<p>As often with models embodying some form of "irrationality" (whatever that means), HD does a great job at matching a whole lot of behaviors, but leaves room for rather annoying <a href="http://en.wikipedia.org/wiki/Dutch_book" rel="nofollow">Dutch Book</a> situations (also know as "money pump" situations). These suggest that HD might generate some inaccurate predictions, and induce undesirable behaviors when included in models.</p>

<p>Here is an example. Suppose today is $t=0$ and I owe you a dollar one periods from now at $t=1$. I offer you the following sequence of deals:</p>

<ul>
<li>In $t=0$, I offer to pay <em>you</em> some fee $f_0$ in $t=2$ in order to postpone my payment from $t=1$ to $t=2$.</li>
<li>In $t=1$, I give you the possibility to pay <em>me</em> some fee $f_1$ in $t=2$ in order to get the dollar back in period $t=1$ instead of $t=2$ (assuming you accepted the deal in $t=0$).</li>
</ul>

<p><strong>Case 1 :</strong> you have a classical exponential discount factor $\frac{1}{(1+\delta)^t}$.</p>

<ul>
<li>Period $t=0$ : what is the minimal additional $f_0^*$ I can offer to pay in $t=2$ for you to let me postpone my payment to $t=2$?</li>
</ul>

<p>$$ \begin{align} f_0^* &amp; = (1+\delta)^2 \left[ \frac{1}{(1+\delta)} - \frac{1}{(1+\delta)^2} \right] \\
                       &amp; = \delta \end{align} $$</p>

<ul>
<li>Period $t=1$ : what is the maximal additional $f_1^*$ I can ask you to pay me in $t=2$ in order for you to get your one dollar in $t=1$ instead of $t=2$ ?</li>
</ul>

<p>$$ \begin{align} f_1^* &amp; = (1+\delta)  \left[ 1 - \frac{1}{(1+\delta)} \right] \\
                       &amp; = \delta \end{align} $$</p>

<p>So summing up the two deals : in period $t=0$ I offer to pay you $\delta$ more in $t=2$ to postpone the payment, but then in $t=1$ you promise to pay me $\delta$ more in $t=2$ to get the money earlier. Overall the deals cancel out, I still give you the dollar in $t=1$ and no one owes the other anything in $t=2$.</p>

<p><strong>Case 2 :</strong> you have HD with discount factor $\frac{1}{(1+\delta t)}$.</p>

<ul>
<li>Period $t=0$ :</li>
</ul>

<p>$$ \begin{align} f_0^*(HD) &amp; = (1+ 2\delta) \left[ \frac{1}{(1+\delta)} - \frac{1}{(1+2\delta)} \right] \\
                       &amp; = \frac{\delta}{1+\delta} \end{align} $$</p>

<ul>
<li>Period $t=1$ : </li>
</ul>

<p>$$ \begin{align} f_1^*(HD) &amp; = (1+\delta)  \left[ 1 - \frac{1}{(1+\delta)} \right] \\
                       &amp; = \delta \end{align} $$</p>

<p>Summing up the two deals, I still give you the dollar in $t=1$, but now you owe me an additional $\left[\delta - \frac{\delta}{1+\delta}\right]$ in $t=2$ which is strictly larger than zero for $\delta &gt; 0$. Thus in a sense, I made money out of "nothing" (or some would say I made money out of your time-inconsistent preferences). </p>

<p>Whether this is bothersome for the HD model depends on whether you believe that these kinds of money pumps are likely or unlikely to occur. </p>

<p>==================</p>

<p>In the comments, the OP also asks whether HD would have any influence on risk behavior. I do not think it would. HD denotes a specific way to discount utility but does not impose restrictions on the form of the instantaneous utility function $u(\cdot)$ which determines risk aversion in uncertain environments. </p>

CC BY-SA 3.0
2014-12-04T05:45:36.973Z
2016-02-20T21:47:34.653Z
2016-02-20T21:47:34.653Z
Martin Van der Linden
<p>In reading the <a href="http://en.wikipedia.org/wiki/Hyperbolic_discounting">Wikipedia article about hyperbolic discounting</a>, I read that it says that hyperbolic discounting created strange consequences:</p>

<blockquote>
  <p>But note, the time inconsistency of this behavior has some quite perverse consequences.</p>
</blockquote>

<p>Is it just that people make choices that their future selves would not make? The same article further says this:</p>

<blockquote>
  <p>Individuals using hyperbolic discounting reveal a strong tendency to make choices that are inconsistent over time – they make choices today that their future self would prefer not to have made, despite using the same reasoning.</p>
</blockquote>

<p>What are the consequences of hyperbolic discounting (HD)? (Although they seem to match some observed behavior, does HD produce predictions that are undesirable?)</p>

<p>Also, beside time inconsistency, what are the consequences of hyperbolic discounting, if any?</p>

<p>(Also, any references to good material are always welcome.)</p>

CC BY-SA 3.0
2014-12-03T22:45:45.647Z
Does the section of "Observations" of the linked article not answer your question? If so, why not?
Herr K.
CC BY-SA 3.0
2014-12-03T23:01:39.240Z
Are you referring to, for example, the sentence "Likewise, some have suggested that high-rate hyperbolic discounting (HD) makes unpredictable (gambling) outcomes more satisfying?" This sorta helps. I'm just wondering---if HD can help explain some behavior, does it at the same time produce unreasonable (empirical or theoretical) predictions? For example, does HD mean that people will sometimes be risk loving? In what ways might hyperbolic discounting be at odds with traditional vNM expected utility? I'm trying to clarify my understanding of these concepts.
jmbejara
CC BY-SA 3.0
2014-12-03T23:27:48.163Z
Is time inconsistency the only issue? (Maybe I don't understand how big of an issue it might be.)
jmbejara
CC BY-SA 3.0
2014-12-03T23:38:47.757Z
I guess both (1) and (2) would be useful to know. I can revise the question to include both those.
jmbejara
<decision-theory><preferences>
Consequences of hyperbolic discounting
jmbejara
# Consequences of hyperbolic discounting

