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\begin{center}

 \begin{Large}

 {\bf   Internalities and  Paternalism: Applying the Compensation Criterion to Multiple Selves
across Time   }  \\

\end{Large}


\bigskip

 Social Choice and Welfare,  38(4): 601-615 (2012)

 \bigskip

15 April    2008   \\

\bigskip

Eric Rasmusen \\

\bigskip

{\it Abstract}

\end{center}

  \vspace*{-12pt}


 One reason to call an activity a vice and suppress it  is  that it reduces a person's   future happiness   more than it increases his present happiness. Gruber \& Koszegi (2001) show how  a vice tax can increase a person's welfare  in a  model of multiple selves with hyperbolic preferences across time.  An interself analogy of the compensation criterion can justify a vice ban  whether  preferences are hyperbolic or   exponential, but subject to the caveat that the person has a binding   constraint on borrowing.  The puzzles that intrapersonal altruism raises, however, lend support to  using  the ``Marshallian'' wealth maximization criterion of Friedman (1988)  instead of the Kaldor-Hicks criterion.  




\bigskip

\begin{small}

\noindent
  \hspace*{20pt} Dan R. and Catherine M. Dalton Professor, Department of
Business
Economics and Public Policy, Kelley School of
Business, Indiana University. visitor (07/08), Nuffield College, Oxford
University.  Office: 011-44-1865 554-163 or (01865) 554-163. Nuffield
College,
Room C3, New Road, Oxford, England, OX1 1NF.
Erasmuse@indiana.edu.   \url{http://www.rasmusen.org}. This paper:
 \url{http://www.rasmusen.org/papers/internality-rasmusen.pdf}.

\noindent
Keywords: Internalities, sin tax, moral regulation, Kaldor-Hicks
criterion, time
inconsistency,  hyperbolic preferences.

\noindent
 I thank Kevin Du and  participants in   seminars at Bath, Bern, and East Anglia   
for
their comments, and Nuffield College, Oxford, for its hospitality.


\end{small}

\newpage




\noindent
  {\bf  1.   Introduction}

     A common justification for using taxes or bans to discourage  vice is that although the activity makes a person happy in the present  it will cause him even more unhappiness in the future. Gruber \&  Koszegi (2001) formalize this argument in the context of smoking and show how  cigarette  taxes can raise welfare. They build  a model of multiple selves linked across time by altruism and the addictive consequences of smoking-- called   ``internalities'' between different time-selves of the same person in analogy to ``externalities'' between individuals.   Their model assumes hyperbolic discounting and as  a welfare criterion it maximizes the utility of the first of the multiple selves, though they recognize  that even Pareto optimality could justify paternalistic intervention.

    In this paper I will show that neither hyperbolic discounting nor that  particular welfare criterion is necessary  for  the idea of discouraging vice to protect future selves to be valid. I will use a simple model of one person who is divided into  three selves across time. A key assumption will be that the person  cannot borrow against future income. Given that assumption, suppression of the vice combined with a transfer from a future self to a present self results in a Pareto improvement. In a context where the transfer cannot be made, there is still a Kaldor-Hicks improvement.

 What is  most important in Gruber \&  Koszegi (2001)   is not the hyperbolic discounting  but the idea of the multiple selves and the inefficiency  that can arise from internalities.  If we are prepared to accept a multiple selves model and the idea of basing a welfare function on   multiple selves across time, then paternalism can be desirable even if the person's discounting is orthodoxly exponential.

The argument has  important caveats, because the Kaldor-Hicks argument depends on transfers between the selves that have their own internalities.    It depends on whether the future self has enough income under his control to compensate the present self  and on whether the present, vice-tempted, self is  is willing to accept a  compensating transfer that will increase present consumption at the expense of future consumption.

 After laying out the ideas of multiple selves and time preference and discussing the previous literature, I  will construct a simplified model of the situation in Gruber  \& Koszegi (2001) and show how with quasi-hyperbolic preferences and their social welfare function a ban on smoking raises welfare.   I will then show how the same result can be obtained with   the Pareto criterion and exponential discounting.  All this will be done in a model without borrowing or saving. When these are introduced and are unconstrained, non-exponential discounting does become necessary to justify paternalism under the Kaldor-Hicks criterion.   The Kaldor-Hicks criterion itself, however, becomes somewhat dubious as a measure of efficiency, and so I discuss the alternative of using  willingness to receive   transfers  from  (or make them to) anonymous third parties instead of known trading partners. 

\bigskip


\noindent
  {\bf   2.    Hyperbolic Discounting and Multiple Selves: The Ideas and the
Literature}

    Before setting out the model,   it will be  useful to  discuss hyperbolic
discounting   in general and how it  can justify    paternalism.



    People commonly have positive time preference: they prefer to
consume more now rather than more later.  The standard way to  include
this in  economic analysis is by  having a positive personal discount
rate in the utility function so that consumption earlier adds more to
utility than consumption later, e.g.
$U_{2000} = C_{2000} + \delta_{2001}C_{2001} +
\delta_{2001}\delta_{2002}C_{2002} +
\delta_{2001}\delta_{2002}\delta_{2002}C_{2003}$, where $\delta_t <1$
and we could but need not assume a constant discount factor
$\delta_t=\delta$.  This   functional form is an example of exponential
discounting, whose key feature is that the time subscripts for the
discount factors are objective years rather than ``years in the
future''. As a result, if we view the person's decisions starting one
year in the future his utility function will be:
$U_{2001} =    C_{2001} +  \delta_{2002}C_{2002} +
\delta_{2002}\delta_{2003}C_{2003}$.

  A nice property of exponential discounting is that a person's
consumption path will be time consistent, meaning that if the person
maximizes his utility at time 2000 by the choices
$(C_{2000}^*, C_{2001}^*,C_{2002}^*,C_{2003}^*)$  then he will maximize
them at time 2000 by the same values
$(C_{2001}^*,C_{2002}^*,C_{2003}^*)$ given his reduced wealth as the
result of the consumption in 2000. The person may regret consuming so
much in the year 2000, so his decisions across time are not consistent
in the sense of being the choices he would make ex post,  but  the 2000
consumption is a sunk decision by 2001 anyway. For positive analysis, we
can see what behavior comes from maximizing $U_{2000}$ and ignore the
later utility functions, at least in a world of certainty. For normative
analysis, we do the same.

    An alternative  way a person might have time preference is to have
discount factors whose levels depend  not on the year itself-- 2000,
2001, 2002--  but on how many years in the future the consumption will
occur-- now, one year from now, two years from now. Looking at the
decision made in year 0, the functional form  could be exactly the same,
e.g.
$U_{0} = C_{0} + \delta_{1}C_{1} + \delta_{1}\delta_{2}C_{2} +
\delta_{1}\delta_{2}\delta_{3}C_{3}$, where $\delta_t <1$.  At time 1 (
year 2001) however, the person would maximize
$U_{1} =    C_{1} + \delta_{1} C_{2} + \delta_{1}  \delta_{2} C_{3}$,
not
$U_{1}' =    C_{1} + \delta_{2} C_{2} + \delta_{2}  \delta_{3} C_{3}$.
This is just one of many ways time preference could be non-exponential (
see Rasmusen [2008] for further explanation)
but if the form it take is not exponential, the person's decisions
become time inconsistent. The optimal choices
$(C_{2000}^*, C_{2001}^*,C_{2002}^*,C_{2003}^*)$    from the 2000
utility function will not match the optimal choices  using the 2001
utility function, $(C_{2001}^{**},C_{2002}^{**},C_{2003}^{**})$. This
difference between the effects of absolute time and relativistic time
discounting  has many implications.   For
example, it may be that the person is expecting a big income bonus in
2001. In year 2000, he might want to choose to spread that income's
consumption between 2001 and 2002 because though he highly values year 0
consumption, he is relatively indifferent between years 1 and 2. By the
time   2001 arrives, however, year 2001 {\it is} year 0, and he would
want to consume the entire bonus immediately.

   Much real world behavior seems well explained by this kind of time
inconsistency.  Predictions using non-exponential utility functions are
more complicated, since now we need to carry out the maximization
problem separately for each time period. In addition, a rational person
will foresee at time 0 that his future self will disregard his earlier
wishes, so the problem becomes one of dynamic programming with an eye to
strategically manipulating future selves. It is clear how the analysis
should proceed, however, except perhaps for whether the analyst should
assume that the person is rational enough to realize that he should act
strategically. The analyst should work backwards from the last decision
made, using that as the reaction function for the decisions in the next-
to-last period.

  What to do about  normative analysis is less clear. Although prominent
past economists and modern philosophers disagree (see Broome (1994),
Parfit (1986), and Ramsey (1928)),  modern economists generally accept
positive time preference as a taste like any other in a utility
function, at least in exponential utility functions. Non-exponential
utility raises problems, though.   When a  person's utility function
changes so that he would make different decisions as time passes, which
utility function should be used as  the   welfare function? The  social
planner has to choose between the consumption level the person would
have chosen for 2001 in 2000,    what he would have chosen in
2001,   or   some compromise between the two.  Moreover,
this is a context in which welfare analysis is particularly important.
Quite possibly the laissez faire choices made by solving the dynamic
programming problem are optimal under neither the time 0 utility
function nor any of the utility functions maximized later, because
optimality would require commitment to avoid the time 0 optimal choice
paths  being undone later.


