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\begin{document}

\title{Topics in Dynamic Public Finance}
\author{John Hassler\\Stockholm 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 reservation 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 In the first period, the individual works and chooses how much to
consume of the income, normalized to unity, and how much to save.

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

\item During the second period, the individual can determine how long it takes
to find a job by choosing the reservation wage $y_{n}$ and costly search
effort $c$. 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 gets UI-benefits $b.$These are paid by
taxes on workers.

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

\item Both the unemployment duration and the wage upon rehiring is non-stochastic.
\end{itemize}

Total disposable income in second period if laid off is therefore the
non-stochastic value
\[
\left(  1-\beta\right)  \left(  b-c\right)  +\beta y_{n}\left(  1-\tau\right)
\equiv y_{l}.
\]


In first periods, individuals decide how much to save, $s.$ Interest rate and
subjective discount rate is normalized to zero. If an individual gets laid
off, he consumes his resources, i.e., his disposable income plus savings.

Welfare is
\[
V=u\left(  1-\tau-s\right)  +\alpha u\left(  1-\tau+s\right)  +\left(
1-\alpha\right)  \left(  u\left(  y_{l}+s\right)  \right)  .
\]


Government budget constraint is%
\begin{align*}
\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{align*}
Denoting the \emph{endogenous} total income by $Y\equiv1+\alpha+\left(
1-\alpha\right)  \beta y_{n},$ this implies
\begin{align*}
b  &  =\frac{Y}{\left(  1-\alpha\right)  \left(  1-\beta\right)  }\tau\\
&  \equiv\mu\tau,
\end{align*}
where we note that $\mu$ is \emph{not} a constant, but depends on individual
choices of $y_{n}$ and $c$ 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
\[
\left(  y_{n}+c\right)  \beta_{c}=1-\beta
\]
since social income is
\[
\left(  1-\beta\left(  y_{n},c\right)  \right)  \left(  b-c\right)  +\beta
y_{n}%
\]
implying that the marginal gain of a marginal unit of effort is $\beta
_{c}\left(  y_{n}+c\right)  $ and the cost is $1-\beta.$

The individual instead gains,
\[
y_{n}\left(  1-\tau\right)  +c-b\text{ }%
\]
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 private costs
$-\left(  y_{n}\left(  1-\tau\right)  +c-b\right)  \beta_{y_{n}}.$ The wedges
between private and social costs/benefits imply that both choices will be
distorted in second best.

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


The optimal UI system maximizes solves
\[
\max_{\tau}V\left(  c,y_{n},s,\mu\left(  \tau\right)  ,\tau\right)
\]


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


This is the envelope theorem. Therefore, the first order condition for
maximizing $V$ by choosing $\tau$ is
\[
\frac{dV}{d\tau}=V_{\mu}\frac{d\mu}{d\tau}+V_{\tau}=0,
\]
where%

\begin{align*}
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{align*}
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{align*}
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{align*}
implying
\begin{align*}
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{align*}


Approximating
\[
u^{\prime}\left(  c_{1}\right)  \approx u^{\prime}\left(  c_{u}\right)
+u^{\prime\prime}\left(  c_{u}\right)  \Delta c
\]
where $\Delta c\equiv c_{1}-c_{u}$ is the fall in consumption if becoming
unemployed. The first order condition is then
\begin{align*}
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}+\left(
1-\alpha\right)  \left(  1-\beta y_{n}+\left(  1-\beta\right)  \frac
{Y}{\left(  1-\alpha\right)  \left(  1-\beta\right)  }\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}+\frac{Y}{1-\alpha}\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}+\frac{1+\alpha+\left(  1-\alpha\right)
\beta y_{n}}{\left(  1-\alpha\right)  }\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}%
\end{align*}
Using the definition
\[
\mu\equiv\frac{b}{\tau}=\frac{Y}{\left(  1-\alpha\right)  \left(
1-\beta\right)  }%
\]
we get
\begin{align*}
\frac{u^{\prime\prime}}{u^{\prime}}\frac{\Delta c}{Y}  &  =\frac{\tau}{\mu
}\frac{d\mu}{d\tau}\\
-R_{r}\frac{\Delta c}{c}  &  =E_{\mu,t}Y
\end{align*}
where $E_{\mu,t}$ is the elasticity of $\mu$ with respect to taxes and $R_{r}$
the relative risk aversion coefficient. Recall that $\mu$ is the ratio between
benefits and taxes should be interpreted as the ratio between employment and unemployment.

Note that we should not interpret $Y$ as the aggregate \emph{level }\ of
income since we have normalized the pre-unemployment income to unity. Instead,
it is a measure of employment. Setting $y_{n}\approx1,Y\approx1+\alpha+\left(
1-\alpha\right)  \beta$ which is the time people work. In this simple model,
this is value is overstated since no unemplyment occur in the first period.
More realistically, it should be close to one, giving
\[
R_{r}\frac{\Delta c}{c}=-E_{\mu,t}%
\]


The interpretation is that the welfare loss (the LHS) should optimally be
given by how elastic the ratio of employment to unemployment is with respect
to taxes.

Without moral hazard, $\frac{d\mu}{d\tau}=0=E_{\mu,t},$ in which case
optimality requires $\Delta c=0.$ With moral hazard, higher taxes tends to
reduce $\mu$ since the employment to unemployment falls in in taxes, i.e.,
$\frac{\tau}{\mu}\frac{d\mu}{d\tau}=E_{\mu,t}$ is negative. Therefore,
$\frac{\Delta c}{c}>0.$ We see that $\frac{\Delta c}{c}$ increases if
$\frac{\tau}{\mu}\frac{d\mu}{d\tau}$ is large in absolute terms and falls if
risk aversion is large. Baily claims that $E_{\mu,t}$ is in the order
$0.15-0.4$. With log utility, this is also how much consumption should fall on
entering unemployment.

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 constraints. The model can therefore be
extended to evaluate UI reforms. With a more dynamic model, and in particular
if capital markets are imperfect, it should be noted that one needs to know
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
\[
\sum_{t=0}^{\infty}\left(  \frac{1}{1+r}\right)  ^{t}\left(  u\left(
c_{t}\right)  -e_{t}\right)
\]
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 exogenous 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 moral
hazard problem is that individuals can affect the probability of finding a
job. As in Baily (1978), the individual 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
\[
p\left(  w_{t}^{\ast},e_{t}\right)  =\int_{w_{t}^{\ast}}^{\infty}f\left(
w_{t},e_{t}\right)  dw_{t}%
\]
with
\begin{align*}
p_{w}\left(  w_{t}^{\ast},e_{t}\right)   &  =-f\left(  w_{t},e_{t}\right)
\leq0\text{ and }\\
p_{e}\left(  w_{t}^{\ast},e_{t}\right)   &  >0
\end{align*}
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
\[
u_{t}=\tilde{u}\left(  w_{t}^{\ast},e_{t}\right)  \equiv\frac{1+r}{r}%
\int_{w_{t}^{\ast}}^{\infty}u\left(  w_{t}\right)  \frac{f\left(  w_{t}%
,e_{t}\right)  }{p\left(  w_{t}^{\ast},e_{t}\right)  }dw_{t}%
\]
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{align*}
\tilde{u}_{w}\left(  w_{t}^{\ast},e_{t}\right)   &  \geq0\\
\tilde{u}_{e}\left(  w_{t}^{\ast},e_{t}\right)   &  \geq0.
\end{align*}


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 distribution of
acceptable job offers.

