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\begin{document}

\title{Topics in Dynamic Public Finance}
\author{John Hassler \\
%EndAName
Stockholm University and Fudan University}
\maketitle
\newpage

\section{Optimal unemployment insurance (UI)}

There is a large literature of optimal unemployment insurance. The basic
issue is how to provide the most efficient unemployment insurance when there
is a moral hazard problem. This is arising from an assumption that
unemployed individuals can affect the probability they find (and accept) a
job offer. However, it is costly for the worker to increase this
probability, e.g., because of effort costs, reduced reservatiobn wages or
opportunity costs of time.

\subsection{The semi-static approach to optimal UI}

The basic idea in Baily and Chetty is to simplify the dynamic problem into a
static one. This makes the model simple and tractable also when savings is
allowed. An important lesson is that when savings is allowed, we can use the
drop in consumption at unemployment as a measure of the welfare loss
associated with unemployment. In a dynamic model, this does not work when
there is no market for savings. Why? The trade-off faced by the planner is
to balance the loss of welfare associated with unemployment against the
negative effect on search induced by UI.

\subsubsection{The simplest model following Baily}

\begin{itemize}
\item At period zero, the individual chooses how much to consume of the
income, normalized to unity, and how much to save.

\item In the beginning of period 1, the individual becomes unemployed with
probability $\alpha $ and otherwise keeps his job.

\item During period one, the individual can determine how long it takes to
find a job by choosing the wage $y_{n}$ and costly search effort. A share $%
\beta =\beta \left( c,y_{n}\right) $ of the second period is spent working
in the new job.

\item While unemployed, the individual bets UI-benefits $b.$These are paid
by taxes on workers.

\item Agents have access to a market for precautionary (buffer stock)
savings.
\end{itemize}

Total income second period if laid off is therefore%
\begin{equation*}
\left( 1-\beta \right) \left( b-c\right) +\beta y_{n}\left( 1-\tau \right)
\equiv y_{l}.
\end{equation*}

In first periods, individuals decide how much to save, $s.$ Interest rate
and subjective discount rate is normalized to zero. Welfare is 
\begin{equation*}
V=u\left( 1-\tau -s\right) +\alpha u\left( 1-\tau +s\right) +\left( 1-\alpha
\right) \left( u\left( y_{l}+s\right) \right) .
\end{equation*}

Government budget constraint is%
\begin{eqnarray*}
\left( 1+\alpha +\left( 1-\alpha \right) \beta y_{n}\right) t &=&\left(
1-\alpha \right) \left( 1-\beta \right) b. \\
&\Longrightarrow &b=\frac{\left( 1+\alpha +\left( 1-\alpha \right) \beta
y_{n}\right) }{\left( 1-\alpha \right) \left( 1-\beta \right) }\tau \equiv
\mu \tau
\end{eqnarray*}%
Denoting the \emph{endogenoues} total income by $Y\equiv 1+\alpha +\left(
1-\alpha \right) \beta y_{n},$ this implies 
\begin{eqnarray*}
b &=&\frac{Y}{\left( 1-\alpha \right) \left( 1-\beta \right) }\tau \\
&\equiv &\mu \tau ,
\end{eqnarray*}
where we note that $\mu $ is not a constant, but depends on individual
choices of $y_{n}$ and $\beta $ and thus indirectly on taxes and benefits.
Given the budget constraint and individual choices, we can therefore write $%
\mu =\mu \left( \tau \right) $ (provided there is a solution, which is not
necessarily true for all $\tau .$Explain!)

Note that in first best, $c$ should be chosen to satisfy 
\begin{equation*}
\left( y_{n}+c\right) \beta _{c}=1-\beta
\end{equation*}%
since the marginal gain in aggregate income is $\left( y_{n}+c\right) $ and
the cost is $1-\beta .$ The individual instead gains, 
\begin{equation*}
y_{n}\left( 1-\tau \right) +c-b\text{ }
\end{equation*}%
so the private value of search is lower. Similarly, an increase in $y_{n}$
has benefits $\beta $ and costs $-\left( y_{n}+c\right) \beta _{y_{n}}.$
While private benefits are $\left( 1-\tau \right) \beta $ and provate costs $%
-\left( y_{n}\left( 1-\tau \right) +c-b\right) \beta _{y_{n}}.$

We can now write%
\begin{eqnarray*}
V &=&u\left( 1-\tau -s\right) +\alpha u\left( 1-\tau +s\right) +\left(
1-\alpha \right) \left( u\left( \left( 1-\beta \right) \left( \mu \tau
-c\right) +\beta y_{n}\left( 1-\tau \right) +s\right) \right) \\
V &=&V\left( c,y_{n},s,\mu ,\tau \right)
\end{eqnarray*}

The optimal UI system maximizes solves 
\begin{equation*}
\max_{\tau }V\left( c,y_{n},s,\mu \left( \tau \right) ,\tau \right)
\end{equation*}

Although, $c,y_{n},s$ are affected by $\tau ,$ these effects need not be
taken into account since by individual optimality, 
\begin{equation*}
V_{c}=V_{Y_{n}}=V_{s}=0.
\end{equation*}

This is the envelope theorem. Therefore, the first order condition for
maximizing $V$ by chosing $\tau $ is 
\begin{equation*}
\frac{dV}{d\tau }=V_{\mu }\frac{d\mu }{d\tau }+V_{\tau }=0,
\end{equation*}%
where

\begin{eqnarray*}
V_{\mu } &=&\left( 1-\alpha \right) u^{\prime }\left( c_{u}\right) \left(
1-\beta \right) \tau \\
V_{\tau } &=&-u^{\prime }\left( c_{1}\right) -au^{\prime }\left(
c_{2}\right) -\left( 1-\alpha \right) u^{\prime }\left( c_{u}\right) \beta
y_{n}+\left( 1-\alpha \right) u^{\prime }\left( c_{u}\right) \left( 1-\beta
\right) \mu ,
\end{eqnarray*}%
where $c_{1}=1-\tau -s$ is first period consumption, $c_{2}=1-\tau +s$ is
second period consumption if the job is retained and $c_{u}=\left( 1-\beta
\right) \left( \mu \tau -c\right) +\beta y_{n}\left( 1-\tau \right) +s$ is
second period consumption if the individual lost his job.

Note that by individual savings optimization (the Euler equation)%
\begin{eqnarray*}
u^{\prime }\left( c_{1}\right) &=&au^{\prime }\left( c_{2}\right) +\left(
1-\alpha \right) u^{\prime }\left( c_{u}\right) \\
u^{\prime }\left( c_{1}\right) -\left( 1-\alpha \right) u^{\prime }\left(
c_{u}\right) &=&au^{\prime }\left( c_{2}\right)
\end{eqnarray*}%
implying 
\begin{eqnarray*}
V_{\tau } &=&-u^{\prime }\left( c_{1}\right) -\left( u^{\prime }\left(
c_{1}\right) -\left( 1-\alpha \right) u^{\prime }\left( c_{u}\right) \right)
-\left( 1-\alpha \right) u^{\prime }\left( c_{u}\right) \beta y_{n}+\left(
1-\alpha \right) u^{\prime }\left( c_{u}\right) \left( 1-\beta \right) \mu \\
&=&-2u^{\prime }\left( c_{1}\right) +\left( 1-\alpha \right) \left( 1-\beta
y_{n}+\left( 1-\beta \right) \mu \right) u^{\prime }\left( c_{u}\right) .
\end{eqnarray*}

Approximating 
\begin{equation*}
u^{\prime }\left( c_{1}\right) \approx u^{\prime }\left( c_{u}\right)
+u^{\prime \prime }\left( c_{u}\right) \Delta c
\end{equation*}%
the first order condition is 
\begin{eqnarray*}
0 &=&\left( 1-\alpha \right) u^{\prime }\left( c_{u}\right) \left( 1-\beta
\right) \tau \frac{d\mu }{d\tau }-2\left( u^{\prime }\left( c_{u}\right)
+u^{\prime \prime }\left( c_{u}\right) \Delta c\right) \\
&&+\left( 1-\alpha \right) \left( 1-\beta y_{n}+\left( 1-\beta \right) \mu
\right) u^{\prime }\left( c_{u}\right) \\
2\left( 1+\frac{u^{\prime \prime }}{u^{\prime }}\Delta c\right) &=&\left(
1-\alpha \right) \left( 1-\beta \right) \tau \frac{d\mu }{d\tau }+\left(
1-\alpha \right) \left( 1-\beta y_{n}+\left( 1-\beta \right) \mu \right) \\
2\left( 1+\frac{u^{\prime \prime }}{u^{\prime }}\Delta c\right) &=&\left(
1-\alpha \right) \left( 1-\beta \right) \tau \frac{d\mu }{d\tau }+2 \\
\frac{u^{\prime \prime }}{u^{\prime }}\Delta c &=&\left( 1-\alpha \right)
\left( 1-\beta \right) \tau \frac{d\mu }{d\tau } \\
\frac{u^{\prime \prime }}{u^{\prime }}\frac{\Delta c}{Y} &=&\frac{\tau }{\mu 
}\frac{d\mu }{d\tau }
\end{eqnarray*}

Without moral hazard, $\frac{d\mu }{d\tau }=0,$ in which case optimality
requires $\Delta c=0.$ With moral hazard, higher taxes tends to reduce $\mu $
since the tax dependency ratio falls. $\frac{\tau }{\mu }\frac{d\mu }{d\tau }
$ is thus negative. Since $\frac{u^{\prime \prime }}{u^{\prime }}$ is
negative $\Delta c>0.$ We see that $\frac{\Delta c}{Y}$ increases if $\frac{%
\tau }{\mu }\frac{d\mu }{d\tau }$ is large in absolute terms and falls if $%
\frac{u^{\prime \prime }}{u^{\prime }}$ is large in absolute terms, i.e.,
risk aversion is large.

This approach has been generalized by Chetty showing that we can have
repeated spells of unemployment, uncertain spells of unemployment, value of
leisure, private insurance and borrowing constaints. The model can therefore
be extented to evaluate UI reforms. With a more dynamic model, and in
particular if capital markets are imperfect, it should be noted that one
needs how the whole consumption profile is affected by unemployment. The
drop at entering unemployment may not be enough. Shimer and Werning (2007),
shows that the \emph{reservation wage} can be used as a summary measure of
how bad unemployment is.

In any case, this the model is not suitable to analyze

\begin{enumerate}
\item General equilibrium effects like impacts on wages, search spillovers
and job creation.

\item Interaction with other taxes-fiscal spillovers.