 A useful way to think of the puzzle  that time inconsistency introduces is to treat a person as a series of multiple selves across time,    an idea  used in the first article on time inconsistency, Strotz (1955).\footnote{The question of what ``self'' means has received much attention in philosophy.  In the Symposium, Plato has Diotima say, in speaking of immortality,   
``For only a short time can a living individual be said to be the same individual. For although a human being is said to be the same person from childhood to old age, yet even though he is called the same, he does not at any time possess the same properties. He is continually becoming a new person, loosing parts, portions of hair, flesh, bones, blood, and all the rest of the body. We observe this not only of his body, but also of his psyche. His habits, his character, his beliefs, his desires, his pleasures, his pains, his fears, none of these remain the same. Some flourish, while others vanish.''
 David Hume wrote  in {\it An Enquiry
Concerning the Principles of Morals} (1751, p. 253), ``What ... gives us so
great a propension to ascribe an identity to these successive perceptions, and to
suppose ourselves possest of an invariable and uninterrupted existence thro' the
whole course of our lives?'' Parfit (1982, 1984) developed the problem considerably.  See chapter 2 of Frederick (1999) for an overview of work on identity.}     Think of the single person as a series of different  persons: Self 0, Self 1, Self 2, and so forth, depending on the time period. Time preference is then interpreted as the degree of altruism, with the assumption that people care less about people more distant in the future.  Naturally people will make different choices and  the social planner must somehow compromise between their different desires. We are accustomed to making such tradeoffs among different individuals, so perhaps we can apply this within the individual too.\footnote{Another possibility is to  model a person as having multiple selves simultaneously in time.  In Cowen (1991) an impulsive self struggles with a ``rational'' self.   In   Fudenberg  \&  Levine (2006) a single  long-term self plays a game with a series of short-run selves across time. Here, the self's desires  will be  unambiguous at any one point in time. }



  The  obvious   welfare criterion to apply is what I will call intraself Pareto optimality: Pareto optimality applied across selves.  A pattern of choices is optimal if no change can be made which would make some self better off without making some other self worse off according to the particular selves' utility functions.  This has frequently been applied to multiple selves models (e.g. in  Bernheim \& Rangel (2007)), since it seems unobjectionable, if weak.  It does have some bite, though, and can justify intervention when, for example,  Self 0  does not  save because because Self 1 would consume all the saving, leaving nothing for Self 2. Self 1 might welcome being constrained to not consume  the entire amount saved by Self 0  so that Self 0 would  save a positive amount.

  A stronger normative approach is   what I will call the  ``precommitment criterion'':  maximize Self 0's utility   by seeing what would happen if he could  directly commit to a future course of action, rather than having to try to manipulate his future selves. An attraction of this criterion is that it is  the same as our standard method for exponential utility, simply adding the recognition that the social planner will  have to  somehow constrain the future selves so that they act in accord with Self 0's wishes.   This is what is most used in the literature, almost always in connection with decisions about consumption and savings (see, e.g. Laibson (1997), Harris  \&  Laibson (2001),  and Krusell \&   Smith   (2003) ).

 Gruber  \& Koszegi  (2001) use a model of a divided self to look at whether smoking should be taxed-- a model focussing not on consumption, but on vice. In their model, a person lives for a finite number of periods and must decide   the amount of smoking and  other consumption for each period.  Smoking is addictive, so smoking more now lowers future utility. Discounting is quasi-hyperbolic rather than exponential, so decisions are time inconsistent;  in period 1 the person would deviate from the period 1 smoking level he thought was best in period 0. They  use the precommitment criterion, maximizing Self 0's utility and show that a large tax on cigarettes is optimal. O'Donoghue  \&   Rabin (2003) also  looks at sin taxes with time-inconsistent consumers to make the point that time-consistent consumers may be hurt very little and time-inconsistent consumers helped a great deal by sin taxes. Caplin \& Leahy (2004) and Whitman (2006) criticize this focussing on   the welfare of the ``long-run '' or earliest self-- the ``dictatorship of the present''.    In effect, focussin on the early self what would happen if a person could commit himself to future actions (in this case, cigarette smoking), and then looking at how government taxes could bring consumption closer to that outcome when direct commitment is not possible. For individual consumers they  use the Self 0 utility function; in their comparison of different types of consumers they are implicitly using the Kaldor-Hicks criterion.  Bhattacharya, \& Ladkdawalla (2004)  look instead at  Pareto improvements that can result from government smoking policy.   They suggest the use of voluntary smoking licenses that  a person could purchase to constrain his future smoking. These licenses would subject the future self to high cigarette taxes, but would also  entitle the future self to a lump-sum transfer that would make him better off overall. We will come back to this later in the paper.

 The precommitment criterion  is attractive in contexts where we think that someone's  judgement is best long before a difficult decision needs to be made. It seems reasonable that I will make a better decision about whether to take up smoking if I have to make  the decision a year before I can actually start smoking. But it is equally unattractive in other examples. We would  think it wise for someone to commit to climbing Mount Everest without oxygen a year in advance despite the danger, discomfort, and cost, just so that he can boast about it at a party tonight.  Commitment is  perhaps too powerful an instrument. At least when discounting is exponential we are letting each self decide on his own period's consumption. The precommitment criterion makes Self 0 a dictator over all the selves. One might argue for that on the grounds that   hyperbolic discounting is the result of loss of self- control, so a person should never decide his own period's consumption-- the opposite conclusion  of the freedom argument-- but that is to introduce a new consideration of some preferences being illegitimate.


A   third  possibility  is to
convert the person's hyperbolic utility function to an exponential one
and do what maximizes the utility of Self 0.  Since discounting will
now be exponential, time inconsistency will vanish.  This criterion
accepts ordinary time preference with its exponential discounting, but
rejects hyperbolic discounting as a normative guide, while admitting it
as a positive predictor of behavior.  This would be the right thing to
do if one considers hyperbolic discounting the result of lack of self-
control over immediate decisions, something of a technological
constraint.  If a  person's hand automatically picks a cigarette  up and
lights it without conscious intent, we do not want to give his action
normative significance. Rather, in our welfare maximization we want to
see how we can reduce the damage from his mistakes. Bernheim \& Rangel (
2005)
make the point well:

\begin{small}
\begin{quotation}

American visitors in London suffer numerous injuries
and fatalities because they often look only to the left before stepping
into streets, even though
they know traffic approaches from the right. This is a systematic
pattern; one can't dismiss it as
an isolated incident. A literal application of the revealed preference
compels us to conclude
either that these people simply have a very strong preference look left,
or that they're masochistic.
If we use these revealed preferences for welfare analysis, there's no
legitimate basis for
preventing someone from stepping in front of a truck. And yet, it's safe
to say that, after
recognizing the purpose of the intervention, anyone would be grateful.
The pedestrian's objective
-- to cross the street safely -- is clear, and the decision is plainly a
mistake.

\end{quotation}
\end{small}

  How precisely  to convert the person's hyperbolic utility to an
exponential one would depend on the particular model. In models such as
the quasi-hyperbolic one we will use below in which some one parameter
represents the degree of closeness to exponential discounting  it would be
natural simply to set that parameter equal to one.   We will not use this
criterion in the present paper, since we wish to focus on what to do if we
retain the idea of consumer sovereignty,  but it is a natural one  if we really
believe the person's choices are ``wrong'' and  in effect   their utility
function needs correcting.


A  fourth criterion, closely related to intraself Pareto optimality,  is what I will call the intraself Kaldor-Hicks criterion:   the Kaldor-Hicks criterion applied across selves. This applies our standard criteria for looking at welfare across individuals to welfare within the individual by asking whether any change is possible which has bigger dollar gains to winners than dollar losses to losers.

Equivalently, the intraself Kaldor-Hicks criterion asks  whether a Coasean bargain is conceivable in which Self X  would agree to transfer wealth to Self Y  in exchange for  Self Y agreeing to changing his behavior. The word ``internality'' is used   in analogy to the ``externality''  for spillovers across individuals. It must be a ``real'' internality for intervention to be helpful under the intraself Pareto optimality and Kaldor-Hicks criteria, because some spillovers across selves  are purely distributional and would not give rise to Coasean potential gains from trade. Unlike Pareto optimality, however, Kaldor- Hicks does not require the bargain to be actually carried out in full.  

   An advantage of the intraself Kaldor-Hicks criterion is that it fully accepts consumer sovereignty and the legitimacy of hyperbolic discounting.   It   takes a technique which is standard in analysis across individuals and merely applies it to a new context.  Its motivations, whether utilitarian or that the social planner is merely replacing missing markets, can be carried over to the new context, where they perhaps fit even better, since here  we are not trading off utility between individuals, just across time.  A complication, however,  is that the earlier self cares about the later self's consumption and so is not  necessarily  helped by receiving a transfer from the later self. As we will see later, this helps expose a problem for the motivation of the Kaldor-Hicks criterion. 

   The intraself pareto optimality and Kaldor-Hicks criteria use   the same analysis to discover if laissez faire is optimal. The difference comes in what is regarded as an improvement, a  difference that  is crucial in practice. A  person's future self might be willing to pay 20,000 dollars for his earlier self not to indulge in a vice that the earlier self values at only 5,000 dollars. Kaldor-Hicks says that taxing the vice or banning it would be an improvement, but Pareto optimality says it would not, because the earlier self is hurt. Both criteria would  support banning the vice, taxing the future self, and subsidizing the present self.

One might question whether it is necessary to add the word ``intraself'' to these names. They are, after all, just applying the old criteria to a new context. That is true, but the application is  controversial, and contradicts the standard definitions.  A situation  that is Pareto optimal in the usual sense is not necessarily intraself Pareto optimal, and vice versa.

