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\title{Dynamic Inconsistency}


\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}{Motivation}
	
	Considerations about future outcomes can distort decisions today
	
	\
	
	Healthcare.gov debacle
		\begin{itemize}
			\item GAO reports failure of oversight by Centers for Medicar \& Medicade Services (CMS)
			\its Major problems with work by contractor (CGI Federal) in early 2013
			\its CMS forwent punitive actions in order to ``better collaborate with CGI Federal in completing the work in order to meet the October 1, 2013 launch''
			\end{itemize}

		\
		
		Look at a variety of examples of these dynamic distortions
\end{frame}


\section{Time Inconsistency}
\begin{frame}{Basic Idea}
	
	A benevolent government faces a commitment problem that leads to inefficiency
	
	\ 
	
	Government announces  a Pareto improving  policy that it plans to implement tomorrow
	
	\ 
	
	Anticipating the policy, people make decisions today that  influence the socially optimal policy tomorrow
	
	\ 
	
	As a result, tomorrow, the still-benevolent government implements a different policy than previously announced
	
	\ 
	
	People were wrong to believe the initial promise and there is inefficiency
\end{frame}

\begin{frame}{Examples}
	
 Final Exam

\

Extracting information

\

 Monetary policy
\end{frame}


\begin{frame}{A  Model}
There is a government and a continuum of consumers, each with 1 unit of a consumption good

\

\begin{enumerate}
	\item Government announces a non-binding planned future tax rate on capital  $t_0$. 
	
	\its Consumers each divide the good between consumption ($c$) and savings ($1-c$). Savings
earn a return $r>1$. 

\its Government collects revenues $tr(1-c)$. 
	
	\its Government spends  revenues  on a public good
	\end{enumerate}
	
	\end{frame}
	
\begin{frame}{Technology}
	
	Total public good, given revenues of $tr(1-c)$, is 
	\[G = \gamma tr(1-c)\]
	
	\ 
	
	$\gamma > 1$ (i.e., public goods are worth more than the money spent on them)
	
	\
	
	$r-1 < \gamma < r$ (i.e., the monetary return on private savings is larger than the monetary return on public goods investment, but not too much bigger).
	
\end{frame}

\begin{frame}{Payoffs}
	
A consumer's utility is given by:
\[ \overbrace{c + (1-t)r(1-c)}^{\mbox{Consumption}}+\overbrace{2\sqrt{G}}^{\mbox{Public Goods}}\]

\

Government's payoff is the sum of the consumer payoffs
\end{frame}

\begin{frame}{The Social Optimum}

Money invested instead of consumed generates positive returns, so  social optimum  involves $c = 0$

\

 Given this, what is the first-best tax rate?

\[\max_t 0 + (1-t)\times r \times 1 + 2 \sqrt{G}.\]

\

Substituting $G = \gamma t r (1-c)$:


\[ \max_t 0 + (1-t)r + 2 \sqrt{\gamma t r}\]



\[ t^{FB} = \frac{\gamma}{r}.\]
\end{frame}

\begin{frame}{Equilibrium: What will the government do?}


Suppose that the government has made some promise, $t_0$, and the consumers have all made some choice $c$. 


\[ \max_t c + (1-t)r(1-c) + 2\sqrt{G}.\]


\[ \max_t c + (1-t)r(1-c) + 2\sqrt{\gamma tr(1-c)}.\]


\[t^*(c) = \begin{cases}
\frac{\gamma}{r(1-c)} &\mbox{ if } c \leq 1 - \frac{\gamma}{r}\\
1 &\mbox{ if } c > 1 - \frac{\gamma}{r}\end{cases}\]
\end{frame}


\begin{frame}{How much will a consumer consume?}
Each consumer in this model is infinitesimally small, so an individual's consumption choice has  zero impact on overall consumption (and, thus, taxes) 

\

If consumer believes that overall consumption by the other consumer will be $C$, then  solves:


\[\max_c c + (1-t^*(C))r(1-c) + 2\sqrt{\gamma t^*(C) r(1-C)}.\]


\[ \max_c c\left[1-\left(1-t^*(C)\right)r\right] + (1-t^*(C))r + 2\sqrt{\gamma t^*(C) r(1-C)}.\]

\end{frame}

\begin{frame}{Consumption, continued}

	\[ \max_c c\left[1-\left(1-t^*(C)\right)r\right] + (1-t^*(C))r + 2\sqrt{\gamma t^*(C) r(1-C)}.\]

