\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}<beamer>{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}<beamer>{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}



