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9 8 1 4
Earnings and Wealth Inequality
and Income Taxation: Quantifying
the Trade-offs of Switching to a
Proportional Income Tax in the U.S.
by Ana Castañeda, Javier Díaz-Giménez
and José-Victor Ríos-Rull
FEDERAL RESERVE BANK
OF CLEVELAND
Working Paper 9814
EARNINGS AND WEALTH INEQUALITY AND INCOME TAXATION:
Quantifying the Trade-Offs of Switching to a Proportional Income Tax in the U.S.
by Ana Castañeda, Javier Díaz-Giménez and José-Victor Ríos-Rull
Ana Castañeda is with Intermoney; Javier Díaz-Giménez is at the
Universidad Carlos III de Madrid; and José-Victor Ríos-Rull is at
the University of Pennsylvania. The authors thank seminar
participants at the University of Iowa, the Penn macro lunch group,
The Federal Reserve Banks of Minneapolis and Cleveland
(especially Ed Prescott and Dave Altig), the NBER Summer
Institute, and the 1998 INCAE Conference in Costa Rica. DíazGiménez thanks the DGICYT for grant PB-940378. Ríos-Rull
thanks the National Science Foundation for grant SBR-9309514,
and the University of Pennsylvania Research Foundations.
Working papers of the Federal Reserve Bank of Cleveland are
preliminary materials circulated to stimulate discussion and critical
comment on research in progress. They may not have been subject
to the formal editorial review accorded official Federal Reserve
Bank of Cleveland publications.
The views stated herein are those of the authors and are not
necessarily those of the Federal Reserve Bank of Cleveland or of
the Board of Governors of the Federal Reserve System.
Working papers are now available electronically through the
Cleveland Fed’s home page on the World Wide Web:
http://www.clev.frb.org.
September 1998
EARNINGS AND WEALTH INEQUALITY AND INCOME TAXATION:
Quantifying the Trade-Offs of Switching to a Proportional Income Tax in the U.S.
By Ana Castañeda, Javier Díaz-Giménez and José-Victor Ríos-Rull
This paper quantifies the steady-state aggregate, distributional and mobility effects of switching
the U.S. to a proportional income tax system. As a prerequisite to the analysis, we propose a
theory of earnings and wealth inequality capable of accounting quantitatively for the key
aggregate and inequality facts of the U.S. economy. This theory is based on savings to smooth
uninsured household-specific risk, for dynastic households that also have some life-cycle
characteristics. A suitable calibration of our model economy replicates the U.S. growth facts,
earnings and wealth distributions, the progressivity of the tax system and the size of the U.S.
government. We also solve a similar model economy in which the government levies a
proportional income tax to finance the same flow of government expenditures and public
transfers. Our findings show that in this class of model worlds a switch from the U.S. tax system
to a proportional tax system implies the following trade-offs, i) it increases efficiency as
measured by aggregate output by 4.4%, ii.) it does not increase inequality as measured by the
Gini index of the earnings, and iii.) it increases inequality as measured by the Gini index of the
wealth distribution by 10.4%, and iv.) it changes by little the mobility between the different
earnings and wealth groups.
Earnings and Wealth Inequality and Income Taxation:
Quantifying the Trade-O s of Switching to a Proportional Income
Tax in the U.S.
Ana Casta~neda,1 Javier Daz-Gimenez2 and Jose-Vctor Ros-Rull3
1
2
3
Intermoney
Universidad Carlos III de Madrid
University of Pennsylvania
File: clevmax.tex
Date: September 3, 1998
Status: First citable version.
ABSTRACT
This paper quanti es the steady-state aggregate, distributional and mobility e ects of switching the U.S. to a proportional income tax system. As a prerequisite to the analysis, we
propose a theory of earnings and wealth inequality capable of accounting quantitatively for
the key aggregate and inequality facts of the U.S. economy. This theory is based on savings
to smooth uninsured household-speci c risk, for dynastic households that also have some
life-cycle characteristics. A suitable calibration of our model economy replicates the U.S.
growth facts, earnings and wealth distributions, the progressivity of the tax system and the
size of the U.S. government. We also solve a similar model economy in which the government
levies a proportional income tax to nance the same ow of government expenditures and
public transfers. Our ndings show that in this class of model worlds a switch from the
U.S. tax system to a proportional tax system implies the following trade-o s, i.) it increases
eciency as measured by aggregate output by 4.4%, ii.) it does not increase inequality as
measured by the Gini index of the earnings, and iii.) it increases inequality as measured by
the Gini index of the wealth distribution by 10.4%, and iv.) it changes by little the mobility
between the di erent earnings and wealth groups.
Processed by LATEX at 10 : 06 a:m: of September3; 1998.
Thanks to the comments from participants at seminars in the University of Iowa, the Penn Macro lunch group,
the Federal Reserve Banks of Minneapolis and Cleveland the NBER Summer Institute and the 1998 INCAE
Conference in Costa Rica and, especially, to Dave Altig and Ed Prescott. Daz-Gimenez thanks the DGICYT
for grant PB-940378. Ros-Rull thanks the National Science Foundation for grant SBR-9309514, and the
University of Pennsylvania Research Foundation. Correspondence to Jose-Vctor Ros-Rull, Department of
Economics, 3718 Locust Walk, University of Pennsylvania, Philadelphia, PA 19104, U.S.A., e-mail address:
vr0j@anaga.sas.upenn.edu
1 Introduction
In this paper we ask what are the implications of switching from the current progressive
income tax system to a proportional income tax system. The key consequences of this
change are a reduction of distortions that result from the high marginal income tax rates
and a reduction of the redistributive properties of the tax system. In short, such a change
will bring about more inequality and more eciency. In this paper we measure how much
more.
In order to do this we need a quantitatively satisfactory theory of wealth and income
inequality. In this paper, we provide such a theory. Our theory is based on uninsurable
di erences in earnings possibilities for life-cycle households that care for the wellbeing of their
o -spring and that are subject to a social security system. Our theory does not use di erences
in patience across agents (not even temporary) to account for the large observed di erences
in assets holdings across households. Unlike other studies that generate a distribution of
wealth endogenously using earnings shocks, (Aiyagari (1994), Krusell and Smith (1998),
Krusell and Smith (1997), Casta~neda, Daz-Gimenez, and Ros-Rull (1998), Quadrini (1997),
Huggett (1996), to name a few) we obtain a joint distribution of earnings and wealth that
matches that in the U.S. data very closely even in the presence of the U.S. tax system.