\

Consumer's payoff is strictly increasing in consumption if $t^*(C) > 1- \frac{1}{r}$ and  strictly decreasing in consumption if $t^*(C) < 1 - \frac{1}{r}$



\[ \BR(C,t^*(\cdot)) = \begin{cases}
	c = 0 &\mbox{ if } t^*(C) < 1 - \frac{1}{r}\\
\mbox{anything} &\mbox{ if }t^*(C) = 1 - \frac{1}{r}\\
c =1 &\mbox{ if } t^*(C) > 1 - \frac{1}{r}.
\end{cases}\]
\end{frame}


\begin{frame}{Low taxation is not time consistent}

Suppose the government promises a tax rate of $t_0 < 1-\frac{1}{r}$ 
	\begin{itemize}
		\item  If the consumers believe that promise, then
consume nothing in the first period  ($c=0$)

\its  Since $0 < 1-\frac{\gamma}{r}$, this leads the government to choose $ t^*(0) = \frac{\gamma}{r}$

\its Since $\frac{\gamma}{r} > 1- \frac{1}{r}$, the government is not keeping its promise
to choose a tax rate $t_0 < 1 - \frac{1}{r}$
\end{itemize}

\

Consumers decide to save because of the low promised tax rate But the high savings rate then makes taxation  appealing to the benevolent government because of public goods. 
\end{frame}

\begin{frame}{High taxation is time consistent}
	Suppose the government promises a tax rate of $t_0 > 1- \frac{1}{r}$
		\begin{itemize}
			\item If the consumers believe the promise, then consume everything  ($c = 1$)
			
			\its This makes any tax rate  a best response by the government
		\end{itemize}
	\end{frame}
	
	

	\begin{frame}{Time Inconsistency and Inefficiency}

In equilibrium there is high taxation and low savings

\

The government is benevolent and the social optimum calls
for high savings and intermediate taxes

\

If savings are actually high, the government will (optimally) implement high taxes so as to provide lots of public good


\

Anticipating this, consumers don't save

\

Commitment problems on 
the part of the benevolent government prevent society from achieving the socially optimal savings rate
\end{frame}


\begin{frame}{Time Inconsistency and Externalities}  

Consumer saving exerts positive externalities on other consumers because the tax revenues from 
savings provide public goods

\

 The benevolent government chooses a tax rate internalizing those externalities 

\

Individual consumers do not internalize externalities, so want to under save

\

When there is a lot of saving, the benevolent government wants to tax a lot, but strategic consumers seek to avoid this
\end{frame} 

\begin{frame}{Takeaways}

Government announces a future policy and if the people 
	make choices accordingly,  then the government wants  to implement a different policy.
	
\


Time inconsistency problems can create situations in which even a benevolent government is unable to implement a Pareto-efficient policy. 


\

 One way to think about time inconsistency problems is that governments face a commitment problem vis-a-vis announced policies. 
	


\

 The source of this commitment problem is typically that the government  internalizes an externality that the individual people do not. 
\end{frame}




\begin{frame}{Power and Policy}
	
	Governments may pursue inefficient policies precisely because those policies  preserve the power of some currently powerful group
	
	\
	 
	Policies that appear to be socially beneficial may actually be socially harmful when one considers not just the direct economic consequences of the policy, but also the political consequences
		\begin{itemize}
			\item Another instance of the second best
		\end{itemize}
		
		
\end{frame}	






\section{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.
	
	\
	
	Consider current underfunding of pensions
		\end{frame}	







\section{Policy and Future Power}

\begin{frame}{The Idea}

Political power allows groups to extract benefits

\

Groups want to hold on to political power

\

Advocate for policies that retain group power, even if doing so is an inefficient way to achieve policy goal



\end{frame}

\begin{frame}{Examples}
	
	Farm subsidies
	
	\
	
	Price supports and protectionism for manufacturing
	
	\
	
	Immigration restrictions
	
\end{frame}

\begin{frame}{A Model}
	
	Population of size $N$ divided into two generations
	
	\
	
	New generation has population share $0<\delta<1$
	
	\
	
	Old generation is divided into farmers ($ 1/2 < \lambda< 1$) and manufacturers ($1-\lambda$)
	
