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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
\[
-\left( 1-\beta\left( y_{n},c\right) \right) c+\beta\left( y_{n}%
,c\right) y_{n}%
\]
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 unemployment 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 voluntarily choose $a_{t}=1.$
Allocations that satisfies this are called incentive compatible allocations.
Given an allocation we can compute the expected discounted utility 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}\times0\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 ($V_{t+1}^{u}\frac{1}{E_{t}u^{\prime
}\left( c_{t+1}\right) }\Rightarrow\frac{1}{E_{t}\frac{1}{u^{\prime}\left(
c_{t+1}\right) }}0,$ this sequence is increasing geometrically without
bounds. It is perhaps intuitive that a sequence of consumption taxes that
increases geometrically without bounds is suboptimal. Similarly, $\tau_{k}%
<0,$the consumption tax approaches -100\%. That neither of this is optimal is
really the Chamley-Judd result.
Before proceeding, we note that using the (\ref{eq_FoCRamseyConsumer}) in the
private budget constraint, we get
\begin{equation}
\sum_{t=0}^{\infty}\beta^{t}\left( c_{t}u_{c}\left( c_{t},1-n_{t}\right)
-n_{t}u_{l}\left( c_{t},1-n_{t}\right) \right) =u_{c}\left( c_{0}%
,1-n_{0}\right) \frac{\left( 1-\tau_{k,0}\right) }{\left( 1+\tau
_{c,0}\right) }\left( 1+r_{0}\right) b_{0} \label{eq_StokeyLucas}%
\end{equation}
An allocation that satisfies (\ref{eq_StokeyLucas}) the private budget
constraint and is privately optimal for some sequences of taxes. If it also
satisfies the aggregate budget constraint it is also implementable as a
competitive equilibrium. Note, that there is no taxes or prices here except
the two initial taxes on pre-existing capital.
Nevertheless we can reformulate the Ramsey problem as max%
\begin{align*}
\max U & =\sum_{t=0}^{\infty}\beta^{t}u\left( c_{t},1-n_{t}\right) \\
s.t.\sum_{t=0}^{\infty}\beta^{t}\left( c_{t}u_{c}\left( c_{t},1-n_{t}%
\right) -n_{t}u_{l}\left( c_{t},1-n_{t}\right) \right) & =u_{c}\left(
c_{0},1-n_{0}\right) \frac{\left( 1-\tau_{k,0}\right) }{\left(
1+\tau_{c,0}\right) }\left( 1+r_{0}\right) b_{0}\\
g_{t}+c_{t} & =w_{t}n_{t}%
\end{align*}
Sometimes, it is easier to work with this \emph{direct or primal approach.
}Here, it is then straight-forward to construct the wedges and then taxes that
implement the Ramsey optimal allocation.
\subsection{The Chamley-Judd result}
Now, we only add a production technology using capital. 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. Following the literature, we let the interest rate on
bonds be tax-free. Thus,%
\begin{align*}
g_{t}+b_{t} & =\tau_{k,t}r_{t}k_{t}+\tau_{n,t}w_{t}n_{t}+\frac{b_{t+1}%
}{R_{t}}\\
& =F\left( k_{t},n_{t}\right) -\left( 1-\tau_{k,t}\right) r_{t}%
k_{t}-\left( 1-\tau_{n,t}\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%
\[
c_{t}+k_{t+1}+\frac{b_{t+1}}{R_{t}}=\left( 1-\tau_{n,t}\right) w_{t}%
n_{t}+\left( 1-\tau_{k,t}\right) k_{t}r_{t}+\left( 1-\delta\right)
k_{t}+b_{t}%
\]
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
_{n,t}\right) w_{t}\\
k_{t+1};\lambda_{t} & =\beta\lambda_{t+1}\left( \left( 1-\tau
_{k,t}\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_{n,t}\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_{k,t}\right) r_{t+1}+\left(
1-\delta\right) \right)
\end{align*}
and the last two the no arbitrage condition
\[
R_{t}=\left( 1-\tau_{k,t}\right) r_{t+1}+\left( 1-\delta\right)
\]
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 $\left\{ \tau_{n,t},\tau_{k,t}%
,b_{t}\right\} _{t=0}^{\infty}$ 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_{k,t}\right)
r_{t}k_{t}-\left( 1-\tau_{n,t}\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_{n,t}\right) w_{t}\right) \\
& \left. +\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_{k,t}\right)
r_{t+1}+\left( 1-\delta\right) \right) \right\}
\end{align*}
Now, the first order condition for $k_{t+1}$ is%
\[
\theta_{t}=\beta\psi_{t+1}\left( F_{k}\left( k_{t+1},n_{t+1}\right)
-\left( 1-\tau_{k,t}\right) r_{t+1}\right) -\beta\theta_{t+1}\left(
F_{k}\left( k_{t+1},n_{t+1}\right) +\left( 1-\delta\right) \right)
\]
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_{k,t}\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.
An interesting case is if government spendings are stochastic. With complete
markets, the government should then commit to a tax system that insures them
against this (Chari et al. 1994). If spending needs are large, taxes on
capital should be high and vice versa.