Unlike Hubbard, Skinner, and Zeldes (1995) who have already pointed out the importance
of government policies in shaping the assets decisions of poor households, we use a general
equilibrium model capable of accounting simultaneously for the asset holdings of rich and
poor households. Unlike Krusell and Smith (1998) our households do not face shocks to
their time preference that produce cross-sectional di erences in patience. The key features
that allow us to match the data are the explicit consideration of i.) life cycle features, ii.)
altruistic agents and iii.) a social security system that induces increases in the income of
many people upon retirement.
Our model economy is a heterogenous agent version of the neoclassical growth model
which we calibrate to match the main macro aggregates, the central features of the public
sector and the main inequality properties of the US economy. Once this is done we change
some features of the tax policy and we compute the new steady state. Another important
feature of our model economy is that we explicitly consider leisure. We think that this is
important when trying to measure the distortionary e ects of taxation since taxes distort
the contemporaneous margin between consumption and leisure.
We nd that our model economy generates the inequality in earnings and wealth observed
1
2
See Quadrini and Ros-Rull (1997) for a review of the previous successes and failures of models in
accounting for wealth inequality.
2 Gokhale, Kotliko , Sefton, and Weale (1998) construct an overlapping generation model with unintended
bequests due to incomplete annuatization and mortality risk. They also nd a key role played by social
security in generation wealth inequality, although they only look at the inequality among those households
whose head is around the retirement age.
1
1
in the U.S. We also nd that a switch from the current tax system to one based on a
proportional income tax implies i.) an increase in eciency as measured by aggregate
consumption of 4.1% and by aggregate output of 4.4%, due to an increase in the capital
stock of 11.4% and to an increase in the labor input of .9%, ii.) no increase in inequality
as measured by the Gini index of the earnings, and iii.) an increase in wealth inequality as
measured by the Gini index of the wealth distribution of about 10.4%, which is quite large
given the very high starting value of this index (it now stands at .78) and the fact that it
has an upper bound of 1., and iv.) there is very little change in the mobility between the
di erent earnings and wealth groups.
In our model all di erences in earnings arise from di erences in the realization of a stationary Markovian stochastic process for \wages" and on the decision of how many hours to
work. Agents cannot engage in activities that change the characteristics of this stochastic
process. In particular, we abstract from such things as education acquisition both for the
agent and for other members of his dynasty. We recognize that these issues might be important ingredients to understand the distribution of income and wealth, but we leave them for
future research.
Other researchers have tried to measure the impact of similar tax changes in the context
of di erent general equilibrium models. For example, Altig, Auerbach, Kotliko , Smetters,
and Walliser (1997) use an overlapping generations model with multiple earnings classes.
In their model, a switch to a proportional income tax increases steady state consumption by
6.9%, capital by 7.6% and labor input by 4.8%. Their aggregate ndings di er from ours
mainly in the large response of the labor input to the tax change. Of course, the di erences
between the distributional implications of their model and ours are enormous: in their model
all changes in assets are due to the life-cycle motive which reduces its ability to generate
di erences in inequality.
Section 2 describes the model economies. Section 3 the calibration targets and the details
of how the benchmark model economy with progressive taxation is capable of accounting for
the U.S. earnings and wealth inequality. Section 4 describes how the proportional income
tax model economy compares with the progressive income tax benchmark model economy.
Section 5 concludes.
3
4
With wages we just mean earning opportunities per unit of time and not necessarily a speci c contractual
arrangement.
4 To generate substantial wealth inequality, agents in their model are restricted to leave a certain inheritance to their o -spring that is a function of their earnings' class a la Fullerton and Rogers (1993).
3
2
2 The model economies
The model economies analyzed in this paper are modi ed versions of the stochastic neoclassical growth model with uninsured idiosyncratic risk and no aggregate uncertainty. The
key features of our model economies are the following: i.) they include a large number of
heterogeneous households, ii.) these households face an uninsured, household-speci c shock
to their employment opportunities, iii.) households face a probability of dying and upon
death the household is replaced by another household of the same dynasty. Households are
altruistic towards future members of their dynasty.
5
2.1 The private sector
2.1.1 Population dynamics and information
We assume that at each point in time our model economy is inhabited by a large number,
actually a measure one continuum, of households. Each period some households are born
and some households die. We assume that the measure of the newly-born is the same as the
measure of the deceased and, consequently, the measure of households remains constant.
Agents age stochastically. They are born as adults, this is to say in working age. With
certain constant probability they retire. Upon retirement, each period they can die or stay
retired, again with constant probability. This way of modeling the demographics allows the
introduction of a life-cycle element in a very parsimonious way but it still captures some
features that we deem to be important: namely that the relative age distribution between
working adults and retirees is the the same as in the data, and that periods are relatively
short.
We also assume that each period, each household faces an idiosyncratic random disturbance, that determines its individual employment opportunities.
We use the compact notation st to denote jointly the age and employment opportunity
of an agent. We assume that these disturbances are independent and identically distributed
across households and that they follow a nite state Markov chain with conditional transition
probabilities given by
6
(s0 j s) = P rfst+1 = s0 j st = sg:
(1)
where s; s0 2 S = f1; 2; : : : ; nsg. Finally, we assume that new-born households draw a shock
from the stationary distribution of employment opportunities for the adults that have been
7
Huggett (1993) and Aiyagari (1994) analyze model worlds that are similar to ours. We extend their
model economies in two ways: i.) we include additional dimensions of household heterogeneity and ii.) we
endogenize the choice between labor and leisure.
6 When we calibrate the model economy we choose these probabilities so that they match the average
durations of working age and of retirement.
7 Note that (: s) is the probability of being alive for one more period, which is smaller than one.
5
j
3
alive for more than one period which we denote ,(s).
2.1.2 Employment opportunities.
The household speci c employment process takes values that belong to set s 2 S = f; rg,
where is a J -dimensional vector that describes the earnings opportunities that the houeshold
whose components are described below and r denotes that the agent is retired and incapable
of working. When a household draws shock j we say that the agent is not retired, and we
assume that it receives an endowment of j > 0 eciency labor units per unit of time which
it can either allocate to the aggregate production technology or use as leisure. Variable ,
denotes the invariant distribution of shock s conditional on not being retired, i.e., it denotes
the invariant distribution of employment opportunities.