\item Time varying benefits.
\end{enumerate}

\subsection{The dynamic approach with observable savings}

The seminal paper by Shavel \& Weiss (1979) focuses on the optimal time
profile of benefits. It is a simple infinite horizon discrete time model
where the aim is to maximize utility of a representative unemployed subject
to a government budget constraint. Utility is given by 
\begin{equation*}
\sum_{t=0}^{\infty }\left( \frac{1}{1+r}\right) ^{t}\left( u\left(
c_{t}\right) -e_{t}\right)
\end{equation*}%
where $c_{t}$ is period $t$ consumption and $e_{t}$ is a privately chosen
unobservable effort associated with job search. The subjective discount rate
is $r,$ which is assumed to coincide with an exogeneous interest rate.

It is assumed that the individual has no access to capital markets so $%
c_{t}=b_{t}$ when the individual is unemployed. After regaining employment,
the wage is $w$ forever.

When the individual becomes employed he stays employed for ever for
simplicity. Agents have no access to credit markets (or equivalently,
savings is perfectly monitored and benefits can be made contingent on them)
so the planner can perfectly control the consumption of the individual. The
mortal hazard problem is that individuals can affect the probability of
finding a job. As in Baily (1978), the indvidual controls both the search
effort (here called $e_{t})$ and the reservation wage (here $w_{t}^{\ast }$
).

Given an effort level $e_{t},$ the individual receives one job offer per
period with an associated wage drawn from a distribution with a time
invariant probability density $f\left( w_{t},e_{t}\right) .$ The probability
of finding an acceptable job in period $t$ is thus 
\begin{equation*}
p\left( w_{t}^{\ast },e_{t}\right) =\int_{w_{t}^{\ast }}^{\infty }f\left(
w_{t},e_{t}\right) dw_{t}
\end{equation*}%
with 
\begin{eqnarray*}
p_{w}\left( w_{t}^{\ast },e_{t}\right) &=&-f\left( w_{t},e_{t}\right) \leq 0%
\text{ and } \\
p_{e}\left( w_{t}^{\ast },e_{t}\right) &>&0
\end{eqnarray*}%
where the latter is by assumption.

Let $E_{t}$ be the expected utility of an unemployed individual that choose
optimally a sequence $\left\{ e_{t+s},w_{t+s}^{\ast }\right\} _{s=0}^{\infty
}.$ Define 
\begin{equation*}
u_{t}=u\left( w_{t}^{\ast },e_{t}\right) \equiv \frac{1+r}{r}%
\int_{w_{t}^{\ast }}^{\infty }u\left( w_{t},e_{t}\right) \frac{f\left(
w_{t},e_{t}\right) }{p\left( w_{t}^{\ast },e_{t}\right) }dw_{t}
\end{equation*}%
This is the expected utility from next period, \emph{conditional }on finding
a job this period, which starts next period.\emph{\ }We note that\emph{\ }%
\begin{eqnarray*}
u_{w}\left( w_{t}^{\ast },e_{t}\right) &\geq &0 \\
u_{e}\left( w_{t}^{\ast },e_{t}\right) &\geq &0.
\end{eqnarray*}

The first inequality follows from the fact that \emph{conditional} on
finding a job, wages are higher for higher reservation wages. The second
inequality is by assumption, higher search effort leads to no worse
acceptable job offers.

$E_{t}$ satisfies the standard Bellman equation%
\begin{equation*}
E_{t}=\max_{e_{t},w_{t}^{\ast }}u\left( b_{t}\right) -e_{t}+\frac{1}{1+r}%
\left( p\left( w_{t}^{\ast },e_{t}\right) u\left( w_{t}^{\ast },e_{t}\right)
+\left( 1-p\left( w_{t}^{\ast },e_{t}\right) \right) E_{t+1}\right)
\end{equation*}

The first-order conditions are%
\begin{eqnarray*}
e_{t};\frac{1}{1+r}\left( p_{e}\left( w_{t}^{\ast },e_{t}\right) \left(
u\left( w_{t}^{\ast },e_{t}\right) -E_{t+1}\right) +p\left( w_{t}^{\ast
},e_{t}\right) u_{e}\left( w_{t}^{\ast },e_{t}\right) \right) &=&1 \\
w_{t}^{\ast };-p_{w}\left( w_{t}^{\ast },e_{t}\right) \left( u\left(
w_{t}^{\ast },e_{t}\right) -E_{t+1}\right) &=&p\left( w_{t}^{\ast
},e_{t}\right) u_{w}\left( w_{t}^{\ast },e_{t}\right) .
\end{eqnarray*}

In the first equation, the LHS is the marginal benefit of higher search
effort, coming from a higher probability of finding a job and better jobs if
found. These balances the cost which is 1. In the second equation, the LHS
is the marginal cost of higher reservation wages, coming from a lower
probability of finding a job. The RHS is the gain, coming from better jobs
if accepted.

By the envelope theorem%
\begin{equation*}
\frac{dE_{t}}{dE_{t+1}}=\frac{\partial E_{t}}{\partial E_{t+1}}=\frac{%
1-p\left( w_{t}^{\ast },e_{t}\right) }{1+r}
\end{equation*}%
Now, anything that reduce $E_{t+1}$ will reduce $1-p\left( w_{t}^{\ast
},e_{t}\right) ,$ i.e., make hiring more likely. To see this, note that if $%
E_{t+1}$ falls. 
\begin{eqnarray*}
&&p_{e}\left( w_{t}^{\ast },e_{t}\right) \left( u\left( w_{t}^{\ast
},e_{t}\right) -E_{t+1}\right) +p\left( w_{t}^{\ast },e_{t}\right)
u_{e}\left( w_{t}^{\ast },e_{t}\right) ,\text{ and} \\
&&-p_{w}\left( w_{t}^{\ast },e_{t}\right) \left( u\left( w_{t}^{\ast
},e_{t}\right) -E_{t+1}\right)
\end{eqnarray*}%
both becomes larger if choices are unchanged. In words, the marginal benefit
of searching higher and the marginal cost of setting higher reservation
wages both increase. Thus, a reduction in $E_{t+1}$ increase search effort
and reduce the reservation wage increasing $p$.

Now, we can show that benefits should have a decresing profile.

Proof:

Suppose contrary that $b_{t}=b_{t+1}.$ Then consider an infinitessimal
increase in $b_{t}$ financed by an actuarially fair reduction in $b_{t+1}$,
that is 
\begin{equation*}
db_{t}=-\frac{1-p}{1+r}db_{t+1}>0
\end{equation*}%
where $p\left( w_{t}^{\ast },e_{t}\right) $ is calculated at the initial
(constant) benefit levels. The direct effect on felicitity levels (period
utilities) is 
\begin{eqnarray*}
&&u^{\prime }\left( b_{t}\right) db_{t}+\frac{1-p}{1+r}u^{\prime }\left(
b_{t+1}\right) db_{t+1} \\
&&-u^{\prime }\left( b_{t}\right) \frac{1-p}{1+r}db_{t+1}+\frac{1-p}{1+r}%
u^{\prime }\left( b_{t+1}\right) db_{t+1} \\
&=&0
\end{eqnarray*}%
since $u^{\prime }\left( b_{t}\right) =u^{\prime }\left( b_{t+1}\right) .$By
the envelope theorem, we need not take into acount changes in endogeneous
variables when calculating welfare. Therefore, $E_{t}$ is unchanged. Since $%
u\left( b_{t}\right) $ has increased, $E_{t+1}$ must have fallen. When
calculating the budgetary effects we need to into account the endogeneous
changes on $p.$

Let 
\begin{equation*}
B_{t}=b_{t}+\frac{1-p}{1+r}b_{t+1}
\end{equation*}

Then,%
\begin{eqnarray*}
dB_{t} &=&db_{t}+\frac{1-p}{1+r}db_{t+1}-\frac{dp}{1+r}b_{t+1} \\
&=&-\frac{dp}{1+r}b_{t+1}
\end{eqnarray*}

Since $E_{t+1}$ has fallen, $dp>0.$ Thus $dB_{t}<0.$ I.e., the cost of
providing utility $E_{t}$ has fallen. Equivalently, the insurance is more
efficient than the starting point $b_{t}=b_{t+1}.$

\subsubsection{Extentions}

Hopenhayn and Nicolini extend the model by Shavel \& Weiss in an important
dimension -- it enriches the policy space of the government by allowing
taxation of workers to be contingent on their unemployment history. It is
shown that the government should use this extra way of "punishing"
unemployment. The intuition is that relative to the first best, which is a
constant unemployment benefit, the government must "punish" unemployment.
Doing this by only reducing unemployment benefits is suboptimal, by
spreading the punishment of unsuccesful search over the entire future of the
individual, a more efficient insurance can be achived. I.e., lower cost of
providing a given utility level. It is shown that this may be quantitatively
important. Another contribution is to show that the problem can be
formulated in a recursive way with tha \emph{promised utility }as state
variable.

Using H\&N's notation, we assume that individuals can choose an unobservable
effort level $a_{t}$ that positively effect the hiring probability. In H\&N
1997, it is assumed that $p\left( a_{t}\right) $ is an concave and
increasing function and hiring is an absorbing state with a wage $w$
forever. In H\&N 2005, it is instead assumed that spells are repeated, with
an exogeneous separation probability $s$ and 
\begin{equation*}
p\left( a\right) =\left\{ 
\begin{array}{c}
p\text{ if }a=1 \\ 
0\text{ otherwise}%
\end{array}%
\right.
\end{equation*}%
The individual has a utility function 
\begin{equation*}
E\sum_{t=0}^{\infty }\left( \frac{1}{1+r}\right) ^{t}\left( u\left(
c_{t}\right) -a_{t}\right) .
\end{equation*}

The problem is now written in the following recursive way. In period zero,
the planner gives a consumption level $c_{0},$ prescribes an effort level $%
a_{0}$ and promised continuation utilities $V_{1}^{e}$ and $V_{1}^{u}.$ The
problem of the planner in period zero is to minimize costs of providing a
given expected utility level $V_{0}$ subject to the incentive constraint the
individual volontarily chooses $a_{0}.$ The problem is recursive and at any
node, cost of providing promised utilities are minimized given incentive
constraints. Let $\theta _{t}\in \left\{ 0,1\right\} $ be the employment
status of the individual in period $t,$ where $\theta _{t}=1$ represents
employmen. Let $\theta ^{t}=\left( \theta _{0},\theta _{1},...\theta
_{t}\right) $ be the history of the agent up until period $t.$ The history
of a person that is unemployed in period $t$ is therefore $\theta
^{t-1}\times 0\equiv \theta _{u}^{t}$, and similarly, $\theta ^{t-1}\times
1\equiv \theta _{e}^{t})$

