Question

4. University of Notre Dame is a premier institution that draws students from all over the world to its campus. Although it is privately funded, it aspires to world-class quality and reputation, which are enhanced when out-of-state residents enroll. Data suggest that in-state enrollment can be described by the equation: QI = 25,000 - PI, where QI = in-state enrollment and PI = in-state tuition. Out-of-state enrollment is given by: QN = 13,500 - .5PN.

a. If tuition for in-state students is $14,000 and for out-of-state students is $19,000, what is total enrollment and demand elasticity for each type of student?

b. Suppose that the marginal cost to the university of an additional student is $7,000. Is Major University maximizing profit at its current tuition charges? Explain.  

c. Because of major funding cuts, the university is expecting to reduce its total enrollment to 11,000 students next year. The university is free to set any tuition charges it wishes. If the goal is to maximize total tuition revenue, what should in-state tuition, out-of-state tuition, and respective enrollments be?

Answer 1

Q.4.

The in-state enrolment has been described by the following equation:

Q1 = 25000 – P1 ………. (i)

Where Q1= in-state enrolment, P1= in-state tuition

The out-of-state enrolment is given by the following equation:

QN = 13500 - .5PN………. (ii)

1. In-state Students Tuition = P1 = $14000 and Out-of state students Tuitions= PN = $19000

From equation (i), we get the value of Q1 as

Q1 = 25000 – 14000 = 11000

Similarly, from the equation (ii), we get the value of QN as

QN= 13500 – 0.5*19000 = 4000

The total enrolment = Q1+QN = `11000+4000 = 15000 …. (Ans.)

Demand elasticity for the in-state students = (∂Q/∂P)*P/Q= -1*(14000/11000)= -1.27….. (Ans.)

Demand elasticity for the in-state students = (∂Q/∂P)*P/Q= -0.5*(19000/4000)= -2.375 ……. (Ans.)

2. Marginal Cost of an additional Student = $7000 = MC

The University is said to be maximising its profit when the marginal revenue is greater than or equal to the marginal cost i.e.

MR≥MC

In a perfectly competitive market,

p=MR=AR

Where p is the price

For the in-state students, the MR is equal to $14000 which is greater than the marginal cost of $7000, thus it can be stated that the University is profit maximising at the current in-state tuition charges.

Similarly, for the out-of-state students, the MR is equal to $19000 which is greater than the marginal cost of $7000, thus it can be stated that the University is profit maximising at the current in-state tuition charges.

3. The new total number of enrolment is equal to 11000 in the next year.

The total revenue for the in-state students can be calculated as follows:

Putting the value of Q1 in the formula of revenue, we get for the revenue from the in-state students enrolment as follows:

P1*Q1 = P1* (25000- P1) = 25000*P1 - P1²

The first order differentiation of the above equation and equating it with zero is gives the optimal value as follows:

25000 – 2*P1= 0

P1 = 12500 which is the in-state tuition when revenue is maximised.

From the equation (i), we get

Q1 = 12500 which is the enrolment of the in-state students when revenue is maximised. (Ans.)

Similarly, putting the value of QN in the formula of revenue, we get for the revenue from the out-of-state students enrolment as follows:

PN*QN = PN* (13500- 0.5*PN) = 13500*PN - 0.5PN²

The first order differentiation of the above equation and equating it with zero is gives the optimal value as follows:

13500 – PN= 0

PN = 13500 which is the out-of-state tuition when revenue is maximised.

From the equation (i), we get

QN = 6750 which is the enrolment of the out-of-state students when revenue is maximised. (Ans.)