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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}

\begin{small}





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.







\end{small}

\end{quotation}





Tresch  

p. 39:





\begin{quotation}

\begin{small}

 

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}

\end{quotation}





\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,    $L_1<L_2$, because  Type 2's have higher wages.







The egalitarian social planner maximizes $U(C_1, S_1)$ subject to the feasibility and budget constraints plus  the constraint that 

\begin{equation} \label {e13}

 U(C_1, S_1) = U(C_2, S_2)

\end{equation}



 (This is not quite the Rawlsian objective function, but it amounts to the same thing in the present and most other contexts, and is just  as valid a social objective function.) 



\bigskip

   

\noindent

 {\bf 3.  The  Lump Sum Tax Problem}





Now, 

the government must choose the level $T$ of the lump-sum tax on Type 2 citizens and subsidy to Type 1 citizens.  This is a transfer in terms of the consumption good, and a citizen is not allowed to refuse to pay.   Being restricted to just one policy tool, it seems it may be difficult to control the three outcome variables, $C_1, C_2$,  and $S_1$, that  could be directly controlled in Section 2  (where once those three were chosen, $S_2$ followed from the budget constraint.  



  As $T$ increases, $S_1$ will increase and $S_2 $ will decrease. Lump sum taxes do not create inefficiency, but they are distortionary. The distortion arises because of the income effect. As the subsidy to Type 1 increases, he will consume more and work less. As the tax on Type 2 increases, he will consume less and work more. 



   The optimal $T$ will be strictly positive whether the government is egalitarian or utilitarian. Increasing $T$ slightly above zero will increase $L_2$ slightly. If $L_1$ does not decrease too much (will it?), total output will rise, and consumption of both types can increase. At any rate, $T$ can be increased, increasing the utlity of the Type 1's, until $U_1=U_2$. 



   

 



I think the Stiglitz proposition remains true. In fact, constraining the government to just one instrument does not change the outcome we found earlier-- this constraint is not binding. 



 The government now does not face a budget constraint-- or rather, we subsume it in our assumption that the entire tax of $T$ is transferred rather than some being wasted.   Nor is $T$ constrained directly. The government does face two incentive compatibility constraints, however: 

\begin{equation} \label {e14}

(C_1, S_1) = Argmax 

   U(C_1,S_1) \; subject\; to \; C_1  \leq   (\overline{S}-

S_1)w_1  +T, C_1  \geq 0,   S_1 \in [0, \overline{S}] 

\end{equation} 



\begin{equation} \label {e15}

 (C_2, S_2) = Argmax 

   U(C_2,S_2) \; subject\; to \; C_2  \leq   (\overline{S}-

S_2)w_2  +T, C_2  \geq 0,   S_2 \in [0, \overline{S}] 

\end{equation} 



   



   



The utilitarian  social planner maximizes  by choice of $T$, 

\begin{equation} \label {e16}

    U(C_1,S_1) + U(C_2, S_2) 

\end{equation} 

 subject to the two incentive compatibility constraints. 

  

 The two incentive compatibility constraints require that 



\begin{equation} \label {e17}

U^1_C - \lambda_1 =0

\end{equation} 

 and 

\begin{equation} \label {e18}

U^1_C -\lambda_1 w_1 =0

\end{equation} 

 so 

\begin{equation} \label {e19}

U^1_C   = \frac{U^1_S}{w_1} 

\end{equation} 



and also that 



\begin{equation} \label {e20}

U^2_C - \lambda_2   =0

\end{equation} 

 and 

\begin{equation} \label {e21}

U^2_S - \lambda_2 w_2 =0

\end{equation} 

so

 \begin{equation} \label {e22}

  U^2_C =  \frac{U^2_S}{w_2}

\end{equation} 

 

 Thus, we already will have two of the three equations in the Social Planner's Problem satisfied. We are left with one equation to be satisfied, and one control variable, $T$, which looks quite feasible. 

The two equations already satisfied restrict us to a one-dimensional set of $(U^1, U^2)$ combinations, monotonic in two dimensional space. The variable $T$ can be chosen to pick any of these. So we can pick the same utilitarian one as the SOcial Planner did, or the same Egalitarian one. 

 

  

 These may both be special cases, in a sense though. I'll have to explore it. It is very helpful that the  players' individual optimization condition-- trading off consumption against effort-- are independent of each other. An objective mixing equality and high utility might give a different result.  

 



A third thing to look at: equal political power of each group. Look at

it as a Nash bargaining problem. Also: what happens if one group  has

all the power?



A DIFFERNET MODEL: 



 The utilitarian result: depends on equal consuming power. If it is

ability to consume that differs across individuals, then we get a

different result. Do this: same labor supply,b ut different consumption

prices.



The standard case is a bit strange, because it has the rich being better

at producing output, but equal at consuming leisure and consumptino.



If someone has high marginal product at work, wouldn't he also have it

at home, in his leisure? Consider, for example, home work. Although not

paid, productivity clearly differs.



 How would we model this? That leisure enters the utility function at a

different rate for talented and untalented-- at alpha*l instead of l.



Lump sum taxes will, of course, induce rentseeking too. By analogy with

monopoly, these may be the biggest losses.

















   

\noindent

 {\bf References}







Non-Leaky Buckets: Optimal Redistributive Taxation and Agency Costs,

with Andrew Lyon,

Journal of Public Economics 58 (1995), 365-90.

Abstract. Economists generally have argued that income redistribution

comes at a cost in

aggregate income. We provide a counter-example in a model where private

information gives rise to

incentive constraints. In the model, a wage tax creates the usual

distortion in labor-leisure choices,

but the redistributive grants that it finances reduce a distortion in

investment in higher education. We

prove that simple redistributive policies can yield Pareto improvements

and increase aggregate

income. Indeed, redistributive policies are more effective in increasing

efficiency than corrective

taxes or subsidies under most circumstances where higher education is

beyond the reach of the poor.



Salanie The economics of taxation



 Tresch public ecnomics a normatiav theory 2nd edition



Myles Public Economics



Jha modern public eocnomics



 





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