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


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


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%
\[
\frac{dE_{t}}{dE_{t+1}}=\frac{\partial E_{t}}{\partial E_{t+1}}=\frac
{1-p\left(  w_{t}^{\ast},e_{t}\right)  }{1+r}%
\]
Now, we will show the important results that anything that reduces next
periods unemployment value $E_{t+1}$ 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{align*}
&  p_{e}\left(  w_{t}^{\ast},e_{t}\right)  \left(  \tilde{u}\left(
w_{t}^{\ast},e_{t}\right)  -E_{t+1}\right)  +p\left(  w_{t}^{\ast}%
,e_{t}\right)  \tilde{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{align*}
both becomes larger if choices are unchanged. In words, the marginal benefit
of searching harder 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 use this to show the key result that benefits should have a
decreasing 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
\[
db_{t}=-\frac{1-p}{1+r}db_{t+1}>0
\]
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{align*}
&  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{align*}
since $u^{\prime}\left(  b_{t}\right)  =u^{\prime}\left(  b_{t+1}\right)  .$By
the envelope theorem, we need not take into account changes in endogenous
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 endogenous
changes on $p.$

Let
\[
B_{t}=b_{t}+\frac{1-p}{1+r}b_{t+1}%
\]


Then,%
\begin{align*}
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{align*}


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{Extensions}

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 unsuccessful search over the entire future of the
individual, a more efficient insurance can be achieved. 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 the \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 affects 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
exogenous separation probability $s$ and
\[
p\left(  a\right)  =\left\{
\begin{array}
[c]{c}%
p\text{ if }a=1\\
0\text{ otherwise}%
\end{array}
\right.
\]
which is the assumption we make here.

The individual has a utility function
\[
E\sum_{t=0}^{\infty}\left(  \frac{1}{1+r}\right)  ^{t}\left(  u\left(
c_{t}\right)  -a_{t}\right)  .
\]


Let $\theta_{t}\in\left\{  0,1\right\}  $ be the employment status of the
individual in period $t,$ where $\theta_{t}=1$ represents employment. 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 $\theta^{t-1}\times0=\left(  \theta
_{0},\theta_{1},...\theta_{t},0\right)  \equiv\theta_{u}^{t}$, and similarly,
$\theta^{t-1}\times1\equiv\theta_{e}^{t}).$

An allocation is now defined as a rule that assigns consumption and effort as
a function of $\theta^{t}$ at every point in time and for every possible
history, $c_{t}=c\left(  \theta^{t}\right)  .$ We focus on allocations where
$a_{t}=1$. Individuals must be induced to volontarily choose $a_{t}=1.$
Allocations that satisfies this are called incentive compatible allocations.

Given an allocation we can compute the expected discouted utililty at every
point in time for every possible history, $V_{t}=V\left(  \theta^{t}\right)
.$ The problem is now to choose the allocation that minimizes the cost of
giving some fixed initial utility level to the representative individual. This
problem can be written in a recursive way. In period zero, the planner gives a
consumption level $c_{0},$ prescribes an effort level $a_{0}$ (=1) and
promised continuation utilities $V_{1}^{e}\equiv V\left(  \theta_{e}%
^{1}\right)  $ and $V_{1}^{u}=V\left(  \theta_{u}^{1}\right)  .$ 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 voluntarily chooses $a_{0}.$ The problem is recursive and at any
node, costs of providing promised utilities are minimized given incentive constraints

The problem of the unemployed individual is also recursive. -- as unemployed,
maximized utility is (the agent only controls $a_{t})$
\[
V\left(  \theta_{u}^{t}\right)  =u\left(  c_{t}\right)  -1+\frac{1}%
{1+r}\left(  pV\left(  \theta_{u}^{t}\times1\right)  +\left(  1-p\right)
V\left(  \theta_{u}^{t}\times0\right)  \right)
\]
with the incentive constraint%
\[
\frac{1}{1+r}p\left(  V\left(  \theta_{e}^{t+1}\right)  -V\left(  \theta
_{u}^{t+1}\right)  \right)  \geq1.
\]


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. Similarly, let
$C\left(  V_{t}\right)  $ denote the minimal cost of providing utility $V$ to
an unemployed (are these function changing over time?). $W$ satisfies%
\begin{align*}
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{align*}
where$V_{t}=V\left(  \theta_{e}^{t}\right)  ,$ $c_{t}=c\left(  \theta_{e}%
^{t}\right)  ,$ $V_{t+1}^{e}=V\left(  \theta_{e}^{t}\times1\right)  $ and
$V_{t+1}^{u}=V\left(  \theta_{e}^{t}\times0\right)  .$

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

$C$ satisfies%
\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)  \geq1,\\
V_{t}=u\left(  c_{t}\right)  -1+\frac{1}{1+r}\left(  pV_{t+1}^{e}+\left(
1-p\right)  V_{t+1}^{u}\right)  .
\end{gather*}
where $V_{t}=V\left(  \theta_{u}^{t}\right)  ,$ $c_{t}=c\left(  \theta_{u}%
^{t}\right)  ,$ $V_{t+1}^{e}=V\left(  \theta_{u}^{t}\times1\right)  $ and
$V_{t+1}^{u}=V\left(  \theta_{u}^{t}\times0\right)  .$

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{align}
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}\nonumber\\
C^{\prime}\left(  V_{t+1}^{u}\right)   &  =\delta_{t}^{e}.\nonumber
\end{align}


The envelope condition is
\[
W^{\prime}\left(  V_{t}\right)  =\delta_{t}^{e}=\frac{1}{u^{\prime}\left(
c_{t}\right)  }=W^{\prime}\left(  V_{t+1}^{e}\right)  =C^{\prime}\left(
V_{t+1}^{u}\right)  .
\]


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. Since $W^{\prime}\left(  V_{t}\right)  =C^{\prime
}\left(  V_{t+1}^{u}\right)  ,$marginal marginal utility does not change if
the person becomes unemployed, i.e., consumption does not change upon loosing
his job either. This is due to the fact that there is no moral hazard problem
on the job and full insurance is therefore optimal.\footnote{From now, I will
mostly skip writing out the explicit dependence on history, hopefully without
creating confusion.}

When the agent is unemployed, the FOC and envelope conditions are%
\begin{align*}
1  &  =\delta_{t}^{u}u^{\prime}\left(  c_{t}\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{align*}


Giving%
\begin{align}
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}\nonumber\\
C^{\prime}\left(  V_{t+1}^{u}\right)   &  =\frac{1}{u^{\prime}\left(
c_{t}\right)  }-\gamma_{t}\frac{p}{1-p}\nonumber
\end{align}


\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{align*}
W^{\prime}\left(  V_{t+1}^{e}\right)   &  >C^{\prime}\left(  V_{t}\right)
>C^{\prime}\left(  V_{t+1}^{u}\right)  ,\\
\frac{1}{u^{\prime}\left(  c\left(  \theta_{u}^{t}\times1\right)  \right)  }
&  >\frac{1}{u^{\prime}\left(  c\left(  \theta_{u}^{t}\right)  \right)
}>\frac{1}{u^{\prime}\left(  c\left(  \theta_{u}^{t}\times1\right)  \right)
}\\
c\left(  \theta_{u}^{t}\times1\right)   &  >c\left(  \theta_{u}^{t}\right)
>c\left(  \theta_{u}^{t}\times1\right)
\end{align*}


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 successively 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 $\frac{1}{1+r}p\left(  V_{t+1}^{e}-V_{t+1}^{u}\right)  \geq1$
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}), $W^{\prime}\left(
V_{t+1}^{e}\right)  =\frac{1}{u^{\prime}\left(  c_{t+1}\right)  },$ where
$c_{t+1}=c\left(  \theta_{t+1}^{e}\right)  $ 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 $c_{t}=c\left(  \theta_{t}\right)  )$. 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{align*}
\frac{1}{u^{\prime}\left(  c\left(  \theta_{u}^{t}\right)  \right)  }  &
=p\frac{1}{u^{\prime}\left(  c\left(  \theta_{u}^{t}\times1\right)  \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{align*}


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%
\[
\frac{1}{u^{\prime}\left(  c_{t}\right)  }\frac{1+r}{1+\rho}=E_{t}\frac
{1}{u^{\prime}\left(  c_{t+1}\right)  }.
\]
}. Note the difference between this and the standard Euler equation.
\[
u^{\prime}\left(  c_{t}\right)  =E_{t}u^{\prime}\left(  c_{t+1}\right)  .
\]


