Suppose a new spa is going to open on SFU Burnaby campus. The owner does not kno

Question

Suppose a new spa is going to open on SFU Burnaby campus. The owner does not know

much about economics and therefore hires you as a consultant. Your job is to write a

report that answers the following questions. You should include all your calculations in

the report, in case the spa owner wants to check them. Suppose customers can get any

fraction of a full-massage.

These are the facts that the owner has gathered: Marginal cost of a massage for the spa

is $20. There are two potential groups of customers: students and university employees.

Each student’s inverse demand for massages is given by: Ps = 705Qs. Each employee’s

inverse demand for massages is given by: Pe = 80 3Qe. There are equal numbers of

potential customers in each group.

(a) If the spa serves students only, what is the profit maximizing price per massage? How

many massages does a student get? What is the loss in efficiency due to monopoly

of the spa on campus?

(b) If the spa owner wants to run a membership program for the students only, such that

each student has to pay both a membership fee and a price per massage, what would

the profit maximizing price for a massage be? How much should the membership fee

be? Is the number of massages that a student gets efficient? What is the student’s

utility?

(c) From now on, we will consider serving both students and employees. Suppose the

owner wants to charge a uniform price for students and employees. Find the aggre-
gate demand and marginal revenue functions. What is the profit maximizing uniform

price? How many massages will each group get? What is the owner’s profit?

(d) If the owner wants to price discriminate, but cannot distinguish students from em-
ployees, what kind of pricing policy do you suggest? Do customers get efficient

number of massages? please explain briefly but precisely.

How many types of constraints does the profit maximization problem have? Explain

verbally what purpose each type of constraints serve. Which of the constraints bind

and which don’t?

(e) If the students can be distinguished from employees (possibly by checking IDs), what

price should be charged for students? What price should be charged for employees?

How many massages does each get? What is the owner’s profit?

Explanation / Answer

a) If the spa serves students only, (and it is the only one of its kind in the campus, establishing a monopoly), then it would be maximizing its profit by producing a level of output at which marginal cost equals marginal revenue.

Marginal cost is given as $20, and marginal revenue is the derivative of total revenue. Total revenue is Price*Qunatity so TR = P*Q = 70Q5Q2. MR = dTR/dQ = 70 - 10Q

70 - 10Q = 20

50 = 10Q

Q = 5, Ps = 70 - 5*5 = $45

Hence, a total of five massages will be given to a student and the profit maximizing price will be $45 per massage

In comparison to monopoly, a competitive spa would have equated marginal cost with price.

MC = P

20 = 70 - 5Q

Q = 10, Price = $20 per massage.

So there is an efficiency loss since as a monopoly, the spa is charging a higher price and giving fewer massages per student. The amount of efficiency loss is the area of the deadweight loss which is 1/2*(difference in quantity)*(difference in price) = 1/2*(5)*(25) = $62.5

b) If there is one type of consumer (and all consumers have the same demand curve), then you can capture all the consumer surplus by setting price equal to marginal cost and setting the fixed fee equal to the consumer surplus for an individual consumer.

So If the spa owner wants to run a membership program for the students only, such that each student has to pay both a membership fee and a price per massage, the profit maximizing price for a massage be must contain a fee and price per massage equal to margianl cost which is $20.

Find the consumer surplus at P = MC, at which the optimum number of massages will be 10.

CS = 1/2*10*(70-20) = $250

So the membership fee should be $250 and price per massage should be $20. The number of massages that a student gets is efficient. Student’s utility is zero

c)

The aggregate demand function will be the sum of the two demand function

Pe + Ps = 150 -3Qe - 5Qs

P = 150 - 8Q

Marginal revenue function is dTR/dQ = 150 - 16Q.

Profit maximizing uniform price is determined when MR = MC

150 - 16Q = 20

130 = 16Q

Q = Approximatley 8 (8.125) , Price = $85 per massage

There will be no massages for any group since the price charged is greater than student's and employees' choke price.

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