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\title{Summing Up Social Dilemmas}


\author{}
% Ethan Bueno de Mesquita\\ \small{Harris School of Public Policy Studies}\\ \small{University of Chicago}}




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\begin{document}

\begin{frame}
  \titlepage
\end{frame}

\begin{frame}
	\begin{figure}
		\includegraphics[width= \textwidth]{figures/summing_up_table.pdf}
	\end{figure}
\end{frame}


\begin{frame}{Social Dilemmas and Governance}

Each of our social dilemmas also happens within government

\

Externalities and interest groups

\

Coordination failure in the bureaucracy

\

Commitment problems and fiscal policy

\

Let's see a couple examples
\end{frame}

\section{Externalities and Interest Groups}

\begin{frame}{A Model of Interest Groups}
	
Factory owner and $N$ citizens invest in lobbying

\

Each hour of lobbying costs \$100 

\

If the citizens do $C$ hours of lobbying and  factory owner does $F$ regulator sides with the citizens with probability 
	\[ \frac{C}{C+F} \]
	
	\
	
	If  citizens win, each  benefits  $b > 0$. If factory oner wins, she  benefits $\pi$
	
	\[b < \pi < N b\]
	\end{frame}
	
	\begin{frame}{Citizen's Best Response}
		
		If citizen $i$ believes other citizens all invest $c$ and owner invests $F$, then solves
		\[ \max_{c_i} \left(\frac{c_i +  (N-1) c}{c_i + (N-1)  c + F} \right) b - 100c_i\]
	
	\
	
			\[ \BR_i(c,F) = \frac{\sqrt{b F}}{10}- F - (N-1)c\]
	
	
	\
	
		Each citizen will make the same contribution 
		\[ \BR_i(F) = \frac{\sqrt{b F}}{10}- F - (N-1)\BR_i(F)\]
		
		\
		
	\[\BR_i(F) = \frac{\sqrt{bF}-10F}{10N}\]		
	\end{frame}
	
	
	\begin{frame}{Factory Owner's Best Response}
		
		If the factory owner believes  citizens  purchase a total of $C$ hours
		\begin{equation*} \max_F \left(\frac{F}{C + F}\right)\pi - 100F\end{equation*}
			
			\
			
				\[ \BR_f(C) = \frac{\sqrt{C\pi} - 10C}{10}.\]
		\end{frame}
		
		\begin{frame}{Equilibrium}
				\[\BR_i(F) = \frac{\sqrt{bF}-10F}{10N}\]	
				
				\
				
			\[ \BR_f(C) = \frac{\sqrt{C\pi} - 10C}{10}.\]
			
		\
		
		\[c^* = \frac{b^2 \pi}{100(b+\pi)^2 N} \ \ \ \ \  \mbox{and} \ \ \ \ \ \ F^* = \frac{b \pi^2}{100(b+\pi)^2}.\]
\end{frame}

\begin{frame}{Who Wins?}
	
		\[C^* =  Nc^* = \frac{b^2 \pi}{100(b+\pi)^2 }\]
	
	\
	
		\[ F^* = \frac{b \pi^2}{100(b+\pi)^2}\]
		
		\
		
		
		Since $\pi > b$, factory owner lobbies more. Citizens win with probability 
		\[ \frac{C^*}{C^* + F^*} = \frac{\frac{b^2 \pi}{100(b+\pi)^2 }}{\frac{b^2 \pi}{100(b+\pi)^2 }+\frac{b \pi^2}{100(b+\pi)^2 }} = \frac{b}{b+\pi} < 1/2.\]
\end{frame}

\begin{frame}{An Example}
	
Suppose $b = 1000$, $N = 100,000$ and $\pi = 1,000,000$

\

Citizens' total value of stopping pollution is \$100,000,000, while  factory owner's value of polluting  is only \$1,000,000

\

Probability citizens win is 
		\[  \frac{1000}{1000+1,000,000} = \frac{1}{1001}.\]
	\end{frame}
	
	\begin{frame}{Concentrated vs.\ Diffuse Interests}
		
		Diffuse interests are hampered by internal externalities problems
		
		\
		
		This makes it hard to organize in support of even very important issues
		
		\
		
		All else equal, concentrated interests (fewer people) are better able to wield political power than concentrated interests
	\end{frame}


\section{Commitment Problems and Fiscal Manipulation}

\begin{frame}{The Model}

Three players: a voter, a left-wing politician, a right-wing politician

\

Two periods

\

 Prior to each period, voter elects a politician
 
 
 \
 
  During each period, there is a   budget of size $1$. 
  
  \
  
  In period 1,  politician in office can borrow $b \in (0,1)$, which must be paid back in period 2 
  \end{frame}
  
  \begin{frame}{Policy}
	  
	  In each period, budget can be spent on right-wing agenda ($R$) or left-wing agenda ($L$)
	 	  
	  \
	  
	 In each period, one of these two agendas is more productive (this is observed before election)	
	 
	 
	 \
	 
	 Value to voter of money spent on the more productive agenda is $\lambda \in \left(\frac{1}{2},1\right)$, while  value of money spent on less productive agenda is $1-\lambda$ 
	 
	 \
	 
	 Politician always values money spent on her agenda at $\lambda$ and other ideological agenda at $1-\lambda$
	 
	
\end{frame}

\begin{frame}{Stakes}
	
	 
		 In  period $t$, the stakes of public policy are $\alpha_t$ (equally likely 
	to be any real number between 0 and 1)

	\

		The value of $\alpha_t$ is observed after the election, but before policy is set
		
		\end{frame}
		
	

\begin{frame}{Optimal Borrowing}

If borrow, expected voter welfare is:

\[ \hskip -20pt U_V(\mbox{borrow}|\alpha_1) =  \overbrace{\alpha_1 \lambda (1+b)}^{\mbox{1st Period Welfare}} + \overbrace{\frac{1}{2}\lambda (1-b)}^{\mbox{Expected 2nd Period Welfare}}\]

\


If don't borrow, expected voter welfare is:
\[ U_V(\mbox{don't borrow}|\alpha_1) = \alpha_1 \lambda + \frac{1}{2}\lambda\]

\

Voter welfare maximized by borrowing if
\[ \alpha_1 > \frac{1}{2}\]
\end{frame}



\begin{frame}{Equilibrium Borrowing}
	
	Politician's expected payoff if she borrows:
\[ U_1(\mbox{borrow}|\alpha_1) = \alpha_1 \lambda (1+b)+ \frac{1}{2}\bigg(p \lambda (1-b) + (1-p)(1-\lambda)(1-b)\bigg)\]

\


Politician's expected payoff if she doesn't borrow:
\[ U_1(\mbox{don't borrow}|\alpha_1) = \alpha_1 \lambda + \frac{1}{2}\bigg(p\lambda + (1-p)(1-\lambda)\bigg)\]

\

Borrow in equilibrium if
\[ \alpha_1 > \frac{p\lambda + (1-p)(1-\lambda)}{2\lambda}\]


\

Politicians borrow too much from future in equilibrium
\end{frame}
		
		
\begin{frame}{Dynamics and Fiscal Distortions}
	
	Because of dynamic concerns, politicians over emphasize the present
	
	\
	
	This can be because of partisan issues (as in our model), various other kinds of risk, individual vs.\ party interests, etc.
	
	\
	
	Think of current problems with unfunded pensions
		\end{frame}	
\end{document}