Week 7. Intertemporal Choice PDF

Behavioural Economics mark.hurlstone @uwa.edu.au Behavioural Economics Outline Intertemporal Choice Exponential PSYC3310: Specialist Topics In Psychology Discounting Discount Factor Utility Streams Mark Hurlstone Delta Mo...

Behavioural Economics mark.hurlstone @uwa.edu.au Behavioural Economics Outline Intertemporal Choice Exponential PSYC3310: Specialist Topics In Psychology Discounting Discount Factor Utility Streams Mark Hurlstone Delta Model Implications Univeristy of Western Australia Indifference Discount Rates Limitations Seminar 7: Intertemporal Choice Hyperbolic Discounting Beta-delta model CSIRO-UWA Behavioural BEL Present-Bias Strengths & Economics Limitations Laboratory [email protected] Behavioural Economics  Today Behavioural Economics mark.hurlstone Examine preferences (4), time (2), and utility @uwa.edu.au maximisation (1) in standard model) Outline (1) (2) (3) (4) Intertemporal Choice Exponential Discounting Discount Factor Utility Streams Delta Model Implications Indifference Discount Rates Intertemporal choice—the exponential discounting Limitations model Hyperbolic Discounting anomalies in the standard Model Beta-delta model Present-Bias behavioural economic alternative—quasi-hyperbolic Strengths & Limitations discounting [email protected] Behavioural Economics  Today Behavioural Economics mark.hurlstone Examine preferences (4), time (2), and utility @uwa.edu.au maximisation (1) in standard model) Outline (1) (2) (3) (4) Intertemporal Choice Exponential Discounting Discount Factor Utility Streams Delta Model Implications Indifference Discount Rates Intertemporal choice—the exponential discounting Limitations model Hyperbolic Discounting anomalies in the standard Model Beta-delta model Present-Bias behavioural economic alternative—quasi-hyperbolic Strengths & Limitations discounting [email protected] Behavioural Economics  Intertemporal choice Behavioural Economics mark.hurlstone Time is important in most decisions because the choices we @uwa.edu.au make will have future consequences Outline Intertemporal choices relate to decisions involving Intertemporal Choice trade-offs between costs and benefits occurring in different Exponential time periods e.g., Discounting Discount Factor when purchasing a 1-year warranty for a new tablet Utility Streams Delta Model computer, you are choosing between a certain loss now Implications Indifference and the possibility of suffering a loss later Discount Rates Limitations Some decisions have immediate benefits and deferred costs Hyperbolic (e.g., movie with friends vs. clean house) Discounting Beta-delta model Others have immediate costs and deferred benefits (e.g., Present-Bias Strengths & Limitations comfortable retirement vs. new car) [email protected] Behavioural Economics  Time discounting Behavioural Economics mark.hurlstone @uwa.edu.au People tend to be impatient—they prefer immediate Outline rewards to delayed rewards Intertemporal $100 today is preferred to $100 tomorrow; $1000 today Choice is preferred to $1000 next year Exponential Discounting Discount Factor When things in the future do not give you as much Utility Streams Delta Model utility—from the point of view of today—as things that Implications happen today, we say you discount the future Indifference Discount Rates the general term is time discounting Limitations Hyperbolic The extent to which you discount the future is a matter Discounting Beta-delta model of preference—known as time preference Present-Bias Strengths & Limitations [email protected] Behavioural Economics  Exponential discounting: Discount factor Behavioural Economics The standard model explains the fact that people prefer their mark.hurlstone @uwa.edu.au money sooner rather than later in terms of exponential Outline discounting Intertemporal Suppose that u > 0 is the utility you derive from receiving a Choice dollar today Exponential Discounting Discount Factor From your current point of view (viz. today) the utility of Utility Streams Delta Model receiving a dollar tomorrow is less than u Implications Indifference