(Asked by: username_0 on 2014-12-03)

In reading the [Wikipedia article about hyperbolic discounting](http://en.wikipedia.org/wiki/Hyperbolic_discounting), I read that it says that hyperbolic discounting created strange consequences:

> But note, the time inconsistency of this behavior has some quite perverse consequences.

Is it just that people make choices that their future selves would not make? The same article further says this:

> Individuals using hyperbolic discounting reveal a strong tendency to make choices that are inconsistent over time – they make choices today that their future self would prefer not to have made, despite using the same reasoning.

What are the consequences of hyperbolic discounting (HD)? (Although they seem to match some observed behavior, does HD produce predictions that are undesirable?)

Also, beside time inconsistency, what are the consequences of hyperbolic discounting, if any?

(Also, any references to good material are always welcome.)

**Tags:** decision-theory, preferences
> **Comments:**
> **username_1**: Does the section of "Observations" of the linked article not answer your question? If so, why not?  
> **username_0**: Are you referring to, for example, the sentence "Likewise, some have suggested that high-rate hyperbolic discounting (HD) makes unpredictable (gambling) outcomes more satisfying?" This sorta helps. I'm just wondering---if HD can help explain some behavior, does it at the same time produce unreasonable (empirical or theoretical) predictions? For example, does HD mean that people will sometimes be risk loving? In what ways might hyperbolic discounting be at odds with traditional vNM expected utility? I'm trying to clarify my understanding of these concepts.  
> **username_0**: Is time inconsistency the only issue? (Maybe I don't understand how big of an issue it might be.)  
> **username_0**: I guess both (1) and (2) would be useful to know. I can revise the question to include both those.  

---

## 1 answers

### Answer by username_2 on 2014-12-04. Score: 8

As often with models embodying some form of "irrationality" (whatever that means), HD does a great job at matching a whole lot of behaviors, but leaves room for rather annoying [Dutch Book](http://en.wikipedia.org/wiki/Dutch_book) situations (also know as "money pump" situations). These suggest that HD might generate some inaccurate predictions, and induce undesirable behaviors when included in models.

Here is an example. Suppose today is $t=0$ and I owe you a dollar one periods from now at $t=1$. I offer you the following sequence of deals:

* In $t=0$, I offer to pay *you* some fee $f\_0$ in $t=2$ in order to postpone my payment from $t=1$ to $t=2$.
* In $t=1$, I give you the possibility to pay *me* some fee $f\_1$ in $t=2$ in order to get the dollar back in period $t=1$ instead of $t=2$ (assuming you accepted the deal in $t=0$).

**Case 1 :** you have a classical exponential discount factor $\frac{1}{(1+\delta)^t}$.

* Period $t=0$ : what is the minimal additional $f\_0^\*$ I can offer to pay in $t=2$ for you to let me postpone my payment to $t=2$?

$$ \begin{align} f\_0^\* & = (1+\delta)^2 \left[ \frac{1}{(1+\delta)} - \frac{1}{(1+\delta)^2} \right] \\
& = \delta \end{align} $$

* Period $t=1$ : what is the maximal additional $f\_1^\*$ I can ask you to pay me in $t=2$ in order for you to get your one dollar in $t=1$ instead of $t=2$ ?

$$ \begin{align} f\_1^\* & = (1+\delta) \left[ 1 - \frac{1}{(1+\delta)} \right] \\
& = \delta \end{align} $$

So summing up the two deals : in period $t=0$ I offer to pay you $\delta$ more in $t=2$ to postpone the payment, but then in $t=1$ you promise to pay me $\delta$ more in $t=2$ to get the money earlier. Overall the deals cancel out, I still give you the dollar in $t=1$ and no one owes the other anything in $t=2$.

**Case 2 :** you have HD with discount factor $\frac{1}{(1+\delta t)}$.

* Period $t=0$ :

$$ \begin{align} f\_0^\*(HD) & = (1+ 2\delta) \left[ \frac{1}{(1+\delta)} - \frac{1}{(1+2\delta)} \right] \\
& = \frac{\delta}{1+\delta} \end{align} $$

* Period $t=1$ :

$$ \begin{align} f\_1^\*(HD) & = (1+\delta) \left[ 1 - \frac{1}{(1+\delta)} \right] \\
& = \delta \end{align} $$

Summing up the two deals, I still give you the dollar in $t=1$, but now you owe me an additional $\left[\delta - \frac{\delta}{1+\delta}\right]$ in $t=2$ which is strictly larger than zero for $\delta > 0$. Thus in a sense, I made money out of "nothing" (or some would say I made money out of your time-inconsistent preferences).

Whether this is bothersome for the HD model depends on whether you believe that these kinds of money pumps are likely or unlikely to occur.

==================

In the comments, the OP also asks whether HD would have any influence on risk behavior. I do not think it would. HD denotes a specific way to discount utility but does not impose restrictions on the form of the instantaneous utility function $u(\cdot)$ which determines risk aversion in uncertain environments.


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