  Closely related to the Kaldor-Hicks criterion is the idea of using a social welfare function to determine what is optimal. A welfare weight could be assigned to each self, and actions and consumption chosen to maximize the sum of the weighted utilities.	The outcome will be one of the  intraself pareto optima. Unlike the Kaldor-Hicks criteria, the social welfare function does not take into account an initial allocation of decision rights to the various selves. Instead of specifying the property rights over actions, the analyst specifies the welfare weights. Constraints on allocating property rights (e.g., borrowing not allowed, so the first self cannot consume everything) can be introduced as constraints on the social planner.  Caplin \& Leahy (2004) take this approach in their article on optimal discounting, in addition assuming that people care about and discount past consumption, not just future. We will use the Kaldor-Hicks approach  here using the natural assumption that each self controls decisions made during his own existence, but  we will also discuss the initial allocation of property rights. We will not, however, be questioning the legitimacy of the rates of time preference, Caplin \& Leahy's main interest. Rather than the allocation of felicity generally or of consumption, we will focus on possibly suboptimal actions that have a direct effect across perios.



\bigskip

\noindent
{\bf 3.  The Model}

A person lives for  three periods labelled 0, 1, and 2.    He receives an endowment of $W$  in  each period.We will denote his   consumption of the single good by  $C_0$, $C_1$, and $C_2$.   Our person can save at interest rate $r$,  but he cannot borrow. We will denote wealth in each period as $W_0$, $W_1$, $W_2$, where $W_0 = W  $ and the later wealths depend on earlier saving.

 In period 1, the person  chooses between  smoking ($X=1$) or not smoking  ($X= 0$). We will assume smoking costs no money, for simplicity.\footnote{The  reader dissatisfied with smoking having no effect on consumption may replace smoking with a juvenile, self-inflicted, tattoo or  with a disease-prone sexual practice.}     If the person chooses to smoke  he receives 1  unit of utility in period 1  and loses $\alpha $ in period 2, where we assume that $ \alpha >1$ so  the loss is bigger than the gain.

  Total utility is given by the following function of the three consumption choices and the smoking choice:
\begin{equation} \label{u0}
U_0 = U(C_0) +  \beta\delta(X +U(C_1)) + \beta\delta^2(-\alpha X
+U(C_2)),
\end{equation}
 with
$0 \leq \beta<1$ and
\begin{equation} \label{consume-early}
  0 \leq    \delta< \frac{1}{1+r}.
\end{equation}
  This   ``consume-early'' assumption  \eqref{consume-early}  ensures that the person's rate of time preference is greater than the interest rate, an assumption useful for the first part of this paper, when we will focus on what happens when the person does not save any of his income. A later section will briefly examine the  opposite, ``consume-late'' assumption. 


We will      assume for the first part of the paper that  utility is linear in
consumption ($U(C)=C$).   Relaxing this assumption and the assumption of equal  endowments $W$  later  will
introduce the complication  of  ``bequests'' motivated
by    consumption smoothing.




The  utility function is quasi-hyperbolic in the style of Laibson
(1997): our person has an exponential discount factor of
$0 \leq \delta< \frac{1}{1+r}$ (so his rate of time preference is
greater than the market interest rate)  and a hyperbolic adjustment
parameter of $0 \leq \beta<1$.     If $\beta=0$ he does not care at all
about the welfare of his future self; if $\beta=1$ he has a standard
exponential-discounting utility function.\footnote{Phelps \& Pollack (
1968) had  used the quasi-hyperbolic form to discount generations of
people over time.   Laibson (1997) applied it to within the self  and
named it ``quasi-hyperbolic''. True ``hyperbolic'' discounting, proposed
by Chung  \&   Herrnstein (1961) in connection with a particular theory
of behavior, has similar qualitative features but is more complicated to
work with. See Angeletos, Laibson, Repetto, Tobacman  \& Weinberg (2001)
for a good brief explanation of the differences.}

A common distinction in the behavioral economics literature is between sophisticated consumers, who  are fully aware that they are time-inconsistent  and adjust their behavior for that, and naive consumers who are not aware that their future self may choose differently.    The model here is simple enough for  the knowledge of time-inconsistency not to matter, but we will assume that consumers do have rational beliefs about what future consumers will do.

   We will  view this person as consisting of three people, Self 0, Self 1, and Self 2. Self 0's utility function, defined over the  present and future decisions viewed from Period 0,  we have just seen. Self 1 and Self 2 have the following utility functions:
\begin{equation} \label{u1}
 U_1 = X +U(C_1) + \beta\delta(-\alpha X +U(C_2)),
\end{equation}
and
\begin{equation}
 U_2 = -\alpha  X  + U(C_2).
\end{equation}


  The motivation for the model is to set up as simply as possible three periods
when smoking: (a)  has no immediate
effect on utility (Period 0), (b)  has a positive effect (Period
1),  and  (c)   has   a negative effect (Period 2). Thus, Self 0 is  close to
what some would call ``the long-run consumer'' who has preferences but chooses
no action, except that he does choose his own consumption.

 The utility functions I have just defined assume that Self 0 cares about Self
1 only through  actions that occur in Self 1's period, and about Self 2 only
through actions that occur in Self 2's period. Self 0 does not care that Self 1
derives
utility from Self 2's consumption. If he did, Self 2's consumption would count
double in Self 0's utility function. That is plausible if Self 0's utility
arises from thinking of how happy he will feel in the future instead of from
thinking of how happy particular actions will be making him in the future.
Caplin \& Leahy (2004) cite Loewenstein (1992) as saying this is the Jevons
approach to utility. Here, we are instead using what he calls the  Samuelson
Sameuelson approach to utility. The difference could be interpreted in terms of
discount factors. Here, I take the Samuelson approach as better reflecting the
idea that each self is like a person deciding on his future actions, and because
the Jevons approach makes most sense when memory as well as anticipation affect
utility (as Caplin\& Leahy [2004] do assume).

Memory yields no  utility in our utility function here.   In period 1 the person  derives no pleasure from remembering his consumption in period 0. For our purpose of looking at the  amoking decision, including memory would make little difference,  so long as we assume that memory yields less utility than the  event  remembered,  though it would add  many extra terms to the equations. It would make smoking somewhat more attractive, since in Period 2 the person would derive utility from Period 1's pleasure from smoking.  





\bigskip

\noindent
{\bf   Positive Analysis: The Laissez Faire Equilibrium Choices }

     Before thinking about welfare, we must work out what choices our three selves will make.   Self 0 controls the value of $C_0$, Self 1 controls $X$ and $C_1$, and Self 2 controls $C_2$. In a sense, this distribution of power, not the equal endowments across time,  is the true distribution of property rights. 

Self 2 would  consume his entire wealth, so $C_2 = W + (1+r) (W_1- C_1)$. If he could choose variables from earlier periods, he would choose $X=0$, $C_0=0$, and $C_1=0$: a ban on smoking,  and zero consumption in the earlier periods so he would have more consumption himself.

Self 1 would maximize (\ref{u1})  by choice of $X$ and $C_1$ subject to the constraint that $C_2 = W + (1+r) (W_1- C_1)$. Under our assumption that utility is linear  the first order solution would not apply, because he would have a corner solution and he would choose to consume his entire wealth: $C_1=W_1$. He derives utility from Self 2's consumption, but it is discounted by $\beta \delta$, which is less than $1/(1+r)$ by  the consume-early assumption.     He would choose $X=1$ if $1- \beta \delta \alpha >0$, i.e. if
\begin{equation} \label{b1}
  \alpha<\frac{1}{\delta\beta},
\end{equation}
 which we will henceforth assume.

If he could control $C_0$ and $C_2$, Self 1 would choose $C_0=0$ to
increase his own consumption,  and he  would choose $C_2=W_2$.

Self 0, like Self 1, is subject to the consume-early assumption and  so
will choose $C_0=W$.

 If Self 0 could control $C_1$ he would   set $C_1=W$, just as Self 1
would  because of the consume-early
assumption. He would choose $X=1$ if
$\beta\delta- \beta\delta^2\alpha >0$, i.e. if $1-  \delta \alpha>0$. so
he would  choosing   smoking for    period 1 if
\begin{equation} \label{c3}
\alpha<\frac{1}{\delta}
\end{equation}

To sum up: in equilibrium  Self 0 would choose $C_0=W$, Self 1 would choose
$C_1=W, X=0$ if  $\alpha \geq 1/(\delta\beta)$    and Self 2 would choose $C_2=
W$. Consumption would be equal in all periods and  Self 1 would smoke.




\bigskip
\noindent
{\bf 4.  Welfare Analysis: The Precommitment Criterion }

   We have   seen that the three selves have contradictory preferences for  the
smoking  and consumption choices. When Selves 0 and 2 want  to ban smoking   but
Self 1 wants to use it,  what should the social planner do?



