Macroeconomic Theory II (G31.1026) — Spring 2010

Heterogeneity in MacroeconomicsGianluca Violante

General Information• Lecture Times and Location: Monday and Wednesday 9:45-11:45 am, in room 517.The last class is on May 3rd.

• Office Hours: Tuesday 11:30-12:30 in my office, room 712. You can always contact meby phone at extension 29771 or by email <glv2@nyu.edu> to arrange an alternativetime.

• TA and Tutorials: The TA is Shengxing Zhang who can be reached at extension 87578or by e-mail <shengxing.zhang@nyu.edu>. He sits in room 818. His office hours areWednesday 3:30-5:30 pm. Weekly tutorials are held on Friday 9:30-11:30 am in Room517.

• Homework: There will be weekly problem sets that are required for a passing grade.You are allowed to cooperate with other students in the class, but every student has tohand in his/her own uniquely written assignment.

• Examination: There will be a written final examination on Wednesday May 5th, 9:30-11:30 am. The final exam does not include material in the last week of class (but thecore exam will).

• Course website: The webpage of the course, where announcements, homeworks, lecturenotes and additional readings will be posted is

http : //www.econ.nyu.edu/user/violante/macrotheory.htm

Please, make sure to check it regularly.

Course Summary and Objectives• Summary: This section of the course is devoted to studying economies where agentsare heterogeneous. These models are very helpful to analyze questions pertaining todistribution of resources, inequality, and the effects of policies. We will start with some“aggregation theorems” in complete markets economies where a representative agent stillexists. Next, we will move towards economies with “incomplete markets” where agentscan only borrow and save through a risk-free bond. We begin by characterizing in detailthe individual problem. Next, we proceed to the description of the stationary equilibrium.Then, we study an incomplete-markets model with aggregate shocks. The last classes aredevoted to defining economies where there is default in equilibrium, and economies withheterogeneous firms.

1

• Objectives: The aim of this section of the course is twofold: 1) to become familiar withthis important class of macroeconomic models, and 2) to learn how to solve numericallyfor the equilibrium of these model economies, in order to perform quantitative research.

Reading Material• Textbooks: The main textbook is Recursive Macroeconomic Theory, by Lars Ljungqvistand Tom Sargent, MIT Press, second edition, 2004 (denoted by LS below). You will alsouse Recursive Methods in Economic Dynamics, by Stokey, Lucas and Prescott, HarvardUniversity Press, 1989. In the recitation of Friday 26th, Shengxing will teach some basicconcepts of measure theory from SLP, chapters 7, 8.1,11.1,11.2 and 12.4. We need themfrom class 6 onwards.

• Other readings: I will make lecture notes and links to papers available online on mywebpage, as we go along.

Course Outline (by class)

1. Heterogeneity in the Neoclassical Growth Model with Complete Markets (LS,8.5.3)We discuss the assumptions on fundamentals under which, although households are het-erogeneous in preferences and endowments, a representative agent exists. And we applythese results to the neoclassical growth model. We discuss quickly (since you have alreadyseen it with Tom) the Negishi method. This methodology allows to calculate the com-petitive equilibrium prices and allocations of complete markets economies (in particular,economies for which the first welfare theorem holds) with heterogeneous households. Thismethod proves to be particularly useful for those economies where aggregation does nothold, hence we cannot use the representative agent. We present one illustration of thismethod based on a paper by Maliar and Maliar.

• Chatterjee, Satyajit (1994) "Transitional dynamics and the distribution of wealth ina neoclassical growth model", Journal of Public Economics

• Maliar, Lilia and Serguei Maliar (2001) "Heterogeneity in capital and skills in aneoclassical stochastic growth model", Journal of Economic Dynamics and Control

• Maliar, Lilia and Serguei Maliar (2003), "The Representative Consumer in the Neo-classical Growth Model with Idiosyncratic Shocks", Review of Economic Dynamics

2. The Income Fluctuation Problem I (LS, 16.1-16.4, 17.13)We discuss the empirical implications of full-insurance for consumption. We review thePermanent Income Hypothesis and we apply it to characterize the consumption-savingproblem of a single-agent who faces a stochastic income stream and can trade only arisk-free bond. We introduce the notion of precautionary savings and relate it to theconvexity of marginal utility (prudence).

2

• Mace, Barbara (1991); "Full Insurance in the Presence of Aggregate Uncertainty,"Journal of Political Economy

• Cochrane, John (1991); "A Simple Test of Consumption Insurance," Journal ofPolitical Economy

• Hall, Robert (1978); "Stochastic Implications of the Life Cycle-Permanent IncomeHypothesis: Theory and Evidence", Journal of Political Economy

• Leland, Haynes (1968); "Saving and Uncertainty: the Precautionary Demand forSaving", Quarterly Journal of Economics

• Sibley, David (1975); "Permanent and Transitory Income Effects in a Model of Op-timal Consumption with Wage Income Uncertainty", Journal of Economic Theory(only section VI)

• Blundell, Richard and Ian Preston (1998); "Consumption Inequality and IncomeUncertainty," Quarterly Journal of Economics

3. The Income Fluctuation Problem II (LS, 16.5-16.8, 17.3-17.5)We introduce borrowing constraints and show that precautionary savings can arise evenwithout prudence as long as borrowing constraints may bind in some state of the world.We then derive an important condition on the interest rate that guarantees that theoptimal individual consumption sequence is bounded above, in presence of income uncer-tainty.

