Question

Alvin and Bobbi each have wealth equal to $100, and each is presented with the same investment opportunity. If the investment is successful then the individual will have wealth equal to $1000. If the investment is a failure then the individual will have wealth equal to $0. The probability of success is 0.10.

Alvin and Bobbi must choose independently whether to make the investment or not. Use expected utility theory to answer the following questions. In this problem all utility functions are defined in terms of wealth, W.

(a) Alvin's preferences are described by the utility function  U = W.

Assuming Alvin acts to maximize utility, will Alvin choose to make this investment? (yes/no/indifferent)

(b) Bobbi's preferences are described by the utility function  U ( W ) = √ W.

Assuming Bobbi acts to maximize utility, will Bobbi choose to make this investment? (yes/no/indifferent)

Now suppose the investment opportunity changes such that the new probability of success is 0.5. All other details remain the same

(c) Alvin's preferences are described by the utility function  U = W.

Assuming Alvin acts to maximize utility, will Alvin choose to make this investment? (yes/no/indifferent)

(d) Bobbi's preferences are described by the utility function  U ( W ) = √ W.

Assuming Bobbi acts to maximize utility, will Bobbi choose to make this investment? (yes/no/indifferent)

(e) Compare your responses in parts (a) and (c). Does Alvin change his behavior? Why or why not? Explain

(f) Compare your responses in parts (b) and (d). Does Bobbi change her behavior? Why or why not? Explain.

Answer 1

So, if investment is successful, wealth = $1000, if it fails = $0, and if investment is not gone for, then wealth = initial wealth = $100

The decison is taken by Bob and Alvin based on their expected utilities, Expected Utility Theory estimates the likely utility of a decision, and chooses that action which maximizes his/her utility. The expected utility is computed by multiplying the utilities by their respective probabilities. Probability of success of investment = 0.1, probability of failure = 0.9

a) Alvin's : U(W) = W, E(U(W)) = p * U(W)

Expected utility of choosing investment = 0.1 * 1000 + 0.9 * 0 = 100

Expected utility of not going for investment = 1* 100 = 100 , with certainty

So, Alvin will be indifferent between the choices of going for investment and not going for investment.

b) Bobbi's: U (W) = ,

Expected utility of choosing investment = 0.1 * + 0.9 * 0

= 3.16

Expected utility of not choosing investment = 1 * = 10 , with certainty

So, expected utility for not choosing investment = 10 > 3.16 = expected utility of choosing investment. So, Bobbi will choose not to invest.

Now, the probability of success increases to 0.5 from 0.1. Now:

c) Alvin's : U(W) = W,

Expected utility of choosing investment = 0.5 * 1000 + 0.5 * 0 = 500

Expected utility of not going for investment = 1* 100 = 100

So, expected utility for not choosing investment = 100 < 500 = expected utility of choosing investment. So, Alvin will choose to go for investment.

d) Bobbi's: U (W) = ,

Expected utility of choosing investment = 0.5 * + 0.9 * 0

= 15.8

Expected utility of not choosing investment = 1 * = 10 ,

So, expected utility for not choosing investment = 10 < 15.8 = expected utility of choosing investment. So, Bobbi will choose to invest.

e) Now, when we compare the answers found in parts a) and c), we see that the decisions of Alvin change. Earlier Alvin was indifferent between both the options, he may or may not have gone for the investment, but later, he surely chooses to go for investment. This is because the probability of the success of the investment increases. Earlier, there was only 10% of the investment being successful, but now there is a 50% chance of it being successful. This is why the expected utility from the investment has increased for Alvin.

f) Similarly, earlier Bobbi was sure to not go for the investment, but now she is sure to choose the investment option. This is because the probability of the success of the investment increases. Earlier, there was only 10% of the investment being successful, but now there is a 50% chance of it being successful. This is why the expected utility from the investment has increased for Bobbi.