The standard approach of making investment decision is that the policy maker compares investment opportunities by their net present values (NPVs), i.e., by the sum of their discounted expected cash flows. If we assume that investment decisions are taken by a group of people, rather than by a single individual, the model must incorporate this evidence by allowing for group diversity which is reflected by differences in opinion about the appropriate method of discounting. In addition to group disagreement, we need to allow for the possibility that each individual group member has some uncertainty about what discount rate to use.
Weighted Discount functions
The group’s weighting distribution $F(r)$ assigns a weight to each member or, equivalently, to each (positive) discount rate r. Member $i$ $(i = 1,…,n)$ favors discount rate $r_i$ and thus her discount function is given by $e^{−r_it}$. Letting $f$ denote the probability mass function of $F$, the group discount function (also weighted discount function) is given by
Definition Let $h : [0,\infty) \to (0,1]$ be strictly decreasing with $h(0) = 1$. $h$ is a weighted discount function if there exists a (cumulative) distribution function $F(r)$ (or a density function $f(r)$) concentrated on $[0,\infty)$ such that
$F$ is called the weighting distribution of $h$.
Example (pseudo-exponential discount function)
where $\lambda>0$, $r>0$, $0<\delta<1$. It gives the weighted average of just two opinions.
The following well-known theorem is attributed to Bernstein (1928), and provides a necessary and sufficient condition for a given discount function to be a weighted discount function.
Theorem (Bernstein’s Theorem) A discount function $h$ is a weighted discount function if and only if it is continuous on $[0,\infty)$, infinitely differentiable on $(0,\infty)$, and satisfies $(−1)^nh^{(n)}(t) \geq 0$, for all non-negative integers $n$ and for all $t > 0$.
Bernstein’s theorem can be used to verify that some commonly used discount functions are indeed weighted discount function. While every weighting distribution $F$ defines a discount function $h^F$, it may not be possible to recover the weighting distribution $F$ in closed form. Even though we may not be able to obtain $h^F$ as the group discount function of any nice distribution function $F$, but we can still study its implications for investment behavior and draw inference on the diversity of the group it represents.
Yet there are also some weighting distributions that lead to nice weighted discount functions (and vice versa). In particular, the generalized hyperbolic discount function with parameters $\alpha > 0$, $\beta > 0$ can be written as
where $f(r;k,\theta)=\frac{r^{k-1}e^{-r/\theta}}{\theta^k\Gamma(k)}$ denotes the density function of the Gamma distribution with parameters $k$ and $\theta$ and where $\Gamma(k)$ denotes the Gamma function evaluated at $k$, i.e., $\Gamma(k)=\int_0^\infty x^{k-1}e^{-x}dx$.
Claim: There is an economically important observation results from the mathematically simple fact that any weighted average of weighted discount functions is also a weighted discount function.
This implies that the above results are not restricted to groups of exponential discounters, each of whom is certain about what discount rate to use. Instead, we may consider groups with some or all members being with the same weighted discounting, because the latter are also weighted discount functions. Likewise, we may consider groups of individual discounters, each of whom is uncertain about the discount rate to use. Then let us prove the claim.
Consider an individual, $i$, with weighted discount function
where $F(r,i)$ denotes the weighting distribution used to build the discount function $h(t;i)$. Note that a non-degenerate $F(r;i)$ may reflect individual $i$’s bounded rationality. For example, if $F(r;i)$ is Gamma-distributed, we know that individual $i$ is a generalized hyperbolic discounter. We are interested in the discount function of a group of such individuals, which may consist of a finite number or a continuum of individuals. Let us denote the discount function of this group by $\overline h(t)$. The weights of the individual group members are given by the weighting distribution $G(i)$. In general case, we have $\overline h(t)=\int_0^\infty h(t;i)dG(i)$.
Proposition The group discount functions of weighted discounters are weighted discount functions. In particular, there exists a distribution $\overline F$ over exponential discount rates $r$ such that
Moreover, $\overline F$ can be computed as $\overline F(r)=\int_0^\infty F(r;i)dG(i)$.
Group Diversity
The main result is that a group has a uniformly greater discount factor than another group (i.e., discounts less heavily at any future time) if and only if it is more diverse. Greater group diversity is captured by a more dispersed weighting distribution $F$ of the corresponding weighted discount function $h^F$, and is defined using stochastic dominance.
Assumption: All weighting distributions have their support contained in $[a,b]$, $0 \leq a \leq b < \infty$.