One approach is to privilege the earliest self.  The social planner's role is
then   to act  to allow the person to commit to all his future actions and
thereby avoid the consequences of  time inconsistency.  We will call this the
precommitment criterion.\footnote{There are two variants of the precommitment criterion. In one, it is literally the earliest self whose utility is maximized. In the other, the modeller creates a ``long-run self'' who exists before Self 0 and who derives no utility from present sensations, only from future sensations. The long-run self approach can be attractive in the context of quasi-hyperbolic utility when the first ``real'' self is making consumption decisions, because the long-run self is effectively a self with exponential utility (he might have a $\beta$ term in his utility function,  but since he takes no immediate  actions his present-orientedness is irrelevant).  In our present context, the difference between the two variants is unimportant, since the smoking action occurs in Period 1, after Period 0. }



 \noindent
  {\it  The Precommitment Criterion: A change in the value of the choice
variables is an improvement if it increases  the utility   of Self 0.}

Using the precommitment criterion,
the laissez faire outcome fails to maximize welfare if
$\alpha \in (1/\delta, 1/ \delta \beta) $ because Self 0's optimality
condition (\ref{c3}) and Self 1's optimality condition (\ref{b1})
conflict.  The optimality conditions differ    whenever there is hyperbolic
discounting and  $ \beta < 1$.

   Self 1 trades off the present  period 1  against the future period 2  at  the
high  rate of 1 to $\beta \delta$, whereas  Self 0 trades off the future period
1 against the far-future period 2 at the lower rate of $\delta$.  The problem is
not that sensations further in the future should not be treated as less
important-- a low $\delta$  by itself would not introduce a reason for
paternalism-- but quasi-hyperbolic discounting makes the present period count
too heavily.


If  discounting is exponential, i.e. $\beta=1$, then there is no time
inconsistency and the precommitment criterion will  not call for a paternalistic
ban on smoking.  Self 1 will make the correct smoking decision even without
coercion.   It is still true that smoking hurts Self 2, possibly by much more
than it helps Self 1,  but if discounting via the exponential discount rate
$\delta$ is heavy enough that Self 1 would choose smoking then it is heavy
enough that Self 0 cares much more about period 1 utility from smoking than
period 2 disutility.




      The precommitment criterion can also be applied to consumption
rates, but it would suggest no change from laissez faire. The positive analysis
above showed that Self 0 wants zero savings and
$C_0=C_1=C_2=W$ anyway,  which precommitment is unnecessary to attain.
Borrowing would be a different matter, but that is ruled out for the moment by
assumption.

\bigskip

   Thus, using the precommitment criterion, a paternalistic ban on smoking might be desirable  to avoid  the consequences of the  high  value placed on present satisfaction. Self 0, like Self 1, places high value on present satisfaction, but the smoking decision is in period 1, so Self 0 wishes to override Self 1's desires because of the harm they cause Self 2.

The objection to the precommitment criterion is that to  follow  the desires of Self 0  is arbitrary. If followed literally, it is no use at all:  one period later, Self 1 will have become ``Self 0'' and the social planner would have to undo the earlier decisions to follow the desires of the new current self. Of course, the cirterion is not meant to be taken literally: what is special about Self 0 is   that he is making a choice about something before it becomes a present decision.   The idea is  that the present-time orientation of a hyperbolically discounting person is illegitimate and we are willing to override consumer sovereignty. That is not an unreasonable argument, but it an uncomfortable one for an economist.  We do not ordinarily say that tastes are illegitimate.    It moreover disregards the desires of Self 2, the post-action self, except insofar as Self 0 cares about them.

\bigskip




\noindent
 {\bf 5.  Welfare Analysis:  The  Kaldor-Hicks Criterion}

  As we have seen, a smoking ban can be optimal under the precommitment
criterion in a multiple selves model. Once we have adopted the idea of multiple
selves, however, defending the precommitment criterion becomes a challenge.
 An alternative is to apply our standard welfare criteria, but to  within the
self. First, we can require Pareto optimality: that a  change is an improvement
only if it benefits all the selves.  Second, we could apply the Kaldor-Hicks
criterion:  requiring    that a change would benefit all selves if appropriate
cash transfers were made, but not requiring the transfers themselves.  If we
accept this logic for policy consequences between different individuals, it
seems all the more acceptable within one individual, since we are no longer
trading off utilities interpersonally.


\noindent
  {\it  The Intraself  Pareto Optimality
Criterion:  A change in the value of the choice variables is an
improvement if it increases  the utility   of
at least one  self and reduces   the utility
  of none. }




\noindent
  {\it  The Intraself Kaldor-Hicks
Criterion:   A change in the value of the choice variables is an
improvement if it could be combined with  money transfers     such that the  combined changes  would increase    the utility of at least one  self and reduce     the utility   of none.     }

 The intraself Kaldor-Hicks criterion takes the laissez faire outcome as the base   and asks whether the winners from a change would  be willing to compensate the losers.\footnote{This is  the version of Kaldor (1939) version; in  the  version  of Hicks (1939) we ask whether the losers would be unwilling to pay the winners to prevent the change; in Scitovsky (1941)  a change must pass both tests.  The idea is even older. See 
 Chipman  \&   Moore (1978) for discussion of its history, including, e.g.,  discussion of Pareto's  and Pigou's views.    The answers in the model will change depending on which version is used, as we will discuss below.  } This captures the idea with which we started that a vice causes more future disutility than it causes  present utility.

Like the precommitment criterion, the  intraself Kaldor-Hicks criterion does not suggest changing consumption  from laissez faire.  There is no way to hypothetically raise the utility of  any two  of our sequential persons   by requiring them to save for a future self.

The smoking choice   $X$    adds $(X - \beta \delta \alpha X)$ to Self 1's utility  and    $- \alpha X$ to Self 2's. Would Self 2 pay enough to change Self 1's decision?   Selves 1 and 2 never meet, so the transaction is not possible, but Kaldor-Hicks only requires a potential Pareto improvement, not an actual one. There is a complication, however: Self 1 cares about Self 2's consumption level, so  a dollar  payment from Self 2 to Self 1 raises Self 1's utility by less than one dollar. Moreover, Self 0's utility is affected by any transaction between Self 1 and Self 2.

  Since Selves 0 and 2 both  might  benefit from the smoking ban, let us imagine both of them making payments to Self 1 in exchange for the ban, payments $P_0$ and $P_2$ as measured  in giving-period dollars.  If we recognize that $U(C_t) = W_t$ for each period, the differences between the  utility functions  when  the smoking bargain is made and when it is not are
\begin{equation} \label{u0s}
U_0 = \left\{\left[W-  P_0  \right] +  \beta\delta \left(  W + (1+r)P_0
+ \frac{P_2}{1+r} \right) +
\beta
\delta^2( W-   P_2) \right\} - \{ W +  \beta\delta(   W +1    ) +
\beta
\delta^2( W  - \alpha   )   \}
\end{equation}
\begin{equation} \label{u1s}
 U_1 = \{ W + (1+r) P_0   +\frac{ P_2}{1+r}   + \beta\delta( W-   P_2)\}
- \{ W+1 +
\beta\delta( W-   \alpha) \}
\end{equation}
and
\begin{equation}
 U_2 =  \{ W-   P_2\} - \{ W - \alpha \}.
\end{equation}

  Self 0's net benefit from the bargain to ban smoking is
\begin{equation} \label{uv}
 Net \; Benefit (Self \;0) =  -P_0+ \beta\delta \left( \left(1+r \right)
P_0 + \frac{P_2}{1+r} -1 \right)
  +   \beta\delta^2  (-P_2    +  \alpha)
\end{equation}

    Self 1's net benefit is
\begin{equation} \label{uw}
  Net \; Benefit (Self \;1)=  (1+r)P_0  + \frac{P_2}{1+r} - 1 + \beta
\delta   (- P_2 +
  \alpha)
\end{equation}
Self 2's is
\begin{equation} \label{usd}
 Net \; Benefit (Self \;2)=   -   P_2 + \alpha
\end{equation}



 We can find the necessary and
sufficient condition for a  Pareto improvement by picking $P_0$ and
$P_2$ to make Selves 1 and 2 indifferent about the bargain  and then
seeing if Self 0's utility rises under  the resulting $P_0$.   Self  2 would be
indifferent if
\begin{equation} \label{usda}
P_2 =  \alpha.
\end{equation}

 Substituting for $P_2$  from  (\ref{usda})  into Self 1's net benefit equation
(\ref{uw}), we
would find that Self 1's net benefit was zero  if Self 0 paid him
\begin{equation}
P_0 =  \frac{1 + r-\alpha} {(1+r)^2}.
\end{equation}

      Substituting  these values of $P_0$ and $P_2$  into Self 0's net benefit
 equation, (\ref{uv}) tells us that Self 0 will derive higher utility from
paying $P_0$ to ban smoking than from blocking the transaction if
\begin{equation} \label{a2}
  \alpha  > 1+r.
\end{equation}

   Unlike the precommitment  criterion, the intraself Kaldor-Hicks criterion does not rely on conflicting time preferences to justify helping a future self, but on low interest rates. It is relying not on the altruism of earlier selves to generate utility, but on the willingness of the future self to sacrifice consumption. The altruism of earlier selves still helps make a smoking ban more attractive, but it is no longer a necessary condition. This can be seen by setting $\beta=0$, so the earlier selves care nothing for the later self. The precommitment criterion no longer has anything to say about a smoking ban, since Self 0  cares nothing about either Self 1 or Self 2.   The intraself Kaldor- Hicks criterion, however, would ban smoking if $\alpha > 1+r$, which is to say if Self 2 loses more from Self 1's smoking  than can be made up for  by the rate of interest.

 We can think of there being market failure within the self,     a missing market for trade of future money for present abstention. If the social planner created that market, he would create gains from trade, and both buyer and seller would benefit when Self 1 sells a smoking ban to Self 2.