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


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
\[
\frac{1}{c_{t}}=\frac{1}{E_{t}c_{t+1}}\Rightarrow c_{t}=E_{t}c_{t+1},
\]
while the Euler equation, guiding private preferences, implies the privately
optimal consumption $c_{t}^{\ast}$ given future consumption is%
\[
c_{t}^{\ast}=\frac{1}{E_{t}\left(  \frac{1}{c_{t+1}}\right)  }<E_{t}c_{t+1}.
\]


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: risk premium 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 $\tau_{t+1}^{u}=\tau^{u}\left(  a_{t}\right)
.$ Then, to have the individual Euler equation satisfied, we need%
\begin{align}
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) \label{eq_Eulerexample}\\
\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) \nonumber
\end{align}


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


Using (\ref{eq_inverseEulerex}) to replace $c_{t}$\ in (\ref{eq_Eulerexample})
together with (\ref{eq_Budgetbalance}) yields$\allowbreak$%
\begin{align*}
\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)  }=\frac{p\Delta c_{t+1}}%
{E_{t}c_{t+1}}\\
\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)  }=-\frac{\left(
1-p\right)  \Delta c_{t+1}}{E_{t}c_{t+1}}%
\end{align*}


These tax rates leads to both the Euler and the inverse Euler equation being
satisfied. 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.

\textbf{Result}: Rendahl (2007)

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

%TCIMACRO{\FRAME{itbpFU}{5.4561in}{4.1053in}{0in}{\Qcb{From Pontus Rendahl 2007
%(Job market paper)}}{}{slide9.emf}{\special{ language "Scientific Word";
%type "GRAPHIC";  maintain-aspect-ratio TRUE;  display "USEDEF";
%valid_file "F";  width 5.4561in;  height 4.1053in;  depth 0in;
%original-width 10.0016in;  original-height 7.5135in;  cropleft "0";
%croptop "1";  cropright "1.0002";  cropbottom "0";
%filename 'figures/Slide9.EMF';file-properties "XNPEU";}}}%
%BeginExpansion
{\parbox[b]{5.4561in}{\begin{center}
\includegraphics[
trim=0.000000in 0.000000in -0.002001in 0.000000in,
natheight=7.513500in,
natwidth=10.001600in,
height=4.1053in,
width=5.4561in
]%
{figures/Slide9.emf}%
\\
From Pontus Rendahl 2007 (Job market paper)
\end{center}}}%
%EndExpansion


\subsection{The Dynamic approach with unobservable saving}

An key assumption in the previous subsection was that the planner can control
the consumption level of the individual at all times, the only unobservable is
search effort. In reality this assumption seems questionable, given the
existence of alternative means of income, capital markets, insurance within an
extended family and durable goods.

In this subsection, we assume that the planner cannot control the consumption
of the individual -- she has access to a perfect market for lending and
borrowing at a fixed interest rate and her wealth is unobservable. Of course,
this extreme is perhaps equally unrealistic and the truth might be somewhere
in between.

An immediate problem is that seach decisions in this setting might depend on
the unobservable wealth level. Making sure that there is always an incentive
to search might then be unfeasible in general. In one special case, the seach
decision is not dependent on wealth, when individuals have CARA utility. This
is the way we go here. Furthermore, we simplify by assuming that seach is
either one or zero.

Individuals maximize their intertemporal utility, given by%

\[
E\int_{0}^{\infty}e^{-rt}U\left(  c_{t}\right)  dt,
\]
where
\[
U\left(  c_{t}\right)  \equiv-e^{-\gamma c_{t}}.
\]


The purpose of the planner is to maximize time zero welfare of an employed
agent subject to

\begin{enumerate}
\item budget balance expressed as actuarial fairness, i.e., that the expected
discounted value of tax payments equals that of benefits (note that this is
not the same as a budget balance in a pay-as-you-go system) and to

\item the constraint that agents volontarily search.
\end{enumerate}

Without loss of generality, we let individuals pay lump-sum taxes, denoted
$\tau$, implying that
\begin{equation}
\dot{A}_{t}=rA_{t}+y-c_{t}-\tau,\label{AssetLoM}%
\end{equation}
where $y=w$ if the individual is employed, $y=b-s$, if the individual is
unemployed and search and $y=b$ if the individual is unemployed without
searching. Define the average discounted probabilities (ADP's) of being
unemployed (in state 2) as
\begin{align*}
\Pi_{2}  & \equiv r\int_{0}^{\infty}e^{-rt}\mu_{2,t}dt\\
& =,
\end{align*}
where $\mu_{2,t}$ is the probabilities of being unemployed at time $t$,
respectively, conditional on being employed at time zero, provided that
unemployed search for a job.

Actuarial fairness the UI system is now a simple linear function of the
benefits%
\begin{equation}
\tau=\Pi_{2}b \label{taxes}%
\end{equation}


Under constant absolute risk aversion and stationary income uncertainty, the
value functions for the two states $j\in\left\{  1,2\right\}  $ can be
separated
\begin{equation}
V\left(  A_{t},j\right)  =W\left(  A_{t}\right)  \tilde{V}_{j}\left(
\tau,b\right)  , \label{valuefunctions}%
\end{equation}
where
\begin{align}
W\left(  A_{t}\right)   &  \equiv\frac{e^{-\gamma A_{t}}}{r}%
\label{eq_valuefunctionparts}\\
\tilde{V}_{j}  &  \equiv-e^{-\gamma c_{j}},\nonumber
\end{align}
and $\sigma_{j}$ are state-dependent consumption constants such that the state
dependent consumption functions are
\begin{equation}
c_{j}\left(  A_{t}\right)  =rA_{t}+\sigma_{j}. \label{consumption}%
\end{equation}


The consumption constants $\sigma_{j}$ are nonlinear functions of income in
all states and thus, depend on the planner choice variables $\tau,$ and $b$.
The constants are found as the unique solutions to the Bellman equations for
each state:
\begin{align}
\sigma_{1}  &  =w-\tau-\frac{q\left(  e^{\gamma\Delta_{2}}-1\right)  }{\gamma
r},\label{Cconstants}\\
\sigma_{2}  &  =b-s-\tau+\frac{h\left(  1-e^{-\gamma\Delta_{2}}\right)
}{\gamma r},\nonumber
\end{align}
where $q$ is the exogeneous hiring rate, $h$ is the hiring rate if the agent
search actively and%
\begin{equation}
\Delta_{2}\equiv\sigma_{1}-\sigma_{2}.\nonumber
\end{equation}


Let us derive these results;

Conjecturing that the value functions are $-\frac{1}{r}e^{-\gamma\left(
rA_{t}+\sigma_{j}\right)  },$ we can write the Bellman equations for the
enployed as%
\begin{align*}
-\frac{1}{r}e^{-\gamma\left(  rA_{t}+\sigma_{1}\right)  } &  =\max_{\sigma
}-e^{-\gamma\left(  rA_{t}+\sigma\right)  }dt-\left(  1-rdt\right)  \left(
1-qdt\right)  \frac{1}{r}e^{-\gamma\left(  rA_{t+dt}+\sigma_{1}\right)  }\\
&  -\left(  1-rdt\right)  qdt\frac{1}{r}\left[  e^{-\gamma\left(
rA_{t+dt}+\sigma_{2}\right)  }\right]  .
\end{align*}
Using the budget constraint, $A_{t+dt}=A_{t}+r\left(  w-\tau-\sigma\right)
dt$, and dividing by $e^{-\gamma rA_{t}},$ this becomes%
\begin{align*}
-\frac{1}{r}e^{-\gamma\sigma_{1}} &  =\max_{\sigma}-e^{-\gamma\sigma
}dt-\left(  1-rdt\right)  \left(  1-qdt\right)  \frac{1}{r}e^{-\gamma\left(
r\left(  w-\tau-\sigma\right)  dt+\sigma_{1}\right)  }\\
&  -\left(  1-rdt\right)  qdt\frac{1}{r}\left[  e^{-\gamma\left(  r\left(
w-\tau-\sigma\right)  dt+\sigma_{2}\right)  }\right]  .
\end{align*}
Using the first-order linear approximation, $e^{-\gamma\left(  r\left(
w-\tau-\sigma\right)  dt+\sigma_{1}\right)  }\approx$ $e^{-\gamma\sigma_{1}%
}-\gamma r\left(  w-\tau-\sigma\right)  dte^{-\gamma\sigma_{1}}$, adding
$\frac{1}{r}e^{-\gamma\sigma_{1}}$ to both sides, dividing by $dt$ and letting
$dt$ approach zero, yields%
\begin{align}
0 &  =\max_{\sigma}\left\{  -re^{-\gamma\left(  \sigma-\sigma_{1}\right)
}+r+\gamma r\left(  w-\tau-\sigma\right)  \right\}  \label{Bellman1}\\
&  +q\left(  1-e^{-\gamma\left(  \sigma_{2}-\sigma_{1}\right)  }\right)
\nonumber
\end{align}


