<p>To see that exponential discounting is (more or less) the only time consistent manner to discount the future, consider a decision maker obtaining a utility (payoff) level at period <span class="math-container">$t$</span> and at period <span class="math-container">$t+1$</span>. Assume the total value of these is evaluated according to the value of:
<span class="math-container">$$
\alpha(t) u_t + \alpha(t+1) u_{t+1},
$$</span>
where <span class="math-container">$u_t$</span>, <span class="math-container">$u_{t+1}$</span> are the utilities obtained in periods <span class="math-container">$t$</span> and <span class="math-container">$t+1$</span> and and <span class="math-container">$\alpha(t), \alpha(t+1)$</span> are the periods <span class="math-container">$t$</span> and <span class="math-container">$t+1$</span> discount rates. Note that this assumes additive separability over time and multiplicative separability of utility and discounting (time preference).</p>
<p>Assume that the consumer is indifferent between the different payoffs <span class="math-container">$(u_t, u_{t+1})$</span> and <span class="math-container">$(v_t, v_{t+1})$</span> (say, for example say <span class="math-container">$u_t &gt; v_t$</span> and <span class="math-container">$v_{t+1} &gt; u_{t+1}$</span>).</p>
<p>Then:
<span class="math-container">$$
\alpha(t) u_t + \alpha(t+1) u_{t+1} = \alpha(t) v_t + \alpha(t+1) v_{t+1}.
$$</span>
Now assume that one period has passed. If the decision maker is time consistent, she still thinks <span class="math-container">$(u_t, u_{t+1})$</span> is equally good as <span class="math-container">$(v_t, v_{t+1})$</span>, so:
<span class="math-container">$$
\alpha(t-1) u_t + \alpha(t) u_{t+1} = \alpha(t-1) v_t + \alpha(t) v_{t+1}.
$$</span>
Using these two conditions, we find that:
<span class="math-container">$$
\frac{\alpha(t+1)}{\alpha(t)} = \frac{\alpha(t)}{\alpha(t-1)} \left(= \frac{u_t - v_t}{v_{t+1} - u_{t+1}}\right).
$$</span>
We can generalize this (by waiting more periods) and show that this condition must hold for any value of <span class="math-container">$t$</span>. In other words, the ratio <span class="math-container">$\frac{\alpha(t)}{\alpha(t-1)}$</span> should be a constant independent of <span class="math-container">$t$</span>. Let's call this number <span class="math-container">$\delta$</span>.</p>
<p>Then we have:
<span class="math-container">$$
\begin{align*}
\alpha(t) &amp;= \frac{\alpha(t)}{\alpha(t-1)} \frac{\alpha(t-1)}{\alpha(t-2)} \ldots \frac{\alpha(1)}{\alpha(0)} \alpha(0),\\
&amp;= \underbrace{\delta\, \delta \ldots \delta}_{t \text{ times }} \, \alpha(0),\\
&amp;= \alpha(0) \delta^t.
\end{align*}
$$</span>
Normalizing <span class="math-container">$\alpha(0) = 1$</span>, we obtain that <span class="math-container">$\alpha(t) = \delta^t$</span>, giving exponential discounting.</p>

CC BY-SA 4.0
2024-03-23T09:54:36.147Z
To expand upon this answer, if one wants to have continuous-time discounting, repeat the same argument for $\alpha(t+s)$ and $\alpha(t)$. Then the ratio should only depend on the relative delay $s$, i.e., $\alpha(t+s)/\alpha(t) = f(s)$. This quickly reduces to the known functional equation $\alpha(t+s) = \alpha(t) \alpha(s)$ under the assumption $\alpha(0) = 1$.
marmistrz
CC BY-SA 4.0
2024-03-20T06:42:05.327Z
2024-03-20T06:42:05.327Z
tdm
<p>Any discounting that yields time consistent preferences can be considered rational discounting.</p>
<p>To check if the intertemporal utility with <em>any</em> discount factor is time consistent, you need to check if agent would like to deviate from their action plan in subsequent time periods.</p>
<p>For example, if we have result where ex ante in period 1, 2 and 3, player would choose some <span class="math-container">$x_1^*=a$</span>, <span class="math-container">$x_2^*=b$</span> and <span class="math-container">$x_3^*=c$</span>, and  when we move to period 2, <span class="math-container">$x_2^*=b$</span> and <span class="math-container">$x_3^*=c$</span> are still optimal, and in period 3 <span class="math-container">$x_3^*=c$</span> is still optimal we have rational time preference and you could call any associated discount factor rational.</p>
<p>An example of such rational preferences that you probably think about is a preference given by utility:</p>
<p><span class="math-container">$$U(x_t) = \sum_{t=1}^n \delta^{t-1} u(x_t) $$</span></p>
<p>You can verify for yourself that for the utility above the plan of an agent will not change in any time period. Proof of it is trivial, you should attempt it on your own. If you find it too difficult, start by proving it for t=2. Extending it to arbitrary t should be natural afterwards.</p>

CC BY-SA 4.0
2024-03-19T21:47:05.927Z
2024-03-19T21:47:05.927Z
1muflon1
<p>I have repeatedly heard a claim that only an exponential discounting is the rational one, however I've never seen the proof. Why is it so?</p>
<p>Intuitively, we would expect exponential discount since the economic growth (and compound interest) is exponential, but it's merely an intuition.</p>
<p>I'd also appreciate a reference that I could cite.</p>

CC BY-SA 4.0
2024-03-19T13:52:57.170Z
Hi @marmistrz, only a comment. I think rationality is related to time-consistency is the context you mention. Exponential discounting implies that we're consistent in our approach to discounting value by a constant. If I offer you a game of soccer in a year, it has a factor less value than the game today. If offer you a game of soccer in two years, a factor is applied again to the already discounted game in one year. It is rational in the sense that remains a consistently programmed behaviour. Something like hyperbolic discounting is time-inconsistent by comparison.
EB3112
CC BY-SA 4.0
2024-03-19T13:58:51.817Z
https://chris-said.io/2018/02/04/hyperbolic-discounting/
EB3112
CC BY-SA 4.0
2024-03-22T23:21:11.090Z
That's a great article, thanks!
marmistrz
<mathematical-economics><discount-rate><proof><rationality><proof-reference>
Why is exponential discounting considered rational?
marmistrz
# Why is exponential discounting considered rational?

