\documentclass[12pt]{article}% \usepackage{amsfonts} \usepackage{amsmath,fullpage,doublespace}% \usepackage{amsmath}% \setcounter{MaxMatrixCols}{30}% \usepackage{amssymb}% \usepackage{graphicx} %TCIDATA{OutputFilter=latex2.dll} %TCIDATA{Version=5.50.0.2953} %TCIDATA{CSTFile=40 LaTeX article.cst} %TCIDATA{Created=Tuesday, September 11, 2007 11:03:29} %TCIDATA{LastRevised=Tuesday, October 23, 2007 18:12:45} %TCIDATA{} %TCIDATA{} %TCIDATA{BibliographyScheme=Manual} %TCIDATA{} %TCIDATA{Language=American English} %TCIDATA{ComputeDefs= %$\theta=1.1$ %} %BeginMSIPreambleData \providecommand{\U}{\protect\rule{.1in}{.1in}} %EndMSIPreambleData \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{result}[theorem]{Result} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \begin{document} \title{Topics in Dynamic Public Finance} \author{John Hassler\\Stockholm University and Fudan University} \maketitle \newpage \section{Optimal unemployment insurance (UI)} There is a large literature of optimal unemployment insurance. The basic issue is how to provide the most efficient unemployment insurance when there is a moral hazard problem. This is arising from an assumption that unemployed individuals can affect the probability they find (and accept) a job offer. However, it is costly for the worker to increase this probability, e.g., because of effort costs, reduced reservation wages or opportunity costs of time. \subsection{The semi-static approach to optimal UI} The basic idea in Baily and Chetty is to simplify the dynamic problem into a static one. This makes the model simple and tractable also when savings is allowed. An important lesson is that when savings is allowed, we can use the drop in consumption at unemployment as a measure of the welfare loss associated with unemployment. In a dynamic model, this does not work when there is no market for savings. Why? The trade-off faced by the planner is to balance the loss of welfare associated with unemployment against the negative effect on search induced by UI. \subsubsection{The simplest model following Baily} \begin{itemize} \item In the first period, the individual works and chooses how much to consume of the income, normalized to unity, and how much to save. \item In the beginning of the second period, the individual becomes unemployed with probability $1-\alpha$ and otherwise keeps his job. \item During the second period, the individual can determine how long it takes to find a job by choosing the reservation wage $y_{n}$ and costly search effort $c$. A share $\beta=\beta\left( c,y_{n}\right)$ of the second period is spent working in the new job. \item While unemployed, the individual gets UI-benefits $b.$These are paid by taxes on workers. \item Agents have access to a market for precautionary (buffer stock) savings. \end{itemize} Total income in second period if laid off is therefore% $\left( 1-\beta\right) \left( b-c\right) +\beta y_{n}\left( 1-\tau\right) \equiv y_{l}.$ In first periods, individuals decide how much to save, $s.$ Interest rate and subjective discount rate is normalized to zero. Welfare is $V=u\left( 1-\tau-s\right) +\alpha u\left( 1-\tau+s\right) +\left( 1-\alpha\right) \left( u\left( y_{l}+s\right) \right) .$ Government budget constraint is% \begin{align*} \left( 1+\alpha+\left( 1-\alpha\right) \beta y_{n}\right) t & =\left( 1-\alpha\right) \left( 1-\beta\right) b.\\ & \Longrightarrow b=\frac{\left( 1+\alpha+\left( 1-\alpha\right) \beta y_{n}\right) }{\left( 1-\alpha\right) \left( 1-\beta\right) }\tau \equiv\mu\tau \end{align*} Denoting the \emph{endogenous} total income by $Y\equiv1+\alpha+\left( 1-\alpha\right) \beta y_{n},$ this implies \begin{align*} b & =\frac{Y}{\left( 1-\alpha\right) \left( 1-\beta\right) }\tau\\ & \equiv\mu\tau, \end{align*} where we note that $\mu$ is not a constant, but depends on individual choices of $y_{n}$ and $\beta$ and thus indirectly on taxes and benefits. Given the budget constraint and individual choices, we can therefore write $\mu =\mu\left( \tau\right)$ (provided there is a solution, which is not necessarily true for all $\tau.$Explain!) Note that in first best, $c$ should be chosen to satisfy $\left( y_{n}+c\right) \beta_{c}=1-\beta$ since the marginal gain in aggregate income is $\left( y_{n}+c\right)$ and the cost is $1-\beta.$ The individual instead gains, $y_{n}\left( 1-\tau\right) +c-b\text{ }%$ so the private value of search is lower. Similarly, an increase in $y_{n}$ has benefits $\beta$ and costs $-\left( y_{n}+c\right) \beta_{y_{n}}.$ While private benefits are $\left( 1-\tau\right) \beta$ and private costs $-\left( y_{n}\left( 1-\tau\right) +c-b\right) \beta_{y_{n}}.