Similarly, for the unemployed, we obtain
\begin{align}
0 &  =\max_{\sigma}\left\{  -re^{-\gamma\left(  \sigma-\sigma_{2}\right)
}+\gamma r\left(  b_{2}-s-\tau-\sigma\right)  \right\}  \label{Bellman2}\\
&  +r+h-he^{-\gamma\left(  \sigma_{1}-\sigma_{2}\right)  }\nonumber
\end{align}


The right hand sides of  (\ref{Bellman1}) and (\ref{Bellman2}) are maximized
at $\sigma=\sigma_{j},$ implying that these values maximize the RHS's of the
Bellman equations.

Substituting $\sigma_{1}$ and $\sigma_{2}$ respectively for $\sigma$ in
(\ref{Bellman1}) and (\ref{Bellman2}) solves the maxima. Finally, solving for
gives the $\sigma_{j}^{s}$ gives (\ref{Cconstants}), which by construction
then solves the Bellman equations.

Clearly, the objective of the planner is now to maximize $\sigma_{1},$ from
which also follows time consistency -- the welfare of employed at all times is maximized.

The first step is now to derive an expression for $\sigma_{1}$ in terms of
$\Delta_{2}$ where the budget constraint (\ref{taxes}) is used to replace the
tax rate$.$ For this purpose, we subtract the second line of (\ref{Cconstants}%
) from the first and solve for $b.$ Then, we use this expression in the budget
constraint $\tau=\Pi_{2}b$ and substitute for $\tau$ in the first line of
(\ref{Cconstants}). This yields%
\begin{equation}
\sigma_{1}=\kappa+\Pi_{2}\left(  \Delta_{2}-\frac{he^{-\gamma\Delta_{2}}%
}{\gamma r}\right)  -\left(  1-\Pi_{2}\right)  q\frac{e^{\gamma\Delta_{2}}%
}{\gamma r}, \label{eq_objective2state}%
\end{equation}
where $\kappa$ is a constant\textbf{, }independent of the choice variables.
Straightforward calculus shows that (\ref{eq_objective2state}) defines
$\sigma_{1}$ as a concave function of $\Delta_{2}$ with a unique maximum at 0.
The reason for\textbf{ }$\sigma_{1}$ being\textbf{ }maximized at $\Delta
_{2}=0$ is obvious -- when actuarial insurance is available, full insurance
maximizes utility. However, $\Delta_{2}=0$ is not incentive compatible.
Searching moving will not occur voluntarily. Now, as in Baily approach, we can
use the consumption fall upon separation, $\Delta_{2}$, to evaluate the gain
by finding employment.

If the unemployed agent shirks she is unemployed for ever, getting an income
$b-$ $\tau$ and a utility%

\[
-\frac{1}{r}e^{-\gamma rA_{t}}e^{-\gamma\left(  b-\tau\right)  }.
\]


The utility if\textbf{ }the individual instead searches is
\[
-\frac{1}{r}e^{-\gamma rA_{t}}e^{-\gamma\sigma_{2}}.
\]


To induce search, we clearly need
\[
\sigma_{2}\geq b-\tau.
\]


Note that the consumption of the unemployed who search is $rA_{t}+\sigma_{2}.$
Furthermore, her total income net of search costs is $rA_{t}+b-\tau-s.$
Therefore, the search condition implies consumption to be strictly higher than
income. Over time, the unemployed depletes\textbf{ }her assets and consumption
therefore falls\textbf{, }despite the benefits being\textbf{ }constant. The
celebrated result by Shawell-Wais and Hopenhayn-Nicolini that consumption
should optimally fall over the unemployment spell when the insurer can
fully\textbf{ }control\textbf{ }consumption (no hidden savings) is therefore
mimicked in this case, where hidden savings are allowed.

The final part is now to express the search constraint in terms of the
consumption difference $\Delta_{2}.$ Using the second line of
(\ref{Cconstants})$,$ the search constraint can be written
\begin{equation}
\Delta_{2}\geq-\frac{\ln\left(  1-\gamma r\frac{s}{h}\right)  }{\gamma},
\label{eq_IC2}%
\end{equation}
which we label the \emph{IC2-condition}. We depict this in Figure 1,%

%TCIMACRO{\FRAME{ftbpF}{5.4457in}{4.1053in}{0pt}{}{}{slide1.emf}%
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\begin{figure}
[ptb]
\begin{center}
\includegraphics[
natheight=7.569700in,
natwidth=10.056000in,
height=4.1053in,
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{figures/Slide1.emf}%
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%EndExpansion


A higher $r$ and $s$ and lower $h$ reduce the value of searching, and shifts
the constraint to the right.

Finally, we can solve for the value of $b$ that makes the IC2 condition bind
exactly. Take the difference between the equations in (\ref{Cconstants}) set
it to $-\frac{\ln\left(  1-\gamma r\frac{s}{h}\right)  }{\gamma}$ and solve
for $b$, which gives%

\begin{align*}
-\frac{\ln\left(  1-\gamma r\frac{s}{h}\right)  }{\gamma}  &  =w-\tau
-\frac{q\left(  e^{-\ln\left(  1-\gamma r\frac{s}{h}\right)  }-1\right)
}{\gamma r}-\left(  b-s-\tau+\frac{h\left(  1-e^{\ln\left(  1-\gamma r\frac
{s}{h}\right)  }\right)  }{\gamma r}\right) \\
b  &  =w+s+\frac{\ln\left(  1-\gamma r\frac{s}{h}\right)  }{\gamma}%
-\frac{q\left(  -\frac{h}{-h+\gamma rs}-1\right)  }{\gamma r}-\frac{h\left(
1--\frac{-h+\gamma rs}{h}\right)  }{\gamma r}\\
&  =w+\frac{\ln\left(  1-\gamma r\frac{s}{h}\right)  }{\gamma}-\frac
{sq}{h-\gamma rs}%
\end{align*}


In Hassler\&Rodriguez (2008), we extend this model and show that it is useful
to analyze multiple incentive constraints. It is immediate to show that
benefits should optimally be constant over time. This since the incentive
constraint does not change over time. We also introduce multiple incentive
constraints, showing that if there is also a need to induce some individuals
to move to find a job, this is optimally done with an initial period of low
benefits.\newpage

\section{Optimal taxation -- the Ramsey approach}

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

To provide some intuition already before providing the famous Chamley \& Judd
result consider the following simple model.

\textbf{Preferences}

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

\[
U=\sum_{t=0}^{\infty}\beta^{t}u\left(  c_{t},1-n_{t}\right)  .
\]


\textbf{Technology}

Output is produced by labor only on a competitive 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 government needs to finance its consumption by tax revenues. For
simplicity, we have already assumed that its consumption does not interfere
with the individuals private problem. We will assumed that the government
cannot finance its consumption by lump sum 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, $\tau_{s}.$ Wages are exogeous  at rate
$w_{t}.$

The representative indivuals budget constraint in the first and second period
is %

\begin{align*}
c_{0}\left(  1+\tau_{c,0}\right)  +k_{1}+b_{1}  & =\left(  1-\tau
_{l,0}\right)  w_{0}n_{0}+\left(  1-\tau_{s,0}\right)  \left(  1+r\right)
\left(  k_{0}+b_{0}\right)  \\
c_{1}\left(  1+\tau_{c,0}\right)  +k_{2}+b_{2}  & =\left(  1-\tau
_{l,1}\right)  w_{1}n_{1}+\left(  1-\tau_{s,1}\right)  \left(  1+r\right)
\left(  k_{1}+b_{1}\right)
\end{align*}


where $k$ is phyiscal savings and $b$ government bonds. The interest rate on
bonds coincides with the rate on storage.