2.1.3 Preferences
We assume that households only derive utility from their consumption and leisure when they
are alive and that they order their random streams of these goods according to
1
X
t=0
8
<X
t
:s (st+1 jst )u(ct ; ` , lt ; st ) +
t+1
(1 , (:jst))
9
=
V (st+1 ; kt+1 ), (st+1 );
2e
X
st+1
(2)
where u is a continuous and strictly concave utility function; 0 < < 1 is the time-discount
factor; ct 0 is household consumption, ` is the households' endowment of productive time,
and lt is labor. Hence, ` , lt is time allocated by the household to non-market activities,
which we call leisure. The second term in equation (2) describes the utility derived by a
household from its bequests. Parameter measures the households concern for the welfare
of its o -spring, and to avoid a cumbersome representation of the utility of the household
induced by its o -spring, we use the compact notation of V (st ; kt ) to denote the utility
of a newborn with idiosyncratic shock st and with wealth given by kt .
+1
+1
+1
+1
2.1.4 Production possibilities.
We assume that aggregate output, Yt, depends on aggregate capital, Kt , and on the aggregate
labor input, Lt , through a constant returns to scale aggregate production function, Yt =
f (Kt ; Lt ). We also assume that the capital stock depreciates at a constant rate .
2.2 The government sector
We assume that the government in our model economies taxes household income and estates
and that it uses the proceeds of taxation to make real transfers to retired households and to
nance government consumption. We assume that income taxes are described by function
(y ), where y denotes income, that estate taxes are described by function E (k), where k
4
denotes wealth, and that transfers to the retirees are described by !(s) where s denotes the
realization of the household-speci c shock. In our model economies, therefore, a government
policy rule is a speci cation of f (y), E (k), !(s)g and the process on government consumption, G. Since we also assume that the government must balance its budget every period,
these policies must satisfy the following restriction:
Gt +
t
= Tt
(3)
where t and Tt denote, respectively, aggregate transfers and aggregate tax revenues.
Note that under these assumptions in our model economy social security takes the form
of transfer to households with state s = r. This implies that the transfer does not depend
on past contributions of the worker.
8
2.3 Market arrangements
We assume that there are no insurance markets for the household-speci c shock, s. To bu er
their streams of consumption against these shocks, households can accumulate assets in the
form of real capital. Moreover, household capital asset holdings are restricted to belonging
to a nite set K. This restriction can be understood as a form of liquidity constraints.
We also assume that rms rent factors of production from households in competitive spot
markets. Consequently, factor prices are given by the corresponding marginal productivities.
9
10
11
2.3.1 Initial endowment and liquidation of assets
We assume that the model economy households are born with an initial endowment of assets
which they inherit from their parents. When their time comes to die, households do so
overnight, after the current-period labor, consumption, investment, and savings have taken
place. Early in the following morning, their estates are liquidated. A fraction 1 , E of
their assets, if any, is inherited by the deceased agent's o -spring. The remaining assets are
transformed into the current period composite good and are taxed away by the government.
We make this assumption for technical reasons. To discriminate between households according to their
past contributions to a social security system requires the inclusion of a second asset-type state variable in
the individual problem which makes it very costly to solve. The computational costs are already very high
in this paper due to the fact that we solve a very large number of model economies in our quest for the
appropriate calibration.
9 This is the key feature of this class of model worlds. When insurance markets are allowed to operate,
this economy collapses to a standard representative agent model, as long as the right initial conditions hold.
10 Aiyagari (1994) shows that in this class of incomplete market economies, the requirement that debt has
to be repaid imposes a lower bound on the set of assets holdings endogenously.
11 In this class of model economies rms do not play any intertemporal role for two main reasons: rst,
they do not make pro ts and, second, they cannot be used by the households who own them to substitute
for insurance by choosing non-pro t maximizing strategies.
8
5
2.4 Equilibrium
In this paper we consider recursive, i.e. stationary Markov, equilibria only. This equilibrium
concept might exclude some other types of equilibria such as those that include arrangements that implement history dependent allocations such as those described, for instance, in
Atkeson and Lucas (1992) and Atkeson and Lucas (1995). The reason for this assumption is
that in this paper we are interested in the aggregate consequences of a speci c set of market
arrangements, but we do not attempt to account for the reasons that justify the existence
of those markets. Furthermore, in this paper we consider steady states only.
Each period, the economy-wide state is the measure of households, x, de ned over B, an
appropriate family of subsets of fK S g.
12
13
2.4.1 The households decision problem
The household state variable is the pair (k; s) which includes the beginning-of-period capital
stock, k, and the realization of the household-speci c process, s. The dynamic program
solved by a household alive in state (k; s) is the following:
v (k; s) =
max
c0;k0 2K;0l`
u(c; ` , l; s) +
8
9
<X
=
X
0 ; s0 ) (s0 js) + [1 , (:js)]
0 (1 , ); s0 ), (s0 )
v
(
k
V
(
k
E
:
;
s0 2e
s0
(4)
s.t.:
= y , (y) + k(1 , )
y = kr + l(k; s)ws + ! (s)
c + k0
where v denotes the households' value function, and r and w denote the factor prices. Since
the households' decision problem is a nite-state, discounted dynamic program, it can be
shown that an optimal stationary Markov plan that solves the problem always exists.
2.4.2 De nition of equilibrium
A steady state equilibrium for this economy is a household policy, fc(k; s), k0 (k; s), l(k, s)g,
a pair of household value functions v(k; s), and V (k; s), a government policy, f (y), E , !(s),
Moreover, in a recent paper Cole and Kocherlakota (1997) have shown that hidden storage makes the
allocation implied by trades with one asset the one that arises in environment like the one in this paper. In
other words, our market structure seems to be the \right one" under certain observability assumptions.
13 Note that we do not need to keep track of household names since the decisions of households in the same
individual state are always the same.
12
6
Gg,
a stationary measure of households, x, a vector of time invariant prices, (r; w), and a
vector of aggregates, (K; L; T; ) such that:
i.) Factor inputs, tax revenues and total transfers are obtained aggregating over households.
K
=
T
=
Z
K;S
Z
K;S
L=
k dx;
(y ) dx +
Z
K;S
Z
K;S
l(k; s) s dx
(5)
E k0 (k; s) [1 , (: j s)]dx;
=
Z
K;S
! (s) dx
(6)
where household income, y(k; s), is de ned above.
ii.) Given x, K , L, r and w, the household policy solves the households' decision problem
described in (4), and factor prices are factor marginal productivities:
r = f1 (K; L) + (1 , ) and w = f2 (K; L) :
(7)
iii.) The utility of a newborn, V (k; s), is the same as that of an older agent, v(k; s).
