\makeatletter \let\ifGm@compatii\relax \makeatother \documentclass[12pt]{beamer} %use for presentation % \documentclass[12pt,handout]{beamer} %use for handouts \setbeameroption{show notes} %uncomment to see notes \usefonttheme{serif} \usefonttheme{structuresmallcapsserif} \newcommand{\its}{\vskip 10pt \item} \newcommand{\pits}{\vskip 10pt \pause \item} \newcommand{\st}{\operatorname{st}} \newcommand{\var}{\operatorname{var}} \newtheorem{prop}{Proposition} \newcommand{\R}{\mathbb{R}} \newcommand{\E}{\mathbb{E}} \newcommand{\BR}{\operatorname{BR}} %\newtheorem{theorem}{Theorem} \newcommand{\un}{\underline} \newcommand{\ov}{\overline} \newcommand{\gp}{\succ} \newcommand{\lp}{\prec} \newcommand{\gep}{\succsim} \newcommand{\lep}{\precsim} \newcommand{\T}{\ensuremath{\mbox{\footnotesize\textsc{occupation}}}} \renewcommand{\S}{\ensuremath{\mbox{\footnotesize\textsc{suicide}}}} %\newcommand{\R}{\ensuremath{\mbox{\footnotesize\textsc{religious}}}} \usepackage{graphicx} \usepackage{ragged2e} % ----------------------------------------- % Center the Frame Title % ----------------------------------------- \setbeamertemplate{frametitle} { \begin{centering} \vspace{0.1in} \insertframetitle \par \end{centering}} % ----------------------------------------- % Number the slides % ----------------------------------------- \setbeamertemplate{footline}[frame number] % ----------------------------------------- % Get rid of the irritating navigation bar % ----------------------------------------- \setbeamertemplate{navigation symbols}{} % ----------------------------------------- % Begin Document % ----------------------------------------- \title[Externalities] % (optional, use only with long paper titles) {Externalities} \author{} % Ethan Bueno de Mesquita\\ \small{Harris School of Public Policy Studies}\\ \small{University of Chicago}} \date[]% (optional, should be abbreviation of conference name) % Delete this, if you do not want the table of contents to pop up at % the beginning of each subsection: \AtBeginSection[] { \begin{frame}{Outline} \tableofcontents[currentsection,currentsubsection] \end{frame} } % Delete this, if you do not want the table of contents to pop up at % the beginning of each subsection: \AtBeginSubsection[] { \begin{frame}{Outline} \tableofcontents[currentsection,currentsubsection] \end{frame} } \begin{document} \begin{frame} \titlepage \end{frame} \section{Introduction} \begin{frame}{Key Ideas} What is an externality? \ Externalities create opportunities for Pareto improving policy \ Externalities require ``active'' and ongoing policy interventions \ The optimal (\emph{second best}) policy intervention involves trade-offs \end{frame} \begin{frame}{Externalities} Person $A$'s action imposes an \emph{externality} on person $B$ if $A$'s action affects $B$'s payoffs \ Suppose Person $A$ can take action $a$ or not take action $a$ \ Person $A$'s action $a$ imposes a \emph{negative externality} on Person $B$ if $B$'s payoff is always lower when $A$ does $a$ than when $A$ does not do $a$ \begin{itemize} \item Positive externalities are defined analogously \end{itemize} \end{frame} \begin{frame}{The General Principle} People do not \emph{internalize} their externalities \begin{itemize} \item Externalities do not influence decision making \end{itemize} \ Relative to the utilitarian optimum \begin{itemize} \item Too little of actions with positive externalities \its Too much of actions with negative externalities \end{itemize} \end{frame} \begin{frame}{The Policy Challenge} Get people to internalize their externalities \begin{itemize} \item Persuasion \its Subsidies/taxes \its Regulation \its Decentralized mechanisms \end{itemize} \ Weight the benefits of internalizing externalities against the costs of the policy intervention \end{frame} \section{Public Goods: Positive Externalities} \begin{frame}{What is a Public Good?