The zero capital income tax result 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}
\subsubsection{A numerical appproach}
Here we follow Klein and Rios-Rull 2003. Consider a stochastic economy
productivity is $z\left( s^{t}\right) $ and government consumption is
$g\left( s^{t}\right) $ where $s^{t}$ is the history of a shock that in
every period belongs to a finite element set $S.$ The shock follows a Markov
chain with transition matrix $\Gamma.$ The representative individuals has
utility given by
\[
E\sum_{t}^{\infty}\beta^{t}u\left( c_{t},h_{t}\right)
\]
and the aggregate resource constraint is
\[
F\left( K\left( s^{t-1}\right) ,H\left( s^{t}\right) ,z\left(
s^{t}\right) \right) +\left( 1-\delta\right) K\left( s^{t-1}\right)
=C\left( s^{t}\right) +K\left( s^{t}\right) +g\left( s^{t}\right)
\]
Individual budget constraints are%
\[
c_{t}+k_{t+1}=\left( 1-\tau_{t}\right) w_{t}h_{t}+\left( 1+r_{t}\left(
1-\theta_{t}\right) \right) k_{t}%
\]
where lower case variables denoted individual and a balanced budget constraint
is imposed on the government%
\[
\theta_{t}k_{t}r_{t}+\tau_{t}w_{t}h_{t}=g_{t}%
\]
If the government could set $\theta_{t}$ at $t,$ this would be an
\emph{ex-post }lump-sum tax. Klein and Rios-Rull assume a limited commitment,
i.e., that taxes are set for the next period. To find a time consistent
solution, we require that the policy the government follows is of Markov type,
i.e., it is a function of the set of state variables only. These are%
\[
\left\{ g,z,K,\theta\right\} \equiv x
\]
Using budget balance, a policy rule is then
\[
\theta_{t+1}=\psi\left( x_{t}\right)
\]
We then define a recursive competitive equilibrium in the standard way, noting
that the value function $v$ depends on the policy rule%
\[
v\left( x,k;\psi\right) .
\]
Assuming the government is benevolent, it assesses welfare according to
\[
V\left( x;\psi\right) =v\left( x,k;\psi\right) .
\]
We can also define the competitive equilibrium and its value function in case
the government decides next periods tax to $\theta^{\prime}$ and following
government follow $\psi\left( x\right) .$ The value function is then
\[
\hat{v}\left( x,\theta^{\prime},k;\psi\right)
\]
The associated welfare of the government is
\[
\hat{V}\left( x,\theta^{\prime};\psi\right)
\]
Now define the current maximizing policy as%
\[
\Psi\left( x;\psi\right) =\arg\max_{\theta^{\prime}}\hat{V}\left(
x,\theta^{\prime};\psi\right)
\]
A Markov perfect optimal tax policy then satisfies the fixed-point
requirement
\[
\Psi\left( x;\psi\left( x\right) \right) =\psi\left( x\right) ,
\]
i.e., if the government expects coming government to use $\psi$ it is optimal
for itself to use $\psi.$
Klein and Rios-Rull use log utility and assume government consumption and
productivity can each take on two different values respectively. They
calibrate the model to US, data. Average $g$ is 20\%, varying 1.6\% points up
or down and an autocorrelation of .66. Productivity has a standard deviation
of 2.4\% with autocorrelation .88.
Comparing the commitment and no commitment they find that in commitment
expected capital income tax rates are (almost) zero but with a standard
deviation of 18\%. having a strong positive correlation with $g$ and a strong
negative with $z.$ Labor income taxes are 31\% and almost fixed.
With 1 years commitment only, the average capital income tax rate is 65\% with
a standard deviation of 11\%. It is positively correlated with $g,$ but less
than with full commitment. Labor income tax rates are 12\% on average with a
standard deviation of 3\%. Output is 14\% lower than under commitment and
somewhat less volatile.
Also 4 years commitment produce high average tax rates on capital income 36\%.
\subsubsection{A time-consistent taxation problem with an analytical solution}
The model economy is populated by a continuum one of dynasties of two-period
lived agents. In the first period of their lives, agents undertake an
investment in human capital. The cost of investment to each individual is
$e^{2}$, and the return is spread over two periods. In particular, the
individual earn labor earning equal to $e\cdot w$ in the first period of her
life, and $e\cdot w\cdot z$ in the second period. $z\leq1$ captures the fact
that agents retire within the second period of their life.
Dynasties derive utility from the consumption of a private and a public good.
