Stochastic dynamic utilities and intertemporal preferences

Abstract

We propose an axiomatic approach which economically underpins the representation of dynamic intertemporal decisions in terms of a utility function, which randomly reacts to the information available to the decision maker throughout time. Our construction is iterative and based on time dependent preference connections, whose characterization is inspired by the original intuition given by Debreu’s State Dependent Utilities (1960).

Appendices

On $\star$-continuity

Throughout this section we fix a probability space $(\varOmega ,{\mathcal G},{\mathbb P})$ and a random field $\phi :{\mathbb R}\times \varOmega \rightarrow {\mathbb R}$ such that for each $f\in \mathscr {L}^\infty ({\mathcal G})$ the map $\omega \mapsto \phi (f(\omega ),\omega )$ is ${\mathcal G}$-measurable and for any $\omega$, $x\mapsto \phi (x,\omega )$ is non decreasing. For any $f\in \mathscr {L}^\infty ({\mathcal G})$ we set

$$ \phi (f(\omega )^+,\omega )=\mathop {\hbox {inf}}\limits _{n\in \mathbb {N}}\phi (f(\omega )+1/n,\omega ) \text { and } \phi (f(\omega )^-,\omega )=\sup _{n\in \mathbb {N}}\phi (f(\omega )-1/n,\omega )$$

and define the following sets:

$$\begin{aligned} RD_f&= \{\omega \in \varOmega : \left( \phi (f(\omega )^+,\omega )-\phi (f(\omega ),\omega )\right)>0\} \\ LD_f&= \{\omega \in \varOmega : \left( \phi (f(\omega ),\omega )-\phi (f(\omega )^-,\omega )\right)>0\} \\ D_f&= \{\omega \in \varOmega : \left( \phi (f(\omega )^+,\omega )-\phi (f(\omega )^-,\omega )\right) >0\} \end{aligned}$$

We now prove a useful lemma which allows to give a well-posed definition of continuity for random fields.

Lemma 4

For each $f\in \mathscr {L}^\infty ({\mathcal G})$ the sets $RD_f$, $LD_f$, $D_f$, defined above, are ${\mathcal G}$-measurable.

Proof

Observe that the set $RD_f$ can be written as:

$$\begin{aligned} RD_f&= \bigcap _{n\in \mathbb {N}}\bigcup _{m\in \mathbb {N}}\left\{ \omega \in \varOmega : \left( \phi \left( f(\omega )+\frac{1}{n},\omega \right) -\phi (f(\omega ),\omega ) \right) > \frac{1}{m}\right\} \\&= \bigcap _{n\in \mathbb {N}}\bigcup _{m\in \mathbb {N}} \left[ \phi \left( f(\cdot )+\frac{1}{n},\cdot \right) -\phi \left( f(\cdot ),\cdot \right) \right] ^{-1} \left( \frac{1}{m},+\infty \right) \end{aligned}$$

which is ${\mathcal G}$-measurable by measurability of the function

$$\begin{aligned} \omega \rightarrow \phi _n(\omega ) = \phi \left( f(\omega )+\frac{1}{n},\omega \right) -\phi (f(\omega ),\omega ). \end{aligned}$$

Clearly a similar argument shows that $LD_f\in {\mathcal G}$. Finally, $D_f = LD_f \cup RD_f \in {\mathcal G}$. $\square$

Definition 3

The random fields $\phi$ is $\star$-continuous if ${\mathbb P}(D_f)=0$ for every $f\in \mathscr {L}^\infty ({\mathcal G})$.

Remark 11

Observe that the set $D_f$ defined in Lemma 4 can be interpreted as:

$$\begin{aligned} D_f = \left\{ \omega \in \varOmega : f(\omega ) \text { is a point of discontinuity of the function } \phi (\cdot ,\omega )\right\} \end{aligned}$$

In particular for any sequence $\{f_n\}_{n\in \mathbb {N}}\subset \mathscr {L}^\infty ({\mathcal G})$ such that $f_{n}(\omega )\rightarrow f(\omega )$ we have $\phi (f_n(\omega ),\omega )\rightarrow \phi (f(\omega ),\omega )$ for any $\omega \in (D_f)^c$. Moreover it follows that the definition of $\star$-continuity is well posed as the set $D_f$ is measurable by Lemma 4.

