Metropolitan Bus Service

You are an analyst for a metropolitan transportation authority. You are asked if it would improve efficiency to buy more buses, and if so, how many more should be bought. Currently there are eighty buses in the fleet. The operating cost of a bus is $30 during the day and $60 during the night, when higher wages must be paid to drivers and other workers. The daily capital cost of the bus, whether or not it is used, is $10. The demand for buses aggregated over persons and stops during the 12 hours of day and night, respectively D and N, are:

                        ??_??=160-????

                        ??_??=80-????

What is the efficient number of buses? What prices should be changed to induce efficient ridership? Will all of the buses be used at night?

(Optional) Now, suppose that a profit maximising private company operated the bus service instead. How many buses would be in the fleet? Would the company operate buses during the night? What prices would be charged during the day and the night and how much profit would the company make?

Randomized Response

Randomized response (following Wikipedia) is a research method used in structured survey interviews. It was first proposed by S. L. Warner in 1965 and later modified by B. G. Greenberg in 1969. It allows respondents to maintaining confidentiality while responding to sensitive issues (such as participating in criminal behavior). Chance decides, unknown to the interviewer, whether the question is to be answered truthfully, or "yes", regardless of the truth.

Suppose that the Canada Revenue Agency (CRA) were to conduct a number of community tall hall meetings on the underground economy. Part of the meeting involves using a raffle ticket barrel. The participants watch CRA officials write "I have participated in the Underground Economy" on 20 index cards and "I have not participated in the Underground Economy" on 80 index cards. Each index card is then placed in its own envelope and added to the raffle ticket barrel.

One by one, each participant comes forward and draws an envelope from the barrel. The participant reads the statement on the card (without the CRA official or anyone else seeing what is written on the card) and answers (truthfully) Yes or No. The card is put back in the envelope and returned to the barrel. The CRA official tallies the results.

Based on the final results of one third Yes responses, the CRA official declares that participation in the underground economy is around 78%. Please explain.