You have just started work for a small company, FitCo, that develops private fitness clubs in small towns. FitCo 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 . FitCo 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 FitCo has opened 24 such clubs in different towns.
Your new boss, Sarah, gives you a copy of an Excel spreadsheet containing data collected last year on FitCo’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. FitCo 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.
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. Sarah also tells you to be explicit about any important limitations your analysis
might have.
Data for FitCo Clubs in 2008
Price (P) Demand (Q) Income (M) Population (N) Revenue Fixed cost Profit
Town A $62 2875 $42,000 24000 $178,250 $101,000 $77,250
Town B $57 2908 $35,000 30000 $165,756 $64,000 $101,756
Town C $84 3472 $65,000 25000 $291,648 $137,000 $154,648
Town D $63 3263 $45,000 28000 $205,569 $107,000 $98,569
Town E $60 2823 $38,000 19000 $169,380 $77,000 $92,380
Town F $46 2502 $26,000 22000 $115,092 $61,000 $54,092
Town G $64 2977 $37,000 31000 $190,528 $68,000 $122,528
Town H $58 3309 $41,000 28000 $191,922 $78,000 $113,922
Town I $63 2596 $44,000 17000 $163,548 $70,000 $93,548
Town J $55 2517 $29,000 26000 $138,435 $52,000 $86,435
Town K $83 3542 $68,000 13000 $293,986 $166,000 $127,986
Town L $61 3132 $50,000 12000 $191,052 $113,000 $78,052
Town M $52 2467 $27,000 23000 $128,284 $32,000 $96,284
Town N $66 3881 $51,000 32000 $256,146 $89,000 $167,146
Town O $60 2882 $42,000 21000 $172,920 $66,000 $106,920
Town P $78 3629 $56,000 37000 $283,062 $125,000 $158,062
Town Q $46 2931 $32,000 15000 $134,826 $67,000 $67,826
Town R $55 2630 $28,000 28000 $144,650 $40,000 $104,650
Town S $55 3381 $41,000 18000 $185,955 $66,000 $119,955
Town T $30 4643 $41,000 23000 $139,290 $84,000 $55,290
Town U $41 4945 $56,000 26000 $202,745 $114,000 $88,745
Town V $28 4784 $37,000 32000 $133,952 $56,000 $77,952
Town W $22 3311 $24,000 24000 $72,842 $36,000 $36,842
Town X $43 4066 $42,000 18000 $174,838 $96,000 $78,838
What is there about this assignment that you don't understand?
Do you have any particular question?
To complete the tasks assigned, we will need to perform several calculations and analyses using the data provided in the Excel spreadsheet for FitCo's 24 existing clubs.
1. Estimating the empirical demand function:
To estimate the demand function, we need to perform a linear regression analysis using the data on price, average income, and the size of the town's population. In Excel, we can use the "Data Analysis" tool and select "Regression." The dependent variable will be the demand for one-month memberships (Q), and the independent variables will be price (P), average income (M), and population (N).
2. Interpreting the estimated demand function:
After performing the linear regression analysis, we will obtain the equation of the estimated demand function. The demand function will take the form: Q = a + bP + cM + dN, where a, b, c, and d are the estimated coefficients. We can interpret the coefficient of price (b) as the elasticity of demand with respect to price. A negative coefficient indicates an inverse relationship between price and demand.
3. Calculating the point price and income elasticities of demand in Town D:
Using the estimated demand function, we can substitute the values of price, income, and population for Town D to calculate the point price elasticity and income elasticity of demand. The formulas for point price elasticity and income elasticity are as follows:
Price elasticity of demand (PED) = (ΔQ/Q) / (ΔP/P)
Income elasticity of demand (YED) = (ΔQ/Q) / (ΔM/M)
We can set ΔP and ΔM to zero since we are calculating point elasticities. Substituting the values for Town D into these formulas will give us the elasticities.
4. Determining if the price charged in Town H and Town W was profit-maximizing:
To determine if the price charged in Town H and Town W was profit-maximizing, we can use the formula for marginal revenue (MR) and set it equal to marginal cost (MC). If the price charged is above the profit-maximizing price, the profit will increase by decreasing the price. If the price is below the profit-maximizing price, the profit will increase by increasing the price. We can compare the actual price charged to the profit-maximizing price for both towns.
5. Selecting the site for the 25th club and determining the appropriate price:
To select the site for the 25th club, we need to consider the annual fixed costs of running the club in both towns, as well as the potential demand based on their populations. We can calculate the breakeven point by dividing the fixed costs by the contribution margin (price minus variable cost per membership). We can then compare the breakeven points for both towns and consider the potential profitability.
6. Writing a short report summarizing the results:
Based on the analysis conducted, we should summarize the results in a report. The report should include the findings from estimating the demand function, interpreting the demand function, calculating elasticities for Town D, determining if the price charged in Town H and Town W was profit-maximizing, and recommending the site and price for the 25th club. It is important to highlight any limitations of the analysis and provide a concise summary of the technical details in the appendix to make it accessible to Sarah, who has limited formal knowledge of the subject matter.