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\begin{document}
\title{Topics in Dynamic Public Finance}
\author{John Hassler\\Stockholm University}
\maketitle
\section{New Public Finance -- the Mirrlees approach}
The purpose of this part of the course is to study some simle public finance
problems when there is heterogeneity in the population.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. The
general problem is to redistribute and provide some public good. We will start
by static examples and then go to some dynamic.
\subsection{A standard static example Public Finance problem}
We assume there is unit mass of individuals with different productivity,
denoted $\theta.$ We normalize the average productivity to unity. Individuals
derive utility from consumption, leisure and a public good. They have a
standard utility function given by
\begin{equation}
U=u\left( c\right) -v\left( n\right) +\Gamma\left( G\right)
\label{eq_utility}%
\end{equation}
where $c$ is consumption, $n$ is labor supply (in units of effort or hours),
$G$ is a public good $u$ and $\Gamma$ are concave functions and $v$ is
convex.\footnote{An alternative interpretation here is that $\Gamma\left(
G\right) $ represents transfers to some non-working individuals.} We assume
additive separability between private and public goods in order to abstract
from differences between individual's taste for the public good coming from
differences in their private consumption.
Often we will use the standard utility functions%
\begin{align*}
u\left( c\right) & =\frac{c^{1-\rho}}{1-\rho}\\
v\left( n\right) & =\frac{n^{1+\frac{1}{\gamma}}}{1+\frac{1}{\gamma}}%
\end{align*}
$\rho$ is the constant relative risk aversion coefficient and $\gamma$ the
Frish labor supply elasticity (labor supply wage elasticity at constant
marginal utility of consumption). Individuals are different in their
productivities, denoted $\theta.$ We let $f\left( \theta\right) $ is the
density of individual productivities. The aggregate resource constraint is
then
\[
\int_{0}^{\infty}f\left( \theta\right) \left( \theta n_{\theta}-c_{\theta
}\right) d\theta-G=0
\]
where $n_{\theta}$ and $c_{\theta}$ represents effort and consumption by the
type with productivity $\theta.$ The planning problem is
\[
\int_{0}^{\infty}f\left( \theta\right) \left( u\left( c_{\theta}\right)
-\nu\left( n_{\theta}\right) +\Gamma\left( G\right) \right)
+\lambda\left( \theta n_{\theta}-c_{\theta}-G\right) d\theta
\]
FOC;%
\begin{align*}
c_{\theta};u^{\prime}\left( c_{\theta}\right) & =\lambda\\
n_{\theta};v^{\prime}\left( n_{\theta}\right) & =\lambda\theta\\
\Gamma^{\prime}\left( G\right) & =\lambda
\end{align*}
As we see, everyone consumes the same, but works depending on productivity.
Since $v$ is convex, $v^{\prime}\left( n\right) $ is increasing and so is
the inverse function $v^{-1\prime}\left( n_{\theta}\right) .$Thus, higher
produtivity individuals work more. With the utility specification above, we
have
\[
n_{\theta}=\left( \lambda\theta\right) ^{\gamma}%
\]
With a Frisch elasticity of unity, labor supply is then proportional to
productivity, not depending on the consumption elasticity. Why? An interesting
implication of this is that a mean preserving spread in the distribution of
productivities has no effect on consumption of private and public goods. With
a lower elasticity, labor supply is concave in productivity implying that a
mean preserving spread reduces labor supply.
Now, we have not yet looked at whether this allocation can be decentralized.
Under which circumstances would it be possible?
The standard approach in PF is to endow the government with a set of arbitray
tools to affect the market allocation and then let the government maximize
over those. Such a problem is sometimes called a \emph{Ramsey problem.}
Let's consider a very typical instrument. Namely a linear income tax $\tau$.
We then need to calculate the decentralized allocation as a function of
$\tau.$ Now, each individual is choosing his labor supply to solve%
\begin{align*}
& \max\left( u\left( c_{\theta}\right) -v\left( n_{\theta}\right)
+\Gamma\left( G\right) \right) \\
s.t.c_{\theta} & =\left( 1-\tau\right) \theta n_{\theta}%
\end{align*}
Note that we assume $\frac{\partial G}{\partial n_{\theta}}=0,$ why?
Define the solution
\[
n_{\theta}^{\ast}\left( \tau\right) \equiv\arg\max\left( u\left( \left(
1-\tau\right) \theta n_{\theta}\right) -v\left( n_{\theta}\right)
+\Gamma\left( G\right) \right)
\]
and the indirect private utility
\[
V_{\theta}\left( \tau\right) \equiv u\left( \left( 1-\tau\right) \theta
n_{\theta}^{\ast}\left( \tau\right) \right) -v\left( n_{\theta}^{\ast
}\left( \tau\right) \right) .
\]
We can now set up the Ramsey problem as%
\begin{align*}
L & \equiv\int_{0}^{\infty}f\left( \theta\right) \left( V_{\theta}\left(
\tau\right) +\Gamma\left( G\right) \right) d\theta\\
s.t.0 & =\int_{0}^{\infty}f\left( \theta\right) \left( \theta n_{\theta
}\left( \tau\right) \tau\right) d\theta-G
\end{align*}
The first order condition for $\tau$ and $G$ are%
\begin{align*}
\frac{\partial}{\partial\tau}\int_{0}^{\infty}f\left( \theta\right)
V_{\theta}\left( \tau\right) d\theta & =-\lambda\frac{\partial}%
{\partial\tau}\int_{0}^{\infty}\theta n_{\theta}\left( \tau\right) \tau
d\theta\\
\Gamma^{\prime}\left( G\right) & =\lambda
\end{align*}
where $\lambda$ is the shadow constraint on the resource constraint. This
yields,
\[
\frac{-\frac{\partial}{\partial\tau}\int_{0}^{\infty}f\left( \theta\right)
V_{\theta}\left( \tau\right) d\theta}{\frac{\partial}{\partial\tau}\int%
_{0}^{\infty}\theta n_{\theta}\left( \tau\right) \tau d\theta}%
=\Gamma^{\prime}\left( G\right)
\]
For later use we note that we can write this as%
\[
\frac{-\int_{0}^{\infty}f\left( \theta\right) \frac{\partial V_{\theta
}\left( \tau\right) }{\partial\tau}d\theta}{\int_{0}^{\infty}f\left(
\theta\right) \frac{\partial\left( \theta n_{\theta}\left( \tau\right)
\tau\right) }{\partial\tau}d\theta}=\Gamma^{\prime}\left( G\right)
\]
Define aggregate private indirect utility
\[
U\left( \tau\right) \equiv\int_{0}^{\infty}f\left( \theta\right)
V_{\theta}\left( \tau\right) d\theta
\]
and
\[
R\left( \tau\right) \equiv\int_{0}^{\infty}f\left( \theta\right) \left(
\theta h_{\theta}\left( \tau\right) \tau\right) d\theta
\]
is government reveneue.
