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\title[Pareto Concepts] % (optional, use only with long paper titles)
{Pareto Concepts}


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




\date[]% (optional, should be abbreviation of conference name)

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\begin{document}

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\begin{frame}{The Project}
	
	
	Consider something much less ambitious than complete agreement on what we mean by good policy
	
	\
	
	Identify limited instances of unequivocally good policy
		\begin{itemize}
			\item Makes some people better off and no one worse off
		\end{itemize}
	\end{frame}




	

\section{Pareto Concepts}


\begin{frame}{Utility Functions}
	
	Each individual's preferences can be represented with a \textbf{utility function}
	
	\
	
	The utility function, $U_i$, represents person $i$'s  preferences if:
		\begin{enumerate}
			\item If person $i$ preferes a policy $x$ to another policy $y$, then $U_i(x) > U_i(y)$
			\its If person $i$ is indifferent between $x$ and $y$, then $U_i(x) = U_i(y)$
			\end{enumerate}
			
	\end{frame}
	
\begin{frame}{Pareto Dominance}
	
	A policy $x$ \emph{\textbf{Pareto dominates}} another policy $y$ if two
	conditions are satisfied:
	    \begin{enumerate}
	        \item No one strictly prefers $y$ to $x$---that is, for all $i$, $U_i(x) \geq U_i(y)$.

	        \its At least one person strictly prefers $x$ to $y$---that is, for at least one $i$, $U_i(x) > U_i(y)$.
	    \end{enumerate}
	
	\
	
	If one policy Pareto dominates another, everyone should be able to agree that policy is better
\end{frame}

\begin{frame}{Pareto Improvement}

	 The move from a policy $y$ to an alternative policy $x$ is a \emph{\textbf{Pareto
	improvement}} if $x$ Pareto dominates $y$.

\

	From any reasonable  welfarist perspective, a policy change that is a  Pareto improvement is  unambiguously good
\end{frame}

\begin{frame}{Pareto Efficiency}
	
	A policy $x$ is \emph{\textbf{Pareto efficient}} if no other policy Pareto dominates it. 
	
	\
	
	A policy $x$ is \emph{\textbf{Pareto inefficient}} if at least one other policy Pareto dominates it. 
	
	\
	
	Pareto efficiency is important for two reasons
		\begin{enumerate}
			\item The set of policies from which there is no unambiguously good policy move
			
			\its We know a lot about how to achieve Pareto efficiency
		\end{enumerate}
	\end{frame}


\section{In Class Practice}
\begin{frame}{Pareto Efficiency and Pareto Improvements}
	
		
		\[ U_1(x) = 5 \ \ \ U_1(y) = 2 \ \ \ U_1(z) = 4\]

		\[ U_2(x) = 1 \ \ \ U_2(y) = 3 \ \ \ U_2(z) = 7\]

		\[ U_3(x) = 4 \ \ \ U_3(y) = 1 \ \ \ U_3(z) = 1\]

\

 \begin{itemize}
	\item What are the Pareto efficient policies?
	\pits Is there a Pareto improvement from $x$\pause, $y$\pause, $z$?
	\end{itemize}
\end{frame}


\begin{frame}{Pareto Efficiency and Pareto Improvements$^2$}
	
		
		\[ U_1(x) = 3 \ \ \ U_1(y) = 4 \ \ \ U_1(z) = 4\]

		\[ U_2(x) = 3 \ \ \ U_2(y) = 4 \ \ \ U_2(z) = 1\]

		\[ U_3(x) = 3 \ \ \ U_3(y) = 4 \ \ \ U_3(z) = 2\]

\

 \begin{itemize}
	\item What are the Pareto efficient policies?
	\pits Is there a Pareto improvement from $x$\pause, $y$\pause, $z$?
	\end{itemize}
\end{frame}

\begin{frame}{Pareto Efficiency and Pareto Improvements$^3$}
	
		
		\[ U_1(x) = 1 \ \ \ U_1(y) = 2 \ \ \ U_1(z) = 3\]

		\[ U_2(x) = 3 \ \ \ U_2(y) = 2 \ \ \ U_2(z) = 0\]

		\[ U_3(x) = -2 \ \ \ U_3(y) = 4 \ \ \ U_3(z) = 0\]

\

  \begin{itemize}
 	\item What are the Pareto efficient policies?
 	\pits Is there a Pareto improvement from $x$\pause, $y$\pause, $z$?
 	\end{itemize}
\end{frame}

\begin{frame}{Pareto Efficiency and Pareto Improvements$^4$}
	
		
		\[ U_1(x) = 3 \ \ \ U_1(y) = 4 \ \ \ U_1(z) = 5\]

		\[ U_2(x) = 3 \ \ \ U_2(y) = 4 \ \ \ U_2(z) = 1\]

		\[ U_3(x) = 3 \ \ \ U_3(y) = 4 \ \ \ U_3(z) = 2\]