The problem of the unemployed individual is also recursive. -- as
unemployed, utility is 
\begin{equation*}
V\left( \theta _{u}^{t}\right) =u\left( c_{t}\right) -1+\frac{1}{1+r}\left(
pV\left( \theta _{e}^{t+1}\right) +\left( 1-p\right) V\left( \theta
_{u}^{t+1}\right) \right)
\end{equation*}%
with the incentive constraint%
\begin{equation*}
\frac{1}{1+r}p\left( V\left( \theta _{e}^{t+1}\right) -V\left( \theta
_{u}^{t+1}\right) \right) \geq 1.
\end{equation*}

Define $W\left( V_{t}\right) $ as the minimum cost for the planner to
provide a given amount of utility $V_{t}$ to an employed. Similarily, let $%
C\left( V_{t}\right) $ denote the mimimal cost of providing utility $V$ to
an unemployed (are these function changing over time?). $W$ satisfies%
\footnote{%
I have here discarded the explicit history dependence. How would it look if
you include it?}%
\begin{eqnarray*}
W\left( V_{t}\right) &=&\min_{c_{t},V_{t+1}^{e},V_{t+1}^{u}}c_{t}-w+\frac{1}{%
1+r}\left( \left( 1-s\right) W\left( V_{t+1}^{e}\right) +sC\left(
V_{t+1}^{u}\right) \right) \\
s.t.V_{t} &=&u\left( c_{t}\right) +\frac{1}{1+r}\left( \left( 1-s\right)
V_{t+1}^{e}+sV_{t+1}^{u}\right) ,
\end{eqnarray*}%
where $V_{t+1}^{e}=V\left( \theta _{e}^{t+1}\right) $ and $%
V_{t+1}^{u}=V\left( \theta _{u}^{t+1}\right) .$

The constraint can be called promise keeping constraint and has a Lagrange
multiplier $\delta _{t}^{e}.$

$C$ satsifies%
\begin{gather*}
C\left( V_{t}\right) =\min_{c_{t},V_{t+1}^{e},V_{t+1}^{u}}c_{t}+\frac{1}{1+r}%
\left( pW\left( V_{t+1}^{e}\right) +\left( 1-p\right) C\left(
V_{t+1}^{u}\right) \right) \\
\text{s.t. }\frac{1}{1+r}p\left( V_{t+1}^{e}-V_{t+1}^{u}\right) \geq 1 \\
u\left( c\right) -1+\frac{1}{1+r}\left( pV_{t+1}^{e}+\left( 1-p\right)
V_{t+1}^{u}\right) =V.
\end{gather*}

The first constraint is the incentive constraint, with an associated
Lagrange multiplier $\gamma _{t}$ and the second is the promised utility
with Lagrange multiplier $\delta _{t}^{u}$.\footnote{%
Note that the Lagrange multipliers depends on the history $\theta _{t}.$}
Given that $u\left( c_{t}\right) $ is concave and $u^{-1}\left( V_{t}\right) 
$ therefore is convex, it is straightforward to show that $C$ and $W$ are
convex functions.

First order conditions when the agent is employed are%
\begin{eqnarray}
1 &=&\delta _{t}^{e}u^{\prime }\left( c_{t}\right)
\label{eq_FOC_employedH&N} \\
W^{\prime }\left( V_{t+1}^{e}\right) &=&\delta _{t}^{e}  \notag \\
C\left( V_{t+1}^{u}\right) &=&\delta _{t}^{e}.  \notag
\end{eqnarray}

The envelope condition is 
\begin{equation*}
W^{\prime }\left( V_{t}\right) =\delta _{t}^{e}=\frac{1}{u^{\prime }\left(
c_{t}\right) }=W^{\prime }\left( V_{t+1}^{e}\right) =C^{\prime }\left(
V_{t+1}^{u}\right) .
\end{equation*}

The fact that $W^{\prime }\left( V_{t}\right) =W^{\prime }\left(
V_{t+1}^{e}\right) $ implies that nothing change for the employed individual
as long as his remains employed.

When the agent is unemployed, the FOC and envelope conditions are%
\begin{eqnarray*}
1 &=&\delta _{t}^{u}u^{\prime }\left( c_{t+1}\right) \\
W^{\prime }\left( V_{t+1}^{e}\right) &=&\gamma _{t}+\delta _{t}^{u} \\
\left( 1-p\right) C^{\prime }\left( V_{t+1}^{u}\right) &=&-\gamma
_{t}p+\delta _{t}^{u}\left( 1-p\right) \\
C^{\prime }\left( V_{t}\right) &=&\delta _{t}^{u}.
\end{eqnarray*}

Giving%
\begin{eqnarray}
C^{\prime }\left( V_{t}\right) &=&\frac{1}{u^{\prime }\left( c_{t}\right) }
\label{eq_FOCH&N} \\
W^{\prime }\left( V_{t+1}^{e}\right) &=&\frac{1}{u^{\prime }\left(
c_{t}\right) }+\gamma _{t}  \notag \\
C^{\prime }\left( V_{t+1}^{u}\right) &=&\frac{1}{u^{\prime }\left(
c_{t}\right) }-\gamma _{t}\frac{p}{1-p}  \notag
\end{eqnarray}

\textbf{Results}

Since the incentive constraint will bind\footnote{%
Prove that it must by assuming that it doesn't and derive the implications
of that.}, $\gamma _{t}>0$ and therefore%
\begin{equation*}
W^{\prime }\left( V_{t+1}^{e}\right) >C^{\prime }\left( V_{t}\right)
>C^{\prime }\left( V_{t+1}^{u}\right) .
\end{equation*}

The result $C^{\prime }\left( V_{t}\right) >C^{\prime }\left(
V_{t+1}^{u}\right) $ and the convexity of $C$ implies that the unemployed
should be made succesivley worse off ($V_{t+1}^{u}<V_{t})$ as long as he
remains unemployed. Since $C^{\prime }\left( V_{t}\right) =\frac{1}{%
u^{\prime }\left( c_{t}\right) }$ this means that consumption must fall.
Furthermore, as the IC-constraint $\frac{1}{1+r}p\left(
V_{t+1}^{e}-V_{t+1}^{u}\right) \geq 1$ binds, if $V_{t+1}^{u}$ keeps falling
as long as the unemployed remains unemployed, so must $V_{t+1}^{e}$ implying
that consumption when becoming employed is lower the lower the agent has
been unemployed.

\subsubsection{The inverse Euler equation.}

Multiplying the second line of (\ref{eq_FOCH&N}) by $p$ and the third by $%
\left( 1-p\right) $ and adding them yields, 
\begin{equation}
\frac{1}{u^{\prime }\left( c_{t}\right) }=pW^{\prime }\left(
V_{t+1}^{e}\right) +\left( 1-p\right) C^{\prime }\left( V_{t+1}^{u}\right) .
\label{eq_InverseEuler}
\end{equation}

Recall that $V_{t+1}^{e}$ is the utility next period if the agent becomes
employed, in which case, by (\ref{eq_FOC_employedH&N}), $W^{\prime }\left(
V_{t+1}^{e}\right) =\frac{1}{u^{\prime }\left( c_{t+1}|_{\theta
_{t+1}=1}\right) },$ where $c_{t+1}|_{\theta _{t+1}=1}$ denotes consumption
in period $t+1$ conditional on the getting a job in $t+1$ (and the history
that led to consumption in $t$ being $c_{t})$. Similarly, $V_{t+1}^{u}$ is
next periods utility if the agent remains unemployed. By (\ref{eq_FOCH&N}), $%
C^{\prime }\left( V_{t+1}^{e}\right) =\frac{1}{u^{\prime }\left(
c_{t+1}|_{\theta _{t+1}=9}\right) },$ where $c_{t+1}|_{\theta _{t+1}=0}$
denotes consumption if the agent remains unemployed. Equation (\ref%
{eq_InverseEuler}) can therefore be written 
\begin{eqnarray*}
\frac{1}{u^{\prime }\left( c_{t}\right) } &=&p\frac{1}{u^{\prime }\left(
c_{t+1}|_{\theta _{t+1}=1}\right) }+\left( 1-p\right) \frac{1}{u^{\prime
}\left( c_{t+1}|_{\theta _{t+1}=0}\right) } \\
\frac{1}{u^{\prime }\left( c_{t}\right) } &=&E_{t}\frac{1}{u^{\prime }\left(
c_{t+1}\right) }.
\end{eqnarray*}

This is the famous "Inverse Euler Equation" (Rogerson, -85 Econometrica)%
\footnote{%
With a difference between subjective and market discount rates ($\rho $ and $%
r$, respectively), we would get%
\begin{equation*}
\frac{1}{u^{\prime }\left( c_{t}\right) }\frac{1+r}{1+\rho }=E_{t}\frac{1}{%
u^{\prime }\left( c_{t+1}\right) }.
\end{equation*}%
}. Note the difference between this and the standard Euler equation. 
\begin{equation*}
u^{\prime }\left( c_{t}\right) =E_{t}u^{\prime }\left( c_{t+1}\right) .
\end{equation*}

The inverse Euler equation has an important implication. To see this, first
note that Jensen's inequality, 
\begin{equation*}
E_{t}\frac{1}{u^{\prime }\left( c_{t+1}\right) }>\frac{1}{E_{t}u^{\prime
}\left( c_{t+1}\right) }\Rightarrow \frac{1}{E_{t}\frac{1}{u^{\prime }\left(
c_{t+1}\right) }}<E_{t}u^{\prime }\left( c_{t+1}\right)
\end{equation*}%
since the inverse function is convex. Using this with the Inverse Euler
equation gives, 
\begin{equation*}
u^{\prime }\left( c_{t}\right) =\frac{1}{E_{t}\frac{1}{u^{\prime }\left(
c_{t+1}\right) }}<E_{t}u^{\prime }\left( c_{t+1}\right) .
\end{equation*}

The fact that $u^{\prime }\left( c_{t}\right) <E_{t}u^{\prime }\left(
c_{t+1}\right) $ in the optimal allocation means that the agent would like
to save more, i.e., he is savings constrained. The incentive constraint
implies that it is optimal to prevent the individual to save as much as he
would like to. Suppose, for example, that utility is logarithmic, then we
have 
\begin{equation*}
\frac{1}{c_{t}}=\frac{1}{E_{t}c_{t+1}}\Rightarrow c_{t}=E_{t}c_{t+1},
\end{equation*}%
while the Euler equation, guiding private preferences, implies the privatly
optimal consumption $c_{t}^{\ast }$ given future consumption is%
\begin{equation*}
c_{t}^{\ast }=\frac{1}{E_{t}\left( \frac{1}{c_{t+1}}\right) }<E_{t}c_{t+1}.
\end{equation*}

The intuition is that with more wealth and higher consumption, it is more
costly to implement the incentive constraint. Thus, the benevolent planner
want to prevent some wealth accumulation. The standard interpretation of
this is that when there are incentive constraints, it may be optimal to tax
the returns to savings. However, it may turn out that this tax is
nevertheless zero in expectation, thus not creating any revenue for the
planner/government (Kocherlakota 2005, Econometrica). How can such a tax
discourage savings? Hint: riskpremium depends on covariance with marginal
utility. Explain!