Defining%
\[
MU\left( \tau\right) \equiv-\frac{dU}{d\tau}/\frac{dR}{d\tau}%
\]
as the marginal aggregate disutility of tax revenues, the optimality condition
can therefore be written as the following equation in $\tau;$%
\begin{equation}
MU\left( \tau\right) =\Gamma^{\prime}\left( R\left( \tau\right) \right)
\label{eq_optimalG}%
\end{equation}
i.e., that the aggregate utility loss of an extra dollar of revenue should be
equal to the marginal value of the public good.
By specifying the functions, we can go further and do a quantitative analysis.
Let's use the specifications above. The problem of each individual is%
\[
\max\frac{\left( \theta\left( 1-\tau\right) n_{\theta}\right) ^{1-\rho}%
}{1-\rho}-\frac{n_{\theta}^{1+\frac{1}{\gamma}}}{1+\frac{1}{\gamma}}%
\]
yielding
\[
n_{\theta}=\left( \theta\left( 1-\tau\right) \right) ^{\frac{\gamma\left(
1-\rho\right) }{1+\rho\gamma}}%
\]
Note that with $\rho=1,$ utility is logaritmic in consumption and labor supply
is independent of taxes and productivity (compare to first best). With $\rho$
smaller (larger) than 1, labour supply increases (decreases) in net wage. Explain!
Using the solution in the two felicity functions yields private indirect
private utility is then
\[
V_{\theta}\left( \tau\right) =\left( \frac{1+\gamma\rho}{\left(
1-\rho\right) \left( \gamma+1\right) }\right) \left( \theta\left(
1-\tau\right) \right) ^{\frac{\left( 1-\rho\right) \left( \gamma
+1\right) }{1+\gamma\rho}}%
\]
and
\[
\frac{\partial V_{\theta}\left( \tau\right) }{\partial\tau}=-\theta
^{\frac{\left( 1-\rho\right) \left( \gamma+1\right) }{1+\gamma\rho}%
}\left( 1-\tau\right) ^{\frac{\left( 1-\rho\right) \left( \gamma
+1\right) }{1+\gamma\rho}-1}.
\]
\textbf{Result .}
\begin{enumerate}
\item \emph{The marginal individual cost of taxation is decreasing
(increasing) in ability }$\theta$\emph{ if }$\rho>\left( <\right) 1.$
\item \emph{The average marginal individual cost of taxation is increasing in
ability inequality if }$\rho>1$\emph{ or smaller than }$\frac{\gamma}%
{2\gamma+1}$.
The first part is obvious, the second follows since $-\frac{dV_{\theta}\left(
\tau\right) }{d\tau}$ is then convex.
\end{enumerate}
From this follows that if $\rho>1,$
\begin{enumerate}
\item[a] agents with ability below that of the representative agent has a
higher marginal utility loss of taxes than the representative agent,
\item[b] the average marginal utility loss of taxation is higher than that of
the median agent (making a representative agent analysis problematic).
\end{enumerate}
We can also compute the marginal revenue from a tax increase for each
individual;%
\begin{align*}
\frac{\partial\left( \theta n_{\theta}\left( \tau\right) \tau\right)
}{\partial\tau} & =\theta n_{\theta}\left( \tau\right) +\theta\tau
\frac{\partial n_{\theta}\left( \tau\right) }{\partial\tau}\\
& =\theta^{\frac{\gamma+1}{1+\gamma\rho}}\left( 1-\tau\right)
^{-\frac{1+2\gamma\rho-\gamma}{1+\gamma\rho}}\left( 1-\tau\frac{1+\gamma
}{1+\gamma\rho}\right) .
\end{align*}
We see this is zero for an interior value of $\tau$ if $\frac{1+\gamma
}{1+\gamma\rho}>1.$ This requires $\rho<1.$ Thus, we a riskaversion larger or
equal to 1, there is no \emph{Laffer curve} maximum. What happens if we use
the tax revenue not for government consumption but for transfers?