V (k; s) = v (k; s):
(8)
iv.) The goods market clears:
Z
K;S
(c(k; s) + k0(k; s)) dx + G f (K; L) + (1 , )K
(9)
v.) The government budget constraint is satis ed:
G+
=T
(10)
vi.) The measure of households is stationary
Z
Z
K0 ;S0
x(K0 ; S0 ) =
K;S
k =k (k;s)
0
0
(s0 js) +
k0 =(1, E )k0 (k;s)
(1 , (:js)) , (s0) dx dk0ds0
(11)
for all (K ; S ) 2 B, and where denotes the indicator function. Appendix 1 describes the
procedure that we use to compute this equilibrium.
0
0
7
3 Calibration
Recall that our strategy is to calibrate the economy to the current tax system, the current
income and wealth distribution, and the key ratios from the national income and product
accounts.
We start describing the calibration targets. Next we describe the functional forms that
we choose, and the parameterization that implements our calibration targets. We end this
section with a description of the quality of the calibration and a discussion of the key elements
that allow us to be able to account so well for the distribution of earnings and wealth. Our
de nition of earnings include labor income only: it does not include either government
transfers or capital income. The sources for the data and the de nitions of all distributional
variables can be found in Daz-Gimenez, Quadrini, and Ros-Rull (1997).
3.1 Quantitative targets
We want the model economy to satisfy certain characteristics that we describe below.
3.1.1 Model period
Time aggregation matters for the cross sectional distribution of ow variables such as earnings. On the other hand, the shorter the time period the larger the wealth to income ratios
and, therefore, the larger the computational costs. The longest time period that is consistent
with the data collection procedures is a year and it is the one we choose.
3.1.2 Main macroeconomic aggregates
We want our output shares in the model economy to mimic those in the U.S. economy. This
means an investment I to output Y ratio, of 16%, a government expenditures G to output
ratio of 19%, and a transfers T r to output ratio of 9%. Unlike the previous two ratios which
do not seem to have any trend in the post World War II U.S. data, transfers have been
steadily growing. Our chosen value is a little lower than that of 14% that shows in the most
recent years. We follow the logic of the analysis of Krusell and Ros-Rull (1997) in identifying
the size of the transfers. We also impose a value of .376 (see Casta~neda et al. (1998) for
details) to the capital share of output. We summarize these statistics in Table 1.
3.1.3 The distribution of earnings
The model economy Lorenz Curve for earnings mimics the U.S. Lorenz Curve for earnings
as depicted in Table 2.
8
Table 1: The main aggregates of the U.S. economy
Y
C
I
G
Tr
U.S. 1.00 .65 .16 .19 .09
3.1.4 The distribution of wealth
The model economy Lorenz Curve for wealth mimics the U.S. Lorenz Curve for wealth as
depicted in Table 2.
Table 2: The earnings and wealth distributions in the U.S.
Bottom
Quintiles
Top
Earnings
-.40
.00
.00
-.40
3.19
12.49
23.33
61.39
12.38
16.37
14.76
0{1
1{5
5{10
0{20
20{40
40{60
60{80
80{100
90{95
95{99
99{100
Gini Index
.63
Wealth
-.52
-.02
.01
-.39
1.74
5.72
13.43
79.49
12.62
23.95
29.55
.78
3.1.5 The U.S. tax system
The model economy tax system mimics key features of the U.S. tax system. We abstract
from property taxes, consumption and excise taxes and from any di erentiation between
capital and labor income for taxation purposes.
Income taxes: Gouveia and Strauss (1994) have characterized the U.S. e ective house-
hold income tax function for 1989 with the following functional form:
(y ) = a0 (y , (y ,a1 + a2 ),1=a1 )
(12)
9
with parameter values a =.258; a =.768; a =.031. Figure 1 shows the implied average and
marginal tax rates for this function.
0
0.5
0.45
0.4
0.35
0.3
Rates 0.25
0.2
0.15
0.1
0.05
0
1
2
U.S. Average Tax Rate
U.S. Marginal Tax Rate
0
50
100
150 200 250 300 350
Income in Thousands of Dollars
400
450
500
Figure 1: The U.S. in 1989 average and marginal tax functions rates according to Gouveia
and Strauss (1994).
With this tax function the degree of progressivity is a function of the units (scale). To
avoid this problem we compute the value of a that equals the tax rates of average per capita
income in the U.S. and in the model. This solves the problem of di erent units. To deal
with the problem that the U.S. has also tax revenues from other sources, we add a source of
proportional income taxation on top of that reported by Gouveia and Strauss (1994) which
amounts to assume that all non income tax government revenue operates as a proportional
tax on income.
2
Estate Taxes: In the model economy the government levies an estate tax, E . We choose
the model economy estate tax function, E (k), to mimic estate taxes in the U.S. economy
which we report in the rst two columns of Table 3 below.
Table 3: The U.S. estate tax
Capital brackets
Tax Rates
0{$600,000
.000
More than $600,000
.5
10
Table 4: Fraction of households in each quintile that remain in the same quintile 5 years
later in the U.S. between 1984 and 1989.
1st
2nd
3rd
4th
5th
Earnings
Wealth
.86
.41
.47
.46
.66
.67
.47
.45
.50
.71
To mimic the U.S. estate taxes we have to translate those $600,000 to model units. Note
that per household income was in 1990 around fty thousand dollars making the limit on
tax free inheritances about twelve times larger. In the model economy the maximum tax
exempt capital, denote it k, is set to be about twelve times average per household income.
Next, we de ne the estate tax function in the following way:
(
1=2(k , k) for k > k
0
for k < k
Estate taxes are an important issue and we will discuss them some more below.
E (k) =
(13)
3.1.6 Relative cross-sectional variance of hours and consumption
We want to match certain characteristics of the cross-sectional distribution of consumption
and hours. In particular, we impose that the standard deviation of consumption is about
four times that of hours work.
3.1.7 Mobility
We want to match the persistence of earnings and wealth as measured by the transition
probabilities of going from certain quintiles in 1984 to others in 1989. Table 4 shows the
mobility properties for households between 1984 and 1989 for both earnings and wealth. The
table reports the fraction of households out of those in each quintile in 1984 that remained in
the same quintile in 1989. Only those households that were present in both periods are taken
into account when constructing the quintiles. The tables is extracted from Daz-Gimenez
et al. (1997) who used the PSID to perform this calculation.
