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Steady State matrix: makrte share for online degree

E-1. The market for on-line degrees and courses is becoming crowded and quality and responsiveness seem to play a role in attracting students to continue in a program. The Academic Dean of Ed Marshal Automotive University (EMAU) is interested in how EMAU is doing in the on-line area. He commissioned a study by a faculty-staff committee to try and compare how EMAU will fair in the next five years.

The study group compared EMAU with other schools with very similar programs - West State U., River Valley U., and State Tech - and added a category that includes all other competition and those who leave on-line education. The study group estimated that new student influx approximated student egress, whether by finishing or dropping on-line education.

Presently EMAU has about 30% of the market with each of the other named schools having approximately 20%, and the "other" category having about 10%. The study group found that EMAU retained only 45% of their students, and from what they could gather, they estimated that of those leaving about 5% went to West State U., 15% went to River Valley, and 15% to State Tech. The other 20% of those leaving either left on-line education or went elsewhere.

From various sources the study group found that retention rate for West State U. was 55%, for River Valley U. and State Tech, 75% each and the "other" category 80%.

The transfer rates for those leaving West State were 5% to EMAU, 10% each to River Valley and State Tech, and 20% to "other." For those leaving River Valley the transfer rates were approximately 0% to EMAU 5% to West State and 10% each to State Tech and "other." Of those not re-enrolling at State Tech the estimated numbers were 5% each for EMAU, West State and River Valley and 10% to "other."

Assuming these numbers to be representative, and that transfers /re-enrollment occur twice yearly, what will be the market share distribution at the end of the next cycle?
What will be the market distribution at the end of the next year?
What will be the market distribution at the end of the 5-year planning period of interest to the Dean?
If changes are not made that effect a change in the enrollment rate, the assumptions for a steady state population (exits=entrants) hold true, and there are no other significant schools entering the competition that changes the numbers for "other" category, what will be the distribution when the competition for students "steady out" (reaches steady state)?

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Please see the attached file for explanation.

The initial state is Q0
Market Share
EMAU 30
WSU 20
RVU 20
STU 20
Others 10

The Transformation Matrix is T
EMAU WSU RVU STU Others
EMAU 0.45 0.05 0 0.05 0.05
WSU 0.05 0.55 0.05 0.05 0.05
RVU 0.15 0.1 0.75 0.05 0.05
STU 0.15 0.1 0.1 0.75 0.05
Others 0.2 0.2 0.1 0.1 0.8

No re-enrollment / transfer rate from others is given. We assume that these students will join the other universities with same probability I.e. 5%

Then in the next cycle the market share distribution will be Q1=T*Q0

Multiplying the matrix we get
EMAU 16
WSU 15
RVU 23
STU 24
Others 22

At the end of the next year the market share will be Q2=T*Q1=T*T*Q0
Multiplying the matrix we get
EMAU 10.25
WSU 12.5
RVU 23.45
STU 25.3
Others 28.5

At the end of five years we will have ...

Solution Summary

This problems shows how to calculate the steady state position for a system which has a constant inflows and outflows.

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