Now assume that $\theta$ is log normal with mean $\mu$ and variance
$\sigma^{2}$, the expectation of $\theta^{x}$ is $e^{\mu x+x^{2}\frac
{\sigma^{2}}{2}}.$ Under the normalization that average productivity is unity,
i.e., $\bar{\theta}\equiv\int_{0}^{\infty}f\left( \theta\right) \theta
d\theta=1,$ we must set%
\[
\mu=-\frac{\sigma^{2}}{2}%
\]
Using this, it is immediate that
\begin{align}
\frac{\partial U\left( \tau\right) }{\partial\tau} & =\int_{0}^{\infty
}f\left( \theta\right) \frac{\partial V_{\theta}\left( \tau\right)
}{\partial\tau}d\theta\label{eq_dU_dtau}\\
& =\left( 1-\tau\right) ^{\frac{\gamma-\rho\left( 1+2\gamma\right)
}{1+\gamma\rho}}e^{-\left( 1-\rho\right) \left( 1+\gamma\right) \frac
{\rho+\gamma\left( 2\rho-1\right) }{\left( 1+\gamma\rho\right) ^{2}}%
\frac{\sigma^{2}}{2}}\nonumber
\end{align}
and%
\begin{align}
\frac{\partial R\left( \tau\right) }{\partial\tau} & =\int_{0}^{\infty
}f\left( \theta\right) \frac{\partial\left( \theta h_{\theta}\left(
\tau\right) \tau\right) }{\partial\tau}d\theta\label{eq_dR_dtau}\\
& =\left( 1-\tau\right) ^{-\frac{1+2\gamma\rho-\gamma}{1+\gamma\rho}%
}\left( 1-\tau\frac{1+\gamma}{1+\gamma\rho}\right) e^{\left( 1-\rho\right)
\gamma\frac{\left( 1+\gamma\right) }{\left( 1+\gamma\rho\right) ^{2}}%
\frac{\sigma^{2}}{2}}%
\end{align}
Then,
\[
MU\left( \tau\right) =\frac{\frac{-\partial U\left( \tau\right) }%
{\partial\tau}}{\frac{\partial R\left( \tau\right) }{\partial\tau}}=\frac
{1}{\left( 1-\tau\frac{1+\gamma}{1+\gamma\rho}\right) }\left(
1-\tau\right) ^{\frac{1-\rho}{1+\gamma\rho}}e^{-\rho\frac{\left(
1-\rho\right) \left( 1+\gamma\right) \left( 1+2\gamma\right) }{\left(
1+\gamma\rho\right) ^{2}}\frac{\sigma^{2}}{2}}%
\]
The standard log-normal distribution has only one parameter ($\sigma$)
determining income dispersion and there is a simple one-to-one mapping between
$\sigma$ and the Gini-coefficient ($\phi$);%
\[
\phi=2\Phi\left( \frac{\sigma}{\sqrt{2}}\right) -1
\]
where $\Phi$ is the standard normal cumulative distribution.\footnote{See
Bourguignon (2003).} In the data, US has a Gini of 0.408, corresponding to
$\sigma=0.7579.$ Sweden has a Gini of 0.25, corresponding to $\sigma
=0.4506.$\footnote{Source: United Nations (2006). Table 15: Inequality in
income or expenditure (PDF). Human Development Report 2006 335. United Nations
Development Programme.} Now we can calculate the marginal cost of linear tax
revenue. In the following graph, I have calculated the cost of revenue with
$\rho=2$ and $\gamma=\frac{1}{2}.$ The utility of public good is set to
$\Gamma\left( G\right) =k\frac{G^{1-\kappa}}{1-\kappa}$ and we use the
budget constraint $G=R\left( \tau\right) .$ We can calibrate $k$ and
$\kappa$ by using spending shares on public goods $\tau_{US}=10.54\%,$ and
$\tau_{SE}=21.83\%$ reported in the Penn Word Table\footnote{Source: Alan
Heston, Robert Summers and Bettina Aten, Penn World Table Version 6.2, Center
for International Comparisons of Production, Income and Prices at the
University of Pennsylvania, September 2006.} for year 2004, giving $k=0.97214$
and $\kappa=0.40219.$
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The downward sloping curves in Figure 1 are $\Gamma^{\prime}\left( R\left(
\tau\right) \right) $ for the US and Swedish parametrizations. In fact,
these curves are basically indistinguishable, indicating that government
revenue as a function of the tax rate are very similar under the two different
parameter sets. Instead, the large difference in chosen tax rates are due to
differences in the utility loss associated with taxation. The upward-sloping
solid curves are $MU\left( \tau\right) $ for the US and Sweden respectively.
In the case of no heterogeneity, depicted by the dashed lines in Figure 2, the
marginal utility cost of taxation is even lower than in Sweden. Therefore,
taxes and public good provision should be higher. In fact, with no inequality,
the optimal tax rate is as high as 29.6\%.
Finally, let's consider if the allocation is pareto efficient. Take a
particular agent $i$ in the productivity distribution. That person pays
$\tau\theta_{i}h_{\theta_{i}}^{\ast}$ in taxes. Suppose now we ask this person
to pay this amount as a lump sum payment but that then his marginal tax is
zero. The problem is therefore%
\begin{align*}
& \max\left( u\left( c_{\theta_{i}}\right) -v\left( n_{\theta_{i}%
}\right) +\Gamma\left( G\right) \right) \\
s.t.c_{\theta} & =\theta n_{\theta_{i}}-T
\end{align*}
where $T$ is exogeneously fixed to $\tau\theta_{i}n_{\theta_{i}}^{\ast}.$
Clearly, the individual could still choose the "old" allocation, call it
$c_{\theta_{i}}^{\ast},h_{\theta_{i}}^{\ast}$ $.$ We know that
\[
\frac{v^{\prime}\left( n_{\theta_{i}}^{\ast}\right) }{u^{\prime}\left(
c_{\theta_{i}}^{\ast}\right) }=\theta_{i}\left( 1-\tau\right)
\]
Therefore, provided $\tau>0,$%
\[
\frac{v^{\prime}\left( n_{\theta_{i}}^{\ast}\right) }{u^{\prime}\left(
c_{\theta_{i}}^{\ast}\right) }<\theta_{i}.
\]
So with a zero marginal tax and lump sum taxes, the individual would choose to
work more (increasing $v^{\prime}$) and consume more (reducing $u^{\prime}).$
That would make him strictly better of and no one else worse off.
The problem, however, is that other people might want to have this deal to. In
particular those with higher productivity. The ones with sufficiently low
productivity would not like to pay $\tau\theta_{i}n_{\theta_{i}}^{\ast}.$ This
suggest a scheme where individuals are given a menue where lower marginal tax
rates and higher transfers and then volontarily sort them selves. This is what
the Mirrlees allocation achieves!