The key property from this table is that the rst and fth quintiles are the most persistent,
with the rst being the most persistent for earnings and the fth being the most persistent
for wealth.
11
3.2 Functional forms
We now turn to the choice of functional forms for the utility and production functions. In an
abuse of terminology, we include the choice of the length of the period as part of the choice
of functional forms.
3.2.1 Preferences
To characterize the household decision problem described in equation (4), we must choose a
form for the utility function. We use a constant relative risk aversion utility function which
is separable in time and in the consumption-leisure decisions.
The utility function of private consumption for workers, s = (see below) is:
,2
c,1
+ B (` , l)
(14)
u(c; l; ) =
1,
1,
This choice is relatively non-standard. It is due to the fact that large shocks are needed to
mimic the cross-sectional di erences in earnings and wealth. With Cobb-Douglas preferences,
the implied variability in hours worked would have been orders of magnitude larger than what
is observed. For retirees, this utility function collapses to that of CRRA in consumption.
Finally, in this exercise we assume that households value the utility of their o -spring in
the same way that they value their utility. Consequently, = 1.
1
1
1
2
3.2.2 Technology
After World War II in the U.S., the real wage has increased at an approximately constant
rate |at least until 1973| and factor income shares have displayed no trend. To account
for these two properties we choose a Cobb-Douglas aggregate production function
f (Kt ; Lt ) = Kt L1t ,
(15)
with depreciation at a constant rate .
3.2.3 Government
We choose the same e ective household income tax function used by Gouveia and Strauss
(1994) described above. We use as functional form for estate taxes a proportional schedule
that applies to all inheritances above a threshold k^. We assume that all government revenue
comes from these two sources only. To accommodate this feature, we add assume that
the taxation from other sources acts like a proportional income tax that is added to that
described by Gouveia and Strauss (1994).
12
3.3 Parameterization of the benchmark model economy
The set of possible parameters that we are free to choose to match the calibration targets is
very large. Also, the set of statistics to match is very large. Its speci c size depends on how
many statistics we want to match.
We can think of the calibration of this model economy as an exercise of solving a system
with a large number of equations and of unknowns. Unfortunately, this is not as simple as it
seems. In general, only linear systems are guaranteed to have (generically) a unique solution
and our model is not linear. Another problem that arises when attempting to calibrate the
model is that there is a large number of inequality constraints that the parameters have to
satisfy. These include the fact that wages are non negative and that that require that the
matrix is indeed a transition matrix (its elements have to be between zero and one). Non
linear equation solvers typically have problems with these type of constraints.
These considerations lead us to use a minimization procedure to calibrate the economy.
We choose the parameters to minimize a weighted sum of the absolute value of the di erences
between our targets and the properties of the model economy where the weights are chosen
to obtain similar relative di erences. The aforementioned targets are those described in
Section 3.1.
The actual parameters chosen are described in the next few Tables.
3.3.1 The employment process
The normalized endowments of eciency labor units are reported in Table 5 and the transition probabilities of the household speci c process are reported in Table 6. Table 5 also
reports the stationary distribution of working-age households.
Table 5: The normalized calibrated endowments of eciency labor units, (s). Relative
productivity and sizes of the employed groups
s= s= s= s=
s=
1
2
3
4
5
(s)
1.
2.92
7.30
35.05 1810.22
,
.236
.451
.279
.034
.00022
These two tables contain 14 independent parameters: the 4 endowments and 10 transition
probabilities. Some transition probabilities are set to zero because the search algorithm
attempted to set to make them negative.
13
Table 6: The transition probabilities for the household-speci c process.
From s
1
.970
.581E-02
.202E-02
.518E-01
.101
.0
1
2
3
4
5
r
2
0.121E-06
.962
.134E-01
.00
.00
.00
To s0 (in %)
3
.819E-02
.100E-01
.953
.278E-01
.571E-01
.00
4
5
.00
.00
.970E-02
.897
.335E-05
.00
.00
.00
.672E-07
.103E-02
.820
.00
r
.222E-01
.222E-01
.222E-01
.222E-01
.222E-01
.944
3.3.2 Preferences, technology and government parameters
Table 7 shows the parameter values that we have chosen for preferences, for technology and
for the government. That same table also reports the average income tax rate, and the
marginal estate tax rate.
Table 7: The benchmark model economy: calibrated preferences and technology parameters
(in yearly terms)
Preferences
Technology
Government
Parameter Value Parameter Value Parameter Value
.89
.092 y
.328
1.50
.376 E
.500
5.50
a
.258
1.70
a
.768
1.00
a
.424
.10
a
.201
1
2
0
1
2
3
3.4 Calibration results
In this section we discuss the results of the calibration exercise.
3.4.1 The aggregate properties of the model economy
Table 8 shows the values of key macroeconomic aggregates of the U.S. economy and of the
benchmark model economy. As it can be noticed, the values of the aggregates in the U.S.
14
and in the model economy are almost exactly identical.
Table 8: The main aggregates of the U.S. economy and the benchmark model economy
Y
C
I
G
Tr
U.S.
1.00 .65 .16 .19 .09
Benchmark 1.00 .64 .17 .19 .09
3.4.2 The distributional properties
In Table 9 we report the key statistics of the earnings distribution in both the the the U.S
(already reported in Table 2) and in the benchmark model economy.
Table 9: The earnings distributions in the U.S. and in the benchmark model economy
U.S.
Model
0{1
-.40
.00
Bottom
1{5
.00
.00
5{10
.00
.00
0{20
-.40
.00
20{40
3.19
3.27
Quintiles 40{60
12.49
14.35
60{80
23.33
20.73
80{100 61.39
61.65
90{95
12.38
10.25
Top
95{99
16.37
16.51
99{100 14.76
15.01
Gini Index
.63
.615
They include eleven points in the Lorenz curve (including the quintiles) and the Gini
index. Note that the distributions of earnings are very similar in both economies. The very
mild di erences arise in the third and fourth quintiles: the share of the third is higher in
the U.S. than in the model economy while the reverse holds for the share of earnings of the
fourth quintile. Also, there is a slightly higher share in the model economy than in the U.S.
economy of those in the 80-90 percentiles. This happens at the expense of those in the 90-95
percentiles for whom the opposite relation holds.