4. Numerical Techniques to Solve the Income Fluctuation ProblemWe present a set of simple numerical techniques to solve for the consumption and savingpolicy functions in the recursive formulation of the income-fluctuation problem for thesingle-agent who self-insures by saving/borrowing through a risk-free bond. In particular,we study a new very fast numerical method, called “endogenous grid method”.

• Tauchen, George (1986); "Finite State Markov Chain Approximations to Univariateand Vector Autoregressions", Economic Letters

• Suen, Richard and Kopecki, Karen (2010); “Finite State Markov-Chain Approxima-tions to Highly Persistent Processes,” Review of Economic Dynamics

• Aruoba B., Fernandez-Villaverde Jesus, and Rubio-Ramirez, Juan (2006); "Compar-ing Solution Methods for Dynamic Equilibrium Economies", Journal of EconomicDynamics and Control

• Carroll, Chris (2006). “The Method of Endogenous Gridpoints for Solving DynamicStochastic Optimization Problems,” Economics Letters

• Barillas, Francisco and Fernandez-Villaverde Jesus (2007). “A Generalization of theEndogenous Grid Method,” Journal of Economic Dynamics and Control

• Judd, Ken (1998); Numerical Methods in Economics, MIT Press, chapters 6-10• Marimon, Ramon and Scott, Andrew (1999); "Computational Methods for the Studyof Dynamic Economies", Oxford University Press

3

• Heer, Burkhard and AlfredMaubner (2005); "DGEModelling, Computational Meth-ods and Applications," Springer

5. The Neoclassical Growth Model with Incomplete Markets (“Bewley Models”)(LS 17.1-17.2, 17.6-17.12)We analyze the equilibrium of a neoclassical growth model populated by a continuum ofagents who face idiosyncratic labor income risk and trade only a risk-free asset. We usethe tools we learned to characterize (as much as possible...) the existence and uniquenessof the invariant distribution.

• Imrohoroglu, Ayse (1989); The Costs of Business Cycles with Indivisibilities andLiquidity Constraints, Journal of Political Economy

• Huggett, Mark (1993); The Risk-Free Rate in Heterogeneous-Agent Incomplete-Insurance Economies, Journal of Economic Dynamics and Control

• Aiyagari, Rao (1994); Uninsured Idiosyncratic Risk and Aggregate Saving, QuarterlyJournal of Economics

• Hopenhayn H. and E. Prescott (1992); Stochastic Monotonicity and Stationary Dis-tributions for Dynamic Economies, Econometrica

6. Some Applications of “Bewley Models”We illustrate how to use this class of self-insurance models to analyze questions relatedto the wealth distribution and to fiscal policy.

• Budria Santiago, Javier Diaz-Gimenez, Vincenzo Quadrini, Victor Rios-Rull (2002);Updated Facts on the US Distributions of Earnings, Income andWealth,MinneapolisFed Quarterly Review

• Floden Martin and Jesper Linde (2001); Idiosyncratic Risk in the U.S. and Sweden:Is there a Role for Government Insurance?, Review of Economic Dynamics

• Aiyagari, Rao and Ellen Mc Grattan (1998); The Optimum Quantity of Debt, Jour-nal of Monetary Economics

7. Constrained efficiency in the Aiyagari modelWe discuss the difference between the first-best allocations and the constrained efficientallocations in the Aiyagari model with self-insurance. We argue that the planner, throughsaving decisions, will manipulate prices in order to raise wages (if the income of the pooris labor intensive), hence redistributing from the lucky-rich to the unlucky-poor. Wealso debate whether macroeconomists should refine the neoclassical growth model withincomplete markets by adding observed channels of insurance (family, bankruptcy laws,public insurance), or whether they should think about the fundamental reasons that limitfull insurance (moral hazard, adverse selection, imperfect enforcement).

• Hong, Jay, Julio Davila, Per Krusell, and Jose-Victor Rios-Rull (2006), Constrainedefficiency in the neoclassical growth model with uninsurable idiosyncratic shocks.

4

8. Transitional Dynamics in the Neoclassical Growth Model with IncompleteMarketsWe study how to compute the transitional dynamics and how to measure correctly thewelfare changes associated to a tax reform.

• Floden, Martin (2001), The Effectiveness of Government Debt and Transfers asInsurance, Journal of Monetary Economics (especially, section 3 on welfare decom-position)

9. Adding Aggregate Risk: A Near-Aggregation Result (LS 17.14.2)We extend the model to add aggregate fluctuations in productivity. We explain how tosolve this model and present the near-aggregation finding of Krusell-Smith.

• Krusell, Per and Tony Smith (1998), Income andWealth Heterogeneity in the Macro-economy, Journal of Political Economy

• Heathcote, Jonathan (2004), Fiscal Policy with Heterogeneous Agents and Incom-plete Markets, Review of Economic Studies

10. Economies with DefaultWe first study an incomplete-market economy where agents face borrowing constraintsthat are tight enough so that they never have the incentive to default in the equilibrium.Then, we formalize a model where agents can default and the financial sector takes intoaccount the default probability and increases the prices of loans accordingly.

• Zhang Harold (1997), Endogenous Borrowing Constraints with Incomplete Markets,Journal of Finance

• Livshits Igor, Jim McGee and Michele Tertilt (2006), Consumer Bankruptcy: AFresh Start, American Economic Review

11. Life-Cycle Economies with Incomplete MarketsWe study a life-cycle version of the standard incomplete-markets model with overlappinggenerations, and an application to optimal Ramsey-style taxation.