$ We can now write% \begin{align*} V & =u\left( 1-\tau-s\right) +\alpha u\left( 1-\tau+s\right) +\left( 1-\alpha\right) \left( u\left( \left( 1-\beta\right) \left( \mu \tau-c\right) +\beta y_{n}\left( 1-\tau\right) +s\right) \right) \\ V & =V\left( c,y_{n},s,\mu,\tau\right) \end{align*} The optimal UI system maximizes solves $\max_{\tau}V\left( c,y_{n},s,\mu\left( \tau\right) ,\tau\right)$ Although, $c,y_{n},s$ are affected by $\tau,$ these effects need not be taken into account since by individual optimality, $V_{c}=V_{Y_{n}}=V_{s}=0.$ This is the envelope theorem. Therefore, the first order condition for maximizing $V$ by choosing $\tau$ is $\frac{dV}{d\tau}=V_{\mu}\frac{d\mu}{d\tau}+V_{\tau}=0,$ where% \begin{align*} V_{\mu} & =\left( 1-\alpha\right) u^{\prime}\left( c_{u}\right) \left( 1-\beta\right) \tau\\ V_{\tau} & =-u^{\prime}\left( c_{1}\right) -au^{\prime}\left( c_{2}\right) -\left( 1-\alpha\right) u^{\prime}\left( c_{u}\right) \beta y_{n}+\left( 1-\alpha\right) u^{\prime}\left( c_{u}\right) \left( 1-\beta\right) \mu, \end{align*} where $c_{1}=1-\tau-s$ is first period consumption, $c_{2}=1-\tau+s$ is second period consumption if the job is retained and $c_{u}=\left( 1-\beta\right) \left( \mu\tau-c\right) +\beta y_{n}\left( 1-\tau\right) +s$ is second period consumption if the individual lost his job. Note that by individual savings optimization (the Euler equation)% \begin{align*} u^{\prime}\left( c_{1}\right) & =au^{\prime}\left( c_{2}\right) +\left( 1-\alpha\right) u^{\prime}\left( c_{u}\right) \\ u^{\prime}\left( c_{1}\right) -\left( 1-\alpha\right) u^{\prime}\left( c_{u}\right) & =au^{\prime}\left( c_{2}\right) \end{align*} implying \begin{align*} V_{\tau} & =-u^{\prime}\left( c_{1}\right) -\left( u^{\prime}\left( c_{1}\right) -\left( 1-\alpha\right) u^{\prime}\left( c_{u}\right) \right) -\left( 1-\alpha\right) u^{\prime}\left( c_{u}\right) \beta y_{n}+\left( 1-\alpha\right) u^{\prime}\left( c_{u}\right) \left( 1-\beta\right) \mu\\ & =-2u^{\prime}\left( c_{1}\right) +\left( 1-\alpha\right) \left( 1-\beta y_{n}+\left( 1-\beta\right) \mu\right) u^{\prime}\left( c_{u}\right) . \end{align*} Approximating $u^{\prime}\left( c_{1}\right) \approx u^{\prime}\left( c_{u}\right) +u^{\prime\prime}\left( c_{u}\right) \Delta c$ where $\Delta c\equiv c_{1}-c_{u}$ is the fall in consumption if becoming unemployed. The first order condition is then \begin{align*} 0 & =\left( 1-\alpha\right) u^{\prime}\left( c_{u}\right) \left( 1-\beta\right) \tau\frac{d\mu}{d\tau}-2\left( u^{\prime}\left( c_{u}\right) +u^{\prime\prime}\left( c_{u}\right) \Delta c\right) \\ & +\left( 1-\alpha\right) \left( 1-\beta y_{n}+\left( 1-\beta\right) \mu\right) u^{\prime}\left( c_{u}\right) \\ 2\left( 1+\frac{u^{\prime\prime}}{u^{\prime}}\Delta c\right) & =\left( 1-\alpha\right) \left( 1-\beta\right) \tau\frac{d\mu}{d\tau}+\left( 1-\alpha\right) \left( 1-\beta y_{n}+\left( 1-\beta\right) \mu\right) \\ 2\left( 1+\frac{u^{\prime\prime}}{u^{\prime}}\Delta c\right) & =\left( 1-\alpha\right) \left( 1-\beta\right) \tau\frac{d\mu}{d\tau}+\left( 1-\alpha\right) \left( 1-\beta y_{n}+\left( 1-\beta\right) \frac {Y}{\left( 1-\alpha\right) \left( 1-\beta\right) }\right) \\ 2\left( 1+\frac{u^{\prime\prime}}{u^{\prime}}\Delta c\right) & =\left( 1-\alpha\right) \left( 1-\beta\right) \tau\frac{d\mu}{d\tau}+\left( 1-\alpha\right) \left( 1-\beta y_{n}+\frac{Y}{\left( 1-\alpha\right) }\right) \\ 2\left( 1+\frac{u^{\prime\prime}}{u^{\prime}}\Delta c\right) & =\left( 1-\alpha\right) \left( 1-\beta\right) \tau\frac{d\mu}{d\tau}+\left( 1-\alpha\right) \left( 1-\beta y_{n}+\frac{1+\alpha+\left( 1-\alpha\right) \beta y_{n}}{\left( 1-\alpha\right) }\right) \\ 2\left( 1+\frac{u^{\prime\prime}}{u^{\prime}}\Delta c\right) & =\left( 1-\alpha\right) \left( 1-\beta\right) \tau\frac{d\mu}{d\tau}+2\\ \frac{u^{\prime\prime}}{u^{\prime}}\Delta c & =\left( 1-\alpha\right) \left( 1-\beta\right) \tau\frac{d\mu}{d\tau}\\ \frac{u^{\prime\prime}}{u^{\prime}}\frac{\Delta c}{Y} & =\frac{\tau}{\mu }\frac{d\mu}{d\tau}\\ \frac{\Delta c}{c} & =\frac{E_{\mu,t}}{-R_{r}}Y \end{align*} where $E_{\mu,t}$ is the elasticity of $\mu$ with respect to taxes and $R_{r}$ the relative risk aversion coefficient. Note that we should not interpret $Y$ as the aggregate \emph{level }\ of income since we have normalized the pre-unemployment income to unity. Assuming that $y_{n}\approx1,Y\approx 1+\alpha+\left( 1-\alpha\right) \beta$ which is the time people work. In this simple model, this is value is overstated since no unemplyment occur in the first period. More realistically, it should be close to one. Without moral hazard, $\frac{d\mu}{d\tau}=0,$ in which case optimality requires $\Delta c=0.