When I was a bachelor student, during my Economics courses, I have learned that taking a loan from the bank is equivalent with accelerating consumption. For example, if Said wants to buy a car that costs 15.000 Euros then he has two major options. First, Said can save money for a certain period of time, let’s assume three hundred Euros each month. After fifty months when the amount needed to buy a car has accumulated in his savings account, Said can go and buy the car. Taking this path, Said has one advantage and one disadvantage. The advantage is that he will pay for the car only the 15.000 Euros that represent the actual cost. The disadvantage is that for five years and two months (fifty months) Said will not have the car.
The second option Said has is to go to a bank and ask for a loan of 15.000 Euros and buy the car as soon as the loan is approved. If he follows this path, Said will have the car as soon as possible, but will have to pay across five years both the 15.000 Euros which represent the cost of the car and the cost of the credit, let’s say about 3000 Euros.
As my former Economics teacher would say, the 3000 Euros Said pays for the loan represent, in fact, a cost for having the car earlier.
Taking the car out of the picture, Said faces the following question: How much would you need to get now in order to give up 18.000 Euros in five years’ time?
In the case of Said buying a car, the answer was 15.000 Euros. However, let’s assume that you are faced with a similar question, namely what is the minimum amount you would accept now in order to give up 10.000 Euros in one year time?
What is your honest answer? Would 8000 be enough? 9000?
From a rational point of view there is a (so called) correct answer to this. You should ask for at least 9803 Euros. This is because if you take this amount and place it in a bank deposit for one year with an interest rate of 2%, after one year you will have 10.000 Euros in your account.
However I am sure that you would have been happy with 9000 and for sure you did not compute the amount using the average interest rate for bank deposits in Euros. Don’t worry, most people ask for less than 9803 Euros and it is quite OK to do so.
You have just learned about time discounting or more accurately inter-temporal discounting. People are willing to give up a part out of a future outcome in order to get it faster. The so called correct answer to the question above, namely 9803 Euros, was obtained through a formula specific for the Discounted Utility Model. In an ideal world the discount rate would be equal to the interest rate on the financial markets. Simply put, in an ideal world, you should have a discount rate equal to the interest rate on the financial market, namely 2%. In the same ideal world, you should have asked for 9803 Euros, and then go immediately to the bank and place them in a deposit for one year, subsequently benefiting of 10.000 Euros in one year from now.
Most of the time, the discount rate is larger than the interest rate on the financial markets. For example if you would have accepted 9000 Euros now in exchange for giving up 10.000 Euros in one year time, then your discount rate would have been 10%, which is significantly larger than the interest rate for one year deposits in Euros, namely 2%.
You might wonder if this is wrong; if it is wrong to have a discount rate of 10% which is larger than the so called correct one of 2%? The answer is that it is not wrong, because the Discounted Utility model was never meant to be the correct one, or in more scientific terms the normative one. When it was proposed by Paul Samuelson in 1937 it was simply a proposition of a theoretical model. The author never claimed that it was the right model for describing or prescribing how decision over time should be made. However, this model was embraced by the scientific community and before long it gained the status of the correct model.
Let’s move away from the debate whether the Discounted Utility model is correct or not and focus on a very interesting aspect of decision making over time. Imagine that I’m asking you the following question:
What is the minimum amount you would accept now in order to give up 10.000 Euros in Four years’ time?
Would you accept 6000 Euros? Probably not… How about 7500? I think that you would be happy with this amount. Right?
The interesting thing that happens when deciding about how much we would want for giving up later rewards is that apparently we don’t hold the discount rate constant. Let me explain a bit more clearly. Let’s assume that you would accept 9000 Euros now in order to give up 10.000 in one year from now. This means that your discount rate is 10%. If we hold this rate constant, you should accept 8100 Euros in order to give up 10.000 in two years’ time. This is 10% less than 9000 Euros. Applying the same rate, you should accept 7290 Euros in order to give up 10.000 in three years’ time. Again, this is 10% less than 8100, which in turn is 10% less than 9000 – the amount you accepted in exchange for giving up 10.000 in one year time. The question was what is the minimum amount you would you accept to give up on 10.000 Euros in Four years’ time. In order to find this out we should discount again with 10% the 7290 Euros sum, leading to the amount of 6561 Euros.
However, most people would not be happy with this amount and would ask for more than 6561 Euros in order to give up on 10.000 Euros in four years’ time. My guess is that the minimum accepted amount would be higher than 6561 Euros, or at least that is what theory tells us. Probably 7500 Euros is closer to the amount you would be happy with in exchange for 10.000 Euros in four years’ time.
The key learning of this thought exercise is that close rewards are discounted more than distant once. Putting things differently, the discount you would be willing to accept in order to get the reward now instead of in 12 months is higher than the discount you would be willing to accept in order to get the reward in 12 months instead of in 24 months. This phenomenon is called hyperbolic discounting.
For a long time, the belief was that the mechanism behind hyperbolic discounting is the decreasing discounting rates. For example if the discount rate for the first year was 10%, then the discount rate for the second year would be lower, say 7%. Similarly, the discount rate for the third year would be even lower, say 5%. Quite recently, this assumption was challenged by a study conducted by Professor Zauberman and colleagues who proved that the mechanism behind hyperbolic discounting is not the decreasing discount rates, but rather the subjective perception of time. In this study the researchers proved that when taking into account the subjective perceptions of time, the discount rate remains constant.
To better understand, let’s focus a bit on what subjective perceptions of time mean. The essence is that a time period of, say, three years is perceived subjectively different from the sum of three one year intervals. Putting this in mathematical sequence, with sp being the subjective perception, it would look like this:
sp(3 years) < sp(1 year) + sp(1 year) + sp(1 year)
According to Professor Zauberman and colleagues, the subjective perception of time explains the difference between the minimum sums accepted to give up on sooner and later rewards. This finding is in line with conclusions on subjective perceptions of other types of values. For example, prospect theory tells us that people perceive probabilities different than their actual value, except for probabilities of zero and one. Similarly the contrast effect tells us that absolute values are judged depending on reference points. According to the study by Professor Zauberman and colleagues, time makes no exception to the rule of relativity.
This post is documented from:
Frederick, S., Loewenstein, G. & O'Donoghue T. (2002), "Time Discounting and Time Preference: A Critical Review," Journal of Economic Literature, 40 (2), 351-401.
Zauberman, G., Kim, B.K., Malkoc, S. A., & Bettman, J. R. (2009). "Discounting time and time discounting: Subjective time perception and intertemporal preferences," Journal of Marketing Research, 46, 543-556.