• Imrohoroglu Ayse, Imrohoroglu Selo, Joines, Douglas (1995), “A life cycle analysisof social security,” Economic Theory

• Huggett, Mark (1996), “Wealth distribution in life-cycle economies.” Journal ofMonetary Economics

• Rios-Rull Jose-Victor (1995), “Models with heterogeneous agents”. In Frontiers ofBusiness Cycle Research, edited by Cooley TF. Princeton, NJ: Princeton UniversityPress.

• Conesa Juan-Carlos, and Dirk Krueger (2006), “On the optimal progressivity of theincome tax code,” Journal of Monetary Economics

5

• Conesa Juan-Carlos, Kitao Sagiri, and Dirk Krueger (2008), “Taxing Capital? Nota Bad Idea After All!” American Economic Review

12. Industry EquilibriumWe use what we have learned about heterogeneous agents economies to study the equi-librium of an industry with firms facing shocks to their productivity level, and withendogenous firm entry and exit. We analyze the impact of firing costs on the averageproductivity of the industry.

• Hopenhayn, Hugo (1992), Entry, Exit and Industry Dynamics in Long-Run Equilib-rium, Econometrica

• Hopenhayn, Hugo and Richard Rogerson (1993), Job Turnover and Policy Evalua-tion: A General Equilibrium Analysis, Journal of Political Economy

13. Industry Equilibrium with International TradeWe extend Hopenhyan’s model to an economy with monopolistic competition which isopen to trade. Firms must pay a fixed cost of exporting to access a foreign market. Weexamine the impact of trade openess on firm selection and aggregate productivity.

• Melitz, Marc (2003). “The Impact of Trade on Intra-Industry Reallocations andAggregate Industry Productivity”, Econometrica

6

1 Heterogeneity in the neoclassical growthmodel withcomplete markets

1.1 Two preliminary results on aggregation of technologies andpreferences

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 that F 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-taking firm of type i solves

max{ki,ni}

©zF¡ki, ni

¢− wni − (r + δ) ki

ª,

with first-order conditions

zFk

¡ki, ni

¢= r + δ, (1)

zFn

¡ki, ni

¢= w,

Recall that by CRS, Fk and Fn are homogenous of degree zero, hence:

Fk (ki, ni)

Fn (ki, ni)=

fk (ki/ni)

fn (ki/ni).

Dividing through the two first-order conditions, we obtain

fk (ki/ni)

fn (ki/ni)=

r + δ

w,

1

and using the fact that the left-hand side is a strictly decreasing function of (ki/ni), we

obtainkini= g

µr + δ

w

¶=

K

N, for every i = 1, 2, ...,M

where capital letters denote averages: every firm 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 all i0s. 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 capital ki0 = κ for all i0s and same preferences

U¡ci0, c

i1, ...

¢=

∞Xt=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

consumers are different only in terms of their initial endowments of wealth. There is no

individual or aggregate uncertainty.

2

1.2.1 The economy

Demographics— The economy is inhabited by N types of infinitely lived agents, indexed

by i = 1, 2, ..., N . Denote by μi the number of agents i and normalize the total number

of agents to one, so that averages and aggregates are the same.

Preferences— Preferences are time separable, defined over streams of consumption,

given by

U =∞Xt=0

βtu¡cit¢,

where the period utility function u belongs to one of the following three classes: log,

power, exponential, i.e.

u (c) =

⎧⎨⎩ln (c̄+ c) with c̄+ c > 0, c̄ ≤ 0(c̄+c)1−σ

1−σ with c̄+ c > 0, c̄ ≤ 0−c̄ exp (−σc) with c̄ > 0

(2)

We impose c̄ ≤ 0 for log and power utility to allow for a subsistence level for consumption,and we impose c̄ > 0 for the exponential case.

Markets and property rights— There are spot markets for the final good (which can

be used for both consumption and investment) and complete financial markets, i.e. there

are no constraints on transfers of income across periods. We follow Chatterjee (1994) in

assigning the property rights on capital to the firm and the ownership of the firm to the

household.1 This is a different arrangement of property rights from the one you are used

to seeing. Here households own shares of the firm.

We will let the initial level of wealth, at date t, differ across agents. Let ait be the

individual wealth of type i at time t. Then, ait = sitAt, where sit is the share of the firm-

value owned by consumer i at time t. By summing both sides of this equation over i and

exploiting the fact thatPN

i=1 μisit = 1 for every t, we obtain that aggregate household

wealth equals the value of the firm. Besides intertemporal trading, there will be no other

securities traded among agents in equilibrium, since there is no risk.

Technology and firm’s problem— The aggregate production technology is Yt =

f (Kt) with f strictly increasing, strictly concave and differentiable. The representative

1If we had chosen to model the firm’s problem as static (i.e. the firm rents capital services fromhouseholds), every argument in this lecture would still hold. You should check this, as well as every otherclaim I make without proving it!

3

firm owns physical capital and makes the investment decision by solving the problem

At = max{Iτ}

∞Xτ=t

µpτpt

¶[f (Kτ)− Iτ ] (3)

s.t.

Kτ+1 = (1− δ)Kτ + Iτ ,

where pt is the time t price of the final good. Let’s define real profits πt ≡ 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 (pτ/pt), the relative price of consumption between time τ and

time t. Recall that pt/pt+1 = 1 + rt+1 where r is the interest rate.

It is useful to compute the first-order condition (FOC) of the firm problem with respect

to Kt+1 by substituting the law of motion for capital into (3) . The problem becomes:

max{Kt+1}

½f (Kt)−Kt+1 + (1− δ)Kt +

pt+1pt[f (Kt+1)−Kt+2 + (1− δ)Kt+1] +

pt+2pt+1

...