15
As far as the wealth distribution is concerned, Table 10 reports the key statistics of the
wealth distribution in both the U.S and in the benchmark model economy. These statistics
are the same as those in Table 2. The only di erence is that the fourth quintile has a
larger share in the model economy than in the U.S. at the expense of the second and third
quintiles. Overall though, we are able to replicate the very large concentration found in the
U.S. distribution of wealth.
Table 10: The wealth distributions in the U.S. and in the benchmark model economy
Bottom
Quintiles
Top
0{1
1{5
5{10
0{20
20{40
40{60
60{80
80{100
90{95
95{99
99{100
Gini Index
U.S.
-.52
-.02
.01
-.39
1.74
5.72
13.43
79.49
12.62
23.95
29.55
Model
.00
.00
.00
.01
.36
3.78
16.60
79.26
11.67
22.81
29.54
.78
.79
3.4.3 Government tax revenues
Figure 2 compares average tax rates in the model economy after normalization of the units
with those reported by Gouveia and Strauss (1994). We can see that the amount of progressivity as indicated by the tax schedule is very close between the U.S. and the benchmark
model economy.
3.4.4 The mobility properties
Table 11 reports the fractions of households of each quintile that remain in the same earnings
and wealth quintiles ve years later. We believe that these statistics capture the most
important features of mobility. Of particular importance are the fractions of households
that remain in the top and bottom quintile.
We see that the performance of the model economy is not as good with respect to mobility
as it was with respect to the other properties that we are interested in. Still, relative large
persistence is found for all quintiles. As in the data, the rst and the fth quintiles are the
16
0.5
0.45
0.4
0.35
0.3
Rates 0.25
0.2
0.15
0.1
0.05
0
U.S. Average Tax Rate
Model Economy Total Tax Rate
Model Economy Income only Tax Rate
0
50
100
150 200 250 300 350
Income in Thousands of Dollars
400
450
500
Figure 2: The U.S. and the model economy average tax functions with and without other
sources of government revenue.
most persistent for both earnings and wealth. With respect to earnings both in the data and
in the model we see that the rst quintile is more persistent than the fth quintile.
However, when we look in more detail, we start seeing di erences between the model
economy and the data. For example, with respect to earnings, the model economy shows too
much persistence in the fth quintile. Also, with respect to wealth the benchmark model
economy is much more persistent than the U.S. data and the relative size of the persistence
between the rst and fth quintiles is reversed: in the model the rst quintile is more
persistent than in the data while the opposite happens with respect to the fth quintile.
The model used in this paper has some life-cycle properties (there is retirement) but
these properties are very stylized. Mobility as recorded by the PSID has a strong life-cycle
component (the age earnings pro le has a very clear inverse U shape, see Auerbach and
Kotliko (1987) or Ros-Rull (1996) for example). Consequently, we should not expect the
model economy to have mobility properties that are very close to those in the data. given
the excessively parsimonious characterization that we have assumed for the life-cycle. We
conjecture that versions of this model that include a more detailed speci cation of the lifecycle will be able to generate mobility statistics that mimic closer those found in the data.
3.4.5 A discussion
Overall, we think that the calibration of the model economy has been very successful. Unlike
previous models, we succeed in replicating the di erent degrees of concentration of earnings
and wealth found in the U.S. economy. There are still a couple of properties of the model
17
Table 11: Mobility in the U.S. and in the benchmark model economy: fraction of households
that remain in the same quintile 5 years later
Earnings
1st
2nd
3rd
4th
5th
Wealth
U.S. Benchmark
Model
.86
.88
.41
.68
.47
.64
.46
.75
.66
.81
U.S. Benchmark
Model
.67
.96
.47
.93
.45
.82
.50
.75
.71
.88
that should be improved. The most important of these relates to the relative share of wealth
of the third and fourth quintiles. We have explored alternative calibrations where we put a
lot more emphasis in replicating the relative share of wealth of the third and fourth quintiles
at the expense of a worst match of other statistics. We found that for these alternative
model economies the ndings are very similar.
To better study mobility issues we think that we should move towards models that have
a much more sophisticated implementation of the age of households. Those models will be
able to simultaneously include mobility due to the life-cycle and to other motives such as
the shocks that we have assumed in this paper.
Finally, there is an issue with the estate tax. Given its current parameterization, the
model economy assumes that there are no ways to avoid inheritance taxes, while we know
that the existence of trusts funds and inter vivos transfers reduces the amount that the
government collects from this source. On the other hand, the model generates a wealth
to output of slightly less than 2., too low relative to the one observed in the U.S. that
is somewhere between 2.5 and 3. We conjecture that versions of the model with a less
unavoidable estate tax are able to increase the wealth to output ratio without changing the
other properties of the model economy.
4 The policy experiments
Once we have a satisfactory implementation of a model economy calibrated to the U.S. tax
system, we proceed to compute the steady state of an alternative economy that has the
following properties:
1. Preferences and technology (including the properties for the employment process) are
identical to those in the benchmark model economy.
18
2. The size of government (expenditures and transfers), in absolute not relative terms, is
identical to the benchmark model.
3. The tax system is di erent. In this model economy the government levies a proportional
tax on income. The estate taxes are unchanged at about 12 times average household
income.
Once we have computed the equilibrium with progressive taxation, computing the one
with proportional taxation is simple. It amounts to solving a system of four equations and
four unknowns. The four equations are the equilibrium condition for the capital labor ratio,
the two conditions on the size of government, and the units condition that determines the
units for the estate tax. The four unknowns are the capital labor ratio, the proportional tax
rate, and the guesses for aggregate wealth and aggregate output.
4.1 The main aggregates in the model economies
Table 12 shows the main aggregates for the benchmark model economy and for the model
economy with proportional income taxation. The table contains the macroeconomic aggreTable 12: The steady state aggregates of the benchmark and the proportional tax model
economies
Y
C
I
G
Tr
Benchmark Proportional
Model
Model % Change
1.000
1.044
4.4%
.635
.661
4.1%
.166
.185
11.4%
.189
.189
.0%
.087
.087
.0%
1.79
1.00
1.00
1.79
32.80
2.627
.639
K
Labor Input
Total Hours
K=Y
Government Revenues / Y
c =c
l =l
1.97
1.009
1.016
1.89
31.63
3.535
.641
11.4%
.9%
1.6%
5.6%
-3.7%
34.5%
.3%
gates used to calibrate the benchmark model economy and also total capital, the labor input,
total hours worked in the market, the wealth to output ratio, the fraction of government
revenues over GDP, and the coecients of variation of both consumption and hours worked.