\bigskip

   We have just seen that if $ \alpha > 1+r$ then  Self 2 would be willing to compensate  Self 1, with the aid of Self 0.  What if Self 2 had to compensate Self 1  without the aid of Self 0? In that case $P_0=0$, and Self 1's net benefit from agreeing not to smoke is, from equation (\ref{uw}),
\begin{equation} \label{uw1}
  Net \; Benefit (Self \;1)=     \frac{P_2}{1+r} - 1 + \beta
\delta   (- P_2 +
  \alpha)
\end{equation}
 This solves to a  necessary payment of
\begin{equation}  \label{necessary}
 P_2 = \frac{(1+r)(1 - \alpha \beta \delta)}{1- \beta \delta (1+r)}.
\end{equation}

     This necessary payment illustrates  three effects of a transfer by Self 2
to Self 1 to stop smoking. First,  Self 1 must be compensated for the direct
effect of not smoking, a loss of 1 to himself that is partly compensated for by
the $\alpha \beta \delta$ in utility to himself from Self 2's benefit. Second,
if $P_2$ is measured in period 2 dollars,   we must multiply it by $(1+r)$ to
get the value in period 1 dollars.  Third,   Self 1 suffers disutility from the
reduction in Self 2's utility that results from Self 2's making the payment, and
Self 2's payment to him must be inflated by  $\frac{ 1}{1- \beta \delta (1+r)}$
to compensate for that. Note that if Self 1 cares only about the present
($\beta \delta =0$) then  the necessary payment is simple:  $ P_2 = 1+r$.  The
complexity arises from Self 1's concern about  period 2 consumption.

  What condition on the harm from smoking $\alpha$ is necessary for Self 2 to be
willing to pay the entire compensation to Self 1?  Self 2 is willing to pay so
long as $P_2 \geq \alpha$. Setting $P_2= \alpha$ in equation (\ref{necessary})
and solving  for $\alpha$, however, yields
$$
\alpha = 1+r,
$$
  exactly the same as the critical value for banning smoking when Self 0 also
paid compensation.

    Why is it that Self 0's benefit from banning smoking has no effect on when
smoking should be banned? One would think that if $\alpha$ is slightly too small
for Self 2 to be willing to pay  enough to Self 1 to stop smoking then Self 0's
additional contribution would tip the balance.  Remember, though, that Self 0
only benefits from a smoking ban if $\alpha > \frac{1}{\beta \delta}$. Thus, the
interval of $\alpha $ for which banning smoking would be optimal but Self 2 is
unwilling to pay the entire compensation is $[ \frac{1}{\beta \delta}, 1+r]$.
This, however, is an empty interval, because by  the consume-early assumption, $\frac{1}{\beta \delta}> 1+r$. Thus, whenever Self 2 is unwilling to pay  enough to Self 1 for him to stop smoking, Self 0 is unwilling to pay him anything at all.

 Conversely, shouldn't Self 0's benefit from smoking affect the optimality condition for smoking? Couldn't it happen that a bargain helps Selves 1 and 2, but to the detriment of Self 0, who does want  Self 1 to smoke if  $ \alpha <\frac{1}{  \delta}$? This cannot happen, because Self 0 also benefits from Self 2's payment to Self 1. If the payment of $P_2$ adds $\frac{1}{1+r}$ to Self 1's consumption utility but takes away 1 from his smoking utility, the net effect is not only zero for Self 1, but for  the  discounted period 1 values that  enter Self 0's utility function, $\beta \delta (X + C_1)$.

\bigskip

 What is efficient  under the intraself Kaldor-Hicks criterion  does  depend  on the distribution of wealth-- that initial allocation of decision rights described above. In the transaction just described, Self 2 pays $\frac{(1+r)(1 - \alpha \beta \delta)}{1- \beta \delta (1+r)}$ to Self 1.  If it is Self 2 alone who pays Self 1, this  requires that Self 2's income be large enough: $W   \geq \frac{(1+r)(1 - \alpha \beta \delta)}{1- \beta \delta (1+r)}$. Otherwise, Self 2 would not pay Self 1 enough to compensate for Self 1's loss of utility from smoking. If it is both Selves 0 and 2 who pay Self 1, then  if Self 2 reaches his budget constraint Self 0 must be able  to pay the remainder.

   If, however, Selves 0 and 2 do  have sufficient wealth to compensate Self 1, we have shown that the internality of harm to Self 2 from Self 1's smoking can justify paternalism under the Kaldor-Hicks criterion even if $\beta=1$ so discounting is exponential. The parameter $\beta$ simply  doesn't matter to whether smoking should be banned.

\noindent
{\it Result 1: In a multiple selves model,  the intraself Kaldor-Hicks criterion
can justify paternalistic banning of a vice even if discounting is exponential.
}


 The  intraself Pareto optimality criterion \label{example}  and  the Kaldor-
Hicks criterion are not the same as their conventional analogs,   so we must be
careful not to misinterpet Result 1. The conventional analogs of the criteria
are implicitly based on the precommitment criterion.

  Suppose that $\beta=0$, so that Self 0  is indifferent about Selves 1 and 2.
It is then Pareto optimal for Self 1 to smoke, because  in maximizing utility
according to Self 0's utility function, which is what is required by
conventional Pareto optimality,    $X=0$ or $X=1$  both  result in the same
utility. Smoking   is not intraself Pareto optimal, however, because both Self 1
and Self 2 can have higher utility if $X=0$ and a suitable transfer is made from
Self 2 to Self 1.

Conversely,   an outcome that is intraself Pareto optimal might not be   Pareto
optimal. Under $\beta=0$ and linear utility, the outcome
  ($C_1=W, C_2=W,C_3=W, X=0$) is intraself Pareto optimal, but  it is not Pareto
optimal because Self 0's utility would be higher  if all the consumption were
transferred to $C_0$.









\bigskip


\noindent
 {\bf 6. Should the Social Planner Permit Borrowing?}


  I have assumed borrowing was impossible. What if  the social planner could replicate borrowing across selves  by a system of age-dependent taxes and transfers? Under laissez faire, each self consumed his entire endowment rather than save. Would they now borrow?

The precommitment  criterion says that the social planner should  permit borrowing, which would transfer all consumption to period 0. Self 0 would choose to consume all three selves' endowments because of the assumption that the rate of time preference is high and the utility functin is linear in consumption.

    The intraself Pareto  and  Kaldor-Hicks criteria  lead to a different result:  they would not  instruct    the government to replicate   borrowing.  Permitting borrowing would help Self 0, but hurt Selfs 1 and 2. Moreover, there    is no exchange of goods for money on which to base a potential Pareto improvement.   These criteria are concerned with  the production and allocation of goods, not with their distribution.

   This illustrates that the  multiple selves paradigm is a bigger change from standard normative economics than one might realize.   The paradigm provides a plausible reason for   banning smoking, but it weakens a conventional economic argument against usury laws.     The argument  for banning smoking is that the ban hurts the present self less than it helps the future self.  If the two could bargain across time, they would do so and both would be better off. The conventional argument against usury  laws is that if someone accepts a loan, he must think that the benefit in current consumption is worth the loss in future consumption. That is just to say that Self 0 benefits from borrowing, however, despite the loss to Self 1 and Self 2. A usury law banning borrowing would hurt Self 0 but benefit Selves 1 and 2, so under the intraself Kaldor-Hicks criterion, we cannot say that the change
is either a gain or a loss.




\bigskip

\noindent
{\bf  The Effect of Borrowing on the Optimality of Smoking }

 I just discussed the implications of borrowing for consumption, but
borrowing has implications for the optimality of smoking   too.
If borrowing possible, that  affects whether
  a Kaldor-Hicks improvement can be made in the smoking decision.
This is due to the income effect created by the possibility of
borrowing, which alters the wealth controlled by each self and thus
affects their willingnesses to pay for smoking changes.

Let us   start from a  position in which  borrowing is possible
without   intervention by the social planner: Selves 0 and 1 may borrow
against future income for present consumption. We will assume they
borrow at the same rate $r$ at which they  can lend.

The laissez faire result  is  that Self 0 would borrow and consume the
entire endowment. Selves 1 and 2 would consume zero, and Self 1 would smoke.
The extreme consumption result occurs because of the assumptions of linear
utility and    the rate of time preference  being    higher than the interest
rate.

Under the precommitment criterion, the  laissez faire  outcome of zero
consumption for Selves 1 and 2  is optimal, but not the outcome of Self
1  smoking   (except in   the extreme case of $\beta=0$, at which
Self 0 is indifferent about Self 1  smoking).   If $\beta$ is
positive   Self 0 would still borrow and consume   all of the
endowments, but  would use the   same critical value of
$\alpha > 1/\delta$ as before  to ban smoking.

    The intraself Pareto improvement and  Kaldor-Hicks criteria,
  agree with the  precommitment criterion  in not
requiring any change from zero consumption by Selves 1 and 2. Any such
change would harm Self 0, and the harm to Self 0 would be exactly
balanced by benefit to the other two selves,   a matter of
redistribution rather than any improvement in social surplus.

  What is more surprising is that unlike when borrowing was impossible,
the  Kaldor-Hicks criterion  now does not  require a ban on smoking whenever the
present value of utility harm exceeds  the utility  benefit ($\alpha > 1+r$).



 There is no scope for mutually beneficial trade of
wealth for wealth. But the possibility of borrowing has also eliminated
the possibility of mutually beneficial trade of wealth for a smoking
ban. As a result of Self 0's borrowing, Selves 1 and 2 have zero
wealth.   Self 2 cannot offer any wealth to Self 1 in
exchange for the smoking ban. Self 0 can. Would he offer enough?

