Question

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. 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 a 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
Qd= 160-Pd
Qn= 80-Pn

what prices should be charged to induce efficient ridership?
a) Pd=45 , Pn=55
b) Pd=40 , Pn=60
c) Pd=70 , Pn=80
d) Pd=60 , Pn=40

Answer 1

option is B

you are asked to improve the operating efficiency of transportation, bus runs day and night and their cost are given we will first calculate over surplus from bus operating in day and in night as well.

this question has problem of profit maximization

d =total number of buses used during day-time

n=ttotal number of buses used during night-time:

Total surplus generated during day time by using d buses + Total surplus generated during night time by using n buses- Cost incurred in operation
= (1/2)d^2 + d(160-d) + (1/2)n^2 + n(80-n) -(30d + 60n + 10d)

= d+160d-d²+ n +80n-n² - 30d- 60n-10d

= d+160d-30d-10d -d²+n +80n-60n-n²

=d+120d-d²+n+20n-n²

and you will get d = 120 and n = 20.

we have found out the no of bus required in night and day by using this equation. and following we have found out their price for day and night.

Qd= 160-Pd or  Pd=-160-QD
Qn= 80-Pn or Pn= 80-Qn

Pd= 160-120=40

Pn= 80-20=60
Hence Pd = 40 , Pn = 60