Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Time Preferences

Marie-Pierre Dargnies January 7, 2015

Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Time Preferences: Why should we care?

Intertemporal choices (decisions involving trade-os among costs and benets occuring at dierent times) are all over: Borrowing, savings Diet, exercise Human capital accumulation

The preferences governing these situations are necessarily important

Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

A motivating example

Mischel et al. (1989) (Science): Participants are 35 preschoolers from Stanford Task: Wait 15 minutes watching a marshmallow. Get another marshmallow ρ(seconds , SAT ) = 0.57 ∗ ∗∗ Ability to delay gratication (maybe time preference) apparently matters for life outcomes

Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

The discounted-utility (DU) model

Model proposed by Paul Samuelson in 1937 Accepted almost instantly as a valid normative standard for public policies and a descriptively accurate representation of actual behaviors... despite Samuelson's reservations Central assumption of the DU model: all motives underlying intertemporal choice can be condensed into a single parameter-the discount rate

Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

The discounted-utility (DU) model (2)

The DU model species a decision maker's intertemporal preferences over consumption proles (ct , ..., cT ) Under the assumptions of completeness, transitivity and continuity, such preferences can be represented by an intertemporal utility function of the form: TP −t 1 k U t (ct , ..., cT ) = D (k )u (ct +k ) where D (k ) = ( 1+ρ ) k =0 u (ct +k ) often interpreted as the person's cardinal instantaneous utility function (her well being in period t + k ) D(k) often interpreted as the person's discount function, the relative weight that she attaches in period t to her well-being in t + k ρ represents the individual's pure rate of time preference (her discount rate) Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Features of the DU model

Integration of new alternatives to existing plans Utility independence: the distribution of utility across time makes no dierence beyond that dictated by discounting Consumption independence: a person's well-being in period t + k is independent of his consumption in any other period Stationary instantaneous utility: the cardinal instantaneous utility function u (cτ ) is constant across time (well-being generated by any activity is the same in dierent periods) Independence of discounting from consumption: the discount function is invariant across all forms of consumption Diminishing marginal utility (u (ct ) is concave) and positive time preference (ρ is positive) Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Other feature of the DU model: Constant discounting and time consistency

Any discount function can be written in the form: kQ −1 1

D (k ) =

(

)

n=0 1+ρn where ρn represents the per-period discount rate for period n Hence, by assuming that the discount function takes the form 1 k D (k ) = ( 1+ρ ) , the DU model assumes a constant per-period discount rate (ρn = ρ for all n) per-period discount rate between periods t and t + 1: D (t )−D (t +1) D (t +1) Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Constant discounting and time consistency

(2) implies time-consistency: later preferences "conrm" earlier ones Formally, for any two consumtion proles (ct , ct +1 , ..., cT ) and (ct0 , ct0 +1 , ..., cT0 ), with ct = ct0 , U t (ct , ct +1 , ..., cT ) ≥ U t (ct0 , ct0+1 , ..., cT0 ) if and only if U t +1 (ct +1 , ..., cT ) ≥ U t +1 (ct0+1 , ..., cT0 )

Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Constant discounting and time consistency

(2) implies time-consistency: later preferences "conrm" earlier ones Formally, for any two consumtion proles (ct , ct +1 , ..., cT ) and (ct0 , ct0 +1 , ..., cT0 ), with ct = ct0 , U t (ct , ct +1 , ..., cT ) ≥ U t (ct0 , ct0+1 , ..., cT0 ) if and only if U t +1 (ct +1 , ..., cT ) ≥ U t +1 (ct0+1 , ..., cT0 ) If in period t one prefers X at τ to Y at τ + d for some τ , then in period t one must prefer X at τ to Y at τ + d for all τ

Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Discounted utility anomalies: Hyperbolic discounting

Declining rate of time preference Thaler (1981) asked subjects to specify the amount of money they would require in 1 month, 1 year, and 10 years to make them indierent to receiving 15$ now Median responses: 20$, 50$, 100$ Imply an average annual discount rate of 345% over a one month horizon, 120% over a 1-year horizon and 19% over a 10-year horizon Other evidence: preferences between two delayed rewards can reverse in favor over the more proximate reward as the time to both rewards diminishes. For example, 110$ in 31 days preferred to 100$ in 30 days but 100$ now preferred to 110$ tomorrow Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Other DU anomalies

The "sign eect" (gains are discounted more than losses)

Thaler (1981) asked subjects to imagine they had received a trac ticket that could be paid now or later and to state how much they would be willing to pay if the payment could be delayed much lower dicount rates than for monetary gains In many studies, subjects found to prefer to incur a loss immediately rather than delay it

The "magnitude eect": small outcomes discounted more than large ones

Thaler's subjects indierent between 15$ now and 60$ in a year, 250$ now and 350$ in a year and 3000$ now and 4000$ in a year This implies discount rates of 139%, 34% and 29% respectively Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Other DU anomalies (2)

The "delay-speedup" asymmetry

Discount rates can be dramatically aected by whether the change in delivery time of an outcome is framed as an acceleration or a delay from some temporal reference point (Loewenstein 1988) Respondents who did not expect to receive a VCR for another year would pay 54$ to receive it now, those who thought they would receive it now demanded 126$ to delay its receipt by a year

Preference for improving sequences

Research studying preferences over sequences of outcomes generally nd that people prefer improving sequences to declining ones (e.g wages) Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Other DU anomalies (3)