\subsection{The static Mirrlees model}
Consider now a simple two type variant of the model above. Furthermore
disregard public good provision. Suppose a share $\pi$ of the population has
high productivity $\theta_{h}$ and the remaining share has productivity
$\theta_{l}\leq\theta_{h}.$ Consider first the first best allocation if the
social welfare function is utilitarian%
\begin{align}
& \max\pi\left( u\left( c_{h}\right) +v\left( n_{h}\right) \right)
+\left( 1-\pi\right) \left( u\left( c_{l}\right) +v\left( n_{l}\right)
\right) \label{eq_oneperiodobjective}\\
s.t.0 & \leq\pi\theta_{h}n_{h}+\left( 1-\pi\right) \theta_{l}n_{l}-\pi
c_{h}-\left( 1-\pi\right) c_{l}\nonumber
\end{align}
where subscripts denote the type, so $c_{h}$, for example, denoted consumption
of the high productivity types.
Denoting the shadow value on the resource constraint by $\lambda,$ we have the
first order conditions%
\begin{align*}
\pi u^{\prime}(c_{h})-\lambda\pi & =0\\
\left( 1-\pi\right) u^{\prime}(c_{l})-\left( 1-\pi\right) \lambda & =0\\
\pi v^{\prime}(n_{h})+\lambda\pi\theta_{h} & =0\\
\left( 1-\pi\right) v^{\prime}(n_{l})+\left( 1-\pi\right) \lambda
\theta_{l} & =0\\
\lambda\left( \pi\theta_{h}n_{h}+\left( 1-\pi\right) \theta_{l}n_{l}-\pi
c_{h}-\left( 1-\pi\right) c_{l}\right) & \geq0
\end{align*}
Clearly the two first constraints imply that
\[
c_{h}=c_{l}%
\]
while the next two implies%
\[
\frac{v^{\prime}(n_{h})}{v^{\prime}(n_{l})}=\frac{\theta_{h}}{\theta_{l}}%
\geq1
\]
that is the marginal disutility of work is higher for the able individuals,
i.e., they work more. Clearly this poses a problem if the planner cannot
observe individual productivity and the effort $h$ the individual puts in.
\emph{The planner is assumed to only observe income and consumption.}
Furthermore,
\begin{align*}
\theta_{h} & =\frac{-v^{\prime}(n_{h})}{u^{\prime}(c_{h})}\\
\theta_{l} & =\frac{-v^{\prime}(n_{l})}{u^{\prime}(c_{l})}%
\end{align*}
with a well-known interpretation.
Consider now the problem of maximizing 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 tell the
individual to provide a given amount of income conditional on the ability an
individual claims to have. So let's consider a situation where each individual
reports her type and the planner then tells her how much income to provide
$y_{i}$ and how much to consume $c_{i}$. Let's call the report $i_{r}.$ The
incentive constraint is then that individuals voluntarily report their true
ability. According to the \emph{revelation principle, }any incentive
compatible allocation can be achived in this way. Thus we can restrict
ourselves to look within the class of allocations that satisfy incentive
constraints. Later, we will discuss how to decentralize that, i.e., construct
a tax-transfer system such that the optimal incentive-compatible allocation is
chosen by the individuals.
The problem is now to solve (\ref{eq_oneperiodobjective}) subject to the
truth-telling constraint
\begin{equation}
u\left( c_{i}\right) +v\left( \frac{y_{i}}{\theta_{i}}\right) \geq
u\left( c_{i_{r}}\right) +v\left( \frac{y_{i_{r}}}{\theta_{i}}\right)
,\forall i_{r},i\in\left\{ h,l\right\} \nonumber
\end{equation}
where we have substituted for $n$ by $y/\theta.$ Note that we always divide by
the true ability. Why?
We will not have both truth-telling constraints binding in the optimal
allocation. We conjecture that truth-telling for the more able person binds.
Why? Let's call the shadow value on that constraint by $\lambda_{I}$ and the
resource constraint $\lambda_{r}.$ The problem is then
\begin{align}
& \max\pi\left( u\left( c_{h}\right) +v\left( \frac{y_{h}}{\theta_{h}%
}\right) \right) +\left( 1-\pi\right) \left( u\left( c_{l}\right)
+v\left( \frac{y_{l}}{\theta_{l}}\right) \right) \\
s.t.0 & \leq\pi y_{h}+\left( 1-\pi\right) y_{l}-\pi c_{h}-\left(
1-\pi\right) c_{l}\nonumber\\
0 & =u\left( c_{h}\right) +v\left( \frac{y_{h}}{\theta_{h}}\right)
-u\left( c_{l}\right) -v\left( \frac{y_{l}}{\theta_{h}}\right)
\end{align}
First order conditions are%
\begin{align*}
\pi u^{\prime}\left( c_{h}\right) -\lambda_{r}\pi+\lambda_{I}u^{\prime
}\left( c_{h}\right) & =0\\
\left( 1-\pi\right) u^{\prime}\left( c_{l}\right) -\lambda_{r}\left(
1-\pi\right) -\lambda_{I}u^{\prime}\left( c_{l}\right) & =0\\
\pi v^{\prime}\left( \frac{y_{h}}{\theta_{h}}\right) \frac{1}{\theta_{h}%
}+\pi\lambda_{r}+\lambda_{I}v^{\prime}\left( \frac{y_{h}}{\theta_{h}}\right)
\frac{1}{\theta_{h}} & =0\\
\left( 1-\pi\right) v^{\prime}\left( \frac{y_{l}}{\theta_{l}}\right)
\frac{1}{\theta_{l}}+\left( 1-\pi\right) \lambda_{r}-\lambda_{I}v^{\prime
}\left( \frac{y_{l}}{\theta_{h}}\right) \frac{1}{\theta_{h}} & =0
\end{align*}
These implies%
\[
\frac{u^{\prime}\left( c_{h}\right) }{u^{\prime}\left( c_{l}\right)
}=\frac{1-\frac{\lambda_{I}}{1-\pi}}{1+\frac{\lambda_{I}}{\pi}}%
\]
Thus, the higher is the $\lambda_{I}$, the larger is the spread in marginal utilities.
