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\begin{titlepage}
\title{
Are Consumption Taxes Really Better Than Income Taxes?
}
\author{Per Krusell, Vincenzo Quadrini, and Jos\'e-V\'\i ctor R\'\i
os-Rull\thanks{JEL Classification: E60, E61, H10. Key words: political
equilibrium, consumption taxes, income taxes. Krusell,
University of Rochester, Institute for International Economic Studies,
and CEPR; Quadrini, University of Pennsylvania and Universitat Pompeu
Fabra; R\'{\i}os-Rull Federal Reserve Bank of Minneapolis and
University of Pennsylvania. Send correspondence to: Research
Department, Federal Reserve Bank of Minneapolis, P.O. Box 291
Minneapolis, Mn 55480-0291, Ph. (612) 335-2888, Fax (612) 340-2366,
e-mail {\tt vr0j@tom.mpls.frb.fed.us}. We thank Stephen Coate, Ellen
McGrattan, Torsten Persson, and participants at seminars/conferences
at the Federal Reserve Bank of Minneapolis, the Federal Reserve Bank
of Richmond, the Institute for International Economic Studies, the
University of Pennsylvania, the University of Rochester, the
University of Toronto, Universidad de Alicante, and Universidad Carlos
III for helpful comments. Krusell and R\'{\i}os-Rull thank the
National Science Foundation for financial support. The views
expressed herein are those of the authors and not necessarily those of
the Federal Reserve Bank of Minneapolis or the Federal Reserve System.
}}
\maketitle
\singlespace
\begin{abstract}
{ We use political-equilibrium theory and the neoclassical growth
model to compare consumption and income tax systems. If government
outlays are used for redistribution through transfers, then
steady-state equilibria in societies that use income taxes are not
necessarily worse in welfare terms, and may even be better. Income
taxes are attractive precisely because they are more distortionary,
since this implies low equilibrium transfer levels. We also find that
switching tax systems typically does not benefit the median voter;
moreover, a change from income to consumption taxes may make everybody
worse off.
% The distortionary effects of taxation are
%taken into account when choosing the level of taxation, resulting in
%lower tax rates in relatively distortionary tax systems. When both
%consumption and income taxes are used and voted on simultaneously,
%more redistribution occurs because there are less distortions than in
%one-tax systems.
}
\end{abstract}
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\end{titlepage}
\newpage
\onehalfspace
%\singlespace
%\newpage
\section{Introduction}
Conventional public finance wisdom argues in favor of consumption
taxes over income taxes.\footnote{See \citeA{Atkinson-Stiglitz-80} and
\citeA{Stiglitz-86} for a summary. \citeA{Summers-81},
\citeA{Auerbach-Kotlikoff-Skinner-83} and
\citeA{Jones-Manuelli-Rossi-93} are examples of positions favorable
to consumption taxes, while \citeA{Browning-Burbidge-90} give a
more agnostic view.} At the same time, most industrialized economies
rely much more on income taxes than on consumption taxes. This
is illustrated in Table \ref{tdsht}, where we tabulate the shares of
government revenue due to consumption, income and social security
taxes (we consider the latter as a form of income taxation) for the
OECD countries. Is the reason for this discrepancy that the economic
models studied in the literature are poor representatives of the
economic reality? Or, is it that taxes are not chosen based on
economic efficiency arguments alone?
We are inclined to believe that
the second of these explanations is the most accurate one.\footnote{It
should be pointed out that the recent results in \citeA{Aiyagari-93b}
are suggestive of an alternative way of rationalizing the absence of
large consumption taxes in the data. He argues that there are
empirically plausible economic environments which call for permanent
positive taxes on capital income.} In particular, political decisions
over taxes invariably have distributional consequences, and as long as
this is the case we should expect outcomes to not only reflect
economic efficiency arguments. Although this view suggests that our
role as policy advisers is limited, it does not make economic analysis
meaningless. What it does suggest, however, is that it may be
important to analyze how economics interacts with the political
process. Accordingly, we study properties of different tax systems in a
context where actual tax levels are not chosen by us, but by the
agents who inhabit our model economies.
Different tax systems imply different types and amounts of
distortions. The voting agents face a trade-off between the amount of
redistribution and the costs associated with it, and it is not obvious
what the resolution of this trade-off is. It might be, as
\citeA{Brennan-Buchanan-77} have suggested for similar
reasons, that more distortionary taxes give better outcomes because
they restrict the amount of government activity undertaken. But it
might also be that with the proper policy instruments (tax systems),
substantial redistribution can be accomplished with little distortion.
In this paper we resolve this tradeoff by making quantitative
comparisons between different tax systems.
In dynamic economies, economic policies are not determined once and
for all but they are continuously changed, or at least allowed to
change. Political parties alternate, and once in charge they implement
policies that can be changed by the party winning the following
election. Therefore, a realistic analysis of fiscal policy has to
consider the political process determining this policy.
Unfortunately, the introduction of an endogenous political mechanism
increases the complexity of the model substantially. There are two
reasons for this. First, it is necessary to study heterogeneous-agent
economies, which increases the state space. In our present model, we
keep this complication minimal by studying an economy with two types
of infinitely-lived agents. Second, the political-equilibrium
analysis requires that we derive each agent's preferences over
policies. This derivation involves a complex forecasting problem: it is
necessary for an agent contemplating different current policy options
to think not only about their respective effect on current prices and
transfers, but on future prices, transfers, and politically determined
policies as well. Here we adopt the rational-expectations methodology:
agents think through all the equilibrium consequences of the different
policies, without making errors in calculation.\footnote{See
\citeA{Muth-61}.} Note, however, that in a sense we are requiring
``more'' rationality from our agents than what needed to be required
from the agents in the original contributions by Lucas and Prescott
(\citeA{Lucas-72},
\citeA{Lucas-Prescott-71}): here, agents also have to correctly
predict what {\em would\/} happen under circumstances which will never
be realized.
The alterations to standard public finance which we consider do not
come for free, of course. For example, we need to specify the fiscal
framework (e.g., what taxes are available) and the political framework
(e.g., what taxes are voted on) and these modeling choices necessarily
introduce some arbitrariness. We confront the taxation problem by
comparing tax {\em systems}, or {\em constitutions}, i.e.\ we
postulate what set of taxes are used and what set of taxes are voted
upon for each economy under study. We first focus on comparing
economies with only one kind of tax; e.g., we compare an economy where
there is only a consumption tax to an economy with only an income tax.
We then extend the analysis to a case in which two taxes are voted
upon simultaneously. We also assume that all constitutions demand
proportional taxation, and that government debt is not allowed as a
policy instrument. Although it is not obvious what features of policy
should be regarded as constitutional in the theory and not subject to
vote, and what should be voted on---we are mainly guided by
tractability and an appeal to realism---this is one of the main
problems of any politico-economic theorizing which is not based on
pure mechanism design.
Our main findings can be summarized as follows:
\begin{itemize}
\item For the case of steady-state equilibria of one-tax
constitutions, income and consumption tax systems are quite similar in
welfare terms (note however, that these welfare comparisons are not of
direct normative interest). Since income taxes are more distortionary,
an income tax system has a lower steady-state level of taxation, and higher
steady-state output.
\item The comparison between two-tax and one-tax systems generally
favor the two-tax system, although it is true here as well that
there will be more transfers with the less distortionary system,
which in this case is the one with more instruments.
\item The tax constitutions we look at have features that should make
them {\bf persist}: if a switch to an alternative constitution is
contemplated, then there will almost always be losers. In particular,
we find in almost all of our examples that the median agents (who are
poorer in our economies) do worse after a change of tax systems, and
that a switch from income to consumption taxes never makes both agents
better off. Thus, if we think of constitutional change as requiring
(at least) a majority, then most of our constitutions will be hard to
overturn.
\item When the purpose of taxation is the provision of public goods or
services and there is no element of pure income transfers between
groups, different results may apply: then, tax systems
based on less distortionary types of taxes can be better.
\end{itemize}
Several recent studies do make the political mechanism endogenous, and
the result which is common among these studies is that the
distribution of agents over income and wealth can be an important
factor determining economic policies, and therefore economic
outcomes.\footnote{Examples of papers in this literature are
\citeA{Alesina-Rodrik-94}, \citeA{Bertola-93},
\citeA{Meltzer-Richard-81}, \citeA{Fernandez-Rogerson-94b},
\citeA{Glomm-Ravikumar-92}, %\citeA{Huffman-93},
\citeA{Kristov-Lindert-McClelland-92}, \citeA{vested},
\citeA{constitutions}, \citeA{jedc}, \citeA{Perotti-93},
\citeA{Persson-Tabellini-94}, and
\citeA{Saint-Paul-Verdier-91,Saint-Paul-Verdier-92}.} In
the context of a standard growth model, however, no attempt has
been made to analyze the equilibrium allocations associated with
different types of taxation when the level of taxes is endogenously
determined. The main reason for this is that in order to analyze the
effect of different tax systems, a richer and more complex model is
required than what is typically studied in this literature. For
example, most of the existing models (i) have limited dynamics, i.e.\
they are considerably simpler than the standard neoclassical growth
model; and/or (ii) abstract from the leisure choice. We make use of
the methods developed in \citeA{constitutions} and \citeA{jedc}, which
are straightforward to amend to allow for a leisure choice. These methods
are computational in nature, since analytical solutions cannot be
obtained for this class of economies.
Our work can be compared to the more standard optimal taxation
literature, which in dynamic environments specifically asks what time
path of taxes maximizes some welfare objective.\footnote{For example,
\citeA{Lucas-Stokey-83} analyze optimal fiscal policy in a stochastic
economy without capital, and \citeA{Chamley-86} considers optimal
capital and labor income taxation in a deterministic economy with
capital accumulation. \citeA{Zhu-92} extends the analysis by studying
the Ramsey taxation problem in an economy with both capital
accumulation and uncertainty. \citeA{Lucas-90} reviews the capital
vs.\ labor income taxation results in light of the new growth theory,
and others (e.g.\ \citeA{Jones-Manuelli-Rossi-93} and
\citeA{Rebelo-Stokey-91}) continue the study of how endogenous growth
models differ with respect to tax prescriptions. For an analysis of
optimal capital and labor income taxation over the cycle, see
\citeA{Chari-Christiano-Kehoe-94}.} There are, however, important
differences between the optimal-taxation approach and ours. First,
one of the concerns of the optimal-taxation literature is debt
management, something that we ignore by requiring budget balance.
Second, we are primarily concerned with determining the total size of
government outlays/transfers, which is typically taken as given in the
literature on optimal taxation. Third and finally, whereas the
optimal-taxation studies are often not concerned with distributional
issues, they are in focus in our study.
Our paper starts with a description of the model in Section
\ref{model}. The analysis starts in Section \ref{some}, where we make
a preliminary characterization of the set of steady states. In Section
\ref{calibration} we describe how we choose our parameter values, and
our numerical analysis proceeds in Section \ref{quantitative}. In this
section, the baseline model is studied: there, all government revenue
is used for transfers. We first look at the properties of the steady
states of different tax systems, and we then study the welfare
properties of tax systems by looking to the transition paths that
follow a change of tax systems. In Section \ref{2tax} we extend our
analysis to economies with two taxes. We look at the provision of
public goods in the following section, Section \ref{public}.
%The last part of the paper, Section \ref{properties}, briefly investigates
%some properties of tax data from the OECD economies. Recall that the
%quantitative implications from our political-economy approach suggest
%that we should be able to correlate economic performance with the
%extent to which consumption taxes are used for raising revenue. We do
%find preliminary evidence in support of the theory: countries which
%use consumption taxes more intensely do have higher tax levels, and
%lower output. Section \ref{conclusions} concludes the paper.
\section{The Model}
\label{model}
The model used is a standard growth model with heterogeneity across
agents in their wealth holdings and/or labor productivity. There is a
continuum of agents of total mass one. Agents are indexed by their
type $i\in{\cal I}\equiv\{1,\cdots,I\}$, with respective fractions
given by $\mu_i$. The per-period utility function is the same for all
types, $u(c,l)={(c^\alpha l^{1-\alpha})^{1-\sigma} - 1\over 1-
\sigma}$, and total utility is the discounted sum of the per-period
utilities, i.e., $\sum_{t=0}^{\infty} \beta^t \,u(c_t,l_t)$.
%A key feature that we introduce into the model is valued leisure. Most
%models that use a political-economy approach with endogenous taxation
%abstract from it, but we consider its inclusion important for three
%reasons: (i) it allows us to explicitly differentiate between labor
%and capital income taxation (without leisure, labor income taxation
%can be costlessly increased without bound); (ii) it has quantitatively
%more interesting implications for the level of taxation; and, as we
%will see, (iii) it is necessary for deriving interesting differences
%between tax systems, since in economies inhabited by agents who do not
%care about leisure there is in a certain sense an equivalence between
%income and consumption taxes.
