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The images provide a problem involving Jacob and Kane, both present-biased agents, and their decisions regarding career choices in physics or basket weaving, based on costs and benefits over time. I'll break this down to help you solve the problem.

### Jacob's Case:

#### Variables:
- **β = 0.8**: Present bias factor for future benefits.
- **δ = 0.95**: Discount factor (how future utility is discounted over time).
- **u(x) = x**: Utility function (utility is equal to the value of x).

#### Costs and Benefits:
- **Physics**:  
  - **Cost**: 5 units for each of the first three periods \(t=1\), \(t=2\), and \(t=3\).  
  - **Benefit**: 20 units at \(t=4\).
  
- **Basket weaving**:  
  - **Cost**: 0 for all periods.  
  - **Benefit**: 2 units at \(t=4\).

Jacob has a **present bias** (β < 1), meaning he undervalues future rewards and focuses more on immediate costs or benefits. To decide whether Jacob chooses physics or basket weaving, we compute the **present value** of both career paths.

#### Physics Career:
The **total cost** for studying physics involves summing the discounted costs for \(t = 1\), \(t = 2\), and \(t = 3\):
\[
\text{Total Cost of Physics} = 5 \times (1 + \beta \cdot \delta + \beta \cdot \delta^2)
\]
The **benefit** for choosing physics comes at \(t=4\), so its present value is:
\[
\text{Benefit of Physics} = 20 \times \beta \cdot \delta^3
\]
The net present value (NPV) for physics is:
\[
\text{NPV for Physics} = 20 \times \beta \cdot \delta^3 - 5 \times (1 + \beta \cdot \delta + \beta \cdot \delta^2)
\]

#### Basket Weaving Career:
Since basket weaving has no cost, the only value is the benefit at \(t=4\), discounted as follows:
\[
\text{NPV for Basket Weaving} = 2 \times \beta \cdot \delta^3
\]

Jacob will choose the option with the higher NPV.

### Kane's Case:

Kane is **sophisticated**, meaning he anticipates his future self’s present bias and can adjust for it at \(t = 1\), but not after that. His values for β and δ are the same as Jacob’s: **β = 0.5**, **δ = 0.95**.

At \(t = 0\), Kane decides based on the future costs and benefits, knowing that at \(t = 1\), he will still have the opportunity to switch his choice. Thus, his decision at \(t = 0\) will be based on comparing the NPV of each option, considering that he can re-evaluate in \(t = 1\).

Kane’s calculations would look similar to Jacob’s, but with a different β value. He will also anticipate his future self’s bias at \(t = 1\), adjusting his decision accordingly.

### Let me know if you would like to proceed with detailed calculations for each decision (Jacob’s and Kane’s) based on the information provided!

Jacob is a naive present-biased agent with β=0.8, δ=0.95, and u(x)=x. Jacob is in Year 12 (t=0) and deciding which course to study at university. He is deciding between physics, which is hard but will deliver a rewarding career, or basket weaving, which is a fun course but has worse career prospects. Jacob can only choose the career for which he studies. The options deliver the following costs and benefits: - Studying physics: cost of 5 in each of t=1, t=2, and t=3 - Studying basket weaving: no cost - Physics career: benefit of 20 at t=4 - Basket weaving career: benefit of 2 at t=4. (a) Jacob’s friend Kane is a sophisticated present-biased agent with β=0.5, δ=0.95, and u(x)=x. Kane is also considering the costs and benefits of a career as a physicist or a basket weaver. He can change his enrolment at t=1, but not at t=2. What does Kane choose at t=0? Explain.

This problem explores the decision-making process of present-biased agents Jacob and Kane, comparing the costs and benefits of studying physics versus basket weaving. Learn how to calculate the net present value (NPV) for each option using present bias, discount factors, and utility functions.

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