Substituting forward yields, %

\[
\sum_{t=0}^{\infty}c_{t}\left(  \frac{1}{1+r}\right)  ^{t}%
%TCIMACRO{\dprod \limits_{s=0}^{t}}%
%BeginExpansion
{\displaystyle\prod\limits_{s=0}^{t}}
%EndExpansion
W_{i,s}+\lim_{t\rightarrow\infty}\left(  \frac{1}{1+r}\right)  ^{t}c_{t}%
%TCIMACRO{\dprod \limits_{s=0}^{t}}%
%BeginExpansion
{\displaystyle\prod\limits_{s=0}^{t}}
%EndExpansion
W_{i,s}\left(  k_{t}+b_{t}\right)  =\sum_{t=0}^{\infty}w_{t}n_{t}%
W_{l,t}\left(  \frac{1}{1+r}\right)  ^{t}%
%TCIMACRO{\dprod \limits_{s=0}^{t}}%
%BeginExpansion
{\displaystyle\prod\limits_{s=0}^{t}}
%EndExpansion
W_{i,s}+\frac{\left(  1-\tau_{t}^{k}\right)  }{\left(  1+\tau_{c,t}\right)
}\left(  1+r\right)  \left(  k_{0}+b_{0}\right)
\]
where
\[
W_{i,t}\equiv\frac{1+\tau_{c,t}}{\left(  1+\tau_{c,t-1}\right)  \left(
1-\tau_{s,t}\right)  }%
\]
and
\[
W_{l,t}\equiv\frac{1-\tau_{l,t}}{1+\tau_{c,t}}%
\]%
\[
U=\sum_{t=0}^{\infty}\beta^{t}u\left(  c_{t},1-n_{t}\right)
\]


First order conditions are
\begin{align*}
\beta^{t}u_{c}\left(  c_{t},1-n_{t}\right)    & =\lambda\left(  \frac{1}%
{1+r}\right)  ^{t}%
%TCIMACRO{\dprod \limits_{s=0}^{t}}%
%BeginExpansion
{\displaystyle\prod\limits_{s=0}^{t}}
%EndExpansion
W_{i,s}\\
\beta^{t}u_{l}\left(  c_{t},1-n_{t}\right)    & =\lambda w_{t}W_{l,t}\left(
\frac{1}{1+r}\right)  ^{t}%
%TCIMACRO{\dprod \limits_{s=0}^{t}}%
%BeginExpansion
{\displaystyle\prod\limits_{s=0}^{t}}
%EndExpansion
W_{i,s}%
\end{align*}


which we can write
\begin{align*}
\frac{u_{l}\left(  c_{t},1-n_{t}\right)  }{u_{c}\left(  c_{t},1-n_{t}\right)
w_{t}}  & =W_{l,t}\\
\frac{u_{c}\left(  c_{t},1-n_{t}\right)  }{u_{c}\left(  c_{0},1-n_{0}\right)
}\left(  \beta\left(  1+r\right)  \right)  ^{t}  & =%
%TCIMACRO{\dprod \limits_{s=0}^{t}}%
%BeginExpansion
{\displaystyle\prod\limits_{s=0}^{t}}
%EndExpansion
W_{i,s}%
\end{align*}


Note here that in the first best, 

\textbf{Individual optimality}

The first order conditions of the individual problem are\footnote{We disregard
the constraint that $i_{1}\geq0,$ otherwise, we could have corner soluitions.}%

\begin{align}
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}}\nonumber\\
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)  }\nonumber\\
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)  }\nonumber
\end{align}


\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{align*}
\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{align*}


Eliminating $\lambda,$ the individual optimality constraints are%
\begin{align*}
\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{align*}
with the solution%

\begin{align}
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)
}\nonumber
\end{align}


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
\[
\max_{\tau_{l},\tau_{c,1},\tau_{c,2}}\ln\left(  1-\tau_{l}\right)  -\ln\left(
1+\tau_{c,1}\right)  +\beta\left(  \ln\left(  1-\tau_{l}\right)  -\ln\left(
1+\tau_{c,2}\right)  \right)
\]
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{align*}
\tau_{c,1};1+\tau_{c,1}  &  =\lambda\frac{w\left(  2+r\right)  }{2\left(
1+\beta\right)  \left(  1+r\right)  }\left(  1-\tau_{l}\right) \\
\tau_{c,2};1+\tau_{c,2}  &  =\lambda\frac{w\left(  2+r\right)  }{2\left(
1+\beta\right)  \left(  1+r\right)  }\left(  1-\tau_{l}\right) \\
\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{align*}


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, as long as $\tau_{l}$ is
properly adjusted. For example, we could choose $\tau_{c,1}=\tau_{c,2}=0,$ in
which case%
\[
\frac{1}{1-\tau_{l}}=\lambda\frac{w\left(  1+\beta\right)  \left(  2+r\right)
}{\left(  1+\beta\right)  2\left(  1+\beta\right)  \left(  1+r\right)  }.
\]


Substituting this into the FOC for or we could set $\tau_{c,1}$ and
$\tau_{c,2}$ yields:%
\begin{align*}
1  &  =\lambda\frac{w\left(  2+r\right)  }{2\left(  1+\beta\right)  \left(
1+r\right)  \left(  1\right)  ^{2}}\frac{\left(  1+\beta\right)  2\left(
1+\beta\right)  \left(  1+r\right)  }{\lambda w\left(  1+\beta\right)  \left(
2+r\right)  }\\
&  =1
\end{align*}
Alternatively, we could set $\tau_{l}=0,$ in which case,
\[
1+\tau_{c,1}=1+\tau_{c,2}=\lambda\frac{w\left(  2+r\right)  }{2\left(
1+\beta\right)  \left(  1+r\right)  }%
\]
which implies that the FOC\ for $\tau_{l}$ also is satisfied at $\tau_{l}=0,$%
\begin{align*}
\frac{1+\beta}{1}  &  =\lambda\frac{w\left(  \left(  \lambda\frac{w\left(
2+r\right)  }{2\left(  1+\beta\right)  \left(  1+r\right)  }\right)
+\beta\left(  \lambda\frac{w\left(  2+r\right)  }{2\left(  1+\beta\right)
\left(  1+r\right)  }\right)  \right)  \left(  2+r\right)  }{2\left(
1+\beta\right)  \left(  1+r\right)  \left(  \lambda\frac{w\left(  2+r\right)
}{2\left(  1+\beta\right)  \left(  1+r\right)  }\right)  ^{2}}\\
&  =1+\beta
\end{align*}


EXPLAIN! This provides \emph{distortion smoothing} -- the labor leisure choice
is distorted equally much in both period. Perhaps, this is not surprising.

Formally, we know $\tau_{c,1}=\tau_{c,2}=\tau_{c}.$ Given this, the first
order conditions are
\begin{align*}
\tau_{c};\frac{1+\tau_{c}}{1-\tau_{l}}  &  =\lambda\frac{w\left(  2+r\right)
}{2\left(  1+\beta\right)  \left(  1+r\right)  }\\
\tau_{l};\frac{1+\tau_{c}}{1-\tau_{l}}  &  =\lambda\frac{w\left(  2+r\right)
}{2\left(  1+\beta\right)  \left(  1+r\right)  }%
\end{align*}


\subsubsection{The general case}

Let us now define%
\begin{align}
\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},\nonumber\\
\frac{\left(  1+\tau_{c,2}\right)  }{\left(  1+\tau_{c,1}\right)  \left(
1-\tau_{s}\right)  }  &  \equiv W_{i}.\nonumber
\end{align}


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


The first two equations are the FOC for labor supply and the third is the
Euler equation (FOC for savings (or $c_{1},c_{2}).$ The fourth is private
budget constraints and the last the aggregate resource constraint.