¾with FOC:

1 =pt+1pt[f 0 (Kt+1) + (1− δ)] . (4)

Household’s problem—Given complete markets, the maximization problem of house-

hold i at time t can therefore be stated as:

max{ciτ}

∞Xτ=t

βτ−tu¡ciτ¢

s.t.∞Xτ=t

pτciτ ≤ pta

it (5)

where ait is the wealth of agent i in term of consumption units at time t. Let λit be the

Lagrange multiplier on the individual i time t Arrow-Debreu budget constraint.

Solution— Consider the log-preferences case. From the FOC of the household problem

at time t with respect to consumption at time τ , we have:

βτ−tu0¡ciτ¢= λitpτ ⇒ βτ−t

µ1

c̄+ ciτ

¶= λitpτ ⇒ ciτ =

βτ−t

λitpτ− c̄. (6)

4

Substituting this FOC into the budget constraint of (5), we can derive an expression

for the multiplier λit:

∞Xτ=t

pτ

µβτ−t

λitpτ− c̄

¶= pta

it

1

λit (1− β)− c̄

∞Xτ=t

pτ = ptaitµ

1

λit

¶= (1− β)

"pta

it + c̄

∞Xτ=t

pτ

#(7)

Let’s now substitute the expression on the last line into equation (6) evaluated at τ = t,

i.e.

cit =1

λitpt− c̄,

in order to solve explicitly for cit:

cit =1

pt

"(1− β) pta

it + (1− β) c̄

∞Xτ=t

pτ

#− c̄

= c̄

"(1− β)

∞Xτ=t

µpτpt

¶− 1#+ (1− β) ait (8)

= Θ¡pt, c̄

¢+ (1− β) ait,

where

Θ¡pt, c̄

¢≡ c̄

"(1− β)

∞Xτ=t

µpτpt

¶− 1#

(9)

is a function of the subsistence level and of the whole price sequence pt = {pt, pt+1, ...}.Thus, we have the optimal individual consumption rule

cit = Θ¡pt, c̄

¢+ (1− β) ait, (10)

which 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 quasi-homothetic, i.e. they have affine Engel curves in wealth: the wealth-

expansion path is linear.2

2When c̄ = 0, preferences are homothetic because the constant in the consumption function becomeszero and Engel curves start at the origin, i.e. they are linear. However, linearity of the wealth-expansionpath is not affected by the constant c̄.

5

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).

1.2.2 Equilibrium aggregate dynamics

Denote aggregate variables with capital letters. From (10), we derive easily that aggregate

consumption only depends on aggregate variables (prices and aggregate wealth), but it is

independent of the distribution of wealth. By summing over i on the LHS and RHS of

(10) with weights μi we arrive at:

Ct = Θ¡pt, c̄

¢+ (1− β)At. (11)

Since equilibrium aggregate consumption Ct has the same form as individual optimal

consumption choice cit, it is clear that (11) can be obtained as the solution to the following

representative agent problem:

max{Cτ}

∞Pτ=t

βτ−tu (Cτ )

s.t.

∞Pτ=t

pτCτ ≤ ptAt

(PP)

which is exactly as in (5) but we have replaced small letters with capital letters. Let’s

make some further progress on the solution. From the FOCs

u0 (Ct)

βu0 (Ct+1)=

ptpt+1

. (12)

From (12) and the FOC for the firm’s problem (4), we obtain the familiar Euler equation

of the neoclassical growth model

u0 (Ct) = βu0 (Ct+1) [f0 (Kt+1) + (1− δ)] . (13)

We can state our first important result:

Result 1.1: If preferences are quasi-homothetic and agents are heterogeneous in ini-

tial endowments, the aggregate dynamics of the neoclassical growth model with complete

markets admit a single-agent representation. Put differently, the dynamics of aggregate

6

quantities and prices are exactly the same as in the standard neoclassical growth model

with representative agent.

This result depends crucially on the linearity of individual optimal consumption with

respect to wealth which, in turn, descends from quasi-homotheticity of preferences. This

type of aggregation, where: 1) preferences of the representative agent are the same as the

ones of the individual agent, and 2) aggregate dynamics do not depend on the distribution

of wealth is called “Gorman aggregation”, from the seminal article by Groman (1953). It

hinges crucially on homothetic utility and heterogeneity only in initial wealth.

Two remarks are in order. First, equation (13) governs the dynamics of capital in

the representative agent growth model where firms rent capital from households, instead

of owning it. Therefore, we have discovered that in complete markets it is irrelevant

whether we attribute property rights on capital to firms (and let households own shares

of the firms) or to workers (and let firms rent capital from households). Second, equation

(13) also governs the dynamics of capital in the social-planner problem. We are still in

complete markets, and the Welfare Theorems hold.

Steady-state— The dynamics of the economy will converge to the steady-state values

of capital stock satisfying the modified golden rule f 0 (K∗) = 1/β − (1− δ). Note now

that in steady-state pt/pt+1 = 1/β for all t, hence from the definition of Θ (pt, c̄) in (9)

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.

To conclude, in the neoclassical growth model with complete markets and where agents

have heterogeneous wealth endowments, the dynamics of the aggregate variables do not

depend on the evolution of the wealth distribution. But is the inverse statement true?

Does the evolution of the wealth distribution across households (i.e., wealth inequality)

depend on the dynamics of aggregate variables (prices and quantities)? We show below

that the answer is: yes, it does.

