Exponential Discounting

Most of us seem to agree that the promise of a dollar in the future is worth less to us than a dollar today, even if the promise is certain to be fulfilled. Economists often assume ‘exponential discounting’, which says that a dollar promised at some time $s$ is worth

$\exp(-\alpha(s - t))$

dollars in hand at time $t.$ The constant $\alpha$ is connected to the ‘interest rate’.

Why are economists so wedded to exponential discounting? The main reason is probably that it’s mathematically simple. But one argument for it goes roughly like this: if your decisions today are to look rational at any future time, you need to use exponential discounting.

In practice, humans, pigeons and rats do not use exponential discounting. So, economists say they are ‘dynamically inconsistent’:

• Wikipedia, Dynamic inconsistency.

In economics, dynamic inconsistency or time inconsistency is a situation in which a decision-maker’s preferences change over time in such a way that a preference can become inconsistent at another point in time. This can be thought of as there being many different “selves” within decision makers, with each “self” representing the decision-maker at a different point in time; the inconsistency occurs when not all preferences are aligned.

I this ‘inconsistent’ could be a misleading term for what’s going on here. It suggests that something bad is happening. That may not be true.

Anyway, some of the early research on this was done by George Ainslie, and here is what he found:

Ainslie’s research showed that a substantial number of subjects reported that they would prefer $50 immediately rather than$100 in six months, but would NOT prefer $50 in 3 months rather than$100 in nine months, even though this was the same choice seen at 3 months’ greater distance. More significantly, those subjects who said they preferred $50 in 3 months to$100 in 9 months said they would NOT prefer $50 in 12 months to$100 in 18 months—again, the same pair of options at a different distance—showing that the preference-reversal effect did not depend on the excitement of getting an immediate reward. Nor does it depend on human culture; the first preference reversal findings were in rats and pigeons.

Let me give a mathematical argument for exponential discounting. Of course it will rely on some assumptions. I’m not claiming these assumptions are true! Far from it. I’m just claiming that if we don’t use exponential discounting, we are violating one or more of these assumptions… or breaking out of the whole framework of my argument. The widespread prevalence of ‘dynamic inconsistency’ suggests that the argument doesn’t apply to real life.

Here’s the argument:

Suppose the value to us at any time $t$ of a dollar given to us at some other time $s$ is $V(t,s).$

Let us assume:

1) The ratio

$\displaystyle{ \frac{V(t,s_2)}{V(t,s_1)} }$

is independent of $t.$ E.g., the ratio of value of a “dollar on Friday” to “a dollar on Thursday” is the same if you’re computing it on Monday, or on Tuesday, or on Wednesday.

2) The quantity $V(t,s)$ depends only on the difference $s - t.$

3) The quantity $V(t,s)$ is a continuous function of $s$ and $t.$

Then we can show

$V(t,s) = k \exp(-\alpha(s-t))$

for some constants $\alpha$ and $k.$ Typically we assume $k = 1$ since the value of a dollar given to us right now is 1. But let’s just see how we get this formula for $V(t,s)$ out of assumptions 1), 2) and 3).

The proof goes like this. By 2) we know

$V(t,s) = F(s-t)$

for some function $F$. By 1) it follows that

$\displaystyle{ \frac{F(s_2 - t)}{F(s_1 - t)} }$

is independent of $t,$ so

$\displaystyle{ \frac{F(s_2 - t)}{F(s_1 - t)} = \frac{F(s_2)}{F(s_1)} }$

or in other words

$F(s_2 - t) F(s_1) = F(s_2) F(s_1 - t)$

Ugh! What next? Well, if we take $s_1 = t,$ we get a simpler equation that’s probably still good enough to get the job done:

$F(s_2 - t) F(t) = F(s_2) F(0)$

Now let’s make up a variable $t' = s_2 - t,$ so that $s_2 = t + t'.$ Then we can rewrite our equation as

$F(t') F(t) = F(t+t') F(0)$

or

$F(t) F(t') = F(t+t') F(0)$

This is beautiful except for the constant $F(0).$ Let’s call that $k$ and factor it out by writing

$F(t) = k G(t)$

Then we get

$G(t) G(t') = G(t+t')$

A theorem of Cauchy implies that any continuous solution of this equation is of the form

$G(t) = \exp(-\alpha t)$

So, we get

$F(t) = k \exp(-\alpha t)$

or

$V(t,s) = k \exp(-\alpha(s-t))$

as desired!

By the way, we don’t need to assume $G$ is continuous: it’s enough to assume $G$ is measurable. You can get bizarre nonmeasurable solutions of $G(t) G(t') = G(t+t')$ using the axiom of choice, but they are not of practical interest.

So, assumption 3) is not the assumption I’d want to attack in trying to argue against exponential discounting. In fact both assumptions 1) and 2) are open to quite a few objections. Can you name some? Here’s one: in real life the interest rate changes with time.

By the way, nothing in the argument I gave shows that $\alpha \ge 0.$ So there could be people who obey assumptions 1)–3) yet believe the promise of a dollar in the future is worth more than a dollar in hand today.

Also, nothing in my argument for the form of $V(t,s)$ assumes that $s \ge t.$ That is, my assumptions as stated also concern the value of a dollar that was promised in the past. So, you might have fun seeing what changes, or does not change, if you restrict the assumptions to say they only apply when $s \ge t.$ The arrow of time seems to be built into economics, after all.

Also, you may enjoy finding the place in my derivation where I might have divided by zero, and figure out to do about that.

If you don’t like exponential discounting—for example, because people use it to argue against spending money now to fight climate change—you might prefer hyperbolic discounting:

• Wikipedia, Hyperbolic discounting.