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{Although 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.

The problem is%

\[
\max_{W_{1},W_{2},W_{i}}\sum_{t=1}^{2}\beta^{t-1}u\left(  c,l\right)
\]
subject to%
\begin{align}
\lambda_{1};\frac{u_{l}\left(  c_{1},l_{1}\right)  }{u_{c}\left(  c_{1}%
,l_{1}\right)  w} &  =W_{1}\\
\lambda_{2};\frac{u_{l}\left(  c_{2},l_{2}\right)  }{u_{c}\left(  c_{2}%
,l_{2}\right)  w} &  =W_{2}\nonumber\\
\lambda_{3};\frac{u_{c}\left(  c_{2},l_{2}\right)  }{u_{c}\left(  c_{1}%
,l_{1}\right)  }\beta\left(  1+r\right)   &  =W_{i}\nonumber\\
\lambda_{4};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}\nonumber\\
\lambda;c_{1}+g+\frac{c_{2}+g}{1+r} &  =w\left(  1-l_{1}\right)
+\frac{w\left(  1-l_{2}\right)  }{1+r}\nonumber
\end{align}
%

\[
\]


\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
consumption 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 achieved. 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 fourth 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
\[
\max_{c_{1},c_{2},l_{1},l_{2}}\sum_{t=1}^{2}\beta^{t-1}u\left(  c,l\right)
\]
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 logarithmic example, (\ref{eq_primalbudget}) becomes%
\begin{align*}
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{align*}
since the $c_{1}=0$ root is irrelevant. Substituting this into the objective
function and taking first order conditions w.r.t. $c_{1}$ and $c_{2}$ yields,%

\[
\max_{c_{1},c_{2},l_{2}}\ln c_{1}+\ln\frac{l_{2}}{2l_{2}\left(  1+\beta
\right)  -\beta}+\beta\left(  \ln c_{2}+\ln l_{2}\right)
\]%
\begin{align*}
\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{align*}%
\[
s.t.\left(  w\left(  1-\frac{l_{2}}{2l_{2}\left(  1+\beta\right)  -\beta
}\right)  +\frac{w\left(  1-l_{2}\right)  }{1+r}-\left(  c_{1}+\frac{c_{2}%
}{1+r}+G\right)  \right)
\]
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%
\[
\sum_{t=0}^{\infty}\beta^{t}u\left(  c_{t},l_{t}\right)
\]


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
competitive. Profit maximization of the representative firm implies%
\begin{align*}
w_{t}  &  =F_{n}\left(  k_{t},n_{t}\right) \\
r_{t}  &  =F_{k}\left(  k_{t},n_{t}\right)
\end{align*}


The government needs to finance an exogenous 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{align*}
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{align*}
where $b_{t}$ is government borrowing and $R_{t}$ is the interest rate on
government bonds.

Households have budget constraints%
\[
c_{t}+k_{t+1}+\frac{b_{t+1}}{R_{t}}=\left(  1-\tau_{t}^{n}\right)  w_{t}%
n_{t}+\left(  1-\tau_{t}^{k}\right)  k_{t}r_{t}+\left(  1-\delta\right)
k_{t}+b_{t}%
\]


First order conditions are:%

\begin{align*}
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{align*}


Clearly, the first three implies
\begin{align*}
\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{align*}
and the last two the no arbitrage condition
\[
R_{t}=\left(  1-\tau_{t+1}^{k}\right)  r_{t+1}+\left(  1-\delta\right)
\]


Transversality conditions are
\begin{align*}
\lim_{T\rightarrow\infty}\left(
%TCIMACRO{\dprod \limits_{i=0}^{T-1}}%
%BeginExpansion
{\displaystyle\prod\limits_{i=0}^{T-1}}
%EndExpansion
R_{i}^{-1}\right)  k_{T+1}  &  =0\\
\lim_{T\rightarrow\infty}\left(
%TCIMACRO{\dprod \limits_{i=0}^{T-1}}%
%BeginExpansion
{\displaystyle\prod\limits_{i=0}^{T-1}}
%EndExpansion
R_{i}^{-1}\right)  \frac{b_{T+1}}{R_{T}}  &  =0
\end{align*}


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{competitive 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 competitive 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{align*}
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{align*}


Now, the first order condition for $k_{t+1}$ is%
\[
\theta_{t}=\beta\psi_{t+1}\left(  F_{k}\left(  k_{t+1},n_{t+1}\right)
-\left(  1-\tau_{t+1}^{k}\right)  r_{t+1}\right)  -\theta_{t+1}\left(
F_{k}\left(  k_{t+1},n_{t+1}\right)  +\left(  1-\delta\right)  \right)
\]
and for $c_{t}$%
\[
u_{c}\left(  c_{t},1-n_{t}\right)  =\theta_{t}%
\]
giving%
\begin{align*}
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{align*}


Suppose there is a steady state of the model, then
\begin{align*}
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{align*}
Private optimality (the Euler equation), implies in steady state
\begin{align*}
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{align*}
giving
\begin{align*}
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}\right)  +u_{c}\\
0 &  =\beta\left(  \psi+u_{c}\right)  \tau^{k}F_{k}%
\end{align*}
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 investment 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 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 occurred 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 maximizes the welfare of a constructed infinitely lived

\subsection{Time consistent taxation}

TBW.\newpage

\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 leisure. 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 exerted to get this income.

Consider a simple two-period example from GTW.

Individual preferences are:%

\[
E\left(  u\left(  c_{1}\right)  +v\left(  n_{1}\right)  +\beta\left(  u\left(
c_{2}\right)  +v\left(  n_{2}\right)  \right)  \right)
\]
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 competitive firms using a linear technology where each
individual $i$ produces
\[
y_{t}\left(  i\right)  =\theta\left(  i\right)  n_{t}\left(  i\right)  .
\]


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}%
%TCIMACRO{\dsum \limits_{i}}%
%BeginExpansion
{\displaystyle\sum\limits_{i}}
%EndExpansion
\left(  y_{1}\left(  i\right)  -c_{1}\left(  i\right)  +%
%TCIMACRO{\dsum \limits_{j}}%
%BeginExpansion
{\displaystyle\sum\limits_{j}}
%EndExpansion
\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}
\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
voluntarily 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{align}
&  u\left(  c_{1}\left(  i\right)  \right)  +v\left(  \frac{y_{1}\left(
i\right)  }{\theta_{1}\left(  i\right)  }\right)  +\beta%
%TCIMACRO{\dsum \limits_{j}}%
%BeginExpansion
{\displaystyle\sum\limits_{j}}
%EndExpansion
\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%
%TCIMACRO{\dsum \limits_{j}}%
%BeginExpansion
{\displaystyle\sum\limits_{j}}
%EndExpansion
\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)
\nonumber
\end{align}
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
\[
\sum_{i}\left(  u\left(  c_{1}\left(  i\right)  \right)  +v\left(  \frac
{y_{1}\left(  i\right)  }{\theta_{1}\left(  i\right)  }\right)  +\beta%
%TCIMACRO{\dsum \limits_{j}}%
%BeginExpansion
{\displaystyle\sum\limits_{j}}
%EndExpansion
\left(  u\left(  c_{2}\left(  i,j\right)  \right)  +v\left(  \frac
{y_{2}\left(  i,j\right)  }{\theta_{2}\left(  i,j\right)  }\right)  \right)
\pi_{2}\left(  j|i\right)  \right)  \pi\left(  i\right)
\]
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 (intratemporal) wedges%
\begin{align*}
\tau_{y_{1}}\left(  i\right)   &  \equiv1+\frac{v^{\prime}\left(  \frac
{y_{1}^{\ast}\left(  i\right)  }{\theta_{1}\left(  i\right)  }\right)
}{\theta_{1}\left(  i\right)  u^{\prime}\left(  c_{1}^{\ast}\left(  i\right)
\right)  },\\
\tau_{y_{2}}\left(  i,j\right)   &  \equiv1+\frac{v^{\prime}\left(
\frac{y_{2}^{\ast}\left(  i,j\right)  }{\theta_{2}\left(  i,j\right)
}\right)  }{\theta_{2}\left(  i,j\right)  u^{\prime}\left(  c_{2}^{\ast
}\left(  i,j\right)  \right)  },
\end{align*}
and the intertemporal wedge%
\[
\tau_{k}\left(  i\right)  \equiv1-\frac{u^{\prime}\left(  c_{1}^{\ast}\left(
i\right)  \right)  }{%
%TCIMACRO{\dsum \limits_{j}}%
%BeginExpansion
{\displaystyle\sum\limits_{j}}
%EndExpansion
\beta Ru^{\prime}\left(  c_{2}\left(  i,j\right)  \right)  \pi_{2}\left(
j|i\right)  }.
\]


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 productivities 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 thruth-telling constraint?.

