Dynamic Strategic Planning - Problem Sets

4. Utility Analysis

 

 

Problem 4.1

Joanna is choosing a job. She has plenty of good offers in her chosen field, with salaries ranging from $50K to $90K per year. Her choice is not obvious, however, because the jobs with the highest salaries tend to offer least or even no vacation days.

Jo's utility functions for salary and vacation are:

Salary ($K/year)

Us

Vacation (weeks)

Uv

50

5

0

10

60

6

1

20

70

8

2

25

80

9

3

30

90

10

4

40

   

5

60

She now feels that the fast paced job in New York ($90K, 0 weeks) is as good as a partnership in her friend's start-up company in which she will either work herself very hard for virtually nothing ($50K, 0 weeks) or, with a 30% chance, make out very well ($90K, 5 weeks). On the other hand, that start-up job would have to have a 40% chance of success to bet out her offer to work in Hawaii ($50K, 5 weeks).

  1. Identify her utility for each of the jobs described above.
  2. Use interpolation to advise Joanna whether she should prefer the Boston office ($70K, 3 weeks) or the Denver job ($60K, 4 weeks).
  3. Using the formula, verify the utilities for all six of the possible outcomes:

U($,W)=Kk$kWU$($)UW(W) + k$U$($) + kWUW(W)

 

Problem 4.2

Alex is willing to pay $100 for a lottery ticket that has a (50:50) chance to win $300. On the other hand, he is prepared to sell his share in a start-up which has a 1/3 chance of being worth $1000 or else nothing) for $200. As regards his automobile insurance, he is glad to pay $100/year to avoid a 40% chance of an accident that year which could cost $300.

  1. Write the indifference statements expressed above.
  2. Plot Alex's utility function.
  3. To what extent, if any, is Alex risk averse?

 

Problem 4.3

(continuation of 2.3)

PDQ is testing a new workstation. If it does meet specifications it has a 60% chance of failing the test. If it does not meet specifications its P (failure) = 10%. Overall, 8 out of 10 workstations meet specifications.

  1. Find P (Meet Spec/Fail Test) using Bayes’ Theorem.
  2. Find P (Meet Spec) if the workstation fails the test the first time, but passes the second and third.
  3. Does PDQ really care about speed or costs?
  4. With changing market conditions for workstations, PDQ may get a sweet deal on the Mars. It is, however, indifferent between buying them at the special price of 40 or waiting. They estimate there is a 40% chance that Mars will soon sell for 30; otherwise they are sure to go back to 60.

    Also, PDQ is indifferent between trading in its unreliable current system, with a (50:50) chance of operating at 6 or 20 MHz, or going for a new system, guaranteed to work at 12 MHz.

  5. Sketch PDQ’s utility functions for speed and cost. Discuss their degree of "risk aversion".
  6. PDQ’s management, when faced with a hypothetical choice between the really fast but expensive Moon system (20 MHz; 60) and a gamble between a P chance at their dream machine (290 MHz; 30) and their current horror (6 MHz; 60), would prefer the gamble as long as P were above 0.6. On the other hand, if the choice were between the cheap (6 MHZ;30) and the gamble, they would take the risk if P ³ K.

  7. Calculate their utility for the Mars machine at the current price (12MHZ; 40).
  8. They see their realistic decision as between the current price Mars, or a 40% chance of a rock bottom price on it (12MHZ; 60). Which is their better choice?
  9. Management wonders whether they should consider hiring an expert who could tell them whether Mars will really drop its price to 30. What is the most utility you would advise them to give up to obtain this service?
  10. What are the implications for all the above if K = 0.4?

 

Problem 4.4

John would be willing to sell the lottery ($500,0.5;-$100) for no less than $100. He would pay up to $50 to play the lottery ($150,0.6;-$50). He would also exchange the lottery ($500, 0.6;$100) for no less than $300.

  1. Write the indifference statements expressed above.
  2. Calculate and draw John’s utility function.
  3. Is John risk averse? why or why not?

 

Problem 4.5

Peter frequently flies between Boston and New York for business. From his experience, the total travel time between the Boston office and the New York headquarters can be quite uncertain. The trip can take as little as 2 hours, or up to 6 hours (depending on weather conditions, airport and city congestion, which NY airport he uses, etc.). The total cost is also variable: it can range from $60 in the best case to $10.

Peter’s utility functions for travel time and cost are given below:

T (hours)

ut(T)

C($)

uc(C)

2

1

60

1

3

0.8

75

0.9

4

0.5

100

0.15

6

0

110

0

Peter feels that a trip takes 2 hours and costs $110 is just as good as an uncertain trip which is equally likely to either take 2 hours and cost only $60, or to end up taking 6 hours and costing $110. He is also indifferent between a sure (6 hours, $60) trip and getting the best case (2 hours, $60) with probability 0.3 or the worst case with 0.7 probability.

  1. What assumptions do you need to make about Peter’s preferences for travel time and cost in order to infer U(T,C) from ut(T) and uc(C)? If these assumptions hold, write a general formula for U(T,C).
  2. Calculate all the coefficients of the formula from a) and write U(T,C) as a function of ut(T) and uc(C).
  3. Help Peter decide between a (4 hours, $60) flight and a (3 hours, $90) flight.

 

Problem 4.6

Given:

U(X) = X (X + 6) / 160, with 0 < X <10

U(Y) = (Y0.5 - 5) / 5, with 25 < Y <100

And the following statements:

(10,25) ~ [(10,100), 0.6; (0,25)]

(0,100) ~ [(10,100), 0.6; (0.25)]

  1. Give the value of U(X,Y) for all four corner points for which the multi-attribute utility is defined.
  2. Find by interpolation U(4,25) and U(4,100).
  3. Write the general formula for U(X,Y).
  4. Solve for the value of K in the formula.
  5. Calculate U(4,100) by using the formula.

 

Problem 4.7

Frances is trying to decide between jobs. Assume that, to a first approximation, her two dominant criteria are salary and average hours of work per week during the year (S, H). She believes she should get at least $60K a year, to justify her graduate education. She also feels that she should not work more than 80 h/week, so she can "have a life".

Her two one-dimensional utility functions are:

Salary ($, K)

U (S)

Hours/wk

U (H)

60

0.0

40

1.0

70

0.1

50

0.9

80

0.3

60

0.7

90

0.5

70

0.3

100

0.9

80

0.0

110

1.0

   

In talking with her you find she is indifferent between a job with an international agency in Geneva (S,H) = (60,40), and a job as a special assistant to a CEO that has a (50:50) chance of being the dream job (S,H) = (110,40) or a real grind (60,80). On the other hand, she is indifferent between a 40% chance that that special assistant job will turn out to be the dream job, and taking a position as a consultant associate (110,80).

  1. By formula, determine if Frances would prefer (80,50) to (90,60).
  2. Make the same assessment by graphical interpolation, showing all work.
  3. Sketch her isovalue lines in the (S,V) plane.