\

 \begin{itemize}
	\item What are the Pareto efficient policies?
	\pits Is there a Pareto improvement from $x$\pause, $y$\pause, $z$?
	\end{itemize}

\end{frame}
		
\section{From Pareto Efficiency to Pareto Improvements}

	\begin{frame}{Efficiency vs.\ Improvement}
		
	
	
	\[ U_1(x) = 5 \ \ \ U_1(y) = 2 \ \ \ U_1(z) = 4\]

	\[ U_2(x) = 1 \ \ \ U_2(y) = 3 \ \ \ U_2(z) = 7\]

	\[ U_3(x) = 4 \ \ \ U_3(y) = 1 \ \ \ U_3(z) = 1\]
		
		\
		
		\begin{itemize}
		\pause \item $y$ is not Pareto efficient
		
				
		\pits $x$ and $z$ are Pareto efficient
		

		
		\pits Move from $y$ to $x$  \emph{not}  Pareto improvement
	\end{itemize}
\end{frame}
		
	\begin{frame}{The Problem}
		
		We care about achieving Pareto improvements
		
		\
		
		We know how to achieve Pareto efficiency
		
		\
		
		Moving from a Pareto inefficient to a Pareto efficient policy need not result in a Pareto improvement
			\begin{itemize}
				\item Distributional concerns
			\end{itemize}
			
			\
			
			Let's see if we can build a bridge between Pareto efficiency and Pareto improvements
		\end{frame}
		
\section{A Model of Policies and Preferences}


\begin{frame}{The Model of Policy}
	
	A policy has two components
		\begin{itemize}
			\item An action 
			
			\its A transfer scheme
		\end{itemize}
		
	\
	
	Think of the action as the policy lever to be pulled
		\begin{itemize}
			\item Free trade, high stakes testing, carbon tax, sanctions
		\end{itemize}
		
		\
		
		The transfer scheme is a redistribution of money
	\end{frame}
	
\begin{frame}{Formalizing the Model}
	
	Society has $n$ people
	
	\
	
	$A$ is the set of possible actions
		
	\
	
	$t = (t_1, t_2, \ldots, t_n)$ is a transfer scheme
		
		
	\begin{itemize}
	 	\item	Individual transfers can be positive or negative
	
		\its Transfer schemes must have \emph{balanced budgets}
		\[ \sum_{i=1}^n t_i = 0\]
	\end{itemize}
	
	\
	
	A policy is a pair $(a,t)$
\end{frame}

\begin{frame}{Individual Preferences}
	
	A person $i$'s preferences over policies, $(a,t)$ are given by that person's utility function, $U_i(a,t)$
	
	\
	
	If person $i$ prefers $(a,t)$ to $(a',t')$, then $U_i(a,t) > U_i(a',t')$
	
	\
	
	If person $i$ is indifferent between $(a,t)$ and $(a',t')$, then $U_i(a,t) = U_i(a',t')$
\end{frame}

\begin{frame}{An Example}
	
	Consider a society made up of two people: The Mayor ($M$) and the Teacher's Union ($T$)
	
	\
	
	There are two actions under consideration
		\begin{itemize}
			\item Using test scores to evaluate teacher performance and determine pay, called \emph{Pay for Performance} ($PP$)
			
			\its Paying teachers solely based on education and seniority, called \emph{Seniority Pay} ($SP$)
		\end{itemize}
		
	\end{frame}
	
	\begin{frame}{Example, continued}
	
	The Mayor prefers pay for performance

	\[ U_M(PP,(0,0)) = 10 \ \ \ \ \  U_M(SP, (0,0)) = 2\]

\

 The Teacher's Union prefers seniority pay
	
	\[ U_T(PP, (0,0)) = 1 \ \ \ \ \ U_T(SP, (0,0)) = 6\]
	
\end{frame}
\begin{frame}{Example, continued$^2$}
		
It may be possible to get both to agree to a move to performance pay by transferring money to the teachers  (e.g., raising the average salary) 

\

Enough of a transfer might compensate the teachers union for the  utility loss from adopting performance pay


\[ U_M(PP,(-10,000,000, \ \  10,000,000)) = 5 \] \[ U_T(PP,(-10,000,000, \ \ 10,000,000)) = 7 \]

	
\end{frame}				
				
				
				
\section{Quasi-Linearity: Building a Bridge}
		
			\begin{frame}{Quasi-Linear Preferences}
				
				Suppose you can separate person $i$'s utility from a policy $(a,t)$ into two components
					\begin{itemize}
						\item Payoff from action $a$ is $v_i(a)$
						\its Payoff from transfer $t_i$ is simply $t_i$ 
					\end{itemize}
				\[ U_i(a,t) = v_i(a) + t_i\]
				