	\
	
	Old generation farmers are a majority of the old generation, but not of the overall population
 
\[ (1-\delta)\lambda < 1/2\]
\end{frame}
	

\begin{frame}{The Game}
Majority of  old generation chooses a policy:
		\begin{enumerate}
			\item Tax manufacturers at rate $t$ and redistribute to the existing farmers
			\item Tax manufacturers at rate $t$ to fund price supports for agriculture
			\item Don't tax manufacturers
			\end{enumerate}
	
		\
	
	The new generation observes the policy choice and then chooses which industry to enter and first period production occurs
	
	\
	
	The new majority chooses a new tax policy  
	
	
	\
	
	Second period production occurs and the game ends
	
\end{frame}

\begin{frame}{Payoffs and Production}
	
	Each player has utility equal to sum of her individual production and transfers
	
	\
	
	Each farmer produces $F$
	
	\
	
	Each manufacturer produces $M > F$ 
	
	\
	
	It is only possible to tax the income from manufacturing
\end{frame}

\begin{frame}{Utilitarian Optimum}
	
	Taxes are not assumed to be distortionary, so taxes and transfers do not directly affect utilitarian payoff
	
	\
	
	Utilitarian payoff is determined by amount produced
	
	\
	
	Manufacturing is more productive than farming, so utilitarian outcome/Pareto efficiency requires that new generation enters manufacturing
	
	\
	
	Any policy choice that leads new generation to become farmers is inefficient
\end{frame}

\subsection{Second Period Policy}
\begin{frame}{Second Period Policy if New Generation Become Farmers}

Farmers remain in majority

\

Implement tax $t$ and redistribute proceeds to farmers (doesn't matter how)

\[ T_F =  \frac{t(1-\delta)(1-\lambda) M}{(1-\delta)\lambda + \delta}.\]

\

Second period payoff for farmers: 
\[ F + T_F\]

Second period payoff for manufacturer:
\[ (1-t)M.\]

\end{frame}

\begin{frame}{Second Period Policy if New Generation Become Manufacturers}
	
	Farmers lose majority
	
	\
	
	No redistribution, so second period payoffs are $F$ and $M$
\end{frame}

\subsection{Sectoral Choice}

\begin{frame}{Following Lump Sum Transfers}

Lifetime payoff to new generation if become farmer:

\[ 2F +T_F\]

\

Lifetime payoff to new generation if become manufacturer:
\[ M(1-t) + M = M(2-t)\]

\

Farm if and only if:
\[  2F + T_F \geq M(2-t)\]

\end{frame}

\begin{frame}{Following Price Subsidies}

Lifetime payoff to new generation if become farmer:
\[ 2\left(F + T_F\right).\]

\

Lifetime payoff to new generation if become manufacturer:
\[ M(2-t).\]

\

Farm if and only if:
\[ 2(F+T_F) \geq M(2-t)\]
\end{frame}

\begin{frame}{Following No Redistribution}

Lifetime payoff to new generation if become farmer:
\[ 2F + T_F\]

\

Lifetime payoff to new generation if become manufacturer:
\[2M\]

\

Farm if and only if:
\[ 2F + T_F \geq 2M\]
\end{frame}


\begin{frame}{Price Subsidies Encourage Farming}

Redistribution:
\[  2F + T_F \geq M(2-t).\]

\

Price support:
\[ 2F+ 2T_F \geq M(2-t).\]


\

No redistribution:
\[ 2F + T_F - 2t\geq M(2-t)\]
\end{frame}

\subsection{Initial Policy}

\begin{frame}{The Trade-Off}
	
	Always do some form of redistribution
	
	\
	
	Price supports make it more likely new generation become farmers, retaining future power
	
	\
	
	Transfers don't have to be shared with the new generation
	
	\
	
	Choose price supports if:
	
	\[ \frac{\delta}{(1-\delta)^2} \leq \lambda(1-\lambda)\]
	
\end{frame}

\begin{frame}{Inefficiency and Commitment}
	
	
	Old farmers lose their redistribution if new generation don't become farmers
	
	\
	
	Hence, they want a policy that incentivizes farming, even though this is inefficient
	
	\
	
	If the new generation could commit to redistribution in second period, old generation would not choose price supports and 
	we would get efficient sectoral sorting
	
	\
	
	Once the new generation chooses manufacturing, they want to get rid of redistribution to farmers
\end{frame}
	

	\begin{frame}{Take Aways}
		In dynamic settings, absence of enforceable contracts and shifting power gives rise to commitment problems

\

		Inefficient behavior ``up the tree"

\

		Institutions for commitment

\

		Contracts

\

		Integration
\end{frame}		
	


\end{document}