Note also that%
\[
u^{\prime}\left( c_{h}\right) \left( 1+\frac{\lambda_{I}}{\pi}\right)
=-v^{\prime}\left( \frac{y_{h}}{\theta_{h}}\right) \frac{1}{\theta_{h}%
}\left( 1+\frac{\lambda_{I}}{\pi}\right)
\]
implying
\[
\theta_{h}=\frac{-v^{\prime}\left( n_{h}\right) }{u^{\prime}\left(
c_{h}\right) }%
\]
while%
\[
-\frac{v^{\prime}\left( \frac{y_{l}}{\theta_{l}}\right) }{u^{\prime}\left(
c_{l}\right) }=\frac{1-\frac{\lambda_{I}}{1-\pi}}{1-\frac{\lambda_{I}%
}{\left( 1-\pi\right) }\frac{v^{\prime}\left( \frac{y_{l}}{\theta_{h}%
}\right) }{v^{\prime}\left( \frac{y_{l}}{\theta_{l}}\right) }\frac
{\theta_{l}}{\theta_{h}}}\theta_{l}<\theta_{l}%
\]
since $1>\frac{v^{\prime}\left( \frac{y_{l}}{\theta_{h}}\right) }{v^{\prime
}\left( \frac{y_{l}}{\theta_{l}}\right) }\frac{\theta_{l}}{\theta_{h}}.$
Thus the labor leisure choice is distorted for the low ability types but not
for the high ability types. The no distortion at the top is a quite general
result when the distribution of abilities is bounded.
Take a simple example where $u\left( c\right) =\ln c$ and $v\left(
n\right) =-\frac{n^{2}}{2}.$ Set $\pi=1/2$ and $\theta_{h}=2,\theta_{l}=1.$
Then, we have%
\begin{align*}
\frac{1}{2}c_{h}^{-1}-\lambda_{r}\frac{1}{2}+\lambda_{I}c_{h}^{-1} & =0\\
\frac{1}{2}c_{l}^{-1}-\lambda_{r}\frac{1}{2}-\lambda_{I}c_{l}^{-1} & =0\\
-\frac{1}{2}n_{h}\frac{1}{2}+\frac{1}{2}\lambda_{r}-\lambda_{I}n_{h}\frac
{1}{2} & =0\\
-\frac{1}{2}n_{l}+\frac{1}{2}\lambda_{r}+\lambda_{I}n_{l}\frac{1}{4} & =0\\
2n_{h}+n_{l}-c_{h}-c_{l} & =0\\
\ln c_{h}-\frac{n_{h}^{2}}{2}-\left( \ln\left( c_{l}\right) -\frac{\left(
\frac{n_{l}}{2}\right) ^{2}}{2}\right) & =0
\end{align*}
The solution is: $n_{l}=0.73338,\lambda_{r}=0.68609,\lambda_{I}=0.12896,c_{h}%
=1.8334,c_{l}=1.0816,n_{h}=1.0908$
Note that $c_{h}n_{h}=2=\theta_{h},$ while $c_{l}n_{l}<1=\theta_{l}.$
In first best, we instead have
\begin{align*}
\frac{1}{2}c_{h}^{-1}-\lambda_{r}\frac{1}{2} & =0\\
\frac{1}{2}c_{l}^{-1}-\lambda_{r}\frac{1}{2} & =0\\
-\frac{1}{2}n_{h}\frac{1}{2}+\frac{1}{2}\lambda_{r} & =0\\
-\frac{1}{2}n_{l}+\frac{1}{2}\lambda_{r} & =0\\
2n_{h}+n_{l}-c_{h}-c_{l} & =0
\end{align*}
with the solution is: $\left\{ \lambda_{r}=0.632\,46,c_{h}=1.\,\allowbreak
581\,1,c_{l}=1.\,\allowbreak581\,1,n_{h}=1.\,\allowbreak264\,9,n_{l}%
=0.632\,46\right\} ,$ in which case $c_{h}n_{h}=\theta_{h}$ and$\ c_{l}%
n_{l}=\theta_{l}.$
\subsubsection{Implementation}
In the simple case discussed above, we can implement the allocation with a
menue of marginal tax rates and transfers. Since the labor-leisure tradeoff is
distorted (not distorted) for the low (high) ability individuals, we need a
tax on labor for only the low ability type. For the low ability type to accept
this, we need to give him a larger lump-sum transfer. Thus, indivduals are
asked to choose either a positive marginal tax and a high transfer or a zero
marginal tax and a smaller transfer (typically negative). Think of the
intution for why this is optimal.
Given that the truth telling constraint is satisfied, individuals solve%
\begin{align*}
& \max\left( u\left( c_{i}\right) +v\left( n_{i}\right) \right) \\
s.t.c_{i} & =\theta_{i}n_{i}\left( 1-\tau_{i}\right) +T_{i}%
\end{align*}
Implying
\[
\theta_{i}\left( 1-\tau_{i}\right) =\frac{-v^{\prime}\left( n_{i}\right)
}{u^{\prime}\left( c_{i}\right) }%
\]
In the example, we then have the two private first-order conditions and two
budget constraints.