The public good is financed with a linear age-independent tax on income,
denoted $\tau_{t}.$
Each period's felicity depends on the total consumption (net of the investment
cost) of the dynasty's member, irrespective of the split of consumption
between the old and the young agent. The preferences of the dynasty which is
alive at $t$ are described by the following linear-quadratic utility functions%
\[
U_{t}=c_{t}+Ag_{t}-e_{t}^{2}+\beta U_{t+1},
\]
where $\beta\in\lbrack0,1)$ is the discount factor, $g_{t}$ denotes the public
good available at $t$ and $A$ is a parameter describing the marginal utility
of the public good. The marginal cost of the public good is unity and we focus
on the case where $A\geq1,$ that will imply that the public good is socially
valuable. Furthemore, we assume that the discount rate, $\left(
1-\beta\right) /\beta$, equals the market interest rate. Given our
assumptions, the savings decisions can be abstracted from, and the welfare of
a dynasty is simply given by the present discounted value of their income net
of investment costs;
\[
U_{t}=\Sigma_{j=0}^{\infty}\beta^{j}\left( \left( 1-\tau_{t+j}\right)
y_{t+j}+Ag_{t+j}-e_{t+j}^{2}\right) ,
\]
where:%
\begin{equation}
y_{t+j}=\left( ze_{t+j-1}+e_{t+j}\right) w, \label{yt}%
\end{equation}
i.e., the gross income accruing to the dynasty at $t+j$, given by the sum of
the labor incomes generated by the parent born at $t+j-1$ and her offspring
born at $t+j$. The parent's human capital depends on her investment at $t+j-1$
($e_{t+j-1}$) while the offspring's human capital depends on her investment at
$t+j$ ($e_{t+j}$). Since agents live for two periods, and the effect of the
human capital investment dies with them, $y_{t}$ only depends on the
realization of two subsequent investments.
Due to a standard free-riding problem, there is not private provision of the
public good. This is instead provided by an agency that will be called
"government" that has access to a technology to turn one unit of revenue into
one unit of public good. The government revenue is collected by taxing agents'
labor income at the flat rate $\tau,$ subject to a balanced budget constraint.
More formally, the government budget constraint requires that $g_{t}\leq
\tau_{t}\left( ze_{t-1}+e_{t}\right) w,$ where, at time $t,$ $e_{t-1}$ is
predetermined. $e_{t},$instead is determined after $\tau_{t}$ is set and in
addition depends on expectations about the future tax rate. In particular, the
optimal investment of a young agent at t is given by
\begin{equation}
e_{t}^{\ast}=e\left( \tau_{t},\tau_{t+1}\right) \equiv\text{max}\left[
0,\frac{1+\beta z-\left( \tau_{t}+\beta z\tau_{t+1}\right) }{2}w\right] .
\label{Effort}%
\end{equation}
This equation shows the distortionary effect of taxation on investment. Note
that taxation at $t+j$ distortj the investment of two generations: that born
at $t+j-1,$ as $e_{t+j-1}^{\ast}=e\left( \tau_{t+j-1},\tau_{t+j}\right) $,
and that born at $t+j,$ as $e_{t+j}^{\ast}=e\left( \tau_{t+j},\tau
_{t+j+1}\right) $).
Letting $e_{t}=e\left( \tau_{t},\tau_{t+1}\right) $ and substituting it in
into the government budget constraint, allows us to express the provision of
public good at $t$ as a function of current and future (one period ahead)
taxes plus the level of investments sunk at $t-1$. More formally:%
\begin{equation}
g_{t}=\tau_{t}\left( ze_{t-1}+e\left( \tau_{t},\tau_{t+1}\right) \right)
w=g\left( \tau_{t},\tau_{t+1},e_{t-1}\right) . \label{tax}%
\end{equation}
Finally, we restrict $\tau_{t}\in\left[ 0,1\right] \forall t$, which implies
that investments, public good provision and private net income ($e_{t}^{\ast
},g_{t}$ and $\left( 1-\tau_{t}\right) y_{t}$) all are non-negative.
Before discussing the Markov equilibrium, let us state the solution to the
full commitment equilibrium\footnote{See Hassler et al (JME 2005).}
\begin{proposition}
\label{prop_eff_riskaverse} The optimal solution to the planner program is%
\begin{equation}
\tau_{t+1}=\max\left\{ 0,\tau^{\ast}-z\left( \tau_{t}-\tau^{\ast}\right)
\right\} <1,
\end{equation}
for $t\geq0$ and%
\[
\tau_{0}=\left\{
\begin{array}
[c]{cc}%
\tau_{0}=\left( 1+\frac{2ze_{-1}}{w\left( 1-\beta z\right) }\right)
\tau^{\ast} & \text{if }e_{-1}\leq\frac{w\left( 1-\beta z\right) }{2z^{2}}\\
\min\left\{ 1,\left( 1+\beta z+\frac{2ze_{-1}}{w}\right) \tau^{\ast
}\right\} & \text{else.}%
\end{array}
\right. ,
\]
where
\[
\tau^{\ast}=\frac{A-1}{2A-1}\in\lbrack0,\frac{1}{2}).
\]
is the steady-state tax rate. If $z<1,$ the Ramsey tax sequence converges
asymptotically in an oscillatory fashion to $\tau^{\ast}.$ If $z=1,$ the
Ramsey tax sequence is a 2-period cycle such that,
\[
\tau_{t}=\left\{
\begin{array}
[c]{cc}%
\tau_{0} & \text{if }t\text{ is even}\\
\max\left\{ 0,2\tau^{\ast}-\tau_{0}\right\} & \text{if }t\text{ is odd.}%
\end{array}
\right.
\]
\end{proposition}
Note that if $e_{-1}=0,$the optimal tax is at the steady state immediately.
With positive $e_{-1},$ the planner wants to tax the pre-installed tax-base
but this implies that also period 0 investments are hurt. To partly offset
this, the planner promises taxes lower than steady state for period 1. But,
there is then incentive to tax investments $e_{1}$ in period 1 a little higher
by setting $\tau_{2}$ above the steady state tax. Oscillating taxes therefore
tends to smooth distortions over time.