Agents are endowed with $\epsilon_i$ efficiency units of labor. They
also hold assets in amounts given by $a_i$.\footnote{As we will see,
all agents of the same type behave the same way in equilibrium,
which allows us to abstract from their names. Obviously, all
endowments of efficiency units need not be different; the same is
true for wealth levels.} We use capital letters to denote the assets
held by all individuals of the same type, allowing us to denote the
distribution of wealth with $A=\{A_i\}_{i\in {\cal I}}\in{\cal R}^I$,
and the distribution of efficiency units of labor with
$\epsilon=\{{\cal E}_i\}_{i\in {\cal I}}\in{\cal R}^I$. Efficiency
units of labor provided to the market, i.e.\ the labor input, which
are given by $N_t=\sum_i \mu_i \epsilon_i (1-l_{it})$, combine with
aggregate capital, given by $K=\sum_i \mu_i A_{i}$, to produce
output through an aggregate production function, $F(K,N)$; we use the
notation $f(A,N)=F(K,N)$ to make explicit the dependence of output on
the distribution of non-human wealth. We assume competition in factor
markets which determines the net-of-depreciation rental price of
capital $r$ and the wage per efficiency unit of labor $w$.
In each period the current tax rate is given and people vote on next
period's tax rate. Agents face either of two tax systems: (i) a
proportional tax on consumption only, which since there is equal
per-capita distribution of the proceeds to all agents results in a
period budget constraint given by $c_{i}(1+\tau^c)+ a_{i}'=r
a_{i} + (1-l_{i})\epsilon_i w+a_{i} +tr^c$; (ii) a
proportional tax on total income only, which results in a period
budget constraint given by $c_{i}+ a_{i}'=\left(r a_{i} +
(1-l_{i}) \epsilon_i w\right)(1-\tau^y)+a_{i} +tr^y$, where
$\tau^c$, and $\tau^y$ are the tax rates for consumption and
income and $tr^c$ and $tr^y$ are the respective transfers. Labor
and capital taxation are also considered, and they involve
straightforward adjustments to the above budget constraints.
Interactions between agents every period determine the policy in place
for the following period.\footnote{In \citeA{constitutions} it is
shown how the fact that the policies are chosen every period is
tangential. The key issue is the amount of real time in between
policy changes. This parameter is set at the calibration stage.} The
mechanism determining what policies are chosen is representative of
the political-economy literature based on majority voting. In
economies with only one policy parameter to be determined, a
single-peakedness condition on the derived preferences for this
parameter is sufficient for implying that the median voter will be
decisive. In calibrated versions of the standard neoclassical growth
environment, we found that the single-peakedness condition is indeed
satisfied. Furthermore, for our parameterizations the median agent has
less than mean income, a feature which characterizes the data, and
this agent will want redistribution even at the cost of some
distortions.
The uses of public funds may be important in the study of taxation.
These include: direct cash transfers with and without dead-weight
losses associated with the management of the tax system; public supply
of private goods on an equal per-capita basis; supply of public goods
having direct impact on agents' utility; and supply of public goods
having direct impact on productivity.\footnote{Several studies assume
that the proceeds from taxation are redistributed as lump-sum
transfers. Examples include \citeA{King-Rebelo-90}, which considers
the effect of progressive taxes on economic growth in a model of
physical and human capital accumulation; \citeA{Bertola-93}, which
studies the functional distribution of income and its importance for
long-run growth in a model with capital externalities; \citeA{jedc},
which determines the equilibrium growth rate when income taxes are
determined by a political mechanism and elections are repeated every
period; and \citeA{constitutions}, which analyzes the impact of
different political and fiscal constitutions on the equilibrium
allocation. Public finance data show that cash transfers are not
pure lump-sum transfers: different agents receive different amounts
of transfers from the public sector. However, public cash transfers
on the whole do redistribute: even though in some countries the
level of cash transfers increases with the level of income, the
ratio of transfers to income decreases as the position of agents in
the income ladder increases (see \citeA{Ruggles-O'Higgins-81a} for
the United States, \citeA{Ruggles-O'Higgins-81b} for the United
Kingdom and \citeA{Saunders-Klau-85} for the OECD
countries).}$^,$\footnote{As long as the provision of the specific
goods supplied is made in a small enough quantity that agents are
indifferent between the transfer of the goods and a direct cash
transfer, this use of public funds is in effect identical to direct
cash transfers.} In this paper our main efforts are concentrated on
the first use of public funds, although we will also consider the
supply of public goods for consumption purposes and as an externality
in production.
The theoretical tools needed to study political equilibria in this
type of environment were developed in \citeA{constitutions}, and we
refer readers interested in a detailed discussion of these tools to
the mentioned paper.
%To solve the problem of the agents, not only processes for prices, but
%also processes for tax rates and for transfers, have to be specified.
\subsection{Equilibria}
We follow \citeA{constitutions} and concentrate on stationary Markov
equilibria. This is accomplished by representing equilibria with
recursive forms, and this representation includes three parts. First,
we postulate a policy as a mapping from the economy-wide state
variables to tax rates and transfers, and we compute the economic
equilibria associated with these policies. Second, we characterize the
economic behavior implied by a one-period deviation from this policy
mapping. Third, we use these deviations to construct preferences over
policies and a political mechanism to aggregate these preferences into
an equilibrium policy. The state variables are the distribution of
non-human wealth, $A\in {\cal R^I}$, and the tax rate inherited from
the past, which is generically denoted by $\tau$. We now describe
these steps in detail for the case of income taxes and lump-sum
transfers. The analysis for other tax systems is similar.
\subsubsection{Economic equilibria given a policy}
Consider a policy function $\tau'=\Psi(A,\tau)$. To avoid excessive
notation, we do not make explicit the implied transfers, but derive
their exact form in each instance. The problem of a given agent of
type $i$ who has wealth $a$ can be written in recursive form as
follows:
\begin{eqnarray}
\label{Joe}
v_i(A,\tau,a;\Psi) &=& \max_{c,a',l}\,\,\,\, u(c,l)+\beta\,\,
v_i(A',\tau',a';\Psi)
\qquad \mbox
{s.t.} \\
\nonumber \\
a' &=& a+\left(a\,r\left(A,N\right) +\left(1-l\right)\epsilon_i
w\left(A,N\right)\right) \left(1-\tau\right)+tr-c, \nonumber \\
tr &=& \tau [f\left(A,N\right)-\delta\sum_j \mu_j A_j],
\nonumber \\
\tau'&=& \Psi(A,\tau), \nonumber \\
A' &=& H(A,\tau;\Psi), \nonumber \\
N &=& N(A,\tau;\Psi). \nonumber
\end{eqnarray}
The functions $w(A,N)$ and $r(A,N)$ are before-tax rental prices of
factors of production and they are determined in competitive factor
markets. The function $H(A,\tau;\Psi)$ is the law of motion of the
distribution of assets that the agents take as given, and the function
$N(A,\tau;\Psi)$ gives the aggregate amount of labor which the agent
also takes as given. The solution to this problem gives next period's
asset holdings as a function $h_i(A,\tau,a;\Psi)$, and leisure as
$l_i(A,\tau,a;\Psi)$. Note that we index value functions, decision
rules, and economy-wide laws of motion with the policy function $\Psi$.
The standard equilibrium conditions in this context are:
\begin{eqnarray}
H_i(A,\tau;\Psi)&=&h_i(A,\tau,A_i;\Psi),\qquad \mbox{ for all }\, i\in
{\cal I}. \\
N(A,\tau;\Psi)&=&\sum_{i\in{\cal I}} \mu_i \epsilon_i
(1-l_i(A,\tau,A_i;\Psi)).
\end{eqnarray}
\subsubsection{Economic equilibria for a one-period policy deviation}
Consider now the following problem for an agent who, given the state
$(A,\tau)$, faces an arbitrary policy $\tau'$ next period, {\em
whereafter the function $\Psi$ will be used to determine the
policy}.
\begin{eqnarray}
\label{Per}
\tilde{v}_i(A,\tau,\tau',a;\Psi) &=& \max_{c,a',l}\,
u(c,l)+\beta\,\,v_i(A',\tau',a';\Psi) \qquad
\mbox {s.t.:} \\
a' &=& a\,+\left(a\,r(A,N) +(1-l)\epsilon_i
w(A,N)\right )(1-\tau)+tr-c, \nonumber \\
tr &=& \tau [f(A,N)-\delta\sum_j \mu_j A_j], \nonumber \\
A' &=& \tilde{H}(A,\tau,\tau';\Psi), \nonumber \\
N &=& \tilde{N}(A,\tau,\tau';\Psi). \nonumber
\end{eqnarray}
The function $\tilde{H}(A,\tau,\tau';\Psi)$ is the law of motion for
the distribution of assets that the agents take as given, and the
function $\tilde{N}(A,\tau,\tau';\Psi)$ determines aggregate
employment. The solution to this problem gives next period's asset
holdings as functions $\tilde{h}_i(A,\tau,\tau',a;\Psi)$ and
$\tilde{l}_i(A,\tau,\tau',a;\Psi))$. The equilibrium conditions in
this context are:
\begin{eqnarray}
\tilde{H}_i(A,\tau,\tau';\Psi)&=&\tilde{h}_i(A,\tau,\tau',A_i;\Psi),
\qquad\mbox{ for all }\, i\in {\cal I}, \\
\tilde{N}(A,\tau,\tau';\Psi)&=&\sum_{i\in{\cal I}} \mu_i \epsilon_i
(1-\tilde{l}_i(A,\tau,\tau',A_i;\Psi)).
\end{eqnarray}
%\noindent bviously, the following conditions have to hold:%
%\begin{eqnarray}
%\tilde{H}(A,\tau,\Psi(A,\tau);\Psi)&=&H(A,\tau;\Psi),\\
%\tilde{N}(A,\tau,\Psi(A,\tau);\Psi)&=&N(A,\tau;\Psi),\\
%\tilde{h}_i(A,\tau,\Psi(A,\tau),A_i;\Psi)&=&h_i(A,\tau,A_i;\Psi),
%\qquad \mbox { for all }\, i\in {\cal I}.
%\end{eqnarray}
\subsubsection{Politico-economic equilibrium}
The function $\tilde{v}_i(A,\tau,\tau',a;\Psi)$ delivers the utility
of a type $i$ agent under tax rate $\tau'$ tomorrow and tax rates
thereafter given by whatever is implied by $\Psi$ and the associated
accumulation of assets: these are the induced preferences over
policies that we are searching for. Hence, with the median voter
referred to as agent $m$, the preferred policy of the median voter
becomes
\begin{equation}
\psi(A,\tau:\Psi)=\mbox{Argmax}_{\tau'}\,\,\,\,
\tilde{v}_m(A,\tau,\tau',A_m;\Psi).
\end{equation}
A politico-economic equilibrium is now a pair of functions $\Psi$ and
$H$ such that, given $\Psi$, $H$ is an economic equilibrium, and such
that $\Psi$ is a political equilibrium, i.e.\ $\Psi=\psi$.
%In what follows we will not look at all possible combinations of
%distributions of wealth and skills, but at a subset which allows us to
%apply the median voter theorem---in the cases that we look at the two
%dimensions of heterogeneity effectively reduce to one. These are
%situations where agents have the same labor efficiency but different
%wealth; the same non-human wealth but different labor efficiency; and
%a situation where all agents have the same ratio of human to non-human
%wealth.
\section{Properties of Steady States}
\label{some}
In this section, we describe some of the key properties of steady
states of the model. The first thing to note is that a steady state
cannot be characterized independently of the whole recursive
equilibrium. The reason for this is that agents need to know the paths
of the economy for all possible tax rates in the following period in
order to evaluate their preferences.\footnote{In \citeA{constitutions}
there is a detailed discussion of this issue.} In particular, to
find a steady state we have to calculate the equilibrium functions $H$
and $\Psi$. A steady state is a solution $A^*,\tau^*$ to the following
system of equations:
\begin{eqnarray}
A^*&=&H(A^*,\tau^*)\\
\tau^*&=&\Psi(A^*,\tau^*).
\end{eqnarray}
It is easy to see that equal distribution and zero taxes is a steady
state by noting that associated with any nonzero level of taxation
there are distortions, but no redistribution. Another property to
notice is that in a steady state there is no net investment and,
therefore, both income and consumption taxes have the same tax base,
i.e.\ a given consumption tax rate implies exactly the same revenue
collection as the same rate applied to income.
We want to compare the set of steady states of economies with
endogenously determined taxes with those of more traditional models
where taxes are determined exogenously. In \citeA{constitutions}
there is a detailed analysis of the implications for economies where
leisure does not enter the utility function; there, it is shown that
with exogenous taxation the distribution of wealth is
irrelevant.\footnote{That analysis follows \citeA{Chatterjee-94}.}
In economies with leisure, the analysis is slightly more complex, since
the steady state level of capital is determined jointly
with the level of work effort. However, the set of distributions of
wealth that are possible as steady states is the same as in the
economy without leisure.
To be more precise, note that any steady state with an exogenous tax
on income can be summarized by
$(F_1(\hat{K},1-\hat{L})-\delta)(1-\tau^y)+1=1/\beta$ and
$(1-\alpha)(F(\hat{K},1-\hat{L})-\delta\hat{K})=\alpha
F_2(\hat{K},1-\hat{L})\hat{L}$, where
$\hat{K}$ is total capital and $\hat{L}$ is total leisure. These
equations can be used to solve for $\hat{K}$ and $\hat{L}$. Given
these totals, the locus of wealth distributions is an
$I-1$-dimensional hyperplane described by the equation $\sum_i \mu_i
A_i= {\hat K}$. For example, when $I=2$ this hyperplane defines a line
with slope equal to $-\mu_1/\mu_2$. Moreover, with the preferences we
assume, agents' ratios of consumption to leisure are proportional to
their labor efficiency. Since consumption is a linear function of
wealth, this also means that the set of steady-state distributions of
leisure choice is an $I-1$-dimensional
hyperplane.\footnote{Everything else given, rich agents work less than
poor agents in this economy.}
It follows from the indeterminacy of steady-state distributions that
any combination of relative labor efficiency levels and relative
wealth levels is possible. In real-world economies, both the
distributions of wages and wealth tend to be skewed to the right, and
wealth distributions tend to be more skewed than wage distributions.
