Given the demand function Q = 10 – 0.1×P, and the total cost function TC = 40 + 10×Q,

calculate the following:

(a) Average revenue (AR),

(b) Total revenue (TR),

(c) Marginal revenue (MR),

(d) Marginal benefit (MB),

(e) Total benefit (TB),

(f) Average cost (AC),

(g) Marginal cost (MC),

(h) Total profit (TP),

(i) Marginal profit (MP),

(j) Total net benefit (TNB),

(k) Marginal net benefit (MNB),

(l) Maximum total profit (TPmax),

(m)Maximum total revenue (TRmax),

(n) Maximum total net benefit

(TNBmax),

(o) Price elasticity value when Q = 6.

Problem 2.

(from Fundamentals of Transportation Systems Analysis. Volume 1: Basic Concepts, by Marvin L.

Manheim. MIT Press, 1979. Problem 1.5(E), page 52-53.)

The country of Freelandia gained independence a few years ago and is mounting a major effort

to promote new agricultural development in previously underdeveloped regions. A trucking

operator in the town of K has previously been providing only local service. Now that a new

major agricultural development program is under way, this operator is considering providing

farm-to-market service to carry agricultural and other natural products from their origin in

locality M to market at K. The distance is 150 miles (one way), with no intermediate major

settlements. After discussions with the local agents of the producers at M, the trucker

estimates that the demand function for shipments from M to K is

𝑉 = 𝑍 + 𝑎0𝑄 − 𝑎1𝑃

TRAN-603 Introduction to Urban Transportation Planning

Homework Assignment #1

2

where V is volume in tons per week, Q is frequency of shipments (per week), P is price charged

per ton, and a0, a1, and Z are parameters. Based upon an average traveling speed of 30 miles

per hour, plus a loading or unloading time of 3 hours at each end, he estimates that he can

manage at the most one round trip every two days, so Q = 3 per week. He also figures that his

costs are related to the mileage he drives per week; his total cost per week is:

𝐶𝑇 = 𝑏0 + 𝑏1𝑚𝑇

where mT = 300Q is the total round-trip mileage driven and b0 and b1 are parameters. The truck

carries 15 tons. He is considering offering an initial frequency of 1 or 2 trips per week at a rate

of $25.00 or $30.00 per ton. Assume b0 = $270, b1 = $0.50, Z = 25, a0 = 13, a1 = 1.

a. For these four combinations of frequency and price, what would be the tonnage carried,

the gross revenues, the total cost, and the net revenue?

b. Which of the four options would be preferred by the operator if his objective where to

maximize net revenue? To minimize costs? To maximize volume carried? Which option

would be preferred by users (shippers)? Can both interests get their first choice

simultaneously? If not, why not?

c. For the proposed service the predominant movement is from M to K; the amount of

freight to be carried in the reverse direction is negligible. There is a possibility of picking

additional cargo at D to go to M; this would incur a detour of 100 miles additional but

could result in an additional load and source of revenue. Would it be profitable for this

operator to make the detour? Discuss qualitativelyAn urban expressway presently carries a peak traffic volume of 2,200 vehicles per hour over three lanes in the peak direction. A simple, approximate service function for this facility is: 𝑡 = 𝑡0 + 𝑏 𝑞 𝑘𝑞𝑐 = 2 + 3 𝑞 3 × 1200 TRAN-603 Introduction to Urban Transportation Planning Homework Assignment #1 3 where k is the number of lanes, qc = 1,200 vehicles per hour per lane, q is the total one-way flow volume in vehicles per hour (q ≤ 0.95kqc), b = 3, and t0 = 2 minutes per mile (see section 7.6). One bus is considered equivalent to about 1.6 automobiles, and the flow stream in the peak hour includes about 60 buses. The demand function for bus transit usage in the corridor is: 𝑉𝐵 = 𝑉0 − 𝑎𝑡𝐵 where tB is bus travel time in minutes, V0 = 4,200 people per hour, and a = 75. a. What is the present travel time for buses and automobiles over the eight-mile expressway? Show that the ridership on the 60 buses is VB = 1,882 persons per hour. b. It is proposed that one auto lane be replaced by a lane for exclusive use of buses, with semi-permanent barriers such as rubber cones between the bus and auto lanes. If the number of buses in the peak hour remained the same, what would their travel time be? (Assume no en route stops and no congestion for buses at exit ramps.) What would be the equilibrium volume of passengers using buses? If the maximum capacity of the buses were 50 passengers, would they be crowded? c. How would the automobile travel times change if one lane were set aside for the exclusive use of buses? d. What assumption is made in this model about the influence of automobile travel times on bus ridership? If you relaxed this assumption, how might the results change?