\paragraph{The Markov allocation (Ramsey allocation without commitment)}
Let us now characterize the optimal time consistent allocation, namely, the
allocation that is chosen by a benevolent planner without access to a
commitment technology. Clearly, the oscillating path described above is not time-consistent.
We will use the recursive formulation of the problem, now assuming that period
$t$ taxes are set in the beginning of period $t,$ and observed before period
$t$ investments are decided. The period $t$ felicity of the planner is given
by%
\begin{align*}
F\left( e_{t-1},\tau_{t},\tau_{t+1}\right) & =\left( 1-\tau_{t}\right)
y_{t}-e\left( \tau_{t},\tau_{t+1}\right) ^{2}+Ag_{t}\\
& =\left( ze_{t-1}+e\left( \tau_{t},\tau_{t+1}\right) \right) \left(
1+\left( A-1\right) \tau_{t}\right) w-e\left( \tau_{t},\tau_{t+1}\right)
^{2},
\end{align*}
where $e_{t-1}$ is pre-determined.
Without committment, the game between the government and the public is not
degenerate. We characterize the equilibrium where $e_{t-1}$ is the only state
variable in period \thinspace$t$ and reputation is not used as a means to
compensate for committment. Thus, taxes are set according to a time-invariant
function $\tau_{t}=T\left( e_{t-1}\right) .$Given this function, individuals
rationally believe that $\tau_{t+1}=T\left( e_{t}\right) $ and individually
rational investment choices must therefore satisfy
\[
e_{t}=\frac{1+\beta z-\left( \tau_{t}+\beta zT\left( e_{t}\right) \right)
}{2}w.
\]
We can now define the equilibrium;
\begin{definition}
\label{def_eq}A time-consistent (Markov) allocation without commitment is
defined as a pair of functions $\left\langle T,I\right\rangle $, where
$T:\left[ 0,\infty\right) \rightarrow\left[ 0,1\right] $ is a public
policy rule, $\tau_{t}=T\left( e_{t-1}\right) ,$ and $I:\left[ 0,1\right]
\rightarrow\lbrack0,\infty)$ is a private investment rule, $e_{t}=I\left(
\tau_{t}\right) $ such that the following functional equations are satisfied,
\end{definition}
\begin{enumerate}
\item $T\left( e_{t-1}\right) =\arg\max_{\tau_{t}}\left\{ F\left(
e_{t-1},\tau_{t},\tau_{t+1}\right) +\beta W\left( e_{t}\right) \right\} $
subject to $e_{t}=I\left( \tau_{t}\right) ,$ $\tau_{t+1}=T\left( I\left(
\tau_{t}\right) \right) ,$
\item $I\left( \tau_{t}\right) =\left( 1+\beta-\left( \tau_{t}+\beta
T\left( I\left( \tau_{t}\right) \right) \right) \right) w/2$,
\item $W\left( e_{t-1}\right) =\max_{\tau_{t}}\left\{ F\left( e_{t-1}%
,\tau_{t},\tau_{t+1}\right) +\beta W\left( e_{t}\right) \right\} $ subject
to $e_{t}=I\left( \tau_{t}\right) ,$ $\tau_{t+1}=T\left( I\left( \tau
_{t}\right) \right) .$
\end{enumerate}
The following Proposition can then be established.
\begin{proposition}
\label{prop_altruism}Assume that either\footnote{This assumption ensures that
the constraint $\tau_{t+1}\leq1$ never binds for $t\geq0.$ Without this
constraint, the analysis would be substantially more complicated, involving
non-continuos policy functions.} $A\leq\frac{z\left( z+1\right) }{\left(
1+\beta\right) z^{2}-1}$ or $\left( 1+\beta\right) z^{2}\leq1.$ Then, the
time-consistent allocation is characterized as follows:
\begin{align*}
T\left( e_{t-1}\right) & =\min\left\{ \bar{\tau}+\alpha_{1}\left(
e_{t-1}-\bar{e}\right) ,1\right\} \\
I\left( \tau_{t}\right) & =\bar{e}-\frac{w}{2+\beta z\alpha_{1}w}\left(
\tau_{t}-\bar{\tau}\right) ,
\end{align*}
where%
\begin{align*}
\bar{e} & =\frac{w\left( 1+\beta z\right) \left( 1-\alpha_{0}\right)
}{2+\alpha_{1}w\left( 1+\beta z\right) }\leq e^{\ast}\\
\bar{\tau} & =\frac{2\alpha_{0}+\alpha_{1}w\left( 1+\beta z\right)
}{2+\alpha_{1}w\left( 1+\beta z\right) }\geq\tau^{\ast}%
\end{align*}
with equalities iff $A=1,$and%
\begin{align*}
\alpha_{1} & =\frac{\sqrt{1+4A\left( A-1\right) \left( 1-\beta
z^{2}\right) }-\left( 1+2\left( 1-\beta z^{2}\right) \left( A-1\right)
\right) }{\beta z\left( A-1\right) \left( 1-\beta z^{2}\right) w}\geq0\\
\alpha_{0} & =\frac{2\left( A-1\right) -\beta z\alpha_{1}w}{2+\left(
A-1\right) \left( 4+\beta z\alpha_{1}w\right) }\geq0\\
\frac{\partial\alpha_{1}}{\partial A} & \geq0,\frac{\partial\alpha_{0}%
}{\partial A}\geq0,\frac{\partial\bar{\tau}}{\partial A}\geq0,\frac
{\partial\bar{e}}{\partial A}\leq0.