First, note that expected utility is not changed.

Second, since utility is changed in parallel 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{align*}
\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{align*}
denote the perturbed consumption levels. The resource expected resource cost
of these are%
\begin{align*}
&  \tilde{c}_{1}\left(  i;\Delta\right)  +%
%TCIMACRO{\dsum \limits_{j}}%
%BeginExpansion
{\displaystyle\sum\limits_{j}}
%EndExpansion
\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)  +%
%TCIMACRO{\dsum \limits_{j}}%
%BeginExpansion
{\displaystyle\sum\limits_{j}}
%EndExpansion
\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{align*}


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

Thus,
\begin{align*}
0  &  =\\
&  =\frac{-\beta}{u^{\prime}\left(  c_{1}^{\ast}\left(  i\right)  \right)  }+%
%TCIMACRO{\dsum \limits_{j}}%
%BeginExpansion
{\displaystyle\sum\limits_{j}}
%EndExpansion
\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{align*}


From Jensen's inequality, we find that
\begin{align*}
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{align*}
\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 $\tau_{k}\left(  i\right)  =0.$

\subsection{A simple logarithmic example: insurance against low ability.}

Suppose in the first period, ability is unity and in the second $\theta>1$ or
$\frac{1}{\theta}$ with equal probability$.$Disregard government consumption
-- set $G_{1}=G_{2}=0$, although non-zero spending is quite easily handled.
The problem is therefore to provide a good insurance against a low-ability
shock when this is not observed.

The first best allocation is the solution to
\begin{align*}
&  \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}}{2R}-c_{1}-\frac{c_{h}+c_{l}}{2R}%
\end{align*}


First order conditions are
\begin{align*}
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{align*}


\subsubsection{A simple example}

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{align*}
\frac{1}{c_{1}}  &  =\lambda\\
\frac{1}{c_{h}}  &  =\lambda\\
\frac{1}{c_{l}}  &  =\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{align*}


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 $y_{1}=\sqrt{\frac{2}{\left(  1+\frac
{1}{2}\left(  \theta^{2}+\theta^{-2}\right)  \right)  }}=n_{1}.$ Therefore,
$n_{h}=\frac{y_{h}}{\theta}=\theta^{2}n_{1}$ and $n_{l}=y_{l}\theta
=\frac{n_{1}}{\theta}.$ 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 voluntarily
choose not to report being low ability. If he misreports, he gets $c_{l}$ and
is asked to produce $y_{l}.$ The constraint is therefore%
\begin{align*}
&  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{align*}%
\begin{align*}
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 c_{l}  &  \geq\frac{y_{h}^{2}-y_{l}^{2}}{2\theta^{2}}%
\end{align*}


We conjecture this is binding. The problem is then
\begin{align*}
&  \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  &  =\ln c_{h}-\ln c_{l}-\frac{y_{h}^{2}-y_{l}^{2}}{2\theta^{2}}.
\end{align*}


Denoting the shadow values by $\lambda_{r}$ and $\lambda_{I}$ the FOCs for the
consumption levels are%
\begin{align*}
c_{1}  &  =\frac{1}{\lambda_{r}}\\
c_{h}  &  =\frac{1+2\lambda_{I}}{\lambda_{r}}\\
c_{l}  &  =\frac{1-2\lambda_{I}}{\lambda_{r}}%
\end{align*}
from which we see
\[
\frac{c_{h}^{\ast}}{c_{1}^{\ast}}=1+2\lambda_{I},\frac{c_{l}^{\ast}}%
{c_{1}^{\ast}}=1-2\lambda_{I}%
\]
and
\[
\tau_{k}\left(  i\right)  \equiv1-\frac{\lambda_{r}}{\frac{\lambda_{r}%
}{1+2\lambda_{I}}\frac{1}{2}+\frac{\lambda_{r}}{1-2\lambda_{I}}\frac{1}{2}%
}=\left(  2\lambda_{I}\right)  ^{2},
\]
: 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{align*}
y_{1}^{\ast}  &  =\lambda_{r}\\
y_{h}^{\ast}  &  =\frac{\lambda_{r}}{1+2\lambda_{I}}\theta^{2}\Rightarrow
y_{h}^{\ast}c_{h}^{\ast}=\frac{\lambda_{r}}{1+2\lambda_{I}}\theta^{2}%
\frac{1+2\lambda_{I}}{\lambda_{r}}=\theta^{2}\\
y_{l}^{\ast}  &  =\frac{\lambda_{r}}{\theta^{4}-2\lambda_{I}}\theta
^{2}\Rightarrow y_{l}^{\ast}c_{l}^{\ast}=\frac{\lambda_{r}}{\theta
^{4}-2\lambda_{I}}\theta^{2}\frac{1-2\lambda_{I}}{\lambda_{r}}=\frac
{1-2\lambda_{I}}{\theta^{2}\left(  1-2\lambda_{I}\theta^{-4}\right)  }%
\end{align*}
\begin{align*}
\tau_{y_{1}}  &  =1-\frac{y_{1}^{\ast}}{\frac{1}{c_{1}^{\ast}}}=0,\\
\tau_{y_{2}}\left(  h\right)   &  =1+\frac{v^{\prime}\left(  \frac{y_{h}%
^{\ast}}{\theta}\right)  }{\theta u^{\prime}\left(  c_{h}^{\ast}\right)
}=1+\frac{-\frac{y_{h}^{\ast}}{\theta}}{\theta\frac{1}{c_{h}^{\ast}}}\\
&  =1+\frac{-\frac{\frac{\lambda_{r}}{1+2\lambda_{I}}\theta^{2}}{\theta}%
}{\theta\frac{1}{\frac{1+2\lambda_{I}}{\lambda_{r}}}}=0
\end{align*}
and%

\begin{align*}
\tau_{y_{2}}\left(  l\right)   &  =1+\frac{v^{\prime}\left(  \theta
y_{l}^{\ast}\right)  }{\frac{1}{\theta}u^{\prime}\left(  c_{l}^{\ast}\right)
}=1+\frac{-\theta y_{l}^{\ast}}{\frac{1}{\theta}\frac{1}{c_{h}^{\ast}}}\\
&  =1+\frac{-\theta\frac{\lambda_{r}}{\theta^{4}-2\lambda_{I}}\theta^{2}%
}{\frac{1}{\theta}\frac{1}{\frac{1-2\lambda_{I}}{\lambda_{r}}}}=2\lambda
_{I}\frac{\theta^{4}-1}{\theta^{4}-2\lambda_{I}}>0
\end{align*}