The effect on Self 0's utility of  a bargain in which he gives $P$ to Self 1 in
exchange for a smoking ban is
\begin{equation}
  -P + \beta \delta (-1+ P) + \beta \delta^2  \alpha
\end{equation}

  The minimum $P$ that Self 1 would accept is $P=\frac{1-\beta \delta \alpha}
{1+r}$. Substituting that $P$ into the expression for Self 0's utility change
and solving for $\alpha$ yields the condition for Self 0's utility to rise from
the bargain:
\begin{equation}
\alpha  >  \frac{1 + \delta \beta r}{\beta \delta (1-\beta \delta + \delta(1+r)
)}
\end{equation}

     This condition is stronger than the precommitment criterion's condition of
$\alpha > 1/\delta$, but weaker than Self 1's condition for not smoking, $\alpha
> 1/(\beta\delta)$.  If $\beta =1$, it is equivalent to $\alpha > 1/\delta$, and
regulation is unnecessary because Self 1 will abstain from smoking whenever Self
0 would be willing to pay him not to smoke.\footnote{ The right-hand-side  is
less than $1/\beta \delta$ because $(1 + \delta \beta r) \beta \delta <  \beta
\delta (1-\beta \delta + \delta(1+r) )$ because $\beta \delta< \beta \delta
(1-\beta \delta + \delta$ and  $(\delta \beta r)  \beta \delta< \delta \beta r$.
The right-hand-side  is at least as great as  $1/\delta$ because $(1 + \delta
\beta r)   \geq  \beta   (1-\beta \delta + \delta(1+r) )$ because $ 1 \geq \beta
(1-\beta \delta + \delta)$ because that reaches a maximum at$\beta =1$,  when
$\beta   (1-\beta \delta + \delta) =1$.  }



\noindent
{\it Result 2: If the person   borrows enough against his future income, the
Kaldor-Hicks criterion might support a smoking ban only if discounting is quasi-
hyperbolic.   }



 Allowing borrowing  has changed the
distribution of wealth. Before, Selves 0, 1, and 2 each controlled $W$
under laissez faire (as well as Self 1 controlling the value of $X$, a
property right which is part of his economic wealth). Now, Self 0
controls all three incomes by being able to borrow them. Self 1 only
controls the value of $X$ and what income Self 0  lets him control
control; Self 2 only controls what income Self 1 lets him control. Thus,
Self 2 is too poor to participate in any Coasean bargain, though Self 0 can
still do so.




  
  



 It is interesting to think about what would happen if we reallocated
economic wealth by allocating the property right of the smoking decision
to Self 2 instead of Self 1 (a purely hypothetical re-allocation, since
Self 2 does not exist at the time of the decision).\footnote{xxx Note to author's self:
``If you asked a random group of economists how to assign property rights in a new society with a literate population so as to maximize the prices (time quantities), explicit and implicit (``shadow''), asking and offer, in the society they would almost certainly begin by giving each mentally competent adult the property rights to his own labor.''
(Posner, 1985, p. 93)}
 Then  smoking   would not
occur under laissez
faire because it would not maximize Self
2's utility. The intraself Kaldor-Hicks criterion would be satisfied,
because Self 1 would have no money   with which to buy
$X=1$ from Self 2.   Nor would Self 1 wish to buy it if he did have $W$ in
money, because the value of smoking to him is less than its cost to Self
2.  Similarly, if Self 0 started by controlling the decision (as he would, in
effect, under the precommitment criterion), Self 1 would be unable to buy the
right to smoke from Self 0, and Kaldor-Hicks would accept a smoking ban as
optimal-- though in that case, if Self 1 had the funds, he would be willing to
buy the right from Self 0 and Self 0 would be willing to sell.    The
criterion's recommendations thus  depend on the initial
allocation of property rights. \footnote{xxx Note to author's self:
 It is their lung too.
 Inheritance taxes==wealth taxes.
   In traditional societies, a man could not alienate his land. It was
his descendants too. Like multiple selves. Entail and Je3wish law.} 



\bigskip

\noindent
{\bf What if the Endowments Are Saved? }

Suppose we drop  the consume-early assumption and replace  it with a ``consume-late'' assumption:
\begin{equation} \label{consume-late}
     \delta \beta > \frac{1}{1+r}.
\end{equation}

In equilibrium Selves 0 and 1 would save their entire income under laissez faire, a result that neither the precommitment criterion nor Kaldor-Hicks would change.

 Self 1 would, however, still smoke, because the effect of smoking on Self 2's utility is unaffected by the rate of interest. And the precommitment criterion would still ban smoking.

Foreshadowing Result 3 below, however, Kaldor-Hicks would no longer support a smoking ban. That is because in our hypothetical transaction, Selves 1 and 2 transfer  consumption to Self 1, but now Self 1 would find this undesirable, a penalty rather than a reward.   Under the consume-late assumption, Self 1 gains utility  of 1 from his own consumption of 1 unit of income, but utility of $\frac{1+r}{\delta \beta}>1 $ from Self 2's consumption.  Self 0 would be passing along his entire endowment to Self 1 anyway, and so could not credibly condition the payment on Self 1 not smoking.   Hence,    potential gains from trade  are absent.





\bigskip


\noindent
{\bf 7. The Model with Concave Utility and Money Transfers across Time}

 So far we have established that internalities can justify paternalistic regulation in a particular model in which borrowing was  exogenously constrained, but that allowing borrowing eliminated the argument for  regulation by leaving the future self without wealth. Under a different assumption, all wealth was saved, and the motive for regulation was eliminated because the present self did not want to reduce the future self's consumption.  Those cross-time consumption decisions were corner solutions because of the assumption of linear utility.        What if  we  introduce a motive for  partial, interior- solution   ``bequests'' from one self to a later self?

   One  might guess from  Barro (1974) and Becker (1974) that bequests would have an important effect.  Barro (1974) applies the idea of ``Ricardian equivalence'' to intergenerational debt. If the government taxes  a future generation  to pay a pension  to the present generation, the direct effect is to make  the present generation  consume more and the future generation less. If the present generation cared enough to leave a bequest to the future generation, however, then when the pension is introduced the bequest will simply be increased to take account of the increase in future taxes.  Becker (1974) introduces the ``Rotten Kid Theorem''  that even an non-altruistic member of a family will act to increase family income  if the family's altruistic head gives transfers to every member of the family.

 Barro's model is simply about transfers of wealth, on which   the Kaldor-Hicks criterion is silent, but he makes the point that bequests can undo the effect of government intervention.  Becker (1974, p. 1078) goes further. Besides his best- known proposition  that each family member will maximize family income if some altruistic member makes transfers to everyone, this paper includes what we might call the  nightlight problem. Imagine an altruistic husband who  must decide whether to read in bed at night, when he knows that his wife would pay ten dollars if he didn't and he would   pay no more than six dollars for the right to read.     Being altruistic,  he cares       about his wife's utility enough to  be making monetary transfers to her. Given that he is going to reduce his own utility to increase his wife's, he wants to do it as efficiently as possible. He will refrain from reading at night,  because if he doesn't mind inflicting ten dollars of harm on his wife, he can do it by giving her ten dollars less to spend.\footnote{Becker also looks at the nightlight problem from the point of view of the wife, in a closer analogy to the Rotten Kid Theorem's result that even  a selfish child will act to increase family income.   If the wife controls the nightlight decision,   she will allow the husband to read at night if that is efficient, because he will compensate her for her lower utility with a bigger cash transfer.}   The nightlight problem is like our smoking example.   Bergstrom (1989) has shown that  the efficiency conclusion  and other extensions of the Rotten Kid Theorem to more than one consumption good   require caveats. If the wife chooses the amount of nightlight in Bergstrom's example, she will choose an inefficiently small amount of light because she knows the husband's choice of monetary transfer will depend on her choice and she acts strategically.

 To see what happens when consumption decisions are interior solutions, let us now give  Self 0 an endowment of $(W+K)$ instead of $W$  and make the utility of consumption strictly concave, a function   $U(C)$ with $U'>0$, $U''<0$.  Let us assume that borrowing is limited to an  amount $B$ in  borrowing period dollars.

 We will be able to establish the main result without analyzing the  consumption and saving decisions fully, because Self 1's decisions are what matter to the result.  Let us imagine a Kaldor-Hicks transfer of $P$ from Self 2 to Self 1 to ban smoking, with $P$ measured in period 2 dollars. This payment will cost Self 2 $ P$  and benefit Self 1 by $P/(1+r)$.  In addition, Self 1 may be borrowing amount  $B$, which will cost Self 2 amount $ (1+r)B$ in consumption.

 First let us look at Self 2's  consumption. If Self 1 borrows the full amount $B$ from Self 2, then   Self 2's consumption will be
 \begin{equation} \label{s1}
 C_2  =  W - (1-X)P  - (1+r)B
 \end{equation}

 If  Self 1 borrows amount $B_1 <B$ or saves,    then Self 2's consumption will
be
 \begin{equation} \label{s2}
\begin{array}{ll}
 C_2 & =  [W   -  X(1-X)P  -   (1+r)B_1]   +  [1+r][W_1 +  B_1 + \frac{(1-X)P}
{1+r}]\\
 \\
 & = W + (1+r) (W_1-C_1).
\end{array}
\end{equation}

   Self 1 is constrained to spend no more than his income on present
consumption, so
\begin{equation} \label{s3}
 C_1  \leq  W_1  + B + \frac{P}{1+r}.
\end{equation}
   If this is binding, then Self 1 is  borrowing the full amount $B$ from Self
2, and Self 2's consumption is given by equation (\ref{s1}). If it is not
binding, then Self 1 is saving or is borrowing less than $B$ and Self 2's
consumption is given by equation (\ref{s2}).

