Thinking and Deciding

DISCOUNTING 481
noted. If you have $100 now, you can invest it and end up with somewhat more a
year from now, say $110. Therefore, you might be indifferent between getting $100
now and getting $110 a year from now, for this reason alone. Similarly, if you need
$100 now, the value of $100 now might be equivalent to $120 a year from now, if
you would otherwise have to borrow the $100 at a rate of 20%.
Another reason why discounting makes sense is that unforeseen events can affect
the utility of money (or any outcome) to an individual. For example, in the interval
between now and when you collect the $100, the economy could be hit by an enormous
wave of inflation that makes the money worthless; you could die; or, to end on
a more pleasant note, you could become a millionaire and have no need for such a
piddling amount. If we make the assumption that each of these events is as likely to
happen at one time as at any other, the same curve results. If you think that there is a
1% chance of such an event’s occurring in the next year, the expected utility of $100
a year from now would be 1% less than it would be otherwise, because there is a 1%
chance that its utility would be zero. The longer you wait, the more likely it is that
one of these things will happen, so the lower the utility of the $100.
So far, we have assumed that you care just as much about your utility in the
future as you do about your immediate utility, but we do not even need to assume
that in order to derive a curve of this form as the normative model. What we do
need to assume is that your relative preference for two future outcomes at different
times depends only on what the outcomes are and on the time difference between
them, but not on the time you make your choice. This is called the principle of delay
independence.
Consider a simpler case. It is 2:59 P.M. A child has a choice of one piece of
candy at 3:00 P.M. or two pieces at 4:00 P.M.. Many children will take the first option.
Suppose that such a child is given the same choice at 2:00 P.M. instead of 2:59 P.M..
That is, the choice is still between one piece at 3:00 P.M. or two pieces at 4:00 P.M..
It is easy to imagine that the child will choose the second option. In this case, the
child has made different choices as a function of the time alone. (Let us put aside
the child’s emotions of frustration and anticipation, for now.) This pattern of choices
is dynamically inconsistent, that is, inconsistent over time. This is nonnormative,
because the time at which the decision is made does not affect the extent to which
the options achieve the child’s goals. If the child’s main goal here is to get as much
candy as possible, for example, then the second option should be chosen in both
cases. Dynamic inconsistency violates the principle of delay independence.
Likewise, your choice between $100 at a certain time and $110 a year later should
not change simply as a function of when you make the choice. If one of these options
achieves your goals better, then you should choose that one, regardless of when you
decide. Unless you learn something that makes a difference between one choice
point and the other — and we have assumed that you do not — such changes are
unwarranted. 5 Note also that this assumption of delay independence does not require
5 Of course, you are free to change your mind, but there is no reason why you should change it in any
particular direction — for example, from the later, larger outcome to the smaller, earlier one. This is the
important point.