Plugging in the numbers and solving yields
\begin{align*}
& \left[ c_{h}n_{h}=\theta_{h}\left( 1-\tau_{h}\right) \right]
_{n_{h}=1.0908,c_{l}=1.081\,6,n_{l}=0.733\,38,c_{h}=1.8334,\theta_{h}%
=2,\theta_{l}=1}\\
& \left[ c_{l}n_{l}=\theta_{l}\left( 1-\tau_{l}\right) \right]
_{n_{h}=1.0908,c_{l}=1.081\,6,n_{l}=0.733\,38,c_{h}=1.8334,\theta_{h}%
=2,\theta_{l}=1}\\
& \left[ c_{h}=\theta_{h}n_{h}\left( 1-\tau_{h}\right) +T_{h}\right]
_{n_{h}=1.0908,c_{l}=1.081\,6,n_{l}=0.733\,38,c_{h}=1.8334,\theta_{h}%
=2,\theta_{l}=1}\\
& \left[ c_{l}=\theta_{l}n_{l}\left( 1-\tau_{l}\right) +T_{l}\right]
_{n_{h}=1.0908,c_{l}=1.081\,6,n_{l}=0.733\,38,c_{h}=1.8334,\theta_{h}%
=2,\theta_{l}=1}%
\end{align*}
The solution is: $\left\{ T_{h}=-0.348\,06,T_{l}=0.499\,87,\tau
_{l}=0.206\,78,\tau_{h}=0\right\} \allowbreak$
Finally, we need to check whether it is necessary to add some non-linearities
in the tex system. Consider the utility if the high transfer, high marginal
tax is chosen by the high ability type. The choice then satisfies%
\begin{align*}
& \left[ c_{h}n_{h}=\theta_{h}\left( 1-\tau_{l}\right) \right]
_{\theta_{h}=2,\tau_{l}=0.20678,T_{h}=-0.34806,T_{l}=0.49987}\\
& \left[ c_{h}=\theta_{h}n_{h}\left( 1-\tau_{l}\right) +T_{l}\right]
_{\theta_{h}=2,\tau_{l}=0.20678,T_{h}=-0.34806,T_{l}=0.49987}%
\end{align*}
with the solution $\left\{ c_{hdev}=1.8559,n_{hdev}=0.85479\right\} .$
Clearly, this gives higher utility and we need to prevent this deviation. This
can be done by having another bracket in the tax system. The following tax
system could then implement the optimal second-best allocation. The indivuals
choose from the following menue;
\begin{enumerate}
\item A lump sum tax $-T_{h}=0.348.$ No marginal income tax.
\item A lum sum transfer $T_{l}=0.500.$ A marginal income tax of $\tau
_{l}=20.7\%$ up to income $n_{l}=0.733.$ Above that, a sufficently high tax
rate to deter any benefit claimant to earn more, e.g., 100\%.
\end{enumerate}
\subsection{Uniform commodity taxation}
An important assumption in the previous subsection was that there is just one
good. In reality, there are many goods, both intermediaries and final goods.
Then, a key issue becomes; Should different goods be taxed at different rates,
i.e., should we use differentiated VAT's? \ If not, we have seen that it does
not matter whether we use a flat consumption tax or a proportional income tax.
One of the most celebrated results in public finance is the Atkinson-Stiglitz
\emph{uniform commodity taxation result }(Atkinson \& Stiglitz, 1972). This
states that under some conditions, most importantly that utility is separable
in leisure and an aggregate of market consumption goods, a uniform tax rate
should be used. Then, it can, as we have discussed above be replaced by a
uniform tax rate on labor income. Loosely speaking, separability means that
utility can be written as a function of a consumption aggregate $g(c),$where
$c$ is a vector $\left[ c_{1},...,c_{n}\right] $ of consumption goods bought
in the market, and labor $n$ (equivalently, leisure) Thus%
\[
\bar{u}\left( c_{1},...c_{n},l\right) =u\left( g\left( c\right)
,n\right) .
\]
As above, productivity is unobserved by the planner and he only observes total
income, not wages. Due to separability, we can separate the consumers problem
in two steps. The last is to maximize $g\left( c\right) $ over the different
consumption goods, given disposable income $\omega$ and the prices $q_{i}$
(including taxes).%
\begin{align}
& \max_{c}g\left( c_{1},...,c_{n}\right) \label{ASSecondStep}\\
s.t.\text{ }\sum_{i}q_{i}c_{i} & \leq\omega\nonumber
\end{align}
This generates demand functions $d_{i}(q,\omega)$ and an associated value
function $h\left( q,\omega\right) \equiv g\left( d(q,\omega)\right) .$ The
latter function $h,$ can be thought of has the optimal consumption aggregate,
given prices and income.
The first step is then to choose labor supply by solving%
\[
\max_{y}u\left( h\left( q,\omega(y)\right) ,\frac{y}{\theta}\right) ,
\]
where $\omega\left( y\right) $ i disposable income given gross income $y.$
Let's follow Boadway and Pestieau (2002) and consider the case were there are
two types, $i\in\left\{ h,l\right\} $ with different planner unobserved
productivities (wages), $\theta_{h}>\theta_{l}.$ We assume that there are two
consumption goods, $c_{1}$ and $c_{2}$ and normalize their relative market
price before taxes to unity. Without loss of generality, we assume the policy
instrument in terms of consumption taxes is the tax on good $2$ and set the
other consumption tax to zero. This is w.o.l.g. since a common tax is
equivalent to a labor income tax. The price on good $2$ faced by consumers is
$1+\tau\equiv q$ implying that the budget constraint of the agent of type $i$
is
\[
\omega_{i}=c_{1}+qc_{2}.
\]
The second step problem (\ref{ASSecondStep}) can now be written%
\begin{equation}
h\left( q,\omega\right) =\max_{c_{2}}g\left( \omega-qc_{2},c_{2}\right)
\end{equation}
giving%
\[
\frac{g_{2}}{g_{1}}=q
\]
and using the envelope theorem, we have
\begin{align}
h_{\omega} & =g_{1},\label{eqASEnvelope}\\
h_{q} & =-g_{1}c_{2}=-h_{w}c_{2}\nonumber
\end{align}
We can now write the planner Lagrangian%
\begin{align*}
L & =\sum_{i=h,l}\pi_{i}u\left( h\left( q,\omega_{i}\right) ,\frac{y_{i}%
}{\theta_{i}}\right) +\lambda_{r}\sum_{i=h,l}\pi_{i}\left( y_{i}+\tau
c_{2}^{i}-\omega_{i}\right) \\
& +\lambda_{I}\left( u\left( h\left( q,\omega_{h}\right) ,\frac{y_{h}%
}{\theta_{h}}\right) -u\left( \left( h\left( q,\omega_{l}\right)
,\frac{y_{l}}{\theta_{h}}\right) \right) \right)
\end{align*}
The first constraint is the budget constraint of the government and the second
is the incentive constraint. We conjecture as above that the high productivity
type must be induced not to falsely report that he is a low productivity type.