Self 1's maximization problem  is therefore  to choose $X$, $C_1$, and $C_2$ to
maximize
\begin{equation} \label{s4}
 U_1(C_1, C_2, X)  = X +U(C_1) + \beta \delta  (-\alpha X + U(C_2))
\end{equation}
 subject to the constraints
 \begin{equation} \label{s5}
\begin{array}{llll}
(a) &   C_1  &\leq    W_2 + B + \frac{(1-X)P}{1+r}& \\
  &  &&\\
(b1)& C_2 &  =    W - (1-X)P  - (1+r)B  & {\rm if \;}  C_1 =    W_1 + B +
\frac{P}{1+r} \\
 &  &&\\
(b2)  &    C_2 & = W + (1+r) (W_1-C_1)& {\rm if \;}  C_1 <    W_1 + B + \frac{P}
{1+r}. \\
\end{array}
\end{equation}

    The first possible case is  when constraint (a) is binding at the solution
to Self 1's problem, so he borrows the full amount $B$. In that case, we can
substitute for $C_2$ from constraint (b1) and write the problem  as   to
maximize, by choice of   $X$,
 \begin{equation} \label{s6}
\begin{array}{l}
  X +U(C_1) + \beta \delta  (-\alpha X + U(W  - (1-X)P  - (1+r)B) \\
\end{array}
\end{equation}

By definition of the case we are considering, $C_1 = W_1 + B + \frac{(1-X)P}
{1+r}$. As for the optimal choice of smoking,  Self 1 will choose to smoke ($X=
1$) if
    \begin{equation} \label{s7}
   1 +U(W_1 + B) + \beta \delta  (-\alpha   + U(W    - (1+r)B)  >
     U( W_1 + B+\frac{P}{1+r}) + \beta \delta    U(W  - P  - (1+r)B).
\end{equation}



    When utility was linear, the marginal utility of income was 1 and  the payment $P$ had to equal $ \frac{(1+r)(1 - \alpha \beta \delta)}{1- \beta \delta (1+r)}$,  in equation (\ref{necessary}).   Now that the marginal utility of income is not constant, the necessary payment $P$ must reflect the fact that for both Self 1 and Self 2  the marginal utility of income is changing with income. Thus, no simple expression is  available for the $P$ necessary to persuade Self 1 to not smoke.

 The second case is the more interesting: what if Self 1 finds it optimal to save,  or to borrow less than $B$? Then  we can  substitute for $C_2$ from constraint (b2) and write the problem  as   to maximize by choice of $C_1$ and $X$
 \begin{equation} \label{s9}
\begin{array}{l}
  X +U(C_1) + \beta \delta  (- \alpha X + U(W + (1+r) (W_1-C_1)). \\
 \end{array}
\end{equation}
 The problem   needs no budget  constraint, because by definition of the case,
income does not constrain the choice of $C_1$. Rather, the desire to save
constrains it.

 In this problem, the condition for choosing $X=1$ is that
 \begin{equation} \label{s10}
\begin{array}{l}
  1  - \beta \delta   \alpha >0, \\
 \end{array}
\end{equation}
and the  first order condition  with respect to $C_1$ is
\begin{equation} \label{s11}
\begin{array}{l}
  U'(C_1) + \beta \delta    (1+r) U'(W + (1+r) (W_1-C_1) =0 \\
 \end{array}
\end{equation}

        The price $P$ appears in neither of these two optimality conditions. Hence,    Self 2's offer of $P$ in exchange for the smoking ban makes no difference. Either Self 1 would abstain from smoking  anyway, or he would  not be persuaded by the offer of money from Self 2.  If discounting is exponential, then it is only Self 2 who would be willing to offer money to Self 1 in exchange for not smoking, because Self 0 shares Self 1's incentives, as we saw earlier. Thus, Kaldor-Hicks no longer can support a smoking ban.

What  if discounting is hyperbolic? Then Self 0 would be willing to pay something to Self 1 for a smoking ban. He would not, however, pay enough. In equilibrium, Self 0 has chosen a particular level of saving such that the marginal utility of a dollar consumed equals the marginal utility from saving that dollar.  The loss to Self 1 from a smoking ban is $(1-\beta \delta \alpha)$ but the gain to Self 0 is the smaller amount    $ (\beta \delta -\beta \delta^2 \alpha)$. Thus, even  if Self 0 viewed the marginal utilities of dollars consumed versus dollars saved as equal, he would be unwilling to make the payment.


\noindent
{\it Result 3: If discounting is exponential and a person  is saving a positive amount or his borrowing is unconstrained, then the intraself Kaldor-Hicks criterion does not support a paternalistic smoking ban. }

  If Self 1  cares enough about Self 2 to pass savings along to him,  then the intraself Kaldor-Hicks criterion does not require Self 1 to  give up smoking for the sake of Self 2.   If Self 1 is selfish enough not to save,    the criterion does  require him to give up smoking.

The  intuition behind Result 3 is that when Self 1 is saving, the   marginal dollar of saving results in period 2 consumption that gives him as much utility (discounted though it is)  as if he spent that dollar on period 1 consumption. He would be hurt by anything that transferred that consumption back to period 1. Thus, if he consumes the Kaldor-Hicks payment given him by Self 2, he is actually worse off.

 Similarly, if Self 1 is borrowing less than the limit, he has decided that though overall he would like to reduce Self 2's consumption to increase his own, at the margin, once he has borrowed the appropriate amount,  his utility is the same from having Self 2 consume   a little as from having himself consume more.

\bigskip

What if discounting is hyperbolic and the person is passing along savings?  The general expression  for  the effect on Self 0's utility of an increase in saving beyond  laissez faire   is too complex to be  illuminating, since it would affect Self 1's saving behavior  and Self 2. A numerical example  will illustrate how regulation can be optimal even when there is savings.    Assume that $W=1$, $K=6$, $r=0$, $B=.2$, and that the utility function has marginal utilities of
\begin{equation} \label{s12}
\begin{array}{lll}
  U'(C ) & = 2  & {\rm  if} \;\; C < 2\\
  & = 1  & {\rm  if} \;\; C >2.\\
 \end{array}
\end{equation}

This person has some incentive to arrange the consumption path so that
consumption equals at least 2 in each period.  We will retain the assumption
that smoking yields 1 in utility to Self 1 and costs $\alpha  \in (1, \frac{1}
{\beta \delta})$ in utility to Self 2.

(1) Suppose $\beta \delta <.5$.  In that case,  high time preference overrides
the incentive to smooth consumption. There is no saving, and Self 2 will pay
Self 1 not to smoke.     The laissez faire outcome will be $C_0=7.2, C_1=1, X=
1$, and $C_2=.8$.  If  Self  0 or 1  deviates by saving more, he will lose
utility at  a rate of 1 per reduced consumption unit and gain it at only a rate
of $2\beta \delta$ from his successor's increased consumption.  Thus, Self  0
will borrow up to the limit of $B=.2$ from  Self  1,  and  Self  1 will borrow
up to the limit from  Self 2. Self 1 will smoke because smoking raises his
utility by 1 directly and reduces it only by $\beta \delta \alpha$ through its
effect on Self 2.

   The Kaldor-Hicks Criterion can support a smoking ban  if $\beta \delta <.5$
even if $\beta=1$ because Self 1 would accept as little as $(.5-\beta \delta
\alpha$  to stop smoking (.5 rather than 1 because his   marginal utility of
consumption  is 2, whereas smoking only increases utility by 1) and  Self  2
would be willing to pay up to  $\alpha/2$, an amount greater than .5 since
$\alpha >1$.  We do not need to resort to payments from Self 0.

(2) Suppose instead that     $.5<\beta \delta<1$, in which case consumption
smoothing  outweighs discounting to some extent.  The laissez faire outcome will
be $C_0=5, C_1=2, X=1$, and $C_2=2$.  If  Self  0 or 1  deviates by saving more,
he will lose utility at  a rate of 1 per increased consumption unit and gain it
at  the lower  rate of $ \beta \delta$ from his successor's increased
consumption.
 If  he  deviates by saving less, he will gain utility at  a rate of 1 per
increased consumption unit and lose it  at  the higher  rate of $2\beta \delta$
from his successor's reduced consumption.

Under laissez faire, Self 1 will smoke because it adds $(1- \beta \delta \alpha)
$ to his utility.

   There is no payment by Self 2 that will persuade Self 1 to stop smoking.  A
payment of $P$ by Self 2 will  have a net effect of $(P  - 2\beta \delta P)<0$
on Self 1's utility,  so Self 1 would refuse the payment even if it were not
contingent on   not smoking.

  Self 0, on the  other hand,  could pay $P$ to Self 1 in addition to the
laissez faire bequest of 3 and Self 1 would have no objection.  The effect on
Self 0's utility  would be to add
\begin{equation} \label{s13}
  -P +  \beta \delta (-1+P) + \beta \delta^2 (\alpha).
\end{equation}


 Self 1  would accept a payment of as little as $P=(1-\beta \delta \alpha)$ to
not smoke. Substituting that for $P$ in   expression \eqref{s13}  tells us that Self
0's utility would rise  after making that payment to stop smoking if
\begin{equation} \label{s14}
 \alpha > \frac{1}{\delta \beta (1- \beta \delta +\delta)},
\end{equation}
 an expression similar to what we found  in conjunction with Result 2 that even
with borrowing  hyperbolic discounting could justify a smoking ban under the
Kaldor-Hicks criterion.

