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\begin{center}

 \begin{Large}

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

\end{Large}


\bigskip



 \bigskip

27 October   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. The present paper shows that  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.




\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. BU 438, 1309 E. 10th Street,
Bloomington, Indiana, 47405-1701. Office: (812) 855-9219.
Fax:812-855-3354. \href{mailto:erasmuse@indiana.edu}{
erasmuse@indiana.edu},  \hspace{5 pt} \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 and the Midwest Law and Economics 2008 meeting
for
their comments, and Nuffield College, Oxford, for its hospitality.


\end{small}

\newpage




\noindent
  {\bf  1.   Introduction}

     A common justification for laws to discourage  vice is that
although the vice  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. They call these consequences     ``internalities'' between
different time-selves of the same person in analogy to
``externalities'' between different people.   Their model assumes
hyperbolic discounting and as  a welfare criterion they maximize  the
utility of the first of the multiple selves.



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





\bigskip

\noindent
{\bf 2.  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.     If the person chooses
to smoke  he receives 1  unit of utility in period 1  and loses
$\alpha $ in period 2, where   $ \alpha >1$ so  the loss
is bigger than the gain.

  Total utility is:
\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}

  The  ``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, where we
will also
  assume   that  utility is
linear in
consumption ($U(C)=C$).




The  utility function includes both the case of  the conventional
exponential discounting  and   quasi-hyperbolic discounting in the
style of Laibson
(1997): our person has an exponential discount factor of
$0  \delta $   and a hyperbolic adjustment
parameter of $\beta$.     If   $\beta=1$ he has a standard
exponential-discounting utility function.
A value of $\beta$ between 0 and 1 leads to the standard features of
hyperbolic discounting: besides preferring earlier consumption to
later, the person puts   special value on immediate consumption: at
period 0 his tradeoff between
consumption in periods 1 and 2, both distant, is different from the
tradeoff he would make in period 1, when period-1 consumption is
immediate.




   This person  onsists of three selves, Self 0,
Self 1, and Self 2. Self 0's utility function is \eqref{u0}.
Self 1 and Self 2 have the  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}


     Self 0 controls the value of $C_0$, Self 1 controls $X$ and
$C_1$, and Self 2 controls $C_2$. 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 $C_2 = W + (1+r) (W_1-
C_1)$.

Self 1 would maximize (\ref{u1})  by choice of $X$ and $C_1$ subject
to the constraint that $C_2 = W + (1+r) (W_1- C_1)$.   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
\begin{equation} \label{b1}
  \alpha<\frac{1}{\delta\beta}.
\end{equation}



Self 0
will choose $C_0=W$ because of the consume-early assumptino.
 If Self 0 could control  smoking, he would choose for Self 1 to smoke
if
\begin{equation} \label{c3}
\alpha<\frac{1}{\delta}
\end{equation}




\bigskip
\noindent
{\bf 3.  Welfare Analysis: The Precommitment Criterion }

   Is the laissez faire outcome optimal?
  An  obvious   welfare criterion   is
intraself Pareto optimality:
  no change can be made which would
make some self better off without making some other self worse off.
This has
frequently been applied to multiple selves models (e.g. in  Bernheim
\& Rangel (2007)).
 The three selves, however,  have contradictory
preferences for
smoking  and consumption. When Selves 0 and 2 do not want smoking but
Self 1 does,  what should the social planner do?


A second  welfare criterion 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.
This is   is    the same criterion
as  we follow in a single-self model,  simply
recognizing that the social planner will  have to  somehow constrain
the future selves to  act in accord with Self 0's wishes.
The precommitment criterion is   commmonly used   in the hyperbolic
discounting
literature, almost always in connection with decisions about
consumption and savings (e.g. Laibson (1997),   Krusell \&   Smith
(2003) ).  It has two variants.  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. 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.



