Appendix A: Integral representation
In this section the trichotomic discounted utility model is extended to an integral representation. In obtaining such a representation, we provide the appropriate foundations for applications with consumption streams, used in finance and macroeconomics. We will make use of the tools developed by Kopylov (
2010). In particular, we exploit only the richness naturally provided by the time dimension. No richness is required of the set of outcomes. Hence, the theory can be applied to any type of outcomes, be they monetary, health related, indivisible goods, and so on.
The set of outcomes is X, time is \(T=[0,\infty )\), and the set of half-open intervals [a, b) is \({\mathscr {T}}\). Consumption streams are \({\mathscr {T}}\)-measurable functions \({\mathbf {x}}:T\rightarrow X\), the set of which is \({\mathscr {C}}\). For a decision time \(d\in T\), let \({\mathscr {C}}_{d}=\{{\mathbf {x}}|_{[d,\infty )}:{\mathbf{x }}\in {\mathscr {C}}\}\) denote the set of consumptions streams restricted to times not earlier than d. A dynamic preference is a collection of static preference relations \({\mathscr {R}}=\{\succcurlyeq _{d}\}_{d\in T}\) where each \(\succcurlyeq _{d}\) is defined over \({\mathscr {C}}_{d}\). A dynamic model\({\mathscr {V}}=\{V_{d}\}_{d\in T}\) is a collection of real-valued functions \(V_{d}:{\mathscr {C}}_{d}\rightarrow {\mathbb {R}}\). A dynamic preference \({\mathscr {R}}\) is represented by \({\mathscr {V}}\) if for each \(\succcurlyeq _{d}\in {\mathscr {R}}\) there is a \(V_{d}\in {\mathscr {V}}\) such that, for all \({\mathbf {x}},{\mathbf {y}}\in {\mathscr {C}}_{d}\), \({\mathbf {x}}\succcurlyeq _{d}{\mathbf {y}}\) if and only if \(V_{d}({\mathbf {x}})\geqslant V_{d}({\mathbf {y}})\).
Some further notation regarding consumption streams is useful. For \({\mathbf {x}},{\mathbf {z}}\in {\mathscr {C}}\) and \(0\leqslant a\leqslant b<\infty \), we use \({\mathbf{x }}[a,b){\mathbf {z}}\) to denote the stream that coincides with \({\mathbf {x}}\) in the interval [a, b), and coincides with \({\mathbf {z}}\) elsewhere. For an outcome \(x\in X\), we use \(\langle x\rangle \) to denote the constant stream that yields outcome x at all points in time. Given a consumption stream \(x\in {\mathscr {C}}\) and \(d\in T\), denote by \({\mathbf {x}}_{d}\) the stream \({\mathbf {x}}_{d}\in {\mathscr {C}}_{d}\) such that \({\mathbf {x}}_{d}(t)={\mathbf {x}}(t-d)\) for all \(t\geqslant d\). The following axioms are assumed:
Axioms
1–
5 are essentially intertemporal analogues of the axioms for subjective expected utility (Kopylov
2010). Axiom
6 is a necessary condition for discounted utility, and is consumption stream version of the condition presented by Halevy (
2015). Under Axioms
1–
6, the following condition is the characterising property of exponential discounting:
Consider two streams that are identical, except on some interval [
a,
b). Time consistency requires that, at any time before this interval, the decision-maker does not reverse previously expressed preferences. Under discounted utility, time consistency holds if and only if exponential discounting holds. Hence, discount functions exhibiting decreasing impatience must, in the context of discounted utility, violate time consistency. Our axiomatisation is phrased in terms of the time consistency properties that remain in the more general model. The following is the consumption stream version of the condition introduced in Sect.
4:
The interpretation is similar to the axiom introduced in Sect.
4. The main result of this section characterises an integral version of trichotomic discounting model in the consumption streams framework:
The representation (
D,
u) in Theorem
5 can be replaced with
\(({\tilde{D}},{\tilde{u}})\) if and only if
\({\tilde{D}}=D\) and
\({\tilde{u}}=au+b\) for some
\(a>0\) and
\(b\in {\mathbb {R}}\). If it happens that
\(\delta _{S}=\delta _{M}=\delta _{L}\), then
S and
M can be chosen arbitrarily. If
\(\delta _{S}=\delta _{M}\) and
\(\delta _{M}\ne \delta _{L}\), then
M is uniquely defined, but
S can be chosen arbitrarily in the interval [0,
M]. If
\(\delta _{S}\ne \delta _{M}\) and
\(\delta _{M}=\delta _{L}\), then
S is uniquely defined, but
M can be chosen arbitrarily in the interval
\([S,\infty )\). If
\(\delta _{S}\ne \delta _{M}\) and
\(\delta _{M}\ne \delta _{L}\), then
S and
M are uniquely defined.