$ With moral hazard, higher taxes tends to reduce $\mu$ since the tax dependency ratio falls. $\frac{\tau}{\mu}\frac{d\mu}{d\tau }=E_{\mu,t}$ is thus negative. Therefore, $\frac{\Delta c}{c}>0.$ We see that $\frac{\Delta c}{c}$ increases if $\frac{\tau}{\mu}\frac{d\mu}{d\tau}$ is large in absolute terms and falls if risk aversion is large. Baily claims that $E_{\mu,t}$ is in the order .15-.4. This approach has been generalized by Chetty showing that we can have repeated spells of unemployment, uncertain spells of unemployment, value of leisure, private insurance and borrowing constraints. The model can therefore be extended to evaluate UI reforms. With a more dynamic model, and in particular if capital markets are imperfect, it should be noted that one needs how the whole consumption profile is affected by unemployment. The drop at entering unemployment may not be enough. Shimer and Werning (2007), shows that the \emph{reservation wage} can be used as a summary measure of how bad unemployment is. In any case, this the model is not suitable to analyze \begin{enumerate} \item General equilibrium effects like impacts on wages, search spillovers and job creation. \item Interaction with other taxes-fiscal spillovers. \item Time varying benefits. \end{enumerate} \subsection{The dynamic approach with observable savings} The seminal paper by Shavel \& Weiss (1979) focuses on the optimal time profile of benefits. It is a simple infinite horizon discrete time model where the aim is to maximize utility of a representative unemployed subject to a government budget constraint. Utility is given by $\sum_{t=0}^{\infty}\left( \frac{1}{1+r}\right) ^{t}\left( u\left( c_{t}\right) -e_{t}\right)$ where $c_{t}$ is period $t$ consumption and $e_{t}$ is a privately chosen unobservable effort associated with job search. The subjective discount rate is $r,$ which is assumed to coincide with an exogenous interest rate. It is assumed that the individual has no access to capital markets so $c_{t}=b_{t}$ when the individual is unemployed. After regaining employment, the wage is $w$ forever. When the individual becomes employed he stays employed for ever for simplicity. Agents have no access to credit markets (or equivalently, savings is perfectly monitored and benefits can be made contingent on them) so the planner can perfectly control the consumption of the individual. The mortal hazard problem is that individuals can affect the probability of finding a job. As in Baily (1978), the individual controls both the search effort (here called $e_{t})$ and the reservation wage (here $w_{t}^{\ast}$ ). Given an effort level $e_{t},$ the individual receives one job offer per period with an associated wage drawn from a distribution with a time invariant probability density $f\left( w_{t},e_{t}\right) .$ The probability of finding an acceptable job in period $t$ is thus $p\left( w_{t}^{\ast},e_{t}\right) =\int_{w_{t}^{\ast}}^{\infty}f\left( w_{t},e_{t}\right) dw_{t}%$ with \begin{align*} p_{w}\left( w_{t}^{\ast},e_{t}\right) & =-f\left( w_{t},e_{t}\right) \leq0\text{ and }\\ p_{e}\left( w_{t}^{\ast},e_{t}\right) & >0 \end{align*} where the latter is by assumption. Let $E_{t}$ be the expected utility of an unemployed individual that choose optimally a sequence $\left\{ e_{t+s},w_{t+s}^{\ast}\right\} _{s=0}^{\infty }.$ Define $u_{t}=\tilde{u}\left( w_{t}^{\ast},e_{t}\right) \equiv\frac{1+r}{r}% \int_{w_{t}^{\ast}}^{\infty}u\left( w_{t}\right) \frac{f\left( w_{t}% ,e_{t}\right) }{p\left( w_{t}^{\ast},e_{t}\right) }dw_{t}%$ This is the expected utility from next period, \emph{conditional }on finding a job this period, which starts next period.\emph{\ }We note that\emph{\ }% \begin{align*} \tilde{u}_{w}\left( w_{t}^{\ast},e_{t}\right) & \geq0\\ \tilde{u}_{e}\left( w_{t}^{\ast},e_{t}\right) & \geq0. \end{align*} The first inequality follows from the fact that \emph{conditional} on finding a job, wages are higher for higher reservation wages. The second inequality is by assumption, higher search effort leads to no worse acceptable job offers. $E_{t}$ satisfies the standard Bellman equation% $E_{t}=\max_{e_{t},w_{t}^{\ast}}u\left( b_{t}\right) -e_{t}+\frac{1}% {1+r}\left( p\left( w_{t}^{\ast},e_{t}\right) \tilde{u}\left( w_{t}^{\ast },e_{t}\right) +\left( 1-p\left( w_{t}^{\ast},e_{t}\right) \right) E_{t+1}\right)$ The first-order conditions are% \begin{align*} e_{t};\frac{1}{1+r}\left( p_{e}\left( w_{t}^{\ast},e_{t}\right) \left( \tilde{u}\left( w_{t}^{\ast},e_{t}\right) -E_{t+1}\right) +p\left( w_{t}^{\ast},e_{t}\right) \tilde{u}_{e}\left( w_{t}^{\ast},e_{t}\right) \right) & =1\\ w_{t}^{\ast};-p_{w}\left( w_{t}^{\ast},e_{t}\right) \left( \tilde{u}\left( w_{t}^{\ast},e_{t}\right) -E_{t+1}\right) & =p\left( w_{t}^{\ast}% ,e_{t}\right) \tilde{u}_{w}\left( w_{t}^{\ast},e_{t}\right) . \end{align*} In the first equation, the LHS is the marginal benefit of higher search effort, coming from a higher probability of finding a job and better jobs if found. These balances the cost which is 1. In the second equation, the LHS is the marginal cost of higher reservation wages, coming from a lower probability of finding a job. The RHS is the gain, coming from better jobs if accepted. By the envelope theorem% $\frac{dE_{t}}{dE_{t+1}}=\frac{\partial E_{t}}{\partial E_{t+1}}=\frac {1-p\left( w_{t}^{\ast},e_{t}\right) }{1+r}%$ Now, anything that reduce $E_{t+1}$ will reduce $1-p\left( w_{t}^{\ast}% ,e_{t}\right) ,$ i.e., make hiring more likely. To see this, note that if $E_{t+1}$ falls. \begin{align*} & p_{e}\left( w_{t}^{\ast},e_{t}\right) \left( \tilde{u}\left( w_{t}% ^{\ast},e_{t}\right) -E_{t+1}\right) +p\left( w_{t}^{\ast},e_{t}\right) \tilde{u}_{e}\left( w_{t}^{\ast},e_{t}\right) ,\text{ and}\\ & -p_{w}\left( w_{t}^{\ast},e_{t}\right) \left( u\left( w_{t}^{\ast}% ,e_{t}\right) -E_{t+1}\right) \end{align*} both becomes larger if choices are unchanged. In words, the marginal benefit of searching higher and the marginal cost of setting higher reservation wages both increase. Thus, a reduction in $E_{t+1}$ increase search effort and reduce the reservation wage increasing $p$. Now, we can show that benefits should have a decreasing profile. Proof: Suppose contrary that $b_{t}=b_{t+1}.$ Then consider an infinitessimal increase in $b_{t}$ financed by an actuarially fair reduction in $b_{t+1}$, that is $db_{t}=-\frac{1-p}{1+r}db_{t+1}>0$ where $p\left( w_{t}^{\ast},e_{t}\right)$ is calculated at the initial (constant) benefit levels. The direct effect on felicitity levels (period utilities) is \begin{align*} & u^{\prime}\left( b_{t}\right) db_{t}+\frac{1-p}{1+r}u^{\prime}\left( b_{t+1}\right) db_{t+1}\\ & -u^{\prime}\left( b_{t}\right) \frac{1-p}{1+r}db_{t+1}+\frac{1-p}% {1+r}u^{\prime}\left( b_{t+1}\right) db_{t+1}\\ & =0 \end{align*} since $u^{\prime}\left( b_{t}\right) =u^{\prime}\left( b_{t+1}\right) .$By the envelope theorem, we need not take into account changes in endogenous variables when calculating welfare. Therefore, $E_{t}$ is unchanged. Since $u\left( b_{t}\right)$ has increased, $E_{t+1}$ must have fallen. When calculating the budgetary effects we need to into account the endogenous changes on $p.$ Let $B_{t}=b_{t}+\frac{1-p}{1+r}b_{t+1}%$ Then,% \begin{align*} dB_{t} & =db_{t}+\frac{1-p}{1+r}db_{t+1}-\frac{dp}{1+r}b_{t+1}\\ & =-\frac{dp}{1+r}b_{t+1}% \end{align*} Since $E_{t+1}$ has fallen, $dp>0.$ Thus $dB_{t}<0.$ I.e., the cost of providing utility $E_{t}$ has fallen. Equivalently, the insurance is more efficient than the starting point $b_{t}=b_{t+1}.$ \subsubsection{Extensions} Hopenhayn and Nicolini extend the model by Shavel \& Weiss in an important dimension -- it enriches the policy space of the government by allowing taxation of workers to be contingent on their unemployment history. It is shown that the government should use this extra way of "punishing" unemployment. The intuition is that relative to the first best, which is a constant unemployment benefit, the government must "punish" unemployment. Doing this by only reducing unemployment benefits is suboptimal, by spreading the punishment of unsuccessful search over the entire future of the individual, a more efficient insurance can be achieved. I.e., lower cost of providing a given utility level. It is shown that this may be quantitatively important. Another contribution is to show that the problem can be formulated in a recursive way with the \emph{promised utility }as state variable. Using H\&N's notation, we assume that individuals can choose an unobservable effort level $a_{t}$ that positively effect the hiring probability. In H\&N 1997, it is assumed that $p\left( a_{t}\right)$ is an concave and increasing function and hiring is an absorbing state with a wage $w$ forever. In H\&N 2005, it is instead assumed that spells are repeated, with an exogenous separation probability $s$ and $p\left( a\right) =\left\{ \begin{array} [c]{c}% p\text{ if }a=1\\ 0\text{ otherwise}% \end{array} \right.$ which is the assumption we make here. The individual has a utility function $E\sum_{t=0}^{\infty}\left( \frac{1}{1+r}\right) ^{t}\left( u\left( c_{t}\right) -a_{t}\right) .