Lecture 11 - hyperbolic & quasi-hyperbolic discounting

OBJECTIVES
Understand hyperbolic discounting and quasi-hyperbolic discounting.
Analyze time inconsistency in decision-making, illustrating how preferences can shift over time and affect choices.
Explore commitment strategies to counteract temptation and improve decision-making outcomes.

HYPERBOLIC DISCOUNTING
Hyperbolic discounting reflects a psychological pattern where individuals display decreasing impatience over time. Initially, the value of immediate rewards is vastly greater than that of delayed ones, but as time progresses, the discount factor actually increases, making future rewards appear more appealing in relative terms.

Mathematical formulation:

• Exponential discounting: D(t) = eta^{t}, where $eta$ represents the discount factor over time.

• Hyperbolic discounting: D(t) = \frac{1}{1 + eta t}, illustrating how preferences can change more dramatically in shorter timeframes.

• Intertemporal utility function:
  U(x{0}, x{1}, …, x{T}) = \frac{x{0}}{D(0)} + \frac{x{1}}{D(1)} + … + \frac{x{T}}{D(T)} where $D(t)$ signifies the discount applied at time $t$. This model assists in evaluating the overall utility derived from consumption at various times, factoring in the diminishing values of future rewards.

COMPARISON OF DISCOUNT FACTORS
Graphical Representation:
The annual discount factor changes based on different periods ($t$):

• Exponential discounting (with eta = 0.85) provides a consistent decay rate, indicating a steady preference for future rewards relative to the present.

• Hyperbolic discounting (with eta = 0.25) reveals the tendency for substantial changes in preference over shorter timeframes, demonstrating a pattern of higher impatience initially, where individuals might opt for smaller immediate rewards over larger future ones.

QUASI-HYPERBOLIC DISCOUNTING
This model introduces present-biased preferences which deviate from standard hyperbolic discounting. These preferences often lead to inconsistency in decision-making when individuals exhibit different choices at varying points in time.
Example scenarios:

• Choosing between $100 today versus $110 tomorrow demonstrates a common behavioral conflict between immediate gratification and the potential benefits of waiting for a larger future reward.

• If asked the same question after 30 days have passed, choices may differ, indicating time-inconsistent behavior as individuals reassess their preferences based on their current circumstances rather than sticking to the prior decision framework.
  Time consistency requirement:
  An individual who prefers future rewards now should theoretically maintain the same preference later; the observed evidence of time inconsistency occurs when this principle is violated, leading to potentially poor decision-making strategies.

PRESENT BIASED PREFERENCES
Example:

• Day 1: Choose D(2) = 0.9 today for a reward and D(30) = 0.85 for the future reward.

• Day 30: Reassessment leads to potential shifts in preference (i.e., a newfound preference for smaller rewards sooner) due to changing perceptions of value or urgency over time.

(β, δ)-PREFERENCES
Model of quasi-hyperbolic discounting with two key parameters:

• Present bias $(eta)$ determines the weight given to immediate rewards over future rewards, creating a mechanism by which preferences can increasingly favor immediate satisfaction.
  Values:

• If eta < 1, preferences lean towards favoring immediate rewards, which significantly increases penalties associated with procrastination.

EXAMPLES FROM HOMEWORK
Different scenarios indicate variations in decision-making as influenced by the values of $(eta)$ and $(
ho)$, where individuals may demonstrate a preference for instant gratification over the benefits of delayed gratification.
Outcomes demonstrate significant variability, influenced heavily by the context and timing of the decisions made, showcasing the complexities of human behavior in economic settings.

TIME INCONSISTENCY
Defined as changing preferences over time, impacting effective planning and decision-making processes.
Important implications for welfare and policy include the necessity of understanding how to structure rewards and consequences so they can align with individuals’ long-term goals, mitigating the detrimental effects of procrastination and impulsive choices.

NAÏVE VS SOPHISTICATED PREFERENCES
Naïve individuals: Unaware of their inconsistent preference tendencies; they often make decisions that do not account for future fluctuations in their preferences, leading to suboptimal outcomes.
Sophisticated individuals: Acknowledge their inconsistencies and can strategize accordingly; however, some may still engage in forms of procrastination or overexertion (termed preproperation), leading to their own form of mismanagement.

COMMITMENT AND TEMPTATION
Sophisticated individuals may employ various commitment devices to limit future options and temptations, enhancing their decision-making process. Examples include signing up for automatic savings plans, committing to structured workout programs, or making public declarations regarding personal goals.
The example of Odysseus illustrates this concept vividly: he bound himself to the mast of his ship in order to resist the enticing temptation of the Sirens’ call, thereby ensuring that he remained committed to his journey.

COMMITMENT DEVICES
Examples include:

• Cutting up credit cards to reduce impulsive spending.

• Deleting distracting apps from smartphones or computers to improve focus.

• Engaging in commitments toward academic or financial goals such as setting up automatic transfers to savings accounts or increasingly rigorous study schedules.

SUMMARY
Understanding hyperbolic and quasi-hyperbolic discounting elucidates the intricacies of how preferences evolve over time and how they impact decision-making strategies.
The study of time inconsistency reveals essential insights, emphasizing the need for tailored approaches to economics and personal decision-making strategies, which can assist individuals in navigating their future choices more successfully and effectively.