In the logarithmic example, suppose individuals can save and borrow a gross
interest rate $r.$ Consider a marginal tax rate that depends on employment
status and last period individual asset holdings, $\tau _{t+1}^{e}=\tau
^{e}\left( a_{t}\right) $ and $\tau _{t+1}^{u}=\tau ^{u}\left( a_{t}\right)
. $ For notational simplicity, let $c_{t+1}|_{\theta _{t+1}=1}\equiv
c_{t+1}^{e}$ and $c_{t+1}|_{\theta _{t+1}=0}\equiv c_{t+1}^{u}.$Then, to
have the individual Euler equation satisfied, we need%
\begin{eqnarray*}
u^{\prime }\left( c_{t}\right) &=&\beta E_{t}u^{\prime }\left(
c_{t+1}\right) \left( 1+r\right) \left( 1-\tau \left( a_{t}\right) \right) \\
\frac{1}{c_{t}} &=&\left( p\frac{1}{c_{t+1}^{e}}\left( 1-\tau
_{t+1}^{e}\right) +\left( 1-p\right) \frac{1}{c_{t+1}^{u}}\left( 1-\tau
_{t+1}^{u}\right) \right)
\end{eqnarray*}

The inverse Euler equation requires%
\begin{equation*}
c_{t}=pc_{t+1}^{e}+\left( 1-p\right) c_{t+1}^{u}
\end{equation*}

Suppose we consider a zero expected tax rate, i.e., $p\tau
_{t+1}^{e}=-\left( 1-p\right) \tau _{t+1}^{u}.$Then, 
\begin{equation*}
\tau _{t+1}^{e}=\frac{-\left( 1-p\right) }{p}\tau _{t+1}^{u}
\end{equation*}%
and the tax rates 
\begin{eqnarray*}
\tau _{t+1}^{u} &=&\frac{p\left( c_{t+1}^{e}-c_{t+1}^{u}\right) }{%
pc_{t+1}^{e}+c_{t+1}^{u}\left( 1-p\right) } \\
\tau _{t+1}^{e} &=&-\frac{\left( 1-p\right) \left(
c_{t+1}^{e}-c_{t+1}^{u}\right) }{pc_{t+1}^{e}+c_{t+1}^{u}\left( 1-p\right) }
\end{eqnarray*}
leads to both the Euler and the inverse Euler equation being satisfied.
Possibly together with lump sum transfers, they can \emph{implement} the
optimal allocation as a private choice of the agents. Note that the tax is 
\emph{negative }in case the agent becomes employed, while positive if he
remains unemployed. That is, it creates a net return that is negatively
correlated with marginal utility. Note also that as long as the individual
remains unemployed, consumption and asset holding falls. A smaller and
smaller difference $c_{t+1}^{e}-c_{t+1}^{u}$ is then required, and the
savings tax can then fall over time. Rendahl (2007) shows that this is a
general result.

\textbf{Result}: Rendahl (2007)

Consider the repeated H\&N economy but where individuals have access to a
safe observable bond. A tax/transfer that only depends on last period asset
holdings and employment status can implement the second-best allocation.
Unemployment benefits falls in the asset position of the agent. Over an
unemployment spell, unemployment benefits increase but consumption falls.

\subsection{The Dynamic approach with unobservable saving}

\section{Optimal taxation -- the Ramsey approach}

\subsection{Optimal taxation under commitment -- the Ramsey problem}

Consider a simple two period model, where individuals choose how much labor
to supply and how much to consume in the two periods. The government must
tax consumption, savings and/or labor to finance its spending needs. There
will be three margins that can be distorted, the labor leisure choice in the
two periods and the relative level of consumption in the two periods.
Perhaps, on might think that optimal taxation should imply that all three
trade-offs should be distorted. As we will see, that turns out not to be the
case. This result can provide some understanding of the important Chamley \&
Judd result which we will derive later.

\textbf{Preferences}

The representative agent has an additively separable utility function in
consumption and leisure,

\begin{equation*}
U\left( c_{1},c_{2},l_{1},l_{2}\right) =\sum_{t=1}^{2}\beta ^{t-1}u\left(
c,l\right) .
\end{equation*}

\textbf{Technology}

Output is produced by labor only on a competetive labor market. One unit of
labor produces $w$ units of the consumption good. The consumption good can
be stored between periods. One unit of the good stored gives $1+r$ units of
the second period, where $r$ is positive or negative. Individuals have one
unit of labor each period to split between work and leisure $l.$

\textbf{Budget constraints}

The governement needs to finance its consumption by tax reveneus. For
simplicity, we have already assumed that its consumption does not interfer
with the individuals private problem. We will assumed that the government
cannot finance its consumption by lumpsum taxation. We do this without
providing an explicit reason within the model. Instead, the government has
at its disposal, a linear labor income tax $\tau _{l,t}$, a consumption tax $%
\tau _{c,t}$ and a tax on savings, $\tau _{s}.$ Individual budget
constraints are therefore

\begin{eqnarray*}
c_{1}\left( 1+\tau _{c,1}\right) +i+b &=&w\left( 1-l_{1}\right) \left(
1-\tau _{l,1}\right) \\
c_{2}\left( 1+\tau _{c,2}\right) &=&w\left( 1-l_{2}\right) \left( 1-\tau
_{l,2}\right) +\left( i+b\right) \left( 1+r\right) \left( 1-\tau _{s}\right)
,
\end{eqnarray*}%
where $i$ is physical investments (stored goods) and $b$ is government
borrowing assumed to require a return $1+r$ before taxes to be held. We can
collapse this to

\begin{equation*}
c_{1}\left( 1+\tau _{c,1}\right) +\frac{c_{2}}{1+r}\frac{1+\tau _{c,2}}{%
\left( 1-\tau _{s}\right) }=w\left( 1-l_{1}\right) \left( 1-\tau
_{l,1}\right) +\frac{w\left( 1-l_{2}\right) \left( 1-\tau _{l,2}\right) }{%
\left( 1+r\right) \left( 1-\tau _{s}\right) }.
\end{equation*}

It turns out that it is convenient to divide this by $\left( 1+\tau
_{c,1}\right) $ and multiply the last term in the RHS by $\frac{1+\tau _{c,2}%
}{1+\tau _{c,2}}.$ We can then write the budget constraint as 
\begin{equation*}
c_{1}+\frac{c_{2}}{1+r}\frac{1+\tau _{c,2}}{\left( 1-\tau _{s}\right) \left(
1+\tau _{c,1}\right) }=w\left( 1-l_{1}\right) \frac{1-\tau _{l,1}}{1+\tau
_{c,1}}+\frac{w\left( 1-l_{2}\right) }{1+r}\frac{1-\tau _{l,2}}{1+\tau _{c,2}%
}\frac{1+\tau _{c,2}}{\left( 1-\tau _{s}\right) \left( 1+\tau _{c,1}\right) }%
.
\end{equation*}

The aggregate resource constraint of the economy is%
\begin{equation}
c_{1}+\frac{c_{2}}{1+r}+G=w\left( 1-l_{1}\right) +\frac{w\left(
1-l_{2}\right) }{1+r}.  \label{eq_resource}
\end{equation}

Do we need to bother about the government budget constraint in addition to
the private and the aggregate?

\textbf{Individual optimality}

The first order conditions of the individual problem are\footnote{%
We disregard the constraint that $i_{1}\geq 0,$ otherwise, we could have
corner soluitions.}

\begin{eqnarray}
c_{1};u_{c}\left( c_{1},l_{1}\right) &=&\lambda  \label{eq_FOCRamsey} \\
l_{1};u_{l}\left( c_{1},l_{1}\right) &=&\lambda w\frac{1-\tau _{l,1}}{1+\tau
_{c,1}}  \notag \\
c_{2};\beta u_{c}\left( c_{2},l_{2}\right) &=&\lambda \frac{\left( 1+\tau
_{c,2}\right) }{\left( 1+r\right) \left( 1-\tau _{s}\right) \left( 1+\tau
_{c,1}\right) }  \notag \\
l_{2};\beta u_{l}\left( c_{2},l_{2}\right) &=&\lambda \frac{w}{1+r}\frac{%
1-\tau _{l,2}}{1+\tau _{c,2}}\frac{1+\tau _{c,2}}{\left( 1-\tau _{s}\right)
\left( 1+\tau _{c,1}\right) }  \notag
\end{eqnarray}

\subsubsection{A simple example with a labor tax and consumption taxes.}

Let us now assume that the government only has access to a constant labor
tax and a consumption tax that is allowed to vary. Also assume for
tractability that $u\left( c,l\right) =\ln c+\ln l$

The first order conditions of the individual problem are then

\begin{eqnarray*}
\frac{1}{c_{1}} &=&\lambda \\
\frac{1}{l_{1}} &=&\lambda w\frac{1-\tau _{l}}{1+\tau _{c,1}} \\
\beta \frac{1}{c_{2}} &=&\lambda \frac{1+\tau _{c,2}}{\left( 1+r\right)
\left( 1+\tau _{c,1}\right) } \\
\beta \frac{1}{l_{2}} &=&\frac{\lambda w}{1+r}\frac{1-\tau _{l}}{1+\tau
_{c,2}}\frac{1+\tau _{c,2}}{1+\tau _{c,1}}
\end{eqnarray*}

Eliminating $\lambda ,$ the individual optimality constraints are%
\begin{eqnarray*}
\frac{l_{1}}{c_{1}}w &=&\frac{1+\tau _{c,1}}{1-\tau _{l}} \\
\frac{c_{2}}{c_{1}\beta \left( 1+r\right) } &=&\frac{1+\tau _{c,1}}{1+\tau
_{c,2}} \\
\frac{l_{2}}{c_{2}}w &=&\frac{1+\tau _{c,2}}{1-\tau _{l}} \\
c_{1}+\frac{c_{2}}{1+r}\frac{1+\tau _{c,2}}{1+\tau _{c,1}} &=&w\left(
1-l_{1}\right) \frac{1-\tau _{l}}{1+\tau _{c,1}}+\frac{w\left(
1-l_{2}\right) }{1+r}\frac{1-\tau _{l}}{1+\tau _{c,2}}\frac{1+\tau _{c,2}}{%
1+\tau _{c,1}}
\end{eqnarray*}%
with the solution

\begin{eqnarray}
c_{1} &=&\frac{\left( 1-\tau _{l}\right) }{\left( 1+\tau _{c,1}\right) }%
\frac{w\left( 2+r\right) }{2\left( 1+\beta \right) \left( 1+r\right) }
\label{eq_simpleallocation} \\
l_{1} &=&\frac{2+r}{2\left( 1+r\right) \left( 1+\beta \right) } \\
c_{2} &=&\beta \frac{\left( 1-\tau _{l}\right) }{\left( 1+\tau _{c,2}\right) 
}\frac{w\left( 2+r\right) }{2\left( 1+\beta \right) } \\
l_{2} &=&\frac{\beta \left( 2+r\right) }{2\left( 1+\beta \right) }  \notag
\end{eqnarray}