Notice also that taking $f \equiv x \in {\mathbb R}$ then $D_x=\{\omega \in \varOmega : \phi (\cdot ,\omega ) \text { is discontinuous in } x\}$. Therefore, the condition ${\mathbb P}(D_x) = 0$ means that for ${\mathbb P}$-a.e. $\omega \in \varOmega$ the map $\phi (\cdot ,\omega )$ is continuous in x . On the other hand, if $\phi$ is ${\mathbb P}-a.s.$ continuous and satisfies the measurability condition of Lemma 4 then it is also $\star$-continuous. Hence, the $\star$-continuity is a notion of continuity which is deeply related to the probability space (in particular, to the $\sigma$-algebra) and is weaker than the ${\mathbb P}$-a.s. continuity of the trajectories but stronger than the ${\mathbb P}$-a.s. continuity at fixed points.

State dependent utilities

As in the rest of the paper $(\varOmega ,{\mathcal F})$ denotes a measurable space and $\mathscr {L}^{\infty }(\mathcal {F})$ is the space of all acts, represented by real valued ${\mathcal F}$-measurable and bounded random variables. We here use the term “act”  in order to match the terminology adopted in [29] on which this Appendix is based. This term must be used with care in order to avoid confusion with the general notion of Anscombe-Aumann acts. Indeed in [1] acts are functions from the state space $(\varOmega ,{\mathcal F})$ to a convex set of lotteries over a consequence set.

In this appendix the preference relation is a binary relation $\succeq$ on ${\mathscr {L}^{\infty }(\mathcal {F})}$ : for f, g $\in {\mathscr {L}^{\infty }(\mathcal {F})}$, if f is preferred to g, write $f \succeq g$. The preference relation satisfies the following axiom:

(A1):

Preference order: if it is reflexive ($\forall f \in {\mathscr {L}^{\infty }(\mathcal {F})}$, $f \sim f$), complete ($\forall f,g \in {\mathscr {L}^{\infty }(\mathcal {F})}$, $f \succeq g$ or $f \preceq g$) and transitive ($\forall f,g,h \in {\mathscr {L}^{\infty }(\mathcal {F})}$ such that $f \succeq g$ and $g \succeq h$ then $f \succeq h$)

Definition 4

A representing function of the preference relation is a function $V : {\mathscr {L}^{\infty }(\mathcal {F})}\rightarrow \mathbb {R}$ which is order-preserving, i.e.,

$$ f\succeq g \iff V(f) \ge V(g). $$

We use the standard conventions: $f\preceq g$ if $g\succeq f$; $f\sim f$ if both $g\succeq f$ and $f\succeq g$; $g\not \sim f$ if either $g\nsucceq f$ or $f\nsucceq g$; $g\succ f$ if $g\succeq f$ but $f \nsucceq g$.

Definition 5

An event $A\in {\mathcal F}$ is null if $f\mathbf {1}_A + g\mathbf {1}_{A^c} \sim g$ $\forall f,g \in {\mathscr {L}^{\infty }(\mathcal {F})}$.We shall denote by $\mathcal {N}(\mathcal {F})$ be the set of null events.

As a consequence a $\succeq$-atom is an element $A\in {\mathcal F}$ such that for every $B\in {\mathcal F}$ with $\varnothing \ne B\subset A$ either B or $A{\setminus } B$ is null.

An event is essential if it belongs to ${\mathcal F}{\setminus } \mathcal {N}(\mathcal {F})$.

We can consider the following additional Axioms:

(A2):

Strictly monotone if $x\mathbf {1}_A + f\mathbf {1}_{A^c} \succ y\mathbf {1}_A + f\mathbf {1}_{A^c}$, for all nonnull events $A\in {\mathcal F}$, for all $f \in {\mathscr {L}^{\infty }(\mathcal {F})}$ and outcomes $x > y$.

(A3):

Sure-thing principle: consider arbitrary $f, g, h \in {\mathscr {L}^{\infty }(\mathcal {F})}$ and $A\in {\mathcal F}$ such that $f\mathbf {1}_A + h\mathbf {1}_{A^c} \preceq g\mathbf {1}_A + h\mathbf {1}_{A^c}$ then for every $c \in {\mathscr {L}^{\infty }(\mathcal {F})}$ we have $f\mathbf {1}_A + c\mathbf {1}_{A^c} \preceq g\mathbf {1}_A + c\mathbf {1}_{A^c}$.