As we see, the wedge for the high ability types is zero, but positive for the
low ability type.\footnote{The wedge, asymptotes to infinity as $\lambda_{I}$
approach $\frac{\theta^{4}}{2}.$ Can you explain?} For later use, we note that%

\begin{align}
y_{1}^{\ast}c_{1}^{\ast}  &  =1\label{ystarcstar}\\
y_{h}^{\ast}c_{h}^{\ast}  &  =\frac{\lambda_{r}}{1+2\lambda_{I}}\theta
^{2}\frac{1+2\lambda_{I}}{\lambda_{r}}=\theta^{2}\nonumber\\
y_{l}^{\ast}c_{l}^{\ast}  &  =\frac{\lambda_{r}}{\theta^{4}-2\lambda_{I}%
}\theta^{2}\frac{1-2\lambda_{I}}{\lambda_{r}}=\frac{1-2\lambda_{I}}{\theta
^{2}\left(  1-2\lambda_{I}\theta^{-4}\right)  }\nonumber
\end{align}


\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
together with type specific transfers implement the allocation in the example.
To do this, consider the individual problem,%
\begin{align*}
&  \max_{c_{1},y_{1},s,y_{h},y_{l},c_{h},c_{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)  -c_{1}-s+T\\
0  &  =y_{h}\left(  1-\tau_{h}\right)  +s\left(  1-\tau_{s,h}\right)
-c_{h}+T_{h}\\
0  &  =y_{l}\left(  1-\tau_{h}\right)  +s\left(  1-\tau_{s,l}\right)
-c_{l}+T_{l}%
\end{align*}
with Lagrange multipliers $\lambda_{1},\lambda_{h}$ and $\lambda_{r}.$

First order conditions for the indviduals are;%
\begin{align}
\frac{1}{c_{1}}  &  =\lambda_{1}\nonumber\\
y_{1}  &  =\lambda_{1}\left(  1-\tau_{1}\right) \nonumber\\
\lambda_{1}  &  =\lambda_{h}\left(  1-\tau_{s,h}\right)  +\lambda_{l}\left(
1-\tau_{l,h}\right) \nonumber\\
\frac{y_{h}}{2\theta^{2}}  &  =\lambda_{h}\left(  1-\tau_{h}\right)
\label{eq_privFOC}\\
\frac{\theta^{2}y_{l}}{2}  &  =\lambda_{l}\left(  1-\tau_{l}\right)
\nonumber\\
\frac{1}{2c_{h}}  &  =\lambda_{h}\nonumber\\
\frac{1}{2c_{l}}  &  =\lambda_{l}\nonumber
\end{align}


Using this, we see that
\[
\frac{1}{c_{1}}=\frac{1}{2c_{h}}\left(  1-\tau_{s,h}\right)  +\frac{1}{2c_{l}%
}\left(  1-\tau_{l,h}\right)
\]


Setting,
\begin{align*}
\tau_{s,h}  &  =-2\lambda_{I}\\
\tau_{s,l}  &  =2\lambda_{I}.
\end{align*}
this gives
\[
\frac{1}{c_{1}}=\frac{1}{2c_{h}}\left(  1+2\lambda_{I}\right)  +\frac
{1}{2c_{l}}\left(  1-2\lambda_{I}\right)
\]
which is satisfied if we plug in the optimal allocation $c_{h}^{\ast}%
=c_{1}^{\ast}\left(  1+2\lambda_{I}\right)  $ and $c_{l}^{\ast}=c_{1}^{\ast
}\left(  1-2\lambda_{I}\right)  $
\[
\frac{1}{c_{1}^{\ast}}=\frac{1+2\lambda_{I}}{2c_{1}^{\ast}\left(
1+2\lambda_{I}\right)  }+\frac{1-2\lambda_{I}}{2c_{1}^{\ast}1-2\lambda_{I}}%
\]


Note that the expected capital income tax rate is zero, but it will make
savings lower than without any taxes.\ Why?

Similarly, by noting from (\ref{ystarcstar}) that in the optimal second best
allocation, we want
\[
y_{1}c_{1}=y_{1}^{\ast}c_{1}^{\ast}=1,
\]
which is implemented by $\tau_{1}=0.$ For the high ability type, the second
best allocation in (\ref{ystarcstar}) is that $y_{h}^{\ast}c_{h}^{\ast}%
=\theta^{2},$ which is implemented by $\tau_{h}=0$ since (\ref{eq_privFOC})
implies that $y_{h}c_{h}=\theta^{2}\left(  1-\tau_{h}\right)  .$

For the low ability type, we want $y_{l}^{\ast}c_{l}^{\ast}=\frac
{1-2\lambda_{I}}{\theta^{2}\left(  1-2\lambda_{I}\theta^{-4}\right)  }$. From
(\ref{eq_privFOC}), we know $y_{l}c_{l}=$ $\frac{1-\tau_{l}}{\theta^{2}},$so
we solve%
\begin{align*}
\frac{1-\tau_{l}}{\theta^{2}}  &  =\frac{1-2\lambda_{I}}{\theta^{2}\left(
1-2\lambda_{I}\theta^{-4}\right)  }\\
&  \Rightarrow\tau_{l}=2\lambda_{I}\frac{\theta^{4}-1}{\theta^{4}-2\lambda
_{I}}.
\end{align*}


Note that if $\lambda_{I}=\frac{1}{2},$ $\tau_{l}=1.$ I.e., the tax rate is
100\%. There is no point going higher than that, so $\lambda_{I}$ cannot be
higher than $\frac{1}{2}.$

Finally, to find the complete allocation, we use the budget constraints of the
private individual and the aggregate resource constraint. This will recover
the transfers $T,T_{h}$ and $T_{l}.$ We should note that $T_{l}>T_{h}$ is
consistent with incentive compatibiity. Why? Because if you claim to be a low
ability type you will have to may a high labor income tax which is bad if you
are high ability and earn a high income. Thus, by taxing high income lower, we
can have a transfer system that transfers more to the low ability types.

\subsubsection{Third best -- laizzes faire}

The allocation in without any government involvements is easily found by
setting all taxes to zero$.$%
\begin{align}
\frac{1}{c_{1}}  &  =\lambda_{1}\nonumber\\
y_{1}  &  =\lambda_{1}\nonumber\\
\lambda_{1}  &  =\lambda_{h}+\lambda_{l}\nonumber\\
\frac{y_{h}}{2\theta^{2}}  &  =\lambda_{h}\label{eq_FOCLaizzesFaire}\\
\frac{\theta^{2}y_{l}}{2}  &  =\lambda_{l}\nonumber\\
\frac{1}{2c_{h}}  &  =\lambda_{h}\nonumber\\
\frac{1}{2c_{l}}  &  =\lambda_{l}\nonumber
\end{align}


Using these and the budget constraints, we get
\begin{align*}
y_{1}  &  =\frac{1}{c_{1}}\\
\frac{1}{c_{1}}  &  =\frac{1}{2c_{h}}+\frac{1}{2c_{l}}\\
\frac{y_{h}}{2\theta^{2}}  &  =\frac{1}{2c_{h}}\\
\frac{\theta^{2}y_{l}}{2}  &  =\frac{1}{2c_{l}}\\
y_{1}  &  =c_{1}+s\\
y_{h}+s  &  =c_{h}\\
y_{l}+s  &  =c_{l}%
\end{align*}
which implies%

\begin{align*}
c_{1}+s  &  =\frac{1}{c_{1}}\\
\frac{1}{c_{1}}  &  =\frac{1}{2c_{h}}+\frac{1}{2c_{l}}\\
c_{h}  &  =\frac{1}{2}s+\frac{1}{2}\sqrt{s^{2}+4\theta^{2}}\\
c_{l}  &  =\frac{\frac{1}{2}s\theta+\frac{1}{2}\sqrt{s^{2}\theta^{2}+4}%
}{\theta}%
\end{align*}


I did not find an analytical solution to this, but setting $\theta=1.1$ I
found the solution $\left\{  c_{1}=0.997\,75,c_{h}=1.\,\allowbreak
102\,3,s=4.\,\allowbreak504\,5\times10^{-3},c_{l}=0.911\,35\right\}  .$

\subsection{Time consistency}

Under the Mirrlees approach, the government announces a menu of taxes or of
consumption baskets. People then make choices that in equilibrium 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 (Roberts,
84). But better equilibria might arise in infinite horizon.


\end{document}