19
We nd that switching to a proportional tax system increases the steady state level of
output by 4.4%. This is due to a much higher steady state level of capital (11.4% more)
and to a higher work e ort that translates into a slight (less than 1%) increase in the labor
input. Obviously, the di erence in the levels of output between the two economies will not
occur upon a switch in policy. They will happen many periods after the policy change once
the transition to the new steady state has been completed (assuming that the new steady
state is stable, a conjecture that cannot be veri ed with the tools at our disposal).
The increase in consumption after the shift to the proportional tax economy is 4.1%,
slightly less than that of output. The rest of the extra output goes to increase investment,
needed since the steady state stock of capital is much larger under proportional income taxation. Note that, by construction, there is no change in the level of government consumption
and, therefore, its share of output is smaller in the economy with proportional income taxes.
Note that given our assumptions about the behavior of the government, implies that public
expenditures and transfers do not change in the proportional income tax model economy.
Note that there is a sizable increase in the stock of capital that is largely responsible
for the increase of steady-state output. This is due to the very small increase of the labor
input which is less than 1.%. As a consequence the wealth to output ratio is higher in the
proportional income taxation, about 6% higher. Total hours worked in the market, on the
other hand, shows a larger increase than the labor input. This can only happen if the increase
in hours worked is done mostly by those with low labor eciency. At rst sight this seems a
contradiction with a switch to proportional income taxation that should reduce the distortion
against work e ort more to high income households than to low income households. However,
the general equilibrium e ects turn out to be strong. There is a positive correlation between
wealth and the best wage shocks. The higher total wealth (and as we will see the fact that
it is more concentrated) induces a strong positive wealth e ect of high wage households that
partly counteracts the substitution e ect arising from the switch to proportional income
taxation.
The total reduction in the size of government as a fraction of output that arises is 3.7.
This is smaller than the increase in output and is due to the fact that taxes are levied on
Net National Product and not on Gross National Product and the increase in the former is
smaller than in the latter due, again, to the small impact on total labor input.
Finally, we see an enormous increase, 35%, of the cross sectional coecient of variation
of household consumption. The hours worked counterpart shows, on the other hand, shows
almost no change. This is a direct implication of the relatively small response of work e ort
to changes in the environment (recall the much larger curvature imposed on the leisure part
of the instantaneous utility function relative to that of consumption) that was imposed in
the calibration stage to obtain a much larger cross sectional variation of consumption than
of hours worked.
20
4.2 Inequality in the model economies
Table 13 reports the key statistics of the distribution of earnings for both model economies.
The central feature of this table is that it seems that there is no change in inequality of
Table 13: The Earnings Distributions in the Benchmark Model Economy and in the Proportional Taxation Model Economy.
Bottom
Quintiles
Top
0{1
1{5
5{10
0{20
20{40
40{60
60{80
80{100
90{95
95{99
99{100
Benchmark
Model
.00
.00
.00
.00
3.27
14.35
20.73
61.65
10.25
16.51
15.01
Proportional
Model
.00
.00
.00
.00
3.31
14.44
20.87
61.38
10.24
16.38
14.80
.615
.613
Gini Index
earnings at all. In other words, the switch to a proportional taxation even though increases
overall work e ort by slightly less than 1.%, it does so in a manner that does not seem to
a ect inequality. This feature is consistent with the very small increase in both the labor
input and total hours that is associated to the proportional income tax model economy.
This means that most agents respond in a similar way to the new income tax. As another
sign of this property of the model economies we can compare the cross sectional standard
deviation of hours. It goes up by :3% in the proportional income tax model economy, an
almost negligible amount. Again we see that the e ect of the change of taxation is very
small and it is also similar for all agents. For the low wage agents there is a mild increase in
the income tax, (substitution e ect going decreasing work e ort) that is counterweighted by
lower asset holdings (wealth e ect increasing work e ort). For the high wage guys (which
are also wealthier), the opposite holds true. Overall, there is a mild increase in work e ort
that has almost no implications for earnings inequality.
Table 14 reports the key statistics of the distribution of wealth for both model economies.
We nd that inequality has increased dramatically. In the benchmark model economy the
third quintile owns a non-negligible amount of wealth, while this is no longer the case in
21
Table 14: The wealth distributions in the benchmark model economy taxation and in the
model economy with proportional taxes
Bottom
Quintiles
Top
0{1
1{5
5{10
0{20
20{40
40{60
60{80
80{100
90{95
95{99
99{100
Benchmark
Model
.00
.00
.00
.01
.36
3.78
16.60
79.26
11.67
22.81
29.54
Proportional
Model
.00
.00
.00
.00
.00
.11
10.44
89.46
11.86
25.77
39.09
.791
.874
Gini Index
the model economy with proportional taxation, where the third quintile owns almost no
wealth. In fact, under proportional taxation, the bottom 60% of asset holders have about 1
per thousand of total wealth. We also nd that under proportional taxes only about 10% of
the population increases their share of wealth, and this group is the group with the highest
wealth. This does not mean that the remaining 90% of the population own less assets, it
means that their share of wealth is smaller. Nevertheless, many households do in fact reduce
their asset holdings (again this is across steady states). This is because the equilibrium
interest rate is smaller and because for some of the low income households the marginal
tax rate has increased. The increase in the Gini Index is enormous, a staggering 10.5%,
especially if we realize that the maximum inequality where one household holds everything
would have implied an increase of 26%.
Given this large increase in wealth inequality and the fact that there is a positive correlation between employment opportunities and wealth it is easy to understand why there
is no increase in earnings inequality. The income and wealth e ects cancel each other for
the labor choice. High wage people face lower marginal income tax rates but are wealthier
inducing a very small e ect in their willingness to work.
In the model economies one can easily calculate many distribution statistics. As an ex14
It should be understood that we refer to the maximum Gini index that can be achieved with non-negative
values of the variable of interest.
