You have just started work for a small company, FitCo, that develops private fit

ID: 1093474 • Letter: Y

Question

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

Explanation / Answer

1. Estimate an empirical demand function for one-month memberships using the data gathered from the firms 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 towns population.)

Q = 2308.53 -49.06 P +0.07M +0.034PP

2. Interpret the estimated demand function for one-month memberships.

The regression equation is as follows:

Q = 2308.53 -49.06 P +0.07M +0.034PP

Where

Q is memberships

P is price

M is income and

PP is population.

Memberships demand rise with income and population, and fall with a rise in price. The law of demand is obeyed (P and Q are inversely related); membership is a non inferior good ( as the coefficient of M is positive). The coefficients show the effect of a unit change in the respective variable on demand. So 1$ rise in income causes demand to rise by 0.07 units.

3. Calculate the point price elasticity of demand and point income elasticity of demand in Town D at the price charged last year.

M = 45000, P = 63 for town D.

Price elasticity = -49.059743 *(63/3263) = -.947

Income elasticity = 0.07038026 *(45000/3263) = .508201492 = 0.971

4. For Town H and Town W (from the Excel spreadsheet) determine whether the price charged last year was above, below or equal to the profit-maximizing price.

As per the data variable costs are zero, so MC =0. In equilibrium MR = MC. To get MR we need TR and a demand function. This is obtained by putting the values of M and PP in the estimated equation. We get

Town H: Q = 2308.53 -49.06 P +0.07M +0.034PP = 6458.53 -49.06P

Or P = 131.65-.0204Q

TR = 131.65Q-.0204Q2

MR = 131.65-.0408Q

Equate this to 0 (=MC)to get Q =3226.7156, P = 65.567

This is more than the 58 that is being charged currently.

Town W: repeat the same exercise to get Q = 4996.53 -49.06P

P = 101.85 -.0204Q

TR = 101.58Q-.0204Q2

MR = 101.58-.0408 Q

Equate to 0 to get Q= 2489.7 P= 50.86

This is also more than the price charged currently.

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.

Using the trend line we can find the demand function in both scenarios:

PP= 22000and M = 60000 so that Q = 7256.53-49.06P

P = 147.91-.0204Q

Tr = 147.91Q-.0204Q*Q

MR = 147.91-.0408Q = MC = 0

So Q= 3625.278 and P = 73.95

TR = 268105.001

Profits=108105.001

II.

Q = 6206.53-49.06P

P = 126.51 -.0204Q

TR = 126.51Q-.0204Q*Q

MR = 126.51-0.0408Q = 0

So Q= 3100.71 P= 84.56

TR = 262492.205

Profits = 192492.205

Site 2 is better as profits are higher

The price must be P= 84.56--as this is the price where MR =MC or profits are being maximized. This is found by equating Mc = 0 with MR as given by the equilibrium condition above.

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REPORT

I. PURPOSE:

The main aim is to find a statistical and quantitative relation between four variables- price, quantity of memberships and two data that can explain quantity demanded- income levels and population size. Theory tells us that demand is function of other three variables discussed- so it is dependent variable. Such relation can be used in forecasting demand and analyzing the source of any change in demand. It reveals the nature of the good in terms of income effect and law of demand and various elasticities. An inelastic demand will allow revenues to rise as price is raised; thus helping in pricing decisions. This can also be used in demand forecasting. Suppose we want to expand the club size/branches. We

can use projected values of income and population to estimate the demand function. Profit calculations are also based on this function.

II. METHOD

We use a simple multiple regression that imposes a linear relation among the independent and dependent variables. The method is least squares method that derives the equation by minimizing the sum of squared errors, where error refers to the difference between actual and estimated value of Q. It makes standard assumptions about the independent variables; they must be independent of each other. Another assumption is that we are below capacity in all centers.

III. ESTIMATES

The equation estimated is Q = 2308.53 -49.06 P +0.07M +0.034PP. it reveals:

The good obeys the law of demand as shown by ve coefficient of P-as P rises Q will fall

The good is non inferior as shown by positive slope of coefficient of M-as M rises Q will rise

As population rises Q will rise. Both M and PP act to shift the demand curve up or down.

If population goes up by 1000 the demand for memberships rises by 34 in a year,(= .034*1000) keeping other variables unchanged.

If the per capita income rises by $100 then demand goes up by 7 units in a year (= .07*100), keeping other variables unchanged.

If price falls by $1. Demand rises by 49.06 units per year.

The model has a high predictive power as R2 = 93.96%. This means that 93.96 % of the variation in Q is explained by these explanatory factors. The rest is due to unknown factors, which are not part of this regression analysis.

P values of all coefficients are low, implying that they are statistically significant individually in explaining the dependent variable memberships..

The same is shown by the F value. It shows that the regression is significant in an overall sense.

Variable costs are zero, so that marginal costs are zero.

Equilibrium will be found in a situation where Marginal revenue = marginal cost =0

in this case.

Elasticity of demand with respect to income and price will depend on the chosen levels of different variables chosen as shown in the exercises above.

Each of the coefficients represents the marginal effect of an independent variable on the dependent variable. So the marginal effect of a change in price is 49.06 units of membership. The marginal effects of income and population stand at .07 and .034 units.

In the two towns H and W we can see that demand is inelastic at given values.

IV. LIMITATIONS

. The biggest limitation is that we are imposinga linear relation among the variables. The relation may be non linear, linearity is imposed by our chosen model. Whether this is true can be partly judged by examining the scatter diagram. As a result, we impose all assumptions that this model needs. For example we assume that the independent variables are independent of each other. In a real world this may be violatedas population grows the incomes may rise due to growing businesses and greater demand. This shows as a positive correlation between independent variables, and can lead to misinterpretation of regression results. Another limitation is that we may be missing on other explanatory variables like the male-female ratio in a town, the state of health care available. Finally, while using this analysis in forecasting and planning new centers (as done above) we need to ensure that the conditions of the proposed site need to be similar to those in our dataset, for us to make sensible predictions. The profile of the new towns must be similar.

The writeup is organized along the following lines. First we briefly look at the purpose behind the exercise, and its possible uses. Next, the method used to estimate demand function for memberships is explained, along with the assumptions needed for the exercise. The third part is most important as it describes the results and conclusions that can be drawn from it in a pointwise manner. Lastly, we enumerate the limitations of our

methods and results.