In our example economies, we examine the role of the relative
distributions of labor efficiency and wealth by varying the
correlation between asset holdings and labor efficiency.
In the economies we study, the endogenous redistribution of resources
among agents affects the selection of taxes dramatically, and it also
affects the distribution of wealth. In particular, the set of steady
states changes character. One way of illustrating the extent of this
effect is to point out that the set of steady states in the 2-agent
case without leisure choice gives a slope of the local linear
approximation to the zero-tax steady-state wealth distribution which
is {\em positive} (i.e.\ far from $-\mu_1/\mu_2$).
In fact, the preferences we assume admit aggregation in the absence of
endogenous policy choice. Therefore, not only the set of steady states
but also the dynamic paths of prices and aggregates are affected by changes
in the initial distribution of a given amount of capital. It is
convenient to use this class of preferences, since it tells us that
any short- or long-run macroeconomic effects of the initial wealth
distribution are due solely to the politics in the model. We turn to
this in Section \ref{quantitative}.
\section{Parameter Selection in Our Example Economies}
\label{calibration}
We posit functional forms for preferences and technologies and use
parameter values that match the standard postwar growth properties of
the U.S.\ economy. This is in the real-business-cycle tradition; our
growth model is very simple, and the growth aspects of the calibration
are not, we hope, controversial. As regards, the wealth distribution,
we preferred to present some different cases in order to highlight the
role of the relative distributions of capital wealth and labor
efficiency for the policy outcomes. We look at three different cases:
(1) equal labor efficiency and differences in non-human wealth; (2)
equal non-human wealth and differences in labor efficiency; and (3)
the same ratio of non-human wealth to the endowment of labor
efficiency units across the two groups of agents.
The production function is Cobb-Douglas, i.e.\ $Y= K^\theta
N^{1-\theta}$, with $\theta=0.36$. Capital depreciates at rate
$\delta$ on an annual basis, and we set it to 0.08.
In our CRRA utility function of a Cobb-Douglas consumption-leisure
index, the coefficient on consumption, $\alpha$, is taken to be 0.33,
and the coefficient of relative risk aversion, $\sigma$, is 2. The
discount rate is set so that the steady state without taxes implies an
interest rate of 4\% annually.
We consider taxes to be chosen one period in advance, and we select
the length of the period to be four years. See \citeA{constitutions}
for an account of the role of the time period in this type of economy.
Finally, the economies we look at have two types of agents of equal
size, and we consider the poorer agent the median voter. Introduction
of more types of agents is straightforward, and \citeA{constitutions}
analyzes a number of such cases in more detail.
\section{A Comparison of Tax Systems}
\label{quantitative}
We now start the analysis based on the computation of
equilibria for the economies described in the calibration
above.\footnote{We do not know whether in general equilibria are
unique for our class of models. However, we have not encountered any
case of multiplicity of equilibria in our computations.}
\subsection{Steady-state analysis}
\label{ststa}
Table $\ref{tdedw}$ shows the steady state levels of taxation and
aggregate output for a variety of economies that differ in the
relative composition of human and non-human wealth among households.
We report the steady-state tax rates, levels of output, and utility levels
of both types of agents for different tax systems (income taxes,
consumption taxes, labor income taxes, and capital income taxes) and
wealth distributions.\footnote{We did not report equilibria for
economies with only capital income taxes in the cases of a very
skewed distribution of wealth; we were unable to compute equilibria
in these cases.} The aggregate output level is reported as a
percentage of the level of output that results in the steady state
where zero taxes are imposed exogenously, and for a variety of spreads
in the distribution. The utility levels are not of direct normative
interest, since the initial conditions differ across steady states.
Also, the utility levels indirectly indicate work effort differences
across tax systems.
A common feature to all economies is that the level of taxation is
increasing with the degree of inequality. This result is standard
in the political-economy literature (see the references above).
Moreover, because taxes have a distortionary effect on the economy,
there is a negative relation between levels of taxation and
aggregate output for all types of taxation.
The first three columns of Table $\ref{tdedw}$ refer to economies
where all agents have the same ratio of human to non-human wealth,
while in the last six columns all agents have either the same labor
efficiency or the same amount of non-human wealth. In the economy with
positive correlation between human and non-human wealth, which is
probably the most empirically relevant case, all agents choose the
same amount of work effort. The main results for these economies (the
first three columns of Table~$\ref{tdedw}$) can be summarized by:
\begin{itemize}
\item Consumption taxes generate lower steady-state output than do
income taxes. The reason for this is that rational agents
internalize the smaller amount of distortion per unit of transfer
associated with consumption taxes, which induces them to choose a
higher level of taxation.
\item Even though it is often thought that consumption and labor
income taxes have similar properties in terms of the distortions
that they generate, this is not the case in our environment. Here,
consumption taxation provides a broader tax base for redistribution
than do labor taxes, making them more attractive for the median
voter. This feature results in higher tax rates for
consumption than for labor income, and hence leads to lower
steady-state output.
\item If capital income taxation is the only way to generate
transfers, then small differences in wealth across agents generate
high tax rates and distortions.\footnote{With our
computational methods this leads to problems when the wealth
differences are substantial; we use linear-quadratic
approximations and cannot impose non-negativity constraints such
as one on investment. Such constraints are likely to be binding in
these cases.}
\end{itemize}
In economies where agents have different ratios of human to non-human
wealth, a new consideration appears: taxation can now not only be used
for direct redistribution through the lump-sum transfers, but it can
also be used to affect relative prices in a way that benefits the
median group at the expense of the other group. This feature is
particularly important in the study of capital and labor taxes, but it
is also present in the cases of income and consumption taxation.
In economies where agents have the same wealth but differ in labor
efficiency, the median agents (the low-efficiency agents) work less
than the non-median agents.\footnote{This result is special to the
type of preferences that we consider.} As a result, the income of
the median agent has a higher share coming from capital than that of
the non-median agent. This means that the relative price changes
triggered by lower aggregate capital---an increase in the rental
rate of capital relative to the wage rate---benefit the median agents.
The middle columns of Table~\ref{tdedw} show the steady-state values
for taxation, output and utilities for these economies. A summary of
the findings is as follows:
\begin{itemize}
\item All tax rates are lower than in the economies where all
agents have the same ratio of human to non-human wealth. This is
because the differences in income between agents are smaller in this
case.
\item Income taxes now lead to lower output than do consumption taxes.
The reason for this is that the reduction of total future capital
has an effect on the relative prices of factors of production that
favors the median agents, and this in turn induces lower output.
income taxes than of consumption taxes.
\item Labor taxes are perfect substitutes for consumption taxes even
though their tax base is different, because the higher revenues of
consumption taxation cannot give any net redistribution since all
agents hold equal amounts of non-human wealth, and because the two
taxes have the same distortionary effects. (The tax rates are only
nominally different because of the form in which they enter the
budget constraint.)
\item Capital income taxation has very interesting properties. First,
it cannot be used to redistribute, just to distort, since all
agents have the same amount of capital. Why, then, do the median
agents want to distort? As said before, the
distortion is not neutral. There is a change in after-tax relative
prices benefitting those with a higher proportion of their income
coming from capital, which in this case means the median households.
\end{itemize}
The third possibility is that agents have the same labor efficiency
but differ in non-human wealth, and here the median agents (the poor
agents) work more than the non-median agents. Because of their lower
asset holdings and the higher work effort, the relative composition of
the median agents' income is thus tilted towards labor income. This
means that a change in the relative price following from a lower
aggregate capital stock benefits the non-median agents. The last three
columns of Table~\ref{tdedw} shows this economy's steady-state values
for output and taxation for all these tax systems. Some of the key
properties of the findings are:
\begin{itemize}
\item Income taxes lead to higher levels of income than do consumption
taxes.
\item Labor income taxes are negative. This is due to
the fact that poorer agents work harder, and, hence, they want to
subsidize labor earnings.
\item Even though the capital income tax rate is quite
high, it is still lower than the one that results in the economy
where all agents have the same ratio of human to non-human wealth.
This comes from the negative effect of the lower capital stock, via
relative prices, on the relative income of the median households.
The same qualitative result holds for income and consumption taxes,
which are also lower in this case.
\end{itemize}
We summarize the key findings of the steady-state analysis of
different tax systems with the following:
\begin{enumerate}
%\renewcommand{\labelenumi}{\bf{\alph } .}
\item[{\bf 1.}] Tax rates are an increasing function of the skewedness
of the income and wealth distribution. This property holds for all tax systems
and all sources of income and wealth differentials.\footnote{For economies with
all agents having the same labor efficiency and different wealth shown
in the last columns of Table~\ref{tdedw}, labor taxes are negative.
Here, the absolute value of the tax rate is increasing with income
concentration.}
\item[{\bf 2.}] Typically, income taxation leads to a higher level of
output than does consumption taxation. This is due to the fact that
income taxation is more distortionary, so that it will tend to be
used less than consumption taxation.\footnote{As stated, this is not
true for economies where agents have the same non-human wealth but
differ in their labor efficiency, but we consider this case more
as a consistency check on our results than as a plausible case.}
\end{enumerate}
\subsection{A comparison with environments where the level of
transfers is exogenous}
\label{cltpr}
In order to highlight the importance of endogenizing the determination
of the level of taxation through the political mechanism, we also
computed the steady-state equilibrium allocations when tax rates are
determined by an exogenous level of transfers. In other words, assume
that the level of transfers is predetermined. Consequently, the level
of taxation must be such that the public budget is balanced. We thus
computed the steady-state equilibria associated with different types
of taxation. As an example, we selected the given amount of
per-capita transfers to equal that amount determined in the
politico-economic equilibrium with income taxes. This type of
analysis is similar to a simplified version of the optimal taxation
approach, and it shows how the introduction of an endogenous political
mechanism can change the views on the preferability of different
systems of taxation. For the case in which all agents have the same
ratio of human to non-human wealth, Table
\ref{texct} shows the steady-state levels of taxation and aggregate
output required to raise the same amount of revenue that the
equilibria with income taxes generate. Output is reported as a
percentage of the level obtained in the case of income taxes. In the
table, we see that consumption taxes are capable of raising this
amount at a much higher level of output.
Of course, the above experiments are not immediately informative about
welfare; we just used the level of output as an indirect measure
of the level of distortions, and as a first pass at the study of the
quantitative properties of the economies that we are interested in.
\subsection{Welfare analysis}
\label{wa1t}
In the welfare analysis of the alternative tax systems it is crucial
to make the comparisons starting from identical initial conditions.
This implies not only a specific wealth distribution, but also an
initial level of taxation. We choose as initial conditions the steady
states that are realized under the different tax systems. In the
first period, the economy still has the taxes associated with the old
tax system set in the previous period, but the following period's tax
rate, which belongs to a different tax system, is now voted on. We
then compute the equilibrium paths associated with the switches to
alternative tax systems and we compare the implied utilities for both
types of agents with those obtained in the steady states. Next, we
compute the constant proportional increase in per-period consumption
that has to be given to the agents when the economy switches tax
systems so that they are indifferent between switching and not
switching. Our procedure implies that a positive (negative) reported
number arises from a welfare loss (gain) when the economy switches tax
systems. We have found all our numerical examples to exhibit
stability, i.e.\ the economy moves from the original steady state
towards a new steady state.
Table \ref{twlltr} shows the findings associated with these
experiments. In the first part of this table, we see that both types
of agents suffer utility losses when switching from a system with
income taxes to one with consumption taxes. This is in clear contrast with
the steady-state comparison. The losses are larger for
the non-median (rich) agents. If we considered the switch from income
taxation to labor taxation, the findings are not so clear: when all
agents have the same ratio of human to non-human wealth, the median
agents realize small welfare losses, while the non-median agents are
better off after the change in tax system. However, when the ratios of
human to non-human wealth are different across types the welfare of
the median agent improves with the change.
The second part of Table \ref{twlltr} shows a switch from the steady
state with consumption taxes to a system with income taxation and to
one with labor taxation. There are small losses for the median agents
and larger gains for the non-median agents, except for the case when
agents have the same labor efficiency but different non-human wealth:
then, both agents gain from changing tax systems.
Finally, the third part of Table \ref{twlltr} shows the properties of
switches from labor taxation to income and consumption taxation. In
this case, the welfare changes are small and entail welfare losses for
the median (except when switching to an income tax system in economies
with the same ratio of human to non-human wealth: here there is a
negligible gain for the median).
We summarize these findings as follows:
\begin{enumerate}
%\renewcommand{\labelenumi}{\bf{\alph } .}
\item There is only one case in which all the groups increase their
welfare after a change in the tax system. This is when consumption
taxes are replaced by income taxes in an economy where all agents
have the same non-human wealth but different labor efficiency.
\item The median agents improve their welfare after a change in tax
systems only in two cases: in the case noted before, and in the case
of a replacement of labor taxation by income taxation in an economy
where all agents have the same ratio of human to non-human wealth
(and in these cases the median agents prefer a change by the
slightest of margins).
\item The two above points lead to the insight that tax systems
have an important stability property: once a tax system is in place
it will be hard to replace with another tax system. This is
particularly true if
the change in tax systems requires some form of qualified majority
(unanimity in our 2-agent case).