7

1.2.3 Equilibrium dynamics of the wealth distribution

From the lifetime budget constraint of agent i at time t

ptcit +

∞Xτ=t+1

pτciτ = pta

it ⇒ ptc

it + pt+1a

it+1 = pta

it (14)

citait+

pt+1ait+1

ptait= 1 ⇒ ait+1

ait=

µptpt+1

¶µ1− cit

ait

¶, (15)

which expresses the growth rate of wealth for type i as a function of her consumption-

wealth ratio.

By aggregating over types in equation (14), we can obtain an equivalent equation at

the aggregate level:Xi

μicit +

µpt+1pt

¶Xi

μiait+1 =Xi

μiait

Ct +

µpt+1pt

¶At+1 = At

At+1

At=

µptpt+1

¶µ1− Ct

At

¶We want to establish conditions under which an individual’s share of total wealth will

grow over time, i.e. sit+1 > sit. First of all, note that:

sit+1sit

> 1 ⇔ ait+1ait

>At+1

At⇔ cit

ait<

Ct

At(16)

Moreover, from equations (10) and (11),

citait=

Θ (pt, c̄)

ait+ (1− β) , and

Cit

At=

Θ (pt, c̄)

At+ (1− β)

and therefore

citait

<Ct

At⇔ Θ (pt, c̄)

ait<

Θ (pt, c̄)

At⇔ Θ

¡pt, c̄

¢ ¡ait −At

¢> 0

and, thus, summarizing we have the following equivalence (i.e., “if and only if”) condition:

sit+1sit

> 1 ⇔ Θ¡pt¢ ¡

ait −At

¢> 0,

which means that whether consumer’s i wealth share is increasing or decreasing over time

depends on 1) the sign of the constantΘ (equal for everyone) and 2) on her relative position

8

in the distribution. For example, if Θ > 0 then for a consumer whose initial wealth is

above average, her share will grow, whereas for a consumer whose initial wealth is below

average, her share will fall. And hence the distribution will become more unequal over

time. Note that the dynamics of the wealth distribution depend on the entire sequence

of prices, hence on the dynamics of aggregate variables in equilibrium.

First of all, in absence of subsistence level, c̄ = 0 we have Θ = 0, and therefore 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 complete markets, homothetic prefer-

ences, heterogeneous endowments, but without subsistence level (c̄ = 0), the wealth distri-

bution remains unchanged along the transition path, i.e. initial conditions in endowments

(and inequality) persist forever.

The intuition is that if c̄ = 0 then ci = (1− β) ai, so the average propensity to

consume, and save, is the same for every agent. Every agent accumulates wealth at the

same rate.

In presence of a subsistence level, the dynamics are more interesting. We now deter-

mine the sign of Θ, through:

Lemma 1.1 (Chatterjee, 1994): The common constant term of the consumption

function Θ (pt, c̄) is greater (less) than zero if and only if the economy is converging from

below (above) to the steady-state, i.e. if Kτ < (>)K∗.

Proof: Suppose the economy grows towards the steady-state, i.e. Kτ < K∗. From

equation (13), the sequence {f 0 (Kτ )} is decreasing and the sequence {pτ/pτ+1} will bedecreasing towards 1/β. Therefore, the sequence {pτ+1/pτ} is increasing towards β, i.e.,

pτ+1/pτ ≤ β for all τ ≥ t where the strict inequality holds at least for some τ . It follows

that

pτ/pt = (pτ/pτ−1) (pτ−1/pτ−2) ... (pt+2/pt+1) (pt+1/pt) < βτ−t.

From the definition of Θ (pt, c̄) in (8), use the above equation to obtain

Θ¡pt, c̄

¢= c̄

"(1− β)

∞Xτ=t

µpτpt

¶− 1#> c̄

"(1− β)

∞Xτ=t

βτ−t − 1#= 0

where the inequality follows from c̄ < 0. QED

9

The implications for the evolution of the wealth distribution in an economy growing

towards the steady-state (the empirically interesting case) are easy to determine, at this

point. In the presence of a subsistence level (c̄ < 0), Θ > 0. From equation (16) this im-

plies that the average propensity to consume (save) declines (increases) with wealth: poor

agents must consume proportionately more out of their wealth to satisfy the subsistence

level. In other words:

Result 1.3: In the neoclassical growth model with complete markets, homothetic pref-

erences, heterogeneous endowments and subsistence level c̄ < 0, as the economy grows

towards the steady-state: (i) the wealth distribution becomes more unequal, 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 wealth

(and consumption) ranking persist forever.

The main conclusion of this lecture is that in this model there is no economic or social

mobility. This is not a good model to understand why some individual are born poor and

make it in life, while other are born rich and end up poor as rats. This is just a model of

castes.

Robustness—We now discuss how robust this result is to two of the key assumptions

made so far in the analysis: 1) all agents have same discount factor β, 2) markets are

complete.

• 1. When agents have different discount factors, then none of the results hold any

longer. Suppose that c̄ = 0 to simplify the analysis. Then, from (10)

cit =¡1− βi

¢ait,

therefore the average propensity to save out of wealth is higher the more pa-

tient is the individual and from (16), 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 becomes degenerate.

2. In absence of markets and trade (autarky), every consumer has access to her

own technology. Each agent i will solve his own maximization problem in

10

isolationmax{ciτ}

∞Pτ=t

βτ−tu (ciτ)

s.t.kiτ+1 = (1− δ) kiτ + f (kiτ)− ciτkit given

with different initial conditions kit. It is easy to see that, independently of

initial conditions, each agent will converge to the same capital stock k∗, hence

in the long-run the distribution of wealth is perfectly equal. Interestingly, we

conclude that less developed financial markets induce less wealth inequality, in

the long-run.