 Gruber  \& Koszegi  (2001) use a model of a divided self to look at
whether smoking should be taxed. A quasi-hyperbolic discounter lives
for a finite
number of periods and must decide   the amount of smoking and  other
consumption for each period.     They  show that a large tax on
cigarettes is optimal if the goal is to maximize Self 0's utility.
O'Donoghue  \&   Rabin (2003) also  look  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.     Bhattacharya  \& Ladkdawalla
(2004)  look instead at interself
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.

What happens in our three-self model if we use the precommitment
criterion?



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    if there is hyperbolic discounting
($\beta < 1$).
  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 distant-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 that quasi-
hyperbolic discounting makes the present period count too heavily.

If  discounting is exponential, i.e. $\beta=1$,  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 smoking utility than period 2 smoking
disutility.





The main objection to the precommitment criterion is its
arbitrariness.  Caplin \& Leahy (2004) and Whitman
(2006) criticize   focussing on   the welfare of the ``long-run ''
or earliest self-- the ``dictatorship of the present''.  Why privilege
Self 0 rather than some later self?   Of course, the criterion 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 we do not ordinarily argue that tastes are illegitimate.
Moreover, the criterion disregards not only the desire of Self 1, the
self making the choice about smoking, but also the desires of Self 2,
the post-action self.

\bigskip




\noindent
 {\bf 4.    The  Kaldor-Hicks Criterion}

  An alternative  to the precommitment criterion is to apply   the
Kaldor-Hicks criterion, but within the self,  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.




 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. This captures
the idea with which we started
that a vice causes more future disutility than it causes  present
utility.

Like the precommitment criterion, it  does not suggest changing
consumption alone from laissez
faire. Because the discount rate exceeds the interest rate, a later
self could not compensate an earlier self for saving more.


The smoking choice    is different. It   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 laissez faire decision of $X=1$?   Selves 1 and 2
never meet, so the
transaction is   impossible, but Kaldor-Hicks only requires a
potential Pareto improvement, not an actual one. There does exist a
complication: 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 also affected by
any
transaction between Self 1 and Self 2.  Thus, when we start to
consider transfers, the question of whether a payment would compensate
is more complicated than it would be between strangers.

  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 to Self 1 of $P_0$ and $P_2$ as measured  in giving-
period dollars.   Since our starting point is  $U(C_t) = W $ in  each
period, the
differences between the  utilities  when  the smoking bargain
is made ($X=0$) and when it is not ($X=1$)  are:
\begin{equation} \label{u0s}
\Delta 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}
\Delta   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}
\Delta  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 a condition for    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 $P_0$. Working
through the algebra yields the necessary condition:\footnote{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 0
must be able  to pay the
remainder.  }
\begin{equation} \label{a2}
  \alpha  > 1+r.
\end{equation}

If the difference  between Self 2's loss from smoking and Self
1's gain exceeds the interest rate, smoking should be banned.

 Nothing in this argument has relied on
hyperbolic discounting. We could have assumed $\beta=1$ and the
conclusion would be the same. Thus we have Result 1.

\noindent
{\it Result 1: The intraself Kaldor-Hicks criterion
can justify paternalistic banning of a vice even if discounting is
exponential.
}


   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.
Result 1  relies 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, still bans  smoking if
$\alpha > 1+r$, which is to say if Self 2 loses more from Self 1's
smoking  than can be compensated 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.

Contrast the conditions under which a smoking ban raises welfare under
our two welfare criteria.


 \begin{equation} \label{ae3}
Precommitment: \alpha \in \left( \frac{1}{\delta }, \frac{1}
{\delta\beta} \right)
\end{equation}

\begin{equation} \label{ae2}
  Kaldor-Hicks:    \alpha  > 1+r.
\end{equation}

In both, greater smoking damage   $\alpha$ makes a ban more
attractive.  Under the precommitment criterion,  the smoking damage
has to be large relative to the time preference discount factor,
because   under hyperbolic discounting  ($\beta <1$) Self 1  attaches
more weight on period 1 actions  than Self  0 would like. Under the
intraself Kaldor-Hicks criterion, the smoking damage has to be large
relative to the interest rate, because it then outweighs the fact that
Self 1 gets less consumption out of having a period 1 dollar than Self
2, able to wait a period and earn interest on it, would get from that
same dollar.