\end{align*}
For all $t,$ the equilibrium law of motion is
\begin{align}
e_{t+1} & =\bar{e}-z_{d}\left( e_{t}-\bar{e}\right) ,
\label{eq_Markovlaw1}\\
\tau_{t+1} & =\bar{\tau}-z_{d}\left( \tau_{t}-\bar{\tau}\right) .
\label{eq_Markovlaw2}%
\end{align}
where
\[
z_{d}\equiv\frac{\alpha_{1}w}{2+\beta\alpha_{1}w}\in\left( 0,z\right) .
\]
Given any $e_{-1}$, the economy converges to a unique steady state such that
$\tau=\bar{\tau}$ and $e=\bar{e}$ following an oscillating path and the
constraint $\tau_{t}\leq1$ iff t=0 and $e_{-1}>\frac{1-\alpha_{0}}{\alpha_{1}%
}$, while $\tau_{t}\geq0$ never binds. \newline
\end{proposition}
The parameter restriction under which the Proposition is stated is a
sufficient condition for the constraint $\tau_{t+1}\leq1$ never to bind for
$t\geq0.$ When this constraint is violated, the equilibrium policy functions
may be discontinuos, making the analysis substantially more involved.
The main findings are that
\begin{enumerate}
\item the Markov allocation implies higher steady-state taxation ($\bar{\tau
}>\tau^{\ast}$) and lower output and investment ($\bar{e}1+\frac{2+z\left(
1-\beta z\right) }{z\left( 2+z\left( 1+\beta\right) \right) }$, as
threshold that decreases in $\beta$ and $z$. I may seem counter intuitive that
a benevolent planner would choose a tax rate that in steady state is on the
wrong side of the Laffer curve. The fact that it can happen is due to lack of
commitment; if $\bar{\tau}>1/2,$ the planner would clearly want to reduce the
steady state tax rate. However, the planner can only control current tax rate
and reducing that leads to higher taxes the next period and the overall effect
of this is to reduce current welfare.
\subsection{}
Here follows a sketch of the proof. The idea of the proof is as follows; Guess
that the optimal policy function is linear in the state variable%
\begin{equation}
\tau_{t+1}=T\left( e_{t}\right) =\alpha_{0}+\alpha_{1}e_{t}, \label{guess1}%
\end{equation}
for the undetermined coefficients $\alpha_{0}$ and $\alpha_{1}.$ Use the guess
to derive the investment rule. Substitute these to into the Bellman equation
for period $t.$ Derive the first-order condition for period $t$ and verify
that it is linear in $e_{t-1}$. Find $a_{0}$ and $\alpha_{1}$ such that the
FOC in period $t$ is satisfied.
The planner felicity in period $t$ is
\[
F\left( e_{t-1},\tau_{t},\tau_{t+1}\right) =\left( ze_{t-1}+e\left(
\tau_{t},\tau_{t+1}\right) \right) \left( 1+\left( A-1\right) \tau
_{t}\right) w-e\left( \tau_{t},\tau_{t+1}\right) ^{2},
\]
Given the guess, the investment decision is $e_{t}=\left( 1+\beta z-\left(
\tau_{t}+\beta z\left( \alpha_{0}+\alpha_{1}e_{t}\right) \right) \right)
w/2,$ implying%
\[
e_{t}=I\left( \tau_{t}\right) =\frac{\left( 1+\beta z\left( 1-\alpha
_{0}\right) \right) w}{2+\beta z\alpha_{1}w}-\frac{w}{2+\beta z\alpha_{1}%
w}\tau_{t}%
\]
and%
\begin{align*}
\tau_{t+1} & =T\left( I\left( \tau_{t}\right) \right) =\bar{\tau}%
+z_{d}\left( \tau_{t}-\bar{\tau}\right) ,\\
e_{t+1} & =I\left( T\left( I\left( e_{t}\right) \right) \right)
=\bar{e}+z_{d}\left( e_{t}-\bar{e}\right) ,
\end{align*}
where
\begin{align}
\bar{\tau} & =\frac{2\alpha_{0}+\alpha_{1}w\left( 1+\beta z\right)
}{2+\alpha_{1}w\left( 1+\beta z\right) },\label{Etau}\\
\bar{e} & =\frac{w\left( 1+\beta z\right) \left( 1-\alpha_{0}\right)
}{2+\alpha_{1}w\left( 1+\beta z\right) },\label{Edef}\\
z_{d} & =-\frac{w\alpha_{1}}{2+\beta z\alpha_{1}w} \label{zd}%
\end{align}
The problem then admits the following recursive formulation:
\begin{align}
W\left( e_{t-1}\right) & =\max_{\tau_{t}}\left\{ F\left( e_{t-1}%
,\tau_{t},\tau_{t+1}\right) +\beta W\left( e_{t}\right) \right\}
,\label{eq_tcrecursive}\\
\text{s.t. }\tau_{t+1} & =\alpha_{0}+\alpha_{1}e_{t},\nonumber\\
