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\begin{document}
\begin{center}
\textbf{\ }
{\Large Axioms of Choice}
\bigskip
\bigskip
Graciela Chichilnisky
Columbia University\bigskip
\bigskip
\textbf{Workshop on Time Preferences}
Copenhagen May 2004\newpage
\end{center}
We propose new axioms for choice over time, and derive the optimality
criteria that they imply.
\bigskip
The axioms treat the present and the future symmetrically, giving a rigorous
meaning to `sustainable development.'
\bigskip The following criteria do not satisfy our axioms
1. Present discounted value
2. Ramsey's criterion
3. Rawlsian rules
4. Overtaking criterion (von Weisacker's)
5. Long run averages \pagebreak
We characterize all the optimality criteria that satisfy our axioms, and
provide examples.
\bigskip
These are a convex combination of `present discounted value' (with constant
or variable discount rates) plus purely finite measures that assign all
weight to the future.
\bigskip
They require a new type of calculus of variations and Euler Lagrange
conditions.\pagebreak\
\begin{center}
\textbf{Choice under Uncertainty}
\bigskip
\end{center}
The axioms apply to choices under uncertainty, providing a symmetric
treatement of small and large probability events.
They resolve the Allais paradox, as well as the `equity premium puzzle'.
\pagebreak
In the `equity premium puzzle' our resolution focuses on market crashes as
well as small probabilities of bankrupcy, both of which exist for equity but
not for bonds.
\bigskip
Such small probability events seem to be underestimated within the standard
expected utility theory.\pagebreak
\begin{center}
\textbf{Discounting}
\end{center}
The present and the future are often linked through `discounting' rates, by
which value tomorrow becomes a fraction of the same value today.\bigskip
$\$1$ in the bank today (or at time $t)$ is deemed equivalent to $%
\$1(1+\lambda )$ tomorrow (or at time $t+1)$, where $\lambda $ is a
`discount factor
\[
0<\lambda <1.
\]%
\pagebreak
Discounting the future leads to ranking a stream of income $%
x=\{x_{t}\}_{t=1,2,...}$ according to its `present discounted value'
\[
W(\alpha )=\sum_{t=1}^{\infty }u(x(t))\lambda ^{-t}
\]
where $0<\lambda <1$ is the `discount factor' and
\[
u(x(t))=\alpha (t)\text{ }
\]
is the utility obtained at time $t$ from $x(t)$ provided by the stream $x$
at time $t$. \pagebreak
\begin{center}
\textbf{Cost Benefit Analysis}\bigskip
\end{center}
Present discounted value is a standard tool enforced often in the US to
select among large federal projects.
\bigskip
It is used to decide whether the present value of the streams of benefits
exceed the streams of costs, called `cost benefit analysis':
\[
B-C(x)=\sum_{t=1}^{\infty }[B(x(t))-C(x(t))]\lambda ^{-t}
\]%
\pagebreak
The problem with `present discounted value' is that it underestimates those
projects that have large initial costs -- but where the positive returns are
in the long term future.
\bigskip
This is typical of large environmental problems such as
1. global warming prevention
2. the disposal of nuclear waste.
\bigskip
Long run benefits are underestimated. \pagebreak
Below we show that presented discounted value is actually `insensitive' to
events in the long run future, as defined below.
Insensitivity to the future conspires against `sustainable development'
defined as `satisfying\newline
the needs of the present, while protecting the needs of the future'
(Brundtland Report, 1987).
The future is to a great extent as important as the past. But this is not so
in standard cost benefit analysis.
What is the alternative?
\begin{center}
\pagebreak
New Axioms
\bigskip
\end{center}
(i) Sensitivity to the future
(ii) Sensitivity to the present
(iii) The ranking of paths is continuos in $L_{\infty \text{ }}$and linear,
which are standard assumptions, satisfied for example by 'present value'
criteria.
We need some definitions.\newpage
\begin{center}
Mathematical Formulation
\end{center}
For each $t=1,2,...,\alpha _{t}\in R$ represents the utility of generation $%
t.$
$\alpha $ $=\alpha _{t}$ and $\beta =\beta _{t}$ , are paths of utility
across time.
These are bounded: $\alpha ,\beta \in F\subset L_{\infty },$ and $%
\sup_{\alpha \in F}\Vert \alpha \Vert _{\infty }W(\beta )\Leftrightarrow W(\alpha ^{\prime })>W(\beta ^{\prime })
\end{equation}%
where $\alpha ^{\prime }$ and $\beta ^{\prime }$ are obtained by modifying
arbitrarily $\alpha $ and $\beta $ beyond some period $T=T(\alpha ,\beta ).$
\begin{center}
Axiom (i) rules out insensitivity to the future\bigskip \pagebreak
\end{center}
A ranking is said to be `insensitive to the present' when
\begin{center}
\begin{equation}
W(\alpha )>W(\beta )\Leftrightarrow W(\alpha ^{\prime })>W(\beta ^{\prime })
\label{two}
\end{equation}%
\bigskip where $\alpha ^{\prime }$ and $\beta ^{\prime }$ are obtained by
modifying arbitrarily $\alpha $ and $\beta $ in any finite number of periods.