\bigskip







\noindent
 {\bf 5. Borrowing and Saving in the Linear-Utility Model}



\noindent
 {\bf  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
  the rate of time preference is high and
utility  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 Selves 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.

   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 the
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,   the benefit in
current consumption must be  worth the loss in future consumption.
That is
just to say that Self 0 benefits from borrowing, however, despite the
loss to Selves 1 and   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.

  We would have to resort to some other
argument to  justify the allocation of property rights among the three
selves. Richard Posner (1985, p. 93) says
``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.''  Here, that would imply a  ban on
borrowing (unless the future self would also favor borrowing), so that
each self would be able to keep (or pass along) the
fruits of his own period's labor.



\bigskip

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

 What is the effect of borrowing on smoking?
The laissez faire result  is  that Self 0 would borrow and consume the
entire endowment,  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,  zero consumption for Selves 1 and
2  is optimal, but whether Self 1 should smoke depends on whether
condition \eqref{ae3} is satisfied.

    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 could offer some wealth to Self 1 in exchange for a smoking
ban. 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} \label{condition1}
\alpha  >  \frac{1 + \delta \beta r}{\beta \delta (1-\beta \delta +
\delta(1+r)
)}
\end{equation}

     Condition \eqref{condition1} 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 can support a smoking ban only if
discounting is quasi-hyperbolic.   }


  The result is even stronger if 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 incomes, and
Self 1 would smoke  if $   \alpha<\frac{1}{\delta\beta}$.
Foreshadowing Result 3 below, however, Kaldor-Hicks would no longer
support a smoking ban. 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 6. The Model with Concave Utility and Money Transfers across
Time}

 So far,  consumption decisions have been 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) (on ``Ricardian
equivalence''  and  intergenerational debt) and    Becker (1974) (on
intrafamily transfers) that bequests
would have an important effect, because both of those authors find
that money transfers can neutralize behavior changes.




Let us now   make the utility of consumption the  strictly
concave
$U(C)$ with $U'>0$, $U''<0$, and $\stackrel{limit}{C \rightarrow 0}
U'(C) = \infty$.  Assume that borrowing is limited
to an  amount $\overline{B}$ in  borrowing period dollars. Let us also
abandon the consume-early assumption.

  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  amount $ P$  and increase
Self 1's consumption  by
$P/(1+r)$.  In addition, Self 1 may be borrowing amount  $B \leq
\overline{B}$, which
will cost Self 2 amount $ (1+r)B$ in consumption.

Let $W_1$ equal Self 1's income of $W$ plus whatever Self 0 saves,
minus whatever Self 1 borrows from him.
   Self 1 is constrained to spend no more than his income on present
consumption, so
\begin{equation} \label{s3}
 C_1  \leq  W_1  + \overline{B} + \frac{P(1-X)}{1+r}.
\end{equation}
   If this is binding,   Self 1 is  borrowing the full amount
$\overline{B}$.
Otherwise, he is saving or   borrowing less than $\overline{B}
$.

Self 2's consumption  equals   his  income of $W$ plus $(1+r)$
times the amount Self 1 saves, $(W_1-C_1)$:
\begin{equation} \label{s3a}
 C_2   = W + (1+r) (W_1-C_1)
\end{equation}


Self 1's maximization problem  is   to choose $X$ and $C_1$  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_1 + \overline{B} + \frac{(1-X)P}{1+r}& \\
 &  &&\\
(b)  &    C_2 & = W + (1+r) (W_1-C_1)&   \\
\end{array}
\end{equation}


  Consider first the case where  constraint (a)  is not binding.
Instead,  Self 1 finds it optimal to save,  or to borrow less than
$\overline{B}$. Then  we can substitute for $C_2$ from constraint (b)
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
income does not constrain the choice of $C_1$.