				\
				
				Two important assumptions
					\begin{itemize}
						\item Payoffs from money are linear, so money $=$ utility 
						
						\its Additive separability of transfers and policy
					\end{itemize}
			\end{frame}
				

			\begin{frame}{Splitting the Problem}
				
				 QL preferences is a model that allows us to split policy problems into two components:
					\begin{enumerate}
						\item \textbf{Efficiency}: Use action to maximize total utility.
						\its \textbf{Distribution}: Use transfers to compensate any losers.
						\end{enumerate}
						
					\
					
\pause				\begin{quotation} \noindent	 Congress created Trade Adjustment Assistance (TAA) in the Trade Expansion Act of 1962 to help workers and firms adjust to dislocation that may be caused by increased trade liberalization. It is justified now, as it was then, on grounds that the government has an obligation to help the 'losers' of policy-driven trade opening.\\--\emph{Congressional Research Service}\end{quotation}
		\end{frame}
		
		
\begin{frame}{QL and Utilitarianism}
				
				Under utilitarianism, prefer $(x,t)$ to $(y,t')$ if and only if

				\[  U_1(x,t) + U_2(x,t) >  U_1(y,t') + U_2(y,t')\]
				
			\pause 	\[ v_1(x) + t_1 + v_2(x) + t_2 > v_1(y) + t_1' + v_2(y) + t_2'\]

\

			\pause	Given a balanced budget, this is equivalent to: 
				
				\[ v_1(x) + v_2(x) > v_1(y) + v_2(y)\]
			
			\
						
			With QL preferences, transfers don't matter for utilitarianism, all that matters is the utility implications of the action
		\end{frame}
		
		\begin{frame}{QL and Pareto Efficiency}
			
			A Pareto efficient policy is one that is not Pareto dominated
			
			\
			
			Suppose there is a policy $(x,t)$ such that there is some other $y \not = x$ with $\sum_{i=1}^n v_i(x) < \sum_{i=1}^n v_i(y)$
			
			\
			
			Under QL, if all budget balanced transfer schemes are possible, $(x,t)$ cannot be Pareto efficient
				\begin{itemize}
					\item Switch action to $y$ and then choose transfers to make up the difference to any losers
					
					\its There is still utility ``left over''
				\end{itemize}
				
				\end{frame}
				
				\begin{frame}{QL, Pareto Efficiency, and Utilitarianism}
					
								
				A policy is only Pareto efficient if it uses the action that maximizes the sum of the $v_i$'s
				
				\
				
				Any policy that does so is Pareto efficient
					\begin{itemize}
						\item Transfers take from one person and give to another, which doesn't affect Pareto efficiency
						\end{itemize}
						
				\
				
				Think about the sum of the $v_i$'s as the amount of ``utility pie"
					\begin{itemize}
						\item One action yields a small pie the other yields a large pie
						
						\its Transfers affect the division of the pie
						\end{itemize}
				\end{frame}
						
			
			\begin{frame}{Any Division of a Small Pie is Pareto Dominated}
				\begin{figure}\begin{center}
				\includegraphics[scale =.5]{figures/pies.pdf}
			\end{center}
				\end{figure}
			\end{frame}
				
		
			\begin{frame}{2 Possible Actions}
						
							Small pie divided 25\% to person 1 and 75\% to person 2
							
							\
							
							Large pie all going to person 1
							
							\
							
							Suppose no transfers are made
							 	\begin{itemize}
									\item Move from the small pie to the large pie is a move to Pareto efficiency
									\its Move from small pie to large pie is not a Pareto improvement
								\end{itemize}
							\end{frame}
									
								\begin{frame}	
									\begin{figure} \begin{center}
									\includegraphics[scale =.5]{figures/pies2.pdf}
									\end{center}
									\end{figure}
								\end{frame}
					
					\begin{frame}{Creating a Pareto Improvement}
							
							If we choose transfers correctly we can always create a Pareto improvement after we move to Pareto efficiency
							
							\
							
							More total utility, so can compensate any distributional losers from action change
						\end{frame}
						
						\begin{frame}
							\begin{figure}\begin{center}
							\includegraphics[scale =.5]{figures/pies-transfers.pdf}
							\end{center}
							\end{figure}
						\end{frame}
						
							\begin{frame}
								\begin{figure}\begin{center}
								\includegraphics[scale =.5]{figures/pie-strict.pdf}
								\end{center}
								\end{figure}
							\end{frame}
							
							
					\begin{frame}{An Example}
						
						2 actions: Free Trade ($FT$) or Protectionism ($P$)
						