\end{enumerate}
\section{Economies with Both Consumption and Income Taxes}
\label{2tax}
It is natural to also ask about the properties of economies in which
two taxes, say an income and a consumption tax, are contemporaneously
voted on. There are well-known problems associated with
multidimensional voting (majority voting may not be
(quasi-)transitive, and the median-voter theorem may not apply).
These problems do not arise in a 2-agent economy in which one
group of agents is larger than the other one---then, all we
need is to maximize the indirect preferences for the more numerous
agent over the tax pair. As before, we let the decisive voter be the
poorer agent.
\subsection{Steady-state analysis for systems with two taxes}
\label{2tst}
In Table $\ref{t2tst}$ we report the steady-state tax rates of the
economies with both consumption and income taxes for different degrees
of income concentration coming from differences in asset holdings,
differences in labor efficiencies and differences in both asset
holdings and labor efficiencies. The key properties that we observe
are:
\begin{itemize}
\item When agents differ in asset holdings, the consumption tax is used for
collecting revenue and the income tax is used for partially offsetting
the distortionary effect of the consumption tax. More specifically,
there will be a positive consumption tax and a {\bf negative} income tax,
producing the following result: (i) a negative income tax partially
offsets the distortionary effect of consumption taxes on the
labor/leisure choice; (ii) at the same time, negative income taxes
increase next period's capital stock and income, which implies a
higher tax base for collection of future revenues and thus future
transfers, as the differences in assets between the groups are
permanent. Thus, the intertemporal distortion that the income tax
creates by subsidizing capital accumulation is somewhat offset, from
the point of view of the (poor) median agent, by the higher future tax
base that it induces.
\item
When agents differ in their non-human wealth (parts 1 and 3 of
Table $\ref{t2tst}$), we have higher income subsidies (a negative
income tax with a higher absolute value) than when agents have the
same non-human wealth. The reason behind this result is that in this
economy the median agents work longer than the richer ones, and
therefore the share of income due to labor is higher. The median
agents thus internalize the role that higher capital tomorrow has in
determining relative prices: higher capital implies higher wages. In
the economy where the ratios of human to non-human wealth are equated
across agents, all agents work the same amount of hours, implying that
everybody receives the same functional composition of income.
This implies that the gains for the median agents from distorting the
relative prices of labor and capital is lower in the case of equal
labor efficiencies.
\item When agents have the same wealth but different labor abilities
(part 2 of Table $\ref{t2tst}$), income taxes are not used and
the properties of the equilibrium are the same as those of the
economies with only consumption taxes. In this case, the distortion
of the intertemporal margin does not change the effective
redistribution that can be implemented through the taxation of
capital income because both types of agents have the same asset
holdings. At the same time, an increase in the relative price of
labor is not worthwhile for the median agents given that their labor
efficiency is low and they work less; in fact, as in the case with a
capital income tax only in the previous section, this argument speaks
for positive income taxes. In sum, however, this effect and the need
to partially offset the labor/leisure distortion created by the
consumption tax will cancel, so from the median's
point of view, non-zero income taxes have a pure distortionary role.
\item The highest transfers occur in the economy where agents
have identical ratios of human to non-human wealth (part 1 of
Table $\ref{t2tst}$), followed by the economy where agents have the
same non-human wealth but different labor efficiency (part 2 of
Table $\ref{t2tst}$), and, finally, by the economy where agents only
differ in non-human wealth (part 3 of Table $\ref{t2tst}$). Given
the parameterization of the model economies, this hierarchy also
corresponds to that of total income.
\item The steady states are associated with at least as high levels of output
as in societies with only one type of taxation.
\end{itemize}
We found the fact that income taxes are never positive (and almost
always negative) in the case when two taxes are voted on quite
striking. This is true even in the case when there is a large
difference among groups in asset holdings. Although this type of tax
focuses more on the asset base, consumption taxes remain more
efficient for the purpose of taxing agents with high asset holdings:
consumption does rise as a function of the asset holding, and taxing this
base is less distortionary.
The fact that there seem to be higher levels of output in the two-tax
steady states than in economies with one type of taxation is
indicative of the desirability of a system with two taxes (thus
casting doubt on
\citeA{Brennan-Buchanan-77}'s general proposition that the more
efficient the government, the worse the resulting economic
performance). However, it cannot be directly used to make welfare comparisons
across tax systems.
To be able to make welfare comparisons we have to perform the same type of
dynamic analysis as in Section \ref{wa1t}.
\subsection{Welfare analysis}
\subsubsection{Switching from a one-tax to a two-tax system}
\label{s1tr2}
The upper part of Table \ref{t1tr2} describes the welfare losses for
the two types of agents that result when the economy starts at a
steady state with one type of tax and it moves to a tax system with
both consumption and income taxes.
We see that a replacement of income taxes with two taxes worsens the
welfare of the non-median agents. The median agent is also worse off
except for the case of equal non-human wealth and differential
labor efficiency. However, the quantitative amount of the welfare
improvement in this latter case is almost zero. This, again,
suggests that if a society is at a point where income taxes are used
but consumption taxes are not, there latter is unlikely to be
introduced. This model thus offers one explanation for the lack of federal
consumption taxes in the United States.
On the other hand, adding income taxes to a society which is in a
steady state with consumption taxes may increase the welfare of all the
agents involved. This occurs when all agents have the same
ratio of human to non-human wealth. As we saw in the previous
subsection, when all agents have the same asset holdings and different
labor efficiencies, income taxes are not used, which implies that the
economy remains in the same position as it was before the
introduction of income taxes; therefore, the welfare of the agents does
not change. When agents have the same labor efficiency and different
non-human wealth, the median agents gain and the non-median agents
lose. We saw in the previous section that in societies with
consumption taxes, a replacement of the existing tax with income taxes
is unlikely. What Table \ref{t1tr2} tells us, instead, is that for
these societies the addition of an income tax to the preexisting
consumption tax is much more likely. Most countries that base most of
their fiscal revenue on income taxes have introduced them later and in
addition to consumption taxes as this model predicts. However, as a
positive theory, the model with simultaneous voting on income and
consumption taxes is problematic since it predicts negative
income taxes.\footnote{It is possible that this result disappears if
one assumes a large enough exogenous source of government revenue.}
Regarding the substitution of labor taxes with the combination of
consumption and income taxation, Table \ref{t1tr2} shows that all
agents lose (except the median agents in the case of an equal ratio of
human to non-human wealth), but the improvements are quantitatively small.
\subsubsection{Switching from a two-tax to a one-tax system}
\label{s2tr1}
The lower part of Table \ref{t1tr2} describes the welfare losses for
the two types of agents that result when the economy is in the steady
state with both consumption and income taxes and it moves to a tax
system with only one tax.
We see that the replacement of a tax system based on both consumption
and income taxes with a system with only one tax always reduces the
welfare of the median agent. This change in the tax system has the
opposite effects on the welfare of the non-median agent in almost all
cases (the only exception is the replacement of the two taxes with a
consumption tax when all agents have the same ratio of human to
non-human wealth; in this case the non-median agents are also worse
off).\footnote{As for the reverse switch of systems, the replacement of
a two-tax system with one with consumption taxes in the case where
all agents have the same asset holdings has no effect on the
equilibrium allocation, as the resulting income tax rate is zero and
the economy maintains its wealth distribution over time.}
\section{Other Roles for Government}
\label{public}
So far, we have been assuming that government outlays are used for
redistributing goods in equal amounts to all agents.
However, many of the government's activities are associated with the
provision of goods and services. In this section we explore the
properties of alternative tax systems when all of the government revenue
is used for purchasing goods, and the political system determines the
levels of spending.\footnote{For brevity of exposition, we do not report the
results of the simulations nor the proofs of the claims we make in the
present section. However, they are available upon request from the authors.}
\subsection{Public provision of private goods}
Public provision of private goods which are highly substitutable with
private consumption include education and certain forms of public
support like health assistance. If the public goods are indeed perfect
substitutes with private goods, and they are distributed on an equal
per-capita basis where no agents have zero private provision of the
good, then the properties of this economy are identical to that with
direct cash transfers.
An interesting variation is the assumption that the government is
inefficient in providing these goods. There are different rationales
for this: costly information acquisition may be necessary, the
incentive structure in the public sector may make it inefficient, and
so on.
To implement the notion of inefficient public provision of private
goods, we consider a widely used
specification for the utility function given by
$u(c,g,l)= \frac{[ (c + \pi g)^\alpha l^{1-\alpha}
]^{1-\sigma}}{1-\sigma}$, where $c$ is private consumption, $g$
public expenditures, $l$ leisure, and $\pi\in [0,1]$ is an index of
efficiency of the public sector.\footnote{A discussion of a variety of
formulations for how public goods affect the economy and their
macroeconomic effects can be found in \citeA{Aschauer-Greenwood-85}.
Examples of studies using
this particular specification are \citeA{Aschauer-85} in testing the
Ricardian Proposition, \citeA{Christiano-Eichenbaum-92} in analyzing
the contribution of government consumption in the generation of
aggregate
fluctuations and \citeA{McGrattan-94} in analyzing the influence of
capital taxes, labor taxes, and government consumption for the
business cycle.}
We compared economies with the same fundamentals but different degrees
of government efficiency, $\pi$. Our findings are parallel to those in
the economies with lump-sum redistribution: economies with higher
efficiency in the government provision of the goods will tend to have
more active governments. For example, an income tax economy where all
agents have the same ratio of human to non-human wealth and where the
median agents have 85\% of the average wealth and which has $\pi=1$
gives a steady-state tax rate of $16.53\%$. However, when $\pi=0.9$,
meaning that the public sector is not perfectly efficient in the
provision of the good, the steady-state tax rate is $7.58\%$.
Correspondingly, steady-state output in the economy with a fully
efficient government economy is only $89.3\%$ of the value in the
economy with an inefficient government. Thus, this constitutes
further evidence that the more efficient the fiscal
instruments of redistribution, the more expanded the public
activity, and the worse the aggregate performance of the economy due
to higher distortionary taxes.\footnote{Of course, this does not
translate into utility terms since the fact that publicly provided
goods are more efficient in generating utility in the case with an
efficient government.}
\subsection{Public provision of public goods}
Another key activity undertaken by the government is the provision of
public goods. It turns out that the properties of the equilibrium of
economies where government outlays are used to provide public goods
depend crucially on the specification of the role these
goods play in utility, and on the ratios of human to non-human wealth for
the different agents. For
example, the public good can enter the utility function in the same
form as the private goods under Cobb-Douglas preferences, i.e.\ as
$u(c,g,l) = \frac{[c^{\alpha_1}g^{\alpha_2}
l^{1-\alpha_1-\alpha_2}]^{1-\sigma}}{1-\sigma}$. In this case,
Engel curves are linear: the shares of expenditures for each good are
independent of the level of income and all agents would like to spend
the same ratio of their income on the public good. This means that
when all agents have the same ratios of human to non-human wealth, they
also have the same preferences over tax rates since now government policy
plays no role for redistribution (both the costs and the benefits are
proportional to their wealth). The
preferred choice balances the distortion that the taxes generate with
the utility they provide and the political problem turns into a pure
optimal-taxation problem: the best taxes are the least distortionary taxes.
When the ratios of human to non-human wealth
differ across agents, the situation changes slightly since now
the contribution to the finance of the public goods is not
proportional to individual incomes (with capital or labor taxes) and
the distortions caused by taxation are not identical for all groups.
Consequently, the preferred tax rates are not the same across types of agents.
The differences, however, are small because the variations across
groups of the tax-induced distortions are also small. This leads to
the finding that the political system generates policies that are very
similar to the ones that we would obtain from optimal-taxation analysis.
If preferences are not Cobb-Douglas over the public good, as in the
case $u(c,g,l)= \frac{[ c^\alpha l^{1-\alpha}
]^{1-\sigma}}{1-\sigma}+\psi(g)$, where $c$ is private consumption,
$l$ leisure, $g$ public spending, and $\psi(\cdot)$ is strictly
increasing and concave.\footnote{If the process generating $g$ is
exogenous, this functional form is particularly convenient because the
agent's maximization problem is not affected by $g$ at the margin. For
this reason it has been used in several studies on taxation. Examples
are \citeA{Aiyagari-Christiano-Eichenbaum-92}, \citeA[sec. II and
III]{Jones-Manuelli-Rossi-93}, \citeA{Chari-Christiano-Kehoe-94}.} Now
the agent's preferences over the optimal level of taxation, and thus
the optimal provision of the public good, depends not only on the
relative sources of income, but also on the absolute value of
individual income. The extent of the differences in preferences across
agents over the amount of the public good then depends on the extent
of the income dispersion. However, in our numerical examples, the
dependence of the equilibrium level of taxation on distribution is not
as strong as in the case of cash transfers, and therefore the steady
state level of output is larger with less distortionary taxes, implying
that consumption taxes are typically better than income taxes.