      We have assumed that   $\alpha < \frac{1}{\delta \beta}$,  but since
$(1-\beta \delta - \delta)>1$,   the critical value of $\alpha$ for Self 0 to be
willing to pay to stop smoking is feasible.  There is a range of values of
$\alpha$-- though  narrow one-- for which Self 1 would not abstain from smoking
unilaterally but would after a payment that Self 0 would be willing to pay.
This does rely on hyperbolic discounting, since if $\beta =1$ our condition for
Self 0 being will to make the payment reduces to $\alpha >1$, the same condition
as for Self 1 to unilaterally abstain from smoking when $\beta=1$.  Thus we have
Result 4.


\noindent
{\it Result 4: If discounting is  hyperbolic and a person  is saving a positive
amount or his borrowing is unconstrained, then the intraself Kaldor-Hicks
criterion can still  support a paternalistic smoking ban. }




\bigskip

\noindent
{\bf 8.   Why Do We Use the Kaldor-Hicks Criterion? }

   The multiple selves model is a fascinating change of perspective for libertarians, or, indeed, for anyone who cares about either freedom of choice or equality.   It is relatively easy to dismiss a multiple self model in which there are two more present selves whose desires conflict.   That model reduces to the old problem of whether a person's  actions should be constrained for his own sake or not, and our affection for equality hardly enters-- we don't feel bad about an impulsive self not being able to consume as much as a disciplined self, who in any case will share the  same pleasurable sensations.   The multiple selves we have been considering, however,  do not exist simultaneously,  do not have moral ranking,  and do not share the benefits and costs equally.

   This application has stretched the Kaldor-Hicks criterion considerably.  The problem is this. The present self might be willing to accept an anonymous payment of \$3,000 to not smoke, and the future self might be willing to pay \$4,000 in present-period dollars. The present self is {\it not} willing, however, to accept \$3,000 from the future self, because he is altruistic and prefers that the future self spend that money, not himself. Thus, because the   present self  cares  so much about the future self, he refuses to help him. In the end, though since the compensation is not actually paid, his concerns about the decline in the future self's consumption don't matter anyway. 

This is a little-known problem,\footnote{Though see Friedman (1988), which  makes the same point in connection with whether transfers from an altruist to someone else will be inefficiently small. }  but it is well known that the   foundations of the compensation criterion---indeed, of the entire idea of economic efficiency and cost-benefit analysis-- are  poorly developed.    I am not here thinking of  the more superficial problems of   whether we use the Kaldor test or the Hicks test, or of the Scitovsky Paradox.  Rather, why do we think that   it is good to remove a tariff that is worth \$1 million to   producers but which consumers would pay \$10 million to remove?    If the consumers actually paid producers \$5 million, there would be a Pareto improvement, and we consequentialist economists would have good reason to support the change.\footnote{Even  Pareto optimality is challenged by non-consequentialists. Someone might believe that the producers have a right to the tariff and that process is important:  they must not only benefit from its removal but must formally agree to it. We need not investigate that here. } If the compensation is not paid, the tariff removal is helping some people at the expense of others. Why are we willing to do that?  


  One answer is that by removing the tariff we have moved from a pareto inferior allocation to a 
 Pareto optimum.  That, however, begs the question. If compensation truly cannot be paid, then the original situation is pareto optimal too. Certainly removal of the tariff is not a pareto improvement. 

 A second answer is that by removing the tariff we have moved
one step closer to a Pareto improvement, and it is better to be closer than further away. Why it is better to be closer is unclear, however, an example illustrates. Suppose    Smith has a dog, which he likes, but he'd like a cat better.
Jones has a cat, which he likes, but he'd like a dog better. Neither
wants to have both a cat and a dog, because the dog will fight the cat. 
A Pareto improvement would be for Smith to send his dog to Jones's house, and Jones to send his cat to Smith's.  If we stop after  Smith sends his dog to Jones's house, though, both Smith and Jones are strictly worse off.  Going partway to the pareto improvement is a pareto worsening. 

A third answer is that it is wrong to focus on just one policy change. If we use the compensation criterion for public policy generally, everyone will be better off than if we did not (see, e.g., Zerbe (2001)). This argument is much like the contractarian argument for involuntary submission to government power:  everyone is better off   overall with a government, even if they grumble about particular government policies. 

   A problem with this third answer is that it relies on a bundling of use of the compensation criterion for all policies. Why not, for example, use the compensation criterion for all policies except for those involving a person's multiple selves over time? Self 1 in our model would applaud this, since he  never wins from application of the compensation criterion. 

  A fourth answer is to fall back on utilitarianism (or common sense?)  and assert that other things equal the world is better if one group of people gains \$10 million and another loses \$1 million. This answer abandons the traditional claim that economics does not admit interpersonal comparisons of utility, but this claim has always been dubious.   We make such comparisons constantly in our daily interactions with family and friends. The     economist's ability to persuade people that efficiency should influence public policy is perhaps based on this too. 

The utilitarian explanation, however, does not require that the payment to the losers from a policy change be made by the winners.  If the objective is to measure willingness to  receive instead of to advance towards a pareto improvement, it should not matter whether the payment comes from the policy winner, from the government, or from some randomly selected victim. 

 In our multiple selves model, or any model with altruism,  it is much simpler to  calculate how much the loser would accept from an anonymous third party than from a winner  towards whom he is linked by altruism. Whether there is saving, borrowing, or neither, all that would matter is the gains and losses from smoking.\footnote{It is important not to think of the test as an auction between winners and losers, however. If Self 1 was saving and valued Self 2's consumption, then he  would want to win an auction in competition against  Self 2  partly to avoid Self 2's having to pay the auctioneer and reduce consumption.}    Thus, I would advocate dropping the Kaldor-Hicks criterion for    the Marshall  criterion defined below.\footnote{I name this ``Marshallian'' following Friedman (1988).  Marshall says in his discussion of consumer surplus in  Book III, Chapter 6 of {\it The  Principles of Economic Analysis}, ``In the same way if we were to neglect for the moment the fact that the same sum of money represents different amounts of pleasure to different people, we might measure the surplus satisfaction which the sale of tea affords, say, in the London market, by the aggregate of the sums by which the prices shown in a complete list of demand prices for tea exceeds its selling price.''     }
 
    
\begin{small}

\noindent
  {\it  The Intraself Kaldor-Hicks
Criterion:   A change in the value of the choice variables is an
improvement if it could be combined with  money transfers     such that the  combined changes  would increase    the utility of at least one  self and reduce     the utility   of none.     }

   


\noindent
  {\it  The Intraself   Marshall
Criterion:   A change in the value of the choice variables is an
improvement if the winners would be willing to pay more to the social planner to implement it than the losers would be willing to pay to prevent it. 
      }
 
\end{small}

   Our standard use of the term ``efficiency'' does not distinguish between these two criteria.  The term  ``wealth maximization'' as used in law and economics, and, notably, by Richard Posner (1979, 1985) is usually framed as  what I call the Marshall criterion, following Friedman (1988).  It  is a simpler idea than the Kaldor-Hicks criterion when altruism enters the picture,  and better represents the utilitarianism that I think  lies behind efficiency's attractiveness. 



 

\bigskip

\noindent
{\bf 9. Concluding Remarks}

 This this paper  I have asked whether applying a version  of the Kaldor-Hicks criterion  to a multiple selves model provides a justification for  intervention by a benevolent social planner. It does provide a justification,  with caveats. A first requirement is that the activity cause more immediate harm than it creates future benefit. If so, this has a ripple effect under dynamic inconsistency because not only does the far future self   suffer more than the near future self benefits, but the present self   suffers sympathetically with the far future self. A second requirement is that the extent of the harm outweigh the real cost of transferring consumption to earlier in time, which is the interest rate ($r$ in the model).    A third requirement  is that if discounting is exponential   the the person not be saving between the time he engages in the harmful activity and the time the harm occurs, or not be borrowing the maximum possible. Otherwise, the earlier self would not value transfers to himself from the future self, and there are no potential gains from trade.   Even if the person engaged in the harmful behavior is making a bequest, however, if discounting is hyperbolic the Kaldor-Hicks criterion can justify restricting his behavior for the sake of selves prior to himself in time who care more than he does for consequences in the far future.

    Thus, we may conclude that hyperbolic discounting and  the  precommitment condition of maximizing the utility function of the pre-action self are not essential for paternalistic arguments for regulating internalities.  For someone who is impatient and constrained in his borrowing, the intraself Kaldor-Hicks criterion can justify regulation even if the person's discounting is exponential. Regulators often are most concerned to regulate the young and poor because they may have poorer information and self-control.  The present model suggests a quite different reason: that they are credit constrained and would be willing to accept payment from their future selves  to abstain from vice.

   Even if the person is saving, the intraself Kaldor-Hicks criterion can justify regulation if discounting is hyperbolic,  but for a narrower range of parameters and for a different reason.  The two reasons for regulation have different motives. When discounting is exponential, regulation  is balancing the vice's harm  to the future self against his willingness to give up consumption. If discounting is  hyperbolic, regulation  is  adjusting   for the pre-vice self's  lack of present-orientedness in balancing the interests of the future selves. Even under hyperbolic discounting, however, the intraself Kaldor-Hicks criterion justifies regulation more narrowly than the precommitment criterion does, because  Kaldor-Hicks requires potential compensation to the regulated self.



\newpage

\bigskip


\noindent
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