1. Customers to Live Theaters, Inc. can be divided into two groups: seniors and everyone else. The inverse demand curves for each of the two groups are given below. The marginal cost (which equals the average variable cost) of serving an additional patron, either senior or everyone else, is equal to $4. Fixed costs are equal to $1000.
Ps = 80 - Qs
Pe = 100 - 2Qe
Where Ps and Pe denote, respectively, the prices charged to seniors and everyone else and Qs and Qe denote the number of seniors and the number of all other customers served.
a. What is Live Theaters' total revenue function? What is its total cost function? Its total profits function?
b. What are the profit maximizing levels of price and output if Live Theaters, Inc. engages in third degree price discrimination? Show that MRe = MRs = MC
c. What are profits associated with this option?
d. If Live Theaters charges one price to all patrons, what would it be? How many customers would it serve? What would be its profits?
2. Computer Products Corp. sells peripheral equipment used by both private businesses and the government. Due to a recession, Computer Product's sales have declined by 100,000 units and it now has 200,000 units of excess capacity. All of its current sales are to private sector customers and each pays $12.00 per unit for the equipment. The sales price is equal to 150% of average variable costs. A government agency has offered to purchase 300,000 units at $10.00 each. If Computer Products accepts the offer, it will not be able to fill 100,000 units of its expected orders from private sector customers over the next few months, although the inability to meet customer demand is not expected to affect future sales.
a. Should Computer Products accept the offer to supply 300,000 units at $10 each to the government agency? What happens to its profits if it accepts the offer?
b. Would your answer change if the inability to meet private sector customer demand reduces sales of 50,000 during this (ignore any effects beyond this period)?
3. Online Tutors offers monthly access to students for either or both math or science tutors. A market study found that the served by Online Tutors can be divided into two types of students: students needing help with math, who are primarily interested in math tutoring but may be interested in some science tutoring, and students who are primarily interested in science tutoring but may need some math tutoring. The study further estimates that there are 300 students who fit the category of "students needing help with math" and 200 students who fit the category of "students needing help with science." Online Tutors estimated demand for each type of tutoring service by type of student is given below. All costs are fixed so that maximizing total revenue is equivalent to maximizing profit.
The demand prices for monthly math and science tutoring services are given below
Type of Student...............................Math Tutoring Only........................Science Tutoring Only
In Need of Math Tutoring............................$100.......................................................$25
In Need of Science Tutoring..........................$50.......................................................$150
a. If the Online Tutoring decides to sell its tutoring services individually, what should it charge for math tutoring? What should it charge for science tutoring?
b. If Online Tutoring offers bundled services, what pricing would you recommend? What happens to its revenues if it uses bundling?
1. First, we note that the profit maximizing rule for anyone is marginal revenue = marginal cost.
We are given:
Ps = 80 - Qs
Pe = 100 - 2Qe
Revenue in the senior market is Ps X Qs = 80Qs - Qs^2
Similarly, revenue in the everyone else market is Pe X Qe = 100Qe - 2Qe^2
Total revenue = sum of both markets = 80Qs - Qs^2 + 100Qe - 2Qe^2
Total cost = $4 X the number of units sold + fixed cost = 4(Qe + Qs) + 1000
Total profit = TR - TC = 76Qs - Qs^2 + 96Qe - 2Qe^2 - 1000
To maximize this function, we take derivatives w.r.t both Qs and Qe and set them to zero.
d(TP)/d(Qs) = 76 - 2Qs = 0 => Qs = 38 => Ps = 42
d(TP)/d(Qe) = 96 - 4Qe = 0 => Qe = 24 => Pe = 52
The marginal revenues are
MRs = d(TR)/d(Qs) = 80 - 2Qs = 80 - 2(38) = 4
MRe = d(TR)/d(Qe) = 100 - 4Qe = 100 - 4(24) = 4
Both are equal to the marginal cost.
Substituting Qs = 38 and Qe = 24 into the total profit function gives TP = 2172.
Now, suppose that only one price (say P) is charged, then we have
P = 80 - Qs => Qs = 80 - P
P = 100 - 2Qe => Qe = 50 - P/2
Total revenue = price X total quantity = P(Qe + Qs) = P(80 - P + 50 - P/2) = P(130 - 1.5P) = 130P - 1.5P^2
Total cost = $4 X the number of units ...
Fixed costs and other financial concepts are investigated in this multi-question answer in 831 words with all calculations displayed.