$ Let $\theta_{t}\in\left\{ 0,1\right\}$ be the employment status of the individual in period $t,$ where $\theta_{t}=1$ represents employment. Let $\theta^{t}=\left( \theta_{0},\theta_{1},...\theta_{t}\right)$ be the history of the agent up until period $t.$ The history of a person that is unemployed in period $t$ is therefore $\theta^{t-1}\times0=\left( \theta _{0},\theta_{1},...\theta_{t},0\right) \equiv\theta_{u}^{t}$, and similarly, $\theta^{t-1}\times1\equiv\theta_{e}^{t}).$ An allocation is now defined as a rule that assigns consumption and effort as a function of $\theta^{t}$ at every point in time and for every possible history, $c_{t}=c\left( \theta^{t}\right) .$ We focus on allocations where $a_{t}=1$. Individuals must be induced to volontarily choose $a_{t}=1.$ Allocations that satisfies this are called incentive compatible allocations. Given an allocation we can compute the expected discouted utililty at every point in time for every possible history, $V_{t}=V\left( \theta^{t}\right) .$ The problem is now to choose the allocation that minimizes the cost of giving some fixed initial utility level to the representative individual. This problem can be written in a recursive way. In period zero, the planner gives a consumption level $c_{0},$ prescribes an effort level $a_{0}$ (=1) and promised continuation utilities $V_{1}^{e}$ and $V_{1}^{u}.$ The problem of the planner in period zero is to minimize costs of providing a given expected utility level $V_{0}$ subject to the incentive constraint the individual voluntarily chooses $a_{0}.$ The problem is recursive and at any node, costs of providing promised utilities are minimized given incentive constraints The problem of the unemployed individual is also recursive. -- as unemployed, maximized utility is (the agent only controls $a_{t})$ $V\left( \theta_{u}^{t}\right) =u\left( c_{t}\right) -1+\frac{1}% {1+r}\left( pV\left( \theta_{u}^{t}\times1\right) +\left( 1-p\right) V\left( \theta_{u}^{t}\times0\right) \right)$ with the incentive constraint% $\frac{1}{1+r}p\left( V\left( \theta_{e}^{t+1}\right) -V\left( \theta _{u}^{t+1}\right) \right) \geq1.$ Define $W\left( V_{t}\right)$ as the minimum cost for the planner to provide a given amount of utility $V_{t}$ to an employed. Similarly, let $C\left( V_{t}\right)$ denote the minimal cost of providing utility $V$ to an unemployed (are these function changing over time?). $W$ satisfies% \begin{align*} W\left( V_{t}\right) & =\min_{c_{t},V_{t+1}^{e},V_{t+1}^{u}}c_{t}% -w+\frac{1}{1+r}\left( \left( 1-s\right) W\left( V_{t+1}^{e}\right) +sC\left( V_{t+1}^{u}\right) \right) \\ s.t.V_{t} & =u\left( c_{t}\right) +\frac{1}{1+r}\left( \left( 1-s\right) V_{t+1}^{e}+sV_{t+1}^{u}\right) , \end{align*} where$V_{t}=V\left( \theta_{e}^{t}\right) ,$ $c_{t}=c\left( \theta_{e}% ^{t}\right) ,$ $V_{t+1}^{e}=V\left( \theta_{e}^{t}\times1\right)$ and $V_{t+1}^{u}=V\left( \theta_{e}^{t}\times0\right) .$ The constraint can be called promise keeping constraint and has a Lagrange multiplier $\delta_{t}^{e}.$ $C$ satisfies% \begin{gather*} C\left( V_{t}\right) =\min_{c_{t},V_{t+1}^{e},V_{t+1}^{u}}c_{t}+\frac {1}{1+r}\left( pW\left( V_{t+1}^{e}\right) +\left( 1-p\right) C\left( V_{t+1}^{u}\right) \right) \\ \text{s.t. }\frac{1}{1+r}p\left( V_{t+1}^{e}-V_{t+1}^{u}\right) \geq1,\\ V_{t}=u\left( c_{t}\right) -1+\frac{1}{1+r}\left( pV_{t+1}^{e}+\left( 1-p\right) V_{t+1}^{u}\right) . \end{gather*} where $V_{t}=V\left( \theta_{u}^{t}\right) ,$ $c_{t}=c\left( \theta_{u}% ^{t}\right) ,$ $V_{t+1}^{e}=V\left( \theta_{u}^{t}\times1\right)$ and $V_{t+1}^{u}=V\left( \theta_{u}^{t}\times0\right) .$ The first constraint is the incentive constraint, with an associated Lagrange multiplier $\gamma_{t}$ and the second is the promised utility with Lagrange multiplier $\delta_{t}^{u}$.\footnote{Note that the Lagrange multipliers depends on the history $\theta_{t}.$} Given that $u\left( c_{t}\right)$ is concave and $u^{-1}\left( V_{t}\right)$ therefore is convex, it is straightforward to show that $C$ and $W$ are convex functions. First order conditions when the agent is employed are% \begin{align} 1 & =\delta_{t}^{e}u^{\prime}\left( c_{t}\right) \label{eq_FOC_employedH&N}% \\ W^{\prime}\left( V_{t+1}^{e}\right) & =\delta_{t}^{e}\nonumber\\ C^{\prime}\left( V_{t+1}^{u}\right) & =\delta_{t}^{e}.\nonumber \end{align} The envelope condition is $W^{\prime}\left( V_{t}\right) =\delta_{t}^{e}=\frac{1}{u^{\prime}\left( c_{t}\right) }=W^{\prime}\left( V_{t+1}^{e}\right) =C^{\prime}\left( V_{t+1}^{u}\right) .