If we consider a public good that acts as an input to production, similar
findings arise.\footnote{For example,
\citeA{Aschauer-89} provides some empirical evidence for this hypothesis:
he shows that the stock of non-military public capital (in particular
that of structures) has a significant effect on private factor
productivities. Other examples of studies that consider the role of
public expenditures as an input to the aggregate production function
are \citeA{Barro-90}, \citeA[sec. IV]{Jones-Manuelli-Rossi-93}, and
\citeA{Alesina-Rodrik-94}.} In particular, we assume the following
specification for the production function: $Y=K^\theta N^{1-\theta}
I_g^\gamma$ where $K$ is private capital, $N$ labor input, and $I_g$ is
public expenditures.\footnote{If $\theta+\gamma\ge 1$, then the model
allow endogenous growth.}
In this case, the relation between distribution and taxation depends
on the type of taxes used to finance public expenditures. More
specifically, with a Cobb-Douglas production function with three
inputs, one of them being a public good, and with income or
consumption taxes, the distribution of income has no effect on the
level of taxation, except for the effect through the tax exemption of
depreciated capital. The reason for this finding is that the increase
in the level of provision of the public good has the effect of
increasing the prices for the services of both capital and labor. This
in turn means that all agents benefit proportionally to their endowment of
capital and labor. Because the cost is also proportional to the
endowment of capital and labor, we obtain agreement over policies
and the standard finding that the less distortionary the taxes are, the
better.
If we assume that tax rates for different kinds of income differ,
then unless the proportion of capital income on labor income is the same
for all agents, the distribution of wealth has an influence on the
equilibrium level of taxation as it affects the relative factor
prices, and hence it affects the various groups of agents differently.
%\section{Some Properties of the Data}
%\label{properties}
%As partial empirical support of our findings, we present two types of
%evidence. Table $\ref{corrdat}$ shows how the size and composition of
%government revenues due to consumption and to income taxes relate to
%the level of output. In this table, we report the cross-correlations
%for per-capita GDP, the ratio of total tax revenue to GDP, the ratio
%of income taxation (including social security) to GDP, the ratio of
%consumption taxation to GDP, and the shares of government revenue due
%to income and to consumption taxation. The correlations are computed
%from a cross-section of OECD countries in 1985. In the table we see that the
%importance of income taxation is positively related to output, both
%measured as the ratio of total income and as a ratio of tax revenue.
%The relative importance of consumption taxation is negatively related to the
%level of output, again both when measured as a ratio to public revenues and
%as ratio to output. The relation between the level of taxation and
%output is, however, negative, even though our model
%is constructed to have the opposite property.
%The other piece of qualitative evidence which can be brought forth in
%support of our present findings is the fact that changes in
%taxation systems are relative infrequent, while changes in the tax
%rates are much more frequent. For example, income taxes
%once instituted have never been removed, and the U.S.\ has never had
%a federal consumption tax.
\section{Conclusions}
\label{conclusions}
In this paper, we have described the properties of a neoclassical
growth model where agents are heterogeneous in asset holdings and/or
labor earnings ability when the level of taxation is determined
through a politico-economic mechanism and the tax proceeds are rebated
as lump-sum transfers. We have shown that, in general, consumption
taxes induce lower output than income taxes as agents internalize the
higher distortionary cost induced by income taxes. Table \ref{tdsht}
provides some support for this finding; the countries with the least
reliance on consumption taxes---Japan, the U.S., and Switzerland---are
all associated with relatively small transfer systems and high output
levels; the reverse is roughly true for the countries with the largest
reliance on consumption taxes. We have also shown that switches from
one tax system to another tend not to increase the welfare of the
median voters. This suggests stability in tax systems, and a
permanence of status quo. This result also holds for tax systems that
allow for the simultaneous taxation of consumption and income, albeit
there are some exceptions in this case. The most important of these
exceptions refers to the case of an addition of income taxes to a
society that only uses consumption taxes; this addition results in
welfare gains for both types of agents.
We have also looked at economies where government outlays are not used
for redistribution but for the provision of public goods. We found
that the determination of the level of taxation through a
politico-economic mechanism tends to lead to the same properties
as those of the standard optimal taxation literature, i.e.,
less distortionary taxes are preferable.
%This shows that
%a key mechanism behind one of the main results of our paper, namely,
%that distortionary taxes are good because they will be less
%extensively used, is the fact that tax revenues are used for
%redistribution.
\newpage
\begin{appendix}
\centerline{{\bf Appendix: Computational Procedure}}
\medskip\noindent
{\bf 1. Algorithm for finding steady states}
\medskip
There are typically an $(I-1)$--dimensional subspace of steady states. We
search for those with a given ratio of asset holdings and labor
efficiencies between the different types of agents. For each ratio,
the search for a steady state involves a search for a tax rate. The
procedure for computing such a tax rate can be described as follows:
\par
\begin{enumerate}
\renewcommand{\labelenumi}{\bf (\roman {enumi})}
\item \label{step 1} Guess a steady state value for tax rate $\tau_0$
and compute the implied stationary values of the other variables,
$S(\tau_0)$.
\item \label{step 2} Let $R_i^0(A,\tau,a,N,A',n,a')$ be a quadratic
approximation to the per-period utility function of agent $i$ around
$S(\tau_0)$.
\item \label{step 3} Fix an initial affine tax policy function $\Psi_0$.
\item \label{step 4} Given $\Psi_0$, use standard methods to solve for
the equilibrium elements associated with the dynamic problem
described in (\ref{Joe}). This step yields linear functions $N_0$,
$H_0$ and quadratic functions $v_{i0}$ for $i=1,..I$.
\item Given $\{v_{i0}\}_{i\in {\cal I}}$, use standard methods to
compute equilibria associated to the dynamic problem described in
(\ref{Per}). This step yields linear functions $\tilde{N}_0$, and
$\tilde{H}_0$, and quadratic functions $\tilde{v}_{i0}$. These
quadratic functions have $\tau\prime$ as an argument.
\item \label{peak} Maximize $\tilde {v}_{i0}$ with respect to $\tau'$
to obtain functions $\psi_{i0}$ describing the preferred tax rate of
the agents. Check for the concavity of the function $\tilde
{v}_{i0}$ with respect to $\tau'$, to ensure that the first-order
conditions deliver a maximum.
\item Use the representative-type condition on the median agent to
obtain the function $\Psi_1$ by letting $\Psi_1(A,\tau) \equiv
\psi_m(A,\tau,A_m;\Psi)$.
\item Compare $\Psi_1$ to $\Psi_0$. If these functions are close
enough, continue to {\bf (ix)}. If not go back to step {\bf (iii)}
and update the guess for the policy function $\Psi_0$. We update to
let the new tax policy function equal $\Psi_1$.
\item Verify that the policy function $\Psi_0$ reproduces the
conjectured tax rate: $\tau_0 = \Psi_0(A,\tau_0)$. If it does not, go
back to step {\bf (i)} and update the guess for $\tau_0$. We update
using $\tau_0=(\tau_0+\Psi_0(A,\tau_0))/2$.
\end{enumerate}
\medskip
\medskip\noindent
{\bf 2. Algorithms for computing transitional dynamics}
\medskip
The second procedure follows steps {\bf (ii)-(ix)} above.
It also involves a separate linearization around each new
point the economy passes through. The slight complication needed is an
additional round of iterations ``within'' step {\bf (ii)}; it is
necessary to ensure that, at each point on the dynamic path, the
points $\{A,\tau,N,A'\}$ and $\{A,\tau,\tau',N,A'\}$ around which the
linearization is made coincide with the equilibrium outcome.
\end{appendix}
\newpage
\singlespace
\NoTitleCaseChange
\bibliographystyle{theapajm}
\bibliography{abbrev,per}
\newpage
\begin{table}[h]
\centerline{
\begin{tabular}{|l||r|r|r|r|}
\hline
& & & & \\
Country &Income &Social &Cons. &GDP per\\
&Tax &Security&Tax&capita\\
& & & & \\
\hline
\hline
& & & & \\
Australia &54.6 & - &28.7 & 8850\\
Austria &26.4 &31.8 &31.0 & 8929\\
Belgium &40.6 &33.1 &22.7 & 9717\\
Canada &44.1 &13.5 &26.1 &12196\\
Denmark &56.8 & 3.8 &32.9 &10884\\
Finland &51.0 & 9.1 &35.7 & 9232\\
France &17.3 &43.3 &28.7 & 9918\\
Germany &34.8 &36.5 &24.6 &10708\\
Greece &17.5 &35.6 &40.0 & 4464\\
Iceland &22.7 & 2.4 &59.5 & 9037\\
Ireland &34.5 &14.8 &42.6 & 5205\\
Italy &36.8 &34.7 &23.6 & 7425\\
Japan &45.8 &30.3 &12.1 & 9447\\
Luxembourg &44.6 &25.2 &23.4 &10540\\
Netherlands &26.3 &44.3 &23.4 & 9092\\
New Zealand &68.7 & - &22.0 & 8000\\
Norway &39.3 &20.6 &36.3 &12623\\
Portugal &25.7 &25.9 &41.3 & 3729\\
Spain &26.2 &41.3 &27.7 & 6437\\
Sweden &42.0 &24.8 &25.4 & 9904\\
Switzerland &40.9 &32.0 &17.5 &10640\\
Turkey &37.0 &14.3 &35.7 & 2533\\
United Kingdom &39.1 &17.6 &29.4 & 8665\\
United States &42.8 &29.4 &15.4 &12532\\
% & & & & \\
\hline
\end{tabular}}
\vspace{0.1cm}
\caption{ Taxation as Percentage of Total Tax Revenue in The
OECD Countries: 1985.\label{tdsht} }
\end{table}
\clearpage
\begin{table}[h]
\centerline{
\begin{tabular}{|l||r|r|r||r|r|r||r|r|r|r|}
\hline
\multicolumn{1}{|l||}{Med./av. wealth} &
0.99&0.85&0.67&1.0&1.0 &1.0 &0.99 &0.85 &0.67\\
\cline{2-10}
\multicolumn{1}{|l||}{Med./av. lab. eff.}
&0.99&0.85&0.67&0.99&0.85&0.67&1.0 &1.0 &1.0 \\
\hline
\hline
{\bf Income}&&&&&&&&&\\
- Tax rate & 1.25& 16.53& 32.11& 1.16& 15.73& 31.30& 0.09& 1.37& 2.90\\
- Output & 98.74& 82.88& 65.87& 98.83& 83.73& 66.77& 99.90& 98.62& 97.07\\
- Med. util. &-1.160&-1.294&-1.463&-1.159&-1.270&-1.406&-1.153&-1.173&-1.199\\
- Non-med. util.&-1.146&-1.109&-1.119&-1.147&-1.121&-1.140&-1.150&-1.135&-1.115\\
\hline
{\bf Consumption}&&&&&&&&&\\
- Tax rate & 1.97& 34.20& 88.21& 1.42& 23.08& 58.28& 0.54& 8.72&21.55\\
- Output & 98.62& 80.48& 61.52& 99.00& 85.94& 70.76& 99.61& 94.18&86.74\\
- Med. util.&-1.160&-1.285&-1.434&-1.158&-1.256&-1.370&-1.153&-1.178&-1.213\\
- Non-med. util.&-1.146&-1.120&-1.170&-1.147&-1.111&-1.120&-1.150&-1.143&-1.142\\
\hline
{\bf Labor }&&&&&&&&&\\
- Tax rate & 1.32& 17.76& 35.02& 1.40& 18.75& 36.83& -0.08& -1.18&-2.26\\
- Output & 99.06& 86.72& 72.36& 99.00& 85.94& 70.76&100.05&100.84&101.86\\
- Med. util.&-1.159&-1.279&-1.434&-1.158&-1.256&-1.370&-1.152&-1.169&-1.192\\
- Non-med.