} \textbf{Non-excludable:} If I have access to the good, so do you \ \textbf{Non-rival:} My use of the good does not diminish your access to it \end{frame} \begin{frame}{Examples} Clean air \ National defense \ Protection from asteroids \ Knowledge \end{frame} \begin{frame}{The Model} $N > 1$ people each decide how much effort to contribute \ Player $i$'s contribution is $e_i$ \ Public good production technology is additive \[ G = e_1 + e_2 + \ldots + e_N\] \ $i$'s cost of effort is $e_i^2$ \end{frame} \begin{frame}{Best Responses} Player $i's$ best response to $\mathbf{e}_{-i}$ is found by solving: \[ \max_{e_i} \overbrace{e_1 + e_2 + \ldots + e_i + \ldots + e_N}^{\mbox{Public Goods}} - \overbrace{e_i^2}^{\mbox{Costs}}\] \ First-order condition \[ 1 - 2 e_i^* = 0\] \[ \BR_i(\mathbf{e}_{-i}) = \frac{1}{2}\] \ Because of the additive technology, player $i$'s best response actually doesn't depend on what anyone else is doing \begin{itemize} \item This wouldn't be true if efforts were complements \end{itemize} \end{frame} \begin{frame}{Nash Equilibrium} The unique equilibrium is all players choose effort $\frac{1}{2}$ \ Player $i$'s equilibrium payoff is: \[ N \times \frac{1}{2} - \left(\frac{1}{2}\right)^2 = \frac{2N - 1}{4}\] \end{frame} \begin{frame}{The First Best} The profile of efforts that maximize the utilitarian welfare is called the \emph{first best} \ The utilitarian welfare is \[ N \times \sum_{i=1}^N e_i - \sum_{i=1}^N e_i^2\] \ The equilibrium efforts are not the first best because people do not internalize their positive externalities \end{frame} \begin{frame}{Finding the First Best} If each player chooses the same effort ($\hat{e}$), each individual's payoff is: \[ \overbrace{N \times \hat{e}}^{\mbox{Public Goods}} - \overbrace{\hat{e}^2}^{\mbox{Costs}}\] \ The first best solves: \[ \max_{\hat{e}} N \left( N \times \hat{e} - \hat{e}^2\right)\] \pause \[ N\left(N - 2e^{FB}\right) = 0 \] \pause \[e^{FB} = \frac{N}{2}\] \end{frame} \begin{frame}{Social Dilemma} Equilibrium effort by each individual is $\frac{1}{2}$ \begin{itemize} \item Individual payoff is $\frac{2N - 1}{4}$ \end{itemize} \ The first best is for each individual to exert effort $\frac{N}{2}$ \begin{itemize} \item Payoff would be $ N \times \frac{N}{2} - \left(\frac{N}{2}\right)^2 = \frac{N^2}{4}$ \end{itemize} \ Each individual is better off if all individuals exert more effort. This is especially true as society gets big. \end{frame} \begin{frame} \begin{figure} \begin{center} \includegraphics[width = \textwidth]{figures/dilemma.pdf} \end{center} \end{figure} \end{frame} \section{Policy Responses} \begin{frame}{Policy Responses} The fundamental problem is the failure to internalize externalities \ Policy makers must look for instruments to get people to internalize externalities \ They must also pay attention to any unintended consequences of using those instruments \end{frame} \subsection{Persuasion} \begin{frame}{Free Riding} Even though payoffs are higher if everyone does first-best effort, no individual has an incentive to behave in this socially optimal way \ Each individual has an incentive to free ride, even if everyone else does social optimum \ \[ (N-1) \times \frac{N}{2} + \frac{1}{2} - \left(\frac{1}{2}\right)^2 = \frac{2N^2 - 2N +1}{4} > \frac{N^2}{4}\] \end{frame} \begin{frame} \begin{figure} \begin{center} \includegraphics[width=\textwidth]{figures/free-ride.pdf} \end{center} \end{figure} \end{frame} \subsection{Pigovian Subsidies and Taxes} \begin{frame}{Pigovian Subsidies in Public Goods Game} Suppose provide a subsidy, $\sigma$, for each unit of effort \ Subsidy must be funded with inefficient taxation \ It costs tax payers $\tau > 1$ dollars for each dollar of revenue the government collects \ Each member of society pays an equal share of the tax burden to fund subsidies for everyone other than herself \end{frame} \begin{frame}{Best Response to Subsidy} \[ \max_{e_i} e_1 + e_2 + \ldots + e_i(1+\sigma) + \ldots + e_N - e_i^2 - \frac{\tau\sigma \sum_{j\not =i} e_j}{N-1}\] \ First-order condition \[ 1 + \sigma - 2 e_i^* = 0\] \[ \BR_i(\mathbf{e}_{-i}, \sigma) = \frac{1+\sigma}{2}\] \ To achieve the first-best level of effort choose $\sigma = N-1$ \begin{itemize} \item People fully internalize externalities \its This is \emph{not} the socially optimal subsidy \end{itemize} \end{frame} \begin{frame}{Direct Regulation vs.