Now, we focus on the FOC for the disposable incomes $\omega_{i}$ and the the
consumer price $q.$ To not have to write out the arguments of all functions,
we use superscript on functions to denote type and hat's on functions denote
for an $h$ type who pretends to be of type $l.$ We then get%
\begin{align*}
\omega_{l};\pi_{l}u_{h}^{l}h_{\omega}^{l}-\lambda_{r}\pi_{l}\left(
1-\tau\frac{\partial c_{2}^{l}}{\partial\omega_{l}}\right) -\lambda_{I}%
\hat{u}_{h}^{h}\hat{h}_{\omega}^{h} & =0\\
\omega_{h};\pi_{h}u_{h}^{h}h_{\omega}^{h}-\lambda_{r}\pi_{h}\left(
1-\tau\frac{\partial c_{2}^{h}}{\partial\omega_{h}}\right) +\lambda_{I}%
u_{h}^{h}h_{\omega}^{h} & =0\\
q;\sum_{i=h,l}\pi_{i}u_{h}^{i}h_{q}^{i}+\lambda_{r}\sum_{i=h,l}\pi_{i}\left(
c_{2}^{i}+\tau\frac{\partial c_{2}^{i}}{\partial q}\right) +\lambda
_{I}\left( u_{h}^{h}h_{q}^{h}-\hat{u}_{h}^{h}\hat{h}_{q}^{h}\right) & =0
\end{align*}
Now, multiply the first equation by $c_{2}^{l}$ and the second by $c_{2}^{h}$
and use (\ref{eqASEnvelope}). Giving%
\begin{align*}
\omega_{l};-\pi_{l}u_{h}^{l}h_{q}^{l}-\lambda_{r}\pi_{l}\left( 1-\tau
\frac{\partial c_{2}^{l}}{\partial\omega_{l}}\right) c_{2}^{l}-\lambda
_{I}\hat{u}_{h}^{h}\hat{h}_{\omega}^{h}c_{2}^{l} & =0\\
\omega_{h};-\pi_{h}u_{h}^{h}h_{q}^{h}-\lambda_{r}\pi_{h}\left( 1-\tau
\frac{\partial c_{2}^{h}}{\partial\omega_{h}}\right) c_{2}^{h}-\lambda
_{I}u_{h}^{h}h_{q}^{h} & =0
\end{align*}
Add these two to the FOC for $q;$ This gives%
\[
\lambda_{r}\tau\sum_{i=h,l}\pi_{i}\left( \frac{\partial c_{2}^{i}}{\partial
q}+\frac{\partial c_{2}^{i}}{\partial\omega_{i}}c_{2}^{i}\right) -\lambda
_{I}\hat{u}_{h}^{h}\left( \hat{h}_{\omega}^{h}c_{2}^{l}+\hat{h}_{q}%
^{h}\right) =0.
\]
Now, consider the parenthesis in the second term, $\hat{h}_{\omega}^{h}%
c_{2}^{l}+\hat{h}_{q}^{h}.$ Spelling out the arguments, we write this%
\[
h_{\omega}\left( q,\omega_{l}\right) c_{2}^{l}+h_{q}\left( \left(
q,\omega_{l}\right) \right) .
\]
From (\ref{eqASEnvelope}) we know this is zero. Recall that this term comes
from the cheating high productivity types, but since he consumes as much of
good 2 as the the low productivity types, the same envelope condition holds.
This would not be the case if also leisure entered in this expression, since
the two types consume different amounts of leisure. We thus end up with
\[
\lambda_{r}\tau\sum_{i=h,l}\pi_{i}\left( \frac{\partial c_{2}^{i}}{\partial
q}+\frac{\partial c_{2}^{i}}{\partial\omega_{i}}c_{2}^{i}\right) =0
\]
Note that $\frac{\partial c_{2}^{i}}{\partial q}+\frac{\partial c_{2}^{i}%
}{\partial\omega_{i}}c_{2}^{i}$ is the derivative of the compensated demand
function for $c_{2},$ i.e., the effect on demand of a marginal increase in the
price $dq$ together with an income transfer of $dqc_{2}.$ Provided this is not
zero, the tax must be zero.
The intuition for the result is that the planner wants to distort only margins
that can help him identify the low productivity individuals (equivalently, the
cheaters). If the marginal rate of substitution is the same for low and high
productivity individuals for some pair of goods, there is no point in
distorting it. One can, of course think of cases where this is not the case.
For example, a cheating high productivity individual consumes a lot of
leisure. Suppose there is one good that is a complement to leisure, like
vacation trips. Such a good should then be taxed higher because it reduces the
value of cheating for the high productivity individual.
A related result to the A-S is the Diamond-Mirrlees production efficiency
result (Diamond \& Mirrlees, 1972). This result states that production, in the
sense the use of different inputs in production, should not be distorted. This
result builds on a similar separability. If consumers care of the final
product, not of how it is produced, distorting production cannot help the
planner doing anything good.
\subsection{The direct approach}
An alternative to the Mirrleesian approach is to work directly with the tax
system and derive optimal properties of that. Saez (2001) show that this can
be done using observed characteristics as labor supply elasticites and the
actual income distribution. To understand the intuition behind the fairly
complicated formulas, consider a tax system $T(y)$ where $y$ is gross income
and $T\left( y\right) $ is the tax payment. Define $\tau\left( y\right)
\equiv T^{\prime}\left( y\right) $ and let $H\left( z\right) $ be the
share of individuals with income at or below $z$, with a density denoted
$h\left( z\right) .$
Consider the effects of a small increase in the marginal tax rate $d\tau$ over
the small intervall $y^{\ast}$ to $y^{\ast}+dy^{\ast}.$ This change is
illustrated in the figure below. Clearly, individuals with income below
$y^{\ast}$ are not affected by the change. Individuals in the interval
$\left[ y^{\ast},y^{\ast}+dy^{\ast}\right] $ face a change in their marginal
tax $\tau,$ but the average tax is (almost) not changed. Thus, there is only a
substitution effect and the change in labor supply depends on the
\emph{compensated }income elasticity. Thus, an increase in that tax rate
reduces labor supply. This is a negative effect seen from the point of view of
a benevolent planner and the importance of it depends on the density of
indivduals $h\left( y^{\ast}\right) $.
