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{\bf Optimal Taxation with Lump Sum Taxes and Perfect Information } \\
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June 14, 2004 \\
\bigskip
Eric Rasmusen \\
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{\it Abstract}
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Lump sum taxes do have effects on labor supply. The social planner's solution by direct choice of labor and leisure is the same as the lump sum taxer's solution by choice of a single tax/transfer amount.
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\begin{small}
\noindent \hspace*{20pt} Indiana University Foundation Professor,
Department of Business Economics and Public Policy, Kelley School of
Business, Indiana University, BU456, 1309 E. 10th Street, Bloomington,
Indiana, 47405-1701. Office: (812) 855-9219. Fax: 812-855-3344
Erasmuse@indiana.edu. \\
Mypage.iu.edu/$\sim$erasmuse,
Mypage.iu.edu/$\sim$erasmuse/papers/lumpsumtaxes-rasmusen.pdf.
\noindent I thank Tillman Klumpp and the IU Game Theory Group for their comments.
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\noindent
{\bf 1. Introduction}
http://economics.about.com/library/glossary/bldef-lump-sum-
tax.htm.
\begin{quotation}
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Definition of The Lump Sum Tax / Lump Sum Taxes: A lump sum tax is a tax
of a fixed amount that has to be paid by everyone regardless of the
level of his or her income. Lump sum taxes are considered efficient
taxes because they do not influence a person’s decision on how much to
work.
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\end{quotation}
Tresch
p. 39:
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Notice two qualities that lump-sum taxes and transfers {\it do not} possess}. First, it is not true that lump-sum redistributions have no effect on economic activity. Any redistribution program can be expected, at the very least, to shift individuals' demands for goods or supplies of factors, with obvious repercussions throughout the entire economy....
\end{small}
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\noindent
{\bf 2. The Model}
All citizens have the same utility function, $U(C, S)$, which increases in a citizen's consumption, $C$, and his leisure, $S$. Each citizen begins with $\overline{S}$ in leisure, which he may use up to produce the consumption good. Type 2 citizens have higher marginal products of labor (``wages'') than type 1 citizens: $w_2 > w_1$. The number of each type of citizen is equal.
It will be convenient to talk sometimes in terms of labor rather than leisure, so let us define $L = \overline{S} -S$.
\noindent
{\bf 2a. Laissez Faire}
With no government intervention, what happens depends on income effects. The Type 2 citizens will work harder and consume more if both leisure and consumption are normal goods, because his wage is higher and the price of consumption is the same for both types. His utility will be higher too-- whether consumption is a normal good or not.
\noindent
{\bf 2b. The Social Planner's Problem}
First, let us follow Stiglitz and allow the government to choose $C_1, C_2, S_1$, and $S_2$. It must obey the budget constraint that
\begin{equation} \label{e1}
C_1 + C_2 = w_1 (\overline{S}-S_1) +w_2 (\overline{S}-S_2).
\end{equation}
and the feasibility constraints that
\begin{equation} \label{e2}
C_1 \geq 0, C_2 \geq 0, S_1 \in [0,\overline{S}] \;and \; S_2 \in [0,
\overline{S}].
\end{equation}
The government might be egalitarian, wishing to maximize utility subject to the constraint that utilities of the two types be equal, or utilitarian, simply maximizing the sum of their utility.
The utilitarian social planner maximizes
\begin{equation} \label {e3}
U(C_1,S_1) + U(C_2, S_2)
\end{equation}
The Lagrangian is
\begin{equation} \label {e4}
U(C_1,S_1) + U(C_2, S_2) - \lambda [C_1 + C_2 - (\overline{S}- S_1)
w_1+ (\overline{S}- S_2)w_2],
\end{equation}
which has first order conditions
\begin{equation} \label {e5}
U^1_C - \lambda =0
\end{equation}
\begin{equation} \label {e6}
U^2_C - \lambda =0
\end{equation}
\begin{equation} \label {e7}
U^1_S -\lambda w_1 =0
\end{equation}
\begin{equation} \label {e8}
U^2_S - \lambda w_2 =0
\end{equation}
Thus, we want
\begin{equation} \label {e9}
U^1_C = U^2_C = \frac{U^1_S}{w_1} = \frac{U^2_S}{w_2}
\end{equation}
To achieve this, we just need three independent equalities to be true:
\begin{equation} \label {e10}
U^1_c = U^2_c
\end{equation}
\begin{equation} \label {e11}
U^1_c = \frac{U^1_S}{w_1}
\end{equation}
and
\begin{equation} \label {e12}
U^2_c = \frac{U^2_S}{w_2}
\end{equation}
We can do that by choice of our three control variables.
Stiglitz Proposition: In the utilitarian regime, if leisure and consumption are both normal goods (which is to say, if they are complements and not subsitutes? ) then $U^1> U^2$.
Proof. Look at the first order conditions.
At the optimum, $L_1