1.2.4 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. From (3), (13) and (10), the set of equations

characterizing the steady-state is:

f 0 (K∗) = 1/β − (1− δ) ,

A∗ =1

1− β[f (K∗)− δK∗]

NXi=1

μiai = A∗,

ci = (1− β) ai, i = 1, 2, ..., N

We therefore have (N + 3) equations and (2N + 2) unknowns³{ci, ai}Ni=1 ,K∗, A∗

´. In

other words, the multiplicity of the steady-state wealth distributions is of order N − 1.3

However, suppose we start from a given wealth distribution at date t = 0 when the

economy 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 our finding in:

Result 1.4: In the steady-state of the neoclassical growth model with N agents, het-

erogeneous initial endowments and homothetic preferences, there is a continuum with

dimension (N − 1) of steady-state wealth distributions. However, given an initial wealth3This means that, if N = 1 (representative agent), the steady-state is unique. If N = 2, there is a

continuum of steady-states of dimension 1, and so on.

11

distribution {ai0}Ni=1 at t = 0, the equilibrium wealth distribution {ait}

Ni=1 in every period

t is uniquely determined, and so is the final 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

zf 0 (K∗) = 1/β − (1− δ) ,

A∗ =1

1− β[zf (K∗)− δK∗] ,

μ1a1 + μ2a2 = A∗

ci = (1− β) ai, i = 1, 2,

So we have 5 equations, but 6 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 level zH is uniquely determined.

Finally, in terms of language, this whole section shows that it is important to distin-

guish “steady-state” from “equilibrium path”. In this economy, the equilibrium path is

always unique (given initial conditions), but the steady-state is not.

2 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 is particularly

useful for those economies where aggregation does not go through and, hence we cannot

use the representative agent.4

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

4In his original paper, Negishi (1960) used this equivalence result to propose a simple way to showexistence of competitive equilibria.

12

where agents are initially endowed with heterogeneous shares {si0}Ni=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 to find the set of weights which correspond to our desired CE allocation. As an aside,

the concept of social welfare was introduced by Samuelson (1956).

2.1 An Example

Consider our neoclassical growth model of section (1.2) with two types of consumers

(N = 2). The agent’s i problem in the decentralized Arrow-Debreu equilibrium can be

written as

max{cit}

∞Xt=0

βtu¡cit¢

s.t.∞Xt=0

ptcit ≤ p0a

i0

where ai0 = si0A0 is the initial wealth endowment, given at t = 0. Let’s assign the

property rights on capital to the firm, so the firm’s problem is exactly the one of the

previous section.

From the FOC of the agent of type i, we obtain

FOC¡cit¢−→ βtu0

¡cit¢= λipt,

where λi is the multiplier on the time zero budget constraint. Thus, putting together the

FOC’s for the two types:u0 (c1t )

u0 (c2t )=

λ1

λ2. (17)

Now, write down the following Negishi planner problem (NP) for our economy

max{c1t ,c2t ,Kt+1}

∞Xt=0

βt£α1u

¡c1t¢+ α2u

¡c2t¢¤

(NP)

s.t.

c1t + c2t +Kt+1 ≤ f (Kt) + (1− δ)Kt

K0 given

13

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

FOC¡cit¢−→ αiβtu0

¡cit¢= θt, i = 1, 2 (18)

FOC (Kt) −→ u0¡c1t¢= βu0

¡c1t+1

¢[f 0 (Kt+1) + (1− δ)] (19)

where θt is the Lagrange multiplier on the planner’s resource constraint at time t. Note

that putting together the first-order conditions for consumption for the two agents we

arrive atu0 (c1t )

u0 (c2t )=

α2

α1, (20)

which tells us that the planner allocates consumption proportionately to the weight it

gives to each consumer (with strictly concave utility).5

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 (20) and (17):

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 u (cit) = log cit, we derived in equation (7) thatµ

1

λi

¶= (1− β) p0a

i0 ⇒ λi =

∙1

(1− β) p0A0

¸1

si0,

therefore we obtain thatα2

α1=

λ1

λ2=

s20s10.

Imposing the natural and innocuous normalization α1 + α2 = 1, we can solve explicitly

for the two weights: α1 = s10 and α2 = s20, i.e., the weights are exactly equal to the initial

wealth shares. The higher is the initial wealth share si0, the lower is the multiplier λi and

the larger is the Pareto weight αi on the Negishi problem: the planner must deliver more

5Note also that the ratio of marginal utility across agents is kept constant in every period (a keyfeatures of complete markets allocations, also called full insurance).

14

to consumption to the agent who has a large initial share of wealth in the decentralized

equilibrium.

Result 1.5: Consider 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 wealth will get a higher weight in the planner’s

problem.

Now, note that using equation (18) we obtain that

αiβtu0 (cit)

αiβt+1u0¡cit+1

¢ = θtθt+1

=⇒ u0 (cit)

βu0¡cit+1

¢ = θtθt+1

.

Substituting this last expression into (19), we arrive at a relationship between the sequence

of capital stocks and the sequence of Lagrange multipliers on the planner’s resource con-

straintθtθt+1

= f 0 (Kt+1) + (1− δ) .

Recall that equation (13) dictating the optimal choice of capital for the firm in the CE

stated thatptpt+1

= f 0 (Kt+1) + (1− δ) .

Hence, we haveθtθt+1

=ptpt+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.6

In conclusion, we have uncovered a tight relation between weights of the NP problem

and initial endowments in the CE and an equivalence between Lagrange multiplier on the

resource constraint of the NP problem and prices in the CE. This strict relationship, that

we have uncovered for the log utility case, is true more in general.

