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Alpha is deciding whether to invest $1 million in a project. There is a 70% chance that the project will be successful, yielding a return of 20% on investment. However, there is a 30% chance that the project will fail, in which case Alpha will only recover 80% of his investment.

1. What is the expected value of investing in the project?

2. Suppose Alpha evaluates the project in accordance with prospect theory. Specifi- cally,

v(x) = (x r)^0.8 .if x r

(r x)^0.8 ,if r > x,

where > 1, r = $1 million is the reference point, and x is the amount of money received by Alpha at the end of the project. The corresponding probability weighting functions are such that = = 0.6.

(a) Argue (mathematically) that Alpha is loss averse.

(b) What is Alphas value of investing in the project if = 2?

(c) Find the value of such that Alpha is indifferent between investing and not investing in the project.

Betas boss assigns him a task on Monday which must be completed before Wednesday. The task takes a total of 10 hours. If Beta works on the task for Xt hours on day t, then he suffers a disutility of Xt^2 on day t. Throughout the problem t {1, 2, 3}, where 1 stands for Monday, 2 for Tuesday, and 3 for Wednesday. Betas time preferences are given by exponential discounting with the discount factor of = 0.8 per day.

1. What is the present value (as measured on Monday) of Betas disutility if he works for 6 hours on Monday and 4 hours on Tuesday?

2. Betas problem is to complete the task before Wednesday in such a way to minimize the present value of his distuility (as measured on Monday). Write down the mathematical version of Betas problem (that is, minimize some function subject to some constraint)

3. How many hours does Beta choose to work the task on Monday?

4. Now suppose that the boss wants to assign a new task to Beta on Tuesday and would therefore like Beta to have more time on Tuesday. She incentivizes Beta by offering him a reward of r3 = 10x1 on Wednesday if Beta works on Monday for x1 hours on the first task. How many hours does Beta choose to work on Monday?

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