 The condition for choosing $X=1$ is
 \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 the  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, as Result 3
states.



\noindent
{\it Result 3: If discounting is exponential and a person's  borrowing
constraint is non-binding (including if he is saving a
positive amount), then the intraself
Kaldor-Hicks criterion does not support a   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.
There is no Kaldor-Hicks inefficiency, but it is not because the
altruist abstains from seemingly inefficient behavior, but because he
will not accept compensation for abstaining.

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.

If constraint (a)   is binding,  there might or might not be a Kaldor-
Hicks transfer that would induce Self 1 to stop smoking.
 Consider what happens if our person care only about the present, so
$\delta =0$. Then $P^*$ is  such that
   \begin{equation} \label{s7}
   1 +U(W_1 + \overline{B})   =
     U\left( W_1 + \overline{B}+\frac{P^*}{1+r}\right).
\end{equation}

Self 2 will be willing to pay $P^*$ if
\begin{equation} \label{s7a}
   U(W -  (1+r) \overline{B} - P^*)  \geq  -\alpha +  U(W -  (1+r)
\overline{B}).
\end{equation}

  If $U''$ is close to zero, so utility is almost linear, and if $r$
is low then the gain in Self 1's consumption utility   from the
transfer of $P^*$ will be close to the loss in Self 0's, and   Self 2
would indeed be willing to pay $P^*$.  If, on the other hand,  the
utility of cosumption  flattens beyond Self 1's initial consumption
level of  $W_1 + \overline{B}$ but increases enough for lower levels
of consumption, then Self 1 will be unwilling to deprive Self 2 of
consumption via the transfer of $P^*$. Thus, there is  a possibility
of regulation being optimal for borrowing-constrained individuals.

This may be easier to see in a numerical example.
 Assume that $W=1$, $K=6$, $r=0$,
$\overline{B}=.2$, and
\begin{equation} \label{s12}
\begin{array}{lll}
  U'(C ) & = 2  & {\rm  if} \;\; C < 2\\
  & = 1  & {\rm  if} \;\; C >2.\\
 \end{array}
\end{equation}



\noindent
(1) Suppose $\beta \delta <.5$. 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.    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. Smoking raises Self 1's
 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, 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 $\alpha >1$.


\bigskip



\noindent
(2) Suppose instead that     $.5<\beta \delta<1$.  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
$2\beta \delta$.



   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.
Self 0's utility  increase by
\begin{equation} \label{s13}
  -P +  \beta \delta (-1+P) + \beta \delta^2 (\alpha).
\end{equation}


 Self 1  would accept   as little as $P=(1-\beta \delta
\alpha)$ to
not smoke. Substituting   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}

    There exists a range of values of $\alpha$ for which Self 1 would
not abstain from smoking unilaterally but would after a payment that
Self 0 would be willing to pay. This  relies on hyperbolic
discounting, since if $\beta =1$condition \eqref{s14} reduces to
$\alpha >1$, the same condition as for Self 1 to unilaterally abstain
from smoking. Thus we have Result 4.


\noindent
{\it {\bf  Result 4: }
The intraself Kaldor-
Hicks
criterion can    support a  smoking ban for someone whose borrowing
constraint is binding  if discounting is hyperbolic.  }





\bigskip

\noindent
{\bf 9. Concluding Remarks}

 Applying  the
Kaldor-Hicks criterion  to a multiple selves model  does provide  a
justification for    a   social planner to ban vice, though   with
caveats. First, the
vice must cause more immediate harm than it creates future
benefit.   Second, the extent of
the harm must outweigh  the    cost of transferring consumption  to an
earlier period.    Third,
  if discounting is exponential, the person must be constrained in his
borrowing.
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  far-future
consequences.





\newpage

\bigskip


\noindent
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