util.&-1.145&-1.109&-1.077&-1.147&-1.111&-1.120&-1.150&-1.131&-1.108\\
\hline
{\bf Capital } &&&&&&&&&\\
- Tax rate & 4.66& & & 1.46& & & 3.23& & \\
- Output & 98.58& & & 99.56& & & 99.03& & \\
- Med. util. &-1.162& & &-1.158& & &-1.155& & \\
- Non-med. util.&-1.148& & &-1.146& & &-1.153& & \\
\hline
\end{tabular}}
\vspace{0.1cm}
\caption{Endogenous cash transfers: steady-state tax rates and
output for a variety of ratios of human to non-human wealth. Tax
rates are percentages, and output levels are percentages relative to
the zero-tax
steady state.\label{tdedw}}
\end{table}
\clearpage
\begin{table}[h]
\centerline{
\begin{tabular}{|l||r|r|r|}
\hline
\multicolumn{1}{|l||}{Med./av. wealth} &
0.99 &0.85 &0.67 \\
\cline{2-4}
\multicolumn{1}{|l||}{Med./av. lab. eff.} &
0.99 &0.85 &0.67 \\
\hline
\hline
{\bf Income} & & & \\
- Tax rate &1.25\% &16.53\% &32.11\% \\
- Output &100.00\% &100.00\% &100.00\% \\
- Med. util. &-1.160 &-1.294 &-1.463 \\
- Non-med. util. &-1.146& -1.109 &-1.119 \\
{\bf Consumption}& & & \\
- Tax rate &1.25\% &15.51\% &26.18\% \\
- Output &100.39\% &108.70\% &128.04\% \\
- Med. util. &-1.159 &-1.271 &-1.414 \\
- Non-med. util. &-1.145&-1.108 &-1.021 \\
{\bf Labor} & & & \\
- Tax rate &1.52\% &20.04\% &39.15\% \\
- Output &100.18\% &102.45\% &104.25\% \\
- Med. util. &-1.160 &-1.282 &-1.441 \\
- Non-med. util. &-1.145&-1.098 &-1.101 \\
{\bf Capital } & & & \\
- Tax rate &6.78\% &70.26\% & \\
- Output &99.16\% &76.40\% & \\
- Med. util. &-1.165 &-1.448 & \\
- Non-med. util. &-1.150&-1.258 & \\
\hline
\end{tabular}}
\vspace{0.1cm}
\caption{Exogenous cash transfers: steady-state tax rates and
output when the level of transfers is exogenous for
different types of taxes. All agents have the same ratio of human to
non-human wealth.\label{texct} }
\end{table}
\clearpage
\begin{table}[h]
\centerline{
\begin{tabular}{|l||c|c|c|}
\hline
%\multicolumn{1}{c||}{} & & &\\
\multicolumn{1}{|c||}{}&Same ratio of human&Same wealth,&Same efficiency,\\
\multicolumn{1}{|c||}{} &to non-human wealth&different efficiency &different wealth\\
\multicolumn{1}{|c||}{{\bf Replacing}} &$\frac{A_m}{A_a}=0.85,\frac{\epsilon_m}{\epsilon_a}=0.85$
&$\frac{A_m}{A_a}=1,\frac{\epsilon_m}{\epsilon_a}=0.85$&
$\frac{A_m}{A_a}=0.85,\frac{\epsilon_m}{\epsilon_a}=1$ \\
& & &\\
\hline
\hline
{\bf Income tax with}& & &\\
% & & &\\
$\quad${\bf Consumption tax} & & &\\
$\quad$ - Median agent & 0.05\% & 0.03\% & 0.10\%\\
$\quad$ - Non-median agent & 1.98\% & 0.21\% & 0.29\%\\
$\quad${\bf Labor tax} & & &\\
$\quad$ - Median agent & 0.01\% & -0.10\% & -0.00\%\\
$\quad$ - Non-median agent & -0.40\% & 0.04\% & -0.01\%\\
% & & &\\
\cline{2-4}
{\bf Consumption tax with}& & &\\
% & & &\\
$\quad${\bf Income tax} & & &\\
$\quad$ - Median agent & 0.38\% & 0.19\% & -0.06\%\\
$\quad$ - Non-median agent & -2.10\% & -0.24\% & -0.26\%\\
$\quad${\bf Labor tax} & & &\\
$\quad$ - Median agent & 0.51\% & 0.21\% & -0.06\%\\
$\quad$ - Non-median agent & -2.35\% & -0.14\% & -0.27\%\\
% & & &\\
\cline{2-4}
{\bf Labor tax with} & & &\\
% & & &\\
$\quad${\bf Income tax} & & &\\
$\quad$ - Median agent & -0.00\% & 0.12\% & 0.00\%\\
$\quad$ - Non-median agent & 0.39\% & -0.06\% & 0.00\%\\
$\quad${\bf Consumption tax}$^*$& & &\\
$\quad$ - Median agent & 0.02\% & 0.07\% & 0.04\%\\
$\quad$ - Non-median agent & 1.20\% & 0.05\% & 0.13\%\\
% & & &\\
\hline
\end{tabular}}
\vspace{0.1cm}
\caption{Welfare losses as percentage-of-consumption flows when the
economy switches
from a steady state with one type of tax to a system with
another type of tax.\newline $^*$Computational difficulties in
this case led us to using ratios of 0.90 rather than
0.85 for the three consumption cases.\label{twlltr} }
\end{table}
\clearpage
%\newpage
\begin{table}[h]
\vspace{0.3cm}
\centerline{
\begin{tabular}{|l||r|r|r|r|r|r|}
\hline
\multicolumn{1}{|l||}{Median/average wealth}
&0.99 &0.85&1.0 &1.0&0.99 &0.85 \\
\cline{2-7}
\multicolumn{1}{|l||}{Median/average lab. eff.}
&0.99 &0.85&0.99 &0.85&1.0 &1.0 \\
\hline
\hline
% & & & & & & \\
{\bf Cons. tax rate} & 4.00& 71.88& 1.42& 23.07& 2.56& 42.96\\
{\bf Income tax rate} & -2.01&-30.77& 0.00& 0.01& -2.02& -32.99\\
{\bf Output } & 99.22& 88.20& 99.00& 85.94&100.22& 102.73\\
{\bf Transfer/output} & 1.54& 31.13& 1.11& 17.99& 0.42& 7.54\\
{\bf Median util.} &-1.158&-1.257&-1.158&-1.256&-1.151&-1.152\\
{\bf Non-median util.}&-1.144&-1.091&-1.147&-1.111&-1.148&-1.121\\
% & & & & & & \\
\hline
\end{tabular}}
\vspace{0.1cm}
\caption{Steady states for economies with cash transfers and two taxes
available. \label{t2tst} }
\end{table}
\clearpage
\begin{table}[htb]
\centerline{
\begin{tabular}{|l||c|c|c|}
\hline
&&&\\
Rel. wealth and lab eff.'s &
$\frac{A_m}{A_a}=0.85,\frac{\epsilon_m}{\epsilon_a}=0.85$ &
$\frac{A_m}{A_a}=1,\frac{\epsilon_m}{\epsilon_a}=0.85$&
$\frac{A_m}{A_a}=0.85,\frac{\epsilon_m}{\epsilon_a}=1$ \\
&&&\\
\hline
\hline
% & & &\\
{\bf Both taxes replace} &&&\\
&&&\\
{\bf Income tax} & & &\\
- Median agent & 0.01\% & -0.00\% & 0.11\% \\
- Non-median agent & 1.29\% & 0.27\% & 0.45\% \\
{\bf Consumption tax} & & &\\
- Median agent & -0.14\% & 0.00\% & -0.04\%\\
- Non-median agent & -0.92\% & 0.00\% & 0.12\%\\
{\bf Labor tax} & & &\\
- Median agent & -0.01\% & 0.12\% & 0.09\%\\
- Non-median agent & 1.84\% & 0.19\% & 0.47\%\\
\hline
\hline
% & & &\\
{\bf Both taxes} &&&\\
{\bf are replaced with} &&&\\
&&&\\
{\bf Income tax} & & &\\
- Median agent & 1.50\% & 0.20\% & 0.59\% \\
- Non-median agent & -1.12\% & -0.24\% & -0.27\% \\
{\bf Consumption tax}$^*$& & &\\
- Median agent & 0.05\% & 0.00\% & 0.05\%\\
- Non-median agent & 0.13\% & 0.00\% & -0.01\%\\
{\bf Labor tax } & & &\\
- Median agent & 1.69\% & 0.21\% & 0.58\% \\
- Non-median agent & -1.29\% & -0.14\% & -0.25\% \\
\hline
\end{tabular}
}
\vspace{0.1cm}
\caption{Welfare losses as percentage-of-consumption flows when the
economy switches
between one- and two-tax systems. \label{t1tr2}}
\end{table}
\clearpage
%\begin{table}[bht]
%\centerline{
%\begin{tabular}{l||c|c|c|}
%%\hline
%% & & &\\
%{\bf Both Taxes } &Same ratio of human&Same wealth, &Same efficiency,\\
%{\bf Are Replaced with} &to non-human wealth&different efficiency
% &different wealth\\
% &$\frac{A_m}{A_a}=0.85,\frac{\epsilon_m}{\epsilon_a}=0.85$
% & $\frac{A_m}{A_a}=1,\frac{\epsilon_m}{\epsilon_a}=0.85$&
% $\frac{A_m}{A_a}=0.85,\frac{\epsilon_m}{\epsilon_a}=1$ \\
% & & &\\
%\hline
%\hline
%% & & &\\
% & & &\\
%{\bf Both taxes} &&&\\
%{\bf are replaced with} &&&\\
% &&&\\
%{\bf Income tax} & & &\\
% - Median agent & 1.50\% & 0.20\% & 0.59\% \\
% - Non-median agent & -1.12\% & -0.24\% & -0.27\% \\
%{\bf Consumption tax}$^*$& & &\\
% - Median agent & 0.05\% & 0.00\% & 0.05\%\\
% - Non-median agent & 0.13\% & 0.00\% & -0.01\%\\
%{\bf Labor tax } & & &\\
% - Median agent & 1.69\% & 0.21\% & 0.58\% \\
% - Non-median agent & -1.29\% & -0.14\% & -0.25\% \\
%% & & &\\
%%\hline
%\end{tabular}}
%\vspace{0.1cm}
%\caption{Welfare losses as percentage-of-consumption flows when the
% economy switches from a steady state with both consumption and
% income taxes to a one-tax system.\newline $^*$Computational
% difficulties in this case led us to using ratios of 0.95 rather than
% 0.85 for the three consumption cases.\label{t2tr1} }
%\end{table}
%\clearpage
%{\samepage{
%\begin{table}[h]
%\centerline{
%\begin{tabular}{l||r|r|r|r|r|r|}
%%\hline
%% & & & & & & \\
% & $\frac{TotTax}{GDP}$ & $\frac{IncTax}{GDP}$ &
% $\frac{Tax}{GDP}$ &
%GDP & $\frac{IncTax}{TaxRev}$ & $\frac{ConsTax}{TaxRev}$ \\
% & & & & & & \\
%\hline
%\hline
% & & & & & & \\
%$\frac{TotTax}{GDP}$ & 1.000 & & & & & \\
% & & & & & & \\
%$\frac{IncTax}{GDP}$ & 0.820 & 1.000 & & & & \\
% & & & & & & \\
%$\frac{ConsTax}{GDP}$ & 0.554 & 0.082 & 1.000 & & & \\
% & & & & & & \\
%GDP & 0.459 & 0.512 & -0.048 & 1.000 & & \\
% & & & & & & \\
%$\frac{IncTax}{TaxRev}$ & 0.201 & 0.692 & -0.610 & 0.365 & 1.000 & \\
% & & & & & & \\
%$\frac{ConsTax}{TaxRev}$ & -0.051 & -0.528 & 0.777 & -0.419 & -0.931 & 1.000\\
% & & & & & & \\
%%\hline
%\end{tabular}}
%\caption{\label{corrdat}}
%\noindent
%Tax, revenue, and output correlations in the OECD Countries:
%1985. Source: {\em
% Revenue Statistics of OECD Member Countries}, OECD, Paris, 1991, and
%\citeA{Summers-Heston-91}.