						\
						
						2 People: Capital ($C$) and Labor ($L$)
						
						\[ v_C(\mbox{FT}) = 12 \ \ \ v_C(\mbox{P}) = 4\]
						\[ v_L(\mbox{FT}) = 2 \ \ \ v_L(\mbox{P}) = 9\]

						\
						
						Free trade is the utilitarian optimum
					\end{frame}
					
					\begin{frame}{Example, continued}
						Suppose status quo is $(P, t_C = 0, t_L = 0)$
												\begin{itemize}
													\item Not Pareto efficient
												\end{itemize}
												
												\
											
											Pareto improving change: $(FT, t_C = -7.5, t_L = 7.5)$
											
											\
											
											Use transfers to compensate Labor for the utility loss associated with free trade
						\end{frame}
						
\section{A Bridge Too Far?}

			\begin{frame}{The Bridge}
				
				We have built a bridge that allows us to first achieve Pareto efficiency and then insure ourselves that we can achieve Pareto improvements
				
				\
				
				This bridge leans on two critical assumptions
					\begin{enumerate}
						\item Correct transfers will be chosen
						\its Quasi-linear preferences
					\end{enumerate}
				\end{frame}
				
				\begin{frame}{Limited Transfers}
					Many factors might get in the way of making transfers
						\begin{itemize}
							\item Technological constraints (collecting transfers is hard or expensive)
							\its Informational constraints (who are the winners and losers)
							\its Economic constraints (transfers may induce other kinds of inefficiency)
							\its Political constraints (what if losers from action change lack power)
						\end{itemize}
						
					\
					
					If transfers are not made, Pareto efficiency on its own need not be normatively compelling
				\end{frame}
				
				\begin{frame}{Non Quasi-Linear Preferences}
					Money is not equal to utility
					
					\
					
					For instance, diminishing marginal utility in money
					
					\
					
					Transfer of equal amount of money from one person to another does not imply transfer of  equal amount of utility
					
					\
					
					Suppose the action hurts those who value money very little
						\begin{itemize}
							\item Have to transfer  a lot of money to make up for utility loss
							\its Take the money from people who value it a lot
							\its See Example 3.4.1 in the book
						\end{itemize}
					\end{frame}
					
				\section{Conclusions}
				
					\begin{frame}{Relationship to Cost-Benefit Analysis}
						
						In cost-benefit, we ask something like whether a proposal, on net, increases total utility relative to status quo
						
						\
						
						A policy that fails cost-benefit can't be a Pareto improvement
						
						\
						
						A policy that passes cost-benefit need not be a Pareto improvement
						
						\
						
						Be cautious about using cost-benefit as a normative criterion for good policy
						
					\end{frame}
					
					\begin{frame}{Are Pareto Improvements Really Uncontroversial}
						
						We've slipped in a welfarist consequentialism
							\begin{itemize}
								\item This is not as bad as having slipped in consequentialism about wealth
							\end{itemize}
						
						\
						
						Pareto improvements are unequivocally good if the only thing we care about is utility
						
						\
						
						There are perfectly sensible normative frameworks that would reject some policies that raise everyone's utility
							\begin{itemize}
								\item Values beyond welfare
							\end{itemize}
							
						\end{frame}
						
						
					\begin{frame}{Take Aways}
						A policy change that is a Pareto improvement is (maybe) unambiguously good
						
						\
						
						We know a lot about how to achieve Pareto efficiency, but a move to efficiency need not be unambiguously good
						
						\
						
						QL model suggests thinking about achieving Pareto improvements in two steps. 
							\begin{enumerate}
								\item Take policy action that improves efficiency (size of pie).
								\its Compensate any `losers' with transfers.
								\end{enumerate}
								
								\end{frame}
								
								\begin{frame}{Takeways$^2$}
								
						There are important caveats
							\begin{itemize}
								\item Will transfers be made? 
								\its Since preferences aren't actually QL, how justified am I in separating efficiency and distribution? 
								\end{itemize}
								
							\
							
													
						Sometimes you might want to pursue policies that are not Pareto improvements
							\begin{itemize}
								\item Distributional effects, political effects
								\its You are on more tenuous normative grounds
							\end{itemize}
							
						\end{frame}
						
						\begin{frame}{Where We Are Going?}
							
							Are there really any Pareto improvements to be had?
							
							\
							
							Social dilemmas
								\begin{itemize}
									\item Fundamental facts about human interactions that create opportunities for good policy
									
									\its We'll need to learn some game theory
															\end{itemize}
															
															\
															
							Governance dilemmas
								\begin{itemize}
									\item The government is not a Pareto improving machine
								\end{itemize}
							\end{frame}
\end{document}