e_{t} & =\frac{\left( 1+\beta z\left( 1-\alpha_{0}\right) \right)
w}{2+\beta z\alpha_{1}w}-\frac{w}{2+\beta z\alpha_{1}w}\tau_{t}.\nonumber
\end{align}
Given the guess, the first-order condition for maximizing the RHS\ of the
Bellman equation is%
\[
\frac{\partial F}{\partial\tau_{t}}+\frac{\partial F}{\partial\tau_{t+1}}%
\frac{d\tau_{t+1}}{d\tau_{t}}+\beta\frac{dW\left( e_{t}\right) }{de_{t}%
}\frac{de_{t}}{d\tau_{t}}=0,
\]
where%
\[
\frac{\partial F}{\partial\tau_{t}}=\left( ze_{t-1}+e\left( \tau_{t}%
,\tau_{t+1}\right) \right) \left( A-1\right) w-\left( \left( 1+\left(
A-1\right) \tau_{t}\right) w-2e\left( \tau_{t},\tau_{t+1}\right) \right)
\frac{w}{2},
\]%
\[
\frac{\partial F}{\partial\tau_{t+1}}=-\beta z\left( \left( 1+\left(
A-1\right) \tau_{t}\right) w-2e\left( \tau_{t},\tau_{t+1}\right) \right)
\frac{w}{2}%
\]
where we have used the fact that%
\[
\frac{\partial e_{t}}{\partial\tau_{t}}=-\frac{w}{2},\frac{\partial e_{t}%
}{\partial\tau_{t+1}}=-\beta z\frac{w}{2},
\]
Using the envelope condition, we obtain%
\[
W^{\prime}\left( e_{t}\right) =\frac{\partial F\left( e_{t},\tau_{t+1}%
,\tau_{t+2}\right) }{\partial e_{t}}=\left( 1+\left( A-1\right) \tau
_{t+1}\right) wz.
\]
which can be expressed in terms of $\tau_{t}$ using the constraints in
(\ref{eq_tcrecursive}). We can then can write the first-order condition as%
\begin{align*}
0 & =\frac{\partial F}{\partial\tau_{t}}+\frac{\partial F}{\partial
\tau_{t+1}}\frac{d\tau_{t+1}}{d\tau_{t}}+\beta W^{\prime}\left( e_{t}\right)
\frac{de_{t}}{d\tau_{t}}\\
0 & =\left( A-\frac{\beta z\alpha_{1}w}{2+\beta z\alpha_{1}w}\right)
e_{t}-\frac{2\left( A-1\right) w}{\left( 2+\beta z\alpha_{1}w\right) ^{2}%
}\tau_{t}+z\left( A-1\right) e_{t-1}\\
& -w\frac{\left( 1+\beta z\right) \left( 2+A\beta z\alpha_{1}w\right)
+2\beta z\alpha_{0}\left( A-1\right) }{\left( 2+\beta z\alpha_{1}w\right)
^{2}}%
\end{align*}
Using the fact that, $e_{t}=\frac{\left( 1+\beta z\left( 1-\alpha
_{0}\right) \right) w}{2+\beta z\alpha_{1}w}-\frac{w}{2+\beta z\alpha_{1}%
w}\tau_{t}$ and the guess $\tau_{t}=\alpha_{0}+\alpha_{1}e_{t-1}$, dividing by
$w$ and collecting terms, this yields%
\begin{align*}
0 & =\left( z\left( A-1\right) -\left( \frac{2A}{\left( 2+\beta
z\alpha_{1}w\right) }+\left( A-1\right) \right) \frac{w\alpha_{1}}{\left(
2+\beta z\alpha_{1}w\right) }\right) e_{t-1}\\
& +\frac{w\left( 1+\beta z\right) }{2+\beta z\alpha_{1}w}\left(
\frac{2A\left( 1-\alpha_{0}\right) }{2+\beta z\alpha_{1}w}-\left(
1+\alpha_{0}\left( A-1\right) \right) \right)
\end{align*}
In order for this condition to be satisfied for all $e_{t-1}$ we need,%
\begin{align}
z\left( A-1\right) -\left( \frac{2A}{\left( 2+\beta z\alpha_{1}w\right)
}+\left( A-1\right) \right) \frac{w\alpha_{1}}{2+\beta z\alpha_{1}w} &
=0\label{effe}\\
\frac{2A\left( 1-\alpha_{0}\right) }{2+\beta z\alpha_{1}w}-\left(
1+\alpha_{0}\left( A-1\right) \right) & =0
\end{align}
A solution for these equations (ignoring the roots that would generate
instability) is: :
\begin{align*}
\alpha_{1} & =\frac{\sqrt{1+4A\left( A-1\right) \left( 1-\beta
z^{2}\right) }-\left( 1+2\left( 1-\beta z^{2}\right) \left( A-1\right)
\right) }{\beta z\left( A-1\right) \left( 1-\beta z^{2}\right) w}\geq0\\
\alpha_{0} & =\frac{2\left( A-1\right) -\beta z\alpha_{1}w}{2+\left(
A-1\right) \left( 4+\beta z\alpha_{1}w\right) }\\
& =\frac{2A\left( A-1\right) \left( 1-\beta z^{2}\right) -\left(
\sqrt{1+4A\left( A-1\right) \left( 1-\beta z^{2}\right) }-1\right)
}{\left( A-1\right) \left( 2A\left( 1-\beta z^{2}\right) +\left(
\sqrt{1+4A\left( A-1\right) \left( 1-\beta z^{2}\right) }-1\right)
\right) }\geq0
\end{align*}
The non-negativity of $\alpha_{0}$ and $\alpha_{1}$ are established by
standard algebra, since, in both the expressions, the numerator and
denominators are both positive.