Above $y^{\ast}+dy^{\ast}$, the marginal income tax rate $\tau$ is unchanged
but the average income is increased by $dy^{\ast}d\tau.$ This has a mechanic
direct effect on reveneues when behavioral changes are disregarded and an
endogenous effect via labor supply that depends on the income elasticity of
labor supply.
Assuming leisure is a normal good, the higher tax increases labor supply above
$y^{\ast}$. Provided the value of government revenue is higher than the value
of private spending for inviduals with income above $y^{\ast},$both these
effects are positive for the planner. The strength of them depends positively
(loosely speaking) on total income above $y^{\ast}+dy^{\ast}$ and therefore on
$(1-H\left( y^{\ast}+dy^{\ast}\right) $.
We have now defined positive and a negative effects of increasing the slope at
$y^{\ast}.$ If $T\left( y\right) $ is optimal, these effects should balance
each other exactly. Furthermore, this should be true at all income levels $y.$
Letting $dy^{\ast}\rightarrow0,$ this then defines a differential equation
that must be satisfied. Together with, e.g., a financing reqirement or any
other condition that pins down the total tax requirements, this defines the
optimal tax. We note that the marginal tax $\tau\left( y\right) $, tends to
be high;
\begin{itemize}
\item if the compensated elasticity at $y$ is low,
\item if $h\left( y\right) $ is low,
\item if total income above $y$, i.e., $\int_{z}^{\infty}yh\left( y\right)
dz,$ is high.
\item if income elasticity above $y$ is high.
\item if the planner's value of money is high relative to the value the
planner attach to marginal income of individuals with income above $y.$
\end{itemize}
%
%TCIMACRO{\FRAME{ftbpF}{4.8285in}{3.628in}{0pt}{}{}{figures.jpg}%
%{\special{ language "Scientific Word"; type "GRAPHIC";
%maintain-aspect-ratio TRUE; display "USEDEF"; valid_file "F";
%width 4.8285in; height 3.628in; depth 0pt; original-width 9.5998in;
%original-height 7.1996in; cropleft "0"; croptop "1"; cropright "1";
%cropbottom "0";
%filename 'C:/Documents and Settings/hasslerj/My Documents/Dropbox/Courses/PubFin/Figures.jpg';file-properties "XNPEU";}%
%}}%
%BeginExpansion
\begin{figure}[ptb]%
\centering
\includegraphics[
natheight=7.199600in,
natwidth=9.599800in,
height=3.628in,
width=4.8285in
]%
{C:/Documents and Settings/hasslerj/My Documents/Dropbox/Courses/PubFin/Figures.jpg}%
\end{figure}
%EndExpansion
If there is a maximum income, the marginal tax rate should be zero there since
no revenue is generated above this income. This is the zero tax rate at the
top result of Mirrlees. In practice, however, we can not easily construct a
system with a specific tax rate at the exact top.
Saez (2001) argues that empirically, shape of the income distribution affects
the optimal tax rate at some each income level $y$ by the term $\frac
{1-H(y)}{yh(y)}.$ The elastic distortion at $y$ depends on the amount of
income generated by individuals with income $y$, i.e., $yh(y).$ If this is
large, distortions are large. Non-distortive revenues depends on how many
individuals earn more than $y,$ i.e., $1-H(Y).$ Saez shows that the empirical
income distribution is close to a Pareto for high income, in which case
$\frac{1-H(y)}{yh(y)}$ is constant. For low incomes, the value is high and for
intermediate it is the lowest. In 1993, the minimum was at \$80 000. The
optimal marginal tax rate is the initially high, decreasing to \$80000 than
increasing to around \$200 000 and then constant. Of course, if we were to
find the maximum, $1-H(y)=0$ and so the tax rate.
\section{New Public Finance -- the dynamic Mirrlees approach}
Let us now consider the dynamic Mirrlees approach to optimal taxation. As
above, individuals are assumed to be different. The only difference is that we
know consider a dynamic environment.
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 large number 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.
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
\[
\frac{c_{h}^{\ast}}{c_{1}^{\ast}}=1+2\lambda_{I},\frac{c_{l}^{\ast}}%
{c_{1}^{\ast}}=1-2\lambda_{I}%
\]
and
\[
\tau_{k}\equiv1-\frac{u^{\prime}\left( c_{1}^{\ast}\right) }{\beta R\left(
\frac{u^{\prime}\left( c_{h}^{\ast}\right) }{2}+\frac{u^{\prime}\left(
c_{l}^{\ast}\right) }{2}\right) }=1-\frac{\lambda_{r}}{\frac{\lambda_{r}%
}{1+2\lambda_{I}}\frac{1}{2}+\frac{\lambda_{r}}{1-2\lambda_{I}}\frac{1}{2}%
}=\left( 2\lambda_{I}\right) ^{2},
\]
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}\\
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}
\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}\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 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.
\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}\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 $c_{1}=0.997\,75,c_{h}=1.102\,3,s=4.\,\allowbreak
504\,5\times10^{-3},c_{l}=0.911\,35,$ $y_{1}=1.0023,y_{h}=1.106\,8,y_{l}%
=0.915\,85.$
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 'MHA87O02.bmp';tempfile-properties "XPR";}}}%
%BeginExpansion
\begin{figure}[ptb]%
\centering
\includegraphics[
natheight=3.454100in,
natwidth=6.558700in,
height=3.4999in,
width=6.6227in
]%
{MHA87O02.bmp}%
\caption{Figure from Golosov \& Tsyvinski (2006)}%
\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 an infinite horizon setting.
\end{document}