We capture this by multiplying the utility of receiving a dollar Discount Rates today by a parameter δ (0 < δ ≤ 1) known as the discount Limitations factor Hyperbolic Discounting Thus, from your current point of view, a dollar tomorrow is Beta-delta model Present-Bias worth δ × u = δu; a dollar the day after tomorrow will be Strengths & Limitations worth δ × δ × u = δ 2 u [email protected] Behavioural Economics  Exponential discounting: Utility streams Behavioural Economics mark.hurlstone @uwa.edu.au In general, we want to be able to evaluate a whole Outline sequence of utilities, that is a utility stream Intertemporal Letting t represent time, we will use t = 0 to represent Choice Exponential today, t = 1 to represent tomorrow, t = 2 to represent the Discounting Discount Factor day after tomorrow, and so on Utility Streams Delta Model We will let ut denote the utility you receive at time t Implications Indifference meaning that: Discount Rates u0 represents the utility you receive today Limitations u1 represents the utility you receive tomorrow Hyperbolic Discounting u2 represents the utility you receive the day after Beta-delta model Present-Bias tomorrow, and so on Strengths & Limitations [email protected] Behavioural Economics  Exponential discounting: Utility streams Behavioural Economics mark.hurlstone @uwa.edu.au Utility streams for different choice options (viz. a, b, c, d) Outline can be represented in table form: Intertemporal Choice t=0 t=1 t=2 Exponential Discounting a 1 0 0 Discount Factor Utility Streams b 0 3 0 Delta Model Implications c 0 0 4 Indifference Discount Rates d 1 3 4 Limitations Hyperbolic Discounting We can determine which option you should choose using Beta-delta model Present-Bias the delta model Strengths & Limitations [email protected] Behavioural Economics  Exponential discounting: The delta model Behavioural Economics mark.hurlstone @uwa.edu.au According to the delta function, the utility U 0 (u) of utility Outline stream u = hu0 , u1 , u2 ,...i from the point of view of time t = 0 Intertemporal Choice is: Exponential Discounting ∞ X Discount Factor Utility Streams U 0 (u) = u0 + δu1 + δ 2 u2 + δ 3 u3 +... = σ t ut (1) Delta Model Implications t=1 Indifference Discount Rates Where δ (0 < δ ≤ 1) is the discount factor which captures Limitations time preference (patience = values close to 1, whereas Hyperbolic Discounting impatience = values close to 0) Beta-delta model Present-Bias Strengths & Limitations [email protected] Behavioural Economics  Exponential discounting: The delta model Behavioural Economics Let’s apply the delta model to the utility streams in the mark.hurlstone @uwa.edu.au table on the earlier slide Outline Assume that δ = 0.9 and each utility stream is Intertemporal evaluated from t = 0 Choice The expected utilities are: Exponential Discounting Discount Factor U 0 (a) = u0 = 1 Utility Streams U 0 (b) = δu1 = 0.9 × 3 = 2.7 Delta Model Implications U 0 (c) = δ 2 u2 = 0.92 × 4 = 3.24 Indifference Discount Rates U 0 (d) = u0 + δu1 + δ 2 u2 = 1 + 2.7 + 3.24 = 6.94 Limitations Hyperbolic If given the choice between all four alternatives, you Discounting Beta-delta model should choose option d Present-Bias Strengths & If given the choice between a, b, and c, you should Limitations choose c [email protected] Behavioural Economics  Exponential discounting: The delta model Behavioural Economics mark.hurlstone @uwa.edu.au What happens if we repeat this process, but this time assume that δ = 0.1? Outline Intertemporal The expected utilities are: Choice U 0 (a) = u0 = 1 Exponential Discounting U 0 (b) = δu1 = 0.1 × 3 = 0.3 Discount Factor Utility Streams U 0 (c) = δ 2 u2 = 0.12 × 4 = 0.04 Delta Model U 0 (d) = u0 + δu1 + δ 2 u2 = 1 + 0.3 + 0.04 = 1.34 Implications Indifference Discount Rates If given the choice between all four alternatives, you Limitations should still choose option d Hyperbolic Discounting But now, if given the choice between a, b, and c, you Beta-delta model Present-Bias should choose a Strengths & Limitations [email protected] Behavioural Economics  Exponential discounting: Implications of discount factor Behavioural Economics mark.hurlstone @uwa.edu.au Outline As this example shows, your discount factor can have a Intertemporal dramatic impact on your choices Choice Exponential If your discount factor is high—viz. close to one—you Discounting Discount Factor exhibit patience and do not discount the future much Utility Streams Delta Model Implications If your discount factor is low—viz. close to zero—you Indifference Discount Rates exhibit impatience and discount the future heavily Limitations You can see how δ captures time preferences Hyperbolic Discounting Beta-delta model Present-Bias Strengths & Limitations [email protected] Behavioural Economics  Exponential discounting: Implications of discount factor Behavioural Economics mark.hurlstone @uwa.edu.au Economists believe that discount factors can be used to Outline Intertemporal explain a great deal of human behaviour Choice Exponential If your discount factor is low, you are are more likely to Discounting Discount Factor spend money, procrastinate, do drugs, and have unsafe Utility Streams Delta Model sex Implications Indifference If your discount factor is high, you are more likely to Discount Rates Limitations save money, plan for the future, say no to drugs and Hyperbolic use protection Discounting Beta-delta model Present-Bias Strengths & Limitations [email protected] Behavioural Economics  Exponential discounting: Indifference Behavioural Economics So far, we have used our knowledge of δ to determine a mark.hurlstone @uwa.edu.au person’s preferences over utility streams Outline Sometimes we want to go the other way—viz. use a person’s Intertemporal preferences to calculate their discount factor Choice Exponential Discounting is measured by getting participants to choose Discounting Discount Factor between an immediate and delayed reward: Utility Streams Delta Model would you prefer $100 today or $110 1-year from now? Implications Indifference would you prefer $100 today or $130 1-year from now? Discount Rates would you prefer $100 today or $160 1-year from now? Limitations and so on... Hyperbolic Discounting Beta-delta model As soon as the participant is indifferent between an Present-Bias Strengths & immediate and delayed reward we can calculate his or her Limitations discount factor [email protected] Behavioural Economics  Exponential discounting: Indifference Behavioural Economics mark.hurlstone @uwa.edu.au Suppose that you are indifferent between a = $100 now, and b = $160 in 1-year Outline Intertemporal Let’s convert the monetary amounts into utilities first: a Choice = 1000.5 = 10; b = 1600.5 = 12.65 Exponential Discounting Given you are indifferent between a and b at time zero, Discount Factor Utility Streams we know that: Delta Model Implications U 0 (a) = U 0 (b) Indifference Discount Rates which implies that 10 = 12.65δ Limitations which is to say that δ = 10/12.65 = 0.79 Hyperbolic Discounting The calculated discount factor indicates that you are Beta-delta model Present-Bias relatively patient Strengths & Limitations [email protected] Behavioural Economics  Exponential discounting: Indifference Behavioural Economics mark.hurlstone @uwa.edu.au Now, let’s suppose that you are indifferent between a = $100 now, and b = $1000 in 1-year Outline Intertemporal Lets again convert the monetary amounts into utilities Choice first: a = 1000.5 = 10; b = 10000.5 = 31.62 Exponential Discounting Given you are indifferent between a and b at time zero, Discount Factor Utility Streams we know that: Delta Model Implications U 0 (a) = U 0 (b) Indifference Discount Rates which implies that 10 = 31.62δ Limitations which is to say that δ = 10/31.62 = 0.32 Hyperbolic Discounting The calculated discount factor indicates that you are Beta-delta model Present-Bias relatively impatient Strengths & Limitations [email protected] Behavioural Economics  Exponential discounting: Discount rates Behavioural Economics Sometimes discounting is expressed in terms of a discount mark.hurlstone rate r rather than a discount factor δ. The conversion is as @uwa.edu.au follows: Outline Intertemporal 1−δ Choice r= (2) δ Exponential Discounting If your discount factor is 0.79 then your discount rate is 0.27. Discount Factor Utility Streams This means you would require an interest rate of 27% to Delta Model Implications delay receiving the $100 (the interest rate would be 212% if Indifference Discount Rates your discount factor is 0.32). Limitations Hyperbolic Knowing r, you can calculate δ as follows: Discounting Beta-delta model Present-Bias 1 Strengths & Limitations δ= (3) 1+r [email protected] Behavioural Economics  Exponential discounting: Limitations Behavioural Economics mark.hurlstone A major shortcoming of this model is that it assumes @uwa.edu.au that people have time consistent preferences: Outline implies that your preferences over two options should Intertemporal not change simply because times passes Choice if you feel (today) that a is better than b, then you felt Exponential Discounting the same way about a and b yesterday, and will feel the Discount Factor Utility Streams same way tomorrow Delta Model Implications Indifference The bad news is that people violate this assumption all Discount Rates the time: Limitations saying you will give up alcohol... Hyperbolic Discounting promising to stop smoking... Beta-delta model Present-Bias purchasing that gym membership... Strengths & Limitations planning to do your homework... [email protected] Behavioural Economics  Exponential Discounting: Limitations Behavioural Economics mark.hurlstone @uwa.edu.au Outline Further shortcomings of the model of exponential Intertemporal discounting: Choice Exponential speaker 1: sign effect, magnitude effect, & temporal Discounting loss aversion Discount Factor Utility Streams speaker 2: delay speed-up asymmetry & preference for Delta Model Implications improving sequences Indifference Discount Rates speaker 3: date-delay effect & violations of Limitations independence Hyperbolic Discounting Beta-delta model Present-Bias Strengths & Limitations [email protected] Behavioural Economics  Quasi-hyperbolic discounting Behavioural Economics mark.hurlstone The evidence we have covered suggests that people do not @uwa.edu.au have time consistent preferences Outline People tend to be patient for long-term gains, but impatient Intertemporal for short-term gains: Choice Exponential on Friday you might plan to do your homework on Discounting Discount Factor Saturday, but when Saturday comes you prefer to do it Utility Streams Delta Model on Monday Implications today you might prefer to reject $100 tomorrow in favour Indifference Discount Rates of $110 the day after, but tomorrow you change your Limitations mind Hyperbolic Discounting We say that there is time inconsistency if someone plans to Beta-delta model Present-Bias do something in the future, but subsequently changes their Strengths & Limitations mind [email protected] Behavioural Economics  Quasi-hyperbolic discounting: Beta-delta model Behavioural Economics mark.hurlstone @uwa.edu.au Outline According to the beta-delta function, the utility U 0 (u) of Intertemporal utility stream u = hu0 , u1 , u2 ,...i from the point of view of time Choice Exponential t = 0 is: Discounting Discount Factor ∞ X 0 2 3 σ t ut (4) Utility Streams Delta Model U (u) = u0 + βδu1 + βδ u2 + βδ u3 +... = u1 + β Implications Indifference t=1 Discount Rates Limitations Where δ is as before, and β is the present bias Hyperbolic Discounting Beta-delta model Present-Bias Strengths & Limitations [email protected] Behavioural Economics  Quasi-hyperbolic discounting: Beta-delta model Behavioural Economics mark.hurlstone @uwa.edu.au The beta-delta model is the same as the delta model, except for the inclusion of the parameter β Outline Intertemporal When β = 1, the model reduces down to the delta model Choice Exponential However, when β < 1, all outcomes beyond the present time Discounting get discounted more than under exponential discounting Discount Factor Utility Streams Delta Model Hence, when β < 1 more weight is given to today than the Implications Indifference future and we say there are present-biased