$ The fact that $W^{\prime}\left( V_{t}\right) =W^{\prime}\left( V_{t+1}% ^{e}\right)$ implies that nothing change for the employed individual as long as his remains employed. In fact, his consumption does not upon loosing his job either. This is due to the fact that there is no moral hazard problem on the job and full insurance is therefore optimal.\footnote{From now, I will mostly skip writing out the explicit dependence on history, hopefully without creating confusion.} When the agent is unemployed, the FOC and envelope conditions are% \begin{align*} 1 & =\delta_{t}^{u}u^{\prime}\left( c_{t+1}\right) \\ W^{\prime}\left( V_{t+1}^{e}\right) & =\gamma_{t}+\delta_{t}^{u}\\ \left( 1-p\right) C^{\prime}\left( V_{t+1}^{u}\right) & =-\gamma _{t}p+\delta_{t}^{u}\left( 1-p\right) \\ C^{\prime}\left( V_{t}\right) & =\delta_{t}^{u}. \end{align*} Giving% \begin{align} C^{\prime}\left( V_{t}\right) & =\frac{1}{u^{\prime}\left( c_{t}\right) }\label{eq_FOCH&N}\\ W^{\prime}\left( V_{t+1}^{e}\right) & =\frac{1}{u^{\prime}\left( c_{t}\right) }+\gamma_{t}\nonumber\\ C^{\prime}\left( V_{t+1}^{u}\right) & =\frac{1}{u^{\prime}\left( c_{t}\right) }-\gamma_{t}\frac{p}{1-p}\nonumber \end{align} \textbf{Results} Since the incentive constraint will bind\footnote{Prove that it must by assuming that it doesn't and derive the implications of that.}, $\gamma_{t}>0$ and therefore% $W^{\prime}\left( V_{t+1}^{e}\right) >C^{\prime}\left( V_{t}\right) >C^{\prime}\left( V_{t+1}^{u}\right) .$ The result $C^{\prime}\left( V_{t}\right) >C^{\prime}\left( V_{t+1}% ^{u}\right)$ and the convexity of $C$ implies that the unemployed should be made successively worse off (V_{t+1}^{u}\frac{1}{E_{t}u^{\prime }\left( c_{t+1}\right) }\Rightarrow\frac{1}{E_{t}\frac{1}{u^{\prime}\left( c_{t+1}\right) }}0, \end{align*} \emph{if and only if there is some uncertainty in }c_{2}^{\ast}.$Note that this uncertainty would come from second period ability being random and the allocation implying that second period consumption depends on the realization of ability. If second period ability is non-random, i.e.,$\pi_{2}\left( j|i\right) =1$for some$j,$then$\tau_{k}\left( i\right) =0.$\subsection{A simple logarithmic example: insurance against low ability.} Suppose in the first period, ability is unity and in the second$\theta>1$or$\frac{1}{\theta}$with equal probability$.$Disregard government consumption -- set$G_{1}=G_{2}=0, although non-zero spending is quite easily handled. The problem is therefore to provide a good insurance against a low-ability shock when this is not observed. The first best allocation is the solution to \begin{align*} & \max_{c_{1},y_{1},c_{h},c_{l},y_{h},y_{l}}u\left( c_{1}\right) +v\left( y_{1}\right) +\beta\left( \frac{u\left( c_{h}\right) +v\left( \frac {y_{h}}{\theta}\right) }{2}+\frac{u\left( c_{l}\right) +v\left( \frac{y_{l}}{\frac{1}{\theta}}\right) }{2}\right) \\ s.t.0 & =y_{1}+\frac{y_{h}+y_{l}}{2R}-c_{1}-\frac{c_{h}+c_{l}}{2R}% \end{align*} First order conditions are \begin{align*} u^{\prime}\left( c_{1}\right) & =\lambda\\ v^{\prime}\left( y_{1}\right) & =-\lambda\\ \beta u^{\prime}\left( c_{h}\right) & =\frac{\lambda}{R}\\ \beta u^{\prime}\left( c_{l}\right) & =\frac{\lambda}{R}\\ \beta v^{\prime}\left( \frac{y_{h}}{\theta}\right) \frac{1}{\theta} & =-\frac{\lambda}{R}\\ \beta v^{\prime}\left( \theta y_{l}\right) \theta & =-\frac{\lambda}{R}% \end{align*} \subsubsection{A simple example} Suppose for example thatu\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 thatc_{1}=c_{h}=c_{l}$while$y_{h}=\theta^{2}y_{1}$and$y_{l}=\frac{y_{1}}{\theta^{2}}$and$y_{1}=\sqrt{\frac{2}{\left( 1+\frac {1}{2}\left( \theta^{2}+\theta^{-2}\right) \right) }}=n_{1}.$Therefore,$n_{h}=\frac{y_{h}}{\theta}=\theta^{2}n_{1}$and$n_{l}=y_{l}\theta =\frac{n_{1}}{\theta}.$Thus, if the individual becomes of high ability in the second period, he should work more but don't get any higher consumption. Is this incentive compatible? We conjecture that the binding incentive constraint is for the high ability type. High has to be given sufficient consumption to make him voluntarily choose not to report being low ability. If he misreports, he gets$c_{l}$and is asked to produce$y_{l}.The constraint is therefore% \begin{align*} & u\left( c_{1}\right) +v\left( y_{1}\right) +\beta\left( \frac{u\left( c_{h}\right) +v\left( \frac{y_{h}}{\theta}\right) }{2}+\frac{u\left( c_{l}\right) +v\left( \theta y_{l}\right) }{2}\right) \\ & \geq u\left( c_{1}\right) +v\left( y_{1}\right) +\beta\left( \frac{u\left( c_{l}\right) +v\left( \frac{y_{l}}{\theta}\right) }{2}% +\frac{u\left( c_{l}\right) +v\left( \theta y_{l}\right) }{2}\right) \end{align*}% \begin{align*} u\left( c_{h}\right) +v\left( \frac{y_{h}}{\theta}\right) & \geq u\left( c_{l}\right) +v\left( \frac{y_{l}}{\theta}\right) \\ \ln c_{h}-\ln c_{l} & \geq\frac{y_{h}^{2}-y_{l}^{2}}{2\theta^{2}}% \end{align*} We conjecture this is binding. The problem is then \begin{align*} & \max_{c_{1},y_{1},c_{h},c_{l},y_{h},y_{l}}\ln\left( c_{1}\right) -\frac{y_{1}^{2}}{2}+\left( \frac{\ln c_{h}-\frac{\left( \frac{y_{h}}% {\theta}\right) ^{2}}{2}}{2}+\frac{\ln c_{l}-\frac{\left( \theta y_{l}\right) ^{2}}{2}}{2}\right) \\ s.t.0 & =y_{1}+\frac{y_{h}+y_{l}}{2}-c_{1}-\frac{c_{h}+c_{l}}{2}\\ 0 & =\ln c_{h}-\ln c_{l}-\frac{y_{h}^{2}-y_{l}^{2}}{2\theta^{2}}. \end{align*} Denoting the shadow values by\lambda_{r}$and$\lambda_{I}the FOCs for the consumption levels are% \begin{align*} c_{1} & =\frac{1}{\lambda_{r}}\\ c_{h} & =\frac{1+2\lambda_{I}}{\lambda_{r}}\\ c_{l} & =\frac{1-2\lambda_{I}}{\lambda_{r}}% \end{align*} from which we see $\frac{c_{h}^{\ast}}{c_{1}^{\ast}}=1+2\lambda_{I},\frac{c_{l}^{\ast}}% {c_{1}^{\ast}}=1-2\lambda_{I}%$ and $\tau_{k}\left( i\right) \equiv1-\frac{\lambda_{r}}{\frac{\lambda_{r}% }{1+2\lambda_{I}}\frac{1}{2}+\frac{\lambda_{r}}{1-2\lambda_{I}}\frac{1}{2}% }=\left( 2\lambda_{I}\right) ^{2},$ : implying a positive intertemporal wedge if the IC constraint binds. The intratemporal wedges are found by analyzing the FOC's for the labor supplies% \begin{align*} y_{1}^{\ast} & =\lambda_{r}\\ y_{h}^{\ast} & =\frac{\lambda_{r}}{1+2\lambda_{I}}\theta^{2}\Rightarrow y_{h}^{\ast}c_{h}^{\ast}=\frac{\lambda_{r}}{1+2\lambda_{I}}\theta^{2}% \frac{1+2\lambda_{I}}{\lambda_{r}}=\theta^{2}\\ y_{l}^{\ast} & =\frac{\lambda_{r}}{\theta^{4}-2\lambda_{I}}\theta ^{2}\Rightarrow y_{l}^{\ast}c_{l}^{\ast}=\frac{\lambda_{r}}{\theta ^{4}-2\lambda_{I}}\theta^{2}\frac{1-2\lambda_{I}}{\lambda_{r}}=\frac {1-2\lambda_{I}}{\theta^{2}\left( 1-2\lambda_{I}\theta^{-4}\right) }% \end{align*} % \begin{align*} \tau_{y_{1}} & =1-\frac{y_{1}^{\ast}}{\frac{1}{c_{1}^{\ast}}}=0,\\ \tau_{y_{2}}\left( h\right) & =1+\frac{v^{\prime}\left( \frac{y_{h}% ^{\ast}}{\theta}\right) }{\theta u^{\prime}\left( c_{h}^{\ast}\right) }=1+\frac{-\frac{y_{h}^{\ast}}{\theta}}{\theta\frac{1}{c_{h}^{\ast}}}\\ & =1+\frac{-\frac{\frac{\lambda_{r}}{1+2\lambda_{I}}\theta^{2}}{\theta}% }{\theta\frac{1}{\frac{1+2\lambda_{I}}{\lambda_{r}}}}=0 \end{align*} and % \begin{align*} \tau_{y_{2}}\left( l\right) & =1+\frac{v^{\prime}\left( \theta y_{l}^{\ast}\right) }{\frac{1}{\theta}u^{\prime}\left( c_{l}^{\ast}\right) }=1+\frac{-\theta y_{l}^{\ast}}{\frac{1}{\theta}\frac{1}{c_{h}^{\ast}}}\\ & =1+\frac{-\theta\frac{\lambda_{r}}{\theta^{4}-2\lambda_{I}}\theta^{2}}% {\frac{1}{\theta}\frac{1}{\frac{1-2\lambda_{I}}{\lambda_{r}}}}=2\lambda _{I}\frac{\theta^{4}-1}{\theta^{4}-2\lambda_{I}}>0 \end{align*} As we see, the wedge for the high ability types is zero, but positive for the low ability type.\footnote{The wedge, asymptotes to infinity as\lambda_{I}$approach$\frac{\theta^{4}}{2}.Can you explain?} For later use, we note that % \begin{align} y_{1}^{\ast}c_{1}^{\ast} & =1\label{ystarcstar}\\ y_{h}^{\ast}c_{h}^{\ast} & =\frac{\lambda_{r}}{1+2\lambda_{I}}\theta^{2}% \frac{1+2\lambda_{I}}{\lambda_{r}}=\theta^{2}\nonumber\\ y_{l}^{\ast}c_{l}^{\ast} & =\frac{\lambda_{r}}{\theta^{4}-2\lambda_{I}}% \theta^{2}\frac{1-2\lambda_{I}}{\lambda_{r}}=\frac{1-2\lambda_{I}}{\theta ^{2}\left( 1-2\lambda_{I}\theta^{-4}\right) }\nonumber \end{align} \subsection{Implementation} It is tempting to interpret the wedges as taxes and subsidies. However, this is not entirely correct since the wedges in general are functions of all taxes. Furthermore, while there is typically a unique set of wedges this is generically not true for the taxes. As we have discussed above, many different tax systems might implement the optimal allocation. One example is the draconian, use 100\% taxation for every choice except the optimal ones. Only by putting additional restrictions is the implementing tax system found. Let us consider a combination if linear labor taxes and savings taxes that together with type specific transfers implement the allocation in the example. To do this, consider the individual problem,% \begin{align*} & \max_{c_{1},y_{1},s,y_{h},y_{l},c_{h},c_{l}}\ln\left( c_{1}\right) -\frac{y_{1}^{2}}{2}+\left( \frac{\ln c_{h}-\frac{\left( \frac{y_{h}}% {\theta}\right) ^{2}}{2}}{2}+\frac{\ln c_{l}-\frac{\left( \theta y_{l}\right) ^{2}}{2}}{2}\right) \\ s.t.0 & =y_{1}\left( 1-\tau_{1}\right) -c_{1}-s+T\\ 0 & =y_{h}\left( 1-\tau_{h}\right) +s\left( 1-\tau_{s,h}\right) -c_{h}+T_{h}\\ 0 & =y_{l}\left( 1-\tau_{h}\right) +s\left( 1-\tau_{s,l}\right) -c_{l}+T_{l}% \end{align*} with Lagrange multipliers\lambda_{1},\lambda_{h}$and$\lambda_{r}.First order conditions for the indviduals