https://file.aiquickdraw.com/mathbot/file/1726379107644rxd49wf2.jpg

### Explanation of the Plot:

The plot shows how the **discounted utility** of the physics career's benefit at \(t = 4\) changes from \(t = 0\) to \(t = 4\) for Jacob, the present-biased agent. 

The key features of the plot are:

- **At \(t = 0\)**: The discounted utility is at its highest, 16. This is because Jacob applies both the \(\beta\)-discount (present bias) and \(\delta\)-discount (future discounting) to the career benefit.
- **At \(t = 4\)**: The discounted utility is 13.03. This is the utility Jacob perceives at the actual time the benefit is received, without further future discounting.
- **Over time**: The discounted utility decreases due to the effect of \(\delta\), which accounts for future discounting. However, the \(\beta\) factor causes a sharper drop between \(t = 0\) and \(t = 1\), reflecting Jacob’s present bias.

The numerical values of discounted utility at each time are:
- \(t = 0\): 16.00
- \(t = 1\): 15.20
- \(t = 2\): 14.44
- \(t = 3\): 13.72
- \(t = 4\): 13.03

Would you like further details on this? 

Here are some follow-up questions that can help you understand this better:

1. How would the plot change if \(\beta\) were closer to 1?
2. What happens if Jacob chooses the basket weaving career instead?
3. How does the cost of studying physics affect the overall decision Jacob will make?
4. How would a different discount rate \(\delta\) affect the plot and decisions?
5. What does present bias imply for Jacob’s career decisions over time?

**Tip**: In decision theory, a smaller \(\beta\) means a stronger present bias, leading individuals to undervalue future rewards.

Jacob is a naive present-biased agent with β=0.8, δ=0.95, and u(x)=x. Jacob is in Year 12 (at 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

Analyze the career decision problem of a present-biased agent deciding between physics and basket weaving. Learn how present bias and time discounting affect Jacob's decision-making process, with step-by-step solutions and utility plots for each career option.

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