Appendix B: Proofs
Proof of Theorem 3
Assume that preferences exhibit the dynamic inverse-S pattern, so that there exists \(x<y\), and \(d_{1}<d_{2}<d_{3}<s<t\), such that: \((s,x)\sim _{d_{3}}(t,y)\), \((s,x)\succcurlyeq _{d_{2}}(t,y)\), and \((s,x)\preccurlyeq _{d_{1}}(t,y)\). If preferences are determined by risk, with a constant hazard rate, then these are equivalent to: \((e^{-\lambda (s-d_{3})},x)\sim ^{r}(e^{-\lambda (t-d_{3})},y)\), \((e^{-\lambda (s-d_{2})},x)\succcurlyeq ^{r}(e^{-\lambda (t-d_{2})},y)\), and \((e^{-\lambda (s-d_{1})},x)\preccurlyeq ^{r}(e^{-\lambda (t-d_{1})},y)\). Define \(p=e^{-\lambda (s-d_{3})}\) , \(q=e^{-\lambda (t-d_{3})}\) , \(\alpha =e^{-\lambda (d_{3}-d_{2})}\) and \(\beta =e^{-\lambda (d_{3}-d_{1})}\) and it follows that risk preferences exhibit the inverse-S pattern. For the converse implication, suppose that risk preferences exhibit the inverse-S pattern, so that there exists \(x<y\), \(0<q<p\leqslant 1\), and \(0<\beta<\alpha <1\), such that: \((p,x)\sim ^{r}(q,y)\), \((\alpha p,x)\preccurlyeq ^{r}(\alpha q,y)\), and \((\beta p,x)\succcurlyeq ^{r}(\beta q,y)\). Fix a decision time \(d_{3}>-\frac{\ln (\beta )}{\lambda }\). Then define \(s=\frac{\lambda d_{3}-\ln (p)}{\lambda }\), \(t=\frac{\lambda d_{3}-\ln (q)}{\lambda }\), \(d_{2}=\frac{\lambda d_{3}+\ln (\alpha )}{\lambda }\) and \(d_{1}=\frac{\lambda d_{3}+\ln (\beta )}{\lambda }\). Then, time preferences, determined by risk preferences, will exhibit the dynamic inverse-S pattern for these particular values. \(\square \)
Proof of Theorem 4
For given \(0\leqslant \sigma<\tau <\infty \), define stationarity in \([\sigma ,\tau )\) by the condition: for all \((s,x),(t,y)\in {\mathbb {R}}_{+}^{2}\) and \(\Delta \) with \(\sigma \leqslant s,s+\Delta ,t,t+\Delta \leqslant \tau \), we have \((s,x)\succcurlyeq _{0}(t,x)\) if and only if \((s+\Delta ,x)\succcurlyeq _{0}(t+\Delta ,y)\). Under discounted utility, time consistency in \([\sigma ,\tau )\) is equivalent to stationarity in \([\sigma ,\tau )\). Suppose that local time consistency holds, so there exists \(0\leqslant \sigma<\tau <\infty \), such that time consistency in (hence stationarity in) \([\sigma ,\tau )\) holds. We will show that D is an exponential function on\([\sigma ,\tau )\). If \(\sigma \) and \(\tau \) are sufficiently close (if \(\sigma >\frac{\tau }{2})\) then, for all \(\sigma \leqslant s,t\leqslant \tau \), we have that \(s+t\) will lie outside of \([\sigma ,\tau )\). Hence, establishing the standard Cauchy functional equation for exponentials, \(D(s+t)=D(s)D(t)\), is not immediately clear. To accomplish this, define a function \({\tilde{D}}:[\sigma ,\tau )\cup [2\sigma ,2\tau )\rightarrow {\mathbb {R}}\) such that \({\tilde{D}}|_{[\sigma ,\tau )}=D\), and \({\tilde{D}}(r)=\{D(s)D(t):r=s+t,\sigma \leqslant s<t\leqslant \tau \}\) for all \(r\in [2\sigma ,2\tau )\).
We now confirm that
\({\tilde{D}}\) is well-defined. To this end, it suffices to show that
\(\sigma \leqslant s,s',t,t'<\tau \) and
\(s+t=s'+t'\) jointly imply
\(D(s)D(t)=D(s')D(t')\). Suppose not, so there exists
\(\sigma \leqslant s,s',t,t'<\tau \) with
\({\tilde{D}}(s){\tilde{D}}(t)\ne {\tilde{D}}(s'){\tilde{D}}(t')\). Let
\(s'=s-\varepsilon \), so that
\(\frac{{\tilde{D}}(s'+\varepsilon )}{{\tilde{D}}(s')}\ne \frac{{\tilde{D}}((t+\varepsilon )}{{\tilde{D}}(t)}\). Given such
\(s',t\), and discounted utility’s continuity and monotonicity properties, there exists
x,
y such that
\((s',x)\sim _{0}(t,y)\), equivalent to
\({\tilde{D}}(s')u(x)={\tilde{D}}(t)u(y)\). Then, we have that
\({\tilde{D}}(s+\varepsilon )u(x)\ne {\tilde{D}}(t+\varepsilon )u(y)\), equivalent to
\((s'+\varepsilon ,x)\not \sim _{0}(t+\varepsilon ,y)\), contradicting stationarity in
\([\sigma ,\tau )\). Hence, the function
\({\tilde{D}}\) is well-defined and, by construction,
\({\tilde{D}}(s+t)={\tilde{D}}(s){\tilde{D}}(t)\) for all
\(t,s\in [\sigma ,\tau )\). By Aczel and Skof (
2007, Note 4, p. 315), there exists constants
\(\alpha >0\) and
\(\delta \in (0,1)\) such that
\({\tilde{D}}(t)=D(t)=\alpha \delta ^{t}\), for all
\(t=[\sigma ,\tau )\).
\(\square \)