Instructions

This file contains detailed answers to the practice questions on time consistency, present bias, and hyperbolic discounting.

Question 1: Time-Consistent Decision-Maker (20 Marks)

Problem Recap:

  • Periods: \(.*?\)
  • Rewards: \(r = (2, 6, 10)\)
  • Costs: \(c = (0, 0, 0)\)
  • Discount factors: \(\beta = 1\) (time-consistent) and \(\delta = 1\).

Answer:

  • A time-consistent decision-maker evaluates future rewards without any present bias (\(\beta = 1\)) and discounts only based on \(\delta\), which here is 1.
  • They will compare the rewards in each period: $ U_1(1) = 2,  U_1(2) = 6,  U_1(3) = 10 $
  • Since there is no cost and rewards are increasing, the DM will choose Period 3 to maximize utility.

Final Answer: The time-consistent DM will complete the task in Period 3.

Question 2: Naifs (20 Marks)

Problem Recap:

  • Periods: \(T = 4\)
  • Rewards: \(r = (4, 6, 7, 12)\)
  • Costs: \(c = (1, 2, 3, 4)\)
  • Discount factors: \(\beta = 0.5\), \(\delta = 1\).

Answer:

  1. Naif’s Plan at Period 1:
    • A naive decision-maker evaluates future rewards using the present bias \(\beta\) but assumes they will follow their current plan. They calculate: \[ U_1(1) = r_1 - c_1 = 4 - 1 = 3, \] \[ U_1(2) = r_2 - c_2 = 0.5 - 0.5 = 3, \] \[ U_1(3) = r_3 - c_3 = 0.5 - 0.5 = 2, \] \[ U_1(4) = r_4 - c_4 = 0.5 - 0.5 = 4. \]
    • The naive DM initially plans to wait until Period 4 to maximize utility.
  2. Naif’s Actual Actions:
    • In Period 2, they re-evaluate and calculate: \[ U_2(2) = r_2 - c_2 = 6 - 2 = 4, \] \[ U_2(3) = r_3 - c_3 = 0.5 - 0.5 = 2, \] \[ U_2(4) = r_4 - c_4 = 0.5 - 0.5 = 4. \]
      • They still plan to wait until Period 4.
    • In Period 3, they re-evaluate and calculate: \[ U_3(3) = r_3 - c_3 = 7 - 3 = 4, \] \[ U_3(4) = r_4 - c_4 = 0.5 - 0.5 = 4. \]
      • Indifference leads them to complete the activity in Period 3.

Final Answer: Naifs plan to wait until Period 4, but they end up completing the activity in Period 3 due to present bias.

Question 3: Sophisticates (20 Marks)

Problem Recap:

  • Periods: \(T = 4\)
  • Rewards: \(r = (3, 7, 9, 15)\)
  • Costs: \(c = (0, 0, 0, 0)\)
  • Discount factors: \(\beta = 0.5\), \(\delta = 1\).

Answer:

  1. Using Backward Induction:
    • In Period 4: \[ U_4(4) = r_4 = 15. \]
    • In Period 3: \[ U_3(3) = r_3 = 9,  U_3(4) = r_4 = 0.5 = 7.5. \]
      • Choose Period 3, as \(9 > 7.5\).
    • In Period 2: \[ U_2(2) = r_2 = 7,  U_2(3) = r_3 = 0.5 = 4.5. \]
      • Choose Period 2, as \(7 > 4.5\).
    • In Period 1: \[ U_1(1) = r_1 = 3,  U_1(2) = r_2 = 0.5 = 3.5. \]
      • Choose Period 1, as \(3.5 > 3\).

Final Answer: Sophisticates will complete the activity in Period 1, anticipating their future selves’ present bias.

Question 4: Mixed Costs and Rewards (20 Marks)

Problem Recap:

  • Periods: \(T = 4\)
  • Rewards: \(r = (8, 12, 14, 18)\)
  • Costs: \(c = (5, 4, 3, 1)\)
  • Discount factors: \(\beta = 0.6\), \(\delta = 1\).

Answer:

  1. Naive DM:
    • Plans to wait until Period 4, but re-evaluates each period.
    • Completes the task in Period 3 when the immediate utility is maximized.
  2. Sophisticated DM:
    • Using backward induction, completes the task in Period 2 to avoid procrastination and maximize utility.

Final Answer: Naive DM completes in Period 3; Sophisticated DM completes in Period 2.

Question 5: Comparison of Naifs and Sophisticates (20 Marks)

Problem Recap:

  • Periods: \(T = 3\)
  • Rewards: \(r = (1, 5, 10)\)
  • Costs: \(c = (6, 4, 1)\)
  • Discount factors: \(\beta = 0.4\), \(\delta = 1\).

Answer:

  1. Naive DM: Procrastinates and cleans the room in Period 3.
  2. Sophisticated DM: Anticipates future procrastination and cleans the room in Period 1.
  3. Time-Consistent DM: Completes the task in Period 3.

Final Answer: - Naive: Period 3 - Sophisticated: Period 1 - Time-Consistent: Period 3.

Bonus: Reflect and Compare

Problem Recap:

Reflect on how the concept of time inconsistency influences decision-making in these scenarios. What are the real-world implications of these models?

Answer:

1. Time Inconsistency: - Time inconsistency occurs because individuals discount future utility disproportionately compared to the present. Naive decision-makers fail to recognize their future preference changes, leading to procrastination and suboptimal decisions. - Sophisticated decision-makers anticipate their time inconsistency and take proactive steps (e.g., acting earlier) to mitigate its effects.

2. Real-World Implications: - Procrastination: Time-inconsistent behavior explains why people delay tasks like filing taxes, exercising, or saving for retirement. - Commitment Devices: Sophisticates often use tools like automatic savings plans, deadlines, or penalties to enforce future actions and overcome procrastination. - Policy Design: Understanding present bias can help policymakers design better interventions, such as default enrollment in retirement plans or incentives for early action.

3. Lessons from Naifs vs. Sophisticates: - Naifs are prone to repeated delays without external help or commitment mechanisms. - Sophisticates demonstrate the value of foresight and planning in counteracting present bias.

Conclusion: By understanding time inconsistency, individuals and policymakers can design strategies to bridge the gap between intentions and actions, improving long-term outcomes.

Total Marks: 100