The University Athletic Facility sells gym memberships to faculty and students.
ID: 1092179 • Letter: T
Question
The University Athletic Facility sells gym memberships to faculty and students. Suppose that faculty demand is qF= 30 - 2pF and student demand is qs = 100 - 10ps. The cost of a membership to the Facility is 5. The facility is able to identify who is a Faculty member and who is a student. What are the prices charged to faculty and students? Calculate each component of total welfare (i.e., faculty consumer surplus, student consumer surplus. Facility profits). Now suppose that the University decides, under pressure from the Economics faculty, to stop the Facility price discriminating. What is the new, single price charged to both faculty and students? What has happened to total output? What has happened to welfare?Explanation / Answer
qs = 100 - 10ps
ps = 10 - qs/10
TRs = ps*qs = 10qs - qs2/10
MRs = 10 - qs/5
qf = 30 - 2pf
pf = 15 - qf/2
TRf = pf*qf = 15qf - qf2/2
MRf = 15 - qf
To maximize profit, the firm will operate such that Marginal Revenue of each segment of customers is equal to the marginal cost.
MRs = MC
10 - qs/5 = 5
qs = 25
ps = 10 - 25/10 = $7.5
MRf = MC
15 - qf = 5
qf = 10
pf = 15 - 10/2 = $10
Student Consumer Surplus = 0.5 * (10 - 7.5) * 25 = $31.25
Faculty Consumer Surplus = 0.5 * (15 - 10) * 10 = $25
Faculty Profit = (10-5)*10 = $50
Student Profit = (7.5 - 5)*25 = $62.5
Total Surplus = 31.25 + 25 + 50 + 62.5 = $168.75
If a single price is charged
qs = 100 - 10p
qf = 30 - 2p
Total q = 130 - 12p
p = 130/12 - q/12
TR = Quantity sold * Price = (130/12 - q/12)*q
MR = 130/12 - q/6
MC = 5
130/12 - q/6 = 5
q = (80/12)*6 = 40
p = 130/12 - 40/12 = 90/12 = $7.5
Total output has increased from 35 to 40.
qs = 100 - 10(7.5) = 25
qf = 30 - 2(7.5) = 15
New Faculty Surplus = 0.5*(15-7.5)*15 = $56.25
New Faculty Profit = (7.5 - 5)*15 = $37.5
New Surplus = 31.25 + 56.25 + 62.5 + 37.5 = $187.5
Welfare has increased as Total Surplus has increased from $168.75 to $187.5
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