%\end{table}
%\vspace{0.1cm}
\end{document}
\begin{table}[h]
\centerline{
\begin{tabular}{|l||r|r|r|r|r|}
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{5}{|c|}{Median Wealth/Average Wealth}\\
\hline
&0.99 &0.95 &0.90 &0.85 &0.67 \\
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{5}{|c|}{Median Labor Efficiency/Average Labor Efficiency}\\
\hline
&0.99 &0.95 &0.90 &0.85 &0.67 \\
\hline
\hline
{\bf Income} & & & & & \\
- Tax Rate &1.253\% &6.023\% &11.502\% &16.527\% &32.111\% \\
- Output &98.738\% &93.879\% &88.191\% &82.879\% &65.872\% \\
{\bf Consumption}& & & & & \\
- Tax Rate &1.973\% &10.272\% &21.639\% &34.204\% &88.210\% \\
- Output &98.620\% &93.211\% &86.699\% &80.483\% &61.525\% \\
{\bf Labor } & & & & & \\
- Tax Rate &1.320\% &6.388\% &12.287\% &17.764\% &35.019\% \\
- Output &99.060\% &95.385\% &90.966\% &86.719\% &72.355\% \\
{\bf Capital } & & & & & \\
- Tax Rate &4.664\% & & & & \\
- Output &98.582\% & & & & \\
\hline
\end{tabular}}
\vspace{0.1cm}
\caption{ Endogenous Cash Transfers: Steady State Tax Rates and
Output associated with a situation where all agents have the same
ration of human to non-human wealth. \label{tdedw}}
\end{table}
\begin{table}[h]
\centerline{
\begin{tabular}{|l||r|r|r|r|r|}
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{5}{|c|}{Median Wealth/Average Wealth}\\
\hline
&1.00 &1.00 &1.00 &1.00 &1.00 \\
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{5}{|c|}{Median Labor Efficiency/Average Labor Efficiency}\\
\hline
&0.99 &0.95 &0.90 &0.85 &0.67 \\
\hline
\hline
{\bf Income } & & & & & \\
- Tax Rate &1.161\% &5.629\% &10.854\% &15.730\% &31.304\% \\
- Output &98.830\% &94.284\% &88.870\% &83.727\% &66.770\% \\
{\bf Consumption} & & & & & \\
- Tax Rate &1.418\% &7.248\% &14.922\% &23.079\% &58.280\% \\
- Output &99.004\% &95.112\% &90.433\% &85.938\% &70.763\% \\
{\bf Labor} & & & & & \\
- Tax Rate &1.398\% &6.758\% &12.985\% &18.753\% &36.826\% \\
- Output &99.004\% &95.112\% &90.432 \%&85.937\% &70.758\% \\
{\bf Capital } & & & & & \\
- Tax Rate &1.457\% &7.111\% &13.811\% & & \\
- Output &99.565\% &97.805\% &95.551\% & & \\
\hline
\end{tabular}}
\vspace{0.1cm}
\caption{ Endogenous Cash Transfers: Steady State Tax Rates and
Output associated with a situation where all agents have the same
wealth but differ in their labor efficiency. \label{tdesw}}
\end{table}
\clearpage
\begin{table}
\centerline{
\begin{tabular}{|l||r|r|r|r|r|}
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{5}{|c|}{Median Wealth/Average Wealth}\\
\hline
&0.99 &0.95 &0.90 &0.85 &0.67 \\
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{5}{|c|}{Median Labor Efficiency/Average Labor Efficiency}\\
\hline
&1.00 &1.00 &1.00 &1.00 &1.00 \\
\hline
\hline
{\bf Income} & & & & & \\
- Tax Rate &0.095\% &0.469\% &0.925\% &1.369\% &2.896\% \\
- Output &99.904\% &99.527\% &99.068\% &98.621\% &97.074\% \\
{\bf Consumption} & & & & & \\
- Tax Rate &0.545\% &2.775\% &5.676\% &8.720\% &21.554\% \\
- Output &99.614\% &98.070\% &96.131\% &94.178\% &86.744\% \\
{\bf Labor} & & & & & \\
- Tax Rate &-0.078\% &-0.393\% &-0.788\% &-1.184\% &-2.264\% \\
- Output &100.055\%&100.278\%&100.557\%&100.836\% &101.862\% \\
{\bf Capital } & & & & & \\
- Tax Rate &3.228\% &16.717\% & & & \\
- Output &99.027\% &94.510\% & & & \\
\hline
\end{tabular}}
\vspace{0.1cm}
\caption{ Endogenous Cash Transfers: Steady State Tax Rates and
Output associated with a situation where all agents have the same
labor efficiency but differ in their wealth. \label{tsedw}}
\end{table}
\begin{table}[h]
\centerline{
\begin{tabular}{|l||r|r|r|r|r|}
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{5}{|c|}{Median Wealth/Average Wealth}\\
\hline
&0.99 &0.95 &0.90 &0.85 &0.67 \\
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{5}{|c|}{Median Labor Efficiency/Average Labor Efficiency}\\
\hline
&0.99 &0.95 &0.90 &0.85 &0.67 \\
\hline
\hline
{\bf Income} & & & & & \\
- Tax Rate &1.253\% &6.023\% &11.502\% &16.527\% &32.111\% \\
- Output &100.00\% &100.00\% &100.00\% &100.00\% &100.00\% \\
{\bf Consumption}& & & & & \\
- Tax Rate &1.250\% & 5.933\% &11.090\% &15.510\% &26.176\% \\
- Output &100.39\% &102.22\% &105.12\% &108.70\% &128.04\% \\
{\bf Labor} & & & & & \\
- Tax Rate &1.525\% &7.321\% &13.961\% &20.039\% &39.152\% \\
- Output &100.18\% &100.87\% &101.69\% &102.45\% &104.25\% \\
{\bf Capital } & & & & & \\
- Tax Rate &6.782\% &29.198\% &50.595\% &70.265\% & \\
- Output &99.16\% &95.37\% &88.90\% &76.40\% & \\
\hline
\end{tabular}}
\vspace{0.1cm}
\caption{ Exogenous Cash Transfers: Value of the steady tax rates and
output associated with a predetermined level of transfers for
different type of taxes. All agents have the same ratio of human to
non-human wealth.\label{texct} }
\end{table}
\clearpage
\begin{table}[h]
\centerline{
\begin{tabular}{|l||c|c|c|}
\hline
& & &\\
&Same ratio of human&Same wealth, &Same efficiency\\
&to non-human wealth&different efficiency &different wealth\\
&$\frac{A_m}{A_a}=0.85,\frac{\epsilon_m}{\epsilon_a}=0.85$
& $\frac{A_m}{A_a}=1,\frac{\epsilon_m}{\epsilon_a}=0.85$&
$\frac{A_m}{A_a}=0.85,\frac{\epsilon_m}{\epsilon_a}=1$ \\
& & &\\
\hline
\hline
{\bf Income Tax}& & &\\
{\bf Replaced with}& & &\\
& & &\\
{\bf Consumption Tax} & & &\\
- Median Agent & 0.05\% & 0.03\% & 0.10\%\\
- Non Median Agent & 1.98\% & 0.21\% & 0.29\%\\
{\bf Labor Tax} & & &\\
- Median Agent & 0.01\% & -0.10\% & -0.00\%\\
- Non Median Agent & -0.40\% & 0.04\% & -0.01\%\\
& & &\\
\hline
{\bf Consumption Tax }& & &\\
{\bf Replaced with}& & &\\
& & &\\
{\bf Income Tax} & & &\\
- Median Agent & 0.38\% & 0.19\% & -0.06\%\\
- Non Median Agent & -2.10\% & -0.24\% & -0.26\%\\
{\bf Labor Tax} & & &\\
- Median Agent & 0.51\% & 0.21\% & -0.06\%\\
- Non Median Agent & -2.35\% & -0.14\% & -0.27\%\\
& & &\\
\hline
{\bf Labor Tax } & & &\\
{\bf Replaced with} & & &\\
& & &\\
{\bf Income Tax} & & &\\
- Median Agent & -0.00\% & 0.12\% & 0.00\%\\
- Non Median Agent & 0.39\% & -0.06\% & 0.00\%\\
{\bf Consumption Tax}$^*$& & &\\
- Median Agent & 0.02\% & 0.07\% & 0.04\%\\
- Non Median Agent & 1.20\% & 0.05\% & 0.13\%\\
& & &\\
\hline
\end{tabular}}
\vspace{0.1cm}
\caption{Welfare losses in consumption flows if the economy switched
from a situation of steady state with only one tax to a system with
another tax system.\newline $^*$ Computational difficulties of
this case lead us to the consideration of ratios of .90 rather than
.85 for the three consumption cases.\label{twlltr} }
\end{table}
\clearpage
\newpage
\begin{table}[h]
\vspace{0.3cm}
\centerline{
\begin{tabular}{|l||r|r|r|r|}
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{4}{|c|}{Median Wealth/Average Wealth}\\
\hline
&0.99 &0.95 &0.90 &0.85 \\
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{4}{|c|}{Median Labor Efficiency/Average Labor Efficiency}\\
\hline
&0.99 &0.95 &0.90 &0.85 \\
\hline
\hline
& & & & \\
{\bf Cons Tax Rate}& 4.001\%& 21.037\%& 44.872\%& 71.876\% \\
{\bf Income Tax Rate}&-2.013\%&-10.110\%&-20.354\%& -30.772\% \\
{\bf Output }& 99.218\%& 96.078\%& 92.134\%& 88.198\% \\
{\bf Trans/Output }& 1.545\%& 8.428\%& 18.730\%& 31.126\% \\
& & & & \\
\hline
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{4}{|c|}{Median Wealth/Average Wealth}\\
\hline
&1.00 &1.00 &1.00 &1.00 \\
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{4}{|c|}{Median Labor Efficiency/Average Labor Efficiency}\\
\hline
&0.99 &0.95 &0.90 &0.85 \\
\hline
\hline
& & & & \\
{\bf Cons Tax Rate}& 1.417\%& 7.244\%& 14.915\%& 23.072\% \\
{\bf Income Tax Rate}& 0.001\%& 0.004\%& 0.006\%& 0.007\% \\
{\bf Output } & 99.004\%& 95.111\%& 90.431\%& 85.936\% \\
{\bf Trans/Output} & 1.105\%& 5.648\%& 11.629\%& 17.987\% \\
& & & & \\
\hline
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{4}{|c|}{Median Wealth/Average Wealth}\\
\hline
&0.99 &0.95 &0.90 &0.85 \\
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{4}{|c|}{Median Labor Efficiency/Average Labor Efficiency}\\
\hline
&1.00 &1.00 &1.00 &1.00 \\
\hline
\hline
& & & & \\
{\bf Cons Tax Rate }& 2.556\%& 13.208\%& 27.494\%& 42.964\% \\
{\bf Income Tax Rate}&-2.023\%&-10.350\%&-21.321\%& -32.989\% \\
{\bf Output } &100.217\%&101.031\%&101.936\%& 102.734\% \\
{\bf Trans/Output} & 0.416\%& 2.203\%& 4.711\%& 7.541\% \\
& & & & \\
\hline
\end{tabular}}
\vspace{0.1cm}
\caption{Steady States of economies with cash transfers and two tax
systems available. \label{t2tst} }
\end{table}
\clearpage
{\samepage
\begin{table}[h]
%\centerline{\bf Welfare Losses from switching to a two tax system}
%\vspace{1.cm}
\centerline{
\begin{tabular}{|l||c|c|c|}
\hline
& & &\\
&Same ratio of human&Same wealth, &Same efficiency\\
{\bf Both Taxes Replace} &to non-human wealth&different efficiency
&different wealth\\
&$\frac{A_m}{A_a}=0.85,\frac{\epsilon_m}{\epsilon_a}=0.85$
& $\frac{A_m}{A_a}=1,\frac{\epsilon_m}{\epsilon_a}=0.
85$&
$\frac{A_m}{A_a}=0.85,\frac{\epsilon_m}{\epsilon_a}=1$
\\
& & &\\
\hline
\hline
& & &\\
{\bf Income Tax} & & &\\
- Median Agent & 0.01\% & -0.00\% & 0.11\% \\
- Non Median Agent & 1.29\% & 0.27\% & 0.45\% \\
{\bf Consumption Tax} & & &\\
- Median Agent & -0.14\% & 0.00\% & -0.04\%\\
- Non Median Agent & -0.92\% & 0.00\% & 0.12\%\\
{\bf Labor Tax} & & &\\
- Median Agent & -0.01\% & 0.12\% & 0.09\%\\
- Non Median Agent & 1.84\% & 0.19\% & 0.47\%\\
& & &\\
\hline
\end{tabular}}
\vspace{0.1cm}
\caption{\bf Welfare losses in consumption flows if the economy switched
from a situation of steady state with one type of taxes to a
system with both consumption and income taxes. \label{t1tr2} }
\end{table}
\begin{table}[h]
\centerline{
\begin{tabular}{|l||c|c|c|}
\hline
& & &\\
{\bf Both Taxes } &Same ratio of human&Same wealth, &Same efficiency\\
{\bf Are Replaced with} &to non-human wealth&different efficiency
&different wealth\\
&$\frac{A_m}{A_a}=0.85,\frac{\epsilon_m}{\epsilon_a}=0.85$
& $\frac{A_m}{A_a}=1,\frac{\epsilon_m}{\epsilon_a}=0.
85$&
$\frac{A_m}{A_a}=0.85,\frac{\epsilon_m}{\epsilon_a}=1$
\\
& & &\\
\hline
\hline
& & &\\
{\bf Income Tax} & & &\\
- Median Agent & 1.50\% & 0.20\% & 0.59\% \\
- Non Median Agent & -1.12\% & -0.24\% & -0.27\% \\
{\bf Consumption Tax}$^*$& & &\\
- Median Agent & 0.05\% & 0.00\% & 0.05\%\\
- Non Median Agent & 0.13\% & 0.00\% & -0.01\%\\
{\bf Labor Tax } & & &\\
- Median Agent & 1.69\% & 0.21\% & 0.58\% \\
- Non Median Agent & -1.29\% & -0.14\% & -0.25\% \\
& & &\\
\hline
\end{tabular}}
\vspace{0.1cm}
\caption{{\bf Welfare losses in consumption flows if the economy switched
from a situation of steady state with both consumption and income
taxes to a one tax system.}\newline $^*$ Computational difficulties of
this case lead us to the consideration of ratios of .95 rather than
.85 for the three consumption cases.\label{t2tr1} }
\end{table}
\clearpage
{\samepage{
\begin{table}[h]
\centerline{
\begin{tabular}{|l||r|r|r|r|r|r|}
\hline
& & & & & & \\
& $\frac{TotTax}{GDP}$ & $\frac{IncTax}{GDP}$ &
$\frac{Tax}{GDP}$ &
GDP & $\frac{IncTax}{TaxRev}$ & $\frac{ConTax}{TaxRev}$ \\
& & & & & & \\
\hline
\hline
& & & & & & \\
$\frac{TotTax}{GDP}$ & 1.000 & & & & & \\
& & & & & & \\
$\frac{IncTax}{GDP}$ & 0.820 & 1.000 & & & & \\
& & & & & & \\
$\frac{ConTax}{GDP}$ & 0.554 & 0.082 & 1.000 & & & \\
& & & & & & \\
GDP & 0.459 & 0.512 & -0.048 & 1.000 & & \\
& & & & & & \\
$\frac{IncTax}{TaxRev}$ & 0.201 & 0.692 & -0.610 & 0.365 & 1.000 & \\
& & & & & & \\
$\frac{ConTax}{TaxRev}$ & -0.051 & -0.528 & 0.777 & -0.419 & -0.931 & 1.000\\
& & & & & & \\
\hline
\end{tabular}}
\caption{\label{tdata} Correlations of Taxes in The OECD Countries:
1985. Source: {\em
Revenue Statistics of OECD Member Countries}, OECD, Paris, 1991, and
\citeA{Summers-Heston-88}.\label{corrdat}
}
\end{table}
\vspace{0.1cm}
}}
\end{document}
Victor,
I computed the utilities of the two agents in in the steady states for tables
2,3,4,5 and 7. Each column below reports:
1) Median Wealth/Average Wealth
2) Median Labor Efficiency/ Average Labor Efficiency
3) TaxRate
4) Utility of Median Agent
5) Utility of Second Agent
In columns 6) and 7) I added to the utilities of columns 4) and 5)
the value of 10 so that the level of the utilities are not negative.
I will send you the appendix when I am finished with it.
Vincenzo.