Let $F^0(x)=F(x)$ and $F^i(x)=\int_a^x F^{i-1}(t)dt$, and we define stochastic dominance as the following.
Definition The distribution $F$ dominates the distribution $G$ in the sense of Nth-order stochastic dominance, denoted by $G \preceq_{NSD}F$, if i) $F^{N-1}(x)\leq G^{N-1}(x)$ for all $a\leq x\leq b$, ii) $F^i(b)\leq G^i(b)$ for $i=1,2,\ldots,N-2$.
Theorem *$G \preceq_{NSD}F\iff$
for all function $u(x)$, s.t. sgn $u^{(n)}(x)=(-1)^{n+1}$ for $n=1,2,\ldots,N$ and $x\in [a,b]$.*
It can be shown that a second-order stochastically dominating distribution must have a higher mean and a lower variance than the distribution that it dominates, with at least one of the two inequalities being strict. Also, we can observe that $G \preceq_{NSD}F\Rightarrow G \preceq_{(N+1)SD}F$. We can define the “weakest” stochastic dominance order as follows.
Definition (Infinity-stochastic dominance.) *$F$ dominates $G$ via infinity-stochastic dominance (denoted as $G \preceq_{\infty SD}F$), if
for all function $u(x)$, s.t. sgn $u^{(n)}(x)=(-1)^{n+1}$ for $n\in \mathbb{N}^+$ and $x\in [a,b]$.*
Now we can introduce the notion of greater group diversity.
Definition (Greater group diversity) Let $F$ and $G$ be weighting distributions of weighted discount functions $h^F$ and $h^G$, respectively. We say that $h^G(G)$ is more diverse than $h^F(F)$ if $G \preceq_{\infty SD}F$.
Note: $G \preceq_{\infty SD}F$ means that $F$ is preferred over $G$ because $F$ has less risk (dispersion) in the stochastic dominance sense.
From the definition of $\infty$-stochastic dominance it follows directly that if $F$ dominates $G$ via stochastic dominance of some order $N$, then $F$ dominates $G$ by $\infty$-stochastic dominance, i.e., $G$ is more diverse than $F$.
Theorem (Greater group diversity and larger discount factors) Suppose $F$ and $G$ are weighting distributions with finite support and the corresponding discount functions are given by $h^G$ and $h^F$, respectively. Then:
The fact that $h^G(t)\geq h^F(t)$ for any $t>0$ is denoted as an uniformly larger discount factor. A larger discount factor implies less discounting, and uniform means that this is the case at any future time. Graphically, it means that the discount function of a more diverse group always lies above that of a less diverse group. We have already known the fact that the pseudo-exponential discount function $h^G$ lies above the exponential discount function $h^F$ at all times. The intuition behind the theorem for this special case is that exponential functions are convex so that, by Jensen’s inequality, averaging two discount factors whose discount rates average to $r=5\%$, results in a discount factor larger than a single exponential discount factor with rate $r=5\%$. For general discount functions, however, comparing discount factors is not equivalent to comparing expected discount rates.
Proof of Theorem: $\Longrightarrow$ Suppose first that $G \preceq_{\infty SD}F$. Let $u(x)=-e^{-xt}$, which is increasing with derivatives that alternate in sign on $[0,\infty)$ for each fixed $t > 0$. Therefore, Definition yields that $h^F(t) \leq h^G(t), \forall t \geq 0$.
$\Longleftarrow$ Now suppose that $h^F(t) \leq h^G(t), \forall t \geq 0$. For any mixed risk averse utility function $u$, the first derivative of $u$ satisfies the assumptions of Bernstein’s Theorem, which requires $(−1)^nu^{(n)}(t) \geq 0$, for all non-negative integers $n$ and for all $t > 0$. Thus, there exists a distribution function $F_u$ such that $u’(t)=\int_0^\infty e^{-ts}dF_u(s)$. It follows that
and thus
Comparing with the analogous expression for $G$, because $h^F(t) \leq h^G(t)$ for all $t \geq 0$ it follows that $G \preceq_{\infty SD}F$.
For weighted discount functions, the above theorem establishes the necessary and sufficient condition for ordering discount factors. Ordering discount factors is equivalent to comparing the weighting distributions in the $\infty$-stochastic dominance sense. We put the property of larger discount factors (rather than greater group diversity) in the center of our mathematical analysis. Greater group diversity should be kept in mind, though, whenever we speak of larger discount factors.
