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\begin{document}
\title{Topics in Dynamic Public Finance}
\author{John Hassler\\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 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.
\end{itemize}
Total income in second period if laid off is therefore%
\[
\left( 1-\beta\right) \left( b-c\right) +\beta y_{n}\left( 1-\tau\right)
\equiv y_{l}.
\]
In first periods, individuals decide how much to save, $s.$ Interest rate and
subjective discount rate is normalized to zero. 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 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
\[
\left( y_{n}+c\right) \beta_{c}=1-\beta
\]
since the marginal gain in aggregate income is $\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
$-\left( y_{n}\left( 1-\tau\right) +c-b\right) \beta_{y_{n}}.$
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
\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}{\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}+\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}\\
\frac{u^{\prime\prime}}{u^{\prime}}\frac{\Delta c}{Y} & =\frac{\tau}{\mu
}\frac{d\mu}{d\tau}\\
\frac{\Delta c}{c} & =\frac{E_{\mu,t}}{-R_{r}}Y
\end{align*}
where $E_{\mu,t}$ is the elasticity of $\mu$ with respect to taxes and $R_{r}$
the relative risk aversion coefficient. Note that we should not interpret $Y$
as the aggregate \emph{level }\ of income since we have normalized the
pre-unemployment income to unity. Assuming that $y_{n}\approx1,Y\approx
1+\alpha+\left( 1-\alpha\right) \beta$ 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.
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. $\frac{\tau}{\mu}\frac{d\mu}{d\tau
}=E_{\mu,t}$ is thus 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 .15-.4.
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 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 mortal
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 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, 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{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 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 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 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
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 $V_{1}^{e}$ 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
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. In fact, his consumption does not 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+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{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%
\[
W^{\prime}\left( V_{t+1}^{e}\right) >C^{\prime}\left( V_{t}\right)
>C^{\prime}\left( V_{t+1}^{u}\right) .
\]
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 ($V_{t+1}^{u}\frac{1}{E_{t}u^{\prime
}\left( c_{t+1}\right) }\Rightarrow\frac{1}{E_{t}\frac{1}{u^{\prime}\left(
c_{t+1}\right) }}0,
\end{align*}
\emph{if and only if there is some uncertainty in }$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}