14
22
ample, Table 15 shows the key statistics of the distribution of consumption across groups. As
Table 15: The distribution of consumption in the benchmark model economy and in the
model economy with proportional taxation
0{1
Bottom 1{5
5{10
0{20
20{40
Quintiles 40{60
60{80
80{100
90{95
Top
95{99
99{100
Benchmark
Model
.23
.94
1.32
6.10
9.41
13.66
19.89
50.94
9.11
13.45
12.11
Proportional Proportional Model
Model
Levels
.22
.23
.86
.90
1.08
1.13
5.16
5.38
7.60
7.93
12.95
13.51
18.76
19.58
55.54
57.96
9.19
9.59
14.82
15.46
15.87
16.56
with the distribution of wealth, proportional taxes only bring about an increase in the share
of consumption for the top 10% of households. A very small number of households experience an increase in their consumption levels. This can be seen by comparing the benchmark
model in the table with the last column, which shows the distribution of consumption of
the proportional income tax model economy measured in the same units as those used for
the benchmark model economy (they are not shares). We see that, besides a tiny increase
shown by the lowest one per cent, only those in the top 10% (plus a few others in the top
10 to 20% group) experience an actual increase in consumption. This nding has important implications. A switch to proportional taxation might reduce the steady state levels of
consumption for a substantial majority of the population.
Still, we cannot, and will not make any normative statement about the desirability of
either tax system. Note that we are comparing steady states, and to compare the welfare
of di erent policies we should take into account the transition paths. The computation of
the transition is a technically very demanding endeavor for relatively simple models (see for
example Auerbach and Kotliko (1987) or Ros-Rull (1994)). For the class of models in this
paper, this type of transition paths exercises are still beyond our computational ability.
4.3 Taxation in the model economies
Figure 3 shows the marginal tax rates of the two economies. Note that mean household
income is around $50; 000 and that the graph also shows for comparison the fraction of
23
income that the government collects in the benchmark model economy. Note that despise
the increase in total output, many households (low income ones) face an increase in taxation
in the new regime.
Benchmark Model Economy Marginal Income Tax Rate
Benchmark Model Economy Government Revenue Rate
Proportional Model Economy Tax Rate
0.5
0.4
Rates
0.3
0.2
0.1
0
0
50
100
150 200 250 300 350
Income in Thousands of Dollars
400
450
500
Figure 3: The benchmark model economy and the proportional income tax model economy
marginal income tax rates.
4.4 Mobility in the model economies
Table 16 shows the fraction of households in each quintile that remain in the same quintile
ve years later. As discussed above, this is a dimension of the benchmark model that we
do not stress. However, we nd that the switch to a proportional taxation system does not
change mobility by much. The only instance in which there is some change is in the fraction
of households that are in the rst earnings quintile in one year and remain in that quintile ve
years later. The table shoes a substantial decrease after a switch towards proportional income
taxation. We believe, however, that this result is not very indicative. The rst households
that have some positive earnings are for both economies not far from the 20% per mark, and
their earnings are minuscule since their hours worked are very tiny. Movements in and out of
the rst quintile are very sensitive to these small changes. As evidence that there is no much
change in the mobility properties, just note that the fraction of households that remain in
the lowest 40% of the model economies is .8366 for the benchmark model economy and .8358
for the proportional income taxation model economy. To summarize, we nd that mobility
is basically una ected by the tax change. We also want to recall that this model was not
designed with the purpose of accounting well for mobility so this ndings, even though they
are important, they are less important than those pertaining macroeconomic aggregates and
24
earnings and wealth inequality.
Table 16: Fraction of households in each quintile that remain in the same quintile 5 years
later in the benchmark model economy and in the model economy with proportional taxation
Earnings
1st
2nd
3rd
4th
5th
Wealth
Benchmark Proportional
Model
Model
.88
.70
.66
.64
.64
.64
.75
.75
.81
.81
Benchmark Proportional
Model
Model
.96
.98
.93
.93
.82
.83
.74
.78
.88
.88
5 Concluding comments
This paper provides a theory of earnings and wealth inequality capable of accounting quantitatively for the inequality observed in the U.S. The key features of our ability to match the
data are the explicit consideration of i.) life cycle features, ii.) altruistic agents and iii.) a
social security system that induces increases in the income of many people upon retirement.
We use the resulting quantitative theory of inequality to measure the e ects from switching from the current, progressive, tax system, to one where income is proportionally taxed.
We compare the steady states of two economies that only di er in the tax system. We nd
that the policy switch implies some important changes. On the one hand, output, wealth
and, to a very little extent, work e ort are higher in the economy with proportional income taxation. On the other hand, in the economy with proportional income taxation, even
though inequality in earnings remains basically unchanged, both wealth and consumption
are much more unequally distributed.
25
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27
Appendix
A Computation
This appendix describes the algorithm that we have used to compute the equilibrium allocations
of the economy. The outline of the algorithm used is the following:
We use a standard non-linear equation solver (speci cally a modi cation of Powell hybrid
method, subroutine DNSQ from the SLATEC package). We use it to search for a zero of a system
of 24 equations and 24 unknowns. The equations include the steady state equilibrium condition as
well as the di erences between the model's statistics and the desired targets. The algorithm typically wants to choose values of the parameters that are not permissible such as negative elements
of the transition matrix . This means that the algorithm e ectively worked as a minimization
routine where the objective is to minimize a weighted sum of the deviations from the targets.
For each speci cation of the parameters computing equilibrium involves three di erent but
important steps.
Step 1: Solve for the decision rules of the agents. This we do by a piecewise linear decision rule
method. The grid is not equally spaced. Given the very large range of possible asset holdings that
we need to achieve the observed wealth concentration, we use a very large number of points. About
1000 grid points per realization of the shock, for a total of 6,000 grid points. At every iteration
in every gridpoint a non-linear equation has to be solved. Monotonicity of this equation and the
assumed piecewise linearity of the decision makes it easy. However, the model has leisure which
adds one extra level of complexity. For every evaluation of the Euler equation another non-linear
equation has to be solved, the contemporaneous rst order condition. Occasionally this rst order
condition does not have a solution, this is the case only when the feasibility constraint on the upper
bound of leisure binds. In this case work e ort is set to zero. See Ros-Rull (1997) for details.
Step 2: Associated to the decision rules there is a Markov process for each agents which satis es
the necessary conditions (see Aiyagari (1994) or Huggett (1995)) for existence of a unique stationary distribution x . We approximate x with the aid of a piecewise linear approximation to its
associated distribution function. Again, the grid for this approximation has many points (about
12,000) and they are particularly close to each other towards the origin. Again, see Ros-Rull (1997)
for details.
Step 3: Compute the statistics (distributional and aggregate) of the model economy. This step
e ectively involves the computation of integrals with respect to the invariant measure x . We
evaluate these integrals directly using our approximation to the distribution function for every
statistic except those that measure mobility. In this latter case we computed the statistics using a
large sample of agents drawn from x . Again see Ros-Rull (1997) for details.
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@FedFRASER