\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.
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 perturbation
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.
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, their 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. Recall that the inverse function theorem says that if
\begin{align*}
u & =u\left( c\right) \text{ and }\\
c & =u^{-1}\left( u\right) ,
\end{align*}
then
\[
\frac{\partial u^{-1}\left( u\right) }{\partial u}=\frac{1}{u^{\prime
}\left( c\right) }=\frac{1}{u^{\prime}\left( u^{-1}\left( u\right)
\right) }.
\]
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*}
which we note is an example of the \emph{inverse Euler equation.}
From Jensen's inequality, we find that
\begin{align*}
u^{\prime}\left( c_{1}^{\ast}\left( i\right) \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,
$n_{h}=\frac{y_{h}}{\theta}=\theta n_{1}$ 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
\[
\frac{c_{h}^{\ast}}{c_{1}^{\ast}}=1+2\lambda_{I},\frac{c_{l}^{\ast}}%
{c_{1}^{\ast}}=1-2\lambda_{I}%
\]
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}\\
y_{l}^{\ast} & =\frac{\lambda_{r}}{\theta^{4}-2\lambda_{I}}\theta^{2}%
\end{align*}
%
\begin{align*}
\tau_{y_{1}} & =1+\frac{v^{\prime}\left( y_{1}^{\ast}\right) }{u^{\prime
}\left( c_{1}^{\ast}\right) }=1-\frac{y_{1}^{\ast}}{\frac{1}{c_{1}^{\ast}}%
}=1-\frac{\lambda_{r}}{\frac{1}{\frac{1}{\lambda_{r}}}}=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}
Before going to the implementation, note that if we eliminate the shadow value
on the resource constraint, we have 7 equations and seven unknowns; Geting rid
of the shadow value on resources, we have 7 conditions and 7 unknowns
\begin{align*}
c_{1} & =\frac{1}{y_{1}},c_{h}=\frac{1+2\lambda_{I}}{y_{1}}\\
c_{l} & =\frac{1-2\lambda_{I}}{y_{1}},y_{h}=\frac{y_{1}}{1+2\lambda_{I}%
}\theta^{2}\\
y_{l} & =\frac{y_{1}}{\theta^{4}-2\lambda_{I}}\theta^{2},\\
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*}
This does not have a nice closed form solution. However, setting $\theta
=1.1,$I numerically found the solution as $c_{1}=0.998\,71,y_{1}%
=1.001\,3,y_{h}=1.108\,9,y_{l}=0.88337,c_{h}=1.091\,2,c_{l}=0.906\,26,\lambda
_{I}=4.628\,6\times10^{-2}.$
As we see, high ability types consume more than low ability types. However,
the former consumes less than their income and the latter more, i.e., there is redistribution.
\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 individuals are;%
\begin{align}
\frac{1}{c_{1}} & =\lambda_{1},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_{s,l}\right) \nonumber\\
\frac{y_{h}}{2\theta^{2}} & =\lambda_{h}\left( 1-\tau_{h}\right)
,\frac{\theta^{2}y_{l}}{2}=\lambda_{l}\left( 1-\tau_{l}\right)
\label{eq_privFOC}\\
\frac{1}{2c_{h}} & =\lambda_{h},\frac{1}{2c_{l}}=\lambda_{l}\nonumber
\end{align}
Using this, we see that
\[
\frac{1}{c_{1}}=\frac{1}{2c_{h}}\left( 1-\tau_{s,h}\right) +\frac{1}{2c_{l}%
}\left( 1-\tau_{s,l}\right)
\]
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. We do
not need to use any transfers in the first period. Thus%
\begin{align*}
T_{h} & =c_{h}-y_{h}-\left( y_{1}-c_{1}\right) \left( 1-\tau_{s,h}\right)
\\
T_{l} & =c_{l}-y_{l}-\left( y_{1}-c_{1}\right) \left( 1-\tau
_{s,l}\right)
\end{align*}
We should note that $T_{l}>T_{h}$ is consistent with incentive compatibility.
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.
