Discuss how utility functions can be determined in practice.

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Question Two

 

A US local authority owns a tramway system; and the tram operators are under pressure to increase passenger numbers. They have to make a decision on whether to lower fares in an attempt to increase passenger numbers.

 

If they decide to reduce fares they will then have to decide whether to launch a radio  advertising campaign to increase awareness of the fare reduction.

 

If fares remain the same then it is estimated that there is a 0.7 probability that the mean number of passengers carried per day over the next year will equal 20 000.  In addition, a 0.3 probability that the number will fall to 15 000.

 

The annual profits associated with these passenger numbers are estimated to be $3million and $1million, respectively.

 

If the fares are reduced, but radio advertising is not used, then it is thought that there is a 0.6 probability that the mean number of passengers carried will increase to 25 000. In addition, a 0.4 probability that the number will increase to 22 000. The resulting profits generated by these passenger numbers are estimated to be $2million and $1.7 million, respectively.

 

Radio advertising of the fare reduction would further the probability of an increase to a mean of 25 000 passengers to 0.8; and reduce the probability that the mean will be 22 000 to 0.2. However, it would reduce the profits associated with these mean passenger numbers by $0.6 million. The tram operating company’s objectives are to [A] maximise passenger numbers and [B] maximise profit.  Note Objective [A] is present as the local authority want to introduce social & environmental benefits {e.g. more people walking, improved air quality, reduced traffic congestion}.

 

(a)   
Utility functions for the mean numbers of passengers carried and the profit have been obtained from the trams operator’s Chief Executive Officer (CEO), as below.

Mean number of passengers

Utility

15 000

0.00

16 200

0.20

17 500

0.40

19 000

0.60

20 000

0.80

22 000

0.95

25 000

1.00

       

Profit ($ million)

Utility

1.0

0.00

1.1

0.20

1.3

0.55

1.4

0.60

1.7

0.75

2.0

0.90

2.5

0.95

3.0

1.00

 

 

 

 

Discuss how utility functions can be determined in practice.

Plot the above utility functions – and provide an interpretation of the plots.    (12 marks)

 

(b)   
The elicitation session revealed that, for the CEO, mean number of passengers and profit are mutually utility independent. You are reminded that, in this case, a two-attribute utility function can be obtained from:

 

u(x1, x2) = k1u(x1) + k2u(x2) + k3u(x1)u(x2)      

 

Where k3 = 1 – k1 – k2

 

The elicitation session also revealed that k1 =0.9 and k2 = 0.6, where the attribute number 1 is the mean number of passengers.

 

Given the above utilities, [i] determine the policy that the tramway should undertake; and [ii] comment on your answer.

(30 marks)

 

(c)   
Provide details (including making reference to the construction industry) of two of the following:

·        
Risk retention                                                                       

·        
Risk avoidance

·        
Risk transfer

·        
Risk reduction

 

(8 marks)

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