Mgt 87540 - Quantitative Research Methods

Problems

California Southern University

MGT 87540: Quantitative Research Methods

November 12, 2017

Problem 14-15 Andy Lange, president of the University of South Wisconsin, is concerned with the declining business at the university book store.  (see problem 14-14 for details). The students tell him that the prices are simply too high. Andy, however, has decided not to lower the prices.  If the same conditions exist, what long-run market shares can Andy expect for the four bookstores?

A state of the system is a vector of four elements. Each is the market share of each bookstore.

The vector is:

[pic 1]

Let the ith state of the system be denoted as si. then next state is si+1

The transition from si to si+1 is given by:

        [pic 2]

Where T is the transition matrix. We start with an even market share for each of the bookstores.

therefore, the initial vector is:

[pic 3]

Thus, after the next period (the next state) the market share will be:

[pic 4]

This means that after this period the university bookstore fell to 0.1875

In Markov processes, the probability state after infinite number of periods is:

        [pic 5]

Since all the entries of the matrix are smaller than 1, T converges to a matrix that is composed by the eigenvector with the corresponding eigenvalue of 1. To find it we need to find the steady state vector of the system:

The steady state probability vector (the state to which the system converges) is given by:

        [pic 6]        

Which leads to the following equations:

        [pic 7]

This leads to the following equations:

[pic 8]

[pic 9]

[pic 10]

[pic 11]

Problem 14-23 John Jones of Bayside Laundry has been providing cleaning and linen service for rental condominiums on the Gulf coast for over 10 years.  Currently, John is servicing 26 condominium developments. John’s two major competitors are Cleanco, which currently services 15 condominium developments, and Beach Services, which performs laundry and cleaning services for 11 condominium developments. Recently, John contacted Bay Bank about a loan to expand his business operations. To justify the loan, John has kept detailed records of his customers and the customers that he received from his two major competitors. During the past year, he was able to keep 18 of his original 26 customers. During the same period, he was able to get 1 new customer from Cleanco and 2 new customers from Beach Services. Unfortunately, John lost 6 of his original customers to Cleanco and 2 of his original customers to Beach Services during the same year. John has also learned that Cleanco has kept 80% of its current customers. He also knows that Beach Services will keep at least 50% of its customers. For John to get the loan from Bay Bank, he needs to show the loan officer that he will maintain an adequate share of the market. The officers of Bay Bank are concerned about the recent trends for market share, and they have decided not to give John a loan unless he will keep at least 35% of the market share in the long run. What types of equilibrium market shares can John expect? If you were an officer of Bay Bank, would you give John a loan?

A matrix of transitional probabilities lets you get from your current state to that of a future state.  The main goal of this matrix is to help you determine where you are going to be at a specified amount of time down the line, regardless of whether that is one month or one year later.  Using historical data, you can determine what percentage of customers are going to stay with Bayside and which ones are going to head to one of the competitors.  You need to know what the current states are and what the anticipated states are going to be in order to calculate an accurate depiction of the business statistics.

To determine whether John Jones is eligible for the loan or not, the following matrix can provide more information about the future:

Bayside                26        18                .70

Cleanco                15        1        6        .80

Beach Services        11        2        2        .50

Bayside retains 70% of their current customers a gains an additional 11% of their customers from Cleanco and Beach Services.  Over the course of time, Bayside will continue to stay at 35% or above for the market shares with just their customers alone and that doesn’t take into consideration the customers they gain from their competitors. If I were an officer at Bay Bank, I would provide John with the loan for his business. He has enough customers of his own that he has retained for many years, so he is only going to continue to grow and flourish.  

Problem 14-27 As a result of a recent survey of students at the University of Wisconsin, it was determined that the university owned bookstore currently has 40% of the market.  (See problem 14-14.) The other three bookstores, Bill’s, College, and Battle’s, each split the remaining initial market share. Given that the state probabilities are the same, what is the market share for the next period given the initial market share? What impact do the initial market shares have on each store next period? What is the impact on the steady state market shares?