Intertemporal  Choice.      Time is important in most economic decisions because the choices we make will have future consequences. Exponential discounting is a very simple way to model our choices over time. All we need to know is the discount factor, and then can easily predict the choice. For that reason exponential discounting is by far the most common way used in economics. The Samuelson (1937) discounted utility (DU) model, together with its axiomatic derivations, defines individual behaviour with respect to time in normative terms. Cash received in the future should be considered today in terms of its present value. In today’s terms, money in the future worth less than today- present value of future cash.

r is the discount rate while 𝛿 = !!!!

is the discount factor. 𝑃𝑉 =  𝑋! + 𝛿𝑋! + 𝛿!𝑋! +⋯+  𝛿!𝑋!

According to the DUM, the present value of discounted utility is:

𝑈! = 𝑢! + 𝛿𝑢!!! + 𝛿!𝑢!!! + 𝛿!𝑢!!!+…..+𝛿!𝑢!!! 𝛿 is interpreted as ‘subjective’ discount factor: measures how the decision maker has a preference for outcomes in the present compared to the future. Typical payoff streams will contain costs as well as rewards. Inter-temporal utility is, therefore, a simple weighted sum of the utility in each period. If 𝛿 < 1  then less weight is given to the utility in a period further away that period is. So the smaller is 𝛿 the more future utility is discounted. Discount factor To better understand exponential discounting we need to see what values for the discount factor seem the most appropriate. In the experimental lab the discount factor can be estimated by giving subjects questions of the form: “Would you prefer £100 today or £150 in a year’s time?” If they answer £100 today then the discount factor is smaller than 0.66 In a study by Benzion, Rapoport and Yagil (1989), subjects were asked questions such as the one displayed above. One thing, however, stands out immediately and that’s the discount factor appears to depend a lot on the context. We can see that the discount factor varies a lot depending on the length of time, amount of money, payment versus receipt, and expedite versus postpone. An individual’s 𝛿 measures their degree of impatience. In the experiment the discount factor is higher the longer it is necessary to wait, suggesting short-term impatience. For example, subjects wanted on average almost as much compensation to postpone receipt six months as to postpone four years. Another thing to note was that the larger the sum of money, the larger the estimated discount factor, suggesting people are more patient for larger amount of money. This is called the absolute magnitude effect. Further, there is also a gain-loss asymmetry where the discount factor is smaller for gains than for losses. For instance, the discount factor is higher in the case of postponing payment (which would require a loss of money) than for postponing receipt (which would involve a gain of money). This is consistent with loss aversion because it suggests that subjects were reluctant to lose money in order to postpone or expedite. Finally, there is a delay-speed asymmetry where the estimated discount factor is higher to postpone than to expedite payment and higher to expedite than to postpone receipt. This is because subjects are willing to pay relatively less money to postpone a payment than they demanded to expedite payment. To really work well, exponential discounting requires that the discount factor should not change depending on the context. Impatience example Would you prefer £80 today or £100 one year from now? Suppose u(x)=x, PV of £100 in one year is 100𝛿. If 𝛿 < 4/5 choose the immediate reward. High 𝛿: patient, will wait to receive higher reward; light discounting Low 𝛿: impatient, will settle for the lower reward today; heavy discounting. Do DMs make decisions that concord with the DUM? In order to make decisions that concord with the DUM, subjects are required to calculate the inter-temporal utility for each option available in the menu of choice, then choose the option with the highest inter-temporal utility. The DUM has several axioms associated with anomalies. Firstly, under the stationarity effect time preferences are constant, but according to Thaler (1981), Olsen (1993) and Lazaro et al. (2002) time preferences maintain an inverse relationship with the time horizon implied in the choice, known as the time effect. Secondly, another anomaly associated with the discounting behaviour is denoted as ‘magnitude effect’. Basically, Andersen et al. (2013) explain that empirical evidence demonstrates that smaller amounts are discounted more than larger amounts. Thirdly, Loewenstein (1988) studied invariability with respect to description and the delay/speed-up asymmetry. Individuals should maintain their preferences when facing the description of an equivalent inter-temporal choice. However, it is

proven that the delay/speed-up asymmetry is anomalous to Samuelson’s DUM and invokes reference dependence and loss aversion. Therefore, Lazaro et al. (2002) explain that the amount required to compensate for delaying a reward is greater than the amount individuals are willing to sacrifice to expedite that benefit in the same time period. Finally, the Samuelson’s DUM is not accurate in capturing people’s behaviour. The discount factor used in the model captures the level of impatience and is constant across time; consequently there is the same level of impatience today or tomorrow when the decision is made. As a consequence of DUM, people do not change their decisions over time, which in reality is not true. Time inconsistency is a well-documented feature of decision-making. People plan to do something, but when it comes to it after the passage of time they change their mind. The DUM states that they should stick with the plans and that their preferences are time consistent. Also, the discount factor should not depend on the magnitude of the payoffs, the discount factor should be the same for gains and losses. Therefore, another model that tries to capture in a better way the present bias in preferences is the quasi-hyperbolic discounting model or 𝛽 − 𝛿 model developed by Laibson (1997).  Hyperbolic  Discounting    People tend to be more impatient over short horizons (smaller discount factors) and less impatient over long horizons (larger discount factor). The DUM hypothesises a constant per-period discount factor, while with hyperbolic discounting the per-period discount factor is allowed to increase over time. This model captures the idea that people prefer immediate rewards and this present bias is captured by a single parameter 𝛽. The quasi-hyperbolic discounting model follows the DUM except that in front of each delta there is a 𝛽  as following:

