quantitative analysis of company expansion

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Smart Health develops private fitness clubs in small towns. Smart Health buys or leases a local hotel or motel, then renovates to provide a gym, swimming pool, sauna,
Jacuzzi, and a small café where patrons can buy juices, smoothies, and other healthy snacks . Smart Health only develops clubs in towns where it has no competitors.
The main product is a one-month membership, which gives patrons unlimited use of the gym and other club facilities. So far Smart Health has opened 24 such clubs in
different towns.

You have just started working for Smart Health, and your new boss, Karen, gives you a copy of an Excel spreadsheet containing data collected last year on Smart
Health’s 24 existing clubs. She asks you to use the data to complete the following important and time-sensitive tasks:

1. Estimate an empirical demand function for one-month memberships using the data gathered from the firm’s 24 clubs. (Assume the demand function is linear. Further
assume that the only variables likely to significantly affect the demand for one-month memberships are price, average income, and the size of the town’s population.)
2. Interpret the estimated demand function for one-month memberships.
3. Calculate the point price elasticity of demand and point income elasticity of demand in Town D at the price charged last year.
4. For Town H and Town W determine whether the price charged last year was above, below or equal to the profit-maximizing price.
5. Smart Health is considering opening a 25th club. The company must choose between one of two towns in which to locate the new club. Both towns have populations of
22,000. However, one of the towns has a relatively high average income of $60,000, while the other has a relatively low average income of $45,000. The annual fixed
costs of running the club in the high income town would be about $160,000, while annual fixed costs of running the club in the low income town would be about $70,000.
Your job is to select the site for the 25th club and to determine the appropriate price for the one-month memberships.
6. Write a short report summarizing the results of the analysis and any recommendations.

You have one week to complete the analysis, interpret the results, and summarize your findings and recommendations in a brief report. Karen tells you that the main
body of the report must be short (a 1000 words at most excluding the title page and any appendices), to the point, and not overly technical. She tells you that her
formal knowledge of mathematics, economics, and statistics is somewhat limited, and asks you to keep that in mind as you write the main body of the report.
Nevertheless, your conclusions and recommendations must be based on a rigorous analysis of the available data and you should provide a concise summary of any technical
details in an appendix. Karen also tells you to be explicit about any important limitations your analysis might have. Finally, Karen tells you that she likes her
reports to be broken up into sections with sensible and self-explanatory headings because it makes them much easier to read and understand.