preferences Discount Rates If you exhibit such preferences, then given the choice Limitations Hyperbolic between a small earlier reward and a bigger, later reward Discounting you will end up choosing the smaller immediate reward (but Beta-delta model Present-Bias regret it afterwards) Strengths & Limitations [email protected] Behavioural Economics  Quasi-hyperbolic discounting: Present-bias Behavioural Economics mark.hurlstone @uwa.edu.au Suppose you are on a diet but have to decide between Outline having a piece of cake at a party on Saturday Intertemporal Choice Eating the cake gives you a utility of 4 Exponential Discounting Discount Factor However, if you eat it, you will have to exercise for four hours Utility Streams on Sunday, giving you a utility of 0 (assuming you are like Delta Model Implications most people) Indifference Discount Rates Alternatively, you could skip the cake, giving you a lowly Limitations utility of 1, but obtain a utility of 6 on Sunday by watching Hyperbolic Discounting back-to-back episodes of The Batchelor Beta-delta model Present-Bias Strengths & Limitations [email protected] Behavioural Economics  Quasi-hyperbolic discounting: Present-bias Behavioural Economics mark.hurlstone @uwa.edu.au Outline Intertemporal Choice Exponential Friday (t = 0) Saturday (t = 1) Sunday (t = 2) Discounting Discount Factor a 0 4 0 Utility Streams Delta Model b 0 1 6 Implications Indifference Discount Rates Limitations Hyperbolic Discounting Beta-delta model Present-Bias Strengths & Limitations [email protected] Behavioural Economics  Quasi-hyperbolic discounting: Present-bias Behavioural Economics Let’s apply the beta-delta model to this example, with β = 0.5 mark.hurlstone @uwa.edu.au and δ = 0.67 Outline On Friday, the utility of eating the cake a and skipping it b is: Intertemporal Choice U 0 (a) = 0 + 0.5 × 0.67 × 4 + 0.5 × 0.672 × 0 = 1.33 Exponential U 0 (b) = 0 + 0.5 × 0.67 × 1 + 0.5 × 0.672 × 6 = 1.67 Discounting Discount Factor Utility Streams On Saturday, the utility of eating the cake a and skipping it b Delta Model Implications is: Indifference Discount Rates U 1 (a) = 4 + 0.5 × 0.67 × 0 = 4 Limitations U 1 (b) = 1 + 0.5 × 0.67 × 6 = 3 Hyperbolic Discounting Beta-delta model On Friday, you would prefer to skip the cake, but come Present-Bias Strengths & Saturday impulsivity causes you to change your Limitations mind—time inconsistency at work [email protected] Behavioural Economics  Quasi-hyperbolic discounting: Strengths & limitations Behavioural Economics mark.hurlstone @uwa.edu.au Quasi-hyperbolic discounting can explain time Outline inconsistent preferences Intertemporal Choice It can account for the fact that people emphasise their Exponential present over their future well-being Discounting Discount Factor Utility Streams It can also account for the fact that people change their Delta Model Implications minds about how to balance the present versus the Indifference Discount Rates future Limitations Thus, it can explain why people intend to diet, stop Hyperbolic Discounting smoking, do homework, and quit drugs, and then fail to Beta-delta model Present-Bias do so Strengths & Limitations [email protected] Behavioural Economics  Quasi-hyperbolic discounting: Strengths & limitations Behavioural Economics mark.hurlstone @uwa.edu.au The model can therefore explain a number of phenomena that are inconsistent with the model of exponential Outline discounting Intertemporal Choice Yet, there are other aspects of the data reviewed by our Exponential Discounting speakers that the model cannot explain, such as the sign Discount Factor effect, preferences for improving sequences, and the Utility Streams Delta Model peak-end rule Implications Indifference Discount Rates The book chapters in the general reading section describe Limitations more elaborate behavioural models that are capable of Hyperbolic providing a more complete account of the data—the chapter Discounting Beta-delta model by Cartwright (2011) provides a nice overview of these Present-Bias Strengths & models Limitations [email protected] Behavioural Economics