********************************************************************
\begin{table}[h]
\centerline{
\begin{tabular}{|l||r|r|r|r|r|}
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{5}{|c|}{Median Wealth/Average Wealth}\\
\hline
&0.99 &0.95 &0.90 &0.85 &0.67 \\
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{5}{|c|}{Median Labor Efficiency/Average Labor Efficiency}\\
\hline
&0.99 &0.95 &0.90 &0.85 &0.67 \\
\hline
\hline
{\bf Income} & & & & & \\
- Tax Rate &1.253\% &6.023\% &11.502\% &16.527\% &32.111\% \\
- Output &98.738\% &93.879\% &88.191\% &82.879\% &65.872\% \\
{\bf Consumption}& & & & & \\
- Tax Rate &1.973\% &10.272\% &21.639\% &34.204\% &88.210\% \\
- Output &98.620\% &93.211\% &86.699\% &80.483\% &61.525\% \\
{\bf Labor } & & & & & \\
- Tax Rate &1.320\% &6.388\% &12.287\% &17.764\% &35.019\% \\
- Output &99.060\% &95.385\% &90.966\% &86.719\% &72.355\% \\
{\bf Capital } & & & & & \\
- Tax Rate &4.664\% & & & & \\
- Output &98.582\% & & & & \\
\hline
\end{tabular}}
\vspace{0.1cm}
\caption{Endogenous cash transfers: steady-state tax rates and
output when all agents have the same
ratio of human to non-human wealth. \label{tdedw}}
\end{table}
\begin{table}[h]
\vspace{0.3cm}
\centerline{
\begin{tabular}{|l||r|r|r|r|}
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{4}{|c|}{Median Wealth/Average Wealth}\\
\hline
&0.99 &0.95 &0.90 &0.85 \\
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{4}{|c|}{Median Labor Efficiency/Average Labor Efficiency}\\
\hline
&0.99 &0.95 &0.90 &0.85 \\
\hline
\hline
& & & & \\
{\bf Cons. Tax Rate}& 4.001\%& 21.037\%& 44.872\%& 71.876\% \\
{\bf Income Tax Rate}&-2.013\%&-10.110\%&-20.354\%& -30.772\% \\
{\bf Output }& 99.218\%& 96.078\%& 92.134\%& 88.198\% \\
{\bf Transfer/Output }& 1.545\%& 8.428\%& 18.730\%& 31.126\% \\
& & & & \\
\hline
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{4}{|c|}{Median Wealth/Average Wealth}\\
\hline
&1.00 &1.00 &1.00 &1.00 \\
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{4}{|c|}{Median Labor Efficiency/Average Labor Efficiency}\\
\hline
&0.99 &0.95 &0.90 &0.85 \\
\hline
\hline
& & & & \\
{\bf Cons. Tax Rate}& 1.417\%& 7.244\%& 14.915\%& 23.072\% \\
{\bf Income Tax Rate}& 0.001\%& 0.004\%& 0.006\%& 0.007\% \\
{\bf Output } & 99.004\%& 95.111\%& 90.431\%& 85.936\% \\
{\bf Transfer/Output} & 1.105\%& 5.648\%& 11.629\%& 17.987\% \\
& & & & \\
\hline
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{4}{|c|}{Median Wealth/Average Wealth}\\
\hline
&0.99 &0.95 &0.90 &0.85 \\
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{4}{|c|}{Median Labor Efficiency/Average Labor Efficiency}\\
\hline
&1.00 &1.00 &1.00 &1.00 \\
\hline
\hline
& & & & \\
{\bf Cons. Tax Rate }& 2.556\%& 13.208\%& 27.494\%& 42.964\% \\
{\bf Income Tax Rate}&-2.023\%&-10.350\%&-21.321\%& -32.989\% \\
{\bf Output } &100.217\%&101.031\%&101.936\%& 102.734\% \\
{\bf Transfer/Output} & 0.416\%& 2.203\%& 4.711\%& 7.541\% \\
& & & & \\
\hline
\end{tabular}}
\vspace{0.1cm}
\caption{Steady states for economies with cash transfers and two taxes
available. \label{t2tst} }
\end{table}
**TABLE 2**
MedWea MedLab TaxRate Util1 Util2 Util3 Util4
INCOME TAXES
0.99 0.99 0.01253 -1.16044 -1.14626 8.83956 8.85374
0.95 0.95 0.06023 -1.19832 -1.13048 8.80168 8.86952
0.90 0.90 0.11502 -1.24591 -1.11693 8.75409 8.88307
0.85 0.85 0.16527 -1.29364 -1.10911 8.70636 8.89089
0.67 0.67 0.32111 -1.46294 -1.11932 8.53706 8.88068
CONSUMPTION TAXES
0.99 0.99 0.01973 -1.15995 -1.14587 8.84005 8.85413
0.95 0.95 0.10272 -1.19571 -1.13029 8.80429 8.86971
0.90 0.90 0.21639 -1.24031 -1.12059 8.75969 8.87941
0.85 0.85 0.34204 -1.28464 -1.12012 8.71536 8.87988
0.67 0.67 0.88210 -1.43372 -1.16958 8.56628 8.83042
LABOR TAXES
0.99 0.99 0.01320 -1.15940 -1.14520 8.84060 8.85480
0.95 0.95 0.06388 -1.19317 -1.12493 8.80683 8.87507
0.90 0.90 0.12287 -1.23581 -1.10540 8.76419 8.89460
0.85 0.85 0.17764 -1.27883 -1.09135 8.72117 8.90865
0.67 0.67 0.35019 -1.43386 -1.07742 8.56614 8.92258
CAPITAL TAXES
0.99 0.99 0.04664 -1.16247 -1.14822 8.83753 8.85178
**TABLE 3**
MedWea MedLab TaxRate Util1 Util2 Util3 Util4
INCOME TAXES
1.00 0.99 0.01161 -1.15898 -1.14734 8.84102 8.85266
1.00 0.95 0.05629 -1.19088 -1.13534 8.80912 8.86466
1.00 0.90 0.10854 -1.23066 -1.12560 8.76934 8.87440
1.00 0.85 0.15730 -1.27014 -1.12089 8.72986 8.87911
1.00 0.67 0.31304 -1.40579 -1.13966 8.59421 8.86034
CONSUMPTION TAXES
1.00 0.99 0.01418 -1.15817 -1.14655 8.84183 8.85345
1.00 0.95 0.07248 -1.18669 -1.13156 8.81331 8.86844
1.00 0.90 0.14922 -1.22192 -1.11845 8.77808 8.88155
1.00 0.85 0.23079 -1.25649 -1.11069 8.74351 8.88931
1.00 0.67 0.58280 -1.37048 -1.11951 8.62952 8.88049
LABOR TAXES
1.00 0.99 0.01398 -1.15817 -1.14655 8.84183 8.85345
1.00 0.95 0.06758 -1.18669 -1.13156 8.81331 8.86844
1.00 0.90 0.12985 -1.22192 -1.11845 8.77808 8.88155
1.00 0.85 0.18753 -1.25650 -1.11069 8.74350 8.88931
1.00 0.67 0.36826 -1.37049 -1.11954 8.62951 8.88046
CAPITAL TAXES
1.00 0.99 0.01457 -1.15821 -1.14645 8.84179 8.85355
1.00 0.95 0.07111 -1.18782 -1.12919 8.81218 8.87081
1.00 0.90 0.13811 -1.22666 -1.10970 8.77334 8.89030
**TABLE 4**
MedWea MedLab TaxRate Util1 Util2 Util3 Util4
INCOME TAXES
0.99 1.00 0.00095 -1.15246 -1.14989 8.84754 8.85011
0.95 1.00 0.00469 -1.15827 -1.14547 8.84173 8.85453
0.90 1.00 0.00925 -1.16554 -1.13999 8.83446 8.86001
0.85 1.00 0.01369 -1.17282 -1.13457 8.82718 8.86543
0.67 1.00 0.02896 -1.19915 -1.11552 8.80085 8.88448
CONSUMPTION TAXES
0.99 1.00 0.00545 -1.15278 -1.15023 8.84722 8.84977
0.95 1.00 0.02775 -1.15992 -1.14743 8.84008 8.85257
0.90 1.00 0.05676 -1.16893 -1.14461 8.83107 8.85539
0.85 1.00 0.08720 -1.17807 -1.14254 8.82193 8.85746
0.67 1.00 0.21554 -1.21293 -1.14245 8.78707 8.85755
LABOR TAXES
0.99 1.00 -0.00078 -1.15222 -1.14965 8.84778 8.85035
0.95 1.00 -0.00393 -1.15708 -1.14426 8.84292 8.85574
0.90 1.00 -0.00788 -1.16320 -1.13756 8.83680 8.86244
0.85 1.00 -0.01184 -1.16936 -1.13091 8.83064 8.86909
0.67 1.00 -0.02264 -1.19220 -1.10763 8.80780 8.89237
CAPITAL TAXES
0.99 1.00 0.03228 -1.15519 -1.15267 8.84481 8.84733
0.95 1.00 0.16717 -1.17459 -1.16284 8.82541 8.83716
**TABLE 5**
MedWea MedLab TaxRate Util1 Util2 Util3 Util4
INCOME TAXES
0.99 0.99 0.01253 -1.16044 -1.14626 8.83956 8.85374
0.95 0.95 0.06023 -1.19832 -1.13048 8.80168 8.86952
0.90 0.90 0.11502 -1.24591 -1.11693 8.75409 8.88307
0.85 0.85 0.16527 -1.29364 -1.10911 8.70636 8.89089
0.67 0.67 0.32111 -1.46294 -1.11932 8.53706 8.88068
CONSUMPTION TAXES
0.99 0.99 0.01250 -1.15930 -1.14513 8.84070 8.85487
0.95 0.95 0.05933 -1.19203 -1.12408 8.80797 8.87592
0.90 0.90 0.11090 -1.23208 -1.10173 8.76792 8.89827
0.85 0.85 0.15510 -1.27149 -1.08214 8.72851 8.91786
0.67 0.67 0.26176 -1.41368 -1.02061 8.58632 8.97939
LABOR TAXES
0.99 0.99 0.01525 -1.15959 -1.14542 8.84041 8.85458
0.95 0.95 0.07321 -1.19412 -1.12640 8.80588 8.87360
0.90 0.90 0.13961 -1.23771 -1.10914 8.76229 8.89086
0.85 0.85 0.20039 -1.28170 -1.09797 8.71830 8.90203
0.67 0.67 0.39152 -1.44144 -1.10103 8.55856 8.89897
CAPITAL TAXES
0.99 0.99 0.06782 -1.16454 -1.15034 8.83546 8.84966
0.95 0.95 0.29198 -1.22276 -1.15424 8.77724 8.84576
0.90 0.90 0.50595 -1.31010 -1.17835 8.68990 8.82165
0.85 0.85 0.70265 -1.44793 -1.25769 8.55207 8.74231
**TABLE 7**
\begin{table}[h]
\vspace{0.3cm}
\centerline{
\begin{tabular}{|l||r|r|r|r|}
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{4}{|c|}{Median Wealth/Average Wealth}\\
\hline
&0.99 &0.95 &0.90 &0.85 \\
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{4}{|c|}{Median Labor Efficiency/Average Labor Efficiency}\\
\hline
&0.99 &0.95 &0.90 &0.85 \\
\hline
\hline
& & & & \\
{\bf Cons. Tax Rate}& 4.001\%& 21.037\%& 44.872\%& 71.876\% \\
{\bf Income Tax Rate}&-2.013\%&-10.110\%&-20.354\%& -30.772\% \\
{\bf Output }& 99.218\%& 96.078\%& 92.134\%& 88.198\% \\
{\bf Transfer/Output }& 1.545\%& 8.428\%& 18.730\%& 31.126\% \\
& & & & \\
\hline
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{4}{|c|}{Median Wealth/Average Wealth}\\
\hline
&1.00 &1.00 &1.00 &1.00 \\
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{4}{|c|}{Median Labor Efficiency/Average Labor Efficiency}\\
\hline
&0.99 &0.95 &0.90 &0.85 \\
\hline
\hline
& & & & \\
{\bf Cons. Tax Rate}& 1.417\%& 7.244\%& 14.915\%& 23.072\% \\
{\bf Income Tax Rate}& 0.001\%& 0.004\%& 0.006\%& 0.007\% \\
{\bf Output } & 99.004\%& 95.111\%& 90.431\%& 85.936\% \\
{\bf Transfer/Output} & 1.105\%& 5.648\%& 11.629\%& 17.987\% \\
& & & & \\
\hline
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{4}{|c|}{Median Wealth/Average Wealth}\\
\hline
&0.99 &0.95 &0.90 &0.85 \\
\hline
\multicolumn{1}{|c||}{ } &
\multicolumn{4}{|c|}{Median Labor Efficiency/Average Labor Efficiency}\\
\hline
&1.00 &1.00 &1.00 &1.00 \\
\hline
\hline
& & & & \\
{\bf Cons. Tax Rate }& 2.556\%& 13.208\%& 27.494\%& 42.964\% \\
{\bf Income Tax Rate}&-2.023\%&-10.350\%&-21.321\%& -32.989\% \\
{\bf Output } &100.217\%&101.031\%&101.936\%& 102.734\% \\
{\bf Transfer/Output} & 0.416\%& 2.203\%& 4.711\%& 7.541\% \\
% & & & & \\
\hline
\end{tabular}}
\vspace{0.1cm}
\caption{Steady states for economies with cash transfers and two taxes
available. \label{t2tst} }
\end{table}