91Ó°ÊÓ

Siblings Sacha and Simon love basketball - and there's nothing better for a basketball fan than March Madness! If they had the chance to attend an NCAA tournament game today, each would value an Elite Eight game at \(\$ 1,000,\) a Final Four game at \(\$ 1,600,\) and a Championship game at \(\$ 1,800 .\) A benevolent aunt offers Sacha and Simon one ticket each to a game of their choosing: Elite Eight, Final Four, or Championship. But there's the hitch; the benevolent aunt is slightly controlling! The siblings can attend an Elite Eight game this year, but if they choose a Final Four game, they'll have to wait one year, and if they want to see the Championship game, they'll have to wait for \(t\) wo years. a. Suppose Sacha discounts each year of waiting at \(10 \%\). Which option will she choose today? b. Suppose Simon discounts hyperbolically so that something a year away is always discounted at \(30 \%,\) while horizons beyond the first year are discounted at \(10 \%\). Which option will Simon choose today? c. Next year, Simon and Sacha's aunt tells them, "You've waited so patiently. If you'd like to see the Final Four game today, you can. Or, you can wait and see next year's Championship game." What will Simon and Sacha do? d. Explain why Sacha's preferences are time-consistent, but Simon's aren't.

Step by step solution

01

Understanding the Problem

We need to evaluate which basketball game option the siblings Sacha and Simon will choose based on the value of the game and their discounting methods. There are three games to consider: Elite Eight (51,000), Final Four (51,600) after one year, and Championship (51,800) after two years.

02

Calculating Sacha's Options

Sacha uses an annual discount rate of 10%. We calculate the present value of each game option using this rate: - **Elite Eight**: No wait, value is 51,000. - **Final Four**: Wait one year, value is calculated as: \[ PV = \frac{1600}{(1 + 0.10)^1} = 1454.55 \]- **Championship**: Wait two years, value is calculated as: \[ PV = \frac{1800}{(1 + 0.10)^2} = 1487.60 \]Sacha will choose the option with the highest present value.

03

Calculating Simon's Options

Simon discounts hyperbolically: 30% for the first year and 10% for the second year. Thus, we calculate:- **Elite Eight**: No wait, value is 51,000. - **Final Four**: Wait one year, value is calculated as: \[ PV = \frac{1600}{(1 + 0.30)^1} = 1230.77 \]- **Championship**: Wait two years, value is calculated using cumulative discounting: \[ PV = \frac{1800}{(1 + 0.30) imes (1 + 0.10)} = 1254.54 \]Simon will choose the option with the highest present value.

04

Next Year Decision for Sacha and Simon

A year passes, and now Sacha can attend the Final Four immediately or wait another year for the Championship. The present value for attending the Final Four immediately is 51,600. Waiting another year for the Championship gives a present value of: - **Sacha**: \[ PV = \frac{1800}{(1 + 0.10)^1} = 1636.36 \] so she will wait. Now, for Simon, the decision is between attending the Final Four immediately for 51,600 or waiting another year for the Championship. \[ PV = \frac{1800}{(1 + 0.10)^1} = 1636.36 \]Simon will also wait.

05

Consistency of Preferences

Sacha's preferences are consistent over time because her discounting is a constant rate of 10% each year, making her decision-making predictable and consistent. Simon has time-inconsistent preferences due to hyperbolic discounting, meaning he values immediate gratification more, leading to changes in preference based on when the options are offered.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discounting in Economics

Discounting in economics is a method used to compare future values with present values. This process helps individuals and businesses decide between options that have different time horizons. Essentially, it's about determining how much a future amount of money is worth right now, given a certain interest or discount rate.

When we talk about discounting, we consider the "time value of money." This idea suggests that money available today is more valuable than the same amount in the future due to its potential earning capacity. By discounting future amounts, we can evaluate various options based on their present value. This allows decision-makers to choose the most economical and beneficial option today, in spite of the time differences in receiving those amounts.

• Helps compare present and future values
• Based on the time value of money
• Aids in decision-making across different time periods

Present Value

Present value (PV) is a key concept when discussing discounting. It's simply the current worth of a future sum of money, given a specific rate of return or discount rate.

To calculate present value, we use the formula:\[ PV = \frac{FV}{(1 + r)^n} \]where \( FV \) is the future value, \( r \) is the discount rate, and \( n \) is the number of periods until the payment is received.

In our example, Sacha uses a 10% discount rate to calculate the present value of the Final Four and Championship games, deciding which option is more valuable to attend today. Similarly, Simon uses his own unique hyperbolic discounting model to find present values.

• Determines current value of future money
• Utilizes a specific discount rate for calculations
• Helps in making today's financial decisions

Hyperbolic Discounting

Hyperbolic discounting is a behavioral economic theory explaining how people underweight long-term rewards. Instead of using a steady rate over time like "exponential" or "constant" discounting, hyperbolic discounting posits a higher rate in the short term that decreases over longer periods.

In the case of Simon, he discounts future rewards at a differing rate compared to Sacha. He applies a 30% discount for rewards one year away, but this considerably drops to 10% for periods extending beyond that. This approach reflects a preference for immediate gratification, thereby exhibiting a steeper initial discounting rate.

Hyperbolic discounting has implications in understanding why individuals often struggle with self-control and why they might choose smaller-sooner over larger-later rewards.

• Highlights preference for immediate benefits
• Uses varying discount rates over time
• Explains time-inconsistent decision-making

Time-Consistency

Time-consistency refers to the extent to which a person's preferences remain the same over time. If someone has time-consistent preferences, their choices today align with future intentions. With time-inconsistent preferences, individuals' future choices can differ from current plans.

In our example, Sacha is a model of time-consistency. She applies a uniform 10% discount rate for each future year. Her calculations and decisions remain consistent over time since her method doesn't change.

Simon's approach is different due to hyperbolic discounting. His preferences shift over time because he places more weight on short-term gratification initially, altering his decisions when faced with future choices. This difference is noteworthy when considering long-term planning and commitment.

• Ensures decision-making stability over time
• Affects future planning and consistency
• Contrasts with time-inconsistent behaviors