The Ramsey problem is now to maximize utility over the tax rates, $\tau
_{l},\tau _{c,1}$ and $\tau _{c,2},$ subject to the resource constraint.
Disregarding constants, this is 
\begin{equation*}
\max_{\tau _{c,1},\tau _{c,.2}}\ln \left( 1-\tau _{l}\right) -\ln \left(
1+\tau _{c,1}\right) +\beta \left( \ln \left( 1-\tau _{l}\right) -\ln \left(
1+\tau _{c,2}\right) \right)
\end{equation*}%
subject to the resource constraint (\ref{eq_resource}) where (\ref%
{eq_simpleallocation}) is used to replace the private choice variables.
First order conditions are 
\begin{eqnarray*}
\tau _{c,1};\frac{1}{1+\tau _{c,1}} &=&\lambda \frac{w\left( 1-\tau
_{l}\right) \left( 2+r\right) }{2\left( 1+\beta \right) \left( 1+r\right)
\left( 1+\tau _{1}\right) ^{2}} \\
\tau _{c,2};\frac{\beta }{1+\tau _{c,2}} &=&\lambda \beta \frac{w\left(
1-\tau _{l}\right) \left( 2+r\right) }{2\left( 1+\beta \right) \left(
1+r\right) \left( 1+\tau _{2}\right) ^{2}} \\
\tau _{l};\frac{1+\beta }{1-\tau _{l}} &=&\lambda \frac{w\left( 1+\tau
_{2}+\beta \left( 1+\tau _{1}\right) \right) \left( 2+r\right) }{2\left(
1+\beta \right) \left( 1+r\right) \left( 1+\tau _{1}\right) \left( 1+\tau
_{2}\right) }
\end{eqnarray*}

As we see, the first order conditions for the consumption taxes are
symmetrical -- implying that it is optimal to set consumption taxes equal in
the two periods. To what level does not matter. EXPLAIN! This provides \emph{%
distortion smoothing} -- the labor leisure choice is distorted equally much
in both period. Perhaps, this is not surprising.

\subsubsection{The general case}

Let us now define%
\begin{eqnarray}
\frac{1-\tau _{l,1}}{1+\tau _{c,1}} &\equiv &W_{1},  \label{eq_wedgedef} \\
\frac{1-\tau _{l,2}}{1+\tau _{c,2}} &\equiv &W_{2},  \notag \\
\frac{\left( 1+\tau _{c,2}\right) }{\left( 1+\tau _{c,1}\right) \left(
1-\tau _{s}\right) } &\equiv &W_{i}.  \notag
\end{eqnarray}

We can then write the system as%
\begin{eqnarray}
\frac{u_{l}\left( c_{1},l_{1}\right) }{u_{c}\left( c_{1},l_{1}\right) w}
&=&W_{1}  \label{eq_generalRamsey} \\
\frac{u_{l}\left( c_{2},l_{2}\right) }{u_{c}\left( c_{2},l_{2}\right) w}
&=&W_{2}  \notag \\
\frac{u_{c}\left( c_{2},l_{2}\right) }{u_{c}\left( c_{1},l_{1}\right) }\beta
\left( 1+r\right) &=&W_{i}  \notag \\
c_{1}+\frac{c_{2}}{1+r}W_{i} &=&w\left( 1-l_{1}\right) W_{1}+\frac{w\left(
1-l_{2}\right) }{1+r}W_{2}W_{i}  \notag \\
c_{1}+g+\frac{c_{2}+g}{1+r} &=&w\left( 1-l_{1}\right) +\frac{w\left(
1-l_{2}\right) }{1+r}  \notag
\end{eqnarray}

Provided $g$ is not too high, this gives a solution for $%
c_{1},c_{2},l_{1},l_{2}$ and one of the tax wedges, as a function of two of
the other wedges and parameters.

\begin{result}
\label{3Wedges}\emph{Altough the government has access to 5 different taxes,
the distortion relative to the first best is a function of the three wedges }%
$W_{1},W_{2}$\emph{\ and }$W_{i}.$
\end{result}

Using result \ref{3Wedges}, we conclude that all tax systems that provide
the same wedges as the one with a constant labor tax and a constant
consumption tax gives the same utility. Provide some examples. Furthermore,
the restriction we imposed, namely $\tau _{l,1}=\tau _{l,2}$ and $\tau
_{s}=0,$ does not reduce welfare. Explain!

Finally, in the optimal allocation $W_{i}=1,$i.e., there is no intertemporal
wedge. With a constant consumption tax, this requires a zero tax on savings.

\subsection{The primal approach}

An often used way of solving the problem is the \emph{primal approach. }The
idea here is to write the problem as the planner directly choosing the
consumtion and labor of the individual. With access to lump sum taxes, the
only constraint for the planner is the resource constraint and first best
will be acheived. With only proportional taxes, incentive compatibility must
be respected. It turns out that we can write this constraint without any
taxes or prices. We do this by substituting the first order constraints of
the individual (first three equations of (\ref{eq_generalRamsey}) into the
private budget constraint (the fourt equation of (\ref{eq_generalRamsey})).
This yields,

\begin{equation}
c_{1}+c_{2}\frac{u_{c}\left( c_{2},l_{2}\right) }{u_{c}\left(
c_{1},l_{1}\right) }\beta =\left( 1-l_{1}\right) \frac{u_{l}\left(
c_{1},l_{1}\right) }{u_{c}\left( c_{1},l_{1}\right) }+\left( 1-l_{2}\right) 
\frac{u_{l}\left( c_{2},l_{2}\right) }{u_{c}\left( c_{2},l_{2}\right) }\frac{%
u_{c}\left( c_{2},l_{2}\right) }{u_{c}\left( c_{1},l_{1}\right) }\beta .
\label{eq_primalbudget}
\end{equation}

The Ramsey problem can then be expressed as 
\begin{equation*}
\max_{c_{1},c_{2},l_{1},l_{2}}\sum_{t=1}^{2}\beta ^{t-1}u\left( c,l\right)
\end{equation*}%
s.t. (\ref{eq_primalbudget}) and (\ref{eq_resource}). As we see, no taxes or
prices (interest rate) enter this problem except through the aggregate
resource constraint.

In our logaritmic example, (\ref{eq_primalbudget}) becomes%
\begin{eqnarray*}
c_{1}\left( 1+\beta \right) &=&c_{1}\left( \frac{1-l_{1}}{l_{1}}+\frac{%
\left( 1-l_{2}\right) }{l_{2}}\beta \right) \\
&\Longrightarrow &l_{1}=\frac{l_{2}}{2l_{2}\left( 1+\beta \right) -\beta }
\end{eqnarray*}%
since the $c_{1}=0$ root is irrelevant. Substituting this into the objective
function and taking first order conditions wrt $c_{1}$ and $c_{2}$ yields,

\begin{equation*}
\max_{c_{1},c_{2},l_{2}}\ln c_{1}+\ln \frac{l_{2}}{2l_{2}\left( 1+\beta
\right) -\beta }+\beta \left( \ln c_{2}+\ln l_{2}\right)
\end{equation*}%
\begin{eqnarray*}
\frac{1}{c_{1}} &=&\lambda \\
\beta \frac{1}{c_{2}} &=&\lambda \frac{1}{1+r} \\
&\Rightarrow &\frac{c_{2}}{c_{1}}=\beta \left( 1+r\right)
\end{eqnarray*}%
\begin{equation*}
s.t.\left( w\left( 1-\frac{l_{2}}{2l_{2}\left( 1+\beta \right) -\beta }%
\right) +\frac{w\left( 1-l_{2}\right) }{1+r}-\left( c_{1}+\frac{c_{2}}{1+r}%
+G\right) \right)
\end{equation*}%
again confirming that the intertemporal margin should be zero, requiring $%
W_{i}=0.$

Often, the focus is on the allocation, i.e., how consumption and leisure is
allocated over time. To go further, one might want to find a tax system that
implements this allocation. As we have seen, there are often many such
systems.

\subsection{The Chamley-Judd result}

There is an infinitely lived representative agent with preferences%
\begin{equation*}
\sum_{t=0}^{\infty }\beta ^{t}u\left( c_{t},l_{t}\right)
\end{equation*}

The household has one unit of labor per period, to be split between leisure $%
l$ and work $n.$The aggregate budget constraint is%
\begin{equation}
c_{t}+g_{t}+k_{t+1}=F\left( k_{t},n_{t}\right) +\left( 1-\delta \right) k_{t}
\label{eq_AggbudgetCJ}
\end{equation}

The production function is constant returns to scale and factor markets are
competetive. Profit maximization of the representative firm implies%
\begin{eqnarray*}
w_{t} &=&F_{n}\left( k_{t},n_{t}\right) \\
r_{t} &=&F_{k}\left( k_{t},n_{t}\right)
\end{eqnarray*}

The government needs to finance an exogeneous stream of expenditures $%
\left\{ g_{t}\right\} _{t}^{\infty }$ using taxes on labor and capital and
can smooth taxes by using a bond. Thus,

\begin{eqnarray*}
g_{t}+b_{t} &=&\tau _{t}^{k}r_{t}k_{t}+\tau _{t}^{n}w_{t}n_{t}+\frac{b_{t+1}%
}{R_{t}} \\
&=&F\left( k_{t},n_{t}\right) -\left( 1-\tau _{t}^{k}\right)
r_{t}k_{t}-\left( 1-\tau _{t}^{n}\right) w_{t}n_{t}+\frac{b_{t+1}}{R_{t}}
\end{eqnarray*}%
where $b_{t}$ is government borrowing and $R_{t}$ is the interest rate on
government bonds.