6Using p0 as the numeraire and imposing the normalizations p0 = θ0 = 1, the above relationshipimplies that pt = θt so equilibrium prices are exactly equal to the shadow prices of consumption in theplanner’s problem.

15

2.2 General application of the Negishi method

In general, without restrictions on preferences, one may not have closed form solutions for

the λ’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}Ni=1. We can describe the algorithm in four steps:

1. In the social planner problem (NP), guess a vector of weights α =©α1, α2, ..., αN

ª.

The normalizationPN

i=1 αi = 1 means that α belongs to the N-dimensional simplex

∆N = {α ∈RN+ :

NXi=1

αi = 1}

and the simplex traces out the entire set of Pareto-optimal allocations.

2. Compute the sequence of allocationsn{cit}

Ni=1 ,Kt

o∞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 N + 2 equations

αiβtu0¡cit¢= θt, i = 1, .., N

NXi=1

μicit +Kt+1 = f (Kt) + (1− δ)Kt

θtθt+1

= f 0 (Kt+1) + (1− δ)

in N + 2 unknowns³{cit}

Ni=1 , Kt+1, θt+1

´. At every t, (Kt, θt) are given, therefore

the Negishi method simplifies enormously the computation of the equilibrium: the

Negishi solution requires solving, for every time t, a small system of equations.

Recall that, instead, to solve for the CE allocations, at every time t one must set

the excess demand function to zero and the excess demand function depends on

the entire price sequence–an infinitely dimensional object. To understand, take

another look at the consumption allocation (10) where Θ (·) depends on the entireprice sequence from t onward.

Instead of guessing (and iterating over) an infinite sequence of prices, one guesses

and iterates over a finite set of weights. With a caveat: even though K0 is given,

θ0 is not. So, one has also to guess a value for θ0. The reason is that, unless you

have the right value for θ0, the system will not be on the saddle-path and capital

16

will diverge. In other words, there is another condition that we need to satisfy in

the growth model, the transversality condition.

3. Exploit the equivalence between prices pt and multipliers θt to verify whether the

time-zero Arrow-Debreu budget constraint of each agent holds exactly at the guessed

vector of weight α. Specifically, for each agent, compute the implicit transfer func-

tion τ i (α) associated with the assumed vector of weights

τ i (α) =∞Xt=0

θtcit

¡αi¢− θ0a

i0, for every i = 1, 2, ..., N (21)

and if (21) holds for agent i with a “greater than” sign, it means that the planner

is giving too much weight to agent i. So, in the next iteration reduce the weight αi

given to agent i. Note one useful property of the transfer functions:

NXi=1

τ i (α) =NXi=1

" ∞Xt=0

θtcit

¡αi¢− θ0a

i0

#=

∞Xt=0

ptCt − p0A0 = 0

since the discounted present value of resources of the economy cannot be greater

than its current wealth. Put differently, recall that from the representative firm

problem

A0 =∞Xt=0

µptp0

¶[f (Kt)− It] =

∞Xt=0

µptp0

¶Ct

which establishes the same result. To conclude, the transfer functions must sum to

zero.

4. Iterate over α until you find the vector of weights α∗ that sets every individual

transfer function τ i (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 across-agents. Thus, we are computing exactly the CE

associated to initial conditions {ai0}Ni=1 .

See also Ljungqvist-Sargent, section 8.5.3, for a discussion of the Negishi algorithm.

2.3 “Non-Gorman” aggregation

Maliar and Maliar (2001, 2003) make use of the Negishi approach to prove a more gen-

eral aggregation result. In their model, agents have non-homothetic preferences in con-

17

sumption and leisure, and are subject to idiosyncratic, but insurable, shocks to labor

endowment.

They prove that the strong Gorman aggregation fails, but one can obtain a weaker

aggregation result. The aggregate dynamics of the model can still be described by a

representative agent (RA). However, the RA’s preferences are different from preferences of

the individual consumer. There exists a new preference shifter for the RA that depends on

the distribution of individual productivity shocks. Therefore, the dynamics of aggregate

variables do depend on the distribution of the shocks.

2.3.1 The Model

Demographics— The economy is inhabited by a continuum of infinitely lived agents,

indexed by i ∈ I ≡ [0, 1]. Denote by μi the measure of agents i in the set I and normalizethe total number of agents to one,

RIdμi = 1, so that averages and aggregates are the

same. Initial heterogeneity is in the dimension of initial wealth endowments.

Uncertainty— Agents are subject to idiosyncratic productivity shocks to skills. Let

εit be the shock of agent i, and suppose shocks are iid, with mean 1, and defined over the

set E. This is not necessary, but it simplifies the notation.

Preferences— Preferences are time separable, defined over strieams of consumption,

given by

U = E0∞Xt=0

βtu¡cit, 1− nit

¢.

where period utility is given by

u (ct, 1− nt) =c1−γt − 11− γ

+A(1− nt)

1−σ − 11− σ

(22)

and note that preferences are not quasi-homothetic, unless σ = γ.

Markets and property rights— There are spot markets for the final good (which

can be used for both consumption and investment) whose price is normalized to one, and

complete financial markets, i.e. agents can trade a full set of state-contingent claims. The

agent’s portfolio is composed by Arrow securities of the type ait+1 (ε) which pay one unit

of consumption at time t+ 1 if the individual’s shock is ε and zero otherwise. Let pt (ε)

the price of this security andRpt (ε) a

it+1 (ε) dε the value of such portfolio for agent i.