Axioms (ii) rules out insensitivity to the present.\newpage Examples \bigskip
\end{center}
Proposition 1. Discounted utility functions such as
\[
W(\alpha )=\sum_{t=1}^{\infty }\lambda ^{-t}\alpha _{t}\text{ }
\]%
or
\[
W(\alpha )=\int_{t=0}^{\infty }e^{-\lambda t}\alpha _{t}\text{ }
\]
are insensitive to the future. This is true even if the discount factor $%
\lambda $ is variable through time, for any $\lambda \in L_{1}^{+}(R^{+}),$
namely $\lambda (t)\geq 0$ $s.t.\int_{R}\lambda (t)<\infty .$
Proof:
Let $\mu $ represent any time discount function such as $\mu (t)=e^{-\lambda
t}$ or more generally, any integrable function $\mu (t)$ in $L_{1}[0,\infty
).$ Then%
\[
\int_{t\in R^{+}}\alpha (t)\mu (t)dt>\int_{t\in R^{+}}\beta (t)\mu (t)dt
\]%
\[
\Leftrightarrow \exists \epsilon >0:\int_{t\in R^{+}}\alpha (t)\mu
(t)dt-\int_{t\in R^{+}}\beta (t)\mu (t)dt>3\epsilon .
\]%
Let $N=N(\alpha ,\beta )>0$ such that
\[
\int_{N}^{\infty }K\mu (t)dt<\epsilon
\]%
where%
\[
K\geqslant \sup_{\gamma \in F,t\in R^{+}}\text{ }(\mid \gamma _{t}\mid )
\]%
If
\[
\alpha (t)=\alpha ^{\prime }(t)\text{ and }\beta (t)=\beta ^{\prime }(t)%
\text{ }a.e.\text{ }\forall tW(\beta )\Rightarrow \int_{t}\alpha ^{\prime }(t)\mu
(t)>\int_{t}\beta ^{\prime }(t)\mu (t)\Rightarrow W(\alpha ^{\prime
})>W(\beta ^{\prime }).
\]%
The reciprocal is immediate.Therefore, by definition, `present discounted
value' is insensitive to the future, which completes the proof.$\blacksquare
$
Present discounted value $W$ is not compatible with sustainable
development.\newpage Example 2.
$W(\alpha )=\lim \inf \alpha _{t}$ is insensitive to the present. It is
ruled out.\bigskip \pagebreak\
\begin{center}
A Representation Theorem\bigskip
\end{center}
Theorem (Chichilnisky 1992, 1996)
\begin{itemize}
\item Prior welfare criteria do not satisfy these axioms.
\item There exist functionals $\Psi :L_{\infty }\rightarrow R$ which satisfy
the three axioms. All such functionals are defined by operators based on
convex combinations of purely and countably additive measures. For example,
if states are discrete indexed by the integers $Z$, there exists $\mu ,$ $%
0<\mu <1,$ and $\forall t,$ $\lambda (t)\geqslant 0$ such that
\begin{equation}
\Psi (\alpha )=\mu \sum_{t=1}^{\infty }\lambda (t)\alpha _{t}+(1-\mu )(\Phi
(\alpha _{t})), \label{four}
\end{equation}%
where $\sum_{1}^{\infty }\lambda (t)<\infty ,$ and where $\Phi $ denotes a
purely finite measure on the integers $Z.$\newpage
\end{itemize}
\begin{center}
Interpretation of $\Psi $\bigskip
\end{center}
The first part of $\Psi $ is an integral operator with a countably additive
kernel $\{\lambda ^{-s}\}_{s\in Z}$ that emphasizes the weight of the
present, such as present discounted value.
The second purely finitely additive part assigns positive weight to the
future.
Both parts are present, so $\Psi $ is sensitive to the present and the long
run future.\pagebreak
The optimization of functionals such as $\Psi $ is not amenable to standard
tools of calculus of variations. This must be redeveloped in new directions.
Some result already exist, see Chichilnisky (1995,6) and Heal (2000).
They give rise to nonautonomous dynamical systems, which are asymptotically
autonomous (Hirsch and Benaim).\pagebreak
\begin{center}
Who is the future?
Sustainability and Global Consciousness
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\end{center}
Chichilnisky, G. ``An Axiomatic Approach to Sustainable Development'' \emph{%
Soc. Choice and Welfare} (1996) 13:321-257.
Chichilnisky, G. ``What is Sustainable Development'' \emph{Land Economics},
1997
Chichilnisky, G. ``Updating Von Neumann Morgenstern Axioms for Choice under
Uncertainty'' The Fields Institute for Mathematical Sciences, 1996, \textit{%
Resource and Energy Economics}, 2000
Chichilnisky, G. and K.Dasol ``An Axiomatic Approach to the Equity Premium
Puzzle'' Columbia University, 2004
Heal, \emph{Valuing the Future}, Columbia University Press, 2002.
Hernstein and J. Milnor ``An Axiomatic Approach to Measurable Utility''
\emph{Econometrica} (1953) 21,:291-297
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