Households have budget constraints%
\begin{equation*}
c_{t}+k_{t+1}+\frac{b_{t+1}}{R_{t}}=\left( 1-\tau _{t}^{n}\right)
w_{t}n_{t}+\left( 1-\tau _{t}^{k}\right) k_{t}r_{t}+\left( 1-\delta \right)
k_{t}+b_{t}
\end{equation*}

First order conditions are:

\begin{eqnarray*}
c_{t};u_{c}\left( c_{t},l_{t}\right) &=&\lambda _{t} \\
l_{t};u_{l}\left( c_{t},l_{t}\right) &=&\lambda _{t}\left( 1-\tau
_{t}^{n}\right) w_{t} \\
k_{t+1};\lambda _{t} &=&\beta \lambda _{t+1}\left( \left( 1-\tau
_{t+1}^{k}\right) r_{t+1}+\left( 1-\delta \right) \right) \\
b_{t+1};\lambda _{t}\frac{1}{R_{t}} &=&\beta \lambda _{t+1}
\end{eqnarray*}

Clearly, the first three implies 
\begin{eqnarray*}
\frac{u_{l}\left( c_{t},l_{t}\right) }{u_{c}\left( c_{t},l_{t}\right) }
&=&\left( 1-\tau _{t}^{n}\right) w_{t} \\
u_{c}\left( c_{t},l_{t}\right) &=&\beta u_{c}\left( c_{t+1},l_{t+1}\right)
\left( \left( 1-\tau _{t+1}^{k}\right) r_{t+1}+\left( 1-\delta \right)
\right)
\end{eqnarray*}%
and the last two the no arbitrage condition 
\begin{equation*}
R_{t}=\left( 1-\tau _{t+1}^{k}\right) r_{t+1}+\left( 1-\delta \right)
\end{equation*}

Transversality conditions are 
\begin{eqnarray*}
\lim_{T\rightarrow \infty }\left( \dprod\limits_{i=0}^{T-1}R_{i}^{-1}\right)
k_{T+1} &=&0 \\
\lim_{T\rightarrow \infty }\left( \dprod\limits_{i=0}^{T-1}R_{i}^{-1}\right) 
\frac{b_{T+1}}{R_{T}} &=&0
\end{eqnarray*}

We can now make the following definitions:

\begin{definition}
A \emph{feasible allocation} is a sequence $\left\{
k_{t},c_{t},l_{t},g_{t}\right\} _{t=0}^{\infty }$ that satisfies the
aggregate budget constraint (\ref{eq_AggbudgetCJ}).
\end{definition}

\begin{definition}
A \emph{price system} is a sequence of prices $\left\{
w_{t},r_{t},R_{t}\right\} _{t=0}^{\infty }$ that is bounded and non-negative.
\end{definition}

\begin{definition}
A \emph{government policy} is a sequence $\left\{ \tau _{t}^{n},\tau
_{t}^{k},b_{t}\right\} _{t=0}^{\infty }$ and perhaps $\left\{ g_{t}\right\}
_{t=0}^{\infty }$ if that can be chosen.
\end{definition}

\begin{definition}
A \emph{competetive equilibrium }is a feasible allocation, a price system
and a government policy such that \newline
\end{definition}

\begin{enumerate}
\item \textit{Given the price system and the government policy, the
allocation solves the maximization problem of the individual and of the firm.%
}

\item \textit{The government budget constraints are satisfied. }
\end{enumerate}

\begin{definition}
The \emph{Ramsey problem} is to choose a competetive equilibrium (i.e.,pick
a particular government policy) that maximizes the welfare of the
representative individual.
\end{definition}

The Lagrangean of the Ramsey problem can be written

\begin{eqnarray*}
L &=&\sum_{t=0}^{\infty }\beta ^{t}\{u\left( c_{t},1-n_{t}\right) \\
&&+\psi _{t}\left( F\left( k_{t},n_{t}\right) -\left( 1-\tau _{t}^{k}\right)
r_{t}k_{t}-\left( 1-\tau _{t}^{n}\right)
w_{t}n_{t}-b_{t}-g_{t}+b_{t+1}/R_{t}\right) \\
&&+\theta _{t}\left( F\left( k_{t},n_{t}\right) +\left( 1-\delta \right)
k_{t}-c_{t}-g_{t}-k_{t+1}\right) \\
&&+\mu _{1,t}\left( u_{l}\left( c_{t},l_{t}\right) -u_{c}\left(
c_{t},l_{t}\right) \left( 1-\tau _{t}^{n}\right) w_{t}\right) \\
&&+\mu _{2,t}\left( u_{c}\left( c_{t},l_{t}\right) -\beta u_{c}\left(
c_{t+1},l_{t+1}\right) \left( 1-\tau _{t+1}^{k}\right) r_{t+1}+\left(
1-\delta \right) \right)
\end{eqnarray*}

Now, the first order condition for $k_{t+1}$ is%
\begin{equation*}
\theta _{t}=\beta \psi _{t+1}\left( F_{k}\left( k_{t+1},n_{t+1}\right)
-\left( 1-\tau _{t+1}^{k}\right) r_{t+1}\right) -\theta _{t+1}\left(
F_{k}\left( k_{t+1},n_{t+1}\right) +\left( 1-\delta \right) \right)
\end{equation*}%
and for $c_{t}$%
\begin{equation*}
u_{c}\left( c_{t},1-n_{t}\right) =\theta _{t}
\end{equation*}%
giving%
\begin{eqnarray*}
u_{c}\left( c_{t},1-n_{t}\right) &=&\beta \psi _{t+1}\left( F_{k}\left(
k_{t+1},n_{t+1}\right) -\left( 1-\tau _{t+1}^{k}\right) r_{t+1}\right) \\
&&+\beta u_{c}\left( c_{t+1},1-n_{t+1}\right) \left( F_{k}\left(
k_{t+1},n_{t+1}\right) +\left( 1-\delta \right) \right) .
\end{eqnarray*}

Suppose there is a steady state of the model, then 
\begin{eqnarray*}
u_{c} &=&\beta \left( \psi \left( F_{k}-\left( 1-\tau ^{k}\right)
F_{k}\right) +u_{c}\left( F_{k}+\left( 1-\delta \right) \right) \right) \\
&=&\beta \left( \psi \tau ^{k}F_{k}+u_{c}\left( F_{k}+\left( 1-\delta
\right) \right) \right) .
\end{eqnarray*}%
Private optimality (the Euler equation), implies in steady state 
\begin{eqnarray*}
u_{c} &=&\beta u_{c}\left( \left( 1-\tau ^{k}\right) F_{k}+\left( 1-\delta
\right) \right) \\
1 &=&\beta \left( F_{k}+\left( 1-\delta \right) -\tau ^{k}F_{k}\right) \\
\frac{1}{\beta }+\tau ^{k}F_{k} &=&F_{k}+\left( 1-\delta \right)
\end{eqnarray*}%
giving 
\begin{eqnarray*}
u_{c} &=&\beta \left( \psi \tau ^{k}F_{k}+u_{c}\left( \frac{1}{\beta }+\tau
^{k}F_{k}\right) \right) \\
&=&\beta \left( \left( \psi +u_{c}\right) \tau ^{k}F_{k}+u_{c}\left( \tau
^{k}F_{k}\right) \right) +u_{c} \\
0 &=&\beta \left( \psi +u_{c}\right) \tau ^{k}F_{k}
\end{eqnarray*}%
requiring $\tau ^{k}=0.$

\subsection{Discussion}

We have shown that also in this simple economy, tax smoothing implies that
the intertemporal margin should not be distorted. We have also found an
equivalence between constant consumption taxes and an investment tax. In an
infinite horizon model, a positive invstment tax in steady state has
implications identical to ever increasing consumption taxes. This can thus
provide some intuition for Chamley \& Judd's result that investment taxes
should not be used in the long run. The result is quite robust. For example
it extends to the case of heterogeneity, if the government wants to use it's
revenues to support some capital poor individuals, it should not tax capital
accumulation in steady state. Here intuition could be that the capital stock
in steady state is elastic enough to imply the the tax incidence of capital
taxes is on workers.

The result also extends to the stochastic case, in which case \emph{expected
taxes }should be zero and not distort savings.

However, it does not go through in some cases:

\begin{enumerate}
\item If there are untaxed factors of production that generate profits and
these factors are strict complements to capital. Then capital should be
taxed (negatively if they are substitutes).

\item If market incompleteness makes people save too much for precautionary
reasons.
\end{enumerate}

In the short run, capital income taxes also collect revenue from sunk
investments. Then, the tax is partly lump sum, which provides an argument
for such taxes early in the planning horizon. But when is that zero? Has it
already occured a long time ago? In any case, we see a time consistency
problem here.

Not also that the long-run maybe quite far out and people alive today might
loose by a policy that maxmizes the welfare of a constructed infinitely lived

\subsection{Time consisten taxation}

\section{New Public Finance -- the Mirrlees approach}

Let us now consider the dynamic Mirrlees approach to optimal taxation. Here,
individuals are assumed to be different. These differences can be either in
their productivity or in their value of lesiure. Such differences imply that
there is differences between individuals in their trade-off between leisure
and work. It is assumed that the government cannot directly observe this
differences, only observe the individuals market choices. For example,
governments observe income, but not the effort excerted to get this income.

Consider a simple two-period example from GTW.

Individual preferences are:

\begin{equation*}
E\left( u\left( c_{1}\right) +v\left( n_{1}\right) +\beta \left( u\left(
c_{2}\right) +v\left( n_{2}\right) \right) \right)
\end{equation*}%
where $c_{t}$ is consumption and $n_{t}$ is labor supply/work effort. $u$ is
increasing and concave and $v$ decreasing and concave. Individuals differ in
their ability, denoted $\theta .$ It is assumed that there is a finite
number $i\in \left\{ 1,2,...,N\right\} $ of ability levels and ability might
change over time. We will interchangeably use type and ability to denote $%
\theta .$ Output is produced in competetive firms using a linear technology
where each individual $i$ produces 
\begin{equation*}
y_{t}\left( i\right) =\theta \left( i\right) n_{t}\left( i\right) .
\end{equation*}

There is a continuum of individuals of a unitary total mass. In the first
period, individuals are given abilities by nature according to a probability
function $\pi _{1}\left( i\right) $. The ability can then change to the
second period. Second period ability is denoted $\theta \left( i,j\right) $
and the transition probability is $\pi _{2}\left( j|i\right) .$

There is a storage technology with return $R$. Finally, the government needs
to finance some spendings $G_{1}$ and $G_{2}$. At first, we analyze the case
of no aggregate uncertainty.

The aggregate resource constraint is 
\begin{equation}
\dsum\limits_{i}\left( y_{1}\left( i\right) -c_{1}\left( i\right)
+\dsum\limits_{j}\frac{y_{2}\left( i,j\right) -c_{2}\left( i,j\right) }{R}%
\pi _{2}\left( j|i\right) \right) \pi _{1}\left( i\right) +K_{1}=G_{1}+\frac{%
G_{2}}{R_{2}}  \label{eq_Resource}
\end{equation}%
where $K_{1}$ is an aggregate initial endowment.

