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(2) STORE WARS
The town of Peacity always has exactly 10,000 residents. By statute, every time new
residents move in or are born an equal number must move away or die.
It used to be that everyone in Peacity bought his or her food at Gracie’s Grocery. But
one day a branch of the national chain store Floormart opened in the neighboring town of
Noville. Floormart is able to offer lower prices and more selection, but doesn’t have the
personal service and local products that characterize Gracie’s.
Starting on opening week, and then for every subsequent one, 15% of the current
Gracie’s customers switch to Floormart because of the exceptional prices. However, it
also turns out that each week 10% of the Peacity residents who were shopping at
Floormart decide to return to the comforts of Gracie’s, despite its higher prices. So for
example, in the first week that Floormart is open, 1500 Peacity residents shop there and
8500 shop at Gracie’s.
The dynamics of these store wars are represented by a directed graph, which shows
how fractions shift between the two stores from week to week:
0.1
0.9
F
G
0.85
0.15
2a) How many Peacity customers shop at each store in the third week after Floormart
opens?
2b) Imagine a slightly different scenario, in which Floormart heavily advertised its
opening and attracted 5000 Peacity customers the first week (so that 5000 shopped at
Gracie’s). In this case how many customers will each store have in the third week,
assuming the weekly transition dynamics remain the same?
2c) Assume the original scenario. Furthermore, Gracie’s needs to average at least
3000 customers a week in order to stay in business. The manager notices that she is
losing customers every week, and asks you to calculate how long she can hang on.
What is your answer (and your explanation)? Would your answer change under the
different scenario of (2b)?
(3) STORE WARS (continued)
The Gracie’s manager hears that Floormart is going to try attracting more customers
with heavy advertising, and that Floormart will also try to retain more customers by
improving service. This could be big trouble for Gracie’s and your friend the manager
hires you to evaluate the situation.
You consider a general transition diagram assuming that Gracie’s loses a fraction “b”
of it’s customers each week, and Floormart loses a fraction “a.” (Both of these fractions
are between 0 and 1, and can be partially influenced by advertising and customer service
at each store.)
a
1-a
F
G
1-b
b
Although it is hard to be precise about how advertising and customer service
affect the values of a and b, it is not too hard to answer the following purely mathematical
question:
(3) What precise inequality involving a and b will guarantee that the steady state
(long-term) number of customers who shop at Gracie’s is at least 3000?
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