Violations of independence and preference for spread Loewenstein and Prelec (1993)

Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Models of hyperbolic discounting

Strotz, rst to suggest that "special attention" be given to declining discount rates Functional  form (Phelps and Pollack (1968), Elster (1979)): 0 D (k ) = 1βδk ifif kk = > 0 with β ≤ 1 We therefore have: TP −t U t (ct , ..., cT ) = u (ct ) + βδk u (ct +k ) k =1 Assumes that the per-period discount rate between now and the next period is 1−βδ βδ

whereas the per-period discount rate between any two future period is 1−δ δ


dynamic inconsistency

Crucial question: is the dynamically inconsistent person aware or not that her preferences will change over time? Two extreme alternatives (Strotz and Pollack): a "naive" person will think her future preferences will be identical to her current ones a "sophisticated" person will correctly predict how her preferences will change over time

One way to identify sophistication is to look for evidence of commitment (eliminate now an inferior option that might be tempting later)

Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Applications of (quasi)-hyperbolic preferences

Overconsumption/ insucient saving (Laibson (1997, 1998)) Procrastination (Fischer (1999) and O'Donoghue and Rabin (1999, 2001)) : a (β, δ) will tend to put o an onerous activity more than she would like to from a prior point of view Addiction (Carrillo (1999), O'Donoghue and Rabin (1999, 2000), Gruber and Koszegi (2000))

Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Applications: Evidence of commitment

Setting deadlines for oneself: Ariely and Wertenbroch (2002) MIT students who have to write 3 papers for a class assigned to 2 dierent conditions: deadline for the 3 papers imposed by instuctor, evenly spaced across the semester students allowed to set their own deadline for each of the 3 papers

In both conditions, penalty for delay: 1 percent per day late Many students in the "free-choice" condition chose to impose deadline on themselves (preference for commitment) Few students chose evenly spaced deadline although those who did not performed worse than those who did Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Application: Golden eggs (Laibson 1997)

Use of illiquid assets (to prevent overconsumption): Golden eggs (Laibson 1997) Consumers with dynamically inconsistent preferences can invest in two instruments, a liquid asset xt and an illiquid asset zt We can then consider each consumer is playing a game with a sequence of temporal selves The use of the illiquid asset will then allow the individual to correct the tendency of his future selves to be present-biased Self t − 1 will control the liquidity available to self t by its allocational choice between the liquid and illiquid asset Implication: the decline in savings in the U.S. since the 1980s caused by the rapid expansion in unsecured credit (credit cards) The introduction of such credit reduced the eectiveness of illiquid assets in disciplining future selves. Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Application: Going to the gym (Dellavigna and Malmendier 2004)

Consumer with (β , δ ) preferences decides whether to go to the gym Going to the gym gives a payo c0 in t=2 Not going results in payo 0 in t=1 and t=2 At t = 0, the consumer wants to go to the gym if: −βδ c + βδ 2 b ≥ 0 ⇔ δ b ≥ c At t = 1, the time-inconsistent consumer will go to the gym if: −c + βδ b ≥ 0 ⇔ βδ b ≥ c A partially naive consumer makes the following forecast as of t = 0: ˆ b ≥ 0 ⇔ βδ ˆ b ≥ c with β ≤ βˆ ≤ 1 −c + βδ He is aware of his quasi-hyperbolic preferences, but underestimates his self-control problem Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Application: Going to the gym (Dellavigna and Malmendier 2004)

Dierence between desired and actual consumption for time-inconsistent consumer: (δ b − c ) − (βδ b − c ) > 0

Dierence between forecasted and actual consumtion: For a totally naive consumer: (δ b − c ) − (βδ b − c )

For a partially naive consumer:

ˆ b − c ) − (βδ b − c ) ≤ (δ b − c ) − (βδ b − c ) (βδ

For a sophisticated consumer: (βδ b − c ) − (βδ b − c ) = 0 rational expectations about future time preferences

Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Habit formation models and reference point models

Habit-formation models

idea that the utility from current consumption can be aected by the level of past consumption (Duesenberry 1952, Pollack, Ryder and Heal) period τ instantaneous utility function: u (cτ , cτ −1 , cτ −2 , ...) 2 where δcδτ δucτ 0 > 0 for τ 0 < τ

Reference-point models

u (cτ , rτ ) = v (cτ − rτ )

where the reference point rτ might depend on past consumption, expectations, social comparison, status quo...

Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Models incorporating utility from anticipation

people derive utility not only from current consumption but also from anticipating future consumption period τ instantaneous utility function: u (cτ ; cτ +1 , cτ +2 , ...) where δδcuτ 0 > 0 for τ 0 > τ If dreading future bad outcomes is a stronger emotion than savoring future good outcomes, utility fom anticipation would generate a sign eect Caplin and Leahy (2001) show that incorporating anxiety into asset-pricing models may help explain the equity premium puzzle (anxiety creates a taste for risk-free assets and an aversion to risky assets) Marie-Pierre Dargnies

Time preferences

Introduction The DU model Models of hyperbolic discounting Other alternative to the DU model

Multiple-self models

Models that view intertemporal choices as the outcome of a conict between multiple selves Thaler and Shefrin (1981) proposed a planner-doer model drawing upon principal-agent theory A series of myopic doers (caring only about their own immediate gratication) interact with a single planner who cares equally about the present and the future

Marie-Pierre Dargnies

Time preferences