are;% \begin{align} \frac{1}{c_{1}} & =\lambda_{1}\nonumber\\ y_{1} & =\lambda_{1}\left( 1-\tau_{1}\right) \nonumber\\ \lambda_{1} & =\lambda_{h}\left( 1-\tau_{s,h}\right) +\lambda_{l}\left( 1-\tau_{l,h}\right) \nonumber\\ \frac{y_{h}}{2\theta^{2}} & =\lambda_{h}\left( 1-\tau_{h}\right) \label{eq_privFOC}\\ \frac{\theta^{2}y_{l}}{2} & =\lambda_{l}\left( 1-\tau_{l}\right) \nonumber\\ \frac{1}{2c_{h}} & =\lambda_{h}\nonumber\\ \frac{1}{2c_{l}} & =\lambda_{l}\nonumber \end{align} Using this, we see that $\frac{1}{c_{1}}=\frac{1}{2c_{h}}\left( 1-\tau_{s,h}\right) +\frac{1}{2c_{l}% }\left( 1-\tau_{l,h}\right)$ Setting, \begin{align*} \tau_{s,h} & =-2\lambda_{I}\\ \tau_{s,l} & =2\lambda_{I}. \end{align*} this gives $\frac{1}{c_{1}}=\frac{1}{2c_{h}}\left( 1+2\lambda_{I}\right) +\frac {1}{2c_{l}}\left( 1-2\lambda_{I}\right)$ which is satisfied if we plug in the optimal allocationc_{h}^{\ast}% =c_{1}^{\ast}\left( 1+2\lambda_{I}\right) $and$c_{l}^{\ast}=c_{1}^{\ast }\left( 1-2\lambda_{I}\right) $$\frac{1}{c_{1}^{\ast}}=\frac{1+2\lambda_{I}}{2c_{1}^{\ast}\left( 1+2\lambda_{I}\right) }+\frac{1-2\lambda_{I}}{2c_{1}^{\ast}1-2\lambda_{I}}%$ Note that the expected capital income tax rate is zero, but it will make savings lower than without any taxes.\ Why? Similarly, by noting from (\ref{ystarcstar}) that in the optimal second best allocation, we want $y_{1}c_{1}=y_{1}^{\ast}c_{1}^{\ast}=1,$ which is implemented by$\tau_{1}=0.$For the high ability type, the second best allocation in (\ref{ystarcstar}) is that$y_{h}^{\ast}c_{h}^{\ast }=\theta^{2},$which is implemented by$\tau_{h}=0$since (\ref{eq_privFOC}) implies that$y_{h}c_{h}=\theta^{2}\left( 1-\tau_{h}\right) .$For the low ability type, we want$y_{l}^{\ast}c_{l}^{\ast}=\frac {1-2\lambda_{I}}{\theta^{2}\left( 1-2\lambda_{I}\theta^{-4}\right) }$. From (\ref{eq_privFOC}), we know$y_{l}c_{l}=\frac{1-\tau_{l}}{\theta^{2}},so we solve% \begin{align*} \frac{1-\tau_{l}}{\theta^{2}} & =\frac{1-2\lambda_{I}}{\theta^{2}\left( 1-2\lambda_{I}\theta^{-4}\right) }\\ & \Rightarrow\tau_{l}=2\lambda_{I}\frac{\theta^{4}-1}{\theta^{4}-2\lambda_{I}% }. \end{align*} Note that if\lambda_{I}=\frac{1}{2},\tau_{l}=1.$I.e., the tax rate is 100\%. There is no point going higher than that, so$\lambda_{I}$cannot be higher than$\frac{1}{2}.$Finally, to find the complete allocation, we use the budget constraints of the private individual and the aggregate resource constraint. This will recover the transfers$T,T_{h}$and$T_{l}.$We should note that$T_{l}>T_{h}$is consistent with incentive compatibiity. Why? Because if you claim to be a low ability type you will have to may a high labor income tax which is bad if you are high ability and earn a high income. Thus, by taxing high income lower, we can have a transfer system that transfers more to the low ability types. \subsubsection{Third best -- laizzes faire} The allocation in without any government involvements is easily found by setting all taxes to zero$.% \begin{align} \frac{1}{c_{1}} & =\lambda_{1}\nonumber\\ y_{1} & =\lambda_{1}\nonumber\\ \lambda_{1} & =\lambda_{h}+\lambda_{l}\nonumber\\ \frac{y_{h}}{2\theta^{2}} & =\lambda_{h}\label{eq_FOCLaizzesFaire}\\ \frac{\theta^{2}y_{l}}{2} & =\lambda_{l}\nonumber\\ \frac{1}{2c_{h}} & =\lambda_{h}\nonumber\\ \frac{1}{2c_{l}} & =\lambda_{l}\nonumber \end{align} Using these and the budget constraints, we get \begin{align*} y_{1} & =\frac{1}{c_{1}}\\ \frac{1}{c_{1}} & =\frac{1}{2c_{h}}+\frac{1}{2c_{l}}\\ \frac{y_{h}}{2\theta^{2}} & =\frac{1}{2c_{h}}\\ \frac{\theta^{2}y_{l}}{2} & =\frac{1}{2c_{l}}\\ y_{1} & =c_{1}+s\\ y_{h}+s & =c_{h}\\ y_{l}+s & =c_{l}% \end{align*} which implies% \begin{align*} c_{1}+s & =\frac{1}{c_{1}}\\ \frac{1}{c_{1}} & =\frac{1}{2c_{h}}+\frac{1}{2c_{l}}\\ c_{h} & =\frac{1}{2}s+\frac{1}{2}\sqrt{s^{2}+4\theta^{2}}\\ c_{l} & =\frac{\frac{1}{2}s\theta+\frac{1}{2}\sqrt{s^{2}\theta^{2}+4}}% {\theta}% \end{align*} I did not find an analytical solution to this, but setting\theta=1.1$I found the solution$\left\{ c_{1}=0.997\,75,c_{h}=1.\,\allowbreak 102\,3,s=4.\,\allowbreak504\,5\times10^{-3},c_{l}=0.911\,35\right\} .\$ \subsection{Time consistency} Under the Mirrlees approach, the government announces a menu of taxes or of consumption baskets. People then make choices that in equilibrium reveal their true types (abilities) to the government. Suppose the government could then re-optimize. Would it like to do this? The problem is more severe in a dynamic setting provided abilities are persistent. Why? In a finite horizon economy, there might only be very bad equilibria (Roberts, 84). But better equilibria might arise in infinite horizon. \end{document}