The problem is now to maximize the utilitarian welfare function subject to
the resource constraints and the incentive constraints, i.e., that
individuals themselves choose labor supply and savings. A way of finding the
second best allocation is to let the planner provide consumption and work
conditional on the ability an individual claims to have (and if relevant,
the aggregate state). Here this is in the first period $c_{1}\left( i\right)
,y_{1}\left( i\right) $ and in the second, $c_{2}\left( i,j\right)
,y_{1}\left( i,j\right) .$ Individuals then report their abilities to the
planner. The strategy of an individual is his first period report and then a
reporting plan as a function of the realized period 2 ability. Let's call
the report $i_{r}$ and $j_{r}\left( j\right) ,$ where the latter is the
report as a function of the true ability. The incentive constraint is then
that individuals volontarily report their true ability. According to the 
\emph{revelation principle, } this always yields the best incentive
compatible allocation. The \emph{truth-telling} constraint is then that 
\begin{eqnarray}
&&u\left( c_{1}\left( i\right) \right) +v\left( \frac{y_{1}\left( i\right) }{%
\theta _{1}\left( i\right) }\right) +\beta \dsum\limits_{j}\left( u\left(
c_{2}\left( i,j\right) \right) +v\left( \frac{y_{2}\left( i,j\right) }{%
\theta _{2}\left( i,j\right) }\right) \right) \pi _{2}\left( j|i\right)
\label{eq_thruthtelling} \\
&\geq &u\left( c_{1}\left( i_{r}\right) \right) +v\left( \frac{y_{1}\left(
i_{r}\right) }{\theta _{1}\left( i\right) }\right) +\beta
\dsum\limits_{j}\left( u\left( c_{2}\left( i_{r},j_{r}\left( j\right)
\right) \right) +v\left( \frac{y_{2}\left( i_{r},j_{r}\left( j\right)
\right) }{\theta _{2}\left( i,j\right) }\right) \right) \pi _{2}\left(
j|i\right)  \notag
\end{eqnarray}%
for any possible reporting strategy $i_{r},j_{r}\left( j\right) $. Note that
the $\theta _{s}$ are the true ones in both sides of the inequality. Note
also that \emph{truth-telling} implies that 
\begin{equation}
u\left( c_{2}\left( i,j\right) \right) +v\left( \frac{y_{2}\left( i,j\right) 
}{\theta _{2}\left( i,j\right) }\right) \geq u\left( c_{2}\left(
i_{r},j_{r}\left( j\right) \right) \right) +v\left( \frac{y_{2}\left(
i_{r},j_{r}\left( j\right) \right) }{\theta _{2}\left( i,j\right) }\right)
\forall j,  \label{eq_IC_statebystate}
\end{equation}%
otherwise utility could be increased by reporting $j_{r}$ if the second
period ability is $j.$ The planning problem is to maximize 
\begin{equation*}
\sum_{i}\left( u\left( c_{1}\left( i\right) \right) +v\left( \frac{%
y_{1}\left( i\right) }{\theta _{1}\left( i\right) }\right) +\beta
\dsum\limits_{j}\left( u\left( c_{2}\left( i,j\right) \right) +v\left( \frac{%
y_{2}\left( i,j\right) }{\theta _{2}\left( i,j\right) }\right) \right) \pi
_{2}\left( j|i\right) \right) \pi \left( i\right)
\end{equation*}%
subject to (\ref{eq_Resource}) and (\ref{eq_thruthtelling}).

Letting stars $^{\ast },$denote optimal allocations. We can now define three
wedges (distortions) that the informational friction may cause. These are
the consumption-leisure (intra-temporal) wedges%
\begin{eqnarray*}
\tau _{y_{1}}\left( i\right) &\equiv &1+\frac{v^{\prime }\left( \frac{%
y_{1}^{\ast }\left( i\right) }{\theta _{1}\left( i\right) }\right) }{%
u^{\prime }\left( c_{1}^{\ast }\left( i\right) \right) }, \\
\tau _{y_{2}}\left( i,j\right) &\equiv &1+\frac{v^{\prime }\left( \frac{%
y_{1}^{\ast }\left( i,j\right) }{\theta _{2}\left( i,j\right) }\right) }{%
u^{\prime }\left( c_{2}^{\ast }\left( i,j\right) \right) },
\end{eqnarray*}%
and the intertemporal wedge%
\begin{equation*}
\tau _{k}\left( i\right) \equiv 1-\frac{u^{\prime }\left( c_{1}^{\ast
}\left( i\right) \right) }{\dsum\limits_{j}\beta Ru^{\prime }\left(
c_{2}\left( i,j\right) \right) \pi _{2}\left( j|i\right) }.
\end{equation*}

Clearly, in absence of government interventions, these wedges would be zero
by perfect competition and the first-order conditions of private
optimization.

\subsection{The inverse Euler equation}

We will now show that if individual productivites are not always constant
over time, the intertemporal wedge will not be zero. The logic is as follows
and similar to what we have done above. In an optimal allocation, the
resource cost (expected present value of consumption) of providing the
equilibrium utility to each type, must be minimized. Consider the following
peturbation around the optimal allocation for a given first period ability
type $i.$ Increase utility by a marginal amount $\Delta $ for all possible
second period types $\left\{ i,j\right\} $ the agent could become. To
compensate, decrease utility by $\beta \Delta $ in the first period.
Clearly, the objective function is not changed. What about the thruthtelling
constraint?.

First, note that expected utility is not changed.

Second, since utility is changed in parallell for all ability levels the
individual could have in the second period, there relative ranking cannot
change. In other words, if we add $\Delta $ to both sides of (\ref%
{eq_IC_statebystate}) it must still be satisfied.

Thus, the incentive constraint is unchanged. However, the resource
constraint is not necessarily invariant to this peturbation. Let 
\begin{eqnarray*}
\tilde{c}_{1}\left( i;\Delta \right) &=&u^{-1}\left( u\left( c_{1}^{\ast
}\left( i\right) \right) -\beta \Delta \right) , \\
\tilde{c}_{2}\left( i,j;\Delta \right) &=&u^{-1}\left( u\left( c_{2}^{\ast
}\left( i,j\right) \right) +\Delta \right)
\end{eqnarray*}%
denote the perturbed consumption levels. The resource expected resource cost
of these are%
\begin{eqnarray*}
&&\tilde{c}_{1}\left( i;\Delta \right) +\dsum\limits_{j}\frac{1}{R}\tilde{c}%
_{2}\left( i,j;\Delta \right) \pi _{2}\left( j|i\right) \\
&=&u^{-1}\left( u\left( c_{1}^{\ast }\left( i\right) \right) -\beta \Delta
\right) +\dsum\limits_{j}\frac{1}{R}u^{-1}\left( u\left( c_{2}^{\ast }\left(
i,j\right) \right) +\Delta \right) \pi _{2}\left( j|i\right) .
\end{eqnarray*}

The first-order condition for minimizing the resource cost over $\Delta $
must be satified at $\Delta =0,$ for the $^{\ast }$ consumption levels to be
optimal.

Thus, 
\begin{eqnarray*}
0 &=& \\
&=&\frac{-\beta }{u^{\prime }\left( c_{1}^{\ast }\left( i\right) \right) }%
+\dsum\limits_{j}\frac{1}{R}\frac{1}{u^{\prime }\left( c_{2}^{\ast }\left(
i,j\right) \right) }\pi _{2}\left( j|i\right) \\
&\Rightarrow &\frac{1}{u^{\prime }\left( c_{1}^{\ast }\left( i\right)
\right) }=E_{1}\frac{1}{\beta Ru^{\prime }\left( c_{2}^{\ast }\left(
i,.\right) \right) }
\end{eqnarray*}

From Jensen's inequality, we find that 
\begin{eqnarray*}
u^{\prime }\left( c_{1}^{\ast }\left( i\right) \right) &<&E\beta Ru^{\prime
}\left( c_{2}^{\ast }\left( i,.\right) \right) \\
&\Rightarrow &\tau _{k}\left( i\right) >0,
\end{eqnarray*}%
\emph{if and only if there is some uncertainty in }$c_{2}^{\ast }.$ Note
that this uncertainty would come from second period ability being random and
the allocation implying that second period consumption depends on the
realization of ability. If second period ability is non-random, i.e., $\pi
_{2}\left( j|i\right) =1$ for some $j,$ then $\tau _{k}\left( i\right) =0.$

Example. Suppose in the first period, ability is unity and in the second $%
\theta >1$ or $\frac{1}{\theta }$ with equal probability$.$

The first best allocation is the solution to 
\begin{eqnarray*}
&&\max_{c_{1},y_{1},c_{h},c_{l},y_{h},y_{l}}u\left( c_{1}\right) +v\left(
y_{1}\right) +\beta \left( \frac{u\left( c_{h}\right) +v\left( \frac{y_{h}}{%
\theta }\right) }{2}+\frac{u\left( c_{l}\right) +v\left( \frac{y_{l}}{\frac{1%
}{\theta }}\right) }{2}\right) \\
s.t.0 &=&y_{1}+\frac{y_{h}+y_{l}}{2}-c_{1}-\frac{c_{h}+c_{l}}{2}
\end{eqnarray*}

First order conditions are 
\begin{eqnarray*}
u^{\prime }\left( c_{1}\right) &=&\lambda \\
v^{\prime }\left( y_{1}\right) &=&-\lambda \\
\beta u^{\prime }\left( c_{h}\right) &=&\frac{\lambda }{R} \\
\beta u^{\prime }\left( c_{l}\right) &=&\frac{\lambda }{R} \\
\beta v^{\prime }\left( \frac{y_{h}}{\theta }\right) \frac{1}{\theta } &=&-%
\frac{\lambda }{R} \\
\beta v^{\prime }\left( \theta y_{l}\right) \theta &=&-\frac{\lambda }{R}
\end{eqnarray*}

Suppose for example that $u\left( c\right) =\ln \left( c\right) $ and $%
v\left( n\right) =-\frac{n^{2}}{2}$ and $\beta =R=1.$ Then, we get 
\begin{eqnarray*}
\frac{1}{c_{1}} &=&\lambda \\
\frac{1}{c_{h}} &=&\lambda \\
\frac{1}{c_{h}} &=&\lambda \\
y_{1} &=&\lambda \\
\frac{y_{h}}{\theta ^{2}} &=&\lambda \\
y_{l}\theta ^{2} &=&\lambda \\
c_{1}+\frac{c_{h}+c_{l}}{2}-y_{1}-\frac{y_{h}+y_{l}}{2} &=&0
\end{eqnarray*}

We see immediately that $c_{1}=c_{h}=c_{l}$ while $y_{h}=\theta ^{2}y_{1}$
and $y_{l}=\frac{y_{1}}{\theta ^{2}}$ and $y_{1}=\sqrt{\frac{2}{\left( 1+%
\frac{1}{2}\left( \theta ^{2}+\theta ^{-2}\right) \right) }}.$ Thus, if the
individual becomes of high ability in the second period, he should work more
but don't get any higher consumption. Is this incentive compatible?