(A3) holds on ${\mathcal S}({\mathcal F})$ if we substitute in the previous statement ${\mathscr {L}^{\infty }(\mathcal {F})}$ with ${\mathcal S}({\mathcal F})$ (as defined in the paragraph Notations).

(A4’):

Norm continuity if $\forall f \in {\mathscr {L}^{\infty }(\mathcal {F})}$ the sets $\{ g \in {\mathscr {L}^{\infty }(\mathcal {F})}: g \succeq f \}$ and $\{ g \in {\mathscr {L}^{\infty }(\mathcal {F})}: f \succeq g \}$ are $\Vert \cdot \Vert _{\infty }$-closed.

Theorem 8

(Debreu 1960, state-dependent expected utility for finite state space) Let ${\mathscr {L}^{\infty }(\mathcal {F})}$ the set of acts and $\succeq$ a preference relation on it. Let the state space $\varOmega = \{ \omega _1, \ldots , \omega _n \}$, where at least three states are nonnull. Then the following two statements are equivalent:

• (i) There exist n continuous functions $V_j : \mathbb {R} \rightarrow \mathbb {R}$, $j = 1, \ldots , n$, that are strictly increasing for all nonnull states and constant for all null states, and such that $\succeq$ is represented by

  $$\begin{aligned} V(f) = \sum _{j=1}^{n} V_j(f(\omega _j)). \end{aligned}$$

  (B.1)

• (ii) $\succeq$ is a norm continuous, strictly monotonic preference order that satisfies the sure thing principle.

The following uniqueness holds for (1) : $W(f) = \sum _{j=1}^{n} W_j(f(\omega _j))$ represent $\succeq$ if and only if there exist $\tau _1, \ldots , \tau _n \in \mathbb {R}$ and $\sigma > 0$ such that $W_j = \tau _j + \sigma V_j$ $\forall j$, implying that $W = \tau + \sigma V$ for $\tau = \tau _1 + \cdots + \tau _n$.

In [29] the previous Theorem is generalized to an infinite state spaces $\varOmega$ when $\varOmega$ contains no atoms. We here recall the integral reformulation of the Debreu representation given in [3] under pointwise continuity.

Definition 6

A preference order is

(A4):

Pointwise continuous if for any uniformly bounded sequence $\{f_n\}\subseteq {\mathscr {L}^{\infty }(\mathcal {F})}$, such that $f_n(\omega )\rightarrow f(\omega )$ for any $\omega \in \varOmega$ then $\forall g \in {\mathscr {L}^{\infty }(\mathcal {F})}$ such that $g \succ f$ (resp. $g \prec f$) $\exists J \in \mathbb {N}$ such that $g \succ f^j$ (resp. $g \prec f^j$) $\forall j > J$.

Theorem 9

([29], Theorem 12 and [3], Theorem 5) Let ${\mathscr {L}^{\infty }(\mathcal {F})}$ be the set of acts and $\succeq$ the preference relation on it. Assume that ${\mathcal F}$ contains at least three disjoint essential events. Then the following two statements are equivalent:

• (i) There exists a countably additive measure $\mathbb {P}$ on $\varOmega$ and a function (the state-dependent utility) $u(\omega , \cdot ) : \mathbb {R} \rightarrow \mathbb {R}$ strictly increasing $\forall \omega \in \varOmega$, such that $\succeq$ is represented by the pointwise continuous integral

  $$ f \rightarrow \int _{\varOmega } u(\omega , f(\omega )) d\mathbb {P}. $$

• (ii) $\succeq$ satisfies: (A1), (A2), (A3) on ${\mathcal S}({\mathcal F})$, (A4).

The following uniqueness holds: the couple $(\mathbb {P},u)$ can be replaced by $(\mathbb {P}^*, u^*)$ if and only if $\mathbb {P}$ and $\mathbb {P}^*$ are equivalent and ${\mathbb P}(u^* = \tau + \sigma \delta u)=1$, where $\tau : \varOmega \rightarrow \mathbb {R}$ is ${\mathcal F}$-measurable, $\sigma > 0$ and $\delta$ is the Radon-Nikodym density function of $\mathbb {P}$ with respect to $\mathbb {P}^*$.