Utility Theory Homework

 

1.  A bettor with utility function u(x) = ln(x), where x is total wealth, has a choice between the following two alternatives:

 

            A         Win $10,000 with probability 0.2

                        Win $1000 with probability 0.8

 

            B         Win $3000 with probability 0.9

                        Lose $2000 with probability 0.1

 

     a. If the bettor currently has $2500, should he choose A or B?

     b. Repeat a, assuming the bettor has $5000.

     c. Repeat a, assuming the bettor has $10,000.

 

2.  Assess your own utility function in two different ways.

     a.  Use the certainty-equivalent approach to assess your utility function for wealth over a range of $100 to $20,000.

     b.  Us the probability-equivalent approach to assess u($1500), u($5600), u($9050), and u(13,700).  Are these assessments consistent with assessment made in part a?  Plot them on the same graph and compare.

 

3.  Utility functions need not relate to $ values.  Here is a problem with 5 abstract outcomes, labeled A thru E, with A the most preferred and E the least preferred.  The DM has made the following assessments:

            - She is indifferent between having C for sure or a lottery in which she wins A with prob .5, and E with prob .5. 

            - She is indifferent between having B for sure, or a lottery in which she wins A with prob .4, and C with prob .6.

            - She is indifferent between these two lotteries:

 

                        1. A 50% chance at B and 50% chance at D

                        2. A 50% chance at A, and 50% chance at E

What are u(A), u(B), u(C), u(D), and u(E)?

 

4.  An orange grower in Florida faces a dilemma.  The weather forecast is for cold weather, and there is a 50% chance that the temperature tonight will be cold enough to freeze and destroy his entire crop, which is worth about $50,000.  He can take two possible actions to try to alleviate his loss if the temperature drops.  First, he could set burners in the orchard; this would cost $5000, but he could still expect to incur damage of approximately $15,000 to $20,000.  Second, he could set up sprinklers to spray the trees.  If the temperature drops, the water would freeze on the fruit and provide some insulation.  This method is cheaper ($2000), but less effective.  With the sprinklers he could expect to incur as much as $25,000 to $30,000 of the loss with no protective action.

     Compare the grower’s options.  Which alternative would you suggest and why?

 

5. Assume you have a ticket that will let you participate in a game of chance (a lottery) that will pay off $10 with a 45% chance (or a 55% chance of getting nothing).  Your friend has a ticket to a different lottery that has a 20% chance of paying $25 (or an 80% chance of paying nothing).  Your friend has offered to let you have his ticket if you will give him your ticket plus one dollar.  Analyze this decision using a decision tree.