To find expressions for the transfers I need to use numerical methods. Using
the results for $\theta=1.1,$ we have
\begin{align*}
T_{h} & =1.0912-1.1089-\left( 1.0013-0.99871\right) \left( 1+2\ast
4.628\,6\times10^{-2}\right) \\
& =-0.0205\\
T_{l} & =0.906\,26-0.88337-\left( 1.0013-0.99871\right) \left(
1-2\ast4.628\,6\times10^{-2}\right) \\
& =0.0205
\end{align*}
\subsubsection{Third best -- laissez 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},y_{1}=\lambda_{1}\nonumber\\
\lambda_{1} & =\lambda_{h}+\lambda_{l}\nonumber\\
\frac{y_{h}}{2\theta^{2}} & =\lambda_{h},\frac{\theta^{2}y_{l}}{2}%
=\lambda_{l}\label{eq_FOCLaizzesFaire}\\
\frac{1}{2c_{h}} & =\lambda_{h},\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 $c_{1}=0.997\,75,c_{h}=1.102\,3,s=4.\,\allowbreak
504\,5\times10^{-3},c_{l}=0.911\,35,$ $y_{1}=1.0023,y_{h}=1.106\,8,y_{l}%
=0.915\,85.$
As we see, consumption is lower in the first period and labor supply is higher
than in second best. Consumption of high ability types is higher and labor
supply lower than in second best. For low ability types, consumption is
actually higher in \emph{laissez faire} but also labor supply. The second
period welfare of low ability types is higher in second best ($-0.285$ vs.
$-0.300\,15).$
\subsubsection{Means tested system}
Suppose now we want to implement the optimal allocation without a savings-tax
but using an asset tested disability transfer instead. That is we set%
\[
T_{l}=\left\{
\begin{array}
[c]{c}%
T_{l}\text{ if }s\leq\bar{s}\\
-\bar{T}\text{ else.}%
\end{array}
\right.
\]
where $\bar{T}$ is sufficiently large to deter savings above $\bar{s}.$ We set
$\bar{s}$ equal to the first best $y_{1}^{\ast}-c_{1}^{\ast}.$ Without a
savings tax, the cap on savings will clearly bind due to the inverse Euler
equation. The problem of the individual is therefore
\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}-\bar{s}+T\\
0 & =y_{h}\left( 1-\tau_{h}\right) +\bar{s}-c_{h}+T_{h}\\
0 & =y_{l}\left( 1-\tau_{l}\right) +\bar{s}-c_{l}+T_{l}%
\end{align*}
First order conditions for the individuals are;%
\begin{align}
c_{1};\frac{1}{c_{1}} & =\lambda_{1}\nonumber\\
y_{1};y_{1} & =\lambda_{1}\left( 1-\tau_{1}\right) \nonumber\\
y_{h};\frac{y_{h}}{2\theta^{2}} & =\lambda_{h}\left( 1-\tau_{h}\right)
\label{eq_privFOC}\\
y_{l};\frac{\theta^{2}y_{l}}{2} & =\lambda_{l}\left( 1-\tau_{l}\right)
\nonumber\\
c_{h};\frac{1}{2c_{h}} & =\lambda_{h}\nonumber\\
c_{l};\frac{1}{2c_{l}} & =\lambda_{l}\nonumber
\end{align}
giving
\begin{align}
1-\tau_{1} & =c_{1}y_{1}\label{eq_FocSimpler}\\
\theta^{2}\left( 1-\tau_{h}\right) & =c_{h}y_{h}\\
\frac{\left( 1-\tau_{l}\right) }{\theta^{2}} & =c_{l}y_{l}\nonumber
\end{align}
We want
\[
1=c_{1}y_{1}\Rightarrow\tau_{1}=0.
\]
We also want
\begin{align}
c_{h}y_{h} & =\theta^{2},\nonumber\\
c_{l}y_{l} & =\frac{1-2\lambda_{I}}{\theta^{2}\left( 1-2\lambda_{I}%
\theta^{-4}\right) }%
\end{align}
requiring
\begin{align*}
\tau_{h} & =0,\\
\tau_{l} & =2\lambda_{I}\frac{\theta^{4}-1}{\theta^{4}-2\lambda_{I}},
\end{align*}
mimicing the results above.
Golosow and Tsyvinski (2006), extend this model and calibrate it to the US.
They assume people live until 75 years and start working at 25. The calibrate
the probability of becoming permanently disabled for each age group. The
problem is substantially simplified by the assumption that disability is
permanent. They find the second best allocation in the same way as we have
done here working backwards from the last period. As here, they show that the
optimal allocation is implementable with transfers with asset limits and taxes
on working people. The able should have zero marginal income taxes as in our
example. In contrast to our example, the low ability types here have zero
labor income and thus face no labor income tax.
An important finding is that asset limits are age dependent and increasing
over (most of) the working life.%
%TCIMACRO{\FRAME{ftbpFU}{6.6227in}{3.4999in}{0pt}{\Qcb{Figure from Golosov \&
%Tsyvinski (2006)}}{}{Figure}{\special{ language "Scientific Word";
%type "GRAPHIC"; maintain-aspect-ratio TRUE; display "USEDEF";
%valid_file "T"; width 6.6227in; height 3.4999in; depth 0pt;
%original-width 6.5587in; original-height 3.4541in; cropleft "0";
%croptop "1"; cropright "1"; cropbottom "0";
%tempfilename 'KTEMJG00.bmp';tempfile-properties "XPR";}}}%
%BeginExpansion
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
natheight=3.454100in,
natwidth=6.558700in,
height=3.4999in,
width=6.6227in
]%
{KTEMJG00.bmp}%
\caption{Figure from Golosov \& Tsyvinski (2006)}%
\end{center}
\end{figure}
%EndExpansion
\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}