\ Pigovian Intervention} Informational requirements \ Monitoring \ Flexibility \ In either case, requires ongoing intervention \end{frame} \section{The Second Best} \begin{frame}{The Idea} Any policy lever government pulls to address one issue has spillover effects in another domain \ Good policy balances these benefits and costs \ The \emph{second best} policy is the socially optimal intervention, given all the constraints and effects \ When policy effects other domains, it is rarely socially optimal to achieve the first best in the given domain \end{frame} \begin{frame}{An Example} Monopolist in a polluting industry \ Break monopoly to improve consumer surplus \ Also increases production, which imposes externalities \ Optimal policy with respect to market power may not be perfect competition because of constraint from the pollution domain \end{frame} \begin{frame}{Second-Best Defined} The \emph{second-best policy} is the policy that maximizes the utilitarian social welfare, taking into consideration all the various effects of the policy. \ The \emph{second-best action} is the action players are induced to take when the second-best policy is implemented. \end{frame} \begin{frame}{Finding the Second-Best Subsidy} Take the equilibrium effort as a function of the subsidy$, \sigma$, as given \ Given the equilibrium effort induced by a given subsidy, $\sigma$, calculate the utilitarian social welfare as a function of $\sigma$ \ Find the $\sigma$ that maximizes that social welfare \begin{itemize} \item The $\sigma$ that does so is the second-best subsidy and the effort it induces is the second-best effort. \end{itemize} \end{frame} \begin{frame}{Individual Welfare under Subsidy} Recall, given a subsidy $\sigma$, each individual will choose effort \[\BR_i(\textbf{e}_{-i},\sigma) = \frac{1 + \sigma}{2} \equiv e^{*,\sigma} \] \ To fund the subsidy, each individual must pay a taxes of \[\frac{(N-1)\times e^{*,\sigma} \times \sigma \times \tau}{N-1} = e^{*,\sigma}\times \sigma \times \tau\] \ \[ u_i(\sigma) = \overbrace{N \times e^{*,\sigma}}^{\mbox{Public Goods}} + \overbrace{e^{*,\sigma}\times \sigma}^{\mbox{$i$'s subsidy}} - \overbrace{(e^{*,\sigma})^2}^{\mbox{$i$'s cost}} - \overbrace{e^{*,\sigma}\times \sigma \times \tau}^{\mbox{$i$'s tax burden}} \] \end{frame} \begin{frame}{Utilitarian Welfare } Utilitarian welfare from subsidy $\sigma$ is \[U(\sigma) = N\times u_i(\sigma)\] \ Substituting for $e^{*,\sigma} = \frac{1+\sigma}{2}$ \[ \hskip -25pt U(\sigma) = N \bigg[\overbrace{N \left(\frac{1+\sigma}{2} \right)}^{\mbox{Public Goods}} + \overbrace{\left(\frac{1+\sigma}{2} \right) \sigma}^{\mbox{$i$'s subsidy}} - \overbrace{\left(\frac{1+\sigma}{2}\right)^2}^{\mbox{$i$'s cost}} - \overbrace{\left(\frac{1+\sigma}{2} \right)\sigma \tau}^{\mbox{$i$'s tax burden}}\bigg]\] \end{frame} \begin{frame}{The Second Best} The second-best subsidy solves \[ \max_\sigma U(\sigma)\] \ \[ \sigma^{SB}= \frac{N-\tau}{2\tau -1}\] \ \[ e^{*,\sigma^{SB}} = \frac{1 + \frac{N-\tau}{2\tau-1}}{2} = \frac{N-1+\tau}{2(2\tau-1)}\] \end{frame} \begin{frame}{Four Points} The more people, the larger is the second-best subsidy because externalities are more important \ \pause The larger is $\tau$, the smaller is the second-best subsidy because the inefficiencies of funding it loom large \ \pause Second-best subsidy is less than the subsidy that induces the first-best effort \[ \frac{N-\tau}{2\tau - 1} < N-1\] \end{frame} \begin{frame} \begin{figure} \begin{center} \includegraphics[width = .55\textwidth]{figures/sb1.pdf}\\ \vskip 20pt \includegraphics[width = .55\textwidth]{figures/sb2.pdf} \end{center} \end{figure} \end{frame} \begin{frame}{Four Points} The more people, the larger is the second-best subsidy because externalities are more important \ The larger is $\tau$, the smaller is the second-best subsidy because the inefficiencies of funding it loom large \ Second-best subsidy is less than the subsidy that induces the first-best effort \[ \frac{N-\tau}{2\tau - 1} < N-1\] \ For any $\tau > 1$, welfare is higher under second-best subsidy than under either no subsidy or subsidy that induces first-best effort \end{frame} \begin{frame} \begin{figure} \begin{center} \includegraphics[width = \textwidth]{figures/welfare11.pdf} \end{center} \end{figure} \end{frame} \begin{frame} \begin{figure} \begin{center} \includegraphics[width = \textwidth]{figures/welfare22.pdf} \end{center} \end{figure} \end{frame} \section{Take Aways} \begin{frame}{Take Aways} Externalities create Pareto inefficiency \ Policy can improve situation by incentivizing people to (partially) internalize their externalities \ Policy intervention must be active and ongoing \ If policy has other costs, the socially optimal policy (second best) does not achieve complete internalization of externalities \end{frame} \end{document}