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.
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.
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.
