In one sense the value of one dollar today is
certainly greater than one dollar in the future precisely because you can buy
something you can really enjoy ** now **as opposed to putting off the
purchase until sometime in the future and enjoying that same wonderful thing

Suppose I really, really love oatmeal raisin cookies. One sits before me now on a plate. I know how good it will taste if I scarf it down now. But suppose instead I contemplate delaying eating the cookie. Suppose I think about eating that same cookie tomorrow instead, or perhaps the day after tomorrow or maybe next week or next month. Clearly there is some discomfort or lessening in satisfaction in my delaying eating this cookie.

We might even be able to quantify the value of my deferment of the immediate pleasure of scarfing down this oatmeal raisin cookies by giving me some hypothetical choices. You might ask me if I would trade eating two cookies tomorrow for eating only one cookie now. Or three cookies a week from now for one now and so on. If you think about it, these kinds of choices exactly reflect the kinds of tradeoffs I have described in time value of money formulas.

You can interpret time value of money formulas as a form of deferred gratification. You trade a smaller amount of money today for a larger amount of money sometime in the future. Now all of the previously described time value formulas for converting future dollars into present dollars have been based on only three variables: the amounts of money involved, the interest or discount rate per time period, the “i,” which is the fundamental conversion factor, and the number of time periods into the future involved in the conversion.

For the previously described time value formulas the interest or discount rate per time period, the conversion factor “i” was considered fixed. It did not matter how large or small the amounts of money involved were. The conversion rate, the “i” remained the same. It also did not matter how many time periods were involved. The same conversion rate “i” was applied to the second period as to the twentieth period, or one hundredth period. It was also assumed that the specific interest rates were determined by market forces that constrained the potential range of choices for the values of “i.”

But when we turn our attention to time preferences
for cookie consumption or even monetary choices where the “i” is not set by, or
constrained by, market forces, well things get more interesting and
complicated. Psychologists and
economists have sought a quantitative description of how people and even
animals weigh time value choices between the current consumption of a reward
and future consumption of a reward^{.}.
A reward can be a cookie or lump sum of cash or any other thing that can bring
immediate satisfaction. The first efforts at a quantified model of time
preferences were called exponential models. Here is the general form of this
model.

Now you might not realize it but this is the same model that we have been applying to all our time value formulas. It looks different because of the “e” term. But all the “e” term does is convert the compounding period from a discrete one like a year or month into a continuous compounding period. All of our formulas up to now have assumed discrete compounding periods that look like this.

This exponential model implies that if I am willing
to trade three cookies one week from now for one cookie today, I should be
willing to trade having one cookie two weeks from now for getting three cookies
three weeks from now. Notice each alternative exchange involves a one-week time
delay in consumption. But it turns out in these kinds of decisions choice preferences reverse. As long as I cannot eat
the cookie ** now**, I will actually prefer getting the three cookies
three weeks from now as compared to having only one cookie two weeks from now.

The model also implies that the amount of dollars or cookies involved should not impact the rate of discounting. If I am indifferent between receiving $10 today and $12 a year from now this implies an interest or conversion rate of 20%. But studies have shown that if the amount of dollars involved are much higher the rate of discounting goes down. In other words, I might also be indifferent to receiving $100 today and $110 a year from now, a conversion rate of 10%.

Yet another somewhat related anomaly is the discovery that gains and losses are discounted at different rates. Specifically, losses are usually discounted at a lower rate than gains. In one study it was found that on average respondents were indifferent between receiving $10 immediately or $21 a year later, a conversion rate of about 200%. But this same group was indifferent between suffering an immediate loss of $10 and losing $15 a year later, a conversion rate of 50%. Some researchers have found that respondents preferred an immediate loss over a delayed loss of equal value.

Because of these pervasive violations of the exponential discounting model behavioral economists have developed alternative models that add a parameter to reflect the fact that conversion rates change over time depending upon the scale and types off tradeoffs I just described.

Now these kind of trade off decisions do not just exist in psychology experiments. There are not uncommon situations where individuals make choices between accepting a lump sum of cash now or a series of future payments. Lottery winners usually are given a choice between an immediate payout versus a series of deferred payments. There are companies that specialize in offering individuals who are receiving payments over time in structural settlements from a personal injury or other legal matter. As you might expect the immediate payouts offered in these situations often have fairly steep discount rates. For example, a winner of a $20 million dollar jackpot might be offered a lump sum of only $10 million or a $1 million dollar payment in each of 20 years. From the standpoint of the lottery this represent a discount rate of about 7.5% There is both empirical and theoretical evidence that the larger the total lottery winning amount the more the immediate cash option is preferred which as noted before violates the prediction that discount rates do not vary with the size of the payoff choices.

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Copyright 2018 Michael Sack Elmaleh