18

Technology and firm’s problem— The aggregate production technology is Yt =

Ztf (Kt, Ht) with f strictly increasing, strictly concave and differentiable. The represen-

tative firm rents capital from households. Ht is aggregate labor input in efficiency units,

i.e. Ht =RIεitn

itdμ

i.

Household problem— For agent i:

max{cit,kit+1,at+1(ε)}

E0

∞Xt=0

βtu¡cit, 1− nit

¢(23)

s.t.

cit + kit+1 +

Zpt (ε) a

it+1 (ε) dε = (1− δ) kit + wtε

itn

it + ait

¡εit¢

ki0, ai0 given

Equilibrium— This is a CM economy. The First Welfare Theorem tells us that the

equilibrium is PO, so we can use a social planner problem to characterize the equilibrium

by applying the Negishi method. The key, as usual, is to find the right weights that

guarantee that allocations are affordable for each agent, given their initial endowments.

Aggregation?— Given that preferences are not homothetic, we know that Gorman’s

strong aggregation concept will not hold. But can we, nevertheless, obtain a RA whose

choices describe the evolution of the aggregate economy? And how the preferences of the

RA look like?

Letting θt be the multiplier on the aggregate feasibility constraint, from the FOC with

respect to individual i in the Negishi planner problem:

αi¡cit¢−γ

= θt (24)

αiA¡1− nit

¢−σ= θtwtε

it

Rearranging gives

cit =

µαi

θit

¶ 1γ

¡1− nit

¢εit =

µAαi

θtwt

¶ 1σ ¡

εit¢1−1/σ

and note that consumption of individual i is proportional to its weight in the social welfare

function. Leisure is directly proportional to its weight (a wealth effect) and inversely

19

proprtional to individual productivity: efficiency arguments induce the planner to make

high-productivity individuals work harder.

Integrating the two FOCs across agents gives

Ct =

ZI

citdμi =

ZI

µαi

θt

¶ 1γ

dμi (25)

1−Ht = 1−ZI

εitnitdμ

i = 1−ZI

µAαi

θtwt

¶ 1σ ¡

εit¢1−1/σ

dμi

Now, note that

cit =

³αi

θt

´ 1γ

RI

³αi

θt

´ 1γdμi

Ct ⇒ cit =(αi)

1γR

I(αi)

1γ dμi

Ct (26)

¡1− nit

¢εit =

µAαi

θtwt

¶ 1σ ¡

εit¢1−1/σ ⇒ ¡

1− nit¢=

µAαi

θtwt

¶ 1σ ¡

εit¢−1/σ

⇒¡1− nit

¢=

(αi)1σ (εit)

−1/σRI(αi)

1σ (εit)

1−1/σdμi

(1−Ht) (27)

Now, consider the social welfare function for the plannerZI

(cit)1−γ − 11− γ

+A(1− nit)

1−σ − 11− σ

dμi

and substitute the two expressions in (26) into the social welfare function:

ZI

αi

"(αi)

1γR

I(αi)1γ dμi

Ct

#1−γ− 1

1− γ+A

"(αi)

1σ (εit)

−1/σRI(α

i)1σ (εit)

1−1/σdμi(1−Ht)

#1−σ− 1

1− σdμi

=

RI α

i(αi)1−γγ dμi∙R

I(αi)1γ dμi

¸1−γC1−γt − 1

1− γ+A

RI α

i(αi)1−σσ (εit)

1−1/σdμi∙R

I(αi)1σ (εit)

1−1/σdμi

¸1−σ (1−Ht)1−σ − 1

1− σ

which yields the utility for the RA

C1−γt − 11− γ

+AXt(1−Ht)

1−σ − 11− σ

where

Xt =

hRI(αi)

1σ (εit)

1−1/σdμiiσhR

I(αi)

1γ dμi

iγ20

Therefore the RA problem which describes the aggregate allocations for this economy

becomes:

max{Ct,Ht,Kt+1}

E0∞Xt=0

βtC1−γt − 11− γ

+AXt(1−Ht)

1−σ − 11− σ

(28)

s.t.

Ct +Kt+1 = (1− δ)Kt + Ztf (Kt, Ht)

K0 given

Some remarks are in order:

1. We have found a RA, but its preferences are not the ones of the individual agent.

Note that we have Ht instead of Nt, and note that we have a new preference shifter

Xt. First reason why this is not Gorman aggregation.

2. The preference shifter, in general, depends on the distribution of shocks and en-

dowments, therefore aggregate dynamics do depend on the distribution. Second

reason why this is not Gorman’s aggregation. Note, however, that the distribution

is exogenous: it’s a simple problem.

3. Suppose γ = σ. Then utility is quasi-homothetic. If there are no skill shocks, but

only differences in endowments, thenXt = 1 andHt = Nt.We are back to Gorman’s

aggregation. If there are idiosyncratic shocks, then Xt 6= 1 and Gorman aggregationfails, which establishes that you need heterogeneity to be only in wealth for Gorman

aggregation to hold.

4. Suppose γ 6= σ. Then utility is not quasi-homothetic. Even if there are no skill

shocks, but only differences in endowments, then Xt depends on the distribution of

endowments and Gorman’s aggregation fails.

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,

as we will argue later in the course. It is mainly a useful theoretical benchmark.

21

2.4 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

[1] Caselli, Francesco and Jaume Ventura (2000), “A representative Consumer Theory of

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

Competitive Economy", Metroeconomica

22

[7] Samuelson, Paul (1956), “Social Indifference Curves,”Quarterly Journal of Economics

[8] Stiglitz, Joseph (1969), “Distribution of Income and Wealth among Individuals”,

Econometrica.

23