𝑈! = 𝑢! + 𝛽𝛿𝑢!!! + 𝛽𝛿!𝑢!!! + 𝛽𝛿!𝑢!!!+…..+𝛽𝛿!𝑢!!! For example, consider a project that costs c but delivers benefits of b the following period. Do it today in period T? Yes, if

−𝑐 + 𝛽𝛿𝑏 > 0 => 𝑐 < 𝛽𝛿𝑏 Plan to do it in period t+s? Yes if

−𝛽𝛿!𝑐 + 𝛽𝛿!!!𝑏 > 0 => 𝑐 < 𝛿𝑏. Since 𝛽 ≤ 1 can be the case that 𝑐 < 𝛿𝑏 (plan to do it tomorrow) but 𝑐 > 𝛽𝛿𝑏 (don’t do it when tomorrow comes). After the passage of time, there is a reverse in preferences, unless the individuals are committed. According to Cartwright (2011) 𝛽  is crucial because it measures the present bias. If 𝛽 = 1 there is no bias, however if 𝛽 < 1 more weight is given to today rather than the future and there is decreasing impatience. People manifest patience while making plans for the future, however this patience decreases when it comes to carrying out the plans. The present is very salient and the cost of doing a planned activity is magnified. Therefore, there are consequences of time inconsistency and they depend whether people know or not they are time inconsistent. Self-control problems such as procrastination1 and prepropreation2 have important consequences. However, the self-control problems might differ between individuals. Therefore, 𝛽 captures an individual’s perception of their present bias or their perception of the value of the true 𝛽. If 𝛽 =  𝛽 then a person’s perception of their self-control problem is correct. This category of individuals is called sophisticates. On the other hand, if 𝛽 = 1 and in reality is not 1, then the individuals are called naïve since they believe they are in complete control. Generally, people are partially naïve or sophisticated, and have knowledge of their self-control problems. In order to deal with self-control problems people tend to use commitments. According to Cartwright (2011) the model of quasi-hyperbolic discounting is a nice simple model that permits to capture aspects of inter-temporal choice, but it doesn’t explain very well the potential benefits of commitment. A model that tries to capture this is developed by Gul and Pesendorfer (2001), they illustrate how someone may make consistent choices, however still desire to commit, in order to avoid the psychological costs of temptation. The quasi-hyperbolic discounting model can explain different types of behaviours where an individual has a costly activity that gives subsequent rewards or beneficial activity that gives subsequent costs. For instance, a very good example is saving. According to Feigenbaum and Findley (2014) many of the popular applications of hyperbolic discounting in economics concern delayed saving for retirement and the over-accumulation of debt. People have an incentive to save, however they are unlikely to do it until the end of the month. For this reason, they could use a commitment to save a certain amount at the beginning of the month and if they try using this money they will feel a cost. For example, this can be done with online saving, but there is a consequence to it. People access the online saving account very easily and it is not such a big cost for them to move the money from one account to another. However, by having a savings account where you have access to just by personally calling the bank or                                                                                                                1  Procrastination represents the action of avoiding realising a costly activity in the present and planning to do it in the future. The activity is not done because the cost is magnified by the present bias.    2  Prepropreation is the action of doing rewarding activities today when individuals should not do them. Thus, the reward is salient and individuals do not think of future costs.    

visiting a branch, the costs of moving money is much higher. This goes beyond just a psychological cost; people incur a real cost by contacting the bank directly. In general, when thinking about saving people have a cost of doing this, but also a benefit. The cost comes immediately, in the present and the benefit in the future. In order to commit to saving, individuals need the benefit to be larger than the cost. For the purpose of understanding better the process, different types of people need to be analysed. If somebody is really impatient, his or her delta discount factor is really low. For that individual, even if the benefit is larger than the cost, they will discount the benefit because it comes tomorrow. This type of individual will never start saving since the cost benefit analysis shows that the cost is bigger than the benefit. A less impatient individual who does not have such a strong present bias will most probably save. In this analysis, it is very important to measure an individual’s perception of their present bias; their perception of self-control contributes to the decision-making process. For instance, if 𝛽 is much larger than their actual 𝛽, the individual will plan to save but will not do it. Further, if people have some knowledge of their self-control problems this could determine them to commit and try to overcome the problems.