(Asked by: username_0 on 2024-03-19)

I have repeatedly heard a claim that only an exponential discounting is the rational one, however I've never seen the proof. Why is it so?

Intuitively, we would expect exponential discount since the economic growth (and compound interest) is exponential, but it's merely an intuition.

I'd also appreciate a reference that I could cite.

**Tags:** mathematical-economics, discount-rate, proof, rationality, proof-reference
> **Comments:**
> **username_1**: Hi @marmistrz, only a comment. I think rationality is related to time-consistency is the context you mention. Exponential discounting implies that we're consistent in our approach to discounting value by a constant. If I offer you a game of soccer in a year, it has a factor less value than the game today. If offer you a game of soccer in two years, a factor is applied again to the already discounted game in one year. It is rational in the sense that remains a consistently programmed behaviour. Something like hyperbolic discounting is time-inconsistent by comparison.  
> **username_1**: https://chris-said.io/2018/02/04/hyperbolic-discounting/  
> **username_0**: That's a great article, thanks!  

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## 2 answers

### Answer by username_2 on 2024-03-20. Score: 5

To see that exponential discounting is (more or less) the only time consistent manner to discount the future, consider a decision maker obtaining a utility (payoff) level at period $t$ and at period $t+1$. Assume the total value of these is evaluated according to the value of:
$$
\alpha(t) u\_t + \alpha(t+1) u\_{t+1},
$$
where $u\_t$, $u\_{t+1}$ are the utilities obtained in periods $t$ and $t+1$ and and $\alpha(t), \alpha(t+1)$ are the periods $t$ and $t+1$ discount rates. Note that this assumes additive separability over time and multiplicative separability of utility and discounting (time preference).

Assume that the consumer is indifferent between the different payoffs $(u\_t, u\_{t+1})$ and $(v\_t, v\_{t+1})$ (say, for example say $u\_t > v\_t$ and $v\_{t+1} > u\_{t+1}$).

Then:
$$
\alpha(t) u\_t + \alpha(t+1) u\_{t+1} = \alpha(t) v\_t + \alpha(t+1) v\_{t+1}.
$$
Now assume that one period has passed. If the decision maker is time consistent, she still thinks $(u\_t, u\_{t+1})$ is equally good as $(v\_t, v\_{t+1})$, so:
$$
\alpha(t-1) u\_t + \alpha(t) u\_{t+1} = \alpha(t-1) v\_t + \alpha(t) v\_{t+1}.
$$
Using these two conditions, we find that:
$$
\frac{\alpha(t+1)}{\alpha(t)} = \frac{\alpha(t)}{\alpha(t-1)} \left(= \frac{u\_t - v\_t}{v\_{t+1} - u\_{t+1}}\right).
$$
We can generalize this (by waiting more periods) and show that this condition must hold for any value of $t$. In other words, the ratio $\frac{\alpha(t)}{\alpha(t-1)}$ should be a constant independent of $t$. Let's call this number $\delta$.

Then we have:
$$
\begin{align\*}
\alpha(t) &= \frac{\alpha(t)}{\alpha(t-1)} \frac{\alpha(t-1)}{\alpha(t-2)} \ldots \frac{\alpha(1)}{\alpha(0)} \alpha(0),\\
&= \underbrace{\delta\, \delta \ldots \delta}\_{t \text{ times }} \, \alpha(0),\\
&= \alpha(0) \delta^t.
\end{align\*}
$$
Normalizing $\alpha(0) = 1$, we obtain that $\alpha(t) = \delta^t$, giving exponential discounting.

> **Comments:**
> **username_0**: To expand upon this answer, if one wants to have continuous-time discounting, repeat the same argument for $\alpha(t+s)$ and $\alpha(t)$. Then the ratio should only depend on the relative delay $s$, i.e., $\alpha(t+s)/\alpha(t) = f(s)$. This quickly reduces to the known functional equation $\alpha(t+s) = \alpha(t) \alpha(s)$ under the assumption $\alpha(0) = 1$.  

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### Answer by username_3 on 2024-03-19. Score: 2

Any discounting that yields time consistent preferences can be considered rational discounting.

To check if the intertemporal utility with *any* discount factor is time consistent, you need to check if agent would like to deviate from their action plan in subsequent time periods.

For example, if we have result where ex ante in period 1, 2 and 3, player would choose some $x\_1^\*=a$, $x\_2^\*=b$ and $x\_3^\*=c$, and when we move to period 2, $x\_2^\*=b$ and $x\_3^\*=c$ are still optimal, and in period 3 $x\_3^\*=c$ is still optimal we have rational time preference and you could call any associated discount factor rational.

An example of such rational preferences that you probably think about is a preference given by utility:

$$U(x\_t) = \sum\_{t=1}^n \delta^{t-1} u(x\_t) $$

You can verify for yourself that for the utility above the plan of an agent will not change in any time period. Proof of it is trivial, you should attempt it on your own. If you find it too difficult, start by proving it for t=2. Extending it to arbitrary t should be natural afterwards.


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