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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 really enjoy now as opposed to putting off the purchase until sometime in the future and enjoying that same wonderful thing later. The theory is that instant gratification is always preferred to delayed gratification. Can time value formulas be invoked to quantify just how much more pleasant instant gratification is compared to delayed gratification is? And is it universally true that delayed gratification is always less pleasant than immediate gratification?

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 delayed gratification 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 parallel the kinds of tradeoffs I have described in time value of money formulas. You can interpret time value of money formulas as a way of measuring deferred gratification. You trade a smaller amount of money today for a larger amount of money sometime in the future.

Now all 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 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 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 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 choices preferences reverse. If 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 only 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 real world 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 owed structural settlements from personal injuries or other legal settlements. A structured settlement is a series of future payouts. As you might expect the immediate payouts offered in these situations often have 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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