We conjecture that the binding incentive constraint is for the high ability
type. High has to be given sufficient consumption to make him volontarily
choose not to report being low ability. If he missreports, he gets $c_{l}$
and is asked to produce $y_{l}.$ The constraint is therefore%
\begin{eqnarray*}
&&u\left( c_{1}\right) +v\left( y_{1}\right) +\beta \left( \frac{u\left(
c_{h}\right) +v\left( \frac{y_{h}}{\theta }\right) }{2}+\frac{u\left(
c_{l}\right) +v\left( \theta y_{l}\right) }{2}\right) \\
&\geq &u\left( c_{1}\right) +v\left( y_{1}\right) +\beta \left( \frac{%
u\left( c_{l}\right) +v\left( \frac{y_{l}}{\theta }\right) }{2}+\frac{%
u\left( c_{l}\right) +v\left( \theta y_{l}\right) }{2}\right)
\end{eqnarray*}%
\begin{eqnarray*}
u\left( c_{h}\right) +v\left( \frac{y_{h}}{\theta }\right) &\geq &u\left(
c_{l}\right) +v\left( \frac{y_{l}}{\theta }\right) \\
\ln c_{h}-\ln \left( c_{l}\right) &\geq &\frac{y_{h}^{2}-y_{l}^{2}}{\theta
^{2}}
\end{eqnarray*}

We conjecture this is binding. The problem is then 
\begin{eqnarray*}
&&\max_{c_{1},y_{1},c_{h},c_{l},y_{h},y_{l}}\ln \left( c_{1}\right) -\frac{%
y_{1}^{2}}{2}+\left( \frac{\ln c_{h}-\frac{\left( \frac{y_{h}}{\theta }%
\right) ^{2}}{2}}{2}+\frac{\ln c_{l}-\frac{\left( \theta y_{l}\right) ^{2}}{2%
}}{2}\right) \\
s.t.0 &=&y_{1}+\frac{y_{h}+y_{l}}{2}-c_{1}-\frac{c_{h}+c_{l}}{2} \\
0 &=&\frac{y_{h}^{2}-y_{l}^{2}}{\theta ^{2}}-\left( \ln c_{h}-\ln \left(
c_{l}\right) \right) .
\end{eqnarray*}

Denoting the shadow values by $\lambda _{r}$ and $\lambda _{I}$ the FOCs for
the consumption levels are%
\begin{eqnarray*}
\frac{1}{c_{1}} &=&\lambda _{r} \\
\frac{1}{c_{h}} &=&\lambda _{r}-\lambda _{l}\frac{1}{c_{h}} \\
\frac{1}{c_{l}} &=&\lambda _{r}+\lambda _{l}\frac{1}{c_{l}}
\end{eqnarray*}%
from which we see 
\begin{equation*}
\frac{1}{c_{1}^{\ast }}=\lambda _{r},\frac{c_{h}^{\ast }}{c_{1}^{\ast }}%
=1+\lambda _{I},\frac{c_{l}^{\ast }}{c_{1}^{\ast }}=1-\lambda _{I}
\end{equation*}%
giving 
\begin{equation*}
\tau _{k}\left( i\right) \equiv 1-\frac{\frac{1}{c_{1}^{\ast }}}{\frac{1}{%
c_{h}^{\ast }}\frac{1}{2}+\frac{1}{c_{l}^{\ast }}\frac{1}{2}}=1-\frac{\frac{1%
}{c_{1}^{\ast }}}{\frac{1}{c_{1}^{\ast }\left( 1+\lambda _{I}\right) }\frac{1%
}{2}+\frac{1}{c_{1}^{\ast }\left( 1-\lambda _{I}\right) }\frac{1}{2}}%
=\lambda _{I}^{2},
\end{equation*}%
implying a positive intertemporal wedge if the IC constraint binds.

The intratemporal wedges are found by analyzing the FOC's for the labor
supplies%
\begin{eqnarray*}
y_{1}^{\ast } &=&\lambda _{r}=\frac{1}{c_{1}^{\ast }} \\
\frac{y_{h}^{\ast }}{\theta ^{2}}\left( 1-2\lambda _{I}\right) &=&\lambda
_{r}=\frac{1+\lambda _{I}}{c_{h}^{\ast }} \\
y_{l}^{\ast }\left( \theta ^{2}+\frac{2\lambda _{I}}{\theta ^{2}}\right)
&=&\lambda _{r}=\frac{1-\lambda _{I}}{c_{l}^{\ast }}
\end{eqnarray*}%
giving%
\begin{eqnarray*}
y_{h}^{\ast } &=&\frac{1}{c_{h}^{\ast }}\frac{\theta ^{2}\left( 1+\lambda
_{I}\right) }{\left( 1-2\lambda _{I}\right) } \\
y_{l}^{\ast } &=&\frac{1-\lambda _{I}}{c_{l}^{\ast }\left( \theta ^{2}+\frac{%
2\lambda _{I}}{\theta ^{2}}\right) }
\end{eqnarray*}%
and 
\begin{eqnarray*}
\tau _{y_{1}} &=&1-\frac{y_{1}^{\ast }}{\frac{1}{c_{1}^{\ast }}}=0, \\
\tau _{y_{2}}\left( h\right) &=&1+\frac{-\frac{y_{h}^{\ast }}{\theta ^{2}}}{%
\frac{y_{h}^{\ast }}{\theta ^{2}}\frac{\left( 1-\lambda _{I}2\right) }{%
1+\lambda _{I}}}=3\frac{-\lambda _{I}}{1-2\lambda _{I}} \\
\tau _{y_{2}}\left( l\right) &=&1+\frac{-y_{l}^{\ast }\theta ^{2}}{%
y_{l}^{\ast }\frac{\left( \theta ^{2}+\lambda _{I}\frac{2}{\theta ^{2}}%
\right) }{1-\lambda _{I}}}=\lambda _{I}\frac{2+\theta ^{4}}{2\lambda
_{I}+\theta ^{4}}
\end{eqnarray*}

Thus, in this case, the high ability type is encouraged to work more while
the low ability type is discouraged. We can understand this by noting that
the high ability type is more willing to take advantage of a subsidy to
work, since he has high productivity and thus low marginal cost of effort.
The subsidy thus helps separating the types.

\subsection{Implementation}

It is tempting to interpret the wedges as taxes and subsidies. However, this
is not entirely correct since the wedges in general are functions of all
taxes. Furthermore, while there is typically a unique set of wedges this is
generically not true for the taxes. As we have discussed above, many
different tax systems might implement the optimal allocation. One example is
the draconian, use 100\% taxation for every choice except the optimal ones.

Only by putting additional restrictions is the implementing tax system
found. Let us consider a combination if linear labor taxes and savings taxes
that implement the allocation in the example. To do this, consider the
individual problem,%
\begin{eqnarray*}
&&\max_{c_{1},y_{1},c_{h},c_{l},y_{h},y_{l}}\ln \left( c_{1}\right) -\frac{%
y_{1}^{2}}{2}+\left( \frac{\ln c_{h}-\frac{\left( \frac{y_{h}}{\theta }%
\right) ^{2}}{2}}{2}+\frac{\ln c_{l}-\frac{\left( \theta y_{l}\right) ^{2}}{2%
}}{2}\right) \\
s.t.0 &=&y_{1}\left( 1-\tau _{1}\right) +\frac{y_{h}}{2}\frac{1-\tau _{h}}{%
1-\tau _{k,h}}+\frac{y_{l}}{2}\frac{1-\tau _{l}}{1-\tau _{k,1}}-c_{1}-\frac{%
c_{h}}{2}\frac{1}{1-\tau _{k,h}}-\frac{c_{l}}{2}\frac{1}{1-\tau _{k,l}}
\end{eqnarray*}

First order conditions are;%
\begin{eqnarray*}
\frac{1}{c_{1}} &=&\lambda \\
y_{1} &=&\frac{1}{c_{1}}\left( 1-\tau _{1}\right) \\
\frac{1}{c_{h}} &=&\frac{1}{c_{1}}\frac{1}{1-\tau _{k,h}} \\
&\Rightarrow &\frac{c_{h}}{c_{1}}=1-\tau _{k,h} \\
\frac{1}{c_{l}} &=&\frac{1}{c_{1}}\frac{1}{1-\tau _{k,l}} \\
&\Rightarrow &\frac{c_{l}}{c_{1}}=1-\tau _{k,l} \\
y_{h}c_{h} &=&\theta ^{2}\left( 1-\tau _{h}\right) \\
y_{l}c_{l} &=&\frac{1-\tau _{l}}{\theta ^{2}}
\end{eqnarray*}

Therefore, the optimal intertemporal ratios $\frac{c_{h}^{\ast }}{%
c_{1}^{\ast }}=1+\lambda _{I}$ and $\frac{c_{l}^{\ast }}{c_{1}^{\ast }}%
=1-\lambda _{I}$ can be implemented by setting 
\begin{eqnarray*}
\tau _{k,h} &=&-\lambda _{I} \\
\tau _{k,l} &=&\lambda _{I}
\end{eqnarray*}%
we can implement the right ratios $\frac{c_{h}}{c_{1}}$ and $\frac{c_{l}}{%
c_{1}}.$ Note that the expected capital income tax rate is zero, but it will
distort savings.\ Why?

Similarly, the optimal intratemporal allocations 
\begin{eqnarray*}
y_{h}^{\ast }c_{h}^{\ast } &=&\frac{\theta ^{2}\left( 1+\lambda _{I}\right) 
}{1-2\lambda _{I}} \\
y_{l}^{\ast }c_{l}^{\ast } &=&\frac{1-\lambda _{I}}{\theta ^{2}+\frac{%
2\lambda _{I}}{\theta ^{2}}}
\end{eqnarray*}
can be implemented by setting 
\begin{eqnarray*}
\frac{1+\lambda _{I}}{1-2\lambda _{I}} &=&1-\tau _{h} \\
\frac{1-\lambda _{I}}{1+\frac{2\lambda _{I}}{\theta ^{4}}} &=&1-\tau _{l}
\end{eqnarray*}
\begin{equation*}
\tau _{l}=\lambda _{I}\frac{2+\theta ^{4}}{2\lambda _{I}+\theta ^{4}},\tau
_{h}=3\frac{\lambda _{I}}{2\lambda _{I}-1}.
\end{equation*}

Combining this system with a lump sum tax to balance the budget, implements
the optimal allocation.

\subsection{Time consistency}

Under the Mirlees approach, the government announces a menu of taxes or of
consumption baskets. People then make choices that in eqauilibrium reveal
their true types (abilities) to the government. Suppose the government could
then re-optimize. Would it like to do this?

The problem is more severe in a dynamic setting provided abilities are
persistent. Why?

In a finite horizon economy, there might only be very bad equilibria
(Robert, 

\end{document}