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2 Portfolio Choice Consider an agent with a wealth w who has to decide how to invest it. She has two choices: to keep in in cash, in which case it will neither grow nor shrink, or to invest it in a startup. There is a 5% chance the startup will grow to 50x the initial investment and a 95% chance it will fail, meaning she will lose her entire investment in it. Her goal is to pick a quantity a to invest in the asset. 1. Write an expression for her wealth in the event that the startup succeeds and in the event that it fails. These are the two 'states of the world' we are interested in for this problem. 2. Assume her utility function is given by ln(w) and that she is an expected utility maximizer. Write her optimization problem. 3. Find the first order condition of the optimization problem. 4. Use the first order condition to the optimization problem to find a formula for the optimal startup investment. 5. Now, suppose that instead of being an expected utility maximizer, she

2 Portfolio Choice Consider an agent with a wealth w who has to decide how to invest it. She has two choices: to keep in in cash, in which case it will neither grow nor shrink, or to invest it in a startup. There is a 5% chance the startup will grow to 50x the initial investment and a 95% chance it will fail, meaning she will lose her entire investment in it. Her goal is to pick a quantity a to invest in the asset. 1. Write an expression for her wealth in the event that the startup succeeds and in the event that it fails. These are the two 'states of the world' we are interested in for this problem. 2. Assume her utility function is given by ln(w) and that she is an expected utility maximizer. Write her optimization problem. 3. Find the first order condition of the optimization problem. 4. Use the first order condition to the optimization problem to find a formula for the optimal startup investment. 5. Now, suppose that instead of being an expected utility maximizer, she

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.8: Linear Programming
Problem 23E: Inventories An electronics store manager stocks from 20 to 30 IBM-compatible computers and from 30...
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2 Portfolio Choice Consider an agent with a wealth w who has to decide how to invest it. She has two choices: to keep in in cash, in which case it will neither grow nor shrink, or to invest it in a startup. There is a 5% chance the startup will grow to 50x the initial investment and a 95% chance it will fail, meaning she will lose her entire investment in it. Her goal is to pick a quantity a to invest in the asset. 1. Write an expression for her wealth in the event that the startup succeeds and in the event that it fails. These are the two 'states of the world' we are interested in for this problem. 2. Assume her utility function is given by In(w) and that she is an expected utility maximizer. Write her optimization problem. 3. Find the first order condition of the optimization problem. 4. Use the first order condition to the optimization problem to find a formula for the optimal startup investment. 5. Now, suppose that instead of being an expected utility maximizer, she makes decisions in a way that is consistent with cumulative prospect the- ory. Discuss how reference points (you'll also have to decide what reference point should be), probability weighting, and loss aversion might each affect her decision.
Transcribed Image Text:2 Portfolio Choice Consider an agent with a wealth w who has to decide how to invest it. She has two choices: to keep in in cash, in which case it will neither grow nor shrink, or to invest it in a startup. There is a 5% chance the startup will grow to 50x the initial investment and a 95% chance it will fail, meaning she will lose her entire investment in it. Her goal is to pick a quantity a to invest in the asset. 1. Write an expression for her wealth in the event that the startup succeeds and in the event that it fails. These are the two 'states of the world' we are interested in for this problem. 2. Assume her utility function is given by In(w) and that she is an expected utility maximizer. Write her optimization problem. 3. Find the first order condition of the optimization problem. 4. Use the first order condition to the optimization problem to find a formula for the optimal startup investment. 5. Now, suppose that instead of being an expected utility maximizer, she makes decisions in a way that is consistent with cumulative prospect the- ory. Discuss how reference points (you'll also have to decide what reference point should be), probability weighting, and loss aversion might each affect her decision.
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