assets.
That is, given initial assets per capita, we converge to
the same steady state irrespectively of the initial distribution of assets.
2. The sequence of equilibrium prices is independent of
the distribution of assets.
4. Transitional dynamics of inequality in ωti
The measure of inequality we will use is the Coefficient of Variation in ωti. This measure is defined as
CV (ωti) = sd(ωti)/ωt, where sd(ωti) stands for the standard deviation across agents, and where ωt stands for
the average over agents.†
† For
a given random variable (rv) x, the standard deviation (sd) is
a measure of disparity (or heterogeneity). Since the sd is affected
by the level of the rv, then we simply normalize the measure by
its mean. This way we can compare distributions even if they
have very different means. Beyond this, it is possible to derive
our results using other measures of inequality such as Lorenz
domination, or the Gini Index (at a higher analytic cost, though).
Remember that

ωti = ait +
∞
X
wt+j

 ∀t,
Qj
j=0 s=0 1 + Rt+s
and so in t + 1 we have that:

i
ωt+1
= ait+1 +
∞
X
wt+1+j
Qj
j=0 s=0 1 + Rt+1+s


(4)
holds. The budget constraint in period t reads cit +
ait+1 = wt + (1 + Rt)ait, which implies that ait+1 = wt +
(1 + Rt)ait − cit, and since cit satisfies Eq. (3), then we
finally obtain
ait+1 = wt + (1 + Rt)ait − (1 − β)(1 + Rt)ωti + c̄Pt.
Insert the expression above in Equation (4) and obtain
i
= wt + (1 + Rt)ait − (1 − β)(1 + Rt)ωti + c̄Pt
ωt+1
P∞
w
+ j=0 Qj t+1+j
,
s=0 1+Rt+1+s
which can be rewritten as:

i
= (1+Rt) ait +
ωt+1
∞
X
wt+j
Qj
j=0 s=0 1 + Rt+s

−(1−β)(1+Rt)ω i+c̄Pt,
t
which we finally write as
i
ωt+1
= β(1 + Rt)ωti + c̄Pt.
(5)
Applying the sd operator on both sides of the previous
expression we get
i
sd(ωt+1
) = β(1 + Rt)sd(ωti).
P i
Dividing by the mean (ωt+1 = (1/N ) i ωt+1) then we
get:
i
)
sd(ωt+1
ωt+1
sd(ωti) ωt
= β(1 + Rt)
,
ωt ωt+1
or that
ωt
i
i
CV (ωt+1) = CV (ωt )β(1 + Rt)
.
ωt+1
It follows from Eq. (5) that
β(1 + Rt)
ωt
ωt+1
=1−
c̄Pt
.
ωt+1
We are now ready to state the main results of this
section.
i
R1. If k0 < k∗ and c̄ = 0, then CV (ωt+1
) = CV (ωti) for
all t.
i
) ≥ CV (ωti)
R2. If k0 < k∗ and c̄ > 0, then CV (ωt+1
for all t. The reason is that over such a transition Pt
is always negative, and so β(1 + Rt)ωt/ωt+1 is always
larger than 1.
We conclude that in the NMG there are non trivial dynamics for inequality in life-time wealth, which are either invariant over time, or monotone.
There is an interesting second twist to the results above,
which has to do with the desirability of policies aiming
at reducing inequality. First, the CE with the representative agent is efficient (as is the CE with heterogeneous agents), and both the steady state and the
transition toward it are independent of the initial distribution of wealth. This means that in a competitive
equilibrium no faster/slower growth is possible by implementing a redistribution of assets, and that no redistribution of assets will ever have any impact on the
steady state level of capital (hence output), consumption, etc. in per capita terms.
A first implication of this fact is that any policy intervention will, at the very least, create inefficiencies.
Hence, the model captures a non favorable trade off between equality and efficiency: less inequality is possible,
but at the cost of reducing economic efficiency.
A second implication of this fact is that there are many
wealth distributions that are compatible with steady
state. In oder words, the NMG has no prescription
regarding inequality at the steady state.
In the current version of the model the prediction is that
either inequality will remain constant over time, or it will
change monotonically (think of what would happen if
the economy was decreasing toward the steady state).
This essentially means that, according to the model,
current levels of inequality are a simple projection of
initial levels of inequality, but nothing else.
We complement the preceding results with two additional implications of the model.
R4. Consider two economies A and B identical in all
respects, except that CVA(ω0i ) > CVB (ω0i ). If c̄ > 0 then
CVA(ωti) > CVB (ωti) for all t.
R5. Consider two economies A and B identical in all
respects, except that K0A < K0B < K ∗. If c̄ > 0 then
CVA(ωti) > CVB (ωti) for all t.
5. Final thoughts
The results we have introduced so far look directly at
life-time wealth. Even if the results are interesting,
it is difficult to empirically observe ω, so it is difficult
to test our theory of competitive inequality dynamics.
What we would need is to be able to develop a theory
but with the focus on observable measures of wealth
and/or income.
Suppose we concentrate on assets ait (this is still difficult to measure, but a way simpler than ω!). Following
the reasoning along the lines above, we would be able
to show that
ait+1 = β(1 + Rt)ait + Dt, ∀t ≥ 0,
where Dt is again an amalgam of future wages and
interest rates that are independent of i. Continuing
with the same sort of reasoning we developed before,
it is possible to show that
Kt
i
i
CV (at+1) = CV (at)β(1 + Rt)
.
Kt+1
Interestingly enough, one is also able to show that in
the NMG, if K0 < K ∗, and it is not too far away from
the steady state, then β(1 + Rt) KKt < 1 for all c̄ > 0
t+1
and c̄ ≤ 0.
Hence, the conclusion is that close to the steady state,
inequality in assets is decreasing over time during a
transition from below.
Finally, it is also interesting to think of the “agents”
in the previous model as countries, or perhaps as “regions” in a country. An important conclusion under
this interpretation is that, if there is a well integrated
capital market, then the regions (or countries) will converge to its own steady state, and in the long run, there
will be differences in wealth that will persist forever.
That is, in the long run there will be poor and rich
regions/countries.
Think for a moment of the possibility of closing down
the asset market, so that each region is left in isolation.
Then the model looks like a collection of N different
regions. The initial differences in assets are simply different initial conditions for capital, but the NMG predicts that there is a unique steady state. This means
that in the long run, all economies (regions/countries)
would be identical! simply by closing down the capital
market.
The intriguing issue is, therefore, Why don’t we observe
something like this in actual economies?
References
Here are a few references that may of your interest.
1. Alvarez, M. J., Dı́az, A.: Minimum consumption and
transitional dynamics in wealth distribution. Journal of
Monetary Economics Vol. 52, (2005).
2. Caselli, F.,Ventura, J.: A Representative consumer
theory of distribution. The American Economic Review
90, p. 909 - 926, (2000).
3. Chatterjee, S.: Transitional dynamics and the distribution of wealth in a neoclassical growth model. Journal of Public Economics 54, p. 97 - 119, (1994).
4. Chatterjee, S., Ravikumar, B.: Minimum consumption requirements: theoretical and quantitative implications for growth and distribution. Macroeconomic
Dynamics 3, p. 482 - 505, (1999).
5. Sorger, G.: Income and wealth distribution in a simple model of growth. Economic Theory 16, pp. 23-42,
(2000).
The starting paper is (3), and it was further developed
by (2) (continuous time) and (4). The reference in (1)
uses similar methods to ours, and conducts a quantitative investigation of the theory. The reference in (5)
introduces endogenous leisure.